A Generalization of Bernoulli's Theorem

The conservation of potential vorticity Q can be expressed as ∂(ρQ/∂t + ∇·J = 0, where J denotes the total flux of potential vorticity. It is shown that J is related under statistically steady conditions to the Bernoulli function B by JθB ,where θ is the potential temperature. This relation is valid even in the nonhydrostatic limit and in the presence of arbitrary nonconservative forces (such as internal friction) and heating rates. In essence, it can be interpreted as a generalization of Bernoulli's theorem to the frictional and diabatic regime. The classical Bernoulli theorem—valid for inviscid adiabatic and steady flows—states that the intersections of surfaces of constant potential temperature and constant Bernoulli function yield streamlines. In the presence of frictional and diabatic effects, these intersections yield the flux lines along which potential vorticity is transported.