This model of the Lorenz equations was invented by Willem Malkus and Lou Howard in the 1970s.
![[A simple waterwheel]](image/wwheel1.gif)
It is a simple waterwheel with leaky cups. Water is poured steadily from the top. If the flow rate is slower than the leakage rate of the cups, rotation will never occure. If the inflow is increased, the cups get heavy enough to start then wheel turning in either direction (left image). Because of symmetry of the wheel, rotation in both directions is possible. Eventually it settles in a steady rotation (mid image). If inflow is increased still further, the rotation of the wheel can be destabilized. The motion becomes chaotic; the wheel rotates several turns until the cup get too heavy to be carried over the top. Then the wheel slows down and may even reverse its direction (right image). The It keeps changing direction erratically. It is in fact impossible to say, after how many tuns the wheel will change its direction [Stro95].
Malkus delveloped a more sophisticated model at MIT which allows to adjust
several parameters. It has a brake to reduce torque momentum and it is
also possible to adjust tilt to control the influence of gravity.
To create a mathematical model of the system, the following parameters have to be considered:
Setting these parameters into proper relations one can find a system of equations describing the rotational speed of the waterwheel.
Looking at the parameters of the waterwheel, there are some, that tend to spin the wheel, which are gravity and inflow and others, that tend to stop it. Latter are leakage and damping. The ratio between these paramters is used to define the Raleigh number. The magnitude of the Raleigh number determines the behaviour of the waterwheel. For instance a Raleigh number being less than 1 indicates, that energy loss is greater than energy inflow. Therefore no rotation occurs.
The Raleigh number also appears in parts of fluid dynamics where a layer of fluid is heated from below. Here it is proportional to the difference in temperature from bottom to top. For small temperature gradients, heat is conducted vertically, but the fluid remains motionless. When the Raleigh number increases, the hot fluid begins to rise, whereas the cold fluid begins to sink. This sets up convection rolls completely analogous to the steady rotation of the chaotic waterwheel. Further increasing the Raleigh number, the convection rolls loose also their stability as the waterwheel [Stro95].
The analogy between the chaotic waterwheel and the fluid dynamics directly leads to the Lorenz Equations .