# 8.6.2. SEAICE Package¶

Authors: Martin Losch, Dimitris Menemenlis, An Nguyen, Jean-Michel Campin, Patrick Heimbach, Chris Hill and Jinlun Zhang

## 8.6.2.1. Introduction¶

Package seaice provides a dynamic and thermodynamic interactive sea ice model.

CPP options enable or disable different aspects of the package
(Section 8.6.2.2). Run-time options, flags, filenames and
field-related dates/times are set in `data.seaice`

(Section 8.6.2.3).
A description of key subroutines is given in
Section 8.6.2.5.
Available diagnostics output is listed in
Section 8.6.2.6.

## 8.6.2.2. SEAICE configuration and compiling¶

### 8.6.2.2.1. Compile-time options¶

As with all MITgcm packages, SEAICE can be turned on or off at compile time (see Section 3.5)

- using the
`packages.conf`

file by adding`seaice`

to it - or using genmake2 adding
`-enable=seaice`

or`-disable=seaice`

switches **required packages and CPP options**: seaice requires the external forcing package pkg/exf to be enabled; no additional CPP options are required.

Parts of the seaice code can be enabled or disabled at compile time via CPP preprocessor flags. These options are set in SEAICE_OPTIONS.h. Table 8.16 summarizes the most important ones. For more options see SEAICE_OPTIONS.h.

CPP option | Default | Description” |
---|---|---|

SEAICE_DEBUG | #undef | enhance STDOUT for debugging |

SEAICE_ALLOW_DYNAMICS | #define | sea ice dynamics code |

SEAICE_CGRID | #define | LSR solver on C-grid (rather than original B-grid) |

SEAICE_ALLOW_EVP | #define | enable use of EVP rheology solver |

SEAICE_ALLOW_JFNK | #define | enable use of JFNK rheology solver |

SEAICE_ALLOW_KRYLOV | #define | enable use of Krylov rheology solver |

SEAICE_LSR_ZEBRA | #undef | use a coloring method for LSR solver |

SEAICE_EXTERNAL_FLUXES | #define | use pkg/exf-computed fluxes as starting point |

SEAICE_ZETA_SMOOTHREG | #define | use differentiable regularization for viscosities |

SEAICE_DELTA_SMOOTHREG | #undef | use differentiable regularization for \(1/\Delta\) |

SEAICE_ALLOW_BOTTOMDRAG | #undef | enable grounding parameterization for improved fastice in shallow seas |

SEAICE_ITD | #undef | run with dynamical sea Ice Thickness Distribution (ITD) |

SEAICE_VARIABLE_SALINITY | #undef | enable sea ice with variable salinity |

ALLOW_SITRACER | #undef | enable sea ice tracer package |

SEAICE_BICE_STRESS | #undef | B-grid only for backward compatiblity: turn on ice-stress on ocean |

EXPLICIT_SSH_SLOPE | #undef | B-grid only for backward compatiblity: use ETAN for tilt computations rather than geostrophic velocities |

## 8.6.2.3. Run-time parameters¶

Run-time parameters (see Table 8.17) are set in data.seaice (read in pkg/seaice/seaice_readparms.F).

### 8.6.2.3.1. Enabling the package¶

seaice package is switched on/off at runtime by setting useSEAICE `= .TRUE.`

in `data.pkg`

.

### 8.6.2.3.2. General flags and parameters¶

Table 8.17 lists most run-time parameters.

Name | Default value | Description |
---|---|---|

SEAICEwriteState | FALSE | write sea ice state to file |

SEAICEuseDYNAMICS | TRUE | use dynamics |

SEAICEuseJFNK | FALSE | use the JFNK-solver |

SEAICEuseTEM | FALSE | use truncated ellipse method |

SEAICEuseStrImpCpl | FALSE | use strength implicit coupling in LSR/JFNK |

SEAICEuseMetricTerms | TRUE | use metric terms in dynamics |

SEAICEuseEVPpickup | TRUE | use EVP pickups |

SEAICEuseFluxForm | TRUE | use flux form for 2nd central difference advection scheme |

SEAICErestoreUnderIce | FALSE | enable restoring to climatology under ice |

SEAICEupdateOceanStress | TRUE | update ocean surface stress accounting for sea ice cover |

SEAICEscaleSurfStress | TRUE | scale atmosphere and ocean-surface stress on ice by concentration (AREA) |

SEAICEaddSnowMass | TRUE | in computing seaiceMass, add snow contribution |

useHB87stressCoupling | FALSE | turn on ice-ocean stress coupling following |

usePW79thermodynamics | TRUE | flag to turn off zero-layer-thermodynamics for testing |

SEAICEadvHeff | TRUE | flag to turn off advection of scalar variable HEFF |

SEAICEadvArea | TRUE | flag to turn off advection of scalar variable AREA |

SEAICEadvSnow | TRUE | flag to turn off advection of scalar variable HSNOW |

SEAICEadvSalt | TRUE | flag to turn off advection of scalar variable HSALT |

SEAICEadvScheme | 77 | set advection scheme for seaice scalar state variables |

SEAICEuseFlooding | TRUE | use flood-freeze algorithm |

SEAICE_no_slip | FALSE | use no-slip boundary conditions instead of free-slip |

SEAICE_deltaTtherm | dTtracerLev (1) | time step for seaice thermodynamics (s) |

SEAICE_deltaTdyn | dTtracerLev (1) | time step for seaice dynamics (s) |

SEAICE_deltaTevp | 0.0 | EVP sub-cycling time step (s); values \(>\) 0 turn on EVP |

SEAICEuseEVPstar | FALSE | use modified EVP* instead of EVP, following [lemieux12] |

SEAICEuseEVPrev | FALSE | “revisited form” variation on EVP*, following [bouillon13] |

SEAICEnEVPstarSteps | unset | number of modified EVP* iterations |

SEAICE_evpAlpha | unset | EVP* parameter (non-dim.), to replace 2*SEAICE_evpTauRelax/SEAICE_deltaTevp |

SEAICE_evpBeta | unset | EVP* parameter (non-dim.), to replace SEAICE_deltaTdyn/SEAICE_deltaTevp |

SEAICEaEVPcoeff | unset | largest stabilized frequency for adaptive EVP (non-dim.) |

SEAICEaEVPcStar | 4.0 | aEVP multiple of stability factor (non-dim.), see [kimmritz16] \(\alpha * \beta = c^\ast * \gamma\) |

SEAICEaEVPalphaMin | 5.0 | aEVP lower limit of alpha and beta (non-dim.), see [kimmritz16] |

SEAICE_elasticParm | 0.33333333 | EVP parameter \(E_0\) (non-dim.), sets relaxation timescale SEAICE_evpTauRelaxtau = SEAICE_elasticParm * SEAICE_deltaTdyn |

SEAICE_evpTauRelax | dTtracerLev (1) * SEAICE_elasticParm | relaxation time scale \(T\) for EVP waves (s) |

SEAICE_OLx | OLx - 2 | overlap for LSR-solver or preconditioner, \(x\)-dimension |

SEAICE_OLy | OLy - 2 | overlap for LSR-solver or preconditioner, \(y\)-dimension |

SEAICEnonLinIterMax | 2/10 | maximum number of non-linear (outer loop) iterations (LSR/JFNK) |

SEAICElinearIterMax | 1500/10 | maximum number of linear iterations (LSR/JFNK) |

SEAICE_JFNK_lsIter | (off) | start line search after “lsIter” Newton iterations |

SEAICEnonLinTol | 1.0E-05 | non-linear tolerance parameter for JFNK solver |

JFNKgamma_lin_min | 0.10 | minimum tolerance parameter for linear JFNK solver |

JFNKgamma_lin_max | 0.99 | maximum tolerance parameter for linear JFNK solver |

JFNKres_tFac | unset | tolerance parameter for FGMRES residual |

SEAICE_JFNKepsilon | 1.0E-06 | step size for the FD-gradient in s/r seaice_jacvec |

SEAICE_dumpFreq | dumpFreq | dump frequency (s) |

SEAICE_dump_mdsio | TRUE | write snapshot using /pkg/mdsio |

SEAICE_dump_mnc | FALSE | write snapshot using /pkg/mnc |

SEAICE_initialHEFF | 0.0 | initial sea ice thickness averaged over grid cell (m) |

SEAICE_drag | 1.0E-03 | air-ice drag coefficient (non-dim.) |

OCEAN_drag | 1.0E-03 | air-ocean drag coefficient (non-dim.) |

SEAICE_waterDrag | 5.5E-03 | water-ice drag coefficient (non-dim.) |

SEAICE_dryIceAlb | 0.75 | winter sea ice albedo |

SEAICE_wetIceAlb | 0.66 | summer sea ice albedo |

SEAICE_drySnowAlb | 0.84 | dry snow albedo |

SEAICE_wetSnowAlb | 0.70 | wet snow albedo |

SEAICE_waterAlbedo | 0.10 | water albedo (not used if #define SEAICE_EXTERNAL_FLUXES) |

SEAICE_strength | 2.75E+04 | sea ice strength constant \(P^{\ast}\) (N/m^{2}) |

SEAICE_cStar | 20.0 | sea ice strength constant \(C^{\ast}\) (non-dim.) |

SEAICE_rhoAir | 1.3 (or pkg/exf value) | density of air (kg/m^{3}) |

SEAICE_cpAir | 1004.0 (or pkg/exf value) | specific heat of air (J/kg/K) |

SEAICE_lhEvap | 2.5E+06 (or pkg/exf value) | latent heat of evaporation (J/kg) |

SEAICE_lhFusion | 3.34E+05 (or pkg/exf value) | latent heat of fusion (J/kg) |

SEAICE_dalton | 1.75E-03 | ice-ocean transfer coefficient for latent and sensible heat (non-dim.) |

useMaykutSatVapPoly | FALSE | use Maykut polynomial to compute saturation vapor pressure |

SEAICE_iceConduct | 2.16560E+00 | sea ice conductivity (W m^{-1} K^{-1}) |

SEAICE_snowConduct | 3.10000E-01 | snow conductivity (W m^{-1} K^{-1}) |

SEAICE_emissivity | 0.970018 (or pkg/exf value) | longwave ocean surface emissivity (non-dim.) |

SEAICE_snowThick | 0.15 | cutoff snow thickness to use snow albedo (m) |

SEAICE_shortwave | 0.30 | ice penetration shortwave radiation factor (non-dim.) |

SEAICE_saltFrac | 0.0 | salinity newly formed ice (as fraction of ocean surface salinity) |

SEAICE_frazilFrac | 1.0 (or computed from other parms) | frazil to sea ice conversion rate, as fraction (relative to the local freezing point of sea ice water) |

SEAICEstressFactor | 1.0 | scaling factor for ice area in computing total ocean stress (non-dim.) |

HeffFile | unset | filename for initial sea ice eff. thickness field HEFF (m) |

AreaFile | unset | filename for initial fraction sea ice cover AREA (non-dim.) |

HsnowFile | unset | filename for initial eff. snow thickness field HSNOW (m) |

HsaltFile | unset | filename for initial eff. sea ice salinity field HSALT
(g/m^{2}) |

LSR_ERROR | 1.0E-04 | sets accuracy of LSR solver |

DIFF1 | 0.0 | parameter used in advect.F |

HO | 0.5 | lead closing parameter \(h_0\) (m); demarcation thickness between thick and thin ice which determines partition between vertical and lateral ice growth |

MIN_ATEMP | -50.0 | minimum air temperature (^{o}C) |

MIN_LWDOWN | 60.0 | minimum downward longwave (W/m^{2}) |

MIN_TICE | -50.0 | minimum ice temperature (^{o}C) |

IMAX_TICE | 10 | number of iterations for ice surface temperature solution |

SEAICE_EPS | 1.0E-10 | a “small number” used in various routines |

SEAICE_area_reg | 1.0E-5 | minimum concentration to regularize ice thickness |

SEAICE_hice_reg | 0.05 | minimum ice thickness (m) for regularization |

SEAICE_multDim | 1 | number of ice categories for thermodynamics |

SEAICE_useMultDimSnow | TRUE | use same fixed pdf for snow as for multi-thickness-category ice |

The following dynamical ice thickness distribution and ridging parameters in Table 8.18 are only active with #define SEAICE_ITD. All parameters are non-dimensional unless indicated.

Name | Default value | Description |
---|---|---|

useHibler79IceStrength | TRUE | use [hib79] ice strength; do not use [rothrock:75] with #define SEAICE_ITD |

SEAICEsimpleRidging | TRUE | use simple ridging a la [hib79] |

SEAICE_cf | 17.0 | scaling parameter of [rothrock:75] ice strength parameterization |

SEAICEpartFunc | 0 | use partition function of [thorndike:75] |

SEAICEredistFunc | 0 | use redistribution function of [hib80] |

SEAICEridgingIterMax | 10 | maximum number of ridging sweeps |

SEAICEshearParm | 0.5 | fraction of shear to be used for ridging |

SEAICEgStar | 0.15 | max. ice conc. that participates in ridging [thorndike:75] |

SEAICEhStar | 25.0 | ridging parameter for [thorndike:75], [lipscomb:07] |

SEAICEaStar | 0.05 | similar to SEAICEgStar for [lipscomb:07] participation function |

SEAICEmuRidging | 3.0 | similar to SEAICEhStar for [lipscomb:07] ridging function |

SEAICEmaxRaft | 1.0 | regularization parameter for rafting |

SEAICEsnowFracRidge | 0.5 | fraction of snow that remains on ridged ice |

SEAICEuseLinRemapITD | TRUE | use linear remapping scheme of [lipscomb:01] |

Hlimit | unset | nITD+1-array of ice thickness category limits (m) |

Hlimit_c1, Hlimit_c2, Hlimit_c3 | 3.0, 15.0, 3.0 | when Hlimit is not set, then these parameters determine Hlimit from a simple function following [lipscomb:01] |

## 8.6.2.4. Description¶

The MITgcm sea ice model is based on a variant of the viscous-plastic (VP) dynamic-thermodynamic sea ice model (Zhang and Hibler 1997 [zhang97]) first introduced in Hibler (1979) and Hibler (1980) [hib79][hib80]. In order to adapt this model to the requirements of coupled ice-ocean state estimation, many important aspects of the original code have been modified and improved, see Losch et al. (2010) [losch:10]:

- the code has been rewritten for an Arakawa C-grid, both B- and C-grid variants are available; the C-grid code allows for no-slip and free-slip lateral boundary conditions;
- three different solution methods for solving the nonlinear momentum equations have been adopted: LSOR (Zhang and Hibler 1997 [zhang97]), EVP (Hunke and Dukowicz 1997 [hun97]), JFNK (Lemieux et al. 2010 [lemieux10], Losch et al. 2014 [losch:14]);
- ice-ocean stress can be formulated as in Hibler and Bryan (1987) [hibler87] or as in Campin et al. (2008) [cam:08];
- ice variables are advected by sophisticated, conservative advection schemes with flux limiting;
- growth and melt parameterizations have been refined and extended in order to allow for more stable automatic differentiation of the code.

The sea ice model is tightly coupled to the ocean compontent of the MITgcm. Heat, fresh water fluxes and surface stresses are computed from the atmospheric state and, by default, modified by the ice model at every time step.

The ice dynamics models that are most widely used for large-scale climate studies are the viscous-plastic (VP) model (Hilber 1979 [hib79]), the cavitating fluid (CF) model (Flato and Hibler 1992 [fla92]), and the elastic-viscous-plastic (EVP) model (Hunke and Dukowicz 1997 [hun97]). Compared to the VP model, the CF model does not allow ice shear in calculating ice motion, stress, and deformation. EVP models approximate VP by adding an elastic term to the equations for easier adaptation to parallel computers. Because of its higher accuracy in plastic solution and relatively simpler formulation, compared to the EVP model, we decided to use the VP model as the default dynamic component of our ice model. To do this we extended the line successive over relaxation (LSOR) method of Zhang and Hibler (1997) [zhang97] for use in a parallel configuration. An EVP model and a free-drift implementation can be selected with run-time flags.

### 8.6.2.4.1. Compatibility with ice-thermodynamics package pkg/thsice¶

By default pkg/seaice includes the original so-called zero-layer thermodynamics with a snow cover as in the appendix of Semtner (1976) [sem76]. The zero-layer thermodynamic model assumes that ice does not store heat and, therefore, tends to exaggerate the seasonal variability in ice thickness. This exaggeration can be significantly reduced by using Winton’s (Winton 2000 [win00]) three-layer thermodynamic model that permits heat storage in ice.

The Winton (2000) sea-ice thermodynamics have been ported to MITgcm; they currently
reside under pkg/thsice, described in Section 8.6.1.
It is fully compatible with the packages seaice and exf.
When turned on together with seaice, the zero-layer
thermodynamics are replaced by the Winton thermodynamics. In order to use
package seaice with the thermodynamics of pkg/thsice,
compile both packages and turn both package on in `data.pkg`

; see an example
in verification/global_ocean.cs32x15/input.icedyn. Note, that once
thsice is turned on, the variables and diagnostics
associated to the default thermodynamics are meaningless, and the diagnostics
of thsice must be used instead.

### 8.6.2.4.2. Surface forcing¶

The sea ice model requires the following input fields: 10 m winds, 2 m air temperature and specific humidity, downward longwave and shortwave radiations, precipitation, evaporation, and river and glacier runoff. The sea ice model also requires surface temperature from the ocean model and the top level horizontal velocity. Output fields are surface wind stress, evaporation minus precipitation minus runoff, net surface heat flux, and net shortwave flux. The sea-ice model is global: in ice-free regions bulk formulae (by default computed in package exf) are used to estimate oceanic forcing from the atmospheric fields.

### 8.6.2.4.3. Dynamics¶

The momentum equation of the sea-ice model is

where \(m=m_{i}+m_{s}\) is the ice and snow mass per unit area; \(\mathbf{u}=u\mathbf{i}+v\mathbf{j}\) is the ice velocity vector; \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are unit vectors in the \(x\), \(y\), and \(z\) directions, respectively; \(f\) is the Coriolis parameter; \(\mathbf{\tau}_\mathrm{air}\) and \(\mathbf{\tau}_\mathrm{ocean}\) are the wind-ice and ocean-ice stresses, respectively; \(g\) is the gravity accelation; \(\nabla\phi(0)\) is the gradient (or tilt) of the sea surface height; \(\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}\) is the sea surface height potential in response to ocean dynamics (\(g\eta\)), to atmospheric pressure loading (\(p_{a}/\rho_{0}\), where \(\rho_{0}\) is a reference density) and a term due to snow and ice loading ; and \(\mathbf{F}=\nabla\cdot\sigma\) is the divergence of the internal ice stress tensor \(\sigma_{ij}\). Advection of sea-ice momentum is neglected. The wind and ice-ocean stress terms are given by

where \(\mathbf{U}_\mathrm{air/ocean}\) are the surface winds of the atmosphere and surface currents of the ocean, respectively; \(C_\mathrm{air/ocean}\) are air and ocean drag coefficients; \(\rho_\mathrm{air/ocean}\) are reference densities; and \(R_\mathrm{air/ocean}\) are rotation matrices that act on the wind/current vectors.

### 8.6.2.4.4. Viscous-Plastic (VP) Rheology¶

For an isotropic system the stress tensor \(\sigma_{ij}\) (\(i,j=1,2\)) can be related to the ice strain rate and strength by a nonlinear viscous-plastic (VP) constitutive law:

The ice strain rate is given by

The maximum ice pressure \(P_{\max}\), a measure of ice strength, depends on both thickness \(h\) and compactness (concentration) \(c\):

with the constants \(P^{\ast}\) (run-time parameter SEAICE_strength) and \(C^{\ast}\) (run-time parameter SEAICE_cStar). The nonlinear bulk and shear viscosities \(\zeta\) and \(\eta\) are functions of ice strain rate invariants and ice strength such that the principal components of the stress lie on an elliptical yield curve with the ratio of major to minor axis \(e\) equal to \(2\); they are given by:

with the abbreviation

\[\Delta = \left[ \left(\dot{\epsilon}_{11}+\dot{\epsilon}_{22}\right)^2 + e^{-2}\left( \left(\dot{\epsilon}_{11}-\dot{\epsilon}_{22} \right)^2 + \dot{\epsilon}_{12}^2 \right) \right]^{\frac{1}{2}}\]

The bulk viscosities are bounded above by imposing both a minimum \(\Delta_{\min}\) (for numerical reasons, run-time parameter SEAICE_deltaMin is set to a default value of \(10^{-10}\,\text{s}^{-1}\), the value of SEAICE_EPS) and a maximum \(\zeta_{\max} = P_{\max}/(2\Delta^\ast)\), where \(\Delta^\ast=(2\times10^4/5\times10^{12})\,\text{s}^{-1}\) \(= 2\times10^{-9}\,\text{s}^{-1}\). Obviously, this corresponds to regularizing \(\Delta\) with the typical value of SEAICE_deltaMin \(= 2\times10^{-9}\). Clearly, some of this regularization is redundant. (There is also the option of bounding \(\zeta\) from below by setting run-time parameter SEAICE_zetaMin \(>0\), but this is generally not recommended). For stress tensor computation the replacement pressure \(P = 2\,\Delta\zeta\) is used so that the stress state always lies on the elliptic yield curve by definition.

Defining the CPP-flag SEAICE_ZETA_SMOOTHREG in SEAICE_OPTIONS.h before compiling replaces the method for bounding \(\zeta\) by a smooth (differentiable) expression:

where \(\Delta_{\min}=10^{-20}\,\text{s}^{-1}\) should be chosen to avoid divisions by zero.

### 8.6.2.4.5. LSR and JFNK solver¶

In matrix notation, the discretized momentum equations can be written as

The solution vector \(\mathbf{x}\) consists of the two velocity components \(u\) and \(v\) that contain the velocity variables at all grid points and at one time level. The standard (and default) method for solving Eq. (8.6) in the sea ice component of MITgcm is an iterative Picard solver: in the \(k\)-th iteration a linearized form \(\mathbf{A}(\mathbf{x}^{k-1})\,\mathbf{x}^{k} = \mathbf{b}(\mathbf{x}^{k-1})\) is solved (in the case of MITgcm it is a Line Successive (over) Relaxation (LSR) algorithm). Picard solvers converge slowly, but in practice the iteration is generally terminated after only a few nonlinear steps and the calculation continues with the next time level. This method is the default method in MITgcm. The number of nonlinear iteration steps or pseudo-time steps can be controlled by the run-time parameter SEAICEnonLinIterMax (default is 2).

In order to overcome the poor convergence of the Picard-solver, Lemieux et al. (2010) [lemieux10] introduced a Jacobian-free Newton-Krylov solver for the sea ice momentum equations. This solver is also implemented in MITgcm (see Losch et al. 2014 [losch:14]). The Newton method transforms minimizing the residual \(\mathbf{F}(\mathbf{x}) = \mathbf{A}(\mathbf{x})\,\mathbf{x} - \mathbf{b}(\mathbf{x})\) to finding the roots of a multivariate Taylor expansion of the residual \(\mathbf{F}\) around the previous (\(k-1\)) estimate \(\mathbf{x}^{k-1}\):

with the Jacobian \(\mathbf{J}\equiv\mathbf{F}'\). The root \(\mathbf{F}(\mathbf{x}^{k-1}+\delta\mathbf{x}^{k})=0\) is found by solving

for \(\delta\mathbf{x}^{k}\). The next (\(k\)-th) estimate is given by \(\mathbf{x}^{k}=\mathbf{x}^{k-1}+a\,\delta\mathbf{x}^{k}\). In order to avoid overshoots the factor \(a\) is iteratively reduced in a line search (\(a=1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\)) until \(\|\mathbf{F}(\mathbf{x}^k)\| < \|\mathbf{F}(\mathbf{x}^{k-1})\|\), where \(\|\cdot\|=\int\cdot\,dx^2\) is the \(L_2\)-norm. In practice, the line search is stopped at \(a=\frac{1}{8}\). The line search starts after SEAICE_JFNK_lsIter nonlinear Newton iterations (off by default).

Forming the Jacobian \(\mathbf{J}\) explicitly is often avoided as “too error prone and time consuming”. Instead, Krylov methods only require the action of \(\mathbf{J}\) on an arbitrary vector \(\mathbf{w}\) and hence allow a matrix free algorithm for solving (8.8). The action of \(\mathbf{J}\) can be approximated by a first-order Taylor series expansion:

or computed exactly with the help of automatic differentiation (AD) tools. SEAICE_JFNKepsilon sets the step size \(\epsilon\).

We use the Flexible Generalized Minimum RESidual (FMGRES) method with right-hand side preconditioning to solve (8.8) iteratively starting from a first guess of \(\delta\mathbf{x}^{k}_{0} = 0\). For the preconditioning matrix \(\mathbf{P}\) we choose a simplified form of the system matrix \(\mathbf{A}(\mathbf{x}^{k-1})\) where \(\mathbf{x}^{k-1}\) is the estimate of the previous Newton step \(k-1\). The transformed equation (8.8) becomes

The Krylov method iteratively improves the approximate solution to (8.10) in subspace (\(\mathbf{r}_0\), \(\mathbf{J}\mathbf{P}^{-1}\mathbf{r}_0\), \((\mathbf{J}\mathbf{P}^{-1})^2\mathbf{r}_0\), \(\dots\), \((\mathbf{J}\mathbf{P}^{-1})^m\mathbf{r}_0\)) with increasing \(m\); \(\mathbf{r}_0 = -\mathbf{F}(\mathbf{x}^{k-1}) -\mathbf{J}(\mathbf{x}^{k-1})\,\delta\mathbf{x}^{k}_{0}\) is the initial residual of (8.8); \(\mathbf{r}_0=-\mathbf{F}(\mathbf{x}^{k-1})\) with the first guess \(\delta\mathbf{x}^{k}_{0}=0\). We allow a Krylov-subspace of dimension \(m=50\) and we do allow restarts for more than 50 Krylov iterations. The preconditioning operation involves applying \(\mathbf{P}^{-1}\) to the basis vectors \(\mathbf{v}_0, \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_m\) of the Krylov subspace. This operation is approximated by solving the linear system \(\mathbf{P}\,\mathbf{w}=\mathbf{v}_i\). Because \(\mathbf{P} \approx \mathbf{A}(\mathbf{x}^{k-1})\), we can use the LSR-algorithm already implemented in the Picard solver. Each preconditioning operation uses a fixed number of 10 LSR-iterations avoiding any termination criterion. More details and results can be found in Losch et al. (2014) [losch:14]).

To use the JFNK-solver set SEAICEuseJFNK `= .TRUE.,`

in the namelist file
`data.seaice`

; `#define`

SEAICE_ALLOW_JFNK in
SEAICE_OPTIONS.h and we recommend using a
smooth regularization of \(\zeta\) by `#define`

SEAICE_ZETA_SMOOTHREG
(see above) for better convergence. The nonlinear Newton iteration is terminated when
the \(L_2\)-norm of the residual is reduced by \(\gamma_{\mathrm{nl}}\)
(run-time parameter SEAICEnonLinTol `= 1.E-4,`

will already lead to expensive simulations)
with respect to the initial norm: \(\|\mathbf{F}(\mathbf{x}^k)\| <
\gamma_{\mathrm{nl}}\|\mathbf{F}(\mathbf{x}^0)\|\).
Within a nonlinear iteration, the linear FGMRES solver is terminated
when the residual is smaller than \(\gamma_k\|\mathbf{F}(\mathbf{x}^{k-1})\|\) where \(\gamma_k\) is determined by

so that the linear tolerance parameter \(\gamma_k\) decreases with the nonlinear Newton step as the nonlinear solution is approached. This inexact Newton method is generally more robust and computationally more efficient than exact methods. Typical parameter choices are \(\gamma_0 =\) JFNKgamma_lin_max \(= 0.99\), \(\gamma_{\min} =\) JFNKgamma_lin_min \(= 0.1\), and \(r =\) JFNKres_tFac \(\times\|\mathbf{F}(\mathbf{x}^{0})\|\) with JFNKres_tFac \(= 0.5\). We recommend a maximum number of nonlinear iterations SEAICEnewtonIterMax \(= 100\) and a maximum number of Krylov iterations SEAICEkrylovIterMax \(= 50\), because the Krylov subspace has a fixed dimension of 50 (but restarts are allowed for SEAICEkrylovIterMax \(> 50\)).

Setting SEAICEuseStrImpCpl `= .TRUE.,`

turns on “strength implicit
coupling” (see Hutchings et al. 2004 [hutchings04]) in the LSR-solver and in the
LSR-preconditioner for the JFNK-solver. In this mode, the different contributions of the stress
divergence terms are reordered so as to increase the diagonal dominance of the system matrix.
Unfortunately, the convergence rate of the LSR solver is increased only slightly,
while the JFNK-convergence appears to be unaffected.

### 8.6.2.4.6. Elastic-Viscous-Plastic (EVP) Dynamics¶

Hunke and Dukowicz (1997) [hun97] introduced an elastic contribution to the strain rate in order to regularize (8.3) in such a way that the resulting elastic-viscous-plastic (EVP) and VP models are identical at steady state,

The EVP-model uses an explicit time stepping scheme with a short timestep. According to the recommendation in Hunke and Dukowicz (1997) [hun97], the EVP-model should be stepped forward in time 120 times (SEAICE_deltaTevp = SEAICE_deltaTdyn /120) within the physical ocean model time step (although this parameter is under debate), to allow for elastic waves to disappear. Because the scheme does not require a matrix inversion it is fast in spite of the small internal timestep and simple to implement on parallel computers. For completeness, we repeat the equations for the components of the stress tensor \(\sigma_{1} = \sigma_{11}+\sigma_{22}\), \(\sigma_{2}= \sigma_{11}-\sigma_{22}\), and \(\sigma_{12}\). Introducing the divergence \(D_D = \dot{\epsilon}_{11}+\dot{\epsilon}_{22}\), and the horizontal tension and shearing strain rates, \(D_T = \dot{\epsilon}_{11}-\dot{\epsilon}_{22}\) and \(D_S = 2\dot{\epsilon}_{12}\), respectively, and using the above abbreviations, the equations (8.12) can be written as:

Here, the elastic parameter \(E\) is redefined in terms of a damping timescale \(T\) for elastic waves

\(T=E_{0}\Delta{t}\) with the tunable parameter \(E_0<1\) and the external (long) timestep \(\Delta{t}\). \(E_{0} = \frac{1}{3}\) is the default value in the code and close to what and recommend.

To use the EVP solver, make sure that both `#define`

SEAICE_CGRID and
`#define`

SEAICE_ALLOW_EVP are set in
SEAICE_OPTIONS.h
(both are defined by default). The solver is turned on by setting the sub-cycling time
step SEAICE_deltaTevp to a value larger than zero. The choice of
this time step is under debate. Hunke and Dukowicz (1997) [hun97] recommend order 120
time steps for the EVP solver within one model time step
\(\Delta{t}\) (deltaTmom). One can also choose order 120 time
steps within the forcing time scale, but then we recommend adjusting
the damping time scale \(T\) accordingly, by setting either SEAICE_elasticParm (\(E_{0}\)),
so that \(E_{0}\Delta{t}=\) forcing time scale, or directly SEAICE_evpTauRelax (\(T\))
to the forcing time scale. (**NOTE**: with the improved EVP variants of the next section,
the above recommendations are obsolete. Use mEVP or aEVP instead.)

### 8.6.2.4.7. More stable variants of Elastic-Viscous-Plastic Dynamics: EVP*, mEVP, and aEVP¶

The genuine EVP scheme appears to give noisy solutions (see Hunke 2001, Lemieux et al. 2012, Bouillon et a1. 2013 [hun01][lemieux12][bouillon13]). This has led to a modified EVP or EVP* (Lemieux et al. 2012, Bouillon et a1. 2013, Kimmritz et al. 2015 [lemieux12][bouillon13][kimmritz15]); here, we refer to these variants by modified EVP (mEVP) and adaptive EVP (aEVP). The main idea is to modify the “natural” time-discretization of the momentum equations:

where \(n\) is the previous time step index, and \(p\) is the previous sub-cycling index. The extra “intertial” term \(m\,(\mathbf{u}^{p+1}-\mathbf{u}^{n})/\Delta{t})\) allows the definition of a residual \(|\mathbf{u}^{p+1}-\mathbf{u}^{p}|\) that, as \(\mathbf{u}^{p+1} \rightarrow \mathbf{u}^{n+1}\), converges to \(0\). In this way EVP can be re-interpreted as a pure iterative solver where the sub-cycling has no association with time-relation (through \(\Delta{t}_{\mathrm{EVP}}\)). Using the terminology of Kimmritz et al. 2015 [kimmritz15], the evolution equations of stress \(\sigma_{ij}\) and momentum \(\mathbf{u}\) can be written as:

\(\mathbf{R}\) contains all terms in the momentum equations except for the rheology terms and the time derivative; \(\alpha\) and \(\beta\) are free parameters (SEAICE_evpAlpha, SEAICE_evpBeta) that replace the time stepping parameters SEAICE_deltaTevp (\(\Delta{t}_{\mathrm{EVP}}\)), SEAICE_elasticParm (\(E_{0}\)), or SEAICE_evpTauRelax (\(T\)). \(\alpha\) and \(\beta\) determine the speed of convergence and the stability. Usually, it makes sense to use \(\alpha = \beta\), and SEAICEnEVPstarSteps \(\gg (\alpha,\,\beta)\) (Kimmritz et al. 2015 [kimmritz15]). Currently, there is no termination criterion and the number of mEVP iterations is fixed to SEAICEnEVPstarSteps.

In order to use mEVP in MITgcm, set SEAICEuseEVPstar `= .TRUE.,`

in `data.seaice`

. If SEAICEuseEVPrev `=.TRUE.,`

the actual form of
equations (8.17) and (8.18) is used with fewer
implicit terms and the factor of \(e^{2}\) dropped in the stress
equations (8.14) and
(8.15). Although this modifies the original
EVP-equations, it turns out to improve convergence (Bouillon et al. 2013 [bouillon13]).

Another variant is the aEVP scheme (Kimmritz et al. 2016 [kimmritz16]), where the value of \(\alpha\) is set dynamically based on the stability criterion

with the grid cell area \(A_c\) and the ice and snow mass \(m\).
This choice sacrifices speed of convergence for stability with the
result that aEVP converges quickly to VP where \(\alpha\) can be
small and more slowly in areas where the equations are stiff. In
practice, aEVP leads to an overall better convergence than mEVP (Kimmritz et al. 2016 [kimmritz16]).
To use aEVP in MITgcm set SEAICEaEVPcoeff \(= \tilde{c}\);
this also sets the default values of SEAICEaEVPcStar (\(c=4\))
and SEAICEaEVPalphaMin (\(\alpha_{\min}=5\)). Good convergence
has been obtained with these values (Kimmritz et al. 2016 [kimmritz16]):
SEAICEaEVPcoeff \(= 0.5\), SEAICEnEVPstarSteps \(= 500\),
SEAICEuseEVPstar `= .TRUE.`

, SEAICEuseEVPrev `= .TRUE.`

.

Note, that probably because of the C-grid staggering of velocities and stresses, mEVP may not converge as successfully as in Kimmritz et al. (2015) [kimmritz15], see also Kimmritz et al. (2016) [kimmritz16], and that convergence at very high resolution (order 5 km) has not been studied yet.

### 8.6.2.4.8. Truncated ellipse method (TEM) for yield curve¶

In the so-called truncated ellipse method the shear viscosity \(\eta\) is capped to suppress any tensile stress:

To enable this method, set `#define`

SEAICE_ALLOW_TEM in
SEAICE_OPTIONS.h and turn
it on with SEAICEuseTEM in `data.seaice`

.

### 8.6.2.4.9. Ice-Ocean stress¶

Moving sea ice exerts a stress on the ocean which is the opposite of
the stress \(\mathbf{\tau}_\mathrm{ocean}\) in
(8.2). This stress is applied directly to the surface
layer of the ocean model. An alternative ocean stress formulation is
given by Hibler and Bryan (1987) [hibler87]. Rather than applying
\(\mathbf{\tau}_\mathrm{ocean}\) directly, the stress is derived from
integrating over the ice thickness to the bottom of the oceanic
surface layer. In the resulting equation for the *combined* ocean-ice
momentum, the interfacial stress cancels and the total stress appears
as the sum of windstress and divergence of internal ice stresses:
\(\delta(z) (\mathbf{\tau}_\mathrm{air} + \mathbf{F})/\rho_0\), see also
Eq. (2) of Hibler and Bryan (1987) [hibler87]. The disadvantage of this formulation is
that now the velocity in the surface layer of the ocean that is used
to advect tracers, is really an average over the ocean surface
velocity and the ice velocity leading to an inconsistency as the ice
temperature and salinity are different from the oceanic variables. To
turn on the stress formulation of Hibler and Bryan (1987) [hibler87], set
useHB87StressCoupling `=.TRUE.`

, in `data.seaice`

.

### 8.6.2.4.10. Finite-volume discretization of the stress tensor divergence¶

On an Arakawa C grid, ice thickness and concentration and thus ice strength \(P\) and bulk and shear viscosities \(\zeta\) and \(\eta\) are naturally defined a C-points in the center of the grid cell. Discretization requires only averaging of \(\zeta\) and \(\eta\) to vorticity or Z-points (or \(\zeta\)-points, but here we use Z in order avoid confusion with the bulk viscosity) at the bottom left corner of the cell to give \(\overline{\zeta}^{Z}\) and \(\overline{\eta}^{Z}\). In the following, the superscripts indicate location at Z or C points, distance across the cell (F), along the cell edge (G), between \(u\)-points (U), \(v\)-points (V), and C-points (C). The control volumes of the \(u\)- and \(v\)-equations in the grid cell at indices \((i,j)\) are \(A_{i,j}^{w}\) and \(A_{i,j}^{s}\), respectively. With these definitions (which follow the model code documentation except that \(\zeta\)-points have been renamed to Z-points), the strain rates are discretized as:

so that the diagonal terms of the strain rate tensor are naturally defined at C-points and the symmetric off-diagonal term at Z-points. No-slip boundary conditions (\(u_{i,j-1}+u_{i,j}=0\) and \(v_{i-1,j}+v_{i,j}=0\) across boundaries) are implemented via “ghost-points”; for free slip boundary conditions \((\epsilon_{12})^Z=0\) on boundaries.

For a spherical polar grid, the coefficients of the metric terms are \(k_{1}=0\) and \(k_{2}=-\tan\phi/a\), with the spherical radius \(a\) and the latitude \(\phi\); \(\Delta{x}_1 = \Delta{x} = a\cos\phi \Delta\lambda\), and \(\Delta{x}_2 = \Delta{y}=a\Delta\phi\). For a general orthogonal curvilinear grid, \(k_{1}\) and \(k_{2}\) can be approximated by finite differences of the cell widths:

The stress tensor is given by the constitutive viscous-plastic relation \(\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} + [(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 ]\delta_{\alpha\beta}\) . The stress tensor divergence \((\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}\), is discretized in finite volumes . This conveniently avoids dealing with further metric terms, as these are “hidden” in the differential cell widths. For the \(u\)-equation (\(\alpha=1\)) we have:

with

Similarly, we have for the \(v\)-equation (\(\alpha=2\)):

with

Again, no-slip boundary conditions are realized via ghost points and \(u_{i,j-1}+u_{i,j}=0\) and \(v_{i-1,j}+v_{i,j}=0\) across boundaries. For free-slip boundary conditions the lateral stress is set to zeros. In analogy to \((\epsilon_{12})^Z=0\) on boundaries, we set \(\sigma_{21}^{Z}=0\), or equivalently \(\eta_{i,j}^{Z}=0\), on boundaries.

### 8.6.2.4.11. Thermodynamics¶

**NOTE: THIS SECTION IS STILL NOT COMPLETE**

In its original formulation the sea ice model uses simple 0-layer thermodynamics following the appendix of Semtner (1976) [sem76]. This formulation neglects storage of heat, that is, the heat capacity of ice is zero, and all internal heat sources so that the heat equation reduces to a constant conductive heat flux. This constant upward conductive heat flux together with a constant ice conductivity implies a linear temperature profile. The boundary conditions for the heat equations are: at the bottom of the ice \(T|_{bottom} = T_{fr}\) (freezing point temperature of sea water), and at the surface: \(Q_{top} = \frac{\partial{T}}{\partial{z}} = (K/h)(T_{0}-T_{fr})\), where \(K\) is the ice conductivity, \(h\) the ice thickness, and \(T_{0}-T_{fr}\) the difference between the ice surface temperature and the water temperature at the bottom of the ice (at the freezing point). The surface heat flux \(Q_{top}\) is computed in a similar way to that of Parkinson and Washington (1979) [parkinson:79] and Manabe et al. (1979) [manabe:79]. The resulting equation for surface temperature is

where \(\epsilon\) is the emissivity of the surface (snow or ice), \(Q_{S/LW\downarrow}\) the downwelling shortwave and longwave radiation to be prescribed, and \(Q_{LW\uparrow}=\epsilon\sigma_B T_{0}^4\) the emitted long wave radiation with the Stefan-Boltzmann constant \(\sigma_B\). With explicit expressions in \(T_0\) for the turbulent fluxes of latent and sensible heat

(8.21) can be solved for
\(T_0\) with an iterative Ralphson-Newton method, which usually
converges very quickly in less that 10 iterations. In these equations, \(\rho_\mathrm{air}\) is the air density (parameter SEAICE_rhoAir), \(C_E\) is the ice-ocean transfer coefficient for sensible and latent heat (parameter SEAICE_dalton), \(\Lambda_v\) and \(\Lambda_f\) are the latent heat of vaporization and fusion, respectively (parameters SEAICE_lhEvap and SEAICE_lhFusion), and \(c_p\) is the specific heat of air (parameter SEAICE_cpAir). For the latent heat \(Q_{LH}\) a choice can be made between the old polynomial expression for saturation humidity \(q_\mathrm{sat}(T_0)\) (by
setting useMaykutSatVapPoly to `.TRUE.`

) and the default
exponential relation approximation that is more accurate at low
temperatures.

In the zero-layer model of Semtner (1976) [sem76], the
conductive heat flux depends strongly on the ice thickness
\(h\). However, the ice thickness in the model represents a mean
over a potentially very heterogeneous thickness distribution. In order
to parameterize a sub-grid scale distribution for heat flux
computations, the mean ice thickness \(h\) is split into \(N\)
thickness categories \(H_{n}\) that are equally distributed
between \(2h\) and a minimum imposed ice thickness of
\(5\,\text{cm}\) by \(H_n= \frac{2n-1}{7}\,h\) for
\(n\in[1,N]\). The heat fluxes computed for each thickness
category are area-averaged to give the total heat flux (see Hibler
1984 [hibler84]). To use this thickness category
parameterization set SEAICE_multDim to the number of
desired categories in `data.seaice`

(7 is a good guess, for anything
larger than 7 modify SEAICE_SIZE.h). Note that this requires different
restart files and switching this flag on in the middle of an
integration is not advised. As an alternative to the flat
distribution, the run-time parameter SEAICE_PDF (1D-array of
lenght nITD) can be used to prescribe an arbitrary
distribution of ice thicknesses, for example derived from observed
distributions (Castro-Morales et al. 2014
[castro-morales14]). In order to include the ice thickness
distribution also for snow, set SEAICE_useMultDimSnow ```
=
.TRUE.
```

(this is the default); only then, the parameterization of
always having a fraction of thin ice is efficient and generally
thicker ice is produced (see Castro-Morales et al. 2014 [castro-morales14]).

The atmospheric heat flux is balanced by an oceanic heat flux from below. The oceanic flux is proportional to \(\rho\,c_{p}\left(T_{w}-T_{fr}\right)\) where \(\rho\) and \(c_{p}\) are the density and heat capacity of sea water and \(T_{fr}\) is the local freezing point temperature that is a function of salinity. This flux is not assumed to instantaneously melt or create ice, but a time scale of three days (run-time parameter SEAICE_gamma_t) is used to relax \(T_{w}\) to the freezing point. The parameterization of lateral and vertical growth of sea ice follows that of Hibler (1979) and Hibler (1980) [hib79][hib80]; the so-called lead closing parameter \(h_{0}\) (run-time parameter HO) has a default value of 0.5 meters.

On top of the ice there is a layer of snow that modifies the heat flux and the albedo (Zhang et al. 1998 [zha98a]). Snow modifies the effective conductivity according to

where \(K_s\) is the conductivity of snow and \(h_s\) the snow thickness. If enough snow accumulates so that its weight submerges the ice and the snow is flooded, a simple mass conserving parameterization of snowice formation (a flood-freeze algorithm following Archimedes’ principle) turns snow into ice until the ice surface is back at \(z=0\) (see Leppäranta 1983 [leppaeranta83]). The flood-freeze algorithm is turned on with run-time parameter SEAICEuseFlooding=.TRUE..

### 8.6.2.4.12. Advection of thermodynamic variables¶

Effective ice thickness (ice volume per unit area, \(c\cdot{h}\)), concentration \(c\) and effective snow thickness (\(c\cdot{h}_{s}\)) are advected by ice velocities:

where \(\Gamma_X\) are the thermodynamic source terms and \(D_{X}\) the diffusive terms for quantities \(X=(c\cdot{h}), c, (c\cdot{h}_{s})\). From the various advection schemes that are available in MITgcm, we recommend flux-limited schemes to preserve sharp gradients and edges that are typical of sea ice distributions and to rule out unphysical over- and undershoots (negative thickness or concentration). These schemes conserve volume and horizontal area and are unconditionally stable, so that we can set \(D_{X}=0\). Run-time flags: SEAICEadvScheme (default=77, is a 2nd-order flux limited scheme), DIFF1 = \(D_{X}/\Delta{x}\) (default=0).

The MITgcm sea ice model provides the option to use the thermodynamics
model of Winton (2000) [win00], which in turn is based on the 3-layer model of
Semtner (1976) [sem76] which treats brine content by means of enthalpy
conservation; the corresponding package thsice is described in
section Section 8.6.1. This scheme requires additional state
variables, namely the enthalpy of the two ice layers (instead of
effective ice salinity), to be advected by ice velocities. The
internal sea ice temperature is inferred from ice enthalpy. To avoid
unphysical (negative) values for ice thickness and concentration, a
positive 2nd-order advection scheme with a SuperBee flux limiter (Roe 1985
[roe:85]) should be used to advect all sea-ice-related quantities
of the Winton (2000) [win00] thermodynamic model (run-time flag
thSIceAdvScheme \(= 77\) and thSIce_diffK \(= D_{X} = 0\)
in `data.ice`

, defaults are 0). Because of the nonlinearity of the
advection scheme, care must be taken in advecting these quantities:
when simply using ice velocity to advect enthalpy, the total energy
(i.e., the volume integral of enthalpy) is not
conserved. Alternatively, one can advect the energy content (i.e.,
product of ice-volume and enthalpy) but then false enthalpy extrema
can occur, which then leads to unrealistic ice temperature. In the
currently implemented solution, the sea-ice mass flux is used to
advect the enthalpy in order to ensure conservation of enthalpy and to
prevent false enthalpy extrema.

### 8.6.2.4.13. Dynamical Ice Thickness Distribution (ITD)¶

The ice thickness distribution model used by MITgcm follows the implementation in the Los Alamos sea ice model CICE (https://github.com/CICE-Consortium/CICE). There are two parts to it that are closely connected: the participation and ridging functions that determine which thickness classes take part in ridging and which thickness classes receive ice during ridging based on Thorndike et al. (1975) [thorndike:75], and the ice strength parameterization by Rothrock (1975) [rothrock:75] which uses this information. The following description is slightly modified from Ungermann et al. (2017) [ungermann:17]. Verification experiment seaice_itd uses the ITD model.

#### Distribution, participation and redistribution functions in ridging¶

When SEAICE_ITD is defined in SEAICE_OPTIONS.h, the ice thickness is described by the ice thickness distribution \(g(h,\mathbf{x},t)\) for the subgrid-scale (see Thorndike et al. 1975 [thorndike:75]), a probability density function for thickness \(h\) following the evolution equation

Here \(f=\frac{\mathrm{d} h}{\mathrm{d} t}\) is the thermodynamic growth rate and \(\Psi\) a function describing the mechanical redistribution of sea ice during ridging or lead opening.

The mechanical redistribution function \(\Psi\) generates open water in divergent motion
and creates ridged ice during convergent motion. The ridging process depends on total strain
rate and on the ratio between shear (run-time parameter SEAICEshearParm) and divergent strain.
In the single category model, ridge formation is treated implicitly by limiting the ice concentration
to a maximum of one (see Hibler 1979 [hib79]), so that further volume increase in convergent motion
leads to thicker ice. (This is also the default for ITD models; to change from the default, set
run-time parameter SEAICEsimpleRidging `=.FALSE.`

in `data.seaice`

).
For the ITD model, the ridging mode in convergence

gives the effective change for the ice volume with thickness between \(h\) and \(h+\textrm{d} h\) as the normalized difference between the ice \(n(h)\) generated by ridging and the ice \(a(h)\) participating in ridging.

The participation function \(a(h) = b(h)g(h)\) can be computed either following Thorndike et al. (1975) [thorndike:75] (run-time parameter SEAICEpartFunc =0) or Lipscomb et al. (2007) [lipscomb:07] (SEAICEpartFunc =1), and similarly the ridging function \(n(h)\) can be computed following Hilber (1980) [hib80] (run-time parameter SEAICEredistFunc =0) or Lipscomb et al. (2007) [lipscomb:07] (SEAICEredistFunc =1). As an example, we show here the functions that Lipscomb et al. (2007) [lipscomb:07] suggested to avoid noise in the solutions. These functions are smooth and avoid non-differentiable discontinuities, but so far we did not find any noise issues as in Lipscomb et al. (2007) [lipscomb:07].

With SEAICEpartFunc =1 in `data.seaice`

, the participation function with the
relative amount of ice of thickness \(h\) weighted by an exponential function

where \(G(h)=\int_0^h g(h) \textrm{d} h\) is the cumulative thickness distribution function, \(b_0\) a normalization factor, and \(a^*\) (SEAICEaStar) the exponential constant that determines which relative amount of thicker and thinner ice take part in ridging.

With SEAICEredistFunc =1 in `data.seaice`

, the ice generated by ridging is calculated as

where the density function \(\gamma(h_1,h)\) of resulting thickness \(h\) for ridged ice with an original thickness of \(h_1\) is taken as

for \(h \geq h_{\min}\), with \(\gamma(h_1,h)=0\) for \(h < h_{\min}\). In this parameterization, the normalization factor \(k=\frac{h_{\min} + \lambda}{h_1}\), the e-folding scale \(\lambda = \mu h_1^{1/2}\) and the minimum ridge thickness \(h_{\min}=\min(2h_1,h_1 + h_{\textrm{raft}})\) all depend on the original thickness \(h_1\). The maximal ice thickness allowed to raft \(h_{\textrm{raft}}\) is constant (SEAICEmaxRaft, default =1 m) and \(\mu\) (SEAICEmuRidging) is a tunable parameter.

In the numerical model these equations are discretized into a set of \(n\)
(nITD defined in SEAICE_SIZE.h) thickness categories employing the
delta function scheme of Bitz et al. (2001) [bitz:01]. For each thickness category in an ITD configuration,
the volume conservation equation (8.22) is evaluated using
the heat flux with the category-specific values for ice and snow thickness,
so there are no conceptual differences in the
thermodynamics between the single category and ITD configurations.
The only difference is that only in the thinnest category the creation
of new ice of thickness \(H_0\) (run-time parameter HO) is possible,
all other categories are limited to basal growth. The conservation of ice
area is replaced by the evolution equation of the ITD (8.23) that is discretized
in thickness space with \(n+1\) category limits given by run-time parameter Hlimit.
If Hlimit is not set in `data.seaice`

, a simple recursive formula following Lipscomb (2001)
[lipscomb:01] is used to compute Hlimit:

with \(H_\mathrm{limit}(0)=0\,\text{m}\) and \(H_\mathrm{limit}(n)=999.9\,\text{m}\). The three constants are the run-time parameters Hlimit_c1, Hlimit_c2, and Hlimit_c3. The total ice concentration and volume can then be calculated by summing up the values for each category.

#### Ice strength parameterization¶

In the default approach of equation (8.4), the ice strength is parameterized following Hibler (1979) [hib79] and \(P\) depends only on average ice concentration and thickness per grid cell and the constant ice strength parameters \(P^{\ast}\) (SEAICE_strength) and \(C^{\ast}\) (SEAICE_cStar). With an ice thickness distribution, it is possible to use a different parameterization following Rothrock (1975) [rothrock:75]

by considering the production of potential energy and the frictional energy loss in ridging.
The physical constant \(C_p = \rho_i (\rho_w - \rho_i) \hat{g} / (2 \rho_w)\) is a
combination of the gravitational acceleration \(\hat{g}\) and the densities \(\rho_i\),
\(\rho_w\) of ice and water, and \(C_f\) (SEAICE_cf) is a scaling factor relating
the amount of work against gravity necessary for ridging to the amount of work against friction.
To calculate the integral, this parameterization needs information about the ITD in each grid cell,
while the default parameterization (8.4) can be used for both ITD and single thickness category models.
In contrast to (8.4), which is based on the plausible assumption that thick
and compact ice is stronger than thin and loose drifting ice, this parameterization (8.24)
clearly contains the more physical assumptions about energy conservation.
For that reason alone this parameterization is often considered to be more physically realistic than (8.4),
but in practice, the success is not so clear (Ungermann et al. 2007 [ungermann:17]).
Ergo, the default is to use (8.4); set useHibler79IceStrength `=.FALSE.`

in `data.seaice`

to change this behavior.

## 8.6.2.5. Key subroutines¶

Top-level routine: pkg/seaice/seaice_model.F

```
C !CALLING SEQUENCE:
c ...
c seaice_model (TOP LEVEL ROUTINE)
c |
c |-- #ifdef SEAICE_CGRID
c | SEAICE_DYNSOLVER
c | |
c | |-- < compute proxy for geostrophic velocity >
c | |
c | |-- < set up mass per unit area and Coriolis terms >
c | |
c | |-- < dynamic masking of areas with no ice >
c | |
c | |
c | #ELSE
c | DYNSOLVER
c | #ENDIF
c |
c |-- if ( useOBCS )
c | OBCS_APPLY_UVICE
c |
c |-- if ( SEAICEadvHeff .OR. SEAICEadvArea .OR. SEAICEadvSnow .OR. SEAICEadvSalt )
c | SEAICE_ADVDIFF
c |
c | SEAICE_REG_RIDGE
c |
c |-- if ( usePW79thermodynamics )
c | SEAICE_GROWTH
c |
c |-- if ( useOBCS )
c | if ( SEAICEadvHeff ) OBCS_APPLY_HEFF
c | if ( SEAICEadvArea ) OBCS_APPLY_AREA
c | if ( SEAICEadvSALT ) OBCS_APPLY_HSALT
c | if ( SEAICEadvSNOW ) OBCS_APPLY_HSNOW
c |
c |-- < do various exchanges >
c |
c |-- < do additional diagnostics >
c |
c o
```

## 8.6.2.6. SEAICE diagnostics¶

Diagnostics output is available via the diagnostics package (see Section 9.1). Available output fields are summarized in the following table:

```
---------+----------+----------------+-----------------
<-Name->|<- grid ->|<-- Units -->|<- Tile (max=80c)
---------+----------+----------------+-----------------
sIceLoad|SM U1|kg/m^2 |sea-ice loading (in Mass of ice+snow / area unit)
---
SEA ICE STATE:
---
SIarea |SM M1|m^2/m^2 |SEAICE fractional ice-covered area [0 to 1]
SIheff |SM M1|m |SEAICE effective ice thickness
SIhsnow |SM M1|m |SEAICE effective snow thickness
SIhsalt |SM M1|g/m^2 |SEAICE effective salinity
SIuice |UU M1|m/s |SEAICE zonal ice velocity, >0 from West to East
SIvice |VV M1|m/s |SEAICE merid. ice velocity, >0 from South to North
---
ATMOSPHERIC STATE AS SEEN BY SEA ICE:
---
SItices |SM C M1|K |Surface Temperature over Sea-Ice (area weighted)
SIuwind |UM U1|m/s |SEAICE zonal 10-m wind speed, >0 increases uVel
SIvwind |VM U1|m/s |SEAICE meridional 10-m wind speed, >0 increases uVel
SIsnPrcp|SM U1|kg/m^2/s |Snow precip. (+=dw) over Sea-Ice (area weighted)
---
FLUXES ACROSS ICE-OCEAN INTERFACE (ATMOS to OCEAN FOR ICE-FREE REGIONS):
---
SIfu |UU U1|N/m^2 |SEAICE zonal surface wind stress, >0 increases uVel
SIfv |VV U1|N/m^2 |SEAICE merid. surface wind stress, >0 increases vVel
SIqnet |SM U1|W/m^2 |Ocean surface heatflux, turb+rad, >0 decreases theta
SIqsw |SM U1|W/m^2 |Ocean surface shortwave radiat., >0 decreases theta
SIempmr |SM U1|kg/m^2/s |Ocean surface freshwater flux, > 0 increases salt
SIqneto |SM U1|W/m^2 |Open Ocean Part of SIqnet, turb+rad, >0 decr theta
SIqneti |SM U1|W/m^2 |Ice Covered Part of SIqnet, turb+rad, >0 decr theta
---
FLUXES ACROSS ATMOSPHERE-ICE INTERFACE (ATMOS to OCEAN FOR ICE-FREE REGIONS):
---
SIatmQnt|SM U1|W/m^2 |Net atmospheric heat flux, >0 decreases theta
SIatmFW |SM U1|kg/m^2/s |Net freshwater flux from atmosphere & land (+=down)
SIfwSubl|SM U1|kg/m^2/s |Freshwater flux of sublimated ice, >0 decreases ice
---
THERMODYNAMIC DIAGNOSTICS:
---
SIareaPR|SM M1|m^2/m^2 |SIarea preceeding ridging process
SIareaPT|SM M1|m^2/m^2 |SIarea preceeding thermodynamic growth/melt
SIheffPT|SM M1|m |SIheff preceeeding thermodynamic growth/melt
SIhsnoPT|SM M1|m |SIhsnow preceeeding thermodynamic growth/melt
SIaQbOCN|SM M1|m/s |Potential HEFF rate of change by ocean ice flux
SIaQbATC|SM M1|m/s |Potential HEFF rate of change by atm flux over ice
SIaQbATO|SM M1|m/s |Potential HEFF rate of change by open ocn atm flux
SIdHbOCN|SM M1|m/s |HEFF rate of change by ocean ice flux
SIdSbATC|SM M1|m/s |HSNOW rate of change by atm flux over sea ice
SIdSbOCN|SM M1|m/s |HSNOW rate of change by ocean ice flux
SIdHbATC|SM M1|m/s |HEFF rate of change by atm flux over sea ice
SIdHbATO|SM M1|m/s |HEFF rate of change by open ocn atm flux
SIdHbFLO|SM M1|m/s |HEFF rate of change by flooding snow
SIdAbATO|SM M1|m^2/m^2/s |Potential AREA rate of change by open ocn atm flux
SIdAbATC|SM M1|m^2/m^2/s |Potential AREA rate of change by atm flux over ice
SIdAbOCN|SM M1|m^2/m^2/s |Potential AREA rate of change by ocean ice flux
SIdA |SM M1|m^2/m^2/s |AREA rate of change (net)
---
DYNAMIC/RHEOLOGY DIAGNOSTICS:
---
SIpress |SM M1|N/m |SEAICE strength (with upper and lower limit)
SIzeta |SM M1|kg/s |SEAICE nonlinear bulk viscosity
SIeta |SM M1|kg/s |SEAICE nonlinear shear viscosity
SIsig1 |SM M1|no units |SEAICE normalized principle stress, component one
SIsig2 |SM M1|no units |SEAICE normalized principle stress, component two
SIshear |SM M1|1/s |SEAICE shear deformation rate
SIdelta |SM M1|1/s |SEAICE Delta deformation rate
SItensil|SM M1|N/m |SEAICE maximal tensile strength
---
ADVECTIVE/DIFFUSIVE FLUXES OF SEA ICE variables:
---
ADVxHEFF|UU M1|m.m^2/s |Zonal Advective Flux of eff ice thickn
ADVyHEFF|VV M1|m.m^2/s |Meridional Advective Flux of eff ice thickn
SIuheff |UU M1|m^2/s |Zonal Transport of eff ice thickn (centered)
SIvheff |VV M1|m^2/s |Meridional Transport of eff ice thickn (centered)
DFxEHEFF|UU M1|m^2/s |Zonal Diffusive Flux of eff ice thickn
DFyEHEFF|VV M1|m^2/s |Meridional Diffusive Flux of eff ice thickn
ADVxAREA|UU M1|m^2/m^2.m^2/s |Zonal Advective Flux of fract area
ADVyAREA|VV M1|m^2/m^2.m^2/s |Meridional Advective Flux of fract area
DFxEAREA|UU M1|m^2/m^2.m^2/s |Zonal Diffusive Flux of fract area
DFyEAREA|VV M1|m^2/m^2.m^2/s |Meridional Diffusive Flux of fract area
ADVxSNOW|UU M1|m.m^2/s |Zonal Advective Flux of eff snow thickn
ADVySNOW|VV M1|m.m^2/s |Meridional Advective Flux of eff snow thickn
DFxESNOW|UU M1|m.m^2/s |Zonal Diffusive Flux of eff snow thickn
DFyESNOW|VV M1|m.m^2/s |Meridional Diffusive Flux of eff snow thickn
ADVxSSLT|UU M1|psu.m^2/s |Zonal Advective Flux of seaice salinity
ADVySSLT|VV M1|psu.m^2/s |Meridional Advective Flux of seaice salinity
DFxESSLT|UU M1|psu.m^2/s |Zonal Diffusive Flux of seaice salinity
DFyESSLT|VV M1|psu.m^2/s |Meridional Diffusive Flux of seaice salinity
```

## 8.6.2.7. Experiments and tutorials that use seaice¶

- verification/lab_sea: Labrador Sea experiment
- verification/seaice_obcs, based on lab_sea
- verification/offline_exf_seaice, idealized topography in a zonally re-entrant channel
- verification/seaice_itd, based on offline_exf_seaice, tests ice thickness distribution
- verification/global_ocean.cs32x15, global cubed-sphere-experiment with combinations of pkg/seaice and pkg/thsice
- verification/1D_ocean_ice_column, just thermodynamics