Geoscience Reference
In-Depth Information
p Figure 2.20. Illustration of where convection might be preferentially initiated along a sur-
face boundary. (Top) Surface boundary denoted by solid line and sense of the vertical
circulation along the boundary denoted by the large curved arrow; axes of boundary layer
rolls indicated by the dashed lines, sense of vertical circulations associated with the rolls denoted
by the smaller curved arrows, and places where upward motion along the surface boundary is
greatest are indicated by plus signs; (bottom) radar reflectivity from the WSR-88D radar near
Enid, OK on April 6, 2010 showing boundary layer roll fine lines (where convergence and
upward motion are greatest) intersecting a dryline radar fine line (where convergence and
upward motion are greatest).
where
is the mean density; Q is the heat source; and the Brunt-Va¨ isa¨ la¨
frequency (N) is assumed to be constant for simplicity. All independent variables
(u 0 , v 0 , w 0 , and p 0 ) are eliminated except for buoyancy B, so that
2
t 2
2 B
z 2
þ N 2
2
x 2
2
y 2
2 Q
z 2
@
=@
@
=@
ð@
=@
þ@
=@
Þ B ¼ @=@
t
@
=@
ð 2
:
291 Þ
We impose the following idealized diabatic heating function to represent
condensation heating that is local, is turned on suddenly, and has a maximum at
mid-levels in the atmosphere (the condensation rate varies as vertical velocity,
which is greatest aloft, but also as the absolute amount of available water vapor,
which is greatest at low levels), but is zero at the ground and at the tropopause
(z ¼ D):
Q ¼ Q 0 ð x Þ sin ð mz Þ H ð t Þ
ð 2
:
292 Þ
where
ð x Þ is the Dirac delta function (infinite at x ¼ 0, of infinitesimal width, but
whose integral is unity); H ð t Þ is a unit step function (0 before t ¼ 0, and 1 at and
after t ¼ 0); and m is an integral multiple of
D, so that Q is zero at z ¼ 0 and
D. (Students of electrical engineering make good use of the following properties of
=
and H: the integral of the Dirac delta function up to an arbitrary time t is the
unit step function and the derivative of the unit step function is the Dirac delta
function.) This heating function, for simplicity, is a slab; there are no variations in
y. It is highly idealized because in real life we cannot simply impose a heating
function that is flow independent. Heat sources are intimately related to the flow.
Solutions are of the form
B ð x
;
;
t Þ¼ b ð x
;
t Þ sin mz
ð 2
:
293 Þ
z
so that substituting (2.293) and (2.292) into (2.291) we get the following equation
for b:
2 b
t 2
N 2
m 2
2 b
x 2
@
=@
=
@
=@
¼ Q 0 ð x Þð t Þ
ð 2
:
294 Þ
The solution to (2.294) is
b ¼ Q 0 =
c m H ð c m t j x
ð 2
:
295 Þ
which is a jump in buoyancy traveling in both the x-directions at the speed c m ,
the phase speed of internal gravity waves; that is
c m ¼ N
=
m
ð 2
:
296 Þ
 
Search WWH ::




Custom Search