Effects of Hydraulic Soil Properties on Vegetation Pattern Formation in Sloping Landscapes

Abstract

Current models of vegetation pattern formation rely on a system of weakly nonlinear reaction–diffusion equations that are coupled by their source terms. While these equations, which are used to describe a spatiotemporal planar evolution of biomass and soil water, qualitatively capture the emergence of various types of vegetation patterns in arid environments, they are phenomenological and have a limited predictive power. We ameliorate these limitations by deriving the vertically averaged Richards’ equation to describe flow (as opposed to “diffusion”) of water in partially saturated soils. This establishes conditions under which this nonlinear equation reduces to its weakly nonlinear reaction–diffusion counterpart used in the previous models, thus relating their unphysical parameters (e.g., diffusion coefficient) to the measurable soil properties (e.g., hydraulic conductivity) used to parameterize the Richards equation. Our model is valid for both flat and sloping landscapes and can handle arbitrary topography and boundary conditions. The result is a model that relates the environmental conditions (e.g., precipitation rate, runoff and soil properties) to formation of multiple patterns observed in nature (such as stripes, labyrinth and spots).

Appendix

Integrating the Richards equation (2) over $x_3$ gives

$$\begin{aligned} \frac{\partial W}{\partial t} = q_3(\mathbf x_h,Z,t) - q_3(\mathbf x_h,s_u,t) - \int _Z^{s_u} (\nabla _h \cdot \mathbf {q}_h ) \mathrm {d}x_3 - S_t, \end{aligned}$$

(A1)

where $S_t (\mathbf x_h, t) \equiv \int _Z^{s_u} S (\mathbf x, t) \mathrm {d}x_3$ and $\mathbf {q}_h = (q_1,q_2)^\top$. According to Leibniz rule,

$$\begin{aligned} \int _Z^{s_u} (\nabla _h \cdot \mathbf {q}_h) \mathrm {d}x_3 = \int _Z^{s_u} \sum _{i=1}^2\frac{\partial q_i}{\partial x_i} \mathrm {d}x_3 = \sum _{i=1}^2 \left[ \frac{\partial }{\partial x_i} \int _Z^{s_u} q_i \mathrm {d}x_3 - \frac{\partial s_u}{\partial x_i} \, q_i(\mathbf x_h,s_u,t) \right] . \end{aligned}$$

(A2)

Hence, (A1) yields

$$\begin{aligned} \frac{\partial W}{\partial t} = - \nabla _h \cdot \mathbf {Q}_h -q_3(\mathbf x_h,s_u,t) + q_3(\mathbf x_h,Z,t) + \mathbf {q}_h(\mathbf x_h,s_u,t) \cdot \nabla _h s_u - S_t, \quad \end{aligned}$$

(A3)

where $\mathbf {Q}_h = (Q_1,Q_2)^\top$ is the specific (per unit length) volumetric flow rate, [L$^2$/T], whose components are given by

$$\begin{aligned} Q_i (\mathbf x_h, t) = \int _Z^{s_u} q_i ( \mathbf x, t) \mathrm {d}x_3, \qquad i = 1,2. \end{aligned}$$

(A4)

Next, we rewrite the equation for the soil surface as $\mathcal {F}(\mathbf x) \equiv x_3 - s_u(\mathbf x_h) = 0$. The unit normal vector to this surface is given by

$$\begin{aligned} \mathbf n = \frac{ \nabla \mathcal {F} }{|\nabla \mathcal {F} |} = \frac{ 1 }{|\nabla \mathcal {F} |} \left( - \frac{\partial s_u}{\partial x_1}, - \frac{\partial s_u}{\partial x_2}, 1 \right) ^\top . \end{aligned}$$

(A5)

Hence, the boundary condition (3) takes the form

$$\begin{aligned} \mathbf {q}_h \cdot \nabla _h s_u - q_3 = p |\nabla \mathcal {F} | \quad \text{ on } \quad x_3 = s_u(\mathbf x_h). \end{aligned}$$

(A6)

The boundary condition (4), together with $q_3 = - K_s K_r \dfrac{\partial }{\partial x_3} (x_3 + \psi )$ from the second relation in (2), gives rise to the condition

$$\begin{aligned} q_3 = - K_s K^\star _r \qquad \text{ on } \quad x_3 = Z, \end{aligned}$$

(A7)

where $K_r^\star$ is the relative conductivity value at $x_3 = Z$. Substituting (A6) and (A7) into (A3) yields

$$\begin{aligned} \frac{\partial W}{\partial t} = - \nabla _h \cdot \mathbf {Q}_h + p \sqrt{1 + |\nabla _h s_u|^2} - K_s K_r^\star - S_t, \nonumber \\ \quad Q_i = \int _Z^{s_u} q_i (\mathbf x_h, x_3) \mathrm {d}x_3 , \quad i = 1,2 \end{aligned}$$

(A8)

where $S_t(\mathbf x_h,t)$ is the total rate of water consumption by plants, $[\text {L/T}]$. Substituting the definition of the Darcy flux $\mathbf q$ in (2) into (A4) yields

$$\begin{aligned} Q_i = - K_s\int _Z^{s_u} K_r(\vartheta ) \frac{\partial \psi }{\partial x_i} \mathrm {d}x_3 = - K_s \int _Z^{s_u} D(\vartheta ) \frac{\partial \vartheta }{\partial x_i} \mathrm {d}x_3 , \qquad i = 1,2 \end{aligned}$$

(A9)

where $D(\vartheta )$ is the moisture diffusivity. Let $F(\vartheta ) \equiv \int ^\vartheta _0 D(s)\,\mathrm {d}s$, then

$$\begin{aligned} Q_i = - K_s\int _Z^{s_u} \frac{\partial }{\partial x_i} F (\mathbf x_h, x_3) \mathrm {d}x_3 , \qquad i = 1,2. \end{aligned}$$

(A10)

Using Leibniz rule,

$$\begin{aligned} Q_i / K_s = - \frac{\partial }{\partial x_i} \int _Z^{s_u} \mathrm {d}x_3 \, F (\mathbf x_h, x_3) + F(s_u, Z) \frac{\partial s_u}{\partial x_i}, \end{aligned}$$

(A11)

and substituting into (A8) leads to

$$\begin{aligned} \frac{\partial W}{\partial t} = K_s \nabla _h^2 G - K_s \nabla _h \cdot [F(s_u,Z) \nabla _h s_u] + p \sqrt{1 + |\nabla _h s_u|^2} - K_s K_r^\star - S_t. \end{aligned}$$

(A12)

with

$$\begin{aligned} G = \int _Z^{s_u} F(\mathbf x_h, x_3) \mathrm {d}x_3. \end{aligned}$$

(A13)

We assume that $K_r = \exp \left( \alpha \psi \right)$ and $\vartheta = \exp \left( \alpha \psi \right)$. Then,

$$\begin{aligned} D(\vartheta ) \equiv K_r(\vartheta ) \frac{\mathrm d \psi }{\mathrm d \vartheta } = \frac{1}{\alpha } \quad \text {and}\quad F(\vartheta ) = \frac{\vartheta }{\alpha }. \end{aligned}$$

(A14)

Hence, $G \equiv W / \alpha$, which yields (6).