Ice rheology

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

(4)\[D_{ij} = F(\sigma,T,\omega,P,d)\, \sigma_{ij}'.\]

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]:

(5)\[\sigma_{ij}' = 2 \nu(D,T,\omega,P,d)\,D_{ij}\]

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 Glen-Nye single-power type. For example,

(6)\[F(\sigma,T) = A(T) \sigma^{n-1}\]

is the common temperature-dependent 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

\[A(T) = A \exp(-Q/(R T)).\]

The more elaborate Goldsby-Kohlstedt 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 Glen-Paterson-Budd-Lliboutry-Duval 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 pressure-melting-temperature 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

Command-line 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 Goldsby-Kohlstedt law [67], so option “-ssa_flow_law gk” is an error.

Table 13 Single-power flow laws. Choose the ice rheology using -sia_flow_law and -ssa_flow_law and one of the names in this table. Flow law choices other than gpbld do not use the liquid water fraction \(\omega\) but only the temperature \(T\).
Name Comments and References
gpbld

Glen-Paterson-Budd-Lliboutry-Duval law [68], the enthalpy-based default in PISM [28]. Extends the Paterson-Budd law (below) to positive liquid water fraction. If \(A_{c}(T)\) is from Paterson-Budd then this law returns

\(A(T,\omega) = A_{c}(T) (1 + C \omega),\)

where \(\omega\) is the liquid water fraction, \(C\) is a configuration parameter flow_law.gpbld.water_frac_coeff [default \(C=181.25\)], and \(\omega\) is capped at level flow_law.gpbld.water_frac_observed_limit.

pb

Paterson-Budd law, the cold-mode default. Fixed Glen exponent \(n=3\). Has a split “Arrhenius” term \(A(T) = A \exp(-Q/RT^*)\) where

\(A = 3.615 \times 10^{-13}\, \text{s}^{-1}\, \text{Pa}^{-3},\) \(Q = 6.0 \times 10^4\, \text{J}\, \text{mol}^{-1}\)

if \(T^* < 263\) K and

\(A = 1.733 \times 10^{3}\, \text{s}^{-1}\, \text{Pa}^{-3},\) \(Q = 13.9 \times 10^4\, \text{J}\, \text{mol}^{-1}\)

if \(T^* > 263\) K.

Here \(T^*\) is pressure-adjusted temperature [66].

arr Cold part of Paterson-Budd. Regardless of temperature, the \(A\) and \(Q\) values for \(T^*<263\) K in the Paterson-Budd law apply. This is the flow law used in the thermomechanically-coupled exact solutions run by pismv -test F and pismv -test G [23], [69].
arrwarm Warm part of Paterson-Budd. Regardless of temperature, the \(A\) and \(Q\) values for \(T^*>263\) K in Paterson-Budd apply.
hooke

Hooke law with

\(A(T) = A \exp(-Q/(RT^*) + 3C (T_r - T^*)^\kappa).\)

Fixed Glen exponent \(n=3\) and constants as in [70], [40].

isothermal_glen The isothermal Glen flow law. Here \(F(\sigma) = A_0 \sigma^{n-1}\) with inverse \(\nu(D) = \frac{1}{2} B_0 D^{(1-n)/(2n)}\) 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. Single-power 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 single-power 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 non-trivially 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

\[D_0 = E_{old}\, A\, \sigma_0^{n_{old}} \qquad \text{and} \qquad D_0 = E_{new}\, A\, \sigma_0^{n_{new}}\]

to be true with a new enhancement factor \(E_{new}\). Eliminating \(D_0\) and solving for the new enhancement factor gives

(7)\[E_{new} = E_{old}\, \sigma_0^{n_{old} - n_{new}}.\]

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 3.0e-15 -sia_n 6.0

because \(E_{new} = 3.0 \sigma_0^{3-6} = 3.0 \times (10^5)^{-3}\) from equation (7).

A corresponding formula applies to -ssa_e if the -ssa_n value changes.

Table 14 For all flow laws, an enhancement factor can be added by a runtime option. For the single-power flow laws in Table 13, the (Glen) exponent can be controlled by a runtime option.
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 PISM-PIK.
-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 PISM-PIK.
-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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