Nonlinear wave interaction for broad-banded, open seas - - PowerPoint PPT Presentation

Nonlinear wave interaction for broad-banded,

• pen seas – deterministic and stochastic theory

Raphael Stuhlmeier (based on joint work with David Andrade and Michael Stiassnie)

• 11. March 2019

Structure of this talk:

• 1. Linear theory and the energy density spectrum
• 2. Nonlinear wave-wave interaction
• 3. Stochastic evolution equations

The (linear) water-wave problem

∇2 φ = 0 on −h < z < 0 ηt = φz on z = 0 φt +g η = 0 on z = 0 φz = 0 on z = −h.

The (linear) water-wave problem

Velocity potential ∇2 φ = 0 on −h < z < 0 ηt = φz on z = 0 φt +g η = 0 on z = 0 φz = 0 on z = −h.

The (linear) water-wave problem

Velocity potential ∇2 φ = 0 on −h < z < 0 ηt = φz on z = 0 φt +g η = 0 on z = 0 φz = 0 on z = −h. Free surface

The (linear) water-wave problem

Velocity potential Fluid domain ∇2 φ = 0 on −h < z < 0 ηt = φz on z = 0 φt +g η = 0 on z = 0 φz = 0 on z = −h. Free surface

The Water-Wave Problem

Linear solution

The linear solutions are η = A k sin(kx −ωt) φ = Aω k cos(kx −ωt)cosh(k(h +z)) sinh(kh) with ω2 = gk tanh(kh). Yielding a time-averaged total energy E ∝ A2.

To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η1 = A1 sin(k1x −ω1t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) η2 = A2 sin(k2x −ω2t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt) ηn = An sin(knx −ωnt)

Random-phase & amplitude description of the sea-surface

Practically: view the free surface as a stochastic process η(t) = ∑

i

ai cos(ωit +φ i).

Random-phase & amplitude description of the sea-surface

Practically: view the free surface as a stochastic process η(t) = ∑

i

ai cos(ωit +φ i). ◮ The spectrum FE{η2} allows the calculation of important characteristics of the sea-state, like significant wave height, important e.g. for engineering of marine structures

What is a spectrum?

1 2 3 4 5 6 Frequency [rad/s] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 S(w) [m2 s / rad] Spectral density for Pierson-Moskowitz spectrum with H

m0 = 1 m, T p = 5 s fp = 1.3 [rad/s]

Narrow vs. broad spectrum

f E(f) f E(f) f E(f) t η(t) t η(t) t η(t)

Homogeneity/Stationarity

t0 t0+τ

η

1(t)

ηk(t) η2(t) η3(t) η

4(t)

ηk(t0) ηk(t0+τ)

Homogeneity/Stationarity

The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea

The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving . . . but waves are dynamic, constantly moving as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations as governed by the Euler equations (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.) (as well as wind input, dissipation, etc.)

The (nonlinear) water wave problem

∇2φ = 0 on −h < z < η(x,y,t) ηt +∇(x,y)φ ·∇(x,y)η = φz on z = η(x,y,t) φt + 1 2(∇φ)2 +gη = 0 on z = η(x,y,t) φz = 0 on z = −h.

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= 0 has plane wave solutions Aei(kx−ωt), for some ω(k).

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= εu2 L (A1ei(k1x−ω1t) +A2ei(k2x−ω2t))

• superposition on LHS

∼ A1A2ei((k1±k2)x−(ω1±ω2)t

• product on RHS

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= εu2 L (A1ei(k1x−ω1t) +A2ei(k2x−ω2t))

• superposition on LHS

∼ A1A2ei((k1±k2)x−(ω1±ω2)t

• product on RHS

Resonant forcing

2nd order

• ω1 ±ω2 ±ω3 = 0

k1 ±k2 ±k3 = 0

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= ε2u3 L (

3

∑

i=1

Aiei(kix−ωit))

• superposition on LHS

∼ A1A2A3ei((k1±k2±k3)x−(ω1±ω2±ω3)t

• product on RHS

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= ε2u3 L (

3

∑

i=1

Aiei(kix−ωit))

• superposition on LHS

∼ A1A2A3ei((k1±k2±k3)x−(ω1±ω2±ω3)t

• product on RHS

Resonant forcing

3rd order

• ω1 ±ω2 ±ω3 ±ω4 = 0

k1 ±k2 ±k3 ±k4 = 0

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= εu2 +ε2u3

Deep-water waves:

ωi =

• g|ki|

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= εu2 +ε2u3

Deep-water waves:

ωi =

• g|ki|

k1 ±k2 ±k3 = 0

• |k1|±
• |k2|±
• |k3| = 0

k1 ±k2 ±k3 ±k4 = 0

• |k1|±
• |k2|±
• |k3|±
• |k4| = 0

Evolution equations for weakly nonlinear waves

L (u)

linear, dispersive

= εu2 +ε2u3

Deep-water waves:

ωi =

• g|ki|

k1 ±k2 ±k3 = 0 ✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤ ❤

• |k1|±
• |k2|±
• |k3| = 0

k1 +k2 −k3 −k4 = 0

• |k1|+
• |k2|−
• |k3|−
• |k4| = 0

Degenerate quartet interaction

Picture from a wave-tank perspective (Bonnefoy et al 2016)

The reduced Zakharov equation

i ∂B ∂t2 =

∞

−∞ T0,1,2,3B∗ 1B2B3δ0+1−2−3ei(ω+ω1−ω2−ω3)t2dk1dk2dk3

with B = B(k,t2), δ0+1−2−3 = δ(k+k1 −k2 −k3) and t2 = ε2T.

The reduced Zakharov equation

i ∂B ∂t2 =

∞

−∞ T0,1,2,3B∗ 1B2B3δ0+1−2−3ei(ω+ω1−ω2−ω3)t2dk1dk2dk3

with B = B(k,t2), δ0+1−2−3 = δ(k+k1 −k2 −k3) and t2 = ε2T.

Single mode – Stokes (1847)

For B(k,t) = b(t)δ(k−k0) Zakharov’s equation becomes: i ∂b ∂t = T0000|b|2b. Inverting b(k,t) =

• g

2ω(k) 1/2 ˆ η(k,t)+i ω(k) 2g 1/2 ˆ ψ(k,t) η = a0 cos(k0x −ω0(1+ a2

0k2

2 )t)

The reduced Zakharov equation

i ∂B ∂t2 =

∞

−∞ T0,1,2,3B∗ 1B2B3δ0+1−2−3ei(ω+ω1−ω2−ω3)t2dk1dk2dk3

with B = B(k,t2), δ0+1−2−3 = δ(k+k1 −k2 −k3) and t2 = ε2T.

Two wave-trains – Longuet-Higgins & Phillips (1962)

For two wave trains with wavenumbers kb > ka, and wave slopes εa = kaaa, εb = kbab, the mutual phase correction is: Ωa = ωa

• 1+ 1

2ε2

a + ωb

ωa · k2

a

k2

b

·ε2

b

• ,

Ωb = ωb

• 1+ 1

2ε2

b + ωa

ωb · kb ka ·ε2

a

• .

The reduced Zakharov equation

i ∂B ∂t2 =

∞

−∞ T0,1,2,3B∗ 1B2B3δ0+1−2−3ei(ω+ω1−ω2−ω3)t2dk1dk2dk3

with B = B(k,t2), δ0+1−2−3 = δ(k+k1 −k2 −k3) and t2 = ε2T.

Narrow-banded sea – Zakharov (1968)

For all energy concentrated about a single wavenumber k0 = (k0,0), and η(x,t) = ℜ(A(x,t)ei(k0x−ω(k0)t)) i

• Ax + 2k0

ω0 At

• − 1

4k0 Axx + 1 2k0 Ayy = k3

0|A|2A.

Deterministic instability growth rates from NLS

Im R 1

2 2

• w

k a 2

0 0 0

• FIG. 19. Three-dimensional instability growth rate of a uniform wave train based on the

three-dimensional nonlinear Schrodinger equation.

Deterministic instability growth rates from ZE

How does this nonlinear interaction impact the stochastic description?

Stochastic evolution equations

Ensemble averages

Define the correlation function (for discrete modes) rnm = r(kn,km,t) := b(kn;t)b∗(km;t) = bnb∗

m.

Stochastic evolution equations

Ensemble averages

Define the correlation function (for discrete modes) rnm = r(kn,km,t) := b(kn;t)b∗(km;t) = bnb∗

m.

Deterministic evolution (to third order) in deep water via (ZE): d dt rnm = d dt bnb∗

m+bn

d dt b∗

m

=−i ∑

p,q,r

Tn,p,q,rb∗

pbqbrb∗ mδ qr npei∆qr

npt

+i ∑

p,q,r

Tm,p,q,rbpb∗

qb∗ r bnδ qr mpe−i∆qr

mpt.

The need for closure

What can we do with b∗

qb∗ r bnbp?

The need for closure

What can we do with b∗

qb∗ r bnbp?

We find d dt b∗

qb∗ r bnbp = i ∑ u,v,w

Tq,u,v,wδ v,w

q,u bub∗ vb∗ wb∗ r bnbpe−i∆v,w

q,u t

+i ∑

u,v,w

Tr,u,v,wδ v,w

r,u bub∗ vb∗ wb∗ qbnbpe−i∆v,w

r,u t

−i ∑

u,v,w

Tn,u,v,wδ v,w

n,u b∗ ubvbwb∗ r b∗ qbpei∆v,w

n,u t

−i ∑

u,v,w

Tp,u,v,wδ v,w

p,u b∗ ubvbwb∗ r b∗ qbnei∆v,w

p,u t

Stochastic assumptions

◮ For bi a zero-mean process, we can decompose (writing rpq = bpb∗

q)

b∗

qb∗ r bnbp = κq,r,n,p +rpqrnr +rnqrpr.

Stochastic assumptions

◮ For bi a zero-mean process, we can decompose (writing rpq = bpb∗

q)

b∗

qb∗ r bnbp = κq,r,n,p +rpqrnr +rnqrpr.

◮ If the process is Gaussian κq,r,n,p ≡ 0, as are all higher order cumulants.

Stochastic assumptions

◮ For bi a zero-mean process, we can decompose (writing rpq = bpb∗

q)

b∗

qb∗ r bnbp = κq,r,n,p +rpqrnr +rnqrpr.

◮ If the process is Gaussian κq,r,n,p ≡ 0, as are all higher order cumulants. ◮ Then bub∗

vb∗ wb∗ mbnbp = ruv(rnwrpm +rnmrpw)

+ruw(rnvrpm +rnmrpv) +rum(rnvrpw +rnwrpv)

Stochastic assumptions

◮ For bi a zero-mean process, we can decompose (writing rpq = bpb∗

q)

b∗

qb∗ r bnbp = κq,r,n,p +rpqrnr +rnqrpr.

◮ If the process is Gaussian κq,r,n,p ≡ 0, as are all higher order cumulants. ◮ Then bub∗

vb∗ wb∗ mbnbp = ruv(rnwrpm +rnmrpw)

+ruw(rnvrpm +rnmrpv) +rum(rnvrpw +rnwrpv) ◮ This allows the construction of a hierarchy of equations, starting at O(r 2)

Hasselmann’s kinetic equation

Assuming

• 1. near-Gaussianity (rij = O(ε2), κj,m,n,p = O(ε4), κu,v,w,m,n,p = o(ε6))

Hasselmann’s kinetic equation

Assuming

• 1. near-Gaussianity (rij = O(ε2), κj,m,n,p = O(ε4), κu,v,w,m,n,p = o(ε6))
• 2. homogeneity

rnm = Cnδ(n −m)

Hasselmann’s kinetic equation

Assuming

• 1. near-Gaussianity (rij = O(ε2), κj,m,n,p = O(ε4), κu,v,w,m,n,p = o(ε6))
• 2. homogeneity

rnm = Cnδ(n −m)

• 3. slow variation of the product

CqCr(Cp +Cn)−CnCp(Cq +Cr)

Hasselmann’s kinetic equation

Assuming

• 1. near-Gaussianity (rij = O(ε2), κj,m,n,p = O(ε4), κu,v,w,m,n,p = o(ε6))
• 2. homogeneity

rnm = Cnδ(n −m)

• 3. slow variation of the product

CqCr(Cp +Cn)−CnCp(Cq +Cr) gives: dCn dt =4π ∑

p,q,r

T 2

npqrδnpqrδ(ωn +ωp −ωq −ωr)

×(CqCr(Cp +Cn)−CnCp(Cq +Cr)).

Hasselmann’s kinetic equation

Assuming

• 1. near-Gaussianity (rij = O(ε2), κj,m,n,p = O(ε4), κu,v,w,m,n,p = o(ε6))
• 2. homogeneity

rnm = Cnδ(n −m)

• 3. slow variation of the product

CqCr(Cp +Cn)−CnCp(Cq +Cr) gives: dCn dt =4π ∑

p,q,r

T 2

npqrδnpqrδ(ωn +ωp −ωq −ωr)

×(CqCr(Cp +Cn)−CnCp(Cq +Cr)). ◮ Evolution time-scale t4 = ε4T, exactly resonant interactions.

Generalized kinetic equation

Assuming

• 1. near-Gaussianity
• 2. homogeneity

rnm = Cnδ(n −m)

• 3. slow variation of the product

✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤ CqCr(Cp +Cn)−CnCp(Cq +Cr) gives: d dt Cj = 2 ∑

m,n,p

Tj,m,n,p δ n,p

j,m ℑ

• ei∆n,p

j,mtκj,m,n,p

• .

d dt κj,m,n,p = 2iTj,m,n,p δ n,p

j,m e−i∆n,p

j,mt

CnCp(Cj +Cm)−CjCm(Cn +Cp)

• .

◮ Evolution time-scale t2 = ε2T, nearly resonant interactions.

Alber’s equation

From 2D-NLS (DS) i

• Ax + 2k0

ω0 At

• − 1

4k0 Axx + 1 2k0 Ayy = k3

0|A|2A.

and assuming quasi-Gaussianity, but a spatially-inhomogeneous state, such that P(x,r,t) = A(x+r/2,t)A∗(x−r/2,t) : i ∂P ∂t + 1 2 g k0 ∂P ∂x

• − 1

4 g k3 ∂ 2P ∂x∂rx −2 ∂ 2P ∂y∂ry

• =
• gk5

0P(x,r,t)

• P(x+ r

2,0,t)−P(x− r 2,0,t)

Alber’s equation

In/stability of Gaussian spectrum

From Stiassnie et al (2008). ˜ W nondimensional spectral width, ˜ K nondimensional disturbance wavenumber.

Discrete CSY equation for inhomogeneous sea-states

Discrete autonomous ZE: dbn dt = −iωnbn −i ∑

p,q,r

Tnpqrb∗

pbqbrδ qr np.

Discrete CSY equation for inhomogeneous sea-states

Discrete autonomous ZE: dbn dt = −iωnbn −i ∑

p,q,r

Tnpqrb∗

pbqbrδ qr np.

For rnm(t) = bn(t)b∗

m(t),

drnm dt = irnm (ωm −ωn)+2i

• ∑

p,q,r

Tmpqrrpqrnrδ qr

mp − ∑ p,q,r

Tnpqrrqprrmδ qr

np

• .

I1 = ∑

i

Rii, I2 = ∑

i

kiRii, I3 = ∑

i

|Rii|2 +2∑

i=j

|Rij|2

Equation for inhomogeneous sea-states

Linear stability analysis – general case

Write rnm as rnm = rh

nmδnm +εri nm,

Equation for inhomogeneous sea-states

Linear stability analysis – general case

Write rnm as rnm = rh

nmδnm +εri nm,

drh

nn

dt = 0, and, for the inhomogeneous terms at order ε: 1 2i drnm dt = rnm

• ωm −ωn

2 +∑

p

Tmppmrpp −∑

p

Tnppnrpp

• +∑

p,q

Tmpqnrpqrnnδ mp

qn −∑ p,q

Tnpqmrqprmmδ np

qm.

Equation for inhomogeneous sea-states

Linear stability analysis – general case

The can be written 1 2i dri dt = Ari, An autonomous n2 −n system of ODEs. Solutions look like exp(iAt).

Equation for inhomogeneous sea-states

Degenerate quartet

The simplest case is when we have 2ka = kb +kc.

Equation for inhomogeneous sea-states

Degenerate quartet

The simplest case is when we have 2ka = kb +kc. We will take ka = (1,0), kb = (1+p,q), kc = (1−p,−q)

ka kb kc

Degenerate quartet

dRaa dt = 4iTaabc

• RabRace−i∆t −RbaRcaei∆t

dRbb dt = −2iTbcaa

• RabRace−i∆t −RbaRcaei∆t

dRcc dt = −2iTaabc

• RabRace−i∆t −RbaRcaei∆t

dRab dt = 2i [Rab (2TbabaRaa +other homogeneous terms) +Taabcei∆t (RcaRaa −RbaRcb −RcaRbb)+RcbRac (Tbcbc −Tacac)

• dRac

dt = 2i [Rac (2TcacaRaa +other homogeneous terms) +Taabcei∆t (RbaRaa −RbaRcc −RcaRbc)+RabRbc (Tcbcb −Tabab)

• dRbc

dt = 2i [Rbc (TcaacRaa +other homogeneous terms) +Taabc

• ei∆tRba

2 −e−i∆tR2 ac

• +RacRba (Tcaca −Tbaba)
• .

where ∆ = ωa +ωa −ωb −ωc

Equation for inhomogeneous sea-states

Initial conditions

Complex amplitudes at t = 0 : ba(0) = |ba|eiφa +|µa|eiφ, bb(0) = |bb|eiφb +|µb|eiφ, bc(0) = |bc|eiφc +|µc|eiφ, ◮ phases φa, φb and φ uniformly distributed over [0,2π)

◮ Rab(0) = ba(0)b∗

b(0) = 0.

◮ small inhomogeneity |µi| ≪ |bi|.

Domain of instability

p q 0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 (εa,εb,εc) = (0.15,0.15,0)

ka = (1,0), kb = (1+p,q), kc = (1−p,−q)

Equation for inhomogeneous sea-states

Instabilities – Long-time behavior case

100 200 300 400 500 600 700 800 900 1000 0.5 1 100 200 300 400 500 600 700 800 900 1000 0.2 0.4 |Rab| |Rac| |Rbc| Raa Rbb Rcc

(p,q) = (0.285,0), εa = 0.15, εb = 0.15

Equation for inhomogeneous sea-states

Fluctuations of wave-energy

η = 1 2π ωa 2g

• baeika·x

+ ωb 2g

• bbeikb·x

+ ωc 2g

• bceikc·x

+c.c.

• ,

Thus E = η2 = 1 2π2 ωa 2g raa + ωb 2g rbb + ωc 2g rcc+ √ωaωb 2g rabei(ka−kb)x + √ωaωc 2g racei(ka−kc)x+ √ωbωc 2g rbcei(kb−kc)x +c.c.

Equation for inhomogeneous sea-states

Fluctuations of wave-energy/variance

(εa,εb,εc) = (0.15,0.15,0)

Equation for inhomogeneous sea-states

Fluctuations of wave-energy/variance

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.6 En PDF 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 CDF

(εa,εb,εc) = (0.15,0.15,0)

Inhomogeneous evolution

Impact on rogue wave statistics

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 H0 Probability 2.85 3.2 3.4 3.6 3.8 4 4.275 10

−8

10

−6

10

−4

10

−2

H0 Probability

Probabilities that H/Hrms0 exceed H0 for purely homogeneous (blue) and inhomogeneous seas (green and red).

Concluding remarks

Benefits of the inhomogeneous CSY evolution equations:

• 1. Faster evolution on time-scale Tε−2 rather than KE Tε−4.
• 2. Ability to predict variations in wave–height statistics.
• 3. Modelling without restriction to narrow bandwidth.

Open questions:

• 1. Classification of domains of instability, long–time behavior.
• 2. Behavior for many modes/discretized spectra.
• 3. Physical source of inhomogeneous disturbances.
• 4. Effect of higher–order terms on overall evolution.

Thank you!

References

• 1. D. Andrade, R. S. and M. Stiassnie, On the generalized kinetic

equation for surface gravity waves, blow-up and its restraint, Fluids, 4 (2019), 2.

• 2. R. S. and M. Stiassnie, Nonlinear dispersion for ocean surface waves,
• J. Fluid Mech., 859 (2019), 49-58.
• 3. R. S. and M. Stiassnie, Evolution of statistically inhomogeneous

degenerate water wave quartets, Phil. Trans. R. Soc. A 376 (2018) 20170101.