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).

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I'd like to know the answer to this. Just a small comment: although the horizontal forcing is important, the atmospheric circulation is largely driven by the vertical gradient (i.e. the ground being heated and the upper atmosphere being cooled due to emission of thermal radiation). From memory I think around 10 times more power is dissipated by this than by the horizontal forcing. So if there were no horizontal gradient in the forcing there probably would still be a circulation, it's just that it wouldn't be organised into coherent convection cells in the same way. – Nathaniel Jan 2 '13 at 4:42
I know of some cut-down GCM models, where the Earth's surface is assumed uniform and the atmosphere assumed dry (some are mentioned here for example). However from my point of view these aren't really the simplest possible models - they still contain lots of empirical parameterisations and they're still too complex to really fully understand. The simplest simulation model would probably assume the flow is rotationally symmetric and just simulate a 2D slice along a longitude line, but I don't know of any atmosphere models that look like that. – Nathaniel Jan 2 '13 at 4:51
It's hard for me to understand how the vertical gradient could amplify circulation, so perhaps someone could simply show an airflow simulation for a non-rotating planet with one circulation cell? It seems to me that cooling of the upper atmosphere around the equator would actually slow circulation...but it's all quite complicated, especially modeling the thermal diffusion/convection within the air. Since there's not much interest in this question, I wonder if someone could just give a quantifiable explanation of the Ferrel cell; why would hot "south air" flow under cold "north air"? – bobuhito Jan 7 '13 at 19:45
In order to understand why the vertical gradient amplifies circulation, you should look into Rayleigh-Bénard convection. I think this will help you to get a picture of why the Ferrel cell exists as well. If I get a chance I'll see if I can post an answer along the lines of what you just described, but it will mean spending some time looking into the details, so I can't guarantee I'll get to it. – Nathaniel Jan 8 '13 at 1:56
Thanks, good link. I get it now, but now would think that even a non-rotating earth could have multiple convection cells. So, I feel like simulation is always needed and there's no hope for the "justifiable formula" I asked for. For the simulator, it seems I need to add another parameter for the air's thermal radiation back into space. – bobuhito Jan 9 '13 at 4:02

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.

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If I'm skimming that paper correctly, their constructal model doesn't predict a single cell for >144h rotation periods (they just give references to other papers showing it), right? – bobuhito Jun 11 '13 at 0:09
Yes this is right. It makes qualitatively sense that very fast rotating bodies would destroy complex transverse equator-pole structures. – Stan Won Jun 25 '13 at 7:56