Propagation properties of Rossby waves for latitudinal β-plane variations of f and zonal variations of the shallow water speed

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1 Ann. Geophys., 30, , 2012 doi: /angeo Authors CC Attribution 3.0 License. Annales Geophysicae Propagation properties of Rossby waves for latitudinal β-plane variations of f and zonal variations of the shallow water speed C. T. Duba 1 and J. F. McKenzie 1,2 1 Departent of Matheatics, Statistics and Physics, Durban University of Technology, P.O. Box 1134, Durban, South Africa 2 School of Matheatical Sciences, University of Kwazulu Natal, Private Bag X54001, Durban, 4000, South Africa Correspondence to: C. T. Duba thaadub@dut.ac.za Received: 4 Deceber 2011 Revised: 7 April 2012 Accepted: 17 April 2012 Published: 15 May 2012 Abstract. Using the shallow water equations for a rotating layer of fluid, the wave and dispersion equations for Rossby waves are developed for the cases of both the standard β- plane approxiation for the latitudinal variation of the Coriolis paraeter f and a zonal variation of the shallow water speed. It is well known that the wave noral diagra for the standard id-latitude Rossby wave on a β-plane is a circle in wave nuber k y,k x space, whose centre is displaced β/2ω units along the negative k x axis, and whose radius is less than this displaceent, which eans that phase propagation is entirely westward. This for of anisotropy arising fro the latitudinal y variation of f, cobined with the highly dispersive nature of the wave, gives rise to a group velocity diagra which perits eastward as well as westward propagation. It is shown that the group velocity diagra is an ellipse, whose centre is displaced westward, and whose ajor and inor axes give the axiu westward, eastward and northward southward group speeds as functions of the frequency and a paraeter which easures the ratio of the low frequency-long wavelength Rossby wave speed to the shallow water speed. We believe these properties of group velocity diagra have not been elucidated in this way before. We present a siilar derivation of the wave noral diagra and its associated group velocity curve for the case of a zonal x variation of the shallow water speed, which ay arise when the depth of an ocean varies zonally fro a continental shelf. Keywords. Electroagnetics Wave propagation 1 Introduction The propagation properties of id-latitude Rossby waves on a β-plane are well known Gill, 1982; Pedlosky, The dispersion equation in either its diagnostic for, ω, k plots, or wave noral for Longuet-Higgins, 1964 shows that phase propagation is purely westward and that the waves cannot propagate above a critical frequency, at which the zonal group velocity becoes zero for a wave nuber equal to the inverse Rossby radius. This iplies that for wavelengths less greater than the Rossby radius, the zonal group velocity is eastward westward, while the phase velocity reains westward. This backward property, i.e. phase and group velocities in opposite directions, also anifests itself at a general angle of phase propagation, in that poleward directed rays energy flux direction correspond to equatorward wave noral or phase directions. In this paper we highlight these anisotropic and dispersive properties through the use of the phase and group velocity diagras, in which the forer takes the well known for of a circle, whose centre is displaced westward and the latter, the less well known for, is an ellipse, whose centre is also displaced westward. The iportant paraeters are the wave frequency ω suitably noralized and a paraeter, which is the ratio of the Rossby wave zonal speed at low frequencies and long wavelengths to the shallow water speed. In the next section we use the shallow water equations to derive the wave equation for the syste, which reduces to the classical Rossby wave equation in the low frequency approxiation. This leads to the well-known dispersion equation at id-latitudes and is generalized to include not only the usual β-effect arising fro the latitudinal variation of the Published by Copernicus Publications on behalf of the European Geosciences Union.

2 850 C. T. Duba and J. F. McKenzie: Propagation properties of Rossby waves Coriolis force, but also a topographic β-effect arising fro background variations in the shallow water speed. In Sect. 3 we outline the propagation properties of the idlatitude Rossby wave in ters not only of its well-known wave noral diagra Longuet-Higgins, 1964, but also its phase and group velocity diagras. Of particular interest is the latter, which is norally given as expressions for the zonal and latitudinal group speeds in ters of the wave nuber vectors, which can be reduced to the zonal wave noral as the generating paraeter. At a given frequency we show that the group velocity, V gy,v gx, curve is in fact an ellipse, with centre displaced westward by an aount that depends on the of the planet and the ratio of the wave frequency to the critical frequency, above which Rossby waves are evanescent. The ajor and inor axes yield the axiu zonal, westward, eastward and northward speeds as functions of frequency and. The corresponding phase velocity diagra is a circle, with centre displaced westward. In the liiting case of or infinite Rossby radius, the group velocity curve becoes a parabola, and the phase velocity curve becoes a line indicating constant zonal phase velocity for all directions of propagation liited of course to the 2nd and 3rd quadrants. These results are suppleented by the relation between the ray direction χ and the wave noral angle φ. It is well known that the latitudinal coponents of the phase and group velocities are anti-parallel. Interestingly, this backward property of the Rossby wave has been invoked to explain the dipole-like foration of equatorial easterly jets resulting fro localized equatorial heating and at higher latitudes westerly jets Diaond et al., In Sect. 4 we carry out siilar calculations for a topographic β-effect arising fro zonal variations of the shallow water speed. In this case, it is as if the properties described in Sect. 3 were rotated through π/2 fro west to north. 2 The general Poincaré-Rossby wave equation and the low frequency Rossby wave The linearised shallow water equations in a rotating layer of fluid of depth H x,y ay be written Gill, 1982; Pedlosky, 1987 Q + f Q = c 2 η 1 t η t + divq = 0 2 in which Q = Q x,q y = H u,v is the perturbation horizontal oentu vector, u, v the horizontal velocity, η the displaceent of the surface fro its equilibriu depth H, f = 2 sinθẑ is the Coriolis paraeter rotation frequency and c = gh the shallow water speed. The operations t div on Eq. 1 plus f ties the z coponent of the curl of Eq. 1, and the use of Eq. 2 to eliinate divq in favour of η/ t, and Eq. 1 to eliinate Q in ters of η iediately lead to the following wave equation for the displaceent η: [ 2 t t 2 + f 2 η div c η ] 2 = c 2 f η, z 3 in which f and c ay be functions of x and y. Equation 3 is the general equation for the cobined syste of Poincaré- Rossby waves Pedlosky, In the low-frequency approxiation t << f, this equation reduces to the classical Rossby wave equation: [ ] f 2 t c 2 2 x y 2 η = β η, 4 z in which β β x,β y = c 2 c 2 f since f = 0,β,0. Equation 4 is a generalization of the usual Rossby wave equation see Pedlosky, 1987 to include the topographic β effect through spatial variations in x and y of the shallow water speed c as well as the y variation of f = f 0 + βy in the classical β-plane approxiation of the latitudinal variation of the vertical coponent f ẑ on a spherical planet of radius R where f = f 0 + β 0 y, β 0 = 2 cosθ 0, f 0 = 2 sinθ 0. 6 R Here θ 0 is the latitude at which the β-plane is constructed tangent to the surface at θ 0, and therefore y easures distance northward whilst x is directed eastward. If the background variations of c 2 x,y and f y are assued slow over a wavelength, the wave Eq. 4 adits a local dispersion relation for plane waves, varying as expiωt k x x k y y naely ω = 5 β yk x + β x k y k 2 x + k2 y + f 2 0 /c2 7 or k y β 2 x + k x + β 2 y = β x 2 + β 2 y 2ω 2ω 4ω 2 f 2 0 c 2. 8 For a given ω the latter wave noral for is a circle centred at β y /2ω,β x /2ω of radius given by the square root of the RHS of Eq. 8. The presence of an inhoogeneity in x zonal eastward, dc 2 /dx, gives rise to a displaceent of the usual Rossby wave, representing purely westward propagation k x < 0, northward so as to perit eastward phase Ann. Geophys., 30, , 2012

3 C. T. Duba and J. F. McKenzie: Propagation properties of Rossby waves interediate Ω 2 1 2Ω k x k Fig. 2. The diagnostic diagra ω, k plot for various values of. Fig. The wave noral circle for various values of. β y c/f 2 0 = cosθ 0 sin 2 θ 0 1 2M, 11 propagation k x > 0, as represented by any k pointing northeast with a corresponding ray directed southwest. Thus, a northwest propagation vector k gives rise to a ray directed southwest. Such are the rather peculiar propagation properties of the Rossby wave, which follow fro the theore that the group velocity vector points along the noral to the wave noral curve in the direction of increasing ω Lighthill, We shall now explore these properties in greater detail through the use of figures representing the wave noral diagra and the phase and group velocity diagras at given frequencies for different values of. 3 Propagation properties of the id-latitude Rossby wave: wave noral, phase and group velocity curves for β y = 0 β x = 0 The propagation properties of the Rossby wave have been discussed extensively, for exaple, in the texts by Gill 1982 and Pedlosky Here we develop the group velocity diagra at a fixed ω, which provides the counterpart to the wave noral diagra. In the case where the Rossby wave arises fro inhoogeneity only in the y-direction β x = 0, the diagnostic ω, k plot and the wave noral curves are given by ω = k x k 2 + 1/, 9 and Eq. 8 ay be written k 2 y + k x = 1 2 ω 4 ω 2 1, 10 in which ω is the noralized frequency, ω/ β y c, k, the noralized wave nuber vector k c/β y 1/2 and is given by M = R/c. 12 Here we have assued that there is no latitudinal variation of c 2, and therefore the β effect arises solely fro the variation of f as in Eq. 6; M is a Mach or Froude nuber easuring the equatorial rotation speed R in units of the shallow water speed c, and is in fact the ratio of the Rossby zonal phase speed at low frequencies and long wavelengths to c. Equation 10 is a circle in k y, k x space of radius 1/4 ω 2 1/, whose centre is displaced along the negative k x axis by 1/2 ω units Longuet-Higgins, 1964, as shown in Fig. 1 along with the ω, k plot in Fig. 2. Note that the phase propagation is entirely westward and propagation requires ω < /2. Figure 3 deonstrates the geoetrical relation between the ray direction χ and the wave noral angle φ, both easured fro k x -axis, which shows the two values of χ for any given φ. The relationship between χ and φ is found by expressing χ as noral to the slope of the k y, k x curve, i.e. tanχ = 1/ k y / k x, which yields tanχ = sinφ cosφ/ cos 2 1 φ + 1 ± 1 4ω 2 /cos 2 φ. 13 For a given φ there are two values of χ, as shown in Fig. 4. We also show the liiting case, which yields χ = 2φ lower sign in Eq. 13 and χ = π upper sign. In this liit f 0 2 /c 2 0, and the Rossby wave Eq. 4 reduces to the classic for see e.g. Yagaata, 1976 t 2 ψ = β ψ x 14 where ψ is the strea function corresponding to strict 2-D copressibility. It is for this reason we include a discussion in the liit. Ann. Geophys., 30, , 2012

4 852 C. T. Duba and J. F. McKenzie: Propagation properties of Rossby waves V p 0.5 V g 1 2Ω k x 2. 5 V x Fig. 3. The wave noral displaced circle showing the relation between the ray direction χ given by the arrows pointing towards the centre easured fro k x and the wave noral angle φ between k and k x. Fig. 5. The phase velocity V p a circle and group velocity V g an ellipse diagras for a fixed ω and = Χ 2 Φ Χ Π 2 the phase velocity diagra reduces to the line V px = ω 2 in V py, V px space. The group velocity V g = ω/ k follows fro Eq. 7 in which we put β x = 0 in the for V gx = β k x 2 k y 2 + f 0 2 /c 2 kx 2 + k y 2 + f 0 2 /c 2 2, 17 V gy = 2βk y k x kx 2 + k y 2 + f 0 2 /c Fig. 4. The variation of χ two values with φ for = 2. The phase velocity V p = ω/k, which follows iediately fro Eq. 7, ay be written in the noralized for V py 2 + V px =, in which V p has been noralized with respect to the shallow water speed c. Thus, the phase velocity diagra is a circle of radius ω 2 /, whose origin is displaced westward by /2 units and therefore lies entirely in the regie of westward propagation. The sallest value of the westward V px in which V py = 0 in Eq. 15 is V pxin = 2 + 2, 16 which approaches ω 2 as. Thus, in the liit which corresponds to infinite Rossby radius or Rossby wave speed greatly in excess of the shallow water speed In the classic texts Pedlosky, 1987; Gill, 1972 the group velocity is left in this less than perspicacious paraetric for, in which the k x is the generating paraeter, with k y given in ters of k x fro the dispersion equation. Here we show that the group velocity diagra is in fact an ellipse. This siple result follows fro a few algebraic steps by eliinating the denoinator fro Eqs. 17 and 18 using the dispersion equation, so that in noralized for, Eqs. 17 and 18 becoe V gx = ω / ωk x, 19 V gy = 2 ω 2 k y / k x. 20 It is now straightforward to eliinate k x in favour of V gx fro Eq. 19, which on substitution into the square of Eq. 20 gives directly the group velocity V gx, V gy curve in the for V gy 2 = 4 ω4 1 V 2 gx ω 2 ω2 V gx ω 2 2, 21 Ann. Geophys., 30, , 2012

5 C. T. Duba and J. F. McKenzie: Propagation properties of Rossby waves V west V north V east Ω Fig. 6. The axiu westward, northward and eastward group speeds as a function of frequency ω for = Ω 2 V x Fig. 7. The group velocity a parabola and phase velocity a line diagras for the case. which ay also be written as V gy 2 [ ] 2 4 ω 2 + V gx + = The group velocity curve is an ellipse, whose centre is displaced ω 2 / units along the V gx axis. The phase and group velocity curves are shown in Fig. 5. The southward group velocity is siply a reflection of the northward group velocity in the x-axis. The axiu eastward and westward group speeds are given by V ± gx = 2 1/2 ±1 1/2, 23 and the axiu northward southward group speed also follows as 1/2 V gyax = ± ω 1/2. 24 Exaples of the behaviour of extreal group velocities V pxin, V pxax and V pyax are shown in Fig. 6. Note that the ellipse collapses to the origin as 4 ω 2, at which the wave noral collapses to the point k x = 1/2 ω. In the liiting case in which >> 1, which can prevail quite near the equator θ 0 0, Eq. 21 tends to the parabola V gy = ±2 ω 1 2 V 1/2 gx ω 2, 25 which is shown in Fig. 7 together with the phase velocity diagra, which is siply the line V px = ω 2. It is of soe interest to ephasize that Rossby waves are backward in the sense that the latitudinal coponents of their phase and group velocities are always in opposite directions. This property can be invoked to describe the foration of a dipole pair of jets in the following way: Northward away fro the equator wave energy flux is associated with southward, towards the equator, wave oentu flux; and the opposite in the case of southward directed energy flux, away fro the equator, corresponds to northward towards the equator oentu flux. In other words, a poleward energy flux fro the equator is associated with an equatorward flux of oentu. Hence, Rossby wave dynaics iplies that localized equatorial heating gives rise to equatorial easterly zonal jets. This convergence of equatorial oentu iplies a deficit at higher latitudes such that a westerly jet ust necessarily for there Diaond et al., Propagation properties of the topographic Rossby wave: wave noral, phase and group velocity curves for β x = 0 β y = 0 In this case a zonal inhoogeneity c a function of x dispersion and wave noral curves given by putting β y = 0 in Eqs. 7 and 8, are ω = k y k x 2 + k y /, k x 2 + k y 1 2 = 1 2 ω 4 ω 2 1, 27 in which ω and k have been noralized to β x c and c/β x 1/2, respectively and is now given by c = β x f0 2, 28 Ann. Geophys., 30, , 2012

6 854 C. T. Duba and J. F. McKenzie: Propagation properties of Rossby waves 2. V gx = 2 ω 2 k x / k y V g V p Hence, the group velocity curve is now given by the ellipse 2 V V gx 2 = 4 gy ω4 ω ω2 V gy ω 2 + 2, 33 or V gx 2 [ 4 ω 2 + V gy 2 ] 2 = 2 4, which is shown in Fig. 8. This curve can be obtained by rotating the curve given by Eq. 22 and Fig. 5 through ninety degrees. Thus, the axiu northward, southward and zonal east or west speeds are given by Eqs. 23 and 24 in which the labels x and y are interchanged. V x 5 Suary Fig. 8. The phase and group velocity diagras for topographic Rossby waves where shallow water speed varies zonally for fixed ω and = 2. β x = f 0 c 2 dc 2 dx. 29 The wave noral diagra is again a circle, but now its centre is displaced 1/2 ω units along the positive k y axis. The corresponding phase velocity diagra is given by V py 2 + V px 2 2 = 2, 30 4 which is a circle of radius ω 2 / 1/2 with centre displaced /2 units along the V py axis. Hence, phase propagation is always northward, although as we can now anticipate, there are both northward and southward coponents of the group velocity. It is as if the classical westward Rossby wave phase were rotated through ninety degrees. The calculation of the group velocity follows siilar lines but now with V gy = ω , 31 ω k y The propagation properties of id-latitude Rossby waves are illustrated through the local dispersion equation either in its diagnostic for ω,k plot or as a wave noral curve for given values of ω, both of which descriptions highlight the dispersive and anisotropic nature of the wave. Here we develop this further by showing that the group velocity diagra is an ellipse, whose centre is displaced westward Fig. 5 and whose ajor and inor axis yield the axiu westward, eastward and northward southward group speeds as a function of frequency and the paraeter Fig. 6. This elegant construction replaces the earlier soewhat cubersoe expressions for the group velocity, which are usually given in ters of the zonal and northward wave nubers acting as generating paraeters as in Eqs. 17 and 18. Siilar diagras exist for the case of topographic Rossby waves, in which shallow water speed varies zonally Fig. 8. We ephasize that the backward property of the Rossby wave has been invoked to explain dipole-like foration of equatorial easterly jets and at higher latitudes westerly jets, as discussed by Diaond et al Acknowledgeents. The authors would like to thank the reviewers for the ost useful and constructive coents towards the finalization of this paper. This work was supported by the National Research Foundation of South Africa. A thank you note to Cleens Depers of BlueStallion Technologies who provided technical assistance with the graphics. Topical Editor C. Jacobi thanks L. Debnath and two other anonyous referees for their help in evaluating this paper. References Diaond, P. H., Gurcan, D. D., Hahn, T. S., Miki, K., and Garbet, X.: Moentu theores and the structure of atospheric jets Ann. Geophys., 30, , 2012

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