According to classical theories, the frequencies of internal-gravity waves in stratified rotating fluids must lie between the Brunt-Väisälä frequency (a measure of the vertical density stratification) and the Coriolis frequency (equal to twice the rotation rate about the vertical axis). It is shown here that, in the case of the Earth\u27s rotation where the pole-to-pole axis of rotation is almost everywhere not parallel to the local vertical, the range of realizable frequencies is broader. New formulas are derived for the lower and upper bounds of the frequencies

Cushman-Roisin, Benoit

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Dartmouth College Dartmouth Digital Commons Open Dartmouth: Peer-reviewed articles by Dartmouth faculty Faculty Work 12-9-1996 Lower and Upper Bounds on Internal-Wave Frequencies in Stratified Rotating Fluids Benoit Cushman-Roisin Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa  Part of the Cosmology, Relativity, and Gravity Commons, and the Physics Commons Dartmouth Digital Commons Citation Cushman-Roisin, Benoit, "Lower and Upper Bounds on Internal-Wave Frequencies in Stratified Rotating Fluids" (1996). Open Dartmouth: Peer-reviewed articles by Dartmouth faculty. 2020. https://digitalcommons.dartmouth.edu/facoa/2020 This Article is brought to you for free and open access by the Faculty Work at Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: Peer-reviewed articles by Dartmouth faculty by an authorized administrator of Dartmouth Digital Commons. For more information, please contact dartmouthdigitalcommons@groups.dartmouth.edu. VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996Lower and Upper Bounds on Internal-Wave Frequencies in Stratified Rotating FluidsBenoit Cushman-RoisinThayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755-8000(Received 25 July 1996)According to classical theories, the frequencies of internal-gravity waves in stratified rotating fluidsmust lie between the Brunt-Väisälä frequency (a measure of the vertical density stratification) and theCoriolis frequency (equal to twice the rotation rate about the vertical axis). It is shown here that, inthe case of the Earth’s rotation where the pole-to-pole axis of rotation is almost everywhere not parallelto the local vertical, the range of realizable frequencies is broader. New formulas are derived for thelower and upper bounds of the frequencies. [S0031-9007(96)01859-5]PACS numbers: 47.32.Ff, 92.10.EiThe classical theory of internal-gravity waves in astably stratified fluid [1–3] yields a dispersion relationthat constrains the wave frequency not to exceed the so-called Brunt-Väisälä frequency, a direct measure of thefluid’s density stratification supporting the waves. Ap-plication of this theory to geophysical fluids such as theatmosphere or the ocean requires modification to incorpo-rate the effect of the Earth’s rotation. Traditionally, theinsertion of the Coriolis acceleration is limited to one newterm in each of the horizontal-momentum equations [3–5], and the dispersion relation is augmented by a termdepending on the so-called Coriolis frequency, equal totwice the projection of the rotation vector onto the verti-cal direction. An elementary analysis of this dispersionrelation yields that the wave frequency is now constrainedto lie between the Brunt-Väisälä frequency and the Corio-lis frequency. Since in most geophysical fluids the formeris larger than the latter, the Brunt-Väisälä frequency actsas an upper bound, and the Coriolis frequency as a lowerbound. Almost everywhere on the Earth, however, thepole-to-pole rotation axis is not parallel to the local ver-tical, and the Coriolis acceleration is not limited to thehorizontal plane. This casts doubt on the traditional con-clusion regarding the range of internal-wave frequenciesin geophysical fluids. The purpose of this Letter is to re-formulate the theory of internal-gravity waves in the pres-ence of the full, three-dimensional Coriolis accelerationand thereby to derive new bounds on the range of realiz-able frequencies.Departing from the traditional theory of internal-gravitywaves only by retaining the full, three-dimensional Cori-olis acceleration, we state the linearized equations gov-erning the three-dimensional flow associated with a smallperturbation to a basic state of motionless and hydrostaticdensity stratification [3],≠u≠t2 fy 1 fpw ­ 21r0≠p≠x, (1a)≠y≠t1 fu ­ 21r0≠p≠y, (1b)≠w≠t2 fpu ­ 21r0≠p≠z1 b , (1c)≠u≠x1≠y≠y1≠w≠z­ 0 , (1d)≠b≠t1 N2w ­ 0 , (1e)where the variables areu, y, and w, the velocity com-ponents along thex (eastward),y (northward), andz (up-ward) Cartesian directions,p, the pressure anomaly (equalto the total pressure minus the hydrostatic pressure of thebasic state), andb, the buoyancy anomaly (equal to minusthe gravitational accelerationg times the relative densityanomaly caused by the motion). The coefficients are thetwo Coriolis frequencies, defined asf ­ 2V sinw, fp ­ 2V cosw , (2)where V is the rotation rate of the Earth (nearly2py24 hr) andw is the local latitude, a reference densityr0,and the square of the Brunt-Väisälä frequency defined byN2 ­ 2gr0drszddz, N . 0 , (3)wherer0 1 rszd is the density profile of the basic state.Assuming a limited range of latitudes and a uniformvertical stratification, we may take all coefficients asconstant and seek wave solutions of the form expfislx 1my 1 nz 2 vtdg. Some algebra yields the followingdispersion relation between the frequencyv and the wavenumber componentsl, m, andn:v2 ­N2sl2 1 m2d 1 s fn 1 fpmd2l2 1 m2 1 n2. (4)Note that the groupingfn 1 fpm represents the pro-jection of the wave number onto the rotation axis.To determine the range of realizable frequencies, weseek the extremal values ofv as l, m, andn assume anyreal value between2` and1`. While this may be doneimmediately, it is instructive to consider first the particularcases of existing theories.The simplest of all cases is the one with no rotations f ­ fp ­ 0d and under the assumption of hydrostaticbalance [no≠wy≠t term on the left-hand side of (1c)].The dispersion relation is then reduced tov2 ­ N2l2 1 m2n2, (5)0031-9007y96y77(24)y4903(3)$10.00 © 1996 The American Physical Society 4903VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996and the frequencyv can take any value between zero andinfinity. Reinstating the vertical acceleration≠wy≠t in thevertical-momentum equation (1c), we obtain the correcteddispersion relation [1–3],v2 ­ N2l2 1 m2l2 1 m2 1 n2, (6)which now limits v2 to N2 and thus places an upperbound on the frequency. [Note that higher frequenciesthan N are permitted if the waves are evanescent in thevertical sn2 , 0d, but such waves are no longer calledinternal-gravity waves.]If we further retain the Coriolis terms due to thevertical component of the rotation vectors f fi 0, fp ­0d, we obtain the traditional dispersion relation for internalgravity waves in a stratified rotating fluid [4,5],v2 ­ N2l2 1 m2l2 1 m2 1 n21 f2n2l2 1 m2 1 n2. (7)In this dispersion relation,v2 appears as a weightedaverage betweenN2 and f2, and must therefore liebetween these values. Since in geophysical flows, theCoriolis frequency is typically much smaller than theBrunt-Väisälä frequency,f acts as a lower bound (inmagnitude sincef may be negative), andN as an upperbound. In the hydrostatic limit [neglect of≠wy≠t in (1c)again], the denominatorsl2 1 m2 1 n2 reduce ton2, andthe dispersion relation becomes [6]v2 ­ N2l2 1 m2n21 f2, (8)in which N no longer acts as an upper bound butfstill acts as a lower bound. In sum, the traditionaltheory concludes that the ambient rotation sets a lowerbound on the frequency of internal gravity waves (namely,the Coriolis frequency, equal to twice the projection ofthe rotation vector onto the local vertical), and that thevertical acceleration sets an upper bound (namely, theBrunt-Väisälä frequency).According to the generalized dispersion relation, (4),the traditional theory leading to (7) is inconsistent. Eitherthe horizontal wave numbersl2 1 m2d1y2 is negligiblecompared to the vertical wave numbern, or it is not.If it is, then (7) is not better than (8), and inclusion ofthe vertical acceleration is unnecessary. If it is not, thenfpm is most likely [7] comparable tofn in the numeratorof (4), and not only the vertical acceleration but alsothe Coriolis acceleration must be retained in the vertical-momentum equation.We now seek the extremal values that the frequencyvcan achieve under the most general dispersion relation (4).First putting to zero the derivative ofv with respect tol,we obtaineither l ­ 0 or s fn 1 fpmd2 ­ N2n2. (9)In the second case,v2 becomesN2, and we concludethat N is a stationary value ofv. As we will shortly see,this will be neither a maximum nor a minimum. In thefirst casesl ­ 0d, v2 reduces tov2 ­N2m2 1 s fn 1 fpmd2m2 1 n2­sN2 1 f2p dl2 1 2ffpl 1 f2l2 1 1, (10)where l ­ myn is the single wave number variable onwhich v depends. Setting to zero the derivative ofvwith respect tol yieldsffpl2 2 sN2 2 f2 1 f2p dl 2 ffp ­ 0 , (11)which always admits two real roots,l ­mn­12ffp∑sN2 2 f2 1 f2p d6qsN2 2 f2 1 f2pd2 1 4f2f2p∏. (12)The corresponding stationary values ofv2 arev26 ­12∑sN2 1 f2 1 f2p d6qsN2 2 f2 1 f2pd2 1 4f2f2p∏, (13)for which it is relatively easy to show that the followingrankings hold:0 , v22 , N2 , v21 and 0 , v22 , f2 , v21 .(14)In conclusion, the minimum value ofv is the positiveroot of v22, and its maximum value is the positive rootof v21, while N is an intermediate stationary value. Italso follows thatvmin is lower than the minimum ofjfj and N , and thatvmax is higher than the maximumof jfj and N. In other words, the range of realizablefrequencies is broader than the interval betweenjfj andN on both low and high frequency sides. ForN ¿ V,asymptotic expressions arev2min ø f2s1 2 f2p yN2d andv2max ø N2 1 f2p . This may explain why internal-wavespectra derived from oceanographic data do indeed showa range of frequencies slightly broader than the intervalfrom jfj andN [8], especially at low latitudes.This work was supported by Grant No. N00014-93-1-0391 of the Office of Naval Research to DartmouthCollege.[1] J. S. Turner, Buoyancy Effects in Fluids(CambridgeUniversity Press, Cambridge, 1973), Sec. 2.2.[2] J. Lighthill, Waves in Fluids(Cambridge University Press,Cambridge, 1978), Sec. 4.1.[3] B. Cushman-Roisin,Introduction to Geophysical FluidDynamics (Prentice Hall, Englewood Cliffs, NJ, 1994),Sec. 3-4 and 10-2.4904VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996[4] A. E. Gill, Atmosphere-Ocean Dynamics(Academic Press,New York, 1982), Sec. 8.4.[5] P. K. Kundu, Fluid Mechanics (Academic Press, NewYork, 1990), Sec. 13-14.[6] J. R. Holton, An Introduction to Dynamic Meteorology(Academic Press, New York, 1992), Sec. 7.5.2.[7] Exceptions to this rule are particular cases whenjmj øjlj , jnj (wave numbers nearly confined to the zonal-vertical plane) and whenjfpj ø jfj (near the poles).[8] C. Garrett and W. Munk, Annu. Rev. Fluid Mech.11,339–369 (1979), and references therein.4905

Lower and Upper Bounds on Internal-Wave Frequencies in Stratified Rotating Fluids

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Lower and Upper Bounds on Internal-Wave Frequencies in Stratified Rotating Fluids

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