workshop on tsunami hydrodynamics in a large river...

NEOWAVENon-hydrostatic Evolution of Ocean WAVE

Yoshiki Yamazaki and Kwok Fai Cheung

Department of Ocean and Resources EngineeringUniversity of Hawaii at Manoa, Honolulu, HI, U.S.A.

August 14 ~ 15, 2011Oregon State University, Corvallis, Oregon

Workshop on Tsunami Hydrodynamics in a Large River

1. NEOWAVE● Theoretical Formulation● Numerical Scheme

2. RESULTS AND DISCUSSIONS● Tide● Wave Dispersion

3. CONCLUSIONS AND FUTURE STUDIES

OUTLINE

Governing Equations● Depth-integrated, Non-hydrostatic Equation

• Consideration of Weakly Wave Dispersion through Non-hydrostatic Pressure.(Stelling and Zijlema, 2003; Yamazaki et al., 2009 & 2011)

Numerical Schemes● Semi-implicit, Finite Difference (FD) Model• Explicit Hydrostatic solution• Implicit Non-hydrostatic solution

● Momentum Conserved Advection (MCA) Scheme • Shock Capturing Scheme for FD Models

(Stelling and Duinmeijer, 2003; Yamazaki et al., 2009 & 2011)

● Two-Way, Grid-Nesting Scheme• Implementation of the inter-grid boundary condition to

describe non-hydrostatic and discontinuous flows.

NEOWAVE

Governing Equations● Variables Definition for Free Surface Flow

NEOWAVE

: total water depth (flow depth): surface elevation: still water depth: bottom displacement

where

( )η−+ζ= hD

η

Dζh

h

sea bottom

z

x, y

U, V

W

η

ζ ζ

h

NEOWAVE

( )D

VUVfhyD

qyq

yg

yVV

xVU

tV 22

21

21 +

−η+−ζ∂∂

−∂∂

−∂ζ∂

−=∂∂

+∂∂

+∂∂

Continuity equation

z-momentum equation

y-momentum equation

x-momentum equation

( )D

VUUfhxD

qxq

xg

yUV

xUU

tU 22

21

21 +

−η+−ζ∂∂

−∂∂

−∂ζ∂

−=∂∂

+∂∂

+∂∂

Dq

tW

=∂∂

0)()()(=

∂∂

+∂

∂+

∂η−ζ∂

yVD

xUD

t

Governing Equations● Depth-integrated, Non-hydrostatic Equations in Cartesian Grid

NEOWAVE

( )D

VUVfhyD

qyq

yg

yVV

xVU

tV 22

21

21 +

−η+−ζ∂∂

−∂∂

−∂ζ∂

−=∂∂

+∂∂

+∂∂

Continuity equation

z-momentum equation

y-momentum equation

x-momentum equation

( )D

VUUfhxD

qxq

xg

yUV

xUU

tU 22

21

21 +

−η+−ζ∂∂

−∂∂

−∂ζ∂

−=∂∂

+∂∂

+∂∂

Dq

tW

=∂∂

0)()()(=

∂∂

+∂

∂+

∂η−ζ∂

yVD

xUD

t

Governing Equations● Non-linear, Shallow Water Equations

Vertical Datum The original DEM data’s vertical datum is NAVD

(1). At near river mouth, Astoria, Tongue Point, Columbia River, OR

MHHW 3.305mMTL 2.068m MSL 2.054m MLLW 0.681m NAVD 0.615m

MTL NAVD2.068m -0.615m = 1.453m

(2). At Longview, Columbia River, WA

MHHW 2.148MTL 1.429MSL 1.385MLLW 0.752NAVD -0.764

MTL NAVD1.429m – (–0.764m) = 2.193m

In this BM, we use the average value as(1.453m +2.193m)/2 = 1.823m ~1.8 m

MHHW : Mean Higher-High WaterMTL : Mean Tide LevelMSL : Mean Sea LevelMLLW : Mean Lower-Low WaterNAVD : North American Vertical Datum

Longview

Astoria

NOAA NOS/CO-OPS http://tidesandcurrents.noaa.gov/station_retrieve.shtml?type=Datums

Original Bathymetry Data

Modified Bathymetry Data

Bathymetry Data Modification

Computational Domain • Low Tide

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

80m grid2077 x 1195

• Mean Tide Level

Computational Domain

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

80m grid2077 x 1195

• High Tide

Computational Domain

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

80m grid2077 x 1195

• Low Tide

Computational Domain● Grid Resolution near the Boundaries

• Mean Tide Level

Computational Domain● Grid Resolution near the Boundaries

• High Tide

Computational Domain● Grid Resolution near the Boundaries

f = 0.025 f = 0.018

Bottom Friction● Darcy’s Friction Factor

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

f = 0.0275 f = 0.02

Bottom Friction● Darcy’s Friction Factor

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

— : computed data — : initial condition

Surface elevation (m) Horizontal velocity, u (m/s) Horizontal velocity, v (m/s)

Initial Condition● River Mouth Boundary Conditions

(1) (2)

(1)

(2)

— : High tide — : MTL —: Low tide

Tide Effects● Hydrostatic Solution

(Non-liner Shallow Water Solution)

Surface elevation (m) Horizontal velocity, u (m/s) Horizontal velocity, v (m/s)

— : High tide — : MTL —: Low tide

Tide Effects● Hydrostatic Solution

(Non-liner Shallow Water Solution)

Surface elevation (m) Horizontal velocity, u (m/s) Horizontal velocity, v (m/s)

— : Hydrostatic solution — : Non-hydrostatic solution

Wave Dispersion Effects● Hydrostatic and Non-hydrostatic Solutions

•Mean Tide Level•No Discharge

Surface elevation (m) Horizontal velocity, u (m/s) Horizontal velocity, v (m/s)

Maximum Amplitude • Low Tide (Bathymetry Data)

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

• Low Tide

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

Maximum Amplitude

• Mean Tide Level (Bathymetry Data)

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

Maximum Amplitude

• Mean Tide Level

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

Maximum Amplitude

• High Tide (Bathymetry Data)

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

Maximum Amplitude

• High Tide

Skamokawa

Portland

Intermediateboundary

Bonneville dam

Longview

Rivermouth

Maximum Amplitude

(1). Bottom Friction Tests(2). Grid Refinement Scheme Implementation(3). Natori River for the 2011 Tohoku-oki Tsunami

In this numerical experiment of modeling Columbia River, the effects of wave dispersion, wave breaking, and tide are very minor. The comparison of computed results with difference solutions indicate the non-linear shallow water model is sufficient to model tsunami inundation.

Conclusions and Future Studies● Conclusions

● Future Studies

● Inundation Modeling of Siletz River

Four Cascadia rupture models based on the 2008 National Seismic Hazard Maps.

APPENDIX 1: River Modeling Example

Cheung, K.F., Wei, Y., Yamazaki, Y., and Yim, C.S. (2011). Modeling of 500-year tsunamis for probabilistic design of coastal infrastructure in the Pacific Northwest. Coastal Engineering, 58(10), 970-985.

— : LZ model — : MT model — : BT model — : GA model

Subfault and Slip Distribution

Sea Surface Deformation

APPENDIX 1: River Modeling Example

GA model

● Inundation Modeling of Siletz River

APPENDIX 1: River Modeling Example

Time sequence of Surface elevation (GA model)

Surface elevation at Siletz River Bridge

Surface elevation at Millport Slough Bridge

● Inundation Modeling of Siletz River