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

1) EXTENDING ADCIRC TO GLOBAL MODEL

Currently, tan(ϕ) terms ignored…

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0

0

0

0

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

EQUATIONS ADCIRC NOW SOLVES

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GWCE

Momentum

τ0 : just a constant here

POLE TREATMENT

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stereographic

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 TIDES

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TPXO9 Atlas

ICS = 2 (old) ICS = 22 (new)

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

HURRICANE KATRINA ON GLOBAL MESH

15-min OWI MET

1hr CFSv2 Met

Max Elev.

NWS = -14

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.

SENSITIVITY TO LOCAL MESH REFINEMENT

150 m

500 m

1.5 km

HWMs going clockwise around coastline

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