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b938a3c63b antn* .. _sub_phys_pkg_shelfice:
                 
274298f12b Jeff* SHELFICE Package
                 ----------------
                 
                 Authors: Martin Losch, Jean-Michel Campin
                 
0a8794f5ee Mart* Introduction
274298f12b Jeff* ~~~~~~~~~~~~
                 
                 :filelink:`pkg/shelfice` provides a thermodynamic model for basal melting
                 underneath floating ice shelves.
                 
                 CPP options enable or disable different aspects of the package
0a8794f5ee Mart* (:numref:`shelfice_config`). Run-time options, flags, filenames and
274298f12b Jeff* field-related dates/times are described in :numref:`shelfice_runtime`. A description of key subroutines is given
0a8794f5ee Mart* in :numref:`shelfice_subroutines`. Available diagnostics output is listed in
274298f12b Jeff* :numref:`shelfice_diagnostics`.
                 
                 
                 .. _shelfice_config:
                 
                 SHELFICE configuration
                 ~~~~~~~~~~~~~~~~~~~~~~
                  
                 As with all MITgcm packages, :filelink:`pkg/shelfice` can be turned on or off at compile
                 time:
                 
                 -  using the ``packages.conf`` file by adding ``shelfice`` to it,
                 
                 -  or using :filelink:`genmake2 <tools/genmake2>` adding ``-enable=shelfice`` or ``disable=shelfice`` switches
                 
                 :filelink:`pkg/shelfice` does not require any additional packages, but it will only
                 work with conventional vertical :math:`z`-coordinates (pressure
                 coordinates are not implemented). If you use it together with
                 vertical mixing schemes, be aware that non-local parameterizations
                 are turned off, e.g., such as :filelink:`pkg/kpp`.
                 
                 Parts of the :filelink:`pkg/shelfice` code can be enabled or disabled at compile time
                 via CPP preprocessor flags. These options are set in :filelink:`SHELFICE_OPTIONS.h <pkg/shelfice/SHELFICE_OPTIONS.h>`:
                 
7b8b86ab99 Timo* .. tabularcolumns:: |\Y{.32}|\Y{.1}|\Y{.570}|
                 
                 .. table:: Compile-time parameters
                    :name: tab_phys_pkg_shelfice_compileparms
                 
                    +-----------------------------------------------+---------+----------------------------------------------------------------------------------------------------------------------+
                    | CPP Flag Name                                 | Default | Description                                                                                                          |
                    +===============================================+=========+======================================================================================================================+
                    | :varlink:`ALLOW_SHELFICE_DEBUG`               | #undef  | include code for enhanced diagnostics and debug output                                                               |
                    +-----------------------------------------------+---------+----------------------------------------------------------------------------------------------------------------------+
                    | :varlink:`ALLOW_ISOMIP_TD`                    | #define | include code for for simplified ISOMIP thermodynamics                                                                |
                    +-----------------------------------------------+---------+----------------------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHI_ALLOW_GAMMAFRICT`               | #define | allow friction velocity-dependent transfer coefficient following Holland and Jenkins (1999) :cite:`holland:99`       |
                    +-----------------------------------------------+---------+----------------------------------------------------------------------------------------------------------------------+
274298f12b Jeff* 
                 .. _shelfice_runtime:
                 
0a8794f5ee Mart* SHELFICE run-time parameters
274298f12b Jeff* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                 
7b8b86ab99 Timo* :filelink:`pkg/shelfice` is switched on/off at run time by setting :varlink:`useSHELFICE` to ``.TRUE.`` in file ``data.pkg``.
274298f12b Jeff* Run-time parameters are set in file ``data.shelfice`` (read in :filelink:`pkg/shelfice/shelfice_readparms.F`),as listed below.
                 
                 The data file specifying under-ice topography of ice shelves (:varlink:`SHELFICEtopoFile`) is in meters; upwards is positive,
                 and as for the bathymetry files, negative values are required for topography below the sea-level.
                 The data file for the pressure load anomaly at the bottom of the ice shelves :varlink:`SHELFICEloadAnomalyFile` is in pressure
                 units (Pa). This field is absolutely required to avoid large
0a8794f5ee Mart* excursions of the free surface during initial adjustment processes,
274298f12b Jeff* obtained by integrating an approximate density from the surface at
                 :math:`z=0` down to the bottom of the last fully dry cell within the
                 ice shelf, see :eq:`surfacepressure`. Note however the file :varlink:`SHELFICEloadAnomalyFile` must
                 not be :math:`p_{top}`, but
67dae37801 Eli * :math:`p_{top}-g\sum_{k'=1}^{n-1}\rho_c \Delta{z}_{k'}`, with
                 :math:`\rho_c =` :varlink:`rhoConst`, so that in the absence of a :math:`\rho^{*}`
                 that is different from :math:`\rho_c`, the anomaly is zero.
274298f12b Jeff* 
7b8b86ab99 Timo* .. tabularcolumns:: |\Y{.275}|\Y{.28}|\Y{.455}|
274298f12b Jeff* 
7b8b86ab99 Timo* .. table:: Run-time parameters and default values; all parameters are in namelist group ``SHELFICE_PARM01``
0a8794f5ee Mart*    :name: tab_phys_pkg_shelfice_runtimeparms
                    :class: longtable
                 
7b8b86ab99 Timo*    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | Parameter                              | Default                                    | Description                                                                                             |
                    +========================================+============================================+=========================================================================================================+
                    | :varlink:`useISOMIPTD`                 | FALSE                                      | use simplified ISOMIP thermodynamics on/off flag                                                        |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEconserve`            | FALSE                                      | use conservative form of temperature boundary conditions on/off flag                                    |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEboundaryLayer`       | FALSE                                      | use simple boundary layer mixing parameterization on/off flag                                           |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHI_withBL_realFWflux`       | FALSE                                      | with :varlink:`SHELFICEboundaryLayer`, allow to use real-FW flux                                        |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHI_withBL_uStarTopDz`       | FALSE                                      | with :varlink:`SHELFICEboundaryLayer`, compute uStar from uVel,vVel averaged over top Dz thickness      |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEloadAnomalyFile`     | :kbd:`' '`                                 | initial geopotential anomaly                                                                            |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEtopoFile`            | :kbd:`' '`                                 | filename for under-ice topography of ice shelves                                                        |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEmassFile`            | :kbd:`' '`                                 | filename for mass of ice shelves                                                                        |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEMassDynTendFile`     | :kbd:`' '`                                 | filename for mass tendency of ice shelves                                                               |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICETransCoeffTFile`     | :kbd:`' '`                                 | filename for spatially varying transfer coefficients                                                    |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICElatentHeat`          | 334.0E+03                                  | latent heat of fusion (J/kg)                                                                            |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEHeatCapacity_Cp`     | 2000.0E+00                                 | specific heat capacity of ice (J/kg/K)                                                                  |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`rhoShelfIce`                 | 917.0E+00                                  | (constant) mean density of ice shelf (kg/m\ :sup:`3`)                                                   |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
00f81e6785 Ou W*    | :varlink:`SHELFICEsalinity`            | 0.0E+00                                    | (constant) salinity of ice shelf                                                                        |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
7b8b86ab99 Timo*    | :varlink:`SHELFICEheatTransCoeff`      | 1.0E-04                                    | transfer coefficient (exchange velocity) for temperature (m/s)                                          |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEsaltTransCoeff`      | :varlink:`SHELFICEsaltToHeatRatio` *       | transfer coefficient (exchange velocity) for salinity (m/s)                                             |
                    |                                        | :varlink:`SHELFICEheatTransCoeff`          |                                                                                                         |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEsaltToHeatRatio`     | 5.05E-03                                   | ratio of salinity to temperature transfer coefficients (non-dim.)                                       |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEkappa`               | 1.54E-06                                   | temperature diffusion coefficient of the ice shelf (m\ :sup:`2`\ /s)                                    |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEthetaSurface`        | -20.0E+00                                  | (constant) surface temperature above the ice shelf (:sup:`o`\ C)                                        |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`no_slip_shelfice`            | :varlink:`no_slip_bottom`                  | slip along bottom of ice shelf on/off flag                                                              |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEDragLinear`          | :varlink:`bottomDragLinear`                | linear drag coefficient at bottom ice shelf (m/s)                                                       |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEDragQuadratic`       | :varlink:`bottomDragQuadratic`             | quadratic drag coefficient at bottom ice shelf (non-dim.)                                               |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEselectDragQuadr`     | -1                                         | select form of quadratic drag coefficient (non-dim.)                                                    |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEMassStepping`        | FALSE                                      | recalculate ice shelf mass at every time step                                                           |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEDynMassOnly`         | FALSE                                      | if :varlink:`SHELFICEmassStepping` = TRUE, exclude freshwater flux contribution                         |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEadvDiffHeatFlux`     | FALSE                                      | use advective-diffusive heat flux into ice shelf instead of default diffusive heat flux                 |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEuseGammaFrict`       | FALSE                                      | use velocity dependent exchange coefficients (Holland and Jenkins 1999 :cite:`holland:99`)              |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICE_oldCalcUStar`       | FALSE                                      | use old uStar averaging expression                                                                      |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICEwriteState`          | FALSE                                      | write ice shelf state to file on/off flag                                                               |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICE_dumpFreq`           | :varlink:`dumpFreq`                        | dump frequency (s)                                                                                      |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
                    | :varlink:`SHELFICE_dump_mnc`           | :varlink:`snapshot_mnc`                    | write snapshot using MNC  on/off flag                                                                   |
                    +----------------------------------------+--------------------------------------------+---------------------------------------------------------------------------------------------------------+
0a8794f5ee Mart* 
                 SHELFICE description
274298f12b Jeff* ~~~~~~~~~~~~~~~~~~~~
                 
                 In the light of isomorphic equations for pressure and height
                 coordinates, the ice shelf topography on top of the water column has a
bd7056e9aa Jeff* similar role as (and in the language of Marshall et al. (2004) :cite:`marshall:04`,
274298f12b Jeff* is isomorphic to) the orography and the pressure boundary conditions at
                 the bottom of the fluid for atmospheric and oceanic models in pressure
0bad585a21 Navi* coordinates. The total pressure :math:`p_{\rm tot}` in the ocean can be
274298f12b Jeff* divided into the pressure at the top of the water column
0bad585a21 Navi* :math:`p_{\rm top}`, the hydrostatic pressure and the non-hydrostatic
                 pressure contribution :math:`p_{\rm nh}`:
274298f12b Jeff* 
                 .. math::
0bad585a21 Navi*    p_{\rm tot} = p_{\rm top} + \int_z^{\eta-h} g\,\rho\,dz + p_{\rm nh}
274298f12b Jeff*    :label: pressureocean
                 
                 
                 with the gravitational acceleration :math:`g`, the density
                 :math:`\rho`, the vertical coordinate :math:`z` (positive upwards), and
                 the dynamic sea-surface height :math:`\eta`. For the open ocean,
0bad585a21 Navi* :math:`p_{\rm top}=p_{a}` (atmospheric pressure) and :math:`h=0`. Underneath
274298f12b Jeff* an ice-shelf that is assumed to be floating in isostatic equilibrium,
0bad585a21 Navi* :math:`p_{\rm top}` at the top of the water column is the atmospheric
274298f12b Jeff* pressure :math:`p_{a}` plus the weight of the ice-shelf. It is this
                 weight of the ice-shelf that has to be provided as a boundary condition
                 at the top of the water column (in run-time parameter :varlink:`SHELFICEloadAnomalyFile`). The weight is
                 conveniently computed by integrating a density profile :math:`\rho^*`,
                 that is constant in time and corresponds to the sea-water replaced by
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 181.


 ice, from :math:`z=0` to a “reference” ice-shelf draft at :math:`z=-h` (Beckmann et al. (1999)
                 :cite:`beckmann:99`), so that
                 
                 .. math::
0bad585a21 Navi*    p_{\rm top} = p_{a} + \int_{-h}^{0}g\,\rho^{*}\,dz
274298f12b Jeff*    :label: ptop
                 
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 188.


 Underneath the ice shelf, the “sea-surface height” :math:`\eta` is the
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 189.


 deviation from the “reference” ice-shelf draft :math:`h`. During a model
                 integration, :math:`\eta` adjusts so that the isostatic equilibrium is
                 maintained for sufficiently slow and large scale motion.
                 
0bad585a21 Navi* In MITgcm, the total pressure anomaly :math:`p'_{\rm tot}` which is used
67dae37801 Eli * for pressure gradient computations is defined by subtracting a purely
                 depth dependent contribution :math:`-g\rho_c z` using constant
0bad585a21 Navi* reference density :math:`\rho_c` from :math:`p_{\rm tot}`.
274298f12b Jeff* :eq:`pressureocean` becomes
                 
                 .. math::
0bad585a21 Navi*      p_{\rm tot} = p_{\rm top} - g \rho_c (z+h)  + g \rho_c \eta + \, \int_z^{\eta-h}{ g (\rho-\rho_c) \, dz} + \, p_{\rm nh}
274298f12b Jeff*      :label: pressure
                 
                 and after rearranging
                 
                 .. math::
0bad585a21 Navi*    p'_{\rm tot} = p'_{\rm top} + g \rho_c \eta + \, \int_z^{\eta-h}{g (\rho-\rho_c) \, dz} + \, p_{\rm nh}
274298f12b Jeff* 
0bad585a21 Navi* with :math:`p'_{\rm tot} = p_{\rm tot} + g\,\rho_c\,z` and
                 :math:`p'_{\rm top} = p_{\rm top} - g\,\rho_c\,h`.
                 The non-hydrostatic pressure contribution :math:`p_{\rm nh}`
274298f12b Jeff* is neglected in the following.
                 
0bad585a21 Navi* In practice, the ice shelf contribution to :math:`p_{\rm top}` is computed
274298f12b Jeff* by integrating :eq:`ptop` from :math:`z=0` to the bottom of the
                 last fully dry cell within the ice shelf:
                 
                 .. math::
0bad585a21 Navi*    p_{\rm top} = g\,\sum_{k'=1}^{n-1}\rho_{k'}^{*}\Delta{z_{k'}} + p_{a}
274298f12b Jeff*    :label: surfacepressure
                 
                 where :math:`n` is the vertical index of the first (at least partially)
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 222.


 “wet” cell and :math:`\Delta{z_{k'}}` is the thickness of the
                 :math:`k'`-th layer (counting downwards). The pressure anomaly for
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 224.


 evaluating the pressure gradient is computed in the center of the “wet”
                 cell :math:`k` as
                 
                 .. math::
0bad585a21 Navi*    p'_{k} = p'_{\rm top} + g\rho_{n}\eta +
67dae37801 Eli *    g\,\sum_{k'=n}^{k}\left((\rho_{k'}-\rho_c)\Delta{z_{k'}}
274298f12b Jeff*      \frac{1+H(k'-k)}{2}\right)
                    :label: discretizedpressure
                 
                 where :math:`H(k'-k)=1` for :math:`k'<k` and :math:`0` otherwise.
                 
bd7056e9aa Jeff*  .. figure:: figs/gridschematic.*
                     :width: 80%
                     :align: center
                     :alt: schematic of vertical section of grid
                     :name: shelfice_grid
                 
0a8794f5ee Mart*     Schematic of a vertical section of the grid at the base of an ice shelf. Grid lines are thin;
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 242.


     the thick line is the model’s representation of the ice shelf-water interface. Plus signs mark the position
                     of pressure points for pressure gradient computations. The letters A, B, and C mark specific grid cells for
                     reference. :math:`h_k` is the fractional cell thickness so that :math:`h_k \Delta z_k` is the actual cell thickness.
bd7056e9aa Jeff* 
                 
274298f12b Jeff* Setting :varlink:`SHELFICEboundaryLayer` ``=.TRUE.`` introduces a simple boundary layer that reduces the potential
                 noise problem at the cost of increased vertical mixing. For this purpose
                 the water temperature at the :math:`k`-th layer abutting ice shelf
                 topography for use in the heat flux parameterizations is computed as a
                 mean temperature :math:`\overline{\theta}_{k}` over a boundary layer of
                 the same thickness as the layer thickness :math:`\Delta{z}_{k}`:
                 
                 .. math::
                    \overline{\theta}_{k} = \theta_{k} h_{k} + \theta_{k+1} (1-h_{k})
                    :label: thetabl
                 
                 where :math:`h_{k}\in[0,1]` is the fractional layer thickness of the
bd7056e9aa Jeff* :math:`k`-th layer (see :numref:`shelfice_grid`). The original contributions due to ice shelf-ocean
274298f12b Jeff* interaction :math:`g_{\theta}` to the total tendency terms
                 :math:`G_{\theta}` in the time-stepping equation
                 :math:`\theta^{n+1} = f(\theta^{n},\Delta{t},G_{\theta}^{n})` are
                 
                 .. math::
67dae37801 Eli *    g_{\theta,k}   = \frac{Q}{\rho_c c_{p} h_{k} \Delta{z}_{k}}
274298f12b Jeff*    \text{ and } g_{\theta,k+1} = 0
                    :label: orgtendency
                 
                 for layers :math:`k` and :math:`k+1` (:math:`c_{p}` is the heat
                 capacity). Averaging these terms over a layer thickness
                 :math:`\Delta{z_{k}}` (e.g., extending from the ice shelf base down to
                 the dashed line in cell C) and applying the averaged tendency to cell A
                 (in layer :math:`k`) and to the appropriate fraction of cells C (in
                 layer :math:`k+1`) yields
                 
                 .. math::
67dae37801 Eli *    g_{\theta,k}^*   = \frac{Q}{\rho_c c_{p} \Delta{z}_{k}}
274298f12b Jeff*    :label: tendencyk
                 
                 .. math::
                    g_{\theta,k+1}^*
67dae37801 Eli *    = \frac{Q}{\rho_c c_{p} \Delta{z}_{k}}
274298f12b Jeff*    \frac{ \Delta{z}_{k} ( 1- h_{k} )}{\Delta{z}_{k+1}}
                    :label: tendencykp1
                 
                 :eq:`tendencykp1` describes averaging over the part of the grid
                 cell :math:`k+1` that is part of the boundary layer with tendency
                 :math:`g_{\theta,k}^*` and the part with no tendency. Salinity is
                 treated in the same way. The momentum equations are not modified.
                 
0a8794f5ee Mart* Three-equations thermodynamics
274298f12b Jeff* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                 
67dae37801 Eli * Freezing and melting form a boundary layer between the ice shelf and the
                 ocean that is represented in the model by an infinitesimal layer used to
                 calculate the exchanges between the ocean and the ice. Melting and
                 freezing at the boundary between saline water and ice imply a freshwater
                 mass flux :math:`q` (:math:`<0` for melting); the relevant heat fluxes
                 into and out of the boundary layer therefore include a diffusive flux
                 through the ice, the latent heat flux due to melting and freezing, and
                 the advective heat that is carried by the freshwater mass flux. There is
                 a salt flux carried by the mass flux if the ice has a non-zero salinity
274298f12b Jeff* :math:`S_I`. Further, the position of the interface between ice and
                 ocean changes because of :math:`q`, so that, say, in the case of melting
67dae37801 Eli * the volume of sea water increases. As a consequence of these fluxes,
                 salinity and temperature are affected.
                 
                 The turbulent tracer exchanges between the infinitesimal boundary layer and
                 the ocean are expressed as diffusive fluxes. Following Jenkins et
                 al. (2001) :cite:`jenkins:01`, the boundary conditions for a tracer take
                 into account that this boundary may not move with the ice ocean interface
                 (for example, in a linear free surface model). The implied upward freshwater
                 flux :math:`q` is therefore included in the boundary conditions for the
                 temperature and salinity equation as an advective flux.
                 
                 The boundary conditions for tracer :math:`X=S,T` (tracer :math:`X` stands
                 for either in-situ temperature :math:`T` or salinity :math:`S`, located at
                 the first interior ocean grid point) in the ocean are expressed as the sum
                 of advective and diffusive fluxes
274298f12b Jeff* 
                 .. math::
67dae37801 Eli *    F_X = (\rho_c \, \gamma_{X} -q ) ( X_{b} - X )
274298f12b Jeff*    :label: jenkinsbc
                 
67dae37801 Eli * where the diffusive flux has been parameterized as a turbulent exchange
                 :math:`\rho_c \, \gamma_{X}( X_{b} - X )` following Holland and
                 Jenkins (1999) :cite:`holland:99` or Jenkins et al. (2001)
                 :cite:`jenkins:01`. :math:`X_b` indicates the tracer in the boundary layer,
                 :math:`\rho_c` the density of seawater (parameter :varlink:`rhoConst`), and
                 :math:`\gamma_X` is the turbulent exchange (or transfer) coefficient
                 (parameters :varlink:`SHELFICEheatTransCoeff` and
                 :varlink:`SHELFICEsaltTransCoeff`), in units of an exchange
                 velocity. In-situ temperature, computed locally from tracer potential
                 temperature, is required here to accurately compute the in-situ freezing
                 point of seawater in order to determine ice melt.
                 
                 The tracer budget for the infinitesimal boundary layer takes the general
                 form:
                 
                 .. math::
                    {\rho_I} K_{I,X} \frac{\partial{X_I}}{\partial{z}}\biggl|_{b}
                    = \rho_c \, \gamma_{X} ( X_{b} - X ) -  q ( X_{b} - X_{I} )
                    :label: jenkinsgenbudget
                 
                 where the LHS represents diffusive flux from the ice evaluated at the
                 interface between the infinitesimal boundary layer and the ice, and the RHS
                 represents the turbulent and advective exchanges between the infinitesimal
                 layer and the ocean and the advective exchange between the boundary layer
                 and the ice (:math:`qX_{I}`, this flux will be zero if the ice contains no
                 tracer: :math:`X_I=0`). The temperature in the boundary layer (:math:`T_b`)
                 is taken to be at the freezing point, which is a function of pressure and
                 salinity, :math:`\rho_I` is ice density (:varlink:`rhoShelfIce`), and
                 :math:`K_{I,X}` the appropriate ice diffusivity.
                 
                 For any material tracer such as salinity, the LHS in :eq:`jenkinsgenbudget`
                 vanishes (no material diffusion into the ice), while for temperature, the
                 term :math:`q\,( T_{b}-T_{I} )` vanishes, because both the boundary layer
                 and the ice are at the freezing point. Instead, the latent heat of freezing
                 is included as an additional term to take into account the conversion of
                 ice to water:
274298f12b Jeff* 
                 .. math::
67dae37801 Eli *    {\rho_I} \, c_{p,I} \, \kappa_{I,T}
                    \frac{\partial{T_I}}{\partial{z}}\biggl|_{b}
                    = c_{p} \, \rho_c \, \gamma_{T} ( T_{b} - T )+ L q.
                    :label: jenkinsheatbudget
                 
                 where :math:`\kappa_{I,T}` is the thermal ice diffusivity
                 (:varlink:`SHELFICEkappa`), :math:`L` is the latent heat of fusion
                 (:varlink:`SHELFICElatentHeat`), :math:`c_{p}` is the specific heat
                 capacity of water (:varlink:`HeatCapacity_Cp`), and :math:`c_{p,I}` the
                 heat capacity of the ice shelf (:varlink:`SHELFICEHeatCapacity_Cp`).  A
                 reasonable choice for :math:`\gamma_T` (:varlink:`SHELFICEheatTransCoeff`),
                 the turbulent exchange coefficient of temperature, is discussed in Holland
                 and Jenkins (1999) :cite:`holland:99` (see
                 :numref:`shelfice_exchange_coeff`).  The temperature at the interface
                 :math:`T_{b}` is assumed to be the in-situ freezing point temperature of
                 sea-water :math:`T_{f}`, which is computed from a linear equation of state:
274298f12b Jeff* 
                 .. math::
67dae37801 Eli *    T_{f} = (0.0901 - 0.0575\ S_{b})
                    - 7.61 \times 10^{-4} p_{b}.
0a8794f5ee Mart*    :label: hellmerfreeze
274298f12b Jeff* 
67dae37801 Eli * where :math:`T_f` is given in :sup:`o`\ C and :math:`p_{b}` is in dBar. In
                 :eq:`jenkinsheatbudget`, the diffusive heat flux at the ice-ocean interface
                 can be appproximated by assuming a linear temperature profile in the ice
                 and approximating the vertical derivative of temperature in the ice as the
                 difference between the ice surface and ice bottom temperatures divided by
                 the ice thickness, so that the left-hand-side of :eq:`jenkinsheatbudget`
                 becomes
274298f12b Jeff* 
                 .. math::
67dae37801 Eli *    {\rho_I} \, c_{p,I} \, \kappa_{I,T}
                    \frac{\partial{T_I}}{\partial{z}}\biggl|_{b}
                    \approx \rho_{I} \, c_{p,I} \, \kappa_{I,T} \frac{(T_{S} - T_{b})}{h}
                    :label: dTdzdiffus
                 
                 where :math:`T_{S}` the (surface) temperature of the ice shelf
                 (:varlink:`SHELFICEthetaSurface`) and :math:`h` is the ice-shelf
                 draft. Alternatively, assuming that the ice is "advected" vertically as
                 implied by the meltflux :math:`q`, the diffusive flux can be approximated
                 as :math:`\min(q,0)\,c_{p,I} (T_{S} - T_{b})` (runtime flag
                 :varlink:`SHELFICEadvDiffHeatFlux`; see Holland and Jenkins, 1999
                 :cite:`holland:99` for details).
                 
                 From the salt budget, the salt flux across the shelf ice-ocean interface is
                 equal to the salt flux due to melting and freezing:
                 
                 .. math::
                    \rho_c \, \gamma_{S} (S - S_{b}) = - q\,(S_{b}-S_{I})
274298f12b Jeff*    :label: hellmersaltbalance
                 
67dae37801 Eli * where :math:`\gamma_S =` :varlink:`SHELFICEsaltToHeatRatio` :math:`*
                 \gamma_T` is the turbulent salinity exchange coefficient.  Note, it is
                 assumed that :math:`\kappa_{I,S} =0`; moreover, the salinity of the ice
                 shelf is generally neglected (:math:`S_{I}=0`).
                 
c717173819 Matt* The budget equations for temperature :eq:`jenkinsheatbudget` (with
                 :eq:`dTdzdiffus`) and salinity
67dae37801 Eli * :eq:`hellmersaltbalance`, together with the freezing point temperature of
                 sea-water :eq:`hellmerfreeze`, form the so-called three-equation-model
                 (e.g., Hellmer and Olbers (1989) :cite:`hellmer:89`, Jenkins et al. (2001)
                 :cite:`jenkins:01`). These equations are solved to obtain :math:`S_b, T_b,
                 q`, which are then substituted into :eq:`jenkinsbc` to obtain the boundary
                 conditions for the temperature and salinity equations of the ocean
                 model. Note that with :math:`S_{I}=0` and :eq:`hellmersaltbalance`, the
                 boundary flux for salinity becomes :math:`F_S = q\,S`, which is the flux
                 that is necessary to account for the dilution of salinity in the case of
                 melting.
                 
                 The three-equation-model tends to yield smaller melt rates than the simpler
                 formulation of the ISOMIP protocol (:numref:`shelfice_isomip`) because the
                 freshwater flux (due to melting) decreases the salinity, which in turn
                 raises the freezing point temperature and thus leads to less melting at the
                 interface. For a simpler thermodynamics model where :math:`S_b` is not
                 computed explicitly, for example as in the ISOMIP protocol, :eq:`jenkinsbc`
                 cannot be applied directly. In this case :eq:`hellmersaltbalance` can be
                 used with :eq:`jenkinsbc` to obtain:
                 
                 .. math:: F_X = q\,(S-S_I)
274298f12b Jeff* 
                 This formulation can be used for all cases for which
67dae37801 Eli * :eq:`hellmersaltbalance` is valid. Further, in this formulation it is
                 obvious that melting (:math:`q<0`) leads to a reduction of salinity.
274298f12b Jeff* 
67dae37801 Eli * The default value of :varlink:`SHELFICEconserve` ``=.FALSE.`` removes the
                 contribution :math:`q\, ( X_{b}-X )` from :eq:`jenkinsbc`, making the
                 boundary conditions non-conservative.
274298f12b Jeff* 
0a8794f5ee Mart* Solving the three-equations system
                 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                 
67dae37801 Eli * There has been some confusion about the three-equations system, so we
                 document the solution in the code here: We use :eq:`hellmerfreeze`
                 :math:`T_{b} = a_{0} S_{b} + \epsilon_{4}` to eliminate :math:`T_{b}`
                 from :eq:`jenkinsheatbudget` with :eq:`dTdzdiffus` and find an
                 expression for the freshwater flux :math:`q`:
0a8794f5ee Mart* 
                 .. math::
                    \begin{aligned}
                    -Lq &= \epsilon_{1} (T - a_{0} S_{b} - \epsilon_{4})
                    + \epsilon_{3} (T_{S} - a_{0} S_{b} - \epsilon_{4}) \\
                    \Leftrightarrow Lq &=  a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}
                      + \epsilon_{q}
                    \end{aligned}
                    :label: solvedmeltrate
                 
                 to be substituted into :eq:`hellmersaltbalance`:
                 
                 .. math::
                    \begin{aligned}
                    \epsilon_{2}\,(S - S_{b}) &= - Lq\,(S_{b}-S_{I})
                    = - (a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}
                      + \epsilon_{q})\,(S_{b}-S_{I}) \\
                    \Leftrightarrow 0 &= a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}^{2}
                    + \{ \epsilon_{q}  - \epsilon_{2}
                      - a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{I} \}\,S_{b}
00f81e6785 Ou W*      + \epsilon_{2}\,S - \epsilon_{q}\,S_{I} \\
                    \Leftrightarrow 0 &= A\,S_{b}^{2} + B\,S_{b} + C \\
                    \Rightarrow S_{b} &= \frac{-B \pm \sqrt{ B^{2} - 4AC }}{2A}
                    \end{aligned}
                 
                 with the abbrevations
                 
                 .. math::
                    \begin{aligned}
                    \epsilon_{1} &= c_{p} \, \rho_c \, \gamma_{T}, \quad
                    \epsilon_{2} = \rho_c L \, \gamma_{S}, \quad
                    \epsilon_{3} = \frac{\rho_{I} \, c_{p,I} \, \kappa}{h}, \quad
                    \epsilon_{4} = b_{0}p + c_{0}, \\
                    \epsilon_{q} &= \epsilon_{1}\,(\epsilon_{4} - T)
                                   + \epsilon_{3}\,(\epsilon_{4} - T_{S}), \\
                    A &= a_{0}\,(\epsilon_{1} + \epsilon_{3}), \quad
                    B = \epsilon_{q} - \epsilon_{2} - a_{0}\,(\epsilon_{1} +
                    \epsilon_{3})\,S_{I}, \quad
                    C = \epsilon_{2}\,S -\epsilon_{q}\,S_{I}.
0a8794f5ee Mart*    \end{aligned}
                 
00f81e6785 Ou W* The smaller non-negative root of the quadratic equation in :math:`S_{b}` is
                 used. By default, the ice shelf salinity :math:`S_{I}` is zero and the
                 quadratic equation simplifies to
0a8794f5ee Mart* 
                 .. math::
                    \begin{aligned}
                    0 &= a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}^{2}
                    + (\epsilon_{q}  - \epsilon_{2}) \,S_{b} + \epsilon_{2}\,S \\
                      S_{b} &= \frac{\epsilon_{2} - \epsilon_{q}\mp
                      \sqrt{(\epsilon_{q}  - \epsilon_{2})^2
b8f9311cc6 Matt*      - 4\, a_{0}\,(\epsilon_{1} + \epsilon_{3})\,\epsilon_{2}\,S}}
0a8794f5ee Mart*      {2\,a_{0}\,(\epsilon_{1} + \epsilon_{3})}
                    \end{aligned}
                 
00f81e6785 Ou W* With :math:`S_b`, the boundary layer temperature :math:`T_b` and the melt rate
                 :math:`q` are known through :eq:`hellmerfreeze` and :eq:`solvedmeltrate`.
0a8794f5ee Mart* 
67dae37801 Eli * .. _shelfice_isomip:
                 
0a8794f5ee Mart* ISOMIP thermodynamics
274298f12b Jeff* ^^^^^^^^^^^^^^^^^^^^^
                 
                 A simpler formulation follows the ISOMIP protocol. The
                 freezing and melting in the boundary layer between ice shelf and ocean
                 is parameterized following Grosfeld et al. (1997) :cite:`grosfeld:97`. In this
67dae37801 Eli * formulation :eq:`jenkinsheatbudget` reduces to
274298f12b Jeff* 
                 .. math::
67dae37801 Eli *    c_{p} \, \rho_c \, \gamma_T (T - T_{b})  = -L q
274298f12b Jeff*    :label: isomipheatbalance
                 
                 and the fresh water flux :math:`q` is computed from
                 
                 .. math::
67dae37801 Eli *    q = - \frac{c_{p} \, \rho_c \, \gamma_T (T - T_{b})}{L}
274298f12b Jeff*    :label: isomipfwflx
                 
67dae37801 Eli * In order to use this formulation, set runtime parameter
                 :varlink:`useISOMIPTD` ``=.TRUE.`` in ``data.shelfice``.
                 
                 .. _shelfice_exchange_coeff:
274298f12b Jeff* 
0a8794f5ee Mart* Exchange coefficients
                 ^^^^^^^^^^^^^^^^^^^^^
                 
                 The default exchange coefficents :math:`\gamma_{T/S}` are constant and
7b8b86ab99 Timo* set by the run-time parameters :varlink:`SHELFICEheatTransCoeff` and
0a8794f5ee Mart* :varlink:`SHELFICEsaltTransCoeff` (see
                 :numref:`tab_phys_pkg_shelfice_runtimeparms`). If
                 :varlink:`SHELFICEuseGammaFrict` ``=.TRUE.``, exchange coefficients
                 are computed from drag laws and friction velocities estimated from
                 ocean speeds following Holland and Jenkins (1999)
67dae37801 Eli * :cite:`holland:99`. This computation can be modified using runtime
                 parameters and the user is referred to
0a8794f5ee Mart* :filelink:`pkg/shelfice/shelfice_readparms.F` for details.
                 
274298f12b Jeff* Remark
                 ^^^^^^
                 
                 The shelfice package and experiments demonstrating its strengths and
67dae37801 Eli * weaknesses are also described in Losch (2008)
                 :cite:`losch:08`. Unfortunately, the description of the
                 thermodynamics in the appendix of Losch (2008) is wrong.
274298f12b Jeff* 
                 .. _shelfice_subroutines:
                 
                 Key subroutines
                 ~~~~~~~~~~~~~~~
                 
                 The main routine is :filelink:`shelfice_thermodynamics.F <pkg/shelfice/shelfice_thermodynamics.F>`
                 but note that :filelink:`/pkg/shelfice` routines are also called when solving the momentum equations.
                 
                 ::
                 
                     C     !CALLING SEQUENCE:
                     C ...
                     C |-FORWARD_STEP           :: Step forward a time-step ( AT LAST !!! )
                     C ...
                     C | |-DO_OCEANIC_PHY       :: Control oceanic physics and parameterization
                     C ...
                     C | | |-SHELFICE_THERMODYNAMICS :: main routine for thermodynamics
                     C                                  with diagnostics
                     C ...
                     C | |-THERMODYNAMICS       :: theta, salt + tracer equations driver.
                     C ...
                     C | | |-EXTERNAL_FORCING_T :: Problem specific forcing for temperature.
                     C | | |-SHELFICE_FORCING_T :: apply heat fluxes from ice shelf model
                     C ...
                     C | | |-EXTERNAL_FORCING_S :: Problem specific forcing for salinity.
                     C | | |-SHELFICE_FORCING_S :: apply fresh water fluxes from ice shelf model
                     C ...
                     C | |-DYNAMICS             :: Momentum equations driver.
                     C ...
                     C | | |-MOM_FLUXFORM       :: Flux form mom eqn. package ( see
                     C ...
                     C | | | |-SHELFICE_U_DRAG  :: apply drag along ice shelf to u-equation
                     C                             with diagnostics
                     C ...
                     C | | |-MOM_VECINV         :: Vector invariant form mom eqn. package ( see
                     C ...
                     C | | | |-SHELFICE_V_DRAG  :: apply drag along ice shelf to v-equation
                     C                             with diagnostics
                     C ...
                     C  o
                 
                 
                 
                 .. _shelfice_diagnostics:
                 
                 SHELFICE diagnostics
                 ~~~~~~~~~~~~~~~~~~~~
                 
                 Diagnostics output is available via the diagnostics package (see
                 :numref:`outp_pack`). Available output fields are summarized as follows:
                 
                 
                 ::
                 
                     ---------+----+----+----------------+-----------------
                      <-Name->|Levs|grid|<--  Units   -->|<- Tile (max=80c)
                     ---------+----+----+----------------+-----------------
                      SHIfwFlx|  1 |SM  |kg/m^2/s        |Ice shelf fresh water flux (positive upward)
                      SHIhtFlx|  1 |SM  |W/m^2           |Ice shelf heat flux  (positive upward)
                      SHIUDrag| 30 |UU  |m/s^2           |U momentum tendency from ice shelf drag
                      SHIVDrag| 30 |VV  |m/s^2           |V momentum tendency from ice shelf drag
                      SHIForcT|  1 |SM  |W/m^2           |Ice shelf forcing for theta, >0 increases theta
                      SHIForcS|  1 |SM  |g/m^2/s         |Ice shelf forcing for salt, >0 increases salt
                 
                 Experiments and tutorials that use shelfice
                 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                 
bd7056e9aa Jeff* See the verification experiment :filelink:`isomip <verification/isomip>` for example usage of :filelink:`pkg/shelfice`.

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