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## Where they come from

The Lorenz equations are not much harder to write down and solve than the systems we have already looked at. The biggest difficulty is actually understanding where they come from. To get there, we first have to understand a bit about Thermal Convection which is just a fancy name for the idea that hot-air rises and cold air sinks. In physics there is a classic problem of convection in a thin layer of fluid that is heated from below (for example a pan with a thin layer of water on a stove). This problem can show a remarkable range of behavior depending on how hard you heat the layer. In general, a parcel of fluid that is hotter than its surroundings will try to do two things. Because it is less dense, it will try to rise like a hot air balloon; however, because it is hot it will also lose heat by cooling. So, if it cools faster than it can move it will just sit there. If it has more heat, or loses it more slowly it will convect. When convection is confined to a thin layer, it can form all kinds of patterns from no flow at all (if you don't heat it enough) to simple rolls to highly turbulent chaotic flow (look at a glass tea kettle some time).

Anyway, the Lorenz equations are just a simplified model of thermal convection in a thin sheet. The principal simplification is that Lorenz, pretended that the fluid velocity could be described by a single roll and that the temperature in the layer could be described by a steady state solution and two time dependent modes. In pictures it looks like

Where W(T) controls the direction and speed of the roll (if W is positive the roll spins clockwise, if W is negative it spins the other direction, the bigger the absolute value of W, the faster it spins). controls the horizontal temperature structure. If is positive the left side of the box is hot, the right side cold (and vice-versa). Finally controls the vertical temperature field. It is always positive but higher values imply that the top of the box is hotter (see the movies below). If both and are zero the the temperature is just layered with hot material at the bottom and cold material on top. The important point of this description is that the spatial structure is assumed to be known everywhere and we only need to solve for the values of through time.

So finally, the Lorenz equations arise from taking the simple roll solution, substituting into the more general equations of convection and coming up with a dynamical system for the three time dependent variables. They're not much to look at but here they are.

This problem has three adjustable parameters ( ) of which we will only worry about the Rayleigh Number r and set the other two to constants.. The Rayleigh number is just a measure of how hard the layer is being heated. If r is less than one, the layer is being heated too gently to convect. Large Rayleigh numbers imply very vigorous convection. The following section will explore the behavior of the Lorenz equations for different values of the Rayleigh number.

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marc spiegelman
Mon Sep 22 21:30:22 EDT 1997