Theoretical Simulation of Free Atmospheric Planetary Waves on an Equatorial Beta-plane

Paper submitte to the J. Theor. Comput. Stu. DRAFT Theoretical Simulation of Free Atmospheric Planetary Waves on an Equatorial Beta-plane Sanro Wellyanto Lubis Department of Geophysics an Meteorology, Bogor Agricultural University, Bogor 668, Inonesia Sonni Setiawan Department of Geophysics an Meteorology, Bogor Agricultural University, Bogor 668, Inonesia Abstract : A simple theoretical moel was evelope to investigate behavior of equatorial planetary waves (EPW) in the Earth s atmosphere. Base on linearize equations of EPW in equatorial -plane, EPW by moe n= has greatest meriional win amplitue at the equator an ecays exponentially when away from the equator (Gaussian ecay). EPW by moe n= has greatest amplitue of meriional win perturbation (v ) at latitue y= ± (nonim) an EPW by moe n= has v equal to zero at latitue y = / (nonim), the peak of amplitue is just outsie the equator. Simulation of Yanai wave results that both zonal win an geopotential fiel have greatest perturbation amplitue at latitue y= ± (u = in Equator) while perturbation of zonal win (u ) an geopotential fiel (Φ ) will be in geostrophic balance at latitue < y < (nonim) or -( gh) / > y > ( gh) / (length imension). Simulation of Kelvin wave results that either zonal win or geopotential fiel has symmetric amplitue an symmetric perturbation relative to Earth s latitue. For special treatments, by moe n=,, an there are two classes of EPW, namely high frequency Poincaré moes waves an the low frequency Rossby moes waves. These waves have special behaviors as functions of n. This paper presents a linear moel of EPW an mathematically goes step-by-step to erive an explain their character on an equatorial beta-plane. Keywors : Equatorial Planetary Waves, Kelvin waves, Yanai waves, Rossby waves, Poincaré waves. E-mail : sanro.lubis@live.com December, INTRODUCTION Theory of atmospheric waves along the equatorial ban is unique, an consiere to be one of the riving atmospheric physical phenomena in the tropics. Various stuies have been conucte to ientify an analyze the behavior of equatorial planetary waves (EPW) which are observe from troposphere to lower stratosphere, such as Kelvin waves, Rossby waves, Yanai waves an Poincaré waves as well [, 6, 8, 9,,,, 7]. Equatorial waves are generate by iabatic heating ue to organize tropical large-scale convective heating in the equatorial belt [6, 9]. Although they o not contain as much energy as other typical tropospheric weather isturbances, EPW causes preominant isturbances in the equatorial atmosphere such as inucing mean-meriional circulation which plays an important role in the heat balance of the equatorial belt, relating to the colness process of the equatorial tropopause an also affecting the patterns of low level moisture convergence that control the istribution of convective heating an convective storms in large longituinal istances [, 6, 9, ]. Regaring the above information, stuy of these waves is substantially important to be evelope. One of the ways is to simulate these waves within the linear shallow water moel on an equatorial -plane. Simulation of EPW helps us to explain how these waves can be observe through relation of atmospheric perturbation elements. Matsuno [7] evelope a moel of quasi-geostrophic motions in the equatorial area by applying a single layer of homogeneous incompressible flui with free surface an assuming that Coriolis parameter to be proportional to the latitue. His work concerne only with the mathematical analyses of the simplifie hyroynamical equations, but it is most interesting to evelop in the EPW simulations. Linzen [] c GFTI & MKI

use the -planes approximation an Laplace s Tial Equation to escribe planetary waves in the equator an mi-latitues. Both of these moels are interesting to be iscusse an simulate, but we must be careful before applying these results in the actual atmospheric isturbances. The aim of this paper is to provie clear overview an analysis of EPW characters by examining horizontal velocity an geopotential perturbations. In this paper analytically wave solutions referencing to Matsuno equations will be foun. The mathematical solutions of EPW will be simulate on an equatorial -plane. Because a complete evelopment of EPW woul be rather complicate, at this time we only simulate the EPW in the free waves form an concentrate on the horizontal structure. The last section of this paper will be close by a conclusion. GOVERNING EQUATIONS Equatorial waves are trappe near the equator, an they propagate in zonal an vertical irections. Mathematical interpretation of EPW s horizontal structure can be simulate by linearise shallow water equations for perturbations on a motionless basic state of mean epth H in -plane. The avantage of the -plane approximation is that it oes not contribute nonlinear terms to the ynamical equations. This equatorial beta-plane approximation is only vali for scales L R, where R is the planetary raius. Briefly, the -plane approximation is a metho to linearize Coriolis force (f) which can be approximate by expaning the latituinal epenence of f in a Taylor series about reference latitue ϕ o, as follows: f(ϕ + ϕ o ) = f( ϕ o ) + + ( ) f ϕ ( ) f ϕ ϕ ϕ=ϕ o ϕ=ϕ o ϕ +... () For the smallest value of ϕ, ϕ = y/r an retaining only the first two terms (neglect the higher orer terms) to yiel: f = f + y () Equation () is calle the mi-latitue -plane approximation, is a Rossby parameter ( = Ωcos R ) which can be also erive from f/ y. For our moel, initial Coriolis force (f o ) can be neglecte at the equator, so f = y is the EPW s -plane approximation. As mentione before, to interpret horizontal structure of EPW, we shoul erive shallow water equations for perturbations on a motionless basic state of mean epth H. Ranall [] explaine that the shallow water equations are the simplest form of the equations of motion that can be use to escribe the horizontal structure of an atmosphere. Shallow water equations can be erive from D momentum equations by ignoring the effects of friction: D u Dt + f ˆk v = ρ p gˆk () [ ] D u ρ Dt + f ˆk v = p ρgˆk () Here, ρ is the ensity of the air, f is Coriolis force, g is gravity force, u is the three-imensional velocity vector, p is air pressure, an D is total erivative operator. By applying the perturbation metho an assuming that flui is incompressible an no flui crosses the mean epth (H) or no share, we can now write the horizontal momentum equation as: [ ] D vh ρ Dt + f ˆk v = ρg h () [ ] D vh Dt + f ˆk v = g h (6) v h is the two-imensional velocity vector (xi+vj). For EPW case, wave movement is in hyrostatic balance. The spee component on the horizontal plane oes not epen on the altitue, thus the continuity equation can be vertically integrate from surface h s = an free surface altitue h f to yiel: t (h f h s ) +. [ v h (h f h s )] = (7) h f t +. [ v h h] = (8) Substitute the -plane approximation into the equations 6 an 8, then we will fin the non-linear shallow water equation in -plane approximation: Du h yv + g Dt x = (9) Dv h + yu + g Dt y = () h t + x (hu) + (hv) y = () u an v are zonal win velocity. In orer to isolate some simple wave forms we nee to linearize the equation 9-. First we consier small perturbations on an initially motionless atmosphere, i.e. basic state wins (u o, v o, w o ) =. Then by applying the perturbation

metho in to equations 9, an, we finally get the linear shallow water equations in equatorial -plane: u t + u u x + v u h vy + g y x = t (ū + u ) + (ū + u ) x (ū + u ) +v y (ū + u ) yv + g x (H + h ) = u t + ū u x yv + g h x = u t yv + Φ x = () v t + u v x + v v h + uy + g y y v t + (ū + u ) v v + v x y = +y (ū + u ) + g y (H + h ) = v t + ū v x + yū + yu + g h y = v t + yu + Φ y = () t (H + h ) + x [(ū + u ) (H + h )] h t + ū h Φ t + gh + y [v (H + h )] = x + H u x + H v y = [ ] u x + v = () y Here, u, v, an h are perturbation of mean zonal flow, meriional flow, an geopotential height respectively. So, linearise shallow water equations in - plane approximation from equations,, an may be briefly written as follows: u t yv + Φ x v t + yu + Φ y Φ [ ] u t + gh x + v y = () = (6) = (7) The above equations are useful to etermine the mathematical structure of EPW.. Mathematical Solution of EPW in the Horizontal Structure To simplify the etermination of the EPW solution, the equation in part -7 must be transforme into a nonim unit form [6, 7]. This transformation is one by efining the characteristic of length unit [L], an time unit [T] in the imensionless variables: x = x [L], y = t = y [L], u = u gh, v = v gh t [T ], Φ = Φ gh, [L] = [V ] [T ] = [T ] gh Here, x*, y*, u*, an v* are nonim units of longitue, latitue, zonal win an meriional win, respectively. Substituting the above nonim units into equation : ( u gh ) (t y [L] v gh + (Φ gh) [T ]) (x [L]) = where, u t [L] gh y v + Φ x = [L] = gh Then, by using the above information, nonimensional forms of u, v, t, an Φ can be written as: ( ) / ( ) / [T ] = = gh c ( ) / ( ) / gh c [L] = = u = u gh, v = v gh t = t (c) /, Φ = Φ gh Here c = gh is the velocity of pure gravity waves. By taking these units, the equations (l-7) are transforme into non-imensional -plane forms: (c) / (u ( ) / c) c t y v c ( ) / ( Φ c ) + c x = / / u c t / c / y v + / / Φ c x = u t y v + Φ x = (8)

(c) / (v ( ) / c) c t + y u c ( ) / ( Φ c ) + c y = +c [ ( c / / v c t + / c / y u + / / Φ c y = v t + y u + Φ y = (9) ) / (u c) x + / / Φ c t + / c / Φ t (c) / ( Φ c ) t ( ) ] / (v c) c y = [ ] u x + v y = + u x + v y = () or in general, the non-imensional linearise shallow water equations in equatorial -plane forms may be written as follows: u Φ yv + t x v Φ + yu + t y Φ t + u x + v y = () = () = () The EPW mathematical solution can be represente in the form of ecaying waves in the meriional irection. Assuming waves act in the y-irection for the perturbations an propagate in the east-west irection [, 9]: u v Φ = û (y) ˆv (y) ˆΦ (y) exp [i (kx ωt)] () Here, k is zonal wavenumber an ω is frequency. The above equation comes from the Fourier component assuming that wave has a regularity in space an time so it is possible to be represente in the sinusoial Fourier forms. If we substitute the equation () into equation (-), we will fin the first-orer orinary ifferential equations in the y irection for meriional structure of amplitue perturbations û, ˆv an, ˆΦ as follows: iωû yˆv + ik ˆΦ = () iωˆv + yû + ˆΦ y = (6) iω ˆΦ + ikû + ˆv y = (7) Fin the solutions for û, ˆv an, ˆΦ by applying some elimination techniques: an, iωû yˆv + ik ˆΦ = iωk ˆΦ + ik û + k ˆv y = [ û = i (k ω ωyˆv k ˆv ] ) y iωkû kŷv + ik ˆΦ = iω ˆv ˆΦ + iωkû + ω y = [ ˆΦ = i (k ω kyˆv ω ˆv ] ) y (8) (9) Substitute the equation (8) an (9) into equation () so we will fin secon-orer orinary ifferential equation for ˆv(y): ˆv (ω y + k kω ) y ˆv = () or in imensional form, equation can be written as: ˆv [( ω y + gh k k ) y ] ˆv = () ω gh The above equations can be simplifie into first-orer ifferential equations in orer to solve epening solution of ˆv to y: ˆv y yˆv = Cˆv () Because the equatorial -plane approximation is not vali beyon ± o away from the equator we have to confine the solutions close to the equator if they are to be goo approximations of the exact solutions, thus the bounary conition for this equation is: ˆv, if y ± C is (ω k k ω ), also known as ispersion relationship of EPW, thus we can moify the equation in orer to fin the recursive relationship: (D y) (D + y) ˆv C = (C ) ˆv C () (D + y) (D y) ˆv C = (C + ) ˆv C ()

D is a ifferential operator (/y), then multiply equation by (D + y) an equation by (D y): (D + y) (D y) [(D + y) ˆv C ] = (C ) [(D + y) ˆv C ] () (D y) (D + y) [(D y) ˆv C ] = (C + ) [(D y) ˆv C ] (6) If equation 6 is compare to equation we will fin a conclusion that both of these equations will have the same form if inex of C in equation replace by C +. (D y) (D + y) ˆv C+ = (C + ) ˆv C+ (7) An finally the recursive relationship is: ˆv C+ = (D y) ˆv C (8) Use this recursive equation in equation : (D + y) ˆv C+ = (C + ) ˆv C (9) Equation 9 shows that the solution of ˆv is epening on C inex. For C =, the solution of ˆv is: ( ) y + y ˆv = ˆv = e y () Then we shoul try to etermine the other values of ˆv by using the recursive relation an equation : ( ) ( ) ˆv = y y ˆv, ˆv = y ( ) y ˆv ( ) ˆv = y y ˆv, ˆv = y ( ) y ˆv ( ) ˆv 6 = y y ˆv, ˆv 7 = y ( ) y ˆv ( ) ˆv 8 = y y ˆv, ˆv 9 = y y ˆv from these solutions, mathematically ˆv has the value only for o s inex or (n + ), or briefly can be written as follows: ( ) n ˆv n+ = y y e y () Remember that C from equation is n +, where n =,,,.., thus the equation can be written in a new form: ˆv y yˆv = (n + ) ˆv () which means that the ifferential form in equation has a solution if: ω k k = n + () ω or in the imensional form: [( ω gh k k ) ] gh = n + () ω where n= -,,,,.., epening solution of ˆv to y in equation can be simplifie as follows: ( ) y y ˆv = e ) (e y y ˆv y ( y y ( y y ( y y ( y y = f (y) () ( ) y e y ˆv = f (y) (6) ( ) y e y ˆv = f (y) (7) ( ) y e y ˆv = f (y) (8) ( ) y e y ˆv = f (y) (9) ) ˆv = e y ) ˆv = e y ) ˆv = e y ) ˆv = e y ( ) n ( ) y y ˆv = e n y y n e y ˆv () by using the equation, the equation can be reuce to: ( ˆv n = e y n y n e y) () n + s inex on ˆv can be replace by n (n =,,,..) to have a uniform notation as a ifferential orer. The amplitue of the wave is a positive number an then the equation () can be written: ( ˆv n = ( ) n e y n y n e y) () The above equation () can be moifie as follows: ( ( ) n e y ˆv n = ( ) n e y n y n e y) () The right sie of last equation is the Hermite Polynomial functions of orer n, which satisfy: ( H n (y) = ( ) n e y n y n e y) ()

6 because H n = ( ) n e y ˆv n, so : Figure : Hermite polynomial (Hn) of orer n. If it s evaluate for each n from equation (), few of these polynomials have the values as follows: H n (y) H (y) = H (y) = y H (y) = y H (y) = 8y y H (y) = 6y 8y + H (y) = y 6y +y By using Hermite notation (), then our equation set becomes: ˆv n = e y H n (y) () Hn (y) strongly etermines the istribution of the meriional component of horizontal win an geopotential fiel perturbation in the EPW problems. Accoring to Setiawan [6], orer-n in EPW is inepenent (orthogonal) to the other n orers, it s because the Hermite polynomials functions are orthogonal with respect to e y : for n m : for n=m : or, + + + e y H n (y) H m (y) y = (6) e y H n (y) H m (y) y = n.n! π (7) e y H n (y) H m (y) y = n.n! π.δ mn (8) + ˆv n (y) ˆv m (y) y = n.n! π.δ mn (9) n an m are inex of Hermite polynomials let say first an secon inex, δ mn is Kronecker elta for positive integers m an n (δ mn = if m n, an δ mn = if m = n ). Orer-n shows how the wave moving system works in a meium, thus n correspons to the number of noes in the meriional velocity profile in the omain y < or it s also calle the moe of EPW. Next is etermining the set of solutions for the horizontal structure of EPW (non-imensional). Differentiate Hn (y) in equation () to y an using Leibniz notation rules in simplifying the equation: y H n (y) = ( ) n ye y n ( y n e y) ( ) n ye y n y ye y n e y ( ) n e y n n e y yn (6) We will fin some solutions of Hermite s ifferential which can be use to solve horizontal structure equations of EPW: y H n (y) = n ( ) n e y y H n (y) = nh n (y) (6) y H n (y) = n (n ) H n (y) (6) n e y yn Using equation (6) into the equation (8) an (9), we will fin other solution forms of EPW s amplitue in the zonal an geopotential component : û n = = = ˆΦ n = = = i (k ω ) ( ωyˆv n k ˆv n y ) [ ωye y H n k y (e y H n ) ] i (k ω ) i (k ω ) [(ω + k) yh n nkh n ] e y ( i (k ω ) kyˆv n ω ˆv n y ) [ kye y H n ω y (6) (e y H n ) ] i (k ω ) i (k ω ) [(ω + k) yh n nωh n ] e y (6) 6

7 with n=,,,,..., an ω ±k. By inserting the value of each amplitue (û, ˆv, an ˆΦ) into the equation () an taking the real part, we finally fin the general equations of EPW in horizontal structure (nonim) as follows: u n (x, y, t) = (k ω ) [(ω + k) yh n nkh n ] e y sin (kx ωt) (6) v n (x, y, t) = e y H n cos (kx ωt) (66) Φ n (x, y, t) = (k ω ) [(ω + k) yh n nωh n ] e y sin (kx ωt) (67) with n=,,,,..., an ω ±k.thus, solutions of EPW are propagations of the system forme by the components u, v, an Φ.. Frequency Bounary of EPW Taking the relation ispersion in equation as a quaratic equation in k: k + k ω + [ n + ω ] = (68) The above equation has various real roots of k, which means that the iscriminant of equation 68 must be positive: ω + (n + ) ω + > (69) Thus, the solutions of frequency bounary in nonimensional forms are: ω > (n + ) + (n + ) (7) or, ω < (n + ) (n + ) (7) (n + ) These solutions provie a frequency bounary for the existence of EPW.. Yanai Wave The case n = is a special case corresponing to the mixe Rossby-gravity or Yanai waves. For moe case n = the general equations for horizontal structure of Yanai wave (nonim) can be prouce by using equation 6-67 as follows: u y n (x, y, t) = k ω e y sin (kx ωt) (7) Φ y n (x, y, t) = k ω e y sin (kx ωt) (7) v n (x, y, t) = e y cos (kx ωt) (76) If n=, the ispersion relationship of EPW becomes: ( ω k k ) = n + ω ( ω k k ) = (77) ω then, this ispersion relationship (77) can be factore: (ω + k) ( ω kω ) = (78) Equation 78 is a cubic equation in ω, so we will have three roots for ω when n an k are specifie, namely: ω = k (79) ω = k + k ω = k k +, k > (8) +, k > an k ± (8) Equation 8 is corresponing to eastwar moving Poincaré wave an equation 8 can be separate into imensional form as follows: ω = k ω = k gh gh ( + k gh { k if large k; ( gh ) if k. ) (8) (8) From the equation 8 we can conclue that westwar moving of EPW by moe n = has mixe character. When k is large, they behave like westwar Poincaré waves an when k is small or k, they behave an in imensional forms can be written as: ( ω > (n + ) + ) (n + ) like westwar Rossby waves. Thus these waves are gh (7) calle as mixe Rossby-gravity waves or commonly known as Yanai waves. The phase spee of this wave in imensional form can be written as: or, [ ( ) ω < (n + ) c = ω gh (7) k = k + ( + ) ] k (8) gh Yanai waves are westwar propagating (ω/k < ) relative to the mean zonal flow an ispersive (from figure ). 7

8 Wave Frequency (Nonim).... Easterly (w/k >) Westerly (w/k <) imensional forms as follows: ( ) û (y) = û o exp y k ω ˆΦ (y) = ( ) gh û o exp y k ω (9) (9) Then phase spee can be written in imensional form as: c = gh (9).... Figure : waves.. Kelvin Wave Zonal Wave Number (Nonim) Dispersion curve of westwar moving Yanai Kelvin waves are a special case when the meriional velocity vanishes everywhere ientically (v = ) an equations -7 reuce to: The secon solution of 89 an 9, the one that propagates westwar, woul grow exponentially for large y an as such is isregare. The solution of zonal win an geopotential fiel perturbations associate with Kelvin waves in nonim forms are: u n (x, y, t) = e y cos (kx ωt) (9) Φ n (x, y, t) = ω k e y cos (kx ωt) (96) Thus, Kelvin waves are eastwar propagating (ω/k > iωû + ik ˆΦ = (8) yû + ˆΦ y = (86) iω ˆΦ + ikû = (87) Eliminating the geopotential component by substituting the equation 8 into equation 87, we fin: ω = ±k (88) Wave Frequency (Nonim) Easterly (w/k > ) Westerly (w/k < ) or in imensional form: ω = ±k gh (89) It s known as the ispersion relation for frequency ω an wavenumber k. If equation 8 is combine to equation 86 we will fin a first-orer orinary ifferential equation for the meriional amplitue structure of û: The solution of this equation is: û ± yû = (9) y û (y) = e y (9) The vali solution of 89 an 9 ecays for a large y only when the negative sign in the exponential function is chosen (our bounary conitions are that the solution must ecay as y ). It is calle the solution of Kelvin waves an can then be written in.... Zonal Wave Number (Nonim) Figure : Dispersion curve of eastwar moving Kelvin wave. ) relative to the mean zonal flow an non-ispersive (figure ). They are a form of gravity waves (9).. Equatorial Rossby an Poincaré Waves The solutions of () that ecay far away from the equator exist only when an important constraint connecting its coefficients is satisfie: ω k k = n +, n =,,,... (97) ω The above coefficients are a ispersion relationship in ω cubic equation where the solutions associate with eastwar an westwar propagating waves. 8

9 For high frequencies we can neglect k/ω in 97 to obtain: ω k k ω ω k = n +, n =,,,... (98) an the solutions are: ω, = ± k + n + (99) This is a ispersion relation for equatorially trappe Poincaré waves. They propagate in either the eastwar (+) or westwar (-) irection. The phase spee of Poincaré waves in imensional form is: c, = ±c + k (n + ) () gh Where c is pure gravity velocity an remember that meriional moe number for Poincaré waves is n =,,... an the frequency bounary shoul be vali in range equation 7 or 7. These wave groups are similar to gravity-inertial waves in mi-latitues []. For low frequencies we can neglect ω in (97) to obtain: ω k k ω k k = n +, n =,,... () ω an its solution is: k ω = k + n + () This is a ispersion relation for equatorially trappe Rossby waves. They propagate only in the westwar (ω/k < ) irection. The phase spee of Rossby waves in imensional form is: c = k ( + ) k (n + ) () gh The phase spee is inversely proportional to the square of the horizontal wavenumber. Meriional moe number for Rossby waves is n =,,... an the frequency bounary shoul be vali in range equation 7 or 7. These waves are similar to their counterparts in the mi-latitues an critically epen on the -effect. Eastwar propagation occurs when they have a high wave number []. Thus, both Poincaré an Rossby waves are ispersive waves. The perturbation solutions for Poincaré an Rossby waves are epening on moe-n an their frequency. In general eigenfunctions for EPW by moe n =,,... are: Wave Frequency (Nonim) Yanai n= Rossby n= Rossby n= Rossby n= Kelvin n= EGW n= EGW n= EGW n= EGW n= EGW n= WGW n= WGW n= WGW n= WGW n= WGW n=.... Zonal Wave Number (Nonim) Figure : The ispersion relation for free equatorial planetary waves (EPW) in nonimensionalize wavenumber an frequency (Kelvin waves, Yanai waves, Rossby waves, East Poincaré waves (EGW) an West Poincaré waves (WGW). for moe-n=, u n (x, y, t) = [ y (k ω (ω + k) k ] ) e y sin (kx ωt) () Φ [ n (x, y, t) = y (k ω (ω + k) ω ] ) e y sin (kx ωt) () v n (x, y, t) = ye y cos (kx ωt) (6) for moe-n=, u n (x, y, t) = [( y (k ω y ) (ω + k) 8yk ] ) e y sin (kx ωt) (7) Φ [( n (x, y, t) = y (k ω y ) (ω + k) 8yω ] ) e y sin (kx ωt) (8) v n (x, y, t) = ( y ) e y cos (kx ωt) (9) an for moe-n=, u n (x, y, t) = (k ω ) [( 8y y ) (ω + k) ( y ) k ] e y sin (kx ωt) () Φ n (x, y, t) = (k ω ) [( 8y y ) (ω + k) ( y ) ω ] e y sin (kx ωt) () v n (x, y, t) = ( 8y y ) e y cos (kx ωt) () 9

THEORETICAL SIMULATIONS As we mentione above that the theoretical simulations of equatorial planetary waves will be esigne base on shallow water equations on a motionless basic state of mean epth H in -plane. The simulations are evelope in a barotropic moel. The simplest conition of barotropy is given by assuming that we have a homogeneous moel atmosphere. A single layer of homogeneous, incompressible flui with a free surface has been applie an it is assume that Coriolis parameter is proportional to the latitue. centere at the equator) an ecays moving away from the equator, while zonal win perturbations have their greatest amplitue at latitue y = ± (nonim) an zero aroun the equatorial latitue belt, as well as to geopotential fiels (figure 6a). Yanai waves amplitue for zonal win, meriional win, an geopotential fiel component are ientically symmetric with ifferent forms. For this case amplitue of the meriional win is Gaussian symmetry an amplitue of zonal win an geopotential fiel are symmetric with two peaks at latitue y = ± (nonim). Base on meriional amplitue simulations, in general for EPW n = perturbation of meriional win associate with existence of EPW has the greatest meriional win amplitue at the equator an Gaussian ecays when away from the equator. EPW by moe n= has greatest amplitue of meriional win perturbation (v ) at latitue y = ± (nonim) an v equal to zero at the equator while EPW by moe n =, amplitue of v equal to zero at latitue y = ±/ (nonim) an greatest amplitues outsie the tropics. The istributions are graphically shown in the figure. The waves are trappe in Figure : Experiment of meriional win amplitue associate with EPW in the north-south irection for moe n =, an. meriional irection (figure ). This trapping occurs symmetrically north an south of the equator, so that the equatorial region becomes a waveguie. The further simulations of EPW will be iscusse below in the special moes of free waves such as Yanai waves, Kelvin waves, Poincaré waves an even Rossby waves.. Yanai an Kelvin Waves Simulations The peak of meriional win perturbations of Yanai waves will occur at the equator (Gaussian istribution Figure 6: Experiment of zonal win amplitue (re) an meriional win amplitue (yellow) associate with Yanai waves in the north-south irection for Yanai waves moe n = (a). Zonal win amplitue (blue) associate with Kelvin waves (b) with k =. an ω = -.66. For Kelvin waves case, the waves are only corresponing to symmetry zonal win an geopotential height while perturbation of meriional win (southerly velocity component) is ientically zero at about the equator (figure 6b). The peak amplitue of zonal win perturbation is at the equator an ecays in a Gaussian form when away from the equator. The shape of zonal win an geopotential fiel amplitue are symmetrical Gaussian. The character of zonal

Latitue (Nonim) n=; k=...... Latitue (Nonim) n=-; k=..8.6..... 6 8 Longitue (Nonim).. 6 8 Longitue (Nonim).6.8 n=; k=... n=-; k=..8.6. Latitue (Nonim).. Latitue (Nonim).. 6 8 Longitue (Nonim).. 6 8 Longitue (Nonim)..6.8 Latitue (Nonim) n=; k=........ Latitue (Nonim) n=-, k=..8.6..... 6 8 Longitue (Nonim).. 6 8 Longitue (Nonim).6.8 Figure 7: Simulation of geopotential fiel an horizontal win perturbations corresponing to Yanai waves moe n= with zonal wavenumber (k)=.,=., an., respectively. Figure 8: Simulation of geopotential fiel an horizontal win perturbations corresponing to Kelvin waves moe n= - with zonal wavenumber (k)=.,=., an., respectively. win perturbation in Kelvin waves is similar to meriional win perturbation of Yanai waves. The perturbations of three components (u, v an Φ ) associate with Yanai an Kelvin waves are simulate in figure 7. From the results of simulation, in general it is clear that the characters of Yanai or Rossby-Gravity waves have a maximum oscillation of the meriional win perturbation at the equator an exponentially ecays (Gaussian) when away from the equator. The ecaying is characterize by a weakening of the meriional win perturbations (look at the arrows shape), while the zonal win is zero at the equator. In the higher latitues y > an y < perturbation of zonal win (u ) an geopotential fiel (Φ ) will be approximately in geostrophic balance while near the equator ageostrophic win components preominate. The value of y omain can be written in imensional form as follows: ( ) ( ) gh < y < gh Figure 7 is Yanai waves simulation with t =, waves with larger k values will prouce more pressure cells where high pressure an low pressure forme are not symmetric to the equator (regularly changes to the longitue). The oscillations of zonal win an geopotential height are antisymmetric but the oscillations of meriional win are symmetric. In aition, the number of cells that forme showe that

a higher k value (zonal wavenumber) will prouce more antisymmetric zonal waves (s), but when k increases, the wavelength (λ) will ecrease. If k =. there are only a pair of high-pressure cell an a pair of low pressure cells (s = ) that sprea in a circumference of the earth s atmosphere while if k = an k =. they increase to two (s = ) an three times (s = ) of k =.. These waves move towar the west an o not have a constant form when they propagate (ispersive). In the case of free Kelvin waves, Figure 8-f escribe some simulations of the horizontal perturbations of geopotential an horizontal win velocity for Kelvin wave n = with k =.,., an.. These provie information that the existence of Kelvin waves are characterize by isturbances of zonal wins which are stronger in the equator an ecay when away from the equator, an there is no oscillation of meriional win component. A higher k value (zonal wave number) will prouce more symmetric zonal wave numbers (s) but the wavelength (λ) is small. These waves move towar the east an have a constant form when they propagate. The phase spee of these waves is easterly relative to the mean zonal flow. Typical values of the Kelvin wave spee are in range c - m/s in troposphere (corresponing to H = - m) with higher values in mile atmosphere. The existence of Kelvin waves in the atmosphere will change the average of zonal win profile in latitue (figure 9). Figure 9: Experiment of mean zonal win profile ue to Kelvin waveriving (nonim).. Equatorial Rosbby Waves an Poincaré Waves Simulations Rossby waves propagate only to the west, whereas their energy (group velocity) may propagate to the east or west (ispersive). Further, they are escribe in figure. Rossby waves with eigensolution for moe n = are characterize by the geostrophic relationship between pressure (geopotential fiels) an velocity wins. The strong zonal velocity is foun along the equator. Rossby waves will be in geostrophic balance between the pressure graient an win at omain y >. an y <. (nonim). Meriional win perturbations vanish along the equator. Zonal win (u ) an geopotential fiel (Φ ) perturbations are symmetric relative to the latitue but the meriional win perturbation (v ) is antisymmetric. For Rossby waves with eigensolution for moe n = are characterize by the geostrophic relationship between pressure (surface elevation) an velocity fiels at omain omain y >. an y <. (nonim). The ifferences from n= are these waves o not have win fiel perturbation along the equator, the maximum peak for zonal win perturbation is approximately at y = ±, v is symmetric, an both of u an Φ are antisymmetric relative to latitue. Rossby waves by moe n = are similar to eigensolution of n =, but for this case the peak of u are approximately in regions namely y=-,, an. In general, we can conclue that Rossby waves are quasi geotrosphic motions, an if we assume for a small k, the Rossby waves will be non-ispersive waves. Consequently, the phase spee of Rossby waves of ifferent moes is c/, c/, c/7. (eq.) or phase spee for n = > n = > n =. In other wors, the higher-orer Rossby moes are much slower an phase spee of Rossby waves < Kelvin waves. Eastwar Poincaré or east propagating Inertiogravity waves have phase an group velocities towar the east. From our simulations (figure ), Poincaré waves by moe n = are characterize by symmetric geopotential height an zonal win perturbations. Meriional win perturbations vanish along the equator an appear at omain about y± (antisymmetry). These waves are also characterize by relationships between Pressure an velocity fiels. For case moe n=, Poincaré waves have symmetric meriional win perturbations about the equator an no evience of zonal win perturbations. For case n =, is similar to n =, but number of the pressure cells increase to be cells relative to Earth s latitue. The higher-orer East propagating Poincaré moes are much faster an phase spee of these waves are more than Kelvin waves or in other wors phase spee of eastwar Poincaré waves > Kelvin waves > Rossby waves.

n=. Latitue (Nonim). 6 8 Longitue (Nonim) n= 6 6 6 8 Longitue (Nonim) n= Latitue (Nonim) 6 8 Longitue (Nonim) Figure : Simulation of horizontal win an geopotential fiel perturbations corresponing to equatorial Rossby waves moe n=,, an.

n= Latitue (Nonim).. 6 8 Longitue (Nonim) n=.. Latitue (Nonim). 6 8 Longitue (Nonim). n= 6 8 Longitue (Nonim) Figure : Simulation of horizontal win an geopotential fiel perturbations corresponing to equatorial Eastwar Poincaré waves moe n=,, an.

n=.. 6 8 Longitue (Nonim) n=.... 6 8 Longitue (Nonim) n= 6 8 Longitue (Nonim) Figure : Simulation of horizontal win an geopotential fiel perturbations corresponing to equatorial Westwar Poincaré waves moe n=,, an.

6. Amplitue (Nonim).. Amplitue (Nonim).. Amplitue (Nonim).. Latitue (Nonim) Longitue (Nonim) Latitue (Nonim) Longitue (Nonim) Latitue (Nonim) Longitue (Nonim) Kelvin n=, k=. Kelvin n=, k=. Kelvin n=, k=..... Amplitue (Nonim) Amplitue (Nonim)..... Amplitue (Nonim)..... Latitue (Nonim) Yanai n=, k=. Longitue (Nonim) Latitue (Nonim) Yanai n=, k=. Longitue (Nonim) Latitue (Nonim) Yanai n=, k=. Longitue (Nonim). Amplitue (Nonim).. Amplitue (Nonim) Amplitue (Nonim). Latitue (Nonim) Rossby n = Longitue (Nonim) Latitue (Nonim) Rossby n= Longitue (Nonim) Latitue (Nonim) Rossby n= Longitue (Nonim).. Amplitue (Nonim).. Amplitue (Nonim)... Amplitue (Nonim). Latitue (Nonim) Eastwar Poincare n= Longitue (Nonim) Latitue (Nonim) Eastwar Poincare n= Longitue (Nonim) Latitue (Nonim) Eastwar Poincare n= Longitue (Nonim). Amplitue (Nonim).. Amplitue (Nonim).... Amplitue (Nonim). Latitue (Nonim) Westwar Poincare n= Longitue (Nonim) Latitue (Nonim) Westwar Poincare n= Longitue (Nonim) Latitue (Nonim) Westwar Poincare n= Longitue (Nonim) Figure : Simulation of surface elevation corresponing to Kelvin waves, Yanai waves, Rossby waves, eastwar Poincaré waves an westwar Poincaré waves, respectively. 6

7 Westwar Poincaré waves have phase velocity towar the west an group velocity also towar the west except for very low zonal wavenumbers. Further, they are escribe in figure. West propagating Poincaré waves with eigensolution for moe n= are characterize by strong zonally symmetric component, as well as the meriional win perturbation vanishes along the equator. Both zonal win an geopotential fiel perturbations are symmetric but meriional win perturbations are antisymmetric when away from the equator. Simulations of westwar Poincaré waves with eigensolution moe n = an are similar to eastwar Poincaré waves. The ifferences are in istribution of horizontal win irections an forms of geopotential fiels relative to latitue. Strong horizontal win perturbations are near the omain < y < an < y < (nonim). Simulation of surface elevation corresponing to Kelvin waves, Yanai waves, Rossby waves, eastwar Poincaré waves an westwar Poincaré waves are escribe in figure. CONCLUSIONS From these simulations we can conclue that k an ω are parameters which control the type of wave patterns. Eigenfunction graually changes if there is a ifferences between k an ω. Further, the general solutions of EPW are forme by functions-n, so if n is an o number then v is an o function an u an Φ are even functions with respect to y. Consequently these waves have symmetric components of u an zero or antisymmetric components of v relative to latitue, for example Rossby n =,, Kelvin n =, eastwar Poincaré waves n =,, an westwar Poincaré waves n =,. If n is even number, each components are reverse, for example Yanai n =, Rossby n =, eastwar Poincaré waves n = an westwar Poincaré waves n =. ACKNOWLEDGMENTS Sanro woul like to acknowlege his gratefulness to his friens whom willing to rea an review his journal, Laurel McCoy (Atmospheric Sciences, University of Missouri-Columbia, USA), Nina Kargapolova (Institute of Computational Mathematics an Mathematical Geophysics, Russia), an Davesh Vashishtha (University of Delhi, Inia). [] A.V. Feorov an J.N. Brown, Equatorial Waves, Elsevier Lt, Yale University, New Haven, USA (9). [] C.F.M. Raupp an P.L.S. Dias, Atmos. Sciences. 6 () 6. [] C.J. Nappo, An Introuction to Atmospheric Gravity Waves, Elsevier Inc, San Diego () 6. [] D.A. Ranall. The Shallow Water Equations. Colorao, http:// kiwi. atmos. colostate. eu/ group/ ave/ QuickStuies. htm, Department of Atmospheric Science, Colorao State University (6). [6] I.M. Dima an J.M. Wallace, Atmos. Sciences. 6 (6) 86. [7] J.M. Wallace an V.E. Kousky, Atmos. Sciences. (968) 9. [8] J.R. Holton an R.S. Linzen, Monthly Weather Review. 96 (968) 8. [9] J.R. Holton, An Introuction to Dynamic Meteorology (Elsevier Inc, USA, ), r E. (). [] M.T. Boehm an S. Lee, Atmos. Sciences. 6 () 7. [] M. Yanai an T. Maruyama, Metor. Soc. Jap. (966) 9. [] M. Yanai an M. Murakami, Atmos. Sciences. 8 (97). [] Norton, Atmos. Sciences. 6 (6). [] R.D. Linzen, Monthly Weather Review. 9 (967) 7. [] R.S. Linzen, Atmos. Sciences. 8 (97) 69. [6] S. Setiawan, Perumusan Struktur Horizontal an Struktur Vertikal Gelombang Ekuatorial Planeter. Departemen Geofisika an Meteorologi FMIPA IPB (). [7] T. Matsuno, Metor. Soc. Jap. (966). REFERENCES [] A.M. Kerr-Munslow an W.A. Norton, Atmos. Sciences. 6 (6). 7