freak waves: beyond the breather solutions of nls equation · breather solutions of nls equation m....

FREAK WAVES: BEYOND THE BREATHER SOLUTIONS OF NLS

EQUATION

M. Onorato

Collaborators: A. Toffoli, A. Iafrati, G. Cavaleri, L. Bertotti, E. Bitner-Gregersen, A. Babanin, P. Janssen, N. Mori

UNIVERSITA’ DI TORINO Dipartimento di Fisica

OUTLINE OF THE PRESENTATION

1)  Quick review on modulational instability and breather solutions of the NLS equation

2)  New approach on studying the modulational instability using Navier-Stokes equation

3)  The case of the Louis-Majesty accident: crossing seas conditions

-0.2-0.1

00.10.2

180 190 200 210 220 230 240

Ampl

itude

(m

)

TIME (s)

-0.2-0.1

00.10.2

180 190 200 210 220 230 240

Ampl

itude

(m

)

TIME (s)

-0.2-0.1

00.10.2

180 190 200 210 220 230 240

Ampl

itude

(m

)

TIME (s)

MODULATIONAL INSTABILITY: Experimental result

BREATHERS: EXACT SOLUTION OF THE NLS N. Akhmediev,et al. (1987)

Two remarks:

The solution depends on steepness and N

1)

2)

MAXIMUM AMPLITUDE

1) It depends on the product εN

2) Maximum amplitude is 3 -> The Peregrine solution

Such solutions have been tested experimentally in a number of wave tanks and fully nonlinear computations

N=5 ε=0.1

THE AKHMEDIEV SOLUTION

OCEAN WAVES ARE CHARACTERIZED BY JONSWAP SPECTRUM

Example: N=3, ε= 0.1 – Wave group is stable

 BREATHERS ARE RARE OBJECTS

BASED ON THIS BASIC IDEA OF MODULATIONAL INSTABILITY, AFTER YEARS OF THEORETICAL W O R K , N U M E R I C A L S I M U L AT I O N S A N D EXPERIMENTAL WORK, THE FOLLOWING QUANTITY is NOW COMPUTED OPERATIONALLY AT THE E.C.M.W.F.:

€

kurtosis = 3+π3BFI2D

2 +18ε 2

α is a fitting constant (Mori et al. JPO 2011)

FREE MODES

BOUND MODES NORMAL VALUE

€

BFI2D =BFI

1+α Δθ 2 /(Δω /ω 0)2

MODULATIONAL INSTABILITY FROM NAVIER-STOKES SIMULATIONS The numerical method has been developed by A. Iafrati (Iafrati 2010, JFM)

Initial condition:

Computational domain:

DISSIPATION

CONCLUSIONS

1)  Modulational instability does not imply always rogue waves

2)  Wave breaking due to modulational instability may result in a dissipation of energy larger in the air than in the water

3)  During the breaking process, dipoles are formed

4)  Dipoles can reach the height of the wave length

THE ACCIDENT

On March 03, 2010 at 15:20 the Louis Majesty has been hit by a wave at deck n. 5 which is 16 m from the undisturbed sea-level

The wave broke the glass windshields in the forward section on deck five

Rogue waves in crossing seas: The Louis Majesty accident

Cavaleri et al. JGR 2012

THE FORECASTING MODEL

The wave fields are the result of the forecasting of Nettuno model from the Italian National Meteorological Service

Resolution of the meteorological model: 7 km

Resolution of the wave model in space (WAM): 1/200

Spectral Resolution of the wave model: number of frequencies: 30 number of directions: 36

DIRECTIONAL SPECTRUM Time 12:00

DIRECTIONAL SPECTRUM Time 13:00

THE DIRECTIONAL SPECTRA Time 14:00

DIRECTIONAL SPECTRUM Time 15:00

THE DIRECTIONAL SPECTRA Time 16:00

THE DIRECTIONAL SPECTRUM Time 17:00

CROSSING SEAS: THE SIMPLEST CASE

ky

kx

θ

A=(k,l)

B=(k,-l)

COUPLED NONLINEAR SHRODINGER EQUATION

Zakharov equation

€

b(k) = A(k − kA)e−iω(kA )t + B(k − kB)e

−iω(kB )t€

i∂bo∂t

=ω obo + T0,1,2,3b1*∫ b2b3δ ko + k1 − k2 − k3( )dk1dk2dk3

• consider the following decomposition

• suppose that both spectral distribution are narrow banded

with

€

kA = (k, l) k B = (k,−l)

€

∂A∂t

+Cx∂A∂x

+Cy∂A∂y

− i α ∂2A∂x 2

+ β∂ 2A∂y 2

− γ∂ 2A∂x∂y

⎡

⎣ ⎢

⎤

⎦ ⎥ + i ξ A

2 + 2ζB 2[ ]A = 0

∂B∂t

+Cx∂B∂x

−Cy∂B∂y

− i α ∂2B∂x 2

+ β∂ 2B∂y 2

+ γ∂ 2B∂x∂y

⎡

⎣ ⎢

⎤

⎦ ⎥ + i ξB

2 + 2ζ A 2[ ]B = 0

COUPLED NLS EQUATIONS

Coefficients are a function of k and l

• Consider perturbations only a function of kx

ky

kx

θ

€

∂A∂t

− iα ∂2A∂x 2

+ i ξ A 2+ 2ζ B 2[ ]A = 0

∂B∂t

− iα ∂2B∂x 2

+ i ξ B 2+ 2ζ A 2[ ]B = 0

AKHMEDIEV BREATHER SOLUTION

€

A(x, t) = c1ψ(x, t) B(x,t) = c2ψ(x,t)Exp[iδ]Look for a solution of the form:

For the Louis Majestic case

€

c1 ≈ c2

€

ψ(x, t) satisfies a standard NLS equation

Parameters of the solution

€

ψ0 =1.7 mfA = fB = 0.1 HzεA = εB = 0.07N = 4

SUMMARY OF THE RESULTS

For θ <35.30, dispersive and both nonlinear terms have the same sign

The ratio between nonlinearity and dispersion becomes larger as θ approaches 35.30 (this is valid for both self-interaction and cross-interaction nonlinearity)

The cross-interaction nonlinearity is stronger than the self Interaction one for angles between 00 and 26.70

ANALYSIS OF THE COEFFICIENTS

α= dispersive term ξ= self-interaction term ζ=cross-interaction term

ANALYSIS OF THE COEFFICIENTS

α= dispersive term ξ= self-interaction term ζ=cross-interaction term

ANALYSIS OF THE COEFFICIENTS

α= dispersive term ξ= self-interaction term ζ=cross-interaction term

DISPERSION RELATION FOR PERTURBATION

€

Ω = αK 2 2(ξ + 2ζ )A02 +αK 2[ ]

AMPLIFICATION FACTOR FOR BREATHER SOLUTIONS

€

AmaxA0

=1+ 2 1− αξ + 2ζ

κ 2

εN

⎛

⎝ ⎜

⎞

⎠ ⎟

2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/ 2

N = number of waves under the envelope ε = initial steepness κ = modulus of the wave number

AMPLIFICATION FACTOR

GROWTH RATE

SUMMARY OF THE RESULTS:

The maximum amplification is for θ ->35.30 and large N

The maximum growth rate is for θ=00 and N≈3

CONSIDERATIONS:

Extreme waves are the result of a maximum amplification factor in a reasonable time scale

WE EXPECT LARGE EXTREME WAVE ACTIVITY AT ANGLES OF θ≈200- 300

EXPERIMENTS: MARINTEK FACILITY

€

E(ω,θ) = E1(ω,θ) + E2(ω,θ)

E1(ω,θ) =αg2

ω 5 Exp −54ω p

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

4⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ γExp −(ω−ω p )

2 /(2σ 2ω p2 )[ ]δ(θ −θ0)

E2(ω,θ) =αg2

ω 5 Exp −54ω p

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

4⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ γExp −(ω−ω p )

2 /(2σ 2ω p2 )[ ]δ(θ + θ0)

DESCRIPTION OF THE EXPERIMENT SUM OF TWO JONSWAP SPECTRA:

with

NUMERICAL SIMULATIONS

HIGHER ORDER SPECTRAL METHOD (THIRD ORDER IN NONLINEARITY)

BOX PERIODIC IN x AND y COORDINATES

INITIAL CONDITIONS PROVIDED BY TWO JONSWAP SPECTRA TRAVELLING AT AN ANGLE

RESULTS ON MAXIMUM KURTOSIS

0 10 20 30 40 50 θ

PROBABILITY OF EXCEEDENCE AT THE TIME OF THE ACCIDENT

Linear case:

PROBABILITY OF EXCEEDENCE AT THE TIME AND PLACE OF THE ACCIDENT

Hs=5.11 m

10-5

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12 14 16

RAYLEIGHMODIFIED RAYLEIGH

Exce

eden

ce p

roba

bilit

y

H (m)

WAITING TIME AT THE TIME AND PLACE OF THE ACCIDENT

Hs=5.11 m

10-1

100

101

102

103

0 2 4 6 8 10 12 14 16

RAYLEIGHMODIFIED RAYLEIGH

WAI

TING

TIM

E (M

INUT

ES)

H (m)

Limitations of the computation