Weather involves many scales of distance, time, and energy. Describing weather means making decisions about what range of scales to focus on.
Key physics involves thermal convection which reflects an interplay of gravity, thermal expansion, viscosity and thermal conduction. This can lead to Rayleigh-Bénard convection and convection cells.
There are multiple scales and they are associated with multiple entities:
-the molecules that make up the fluid
-small volumes of fluid that are in local thermodynamic equilibrium with a well-defined temperature, density, and velocity
-individual convection cells (rolls)
-collections of cells.
At each scale, the corresponding entities can be viewed as emerging from the interacting entities at the next smallest scale. Hence, they are collective degrees of freedom.
In principle, a complete description, including the transition to turbulence, is given by the equations of fluid dynamics, including the Navier-Stokes equation. Despite the apparent simplicity of these equations, making definitive predictions from them remains elusive.
A famous toy model was studied by the meteorologist Edward Lorenz in 1963, in a seminal paper, "Deterministic Nonperiodic Flow." Under the restrictive conditions of considering the dynamics of a single convection roll the model can be derived from the full hydrodynamic equations.
Lorenz's study stimulated the field of chaos theory, and is beautifully described in James Gleick's book Chaos: The Making of a New Science.
Here, I discuss Lorenz's model in the context of emergence.
The model consists of (just) three coupled ODEs (ordinary differential equations):
The variables x(t), y(t), and z(t) describe, respectively, the amplitude of the velocity mode, the temperature mode, and the mode measuring the heat flux Nu, the Nusselt number. x and y characterize the roll pattern.
There are three dimensionless parameters in the model: r, sigma, and b.
r is the ratio of the temperature difference between the hot and cold plate, to its critical value for the onset of convection. It can also be viewed as the ratio of the Rayleigh number to its critical value.
sigma is the Prandtl number, the ratio of the kinematic viscosity to the thermal diffusivity. Sigma is about 0.7 in air and 7 in water. Lorenz used sigma = 10.
b is of order unity and conventionally taken to have the value 8/3. It arises from the nonlinear coupling of the fluid velocity and temperature gradient in the Boussinesq approximation.
The model is a toy model because for values of r larger than r_c (defined below) "the three mode approximation for the PDEs describing thermal convection... ceased to be physically
realistic, but mathematically the model now starts to show its most fascinating properties,.."
Novelty
The model has several distinct types of long-time dynamics: stable fixed points (no convection), limit cycles (convective rolls), and most strikingly a chaotic strange attractor (represented below). The chaos is reflected in the sensitive dependence on initial conditions.
Discontinuities
Quantitative changes lead to qualitative changes. For r < 1, no convection occurs. For r > 1, convective rolls develop, but these become unstable for
and a strange attractor develops.
Phase diagram
Lorenz only considered one set of parameter values [r =28, sigma=10, and b=8/3]. This was rather fortunate, because then strange attractor was waiting to be discovered.
The phase diagram maps out the qualitatively different behaviours that occur as a function of sigma (vertical axis) and r (horizontal axis).
Different phases are the fixed points P± associated with convective rolls (black), orbits of period 2 (red), period 4 (green), period 8 (blue), and chaotic attractors (white).
H. R. DULLIN, S. SCHMIDT, P. H. RICHTER, and S. K. GROSSMANN
The details of the molecular composition of the fluid and the intermolecular interactions are irrelevant beyond how they determine the three parameters in the model. Hence, qualitatively similar behaviour can occur in systems with a wide range of chemical compositions and physical properties.
Unpredictability
Although the system of three ODEs is simple, discovery of the strange attractor and the chaotic dynamics was unanticipated. Furthermore, the dynamics in the chaotic regime are unpredictable, given the sensitivity to initial conditions.
Top-down causation
The properties and behaviour of the system are not just determined by the properties of the molecules and their interactions. The external boundary conditions, the applied temperature gradient and the spatial separation L of the hot and cold plates, are just as important in determining the dynamics of the system, including motion as much smaller length scales.
