# Waves in the giant planet atmospheres

We saw in Section 5.3.1 that turbulent motion arises when parcels that are deflected from their original positions feel forces that pull them further from their equilibrium positions. In some circumstances, however, displaced parcels may feel forces that force them to return to their equilibrium positions. Such forces give rise to a wide range of wave motions that are clearly observed in all planetary atmospheres, including the giant planets.

The two main properties of planetary atmospheres that may lead to waves (ignoring sound waves) are vertical stratification and the rotation of the atmosphere (Andrews et al., 1987). Internal gravity waves result directly from stable stratification, while larger scale inertia-gravity waves result from a combination of stratification and Coriolis effects. Planetary, or Rossby waves, result from the polar gradient in planetary vorticity (the ^-effect) and the conservation of potential vorticity. To deal with the whole spectrum of waves fully is beyond the scope of this book and the reader is referred to books such as Andrews (2000), Andrews et al. (1987), Holton (1992), or Houghton (1986). Historically, the equations of motion have first been linearized by separating parameters such as pressure and wind into their mean and perturbation components and then looking for waves with small enough amplitude that the linearizing approximation still holds. This linear wave theory is explored in detail by Andrews et al. (1987). A wide spectrum of waves is predicted under different assumed conditions, including waves that are free to travel in all directions and others that are trapped in certain latitude bands. A particularly important example of the latter for study of giant planet dynamics, are the so-called equatorially trapped waves

(Allison, 1990; Andrews et al., 1987). More recently, advances in computation have made it possible to search for wave motion in the full nonlinear primitive equations and thus look for waves with large amplitudes (e.g., Dowling et al., 1998). However, in this section we will summarize the main features of linear waves that are predicted to occur under different conditions in giant planet atmospheres.

### Gravity waves and radius of deformation

Consider a parcel of air at a certain height and at the same temperature as the surroundings T0, which is moved vertically and adiabatically from its equilibrium position by a small distance Sz. If no condensation occurs then the new temperature of the parcel will be T1 = T0 — rd Sz where rd is the dry adiabatic lapse rate. The temperature of the surrounding atmosphere, however, will be Te1 = T0 + (dT/dz) Sz, where dT /dz is the background lapse rate, and if the two are different the parcel will feel a buoyancy force of

where p1 is the density of the parcel; pe1 is the density of environmental air at the same altitude; and v is the parcel's volume. Hence, the parcel's acceleration is given by