is defined by
where U is a characteristic velocity scale, f the Coriolis parameter, and L a characteristic length scale. If the Rossby number is large, then the effect of the Earth's rotation on the phenomenon in question can be neglected. This is also called the Kibel number.
In a homogeneous layer of fluid the
barotropic Rossby radius 
is
given by
where c is the gravity wave propagation velocity
, g the
gravitational acceleration,
H the water depth, and
f the Coriolis parameter.
In the deep ocean where H is 4 or 5 km, the baroclinic radius is
around 2000 km, but on the continental shelves with depths
closer to 50 to 100 m it is around 200 km.
In a stratified fluid the baroclinic Rossby radius is similarly computed, except that c is now the wave speed of the nth baroclinic mode as would be found in a normal mode decomposition of the system. The baroclinic radius is a natural scale in the ocean associated with boundary phenomena such as boundary currents, fronts, and eddies. The first mode baroclinic radius is typically around 10-30 km in the ocean. See Gill (1982).
-effect of latitudinal variation of
the local vertical component of the earth's angular
rotation vector, i.e. the Coriolis force. In the atmosphere
they are easily observed as the large-scale meanders of the
mid-latitude jet stream that are responsible for prevailing
seasonal (via blocking) and
day-to-day weather patterns. They are more difficult to
detect in the ocean as their sea surface height signature
is on the order of 10 cm, their propagation speeds of order
10 cm/s, and their wavelengths hundreds to thousands of
kilometers.
Rossby waves in the ocean are responsible for establishing the westward intensification of circulation gyres, the Gulf Stream being one example of this. They are also the dynamic mechanism for the transient adjustment of the ocean to changes in large-scale atmospheric forcing, e.g. information is transmitted from the tropical oceans to mid- and high-latitudes via Rossby waves acting in concert with coastal trapped waves. They are generated by wind and buoyancy forcing at the eastern boundaries and over the ocean interior. They are also known to be generated by perturbations along the eastern boundaries caused by coastal trapped waves originating at low latitudes. They subsequently freely propagate away from their source regions.
Standard theory derives the properties of freely propagating Rossby waves from the linearized equations of motion for large-scale, low-frequency motion about a state of rest, which yields an equation for normal modes. These normal modes can be found by specifying surface and bottom boundary conditions and solving an eigenvalue problem that depends only on the local stratification. There are an infinite number of wave modes ordered by decreasing phase speed, which are westward for all modes. Solutions for low frequencies and long wavelengths are zonally nondispersive, i.e. the phase speed is independent of the wavelength.
The lowest mode is the barotropic mode. It is uniform vertically and propagates across an ocean basin in about a week. The next gravest, or first baroclinic, mode is surface intensified, depends strongly on the stratification profile, has a velocity profile that changes sign at the depth of the thermocline, and takes months to cross the same basin as the first mode does in a week. The surface height variations of this mode are mirrored as thermocline depth variations of the opposite sign, which are also about three orders of magnitude larger, i.e. a 5 cm surface elevation variation would correspond to a 50 m depression in the thermocline. See Platzman (1968), Dickinson (1978) and Kuo (1973).