Ice Sheet Models in Climate

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(1)Ice Sheet Models in Climate Modeling. Dan Martin Lawrence Berkeley National Laboratory. May 20, 2013.

(2) Outline: 1.. Why do we care about ice sheets?. 2.. What does an ice sheet model look like?. a. Example: The BISICLES model 3.. How do we link to a climate model?. a. Example: CESM.

(3) Why do we care?.

(4) Sea-level rise (SLR) and coastal flooding  . . U.S. flood losses from 1978–2010 were $38 billion (NFIP 2010). About 53% of the U.S. population lives in coastal counties, with 3.7 million people within 1 m of high tide (Strauss et al. 2012). Sea-level rise greatly increase the odds of damaging floods from storm surges (Tebaldi et al. 2012).  Analyzed 55 U.S. coastal sites with tide gauge records  By 2030, two-thirds of these sites will have at least doubled probability of century-scale floods.  These projections are based on mean sea-level change alone, without changes in frequency or intensity of storms..

(5) Sea-level change since the last interglacial. IPCC AR4; from Waelbroeck et al. 2002. • Global mean sea level rose by 120 m from 20 ka to 6 ka • Sea level was 6–10 m higher during the Last Interglacial (125 ka)..

(6) Sea-level change over the past two millennia. Reconstructed sea level at coastal sites in N. Carolina (Kemp et al. 2011). SLR rate = 2.1 mm/yr since late 19th century; fastest in 2000 years.

(7) Sea-level rise over the past two decades. Global mean sea-level rise from satellite altimetry (Nerem et al. 2010). SLR rate = 3.1 ± 0.4 mm/yr, 1993–2012.

(8) Church et al., GRL, 38 (2011).

(9) Regional sea-level variations. Univ of Colorado 2012_rel2. −15 −12 −9. −6. −3. 0. 3. 6. 9. 12. 15. mm/yr. Sea-level trends from satellite altimetry, 1993–2012 (Nerem et al. 2010). Regional variations are large compared to the global mean, but are likely to be smoothed out over longer time scales..

(10) Global sea level budget • Ocean thermal expansion: ~1 mm/yr • Glaciers and ice caps: ~1 mm/yr • Ice sheets: ~1 mm/yr • Greenland 0.6 mm/yr • Antarctica 0.4 mm/yr. • Terrestrial storage:. ~0 mm/yr. • Dam retention -0.3 mm/yr • Groundwater depletion 0.3 mm/yr The ice sheet contribution has roughly doubled since 2000 and will likely continue to increase.. Greenland ice mass loss Antarctic ice mass loss (Velicogna 2009).

(11) Global sea level budget The global budget appears to be closed, but large uncertainties remain. • Jacob et al. 2012: • GRACE gravity data suggest a glacier contribution of only 0.4 mm/yr, 2003–2010 (excluding glaciers peripheral to ice sheets) • Total land-ice contribution ~1.5 mm/yr • Pokhrel et al. 2012: • Terrestrial storage contributed ~0.8 mm/yr, 1961– 2003 (mainly from groundwater depletion).

(12) Greenland Ice Sheet  . . 7 m sea-level equivalent Accumulation balanced by surface runoff and iceberg calving Increasing mass loss (~200 Gt/yr) since late 1990s from increased surface melting, combined with thinning and acceleration of large outlet glaciers Greenland winter flow speed (Greenland Ice Mapping Project).

(13) Antarctic Ice Sheet . . . 60 m sea-level equivalent (~5 m in marine-grounded parts of West Antarctica) Accumulation balanced by flow into floating ice shelves; little surface melting Increasing mass loss (~150 Gt/yr) from West Antarctica and the Antarctic Peninsula, triggered by warm ocean water reaching the base of ice shelves. Antarctic ice flow speed (Rignot et al. 2011).

(14) Glaciers and ice caps • 200,000+ glaciers and ice caps worldwide • Only 0.6 m sea-level equivalent (Radic & Hock 2010), but short response times • Most glaciers are out of balance with the climate and are retreating • Total mass loss (~350 Gt/yr) has usually been estimated by upscaling observations from a few dozen glaciers. Modeled surface mass budget, Canadian Archipelago, 2003– 2009 (Gardner et al. 2011).

(15) 21st century sea-level projections • IPCC AR4: 0.2–0.6 m, 2m excluding ice-sheet dynamic feedbacks • Semi-empirical models: 0.5–1.5 m (based on simple 1 m relationships between SLR and temperature/radiative forcing). • Pfeffer et al. 2008: 0.8-2 m (kinematic constraints for ice sheets). AR4. 0 Projected 21st century sea-level rise (Rahmstorf 2010). • Semi-empirical models project more SLR than physics-based models used in AR4; not clear why. • More realistic physical models are needed to better constrain the uncertainty (especially ice sheet models coupled to climate models)..

(16) Regional sea-level variations Sea-level rise is not spatially uniform. During the 21st century, regional variations will likely be comparable to the global signal: • Ocean dynamic changes • Fast changes caused by land-ice mass loss • Glacial isostatic adjustment (GIA) • Coastal erosion and subsidence.

(17) Regional variations from ice-sheet mass loss Ice-sheet mass loss results in instantaneous elastic rebound and changes in self gravity and the Earth’s rotation.  Migration of water away from shrinking ice sheets will tend to stabilize marine ice. . Relative sea-level change from retreat of the West Antarctic Ice Sheet (left) and Greenland Ice Sheet (right) (Mitrovica et al. 2011)..

(18) Putting it all together Slangen et al. 2012: Used climate model simulations (A1B) to generate maps of 21st century sea-level contributions from • ocean steric effects • retreat of glaciers and ice sheets (with gravitational adjustments) • glacial isostatic adjustment (small globally but important locally).

(19) Putting it all together Ice sheets. Glaciers. Steric effects GIA 21st century sea-level projections (Slangen et al. 2012).

(20) Putting it all together . The ice-sheet contribution in these projections is based on AR4 and is probably too low.. . The projections could be improved by using physicsbased ice sheet models, coupled to global climate models..

(21) Marine Ice Shelves: Larsen B Breakup (2002). January 31, 2002.

(22) Marine Ice Shelves: Larsen B Breakup (2002). February 17, 2002.

(23) Marine Ice Shelves: Larsen B Breakup (2002). February 23, 2002.

(24) Marine Ice Shelves: Larsen B Breakup (2002). March 5, 2002.

(25) Marine Ice Shelves: Larsen B Breakup (2002) . 3,250 square kilometers (1,250 square miles). Breakup took about 1 month  Results: Larsen A and B glaciers .  abrupt acceleration, about 300% on average  mass loss went from 2–4 gigatonnes per year in 1996 and 2000 (a gigatonne is one billion metric tonnes), to between 22 and 40 gigatonnes per year in 2006. . Likely due to exceptionally warm summer  Melt pools on surface –- surface melting  Warm ocean temperatures in the Weddell Sea. Not the last – Wilkins (2008-2009).

(26) Antarctic Marine Ice Sheet Instability. Figures courtesy of C. Hulbe (PSU). LeBrocq et al., 2010.

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(28) What does an ice sheet model look like?.

(29) Image source: http://www.nasa.gov/images/content/53743main_atmos_circ.jpg.

(30) H &    UH  b& m t. Basal Melting: climate model (ocean / sea ice). Flux Divergence: Ice flow model. Image source: http://www.nasa.gov/images/content/53743main_atmos_circ.jpg. Sfc Mass Balance: climate model (atmos. / land).

(31) Components of a Land Ice Model. (1) Conservations Equations. (2) Coupling to Climate. (3) Physics.

(32) Components of a Land Ice Model. (1) Conservations Equations. “dynamical core” of model (2) Coupling to Climate (external forcing). reanalysis / proxy data, climate model (3) Physics physical processes through submodels, parameterizations, etc. (everything else).

(33) (1) Conservation Equations (dynamical core) Conservation of mass:. H &    UH  b& m t Conservation of energy:.  ij &ij  k  T     T   u  T  t c  c  Conservation of momentum:.  ij. P   gi  0 x j xi.

(34) (2) Climate Coupling: Mass Balance Terms. H &    UH  b& m t. momentum balance (dycore). coupling to atmos. and land model. coupling to ocean model.

(35) (3) Physics (everything else). Constitutive laws. Basal sliding submodels / parameterizations Surface and subglacial Hydrology Iceberg calving. etc ….

(36) Models and Approximations Physics: Non-Newtonian viscous flow: 𝜇(𝜖 2 ,T) = A(T)(𝜖 2 ) . (1−𝑛) 2. Full-Stokes  Best fidelity to ice sheet dynamics  Computationally expensive (full 3D coupled nonlinear elliptic equations). . Approximate Stokes  Use scaling arguments to produce simpler set of equations  Common expansion is in ratio of vertical to horizontal length scales (𝜀 =  E.g. Blatter-Pattyn (most common “higher-order” model), accurate to  Still 3D, but solve simplified elliptic system (e.g. 2 coupled equations). . Depth-integrated  “Shallow Ice” and “Shallow-Shelf” approximations (accurate to O(𝜀) )  Special case of approximate Stokes with 2D equation set  Easiest to work with computationally, generally less accurate. . [ℎ] [𝑙] O(𝜀 2 ).

(37) Berkeley-ISICLES (BISICLES) . DOE ISICLES-funded project to develop a scalable adaptive mesh refinement (AMR) ice sheet model/dycore  Local refinement of computational mesh to improve accuracy. . Use Chombo AMR framework to support block-structured AMR    . Support for AMR discretizations Scalable solvers Developed at LBNL DOE ASCR supported (FASTMath). Interface to CISM as an alternate dycore.  Collaboration with LANL and Bristol (U.K.)  Continuation in SciDAC-funded PISCEES. . (Figure courtesy of Ann Almgren).

(38) Why is this useful?. (another BISICLE for another fish?)  Ice sheets -- Localized regions where high resolution needed to accurately resolve ice-sheet dynamics (500 m or better at grounding lines)  Antarctica is really big – too big to resolve at that level of resolution.  Large regions where such fine resolution is unnecessary (e.g. East Antarctica)  Well-suited for adaptive mesh refinement (AMR)  Problems still large: need good parallel efficiency  Dominated by nonlinear coupled elliptic system for ice velocity solve: good linear and nonlinear solvers. Rignot et al., Science, 333 (2011).

(39) “L1L2” Model (Schoof and Hindmarsh, 2010). . Uses asymptotic structure of full Stokes system to construct a higher-order approximation  Expansion in =. 𝐻 𝐿. and. l=. 𝜏𝑠ℎ𝑒𝑎𝑟 (ratio of shear & normal stresses) 𝜏𝑛𝑜𝑟𝑚𝑎𝑙. • Large l: shear-dominated flow • Small l: sliding-dominated flow.  Computing velocity to 𝑂(𝜀 2 ) only requires τ to 𝑂(𝜀)  Computationally much less expensive -- enables fully 2D vertically integrated discretizations. (can reconstruct 3d) . Similar formal accuracy to Blatter-Pattyn 𝑂(𝜀 2 )  Recovers proper fast- and slow-sliding limits: −1. • SIA (1 ≪ 𝜆 ≤ 𝜀 𝑛 ) -- accurate to 𝑂(𝜀 2 𝜆𝑛−2 ) • SSA (𝜀 ≤ 𝜆 ≤ 1) – accurate to 𝑂(𝜀 2 ).

(40) Discretizations . Baseline model is the one used in Glimmer-CISM:  Logically-rectangular grid, obtained from a time-dependent uniform mapping.  2D equation for ice thickness, coupled with 2D steady elliptic equation for the horizontal velocity components. The vertical velocity is obtained from the assumption of incompressibility.  Advection-diffusion equation for temperature.. . . H  b    Hu t. T k 2  T   T  uT   w  t c c  z. Use of Finite-volume discretizations (vs. Finite-difference discretizations) simplifies implementation of local refinement. Software implementation based on constructing and extending existing solvers using the Chombo libraries..

(41) “L1L2” Model (Schoof and Hindmarsh, 2010), cont. . Can construct a computationally efficient scheme: 1. Approximate constitutive relation relating 𝑔𝑟𝑎𝑑 𝑢 and stress field 𝜏 with one relating 𝑔𝑟𝑎𝑑(𝑢 𝑧=𝑏 ), vertical shear stresses 𝜏𝑥𝑧 and 𝜏𝑥𝑧 given by the SIA / lubrication approximation and other components 𝜏𝑥𝑥 𝑥, 𝑦, 𝑧 , 𝜏𝑥𝑦 𝑥, 𝑦, 𝑧 , etc 2. leads to an effective viscosity 𝜇(𝑥, 𝑦, 𝑧) which depends only on 𝑔𝑟𝑎𝑑(𝑢 and 𝑔𝑟𝑎𝑑 𝑧𝑠 , ice thickness, etc. 𝑧=𝑏 ). 3. Momentum equation can then be integrated vertically, giving a nonlinear, 2D, elliptic equation for 𝑢 𝑧=𝑏 (𝑥, 𝑦) 4.. 𝑢(𝑥, 𝑦, 𝑧) can be reconstructed from 𝑢. 𝑧=𝑏 (𝑥, 𝑦).

(42) Modified “L1L2” Model (SSA*) . Can construct a computationally efficient scheme: 1. Approximate constitutive relation relating 𝑔𝑟𝑎𝑑 𝑢 and stress field 𝜏 with one relating 𝑔𝑟𝑎𝑑(𝑢 𝑧=𝑏 ), vertical shear stresses 𝜏𝑥𝑧 and 𝜏𝑥𝑧 given by the SIA / lubrication approximation and other components 𝜏𝑥𝑥 𝑥, 𝑦, 𝑧 , 𝜏𝑥𝑦 𝑥, 𝑦, 𝑧 , etc 2. leads to an effective viscosity 𝜇(𝑥, 𝑦, 𝑧) which depends only on 𝑔𝑟𝑎𝑑(𝑢 and 𝑔𝑟𝑎𝑑 𝑧𝑠 , ice thickness, etc. 𝑧=𝑏 ). 3. Momentum equation can then be integrated vertically, giving a nonlinear, 2D, elliptic equation for 𝑢 𝑧=𝑏 (𝑥, 𝑦). 4.. 𝑢(𝑥, 𝑦, 𝑧) can be reconstructed from 𝑢. 4. Use 𝑢(𝑥, 𝑦, 𝑧) = 𝑢. 𝑧=𝑏 (𝑥, 𝑦). 𝑧=𝑏 (𝑥, 𝑦). (neglect vertical shear in flux velocity).

(43) BISICLES Results - MISMIP3D Experiment P75R: (Pattyn et al (2011)      .  . Begin with steady-state (equilibrium) grounding line. Add Gaussian slippery spot perturbation at center of grounding line Ice velocity increases, GL advances. After 100 years, remove perturbation. Grounding line should return to original steady state. Figures show AMR calculation:  ∆𝑥0 = 6.5𝑘𝑚 base mesh,  5 levels of refinement  Finest mesh ∆𝑥4 = 0.195𝑘𝑚.  t = 0, 1, 50, 101, 120, 200 yr Boxes show patches of refined mesh. GL positions match Elmer (full-Stokes).

(44) MISMIP3D (cont): L1L2 (SSA*) Spatial Resolution. • Very fine (~200 m) resolution needed to achieve reversability!.

(45) MISMIP3D: SSA vs. “L1L2” or “SSA*”. • Direct comparison of SSA vs. SSA* • (fully resolved spatially, same numerics, etc) • Note difference in steady-state GL positions.

(46) Simple Rheology/Damage model. Pine Island Glacier. • • • •. Filchner-Ronne Ice Shelf. Ross Ice Shelf. Solve control problem for ice initial condition Include new parameter 𝜑 which multiplies viscosity 𝜑 < 1 (blue) = softening 𝜑 > 1 (red) = hardening.

(47) BISICLES Results – Ice2Sea Amundsen Sea . Study of effects of warm-water incursion into Amundsen Sea.. . Results from Payne et al, (2012), submitted.. . Modified 1996 BEDMAP geometry (Le Brocq 2010), basal traction and damage coefficients to match Joughin 2010 velocity.. . Background SMB and basal melt rate chosen for initial equilibrium.. . SMB held fixed.. . Perturbations in the form of additional subshelf melting:  derived from FESOM circumpolar deep water  ~5 m/a in 21st Century,  ~25 m/a in 22nd Century..

(48) Ice2Sea Amundsen (cont).

(49) Ice2Sea Amundsen (cont) . Need at least 2 km resolution to get any measurable contribution to SLR.. . Appears to converge at first-order in ∆x. SLR vs. year, Amundsen Sea Sector.

(50) Ice2Sea Amundsen (cont) – Thwaites? • In 400 year run, Thwaites destabilizes as well. • Essentially same forcing as previous run, subshelf melting held constant past 2200. • More melting: extra 5 m/a • Thwaites is very stable, until it tips..

(51) Ice2Sea Amundsen, (cont).

(52) Continental-scale: Antarctica • Ice2sea geometry • Temperature field from Pattyn and Gladstone.

(53) Antarctica (Ice2Sea) • Refinement based on Laplacian(velocity), grounding lines • 5 km base mesh with 3 levels of refinement • base level (5 km): 409,600 cells (100% of domain) • level 1 (2.5 km): 370,112 cells (22.5% of domain) • Level 2 (1.25 km): 955,072 cells (14.6% of domain) • Level 3 (625 m): 2,065,536 cells (7.88% of domain).

(54) Alternative approach: Variable-resolution grids • MPAS - Model for Prediction Across Scales: A climate modeling framework that supports dynamical cores on unstructured Voronoi (SCVT) meshes • Allows high resolution in regions of interest (e.g., ice streams, grounding lines, sub-ice-shelf cavities); reduces number of grid cells by ~10x • MPAS ice sheet model development started using methods developed at LANL and NCAR • MPAS will be used to interface CISM with FSU/USC HO and Stokes dycores (upcoming talk by M. Perego) Left: Voronoi mesh for the Greenland ice sheet Right: Global variable-resolution mesh for POP, 120 K nodes.

(55) MPAS: Variable-resolution grids. Ringler et al., Ocean Dyn. (2008). Figure / photos courtesy of M. Hoffman (LANL).

(56) Variable-resolution grids. Ringler et al., Ocean Dyn. (2008).

(57) Subglacial Hydrology: Velocity vs. Subglacial Water Pressure. Catania et al., C11A-03, Monday 8:30 am.

(58) Subglacial Hydrology Modeling.

(59) Subglacial Hydrology Modeling. Total flux Distributed Channelized. Flowers, Hydrol. Process. 22 (2008).

(60) How Do We Link to a Climate Model?.

(61) Chain of inference Atmosphere–ocean general circulation model Land-ice surface mass balance. Global mean ocean expansion, local steric sea-level anomalies. Glacier model. Ice sheet model. Sea-level (gravity) model. Ocean steric contribution Glacial isostatic adjustment. Subsidence & other local processes. Local and regional estimates of future sea-level change (Adapted from Slangen et al. 2012).

(62) Ice sheets in CESM 1.0 Land -> Ice sheet (10 classes)   . Ice sheet -> Land (10 classes). Surface mass balance Surface elevation Surface temperature.   . Ice fraction and elevation Runoff and calving fluxes Heat flux to surface. Atmosphere Land surface. (Ice sheet surface mass balance). Coupler. Ice sheet (Dynamics). Ocean. Sea Ice.

(63) Ice sheet surface mass balance in CESM . . Traditional approach: Pass Ice sheet temperature and precipitation grid cell fields to the ice sheet model and compute the mass balance using a positive-degree-day scheme. CESM computes the SMB in the land model (CLM) on a coarse (~100 km) grid in 10 elevation classes  Cost savings (~1/10 as many columns)  Energetic consistency  Avoid code duplication  Surface albedo changes feed back on the atmosphere. Land grid cell.

(64) What’s missing? 1. Improved models of glaciers and ice caps 2. More realistic ice sheet models. 3. Coupling of ice-sheet and ocean models 4. Integration of different sea-level components into useful predictions for stakeholders.

(65) What’s missing? 1. Improved models of glaciers and ice caps • GIC are projected to contribute ~12 ± 4 cm of SLR by 2100 (Radic & Hock 2011). • Calibrated mass-balance model forced by GCM temperature and precipitation. ESRI Digital Chart of the World, World Glacier Monitoring Service.

(66) Glaciers and ice caps in CESM • Too many glaciers and too little data to model the dynamics of each glacier • Instead, use a statistical model based on relationships between glacier area and thickness • Glacier area and thickness evolve together; retreat to higher elevations is an important feedback • Complete glacier outlines now available from Randolph Glacier Inventory. • Force model with glacier surface mass balance b(z) from a climate model • Now developing a glacier model for the Regional Arctic System Model (RASM); could be used in CESM • Need to improve and validate CLM’s surface mass balance scheme for glaciers (downscaling, ice albedo).

(67) Ice-ocean coupling challenges • The ocean must be able to circulate beneath ice shelves, exchanging heat and mass at the shelf boundary. • The upper boundary of the ocean must move up and down as ice shelves advance and retreat.. • We must simulate complex interactions at small spatial scales (< 10 km) in regions that are sparsely observed. • Simulations are expensive and hard to validate • The ice-sheet/ocean system must be spun up to steady state with little or no drift. Spinup times are ~103 yr for the ocean, ~104 – 105 yr for ice sheets..

(68) Ice-sheet/ocean interface in POP As part of the DOE IMPACTS project on abrupt climate change, we have developed POP2X, which includes cavities beneath ice shelves.. ice shelf. See Asay-Davis talk in LIWG on Thursday for more details. Ice/ocean boundary defined by partial-top cells (analogous to partial-bottom cells; based on Losch 2008). Ice. Ocean.

(69) Future work: Moving boundaries Immersed Boundary Method includes ghost cells adjacent to boundary implicit representation of sloped interface geometry as ice sheet retreats, ghost cells become new ocean cells no partial cells, so never have infinitesimally thin cells.

(70) Greenland Ice Sheet Internal Layers. Radar data courtesy of CReSIS Layer tracing and figure courtesy of Joe MacGregor (UTIG) See also J. MacGregor talk C12B-03, 10:30 am Monday.

(71) Acknowledgements:     . US Department of Energy Office of Science (ASCR) funded BISICLES project US Department of Energy Office of Science (ASCR/BER) SciDAC applications program (PISCEES) Steph Cornford, Tony Payne at the University of Bristol Bill Lipscomb, Doug Ranken, Stephen Price (LANL) Mark Adams (Columbia University).

(72) Extras.

(73) Parallel scaling, Antarctica benchmark. (Preliminary scaling result – includes I/O and serialized initialization).

(74) Nonlinear Solvers  Most computational effort spent in nonlinear ice velocity solve..  Picard iteration: • Robust • Simple to implement • Slow (but steady) convergence  Jacobian-free Newton-Krylov (JFNK): • More complex to implement • Works best with decent initial guess • Rapid convergence • Well-suited for Chombo AMR elliptic solvers  Approach – use Picard iteration initially, then switch to JFNK when convergence slows.

(75) Nonlinear Solvers (cont).

(76) Linear Solvers – GAMG vs. Geometric MG.

(77) New FAS Multigrid nonlinear solver . Full Approximation Storage (FAS) – nonlinear multigrid. . Picard, JFNK:  linear solver nested inside of nonlinear one  Linear Multigrid solvers (residual-correction form) work well.. . FAS Multigrid – fully nonlinear solver (no outer solver)  Can outperform JFNK/MG  More robust (don’t need good initial guess)  Simpler to implement and maintain  Nonlinear convergence similar to MG linear convergence.

(78) FAS Multigrid nonlinear solver (cont). Solution time for 8 cores on linux desktop: FAS-MG: 715 s JFNK-MG: 1128 s.

(79) Embedded Boundary (EB) for Grounding Lines  Embedded Boundary (EBChombo) • Currently force GL and ice margins to cell faces. • “Stair-step” discretization Known to be inadequate from experience with Stefan Problem in other contexts!. • Use Chombo Embedded-boundary support to improve discretization of GL’s and ice margins.. • Can solve as a Stefan Problem, with appropriate jump conditions enforced at grounding line. (as in Schoof, 2007).

(80) CISM/BISICLES coupling update . Extension of existing serial coupled code to work in the fully distributed case (no serial bottlenecks). . CISM (F90) to BISICLES (C++). . General API design for coupling alternate dynamical cores into CISM. . Moderately complex build process (working to streamline). . CISM owns “main”, problem setup, coupler to CESM, other physics (like isostasy, hydrology, etc). . CISM hands control to BISICLES, which evolves the ice sheet. . BISICLES passes fields like thickness, velocity back to CISM.

(81) CISM/BISICLES coupling update (cont).

(82) Temporal Discretization Update equation for H:. 𝜕𝐻 𝜕𝑡. + 𝛻 ∙ 𝑢𝐻 = 𝑆. “looks” like hyperbolic advection equation (explicit scheme, Courant stability -- ∆𝑡 ∝ ∆𝑥) Velocity field has 𝛻𝐻 piece – diffusion equation for H (∆𝑡 ∝ ∆𝑥 2 !).  . Strategy (Cornford) – try to factor out diffusive flux and discretize as an advection-diffusion equation:.  . 𝐹 = 𝑢𝐻 = 𝐹𝑎𝑑𝑣𝑒𝑐𝑡𝑖𝑣𝑒 + 𝐹𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑒. . 𝐹𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑒 = −𝐷 𝛻𝐻. . Now solve:  . 𝜕𝐻 𝜕𝑡. + 𝛻 ∙ 𝐹𝑎𝑑𝑣𝑒𝑐𝑡𝑖𝑣𝑒 = 𝛻 ∙ (𝐷 𝛻𝐻) + 𝑆. Advective fluxes: explicit update using unsplit 2nd Order PPM scheme Diffusive fluxes: implicit update (Backward Euler for now).

(83) Temporal Discretization (cont)  . Test case based on ISMIP-HOM A geometry ∆𝑥 = 2.5 𝑘𝑚, ∆𝑡𝐶𝐹𝐿 = 5 a Explicit advance. . Semi-implicit advance. Unfortunately, still run into stability issues finer than ∆𝑥 < 0.5 𝑘𝑚.

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