WAVE IN SHALLOW WATER: THEORY AND NUMERICAL SIMULATIONS

Miguel Onorato Università di Torino, Dip. Fisica Generale – ITALY

Collaborators:

D. Proment, A. Osborne, P. Janssen, D. Resio

Operational forecasting is based on the numerical integration of the water wave kinetic equation (Hasselmann equation, 1962)

"N v 1 + # $ (C N ) = S + S + S "t g 1 NL diss In where

2 S T ( k) ( )[N N (N N ) N N (N N )]dk NL = & 1,2,3,4 " # " #$ 3 4 1 + 2 % 1 2 3 + 4 2,3,4

!

Sdiss = "# D (k)N(k)

! SIn = "(k)N(k)

!

! MOTIVATION

WAVE MODELING IN SHALLOW WATER

Common thinking in the wave forecasting community: a) FOUR-WAVE RESONANT INTERACTIONS IN DEEP /INTERMEDIATE WATER

b) THREE -WAVE INTERACTIONS (NEVER RESONANT) IN SHALLOW WATER i) deterministic modeling (Boussinesq or higher order) ii) evolution equations for spectra and bi-spectra based on Boussinesq- like equations i) Are the four-wave interactions relevant in shallow water? ii) Does an irreversible transfer of occur in flat-bottom shallow water ?

METHODOLOGY

Theoretical analysis of the oldest, flat-bottom shallow water model: THE BOUSSINESQ EQUATIONS

"t + # $ [(" + h)u] = 0 h 2 u + u$ #u + g#" % ## $ u = 0 t 3 t

! SUMMARY OF THE RESULTS i) A FOUR-WAVE HASSELMANN-LIKE EQUATION (BOUSSINESQ KINETC EQUATION) HAS BEEN DERIVED FROM THE BOUSSINESQ EQUATIONS ii) BOUSSINESQ KINETC EQUATION CORRESPONDS TO THE ARBITRARY DEPTH HASSELMANN EQUATION WHEN THE LIMIT OF SMALL DEPTH IS PROPERLY TAKEN iii) THE ESTIMATED TIME SCALE OF THE FOUR-WAVE NONLINEAR INTERACTIONS IN SHALLOW WATER IS 1 % kh( 4 1 % 1 ( 4 " ~ ' * ~ ' * NL # & $ ) # & a/h)

! iv) BASED ON DIMENSIONAL ARGUMENTS, DIRECT SOLUTION IS FOUND TO BE OF THE FORM OF E(k) ~ k"4 / 3

v) RESULTS ON DIRECT CASCADE ARE CONFIRMED BY DIRECT NUMERICAL COMPUTATIONS OF THE BOUSSINESQ EQUATIONS. ! vi) EXPERIMENTAL INDICATION OF CASCADE IN SHALLOW WATER (see also Smith and Vincent 2003, Kaihatu et al. 2007 for shoaling waves)

References: Janssen and M.O., J. Phys. (2007) M.O. et al JFM (2009) Zakharov, Europ. J. Mechanics B/Fluids (1999) Zakharov, in Nonlinear Waves and Weak Turbulence (1998)

Derivation of the Boussinesq kinetic equations

"t + # $ [(" + h)u] = 0 h 2 u + u$ #u + g#" % ## $ u = 0 t 3 t

i) WRITE THE EQUATION IN FOURIER SPACE AND INTRODUCE THE COMPLEX AMPLITUDE ! 1/ 2 1/ 2 # g & # " & a(k,t) = )(k,t) + i* k +(k,t) % ( k% ( $ 2"k ' $ 2g'

k "(k) = gh 2 2 2 2 1/ 2 "(k) =1+ k h /3 with (1+ k h /3) !

! !

ii) USE OF THE MULTIPLE SCALE EXPANSION:

2 •Introduce a slow time scale: " = # t

•Look for a solution of the form:

(1) 2 (2) ak (t,") = bk (t,!") + #bk (t,") + # bk (t,") + ...

"b0 2 (B ) * + i#0b0 = $i% ' T0123b1 b2b3&(k0 + k1 $ k2 $ k3)dk123 ! "t

! Relation with the Hasselmann equation

(B ) (ad ) T0123 = LimT0123 kh "0

1.5

3 T /k 0000

0 !

-1.5 Arbitrary depth

-3 Boussinesq

0 0.5 1 1.5 2 2.5 3 3.5 4 k h

Narrow band limits of the Boussinesq-Zakharov equation

•Long crested case (one dimensional propagation):

Defocousing Nonlinear Schroedinger equation in shallow water

•Weakly two-dimensional case:

Shallow water limit of the Davey-Stewartson

BOTH EQUATIONS ARE INTEGRABLE!!

NO ENERGY TRANSFER!!

iii) STATISTICAL DESCRIPTION OF THE SHALLOW WATER ZAKHAROV EQUATION

Hypothesis: i) Homogeneity ii) Quasi-Gaussian approximation

2 $ ' "N0 (B ) 1 1 1 1 = , (T0123 ) N0N1N2N3& + # # )* (+0 + +1 #+2 #+3 )*(k0 + k1 # k2 # k3 )dk123 "t % N0 N1 N2 N3 (

where ! * < a(ki,t)a (k j ,t) >= N(ki,t)"(ki # k j )

!

NONLINEAR TIME SCALE

2 $ ' "N0 (B ) 1 1 1 1 = , (T0123 ) N0N1N2N3& + # # )* (+0 + +1 #+2 #+3 )*(k0 + k1 # k2 # k3 )dk123 "t % N0 N1 N2 N3 (

(Β) 2 −1 2 4 1/τNL ∼ (T ) ω Ν k !

with T(Β)∼ k2/h 1 % kh( 4 1 % 1 ( 4 " ~ ' * ~ ' * NL # & $ ) # & a/h)

! Power law solutions of the Boussinesq kinetic equation

Hypothesis: existence of an equilibrium range in Fourier space where there is a constant flux of energy, independent of k

Dimensional analysis of the kinetic equation

ΔN/Δt ∼ T(Β)2ω−1Ν3k4 with T(Β)∼ k2/h

Flux of energy Π: Π∼k2ΔΕ/Δt∼ Ν3k10 must be independent of k, therefore E∼k-4/3 (in agreement with Zakharov 1999)

Numerical simulations of Boussinesq equations

10-2

10-3

10-4

10-5

10-6 16 T SPECTRUM 48 T 10-7 80 T 160 T

WAVE 1600 T 10-8 -4/3 K 10-9 0.01 0.1 1 kh

EFFECTS OF REGULAR DISCTRETIZATION: LACK OF EXACT Example:

-1 h=1 m, kp=0.2 m -1 Δkx=0.01 m -1 Δky=0.001 m k1= kp, k2=kp

IN A 256x256 GRID THERE ARE NOT EXACT RESONANCES!

−4 |ω1+ω2−ω3−ω4|<10 ω1 2 10-2

1.5 10-2

1 10-2

5 10-3

Ky 0 100

-5 10-3

-1 10-2

-1.5 10-2

-2 10-2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Kx

−3 |ω1+ω2−ω3−ω4|<10 ω1

3 10-2

2 10-2

1 10-2

Ky 0 100

-1 10-2

-2 10-2

-3 10-2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Kx Some experimental indication of shallow water cascade

(from Smith and Vincent JGR 2003)

Some indication of shallow water cascade

(from Kaihatu et al. JGR 2007)

From Resio and Long (JGR 2006)

10-2

10-3

10-4

-5 WAVE SPECTRUM 10

10-6 k-4/3 10-7 0.01 0.1 1 10 100 k h

THREE WAVE INTERACTIONS CAN BE ASYMPTOTYCALLY RESONANT

k3=k1- k2 ω3~ ω1-ω2

If k1 is close to k2 then

2 ω3+ ω2-ω1~c0( k2- k1)(kh) ~0

CONCLUSIONS

•The Boussinesq kinetic equation is equivalent to the shallow water limit of the Hasselmann equation

• Four-wave resonant interaction can be relevant in shallow water

•Power law solutions of the Boussinesq kinetic have been confirmed by numerical simulations

•There is some indication that this solution can be observed in the surf-zone

•Infragravity waves in Currituck Sound show also power law k-4/3