Asymptotic Analyses for Atmospheric Flows and ... - FU Berlin, FB MI

778 ZAMM · Z. angew. Math. Mech. 80 (2000) 11-12
Small scale dynamics:
On the small scales comparable to h sc it is found that
∇ ⃗x p (0) = ∇ ⃗ξ p (0) = ∇ ⃗x p (1) ≡ 0 so that p (0) = p(z,t) , p (1) = p (1) ( ⃗ ξ,z,t) . (49)
Consistent with observations, temporal variations of the background pressure p are suppressed, so that p = p(z).
Next one recovers the pseudo-incompressible flow equations for stratified flows from (37), except for one
modification concerning the influence of long wavelength pressure gradients. The horizontal momentum equation
now reads
⃗u t + ⃗u ·∇ ⃗x ⃗u + w⃗u z + 1 ρ ∇ ⃗x p (2) = ⃗ D Ro
u − 1 ρ ∇ ⃗ ξ
p (1) , (50)
where we have included the Coriolis terms in ⃗ D Ro
u
to abbreviate the notation. One may expect to find the fully
three-dimensional anelastic approximation from (39), plus the long wave pressure gradient term as in (50), if
variations of the potential temperature are assumed to be of order O(M 2 ) only. This case was not discussed in [5].
The only new effect on the small scales is thus the pressure gradient term 1 ρ ∇ ξ ⃗ p (1) , which is responsible for
a bulk acceleration of the fluid. Notice that the ⃗x-dependence of the second order pressure p (2) is determined by a
small scale divergence constraint analogous to (37) 2 .
Large scale dynamics:
In a standard fashion, the relevant equation for the dynamics on the 1 M l refscale
are obtained by horizontally averaging over the small scale coordinate ⃗x. The ⃗x-averaging operator will be
abbreviated by 〈·〉. The large scale dynamics involves three variables, namely the long wave pressure mode p (1) ,
the averaged vertical component of vorticity ζ = ⃗ k ·∇ ⃗ξ
×〈⃗u (0) 〉, and the averaged first order vertical velocity
w (1) = 〈w (1) 〉. The long wavelength dynamical equations for the considered regime read
( ∂2
∂t 2 − c2 ∇ ⃗ξ 2 ) p (1) = (1− c 2 ∂ ∂z ) ∂ ∂t (ρ w(1) ) − c 2 ζ + Q p (1) ,
N 2 c 2 (ρ w (1) ) = −(1 + c 2 ∂ ∂z ) ∂ ∂t
p (1) + Q ρw (1) ,
(51)
∂
∂t ζ = 1
c 2 ∂
∂t p(1) + ∂ ∂z (ρ w(1) ) + Q ζ
.
Here Q i are inhomogeneous terms that combine the effects of small scale averages of nonlinear terms, and of the
momentum and energy inhomogeneities ⃗ D v ,D p from (1). The system in (51) supports internal gravity waves, long
wave acoustics, and the Lamb wave. It represents the link between the multiple length scale asymptotics from [5]
and classical analyses for small perturbations from a state of rest.
We notice also that the in the stationary limit (∂/∂t ≡ 0) the first equation in (51) reduces to the pressure
Poisson equation from the quasi-geostrophic theory obtained by applying the horizontal divergence ∇ ||
· (·) to
the geostrophic balance equation (6). Thus we verify that the Coriolis effects achieve dominant importance for
quasi-steady flow on the larger ⃗ ξ-scale as announced earlier.