Computational Methods for Storm Surge

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Columbia University

Kyle T. Mandli

Department of Applied Physics and Applied Mathematics

Kyle T. Mandli

Storm Surge2

Source: Jocelyn Augustino / FEMA - http://www.fema.gov/photdata/original/38891.jpg

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Storm Surge

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Wind

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Storm Surge Generation

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Storm Surge Forcing

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rPAAtmospheric Pressure

(2⌦ sin�)⇥ uCoriolis

Short (deep) water waves

Bottom Friction

Wind

�b = hugn2

h4/3

�u2 + v2

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Storm Surge Model

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ht

+ (hu)x

+ (hv)y

= 0

�ghbx

+ fhv � h

⇢(P

A

)x

+1

⇢(⌧

sx

� ⌧bx

)

(hu)t

+

✓hu2 +

1

2gh2

◆

x

+ (huv)y

=

�ghby � fhu� h

⇢(PA)y +

1

⇢(⌧sy � ⌧by)

(hv)t + (huv)x +

�hv2 +

1

2gh2

�

y

=

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Storm Representation

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Montage of Hurricane Ike. Image courtesy of CIMMS: http://cimss.ssec.wisc.edu/tropic2

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Holland Hurricane Model

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Holland, G. J. An Analytic Model of the Wind and Pressure Profiles in Hurricanes. Monthly Weather Review 108, 1212-1218 (1980)

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Questions9

Hansen, J., R. Ruedy, M. Sato, and K. Lo, 2010: Global surface temperature change, Rev. Geophys., 48, RG4004, doi:10.1029/2010RG000345.

GISTEMP Team, 2015: GISS Surface Temperature Analysis (GISTEMP). NASA Goddard Institute for Space Studies. Dataset accessed 2015-10-13 at http://data.giss.nasa.gov/gistemp/.

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How does sea-level rise effect surge?

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Geophysical Fluid Dynamics Laboratory - Climate Impact of Quadrupling CO2, Princeton, NJ, USA: NOAA GFDL, http://www.gfdl.noaa.gov/climate-impact-of-quadrupling-co2

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NASA and NHC

Will dangerous storms become more frequent?

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NOAA

Will dangerous storms become more powerful?

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Can we forecast events?13

NOAA - NHC

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Can we quantify uncertainty?

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7200 10800 14400−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2Gauge 21401

Time (s)

Wate

r su

rface

ele

vatio

n (

m)

µ

µ ± 2σ

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How do we protect ourselves?

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Donar Reiskoffer

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Can we protect ourselves?16

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Needed Capabilities

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Sea-Level RiseChange in storm frequency and intensity

OptimalityRegion implications

Ensembles ~ 106

Disparate scalesBathymetry

Protection Strategies

RequirementsClimate

Forecasting

Uncertainty in storm forecastsCritical Decision Making

Protection

Approach

Adaptive Mesh Refinement

H-box MethodsSub-scale Models

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Things we are doing

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Jeff Schmaltz, MODIS Rapid Response Team, NASA/GSFC

Adaptive Mesh Refinement

Sub-Scale Barrier Modeling

Two-Layer Shallow Water

Air-Sea Interaction

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Adaptive Mesh Refinement19

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Adaptive Discretization

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Level 3

Level 2

Level 1x

y

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GeoClaw

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www.clawpack.org

Randy LeVeque(U. Washington)

Marsha Berger (NYU)

Dave George(USGS)

Berger, M. J., George, D. L., LeVeque, R. J. & Mandli, K. T. The GeoClaw software for depth-averaged flows with adaptive refinement. Advances in Water Resources 34, 1195–1206 (2011).

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Chile 2010 Tsunami

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Chile 2010 Tsunami

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Hurricane Ike

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NASA

Level Refinement Factor Resolution (m)

1 25250

2 2 12600

3 2 6300

4 2 3150

5 6 525

6 4 130

7 4 33

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Domain and Friction

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0

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ADCIRC Comparison

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Hope, M. E. et al. Hindcast and validation of Hurricane Ike (2008) waves, forerunner, and storm surge. J. Geophys. Res. Oceans 118, 1–37 (2013).

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Gauge Locations

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Gauge Comparisons

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Gauge Comparisons

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Gauge Comparisons

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Gauge Comparisons

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Patch Statistics

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Timings

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Package Bathymetry Computer Cores Wall Time Core Time

ADCIRC Fine Stampede 4000 35 minutes 2333 hours

GeoClaw Fine Stampede 16 2 hours 32 hours

GeoClaw Coarse Laptop 4 2 hours 8 hours

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Surge Protection38

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Surge Protection

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Where do we put protection?

How high does it need to be?

Is it regionally destructive?

Is this optimal?

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Flood Protection

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MRGO and the Intracoastal Waterway and Industrial (IHNC) Canals, Eastern New Orleans, Louisiana (New Orleans, U.S. Army Corp of Engineers)

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Flood Protection

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The entrance the 168th Street subway station in New York City. Taken by Seidenstud - 2006

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Failure is Bad

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Representation of Barriers

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Jiao Li

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Riemann Solver

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Riemann Solver

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Riemann Solver

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Riemann Solver

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Riemann Solver

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Riemann Solver

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Redistribution of Waves

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f(Qk+1)� f(Qk) =MwX

p=1

�pk+1/2r

p ⌘MwX

p=1

Zpk+1/2

Z1 = �1r1

Z2 = �2r2 Z3 = �3r3

Z4 = �4r4

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Redistribution of Waves

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b� = [�1 + �1, 0, 0,�4 + �2]

R · b� = R · �

(�1 = (s4�s2)�2+(s4�s3)�3

s4�s1 ,

�2 = (s2�s1)�2+(s3�s1)�3

s4�s1 .

R = [r1, r2, r3, r4], � = [�1,�2,�3,�4]

Z1 = (�1 + �1)r1 Z4 = (�4 + �4)r4

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Riemann Solver

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Riemann Solver

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Dry Test

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Comparisons

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Under Pressure

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⇢g(⌘ � z)

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Water on the Wall

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Use other side of wall as ghost fluid depth

Use averages of states

Qm =QL

k+1/2 +QRk+1/2

2QL

W = Qm � (BW �BL)

QRW = Qm � (BW �BR)

QLW = QL

k+1/2 � (BW �BL)

QRW = QR

k+1/2 � (BW �BR)

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Ghost Water

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The CFL Strikes Back

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H-Box

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H-box: 1D

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Wave-Propagation Prospective

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Z�i+1/2

Z�i�1/2 Z�

i+3/2 Z+i+3/2Z+

i+1/2

Z+i�1/2

�t1

�t2

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Wave-Propagation Prospective

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Z�i�1/2 Z+

i+3/2

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Wave-Propagation Prospective

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Z�i�1/2 Z+

i+3/2

Update

Q⇤i Q⇤

i+1

Solve

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Wave-Propagation Prospective

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Z�i�1/2 Z+

i+3/2

Q⇤i Q⇤

i+1

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H-Box Methods

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i i+ 1

j

j + 1

j � 1

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H-Box Methods

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i i+ 1

j

j + 1

j � 1

i i+ 1

j

j + 1

j � 1

Normal Transverse

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Outlook

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Storm Surge Forecasting

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Mandli, K. T. & Dawson, C. N. Adaptive Mesh Refinement for Storm Surge. Ocean Modelling 75, 36–50 (2014).

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UQ and Data Assimilation70

Sraj, I., Mandli, K. T., Knio, O. M., Dawson, C. N., & Hoteit, I. Uncertainty Quantification and Inference of Manning’s Friction Coefficient using DART Buoy Data during the Tohoku Tsunami. Ocean Modelling (2014).

7200 10800 14400−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2Gauge 21401

Time (s)

Wa

ter

surf

ace

ele

vatio

n (

m)

µ

µ ± 2σ

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Adding Depth to the Shallow Water Equations

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Two-Layer Shallow Water

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Mandli, K. T. A Numerical Method for the Two Layer Shallow Water Equations with Dry States. Ocean Modelling 72, 80–91 (2013).Mandli, K. T. A Numerical Method for the Two Layer Shallow Water Equations with Dry States. Ocean Modelling 72, 80–91 (2013).

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Hyperbolicity

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frequency, f10010−1

Primarydisturbingforce

10−210−310−410−5 101

Wind

Storm systems, earthquakes

Sun, Moon

Energy,E(f)

· · ·

24hr

12hr

5min

30sec

1sec

0.1secPeriod

“Wind-sea” spectrum

(see Figure 3)

Primaryrestoringforce

Gravity

Coriolis force Surface tension

0.25sec

Swell

Spectral models

Air-Sea Waves74

E. Kubatko, adapted from Munk, W. H. Origin and generation of waves. Coastal Engineering Proceedings (1950).Colton, C. J. , Mandli K.T., Kubatko, E., Fractally homogeneous, air-sea turbulence with Frequency-integrated, wind-driven gravity waves. Submitted to Ocean Modelling.

time8/1 8/2 8/3 8/4 8/5 8/6 8/7 8/8 8/9 8/10 8/11 8/12 8/13 8/14 8/15 8/16 8/17 8/18 8/19 8/20 8/21 8/22 8/23 8/24 8/25 8/26 8/27 8/28 8/29 8/30 8/31 9/1

Hs(m

)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Lake Erie August 2011

Buoy 45005Freq-int (5.13 mins)SWAN (207 mins)Observed

SWAN - 3.45 hoursMoment Eqs. - 5.13 minutes

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Dirty AMR Secrets75

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Ongoing Work

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Burstedde, C., Calhoun, D. A., Mandli, K. & Terrel, A. R. ForestClaw: Hybrid forest-of-octrees AMR for hyperbolic conservation laws. in ParCo 2013

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Discipline Problem descrip-tion

Representativesolver

Solving approach Dominant kernel Dominant kerneltime

Fluid dynamics 3D transonic tran-sient laminar vis-cous flow

SPEC CPU2006410.bwaves (test)

finite difference discretization with im-plicit time stepping on the compressible,viscous Navier-Stokes equations

Bi-CGstab 76.68 + 11.74%

Magneto-hydrodynamics

2D Hartmannproblem

OpenFOAM finite difference discretization on incom-pressible, viscous Navier-Stokes equa-tion, coupled with Maxwell’s equations

preconditioned CG 45.78%

Fluid dynamics lid-driven cavityflow

OpenFOAM finite volume discretization on incom-pressible, viscous Navier-Stokes equa-tions

preconditioned CG 13.06%

Engineering me-chanics

Cook’s membrane deal.II finite element discretization with nonlin-ear spring forces

Solving Helmholtz elliptic PDE usingpreconditioned SOR and CG

15.3%

Table 1: Function profile of some nonlinear PDE solvers. In all solvers the dominant kernel is a system of equation solvingsubroutine working on a nonlinear system. The computing time is more concentrated in linear algebra when space is discretizedusing structured grids such as finite difference. Finite volume and finite elements are less structured. The resulting less regularmemory accesses shift computation time away from solving systems of equations.

∫∫

u

f ’(u) =2u+1

δ ≈ f(u)/f ’(u)

DAC

DAC

f(u) =u2+u-RHS

-

-

FUNCTION

QUOTIENT

u integrator δ integrator

DERIVATIVE

Figure 2: An analog circuit for continuous Newton’s methodfor solving Equation 1. Clockwise from the center left, themajor blocks are analog function units: an integrator forholding the present guess of u, a block for evaluating thederivative, a block for doing division using gradient descent,and a block for evaluating the nonlinear function. Integratorsare physically realized using capacitors. Constant values aregenerated using digital-to-analog converters (DACs). Num-bers are summed by joining wires to sum currents. Numbersare represented as analog current and voltage, and the cir-cuit values change continuously in time, with no concept ofclock cycles or steps.

The quotient would be negated and fed to the input of theu(t) integrator as du

dt , the rate of change of u. The valueschange continuously, without any notion of clock cycles ortime steps, and so the whole system is described as an ODE.When the continuous Newton’s method converges, the in-puts to the integrators tend toward zero, the output of theintegrators are steady, and at that point we can measure theoutput using analog-to-digital converters.

Semilinear PDE test case: We test the continuous New-ton’s method running on an analog accelerator to solve Equa-tion 1. We know the solutions should be the roots of aquadratic equation with the closed form solution:

u =�1±

p1+4RHS2

The square root causes real solutions to exist only whenRHS ��0.25.

Analog accelerator result: The leftmost panel of Fig-

ure 3 shows the analog solution compared to the correct so-lution, which has the shape of half of a parabola. Lookingat the closed form solution, we’d expect two valid solutionswhen RHS>� 1

4 . The analog accelerator returns only one ofthe two expected results, depending on the configuration ofthe accelerator. The bottom half of the plot needs the deriva-tive to be less than zero, which is not compatible with theway we find the quotient. To allow the derivative be nega-tive, we can modify the Newton’s method to be

up+1 = up �� f (up)

� f 0(up)

The continuous Newton’s method then returns the bottomhalf the parabola.

Initial conditions: When we use the analog accelerator,we need a correct choice of the initial conditions for the twointegrators, one storing r0 and the other storing the quotientf (r0)f 0(r0)

. We show the influence of these choices in the centerand right panels of Figure 3. Viewed edge-on, these picturesshow the parabolic shape in the left panel of Figure 3; on thenew third axis, we plot the choice of initial conditions. Thecenter panel of Figure 3 shows that the initial condition onthe quotient integrator has no special meaning, and a neu-tral choice of zero offers the best guarantee of convergingto a solution. So in the case studies in the rest of this pa-per we will initialize the quotient integrators to zero. Theright panel of Figure 3 shows that the best initial conditionon the integrator containing ~u = r0 is a value close to theactual solution. Luckily, in the context of solving nonlinearPDEs, there is usually a good initial guess coming from theboundary conditions of the PDE.

3.3 Nonlinear root-finding: hybrid analog &digital

Thus far, we have considered solving nonlinear equationsin digital and analog separately. The digital methods allowuse of binary floating point numbers, which have higher pre-cision and accuracy, but encounter problems in selecting aNewton step size. The analog methods have more reliabil-ity, but the computational results have lower accuracy andprecision. A standalone analog accelerator may be useful ingraphics and embedded systems applications where approxi-mate solutions for nonlinear PDEs may be useful. But nearly

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Analog/Digital Computing77