We aim to derive a (2+ 1)-dimensional variable coefficient KdV (3D VCKdV) equation from the barotropic and quasi-geostrophic potential vorticity equation
without dissipation on a beta-plane.
Momentum equation can be written into vorticity equation
in the following form:
after calculation, the dimensionless vorticity equation
While, as we know, the (2 + l)-dimensional model can be applied to reflect the evolution of envelope Rossby solitary waves in a plane, which is more suitable for the real ocean and atmosphere, in this paper, by using the method of multiple scales and perturbation method, starting from the barotropic atmospheric vorticity equation
, we will derive the coupled (2 + 1)-dimensional nonlinear Schrodinger equations for envelope Rossby solitary waves in Section 2.
Under some assumptions the Rossby wave dynamics at the so-called equivalent-barotropic level of the atmosphere can be described by the following vorticity equation
Courtier, "Variational assimilation of meteorological observations with the adjoint vorticity equation
. I: theory," Quarterly Journal of the Royal Meteorological Society, vol.
Interestingly Navier-Stokes equation which implies vorticity equation
can also be rewritten in terms of Yukawa equation ;
Vertically oriented vortex tubes motion can be expressed by the vorticity equation
in the z-direction  as d[zeta]/dt = [zeta] x [partial derivative]w/[partial derivative]x + [eta] x [partial derivative]w/[partial derivative]y + [zeta] x [partial derivative]w/[partial derivative]z, (1) where ([xi], [eta], [zeta]) are vorticities in the (x, y, z) directions, respectively, and w is the z-component of velocity.
Rossby waves are large-scale atmospheric movement and satisfy the dimensionless quasigeostrophic potential vorticity equation
as follows [17, 18]:
can be derived using horizontal momentum equations.
Carey, "High-order compact scheme for the steady stream-function vorticity equations
," International Journal for Numerical Methods in Engineering, vol.
Another basis can be developed by considering the momentum, continuity, and vorticity equations
. For steady, incompressible, low-Reynolds-number flow in a passage having nearly parallel walls, the Navier-Stokes equations become