Los2006a

Los2006a
This work has been accepted for publication in JPO. Copyright may be transferred without further notice, and this version
may no longer be accessible.
Adjoint Sensitivity of an Ocean General Circulation Model to Bottom
Topography
M ARTIN L OSCH
Alfred-Wegener-Institut für Polar- und Meeresforschung Bremerhaven, Germany
PATRICK H EIMBACH
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge (MA), USA
ABSTRACT
Bottom topography, or more generally the geometry of the ocean basins, is an important ingredient in numerical ocean modeling. With the help of an adjoint model, it is shown that scalar diagnostics or objective functions
in a coarse resolution model, such as the transport through Drake Passage, the strength of the Atlantic meridional
overturning circulation, the Deacon cell, and the meridional heat transport across 32◦ S, are sensitive to bottom
topography as much as they are to surface boundary conditions. For example, adjoint topography sensitivities
of the transport through Drake Passage are large in choke point areas such as the Crozet-Kerguélen Plateau and
south of New Zealand; the Atlantic meridional overturning circulation is sensitive to topography in the western
boundary region of the North Atlantic Ocean and along the Scotland-Iceland ridge. Many sensitivities are connected to steep topography and can be interpreted in terms of bottom form stress, that is, the product of bottom
pressure and topography gradient. The adjoint sensitivities are found to agree to direct perturbation methods
with deviations smaller than 30% for significant perturbations on time scales of 100 years, so that the assumption of quasi-linearity that is implicit to the adjoint method holds. The horizontal resolution of the numerical
model affects the sensitivities to bottom topography, but large scale patterns and the overall impact of changes in
topography appear to be robust. The relative impact of changes in topography and surface boundary conditions
on the model circulation is estimated by multiplying the adjoint sensitivities with assumed uncertainties. If the
uncertainties are correlated in space, changing the surface boundary conditions has a larger impact on the scalar
diagnostics than topography does, but the effects can locally be on the same order of magnitude if uncorrelated
uncertainties are assumed. In either case, bottom topography variations within their prior uncertainties affect
the solution of an ocean circulation model. To this extent, including topography in the control vector can be
expected to compensate for identifiable model errors, and thus improve the solution of estimation problems.
——————–
1. Introduction
trols. They each have associated uncertainties which can be
significant. The result of an OGCM is sensitive to changes
in all of these fields and parameters (e.g., Bryan 1987, Losch
et al. 2004, Marsland and Wolff 2001, Simmons et al. 2004).
Forcing fields, initial conditions, and parameters are derived
either from measurements, reanalysis data products, estimates by different models or, if there is no information available at all, from tuning exercises such that the resulting circulation is consistent with available observations (within assumed error bars) or meets some subjective norms (expectations) of the investigator. From this perspective it is important to know which parameters influence the solution significantly, so that future efforts can be directed at estimating or
measuring these parameters with higher precision. Numer-
Numerical ocean general circulation models (OGCM)
consist of a set of discretized partial differential equations
for a set of prognostic variables (the numerical ocean state)
which are solved subject to initial conditions and boundary
conditions (lateral and surface boundary conditions, surface
forcing); the solution also depends on a number of model parameters (e.g., diffusivity and viscosity parameters). These
quantities are referred to as independent parameters, or conCorresponding author address:
M. Losch, Alfred-Wegener-Institut für Polar- und Meeresforschung Postfach 120161, 27515 Bremerhaven, Germany.
([email protected])
1
2
ous studies have explored the sensitivity of certain features
of the circulation, for example the strength of the Antarctic
Circumpolar Current (ACC), the Atlantic overturning circulation, to most of the above parameters or how closely the
model result reproduces individual measurements or globally averaged, say, temperature profiles (e.g., Gent et al.
2001, Large et al. 1997).
The representation of the computational domain has so far
received little attention in the context of uncertainty assessment of “model parameters”. Although the role of topography in numerical models has been recognized for a long time
(e.g., Adcroft et al. 1997, Holland 1973, Marshall 1995a, Olbers 1998, Pacanowski and Anand 1998), there is no general,
objective way of representing topographic sub-grid scales on
a computational grid in a way that leads to the “best possible” flow field. Topography may thus undergo a considerable amount of ad-hoc tuning before it is introduced into the
OGCM. Smoothing a given topography is a widely used procedure to remove too steep gradients (Penduff et al. 2002),
“holes” are filled in, straits are deepened, opened, or closed,
in order to make the model produce a “more realistic” result
(Large et al. 1997). In the extreme case, topography may
be changed significantly so as to compensate for lacking numerics or resolution in deriving a consistent flow field. Other
parameters such as the boundary conditions that apply along
the lateral solid walls are equally uncertain in global ocean
models (Adcroft and Marshall 1998).
Another source of uncertainty is the bathymetric data
sets themselves. As an example, consider the difference
between two well-known data sets: ETOPO5 (Sloss 1988)
and its successor ETOPO2 (http://www.ngdc.noaa.
gov/mgg/fliers/01mgg04.html), which is in large
parts based on the satellite derived topography of Smith and
Sandwell (1997). The topographies implied by the two data
sets are profoundly different (Fig. 1a). For visualization, the
original data sets have been smoothed with a radial block
mean filter of radius 112 km (≈ 1◦ spherical distance). The
root-mean-square (rms) difference on a 1/3◦ -grid over the
oceans is still 131 m and the maximum differences are as
large as 2000 m. On large scales the topographies differ by
approximately 100 m, for example south of Australia in the
Southern Ocean. Such differences are significant since they
can be represented in ocean models. As a reference, Fig. 1b
depicts the topography used in the present study.
State estimation based on control theory provides a natural framework for accounting for parameter uncertainties of
the type discussed above (see e.g., Wunsch 1996). In this
context Losch and Wunsch (2003) have explored the possibility of using bottom topography as a control variable in
a barotropic shallow water model. The scope of this paper is to extend their study, and to investigate the sensitivity of the state of a given OGCM, here the MITgcm (Marshall et al. 1997), to the representation of bottom topography. In particular, (a) we show that bottom topography sensitivities in our OGCM can be computed with the help of
the adjoint method, making bottom topography a possible
a) difference between ETOPO2 and ETOPO5 [m]
500
60oN
30oN
0o
0
30oS
o
60 S
o
0
o
90 E
o
180 W
o
90 W
o
0
b) model topography [m]
−500
0
60oN
−1000
o
30 N
−2000
o
−3000
0
−4000
30oS
−5000
60oS
o
0
o
90 E
o
180 W
o
90 W
o
0
−6000
F IG . 1. a) Equal area map (Gall-Peters projection) of the difference between ETOPO2 and ETOPO5 topographies, smoothed with
a 1◦ radial block average (in units of meters). b) Model topography
H
control variable in state estimation, (b) we quantify bottom
topography sensitivities and show that they can have a significant impact on the modeled ocean circulation, (c) we put
bottom topography sensitivities into context to momentum
and buoyancy forcing sensitivities to assess their relative impact. The model configuration used in this work is limited by
coarse resolution and crude parameterizations of sub-gridscale processes. Nevertheless, the OGCM is a fully fledged
baroclinic model. In the context of sensitivity studies it has
been used in this configuration in several recent studies (e.g.,
Ferreira et al. 2005, Huang et al. 2003, Losch et al. 2004).
The implementation of bottom topography control will be
shown to be significantly more complex than (i) bottom topography control in a shallow water model, and (ii) initial
condition, boundary forcing, or mixing controls in a baroclinic OGCM. Therefore, our work should be viewed as an
early step towards establishing topography as a control variable in an OGCM, showing the potential of this approach.
Our study hinges on the use of the adjoint method (see
e.g., Wunsch 1996). This method enables us to compute,
simultaneously and in a single adjoint integration, all linear
sensitivities of a scalar-valued model diagnostics J (often referred to as objective function, probing function, or cost function) to changes in a given set control variables u (for example, surface forcing, topography). The sensitivity is given
by the gradient ∇u J in multi-dimensional control space.
In the present context, J might represent volume transport
through Drake Passage or the strength of the Atlantic merid-
3
ional overturning circulation. This method has both technical and conceptual advantages over direct perturbation approaches but requires a separate (tedious) code development
effort. Before the advent and increasing sophistication of
algorithmic/automatic differentiation (AD) tools during the
1990’s this development inevitably was inexact (e.g. Sirkes
and Tzipermann 1997) and lagged behind the forward code
development. Early OGCM studies based on hand-written
adjoints include those of Tziperman and Thacker (1989),
Sheinbaum and Anderson (1990) and Marotzke and Wunsch
(1993). Carl Wunsch’s foresight at the outset of the MITgcm
development that forward and adjoint model code ought to
be developed in parallel by exploiting the new technology
of AD was key to what has evolved into the “Estimating the
Circulation and Climate of the Oceans” (ECCO) Consortium
and has lead to the adjoint capabilities of the MITgcm today.
In the following we briefly review recent OGCM applications of the adjoint method in the context of sensitivity studies (a), state estimation (b), and bottom topography control
(c).
(a) The first sensitivity study using the MITgcm adjoint was undertaken by Marotzke et al. (1999), who investigated the (tangent) linear sensitivity of the meridional heat transport in the Atlantic. Bugnion (2001) and
Köhl (2005) extended this study to climate time scales,
and to higher-resolution, respectively. Other examples are
the study of SST anomalies in the North Atlantic (Junge
and Haine 2001), midlatitude–ENSO teleconnection mechanisms (Galanti and Tziperman 2003, Galanti et al. 2002), the
deep ocean heat content (Huang et al. 2003), water mass origins and pathways in the Pacific (Fukumori et al. 2004), the
carbon sequestration efficiency (Hill et al. 2004) in ocean
models, optimal observing system design (Köhl and Stammer 2004), sensitivity of ocean biological productivity to
iron fluxes (Dutkiewicz et al. 2006), and impact of open
boundary conditions in a regional North Atlantic circulation
(Ayoub 2005). Other groups have developed adjoint components of their OGCMs as well, and provided interesting applications of the adjoint method, for example, Galanti et al.
(2002, using MOM), Weaver et al. (2003, using OPA), and
Moore et al. (2004, using ROMS).
(b) With their publication of a dynamically consistent, decadal, quasi-global state estimate (see http://
www.ecco-group.org/), Stammer et al. (2002, 2003)
demonstrated the practicality of solving the estimation problem of a time-dependent ocean state as laid out in Wunsch
(1996) (see also Wunsch and Heimbach 2006). Subsequent
global and regional-scale studies include those of Köhl and
Willebrand (2003), Ferron and Marotzke (2003), Lea et al.
(2006), and Gebbie et al. (2006). An important extension
of the state estimation which adds transient tracers such as
chlorofluorocarbons (CFCs) as observational constraints was
undertaken by Li and Wunsch (2003, 2004).
(c) Adjoint models have been used to study bottom topography as a control variable in tidal modeling and gravity
wave dynamics in shallow water models of shelf seas (Das
and Lardner 1991, 1992, Heemink et al. 2002, Lardner et al.
1993, ten Brummelhuis et al. 1993). In idealized scenarios, Losch and Wunsch (2003) successfully optimized bottom topography in the limit of nearly-geostrophic dynamics
by assimilating (synthetic) sea surface height data. Here we
extend those approaches to a fully fledged OGCM, but restrict ourselves to adjoint sensitivity analysis. The adjoint
sensitivity analysis of a realistic ocean model helps to identify physical processes within the model system and already
allows us address questions in an oceanographic context.
In the future, bottom topography will likely be included
in the control vector of such state estimation studies.
2. MITgcm and its adjoint
The M.I.T. General Circulation Model (MITgcm) is a
general purpose grid-point algorithm that solves the Boussinesq form of the Navier-Stokes equations for an incompressible fluid, hydrostatic or fully non-hydrostatic, with a spatial
finite-volume discretization on a curvilinear computational
grid (in the present context on a three-dimensional longitude,
latitude, depth grid). The model algorithm is described in
Marshall et al. (1997); for online documentation and access
to the model code, see MITgcm Group (2002).
The tangent linear and adjoint model of MITgcm are
generated via automatic/algorithmic differentiation (AD)
(Griewank 2000). To this end, the MITgcm has been adapted
for use with the Tangent linear and Adjoint Model Compiler
(TAMC), and its successor TAF (Transformation of Algorithms in Fortran, Giering and Kaminski 1998) to provide
exact, efficient tangent-linear and adjoint code (Heimbach
et al. 2002, 2005, Marotzke et al. 1999). The adjoint model
framework incorporates the ECCO state estimation infrastructure, but only part of it here because we restrict ourselves
to sensitivity studies.
a. Some aspects of bottom topography control
The starting point for our adjoint code development was
the adjoint model used in previous sensitivity and state estimation studies which computes the local gradient of the objective function with respect to a (by now “standard”) control
vector comprising initial temperature and salinity as well as
air-sea buoyancy and momentum flux fields (other control
variables are available, but do not play a role here).
The procedure for adding topography to the control vector
is based on the “partial cell” treatment in the MITgcm (Adcroft et al. 1997). This technique enables continuous variation of the fractional volume per grid cell between zero and
one with respect to the vertical component and — apart from
its original motivation to represent topography more accurately without increasing the vertical resolution — ensures
differentiability in the cells abutting topography. Total depth
H is then written as the sum over all tracer cells
X
H(i, j) =
hc (i, j, k) ∆rf (k),
(1)
k
4
where ∆rf (k) denotes the full-cell thickness of layer k and
hc (i, j, k) the fractional cell thickness at the tracer point of
the C-grid. Denoting the bottom layer at (i, j) as klow , the
hc (i, j, k) assume values


for k < klow
 = 1,
hc (i, j, k)
= hlow (i, j) ∈ (0, 1] for k = klow (2)


= 0,
for k > klow .
From these, fractional thicknesses are derived for the
cell faces which carry the zonal (u at W -face) and
meridional (v at S-face) velocity points on the C-grid,
hw (i, j, k), hs (i, j, k), respectively. The fractional elements
hx (i, j, k), x = c, w, s are central to the finite-volume spatial discretization. For more details on partial cells we refer
to Adcroft et al. (1997) and MITgcm Group (2002).
The variation of Eq. (2) yields δhc (i, j, k)
=
δhlow (i, j) δk klow , with the Kronecker delta symbol δk klow ,
and thus variation of Eq.(1) reduces to
X
δhc (i, j, k) ∆rf (k) = δhlow (i, j) ∆rf (klow )
δH(i, j) =
k
Thus, while formulating the control problem in terms of the
full depth H and 3-dimensional field hc (i, j, k), when taking
the derivative all terms in the sum (1) vanish except the bottom layer cell k = klow . The source of the adjoint bottom
topography sensitivity is thus confined to the bottom layer,
and propagated into the interior solely by adjoint variables of
the time-dependent flow field (in the language of AD, while
the interior (k 6= klow ) cell thicknesses are active, their sensitivities evaluate to zero, and only k = klow is effectively
active).
The tangent-linear model (the adjoint of which is very
closely related, see e.g., Giering and Kaminski 1998) depends on the particular choice of which fields are considered as control variables, that is, whether a field is considered constant (passive) or variable (active) with respect to
differentiation.
Consider a tracer cell face area normal to the x (zonal)
direction, Ax = ∆y ∆rf hw , with meridional arc length ∆y.
Then, the zonal volume transport U through the cell face Ax
is given by
U = u Ax
(3)
In the absence of bottom topography control, this represents
a linear expression in u with constant Ax , and linearization
trivially yields δU = δu Ax . However, if hc is allowed to
vary, then hw varies and Eq.(3) is quadratic, so that linearization requires the product rule, thus
δU = δu Ax + u δAx
(4)
where δAx = ∆y ∆rf δhw . Recalling Eq.(2), we note that
δAx = 0, except at k = klow . All advective terms are
affected and will propagate changes in fractional cell thicknesses. In a similar fashion, linearization of viscous fluxes
(Laplacian and biharmonic friction), bottom and side drag
terms will be modified.
Modifications in the MITgcm to account for the activity
of partial cells (which are passive parameters if bottom topography is not a control) are mostly straightforward, with
one exception: The conjugate gradient elliptic solver loses
its self-adjoint property which had made it easy to deal with
previously (Marotzke et al. 1999). We point out that changes
necessary in the model to recover a correct adjoint were substantial, and qualitatively different from code modifications
to accommodate the “standard” control variables. Nevertheless, we confine this technical discussion to the appendix.
3. Adjoint Sensitivity Simulations to Bottom
Topography
In this first study of adjoint bottom topography sensitivities of a fully-fledged baroclinic OGCM, we restrict ourselves to a coarse-resolution setup. This enables us to conduct a larger (and longer) number of simulations as well as
longer simulations to elicit general magnitudes and patterns
of sensitivities. The configuration of the OGCM is very similar to that of Ferreira et al. (2005). It covers the globe from
80◦ S to 80◦ N with a horizontal resolution of 4◦ × 4◦ and 15
layers with thicknesses ranging from 50 m near the surface
to 690m near the bottom. To concentrate on bottom topography effects, the model is forced with annual means of wind
forcing fields (Trenberth et al. 1990) and heat and freshwater
fluxes (Jiang et al. 1999) for most experiments. No temperature or salinity relaxation to climatology is applied. The
topography is derived from ETOPO2 by horizontal smoothing with an averaging kernel with a radius of 230 km and
interpolated to the 4◦ -grid. The only additional editing of
topography involves the Drake Passage: it is deepened by
resetting 12 grid points to 3870 m and and 2 to 4510 m, to allow a more realistic transport through the strait in the coarse
resolution model. These modifications do affect the circulation, but do not alter the qualitative picture of our sensitivity
results. The topography used is shown in Fig. 1b.
To avoid problems with vigorous initial adjustment processes, all model integrations are started from a 2000 year
spin-up integration that was initialized with temperature and
salinity climatologies (Levitus and Boyer 1994, Levitus et al.
1994). Again, this choice has little impact on the overall patterns of the sensitivities. For computational efficiency, all integrations use asynchronous time stepping with a tracer time
step of 2 days and a momentum time step of 20 minutes.
Preliminary test showed that neither the horizontal distribution nor the magnitude of the sensitivity maps are greatly
affected by this choice (not shown) as long as the integration
times are longer than gravity-wave time scales.
For comparison, we use a configuration with a horizontal
resolution of 2◦ and 23 vertical layers ranging from 10 m
near the surface to 500 m at depth, similar to the one used
in Stammer et al. (2002), and driven with the same surface
forcing as the 4◦ -model.
5
TABLE 1. Summary of adjoint sensitivity calculations
1 year
10 years
100 years
500 years
1000 years
DPT4D1Y
DPT4D10Y
DPT4D100Y
DPT4D500Y
DPT4D1000Y
MOC4D1Y
MOC4D10Y
MOC4D100Y
DTP4D500Y
MOC4D1000Y
SOO4D1Y
SOO4D10Y
SOO4D100Y
DTP4D500Y
SOH4D1Y
SOH4D10Y
SOH4D100Y
SOH4D500Y
DPT2D1Y
DPT2D10Y
MOC2D1Y
MOC2D10Y
a. Drake Passage Transport
The volume transport through Drake Passage averaged
over the last year of the 100-year-integration is chosen as
a scalar measure (objective function) for the strength of
the ACC. In our model simulation this transport is low
(76 Sv), which is in part attributed to the surface forcing
(with monthly-varying climatological fluxes and sea surface temperature restoring the transport rises to 102 Sv, not
shown). Fig. 2a shows maps of the sensitivity of the Drake
Passage transport with respect to bottom topography for an
integration of 100 years.
The sensitivity to bottom topography is localized and the
global map is not smooth. In general, the flow is affected by
topography gradients, and small variations in absolute depth
are less important. For example, in a zonal integral along
a latitude circle (or a closed streamline), momentum input
by surface wind stress is balanced to a good approximation
by topographic or bottom form stress (bottom pressure times
horizontal topography gradient pb ∇H, Olbers 1998). Overlaying the topography with sensitivity to topography shows
that changing the topography according to the sign of the
sensitivity generally tends to reduce the zonal topographic
gradients. As an example a section along 50◦ S is shown in
Fig. 3. Smoother topography leads to less bottom form stress
and allows the flow to accelerate.
In coarse-resolution models as discussed here, topography gradients cannot be realistically resolved in general, and
are often represented by two adjacent grid points. Therefore,
if reducing the topographic gradients has an effect on the objective function, adjacent grid points can have sensitivities
0.5
0
0
−1000
−0.5
−2000
−1
−3000
−1.5
−4000
−2
−5000
−2.5
0
50
100
150
200
longitude [°E]
250
300
350
depth [m]
We have conducted a series of adjoint sensitivity calculations for a set of dynamically interesting scalar model diagnostics (objective functions) which are subject of frequent
oceanographic and climate analyses. A summary is given
in Table 1. In the following we discuss mainly experiments
with (adjoint) integration times of 100 years.
sensitivity [10−3Sv/m]
objective function
(4◦ resolution)
Drake Passage transport
Atlantic Meridional
Overturning
Southern
Ocean
overturning
Southern Ocean heat
uptake
(2◦ resolution)
Drake Passage transport
Atlantic Meridional
Overturning
−6000
F IG . 3. Depth (thin dashed line, in m) and sensitivity (thick solid
line, in Sv/m) of the Drake Passage transport to topography along
50◦ S.
with opposite signs (cf. Fig. 3). If the topography is resolved
by the grid, these dipole pattern are still present, but on larger
scales than the grid scale (see Figure 3 in Losch and Wunsch 2003). In Section 4 we will see that the sensitivities do
depend on the grid resolution. With higher resolution they
are concentrated near continental slopes and generally over
steep topography.
The sensitivity to topography is concentrated at a few
“choke points” in the Southern Ocean with steep topography: Drake Passage, Crozet-Kerguélen Plateau, and the
Indian Southpolar Ridge/Macquarie Ridge/New Zealand
Plateau area south of New Zealand. Decreasing the height
H of the topography in the latter two regions leads to higher
Drake Passage transports as one would expect from simple geostrophic contour considerations: “flatter” topography
leads to less blocked geostrophic contours, which in turn allows a higher transport. In a different language, the momentum input by surface stress is balanced in the closed stream-
6
a) Drake Passage, ∂φ/∂H [10−3Sv/m]
b) Atlantic MOC, ∂φ/∂H [10−4Sv/m]
60oN
60oN
o
30 N
1
o
30 N
0.8
o
o
0
0
30 S
o
30 S
o
o
60 S
0o
0.6
o
90oE
180oW
90oW
0o
0.4
60 S
0o
90oE
180oW
90oW
°
c) Deacon Cell, ∂φ/∂H [10−5Sv/m]
0o
0.2
0
−6
d) heat transport across 32 S, [10 PW/m]
60 N
o
60 N
o
−0.2
30oN
30oN
−0.4
0o
0o
30oS
30oS
−0.6
−0.8
o
60 S
0o
−1
o
90oE
180oW
90oW
0o
60 S
0o
90oE
180oW
90oW
0o
F IG . 2. Adjoint sensitivity with respect to topography of a) the volume transport through Drake Passage, b) the Atlantic meridional
overturning circulation (MOC), c) the Deacon cell, and d) the heat transport across 32◦ S. In all cases, the integration period is 100 years.
line average by bottom form stress, that is, by the average
of the product of bottom pressure and the gradient of topography (Olbers 1998). Decreasing the horizontal (alongstreamline) topography gradient reduces the form stress term
so less momentum taken out of the system than added, thus
increasing the transport. In the Drake Passage the situation is not so obvious, but the sensitivities can be understood using the same arguments. The ACC turns north onto
the Argentinean Plateau just after the Drake Passage. Decreasing the topographic gradient along its streamlines can
be achieved either by increasing the height of the topography (positive sensitivity) directly in the passage and across
the Scotia Ridge where the ocean is deepest or by decreasing the height near South Georgia and at the southern rim of
the shallow Argentinean Plateau. Therefore the sensitivities
have opposite signs in this region. The positive sensitivities
at 50–46◦W and 50◦ S correspond to a sharp drop in topography (cf. Fig. 3).
b. Atlantic Meridional Overturning Stream Function
The adjoint sensitivity of the strength of the Atlantic
meridional overturning circulation (MOC) averaged over the
last year of a 100 year integration is shown in Fig. 2b. We
define the strength of the MOC as the maximum value of the
overturning stream function at 32◦ N, which happens to be at
1080 m depth (the bottom of the 7th model layer). The MOC
is somewhat overestimated with 29 Sv; we attribute this
strong overturning mainly to inappropriate surface boundary
conditions (constant heat and fresh water fluxes, no restoring), the coarse resolution, and the parameterization of convection by increased vertical diffusivity.
The peak values of the sensitivity to topography are
8 × 10−3 Sv m−1 but the color scale in Fig. 2b is chosen to emphasize smaller values in the Southern Ocean.
Strong topography sensitivities on the order of 10−3 Sv m−1
near Newfoundland and along the path of the Gulf Stream
and the North Atlantic current show that the position of
both the upper branch of the overturning circulation (the
Gulf Stream/North Atlantic Current) and the lower branch
(Deep Western Boundary Current) in part determine the
overturning strength (see also Fig. 5 in Section 4 Section c).
Both branches are boundary currents (barely resolved in the
present configuration) that are strongly influenced by topography. Because of the chosen color scale, many second order
features appear in the North Atlantic that are difficult to interpret. In Section 4 we will see that these patterns depend
strongly on the horizontal resolution.
The MOC sensitivities in the Southern Ocean are weaker
than Drake Passage transport sensitivities in the Southern
ocean (note that the color scale is different by one order of
magnitude to account for the much weaker MOC sensitivities) because (i) the Southern Ocean has a only a remote
7
influence on the MOC (diffusive processes weaken this influence), but a more direct influence on ACC transport, and
(ii) the time scale of 100 years may not be not long enough
to propagate the entire adjoint signal from the North Atlantic
to the Southern Ocean. However, the topography sensitivities in the Southern Ocean have a pattern that is similar to
that of the Drake Passage transport (Fig. 2a) suggesting that
a stronger overturning is connected to a stronger ACC, as
proposed by Toggweiler and Samuels (1995). Zonal wind
stress sensitivity patterns in the Southern Ocean (not shown)
are also similar to the Drake Passage case (see Section 5) and
thereby support this interpretation.
c. Overturning and Meridional Heat Transport in the
Southern Ocean
The Southern Ocean overturning or Deacon cell (e.g.,
Döös and Webb 1994, Speer et al. 1999) is the secondary circulation in the Southern Ocean (the ACC being the primary
one). In the zonal Eulerian mean, it consists of a northward
Ekman transport, which is the main driver of the Deacon cell,
a downwelling branch north of the Drake Passage latitude,
a deep return flow, below the topography and geostrophically balanced, and an upwelling branch around Antarctica
which ventilates deep waters from the north. This overturning circulation has been subject to many studies (see Olbers
et al. 2004, for a review). In coarse resolution models with
Gent and McWilliams (1990, GM) parameterization of the
eddy fluxes the Deacon cell is eliminated as a tracer transport
agent (Borowski et al. 2002), but in our configuration without GM-parameterization, the Deacon cell is rather strong
(44 Sv).
Connected to the overturning circulation, the surface heat
flux in the Southern Ocean cannot be observed directly.
Models and indirect data have been used to estimate the heat
budget in the Southern Ocean (e.g., Ganachaud and Wunsch
2000, Sloyan and Rintoul 2001, Trenberth and Caron 2001,
Wunsch 2005). Flux estimates range from −0.6 to −0.9 PW
across 32◦ S implying a poleward flux of heat (Trenberth and
Caron 2001, Wunsch 2005).
In our simplified model, the heat flux at the surface is
prescribed. In a steady state and in the absence of further
heat sources, the northward heat transport across a given latitude is balanced by the integrated surface heat flux north
of that latitude. The forcing data (Jiang et al. 1999) used
in this study imply a mean southward transport of 0.45 PW
across 32◦ S consistent with the observed value in the model,
and within errors of estimates of Wunsch (2005) and references cited therein. Also the partition of the transport
between the ocean basin shows the observed pattern with
northward transport in the Atlantic (0.57 PW) and Pacific
Oceans (0.23 PW) and southward transport of 1.25 PW in the
Indian Ocean (e.g., Ganachaud and Wunsch 2000, Sloyan
and Rintoul 2001).
The sensitivity of the Deacon cell, which we define as the
maximum overturning strength at 48◦ S and 50 m depth (bot-
tom of the top layer of the model), is shown in Fig. 2c. The
pattern of the overturning sensitivity to topography is similar to that for the Drake Passage transport (Fig. 2a) but often
with reversed sign. (Note also that the sensitivities are two
orders of magnitude smaller than those for the Drake Passage
transport.) For example increasing the height of topography (i.e., making the ocean shallower) south of New Zealand
or at the Crozet-Kerguélen Plateau increases the overturning
strength but decreases the Drake Passage transport. On the
other hand, the sensitivity along the mid-ocean ridges has
the same sign in most places, so that increasing the height
of the mid-ocean ridge in the South Pacific again decreases
the Drake Passage transport, but now also decreases the Deacon cell. We leave the interpretation of this ambiguous result
until we have a solution with higher resolution and a better
parameterization of eddy transfer processes.
The sensitivity of the advective heat transport across
32 ◦ C to bottom topography is shown in Fig. 2d. Note that
the heat transport sensitivity is measured in PW m−1 , while
the overturning sensitivity is in Sv m−1 . In spite of the different units, the sensitivity to topography is strong in locations similar to the ones for Deacon cell and heat transport,
supporting the connection between these quantities. However, more often than not, the sign of the heat transport sensitivity is opposite to that of the overturning sensitivity, because a stronger general circulation increases the overturning
and the southward heat transport, hence decreases the northward heat transport. Therefore, weaker western boundary
currents, the main agents of southward heat transport in the
Southern Ocean increases the northward heat transport.
4. Linearity, time scales, and horizontal resolution
In the following we assess the degree to which the implicit assumption of linearity holds, and investigate the dependency of the sensitivity patterns with respect to integration times, and horizontal resolution.
a. Finite difference perturbation experiments
The adjoint method implies linearization about the state
of the model so that the adjoint sensitivities are always linear
sensitivities. The assumption of quasi-linearity is tested in
the following by a small sample of perturbation experiments.
It is important to understand the difference between what
is usually called gradient checks and testing quasi-linearity.
Gradient checks test individual elements of the adjointgenerated gradient (∇ui J)adj. against finite-difference integrations in the limit of small perturbations, , in the corresponding control vector element and for short integration
times, (∇ui J)f.d. = [J(ui + ) − J(ui )]/. We performed
various gradient checks to verify the accuracy of the automatically generated adjoint code, but will omit technical details. Instead we focus here on testing quasi-linearity: we
use large finite perturbations and long time scales to assess
to what degree the assumption of linearity, that is implicitly
8
made when dealing with adjoint sensitivities, is valid.
We choose four different points where we add a perturbation of 100 m to the height of the topography (making the
ocean locally shallower): Indonesian Throughflow (130◦E,
2◦ S), Macquarie Ridge (166◦E, 54◦ S), exit of Labrador
Sea (42◦ W, 54◦ N), and a point in the North Pacific Ocean
(142◦ W, 38◦ N) with very low sensitivity. We compare the
resulting differences to a reference run with the original topography to the linear adjoint sensitivities in Table 2.
Adjoint and finite-difference sensitivities give the same
order of magnitude transport differences. The adjoint sensitivity results differ by less than 30% from the perturbation
results except for cases when the sensitivity is very small and
the implied transport changes smaller than 0.001 Sv. Decreasing the integration time decreases the differences further (e.g., to order 5% for 1 year integrations). We conclude
that on time scales of 100 years, the adjoint sensitivities can
be used to estimate the impact of finite topography uncertainties on scalar model diagnostics within roughly 30%.
b. Different Time Scales
Fig. 4 shows that the sensitivity to topography changes
with the length of the integration: as an example, we choose
as the objective function the meridional heat transport across
32◦ S, averaged over the last year of the integration and integrate the model for 1, 10, 100, and 500 years. The topography sensitivity changes in pattern and magnitude for the
first three integrations, but there are only small differences
between the 100 year and the 500 year run, mostly restricted
to regions far away from 32◦ S. Again, the effect of local vs.
remote effects is visible, the former playing a dominant role
for short integration times (1 year), the latter increasing in
influence with increasing time scales (100, 500 years).
In forward perturbation experiments such as the ones in
Section 3 a, we found that the difference in, for example,
temperature between two simulations, albeit two to three orders of magnitude smaller than the signal, has not reached a
steady state after 500 years. Because the model includes a
parameterization of convection, we expect non-linear behavior to maintain small differences between the experiments,
even in our coarse configurations: small differences in buoyancy cause different convective depths, that lead to new
anomalies that are propagated by Rossby and Kelvin waves.
Therefore, small changes in sensitivity near the “measurement” area along 32◦ S, where the objective function is computed, are not surprising even after long time scales, but are
a consequence of the internal variability of the model.
We conclude that after 100 years the sensitivities near the
location of the “measurement” have nearly equilibrated and
that the sensitivities of the 100 year-integrations are representative of principle sensitivity patterns.
c. Higher Resolution
The sensitivity to bottom topography is expected to
depend in part on the horizontal resolution — because
with varying resolution, the topographic gradients and the
sub-grid-scale processes that are implicitly parameterized,
change. Therefore, increasing the resolution from 4◦ to 2◦
should decrease the sensitivity. Doubling the resolution requires a shorter time step so that the overall computational
cost (wall clock time and memory requirements) for a 2◦ experiment is at least 16 times that of the 4◦ experiment. For
this reason, we compare only shorter 1 and 10-year integrations of a 4◦ experiment to a 2◦ experiment in Fig. 5 for the
objective function Atlantic meridional overturning stream
function.
As expected the sensitivities in the 2◦ experiments and in
the 4◦ experiments are different. For example, the sensitivity of the Drake Passage transport over the Crozet-Kerguélen
Plateau is less pronounced in the 2◦ resolution run, because
at this resolution the topography resolves a few passages below 2000 m depth and hence the ACC feels less topographic
stress: the ACC is steered around the Kerguèlen Plateau and
not across it as in the coarser and more viscous experiment
(not shown).
The MOC sensitivities in the North Atlantic Ocean are
much finer in the the 2◦ run. As the continental margins
and the boundary currents become sharper, so does the sensitivity to topography. The sensitivity in the interior of the
North Atlantic basin that is visible in the 4◦ run has vanished,
pointing towards too strong horizontal viscosity/diffusion in
the coarser experiment: with strong internal friction, the
boundary currents are more spread-out and are also affected
by topography in the interior.
In spite of these differences, several important patterns
do appear to be robust. For example, the sensitivities in both
configurations are pronounced along the western boundary
of the North Atlantic and after 10 years also in the Denmark
Strait and over the Scotland-Iceland Ridge. This similarity
is encouraging and shows that the 4◦ runs do contain relevant
sensitivity patterns.
From the graphs in Fig. 5, the point-wise sensitivities in
the coarse runs appear to be larger in both amplitude and spatial extent. To estimate the overall effect of the sensitivities,
the sensitivities ∂φ/∂ui at position i are multiplied by the
uncertainties ∆ui :
∆φi =
∂φ
∆ui .
∂ui
(5)
In the limit of uncorrelated uncertainties, the point-wise contributions ∆H ∂φ/∂H are added geometrically
pP according
2
to uncorrelated error propagation (∆φ =
i ∆φi ): the
effect of an uncorrelated global topography uncertainty of
∆H = 100 m would result in an uncertainty of the transport through Drake Passage by 1 Sv in the 2◦ experiment
and 0.6 Sv in the 4◦ experiment. For the meridional overturning strength, the estimated uncertainties are 0.4 SV and
0.6 Sv respectively. In the other limit of perfect correlation,
9
TABLE 2. Comparison of implied transport difference from linear adjoint sensitivities ∆φadj = ∆H · (∂φ/∂H) and perturbation
experiments ∆φfwd = φ(H + ∆H) − φ(H) for 100 year integrations. The relative difference is [(∆φadj − ∆φfwd )/∆φfwd ] × 100.
130◦ E, 2◦ S 166◦ E, 54◦ S 142◦ W, 38◦ N
Drake Passage transport (DPT4D100Y)
∆φfd [Sv]
0.0822
−0.1885
−2 × 10−6
∆φadj [Sv]
0.0632
−0.1533
−7 × 10−7
rel. diff. [%]
23
19
69
Atlantic meridional overturning circulation (MOC4D100Y)
∆φfd [Sv]
0.0386
−0.0139
6 × 10−7
∆φadj [Sv]
0.0286
−0.0141
3 × 10−7
rel. diff. [%]
26
1
46
“Deacon Cell” (SOO4D100Y)
∆φfd [Sv]
0.0026
0.0092
−7 × 10−7
∆φadj [Sv]
0.0023
0.0096
−8 × 10−8
rel. diff. [%]
10
5
89
Heat transport across 32◦ S (SOH4D100Y)
∆φfd [Sv]
0.0019
−0.0006
−3 × 10−7
∆φadj [Sv]
0.0010
−0.0007
−7 × 10−8
rel. diff. [%]
44
21
80
that is by simply adding all ∆φi in Eq. (5), the implied uncertainties are −4.4 Sv for the 2◦ experiment and −2.0 Sv
for the 4◦ experiment for the Drake Passage case. For the
MOC, we have 0.16 Sv for the 2◦ experiment, −1.6 Sv for
the 4◦ experiment. These numbers are ambiguous and do
not support the hypothesis that the overall sensitivity to topography is smaller in high resolution models. All that can
be said is, that the sensitivities to topography do change with
resolution, and more experiments are needed to explore the
sensitivities at higher resolutions.
5. Importance Relative to Momentum and Buoyancy
Forcing
To put adjoint sensitivities with respect to bottom topography into perspective, we show the sensitivity of the Drake
Passage transport to the control variables zonal and meridional wind stress, heat and fresh water flux in Fig. 6.
The zonal wind stress sensitivity map shows that increasing the zonal wind stress in the Southern Ocean, in particular in Drake Passage itself, south of Africa and Australia,
increases the transport through the Drake Passage (positive
sensitivity). This is accomplished by an increased Ekman
transport piling up surface waters at northern boundaries
(Drake Passage, Africa, and Australia). The increased surface height gradient then accelerates the ACC. Less northward (i.e., more southward) meridional wind stress along
the Pacific coast of South America moves less water out of
(“pushes” more water into) the Antarctic Circumpolar Current, thus increasing its strength (negative sensitivity). The
northern hemisphere wind stress does not seem to affect the
ACC as much on the time scale of 100 years.
Heat and fresh water fluxes are converted to buoyancy
42◦ W, 54◦ N
−0.0250
−0.0253
1
−0.0590
−0.0611
4
0.0001
0.0001
33
0.0003
0.0002
19
fluxes by scaling with gαθ /(ρ0 cw ) and gβS S0 , respectively.
Upward flux (out of the ocean) is positive. The parameters are the gravity acceleration g = 9.81 m s−2 , the thermal expansion coefficient αθ = 2 × 10−4 K−1 , the saline
contraction coefficient βS = 7.4 × 10−4 , the heat capacity
of water cw = 3994 J kg−1 K−1 , a reference density ρ0 =
1035 kg m−3 , and a reference salinity S0 = 35. In buoyancy flux units, it becomes immediately clear that heat and
freshwater fluxes have very similar effects on the Drake Passage transport, and their effect can be interpreted as a general
sensitivity to buoyancy fluxes. The physical interpretation is
also intuitive: decreasing the outgoing buoyancy flux north
of approximately 50◦ S increases the buoyancy of the surface
layers, thus the stratification; south of 50◦ S increased upward buoyancy flux decreases the buoyancy around Antarctica. Both mechanisms increase the meridional density gradient, which then accelerates the ACC. The sensitivity in the
northern hemisphere is much larger (more negative) in the
Atlantic Ocean than in the remaining oceans pointing to an
intimate link between the Atlantic Overturning Circulation
and the strength of the ACC.
In the Southern Ocean, mean wind stress values are on
the order of 0.1 N m−2 , surface buoyancy fluxes range from
−5 to +5×10−8 m2 s−3 . Assume uncertainties as large as
50% for wind stress and buoyancy fluxes; uncertainty of topography is estimated at 100 m (locally the uncertainties can
be much larger, cf. Fig. 1a). From Fig. 6, we read off a zonal
wind stress sensitivity of 2 Sv/(N m−2 ) in Drake Passage, a
buoyancy flux sensitivity of 1×107 Sv/(m2 s−3 ) in the Atlantic Ocean, and from Fig. 2 a topography sensitivity of
2×10−3 Sv/m near the Crozet-Kerguèlen Plateau. According to Eq. (5), changing the control variables in these points
by their respective large uncertainties changes the Drake
10
1 year
10 years
60oN
60oN
30oN
30oN
1
0.8
o
0
0
30 S
o
30 S
o
o
60 S
o
0
0.6
o
o
90 E
o
180 W
o
90 W
o
0
0.4
60 S
o
0
o
90 E
o
180 W
o
90 W
o
0.2
0
0
100 years
500 years
60oN
60oN
−0.2
30oN
30oN
−0.4
0o
0o
∂φ/∂H [10−6 PW/m]
o
−0.6
−0.8
o
30 S
o
o
60 S
o
0
30 S
60 S
o
0
−1
o
o
90 E
o
180 W
o
90 W
o
0
o
90 E
o
180 W
o
90 W
o
0
F IG . 4. Adjoint sensitivity of heat transport across 32◦ S to topography for different time scales.
Passage transport by 0.1 Sv (wind stress), 0.25 Sv (buoyancy flux), and 0.2 Sv (topography), that is, by comparable amounts. Clearly, these rough estimates refer to pointwise changes. If we were to estimate the effect of, for
instance, large scale wind stress fluctuations in the Southern Ocean, the point-wise effects would accumulate, depending on the error correlation and the topographic effects would be smaller, because uncertainties in topography
are likely to be uncorrelated. In the case of perfect correlation, increasing the zonal wind stress south of 38◦ S by
0.05 N m−2 implies an integral linear increase of the Drake
Passage transport by 20 Sv consistent with the dramatic increase in transport that Gent et al. (2001) observe when they
add 0.05×secant(latitude) N m−2 to the zonal wind stress
(their experiment H). Increasing the heat flux south of 52◦ S
by 10 W m−2 and assuming perfect correlation increases the
transport by 3.0 Sv. Assuming a systematic bias of +100 m
south of 38◦ S leads to a decrease in transport by 2.1 Sv.
In the other limit of uncorrelated uncertainties wind stress
uncertainties of 0.05 N m−2 south of 38◦ S lead to 1 Sv
Drake Passage transport uncertainty, heat flux uncertainties of 10 W m−2 to 0.5 Sv, and topographic uncertainties of
100 m south of 38◦ S to 0.6 Sv. We note, that both locally
and on the large scale uncertainties in bottom topography
can have the same size effect on the Drake Passage transport
as uncertainties in surface forcing fields. All effects on the
transport have measurable magnitudes and therefore should
be controllable. Uncertainties in surface forcing fields are
more likely to be correlated in space than topography uncertainties, and the overall effect of surface wind stress uncertainties is estimated to be larger than that of the topography
uncertainties.
6. Discussion
Numerical ocean models are sensitive to the geometry
of the model domain. This sensitivity can be qualified and
quantified with the help of an adjoint model that computes
the gradient of scalar objective functions (model diagnostics) with respect to the control variables.
We found in a coarse grid configuration of the MITgcm,
that the sensitivity with respect to bottom topography is locally equally important to the mean circulation of the model
as are surface boundary conditions. Many interesting aspects about the model dynamics are revealed in the sensitivity maps. We have briefly touched upon this topic to illustrate the power of the adjoint sensitivities. To the extent that
the model represents real processes, the conclusions about
the model dynamics can be transfered to the real ocean.
For future applications of the presented adjoint technology, one can imagine, for example, an objective function that
is directly related to the position of the Gulf Stream separation point, thereby finding a way to represent a proper Gulf
Stream separation in a coarse model by systematically tuning the topography. Or, extending the work of Köhl (2005), a
study on the impact of the topography of the Denmark Strait
11
1 year
80oN
60 N
40oN
40oN
20oN
20oN
0o
0o
20oS
20oS
2
o
o
o
o
o
100 W 75 W 50 W 25 W
o
0
o
25 E
1.5
1
0.5
o
o
o
o
100 W 75 W 50 W 25 W
o
0
o
25 E
0
10 years
80oN
10 years
80oN
Sv/m]
o
−4
60 N
∂φ/∂H [10
1 year
80oN
−0.5
60oN
60oN
40oN
40oN
o
20 N
−1
o
20 N
−1.5
o
o
0
0
20oS
20oS
o
o
o
o
100 W 75 W 50 W 25 W
o
0
o
25 E
−2
o
o
o
o
100 W 75 W 50 W 25 W
o
0
o
25 E
F IG . 5. Comparison of adjoint sensitivity to topography between 4◦ (left column) and 2◦ -resolution (right column) runs for 1 year (top
row) and 10 years (bottom row) for the Atlantic meridional overturning stream function.
and the Scotland-Iceland-Ridge could be modified from the
traditional approach of changing the topography locally and
observing the changes in circulation (e.g., Beismann and
Barnier 2004) to asking directly, which control variables affect, say, the Atlantic meridional overturning stream function. Due to the limited resolution of our experiments, we
were only able to point to regions where topography matters.
In our simulations, many details are different from realistic scenarios, and necessarily so are the adjoint sensitivities.
In the following we name a few.
For computational efficiency, and because this study is in
part a proof of concept, we chose a coarse resolution grid,
similar to other studies (e.g., Ferreira et al. 2005). We have
shown that many of the sensitivity features remain when
the resolution is doubled, but have also shown that other
features start to change qualitatively (e.g., emergence of a
sharper western boundary current structure, and ScotlandIceland Ridge with increasing resolution). Care must therefore be taken to interpret sensitivities within the context of
the numerical model and configuration (the issue of convergence of a model towards a solution with increasing reso-
lution, may be extended to the question of convergence of
adjoint sensitivities, a difficult topic in the context of interpreting adjoint sensitivities in highly nonlinear regimes). In
general, we expect that the maps of sensitivity to topography becomes smoother and thus more straightforward to interpret when the resolution increases sufficiently to resolve
steep topography gradients. Finally, at eddy permitting resolution, when according to Penduff et al. (2002) roughness of
topography can actually lead to less realistic solutions, the
sensitivity to topography may be completely different than
in our experiments.
With respect to the forcing, we note that imposing constant surface fluxes that are independent of the underlying
ocean state is clearly unrealistic. We have chosen this type
of idealized boundary condition to facilitate the interpretation of the flux sensitivities. Allowing some feedback between ocean model and surface boundary conditions, for example, by mixed boundary conditions (prescribed fluxes in
combination with temperature restoring), both improves the
solution and leads to different flux sensitivities, but we found
in preliminary experiments that neither restoring boundary
conditions nor a seasonal cycle in the boundary conditions
12
2
2
a) ∂φ/∂τx [Sv m /N]
b) ∂φ/∂τy [Sv m /N]
60 N
o
60 N
30 N
o
30 N
0o
0o
30oS
30oS
60oS
0o
−2
o
o
o
90 E
−1
o
180 W
o
90 W
0
0
1
2
7
3
o
60oS
0o
−2
2
90oW
0o
0
1
2
3
2
d) ∂φ/∂EmP [10 Sv s /m ]
60oN
60oN
30oN
30oN
o
o
0
0
o
30 S
o
30 S
o
−2
−1
180oW
7
c) ∂φ/∂Q [10 Sv s /m ]
60 S
o
0
90oE
o
o
90 E
−1
o
180 W
o
o
90 W
0
0
1
2
60 S
o
0
−2
o
90 E
−1
180 W
o
90 W
o
0
o
0
1
2
F IG . 6. Adjoint sensitivity with respect to surface boundary conditions of the volume transport through Drake Passage after 100 years:
a) zonal wind stress τx , b) meridional wind stress τy , c) heat flux Q, d) fresh water flux EmP . Heat and fresh water flux sensitivities are
rescaled into units of buoyancy flux sensitivity.
have a fundamental effect on the sensitivity to topography
(not shown). Using atmospheric state variables such as air
temperature, specific humidity and wind speed as control
variables, from which surface fluxes are computed via bulk
formulae, captures the feedback between atmosphere and
ocean even better (as well as the direct coupling between
momentum and buoyancy fluxes) and may lead to more realistic sensitivities (efforts in this regards are currently under
way in the context of improving the ECCO-like state estimation systems).
For technical reasons, the current model configuration
does not include a modern eddy transfer parameterization
such as that of Gent and McWilliams (1990), although in
principle this is possible (Ferreira et al. 2005, who used an
adjoint model with GM-parameterization but without topography as a control variable). Parameterizing turbulent mixing by horizontal diffusion often leads to spurious convection in the Southern Ocean, so the processes in the model
that maintain the density structure in the Southern Ocean are
different from those in the real world. The southern over-
turning cell is most likely affected by this parameterization
so that particularly the interpretation of the Deacon cell sensitivities in Section 3 c should not be generalized. On the
other hand, the impact of the topography on the near surface
flow depends significantly on the oceanic stratification (Marshall 1995b), and one expects a more pronounced impact of
the topography at high latitudes regions where the stratification is weak.
Other parameterizations, such as implicit diffusion for
deep convection, could also be replaced by more realistic
schemes. Parameterizing vertical mixing as a function of topography roughness, following Simmons et al. (2004), adds
a further connection between topography and the general circulation. We expect the topography sensitivities to change
when including such a parameterization.
Flow near the bottom tends to be affected more strongly
by topography than near-surface currents that are baroclinically shielded from the effects of topography (Marshall and
Stephens 2001). Either the explicit resolution of such bottom or slope currents or their efficient parameterization, for
13
example by a bottom boundary layer model (Beckmann and
Doescher 1997, Campin and Goosse 1999) might change the
behavior of the ocean model and its sensitivity to bottom topography.
The present findings have consequences for the choice
of control variables (the control vector) in ocean state estimation studies. The control variables determine the solution of a model. In state estimation the elements of the
control vector are systematically adjusted within prescribed
error bounds to fit the model variables to observations. In
most of the current state estimation studies the control vector consists of surface boundary conditions and initial conditions. With our findings that bottom topography affects
the solution of an OGCM measurably and, at least locally,
as much as surface boundary conditions, it is natural to include bottom topography in the control vector and allow it to
adjust within prescribed (probably small) limits in the framework of state estimation. Losch and Wunsch (2003) demonstrated the feasibility of this idea in a single-layer, nearlygeostrophic model so that there do not appear to be any fundamental difficulties in doing so. Other parameters, such as
mixing coefficients (Stammer 2005), eddy stresses (Ferreira
et al. 2005), or lateral slip boundary conditions affect the solution as well and should also be considered. Topography,
diffusivity, and lateral boundary conditions promise to be a
challenging combination, both technically and dynamically,
of control variables in a general circulation model.
Acknowledgments.
We thank Carl Wunsch who initiated this work for inspiration and support. ML is grateful
for the hospitality of the MIT on several occasions. Sergey
Danilov provided many helpful comments. This work was
supported in part by the National Oceanographic Partnership
Program (NOPP; NASA, NOAA, NSF) through the ECCO
Consortium and its follow-on ECCO-GODAE, and by NSF
through the ITR program.
APPENDIX
A. Technical issues of topography as a control variable
While relying on automatic adjoint code generation, some
important issues needed to be addressed. We spare the reader
the technical details, but highlight a few conceptual aspects.
The surface pressure field at each time step is found by
an implicit scheme. This scheme involves an elliptic matrixoperator and the corresponding equations are solved iteratively. For constant matrix coefficients this operator is selfadjoint, but the coefficients depend on the topography; if bottom topography is a control variable, the operator looses its
self-adjoint properties and derivative code for this iterative
solver must be generated. Under certain circumstances, the
implicit function theorem may be invoked to show that a corresponding iterative solver for the adjoint equations can be
derived. However, Giles (2002) shows that straightforward
AD may fail in providing an efficient solver. Here, we chose
to store intermediate results of the iteration at each time step.
In the future, this costly approach may be replaced by a more
efficient and generally stable solution.
There is a certain freedom in choosing the control variable topography on a C-grid: the topographic height at either the pressure/tracer-points or the velocity points. Because, on a C-grid there are two velocity points per grid cell,
the latter choice doubles the number of topography related
control variables. It also changes the solution algorithm,
implying shaved or cut cells instead of the standard partial
cells (Adcroft et al. 1997). Currently, we chose to use topographic height at the pressure/tracers points as control variables instead. This choice involves many non-continuous
(non-differentiable) function calls (FORTRAN MIN/MAX
functions) when generating the fractional volume cells. The
MIN/MAX functions can be replaced by quasi-continuous
approximations in order to ensure differentiability. In practice, doing so changes the resulting sensitivities only locally.
REFERENCES
Adcroft, A., and D. Marshall, 1998: How slippery are piecewiseconstant coastlines in numerical ocean models? Tellus, 50A,
95–108.
Adcroft, A., C. Hill, and J. Marshall, 1997: Representation of topography by shaved cells in a height coordinate ocean model.
Mon. Wea. Rev., 125(9), 2293–2315.
Ayoub, N., 2005: Estimation of boundary values in a North Atlantic
circulation model using an adjoint method. Ocean Modelling,
in press.
Beckmann, A., and R. Doescher, 1997: A method for improved
representation of dense water spreading over topography in
geopotential-coordinate models. J. Phys. Oceanogr., 27, 581–
591.
Beismann, J.-O., and B. Barnier, 2004: Variability of the meridional overturning circulation of the North Atlantic: Sensitivity
to overflows of dense water masses. Ocean Dynamics, 54(1),
92–106.
Borowski, D., R. Gerdes, and D. Olbers, 2002: Thermohaline and
wind forcing of a circumpolar channel with blocked geostrophic
contours. J. Phys. Oceanogr., 32(9), 2520–2540.
Bryan, F., 1987: Parameter sensitivity of primitive equation ocean
general circulation models. J. Phys. Oceanogr., 17(7), 970–985.
Bugnion, V., 2001: Driving the ocean’s overturning: An adjoint
sensitivity study, Ph.D. thesis, MIT.
Campin, J.-M., and H. Goosse, 1999: Parameterization of densitydriven downsloping flow for a coarse-resolution ocean model in
z-coordinate. Tellus, 51A(412-430).
Das, S. K., and R. W. Lardner, 1991: On the estimation of parameters of hydraulic models by assimilation of periodic tidal data.
J. Geophys. Res., 96(C8), 15,187–15,196.
Das, S. K., and R. W. Lardner, 1992: Variational parameter estimation for a two-dimensional numerical tidal model. Int. J. Numer.
Methods Fluids, 15(3), 313–327.
Döös, K., and D. J. Webb, 1994: The Deacon cell and the other
meridional cells of the Southern Ocean. J. Phys. Oceanogr., 24,
429–442.
Dutkiewicz, S., M. J. Follows, P. Heimbach, and J. C. Marshall,
2006: Controls on ocean productivity and air-sea carbon flux:
an adjoint model sensitivity study. Geophys. Res. Lett., 33(2),
L02,603, doi:10.1029/2005GL024987.
14
Ferreira, D., J. Marshall, and P. Heimbach, 2005: Estimating eddy
stresses by fitting dynamics to observations using a residualmean ocean circulation model. J. Phys. Oceanogr., 35(10),
1891–1910.
Ferron, B., and J. Marotzke, 2003: Impact of 4D-variational data
assimilation of WOCE hydrography on the meridional circulation of the Indian Ocean. Deep-Sea Res. II, 50, 2005–2021.
Fukumori, I., T. Lee, B. Cheng, and D. Menemenlis, 2004: The
origin, pathway, and destination of Niño-3 water estimated by
a simulated passive tracer and its adjoint. J. Phys. Oceanogr.,
34(3), 582–604.
Galanti, E., and E. Tziperman, 2003: A midlatitude–ENSO teleconnection mechanism via baroclinically unstable long Rossby
waves. J. Phys. Oceanogr., 33, 1877–1887.
Galanti, E., E. Tziperman, M. Harrison, A. Rosati, R. Giering, and
Z. Sirkes, 2002: The equatorial thermocline outcropping - A
seasonal control on the tropical Pacific ocean-atmosphere instability. J. Climate, 15, 2721–2739.
Ganachaud, A., and C. Wunsch, 2000: Improved estimates of
global ocean circulation, heat transport and mixing from hydrographic data. Nature, 408, 453–457.
Gebbie, G., P. Heimbach, and C. Wunsch, 2006: Strategies for
nested and eddy-resolving state estimation. J. Geophys. Res.,
submitted.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean
circulation models. J. Phys. Oceanogr., 20(1), 150–155.
Gent, P. R., W. G. Large, and F. O. Bryan, 2001: What sets the mean
transport through Drake Passage? J. Geophys. Res., 106(C2),
2693–2712.
Giering, R., and T. Kaminski, 1998: Recipes for adjoint code construction. ACM Trans. Math. Softw., 24(4), 437–474.
Giles, M. B., 2002: On the iterative solution of adjoint equations.
Automatic Differentiation of Algorithms: From Simulation to
Optimization, G. Corliss, C. Faure, A. Griewank, L. Hascoet,
and U. Naumann, Eds., Springer-Verlag, pp. 145–152.
Griewank, A., 2000: Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation. vol. 19 of Frontiers in
Applied Mathematics, SIAM, 369 pp.
Heemink, A. W., E. E. A. Mouthaan, M. R. T. Roest, E. A. H.
Vollebregt, K. B. Robaczewska, and M. Verlaan, 2002: Inverse
3D shallow water flow modelling of the continental shelf. Cont.
Shelf Res., 22(3), 465–484.
Heimbach, P., C. Hill, and R. Giering, 2002: Automatic generation of efficient adjoint code for a parallel Navier-Stokes solver.
Computational Science – ICCS 2002, J.J. Dongarra, P.M.A.
Sloot and C.J.K. Tan, Ed., vol. 2331, part 3 of Lecture Notes
in Computer Science, Springer-Verlag, pp. 1019–1028.
Heimbach, P., C. Hill, and R. Giering, 2005: An efficient exact
adjoint of the parallel MIT general circulation model, generated
via automatic differentiation. Future Generation Computer Systems (FGCS), 21(8), 1356–1371.
Hill, C., V. Bugnion, M. Follows, and J. Marshall, 2004: Evaluating
carbon sequestration efficiency in an ocean circulation model
by adjoint sensitivity analysis. J. Geophys. Res., 109(C11),
C11,005, doi:10.1029/2002JC001598.
Holland, W. R., 1973: Baroclinic and topographic influences on the
transport in western boundary currents. Geophys. Fluid Dyn., 4,
187–210.
Huang, B., P. H. Stone, and C. Hill, 2003:
Sensitivities of
deep-ocean heat uptake and heat content to surface fluxes and
subgrid-scale parameters in an ocean general circulation model
with idealized geometry. J. Geophys. Res., 108(C1), 3015,
doi:10.1029/2001JC001218.
Jiang, S., P. H. Stone, and P. Malanotte-Rizzoli, 1999: An assessment of the Geophysical Fluid Dynamics Laboratory ocean
model with coarse resolution: Annual-mean climatology. J.
Geophys. Res., 104(C11), 25,623–25,645.
Junge, M. M., and T. W. N. Haine, 2001: Mechanisms of North Atlantic wintertime sea surface temperature anomalies. J. Climate,
14(24), 4560–4572.
Köhl, A., 2005: Anomalies of the meridional overturning: Mechanisms in the North Atlantic. J. Phys. Oceanogr., 35(8), 1455–
1472.
Köhl, A., and D. Stammer, 2004: Optimal observations for variational data assimilation. J. Phys. Oceanogr., 34(3), 529–542.
Köhl, A., and J. Willebrand, 2003: Variational assimilation of
SSH variability from TOPEX/POSEIDON and ERS1 into an
eddy-permitting model of the North Atlantic. J. Geophys. Res.,
108(C3), 3092, doi:10.1029/2001JC000982.
Lardner, R. W., A. H. Al-Rabeh, and N. Gunay, 1993: Optimal
estimation of parameters for a two-dimensional hydrodynamical
model of the Arabian Gulf. J. Geophys. Res., 98(C10), 18,229–
18,242.
Large, W. G., G. Danabasoglu, S. C. Doney, and J. C. McWilliams,
1997: Sensitivity of surface forcing and boundary layer mixing
in a global ocean model: Annual-mean climatology. J. Phys.
Oceanogr., 27(11), 2418–2447.
Lea, D. J., T. W. N. Haine, and R. F. Gasparovic, 2006: Observability of the Irminger Sea circulation using variational data assimilation. Quart. J. Roy. Meteor. Soc..
Levitus, S., and T. Boyer, 1994: World Ocean Atlas 1994. Volume
4: Temperature. NOAA, NOAA Atlas NEDSIS 4.
Levitus, S., R. Burgett, and T. Boyer, 1994: World Ocean Atlas
1994. Volume 3: Salinity. NOAA, NOAA Atlas NEDSIS 3.
Li, X., and C. Wunsch, 2003:
Contraining the North Atlantic circulation between 4.5◦ s and 39.5◦ n with transient
tracer observations.
J. Geophys. Res., 108(C10), 3318,
doi:10.1029/2002JC001765.
Li, X., and C. Wunsch, 2004: An adjoint sensitivity study of
chlorofluorocarbons in the North Atlantic. J. Geophys. Res.,
109(C1), C01,007, doi:10.1029/2003JC002014.
Losch, M., and C. Wunsch, 2003: Bottom topography as a control
parameter in an ocean circulation model. J. Atmos. Oceanic
Technol., 20(11), 1685–1696.
Losch, M., A. Adcroft, and J.-M. Campin, 2004: How sensitive
are coarse general circulation models to fundamental approximations in the equations of motion? J. Phys. Oceanogr., 34(1),
306–319.
Marotzke, J., and C. Wunsch, 1993: Finding the steady state of
a general circulation model through data assimilation: Application to the North Atlantic Ocean. J. Geophys. Res., 98(C11),
20,149–20,167.
Marotzke, J., R. Giering, K. Q. Zang, D. Stammer, C. Hill, and
T. Lee, 1999: Construction of the adjoint MIT ocean general
circulation model and application to Atlantic heat transport sensitivity. J. Geophys. Res., 104(C12), 29,529–29,547.
Marshall, D., 1995a: Influence of topography on the large-scale
ocean circulation. J. Phys. Oceanogr., 25(7), 1622–1635.
Marshall, D., 1995b: Topographic steering of the Antarctic Circumpolar Current. J. Phys. Oceanogr., 25(7), 1635–1650.
15
Marshall, D. P., and J. C. Stephens, 2001: On the insensitivity of
the wind-driven circulation to bottom topography. J. Mar. Res.,
59(1), 1–27.
Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey,
1997: A finite-volume, incompressible Navier Stokes model
for studies of the ocean on parallel computers. J. Geophys. Res.,
102(C3), 5753–5766.
Marsland, S., and J.-O. Wolff, 2001: On the sensitivity of Southern
Ocean sea ice to the surface freshwater flux: A model study. J.
Geophys. Res., 106(C2), 2723–2741.
MITgcm Group, 2002: MITgcm Release 1 Manual. MIT/EAPS,
(online documentation), http://mitgcm.org/sealion/
online_documents/manual.html.
Moore, A. M., H. G. Arango, E. D. Lorenzo, B. D. Cornuelle, A. J.
Miller, and D. J. Neilson, 2004: A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint
of a regional ocean model. Ocean Modelling, 7(1–2), 227–258.
Olbers, D., 1998: Comments on “On the obscurantist physics of
‘form drag’ in theorizing about the Circumpolar Current”. J.
Phys. Oceanogr., 28(8), 1647–1654.
Olbers, D., D. Borowski, C. Völker, and J.-O. Wolff, 2004: The dynamical balance, transport and circulation of the Antarctic Circumpolar Current. Antarctic Science, 16, 439–470.
Pacanowski, R. C., and G. Anand, 1998: Transient response in a
z-level ocean model that resolves topography with partial cells.
Mon. Wea. Rev., 126(12), 3248–3270.
Penduff, T., B. Barnier, M.-A. Kerbiriou, and J. Verron, 2002:
How topographic smoothing contributes to differences between
the eddy flows simulated by sigma- and geopotential-coordinate
models. J. Phys. Oceanogr., 32(1), 122–137.
Sheinbaum, J., and D. L. T. Anderson, 1990: Variational assimilation of XBT data. part 1. J. Phys. Oceanogr., 20(5), 672–688.
Simmons, H. L., S. R. Jayne, L. C. St. Laurent, and A. J. Weaver,
2004: Tidally driven mixing in a numerical model of the ocean
general circulation. Ocean Modelling, 6(3–4), 245–263.
Sirkes, Z., and E. Tzipermann, 1997: Finite difference of adjoint
or adjoint of finite difference? Mon. Wea. Rev., 125(12), 3373–
3378.
Sloss, P. W., 1988: Data announcement 88-MGG-02, digital relief
of the surface of the Earth, NOAA, National Geophysical Data
Center, Boulder, CO.
Sloyan, B. M., and S. Rintoul, 2001: The Southern Ocean limb
of the global deep overturning circulation. J. Phys. Oceanogr.,
31(1), 143–173.
Smith, W. H. F., and D. T. Sandwell, 1997: Global seafloor topography from satellite altimetry and ship depth soundings. Science,
277, 1957–1962.
Speer, K. G., S. R. Rintoul, and B. M. Sloyan, 1999: The diabatic
Deacon cell. J. Phys. Oceanogr., 30(12), 3212–3222.
Stammer, D., 2005: Adjusting internal model errors through ocean
state estimation. J. Phys. Oceanogr., 35(6), 1143–1153.
Stammer, D., C. Wunsch, R. Giering, C. Eckert, P. Heimbach,
J. Marotzke, A. Adcroft, C. N. Hill, and J. Marshall, 2002:
The global ocean circulation during 1992–1997, estimated from
ocean observations and a general circulation model. J. Geophys.
Res., 107(C9), 3118, doi:10.1029/2001JC000888.
Stammer, D., C. Wunsch, R. Giering, C. Eckert, P. Heimbach,
J. Marotzke, A. Adcroft, C. Hill, and J. Marshall, 2003: Volume, heat and freshwater transports of the global ocean circulation 1992–1997, estimated from a general circulation model
constrained by WOCE data. J. Geophys. Res., 108(C1), 3007,
doi:10.1029/2001JC001115.
ten Brummelhuis, P. G. J., A. W. Heemink, and H. F. P. van den
Boogaard, 1993: Identification of shallow sea models. Int. J.
Numer. Methods Fluids, 17(8), 637–665.
Toggweiler, J. R., and B. Samuels, 1995: Effect of Drake Passage
on the global thermohaline circulation. Deep-Sea Res. I, 42(4),
477–500.
Trenberth, K. E., and J. M. Caron, 2001: Estimates of meridional
atmosphere and ocean heat transports. J. Climate, 14(8), 3433–
3443.
Trenberth, K. E., J. G. Olson, and W. G. Large, 1990: The mean
annual cycle in global ocean wind stress. J. Phys. Oceanogr.,
20, 1742–1760.
Tziperman, E., and W. C. Thacker, 1989:
An optimalcontrol/adjoint-equations approach to studying the oceanic general circulation. jpo, 19(10), 1471–1485.
Weaver, A. T., J. Vialard, and D. L. T. Anderson, 2003: Three- and
four-dimensional variational assimilation with an ocean general
circulation model of the tropical Pacific Ocean. part 1: Formulation, internal diagnostics and consistency checks. Mon. Wea.
Rev., 131(t), 1360–1378.
Wunsch, C., 1984: Acoustic tomography and other answers. ”It’s
the water that makes you drunk.” A celebration in Geophysics
and Oceanography – 1982. In honor of Walter Munk on his 65th
birthday., Scripps Institution of Oceanography Reference Series
84-5, Scripps Institution of Oceanography, of California, San
Diego, pp. 47–62.
Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.
Wunsch, C., 2005: The total meridional heat flux and its oceanic
and atmospheric partition. J. Climate, 18(21), 4374–4380.
Wunsch, C., and P. Heimbach, 2006: Practical global oceanic state
estimation. Physica D, submitted.
Printed February 18, 2006.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement