Geoscience Reference
In-Depth Information
not tell us anything about the geometry of the cells. The simplest cells would be
equilateral triangles, more complicated cells would be squares, and the most com-
plicated would be pentagons, etc.; the problem is like that of filling a bathroom
floor with equilateral tiles. If j is the number of sides of a regular polygon, then
the angle at each vertex of the polygon is
j Þ . If the polygons fit together
evenly as in a jigsaw puzzle, then the angle must divide into 2
ð 1 2
=
an integral number
of times, so that
1 2
=
j Þ¼ m
ð 2
:
247 Þ
2
where m is an integer. From (2.247), we note that
m ¼ 2j
j 2 Þ¼ 2 þ 4
j 2 Þ
ð 2
:
248 Þ
For j ¼ 3, 4, and 6, m is an integer (5 does not work); if j 2
6), m
cannot be an integer. So, equilateral triangles, squares, and hexagons are the only
possible patterns of cells.
The free-slip boundary conditions at the plates we considered were easy
mathematically to deal with. The analysis with no-slip boundary conditions is
more complicated because the relatively simple sinusoidal variation in z for W (cf.
(2.233)) does not satisfy the boundary conditions. 7 Much work has been done
solving the Rayleigh-Be ´ nard problem using other boundary conditions, and the
reader is referred to Emanuel (1994) and Chandrasekhar (1961) for the gruesome
details, which are in many instances rather complicated. It has been found that the
critical Rayleigh number and the critical horizontal wave number (and the critical
horizontal wavelength) vary for the different cases, so boundary conditions do
indeed matter. The case of mixed boundary conditions (i.e., of no slip at the
bottom, but free slip at the top) perhaps most closely mimic what happens in the
real atmosphere, where there is indeed a rigid surface and the top of the heated
boundary layer is not a like a rigid plate.
In the case when cold air flows over a relatively warm water surface and is
heated from below, then the lower boundary condition must include the inter-
action of the air with the water, which is certainly different from what happens
over a rigid land surface. Of course, flow over non-level ground also adds a
complication.
Yet another variation on the problem is that of what happens when vertical
heat flux is held fixed at the boundaries, rather than the temperature. In this case,
the thermal conductivity of the plates is relatively low and mechanically induced
fluxes of heat and momentum from the surface dominate near the surface. The
boundary condition involving B is then that
>
4 (i.e., if j
>
z ¼ 0atz ¼ 0 and 1, rather
than r h B ¼ 0. While variations on the problem are interesting in their own right,
they will not be considered further in this text, mainly because the theories, which
are complicated mathematically, are still for ultra-simple cases: they are linear, and
moisture and cloud microphysics are not even considered. So, we may consider 16
possible combinations of boundary conditions: free slip or no slip at each bound-
ary and perfectly conducting or insulating, yielding free slip at both the top and
bottom, no slip at the top and bottom, free slip at the top and no slip at the
@
B
=@
7 When the no slip boundary condition is imposed, the first derivative of W with respect to
height vanishes at the upper and lower boundaries.
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