Internal solitary waves in the ocean: Analysis using the ...christov.tmnt-lab.org/downloads/IMACS_Waves2007_talk.pdfnonlinear evolution equations termed the (inverse) scattering transform

Internal solitary waves in the ocean:Analysis using the periodic, inverse scattering transform

Ivan Christov*

Department of Mathematics,Texas A&M University, College Station, Texas 77843-3368

[email protected]

Special Session on Nonlinear Wave Phenomena in the Physical Sciences:Some Recent Studies

5th IMACS International Conference on Nonlinear Evolution Equationsand Wave Phenomena: Computation and Theory,

Athens, GeorgiaApril 16, 2007

*Supported by ONR/NRL funding and a Travel Award from the organizers.Ivan Christov (TAMU) Internal Solitons & the IST IMACS Waves 2007 1 / 14

http://www.math.tamu.edu/~christov/

mailto:[email protected]

The Big Picture

Motivation:Linear Fourier analysis often proves ineffective for the analysis ofinherently nonlinear physical phenomena.Nonlinearity plays a crucial part in real-world problems.

Background:GGKM 1967 ... AKNS 1973 ... Flaschka, McLaughlin / Dubrovin,Matveev, Novikov 1976: Developed a general method for solvingnonlinear evolution equations termed the (inverse) scattering transform.Osborne ca.1980–2000: Application of the scattering transform for theperiodic KdV eq. to the analysis of oceanographic data — the so-callednonlinear Fourier analysis.

Outline of the talk:1 Modeling propagation of internal solitons in the ocean: the

Korteweg–de Vries equation.2 The inverse scattering transform as a nonlinear Fourier analysis.3 Numerics of the discrete scattering transform.4 Analysis of internal solitary wave trains in the Yellow Sea using the

discrete scattering transform.

Ivan Christov (TAMU) Internal Solitons & the IST IMACS Waves 2007 2 / 14

Statement of the Problem and Notation

A model for internal solitary waves (Osborne & Burch 1980):

KdV Eq.: ηt + c0ηx + αηηx + βηxxx = 0, (x , t) ∈ [0, L]× (0,∞);

IBC:

{η(x , t = 0) = η0(x), x ∈ [0, L],

η(x + L, t) = η(x , t), (x , t) ∈ [0, L]× [0,∞).

c

h1

h2

!1

!2

Free SurfaceIsopycnal "!x,t"

Internal Solitons

The physical parameters are

c0 '

√g

∆ρ

ρ

(h1h2

h1 + h2

),

α ' −3c0

2

(h2 − h1

h1h2

),

β ' 1

6c0h1h2,

∆ρ := ρ2 − ρ1, ρ ' ρ1 ' ρ2.


Fourier Analysis of Linear PDEs

Consider the linearized KdV equation (i.e., α = 0):

ηt + c0ηx + βηxxx = 0.

We can find its exact solution in terms of a Fourier Series:

1 “Discrete Fourier Transform” (DFT): Find the spectrum {cj ;φj}:cj =

√a2j + b2

j , φj = arctan (−bj/aj), where aj and bj are the usual

Fourier coefficients.

2 “Inverse Discrete Fourier Transform” (IDFT): Construct the solutionto the linear PDE from the spectrum {cj ;φj}:

η(x , t) =a0

2+

N∑j=1

cj cos(kjx − ωj t + φj),

where ωj = c0kj − βk3j from the dispersion relation for the PDE.


“Fourier Analysis” of Integrable Nonlinear PDEs

Going back to the (nonlinear) KdV equation:

ηt + c0ηx + αηηx + βηxxx = 0,

we can find its exact solution via the Scattering Transform:

1 Direct Scattering Transform (DST): Solve the associated Schrodingereigenvalue problem: [

− d2

dx2− λη(x , 0)

]ψ = Eψ,

where λ = α6β is the nonlinearity to dispersion ratio, to get the

discrete spectrum, aka “scattering data,” {Ej ;µj}.2 Inverse Scattering Transform (IST): Construct the nonlinear Fourier

series from the spectrum {Ej ;µj} using hyperelliptic functions or theRiemann Θ-function.

? Together these two steps constitute the periodic, inverse scatteringtransform (PIST) for the KdV eq.


The Nonlinear Fourier Series

In terms of the hyperelliptic (aka Abelian) functions we have

η(x , t) =1

λ

−E1 +N∑

j=1

2µj(x , t)− E2j − E2j+1

.

All nonlinear waves and their interactions are obtained from this linearsuperposition!

In the small amplitude limit, maxx ,t |µj(x , t)| � 1, we haveµj(x , t) ∼ cos(x − ωj t + φj) — we get the ordinary Fourier series!

If there are no interactions (N = 1, i.e., just one wave), we haveµ(x , t) = cn2(x − ωt + φ|m), which is a Jacobian elliptic functionwith modulus m (a cnoidal wave).

The amplitudes of the nonlinear oscillations are given by

Aj =

{2λ(Eref − E2j), for solitons;12λ(E2j+1 − E2j), otherwise (radiation).


On Moduli and Wave Numbers

In the hyperelliptic representation, we have kj = 2πj/L as in Fouriertheory (the wave numbers are “commensurable”).

We can compute the elliptic modulus mj of the hyperelliptic functions,which is called the soliton index, from the discrete spectrum as

mj =E2j+1 − E2j

E2j+1 − E2j−1, 1 ≤ j ≤ N.

Then, the oscillations fall into three general categories:

mj & 0.99 ⇒ solitons [note cn2(x |mj = 1) = sech2(x)],mj & 0.5 ⇒ nonlinearly interacting cnoidal waves, “less nonlinear”than the solitons (e.g., Stokes waves),mj � 1.0 ⇒ radiation [note cn2(x |mj = 0) = cos2(x)].


Numerics of the DST (I)

For an e-value problem with periodic potential, we can introduce an“appropriate” basis of eigenfuctions and appeal to Floquet’s theoremto obtain

Φ(x + L,E ) = α(x ,E )Φ(x ,E ),

where α is the monodromy matrix and, e.g., Φ =

(φ φx

φ∗ φ∗x

).

The main spectrum is those Ej s.t. ∆(Ej) := 12 tr α(x ,Ej) = ±1.

The auxiliary spectrum is those µj s.t. α21(x , µj) = 0.

Q: What is a numerically-accessible quantity that we can us tocompute the monodromy matrix and the discrete eigenvalues {Ej , µj}of our Schrodinger e-value problem?


Numerics of the DST (II)

A: Rewrite the Schrodinger e-value problem as a first-order system:

d

dxΨ(x ,E ) = B(x ,E )Ψ(x ,E ), B(x ,E ) =

(0 1

−q(x ,E ) 0

),

where Ψ := (ψ,ψx)> and q(x ,E ) := λη(x , 0) + E .

Integrating we have Ψ(x ,E ) = exp{∫

B(x ,E ) dx}.Now, η(x , 0) is usually a discrete data set, i.e. its values are onlyknown at x = xj := j∆x , j ∈ {0, . . . ,M − 1}.Then, we immediately have that Ψ(xj+1,E ) = e∆xB(xj ,E)Ψ(xj ,E ).

Iterating this relation gives Ψ(x + L,E ) = M(x ,E )Ψ(x ,E )∀x ∈ [0, L], where M(x ,E ) :=

∏0j=M−1 e∆xB(xj ,E) is the so-called

scattering matrix.

Finally, it can be shown that 12 tr α ≡ 1

2 tr M and α21 ≡ M21!⇒ We’ve found a numerically-accessible analogue to the monodromymatrix.


Numerics of the DST (III)

Bad news: finding the ±1 crossings of the Floquet discriminant∆(E ) ≡ 1

2 tr M and the zero crossings of M21(E ) is no easy task.

Osborne 1994 suggests to use “accounting functions” to count thezeros of M11(E ), then find the latter’s zeros along with those ofM21(E ) to get a bracketing of the ±1 crossings of ∆(E ).

Problem 1: The accounting functions suffer numerical instability atthe scales the spectrum is located at for the internal soliton model.

Problem 2: Computing all the zeros of M11(E ) is a lot of extra work.

Idea: Sample ∆(E ) and M21(E ), find brackets for their zerocrossings. Then, use the following facts:

1 M21(E ) = 0 exactly once between ±1/± 1 crossings of ∆(E ),2 ∆(E ) = 0 exactly once between ±1/∓ 1 crossings ∆(E ).

Finally, keeping the roots bracketed during the root-finding is critical.⇒ Use the regula falsi method; it guarantees the latter and can havesuperlinear convergence unlike the bisection method (Osborne 1994).


Visualizing the Numerics of the DST

-4µ 10-6

-3µ 10-6

-2µ 10-6

-1µ 10-6 0 1µ 10

-6

E Hm-2L

-40

-20

0

20

40

The zero crossings of M21(E ) (the blue oscillations) determine thenumber of the degrees of freedom in the DST spectrum; eachcrossing is a µj value.

Consecutive +1/+ 1 and −1/− 1 crossings of ∆(E ) (the redoscillations) determine the open bands of the DST spectrum; eachcrossing is an Ej value.


Internal Solitary Waves in the Yellow Sea

60 80 100 120 140 160 180 200

Range Hkm L

42.5

45

47.5

50

52.5

55

57.5

60

htpe

DHm

L

100 110 120 130 140 150 160 170

x Hkm L

-15

-12.5

-10

-7.5

-5

-2.5

h 0HxLHm

L

(⇐) Internal solitary waves in the Yellow Sea from a numericalsimulations of solitary-wave generation by bottom topography, usingthe nonhydrostatic Lamb model, by Chin-Bing et al. (2003). [Depthmeasured from the sea bottom.]

(⇒) The two (middle) wave packets we are interested in; part of theσt = 22.0 isopycnal, which is located at a depth of ≈ 12m. [Depthmeasured from the undisplaced isopycnal.]


Fourier Analysis of the Internal Solitary Waves

0 0.01 0.02 0.03 0.04

2p jêL Hm-1L

0

0.1

0.2

0.3

0.4

A j

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

2p jêL Hm-1L

-8

-6

-4

-2

0

The FFT spectrum. The DST spectrum.

Three things to note about the DST spectrum:

the narrower (and distinct) range of wave numbers predicted,the nonlinear non-soliton waves present in the spectrum,there are a lot fewer oscillations modes predicted.

Point: The DST can find solitons in the data set — no guess worknecessary.


Selected Bibliography

Ablowitz & Segur, Solitons and the Inverse Scattering Transform.SIAM, 1981.

Osborne & Burch, “Internal Solitons in the Andaman Sea.”Science 208 (1980) 451.

Osborne, “Numerical construction of nonlinear wave-trainsolutions of the periodic Korteweg–de Vries equation.”Phys. Rev. E 48 (1993) 296.

Osborne, “Automatic algorithm for the numerical inversescattering transform of the Korteweg–de Vries equation.”Math. Comput. Simul. 36 (1994) 431.

Chin-Bing, et al., “Analysis of coupled oceanographic andacoustic soliton simulations in the Yellow Sea ... ”Math. Comput. Simul. 62 (2003) 11.


Documents

Internal solitary waves in the ocean: Analysis using the ...christov.tmnt-lab.org/downloads/IMACS_Waves2007_talk.pdfnonlinear evolution equations termed the (inverse) scattering transform