Quasi-equilibrium Theory of Small Perturbations to Radiative- Convective Equilibrium States • See “CalTech 2005” paper on web site • Free troposphere assumed to have moist adiabatic lapse rate (s* does not vary with height • Boundary layer quasi-equilibrium applies
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Quasi-equilibrium Theory of Small Perturbations to Radiative-
Convective Equilibrium States• See “CalTech 2005” paper on web site• Free troposphere assumed to have moist
adiabatic lapse rate (s* does not vary with height
• Boundary layer quasi-equilibrium applies
Basis of statistical equilibrium physics
• Dates to Arakawa and Schubert (1974)• Analogy to continuum hypothesis:
Perturbations must have space scales >> intercloud spacing
• TKE consumption by convection ~ CAPE generation by large scale
• Numerical models on the verge of simulating clouds + large-scale waves
• We further assume convective criticality
Implications of the moist adiabatic lapse rate for the structure of
tropical disturbances• Approximate moist adiabatic condition as
that of constant saturation entropy:
• Assume hydrostatic perturbations:0 0
** ln ln vp d
L qpTs c RT p T⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
' 'pφ α∂
= −∂
• Maxwell’s relation:
• Integrate:
*
' * ' * '* p s
Ts ss pαα
⎛ ⎞∂ ∂⎛ ⎞= = ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
( )' '( , , ) ( , , ) * 'b x y t T x y t T sφ φ= + −
Only barotropic and first baroclinic mode survive
This implies, through the linearized momentum equations, e.g.
u fvt x
φ∂ ∂= − +
∂ ∂
that the horizontal velocities may be partitioned similarly:
( )( )
( , , ) ( , , ) *( , , );
( , , ) ( , , ) *( , , ).b
b
u u x y t T x y t T u x y t
v v x y t T x y t T v x y t
= + −
= + −
Implications for vertical structure of vertical velocity
u vp x yω ⎛ ⎞∂ ∂ ∂= − +⎜ ⎟∂ ∂ ∂⎝ ⎠
Integrate:
( ) ( )( )0
0 0* *' .
pb bp
u v u vp p p p T Tdpx y x y
ω⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂
= − + − − − +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∫
At tropopause:
( )0b b
t tu vp px y
ω⎛ ⎞∂ ∂
= − +⎜ ⎟∂ ∂⎝ ⎠This implies that if a rigid lid is imposed at the tropopause, the divergence of the barotropicvelocities must vanish and the barotropiccomponents therefore satisfy the barotropicvorticity equation: