1. VIC Frozen Soils Simulation in Permafrost Regions: Modifications, Improvements, and Testing Jennifer Adam Amanda Tan May 2, 2007 Hydro Group Seminar

2. Talk Overview 1. Motivation 2. Model Description 3. Model Improvements 4. Study Domain 5. Testing of Model Improvements 6. Conclusions

3. Motivation

4. 1. Precipitation/Streamflow Trend Inconsistency in Permafrost Basins (e.g. Aldan River) 1940 1960 1980 2000 Adam and Lettenmaier (2007) Trend, mm/year -2 Precipitation Streamflow

5. WinterSummerWinterSummer Permafrost Definition (loose) Seasonally Frozen Ground Permafrost Active Layer Permafrost Layer cm km cm m

6. 2. Active Layer Depth Simulated Poorly  Snow cover extent, lake freeze and break-up dates, streamflow climatology – satisfactory  Permafrost active layer depth – unsatisfactory Su et al. (2005)

7. 3. Historical Streamflow Trends Not Captured in Permafrost Basins Observed Simulated Yenisei Lena Q, 10 3 m 3 s -1 1940 1960 1980 2000 Ob’ Q, 10 3 m 3 s -1 1940 1960 1980 Permafrost Basins Non-Permafrost Basin

8. Model Description

10. Frozen Soils Simulation: Heat Equation (Cherkauer et al. 1999) With parameterizations for C s, κ, and θ i Term 1: Heat Storage (Time) Term 2: Vertical Heat Conduction (Space) Term 3: Latent Heat (Time)

11. VIC Frozen Soils Algorithm  Cherkauer and Lettenmaier (1999) finite difference algorithm  solving of thermal fluxes through soil column  infiltration/runoff response adjusted to account for effects of soil ice content  parameterization for spatial distribution of frost  tracks multiple freeze/thaw layers  can use either “zero flux” or “constant temperature” bottom boundary (Cherkauer et al. 1999)

12. Current Implementation (Cherkauer et al. 1999)

13. Spatial Frost Algorithm  To produce a spatial distribution of ice content and subsequently soil moisture drainage. (Cherkauer et al. 2003)

14. Frozen Soils Options… WOW!  NODES 18 # number of soil thermal nodes  FULL_ENERGY TRUE# run full energy balance mode  GRND_FLUXTRUE # solve surface energy balance  FROZEN_SOIL TRUE # run frozen soils  QUICK_FLUX FALSE # Liang et al. 1999, otherwise Cherkauer et al. 1999  QUICK_SOLVE FALSE # Cherkauer et al. 1999 for final step only, Liang et al. 1999 for rest  NOFLUX TRUE # zero flux bottom boundary  IMPLICIT TRUE # uses implicit solver (NEW)  EXP_TRANS TRUE # exponential grid transform (NEW)  QUICK_FS FALSE # linear equations for max unfrozen water content (not tested?) QUICK_FS_TEMPS 7  SPATIAL_FROST TRUE # sub-grid frost distribution FROST_SUBAREAS 10  Dp15# damping depth (m)  Tave-4.2# temp. at damping depth (°C)

15. Changes and Improvements 1. Zero Flux Bottom Boundary Parameterization 2. Exponential Grid Transformation 3. Implicit Solver 4. Patch for the “Cold Nose” Problem

16. 1. Soil Temperature Sensitivity to Bottom Boundary Specification Soil Temperature, ° C 1930 1931 1932 Constant T BB Dp = 4 m Tb,init = -12 °C 10 0 -10 -20 Zero Flux BB Dp = 4 m Tb,init = -12 °C Zero Flux BB Dp = 15 m Tb,init = -3 °C Soil Surface (Top Boundary) Soil Bottom Boundary

17. BB Temperature Initialization Annual Soil Temperature @ 3.2 m,  C -6 -4 -2 0 2 4 6 A) Frauenfield et al. 2004 station data B) Interpolated station data

18. 2. Exponential Grid Transformation Linear Distribution Exponential Distribution Parameters: Dp=15m, Nnodes=18 z 1 =soil depth 1 z 2 =2·soil depth 1 z i =linearly distributed to Dp Problem: discontinuity in Δz between nodes 2 and 3 z i =exp(b·i)+c Problem: discontinuity in Δz between all nodes 1 m 3 m 5 m 7 m 9 m 11 m 13 m 15 m i=0 i=Nnodes-1

19. Exponential Grid Transformation, continued Transform spatial derivatives only (temporal derivatives are unaffected) Expand heat conduction term (chain rule), because κ varies with z

20. Exponential Grid Transformation, continued Introduce new space variable, η (T will vary exponentially with z, but linearly with η) Develop transform function, η=f(z) (chain rule)

21. Exponential Grid Transformation, continued  Determine partial derivatives (in η)  Substitute partial derivatives with η into heat conduction terms new elements

22. Exponential Grid Transformation, continued  Determine constants, b and c Boundary condition 1: z(η=Nnodes-1) = Dp Boundary condition 2: z(η=0) = 0 Therefore: c = -1  Solve in η-space (because the η-nodes are linearly distributed, the finite difference assumptions are not compromised)  Map temperatures in η-space back to z-space  Recalculate C s, κ, and θ i as a function of T for each z-node

23. 3. Implicit Solvers Time Space (Thermal Nodes) t-1 t 0 1 2 3 4 5 6 Explicit (forward in time) Implicit (backward in time) known unknown 7 explicit equations solved independently 7 implicit equations solved simultaneously Time Space (Thermal Nodes) t-1 t 0 1 2 3 4 5 6

24. Stability Issues  Stability of convergence Implicit: unconditionally stable Explicit: satisfy the Courant-Friedrichs-Lewy Condition λ≤1/4 for no oscillating errors λ=1/6 to minimize truncation error (solution: make Δt≤1hr or Δz≥0.2m )  Ability to find a solution Explicit: not an issue with physically reasonable values (root_brent is very robust!) Implicit: often is unable to find a solution at the initial formation of ice in the soil column =1 to 1.5 for Δt=3hr, Δz=0.1m

25. Implicit VIC Frozen Soils Algorithm  Newton-Raphson method to solve non-linear system of simultaneous equations (Ming Pan) New Functions (mainly from Numerical Recipes)  solve_T_profile_implicit (replaces solve_T_profile)  fda_heat_eqn (replaces soil_thermal_eqn)  newt_raph (replaces root_brent)  fdjac3 (approximation for Jacobian)  tridiag (solves tridiagonal linear system) Modifications from original code  Merged to VIC 4.1.0 from VIC 4.0.3  Added NOFLUX and EXP_TRANS options  Nodal updating of C s, κ, and θ i as a function of T during iteration  Allowed for time-varying C s :  When unable to find a solution, defers to explicit solver for that time-step

26. 4. “Cold Nose” Problem Time Step Temperature, °C Soil Top Boundary Soil Bottom Boundary  Run-away temperatures in near-surface thermal nodes  The coldest node becomes colder and breaks the 2 nd law of thermodynamics, e.g. “Heat cannot of itself pass from a colder to a hotter body”  VIC crashes – error statement is “increase SOIL_DT” or “increase SURF_DT”  Occurs for all versions of VIC when using the finite- difference scheme, with all modes (implicit/explicit, noflux, exp/linear, etc…)

27. Explanation Heat Equation: Term1 Conduction: Term2 chain rule As T decreases, κ increases (especially if θ i increases) Therefore, at a “cold nose”: but If |Term1|>|Term2|, heat flows from the cold node T at that node escape towards -∞

28. The “Cold Nose” Patch Finite Difference Approx. Actual  Is calculation being made for the two near-surface nodes?  Is the node fall on a “cold nose”?  Is Term1 greater than Term2 in absolute value?  THEN Term1 = 0  To allow for some lenience, can also check that |TL-TU| > 5 i=0 i=1 i=2 i=3 -10 0 -50 Temperature, °C Depth, cm 10 20 30 -15 T TU TL

29. Summary of Changes Zero Flux Bottom Boundary Parameterization Change in implementation, not in code Involves also increasing Dp and Nnodes Necessary for climate change studies in permafrost regions Exponential Grid Transformation Allows for closer node spacing near surface Solves problem of discontinuity in Δz Implicit Solver Should give more accurate solution No convergence instability (no wildly wrong results) Nodal updating, change of C s with time Defers to explicit when no solution is found Should give lower simulation time, but doesn’t… Patch for the “Cold Nose” Problem Inelegant, but it works (exists in implicit and explicit modes) Does this problem go away if Δz becomes infinitesimally small?

30. Study Area: Aldan River Basin Arctic Ocean

31. Lena River Basin Forest Shrubland Savanna Grassland Wetland Cropland Urban Barren Tundra Revenga et al. 1998 Aldan Tributary 700,000 km 2

32. Permafrost Distribution Continuous, 90-100% Discontinuous, 50-90% Sporadic, 10-50% Seasonally Frozen Ground Isolated, <10% Brown et al. 1998 Aldan: 89% continuous 10% discontinuous 1% sporadic

33. Model Testing

34. In-Situ Observations Soil Moisture Precipitation Soil Temperature Snow Depth Air Temperature

35. Simulations conducted RunParametersSimulation Time 1 (Base- line) Optimum parameters: Damping depth, Dp = 15m Nodes = 18 Exponential transformation Implicit solution No Flux bottom boundary 70:31:32 2Damping depth, Dp = 10m55:24:19 3Implicit = False61:24:24 4Exp_Trans = False23:28:45 5No_Flux = False No. of nodes = 5, Dp = 4m (Su et al. 2005 set-up/Traditional) 17:29:43 6Frost subareas = 1037:44:42

36. Comparison of simulation time

37. Soil Moisture Comparison Run #Parameter Changed Bias (mm) Difference from Baseline (mm) 1Baseline-0.732- 2Damping depth decreased 3.2924.024 3Implicit solution off -3.793-3.007 4Exponential transform off 2.8913.623 5Su et al. (2005)1.9732.705 6Increasing frost subareas 3.4274.159

38. Soil Moisture Comparison, Optimized Run (Run 1) --- Observed (Liquid) --- Total Soil Moisture --- Liquid Water --- Ice Content

39. Soil Temperature Comparison Run # Parameter Changed BiasDifference from Baseline 1Baseline0.953- 2Damping depth decreased 1.7560.803 3Implicit solution off7.1435.387 4Exponential transform off 3.8442.891 5Su et al. (2005)3.7481.208 6Increasing frost subareas 2.3481.395

40. Soil Temperature Comparison, Optimized Run (Run 1)

41. Streamflow comparison

42. Streamflow Comparison

43. Conclusions

44.  Optimum set of parameters:  Damping depth = 15m  Number of nodes = 18  Implicit solution  Exponential distribution of thermal nodes  No flux bottom boundary  Traditional setup  Constant flux, Dp=4m, 5 nodes  Gives lowest simulation time  Not suitable for climate change/permafrost studies  When using the optimized parameters, calibration for streamflow is required