Finding the Stokes Wave: Low Steepness to Highest Wave

Free Surface Hydrodynamics Numerical Simulations and More

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich

Department of Mathematics University of Arizona, Tucson, Arizona, 85721 [email protected]

TexAMP 2014 November 21-23

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Free Surface Hydrodynamics Numerical Simulations and More Outline

1 Free Surface Hydrodynamics Formulation of Problem Hamiltonian Formalism Equations of Motion Finite Amplitude Travelling Waves

2 Numerical Simulations and More Travelling Wave solution a.k.a Stokes Wave Parameter Oscillation Evolution Problem

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Contents

1 Free Surface Hydrodynamics Formulation of Problem Hamiltonian Formalism Equations of Motion Finite Amplitude Travelling Waves Travelling Wave solution a.k.a Stokes Wave Parameter Oscillation Evolution Problem

Figure : Nearly plane waves in the wake of a boat in Maas-Waal Canal.

Consider a potential flow of ideal fluid in a seminfinite strip (x, y) ∈ [−π, π]x[−∞, η(x, t)] periodic in x-variable.

∇2Φ = 0

with BC and domain deﬁned by: ∂η ∂η ∂Φ ∂Φ + − + = 0 ∂t ∂x ∂x ∂y y=η(x,t) ∂Φ 1 2 + (∇Φ) + gη = 0 ∂t 2 y=η(x,t)

∂Φ = 0 ∂y y→−∞

We introduce the conformal map z(w, t) = w +z ˜(w, t), so that in new variable w = u + iv ﬂuid occupies region [−π, π]x[−∞, 0].

− Analyticity of a functionz ˜(w) in C imposes a relation between real and imaginary parts ofz ˜ on the free surface v = 0: z˜(u) =x ˜(u) + iy(u) =x ˜(u) + iHˆ x˜(u) = 2Pˆx˜ ˆ ˆ 1 ˆ where H is Hilbert operator, and P = 2 1 + iH is projector.

In the absence of viscosity, free surface hydrodynamics constitute a Hamiltonian system with:

1 ZZ η(x) g Z H = ∇Φ2 dy dx + η2 dx 2 −∞ 2 1 Z g Z H = ΦΦ | du + y 2x du 2 v v=0 2 u Introduce complex potential Π(w) = Φ(w) + iΘ(w) analytic in − C . Deﬁne ψ(u) = Φ(u, v = 0) and we have: Θ(u) = Hˆ ψ(u) ˆ Φv |v=0 = − Θu|v=0 = −Hψu

at v = 0, where −Hˆ ∂u is Dirichlet-Neumann operator. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Formulation of Problem Free Surface Hydrodynamics Hamiltonian Formalism Numerical Simulations and More Equations of Motion Finite Amplitude Travelling Waves Equations of motion

Equations of motion are found from extremizing action S: Z Z L = ψηt du − H, S = L dt

And are:

2 2 ! ψ − Hˆ ψu ˆ u ˆ Hψu ψt + gy = − 2 + ψuH 2 2|zu| |zu| ˆ ˆ Hψu zt = zu(H − i) 2 |zu| The results of simulation of a ﬂavour of these equations is described in further sections.

A Stokes wave is a fully nonlinear wave propagating over one dimensional free surface of an ideal ﬂuid. The deﬁning parameters of a Stokes wave is steepness. The steepness is measured as a ratio of crest to trough height H over a wavelength λ

A solution propagating with ﬁxed velocity c:

z(u, t) = u + z(u − ct) ψ(u, t) = ψ(u − ct)

satisﬁes: 2 c ˆ 1 ˆ 2 ˆ 2 k − 1 y − ky + yky = 0 c0 2 here c = pg/k - phase velocity of linear gravity waves, and 0 √ 0 kˆ = |k| = −∆

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Contents

Formulation of Problem Hamiltonian Formalism Equations of Motion Finite Amplitude Travelling Waves

2 Numerical Simulations and More Travelling Wave solution a.k.a Stokes Wave Parameter Oscillation Evolution Problem

We solve equation: 2 c ˆ 1 ˆ 2 ˆ 2 k − 1 y − ky + yky = 0 c0 2 with Newton Conjugate-Gradient iterations in Fourier space: ∗ (n) (n) (n) Write exact solution yk = yk + δyk and linearize at yk : ˆ (n) (n) ˆ (n) ˆ (n) L0 yk + δyk = L0yk + L1δyk = 0 Solve linearized system: ˆ (n) ˆ (n) L1δyk = −L0yk (n+1) (n) (n) yk = yk + δyk These simulations were performed in quadruple precision. Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Stokes Waves

H/λ = 0.140984 -5 0.4 H/λ = 0.135748 10 H/λ = 0.127549 0.2 10-15 λ | 0 k ~ (x)/ |z η 10-25 -0.2 H/λ = 0.140984 -35 H/λ = 0.135748 -0.4 10 H/λ = 0.127549

-0.5 -0.25 0 0.25 0.5 0 -4000 -8000 -12000 -16000 x/λ k

Figure : Stokes waves computed with Newton-CG method (left) and their spectra (right). Simulations with N = 2048 (blue), N = 4096 (green) and N = 4194304 (orange) Fourier modes. Black dashed lines are asymptotic decay predicted by theory.

− Recall thatz ˜(u) is analytic in C , but does it have an analytic continuation into the upper half plane?

− Recall thatz ˜(u) is analytic in C , but does it have an analytic continuation into the upper half plane? Yes! Stokes wave can be written as a Cauchy integral over branch cut extending from ivc to +i∞

Z π zk = z˜(u) exp(−iku) du −π + We deform the integration contour into C and assume the local behaviour about ivc to be:

β z(w) ∼ (w − ivc )

It is easy to show that asymptotically

−β−1 zk → |k| exp(−|k|vc ) k → −∞

1 Result from 1973 by M. Grant shows that β = 2 and it is supported by analysis of the spectra we have found.

2 1 10-2 1.6 10-4 (H - H )/λ -6 max

1.2

c 10 v -10-1 -10-3 -10-5 0.8

0.4 numerics 0 0 0.03 0.06 0.09 0.12 λ H/λ Hmax/

Figure : Position of the singularity vc as a function of wave steepness. H Our estimate for steepness of highest max = 0.1410633 ± 4 · 10−7 λ

The limiting Stokes wave a.k.a wave of greatest height has a jump 2π in derivative ηx and forms a 3 angle on the surface. The singularity of limiting Stokes appears as a result of coalescence of more than one singularity, e.g.:

1/2 2/3 z(w) ∼ f (w)(w − ivc ) + h.o.t. → w

as vc → 0

where f (w) is some regular function. Finding the limiting Stokes wave from presented equations faces several major diﬃculties.

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave Divergent!

Travelling Wave solution a.k.a Stokes Wave Free Surface Hydrodynamics Parameter Oscillation Numerical Simulations and More Evolution Problem Everwidening Spectrum

The Asymptotics of Fourier coeﬃcients as k → ∞: 1 y ∼ exp (−|k|v ) k |k|3/2 c 1 (kyˆ ) ∼ exp (−|k|v ) k |k|1/2 c

Take a limit vc → 0 and: 1 (kyˆ ) ∼ k |k|1/2

The Asymptotics of Fourier coeﬃcients as k → ∞: 1 y ∼ exp (−|k|v ) k |k|3/2 c 1 (kyˆ ) ∼ exp (−|k|v ) k |k|1/2 c

Take a limit vc → 0 and: 1 (kyˆ ) ∼ k |k|1/2

Divergent!

1.1 1.093

1.08 1.0928 1.06 1.0926 1.04 wave speed 1.0924 1.02 L-H 1975 1 Schwartz 1979 1.0922 0.04 0.08 0.12 0.16 0.138 0.139 0.14 0.141 0.142 H/λ H/λ

1 Study of Fourier spectrum: Accurate, but only allows to study the singularities closest to the real axis 2 Construction of Padéinterpolant and analysis of its poles: Straight-forward construction of Padéinterpolant suffers from catastrophic loss of precision in finite digit arithmetic. Solution: Alpert-Greengard-Hagströmalgorithm(AGH) can construct Padéapproximation using many points on the grid.

We expand periodic interval u ∈ [−π, π] to inﬁnite interval ζ ∈ (−∞, ∞) with auxiliary transform: w ζ = tan 2 Applying Pad´eapproximation we observe that Stokes wave is a branch cut:

d Z 1 X αk ρ(χ)dχ z˜(u) = z(u) − u ≈ u ≈ u tan − iχ χc tan − iχ k=1 2 k 2 The branch cut spans along the positive imaginary axis from point iχc to +i∞

0.8 approximation numerics -5 10 0.7 analytic

0.6 10-10 0.5

-15 ρ 0.4 10 Max |Err| 0.3

10-20 0.2

0.1 -25 10 (a) (b) 0 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 N, number of poles tanh(v/2)

Figure : (a) Maximum of absolute error between the Stokes wave solution and its approximation by poles as a function of the number of poles; (b) Reconstructed jump on the branch cut using residues and position of poles from AGH algorithm

0.8 0.7 0.6 0.5

ρ 0.4 0.3 0.2 H/λ = 0.1409535 0.1 H/λ = 0.1255102 H/λ = 0.1173340 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 tanh(v/2)

The equations on ψ(u, t) and z(u, t) are not optimal for numerical simulations, instead equations are formulated in terms of R(u, t) = 1 and V (u, t) = iψu : zu zu

0 0 Rt = i UR − U R 0 0 Vt = i UV − B R + g (R − 1)

ˆ 1 ˆ it is convenient to introduce projection operator P = 2 (1 + iH), ˆ ! 2 ˆ −Hψu ˆ |Φu| then U = 2P 2 and B = P 2 |zu| |zu|

0.45 numerics, no hyperviscosity 0.4 numerics, with hyperviscosity 0.35 prediction

0.3

0.25 (t) c v 0.2

0.15

0.1

0.05

0 0 0.2 0.4 0.6 0.8 1 t, time

Analytical properties of Stokes wave are fully determined by a single branch cut. High steepness waves have been constructed numerically. The prediction of oscillatory approach to limiting wave was conﬁrmed, with several of the oscillations being well-resolved. Closed integral equations on ρ(χ) were found.

Sergey Dyachenko*, Pavel Lushnikov and Alexander Korotkevich Finding the Stokes Wave: Low Steepness to Highest Wave