Study of Propagation Properties of Rossby Waves in the Atmosphere and Relationship Between the Phase Velocity and the Group Velocity

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1 Aerican Journal of Manageent Science and Engineering 17; (6): htt:// doi: /j.ajse ISSN: X (Print); ISSN: (Online) Study of Proagation Proerties of Rossby Waes in the Atoshere and Relationshi Between the Phase Velocity and the Grou Velocity Mohaed Anwar Batal, Sawsan Othan Deartent of Physic, Aleo Uniersity, Aleo City, Syria Eail address: (M. A. Batal), (S. Othan) To cite this article: Mohaed Anwar Batal, Sawsan Othan. Study of Proagation Proerties of Ross by Waes in the Atoshere and Relationshi Between the Phase Velocity and the Grou Velocity. Aerican Journal of Manageent Science and Engineering. Vol., No. 6, 17, doi: /j.ajse Receied: March 3, 17; Acceted: May 11, 17; Published: January 3, 18 Abstract: Using Rossby wae equations, the disersion equation is deeloed. The wae noral diagra for Rossby waes on a beta lane is a circle in wae nuber (k, l) sace whose center is dislaced along the negatie X axis, and whose radius is less than this dislaceent, which eans that hase roagation is entirely westward. The hase elocity diagra is a circle whose center is dislaced along the negatie X axis, the grou elocity diagra is an ellise whose center is dislaced westward and whose ajor and inor axes gie the axiu all the directions grou seeds as function of the frequency and araeter. Keywords: Rossby Waes, Planetary Waes, Phase Velocity, Grou Velocity, Barotroic Fluid, Baroclinic Fluid 1. Introduction The wae tye that is of ost iortance for large-scale eteorological rocesses is the Rossby wae, or lanetary wae. In an iniscid barotroic fluid of constant deth (where the diergence of the horizontal elocity ust anish), the Rossby wae is an absolute orticity-consering otion that owes its existence to the ariation of the Coriolis araeter with latitude, the so-called -effect. More generally, in baroclinic atoshere, the Rossby wae is a otential orticity consering otion that owes its existence to the isentroic gradient of otential orticity [1]. Rossby waes lay a central role in geohysical fluid dynaics and dynaical eteorology, articularly in the dynaics of quasi-geostrohic flow []. Their reflection roerties at coast lines hel to exlain certain features of western boundary currents [3]. The anisotroic and disersie roagation roerties of Rossby waes (in articular the backward roerty in which hase and grou elocities are oosite in the N-S direction) hae been inoked, together with equatorial heating, to exlain diole-like foration of equatorial easterly jets accoanied, at higher latitudes, by westerly jets [4]. In addition, an inerse turbulent cascade together with Rossby waes can lead to the foration of zonal flows []. These ideas hae been further deeloed recently [5] in odels of an eddy-drien jet. Rossby wae roagation can be understood in a qualitatie fashion by considering a closed chain of fluid arcels initially aligned along a circle of latitude [1]. The ajor tye of lanetary scale waes can be classified into seeral cases, based on energy source or horizontal and ertical structure. Based on energy source, of lanetary scale waes can be diided into free and forced odes. The free odes are the noral oscillations in the atoshere. They can be excited by sall rando forcing and are liited in alitude by dissiation. The free lanetary (rossby) odes are global extent and with eriods of about, 5, 1 and 16 days [6, 7]. The forced Rossby odes can be excited by flow oer toograhy or by land-sea heating contrasts. Based on horizontal structure classification, lanetary waes can be diided into two, global odes and equatorial odes. The global odes are the lanetary waes can roagate both eridionally and zonally. Rossby waes are the ost iortant global odes. The equatorial odes are traed in the equatorial wae-guide and roagate zonally along the equator. Kelin and Rossby-graity waes are the ost

2 Aerican Journal of Manageent Science and Engineering 17; (6): iortant equatorial odes. In addition to the aboe two classification, lanetary waes can be diided into external and internal odes fro their ertical structure. External odes are ertically traed odes and the energy density of such odes decays exonentially in the ertical. Internal odes are ertically roagating (hase surfaces tilt with height) odes and they can transfer oentu and energy ertically oer any scale heights. The forced Rossby waes, Kelin waes, Rossby-graity waes and Graity waes can roagate ertically and transfer energy and oentu between the lower and uer atoshere under suitable background conditions [8]. Rossby waes lay a ital role in the dynaics of the iddle atoshere. They riarily resonsible for the asyetries in the olar ortex, sudden waring in the stratoshere, forcing of the uasi-biennial Oscillation (OBO), Sei-Annual Oscillation (SAO). Meridionally roagating Rossby waes carry angular oentu between the troics and iddle latitudes. Planetary waes can reach large alitudes if they roagate uwards to the MLT region. Vertical roagation is deterined by wae/ean flow interactions [9]. Most traeling waes are westward roagating, and so they can roagate through the eastward winds of the winter stratoshere, but not the westward winds of the suer stratoshere. The continental configuration and toograhy of the northern heishere roduces ore lanetary wae actiity than in the southern heishere. This lanetary wae actiity acts to cause sudden stratosheric waring and break down the olar ortex at the end of winter in the northern heishere. In contrast, the southern olar ortex is ore stable and there has only been one ajor sudden stratosheric waring in the southern heishere, which occurred in. [7] define a y coordinate y = a( ϕ ϕ), and an x coordinate x = aλ, where λ Is longitude. Since f = f ( ϕ) = Ω sinϕ, in the ( x, y) syste it becoes f = f ( y). Assuing that the width of the stri is sall enough, we can aroxiate f ( ϕ) as Taylor series about the central latitude: df f ( ϕ) f ( ϕ) + ( ϕ ϕ)( )( ϕ) +... dϕ Where f ( ϕ) sin ϕ, df. dϕ cosϕ for y, we get = Ω = Ω. Substituting f ( y) = f + y Ω Where f = Ωsinϕ and = cos ϕ. For a latitude a of π, = s. the sign of f changes fro 4 north to south heishere, Is always ositie (since f always increase northward)[1]. Figure shows Rossby waes in the atoshere.. The Plane We saw in the deriation of the barotroic orticity equation the otential iortance of the fact that the Coriolis araeter aries with latitude, a consequence of sherical geoetry. Howeer, dealing with sherical geoetry is (a little) ore colicated than with lanar geoetry, so it is coon to reresent a stri of the shere-liited in latitude but going all the way around the world in longitude-as a lane, as in Figure 1. Figure. Shows Rossby waes in the atoshere at 5 b ressure. 3. Purose of Study Study of relationshi between the hase elocity and the grou elocity of Rossby waes in the atoshere. 4. Results and Discussion 4.1. The Disersion Equation To get the disersion equation for Rossby waes lets deend on the quasi-geostrohic equations: Figure 1. Shows the lane. Let us consider a stri centered on longitude ϕ, and D u f y = (1) g g a g D + f u + yu = () g g a g

3 1 Mohaed Anwar Batal and Sawsan Othan: Study of Proagation Proerties of Rossby Waes in the Atoshere and Relationshi Between the Phase Velocity and the Grou Velocity ua a wa = x y z (3) gρ Dg ( ) + N wa = (4) ρ Dg = + ug + g t x y Where: Where u = u u, =, w = w are the difference a g a g a between the true elocity and the geostrohic flow, N is the square of the buoyancy frequency, ρ is reference alue of the background density, ρ is erturbation in density, g is acceleration graity, f is Coriolis araeter. Where f = Ω sinϕ Geostrohic balance can be exressed by: ψ ψ u u g = g = y x Where ψ = is the geostrohic strea function, is fρ the erturbation in ressure. Hydrostatic balance can be written as: Take x (1), y ρ = f ρ g () to obtain: ψ z D u f y y g g a g = D f u yu x g g + a + g = Subtraction (7) fro (6) and using the ass-continuity equation (3) to obtain: wa Dgζ = f z (5) (6) (7) ug g ψ ψ ζ = f + y + = f + y (8) y x x y is the Z coonent of the absolute orticity associated with the geostrohic flow. Using equations (4) and(5) the ertical elocity can be exressed as: w w a a gρ = Dg ( ) ρ N f ψ = Dg ( ) N z Substitution of this exression into the orticity equation (8) and further careful aniulation lead finally to the quasigeostrohic otential orticity equation: Dgq = f ψ q = ζ + ( ) z N z f y x y z N ψ ψ ψ q = f ( ) z (9) (1) Let us consider sall-alitude disturbances to a unifor zonal background flow ( u,,), whereu is a constant; by this unifor flow corresonds to a geostrohic strea function ψ = uy. For the total flow (the background lus a sall disturbance): ψ = uy + ψ (11) Substitute into the equation (1) and neglect ters that are quadratic in ψ. The quasi-geostrohic otential orticity equation for this flow is: Where Γ is the ellitic oerator. q = f + y + Γ ψ (1) f Γ = ( ) x y z N z In addition, the quasi-geostrohic otential orticity equation linearizes to: Let us take ψ ( + u ) Γ ψ + = t x x (13) N to be constant and look for lane-wae solutions to this equation by substituting: [ i kx ly z t ] ψ = Reψˆ ex ( ) Where ψ ˆ is a colex alitude, k, l, are waenubers on X, Y, Z. We obtain the disersion relation for Rossby waes: = ku k f N k l (14) Whenu =, the disersion relation for Rossby waes [1]:

4 Aerican Journal of Manageent Science and Engineering 17; (6): = k f N k l (15) 4.3. The Grou Velocity The grou elocity g = can be written K Or l f ( k ) 4 N = (16) The latter wae noral for is a circle centred at (,) and radius gien by 4.. The Phase Velocity The hase elocity f N 4. = can be written: K 4 y x + ( + ) = (1 ) (17) 4 N Where =, = and f The hase elocity diagra is a circle of radius 4 (1 ) whose origin is dislaced westward by units and therefore lies entirely in the regie of Westward roagation. It reresented in Figure 3. The sallest alue of the westward y = : Which aroaches 4 ( x ) in = + 1 (18) ( x ) in = when gx = f l N f N ( k ( + )) gy ( k l ) = kl f N ( k l ) (19) () by eliinating the denoinator fro equations (19) and () using the disersion equation, so that in noralized for, equations (19) and () becoe gx 1 = ( + ) (1) k gy l = ( ) () k It is now straightforward to eliinate k in faour of gx fro equation (1), which on substitution into the square of equation () gies directly the grou elocity cure in the for 4 gx gx gy = 4 (1 ) ( ) This equation can be written gy gx + = (1 ) (1 ) 4 4 (3) (4) The southward grou elocity is sily a reflection of the northward grou elocity in the x-axis. Equation (4) can be written gy 4 gx d (1 ) / b = 1 (5) 4 4 Where b = (1 ), d = (1 ) 4 Equation (5) can be written in the olar for g = (6) 1 + ecos χ Figure 3. Shows the hase elocity when = 4, =.6.. Where 4 1/ d 4 1/ (1 ), e 1 (1 ) d = = = = b b

5 3 Mohaed Anwar Batal and Sawsan Othan: Study of Proagation Proerties of Rossby Waes in the Atoshere and Relationshi Between the Phase Velocity and the Grou Velocity The shift along the negatie x axis In ters of 4 e (1 ) = (7) 1 e and, the ellise (6) ay be written g 1/ 4 = / 1 + cos χ (8) The ellise collases to the origin as 4 at which 1 the wae noral collases to the oint k =. In the liiting case in which >> 1, which can reail quite near the equator, Equation (3) tends to the arabola [11]. gy Which reresented in Figure 6. gx = ± (1 ) (31) The grou elocity diagra is a ellise whose origin is 4 dislaced westward by (1 ) units along gx. It reresented in Figure 4. Figure 6. Shows the grou elocity (a arabola) and hase elocity (a line) Diagras for the case. Figure 4. Shows The grou elocity when = 4, =.6. The axiu eastward and westward grou seeds are gien by 4 1/ 4 ± 1/ gx = (1 ) ± 1 (1 ) (9) and the axiu northward (southward) grou seed also follows as gy 1/ 4 1/ = ± (1 ) (3) Exales of the behaior of external grou elocities are shown in Figure 5. Figure 5. Shows the axiu northward and eastward grou Seeds as a function of frequency for the case = 4. It is of soe interest to ehasize that Rossby waes are backward in the sense that the latitudinal coonents of their hase and grou elocities are always in oosite directions. This roerty can be inoked to describe the foration of a diole air of jets in the following way: Northward (away fro the equator) wae energy flux is associated with southward, towards the equator, wae oentu flux; and the oosite in the case of southward directed energy flux, away fro the equator, corresonds to northward (towards the equator) oentu flux. In other words, a ole ward energy flux fro the equator is associated with an equator ward flux of oentu. Hence, Rossby wae dynaics ilies that localized equatorial heating gies rise to equatorial easterly zonal jets. This conergence of equatorial oentu ilies a deficit at higher latitudes such that a westerly jet ust necessarily for there [1]. 5. Conclusion the hase seed is negatie, so the hase of rossby waes always roagates westward. Since is a nonlinear function of k, Rossby waes are disersie. The agnitude of the grou elocity is, tyically, greater for the westward roagating long waes than for the eastward roagating short waes. The latitudinal coonents for Rossby waes of their hase and grou elocities are always in oosite direction.

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