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MODELING THE EARTH, MESH RESOLUTION EFFECTS, AND REMOVING TIME STEP CONSTRAINTS
William Pringle Computational Hydraulics LabDam Wirasaet Civil & Env. Eng. & Earth Sci.Joannes Westerink University of Notre DameKeith Roberts
OVERVIEW OF COASTAL OCEAN MODELS
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• ADCIRC is considered only a coastal model
• Continuous GalerkinFinite-Element method (CG-FEM)
• Uses GWCEformulation to avoid high-frequency oscillations in CG-FEM method
AIMS
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1) Extend ADCIRC to a global modela. Show improvement over old versionb. Mesh resolution sensitivities
2) Improve stability and mass-conservation issues associated with GWCE formulation
• Note: CG-FEM method is nice because we can keep 2nd orderaccuracy and have a lot of tolerance for skewed elements(Fringer et al., 2019)
/R/G
CURRENT ADCIRC MODEL EQUATIONS
• Main problem:
Solving this term in continuity eq. directly is difficult with CG-FEM due to nonlinearity of the ϕ dependent terms..
• This expansion eliminates nonlinearity but the tan(ϕ) term is extremely stiff for the numerical method so it has just been ignored…
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0
SOLUTION BY REFORMULATION
• Use an arbitrary cylindrical projection to map (λ,ϕ) onto (x,y):(Select desired p = 0, 1, 2)
• Multiply continuity by cosp(ϕ) [= 1 when p = 0]:
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(λ0,ϕ 0) is arbitrary origin
this is just a constant
Continuity in a nice form to solve!
ICS = 20
ICS = 21
ICS = 22
ICS = 2 sans tan(ϕ) terms
POLE TREATMENT
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• Set ICS to a negative value, ex.: ICS = -22
• Supply a fort.rotmfile, ex.:znorth_in_spherical_coors114.16991 0.77432
https://wiki.adcirc.org/wiki/Fort.rotm
https://wiki.adcirc.org/wiki/ICS
FIRST: TEST TIDES ON A HIGHLY RESOLVED MESH
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• Automatically generated by OceanMesh2D
See: Example_7_Global.m
• 6 million vertices
• 1.5 km to 25 km resolution
• Highly resolved along topographic gradients
Mesh rotated to have Greenland-Antarctica poles
M2 RMSE
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ICS = 2 (old)
ICS = 2 (old)
ICS = 22 (new)
ICS = 22 (new)
[M2, N2, S2, K1, O1]
(Median = 6.67 cm)
SENSITIVITY TO MESH RESOLUTION
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K
edgefx options in OceanMesh2DMinEle = ‘h0’TLS = ‘slp’FL = ‘fl’
Pretty good
SENSITIVITY TO LOCAL MESH REFINEMENT
Max Elev.
HWMs
500 m – 1.5 km 150 m – 500 m
150 m
500 m
1.5 km
Max Elev. Max Elev.
2) IMPROVING STABILITY AND MASS CONSERVATION ISSUES
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• What is the stability criteria for the GWCE? Let’s check
1D linear case:
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I follow method outlined in Kinnmark (1986) monograph based on Routh-Hurwitz criterion.
1) Make assumption on relationship between α1, α2, α3
(A00, B00, C00):e.g., 0.35, 0.30, 0.35. Kinnmark (1986) analyzed this one
i.e., TAU0 must be less than linear bottom friction everywhere! (which is impossible for quadratic friction as U → 0)
𝝉𝒃 = 𝑪𝒇
𝑼
𝑯
Kinnmark, I. P. E.: The Shallow Water Wave Equations: Formulation, Analysis and Application, edited by C. A. Brebbia and S. A. Orszag, Springer Berlin Heidelberg, Berlin, Heidelberg., 1986.
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Alternative relationship between α1, α2, α3 (A00, B00, C00):
i.e., criteria between TAU0 and bottom friction vanishes!
Example:Select κ = 0.5, consistent mass-matrix: m = 2/3.A00, B00, C00 = 0.5, 0.5, 0TAU0 < 8/(3∆t)
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Timings on 480 processors: Global model using the improved choice of A00, B00, C00, TAU0
• Old Hurricane Katrina simulations (Dietrich et al., 2011):DT = 1 sec, 60-minper simulation dayon 480 processors
90-sec
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WHY WE SHOULD USE CONSTANT TAU0
Mass conservation
Spatially/temporally varying
Spatially varying
constant
SUMMARY
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1) Extended ADCIRC to a global modela. Tide solutions with ICS = -22 clearly improved from old version (ICS = 2)b. Resolution experiments show that global tide solutions decay beyond
~3 km min, and topographic slopes need to be well-resolved. Local refinement generally decreases open ocean max. storm tide elevations
2) Improved stability and mass-conservation issues associated with GWCE formulationa. Stability analysis shows that A00, B00, C00 = 0.5, 0.5, 0 is non-Courant
limited (in the linear sense) if TAU0 < 8/(3∆t)
Hurricane Katrina Examples: Global 1.5 km mesh runs with ∆t = 120 sLocally refined 500 m runs with ∆t = 120 sLocally refined 150 m mesh runs with ∆t = 50 s
b. From mass-conservation point of view we should use constant TAU0
Also see: https://wiki.adcirc.org/wiki/IMfor info on new option for fully implicit gravity wave term (IM Digit 6 = 3)
Pringle, W. J., et al. Global Storm Tide Modeling with ADCIRC v55 : Unstructured
Mesh Design and Performance, will submit to Geosci. Model. Dev.
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https://wpringle.github.io/GLOCOFFS/
Continuous 5-day forecasts every 6 hrs
Since April, 20191330+ commits