where on the right-hand side we have substituted the expression for density in terms of temperature and pressure, assuming that the gas is ideal. Substituting for T1 and Te1 in terms of lapse rates and displacements, and rearranging, we find d2(Sz) g (dT )

In a statically stable atmosphere, N| is positive and thus Equation (5.50) represents simple harmonic oscillation of the parcel about its equilibrium position. The oscillation angular frequency, NB, is known as the buoyancy, or Brunt-Vâisâlâ frequency. If NB is negative, then the solution of Equation (5.50) is exponential in z and thus the atmosphere is unstable. It can be shown (Andrews, 2000) that Equation (5.51) may be further simplified to a form incorporating potential temperature and static stability

If the horizontal direction is also considered, it is easily shown that these gravity waves propagate horizontally as well as vertically (e.g., Andrews, 2000; Houghton, 1986). Vertical perturbations in the lower troposphere, such as those generated by convection, may initiate such oscillations in the stable part of the atmosphere above the radiative-convective boundary. The resulting waves propagate both horizontally and vertically, and their amplitude is found to increase exponentially with height since their energy flux, which is conserved, is proportional to pv2, where p is the density and v0 is the velocity amplitude of the wave. Small-scale, high-frequency waves do not feel the effects of the Coriolis force and are called internal gravity waves. Larger scale, lower frequency waves, however, are affected by the planet's rotation and are called inertia-gravity waves. The length scale at which Coriolis effects become important is defined by the radius of deformation, a = c/f (sometimes called the Rossby radius), where c is the phase speed of the wave and f is the Coriolis parameter (Gill, 1982). It should be noted that there is not a single radius of deformation, but in fact an individual deformation radius for every particular class and mode of wave since the phase speed c depends on wavenumber, and also on vertical stratification through the Brunt-Vaisala frequency. For atmospheres where the geostrophic approximation applies it is found that the energy of shortwavelength disturbances is mainly in the form of kinetic energy, while the energy of longwave-length disturbances is mainly in the form of potential energy. Both are equal at the scale of the radius of deformation. This scale is also sometimes called the "synoptic scale''. The radius of deformation may also be thought of as the "preferred" horizontal length of disturbances in the sense that waves arising from baroclinic instabilities (Section 5.3.1) grow most rapidly (and are thus most visible) when their horizontal dimension is of this radius. For a continuously stratified atmosphere, a useful expression for the mean radius of deformation (Gierasch and Conrath, 1993) is a = NBH/f, where H is the scale height of the atmosphere.

Gravity waves are thought to be the dominant source of vertical eddy mixing in the stratospheres of the giant planets. At very high altitudes the waves 'break' as described in Chapter 4, much as surface water waves on the lakes and seas break when they reach beaches. At altitudes where the gravity waves break, the momentum of the wave is transferred to the momentum of the mean flow.

Kelvin waves

Kelvin waves are a special class of gravity waves that are found to move eastwards relative to the mean zonal flow, but only at latitudes close to the equator. Air moving in the atmospheres of rotating planets is affected by the Coriolis force, which for prograde-spinning planets deflects the air to the right in the northern hemisphere and to the left in the southern hemisphere. Consider a pressure disturbance moving eastwards relative to the mean zonal flow, which is symmetric about the equator. The Coriolis force is zero at the equator, but increases with latitude. Hence, the part of the disturbance to the north of the equator is deflected south, while that to the south of the equator is deflected north, which by conservation of mass increases the pressure at the equator. Eventually, the equatorial pressure rises to a level sufficient to balance the Coriolis ''compression'' and the disturbance spreads latitudinally again before the cycle repeats. These equatorially trapped Kelvin waves are observed in the Earth's atmosphere and may also be important at equatorial latitudes in the giant planets. This description is very simplistic and more detailed treatments are summarized by Allison (1990) and Andrews et al. (1987). The amplitude of these waves is calculated to diminish away from the equator with an exponential decay length Leq, known as the equatorial deformation radius where c is the phase speed. A key diagnostic feature of equatorially trapped Kelvin waves is that the meridional velocity across the equator is zero.

Rossby or planetary waves

We saw in Section 5.2.1 that potential vorticity is conserved by an atmosphere where friction and diabatic heating are negligible. Consider a parcel of air of height h at some latitude 0 in the northern hemisphere, which initially has zero relative vorticity C. Suppose the parcel is displaced northwards. From Section 5.2.1 we know that its potential vorticity (C + f )/h must be conserved. However, the Coriolis parameter f increases as the parcel moves north and thus the absolute vorticity C must decrease in order to compensate. Hence, the parcel gains negative (clockwise) relative vorticity. Similarly a parcel displaced southwards gains positive (anti-clockwise) relative vor-ticity. This conservation mechanism gives rise to Rossby, or planetary, waves at latitudes where the air stream is moving in the eastward direction. Air deflected to the south will gain positive relative vorticity and thus turn towards the left and then northwards. After crossing the starting latitude, the air will gain negative relative vorticity and thus turn towards the right and then southwards and so on, setting up a stable wave. To outline how we may show this analytically, consider the momentum equations in pressure coordinates for frictionless flow, where vertical motion is neglected (Houghton, 1986):

dt dx dy J dx d d d\ dz dt+ udX + V d~y) V-fu = g¥y ■ (5'55)

Differentiating Equation (5.55) with respect to x and differentiating Equation (5.54) with respect to y and then subtracting gives another form of the vorticity equation

We now substitute the following mean and perturbation values: u = u + u',

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