Two-Dimensional and Quasi-Geostrophic Turbulence

Abstract

Let us begin by considering a fluid of uniform density ρ₀ in a frame which may be rotating with a constant rotation $\overrightarrow\Omega$. 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 $\overrightarrow \Omega$, and look for two-dimensional solutions $\overrightarrow u$(x,y,t) 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 $\mathop \Omega \limits^ \to )$ and vertical components of the velocity. The continuity equation implies that the velocity field is horizontally non divergent

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

(IX-1-1)

and hence there exists a stream function $\varphi \left( {x,y,t} \right)$ such that

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

(IX-1-2)