Breaking of Progressive Gravity Surface Wave

Dyachenko A.I.

Landau Institute for Theoretical Physics RAS

Equations

[figure: x; y; η(x, t); z] potential irrotational flow

△ϕ(x, y, t) = 0

Boundary conditions: [ \begin{matrix}\frac{∂ϕ}{∂t} + \frac{1}{2}|∇ϕ|^{2} + gη = \frac{P}{ρ} , \\ \frac{∂η}{∂t} + η_{x}ϕ_{x} = ϕ_{y} \end{matrix} | at y = η(x, t).

\begin{aligned}\frac{∂ϕ}{∂y} & = & 0, y → −∞, \\ \frac{∂ϕ}{∂x} & = & 0, |x| → ∞. \end{aligned}

Equations

[figure: x; y; η(x, t); z] potential irrotational flow

△ϕ(x, y, t) = 0

Boundary conditions: [ \begin{matrix}\frac{∂ϕ}{∂t} + \frac{1}{2}|∇ϕ|^{2} + gη = \frac{P}{ρ} , \\ \frac{∂η}{∂t} + η_{x}ϕ_{x} = ϕ_{y} \end{matrix} | at y = η(x, t).

\begin{aligned}\frac{∂ϕ}{∂y} & = & 0, y → −∞, \\ \frac{∂ϕ}{∂x} & = & 0, |x| → ∞. \end{aligned}

Hamiltonian and Lagrangian

Hamiltonian H is the total energy of the fluid H = T + U

\begin{aligned}T & = & \frac{1}{2} ∫_{−∞}^{∞} dx ∫_{−η}^{η}(∇Φ)^{2}dy , \\ U & = & \frac{g}{2} ∫ η^{2}dx . \end{aligned} \begin{aligned}\frac{∂η}{∂t} & = & \frac{δH}{δΨ} , \\ \frac{∂Ψ}{∂t} & = & −\frac{δH}{δη} , \\ Ψ(x, t) & = & Φ(x, y, t)|_{y=η} \end{aligned}

Action S = ∫ Ldt

δS = 0,

Lagrangian

L = ∫ Ψ\frac{∂η}{∂t} dx − H .

Breaking of Progressive Gravity Surface Wave – p. 3

Hamiltonian and Lagrangian

Hamiltonian H is the total energy of the fluid H = T + U

\begin{aligned}T & = & \frac{1}{2} ∫_{−∞}^{∞} dx ∫_{−η}^{η}(∇Φ)^{2}dy , \\ U & = & \frac{g}{2} ∫ η^{2}dx . \end{aligned} \begin{aligned}\frac{∂η}{∂t} & = & \frac{δH}{δΨ} , \\ \frac{∂Ψ}{∂t} & = & −\frac{δH}{δη} , \\ Ψ(x, t) & = & Φ(x, y, t)|_{y=η} \end{aligned}

Action S = ∫ Ldt

δS = 0,

Lagrangian

L = ∫ Ψ\frac{∂η}{∂t} dx − H .

Breaking of Progressive Gravity Surface Wave – p. 3

Conformal Mapping

Let us apply the conformal mapping of the domain on the plane z = x + iy,

−∞ < x < ∞, −∞ < y ≤ η(x, t) ,

to the lower half-plane,

−∞ < u < ∞, −∞ < v ≤ 0 ,

on the plane ω = u + iv.

[figure: x; y; η(x, t); z; u; v; w]

Breaking of Progressive Gravity Surface Wave – p. 4

Conformal Mapping

After this transformation, the shape of the surface is given parametrically by

y = y(u, t), x = u + x̃(u, t) .

Functions y and x̃ are coupled by the relations:

y = Ĥx̃ x̃ = −Ĥy .

Here Ĥ is the Hilbert transformation,

Ĥf(u) = \frac{1}{π} P.V. ∫_{−∞}^{∞} \frac{f(u′)}{(u′ − u)}du′ .

For Fourier harmonics y_{k} = isign(k)x_{k}.

Breaking of Progressive Gravity Surface Wave – p. 5

Implicit Equations

Lagrangian can be expressed as follows,

L = ∫_{−∞}^{∞} Ψ(y_{t}x_{u}−x_{t}y_{u})du+\frac{1}{2} ∫_{−∞}^{∞} (ΨĤΨ_{u}du − gy^{2}x_{u}) du+∫_{−∞}^{∞} f(y−Ĥx̃)du .

Here f is the Lagrange multiplier. Hamilton’s principle,

\frac{δS}{δΨ} = 0 , \frac{δS}{δy} = 0, and \frac{δS}{δx} = 0 .

gives the following equations,

y_{t}x_{u} − x_{t}y_{u} = −ĤΨ_{u} ,
Ψ_{t}x_{u} − Ψ_{u}x_{t} + gyx_{u} = Ĥ(Ψ_{t}y_{u} − Ψ_{u}y_{t} + gyy_{u}) .

Ovsyannikov, 1973

Breaking of Progressive Gravity Surface Wave – p. 6

Explicit Equations

If conformal mapping has been applied then it is naturally introduce complex analytic functions

z = x + iy, and complex velocity potential Φ = Ψ + ĤΨ.

z_{t} = iUz_{u},
Φ_{t} = iUΦ_{u} − P̂(\frac{|Φ_{u}|^{2}}{|z_{u}|^{2}} ) + ig(z − u).

U is a complex transport velocity:

U = P̂(\frac{−ĤΨ_{u}}{|z_{u}|^{2}} ). u → w

Projector operator P̂(f) = \frac{1}{2}(1 + iĤ)(f).

Breaking of Progressive Gravity Surface Wave – p. 7

Classical variables Ψ, η

H = \frac{1}{2} ∫_{−∞}^{∞} ΨĜ(η)Ψdx + \frac{g}{2} ∫_{−∞}^{∞} η^{2}dx

Normal complex variable a_{k}:

η_{k} = \sqrt{\frac{ω_{k}}{2g}} (a_{k}+a_{−k}^{∗}) ψ_{k} = −i\sqrt{\frac{2g}{ω_{k}}} (a_{k}−a_{−k}^{∗}) ω_{k} = \sqrt{gk}

a_{k} satisfies the equation

\frac{∂a_{k}}{∂t} + i\frac{δH}{δa_{k}^{∗}} = 0,

a_{k} → b_{k}, excluding nonresonant terms.

Breaking of Progressive Gravity Surface Wave – p. 8

Classical variables b_{k}

Hamiltonian without cubic terms:

H = ∫ ω_{k}b_{k}b_{k}^{∗}dk+\frac{1}{2} ∫ T_{kk_{1}}^{k_{2}k_{3}}b_{k}^{∗}b_{k_{1}}^{∗}b_{k_{2}}b_{k_{3}}δ_{k+k_{1}−k_{2}−k_{3}}dkdk_{1}dk_{2}dk_{3}+…;

k + k_{1} = k_{2} + k_{3}, ω_{k} + ω_{k_{1}} = ω_{k_{2}} + ω_{k_{3}},
T_{kk_{1}}^{k_{2}k_{3}} = 0!
T_{k_{4}k_{5}}^{k_{1}k_{2}k_{3}} = \frac{2}{g^{1/2}π^{3/2}} \sqrt{\frac{ω_{k_{1}}ω_{k_{2}}ω_{k_{3}}}{ω_{k_{4}}ω_{k_{5}}}} \frac{k_{1}k_{2}k_{3}k_{4}k_{5}}{max(k_{1},k_{2},k_{3})}

Breaking of Progressive Gravity Surface Wave – p. 9

Classical variables b_{k}

Hamiltonian without cubic terms:

H = ∫ ω_{k}b_{k}b_{k}^{∗}dk+\frac{1}{2} ∫ T_{kk_{1}}^{k_{2}k_{3}}b_{k}^{∗}b_{k_{1}}^{∗}b_{k_{2}}b_{k_{3}}δ_{k+k_{1}−k_{2}−k_{3}}dkdk_{1}dk_{2}dk_{3}+…;

k + k_{1} = k_{2} + k_{3}, ω_{k} + ω_{k_{1}} = ω_{k_{2}} + ω_{k_{3}},
T_{kk_{1}}^{k_{2}k_{3}} = 0!
T_{k_{4}k_{5}}^{k_{1}k_{2}k_{3}} = \frac{2}{g^{1/2}π^{3/2}} \sqrt{\frac{ω_{k_{1}}ω_{k_{2}}ω_{k_{3}}}{ω_{k_{4}}ω_{k_{5}}}} \frac{k_{1}k_{2}k_{3}k_{4}k_{5}}{max(k_{1},k_{2},k_{3})}

Breaking of Progressive Gravity Surface Wave – p. 9

Cubic Equations

It turned out, that the equations can be simplified just by changing variables. Introduce instead of z(w, t) and Φ(w, t) another analytic functions R(w, t) and V (w, t)

R = \frac{1}{z_{w}} , Φ_{w} = −iV z_{w}.

R_{t} = i [UR′ − U′R] ,
V_{t} = i [UV′ − RP̂(V V̄)′] + g(R − 1).

Complex transport velocity U is defined via P̂

U = P̂(V R̄ + V̄ R).

Breaking of Progressive Gravity Surface Wave – p. 10

Setup the problem

The shape of stationary progressive wave is given by:

y = \frac{c^{2}}{2g} (1 − \frac{1}{|z_{w}|^{2}} ),

while Φ is related to the surface as

Φ = −c(z − w), V = ic(R − 1). The amplitude of the wave \frac{h}{L} ≃ 0.088,

For the sharp peaked limiting wave \frac{h}{L} ≃ 0.141.

Breaking of Progressive Gravity Surface Wave – p. 11

Setup the problem

Put 10 such waves in the periodic domain of 2π. Add long scale perturbation to this wave train

V_{k} = ic(R_{k} − 1) + ϵδ_{1,k}.

Setup the problem

Put 10 such waves in the periodic domain of 2π. Add long scale perturbation to this wave train

V_{k} = ic(R_{k} − 1) + ϵδ_{1,k}.

Simulation

Figure 3: Typical profile of the wave train

Breaking of Progressive Gravity Surface Wave – p. 13

Simulation

Figure 4: Typical spectra of |R_{k}|^{2} and |V_{k}|^{2}.

Breaking of Progressive Gravity Surface Wave – p. 14

Wave breaks

Figure 5: Wave breaks.

Breaking of Progressive Gravity Surface Wave – p. 15

Slope of the surface

Figure 6: Sharp wedge with the angle \frac{π}{3}

Breaking of Progressive Gravity Surface Wave – p. 16

Power spectrum

Figure 7: Spectrum of |R_{k}|^{2} and k^{− \frac{8}{3}} and k^{− \frac{10}{3}} .

? ? ?

Sharp peaked stationary gravity wave at the crest

z ∼ i\frac{c^{2}}{2g} +(\frac{9}{4} \frac{c^{2}}{g} )^{\frac{1}{3}} [−iw^{2}]^{\frac{1}{3}} +ϵw^{α}+… ( ^{3} \sqrt{−i} = e^{−i \frac{π}{6}} )

This is well known result of Stokes with the angle - \frac{2π}{3}. For breaking wave numerics shows - \frac{π}{3} wedge.

z ∼ iA(t) + B(t)[−i(w − a(t))(w − b(t))]^{\frac{1}{3}} + … Root branch that corresponds to the numerics is

z = \{ \begin{matrix}−i|u|^{1/3}, missing real part & u ≤ 0; \\ |u|^{1/3}(\frac{\sqrt{3}}{2} − \frac{i}{2}), & u > 0, \end{matrix}

Breaking of Progressive Gravity Surface Wave – p. 18

? ? ?

Sharp peaked stationary gravity wave at the crest

z ∼ i\frac{c^{2}}{2g} +(\frac{9}{4} \frac{c^{2}}{g} )^{\frac{1}{3}} [−iw^{2}]^{\frac{1}{3}} +ϵw^{α}+… ( ^{3} \sqrt{−i} = e^{−i \frac{π}{6}} )

This is well known result of Stokes with the angle - \frac{2π}{3}. For breaking wave numerics shows - \frac{π}{3} wedge.

z ∼ iA(t) + B(t)[−i(w − a(t))(w − b(t))]^{\frac{1}{3}} + … Root branch that corresponds to the numerics is

z = \{ \begin{matrix}−i|u|^{1/3}, missing real part & u ≤ 0; \\ |u|^{1/3}(\frac{\sqrt{3}}{2} − \frac{i}{2}), & u > 0, \end{matrix}

Breaking of Progressive Gravity Surface Wave – p. 18

? ? ?

Wave profile in (x, y) coordinates:

\{ \begin{matrix}y^{3} = x, & x ≤ 0; \\ y = \frac{1}{\sqrt{3}}x, & x > 0. \end{matrix}

“Overturning” takes place only near the wave top, main shape of the wave looks like Stokes wave. Distance between branch points a and b is small with respect to wavelength.