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Behavior of the Lorenz equations

 

Table 1 shows the behavior of the Lorenz Equations for increasing values of heating. The first figure in each row shows the value of each of the variables W (black line), tex2html_wrap_inline1262 (red line) and tex2html_wrap_inline960 (green line) as a function of time. The second figure shows a modal reconstruction of the temperature field at time t=2 (which is where the movies begin). At low values of heating r<1, the initial temperature perturbation cools too fast and the layer stop convecting. For r=1.1 the roll convects at a steady velocity clockwise. If we had started the problem with a hotter right hand side ( tex2html_wrap_inline1262 negative), then we would have a roll that rotated the other way. For this problem, as we increase the Rayleigh number to just shy of 25, all initial conditions will eventually settle down to one of two steady states, a roll that either rotates clockwise or counter clockwise. As the Rayleigh number is increased, it takes longer and longer to settle down and the final speed of the roll gets higher and higher. At a critical Rayleigh number (of 24.74 for this problem), however, the roll becomes unstable and we enter the chaotic regime where the roll continuously oscillates in time and flips direction in a predictable, yet unpredictable way. Welcome to Chaos

What is surprising about this problem is that these equations are deterministic in the sense that for any value of the three parameters, we know exactly how the behavior will change. Yet if we solve the problem far enough in time, it turns out that for some cases it is impossible to predict where it will end up. This was Lorenz's enormous contribution, before these equations it was thought that if you knew enough, the future was predictable. At least in this problem, that is not true. Time to look at these yourself

 

Rayleigh Number, Comments Time Series Modal Reconstruction
r=0.5 Stable (no convection) tex2html_wrap1330 tex2html_wrap1332 movie
r=1.1 Stable (steady convection) tex2html_wrap1334 tex2html_wrap1336 movie
r=12 Stable (faster steady convection) tex2html_wrap1338 tex2html_wrap1340 movie
r=24 just sub-chaotic tex2html_wrap1342 tex2html_wrap1344 movie
r=28 Unstable chaos tex2html_wrap1346 tex2html_wrap1348 movie
Table 1: Behavior of the Lorenz Equations as a function of Rayleigh Number r ( tex2html_wrap_inline950 , b=8/3), all problems in the figure began with initial conditions W=0, tex2html_wrap_inline956 , tex2html_wrap_inline958 which is just a slightly warmer left side of the box.

 


next up previous
Next: Problem 1: Lorenz in Up: Lab 3: Onward to Previous: Where they come from

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