Blow-up of Solutions in System of Atmosphere Dynamics

Abstract

To discuss processes of dynamics of baroclinic atmosphere in adiabatic approximation under quasi-static condition we consider the two-dimensional system of equations obtained by averaging over height [1]:

M(∂_tV + (V, ∇) V) = -∇P + lMV⊥ + б Mr, (1-2)

∂ _t M+div(MV)=0, (3)

∂ _t P+(V, ∇P)+kPdivV =0, (4)

Here $V = ({V_1},{V_2}) = \bar v,P = \hat p,M = \hat \rho ,(\hat \psi = \smallint _0^\infty \psi dz,\bar \psi = \tfrac{1}{{\hat \rho }}\smallint _0^\infty \rho \psi dz for some given function \psi )$, r is the radius-vector of point in R.², V_⊥ = (V₂, -V₁), p≥ 0 - the density, p≥ 0 - the pressure V- the velocity vector, $k = \frac{{2\gamma - 1}}{\gamma },$ r - specific heat ratio (r > 1), 1 = 2w² sin Ø - the Coriolis parameter, w = const - the angular velocity of Earth rotation, Ø - the latitude. For the processes of middle scale the value of 1 can be considered approximately as the constant. Note that if r > 1, then ir ∈ (1, 2), and lim _{r →∞} k = 2. The parameterбmay be equal w² or zero in dependence on the centrifugal force is taken into account or not.