07_DW_ADCIRC_2018v2.pdf

(1)

Using ADICRC with Sponge Layers as an Open-ocean

Boundary Treatment in Storm-Surge Models

D. Wirasaet1_{, W. Pringle}1_{, J. Gonzalez}2_{, A. Suhardjo}1_,

B. Joyce1_{, and J. J. Westerink}1

1_{Computational Hydraulics Laboratory}

Department of Civil and Environmental Engineering and Earth Sciences University of Notre Dame, IN

2_{CARICOOS, PR}

(2)

Motivation

Schematic diagram of Southern Louisiana, East Coast, and Puerto Rico Coastal hydrodynamic flow model

Model Domain

ΓOpen Boundary

Model driver:

River Winds Tides

High-resolution in geographical regions of interest.

(3)

Motivation

Surface water level of a Tidal run att= 8.14days

Applying OBC could result in unfavorable effects.

(4)

Motivation

Surface water level of a Tidal run att= 8.14days

Applying OBC could result in unfavorable effects.

(5)

Absorbing-layer type approaches

Extend the computation domain andforcethe computed solution to match a known reference solution.

Two approaches that are widely used:

Perfectly Matched Layers (PMLs) [Hu,Comp & Fluid, 1998, Hesthaven,JCP, 1998]

Minimize reflection

Introduce extra equations of unknown auxiliary variables.

Sponge Layers [Israeli and Orszag,JCP, 1981, Bodony,JCP, 2006, Lavelle et al., Ocean Modelling, 2008]

Do not ensure perfect transmission

(6)

Shallow Water Equations (SWEs) with a Sponge Layer

SWEs in the Generalized Wave Continuity Equation (GWCE) form:

∂2η

∂t2 +τ0

∂η

∂t +∇ ·J−uH· ∇τ0=−τ0σ(x)(η−ηc) − σ(x)

∂η

∂t −

∂ηc

∂t

(1)

∂u

∂t =−(u· ∇)u−fk×u−g∇η−τbu− σ(x)(u−uc) (2)

J=H×(RHS of (2)) +u∂η

∂t +τouH (3)

η= surface elevation g= gravitational acceleration

H=η+btotal water depth f = Coriolis parameter

b= bathymetric depth τb= a bottom stress coefficient

u= (U,V) depth-avg. velocities τ0= GWCE constant

σ(x) - time-independent spatially-varying absorption coefficients;

(7)

Note on Sponge Layer Coefficients

In this work, the following profileσ(x) is considered:

σ(x) =σm

_x

L

n

, whereσm=−

(n+ 1)√gHlog(1/F)

L(xc/L)n+1

(4)

For 1-D linear SWE withf = 0,τb= 0,ηc =uc = 0, the exact solution in sponge zone behaves like

η+

s H

gu∼exp(−γ(x)), whereγ(x) =

x

Z

0

σ(s)

√

gHds.

⇒The sponge profile (4) leads to the out-going wave being reduced by a factor

of F at xc into the sponge layer.

a

Out−going wave

c

a/F

0 X

(8)

Sponge Layers in Tide models: Indian Ocean and Western Pacific

M2 Tides

ADCIRC with OBC TPXO8

(9)

Sponge Layers in Tide models: Indian Ocean and Western Pacific

M2 Tides

ADCIRC with Sponge Layer TPXO8

(10)

Wind-Forced Models

Influence of wind forcing and atmospheric pressure:

LHS of Momentum Eqs =−g∇η− 1

ρ0

∇pa+

1 ρ0H

τs +. . . (5)

wherepa(x,t) is atmosphere pressure andτs the wind stresses,

τs = (τsx, τsy)≈ρaCD|ug|ug, ug(x,t)–10m wind velocity

(11)

Open-Ocean boundary treatments (not consider tide)

1 _{The inverted barometer effect as the BCs on the open-ocean boundary}

(OBCIB):

η_OB=

pao−pa

ρ0g

(6)

wherepao is the background pressure.

2 In modeling with sponge layers, since the solution in sponge zones are

unknown, the reference solution is set to zero

ηc(x,t) = 0, uc(x,t) =0 (7)

Treating the atmospheric forcing terms in sponge zones:

• SPG-A:τs and∇pa are included in the sponge zones.

(12)

Test problem

1 24 30 36 42 48 54 60 66 coast coast Ω x

Ω= [−L/2, L/2]×[0, L/2] Strom Track (Elapsed time in hours)

bathymetry

0 100 200 300 400 500

y (km) -2000 -1500 -1000 -500 0 -b (m)

Winds (fraction of geostrophic ):

pa=pao−δpaexp

−2(X 2₊_Y2₎

R2

c

, (ug,vg) =

α fρa

−∂pa

∂y ,

∂pa

∂x

X =x−x0−u0t, Y =y−y0−v0t

L= 1000 km;pao= 1.013 bar;δpa= 50 mbar;Rc = 400 km ;

p

u2

0+v02= 15 m/s;f = 1.2×10−4 m-1;ρa= 2.25 kg/m3;σ= 0.4.

Minimum pressureat the storm center = 0.963 bar;maximum wind velocity≈

(13)

Numerical Experiments

Nonlinear 2D SWEs.

Quadratic bottom frictionτb=cd|u|u,cd= 0.0025.

Grid size≈20km. Sponge-region width = 100km.

-6 -4 -2 0 2 4 6

x _×₁₀5

0 1 2 3 4 5

y

×105

Semi-implicit in time with ∆t = 60. No ramping.

Solution for a run on an extended domain (Ω = [−Le,Le]×[0,L],

(14)

Surface elevation & Velocity

Ref OBC-IB SPG-B

t = 36 hr

0.2

0.4

-5 0 5

x ×105

0 1 2 3 4 5 y

×105 t = 36 hr

0

0.2 0.4

-5 0 5

x ×105

0 1 2 3 4 5 y

×105 t = 36 hr

0 0

0

0.2

-5 0 5

x ×105

0 1 2 3 4 5 y ×105

t = 48 hr

0 0.2 0.2 0.2 0.2 0.4

0.40.6 0.81 1.21.41.61.8 0.6

-5 0 5

x ×105 0 1 2 3 4 5 y

×105 t = 48 hr

0 0 0 0.2 0.2 0.2 0.2 0.2 0.4

0.4 0.81 1.21.41.6 0.6

-5 0 5

x ×105 0 1 2 3 4 5 y

×105 t = 48 hr

0 0 0 0.2 0.2 0.2 0.2 0.4

0.40.6 0.81 1.21.41.6 0.6

-5 0 5

x ×105 0 1 2 3 4 5 y

×105

t = 60 hr

0 0 0 0 0 0.2

0.2 0.4_{0.6 0.81 1.21.4}

1.61.82

-5 0 5

x ×105 0 1 2 3 4 5 y

×105 t = 60 hr

-0.20 0 0 0 0 0 0 0 0.2 0.2

0.4 0.60.811.21.41.61.82

-5 0 5

x ×105 0 1 2 3 4 5 y

×105 t = 60 hr

0 0 0 0 0 0 0.2 0.2 0.4 0.60.811.21.41.61.82

-5 0 5

x ×105 0 1 2 3 4 5 y

(15)

Global integral quantities

• Excess massRηdx/|Ω| •EnergyR(gη+ 1/2H|u|2₎_d_x_/_|_Ω_|

10 20 30 40 50 60 70

tt (hr) -0.1 0 0.1 0.2 0.3 Excess mass Ref OBC-IB SPGA SPGB

10 20 30 40 50 60 70 tt (hr) 0 2 4 6 8 10 12 14 Mechanical energy Ref OBC-IB SPGA SPGB

•qR

(η−ηref)2dx/|Ω| •

q R

|u−uref|2dx/|Ω|

10 20 30 40 50 60 70

tt (hr) 0 0.5 1 1.5 2 2.5 3 | ζ - ζref |/A ×10-7 OBC-IB SPGA SPGB

10 20 30 40 50 60 70

tt (hr) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

| u - u

ref

|/A

×10-7

(16)

Storm-surge Hindcast: Hurricane Maria

(17)

PRVI Grid with a sponge layer

Original PRVI Model:

Good resolution along the East coast and the northern Gulf of Mexico

Focus on the Caribbean, especially, around Puerto Rico and Virgin Islands (PRVI).

fine resolution carefully selecting Mannings’n values.

(18)

PRVI Grid with a sponge layer

PRVI with Sponge Layer:

Good resolution along the East coast and the northern Gulf of Mexico

Focus on the Caribbean, especially, around Puerto Rico and Virgin Islands (PRVI).

fine resolution carefully selecting Mannings’n values.

(19)

Model details

Model forcing: winds, atmospheric pressure, and tides.

Wind stresses and atmospheric pressure are based on wind data of Oceanweather, Inc (OWI).

Both tidal potential functions (TIP) and self attraction and loading (SAL) are included.

Dissipation: Mannings’n bottom friction and linear bottom friction parameterizing internal tides.

LHS of Momentum Eq.=. . .+ gn

2

H1/3 |u|

Hu+Fitu+. . .

In sponge zones, the reference solution is set to the TPXO8-Atlas global tidal solution:

sc(x,t) =

X

n

fnAs,ncos(ωs,nt−φs,n+θn), s=ζ,u,v (8)

wherefn,θn- nodal factor and equilibrium argument. 8 principal

(20)

Some run parameters

Run period Sep 9-23, 2017 (14 days) Spin-up time 5 days

Time step 1 second, semi-implicit mode

Bottom friction Mannings’n values: from 0.016 to 0.025 in open water with lower limitcd= max(gn2/H1/3,0.002)

phy. parameter f = 1.45×10−4_sin(_φ_{) 1/s,}_g_{= 9}_._{81 m/s}2_, p0a= 1.013 bar

Wind stress Garratt’s formula forCD.

gn2_/_H4/3

(21)

Water elevation & Wind Velocity

(22)

Water elevation & Wind Velocity

(23)

Water elevation & Wind Velocity

(24)

Water elevation & Wind Velocity

(25)

Water elevation & Wind Velocity

(26)

Water level at observation station

(27)

Water level at observation station

(28)

Water level at observation station

(29)

Water level at observation station

(30)

Summary

Implement and assess the use of sponge layers as a boundary treatment in the ADCIRC solver.

The approaches yields accurate solution and are rather robust for linear and nonlinear SWE.

The placement of the open boundaries in a basin-scale tide model becomes less of an issue in comparison to OBCs.

Circumvent a failure that happens near the open boundary in certain models including the advection term.

Challenges in using a sponge layer for realistic applications involve determining reference solutions (ηc,uc), a length of the layer, and the sponge coefficientσ(x).

The sponge layers with a global inverse tidal solution as a reference solution perform rather well in a realistic basin-scale tide model.

The performance of the sponge layers to wind-forced flow depends on problems considered.