The quasi-geostrophic omega equation (see Bluestein, Volume I; eq. 5.6.14

The quasi-geostrophic omega equation (see Bluestein, Volume I; eq. 5.6.14 (2 + (f02 /)2/∂p2) = (-f0 /)/p{-vgp(g + f)} + (-R/p)2p{vgpT} + (-R/p)2p{1/Cp (dQ/dt)} + (-f0 /)/p{k   x F}

Multiplying the omega equation by : (2 + (f02)2/∂p2) = (-f0)/p{-vgp(g + f)} + (-R/p)2p{vgpT} + (-R/p)2p{1/Cp (dQ/dt)} + (-f0 )/p{k   x F}=FF=  x all forcing terms

The omega equation is now:(Ada+B)da=-FF Where Ada=2(2/L)2(R/p)[(dT/dp)da -(T/p) ] B=(8 f02)/(p)2

If the atmosphere is neutral, then Ada= 0 And the omega equation becomes: BN=-FF Therefore, the effect of hydrostatic stability is to make da smaller

The extent to which da becomes smaller is represented by N/da=(Ada+B)/B

Consider a uniformly saturated atmosphere: T/t =-v • T - ma[ (T/p) -(dT/dp)ma] And dT/dt= ma (dT/dp)ma

The omega equation now becomes: (Ama+B)ma= -FF, where Ama = 2(2/L)2(R/p) [(dT/dp)ma -(T/p)]

The ratio of ma to da is ma/da = (Ada+B)/(Ama+B)

Consider our three previous air mass soundings (MT, MP, and CP): With respective 1000 mb temperatures of 20, 10, and 0 deg C: Ada = 2(2/L)2(R/pp) [Tda -T] Ama = 2(2/L)2(R/pp) [Tma -T]

N/da=(Ada+B)/BT1000=20 C, T250=-45 C T250,da=-76.5 C; T250,ma=-48.5 C T(deg C)\L(103km) /2  2  20 5.3 2.1 1.3 10 5.6 2.2 1.3 0 5.9 2.2 1.3

ma/da=(Ada+B)/(Ama+B) T(deg C)\L(103km) /2  2  20 3.6 1.9 1.2 10 1.6 1.3 1.1 0 1.2 1.1 1.1

For dry processes: Static stability inhibition is independent of temperature, and is most prominent for small disturbances

For moist processes During the winter, intense, small scale cyclones form over warm, moist oceans, but rarely over land

The quasi-geostrophic omega equation (see Bluestein, Volume I; eq. 5.6.14