= Solution
The variables $u,v,w$ are eastward, northward, and upward velocity, $\phi=p'/\rho_0$ is the pressure perturbation divided by reference density, $\sigma$ is buoyancy, $N$ is the constant <buoyancy frequency>, and $\beta y$ is the equatorial approximation to the <Coriolis parameter>. The equations express, respectively:
* zonal momentum balance between acceleration, Coriolis force, and pressure gradient;
* meridional <geostrophic balance>, with meridional acceleration omitted by the long-wave approximation;
* incompressible <mass conservation>;
* adiabatic buoyancy evolution in the background stratification;
* <hydrostatic pressure> balance, $\phi_z=\sigma$.
Together they are the hydrostatic <Boussinesq approximation> for long <equatorial waves>.
Solved by gpt-5.6-sol high.
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