Physical interpretation of the Lorenz system

Question

The Lorenz equations $$ \frac{dx}{dt} = \sigma(y-x);\\ \frac{dy}{dt} = x(\rho-z)-y;\\ \frac{dz}{dt} = xy - \beta z $$ were (I believe) the first set of nonlinear equations known to exhibit chaotic behaviour. They were derived as a simplified model of Rayleigh-Bénard convection in the Atmosphere.

It's easy to find material on their properties from a purely mathematical point of view, but comprehensible material on their original physical meaning is harder to come by. Perhaps this is partly because the derivation is fairly advanced, being based (as I understand it) on a Fourier transformation of an approximation to the Navier-Stokes equations. I would like to know whether it's possible to get a good basic physical intuition of the physical interpretation of the variables and parameters of these equations without having to understand all of these mathematical details.

I'm particularly interested in the heat transfer properties of Rayleigh-Bénard convection. If the Lorenz equations are a model of convection cells then perhaps one of the parameters can be interpreted as the temperature gradient, and perhaps the instantaneous heat flux can be calculated from the variables. Is is possible to interpret the equations in such a way?

_{Note: I know that the Lorenz-Malkus waterwheel is a physical interpretation of the Lorenz equations, and I can understand how the equations relate to the waterwheel --- but I can't quite understand how the waterwheel relates to thermally-driven convection! The issue is that the water wheel has a constant inflow of water whereas (as I understand it) the convective system Lorenz originally modelled has a constant temperature gradient and a variable heat flow rate. An answer that explains how the thermal convection scenario maps to the water wheel would be just as good as one that explains how it maps to the Lorenz equations themselves.}

Answer 1

Indeed, in Kundu & Cohen, Fluid mechanics, the derivation of Lorenz system is rather sketchy. So let us turn to the source. Original paper by E.N. Lorenz is:

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences, 20(2), 130-141. doi, Free pdf.

(12000+ citations!)

The text is quite accessible, so for all the details I suggest you look at it. Here, let us just walk through the basics.

We start with a 2D free convection problem in a layer of uniform depth $h$ where the temperature difference is maintained between upper and lower boundaries. The model uses Boussinesq approximation of buoyancy due to thermal expansion. Velocity field is parametrized by the stream function $\psi$, while thermal properties are given by the function $T'$ of temperature difference between the actual fluid temperature and the linear temperature profile present in the absence of convection.

The linearized system of equations arising from this model is classic (Rayleigh, 1916) example of instability developing in a static fluid. The calculations for the (in)stability analysis could be found for example in Kundu & Cohen. The static solution becomes unstable if dimensionless parameter, Rayleigh number, exceeds the critical value $\mathrm{Ra}_{cr} = 27 \pi^4/4$.

Lorenz work introduces non-linearity. Building upon results by Saltzman:

Saltzman, B. (1962). Finite amplitude free convection as an initial value problem-I. Journal of the Atmospheric Sciences, 19(4), 329-341. doi, Free pdf.

which gave the general Fourier expansion (or Galerkin approximation) for the full convection problem, Lorenz truncated the system by keeping the lowest modes: \begin{align*} \psi &∝ X(t) \cos πz \sin kx,\\ T' &∝ Y(t) \cos πz \cos kx + Z(t) \sin 2πz, \end{align*} The flow pattern correspond to the simplest Bénard cells pattern:

So the physical meaning of variables $X$, $Y$, $Z$ is quite straightforward: $X$ is proportional to the intensity of convective motion, $Y$ is proportional to the temperature difference between the ascending and descending currents (same signs of $X$ and $Y$ mean that warmer fluid is rising), and $Z$ is proportional to the distortion of the average vertical profile of temperature from linearity, positive $Z$ means strong gradients near boundaries.

Parameters of the Lorenz system $\sigma$, $\rho$, $\beta$, are expressed as dimensionless quantities from the model. $\sigma$ is the Prandtl number, $\rho = \mathrm{Ra}/\mathrm{Ra}_{cr}$, and $\beta = 4π^2/(π^2 + k^2)$. The solution to this system is expected to approximate the behaviour of the full problem only while Rayleigh number is only slightly supercritical.

For the heat transfer in this model I refer you to Saltzman's paper, which contains expressions for that, but from the form of functions $\psi$ and $\theta$ we can see that heat transfer through convection would be proportional to $XY$, while overall additional heat flux proportional to $\partial_z T'$ taken at a boundary where $z$-component of velocity is zero, would be proportional to (averaged) value of $Z$. (Note that modification to $T'$ by Y part disappears after averaging over $x$).