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view on githubraw file Latest commit 204068be on 2025-11-16 04:31:18 UTCadc83e5d7b Mart*0001 .. _sub_phys_pkg_seaice: 0002 0003 SEAICE Package 258fe29c91 Jeff*0004 ************** adc83e5d7b Mart*0005 0006 Authors: Martin Losch, Dimitris Menemenlis, An Nguyen, Jean-Michel c512e371cc drin*0007 Campin, Patrick Heimbach, Chris Hill, Jinlun Zhang, and Damien Ringeisen adc83e5d7b Mart*0008 0009 .. _ssub_phys_pkg_seaice_intro: 0010 0011 Introduction 258fe29c91 Jeff*0012 ============ adc83e5d7b Mart*0013 c512e371cc drin*0014 Package :filelink:`seaice <pkg/seaice>` provides a dynamic and thermodynamic 0015 interactive sea ice model. adc83e5d7b Mart*0016 0017 CPP options enable or disable different aspects of the package 9986b4a53e Jeff*0018 (:numref:`ssub_phys_pkg_seaice_config`). Run-time options, flags, filenames and c512e371cc drin*0019 field-related dates/times are set in ``data.seaice`` 0020 (:numref:`ssub_phys_pkg_seaice_runtime`). A description of key subroutines is 0021 given in :numref:`ssub_phys_pkg_seaice_subroutines`. Available diagnostics 0022 output is listed in :numref:`ssub_phys_pkg_seaice_diagnostics`. adc83e5d7b Mart*0023 61f2157921 Oliv*0024 .. _ssub_phys_pkg_seaice_config: adc83e5d7b Mart*0025 61f2157921 Oliv*0026 SEAICE configuration and compiling 258fe29c91 Jeff*0027 ================================== adc83e5d7b Mart*0028 61f2157921 Oliv*0029 Compile-time options 258fe29c91 Jeff*0030 -------------------- adc83e5d7b Mart*0031 c512e371cc drin*0032 As with all MITgcm packages, SEAICE can be turned on or off at compile time 0033 (see :numref:`building_code`) adc83e5d7b Mart*0034 c512e371cc drin*0035 - using the ``packages.conf`` file by adding ``seaice`` to it adc83e5d7b Mart*0036 c512e371cc drin*0037 - or using :filelink:`genmake2 <tools/genmake2>` adding ``-enable=seaice`` or 0038 ``-disable=seaice`` switches adc83e5d7b Mart*0039 c512e371cc drin*0040 - **required packages and CPP options**: 0041 :filelink:`seaice <pkg/seaice>` requires the external forcing package 0042 :filelink:`pkg/exf` to be enabled; no additional CPP options are required. adc83e5d7b Mart*0043 0044 c512e371cc drin*0045 Parts of the :filelink:`seaice <pkg/seaice>` code can be enabled or disabled at 0046 compile time via CPP preprocessor flags. These options are set in 0047 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>`. 258fe29c91 Jeff*0048 :numref:`tab_phys_pkg_seaice_cpp` summarizes the most important ones. For more 382462ccb5 Mart*0049 options see :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>`. Note 0050 that defining :varlink:`SEAICE_BGRID_DYNAMICS` turns on legacy code and thus 0051 automatically undefines more recent features, see :filelink:`SEAICE_OPTIONS.h 0052 <pkg/seaice/SEAICE_OPTIONS.h>` for details. 258fe29c91 Jeff*0053 0054 .. tabularcolumns:: |\Y{.375}|\Y{.1}|\Y{.55}| adc83e5d7b Mart*0055 c512e371cc drin*0056 .. csv-table:: Some of the most relevant CPP preprocessor flags in the :filelink:`seaice <pkg/seaice>` package. 258fe29c91 Jeff*0057 :header: "CPP option", "Default", Description" 0058 :widths: 30, 10, 60 bc3b9fecef Mart*0059 :name: tab_phys_pkg_seaice_cpp 0060 258fe29c91 Jeff*0061 :varlink:`SEAICE_DEBUG`, #undef, enhance STDOUT for debugging 382462ccb5 Mart*0062 :varlink:`SEAICE_CGRID`, #define, use sea ice dynamics on C-grid 258fe29c91 Jeff*0063 :varlink:`SEAICE_ALLOW_EVP`, #define, enable use of EVP rheology solver 0064 :varlink:`SEAICE_ALLOW_JFNK`, #define, enable use of JFNK rheology solver 0065 :varlink:`SEAICE_ALLOW_KRYLOV`, #define, enable use of Krylov rheology solver c512e371cc drin*0066 :varlink:`SEAICE_ALLOW_TEM`, #undef, enable use of the truncated ellipse method (TEM) and coulombic yield curve 0067 :varlink:`SEAICE_ALLOW_MCS`, #undef, enable use of Mohr-Coulomb yield curve with shear flow rule 0068 :varlink:`SEAICE_ALLOW_MCE`, #undef, enable use of Mohr-Coulomb yield curve with elliptical plastic potential 0069 :varlink:`SEAICE_ALLOW_TD`, #undef, enable use of teardrop and parabolic Lens yield curves with normal flow rules 258fe29c91 Jeff*0070 :varlink:`SEAICE_LSR_ZEBRA`, #undef, use a coloring method for LSR solver a4e168e012 antn*0071 :varlink:`SEAICE_ALLOW_FREEDRIFT`, #undef, enable solve approximate sea ice momentum equation and bypass solving for sea ice internal stress 258fe29c91 Jeff*0072 :varlink:`SEAICE_EXTERNAL_FLUXES`, #define, use :filelink:`pkg/exf`-computed fluxes as starting point 0073 :varlink:`SEAICE_ZETA_SMOOTHREG`, #define, use differentiable regularization for viscosities 14673ec2d0 Mart*0074 :varlink:`SEAICE_DELTA_SMOOTHREG`, #undef, use differentiable regularization :math:`\Delta_{\mathrm{reg}}=\sqrt{\Delta^2+\Delta_{\min}}` instead of :math:`\max`-function for :math:`1/\Delta_{\mathrm{reg}}` 258fe29c91 Jeff*0075 :varlink:`SEAICE_ALLOW_BOTTOMDRAG`, #undef, enable grounding parameterization for improved fastice in shallow seas 5bb179ddc2 Mart*0076 :varlink:`SEAICE_ALLOW_SIDEDRAG`, #undef, enable lateral drag parameterization for improved fastice along coastlines and islands 382462ccb5 Mart*0077 :varlink:`SEAICE_BGRID_DYNAMICS`, #undef, use sea ice dynamics code on legacy B-grid; most of the previous flags are not available with B-grid 0078 :varlink:`SEAICE_BICE_STRESS`, #undef, B-grid only for backward compatiblity: turn on ice-stress on ocean; defined by default if :varlink:`SEAICE_BGRID_DYNAMICS` is defined 0079 :varlink:`EXPLICIT_SSH_SLOPE`, #undef, B-grid only for backward compatiblity: use ETAN for tilt computations rather than geostrophic velocities; defined by default if :varlink:`SEAICE_BGRID_DYNAMICS` is defined 0080 :varlink:`SEAICE_LSRBNEW`, #undef, FV discretization for B-grid 258fe29c91 Jeff*0081 :varlink:`SEAICE_ITD`, #undef, run with dynamical sea Ice Thickness Distribution (ITD) 0082 :varlink:`SEAICE_VARIABLE_SALINITY`, #undef, enable sea ice with variable salinity e2fbc60f23 Jeff*0083 :varlink:`SEAICE_CAP_ICELOAD`, #undef, enable to limit seaice load (:varlink:`siceLoad`) on the sea surface 258fe29c91 Jeff*0084 :varlink:`ALLOW_SITRACER`, #undef, enable sea ice tracer package a4e168e012 antn*0085 :varlink:`SEAICE_USE_GROWTH_ADX`, #undef, use of adjointable but more simplified sea ice thermodynamics model in :filelink:`seaice_growth_adx.F <pkg/seaice/seaice_growth_adx.F>` instead of :filelink:`seaice_growth.F <pkg/seaice/seaice_growth.F>` adc83e5d7b Mart*0086 61f2157921 Oliv*0087 .. _ssub_phys_pkg_seaice_runtime: adc83e5d7b Mart*0088 2c231b0ebd Mart*0089 Run-time parameters 9986b4a53e Jeff*0090 =================== adc83e5d7b Mart*0091 9986b4a53e Jeff*0092 Run-time parameters (see :numref:`tab_phys_pkg_seaice_runtimeparms`) are set in 0bad585a21 Navi*0093 ``data.seaice`` (read in :filelink:`pkg/seaice/seaice_readparms.F`). adc83e5d7b Mart*0094 0095 Enabling the package 258fe29c91 Jeff*0096 -------------------- adc83e5d7b Mart*0097 c512e371cc drin*0098 :filelink:`seaice <pkg/seaice>` package is switched on/off at run-time by dc26f158aa Mart*0099 setting :varlink:`useSEAICE` ``= .TRUE.,`` in ``data.pkg``. adc83e5d7b Mart*0100 0101 General flags and parameters 258fe29c91 Jeff*0102 ---------------------------- adc83e5d7b Mart*0103 9986b4a53e Jeff*0104 :numref:`tab_phys_pkg_seaice_runtimeparms` lists most run-time parameters. adc83e5d7b Mart*0105 258fe29c91 Jeff*0106 .. tabularcolumns:: |\Y{.275}|\Y{.20}|\Y{.525}| adc83e5d7b Mart*0107 9986b4a53e Jeff*0108 .. table:: Run-time parameters and default values 258fe29c91 Jeff*0109 :class: longtable adc83e5d7b Mart*0110 :name: tab_phys_pkg_seaice_runtimeparms 0111 258fe29c91 Jeff*0112 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0113 | Name | Default value | Description | 0114 +====================================+==============================+=========================================================================+ c61841e2fd Jeff*0115 | :varlink:`SEAICEwriteState` | FALSE | write sea ice state to file | 258fe29c91 Jeff*0116 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0117 | :varlink:`SEAICEuseDYNAMICS` | TRUE | use dynamics | 0118 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0119 | :varlink:`SEAICEuseJFNK` | FALSE | use the JFNK-solver | 0120 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c512e371cc drin*0121 | :varlink:`SEAICEuseTEM` | FALSE | use truncated ellipse method or coulombic yield curve | 0122 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0123 | :varlink:`SEAICEuseMCS` | FALSE | use the Mohr-Coulomb yield curve with shear flow rule | 0124 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0125 | :varlink:`SEAICEuseMCE` | FALSE | use the Mohr-Coulomb yield curve with elliptical plastic potential | 0126 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0127 | :varlink:`SEAICEuseTD` | FALSE | use the teardrop yield curve with normal flow rule | 0128 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0129 | :varlink:`SEAICEusePL` | FALSE | use the parabolic Lens yield curve with normal flow rule | 258fe29c91 Jeff*0130 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0131 | :varlink:`SEAICEuseStrImpCpl` | FALSE | use strength implicit coupling in LSR/JFNK | 0132 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0133 | :varlink:`SEAICEuseMetricTerms` | TRUE | use metric terms in dynamics | 0134 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0135 | :varlink:`SEAICEuseEVPpickup` | TRUE | use EVP pickups | 0136 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ a4e168e012 antn*0137 | :varlink:`SEAICEuseFREEDRIFT` | FALSE | solve approximate momentum equation, bypassing rheology | 0138 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c61841e2fd Jeff*0139 | :varlink:`SEAICEuseFluxForm` | TRUE | use flux form for 2nd central difference advection scheme | 258fe29c91 Jeff*0140 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0141 | :varlink:`SEAICErestoreUnderIce` | FALSE | enable restoring to climatology under ice | 0142 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0143 | :varlink:`SEAICEupdateOceanStress` | TRUE | update ocean surface stress accounting for sea ice cover | 0144 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ dc26f158aa Mart*0145 | :varlink:`SEAICEscaleSurfStress` | TRUE | scale atmosphere and ocean-surface stress on ice by concentration (AREA)| 258fe29c91 Jeff*0146 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0147 | :varlink:`SEAICEaddSnowMass` | TRUE | in computing seaiceMass, add snow contribution | 0148 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0149 | :varlink:`useHB87stressCoupling` | FALSE | turn on ice-ocean stress coupling following | 0150 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0151 | :varlink:`usePW79thermodynamics` | TRUE | flag to turn off zero-layer-thermodynamics for testing | 0152 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0153 | :varlink:`SEAICEadvHeff` | TRUE | flag to turn off advection of scalar variable :varlink:`HEFF` | 0154 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0155 | :varlink:`SEAICEadvArea` | TRUE | flag to turn off advection of scalar variable :varlink:`AREA` | 0156 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0157 | :varlink:`SEAICEadvSnow` | TRUE | flag to turn off advection of scalar variable :varlink:`HSNOW` | 0158 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0159 | :varlink:`SEAICEadvSalt` | TRUE | flag to turn off advection of scalar variable :varlink:`HSALT` | 0160 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0161 | :varlink:`SEAICEadvScheme` | 77 | set advection scheme for seaice scalar state variables | 0162 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0163 | :varlink:`SEAICEuseFlooding` | TRUE | use flood-freeze algorithm | 0164 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ a4e168e012 antn*0165 | :varlink:`SINegFac` | 1.0 | over/undershoot factor for seaice advective term in forward/adjoint | 0166 | | | (SEAICE_USE_GROWTH_ADX only) | 0167 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0168 | :varlink:`SEAICE_no_slip` | FALSE | use no-slip boundary conditions instead of free-slip | 0169 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0170 | :varlink:`SEAICE_deltaTtherm` | :varlink:`dTtracerLev` (1) | time step for seaice thermodynamics (s) | 0171 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0172 | :varlink:`SEAICE_deltaTdyn` | :varlink:`dTtracerLev` (1) | time step for seaice dynamics (s) | 0173 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0174 | :varlink:`SEAICE_deltaTevp` | 0.0 | EVP sub-cycling time step (s); values :math:`>` 0 turn on EVP | 0175 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ dc26f158aa Mart*0176 | :varlink:`SEAICEuseEVPstar` | TRUE | use modified EVP\* instead of EVP, following :cite:`lemieux:12` | 258fe29c91 Jeff*0177 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ dc26f158aa Mart*0178 | :varlink:`SEAICEuseEVPrev` | TRUE | "revisited form" variation on EVP\*, following :cite:`bouillon:13` | 258fe29c91 Jeff*0179 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0180 | :varlink:`SEAICEnEVPstarSteps` | unset | number of modified EVP\* iterations | 0181 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0182 | :varlink:`SEAICE_evpAlpha` | unset | EVP\* parameter (non-dim.), to replace | 0183 | | | 2*\ :varlink:`SEAICE_evpTauRelax`\ /\ :varlink:`SEAICE_deltaTevp` | 0184 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0185 | :varlink:`SEAICE_evpBeta` | unset | EVP\* parameter (non-dim.), to replace | 0186 | | | :varlink:`SEAICE_deltaTdyn`\ /\ :varlink:`SEAICE_deltaTevp` | 0187 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0188 | :varlink:`SEAICEaEVPcoeff` | unset | largest stabilized frequency for adaptive EVP (non-dim.) | 0189 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c512e371cc drin*0190 | :varlink:`SEAICEaEVPcStar` | 4.0 | aEVP multiple of stability factor (non-dim.), see :cite:`kimmritz:16` | 258fe29c91 Jeff*0191 | | | :math:`\alpha * \beta = c^\ast * \gamma` | 0192 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c512e371cc drin*0193 | :varlink:`SEAICEaEVPalphaMin` | 5.0 | aEVP lower limit of alpha and beta (non-dim.), see :cite:`kimmritz:16` | 258fe29c91 Jeff*0194 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 204068be6a R. S*0195 | :varlink:`SEAICE_evpAreaReg` | -1.0 | a minimun ice fraction for regularizations of denomU/V in EVP; | 0196 | | | off by default (-1), 1.E-5 is a useful value for high-res simulation | 0197 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0198 | :varlink:`SEAICE_elasticParm` | 0.33333333 | EVP parameter :math:`E_0` (non-dim.), sets relaxation timescale | c512e371cc drin*0199 | | | :varlink:`SEAICE_evpTauRelax` = | 258fe29c91 Jeff*0200 | | | :varlink:`SEAICE_elasticParm` * :varlink:`SEAICE_deltaTdyn` | 0201 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0202 | :varlink:`SEAICE_evpTauRelax` | :varlink:`dTtracerLev` (1) * | relaxation time scale :math:`T` for EVP waves (s) | 0203 | | :varlink:`SEAICE_elasticParm`| | 0204 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0205 | :varlink:`SEAICE_OLx` | :varlink:`OLx` - 2 | overlap for LSR-solver or preconditioner, :math:`x`-dimension | 0206 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0207 | :varlink:`SEAICE_OLy` | :varlink:`OLy` - 2 | overlap for LSR-solver or preconditioner, :math:`y`-dimension | 0208 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c61841e2fd Jeff*0209 | :varlink:`SEAICEnonLinIterMax` | 2/10 | maximum number of non-linear (outer loop) iterations (LSR/JFNK) | 258fe29c91 Jeff*0210 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c61841e2fd Jeff*0211 | :varlink:`SEAICElinearIterMax` | 1500/10 | maximum number of linear iterations (LSR/JFNK) | 258fe29c91 Jeff*0212 +------------------------------------+------------------------------+-------------------------------------------------------------------------+** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 214.
0213 | :varlink:`SEAICE_JFNK_lsIter` | (off) | start line search after “lsIter†Newton iterations | 0214 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c704c5a1ef Mart*0215 | :varlink:`SEAICE_JFNK_lsLmax` | 4 | maximum number of line search steps | 0216 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0217 | :varlink:`SEAICE_JFNK_lsGamma` | 0.5 | line search step size parameter | 0218 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0219 | :varlink:`SEAICEnonLinTol` | 1.0E-05 | non-linear tolerance parameter for JFNK solver | 0220 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0221 | :varlink:`JFNKgamma_lin_min` | 0.10 | minimum tolerance parameter for linear JFNK solver | 0222 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0223 | :varlink:`JFNKgamma_lin_max` | 0.99 | maximum tolerance parameter for linear JFNK solver | 0224 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0225 | :varlink:`JFNKres_tFac` | unset | tolerance parameter for FGMRES residual | 0226 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0227 | :varlink:`SEAICE_JFNKepsilon` | 1.0E-06 | step size for the FD-gradient in s/r seaice_jacvec | 0228 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0229 | :varlink:`SEAICE_dumpFreq` | dumpFreq | dump frequency (s) | 0230 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0231 | :varlink:`SEAICE_dump_mdsio` | TRUE | write snapshot using :filelink:`/pkg/mdsio` | 0232 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0233 | :varlink:`SEAICE_dump_mnc` | FALSE | write snapshot using :filelink:`/pkg/mnc` | 0234 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0235 | :varlink:`SEAICE_initialHEFF` | 0.0 | initial sea ice thickness averaged over grid cell (m) | 0236 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0237 | :varlink:`SEAICE_drag` | 1.0E-03 | air-ice drag coefficient (non-dim.) | 0238 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0239 | :varlink:`OCEAN_drag` | 1.0E-03 | air-ocean drag coefficient (non-dim.) | 0240 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0241 | :varlink:`SEAICE_waterDrag` | 5.5E-03 | water-ice drag coefficient (non-dim.) | 0242 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0243 | :varlink:`SEAICE_dryIceAlb` | 0.75 | winter sea ice albedo | 0244 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0245 | :varlink:`SEAICE_wetIceAlb` | 0.66 | summer sea ice albedo | 0246 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0247 | :varlink:`SEAICE_drySnowAlb` | 0.84 | dry snow albedo | 0248 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0249 | :varlink:`SEAICE_wetSnowAlb` | 0.70 | wet snow albedo | 0250 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0251 | :varlink:`SEAICE_waterAlbedo` | 0.10 | water albedo (not used if #define :varlink:`SEAICE_EXTERNAL_FLUXES`) | 0252 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0253 | :varlink:`SEAICE_strength` | 2.75E+04 | sea ice strength constant :math:`P^{\ast}` (N/m\ :sup:`2`) | 0254 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0255 | :varlink:`SEAICE_cStar` | 20.0 | sea ice strength constant :math:`C^{\ast}` (non-dim.) | 0256 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ aa30c76f3a Dami*0257 | :varlink:`SEAICE_eccen` | 2.0 | VP rheology ellipse aspect ratio :math:`e` | 0258 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c512e371cc drin*0259 | :varlink:`SEAICE_eccfr` | = :varlink:`SEAICE_eccen` | sea ice plastic potential ellipse aspect ratio :math:`e_G` | 0260 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0261 | :varlink:`SEAICEmcMU` | 1.0 | slope of the Mohr-Coulomb yield curve | 0262 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0263 | :varlink:`SEAICEpressReplFac` | 1.0 | use replacement pressure (0.0-1.0) | 0264 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0265 | :varlink:`SEAICE_tensilFac` | 0.0 | tensile factor for the yield curve | 0266 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0267 | :varlink:`SEAICE_rhoAir` | 1.3 (or | density of air (kg/m\ :sup:`3`) | 0268 | | :filelink:`pkg/exf` value) | | 0269 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0270 | :varlink:`SEAICE_cpAir` | 1004.0 (or | specific heat of air (J/kg/K) | 0271 | | :filelink:`pkg/exf` value) | | 0272 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0273 | :varlink:`SEAICE_lhEvap` | 2.5E+06 (or | latent heat of evaporation (J/kg) | 0274 | | :filelink:`pkg/exf` value) | | 0275 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0276 | :varlink:`SEAICE_lhFusion` | 3.34E+05 (or | latent heat of fusion (J/kg) | 0277 | | :filelink:`pkg/exf` value) | | 0278 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c61841e2fd Jeff*0279 | :varlink:`SEAICE_dalton` | 1.75E-03 | ice-ocean transfer coefficient for latent and sensible heat (non-dim.) | 258fe29c91 Jeff*0280 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ de1b16b92a Jeff*0281 | :varlink:`useMaykutSatVapPoly` | FALSE | use Maykut polynomial to compute saturation vapor pressure | 0282 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0283 | :varlink:`SEAICE_iceConduct` | 2.16560E+00 | sea ice conductivity (W m\ :sup:`-1` K\ :sup:`-1`) | 0284 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0285 | :varlink:`SEAICE_snowConduct` | 3.10000E-01 | snow conductivity (W m\ :sup:`-1` K\ :sup:`-1`) | 0286 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0287 | :varlink:`SEAICE_emissivity` | 0.970018 (or | longwave ocean surface emissivity (non-dim.) | 0288 | | :filelink:`pkg/exf` value) | | 0289 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0290 | :varlink:`SEAICE_snowThick` | 0.15 | cutoff snow thickness to use snow albedo (m) | 0291 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0292 | :varlink:`SEAICE_shortwave` | 0.30 | ice penetration shortwave radiation factor (non-dim.) | 0293 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0294 | :varlink:`SEAICE_saltFrac` | 0.0 | salinity newly formed ice (as fraction of ocean surface salinity) | 0295 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0296 | :varlink:`SEAICE_frazilFrac` | 1.0 (or | frazil to sea ice conversion rate, as fraction | 0297 | | computed from other parms) | (relative to the local freezing point of sea ice water) | 0298 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0299 | :varlink:`SEAICEstressFactor` | 1.0 | scaling factor for ice area in computing total ocean stress (non-dim.) | 0300 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0301 | :varlink:`HeffFile` | unset | filename for initial sea ice eff. thickness field :varlink:`HEFF` (m) | 0302 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0303 | :varlink:`AreaFile` | unset | filename for initial fraction sea ice cover :varlink:`AREA` (non-dim.) | 0304 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0305 | :varlink:`HsnowFile` | unset | filename for initial eff. snow thickness field :varlink:`HSNOW` (m) | 0306 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0307 | :varlink:`HsaltFile` | unset | filename for initial eff. sea ice salinity field :varlink:`HSALT` | 0308 | | | (g/m\ :sup:`2`) | 0309 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ dc26f158aa Mart*0310 | :varlink:`LSR_ERROR` | 1.0E-05 | sets accuracy of LSR solver | 258fe29c91 Jeff*0311 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0312 | :varlink:`DIFF1` | 0.0 | parameter used in advect.F | 0313 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0314 | :varlink:`HO` | 0.5 | lead closing parameter :math:`h_0` (m); demarcation thickness between | 0315 | | | thick and thin ice which determines partition between vertical and | 0316 | | | lateral ice growth | 0317 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0318 | :varlink:`MIN_ATEMP` | -50.0 | minimum air temperature (:sup:`o`\ C) | 0319 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0320 | :varlink:`MIN_LWDOWN` | 60.0 | minimum downward longwave (W/m\ :sup:`2`) | 0321 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0322 | :varlink:`MIN_TICE` | -50.0 | minimum ice temperature (:sup:`o`\ C) | 0323 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0324 | :varlink:`IMAX_TICE` | 10 | number of iterations for ice surface temperature solution | 0325 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0326 | :varlink:`SEAICE_EPS` | 1.0E-10 | a "small number" used in various routines | 2c231b0ebd Mart*0327 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 14673ec2d0 Mart*0328 | :varlink:`SEAICE_deltaMin` | :varlink:`SEAICE_EPS` | minimum to regularize :math:`\Delta` | 0329 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0330 | :varlink:`SEAICE_area_reg` | 1.0E-5 | minimum concentration to regularize ice thickness | 0331 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0332 | :varlink:`SEAICE_hice_reg` | 0.05 | minimum ice thickness (m) for regularization | 0333 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0334 | :varlink:`SEAICE_multDim` | 1 | number of ice categories for thermodynamics | 0335 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0336 | :varlink:`SEAICE_useMultDimSnow` | TRUE | use same fixed pdf for snow as for multi-thickness-category ice | 0337 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 5bb179ddc2 Mart*0338 | :varlink:`SEAICEbasalDragK1` | 8.0 | basal drag parameter K\ :sub:`1` :cite:`lemieux:15` | 0339 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0340 | :varlink:`SEAICEbasalDragK2` | 0.0 | basal drag parameter K\ :sub:`2` | 0341 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0342 | :varlink:`SEAICE_cBasalStar` | :varlink:`SEAICE_cStar` value| basal drag parameter (no units) | 0343 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0344 | :varlink:`SEAICEbasalDragU0` | 5.E-5 | basal drag parameter (m/s) | 0345 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0346 | :varlink:`SEAICESideDrag` | 0.0 | lateral drag coefficient :cite:`liu:22` | 0347 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0348 | :varlink:`uCoastLineFile` | unset | filename for coastline length for u-equation | 0349 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0350 | :varlink:`vCoastLineFile` | unset | filename for coastline length for v-equation | 0351 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 258fe29c91 Jeff*0352 0353 c512e371cc drin*0354 The following dynamical ice thickness distribution and ridging parameters in 0355 :numref:`tab_phys_pkg_seaice_ridging` are only active with #define 0356 :varlink:`SEAICE_ITD`. All parameters are non-dimensional unless indicated. 258fe29c91 Jeff*0357 0358 .. tabularcolumns:: |\Y{.275}|\Y{.20}|\Y{.525}| 0359 0360 .. table:: Thickness distribution and ridging parameters 0361 :name: tab_phys_pkg_seaice_ridging 0362 0363 0364 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0365 | Name | Default value | Description | 0366 +====================================+==============================+=========================================================================+ c512e371cc drin*0367 | :varlink:`useHibler79IceStrength` | TRUE | use :cite:`hibler:79` ice strength; do not use :cite:`rothrock:75` | 258fe29c91 Jeff*0368 | | | with #define :varlink:`SEAICE_ITD` | 0369 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c512e371cc drin*0370 | :varlink:`SEAICEsimpleRidging` | TRUE | use simple ridging a la :cite:`hibler:79` | 258fe29c91 Jeff*0371 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0372 | :varlink:`SEAICE_cf` | 17.0 | scaling parameter of :cite:`rothrock:75` ice strength parameterization | 0373 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0374 | :varlink:`SEAICEpartFunc` | 0 | use partition function of :cite:`thorndike:75` | 0375 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c512e371cc drin*0376 | :varlink:`SEAICEredistFunc` | 0 | use redistribution function of :cite:`hibler:80` | 258fe29c91 Jeff*0377 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0378 | :varlink:`SEAICEridgingIterMax` | 10 | maximum number of ridging sweeps | 0379 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0380 | :varlink:`SEAICEshearParm` | 0.5 | fraction of shear to be used for ridging | 0381 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0382 | :varlink:`SEAICEgStar` | 0.15 | max. ice conc. that participates in ridging :cite:`thorndike:75` | 0383 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0384 | :varlink:`SEAICEhStar` | 25.0 | ridging parameter for :cite:`thorndike:75`, :cite:`lipscomb:07` | 0385 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0386 | :varlink:`SEAICEaStar` | 0.05 | similar to :varlink:`SEAICEgStar` for | 0387 | | | :cite:`lipscomb:07` participation function | 0388 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0389 | :varlink:`SEAICEmuRidging` | 3.0 | similar to :varlink:`SEAICEhStar` for | 0390 | | | :cite:`lipscomb:07` ridging function | 0391 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 2c231b0ebd Mart*0392 | :varlink:`SEAICEmaxRaft` | 1.0 | regularization parameter for rafting | 258fe29c91 Jeff*0393 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0394 | :varlink:`SEAICEsnowFracRidge` | 0.5 | fraction of snow that remains on ridged ice | 0395 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0396 | :varlink:`SEAICEuseLinRemapITD` | TRUE | use linear remapping scheme of :cite:`lipscomb:01` | 0397 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ c61841e2fd Jeff*0398 | :varlink:`Hlimit` | unset | nITD+1-array of ice thickness category limits (m) | 258fe29c91 Jeff*0399 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 0400 | :varlink:`Hlimit_c1`, | 3.0, | when :varlink:`Hlimit` is not set, then these parameters | 0401 | :varlink:`Hlimit_c2`, | 15.0, | determine :varlink:`Hlimit` from a simple function | 0402 | :varlink:`Hlimit_c3` | 3.0 | following :cite:`lipscomb:01` | 0403 +------------------------------------+------------------------------+-------------------------------------------------------------------------+ 2c231b0ebd Mart*0404 adc83e5d7b Mart*0405 0406 .. _ssub_phys_pkg_seaice_descr: 0407 0408 Description 258fe29c91 Jeff*0409 =========== adc83e5d7b Mart*0410 c512e371cc drin*0411 The MITgcm sea ice model is based on a variant of the viscous-plastic (VP) 0412 dynamic-thermodynamic sea ice model (Zhang and Hibler 1997 :cite:`zhang:97`) 0413 first introduced in Hibler (1979) and Hibler (1980) 0414 :cite:`hibler:79,hibler:80`. In order to adapt this model to the requirements 0415 of coupled ice-ocean state estimation, many important aspects of the original 0416 code have been modified and improved, see Losch et al. (2010) :cite:`losch:10`: adc83e5d7b Mart*0417 0418 - the code has been rewritten for an Arakawa C-grid, both B- and C-grid c512e371cc drin*0419 variants are available; the C-grid code allows for no-slip and free-slip 0420 lateral boundary conditions; adc83e5d7b Mart*0421 0422 - three different solution methods for solving the nonlinear momentum c512e371cc drin*0423 equations have been adopted: LSOR (Zhang and Hibler 1997 :cite:`zhang:97`), 0424 EVP (Hunke and Dukowicz 1997 :cite:`hunke:97`), 0425 JFNK (Lemieux et al. 2010 :cite:`lemieux:10`, Losch et al. 2014 0426 :cite:`losch:14`); adc83e5d7b Mart*0427 258fe29c91 Jeff*0428 - ice-ocean stress can be formulated as in Hibler and Bryan (1987) c512e371cc drin*0429 :cite:`hibler:87` or as in Campin et al. (2008) :cite:`campin:08`; adc83e5d7b Mart*0430 0431 - ice variables are advected by sophisticated, conservative advection 0432 schemes with flux limiting; 0433 0434 - growth and melt parameterizations have been refined and extended in 0435 order to allow for more stable automatic differentiation of the code. 0436 0437 The sea ice model is tightly coupled to the ocean compontent of the c512e371cc drin*0438 MITgcm. Heat, fresh water fluxes and surface stresses are computed from the 0439 atmospheric state and, by default, modified by the ice model at every time 0440 step. 0441 0442 The ice dynamics models that are most widely used for large-scale climate 0443 studies are the viscous-plastic (VP) model (Hilber 1979 :cite:`hibler:79`), the 0444 cavitating fluid (CF) model (Flato and Hibler 1992 :cite:`flato:92`), and the 0445 elastic-viscous-plastic (EVP) model (Hunke and Dukowicz 1997 :cite:`hunke:97`). 0446 Compared to the VP model, the CF model does not allow ice shear in calculating 0447 ice motion, stress, and deformation. EVP models approximate VP by adding an 0448 elastic term to the equations for easier adaptation to parallel 0449 computers. Because of its higher accuracy in plastic solution and relatively 0450 simpler formulation, compared to the EVP model, we decided to use the VP model 0451 as the default dynamic component of our ice model. To do this we extended the 0452 line successive over relaxation (LSOR) method of Zhang and Hibler (1997) 0453 :cite:`zhang:97` for use in a parallel configuration. An EVP model and a 9986b4a53e Jeff*0454 free-drift implementation can be selected with run-time flags. adc83e5d7b Mart*0455 a4e168e012 antn*0456 :filelink:`pkg/seaice` includes the original so-called zero-layer 0457 thermodynamics with a snow cover as in the appendix of Semtner (1976) 0458 :cite:`semtner:76`. Two versions of this zero-layer thermodynamic code exist, 0459 with a more developed version :filelink:`seaice_growth.F 0460 <pkg/seaice/seaice_growth.F>` and a simplified version 0461 :filelink:`seaice_growth_adx.F <pkg/seaice/seaice_growth_adx.F>` based on 0462 Fenty (2013) :cite:`fenty:13` that excludes physics such as ITD, treatment for 0463 sublimation, and frazil ice but provides a stable sea ice adjointable with 0464 physical sensitivity. When the seaice_growth_adx code is enabled (by defining 0465 :varlink:`SEAICE_USE_GROWTH_ADX` in :filelink:`SEAICE_OPTIONS.h 0466 <pkg/seaice/SEAICE_OPTIONS.h>`), the regularization parameter 0467 :varlink:`SINegFac` is set to zero in adjoint mode to disable the potential 0468 propagation of unphysical terms associated with sea ice dynamics. 0469 adc83e5d7b Mart*0470 0471 .. _para_phys_pkg_seaice_thsice: 0472 258fe29c91 Jeff*0473 Compatibility with ice-thermodynamics package :filelink:`pkg/thsice` 0474 -------------------------------------------------------------------- adc83e5d7b Mart*0475 a4e168e012 antn*0476 The zero-layer thermodynamic model assumes that ice does c512e371cc drin*0477 not store heat and, therefore, tends to exaggerate the seasonal variability in 0478 ice thickness. This exaggeration can be significantly reduced by using Winton's 0479 (Winton 2000 :cite:`winton:00`) three-layer thermodynamic model that permits 0480 heat storage in ice. 0481 0482 The Winton (2000) sea-ice thermodynamics have been ported to MITgcm; they 0483 currently reside under :filelink:`pkg/thsice`, described in 0484 :numref:`sub_phys_pkg_thsice`. It is fully compatible with the packages 0485 :filelink:`seaice <pkg/seaice>` and :filelink:`exf <pkg/exf>`. When turned on 0486 together with :filelink:`seaice <pkg/seaice>`, the zero-layer thermodynamics 0487 are replaced by the Winton thermodynamics. In order to use package 0488 :filelink:`seaice <pkg/seaice>` with the thermodynamics of 0489 :filelink:`pkg/thsice`, compile both packages and turn both package on in 0490 ``data.pkg``; see an example in 0491 :filelink:`verification/global_ocean.cs32x15/input.icedyn`. Note, that once 258fe29c91 Jeff*0492 :filelink:`thsice <pkg/thsice>` is turned on, the variables and diagnostics 0493 associated to the default thermodynamics are meaningless, and the diagnostics 0494 of :filelink:`thsice <pkg/thsice>` must be used instead. adc83e5d7b Mart*0495 0496 .. _para_phys_pkg_seaice_surfaceforcing: 0497 0498 Surface forcing 258fe29c91 Jeff*0499 --------------- adc83e5d7b Mart*0500 c512e371cc drin*0501 The sea ice model requires the following input fields: 10 m winds, 2 m air 0502 temperature and specific humidity, downward longwave and shortwave radiations, 0503 precipitation, evaporation, and river and glacier runoff. The sea ice model 0504 also requires surface temperature from the ocean model and the top level 0505 horizontal velocity. Output fields are surface wind stress, evaporation minus 0506 precipitation minus runoff, net surface heat flux, and net shortwave flux. The 0507 sea-ice model is global: in ice-free regions bulk formulae (by default computed 0508 in package :filelink:`exf <pkg/exf>`) are used to estimate oceanic forcing from 0509 the atmospheric fields. adc83e5d7b Mart*0510 a4e168e012 antn*0511 .. _ssub_phys_pkg_seaice_dynamics: adc83e5d7b Mart*0512 0513 Dynamics a4e168e012 antn*0514 ======== adc83e5d7b Mart*0515 0516 The momentum equation of the sea-ice model is 0517 0518 .. math:: 0bad585a21 Navi*0519 m \frac{D\mathbf{u}}{Dt} = -mf\hat{\mathbf{k}}\times\mathbf{u} + 9c29098ece Jeff*0520 \mathbf{\tau}_\mathrm{air} + \mathbf{\tau}_\mathrm{ocean} 258fe29c91 Jeff*0521 - m \nabla{\phi(0)} + \mathbf{F} adc83e5d7b Mart*0522 :label: eq_momseaice 0523 14673ec2d0 Mart*0524 where :math:`m=m_{i}+m_{s}` is the ice and snow mass per unit area. The ice 0525 mass per grid cell is :math:`m_i=\rho_{\mathrm{ice}} h\,c` with the mean ice 0526 density :math:`\rho_{\mathrm{ice}}` and the mean thickness :math:`h\,c = ` 0527 volume per grid cell area that is the product of the actual thickness :math:`h` 0528 of the ice covered part of the cell and the fractional ice cover :math:`c = 0529 [0,1]`, sloppily also called ice concentration. A similar relationship defines 0530 the snow mass per grid cell :math:`m_s`. 0531 :math:`\mathbf{u}=u\hat{\mathbf{i}}+v\hat{\mathbf{j}}` is the ice velocity 0532 vector; :math:`\hat{\mathbf{i}}`, :math:`\hat{\mathbf{j}}`, and 0533 :math:`\hat{\mathbf{k}}` are unit vectors in the :math:`x`, :math:`y`, and 0534 :math:`z` directions, respectively; :math:`f` is the Coriolis parameter; 0535 :math:`\mathbf{\tau}_\mathrm{air}` and :math:`\mathbf{\tau}_\mathrm{ocean}` are 0536 the wind-ice and ocean-ice stresses, respectively; :math:`g` is the gravity 0537 accelation; :math:`\nabla\phi(0)` is the gradient (or tilt) of the sea surface 0538 height; :math:`\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}` is the sea 0539 surface height potential in response to ocean dynamics (:math:`g\eta`), 0540 atmospheric pressure loading (:math:`p_{a}/\rho_{0}`, where :math:`\rho_{0}` is 0541 a reference density), and a term due to snow and ice loading; and 0542 :math:`\mathbf{F}= \nabla \cdot\sigma` is the divergence of the internal ice 0543 stress tensor :math:`\sigma_{ij}`. Advection of sea-ice momentum is 0544 neglected. The wind and ice-ocean stress terms are given by adc83e5d7b Mart*0545 0546 .. math:: 0547 \begin{aligned} 9c29098ece Jeff*0548 \mathbf{\tau}_\mathrm{air} = & \rho_\mathrm{air} C_\mathrm{air} c512e371cc drin*0549 |\mathbf{U}_\mathrm{air} -\mathbf{u}| R_\mathrm{air} 0550 (\mathbf{U}_\mathrm{air} - \mathbf{u}) \\ 9c29098ece Jeff*0551 \mathbf{\tau}_\mathrm{ocean} = & \rho_\mathrm{ocean}C_\mathrm{ocean} 0552 |\mathbf{U}_\mathrm{ocean}-\mathbf{u}| c512e371cc drin*0553 R_\mathrm{ocean}(\mathbf{U}_\mathrm{ocean} - \mathbf{u}) adc83e5d7b Mart*0554 \end{aligned} 0555 c512e371cc drin*0556 where :math:`\mathbf{U}_\mathrm{air/ocean}` are the surface winds of the 0557 atmosphere and surface currents of the ocean, respectively; 0558 :math:`C_\mathrm{air/ocean}` are air and ocean drag coefficients; 9c29098ece Jeff*0559 :math:`\rho_\mathrm{air/ocean}` are reference densities; and 0560 :math:`R_\mathrm{air/ocean}` are rotation matrices that act on the wind/current adc83e5d7b Mart*0561 vectors. 0562 0563 .. _para_phys_pkg_seaice_VPrheology: 0564 0565 Viscous-Plastic (VP) Rheology 258fe29c91 Jeff*0566 ----------------------------- adc83e5d7b Mart*0567 c512e371cc drin*0568 For an isotropic system the stress tensor :math:`\sigma_{ij}` (:math:`i,j=1,2`) 0569 can be related to the ice strain rate and strength by a nonlinear 0570 viscous-plastic (VP) constitutive law: adc83e5d7b Mart*0571 0572 .. math:: 2c231b0ebd Mart*0573 \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} 258fe29c91 Jeff*0574 + \left[\zeta(\dot{\epsilon}_{ij},P) - 2c231b0ebd Mart*0575 \eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij} 258fe29c91 Jeff*0576 - \frac{P}{2}\delta_{ij} 0577 :label: eq_vpequation adc83e5d7b Mart*0578 0579 The ice strain rate is given by 0580 0581 .. math:: 2c231b0ebd Mart*0582 \dot{\epsilon}_{ij} = \frac{1}{2}\left( adc83e5d7b Mart*0583 \frac{\partial{u_{i}}}{\partial{x_{j}}} + 258fe29c91 Jeff*0584 \frac{\partial{u_{j}}}{\partial{x_{i}}}\right) adc83e5d7b Mart*0585 c512e371cc drin*0586 The maximum ice pressure :math:`P_{\max}` (variable :varlink:`PRESS0` in the 0587 code), a measure of ice strength, depends on both thickness :math:`h` and 0588 compactness (concentration) :math:`c`: adc83e5d7b Mart*0589 0590 .. math:: 0452697f42 Oliv*0591 :label: eq_icestrength adc83e5d7b Mart*0592 3b6b5ca15d Mart*0593 P_{\max} = P^{\ast}c\,h\,\exp\{-C^{\ast}\cdot(1-c)\}, adc83e5d7b Mart*0594 2c231b0ebd Mart*0595 with the constants :math:`P^{\ast}` (run-time parameter 9986b4a53e Jeff*0596 :varlink:`SEAICE_strength`) and :math:`C^{\ast}` (run-time parameter 14673ec2d0 Mart*0597 :varlink:`SEAICE_cStar`). Note that Hibler (1979) :cite:`hibler:79` defines 0598 :math:`h` as the "mean thickness" or an "equivalent ice thickness" for mass, 0599 which is :math:`c\,h` with our definitions. By default, :math:`P` (variable 0600 :varlink:`PRESS` in the code) is the replacement pressure c512e371cc drin*0601 0602 .. math:: 0603 :label: eq_pressrepl 0604 0605 P = (1-k_t)\,P_{\max} \left( (1 - f_{r}) 0bad585a21 Navi*0606 + f_{r} \frac{\Delta}{\Delta_{\rm reg}} \right) c512e371cc drin*0607 14673ec2d0 Mart*0608 where :math:`f_{r}` is a run-time parameter :varlink:`SEAICEpressReplFac` 0bad585a21 Navi*0609 (default = 1.0), and :math:`\Delta_{\rm reg}` is a regularized form of c512e371cc drin*0610 :math:`\Delta = \left[ \left(\dot{\epsilon}_{11}+\dot{\epsilon}_{22}\right)^2 + 0611 e^{-2}\left( \left(\dot{\epsilon}_{11}-\dot{\epsilon}_{22} \right)^2 + 14673ec2d0 Mart*0612 4\,\dot{\epsilon}_{12}^2 \right) \right]^{\frac{1}{2}}`. By default 0613 :math:`\Delta_{\mathrm{reg}}=\max(\Delta,\Delta_{\min})`. If CPP-flag 0614 :varlink:`SEAICE_DELTA_SMOOTHREG` is defined, 0615 :math:`\Delta_{\mathrm{reg}}=\sqrt{\Delta^2+\Delta^2_{\min}}`. Run-time 0616 parameter :varlink:`SEAICE_deltaMin` :math:`= \Delta_{\min} = 10^{-10}` by 0617 default. c512e371cc drin*0618 0619 The tensile strength factor :math:`k_t` (run-time parameter 0620 :varlink:`SEAICE_tensilFac`) determines the ice tensile strength :math:`T = 0621 k_t\cdot P_{\max}`, as defined by König Beatty and Holland (2010) 0622 :cite:`konig:10`. :varlink:`SEAICE_tensilFac` is zero by default. 0623 0624 Different VP rheologies can be used to model sea ice dynamics. The different 0625 rheologies are characterized by different definitions of the bulk and shear 0626 viscosities :math:`\zeta` and :math:`\eta` in :eq:`eq_vpequation`. The 0627 following :numref:`tab_phys_pkg_seaice_rheologies` is a summary of the 0628 available choices with recommended (sensible) parameter values. All the 0629 rheologies presented here depend on the ice strength :math:`P` 0630 :eq:`eq_pressrepl`. 0631 0632 .. tabularcolumns:: |\Y{.275}|\Y{.450}|\Y{.275}| 0633 0634 .. table:: Overview over availabe sea ice viscous-plastic rheologies 0635 :class: longtable 0636 :name: tab_phys_pkg_seaice_rheologies 0637 0638 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0639 | Name | CPP flags | Run-time flags (recommended value) | 0640 +=======================================+=======================================+====================================================+ 0641 | :ref:`rheologies_ellnfr` | None (default) | - :varlink:`SEAICE_eccen` (= 2.0) | 0642 | | | - :varlink:`SEAICE_tensilFac` (= 0.0) | 0643 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0644 | :ref:`rheologies_ellnnfr` | None | - :varlink:`SEAICE_eccen` (= 2.0) | 0645 | | | - :varlink:`SEAICE_eccfr` (< 2.0) | 0646 | | | - :varlink:`SEAICE_tensilFac` (= 0.0) | 0647 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0648 | :ref:`rheologies_TEM` | :varlink:`SEAICE_ALLOW_TEM` | - :varlink:`SEAICEuseTEM` (=.TRUE.) | 0649 | | | - :varlink:`SEAICE_eccen` (= 1.4) | 0650 | | | - :varlink:`SEAICE_eccfr` (< 1.4) | 0651 | | | - :varlink:`SEAICE_tensilFac` (= 0.05) | 0652 | | | - :varlink:`SEAICEmcMU` (= 0.6 to 0.8) | 0653 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0654 | :ref:`rheologies_MCE` | :varlink:`SEAICE_ALLOW_MCE` | - :varlink:`SEAICEuseMCE` (=.TRUE.) | 0655 | | | - :varlink:`SEAICE_eccen` (= 1.4) | 0656 | | | - :varlink:`SEAICE_eccfr` (< 1.4) | 0657 | | | - :varlink:`SEAICE_tensilFac` (= 0.05) | 0658 | | | - :varlink:`SEAICEmcMU` (= 0.6 to 0.8) | 0659 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0660 | :ref:`rheologies_MCS` | :varlink:`SEAICE_ALLOW_MCS` | - :varlink:`SEAICEuseMCS` (=.TRUE.) | 0661 | | | - :varlink:`SEAICE_tensilFac` (= 0.05) | 0662 | | | - :varlink:`SEAICEmcMU` (= 0.6 to 0.8) | 0663 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0664 | :ref:`rheologies_TD` | :varlink:`SEAICE_ALLOW_TD` | - :varlink:`SEAICEuseTD` (=.TRUE.) | 0665 | | | - :varlink:`SEAICE_tensilFac` (= 0.025) | 0666 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0667 | :ref:`rheologies_PL` | :varlink:`SEAICE_ALLOW_TD` | - :varlink:`SEAICEusePL` (=.TRUE.) | 0668 | | | - :varlink:`SEAICE_tensilFac` (= 0.025) | 0669 +---------------------------------------+---------------------------------------+----------------------------------------------------+ 0670 0671 0672 **Note:** With the exception of the default rheology and the TEM (with 0673 :varlink:`SEAICEmcMU` : :math:`\mu=1.0`), these rheologies are not implemented 0674 in EVP (:numref:`para_phys_pkg_seaice_EVPdynamics`). 0675 0676 .. _rheologies_ellnfr: 0677 0678 Elliptical yield curve with normal flow rule 0679 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0680 0681 The default rheology in the sea ice module of the MITgcm implements the widely 0682 used elliptical yield curve with a normal flow rule :cite:`hibler:79`. For 0683 this yield curve, the nonlinear bulk and shear viscosities :math:`\zeta` and 0684 :math:`\eta` are functions of ice strain rate invariants and ice strength such 0685 that the principal components of the stress lie on an elliptical yield curve 0686 with the ratio of major to minor axis :math:`e = 2.0` (run-time parameter 0687 :varlink:`SEAICE_eccen`); they are given by: adc83e5d7b Mart*0688 0689 .. math:: 0690 \begin{aligned} 14673ec2d0 Mart*0691 \zeta =& \min\left(\frac{(1+k_t)P_{\max}}{2\Delta_\mathrm{reg}}, adc83e5d7b Mart*0692 \zeta_{\max}\right) \\ c512e371cc drin*0693 \eta =& \frac{\zeta}{e^2} 0694 \end{aligned} 0695 :label: eq_zetareg 258fe29c91 Jeff*0696 0697 0698 with the abbreviation 0699 0700 .. math:: 0701 \Delta = \left[ c61841e2fd Jeff*0702 \left(\dot{\epsilon}_{11}+\dot{\epsilon}_{22}\right)^2 0703 + e^{-2}\left( \left(\dot{\epsilon}_{11}-\dot{\epsilon}_{22} \right)^2 14673ec2d0 Mart*0704 + 4\,\dot{\epsilon}_{12}^2 \right) c61841e2fd Jeff*0705 \right]^{\frac{1}{2}} adc83e5d7b Mart*0706 0707 The bulk viscosities are bounded above by imposing both a minimum 14673ec2d0 Mart*0708 :math:`\Delta_{\min}` and replacing :math:`\Delta` by the regularized version 0709 :math:`\Delta_\mathrm{reg}` (for historical reasons, run-time parameter c512e371cc drin*0710 :varlink:`SEAICE_deltaMin` is set to a default value of 0711 :math:`10^{-10}\,\text{s}^{-1}`, the value of :varlink:`SEAICE_EPS`) and a 0712 maximum :math:`\zeta_{\max} = P_{\max}/(2\Delta^\ast)`, where 14673ec2d0 Mart*0713 :math:`\Delta^\ast=(2\times10^4/5\times10^{12})\,\text{s}^{-1} = 0714 2\times10^{-9}\,\text{s}^{-1}` (:varlink:`SEAICE_zetaMaxFac` 0715 :math:`=\frac{1}{2\Delta^\ast}`). Obviously, this corresponds to regularizing c512e371cc drin*0716 :math:`\Delta` with the typical value of :varlink:`SEAICE_deltaMin` :math:`= 0717 2\times10^{-9}`. Clearly, some of this regularization is redundant. (There is 0718 also the option of bounding :math:`\zeta` from below by setting run-time 0719 parameter :varlink:`SEAICE_zetaMin` :math:`>0`, but this is generally not 0720 recommended). For stress tensor computation the replacement pressure :math:`P = 0721 2\,\Delta\zeta` is used so that the stress state always lies on the elliptic 0722 yield curve by definition. adc83e5d7b Mart*0723 258fe29c91 Jeff*0724 Defining the CPP-flag :varlink:`SEAICE_ZETA_SMOOTHREG` in c512e371cc drin*0725 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` before compiling 0726 replaces the method for bounding :math:`\zeta` by a smooth (differentiable) 0727 expression: adc83e5d7b Mart*0728 0729 .. math:: 258fe29c91 Jeff*0730 \begin{split} c512e371cc drin*0731 \zeta &= \zeta_{\max}\tanh\left(\frac{(1+k_t)P_{\max}}{2\, 14673ec2d0 Mart*0732 \Delta_\mathrm{reg} \,\zeta_{\max}}\right)\\ c512e371cc drin*0733 &= \frac{(1+k_t)P_{\max}}{2\Delta^\ast} 14673ec2d0 Mart*0734 \tanh\left(\frac{\Delta^\ast}{\Delta_\mathrm{reg}}\right) 258fe29c91 Jeff*0735 \end{split} 0736 :label: eq_zetaregsmooth adc83e5d7b Mart*0737 c61841e2fd Jeff*0738 where :math:`\Delta_{\min}=10^{-20}\,\text{s}^{-1}` should be chosen to avoid adc83e5d7b Mart*0739 divisions by zero. 0740 c512e371cc drin*0741 In this default formulation the yield curve does not allow isotropic tensile 0742 stress, that is, sea ice can be "pulled apart" without any effort. Setting the 0743 parameter :math:`k_t` (:varlink:`SEAICE_tensilFac`) to a small value larger 0744 than zero, extends the yield curve into a region where the divergence of the 0745 stress :math:`\sigma_{11}+\sigma_{22} > 0` to allow some tensile stress. 0746 0747 Besides this commonly used default rheology, a number of a alternative 0748 rheologies are implemented. Some of these are experiemental and should be used 0749 with caution. 0750 0751 .. _rheologies_ellnnfr: 0752 0753 Elliptical yield curve with non-normal flow rule 0754 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0755 0756 Defining the run-time parameter :varlink:`SEAICE_eccfr` with a value different 0757 from :varlink:`SEAICE_eccen` allows one to use an elliptical yield curve with a 0758 non-normal flow rule as described in Ringeisen et al. (2020) 0759 :cite:`ringeisen:20`. In this case the viscosities are functions of 0760 :math:`e_F` (:varlink:`SEAICE_eccen`) and :math:`e_G` 0761 (:varlink:`SEAICE_eccfr`): 0762 0763 .. math:: 0764 \begin{aligned} 0765 \zeta &= \frac{P_{\max}(1+k_t)}{2\Delta} \\ 0766 \eta &= \frac{\zeta}{e_G^2} = \frac{P_{\max}(1+k_t)}{2e_G^2\Delta} 0767 \end{aligned} 0768 0769 with the abbreviation 0770 0771 .. math:: 0772 \Delta = \sqrt{(\dot{\epsilon}_{11}-\dot{\epsilon}_{22})^2 0773 +\frac{e_F^2}{e_G^4}((\dot{\epsilon}_{11} 14673ec2d0 Mart*0774 -\dot{\epsilon}_{22})^2+4\,\dot{\epsilon}_{12}^2)}. c512e371cc drin*0775 0776 Note that if :math:`e_G=e_F=e`, these formulae reduce to the normal flow rule. 0777 0778 .. _rheologies_TEM: 0779 0780 Truncated ellipse method (TEM) for elliptical yield curve 0781 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0782 0783 In the so-called truncated ellipse method, the shear viscosity :math:`\eta` is 0784 capped to suppress any tensile stress: 0785 0786 .. math:: 0787 \eta = \min\left(\frac{\zeta}{e^2}, 0788 \frac{\frac{(1+k_t)\,P_{\max}}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} 0789 {\sqrt{\max(\Delta_{\min}^{2},(\dot{\epsilon}_{11}-\dot{\epsilon}_{22})^2 0790 +4\dot{\epsilon}_{12}^2})}\right). 0791 :label: eq_etatem 0792 0793 To enable this method, set ``#define`` :varlink:`SEAICE_ALLOW_TEM` in 0794 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and turn it on with dc26f158aa Mart*0795 :varlink:`SEAICEuseTEM` ``=.TRUE.,`` in ``data.seaice``. This parameter c512e371cc drin*0796 combination implies the default of :varlink:`SEAICEmcMU` :math:`= 1.0`. 0797 0798 Instead of an ellipse that is truncated by constant slope coulombic limbs, this 0799 yield curve can also be seen as a Mohr-Coulomb yield curve with elliptical flow 0800 rule that is truncated for high :math:`P` by an ellipse. As a consequence, the 0801 Mohr-Coulomb slope :varlink:`SEAICEmcMU` can be set in ``data.seaice`` to 0802 values :math:`\ne 1.0`. This defines a coulombic yield curve similar to the 0803 ones shown in Hibler and Schulson (2000) :cite:`hibler:00` and Ringeisen et 0804 al. (2019) :cite:`ringeisen:19`. 0805 0806 For this rheology, it is recommended to use a non-zero tensile strength, so set 0807 :varlink:`SEAICE_tensilFac` :math:`=k_{t}>0` in ``data.seaice``, e.g., :math:`= 0808 0.05` or 5%. 0809 0810 .. _rheologies_MCE: 0811 0812 Mohr-Coulomb yield curve with elliptical plastic potential 0813 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0814 0815 To use a Mohr-Coulomb rheology, set ``#define`` :varlink:`SEAICE_ALLOW_MCE` in 0816 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and dc26f158aa Mart*0817 :varlink:`SEAICEuseMCE` ``= .TRUE.,`` in ``data.seaice``. This Mohr-Coulomb c512e371cc drin*0818 yield curve uses an elliptical plastic potential to define the flow rule. The 0819 slope of the Mohr-Coulomb yield curve is defined by :varlink:`SEAICEmcMU` in 0820 ``data.seaice``, and the plastic potential ellipse aspect ratio is set by 0821 :varlink:`SEAICE_eccfr` in ``data.seaice``. For details of this rheology, see 0822 https://doi.org/10.26092/elib/380, Chapter 2. 0823 0824 For this rheology, it is recommended to use a non-zero tensile strength, so set 0825 :varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`= 0.05` 0826 or 5%. 0827 0828 .. _rheologies_MCS: 0829 0830 Mohr-Coulomb yield curve with shear flow rule 0831 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0832 0833 To use the specifc Mohr-Coulomb rheology as defined first by Ip et al. (1991) 0834 :cite:`ip:91`, set ``#define`` :varlink:`SEAICE_ALLOW_MCS` in 0835 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and dc26f158aa Mart*0836 :varlink:`SEAICEuseMCS` ``= .TRUE.,`` in ``data.seaice``. The slope of the c512e371cc drin*0837 Mohr-Coulomb yield curve is defined by :varlink:`SEAICEmcMU` in 0838 ``data.seaice``. For details of this rheology, including the tensile strength, 0839 see https://doi.org/10.26092/elib/380, Chapter 2. 0840 0841 For this rheology, it is recommended to use a non-zero tensile strength, so set 0842 :varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`= 0.05` 0843 or 5%. 0844 0845 **WARNING: This rheology is known to be unstable. Use with caution!** 0846 0847 .. _rheologies_TD: 0848 0849 Teardrop yield curve with normal flow rule 0850 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0851 0852 The teardrop rheology was first described in Zhang and Rothrock (2005) 0853 :cite:`zha:05`. Here we implement a slightly modified version (See 0854 https://doi.org/10.26092/elib/380, Chapter 2). 0855 0856 To use this rheology, set ``#define`` :varlink:`SEAICE_ALLOW_TEARDROP` in 0857 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and dc26f158aa Mart*0858 :varlink:`SEAICEuseTD` ``= .TRUE.,`` in ``data.seaice``. The size of the yield c512e371cc drin*0859 curve can be modified by changing the tensile strength, using 0860 :varlink:`SEAICE_tensFac` in ``data.seaice``. 0861 0862 For this rheology, it is recommended to use a non-zero tensile strength, so set 0863 :varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`= 0864 0.025` or 2.5%. 0865 0866 .. _rheologies_PL: 0867 0868 Parabolic lens yield curve with normal flow rule 0869 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0870 0871 The parabolic lens rheology was first described in Zhang and Rothrock (2005) 0872 :cite:`zha:05`. Here we implement a slightly modified version (See 0873 https://doi.org/10.26092/elib/380, Chapter 2). 0874 0875 To use this rheology, set ``#define`` :varlink:`SEAICE_ALLOW_TEARDROP` in 0876 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and dc26f158aa Mart*0877 :varlink:`SEAICEusePL` ``= .TRUE.,`` in ``data.seaice``. The size of the yield c512e371cc drin*0878 curve can be modified by changing the tensile strength, using 0879 :varlink:`SEAICE_tensFac` in ``data.seaice``. 0880 0881 For this rheology, it is recommended to use a non-zero tensile strength, so set 0882 :varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`= 0883 0.025` or 2.5%. 0884 adc83e5d7b Mart*0885 .. _para_phys_pkg_seaice_LSRJFNK: 0886 0887 LSR and JFNK solver 258fe29c91 Jeff*0888 ------------------- adc83e5d7b Mart*0889 c512e371cc drin*0890 In matrix notation, the discretized momentum equations can be written as adc83e5d7b Mart*0891 0892 .. math:: 0893 :label: eq_matrixmom 2c231b0ebd Mart*0894 c61841e2fd Jeff*0895 \mathbf{A}(\mathbf{x})\,\mathbf{x} = \mathbf{b}(\mathbf{x}). adc83e5d7b Mart*0896 c512e371cc drin*0897 The solution vector :math:`\mathbf{x}` consists of the two velocity components 0898 :math:`u` and :math:`v` that contain the velocity variables at all grid points 0899 and at one time level. The standard (and default) method for solving 0900 Eq. :eq:`eq_matrixmom` in the sea ice component of MITgcm is an iterative 0901 Picard solver: in the :math:`k`-th iteration a linearized form adc83e5d7b Mart*0902 :math:`\mathbf{A}(\mathbf{x}^{k-1})\,\mathbf{x}^{k} = c512e371cc drin*0903 \mathbf{b}(\mathbf{x}^{k-1})` is solved (in the case of MITgcm it is a Line 0904 Successive (over) Relaxation (LSR) algorithm). Picard solvers converge slowly, 0905 but in practice the iteration is generally terminated after only a few 0906 nonlinear steps and the calculation continues with the next time level. This 0907 method is the default method in MITgcm. The number of nonlinear iteration steps 0908 or pseudo-time steps can be controlled by the run-time parameter dc26f158aa Mart*0909 :varlink:`SEAICEnonLinIterMax`. This parameter's default is 2, but using a 0910 number of at least 10 is recommended for better solutions that are converged at 0911 least in an energy norm sense (Zhang and Hibler 1997) :cite:`zhang:97`. c512e371cc drin*0912 dc26f158aa Mart*0913 In order to overcome the poor convergence of the Picard solver, Lemieux et c512e371cc drin*0914 al. (2010) :cite:`lemieux:10` introduced a Jacobian-free Newton-Krylov solver 0915 for the sea ice momentum equations. This solver is also implemented in MITgcm 0916 (see Losch et al. 2014 :cite:`losch:14`). The Newton method transforms 0917 minimizing the residual :math:`\mathbf{F}(\mathbf{x}) = 0918 \mathbf{A}(\mathbf{x})\,\mathbf{x} - \mathbf{b}(\mathbf{x})` to finding the 0919 roots of a multivariate Taylor expansion of the residual :math:`\mathbf{F}` 0920 around the previous (:math:`k-1`) estimate :math:`\mathbf{x}^{k-1}`: adc83e5d7b Mart*0921 0922 .. math:: 258fe29c91 Jeff*0923 \mathbf{F}(\mathbf{x}^{k-1}+\delta\mathbf{x}^{k}) = 0924 \mathbf{F}(\mathbf{x}^{k-1}) + \mathbf{F}'(\mathbf{x}^{k-1}) 0925 \,\delta\mathbf{x}^{k} adc83e5d7b Mart*0926 :label: eq_jfnktaylor 0927 c512e371cc drin*0928 with the Jacobian :math:`\mathbf{J}\equiv\mathbf{F}'`. The root 0929 :math:`\mathbf{F}(\mathbf{x}^{k-1}+\delta\mathbf{x}^{k})=0` is found by solving adc83e5d7b Mart*0930 0931 .. math:: 258fe29c91 Jeff*0932 \mathbf{J}(\mathbf{x}^{k-1})\,\delta\mathbf{x}^{k} = 0933 -\mathbf{F}(\mathbf{x}^{k-1}) adc83e5d7b Mart*0934 :label: eq_jfnklin 0935 c512e371cc drin*0936 for :math:`\delta\mathbf{x}^{k}`. The next (:math:`k`-th) estimate is given by c704c5a1ef Mart*0937 :math:`\mathbf{x}^{k}=\mathbf{x}^{k-1}+(1-\gamma_{\mathrm{LS}})^{l} 0938 \,\delta\mathbf{x}^{k}`. 0939 0940 By default :math:`l=0`, but in order to avoid overshoots, the step size factor 0941 :math:`(1-\gamma_{\mathrm{LS}})^{l}` with :math:`\gamma_{\mathrm{LS}}<1` can be 0942 iteratively reduced in a line search with :math:`l=0,1,2,\ldots` until c512e371cc drin*0943 :math:`\|\mathbf{F}(\mathbf{x}^k)\| < \|\mathbf{F}(\mathbf{x}^{k-1})\|`, where c704c5a1ef Mart*0944 :math:`\|\cdot\|=\int\cdot\,dx^2` is the :math:`L_2`-norm. The line search 0945 starts after :varlink:`SEAICE_JFNK_lsIter` nonlinear Newton iterations (off by 0946 default) to allow for full Newton steps at the beginning of the iteration. If 0947 the line search is turned on by setting :varlink:`SEAICE_JFNK_lsIter` to a 0948 non-negative value in ``data.seaice``, by default, the line search with 0949 :math:`\gamma_\mathrm{LS}=\frac{1}{2}` (runtime parameter 0950 :varlink:`SEAICE_JFNK_lsGamma`) is stopped after :math:`L_{\max}=4` (runtime 0951 parameter :varlink:`SEAICE_JFNK_lsLmax`) steps. c512e371cc drin*0952** Warning **
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0953 Forming the Jacobian :math:`\mathbf{J}` explicitly is often avoided as “too** Warning **
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0954 error prone and time consumingâ€. Instead, Krylov methods only require the 0955 action of :math:`\mathbf{J}` on an arbitrary vector :math:`\mathbf{w}` and 0956 hence allow a matrix free algorithm for solving :eq:`eq_jfnklin`. The action of 0957 :math:`\mathbf{J}` can be approximated by a first-order Taylor series 0958 expansion: adc83e5d7b Mart*0959 0960 .. math:: 258fe29c91 Jeff*0961 \mathbf{J}(\mathbf{x}^{k-1})\,\mathbf{w} \approx 0962 \frac{\mathbf{F}(\mathbf{x}^{k-1}+\epsilon\mathbf{w}) 0963 - \mathbf{F}(\mathbf{x}^{k-1})} \epsilon adc83e5d7b Mart*0964 :label: eq_jfnkjacvecfd 0965 0966 or computed exactly with the help of automatic differentiation (AD) 258fe29c91 Jeff*0967 tools. :varlink:`SEAICE_JFNKepsilon` sets the step size :math:`\epsilon`. adc83e5d7b Mart*0968 258fe29c91 Jeff*0969 We use the Flexible Generalized Minimum RESidual (FMGRES) method with c512e371cc drin*0970 right-hand side preconditioning to solve :eq:`eq_jfnklin` iteratively starting 0971 from a first guess of :math:`\delta\mathbf{x}^{k}_{0} = 0`. For the 0972 preconditioning matrix :math:`\mathbf{P}` we choose a simplified form of the 0973 system matrix :math:`\mathbf{A}(\mathbf{x}^{k-1})` where 0974 :math:`\mathbf{x}^{k-1}` is the estimate of the previous Newton step 0975 :math:`k-1`. The transformed equation :eq:`eq_jfnklin` becomes adc83e5d7b Mart*0976 0977 .. math:: 0978 \mathbf{J}(\mathbf{x}^{k-1})\,\mathbf{P}^{-1}\delta\mathbf{z} = 0979 -\mathbf{F}(\mathbf{x}^{k-1}), \quad\text{with} \quad 258fe29c91 Jeff*0980 \delta{\mathbf{z}} = \mathbf{P}\delta\mathbf{x}^{k} 0981 :label: eq_jfnklinpc adc83e5d7b Mart*0982 c512e371cc drin*0983 The Krylov method iteratively improves the approximate solution to 0984 :eq:`eq_jfnklinpc` in subspace (:math:`\mathbf{r}_0`, 0985 :math:`\mathbf{J}\mathbf{P}^{-1}\mathbf{r}_0`, 2c231b0ebd Mart*0986 :math:`(\mathbf{J}\mathbf{P}^{-1})^2\mathbf{r}_0`, :math:`\dots`, c512e371cc drin*0987 :math:`(\mathbf{J}\mathbf{P}^{-1})^m\mathbf{r}_0`) with increasing :math:`m`; 0988 :math:`\mathbf{r}_0 = -\mathbf{F}(\mathbf{x}^{k-1}) 0989 -\mathbf{J}(\mathbf{x}^{k-1})\,\delta\mathbf{x}^{k}_{0}` is the initial 0990 residual of :eq:`eq_jfnklin`; 0991 :math:`\mathbf{r}_0=-\mathbf{F}(\mathbf{x}^{k-1})` with the first guess dc26f158aa Mart*0992 :math:`\delta\mathbf{x}^{k}_{0}=0`. We allow a Krylov subspace of dimension \ c512e371cc drin*0993 :math:`m=50` and we do allow restarts for more than 50 Krylov iterations. The 0994 preconditioning operation involves applying :math:`\mathbf{P}^{-1}` to the 0995 basis vectors :math:`\mathbf{v}_0, \mathbf{v}_1, \mathbf{v}_2, \ldots, 0996 \mathbf{v}_m` of the Krylov subspace. This operation is approximated by solving 0997 the linear system :math:`\mathbf{P}\,\mathbf{w}=\mathbf{v}_i`. Because 0998 :math:`\mathbf{P} \approx \mathbf{A}(\mathbf{x}^{k-1})`, we can use the dc26f158aa Mart*0999 LSR algorithm already implemented in the Picard solver. Each preconditioning 1000 operation uses a fixed number of 10 LSR iterations avoiding any termination c512e371cc drin*1001 criterion. More details and results can be found in Losch et al. (2014) 1002 :cite:`losch:14`). 1003 dc26f158aa Mart*1004 To use the JFNK solver set :varlink:`SEAICEuseJFNK` ``= .TRUE.,`` in the c512e371cc drin*1005 namelist file ``data.seaice``; ``#define`` :varlink:`SEAICE_ALLOW_JFNK` in 1006 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and we recommend 1007 using a smooth regularization of :math:`\zeta` by ``#define`` 1008 :varlink:`SEAICE_ZETA_SMOOTHREG` (see above) for better convergence. The 1009 nonlinear Newton iteration is terminated when the :math:`L_2`-norm of the 1010 residual is reduced by :math:`\gamma_{\mathrm{nl}}` (run-time parameter 1011 :varlink:`SEAICEnonLinTol` ``= 1.E-4,`` will already lead to expensive 1012 simulations) with respect to the initial norm: 1013 :math:`\|\mathbf{F}(\mathbf{x}^k)\| < 1014 \gamma_{\mathrm{nl}}\|\mathbf{F}(\mathbf{x}^0)\|`. Within a nonlinear 1015 iteration, the linear FGMRES solver is terminated when the residual is smaller 1016 than :math:`\gamma_k\|\mathbf{F}(\mathbf{x}^{k-1})\|` where :math:`\gamma_k` is 1017 determined by adc83e5d7b Mart*1018 1019 .. math:: 2c231b0ebd Mart*1020 \gamma_k = 1021 \begin{cases} 1022 \gamma_0 &\text{for $\|\mathbf{F}(\mathbf{x}^{k-1})\| \geq r$}, \\ 258fe29c91 Jeff*1023 \max\left(\gamma_{\min}, 1024 \frac{\|\mathbf{F}(\mathbf{x}^{k-1})\|} 2c231b0ebd Mart*1025 {\|\mathbf{F}(\mathbf{x}^{k-2})\|}\right) 258fe29c91 Jeff*1026 &\text{for $\|\mathbf{F}(\mathbf{x}^{k-1})\| < r$,} 1027 \end{cases} 1028 :label: eq_jfnkgammalin adc83e5d7b Mart*1029 c512e371cc drin*1030 so that the linear tolerance parameter :math:`\gamma_k` decreases with the 1031 nonlinear Newton step as the nonlinear solution is approached. This inexact 1032 Newton method is generally more robust and computationally more efficient than 1033 exact methods. Typical parameter choices are :math:`\gamma_0 =` 1034 :varlink:`JFNKgamma_lin_max` :math:`= 0.99`, :math:`\gamma_{\min} =` 1035 :varlink:`JFNKgamma_lin_min` :math:`= 0.1`, and :math:`r =` 258fe29c91 Jeff*1036 :varlink:`JFNKres_tFac` :math:`\times\|\mathbf{F}(\mathbf{x}^{0})\|` with 1037 :varlink:`JFNKres_tFac` :math:`= 0.5`. We recommend a maximum number of c512e371cc drin*1038 nonlinear iterations :varlink:`SEAICEnewtonIterMax` :math:`= 100` and a maximum 1039 number of Krylov iterations :varlink:`SEAICEkrylovIterMax` :math:`= 50`, 1040 because the Krylov subspace has a fixed dimension of 50 (but restarts are 1041 allowed for :varlink:`SEAICEkrylovIterMax` :math:`> 50`). adc83e5d7b Mart*1042 dc26f158aa Mart*** Warning **
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1043 Setting :varlink:`SEAICEuseStrImpCpl` to ``.TRUE.`` turns on “strength implicit** Warning **
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1044 coupling†(see Hutchings et al. 2004 :cite:`hutchings:04`) in the LSR solver 1045 and in the LSR preconditioner for the JFNK solver. In this mode, the different c512e371cc drin*1046 contributions of the stress divergence terms are reordered so as to increase 1047 the diagonal dominance of the system matrix. Unfortunately, the convergence dc26f158aa Mart*1048 rate of the LSR solver is increased only slightly, while the JFNK convergence c512e371cc drin*1049 appears to be unaffected. adc83e5d7b Mart*1050 1051 .. _para_phys_pkg_seaice_EVPdynamics: 1052 1053 Elastic-Viscous-Plastic (EVP) Dynamics 258fe29c91 Jeff*1054 -------------------------------------- adc83e5d7b Mart*1055 c512e371cc drin*1056 Hunke and Dukowicz (1997) :cite:`hunke:97` introduced an elastic contribution 1057 to the strain rate in order to regularize :eq:`eq_vpequation` in such a way 1058 that the resulting elastic-viscous-plastic (EVP) and VP models are identical at 1059 steady state, adc83e5d7b Mart*1060 1061 .. math:: 258fe29c91 Jeff*1062 \frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} + 2c231b0ebd Mart*1063 \frac{1}{2\eta}\sigma_{ij} 1064 + \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij} 258fe29c91 Jeff*1065 + \frac{P}{4\zeta}\delta_{ij} 1066 = \dot{\epsilon}_{ij}. adc83e5d7b Mart*1067 :label: eq_evpequation 1068 dc26f158aa Mart*1069 The EVP model uses an explicit time stepping scheme with a short timestep. c512e371cc drin*1070 According to the recommendation in Hunke and Dukowicz (1997) :cite:`hunke:97`, 258fe29c91 Jeff*1071 the EVP-model should be stepped forward in time 120 times 1072 (:varlink:`SEAICE_deltaTevp` = :varlink:`SEAICE_deltaTdyn` /120) within the c512e371cc drin*1073 physical ocean model time step (although this parameter is under debate), to 1074 allow for elastic waves to disappear. Because the scheme does not require a 258fe29c91 Jeff*1075 matrix inversion it is fast in spite of the small internal timestep and simple 1076 to implement on parallel computers. For completeness, we repeat the equations c512e371cc drin*1077 for the components of the stress tensor :math:`\sigma_{1} = 1078 \sigma_{11}+\sigma_{22}`, :math:`\sigma_{2}= \sigma_{11}-\sigma_{22}`, and 1079 :math:`\sigma_{12}`. Introducing the divergence :math:`D_D = 1080 \dot{\epsilon}_{11}+\dot{\epsilon}_{22}`, and the horizontal tension and 1081 shearing strain rates, :math:`D_T = \dot{\epsilon}_{11}-\dot{\epsilon}_{22}` 1082 and :math:`D_S = 2\dot{\epsilon}_{12}`, respectively, and using the above 1083 abbreviations, the equations :eq:`eq_evpequation` can be written as: adc83e5d7b Mart*1084 1085 .. math:: 258fe29c91 Jeff*1086 \frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} + 1087 \frac{P}{2T} = \frac{P}{2T\Delta} D_D 1088 :label: eq_evpstresstensor1 0452697f42 Oliv*1089 1090 .. math:: 258fe29c91 Jeff*1091 \frac{\partial\sigma_{2}}{\partial{t}} + \frac{\sigma_{2} e^{2}}{2T} 1092 = \frac{P}{2T\Delta} D_T 1093 :label: eq_evpstresstensor2 0452697f42 Oliv*1094 1095 .. math:: 258fe29c91 Jeff*1096 \frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} 1097 = \frac{P}{4T\Delta} D_S 1098 :label: eq_evpstresstensor12 adc83e5d7b Mart*1099 1100 Here, the elastic parameter :math:`E` is redefined in terms of a damping 1101 timescale :math:`T` for elastic waves 1102 258fe29c91 Jeff*1103 .. math:: E=\frac{\zeta}{T} adc83e5d7b Mart*1104 c512e371cc drin*1105 :math:`T=E_{0}\Delta{t}` with the tunable parameter :math:`E_0<1` and the 1106 external (long) timestep :math:`\Delta{t}`. :math:`E_{0} = \frac{1}{3}` is the dc26f158aa Mart*1107 default value in the code and close to what Hunke and Dukowicz (1997) 1108 :cite:`hunke:97` recommend. c512e371cc drin*1109 dc26f158aa Mart*1110 We do not recommend to use the EVP solver in its original form. Instead, use 1111 mEVP or aEVP instead (see :numref:`para_phys_pkg_seaice_EVPstar`). If you 1112 really need to use the original EVP solver, make sure that both ``#define`` 1113 :varlink:`SEAICE_CGRID` and ``#define`` :varlink:`SEAICE_ALLOW_EVP` are set in c512e371cc drin*1114 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` (both are defined by dc26f158aa Mart*1115 default). By default, the runtime parameters :varlink:`SEAICEuseEVPstar` and 1116 :varlink:`SEAICEuseEVPrev` are set to ``.TRUE.``, which already improves the 1117 behavoir of EVP, but for the original EVP they should be set to ``.FALSE.``. The 1118 solver is turned on by setting the sub-cycling time step c512e371cc drin*1119 :varlink:`SEAICE_deltaTevp` to a value larger than zero. The choice of this dc26f158aa Mart*1120 time step is under debate. Hunke and Dukowicz (1997) :cite:`hunke:97` recommend 1121 order 120 time steps for the EVP solver within one model time step 258fe29c91 Jeff*1122 :math:`\Delta{t}` (:varlink:`deltaTmom`). One can also choose order 120 time c512e371cc drin*1123 steps within the forcing time scale, but then we recommend adjusting the 1124 damping time scale :math:`T` accordingly, by setting either 1125 :varlink:`SEAICE_elasticParm` (:math:`E_{0}`), so that :math:`E_{0}\Delta{t}=` 1126 forcing time scale, or directly :varlink:`SEAICE_evpTauRelax` (:math:`T`) to 1127 the forcing time scale. (**NOTE**: with the improved EVP variants of the next 1128 section, the above recommendations are obsolete. Use mEVP or aEVP instead.) adc83e5d7b Mart*1129 1130 .. _para_phys_pkg_seaice_EVPstar: 1131 1c8cebb321 Jeff*1132 More stable variants of Elastic-Viscous-Plastic Dynamics: EVP\*, mEVP, and aEVP 1133 ------------------------------------------------------------------------------- adc83e5d7b Mart*1134 c512e371cc drin*1135 The genuine EVP scheme appears to give noisy solutions (see Hunke 2001, Lemieux 1136 et al. 2012, Bouillon et a1. 2013 1137 :cite:`hunke:01,lemieux:12,bouillon:13`). This has led to a modified EVP or 1138 EVP\* (Lemieux et al. 2012, Bouillon et a1. 2013, Kimmritz et al. 2015 1139 :cite:`lemieux:12,bouillon:13,kimmritz:15`); here, we refer to these variants 1140 by modified EVP (mEVP) and adaptive EVP (aEVP). The main idea is to modify the** Warning **
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1141 “natural†time-discretization of the momentum equations: adc83e5d7b Mart*1142 1143 .. math:: 258fe29c91 Jeff*1144 m\frac{D\mathbf{u}}{Dt} \approx 1145 m\frac{\mathbf{u}^{p+1}-\mathbf{u}^{n}}{\Delta{t}} + 1146 \beta^{\ast}\frac{\mathbf{u}^{p+1}-\mathbf{u}^{p}}{\Delta{t}_{\mathrm{EVP}}} adc83e5d7b Mart*1147 :label: eq_evpstar 1148 c512e371cc drin*1149 where :math:`n` is the previous time step index, and :math:`p` is the previous** Warning **
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1150 sub-cycling index. The extra “intertial†term 1151 :math:`m\,(\mathbf{u}^{p+1}-\mathbf{u}^{n})/\Delta{t})` allows the definition 1152 of a residual :math:`|\mathbf{u}^{p+1}-\mathbf{u}^{p}|` that, as 1153 :math:`\mathbf{u}^{p+1} \rightarrow \mathbf{u}^{n+1}`, converges to 1154 :math:`0`. In this way EVP can be re-interpreted as a pure iterative solver 1155 where the sub-cycling has no association with time-relation (through dc26f158aa Mart*1156 :math:`\Delta{t}_{\mathrm{EVP}}`). With the setting of 1157 :varlink:`SEAICEuseEVPstar` to ``.TRUE.`` (default), this form of EVP is used. 1158 Using the terminology of Kimmritz et al. 2015 :cite:`kimmritz:15`, the evolution 1159 equations of stress :math:`\sigma_{ij}` and momentum :math:`\mathbf{u}` can be 1160 written as: adc83e5d7b Mart*1161 1162 .. math:: 258fe29c91 Jeff*1163 \sigma_{ij}^{p+1}=\sigma_{ij}^p+\frac{1}{\alpha} 1164 \Big(\sigma_{ij}(\mathbf{u}^p)-\sigma_{ij}^p\Big), 1165 \phantom{\int} 1166 :label: eq_evpstarsigma 0452697f42 Oliv*1167 1168 .. math:: 258fe29c91 Jeff*1169 \mathbf{u}^{p+1}=\mathbf{u}^p+\frac{1}{\beta} 0bad585a21 Navi*1170 \Big(\frac{\Delta t}{m} \nabla \cdot\boldsymbol{\sigma}^{p+1}+ 258fe29c91 Jeff*1171 \frac{\Delta t}{m}\mathbf{R}^{p}+\mathbf{u}_n c61841e2fd Jeff*1172 -\mathbf{u}^p\Big) 258fe29c91 Jeff*1173 :label: eq_evpstarmom adc83e5d7b Mart*1174 c512e371cc drin*1175 :math:`\mathbf{R}` contains all terms in the momentum equations except for the 1176 rheology terms and the time derivative; :math:`\alpha` and :math:`\beta` are 1177 free parameters (:varlink:`SEAICE_evpAlpha`, :varlink:`SEAICE_evpBeta`) that 1178 replace the time stepping parameters :varlink:`SEAICE_deltaTevp` 1179 (:math:`\Delta{t}_{\mathrm{EVP}}`), :varlink:`SEAICE_elasticParm` 1180 (:math:`E_{0}`), or :varlink:`SEAICE_evpTauRelax` (:math:`T`). :math:`\alpha` 1181 and :math:`\beta` determine the speed of convergence and the 1182 stability. Usually, it makes sense to use :math:`\alpha = \beta`, and 1183 :varlink:`SEAICEnEVPstarSteps` :math:`\gg (\alpha,\,\beta)` (Kimmritz et 1184 al. 2015 :cite:`kimmritz:15`). Currently, there is no termination criterion and 1185 the number of mEVP iterations is fixed to :varlink:`SEAICEnEVPstarSteps`. adc83e5d7b Mart*1186 dc26f158aa Mart*1187 In order to use mEVP in MITgcm, compile with both ``#define`` 1188 :varlink:`SEAICE_CGRID` and ``#define`` :varlink:`SEAICE_ALLOW_EVP` in 1189 :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` (default) and make 1190 sure that :varlink:`SEAICEuseEVPstar` ``= .TRUE.,`` (default) in ``data.seaice``. 1191 By default :varlink:`SEAICEuseEVPrev` is set to ``.TRUE.`` and the 1192 actual form of equations :eq:`eq_evpstarsigma` and :eq:`eq_evpstarmom` is used 1193 with fewer implicit terms and the factor of :math:`e^{2}` dropped in the stress 1194 equations :eq:`eq_evpstresstensor2` and :eq:`eq_evpstresstensor12`. Although 1195 this modifies the original EVP equations, it turns out to improve convergence c512e371cc drin*1196 (Bouillon et al. 2013 :cite:`bouillon:13`). adc83e5d7b Mart*1197 dc26f158aa Mart*1198 The aEVP scheme is an enhanced variant of mEVP (Kimmritz et al. 2016 1199 :cite:`kimmritz:16`), where the value of :math:`\alpha` is set dynamically based 1200 on the stability criterion adc83e5d7b Mart*1201 1202 .. math:: c61841e2fd Jeff*1203 \alpha = \beta = \max\left( \tilde{c} \pi\sqrt{c \frac{\zeta}{A_{c}} 258fe29c91 Jeff*1204 \frac{\Delta{t}}{\max(m,10^{-4}\,\text{kg})}},\alpha_{\min} \right) 0452697f42 Oliv*1205 :label: eq_aevpalpha adc83e5d7b Mart*1206 c512e371cc drin*1207 with the grid cell area :math:`A_c` and the ice and snow mass :math:`m`. This 1208 choice sacrifices speed of convergence for stability with the result that aEVP 1209 converges quickly to VP where :math:`\alpha` can be small and more slowly in 1210 areas where the equations are stiff. In practice, aEVP leads to an overall dc26f158aa Mart*1211 better convergence than mEVP (Kimmritz et al. 2016 :cite:`kimmritz:16`). To use 1212 aEVP in MITgcm set :varlink:`SEAICEaEVPcoeff` :math:`= \tilde{c}` 1213 (see :eq:`eq_aevpalpha`; default is unset); this also 1214 sets the default values of :varlink:`SEAICEaEVPcStar` (:math:`c=4`) and c512e371cc drin*1215 :varlink:`SEAICEaEVPalphaMin` (:math:`\alpha_{\min}=5`). Good convergence has 1216 been obtained with these values (Kimmritz et al. 2016 :cite:`kimmritz:16`): adc83e5d7b Mart*1217 dc26f158aa Mart*1218 :: 1219 1220 SEAICEaEVPcoeff = 0.5, 1221 SEAICEnEVPstarSteps = 500, 1222 # The following two parameters are required by mEVP and aEVP, 1223 # but they are TRUE by default: 1224 SEAICEuseEVPstar = .TRUE., 1225 SEAICEuseEVPrev = .TRUE., 1226 1227 Because of the C-grid staggering of velocities and c512e371cc drin*1228 stresses, mEVP may not converge as successfully as in Kimmritz et al. (2015) dc26f158aa Mart*1229 :cite:`kimmritz:15`, see also Kimmritz et al. (2016) :cite:`kimmritz:16`. 1230 Convergence at very high resolution (order 5 km) has not yet been studied. adc83e5d7b Mart*1231 1232 .. _para_phys_pkg_seaice_iceoceanstress: 1233 1234 Ice-Ocean stress 258fe29c91 Jeff*1235 ---------------- adc83e5d7b Mart*1236 c512e371cc drin*1237 Moving sea ice exerts a stress on the ocean which is the opposite of the stress 1238 :math:`\mathbf{\tau}_\mathrm{ocean}` in :eq:`eq_momseaice`. This stress is 1239 applied directly to the surface layer of the ocean model. An alternative ocean 1240 stress formulation is given by Hibler and Bryan (1987) 1241 :cite:`hibler:87`. Rather than applying :math:`\mathbf{\tau}_\mathrm{ocean}` 1242 directly, the stress is derived from integrating over the ice thickness to the 1243 bottom of the oceanic surface layer. In the resulting equation for the 1244 *combined* ocean-ice momentum, the interfacial stress cancels and the total 1245 stress appears as the sum of windstress and divergence of internal ice 1246 stresses: :math:`\delta(z) (\mathbf{\tau}_\mathrm{air} + \mathbf{F})/\rho_0`, 1247 see also Eq. (2) of Hibler and Bryan (1987) :cite:`hibler:87`. The disadvantage 1248 of this formulation is that now the velocity in the surface layer of the ocean 1249 that is used to advect tracers, is really an average over the ocean surface adc83e5d7b Mart*1250 velocity and the ice velocity leading to an inconsistency as the ice c512e371cc drin*1251 temperature and salinity are different from the oceanic variables. To turn on 1252 the stress formulation of Hibler and Bryan (1987) :cite:`hibler:87`, set dc26f158aa Mart*1253 :varlink:`useHB87StressCoupling` ``=.TRUE.,``, in ``data.seaice``. adc83e5d7b Mart*1254 1255 .. _para_phys_pkg_seaice_discretization: 1256 1257 1258 Finite-volume discretization of the stress tensor divergence 258fe29c91 Jeff*1259 ------------------------------------------------------------ adc83e5d7b Mart*1260 c512e371cc drin*1261 On an Arakawa C grid, ice thickness and concentration and thus ice strength 1262 :math:`P` and bulk and shear viscosities :math:`\zeta` and :math:`\eta` are 1263 naturally defined a C-points in the center of the grid cell. Discretization 1264 requires only averaging of :math:`\zeta` and :math:`\eta` to vorticity or 1265 Z-points (or :math:`\zeta`-points, but here we use Z in order avoid confusion 1266 with the bulk viscosity) at the bottom left corner of the cell to give 1267 :math:`\overline{\zeta}^{Z}` and :math:`\overline{\eta}^{Z}`. In the following, 1268 the superscripts indicate location at Z or C points, distance across the cell 1269 (F), along the cell edge (G), between :math:`u`-points (U), :math:`v`-points 1270 (V), and C-points (C). The control volumes of the :math:`u`- and adc83e5d7b Mart*1271 :math:`v`-equations in the grid cell at indices :math:`(i,j)` are 1272 :math:`A_{i,j}^{w}` and :math:`A_{i,j}^{s}`, respectively. With these 1273 definitions (which follow the model code documentation except that c512e371cc drin*1274 :math:`\zeta`-points have been renamed to Z-points), the strain rates are 1275 discretized as: adc83e5d7b Mart*1276 1277 .. math:: 1278 \begin{aligned} 1279 \dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag 2c231b0ebd Mart*1280 => (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} 1281 + k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\ adc83e5d7b Mart*1282 \dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag 2c231b0ebd Mart*1283 => (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} 1284 + k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ adc83e5d7b Mart*1285 \dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl( 1286 \partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1} 1287 \biggr) \\ \notag 1288 => (\epsilon_{12})_{i,j}^Z &= \frac{1}{2} 2c231b0ebd Mart*1289 \biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V} adc83e5d7b Mart*1290 + \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag 1291 &\phantom{=\frac{1}{2}\biggl(} 1292 - k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} 1293 - k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} c512e371cc drin*1294 \biggr), 1295 \end{aligned} adc83e5d7b Mart*1296 c512e371cc drin*1297 so that the diagonal terms of the strain rate tensor are naturally defined at 1298 C-points and the symmetric off-diagonal term at Z-points. No-slip boundary 1299 conditions (:math:`u_{i,j-1}+u_{i,j}=0` and :math:`v_{i-1,j}+v_{i,j}=0` across** Warning **
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1300 boundaries) are implemented via “ghost-pointsâ€; for free slip boundary 1301 conditions :math:`(\epsilon_{12})^Z=0` on boundaries. adc83e5d7b Mart*1302 1303 For a spherical polar grid, the coefficients of the metric terms are 1304 :math:`k_{1}=0` and :math:`k_{2}=-\tan\phi/a`, with the spherical radius c512e371cc drin*1305 :math:`a` and the latitude :math:`\phi`; :math:`\Delta{x}_1 = \Delta{x} = 1306 a\cos\phi \Delta\lambda`, and :math:`\Delta{x}_2 = \Delta{y}=a\Delta\phi`. For 1307 a general orthogonal curvilinear grid, :math:`k_{1}` and :math:`k_{2}` can be 1308 approximated by finite differences of the cell widths: adc83e5d7b Mart*1309 1310 .. math:: 1311 \begin{aligned} 1312 k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}} 1313 \frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\ 1314 k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}} 1315 \frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\ 1316 k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}} 1317 \frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\ 1318 k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}} c512e371cc drin*1319 \frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}} 1320 \end{aligned} adc83e5d7b Mart*1321 1322 The stress tensor is given by the constitutive viscous-plastic relation 1323 :math:`\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} + c512e371cc drin*1324 [(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 ]\delta_{\alpha\beta}` . The 1325 stress tensor divergence :math:`(\nabla\sigma)_{\alpha} = 1326 \partial_\beta\sigma_{\beta\alpha}`, is discretized in finite volumes . This** Warning **
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1327 conveniently avoids dealing with further metric terms, as these are “hidden†in 1328 the differential cell widths. For the :math:`u`-equation (:math:`\alpha=1`) we 1329 have: adc83e5d7b Mart*1330 1331 .. math:: 1332 \begin{aligned} 1333 (\nabla\sigma)_{1}: \phantom{=}& 1334 \frac{1}{A_{i,j}^w} c512e371cc drin*1335 \int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21}) 1336 \,dx_1\,dx_2 \\\notag adc83e5d7b Mart*1337 =& \frac{1}{A_{i,j}^w} \biggl\{ c512e371cc drin*1338 \int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl|_{x_{1}}^{x_{1} 1339 +\Delta{x}_{1}} 1340 + \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl|_{x_{2}}^{x_{2} 1341 +\Delta{x}_{2}} adc83e5d7b Mart*1342 \biggr\} \\ \notag 1343 \approx& \frac{1}{A_{i,j}^w} \biggl\{ 1344 \Delta{x}_2\sigma_{11}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} 1345 + \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} 1346 \biggr\} \\ \notag 1347 =& \frac{1}{A_{i,j}^w} \biggl\{ 1348 (\Delta{x}_2\sigma_{11})_{i,j}^C - 2c231b0ebd Mart*1349 (\Delta{x}_2\sigma_{11})_{i-1,j}^C adc83e5d7b Mart*1350 \\\notag 1351 \phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{} 1352 + (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z c512e371cc drin*1353 \biggr\} 1354 \end{aligned} adc83e5d7b Mart*1355 1356 with 1357 1358 .. math:: 1359 \begin{aligned} 1360 (\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+} 1361 \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} 1362 \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag 1363 &+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} 1364 k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag 1365 \phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} 1366 \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag 1367 \phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} 1368 k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag 1369 \phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\ 1370 (\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+} 1371 \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} 1372 \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag 1373 & + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} 1374 \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag 2c231b0ebd Mart*1375 & - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} adc83e5d7b Mart*1376 k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag 2c231b0ebd Mart*1377 & - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} c512e371cc drin*1378 k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} 1379 \end{aligned} adc83e5d7b Mart*1380 1381 Similarly, we have for the :math:`v`-equation (:math:`\alpha=2`): 1382 1383 .. math:: 1384 \begin{aligned} 1385 (\nabla\sigma)_{2}: \phantom{=}& 1386 \frac{1}{A_{i,j}^s} c512e371cc drin*1387 \int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22}) 1388 \,dx_1\,dx_2 \\\notag adc83e5d7b Mart*1389 =& \frac{1}{A_{i,j}^s} \biggl\{ c512e371cc drin*1390 \int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl|_{x_{1}}^{x_{1} 1391 +\Delta{x}_{1}} 1392 + \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl|_{x_{2}}^{x_{2} 1393 +\Delta{x}_{2}} adc83e5d7b Mart*1394 \biggr\} \\ \notag 1395 \approx& \frac{1}{A_{i,j}^s} \biggl\{ 1396 \Delta{x}_2\sigma_{12}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} 1397 + \Delta{x}_1\sigma_{22}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} 1398 \biggr\} \\ \notag 1399 =& \frac{1}{A_{i,j}^s} \biggl\{ 1400 (\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z 1401 \\ \notag 1402 \phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{} 1403 + (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C 1404 \biggr\} \end{aligned} 1405 1406 with 1407 1408 .. math:: 1409 \begin{aligned} 1410 (\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+} 1411 \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} 2c231b0ebd Mart*1412 \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} adc83e5d7b Mart*1413 \\\notag & 1414 + \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} 1415 \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\\notag 1416 &- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} 2c231b0ebd Mart*1417 k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} adc83e5d7b Mart*1418 \\\notag & 1419 - \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} 1420 k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag 1421 (\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+} 1422 \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} 1423 \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag 1424 &+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} 1425 k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag 1426 & + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} 1427 \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag 1428 & + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} 1429 k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag 1430 & -\Delta{x}_{i,j}^{F} \frac{P}{2}\end{aligned} 1431 258fe29c91 Jeff*1432 Again, no-slip boundary conditions are realized via ghost points and adc83e5d7b Mart*1433 :math:`u_{i,j-1}+u_{i,j}=0` and :math:`v_{i-1,j}+v_{i,j}=0` across c512e371cc drin*1434 boundaries. For free-slip boundary conditions the lateral stress is set to 1435 zeros. In analogy to :math:`(\epsilon_{12})^Z=0` on boundaries, we set 1436 :math:`\sigma_{21}^{Z}=0`, or equivalently :math:`\eta_{i,j}^{Z}=0`, on 1437 boundaries. adc83e5d7b Mart*1438 a4e168e012 antn*1439 .. _ssub_phys_pkg_seaice_thermodynamics: adc83e5d7b Mart*1440 1441 Thermodynamics a4e168e012 antn*1442 ============== adc83e5d7b Mart*1443 c61841e2fd Jeff*1444 **NOTE: THIS SECTION IS STILL NOT COMPLETE** adc83e5d7b Mart*1445 c512e371cc drin*1446 In its original formulation the sea ice model uses simple 0-layer 1447 thermodynamics following the appendix of Semtner (1976) 1448 :cite:`semtner:76`. This formulation neglects storage of heat, that is, the 1449 heat capacity of ice is zero, and all internal heat sources so that the heat 1450 equation reduces to a constant conductive heat flux. This constant upward 1451 conductive heat flux together with a constant ice conductivity implies a linear 1452 temperature profile. The boundary conditions for the heat equations are: at the 0bad585a21 Navi*1453 bottom of the ice :math:`T|_{\rm bottom} = T_{\rm fr}` (freezing point temperature of 1454 sea water), and at the surface: :math:`Q_{\rm top} = 1455 \frac{\partial{T}}{\partial{z}} = (K/h)(T_{0}-T_{\rm fr})`, where :math:`K` is the 1456 ice conductivity, :math:`h` the ice thickness, and :math:`T_{0}-T_{\rm fr}` the c512e371cc drin*1457 difference between the ice surface temperature and the water temperature at the 1458 bottom of the ice (at the freezing point). The surface heat flux 0bad585a21 Navi*1459 :math:`Q_{\rm top}` is computed in a similar way to that of Parkinson and c512e371cc drin*1460 Washington (1979) :cite:`parkinson:79` and Manabe et al. (1979) 1461 :cite:`manabe:79`. The resulting equation for surface temperature is adc83e5d7b Mart*1462 c61841e2fd Jeff*1463 .. math:: 1464 \begin{aligned} 0bad585a21 Navi*1465 \frac{K}{h}(T_{0}-T_{\rm fr}) &= Q_{\rm SW\downarrow}(1-\mathrm{albedo}) \\ 1466 & + \epsilon Q_{\rm LW\downarrow} - Q_{\rm LW\uparrow}(T_{0}) \\ 1467 & + Q_{\rm LH}(T_{0}) + Q_{\rm SH}(T_{0}), c61841e2fd Jeff*1468 \end{aligned} 1469 :label: eq_zerolayerheatbalance 2c231b0ebd Mart*1470 c61841e2fd Jeff*1471 where :math:`\epsilon` is the emissivity of the surface (snow or ice), 0bad585a21 Navi*1472 :math:`Q_{\rm S/LW\downarrow}` the downwelling shortwave and longwave radiation to 1473 be prescribed, and :math:`Q_{\rm LW\uparrow}=\epsilon\sigma_B T_{0}^4` the emitted c512e371cc drin*1474 long wave radiation with the Stefan-Boltzmann constant :math:`\sigma_B`. With 1475 explicit expressions in :math:`T_0` for the turbulent fluxes of latent and 1476 sensible heat c61841e2fd Jeff*1477 1478 .. math:: 2c231b0ebd Mart*1479 \begin{aligned} 0bad585a21 Navi*1480 Q_{\rm LH} &= \rho_\mathrm{air} C_E (\Lambda_v + \Lambda_f) 2c231b0ebd Mart*1481 |\mathbf{U}_\mathrm{air}| 1482 \left[ q_\mathrm{air} - q_\mathrm{sat}(T_0)\right] \\ 0bad585a21 Navi*1483 Q_{\rm SH} &= \rho_\mathrm{air} c_p C_E |\mathbf{U}_\mathrm{air}| 2c231b0ebd Mart*1484 \left[ T_\mathrm{10m} - T_{0} \right], 1485 \end{aligned} c61841e2fd Jeff*1486 0bad585a21 Navi*1487 :eq:`eq_zerolayerheatbalance` can be solved for :math:`T_0` with an iterative 1488 Ralphson-Newton method, which usually converges very quickly in less that 10 1489 iterations. In these equations, :math:`\rho_\mathrm{air}` is the air density 1490 (parameter :varlink:`SEAICE_rhoAir`), :math:`C_E` is the ice-ocean transfer 1491 coefficient for sensible and latent heat (parameter :varlink:`SEAICE_dalton`), c512e371cc drin*1492 :math:`\Lambda_v` and :math:`\Lambda_f` are the latent heat of vaporization and 0bad585a21 Navi*1493 fusion, respectively (parameters :varlink:`SEAICE_lhEvap` and 1494 :varlink:`SEAICE_lhFusion`), and :math:`c_p` is the specific heat of air 1495 (parameter :varlink:`SEAICE_cpAir`). For the latent heat :math:`Q_{\rm LH}` a 1496 choice can be made between the old polynomial expression for saturation 1497 humidity :math:`q_\mathrm{sat}(T_0)` (by setting 1498 :varlink:`useMaykutSatVapPoly` to ``.TRUE.``) and the default exponential 1499 relation approximation that is more accurate at low temperatures. c512e371cc drin*1500 1501 In the zero-layer model of Semtner (1976) :cite:`semtner:76`, the conductive 1502 heat flux depends strongly on the ice thickness :math:`h`. However, the ice 1503 thickness in the model represents a mean over a potentially very heterogeneous 1504 thickness distribution. In order to parameterize a sub-grid scale distribution 14673ec2d0 Mart*1505 for heat flux computations, the ice thickness :math:`h` is split into c512e371cc drin*1506 :math:`N` thickness categories :math:`H_{n}` that are equally distributed 1507 between :math:`2h` and a minimum imposed ice thickness of :math:`5\,\text{cm}` 1508 by :math:`H_n= \frac{2n-1}{7}\,h` for :math:`n\in[1,N]`. The heat fluxes 1509 computed for each thickness category are area-averaged to give the total heat 1510 flux (see Hibler 1984 :cite:`hibler:84`). To use this thickness category 1511 parameterization set :varlink:`SEAICE_multDim` to the number of desired 1512 categories in ``data.seaice`` (7 is a good guess, for anything larger than 7 1513 modify :filelink:`SEAICE_SIZE.h <pkg/seaice/SEAICE_SIZE.h>`). Note that this 1514 requires different restart files and switching this flag on in the middle of an 1515 integration is not advised. As an alternative to the flat distribution, the 1516 run-time parameter :varlink:`SEAICE_PDF` (1D-array of lenght :varlink:`nITD`) 1517 can be used to prescribe an arbitrary distribution of ice thicknesses, for 1518 example derived from observed distributions (Castro-Morales et al. 2014 1519 :cite:`castro-morales:14`). In order to include the ice thickness distribution dc26f158aa Mart*1520 also for snow, set :varlink:`SEAICE_useMultDimSnow` to ``.TRUE.`` (this is the c512e371cc drin*1521 default); only then, the parameterization of always having a fraction of thin 1522 ice is efficient and generally thicker ice is produced (see Castro-Morales et 1523 al. 2014 :cite:`castro-morales:14`). 1524 1525 The atmospheric heat flux is balanced by an oceanic heat flux from below. The 1526 oceanic flux is proportional to :math:`\rho\,c_{p}\left(T_{w}-T_{fr}\right)` 1527 where :math:`\rho` and :math:`c_{p}` are the density and heat capacity of sea 0bad585a21 Navi*1528 water and :math:`T_{\rm fr}` is the local freezing point temperature that is a c512e371cc drin*1529 function of salinity. This flux is not assumed to instantaneously melt or 1530 create ice, but a time scale of three days (run-time parameter 1531 :varlink:`SEAICE_gamma_t`) is used to relax :math:`T_{w}` to the freezing 1532 point. The parameterization of lateral and vertical growth of sea ice follows 1533 that of Hibler (1979) and Hibler (1980) :cite:`hibler:79,hibler:80`; the 1534 so-called lead closing parameter :math:`h_{0}` (run-time parameter 1535 :varlink:`HO`) has a default value of 0.5 meters. 1536 1537 On top of the ice there is a layer of snow that modifies the heat flux and the 1538 albedo (Zhang et al. 1998 :cite:`zha:98`). Snow modifies the effective 1539 conductivity according to adc83e5d7b Mart*1540 1541 .. math:: \frac{K}{h} \rightarrow \frac{1}{\frac{h_{s}}{K_{s}}+\frac{h}{K}}, 1542 1543 where :math:`K_s` is the conductivity of snow and :math:`h_s` the snow c512e371cc drin*1544 thickness. If enough snow accumulates so that its weight submerges the ice and 1545 the snow is flooded, a simple mass conserving parameterization of snowice** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 1547.
1546 formation (a flood-freeze algorithm following Archimedes’ principle) turns snow 1547 into ice until the ice surface is back at :math:`z=0` (see Leppäranta 1983 1548 :cite:`leppaeranta:83`). The flood-freeze algorithm is turned on with run-time dc26f158aa Mart*1549 parameter :varlink:`SEAICEuseFlooding` set to ``.TRUE.``. adc83e5d7b Mart*1550 1551 .. _para_phys_pkg_seaice_advection: 1552 1553 Advection of thermodynamic variables 258fe29c91 Jeff*1554 ------------------------------------ adc83e5d7b Mart*1555 14673ec2d0 Mart*1556 Mean ice thickness (ice volume per unit area, :math:`c h`, model variable 1557 :varlink:`HEFF`, which implies the misleading name "effective thickness"), 1558 concentration :math:`c` (model variable :varlink:`AREA`) and mean snow 1559 thickness (:math:`c h_s`, model variable :varlink:`HSNOW`) are advected by ice 1560 velocities: adc83e5d7b Mart*1561 1562 .. math:: 258fe29c91 Jeff*1563 \frac{\partial{X}}{\partial{t}} = 0bad585a21 Navi*1564 - \nabla \cdot\left(\mathbf{u}\,X\right) + \Gamma_{X} + D_{X} 0452697f42 Oliv*1565 :label: eq_advection adc83e5d7b Mart*1566 c512e371cc drin*1567 where :math:`\Gamma_X` are the thermodynamic source terms and :math:`D_{X}` the 14673ec2d0 Mart*1568 diffusive terms for quantities :math:`X= c h, c, c h_s` or any other tracer, 1569 such as sea ice salinity. From the various advection schemes that are available 1570 in MITgcm, we recommend flux-limited schemes (runtime flag 1571 :varlink:`SEAICEadvScheme`; default=77, a 2nd-order flux limited scheme) to 1572 preserve sharp gradients and edges that are typical of sea ice distributions 1573 and to rule out unphysical over- and undershoots (negative thickness or 1574 concentration). These schemes conserve volume and horizontal area and are 1575 unconditionally stable, so that we can set :math:`D_{X}=0` (runtime flag 1576 :varlink:`DIFF1` = :math:`D_{X}/\Delta{x}`; default=0). adc83e5d7b Mart*1577 c512e371cc drin*1578 The MITgcm sea ice model provides the option to use the thermodynamics model of 1579 Winton (2000) :cite:`winton:00`, which in turn is based on the 3-layer model of 1580 Semtner (1976) :cite:`semtner:76` which treats brine content by means of 1581 enthalpy conservation; the corresponding package :filelink:`thsice 1582 <pkg/thsice>` is described in section :numref:`sub_phys_pkg_thsice`. This 1583 scheme requires additional state variables, namely the enthalpy of the two ice 1584 layers (instead of effective ice salinity), to be advected by ice 1585 velocities. The internal sea ice temperature is inferred from ice enthalpy. To 1586 avoid unphysical (negative) values for ice thickness and concentration, a 258fe29c91 Jeff*1587 positive 2nd-order advection scheme with a SuperBee flux limiter (Roe 1985 c512e371cc drin*1588 :cite:`roe:85`) should be used to advect all sea-ice-related quantities of the 1589 Winton (2000) :cite:`winton:00` thermodynamic model (run-time flag 1590 :varlink:`thSIceAdvScheme` :math:`= 77` and :varlink:`thSIce_diffK` :math:`= 1591 D_{X} = 0` in ``data.ice``, defaults are 0). Because of the nonlinearity of the 1592 advection scheme, care must be taken in advecting these quantities: when simply 1593 using ice velocity to advect enthalpy, the total energy (i.e., the volume 1594 integral of enthalpy) is not conserved. Alternatively, one can advect the 1595 energy content (i.e., product of ice-volume and enthalpy) but then false 1596 enthalpy extrema can occur, which then leads to unrealistic ice temperature. In 1597 the currently implemented solution, the sea-ice mass flux is used to advect the 1598 enthalpy in order to ensure conservation of enthalpy and to prevent false 1599 enthalpy extrema. adc83e5d7b Mart*1600 dce651c1fb Mart*1601 .. _para_phys_pkg_seaice_itd: 1602 1603 Dynamical Ice Thickness Distribution (ITD) 258fe29c91 Jeff*1604 ------------------------------------------ dce651c1fb Mart*1605 258fe29c91 Jeff*1606 The ice thickness distribution model used by MITgcm follows the implementation 1607 in the Los Alamos sea ice model CICE (https://github.com/CICE-Consortium/CICE). c512e371cc drin*1608 There are two parts to it that are closely connected: the participation and 1609 ridging functions that determine which thickness classes take part in ridging 1610 and which thickness classes receive ice during ridging based on Thorndike et 1611 al. (1975) :cite:`thorndike:75`, and the ice strength parameterization by 1612 Rothrock (1975) :cite:`rothrock:75` which uses this information. The following 1613 description is slightly modified from Ungermann et al. (2017) 1614 :cite:`ungermann:17`. Verification experiment :filelink:`seaice_itd 1615 <verification/seaice_itd>` uses the ITD model. dce651c1fb Mart*1616 1617 Distribution, participation and redistribution functions in ridging 258fe29c91 Jeff*1618 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ dce651c1fb Mart*1619 c512e371cc drin*1620 When :varlink:`SEAICE_ITD` is defined in :filelink:`SEAICE_OPTIONS.h 1621 <pkg/seaice/SEAICE_OPTIONS.h>`, the ice thickness is described by the ice 1622 thickness distribution :math:`g(h,\mathbf{x},t)` for the subgrid-scale (see 1623 Thorndike et al. 1975 :cite:`thorndike:75`), a probability density function for 1624 thickness :math:`h` following the evolution equation dce651c1fb Mart*1625 1626 1627 .. math:: 0bad585a21 Navi*1628 \frac{\partial g}{\partial t} = - \nabla \cdot (\mathbf{u} g) - \frac{\partial}{\partial h}(fg) + \Psi. dce651c1fb Mart*1629 :label: eq_itd 1630 1631 c512e371cc drin*1632 Here :math:`f=\frac{\mathrm{d} h}{\mathrm{d} t}` is the thermodynamic growth 1633 rate and :math:`\Psi` a function describing the mechanical redistribution of 1634 sea ice during ridging or lead opening. dce651c1fb Mart*1635 c512e371cc drin*1636 The mechanical redistribution function :math:`\Psi` generates open water in 1637 divergent motion and creates ridged ice during convergent motion. The ridging 1638 process depends on total strain rate and on the ratio between shear (run-time 1639 parameter :varlink:`SEAICEshearParm`) and divergent strain. In the single 1640 category model, ridge formation is treated implicitly by limiting the ice 1641 concentration to a maximum of one (see Hibler 1979 :cite:`hibler:79`), so that 1642 further volume increase in convergent motion leads to thicker ice. (This is 1643 also the default for ITD models; to change from the default, set run-time dc26f158aa Mart*1644 parameter :varlink:`SEAICEsimpleRidging` ``=.FALSE.,`` in ``data.seaice``). For c512e371cc drin*1645 the ITD model, the ridging mode in convergence dce651c1fb Mart*1646 1647 .. math:: 1648 \omega_r(h)= \frac{-a(h)+n(h)}{N} 2c231b0ebd Mart*1649 c512e371cc drin*1650 gives the effective change for the ice volume with thickness between :math:`h` 1651 and :math:`h+\textrm{d} h` as the normalized difference between the ice 1652 :math:`n(h)` generated by ridging and the ice :math:`a(h)` participating in 1653 ridging. 1654 1655 The participation function :math:`a(h) = b(h)g(h)` can be computed either 1656 following Thorndike et al. (1975) :cite:`thorndike:75` (run-time parameter 1657 :varlink:`SEAICEpartFunc` =0) or Lipscomb et al. (2007) :cite:`lipscomb:07` 1658 (:varlink:`SEAICEpartFunc` =1), and similarly the ridging function :math:`n(h)` 1659 can be computed following Hilber (1980) :cite:`hibler:80` (run-time parameter 1660 :varlink:`SEAICEredistFunc` =0) or Lipscomb et al. (2007) :cite:`lipscomb:07` 1661 (:varlink:`SEAICEredistFunc` =1). As an example, we show here the functions 1662 that Lipscomb et al. (2007) :cite:`lipscomb:07` suggested to avoid noise in the 1663 solutions. These functions are smooth and avoid non-differentiable 1664 discontinuities, but so far we did not find any noise issues as in Lipscomb et 1665 al. (2007) :cite:`lipscomb:07`. 1666 dc26f158aa Mart*1667 With :varlink:`SEAICEpartFunc` ``= 1,`` in ``data.seaice``, the participation c512e371cc drin*1668 function with the relative amount of ice of thickness :math:`h` weighted by an 1669 exponential function dce651c1fb Mart*1670 1671 .. math:: 1672 b(h) = b_0 \exp [ -G(h)/a^*] 2c231b0ebd Mart*1673 c512e371cc drin*1674 where :math:`G(h)=\int_0^h g(h) \textrm{d} h` is the cumulative thickness 1675 distribution function, :math:`b_0` a normalization factor, and :math:`a^*` 1676 (:varlink:`SEAICEaStar`) the exponential constant that determines which 1677 relative amount of thicker and thinner ice take part in ridging. dce651c1fb Mart*1678 dc26f158aa Mart*1679 With :varlink:`SEAICEredistFunc` ``= 1,`` in ``data.seaice``, the ice generated by c512e371cc drin*1680 ridging is calculated as dce651c1fb Mart*1681 1682 .. math:: 1683 n(h) = \int_0^\infty a(h_1)\gamma(h_1,h) \textrm{d} h_1 1684 c512e371cc drin*1685 where the density function :math:`\gamma(h_1,h)` of resulting thickness 1686 :math:`h` for ridged ice with an original thickness of :math:`h_1` is taken as dce651c1fb Mart*1687 1688 .. math:: c512e371cc drin*1689 \gamma(h_1, h) = \frac{1}{k \lambda} 1690 \exp\left[{\frac{-(h-h_{\min})}{\lambda}}\right] 1691 1692 for :math:`h \geq h_{\min}`, with :math:`\gamma(h_1,h)=0` for :math:`h < 1693 h_{\min}`. In this parameterization, the normalization factor 1694 :math:`k=\frac{h_{\min} + \lambda}{h_1}`, the e-folding scale :math:`\lambda = 1695 \mu h_1^{1/2}` and the minimum ridge thickness :math:`h_{\min}=\min(2h_1,h_1 + 1696 h_{\textrm{raft}})` all depend on the original thickness :math:`h_1`. The 1697 maximal ice thickness allowed to raft :math:`h_{\textrm{raft}}` is constant 1698 (:varlink:`SEAICEmaxRaft`, default =1 m) and :math:`\mu` 1699 (:varlink:`SEAICEmuRidging`) is a tunable parameter. 258fe29c91 Jeff*1700 1701 In the numerical model these equations are discretized into a set of :math:`n` c512e371cc drin*1702 (:varlink:`nITD` defined in :filelink:`SEAICE_SIZE.h 1703 <pkg/seaice/SEAICE_SIZE.h>`) thickness categories employing the delta function 1704 scheme of Bitz et al. (2001) :cite:`bitz:01`. For each thickness category in 1705 an ITD configuration, the volume conservation equation :eq:`eq_advection` is 1706 evaluated using the heat flux with the category-specific values for ice and 1707 snow thickness, so there are no conceptual differences in the thermodynamics 1708 between the single category and ITD configurations. The only difference is 1709 that only in the thinnest category the creation of new ice of thickness 1710 :math:`H_0` (run-time parameter :varlink:`HO`) is possible, all other 1711 categories are limited to basal growth. The conservation of ice area is 1712 replaced by the evolution equation of the ITD :eq:`eq_itd` that is discretized 1713 in thickness space with :math:`n+1` category limits given by run-time parameter 1714 :varlink:`Hlimit`. If :varlink:`Hlimit` is not set in ``data.seaice``, a 1715 simple recursive formula following Lipscomb (2001) :cite:`lipscomb:01` is used 1716 to compute :varlink:`Hlimit`: 258fe29c91 Jeff*1717 1718 .. math:: c512e371cc drin*1719 H_\mathrm{limit}(k) = H_\mathrm{limit}(k-1) + \frac{c_1}{n} 1720 + \frac{c_1 c_2}{n} [ 1 + \tanh c_3 (\frac{k-1}{n} - 1) ] 258fe29c91 Jeff*1721 0bad585a21 Navi*1722 with :math:`H_\mathrm{limit}(0)=0` m and 1723 :math:`H_\mathrm{limit}(n)=999.9` m. The three constants are the c512e371cc drin*1724 run-time parameters :varlink:`Hlimit_c1`, :varlink:`Hlimit_c2`, and 1725 :varlink:`Hlimit_c3`. The total ice concentration and volume can then be 1726 calculated by summing up the values for each category. dce651c1fb Mart*1727 1728 Ice strength parameterization 258fe29c91 Jeff*1729 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ dce651c1fb Mart*1730 c512e371cc drin*1731 In the default approach of equation :eq:`eq_icestrength`, the ice strength is 1732 parameterized following Hibler (1979) :cite:`hibler:79` and :math:`P` depends 1733 only on average ice concentration and thickness per grid cell and the constant 1734 ice strength parameters :math:`P^{\ast}` (:varlink:`SEAICE_strength`) and 1735 :math:`C^{\ast}` (:varlink:`SEAICE_cStar`). With an ice thickness 1736 distribution, it is possible to use a different parameterization following 1737 Rothrock (1975) :cite:`rothrock:75` dce651c1fb Mart*1738 1739 .. math:: 258fe29c91 Jeff*1740 P = C_f C_p \int_0^\infty h^2 \omega_r(h) \textrm{d}h dce651c1fb Mart*1741 :label: eq_rothrock 1742 c512e371cc drin*1743 by considering the production of potential energy and the frictional energy 1744 loss in ridging. The physical constant :math:`C_p = \rho_i (\rho_w - \rho_i) 1745 \hat{g} / (2 \rho_w)` is a combination of the gravitational acceleration 1746 :math:`\hat{g}` and the densities :math:`\rho_i`, :math:`\rho_w` of ice and 1747 water, and :math:`C_f` (:varlink:`SEAICE_cf`) is a scaling factor relating the 1748 amount of work against gravity necessary for ridging to the amount of work 1749 against friction. To calculate the integral, this parameterization needs 1750 information about the ITD in each grid cell, while the default 1751 parameterization :eq:`eq_icestrength` can be used for both ITD and single 1752 thickness category models. In contrast to :eq:`eq_icestrength`, which is based 1753 on the plausible assumption that thick and compact ice is stronger than thin 1754 and loose drifting ice, this parameterization :eq:`eq_rothrock` clearly 1755 contains the more physical assumptions about energy conservation. For that 1756 reason alone this parameterization is often considered to be more physically 1757 realistic than :eq:`eq_icestrength`, but in practice, the success is not so 1758 clear (Ungermann et al. 2007 :cite:`ungermann:17`). Ergo, the default is to dc26f158aa Mart*1759 use :eq:`eq_icestrength`; set :varlink:`useHibler79IceStrength` ``=.FALSE.,`` in c512e371cc drin*1760 ``data.seaice`` to change this behavior. dce651c1fb Mart*1761 5b18319545 Jeff*1762 Known issues and work-arounds 1763 ============================= 1764 1765 - An often encountered problem in long simulations with sea ice models is 1766 (local) perpetually increasing sea ice (plus snow) height; this is 1767 problematic when using a non-linear free surface and dc26f158aa Mart*1768 :varlink:`useRealFreshWaterFlux` set to ``.TRUE.``, because the mass of the sea ice 5b18319545 Jeff*1769 places a load on the sea surface, which if too large, can cause the surface 1770 cells of the model to become too thin so that the model eventually stops with 1771 an error message. Usually this problem occurs because of dynamical ice growth 1772 (i.e., convergence and ridging of ice) or simply too much net precipitation 1773 with insufficient summer surface melting. If the problem is dynamical in 1774 nature (e.g., caused by ridging in a deep inlet), the first step to try is to 1775 turn off the replacement pressure method (:varlink:`SEAICEpressReplFac` = 0; 1776 in :numref:`para_phys_pkg_seaice_VPrheology`); turning this off provides 1777 resistance against additional growth due to further ridging, because the ice 1778 pressure :math:`P` is no longer reduced as :math:`\Delta\rightarrow 0` in 1779 nearly motionless thick ice :eq:`eq_pressrepl`. If this does not solve the 1780 problem, a somewhat more radical yet effective approach is simply to cap the 1781 sea ice load on the free surface by defining the CPP option 1782 :varlink:`SEAICE_CAP_ICELOAD`. This option effectively limits the sea ice 1783 load (variable :varlink:`sIceLoad`) to a mass of 1/5 of the the top grid cell 1784 depth. If desired, this limit can be changed in routine 1785 :filelink:`seaice_growth.F <pkg/seaice/seaice_growth.F>` where variable 1786 :varlink:`heffTooHeavy` is assigned. dc26f158aa Mart*1787 61f2157921 Oliv*1788 .. _ssub_phys_pkg_seaice_subroutines: adc83e5d7b Mart*1789 1790 Key subroutines 258fe29c91 Jeff*1791 =============== adc83e5d7b Mart*1792 258fe29c91 Jeff*1793 Top-level routine: :filelink:`pkg/seaice/seaice_model.F` adc83e5d7b Mart*1794 1795 :: 1796 1797 1798 C !CALLING SEQUENCE: 1799 c ... 1800 c seaice_model (TOP LEVEL ROUTINE) 1801 c | 1802 c |-- #ifdef SEAICE_CGRID 1803 c | SEAICE_DYNSOLVER 1804 c | | 1805 c | |-- < compute proxy for geostrophic velocity > 1806 c | | 1807 c | |-- < set up mass per unit area and Coriolis terms > 1808 c | | 1809 c | |-- < dynamic masking of areas with no ice > 1810 c | | 1811 c | | 1812 c | #ELSE 1813 c | DYNSOLVER 1814 c | #ENDIF 1815 c | 2c231b0ebd Mart*1816 c |-- if ( useOBCS ) adc83e5d7b Mart*1817 c | OBCS_APPLY_UVICE 1818 c | 1819 c |-- if ( SEAICEadvHeff .OR. SEAICEadvArea .OR. SEAICEadvSnow .OR. SEAICEadvSalt ) 1820 c | SEAICE_ADVDIFF 1821 c | 1822 c | SEAICE_REG_RIDGE 1823 c | 2c231b0ebd Mart*1824 c |-- if ( usePW79thermodynamics ) adc83e5d7b Mart*1825 c | SEAICE_GROWTH 1826 c | 2c231b0ebd Mart*1827 c |-- if ( useOBCS ) adc83e5d7b Mart*1828 c | if ( SEAICEadvHeff ) OBCS_APPLY_HEFF 1829 c | if ( SEAICEadvArea ) OBCS_APPLY_AREA 1830 c | if ( SEAICEadvSALT ) OBCS_APPLY_HSALT 1831 c | if ( SEAICEadvSNOW ) OBCS_APPLY_HSNOW 1832 c | 1833 c |-- < do various exchanges > 1834 c | 1835 c |-- < do additional diagnostics > 1836 c | 1837 c o 1838 61f2157921 Oliv*1839 .. _ssub_phys_pkg_seaice_diagnostics: adc83e5d7b Mart*1840 1841 SEAICE diagnostics 258fe29c91 Jeff*1842 ================== adc83e5d7b Mart*1843 c512e371cc drin*1844 Diagnostics output is available via the diagnostics package (see 1845 :numref:`sub_outp_pkg_diagnostics`). Available output fields are summarized in 1846 the following table: d25560575e Oliv*1847 1848 .. code-block:: text 1849 1850 ---------+----------+----------------+----------------- 1851 <-Name->|<- grid ->|<-- Units -->|<- Tile (max=80c) 1852 ---------+----------+----------------+----------------- 1853 sIceLoad|SM U1|kg/m^2 |sea-ice loading (in Mass of ice+snow / area unit) 1854 --- 1855 SEA ICE STATE: 1856 --- 1857 SIarea |SM M1|m^2/m^2 |SEAICE fractional ice-covered area [0 to 1] 1858 SIheff |SM M1|m |SEAICE effective ice thickness 1859 SIhsnow |SM M1|m |SEAICE effective snow thickness 1860 SIhsalt |SM M1|g/m^2 |SEAICE effective salinity 1861 SIuice |UU M1|m/s |SEAICE zonal ice velocity, >0 from West to East 1862 SIvice |VV M1|m/s |SEAICE merid. ice velocity, >0 from South to North 1863 --- 1864 ATMOSPHERIC STATE AS SEEN BY SEA ICE: 1865 --- 1866 SItices |SM C M1|K |Surface Temperature over Sea-Ice (area weighted) 1867 SIuwind |UM U1|m/s |SEAICE zonal 10-m wind speed, >0 increases uVel 1868 SIvwind |VM U1|m/s |SEAICE meridional 10-m wind speed, >0 increases uVel 1869 SIsnPrcp|SM U1|kg/m^2/s |Snow precip. (+=dw) over Sea-Ice (area weighted) 1870 --- 1871 FLUXES ACROSS ICE-OCEAN INTERFACE (ATMOS to OCEAN FOR ICE-FREE REGIONS): 1872 --- 1873 SIfu |UU U1|N/m^2 |SEAICE zonal surface wind stress, >0 increases uVel 1874 SIfv |VV U1|N/m^2 |SEAICE merid. surface wind stress, >0 increases vVel 1875 SIqnet |SM U1|W/m^2 |Ocean surface heatflux, turb+rad, >0 decreases theta 1876 SIqsw |SM U1|W/m^2 |Ocean surface shortwave radiat., >0 decreases theta 1877 SIempmr |SM U1|kg/m^2/s |Ocean surface freshwater flux, > 0 increases salt 1878 SIqneto |SM U1|W/m^2 |Open Ocean Part of SIqnet, turb+rad, >0 decr theta 1879 SIqneti |SM U1|W/m^2 |Ice Covered Part of SIqnet, turb+rad, >0 decr theta 1880 --- 1881 FLUXES ACROSS ATMOSPHERE-ICE INTERFACE (ATMOS to OCEAN FOR ICE-FREE REGIONS): 1882 --- 1883 SIatmQnt|SM U1|W/m^2 |Net atmospheric heat flux, >0 decreases theta 1884 SIatmFW |SM U1|kg/m^2/s |Net freshwater flux from atmosphere & land (+=down) 1885 SIfwSubl|SM U1|kg/m^2/s |Freshwater flux of sublimated ice, >0 decreases ice 1886 --- 1887 THERMODYNAMIC DIAGNOSTICS: 1888 --- 1889 SIareaPR|SM M1|m^2/m^2 |SIarea preceeding ridging process 1890 SIareaPT|SM M1|m^2/m^2 |SIarea preceeding thermodynamic growth/melt 1891 SIheffPT|SM M1|m |SIheff preceeeding thermodynamic growth/melt 1892 SIhsnoPT|SM M1|m |SIhsnow preceeeding thermodynamic growth/melt 1893 SIaQbOCN|SM M1|m/s |Potential HEFF rate of change by ocean ice flux 1894 SIaQbATC|SM M1|m/s |Potential HEFF rate of change by atm flux over ice 1895 SIaQbATO|SM M1|m/s |Potential HEFF rate of change by open ocn atm flux 1896 SIdHbOCN|SM M1|m/s |HEFF rate of change by ocean ice flux 1897 SIdSbATC|SM M1|m/s |HSNOW rate of change by atm flux over sea ice 1898 SIdSbOCN|SM M1|m/s |HSNOW rate of change by ocean ice flux 1899 SIdHbATC|SM M1|m/s |HEFF rate of change by atm flux over sea ice 1900 SIdHbATO|SM M1|m/s |HEFF rate of change by open ocn atm flux 1901 SIdHbFLO|SM M1|m/s |HEFF rate of change by flooding snow 1902 SIdAbATO|SM M1|m^2/m^2/s |Potential AREA rate of change by open ocn atm flux 1903 SIdAbATC|SM M1|m^2/m^2/s |Potential AREA rate of change by atm flux over ice 1904 SIdAbOCN|SM M1|m^2/m^2/s |Potential AREA rate of change by ocean ice flux 1905 SIdA |SM M1|m^2/m^2/s |AREA rate of change (net) 1906 --- 1907 DYNAMIC/RHEOLOGY DIAGNOSTICS: 1908 --- b8665dacca Mart*1909 SIpress |SM M1|N/m |SEAICE strength (with upper and lower limit) 1910 SIzeta |SM M1|kg/s |SEAICE nonlinear bulk viscosity 1911 SIeta |SM M1|kg/s |SEAICE nonlinear shear viscosity 1912 SIsig1 |SM M1|no units |SEAICE normalized principle stress, component one 1913 SIsig2 |SM M1|no units |SEAICE normalized principle stress, component two 1914 SIshear |SM M1|1/s |SEAICE shear deformation rate 1915 SIdelta |SM M1|1/s |SEAICE Delta deformation rate 1916 SItensil|SM M1|N/m |SEAICE maximal tensile strength d25560575e Oliv*1917 --- 1918 ADVECTIVE/DIFFUSIVE FLUXES OF SEA ICE variables: 1919 --- 1920 ADVxHEFF|UU M1|m.m^2/s |Zonal Advective Flux of eff ice thickn 1921 ADVyHEFF|VV M1|m.m^2/s |Meridional Advective Flux of eff ice thickn 1922 SIuheff |UU M1|m^2/s |Zonal Transport of eff ice thickn (centered) 1923 SIvheff |VV M1|m^2/s |Meridional Transport of eff ice thickn (centered) 1924 DFxEHEFF|UU M1|m^2/s |Zonal Diffusive Flux of eff ice thickn 1925 DFyEHEFF|VV M1|m^2/s |Meridional Diffusive Flux of eff ice thickn 1926 ADVxAREA|UU M1|m^2/m^2.m^2/s |Zonal Advective Flux of fract area 1927 ADVyAREA|VV M1|m^2/m^2.m^2/s |Meridional Advective Flux of fract area 1928 DFxEAREA|UU M1|m^2/m^2.m^2/s |Zonal Diffusive Flux of fract area 1929 DFyEAREA|VV M1|m^2/m^2.m^2/s |Meridional Diffusive Flux of fract area 1930 ADVxSNOW|UU M1|m.m^2/s |Zonal Advective Flux of eff snow thickn 1931 ADVySNOW|VV M1|m.m^2/s |Meridional Advective Flux of eff snow thickn 1932 DFxESNOW|UU M1|m.m^2/s |Zonal Diffusive Flux of eff snow thickn 1933 DFyESNOW|VV M1|m.m^2/s |Meridional Diffusive Flux of eff snow thickn ba0b047096 Mart*1934 ADVxSSLT|UU M1|(g/kg).m^2/s |Zonal Advective Flux of seaice salinity 1935 ADVySSLT|VV M1|(g/kg).m^2/s |Meridional Advective Flux of seaice salinity 1936 DFxESSLT|UU M1|(g/kg).m^2/s |Zonal Diffusive Flux of seaice salinity 1937 DFyESSLT|VV M1|(g/kg).m^2/s |Meridional Diffusive Flux of seaice salinity d25560575e Oliv*1938 adc83e5d7b Mart*1939 1940 Experiments and tutorials that use seaice 258fe29c91 Jeff*1941 ========================================= 1942 1943 - :filelink:`verification/lab_sea`: Labrador Sea experiment 1944 - :filelink:`verification/seaice_obcs`, based on :filelink:`lab_sea <verification/lab_sea>` c512e371cc drin*1945 - :filelink:`verification/offline_exf_seaice`, idealized topography in a zonally re-entrant channel, tests solvers and rheologies 258fe29c91 Jeff*1946 - :filelink:`verification/seaice_itd`, based on :filelink:`offline_exf_seaice <verification/offline_exf_seaice>`, tests ice thickness distribution 1947 - :filelink:`verification/global_ocean.cs32x15`, global cubed-sphere-experiment with combinations of :filelink:`pkg/seaice` and :filelink:`pkg/thsice` 1948 - :filelink:`verification/1D_ocean_ice_column`, just thermodynamics
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