Eliassen-Palm Theory

Transcription

1 Eliassen-Palm Theory David Painemal MPO611 April 2007 I. Introduction The separation of the flow into its zonal average and the deviations therefrom has been a dominant paradigm for analyses of the general circulation. Eliassen-Palm (EP) theory provides an elegant framework with which to understand the eddy-mean flow interaction. Since Eliassen and Palm (1960) paper, several studies have extended the use of this theory to include the effect of humidity (Stone and Salustri, 1984), use of isentropic coordinates (e. g. Andrews et al. 1987), applications in numerical models, and applications in the middle and upper troposphere (e.g. Trenberth 1986). This review intends to give the basic theoretical formulation of the problem. Most of the paper is focused on interpreting the EP flux. The mathematical formulation (a quasi-geostrophic formulation) is given in section 2. The Eliassen-Palm theorem (and the generalized theorem) is proved in section 3. In the last section is described a diagnostic study using EP flux applied to Southern Hemisphere. The reader is invited to follow the references in order to get a more complete mathematical treatment of the problem. 1

2 II. Basic equations Following Andrews et al. (1986), we express the quasi-geostrophic equation as follows: u t + uu x + vu y " fv 1 " #yv + $ x = "X v t + uv x + vv y + fu 1 + #yu + $ y = "Y "% + $ p = 0 (1) % t + +&% p = "Q u (1)x + v (1)y + & p = 0 Where: u a = "u/"a,u ab = " 2 u /"a"b. The terms X, -Y and Q represent arbitrary body forces and heating, whose possible physical causes may be left specified. The parameter with subscript 1, represent the first order approximation, and variables without subscripts represent the zero order (geostrophic) approximation. Averaging the equations we obtain: u t " fv = "(u'v') y " X v t + fu + # y = "(v' 2 ) y "Y "$ + p z = 0 $ t + % $ p = "(v'$') y " Q v y + % p = 0 (2) The overbar denotes the usual zonal average and prime the departure therefrom. Using the transformed Eulerian mean (Andrews and McIntyre, 1976), we can define 2 new variables: "* = " + (v'#'/# p ) y, v * = v 1 $ (v'#'/# p ) p (3) As explained by Holton (2004), this transformation takes account of the fact that within the set of equation 3 there tends to be a strong cancellation between the eddy heat flux convergence and the adiabatic cooling (thermodynamic equation), while the diabatic 2

3 heating term is a small residual. Since in the mean an air parcel will rise to a higher equilibrium altitude only if its potential temperature is increased by adiabatic heating, it is the residual meridional circulation associated with adiabatic processes that is directly related to the mean meridional flow. Writing the equations in terms of the transformed eulerian mean: "u "t # f v * #X = $ % F f u p # R& y = 0 v y *+' p* = 0 "& + & p' *#Q = 0 "t (4) In this new set of equations the EP flux naturally appears. The vector F or the Eliassen-Palm vector is defined: F = { F (y),f ( p ) } F (y ) = "v'u' F ( p) = f v'#'/# p (5) " # F $ %F (y ) %y + %F ( p) %p (6) X and Q correspond to the Eulerian friction and heating. R is the gas constant times (p 0 / p) 1/(c p / c v ) p 0 "1. Equations 4 comprise a complete set of equations for the mean state described in terms of the quantities { u,v*,"*,#}. Thus under the QG framework, we can interpret " # F as the internal forcing of the mean state of the disturbances. Note if X,Q and " # F vanish, the equations permit a steady mean state which u t, " t, v * and " * are all zero: this is Charney and Drazin nonacceleration theorem. A more general derivation of EP flux based on Boussinesq, hydrostatic and primitive equations of motion for a beta-plane, can be found in Andrews and McIntyre (1976). 3

4 III. Eliassen-Palm theorem The EP theorem states that under nonaccelerations conditions " # F = 0. Nonaccelerations conditions mean: i. Strictly conservative eddy and mean-flow dynamics. ii. Constant eddy amplitude, measured in a Lagrangian sense involving north-south air parcels displacements. iii. Steady mean flow, in both Eulerian and Lagrangian sense. In order to prove the EP theorem, we use the potential vorticity equation. If the motion is quasi-geostrophic we have conservation of potential vorticity (q), then: mean, we get: q t + uq x + vq y = 0. (7) As the geostrophic flow is non-divergent, we get: q t + (uq) x + (vq) y = 0. (8) Using the decomposition of q = q + q' and replacing into (8) and taking the zonal q t + (v'q') y = 0 (9) Under nonaccceleration conditions the mean state is steady, then (v'q') y = 0 where (v'q') is independent of latitude. If it is assumed (v'q') vanishes at latitude y=y N, then (v'q') must be zero. The next step is to relate (v'q') to the divergence of the EP flux. The eddy potential vorticity (PV) is: q'= v' x "u' y + f ( #' ) p (10) # p In order to find a useful relationship between (v'q') and EP flux, we expand the first term: 0} v'q' = v'2 x 2 " v'u' y + v' f ( #' ) p = "v'u' y + v' f ( #' ) p (11) # p # p 4

5 Considering the fact that the wind is non-divergent: u' x +v' y = 0, we express the wind in terms of the streamfunction: fv'= "# "x, fy'= "#$. It is possible to re-arrange the #y terms of the equation (11) (making use of the streamfunction) to get exactly: v'q' = ("v'u') y + ( f v'#'/# p) = $ % F (For more details, the reader is suggested to read Holton, 2004, who uses a z-coordinate to prove this relationship). We have proved that EP divergence is the QG eddy potential vorticity flux. Thus, under quasigeostrophic approximations and nonaccelerations conditions, we have proved EP theorem, i.e. " # F = 0. In absence of heating and friction in a stationary wave field, the fluxes of heat and momentum produce mean meridional circulations which precisely cancel the tendency of the fluxes to alter the mean state (Pedlosky, 1987). As emphasized by Edmon et al (1980), EP flux should be considered a fundamental diagnostics because it relates the wave-action and group velocity concept. McIntire (1976) pointed out that whenenver the eddy dynamics is wavelike, F may be regarded as a measure of the net rate of transfer of wave activity from one latitude and height to another. Considering this interpretation of F, the conservation equation related to F can be written. "A "t + # $ F = D (12) Where D is zero for conservative motion (no dissipation or generation of the waves by diabatic or frictional effects). A can be regarded as a conservable measure of local wave activity and we will call it the density of EP wave activity. This result was derived by McIntyre (1976) as a special case of the so called generalized Eliassen-Palm relation. Based on McIntyre (1976), we can express the local wave activity (A) for waves of small amplitude as: 5

6 A " 1 2 q'2 /q y (13) If the dissipation is neglected, we can get this relationship from the QG potential vorticity equation (eq. 7), dividing by q y and using the identity v'q' = " # F Edmon et al (1980) states if the dynamics of the flow is Rossby wavelike, with q y positive, the pattern of arrows representing F is a convenient measure of net propagation from one height and latitude to another. As it was obtained from equations (4), the divergence of F represent a zonal force on the mean state comprising the total effect of the eddies. Considering planetary waves of small enough latitudinal and vertical wavelength, we can prove: F = ca (14) Where c is the local group velocity projected onto the meridional plane. Equation 14 shows that F is parallel to the meridional projection of c, under conditions for which c is meaningful. III.2 EP flux and eddy forcing and the treatment of condensation heating Stone and Salustri (1984) generalize the EP flux for QG motion to include largescale eddy forcing of condensation heating. Recalling that EP flux only represents the internal eddy forcing of the zonal mean. Stone and Salustri (1984) following Edmon et al (1980) define a generalized EP flux including humidity effects in spherical coordinates: F = { F (" ),F ( p ) } F (" ) = #acos(")v'u' F ( p ) = acos(") f v'($'+h'l /c p /$ p (14) Where L is latent heath, c p is the specific heat at constant pressure and h is the specific humidity. 6

7 When comparing equations (5) and (14), besides the difference due to different coordinate system used, we note that the effect of the humidity (h) only modifies the vertical component of the EP flux. IV. General applications of EP theory: Wave-mean flow interaction in the Southern Hemisphere. An interesting application of Eliassen-Palm theory can be found in Hartmann Mechoso and Yamazaki (1983). Using EP vectors and divergence, they characterized the wave-mean flow interaction processes in the Southern Hemisphere during the winter of The zonal mean wind averaged for the individual months of June, July and August 1979 is shown in Fig. 1. The presence of a stratospheric jet with a southward displacement is apparent. The time evolution of the zonal-mean wind and temperature at 2 mb are shown in Fig 2. The same displacement and intensification poleward of the jet stream is observed. These changes are accompanied with and increasing of the temperature in the extratropics. 7

8 Figure 1: Latitude-height contour plots of the zonal-mean geostrophic wind averaged over the months of a) June and b) July and c) August. Contours interval 5 m/s. Figure 2: Latitude-time contour plots at 2mb of a) zonal-mean geostrophic wind (contour interval 10 m/s) and b) the zonal mean temperature (contour interval 5K). flux to: With a general description of the wind and temperature profile, the authors use the EP - Detect the regions with a considerable eddy activity. 8

9 - Quantify the acceleration/deceleration of the internal forcing in the mean zonal flow. - Understand the contribution of the different zonal wavenumber in the eddy activity and the role of the transient eddies at different scales. - Establish a link between eddy forcing and baroclinic instability. Using a similar theoretical formulation given by Edmon et al. (1980), The EP divergence and vector was calculated. Eliassen-Palm flux diagrams averaged over the months of June, July and August 1979 are shown in figure 3. There is a strong upward E- P flux in the lower troposphere and a strong convergence of E-P flux and deceleration of the mean flow by eddies. In the stratosphere the driving of the mean flow exhibits a dipole structure with acceleration of the mean flow on the poleward flank of the stratosphere jet and deceleration equatorward of the jet core. The reader can compare the figure 1 and 3 to verify the acceleration/deceleration role of the divergence of EP flux over the zonal mean flow. Much of the accelerations of the zonal wind in the upper troposphere during 1979 winter was provided by zonal wave number 1 (figure 4). A comparison between figure 3 and 4 reveals the importance of the wave number 1 contribution. 9

10 Fig. 3: Eliassen-Palm flux diagrams for the Southern Hemisphere averaged for the months of a) June, b) July and c) August The E-P flux vectors are represented by arrows whose lengths are scaled relative to the arrows above the figures. The lengths of these reference arrows are equivalent to kgm/s. Contours indicate convergence of EP flux. At 1000 mb and above the scale vector reduces by a factor of 10 so that the arrows in the stratosphere may more easily be seen. Fig. 4: As in Fig. 3 except for the wavenumber 1 only. The scale of the arrow corresponds to 2x10 15 kgm/s. Figure 6: EP flux diagram for the time-averaged eddies during the period 14 May-13 September Scaling factor for the vectors is 2x10 15 kgm/s. 10

11 Figure 7: EP flux diagram for the transient components of zonal wavenumber (a)1-3, (b) 4-5, and (c) 6-10 for the period 14 May-13 September The scaling factor is 2x In order to represent the stationary behavior of the atmosphere, it was computed the averaged EP flux and divergence for the period 14 May-13 September 1979 (Fig. 6). It is clear again the important contribution given by wavenumber 1. However observing the small values of acceleration we can conclude that the stationary waves have a small contribution to the driving of the tropospheric flow. Additionally in the stratosphere, the stationary waves provides an acceleration of 2m/s/day. EP diagrams flux for the transient waves (defined as the deviation on a particular day from the time mean for the entire period) separated in groups of zonal wavenumbers are presented in figure 7. We can observe that the waves 1-3 produce almost all of the upward wave flux at 20 km and above and all the eddy-driving of the mean flow in the upper stratosphere. The lowest zonal wavenumbers also contribute a substantial mean deceleration of 4/m/s/day at 65 S. The EP flux diagram for waves 4-5 (fig.7.b) shows upward flux at 850 mb which is larger than the upward flux at 850 mb in waves 1-3. The zonal flow acceleration in the upper stratosphere by waves 4-10 negligible. 11

12 Figure 8: Latitude-height contour plot of the temporal correlation coefficient between wave driving (EP flux divergence) and mean zonal wind acceleration. Contour interval is 0.1. Values larger than 40% are shaded. A zonal cross section of the correlation coefficient between the large scale eddy forcing and the mean zonal wind acceleration is shown in figure 8. The correlation reaches maxima of about 0.6 in the regions of the tropospheric and stratospheric jet streams. Relative minima occur in the lower troposphere, the lower stratosphere, and the polar and equatorial stratosphere. As it was proved previously, the divergence of the EP flux is equal to the meridional flux of potential vorticity. A downgradient flux of potential vorticity in the troposphere occurs in association with baroclinic instability. Even though these results do not show accurately the EP fluxes at the surface, it has been hypothesized that relevant EP convergence is localized below 850 hpa is associated with baroclinically unstable waves. Trenberth (1986) found EP flux divergence above 850 mb (decerating the westerlies above 850), and convergence below 850 mb and accelerating the westerlies below there through the induced Ferrel cell and action of Coriolis torque. 12

13 Conclusion EP flux and transformed Eulerian flow have proved to be a powerful and practical way of viewing the dynamics of eddies on a zonal flow. The EP flux diagnosis does not depend on the restrictive assumptions originally given by Eliassen and Palm (1961). Eddy and mean zonal flow changes in the Southern Hemisphere during the 1979 winter have been described. From these results, we can conclude that EP is particularly useful to study the evolution of the tropospheric and stratospheric jet and their interactions with the transient eddies. References Andrews D. G., J. Holton and C. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press Andrews, D. G. and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos Sci., 33, Edmon, H. J., Jr., B. J. Hoskins and M. E. McIntyre, 1980: Eliassen-Palm crosssections for the troposphere. J. Atmos. Sci., 37, Eliassen, A., N. and E. Palm, 1961: On the tansfer of energy in stationary mountain wave. Geof. Pub., 22, Hartmann, D. L., C. R. Mechoso and K. Yamazaki, 1984: Observations of wavemean flow interactions in the Southern Hemisphere. J. Atmos. Sci., 41, Holton, J. R., 2004: An introduction to Dynamic Meteorology, 4 th Ed. Academic Press. Palmer T., N., 1982: Properties of the Eliassen-Palm flux for planetary scla motions. J. Atmos. Sci., 39,

14 Stone P. H. and G. Salustri, 1984: Generalization of the Quasi-Geostrophic Eliassen-Palm flux to include eddy forcing of condensation heating. J. Atmos. Sci., 41, Trenberth, K., E., 1986: An assessment of the impact of transient eddies on the zonal flow during a blocking episode using localized Eliassen-Palm flux diagnostics, J. Atmos. Sci., 43,