Atmospheric Circulation What is the simplest simulatable model giving our rotating earth its 3 circulation cells (Hadley, Ferrel, Polar)?
The model should also show 1 circulation cell if the earth's rotation were stopped (or in a case like Venus), and
no circulation at all if radiation were somehow uniform to/from the earth (instead of true equator-heating due
to the sun's point source).  By the way, it would be nice if the same model could show 8 circulation cells in a
case like Jupiter.
I expect the earth's $23^{\circ}$ tilt can be ignored.  The important parameters seem to be the planet's radius,
the planet's rotation speed, the planet's mass, the atmosphere's mass, the atmosphere's molar density
(treated as an ideal gas with energy absorption from the earth's surface, but not directly from the sun's rays),
and the atmosphere's viscosity.  I hope the earth's surface could simply be modelled with a constant
temperature profile, increasing from the pole to the equator (or, I wonder if daily thermal cycling of this
surface temperature profile might be needed to generate multiple circulation cells).
Anyway, using parameters like these, I'm hoping to see a justifiable formula giving 3 (I can't yet accept the
qualitative "Hadley cell air falls with inertia and friction" explanations on the internet).
 A: The simplest explanation and analysis is here : http://home.uevora.pt/~ahr/D.pdf
It is based on the constructal law. Constructal law promoted by Bejan&al, belongs to the family of variational principles but deals with macroscopic flow distributions.
According to Bejan, the constructal law belongs to the laws of nature close to but different from the 2nd law of thermodynamics.
The constructal law says that in any flow system, the flow will be organised in such a way that the access of every part of the system to the flow configuration will be maximized. This looks close to ideas about maximum/minimum entropy production but is in reality different.
To be complete, the existence of a constructal law as a law of nature is generally not accepted by fluid mechanics specialists. Personally I stay agnostic - the law status is not demonstrated yet the results obtained by applying this principle are correct in most cases.
To summarize this paper which is fairly long and deals also with diurnal heat transfer betweeen the hot daily half and cold night half, I highlight :


*

*the number of cells depends on rotation speed and on the equator-pole temperature difference. For slow rotations (> 144 h) a single cell develops. For fast rotations (< 24 h) 3 cells develop. For fixed rotation speed the number of cells is farther modulated by the temperature difference. At 24 hours and 20°C only one cell is present, for 60°C (real Earth) 3 cells are present. For 130°C the polar cell disappears.

*Bejan considers a heat source (equatorial band) of surface AH and a heat sink (polar cap) of surface AL with fractions AH/A = x and AL/A = 1-x. The largest part of the paper is then used to define the characteristics of the convective 3D flow from AH to AL.

*Once the flows characterised, Bejan applies the constructal law which says that the flows will configure themselves in such a way that the heat transport Q (equator -> pole) is maximized.
e.g (dQ/dx)at constant TH = 0 and (dQ/dx) at constant TL = 0. This yields 2 solutions for x thus 2 partitions of the surface. In 1 the source is the Hadley cell and the sinks are (Ferrel + Pole) cells while in 2 the sources are (Hadley + Ferrel) cells while the sink is the Pole cell. Hence the system maximizes the heat transfer with 3 cells where the middle one plays a role of intermediary - sink to one and source to the other.

*Regardless whether one believes in constructal law as a natural law or an approximation of a yet unproven and/or unknown law, i find the paper interesting, consistent and the results are confirmed by observation. Using the model on Jupiter would not be easy because the largest part of the model deals with dynamical properties of the fluids (shear, turbulence etc) which are not known for Jupiter's atmosphere whose depths are totally different from the Earth.
