The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 81

the viscosity decreases the Rayleigh number, which reduces it from its critical
value.
The student is advised to look elsewhere (Emanuel's text and the original
journal articles) for more detailed and thorough analyses of what happens when
there are free-slip boundary conditions and the analyses of what happens when
there are no-slip boundary conditions, etc. These treatments are more complicated
and not worth the time spent on them for the purposes of this text.

2.9.3 Convection in a linearly sheared atmosphere without rotation

We now consider the case of Rayleigh-Be´ nard convection when there is no
background rotation, but there is vertical shear in the basic state. This vertical
shear may be due to the thermal wind or due to surface friction or to both.
Suppose, for simplicity, that vertical shear is constant and in the x-direction only,
so that

H Þ z þ u 0

u ¼ð U 0 = ð 2 : 275 Þ

The unscaled equations, with the prime notation dropped for the perturbation
quantities, are as follows:

2

½@=@ t þð U 0 = H Þ z @=@ x r u ¼ 1 = @ p =@ x ð 2 : 276 Þ

2

½@=@ t þð U 0 = H Þ z @=@ x r v ¼ 1 = @ p =@ y ð 2 : 277 Þ

2

½@=@ t þð U 0 = H Þ z @=@ x r w ¼ 1 = @ p =@ z þ B ð 2 : 278 Þ

2

½@=@ t þð U 0 = H Þ z @=@ x r B ¼ w ð 2 : 279 Þ

@ u =@ x þ@v=@ y þ@ w =@ z ¼ 0 ð 2 : 280 Þ

We scale each space variable as in (2.217) and time as in (2.219); wind variables
are scaled as the space scale divided by the time scale ( H). Pressure is scaled

such that the pressure gradient term is on the order of the inertial acceleration.
Buoyancy is scaled using its definition (2.211) and multiplying by the dimension-
less Prandtl number (2.221). The scaled equations are given by

=

2

ð@=@ t þ Re z @=@ x r Þ u ¼@ p =@ x Re w ð 2 : 281 Þ

2

ð@=@ t þ Re z @=@ x r Þv ¼@ p =@ y ð 2 : 282 Þ

2

ð@=@ t þ Re z @=@ x r Þ w ¼@ p =@ z þ Ra B ð 2 : 283 Þ

2

ð@=@ t þ Re z @=@ x r Þ B ¼ w ð 2 : 284 Þ

@ u =@ x þ@v=@ y þ@ w =@ z ¼ 0 ð 2 : 285 Þ

where Re ¼ U 0 H and all the dependent variables are of order one.

There have been numerous (mathematically complicated) analytical studies of
the stability of the above equations. Asai's seminal work around 1970 demon-
strated that when the mean state is weakly stable or relatively weakly statically
unstable (e.g., when strongly heated from below) the most unstable modes (when

=