Rossbygravity waves are equatoriallytrapped waves (much like Kelvin waves), meaning that they rapidly decay as their distance increases away from the equator (along as the BruntVaisala frequency does not remain constant). These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius.^{[1]} They always carry energy eastward, but, oddly, their 'crests' and 'troughs' may propagate westward if their periods are long enough. The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H ^{[2]}. Because the Coriolis parameter (f = 2Ωsin(θ) where Ω is the angular velocity of the earth, 7.2921 10^{5} rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta plane” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and "β" is the the variation of the coriolis parameter with latitude, .^{[3]} With the inclusion of this approximation, the primitive equations become (neglecting friction):
These 3 equations can be separated and solved using solutions in the form of zonallypropagating waves, which are analogous to exponential solutions with a dependence on x and t and the inclusion of structure functions that vary in the ydirection:
Once the frequency relation is formulated in terms of ω, the
angular frequency, the problem can be solved with 3 distinct
solutions. These three solutions correspond to the
equatoriallytrapped gravity wave, the equatoriallytrapped Rossby
wave and the mixed Rossbygravity wave (which has some of the
characteristics of the former two) ^{[3]}.
It is important to note that equatorial gravity waves can be either
westward or eastwardpropagating and correspond to n=1 (same as
for the equatoriallytrapped Rossby wave) on a dispersion relation
diagram ("wk" diagram). At n=0 on a dispersion relation diagram,
the mixed Rossbygravity waves can be found where for large,
positive zonal wave numbers (+k), the solution behaves like a
gravity wave; but for large, negative zonal wave numbers (k), the
solution appears to be a Rossby wave (hence the term Rossbygravity
waves).^{[1]} As
mentioned earlier, the group velocity (or energy packet/dispersion)
is always directed toward the east with a maximum for short waves
(gravity waves).^{[1]}
As previously stated, the mixed Rossbygravity waves are equatoriallytrapped waves unless the buoyancy frequency remains constant, introducing an additional vertical wave number to complement the zonal wave number and angular frequency. If this BruntVaisala frequency does not change, then these waves become verticallypropagating solutions.^{[1]} On a typical "m,k" dispersion diagram, the group velocity (energy) would be directed at right angles to the n=0 (mixed Rossbygravity waves) and n=1 (gravity or Rossby waves) curves and would increase in the direction of increasing angular frequency.^{[1]} Typical group velocities for each component are the following: 1 cm/s for gravity waves and 2 mm/s for planetary (Rossby) waves.^{[1]}
These verticallypropagating mixed Rossbygravity waves were first observed in the stratosphere as westwardpropagating mixed waves by M. Yanai.^{[4]} They had the following characteristics: 45 days, horizontal wavenumbers of 4 (4 waves circling the earth, corresponding to wavelengths of 10,000 km), vertical wavelengths of 48 km, and upward group velocity.^{[1]} Similarly, westwardpropagating mixed waves were also found in the Atlantic Ocean by Weisberg et al. (1979) with periods of 31 days, horizontal wavelengths of 1200 km, vertical wavelengths of 1 km, and downward group velocity.^{[1]} Also, the verticallypropagating gravity wave component was found in the stratosphere with periods of 35 hours, horizontal wavelengths of 2400 km, and vertical wavelengths of 5 km.^{[1]}
