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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 potenJal flow) Computational Cost e.g., Truchas (Los Alamos Lab, USA) Short • Boussinesq Model (H2D) O(days-months) - Based on Boussinesq-type Equaons for weakly Accuracy wave 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 u2 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 soluJon 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 2 OO()= 1,( )= 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 subsJtuJng the higher order terms by an Order one relaon, e.g., ∂u = +O(,2 ) ∂t The resulJng equaons have slightly different dispersion characterisJcs. 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). AddiJonally, Jme integraon would require a Dt6 accurate scheme. • It is difficult to specify boundary condiJons. 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 tradiJonal 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 Bo]om fricJon: 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 analyJcal 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 mulJ-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 pracJcal 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 connecJng 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 enJre 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 Jme histories of the horizontal and verJcal 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 resulJng tsunami wave height and propagaon Jme 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 .
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