Multiplicative jump processes and applications to leaching of salt and contaminants in the soil

We consider simple systems driven multiplicatively by white shot noise, which appear in the modeling of the dynamics of soil nutrients and contaminants. The dynamics of these systems is analyzed in two ways: solving a hierarchy of linear ordinary differential equations for the moments, which gives a time scale of convergence of the stationary probability density function; and characterizing the crossing properties, such as the mean first-passage time and the mean frequency of level crossing. These results are readily applicable to the study of geophysical systems, such as the problem of accumulation of salt in the root zone, i.e., soil salinization.

Figure 1

Two trajectories of Eq. (4), with initial conditions above and below $x^{☆}=a/b$. The parameters are $a=b=c=1$, $\lambda =1.5$, and $\gamma =2.0$.

Figure 2

Probability distribution function of $x$, according to Eq. (11). (a) Colored shapes represent $p_x\left(x\right)$ calculated at the grid points of parameter space $\lambda /b$ versus $\gamma /c$. (b) Blowup of $p_x\left(x\right)$ calculated for three values of $(\gamma /c,\lambda /b)$: $(3.0,1.5)$ in red (black), $(0.5,1.5)$ in orange (dark gray), and $(0.5,2.5)$ in yellow (light gray). The red (black) dashed and dotted lines show the location of the mode and the mean, respectively, for $(\gamma /c=3.0,\lambda /b=1.5)$.

Figure 3

(a) Evolution of the first six normalized raw moments of $\chi$. The curves for $n=1,2,3$ are according to Eqs. (19). (b) Evolution of the mean, standard deviation, skewness, and excess kurtosis. Dashed lines denote analytical solutions and solid lines denote numerical solutions, calculated from an ensemble of $10000$ simulations. The parameters are $a=b=c=1$, $\gamma =3.5$, $\lambda =1.5$, and $m_i=0$.

Figure 4

(a) Simulations of trajectories as a function of time, according to Eq. (4). In green (blue), all trajectories start at $x_0=0.2$ ($x_0=1.8$) and are calculated until they reach the upper (lower) threshold $x_c=0.8$ ($x_c=1.0$). (b) and (c) Mean first-passage time as a function of starting point $x_0$, for processes with crossing thresholds $x_c=0.8$ and $1.0$, respectively. Solid lines denote analytical results and open circles denote numerical simulations carried out for ensembles of $10000$ runs. (d) Successive upcrossings of the threshold $x_c$. (e) Mean frequency of upcrossing $\nu$ as a function of the critical crossing threshold $x_c$. The solid line denotes analytical result shown in Eq. (21) and open circles denote numerical simulations carried out for ensembles of 5000 runs. The parameters are $a=b=c=1$ and (a)–(c) $\gamma =1.0$ and $\lambda =2.0$ and (d) and (e) $\gamma =2.0$ and $\lambda =2.0$.

Figure 5

(a) The PDF $p_x\left(x\right)$ for decreasing values of the parameter $b$ ($d^{-1}$) and the PDF $p_g\left(x\right)$ (thick line). (b) The thin gray lines denote the simulated dynamics of the salt mass, while the analytical dynamics for the mean salinity is shown in black, enclosed by one standard deviation in red dashed lines. (c) The steady-state PDF $p_g\left(x\right)$ is denoted by the black line and the histogram shows numerical results. (d) Crossing time as function of salinity threshold. The black line and circles correspond to analytical and numerical calculations of the MFPT, the green dashed line denotes the time $t_c$ for the mean to cross the threshold, according to Eq. (25), and the vertical dotted lines indicate the two salinity threshold levels considered, for pepper and tomato. The parameters are $a=0.054$ g $m^{-2}$ $d^{-1}$, $c=9.26\times {10}^{-2}$ ${\mathrm{cm}}^{-1}$ $d^{-1}$, $\gamma =0.93$ ${\mathrm{cm}}^{-1}$, and $\lambda =1.18\times {10}^{-2}$ $d^{-1}$.