For a general vertical coordinate s, the continuity equation is [Kasahara (1974)]:

· Transformation to an appropriate terrain-following coordinate gives the continuous form on which the New Dynamics is based.

· Omitting the local time derivative term gives various acoustically-filtered (or incompressible) forms. The precise form depends on:

¨ what we assume about r in the remaining terms;

¨ what approximations we make in other equations;

¨ whether we insist on energy conservation; and

¨ whether we choose to start from a different form of

the continuity equation!

The resulting equation sets are called – for example – Boussinesq, anelastic, modified anelastic, hybrid anelastic, pseudo-incompressible, or simply given their entrants’ names (e.g. Dutton–Fichtl, Lipps–Hemler, Ogura–Phillips, Wilhelmson–Ogura). See Davies, Staniforth, Wood and Thuburn (2003) QJRMS, in press

2. Putting s = p = pressure, and making approximations that are good if gives [Miller (1974)]:

This is the same as the familiar hydrostatic form, but no hydrostatic assumption is made. A diagnostic relation of the form (valid if ) is applied. [See White (1989). A similar stratagem was used to represent some of the Coriolis and metric terms in the Old Dynamics; see White and Bromley (1995).]

There are no vertically-propagating acoustic modes.

3. Putting s = p = hydrostatic pressure, gives [Laprise (1992)]

Note that no hydrostatic or incompressibility assumption is made: p is simply defined by . It is a nice result that the unapproximated continuity equation takes such a simple form when p is used as vertical coordinate.

See Wood and Staniforth, QJRMS January (C) 2003.

Tabular summary