Tsunami modeling

Philip L-F. Liu Class of 1912 Professor School of Civil and Environmental Engineering Cornell University Ithaca, NY USA

PASI 2013: Tsunamis and storm surges Valparaiso, Chile January 2-13, 2013

1/17/13 1 Mathematical models for tsunami waves

1/17/13 2 Model Types

• Nonlinear Shallow Water Equations

– Linear Shallow Water Equations

• Boussinesq-type Equations

• Computational Fluid Dynamics – RANS, SPH, LES, DNS

1/17/13 3 CFD models o Navier-Stokes equations

where is the velocity vector and p the pressure. o Boundary conditions and initial conditions 1. On the free surface, zxyt = (, ,) the kinematic boundary condition Dz()∂∂∂ =++=0, or uvw Dt∂∂∂ t x y

1/17/13 4 2. On the free surface, the dynamic boundary condition requires:

pp==atm 0, on z = ( xyt , , ) 3. On the seafloor, zhxyt = (, ,) , the kinematic boundary condition

Dz() h∂∂∂ h h h =++=0, or uv w Dt∂∂∂ t x y If the time history of the seafloor displacement is prescribed, this boundary condition is linear.

1/17/13 5 o Initial conditions 1. If the time history of the seafloor displacement is prescribed, the initial conditions can be given as: ==0,u 0, at t = 0. 2. If seafloor is stationary, the bottom boundary condition become ∂∂hh uv+== wzhxy, on ( , ) ∂∂xy The initial free surface displacement needs to be prescribed,

(xyt , ,== 0)0 ( xy , ) with u ( xyzt , , , == 0) 0,

where 0 (, xy ) mimics the final seafloor displacement after an earthquake.

1/17/13 6 Approximated long-wave models

o Long-wave assumptions o Depth-integrated equations (reduce 3D to 2D) • Boussinesq-type equations • Shallow-water equations

1/17/13 7 Brief Review on Tsunami Models

• O(months-years) N-S model (3-D) HIGH  Based on Navier-Stokes Equaons (or Euler Equaons for potenal flow) Computational Cost Computational

e.g., Truchas (Los Alamos Lab, USA) Short • Boussinesq Model (H2D) O(days-months) -  Based on Boussinesq-type Equaons for weakly wave Accuracy dispersive waves, e.g. FUNWAVE (Wei and Kirby, 1995; Kirby et al, 1998) CoulWave (Lyne and Liu, 2004) • SWE Model (H2D) O(hours)  based on Shallow Water Equaons for non-dispersive waves, e.g., COMCOT (Liu et al,1995), MOST (Titov et al, 1997) Anuga (Australia) Dispersion is neglected!! LOW TSUNAMI (ARSC, UAF)

1/17/13 8 Long wave approximaons

Consider small amplitude single harmonic progressive wave:

= Aeikx() t gkAcosh k ( z+ h ) uxzt(,,)= eikx() t , cosh kh igkAsinh k ( z+ h ) wxzt(,,)= eikx() t , cosh kh h/L>1/2 coshkz (+ h ) pg= cosh kh g Ckh== tanh kk h/L>1/2

1124 coshkz (+=+ h ) 1[ kz ( + h )] +[ kz ( + h )] + 2 24 1 3 sinhkz (+= h )[ kz ( + h )] +[ kz ( + h )] + 6

1/17/13 9 Shallow water wave equations

• Assumptions: 1. Horizontal scales (i.e., wavelength) are much longer than the vertical scale (i.e., the water depth). Accurate only for very long waves, kh < 0.25 (wavelength > 25 water depths) 2. Ignore the viscous effects.

• Consequences: 1. The vertical velocity, w, is much smaller than the horizontal velocity, (u,v). 2. The pressure is hydrostatic, pg = () z. 3. The leading horizontal velocity is uniform in water depth.

1/17/13 10 • Nonlinear shallow-water wave equations Conservation of mass: ∂∂ ∂ [++hhuhv] ( +) +( +) =0 ∂∂tx ∂ y Momentum equations: ∂u +uug +g =0 t ∂ • Linear shallow–water wave equations ∂ [ ++hh] g[u ] =0 t ∂ ∂u +g =0 ∂t 1/17/13 11 Special case: One-dimensional and constant depth ∂∂ ++=( hu) 0, ∂∂tx ∂∂∂uu ++ug =0. ∂∂txx ∂ which can be rewritten as ∂∂ ++(uC) ( u +20, C) = ∂∂tx ∂∂ +(uC) ( u 2 C) = 0. tx ∂∂ where . Cg=+( h) dx uC±=± are Riemann invariants along charateristics along u 2 C . dt

1/17/13 12 Linear shallow water wave equation

∂∂ u +=h 0, ∂∂tx ∂∂u +=g 0. ∂∂tx

which can be combined into a wave equaon ∂∂22 =gh 0 ∂∂tx22

The general soluon is =±(),x Ct C = gh

Wave form does not change! NO dispersion!

1/17/13 13 Boussinesq equations (Peregrine, 1967; Ngowu, 1993)

o Pressure field is not hydrostatic: quadratic o Horizontal velocities are not vertically uniform

2 uxzt(,,)=++ Axt (,) zBxt (,) z Cxt (,)

should be small compared to A(x,t)

z Accurate for long and x u(x,z,t) intermediate depth waves, kh < 3 (wavelength > 2 water depths) Functions B, C lead to 3rd order spatial derivatives in model

1/17/13 14 High-Order Boussinesq Equations (Gobbi et al., 2000)

2 uxzt(,,)=+ Axt (,) z * Bxt (,) + z * Cxt (,) 34 ++zDxtzExt * ( , ) * ( , )

Should be small compared to B,C group

z - Accurate for long, x intermediate, and moderately deep waves, kh < 6 (wavelength > 1 water depth) - Functions D, E lead to 5th order spatial derivatives in model

1/17/13 15 Derivation of depth-integrated wave equations

. Normalize the conservation equations and the associated boundary conditions

. Perturbation analysis h ;1 = 0 = L0 . Integrate the governing equations and employ o the irrotational condition o the boundary conditions

1/17/13 16 Boussinesq-type equations OO()= 1,( 2 )= 1

1/17/13 17 Boussinesq equaons

OOkh()== (2 );

11∂∂h +++(hhO)uu + 23 =( ) (,, 422 ) ∂∂tt 3

2 ∂∂u 2422h +++uu ( u) hhO( u) =(, ,) ∂∂tt2

Different Boussinesq equaon forms can be derived by using different Representave velocity or by substung the higher order terms by an Order one relaon, e.g., ∂u = +O(,2 ) ∂t The resulng equaons have slightly different dispersion characteriscs.

1/17/13 18 Modeling the dispersion effects

• To apply the depth-integrated equaons for shorter waves, higher order terms can be included. However, it more difficult to solve the high-order model – Momentum equaon: ∂∂uu ∂5 u +++=uC.... 0 ∂∂tx1 ∂ x5 – To solve consistently, numerical truncaon error (Taylor series error) for leading term must be less important than included terms. • For example: 2nd order in space finite difference: ∂ux(,) t ux (+ xt ,) ux ( xt ,) x2 ∂3 ux (,) t oo= o o ∂xx26∂ x3 • High-order model requires use of 6-point difference formulas (Dx6 accuracy). Addionally, me integraon would require a Dt6 accurate scheme. • It is difficult to specify boundary condions.

1/17/13 19 COULWAVE (Cornell University Long and Intermediate Wave Modeling) A Multiple Layers Approach

• Divide water column into arbitrary layers – Each layer governed by an independent velocity profile, each in the same form as tradional Boussinesq models: z

x 2 uz1111()=+ A BzCz + z 1 uz11()= uz 21 ()

2 uz2222()=+ A BzCz +

z2 uz22()= uz 32 ()

2 uz3333()=+ A BzCz +

1/17/13 20 COULWAVE (Cornell University Long and Intermediate Wave Modeling)

• Regardless of # of layers, highest order of derivation is 3 • The more layers used, the more accurate the model • Any # of layers can be used – 1-Layer model = Boussinesq model – Numerical applications of 2-Layer model to be discussed • Location of layers will be optimized for good agreement with known, analytic properties of water waves

2 uz1111()=+ A BzCz +

2 uz2222()=+ A BzCz +

2 uz3333()=+ A BzCz +

1/17/13 21 COULWAVE . Governing equations: multi-layer, fully-nonlinear

COULWAVE

Linear theory high-order, single-layer result . Numerical scheme: predictor-corrector method o Predictor: explicit 3rd-order Adams-Bashforth o Corrector: implicit 4th-order Adams-Moulton 1/17/13 22 Linear Dispersion Properties of Multi-Layer Model • Compare phase and group velocity with linear theory 1.2 Boussinesq High-Order 1.15Equaons Boussinesq Two-Layer Model (One-Layer Equaons 1.1 Four-Layer Model) linear Three-Layer Model Model 1.05 C / C

1

0.95 5 10 15 20 25 30 35 kh Boussinesq High-Order Equaons Boussinesq 1.2 Equaons

1.15 Two-Layer Model

1.1 linear Three-Layer Model Four-Layer Model 1.05 Cg / Cg

1

0.95 5 10 15 20 25 30 35 1/17/13 23 kh General remarks on tsunami modeling by long-wave equations . Add-hoc dissipation mechanism o Boom fricon: quadrac form o Wave breaking: parameterized model . Controlling wave mechanism o Near the earthquake source: linear, dispersive or non-dispersive waves o Deep ocean: linear, non-dispersive waves o Close to the shore: nonlinear, non-dispersive waves (Observaon from the wave gauge data, and, the analycal study of one-dimensional problem)

1/17/13 24 Tsunami modeling packages . Boussinesq-type equaons model o COULWAVE (Cornell University Long and Intermediate Wave Modeling Package) o An improved mul-layer approach . Shallow water equaons model o COMCOT (Cornell Mul-grid Coupled Tsunami Model) o Covers tsunami generaon, propagaon, and wave runup/rundown on coastal regions o If desired, physical frequency dispersion can be mimicked by the numerical dispersion o This is a more praccal choice for the tsunami simulaon 1/17/13 25 Numerical simulaon of tsunami waves . Governing equaons (COMCOT)

o Linear model: deep ocean o Nonlinear model: shallower region 1/17/13 26 COMCOT: numerical scheme . Explicit leap-frog Finite Differencing Method

center of sub-level grid . Nested grid center of parent grid volume flux of sub-level grid

connecng boundary volume flux of parent grid

1/17/13 27 Inputs (inial condions) for the tsunami wave modeling

. Earthquake-generated seafloor displacement o Impulsive model • The seafloor deforms instantaneously and the enre fault line ruptures simultaneously • The sea surface follows the seafloor deformaon instantaneously o Transient model • The seafloor deformaon and the rupture along the fault line are both described as transient processes • Require me histories of the horizontal and vercal seafloor displacements

1/17/13 28 Fault plane models . Impulsive vs. transient models o Transient models are not always available o The speed of the rupture is orders of magnitude faster than the tsunami wave propagaon speed o Advanced impulsive models can provide detailed spaal variaons of the final seafloor displacement o No significant difference in the resulng tsunami wave height and propagaon me from these two types of fault plane models • Analysis arguments: Kajiura (1963); Momoi (1964-5); Tuck and Hwang (1972) • Numerical experiments: Wang and Liu (2006) 1/17/13 29 Numerical simulaons of tsunami waves . Source: impulsive model . Deep ocean: linear shallow water equaons . Shallower region (connental shelf, coastal area): nonlinear shallow water equaons . Effects of frequency dispersion o Near the source region o Over a very long traveling distance . Shoreline: moving boundary scheme o Various numerical techniques o Esmaon on the runup (inundaon) 1/17/13 30 Moving boundary algorithms . Staircase representaon (COMCOT)

. Extrapolaon model (COULWAVE)

1/17/13 31 Extrapolaon scheme for the moving shoreline

-3 -2 -1 i = 0 1 2

extrapolaon

governing equaons

1/17/13 32 Staircase shoreline

1/17/13 33 Validaon of Runup Algorithm (Lyne, Wu and Liu 2002, Coastal Engineering)

• Runup of solitary wave around a circular island – Experimental data taken from Liu et al. (J.F.M. 1995)

• Physical setup: Elevation (m) – Sll water depth = 0.32 m – Slope of side walls = 1:4 – Depth profile 

• Numerical simulaon of conical island runup: – Wave amplitude = 0.028 m – Sll water depth = 0.32 m – Beach slope = 1:4 – Dx = 0.1 m

3D animaon of runup of solitary wave around a circular island Validation of Runup Algorithm

experimental Time series comparisons numerical 1 2

3 4 1 2 island 4

Incident wave 3 direction o 1960 Chilean Tsunami

The epicenter of the 1960 Chilean earthquake was located about 100 km offshore of the Chilean coast. The fault zone was roughly 800 km long and 200 km wide, and the displacement of the fault was 24 m. The orientation of the fault was N10 E. The focal depth of the slip was estimated at 53 km with a 90 degree slip angle and a 10 degree dip angle. Using these estimated fault parameters, we can calculate the initial free-surface displacement (Mansinha and Smylie, 1971). The wavelength of the initial tsunami form was roughly 1,000 km and the wave height was roughly 10 m.

1960 Chilean Tsunami Inundaon in Hilo Bay Some common difficules in using depth-integrated wave equaons

• Most of numerical algorithms are dissipave, especially the moving shoreline algorithms; • Most of models do not include wave breaking; • Most of models specify boom fricon coefficients and wave breaking parameters empirically with limited validaon; • Depth-integrated wave equaons can not adequately address the wave-structure interacon issues. Other open issues

• Coupling the hydrodynamic models with sediment transport models • Coupling the hydrodynamic models with debris flow models • Coupling the hydrodynamic models with soil (foundaon) dynamic models 3D/2D Numerical Modeling of Tsunamis in Nearshore Environment and Their Interacon with Structures

u =0

∂( u) T + (uu) = p + % + g %=+( uu) ∂t Turbulence Models

Approximate Exact Equations Equations

Time Average Spatially Filter DNS

Do not use More Boussinesq LES Physics Use Boussinesq

Reynolds Stress Model (6 new PDEs)

Less Work 2 Equation Models (k-epslion) 1 Equation Models (k-epslion) Algebraic Model