en>Gro-Tsen: /* Definition */ fix thinko in example of valuation over the Gaussian rationals

2013-09-24T15:46:16Z

Definition: fix thinko in example of valuation over the Gaussian rationals

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			* 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 3 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" /> It is important to note that 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'' = 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 (−''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'' = 0 (mixed Rossby-gravity waves) and ''n'' = 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 cm/s for gravity waves and 2 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 km), vertical wavelengths of 4–8 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 km, vertical wavelengths of 1 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 km, and vertical wavelengths of 5 km.<ref name="Gill" />

			==References==
			{{reflist}}

			{{physical oceanography}}

			[[Category:Physical oceanography]]

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en>Gro-Tsen: /* Definition */ fix thinko in example of valuation over the Gaussian rationals

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