Effects of aerosols in making artificial rain: a modeling study

Abstract

Rain is an important phenomenon occurring in nature. Nowadays, in some places across the world, less amount of rain is observed, which affects human population in several ways, like shortage of drinking water, less production from agriculture fields, increase in pollution level, etc. Keeping this point in view, in this paper, a mathematical model for making artificial rain is proposed and analyzed. In the modeling process, it is assumed that water vapors are continuously formed in the atmosphere and are not condensed enough to form clouds. Further, it is assumed that aerosols (mixture of two or more) are introduced in the atmosphere to increase the process of nucleation of cloud droplets from water vapors and changing them into raindrops. The qualitative analysis of the proposed model is presented using stability theory of differential equations. It is found that model exhibits only one nonnegative equilibrium. The sufficient conditions for stability of the equilibrium have been obtained. Numerical simulation is carried out to see the effect of key parameters on the process leading to rainfall.

Appendix

1.1 1: Proof of Theorem 1

We linearize the system (1) about $E^*$ by using the following transformations

$$\begin{aligned} C_v=C_v^*+C_{v1}, C_d=C_d^*+C_{d1}, C_r=C_r^*+C_{r1}, C_h=C_h^*+C_{h1}. \end{aligned}$$

(18)

Now consider the following positive definite function

$$\begin{aligned} V=\frac{1}{2}(C_{v1}^2+m_1C_{d1}^2+m_2C_{r1}^2+m_3C_{h1}^2) \end{aligned}$$

(19)

(where $m_1 , m_2 , m_3$ are some positive constants to be chosen appropriately). Differentiating above equation with respect to t along the solutions of linearized system of (1), we get

$$\begin{aligned} \frac{dV}{dt}=- & {} (\theta _0+\theta _1C_h^*)C_{v1}^2-m_1(\lambda _0+\lambda _1C_h^*)C_{d1}^2-m_2r_0C_{r1}^2 -m_3a^*C_{h1}^2 \nonumber \\+ & {} C_{v1}C_{h1}\left[ -\theta _1C_v^*-m_3\theta _1C_h^*\right] \nonumber \\+ & {} C_{v1}C_{d1}\left[ m_1(\lambda +\pi _1\theta _1C_h^*)\right] \nonumber \\+ & {} C_{d1}C_{h1}\left[ m_1(\pi _1\theta _1C_v^*-\lambda _1C_d^*)-m_3\lambda _1C_h^*\right] \nonumber \\+ & {} C_{d1}C_{r1}\left[ m_2(r+\pi _2\lambda _1C_h^*)\right] \nonumber \\+ & {} C_{r1}C_{h1}\left[ m_2\pi _2\lambda _1C_d^*-m_3\lambda _2C_h^*\right] . \end{aligned}$$

(20)

Choosing $m_2=m_1\frac{\pi _1\theta _1\lambda _2C_v^*}{\pi _2\lambda _1^2C_d^*}$ and $m_3=m_1\frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}$, $\frac{dV}{dt}$is simplified as:

$$\begin{aligned} \frac{dV}{dt}=- & {} (\theta _0+\theta _1C_h^*)C_{v1}^2-m_1(\lambda _0+\lambda _1C_h^*)C_{d1}^2 -m_1\frac{\pi _1\theta _1\lambda _2r_0C_v^*}{\pi _2\lambda _1^2C_d^*}C_{r1}^2\nonumber \\- & {} m_1\frac{\pi _1\theta _1a^*C_v^*}{\lambda _1C_h^*}C_{h1}^2 \nonumber \\+ & {} C_{v1}C_{h1}\left[ -\theta _1C_v^*\right] \nonumber \\+ & {} C_{v1}C_{h1}\left[ -m_1\frac{\pi _1\theta _1^2C_v^*}{\lambda _1}\right] \nonumber \\+ & {} C_{v1}C_{d1}\left[ m_1(\lambda +\pi _1\theta _1C_h^*)\right] \nonumber \\+ & {} C_{d1}C_{h1}\left[ -m_1\lambda _1C_d^*\right] \nonumber \\+ & {} C_{d1}C_{r1}\left[ m_1\frac{\pi _1\theta _1\lambda _2C_v^*}{\pi _2\lambda _1^2C_d^*}(r+\pi _2\lambda _1C_h^*)\right] . \end{aligned}$$

(21)

Now $\frac{dV}{dt}$ will be negative definite provided the conditions (11), (12) and (13) are satisfied.

1.2 2: Proof of Theorem 2

To prove this theorem we consider the following positive definite function about $E^*$,

$$\begin{aligned} U=\frac{1}{2}\left[ (C_v-C_v^*)^2+k_1(C_d-C_d^*)^2+k_2(C_r-C_r^*)^2+k_3(C_h-C_h^*)^2\right] \end{aligned}$$

(22)

(where $k_1 , k_2 , k_3$ are some positive constants to be chosen appropriately). Differentiating above equation with respect to t along the solutions of system (1) and making simple algebraic manipulations, we get

$$\begin{aligned} \frac{dU}{dt}=- & {} k_3(\theta _1C_v+\lambda _1C_d+\lambda _2C_r)(C_h-C_h^*)^2\nonumber \\- & {} (\theta _0+\theta _1C_h^*)(C_v-C_v^*)^2-k_1(\lambda _0+\lambda _1C_h^*)(C_d-C_d^*)^2\nonumber \\- & {} k_2r_0(C_r-C_r^*)^2-k_3\delta _h(C_h-C_h^*)^2\nonumber \\+ & {} (C_v-C_v^*)(C_h-C_h^*)\left[ -\theta _1C_v\right] \nonumber \\+ & {} (C_v-C_v^*)(C_h-C_h^*)\left[ -k_3\theta _1C_h^*\right] \nonumber \\+ & {} (C_v-C_v^*)(C_d-C_d^*)\left[ -k_1(\lambda +\pi _1\theta _1C_h)\right] \nonumber \\+ & {} (C_d-C_d^*)(C_h-C_h^*)\left[ k_1(\pi _1\theta _1C_v^*-\lambda _1C_d)-k_3\lambda _1C_h^*\right] \nonumber \\+ & {} (C_d-C_d^*)(C_r-C_r^*)\left[ k_2(r+\pi _2\lambda _1C_h)\right] \nonumber \\+ & {} (C_r-C_r^*)(C_h-C_h^*)\left[ k_2\pi _2\lambda _1C_d^*-k_3\lambda _2C_h^*\right] . \end{aligned}$$

(23)

Choosing $k_2=k_1\frac{\pi _1\theta _1\lambda _2C_v^*}{\pi _2\lambda _1^2C_d^*}$ and $k_3=k_1\frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}$, $\frac{dU}{dt}$is simplified as

$$\begin{aligned} \frac{dU}{dt}=- & {} k_1\frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}(\theta _1C_v+\lambda _1C_d +\lambda _2C_r)(C_h-C_h^*)^2\nonumber \\- & {} (\theta _0+\theta _1C_h^*)(C_v-C_v^*)^2-k_1(\lambda _0+\lambda _1C_h^*)(C_d-C_d^*)^2\nonumber \\- & {} k_1\frac{\pi _1\theta _1\lambda _2C_v^*}{\pi _2\lambda _1^2C_d^*}r_0(C_r-C_r^*)^2 -k_1\frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}\delta _h(C_h-C_h^*)^2\nonumber \\+ & {} (C_v-C_v^*)(C_h-C_h^*)\left[ -\theta _1C_v\right] \nonumber \\+ & {} (C_v-C_v^*)(C_h-C_h^*)\left[ -k_1\frac{\pi _1\theta _1^2C_v^*}{\lambda _1}\theta _1C_h^*\right] \nonumber \\+ & {} (C_v-C_v^*)(C_d-C_d^*)\left[ -k_1(\lambda +\pi _1\theta _1C_h)\right] \nonumber \\+ & {} (C_d-C_d^*)(C_h-C_h^*)\left[ -k_1\lambda _1C_d\right] \nonumber \\+ & {} (C_d-C_d^*)(C_r-C_r^*)\left[ k_1\frac{\pi _1\theta _1\lambda _2C_v^*}{\pi _2\lambda _1^2C_d^*} (r+\pi _2\lambda _1C_h)\right] . \end{aligned}$$

(24)

Now $\frac{dU}{dt}$ will be negative definite inside the region of attraction provided the following conditions are satisfied:

$$\begin{aligned}&\left[ \theta _1\frac{Q_v}{\theta _0}\right] ^2<\frac{4}{9}k_1\delta _h \frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}(\theta _0+\theta _1C_h^*) \end{aligned}$$

(25)

$$\begin{aligned}&k_1\left[ \frac{\pi _1\theta _1^2C_v^*}{\lambda _1}\right] ^2<\frac{4}{9}\delta _h \frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}(\theta _0+\theta _1C_h^*) \end{aligned}$$

(26)

$$\begin{aligned}&k_1\left[ \lambda +\pi _1\theta _1\frac{Q_h}{\delta _h}\right] ^2<\frac{4}{9}(\theta _0+\theta _1C_h^*) (\lambda _0+\lambda _1C_h^*) \end{aligned}$$

(27)

$$\begin{aligned}&(\lambda _1R_d)^2<\frac{4}{9}\delta _h\frac{\pi _1\theta _1C_v^*}{\lambda _1C_h^*}(\lambda _0+\lambda _1C_h^*) \end{aligned}$$

(28)

$$\begin{aligned}&\frac{\pi _1\theta _1\lambda _2C_v^*}{\pi _2\lambda _1^2C_d^*} \left[ r+\frac{\pi _2\lambda _1Q_h}{\delta _h}\right] ^2<\frac{4}{3}r_0(\lambda _0+\lambda _1C_h^*) \end{aligned}$$

(29)

From above inequalities (25), (26) and (27), we may choose $k_1>0$ if the following condition is satisfied:

$$\begin{aligned} \lambda _1\theta _1C_v^*C_h^*<\frac{16}{81}\pi _1\theta _0^2\delta _h \min \left[ \frac{\lambda _1\delta _h}{\pi _1\theta _1^3C_v^*C_h^*}, \frac{\lambda _0+\lambda _1C_h^*}{\lambda +\pi _1\theta _1\frac{Q_h}{\delta _h}}\right] . \end{aligned}$$

(30)

Here we have used the fact that $(\theta _0+\theta _1C_h^*)=Q_v/C_v^*$. Thus, the system (1) is globally stable inside the region of attraction if the conditions (14), (15) and (16) are satisfied.