Weertman-style sliding law¶

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.

The “Weertman-type sliding law” ([55], equations 5.35 and 5.91) has the form

\[\begin{split}\mathbf{u}_s = \begin{cases} \mathbf{0}, & T_b < T_m, \\ -C_b(\rho g H)^{p-q}|\nabla h|^{p-1}\nabla h, & T_b = T_m, \end{cases}\end{split}\]

\(T_b\) is the ice temperature, and \(T_m\) is the pressure-melting temperature. The constant \(C_b\) and exponents \(p\) and \(q\) are tuning parameters.

The particular form implemented in PISM comes from equation 5 in [71]:

(5)¶\[\mathbf{u}_s = -\frac{2 A_s \beta_c (\rho g H)^{n}}{N - P} |\nabla h|^{n-1} \nabla h.\]

Table 11 Notation used in (5)¶
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 (5) 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 to set \(k\) and \(A_s\), respectively. Default values come from [71].

Parameters¶

Prefix: stress_balance.weertman_sliding.

A (1.8e-16 Pa-3 year-1 m-2) Sliding parameter in the Weertman-style sliding parameterization [71]
k (0.2) The ratio of the basal water pressure and the ice overburden pressure in the Weertman-style sliding parameterization.