Propagation of internal solitary waves over

topography: deformation and disintegration

Roger GrimshawDepartment of Mathematical Sciences, Loughborough

University , UK

in collaboration with:Efim PelinovskyTatiana Talipova

Presentation for BIRS workshop on CoordinatedMathematical Modeling of Internal Waves, April 5-9 2010

Internal solitary waves: Ocean

Internal solitary waves: Atmosphere

Morning Glory Waves: Burketown

Internal solitary waves: Atmosphere

Morning Glory Waves: Sable Island

1 of 1 05/04/2010 18:51

Internal solitary waves: Ocean

Typically internal solitary waves are long nonlinear waves.

z!z=

z=-h

The background state is the density ρ0(z) and horizontalcurrent u0(z), while the buoyancy frequency is given by

ρ0N2 = −gρ0z . (1)

Then seek an asymptotic solution for the vertical particledisplacement ζ = ζ(x , z , t), based on the assumption that thewaves are long and weakly nonlinear.

Korteweg-de Vries equation

ζ ∼ ε2A(X ,T )φ(z) , X = ε(x − ct),T = ε3t , (2)

where the modal function φ(z) satisfies

{ρ0(c − u0)2φz}z + ρ0N2φ = 0 , for − h < z < 0 , (3)

φ = 0 at z = −h , (c − u0)2φz = gφ at z = 0 , (4)

and the amplitude satisfies the Korteweg-de Vries equation

AT + µAAX + δAXXX = 0 . (5)

The coefficients µ and δ are given by

Iµ = 3

∫ 0

−h

ρ0(c − u0)2φ3z dz , (6)

I δ =

∫ 0

−h

ρ0(c − u0)2φ2 dz , (7)

I = 2

∫ 0

−h

ρ0(c − u0)φ2z dz . (8)

Extended Korteweg-de Vries equation

A particularly important special case arises when the nonlinearcoefficient µ (5) is close to zero. In this situation, a cubicnonlinear term is needed and the KdV equation (5) is replacedby the extended KdV (Gardner) equation,

AT + µAAX + ε2µ1A2AX + δAXXX = 0 . (9)

For µ ≈ 0, a rescaling is needed and the optimal choice is toassume that µ is 0(ε), and then replace A with A/ε. In effectthe amplitude parameter is ε in place of ε2. Henceforth weassume that this rescaling has been done. Like the KdVequation, (9) is integrable, and has solitary wave solutions.There are two independent forms of the eKdV equation (9),depending on the sign of δµ1.

Solitary waves of the eKdV equation

A =H

1 + B cosh K (X − VT ), (10)

where V =µH

6= δK 2 , B2 = 1 +

6δµ1K 2

µ2, (11)

with a single parameter B . For δµ1 < 0, 0 < B < 1, andthe family ranges from small-amplitude waves of KdV-type(“sech2”-profile) ( B → 1) to a limiting flat-topped wave ofamplitude −µ/µ1 (B → 0), the so-called “table-top” wave.For δµ1 > 0 there are two branches; one has 1 < B <∞ andranges from small-amplitude KdV-type waves (B → 1), tolarge waves with a “sech”-profile (B →∞). The otherbranch, −∞ < B < 1, has the opposite polarity and rangesfrom large waves with a “sech”-profile to a limiting algebraicwave of amplitude −2µ/µ1. Waves with smaller amplitudes donot exist, and are replaced by breathers .

Solitary waves of the eKdV equation

-6 -3 0 3 6

x

0

4

8

12

.

-0.8 -0.4 0 0.4 0.8

x

-80

-60

-40

-20

0

20

40

60

.

A

A

Variable coefficients

The derivation sketched so far is for a waveguide with constantproperties in the horizontal direction. But, in the ocean thereis varying depth, and variations in the basic statehydrology and background currents. These effects can beformally incorporated into the theory by supposing that thebasic state is a function of the slow variable χ = ε3x . That is,h = h(χ), u0 = u0(χ, z) with a corresponding vertical velocityfield ε3w0(z , χ), a density field ρ0(z , χ) a correspondingpressure field p0(χ, z) and a free surface displacement η0(χ).With this scaling, the slow background variability enters theasymptotic analysis at the same order as the weakly nonlinearand weakly dispersive effects. An asymptotic analysisanalogous to that sketched above then produces a variablecoefficient extended KdV equation. The modal system isagain defined by (3, 4), but now c = c(χ) and φ = φ(z , χ),where the χ-dependence is parametric.

Variable-coefficient extended KdV equation

With all small parameters removed, this is

Aτ + αAAξ + α1A2Aξ + λAξξξ = 0 . (12)

τ =

∫ χ dχ

c, ξ = t − τ , (13)

where the original amplitude A has been replaced by√

Q A. Qis the linear magnification factor, defined so that QA2 is thewave action flux. The coefficients α(τ), α1(τ), δ(τ) andQ(τ) are given by

α =µ

cQ1/2, α1 =

µ1

cQ, λ =

δ

c3, Q = c2I . (14)

Slowly-varying solitary wave

The evKdV equation (12) possesses two relevant conservationlaws, ∫ ∞

−∞A dx = constant , (15)∫ ∞

−∞A2 dx = constant , (16)

representing conservation of “mass” and “momentum”respectively (more strictly an approximate representation ofthe physical mass and wave action flux).

The slowly-varying solitary wave is then defined as before ,but its parameter B(τ) now varies slowly in a mannerdetermined by conservation of momentum (16). Mass isconserved by the generation of a trailing shelf.

Slowly-varying solitary wave

G (B) = constant| α31

λα2|1/2 , (17)

where G (B) = |B2 − 1|3/2∫ ∞

−∞

du

(1 + B cosh u)2.

The integral term in G (B) can be explicitly evaluated, and sothese expressions provide explicit formulas for the variation ofB(τ) as the environmental parameters vary.But since the conservation of momentum completely definesthe slowly-varying solitary wave, total mass (15) is conservedby a trailing shelf (linear long wave) whose amplitude Ashelf

at the rear of the solitary wave is

VAshelf = −∂Msol

∂τ, Msol =

∫ ∞

∞Asol dξ , (18)

and where Asol is the solitary wave solution.

Critical point α = 0, α1 < 0

The adiabatic expressions (17, 18) show that the criticalpoints where α = 0 (or where α1 = 0) are sites where we mayexpect a dramatic change in the wave structure. Suppose firstthat α passes through zero, but that α1 < 0, 0 < B < 1 atthe critical point α = 0. Then as α→ 0, it follows from (17)that B → 0 and the wave approaches the limiting “table-top”wave. But in this limit, K ∼ |α|, and so the amplitudeapproaches the limiting value −α/α1. Thus the waveamplitude decreases to zero, but the mass M0 of the solitarywave grows as |α|−1 and the amplitude A1 of the trailing shelfgrows as 1/|α|4. Essentially the trailing shelf passes throughthe critical point as a disturbance of the opposite polarity tothat of the original solitary wave, which then being in anenvironment with the opposite sign of α, can generate a trainof solitary waves of the opposite polarity, riding on apedestal of the same polarity as the original wave.

Critical point α = 0, α1 = 0: KdV case

δ = 1, α1 = 0 and α varies from −1 to 1 (that is thevariable-coefficient KdV equation). The upper panel is whenα = 0 and the lower panel is when α = 1. This is conversionof a depression wave into a train of elevation wavesriding on a negative pedestal.

520 560 600 640 680 720ξ

-8

-6

-4

-2

0

2

4

520 560 600 640 680 720ξ

-4

-2

0

2

4

6

A

A

Critical point α = 0, α1 < 0: eKdV case

δ = 1, α1 = −0.083 and α varies from 1 to −1, that is, thevariable coefficient eKdV equation, with a negative cubicnonlinear coefficient. This shows the conversion of anelevation “table-top” wave into a depression “table-top”wave, riding on a positive pedestal .

α = + 1

α = 0

α = − 1

500 550 600-15

0

15450 500 550

-15

0

15A

500 550 600

ξ

-15

0

15

Critical point α = 0, α1 > 0: eKdV case

Next, let us suppose that at the critical point where α = 0,α1 > 0. In this case, 1 < |B | <∞ and there are the twosub-cases to consider, B > 1 or B < −1, when the the solitarywave has the same or opposite polarity to α. Then, asα→ 0, |B | → ∞ as |B | ∼ 1/|α|. It follows from (11) thatthen K ∼ 1, D ∼ 1/|α|, a ∼ 1,M0 ∼ 1. It follows that thewave adopts the “sech”-profile, but has finite amplitude, andso can pass through the critical point α = 0 withoutdestruction. But the wave changes branches from B > 1to B < −1 as |B | → ∞, or vice versa. An interestingsituation then arises when the wave belongs to the branchwith −∞ < B < −1 and the amplitude is reducing. If thelimiting amplitude of −2α/α1 is reached, then there can be nofurther reduction in amplitude for a solitary wave, and insteada breather will form.

Critical point α = 0, α1 > 0: eKdV case

δ = 1, α1 = 0.3 and α varies from 1 to −1 for −T < τ < T ,that is, the variable coefficient eKdV equation, with a positivecubic nonlinear coefficient. This shows the adiabatic evolutionof an elevation wave from τ = −T to τ = T , where itsamplitude is too small, and so the wave becomes a breather.

T

2T

4T

-15

0

15A

-15

0

15

140 150 160

ξ

-15

0

15

Wave propagation, deformation and disintegration

Simulation of veKdV (12) of the passage of an initial solitarywave of depression across the North West Shelf ofAustralia.

Wave propagation, deformation and disintegration:

NWS, initial depression wave of 15m amplitude

Wave propagation, deformation and disintegration

Simulation using (12) of the passage of an initial solitary waveof depression across the Malin shelf off west coast ofScotland.

Wave propagation, deformation and disintegration:

Malin Shelf, fission, initial amplitude of 21m

Wave propagation, deformation and disintegration

Simulation using (12) of the passage of an initial solitary waveof depression across the Arctic shelf off north coast ofRussia.

Wave propagation, deformation and disintegration:

Arctic Shelf, adiabatic, initial amplitude of 13m

World map of KdV coefficients

a) speed of propagation, c (m/s)

b) dispersion coefficient, δ (m3/s)

World map of KdV coefficients

c) coefficient of quadratic nonlinearity, μ (s-1)

d) coefficient of cubic nonlinearity, μ1 (m-1s-1)

Issues

• Strong nonlinearity: KdV, eKdV formally valid only forweak nonlinearity, but do capture the nonlinearity/dispersionbalance.

• Stability: Localized shear instability, and/or globalinstability.

• Frictional decay: Bottom friction, CD |A|A, or interior“eddy viscosity” friction.

• Transverse effects, earth’s rotation: Rotation-modifiedKadomtsev-Petviashvili (KP) equation, with rotation there areno KdV-type solitary waves.

• Mixing: Localized through shear or convective instability, orspread over the wave path through topographic effects.

References

Grimshaw, R., Pelinovsky, E. and Talipova, T. (2007).Modeling internal solitary waves in the coastal ocean. Surveysin Geophysics, 28, 273-298.

Grimshaw, R., Pelinovsky, E., Talipova, T. and Kurkin, A.(2004). Simulation of the transformation of internal solitarywaves on oceanic shelves. J. Phys. Ocean., 34 , 2774-2779.

Grimshaw, R. (2001). Internal solitary waves. EnvironmentalStratified Flows, ed. R. Grimshaw, Kluwer, Boston, Chapter 1,1-29.

The end, thank you