Buoyancy perturbation ($b^{\prime }$)

A buoyancy perturbation is the departure $b^{\prime }$ from a prescribed basic buoyancy field. In a fluid with constant buoyancy frequency $N$, vertical velocity $w^{\prime }$ changes it according to $({\partial }_t+\mathbf{U}\cdot \nabla )b^{\prime }=-N^2{w}^{\prime }$.

In a radially stratified shearing sheet, one can normalize the scalar buoyancy perturbation by writing $b_x=-N^2\theta$ and $D_t\theta =u_x+\xi {\nabla }^2\theta$. With this convention $\theta$ has dimensions of length, rather than temperature; $\xi$ is thermal diffusivity.