Analysis of Barotropic Model

Abstract

In this chapter we shall consider the following equations of the barotropic atmosphere on rotating sphere S² ⊂ R³ with radius a:

$$\frac{\partial }{{\partial t}}\Delta \psi + J\left( {\psi ,\Delta \psi + l} \right) = - \sigma \Delta \psi + v\mathop \Delta \nolimits^2 \psi + f,\quad \psi \left| {_{t = 0} } \right.\; = \;\psi _0 ,$$

(3.1)

and

$$\frac{\partial }{{\partial t}}\Delta \psi + J\left( {\psi ,\Delta \psi + l} \right) = - \sigma \Delta \psi + v\mathop {\left( { - \Delta \,} \right)}\nolimits^{s + 1} \psi + f,\quad \psi \left| {_{t = 0} } \right.\; = \;\psi _0 ,$$

(3.2)

where ψ is the stream-function, J(a,b) is the Jacobian, l = 2Ω sin φ, s ≥ 1, while the parameters Ω > 0 and σ ≥ 0 have the dimensions [Ω] = T⁻¹, [σ] = T⁻¹, respectively.