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), 
(red line) and
(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 ( 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) |  |  movie |
| r=1.1 Stable (steady convection) |  |  movie |
| r=12 Stable (faster steady convection) |  |  movie |
| r=24 just sub-chaotic |  |  movie |
| r=28 Unstable chaos |  |  movie |
, b=8/3), all problems in the
figure began with initial conditions W=0, 
, 
which is just a slightly warmer left side of the box.