2.1 Primitive equations model of GFD

The hydrostatic primitive equations can be written in standard notation as follows:

     (2.1)

     (2.2)

     (2.3)

Here ? is density, p is pressure, the subscript “h” denotes the horizontal part of the fields, or operators which depend on and t. The velocity field is three-dimensional: , as well as the operator nabla . These equations are written on the tangent plane to a sphere rotating with angular velocity ?. The coordinate in the meridional direction is y (northward in the geophysical context), the coordinate in the zonal direction (eastward) is x, and z is the vertical coordinate in the direction normal to the tangent plane, i.e. opposite to the effective gravity , which includes a (small) correction due to the centrifugal acceleration (see e.g. Holton, 1979; the asterisk will be omitted from now on). is the normal unit vector in z-direction. The Coriolis parameter f entering the horizontal momentum equation is equal to , if the f-plane approximation is used, and to , if the ?-plane approximation is used. Here ? is the contact point latitude which is considered as fixed or variable in the first and second approximations, respectively. Equations (2.1)–(2.3) are non-dissipative. They express momentum and mass conservation of the fluid. Molecular (or turbulent) viscosity and diffusivity may be easily introduced in the r.h.s. of the momentum and mass equations. It should be noted, however that the Reynolds numbers for free (i.e. far from boundary layers) atmospheric and oceanic motions of synoptic scale are extremely high (Pedlosky, 1982; Holton, 1979). This is why, being interested in what follows mostly in the intrinsic dynamics of the system, and not in the forced-dissipative one, we will almost never use the dissipative form of equations.

We will work for simplicity with a rigid lid and flat bottom boundary conditions for equations (2.1)–(2.3):

     (2.4)

which are of frequent use in GFD. Other upper boundary conditions, like free surface boundary, or prescribed form of the pressure (geopotential) function at a given vertical level (cf. e.g. Holton, 1979) may be used as well. In (2.3) the continuity equation

     (2.5)

is replaced by the incompressibility condition and mass conservation by each fluid parcel. Such (Boussinesq) approximation is totally justified in the oceanic context taking into account the negligible compressibility of water. Further simplification may be made in this case, using the fact that in the ocean

     (2.6)

Here is a constant, is a background oceanic stratification, and represents the spatio-temporal density variations. Then, the variable part of the density may be neglected in the horizontal momentum equations, and we get

     (2.7)

where the geopotential is introduced. Remarkably, the same equations, up to the change of sign in the hydrostatic equation, arise in the atmospheric context if the so-called pseudo-height Z, which was first introduced by Hoskins and Bretherton (1972), is used as the vertical coordinate:

     (2.8)

Pseudo-height is a modified pressure coordinate (see Holton, 1979, for pressure coordinates in the atmosphere). The constants are given by , and . In these definitions, and are (constant) surface pressure and density, and are the specific heats of air at constant volume and constant pressure respectively. The variable ? has a different physical meaning in the atmosphere: it becomes up to a sign the so-called potential temperature ? which is directly related to the entropy density of air parcels (cf. Holton, 1979). The pseudo-height Z and the physical height z are related as follows: .

2.2 Vertical averaging of the primitive equations and multi-layer RSW models

The derivation of the RSW model by vertical averaging was proposed in the classical works of Jeffreys (1925) in the linear approximation, and Obukhov (1949) in the nonlinear case. We follow below the method of Obukhov by generalizing it to the multi-layer case. To lose no generality we will consider a general case of equations (2.1), together with the continuity equation (2.5), and the hydrostatic balance equation (2.2). Equations (2.1), (2.5) may be rewritten as evolution equations of the horizontal momentum density:

     (2.9)

     (2.10)

As a preliminary step we integrate these equations in the vertical direction between a pair of material surfaces . By definition

     (2.11)

By using (2.11) and the formula

     (2.12)

which is valid for any function F, we get from (2.9)

     (2.13)

Analogously, from (2.10)...