View source for Physics:Rossby-gravity waves

{{short description|Equatorially trapped waves that carry energy eastwards}}
'''Rossby-gravity waves''' are equatorially trapped waves (much like Kelvin waves), meaning that they rapidly decay as their distance increases away from the equator (so long as the Brunt–Vaisala frequency does not remain constant).  These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial [[Earth:Rossby radius of deformation|Rossby deformation radius]].<ref name="Gill">Gill, Adrian E., 1982: ''Atmosphere-Ocean Dynamics,'' International Geophysics Series, Volume 30, Academic Press, 662 pp.</ref>  They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

==Derivation==

The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H.<ref name="Zhang">Zhang, Dalin, 2008: Personal Communication, “Waves in Rotating, Homogeneous Fluids,” University of Maryland, College Park.</ref>  Because the Coriolis parameter (''f''&nbsp;=&nbsp;2Ω&nbsp;sin(''θ'') where Ω is the [[Physics:Angular velocity|angular velocity]] of the earth, 7.2921&nbsp;&times;&nbsp;10<sup>−5</sup>&nbsp;rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial [[Physics:Beta plane|beta plane]]” approximation must be made.  This approximation states that ''f'' is approximately equal to β''y'', where ''y'' is the distance from the equator and β is the variation of the Coriolis parameter with latitude, <math>\frac{\partial f}{\partial y} = \beta</math>.<ref name="Holton">Holton, James R., 2004: ''An Introduction to Dynamic Meteorology.''  Elsevier Academic Press, Burlington, MA, pp. 394–400.</ref>  With the inclusion of this approximation, the [[Earth:Primitive equations|primitive equations]] become (neglecting friction):

* the continuity equation (accounting for the effects of horizontal convergence and divergence and written with geopotential height):

::<math>\frac{\partial \phi}{\partial t} + c^2 \left ( \frac{\partial v}{\partial y} + \frac{\partial u}{\partial x} \right ) = 0</math>

* the U-momentum equation (zonal wind component):

::<math>\frac{\partial u}{\partial t} - v \beta y = -\frac{\partial \phi}{\partial x}</math>

* the V-momentum equation (meridional wind component):

::<math>\frac{\partial v}{\partial t} + u \beta y = -\frac{\partial \phi}{\partial y}</math>.<ref name="Zhang" />

These three equations can be separated and solved using solutions in the form of zonally propagating waves, which are analogous to exponential solutions with a dependence on ''x'' and ''t'' and the inclusion of structure functions that vary in the ''y''-direction:

:<math> \begin{Bmatrix}u, v, \phi \end{Bmatrix} = \begin{Bmatrix}\hat u(y), \hat v(y), \hat \phi(y) \end{Bmatrix} e^{i(k x - \omega t)} </math>.<ref name="Zhang" />

Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with three distinct solutions.  These three solutions correspond to the equatorially trapped gravity wave, the equatorially trapped Rossby wave and the mixed Rossby-gravity wave (which has some of the characteristics of the former two) .<ref name="Holton" />  Equatorial gravity waves can be either westward- or eastward-propagating, and correspond to n=1 (same as for the equatorially trapped Rossby wave) on a dispersion relation diagram ("w-k" diagram).  At ''n''&nbsp;=&nbsp;0 on a dispersion relation diagram, the mixed Rossby-gravity 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 (&minus;''k''), the solution appears to be a Rossby wave (hence the term Rossby-gravity waves).<ref name="Gill" />  As mentioned earlier, the group velocity (or energy packet/dispersion) is always directed toward the east with a maximum for short waves (gravity waves).<ref name="Gill" />

==Vertically propagating Rossby-gravity waves==

As previously stated, the mixed Rossby-gravity waves are equatorially trapped waves unless the buoyancy frequency remains constant, introducing an additional vertical wave number to complement the zonal wave number and angular frequency.  If this Brunt–Vaisala frequency does not change, then these waves become vertically propagating solutions.<ref name="Gill" />  On a typical "''m'',''k''" dispersion diagram, the group velocity (energy) would be directed at right angles to the ''n''&nbsp;=&nbsp;0 (mixed Rossby-gravity waves) and ''n''&nbsp;=&nbsp;1 (gravity or Rossby waves) curves and would increase in the direction of increasing angular frequency.<ref name="Gill" />  Typical group velocities for each component are the following: 1&nbsp;cm/s for gravity waves and 2&nbsp;mm/s for planetary (Rossby) waves.<ref name="Gill" />  

These vertically propagating mixed Rossby-gravity waves were first observed in the stratosphere as westward-propagating mixed waves by M. Yanai.<ref>Yanai, M. and T. Maruyama, 1966: Stratospheric wave disturbances propagating over the equatorial pacific. J. Met. Soc. Japan, 44, 291–194.</ref>  They had the following characteristics: 4–5 days, horizontal wavenumbers of 4 (four waves circling the earth, corresponding to wavelengths of 10,000&nbsp;km), vertical wavelengths of 4–8&nbsp;km, and upward group velocity.<ref name="Gill" />  Similarly, westward-propagating mixed waves were also found in the Atlantic Ocean by Weisberg et al. (1979) with periods of 31 days, horizontal wavelengths of 1200&nbsp;km, vertical wavelengths of 1&nbsp;km, and downward group velocity.<ref name="Gill" />  Also, the vertically propagating gravity wave component was found in the stratosphere with periods of 35 hours, horizontal wavelengths of 2400&nbsp;km, and vertical wavelengths of 5&nbsp;km.<ref name="Gill" />

==See also==
*[[Rossby wave]]
*[[Earth:Equatorial Rossby wave|Equatorial Rossby wave]]


==References==
{{reflist}}

{{physical oceanography}}

[[Category:Physical oceanography]]
[[Category:Gravity waves]]

{{Sourceattribution|Rossby-gravity waves}}