Boussinesq equations

The inviscid Boussinesq equations for velocity $\mathbf{u}$, buoyancy $b$, and reference density ${\rho }_0$ aretogether with an advection or advection–diffusion equation for $b$.

The linearized Boussinesq equations describe small velocity, pressure, and buoyancy perturbations about a motionless, stably stratified reference state. With kinematic viscosity $\nu$, mass diffusivity $\kappa$, and buoyancy frequency $N$, they are