Basal Friction

Introduction

All friction laws in ISSM are implemented as:

$Equation 1$

where $Equation 4$ is the effective pressure, $Equation 3$ and $Equation 2$ are the basal stress and sliding velocities respectively. The friction laws described below are describing the norm of the basal stress for simplicity but all implementations are such that the oppose motion (i.e. the direction of the basal stress is the opposite of $Equation 5$ ).

Most friction laws use a switch to define how the effective pressure, $Equation 6$ is calculated: md.friction.coupling:

0: $Equation 7$ uniform sheet (negative water pressure ok, default)
1: $Equation 9$ , so that $Equation 8$ is equal to the overburden pressure
2: $Equation 11$ . Same as 0, but $Equation 10$
3: Use effective pressure prescrived in md.friction.effective_pressure
4: Use effective pressure dynamically calculated by the hydrology model (i.e., fully coupled)

Budd Friction law (friction)

The default friction law is defined as [Paterson1994] (p 151):

$Equation 12$

where:

$Equation 13$ is the basal velocity magnitude
$Equation 14$ is the basal stress magnitude
$Equation 15$ is the effective pressure
$Equation 17$ and $Equation 16$ are friction law exponents

In ISSM, this friction law is implemented in terms of basal stress, following [Budd1979]:

$Equation 18$

where:

$Equation 19$ friction coefficient
$Equation 21$ and $Equation 20$ are friction law exponents:

$Equation 22$

This friction law can be selected as follows:

>> md.friction = friction();

The following fields need to be specified:

md.friction.coefficient: friction coefficient
md.friction.p: p exponent
md.friction.q: q exponent

Weertman Friction law (weertmanfriction)

The Weertman friction [Weertman1957] law reads:

$Equation 23$

$Equation 24$ is a friction coefficient (variable in space)
$Equation 25$ is a friction law exponent

In ISSM, this friction law is implemented in terms of basal stress:

$Equation 26$

This friction law can be selected as follows:

>> md.friction = frictionweertman();

One can display the following fields by running:

>> md.friction

md.friction.C: friction coefficient
md.friction.m: m exponent

Coulomb-limited sliding 1 (frictioncoulomb)

$Equation 27$

Regularized Coulomb-limited sliding 1 (frictionregcoulomb)

Sliding law from [Joughin2019]:

$Equation 28$

Coulomb-limited sliding 2 (frictioncoulomb2)

Coulomb-limited sliding law used in MISMIP+ [Cornford2020]:

$Equation 29$

where $Equation 30$ . Note that this friction law is exactly the same as frictionschoof described below, with $Equation 31$ .

Regularized Coulomb-limited sliding 2 (frictionregcoulomb2)

Sliding law from [Helanow2021]:

$Equation 32$

Friction Tsai (frictiontsai)

from [Tsai2015]:

$Equation 33$

Friction Schoof (frictionschoof)

from [Schoof2005,Gagliardini2007] (note that we use $Equation 34$ to make sure it is a positive number):

$Equation 35$

Friction PISM (frictionpism)

Under construction

Thin water layer friction law (frictionwaterlayer)

The thin water layer friction law is similar to the default friction law except that the effective pressure includes a specified layer of water at the bed:

$Equation 36$