This Demonstration shows the solutions of a model describing the relationship between atmospheric aerosol particles, cloud depth, and rainfall.

Three equations describe the model [1]:

,

,

,

where

is the cloud depth,

represents aerosol drop concentration, and

is the rain rate. The model also depends on five parameters:

and

represent maximum values of cloud depth and aerosol drop concentration,

and

are system time constants, and

is a delay term that is a function of the state of the cloud before rain forms;

,

, and

are system-dependent measured constants.

The solutions show damped oscillations leading to a steady state as well as oscillations without a steady state. Solutions to the model equations tend to approach steady state more rapidly with larger values of

, so that

can be viewed as a damping parameter of the cloud-rain coupled oscillator. The probability of oscillation around a steady state increases as the delay time

increases, allowing the cloud to attain more significant depth before rain starts to develop. To follow the trajectory of the solutions, you can vary the time window, the delay time

, and the maximum cloud depth and drop concentration,

and

.