J.S. Allen, Peter R. Gent and Darryl D. Holm- On Kelvin Waves in Balance Models

8/3/2019 J.S. Allen, Peter R. Gent and Darryl D. Holm- On Kelvin Waves in Balance Models

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2060 VOLUME 27J 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

1997 American Meteorological Society

On Kelvin Waves in Balance Models

J. S. ALLEN

College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

PETER R. GENT

National Center for Atmospheric Research, Boulder, Colorado

DARRYL D. HOLM

Los Alamos National Laboratory, Los Alamos, New Mexico

2 December 1996 and 10 March 1997

ABSTRACT

The f-plane linear shallow-water equations support coastal Kelvin waves. These waves propagate along thecoast and have zero velocity normal to the coast. It is shown that the balance equations also support coastalKelvin waves, but these waves differ depending upon the boundary conditions imposed. Three different boundaryconditions and resulting Kelvin wave approximations are examined. It is shown that one set of boundaryconditions gives balance-model Kelvin waves that are closer to those of the shallow-water equations than theother two boundary conditions.

1. Introduction

The balance equations for a continuously stratified,rotating fluid were first systematically derived by Lorenz(1960). He showed that they are truncations of the prim-itive equation vorticity and divergence equationsthrough two orders in the usual midlatitude Rossbynumber scaling and that they have a global energy con-servation. Gent and McWilliams (1983) developed con-sistent boundary conditions for this model in a boundeddomain. Allen (1991) proposed an extension to thismodel that can be written easily as a momentum equa-tion and has a potential vorticity conservation law inaddition to energy conservation. He also formulated adifferent set of boundary conditions than those in Gentand McWilliams (1983). Holm (1996) derived this ex-tended balance model from an asymptotic expansion ofHamiltons principle, demonstrated its Hamiltonianstructure, and proposed a companion, but slightly dif-ferent, balance model in isopycnal coordinates.

To clarify the issue of appropriate boundary condi-tions for the balance equations, we obtain analyticalsolutions for midlatitude Kelvin waves in the balanceequation approximation and compare them to the linear

Corresponding author address: Dr. John S. Allen, College ofOceanographic and Atmospheric Sciences, Oregon State University,104 Ocean Admin. Bldg., Corvallis, OR 97331-5503.E-mail: [email protected]. edu

shallow-water model Kelvin wave solution. We find thatthe Kelvin waves resulting from Allens (1991) bound-ary conditions, which were also utilized in numericalexperiments by Allen et al. (1990), are a better approx-imation to the shallow-water Kelvin waves than those

obtained with the boundary conditions of Gent andMcWilliams (1983), which were also employed byHolm (1996). The shallow-water Kelvin wave solutionis given in section 2, and the various balance equationapproximations to the Kelvin wave are given in section3. The conclusions are briefly stated in section 4.

2. Linear shallow-water equations

We consider the linearized shallow-water equationsin a fluid of average depth Hand a free surface elevationof. The equations are

u fk u g 0, (2.1)t

H u 0, (2.2)t

where u is the horizontal velocity, k is a unit verticalvector, f is the constant Coriolis frequency, g is thegravitational constant, t is time, and subscript t denotespartial differentiation. These equations imply conser-vation of a linearized potential vorticity, in the form

[Hk u f]t

0. (2.3)

The energy equation is

Et

g (u) 0, (2.4a)


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where

12E [u u (g/H) ]. (2.4b)

2

Thus, in bounded domains the boundary condition of

no normal flow at the domain boundary ensures globalenergy conservation, that is, conservation of the areaintegral of E.

We now consider an eastern ocean boundary locatedat x 0. The f-plane equations support coastal Kelvinwaves that travel northward or southward in the North-ern and Southern Hemispheres, respectively. The North-ern Hemisphere Kelvin wave solution is given by

u 0, (2.5a)

(, ) (c, H) exp[f x/c i(ly t)] (2.5b,c)

cl, (2.6)

where c (gH)1/2 is the gravity wave speed and where

the dispersion relation (2.6) implies that the waves arenondispersive. The zonal decay scale of the waves awayfrom the boundary is equal to c/f.

3. Linear balance equations

The linear balance equations are an approximation tothe shallow-water equations that retain the time deriv-ative of the rotational component of the velocity only.They take the form

uRt fk u g 0, (3.1)

where

u uR uD k , (3.2)in addition to the exact linear height equation (2.2).Equations for the vertical component of vorticity andthe horizontal divergence are typically considered inplace of the momentum equations (3.1). In terms of thevariables (, , ), these are

2 2 f 0, (3.3a)t

2 2f g , (3.3b)

while (2.2) is

t

H2 0. (3.4)

No time derivative appears in the divergence equation

(3.3b) and this filters out high-frequency gravityinertialwaves. A relevant point, evident from the governingequations (3.3a,b) and (3.4), is that arbitrary solutionsof 2 0 and 2 0 may be added to and ,respectively, without affecting the satisfaction of(3.3a,b) and (3.4). Proper boundary conditions are nec-essary to ensure appropriate solutions for and .

The linear balance equations conserve a linearizedpotential vorticity,

[Hk uR

f]t

0, (3.5)

which, since k u k uR, is identical to theshallow-water equation (2.3). The energy equation is

ERt

g (u) (uRt

) 0, (3.6a)

where

1 2E [u u (g/H) ]. (3.6b) R R R2

In bounded domains, the area integral of ER

will beconserved if, at the boundary, there is no normal flow

u n 0, (3.7)

and if

u n ds 0, (3.8) RtC

where n is the outward pointing unit vector normal tothe boundary, and the integral in (3.8) is around the

boundary contour C. Thus, global conservation of theenergy E

Rcannot be assured merely by the boundary

condition of no normal flow at the domain boundaryand a second boundary condition is needed. This isstandard for the balance equations because the speci-fication of unique solutions for both and requiresan extra boundary condition in addition to (3.7) (Gentand McWilliams 1983; Allen et al. 1990).

Global energy conservation in Eq. (3.6) can be as-sured by at least three different choices of boundaryconditions. They are

(A) Allen (1991)

u n 0, (3.9a)

(g fk uR) n 0, (3.9b)

(B) Gent and McWilliams (1983) and Holm (1996)

u n 0, uR

n 0, (3.10a,b)

(which implies uD n 0), and

(C)

u n 0, 0. (3.11a,b)

Note that the Allen (1991) conditions (A) ensure con-servation of energy E

Rsince (3.1) and (3.9b) imply that,

on the boundary,

uRt

n f(k ) n 0, (3.12)

so that (3.8) is satisfied because the integrand is a perfectdifferential.

With boundary conditions (A), the Kelvin wave so-lution at an eastern ocean boundary is

u 0, (3.13a)

(, ) (f/k, H) exp[kx i(ly t)], (3.13b,c)

f l/k, (3.14)

where


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2062 VOLUME 27J 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

FIG. 1. Dispersion relations for the linear shallow-water equationsSWE (2.6) and for the linear balance equations with boundary con-ditions A (3.14), B (3.18), and C (3.23b).

k (l2 f2/c2)1/2, (3.15)

and where, for definiteness in what follows, we assumel 0. In comparison to the shallow-water equation(2.5), this Kelvin wave solution also has zero zonalvelocity and a single decay scale away from the coastequal to 1/k. The dispersion relation (3.14) is plotted inFig. 1 along with the shallow-water relation (2.6). Inthe small wavenumber low-frequency limit, the A dis-

persion relation (3.14) gives

12 2 2 cl 1 c l /f for l K f/c, (3.16) 2

which is second-order accurate in this important limit.Likewise, the error in the decay scale 1/k and in theamplitude of is small O[(cl/f) 2] for l K f/c. As shownin Fig. 1, /f 1 for l k f/c so the frequency is finiteas the meridional wavenumber becomes very large.

Boundary conditions B impose that the normal com-ponents of both the rotational and divergent velocitiesare zero. They are discussed in Gent and McWilliams(1983), where they are labeled choice 1, and by Holm

(1996) in his derivation of the balance equations basedon Hamiltonian dynamics. The choice 2 boundary con-ditions discussed in Gent and McWilliams (1983) areequivalent to choice 1 in this linear f-plane model. Fora coastal Kelvin wave, boundary conditions B implythat the solution cannot be written purely in terms ofthe spatial function exp(kx). Additional componentsmust be added to and that satisfy 2 0 and 2 0, respectively, and so have an exp(lx) spatial de-pendence. The eastern boundary Kelvin wave solutionthat satisfies boundary condition B is given by

2c lu i exp[i(ly t)]

2f

[exp(kx) exp(lx)], (3.17a)

2c exp[i(ly t)]

2f

[(f k l) exp(kx) l exp(lx)], (3.17b)

H exp[kx i(ly t)], (3.17c)

f l/(k l). (3.18)

The dispersion relation in (3.18) is plotted in Fig. 1 andhas two undesirable features compared to the dispersionrelation (3.14) for boundary conditions A. The first isthat in the small wavenumber, low-frequency limit thefrequency is only first-order accurate, that is,

cl(1 cl/f ) for l K f/c, (3.19)

and the second is that the frequency becomes very largeas the meridional wavenumber gets large; that is, 2(cl)2/ffor l kf/c. In addition, compared to the shallow-water solution (2.5), this solution has zero zonal velocityonly at the boundary and has two decay scales awayfrom the coast equal to 1/k and 1/l. In the small merid-ional wavenumber limit l K f/c, however, the relativemagnitude of the component of with decay scale 1/lis small O[(cl/f)2], as is the magnitude of the nonzerozonal velocity. It is clear, nevertheless, that in the limitl K f/c, that is, for K f, the coastal Kelvin wave inthe balance equations with boundary conditions A is abetter approximation to the shallow-water Kelvin wavethan that with boundary conditions B.

Boundary conditions C impose that the normal com-ponent of velocity and the divergent potential are zeroon the boundary. The eastern boundary Kelvin wavesolution that satisfies boundary conditions C is given by

2icu (k f l) exp[i(ly t)]

2f

[exp(kx) exp(lx)], (3.20a)

2c exp[i(ly t)]

2f

[(f k l) exp(kx)

(k f l) exp(lx)], (3.20b) H exp[kx i(ly t)], (3.20c)

2(f ) f l/(k l). (3.21)

The dispersion relation (3.21) has two roots

11/2 f[1 (1 4l/(k l)) ], (3.22)

2

where, for l K f/c,


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f(1 cl/f ), (3.23a)

12 2 2 cl 1 c l /f . (3.23b)

2Thus, the solution with

corresponds closely to a

Kelvin wave while the solution with

is a spuriousmode of oscillation, which is an unfavorable aspect ofboundary conditions C. The dispersion relation (3.23b)for

is plotted in Fig. 1. Compared to the shallow-

water solution (2.5), the

solution shares the disad-vantages of the B boundary condition solution in havingthe zonal velocity zero only at the boundary and havingtwo decay scales away from the boundary. However, inthe limit l K f/c, the nonzero zonal velocity in the

solution is small O[(cl/f)3] as is the relative magnitudeof the component ofwith decay scale 1/l. Finally, notethat the frequency becomes large as the meridionalwavenumber gets large; that is,

2cl for l k f/c.

4. Conclusions

The midlatitude coastal Kelvin waves are well rep-resented in linearized balance models for l K f/c, orequivalently K f, using the boundary conditions for-mulated by Allen (1991) when he proposed the balanceequations based on momentum. The Kelvin waves are

not as well represented in this limit using other boundaryconditions including the choice 1 of Gent and Mc-Williams (1983), which sets the normal components ofboth the rotational and divergent velocities to zero.

Acknowledgments. This note was written while all ofthe authors were enjoying the hospitality of the IsaacNewton Institute of the University of Cambridge in En-gland. For J.S.A. the work was partially supported byN SF G ra nt O CE -9 31 43 17 a nd b y O NR G ra ntN00014-93-1-1301. The National Center for Atmo-spheric Research is sponsored by the National ScienceFoundation.

REFERENCES

Allen, J. S., 1991: Balance equations based on momentum equationswith global invariants of potential enstrophy and energy. J. Phys.Oceanogr., 21, 265276., J. A. Barth, and P. A. Newberger, 1990: On intermediate models

for barotropic continental shelf and slope flow fields. Part III:Comparison of numerical model solutions in periodic channels. J. Phys. Oceanogr., 20, 19491973.

Gent, P. R., and J. C. McWilliams, 1983: Consistent balanced modelsin bounded and periodic domains. Dyn. Atmos. Oceans, 7, 6793.

Holm, D. D., 1996: Hamiltonian balance equations. Physica D., 98,379414.

Lorenz, E. N., 1960: Energy and numerical weather prediction. Tellus,12, 364373.

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J.S. Allen, Peter R. Gent and Darryl D. Holm- On Kelvin Waves in Balance Models