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{array}{r} u_{s} = {\begin{cases} 0, & T_{b} < T_{m}, \\ - C_{b} (ρ g H)^{p - q} | \nabla h |^{p - 1} \nabla h, & T_{b} = T_{m}, \end{cases} \end{array}

$T_{b}$ 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]:

(6)¶

u_{s} = - \frac{2 A_{s} β_{c} (ρ g H)^{n}}{N - P} | \nabla h |^{n - 1} \nabla h .

Table 12 Notation used in (6)¶
Variable	Meaning
$H$	ice thickness
$h$	ice surface elevation
$n$	flow law exponent
$g$	acceleration due to gravity
$ρ$	ice density
$N$	ice overburden pressure, $N = ρ g H$
$P$	basal water pressure
$A_{s}$	sliding parameter
$β_{c}$	“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 (6) to

u_{s} = - \frac{2 A_{s}}{1 - k} (ρ 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.