While coupled models in Earth system science have been in existence for decades and such coupled models are often viewed as single entities (ocean–atmosphere general circulation models, for example), the field of coupled ice sheet–ocean modelling is relatively young. FISOC is intended as a framework for coupling independent models rather than as a coupled model in itself. Building and running a coupled ice sheet–ocean model is currently more complex than building and running both an ice and an ocean model independently. FISOC aims to minimize the additional complexity.

The ice and ocean components may use their standard runtime input files, and their paths are set in a FISOC runtime configuration file, along with information about time stepping and variables to be exchanged.

FISOC adopts the hierarchical modular structure of the Earth System Modelling Framework. The FISOC code structures are summarized in Fig. . A top-level executable is called a FISOC parent module (this could in principle also be embedded within a larger coupled model framework). The parent module coordinates calling of the ice, ocean and regridding components. Regridding is one of the reasons to make use of ESMF, described further in Sect. . The ice and ocean components are independent models that are not included in the FISOC code repository and compiled as libraries to be called by FISOC during runtime. On each side (ice and ocean) of the coupling is a model-specific wrapper, whose main runtime functions are as follows:

call the component's initialize, run, and finalize routines as required;

Further processing of variables (such as calculating rates of change) is implemented by the ice and ocean generic code modules.

Incorporating a new ice or ocean component into FISOC can be straightforward, depending on the existing level of ESMF compatibility of the new component. Models able to provide mesh information and variables in ESMF data structures can be very easily built in to FISOC. The only coding required for a new component is a new model-specific wrapper in the FISOC repository. Copying an existing wrapper can be a viable starting point.

FISOC is designed to facilitate swapping between different ocean or ice components. Currently two different ocean components and one ice component are available through FISOC. Table  summarizes components currently coupled into FISOC. In some cases, a non-standard build of the component is required for FISOC compatibility, and these cases are described in the FISOC manual, which can be obtained through the FISOC repository (Sect. ).

The ice component Elmer/Ice is a powerful, flexible, state-of-the-art ice dynamic model.

The Regional Ocean Modeling System (ROMS;) is a 3-D terrain-following, sigma coordinate ocean model that has already been adapted to use in ice shelf cavities . The module for ice shelf cavities implemented in the Finite Volume Community Ocean Model (FVCOM;) provides non-hydrostatic options and a horizontally unstructured mesh that lends itself to refinement and may be more suited to small-scale processes such as ice shelf channels .

As stated above, FISOC provides coupling on a horizontal plane onto which the lower surface of an ice shelf can be projected. It is this plane on which ice and ocean properties are exchanged through the FISOC. Adapting the FISOC code to handle a vertical ice cliff is expected to be straightforward and would be desirable for application to the Greenland ice sheet. More complex 3-D ice–ocean interface geometries are challenging not only for FISOC but also for the current generation of ice sheet and ocean models.

FISOC has access to all the runtime regridding options provided by ESMF. These include nearest-neighbour options, conservative options, patch recovery and bilinear regridding. These options are available for structured grids and unstructured meshes. FISOC requires that both ice and ocean components define their grid or mesh on the same coordinate system and that both components use the same projection. All FISOC simulations to date have used a Cartesian coordinate system (i.e. all components have so far used Cartesian coordinates).

Our current FISOC setup does not meet the requirements for all forms of ESMF regridding. Specifically, the conservative methods, when an unstructured mesh is involved, require that field values are defined on elements and not on nodes. Elmer, by default, provides field values on nodes but can also provide element-wide values or values on integration points within elements. We will need to either map nodal values to element values or utilize element-type variables in order to use conservative regridding, and this is intended as a future development.

When using FISOC to couple Elmer/Ice to ROMS, the ROMS grid extends beyond the Elmer/Ice mesh. This is due to ROMS using a staggered grid (Arakawa C-grid) and ghost cells extending beyond the active domain. This necessitates the use of extrapolation. ESMF regridding methods provide options for extrapolation, which are used here. Simulations in the current study use either nearest “source to destination” (STOD, a form of nearest neighbour) regridding or use bilinear interpolation (in which case nearest STOD is used only for destination points that lie outside the source domain).

We use subscripts with square brackets, [X], where X is either O (ocean component) or I (ice component), to denote a variable that exists in both ice and ocean components with the same physical meaning but potentially different values due to being represented on different grids or meshes.

The timescales for sub-shelf cavity circulation behaviour are in general much shorter than the timescales for ice flow and geometry evolution (typically minutes to days instead of years to centuries). Typical time step sizes are correspondingly smaller for ocean models (seconds to minutes) than for ice sheet models (days to months). A single ice sheet model time step, if the Stokes equations are solved in full, will typically require orders of magnitude more computational time than a single ocean time step. Due to the combination of these two reasons the ice and ocean components of FISOC will in general use different time steps, with the ice time step size being much larger. We define relevant terminology for coupling timescales below.

Fully synchronous coupling. The ice and ocean components have the same time step size, and they exchange variables every time step.

In the current study, FISOC sets the coupling interval as equal to the ice component time step size. This is an exact multiple of the ocean model time step size. More generally (for potential future experiments), FISOC calls each component for a fixed time period and allows the component to determine its own time stepping within that period. In principle, adaptive time stepping could be implemented within this framework as long as each component runs for the required amount of simulated time. FISOC does not currently provide an option to vary the coupling interval during a simulation, but this could be implemented if needed.

FISOC is flexible with regard to time processing of ocean or ice variables. It is possible to cumulate variables, calculate averages, or use snapshots. In the current study, the ocean components (both ROMS and FVCOM) calculate averaged basal melt rates over the coupling interval and pass these averages through FISOC to the ice component. In the current study, as the ice component time step size is equal to the coupling interval for all simulations, no time processing of ice component variables is needed.

In principle, FISOC supports all three synchronicity options, though fully synchronous coupling is not practical to achieve when solving the Stokes equations for the ice component. The experiments carried out for this paper use semi-synchronous coupling with cavity geometry evolution as described in Sect. .

and implement fully synchronous coupling, whereas and implement asynchronous coupling with ocean restarts.

The evolution of cavity geometry under the ice shelf, defined by a reference ice draft, zd (positive upward), and grounding line location, is calculated by the ice component forced by the melt rates passed from the ocean component. We refer to zd as a “reference” ice draft because the ocean component may further modify the ice draft according to the dynamic pressure field. The ocean component's “free surface” variable, ζ, represents the height of the upper surface of the ocean domain relative to a mean sea level for the open ocean. Under the ice shelf, ζ represents the deviation of the upper surface of the ocean domain relative to the reference ice draft zd (similar to ). To summarize the meaning of the key variables, zd[I] is the reference ice draft computed by the ice component, zd[O] is the same but regridded for the ocean component and (zd[O]+ζ) is the actual ice draft according to the ocean component.

Given the potential for non-synchronicity of the ice and ocean component time stepping, several methods are implemented in FISOC for the ocean to update its representation of zd. All the processing options described below are applied on the ocean grid after the ice component representation of ice geometry has been regridded (i.e. zd[I] regridded to zd[O]).

Most recent ice. The simplest option is that the ocean component uses the ice draft directly from the most recent ice component time step. If fully synchronous coupling is used, this option should be chosen. The main disadvantage of this approach for semi-synchronous or asynchronous coupling is that due to the much longer time step of the ice component the ocean component will experience large, occasional changes in ice draft instead of smoothly evolving ice draft. This could be both physically unrealistic and potentially numerically challenging for the ocean component.

Rate. The vertical rate of change of ice draft, dzddt, is calculated by FISOC after each ice component time step using the two most recent ice component time steps. If we assume that the ice component completes a time step at time t, the rate at this time is given by 1dzd[O,t]dt=zd[I,t]-zd[I,t-ΔtI]ΔtI, where zd[O,t] is the ocean component's reference ice draft at time t, zd[I,t] is the ice component's reference ice draft at time t, zd[I,t-ΔtI] is the ice component's reference ice draft at time t-ΔtI and ΔtI is the ice component time step size. This rate of change is used by the ocean component to update the cavity geometry until the next ice component time step completes. In this sense the ocean component lags the ice component as mentioned in Sect. . This approach provides temporally smooth changes to the ocean representation of the ice draft but has the potential for the ice and ocean representations to diverge over time as a result of regridding artefacts.

Corrected rate. This is the same as above, except that a drift correction is applied to ensure that ice and ocean representations of cavity geometry do not diverge: 2dzd[O,t]dt=zd[I,t]-zd[I,t-ΔtI]+fcavzd[I,t]-zd[O,t]ΔtI, where fcav is a cavity correction factor between 0 and 1. Equation () is applied at coupling time steps, and the calculated rate of cavity change is then held constant during ocean component evolution until the coupling interval completes. Conceptually, this option prioritizes ice–ocean geometry consistency over mass conservation.

Linear interpolation. The ocean representation of the ice draft is given by temporal linear interpolation between the two most recent ice sheet time steps. This imposes additional lag of the ocean component behind the ice component.

The above options are all implemented in FISOC, but only the “rate” and “corrected rate” approaches are used in the current study.

The cavity geometry may be initialized independently by ice and ocean components. In this case, the user must ensure consistency. It is also possible for the cavity geometry from the ice component to be imposed on the ocean component during FISOC initialization. This ensures consistency.

Handling cavity evolution is a little more complicated in the case of an evolving grounding line, as discussed in Sect.  below.

Exchange of heat at the ice–ocean interface is handled within the ocean model. Like many ocean models, FVCOM and ROMS adopt the three-equation formulation for thermodynamic exchange . This parameterization assumes that the interface is at the in situ pressure freezing point and that there is a heat balance and salt balance at the interface. Both ROMS and FVCOM assume constant turbulent transfer coefficients for scaling the heat and salt fluxes through the interface, with thermal and saline exchange velocities calculated as the product of these coefficients with friction velocity. Further details of the ROMS- and FVCOM-specific implementations of the three-equation formulation are given by and , respectively. An ablation or melt rate is calculated for each ocean model grid cell, which is then passed to FISOC as a boundary condition for the lower surface of the ice model at the coupling time interval.

Internally, both ocean models account for the thermodynamic effect of basal melting by imposing virtual heat and salt fluxes within a fixed geometry at each ocean model time step to mimic the effects of basal melting, rather than employing an explicit volume flux at the ice–ocean interface. Independent of this, a geometry change is passed back from the ice model through FISOC at after each coupling interval (including the effect of melting and freezing, as well as any ice dynamical response), which is used to update the ocean component cavity shape (Sect. ).

For some applications, conductive heat fluxes into the ice shelf due to vertical temperature gradients in the ice at the ice–ocean interface are required by the three-equation parameterization to calculate the flux balance at the ice ocean interface. While ice–ocean thermodynamic parameterizations in ocean-only models must make an assumption about this temperature gradient, FISOC can pass the temperature gradient from the ice component directly to the ocean component. This feature is not demonstrated in the current study but will be properly tested in future studies.

Non-zero basal melt rates may be calculated by the ocean component in regions that are defined as grounded by the ice model. This could occur due to isolated patches of ungrounding upstream of the grounding line or discrepancies between the ice and ocean component's representation of the grounded region. Basal melt rates are masked using the ice component's grounded mask before being applied within the ice component. This has the potential to impact on mass conservation in the coupled system. Future studies utilizing conservative regridding will ensure that passing masked field variables between components remains conservative.

Aside from the geometry evolution, an ocean boundary condition for pressure at the ice–ocean interface, Pinterface, must be provided to the ocean component. FISOC can pass pressure directly from ice to ocean components. However, using actual ice overburden directly as an upper ocean boundary condition results in higher horizontal pressure gradients at the grounding line (for dry cells, see Sect. ) than ocean models can typically handle . In the current study, the ocean component uses the reference ice draft (see Sect. ) to estimate floatation pressure. ROMS assumes a constant reference ocean density: 3Pinterface=-gρorzd[O], where g is acceleration due to gravity, ρor is a reference ocean density and zd[O] is the ocean representation of ice draft (positive upward). For the current study, all simulations with ROMS use ρor=1027kgm-3. FVCOM assumes a constant vertical ocean density gradient following : 4Pinterface=-gρo1+0.5dρodzzd[O]zd[O], where ρo is ocean water density, ρo1 is ocean water density of the top ocean layer, and the vertical ocean water density gradient, dρodz, is set to 8.3927×10-4kgm-4.

Grounding line movement in FISOC requires that both ice and ocean components support it. Numerical convergence issues place constraints in terms of mesh resolution for representing grounding line movement in ice sheet models . While FISOC allows ice draft to be passed to the ocean component (Sect. ), FISOC does not impose the ice component grounding line position on the ocean component. Instead, the ocean component uses the evolving cavity geometry to evolve the grounding line.

A recent ice–ocean coupling study used a “thin film” approach to allow grounding line movement. A thin passive water layer is allowed to exist under the grounded ice, and an activation criterion is imposed to allow the layer to inflate to represent grounding line retreat. The current study takes a conceptually similar approach, modifying the existing wetting and drying schemes independently in both ROMS and FVCOM. “Dry” cells are used for the passive water column under grounded ice, and “wet” cells are used for the active water column under floating ice or the open ocean. The wet–dry mask is two dimensional, so while it is conventional to talk about dry or wet cells, this actually refers to dry or wet columns. The grounding line evolves in the two horizontal dimensions and is represented in the ocean component as the vertical surface between dry and wet columns.

The original criterion in both ROMS and FVCOM for a cell to remain dry is given by 5ζ-zb<Dcrit, where zb is the bottom boundary depth (bathymetry, or bedrock depth, positive upward), and Dcrit is a critical water column thickness for wet or dry activation. Dcrit is a parameter to be set by the user (typical values lie between 1 to 20 m). Thus, cells with a water column thickness less than Dcrit are designated as dry. Flux of water into dry cells is allowed, but flux of water out of dry cells is prevented.

The FVCOM criterion for an element to be dry has been modified for the presence of a marine ice sheet and ice shelf system as follows: 6-zb+zd[O]<Dcrit. This is a purely geometric criterion based entirely on the geometry determined by the ice component. The ROMS criterion for a cell to be dry has been modified for the presence of a marine ice sheet and ice shelf system as follows: 7ζ-zb-(zs[O]-zd[O]-Dcrit)ρiρor≤0, where zs[O] is the ocean representation of ice sheet and ice shelf upper surface height. zs[O] is needed in this equation because the floatation assumption cannot be made for grounded ice. This equation essentially compares ζ against the height above buoyancy of the grounded ice. In other words, if the dynamic variations in ocean pressure are sufficient to overcome the higher ice pressure due to the positive height above buoyancy, the cell can become ungrounded. The conceptual difference between the FVCOM and ROMS wetting criteria is that ROMS allows dynamic ocean pressure variations to make minor grounding line adjustments relative to the grounding line determined by the ice geometry, whereas FVCOM uses just the ice geometry to determine grounding line position.

FISOC allows the ice component to pass any geometry variables to the ocean, such as ice draft, ice thickness, upper surface elevation or the rates of change of any of these variables. In the event that geometry variables other than zd are passed to the ocean, the same processing method is used as for zd, as described in Sect. . In the current study, dzddt is passed to the ocean component, and in one case both dzddt and dzsdt are passed (details in Sect. ). When dzsdt is passed, dzsdt is processed the same way as dzddt. If the grounding line problem is solved and if zd is processed for passing to the ocean using the corrected rate method, Eq. () is modified to account for the dry water column thickness, which is initialized to Dcrit. The correction term changes from fcavzd[I,t]-zd[O,t] to fcavmax⁡(zd[I,t],zb+Dcrit)-zd[O,t].

There are no connectivity restrictions on wetting and drying for either of the ocean components in the current study. This means that it is possible for individual cells or regions containing multiple cells that are upstream of the grounding line to become wet (i.e. to unground). This occurs on small spatial and temporal scales in ROMS (individual cells a short distance upstream of the grounding line sometimes become temporarily wet) but not at all in FVCOM (likely due to choice of wetting criterion).

Verification experiment 1 (VE1) is a simple experiment in which a linearly sloping ice shelf is allowed to adjust toward steady state. The experiment is not run long enough to attain steady state but is long enough to demonstrate the evolution of the coupled system; see Table  for run length and a summary of other model choices and parameter values used in VE1.

All ice and ocean vertical side boundaries are closed: there is no flow in or out of the domain. There is mass exchange between the ice and ocean (and therefore also heat exchange). The coupling centres on the evolution of ice geometry: the ocean component passes an ice shelf basal melt rate to the ice component, and the ice component passes a rate of change of ice draft to the ocean component.

We expect adjustment toward a uniform-thickness ice shelf to occur by the following two mechanisms.

Ice dynamics. The gravitational driving force will tend to cause flow from thicker to thinner regions.

Verification experiment 2 (VE2) is a modified version of VE1 but with part of the region grounded and a net ice flow through the domain allowed. The setup is identical to VE1 except where stated otherwise in this section. This experiment aims to combine design simplicity with an evolving grounding line rather than to represent a system directly analogous to a real-world example.

Figure  summarizes the coupled system state at the start and end of simulation VE1_ER (see also Table  for a summary of the experiments). After the first coupling interval (10 d), the ocean component demonstrates a vigorous overturning circulation and high melt rates, especially in the deeper part of the domain. After the last coupling interval (100 years) the combination of melting and ice flow has caused a redistribution of the ice shelf, with an overall reduction in the along-domain gradients. The melt rates and overturning circulation are much weaker than at the start.

The ocean circulation throughout the simulation is predominantly a buoyancy-driven overturning along the domain, with very little cross-domain flow. The peak ocean flow speeds are always located at the top of the ocean domain directly under the ice shelf, where a fast, shallow buoyancy-driven flow from deeper to shallower ice draft is balanced by a much deeper return flow.

Figure  shows the evolution over time of the total mass of both ice and ocean components and the total coupled system from experiments VE1_ER and VE1_EF. Note that both ocean models employ the Boussinesq approximation and that the mass in Fig.  is calculated as volume multiplied by the reference ocean density from Table . Relatively rapid mass transfer from the ice to the ocean occurs during the first few years as the relatively warm ocean water transfers its energy to the ice. After this initial period of net melting, the ocean water temperature is close to freezing point and a long-term freezing trend can be seen that is stronger and more sustained in the ROMS ocean component than FVCOM. In a physically realistic coupled system, the ice and ocean would come into thermodynamic equilibrium and the spatial net mass transfer would approach zero.

The net mass change of the coupled system is more than an order of magnitude smaller than the mass change of the individual components for both experiments VE1_ER and VE1_EF. The current study does not use conservative regridding (Sect. ), and therefore machine precision conservation is not expected. There are additional potential sources of error. The lag of the ocean component behind the ice component (Sect. ) will cause a similar lag in total mass evolution. Use of the corrected rate cavity option (Sect. ) prioritizes geometry consistency between components above mass conservation. The aim of analysing mass conservation in the current study is to ensure that the cumulative impact of these potential error sources is small compared to the signal. This has been achieved, and it will be possible to quantify and minimize or eliminate all sources of error in future studies using conservative regridding methods.

This is a partially grounded experiment in which the ice component boundaries are not closed, a through-domain flow of ice is allowed and the grounding line is allowed to evolve in the coupled system (described in Sect. ). While the initial slope of the lower surface of the ice shelf is the same in both VE1 and VE2, the open inflow and outflow boundaries in the ice component and the relatively shallower ice in the grounded region both lead to a shelf that is much shallower in slope for VE2 than for VE1 for most of the simulation period. Figure  illustrates the shape of the ice sheet or ice shelf at the start of the simulation and after 25 years (from simulation VE2_ER). Note that the ice outflow boundary is more active than the inflow, with the flux into the domain through the inflow boundary remaining small and positive throughout the simulation. The ice draft is deepest in the middle of the domain, at around 30 km downstream (in terms of ice flow direction) from the grounding line. The ice draft impacts on circulation and melt, with the strong overturning of VE1_ER not present here. Melting occurs under the deepest ice, with refreezing elsewhere (Fig. ).

Comparing the coupled simulation VE2_ER to the ice-only simulation (not shown) where the only difference is that the ice component features zero basal melt, it might be expected that the coupled simulation would exhibit a significantly thinner ice shelf due to melting. However, the ice dynamics partially compensate for this in terms of the ice geometry: the melt-induced thinning leads to acceleration in the ice, and the thickness difference is smaller than expected. However, this should not be interpreted as a stabilizing feedback response of ice dynamics to ocean-induced melting, as the increased ice flow would tend to drain the grounded ice more quickly, potentially triggering marine ice sheet instability . Instead this effect may tend to partially mask an ocean-induced ice sheet destabilization if the observational focus is on ice shelf geometry.

As described in Sect. , the ice and ocean component each evolve the grounding line on their own time step and on their own grid or mesh. There is potential for discrepancy between ice and ocean grounded area due to method of cavity evolution (Sect. ), regridding errors, the inherent differences between grids or meshes, and the methods used to determine grounding line position. While ice geometry is a key determinant of grounding line position, the ice component also tests for a contact force and the ocean component ROMS tests height above buoyancy against the free surface variable ζ (Sect. ). Here we look at consistency of grounded area between components.

The evolution of grounded area in both ice and ocean components is shown in Fig.  for simulation VE2_ER. While the ice component employs an unstructured mesh of triangular elements (on the lower surface of the 3-D ice body), the ocean component employs a regular grid of square cells. The ocean component appears to exhibit a step-like reduction over time of grounded area. This is due to the row-by-row manifestation of grounding line retreat in the ocean component due to the alignment of grid rows with the linear down-sloping geometry. Grounding line retreat starts at the lateral edges of a row (ungrounding near the sidewall boundary), and the “wetting” of dry cells propagates toward the centre of the row. This step-like behaviour (with the spacing of the green lines in Fig.  indicating the total area of a row of cells) explains the main difference between ice and ocean grounded area. The evolution of grounded area is shown in Fig.  for simulation VE2_EF. Behaviour is similar to VE2_ER.

The initial rapid reduction in grounded area is due to the initial geometry. A region immediately upstream of the grounding line is initially very lightly grounded, and this region quickly becomes floating. The ocean component lags the ice component in this ungrounding, as can be seen in the first part of the difference plot in Figs.  and . This lag is in part due to the rate and corrected rate cavity update methods, in which the ocean component uses the most recent two ice component outputs to calculate a rate of change of geometry. This inevitably causes the ocean component to lag by approximately one coupling interval. The discrepancy may also be in part due to the fact that the region in question is close to floatation, and thus the threshold for dry cells to become wet is highly sensitive to ζ, at least for the ROMS implementation. In both experiments, the ice–ocean grounded area discrepancy has a tendency to reduce over time.

The computational time spent in both the ice and ocean components was measured for simulation VE2_ER. The ice component is more expensive than the ocean component during the first coupling interval but is significantly cheaper thereafter. Total time spent in the ice component over the 46-year simulation is approximately one-third that spent in the ocean component. The computational time spent within the central coupling code (calling routines and regridding) was negligible compared to time spent in ice and ocean components. This is with a 10 d coupling interval. If fully synchronous coupling is approached (i.e. if the coupling interval approaches the ocean time step size), the ice component will become much more expensive and it is possible the central coupling code may become significant. We do not anticipate fully synchronous ice–ocean coupling to become practical in the near future, at least not if the ice component directly solves the Stokes equations without simplifying assumptions, as is the case in the current study. The fully synchronous coupling of and is achieved by using the “shallow shelf approximation” for the ice component and running both components on the same grid.