Abnormal Amplification of Long Waves in the Coastal Zone Ira Didenkulova & Efim Pelinovsky Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Nizhny Novgorod, Russia Wave Engineering Laboratory, Institute of Cybernetics, Tallinn, Estonia

Which bottom profile provides maximal amplification? Motivation: Role of Each Factor Which bottom profile provides maximal amplification?

Simplified Linear Theory of 1D Shallow Water Waves - Wave Speed (x,t) – Water Displacement h(x) – Water Depth

“Non-Reflected” Beach Seek Solution of Wave Equation Two unknown Functions: A and 

Exact Separation - wavenumber where

Energy Flux Conservation One equation is integrated exactly Energy Flux Conservation

No simple than Initial Wave Equation Second Equation can not be integrated generally It is a Variable-Coefficient 2d Order Equation No simple than Initial Wave Equation

If Depth varies smoothly – WKB Approach eikonal together

Asymptotic Solution for h(x) Described slowly varied propagated wave Reflection – beyond asymptotic method As exp(-1/) Mathematics: Theory of catastrophes, caustics, Maslov operator, ray approach…. Arnold, Maslov, Berry, Dobrokhotov, ……

Is it Propagated Wave???? To solve: Second Equation can not be integrated generally To solve: 1. Existing Analytical Solutions from Books 2. Find h(x) through A(x) – 1st order equation Is it Propagated Wave????

Is it Propagated Wave???? V. Ginzburg – Nobel Laureate in Physics (2003) Wave propagation in plasma (1968)-YES L.Brekhovskikh – Leader and Head of Russian Oceanography and Acoustics Wave propagation in layer media-NO

Overdetermined System Try to keep Features of Pure Propagated Wave Overdetermined System

“Non-Reflected” Beach together with gives h(x) ~ x4/3

“Non-Reflected” Beach Propagated Wave Impulse posses a shape But it is a singular solution at x = 0 (h = 0)

Velocity Field WKB amplitude

Non-bounded velocity Big depth

It is a Solution, but is it a Wave??? Physical Solution Vanishing on the Ends Sign-variable pulse It is a Solution, but is it a Wave???

Reduction to constant-coefficient wave equation The solution reduces

If It proves uniqueness of exact travelling wave solutions in inhomogeneous media

As a result, the general solution (Cauchy problem) can be founded Natural condition on the shoreline – boundness of water displacement As a result, the general solution (Cauchy problem) can be founded

where

Piston Model of Wave Generation

If initial disturbance is sign-variable

No current

Sign – constant initial disturbance

current

Zoom

Runup on beach x4/3 Bounded on shore x = 0 (runup)

Velocity Field on Shoreline But discharge

Soliton Runup Plane Beach

“Non-Reflected” Beach with Reflection Coefficient:

Reflection from “Non-Reflected” Beach

“Non-Reflected” Beach Pulse Reflection from “Non-Reflected” Beach i = d/dt Operator form

Impulse Reflection from “Non-reflected” Beach

Impulse Reflection from “Non-reflected” Beach

Tail - Distributed Reflection

For any Pulses with vanishing ends “Pure” Distributed Reflection “Non-reflected” Pulse

From follows From boundary condition on jog follows as obtained early

Nonlinear Traveling Waves in Strongly Inhomogeneous Media Example: inclined channel of parabolic cross-section

Basic Equations - water displacement, u - depth-averaged flow, S - variable water cross-section of the channel For parabolic channel

Linear Waves If

Nonlinear Theory

And final linear system Legendre (Hodograph) Transformation New variables And final linear system

Nonlinear Traveling Wave

Deformation of the wave shape in approaching wave: blue dashed and red solid lines correspond to an incident wave and the wave near the shoreline respectively

Variation of the positive (red) and negative (blue) amplitudes with distance; black solid line corresponds to the linear Green’s law

Shapes of water displacement (red) and velocity (black) near the shoreline

“Non-Reflected” Potential allows Conclusions: “Non-Reflected” Potential allows To be benchmarks 2. To give simple algorithm to compute wave propagation above complicated relief, 3. To understand better the wave physics regimes

E. Pelinovsky Hydrodynamics of Tsunami Waves Nonlinear Dynamics of Tsunami Waves E. Pelinovsky Hydrodynamics of Tsunami Waves http://www.appl.sci-nnov.ru/biblio/b3.html

Springer, 2008 Springer, 2009