Geoscience Reference
In-Depth Information
The smallest Rayleigh number possible, the critical Rayleigh number (Ra
c
),
is
found for the n
¼
1 mode; that is
Ra
c
¼½ð
k
2
2
3
2
k
2
þ
Þ
þ
T
0
=
ð
2
:
266
Þ
Recall that without rotation (2.241) for n
¼
1
Ra
c
¼ð
k
2
2
3
k
2
þ
Þ
=
ð
2
:
267
Þ
2
H
4
2
It
0, the critical Rayleigh number is
increased by rotation, so that the effect of rotation is stabilizing. The reader is left
to ponder whether convective storms that rotate (e.g., supercells, to be discussed
later) are more stable than those that do not and whether or not that rotation is
therefore responsible for their relative longevity.
The wavenumber at the minimum Rayleigh number, the critical Rayleigh
number, for the n
¼
1 mode, is found by differentiating (2.266) with respect to k
and setting the resultant equation to zero
follows
that since T
0
¼ð
2
OÞ
=
>
k
¼
0
¼½ð
k
2
2
2
2
ð
2k
2
2
4
@
Ra
c
=@
þ
Þ=
=
1
Þ
T
0
=
ð
2
:
268
Þ
We let
r
¼
k
2
2
=
ð
2
:
269
Þ
so that (2.268) can be written as
2
4
ð
r
þ
1
Þ
ð
2r
1
Þ¼
T
0
=
ð
2
:
270
Þ
is 2r
3
þ
lower order terms. So, as the RHS of
The LHS of
(2.270)
(2.270)
4
. Then the first mode to become unstable for large
Taylor number, the critical mode is r
c
:
4
!1
,2r
3
T
0
=
!
T
0
=
4
1
=
3
lim
T
0
!1
r
c
¼ð
T
0
=
2
Þ
ð
2
:
271
Þ
From the definition of the Rayleigh number (2.222) it follows that
H
4
c
¼
Ra
c
=
ð
2
:
272
Þ
Meanwhile, from (2.266), (2.271), and (2.269), it follows that for the n
¼
1 mode
when T
0
!1
,
h
i
=
1
=
3
T
2
=
3
0
3
4
H
4
c
¼
2
ð
2
Þ
ð
2
:
273
Þ
3
in (2.266) when T
0
!1
.
Substituting for the definition of T
0
(2.256) in (2.273), it follows that
2
and k
6
2
where it is recognized that T
0
ð
Þ
4
1
=
3
4
=
3
1
=
3
3
c
¼
2
ð
2
Þ
ð
2
O=
H
Þ
ð
2
:
274
Þ
Recall that
represents the temperature gradient between the lower and upper
plate. Then for a very large Taylor number, when the coecient of viscosity
increases, the critical temperature gradient decreases. In other words, viscosity is
destabilizing when rotation is very important. This situation is in sharp contrast to
the non-rotating case, when viscosity is stabilizing (cf. (2.222)), in that increasing
Search WWH ::
Custom Search