Rossby radius

Ross Bannister, July 2004

Kalnay gives the definition of large and small horizontal and vertical scales. The linearized shallow-water is,

where is the Coriolis parameter, is streamfunction, is the acceleration due to gravity, is the mean depth of the fluid and is the geopotential. The geostrophic relation is,

Making a perturbation in geopotential only, , at some initial time gives rise to the perturbation in potential vorticity, ,

After a period of time, geostrophic adjustment causes the unbalanced part of this initial perturbation to dissipate. The part remaining is in geostrophic balance, Eq. (2). Potential vorticity will however be conserved,

and substituting from Eq. (2) gives,

Let us assume a form of the resulting geostrophically balanced geopotential,

where are wavenumbers, are, respectively, longitude and latitude distances, and is the radius of the Earth. The total wavenumber is, : . This is a dimensionless integer, as used in the spectral expansion on the sphere. Substituting Eq. (6) into (5) yields,

Equating Eqs. (3) and (7) allows the geostrophic geopotential to be related to the initial geopotential,

where the Rossby radius of deformation is,

In the case of (small horizontal scales and large vertical scales), the geostrophically adjusted geopotential will be much smaller than the initial geopotential. In the other extreme, (large horizontal scales small vertical scales), the geostrophically adjusted geopotential will be equal to the initial geopotential.

We would like to derive a partition between these regimes. The partition is,

In Eq. (9), let the height scale, be related to the vertical mode index by , where is the vertical height of the model, and is mode index. Putting this, Eq. (10) and Eq. (9) together results in a expression for the partition , between the two regimes (given a ),