The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 74

represents the ratio of the convective vertical flux of buoyancy to the molecular
vertical flux of buoyancy. When the buoyancy is the same order of magnitude as
the inertial acceleration the flow is turbulent. For both laminar and turbulent
flows, the Rayleigh number varies monotonically as the Reynolds number and the
Prandtl number. The Reynolds number expresses the relative importance of the
inertial acceleration to the viscous acceleration, and is an indicator of the relative
amount of turbulent flow to laminar flow. 6 The Rayleigh and Prandtl numbers
determine the type of ensuing motions. We are not going to find the actual exact
solutions to (2.220), but instead just seek out the conditions under which we have
growing solutions (i.e., solutions that are unstable). To do so, we look for solu-
tions of the form

ð 1 ð 1

e ! t e i ð k x þ k y Þ dk x dk y

w ¼ W ð z Þ Re ð 2 : 223 Þ

1 1

where the complex growth rate ! i and k x and k y are horizontal wave

numbers in the x and y-directions, where k 2

! ¼ ! r þ i

¼ k x þ k y . This form of solution is a
Fourier series for the horizontal part of the solution; the vertical and horizontal
parts of the solution are separable. We now substitute (2.223) into (2.220) and get
the following ''characteristic value equation'':

! i Þþ k 2 d 2 dz 2 ! i þ k 2 d 2 dz 2 ð k 2 d 2 dz 2 Þ W ¼ Ra k 2 W

ð 2

½ð! r þ i = ½! r þ i = =

: 224 Þ

This is a sixth-order ordinary differential equation for W with respect to z; six
boundary conditions are needed to solve for W. Solutions may be found for
certain combinations of

, and Ra.
The kinematic boundary condition is applied at the top plate and the bottom
plate, which means that

! r , ! i , k,

W ¼ 0 tz ¼ 0 and 1 ð 2 : 225 Þ

This boundary condition means that there is no flow (w ¼ 0) of fluid from or into
each plate. We have a choice of two commonly used boundary conditions on the
horizontal flow at the top and bottom plates: ''free slip'' (zero stress in which there
is no friction layer; free slip is perhaps better than ''pink slip'' or ''just fire the
problem!'') or ''no slip''. Let us consider the simpler one (simpler in terms of the
resulting mathematics), the free-slip boundary condition, in which there is no
stress at the plates, so that the fluid is allowed to move by the stationary plate. In
the real atmosphere, however, there does tend to be some stress (vertical shear) at
the ground. Since the stress is proportional to the vertical gradient in the flow, the
vertical derivatives of the flow are constrained to be zero, so that

@ u =@ z ¼ @v=@ z ¼ 0 tz ¼ 0 and 1 ð 2 : 226 Þ

From the Boussinesq continuity equation (2.35) we know that

@ u =@ x þ@v=@ y ¼@ w =@ z ð 2 : 227 Þ

6 In this interpretation of the Rayleigh number, T b T t ¼ T 0 .