John P. Boyd- The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow

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NOTES AND CORRESPONDENCE

The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow

JOHN P. BOYD

University of Michigan, Ann Arbor, Michigan

(Manuscript received 27 May 2004, in final form 15 September 2004)

ABSTRACT

The effects of latitudinal shear on equatorial Kelvin waves in the one-and-one-half-layer model areexamined through a mixture of perturbation theory and numerical solutions. For waves proportional toexp(ikx), where k is the zonal wavenumber and x is longitude, earlier perturbation theories predictedarbitrarily large distortions in the limit k → . In reality, the distortions are always finite but are very

different depending on the sign of the equatorial jet. When the mean jet is westward, the Kelvin wavebecomes very, very narrow. When the mean jet flows eastward, the Kelvin wave splits in two with peaks welloff the equator and exponentially small amplitude at the equator itself. The phase speed is always a boundedfunction of k, asymptotically approaching a constant. This condition has important implications for thenonlinear behavior of Kelvin waves. Strong nonlinearity cannot be balanced by contracting longitudinalscale, as in the author’s earlier Korteweg–deVries theory for equatorial solitons: for sufficiently largeamplitude, the Kelvin wave must always evolve to a front.

1. Introduction

The tropical ocean and atmosphere are filled withstrong latitudinally and vertically varying mean flows.Many authors have explored the effects of shear onequatorially trapped waves, including Lindzen (1970,1971, 1972), Holton (1970), Philander (1976), Boyd(1978a, b), McPhaden and Knox (1979), Boyd andChristidis (1982, 1983, 1987) Ripa and Marinone(1983), McPhaden, Proehl and Rothstein (1986),Brossier (1987), Zhang and Webster (1989), Johnsonand McPhaden (1993), Proehl (1996, 1998), Boyd andNatarov (1998), Masina and Philander (1999), Masina,Philander and Bush (1999), Natarov and Boyd (2001,2002), Chelton et al. (2003), and so on. However, somepuzzles remain.

Boyd (1978a, b) derived perturbative expansions inpowers of the shear for the equatorial Kelvin wave.Although accurate for moderate zonal wavenumber k,the corrections are proportional to k times the strengthof the shear. This condition implies that the theory hasan “ultraviolet catastrophe,” to borrow a phrase fromquantum electrodynamics, in the sense that the correc-tions become infinitely large as k → .

This breakdown of perturbation theory is particularlydisconcerting because a good understanding of theshort-wave behavior of Kelvin waves is necessary fortheories of nonlinear Kelvin dynamics. Boyd (1984)showed that in the presence of latitudinally varying

mean currents, the Kelvin wave is dispersive: nonlin-earity can balance this dispersion to create solitarywaves of Korteweg–deVries (KdV) type. The forma-tion of Kelvin solitons was later verified in numericalone-and-one-half-layer models by Long and Chang(1990) and Boyd (1998).

The Korteweg–deVries theory predicts that solitonsof arbitrarily large amplitude will form: as the height of a crest increases, the soliton becomes narrower to in-crease the dispersion proportionately. However, thisoutcome is not what actually happens. Chen and Boyd(2002) showed that solitary waves exist only below afinite-amplitude threshold; larger waves invariablybreak. Further numerical experiments reported in Boyd

(2003, 2004, manuscript submitted to Math. Comput .)suggest that the limiting solitary wave, the boundarybetween breaking and nonbreaking behavior, is a so-called corner wave—that is, a wave with a discontinu-ous slope at the crest.

The reason for this discrepancy between Korteweg–deVries theory and reality is that the KdV theory is along-wave theory: the linear wave dispersion is approxi-mated by

c c0

bk2 1

Corresponding author address: John P. Boyd, Department of Atmospheric, Oceanic, and Space Sciences, University of Michi-gan, 2455 Hayward Ave., Ann Arbor, MI 48109-2143.E-mail: [email protected]

1138 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 35

© 2005 American Meteorological Society



for some constants c0 and b. This equation unrealisti-cally predicts an infinite phase speed as k→ and alsoa dispersion that is unbounded.

One should hardly be shocked that the KdV theory,which assumes that the wave is weakly nonlinear and of large zonal scale, should spin fairy tales about Kelvin

waves that are strongly nonlinear, or of small zonalscale. But what does happen to Kelvin waves in theshort-wave limit?

In this note, I derive two different perturbation ex-pansions for linear Kelvin waves to provide a partialanswer.

The model is the one-and-one-layer model—that is,the shallow-water wave equations in the equatorialbeta-plane approximation. For an equation of zonalwavenumber k and phase speed c, the equations, lin-earized about a mean east–west current U ( y) and amean depth field 1 ( y), are

ikU cu U y y ik 0, 2

ikU c yu y 0, and 3

ikU c ik1 yu 1 y y

y 0, 4

where the mean flow and height field are in geostrophicbalance, y yU ( y), and subscript y denotes differ-entiation with respect to latitude. Note that the equa-tions have been nondimensionalized in the fashionusual in equatorial oceanography so that the unper-turbed Kelvin wave has a phase speed of 1 and u exp[(1/2) y2]. Using the equivalent depth of the first

baroclinic mode of the ocean in the nondimensionaliza-tion gives a length scale of about 300 km, a time scale of about 1.7 days, and a velocity scale of about 2 m s1.

This eigenproblem was numerically solved by dis-cretizing in rational Chebyshev functions TB( y; L),where L 2, and then the algebraic eigenvalue prob-lem was solved using the QZ method (Boyd 2001). Onehundred basis functions for each unknown were suffi-cient to give at least six decimal places of accuracy forall cases illustrated.

Although a number of profiles were studied, thetheories shall be illustrated primarily through a simpleGaussian jet:

U y S exp y2

, with y 1 2U y. 5

There are two dramatically different kinds of behaviordepending on the sign of S. When S 0, a westward jet,the Kelvin wave becomes very, very narrow as k → .When S 0, the Kelvin wave splits in two and haspeaks well off the equator with an exponentially smallamplitude at the equator. These drastically differentbehaviors require separate perturbation theories, whichare given in the next two sections, respectively.

2. Multiple-scale perturbation theory forwestward jets

a. Symmetric jets

For technical reasons explained in section 2b, atten-tion shall be initially restricted to jets that are symmet-

ric with respect to the equator, that is, that have theproperty that U ( y) U ( y) for all y. Numerical solu-tion of the eigenvalue problem showed that as k → ,the Kelvin wave became narrower and narrower, u ,and decreased as roughly O(k1/2) relative to the othertwo fields. This result suggested applying the method of multiple scales with the scalings defined below.

Define a new coordinate such that the Kelvin wavehas O(1) width in Y:

Y k1 2 y↔ y Y k1 2,d

dy k1 2

d

dY . 6

Because the Kelvin wave becomes narrower and nar-rower as k→ , it is legitimate to approximate U and

by power series approximations because the wave hasnegligible amplitude wherever the series is inaccurate:

U 0 2 y2 0 k1

2Y 2. 7

To satisfy geostrophic balance, constant y yU ( y) dy, one must have

0 1 20 y2 O y4 0 1 2k1

0Y 2.

8

Expand

c c0 k1 2c1 k1c2, 9

u u0Y k1 2

u1Y k1

u2Y , 10

k1 2 0Y k1

1Y k3 2 2Y , and 11

p0Y k1 2 p1Y k1 p2Y . 12

Note that the expansion for begins with O(k1/2).Rewriting the linearized shallow water wave equationsin Y , substituting the expansions of c and u, , and ,and matching powers gives

c 0 1 0 k1c2, 13

p0Y 1 0u0Y , 0Y iu0,Y , 14

p1Y 1 0u1Y , 1Y iu1,Y , 15

p2Y 1 0u2Y u0Y 2Y 2 c2, and

16

2Y iu2,Y i22 1Y

1 0

u0 y. 17

The residual of the height equation at second orderdoes not yield u2(Y ), which is determined at higherorder, but instead gives the differential equation

JUNE 2005 N O T E S A N D C O R R E S P O N D E N C E 1139



u0,YY 21

1 0

c2

22

1 0

1

2

0

1 0Y 2u0 0. 18

This eigenproblem has the solution

u0Y exp Y 2, 19

c2 1 0, and 20

1

2 22

1 0

1

2

0

1 0

. 21

Figure 1 illustrates a typical case.Another important conclusion is that the dispersion

of the Kelvin wave is not arbitrarily large as k → ;instead, it asymptotically approaches a constant plus acorrection that diminishes as O(1/k).

b. Unsymmetric jets

The solution above is special in that the mean flowwas assumed to be symmetric with respect to the equa-tor. Asymmetry, and a nonzero shear at the equator,

introduce an additional effect: the peak is shifted awayfrom the equator.

When dU /dy(0) 0, the numerical studies show thatthe peak of the Kelvin wave is shifted by an amount y sthat is independent of k to lowest order. Because thisshift is not a perturbative effect in k, unsymmetric jetscan be studied analytically only by a double series (inwhich both k1/2 and the strength S of the shear areindependent small parameters). For this reason, I shallnot give a perturbative treatment, but instead shall il-

lustrate the situation with a typical numerical case inFig. 2.

One crucial point is that all the profiles are roughlycentered on y 1, independent of k. The other is thatthe Kelvin wave again becomes narrower and narroweras k→ . Thus, the only qualitative difference from thesymmetric jet is the k-independent shift in latitude.

Boyd (1978a,b) and Boyd and Christidis (1982)showed that equatorial shear shifts the equator forgravity waves: in a linear shear U y, the wave has astructure similar to that in the absence of shear exceptthat the center of the disturbance is moved to y /2.Thus, a shear-induced shift in the center of the Kelvinwave is hardly a surprise.

However, the mechanism for short Kelvin waves isdifferent. The profile in Fig. 2 has dU /dy 0, and so agravity wave would be shifted into the Southern Hemi-sphere. The short Kelvin wave is shifted into the North-ern Hemisphere so that its peak lies near the negativemaxima of the jet.

3. Eastward jets

Figure 3 shows what happens for the same symmetric

jet as in Fig. 1, but with the sign of the mean flowreversed. Instead of becoming more and more concen-trated around the equator, the Kelvin wave becomesexpelled from the equator: a single peak (in the absenceof the jet) divides into two with, as k→ , no amplitudeat the location of the jet.

Because the Kelvin wave is wide, it is not possible toapply the method of multiple scales or to approximateU ( y) by a power series in y. The crucial observation isthat, because the wave, as k → , is found only on the

FIG. 1. Profiles of Kelvin wave zonal velocity u for k 10,20, . . . , 100 for the symmetric westward jet U ( y) (2/5)exp( y2). The curves are not labeled, but they become narrower

and narrower as k increases. For comparison, the shape of theKelvin wave in the absence of shear is shown as the dashed line.

FIG. 2. Profiles of u for k 10, 20, . . . , 100 for the antisym-metric jet U ( y) [(3/4) y (1/8) y3] exp( y2/4). The profiles arenot labeled, but they become narrower and narrower as k in-creases. For comparison, the shape of the Kelvin wave in theabsence of shear is shown as the dashed line.




fringes of the jet, one may pretend that the jet is smallwith an amplitude inversely proportional to 1/k:

U y k2U ˜ y and y k2̃ y. 22

In a similar way, some experimentation shows that aconsistent theory results if

c 1 k2c2, 23

u u0 y k2u2 y, 24

k1 0 y k2 1 y, and 25

u0 y k22 y. 26

Note that once again the expansion for begins with aninverse power of k, unlike the other fields. Note alsothat u and are identical to lowest order, though theyare different at O(1/k2).

Matching powers of k, the north–south momentumequation gives

0 i yu0 y u0, y and 1 y 0, 27

where subscript y denotes y differentiation as before.At O(1 /k2), the sum of the residuals in the x-momentum equation and height equation is the homo- geneous differential equation

u0, yy 2c2 1 y2 2k2U y k2 yu0 0, 28

where the scaled mean wind and height, U ˜ and ̃ , havebeen replaced by the equivalent expression in U , ,and k.

In general, this eigenproblem can only be solved nu-merically with c2 as the eigenvalue. However, this

eigenproblem contains only a single unknown (u0),whereas the original system of equations has three (u, , ).

Figure 4 shows there is good agreement betweentheory and numerical solution for large k. The eigen-value of the one-unknown problem for this case is c2 6.61; this implies c 1 c2/k2 1.000 66, which com-pares favorably with the exact numerical phase speed,c 1.000 63.

The theory also shows that, as k → , the phasespeed varies only very slowly with k. Thus the Kelvin

wave, instead of the unbounded dispersion predicted bythe Korteweg–deVries Kelvin soliton theory, becomesnondispersive again in the limit k → .

4. Conclusions

The arbitrarily strong dispersion and arbitrarily largecorrections of the older, small-and-moderate k pertur-bation theories do not describe the Kelvin waves as k→. Instead, the Kelvin wave in a shear flow comes toresemble, as the zonal wavelength decreases, more andmore the Kelvin wave in the absence of a mean flow inthe sense that, for both cases described here, u andthe phase speed c becomes independent of k.

Current-day observing systems are inadequate tomeasure very short Kelvin waves, and so one cannotmake comparisons with observations. However, theshort-wave behavior of Kelvin waves has observableconsequences. Numerical experiments have confirmedthat weakly nonlinear Kelvin waves do form solitarywaves and cnoidal waves as predicted in Boyd (1984).However, moderately large Kelvin waves invariablyevolve to fronts and breaking. The reason is that, asshown here, the dispersion of Kelvin waves does not

FIG. 4. Comparison of u( y) with u0( y) as given by perturbationtheory for k 100 for the symmetric eastward jet U ( y) (2/5)exp( y2). The exact solution is solid; u0( y) is dashed.

FIG. 3. Zonal velocity for k 10, 20, . . . , 100 for the symmetriceastward jet U ( y) (2/5) exp( y2), which is shown as the curvelabeled with circles. The curves are not labeled, but they move

farther and farther to the right as k increases.

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become arbitrarily large as the zonal scale decreases,but on the contrary weakens.

Any Kelvin wave large enough to be interesting istherefore likely to be a breaking wave or an undularbore.

Acknowledgments. This work was supported by NSFGrants OCE9521133 and OCE9986368. I thank DennisMoore and the other reviewer for their suggestions.

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——, and A. Natarov, 1998: A Sturm-Liouville eigenproblem of the fourth kind: A critical latitude with equatorial trapping.Stud. Appl. Math., 101, 433–455.

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John P. Boyd- The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow