Instabilities in Some Shallow Water Models

Instabilities in some shallow water models

Vera Mikyoung Hur University of Illinois at Urbana-Champaign

with Jared Bronski, Mat Johnson, Ashish Pandey, Leeds Tao A long time ago in a galaxy far, far away ...

[Levi-Civita; 1925] and, independently, [Nekrasov; 1921] proved the existence of periodic traveling waves (in an irrotational flow of infinitely deep water) — namely, Stokes waves — when the amplitude to wavelength ratio is sufficiently small.

[Krasovskii; 1960-61] proved the existence of Stokes waves subject to that the maximum slope < 30◦.

Therefore, no doubt has remained that Stokes waves are theoretically possible as states of perfect dynamic equilibrium.

By spring of 1965, however, Benjamin had trouble reproducing Stokes waves experimentally and believed that they be unstable.... Disintegration of a train of Stokes waves

An oscillating plunger generates a train of waves, of

wavelength 2.3m in water 7.6m deep, traveling away

from the observer in a large towing basin at the Ship

Division of the National Physical Laboratory. The

upper photograph shows, close to the wave maker, a

regular pattern of plane waves except for small-scale

roughness. In the lower photograph, some 60m (28

wavelengths) farther along the tank, the same wave

train has suﬀered drastic distortion. The instability was

triggered by imposing on the motion of the wave maker

a slight modulation at the unstable side-band

frequencies; but the same disintegration occurs

naturally over a somewhat longer distance. BF instability: the beginning and the present

The research in modulational instability started in the 1960s by Benjamin, Whitham, Lighthill, Zakharov,..., simultaneously but independently — “the idea was emerging when the time was ripe.”

Stokes waves would be unstable to sideband perturbations if (wave number)·(undisturbed water depth)> 1.36...

[Bridges and Mielke; 1995] studied the instability in a rigorous fashion. (Many earlier arguments, while perfectly correct, are hard to analytically justify.)

But the proof breaks down in the inﬁnite depth case. The full structure of the spectrum of the linearized problem is open. MI and variational structure

[Bronski and H.; 2014] derives a modulational instability index for periodic traveling waves of

2 ut + Mux + (u )x = 0, Mdu(k) = m(k)ˆu(k). (?)

It combines the dispersion relation in the linear motion and the quadratic power-law nonlinearity.

2 2 In case m(k) = k (M = −∂x ),(?) recovers the KdV equation.

(?) is nonlocal unless m is a polynomial of ik. Examples include the Benjamin-Ono equation, for which m(k) = |k|, and more generally, the fractional KdV equation, α for which m(k) = |k| , −1 6 α 6 2. Breaking of shallow water waves [Whitham; 1974] said ”the breaking phenomenon is one of the most intriguing long-standing problems of water wave theory.” The shallow water equations

ht + ux + (uh)x = 0, ut + hx + uux = 0 explain breaking. But it goes too far — all solutions with an increase of elevation break.

In a simple theory including dispersion eﬀects, by the way q tanh k 1 2 4 cWW (k) = ± k = ± 1 − 6 k + O(k ), namely the KdV equation 1 2 3 ut + (1 + 6 ∂x )ux + 2 uux = 0, it, in turn, goes too far — no solutions break.

Whitham in 1967 proposed

3 q tanh k ut + Mux + 2 uux = 0, Mdu(k) = k uˆ(k). MI: setup

Consider the growing mode problem for the linearized equation

µv = ∂x (M − 2u − c)v =: L(u)v.

We say u is spectrally unstable if specL2(R)(L(u)) 6⊂ iR. NB. v needs not be T -periodic — a sideband perturbation.

By Floquet theory, [ −iξx iξx spec 2 (L) = spec 2 (L ), L = e Le . L (R) Lper ([0,T ]) ξ ξ ξ∈(−1/2,1/2]

The spectrum of Lξ consists merely of discrete eigenvalues. Jordan block structure at the origin

Assume (N1) u is not constant, 2 (N2) ker(δ E) = span{ux }, (N3) G := Mc Pa − MaPc 6= 0. Then, † † Lv1 := Lua = 0, L w1 := L (Mc u − Pc ) = 0, † † −1 Lv2 := Lux = 0, L w2 := L (∂x (Mauc − Mc ua)) = w3, † † Lv3 := Luc = v2, L w3 := L (Pa − Mau) = 0,

and hwj , vk i = Gδjk .

Therefore zero is a generalized eigenvalue of L0 = L with algebraic multiplicity three and geometric multiplicity two. Perturbation and MI

For |ξ| 1 1 2 Lξ = L0 + iξL1 − 2 ξ L2 + ··· ,

where L0 = L0, L1 = [L0, x], L2 = [L1, x], ....

Consider the perturbation problem 1 2 µφ = (L0 + iξL1 − 2 ξ L2)φ, where 2 µ = iξµ1 − ξ µ2 + ··· , φ = φ0 + iξφ1 + ··· The perturbation problem admits a complex eigenvalue for µ1 provided that   0 −Gc32 b33 D = −1 b32 0  b22 −Gc22 − b22b32 b23 admits a complex eigenvalue, where

b22 = −hw1, L1v3i,

b23 = hw1, L1v1i, −1 1 c22 = −hw1, L1L0 L1v2i + 2 hw1, L2v2i, b32 = hw2, L1v2i + hw3, L1v3i,

b33 = hw1, L1v1i, −1 1 c32 = −hw3, L1L0 L1v2i + 2 hw3, L2v2i. bmn’s and cmn’s may be expressed in terms of K, U, P, M as functions of c, a, T . α e.g. If m(k) = |k| , hw2, L1v2i = −α(Mc Ua − MaUc − cG). Application 1: fKdV

In the case of the Benjamin-Ono equation, i.e. α = 1, √ 2π/T 2π c2−4a−(2π/T )2 √ u(x; c, a, T ) = − 1 ( c2 − 4a + c), T q c2−4a 2 c2−4a−(2π/T )2 − cos(2πx/T ) where c < 0 and c2 − 4a − (2π/T )2 > 0,

−πT (πT )2(1 − (2π/cT )2) 0  D =  1 πT 0  . 2π2 0 πT

Eigenvalues are πT and ±πT p2 − (2π/cT )2. Therefore all periodic traveling waves are modulationally stable. Something weird

We examined for c = −5, a = 0, k = 4, and for parameter values in the range 1 < α < 2 (BO-KdV) and, interestingly, found a modulationally unstable wave at α ≈ 1.025.

the imaginary part of the the wave proﬁle at α ≈ 1.025, eigenvalue as a function of α the onset of instability

To compare, small-amplitude waves are modulationally unstable if and only if α < 1. Application 2: the Whitham equation r tanh k If m(k) = , a small-amplitude, 2π/k-periodic traveling k wave is 1 1 cos(2z) u(k, a, b)(z) =b(1 − m(k)) + a cos z + a2 + + ... 2 m(k) − 1 m(k) − m(2k) 1 1 c(k, a, b) =m(k) + 2b(1 − m(k)) + a2 + + ... m(k) − 1 m(k) − m(2k) for |a|, |b| 1, z = kx.

[H. and Johnson, 2015] proved that it is modulationally unstable if (km(k))00((km(k))0 − m(0))i (k) ind(k) = 3 < 0, m(k) − m(2k) 0 where i3(k) = 2(m(k) − m(2k)) + (km(k)) − m(0). It happens when k > 1.14...; otherwise spectrally stable to square integrable perturbations. Recall (km(k))00((km(k))0 − m(0))i (k) i (k)i (k)i (k) ind(k) = 3 =: 1 2 3 , m(k) − m(2k) i4(k) and i3(k) = 2(m(k) − m(2k)) + i2(k). Modulational stability changes when (1) i1(k) = 0; the group velocity is the minimum, (2) i2(k) = 0; the group velocity matches the limiting phase speed, (3) i3(k) = 0; the Benjamin-Feir instability index vanishes. (4) i4(k) = 0; the fundamental harmonic is resonant with the second harmonic. s T tanh(kd) With surface tension m(k) = gd 1 + k2 , gd2 k T > 0 is the coeﬃcient of surface tension. 2 If T /gd < 1/3, in(k), n = 1, 2, 3, 4, changes its sign once. 2 If T /gd > 1/3, i1(k), i2(k), i4(k) > 0, i3(k) vanishes once. Bi-directional or Boussinesq-Whitham equations [Deconinck and Oliveras; 2011] numerically observed high frequency instabilities, concluding all Stokes waves are unstable! Explain.

[H. and Pandey, in progress] proved that a small-amplitude, 2π/k-periodic traveling wave of 2 ht + ux + (uh)x = 0, ut + M hx + uux = 0 is modulationally unstable if k > 1.62.... But we have not observed high frequency instabilities.

2 3 2 htt −M hxx − 2 (h )xx = 0 is linearly ill-posed in the periodic setting. [Deconinck and Trichtchenko; 2015] numerically observed high-frequency instabilities, though.

−2 3 2 M htt − hxx − 2 (h )xx = 0 fails to explain the BF instability. Including the eﬀects of surface tension,

bi-directional Whitham uni-directional Whitham the water wave problem

[H. and Tao, 2015] proved wave breaking for 2 ht + ux + uhx = 0, ut + M hx + uux = 0.