tstress-balance-models.rst - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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tstress-balance-models.rst (7131B)
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1 .. _sec-stress-balance-models:
2
3 Two stress balance models: SIA and SSA
4 --------------------------------------
5
6 At each time-step of a typical PISM run, the geometry, temperature, and basal strength of
7 the ice sheet are included into stress (momentum) balance equations to determine the
8 velocity of the flowing ice. The "full" stress balance equations for flowing ice form a
9 non-Newtonian Stokes model :cite:`Fowler`. PISM does not attempt to solve the Stokes equations
10 themselves, however. Instead it can numerically solve, in parallel, two different shallow
11 approximations which are well-suited to ice sheet and ice shelf systems:
12
13 - the non-sliding shallow ice approximation (SIA) :cite:`Hutter`, also called the "lubrication
14 approximation" :cite:`Fowler`, which describes ice as flowing by shear in planes parallel to
15 the geoid, with a strong connection of the ice base to the bedrock, and
16 - the shallow shelf approximation (SSA) :cite:`WeisGreveHutter`, which describes a
17 membrane-type flow of floating ice :cite:`Morland`, or of grounded ice which is sliding over
18 a weak base :cite:`MacAyeal`, :cite:`SchoofStream`.
19
20 The SIA equations are easier to solve numerically than the SSA, and easier to parallelize,
21 because they are local in each column of ice. Specifically, they describe the vertical
22 shear stress as a local function of the driving stress :cite:`Paterson`. They can confidently
23 be applied to those grounded parts of ice sheets for which the basal ice is frozen to the
24 bedrock, or which is minimally sliding, and where the bed topography is relatively
25 slowly-varying in the map-plane :cite:`Fowler`. These characteristics apply to the majority (by
26 area) of the Greenland and Antarctic ice sheets.
27
28 We solve the SIA with a non-sliding base because the traditional :cite:`Greve`,
29 :cite:`HuybrechtsdeWolde`, :cite:`PayneBaldwin` addition of ad hoc "sliding laws" into the SIA
30 stress balance, and especially schemes which "switch on" at the pressure-melting
31 temperature :cite:`EISMINT00`, have bad continuum :cite:`Fowler01` and numerical (see
32 :cite:`BBssasliding`, appendix B) modeling consequences.
33
34 The SSA equations can confidently be applied to large floating ice shelves, which have
35 small depth-to-width ratio and negligible basal resistance :cite:`Morland`,
36 :cite:`MorlandZainuddin`. The flow speeds in ice shelves are frequently an order-of-magnitude
37 higher than in the non-sliding, grounded parts of ice sheets.
38
39 Terrestrial ice sheets also have fast-flowing grounded parts, however, called "ice
40 streams" or "outlet glaciers" :cite:`TrufferEchelmeyer`. Such features appear at the margin of,
41 and sometimes well into the interior of, the Greenland :cite:`Joughinetal2001` and Antarctic
42 :cite:`BamberVaughanJoughin` ice sheets. Describing these faster-flowing grounded parts of ice
43 sheets requires something more than the non-sliding SIA. This is because adjacent columns
44 of ice which have different amounts of basal resistance exert strong "longitudinal" or
45 "membrane" stresses :cite:`SchoofStream` on each other.
46
47 In PISM the SSA may be used as a "sliding law" for grounded ice which is already modeled
48 everywhere by the non-sliding SIA :cite:`BBssasliding`, :cite:`Winkelmannetal2011`. For
49 grounded ice, in addition to including shear in planes parallel to the geoid, we must
50 balance the membrane stresses where there is sliding. This inclusion of a membrane stress
51 balance is especially important when there are spatial and/or temporal changes in basal
52 strength. This "sliding law" role for the SSA is in addition to its more obvious role in
53 ice shelf modeling. The SSA plays both roles in a PISM whole ice sheet model in which
54 there are large floating ice shelves (e.g. as in Antarctica :cite:`Golledgeetal2012ant`,
55 :cite:`Martinetal2011`, :cite:`Winkelmannetal2011`; see also :ref:`sec-ross`).
56
57 The "SIA+SSA hybrid" model is recommended for most whole ice sheet modeling purposes
58 because it seems to be a good compromise given currently-available data and computational
59 power. A related hybrid model described by Pollard and deConto :cite:`PollardDeConto` adds the
60 shear to the SSA solution in a slightly-different manner, but it confirms the success of
61 the hybrid concept.
62
63 By default, however, PISM does not turn on (activate) the SSA solver. This is because a
64 decision to solve the SSA must go with a conscious user choice about basal strength. The
65 user must both use a command-line option to turn on the SSA (e.g. option ``-stress_balance
66 ssa``; see section :ref:`sec-stressbalance`) and also make choices in input files and
67 runtime options about basal strength (see section :ref:`sec-basestrength`). Indeed,
68 uncertainties in basal strength boundary conditions usually dominate the modeling error
69 made by not including higher-order stresses in the balance.
70
71 When the SSA model is applied a parameterized sliding relation must be chosen. A
72 well-known SSA model with a linear basal resistance relation is the Siple Coast
73 (Antarctica) ice stream model by MacAyeal :cite:`MacAyeal`. The linear sliding law choice is
74 explained by supposing the saturated till is a linearly-viscous fluid. A free boundary
75 problem with the same SSA balance equations but a different sliding law is the Schoof
76 :cite:`SchoofStream` model of ice streams, using a plastic (Coulomb) sliding relation. In this
77 model ice streams appear where there is "till failure" :cite:`Paterson`, i.e. where the basal
78 shear stress exceeds the yield stress. In this model the location of ice streams is not
79 imposed in advance.
80
81 As noted, both the SIA and SSA models are *shallow* approximations. These equations are
82 derived from the Stokes equations by distinct small-parameter arguments, both based on a
83 small depth-to-width ratio for the ice sheet. For the small-parameter argument in the SIA
84 case see :cite:`Fowler`. For the corresponding SSA argument, see :cite:`WeisGreveHutter`
85 or the appendices of :cite:`SchoofStream`. Schoof and Hindmarsh :cite:`SchoofHindmarsh`
86 have analyzed the connections between these shallowest models and higher-order models,
87 while :cite:`GreveBlatter2009` discusses ice dynamics and stress balances comprehensively.
88 Note that SIA, SSA, and higher-order models all approximate the pressure as hydrostatic.
89
90 Instead of a SIA+SSA hybrid model as in PISM, one might use the Stokes equations, or a
91 "higher-order" model (i.e. less-shallow approximations :cite:`Blatter`, :cite:`Pattyn03`),
92 but this immediately leads to a resolution-versus-stress-inclusion tradeoff. The amount of
93 computation per map-plane grid location is much higher in higher-order models, although
94 careful numerical analysis can generate large performance improvements for such equations
95 :cite:`BrownSmithAhmadia2013`.
96
97 Time-stepping solutions of the mass conservation and energy conservation equations, which
98 use the ice velocity for advection, can use any of the SIA or SSA or SIA+SSA hybrid stress
99 balances. No user action is required to turn on these conservation models. They can be
100 turned off by user options ``-no_mass`` (ice geometry does not evolve) or ``-energy none``
101 (ice enthalpy and temperature does not evolve), respectively.