8.6.3. SHELFICE Package¶

Authors: Martin Losch, Jean-Michel Campin

8.6.3.1. 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 8.6.3.2). Run-time options, flags, filenames and field-related dates/times are described in Section 8.6.3.3. A description of key subroutines is given in Section 8.6.3.5. Available diagnostics output is listed in Section 8.6.3.6.

8.6.3.2. 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:

CPP Flag Name	Default	Description
ALLOW_SHELFICE_DEBUG	#undef	include code for enhanced diagnostics and debug output
ALLOW_ISOMIP_TD	#define	include code for for simplified ISOMIP thermodynamics
SHI_ALLOW_GAMMAFRICT	#define	allow friction velocity-dependent transfer coefficient following Holland and Jenkins (1999) [HJ99]

8.6.3.3. 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 (8.30). Note however the file SHELFICEloadAnomalyFile must not be \(p_{top}\), but \(p_{top}-g\sum_{k'=1}^{n-1}\rho_{0}\Delta{z}_{k'}\), with \(\rho_{0}=\) rhoConst, so that in the absence of a \(\rho^{*}\) that is different from \(\rho_{0}\), the anomaly is zero.

Table 8.20 Run-time parameters and default values¶
Parameter	Group	Default	Description
useISOMIPTD	SHELFICE_PARM01	FALSE	use simplified ISOMIP thermodynamics on/off flag
SHELFICEconserve	SHELFICE_PARM01	FALSE	use conservative form of temperature boundary conditions on/off flag
SHELFICEboundaryLayer	SHELFICE_PARM01	FALSE	use simple boundary layer mixing parameterization on/off flag
SHI_withBL_realFWflux	SHELFICE_PARM01	FALSE	with SHELFICEboundaryLayer, allow to use real-FW flux
SHI_withBL_uStarTopDz	SHELFICE_PARM01	FALSE	with SHELFICEboundaryLayer, compute uStar from uVel,vVel averaged over top Dz thickness
SHELFICEloadAnomalyFile	SHELFICE_PARM01	`' '`	initial geopotential anomaly
SHELFICEtopoFile	SHELFICE_PARM01	`' '`	filename for under-ice topography of ice shelves
SHELFICEmassFile	SHELFICE_PARM01	`' '`	filename for mass of ice shelves
SHELFICEMassDynTendFile	SHELFICE_PARM01	`' '`	filename for mass tendency of ice shelves
SHELFICETransCoeffTFile	SHELFICE_PARM01	`' '`	filename for spatially varying transfer coefficients
SHELFICElatentHeat	SHELFICE_PARM01	334.0E+03	latent heat of fusion (J/kg)
SHELFICEHeatCapacity_Cp	SHELFICE_PARM01	2000.0E+00	specific heat capacity of ice (J/kg/K)
rhoShelfIce	SHELFICE_PARM01	917.0E+00	(constant) mean density of ice shelf (kg/m³)
SHELFICEheatTransCoeff	SHELFICE_PARM01	1.0E-04	transfer coefficient (exchange velocity) for temperature (m/s)
SHELFICEsaltTransCoeff	SHELFICE_PARM01	5.05E-03 \(\times\) SHELFICEheatTransCoeff	transfer coefficient (exchange velocity) for salinity (m/s)
SHELFICEkappa	SHELFICE_PARM01	1.54E-06	temperature diffusion coefficient of the ice shelf (m²/s)
SHELFICEthetaSurface	SHELFICE_PARM01	-20.0E+00	(constant) surface temperature above the ice shelf (^oC)
no_slip_shelfice	SHELFICE_PARM01	no_slip_bottom	slip along bottom of ice shelf on/off flag
SHELFICEDragLinear	SHELFICE_PARM01	bottomDragLinear	linear drag coefficient at bottom ice shelf (m/s)
SHELFICEDragQuadratic	SHELFICE_PARM01	bottomDragQuadratic	quadratic drag coefficient at bottom ice shelf (non-dim.)
SHELFICEselectDragQuadr	SHELFICE_PARM01	-1	select form of quadratic drag coefficient (non-dim.)
SHELFICEMassStepping	SHELFICE_PARM01	FALSE	recalculate ice shelf mass at every time step
SHELFICEDynMassOnly	SHELFICE_PARM01	FALSE	if SHELFICEmassStepping = TRUE, exclude freshwater flux contribution
SHELFICEadvDiffHeatFlux	SHELFICE_PARM01	FALSE	use advective-diffusive heat flux into ice shelf instead of default diffusive heat flux
SHELFICEuseGammaFrict	SHELFICE_PARM01	FALSE	use velocity dependent exchange coefficients (Holland and Jenkins 1999 [HJ99])
SHELFICE_oldCalcUStar	SHELFICE_PARM01	FALSE	use old uStar averaging expression
SHELFICEwriteState	SHELFICE_PARM01	FALSE	write ice shelf state to file on/off flag
SHELFICE_dumpFreq	SHELFICE_PARM01	dumpFreq	dump frequency (s)
SHELFICE_dump_mnc	SHELFICE_PARM01	snapshot_mnc	write snapshot using MNC on/off flag

8.6.3.4. 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 \(p_{tot}\) in the ocean can be divided into the pressure at the top of the water column \(p_{top}\), the hydrostatic pressure and the non-hydrostatic pressure contribution \(p_{NH}\):

(8.27)¶\[p_{tot} = p_{top} + \int_z^{\eta-h} g\,\rho\,dz + p_{NH}\]

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, \(p_{top}=p_{a}\) (atmospheric pressure) and \(h=0\). Underneath an ice-shelf that is assumed to be floating in isostatic equilibrium, \(p_{top}\) 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

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

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 \(p'_{tot}\) which is used for pressure gradient computations is defined by substracting a purely depth dependent contribution \(-g\rho_{0}z\) with a constant reference density \(\rho_{0}\) from \(p_{tot}\). (8.27) becomes

(8.29)¶\[p_{tot} = p_{top} - g \rho_0 (z+h) + g \rho_0 \eta + \, \int_z^{\eta-h}{ g (\rho-\rho_0) \, dz} + \, p_{NH}\]

and after rearranging

\[p'_{tot} = p'_{top} + g \rho_0 \eta + \, \int_z^{\eta-h}{g (\rho-\rho_0) \, dz} + \, p_{NH}\]

with \(p'_{tot} = p_{tot} + g\,\rho_0\,z\) and \(p'_{top} = p_{top} - g\,\rho_0\,h\). The non-hydrostatic pressure contribution \(p_{NH}\) is neglected in the following.

In practice, the ice shelf contribution to \(p_{top}\) is computed by integrating (8.28) from \(z=0\) to the bottom of the last fully dry cell within the ice shelf:

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

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

(8.31)¶\[p'_{k} = p'_{top} + g\rho_{n}\eta + g\,\sum_{k'=n}^{k}\left((\rho_{k'}-\rho_{0})\Delta{z_{k'}} \frac{1+H(k'-k)}{2}\right)\]

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

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 \(\Delta{z}_{k}\):

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

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

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

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

(8.34)¶\[g_{\theta,k}^* = \frac{Q}{\rho_{0} c_{p} \Delta{z}_{k}}\]

(8.35)¶\[g_{\theta,k+1}^* = \frac{Q}{\rho_{0} c_{p} \Delta{z}_{k}} \frac{ \Delta{z}_{k} ( 1- h_{k} )}{\Delta{z}_{k+1}}\]

(8.35) 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.

8.6.3.4.1. Three-equations thermodynamics¶

Freezing and melting form a boundary layer between ice shelf and ocean. Phase transitions at the boundary between saline water and ice imply the following fluxes across the boundary: the freshwater mass flux \(q\) (\(<0\) for melting); the heat flux that consists of the diffusive flux through the ice, the latent heat flux due to melting and freezing and the heat that is carried by the mass flux; and the salinity that is 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 salinity and temperature are modified.

The turbulent exchange terms for tracers at the ice-ocean interface are generally expressed as diffusive fluxes. Following Jenkins et al. (2001) [JHH01], the boundary conditions for a tracer take into account that this boundary is not a material surface. The implied upward freshwater flux \(q\) (in mass units, negative for melting) is included in the boundary conditions for the temperature and salinity equation as an advective flux:

(8.36)¶\[{\rho}K\frac{\partial{X}}{\partial{z}}\biggl|_{b} = (\rho\gamma_{X}-q) ( X_{b} - X )\]

where tracer \(X\) stands for either temperature \(T\) or salinity \(S\). \(X_b\) is the tracer at the interface (taken to be at freezing), \(X\) is the tracer at the first interior grid point, \(\rho\) is the density of seawater, and \(\gamma_X\) is the turbulent exchange coefficient (in units of an exchange velocity). The left hand side of (8.36) is shorthand for the (downward) flux of tracer \(X\) across the boundary. \(T_b\), \(S_b\) and the freshwater flux \(q\) are obtained from solving a system of three equations that is derived from the heat and freshwater balance at the ice ocean interface.

In this so-called three-equation-model (e.g., Hellmer and Olbers (1989) [HO89], Jenkins et al. (2001) [JHH01]) the heat balance at the ice-ocean interface is expressed as

(8.37)¶\[c_{p} \rho \gamma_T (T - T_{b}) +\rho_{I} c_{p,I} \kappa \frac{(T_{S} - T_{b})}{h} = -Lq\]

where \(\rho\) is the density of sea-water, \(c_{p} = 3974 \, \text{J kg}^{-1} \text{K}^{-1}\) is the specific heat capacity of water and \(\gamma_T\) the turbulent exchange coefficient of temperature. The value of \(\gamma_T\) is discussed in Holland and Jenkins (1999) [HJ99]. \(L = 334000 \, \text{J kg}^{-1}\) is the latent heat of fusion. \(\rho_{I} = 920 \, \text{kg m}^{-3}\), \(c_{p,I} = 2000 \, \text{J kg}^{-1} \text{K}^{-1}\), and \(T_{S}\) are the density, heat capacity and the surface temperature of the ice shelf; \(\kappa=1.54\times10^{-6} \, \text{m}^2 \text{s}^{-1}\) is the heat diffusivity through the ice-shelf and \(h\) is the ice-shelf draft. The second term on the right hand side describes the heat flux through the ice shelf. A constant surface temperature \(T_S=-20^{\circ}\text{C}\) is imposed. \(T\) is the temperature of the model cell adjacent to the ice-water interface. 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

(8.38)¶\[T_{f} = (0.0901 - 0.0575\ S_{b})^{\circ} - 7.61 \times 10^{-4}\frac{\text{K}}{\text{dBar}}\ p_{b}\]

with the salinity \(S_{b}\) and the pressure \(p_{b}\) (in dBar) in the cell at the ice-water interface. From the salt budget, the salt flux across the shelf ice-ocean interface is equal to the salt flux due to melting and freezing:

(8.39)¶\[\rho \gamma_{S} (S - S_{b}) = - q\,(S_{b}-S_{I})\]

where \(\gamma_S = 5.05\times10^{-3}\gamma_T\) is the turbulent salinity exchange coefficient, and \(S\) and \(S_{b}\) are defined in analogy to temperature as the salinity of the model cell adjacent to the ice-water interface and at the interface, respectively. Note, that the salinity of the ice shelf is generally neglected (\(S_{I}=0\)). (8.37) to (8.39) can be solved for \(S_{b}\), \(T_{b}\), and the freshwater flux \(q\) due to melting. These values are substituted into expression (8.36) to obtain the boundary conditions for the temperature and salinity equations of the ocean model. This formulation tends to yield smaller melt rates than the simpler formulation of the ISOMIP protocol because the freshwater flux due to melting decreases the salinity which 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, (8.36) cannot be applied directly. In this case (8.39) can be used with (8.36) to obtain:

\[\rho{K}\frac{\partial{S}}{\partial{z}}\biggl|_{b} = q\,(S-S_I)\]

This formulation can be used for all cases for which (8.39) 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 (8.36), making the boundary conditions for temperature non-conservative.

8.6.3.4.2. 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 (8.38) \(T_{b} = a_{0} S_{b} + \epsilon_{4}\) to eliminate \(T_{b}\) from (8.37) and find an expression for the freshwater flux \(q\):

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

to be substituted into (8.39):

\[\begin{split}\begin{aligned} \epsilon_{2}\,(S - S_{b}) &= - Lq\,(S_{b}-S_{I}) = - (a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b} + \epsilon_{q})\,(S_{b}-S_{I}) \\ \Leftrightarrow 0 &= a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{b}^{2} + \{ \epsilon_{q} - \epsilon_{2} - a_{0}\,(\epsilon_{1} + \epsilon_{3})\,S_{I} \}\,S_{b} + \epsilon_{2}\,S - \epsilon_{q}\,S_{I} \end{aligned}\end{split}\]

where the abbrevations \(\epsilon_{1} = c_{p} \rho \gamma_{T}\), \(\epsilon_{2} = \rho L \gamma_{S}\), \(\epsilon_{3} = \frac{\rho_{I} c_{p,I} \kappa}{h}\), \(\epsilon_{4}=b_{0}p + c_{0}\), \(\epsilon_{q} = \epsilon_{1}\,(\epsilon_{4} - T) + \epsilon_{3}\,(\epsilon_{4} - T_{S})\) have been introduced. The quadratic equation in \(S_{b}\) is solved and the smaller non-negative root is used. In the MITgcm code, the ice shelf salinity \(S_{I}\) is always zero and the quadratic equation simplifies to

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

With \(S_b\), the boundary layer temperature \(T_b\) and the melt rate \(q\) are known through (8.38) and (8.40).

8.6.3.4.3. 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 (8.37) reduces to

(8.41)¶\[c_{p} \rho \gamma_T (T - T_{b}) = -Lq\]

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

(8.42)¶\[q = - \frac{c_{p} \rho \gamma_T (T - T_{b})}{L}\]

In order to use this formulation, set run-time parameter useISOMIPTD =.TRUE. in data.shelfice.

8.6.3.4.4. Exchange coefficients¶

The default exchange coefficents \(\gamma_{T/S}\) are constant and set by the runtime parameters SHELFICEheatTransCoeff and SHELFICEsaltTransCoeff (see Table 8.20). 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 user is referred to S/R pkg/shelfice/shelfice_readparms.F for details.

8.6.3.4.5. Remark¶

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

8.6.3.5. Key subroutines¶

The main routine is shelfice_thermodynamics.F but note that /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

8.6.3.6. 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

8.6.3.7. Experiments and tutorials that use shelfice¶

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