gmd 7 663 2014

gmd 7 663 2014
Model Development
Open Access
Geosci. Model Dev., 7, 663–693, 2014
© Author(s) 2014. CC Attribution 3.0 License.
The Finite Element Sea Ice-Ocean Model (FESOM) v.1.4:
formulation of an ocean general circulation model
Q. Wang, S. Danilov, D. Sidorenko, R. Timmermann, C. Wekerle, X. Wang, T. Jung, and J. Schröter
Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany
Correspondence to: Q. Wang ([email protected])
Received: 10 June 2013 – Published in Geosci. Model Dev. Discuss.: 23 July 2013
Revised: 23 January 2014 – Accepted: 11 March 2014 – Published: 30 April 2014
Abstract. The Finite Element Sea Ice-Ocean Model (FESOM) is the first global ocean general circulation model
based on unstructured-mesh methods that has been developed for the purpose of climate research. The advantage of
unstructured-mesh models is their flexible multi-resolution
modelling functionality. In this study, an overview of the
main features of FESOM will be given; based on sensitivity
experiments a number of specific parameter choices will be
explained; and directions of future developments will be outlined. It is argued that FESOM is sufficiently mature to explore the benefits of multi-resolution climate modelling and
that its applications will provide information useful for the
advancement of climate modelling on unstructured meshes.
Climate models are becoming increasingly important to
a wider range of users. They provide projections of anthropogenic climate change; they are extensively used for subseasonal, seasonal and decadal predictions; and they help us
to understand the functioning of the climate system.
Despite substantial progress in climate modelling, even
the most sophisticated models still show substantial shortcomings. Simulations of the Atlantic meridional overturning circulation, for example, still vary greatly in strength
and pattern between climate models (Randall et al., 2007).
Furthermore, many climate models still show substantial
problems when it comes to simulating the observed path
of the Gulf Stream. It is increasingly being recognized that
a lack of sufficiently high spatial resolution is one of the
main causes of the existing model shortcomings. In fact,
many climate-relevant processes are too small scale in nature
to be explicitly resolved by state-of-the-art climate models on available supercomputers and therefore need to be
parametrized (Griffies, 2004; Shukla et al., 2009; Jakob,
In recent years a new generation of ocean models that employ unstructured-mesh methods has emerged. These models allow the use of high spatial resolution in dynamically active regions while keeping a relatively coarse resolution otherwise. It is through this multi-resolution flexibility that unstructured-mesh models provide new opportunities to advance the field of climate modelling. Most existing unstructured-mesh models have dealt with coastal or
regional applications (e.g. Chen et al., 2003; Fringer et al.,
2006; Zhang and Baptista, 2008).
This paper focuses on the setting of unstructured-mesh
models for global applications, that is, on configurations
more geared towards climate research applications. More
specifically, the latest version of the Finite Element Sea IceOcean Model (FESOM) – the first mature, global sea-ice
ocean model that employs unstructured-mesh methods – will
be described. The fact that FESOM solves the hydrostatic
primitive equations for the ocean and comprises a finite element sea ice module makes it an ideal candidate for climate
research applications. FESOM has been developed at the Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research (AWI) over the last 10 yr (Danilov et al., 2004;
Wang et al., 2008; Timmermann et al., 2009).
The present study is meant to give a thorough overview of
FESOM in the context of global ocean modelling. Providing
the climate modelling community with such an overview of
the most mature global multi-resolution sea-ice ocean model
seems justified given that unstructured mesh modelling does
not feature yet in standard textbooks on ocean modelling.
Published by Copernicus Publications on behalf of the European Geosciences Union.
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Furthermore, the field is advancing so rapidly that details
of the implementation of FESOM described in previous papers (Danilov et al., 2004; Wang et al., 2008; Timmermann
et al., 2009) are already outdated. It is also expected that
other modelling groups working on the development of similar models (e.g. Ringler et al., 2013) will benefit from a detailed overview of our implementation of unstructured-mesh
methods in global models. Finally, we expect that the present
study, which also entails details on parametrizations and
model tuning, will stimulate discussions and therefore ultimately advance the development of multi-resolution models.
The basic numerical formulation of FESOM including
spatial and temporal discretization is described in Sect. 2.
Section 3 represents the key elements of FESOM which are
fundamental in formulating ocean climate models. This section partly takes a review form, describing various physical
parametrizations and numerical methods presented in the literature. A short summary will be given in the last section.
A brief historical review of FESOM’s development is given
in the Appendix.
Fig. 1. Schematic of horizontal discretization with the illustration
of basis functions used in FESOM. The stencil mentioned in the
text consists of seven nodes for node i in the example shown in this
Fig. 11. Schematic of horizontal discretization with the illustr
2 Numerical core of FESOM
of basis functions used in FESOM. The stencil mentioned in
text consists of seven nodes for node i in the example shown in
2.1 Spatial discretization
the shaved cells used by Adcroft et al. (1997). Keeping the 3Here we briefly explain the implementation of the
finite element method in FESOM. For a detailed description of the implementation see Wang et al. (2008). The variational formulation with the FE method involves two basic steps. First, the
partial differential equations (primitive equations) are multiplied by a test function and integrated over the model domain. Second, the unknown variables are approximated with
a sum over a finite set of basis functions. FESOM uses the
combination of continuous, piecewise linear basis functions
in two dimensions for surface elevation and in three dimensions for velocity and tracers. For example, sea surface elevation η is discretized using basis functions Mi as
ηi Mi ,
where ηi is the discrete value of η at grid node i of the 2-D
computational mesh. The test functions are the same as basis
functions, leading to the standard Galerkin formulation.
In two dimensions FESOM uses triangular surface
meshes. Figure 1 shows the schematic of 2-D basis functions
on a triangular mesh. The basis function Mi is equal to one at
grid node i and goes linearly to zero at its neighbour nodes;
it equals zero outside the stencil formed by the neighbour
nodes. The 3-D mesh is generated by dropping vertical lines
starting from the surface 2-D nodes, forming prisms which
are then cut into tetrahedral elements (Fig. 2). Except for layers adjacent to sloping ocean bottom each prism is cut into
three tetrahedra; over a sloping bottom not all three tetrahedra are used in order to employ shaved cells, in analogy to
Geosci. Model Dev., 7, 663–693, 2014
D grid nodes vertically aligned (i.e. all 3-D nodes have their
corresponding 2-D surface nodes above them) is necessitated
by the dominance of the hydrostatic balance in the ocean.
For a finite element discretization the basis functions
for velocity and pressure (surface elevation in the hydrostatic case) should meet the so-called LBB condition
(Ladyzhenskaya, 1969; Babuska, 1973; Brezzi, 1974), otherwise spurious pressure modes can be excited. These modes
are similar to the pressure modes of Arakawa A and B grids
(Arakawa, 1966). The basis functions used in FESOM for
velocity and pressure do not satisfy the LBB condition, so
some measures to stabilize the code against spurious pressure modes are required. Note that pressure modes on unstructured meshes are triggered more easily than in finitedifference models and robust stabilization is always needed.
In the early model version the Galerkin least squares
(GLS) method proposed by Codina and Soto (1997) was used
to solve the difficulty related to the LBB condition. In the current model version the GLS method is replaced by a pressure
projection method described by Zienkiewicz et al. (1999) to
circumvent the LBB condition. With the GLS method the iterative solver needs to solve the surface elevation equation
and the vertically integrated momentum equations together
(Danilov et al., 2004), whereas with the pressure projection
method the solution of surface elevation is separated and no
barotropic velocity is introduced (Wang et al., 2008). Therefore using the pressure projection method reduces the computational cost. It also leads to a more consistent code, as in
the GLS case the horizontal velocity and vertically integrated
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
waves, and needs iterative solvers. The Coriolis force term
uses the semi-implicit method to well represent inertial oscillations.
The default tracer advection scheme is an explicit fluxcorrected-transport (FCT) scheme (Sect. 3.5). The GM
parametrization is incorporated into the model with the Euler forward method (see Sect. 3.10 for the description of the
GM parametrization), while the vertical diffusivity uses the
Euler backward method for the same reason as for vertical
An external iterative solver is called for solving the surface
elevation equation. The final momentum and tracer equaFig.12.
2. Schematic
Schematic of
of spatial
each tions have only matrices of time derivative terms on the leftsurfacetriangle
tetra- hand side of the equations, which can be relatively efficiently
(b). Except
for layers
to sloping
(b). Except
for layers
to sloping
each solved. Overall the dynamics and thermodynamics in the
model are staggered in time with a half time step. That is,
prism is cut to three tetrahedra.
the new velocity is used to advect tracers, and the updated
temperature and salinity are then used to calculate density.
horizontal velocity cannot be in the same functional space
in the presence of bottom topography, leading to projection
2.3 Model efficiency
errors. The Appendix provides an overview on the development history of FESOM.
Models formulated on unstructured meshes are slower than
their structured-mesh counterparts because of indirect mem2.2 Temporal discretization
ory addressing and increased number of numerical operations. If an unstructured-mesh model is ns times slower
The advection term in the momentum equation is solved with
than typical structured-mesh models, simulations using it can
the so-called characteristic Galerkin method (Zienkiewicz
computationally benefit from its unstructured-mesh funcand Taylor, 2000), which is effectively the explicit secondtionality when the fine resolution region occupies less than
order finite-element Taylor–Galerkin method. The method is
1/ns portion of the total computational area and most of the
based on taking temporal discretization using Taylor expancomputational degrees of freedom are confined there. In the
sion before applying spatial discretization. Using this method
course of FESOM development we have chosen spatial and
with the linear spatial discretization as mentioned above, the
temporal discretization schemes by taking both model acculeading-order error of the advection equation is still second
racy and efficiency into account. FESOM is about 10 times
order and generates numerical dispersion (Durran, 1999),
slower than a typical structured-mesh model (Danilov et al.,
thus requiring friction for numerical stability.
2008). In its practical applications we therefore limit the loThe horizontal viscosity is solved with the explicit Euler
cally refined region to less than 10 % of the total domain
forward method (Sect. 3.9). The vertical viscosity is solved
area (e.g. Q. Wang et al., 2012; Timmermann et al., 2012;
with the Euler backward method because the forward time
Timmermann and Hellmer, 2013; Wekerle et al., 2013; Wang
stepping for vertical viscosity is unstable with a typical veret al., 2013). In these applications mesh resolutions are in1
tical resolution and time step. To ensure solution efficiency,
creased locally by factors ranging from 8 to 60 and most
we solve the implicit vertical mixing operators separately
computational grid nodes are located inside the refinement
from other parts of the momentum and tracer equations. The
region, ensuring great computational benefit.
surface elevation is solved implicitly to damp fast gravity
The run-time memory access in the current FESOM ver1 The stability of an explicit Euler forward time stepping method
sion, hindered by its 1-D storage of variable arrays, is one
of the bottlenecks for model efficiency. Some software engi≤ 1/2, where Av is the vertical
for vertical viscosity requires Av 1t
neering work is required in the future to identify the potential
mixing coefficient. A typical time step of 1t = 40 min and a surface
vertical resolution of 1z = 10 m require Av ≤ 0.02 m2 s−1 . Vertical
in improving memory access efficiency. Other unstructuredmixing coefficients in the surface boundary layer obtained from the
mesh numerical methods have shown potential in developKPP scheme (see Sect. 3.6) can readily be higher than this value
ing more efficient ocean models (for a review see Danilov,
2013). In our model development team, we reserve some
2 To guarantee global conservation the vertical diffusion equations are solved using the finite element method. For continuous basis functions the discretized vertical diffusion equations involve horizontal connections through the mass matrix with the time derivative
term, and they cannot be solved efficiently. We chose to lump the
matrix of time derivative terms (lumping means to sum all entries in
a row to the diagonal in the matrix). This leads to a decoupled equation for each water column, which can be very efficiently solved.
3 Implicit advection and lateral friction would require iterative
solvers with pre-conditioning in every time step, thus slowing down
the solution.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
human resources for research on different numerical methods, although currently the main effort is on FESOM development and its climate scale applications.
Key elements of the model
Different numerical and parametrization schemes are available in FESOM. A detailed description of all the available
features of FESOM is beyond the scope of the present study.
Here the focus will be on those model elements that are crucial for climate scale applications. Illustrations of model sensitivity to different choices are presented here only for some
of the model elements. A relatively complete description of
the key model elements does not allow us to pursue thorough
sensitivity studies for all of them here.
When configuring an OGCM for the purpose of climate
research, choices for numerical and parametrization schemes
are made according to the model developers’ practical experience as well as knowledge from the literature. For the
sake of the paper we provide a review form in some parts
of the model description in this section. The review details
the background for making our choices and importantly lists
recommendations from previous studies which can improve
model integrity. In our future work it will be valuable to evaluate those recommended options that have not been tested
with FESOM yet. We also note that the reviews made in this
section are not complete in themselves as they are organized
following our own model development issues considered so
far. For a thorough review on ocean model fundamentals, see
Griffies (2004).
All sensitivity tests presented in this section are based
on atmospheric forcing fields taken from the inter-annual
CORE-II data provided by Large and Yeager (2009). All simulations were carried out for 60 yr over the period of forcing
provided by Large and Yeager (2009).
Two-dimensional mesh
FESOM uses spherical coordinates, so the meridional and
zonal velocities would be poorly approximated on a triangle
covering the North Pole. To avoid the singularity a spherical coordinate system with the north pole over Greenland
(40◦ W, 75◦ N) is used.4
FESOM uses triangular surface meshes. There are a few
free triangle-mesh generators available, including DistMesh
(Persson and Strang, 2004) and Triangle (Shewchuk, 1996).
The mesh quality, the extent to which the triangles are close
to equilateral ones, can be further improved after mesh generation by relaxation of grid point locations. An abrupt change
in resolution can lead to bad triangles (with too small/big inner angles and very different edge lengths) thus degrading
the quality of meshes, so a transition zone between high and
4 The three Euler angles for performing the rotation are (50◦ ,
15◦ , −90◦ ) with the z–x–z rotation convention.
Geosci. Model Dev., 7, 663–693, 2014
Fig. 3.13.
of example
an example
of an
is used
all simulations
this work.
all in
in thisinwork.
coarse resolutions is generally introduced. The resolution of
a triangle is defined by its minimum height.
In practical applications with limited computational resources, we keep the horizontal mesh resolution coarse in
most parts of the global ocean (for example, at nominal one
degree as used in popular climate models), and refine particularly chosen regions. The equatorial band with meridionally narrow currents and equatorially trapped waves requires
higher resolution on the order of 1/3◦ (Latif et al., 1998;
Schneider et al., 2003). Figure 3 shows the horizontal resolution of an example mesh with increased resolution in the
equatorial band. It is nominal one degree in most parts of the
ocean and increases to 24 km poleward of 50◦ N. The resolution of this mesh is designed in terms of both kilometre (north
of 50◦ N) and geographical degree (south of 50◦ N) as a particular example, although it is not necessarily so in general
cases. Thanks to the flexibility of unstructured meshes, one
can fully avoid the imprint of geographic coordinates and design meshes based on distances along the spherical surface.
Many key passages between ocean basins such as Indonesian Throughflow, Bering Strait and Canadian Arctic
Archipelago (CAA) are important for basin exchange but
narrow and difficult to resolve in global models. Figure 4
shows an example mesh which was configured to study the
Arctic Ocean freshwater circulation (Wekerle et al., 2013).
This global mesh uses a coarse resolution south of 50◦ N and
increases resolution to 24 km poleward, with the CAA region
resolved at a 5 km scale. Traditional climate models cannot
resolve the straits as narrow as in the CAA (27–53 km wide at
the narrowest locations in the two largest CAA passages), so
they are usually widened and/or deepened to allow adequate
throughflow. However, such empirical treatment of CAA results in a very large range in the simulated CAA freshwater
transport among a set of state-of-the-art ocean models (Jahn
et al., 2012). The improved simulation by explicitly resolving the narrow straits with a global FESOM setup indicates
the potential of unstructured meshes in representing narrow
strait throughflows in global models (Wekerle et al., 2013).
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
It is worth mentioning that the model time step size is constrained by the finest resolution on the mesh.5 If the number
of grid points in the high-resolution regions is only a few percent of the total grid points, we cannot enjoy the advantage
of using FESOM because the overall time step size has to be
set small. Therefore, in practice, increasing horizontal resolution in narrow straits is usually implemented in applications
when some part of the ocean basin is also locally refined.
In such applications a large portion of the computational
grid points are in fine-resolution areas. To benefit from the
multi-resolution capability even in cases when only a very
small portion of the computational grid points have locally
increased resolutions, multirate time stepping schemes are
needed. Seny et al. (2013) gave an example of such schemes
applied in a discontinuous Galerkin model.
Vertical coordinates and discretization
The choice of vertical coordinates or vertical grids is one
of the most important aspects in the design of ocean circulation models (Griffies et al., 2000). Coordinated projects
have been carried out to study the performance of different
types of vertical discretization, – for example, DAMEE-NAB
(Chassignet et al., 2000) and DYNAMO (Willebrand et al.,
2001). Different vertical coordinates are used in ocean models and each of them has its own advantages and disadvantages (Haidvogel and Beckmann, 1999; Griffies, 2004).
FESOM uses z coordinates (also called geopotential coordinates) in the vertical. The primitive equations are discretized on z coordinates without coordinates transformation, while sigma or more general vertical grids can be conveniently used because of the FE formulation (Wang et al.,
2008). The 2-D mesh and 3-D discretization (including grid
types and vertical resolution) are set during the mesh generation stage off-line before carrying out simulations.
Similar to traditional sigma grid models, truncation errors
in computing hydrostatic pressure gradient exist on sigma
grids in FESOM (see Sect. 3.4). The error in hydrostatic pressure gradient can be reduced by introducing high-order interpolations, but it is not trivial and can potentially degrade
the solutions in climate scale simulations. Therefore, z-level
grids are recommended in setting up ocean climate models.
We use shaved cells over the ocean bottom on z-level grids to
better represent the gentle topographic slopes (Adcroft et al.,
1997; Wang et al., 2008). A faithful representation of bottom
topography by using shaved (or partial) cells can generally
improve the integrity of ocean model simulations (MaierReimer et al., 1993; Adcroft et al., 1997; Pacanowski and
Gnanadesikan, 1998; Myers and Deacu, 2004; Barnier et al.,
5 While an implicit advection scheme is stable in terms of tempo-
ral discretization, it is not necessarily accurate if the Courant number (U 1t/1, where U is the current speed, 1t is the time step and
1 is the grid spacing) is large.
Fig. 4.
of of
an an
the the
in stereographic
the stereographic
(a) (a)
in the
and (b)and
the CAA
of theto
Combining z levels in the bulk of the ocean with sigma
grids in shelf regions of interest is a viable alternative to zlevel grids. A schematic of such hybrid grids is shown in
Fig. 5. These types of grids are used in FESOM when the
ice cavities are included in the model (Timmermann et al.,
2012). This hybrid grid is similar to the generalized coordinate used in POM (Princeton Ocean Model) described by
Ezer and Mellor (2004). As illustrated in Fig. 5, sigma grids
are used under ice shelf cavities and along the continental
shelf around the Antarctic, while the z-level grids are used
in all other parts of the global ocean to minimize pressure
gradient errors.
There are a few reasons for using sigma grids in these
marginal seas. Sigma grids offer the flexibility in vertical refinement (near the ocean surface, near the bottom, or in the
whole column over the shallow continental shelves). The iceshelf–ocean interactions can be better represented with vertically refined resolution near the ocean surface. Increased vertical resolution is also beneficial for representing continental
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Ice Shelf
Bottom topography
A blend of several bottom topography data sets is used
to provide the bottom topography for FESOM. North of
69◦ N the 2 km resolution version (version 2) of International Bathymetric Chart of the Arctic Oceans (IBCAO version 2, Jakobsson et al., 2008) is used, while south of 64◦ N
the 1 min resolution version of General Bathymetric Chart
of the Oceans (GEBCO) is used. Between 64◦ N and 69◦ N
the topography is taken as a linear combination of the two
data sets. The ocean bottom topography, lower ice surface
height and ice shelf grounding line in the ice cavity regions
around the Antarctica are derived from the one minute Re−4
fined Topography data set (Rtopo-1, Timmermann et al.,
which is based on the BEDMAP version 1 data set
Distance (Km)
(Lythe et al., 2001).7
The raw topography data are used to determine the ocean
Fig. 5. Schematic of a hybrid vertical grid. The sigma grid is used in
. 15. Schematic
a ice
grid is used They are bilinearly interpolated to the model grid
shelf modelling
for grids.
the Antarctic
points. After the raw topography is interpolated to the model
the ice
and along
the continental
shelf; Antarctic
z-level grids are
for the
grid,logrid scale smoothing of topography is applied to get rid
used in other parts of the ocean.
ed under the ice shelf and along the continental shelf. Z-levelscale noise. The smoothing for each 2-D node is performed over its 2-D stencil (consisting of all 2-D nodes conds are used in other parts of the ocean.
nected by edges with it).8 Preliminary smoothing on coarse
shelf and ocean basins exchange processes, including dense
intermediate meshes should be avoided because it may be too
shelf water outflow and circumpolar water inflow; z-level
strong for the fine part of the model mesh. One may choose to
grids with shaved cells under ice shelves are found to be useexplicitly resolve narrow ocean straits using locally increased
ful in simulating the ocean circulation in ice cavities (Losch,
resolution in some cases and apply manual mesh and topog2008), while the merits of sigma grids are flexible vertical
raphy modification at unresolved straits in other cases. Modresolution and less grid scale noise, thus less spurious mixellers need to decide how to treat individual narrow straits
depending on the research interest and overall mesh design.
On the z-level grid the vertical resolution is usually set
The topography is bilinearly interpolated from the data
finer in the upper 100–200 m depth to better resolve the surgrid
with fine resolution (2 km and 1 min) to model grids.
face boundary layer and becomes coarser with depth. Shaved
model resolution is much lower than the topography
cells are generally used at the bottom. In the region of sigma
adequate smoothing of model topography
grids the vertical resolution is set depending on scientific incan
impact on the simulated ocean circulaterest, for example, increasing the near-surface resolution untion.
to repeat the grid scale smoothing
der the ice shelf and the near-bottom resolution where contiseveral
and b show the bottom topography
nental shelf and basin water mass exchange is important. The
three and one times respecvertical resolution distribution function of Song and Haidvotively
in Fig. 3. Sensitivity expergel (1994) is used in the mesh generator for adjusting the
topography indicate that
sigma grid resolution.
the former leads to a more realistic ocean circulation (Fig. 7).
The barotropic stream functions with the two versions
of model topography are different mainly in the Southern
Ocean, along the western boundary and in the North Atlantic
Depth (Km)
6 A convenient recipe is to define the sigma levels as
7 Improved ice bed, surface and thickness data sets for Antarctica
z(k) = hmin s(k) + (H − hmin ){(1 − θb )sinh(θ s(k))/sinh(θ) +
θb [tanh(θ (s(k) + 0.5)) − tanh(θ/2)]/(2tanh(θ/2))},
(1 ≤ k ≤ N ) is the vertical layer index, hmin is the minimum
depth in the sigma grid region, H is the water column thickness,
s(k) = −(k − 1)/(N − 1) and N is the number of vertical layers
(Song and Haidvogel, 1994). 0 ≤ θ ≤ 20 and 0 ≤ θb ≤ 1 are the
tuning parameters for designing the vertical discretization. Larger
θ leads to more refined near-surface layers and if θb approaches 1
resolution at the bottom is also refined. A transition zone is required
to smoothly connect the sigma and z-level grids.
(BEDMAP2, Fretwel et al., 2013) and a new bathymetry data set
for the Arctic Ocean (IBCAO3, Jakobsson et al., 2012) have been
released recently and their impact on model simulations compared
to previous data sets need to be tested in sensitivity studies.
8 The smoothing at node i is a weighted mean over its stencil.
The neighbour nodes have a weight one and node i has a weight
2n, where n is the total number of neighbour nodes. As an abrupt
change in mesh resolution is avoided, the variation in distance between node i and neighbour nodes is not accounted for in the
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Fig. 16.
after three
three iterations
iterations of
of stencil filter
6. Bottom
and (b) after a single iteration.
subpolar gyre. In the Weddell Sea, the maximum transport
in observational estimates is 29.5 Sv along the transect between the northern tip of the Antarctic Peninsula and Kapp
Norvegia (Fahrbach et al., 1994) and more than 60 Sv at the
Greenwich Meridian (Schroeder and Fahrbach, 1999). Observations suggest mean southward transport at the Labrador
Sea exit at 53◦ N ranging from 37 Sv (Fischer et al., 2004)
to 42 Sv (Fischer et al., 2010). Circulations in the Southern Ocean and Labrador Sea are dynamically controlled by
the Joint Effect of Baroclinicity and Relief (JEBAR, Olbers
et al., 2004; Eden and Willebrand, 2001), so the model results have large sensitivity to the treatment of bottom topography in these places. Because of the relatively coarse mesh
(Fig. 3), maximum barotropic transport in these regions is
weaker than observed in both simulations. The maximum
gyre transport in Weddell Sea and North Atlantic subpolar
gyre is about 34 Sv and 28 Sv respectively for the topography shown in Fig. 6a (see Fig. 7a). If topography smoothing
is applied only once, the transport is lower by about 10 Sv in
both regions (see Fig. 7b).
Over the terrain-following part of the mesh the topography is smoothed by adjusting the slope parameter r0 (also
called Beckmann and Haidvogel number, Beckmann and
Haidvogel, 1993) and the hydrostatic inconsistency number
7. (a)
function (Sv)
Fig. 17.
(Sv) in
in aa simulation
simulation with
with the
6a and
topography of Fig. 16a
(b) its
its difference
difference from
from aa run
run with
with the
are are
the mean
over the
topography of
the mean
of total
yr simulations.
yr of60
60 yr simulations.
r1 (also called Haney number, Haney, 1991).9 The smoothing helps to alleviate hydrostatic pressure gradient errors and
maintain numerical stability. In practice we recommend the
criteria r0 ≤ 0.2 and r1 ≤ 3. The smoothing is done on each
2-D stencil starting from the shallowest grid point until the
deepest grid point. This procedure is repeated until the criteria are satisfied throughout the mesh.
Hydrostatic pressure gradient
Care should be taken in the calculation of the hydrostatic
pressure gradient on sigma grids. Pressure gradient errors
are not avoidable when the sigma grid surface deviates
from the geopotential coordinate, but can be reduced with
carefully designed numerics (Shchepetkin and McWilliams,
2003). A few measures are taken to reduce the errors in FESOM. The widely used method of exchanging the sequence
o and r1 are defined on edges between two neighbouring
|H −H |
nodes. r0 = Hi +Hj , where H is the water column thickness,
and subscripts i and j indicate two neighbouring nodes. r1 =
zi (k)+zi (k+1)−zj (k)−zj (k+1)
zi (k)+zj (k)−zi (k+1)−zj (k+1) , where z(k) is the vertical coordinate
at layer k. The smoothing is done for r0 first and then for r1 ,
over stencils starting from the shallowest depth to the deepest. The
smoothing procedure usually needs to be repeated to satisfy the criteria throughout the mesh.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
of integration and differentiation is employed (Song, 1998;
Song and Wright, 1998). The horizontal derivatives of in situ
density are taken first and pressure gradient forces are calculated then. In this way pressure gradient force errors are
reduced but still present because of truncation errors in representing density with linear functions. The second measure
is to use high-order interpolation in the vertical to interpolate
density to a common depth for computing the density gradient. The idea is discussed and assessed in Wang et al. (2008).
In practice more measures are taken to control the pressure
gradient errors on sigma grids. A common additional recipe
is to apply topography smoothing to satisfy the criteria for
r0 and r1 as described in Sect. 3.3. Increasing resolution also
helps to reduce pressure gradient errors (Wang et al., 2008).
Furthermore, the sigma grid as a part of the hybrid grid is
only used around the Antarctic continental shelf and under
ice shelves (Sect. 3.2), where we commonly use increased
resolution to resolve small geometrical features.
Tracer advection
The commonly used tracer advection scheme in FESOM
is an explicit second-order flux-corrected-transport (FCT)
scheme. The classical FCT version described by Löhner
et al. (1987) is employed as it works well for transient problems. The FCT scheme preserves monotonicity and eliminates overshoots, a property useful for maintaining numerical
stability on eddying scales. Upon comparison to a secondorder scheme without flux limiter and an implicit second order scheme in idealized 2-D test cases, at coarse resolution
the FCT scheme tends to slightly reduce local maxima even
for a smooth field, but it well represents a sharp front and
shows least dispersion errors in general (Wang, 2007).
Advection schemes should be able to provide adequate
dissipation on grid scales and keep large scales less dissipated. Griffies and Hallberg (2000) show that it is important to adequately resolve the admitted scales of motion
in order to maintain a small amount of spurious diapycnal
mixing in z-coordinate models with commonly used advection schemes. They find that spurious diapycnal mixing can
reach more than 10−4 m2 s−1 depending on the advection
scheme and the flow regime.10 Ilicak et al. (2012) demonstrate that spurious dianeutral transport is directly proportional to the lateral grid Reynolds number. Our preliminary
tests show that the effective spurious diapycnal mixing associated with the FESOM FCT scheme is similarly high as
shown in Griffies and Hallberg (2000). Systematic research is
needed for exploring alternative transport schemes and limiters and for investigating the dependence on the Reynolds
number, especially in the context of mesh irregularity.
10 The tests by Jochum (2009) indicate that relatively large spuri-
ous mixing occurs locally in practice as varying background diffusivity at the order of 0.01 × 10−4 m2 s−1 still produces difference
in coarse model results.
Geosci. Model Dev., 7, 663–693, 2014
Diapycnal mixing
Diapycnal mixing in the ocean has a strong impact on the
dynamics of the ocean circulation and on the climate system as a whole (e.g. Bryan, 1987; Park and Bryan, 2000;
Wunsch and Ferrari, 2004). The mixing processes are not resolved in present ocean models and need to be parametrized.
The k-profile parametrization (KPP) proposed by Large et al.
(1994) provides a framework accounting for important diapycnal mixing processes, including wind stirring and buoyancy loss at the surface, non-local effects in the surface
boundary layer, shear instability, internal wave breaking and
double diffusion. Previous studies (Large et al., 1997; Gent
et al., 1998) suggest that the KPP scheme is preferable in
climate simulations. It is implemented in many current climate models. It is also used in FESOM for large-scale simulations.11
Mixing induced by double diffusion (due to salt fingering
and double diffusive convection) was found to have a relatively small impact on the mixed layer depth (Danabasoglu
et al., 2006) and upper ocean temperature and salinity
(Glessmer et al., 2008) in sensitivity studies, although its impact (mainly from salt fingering) on biogeochemical properties is pronounced and cannot be neglected in ecosystem
modelling (Glessmer et al., 2008). The double diffusion mixing scheme modified by Danabasoglu et al. (2006) is implemented in FESOM.
Diapycnal mixing from barotropic tides
Mixing due to shear instability is parametrized as a function
of Richardson number (Large et al., 1994). To include the
mixing from barotropic tides interacting with ocean bottom,
especially in the relatively shallow continental shelf regions,
the tidal speed is accounted for in the computation of the
Richardson number as proposed by Lee et al. (2006). As the
tidal speed is large along the coast (Fig. 8), the Richardson
number is small and vertical mixing is large in these regions.
The original tidal mixing scheme of Lee et al. (2006) leads to
too strong vertical mixing even far away vertically from the
ocean bottom, as manifested by unrealistic winter polynyas
in the central Weddell Sea in our simulations (not shown).
The exponential decay as a function of distance from the
ocean bottom suggested by Griffies (2012) is implemented
in FESOM. It helps to remove the spurious large mixing.12
11 Other mixing schemes are also used in climate models. For example, the current version of the MPIOM-ECHAM6 Earth System
Model (Jungclaus et al., 2013) uses the Pacanowski and Philander
(1981) scheme.
12 The parametrization of mixing from barotropic tides interacting with the continental shelf is given by κvtidal = κmax (1 +
σ Ri)−p exp−(H −|z|)/ztide , where κmax = 5 × 10−3 m2 s−1 , σ = 3,
p = 1/4, Ri is the Richardson number based on tidal speed (see
Lee et al., 2006, for details), H is the water column thickness, and
ztide is an exponential decay length scale (proportional to tidal speed
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Fram Strait Freshwater Transport
M2 Speed [cm/s]
60 N
30 N
30 S
90 S o
180 W
no tidal mixing
with tidal mixing
60 S
Freshwater Transport (mSv)
90 N
120 W
60 W
60 E
120 E
180 W
M2 tidal
tidal speed
speed map (cm
High speed is mainly located
8. M2
).). High
in shallow
shallow shelf
shelf regions,
regions, including
including some
some Arctic
Arctic and North Atlantic
coastal regions.
The barotropic tidal mixing was found to be useful as it
assists in the horizontal spreading of river water at certain
river mouths (Griffies et al., 2005). Our simulation shows
that this is the case especially for the Arctic river runoff. The
increased horizontal spreading of river water from the Arctic marginal seas leads to an increase in the freshwater flux
at Fram Strait (see Fig. 9). The freshwater transport at Fram
Strait shows the largest spread among the four Arctic gates
in a recent Arctic Ocean model intercomparison, partly attributed to uncertainties in simulated salinity in the western
Arctic Ocean (Jahn et al., 2012). Due to its impact on river
water spreading and salinity, tidal mixing is among the key
processes that need to be investigated to understand the reported model biases.
Mixing due to barotropic tides has a large-scale impact
as it reduces the Atlantic Meridional Overturning Circulation (AMOC) (Fig. 10). The Labrador Sea Water production
and AMOC are more sensitive to the freshwater exported
through Fram Strait than through Davis Strait (Wekerle et al.,
2013). The increased freshwater export through Fram Strait
is an important mechanism through which tidal mixing can
consequently weaken the Labrador Sea deep convection and
AMOC, while there could be other relevant processes. It
should be noticed that the tests shown here are carried out
with an ocean-alone model and surface salinity restoring to
climatology is enforced. Salinity restoring can provide a local salt sink/source. We speculate that the impact of tidal
mixing on the Arctic freshwater export and large-scale circulation is also significant in coupled climate models (without surface salinity restoring). To include the impact of tides,
Mueller et al. (2010) added a tidal model into a coupled climate model (MPIOM-ECHAM5). In this case the tidal velocity is simulated and tidal mixing is explicitly taken into
times the M2 tide period). The exponential decay term was introduced by Griffies (2012) and does not exist in the original formula
of Lee et al. (2006).
9. Time
Fig. 19.
Time series
series of
of Fram
Fram Strait
Strait freshwater transport (m Sv
The impact of barotropic tidal mixing is illustrated.
account through the dependence of mixing coefficients on the
Richardson number. They also reported that tides have pronounced influence on the ocean circulation, including weakening of the Labrador Sea deep convection.
Diapycnal mixing associated with internal wave
energy dissipation
The background vertical diffusivity in the KPP scheme represents the mixing due to internal wave breaking, which provides mechanical energy to lift cold water across the thermocline and increase the potential energy of water, thus sustaining the large-scale overturning circulation (Huang, 1999;
Wunsch and Ferrari, 2004). Wind and tides are the main energy sources for this mechanical energy in the abyss (Munk
and Wunsch, 1998). Observational estimates indicate that the
diapycnal diffusivity is of the order of 0.12 ± 0.02 − 0.17 ±
0.02 × 10−4 m2 s−1 (Ledwell et al., 1993, 1998) in the subtropical Atlantic pycnocline and 0.15 ± 0.07 × 10−4 m2 s−1
in the North Pacific Ocean (Kelley and Van Scoy, 1999), and
much smaller values were observed near the equator (Gregg
et al., 2003). In the deep ocean the diffusivity is observed
to be small (0.1 × 10−4 m2 s−1 ) over smooth topography and
much larger (1 − 5 × 10−4 m2 s−1 ) near the bottom in regions
of rough topography (Polzin et al., 1997; Toole et al., 1997;
Ledwell et al., 2000; Morris et al., 2001; Laurent et al., 2012).
A modified version of the Bryan and Lewis (1979) background vertical tracer diffusivity is used poleward of 15◦ in
the model formulation with FESOM (Fig. 11).13 The minimum value is 0.1 × 10−4 m2 s−1 at the surface and the maximum value is 1.1 × 10−4 m2 s−1 close to the ocean bottom.
Motivated by observations (Gregg et al., 2003) the magnitude of this vertical profile is made one order smaller within
the ±5◦ latitude range (0.01 × 10−4 m2 s−1 at ocean surface)
and increased linearly to the off-equator value at 15◦ N/S.
Using a coupled climate model Jochum et al. (2008) found
13 The background vertical tracer diffusivity poleward of 15◦ N/S
is computed as a function of depth {0.6 + 1.0598/π × atan[4.5 ×
10−3 × (|z| − 2500.0)]} × 10−4 .
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Off−Equator Background Vertical Diffusivity
Depth (km)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Background Vertical Diffusivity (1e−4 m2/s)
1.1 1.2
Background vertical
vertical diffusivity
of 15
of 15
Atlantic results in their particular tests. Anyway, observations for diapycnal diffusivity have motivated the utilization
of more realistic diffusivity values in present climate models
(see e.g. Danabasoglu et al., 2012).
The importance of the Arctic Ocean in the climate system especially in a warming world and the reported difficulty in robustly representing the surface and deep circulation in the Arctic Ocean in state-of-the-art ocean models
(e.g. Karcher et al., 2007; Zhang and Steele, 2007; Jahn
et al., 2012) warrant research on improving numerical models including diapycnal mixing parametrizations. The diapycnal mixing in the halocline in the central Arctic Ocean
is small compared to mid-latitude, largely due to the presence of sea ice (Rainville and Winsor, 2008; Fer, 2009). The
Fig. 10. (a) Atlantic Meridional Overturning Circulation (AMOC)
diapycnal diffusivity is 0.05 ± 0.02 × 10−4 m2 s−1 averaged
Fig. 110.
(a) function
in a run without
mixing due to
over 70–220 m depth in the Amundsen Basin and as low as
a the
to × 10−4 m2 s−1 in the upper cold halocline (Fer, 2009).
barotropic tides
a run with the tidal
mix- due0.01
ing. The
10 yr mean in
60 yrasimulations.
run with the tidal mixA small background vertical diffusivity of 0.01×10−4 m2 s−1
ing. The results are the last 10 yr mean in 60 yr simulations. was used in the KPP scheme and found to be optimal in some
regional Arctic models (Zhang and Steele, 2007; Nguyen
et al., 2009). The decrease in diapycnal diffusivity in the Arcthat using the small background vertical diffusivity (0.01 ×
Ocean was taken into account in present climate models
10−4 m2 s−1 ) in the equatorial band improves the tropical
(e.g. Jochum et al., 2013). In our practice we found that usprecipitation, although the improvement is only minor coming this small value indeed improves the representation of the
pared to existing biases. We use a constant background versummer warm layer, but it increases the misfit of the halotical viscosity of 10−4 m2 s−1 , and there is no observational
cline (too fresh in the upper halocline and too saline in the
justification for this value.
lower halocline in the model) and leads to too low freshwaEnhanced vertical mixing in the thermocline arising from
ter export through Fram Strait. Therefore, this local tuning
Parametric Subharmonic Instability (PSI) of the M2 tide at
of background diapycnal diffusivity for the Arctic Ocean is
the 28.9◦ N/S band (Tian et al., 2006; Alford et al., 2007;
not adopted in FESOM. Presumably, using a more realistic
Hibiya et al., 2007) is not accounted for in our model formuvertical profile of diapycnal diffusivity with a range 0.01–
lation. The model sensitivity study by Jochum et al. (2008)
0.1 × 10−4 m2 s−1 in the halocline as suggested by observashows that increasing the off-equator background vertical
tions (Fer, 2009) can more adequately simulate the Arctic
diffusivity in the thermocline toward the observational estiOcean circulation. This hypothesis has not been tested yet.
mate (0.17×10 m s instead of 0.1×10 m s ) or accounting for the mixing arising from PSI worsens the North
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Improved understanding of mixing processes in the ocean
has led to a parametrization of abyssal mixing induced by
internal wave breaking associated with baroclinic tidal energy (St Laurent et al., 2002; Simmons et al., 2004). Concentrating intense mixing above rough topography where major tidal energy dissipates was found to be preferable for
representing deep ocean stratification and Southern Ocean
heat uptake in climate models (Saenko, 2006; Exarchou
et al., 2013). The model sensitivity study by Jayne (2009)
shows that using the tidal mixing parametrization proposed
by St Laurent et al. (2002) can significantly enhance the
deep cell of the meridional overturning circulation (MOC)
in comparison with only using an ad hoc background vertical diffusivity, although the upper cell of the MOC and
the poleward heat transport (the often used diagnostics for
adjudging climate models) are not strongly affected by this
parametrization. Present climate and earth system models
tend to use the St Laurent et al. (2002) parametrization instead of the Bryan and Lewis (1979) type of background
diffusivity (e.g. Danabasoglu et al., 2012; Delworth et al.,
2012; Dunne et al., 2012). The merit of the abyssal diapycnal mixing parametrization of St Laurent et al. (2002) is
that it is based on energy conservation and is more consistent with physical principles. Compared to the tidal mixing
scheme by Lee et al. (2006) which tends to increase vertical diffusivity in regions of low Richardson numbers (particularly the continental shelf regions), the parametrization
of St Laurent et al. (2002) and Simmons et al. (2004) allows
for enhanced tidal mixing in deep ocean regions. Comparing
these different approaches remains for our future work.
Energy associated with mesoscale eddies is another importance source for turbulent mixing. Saenko et al. (2012)
have recently investigated the individual effects of the tide
and eddy dissipation energies on the ocean circulation. They
showed that the overturning circulation and stratification
in the deep ocean are too weak when only the tidal energy maintains diapycnal mixing. With the addition of the
eddy dissipation, the deep-ocean thermal structure became
closer to that observed and the overturning and stratification in the abyss became stronger. Jochum et al. (2013) developed a parametrization for wind-generated near-inertial
waves (NIWs) and found that tropical sea surface temperature and precipitation and mid-latitude westerlies are sensitive to the inclusion of NIW in their climate model. They
concluded that because of its importance for global climate
the uncertainty in the observed tropical NIW energy needs
to be reduced. Presumably the recent progress in the understanding of diapycnal mixing processes will increase model
overall fidelity when practical parametrizations for these processes are taken into account.
Penetrative short-wave radiation
The infrared radiation from the solar heating is almost completely absorbed in the upper 2 m water column, while the
ultraviolet and visible part of solar radiation (wavelengths
< 750 nm) can penetrate deeper into the ocean depending on
the ocean colour. In the biologically unproductive waters of
the subtropical gyres, solar radiation can directly contribute
to the heat content at depths greater than 100 m. Adding all
the solar radiation to the uppermost cell in ocean models with
a vertical resolution of 10 m or finer can overheat the ocean
surface in regions where the penetration depth is deep in reality.
Many sensitivity studies have shown that adequately accounting for short-wave radiation penetration and its spacial
and seasonal variation is important for simulating sea surface temperature (SST) and mixed layer depth at low latitude,
equatorial undercurrents, and tropical cyclones and El Niño–
Southern Oscillation (ENSO) (Schneider and Zhu, 1998;
Nakamoto et al., 2001; Rochford et al., 2001; Murtugudde
et al., 2002; Timmermann et al., 2002; Sweeney et al., 2005;
Marzeion et al., 2005; Anderson et al., 2007; BallabreraPoy et al., 2007; Zhang et al., 2009; Jochum et al., 2010;
Gnanadesikan et al., 2010; Liu et al., 2012). The solar radiation absorption is influenced by ocean colour on a global
scale, so the bio-physical feedbacks have a global impact on
the simulated results, including sea-ice thickness in the Arctic Ocean and the MOC (Lengaigne et al., 2009; Patara et al.,
One traditional way to account for the spatial variation of
short-wave penetration in climate models is to use spatially
varying attenuation depths in an exponential penetration profile, which was found to be preferable compared to using
a constant depth (Murtugudde et al., 2002). The seasonal
variability of the attenuation depth plays an important role in
the interannual variability in the tropical Pacific (BallabreraPoy et al., 2007). The interannual variability in short-wave
absorption was also found to be important in representing
ENSO in climate models (Jochum et al., 2010).
We use the short-wave penetration treatment as suggested
by Sweeney et al. (2005) and Griffies et al. (2005). The optical model of Morel and Antoine (1994) is used to compute
visible light absorption.14 The chlorophyll seasonal climatology of Sweeney et al. (2005) (see Fig. 12) is used in the
computation. The visible light attenuation profile is obtained
from the optical model, and the difference between two vertical grid levels is used to heat the cells in between. Sweeney
et al. (2005) show that the optical models of Morel and Antoine (1994) and Ohlmann (2003) produce a relatively small
difference in their ocean model.
14 The attenuation profile of downward radiation in the vis0 (x, y)(V ez/ζ1 +
ible band is computed via IVIS (x, y, z) = IVIS
V2 ez/ζ2 ), where V1 , V2 , ζ1 and ζ2 are computed from an empirical relationship as a function of chlorophyll a concentration as sug0 is 54 % of the downward
gested by Morel and Antoine (1994). IVIS
solar radiation to the ocean, and the other part is infrared radiation
and is directly added to the ocean surface.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Chlorophyll Concentration [mg/m ]
60 N
30 S
60 S
90 S o
180 W
120 W
60 W
60 E
120 E
180 W
Fig. 112.
12. Annual
chlorophyll concentration
)) climaFig.
Annual mean Chlorophyll
concentration (mg m
tology. Clear
Clear waters are mainly in the subtropical basins and high
chlorophyll concentration
concentration is
is seen
seen in
in regions
regions with
with high
high level
level of biological activity.
activity. The data are from Sweeney et al. (2005).
In some Earth System Models ecosystem models are used
to better represent chlorophyll fields and the bio-physical
feedbacks (Lengaigne et al., 2009; Loeptien et al., 2009;
Jochum et al., 2010; Patara et al., 2012). Prognostic biogeochemistry is potentially beneficial in improving the fidelity of
climate prediction through adaptive bio-physical feedbacks.
An ecosystem model (REcoM, Schartau et al., 2007) has just
been coupled to FESOM but is not included in the short-wave
penetration parametrization in the present model version.
Vertical overturning
The hydrostatic approximation necessitates the use of
a parametrization for unresolved vertical overturning processes. One approach is to use the convection adjustment
schemes from Cox (1984) or Rahmstorf (1993). The latter
scheme can efficiently remove all static instability in a water
column. Another approach is to use a large vertical mixing
coefficient (e.g. 10 m2 s−1 ) to quickly mix vertically unstable water columns and it is employed in FESOM.15 As indicated by Klinger et al. (1996), using a large but finite vertical
mixing coefficient can improve the simulation compared to
instantaneous convection adjustment. Note that the vertical
diffusion approach can only be realized through implicit time
Horizontal viscosity
Horizontal momentum friction in ocean models is employed
mainly for practical computational reasons and not motivated
15 Super-parametrization as an alternative is found to be greatly
superior to the convection adjustment parametrization at much less
computational cost than running non-hydrostatic models (Campin
et al., 2011). Its potential in climate modelling needs to be explored.
16 Implicit time stepping methods for vertical diffusion are needed
in general. See footnote 1.
Geosci. Model Dev., 7, 663–693, 2014
by first principles (see the review of Griffies et al., 2000). As
a numerical closure, horizontal friction is intended to suppress grid noise associated with the grid Reynolds number
and to resolve viscous boundary currents (Bryan et al., 1975;
Large et al., 2001; Smith and McWilliams, 2003).17 In practice, horizontal friction in climate models is kept as small as
feasible provided the grid noise is at an acceptable level and
the western boundary layers are properly resolved.
Both Laplacian and biharmonic momentum friction operators are used in large-scale ocean simulations, and there
is no first principle motivating either form. With respect to
the dissipation scale-selectivity, the biharmonic operator is
favourable compared to the Laplacian operator as it induces
less dissipation at the resolved scales and concentrates dissipation at the grid scale (Griffies and Hallberg, 2000; Griffies,
2004). Large et al. (2001) and Smith and McWilliams (2003)
proposed an anisotropic viscosity scheme by distinguishing
the along and cross-flow directions in strong jets in order to
reduce horizontal dissipation while satisfying the numerical
constraints. Larger zonal viscosities were used in the equatorial band to maintain numerical stability in the presence of
strong zonal currents, and larger meridional viscosities were
employed along the western boundaries to resolve the Munk
boundary layer (Munk, 1950), while the meridional viscosity remained small in the equatorial band to better capture
the magnitude and structure of the equatorial current. This
approach was adopted in the previous GFDL climate model
(Griffies et al., 2005), while isotropic viscosities are restored
in a new GFDL Earth System Model to “allow more vigorous tropical instability wave activity at the expense of adding
zonal grid noise, particularly in the tropics” (Dunne et al.,
Different choices for viscosity values were made in different ocean climate models. One choice is to use the prescribed viscosity. Due to the convergence of the meridians
grid resolution also varies on structured meshes in some traditional models. To avoid numerical instability associated
with large viscosity in an explicit time stepping scheme, viscosity is often scaled with a power of the grid spacing (e.g.
Bryan et al., 2007; Hecht et al., 2008). Another approach is to
use flow-dependent Smagorinsky viscosities (Smagorinsky,
1963, 1993). The Smagorinsky viscosity is proportional to
the local horizontal deformation rate times the squared grid
spacing in the case of the Laplacian operator. It is enhanced
in regions of large horizontal shear, thus providing increased
17 In the case of Laplacian viscosity the grid Reynolds number
is defined as Re = U 1/A, where U is the speed of the currents, 1
is the grid resolution, and A is the viscosity. In one dimension, the
centred discretization of momentum advection requires Re < 2 (or
A > U√1/2) to suppress the dispersion errors. The second constraint
A > ( 3/π)3 β13 ensures that the frictional western boundary is
resolved by at least one grid point (Bryan et al., 1975). Here β is
the meridional gradient of the Coriolis parameter. Additionally, an
explicit time stepping (Euler forward) method enforces an upper
bound for horizontal viscosity, i.e. A < 12 /(21t).
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
dissipation where it is required to maintain stability. Its dependence on grid spacing eliminates the requirement for additional scaling as done for a priori specified viscosities.
We use the biharmonic friction with a Smagorinsky viscosity in FESOM large-scale simulations. Griffies and Hallberg (2000) provide a thorough review on this scheme.
As linear (first order) basis functions are used in FESOM,
a direct formulation of the biharmonic operator cannot be
achieved. Therefore, a two-step approach (first evaluating
nodal Laplacian operators, then constructing the biharmonic
operators) is used as described by Wang et al. (2008). The
biharmonic Smagorinsky viscosity is computed as Laplacian
Smagorinsky viscosity times 12 /8 as suggested from the linear stability analysis (Griffies and Hallberg, 2000),18 where
1 is the local grid resolution. The dimensionless scaling parameter in the computation of Smagorinsky viscosity is set
equal to π in our practice. To resolve the western boundaries, a minimum biharmonic viscosity of β15 is set at the
four grid points close to the western boundaries, where β is
the meridional gradient of the Coriolis parameter. When grid
resolution is increased along the western boundaries and the
velocity becomes more vigorous, the western boundary constraint becomes less stringent than the Reynolds constraints
(see the discussion in Griffies and Hallberg, 2000).
Figure 13a shows the sensitivity of barotropic stream functions to the two forms of friction operators (biharmonic and
Laplacian). The Smagorinsky viscosity is used in both simulations. The difference of barotropic stream functions is seen
mainly in the Southern Ocean and northern North Atlantic. In
both regions the biharmonic viscosity leads to a stronger circulation, by about 4 Sv for the Antarctic Circumpolar Current
(ACC) and 8 Sv for the subpolar gyre in the North Atlantic.
Strong and narrow currents are sensitive to the form of friction operators. The strength of North Atlantic Current, South
Atlantic Current and Pacific equatorial current is enhanced
by 2–4 Sv for the biharmonic case. Local impacts along ACC
near topographic features or where the current narrows are
also visible.
Consistent with the sensitivity study of Jochum et al.
(2008), the boundary currents off east Greenland and west
Greenland are enhanced with reduced momentum friction by
using the biharmonic operator, thus increasing warm, saline
Irminger Current water inflow to the Labrador Sea and decreasing Labrador Sea sea-ice area (not shown).19 Enhanced
18 For centred differences in space and forward difference in time
12 for the Laplacian
in a 2-D case, the stability requirement is A < 41t
1 for the biharmonic operator, thus a ratio of
operator and B < 321t
1 /8 between B and A, where A is the Laplacian viscosity, B is
the biharmonic viscosity, 1t is the time step and 1 is the horizontal
19 Although a similar conclusion is obtained, our sensitivity tests
are different from that of Jochum et al. (2008). They reduce momentum dissipation by replacing the combination of background
and Smagorinsky viscosity by only the background one, while we
Fig. 13.
function difference
(Sv) (Sv)
runs Laplacian
with Laplacian
and biharmonic
and biharmonic
(the latter
latter minus
betweenaa run
run with
the former).
biharmonic viscosity scaled with
with third
third power
power of
of the
the horizontal
horizontal resresolution and a run with biharmonic
biharmonic Smagorinsky
Smagorinsky viscosity
viscosity (the
(the latter
minus the
10 mean
yr mean
in yr
60 yr
are are
the the
10 yr
in 60
fraction of Atlantic Water in the Labrador Sea weakens the
stratification and results in stronger deep convection there,
thus an enhanced AMOC upper cell (by about 1 Sv, see
Fig. 14a). The increase in subpolar gyre strength (Fig. 13a)
is associated with increased density in the Labrador Sea resulting from enhanced Atlantic Water inflow and deep water
More Atlantic Water accumulates south of the Greenland–
Scotland Ridge (GSR) in the case of Laplacian friction, leading to stronger deep convection north of 60◦ N, which is manifested by the strengthening of the overturning circulation at
intermediate depth between 50–60◦ N (Fig. 14a). The commonly called Labrador Sea Water feeding the Deep Western
Boundary Current has its origin both in the Labrador Sea and
south of GSR including the Irminger Sea (Pickart et al., 2003;
Vage et al., 2008; de Jong et al., 2012). Deep mixed layers
indicate the presence of deep convection in both regions in
both simulations, but reduced dissipation in the biharmonic
case favours it in the Labrador Sea. Since reduced dissipation
drives both the AMOC and subpolar gyre strength toward
compare the Laplacian friction to the biharmonic friction, with the
latter having less dissipation on resolved scales.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Fig. 14. The same as Fig. 13 but for AMOC (Sv).
Fig. 114.
The same as Fig. 113 but for AMOC (Sv).
observations in our test, we choose to use the low dissipative
biharmonic friction operator. Different regions of the global
ocean were analysed in viscosity sensitivity experiments by
Jochum et al. (2008), and generally improved ocean circulations were observed in their coupled climate model with reduced dissipation (at the expense of an increase in numerical
Figures 13b and 14b compare the impact of Smagorinsky
and flow-independent, prescribed viscosities. In both simulations the biharmonic friction is used. In the prescribed
viscosity case, viscosity is a function of the cubed grid
resolution, B0 13 /130 , where B0 = −2.7 × 1013 m4 s−1 and
10 = 112 km. No obvious instability is visible in both simulations. The difference in the large-scale circulation between
the two simulations is clearly less significant than for the
two friction operators. With the Smagorinsky viscosity the
barotropic stream function is higher by 2–3 Sv in the central Labrador Sea. The increase in the AMOC upper cell is
also rather small (0.1–0.2 Sv). The small difference between
Geosci. Model Dev., 7, 663–693, 2014
the two simulations is not unexpected because the prescribed
viscosity is relatively small.
Further evaluation of the impact of momentum dissipation
on the large-scale circulation still needs to be pursued, especially for eddy-permitting and eddy-resolving simulations.
For example, an intermediate value of biharmonic viscosity
(0.5B0 13 /130 ) is found to produce good Gulf Stream separation and realistic North Atlantic Current penetration into
the Northwest Corner region in eddy resolving (0.1◦ ) simulations by Bryan et al. (2007). They got a southward displaced Gulf Stream separation with a lower viscosity and
large SST errors at the subtropical–subpolar gyre boundary
with a higher viscosity, indicating that only a small range
in the parameter space exists for tuning their eddy-resolving
model. To overcome the problems of too early separation of
the Gulf Stream with a small biharmonic viscosity and establishment of a permanent eddy north of Cape Hatters with
a large biharmonic viscosity, Chassignet and Garraffo (2001)
and Chassignet and Marshall (2008) recommend to jointly
use biharmonic and Laplacian viscosity in eddy resolving
models, with both values smaller than those when only one
friction form is used. They found that by combining the two
operators it is possible to retain the scale selectiveness of the
biharmonic operator and to provide useful damping at larger
scales, the latter of which helps to eliminate the wrong permanent eddy. Some recent eddy permitting coupled climate
models have chosen to use the biharmonic friction operator
(Farneti et al., 2010; Delworth et al., 2012). Providing a unified closure for momentum dissipation valid in various dynamical situations and on meshes refined in different ways
and in different regions remains an important and challenging task in developing unstructured-mesh models.
In the sigma grid region in the case of using a hybrid grid,
we apply momentum friction along the sigma grid slope to
maintain numerical stability.20 For unknown reasons the biharmonic operator turns out to be unstable even if it acts
along the sigma grid slope. The two-step implementation of
the biharmonic operator is presumably the cause of this problem. Therefore we currently employ the Laplacian operator
on sigma grids together with the Smagorinsky viscosity.
Eddy mixing and stirring
Much of the mixing induced by mesoscale eddies is oriented
along locally referenced potential density surfaces (neutral
surfaces, McDougall, 1987), which has motivated the utilization of rotated tracer diffusion (Redi, 1982; Olbers et al.,
1985; Griffies et al., 1998).21 The use of isoneutral diffusion
20 As the sigma grid slope can be very steep, a horizontal friction
imposes a large component perpendicular to the grid slope, which
can readily lead to instability even with a very small time step in the
case of a forward time stepping.
21 Neutral diffusion is described by Laplacian operators in ocean
climate models, although eddy-resolving models use neutral biharmonic operators to add dissipation at grid scales to maintain numer-
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
(often called Redi diffusion in the literature) significantly reduces the unphysical diapycnal mixing associated with horizontal diffusion (Veronis, 1975; Böning et al., 1995), thus
improving model integrity. In the bulk of the ocean interior
the neutral slope is small, which motivates the application of
the small slope approximation to simplify the diffusion tensor (Gent and McWilliams, 1990).
Gent and McWilliams (1990) and Gent et al. (1995) (referred to as GM90 hereafter) provided a closure to represent the adiabatic stirring effects of mesoscale eddies. They
suggested a form of eddy-induced bolus velocity for zcoordinate models by considering the reduction of available
potential energy through baroclinic instability. This bolus
velocity is added in tracer equations to advect tracers together with resolved velocity. The implementation of the GM
parametrization in z-coordinates models significantly improves the model results including temperature distribution,
heat transport and especially deep convection (Danabasoglu
et al., 1994; Danabasoglu and McWilliams, 1995; Robitaille
and Weaver, 1995; Duffy et al., 1995, 1997; England, 1995;
England and Hirst, 1997; Hirst and McDougall, 1996, 1998;
Hirst et al., 2000). The eddy-induced velocity as given by
GM90 is v ∗ = −∂z (κgm S) + ẑ∇ · (κgm S), where κgm is the
GM thickness diffusivity and S is the neutral slope. It involves computing the derivative of the thickness diffusivity
and neutral slope and appears to be noisy in numerical realizations, which is one of the factors that motivated the derivation of the skew flux formulation for eddy-induced transport
in Griffies (1998). Using the skew flux formulation also unifies the tracer mixing operators arising from Redi diffusion
and GM stirring. It turns out to be very convenient in its implementation in the variational formulation in FESOM. Overall, the small slope approximation for neutral diffusion (Gent
and McWilliams, 1990) and the skew diffusion form for eddy
stirring (Griffies et al., 1998) are the standard neutral physics
options in FESOM.
In FESOM there is a caveat with hybrid grids, for which
the sigma grid is used around the Antarctic coast when
ice shelf cavities are present. Because sigma grid slopes
and neutral slopes are very different, using neutral physics
parametrization will lead to numerical instability on sigma
grids. For the moment we use along-sigma diffusivity. The
roles played by mesoscale eddies on continental shelf and in
ice cavities around the Antarctic are unclear. In practice we
use high resolution (∼ 10 km or finer) in the sigma grid region, which can eliminate the drawback to some extent.
constrained to transport horizontally, horizontal diffusion
is more physical and should be applied (Treguier et al.,
1997). This idea has been commonly taken in ocean climate model practice (e.g. Griffies, 2004; Griffies et al., 2005;
Danabasoglu et al., 2008). We use the mixed layer depth
(MLD) as an approximation of the surface diabatic boundary layer depth, within which horizontal diffusion is applied.
The MLD is defined as the shallowest depth where the interpolated buoyancy gradient matches the maximum buoyancy
gradient between the surface and any discrete depth within
that water column (Large et al., 1997).
The diffusion tensor is not bounded as the neutral slopes
increase, so numerical instability can be incurred when the
neutral slopes are very steep. Therefore, the exponential tapering function suggested by Danabasoglu and McWilliams
(1995) is applied to the diffusion tensor to change neutral
diffusion to horizontal diffusion in regions of steep neutral
slopes.22 The same tapering function is also applied to the
GM thickness diffusivity κgm below the MLD to avoid unbounded eddy velocity v ∗ , which is proportional to the gradient of neutral slopes.
Within the surface boundary layer we treat the skew flux
as implemented by Griffies et al. (2005). The product (κgm S)
is linearly tapered from the value at the base of the surface
boundary layer to zero at the ocean surface, as suggested by
Treguier et al. (1997). A linear function of κgm S with depth
means that the horizontal eddy velocity u∗ = −∂z (κgm S) is
vertically constant in the surface boundary layer. Maintaining
an eddy-induced transport in the boundary layer is supported
by the fact that baroclinic eddies are active in deep convection regions (see the review by Griffies, 2004). Indeed, simulations parametrizing eddy-induced velocity in the surface
boundary layer show significant improvements compared to
a control integration that tapers the effects of the eddies as the
surface is approached (Danabasoglu et al., 2008). Our implementation of the GM eddy flux near the surface is different
from that of Griffies et al. (2005) with respect to the definition of the boundary layer. We define the surface diabatic
boundary layer depth using the MLD definition of Large et al.
(1997), while the boundary layer base is set to where the
magnitude of the slope S in either horizontal direction is just
greater than a critical value in Griffies et al. (2005).
Using the MLD to define the surface diabatic layer eliminates the requirement to choose a critical neutral slope for
defining the boundary layer. Additionally, tests with FESOM
show that boundary layer depth fields defined via critical
neutral slopes are less smooth than MLD, which is possibly
Diabatic boundary layer
Neutral diffusion represents mesoscale mixing in the adiabatic ocean interior. In the surface diabatic boundary layer
where eddies reaching the ocean surface are kinematically
ical stability (Roberts and Marshall, 1998). Griffies (2004) provides
a thorough review on the properties of biharmonic operators and explains why Laplacian operators are preferable for tracer diffusion.
22 The tapering function is only applied to the off-diagonal en-
tries of the diffusion tensor, thus maintaining a horizontal diffusion when these entries are tapered to zero. The function suggested by Danabasoglu and McWilliams (1995) is f (S) = 0.5(1 +
tanh( S maxS −|S| )), where S d = 0.001 is the width scale of the taperd
ing function and S max = 0.05 is the cut-off value beyond which f
decreases to zero rapidly.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
linked to the fact that static instability is not completely
removed instantaneously through the large but finite vertical diffusivity (Sect. 3.8). Based on theoretical consideration, Ferrari et al. (2008) proposed an eddy parametrization
for the near-boundary regions. They introduced a transition
layer connecting the quasi-adiabatic interior and the turbulent boundary layer where eddy-induced velocity is parallel
to the boundary. A simplified version of this parametrization
was implemented by Danabasoglu et al. (2008). Our implementation is the same as the case with vanishing transition
layers in Danabasoglu et al. (2008), who reported only minor difference induced by nontrivial transition layers.
Eddy-induced bolus velocity can go infinite if the neutral
slope is not limited from above. In addition to the exponential
tapering function being applied below the surface boundary
layer as mentioned above, the magnitudes of neutral slopes
at the base of the surface boundary layer are also constrained
below a critical value S max to ensure finite bolus velocity
without incurring numerical instability in the surface boundary layer. Sensitivity tests show that using a critical slope
larger than 0.1 can entail numerical instability sometimes.
Another consideration for the choice of S max is associated
with the tapering function for the ocean interior (see footnote 22). The magnitude of neutral slopes is mostly less than
0.01 below the surface layer, so we have empirically chosen S max = 0.05. As suggested by Gerdes et al. (1991) and
Griffies et al. (2005) the GM parametrization is changed back
to horizontal diffusion at grid points adjacent to ocean bottom
to avoid overshoots in tracer fields.
Neutral and thickness diffusivity
Methods to specify neutral diffusivity and thickness diffusivity differ among ocean climate models, including using
a constant value (Danabasoglu et al., 2006), horizontally
varying diffusivity depending on vertically averaged flow
fields (Visbeck et al., 1997; Griffies et al., 2005), and diffusivity varying in three dimensions depending on flow fields
(Danabasoglu and Marshall, 2007; Eden et al., 2009). Estimates from observations (e.g. Ledwell et al., 1998; Bauer
et al., 1998; Sundermeyer and Price, 1998; Zhurbas and
Oh, 2003; Marshall et al., 2006) and high-resolution ocean
models (e.g. Bryan et al., 1999; Eden, 2006; Eden et al.,
2007) have revealed pronounced variability of eddy diffusivity in space and time. Many numerical and theoretical studies have focused on the prescription for the vertical variation (Danabasoglu and McWilliams, 1995; Killworth, 1997;
Treguier, 1999) and horizontal variation (Held and Larichev,
1996; Visbeck et al., 1997; Griffies et al., 2005) of eddy
diffusivity. More recent efforts in prescribing eddy diffusivity have focused on schemes of eddy diffusivity varying in
three dimensions and time. Motivated by the finding that
the squared buoyancy frequency (N 2 ) shows a vertical structure similar to the diagnosed diffusivity (Ferreira et al., 2005;
Eden, 2006; Eden et al., 2007; Ferreira and Marshall, 2006),
Geosci. Model Dev., 7, 663–693, 2014
Danabasoglu and Marshall (2007) have investigated the impacts of using diffusivity proportional to N 2 in model simulations. Eden and Greatbatch (2008) proposed a closure for
eddy thickness diffusivity consisting of a prognostic equation
for the eddy kinetic energy and an eddy length scale.
The N 2 -dependent thickness diffusivity as suggested
by Ferreira et al. (2005) and Ferreira and Marshall
(2006) was implemented in the NCAR Community Climate System Model (CCSM3), and it leads to improved
results with respect to observations compared to using a constant diffusivity (Danabasoglu and Marshall,
2007). It is further used in the updated NCAR climate model CCSM4 (Danabasoglu et al., 2012). Currently this approach is also used in FESOM simulations.
The thickness diffusivity is calculated as κgm (x, y, z, t) =
κref (x, y)N (x, y, z, t)2 /Nref (x, y, t)2 , where κref (x, y) is
a reference diffusivity at horizontal location (x, y) and
N (x, y, z, t) is the local buoyancy frequency. Nref (x, y, t)
is the reference buoyancy frequency taken just below
2 is the
MLD, provided that N 2 > 0 there. Otherwise Nref
first stable N below MLD. Following Danabasoglu and
2 is constrained by N
Marshall (2007) the ratio N 2 /Nref
min ≤
N /Nref ≤ 1, where Nmin sets the lower bound for diffusivity. The neutral diffusivity is set equal to the thickness diffusivity below the MLD. Within the MLD the linear tapering is
applied to eddy skew flux and the horizontal diffusivity is set
to the reference diffusivity.
The reference diffusivity κref (x, y) is set to a constant
(1500 m2 s−1 ) for regions where horizontal resolution is
coarser than 50 km, and scaled down when resolution is
finer.23 We prescribed the scaling function for diffusivity
based on experience obtained so far. Sensitivity tests with
25 km resolution in the Arctic Ocean (where the first baroclinic Rossby radius is less than 8 km in the Eurasian Basin
as derived from climatology data – Qun Li, personal communication, 2012) show that using a neutral diffusivity larger
than 50 m2 s−1 leads to a too diffused boundary currents in
the Arctic Atlantic Water layer.24 Therefore we reduce the
reference diffusivity rapidly from 1500 m2 s−1 at 50 km resolution to 50 m2 s−1 at 25 km resolution. The ratio of the first
Rossby radius to the grid scale (λ1 /1) is a pertinent control
parameter for scaling mesoscale eddy diffusivity.25 Using it
to construct a scaling function may need case-based tuning in
23 The reference diffusivity [m2 s−1 ] is set to 1500 for 1 ≥ 50,
50+58(1−25) for 25 ≤ 1 < 50, and 50(1/25)2 for 1 < 25. Here
1 is local horizontal resolution with unit km.
24 Other parametrizations such as the anisotropic GM
parametrization suggested by Smith and Gent (2004) and the
Neptune parametrization (Maltrud and Holloway, 2008; Holloway
and Wang, 2009) could improve the solution of the Arctic boundary
currents. These options need to be explored in the future.
25 The recent work of Hallberg (2013) provides insight into this
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Mean Thickness Diffusivity at 300 m Depth
90 N
60 N
30 N
30 S
60 S
90 S o
180 W
120 W
60 W
60 E
120 E
180 W
years mean
Fig. 115.
15. The
mean thickness
thickness diffusivity (m
)] atat 300
on the
the reference
depth. The
The simulation
simulation is carried out on
Fig. 13.
multi-resolution climate simulations and remains a research
topic for FESOM applications.26
We set Nmin = 0.2, meaning that the diffusivity is confined above 300 m2 s−1 in regions with resolution coarser
than 50 km. The time mean thickness diffusivity at 300 m
depth on the reference mesh (Fig. 3) is shown in Fig. 15.
Largest values are found in regions where intense eddy activity is expected, including the ACC and western boundary
currents. Due to the resolution dependence of the reference
diffusivity, reduced values are found in the equatorial band
and north of 50◦ N where the grid spacing is small (Fig. 3).
As also noticed in Danabasoglu and Marshall (2007), the
diffusivity scheme produces undesirable large values in the
eastern South Pacific. The zonal-mean distribution (Fig. 16)
shows that the diffusivity decreases from the base of surface
diabatic layer downwards as expected from vertical distribution of squared buoyancy frequency. Largest values are found
in the Tropics just below the diabatic layer, with the value in
the equatorial band scaled down due to higher horizontal resolution. Deep penetration of large diffusivity occurs at midto high latitude on both hemispheres, while the deepest penetration is in the Southern Ocean. The deep reaching high diffusivity north of 60◦ N in Danabasoglu and Marshall (2007)
(their Fig. 1, associated with deep convection regions in the
26 In an eddying regime, eddies can transfer tracer variance to the
grid scale, and this variance must be dissipated without inducing
spurious diapycnal mixing, with the neutral diffusion operator being
a possible numerical dissipation form (Roberts and Marshall, 1998;
Griffies and Hallberg, 2000; Griffies, 2004). In practice the choice
of diffusivity depends on the advection scheme used in the model.
By using an improved advection scheme the GM parametrization
was completely turned off in ocean-eddy-permitting climate simulations in Farneti et al. (2010) and Farneti and Gent (2011).
Zonal mean temporal mean
mean thickness
thickness diffusivity
diffusivity (m
16. Zonal-mean
The mean surface boundary layer
plotted. The results
yr simulation.
yr mean
mean in a 60 yr
North Atlantic) is absent in Fig. 16 because the reference diffusivity is scaled to about 50 m2 s−1 on our reference mesh.
Nmin turns out to be one of the key tuning parameters in
the calculation of diffusivity. The residual meridional overturning stream functions (Eulerian mean plus eddy contribution parametrized by thickness diffusivity) in the Southern Ocean from simulations with Nmin = 0.2 and 0.1 are
shown in Fig. 17a and b, respectively. Both the Deacon cell
and Antarctic Bottom Water (AABW) cell show very similar
structures between the two simulations. State estimate using
an adjoint eddy-permitting (1/6◦ ) model by Mazloff et al.
(2010) shows a Southern Ocean Ekman transport of about
31 Sv, a Deacon cell in depth space (Doos and Webb, 1994)
reaching more than 3000 m depth and a maximum AABW
transport of about 16 Sv (their Fig. 10a). Both simulations
reasonably reproduce the meridional overturning circulation
structure reported by Mazloff et al. (2010), with some underestimation of the circulation strength of both bottom water and intermediate water. Figures 17c and d compare the
parametrized eddy meridional overturning stream functions
between the two simulations. With a decrease of Nmin from
0.2 to 0.1, the eddy MOC maximum reduces from about
10 Sv to 5 Sv. Although decreasing Nmin (the lower bound of
diffusivity) can have impacts on the diffusivity mainly below
1500 m depth for the Southern Ocean region (see Fig. 16),
the weakening of the eddy meridional overturning is over
the whole water column. Along with the weakening of the
eddy-induced transport, the AABW transport also weakens
(Fig. 17a and b), consistent with other model results (Farneti
and Gent, 2011).
Mesoscale eddies in the Southern Ocean can buffer the
ocean response to atmospheric changes (Meredith and Hogg,
2006; Hallberg and Gnanadesikan, 2006; Böning et al.,
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Fig. 17. Residual MOC
(a) a reference
run and
with a smaller
MOC [Sv]
run andN(b)
run (c), (d) are the same as (a), (b), respectively, but
min a(0.1).
for the eddy MOC. with a smaller Nmin (0.1). (c), (d) are the same as (a), (b), respectively, but for the eddy MOC.
2008; Farneti et al., 2010; Viebahn and Eden, 2010; Jones
et al., 2011; Abernathey et al., 2011; Meredith et al., 2012;
Morrison and Hogg, 2013; Munday et al., 2013), so assessing and improving their parametrization in climate models is critically important. It is possible to use the present
GM parametrizations to produce a response of the Southern Ocean to changing wind stress in coarse climate models
that is broadly consistent with what is seen in eddying ocean
models (e.g. Gent and Danabasoglu, 2011). However, eddy
parametrizations have only demonstrated some success in reproducing the eddy compensation but not the eddy saturation.27 As remarked by Munday et al. (2013), eddy compensation is achieved at the expense of being not able to realize
the eddy-saturated regime using the parametrization. Hence
they suggested that parametrizations with a prognostic eddy
kinetic energy (EKE) variable (Eden and Greatbatch, 2008;
Marshall and Adcroft, 2010), which can be tied directly to
27 Eddy saturation refers to the phenomenon that ACC transport
shows limited response to increased wind stress. It can be explained
by a rough balance between the tendency for Ekman transport to
steepen isopycnals and for eddies to flatten them. Eddy compensation refers to the phenomenon that changes in eddy-induced MOC
can partially compensate those of Ekman transport. These two phenomena are linked but with dynamical distinction (e.g. Morrison
and Hogg, 2013).
Geosci. Model Dev., 7, 663–693, 2014
wind stress, be preferable schemes for thickness diffusivity.
However, schemes such as the one proposed by Eden and
Greatbatch (2008), while probably a better way to go, assume that the time evolution of EKE can be parametrized
in a model. This is not a trivial task as the relationship of
changes in EKE to changes in forcing is one of the big unsolved problems.
Attention has been paid to the practical implementation of traditional GM parametrizations. For example,
Gnanadesikan et al. (2007) and Farneti and Gent (2011)
found that model results are very sensitive to the critical neutral slope in their simulations; some simulated features can
be improved at the expense of worsening some other features when increasing the critical neutral slope. These findings also indicate that much research is still required for
mesoscale eddy parametrizations. Ferrari et al. (2010) propose a parametrization for mesoscale eddy transport which
solves a boundary value problem for each vertical ocean column. They show that this scheme works robustly and performs as well as their implementation of the more conventional GM scheme. Further study is required to explore its
potential in increasing the fidelity of ocean model simulations.
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
River runoff distribution
River discharge is one of the important processes that redistribute water masses in the earth system. For example,
the Arctic Ocean is the largest freshwater reservoir in the
global ocean, with 38 % of its freshwater source provided by
river runoff (Serreze et al., 2006). Faithfully representing the
circulation of freshwater supplied by rivers in ocean models is important, but depends on the numerical treatment of
the runoff. As reported in Griffies et al. (2005), adding river
runoff into the surface grid cell can lead to too much freshwater on the surface stabilizing the water column. This motivated them to insert river runoff into the upper four model
grid cells. This approach is a parametrization for unresolved
processes that can influence river runoff distribution in reality, including tidal mixing. Due to the current model numerics we did not implement this approach in FESOM. The
diapycnal mixing parametrization for barotropic tides proposed by Lee et al. (2006) is a remedy we use to improve
river runoff representation, as mentioned in Sect. 3.6.1.
Another approach used in climate models is to spread river
runoff over a wide region near river mouths (Danabasoglu
et al., 2006). This approach is expected to remedy possibly under-resolved spreading of river runoff at coarse resolution, for example by eddies. Using a high-resolution model
McGeehan and Maslowski (2011) showed how eddies transport water masses of the Labrador Sea boundary current into
the gyre interior. The first baroclinic Rossby radius of deformation λ1 is typically small in coastal regions. For example,
λ1 is of the order of 3 km on the western Arctic shelf, so
resolving mesoscale eddies cannot be afforded even in regional models. Arguably, adding river runoff over a wide region can be a poor man’s approach to account for the underresolved processes that facilitate freshwater penetration to
ocean basins.
Typically we distribute river runoff around river mouths
using a linear function decreasing from one at the river
mouths to zero at 400 km distance. Figure 18a shows the
river runoff distribution of the long-term climatology derived
from Dai et al. (2009). We carry out sensitivity tests using
this distribution (reference run) and another one where the
river runoff is distributed within 100 km distance from river
mouths (sensitivity run, Fig. 18b). The difference between
the two experiments for salinity at surface and 100 m depth
is shown in Fig. 19a and b. As expected, the largest difference
is close to river mouths where the difference is directly enforced, with lower salinity immediately at river mouths and
higher salinity around them in the sensitivity run.
The impact of applying different river runoff distribution on the Arctic basin develops with time. The salinity in
the halocline is characterized by local difference patches of
±0.1 psu in the Arctic basin (Fig. 19b). The changes of salinity between the two runs are nonuniform with both positive
and negative signs, implying that associated changes in local
circulation are also responsible for the observed difference
Fig. 18.
s−1 ). (a) The
river runoff is distributed over 400 km distance from river mouths
using a linear distribution function. (b) The same as (a) but over
100 km.
in salinity. To assess the significance of the impact from adjusting the river runoff distribution, we compared the difference in salinity in the halocline to that induced by adjusting
background vertical diffusivity (changing from the currently
used value 0.1 × 10−4 m2 s−1 to 0.01 × 10−4 m2 s−1 as used
by Nguyen et al. (2009) and Zhang and Steele (2007), see
discussion in Sect. 3.6.2). We found that the difference obtained here is a few times smaller. Further analysis shows
that the freshwater export flux remains almost intact for both
Fram Strait and CAA in the sensitivity run, so the impact
of applying different river runoff distribution in the Arctic
Ocean is mainly limited to the Arctic basin.
Both the temperature and salinity in the North Atlantic
subpolar gyre are increased in the sensitivity run (Fig. 19).
This is consistent with the strengthening of the AMOC upper cell (Fig. 20), which increases the supply of warm, saline
Atlantic Water to the subpolar gyre. As the freshwater export from the Arctic Ocean remains the same, the changes
in AMOC are linked to modified river runoff distribution
along the North American and Greenland coasts. This is not
an unexpected impact as confining river runoff more to the
coast will facilitate deep convection in the Labrador Sea thus
a stronger AMOC. Although the impact of a wide spreading
of river runoff is moderate as tested here, we keep this option
in the model.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element
Sea ice-Ocean
Q. Wang
et al.: TheModel
Element Sea Ice-Ocean Model (FESOM)
Fig. 19. The salinity difference between runs with river runoff distributed over 400 km and 100 km (the latter minus the former) for (a) surface
Fig. 119. The salinity difference between runs with river runoff disand (b) 100 m depth. (c), (d) The same as (a), (b), respectively, but for temperature. The results are the last 10 yr mean in 60 yr simulations.
tributed over 400 km and 100 km (the latter minus former) for (a)
surface and (b) 100 m depth. (c), (d) The same as (a), (b), respectively, but for temperature. The results are the last 10 yr mean in
3.12 Free surface formulation
different from the reference salinity and the dilution effect
60 yr simulations.
While rigid-lid ocean models are becoming obsolete, ocean
models with the free-surface method and fixed volume for
tracer budget have been widely used during the last decade.
In the latter type of models, the sea surface height equation
has a free-surface algorithm whereas tracers cannot experience dilution or concentration associated with ocean volume
changes. To have the impact of surface freshwater flux (evaporation, precipitation, river runoff, and freshwater associated
with ice/snow thermodynamics) on salinity, a virtual salt flux
has to be introduced for the salinity equation as a surface
boundary condition. In reality there are no salt changes in
the ocean except for changes through storage of salt in sea
ice. Therefore the virtual salt flux formulation is unphysical,
although it parametrizes most of the effect of surface freshwater flux on surface salinity.
The virtual salt flux is given by F virtual, salt = sref qw , where
qw is total surface freshwater flux and sref is a reference salinity. If sref is set to a constant, the ocean salt is conserved
upon that the global integral of qw is zero. A problem with
this choice is that the local sea surface salinity can be very
Geosci. Model Dev., 7, 663–693, 2014
of surface freshwater on salinity cannot be well represented
in the model. When the local sea surface salinity is far from
the reference salinity, virtual salt flux formulation can lead to
too small or too large salinity and thus model blowup. Another choice for calculating the virtual salt flux is to use local
sea surface salinity as the reference salinity. In this case the
local feedback on salinity from freshwater flux is properly
simulated, but the total salt conservation in the ocean is not
automatically ensured even when the global integral of qw is
zero. One practical remedy for this is to calculate the global
integral of virtual salt flux and remove it by subtracting its
global mean over the globe during the model runtime. Effectively this remedy alters local salinity unphysically and might
spoil model integrity on climate scales.
In a full free-surface formulation the ocean volume
changes with the vertical movement of surface grid points
and tracer concentration directly reacts to these changes. The
virtual salt flux is not required and the surface water flux is
accounted for in the sea surface height equation. Although
the comparison by Yin et al. (2010) shows that the difference
between the results using virtual salt fluxes and freshwater
Q. Wang
et al.: The Finite Element Sea ice-Ocean Model (FESOM)
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model
AMOC Difference (Sv)
Depth (km)
in the response to wind as shown by Campin et al. (2008),
while the importance of such high-frequency variability on
climate scales is unclear. The freshwater flux boundary condition, however, is not influenced by applying this constraint.
At the moment the model is still not updated to support full
free-surface on hybrid grids.
By now all published and most on-going applications of
FESOM have been performed with the virtual salt flux formulation. The current coupling of FESOM to an atmospheric
model also uses the virtual salt flux option in the on-going
model evaluation. For new applications and for the coupled
model in a later stage the physically more consistent full freesurface formulation is the recommended option.
of aofrun
Fig. 20.
a run
400 km
from afrom
run with
a 100
kma distribution
distance. The
km distance
a run
100 km distribution
are the
10 yr
in yr
a 60
yr simulation.
last 10
in a 60 yr simulation.
fluxes is statistically insignificant in both unforced control
runs and water-hosing runs with freshwater forcing resembling future projections, the practice of employing virtual
salt fluxes is physically compromised, prompting the trend to
full free-surface formulation. Indeed, the majority of present
ocean climate models are using full free-surface formulation
(see, for example, the models used in Danabasoglu et al.,
FESOM has taken the free-surface formulation with fixed
ocean volume since its first construction (Danilov et al.,
2004). The basic numerical core of the current model version was finished in 2009 and still assumed fixed ocean volume (Wang et al., 2008; Timmermann et al., 2009). Because
of human limitations it was another two years before the full
free-surface formulation was updated in the model. The full
free-surface algorithm uses the arbitrary Lagrangian Eulerian
(ALE) formulation (Farhat et al., 2001; Formaggia and Nobile, 2004; Nithiarasu, 2005; Badia and Codina, 2006). An
example of ALE formulation implementation in a finite element ocean model is described by White et al. (2008). In
principle, the ALE formulation allows vertical movement of
all grid layers (an analogue to the z∗ coordinate). However,
matrices and derivatives need to be updated when the mesh
geometry is changed, which is costly, so we only allow the
surface grid points to move. Tests show that only the moving
surface layer in the full free-surface formulation increases
about 10 % of the total computation time on typical meshes.
The drawback of only moving the surface grid points is that
the sea ice loading is constrained by the first layer thickness. We limit the loading from sea ice to half of the first
layer thickness to make sure that the first layer thickness
will not vanish. Limiting ice load cannot realistically represent oceanic variability associated with ice-loading effects
Ice shelf model
Ice sheets are an important component of the earth system.
They should be adequately taken into account in order to
predict and understand sea level rise. Ice shelves provide an
important interface between the Antarctic Ice Sheet and the
surrounding ocean. Ice shelf basal melting feeds the AABW
and modifies the ice shelves, the latter of which can potentially influence the ice sheet dynamics. FESOM has an ice
shelf component which can explicitly simulate the ocean dynamics in sub-ice-shelf cavities and ice-shelf–ocean interaction (Timmermann et al., 2012).
A three-equation system is used to compute the temperature and salinity in the boundary layer between ice and
ocean and the melt rate at the ice shelf base as proposed by
Hellmer and Olbers (1989) and refined by Holland and Jenkins (1999). Turbulent fluxes of heat and salt are computed
with coefficients depending on the friction velocity following the work of Jenkins (1991). To initialize temperature and
salinity in ice cavities we take the temperature and salinity
profiles at the nearest ice shelf edge. Enough spin-up time
(O(20) yr) is necessary to adjust the hydrography under the
ice shelf. By now we assume a steady state for ice shelf thickness and cavity geometry; basal mass loss is assumed to be
in equilibrium with surface accumulation and the divergence
of the ice shelf flow field. Investigating the impact of varying
cavity geometry and grounding-line migration is an active
research topic.
The numerical formulation of the ice shelf model is summarized here. Locally refined resolution is needed to resolve
small ice shelves; sigma grids are used for ice cavities and
surrounding continental shelf regions (Sect. 3.2), with measures to control pressure gradient errors (Sect. 3.4); bottom
topography and ice shelf draft data with improved quality
compiled by Timmermann et al. (2010) are used (Sect. 3.3).
As shown by Timmermann et al. (2012) and Timmermann
and Hellmer (2013), basal mass fluxes are in most cases realistic from the model, but differences from observations suggest that further improvement is still desirable. The major
issues are linked to the utilization of sigma grids, including
possible distortion to flow dynamics caused by smoothing
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
of bottom bathymetry and ice shelf draft required for reducing pressure gradient errors, requirement for individual numerical formulation of GM parametrization (Sect. 3.10), and
missing support for the full free-surface option in the sigma
grid region (Sect. 3.12). These aspects remain to be explored
and improved.
Unstructured-mesh models open new horizons for climate
modelling: local dynamics can be better resolved with locally increased resolution without traditional nesting and the
improved local processes can provide feedback to the largescale circulation. In this paper we give an overview about
the formulation of FESOM, which is the first ocean general circulation model that uses unstructured meshes and
therefore makes it possible to carry out multi-resolution
simulations. We described the key model elements including the two-dimensional mesh, vertical discretization, bottom topography, pressure gradient calculation, tracer advection scheme, diapycnal mixing parametrization, penetrative
short-wave treatment, convection adjustment, horizontal momentum friction, GM parametrization, river runoff distribution, free-surface formulation and ice shelf modelling. The
progress reported here is a result of our own continuing
model development efforts as well as the recent advances
by the ocean modelling community in general. The model
version described here is the standard version for our ocean
stand-alone studies and is employed as the sea-ice ocean
component of a new coupled climate model (Sidorenko et al.,
Along with the model description we briefly reviewed
some of the key elements related to ocean climate models.
Discussions on the knowledge gained in the community provide the guideline for making choices in constructing our
model. Griffies (2004) has provided a thorough review on
ocean model fundamentals. Due to limitations in resources
we did not implement or test all numerical and parametrization options recommended in other studies. Investigations to
further improve numerical and physical schemes are required
as outlined in Sect. 3. There are other model components
that should be updated in FESOM – for example, the overflow parametrization (there are different schemes suggested
in previous studies, e.g. Beckmann and Döscher, 1997;
Danabasoglu et al., 2010). Parametrizations with scale selectivity are critically important in unstructured-mesh models. We are only beginning to explore the multi-resolution aspects of parametrizations. More sensitivity studies and multiresolution tests are desirable to improve the formulation and
implementation of such parametrizations. We note that broad
collaborations, like the ongoing international joint project
COREs (Griffies et al., 2009; Danabasoglu et al., 2014), are
helpful to identify common issues in present state-of-the-art
Geosci. Model Dev., 7, 663–693, 2014
ocean models and consolidate efforts in ocean model development.
In summary, we would argue that unstructured-mesh seaice ocean models have matured substantially in recent years.
Consequently, they have become attractive options for simulating multi-resolution aspects of the climate system. First
climate-relevant applications are appearing (e.g. X. Wang
et al., 2012; Hellmer et al., 2012; Timmermann et al., 2012;
Timmermann and Hellmer, 2013; Wekerle et al., 2013; Jung
et al., 2014). However, further research is urgently required
to explore the full potential of multi-resolution modelling in
climate research. Large model uncertainty as shown in the
previous IPCC reports and recent COREs model intercomparisons (Griffies et al., 2009; Danabasoglu et al., 2014) indicates that model development requires long-term continuous
efforts in the broad modelling community; both international
collaboration and individual effort from each model development group are necessary to advance the field.
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Appendix A
Overview on the development history of FESOM
The first FESOM version (version 1.1) was documented by
Danilov et al. (2004). In that version the model used the GLS
stabilization which required to introduce the barotropic velocity. It needed to be solved for together with the sea surface
elevation (as mentioned in Sect. 2.1). Advection and friction
operators for both momentum and tracers were implicit in
time, so iterative solvers were called for all equations and
particularly needed to be pre-conditioned in every time step.
Overall, the approach proved to be too slow for climate scale
The issue of model efficiency prompted the model development team to pursue different numerical formulations (version 1.2, Wang et al., 2008). With the purpose to build up
a more efficient and robust ocean model, the pressure projection method was adopted to decouple the solution of surface elevation and velocity, the momentum advection and lateral diffusivity and viscosity terms were changed to explicit
schemes, the FCT tracer advection scheme was introduced,
the hybrid grid functionality was developed, and some physical parameterizations were incorporated. All these features
are kept in the current model version. Further experience was
obtained through the work of Timmermann et al. (2009, version 1.3), who concentrated on coupling a finite-element seaice model to the ocean code. Since that work was initiated
before the work reported in Wang et al. (2008), it was based
on a preliminary ocean model code void of most of model
updates except for the pressure projection method.
The explored model features from Wang et al. (2008) and
Timmermann et al. (2009) have been combined afterwards
(version 1.4). For the sake of model development the prism
elements (see Fig. 2a) were used in Wang et al. (2008). In
prisms basis functions are bilinear (horizontal by vertical) on
z-level grids, which allows one to carry out analytical computations of integrals. They deviate from bilinear on general
meshes (like sigma grids or shaved prisms) and require to use
slower quadrature rules. The code should handle these situations separately for the purpose of high numerical efficiency
and turns out to be inconvenient to maintain. In contrast,
tetrahedral elements always allow for analytical integration.
The final model hence uses tetrahedral elements as illustrated
in Fig. 2b. The new model version is about 10 times faster
than the early version described in Danilov et al. (2004).
There have been a few accomplishments with FESOM
development since the last FESOM reports in Wang et al.
(2008) and Timmermann et al. (2009). First, a finalized
model version combining features obtained during the past
development phase is released. It remains stable with respect
to its dynamical core over past three years and is recently
employed in a coupled climate model. Second, the functionality of modelling ice shelves is added (Sect. 3.13), which
utilizes a hybrid grid (Sect. 3.2). Third, the full free-surface
formulation is added (Sect. 3.12). This is the recommended
option for future applications. Finally, in contrast to the earlier development phase when our focus was mostly on the
numerical core, more attention is paid to verifying parameterizations and evaluating global models (Sidorenko et al.,
2011; X. Wang et al., 2012; Scholz et al., 2013). Although
the development of FESOM has reached a milestone, much
research is still required on the route of our model development as outlined along the discussions in Sect. 3.
Geosci. Model Dev., 7, 663–693, 2014
Q. Wang et al.: The Finite Element Sea Ice-Ocean Model (FESOM)
Acknowledgements. During our model development we got
valuable support from colleagues in the AWI computer centre,
including Sven Harig, Lars Nerger, Annika Fuchs, Alexey Androsov, Natalja Rakowsky, Wolfgang Hiller, Stephan Frickenhaus
and many others. Many model users provided useful feedbacks
which helped improve the code. We thank Richard Greatbatch
and two anonymous reviewers for helpful comments on the paper.
Part of the work was financially supported by Helmholtz Climate
Initiative REKLIM (Regional Climate Change). The computational
resources for this work were provided through the North-German
Supercomputing Alliance (HLRN).
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