adaptiveEVP r2

adaptiveEVP r2
2
The adaptive EVP method for solving the sea ice
momentum equation
3
Madlen Kimmritza,∗, Sergey Danilova,b , Martin Loscha
1
a Alfred
4
5
b A.
Wegener Institute, Bussestrasse 24, D-27570 Bremerhaven
M. Obukhov Institute of Atmospheric Physics RAS, Moscow, Russia
6
Abstract
7
Stability and convergence of the modified EVP implementation of the visco-
8
plastic sea ice rheology by Bouillon et al., Ocean Modell., 2013, is analyzed on
9
B- and C-grids. It is shown that the implementation on a B-grid is less restrictive
10
with respect to stability requirements than on a C-grid. On C-grids convergence
11
is sensitive to the discretization of the viscosities. We suggest to adaptively
12
vary the parameters of pseudotime subcycling of the modified EVP scheme
13
in time and space to satisfy local stability constraints. This new approach
14
generally improves the convergence of the modified EVP scheme and hence its
15
numerical efficiency. The performance of the new “adaptive EVP” approach is
16
illustrated in a series of experiments with the sea ice component of the MIT
17
general circulation model (MITgcm) that is formulated on a C-grid.
18
Keywords: VP rheology, EVP rheology, Sea ice, MITgcm, B-grid, C-grid,
19
adaptive relaxation parameter
20
2010 MSC: 00-01, 99-00
21
1. Introduction
22
The viscous-plastic (VP) rheology (Hibler III, 1979), connecting sea ice de-
23
formation rates with ice stresses, forms the basis of most climate sea-ice models.
24
The resulting set of equations of ice dynamics is very stiff and thus calls for the
∗ Corresponding author (now affiliated to Nansen Environmental and Remote Sensing Center and Bjerknes Centre for Climate Research, Bergen, Norway)
Email addresses: [email protected] (Madlen Kimmritz),
[email protected] (Sergey Danilov), [email protected] (Martin Losch)
Preprint submitted to Ocean Modelling
March 11, 2016
25
design of efficient solution methods to avoid the restriction to very small time
26
steps in standard explicit methods. Partial linearization allows the stiff part
27
of the problem to be treated implicitly, but requires iterative solvers (Zhang
28
and Hibler, 1997). Although this linearization lifts the time step restriction, it
29
requires many (Picard) iterations to recover the full nonlinear solution. Tra-
30
ditionally only a few Picard iterations are made and convergence is sacrificed
31
(Lemieux and Tremblay, 2009). This motivated the development of fully non-
32
linear Jacobian-free Newton-Krylov (JFNK) solvers (Lemieux et al., 2010, 2012,
33
Losch et al., 2014). They converge faster than previous methods but still remain
34
an expensive solution.
35
The elastic-viscous-plastic (EVP) method is an alternative to implicit meth-
36
ods. It relaxes the time step limitation of the explicit VP method by introduc-
37
ing an additional (artificial, not physically motivated) elastic term to the stress
38
equations. This allows a fully explicit time stepping scheme with much larger
39
time steps than possible for the VP method (Hunke and Dukowicz, 1997, Hunke,
40
2001), but still requires subcycling within the external time step commonly set
41
by the ocean model. The effects of the additional elasticity term, however, are
42
reported to lead to noticeable differences in the deformation field, and result
43
in solutions with smaller viscosities and weaker ice (e.g., Lemieux et al., 2012,
44
Losch et al., 2010, Losch and Danilov, 2012, Bouillon et al., 2013).
45
In many cases, these effects are linked to the violation of local stability limits
46
(analogous to the Courant number constraint for advection) associated with the
47
explicit time stepping scheme of the subcycling process (Hunke and Dukowicz,
48
1997, Hunke, 2001). Their most frequent manifestation is grid-scale noise in the
49
ice velocity derivatives and hence in ice viscosities, in particular, on meshes with
50
fine or variable resolution (Losch and Danilov, 2012) (the numerical code may
51
remain stable and simulate smooth fields of ice concentration and thickness). In
52
an attempt to improve the performance of the EVP method, a modification of
53
the time-discrete model was proposed by adding an inertial time stepping term
54
to the momentum balance (Lemieux et al., 2012). This mEVP (modified EVP)
55
method was reformulated by Bouillon et al. (2013) as a “pseudotime” iterative
2
56
scheme. By construction, it should lead to solutions that are identical to those
57
of the VP method provided the scheme is stable and runs to convergence. The
58
analysis of mEVP for a simplified one-dimensional (1D) case suggests that the
59
stability is defined by a single parameter that depends on the resolution, the
60
time step, the ice viscosity, and on the relaxation parameters of the pseudotime
61
stepping (Bouillon et al., 2013, Kimmritz et al., 2015),.
62
Although the 1D analysis is expected to be valid at least qualitatively in
63
two dimensions (2D), there are a few aspects that are not covered by the 1D
64
analysis: the velocity and stress divergence vectors are not collinear in 2D;
65
velocities are staggered in space (on a C-grid) but are collocated on a B-grid,
66
so that on a C-grid one works with normal velocity components rather than the
67
full velocity vector (as on the B-grid); on C-grids the components of the strain
68
rate tensor and the stress components are not collocated. These aspects affect
69
the convergence properties of the method. Several C-grid implementations have
70
been suggested in literature (e.g. Bouillon et al., 2013, Lemieux et al., 2012,
71
Losch et al., 2010).
72
This work extends the analysis of Kimmritz et al. (2015) by exploring the
73
impact of space discretizations on the stability properties of the mEVP method.
74
Motivated by this analysis we propose a new adaptive EVP implementation
75
(aEVP). In this scheme the parameters of the pseudotime stepping are locally
76
adjusted in each pseudotime subcycle in order to ensure stability. In simple
77
experiments we demonstrate that this scheme leads to a significant improvement
78
of the convergence properties.
79
The article is organized as follows: In Section 2 we briefly review the gov-
80
erning equations, the mEVP scheme as formulated in Bouillon et al. (2013) and
81
its discretization on B- and C-grids. We continue with the stability analysis
82
of the linearized 2D equations in Section 3, and introduce the aEVP method
83
and explore its stability properties in Section 4. In Section 5, we illustrate
84
our results in experiments performed with the sea ice component of an ocean
85
general circulation model (MITgcm, see the source code at http://mitgcm.org).
86
Conclusions and outlook are given in Section 6.
3
87
88
2. Model description
The horizontal momentum balance of sea ice is written as
89
m(∂t + f k×)u = aτ − Cd aρo (u − uo )|u − uo | + F − mg∇H.
90
Here m is the ice (plus snow) mass per unit area, f is the Coriolis parameter and
91
k the vertical unit vector, a the ice concentration, u and uo the ice and ocean
92
velocities, ρo is the ocean water density, τ the wind stress, H the sea surface
93
elevation, g the acceleration due to gravity and Fl = ∂σkl /∂xk the divergence of
94
the internal stress tensor σkl (with indices k, l denoting x1 and x2 directions).
95
96
97
98
We follow Bouillon et al. (2013) in writing the VP constitutive law as
P
1
σkl (u) =
(˙d − ∆)δkl + 2 (2˙kl − ˙d δkl ) ,
2(∆ + ∆min )
e
with
˙kl
1/2
1 2
2
∆ = ˙d + 2 ˙s
.
e
1
= (∂k ul + ∂l uk ) ,
2
(1)
(2)
(3)
99
The stress tensor σ(u) is symmetric, i.e. σ12 (u) = σ21 (u). The term ˙d = ˙kk
100
describes the divergence, and ˙s = ((˙11 − ˙22 )2 + 4˙212 )1/2 is the shear. The
101
parameter e = 2 is the ratio of the major axes of the elliptic yield curve. Note
102
that the use of the replacement pressure, (∆/(∆ + ∆min ))P (Hibler III and
103
Ip, 1995) in the formulation of the VP constitutive law (2) ensures that the
104
stress state is on an elliptic yield curve even when ∆ . ∆min . The ice strength
105
P is parameterized as P = hP ∗ e−c
∗
∗
(1−a)
, where h is the mean thickness of
∗
and c are set to P ∗ = 27500 Nm−2 and
106
the grid cell, and the constants P
107
c∗ = 20. For future reference we introduce the bulk and shear viscosities ζ =
108
0.5 P/(∆ + ∆min ) and η = ζ/e2 .
109
2.1. The mEVP scheme as a pseudotime iterative scheme
110
The difficulty in integrating (1) is the stiff character of the stress term, which
111
requires prohibitively small time steps in an explicit time stepping scheme. The
112
traditional approach is either implicit (Zhang and Hibler, 1997) where viscosities
113
are estimated at the previous nonlinear iteration and several iterations are made,
4
114
or explicit, through the EVP formulation (Hunke and Dukowicz, 1997, Hunke
115
and Lipscomb, 2008) where adding a pseudo-elastic term reduces the time step
116
limitations. A discussion of the convergence issues can be found, for instance,
117
in Bouillon et al. (2013), Kimmritz et al. (2015) and is not repeated here.
118
The suggestion by Bouillon et al. (2013) is equivalent, up to details of treating
119
the Coriolis and the ice-ocean drag terms, to formulating the mEVP method as:
122
1
σ(up ) − σ p ,
α
1 ∆t
∆t p+1/2
up+1 − up =
∇ · σ p+1 +
R
+ un − up .
β m
m
123
In (5), R sums all the terms in the momentum equation except for the rheol-
124
ogy and the time derivative, ∆t is the external time step of the sea ice model
125
commonly set by the ocean model, the index n labels the time levels of the
126
model time, and the index p is that of pseudotime (subcycling step number).
127
The Coriolis term in Rp+1/2 is treated implicitly in our B-grid implementation,
128
but is explicit on the C-grid, and the ice-ocean stress term is linearly-implicit
129
(Cd ρo |uo − up |(uo − up+1 )). The term σ(up ) in (4) implies that the stresses
130
are estimated by (2) based on the velocity of iteration p, and σ p is the variable
131
of the pseudotime iteration. The relaxation parameters α and β in (4) and (5)
132
are chosen to satisfy stability constraints, see Bouillon et al. (2013), Kimmritz
133
et al. (2015). They replace the terms 2T /∆te and (β ∗ /m)(∆t/∆te ), where T
134
is the elastic damping time scale and ∆te the subcycling time step of standard
135
EVP formulation; the parameter β ∗ was introduced in Lemieux et al. (2012). If
136
(4) and (5) are iterated to convergence, their left hand sides can be set to zero
137
leaving the VP solution:
120
121
σ p+1 − σ p =
(4)
(5)
138
m
un+1 − un = ∇ · σ(un+1 ) + R∗ ,
∆t
139
with R∗ = limp→∞ Rp+1/2 and un+1 = limp→∞ up . While one may introduce a
140
convergence criterion to determine the number of iteration steps, historically, the
141
actual number of pseudotime iterations N is selected experimentally to ensure
142
the accuracy needed. The new velocity un+1 at time step n + 1 is estimated at
5
(6)
143
the last pseudotime step p = N . The initial values for p = 1 are taken from the
144
previous time step n.
145
2.2. Spatial discretizations
146
We consider discretizations on Arakawa B- and C- grids that are commonly
147
used in sea-ice models. The positions of variables on these grids are depicted in
Figure 1. Note, that in this section (i, j) is used as mesh indices. For simplicity
variables
B-grid
C-grid
scalars
c
c
velocities (u, v)
z
(u,v)
ε̇kk , σkk
c
c
ζ, η in bulk stress definition
c
c
ε̇12 , σ12
c
z
η in shear stress definition
c
z
Figure 1: On the left hand side the location of the cell points are sketched: c is the cell
center (square symbol), z a vertex (circle), u and v the velocity points on a C-grid. All
points in the dashed box are indexed with the same index pair (i, j). The table on the right
hand side displays the location of the variables on B- and C-grids. Scalar quantities are ice
concentration, ice mass, ice strength and sea surface elevation.
148
149
we use Cartesian coordinates and uniform grids with cell widths ∆x1 and ∆x2 .
150
The complete discretization on general orthogonal curvilinear grids can be found
151
in Bouillon et al. (2009) and Losch et al. (2010). For convenience we introduce
152
the notation
153
δ1 φi,j = φi,j − φi−1,j ,
δ2 φi,j = φi,j − φi,j−1 ,
1
154
155
2
φi,j = (φi,j + φi+1,j )/2,
φi,j = (φi,j + φi,j+1 )/2
1,2
156
for a quantity φ at a cell with index (i, j). An expression of the form φi,j
157
defines the successive application of both directional averaging operators on φ.
158
Note, that the location of the discretized derivatives depends on the respective
159
grid arrangement of variables.
6
160
The strain rates on a B-grid are given by
2
1
162
163
(ε̇11 )ij = δ1 (u1 )i+1,j ∆x−1
(ε̇22 )ij = δ2 (u2 )i,j+1 ∆x−1
1 ,
2 ,
1
2
1
−1
(ε̇12 )ij =
δ2 (u1 )i,j+1 ∆x−1
,
2 + δ1 (u2 )i+1,j ∆x1
2
164
u1 and u2 denote the first and the second velocity component, respectively. On
165
C-grid, the definition of the strain rates is the same as on the B-grid but without
166
the averaging step. In the B-grid arrangement, the divergence of the stress
167
tensor, which contributes as a forcing in the momentum balance, is reconstructed
168
on nodes as (k = 1, 2 for the two sea ice momentum equations)
161
2
1
169
−1
((∇ · σ)k )i,j = δ1 (σ1k )i,j−1 ∆x−1
1 + δ2 (σk2 )i−1,j ∆x2 .
170
On a C-grid, the vector quality of the divergence is lost. Instead it is given on
171
u and v points by
172
((∇ · σ)1 )i,j = δ1 (σ11 )i,j ∆x−1
1
+ δ2 (σ12 )i,j+1 ∆x−1
2 ,
173
174
−1
((∇ · σ)2 )i,j = δ1 (σ12 )i+1,j ∆x−1
1 + δ2 (σ22 )i,j+1 ∆x2 .
175
In the B-grid framework all derivatives include averaging but are collocated and
176
share the same stencil. There is no immediate averaging of velocity derivatives
177
for C-grid discretizations. While this results in a smaller stencil, the tensor
178
components and derivatives are defined at different locations. For this reason
179
we still need averaging for the determination of ∆ and hence for computing the
180
viscosities η and ζ.
181
Further steps in the B-grid arrangement are straightforward. On C-grids,
182
there is some freedom in computing the viscosities. More precisely, since the
183
bulk and shear stresses are defined at different locations, we also need to de-
184
fine viscosities on these different locations. We consider two options. One is
185
introduced in Bouillon et al. (2013), the other one is the current default imple-
186
mentation in the sea ice component of the MITgcm (Losch et al., 2010, see the
187
source code at http://mitgcm.org).
188
The discretization of ∆ on cell centers coincides in both cases; the con-
189
tributing square of the shear strain rate is formulated as a weighted average of
7
190
its adjacent nodal values. Since we treat ∆x1 and ∆x2 as constants, it reduces
191
to (ε̇212 ) . A formulation on more general grids can be found in Bouillon et al.
192
(2013). The definition of the nodal shear viscosity differs in the two cases: While
193
in Bouillon et al. (2013) it is given as the average values of the adjacent cells,
194
the MITgcm counterpart aims to keep the stencil of the single contributions as
195
small as possible. Denoting the former approach as C1 and the latter as C2 the
196
shear viscosities at nodal points are given as
12
197
198
199
(C1)
ηi−1,j−1 12
12
Pi−1,j−1 / 2e2 (∆zij + ∆min )
(C2)
with nodal value
1/2
12
12
∆zij = (ε̇11 + ε̇22 )2i−1,j−1 + e−2 (ε̇11 − ε̇22 )2i−1,j−1 + 4(ε̇12 )2ij
.
200
In an attempt to circumvent the ambiguity in the definition of the viscosities,
201
we also considered an approach that first reconstructs full velocities to B-grid
202
locations, then computes stresses and their divergence on B-grid and projects the
203
result to the C-grid locations. Its excessive averaging and lack of commutability
204
of derivatives, accompanied by unfavorable mathematical properties and very
205
poor stability, however, forced us to discard it.
206
3. Stability analysis
207
We begin with generalizing the linear analysis of Kimmritz et al. (2015) to
208
two dimensions. We will see that despite added complexity and the fact that
209
the vectors of velocity and stress divergence are not collinear, the stability still
210
depends on parameters that are similar to that of the 1D case and that the
211
C-grid discretization is less stable than B-grid discretization. Similar to the
212
1D analysis we will assume that P and ∆ = ∆min are constant, and drop un
213
and Rp+1/2 (under these assumptions C1 and C2 are similar). In order to add
214
stability to the scheme, we take the last term σ p in (4) and the last term up in
8
215
(5) implicitly:
α
ζ
p
σkl
+
(1 − e−2 )∇ · up δkl + 2e−2 ε̇pkl ,
α+1
α+1
β
1 ∆t
p
=
u +
∇ · σ p+1 .
β+1
β+1 m
216
p+1
σkl
=
(7)
217
up+1
(8)
218
219
For the linear analysis we focus on a single Fourier harmonic in space
220
(σ p (x), up (x))T = vp eikx
221
p
p
p
(x), up1 (x), up2 (x)) and vector vp ∈ C5 .
(x), σ22
(x), σ12
with (σ p (x), up (x)) = (σ11
222
After inserting expression (9) in equations (7) and (8) they reduce to a system
223
of five equations for the components of vp . In matrix form, they read
(9)
224
vp+1 = A vp
225
with the 5 by 5 matrix A that corresponds to the operators on the right hand
226
side of (7) and (8) and also incorporates the dependence on the wave vector
227
k. The related iterative scheme converges if vp decays as p tends to infinity.
228
Introducing the amplification factor λ as vp+1 = λvp , we see that such a solution
229
is only possible if λ is an eigenvalue of the matrix A (with the eigenvector vp ).
230
There are five complex-valued solutions λi . The formal stability condition of
231
the discrete equations is |λi | ≤ 1 for all i = 1...5. But, in analogy to the 1D case,
232
we argue that the more restrictive condition, |λi | < 1 and |ϕi | 1, where ϕi is
233
the phase of λi , has to be imposed due to the nonlinearity of the full equations
234
(Kimmritz et al., 2015). Because of the fifth order of the characteristic equation,
235
we will explore the behavior of its roots numerically.
236
237
238
239
Using the notation
dσ =
α
,
α+1
du =
ζ
,
α+1
cσ =
the matrix A can be written as
9
1 ∆t
,
β+1 m
cu =
β
.
β+1




0


A=
0


cσ dσ ψx1

0
240
241
242
243
dσ
0
0
e1 ψx1
dσ
0
e4 ψx2
0
dσ
e2 ψx1
cσ dσ ψx2
0
a1
cσ dσ ψx1
cσ dσ ψx2
e3 ψx1 x2
e2 ψx2



e4 ψx1 


e1 ψx2  ,


e3 ψx1 x2 

a2
(10)
where e1 = du (1 + e−2 ), e2 = du (1 − e−2 ), e3 = cσ du , e4 = du e−2 , and al =
cu + cσ du (1 + e−2 )ψxl xl + e−2 ψxl∗ xl∗ with l ∈ {1, 2}, l∗ = 1 for l = 2 and vice
244
versa. On a B-grid, the remaining terms (stemming from derivatives) take the
245
form
246
247
248
249
ψxl = 2i sin(0.5kl ∆xl ) cos(0.5kl∗ ∆xl∗ )/∆xl ,
ψxl xl = (cos(kl ∆xl ) cos(kl∗ ∆xl∗ ) + cos(kl ∆xl ) − cos(kl∗ ∆xl∗ ) − 1) /∆x2l∗ ,
ψx1 x2 = − (sin(k1 ∆x1 ) sin(k2 ∆x2 )) /(∆x1 ∆x2 ).
250
Averaging, intrinsic to the derivatives on a B-grid, leads to additional cosine
251
multipliers, so that derivatives always depend on both components of the wave
252
number. In contrast, on a C-grid the derivatives only depend on the wave
253
numbers related to their directions:
ψxl = 2i sin(0.5kl ∆xl )/∆xl ,
254
ψxl xl = 2 (cos(kl ∆xl ) − 1) /∆x2l ,
255
ψx1 x2 = −4 (sin(0.5k1 ∆x1 ) sin(0.5k2 ∆x2 )) /(∆x1 ∆x2 ) .
256
257
258
Setting either k1 or k2 to zero reduces the system to the 1D case where B- and
259
C-grids coincide. Since we assumed a constant value for ∆, there is no difference
260
between the two implementations (C1 and C2) on the C-grid.
261
3.1. General considerations
262
Throughout this section we use ∆x = ∆x1 = ∆x2 . Since the strongest
263
pseudotime step limitations are expected at the largest resolved wave numbers
264
we choose
265
n
o
(k1 , k2 ) ∈ π ∆x−1 (cos φ, sin φ) φ ∈ [0, 1] · 2π .
10
(11)
266
We set ∆x = 105 m, ∆t = 3600 s, a = 1, m = 1 m, ∆ = 2·10−7 s−1 , and α = β.
267
Figure 2 plots the eigenvalues on a B-grid and on a C-grid for α = β ∈ {140, 500}
268
and various angles φ between the horizontal waves (see also (11)). In the plots
269
we additionally depicted the unit circle in order to highlight the magnitudes and
270
phases of the eigenvalues. In agreement with Kimmritz et al. (2015), both the
271
magnitudes of the phases ϕ and the magnitudes |λ| are controlled by α and β.
272
The larger α and β, the closer are the eigenvalues to the stable region close to
1. There is always an eigenvalue with zero phase, which corresponds to motions
Figure 2: Eigenvalues of the system matrix A for the B- and the C-grid for α = β = 140
(graphs (a) and (b)) and α = β = 500 (graphs (c) and (d)). The wave numbers (k1 , k2 ) are
given by equation (11) with angle φ varying between 0 and π/4 with increments of 0.005. The
grey circle denotes the unit circle around the origin. In the stable cases, the differences of the
eigenvalues from the unit circles are 1/α, see also Table 1. For α = β = 140 on C grid, the
magnitudes of the eigenvalues, |λ|, exceed 1 for φ > 0.154π.
273
274
that are little affected by the sea ice stresses. The other four eigenvalues appear
275
in complex conjugate pairs if the solution is stable (they may become real-valued
276
for larger ∆ or smaller wave numbers). The maximum phase is larger for the
277
eigenvalues on the C-grid indicating that the C-grid implementation is more
278
susceptible to instability than the B-grid discretization. We assume that the
279
additional averaging on the B-grid improves the stability of the scheme. For
280
instance, the case α = β = 140 is unstable on the C-grid, but stable on the
11
281
B-grid. At the onset of instability, two complex valued eigenvalues coincide at
282
-1 and diverge from this point along the real axis for increasing angles φ. The
283
eigenvalues in a stable situation have magnitudes of α/(1 + α) < 1 (Table 1). In
284
the numerical analysis, we observed eigenvalues with magnitudes of α/(1 + α)
and β/(1 + β) for α 6= β.
α = β = 140
max{|λ|}
B-grid
0.993(∗)
C-grid
2.638
max{ϕ}
α = β = 500
max{|λ|}
max{ϕ}
0.69 π
0.998(∗)
0.16 π
π
0.998(∗)
0.20 π
Table 1: Eigenvalues for α = β ∈ {140, 500} with maximum absolute value or phase on a
B-grid and on a C-grid as depicted in Figure 2. The symbol (∗) indicates, that all eigenvalues
of the 5 times 5 matrix have the same magnitude (α/(1 + α)).
285
286
Figure 3 presents the dependence of the maximum phase of the eigenvalues
287
on the governing parameters for α = β = 250. There is only a weak sensitivity
288
of max{ϕ} on the ice mass m (not shown). Lower values of ∆, higher resolution
289
in space, and higher ice concentrations lead to larger phases in the eigenvalues
290
and thus to a less stable system in agreement with previous stability analyses
291
(Kimmritz et al., 2015). For very fine meshes it is important to note that
292
increasing the mesh resolution while scaling the time resolution at the same
293
rate (∆t ∼ ∆x) makes the scheme unstable (Fig. 3(c)), but when the time step
294
is reduced proportionally to the square of the spatial resolution (∆t ∼ ∆x2 ),
295
the scheme remains stable (Fig. 3(f)) in agreement with the stability constraint
296
derived in Kimmritz et al. (2015). Reduced grid spacing ∆x with constant
297
time step ∆t (Fig. 3(d)), which is a typical situation for models with locally
298
refined meshes, leads to lower stability. Thus, the graphs in Figure 3 indicate
299
a proper (i.e. stability preserving) scaling of ∆t for mesh refinements or for
300
meshes with strongly varying resolution. In all cases, the phase is slightly larger
301
on the C-grid than on the B-grid.
12
Figure 3: Dependence of the maximum phase of the eigenvalues (larger phase implies less
stability) on ∆ (a), on ice concentration a (b), on ∆t, which scales at the same rate as ∆x
with initial (∆t, ∆x) = (3600 s, 105 m) (c), on ∆x with fixed ∆t (d), on ∆t with constant
∆x (e) and ∆t which scales with ∆x2 with initial (∆t, ∆x) = (3600 s, 105 m) (f) on a B-grid
(black line) and on a C-grid (grey dashed line). For small ice concentrations a, the phase is
small, because the ice strength P is small.
302
4. The adaptive EVP method
303
The choice of parameters α and β is the key for providing stability of the
304
solution. Based on the 1D analysis, Kimmritz et al. (2015) proposed to select α
305
and β so that αβ γ, where γ = k 2 P ∆t/(2∆m), with k 2 < (π/∆x)2 , governs
306
stability. The regimes that are challenging for stability of the iterative process
307
are those when γ is large and thus controls the phase (frequency) of the pseudo-
308
time iteration. The results shown in Figure 3 and additional computations (not
309
shown) suggest that in 2D the largest phase is controlled by the same parameter
310
γ for a fixed wave vector direction as in the 1D case; Figure 2 also indicates
311
that the 2D character of the problem implies some additional dependence on
312
the wave vector direction.
313
Keeping α and β sufficiently large to provide stability has the downside that
13
314
the speed of convergence is slowed down and a large number of pseudotime
315
steps N is required (N > α, β) to reach convergence. In practice, very large
316
α and β are only required in regions where viscosities (P/2∆) are large or the
317
mesh resolution is high, while keeping them large outside of these regions only
318
deteriorates convergence. A solution to this dilemma is making α and β variable
319
in space and time, which is possible because mEVP, as opposed to the standard
320
EVP approach, fully detaches α and β from the external time stepping scheme.
321
We now introduce an approach which makes use of this possibility.
Motivated by the fact that γ = k 2 P ∆t/(2∆m) controls stability, we write it
322
323
324
as
γ=ζ
c ∆t
Ac m
325
and require that αβ γ. Here, Ac denotes the area of the local 2D grid cell
326
and constant c is a numerical factor such that the term c/Ac accounts for the
327
contribution due to the eigenvalue k 2 of the Laplacian operator, see Kimmritz
328
et al. (2015), which has the upper limit of π 2 /Ac . While this implies an upper
329
bound of π 2 for c, c can be much smaller if ice remains smooth on the grid
330
scale. In practice, the value of c depends on forcing, geometry of boundaries
331
and on resolution and has to be selected experimentally. In most cases, when
332
the solution is stable, there is no grid-scale noise so that c can be smaller than
333
π 2 by an order of magnitude. On finer meshes the geometrical complexity of
334
solutions may be locally increased (e.g. Losch et al., 2014), which may require
335
using c closer to its upper bound.
336
In order to satisfy the stability requirement, we choose
337
α = β = (c̃γ)1/2
338
with the empirical scaling factor c̃. It should be sufficiently large to preserve
339
stability, but just large enough to ensure convergence as fast as possible. The
340
parameters c and c̃ can easily be combined into a single parameter, but we keep
341
them separate here to emphasize their origin.
(12)
342
For instance, with c = (0.5π)2 and c̃ = 4, the phases of the eigenvalues,
343
independently of the magnitudes of ∆, ∆t or ∆x, reach values of about 0.86π
14
344
on a C-grid and of 0.71π on a B-grid. Since the mean ice thickness enters
345
both ice mass m and ice strength P , it has no effect on stability. Lowering the
346
ice concentrations leads to lower maximum phases of the eigenvalues. This is
347
due to the small exponential factor in the ice strength P for ice concentrations
348
much smaller than 1. This factor makes γ small, so that it does not govern the
349
behavior of the eigenvalues because the contributions from the internal stress
350
also become small with small ice concentrations. Since α−1 and β −1 play the
351
role of the subcycling time steps (in units of ∆t), α and β should be bounded
352
from below to ensure a sufficient accuracy of the subcycling. This adaptive
353
approach thus guarantees stability of the iterative scheme independent of the
354
problem parameters.
355
In this approach, places where α and β are large because of large values
356
of γ will be characterized by slower convergence, but will remain stable. We
357
suggest to select the number of pseudotime steps N = const so as to provide the
358
convergence over a dominant fraction of the domain (where γ is moderate). The
359
convergence in local regions with high α and β will be sacrificed in favor of faster
360
code performance. It may still be recovered over several external time steps. It
361
is also expected that places with high α and β are those where ice velocities are
362
small, so that incurring errors in the ice distribution are not necessarily large.
363
If this approach is adopted, N has to be selected experimentally.
364
Finally, we would like to point out that the eigenvalue analysis revealed (not
365
shown), that setting α 6= β by splitting γ in constituent multipliers generally
366
requires an individual scaling of α and β if the resolution in time or space is
367
varied. We do not consider this case here.
368
So far we were guided by the results of the linear analysis. We turn to
369
numerical experiments to study the behavior of the adaptive EVP method in
370
the nonlinear case.
15
371
5. Numerical experiments
372
In this section we explore the convergence of the full sea ice momentum
373
equation on B- and C-grids. We will demonstrate that the discretization details
374
of the viscosities on a C-grid influences the convergence of the mEVP method
375
to the extent that it even may lose convergence. We will also demonstrate that
376
the adaptive approach generally leads to improved convergence compared to
377
simulations with constant α and β.
378
5.1. Experimental setup
379
The simple model configuration with a Lx1 × Lx2 = 1280 km × 1280 km
380
domain and a Cartesian grid with a constant grid size of 16 km follows that of
381
Hunke (2001), but without topography in the model interior. The sea ice is
382
driven by the ocean currents with the velocity (in m/s)
383
384
u0 = 0.1(2x2 − x2,min )/Lx2
v0 = −0.1(2x1 − x1,min )/Lx1
and wind stress
385
τ = Ca ρa ua |ua |
386
with atmospheric drag coefficient Ca = 2.25 · 10−3 , air density ρa and wind
387
velocity (in m/s)
388
389
390
ua = 5 + (sin(2πt/T ) − 3) sin(2πx1 /Lx1 ) sin(πx2 /Lx2 ) ,
va = 5 + (sin(2πt/T ) − 3) sin(2πx2 /Lx2 ) sin(πx1 /Lx1 ) ,
391
with T = 4 days. Initially, the ice is 2 m thick and the ice concentration
392
increases linearly from 0 in the west to 1 in the east, so that the mean ice
393
thickness h varies from 0 to 2 m. The mean wind pushes the ice into the
394
northeast corner where it gradually piles up until it becomes sufficiently thick
395
to be stopped. We will use ∆min = 2 · 10−9 s−1 (Hibler III, 1979).
396
5.2. Convergence of B- and C-grid discretizations of the mEVP method
397
We start with an examination of convergence and stability of the mEVP
398
scheme on B- and C-grids. It suffices to consider the first external time level
16
399
(Kimmritz et al., 2015). Recall the C1 and C2 discretizations of the shear
400
viscosities at nodal points. In the C1 case, the nodal shear viscosity is the
401
average of the shear viscosities defined at adjacent cells; in the C2 case, it is
402
computed with fewest possible averages of the contributing variables. Figure 4
403
404
plots the residuals:
1/2

p 2
p 2
2 p+1
X
X α2 |σ p+1
β
|u
−
u
|
−
σ
|
ij
ij 
ij
ij

+
,
2 |σ 2 − σ 1 |2
2 |u2 − u1 |2
α
β
ij
ij
ij
ij
i,j
i,j
405
of the subcycling at the first time level for B-, C1- and C2-grid discretizations.
406
We weighted the single contributions in the definition of the residual by the
407
inverse of the first residuals of the subcycling in order to put each of the contributions on equal footing. Convergence within numerical working precision is
Figure 4: Residuals of the first time level of the full nonlinear problem for the B-, the C1and the C2-grid discretization, and for different choices of α = β ((a) α = β = 250, (b)
α = β = 500).
408
409
reached for α = β = 250 after 0.75 · 104 subcycling steps only in the B-grid case,
410
and for α = β = 500 after 1.5 · 104 subcycling steps for the B-grid discretization
411
and the C1-grid case. The C2-grid discretization does not converge in any case.
412
(Note that Lemieux and Tremblay (2009) also needed O(104 ) nonlinear steps in
413
their Picard iteration.) We cannot give a rigorous explanation for this behavior,
414
but we hypothesize that the viscosity computation in the C2-case prevents the
415
discrete analogue of (2) to be satisfied exactly. For the remaining schemes we
416
recover the expected behavior (see also Kimmritz et al., 2015): higher values of
417
α and β guarantee stability but slow down the speed of convergence. In agree-
418
ment with our analysis above, the stability constraints appear to be stricter for
17
419
the C1-discretization than for the B-grid discretization. However, if the C1-
420
grid scheme converges, its convergence rate is only marginally slower than the
421
convergence rate of the B-grid scheme.
422
5.3. Convergence of B- and C-grid discretizations of the aEVP method
423
In Figure 5 we compare the convergence rates of the aEVP approach with α
424
and β computed by (12) to the mEVP scheme (Bouillon et al., 2013, Kimmritz
425
et al., 2015) with fixed α = β = 500. The parameters for the aEVP scheme are
426
set to c = (0.01π)2 , c̃ = 4, and (α, β) ≥ 5. Note, that we set c to a very small
427
value. This implies, that we deal with scales that are two orders of magnitude
428
larger than the grid scale and thus consider basin scale. It can only reflect the
429
fact that within the first time step there is still no detail in the velocity field and
430
thus allows us to use this small value. On later time levels we expect a larger
variety of scales in the velocity field, which requires larger values for c.
Figure 5: Residuals in the subcycling on the first time level for different discretizations. Graph
(a) plots the entire convergence behavior, graph (b) is a zoom into the first 500 subcycling
steps.
431
432
As in the mEVP case, the B-grid and the C1-grid discretizations lead to
433
convergence, but the C2-case does not converge. Convergence in the C2-case is
434
also not gained for different settings of c and c̃ (not shown). The convergence of
435
the adaptive approach for the B- and the C1-grid case is faster than for mEVP
436
by a factor of 3, but the final residual for the C1-grid is slightly larger than for
437
the mEVP scheme. As in practice the affordable number of subcycling steps
438
is probably 500 or less (Kimmritz et al., 2015), we concentrate on the residual
18
439
development over the first 500 sybcycling steps in Figure 5 (b). Compared to
440
the mEVP approach we see a reduction in the residual of more than one order
441
of magnitude in the convergent cases. Even for the C2-case the residuals are
442
smaller for aEVP. In agreement with our theoretical analysis, there are more
443
oscillations in the residuals for the C1-grid case than for the B-grid case.
444
Errors may accumulate over finite time intervals. We simulate the ice evo-
445
lution over one month with N = 500 subcycling steps and examine the perfor-
446
mance of the aEVP scheme implemented now in the MITgcm with the C1-grid
447
arrangement. Because of the oscillatory decrease of the residuals at the first
448
time level we use a larger stabilizing parameter c = (0.5π)2 . In the beginning
449
of the subcycling at time level 1440 the residuals in the momentum and in the
450
stress equations of the aEVP scheme are almost an order of magnitude smaller
451
than the ones of the mEVP scheme with α = β = 500 (Figure 6 (a)). The resid-
452
uals of the momentum equations in both schemes decrease at a similar rate in
453
both schemes. In the subcycling of the mEVP scheme the residuals of the stress
454
equations converge with a rate, which is similar to the rate of the momentum
455
equations. The residual of the stress equations in the aEVP scheme increases
456
in the first 20 subcycling steps, which might be explained by the adaptation of
457
the α field to the updated fields on the new time level. After this ’initial’ phase
458
the residual in the stress equations decreases at an increased rate, such that
459
at the end of the subcycling the residual of the stress equations in the aEVP
460
sheme is about 1.5 orders smaller than the residual of the stress equations in the
461
mEVP scheme. At the end of the subcycling at time level 1440, α (and thus β)
462
is very small (α = 5) in the large region of weak ice (Figure 6 (b)). Kimmritz
463
et al. (2015) demonstrated that the number of subcycling steps to reach full
464
convergence for the given example is of the order of NEV P = 40α. Thus, we
465
can presume, that the scheme reached full convergence in those regions with
466
N = 500 subcycling steps. However, since α−1 and β −1 define the pseudotime
467
step in units of ∆t, too small values may lead to a loss of accuracy of the pseu-
468
dotime iterations. Thus we recommend to always impose lower bounds for α
469
and β.
19
Figure 6: (a) Residual development for the subcycling at time level 1440 of the aEVP scheme
with c = (0.5π)2 and c̃ = 4, and the mEVP scheme with α = β = 500. The residuals in the
P
2 |up+1 − up |2 )1/2 , the residuals in the
momentum equations (res(mom)) are given by ( ij βij
P
stress equations (res(stress)) are computed as ( ij α2ij |σ p+1 − σ p |2 )1/2 . (b) The α field at
the end of the subcycling at time level 1440 of the aEVP scheme with 500 subcycling steps.
470
Beside sufficient accuracy, the aEVP scheme should guarantee smoothness
471
of the solution. According to Kimmritz et al. (2015), the corresponding mEVP
472
scheme with α = β = 250 shows noise in the divergence field. Figure 6 indicates
473
that in the aEVP scheme large values of α are only used in a small region in the
474
lower right corner of the domain where the ice is strong. Outside this region, α
475
ranges between 200 and 300 over the area with ice concentrations between 0.8
476
and 1.
477
To evaluate the aEVP scheme we use a converged VP solution determined
478
with the JFNK solver of the MITgcm (Losch et al., 2014) with a C1-grid dis-
479
cretization and a residual reduction of order 10−9 in each time step as reference
480
solution, and also consider solutions of the mEVP scheme with α = β = 500
481
to illustrate the improvements through adaptivity. We note that the solutions
482
of the mEVP scheme with α = β = 500 and N = 20000 (full convergence)
483
coincide with the solutions determined with the JFNK solver, but N as large as
484
this would be too expensive for practical applications (climate simulations). To
485
examine the effect of the lower bounds of α and β in the adaptive scheme with
486
N = 500, 300 and 200 subcycling steps, we explore the cases (α, β) ≥ 5 and
487
(α, β) ≥ 50. In Figure 7 we present the deviations in the divergence field from
20
488
the reference solution after one month of integration. We note, that the results
489
in the ∆ field, the ice concentration and ice thickness are of similar quality (not
490
shown). The aEVP and mEVP schemes have been run with N = 500, 300 and
491
200 subcycling steps (columns from left to right). The black lines in the graphs
492
mark the boundary with ice concentration of 0.01. The regions left of them
493
correspond to open water. The errors seen there are of little relevance and will
494
not be discussed.
495
Compared to the mEVP solution with N = 500 subcycling steps, any of the
496
aEVP solutions leads to a remarkable reduction in the errors of the adaptive
497
scheme even for the case of N = 200 subcycling steps. According to Figure 7
498
the aEVP scheme shows virtually no errors in the area covered with ice for
499
N = 500. The errors increase only slightly for N = 300 and even for the case
500
of N = 200 they remain small and are much smaller than the errors for mEVP.
501
For N = 200, the residuals of the aEVP scheme in regions with strong ice
502
show noisy behavior for the lower bound for α and β of 5 (graphs (a) – (c)). This
503
noise vanishes when we increase the lower bound to 50 (graphs (d) – (f)). We
504
relate the emergence of noise in the first case to an excessively large pseudotime
505
step and hence reduced pseudotime iteration accuracy. These errors accumulate
506
already in the early stage of the simulation. A lower bound substantially larger
507
than 50, however, is not advisable as it may have adverse effects on the conver-
508
gence in large parts of the ice covered regions thus jeopardizing the benefits of
509
the aEVP scheme.
510
6. Conclusion and Outlook
511
The present work has two main results: First, the modified EVP scheme
512
(Bouillon et al., 2013) is less stable on a C-grid, than on a B-grid, and con-
513
vergence of the scheme on a C-grid is sensitive to the implementation of the
514
viscosities. Second, we introduced the new adaptive EVP scheme, which locally
515
respects stability constraints as derived in (Kimmritz et al., 2015), and shows
516
improved convergence properties while guaranteeing stability in regions with
21
517
higher stability constraints.
518
The main advantage of the mEVP implementation (Bouillon et al., 2013)
519
of the commonly used viscous-plastic rheology over the traditional EVP imple-
520
mentation (Hunke and Dukowicz, 1997) is the decoupling of the parameters of
521
the subcycling from the external time stepping. The mEVP is formulated as a
522
pseudotime solver of ice dynamics with the VP rheology. Convergent solutions
523
can only be obtained if the iterative process is numerically stable. In this paper
524
we elucidated the sensitivity of the convergence of the mEVP approach to the
525
detail of numerical discretization. An elementary eigenvalue analysis revealed
526
that the mEVP implementation on a B-grid is more stable than on a C-grid.
527
If both schemes are stable and converge, their convergence rates are compara-
528
ble. The convergence on C-grids, however, is sensitive to the implementation of
529
the viscosities. We considered two versions of implementation that have been
530
suggested in literature; one of them (C2) does not converge to the VP solution
531
and is always contaminated by noise, while the other (C1) does so under stable
532
conditions. The lack of convergence for the C2 implementation might be related
533
to its lack of energy consistency (Bouillon et al., 2013). A rigorous explanation
534
for this behavior is still missing, but we hope that this result on its own provides
535
an important message to modellers.
536
In our earlier work we showed that, on the one hand the mEVP parameters
537
α and β need to be sufficiently large to ensure stability. They define the fre-
538
quency of the numerical oscillations. The requirement γ/(αβ) 1 limits the
539
frequency of these oscillations to sufficiently low values to be well represented
540
by the pseudotime iterations. On the other hand, large values of α and β ne-
541
cessitate a large number of subcycling steps to reach convergence, which makes
542
the scheme very expensive for practical applications (long climate simulations).
543
Emphasizing the dependence of γ on the mesh resolution we pointed out that
544
the tendency to use finer meshes in large-scale ocean modelling implies larger
545
values of γ, hence larger values for α, β and N . This would increase the com-
546
putational cost of sea ice codes further. This argument is valid for any change
547
in the model parameters that effects an increase in γ.
22
548
The main point of the present study is the new adaptive implementation
549
of the mEVP approach. Instead of being constant, the parameters α and β
550
are locally adjusted at each pseudotime step (12). The (constant) number of
551
iterations N is selected experimentally so as to provide reasonable accuracy
552
everywhere in the ice covered domain.
553
By choosing α and β adaptively we guarantee global stability. Since the
554
adaptive α and β are relatively low in wide areas of the ice covered domain,
555
convergence in those regions is improved with respect to the mEVP method.
556
Our test experiments reveal a substantial error reduction in the aEVP solutions
557
compared to the mEVP solutions even for smaller N . This is a big gain in terms
558
of computational costs. In preliminary tests, 500 subcycling steps already raised
559
the cost of the sea ice component to about 50% of the ocean model, which is
560
undesirably large. In a next step, the aEVP approach has to be applied to a
561
realistic scenario in order to test the overall performance and to learn about
562
admissible N . This will be the subject of a companion paper.
563
The aEVP approach can be especially useful for models that are based on
564
locally refined meshes, as it guarantees stability in the most refined areas. It
565
will also lead to advantages in areas where the ice is weak or of relatively low
566
concentration by reducing α and β and hence improving convergence there.
567
The new adaptive approach can be further augmented in several ways. The
568
version described here still contains parameters that have to be selected exper-
569
imentally, yet they can be estimated at run time. For instance, the factor c
570
can be assessed through the local smoothness of the velocity field. The other
571
question is the optimal choice of N based on the information of the distribution
572
of α and β. While it is difficult to change N during the subcycling, it is possible
573
to select different N at different external time steps. These opportunities will
574
be explored in future work.
575
In closing, we like to point out that there are other recently published sea
576
ice rheologies that also involve elasticity, such as the elastic plastic anisotropic
577
rheology (Tsamados et al., 2013) or the elasto brittle approach (Girard et al.,
578
2009, 2011, Bouillon and Rampal, 2015). If the appropriate schemes are solved
23
579
explicitly through pseudotime stepping, a stability analysis similar to ours or to
580
Kimmritz et al. (2015) may serve as a basis for designing an approach similar
581
to the aEVP.
582
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Presentation of the dynamical core of
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J. Zhang and W.D Hibler. On an efficient numerical method for modeling sea
ice dynamics. J. Geophys. Res., 102:8691–8702, 1997.
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Figure 7: Differences in the divergence field between the reference solution and the aEVP
solution with the lower bound (α, β) > 5 for (a) N = 500, (b) N = 300 and (c) N = 200.
Graphs (d)–(f): Same as (a)–(c) with the lower bound (α, β) > 50. Graphs (g)–(i): Differences
in the divergence field between the reference solution and the mEVP solution with α = β = 500
and (g) N = 500, (h) N = 300 and (i) N = 200. The black lines are the isolines of ice
concentration for a = 0.01. All of these runs use the C1-grid formulation.
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