Controlling basal strength

When using option -stress_balance ssa+sia, the SIA+SSA hybrid stress balance, a model for basal resistance is required. This model for basal resistance is based, at least conceptually, on the hypothesis that the ice sheet is underlain by a layer of till [83]. The user can control the parts of this model:

  • the so-called sliding law, typically a power law, which relates the ice base (sliding) velocity to the basal shear stress, and which has a coefficient which is or has the units of a yield stress,
  • the model relating the effective pressure on the till layer to the yield stress of that layer, and
  • the model for relating the amount of water stored in the till to the effective pressure on the till.

This subsection explains the relevant options.

The primary example of -stress_balance ssa+sia usage is in section Getting started: a Greenland ice sheet example of this Manual, but the option is also used in sections MISMIP, MISMIP3d, and Example: A regional model of the Jakobshavn outlet glacier in Greenland.

In PISM the key coefficient in the sliding is always denoted as yield stress \(\tau_c\), which is tauc in PISM output files. This parameter represents the strength of the aggregate material at the base of an ice sheet, a poorly-observed mixture of pressurized liquid water, ice, granular till, and bedrock bumps. The yield stress concept also extends to the power law form, and thus most standard sliding laws can be chosen by user options (below). One reason that the yield stress is a useful parameter is that it can be compared, when looking at PISM output files, to the driving stress (taud_mag in PISM output files). Specifically, where tauc \(<\) taud_mag you are likely to see sliding if option -stress_balance ssa+sia is used.

A historical note on modeling basal sliding is in order. Sliding can be added directly to a SIA stress balance model by making the sliding velocity a local function of the basal value of the driving stress. Such an SIA sliding mechanism appears in ISMIP-HEINO [84] and in EISMINT II experiment H [19], among other places. This kind of sliding is not recommended, as it does not make sense to regard the driving stress as the local generator of flow if the bed is not holding all of that stress [22], [41]. Within PISM, for historical reasons, there is an implementation of SIA-based sliding only for verification test E; see section Verification. PISM does not support this SIA-based sliding mode in other contexts.

Choosing the sliding law

In PISM the sliding law can be chosen to be a purely-plastic (Coulomb) model, namely,

(8)\[|\boldsymbol{\tau}_b| \le \tau_c \quad \text{and} \quad \boldsymbol{\tau}_b = - \tau_c \frac{\mathbf{u}}{|\mathbf{u}|} \quad\text{if and only if}\quad |\mathbf{u}| > 0.\]

Equation (8) says that the (vector) basal shear stress \(\boldsymbol{\tau}_b\) is at most the yield stress \(\tau_c\), and that only when the shear stress reaches the yield value can there be sliding. The sliding law can, however, also be chosen to be the power law

(9)\[\boldsymbol{\tau}_b = - \tau_c \frac{\mathbf{u}}{u_{\text{threshold}}^q |\mathbf{u}|^{1-q}},\]

where \(u_{\text{threshold}}\) is a parameter with units of velocity (see below). Condition (8) is studied in [38] and [85] in particular, while power laws for sliding are common across the glaciological literature (e.g. see [71], [49]). Notice that the coefficient \(\tau_c\) in (9) has units of stress, regardless of the power \(q\).

In both of the above equations (8) and (9) we call \(\tau_c\) the yield stress. It corresponds to the variable tauc in PISM output files. We call the power law (9) a “pseudo-plastic” law with power \(q\) and threshold velocity \(u_{\text{threshold}}\). At the threshold velocity the basal shear stress \(\boldsymbol{\tau}_b\) has exact magnitude \(\tau_c\). In equation (9), \(q\) is the power controlled by -pseudo_plastic_q, and the threshold velocity \(u_{\text{threshold}}\) is controlled by -pseudo_plastic_uthreshold. The plastic model (8) is the \(q=0\) case of (9).

See Table 16 for options controlling the choice of sliding law. The purely plastic case is the default; just use -stress_balance ssa+sia to turn it on. (Or use -stress_balance ssa if a model with no vertical shear is desired.)

Warning

Options -pseudo_plastic_q and -pseudo_plastic_uthreshold have no effect if -pseudo_plastic is not set.

Table 16 Sliding law command-line options
Option Description
-pseudo_plastic Enables the pseudo-plastic power law model. If this is not set the sliding law is purely-plastic, so pseudo_plastic_q and pseudo_plastic_uthreshold are inactive.
-plastic_reg (m/a) Set the value of \(\epsilon\) regularization of the plastic law, in the formula \(\boldsymbol{\tau}_b = - \tau_c \mathbf{u}/\sqrt{|\mathbf{u}|^2 + \epsilon^2}\). The default is \(0.01\) m/a. This parameter is inactive if -pseudo_plastic is set.
-pseudo_plastic_q Set the exponent \(q\) in (9). The default is \(0.25\).
-pseudo_plastic_uthreshold (m/a) Set \(u_{\text{threshold}}\) in (9). The default is \(100\) m/a.

Equation (9) is a very flexible power law form. For example, the linear case is \(q=1\), in which case if \(\beta=\tau_c/u_{\text{threshold}}\) then the law is of the form

\[\boldsymbol{\tau}_b = - \beta \mathbf{u}\]

(The “\(\beta\)” coefficient is also called \(\beta^2\) in some sources (see [37], for example).) If you want such a linear sliding law, and you have a value \(\beta=\) beta in \(\text{Pa}\,\text{s}\,\text{m}^{-1}\), then use this option combination:

-pseudo_plastic \
-pseudo_plastic_q 1.0 \
-pseudo_plastic_uthreshold 3.1556926e7 \
-yield_stress constant -tauc beta

This sets \(u_{\text{threshold}}\) to 1 \(\text{m}\,\text{s}^{-1}\) but using units \(\text{m}\,\text{a}^{-1}\).

More generally, it is common in the literature to see power-law sliding relations in the form

\[\boldsymbol{\tau}_b = - C |\mathbf{u}|^{m-1} \mathbf{u},\]

where \(C\) is a constant, as for example in sections MISMIP and MISMIP3d. In that case, use this option combination:

-pseudo_plastic \
-pseudo_plastic_q m \
-pseudo_plastic_uthreshold 3.1556926e7 \
-yield_stress constant \
-tauc C

Determining the yield stress

Other than setting it to a constant, which only applies in some special cases, the above discussion does not determine the yield stress \(\tau_c\). As shown in Table 17, there are two schemes for determining \(\tau_c\) in a spatially-variable manner:

  • -yield_stress mohr_coulomb (the default) determines the yields stress by models of till material property (the till friction angle) and of the effective pressure on the saturated till, or
  • -yield_stress constant allows the yield stress to be supplied as time-independent data, read from the input file.

In normal modelling cases, variations in yield stress are part of the explanation of the locations of ice streams [38]. The default model -yield_stress mohr_coulomb determines these variations in time and space. The value of \(\tau_c\) is determined in part by a subglacial hydrology model, including the modeled till-pore water amount tillwat (section Subglacial hydrology), which then determines the effective pressure \(N_{till}\) (see below). The value of \(\tau_c\) is also determined in part by a material property field \(\phi=\) tillphi, the “till friction angle”. These quantities are related by the Mohr-Coulomb criterion [71]:

(10)\[\tau_c = c_{0} + (\tan\phi)\,N_{till}.\]

Here \(c_0\) is called the “till cohesion”, whose default value in PISM is zero (see [38], formula (2.4)) but which can be set by option -till_cohesion.

Option combination -yield_stress constant -tauc X can be used to fix the yield stress to have value \(\tau_c = X\) at all grounded locations and all times if desired. This is unlikely to be a good modelling choice for real ice sheets.

Table 17 Command-line options controlling how yield stress is determined
Option Description
-yield_stress mohr_coulomb The default. Use equation (10) to determine \(\tau_c\). Only effective if -stress_balance ssa or -stress_balance ssa+sia is also set.
-till_cohesion Set the value of the till cohesion (\(c_{0}\)) in the plastic till model. The value is a pressure, given in Pa.
-tauc_slippery_grounding_lines If set, reduces the basal yield stress at grounded-below-sea-level grid points one cell away from floating ice or ocean. Specifically, it replaces the normally-computed \(\tau_c\) from the Mohr-Coulomb relation, which uses the effective pressure from the modeled amount of water in the till, by the minimum value of \(\tau_c\) from Mohr-Coulomb, i.e. using the effective pressure corresponding to the maximum amount of till-stored water. Does not alter the reported amount of till water, nor does this mechanism affect water conservation.
-plastic_phi (degrees) Use a constant till friction angle. The default is \(30^{\circ}\).
-topg_to_phi (list of 4 numbers) Compute \(\phi\) using equation (11).
-yield_stress constant Keep the current values of the till yield stress \(\tau_c\). That is, do not update them by the default model using the stored basal melt water. Only effective if -stress_balance ssa or -stress_balance ssa+sia is also set.
-tauc Directly set the till yield stress \(\tau_c\), in units Pa, at all grounded locations and all times. Only effective if used with -yield_stress constant, because otherwise \(\tau_c\) is updated dynamically.

We find that an effective, though heuristic, way to determine \(\phi=\) tillphi in (10) is to make it a function of bed elevation [59], [3], [30]. This heuristic is motivated by hypothesis that basal material with a marine history should be weak [4]. PISM has a mechanism setting \(\phi =\) tillphi to be a piecewise-linear function of bed elevation. The option is

-topg_to_phi phimin,phimax,bmin,bmax
\[\begin{split}\newcommand{\Diff}[2]{ \frac{\mathrm{d}#1}{\mathrm{d}#2} } \newcommand{\diff}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\var}[2]{ {#1}_{\text{#2}} } \newcommand{\h}[1]{ \var{h}{#1} } \newcommand{\T}[1]{ \var{T}{#1} } \newcommand{\m}[1]{ \var{m}{#1} } \newcommand{\ms}[1]{ \var{m^{*}}{#1} } \newcommand{\psw}{p_{\text{ocean}}} \newcommand{\pice}{p_{\text{ice}}} \newcommand{\pmelange}{p_{\text{melange}}} \newcommand{\n}{\mathbf{n}} \newcommand{\nx}{\n_{x}} \newcommand{\ny}{\n_{y}} \newcommand{\phimin}{\phi_{\mathrm{min}}} \newcommand{\phimax}{\phi_{\mathrm{max}}} \newcommand{\bmin}{b_{\mathrm{min}}} \newcommand{\bmax}{b_{\mathrm{max}}} \newcommand{\bq}{\mathbf{q}} \newcommand{\Up}[2]{\operatorname{Up}\left(#1\big|#2\right)} \newcommand{\uppair}[2]{\left\{\begin{matrix} #1 \\ #2 \end{matrix}\right\}}\end{split}\]

Thus the user supplies 4 parameters: \(\phimin\), \(\phimax\), \(\bmin\), \(\bmax\), where \(b\) stands for the bed elevation. To explain these, we define \(M = (\phimax - \phimin) / (\bmax - \bmin)\). Then

(11)\[\begin{split}\phi(x,y) = \begin{cases} \phimin, & b(x,y) \le \bmin, \\ \phimin + (b(x,y) - \bmin) \,M, & \bmin < b(x,y) < \bmax, \\ \phimax, & \bmax \le b(x,y). \end{cases}\end{split}\]

It is worth noting that an earth deformation model (see section Earth deformation models) changes \(b(x,y)=\mathrm{topg}\) used in (11), so that a sequence of runs such as

pismr -i foo.nc -bed_def lc -stress_balance ssa+sia -topg_to_phi 10,30,-50,0 ... -o bar.nc
pismr -i bar.nc -bed_def lc -stress_balance ssa+sia -topg_to_phi 10,30,-50,0 ... -o baz.nc

will use different tillphi fields in the first and second runs. PISM will print a warning during initialization of the second run:

* Initializing the default basal yield stress model...
  option -topg_to_phi seen; creating tillphi map from bed elev ...
PISM WARNING: -topg_to_phi computation will override the 'tillphi' field
              present in the input file 'bar.nc'!

Omitting the -topg_to_phi option in the second run would make PISM continue with the same tillphi field which was set in the first run.

Determining the effective pressure

When using the default option -yield_stress mohr_coulomb, the effective pressure on the till \(N_{till}\) is determined by the modeled amount of water in the till. Lower effective pressure means that more of the weight of the ice is carried by the pressurized water in the till and thus the ice can slide more easily. That is, equation (10) sets the value of \(\tau_c\) proportionately to \(N_{till}\). The amount of water in the till is, however, a nontrivial output of the hydrology (section Subglacial hydrology) and conservation-of-energy (section Modeling conservation of energy) submodels in PISM.

Following [86], based on laboratory experiments with till extracted from an ice stream in Antarctica, [87] propose the following parameterization which is used in PISM. It is based on the ratio \(s=W_{till}/W_{till}^{max}\) where \(W_{till}=\) tillwat is the effective thickness of water in the till and \(W_{till}^{max}=\) hydrology.tillwat_max is the maximum amount of water in the till (see section Subglacial hydrology):

(12)\[N_{till} = \min\left\{P_o, N_0 \left(\frac{\delta P_o}{N_0}\right)^s \, 10^{(e_0/C_c) \left(1 - s\right).}\right\}\]

Here \(P_o\) is the ice overburden pressure, which is determined entirely by the ice thickness and density, and the remaining parameters are set by options in Table 18. While there is experimental support for the default values of \(C_c\), \(e_0\), and \(N_0\), the value of \(\delta=\) basal_yield_stress.mohr_coulomb.till_effective_fraction_overburden should be regarded as uncertain, important, and subject to parameter studies to assess its effect.

Table 18 Command-line options controlling how till effective pressure \(N_{till}\) in equation (10) is determined
Option Description
-till_reference_void_ratio \(= e_0\) in (12), dimensionless, with default value 0.69 [86]
-till_compressibility_coefficient \(= C_c\) in (12), dimensionless, with default value 0.12 [86]
-till_effective_fraction_overburden \(= \delta\) in (12), dimensionless, with default value 0.02 [87]
-till_reference_effective_pressure \(= N_0\) in (12), in Pa, with default value 1000.0 [86]

Previous Up Next