Ice rheology¶
Contents
The “rheology” of a viscous fluid refers to the relation between the applied stress and the resulting deformation, the strain rate. The models of ice rheology available in PISM are all isotropic [39]. A rheology in this class is described by a “flow law”, which is, in the most general case in PISM, a function \(F(\sigma,T,\omega,P,d)\) in the “constitutive relation” form
Here \(D_{ij}\) is the strain rate tensor, \(\sigma_{ij}'\) is the stress deviator tensor, \(T\) is the ice temperature, \(\omega\) is the liquid water fraction, \(P\) is the pressure, \(d\) is the grain size, and \(\sigma^2 = \frac{1}{2} \\sigma_{ij}'\_F = \frac{1}{2} \sigma_{ij}' \sigma_{ij}'\) defines the second invariant \(\sigma\) of the stress deviator tensor.
Form (4) of the flow law is used in the SIA, but the “viscosity” form of a flow law, found by inverting the constitutive relation (4), is needed for ice shelf and ice stream (SSA) flow [22]:
Here \(\nu(D,T,\omega,P,d)\) is the “effective viscosity” and \(D^2 = \frac{1}{2} D_{ij} D_{ij}\).
Most of the flow laws in PISM are of GlenNye singlepower type. For example,
is the common temperaturedependent Glen law [66], [23] (which has no dependence on liquid water fraction, pressure, or grain size). If the ice softness \(A(T)=A_0\) is constant then the law is isothermal, whereas if there is dependence on temperature then \(A(T)\) is usually a generalization of “Arrhenius” form
The more elaborate GoldsbyKohlstedt law [67] is a function \(F(\sigma,T,P,d)\), but in this case the function \(F\) cannot be factored into a product of a function of \(T,P,d\) and a single power of \(\sigma\), as in form (6).
There is only one choice for the flow law which takes full advantage of the enthalpy mode of PISM, which is the thermodynamical modeling (i.e. conservation of energy) default. Namely the GlenPatersonBuddLliboutryDuval flow law [28], [68], [66], which is a function \(F(\sigma,T,\omega,P)\). This law is the only one in the literature where the ice softness depends on both the temperature and the liquid water fraction, so it parameterizes the (observed) softening of pressuremeltingtemperature ice as its liquid fraction increases. One can use this default polythermal law or one may choose among a number of “cold ice” laws listed in Table 13 which do not use the liquid water fraction.
All flow law parameters can be changed using configuration parameters; see section
PISM’s configuration parameters and how to change them and the implementation of flow laws in the Source Code Browser. Note that different flow laws have different numbers of parameters, but
all have at least two parameters (e.g. \(A_0\) and \(n\) in isothermal_glen
). One can
create a new, and reasonably arbitrarily, scalar function \(F\) by modifying source code;
see source files in src/base/rheology/
.
Choosing the flow laws for SIA and SSA stress balances¶
Commandline options sia_flow_law
and ssa_flow_law
choose which flow law
is used by the SIA and SSA stress balances, respectively. Allowed arguments are listed in
Table 13 below. Viscosity form (5) is not known for the
GoldsbyKohlstedt law [67], so option “ssa_flow_law gk
” is an
error.
Name  Comments and References 

gpbld 
GlenPatersonBuddLliboutryDuval law [68], the enthalpybased default in PISM [28]. Extends the PatersonBudd law (below) to positive liquid water fraction. If \(A_{c}(T)\) is from PatersonBudd then this law returns
where \(\omega\) is the liquid water fraction, \(C\) is a configuration parameter

pb 
PatersonBudd law, the coldmode default. Fixed Glen exponent \(n=3\). Has a split “Arrhenius” term \(A(T) = A \exp(Q/RT^*)\) where
if \(T^* < 263\) K and
if \(T^* > 263\) K. Here \(T^*\) is pressureadjusted temperature [66]. 
arr 
Cold part of PatersonBudd. Regardless of temperature, the \(A\) and \(Q\) values for
\(T^*<263\) K in the PatersonBudd law apply. This is the flow law used in the
thermomechanicallycoupled exact solutions run by pismv test F and
pismv test G [23], [69]. 
arrwarm 
Warm part of PatersonBudd. Regardless of temperature, the \(A\) and \(Q\) values for \(T^*>263\) K in PatersonBudd apply. 
hooke 
Hooke law with

isothermal_glen 
The isothermal Glen flow law. Here \(F(\sigma) = A_0 \sigma^{n1}\) with inverse \(\nu(D) = \frac{1}{2} B_0 D^{(1n)/(2n)}\) where \(A_0\) is the ice softness and \(B_0=A_0^{1/n}\) is the ice hardness. 
gk 
This law has a combination of exponents from \(n=1.8\) to \(n=4\)
[67]. It can only be used by the SIA stress balance. Because it has
more than one power, option sia_n has no effect, though sia_e works as
expected. This law does not use the liquid water fraction, but only the
temperature. 
Choose enhancement factor and exponent¶
An enhancement factor can be added to any flow law through a runtime option. Singlepower laws also permit control of the flow law exponent through a runtime option.
Options sia_e
and ssa_e
set flow enhancement factors for the SIA and SSA
respectively. Option sia_e
sets “\(e\)” in \(D_{ij} = e\, F(\sigma,T,\omega,P,d)\,
\sigma_{ij}',\) in equation (4). Option ssa_e
sets “\(e\)” in the
viscosity form so that \(\sigma_{ij}' = e^{1/n}\, 2\, \nu(D,T,\omega,P,d)\, D_{ij}.\)
Options sia_n
and ssa_n
set the exponent when a singlepower flow law is
used (see Table 13). Simply changing to a different value from the default
\(n=3\) is not recommended without a corresponding change to the enhancement factor,
however. This is because the coefficient and the power are nontrivially linked when a
power law is fit to experimental data [71], [66].
Here is a possible approach to adjusting both the enhancement factor and the exponent. Suppose \(\sigma_0\) is preferred as a scale (reference) for the driving stress that appears in both SIA and SSA models. Typically this is on the order of one bar or \(10^5\) Pa. Suppose one wants the same amount of deformation \(D_0\) at this reference driving stress as one changes from the old exponent \(n_{old}\) to the new exponent \(n_{new}\). That is, suppose one wants both
to be true with a new enhancement factor \(E_{new}\). Eliminating \(D_0\) and solving for the new enhancement factor gives
It follows, for example, that if one has a run with values
sia_e 3.0 sia_n 3.0
then a new run with exponent \(n=6.0\) and the same deformation at the reference driving stress of \(10^5\) Pa will use
sia_e 3.0e15 sia_n 6.0
because \(E_{new} = 3.0 \sigma_0^{36} = 3.0 \times (10^5)^{3}\) from equation (7).
A corresponding formula applies to ssa_e
if the ssa_n
value changes.
Option  Configuration parameter  Comments 

sia_e (1.0) 
stress_balance.sia.enhancement_factor 
Note (see the supplement of [59]) used \(3.0\) for Greenland ice sheet simulations while [3] used \(4.5\) for simulations of the Antarctic ice sheet with PISMPIK. 
sia_n (3.0) 
stress_balance.sia.Glen_exponent 
See text and eqn (7) to also set sia_e if sia_n changes. 
ssa_e (1.0) 
stress_balance.ssa.enhancement_factor 
Note [3] used \(0.512\) for simulations of the Antarctic ice sheet with PISMPIK. 
ssa_n (3.0) 
stress_balance.ssa.Glen_exponent 
See text and eqn (7) to also set ssa_e if ssa_n
changes. 
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