B. Wang- Kelvin Waves

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KELVIN WAVES

B Wang, University of Hawaii, Honolulu, HI, USA

Copyright 2002 Elsevier Science Ltd. All Rights Reserved.

doi:10.1006/rwas.2002.0191

Wang, B

University of Hawaii,

Department of Meteorology,

2525 Correa Road,

Honolulu, HI 96822, USA

Introduction

The Kelvin wave is a large-scale wave motion of greatpractical importance in the Earths atmosphere andocean. Discovered by Sir William Thompson (wholater became Lord Kelvin)in 1879, the Kelvin wave is aspecial type of gravity wave that is affected by theEarths rotation and trapped at the Equator or alonglateral vertical boundaries such as coastlines ormountain ranges. The existence of the Kelvin waverelies on (a) gravity and stable stratification forsustaining a gravitational oscillation, (b) significantCoriolis acceleration, and (c) the presence of verticalboundaries or the equator. The unique feature of theKelvin wave is its unidirectional propagation. The

Kelvin wave moves equatorward along a westernboundary, poleward along an eastern boundary, andcyclonically around a closed boundary (counterclock-wise in the Northern Hemisphere and clockwise in theSouthern Hemisphere). The wave amplitude is largestat the boundary and decays exponentially withdistance from it. At the Equator, Kelvin waves alwayspropagate eastward, reaching their maximum magni-tude at the Equator and decaying exponentially withincreasing latitude.

There are two basic types of Kelvin waves: bound-ary trapped and equatorially trapped. Each type of

Kelvin wave may be further subdivided into surfaceand internal Kelvin waves. Surface, or barotropic,waves penetrate the entire depth of the fluid. Kelvinwaves also appear within the stably stratified oceanand atmosphere, and are called internal, or baroclinic,Kelvin waves. Internal Kelvin waves are often found ina layer with large density gradients; the densitygradient acts as an interface that allows the existenceof internal gravity waves. Examples of such densitygradients are the oceanic thermocline (a layer of largevertical temperature gradient separating a shallowlayer of warm surface water about 50200 m deep and

a much deeper layer of cold water below) and the

lower edge of an atmospheric inversion, a layer inwhich temperature increases with height. Like gravitywaves, Kelvin waves can also propagate vertically in a

continuously stratified geophysical fluid.Atmospheric Kelvin waves play an important role in

the adjustment of the tropical atmosphere to convec-tive latent heat release, in the stratospheric quasi-biennial oscillation, and in the generation and main-tenance of the MaddenJulian Oscillation. OceanicKelvin waves play a critical role in tidal motion, in theadjustment of the tropical ocean to wind stress forcing,and in generating and sustaining the El Nino SouthernOscillation.

Boundary-Trapped Kelvin Waves

Surface Kelvin Waves

The mechanism and properties of the Kelvin wave canbe illustrated by considering a horizontally propagat-ing Kelvin wave in a rotating fluid of uniform finitedepth H, where His small compared to the horizontalextent of the fluid. The fluid has homogeneous densityand a free surface, and is confined by a vertical lateralboundary. Such an idealized model is referred to ingeophysical fluid dynamics as a shallow water model.

The lateral bounding wall prohibits flow across theboundary, and this absence of transverse motion withrespect to the lateral boundary is a defining character-istic of Kelvin waves. Fluid parcels (elements) areconstrained to move in a vertical plane parallel to thelateral boundary in the neighborhood of the boundary.Thus, the horizontal longshore (along the boundary)component of the Coriolis force must vanish. Conse-quently, the wave motion at the lateral boundary, andin anyparallel vertical plane,is exactlythe same as thatin a nonrotating system, i.e., the shallow water gravitywave (Figure 1). The wave travels along the boundary

with the shallow water gravity wave speed C deter-mined by the square root of the product of thegravitational acceleration g and the depth of the

fluid, C gH1=2. The shape of the wave in thelongshore direction is arbitrary and is conserved as thewave travels. This implies that the surface Kelvin waveis nondispersive, and that the wave energy is trans-mitted at the speed of the shallow water gravity wave.Because the Kelvin wave solution in any vertical planeparallel to the lateral boundary is identical to that ofthe nonrotating case, the energy of a Kelvin wave ispartitioned equally between kinetic and potential

energy.

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A fundamental difference between the Kelvin waveand the two-dimensional gravity wave is that theKelvin wave can propagate in only one direction,rather than in two opposite directions. This is due tothe constraints of the Earths rotation and the presence

of the lateral boundary. Rotation modifies the flow bypiling up fluid against the lateral boundary throughEkman transport, producing an offshore (normal tothe boundary) pressure gradient force associated withthe surface elevation. Since offshore motion is pro-hibited by the presence of the boundary, the offshorepressure gradient force must balance the Coriolis forceassociated with the longshore flow. For this reason, themotion is referred to as semigeostrophic (i.e., geo-strophic balance is reached in only one direction). Adirect consequence of this geostrophic balance is theexponential decay of the longshore velocity and

surface height with distance from the lateral boundary(Figure 1). The Kelvin wave amplitude is significantonly within an e-folding distance of the order of theRossby radius of deformation R from the lateralboundary. Thisimportant length scale is defined by theratio of the gravity wave speed, C, over the absolutevalue of the Coriolis parameter f. Over this charac-teristic distance, the tendency of the gravitationalforce to flatten the free surface is balanced by thetendency of the Coriolis force to deform the surface.This is possible only for a wave traveling in thedirection along which the lateral boundary (where the

wave has maximum amplitude) is always on the right

in the Northern Hemisphere and on the left in theSouthern Hemisphere. Therefore, the effects of rota-tion and the lateral boundary determine the unidirec-tional propagation and the trapped behavior of theKelvin wave.

Internal Kelvin Waves

The horizontally propagating internal, or baroclinic,Kelvin wave behaves in the same manner as the surfacewave except that the motion varies with depth. Mostfrequently, it occurs at a stable interface (or a thin layerwith large vertical density gradients) separating tworelatively homogeneous layers. For internal Kelvinwaves,thepressuregradientforce normalto the lateralboundary arises from the tilt of the interface and isbalanced by the Coriolis force associated with thevertical differential flow parallel to the boundary. Theinternal Kelvin wave speed depends on the densitydifference across the interface and is normally muchslower than that of surface Kelvin waves. In the ocean,the typical speed for internal coastal Kelvin waves is ofthe order of 1 m s1 and the Rossby radius ofdeformation is of the order of 10km in the mid-latitudes. Evidence for coastal Kelvin wave propaga-tion along the eastern boundary of the Pacific has beenobserved in coastal sea level and temperature records.

In the atmosphere, boundary trapped Kelvin wavesoccur primarily in the form of internal waves. They areoften found along the edge of a plateau or a mountainrange, such as the coast of South Africa, the west coastof California, and the eastern flank of the TibetanPlateau. These internal Kelvin waves arecreated near astable inversion layer (often located at the top of theboundary layer) against steep topography. The elevat-ed plateau or mountain range rises above the inver-sion, forming a lateral boundary for the air of thelower layer. Energy is prevented from escaping verti-cally by the inversion and prevented from escapinglaterally by the topography. The maximum distur-bance intensity of these waves is found near the coastor plateau and decreases exponentially in intensity

away from the coast or plateau. The offshore extent ofthe waves depends on both the thermal structure andtopography. When the topography is steep, the depthof the inversion determines the Rossby radius for theatmospheric Kelvin wave. A typical value for atmos-pheric internal Kelvin waves is on an order of1000 km.

Equatorially Trapped Kelvin Waves

Matsuno in 1966 showed that the eastward-propa-gating Kelvin wave is a possible free solution to the

perturbation equations of the shallow water model on

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F0001 Figure 1 Northern hemisphere Kelvin waves on opposite sides

of a channel that is wide comparedwiththe Rossby radius. In each

vertical plane parallel to the coast, the currents (shown by arrows)

are entirelywithinthe plane and areexactlythe sameas those for a

long gravity wave in a nonrotating channel. However, the surface

elevation varies exponentially with distance from the coast in order

to give a geostrophic balance. This means Kelvinwavesmove with

the coast on their right in the Northern Hemisphere and on their left

in the Southern Hemisphere. (From Mortimer, 1977.)

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an equatorial b-plane (b is the meridional gradient ofthe Coriolis parameter), provided that the meridionalvelocity vanishes. This type of wave is called anequatorial Kelvin wave, so named because it isextremely similar in character to coastally trappedKelvin waves, with the Equator serving as a boundary.Like the coastal Kelvin wave, the propagation of anequatorial Kelvin wave is unidirectional, i.e., eastwardonly. In each vertical plane parallel to the equatorialvertical plane, the motion of fluid particles is preciselythe same as that in a shallow water gravity wave(Figure 2). The Kelvin wave propagates without

dispersion at the speed C gH1=2, as for nonrotat-ing gravity waves. Because the Coriolis parameterchanges sign at the Equator, eastward flow occurringon both sides of the Equator would induce equator-ward Ekman mass transport, piling up fluid at theEquator and generating a meridionalpressure gradient

force. In this sense, the Equator acts as a lateral wall.The Earths rotation links the motion in each latitu-dinal plane, because momentum conservation in the

northsouth direction requires a geostrophic balancebetween the eastward velocity and the meridionalpressure gradient force. This geostrophic balanceresults in the perturbation zonal velocity reaching amaximum on the Equator and decaying with increas-ing distance from the Equator (Figure 2). This ispossible only for eastward-traveling waves. Thus,equatorial Kelvin waves are eastward-propagatingand have zonal velocity and pressure perturbationsthat vary with latitude as Gaussian functions centeredon the Equator (Figure 2). The e-folding distance fordecay with increasing latitude is given by

Rc C=2b1=2, where C gH1=2 is the gravitywave speed and b is the meridional gradient of theCoriolis parameter at the equator. Rc is called theequatorial Rossby radius of deformation, because ofits relationship with the decay scale for the case ofconstant Coriolis parameter. The same analysis can be

applied to baroclinic waves in both atmosphere andocean, with H being interpreted as the equivalentdepth. Typical values of the internal gravity wavespeed, C, for the tropical atmosphere are 208 0 m s1, giving an equatorial Rossby radius ofbetween 6 and 12 degrees of latitude. For baroclinicocean waves, appropriate values of Care typically inthe range of 23 m s1, so that the equatorial Rossbyradius is 200250 km.

Vertically Propagating Kelvin Waves

In general, the Earths rotation traps planetary-scalegravity waves in the troposphere unless the frequencyof the wave is greater than the Coriolis frequency(about (1 day)1). For this reason, mid-latitude syn-optic waves are generally unable to penetrate signif-icantly into the stratosphere. However, near theEquator, the dramatic decrease in the Coriolis param-eter allows these longer-period waves to propagatevertically. Vertically propagating Kelvin waves havebeen identified in both the equatorial atmosphere andocean.

Vertically propagating Kelvin waves can be illus-

trated by considering a continuously stratified fluidwith a constant buoyancy frequency in a semi-infinitevertical domain near a lateral boundary or in thevicinity of the Equator. For simplicity, consider a linearequatorial b-plane model. Solutions can be obtainedusing the normal-mode technique by neglecting me-ridional perturbations. In a vertical section along theequator, or in any parallel vertical plane, the motion ofvertically propagating equatorial Kelvin waves sharesthe properties of an ordinary vertically propagatinggravity wave. A vertical cross-section of the perturba-tion motion, pressure, and temperature structure for

vertically propagating Kelvin waves is shown in Figure

(n=1), k=1, Kelvin

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F0002 Figure 2 The theoretical equatorially trapped Kelvin wave

solution to the linear shallow water equations on an equatorial b-

plane for a nondimensional zonal wavenumber 1. Hatching is for

convergence and shading for divergence, with a 0.6 unit interval

between successive levels of hatching or shading, and with the

zero divergence contour omitted. Unshaded contours are for

geopotential, with a contour interval of 0.5units. Negative contours

aredashed andthe zero contour is omitted. Thelargest wind vector

is 2.3 units, as marked. The dimensional scales are as in Matsuno

(1966). (Reproducedwith permissionfrom Wheeler M, etal. (2000)

Large-scale dynamical fields associated with collectively coupled

equatorial waves. Journal of the AtmosphericSciences57(5):613

640.)

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3. The local change in temperature is due to adiabaticwarming or cooling, so that the temperature oscilla-tion leads the vertical (zonal) wind and pressure

oscillations by a quarter-cycle.Since stratospheric Kelvin waves are forced frombelow by disturbances in the troposphere, the waveenergy propagation must have an upward component.According to theory, Kelvin waves become dispersivein the presence of vertical propagation, and theverticalcomponent of their phase velocity is always oppositeto that of their group velocity (the velocity at which thewave energy is transmitted). Thus, the phase velocitymust have a downward component. The condition ofeastward propagation due to equatorial trappingrequires that the vertical wavenumber be negative.

Hence, an eastward- and downward-propagatingequatorial Kelvin wave has constant phase lines thattilt eastward with height (Figure 3).

Because the variation in zonal wind depends on thepressure gradient force, the highest zonal pressuregradient precedes the largest westerly acceleration by aquarter-wavelength, and the zonal wind and pressurewaves coincide. This creates an upward flux of waveenergy. An individual parcel moves up along the tiltedphase line, bringing westerly momentum upward, andmoves down, bringing easterly momentum down-ward. Thus, the Kelvin wave transports westerly

momentum upward.

Significance of Kelvin Waves in the

Atmosphere and Ocean

Oceanic Kelvin Waves

Kelvin waves are essential in the description of oceantides. For a deep ocean H 5 km at 301 N, the

Rossby radius for barotropic Kelvin waves is about3000 km. Continental shelf regions normally extendabout a hundred kilometers seaward; hence, a steepcontinental slope is practically indistinguishable froma vertical boundary at the scale of the Rossby radius.Thus, a barotropic Kelvin wave extends far from thecoast and occupies a substantial fraction of a typicalocean. Much of the energy of tide waves travelingalong continents is transmitted in the form of baro-tropic Kelvin waves with a speed of about 200 m s1.For instance, along the coast of California more thantwo-thirds of the semidiurnal and half the diurnal tidal

amplitudes can be accounted for by traveling baro-tropic Kelvin waves. For shallow seas and coastalwaters, the Rossby radius is about 200 km. When aKelvin wave moves through a region in which the fluiddepth or the Coriolis parameter varies and the waveenergy flux remains constant, the amplitude of the

wave varies in proportion to f=H1=2. Thus, wave

amplitude increases when Kelvin waves move intoshallow water. In coastal regions, Kelvin waves canalso be generated as storm surges are diffracted byvertical boundaries and scattered by irregular coast-

lines. Variable longshore winds and atmosphericpressure gradients acting on the sea surface are alsopossible energy sources for oceanic Kelvin waves.

Internal coastal Kelvin waves can be generated bywind-induced, time-dependent coastal upwelling.Coastal upwelling (downwelling) is caused by anEkman mass flux transported offshore (onshore) andforced by longshore winds. The disturbances can thenpropagate along the coast as boundary-trapped inter-nal Kelvin waves. Therefore, the amount of upwellingdepends not only on local wind forcing but also on theforcing that generated thewaves at an earlier time. In a

lake, during strong wind forcing, the region ofupwelling progresses cyclonically (in the NorthernHemisphere) at the speed of an internal Kelvin wave.

Equatorial Kelvin waves play a critical part inthermocline adjustment. As the equatorial oceancirculation responds to wind stress forcing, equatorialKelvin waves transmit signals rapidly from the west-ern to the eastern extremities of the ocean basin.Internal Kelvin waves propagating along the therm-ocline take about two months to cross the entireequatorial Pacific. These equatorial Kelvin waves playan essential role in the El Nino Southern Oscillation

(ENSO) cycle. First, they support a positive feedback

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F0003 Figure 3 Longitudeheight section along the Equator showing

pressure, temperature, and wind perturbations for a thermallydamped Kelvin wave. Heavy wavy lines indicate material lines;

shortbluntarrowsshowphasepropagation. Areas of high pressure

are shaded. Length of the small thin arrows is proportional to the

wave amplitude, which decreases with height owing to damping.

The large shaded arrow indicates the net mean flow acceleration

owingto the wavestress divergence. (Reproducedwith permission

from Holton JR (1992) Introduction to Dynamic Meteorology, 3rd

edn. San Diego: Academic Press.)

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between the central Pacific zonal wind and easternPacific sea surface temperature (SST) anomalies. Awesterly wind anomaly excites downwelling Kelvinwaves, which propagate into the eastern Pacific,suppressing the thermocline and causing the SST torise; this, in turn, enhances the central Pacific westerlywind anomaly by increasing the eastward pressuregradient force in the atmosphere. This positive feed-back provides a development mechanism for ENSOSST warming. Second, the cyclonic wind stress curlassociated with the central Pacific westerly windanomaly can induce upwelling oceanic Rossby wavesthat propagate westward. These waves are eventuallyreflected at the western ocean boundary, generating

upwelling equatorial Kelvin waves, which propagateinto the eastern Pacific and offset the warming byenhancing vertical cold advection. This negativefeedback provides a mechanism for turning the cou-pled system to its opposite (La Nina) phase andsustaining the ENSO cycle. In addition, the atmos-pheric intraseasonal wind forcing continuously gen-erates equatorial Kelvin waves whose nonlinearrectification to the mean state may also contribute tothe eastern Pacific warming.

Atmospheric Equatorial Kelvin Waves

The atmospheric equatorial Kelvin wave is one of thecritical wave motions in the response of the tropical

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F0004 Figure 4 The solution of the forced shallow water equations for heating (or evaporation) that is confined to the range of longitudes

jxjo2ae|.The distributionwithlatitudehas a maximum northof theEquator.The arrowsin (A)give thehorizontalvelocityfieldand thesolidcontours in the upper panel give the vertical velocity, which has a distribution close to that of the heating function. The motion is upward

within the contours north of the Equator with a maximum near x 0, y ae. The contours in the lower panel are pressure contours, andthe axesare labeled in units of ae. (B) The meridional circulation when theresponse is interpreted as a baroclinic response to heating with

sinusoidal distribution in the vertical. The upper panel gives the zonal flow (E for easterlies, W for westerlies) and the lower panel the

meridional flow (Hadley circulation). (C) The meridionally averaged zonal flow (Walker circulation) with the same interpretation, i.e., as a

baroclinic response. (Reproduced with permission from Gill AE (1982) AtmosphereOcean Dynamics. New York: Academic Press.)

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atmospheric circulation to a heat source (Figure 4).When an imposed heating centered on the Equator isswitched on at some initial time, Kelvin waves carryinformation rapidly eastward, thereby creating east-erly trade winds in that region and forming a Walkercell (rising motion over the heat source region andsinking motion to its east). Internal equatorial Kelvinwaves traveling with typical speeds of 2080 m s1

are an effective means by which the equatorialatmosphere becomes homogenized in the zonal direc-tion. The easterly winds are in geostrophic equilibri-um, so that there is a trough along the Equator, withthe winds along the Equator flowing directly down thepressure gradient (Figure 4). The westward-propagat-ing Rossby wave regime to the west of the forcingregion is about one-third the size of the Kelvin waveregime because Rossby waves travel at one-third of theKelvin wave speed. The equatorial westerlies betweenthe symmetric Rossby waves provide inflow into theheating region, and are in geostrophic equilibrium, sothat a relative ridge appears along the Equator.Meanwhile, the flow converges toward the Equator.If atmospheric damping is taken into account, thissimple model can largely explain the steady-stateatmospheric response to an imposed heat source.

Kelvin and mixed Rossby-gravity waves are thepredominant disturbances in the equatorial strato-sphere, and play a critical role in the stratosphericcirculation through their vertical transport of energyand momentum. Stratospheric Kelvin waves are

excited by oscillations in the large-scale convective

heating pattern in the troposphere, and are a source ofwesterly momentum for the QBO. The QBO is a zonalwind oscillation in the equatorial stratosphere, andpropagates downward with a period of about 2430months. Figure 5 shows an example of the zonal windoscillations caused by the passage of Kelvin waves nearthe Equator. The descent of the westerly phase of theQBO is shown in the figure. At each level, there is anincrease of the zonal wind with time. Superposed onthis secular trend is a fluctuating Kelvin wave compo-nent with a period of about 12 days and a verticalwavelength of about 1012 km. As vertically propa-gating Kelvin waves carry westerly momentum up-ward, they are damped by radiative cooling, small-scale turbulence, and critical level interaction. As thewaves are damped, they lose momentum and acceler-ate the westerly mean flow. The damping depends onthe Doppler-shifted wave frequency. As the Doppler-shifted frequency decreases, the vertical component ofthe group velocity also decreases, and a longer time isavailable for the wave energy to be damped. Hence,westerly Kelvin waves tend to be damped preferen-tially in the westerly shear zone where their Doppler-shifted frequencies decrease with height. The associ-ated momentum flux convergence produces westerlyacceleration of the mean flow, causing the westerlyshear zone to descend. A similar argument is valid forthe downward propagation of the easterly phase of theQBO through the action of Rossby-gravity waves.

A peak in the variability of the tropical atmosphere

appears in the 30- to 60-day period range, and is

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F0005 Figure 5 Timeheight section of zonal wind at Canton Island (31 S). Isotachs at intervals of 5 m s1. Westerlies are shaded.

(Reproduced with permission from Holton JR (1992) Introduction to Dynamic Meteorology, 3rd edn. San Diego:Academic Press;original

courtesy of J. M. Wallace and V. E. Kousky.)

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known as the MaddenJulian oscillation (MJO).Madden and Julian found that a 30- to 60-dayoscillation in zonal winds is in approximate geo-strophic balance with varying pressure maxima andminima centered at the Equator. A low-level low-pressure anomaly is accompanied by low-level easterlyanomalies. The pressure and zonal wind are out ofphase in the upper troposphere. The wave patternsmove eastward along the Equator. At the Equator, themeridional winds appear to be insignificant. Theamplitude of the oscillation decays with distance awayfrom the Equator. These features aresimilar to those ofinternal equatorial Kelvin waves except that the largevertical scale of the MJO implies a faster phase speedthan is observed. Arguments involving coupling withequatorial westward-traveling Rossby waves andinteraction with the release of latent heat in thedisturbances as well as viscous damping have beeninvoked to explain the observed slow phase speed.

See also

Dynamic Meteorology: Waves and Instabilities (0141).

El Nino and the Southern Oscillation: Observation

(0148); Theory (0149). Middle Atmosphere: Quasi-

Biennial Oscillation (0232). Ocean Circulation: Surface/

Wind Driven Circulation (0280). Rossby Waves (0346).

Tropical Meteorology: Equatorial Waves (0414); Intra-

seasonal Oscillation (Madden-Julian Oscillation) (0415).

Further ReadingGill AE (1982) AtmosphereOcean Dynamics. New York:

Academic Press.Holton JR (1992) Introduction to Dynamic Meteorology,

3rd edn. San Diego: Academic Press.Pedlosky J (1987) Geophysical Fluid Dynamics. Berlin:

Springer-Verlag.LeBlonde PH and Lawrence AM (1978) Waves in Ocean.

New York: Elsevier.Cushman and Roison (1994) An Introduction to Geophysi-

cal Fluid Dynamics. Englewood Cliffs: Prentice-Hall.Matsuno T (1966) Quasi-geostrophic motion in the equa-

torial area. Journal of the Meteorological Society of

Japan 44: 2542.Philander SG (1990) El Nino, La Nina and the Southern

Oscillation. New York: Academic Press.

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B. Wang- Kelvin Waves