The GeoClaw Software for Tsunamis and Other Hazardous Flows · PDF fileThe GeoClaw Software for Tsunamis and Other Hazardous Flows ... Originally f77 with Matlab graphics. ... Tsunami

The GeoClaw Software for Tsunamisand Other Hazardous Flows

Randall J. LeVequeDepartment of Applied Mathematics

University of Washington

GeoClaw: www.clawpack.org/geoclaw

R. J. LeVeque LJLL, Paris VI, Dec. 3, 2010

http://www.clawpack.org/geoclaw

Collaborators

David George, University of WashingtonMendenhall postdoctoral Fellow at theUSGS Cascades Volcano Observatory (CVO)

Marsha Berger, Courant Institute, NYU

Roger Denlinger and Dick Iverson,USGS Cascades Volcano Observatory (CVO)

David Alexander and William Johnstone,Spatial Vision Group, Vancouver, BC

Barbara Lence, Civil Engineering, UBC

Harry Yeh, Civil Engineering, OSU

Numerous other students and colleagues

Supported in part by NSF, ONR


Outline

• Clawpack and GeoClaw software

• Malpasset dam break 1959

• Tsunami simulations

• Finite volume methods, Riemann solvers

• Well-balanced methods


Depth-averaged models

Reduce three-dimensional free surface problem to...

Two-dimensional Shallow Water (St. Venant) Equations

ht + (hu)x + (hv)y = 0

(hu)t +(hu2 +

12gh2)x

+ (huv)y = −ghBx(x, y)

(hv)t + (huv)x +(hv2 +

12gh2)y

= −ghBy(x, y)

where (u, v) are velocities in the horizontal directions (x, y),

B(x, y) = bathymetry (underwater topography),12gh

2 = hydrostatic pressure.



Advantages:• 2D rather than 3D

Often critical for realistic geophysical flowsVastly different spatial scales, e.g. ocean to harborNeed Adaptive Mesh Refinement even in 2D!

• No free surface η(x, y, t).

Possible problems:• When is this valid?• What if fluid is not homogeneous, or

shallow water assumptions don’t hold?• Often still a free boundary in the x-y domain,

at the shoreline or at the margins of the flow.• Small perturbations to steady state hard to capture.



Advantages:• 2D rather than 3D

Often critical for realistic geophysical flowsVastly different spatial scales, e.g. ocean to harborNeed Adaptive Mesh Refinement even in 2D!

• No free surface η(x, y, t).

Possible problems:• When is this valid?• What if fluid is not homogeneous, or

shallow water assumptions don’t hold?• Often still a free boundary in the x-y domain,

at the shoreline or at the margins of the flow.• Small perturbations to steady state hard to capture.


CLAWPACK — Conservation Laws Package

• Open source, 1d, 2d, 3d www.clawpack.org

• Originally f77 with Matlab graphics.• Moving to f95 with Python.• Adaptive mesh refinement.• OpenMP and MPI.

User supplies:• Riemann solver, splitting data into waves and speeds

(Need not be in conservation form)

• Boundary condition routine to extend data to ghost cellsStandard bc1.f routine includes many standard BC’s

• Initial conditions — qinit.f

• Source terms — src1.f


http://www.clawpack.org

GeoClaw Software

TsunamiClaw: Dave George’s code based on Clawpack.

Recently made more general as GeoClaw.

Currently includes:• 2d library for depth-averaged flows over topography.• Well-balanced Riemann solvers that handle dry cells.• General tools for dealing with multiple data sets at different

resolutions.• Tools for specifying regions where refinement is desired.• Graphics routines (Matlab currently, moving to Python).• Output of time series at gauge locations or on fixed grids.


Shallow water equations with bathymetry B(x, y)

ht + (hu)x + (hv)y = 0

(hu)t +(hu2 +

12gh2

)x

+ (huv)y = −ghBx(x, y)

(hv)t + (huv)x +(hv2 +

12gh2

)y

= −ghBy(x, y)

Some issues:

• Delicate balance between flux divergence and bathymetry:h varies on order of 4000m, rapid variations in oceanWaves have magnitude 1m or less.

• Cartesian grid used, with h = 0 in dry cells:Cells become wet/dry as wave advances on shoreRobust Riemann solvers needed.

• Adaptive mesh refinement crucialInteraction of AMR with source terms, dry states


Malpasset Dam Failure

Catastrophic failure in 1959


Malpasset Dam Failure


Modeling work by David George, using GeoClaw

Coarse: 400m cell side, Level 2: 50m, Level 3: 12m, Level 4: 3m




















Malpasset survey locations


Malpasset survey locations


Grid convergence study

Water depth gauge at location P2 computed with two differentresolutions (using 4 levels or only 3):


Tsunamis

Generated by• Earthquakes,• Landslides,• Submarine landslides,• Volcanoes,• Meteorite or asteroid impact

There were 97 significant tsunamis during the 1990’s,causing 16,000 casualties.

There have been approximately 28 tsunamis with run-upgreater than 1m on the west coast of the U.S. since 1812.


Tsunamis

• Small amplitude in ocean (< 1 meter) but can grow to10s of meters at shore.

• Run-up along shore can inundate 100s of meters inland

• Long wavelength (as much as 200 km)

• Propagation speed√gh (much slower near shore)

• Average depth of Pacific or Indian Ocean is 4000 m=⇒ average speed 200 m/s ≈ 450 mph


Sumatra event of December 26, 2004Magnitude 9.1 quake near Sumatra, where Indian tectonic plateis being subducted under the Burma platelet.

Rupture along subduction zone≈ 1200 km long, 150 km wide

Propagating at ≈ 2 km/sec (for ≈ 10 minutes)

Fault slip up to 15 m, uplift of several meters.(Fault model from Caltech Seismolab.)

www.livescience.com


Tsunami simulations

Adaptive mesh refinement is essential

Zoom on Madras harbors with 4 levels of refinement:• Level 1: 1 degree resolution (∆x ≈ 110 km)• Level 2 refined by 8.• Level 3 refined by 8: ∆x ≈ 1.6 km (only near coast)• Level 4 refined by 64: ∆x ≈ 25 meters (only near Madras)

Factor 4096 refinement in x and y.

Less refinement needed in time since c ≈ √gh.

Runs in a few hours on a laptop. Movie


Satellite data

Jason-1 Satellite passed over the Indian Ocean during thetsunami event.

Surface height on two passes(one a week before)

Disparity shows tsunami:


Comparison of simulation with satellite data


Cascadia subduction fault

• 1200 km long off-shore fault stretching from northern California tosouthern Canada.

• Last major rupture: magnitude 9.0 earthquake on January 26, 1700.

• Tsunami recorded in Japan with run-up of up to 5 meters.

• Historically there appear to be magnitude 8 or larger quakes every 500years on average.


Tsunami Deposits

From: K. Jankaew, B. F. Atwater, Y. Sawai, M. Choowong, T. Charoentitirat,M. E. Martin and A. Prendergast, Medieval forewarning of the 2004 IndianOcean tsunami in Thailand, Nature 455 (2008), 1228–1231


Tsunami Deposits

From: J. Bourgeois, Chapter 3 of The Sea, Volume 15:Tsunamis, Harvard University Press, 2009.


Cascadia event simulations

Magnitude 9.0 earthquake similar to 1700 event.

Dave Alexander, Bill Johnstone, SpatialVision, Vancouver, BC

Barbara Lence, Civil Engineering, UBC

Movies:

Vancouver Island and Olympic Penninsula

Ucluelet


Hazard Study for Tofino, BC








Comparison to NOAA model

Thanks to Tim Walsh (WA State DNR)


Tsunami from 27 Feb 2010 quake off Chile


Cross section of Atlantic Ocean & tsunami


Inundation of Hilo, Hawaii

Using 5 levels of refinement with ratios 8, 4, 16, 32.

Resolution ≈ 160 km on Level 1 and ≈ 10m on Level 5.

Total refinement factor: 214 = 16, 384 in each direction.

With 15 m displacement at fault:



Inundation of Hilo, Hawaii

Using 5 levels of refinement with ratios 8, 4, 16, 32.

Resolution ≈ 160 km on Level 1 and ≈ 10m on Level 5.

Total refinement factor: 214 = 16, 384 in each direction.




Godunov’s Method for qt + f(q)x = 0

1. Solve Riemann problems at all interfaces, yielding wavesWpi−1/2 and speeds spi−1/2, for p = 1, 2, . . . , m.

Riemann problem: Original equation with piecewise constantdata.


Wave-propagation viewpoint

For linear system qt+Aqx = 0, the Riemann solution consists of

wavesWp propagating at constant speed λp.λ2∆t

W1i−1/2

W1i+1/2

W2i−1/2

W3i−1/2

Qi −Qi−1 =m∑p=1

αpi−1/2rp ≡

m∑p=1

Wpi−1/2.

Qn+1i = Qni −

∆t∆x[λ2W2

i−1/2 + λ3W3i−1/2 + λ1W1

i+1/2

].


Upwind wave-propagation algorithm

Qn+1i = Qni −

∆t∆x

m∑p=1

(λp)+Wpi−1/2 +

m∑p=1

(λp)−Wpi+1/2

or

Qn+1i = Qni −

∆t∆x

[A+∆Qi−1/2 +A−∆Qi+1/2

].

where the fluctuations are defined by

A−∆Qi−1/2 =m∑p=1

(λp)−Wpi−1/2, left-going

A+∆Qi−1/2 =m∑p=1

(λp)+Wpi−1/2, right-going


Upwind wave-propagation algorithm

Qn+1i = Qni −

∆t∆x

m∑p=1

(spi−1/2)+Wpi−1/2 +

m∑p=1

(spi+1/2)−Wpi+1/2

where

s+ = max(s, 0), s− = min(s, 0).

Note: Requires only waves and speeds.

Applicable also to hyperbolic problems not in conservation form.

For qt + f(q)x = 0, conservative if waves chosen properly,e.g. using Roe-average of Jacobians.

Great for general software, but only first-order accurate (upwindmethod for linear systems).


Wave-propagation form of high-resolution method

Qn+1i = Qni −

∆t∆x

m∑p=1

(spi−1/2)+Wpi−1/2 +

m∑p=1

(spi+1/2)−Wpi+1/2

− ∆t

∆x(F̃i+1/2 − F̃i−1/2)

Correction flux:

F̃i−1/2 =12

Mw∑p=1

|spi−1/2|(

1− ∆t∆x|spi−1/2|

)W̃pi−1/2

where W̃pi−1/2 is a limited version ofWp

i−1/2 to avoid oscillations.

(Unlimited waves W̃p =Wp =⇒ Lax-Wendroff for a linearsystem =⇒ nonphysical oscillations near shocks.)


Summary of wave propagation algorithms

For qt + f(q)x = 0, the flux difference

A∆Qi−1/2 = f(Qi)− f(Qi−1)

is split into:

left-going fluctuation: A−∆Qi−1/2, updates Qi−1,right-going fluctuation: A+∆Qi−1/2, updates Qi,Waves: Qi −Qi−1 =

∑αprp =

∑Wp

Often take A±∆Qi−1/2 =∑

(sp)±Wp.

f-wave formulation: Bale, RJL, Mitran, Rossmanith, SISC 2002

f-waves: f(Qi)− f(Qi−1) =∑βprp =

∑Zp


(sgn(sp))±Zp.

In either case, limiters are applied to waves or f-waves for usein high-resolution correction terms.




A∆Qi−1/2 = f(Qi)− f(Qi−1)

is split into:


∑αprp =

∑Wp


(sp)±Wp.



∑Zp


(sgn(sp))±Zp.





A∆Qi−1/2 = f(Qi)− f(Qi−1)

is split into:


∑αprp =

∑Wp


(sp)±Wp.



∑Zp


(sgn(sp))±Zp.



Incorporating source term in f-waves

qt + f(q)x = ψ(q)σx(x)

Concentrate source at interfaces: Ψi−1/2(σi − σi−1)

Split f(Qi)− f(Qi−1)− (σi − σi−1)Ψi−1/2 =∑

pZpi−1/2

Use these waves in wave-propagation algorithm.

Steady state maintained:

If f(Qi)−f(Qi−1)∆x = Ψi−1/2

(σi−σi−1)∆x then Zp ≡ 0

Near steady state:

Deviation from steady state is split into waves and limited.


Incorporating source term in f-waves

qt + f(q)x = ψ(q)σx(x) =⇒ Ψi−1/2(σi − σi−1)

Question: How to average ψ(q) between cells to get Ψi−1/2?

For some problems (e.g. ocean-at-rest) can simply usearithmetic average.

Ψi−1/2 =12

(ψ(Qi−1) + ψ(Qi)).


Shallow water equations with bathymetry B(x)

ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)

Ocean-at-rest equilibrium:

ue ≡ 0, he(x) +B(x) ≡ η̄ = sea level.

UsingΨi−1/2 = −g

2(hi−1 + hi)

gives exactly well-balanced method, but only because hydrostatic pressure isquadratic function of h:

f(Qi)− f(Qi−1)−Ψi−1/2(Bi −Bi−1) =

=(

12gh2

i −12gh2

i−1

)+g

2(hi−1 + hi)(Bi −Bi−1)

=g

2(hi−1 + hi)((hi +Bi)− (hi−1 +Bi−1))

= 0 if hi +Bi = hi−1 +Bi−1 = η̄.



ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)




2(hi−1 + hi)


f(Qi)− f(Qi−1)−Ψi−1/2(Bi −Bi−1) =

=(

12gh2

i −12gh2

i−1

)+g

2(hi−1 + hi)(Bi −Bi−1)

=g

2(hi−1 + hi)((hi +Bi)− (hi−1 +Bi−1))

= 0 if hi +Bi = hi−1 +Bi−1 = η̄.



ht + (hu)x = 0

(hu)t +(hu2 +

12gh2)x

= −ghBx(x)




2(hi−1 + hi)


f(Qi)− f(Qi−1)−Ψi−1/2(Bi −Bi−1) =

=(

12gh2

i −12gh2

i−1

)+g

2(hi−1 + hi)(Bi −Bi−1)

=g

2(hi−1 + hi)((hi +Bi)− (hi−1 +Bi−1))

= 0 if hi +Bi = hi−1 +Bi−1 = η̄.


Some other applications of depth-averaged eqns

Shallow water equations:• Storm surges, hurricanes (K. Mandli)• River flooding• Dam breaks (D. George)

More complex flows:• Flow on steep terrain (D. George, K. Mandli)• Debris flows (D. George, R. Iverson, R. Denlinger)• Lahars, landslides and avalanches• Multi-layer, internal waves (J. Kim, K. Mandli)• Underwater landslides / tsunami generation (J. Kim)• Pyroclastic flows and surges (M. Pelanti)• Sediment transport• Lava flows


Some other applications of depth-averaged eqns

Shallow water equations:• Storm surges, hurricanes (K. Mandli)• River flooding• Dam breaks (D. George)

More complex flows:• Flow on steep terrain (D. George, K. Mandli)• Debris flows (D. George, R. Iverson, R. Denlinger)• Lahars, landslides and avalanches• Multi-layer, internal waves (J. Kim, K. Mandli)• Underwater landslides / tsunami generation (J. Kim)• Pyroclastic flows and surges (M. Pelanti)• Sediment transport• Lava flows


Some references

GeoClaw: www.clawpack.org/geoclaw

Recent papers with references and codes:www.clawpack.org/papers/

• A Well-Balanced Path-Integral f-wave Method for HyperbolicProblems with Source Terms , to appear in J. Sci. Comput.

• The GeoClaw software for depth-averaged flows withadaptive refinement, by M. J. Berger, D. L. George, RJL,and K. M. Mandli,

www.clawpack.org/links/awr10/

• Tsunami modeling with adaptively refined finite volumemethods, by RJL, D.L. George, and M.J. Berger, ActaNumerica 2011,

www.clawpack.org/links/an11/


http://www.clawpack.org/geoclaw

http://www.clawpack.org/papers/

http://www.amath.washington.edu/~rjl/pubs/wbfwave10/index.html

http://www.amath.washington.edu/~rjl/pubs/wbfwave10/index.html

http://www.clawpack.org/links/awr10/

http://www.clawpack.org/links/an11/


Also in Vancouver:

Links:http://www.sfu.ca/WAVES/http://www.iciam2011.com/


http://www.sfu.ca/WAVES/

http://www.iciam2011.com/

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The GeoClaw Software for Tsunamis and Other Hazardous Flows · PDF fileThe GeoClaw Software for Tsunamis and Other Hazardous Flows ... Originally f77 with Matlab graphics. ... Tsunami