Although the atmospheres of these planets are very deep, the magnitude of the gravitational force greatly exceeds anything else and so the flow is to a first approximation two-dimensional. Under such conditions, any turbulence that is generated in the flow has the counter-intuitive property of being converted into larger and larger scale eddies. This is exactly the opposite of what occurs in more familiar three-dimensional turbulence, where turbulence cascades into smaller and smaller scales, a classic example being the break-up and dispersion of a smoke ring. The process is called the "backwards energy cascade" (Charney, 1971) and has fundamental consequences for planetary atmospheres. The energy spectrum of two-dimensional turbulence is found to follow the classic Kolmogorov scaling E(k) a k~5/3 (Read, 2001). Turbulence in a planetary atmosphere may arise through a number of mechanisms such as static instability, where the atmosphere is unstable to convective overturning, or in cases where there is excessive horizontal or vertical wind shear. In general, it is found that once turbulence is initiated, the associated energy may be dissipated either due to friction, or transfer of energy to the mean flow. It can be shown (Andrews et al., 1987) that many forms of turbulence will persist provided that a quantity known as the Richardson number, Ri, is less than approximately 1, where

and where g is gravitational acceleration; T is the static stability described below in Equation (5.42); T is temperature; and u is the zonal (east-west) wind. In addition to assessing turbulence, the Richardson number provides a very useful measure of the dominant heat-transporting modes on planets as described by Stone (1976). For example, negative values of Ri (hence persistent turbulence) are associated with static instability and free convection, while large positive numbers are associated with Hadley cell circulations. A summary diagram of the different heat transfer modes associated with different Ri on the giant planets is shown in Figure 5.7 (after Stone, 1976).

For the giant planets, assuming geostrophic conditions and barotropic flow, turbulent eddies are predicted to grow and absorb smaller disturbances until their size reaches the Rhines length Lp (Rhines, 1973, 1975)

(where U is mean wind speed and the ft-parameter is defined below in Equation 5.46) when the disturbance is converted to planetary waves and the backwards energy cascade terminates. Thus, eddies have a maximum scale and this is likely to be related to the zonal structure scale (see Section 5.4.1). This property provides one link

Static Instability_

Static Stability

Free Convection_ , Forced Convection inertial Instability Boroclinic Instability Hadley Regime

Small Scale Turbulence ( L S, H ) I Large Scale Tur bulence ( H< L < R )

~-2 0 heating from below

~ IOOO Ri differential heating

Figure 5.7. Stone's (1976) regime diagram of the main characteristic atmospheric motion as a function of the Richardson number (Ri). From Stone (1976). Reprinted by permission of the University of Arizona Press. © Arizona Board of Regents.

between U and L for the giant planets, but another condition is required to uniquely determine both the length and velocity scales. The other condition may be provided by the radius of deformation, which is the length scale at which Coriolis effects become significant (described in Section 5.3.2) and which is found to be similar to the width of the zones on Jupiter (Gierasch and Conrath, 1993), although this conclusion depends on the assumed static stability.

We will now outline some of the instability mechanisms which may, or may not, be responsible for turbulence in the giant planet atmospheres, and which are also summarized for convenience in Table 5.1 (after Stone, 1976).

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