1 On Derivation and Interpretation of Kuo Eliassen Equation Jun-Ichi Yano 1 1 GAME/CNRM, Météo-France and CNRS, 31057 Toulouse Cedex, France Manuscript submitted 22 September 2010 The Kuo Eliassen equation provides a balance condition for both tropical cyclone like vortex systems as well as zonally symmetric meridional circulations. This condition is examined with the former application more in mind. The condition is derived more pedagogically based on the bounded derivative method. Some physical interpretations as well as basic mathematical remarks on this condition are provided. Analogy with quasi geostrophic system is also remarked. 1. Introduction Balanced vortex dynamics on the f plane is originally introduced by Eliassen (1952), and is investigated by Shapiro and Willoughby (1982), Schubert and Hack (1982), and many others. Schubert and McNoldy (2010, SM10 hereinafter) develop a physical interpretation of this system. The present note, being inspired by SM10, attempts to provide further interpretations on this system from a more basic level. In considering this problem, a balance condition (Eq. 2.10 of SM10) arises from a consistency of the system. A similar balance condition can be derived in considering zonally symmetric meridional circulations, as originally considered by Kuo (1956). For this reason, this balance condition is sometimes called Kuo Eliassen equation (e.g., Krishnamurti et al. 1994, Chang 1996). Two version are, of course, different in details mainly due to difference of horizontal scales of the two types of circulations. The present note considers a version for the balanced vortex dynamics as considered by SM10. By presenting three different types of analytical solutions, SM10 presents interpretations of Rossby length and depth arise in Kuo Eliassen equation. The goal of the present notes is, in turn, to examine the derivation of this balance equation and then offer further interpretations of this equation under further approximations. The next section examines the derivation of Kuo Eliassen equation from a point of view of the bounded derivative method (Kreiss 1980, Browning et al. 1980). Further interpretations of this equation is attempted from a more basic level than SM10 in Sec. 3. Some mathematical issues are discussed separately in Sec. 4 in order to better understand exact analytical solutions provided by SM10. Finally, Sec. 5 points out analogy between the Kuo Eliassen equation and the quasi geostrophic model. 2. Derivation of Kuo Eliassen Equation Kuo Eliassen equation is derived by SM10 as their Eq. (2.10). Starting point for deriving this equation is a balance vortex dynamics equation system given by their Eq. (2.1) (2.5). In order to obtain insights on the derivation of this equation, I recover neglected weak local time tendencies in their equations for azimuthal (Eq. 2.1) and vertical (Eq. 2.3) velocities with their smallness indicated by a parameter ǫ. Thus, the system in concern is given by ǫ u t f av = φ r, (1a) v t + ζ au + w v z = 0 ǫ w t + φ z = g θ 0 θ, (1b) (1c) (ru) rr + (ρw) = 0, (1d) ρz θ t + uθ r + wθ z = θ. (1e) To whom correspondence should be addressed. Jun-Ichi Yano, CNRM, Météo-France, 42 av Coriolis, 31057 Toulouse Cedex, France. e-mail: jun-ichi.yano@zmaw.de.
2 Yano Here, f a = f +v/r is an absolute Coriolis parameter felt by the radial wind v, and ζ a = f + (rv)/rr is the absolute vorticity, which also works as an effective Croiolis parameter for the azimuthal wind u. Note otherwise the identical notations as in SM10 are adopted here. This set (Eq. 1.a e) is obtained by partially recovering O(ǫ) terms from Eq. (1.1) (1.4) of Shapiro and Willoughby (1982). The parameter ǫ can be considered as a measure of the ratio of the radial and vertical velocities to the azimuthal velocity. The purpose here is to indicate the main terms neglected in the balanced vortex dynamics: acceleration of azimuthal and vertical velocities. Then we ask a question what conditions ensure the neglect of those terms. More terms of O(ǫ) from the original full system may be recovered, but only with an expense of making the following derivative more involved. Only the essential terms are retained in order to make the point as lucid as possible. A series of balance conditions can be obtained by writing down a linear tendency equation for the vorticity, ζ ϕ = u/z w/r, in the azimuthal direction. Note that an arising question is naturally linear by already neglecting the advection terms in Eq. (1a) and (1c). For this purpose, we take z derivative and r derivative on Eq. (1a) and (1c), respectively, and take a difference: ǫ ζ ϕ t = f av z g θ 0 θ r. It is important to note that this linear vorticity tendency equation is satisfied even after taking time derivatives any number of times. Thus, by taking time derivatives for n times on the above, we obtain a series of the conditions: ǫ n+1 ζ ϕ n = tn+1 t n (f av z g θ θ 0 r ) for n = 0, 1,. This series states that the right hand side must remain always small enough in order that the acceleration (as defined as any order of time derivatives) of vorticity (left hand side) is well bounded to O(ǫ) as required for the balanced dynamics. The condition reduces in the limit ǫ 0 to: n av t n(f z g θ ) = 0. (2) θ 0 r The method for posing a series of constraints (2) with n = 0, 1, is called the bounded derivative in the context of the model initialization problem (Kreiss 1980, Browning et al. 1980). For the present purpose, suffices it to pose the condition (2) for n = 0 and 1. With n = 0 in Eq. (2), we recover the thermal wind balance given as the second equality in Eq. (2.8) of SM10. With n = 1, we recover Kuo Eliassen equation given by (2.10) of SM10. This condition states that the acceleration of the azimuthal Coriolis force (the first term) must balance with that of buoyancy (the second term) in order that the system is always balanced (or bounded to O(ǫ)). The acceleration rate of the Coriolis force, f a v, is evaluated as f a v t = (f a + v r )v t = (f a+ v r )(ζ au+w v z ) ˆf 2 u (3a) Here, the last approximate equality is obtained by neglecting the advection term. An effective Coriolis parameter ˆf is introduced by ˆf 2 = (f a + v r )ζ a (cf., Vigh and Schubert 2009). Note that ˆf reduces to the Coriolis parameter itself, f, in linear limit. Here, Eq. (1a) says that the azimuthal wind u is accelerated by the Coriolis force f a v with a rate characterized by the absolute Coriolis parameter f a. According to Eq. (3a), the Coriolis force is, in turn, accelerated by a rate characterized by another effective Coriolis factor, f a + v/r. As the whole, the azimuthal wind u is doubly accelerated by a rate proportional the square of the effective Coriolis parameter, ˆf: 2 u t 2 ˆf 2 u This balance leads to an oscillation tendency of the azimuthal wind, u, with the inertial frequency ˆf f. On the other hand, the acceleration rate of buoyancy is estimated by the thermodynamic equation (1e) as g θ θ 0 t g θ θ z w g θ θ = N 2 w g θ θ. (3b) The approximate equality above is again obtained by neglecting the horizontal advection. As the first term in the middle expression shows, the buoyancy acceleration is primarily dictated by the vertical heat advection. That leads to an oscillation tendency of the vertical velocity, w, by the Vaisala frequency, N = (gθ/θz) 1/2 : 2 w t 2 N2 w Substitution of Eq. (3a, b) into Eq. (2) with n = 1 leads to z ( ˆf 2 u) r (N2 w) g θ θ 0 r. JAMES-D
Kuo Eliassen Equation 3 Consequently, in order to maintain the balance condition, horizontal acceleration tendency with the inertial frequency ˆf f (first term) must balance with vertical acceleration tendency with the Vaisala frequency N. By further substituting the definition of the streamfucntion, ψ, given by Eq. (2.9) of SM10 into the above, we obtain an approximate form of the Kuo Eliassen equation: [ r N 2 ρ (1 r r r) + z ˆf 2 ρ z ]ψ g θ 0 θ r. (4) Eq. (4) agrees with Eq. (2.10) of SM10 when the definitions of the coefficients, A = N 2 /ρ and C = ˆf 2 /ρ, as given by their Eq. (2.8), are substituted, and also setting B 0. It is interesting to note that neglect of advection has an equivalent effect as neglecting the baroclinicity of the system. 3. Interpretation of Kuo Eliassen Equation Interpretation of Kuo Eliassen equation is further prompted by assuming the coefficients, A and C, are constants, as assumed by SM10, as well as neglecting the curvature effects in the radial direction, by setting (1/r)(/r)r /r. As a result, Eq. (4) reduces to [ 1ˆf2 2 r 2 + 1 N 2 2 z 2]ψ 1 (N ˆf) 2 g θ 0 θ r. (5) In order to obtain a simple interpretation of Eq. (5), let us assume that the secondary circulation is approximately represented by single horizontal and vertical wavenumbers, k and m, respectively. Let us also concentrate on the homogeneous problem by setting θ = 0 by following the main focus in SM10. Then Eq. (5) furthermore reduces to: [( kˆf ) 2 + ( m N )2 ]ψ 0 The equation has a simple interpretation that the horizontal phase velocity c H = ˆf/k due to the inertial oscillation must match with the vertical phase velocity c V = N/m due to the buoyancy oscillation to the order of magnitude, i.e., c H c V (6) or ˆf N k m. (7) When Eq. (7) is solved for the horizontal wavenumber, k for a given vertical wavenumber, m, we obtain an inverse horizontal wavenumber: k 1 1 m (Ṋ f ), (8a) which is the Rossby length (or Rossby radius of deformation). Alternatively, when Eq. (7) is solved for the vertical wavenumber, m for a given horizontal wavenumber, k, we obtain an inverse vertical wavenumber: m 1 1 k ( ˆf N ), (8b) which is the Rossby depth. Consequently, Rossby length and depth (Eq. 8a, b) are interpreted as scales that the matching of the two phase velocities is satisfied (Eq. 6) for given vertical and horizontal scales, so that the secondary circulation can be maintained in a manner consistent with the balance condition for the vortical motion. Here, many readers may wonder why I did not put Eq. (6) instead more precisely as c H ±ic V. The purpose above is merely to show that we can understand the meaning of Rossby length and depth by a very simple order of magnitude argument. For such a purpose, even the question of whether c H and c V are real or imaginary does not count. These mathematically more subtle issues are discussed in the next section in order to better understand the analytical solutions presented by SM10 from a wider perspective. 4. Mathematical Remarks From mathematical point of view, Eq. (7) suggests when either the horizontal coordinate is stretched by a rate, N/ ˆf, or alternatively the vertical coordinate is stretched by a rate, ˆf/N, Eq. (5) reduces to a Poisson problem: [ 2 2 + r 2 z 2]ψ g θ ˆα2 θ 0 r. with the definition of a constant ˆα depending on which coordinate is actually stretched. The star implies stretched coordinates. The stretching factor Γ = N/ ˆf plays a major role in analytical solutions of SM10. See their Eq. (3.11) and (4.5) especially. For better elucidation, here, again, let us seek a solution with local wavenumbers, k and m for a situation
4 Yano without diabatic heating, i.e., θ = 0. The above equation, then reduces to: k 2 + m 2 = 0. (9) The result (9) means that when a real wavenumber, k, is assumed in the horizontal direction, a vertical structure of the solution is evanescent (i.e., m 2 < 0), and vice versa. It is important to note that this mathematical structure of the problem is also reflected on the full problem with the cylindrical coordinates as considered by SM10. In Sec. 3, SM10 seek a solution under a Fourier expansion in the vertical direction. This corresponds to the case in Eq. (9) that considering a real wavenumber, m, in the vertical direction. Thus, the solution in the horizontal direction must be evanescent. It is reflected on the fact that SM10 found a radial structure described by modified Bessel functions. Those functions are essentially evanescent in their behavior (cf., Olver 1974). On the other hand, in Sec. 4, SM10 seek a solution assuming a Bessel function form in the radial direction. Being equivalent of assuming a wave form in the radial direction, the vertical structure of the solution become evanescent as expected from Eq. (9), and as found in their solution (Eq. 4.6). The relation (9) suggests that the secondary circulation can locally be wave like either in the radial or the vertical direction only. Consistency of this conclusion with the full solution with the cylindrical coordinates in SM10 furthermore suggests that an overall character of the secondary circulation solution can well be inferred without explicitly taking into account the curvature effects of the system. 5. Analogy with the quasi geostrophic system Finally, it would be helpful to recall that a similar issue with Rossby length and depth is also found in the quasi geostrophic system. The quasi geostrophic system is dictated by a conservation law of a potential vorticity q: ( t + v H H )q = 0 with the horizontal velocity v H = (u x, u y ) under geostrophy: u x = 1 φ f y, u y = 1 f φ x with the Cartesian coordinates (x, y). Here, the potential vorticity, q, is defined by q = 1 f Hφ + f ρ z ( ρ N 2 φ) (10) z under the notations of SM10. A similarity between Eq. (10) and Eq. (4) is hard to miss. By assuming ρ/n 2 constant, Eq. (10) further reduces to: q = f( 1 f 2 H + 1 N 2 2 z 2)φ, which can be more directly compared with Eq. (5). Note that the horizontal Laplacian H above plays the same role as 2 /r 2 in Eq. (5). Thus, the behavior of the quasi geostrophic system can be understood, to some extent, in analogy with that of the Kuo Eliassen equation discussed above. Important role of the Rossby length, 1/m(N/f), in the quasi geostrophic dynamics is hard to overemphasize (cf., Pedlosky 1987). This knowledge would also help to further understand the Kuo Eliassen equation. References Browning, G., A. Kasahara, H O. Kreiss, 1980: Initialization of the primitive equations by the bounded derivative method. J. Atmos. Sci., 37, 1424 1436. Chang, E. K. M., 1996: Mean meridional circulation driven by eddy forcings of different timescales. J. Atmos. Sci., 53, 113 125. Eliassen, A., 1952: Slow thermally or frictionally controlled meridional circulations in a circular vortex. Astrophysica Norvegica, 5, 19 60. Kreiss H. O., 1980: Problems with different time scales for partial-differential equations. Comm. Pure Appl. Math., 33, 399 439. Krishnamurti T. N., H. S. Bedi, D. Oosterhof, V. Hardiker, 1994: The formation of Hurricane-Frederic of 1979. Mon. Wea. Rev., 122, 1050 1074. Kuo, H-L., 1956: Forced and free meridional circulations in the atmosphere. J. Meteor., 13, 561 568 Olver, F. W. J., 1974: Asymptotics and Special Functions, Academic Press, 572pp. Pedlosky, J., 1987: Geophysical Fluid Dynamics. 710pp, Springer Verlag. 2nd Ed., Schubert, W. H., and J. J. Hack, 1982: Inertial Stability and Tropical Cyclone Development. J. Atmos. Sci., 39, 1687 1697 Schubert, W. H., and B. D. McNoldy, 2010: Application of the concepts of Rossby length and Rossby depth to tropical cyclone dynamics. accepted to J. Adv. Model. Earth Syst. JAMES-D
Kuo Eliassen Equation 5 Shapiro, L. J., and H. E. Willoughby, 1982: The response of balance hurricane to local sources of heat and momentum. J. Atmos. Sci., 39, 378 394. Vigh, J. L., and W. H. Schubert, 2009: Rapid development of the tropical cyclone warm core. J. Atmos. Sci., 66, 3335 3350.