# CICE: the Los Alamos Sea Ice Model Documentation and Software

CICE: the Los Alamos Sea Ice Model Documentation and Software User’s Manual Version 4.0 LA-CC-06-012 Elizabeth C. Hunke and William H. Lipscomb T-3 Fluid Dynamics Group, Los Alamos National Laboratory Los Alamos NM 87545 November 21, 2008 Contents 1 Introduction 2 2 Coupling with other climate model components 2.1 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 3 Model components 3.1 Horizontal transport . . . . . . . . . . . . . 3.1.1 Reconstructing area and tracer fields 3.1.2 Locating departure triangles . . . . 3.1.3 Integrating fields . . . . . . . . . . 3.1.4 Updating state variables . . . . . . 3.2 Transport in thickness space . . . . . . . . 3.3 Mechanical redistribution . . . . . . . . . . 3.4 Dynamics . . . . . . . . . . . . . . . . . . 3.5 Thermodynamics . . . . . . . . . . . . . . 3.5.1 Thermodynamic surface forcing . . 3.5.2 New temperatures . . . . . . . . . 3.5.3 Growth and melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10 12 14 17 18 18 21 23 26 26 29 32 Numerical implementation 4.1 Directory structure . . . . . . . . . . 4.2 Grid, boundary conditions and masks 4.3 Initialization and coupling . . . . . . 4.4 Choosing an appropriate time step . . 4.5 Model output . . . . . . . . . . . . . 4.6 Execution procedures . . . . . . . . . 4.7 Performance . . . . . . . . . . . . . . 4.8 Adding things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 34 37 39 40 41 42 44 46 4 . . . . . . . . . . . . . . . . . . . . . . . . 1 4.8.1 4.8.2 4.8.3 5 1 Timers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 History fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Tracers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Troubleshooting 5.1 Initial setup . . . . . . . 5.2 Slow execution . . . . . 5.3 Debugging hints . . . . . 5.4 Known bugs . . . . . . . 5.5 Interpretation of albedos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 48 49 49 50 50 Acknowledgments and Copyright 50 Table of namelist options 52 Index of primary variables and parameters 56 General Index 68 Bibliography 71 Introduction The Los Alamos sea ice model (CICE) is the result of an effort to develop a computationally efficient sea ice component for a fully coupled atmosphere-ice-ocean-land global climate model. It was designed to be compatible with the Parallel Ocean Program (POP), an ocean circulation model developed at Los Alamos National Laboratory for use on massively parallel computers [38, 9, 10]. The current version of the model has been enhanced greatly through collaborations with members of the Community Climate System Model (CCSM) Polar Climate Working Group, based at the National Center for Atmospheric Research (NCAR), and researchers at the U.K. Met Office Hadley Centre. CICE has several interacting components: a thermodynamic model that computes local growth rates of snow and ice due to vertical conductive, radiative and turbulent fluxes, along with snowfall; a model of ice dynamics, which predicts the velocity field of the ice pack based on a model of the material strength of the ice; a transport model that describes advection of the areal concentration, ice volumes and other state variables; and a ridging parameterization that transfers ice among thickness categories based on energetic balances and rates of strain. Additional routines prepare and execute data exchanges with an external “flux coupler,” which then passes the data to other climate model components such as POP. This model release is CICE version 4.0, available from http://climate.lanl.gov/Models/CICE/. It replaces CICE 3.14, which was released in August 2006. Although the model physics is similar to that of version 3.14, the code has changed substantially: • A multiple-scattering radiative scheme, referred to as “Delta-Eddington,” is optional. • A basic melt pond parameterization takes advantage of the Delta-Eddington radiative scheme. • A passive ice age tracer is available, and new tracers can be added easily. • With incremental remapping advection, the user can specify the geometric area flux across each cell edge, so that the divergence computed from the momentum equation and the divergence implied by the remapping can be made identical. Introduction 3 • The snow model now allows more than one layer. • A zero-layer thermodynamics option has been added along with an option to not compute the surface temperature. Similarly, wind stress may be provided to the model instead of being calculated internally. These options allow the alternative coupling strategy employed in the Hadley Centre GCM. • The World Meteorological Organization definition of ice thickness categories is available for the ice thickness distribution. • The infrastructure has been overhauled and updated as in POP. Most notably, this includes “subblocking,” where the grid is first divided into blocks of grid cells that are distributed across processors for optimal load balancing. Better cache utilization may also increase model efficiency on some architectures. For serial (single processor) runs, subblocking may still improve performance with cache because all-land blocks are eliminated. • A novel application of the subblocking infrastructure involves distributing the blocks via space-filling curves, implemented in ice spacecurve.F90 [7]. • Blocks may include “padding” cells outside the physical domain, to allow uneven domain decomposition. • The code now runs on global tripole grids in addition to our standard, displaced-pole grids. • Regional grids with open boundaries are now supported, with basic restoring along lateral grid boundaries in a new module, ice restoring.F90. • Support for netCDF grid files has been added, and all output may be written in binary form if netCDF libraries are unavailable. • File units are allocated dynamically. • Options for other atmosphere and ocean forcing data sets are available in ice forcing.F90. • Coupling options have been generalized. • Coupling is now assumed to occur at the beginning/end of the timestep, rather than in the middle of the thermodynamics, simplifying the code considerably. • Several history fields and namelist options have been added. Generally speaking, subroutine names are given in italic and file names are boldface in this document. Symbols used in the code are typewritten, while corresponding symbols in this document are in the math font which, granted, looks a lot like italic. A comprehensive index, including a glossary of symbols with many of their values, appears at the end. The organization of this software distribution is described in Section 4.1; most files and subroutines referred to in this documentation are part of the CICE code found in cice/source/, unless otherwise noted. After many years “CICE” has finally become an acronym, for ”Community Ice CodE.” We still pronounce the name as “sea ice,” but there has been a small grass-roots movement underway to alter the model name’s pronunciation from “sea ice” to what an Italian might say, ch¯e0 -ch¯a or “chee-chay.” Others choose to say s¯ıs (English, rhymes with “ice”), s¯es (French, like “cease”), or sh¯e-¯ı-soo (“Shii-aisu,” Japanese). 4 Introduction z◦ ~a U Qa ρa Θa Ta Fsw ↓ FL↓ Frain Fsnow ~τa Fs Fl FL↑ Fevap α Tsfc Atmosphere Ocean Provided by the flux coupler to the sea ice model Atmosphere level height Ffrzmlt Freezing/melting potential Wind velocity Tw Sea surface temperature Specific humidity S Sea surface salinity Air density ∇H◦ Sea surface slope ~w Air potential temperature U Surface ocean currents Air temperature Shortwave radiation (4 bands) Incoming longwave radiation Rainfall rate Snowfall rate Provided by the sea ice model to the flux coupler Wind stress Fsw ⇓ Penetrating shortwave Sensible heat flux Fwater Fresh water flux Latent heat flux Fhocn Net heat flux to ocean Outgoing longwave Fsalt Salt flux Evaporated water ~τw Ice-ocean stress Surface albedo (4 bands) Surface temperature ai Ice fraction Taref 2 m reference temperature (diagnostic) Qref 2 m reference humidity (diagnostic) a Fswabs Absorbed shortwave (diagnostic) Table 1: Data exchanged between the CCSM flux coupler and the sea ice model. 5 2 Coupling with other climate model components The sea ice model exchanges information with the other model components via a flux coupler. CICE has been coupled into numerous climate models with a variety of coupling techniques. This document is oriented primarily toward the CCSM Flux Coupler [23] from NCAR, the first major climate model to incorporate CICE. The flux coupler was originally intended to gather state variables from the component models, compute fluxes at the model interfaces, and return these fluxes to the component models for use in the next integration period, maintaining conservation of momentum, heat and fresh water. However, several of these fluxes are now computed in the ice model itself and provided to the flux coupler for distribution to the other components, for two reasons. First, some of the fluxes depend strongly on the state of the ice, and vice versa, implying that an implicit, simultaneous determination of the ice state and the surface fluxes is necessary for consistency and stability. Second, given the various ice types in a single grid cell, it is more efficient for the ice model to determine the net ice characteristics of the grid cell and provide the resulting fluxes, rather than passing several values of the state variables for each cell. These considerations are explained in more detail below. The fluxes and state variables passed between the sea ice model and the CCSM flux coupler are listed in Table 1. By convention, directional fluxes are positive downward. The ice fraction ai (aice)1 is the total fractional ice coverage of a grid cell. That is, in each cell, ai = 0 if there is no ice ai = 1 if there is no open water 0 < ai < 1 if there is both ice and open water, where ai is the sum of fractional ice areas for each category of ice. The ice fraction is used by the flux coupler to merge fluxes from the ice model with fluxes from the other components. For example, the penetrating shortwave radiation flux, weighted by ai , is combined with the net shortwave radiation flux through ice-free leads, weighted by (1 − ai ), to obtain the net shortwave flux into the ocean over the entire grid cell. The flux coupler requires the fluxes to be divided by the total ice area so that the ice and land models are treated identically (land also may occupy less than 100% of an atmospheric grid cell). These fluxes are “per unit ice area” rather than “per unit grid cell area.” In some coupled climate models (for example, recent versions of the U.K. Hadley Centre model) the surface air temperature and fluxes are computed within the atmosphere model and are passed to CICE. In this case the logical parameter calc Tsfc in ice therm vertical is set to false. The fields fsurfn (the net surface heat flux from the atmosphere), flatn (the surface latent heat flux), and fcondtopn (the conductive flux at the top surface) for each ice thickness category are copied or derived from the input coupler fluxes and are passed to the thermodynamic driver subroutine, thermo vertical. At the end of the time step, the surface temperature and effective conductivity (i.e., thermal conductivity divided by thickness) of the top ice/snow layer in each category are returned to the atmosphere model via the coupler. Since the ice surface temperature is treated explicitly, the effective conductivity may need to be limited to ensure stability. As a result, accuracy may be significantly reduced, especially for thin ice or snow layers. A more stable and accurate procedure would be to compute the temperature profiles for both the atmosphere and ice, together with the surface fluxes, in a single implicit calculation. This was judged impractical, however, given that the atmosphere and sea ice models generally exist on different grids and/or processor sets. 2.1 Atmosphere The wind velocity, specific humidity, air density and potential temperature at the given level height z◦ are used to compute transfer coefficients used in formulas for the surface wind stress and turbulent heat fluxes 1 Typewritten equivalents used in the code are described in the index. 6 Coupling with other climate model components ~τa , Fs , and Fl , as described below. Wind stress is arguably the primary forcing mechanism for the ice motion, although the ice–ocean stress, Coriolis force, and slope of the ocean surface are also important [41]. The sensible and latent heat fluxes, Fs and Fl , along with shortwave and longwave radiation, Fsw ↓ , FL↓ and FL↑ , are included in the flux balance that determines the ice or snow surface temperature when calc Tsfc = true. As described in Section 3.5, these fluxes depend nonlinearly on the ice surface temperature Tsfc . The balance equation is iterated until convergence, and the resulting fluxes and Tsfc are then passed to the flux coupler. The snowfall precipitation rate (provided as liquid water equivalent and converted by the ice model to snow depth) also contributes to the heat and water mass budgets of the ice layer. Melt ponds generally form on the ice surface in the Arctic and refreeze later in the fall, reducing the total amount of fresh water that reaches the ocean and altering the heat budget of the ice; this version includes a simple melt pond parameterization. Rain and all melted snow end up in the ocean. Wind stress and transfer coefficients for the turbulent heat fluxes are computed in subroutine atmo boundary layer following [23]. For clarity, the equations are reproduced here in the present notation. The wind stress and turbulent heat flux calculation accounts for both stable and unstable atmosphere-ice boundary layers. Define the “stability” κgz◦ Θ∗ Q∗ Υ = ∗2 + , u Θa (1 + 0.606Qa ) 1/0.606 + Qa where κ is the von Karman constant, g is gravitational acceleration, and u∗ , Θ∗ and Q∗ are turbulent scales for velocity, temperature and humidity, respectively: ~ u∗ = cu U a Θ∗ = cθ (Θa − Tsfc ) ∗ Q (1) = cq (Qa − Qsfc ) . The wind speed has a minimum value of 1 m/s. We have ignored ice motion in u∗ , and Tsfc and Qsfc are the surface temperature and specific humidity, respectively. The latter is calculated by assuming a saturated surface, as described in Section 3.5.1. The exchange coefficients cu , cθ and cq are initialized as κ ln(zref /zice ) and updated during a short iteration, as they depend upon the turbulent scales. Here, zref is a reference height of 10 m and zice is the roughness length scale for the given sea ice category. Υ is constrained to have magnitude less than 10. Further, defining χ = (1 − 16Υ)0.25 and χ ≥ 1, the “integrated flux profiles” for momentum and stability in the unstable (Υ < 0) case are given by h i π ψm = 2 ln [0.5(1 + χ)] + ln 0.5(1 + χ2 ) − 2 tan−1 χ + , 2 h i 2 ψs = 2 ln 0.5(1 + χ ) . In a departure from the parameterization used in [23], we use profiles for the stable case following [22], ψm = ψs = − [0.7Υ + 0.75 (Υ − 14.3) exp (−0.35Υ) + 10.7] . The coefficients are then updated as cu 1 + cu (λ − ψm ) /κ cθ = 1 + cθ (λ − ψs ) /κ = c0θ c0u = c0θ c0q 2.2 Ocean 7 where λ = ln (z◦ /zref ). The first iteration ends with new turbulent scales from equations (1). After five iterations the latent and sensible heat flux coefficients are computed, along with the wind stress: Cl = ρa (Lvap + Lice ) u∗ cq Cs = ρa cp u∗ c∗θ + 1, ~a ρa u∗2 U ~τa = , ~ a| |U where Lvap and Lice are latent heats of vaporization and fusion, ρa is the density of air and cp is its specific heat. Again following [22], we have added a constant to the sensible heat flux coefficient in order to allow some heat to pass between the atmosphere and the ice surface in stable, calm conditions. The atmospheric reference temperature Taref is computed from Ta and Tsfc using the coefficients cu , cθ and cq . Although the sea ice model does not use this quantity, it is convenient for the ice model to perform this calculation. The atmospheric reference temperature is returned to the flux coupler as a climate diagnostic. The same is true for the reference humidity, Qref a . Additional details about the latent and sensible heat fluxes and other quantities referred to here can be found in Section 3.5.1. 2.2 Ocean New sea ice forms when the ocean temperature drops below its freezing temperature, Tf = −µS, where S is the seawater salinity and µ = 0.054 ◦ /psu is the ratio of the freezing temperature of brine to its salinity. The ocean model performs this calculation; if the freezing/melting potential Ffrzmlt is positive, its value represents a certain amount of frazil ice that has formed in one or more layers of the ocean and floated to the surface. (The ocean model assumes that the amount of new ice implied by the freezing potential actually forms.) In general, this ice is added to the thinnest ice category. The new ice is grown in the open water area of the grid cell to a specified minimum thickness; if the open water area is nearly zero or if there is more new ice than will fit into the thinnest ice category, then the new ice is spread over the entire cell. If Ffrzmlt is negative, it is used to heat already existing ice from below. In particular, the sea surface temperature and salinity are used to compute an oceanic heat flux Fw (|Fw | ≤ |Ffrzmlt |) which is applied at the bottom of the ice. The portion of the melting potential actually used to melt ice is returned to the coupler in Fhocn . The ocean model adjusts its own heat budget with this quantity, assuming that the rest of the flux remained in the ocean. In addition to runoff from rain and melted snow, the fresh water flux Fwater includes ice meltwater from the top surface and water frozen (a negative flux) or melted at the bottom surface of the ice. This flux is computed as the net change of fresh water in the ice and snow volume over the coupling time step, excluding frazil ice formation and newly accumulated snow. Setting the namelist option update ocn f to true causes frazil ice to be included in the fresh water and salt fluxes. There is a flux of salt into the ocean under melting conditions, and a (negative) flux when sea water is freezing. However, melting sea ice ultimately freshens the top ocean layer, since the ocean is much more saline than the ice. The ice model passes the net flux of salt Fsalt to the flux coupler, based on the net change in salt for ice in all categories. In the present configuration, ice ref salinity is used for computing the salt flux, although the ice salinity used in the thermodynamic calculation has differing values in the ice layers. A fraction of the incoming shortwave Fsw ⇓ penetrates the snow and ice layers and passes into the ocean, as described in Section 3.5.1. Many ice models compute the sea surface slope ∇H◦ from geostrophic ocean currents provided by an ocean model or other data source. In our case, the sea surface height H◦ is a prognostic variable in POP—the 8 Model components flux coupler can provide the surface slope directly, rather than inferring it from the currents. (The option of computing it from the currents is provided in subroutine evp prep.) The sea ice model uses the surface layer ~ w to determine the stress between the ocean and the ice, and subsequently the ice velocity ~u. This currents U stress, relative to the ice, h ~ u ~τw = cw ρw U w −~ ~ w − ~u cos θ + kˆ × U ~ w − ~u sin θ U i is then passed to the flux coupler (relative to the ocean) for use by the ocean model. Here, θ is the turning angle between geostrophic and surface currents, cw is the ocean drag coefficient, ρw is the density of seawater (dragw = cw ρw ), and kˆ is the vertical unit vector. The turning angle is necessary if the top ocean model layers are not able to resolve the Ekman spiral in the boundary layer. If the top layer is sufficiently thin compared to the typical depth of the Ekman spiral, then θ = 0 is a good approximation. Here we assume that the top layer is thin enough. 3 Model components The Arctic and Antarctic sea ice packs are mixtures of open water, thin first-year ice, thicker multiyear ice, and thick pressure ridges. The thermodynamic and dynamic properties of the ice pack depend on how much ice lies in each thickness range. Thus the basic problem in sea ice modeling is to describe the evolution of the ice thickness distribution (ITD) in time and space. The fundamental equation solved by CICE is [43]: ∂g ∂ = −∇ · (gu) − (f g) + ψ, ∂t ∂h (2) ∂ ∂ where u is the horizontal ice velocity, ∇ = ( ∂x , ∂y ), f is the rate of thermodynamic ice growth, ψ is a ridging redistribution function, and g is the ice thickness distribution function. We define g(x, h, t) dh as the fractional area covered by ice in the thickness range (h, h + dh) at a given time and location. Equation (2) is solved by partitioning the ice pack in each grid cell into discrete thickness categories. The number of categories can be set by the user, with a default value NC = 5. (Five categories, plus open water, are generally sufficient to simulate the annual cycles of ice thickness, ice strength, and surface fluxes [3, 25].) Each category n has lower thickness bound Hn−1 and upper bound Hn . The lower bound of the thinnest ice category, H0 , is set to zero. The other boundaries are chosen with greater resolution for small h, since the properties of the ice pack are especially sensitive to the amount of thin ice [29]. The continuous function g(h) is replaced by the discrete variable ain , defined as the fractional area covered by ice in the P C thickness range (Hn−1 , Hn ). We denote the fractional area of open water by ai0 , giving N n=0 ain = 1 by definition. Category boundaries are computed in init itd using one of several formulas, summarized in Table 2. Setting the namelist variable kcatbound equal to 0 or 1 gives lower thickness boundaries for any number of thickness categories NC . Table 2 shows the boundary values for NC = 5. A third option specifies the boundaries based on the World Meteorological Organization classification; the full WMO thickness distribution is used if NC = 7; if NC = 5 or 6, some of the thinner categories are combined. The original formula (kcatbound = 0) is the default because it was used to create the restart files included with the code distribution. Users may substitute their own preferred boundaries in init itd. In addition to the fractional ice area, ain , we define the following state variables for each category n: • vin , the ice volume, equal to the product of ain and the ice thickness hin . • vsn , the snow volume, equal to the product of ain and the snow thickness hsn . Model components 9 distribution kcatbound NC category 1 2 3 4 5 6 7 original 0 5 0.00 0.64 1.39 2.47 4.57 round WMO 1 2 5 5 6 lower bound (m) 0.00 0.00 0.00 0.60 0.30 0.15 1.40 0.70 0.30 2.40 1.20 0.70 3.60 2.00 1.20 2.00 7 0.00 0.10 0.15 0.30 0.70 1.20 2.00 Table 2: Lower boundary values for thickness categories, in meters, for the three distribution options (kcatbound). In the WMO case, the distribution used depends on the number of categories used. • eink , the internal ice energy in layer k, equal to the product of the ice layer volume, vin /Ni , and the ice layer enthalpy, qink . Here Ni is the total number of ice layers, with a default value Ni = 4, and qink is the negative of the energy needed to melt a unit volume of ice and raise its temperature to 0◦ C; it is discussed in Section 3.5. (NOTE: In the current code, ei < 0 and qi < 0 with ei = vi qi .) • esnk , the internal snow energy in layer k, equal to the product of the snow layer volume, vsn /Ns , and the snow layer enthalpy, qsnk , where Ns is the number of snow layers. (Similarly, es < 0 in the code.) Earlier versions of CICE had a single snow layer, but multiple layers are now allowed. The default value is Ns = 1. • Tsfn , the surface temperature • τage , the volume-weighted mean ice age. Since the fractional area is unitless, the volume variables have units of meters (i.e., m3 of ice or snow per m2 of grid cell area), and the energy variables have units of J/m2 . Both Tsfn and τage are assigned to the tracer array trcrn, which consists of Ntr fields. By default, Ntr = 2, but users may create any number of additional tracer fields. The three terms on the right-hand side of (2) describe three kinds of sea ice transport: (1) horizontal transport in (x, y) space; (2) transport in thickness space h due to thermodynamic growth and melting; and (3) transport in thickness space h due to ridging and other mechanical processes. We solve the equation by operator splitting in three stages, with two of the three terms on the right set to zero in each stage. We compute horizontal transport using the incremental remapping scheme of [8] as adapted for sea ice by [26]; this scheme is discussed in Section 3.1. Ice is transported in thickness space using the remapping scheme of [25], as described in Section 3.2. The mechanical redistribution scheme, based on [43], [35], [15], [12], and [27], is outlined in Section 3.3. To solve the horizontal transport and ridging equations, we need the ice velocity u, and to compute transport in thickness space, we must know the the ice growth rate f in each thickness category. We use the elastic-viscous-plastic (EVP) ice dynamics scheme of [18], as modified by [6], [16], [19] and [20], to find the velocity, as described in Section 3.4. Finally, we use the thermodynamic model of [4], discussed in Section 3.5, to compute f . The order in which these computations are performed in the code itself was chosen so that quantities sent to the coupler are consistent with each other and as up-to-date as possible. In earlier versions of CICE, fluxes and state variables were sent to the coupler during the timestep, after initial thermodynamic surface 10 Model components fluxes had been computed but before the rest of the thermodynamics and dynamics calculations. This was done to improve load balancing in a concurrent coupling scheme (model components ran simultaneously rather than sequentially). CCSM no longer requires this and so the coupling is now done at the end of the timestep, simplifying the code a great deal. However, the Delta-Eddington radiative scheme computes albedo and shortwave components simultaneously, and in order to have the most up-to-date values available for the coupler at the end of the timestep, the order of radiation calculations needed to be shifted. Albedo and shortwave components are computed after the ice state has been modified by both thermodynamics and dynamics, so that they are consistent with the ice area and thickness at the end of the step when sent to the coupler. However, they are computed using the downwelling shortwave from the beginning of the timestep. Rather than recompute the albedo and shortwave components at the beginning of the next timestep using new values of the downwelling shortwave forcing, the shortwave components computed at the end of the last timestep are scaled for the new forcing. 3.1 Horizontal transport We wish to solve the continuity or transport equation, ∂ain + ∇ · (ain u) = 0, ∂t (3) for the fractional ice area in each thickness category n. Equation (3) describes the conservation of ice area under horizontal transport. It is obtained from (2) by discretizing g and neglecting the second and third terms on the right-hand side, which are treated separately (Sections 3.2 and 3.3). There are similar conservation equations for ice volume, snow volume, ice energy and snow energy: ∂vin + ∇ · (vin u) ∂t ∂vsn + ∇ · (vsn u) ∂t ∂eink + ∇ · (eink u) ∂t ∂esnk + ∇ · (esnk u) ∂t = 0, (4) = 0, (5) = 0, (6) = 0. (7) In addition, there are one or more equations describing tracer transport, whose values are contained in the trcrn array. These equations typically have one of the following three forms ∂ (ain Tn ) + ∇ · (ain Tn u) = 0, ∂t ∂ (vin Tn ) + ∇ · (vin Tn u) = 0, ∂t ∂ (vsn Tn ) + ∇ · (vsn Tn u) = 0. ∂t (8) (9) (10) Equation (8) describes the transport of surface temperature, whereas (9) and (10) describe the transport of passive tracers such as volume-weighted ice age and snow age. Each tracer field is given an integer index, trcr depend, which has the value 0, 1, or 2 depending on whether the appropriate conservation equation is (8) (9), or (10), respectively. The total number of tracers is Ntr ≥ 1. In the default configuration there are two tracers: surface temperature and volume-weighted ice age. A melt pond volume tracer that is also available. Users may add any number of additional tracers that are transported conservatively provided that trcr depend is defined appropriately. See Section 4.8.3 for guidance on adding tracers. Horizontal transport 11 The age of the ice is treated as an ice-area tracer (trcr depend = 0). It is initialized at 0 when ice forms as frazil, and the ice ages the length of the timestep during each timestep. Freezing directly onto the bottom of the ice does not affect the age, nor does melting. Mechanical redistribution processes and advection alter the age of ice in any given grid cell in a conservative manner following changes in ice area. The sea ice age tracer is validated in [17]. By default, ice and snow are assumed to have constant densities, so that volume conservation is equivalent to mass conservation. Variable-density ice and snow layers can be transported conservatively by defining tracers corresponding to ice and snow density, as explained in the introductory comments in ice transport remap.F90. Prognostic equations for ice and/or snow density may be included in future model versions but have not yet been implemented. Two transport schemes are available: upwind and the incremental remapping scheme of [8] as modified for sea ice by [26]. The remapping scheme has several desirable features: • It conserves the quantity being transported (area, volume, or energy). • It is non-oscillatory; that is, it does not create spurious ripples in the transported fields. • It preserves tracer monotonicity. That is, it does not create new extrema in the thickness and enthalpy fields; the values at time m + 1 are bounded by the values at time m. • It is second-order accurate in space and therefore is much less diffusive than first-order schemes (e.g., upwind). The accuracy may be reduced locally to first order to preserve monotonicity. • It is efficient for large numbers of categories or tracers. Much of the work is geometrical and is performed only once per grid cell instead of being repeated for each quantity being transported. The time step is limited by the requirement that trajectories projected backward from grid cell corners are confined to the four surrounding cells; this is what is meant by incremental remapping as opposed to general remapping. This requirement leads to a CFL-like condition, max |u|∆t ≤ 1. ∆x For highly divergent velocity fields the maximum time step must be reduced by a factor of two to ensure that trajectories do not cross. However, ice velocity fields in climate models usually have small divergences per time step relative to the grid size. The remapping algorithm can be summarized as follows: 1. Given mean values of the ice area and tracer fields in each grid cell, construct linear approximations of these fields. Limit the field gradients to preserve monotonicity. 2. Given ice velocities at grid cell corners, identify departure regions for the fluxes across each cell edge. Divide these departure regions into triangles and compute the coordinates of the triangle vertices. 3. Integrate the area and tracer fields over the departure triangles to obtain the area, volume, and energy transported across each cell edge. 4. Given these transports, update the state variables. Since all scalar fields are transported by the same velocity field, step (2) is done only once per time step. The other three steps are repeated for each field in each thickness category. These steps are described below. 12 3.1.1 Model components Reconstructing area and tracer fields First, using the known values of the state variables, the ice area and tracer fields are reconstructed in each grid cell as linear functions of x and y. For each field we compute the value at the cell center (i.e., at the origin of a 2D Cartesian coordinate system defined for that grid cell), along with gradients in the x and y directions. The gradients are limited to preserve monotonicity. When integrated over a grid cell, the reconstructed fields must have mean values equal to the known state variables, denoted by a ¯ for fractional ˜ area, h for thickness, and qˆ for enthalpy. The mean values are not, in general, equal to the values at the cell center. For example, the mean ice area must equal the value at the centroid, which may not lie at the cell center. Consider first the fractional ice area, the analog to fluid density ρ in [8]. For each thickness category we construct a field a(r) whose mean is a ¯, where r = (x, y) is the position vector relative to the cell center. That is, we require Z a dA = a ¯ A, (11) A where A = R A dA is the grid cell area. Equation (11) is satisfied if a(r) has the form a(r) = a ¯ + αa h∇ai · (r − ¯ r), (12) where h∇ai is a centered estimate of the area gradient within the cell, αa is a limiting coefficient that enforces monotonicity, and ¯ r is the cell centroid: 1 ¯ r= A Z r dA. A It follows from (12) that the ice area at the cell center (r = 0) is ac = a ¯ − ax x − ay y, where ax = αa (∂a/∂x) and ay = αa (∂a/∂y) are the limitedR gradients in the x and y directions, respecR tively, and the components of ¯ r, x = A x dA/A and y = A y dA/A, are evaluated using the triangle integration formulas described in Section 3.1.3. These means, along with higher-order means such as x2 , xy, and y 2 , are computed once and stored. Next consider the ice and snow thickness and enthalpy fields. Thickness is analogous to the tracer concentration T in [8], but there is no analog in [8] to the enthalpy. The reconstructed ice or snow thickness h(r) and enthalpy q(r) must satisfy Z ˜ A, a h dA = a ¯h (13) ˜ qˆ A, a h q dA = a ¯h (14) A Z A ˜ = h(˜r) is the thickness at the center of ice area, and qˆ = q(ˆr) is the enthalpy at the center of ice or where h snow volume. Equations (13) and (14) are satisfied when h(r) and q(r) are given by ˜ + αh h∇hi · (r − ˜ h(r) = h r), (15) q(r) = qˆ + αq h∇qi · (r − ˆ r), (16) where αh and αq are limiting coefficients. The center of ice area, ˜ r, and the center of ice or snow volume, ˆ r, are given by Z 1 ˜ r= a r dA, a ¯A A Horizontal transport 13 1 a h r dA. ˜ a ¯hA A Evaluating the integrals, we find that the components of ˜ r are Z ˆ r= x ˜= ac x + ax x2 + ay xy , a ¯ y˜ = ac y + ax xy + ay y 2 , a ¯ and the components of ˆ r are x ˆ= c1 x + c2 x2 + c3 xy + c4 x3 + c5 x2 y + c6 xy 2 , ˜ a ¯h yˆ = c1 y + c2 xy + c3 y 2 + c4 x2 y + c5 xy 2 + c6 y 3 , ˜ a ¯h where c1 ≡ ac hc , c2 ≡ ac hx + ax hc , c3 ≡ ac hy + ay hc , c4 ≡ ax hx , c5 ≡ ax hy + ay hx , c6 ≡ ay hy . From (15) and (16), the thickness and enthalpy at the cell center are given by ˜ − hx x hc = h ˜ − hy y˜, qc = qˆ − qx x ˆ − qy yˆ, where hx , hy , qx and qy are the limited gradients of thickness and enthalpy. The surface temperature is treated the same way as ice or snow thickness, but it has no associated enthalpy. Tracers obeying conservation equations of the form (9) and (10) are treated in analogy to ice and snow enthalpy, respectively. ¯ j) denotes the mean value of some field in grid We preserve monotonicity by van Leer limiting. If φ(i, ¯ cell (i, j), we first compute centered gradients of φ in the x and y directions, then check whether these gradients give values of φ within cell (i, j) that lie outside the range of φ¯ in the cell and its eight neighbors. Let φ¯max and φ¯min be the maximum and minimum values of φ¯ over the cell and its neighbors, and let φmax and φmin be the maximum and minimum values of the reconstructed φ within the cell. Since the reconstruction is linear, φmax and φmin are located at cell corners. If φmax > φ¯max or φmin < φ¯min , we multiply the unlimited gradient by α = min(αmax , αmin ), where ¯ ¯ αmax = (φ¯max − φ)/(φ max − φ), ¯ ¯ αmin = (φ¯min − φ)/(φ min − φ). Otherwise the gradient need not be limited. Earlier versions of CICE (through 3.14) computed gradients in physical space. In version 4.0, gradients are computed in a scaled space in which each grid cell has sides of unit length. The origin is at the cell center, and the four vertices are located at (0.5, 0.5), (-0.5,0.5),(-0.5, -0.5) and (0.5, -0.5). In this coordinate system, several of the above grid-cell-mean quantities vanish (because they are odd functions of x and/or y), but they have been retained in the code for generality. 14 Model components Figure 1: In incremental remapping, conserved quantities are remapped from the shaded departure region, a quadrilateral formed by connecting the backward trajectories from the four cell corners, to the grid cell labeled H. The region fluxed across the north edge of cell H consists of a triangle (abc) in the N W cell and a quadrilateral (two triangles, acd and ade) in the N cell. 3.1.2 Locating departure triangles The method for locating departure triangles is discussed in detail by [8]. The basic idea is illustrated in Figure 1, which shows a shaded quadrilateral departure region whose contents are transported to the target or home grid cell, labeled H. The neighboring grid cells are labeled by compass directions: N W , N , N E, W , and E. The four vectors point along the velocity field at the cell corners, and the departure region is formed by joining the starting points of these vectors. Instead of integrating over the entire departure region, it is convenient to compute fluxes across cell edges. We identify departure regions for the north and east edges of each cell, which are also the south and west edges of neighboring cells. Consider the north edge of the home cell, across which there are fluxes from the neighboring N W and N cells. The contributing region from the N W cell is a triangle with vertices abc, and that from the N cell is a quadrilateral that can be divided into two triangles with vertices acd and ade. Focusing on triangle abc, we first determine the coordinates of vertices b and c relative to the cell corner (vertex a), using Euclidean geometry to find vertex c. Then we translate the three vertices to a coordinate system centered in the N W cell. This translation is needed in order to integrate fields (Section 3.1.3) in the coordinate system where they have been reconstructed (Section 3.1.1). Repeating this process for the north and east edges of each grid cell, we compute the vertices of all the departure triangles associated with each cell edge. Figure 2, reproduced from [8], shows all possible triangles that can contribute fluxes across the north edge of a grid cell. There are 20 triangles, which can be organized into five groups of four mutually exclusive triangles as shown in Table 3. In this table, (x1 , y1 ) and (x2 , y2 ) are the Cartesian coordinates of the departure points relative to the northwest and northeast cell corners, respectively. The departure points are joined by a straight line that intersects the west edge at (0, ya ) relative to the northwest corner and intersects the east edge at (0, yb ) relative to the northeast corner. The east cell triangles and selecting conditions are identical except for a rotation through 90 degrees. This scheme was originally designed for rectangular grids. Grid cells in CICE actually lie on the surface of a sphere and must be projected onto a plane. The projection used in CICE 4.0 maps each grid cell to a square with sides of unit length. Departure triangles across a given cell edge are computed in a coordinate system whose origin lies at the midpoint of the edge and whose vertices are at (-0.5, 0) and (0.5, 0). Inter- Horizontal transport 15 Triangle group Triangle label Selecting logical condition 1 NW NW1 W W2 ya ya ya ya > 0 and y1 < 0 and y1 < 0 and y1 > 0 and y1 ≥ 0 and x1 ≥ 0 and x1 < 0 and x1 < 0 and x1 <0 <0 <0 <0 2 NE NE1 E E2 yb yb yb yb > 0 and y2 < 0 and y2 < 0 and y2 > 0 and y2 ≥ 0 and x2 ≥ 0 and x2 < 0 and x2 < 0 and x2 >0 >0 >0 >0 3 W1 NW2 E1 NE2 ya < 0 and y1 ≥ 0 and x1 < 0 ya > 0 and y1 < 0 and x1 < 0 yb < 0 and y2 ≥ 0 and x2 > 0 yb > 0 and y2 < 0 and x2 > 0 4 H1a N1a H1b N1b ya yb ≥ 0 and ya + yb < 0 ya yb ≥ 0 and ya + yb > 0 ya yb < 0 and y˜1 < 0 ya yb < 0 and y˜1 > 0 5 H2a N2a H2b N2b ya yb ≥ 0 and ya + yb < 0 ya yb ≥ 0 and ya + yb > 0 ya yb < 0 and y˜2 < 0 ya yb < 0 and y˜2 > 0 Table 3: Evaluation of contributions from the 20 triangles across the north cell edge. The coordinates x1 , x2 , y1 , y2 , ya , and yb are defined in the text. We define y˜1 = y1 if x1 > 0, else y˜1 = ya . Similarly, y˜2 = y2 if x2 < 0, else y˜2 = yb . 16 Model components N NW NW W N 1 W 2 Home NE NW NE E E NE N2a 1 N1a 2 H1a W E H2a Home (a) (b) N NE NW 1 N1b H1b W N2b H2b 2 E Home (c) NW W N 1 NW1 W1 2 Home NE1 E1 NE NW E W (d1) NW2 W2 1 N 2 NE2 NE E2 Home E (d2) Figure 2: The 20 possible triangles that can contribute fluxes across the north edge of a grid cell. section points are computed assuming Cartesian geometry with cell edges meeting at right angles. Let CL and CR denote the left and right vertices, which are joined by line CLR. Similarly, let DL and DR denote the departure points, which are joined by line DLR. Also, let IL and IR denote the intersection points (0, ya ) and (0,yb ) respectively, and let IC = (xc , 0) denote the intersection of CLR and DLR. It can be shown that ya , yb , and xc are given by xCL (yDM − yDL ) + xDM yDL − xDL yDM xDM − xDL , xCR (yDR − yDM ) − xDM yDR + xDR yDM = x − xDM , DR xDR − xDL = xDL − yDL yDR − yDL ya = yb xc Each departure triangle is defined by three of the seven points (CL, CR, DL, DR, IL, IR, IC). Given a 2D velocity field u , the divergence ∇ · u in a given grid cell can be computed from the local velocities and written in terms of fluxes across each cell edge: 1 ∇·u= A uN E + uSE 2 LE + uN W + uSW 2 LW + uN E + uN W 2 LN + uSE + uSW 2 LS , (17) where L is an edge length and the indices N, S, E, W denote compass directions. Equation (17) is equivalent to the divergence computed in the EVP dynamics (Section 3.4). In general, the fluxes in this expression are not equal to those implied by the above scheme for locating departure regions. For some applications it may be desirable to prescribe the divergence by prescribing the area of the departure region for each edge. This can be done in CICE 4.0 by setting l fixed area = true in ice transport driver.F90 and passing the prescribed departure areas (edgearea e and edgearea n) into the remapping routine. An extra triangle is then constructed for each departure region to ensure that the total area is equal to the prescribed value. This idea was suggested and first implemented by Mats Bentsen of the Nansen Environmental and Remote Sensing Center (Norway), who applied an earlier version of the CICE remapping scheme to an ocean model. The implementation in CICE 4.0 is somewhat more general, allowing for departure regions lying on both Horizontal transport 17 sides of a cell edge. The extra triangle is constrained to lie in one but not both of the grid cells that share the edge. Since this option has yet to be fully tested in CICE, the current default is l fixed area = false. We made one other change in the scheme of [8] for locating triangles. In their paper, departure points are defined by projecting cell corner velocities directly backward. That is, xD = −u ∆t, (18) where xD is the location of the departure point relative to the cell corner and u is the velocity at the corner. This approximation is only first-order accurate. Accuracy can be improved by estimating the velocity at the midpoint of the trajectory. 3.1.3 Integrating fields Next, we integrate the reconstructed fields over the departure triangles to find the total area, volume, and energy transported across each cell edge. Area transports are easy to compute since the area is linear in x and y. Given a triangle with vertices xi = (xi , yi ), i ∈ {1, 2, 3}, the triangle area is AT = 1 |(x2 − x1 )(y3 − y1 ) − (y2 − y1 )(x3 − x1 )| . 2 (19) The integral Fa of any linear function f (r) over a triangle is given by Fa = AT f (x0 ), (20) where x0 = (x0 , y0 ) is the triangle midpoint, x0 = 3 1X xi . 3 i=1 (21) To compute the area transport, we evaluate the area at the midpoint, a(x0 ) = ac + ax x0 + ay y0 , (22) and multiply by AT . By convention, northward and eastward transport is positive, while southward and westward transport is negative. Equation (20) cannot be used for volume transport, because the reconstructed volumes are quadratic functions of position. (They are products of two linear functions, area and thickness.) The integral of a quadratic polynomial over a triangle requires function evaluations at three points, Fh = 3 AT X f x0i , 3 i=1 (23) where x0i = (x0 + xi )/2 are points lying halfway between the midpoint and the three vertices. [8] use this formula to compute transports of the product ρ T , which is analogous to ice volume. Equation (23) does not work for ice and snow energies, which are cubic functions—products of area, thickness, and enthalpy. Integrals of a cubic polynomial over a triangle can be evaluated using a four-point formula [42]: " F q = AT 3 9 25 X − f (x0 ) + f (x00i ) 16 48 i=1 # (24) where xi 00 = (3x0 + 2xi )/5. To evaluate functions at specific points, we must compute many products of the form a(x) h(x) and a(x) h(x) q(x), where each term in the product is the sum of a cell-center value and two displacement terms. In the code, the computation is sped up by storing some sums that are used repeatedly. 18 3.1.4 Model components Updating state variables Finally, we compute new values of the state variables in each ice category and grid cell. The new fractional ice areas a0in (i, j) are given by a0in (i, j) = ain (i, j) + FaE (i − 1, j) − FaE (i, j) + FaN (i, j − 1) − FaN (i, j) A(i, j) (25) where FaE (i, j) and FaN (i, j) are the area transports across the east and north edges, respectively, of cell (i, j), and A(i, j) is the grid cell area. All transports added to one cell are subtracted from a neighboring cell; thus (25) conserves total ice area. The new ice volumes and energies are computed analogously. New thicknesses are given by the ratio of volume to area, and enthalpies by the ratio of energy to volume. Tracer monotonicity is ensured because R a h dA , h = RA 0 A a dA R a h q dA q = RA , 0 Aah h0 dA q0 where and are the new-time thickness and enthalpy, given by integrating the old-time ice area, volume, and energy over a Lagrangian departure region with area A. That is, the new-time thickness and enthalpy are weighted averages over old-time values, with non-negative weights a and ah. Thus the new-time values must lie between the maximum and minimum of the old-time values. 3.2 Transport in thickness space Next we solve the equation for ice transport in thickness space due to thermodynamic growth and melt, ∂g ∂ + (f g) = 0, (26) ∂t ∂h which is obtained from (2) by neglecting the first and third terms on the right-hand side. We use the remapping method of [25], in which thickness categories are represented as Lagrangian grid cells whose boundaries are projected forward in time. The thickness distribution function g is approximated as a linear function of h in each displaced category and is then remapped onto the original thickness categories. This method is numerically smooth and is not too diffusive. It can be viewed as a 1D simplification of the 2D incremental remapping scheme described above. We first compute the displacement of category boundaries in thickness space. Assume that at time m m the ice areas am n and mean ice thicknesses hn are known for each thickness category. (For now we omit the subscript i that distinguishes ice from snow.) We use a thermodynamic model (Section 3.5) to compute the new mean thicknesses hm+1 at time m + 1. The time step must be small enough that trajectories do n m+1 m+1 not cross; i.e., hn < hn+1 for each pair of adjacent categories. The growth rate at h = hn is given m )/∆t. By linear interpolation we estimate the growth rate F at the upper category by fn = (hm+1 − h n n n boundary Hn : fn+1 − fn (Hn − hn ). Fn = fn + hn+1 − hn If an or an+1 = 0, Fn is set to the growth rate in the nonzero category, and if an = an+1 = 0, we set Fn = 0. The temporary displaced boundaries are given by Hn∗ = Hn + Fn ∆t, n = 1 to N − 1 Transport in thickness space 19 The boundaries must not be displaced by more than one category to the left or right; that is, we require Hn−1 < Hn∗ < Hn+1 . Without this requirement we would need to do a general remapping rather than an incremental remapping, at the cost of added complexity. Next we construct g(h) in the displaced thickness categories. The ice areas in the displaced categories are am+1 = am n n , since area is conserved following the motion in thickness space (i.e., during vertical ice m+1 . For conciseness, define growth or melting). The new ice volumes are vnm+1 = (an hn )m+1 = am n hn ∗ ∗ HL = Hn−1 and HR = Hn and drop the time index m + 1. We wish to construct a continuous function g(h) within each category such that the total area and volume at time m + 1 are an and vn , respectively: Z HR g dh = an , (27) h g dh = vn . (28) HL Z HR HL The simplest polynomial that can satisfy both equations is a line. It is convenient to change coordinates, writing g(η) = g1 η + g0 , where η = h − HL and the coefficients g0 and g1 are to be determined. Then (27) and (28) can be written as η2 g1 R + g0 ηR = an , 2 3 ηR η2 + g0 R = an ηn , 3 2 where ηR = HR − HL and ηn = hn − HL . These equations have the solution g1 2ηR − ηn , 3 (29) 12an ηR g1 = 3 ηn − . 2 ηR (30) g0 = 6an 2 ηR Since g is linear, its maximum and minimum values lie at the boundaries, η = 0 and ηR : g(0) = 6an 2 ηR 2ηR − ηn = g0 , 3 6an g(ηR ) = 2 ηR ηR ηn − . 3 (31) (32) Equation (31) implies that g(0) < 0 when ηn > 2ηR /3, i.e., when hn lies in the right third of the thickness range (HL , HR ). Similarly, (32) implies that g(ηR ) < 0 when ηn < ηR /3, i.e., when hn is in the left third of the range. Since negative values of g are unphysical, a different solution is needed when hn lies outside the central third of the thickness range. If hn is in the left third of the range, we define a cutoff thickness, HC = 3hn − 2HL , and set g = 0 between HC and HR . Equations (29) and (30) are then valid with ηR redefined as HC − HL . And if hn is in the right third of the range, we define HC = 3hn − 2HR and set g = 0 between HL and HC . In this case, (29) and (30) apply with ηR = HR − HC and ηn = hn − HC . Figure 3 illustrates the linear reconstruction of g for the simple cases HL = 0, HR = 1, an = 1, and hn = 0.2, 0.4, 0.6, and 0.8. Note that g slopes downward (g1 < 0) when hn is less than the midpoint thickness, (HL + HR )/2 = 1/2, and upward when hn exceeds the midpoint thickness. For hn = 0.2 and 0.8, g = 0 over part of the range. Finally, we remap the thickness distribution to the original boundaries by transferring area and volume between categories. We compute the ice area ∆an and volume ∆vn between each original boundary Hn and 20 Model components 3.5 3.0 h = 0.2 h = 0.4 h = 0.6 h = 0.8 2.5 g(h) 2.0 1.5 1.0 0.5 0.0 0 0.2 0.4 0.6 0.8 1 Ice thickness Figure 3: Linear approximation of the thickness distribution function g(h) for an ice category with left boundary HL = 0, right boundary HR = 1, fractional area an = 1, and mean ice thickness hn = 0.2, 0.4, 0.6, and 0.8. displaced boundary Hn∗ . If Hn∗ > Hn , ice moves from category n to n + 1. The area and volume transferred are Z ∗ Hn ∆an = g dh, (33) h g dh. (34) Hn ∗ Hn Z ∆vn = Hn If Hn∗ < HN , ice area and volume are transferred from category n + 1 to n using (33) and (34) with the limits of integration reversed. To evaluate the integrals we change coordinates from h to η = h − HL , where HL is the left limit of the range over which g > 0, and write g(η) using (29) and (30). In this way we obtain the new areas an and volumes vn between the original boundaries Hn−1 and Hn in each category. The new thicknesses, hn = vn /an , are guaranteed to lie in the range (Hn−1 , Hn ). If g = 0 in the part of a category that is remapped to a neighboring category, no ice is transferred. Other conserved quantities are transferred in proportion to the ice volume ∆vin . (We now use the subscripts i and s to distinguish ice from snow.) For example, the transferred snow volume is ∆vsn = vsn (∆vin /vin ), and the transferred ice energy in layer k is ∆eink = eink (∆vin /vin ). The left and right boundaries of the domain require special treatment. If ice is growing in open water at a rate F0 , the left boundary H0 is shifted to the right by F0 ∆t before g is constructed in category 1, then reset to zero after the remapping is complete. New ice is then added to the grid cell, conserving area, volume, and energy. If ice cannot grow in open water (because the ocean is too warm or the net surface energy flux is downward), H0 is fixed at zero, and the growth rate at the left boundary is estimated as F0 = f1 . If F0 < 0, all ice thinner than ∆h0 = −F0 ∆t is assumed to have melted, and the ice area in category 1 is reduced accordingly. The area of new open water is Z ∆a0 = ∆h0 g dh. 0 The right boundary HN is not fixed but varies with hN , the mean ice thickness in the thickest category. Given hN , we set HN = 3hN − 2HN −1 , which ensures that g(h) > 0 for HN −1 < h < HN and g(h) = 0 for h ≥ HN . No ice crosses the right boundary. 3.3 Mechanical redistribution 21 If the ice growth or melt rates in a given grid cell are too large, the thickness remapping scheme will not work. Instead, the thickness categories in that grid cell are treated as delta functions following [3], and categories outside their prescribed boundaries are merged with neighboring categories as needed. For time steps of less than a day and category thickness ranges of 10 cm or more, this simplification is needed rarely, if ever. 3.3 Mechanical redistribution The last term on the right-hand side of (2) is ψ, which describes the redistribution of ice in thickness space due to ridging and other mechanical processes. The mechanical redistribution scheme in CICE is based on [43], [35], [15], [12], and [27]. This scheme converts thinner ice to thicker ice and is applied after horizontal transport. When the ice is converging, enough ice ridges to ensure that the ice area does not exceed the grid cell area. First we specify the participation function: the thickness distribution aP (h) = b(h) g(h) of the ice participating in ridging. (We use “ridging” as shorthand for all forms of mechanical redistribution, including rafting.) The weighting function b(h) favors ridging of thin ice and closing of open water in preference to ridging of thicker ice. There are two options for the form of b(h). If krdg partic = 0 in the namelist, we follow [43] and set ( G(h) 2 ∗ G∗ (1 − G∗ ) if G(h) < G b(h) = (35) 0 otherwise where G(h) is the fractional area covered by ice thinner than h, and G∗ is an empirical constant. Integrating aP (h) between category boundaries Hn−1 and Hn , we obtain the mean value of aP in category n: 2 Gn−1 + Gn (Gn − Gn−1 ) 1 − , G∗ 2G∗ aP n = (36) where aP n is the ratio of the ice area ridging (or open water area closing) in category n to the total area ridging and closing, and Gn is the total fractional ice area in categories 0 to n. Equation (36) applies to categories with Gn < G∗ . If Gn−1 < G∗ < Gn , then (36) is valid with G∗ replacing Gn , and if Gn−1 > G∗ , then aP n = 0. If the open water fraction a0 > G∗ , no ice can ridge, because “ridging” simply reduces the area of open water. As in [43] we set G∗ = 0.15. If the spatial resolution is too fine for a given time step ∆t, the weighting function (35) can promote numerical instability. For ∆t = 1 hour, resolutions finer than ∆x ∼ 10 km are typically unstable. The instability results from feedback between the ridging scheme and the dynamics via the ice strength. If the strength changes significantly on time scales less than ∆t, the viscous-plastic solution of the momentum equation is inaccurate and sometimes oscillatory. As a result, the fields of ice area, thickness, velocity, strength, divergence, and shear can become noisy and unphysical. A more stable weighting function was suggested by [27]: b(h) = exp[−G(h)/a∗ ] a∗ [1 − exp(−1/a∗ )] (37) When integrated between category boundaries, (37) implies aP n = exp(−Gn−1 /a∗ ) − exp(−Gn /a∗ ) 1 − exp(−1/a∗ ) (38) This weighting function is used if krdg partic = 1 in the namelist. From (37), the mean value of G for ice participating in ridging is a∗ , as compared to G∗ /3 for (35). For typical ice thickness distributions, 22 Model components setting a∗ = 0.05 with krdg partic = 1 gives participation fractions similar to those given by G∗ = 0.15 with krdg partic = 0. See [27] for a detailed comparison of these two participation functions. Thin ice is converted to thick, ridged ice in a way that reduces the total ice area while conserving ice volume and internal energy. There are two namelist options for redistributing ice among thickness categories. If krdg redist = 0, ridging is distributed uniformly between √ice∗ of thickness hn forms ridges whose area ∗ Hmin = 2hn and Hmax = 2 H hn , as in [15]. The default value of H is 25 m, as in earlier versions of CICE. Observations suggest that H ∗ = 50 m gives a better fit to first-year ridges [1], although the lower value may be appropriate for multiyear ridges [12]. The ratio of the mean ridge thickness to the thickness of ridging ice is kn = (Hmin + Hmax )/(2hn ). If the area of category n is reduced by ridging at the rate rn , the area of thicker categories grows simultaneously at the rate rn /kn . Thus the net rate of area loss due to ridging of ice in category n is rn (1 − 1/kn ). The ridged ice area and volume are apportioned among categories in the thickness range (Hmin , Hmax ). The fraction of the new ridge area in category m is area fm = HR − HL , Hmax − Hmin (39) where HL = max(Hm−1 , Hmin ) and HR = min(Hm , Hmax ). The fraction of the ridge volume going to category m is (HR )2 − (HL )2 vol fm = . (40) (Hmax )2 − (Hmin )2 This uniform redistribution function tends to produce too little ice in the 3–5 m range and too much ice thicker than 10 m [1]. Observations show that the ITD of ridges is better approximated by a negative exponential. Setting krdg redist = 1 gives ridges with an exponential ITD [27]: gR (h) ∝ exp[−(h − Hmin )/λ] (41) for h ≥ Hmin , with gR (h) = 0 for h < Hmin . Here, λ is an empirical e-folding scale and Hmin = 2hn 1/2 (where hn is the thickness of ridging ice). We assume that λ = µhn , where µ is a tunable parameter 1/2 with units m1/2 . Thus the mean ridge thickness increases in proportion to hn , as in [15]. The default value µ = 4.0 m1/2 gives λ in the range 1–4 m for most ridged ice. Ice strengths with µ = 4.0 m1/2 and krdg redist = 1 are roughly comparable to the strengths with H ∗ = 50 m and krdg redist = 0. From (41) it can be shown that the fractional area going to category m as a result of ridging is area fm = exp[−(Hm−1 − Hmin )/λ] − exp[−(Hm − Hmin )/λ]. (42) The fractional volume going to category m is vol fm = (Hm−1 + λ) exp[−(Hm−1 − Hmin )/λ] − (Hm + λ) exp[−(Hm − Hmin )/λ] . Hmin + λ (43) Equations (42) and (43) replace (39) and (40) when krdg redist = 1. Internal ice energy is transferred between categories in proportion to ice volume. Snow volume and internal energy are transferred in the same way, except that a fraction of the snow may be deposited in the ocean instead of added to the new ridge. The net area removed by ridging and closing is a function of the strain rates. Let Rnet be the net rate of area loss for the ice pack (i.e., the rate of open water area closing, plus the net rate of ice area loss due to ridging). Following [12], Rnet is given by Rnet = Cs (∆ − |DD |) − min(DD , 0), 2 (44) 3.4 Dynamics 23 where Cs is the fraction of shear dissipation energy that contributes to ridge-building, DD is the divergence, and ∆ is a function of the divergence and shear. These strain rates are computed by the dynamics scheme. The default value of Cs is 0.25. P Next, define Rtot = N n=0 rn . This rate is related to Rnet by " Rnet = aP 0 + N X aP n n=1 1 1− kn # Rtot . (45) Given Rnet from (44), we use (45) to compute Rtot . Then the area ridged in category n is given by arn = rn ∆t, where rn = aP n Rtot . The area of new ridges is arn /kn , and the volume of new ridges is arn hn (since volume is conserved during ridging). We remove the ridging ice from category n and use (39) and (40) (or 42) and (43)) to redistribute the ice among thicker categories. Occasionally the ridging rate in thickness category n may be large enough to ridge the entire area an during a time interval less than ∆t. In this case Rtot is reduced to the value that exactly ridges an area an during ∆t. After each ridging iteration, the total fractional ice area ai is computed. If ai > 1, the ridging is repeated with a value of Rnet sufficient to yield ai = 1. The ice strength P may be computed in either of two ways. If the namelist parameter kstrength = 0, we use the strength formula from [14]: P = P ∗ h exp[−C(1 − ai )], (46) where P ∗ = 27, 500 N/m and C = 20 are empirical constants, and h is the mean ice thickness. Alternatively, setting kstrength = 1 gives an ice strength closely related to the ridging scheme. Following [35], the strength is assumed proportional to the change in ice potential energy ∆EP per unit area of compressive deformation. Given uniform ridge ITDs (krdg redist = 0), we have P = Cf Cp β NC X " −aP n h2n n=1 aP n + kn (Hnmax )3 − (Hnmin )3 3(Hnmax − Hnmin ) !# , (47) where CP = (g/2)(ρi /ρw )(ρw − ρi ), β = Rtot /Rnet > 1 from (45), and Cf is an empirical parameter that accounts for frictional energy dissipation. Following [12], we set Cf = 17. The first term in the summation is the potential energy of ridging ice, and the second, larger term is the potential energy of the resulting ridges. The factor of β is included because aP n is normalized with respect to the total area of ice ridging, not the net area removed. Recall that more than one unit area of ice must be ridged to reduce the net ice area by one unit. For exponential ridge ITDs (krdg redist = 1), the ridge potential energy is modified: P = Cf Cp β NC X −aP n h2n + n=1 aP n 2 Hmin + 2Hmin λ + 2λ2 kn (48) The energy-based ice strength given by (47) or (48) is more physically realistic than the strength given by (46). However, use of (46) is less likely to allow numerical instability at a given resolution and time step. See [27] for more details. 3.4 Dynamics The elastic-viscous-plastic (EVP) model represents a modification of the standard viscous-plastic (VP) model for sea ice dynamics [14]. It reduces to the VP model at time scales associated with the wind forcing, while at shorter time scales the adjustment process takes place by a numerically more efficient elastic wave mechanism. While retaining the essential physics, this elastic wave modification leads to a fully explicit numerical scheme which greatly improves the model’s computational efficiency. 24 Model components The EVP sea ice dynamics model is thoroughly documented in [18], [16], [19] and [20]. Simulation results and performance of this model have been compared with the VP model in realistic simulations of the Arctic [21]. Here we summarize the equations and direct the reader to the above references for details. The numerical implementation in this code release is that of [19] and [20]. The force balance per unit area in the ice pack is given by a two-dimensional momentum equation [14], obtained by integrating the 3D equation through the thickness of the ice in the vertical direction: m ∂u = ∇ · σ + ~τa + ~τw − kˆ × mf u − mg∇H◦ , ∂t (49) where m is the combined mass of ice and snow per unit area and ~τa and ~τw are wind and ocean stresses, respectively. The strength of the ice is represented by the internal stress tensor σij , and the other two terms on the right hand side are stresses due to Coriolis effects and the sea surface slope. The parameterization for the wind and ice-ocean stress terms must contain the ice concentration as a multiplicative factor to be consistent with the formal theory of free drift in low ice concentration regions. A careful explanation of the issue and its continuum solution is provided in [20] and [6]. For convenience we formulate the stress tensor σ in terms of σ1 = σ11 + σ22 , σ2 = σ11 − σ22 , and introduce the divergence, DD , and the horizontal tension and shearing strain rates, DT and DS respectively. The internal stress tensor is determined from a regularized version of the VP constitutive law, 1 ∂σ1 σ1 P + + E ∂t 2ζ 2ζ 1 ∂σ2 σ2 + E ∂t 2η 1 ∂σ12 σ12 + E ∂t 2η = DD , (50) = DT , (51) = 1 DS , 2 (52) where DD = ˙11 + ˙22 , (53) = ˙11 − ˙22 , (54) DT DS = 2˙12 , (55) ! 1 ∂ui ∂uj + , 2 ∂xj ∂xi P , ζ = 2∆ P η = , 2∆e2 1/2 1 2 ∆ = DD + 2 DT2 + DS2 , e ˙ij = and P is a function of the ice thickness and concentration, described in Section 3.3. The dynamics component employs a “replacement pressure” (see [13], for example), which serves to prevent residual ice motion due to spatial variations of P when the rates of strain are exactly zero. Several changes have been made to the EVP model since the original release. In the earlier version, the viscosities were held fixed while the stress and momentum equations were subcycled with the smaller time step dte. The reason for implementing the EVP model in this way was to reproduce the results of the original VP model as closely as possible. When solved with time steps of several hours or more, the VP model suffers a linearization error associated with the viscosities, which are lagged over the time step [16]. Dynamics 25 This led to principal stress states that were widely scattered outside the elliptical yield curve in both models [21]. We have addressed this problem by updating the viscosities during the subcycling, so that the entire dynamics component is subcycled within the time step. Taken alone, this change would require an increased number operations to compute the viscosities. However, the new dynamics component is roughly as efficient as the earlier version due to a change in the definition of the elastic parameter E. E is now defined in terms of a damping timescale T for elastic waves, ∆te < T < ∆t, as ζ E= , T where T = E◦ ∆t and E◦ (eyc) is a tunable parameter less than one, as before. The stress equations (50–52) become ∂σ1 σ1 P P + + = DD , ∂t 2T 2T 2T ∆ ∂σ2 e2 σ2 P + = DT , ∂t 2T 2T ∆ P ∂σ12 e2 σ12 + = DS . ∂t 2T 4T ∆ All coefficients on the left-hand side are constant except for P , which changes only on the longer time step ∆t. This modification compensates for the decreased efficiency of including the viscosity terms in the subcycling. (Note that the viscosities do not appear explicitly.) Choices of the parameters used to define E, T and ∆te are discussed in Section 4.4. A different discretization of the stress terms ∂σij /∂xj in the momentum equation is now used, which enabled the discrete equations to be derived from the continuous equations written in curvilinear coordinates. In this manner, metric terms associated with the curvature of the grid were incorporated into the discretization explicitly. We no longer use a “triangle discretization,” in which the strain rates and stresses were constant over each of four subtriangles within each grid cell, and instead use a bilinear approximation for the velocities and stresses. Details pertaining to the spatial discretization are found in [19]. The momentum equation is discretized in time as follows. First, for clarity, the two components of (49) are ∂u ∂σ1j ∂H◦ m = + τax + ai cw ρw |Uw − u| [(Uw − u) cos θ − (Vw − v) sin θ] + mf v − mg , ∂t xj ∂x ∂σ2j ∂v ∂H◦ m = + τay + ai cw ρw |Uw − u| [(Uw − u) sin θ − (Vw − v) cos θ] − mf u − mg . ∂t xj ∂y In the code, vrel = ai cw ρw Uw − uk , where k denotes the subcycling step. The following equations illustrate the time discretization and define some of the other variables used in the code. k+1 ∂σ1j ∂H◦ m + vrel cos θ uk+1 −(mf + vrel sin θ) v k+1 = + τax − mg +vrel (Uw cos θ − Vw sin θ) + uk , | {z } | {z } ∆t xj | ∆t {z } {z ∂x} | {z } | ccb waterx m cca strintx forcex k+1 m ∂σ2j ∂H◦ m (mf + vrel sin θ) uk+1 + + vrel cos θ v k+1 = + τay − mg +vrel (Uw sin θ + Vw cos θ) + v k , | {z } | {z } ∆t x ∂x ∆t j | {z } {z } | {z } | ccb watery cca strinty forcey and vrel·waterx(y) = taux(y). We solve this system of equations analytically for uk+1 and v k+1 . When the subcycling is finished for each (thermodynamic) time step, the ice-ocean stress must be constructed from taux(y) and the terms containing vrel on the left hand side of the equations. This is done in subroutine evp finish. 26 3.5 Model components Thermodynamics The thermodynamic sea ice model is based on [32] and [4], and is described more fully in [24]. For each thickness category the model computes changes in the ice and snow thickness and vertical temperature profile resulting from radiative, turbulent, and conductive heat fluxes. The ice has a temperature-dependent specific heat to simulate the effect of brine pocket melting and freezing. Each thickness category n in each grid cell is treated as a horizontally uniform column with ice thickness hin = vin /ain and snow thickness hsn = vsn /ain . (Henceforth we omit the category index n.) Each column is divided into Ni ice layers of thickness ∆hi = hi /Ni and Ns snow layers of thickness ∆hs = hs /Ns . The surface temperature (i.e., the temperature of ice or snow at the interface with the atmosphere) is Tsf , which cannot exceed 0◦ C. The temperature at the midpoint of the snow layer is Ts , and the midpoint ice layer temperatures are Tik , where k ranges from 1 to Ni . The temperature at the bottom of the ice is held at Tf , the freezing temperature of the ocean mixed layer. All temperatures are in degrees Celsius unless stated otherwise. The vertical salinity profile is prescribed and is unchanging in time. The snow is assumed to be fresh, and the midpoint salinity Sik in each ice layer is given by a 1 Sik = Smax [1 − cos(πz ( z+b ) )], 2 (56) where z ≡ (k − 1/2)/Ni , Smax = 3.2 psu, and a = 0.407 and b = 0.573 are determined from a leastsquares fit to the salinity profile observed in multiyear sea ice by [36]. This profile varies from S = 0 at the top surface (z = 0) to S = Smax at the bottom surface (z = 1) and is similar to that used by [32]. Equation (56) is fairly accurate for ice that has drained at the top surface due to summer melting. It is not a good approximation for cold first-year ice, which has a more vertically uniform salinity because it has not yet drained. However, the effects of salinity on heat capacity are small for temperatures well below freezing, so the salinity error does not lead to significant temperature errors. Each ice layer has an enthalpy qik , defined as the negative of the energy required to melt a unit volume of ice and raise its temperature to 0◦ C. Because of internal melting and freezing in brine pockets, the ice enthalpy depends on the brine pocket volume and is a function of temperature and salinity. Since the salinity is prescribed, there is a one-to-one relationship between temperature and enthalpy. We can also define a snow enthalpy qs , which depends on temperature alone. Expressions for the enthalpy are derived in Section 3.5.3. Given surface forcing at the atmosphere-ice and ice-ocean interfaces along with the ice and snow thicknesses and temperatures/enthalpies at time m, the thermodynamic model advances these quantities to time m + 1. The calculation proceeds in two steps. First we solve a set of equations for the new temperatures, as discussed in Section 3.5.2. Then we compute the melting, if any, of ice or snow at the top surface, and the growth or melting of ice at the bottom surface, as described in Section 3.5.3. We begin by describing the surface forcing parameterizations, which are closely related to the ice and snow surface temperatures. 3.5.1 Thermodynamic surface forcing The net energy flux from the atmosphere to the ice (with all fluxes defined as positive downward) is F0 = Fs + Fl + FL↓ + FL↑ + (1 − α)(1 − i0 )Fsw , where Fs is the sensible heat flux, Fl is the latent heat flux, FL↓ is the incoming longwave flux, FL↑ is the outgoing longwave flux, Fsw is the incoming shortwave flux, α is the shortwave albedo, and i0 is the fraction of absorbed shortwave flux that penetrates into the ice. The albedo may be altered by the presence of melt ponds. Thermodynamics 27 Shortwave radiation: Delta-Eddington Two methods for computing albedo and shortwave fluxes are available, the default (“ccsm3”) method, described below, and a multiple scattering radiative transfer scheme that uses a Delta-Eddington approach. “Inherent” optical properties (IOPs) for snow and sea ice, such as extinction coefficient and single scattering albedo, are prescribed based on physical measurements; reflected, absorbed and transmitted shortwave radiation (“apparent” optical properties) are then computed for each snow and ice layer in a self-consistent manner. Absorptive effects of inclusions in the ice/snow matrix such as dust and algae can also be included, along with radiative treatment of melt ponds and other changes in physical properties, for example granularization associated with snow aging. The present code includes a simple parameterization of surface melt ponds. The Delta-Eddington formulation is described in detail in [5]. Since publication of this technical paper, a number of improvements have been made to the Delta-Eddington scheme, including a surface scattering layer and internal shortwave absorption for snow, generalization for multiple snow layers and more than four layers of ice, and updated IOP values. The melt pond parameterization is designed to be used in conjunction with the Delta-Eddington shortwave scheme. Melt pond volume is carried on each ice thickness category as an area tracer. Defined simply as the product of pond area, ap , and depth, hp , the melt pond volume, vp , grows through addition of ice or snow melt water or rain water, and shrinks when the ice surface temperature becomes cold, ρi ρs ∆t + dhs + Frain , ρw ρw ρw !# max (Tp − Tsf c , 0) , Tp pond growth : vp0 = vp (t) + r1 dhi " pond contraction : vp (t + ∆t) = vp0 exp r2 where dhi and dhs represent ice and snow melt at the top surface of each thickness category, r1 = 0.1 specifies the fraction of available liquid water captured by the ponds, and r2 = 0.01. Here, Tp is a reference temperature equal to -2◦ C. Pond depth is assumed to be a linear function of the pond fraction (hp = 0.8ap ) and is limited by the category ice thickness (hp ≤ 0.9hi ). Ponds are allowed only on ice at least 10 cm thick. The namelist parameters R ice and R pnd adjust the albedo of bare or ponded ice by the product of the namelist value and one standard deviation. For example, if R ice = 0.1, the albedo increases by 0.1σ. Similarly, setting R snw = 0.1 decreases the snow grain radius by 0.1σ (thus increasing the albedo). See [5] for details; the melt pond and Delta-Eddington parameterizations are further explained and validated in a manuscript currently in preparation [2]. Shortwave radiation: CCSM3 In the parameterization used in the previous version of the Community Climate System Model (CCSM3), the albedo depends on the temperature and thickness of ice and snow and on the spectral distribution of the incoming solar radiation. Albedo parameters have been chosen to fit observations from the SHEBA field experiment. For Tsf < −1◦ C and hi > 0.5 m, the bare ice albedo is 0.78 for visible wavelengths (< 700 nm) and 0.36 for near IR wavelengths (> 700 nm). As hi decreases from 0.5 m to zero, the ice albedo decreases smoothly (using an arctangent function) to the ocean albedo, 0.06. The ice albedo in both spectral bands decreases by 0.075 as Tsf rises from −1◦ C to 0◦ C. The albedo of cold snow (Tsf < −1◦ C) is 0.98 for visible wavelengths and 0.70 for near IR wavelengths. The visible snow albedo decreases by 0.10 and the near IR albedo by 0.15 as Tsf increases from −1◦ C to 0◦ C. The total albedo is an area-weighted average of the ice and snow albedos, where the fractional snow-covered area is fsnow = hs , hs + hsnowpatch and hsnowpatch = 0.02 m. The envelope of albedo values is shown in Figure 4. This albedo formulation incorporates the effects of melt ponds implicitly; the explicit melt pond parameterization is not used in this case. 28 Model components Figure 4: Albedo as a function of ice thickness and temperature, for the two extrema in snow depth, for the default (CCSM3) shortwave option. Maximum snow depth is computed based on Archimedes’ Principle for the given ice thickness. These curves represent the envelope of possible albedo values. The net absorbed shortwave flux is Fswabs = (1 − αj )Fsw↓j , where the summation is over four radiative categories (direct and diffuse visible, direct and diffuse near infrared). The flux penetrating into the ice is I0 = i0 Fswabs , where i0 = 0.70 (1 − fsnow ) for visible radiation and i0 = 0 for near IR. Radiation penetrating into the ice is attenuated according to Beer’s Law: P I(z) = I0 exp(−κi z), (57) where I(z) is the shortwave flux that reaches depth z beneath the surface without being absorbed, and κi is the bulk extinction coefficient for solar radiation in ice, set to 1.4 m−1 for visible wavelengths [11]. A fraction exp(−κi hi ) of the penetrating solar radiation passes through the ice to the ocean (Fsw ⇓ ). Longwave radiation, turbulent fluxes While incoming shortwave and longwave radiation are obtained from the atmosphere, outgoing longwave radiation and the turbulent heat fluxes are derived quantities. Outgoing longwave takes the standard blackbody form, FL↑ = σ TsfK 4 , where = 0.95 is the emissivity of snow or ice, σ is the Stefan- TsfK Boltzmann constant and is the surface temperature in Kelvin. (The longwave fluxes are partitioned such that FL↓ is absorbed at the surface, the remaining (1 − ) FL↓ being returned to the atmosphere via FL↑ .) The sensible heat flux is proportional to the difference between air potential temperature and the surface temperature of the snow or snow-free ice, Fs = Cs Θa − TsfK . Cs and Cl (below) are nonlinear turbulent heat transfer coefficients described in Section 2.1. Similarly, the latent heat flux is proportional to the difference between Qa and the surface saturation specific humidity Qsf : Fl = Cl (Qa − Qsf ) , Qsf K = (q1 /ρa ) exp(−q2 /Tsf ), Thermodynamics 29 K is the surface temperature in Kelvin, and ρ is the where q1 = 1.16378 × 107 kg/m3 , q2 = 5897.8 K, Tsf a surface air density. The net downward heat flux from the ice to the ocean is given by [30]: Fbot = −ρw cw ch u∗ (Tw − Tf ), (58) where ρw is the density of seawater, cw is the specific heat of seawater, ch = 0.006 is a heat transfer p coefficient, u∗ = |~τw | /ρw is the friction velocity, and Tw is the sea surface temperature. The minimum value of u∗ depends on whether the model is run coupled; lack of currents in uncoupled runs means that not enough heat is available to melt ice in the standard formulation. We have u∗ min = 5 × 10−3 for coupled runs and 5 × 10−2 for uncoupled runs. Fbot is limited by the total amount of heat available from the ocean, Ff rzmlt . Additional heat, Fside , is used to melt the ice laterally following [31] and [40]. Fbot and the fraction of ice melting laterally are scaled so that Fbot + Fside ≥ Ff rzmlt in the case that Ff rzmlt < 0 (melting). 3.5.2 New temperatures An option for zero-layer thermodynamics [37] is available in this version of CICE by setting the namelist parameter heat capacity to false and changing the number of ice layers, nilyr, in ice domain size.F90 to 1. In the zero-layer case, the ice is fresh and the thermodynamic calculations are much simpler than in the default configuration, which we describe here. m at time m, we solve a set of finite-difference equations to Given the temperatures Tsfm , Tsm , and Tik obtain the new temperatures at time m + 1. Each temperature is coupled to the temperatures of the layers immediately above and below by heat conduction terms that are treated implicitly. For example, the rate of change of Tik depends on the new temperatures in layers k −1, k, and k +1. Thus we have a set of equations of the form Ax = b, (59) where A is a tridiagonal matrix, x is a column vector whose components are the unknown new temperatures, and b is another column vector. Given A and b, we can compute x with a standard tridiagonal solver. There are four general cases: (1) Tsf < 0◦ C, snow present; (2) Tsf = 0◦ C, snow present; (3) Tsf < 0◦ C, snow absent; and (4) Tsf = 0◦ C, snow absent. For case 1 we have one equation (the top row of the matrix) for the new surface temperature, Ns equations for the new snow temperatures, and Ni equations for the new ice temperatures. For cases 2 and 4 we omit the equation for the surface temperature, which is held at 0◦ C, and for cases 3 and 4 we omit the snow temperature equations. Snow is considered absent if the snow depth is less than a user-specified minimum value, hs min. (Very thin snow layers are still transported conservatively by the transport modules; they are simply ignored by the thermodynamics.) The rate of temperature change in the ice interior is given by [32]: ∂Ti ∂ ∂Ti ρi ci = Ki ∂t ∂z ∂z − ∂ [Ipen (z)], ∂z (60) where ρi = 917 kg/m3 is the sea ice density (assumed to be uniform), ci (T, S) is the specific heat of sea ice, Ki (T, S) is the thermal conductivity of sea ice, Ipen is the flux of penetrating solar radiation at depth z, and z is the vertical coordinate, defined to be positive downward with z = 0 at the top surface. If shortwave = ‘default’, the penetrating radiation is given by Beer’s Law: Ipen (z) = I0 exp(−κi z), where I0 is the penetrating solar flux at the top ice surface and κi is an extinction coefficient. If shortwave = ‘dEdd’, then solar absorption is computed by the Delta-Eddington scheme. 30 Model components The specific heat of sea ice is given to an excellent approximation by [34] ci (T, S) = c0 + L0 µS , T2 (61) where c0 = 2106 J/kg/deg is the specific heat of fresh ice at 0◦ C, L0 = 3.34 × 105 J/kg is the latent heat of fusion of fresh ice at 0◦ C, and µ = 0.054 deg/psu is the ratio between the freezing temperature and salinity of brine. Following [45], the thermal conductivity is given by Ki (T, S) = K0 + βS , T (62) where K0 = 2.03 W/m/deg is the conductivity of fresh ice and β = 0.13 W/m/psu is an empirical constant. Experimental results [44] suggest that (62) may not be a good description of the thermal conductivity of sea ice. In particular, the measured conductivity does not markedly decrease as T approaches 0◦ C, but does decrease near the top surface (regardless of temperature). We have not tested alternative parameterizations but will likely do so in the future. The equation for temperature changes in snow is analogous to (60), with ρs = 330 kg/m3 , cs = c0 , and Ks = 0.30 W/m/deg replacing the corresponding ice values. If shortwave = ‘default’, then the penetrating solar radiation is equal to zero for snow-covered ice, since most of the incoming sunlight is absorbed near the top surface. If shortwave = ‘dEdd’, however, then Ipen is nonzero in snow layers. We now convert (60) to finite-difference form. The resulting equations are second-order accurate in space, except possibly at material boundaries, and first-order accurate in time. Before writing the equations in full we give finite-difference expressions for some of the terms. First consider the terms on the left-hand side of (60). We write the time derivatives as ∂T T m+1 − T m = , ∂t ∆t where T is the temperature of either ice or snow and m is a time index. The specific heat of ice layer k is approximated as L0 µSik cik = c0 + m m+1 , (63) Tik Tik which ensures that energy is conserved during a change in temperature. This can be shown by using (61) to m to T m+1 ; the result is c (T m+1 − T m ), where c is given by (63). The specific integrate ci dT from Tik ik ik ik ik ik m+1 heat is a nonlinear function of Tik , the unknown new temperature. We can retain a set of linear equations, m+1 m and then iterating the solution, updating T m+1 in (63) with each however, by initially guessing Tik = Tik ik iteration until the solution converges. Next consider the first term on the right-hand side of (60). The first term describes heat diffusion and is discretized for a given ice or snow layer k as ∂ ∂T K ∂z ∂z = i 1 h ∗ m+1 m+1 ∗ Kk (Tk−1 − Tkm+1 ) − Kk+1 (Tkm+1 − Tk+1 ) , ∆h (64) where ∆h is the layer thickness and Kk is the effective conductivity at the upper boundary of layer k. This discretization is centered and second-order accurate in space, except at the boundaries. The flux terms on the right-hand side (RHS) are treated implicitly; i.e., they depend on the temperatures at the new time m + 1. The resulting scheme is first-order accurate in time and unconditionally stable. The effective conductivity K ∗ at the interface of layers k − 1 and k is defined as Kk∗ = 2Kk−1 Kk , Kk−1 hk + Kk hk−1 Thermodynamics 31 which reduces to the appropriate values in the limits Kk Kk−1 (or vice versa) and hk hk−1 (or vice versa). The effective conductivity at the top (bottom) interface of the ice-snow column is given by K ∗ = 2K/∆h, where K and ∆h are the thermal conductivity and thickness of the top (bottom) layer. The second term on the RHS of (60) is discretized as ∂ τk−1 − τk Ik [Ipen (z)] = I0 = ∂z ∆h ∆h where τk is the fraction of the penetrating solar radiation I0 that is transmitted through layer k without being absorbed. We now construct a system of equations for the new temperatures. For Tsf < 0◦ C we require F0 = Fct , (65) where Fct is the conductive flux from the top surface to the ice interior, and both fluxes are evaluated at time m + 1. Although F0 is a nonlinear function of Tsf , we can make the linear approximation dF0 dTsf F0m+1 = F0∗ + !∗ (Tsfm+1 − Tsf∗ ), where Tsf∗ is the surface temperature from the most recent iteration, and F0∗ and (dF0 /dTsf )∗ are functions of Tsf∗ . We initialize Tsf∗ = Tsfm and update it with each iteration. Thus we can rewrite (65) as F0∗ + dF0 dTsf !∗ (Tsfm+1 − Tsf∗ ) = K1∗ (Tsfm+1 − T1m+1 ), Rearranging terms, we obtain " dF0 dTsf !∗ # − K1∗ Tsfm+1 + K1∗ T1m+1 = dF0 dTsf !∗ Tsf∗ − F0∗ , (66) the first equation in the set of equations (59). The temperature change in ice/snow layer k is ρk ck (Tkm+1 − Tkm ) 1 m+1 = [K ∗ (T m+1 − Tkm+1 ) − Kk+1 (Tkm+1 − Tk+1 )], ∆t ∆hk k k−1 (67) where T0 = Tsf in the equation for layer 1. In tridiagonal matrix form, (67) becomes m+1 m+1 −ηk Kk Tk−1 + [1 + ηk (Kk + Kk+1 )] Tkm+1 − ηk Kk+1 Tk+1 = Tkm + ηk Ik , (68) where ηk = ∆t/(ρk ck ∆hk ). In the equation for the bottom ice layer, the temperature at the ice-ocean interface is held fixed at Tf , the freezing temperature of the mixed layer; thus the last term on the LHS is known and is moved to the RHS. If Tsf = 0◦ C, then there is no surface flux equation. In this case the first equation in (59) is similar to (68), but with the first term on the LHS moved to the RHS. These equations are modified if Tsf and Fct are computed within the atmospheric model and passed to CICE (calc Tsfc = false; see Section 2). In this case there is no surface flux equation. The top layer temperature is computed by an equation similar to (68) but with the first term on the LHS replaced by η1 Fct and moved to the RHS. The main drawback of treating the surface temperature and fluxes explicitly is that the solution scheme is no longer unconditionally stable. Instead, the effective conductivity in the top layer must satisfy a diffusive CFL condition: ρch K∗ ≤ . ∆t 32 Model components For thin layers and typical coupling intervals (∼ 1 hr), K ∗ may need to be limited before being passed to the atmosphere via the coupler. Otherwise, the fluxes that are returned to CICE may result in oscillating, highly inaccurate temperatures. The effect of limiting is to treat the ice as a poor heat conductor. As a result, winter growth rates are reduced, and the ice is likely to be too thin (other things being equal). The values of hs min and ∆t must therefore be chosen with care. If hs min is too small, frequent limiting is required, but if hs min is too large, snow will be ignored when its thermodynamic effects are significant. Likewise, infrequent coupling requires more limiting, whereas frequent coupling is computationally expensive. This completes the specification of the matrix equations for the four cases. We compute the new temperatures using a tridiagonal solver. After each iteration we check to see whether the following conditions hold: 1. Tsf ≤ 0◦ C. 2. The change in Tsf since the previous iteration is less than a prescribed limit, ∆Tmax . 3. F0 ≥ Fct . (If F0 < Fct , ice would be growing at the top surface, which is not allowed.) 4. The rate at which energy is added to the ice by the external fluxes equals the rate at which the internal ice energy is changing, to within a prescribed limit ∆Fmax . We also check the convergence rate of Tsf . If Tsf is oscillating and failing to converge, we average temperatures from successive iterations to improve convergence. When all these conditions are satisfied—usually within two to four iterations for ∆Tmax ≈ 0.01◦ C and ∆Fmax ≈ 0.01 W/m2 —the calculation is complete. 3.5.3 Growth and melting We first derive expressions for the enthalpy q. The enthalpy of snow (or fresh ice) is given by qs (T ) = −ρs (−c0 T + L0 ). Sea ice enthalpy is more complex, because of brine pockets whose salinity varies inversely with temperature. The specific heat of sea ice, given by (61), includes not only the energy needed to warm or cool ice, but also the energy used to freeze or melt ice adjacent to brine pockets. Equation (61) can be integrated to give the energy δe required to raise the temperature of a unit mass of sea ice of salinity S from T to T 0 : δe (T, T 0 ) = c0 (T 0 − T ) + L0 µS 1 1 − 0 . T T If we let T 0 = Tm ≡ −µS, the temperature at which the ice is completely melted, we have δe (T, Tm ) = c0 (Tm − T ) + L0 Tm 1− . T Multiplying by ρi to change the units from J/kg to J/m3 and adding a term for the energy needed to raise the meltwater temperature to 0◦ C, we obtain the sea ice enthalpy: qi (T, S) = −ρi c0 (Tm − T ) + L0 Tm 1− T − cw Tm . (69) Note that (69) is a quadratic equation in T . Given the layer enthalpies we can compute the temperatures using the quadratic formula: √ −b − b2 − 4ac T = , 2a Thermodynamics 33 where a = c0 , b = (cw − c0 ) Tm − qi − L0 , ρi c = L0 Tm . The other root is unphysical. Melting at the top surface is given by ( q δh = (F0 − Fct ) ∆t if F0 > Fct 0 otherwise (70) where q is the enthalpy of the surface ice or snow layer (recall that q < 0) and δh is the change in thickness. If the layer melts completely, the remaining flux is used to melt the layers beneath. Any energy left over when the ice and snow are gone is added to the ocean mixed layer. Ice cannot grow at the top surface due to conductive fluxes; however, snow-ice can form. New snowfall is added at the end of the thermodynamic time step. Growth and melting at the bottom ice surface are governed by q δh = (Fcb − Fbot ) ∆t, (71) where Fbot is given by (58) and Fcb is the conductive heat flux at the bottom surface: Fcb = Ki,N +1 (TiN − Tf ). ∆hi If ice is melting at the bottom surface, q in (71) is the enthalpy of the bottom ice layer. If ice is growing, q is the enthalpy of new ice with temperature Tf and salinity Smax . This ice is added to the bottom layer. If the latent heat flux is negative (i.e., latent heat is transferred from the ice to the atmosphere), snow or snow-free ice sublimates at the top surface. If the latent heat flux is positive, vapor from the atmosphere is deposited at the surface as snow or ice. The thickness change of the surface layer is given by (ρLv − q)δh = Fl ∆t, (72) where ρ is the density of the surface material (snow or ice), and Lv = 2.501 × 106 J/kg is the latent heat of vaporization of liquid water at 0◦ C. Note that ρLv is nearly an order of magnitude larger than typical values of q. For positive latent heat fluxes, the deposited snow or ice is assumed to have the same enthalpy as the existing surface layer. After growth and melting, the various ice layers no longer have equal thicknesses. We therefore adjust the layer interfaces, conserving energy, so as to restore layers of equal thickness ∆hi = hi /Ni . This is done by computing the overlap ηkm of each new layer k with each old layer m: ηkm = min(zm , zk ) − max(zm−1 , zk−1 ), where zm and zk are the vertical coordinates of the old and new layers, respectively. The enthalpies of the new layers are qk = N 1 Xi ηkm qm . ∆hi m=1 34 Numerical implementation At the end of the time step we check whether the snow is deep enough to lie partially below the surface of the ocean (freeboard). From Archimedes’ principle, the base of the snow is at sea level when ρi h i + ρs h s = ρw h i . Thus the snow base lies below sea level when h∗ ≡ hs − (ρw − ρi )hi > 0. ρs In this case we raise the snow base to sea level by converting some snow to ice: δhs = δhi = −ρi h∗ , ρw ρs h ∗ . ρw In rare cases this process can increase the ice thickness substantially. For this reason snow-ice conversions are postponed until after the remapping in thickness space (Section 3.2), which assumes that ice growth during a single time step is fairly small. Lateral melting is accomplished by multiplying the state variables by 1−rside , where rside is the fraction of ice melted laterally, and adjusting the ice energy and fluxes as appropriate. 4 Numerical implementation CICE is written in FORTRAN90 and runs on platforms using UNIX, LINUX, and other operating systems. The code is parallelized via grid decomposition with MPI and includes some optimizations for vector architectures. A second, “external” layer of parallelization involves message passing between CICE and the flux coupler, which may be running on different machines in a distributed system. The parallelization scheme for CICE was designed so that MPI could be used for the coupling along with either MPI or no parallelization internally. The internal parallelization method is set at compile time with the NTASK definition in the compile script. Message passing between the ice model and the CCSM flux coupler is accomplished with MPI, regardless of the type of internal parallelization used for CICE, although the ice model may be coupled to another system without using MPI. 4.1 Directory structure The present code distribution includes make files, several scripts and some input files. The main directory is cice/, and a run directory (rundir/) is created upon initial execution of the script comp ice. One year of atmospheric forcing data is also available from the code distribution web site (see the README file for details). cice/ README v4.0 basic information bld/ makefiles Macros.hOSi.hSITEi.hmachinei macro definitions for the given operating system, used by Makefile.hOSi Makefile.hOSi primary makefile for the given operating system (hstdi works for most systems) Directory structure 35 makedep.c perl script that determines module dependencies clean ice script that removes files from the compile directory comp ice script that sets up the run directory and compiles the code csm share/ modules based on “shared” code in CCSM shr orb mod.F90 orbital parameterizations doc/ documentation cicedoc.pdf this document PDF/ PDF documents of numerous publications related to CICE drivers/ institution-specific modules cice4/ official driver for CICE v.4 (LANL) CICE.F90 main program CICE.F90 debug debugging version of CICE.F90 CICE FinalMod.F90 routines for finishing and exiting a run CICE InitMod.F90 routines for initializing a run CICE RunMod.F90 main driver routines for time stepping ice constants.F90 physical and numerical constants and parameters esmf/ Earth System Modeling Framework driver (www.esmf.ucar.edu) CICE.F90 main program CICE ComponentMod.F90 subroutinized version of CICE.F90 for direct coupling CICE FinalMod.F90 routines for finishing and exiting a run CICE InitMod.F90 routines for initializing a run CICE RunMod.F90 main driver routines for time stepping ice constants.F90 physical and numerical constants and parameters ice.log.hOSi.hSITEi.hmachinei sample diagnostic output files input templates/ input files that may be modified for other CICE configurations gx1/ h1◦ i displaced pole grid files global gx1.grid h1◦ i displaced pole grid (binary) global gx1.kmt h1◦ i land mask (binary) ice.restart file pointer for restart file name ice in namelist input data (data paths depend on particular system) iced gx1 v4.0 kcatbound0 restart file used for initial condition gx3/ h3◦ i displaced pole grid files global global global global gx3.grid h3◦ i displaced pole grid (binary) gx3.kmt h3◦ i land mask (binary) gx3.grid.nc h3◦ i displaced pole grid (netCDF) gx3.kmt.nc h3◦ i land mask (netCDF) 36 Numerical implementation ice.restart file pointer for restart file name ice in namelist input data (data paths depend on particular system) iced gx3 v4.0 kcatbound0 restart file used for initial condition run ice.hOSi.hSITEi.hmachinei sample script for running on the given operating system mpi/ modules that require MPI calls ice boundary.F90 boundary conditions ice broadcast.F90 routines for broadcasting data across processors ice communicate.F90 routines for communicating between processors ice exit.F90 aborts or exits the run ice gather scatter.F90 gathers/scatters data to/from one processor from/to all processors ice global reductions.F90 global sums, minvals, maxvals, etc., across processors ice timers.F90 timing routines serial/ same modules as in mpi/ but without MPI calls source/ general CICE source code ice age.F90 handles most work associated with the age tracer ice atmo.F90 stability-based parameterization for calculation of turbulent ice-atmosphere fluxes ice blocks.F90 for decomposing global domain into blocks ice calendar.F90 keeps track of what time it is ice diagnostics.F90 miscellaneous diagnostic and debugging routines ice distribution.F90 for distributing blocks across processors ice domain.F90 decompositions, distributions and related parallel processing info ice domain size.F90 domain and block sizes ice dyn evp.F90 elastic-viscous-plastic dynamics component ice fileunits.F90 unit numbers for I/O ice flux.F90 fluxes needed/produced by the model ice forcing.F90 routines to read and interpolate forcing data for stand-alone ice model runs ice grid.F90 grid and land masks ice history.F90 netCDF or binary output routines ice init.F90 namelist and initializations ice itd.F90 utilities for managing ice thickness distribution ice kinds mod.F90 basic definitions of reals, integers, etc. ice mechred.F90 mechanical redistribution component (ridging) ice meltpond.F90 melt pond parameterization ice ocean.F90 mixed layer ocean model ice orbital.F90 orbital parameters for Delta-Eddington shortwave parameterization ice read write.F90 utilities for reading and writing files 4.2 Grid, boundary conditions and masks 37 ice restart.F90 read/write core restart file ice restoring.F90 basic restoring for open boundary conditions ice shortwave.F90 shortwave and albedo parameterizations ice spacecurve.F90 space-filling-curves distribution method ice state.F90 essential arrays to describe the state of the ice ice step mod.F90 routines for time stepping the major code components ice therm itd.F90 thermodynamic changes mostly related to ice thickness distribution ice therm vertical.F90 vertical growth rates and fluxes ice transport driver.F90 driver for horizontal advection ice transport remap.F90 horizontal advection via incremental remapping ice work.F90 globally accessible work arrays rundir/ execution or “run” directory created when the code is compiled using the comp ice script (gx3) cice code executable compile/ directory containing object files, etc. grid horizontal grid file from cice/input templates/gx3/ ice.log.[ID] diagnostic output file ice in namelist input data from cice/input templates/gx3/ hist/iceh.[timeID].nc output history file kmt land mask file from cice/input templates/gx3/ restart/ restart directory iced gx3 v4.0 kcatbound0 initial condition from cice/input templates/gx3/ ice.restart file restart pointer from cice/input templates/gx3/ run ice batch run script file from cice/input templates/ 4.2 Grid, boundary conditions and masks The spatial discretization is specialized for a generalized orthogonal B-grid as in [33] or [39]. The ice and snow area, volume and energy are given at the center of the cell, velocity is defined at the corners, and the internal ice stress tensor takes four different values within a grid cell; bilinear approximations are used for the stress tensor and the ice velocity across the cell, as described in [19]. This tends to avoid the grid decoupling problems associated with the B-grid. EVP is available on the C-grid through the MITgcm code distribution, http://mitgcm.org/cgi-bin/viewcvs.cgi/MITgcm/pkg/seaice Since ice thickness and thermodynamic variables such as temperature are given in the center of each cell, the grid cells are referred to as “T cells.” We also occasionally refer to “U cells,” which are centered on the northeast corner of the corresponding T cells and have velocity in the center of each. The velocity components are aligned along grid lines. The user has several choices of grid routines: popgrid reads grid lengths and other parameters for a nonuniform grid (including tripole grids), and rectgrid creates a regular rectangular grid. The panarctic grid is a regional configuration [28]. The input files global gx3.grid and global gx3.kmt contain the h3◦ i POP grid and land mask; global gx1.grid and global gx1.kmt contain the h1◦ i grid and land mask. These are binary unformatted, direct access files produced on an SGI (Big Endian). If you are using an incompatible 38 Numerical implementation Figure 5: Grid parameters for a sample one-dimensional, 20-cell global domain decomposed into four local subdomains. Each local domain has one ghost cell on each side, and the physical portion of the local domains are labeled ilo:ihi. The parameter nx block is the total number of cells in the local domain, including ghost cells, and the same numbering system is applied to each of the four subdomains. (Little Endian) architecture, choose rectangular instead of displaced pole in ice in, or follow procedures as for coyote (hOSi.hSITEi.hmachinei = Linux.LANL.coyote). There are netCDF versions of the gx3 grid files available. In general, the global gridded domain is nx global×ny global, while the subdomains used in the block distribution are nx block×ny block. The physical portion of a subdomain is indexed as [ilo:ihi,jlo:jhi], with nghost “ghost” or “halo” cells outside the domain used for boundary conditions. These parameters are illustrated in Figure 5 in one dimension. The routines global scatter and global gather distribute information from the global domain to the local domains and back, respectively. If MPI is not being used for grid decomposition in the ice model, these routines simply adjust the indexing on the global domain to the single, local domain index coordinates. Although we recommend that the user choose the local domains so that the global domain is evenly divided, if this is not possible then the furthest east and/or north blocks will contain nonphysical points (“padding”). These points are excluded from the computation domain and have little effect on model performance. The user chooses a block size BLCKX×BLCKY and the number of processors NTASK in comp ice. Parameters in the domain nml namelist in ice in determine how the blocks are distributed across the processors, and how the processors are distributed across the grid domain. Recommended combinations of these parameters for best performance are given in Section 4.7. The script comp in computes the maximum number of blocks on each processor for typical Cartesian distributions, but for non-Cartesian cases MXBLCKS may need to be set in the script. The code will print this information to the log file before aborting, and the user will need to adjust MXBLCKS in comp ice and recompile. The code will also print a warning if the maximum number of blocks is too large. Although this is not fatal, it does require excess memory. A loop at the end of routine create blocks in module ice blocks.F90 will print the locations for all of the blocks on the global grid if dbug is set to be true. Likewise, a similar loop at the end of routine create local block ids in module ice distribution.F90 will print the processor and local block number for each block. With this information, the grid decomposition into processors and blocks can be ascertained. The dbug flag must be manually set in the code in each case (independently of the dbug flag in ice in), as there may be hundreds or thousands of blocks to print and this information should be needed only rarely. This information is much easier to look at using a debugger such as Totalview. Open boundary conditions are the default in CICE; the physical domain can still be closed using the 4.3 Initialization and coupling 39 land mask. In our bipolar, displaced-pole grids, one row of grid cells along the north and south boundaries is located on land, and along east/west domain boundaries not masked by land, periodic conditions wrap the domain around the globe. CICE can be run on regional grids with open boundary conditions. Except for variables describing grid lengths, non-land ghost cells along the grid edge are filled using Neumann (reflecting) conditions, as would be appropriate for an open boundary in mid-ocean. The ice state can be restored to data values along open boundaries, also; restoring is not needed if the boundaries are only an ice sink (the ice simply flows out), but it is needed for an ice source. The namelist variable restore ice turns this functionality on and off and currently uses the restoring timescale trestore (also used for restoring ocean sea surface temperature in stand-alone ice runs). This implementation is only intended to provide the “hooks” for a more sophisticated treatment; the rectangular grid option can be used to test this configuration. Much of the infrastructure used in CICE, including the boundary routines, is adopted from POP. The boundary routines perform boundary communications among processors when MPI is in use and among blocks whenever there is more than one block per processor. A land mask hm (Mh ) is specified in the cell centers, with 0 representing land and 1 representing ocean cells. A corresponding mask uvm (Mu ) for velocity and other corner quantities is given by Mu (i, j) = min{Mh (l), l = (i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1)}. The logical masks tmask and umask (which correspond to the real masks hm and uvm, respectively) are useful in conditional statements. In addition to the land masks, two other masks are implemented in evp prep in order to reduce the dynamics component’s work on a global grid. At each time step the logical masks ice tmask and ice umask are determined from the current ice extent, such that they have the value “true” wherever ice exists. They also include a border of cells around the ice pack for numerical purposes. These masks are used in the dynamics component to prevent unnecessary calculations on grid points where there is no ice. They are not used in the thermodynamics component, so that ice may form in previously ice-free cells. Like the land masks hm and uvm, the ice extent masks ice tmask and ice umask are for T cells and U cells, respectively. Two additional masks are created for the user’s convenience: lmask n and lmask s can be used to compute or write data only for the northern or southern hemispheres, respectively. Special constants (spval and spval dbl, each equal to 1030 ) are used to indicate land points in the history files and diagnostics. 4.3 Initialization and coupling The ice model’s parameters and variables are initialized in several steps. Many constants and physical parameters are set in ice constants.F90. Namelist variables (Table 7), whose values can be altered at run time, are handled in input data and other initialization routines. These variables are given default values in the code, which may then be changed when the input file ice in is read. Other physical constants, numerical parameters, and variables are first set in initialization routines for each ice model component or module. Then, if the ice model is being restarted from a previous run, some variables are read and reinitialized in restartfile. Finally, albedo and other quantities dependent on the initial ice state are set. Some of these parameters will be described in more detail in Table 7. Three namelist variables control model initialization, ice ic, runtype, and restart, as described in Table 4. It is possible to do an initial run from a file filename in two ways: (1) set runtype = ‘initial’, restart = true and ice ic = filename, or (2) runtype = ‘continue’ and pointer file = ./restart/ice.restart file where ./restart/ice.restart file contains the line “./restart/filename”. The first option is convenient when repeatedly starting from a given file when subsequent restart files have been written. 40 Numerical implementation ice ic none default filename initial/false no ice SST/latitude dependent no icec runtype/restart initial/true no iceb SST/latitude dependentb start from filename continue/true (or falsea ) restart using pointer file restart using pointer file restart using pointer file Table 4: Ice initial state resulting from combinations of ice ic, runtype and restart. restart is reset to true. b restart is reset to false. c ice ic is reset to ‘none.’ a If false, MPI is initialized in init communicate for both coupled and stand-alone MPI runs. The ice component communicates with a flux coupler or other climate components via external routines that handle the variables listed in Table 1. For stand-alone runs, routines in ice forcing.F90 read and interpolate data from files, and are intended merely to provide guidance for the user to write his or her own routines. Whether the code is to be run in stand-alone or coupled mode is determined at compile time, as described below. 4.4 Choosing an appropriate time step The time step is chosen based on stability of the transport component (both horizontal and in thickness space) and on resolution of the physical forcing. CICE allows the dynamics, advection and ridging portion of the code to be run with a shorter timestep, ∆tdyn (dt dyn), than the thermodynamics timestep ∆t (dt). In this case, dt and the integer ndyn dt are specified, and dt dyn = dt/ndyn dt. A conservative estimate of the horizontal transport time step bound, or CFL condition, under remapping yields min (∆x, ∆y) ∆tdyn < . 2 max (u, v) Numerical estimates for this bound for several POP grids, assuming max(u, v) = 0.5 m/s, are as follows: √ grid label N pole singularity dimensions min ∆x · ∆y max ∆tdyn gx3 Greenland 100 × 116 39 × 103 m 10.8 hr gx1 Greenland 320 × 384 18 × 103 m 5.0 hr p4 Canada 900 × 600 6.5 × 103 m 1.8 hr As discussed in section 3.3 and [27], the maximum time step in practice is usually determined by the time scale for large changes in the ice strength (which depends in part on wind strength). Using the strength parameterization of [35], as in eq. 47, limits the time step to ∼30 minutes for the old ridging scheme, and to ∼2 hours for the new scheme, assuming ∆x = 10 km. Practical limits may be somewhat less, depending on the strength of the atmospheric winds. Transport in thickness space imposes a similar restraint on the time step, given by the ice growth/melt rate and the smallest range of thickness among the categories, ∆t < min(∆H)/2 max(f ), where ∆H is the distance between category boundaries and f is the thermodynamic growth rate. For the 5-category ice thickness distribution used as the default in this distribution, this is not a stringent limitation: ∆t < 19.4 hr, assuming max(f ) = 40 cm/day. The dynamics component is subcycled ndte (N ) times per dynamics time step so that the elastic waves essentially disappear before the next time step. The subcycling time step (∆te ) is thus dte = dt dyn/ndte. A second parameter, E◦ (eyc), must be selected, which defines the elastic wave damping timescale T , described in Section 3.4, as eyc*dt dyn. The forcing terms are not updated during the subcycling. Given 4.5 Model output 41 the small step (dte) at which the EVP dynamics model is subcycled, the elastic parameter E is also limited by stability constraints, as discussed in [18]. Linear stability analysis for the dynamics component shows that the numerical method is stable as long as the subcycling time step ∆te sufficiently resolves the damping timescale T . For the stability analysis we had to make several simplifications of the problem; hence the location of the boundary between stable and unstable regions is merely an estimate. In practice, the ratio ∆te : T : ∆t = 1 : 40 : 120 provides both stability and acceptable efficiency for time steps (∆t) on the order of 1 hour. Note that only T and ∆te figure into the stability of the dynamics component; ∆t does not. Although the time step may not be tightly limited by stability considerations, large time steps (eg., ∆t = 1 day, given daily forcing) do not produce accurate results in the dynamics component. The reasons for this error are discussed in [18]; see [21] for its practical effects. The thermodynamics component is stable for any time step, as long as the surface temperature Tsf c is computed internally. 4.5 Model output Model output data is averaged over the period given by histfreq and histfreq n, and written to binary or netCDF files prepended by history file in ice in. That is, if history file = ‘iceh’ then the filenames will have the form iceh.[timeID].nc or iceh.[timeID].da, depending on the history format chosen. The netCDF history files are now CF-compliant; header information for data contained in the netCDF files is displayed with the command ncdump -h filename.nc. With binary files, a separate header file is written with equivalent information. Standard fields are output according to settings in the icefields nml namelist in ice in.The user may add (or subtract) variables not already available in the namelist by following the instructions in section 4.8.2. If write ic is set to T in ice in, a snapshot of the same set of history fields at the start of the run will be written to the history directory in iceh ic.[timeID].nc(da). The normalized principal components of internal ice stress are computed in principal stress and written to the history file. This calculation is not necessary for the simulation; principal stresses are merely computed for diagnostic purposes and included here for the user’s convenience. Like histfreq, the parameters diagfreq and diagfreq n can be used to regulate how often output is written to a log file. The log file unit to which diagnostic output is written is set in ice fileunits.F90. If diag type = ‘stdout’, then it is written to standard out (or to ice.log.[ID] if you redirect standard out as in run ice); otherwise it is written to the file given by diag file. In addition to the standard diagnostic output (maximum area-averaged thickness, velocity, average albedo, total ice area, and total ice and snow volumes), the namelist options print points and print global cause additional diagnostic information to be computed and written. print global outputs global sums that are useful for checking global conservation of mass and energy. print points writes data for two specific grid points. Currently, one point is near the North Pole and the other is in the Weddell Sea; these may be changed in ice diagnostics.F90. A binary unformatted file is created that contains all of the data that CICE needs for a full restart. The filename begins with the character string dumpfile, and the restart dump frequency is given by dumpfreq and dumpfreq n. The pointer to the filename from which the restart data is to be read for a continuation run is set in pointer file. Timers are declared and initialized in ice timers.F90, and the code to be timed is wrapped with calls to ice timer start and ice timer stop. Finally, ice timer print writes the results to the log file. The optional “stats” argument (true/false) prints additional statistics. Calling ice timer print all prints all of the timings at once, rather than having to call each individually. Currently, the timers are set up as in Table 5. Section 4.8.1 contains instructions for adding timers. The timings provided by these timers are not mutually exclusive. For example, the column timer (5) 42 Numerical implementation Timer Index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Label Total Step Dynamics Advectn Column Thermo Shortwave Ridging Cat Conv Coupling ReadWrite Diags History Bound the entire run total minus initialization and exit EVP horizontal transport all vertical (column) processes vertical thermodynamics SW radiation and albedo mechanical redistribution transport in thickness space sending/receiving coupler messages reading/writing files diagnostics (log file) history output boundary conditions and subdomain communications Table 5: CICE timers. includes the timings from 6, 7, 8 and 9, and subroutine bound (timer 14) is called from many different places in the code, including the dynamics and advection routines. The timers use MPI WTIME for parallel runs and the F90 intrinsic system clock for single-processor runs. 4.6 Execution procedures To compile and execute the code: in the source directory, 1. Download the forcing data used for testing from the CICE website, http://climate.lanl.gov/Models/CICE/. 2. Create Macros.* and run ice.* files for your particular platform, if they do not already exist (type ‘uname -s’ at the prompt to get hOSi). 3. Alter directories in the script comp ice. 4. Run comp ice to set up the run directory and make the executable ‘cice’. 5. To clean the compile directory and start fresh, alter directories in the script clean ice and execute the script. In the run directory, 1. Alter atm data dir and ocn data dir in the namelist file ice in. 2. Alter the script run ice for your system. 3. Execute run ice. Execution procedures 43 variable options description RES gx3, gx1 grid resolution BINTYPE MPI use MPI for internal parallelization NTASK (integer) total number of processors BLCKX (integer) number of grid cells on each block in the x-direction† BLCKY (integer) number of grid cells on each block in the y-direction† MXBLCKS (integer) maximum number of blocks per processor USE ESMF yes/no for coupling with the Earth System Modeling Framework yes/no for single-column CAM runs CAM ICE NETCDF yes/no use ‘no’ if netCDF library is unavailable DITTO yes/no for reproducible diagnostics † Does not include ghost cells. Table 6: Configuration options available in comp ice. If this fails, see Section 5.1. This procedure creates the output log file ice.log.[ID], and if npt is long enough compared with dumpfreq and histfreq, dump files iced.[timeID] and netCDF (or binary) history output files iceh [timeID].nc (.da). Using the h3◦ i grid, the log file should be similar to ice.log.hOSi, provided for the user’s convenience. These log files were created using MPI on 4 processors on the h3◦ i grid. Several options are available in comp ice for configuring the run, shown in Table 6. If NTASK = 1, then the serial/ code is used, otherwise the code in mpi/ is used. Note that the value of NTASK in comp ice must equal the value of nprocs in ice in. Generally the value of MXBLCKS computed by comp ice is sufficient, but sometimes it will need to be set explicitly, as discussed in Section 4.7. The scripts define a number of environment variables, mostly as directories that you will need to edit for your own environment. $SYSTEM USERDIR, which on machines at Oak Ridge National Laboratory points automatically to scratch space, is intended to be a disk where the run directory resides. The ‘reproducible’ option (DITTO) makes diagnostics bit-for-bit when varying the number of processors. (The simulation results are bit-for-bit regardless, because they do not require global sums or max/mins as do the diagnostics.) This was done mainly by increasing the precision for the global reduction calculations, except for regular double-precision (r8) calculations involving MPI; MPI can not handle MPI REAL16 on some architectures. Instead, these cases perform sums or max/min calculations across the global block structure, so that the results are bit-for-bit as long as the block distribution is the same (the number of processors can be different). CICE namelist variables available for changes after compile time appear in ice.log.* with values read from the file ice in; their definitions are given in Section 5.5. For example, to run for a different length of time, say three days, set npt = 72 in ice in. At present, the user supplies the time step dt, the number of dynamics/advection/ridging subcycles ndyn dt, and the number of evp subcycles ndte, and dte is then calculated in subroutine init evp. The primary reason for doing it this way is to ensure that ndte is an integer. To restart from a previous run, set restart = .true. in ice in. There are two ways of restarting from a given file. The restart pointer file ice.restart file (created by the previous run) contains the name of the last written data file (iced.[timeID]). Alternatively, a filename can be assigned to ice ic in ice in. Consult Section 4.3 for more details. Restarts are exact for MPI or single processor runs. 44 Numerical implementation a b c d e f Figure 6: Distribution of 256 blocks across 16 processors, represented by colors, on the gx1 grid: (a) cartesian, slenderX1, (b) cartesian, slenderX2, (c) cartesian, square-ice (square-pop is equivalent here), (d) rake with block weighting, (e) rake with latitude weighting, (f) spacecurve. Each block consists of 20x24 grid cells, and white blocks consist entirely of land cells. 4.7 Performance Namelist options (domain nml) provide considerable flexibility for finding the most efficient processor and block configuration. Some of these choices are illustration in Figure 6. processor shape chooses between tall, thin processor domains (slenderX1 or slenderX2, often better for sea ice simulations on global grids where nearly all of the work is at the top and bottom of the grid with little to do in between) and close-to-square domains, which maximize the volume to surface ratio (and therefore on-processor computations to message passing, if there were ice in every grid cell). In cases where the number of processors is not a perfect square (4, 9, 16...), the processor shape namelist variable allows the user to choose how the processors are arranged. Here again, it is better in the sea ice model to have more processors in x than in y, for example, 8 processors arranged 4x2 (square-ice) rather than 2x4 (square-pop). The latter option is offered for direct-communication compatibility with POP, in which this is the default. The distribution type options allow standard Cartesian distribution of blocks, redistribution via a ‘rake’ algorithm for improved load balancing across processors, and redistribution based on space-filling curves. The rake and space-filling curve algorithms are primarily helpful when using squarish processor domains where some processors (located near the equator) would otherwise have little work to do. Processor domains need not be rectangular, however. distribution wght chooses how the work-per-block estimates are weighted. The ‘block’ option is the default in POP, which uses a lot of array syntax requiring calculations over entire blocks (whether or not land is present), and is provided here for direct-communication compatibility with POP. The ‘latitude’ Performance 45 Figure 7: (a) CICE timings for 8, 16, 32, and 64 processors of the Linux cluster “coyote,” after 240 1-hr time steps on a 320x384 displaced pole grid. (b) 8-processor timings for various processor shapes (indicated by the upper pair of numbers; eg., 8x1 = ‘slenderX1’) and block sizes. 46 Numerical implementation option weights the blocks based on latitude and the number of ocean grid cells they contain. The rake distribution type is initialized as a standard, Cartesian distribution. Using the work-per-block estimates, blocks are “raked” onto neighboring processors as needed to improve load balancing characteristics among processors, first in the x direction and then in y. Space-filling curves reduce a multi-dimensional space (2D, in our case) to one dimension. The curve is composed of a string of blocks that is snipped into sections, again based on the work per processor, and each piece is placed on a processor for optimal load balancing. This option requires that the block size be chosen such that the number of blocks in the x direction equals the number of blocks in the y direction, and that number must be factorable as 2n 3m 5p where n, m, p are integers. For example, a 16x16 array of blocks, each containing 20x24 grid cells, fills the gx1 grid (n = 4, m = p = 0). If either of these conditions is not met, a Cartesian distribution is used instead. The user provides the total number of processors and the block dimensions in the setup script (comp ice). When moving toward smaller, more numerous blocks, there is a point where the code becomes less efficient; blocks should not have fewer than about 20 grid cells in each direction. Squarish blocks are probably better, again to optimize the volume-to-surface ratio for communications. In general, the following rules-of-thumb seem to work best: • For large numbers of processors: processor shape = ‘slenderX1’ with one block per processor (distribution type and distribution wght do not matter—they default to cartesian). ‘slenderX2’ is not quite as efficient for the same number of processors but is better than either square option, and is sometimes necessary to keep the processor domain width sufficiently large. • For small numbers of processors: distribution type = ‘rake’ or ‘spacecurve’ with distribution wght = ‘latitude’ and processor shape = ‘square-ice’. • The cross-over point, i.e. the number of processors where these choices result in about the same timings, will vary depending on the grid and the processor/architecture. Figure 7 shows the model’s scaling characteristics for up to 64 processors on the Linux cluster “coyote.” These runs were made on the gx1 global grid (320x384, approximately 1 degree resolution), from the restart file included with the code distribution. On the gx1 grid, the scaling is nearly linear up to 64 processors for “slender” (x1 or x2) decompositions. More than 64 processors do not have enough ice work to do on this grid, and the timings do not scale as well. The lower plot shows timings for 8-processor block distributions versus the total number of blocks, illustrating the performance gains that are available by redistributing smaller blocks across processors. Throughout the code, (i, j) loops have been combined into a single loop, often over just ocean cells or those containing sea ice. This was done to reduce unnecessary operations and to improve vector performance. 4.8 4.8.1 Adding things Timers Timing any section of code, or multiple sections, consists of defining the timer and then wrapping the code with start and stop commands for that timer. Printing of the timer output is done simultaneously for all timers. To add a timer, first declare it (timer [tmr]) at the top of ice timers.F90 (we recommend doing this in both the mpi/ and serial/ directories), then add a call to get ice timer in the subroutine init ice timers. In the module containing the code to be timed, call ice timer start(timer [tmr]) at the beginning of the section to be timed, and a similar call to ice timer stop at the end. Be careful not to have one command outside of a loop and the other command inside. Timers can be run for individual blocks, if desired, by including the block ID in the timer calls. Adding things 4.8.2 47 History fields To add a variable to be printed in the history output, in ice history.F90: 1. increase avgsiz 2. add a logical flag for the new field 3. add the logical flag to the namelist (here and also in ice in) 4. assign an index number 5. in init hist, • add field name (vname) • add description (vdesc) • add unit (vunit) • add comment (vcomment) • broadcast the flag • load the iout array with the logical flag • specify a unit conversion factor if necessary 6. increment the field in ice write hist 7. in ice in, add the logical flag to the namelist. In the present release, full category output for seven variables may be written. Whether the variable being added has a category index or not affects how avgsiz and the index number are defined in ice history.F90. For instance, if the field has a category index, increase avgsiz from N + M ∗ NC to N + (M + 1) ∗ NC and add a new index number at the end of the list, following the pattern of the previous lines. If the field is just two-dimensional, increase avgsiz to (N + 1) + M ∗ NC and add a new index number just before n aicen, modifying the following lines as needed. 4.8.3 Tracers For backward compatibility of restart files, each new tracer has its own restart file, generated in its own module, ice [tracer].F90, which also contains as much of the additional tracer code as possible. We recommend that the logical namelist variable tr [tracer] be used for all calls involving the new tracer outside of ice [tracer].F90, in case other users do not want to use that tracer. Two optional tracers are available in the code, ice age and melt pond volume. Age is a volume-weighted tracer, while melt pond volume is an area-weighted tracer. In the absence of sources and sinks, the total mass of a volume-weighted tracer such as soot (kg) is conserved under transport in horizontal and thickness space (the mass in a given grid cell will change), whereas the soot concentration (kg/m) is unchanged following the motion, and in particular, the concentration is unchanged when there is surface or basal melting. The proper units for a volume-weighted mass tracer in the tracer array are kg/m. To add a tracer, follow these steps; the ice age tracer can be used as a pattern. 1. ice domain size.F90: increase ntrcr 2. ice state.F90: define tracer index nt [tracer] 48 5 TROUBLESHOOTING 3. ice [tracer].F90: create initialization, physics, restart routines 4. ice fileunits.F90: add new dump and restart file units 5. ice init.F90: • add logical namelist variable tr [tracer] • increment loop to count tracers based on namelist input • define tracer types (trcr depend = 0 for ice area tracers, 1 for ice volume, 2 for snow volume) 6. CICE InitMod.F90: initialize tracer (includes reading restart file) 7. CICE RunMod.F90, ice step mod.F90: • call routine to write tracer restart file • call physics routines in ice [tracer].F90 8. ice history.F90: add history variables (Section 4.8.2) 9. ice in: add namelist variables to tracer nml and icefields nml 5 5.1 Troubleshooting Initial setup The scrip comp ice is configured so that the files grid, kmt, ice in, run ice, iced gx3 v4.0 kcatbound0 and ice.restart file are NOT overwritten after the first setup. If you wish to make changes to the original files in input templates/ rather than those in the run directory, either remove the files from the run directory before executing comp ice or edit the script. The code may abort during the setup phase for any number of reasons, and often the buffer containing the diagnostic output fails to print before the executable exits. The quickest way to get the diagnostic information is to run the code in an interactive shell with just the command cice for serial runs or “mpirun -np N cice” for MPI runs, where N is the appropriate number of processors (or a command appropriate for your computer’s software). If the code fails to compile or run, or if the model configuration is changed, try the following: • create Macros.*. Makefile.* and run ice.* files for your particular platform, if they do not already exist (type ‘uname -s’ at the prompt and compare the result with the file suffixes; we rename UNICOS/mp as UNICOS for simplicity). • modify the INCLUDE directory path and other settings for your system in the scripts, Macros.* and Makefile.* files. • alter directory paths, file names and the execution command as needed in run ice and ice in. • ensure that nprocs in ice in is equal to NTASK in comp ice. • ensure that the block size NXBLOCK, NYBLOCK in comp ice is compatible with the processor shape and other domain options in ice in 5.2 Slow execution 49 • if using the rake or space-filling curve algorithms for block distribution (distribution type in ice in) the code will abort if MXBLCKS is not large enough. The correct value is provided in the diagnostic output. • if starting from a restart file, ensure that kcatbound is the same as that used to create the file (kcatbound = 0 for the files included in this code distribution). Other configuration parameters, such as ncat, must also be consistent between runs. • for stand-alone runs, check that -Dcoupled is not set in the Macros.* file. • for coupled runs, check that -Dcoupled and other coupled-model-specific (eg., CCSM, popcice or hadgem) preprocessing options are set in the Macros.* file. • edit the grid size and other parameters in comp ice. • remove the compile/ directory completely and recompile. 5.2 Slow execution On some architectures, underflows (10−300 for example) are not flushed to zero automatically. Usually a compiler flag is available to do this, but if not, try uncommenting the block of code at the end of subroutine stress in ice dyn evp.F90. You will take a hit for the extra computations, but it will not be as bad as running with the underflows. 5.3 Debugging hints Several utilities are available that can be helpful when debugging the code. Not all of these will work everywhere in the code, due to possible conflicts in module dependencies. debug ice (CICE.F90) A wrapper for print state that is easily called from numerous points during the timestepping loop (see CICE.F90 debug, which can be substituted for CICE.F90). print state (ice diagnostics.F90) Print the ice state and forcing fields for a given grid cell. dbug = .true. (ice in) Print numerous diagnostic quantities. print global (ice in) If true, compute and print numerous global sums for energy and mass balance analysis. This option can significantly degrade code efficiency. print points (ice in) If true, print numerous diagnostic quantities for two grid cells, one near the north pole and one in the Weddell Sea. This utility also provides the local grid indices and block and processor numbers (ip, jp, iblkp, mtask) for these points, which can be used in conjunction with check step, to call print state. These flags are set in ice diagnostics.F90. This option can be fairly slow, due to gathering data from processors. global minval, global maxval, global sum (ice global reductions.F90) Compute and print the minimum and maximum values for an individual real array, or its global sum. 50 Acknowledgments and Copyright 5.4 Known bugs 1. Fluxes sent to the CCSM coupler may have incorrect values in grid cells that change from an icefree state to having ice during the given time step, or vice versa, due to scaling by the ice area. The authors of the CCSM flux coupler insist on the area scaling so that the ice and land models are treated consistently in the coupler (but note that the land area does not suddenly become zero in a grid cell, as does the ice area). 2. With the standard CCSM radiative scheme (shortwave = ‘default’), a sizable fraction (more than 10%) of the total shortwave radiation is absorbed at the surface but should be penetrating into the ice interior instead. This is due to use of the aggregated, effective albedo rather than the bare ice albedo when snowpatch < 1. 3. The date-of-onset diagnostic variables, melt onset and frz onset, are not included in the restart file, and therefore may be incorrect for the current year if the run is restarted after Jan 1. Also, these variables were implemented with the Arctic in mind and may be incorrect for the Antarctic. 4. The single-processor system clock time may give erratic results on some architectures. 5. History files that contain time averaged data (hist avg = .true. in ice in) will be incorrect if restarting from midway through an averaging period. 6. In stand-alone runs, restarts from the end of ycycle will not be exact. 5.5 Interpretation of albedos Interpretation of the albedos in the history output file is a little tricky. The snow-and-ice albedo, albsni, is merged across categories and then scaled (divided) by the total ice area, as with other variables sent to the CCSM coupler. The diagnostic albedos albice, albsno, and albpnd are merged over categories but not scaled. The latter three history variables represent completely bare or completely snow- or melt-pondcovered ice; that is, they do not take into account the snow or melt pond fraction (albsni does, as does the code itself during thermodyamic computations). This is to facilitate comparison with typical values in measurements or other albedo parameterizations. The melt pond albedo albpnd is only computed for the Delta-Eddington shortwave case. With the Delta-Eddington parameterization, the albedo depends on the cosine of the zenith angle (cos ϕ, coszen) and is zero if the sun is below the horizon (cos ϕ < 0). Therefore time-averaged albedo fields would be low if a diurnal solar cycle is used, because zero values would be included in the average for half of each 24-hour period. To rectify this, a separate counter is used for the averaging that is incremented only when cos ϕ > 0. The albedos will still be zero in the dark, polar winter hemisphere. Acknowledgments and Copyright This work has been supported through the Department of Energy Computer Hardware Applied Mathematics and Model Physics (CHAMMP) program, Climate Change Prediction Program (CCPP), and Scientific Discovery through Advanced Computing (SCIDAC) program, with additional support from the T-3 Fluid Dynamics Group at Los Alamos National Laboratory. Special thanks are due to the following people: • members of the CCSM Polar Climate Working Group, including David Bailey, Cecilia Bitz, Bruce Briegleb, Tony Craig, Marika Holland, John Dennis, Julie Schramm, Bonnie Light and Phil Jones, Acknowledgments and Copyright 51 • Alison McLaren, Ann Keen and others working with the Hadley Centre GCM for testing non-standard model configurations and providing their code to us, • the many researchers who tested beta versions and waited patiently for the official release. c Copyright 2008, LANS LLC. All rights reserved. Unless otherwise indicated, this information has been authored by an employee or employees of the Los Alamos National Security, LLC (LANS), operator of the Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 with the U.S. Department of Energy. The U.S. Government has rights to use, reproduce, and distribute this information. The public may copy and use this information without charge, provided that this Notice and any statement of authorship are reproduced on all copies. Neither the Government nor LANS makes any warranty, express or implied, or assumes any liability or responsibility for the use of this information. Beginning with version 4.0, the CICE code carries Los Alamos Software Release number LA-CC-06-012. 52 Namelist options Table of namelist options variable options/format description setup nml days per year year init istep0 dt npt ndyn dt 360 or 365 yyyy integer seconds integer integer Time number of days in a model year the initial year, if not using restart initial time step number thermodynamics time step length total number of time steps to take number of dynamics/advection/ridging steps per thermo timestep runtype ice ic restart restart dir restart file pointer file dumpfreq dumpfreq n diagfreq diag type diag file print global print points latpnt lonpnt dbug initial continue default none path/file true/false path/ filename prefix pointer filename y m d integer Initialization/Restarting start from ice ic restart using pointer file latitude and sst dependent no ice restart file name initialize using restart file path to restart directory output file for restart dump contains restart filename write restart every dumpfreq n years write restart every dumpfreq n months write restart every dumpfreq n days frequency restart data is written integer eg., 10 stdout file filename true/false true/false real real true/false Model Output frequency of diagnostic output in dt once every 10 time steps write diagnostic output to stdout write diagnostic output to file diagnostic output file (script may reset) print diagnostic data, global sums print diagnostic data for two grid points latitude of (2) diagnostic points longitude of (2) diagnostic points if true, write extra diagnostics Table 7: Namelist options (continued next page). recommended value 365 0 3600. 1 default .true. ‘iced’ y 1 24 .false. .false. .false. Namelist options 53 variable options/format description histfreq y m d h 1 integer true false path/ filename prefix nc bin true/false path/ filename prefix string Model Output, continued write history every histfreq n years write history every histfreq n months write history every histfreq n days write history every histfreq n hours write history every time step frequency history output is written write time-averaged data write snapshots of data path to history output directory output file for history write netCDF history file write direct access, binary file write initial condition path to initial condition directory output file for initial condition label for run (currently CCSM only) nc bin rectangular displaced pole tripole panarctic filename filename 0 1 2 Grid read netCDF grid and kmt files read direct access, binary file defined in rectgrid read from file in popgrid read from file in popgrid read from file in panarctic grid name of grid file to be read name of land mask file to be read original category boundary formula new formula with round numbers WMO standard categories integer slenderX1 slenderX2 square-ice square-pop cartesian rake spacecurve block latitude Domain number of processors to use 1 processor in the y direction (tall, thin) 2 processors in the y direction (thin) more processors in x than y, ∼square more processors in y than x, ∼square distribute blocks in 2D Cartesian array redistribute blocks among neighbors distribute blocks via space-filling curves full block size sets work per block latitude/ocean sets work per block histfreq n hist avg history dir history file history format write ic incond dir incond file runid grid nml grid format grid type grid file kmt file kcatbound domain nml nprocs processor shape distribution type distribution weight Table 7: Namelist options (continued). recommended value .true. ‘iceh’ nc ‘iceh’ bin’ displaced pole ‘grid’ ‘kmt’ 0 54 variable ew boundary type ns boundary type tracer nml tr iage restart age tr pond restart pond ice nml kitd Namelist options options/format description cyclic open closed cyclic open closed tripole Domain, continued periodic boundary conditions in x-direction Neumann boundary conditions in x Dirichlet boundary conditions in x periodic boundary conditions in y-direction Neumann boundary conditions in y Dirichlet boundary conditions in y tripole boundary conditions in y true/false true/false true/false true/false Tracers ice age restart tracer values from file melt ponds restart tracer values from file albicev albicei albsnowv albsnowi R ice 0 1 0 1 integer 0 1 0 1 0 1 remap upwind true false default dEdd default constant 0<α<1 0<α<1 0<α<1 0<α<1 real R pnd R snw real real kdyn ndte kstrength krdg partic krdg redist advection heat capacity shortwave albedo type Physical Parameterizations delta function ITD approximation linear remapping ITD approximation dynamics OFF EVP dynamics number of EVP subcycles ice strength formulation [14] ice strength formulation [35] old ridging participation function new ridging participation function old ridging redistribution function new ridging redistribution function linear remapping advection donor cell advection salinity-dependent thermodynamics zero-layer thermodynamic model NCAR CCSM3 distribution method Delta-Eddington method NCAR CCSM3 albedos four constant albedos visible ice albedo for thicker ice near infrared ice albedo for thicker ice visible, cold snow albedo near infrared, cold snow albedo tuning parameter for sea ice albedo from Delta-Eddington shortwave ... for ponded sea ice albedo ... ... for snow (broadband albedo) ... Table 7: Namelist options (continued). recommended value 1 1 120 1 1 1 remap .true. default default Namelist options variable atmbndy fyear init ycycle atm data format atm data type atm data dir calc strair calc Tsfc precip units Tfrzpt update ocn f oceanmixed ice ocn data format sss data type sst data type ocn data dir oceanmixed file restore sst trestore restore ice icefields nml f hvari 55 options/format default constant yyyy integer nc bin default ecmwf ncar LYq monthly path/ true false true/false mks mm per month mm per sec constant linear S true false true/false nc bin default clim ncar default clim ncar path/ filename true/false integer true/false true/false description Forcing stability-based boundary layer bulk transfer coefficients first year of atmospheric forcing data number of years in forcing data cycle read netCDF atmo forcing files read direct access, binary files constant values defined in the code ECMWF forcing data NCAR bulk forcing data AOMIP/Large-Yeager forcing data monthly forcing data path to atmospheric forcing data directory calculate wind stress and speed read wind stress and speed from files calculate surface temperature liquid precipitation data units (same as MKS units) freezing temperature = -1.8◦ C linear function of salinity profile include frazil water/salt fluxes in ocn fluxes do not include (when coupling with POP) active ocean mixed layer calculation read netCDF ocean forcing files read direct access, binary files constant values defined in the code climatological data POP ocean forcing data constant values defined in the code climatological data POP ocean forcing data path to oceanic forcing data directory data file containing ocean forcing data restore sst to data sst restoring time scale (days) restore ice state along lateral boundaries History Fields write hvari to history Table 7: Namelist options (continued from previous page). recommended value default bin .true. .true. (if uncoupled) bin 56 Index of primary variables and parameters Index of primary variables and parameters This index defines many of the symbols used frequently in the ice model code. Values appearing in this list are fixed or recommended; most namelist parameters are indicated (•) with their default values. For other namelist options, see Table 7. All quantities in the code are expressed in MKS units (temperatures may take either Celsius or Kelvin units). A a min advection ahmax aice extmin aice init aice0 aice(n) albedo type albicei albicev albocn albsnowi albsnowv alpha alv(n)dr(f) alv(n)dr(f) gbm ANGLE ANGLET apondn astar atm data dir atm data format atm data type atmbndy awtidf awtidr awtvdf awtvdr avgsiz minimum area concentration for computing velocity . . . . . . . . . . . . • type of advection algorithm used (‘remap’ or ‘upwind’) . . . . . . . thickness above which ice albedo is constant . . . . . . . . . . . . . . . . . . . minimum value for ice extent diagnostic concentration of ice at beginning of timestep fractional open water area total concentration of ice in grid cell (in category n) • type of albedo parameterization (‘default’ or ‘constant’) • near infrared ice albedo for thicker ice • visible ice albedo for thicker ice ocean albedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • near infrared, cold snow albedo • visible, cold snow albedo floe shape constant for lateral melt . . . . . . . . . . . . . . . . . . . . . . . . . . . albedo: visible (near IR), direct (diffuse) grid-box-mean value of alv(n)dr(f) for conversions between the POP grid and latitude-longitude grids ANGLE converted to T-cells area concentration of melt ponds e-folding scale for participation function . . . . . . . . . . . . . . . . . . . . . . • directory for atmospheric forcing data • format of atmospheric forcing files • type of atmospheric forcing • atmo boundary layer parameterization (‘default’ or ‘constant’) weighting factor for near-ir, diffuse albedo . . . . . . . . . . . . . . . . . . . . weighting factor for near-ir, direct albedo . . . . . . . . . . . . . . . . . . . . . weighting factor for visible, diffuse albedo . . . . . . . . . . . . . . . . . . . . weighting factor for visible, direct albedo . . . . . . . . . . . . . . . . . . . . . number of cell-averaged fields that can be written to history file . 0.001 remap 0.5 m 0.15 0.06 0.66 radians radians 0.05 0.36218 0.63282 0.00182 0.00318 81 Index of primary variables and parameters 57 B bignum block block id block size x(y) blockGlobalID blockLocalID blockLocation blocks ice a large number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . data type for blocks global block number number of cells along x(y) direction of block global block IDs local block IDs processor location of block local block IDs 1030 C chni calc strair calc Tsfc Cf char len char len long check step check umax cldf cm to m coldice coldsnow congel cosw coszen Cp cp air cp ice cp ocn cp wv cp063 cp455 Cs Cstar real(n) • if true, calculate wind stress . . . . . . . . . . . . . . . . . . . • if true, calculate surface temperature . . . . . . . . . . . ratio of ridging work to PE change in ridging . . . . . length of character variable strings . . . . . . . . . . . . . . length of longer character variable strings . . . . . . . . time step on which to begin writing debugging data if true, check for ice speed > umax stab cloud fraction cm to meters conversion . . . . . . . . . . . . . . . . . . . . . . . . value for constant albedo parameterization . . . . . . . value for constant albedo parameterization . . . . . . . basal ice growth cosine of the turning angle in water . . . . . . . . . . . . . cosine of the zenith angle proportionality constant for potential energy specific heat of air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . specific heat of fresh ice . . . . . . . . . . . . . . . . . . . . . . . . specific heat of sea water . . . . . . . . . . . . . . . . . . . . . . specific heat of water vapor . . . . . . . . . . . . . . . . . . . . . diffuse fresnel reflectivity (above) diffuse fresnel reflectivity (below) fraction of shear energy contributing to ridging . . . constant in Hibler ice strength formula . . . . . . . . . . . T T 17. 80 256 0.01 0.70 0.81 m 1. kg/m2 /s2 1005.0 J/kg/K 2106. J/kg/K 4218. J/kg/K 1.81×103 J/kg/K 0.063 0.455 0.25 20. D daidtd daidtt dalb mlt dalb mlti dalb mltv dardg1dt dardg2dt ice area tendency due to dynamics/transport ice area tendency due to thermodynamics [see ice shortwave.F90] . . . . . . . . . . . . . . . . . . . . . . . [see ice shortwave.F90] . . . . . . . . . . . . . . . . . . . . . . . [see ice shortwave.F90] . . . . . . . . . . . . . . . . . . . . . . . rate of fractional area loss by ridging ice rate of fractional area gain by new ridges 1/s 1/s -0.075 -0.100 -0.150 1/s 1/s 58 daymo daycal days per year dbl kind dbug Delta depressT diag file diag type diagfreq distrb info distribution type distribution weight divu divu adv dragio dragw dt dT mlt dte dtei dump file dumpfreq dumpfreq n dxt dxu dyn dt dyt dyu dvidtd dvidtt dvirdgdt Index of primary variables and parameters number of days in one month day number at end of month • number of days in one year definition of double precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . • write extra diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . function of strain rates (see Section 3.4) ratio of freezing temperature to salinity of brine . . . . . . . . . . . • diagnostic output file (alternative to standard out) • where diagnostic output is written . . . . . . . . . . . . . . . . . . . . . . • how often diagnostic output is written (10 = once per 10 dt) block distribution information • ‘cartesian’ or ‘rake’ or ‘spacecurve’ • weighting method used to compute work per block strain rate I component, velocity divergence divergence associated with advection drag coefficient for water on ice drag coefficient for water on ice*ρw . . . . . . . . . . . . . . . . . . . . . . • thermodynamics time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [see ice shortwave.F90] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . subcycling time step for EVP dynamics (∆te ) 1/dte, where dte is the EVP subcycling time step • output file for restart dump • dump frequency for restarts, y, m or d • restart output frequency width of T cell (∆x) through the middle width of U cell (∆x) through the middle dynamics and transport time step (∆tdyn ) height of T cell (∆y) through the middle height of U cell (∆y) through the middle ice volume tendency due to dynamics/transport ice volume tendency due to thermodynamics ice volume ridging rate 365 selected real kind(13) .false. 1/s 0.054 deg/psu stdout 1/s 1/s 0.00536 kg/m3 3600. s 1. deg s 1/s m m s m m m/s m/s m/s E ecci eice(n) emissivity eps11 eps13 eps16 esno(n) evap evp damping ew boundary type eyc yield curve minor/major axis ratio, squared . . . . . . . . . . . . . . . energy of melting of ice per unit area (in category n) emissivity of snow and ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a small number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a small number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a small number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . energy of melting of snow per unit area (in category n) evaporative water flux • if true, use evp damping procedure [16] . . . . . . . . . . . . . . . . . • type of east-west boundary condition coefficient for calculating the parameter E, 0< eyc <1 . . . . . 1/4 J/m2 0.95 10−11 10−13 10−16 J/m2 kg/m2 /s F 0.36 Index of primary variables and parameters 59 F fcondtop(n)( f) fcor blk ferrmax fhocn fhocn gbm field loc center field loc Eface field loc NEcorner field loc Nface field loc noupdate field loc unknown field loc Wface field type angle field type noupdate field type scalar field type unknown field type vector flat floediam flw flwout fm frain frazil fresh fresh gbm frz onset frzmlt frzmlt max fsalt fsalt gbm fsens fsnow fsnowrdg fsurf(n)( f) fsw fswabs fswfac fswthru fswthru gbm fyear fyear final fyear init conductive heat flux Coriolis parameter max allowed energy flux error (thermodynamics) . . . . . net heat flux to ocean grid-box-mean net heat flux to ocean (fhocn) field centered on grid cell . . . . . . . . . . . . . . . . . . . . . . . . . . field centered on east face . . . . . . . . . . . . . . . . . . . . . . . . . . field on northeast corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . field centered on north face . . . . . . . . . . . . . . . . . . . . . . . . . ignore location of field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unknown location of field . . . . . . . . . . . . . . . . . . . . . . . . . . field centered on west face . . . . . . . . . . . . . . . . . . . . . . . . . angle field type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ignore field type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar field type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unknown field type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vector field type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . latent heat flux effective floe diameter for lateral melt . . . . . . . . . . . . . . . incoming longwave radiation outgoing longwave radiation Coriolis parameter * mass in U cell rainfall rate frazil ice growth fresh water flux to ocean grid-box-mean fresh water flux (fresh) day of year that freezing begins freezing/melting potential maximum magnitude of freezing/melting potential . . . . net salt flux to ocean grid-box-mean salt flux to ocean (fsalt) sensible heat flux snowfall rate snow fraction that survives in ridging . . . . . . . . . . . . . . . . . net surface heat flux excluding fcondtop incoming shortwave radiation absorbed shortwave radiation scaling factor to adjust ice quantities for updated data shortwave penetrating to ocean grid-box-mean shortwave penetrating to ocean (fswthru) current data year last data year • initial data year W/m2 1/s 1.×10−3 W/m2 W/m2 W/m2 1 4 2 3 -1 0 5 3 -1 1 0 2 W/m2 300. m W/m2 W/m2 kg/s kg/m2 /s m kg/m2 /s kg/m2 /s W/m2 1000. W/m2 kg/m2 /s kg/m2 /s W/m2 kg/m2 /s 0.5 W/m2 W/m2 W/m2 W/m2 W/m2 60 Index of primary variables and parameters G gravit grid file grid format grid type Gstar gravitational acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • input file for grid info • format of grid files • ‘rectangular’ or ‘displaced pole’ or ‘column’ piecewise-linear ridging participation function parameter . . . . 9.80616 m/s2 0.15 H halo info heat capacity hfrazilmin hi min hicen hin max hist avg histfreq histfreq n history dir history file history format hm hmix hour hp0 hpmin hpondn hs min hsmin hs0 Hstar HTE HTN HTS HTW information for updating ghost cells • if true, use salinity-dependent thermodynamics . . . . . . . . . . . . minimum thickness of new frazil ice . . . . . . . . . . . . . . . . . . . . . . . minimum ice thickness for thinnest ice category . . . . . . . . . . . . ice thickness in category n category thickness limits • if true, write averaged data instead of snapshots . . . . . . . . . . . • units of history output frequency: y, m, w, d or 1 • integer output frequency in histfreq units • path to history output files • history output file prefix • format of history files land/boundary mask, thickness (T-cell) ocean mixed layer depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hour of the year pond depth at which transition to bare ice occurs . . . . . . . . . . . . minimum melt pond depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . melt pond depth minimum thickness for which Ts is computed . . . . . . . . . . . . . . minimum snow depth for Delta-Eddington . . . . . . . . . . . . . . . . . snow depth at which transition to ice occurs (dEdd) . . . . . . . . . determines mean thickness of ridged ice . . . . . . . . . . . . . . . . . . . length of eastern edge (∆y) of T-cell length of northern edge (∆x) of T-cell length of southern edge (∆x) of T-cell length of western edge of (∆y) T-cell T 0.05 m 0.01 m m m T 20. m 0.2 m 0.005 m m 1.×10−4 m 1.×10−4 m 0.03 m 25. m m m m m I i(j) glob i0vis iblkp i(j)block ice ic ice ref salinity icells iceruf global domain location for each grid cell fraction of penetrating visible solar radiation . . . . . . . . . . . . . . . block on which to write debugging data Cartesian i,j position of block • choice of initial conditions (see Table 4) reference salinity for ice-ocean exchanges . . . . . . . . . . . . . . . . . . number of grid cells with specified property (for vectorization) ice surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.70 4. psu 5.×10−4 m Index of primary variables and parameters icetmask iceumask idate idate0 ierr i(j)hi i(j)lo ilyr1 ilyrn incond dir incond file int kind integral order ip, jp istep istep0 istep1 Iswabs ice extent mask (T-cell) ice extent mask (U-cell) the date at the end of the current time step (yyyymmdd) initial date general-use error flag last i(j) index of physical domain (local) first i(j) index of physical domain (local) index of the top layer in each cat (for eicen) index of the bottom layer in each cat (for eicen) • directory to write snapshot of initial condition • prefix for initial condition file name definition of an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . polynomial order of quadrature integrals in remapping . . . . . . local processor coordinates on which to write debugging data local step counter for time loop • number of steps taken in previous run . . . . . . . . . . . . . . . . . . . . total number of steps at current time step shortwave radiation absorbed in ice layers 61 selected real kind(6) 3 0 W/m2 K kappan kappav kcatbound kdyn kg to g kice kimin kitd kmt file krdg partic krdg redist kseaice ksno kstrength visible extinction coefficient in ice, wavelength>700nm . . . . . visible extinction coefficient in ice, wavelength<700nm . . . . . • category boundary formula • type of dynamics (1 = EVP, 0 = off) . . . . . . . . . . . . . . . . . . . . . . kg to g conversion factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thermal conductivity of fresh ice . . . . . . . . . . . . . . . . . . . . . . . . . . . minimum conductivity of saline ice • type of itd conversions (0 = delta function, 1 = linear remap) • input file for land mask info • ridging participation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . • ridging redistribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . thermal conductivity of ice for zero-layer thermodynamics. . . . thermal conductivity of snow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • ice stength formulation (1= Rothrock 1975, 0= Hibler 1979) 17.6 m−1 1.4 m−1 1 1000. 2.03 W/m/deg 0.10 W/m/deg 1 1 1 2.0 W/m/deg 0.30 W/m/deg 1 L l brine flag for brine pocket effects l conservation check if true, check conservation when ridging l fixed area flag for prescribing remapping fluxes latpt • latitude of diagnostic points latt(u) bounds latitude of T(U) grid cell corners Lfresh latent heat of melting of fresh ice = Lsub - Lvap lhcoef transfer coefficient for latent heat northern (southern) hemisphere mask lmask n(s) local id local address of block in current distribution log kind definition of a logical variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degrees N degrees N J/kg kind(.true.) 62 lonpt lont(u) bounds Lsub ltripole grid Lvap Index of primary variables and parameters • longitude of diagnostic points longitude of T(U) grid cell corners latent heat of sublimation for fresh water . . . . . . . . . . . . . . . . flag to signal use of tripole grid latent heat of vaporization for fresh water . . . . . . . . . . . . . . . . degrees E degrees E 2.835× 106 J/kg minimum mass for computing velocity . . . . . . . . . . . . . . . . . . meters to cm conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . constant for lateral melt rate . . . . . . . . . . . . . . . . . . . . . . . . . . . constant for lateral melt rate . . . . . . . . . . . . . . . . . . . . . . . . . . . m2 to km2 conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . task ID for the controlling processor maximum number of blocks per processor maximum thickness of ice that rafts . . . . . . . . . . . . . . . . . . . . . day of the month basal ice melt lateral ice melt snow melt top ice melt threshold for brine pockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . day of year that surface melt begins the month number previous month number m per s to cm per day conversion . . . . . . . . . . . . . . . . . . . . . . . local processor number that writes debugging data e-folding scale of ridged ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . task ID for the current processor 0.01 kg/m2 100. 1.6×10−6 m/s deg−m2 1.36 1×10−6 2.501×106 J/kg M m min m to cm m1 m2 m2 to km2 master task max blocks maxraft mday meltb meltl melts meltt min salin mlt onset month monthp mps to cmpdy mtask mu rdg my task 1. m m m m m 0.1 psu 8.64×106 4. m1/2 N nblocks nblocks tot nblocks x(y) ncat ndte ndyn dt new day new hour new month new year nghost ngroups nhlat nilyr nprocs npt number of blocks on current processor total number of blocks in decomposition total number of blocks in x(y) direction number of ice categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • number of subcycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • number of dynamics/advection steps under thermo . . . . . . flag for beginning new day flag for beginning new hour flag for beginning new month flag for beginning new year number of rows of ghost cells surrounding each subdomain number of groups of flux triangles in remapping . . . . . . . . . northern latitude of artificial mask edge . . . . . . . . . . . . . . . . . number of ice layers in each category . . . . . . . . . . . . . . . . . . . • total number of processors • total number of time steps (dt) 5 120 1 1 5 30◦ S 4 Index of primary variables and parameters ns boundary type nslyr nspint nt htrcri ntilyr ntrace ntrcr ntslyr nu diag nu dump nu dump age nu dump pond nu forcing nu grid nu hdr nu history nu kmt nu nml nu restart nu restart age nu restart pond nu rst pointer nx(y) block nx(y) global nyr 63 • type of north-south boundary condition number of snow layers in each category number of solar spectral intervals tracer index sum of number of ice layers in all categories number of fields being transported number of tracers transported in remapping sum of number of snow layers in all categories unit number for diagnostics output file unit number for dump file for restarting unit number for age dump file for restarting unit number for melt pond dump file for restarting unit number for forcing data file unit number for grid file unit number for binary history header file unit number for history file unit number for land mask file unit number for namelist input file unit number for restart input file unit number for age restart input file unit number for melt pond restart input file unit number for pointer to latest restart file total number of gridpoints on block in x(y) direction number of physical gridpoints in x(y) direction, global domain year number O oceanmixed file oceanmixed ice ocn data dir ocn data format omega opening • data file containing ocean forcing data • if true, use internal ocean mixed layer • directory for ocean forcing data • format of ocean forcing files angular velocity of Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rate of ice opening due to divergence and shear P p001 p01 p027 p05 p055 p1 p111 p15 p166 p2 p222 1/1000 1/100 1/36 1/20 1/18 1/10 1/9 15/100 1/6 1/5 2/9 7.292×10−5 rad/s 1/s 64 p25 p333 p4 p5 p52083 p5625m p6 p666 p75 pi pi2 pih pointer file potT precip units print global print points processor shape prs sig Pstar puny Index of primary variables and parameters 1/4 1/3 2/5 1/2 25/48 -9/16 3/5 2/3 3/4 π 2π π/2 • input file for restarting atmospheric potential temperature • liquid precipitation data units • if true, print global data . . . . . . . . . . . . . . . . . . . . . . . . • if true, print point data . . . . . . . . . . . . . . . . . . . . . . . . . • descriptor for processor aspect ratio replacement pressure ice strength parameter a small positive number . . . . . . . . . . . . . . . . . . . . . . . . . . . N/m 2.75×104 N/m 1×10−11 specific humidity at 10 m deep ocean heat flux for saturated specific humidity over ice . . . . . . . . . . . . . for saturated specific humidity over ocean . . . . . . . . . . 2m atmospheric reference specific humidity kg/kg W/m2 1.16378×107 kg/m3 6.275724×106 kg/m3 kg/kg K F F Q Qa qdp qqqice qqqocn Qref R R ice R pnd R snw r16 kind rad to deg radius rdg conv rdg shear real kind refindx restart restart age restart dir restart file restart pond restore ice • parameter for Delta-Eddington ice albedo • parameter for Delta-Eddington pond albedo • parameter for Delta-Eddington snow albedo definition of quad precision . . . . . . . . . . . . . . . . . . . . . . . degree-radian conversion . . . . . . . . . . . . . . . . . . . . . . . . . earth radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . convergence for ridging shear for ridging definition of single precision real . . . . . . . . . . . . . . . . . . refractive index of sea ice . . . . . . . . . . . . . . . . . . . . . . . . • if true, initialize using restart file instead of defaults • if true, read age restart file • path to restart/dump files • restart file prefix • if true, read melt pond restart file • if true, restore ice state along lateral boundaries selected real kind(26) 180/π 6.37×106 m 1/s 1/s selected real kind(6) 1.310 T Index of primary variables and parameters restore sst rhoa rhofresh rhoi rhos rhow rnilyr rside rsnw fresh rsnw melt rsnw nonmelt rsnw sig runtype • restore sst to data air density density of fresh water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . density of ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . density of snow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . density of seawater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . real(nlyr) fraction of ice that melts laterally freshly fallen snow grain radius . . . . . . . . . . . . . . . . . . . . . . . . melting snow grain radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nonmelting snow grain radius . . . . . . . . . . . . . . . . . . . . . . . . . standard deviation of snow grain radius . . . . . . . . . . . . . . . . . • type of initialization used 65 kg/m3 1000.0 kg/m3 917. kg/m3 330. kg/m3 1026. kg/m3 100. × 10−6 m 1000. × 10−6 m 500. × 10−6 m 250. × 10−6 m S salin saltmax scale factor sec secday shcoef shear shlat shortwave sig1(2) sinw snoice snowpatch spval spval dbl ss tltx(y) sss sss data type sst sst data type Sswabs stefan-boltzmann stop now strairx(y) strairx(y)T strax(y) strength stress12 stressm stressp strintx(y) ice salinity max salinity, at ice base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scaling factor for shortwave radiation components seconds elasped into idate number of seconds in a day . . . . . . . . . . . . . . . . . . . . . . . . . . . . transfer coefficient for sensible heat strain rate II component southern latitude of artificial mask edge . . . . . . . . . . . . . . . . • flag for shortwave parameterization (‘default’ or ‘dEdd’) principal stress components (diagnostic) sine of the turning angle in water . . . . . . . . . . . . . . . . . . . . . . snow-ice formation length scale for parameterizing nonuniform snow coverage special value (single precision) . . . . . . . . . . . . . . . . . . . . . . . . . special value (double precision) . . . . . . . . . . . . . . . . . . . . . . . . sea surface slope in the x(y) direction sea surface salinity • source of surface salinity data sea surface temperature • source of surface temperature data shortwave radiation absorbed in snow layers Stefan-Boltzmann constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . if 1, end program execution stress on ice by air in the x(y)-direction (centered in U cell) stress on ice by air, x(y)-direction (centered in T cell) wind stress components from data ice strength (pressure) internal ice stress, σ12 internal ice stress, σ11 − σ22 internal ice stress, σ11 + σ22 divergence of internal ice stress, x(y) psu 3.2 psu 86400. 1/s 30◦ N 0. m 0.02 m 1030 1030 m/m psu C W/m2 5.67×10−8 W/m2 K4 N/m2 N/m2 N/m2 N/m N/m N/m N/m N/m2 66 strocnx(y) strocnx(y)T strtltx(y) swv(n)dr(f) Index of primary variables and parameters ice-ocean stress in the x(y)-direction (U-cell) ice-ocean stress, x(y)-dir. (T-cell) surface stress due to sea surface slope incoming shortwave radiation, visible (near IR), direct (diffuse) N/m2 N/m2 N/m2 W/m2 air temperature at 10 m area of T-cell area of northern hemisphere T-cells 1/tarea area of southern hemisphere T-cells absolute day number freezing temperature freezing temp of fresh ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • type of freezing temperature (‘constant’ or ‘linear S’) total elapsed time time of last forcing update melting temperature of ice top surface . . . . . . . . . . . . . . . . . . . . . . . puny * tarea latitude of cell center longitude of cell center land/boundary mask, thickness (T-cell) total mass of ice and snow minimum allowed internal temperature . . . . . . . . . . . . . . . . . . . . . melting temperature of ice temperature of constant freezing point parameterization . . . . . . • if true, use ice age tracer • if true, use explicit melt ponds ice tracers tracer dependency on basic state variables 2m atmospheric reference temperature • sst restoring time scale if true, block lies along tripole boundary max allowed Tsf c error (thermodynamics) . . . . . . . . . . . . . . . . . . temperature of ice/snow top surface (in category n) melting temperature of snow top surface . . . . . . . . . . . . . . . . . . . . for saturated specific humidity over ice . . . . . . . . . . . . . . . . . . . . . for saturated specific humidity over ocean . . . . . . . . . . . . . . . . . . . K m2 m2 1/m2 m2 T Tair tarea tarean tarear tareas tday Tf Tffresh Tfrzpt time time forc Timelt tinyarea TLAT TLON tmask tmass Tmin Tmlt Tocnfrz tr iage tr pond trcr trcr depend Tref trestore tripole Tsf errmax Tsfc(n) Tsmelt TTTice TTTocn C 273.15 K s s 0. C m2 radians radians kg/m2 -100. C -1.8 C K days 5.×10−4 deg C 0. C 5897.8 K 5107.4 K U uarea uarear uatm ULAT ULON umask area of U-cell 1/uarea wind velocity in the x direction latitude of U-cell centers longitude of U-cell centers land/boundary mask, velocity (U-cell) m2 m−2 m/s radians radians Index of primary variables and parameters umax stab umin uocn update ocn f ustar min ustar scale uvel uvm ice speed threashold (diagnostics) . . . . . . . . . . . . . . . . min wind speed for turbulent fluxes . . . . . . . . . . . . . . ocean current in the x-direction • if true, include frazil ice fluxes in ocean flux fields minimum friction velocity under ice . . . . . . . . . . . . . scaling factor for ice-ocean heat flux x-component of ice velocity land/boundary mask, velocity (U-cell) 67 1. m/s 1. m/s m/s 5 × 10−3 m/s m/s V vatm vice(n) vicen init vocn vonkar vsno(n) vvel wind velocity in the y direction volume per unit area of ice (in category n) ice volume at beginning of timestep ocean current in the y-direction von Karman constant . . . . . . . . . . . . . . . . . . . . . . . . . . . volume per unit area of snow (in category n) y-component of ice velocity m/s m m m/s 0.4 m m/s value for constant albedo parameterization . . . . . . . . value for constant albedo parameterization . . . . . . . . wind speed allocatable, 2D dbl kind work array allocatable, 2D integer work array allocatable, 2D long integer work array allocatable, 2D real kind work array allocatable, 3D real kind work array (nx block, ny block, max blocks) work array (nx block, ny block) work array if true, write history now • if true, write initial conditions if 1, write restart now 0.68 0.77 m/s W warmice warmsno wind work g1(23) work gi4 work gi8 work gr work gr3 work1(2) worka(bcd) write history write ic write restart Y ycycle yday year init • number of years in forcing data cycle day of the year • the initial year Z zlvl zref zTrf zvir atmospheric level height reference height for stability . . . . . . . . . . . . . . . . . . . . reference height for Tref , Qref . . . . . . . . . . . . . . . . . gas constant (water vapor)/gas constant (air) - 1 . . . m 10. m 2. m 0.606 Index advection, see transport albedo, 4, 10, 27, 37, 39, 50 AOMIP, 55 area, ice, see ice fraction energy, see enthalpy enthalpy, 9, 12, 13, 26, 32–33 evaporation, 4, 33 EVP, see elastic-viscous-plastic dynamics bilinear, 25, 37 blocks, 3, 36, 38–39, 43, 44–46, 49 boundary communication, 39 condition, 37–39, 42 layer, 6–7, 8 thickness category, 8, 18–21 flux coupler, 3, 5–8, 9, 29, 34, 40, 42, 49, 50 fraction, ice, see ice fraction frazil, 7 freeboard, 34 freezing potential, 4, 7 fresh water flux, 4, 6, 7 grid, 3, 14, 25, 35, 36, 37–39, 40 C-grid, 37 categories, thickness, see thickness distribution CCSM, 2, 5, 10, 27, 50 CFL condition, 11, 18, 31, 40 Community Climate System Model, see CCSM concentration, see ice fraction conservation, 5, 10, 11, 18, 20, 22, 30, 33, 41 conservation equation, see transport continuity equation, see transport Coriolis, 6, 24 coupling, see flux coupler currents, ocean, 4, 7, 29 Hadley Centre, 2, 3, 5, 50 halo, 38 height reference, 4–6 sea surface, 7 history, 3, 36, 37, 41, 43, 47, 50 humidity reference, 4, 7 specific, 4–6, 28 damping timescale, 25 Delta-Eddington, see radiation density atmosphere, 4, 5, 7, 29 ice or snow, 29, 33 ocean, 8, 29 diagnostics, 7, 36, 41, 49 distribution block, see blocks thickness, see thickness distribution dynamics elastic-viscous-plastic, see elastic-viscous-plastic ridging, see ridging transport, see transport ice, see individual variables age, 2, 9, 10, 36, 47 fraction, 5, 8–10, 12, 18, 19, 21–24, 50 growth, 8, 18, 20, 32–34 ice-ocean stress, 4, 6, 8, 24 initial condition, 39, 41 internal stress, 23–25, 37, 41 LANL, 2, 50, 51 latent heat, 4, 6, 7, 26, 28, 33 lateral melt, 29, 34 leads, see open water longwave, see radiation, longwave Los Alamos National Laboratory, see LANL Los Alamos National Security, LLC, 51 masks, 37–39 mechanical distribution, see ridging ECMWF, 55 melt pond, 2, 6, 27, 36, 47 elastic melting potential, 4, 7, 33 -viscous-plastic dynamics, 2, 9, 23–25, 36, 39– meltwater, 6, 7, 27, 32 42 mixed layer, 26, 33, 36 momentum equation, 24 waves, 23, 25, 40 68 INDEX monotonicity, 11–13, 18 MPI, 34, 36, 38, 39, 42, 43, 48 69 slope, sea surface, 4, 6, 7, 24 snow, 2–4, 6–8, 10, 12, 20, 22, 24, 26–34, 50 solar, see radiation, shortwave namelist, 3, 36, 52–55 space-filling curve, 3, 37 National Center for Atmospheric Research, see NCAR specific humidity, see humidity, specific NCAR, 2, 5, 54, 55 stability, 5–7, 21, 36, 40–41 state variables, 4, 5, 8, 18, 37 Oak Ridge National Laboratory, 43 strain rate, 2, 23, 24 ocean, 7–8 strength, 2, 23 currents, see currents, ocean stress density, see density, ocean ice-ocean, see ice-ocean stress heat, 4, 7, 29 principal, 25, 41 mixed layer, see mixed layer tensor, see internal stress salinity, see salinity, ocean wind, see wind stress stress, see ice-ocean stress subcycling, 24, 40 surface height, see height, sea surface sublimation, see evaporation surface slope, see slope, sea surface surface height, see height, sea surface temperature, see temperature, ocean open water, 5, 7, 8, 20, 21, 22 temperature, 26–34, 56 atmospheric, 4 Parallel Ocean Program, see POP freezing, 7 parallelization, 34, 36, 43 ice, 9, 26, 33 POP, 2, 3, 7, 37, 39, 40, 55 ocean, 4, 7 potential, 4, 5 radiation reference, 4, 7 Delta-Eddington, 2, 10, 27 surface, 6, 9, 13 longwave, 4, 6, 26, 28 thermodynamics, 3, 26–34 shortwave, 4–7, 10, 26–28, 29, 50 thickness rain, 4, 6, 7 distribution, 2, 8–9, 11, 18–23, 26, 36, 37, 40 reference ice or snow, 7–9, 12–13, 18, 24, 26, 33–34, 37 height, see height, reference space, see transport, thickness humidity, see humidity, reference timers, 36, 41, 46, 50 temperature, see temperature, reference tracer, 9 regional, 39 tracers, 10–11, 11–13, 36, 47–48, 63 remapping transport, 2, 8–21, 37, 40, 42 incremental, 9, 10–21, 37, 40 horizontal, 2, 10–18 linear, see transport, thickness thickness, 18–21 replacement pressure, 24 tripole, 3, 37 reproducible, 43 turbulent fluxes restart, 35–37, 39, 41, 43, 50 latent heat, see latent heat restoring, 37 sensible heat, see sensible heat ridging, 2, 8, 9, 21–23, 36, 42 wind stress, see wind stress salinity ice, 7, 26, 30, 32, 33 upwind, 11 ocean, 4, 7 van Leer, 13 salt, see salinity velocity, ice, 2, 6, 8, 9, 11, 14, 17, 23–25, 37 sensible heat, 4, 6, 7, 26, 28 shortwave, see radiation, shortwave volume, ice or snow, 8–10, 12, 17–19, 22, 23, 37 70 water, open, see open water wind stress, 3–6, 7, 24 velocity, 4–6, 40 WMO, 3, 8 zero-layer model, 3, 29 INDEX REFERENCES 71 References [1] T. 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