Hei2009e

Hei2009e
On the formulation of sea-ice models. Part 2:
Lessons from multi-year adjoint sea ice export
sensitivities through the Canadian Arctic
Archipelago.
Patick Heimbach a,1, Dimitris Menemenlis b, Martin Losch c,
Jean-Michel Campin a and Chris Hill a
a Department
of Earth, Atmospheric, and Planetary Sciences, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
b Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA
c Alfred-Wegener-Institut
für Polar- und Meeresforschung, Postfach 120161, 27515
Bremerhaven, Germany
Abstract
The adjoint of an ocean general circulation model is at the heart of the ocean
state estimation system of the Estimating the Circulation and Climate of the Ocean
(ECCO) project. As part of an ongoing effort to extend ECCO to a coupled ocean/seaice estimation system, a dynamic and thermodynamic sea-ice model has been developed for the Massachusetts Institute of Technology general circulation model
(MITgcm). One key requirement is the ability to generate, by means of automatic
differentiation (AD), tangent linear (TLM) and adjoint (ADM) model code for the
coupled MITgcm ocean/sea-ice system. This second part of a two-part paper de-
Preprint submitted to Elsevier
19 January 2010
scribes aspects of the adjoint model. The adjoint ocean and sea ice model is used
to calculate transient sensitivities of solid (ice & snow) freshwater export through
Lancaster Sound in the Canadian Arctic Archipelago (CAA). The adjoint state provides a complementary view of the dynamics. In particular, the transient, multi-year
sensitivity patterns reflect dominant pathways and propagation timescales through
the CAA as resolved by the model, thus shedding light on causal relationships, in
the model, across the Archipelago. The computational cost of inferring such causal
relationships from forward model diagnostics alone would be prohibitive. The role of
the exact model trajectory around which the adjoint is calculated (and therefore of
the exactness of the adjoint) is exposed through calculations using free-slip vs noslip lateral boundary conditions. Effective ice thickness, sea surface temperature,
and precipitation sensitivities, are discussed in detail as examples of the coupled
sea-ice/ocean and atmospheric forcing control space. To test the reliability of the
adjoint, finite-difference perturbation experiments were performed for each of these
elements and the cost perturbations were compared to those “predicted” by the
adjoint. Overall, remarkable qualitative and quantitative agreement is found. In
particular, the adjoint correctly “predicts” a seasonal sign change in precipitation
sensitivities. A physical mechanism for this sign change is presented. The availability
of the coupled adjoint opens up the prospect for adjoint-based coupled ocean/sea-ice
state estimation.
Key words: NUMERICAL SEA ICE MODELING, VISCOUS-PLASTIC
RHEOLOGY, COUPLED OCEAN AND SEA ICE MODEL, STATE
ESTIMATION, ADJOINT MODELING, CANADIAN ARCTIC
ARCHIPELAGO, SEA-ICE EXPORT, SENSITIVITIES
1
corresponding author, email: [email protected],
ph: +1-617-253-5259, fax: +1-617-253-4464
2
1
1
Introduction
2
This is the second part of a two-part paper (see Losch et al., 2010, for part 1)
3
describing the development of a sea-ice model for use in adjoint-based regional
4
and global coupled ocean/sea-ice state estimation and sensitivity studies. It
5
has been shown (e.g., Marotzke et al., 1999, Galanti et al., 2002, Galanti and
6
Tziperman, 2003, Köhl, 2005, Bugnion et al., 2006a,b, Losch and Heimbach,
7
2007, Moore et al., 2009, Veneziani et al., 2009) that adjoints are very valuable
8
research tools to investigate sensitivities of key model diagnostics with respect
9
to a wide variety of model inputs. Furthermore, increasing sophistication of
10
global-scale as well as regional, polar state estimation systems, which attempt
11
to synthesize observations and models (e.g., Miller et al., 2006, Duliere and
12
Fichefet, 2007, Lisaæter et al., 2007, Stark et al., 2008, Stoessel, 2008, Pan-
13
teleev et al., 2010) call for adequate representation of sea-ice in the model
14
so as to represent relevant processes and to incorporate sea-ice observations
15
in constraining the coupled system. The estimation system developed within
16
the Estimating the Circulation and Climate of the Ocean (ECCO) consortium
17
is based on the adjoint or Lagrange multiplier method (LMM) (e.g., Wun-
18
sch, 2006). It thus relies heavily on the availability of an adjoint model of
19
the underlying general circulation model (Stammer et al., 2002a, Wunsch and
20
Heimbach, 2007, Heimbach and Wunsch, 2007, and references therein).
21
Collectively, the lack, until recently, of an interactive sea-ice component in the
22
ECCO approach, the experience gained (and the success) with the ocean-only
23
problem, the importance of representing polar-subpolar interactions in ECCO-
24
type calculations, and the need to incorporate sea-ice observations, make a
25
compelling case for the development of a new sea-ice model. While many of
3
26
its features are “conventional” (yet for the most part state-of-the-art), the
27
ability to generate efficient adjoint code for coupled ocean/sea-ice simulations
28
by means of automatic (or algorithmic) differentiation (AD: Griewank and
29
Walther, 2008) sets this model apart from existing models. Whereas a few
30
existing models (Kim et al., 2006a,b) allow for the generation of tangent linear
31
code for sea-ice-only model configurations by means of the so-called forward-
32
mode AD, until very recently none of these were capable of producing efficient
33
adjoint code by means of reverse-mode AD, let alone in a coupled ocean/sea-
34
ice configuration, which can propagate sensitivities back and forth between the
35
two components. Such coupled sensitivity propagation is highly desirable as it
36
permits sea-ice and ocean observations to be used as simultaneous constraints
37
on each other, yielding a truly coupled estimation problem.
38
In addition to the coupled ocean and sea ice system described here, one other
39
coupled adjoint system has recently become available for an Arctic configu-
40
ration and was used to isolate dominant mechanisms responsible for the 2007
41
Arctic sea-ice minimum (Kauker et al., 2009). The availability of two adjoint
42
modeling systems holds the prospect (for the first time) to compare adjoint
43
calculations for a specific regional setup using different models. This is a pro-
44
posed future objective within the Arctic Ocean Model Intercomparison Project
45
(AOMIP).
46
The MITgcm sea ice model was described in detail in Part 1. It borrows
47
many components from current-generation sea ice models, but these compo-
48
nents were reformulated on an Arakawa C grid in order to match the MITgcm
49
oceanic grid, and they were modified in many ways to permit efficient and
50
accurate automatic differentiation. Part 1 provided a detailed discussion of
51
the effect on the solution of various choices in the numerical implementation,
4
52
in particular related to sea-ice dynamics. Such sensitivities are structural or
53
configuration-based, rather than exploring a continuous space of control vari-
54
ables, and are best assessed in separate forward calculations. Special emphasis
55
was put on aspects of the sea-ice dynamics, such as the use of different solvers
56
for sea-ice rheology, the formulation of these solvers on an Arakawa B vs C
57
grid, and the use of free-slip vs no-slip lateral boundary conditions. These
58
scenarios provide important baseline trajectories for the adjoint calculations
59
presented here, as they underscore the importance of the underlying state,
60
around which the model is linearized.
61
Part 2 focusses on the adjoint component, its generation by means of AD,
62
its reliability, and on the interpretability of adjoint variables. We investigate
63
sensitivities of sea-ice transport through narrow straits, for which rheology
64
configurations become crucial, and the dependence of adjoint sensitivities on
65
the choices of configuration elements described in Part 1. The power of the
66
adjoint is demonstrated through a case study of sea-ice transport through the
67
Canadian Arctic Archipelago (CAA) measured in terms of its export through
68
Lancaster Sound. Thereby we complement a recent study by Lietaer et al.
69
(2008) that focused on the role of narrow straits in this region in setting the
70
sea-ice mass balance in the Arctic. While Part 1 of the present paper showed
71
that different grids, different rheologies, and different lateral boundary con-
72
ditions lead to considerable differences in the computed sea-ice state, here
73
we show that adjoint sensitivities may differ substantially depending on the
74
baseline trajectory, around which the model is linearized. The present analysis
75
provides important complementary information to the configuration sensitiv-
76
ities of Part 1: it enables us to extend analysis to continuous parameters, it
77
demonstrates the degree of detail the adjoint variables contain, and it exposes
5
78
causal relationships.
79
The remainder of Part 2 is organized as follows: Section 2 provides some details
80
of the adjoint code generation by means of AD. Multi-year transient sensitiv-
81
ities of sea-ice export through the Canadian Arctic Archipelago are presented
82
in Section 3. Extending the analysis of Part 1, we assess the consequences of
83
the choices of lateral boundary conditions on the ensuing model sensitivities
84
for various control variables. Discussion and conclusions are in Section 4.
85
2
86
There is now a growing body of literature on adjoint applications in oceanog-
87
raphy and adjoint code generation via AD. We therefore limit the description
88
of the method to a brief summary. For discrete problems as considered here,
89
the adjoint model operator (ADM) is the transpose of the Jacobian or tangent
90
linear model operator (TLM) of the full (in general nonlinear) forward model
91
(NLM), in this case, the MITgcm coupled ocean and sea ice model. Consider
92
a scalar-valued model diagnostics, referred to as objective function, and an
93
m-dimensional control space (referred to as space of independent variables)
94
whose elements we may wish to perturb to assess their impact on the objective
95
function. In the context of data assimilation the objective function may be the
96
least-square model vs. data misfit, whereas here, we may choose almost any
97
function that is (at least piece-wise) differentiable with respect to the control
98
variables. Here, we shall be focusing on the solid freshwater export through
99
Lancaster Sound.
100
MITgcm adjoint code generation
Two- and three-dimensional control variables used in the present study are
6
Table 1
List of control variables used. The controls are either part of the oceanic (O) or seaice (I) state, or time-varying elements of the atmospheric (A) boundary conditions.
component
variable
dim.
time
O
temperature
3-D
init.
O
salinity
3-D
init.
O
vertical diffusivity
3-D
const.
I
concentration
2-D
init.
I
thickness
2-D
init.
A
air temperature
2-D
2-day
A
specific humidity
2-D
2-day
A
shortwave radiation
2-D
2-day
A
precipitation
2-D
2-day
A
zonal windspeed
2-D
2-day
A
merid. windspeed
2-D
2-day
101
listed in Table 1. They consist of two- or three-dimensional fields of initial
102
conditions of the ocean or sea-ice state, ocean vertical mixing coefficients,
103
and time-varying surface boundary conditions (surface air temperature, spe-
104
cific humidity, shortwave radiation, precipitation, zonal and meridional wind
105
speed). The TLM computes the objective functions’s directional derivatives
106
for a given perturbation direction. In contrast, the ADM computes the the full
107
gradient of the objective function with respect to all control variables. When
7
108
combined, the control variables may span a potentially high-dimensional, e.g.,
109
O(108 ), control space. At this problem dimension, perturbing individual pa-
110
rameters to assess model sensitivities is prohibitive. By contrast, transient
111
sensitivities of the objective function to any element of the control and model
112
state space can be computed very efficiently in one single adjoint model inte-
113
gration, provided an adjoint model is available.
114
Conventionally, adjoint models are developed “by hand” through implement-
115
ing code which solves the adjoint equations (e.g., Marchuk, 1995, Wunsch,
116
1996) of the given forward equations. The burden of developing “by hand” an
117
adjoint model in general matches that of the forward model development. The
118
substantial extra investment often prevents serious attempts at making avail-
119
able adjoint components of sophisticated models. Furthermore, the work of
120
keeping the adjoint model up-to-date with its forward parent model matches
121
the work of forward model development. The alternative route of rigorous ap-
122
plication of AD tools has proven very successful in the context of MITgcm
123
ocean modeling applications.
124
Certain limitations regarding coding standards apply. Although they vary from
125
tool to tool, they are similar across various tools and are related to the abil-
126
ity to efficiently reverse the flow through the model. Work is thus required
127
initially to make the model amenable to efficient adjoint code generation for
128
a given AD tool. This part of the adjoint code generation is not automatic
129
(we sometimes refer to it as semi-automatic) and can be substantial for legacy
130
code, in particular if the code is badly modularized and contains many ir-
131
reducible control flows (e.g., GO TO statements, which are considered bad
132
coding practice anyways).
8
133
It is important to note, nevertheless, that once the tailoring of the model code
134
to the AD code is in place, any further forward model development can be
135
easily incorporated in the adjoint model via AD. Furthermore, the notion of
136
the adjoint is misleading, since the structure of the adjoint depends critically
137
on the control problem posed (a passive tracer sensitivity yields a very different
138
Jacobian to an active tracer sensitivity). A clear example of the dependence
139
of the structure of the adjoint model on the control problem is the extension
140
of the MITgcm adjoint model to a configuration that uses bottom topography
141
as a control variable (Losch and Heimbach, 2007). The AD approach enables
142
a much more thorough and smoother adjoint model extension than would be
143
possible via hand-coding.
144
The adjoint model of the MITgcm has become an invaluable tool for sensitivity
145
analysis as well as for state estimation (for a recent overview and summary, see
146
Heimbach, 2008). AD also enables a large variety of configurations and studies
147
to be conducted with adjoint methods without the onerous task of modifying
148
the adjoint of each new configuration by hand. Giering and Kaminski (1998)
149
discuss in detail the advantages of AD.
150
The AD route was also taken in developing and adapting the sea-ice compo-
151
nent of the MITgcm, so that tangent linear and adjoint components can be ob-
152
tained and kept up to date without excessive effort. As for the TLM and ADM
153
components of the MITgcm ocean model, we rely on the AD tool “Transfor-
154
mation of Algorithms in Fortran” (TAF) developed by Fastopt (Giering and
155
Kaminski, 1998) to generate TLM and ADM code of the MITgcm sea ice
156
model (for details see Marotzke et al., 1999, Heimbach et al., 2005). Note that
157
for the ocean component, we are now also able to generate efficient derivative
158
code using the new open-source tool OpenAD (Utke et al., 2008). Appendix
9
159
A provides details of adjoint code generation for the coupled ocean and sea
160
ice MITgcm configuration.
161
Since conducting this study, further changes to the thermodynamic formula-
162
tion have been implemented, which improve certain aspects of forward and
163
adjoint model behavior. These changes are discussed in detail in Fenty (2010)
164
along with application of the coupled ocean and sea ice MITgcm adjoint to
165
estimating the state of the Labrador Sea during 1996–1997.
166
To conclude this section, we emphasize the coupled nature of the MITgcm
167
ocean and sea ice adjoint. Figure 1 illustrates the relationship between control
168
variables and the objective function J when using the tangent linear model
169
(TLM, left diagram), or the adjoint model (ADM, right diagram). The control
170
space consists of atmospheric perturbations (e.g., surface air temperature δTa
171
and precipitation δp), sea-ice perturbations (e.g., ice concentration δc and ice
172
thickness δh), and oceanic perturbations (e.g., potential temperature δΘ and
173
salinity δS). The left diagram depicts how each perturbation of an element of
174
the control space leads to a perturbed objective function δJ via the TLM. In
175
contrast, the right diagram shows the reverse propagation of adjoint variables
176
or sensitivities labeled with an asterisk (∗ ). The notation reflects the fact that
177
adjoint variables are formally Lagrange multipliers or elements of the model’s
178
co-tangent space (as opposed to perturbations which are formally elements of
179
the model’s tangent space). For example, δ ∗ c refers to the gradient ∂J/∂c. The
180
aim of the diagram is to show (in a very simplified way) two things. First, it
181
depicts how sensitivities of an objective function (e.g., sea ice export as will be
182
defined later) to changes in, e.g., ice concentration ∂J/∂c is affected by changes
183
in, e.g., ocean temperature via the chain rule ∂J/∂Θ = ∂J/∂c · ∂c/∂Θ. The
184
adjoint model thus maps the adjoint objective function state to the adjoint
10
185
sea-ice state, and from there to the coupled adjoint oceanic and surface atmo-
186
spheric state. Second, it can be seen that the ADM maps from a 1-dimensional
187
state (δ ∗ J) to a multi-dimensional state (δ ∗ c, δ ∗ h, δ ∗ Ta , δ ∗ p, δ ∗ Θ, δ ∗ S) whereas
188
the TLM maps from a multi-dimensional state (δc, δh, δTa , δp, δΘ, δS) to a
189
1-dimensional state (δJ). This is the reason why only one adjoint integration
190
is needed to assemble all the gradients of the objective function while one
191
tangent linear integrations per dimension of the control space is needed to as-
192
semble the same gradient. Rigorous derivations can be found in, for example,
193
Chapter 5 of the MITgcm documentation (Adcroft et al., 2002), in Wunsch
194
(2006), or in Giering and Kaminski (1998).
195
3
196
A case study: Sensitivities of sea-ice export through Lancaster
Sound
197
We demonstrate the power of the adjoint method in the context of investigat-
198
ing sea-ice export sensitivities through Lancaster Sound (LS). The rationale
199
for this choice is to complement the analysis of sea-ice dynamics in the pres-
200
ence of narrow straits of Part 1. LS is one of the main paths of sea ice export
201
through the Canadian Arctic Archipelago (CAA) (Melling, 2002, Prinsenberg
202
and Hamilton, 2005, Michel et al., 2006, Münchow et al., 2006, Kwok, 2006).
203
Figure 2 shows the intricate local geography of CAA straits, sounds, and
204
islands. Export sensitivities reflect dominant pathways through the CAA, as
205
resolved by the model. Sensitivity maps provide a very detailed view of various
206
quantities affecting the sea-ice export (and thus the underlying propagation
207
pathways). A caveat of this study is the limited resolution, which is not ad-
208
equate to realistically simulate the CAA. For example, while the dominant
11
δTa
δ ∗ Ta
δp
δ∗p
atmosphere
atmosphere
δc
δΘ
δh
δJ
δ∗c
δ∗h
δ∗J
sea−ice
sea−ice
ocean
ocean
δS
δ∗Θ
δ∗S
Fig. 1. This diagram illustrates how the tangent linear model (TLM, left panel)
maps perturbations in the oceanic, atmospheric, or sea-ice state into a perturbation
of the objective function δJ, whereas the adjoint model (ADM, right panel) maps
the adjoint objective function δ∗ J (seeded to unity) into the adjoint sea-ice state,
which is a sensitivity or gradient, e.g., δ∗ c = ∂J/∂c, and into the coupled ocean and
atmospheric adjoint states. The TLM computes how a perturbation in one input
affects all outputs whereas the adjoint model computes how one particular output
is affected by all inputs.
209
circulation through LS is toward the East, there is a small Westward flow to
210
the North, hugging the coast of Devon Island, which is not resolved in our
211
simulation. Nevertheless, the focus here is on elucidating model sensitivities
212
in a general way. For any given simulation, whether deemed “realistic” or
213
not, the adjoint provides exact model sensitivities, which help inform whether
214
hypothesized processes are actually borne out by the model dynamics. Note
215
that the resolution used in this study is at least as good as or better than the
216
resolution used for IPCC-type calculations.
12
84
N
N
rro
Str
ait
Na
res
d
re
Isla
n
sm
e
h.
nC
mM
Ba
Elle
o
72
Penny Strait
W
Bya
o
0
arti
ure
M'Cl Strait
12
ille
Massey Sound
Ballantyne Str
ait
o
oN
oN
81
78
oN
75
Vis
co
So unt
un Me
d
lv
o
60 W
Prince Gustaf
Adolf Sea
w S Devo
nI
tra
sla
it
Lan nd
cas
ter
S
oun
d
Baffin Bay
10
0o
W
80 oW
Fig. 2. Map of the Canadian Arctic Archipelago with model coastlines and grid
(filled grey boxes are land). The black contours are the true coastlines as taken
from the GSHHS data base (Wessel and Smith, 1996). The gate at 82◦ W across
which the solid freshwater export is computed is indicated as black line.
217
3.1 The model configuration
218
The model domain is similar to the one described in Part 1. It is carved
219
out from the Arctic face of a global, eddy-admitting, cubed-sphere simulation
220
(Menemenlis et al., 2005) but with 36-km instead of 18-km grid cell width,
221
i.e., coarsened horizontal resolution compared to the configuration described
222
in Part 1. The vertical discretization is the same as in Part 1, i.e. the model
223
has 50 vertical depth levels, which are unevenly spaced, ranging from 10 m
224
layer thicknesses in the top 100 m to a maximum of 456 m layer thickness
13
225
at depth. The adjoint model for this configuration runs efficiently on 80 pro-
226
cessors, inferred from benchmarks on both an SGI Altix and on an IBM SP5
227
at NASA/ARC and at NCAR/CSL, respectively. Following a 4-year spinup
228
(1985 to 1988), the model is integrated for an additional four years and nine
229
months between January 1, 1989 and September 30, 1993. It is forced at the
230
surface using realistic 6-hourly NCEP/NCAR atmospheric state variables. The
231
objective function J is chosen as the “solid” freshwater export through LS,
232
at approximately 74◦ N, 82◦ W in Fig. 2, integrated over the final 12-month
233
period, i.e., October 1, 1992 to September 30, 1993. That is,
234
235
J =
1
ρf resh
Z
Sep 93
Oct 92
Z
LS
(ρ h c + ρs hs c) u ds dt,
(1)
236
is the mass export of ice and snow converted to units of freshwater. Further-
237
more, for each grid cell (i, j) of the section, along which the integral
238
is taken, c(i, j) is the fractional ice cover, u(i, j) is the along-channel ice drift
239
velocity, h(i, j) and hs (i, j) are the ice and snow thicknesses, and ρ, ρs , and
240
ρf resh are the ice, snow and freshwater densities, respectively. At the given
241
resolution, the section amounts to three grid points. The forward trajectory of
242
the model integration resembles broadly that of the model in Part 1 but some
243
details are different due to the different resolution and integration period.
244
For example, the differences in annual solid freshwater export through LS as
245
defined in eqn. (1) are smaller between no-slip and free-slip lateral boundary
246
conditions at higher resolution, as shown in Part 1, Section 4.3 (91±85 km3 y−1
247
and 77 ± 110 km3 y−1 for free-slip and no-slip, respectively, and for the C-grid
248
LSR solver; ± values refer to standard deviations of the annual mean) than
249
at lower resolution (116 ± 101 km3 y−1 and 39 ± 64 km3 y−1 for free-slip and
14
R
. . . ds
250
no-slip, respectively). The large range of these estimates emphasizes the need
251
to better understand the model sensitivities to lateral boundary conditions
252
and to different configuration details. We aim to explore these sensitivities
253
across the entire model state space in a comprehensive manner by means of
254
the adjoint model.
255
The adjoint model is the transpose of the tangent linear model operator. It
256
thus runs backwards in time from September 1993 to January 1989. During
257
this integration period, the Lagrange multipliers of the model subject to ob-
258
jective function (1) are accumulated. These Langrange multipliers are the
259
sensitivities, or derivatives, of the objective function with respect to each con-
260
trol variable and to each element of the intermediate coupled ocean and sea
261
ice model state variables. Thus, all sensitivity elements of the model state
262
and of the surface atmospheric state are available for analysis of the tran-
263
sient sensitivity behavior. Over the open ocean, the adjoint of the Large and
264
Yeager (2004) bulk formula scheme computes sensitivities to the time-varying
265
atmospheric state. Specifically, ocean sensitivities propagate to air-sea flux
266
sensitivities, which are mapped to atmospheric state sensitivities via the bulk
267
formula adjoint. Similarly, over ice-covered areas, the sea-ice model adjoint
268
(rather than the bulk formula adjoint) converts surface ocean sensitivities to
269
atmospheric sensitivities.
270
3.2 Adjoint sensitivities
271
The most readily interpretable ice-export sensitivity is that to ice thickness,
272
∂J/∂(hc). Maps of transient sensitivities ∂J/∂(hc) are shown for free-slip
273
(Fig. 3) and for no-slip (Fig. 4) boundary conditions. Each figure depicts four
15
01−Oct−1992
80
75
70
02−Oct−1991
80
o
N
75
o
N
70
o
N
12
o
0 oW
60
o
N
o
N
o
N
12
W
0 oW
o
80 W
100 oW
75
70
80
N
75
o
N
70
o
N
12
o
0 oW
−0.2
60
−0.15
W
o
N
o
N
o
N
12
W
0 oW
60
o
o
100 oW
80 W
−0.1
o
02−Oct−1989
o
100 oW
W
o
80 W
100 oW
02−Oct−1990
80
o
60
−0.05
0
0.05
80 W
0.1
0.15
0.2
Fig. 3. Sensitivity ∂J/∂(hc) in m3 s−1 /m for four different times using free-slip lateral sea ice boundary conditions. The color scale is chosen to illustrate the patterns
of the sensitivities. The objective function (1) was evaluated between October 1992
and September 1993. Sensitivity patterns extend backward in time upstream of the
LS section.
274
sensitivity snapshots of the objective function J, starting October 1, 1992,
275
i.e., at the beginning of the 12-month averaging period, and going back in
276
time to October 2, 1989. As a reminder, the full period over which the adjoint
277
sensitivities are calculated is (backward in time) between September 30, 1993
278
and January 1, 1989.
279
The sensitivity patterns for ice thickness are predominantly positive. The in-
280
terpretation is that an increase in ice volume in most places west, i.e., “up-
281
stream”, of LS increases the solid freshwater export at the exit section. The
16
01−Oct−1992
80
75
70
02−Oct−1991
80
o
N
75
o
N
70
o
N
12
o
0 oW
60
o
N
o
N
o
N
12
W
0 oW
o
80 W
100 oW
75
70
80
N
75
o
N
70
o
N
12
o
0 oW
−0.2
60
−0.15
W
o
N
o
N
o
N
12
W
0 oW
60
o
o
100 oW
80 W
−0.1
o
02−Oct−1989
o
100 oW
W
o
80 W
100 oW
02−Oct−1990
80
o
60
−0.05
0
0.05
80 W
0.1
0.15
0.2
Fig. 4. Same as in Fig. 3 but for no-slip lateral sea ice boundary conditions.
282
transient nature of the sensitivity patterns is evident: the area upstream of
283
LS that contributes to the export sensitivity is larger in the earlier snapshot.
284
In the free-slip case, the sensivity follows (backwards in time) the dominant
285
pathway through Barrow Strait into Viscount Melville Sound, and from there
286
trough M’Clure Strait into the Arctic Ocean
287
ward from Viscount Melville Sound through Byam Martin Channel into Prince
288
Gustav Adolf Sea and through Penny Strait into MacLean Strait.
289
There are large differences between the free-slip and no-slip solutions. By
290
the end of the adjoint integration in January 1989, the no-slip sensitivities
2
2
. Secondary paths are north-
(the branch of the “Northwest Passage” apparently discovered by Robert McClure
during his 1850 to 1854 expedition; McClure lost his vessel in the Viscount Melville
Sound)
17
291
(Fig. 4) are generally weaker than the free slip sensitivities and hardly reach
292
beyond the western end of Barrow Strait. In contrast, the free-slip sensitivities
293
(Fig. 3) extend through most of the CAA and into the Arctic interior, both to
294
the West (M’Clure Strait) and to the North (Ballantyne Strait, Prince Gustav
295
Adolf Sea, Massey Sound). In this case the ice can drift more easily through
296
narrow straits and a positive ice volume anomaly anywhere upstream in the
297
CAA increases ice export through LS within the simulated 4-year period.
298
One peculiar feature in the October 1992 sensitivity maps are the negative
299
sensivities to the East and, albeit much weaker, to the West of LS. The former
300
can be explained by indirect effects: less ice eastward of LS results in less
301
resistance to eastward drift and thus more export. A similar mechanism might
302
account for the latter, albeit more speculative: less ice to the West means that
303
more ice can be moved eastward from Barrow Strait into LS leading to more
304
ice export.
305
The temporal evolution of several ice export sensitivities along a zonal axis
306
through LS, Barrow Strait, and Melville Sound (115◦ W to 80◦ W, averaged
307
across the passages) are depicted in Fig. 5 as Hovmoeller-type diagrams, that
308
is, as two-dimensional plots of sensitivities as a function of longitude and time.
309
Serving as examples for the ocean, sea-ice, and atmospheric forcing compo-
310
nents of the model, we depict, from top to bottom, the sensitivities to ice
311
thickness (hc), to ice and ocean surface temperature (SST), and to precipi-
312
tation (p) for free-slip (left column) and for no-slip (right column) ice drift
313
boundary conditions. The green line marks the starting time (1 Oct. 1992)
314
of the 12-month ice export objective function integration (Eqn. 1). Also in-
315
dicated are times when a perturbation in precipitation leads to a positive
316
(Apr. 1991) or to a negative (Nov. 1991) ice export anomaly (see also Fig.
18
free slip
no slip
Jul
3
93 Jan
2
∂J/∂(hc) [m2 s−1/m]
Jul
92 Jan
Jul
91 Jan
Jul
1
0
−1
−2
90 Jan
Jul
−3
89 Jan
Jul
0.3
93 Jan
0.2
∂J/∂SST [m2 s−1/K]
Jul
92 Jan
Jul
91 Jan
Jul
0.1
0
−0.1
−0.2
90 Jan
Jul
89 Jan
Jul
3
∂J/∂p [103 m2 s−1/ (m s−1)]
93 Jan
Jul
92 Jan
Jul
91 Jan
Jul
90 Jan
Jul
89 Jan
2
1
0
−1
−2
−3
o
110 W
o
100 W
o
90 W
o
110 W
o
100 W
o
90 W
Fig. 5. Time vs. longitude diagrams along the axis of Viscount Melville Sound,
Barrow Strait, and LS. The diagrams show the sensitivities (derivatives) of the solid
freshwater export J through LS (Fig. 2) with respect to ice thickness (hc, top), to
ice and ocean surface temperature (SST, middle), and to precipitation (p, bottom)
for free-slip (left) and for no-slip (right) boundary conditions. J was integrated over
the last year (period above green line). A precipitation perturbation during Apr.
1st. 1991 (dash-dottel line) or Nov. 1st 1991 (dashed line) leads to a positive or
negative export anomaly, respectively. Contours are of the normalized ice strength
19
P/P ∗ . Bars in the longitude axis indicates the flux gate at 82◦ W.
317
8). Each plot is overlaid with contours 1 and 3 of the normalized ice strength
318
P/P ∗ = (hc) exp[−C (1 − c)].
319
The Hovmoeller-type diagrams of ice thickness (top row) and SST (second
320
row) sensitivities are coherent: more ice in LS leads to more export and one
321
way to form more ice is by colder surface temperatures. In the free-slip case
322
the sensitivities spread out in “pulses” following a seasonal cycle: ice can prop-
323
agate eastward (forward in time) and thus sensitivities propagate westward
324
(backwards in time) when the ice strength is low in late summer to early au-
325
tumn (Fig. 6, bottom panels). In contrast, during winter, the sensitivities show
326
little to no westward propagation as the ice is frozen solid and does not move.
327
In the no-slip case the normalized ice strength does not fall below 1 during
328
the winters of 1991 to 1993 (mainly because the ice concentrations remain
329
near 100%, not shown). Ice is therefore blocked and cannot drift eastwards
330
(forward in time) through the Viscount Melville Sound, Barrow Strait, and
331
LS channel system. Consequently, the sensitivities do not propagate westward
332
(backwards in time) and the export through LS is only affected by local ice
333
formation and melting for the entire integration period.
334
It is worth contrasting the sensitivity diagrams of Fig. 5 with the Hovmoeller-
335
type diagrams of the corresponding state variables (Figs. 6 and 7). The sensi-
336
tivities show clear causal connections of ice motion over the years, that is, they
337
expose the winter arrest and the summer evolution of the ice. These causal
338
connections cannot easily be inferred from the Hovmoeller-type diagrams of
339
ice and snow thickness. This example illustrates the usefulness and comple-
340
mentary nature of the adjoint variables for investigating dynamical linkages
341
in the ocean/sea-ice system.
20
free slip
no slip
Jul
4.5
93 Jan
4
Jul
3.5
92 Jan
hc [m]
3
Jul
91 Jan
2.5
2
1.5
Jul
1
90 Jan
0.5
Jul
0
89 Jan
Jul
0.4
93 Jan
0.35
Jul
0.3
h c [m]
92 Jan
s
Jul
91 Jan
0.25
0.2
0.15
Jul
0.1
90 Jan
0.05
Jul
0
89 Jan
Jul
4.5
93 Jan
4
normalized ice strength
Jul
92 Jan
Jul
91 Jan
Jul
90 Jan
3
2.5
2
1.5
1
0.5
Jul
89 Jan
3.5
0
o
110 W
o
100 W
o
90 W
o
110 W
o
100 W
o
90 W
Fig. 6. Hovmoeller-type diagrams along the axis of Viscount Melville Sound, Barrow
Strait, and LS. The diagrams show ice thickness (hc, top), snow thickness (hs c,
middle), and normalized ice strength (P/P ∗ , bottom) for free-slip (left) and for
no-slip (right) sea ice boundary conditions. For orientation, each plot is overlaid
with contours 1 and 3 of the normalized ice strength. Green line is as in Fig. 5.
21
free slip
no slip
Jul
−1.72
93 Jan
−1.725
Jul
−1.73
−1.735
o
SST [ C]
92 Jan
Jul
91 Jan
Jul
−1.74
−1.745
−1.75
90 Jan
−1.755
Jul
−1.76
89 Jan
Jul
32
93 Jan
31.5
Jul
SSS [ppt]
92 Jan
Jul
91 Jan
31
30.5
30
Jul
29.5
90 Jan
Jul
29
89 Jan
−3
Jul
x 10
5
93 Jan
P−E+R [g/m /s]
Jul
2
92 Jan
Jul
91 Jan
Jul
0
90 Jan
Jul
89 Jan
−5
o
110 W
o
100 W
o
90 W
o
110 W
o
100 W
o
90 W
Fig. 7. Same as in Fig. 6 but for SST (top panels), SSS (middle panels), and precipitation minus evaporation plus runoff, P − E + R (bottom panels).
22
342
The sensitivities to precipitation are more complex. To first order, they have
343
an oscillatory pattern with negative sensitivity (more precipitation leads to
344
less export) between roughly September and December and mostly positive
345
sensitivity from January through June (sensitivities are negligible during the
346
summer). Times of positive sensitivities coincide with times of normalized
347
ice strengths exceeding values of 3. This pattern is broken only immediatly
348
preceding the evaluation period of the ice export objective function in 1992.
349
In contrast to previous years, the sensitivity is negative between January and
350
August 1992 and east of 95◦ W.
351
We attempt to elucidate the mechanisms underlying these precipitation sen-
352
sitivities in Section 3.4 in the context of forward perturbation experiments.
353
3.3 Forward perturbation experiments
354
Applying an automatically generated adjoint model under potentially highly
355
nonlinear conditions incites the question to what extent the adjoint sensi-
356
tivities are “reliable” in the sense of accurately representing forward model
357
sensitivities. Adjoint sensitivities that are physically interpretable provide a
358
partial answer but an independent, quantitative test is needed to gain confi-
359
dence in the calculations. Such a verification can be achieved by comparing
360
adjoint-derived gradients with ones obtained from finite-difference perturba-
361
tion experiments. Specifically, for a control variable u of interest, we can read-
362
ily calculate an expected change δJ in the objective function for an applied
363
perturbation δu over domain A based on adjoint sensitivities ∂J/∂u:
23
364
365
δJ =
Z
A
∂J
δu dA
∂u
(2)
366
Alternatively, we can infer the magnitude of the objective perturbation δJ
367
without use of the adjoint. Instead we apply the same perturbation δu to the
368
control space over the same domain A and integrate the forward model. The
369
perturbed objective function is
370
371
δJ = J(u + δu) − J(u).
372
The degree to which Eqns. (2) and (3) agree depends both on the magnitude
373
of perturbation δu and on the length of the integration period.
374
We distinguish two types of adjoint-model tests. First there are finite differ-
375
ence tests performed over short time intervals, over which the assumption of
376
linearity is expected to hold, and where individual elements of the control vec-
377
tor are perturbed. We refer to these tests as gradient checks. Gradient checks
378
are performed on a routine, automated basis for various MITgcm verifica-
379
tion setups, including verification setups that exercise coupled ocean and sea
380
ice model configurations. These automated tests insure that updates to the
381
MITgcm repository do not break the differentiability of the code.
382
A second type of adjoint-model tests is finite difference tests performed over
383
longer time intervals and where a whole area is perturbed, guided by the ad-
384
joint sensitivity maps, in order to investigate physical mechanisms. The exam-
385
ples discussed herein and summarized in Table 2 are of this second type of sen-
386
sitivity experiments. For nonlinear models, the deviations between Eqns. (2)
387
and (3) are expected to increase both with perturbation magnitude as well as
24
(3)
Table 2
Summary of forward perturbation experiments and comparison of adjoint-based
and finite-difference-based objective function sensitivities. All perturbations were
applied to a region centered at 101.24◦ W, 75.76◦ N. The reference value for ice and
snow export through LS is J0 = 69.6 km3 /yr. For perturbations to the time-varying
precipitation p the perturbation interval is indicated by ∆t.
δJ(adj.)
km3 /yr
δJ(f wd.)
km3 /yr
% diff.
0.5 m
0.98
1.1
11
0.5◦ C
-0.125
-0.108
16
1-Apr-91 10 dy
1.6·10−7 m/s
0.185
0.191
3
p
1-Nov-91 10 dy
1.6·10−7 m/s
-0.435
-1.016
57
ATM3
p
1-Apr-91 10 dy -1.6·10−7 m/s
-0.185
-0.071
62
ATM4
p
1-Nov-91 10 dy -1.6·10−7 m/s
0.435
0.259
40
exp.
variable
time
∆t
δu
ICE1
hc
1-Jan-89
init.
OCE1
SST
1-Jan-89
init.
ATM1
p
ATM2
388
with integration time.
389
Comparison between finite-difference and adjoint-derived ice-export perturba-
390
tions show remarkable agreement for initial value perturbations of ice thick-
391
ness (ICE1) or sea surface temperature (OCE1). Deviations between perturbed
392
objective function values remain below 16% (see Table 2). Figure 8 depicts
393
the temporal evolution of perturbed minus unperturbed monthly ice export
394
through LS for initial ice thickness (top panel) and SST (middle panel) pertur-
395
bations. In both cases, differences are confined to the melting season, during
396
which the ice unlocks and which can lead to significant export. Large differ-
397
ences are seen during (but are not confined to) the period during which the
398
ice export objective function J is integrated (grey box). As “predicted” by
25
0.2
0.15
0.1
0.05
0
−0.05
Jan−92
Apr
Jul
Oct
Jan−93
Apr
Jul
Oct
Jan−94
Apr
Jul
Oct
Jan−93
Apr
Jul
Oct
Jan−94
0.01
0
−0.01
−0.02
−0.03
−0.04
Jan−92
0.06
1−Nov−1991 perturbation
1−Apr−1991 perturbation
3
Difference in freshwater export [km /y]
3
Difference in freshwater export [km /y]
3
Difference in freshwater export [km /y]
0.3
0.25
0.04
0.02
0
−0.02
−0.04
−0.06
Jan−92
Apr
Jul
Oct
Jan−93
Apr
Jul
Oct
Jan−94
Fig. 8. Difference in monthly solid freshwater export at 82◦ W between perturbed
and unperturbed forward integrations. From top to bottom, perturbations are initial
ice thickness (ICE1 in Table 2), initial sea-surface temperature (OCE1), and precipitation (ATM1 and ATM2). The grey box indicates the period during which the ice
export objective function J is integrated, and reflects the integrated anomalies in
Table 2.
399
the adjoint, the two curves are of opposite sign and scales differ by almost an
400
order of magnitude.
401
3.4 Sign change of precipitation sensitivities
402
Our next goal is to explain the sign and magnitude changes through time of
403
the transient precipitation sensitivities. To investigate this, we have carried
26
404
out the following two perturbation experiments: (i) an experiment labeled
405
ATM1, in which we perturb precipitation over a 10-day period between April
406
1 and 10, 1991, coincident with a period of positive adjoint sensitivities, and
407
(ii) an experiment labeled ATM2, in which we apply the same perturbation
408
over a 10-day period between November 1 and 10, 1991, coincident with a
409
period of negative adjoint sensitivities. The perturbation magnitude chosen
410
is δu = 1.6 × 10−7 m/s, which is of comparable magnitude with the stan-
411
dard deviation of precipitation. The perturbation experiments confirm the
412
sign change when perturbing in different seasons. We observe good quantita-
413
tive agreement for the April 1991 case and a 50% deviation for the November
414
1991 case. The discrepancy between the finite-difference and adjoint-based
415
sensitivity estimates results from model nonlinearities and from the multi-
416
year integration period. To support this statement, we repeated perturba-
417
tion experiments ATM1 and ATM2 but applied a perturbation with opposite
418
sign, i.e., δu = −1.6 × 10−7 m/s (experiments ATM3 and ATM4 in Table
419
2). For negative δu, both perturbation periods lead to about 50% discrepan-
420
cies between finite-difference and adjoint-derived ice export sensitivities. The
421
finite-difference export changes are different in amplitude for positive and for
422
negative perturbations, confirming that model nonlinearities start to impact
423
these calculations.
424
These experiments constitute severe tests of the adjoint model in the sense
425
that they push the limit of the linearity assumption. Nevertheless, the results
426
confirm that adjoint sensitivities provide useful qualitative, and, within cer-
427
tain limits, quantitative information of comprehensive model sensitivities that
428
cannot realistically be computed otherwise.
429
To investigate in more detail the oscillatory behavior of precipitation sen27
perturbation on 1991/11/01
perturbation on 1991/04/01
Jul
0.1
(hc) (m)
1993 Jan
Jul
1992 Jan
Jul
0
−0.1
1991 Jan
(hsnowc) (m)
Jul
1993 Jan
Jul
1992 Jan
Jul
0.01
0
−0.01
1991 Jan
SST (10−6ϒC)
Jul
1993 Jan
Jul
1992 Jan
Jul
1
0
−1
shortwave (W/m2)
1991 Jan
Jul
1993 Jan
Jul
1992 Jan
Jul
1991 Jan
o
110 W
o
100 W
o
o
90 W
110 W
o
100 W
0.5
0
−0.5
o
90 W
Fig. 9. Same as in Fig. 6 but restricted to the period 1991–1993 and for the differences in (from top to bottom) ice thickness (hc), snow thickness (hsnow c), sea–
surface temperature (SST), and shortwave radiation (for completeness) between a
perturbed and unperturbed run in precipitation of 1.6 × 10−1 m s−1 on November
1, 1991 (left panels) and on April 1, 1991 (right panels). The vertical line marks the
position where the perturbation was applied.
430
sitivities we have plotted differences in ice thickness, snow thicknesses, and
431
SST, between perturbed and unperturbed simulations along the LS axis as a
432
function of time. Figure 9 shows how the small localized perturbations of pre-
433
cipitation are propagated, depending on whether applied during early winter
434
(November, left column) or late winter (April, right column). More precipation
435
leads to more snow on the ice in all cases. However, the same perturbation in
436
different seasons has an opposite effect on the solid freshwater export through
437
LS. Both the adjoint and the perturbation results suggest the following mech28
438
anism to be at play:
439
• More snow in November (on thin ice) insulates the ice by reducing the
440
effective conductivity and thus the heat flux through the ice. This insulating
441
effect slows down the cooling of the surface water underneath the ice. In
442
summary, more snow early in the winter limits the ice growth from above
443
and below (negative sensitivity).
444
• More snow in April (on thick ice) insulates the ice against melting. Short-
445
wave radiation cannot penetrate the snow cover and snow has a higher
446
albedo than ice (0.85 for dry snow and 0.75 for dry ice in our simulations);
447
thus it protects the ice against melting in the spring, more specifically, after
448
January, and it may lead to more ice in the following growing season.
449
A secondary effect is the accumulation of snow, which increases the exported
450
volume. The feedback from SST appears to be negligible because there is little
451
connection of anomalies beyond a full seasonal cycle.
452
We note that the effect of snow vs rain seems to be irrelevant in explaining
453
positive vs negative sensitivity patterns. In the current implementation, the
454
model differentiates between snow and rain depending on the thermodynamic
455
growth rate of sea ice; when it is cold enough for ice to grow, all precipitation
456
is assumed to be snow. The surface atmospheric conditions most of the year in
457
the Lancaster Sound region are such that almost all precipitation is treated as
458
snow, except for a short period in July and August; even then, air temperatures
459
are only slightly above freezing.
460
Finally, the negative sensitivities to precipitation between 95◦ W and 85◦ W
461
during the spring of 1992, which break the oscillatory pattern, may also be
462
explained by the presence of snow: in an area of large snow accumulation
29
463
(almost 50 cm: see Fig. 6, middle panel), ice cannot melt and it tends to block
464
the channel so that ice coming from the West cannot pass, thus leading to less
465
ice export in the next season. The reason why this is true for the spring of
466
1992 but not for the spring of 1991 is that by then the high sensitivites have
467
propagated westward out of the area of thick snow and ice around 90◦ W.
468
4
469
In this study we have extended the MITgcm adjoint modeling capabilities to
470
a coupled ocean and sea-ice configuration. The key development is a dynamic
471
and thermodynamic sea-ice model akin to most state-of-the-art models but
472
that is amenable to efficient, exact, parallel adjoint code generation via au-
473
tomatic differentiation. At least two natural lines of applications are made
474
possible by the availability of the adjoint model: (i) use of the coupled ad-
475
joint modeling capabilities for comprehensive sensitivity calculations of the
476
ocean/sea-ice system at high Northern and Southern latitudes and (ii) exten-
477
sion of the ECCO state estimation infrastructure to derive estimates that are
478
constrained both by ocean and by sea-ice observations.
479
The power of the adjoint method was demonstrated through a multi-year
480
sensitivity calculation of solid freshwater (sea-ice and snow) export through
481
Lancaster Sound in the Canadian Arctic Archipelago (CAA). The region was
482
chosen so as to complement the forward-model study presented in Part 1,
483
which examined the impact of rheology and dynamics on sea-ice drift through
484
narrow straits. The transient adjoint sensitivities reveal dominant pathways of
485
sea-ice propagation through the CAA. They clearly expose causal, time-lagged
486
relationships between ice export and various ocean, sea-ice, and atmospheric
Discussion and conclusion
30
487
variables of the coupled system. The computational cost of establishing all
488
these relationships through pure forward calculations would be prohibitive.
489
The sensitivity patterns (and thus causal relationships) differ substantially,
490
depending on which lateral ice drift boundary condition (free-slip or no-slip) is
491
imposed. Our results indicate that for the coarse-resolution configuration used
492
here the free-slip boundary condition results in swifter ice movement and in a
493
much larger region of influence than does the no-slip boundary condition. Note
494
though that this statement may not hold for simulations at higher resolution.
495
The present calculations confirm some expected responses, for example, the in-
496
crease in ice export with increasing ice thickness and the decrease in ice export
497
with increasing sea surface temperature. They also reveal mechanisms which,
498
although plausible, cannot be readily anticipated. As an example we presented
499
precipitation sensitivities, which exhibit an annual oscillatory behavior, with
500
negative sensitivities prevailing throughout the fall and early winter and pos-
501
itive sensitivities from late winter though spring. This behavior can be traced
502
to the different impact of snow accumulation over ice, depending on the stage
503
of ice evolution. For growing ice, snow accumulation suppresses ice growth
504
(negative sensitivity) whereas for melting ice, snow accumulation suppresses
505
ice melt (positive sensitivity). A secondary effect is the snow accumulation
506
on downstream ice export (positive sensitivity). Differences between snow and
507
rain seem negligible in our case study, since precipitation is in the form of
508
snow for an overwhelming part of the year.
509
Given the automated nature of adjoint code generation and the nonlinearity
510
of the problem when considered over sufficiently long time scales, indepen-
511
dent tests are needed to gain confidence in the adjoint solutions. We have
512
presented such tests in the form of finite difference experiments, guided by
31
513
the adjoint solution, and we compared objective function differences inferred
514
from forward perturbation experiments with differences inferred from adjoint
515
sensitivity information. We found very good quantitative agreement for initial
516
ice thickness and for sea surface temperature perturbations.
517
As described above, sensitivities to precipitation show an annual oscillatory
518
behavior, which is confirmed by forward perturbation experiments. In terms
519
of amplitude, precipitation shows a larger deviation (order of 50%) between
520
adjoint-based and finite-difference-based estimates of ice and snow transport
521
sensitivity through Lancaster Sound. Furthermore, finite difference perturba-
522
tions exhibit an asymmetry between positive and negative perturbations of
523
equal size. This points to the fact that, on multi-year time scales, nonlinear
524
effects can no longer be ignored and it indicates a limit to the usefulness of
525
the adjoint sensitivity information.
526
Given the urgency of understanding cryospheric changes, adjoint applications
527
are emerging as powerful research tools, e.g., the study of Kauker et al. (2009)
528
who attempt to isolate dominant mechanisms responsible for the 2007 Arctic
529
sea-ice minimum, and the study of Heimbach and Bugnion (2009) who demon-
530
strate how to infer Greenland ice sheet volume sensitivities from a large-scale
531
ice sheet adjoint model. The results of the present study encourage application
532
of the MITgcm coupled ocean/sea-ice adjoint system to a variety of sensitivity
533
studies of Arctic and Southern Ocean climate variability. The system has ma-
534
tured to a stage where coupled ocean/sea-ice estimation becomes feasible. For
535
the limited domain of the the Labrador Sea, single-year estimates have indeed
536
successfully been produced by Fenty (2010) for the mid-1990s and mid-2000s,
537
and will be reported elsewhere. Steps both toward a full Arctic and a global
538
system are now within reach. The prospect of using observations of one com32
539
ponent (e.g., daily sea-ice concentration) to constrain the other component
540
(near-surface ocean properties) through the information propagation of the
541
adjoint holds promise in deriving better, dynamically consistent estimates of
542
the polar environments.
543
A
544
TAF (Giering and Kaminski, 1998) and OpenAD (Utke et al., 2008) are source-
545
to-source transformation tools, which take the Fortran source code of the
546
nonlinear parent model (NLM) and generate Fortran code for the derivative
547
model once the control space and objective function have been specified. The
548
specification is an important step. It determines, in part, the structure of the
549
TLM and ADM. For different control problems the TLM and ADM may be
550
different, underlining the advantage of AD over hand-coding. At a basic level,
551
the AD tool knows the derivative expression for all intrinsic Fortran functions
552
(+,-,*,/,SQRT,SIN, etc.) and it readily produces line-by-line tangent linear
553
code. The full tangent linear model is assembled by rigorous application of the
554
chain rule (and the product rule) to the derivative line expressions. The adjoint
555
code can be derived from the line-by-line TLM code, formulated in matrix
556
form, by taking the matrix transpose and putting the resulting equations in
557
code form.
Issues of AD-based adjoint code generation
An example Consider as a simple example the line of code for calculating
the nonlinear bulk viscosity ζ from the shear viscosity η and from the ratio e
33
of the major to minor axis of the elliptical yield curve (Hibler, 1979):
ζ
.
e2
η =
(A.1)
The total derivative is
∂η
∂η
δζ +
δe
∂ζ
∂e
1
2ζ
= 2 δζ − 3 δe.
e
e
δη =
(A.2)
The variables δη, δe, and δζ are perturbations to the NLM state variables and
may be viewed as elements of the TLM state space. Rewriting this in matrix
form,
l+1



 δζ 










 δe 











=
δη













l

1
 δζ 
0 0




0
1
1 −2ζ
e2 e3








 δe 
0










,
(A.3)
,
(A.4)
δη
0
enables easy access to the transpose
l

 ∗ 
δ ζ 








 ∗ 
δ e








 ∗ 

=
δ η
0
1
e2
1
−2ζ
e3
00
0

1





0





l+1

 ∗ 
δ ζ 








 ∗ 
δ e








 ∗ 
δ η
where δ ∗ η, δ ∗ e, and δ ∗ ζ are sensitivities, i.e., elements of the ADM state space
or elements of the dual space to the TLM space. From this the adjoint code
can easily be read-off as follows:
1 ∗
δ η,
e2
2ζ
δ ∗ e = δ ∗ e − 3 δ ∗ η,
e
δ∗ζ = δ∗ζ +
δ ∗ η = 0.
34
(A.5)
558
Note that:
559
• the TLM propagates the impact of perturbing one input component (δη)
560
on all output variables (a directional derivative), here just one scalar-valued
561
objective function,
562
• the ADM accumulates the sensitivities of one output variable (here scalar-
563
valued) to all input components (a gradient),
564
• the required variables are elements of the model state, which are needed
565
to evaluate the derivative expression, including nonlinear functions and con-
566
ditional statements, and for the ADM they need to be available in reverse
567
order,
568
• Eqn. (A.5) states that the shear viscosity sensitivity δ ∗ η impacts the bulk
569
viscosity sensitivity δ ∗ ζ in a linear fashion, whereas it affects the ratio of the
570
elliptic yield curve δ ∗ e nonlinearly.
571
Required variables and checkpointing An important issue is the evalu-
572
ation of nonlinear or conditional expressions. In Eqn. (A.5) the values of e and
573
ζ are required to evaluate the derivative. AD tools solve this problem for TLM
574
generation by interlacing the TLM calculation with the NLM calculation. In
575
this way, the state of e and ζ is known just when it is needed by the TLM.
576
For the ADM the solution is significantly harder since the state of e and ζ are
577
required in reverse order of the NLM execution. Overcoming this discrepancy
578
is at the heart of implementing efficient adjoint code. The approach taken is
579
a blend of two extremes, which are (i) recomputing the required state, or (ii)
580
storing the whole state. For complex models, such as the MITgcm, neither of
35
581
these in their pure form is feasible but an optimal blend, known as adjoint
582
multi-level checkpointing, enables the generation of efficient and exact adjoint
583
code. For TAF, which implements a recompute-all behavior as default, the task
584
consists of targeting active variables in relevant, e.g., nonlinear or conditional,
585
code expressions, whose storing will avoid excessive required recomputations.
586
TAF directives enable the modeler to support TAF, alter its default behavior,
587
and render the adjoint more efficient. A detailed description in the context
588
of the MITgcm is given in Heimbach et al. (2005). Alternative approaches of
589
store-all by default are implemented in other tools (e.g., OpenAD, see Utke
590
et al., 2008).
591
Hand-coded adjoint models are sometimes considered as more efficient and
592
faster in view of the ability of the code developer to explicitly optimize the
593
code. This view needs to be formulated in more detail since it may be mislead-
594
ing in its general form. Significant code optimization can be obtained through
595
relaxing the requirement of provision of the exact model forward state at the
596
time of derivative evaluation. A code developer may decide that certain vari-
597
ables vary sufficiently slowly such that a time-mean (or, in certain applications,
598
an equilibrium state) constitute an appropriate substitute. While this substi-
599
tution leads indeed to significant adjoint model speed-up and/or memory re-
600
duction (omission of required recomputations) the comparison in performance
601
is no longer warranted. This is because similar interventions are possible for
602
AD generated code, in which recomputation or STORE/RESTORE opera-
603
tions may very well be replaced by similar approximations after the adjoint
604
code has been generated. Code efficiently is thus not primarily an AD issue,
605
but an issue of deciding which approximations to the exact linearizations are
606
permissible. These decisions are either made at the outset (for hand-coding),
36
607
or after the fact (for AD). Which of the routes of either simplifying an AD-
608
generated adjoint or extending an approximate hand-coded adjoint is simpler
609
and leads to more efficient adjoint models remains subject to research. Clearly,
610
providing means (e.g. through directives) of prescribing approximation levels
611
to AD tools would be an attractive feature of AD tools, and very useful for
612
large-scale applications.
613
Retaining scalability of the coupled ocean/sea-ice adjoint Another
614
aspect is ensuring scalability of the adjoint code on high performance computer
615
systems. Here again, automatic differentiation provides adjoint code, which
616
implements the same domain decomposition strategy adopted in the forward
617
model. It thus inherits the same parallel modeling approach, and therefore
618
essentially the same scalable code efficiency as the parent model. In terms of
619
across-processor operations, such as exchanging information between processor
620
tiles, global sums, etc., the same set of adjoint primitives can be used that have
621
been developed for the MITgcm ocean component (Heimbach et al., 2005).
622
The main parallel operations are exchanges between processors (send/receive,
623
gather/scatter), as well as global sums (reduce). All of these are linear opera-
624
tions in nature. Therefore there are no fundamental hurdles to parallel adjoint
625
model execution. Adjoint primitives of the parallel support package have been
626
written by hand since no adjoint support of the Message Passing Interface
627
(MPI) is currently available (Heimbach et al., 2005). Nevertheless, efforts are
628
currently under way to extend MPI libraries to include support for adjoint
629
model generation (Utke et al., 2009).
37
630
Iterative solvers and their adjoint Next, we briefly describe the treat-
631
ment of the sea-ice rheology solver. The solver used here is an adaptation
632
of the line successive over-relaxation (LSOR) method of Zhang and Hibler
633
III (1997) to an Arakawa C grid (see Part 1). At the heart of this method
634
is an iterative approach used to solve the momentum equations for ice drift
635
velocities, based on a tridiagonal matrix solver. A challenge is to generate
636
the adjoint of the iterative procedure. A similar issue was encountered in the
637
context of adding bottom topography as a control variable to the MITgcm,
638
which breaks the self-adjoint property of the elliptic pressure solver and which
639
required adjoint code generation for this routine (Losch and Heimbach, 2007).
640
The approach taken here consists of invoking the implicit function theorem
641
in order to simplify the reverse accumulation of sensitivities in terms of re-
642
quired variables during the (reverse) iteration, e.g., Christianson (1998) and
643
Griewank and Walther (2008), chapter 15. Essentially this theorem states that
644
only the variable at the fixed point is required, thus avoiding the potentially
645
memory-intensive storing of the entire intermediate state of the iteration. TAF
646
accommodates this feature via directives that identify a loop in the code as
647
fixed-point iteration (Giering and Kaminski, 1998), and which we use here. We
648
note that caveats exist between analytical derivation of the adjoint equations
649
for implicit functions and its validity for numerical implementation (Giles,
650
2001). Deciding whether the generated code is reliable has to be based, some-
651
what heuristically, on detailed gradient checks, as was done in this study.
652
A note on recent developments in the use of fully implicit method in ocean,
653
sea-ice and land-ice modeling seems warranted. Methods such as Jacobi-free
654
Newton-Krylov (JFNK) methods enable very efficient model integrations us-
655
ing rather long time steps and showing very favorable convergence behavior.
38
656
Most implementations (in particular those aimed at scalable applications) take
657
advantage of black-box solvers such as GMRES, Trilinos or PETSc. In such
658
cases, differentiation through the solvers is either not possible (black-box)
659
or very difficult and not recommended. Instead, use of the knowledge of the
660
solver for the adjoint system of differential equations and implementation of
661
the adjoint solver (usually part of the same black-box package) is preferable.
662
Approximating the adjoint of mixing parameterization schemes
663
Mixing schemes introduce additional nonlinear behavior on various time scales
664
that may cause problems for the adjoint. Generating exact adjoint for most
665
schemes does not per se present a fundamental problem. For example, Marotzke
666
et al. (1999) describe in some detail the adjoint of the convective adjustment
667
scheme. Ferreira et al. (2005) take advantage of the adjoint to estimate eddy-
668
induced stresses in the ocean interior as a way to estimate parameters relevant
669
for eddy-induced mixing.
670
However, with increasing time scales, resolution, nonlinearity of the scheme,
671
or a combination thereof, the use of the adjoint will be prevented due to
672
exponential growth of sensitivities. Approximating the adjoint under such cir-
673
cumstances has been found to be necessary to retain a stable solution. In the
674
present calculation the approximation was made by excluding the adjoint of
675
the non-local K-profile parameterization (KPP) scheme for vertical mixing
676
(Large et al., 1994).
677
Some modifications have recently been made to the sea-ice thermodynamics,
678
in particular to the treatment of sea-ice growth, in order to improve both
679
certain forward model features as well as the adjoint model behavior. These
39
680
changes will be discussed in detail elsewhere (Fenty, 2010).
681
Concluding remarks Many issues of generating efficient exact adjoint
682
sea-ice code are similar to those for the ocean model’s adjoint. Linearizing
683
the model around the exact nonlinear model trajectory is a crucial aspect in
684
the presence of different regimes. For example, is the thermodynamic growth
685
term for sea-ice evaluated near or far away from the freezing point of the ocean
686
surface? Adapting the (parent) model code to support the AD tool in providing
687
exact and efficient adjoint code represents the main workload, initially. For
688
legacy code, this task may become substantial but it is fairly straightforward
689
when writing new code with an AD tool in mind. Once this initial task is
690
completed, generating the adjoint code of a new model configuration takes
691
about 10 minutes.
692
693
Acknowledgements
694
This work is a contribution to the ECCO2 project sponsored by the NASA
695
Modeling Analysis and Prediction (MAP) program and to the ECCO-GODAE
696
project sponsored by the National Oceanographic Partnership Program (NOPP).
697
DM carried out this work at JPL/Caltech under contract with NASA. Com-
698
puting resources were provided by NASA/ARC, NCAR/CSL, and JPL/SVF.
699
Careful reviews by the two anonymous reviewers significantly improved the
700
readability of the paper and are gratefully acknowledged.
40
701
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