Stratospheric Dynamics and Coupling with Troposphere and Mesosphere
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1 WDS'13 Proceedings of Contributed Papers, Part III, 6 66, 13. ISBN MATFYZPRESS Stratospheric Dynamics and Coupling with Troposphere and Mesosphere P. Šácha Charles University in Prague, Faculty of Mathematics and Physics, Prague, Cech Republic. Abstract. The large-scale dynamics of the stratosphere and its interaction and coupling with troposphere are reviewed. Geophysical fluid dynamics methods, quantities suitable for the description of the middle atmosphere circulation and general patterns of the stratospheric circulation are described. Emphasis is placed on the important effects of two-way interaction between waves and mean flow. In the context of simple models experiments, possible dynamical mechanisms of coupling with troposphere are discussed. Finally, future research plans are introduced (preparing datasets, model experiments, coupling with mesosphere). Introduction The atmosphere is conventionally divided into regions according to a vertical temperature gradient. In their fundamental textbook Andrews et al. [1987] have defined the term middle atmosphere as a region from the tropopause to the homopause (1 11 km) containing the stratosphere, mesosphere and lower thermosphere, where the same physical processes govern the dynamics and similar approximations are valid. In this part of atmosphere we can neglect effects of latent heating on one hand and molecular diffusion and electromagnetic forces on the other hand. The understanding of the processes taking place in the middle atmosphere is essential for studying the climate system and could improve the long-range forecast skills [Hardiman and Haynes, 8]. There is also growing evidence that its lowest part, the stratosphere, has the potential to affect significantly conditions at the surface [Haynes, 5]. In the current debate about possible influence of this year s stratospheric sudden warming on the observed negative anomaly in the sea level pressure field over the North Atlantic Ocean plays an important role the question of causality connected with a propagation of information. There are many ways how the stratosphere could affect tropospheric conditions, but in this review our focus will be exclusively on the vertical coupling of dynamics between the stratosphere and the troposphere and consequently on possible dynamical downward links, which can be, for example, responsible for observed signals of solar variability in the troposphere [Haynes, 5]. In the middle atmosphere, the evolution of a system is governed by a three-way interaction between dynamics, physics (microphysics, phase changes, radiative transfer) and chemical processes. By the term dynamics we understand physical processes described by the momentum and thermodynamic equations. Further in this text the emphasis is on processes connected with large-scale flows that fall within the scope of geophysical fluid dynamics. A comprehensive review of the effects of small-scale processes (primarily inertiogravity waves) on the large-scale middle atmospheric dynamics was given by Fritts and Alexander [3]. The review is structured as follows. The first section introduces the governing equations, their simplifications often used in the middle atmosphere, linear perturbation theory and some important and illustrative results of it. Then in the following section different types of averaging are demonstrated with their usage restrictions and advantages. Next the important findings of the two-way interaction theory between waves and mean flow are discussed with effects on the circulation in the middle atmosphere and in troposphere. The last section introduces theoretical ways of downward propagation of the information in the confrontation with observations and numerical modeling studies and subsequently new hypothesis and future plans are introduced. Governing equations and analytical solution Although the Navier Stokes equations are an approximation to some order, they are far more complicated than desirable for middle atmospheric phenomena considered here. As stated, for example, by Andrews et al. [1987], the most general set of equations used in middle atmosphere dynamics are the so called primitive equations, which we get after simplifications of the equations of motion for a gas in a rotating frame. They consist from two momentum balance equations one in onal and one in meridional direction, hydrostatic balance and mass continuity equation and thermodynamical relation between diabatic heating and material rate of change of potential temperature. Derivation, summary of approximations made and discussion of consistency of the hydrostatic primitive equations can be found in White et al. [5]. 6
2 Despite the simplifications, primitive equations are still a complicated set and further approximations are needed to have a chance to obtain some analytical solutions. There are many ways how to simplify the primitive equations. For example, by making suitable geometrical (beta-plane) and dynamical (small Ekman, Rossby and temporal Rossby number) approximations we may obtain, following e.g. Holton (199) or Cushman Roisin (1994), the quasi-geostrophic potential vorticity equation for nonlinear motions in a continuously stratified fluid on a beta-plane which should fully capture the large-scale, slower motions, at least in the extratropical region [Andrews et al., 1987]. Another method, which could help us to guide the governing equations to some analytically resolvable form, is the linear wave theory. This theory helps us to linearie the equations (e.g. neglecting the advective processes). It starts with decomposing of the dependent variables into a mean and wave (perturbation) part. Thus, some kind of averaging which should help us to determine the mean part is needed. Typically, space (Eulerian approach) and time (Lagrangian approach) averaging are used in fluid dynamics and will be discussed in the next section. Particularly, in the middle atmosphere their combination named generalied Lagrangian mean (GLM, for details see Craik [1988]) is often used. The GLM approach is quite technical in nature and we will not discuss it here. After above discussed decomposition of the variables, next step is lineariing the equations in perturbation about the basic state. The rules for lineariation can be found e.g. in Horák and Raidl [7]. Then, after assuming and substituting the wave like solution of the perturbation we may obtain extra information about the perturbation. Depending on approximations and assumptions made during modifying the original set of equations we may get various dispersion relations or structure equations for some wave pattern and thus a lot of different wave types (atmospheric thermal tides, planetary waves, gravity waves, see Andrews et al. [1987]). Now, by applying the linear wave theory on the quasi-geostrophic beta-plane potential vorticity equation we can show an application of the theory and at the same time obtain a very important and illustrative result. Assuming the flow is frictionless and adiabatic, then the quasi-geostrophic potential vorticity is conserved following the horiontal geostrophic flow under the quasi-geostrophic conditions. The right hand side of the equation is null and so we have the linearied quasi-geostrophic beta-plane potential vorticity equation in the form: + u q + v q y =. (1) t x 1 f q = ψ xx + ψ yy + N ρ ψ. () ρ Where u denotes the onal mean flow, N the buoyancy frequency (we assume both to be constant further), q is the disturbance quasi-geostrophic potential vorticity, v is the northward wind velocity disturbance and q y is the basic northward quasi-geostrophic potential vorticity gradient that in our case is reduced to equal beta. After substituting a wave-like solution we obtain a dispersion relation for the vertical wave number m: N β 1 m = ( k l ) +. (3) f u c 4H Here c is the phase speed, k and l the horiontal wave numbers and H is the scale height. Due to the form of the im assumed wave-like solution, where m is situated in a complex antilogarithm e, the wave is evanescent if m is imaginary. Thus the wave and the information may propagate upward only if m is real. This fact gives us the famous Charney Drain condition for vertical propagation which is conventionally written in the form: < u c <. (4) u critical Thus the wave speed must be westward relative to the mean flow but not too much westward [Charney and Drain, 1961]. The critical velocity depends on the properties of surroundings and is also an inverse function of the horiontal wave number. So, if we consider stationary waves (approximately Rossby waves), they could propagate vertically only in westerly and not too strong winds. The longer the waves are in horiontal the stronger eastward mean onal wind is critical for them. Two-way interaction As noted above, there are two main possibilities how to split a motion into a mean part and an oscillatory part. Different theories are connected with both of them. Although the circulation in the middle atmosphere shows great variability with altitude, latitude and longitude, according to climatology of wind, the longitudinal 63
3 variations could be at leading order neglected. So, the natural example of an Eulerian mean in the middle atmosphere is onal mean. To explain the longitudinally averaged state we have to take into account systematic effects of the deviations (waves). The theory of wave mean-flow interaction is an old and highly developed theory. It still provides a useful quantitative framework for understanding the circulation of the middle atmosphere [Haynes, 5]. Nevertheless it has significant limitations (e.g. limitation to small amplitudes and failing for significantly nononal flows) and is therefore advantageous to combine it with the Lagrangian approach. Here the averages are being taken following fluid parcels. It allows larger (but finite) amplitudes of perturbation and may give more detailed information about trajectories and so it is naturally used in transport studies. In this review in all equations onal averaging is utilied and a mean quantity is indicated by an overbar. One of the most important results of the theory of wave mean-flow interaction is Charney Drain nonacceleration theorem. The noninteraction theorem itself has a lot of versions, involving different wave types and different assumptions; a good summary of it is given in Boyd [1976]. In the most illustrative version, it is shown that in the absence of friction, heating and with an assumption of an idealied ambient vertical shear flow small amplitude stationary Rossby wave cannot alter the mean flow. Such a derivation can be found, for example, in Pedlosky [1979]. We will not reproduce it here, but we will emphasie some of its parts and consequences. The springboards for Charney Drain theorem derivation are potential vorticity equation and meridional circulation stream function that are derived from onally averaged and partitioned quasi-geostrophic equations. There are two terms of particular importance in them. The Reynolds stress gradient (or momentum flux caused by waves) and eddy heat flux gradient. For steady waves, the relation between them is: y 1 ρ ( v u ) = v θ ρ S. (5) Using (5) we can write the equation for meridional circulation stream function in the form with linear elliptic operator: L ( χ ) = L θ v ρ S. (6) After solving it we obtain the equality between the components of the meridional circulation and eddy fluxes. So, in the absence of heating and friction and for stationary linear waves, the heat and momentum flux produced by the waves is exactly canceled out by the meridional circulation and the causality is such that in the presence of wave fluxes meridional circulation is excited to act against the effect of these fluxes. Notice that such a cancellation is not possible if the fluxes are changing through the time and hence the restriction to steady waves. So the wave fluxes rather than influencing the onal mean flow directly drive the meridional circulation [Holton, 199]. The physical meaning of this circulation could be such that it immediately maintains geostrophic and hydrostatic balance potentially distorted by the fluxes. But in the real atmosphere friction, diabatic processes and nonstationarity enter this steady state balance, the meridional circulation cannot fully cancel the eddy fluxes and even the relation (5) does not hold any more. Then the onal flow is not longer only a catalyst responsible for the existence of the wave, but is also directly affected by the wave field. This is confirmed by the observations, e.g. when onal winds and temperatures are maintained steady against dissipation over a long period due to the energy source from waves. As firstly noted Dickinson [1969], there are two important mechanisms how the wave can transfer its energy, absorption at the singular lines where c = u and damping and breaking of waves. It is physically sensible to incorporate the two influences (driving meridional circulation and altering mean flow) of the waves into one. This is the idea of the definition of residual mean circulation leading to introduction of Eliassen Palm flux and obtaining the tranformed Eulerian-mean set of equations. The residual mean meridional circulation is defined so that it stays ero if the nonacceleration theorem holds. The Eliassen Palm flux divergence has the dimension of a force and in the quasi-geostrophic beta-plane case it has a physical meaning as a measure of invalidity of relation (5). So it is null under the familiar assumptions of Charney Drain theorem. Under more general conditions we get a so called generalied Eliassen Palm theorem which has the form [Andrews et al., 1987]: A 3 + F = D + O( α ). (7) t Where A is the so called wave-activity density, F is Eliassen Palm flux and D contains the frictional and 3 O α term represents the nonlinear wave effects and vanishes for purely linear waves. The diabatic effects. The ( ) 64
4 interesting fact is that the theorem takes a form of a conservation law for wave properties when the terms on the right hand side of (7) are ero. Set of the transformed Eulerian-mean equations could be find e.g. in Haynes [5]. They describe a coupled response of the wind and temperature fields on the one hand and the residual meridional circulation on the other to applied wave force and heating (that will be questioned in the next section). Haynes [5] in his paper further shows that this set of equations could be solved for the vertical velocity and for the steady state the analytical solution could be found in the form: 1 cosφ w = Ω ( ) ρ, ρ cosφ φ sinφ ( ) G( φ ) d. (8) It implies that in the steady state limit, the vertical velocity at a given level is determined only by the wave forcing G above. This downward control principle has nevertheless the limitation that it puts no constraints on what caused the wave forces (cause and effect question), but it still provides a useful insight into meridional mean circulation. For example, it shows that forces exerted high in the middle atmosphere (upper stratosphere or mesosphere) may influence the vertical velocity and hence temperatures in lower stratosphere. Because the density factor in the integral works against this, it demands the wave forces in the lower and middle stratosphere to be very small, as the may be, for example, in Southern Hemisphere mid-winter [Haynes, 5]. Cause and effect question Longitudinal wave force per unit mass G in the transformed Eulerian-mean equations represents the effect of waves on the mean flow, but is simultaneously a function of the mean flow. In the context of Rossby waves it has to represent propagation, breaking and vortex interaction. However, all these processes are implicitly dependent also on the mean flow properties. Therefore G has to be very complicated (and, as yet, undetermined) function of the mean flow [Haynes, 5]. The distinguishing of what is the cause and the effect from observations is troublesome. To settle the question of causality we need to focus on the communication of information. Many recent papers have presented modelling studies, where the downward propagation of information was studied. Hardiman and Haynes [8] has used one-dimensional mechanistic and three-dimensional general circulation model to study dynamical connections over larger vertical distances. They examined the effect of perturbations to the middle atmosphere on the lower stratosphere because, as they noted, effects of the lower stratosphere on the troposphere are then unsurprising due to the fact that potential vorticity inversion is nonlocal (see the form of eq. (6) elliptic operator). In all the previously mentioned modelling studies the view is taken that if a perturbation at some level leads to a change below that level, then downward propagation of information has occurred. In their paper, Hardiman and Haynes [8] argue that some observed height time patterns should clearly not be interpreted as downward propagation of information but shows that such a propagation is possible thanks to the nonlinear processes and dynamical sensitivity of the stratosphere. 1 In the onally symmetric case the response to upper level perturbation decays naturally as ρ. Nevertheless on the basis of nonlinear nature of the wave-mean flow interaction the stratospheric circulation may exhibit very chaotic behaviour. In such a case the sensitivity to initial condition can enhance the lower level response and the downward propagation of information over larger vertical distances is then possible. Conclusion and discussion A brief review of the most important findings of the interaction theory between waves and mean flow was given. The text is logically separated to cover the chronology of the research of middle atmosphere dynamics. Stratosphere plays an evident and active role in tropospheric variations but the dynamical mechanisms are often in a level of hypothesis. A key outstanding dynamical problem is the determination and prediction of changes in the wave force G [Haynes, 5]. It contains not only the problems connected with nonlinear behaviour but also the inclusion of the contribution of the small scale waves (inertia-gravity waves). This contribution may be especially important in the mesosphere. The author therefore intends to focus on the combination of analyses of GPS Radio Occultation data, which have very high vertical resolution, with experiments on numerical models encompassing the mesosphere to examine the possible coupling of the mesosphere with the stratosphere and subsequently with the troposphere. Acknowledgments. The presented work was supported by the Charles University Grant Agency under Contract
5 References Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics, 489 pp., Elsevier, New York. Boyd, J. P., 1976: The noninteraction of waves with the onally averaged flow on a spherical earth and the interrelationships of eddy fluxes of energy, heat and momentum, J. Atmos. Sci., 33, Charney, J. G. and Drain, P. G., 1961: Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res., 66, Craik, Alex D. D., 1988: Wave interactions and fluid flows. Cambridge University Press. Cushman-Roisin, B., 1994: Introduction to geophysical fluid dynamics Prentice-Hall. Englewood Cliffs, NJ 763. Dickinson, R. E., 1969: Theory of planetary wave-onal flow interaction. J. Atmos. Sci., 6, Fritts D. C., Alexander M. J., 3: Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41: doi:1.19/1rg16. Hardiman, Steven C., and Peter H. Haynes, 8: Dynamical sensitivity of the stratospheric circulation and downward influence of upper level perturbations. J. Geophys. Res.: Atmospheres (1984 1) 113.D3. Haynes, Peter, 5: Stratospheric dynamics. Annu. Rev. Fluid Mech. 37: Holton, J. R., 1974: Forcing of mean flows by stationary waves. J. Atmos. Sci.,31, Holton, 199: An introduction to dynamic meteorology, third edition. Academic press. Horák J., Raidl A., 7: Hydrodynamická stabilita atmosféry a nelineární problémy geofyikální hydrodynamiky, V Prae, 38 s. Pedlosky J., 1979: Geophysical Fluid Dynamics. Springer. 66
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