Aerosol-Cloud-Rain Equations with Time Delay

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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 .


Contributed by: Clay Gruesbeck (February 2013)
Open content licensed under CC BY-NC-SA




[1] I. Koren and G. Feingold, "Aerosol-Cloud-Precipitation System as a Predator-Prey Problem," Proceedings of the National Academy of Sciences of the United States of America, 108(30), pp. 12227–12232.

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