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 S. Bouillon and P. Rampal. 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Numerical convergence of viscousplastic sea ice models. J. Geophys. Res., 114, C05009, 2009. 614 J.-F. Lemieux, B. Tremblay, J. Sedlác̆ek, P. Tupper, St. Thomas, D. Huard, and 615 J.-P. Auclair. Improving the numerical convergence of viscous-plastic sea ice 616 models with the jacobian-free newtonkrylov method. J. Comp. Phys., 229(8): 617 2840–2852, 2010. doi: http://dx.doi.org/10.1016/j.jcp.2009.12.011. 618 J.-F. Lemieux, D. Knoll, B. Tremblay, D.M. Holland, and M. Losch. A com- 619 parison of the Jacobian-free Newton-Krylov method and the EVP model for 620 solving the sea ice momentum equation with a viscous-plastic formulation: a 621 serial algorithm study. J. Comp. Phys., 231(17):5926–5944, 2012. 622 M. Losch and S. Danilov. On solving the momentum equations of dynamic sea 623 ice models with implicit solvers and the elasticviscousplastic technique. Ocean 624 Modelling, 41:42–52, 2012. 625 M. Losch, D. Menemenlis, J.-M. Campin, P. Heimbach, and C. Hill. On the for- 626 mulation of sea-ice models. Part 1: Effects of different solver implementations 627 and parameterizations. Ocean Modelling, 3(1-2):129–144, 2010. 628 M. Losch, A. Fuchs, J.-F. Lemieux, and A. Vanselow. A parallel Jacobian-free 629 NewtonKrylov solver for a coupled sea ice-ocean model. J. Comp. Phys., 257, 630 Part A(0):901–911, 2014. 631 M. Tsamados, D. L. Feltham, and A. V. Wilchinsky. Impact of a new anisotropic 632 rheology on simulations of arctic sea ice. J. Geophys. Res.: Oceans, 118(1): 633 91–107, 2013. doi: 10.1029/2012JC007990. 25 634 635 J. Zhang and W.D Hibler. On an efficient numerical method for modeling sea ice dynamics. J. Geophys. Res., 102:8691–8702, 1997. 26 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. 27

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