Planetary atmospheres, oceans and interiors are dominantly in a state of hydrostatic equilibrium, with most dynamic processes being perturbations of that state. With this in mind, in this chapter, we will separate the elastic and fluid governing equations into static balance and perturbation equations. Static reference-state variables are indicated by a subscript r while perturbations have a prime. Note that reference-state variables often are functions of position – particularly spherical radius or elevation. We begin in § 7.1 with a presentation of the equations governing a static state. Next, in § 7.2 we briefly discuss the structures of the atmosphere, oceans and mantle. Then in § 7.3 we develop a set of equations quantifying small departures from the static state.

Static State

The static state of an elastic solid is simply the reference state with u = 0 and while the static state of a fluid body is v = 0 and again. The equation of conservation of mass is identically satisfied in the static state for both elastic and fluid bodies, while the momentum equation for both is simply the hydrostatic balance

where and z is the upward coordinate.

Thermodynamics tells us that the density is a function of the pressure, temperature and composition. We will ignore compositional effects for the time being and consider ρr = f (pr, T). A necessary condition for a fluid body to be in a static state is obtained by taking the curl of the momentum equation:

Isothermal and isobaric surfaces must coincide in a static (barotropic) state. This is possible in a non-rotating, self-gravitating body, but if the body is rotating and density depends on temperature, a static state is not possible; gravity requires elliptic isobars, while thermal conduction still produces spherical isotherms. This situation occurs, for example, in stellar interiors; they are in continual Eddington-Sweet motion.

With the functional relation between ρr and pr known, it is fairly straightforward to determine the static state of the atmosphere, oceans and Earth's interior. These structures are presented and discussed in the following section.