University:
Boston University
Course:
CAS PY 105 | Physics 1
Academic year:

2020
Views:

107

Pages:

19
Author:

bigbangworldwr29

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 22 Linear Dispersion Properties of Multi-Layer Model • Compare phase and group velocity with linear theory C / Clinear 1.2 High-‐Order Boussinesq EquaJons Boussinesq 1.15EquaJons (One-‐Layer 1.1 Model) Two-‐Layer Model Four-‐Layer Model Three-‐Layer Model 1.05 1 0.95 Boussinesq EquaJons 1.2 5 High-‐Order Boussinesq EquaJons 10 20 kh 25 30 35 Two-‐Layer Model 1.15 Cg / Cglinear 15 1.1 Four-‐Layer Model Three-‐Layer Model 1.05 1 0.95 5 10 15 20 kh 25 30 35 23 General remarks on tsunami modeling by long-wave equations  Add-hoc dissipation mechanism o Bo]om fricJon: quadraJc 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 (ObservaJon from the wave gauge data, and, the analyJcal study of one-‐dimensional problem) 24 Tsunami modeling packages  Boussinesq-‐type equaJons model o COULWAVE (Cornell University Long and Intermediate Wave Modeling Package) o An improved mulJ-‐layer approach  Shallow water equaJons model o COMCOT (Cornell MulJ-‐grid Coupled Tsunami Model) o Covers tsunami generaJon, propagaJon, 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 simulaJon 25 Numerical simula=on of tsunami waves  Governing equaJons (COMCOT) o Linear model: deep ocean o Nonlinear model: shallower region 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 (ini=al condi=ons) 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 deformaJon instantaneously o Transient model • The seaﬂoor deformaJon and the rupture along the fault line are both described as transient processes • Require Jme histories of the horizontal and verJcal seaﬂoor displacements 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 propagaJon speed o Advanced impulsive models can provide detailed spaJal variaJons of the ﬁnal seaﬂoor displacement o No signiﬁcant diﬀerence in the resulJng tsunami wave height and propagaJon 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) 29 Numerical simula=ons of tsunami waves  Source: impulsive model  Deep ocean: linear shallow water equaJons  Shallower region (conJnental shelf, coastal area): nonlinear shallow water equaJons  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 EsJmaJon on the runup (inundaJon) 30 Moving boundary algorithms  Staircase representaJon (COMCOT)  ExtrapolaJon model (COULWAVE) 31 ExtrapolaJon scheme for the moving shoreline -‐3 -‐2 -‐1 i = 0 1 2 extrapolaJon governing equaJons 1/17/13 32 Staircase shoreline 1/17/13 33 Valida&on of Runup Algorithm (Lyne8, 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: – SJll water depth = 0.32 m – Slope of side walls = 1:4 – Depth proﬁle  • Numerical simulaJon of conical island runup: – Wave amplitude = 0.028 m – SJll water depth = 0.32 m – Beach slope = 1:4 – Dx = 0.1 m Elevation (m) • 3D anima=on of runup of solitary wave around a circular island Validation of Runup Algorithm Time series comparisons 1 2 Incident wave direction island 3 4 experimental numerical 1 2 3 4 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 InundaJon in Hilo Bay Some common diﬃcul=es in using depth-‐integrated wave equa=ons • • • • Most of numerical algorithms are dissipaJve, especially the moving shoreline algorithms; Most of models do not include wave breaking; Most of models specify bo]om fricJon coeﬃcients and wave breaking parameters empirically with limited validaJon; Depth-‐integrated wave equaJons can not adequately address the wave-‐structure interacJon 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 (foundaJon) dynamic models 3D/2D Numerical Modeling of Tsunamis in Nearshore Environment and Their Interac=on with Structures u=0 ∂ ( u) + ∂t ( uu ) = p+ %= %+ g ( u+ T u) Turbulence Models Approximate Equations Time Average Do not use Boussinesq Exact Equations Spatially Filter LES DNS More Physics Use Boussinesq Reynolds Stress Model (6 new PDEs) 2 Equation Models (k-epslion) Less Work 1 Equation Models (k-epslion) Algebraic Model

Mathematical Models For Tsunami Waves Coulwave

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock