As we have discussed, Lorenz, in his celebrated paper, puts forth a simple model that describes the behavior of a layer of fluid that is warmer at the bottom than at the top. Such a layer is called a Rayleigh-Bénard layer, since Bénard and Rayleigh were the first to analyze such a layer around a century ago.

It turns out that although the model used by Lorenz provides a more-or-less reasonable description for fluid convection in a shallow layer of water sitting on a warm surface, it does not appear to provide a correct description for convection patterns occurring in the Earth's atmosphere, so that the atmospheric convection diagram shown on the second and succeeding slides is almost certainly incorrect when applied to the Earth's atmosphere. Below are a few thoughts that lead us to this conclusion.

The natural time unit used by Lorenz in his model,
which is the actual time *T** elapsed when
his dimensionless time
variable *τ* increases by one unit, is given by the
expression

where *H* is the height of the fluid layer, *a* is the
ratio of the convection roll height to its width
(chosen by Lorenz to
be ; *a* is
the inverse of what is normally termed the aspect ratio),
and *κ*
is the thermal diffusivity of the fluid.

*H ^{2}*/

Other relevant parameters—properties of the
fluid—include its kinematic viscosity *ν* (the
ratio of the fluid viscosity to its density) and its volume
thermal expansion coefficient *β*.

Two dimensionless parameters are particularly
relevant. They are the Prandtl number
*σ = ν/κ*, and
the Rayleigh number *R _{a}*.
In the Lorenz model (and in Rayleigh's analysis), the
Rayleigh number is given by the expression

where

Convection rolls develop when *R _{a}* exceeds
a critical value

The minimum value of

Parameter | Air | Water |
---|---|---|

κ(m^{2}/sec) |
2.1×10^{-5} |
1.5×10^{-7} |

ν(m^{2}/sec) |
1.5×10^{-5} |
9.8×10^{-7} |

σ(ν/κ) |
0.7 | 6.5 |

β(K^{-1}) |
3.4×10^{-3} |
2.0×10^{-4} |

T*(sec) |
3.2×10^{3}H^{2} |
4.5×10^{5}H^{2} |

R_{a} |
1.1×10^{8}ΔT H^{3} |
1.3×10^{10}ΔT H^{3} |

R_{c} |
658 | 658 |

Here's a table that compares a layer of air with a layer of water, in terms of the various parameters above, all at 20°C:

Now in the Lorenz model, steady convection is
always stable if the Prandtl number
*σ* is less than 1+*b*.
Since *b* is positive, the model
cannot predict chaotic convection in our atmosphere,
for which *σ* = 0.7.

Furthermore, for a layer of air only 1 meter high, the
time scale *T** for convection circulation is on the
order of 3200 seconds, or nearly an hour. If we take
*H* on the order of 10^{4} meters (the scale
height of the atmosphere), *T** becomes approximately
10,000 years.

Finally, the value of the Rayleigh number
*R _{a}* for the atmosphere, even for a layer
only one meter high with the bottom only 0.1°C warmer than
the top, is much larger than is appropriate for the Lorenz model.
A value of

Clearly atmospheric convection phenomena cannot be appropriately described by this simple model.

On the other hand water,
for which (at 20°C)
*σ* = 6.5,
may lie withing the model's range of applicability.
For a layer of water 2 cm high, the
value of *T** is approximately 3 minutes (a reasonable
time), and if Δ*T* = 1°C, the Rayleigh
number
*R _{a}* = 1.0×10

Generally speaking, the region of applicability for the Lorenz model
is rather delicate, covering only a small range of
Δ*T*. In order to observe the convection pattern
of rolls in a pan of water that are described by the model,
it is necessary to start with a very quiet water layer and
heat it from the bottom uniformly and slowly. If one
just puts a pot of water on the stove and turns on the burner
in the normal way, those convection rolls will never
have a chance to form.

In the case of the Earth's atmosphere it would be even more difficult, and not only for the reasons listed above. Too many other perturbations exist, stimulated by the Earth's rotation and by the large temperature differences spanning the Earth's surface, for example, between the equator and the polar regions—perturbations that give rise to the trade winds and the numerous cyclonic storms. However, Lorenz's comments about the unpredictability of our weather do not depend on whether his model is directly applicable to the atmosphere, since in the actual atmosphere, with its more turbulent behavior, any motion of the atmosphere must still be very sensitive to initial conditions. That is, the “butterfly effect” must still prevail.

— Peter Scott