Vertical Mixingeddy Mixing Coefficients

In Chapter 2 we outlined the bulk composition of the outer gaseous envelopes of the giant planets. These compositions refer to the "deep" atmosphere, although for Jupiter and Saturn this refers to pressures only up to 10 bar to 20 bar, and up to approximately 100 bar for Uranus and Neptune. In the upper, observable parts of the atmosphere, the composition of certain gas species vary as a function of height due to processes such as photochemistry and condensation. As we have seen in Section 4.1.3 for ortho-hydrogen/para-hydrogen the composition is also a function of the rate at which air is uplifted from the warm interior since if the upwelling is rapid enough, non-equilibrium "quenched" molecules may be present. Hence, the rate of vertical motion has a major effect on the vertical profiles of composition, and also cloud structure.

Air parcels may be transported vertically in atmospheres by three main mechanisms: (1) convection; (2) atmospheric waves; and (3) turbulence. To understand how these processes affect the measured abundance profiles, consider the continuity equation in the vertical direction (Atreya, 1986; Yung and DeMore, 1999) for a certain gas species which has a number density ni f + £ = (4-)

where fa is the vertical flux of molecules; Pi is the chemical production rate; and Li is the chemical loss rate. If the mean vertical wind speed w is known at a certain location, then the vertical flux of molecules is simply fa = niw. For the organized belt/zone circulation of Jupiter, where zones are interpreted as regions of general upwelling and belts are regions of general subsidence, the mean vertical wind may sometimes be calculated from departures of the temperature from the radiative equilibrium temperature (Gierasch et al., 1986; Conrath et al., 1998). However, even with these general flows, there are superimposed smaller convective events, such as isolated convective plumes observed in Jupiter's North Equatorial Belt, and a good deal of turbulence that tends to mix the air vertically, even in regions which from dynamical models (Chapter 5) appear to be regions of general uplift and subsidence only. Hence, to understand the effects of these processes on the vertical abundance profiles, we need a more general way to parameterize the vertical flux of molecules.

According to Prandtl's mixing length theory (Holton, 1992), a parcel of air displaced vertically will carry the mean abundances of its original level for a characteristic distance l' analogous to the mean free path in molecular diffusion. This displacement will create a turbulent fluctuation in the composition of the new level, whose magnitude will depend on l ' and on the vertical gradient of the mean composition. Thus, the process is very similar to molecular diffusion and may be modeled in the same way if we define an eddy-mixing coefficient K, analogous to the molecular diffusion coefficient D. Using such a model the vertical flux of species i may be calculated as being due both to molecular and eddy diffusion via an equation very similar to Equation (3.17), presented on p. 63

where Di is the molecular diffusion coefficient of the i th component; T is temperature; Ht and Ha are the pressure scale heights of the i th species and bulk atmosphere, respectively (Hi = RT/mtg, Ha = RT/mg); and K is the eddy diffusion coefficient determined via observation. When molecular diffusion dominates, each species tends to its own profile with its own scale height. When eddy diffusion dominates, the gas is well-mixed and has a single bulk scale height. The level where molecular diffusion becomes dominant is called the homopause.

In the stratospheres of the outer planets it is thought that the principal mechanism for eddy mixing is the dissipation and break-up of vertically propagating gravity waves. The kinetic energy of non-dissipative waves should remain constant and this is, by simple analysis, proportional to pa^, where p is the density and av is the amplitude of the velocity. Thus the amplitude of these waves should be roughly proportional to p-0'5 or equivalently n-0'5, where n is the number density (Andrews et al., 1987). Hence, the amplitude of these waves grows rapidly with height and eventually becomes so high that the wave is unstable to either convective instability or shear instability. This leads to wave "breaking" and turbulent mixing of the air. Such waves are generated in the troposphere due to convective turbulence, or for the terrestrial planets from deflection of air over surface features. Small-amplitude waves clearly travel higher into the stratosphere before breaking than high-amplitude waves and thus waves break at a range of altitudes in the stratosphere, providing a source of eddy mixing throughout the region. Gravity waves will be discussed in more detail in Chapter 5. By mixing length theory (Holton, 1992) the eddy-mixing coefficient should be proportional to the mean mixing length and thus we expect that K is proportional to the amplitude of breaking gravity waves and thus K a n-0'5. In the troposphere,

Table 4.2. Measured estimates of the eddy-mixing coefficient in the giant planet atmospheres.


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