8.6.3. SHELFICE Package

Authors: Martin Losch, Jean-Michel Campin Introduction

pkg/shelfice provides a thermodynamic model for basal melting underneath floating ice shelves.

CPP options enable or disable different aspects of the package (Section Run-time options, flags, filenames and field-related dates/times are described in Section A description of key subroutines is given in Section Available diagnostics output is listed in Section SHELFICE configuration

As with all MITgcm packages, pkg/shelfice can be turned on or off at compile time:

  • using the packages.conf file by adding shelfice to it,

  • or using genmake2 adding -enable=shelfice or disable=shelfice switches

pkg/shelfice does not require any additional packages, but it will only work with conventional vertical \(z\)-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 pkg/kpp.

Parts of the pkg/shelfice code can be enabled or disabled at compile time via CPP preprocessor flags. These options are set in SHELFICE_OPTIONS.h:

Table 8.22 Compile-time parameters

CPP Flag Name





include code for enhanced diagnostics and debug output



include code for for simplified ISOMIP thermodynamics



allow friction velocity-dependent transfer coefficient following Holland and Jenkins (1999) [HJ99] SHELFICE run-time parameters

pkg/shelfice is switched on/off at run time by setting useSHELFICE to .TRUE. in file data.pkg. Run-time parameters are set in file data.shelfice (read in pkg/shelfice/shelfice_readparms.F),as listed below.

The data file specifying under-ice topography of ice shelves (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 SHELFICEloadAnomalyFile is in pressure units (Pa). This field is absolutely required to avoid large excursions of the free surface during initial adjustment processes, obtained by integrating an approximate density from the surface at \(z=0\) down to the bottom of the last fully dry cell within the ice shelf, see (8.31). Note however the file SHELFICEloadAnomalyFile must not be \(p_{top}\), but \(p_{top}-g\sum_{k'=1}^{n-1}\rho_c \Delta{z}_{k'}\), with \(\rho_c =\) rhoConst, so that in the absence of a \(\rho^{*}\) that is different from \(\rho_c\), the anomaly is zero.

Table 8.23 Run-time parameters and default values; all parameters are in namelist group SHELFICE_PARM01






use simplified ISOMIP thermodynamics on/off flag



use conservative form of temperature boundary conditions on/off flag



use simple boundary layer mixing parameterization on/off flag



with SHELFICEboundaryLayer, allow to use real-FW flux



with SHELFICEboundaryLayer, compute uStar from uVel,vVel averaged over top Dz thickness


' '

initial geopotential anomaly


' '

filename for under-ice topography of ice shelves


' '

filename for mass of ice shelves


' '

filename for mass tendency of ice shelves


' '

filename for spatially varying transfer coefficients



latent heat of fusion (J/kg)



specific heat capacity of ice (J/kg/K)



(constant) mean density of ice shelf (kg/m3)



(constant) salinity of ice shelf



transfer coefficient (exchange velocity) for temperature (m/s)


SHELFICEsaltToHeatRatio * SHELFICEheatTransCoeff

transfer coefficient (exchange velocity) for salinity (m/s)



ratio of salinity to temperature transfer coefficients (non-dim.)



temperature diffusion coefficient of the ice shelf (m2/s)



(constant) surface temperature above the ice shelf (oC)



slip along bottom of ice shelf on/off flag



linear drag coefficient at bottom ice shelf (m/s)



quadratic drag coefficient at bottom ice shelf (non-dim.)



select form of quadratic drag coefficient (non-dim.)



recalculate ice shelf mass at every time step



if SHELFICEmassStepping = TRUE, exclude freshwater flux contribution



use advective-diffusive heat flux into ice shelf instead of default diffusive heat flux



use velocity dependent exchange coefficients (Holland and Jenkins 1999 [HJ99])



use old uStar averaging expression



write ice shelf state to file on/off flag



dump frequency (s)



write snapshot using MNC on/off flag SHELFICE description

In the light of isomorphic equations for pressure and height coordinates, the ice shelf topography on top of the water column has a similar role as (and in the language of Marshall et al. (2004) [MAC+04], is isomorphic to) the orography and the pressure boundary conditions at the bottom of the fluid for atmospheric and oceanic models in pressure coordinates. The total pressure \(p_{\rm tot}\) in the ocean can be divided into the pressure at the top of the water column \(p_{\rm top}\), the hydrostatic pressure and the non-hydrostatic pressure contribution \(p_{\rm nh}\):

(8.28)\[p_{\rm tot} = p_{\rm top} + \int_z^{\eta-h} g\,\rho\,dz + p_{\rm nh}\]

with the gravitational acceleration \(g\), the density \(\rho\), the vertical coordinate \(z\) (positive upwards), and the dynamic sea-surface height \(\eta\). For the open ocean, \(p_{\rm top}=p_{a}\) (atmospheric pressure) and \(h=0\). Underneath an ice-shelf that is assumed to be floating in isostatic equilibrium, \(p_{\rm top}\) at the top of the water column is the atmospheric pressure \(p_{a}\) 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 SHELFICEloadAnomalyFile). The weight is conveniently computed by integrating a density profile \(\rho^*\), that is constant in time and corresponds to the sea-water replaced by ice, from \(z=0\) to a “reference” ice-shelf draft at \(z=-h\) (Beckmann et al. (1999) [BHT99]), so that

(8.29)\[p_{\rm top} = p_{a} + \int_{-h}^{0}g\,\rho^{*}\,dz\]

Underneath the ice shelf, the “sea-surface height” \(\eta\) is the deviation from the “reference” ice-shelf draft \(h\). During a model integration, \(\eta\) adjusts so that the isostatic equilibrium is maintained for sufficiently slow and large scale motion.

In MITgcm, the total pressure anomaly \(p'_{\rm tot}\) which is used for pressure gradient computations is defined by subtracting a purely depth dependent contribution \(-g\rho_c z\) using constant reference density \(\rho_c\) from \(p_{\rm tot}\). (8.28) becomes

(8.30)\[p_{\rm tot} = p_{\rm top} - g \rho_c (z+h) + g \rho_c \eta + \, \int_z^{\eta-h}{ g (\rho-\rho_c) \, dz} + \, p_{\rm nh}\]

and after rearranging

\[p'_{\rm tot} = p'_{\rm top} + g \rho_c \eta + \, \int_z^{\eta-h}{g (\rho-\rho_c) \, dz} + \, p_{\rm nh}\]

with \(p'_{\rm tot} = p_{\rm tot} + g\,\rho_c\,z\) and \(p'_{\rm top} = p_{\rm top} - g\,\rho_c\,h\). The non-hydrostatic pressure contribution \(p_{\rm nh}\) is neglected in the following.

In practice, the ice shelf contribution to \(p_{\rm top}\) is computed by integrating (8.29) from \(z=0\) to the bottom of the last fully dry cell within the ice shelf:

(8.31)\[p_{\rm top} = g\,\sum_{k'=1}^{n-1}\rho_{k'}^{*}\Delta{z_{k'}} + p_{a}\]

where \(n\) is the vertical index of the first (at least partially) “wet” cell and \(\Delta{z_{k'}}\) is the thickness of the \(k'\)-th layer (counting downwards). The pressure anomaly for evaluating the pressure gradient is computed in the center of the “wet” cell \(k\) as

(8.32)\[p'_{k} = p'_{\rm top} + g\rho_{n}\eta + g\,\sum_{k'=n}^{k}\left((\rho_{k'}-\rho_c)\Delta{z_{k'}} \frac{1+H(k'-k)}{2}\right)\]

where \(H(k'-k)=1\) for \(k'<k\) and \(0\) otherwise.

schematic of vertical section of grid

Figure 8.11 Schematic of a vertical section of the grid at the base of an ice shelf. Grid lines are thin; 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. \(h_k\) is the fractional cell thickness so that \(h_k \Delta z_k\) is the actual cell thickness.

Setting 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 \(k\)-th layer abutting ice shelf topography for use in the heat flux parameterizations is computed as a mean temperature \(\overline{\theta}_{k}\) over a boundary layer of the same thickness as the layer thickness \(\Delta{z}_{k}\):

(8.33)\[\overline{\theta}_{k} = \theta_{k} h_{k} + \theta_{k+1} (1-h_{k})\]

where \(h_{k}\in[0,1]\) is the fractional layer thickness of the \(k\)-th layer (see Figure 8.11). The original contributions due to ice shelf-ocean interaction \(g_{\theta}\) to the total tendency terms \(G_{\theta}\) in the time-stepping equation \(\theta^{n+1} = f(\theta^{n},\Delta{t},G_{\theta}^{n})\) are

(8.34)\[g_{\theta,k} = \frac{Q}{\rho_c c_{p} h_{k} \Delta{z}_{k}} \text{ and } g_{\theta,k+1} = 0\]

for layers \(k\) and \(k+1\) (\(c_{p}\) is the heat capacity). Averaging these terms over a layer thickness \(\Delta{z_{k}}\) (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 \(k\)) and to the appropriate fraction of cells C (in layer \(k+1\)) yields

(8.35)\[g_{\theta,k}^* = \frac{Q}{\rho_c c_{p} \Delta{z}_{k}}\]
(8.36)\[g_{\theta,k+1}^* = \frac{Q}{\rho_c c_{p} \Delta{z}_{k}} \frac{ \Delta{z}_{k} ( 1- h_{k} )}{\Delta{z}_{k+1}}\]

(8.36) describes averaging over the part of the grid cell \(k+1\) that is part of the boundary layer with tendency \(g_{\theta,k}^*\) and the part with no tendency. Salinity is treated in the same way. The momentum equations are not modified. Three-equations thermodynamics

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 \(q\) (\(<0\) 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 \(S_I\). Further, the position of the interface between ice and ocean changes because of \(q\), so that, say, in the case of melting 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) [JHH01], 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 \(q\) is therefore included in the boundary conditions for the temperature and salinity equation as an advective flux.

The boundary conditions for tracer \(X=S,T\) (tracer \(X\) stands for either in-situ temperature \(T\) or salinity \(S\), located at the first interior ocean grid point) in the ocean are expressed as the sum of advective and diffusive fluxes

(8.37)\[F_X = (\rho_c \, \gamma_{X} -q ) ( X_{b} - X )\]

where the diffusive flux has been parameterized as a turbulent exchange \(\rho_c \, \gamma_{X}( X_{b} - X )\) following Holland and Jenkins (1999) [HJ99] or Jenkins et al. (2001) [JHH01]. \(X_b\) indicates the tracer in the boundary layer, \(\rho_c\) the density of seawater (parameter rhoConst), and \(\gamma_X\) is the turbulent exchange (or transfer) coefficient (parameters SHELFICEheatTransCoeff and 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:

(8.38)\[{\rho_I} K_{I,X} \frac{\partial{X_I}}{\partial{z}}\biggl|_{b} = \rho_c \, \gamma_{X} ( X_{b} - X ) - q ( X_{b} - X_{I} )\]

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 (\(qX_{I}\), this flux will be zero if the ice contains no tracer: \(X_I=0\)). The temperature in the boundary layer (\(T_b\)) is taken to be at the freezing point, which is a function of pressure and salinity, \(\rho_I\) is ice density (rhoShelfIce), and \(K_{I,X}\) the appropriate ice diffusivity.

For any material tracer such as salinity, the LHS in (8.38) vanishes (no material diffusion into the ice), while for temperature, the term \(q\,( T_{b}-T_{I} )\) 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:

(8.39)\[{\rho_I} \, c_{p,I} \, \kappa_{I,T} \frac{\partial{T_I}}{\partial{z}}\biggl|_{b} = c_{p} \, \rho_c \, \gamma_{T} ( T_{b} - T )+ L q.\]

where \(\kappa_{I,T}\) is the thermal ice diffusivity (SHELFICEkappa), \(L\) is the latent heat of fusion (SHELFICElatentHeat), \(c_{p}\) is the specific heat capacity of water (HeatCapacity_Cp), and \(c_{p,I}\) the heat capacity of the ice shelf (SHELFICEHeatCapacity_Cp). A reasonable choice for \(\gamma_T\) (SHELFICEheatTransCoeff), the turbulent exchange coefficient of temperature, is discussed in Holland and Jenkins (1999) [HJ99] (see Section The temperature at the interface \(T_{b}\) is assumed to be the in-situ freezing point temperature of sea-water \(T_{f}\), which is computed from a linear equation of state:

(8.40)\[T_{f} = (0.0901 - 0.0575\ S_{b}) - 7.61 \times 10^{-4} p_{b}.\]

where \(T_f\) is given in oC and \(p_{b}\) is in dBar. In (8.39), 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 (8.39) becomes

(8.41)\[{\rho_I} \, c_{p,I} \, \kappa_{I,T} \frac{\partial{T_I}}{\partial{z}}\biggl|_{b} \approx \rho_{I} \, c_{p,I} \, \kappa_{I,T} \frac{(T_{S} - T_{b})}{h}\]

where \(T_{S}\) the (surface) temperature of the ice shelf (SHELFICEthetaSurface) and \(h\) is the ice-shelf draft. Alternatively, assuming that the ice is “advected” vertically as implied by the meltflux \(q\), the diffusive flux can be approximated as \(\min(q,0)\,c_{p,I} (T_{S} - T_{b})\) (runtime flag SHELFICEadvDiffHeatFlux; see Holland and Jenkins, 1999 [HJ99] 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:

(8.42)\[\rho_c \, \gamma_{S} (S - S_{b}) = - q\,(S_{b}-S_{I})\]

where \(\gamma_S =\) SHELFICEsaltToHeatRatio \(* \gamma_T\) is the turbulent salinity exchange coefficient. Note, it is assumed that \(\kappa_{I,S} =0\); moreover, the salinity of the ice shelf is generally neglected (\(S_{I}=0\)).

The budget equations for temperature (8.39) (with (8.41)) and salinity (8.42), together with the freezing point temperature of sea-water (8.40), form the so-called three-equation-model (e.g., Hellmer and Olbers (1989) [HO89], Jenkins et al. (2001) [JHH01]). These equations are solved to obtain \(S_b, T_b, q\), which are then substituted into (8.37) to obtain the boundary conditions for the temperature and salinity equations of the ocean model. Note that with \(S_{I}=0\) and (8.42), the boundary flux for salinity becomes \(F_S = q\,S\), 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 (Section 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 \(S_b\) is not computed explicitly, for example as in the ISOMIP protocol, (8.37) cannot be applied directly. In this case (8.42) can be used with (8.37) to obtain:

\[F_X = q\,(S-S_I)\]

This formulation can be used for all cases for which (8.42) is valid. Further, in this formulation it is obvious that melting (\(q<0\)) leads to a reduction of salinity.

The default value of SHELFICEconserve =.FALSE. removes the contribution \(q\, ( X_{b}-X )\) from (8.37), making the boundary conditions non-conservative. Solving the three-equations system

There has been some confusion about the three-equations system, so we document the solution in the code here: We use (8.40) \(T_{b} = a_{0} S_{b} + \epsilon_{4}\) to eliminate \(T_{b}\) from (8.39) with (8.41) and find an expression for the freshwater flux \(q\):

(8.43)\[\begin{split}\begin{aligned} -Lq &= \epsilon_{1} (T - a_{0} S_{b} - \epsilon_{4}) + \epsilon_{3} (T_{S} - a_{0} S_{b} - \epsilon_{4}) \\ \Leftrightarrow Lq &= a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b} + \epsilon_{q} \end{aligned}\end{split}\]

to be substituted into (8.42):

\[\begin{split}\begin{aligned} \epsilon_{2}\,(S - S_{b}) &= - Lq\,(S_{b}-S_{I}) = - (a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b} + \epsilon_{q})\,(S_{b}-S_{I}) \\ \Leftrightarrow 0 &= a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}^{2} + \{ \epsilon_{q} - \epsilon_{2} - a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{I} \}\,S_{b} + \epsilon_{2}\,S - \epsilon_{q}\,S_{I} \\ \Leftrightarrow 0 &= A\,S_{b}^{2} + B\,S_{b} + C \\ \Rightarrow S_{b} &= \frac{-B \pm \sqrt{ B^{2} - 4AC }}{2A} \end{aligned}\end{split}\]

with the abbrevations

\[\begin{split}\begin{aligned} \epsilon_{1} &= c_{p} \, \rho_c \, \gamma_{T}, \quad \epsilon_{2} = \rho_c L \, \gamma_{S}, \quad \epsilon_{3} = \frac{\rho_{I} \, c_{p,I} \, \kappa}{h}, \quad \epsilon_{4} = b_{0}p + c_{0}, \\ \epsilon_{q} &= \epsilon_{1}\,(\epsilon_{4} - T) + \epsilon_{3}\,(\epsilon_{4} - T_{S}), \\ A &= a_{0}\,(\epsilon_{1} + \epsilon_{3}), \quad B = \epsilon_{q} - \epsilon_{2} - a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{I}, \quad C = \epsilon_{2}\,S -\epsilon_{q}\,S_{I}. \end{aligned}\end{split}\]

The smaller non-negative root of the quadratic equation in \(S_{b}\) is used. By default, the ice shelf salinity \(S_{I}\) is zero and the quadratic equation simplifies to

\[\begin{split}\begin{aligned} 0 &= a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}^{2} + (\epsilon_{q} - \epsilon_{2}) \,S_{b} + \epsilon_{2}\,S \\ S_{b} &= \frac{\epsilon_{2} - \epsilon_{q}\mp \sqrt{(\epsilon_{q} - \epsilon_{2})^2 - 4\, a_{0}\,(\epsilon_{1} + \epsilon_{3})\,\epsilon_{2}\,S}} {2\,a_{0}\,(\epsilon_{1} + \epsilon_{3})} \end{aligned}\end{split}\]

With \(S_b\), the boundary layer temperature \(T_b\) and the melt rate \(q\) are known through (8.40) and (8.43). ISOMIP thermodynamics

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) [GGD97]. In this formulation (8.39) reduces to

(8.44)\[c_{p} \, \rho_c \, \gamma_T (T - T_{b}) = -L q\]

and the fresh water flux \(q\) is computed from

(8.45)\[q = - \frac{c_{p} \, \rho_c \, \gamma_T (T - T_{b})}{L}\]

In order to use this formulation, set runtime parameter useISOMIPTD =.TRUE. in data.shelfice. Exchange coefficients

The default exchange coefficents \(\gamma_{T/S}\) are constant and set by the run-time parameters SHELFICEheatTransCoeff and SHELFICEsaltTransCoeff (see Table 8.23). If SHELFICEuseGammaFrict =.TRUE., exchange coefficients are computed from drag laws and friction velocities estimated from ocean speeds following Holland and Jenkins (1999) [HJ99]. This computation can be modified using runtime parameters and the user is referred to pkg/shelfice/shelfice_readparms.F for details. Remark

The shelfice package and experiments demonstrating its strengths and weaknesses are also described in Losch (2008) [Los08]. Unfortunately, the description of the thermodynamics in the appendix of Losch (2008) is wrong. Key subroutines

The main routine is shelfice_thermodynamics.F but note that /pkg/shelfice routines are also called when solving the momentum equations.

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

Diagnostics output is available via the diagnostics package (see Section 9). 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

See the verification experiment isomip for example usage of pkg/shelfice.