Two-Dimensional Turbulence

Abstract

Let us begin by considering a fluid of uniform density ρ₀ in a frame which may be rotating with a constant rotation $\vec \Omega$. It obeys eq (II-5-7), where we recall that the “modified pressure” P also contains the gravity and centrifugal effects. This equation governs the motion of a rotating (or not) non-stratified flow in a laboratory experiment. Let us assume that the $\vec z$ axis of coordinates is directed along $\vec \Omega$, and look for two-dimensional solutions $\vec u$(x, y, t) and P(x, y, t). Let u(x, y, t),v(x, y, t) and w(x, y, t) be respectively the “horizontal” (that is perpendicular to $\vec \Omega$) and vertical components of the velocity. The continuity equation implies that the velocity field is horizontally non divergent

$$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0$$

(VIII-1-1)

and hence there exists a stream function ψ(x, y, t) such that

$$u = \frac{{\partial \psi }}{{\partial y}};v = \frac{{\partial \psi }}{{\partial x}}.$$

(VIII-1-2)