Shallow shelf approximation (SSA)¶
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 9, 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 10. 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 pseudoyieldstress 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) spatiallyvarying 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, thermodynamicallydetermined (see Modeling conservation of energy).
Option 
Description 


Both finite difference ( 

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 

View the product \(\nu H\) for your simulation as a runtime viewer (section Runtime diagnostic viewers). In a typical Greenland run we see a wide range of values for \(\nu H\) from \(\sim 10^{14}\) to \(\sim 10^{20}\) \(\text{Pa}\,\text{m}\,\text{s}\). 
Option 
Description 


Set the maximum allowed number of Picard (nonlinear) iterations in solving the shallow shelf approximation. 

The Picard iteration computes a verticallyaveraged 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 verticallyaveraged effective viscosity satisfy
\[\(\nu^{(k)}  \nu^{(k1)}) H\_1 \le Z \\nu^{(k)} H\_1\]
where \(Z=\) 

Set the relative change tolerance for the iteration inside the Krylov linear solver used at each Picard iteration. 

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. 
Parameters¶
Prefix: stress_balance.ssa.
Glen_exponent
(3) Glen exponent in ice flow law for SSAcompute_surface_gradient_inward
(no) If yes then use inward firstorder differencing in computing surface gradient in the SSA objects.dirichlet_bc
(no) apply SSA velocity Dirichlet boundary conditionenhancement_factor
(1) Flow enhancement factor for SSAepsilon
(1e+13 Pascal second meter) Initial amount of regularization in computation of product of effective viscosity and thickness (\(\nu H\)). This default value for \(\nu H\) comes e.g. from a hardness for the Ross ice shelf (\(\bar B\)) = 1.9e8 Pa \(s^{1/3}\) [62] and a typical strain rate of 0.001 1/year for the Ross ice shelf, giving \(\nu = (\bar B) / (2 \cdot 0.001^{2/3})\) = 9.49e+14 Pa s ~ 30 MPa year, the value in [63], but with a tiny thickness \(H\) of about 1 cm.fd
.brutal_sliding
(false) Enhance sliding speed brutally.fd
.brutal_sliding_scale
(1) Brutal SSA Sliding Scalefd
.flow_line_mode
(false) Set \(v\) (the \(y\) component of the ice velocity) to zero when assembling the systemfd
.lateral_drag
.enabled
(false) Set viscosity at ice shelf margin next to ice free bedrock as friction parameterizationfd
.lateral_drag
.viscosity
(5e+15 Pascal second) Staggered viscosity used as side friction parameterization.fd
.max_iterations
(300) Maximum number of Picard iterations for the ice viscosity computation, in theSSAFD
objectfd
.max_speed
(300000 km s1) Upper bound for the ice speed computed by theSSAFD
solver.fd
.nuH_iter_failure_underrelaxation
(0.8) In event of “Effective viscosity not converged” failure, use outer iteration rule nuH < nuH + f (nuH  nuH_old), where f is this parameter.fd
.relative_convergence
(0.0001) Relative change tolerance for the effective viscosity in theSSAFD
objectfd
.replace_zero_diagonal_entries
(yes) Replace zero diagonal entries in theSSAFD
matrix with :config:’basal_resistance.beta_ice_free_bedrock’ to avoid solver failures.flow_law
(gpbld
) The SSA flow law.method
(fd
) Algorithm for computing the SSA solution.read_initial_guess
(yes) Read the initial guess from the input file when restarting.strength_extension
.constant_nu
(9.48681e+14 Pascal second) The SSA is made elliptic by use of a constant value for the product of viscosity (nu) and thickness (H). This value for nu comes from hardness (bar B)=1.9e8 \(Pa s^{1/3}\) [62] and a typical strain rate of 0.001 year1: \(\nu = (\bar B) / (2 \cdot 0.001^{2/3})\). Compare the value of 9.45e14 Pa s = 30 MPa year in [63].strength_extension
.min_thickness
(50 meters) The SSA is made elliptic by use of a constant value for the product of viscosity (nu) and thickness (H). At ice thicknesses below this value the product nu*H switches from the normal vertical integral to a constant value. The geometry itself is not affected by this value.
Technical remarks¶
Ice stress balance models ignoring inertia are “diagnostic” models that do not have a “state” evolving through time: the ice velocity is fully determined by ice geometry, basal boundary conditions, and the ice viscosity.
In addition to this, shallow stress balance models (other than the SIA) correspond to nonlinear systems of equations, which means that computing an estimate of the ice velocity is an iterative process starting with a particular initial guess.
The quality of this guess may affect the number of iterations needed and even whether the solver succeeds at all. It also has an influence on the computed solution: if an equation has a unique solution, estimates produced using different initial guesses should be close, but they need not be identical. If an equation has multiple solutions, even a “small” change in the initial guess may give a completely different result.
In the context of a evolutionary model we can usually assume that the change in the state
of the model from one step to the next is “small” and we use (\(u, v\)) estimates from one
time step as the initial guess for the next. To ensure that stopping and restarting a
simulation does not affect results we save these to the output file as variables
u_ssa
and v_ssa
and read them in when restarting a stopped simulation.
One could say that the continuum SSA model does not have a state, but its implementation
does. Set stress_balance
.ssa
.read_initial_guess
to “false” to ignore it during
initialization and use the zero initial guess instead.
Previous  Up  Next 