Variational analysis of landscape elevation and drainage networks

Abstract

Landscapes evolve towards surfaces with complex networks of channels and ridges in response to climatic and tectonic forcing. Here, we analyse variational principles giving rise to minimalist models of landscape evolution as a system of partial differential equations that capture the essential dynamics of sediment and water balances. Our results show that in the absence of diffusive soil transport the steady-state surface extremizes the average domain elevation. Depending on the exponent m of the specific drainage area in the erosion term, the critical surfaces are either minima (0 < m < 1) or maxima (m > 1), with m = 1 corresponding to a saddle point. We establish a connection between landscape evolution models and optimal channel networks and elucidate the role of diffusion in the governing variational principles.

1. Introduction

The dynamic of the land surface elevation is governed by the balance between soil erosion, deposition and production, which are modulated by climatic and tectonic activities [1–4]. When soil erosion is driven by water movement on the surface, landscape evolution gives rise to networks of channels and ridges with scaling laws typical of complex systems [5–8]. Minimalist mathematical models and physical experiments produce surface patterns resembling those observed in nature, helping to shed light on their statistical and scaling behaviour under different conditions of external forcing [1,2,9,10].

The landscape elevation is linked to the surface sediment budget, accounting for soil creep, erosion and uplift [4,11–13]. The erosion at a point is commonly assumed to be a power function of slope and water flux; thus, the sediment budget equation is coupled to an equation for the specific contributing area, resulting from a simplified water continuity equation [14]. This minimalist model produces a channelization sequence, the intensity of which is modulated by the relative magnitude of erosive to diffusive transport, with intriguing analogies with other complex non-equilibrium systems such as fluid turbulence [1,15].

The intertwined channel and ridge networks that emerge from this type of evolution (see figure 1, for example) are suggestive of an optimal transport strategy as observed in the branched regime of the Monge–Kantorovich problem where the cost function favours mass aggregation [16–18]. The optimality principle behind land-surface formation and river networks has been studied under transport-limited conditions and for the area of unchannelled slopes [19,20], and more commonly within the context of so-called optimal channel networks (OCNs) and optimal transportation theory [21–24]. OCNs are the configurations that connect sites over a discretized domain while minimizing the total energy loss of the water flow [21,25]. Depending on the exponent of the drainage area used for the computation of local energy dissipation (denoted by γ), the optimal configurations range from spanning trees, resembling natural river networks for 0 < γ < 1 [25], to spiral networks, for γ < 0 [26].

Figure 1.

The channelized surface generated by numerical solution of equations (2.1) and (2.2), when erosion defined as a power function of slope and specific drainage area is dominant over diffusive soil transport. The surface elevation is denoted by h and h_max is the maximum elevation. Model parameters in equation (2.1) are m = 0.5, D = 0.005 m² y⁻¹, K = m^0.5 y⁻¹ and U = 0.01 m y⁻¹. (Online version in colour.)

The connection between OCNs and landscape evolution models (LEMs) has been investigated in several contributions [21,22,27,28]. Most significantly, it has been shown that OCNs are stationary solutions of a sediment transport law. They correspond to landscapes that accommodate flow towards the steepest slope [21,28]. Such results are based on strong discretization [29], in which the surface is approximated by a network of connected nodes with links to represent the hydraulic connections. The generalization of these findings and the connection to models in the continuous domain have received less attention in the literature.

In this paper, we show that the steady-state landscapes obtained by solving the governing PDEs of sediment and water continuity equations in a continuous domain and under negligible soil diffusion assumption reach an optimal state, defined based on the average domain elevation. We prove that each of the two governing PDEs can be replaced by a corresponding variational principle. By investigating the second variation, we show that the optimal states may be reached by either minimizing or maximizing the average elevation based on the exponent of the specific drainage area in the erosion law. The role of diffusion on the underlying variational principle is also discussed.

2. Landscape evolution models

In LEMs, the land surface elevation h evolves as a function of diffusive soil creep, erosion, deposition and tectonic uplift. When the deposition of mobilized sediment is negligible (detachment-limited condition), the surface evolution can be written according to [4,10,11]

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=D\mathrm{\Delta }h-K{a}^m|\mathrm{\nabla }h|+U,$$ 2.1

where a is the specific upstream catchment area (or catchment area per unit contour length). The diffusive soil creep is given as DΔh and D is the soil diffusivity. The term $K{a}^m|\mathrm{\nabla }h|$ quantifies the erosion (sediment movement due to water flow) with constants m and K. The tectonic uplift rate is denoted by U. In this model, the specific catchment area a is proportional to the water height generated by a steady and spatially uniform rainfall [14]. With the assumption that water moves with constant speed in the direction of the steepest slope [30], a is given by the steady-state continuity equation of water flow over the surface [1,14],

$$\mathrm{\nabla }.\left(a\frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}\right)=-1.$$ 2.2

In a more general case, the erosion term in equation (2.1) is a power function of slope, i.e. $K{a}^m|\mathrm{\nabla }h{|}^n$ [1,10]. Here, we analyse the special case of n = 1; however, the results presented for the steady state in the absence of diffusion can be extended to the general case of n with a simple change of variables (m′ = m/n, K′ = K^1/n and U′ = U^1/n). The coupled equations (2.1) and (2.2) form a closed system. In a domain with a single length scale l (e.g. a square or a semi-infinite stripe of constant width), the behaviour of the system is controlled by the ‘channelization index’ ${\mathcal{C}}_{\mathcal{I}}=K{l}^{m+1}/D$ [14]. For low values of ${\mathcal{C}}_{\mathcal{I}}$, the steady-state surfaces are smooth with no concentrated flow paths. As ${\mathcal{C}}_{\mathcal{I}}$ exceeds a critical threshold, the surface becomes channelized with only first-order non-branched channels. In the infinite stripe case, this threshold is ${{\mathcal{C}}_{\mathcal{I}}}_{cr}\approx 37$ for m = 1, a result derived by linearizing the system of equations around its unchannelled solution for m = 1 [1]. Numerical simulations further showed that higher m values correspond to smaller ${{\mathcal{C}}_{\mathcal{I}}}_{cr}$ [1]. As the value of ${\mathcal{C}}_{\mathcal{I}}$ increases beyond ${{\mathcal{C}}_{\mathcal{I}}}_{cr}$, the surface becomes progressively channelized with more branching. Figure 2 shows the controls of ${\mathcal{C}}_{\mathcal{I}}$ in the level of surface channelization for a set of numerical simulations in a 100 m by 100 m domain. The parameter m also controls the structure of the channel networks: surfaces with higher m values tend to have more branched networks with wider channels and higher bifurcation angles [1].

Figure 2.

Numerical solution of the system of equations (2.1) and (2.2) in a square domain (100 m by 100 m with 1 m grid spacing) for m = 0.5, m = 1 and m = 1.5 with increasing C_I. As the channelization index increases (higher erosion), the surface becomes more dissected with branching channels. The three-dimensional view (b) is shown in figure 1. The numerical scheme uses an implicit solver for the erosion term where the specific drainage area a is approximated by the D_∞ flow direction method [1,31]. The model parameters are D = 0.005 m² y⁻¹ and U = 0.01 m y⁻¹, where K was appropriately selected to get the desired ${\mathcal{C}}_{\mathcal{I}}$ for a given m. The initial elevation field for an LEM model was a tent-shaped surface with 0.1 m elevation at the divide that was perturbed by white noise with zero mean and 0.001 m standard deviation. Before initiating the simulation, this rough surface was processed to fill any local minima to avoid the formation of lakes. (Online version in colour.)

3. Optimal channel networks

As mentioned in the Introduction, the scaling laws of channel networks have been mostly studied in relation to OCNs. These are the configurations that are defined over a lattice of N nodes and locally minimize [22,24,25],

$$J_{\text{OCN}}\hspace{0.17em}[c]=\sum _i{A}_i^{\gamma },$$ 3.1

where γ is a constant and A_i is the drainage area at node i (the total area that drains into node i ), which is uniquely defined based on the connectivity matrix of the configuration c. The previous condition is obtained by assuming a power relation between the local slope and the drainage area at each link (i.e. $|\mathrm{\nabla }h{|}_i\propto A_i^{\gamma -1}$), so that functional (3.1) is proportional to the total dissipated energy of water flow. From the same power relation between the local slope and the drainage area, Rinaldo et al. [32] showed that functional (3.1) is proportional to the average domain elevation of the discretized domain.

OCNs give rise to several statistical scaling laws (e.g. Horton’s laws of stream length and number [33]) observed in the natural river networks [8]. OCNs are commonly generated by numerical optimization of functional (3.1) over a lattice. Starting from an initial loop-less random configuration spanning the whole domain, at each iteration, a random change in the configuration is tried. If this change results in a loop-less configuration, the drainage area is computed from the altered connectivity matrix, which, in turn, gives a new value of functional (3.1). If the change reduces the numerical value of functional (3.1), it is accepted; otherwise, the change is reversed. This process is repeated until a given number of iterations (1000 in this study) with no accepted changes are tried. This approach is referred to as greedy, in the sense that it only accepts or rejects random changes based on the immediate response of the objective functional; therefore, it tends towards a so-called imperfect OCN [22]. Imperfect OCNs are feasible (dynamically accessible) optimal states of the system given the initial conditions and are shown to bear more resemblance to natural networks than the global (or near-global) optimal configuration achieved by more sophisticated optimization algorithms such as simulated annealing [22,25,34].

Figure 3 shows the configurations corresponding to a local minimum of functional (3.1) for different values of γ. The range 0 < γ < 1 leads to branching channels similar to natural rivers [25]. For γ > 1, the configuration minimizing the energy expenditure connects each node to the outlet (domain boundaries in this case) following the shortest path [25]. For γ < 0, loops are preferred and, if one excludes loops, spiral networks emerge [26] with configurations very different from those observed in the natural river networks. The role of γ is reminiscent of the behaviour of the Monge–Kantorovich optimal transport, where congested (similar to figure 3a) or branched (similar to figure 3b) transport may occur depending on the exponent of the Wasserstein distance used to penalize the transport of mass [17,18]. The extended dynamic Monge–Kantorovich formulation proposed by [35] also exhibits a similar transition from congested to branched transport depending on the sub- or super-linear growth of the transport density with respect to the transport flux.

Figure 3.

The configurations that locally minimize functional (3.1) in a 30 m by 30 m domain with 1 m grid spacing. (a) For γ > 1, the optimal configuration follows the shortest path to the outlet (domain boundary). (b) For 0 < γ < 1, tree-like structures emerge as the optimal configurations, which resemble natural river networks. (c) If only loop-less configurations are allowed, γ < 0 results in spiral configurations. These results are achieved by a greedy optimization algorithm starting from a random loop-less initial configuration. The widths on the links are proportional to $\sqrt{A}$. (Online version in colour.)

4. Variational principle for landscape evolution models in the absence of diffusion

(a). The first variation

We first consider the case of negligible diffusive transport (i.e. D = 0 or ${\mathcal{C}}_{\mathcal{I}}\to \mathrm{\infty }$). The case D > 0 is studied in §6. Under the assumption that a steady-state solution exists (dh/dt = 0 in equation (2.1) with zero elevation at the boundaries), we show that this solution corresponds to a critical function of the objective functional

$$J[h]={\int }_{S}h\hspace{0.17em}\text{d}s,$$ 4.1

where the integral is over the domain S and Ω is the domain boundary. In particular, we prove in this section that the first variation of functional (4.1), subject to the constraint of the sediment continuity equation (equation (2.1) at steady state with D = 0), vanishes when the h and a fields satisfy equation (2.2). Functional (4.1) is proportional to the average domain elevation and is equivalent to the mean potential energy of the surface water, which gradually dissipates as it flows towards the domain boundaries.

We first consider functional (4.1) for the landscape elevation. Following the classical approach of the calculus of variation originally developed by Lagrange [36], the constraint of the sediment balance equation can be imposed using a Lagrange multiplier field λ as

$$J[h]={\int }_{S}h+\lambda (-K{a}^m|\mathrm{\nabla }h|+U)\text{d}s.$$ 4.2

In this manner, we ignore the specific area equation, which is replaced by the condition of extremization of functional (4.1), and thus treat a independent of h.

Introducing a variation g of h over the domain S with the boundary conditions g(x ∈ Ω) = 0, the first variation of the functional J in equation (4.2) is

$$\begin{array}{rl}\delta J[h,g]=\frac{\text{d}J[h+ϵg]}{\text{d}ϵ}{|}_{ϵ=0}& =\frac{\text{d}}{\text{d}ϵ}{\int }_S(h+ϵg)+\lambda (-K{a}^m|\mathrm{\nabla }h+ϵ\mathrm{\nabla }g|+U)\text{d}s{|}_{ϵ=0}\\ & ={\int }_{S}g\hspace{0.17em}\text{d}s-K{\int }_S\lambda a^m\frac{\text{d}}{\text{d}ϵ}|\mathrm{\nabla }h+ϵ\mathrm{\nabla }g|\hspace{0.17em}\text{d}s{|}_{ϵ=0}\\ & ={\int }_{S}g\hspace{0.17em}\text{d}s-K{\int }_S\lambda a^m\frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}\cdot \mathrm{\nabla }g\hspace{0.17em}\text{d}s.\end{array}$$ 4.3

Integrating the second term by parts and using the boundary condition g(x ∈ Ω) = 0 leads to

$$\delta J[h,g]={\int }_{S}g[1+K\mathrm{\nabla }\cdot \left(\lambda a^m\frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}\right)]\text{d}s.$$ 4.4

For the critical functions, the first variation must vanish (δJ[h, g] = 0) for any g; thus, the fundamental lemma of calculus of variations [36,37] yields

$$1+K\mathrm{\nabla }\cdot \left(\lambda a^m\frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}\right)=0.$$ 4.5

This shows that equation (4.5) is equivalent to equation (2.2) for λ = (1/K)a^1−m, proving our claim at the beginning of this section. Thus, the a and h fields that extremize functional (4.1) and follow the erosion law given by equation (2.1) with D = 0 and $\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=0$ also satisfy equation (2.2) and vice versa. This means that equation (2.2) in the system of equations governing landscape evolution can be replaced by a variational principle based on the mean elevation over the domain, that is, equation (4.1). This result can also be achieved by writing the Euler–Lagrange equation [37] of the functional (4.2) with the appropriate boundary condition.

(b). Dual formulation

It is interesting to show that the problem has a dual formulation in which the sediment continuity can be replaced by a variational principle based on a functional, constructed with the a field, given by

$$J[a]={\int }_S{a}^{1-m}\hspace{0.17em}\text{d}s.$$ 4.6

Proceeding as before, but now with b defined as a variation of a with the boundary condition b(x ∈ Ω) = 0, the first variation of functional (4.6) with the constraint of equation (2.2) is

$$\begin{array}{rl}\delta J[a,b]=\frac{\text{d}J[a+ϵb]}{\text{d}ϵ}{|}_{ϵ=0}& =\frac{\text{d}}{\text{d}ϵ}{\int }_S{(a+ϵb)}^{1-m}+\lambda [\mathrm{\nabla }\cdot ((a+ϵb)\frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|})+1]\text{d}s{|}_{ϵ=0}\\ & ={\int }_S(1-m)b{(a+ϵb)}^{-m}\hspace{0.17em}\text{d}s+{\int }_S\lambda (\mathrm{\nabla }b\cdot \frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}+b\mathrm{\nabla }\cdot \frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|})\text{d}s{|}_{ϵ=0}\\ & ={\int }_{S}b[(1-m)\hspace{0.17em}{a}^{-m}+\lambda \mathrm{\nabla }\cdot \frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}]\text{d}s+{\int }_S\lambda \mathrm{\nabla }b\cdot \frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}\hspace{0.17em}\text{d}s\\ & ={\int }_{S}b[(1-m)\hspace{0.17em}{a}^{-m}-\mathrm{\nabla }\lambda \cdot \frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}]\text{d}s,\end{array}$$ 4.7

where we used integration by parts with b(x ∈ Ω) = 0 as the boundary condition. The first variation must be zero for any variation b; therefore, we have [36,37]

$$(1-m)\hspace{0.17em}{a}^{-m}-\mathrm{\nabla }\lambda \cdot \frac{\mathrm{\nabla }h}{|\mathrm{\nabla }h|}=0,$$ 4.8

which yields the sediment continuity (equation (2.1) at the steady state with D = 0) for λ = (K(1 − m)/U) h. It is interesting to note that the objective functional (4.6) is equivalent to functional (4.2) at its critical points (refer to appendix Aa for details).

(c). The second variation

We study the second variation at the critical points by writing functional (4.2) as (refer to appendix Aa for details)

$$J[h]={\left(\frac{U}{K}\right)}^{1/m}{\int }_S|\mathrm{\nabla }h{|}^{(m-1)/m}\hspace{0.17em}\text{d}s.$$ 4.9

Defining a variation g with g(x ∈ Ω) = 0 as the boundary condition, the second variation of this functional is

$$\begin{array}{rl}{\delta }^{2}J[h,g]=\frac{{\text{d}}^{2}J[h+ϵg]}{\text{d}{ϵ}^2}{|}_{ϵ=0}& ={\left(\frac{U}{K}\right)}^{1/m}\frac{{\text{d}}^2}{\text{d}{ϵ}^2}{\int }_S|\mathrm{\nabla }h+ϵ\mathrm{\nabla }g{|}^{(m-1)/m}\hspace{0.17em}\text{d}s{|}_{ϵ=0}\\ & =\frac{m-1}{m}{\left(\frac{U}{K}\right)}^{1/m}\frac{\text{d}}{\text{d}ϵ}{\int }_S|\mathrm{\nabla }h+ϵ\mathrm{\nabla }g{|}^{-\frac{m+1}{m}}\mathrm{\nabla }g\cdot (\mathrm{\nabla }h+ϵ\mathrm{\nabla }g)\text{d}s{|}_{ϵ=0}\\ & =\frac{1-m}{m^2}{\left(\frac{U}{K}\right)}^{1/m}{\int }_S|\mathrm{\nabla }g{|}^2|\mathrm{\nabla }h{|}^{-\frac{m+1}{m}}\text{d}s.\end{array}$$ 4.10

Thus, a critical function of J is a minimum for any non-zero variation if and only if [37]

$${\delta }^{2}J[h,g]=\frac{1-m}{m^2}{\left(\frac{U}{K}\right)}^{1/m}{\int }_S|\mathrm{\nabla }g{|}^2|\mathrm{\nabla }h{|}^{-\frac{m+1}{m}}\text{d}s>0.$$ 4.11

From the constraint of the sediment continuity, we have $|\mathrm{\nabla }h|>0$. Given the boundary condition of the variation g (i.e. g(x ∈ Ω) = 0), if $|\mathrm{\nabla }g|=0$, it immediately follows that g ≡ 0. This means that the integral on the RHS of equation (4.11) is positive definite for any non-zero variation g. Thus, the critical functions of the variational problem in equation (4.2) are minima for 0 < m < 1. Following the same line of reasoning, it is easy to show that the critical functions are maxima for m > 1. At m = 1, the critical functions are saddles, having the same value as the objective function. This result exactly follows from integrating equation (2.1) over the domain with area A_s, D = 0 and zero elevation at the boundaries, which yields ${\int }_{S}a|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s=(U/K)A_s$. Given that ${\int }_{S}a|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s={\int }_{S}h\hspace{0.17em}\text{d}s$ (refer to appendix Aa for details), we have ${\int }_{S}h\hspace{0.17em}\text{d}s=(U/K)A_s$, which indicates that steady-state solutions have the same average elevation regardless of the a field.

5. Connection with optimal channel networks

To interpret these results in the context of the OCNs, one should note that the objective functional $\int a^{1-m}\hspace{0.17em}\text{d}s$ defined over a continuous domain can be written on a lattice with dx discretization size as $\text{d}{x}^{m-1}\sum A_i^{1-m}$ using the approximation A_i = a_i dx, where a_i is defined as the centre of grid i [1,31]. Thus, the functional (3.1) is proportional to the functional (4.6) and γ ≡ 1 − m. As discussed earlier, the exponent γ controls the structure of the OCNs. For 0 < γ < 1 (equivalent to 0 < m <1), each spanning tree is a local minimum [26] for which there is a height function such that water flows towards the steepest descent [21]. This is in agreement with our finding in the continuum formulation, which states that the critical functions of functional (4.2) should satisfy a continuity equation for a, constructed upon the assumption of water moving towards the surface gradient. The range γ > 1 (equivalent to m < 0) gives a configuration with the shortest path to the outlet [25], although this case is not physically possible according to the sediment continuity [22].

The numerical solution of the system of equations (2.1) and (2.2) with D = 0 (i.e. C_I > >1) and m > 1 gives surfaces with channel networks resembling those observed in nature (e.g. figure 2j, which shows the steady-state surface for m = 1.5 and ${\mathcal{C}}_{\mathcal{I}}={10}^5$). On the other hand, minimizing functional (3.1) with γ < 0 (equivalent to m > 1) creates loops or spiral paths as shown in figure 3c. The second variation analysis performed earlier addresses this discrepancy by showing that realistic networks for OCNs with γ < 0 (i.e. m > 1) emerge by maximizing equation (4.6) or its discretized form in equation (3.1), whereas for 0 < γ < 1 and γ > 1 (i.e. 0 < m < 1 and m < 0) the objective functional should be minimized.

We confirm this finding by a set of numerical experiments where we minimized equation (4.6) for γ = 0.5 and maximized it for γ = −0.5 in a discretized domain following the greedy optimization algorithm explained in §3. In figure 4, the optimal a fields are compared with the results from the numerical solution of equations (2.1) and (2.2) for m = 0.5 (i.e. γ = 0.5) and m = 1.5 (i.e. γ = −0.5). Both approaches converged to qualitatively similar a fields as shown in figure 4a(ii), b(ii). Figure 4a(i), b(i) shows the distribution of a (exceedance probability) from OCNs and LEMs that contain a very similar power distribution for small a, which is distributed by the finite-size effect for large a. The exponents close to −0.43 as observed for γ = 0.5 (i.e. m = 0.5) are widely reported in the literature as the signature of the feasible optimality, whereas exponents close to −0.5 characterize the (near) global optimal state [8,22].

Figure 4.

Distribution of the specific drainage area a for (a(i)) m = 0.5 and (b(i)) m = 1.5 with a greedy optimization of equation (4.6) (denoted by OCN) and the numerical simulation of the PDEs in equations (2.1) and (2.2) (denoted by LEM). The fitted power functions to the first portion of the distribution from the OCN with the scaling exponents are also shown. Both approaches are applied to the same 100 m by 100 m domain with 1 m grid spacing. The resulting a fields are shown in panels (ii). The optimization scheme was a greedy minimization for m = 0.5 and maximization for m = 1.5, in which random changes in node connectivity that led to better objective functions (smaller for m = 0.5 and larger for m = 1.5) are accepted, until no change occurs for a given number of trials (1000 for these results). (Online version in colour.)

6. The role of diffusion

Soil diffusion transport smooths the surface and tends to prevent the formation of channels [1]. Figure 5 shows the effect of diffusion on the steady-state surfaces achieved by numerically solving the governing equations in a long rectangular domain with zero elevation at the boundaries. Smaller diffusion (higher ${\mathcal{C}}_{\mathcal{I}}$) results in a flatter mean-elevation profile $\overline{h}$. The profiles were computed by averaging the elevation along the x-axis for 100 < x < 600 m to minimize the effect of the side boundaries. The choice of the long domain was meant to approximate a semi-infinite case with parallel boundaries where the distance between two boundaries (l_y = 100 m) is the only dominant length scale in computing ${\mathcal{C}}_{\mathcal{I}}$.

Figure 5.

(a) The effect of diffusion on the mean-elevation profile, denoted by $\overline{h}$, in a long rectangular domain (700 m by 100 m). Higher ${\mathcal{C}}_{\mathcal{I}}$ (relatively smaller diffusion) results in progressively more uniform mean profiles. The normalized h fields for an unchannelized case with ${\mathcal{C}}_{\mathcal{I}}=10$ (b) and a channelized case with ${\mathcal{C}}_{\mathcal{I}}={10}^2$ (c) 10³ (d) and 10⁴ (e) are shown. The model parameters are the same as those reported in figure 2 and m = 0.5. (Online version in colour.)

In this section, we show that one can recover equation (2.2) for the general case (non-zero K and D) by using the sediment continuity equation (2.1) and the variational principle (6.1),

$$J[h]={\int }_{S}h-\frac{D}{K}h\mathrm{\Delta }{a}^{1-m}\hspace{0.17em}\text{d}s.$$ 6.1

Adding the steady-state sediment continuity (equation (2.1)) to functional (6.1) as the constraint with the Lagrange multiplier field λ gives

$$J[h]={\int }_{S}h-\frac{D}{K}h\mathrm{\Delta }{a}^{1-m}+\lambda (D\mathrm{\Delta }h-K{a}^m|\mathrm{\nabla }h|+U)\text{d}s.$$ 6.2

Following similar steps to those for the case D = 0 and using an additional boundary condition on the variation g (i.e. $\mathrm{\nabla }g\cdot n{|}_{\mathrm{\Omega }}=0$ ), the first variation of this functional is

$$(1-\frac{D}{K}\mathrm{\Delta }{a}^{1-m})+K\mathrm{\nabla }\cdot \left(\lambda a^m\frac{\mathrm{\nabla }z}{|\mathrm{\nabla }z|}\right)+D\mathrm{\Delta }\lambda =0,$$ 6.3

which is equivalent to the water balance equation (equation (2.2)) with $\lambda =\frac{1}{K}{a}^{1-m}$. At a stable point of equation (6.2), we have $\lambda =\frac{1}{K}{a}^{1-m}$ and the water and sediment continuity equation also hold; therefore, equation (A 7) is valid and equation (6.2) is

$$J[h]=\frac{D}{K}{\int }_S{a}^{1-m}\mathrm{\Delta }h\hspace{0.17em}\text{d}s-\frac{D}{K}{\int }_{S}h\mathrm{\Delta }{a}^{1-m}\hspace{0.17em}\text{d}s+\frac{U}{K}{\int }_S{a}^{1-m}\hspace{0.17em}\text{d}s.$$ 6.4

Integration by parts of the first two terms and using the boundary condition (h_Ω = 0) leads to

$$J[h]=\frac{U}{K}{\int }_S{a}^{1-m}\hspace{0.17em}\text{d}s+\frac{D}{K}{\int }_{\mathrm{\Omega }}{a}^{1-m}(\mathrm{\nabla }h\cdot n)\hspace{0.17em}\text{d}\omega ,$$ 6.5

where the second integral is a flux at the domain boundary and the first integral is the objective function derived earlier for D = 0, which resembles the objective functional of the OCN theory (equation (3.1)). From a numerical perspective, the functional (6.5) can be used to generate locally optimal configurations; however, the dependence on both h and a makes a numerical solution challenging. In contrast to the case D = 0, where the variational principle can be written only with a single variable (i.e.the specific catchment area a), for D > 0 both elevation and area are present in the objective functional of equation (6.5). When the objective functional is only a function of a, one can easily use a search algorithm to gradually move towards a near-optimal configuration, as is commonly done for OCNs. However, when both h and a appear in the functional, using the OCN-type algorithm may not be straightforward since any change in configuration modifies both a and h.

Another specific case of this problem corresponds to the condition of negligible erosion (K = 0), in which the sediment continuity becomes the Poisson equation and decouples from the water-continuity equation,

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=D\mathrm{\Delta }h+U.$$ 6.6

At steady state, the solution of this equation corresponds to the critical function of the functional (refer appendix Ab for details),

$$J[h]={\int }_{S}h+\frac{D}{2U}|\mathrm{\nabla }h{|}^2\hspace{0.17em}\text{d}s.$$ 6.7

This functional without h in the integrand is the Dirichlet integral and is similar to the variational formulation of the Fokker–Planck equation associated with the Brownian motion [38,39]. By analysing the second variation, it is easy to show that the critical function of this functional is a minimum for D/U > 0 (refer appendix Ab for details).

7. Conclusion

The minimalist LEM considered here has been used to model the essential dynamics of surface evolution to reproduce statistically similar surfaces to those observed in nature [3,40]. On the other hand, networks generated by minimizing the total energy dissipation have been shown to embody several scaling laws observed in natural river networks [8], hinting at a possible connection between these two fundamental theories of landscape evolution [21,22,26,28].

We have shown that at steady state each equation of this LEM can be independently replaced by a variational principle in the absence of the diffusive transport, a finding that proves the tendency of landscapes towards an optimal state. The dual formulation of the variational principle in terms of elevation and specific drainage area resembles the dual (or reciprocal) variational principles in diffusion, elasticity and viscous flow [41–43].

The assumption of negligible diffusion introduces challenges related to the possible development of singularities. However, similarly to the insights derived from the analysis of the Euler equation for inviscid fluids, neglecting diffusion allows a clear derivation of variational principles [44] as well as a clear connection with OCN theory. In addition to steady-state conditions, optimality in the river network should also be expected in slowly evolving landscapes where a quasi-steady state can be assumed after the rapid formation of the drainage network [28].

Interestingly, the properties of the optimal state (critical functions) vary based on a model parameter m (exponent of the specific drainage area in the erosion term). For 0 < m < 1, the optimal states are minima, for m > 1 they are maxima, and at m = 1, which corresponds to a saddle point, the objective functional is independent of the surface connectivity and the specific drainage area. This result is relevant for OCN theory: OCNs are generally associated with a specific range of the parameter γ (i.e. 0 < γ < 1, where γ ≡ 1 − m). However, our results suggest that OCN theory applies to γ < 0, where the objective functional should be maximized.

A. Appendix

(a) Objective functional in the absence of diffusion

In this section, we show that functional (4.2) at its critical points is equivalent to functional (4.6). We begin by substituting λ = (1/K) a^1−m in equation (4.2),

$$J[h]={\int }_S(h-a|\mathrm{\nabla }h|+\frac{U}{K}{a}^{1-m})\text{d}s.$$ A 1

To compute the term ${\int }_{S}a|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s$, it is convenient to map the integral into a curvilinear coordinate system (u, v), where v is in the direction of the surface gradient and u is perpendicular to v (figure 6). Given that the streamlines are defined in the direction of the gradient, the coordinates v and u are tangent to the streamlines and contour lines, respectively. In such an orthogonal curvilinear coordinate, the element of the area is defined as $\text{d}s=\mathcal{J}\hspace{0.17em}\text{d}u\text{d}v$, where $\mathcal{J}$ is the Jacobian defined as [45,46]

$$\mathcal{J}=|x_u{y}_v-x_v{y}_u|,$$ A 2

where the subscripts denote the derivative with respect to that coordinate. The elements of length along the u and v coordinates are also defined as $\text{d}w=\sqrt{E}\hspace{0.17em}\text{d}u$ and $\text{d}l=\sqrt{G}\hspace{0.17em}\text{d}v$, where $E=x_u^2+y_u^2$ and $G=x_v^2+y_v^2$. Given the orthogonality of u and v, we have $\mathcal{J}=\sqrt{EG}$ [45,46].

Figure 6.

Curvilinear coordinate system (u, v), where v is in the direction of the surface gradient and in the direction of streamlines, whereas u is perpendicular to v and in the direction of contourlines. The schematic definitions of contour length w and catchment area A in equation (A 3) are also shown. (Online version in colour.)

The specific catchment area a from equation (2.2) is by definition the catchment area A per unit contour length w for w → 0 [14], which can be transferred to curvilinear coordinates through the following integral equations [46]:

$$a=\underset{w\to 0}{lim}\frac{A}{w}=\frac{1}{\sqrt{E}}{\int }_v\mathcal{J}\hspace{0.17em}\text{d}v.$$ A 3

Using a in equation (A 3) and given that $|\mathrm{\nabla }h|=-(\mathrm{\partial }h/\mathrm{\partial }l)$, where $\text{d}l=\sqrt{G}\hspace{0.17em}\text{d}v$ is the length of the element along v (i.e. along the streamline), the integral ${\int }_{S}ha|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s$ is

$${\int }_{S}a|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s={\int }_u{\int }_v(\frac{1}{\sqrt{E}}{\int }_v\mathcal{J}\hspace{0.17em}\text{d}q)\frac{1}{\sqrt{G}}\frac{\mathrm{\partial }h}{\mathrm{\partial }v}\mathcal{J}\hspace{0.17em}\text{d}v\hspace{0.17em}\text{d}u.$$ A 4

Integrating by parts gives

$${\int }_{S}a|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s={\int }_u[h({\int }_v\mathcal{J}\hspace{0.17em}\text{d}q){|}_0^{v_t}-{\int }_{v}h\frac{\mathrm{\partial }}{\mathrm{\partial }v}({\int }_v\mathcal{J}\hspace{0.17em}\text{d}q)\text{d}v]\text{d}u.$$ A 5

At the initiation point of each streamline (v = 0), we have ${\int }_v\mathcal{J}\hspace{0.17em}\text{d}q=0$. The terminal point (v = v_t) where a streamline flows out of the domain is in fact a point along the boundaries in our case, where we have already enforced h = 0 as the boundary condition; therefore,

$$h({\int }_v\mathcal{J}\hspace{0.17em}\text{d}q){|}_0^{v_t}=0.$$ A 6

The derivative with respect to v in the second term of the integral in equation (A 5) is simplified as

$${\int }_{S}a|\mathrm{\nabla }h|\hspace{0.17em}\text{d}s={\int }_u{\int }_{v}h\mathcal{J}\hspace{0.17em}\text{d}v\text{d}u={\int }_{S}h\hspace{0.17em}\text{d}s.$$ A 7

Thus, equation (A 1) is simplified as (4.6), from which it is easy to write the functional in terms of gradient (equation (4.9)) using the sediment balance equation.

(b) Variations in the absence of erosion

Similarly to the case D = 0 and with the same definition for g, the first variation is defined as

$$\begin{array}{rl}\delta J[h,g]=\frac{dJ[h+ϵg]}{dϵ}{|}_{ϵ=0}& =\frac{\text{d}}{\text{d}ϵ}{\int }_S(h+ϵg)+\frac{D}{2U}|\mathrm{\nabla }h+ϵ\mathrm{\nabla }g{|}^2\hspace{0.17em}\text{d}s{|}_{ϵ=0}\\ & ={\int }_{S}g+\frac{D}{U}\mathrm{\nabla }g\cdot (\mathrm{\nabla }h+ϵ\mathrm{\nabla }g)\hspace{0.17em}\text{d}s{|}_{ϵ=0}\\ & ={\int }_{S}g+\frac{D}{U}\mathrm{\nabla }g\cdot \mathrm{\nabla }h\hspace{0.17em}\text{d}s\\ & ={\int }_{S}g(1+\frac{D}{U}\mathrm{\Delta }h)\text{d}s,\end{array}$$ A 8

while the second variation is

$$\begin{array}{rl}{\delta }^{2}J[h,g]=\frac{d^{2}J[h+ϵg]}{d{ϵ}^2}{|}_{ϵ=0}& =\frac{{\text{d}}^2}{\text{d}{ϵ}^2}{\int }_S(h+ϵg)+\frac{D}{2U}|\mathrm{\nabla }h+ϵ\mathrm{\nabla }g{|}^2\hspace{0.17em}\text{d}s{|}_{ϵ=0}\\ & =\frac{D}{U}{\int }_S|\mathrm{\nabla }g{|}^2\hspace{0.17em}\text{d}s.\end{array}$$ A 9

Data accessibility

The code for generating OCNs is available at https://github.com/MiladHooshyar/Optimal-Channel-Network.

Authors' contributions

M.H. and A.P. designed the research; M.H. performed the research; M.H. and S.A analysed the data. All authors discussed the results and wrote the paper.

Competing interests

We declare we have no competing interest.

Funding

No funding has been received for this article.