Choosing the stress balance

The basic stress balance used for all grounded ice in PISM is the non-sliding, thermomechanically-coupled SIA [23]. For the vast majority of most ice sheets, as measured by area or volume, this is an appropriate model, which is an \(O(\epsilon^2)\) approximation to the Stokes model if \(\epsilon\) is the depth-to-length ratio of the ice sheet [24].

The shallow shelf approximation (SSA) stress balance applies to floating ice. See the Ross ice shelf example in section An SSA flow model for the Ross Ice Shelf in Antarctica for an example in which the SSA is only applied to floating ice.

In PISM the SSA is also used to describe the sliding of grounded ice and the formation of ice streams [22]. Specifically for the SSA with “plastic” (Coulomb friction) basal resistance, the locations of ice streams are determined as part of a free boundary problem of Schoof [38], a model for emergent ice streams within a ice sheet and ice shelf system. This model explains ice streams through a combination of plastic till failure and SSA stress balance.

This SSA description of ice streams is the preferred “sliding law” for the SIA [22], [30]. The SSA should be combined with the SIA, in this way, in preference to classical SIA sliding laws which make the sliding velocity of ice a local function of the basal value of the driving stress. The resulting combination of SIA and SSA is a “hybrid” approximation of the Stokes model [30]. Option -stress_balance ssa+sia turns on this “hybrid” model. In this use of the SSA as a sliding law, floating ice is also subject to the SSA.

Table 9 describes the basic choice of stress balance.

Table 9 The basic choice of stress balance
Option Description
-stress_balance none Turn off ice flow completely.
-stress_balance sia (default) Grounded ice flows by the non-sliding SIA. Floating ice does not flow, so this model is not recommended for marine ice sheets.
-stress_balance ssa Use the SSA model exclusively. Horizontal ice velocity is constant throughout ice columns.
-stress_balance prescribed_sliding Use the constant-in-time prescribed sliding velocity field read from a file set using -prescribed_sliding_file, variables ubar and vbar. Horizontal ice velocity is constant throughout ice columns.
-stress_balance ssa+sia The recommended sliding law, which gives the SIA+SSA hybrid stress balance. Combines SSA-computed velocity, using pseudo-plastic till, with SIA-computed velocity according to the combination in [30]; similar to [22]. Floating ice uses SSA only.
-stress_balance prescribed_sliding+sia Use the constant-in-time prescribed sliding velocity in combination with the non-sliding SIA.

Controlling the SSA stress balance model

If the SSA stress balance is used, a choice of two solvers is available, namely -ssa_method fd (default) or -ssa_method fem. See Table 10, which describes additional controls on the numerical solution of the stress balance equations. If option -ssa_method fd is chosen then several more controls on numerics are available; see Table 11. If the ice sheet being modeled has any floating ice then the user is advised to read section PIK options for marine ice sheets on modeling marine ice sheets.

When using SSA as a “sliding law” one also needs to model the yield stress, or a pseudo-yield-stress in the case of power law sliding (section Controlling basal strength).

The basal yield stress is normally a function of the amount of water stored in the till and a (generally) spatially-varying till strength. The amount of stored basal water is modeled by the subglacial hydrology model (section Subglacial hydrology) based on the basal melt rate which is, primarily, thermodynamically-determined (see Modeling conservation of energy).

Table 10 Choice of, and controls on, the numerical SSA stress balance.
Option Description
-ssa_method [ fd | fem ] Both finite difference (fd; the default) and finite element (fem) versions of the SSA numerical solver are implemented in PISM. The fd solver is the only one which allows PIK options (section PIK options for marine ice sheets). fd uses Picard iteration [22], while fem uses a Newton method. The fem solver has surface velocity inversion capability [60].
-ssa_eps (\(10^{13}\)) The numerical schemes for the SSA compute an effective viscosity \(\nu\) which depends on strain rates and ice hardness (thus temperature). The minimum value of the effective viscosity times the thickness (i.e. \(\nu H\)) largely determines the difficulty of solving the numerical SSA. This constant is added to keep \(\nu H\) bounded away from zero: \(\nu H \to \nu H + \epsilon_{\text{SSA}}\), where \(\epsilon_{\text{SSA}}\) is set using this option. Units of ssa_eps are \(\text{Pa}\,\text{m}\,\text{s}\). Set to zero to turn off this lower bound.
-ssa_view_nuh View the product \(\nu H\) for your simulation as a runtime viewer (section Run-time diagnostic viewers). In a typical Greenland run we see a wide range of values for \(\nu H\) from \(\sim 10^{14}\) to \(\sim 10^{20}\) \(\text{Pa}\,\text{m}\,\text{s}\).
Table 11 Controls on the numerical iteration of the -ssa_method fd solver
Option Description
-ssa_maxi (300) Set the maximum allowed number of Picard (nonlinear) iterations in solving the shallow shelf approximation.
-ssa_rtol (\(10^{-4}\))

The Picard iteration computes a vertically-averaged effective viscosity which is used to solve the equations for horizontal velocity. Then the new velocities are used to recompute an effective viscosity, and so on. This option sets the relative change tolerance for the effective viscosity. The Picard iteration stops when successive values \(\nu^{(k)}\) of the vertically-averaged effective viscosity satisfy

\[\|(\nu^{(k)} - \nu^{(k-1)}) H\|_1 \le Z \|\nu^{(k)} H\|_1\]

where \(Z=\) ssa_rtol.

-ssafd_ksp_rtol (\(10^{-5}\)) Set the relative change tolerance for the iteration inside the Krylov linear solver used at each Picard iteration.
-ssafd_max_speed (\(50 km/yr\)) Limits computed SSA velocities: ice speed is capped at this limit after each Picard iteration of the SSAFD solver. This may allow PISM to take longer time steps by ignoring high velocities at a few troublesome locations.

Controlling the SIA stress balance model

The section Computing ice age describes coupling SIA stress balance to the age of the ice.

The section Surface gradient method covers available methods of computing the surface gradient.

Note

The explicit time stepping of the mass continuity equation in the case of the SIA flow comes with a severe restriction on time step length:

\[\Delta t \le \frac{2 R}{D\left( 1/\Delta x^2 + 1/\Delta y^2 \right)}\]

Here \(D\) is the maximum diffusivity of the SIA flow and \(R\) is time_stepping.adaptive_ratio, a tuning parameter that further reduces the maximum allowed time step length.

The maximum diffusivity \(D\) may be achieved at an isolated grid point near the ice margin. In this case it might make sense to limit the diffusivity of the SIA flow, sacrificing accuracy at a few grid points to increase time step length and reduce the computational cost. Set stress_balance.sia.limit_diffusivity to enable this mechanism.

When stress_balance.sia.limit_diffusivity is false PISM stops as soon as the SIA diffusivity at any grid point exceeds stress_balance.sia.max_diffusivity. We do this to make it easier to detect problematic model configurations: in many cases it does not make sense to continue a simulation if \(D\) is very large.

Weertman-style sliding

Warning

This kind of sliding is, in general, a bad idea. We implement it to simplify comparisons of the “hybrid” model mentioned above to older studies using this parameterization.

PISM implements equation 5 from [72]:

(3)\[\mathbf{u}_s = \frac{2 A_s \beta_c (\rho g H)^{n}}{N - P} |\nabla h|^{n-1} \nabla h.\]
Table 12 Notation used in (3)
Variable Meaning
\(H\) ice thickness
\(h\) ice surface elevation
\(n\) flow law exponent
\(g\) acceleration due to gravity
\(\rho\) ice density
\(N\) ice overburden pressure, \(N = \rho g H\)
\(P\) basal water pressure
\(A_s\) sliding parameter
\(\beta_c\) “constriction parameter” capturing the effect of valley walls on the flow; set to \(1\) in this implementation

We assume that the basal water pressure is a given constant fraction of the overburden pressure: \(P = k N\). This simplifies (3) to

\[\mathbf{u}_s = \frac{2 A_s}{1 - k} ( \rho g H\, |\nabla h| )^{n-1} \nabla h.\]

This parameterization is used for grounded ice where the base of the ice is temperate.

To enable, use -stress_balance weertman_sliding (this results in constant-in-depth ice velocity) or -stress_balance weertman_sliding+sia to use this parameterization as a sliding law with the deformational flow modeled using the SIA model.

Use configuration parameters stress_balance.weertman_sliding.k and stress_balance.weertman_sliding.A tot set \(k\) and \(A_s\), respectively. Default values come from [72].


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