Weather natural attractlet · § 7.4

Convection Cells

A layer of fluid heated from below, turning over in rolls that carry the heat the bottom plate delivers and release none of their own.

Heat a thin layer of fluid from below. At first the heat simply conducts upward and nothing moves. Past a critical heating, the warm, light fluid at the bottom starts to rise, the cool fluid at the top sinks, and the layer organizes into rolls that turn over side by side. The rolls move heat far faster than conduction could. But they carry heat; they do not make it. All of it comes in through the plate below. Cut the heat and the rolls run down in a fraction of a thermal diffusion time. This is the system Lorenz cut down to three equations.

A Slice Through the Layer

1 deep, 5.66 wide, repeating side to side · hot plate below, cold plate above

heating, Rayleigh number—

heat carried, times conduction alone (Nusselt number)—

fastest rising flow (diffusion units)—

rolls—

time (thermal diffusion times)—

growing

A real simulation, on a coarse grid. It solves the two-dimensional Boussinesq equations for the layer on a 64 by 16 grid. Two idealizations: the top and bottom walls are free-slip, which puts onset at a Rayleigh number of 657.5 (real rigid plates give 1708), and the Prandtl number is 1, between air (about 0.7) and water (about 7). The grid is good for onset and moderate heating, not for turbulence. Colour is temperature, orange hot and blue cold; white dots are tracers carried by the flow. Time is the solver’s clock.

Why It Reads as an Attractlet

an authored reading, not a measurement

∂T/∂t + u·∇T = ∇²T
∂ω/∂t + u·∇ω = Pr ∇²ω + Ra Pr ∂T/∂x
T temperature · ω vorticity · Ra heating · Pr = 1

The test on the Weather page asks whether a system’s own activity releases the energy that drives it, or only carries energy delivered to it. The rolls only carry. Heat enters through the bottom plate and leaves through the top, and the rolls move it along. That matches the kernel’s Natural Attractlet (§ 7.4): “a naturally-occurring non-recursive structure that exhibits attractor-like behavior under external driving.” Press Cut the heat: the rolls do not outlast their supply.

The strongest objection. The flow does sharpen the temperature differences that drive it, which looks like a loop. The reading here is that it rearranges heat it is handed rather than releasing any of its own, and that is where the Weather page’s test draws the line. A thunderstorm, by contrast, releases latent heat by its own lifting.

Here the reading and the program agree. The layer is a natural attractlet; the program that runs it is an attractlet model under § 7.4.

The Shape of the Basin

measured on this model

Below onset there is one state, the still layer, and every disturbance dies back into it. Above onset the still layer loses its basin and rolls take over. Close to onset they take a long time to appear: at a Rayleigh number of 700, just above 657.5, the first rolls showed after about 18 thermal diffusion times, against 0.2 at 5,000.

There is more than one way to roll. At a Rayleigh number of 5,000, twelve different random starts settled into 6 rolls seven times and 4 rolls five times. Each pattern has its own basin, and they carry slightly different amounts of heat: 3.68 and 3.86 times conduction. Press Stir it: in eight trials, seven stirred layers came back to the pattern they had and one crossed into the other. Press Start over for a new random start.

Press Cut the heat and every basin is gone at once: the fastest flow falls from about 40 to below 1 within 0.2 thermal diffusion times, and the layer goes back to conducting.

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