Parametric transitions between bare and vegetated states in water-driven patterns

Flow Variability.

Modeling flow variability using a minimalist approach is a challenging task in river science, because of the nontrivial features of the discharge time series Q(t) (e.g., long-term correlation, intermittent behavior, etc.). The Compound Poisson Process (CPP) provides a parsimonious and robust strategy (24) by addressing the stochastic equation ∂tQ = F(t)−Q/τ, in which t is time, τ is the integral temporal scale (namely the integral of the autocorrelation function of Q), F is a shot noise with mean intervals between two pulses equal to τcv2 and mean value Q¯cv2, and Q¯ and cv are the mean and the variation coefficient of the discharge. A sample realization of CPP is reported in Fig. 2A (blue line). At steady state, the PDF of the CPP is Gamma-distributed (Fig. 2 A, Right) and the upcrossing time Tξ+—that is, the average time Q stays above a certain threshold ξ—is known in closed form (25)Tξ+=τeξ/Q¯cv2E1−1cv2,ξQ¯cv2,[1]where E(a,z) is the exponential integral function (26). Eq. 1 will be used later on in the triad interaction.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 2.

CPP (A and B) and alternate bar formation (C and D). (A) Simulated CPP for the river water discharge Q and its probability density function pQ (light blue lines). Inset shows the periodic signal used for the Floquet theory (red lines). (B) Water discharge series for the Isère and the Alpine Rhine (last 4 y reported). Notice the different shape of the PDF, with the sharper Gamma distribution for the Isère. (C) Saturation of bar amplitude during a time ts as described by the Landau Eq. 2. (D) Neutral stability curve discriminating the domain of bar formation (gray area) from the stable uniform solution (k is the longitudinal wavenumber, ds=0.005 and F=0.75).

Morphodynamics.

The second element of the triad interaction—sediment transport—is a threshold process that activates when the flow rate overcomes a lower critical value and becomes morphologically effective when an upper statistically rare flow rate Qf is overcome (Fig. 2A). This so-called formative discharge, an open issue in geomorphology, resets the riverbed by erasing any previous pattern and uprooting all vegetation. The bare flat state so created is the starting point of our analysis.

In a straight river with nonerodible banks, when Q>Qf, the sediment transport morphodynamically triggers the pattern formation of migrating two-dimensional bedforms, called alternate bars (Fig. 1 D and E), which were theoretically studied through linear and weakly nonlinear analyses (27, 28). Most previous studies neglected the role of flow variability, with the notable exceptions of refs. 29 and 30. Starting from infinitesimal perturbations, the bar amplitude A(t) asymptotically saturates to a finite value As in a nonlinear fashion, essentially following a Stuart–Landau equation (Fig. 2C):∂tA=ΩA−σ|A|A2.[2]For our convenience, we define the saturation time ts, such that A(ts) = 0.99As and A(0) = 0.01As, and we stipulate that a condition for bar formation is that the formative event must have a duration at least equal to the saturation time—that is, TQf+≥ts. An evaluation of the complex growth rate Ω and the first Landau coefficient σ is here given through the technique of the Center Manifold Projection (CMP), which, unlike the commonly used multiple scaling theory (28), offers a means to find As even away from the critical conditions but close to the neutral curve (31, 32).

Let us introduce the following scaling quantities: the river half-width B for the horizontal lengths, the depth D0 for the vertical length, U0 = Qf/2BD0 for the velocity, B/U0 for the time scale, and us = (Δ−1)gd for the sediment velocity (Δ, d, and g being the relative density, particle size, and gravitational acceleration, respectively). The governing dimensionless numbers result: the aspect ratio β = B/D0, the relative roughness ds = d/D0, the sediment transport capacity γ = dsus/(1−p)U0 with p the porosity of sediment, and the Froude number F = U0/gD0. With the above scaling and constant formative discharge Qf, momentum and mass conservation lead to the celebrated dispersion relation for bar linear instability (27) and to the neutral condition Ωr = 0 (subscript r refers to the real part). This condition manifests as a marginal curve in the longitudinal wavenumber–aspect ratio plane, with a minimum at the critical point (kc,βc) (see Fig. 2D). The linear analysis also provides the bar wavelength (fastest growing mode). Further details are provided in Methods and SI Appendix, Section S2.

The weakly nonlinear analysis through CMP leads to Eq. 2, from which one obtains As = Ωr/σr and ts∼6.6/Ωr. Furthermore, as the definition of volumetric discharge reads Qf = 2βFg(d/ds)5/2, both the saturation time and the formative discharge are functions solely of the morphodynamic parameters (ds, d, β, F). This allows us to define just two out of the three CPP parameters (Q¯,cv,τ), because the third one is given by the above mentioned condition for bar formation—that is, TQf+(Q¯,cv,τ) = ts. With the aid of Eq. 1, such condition leads toτ∼(tsQf)/(cv2Q¯), for Qf≫μ.[3]

Vegetation Dynamics.

After the formative event is extinguished, the flow decreases to ordinary values, reducing the water level and letting the crests of the recently formed bars emerge. At this point, vegetation may grow on the dried areas depending on flow variability. Vegetation is assumed to develop by following a logistic Holling type III equation, wherein the carrying capacity, K*, can be assumed as a quadratic function of the water depth, K* = K*(D) (15, 33). In fact, fluvial tree species usually perform the maximum growth when the water table is at an optimal depth: At lower stages, the roots cannot reach the water, while at higher stages, water logging occurs, thus reducing respiration and gas exchange in the root zone (34). In addition, the submerged sites experience a decline in vegetation because of anoxic conditions and uprooting induced by flow drag, which is proportional to the water velocity squared (16, 35). After assuming a linear relationship between flow drag and biomass removal and defining ψ as the density of biomass per unit area normalized to the maximum carrying capacity, we have∂ψ∂t=νgψ(K−ψ)−θ(D)νdD|U|2ψ,[4]where νg and νd are dimensionless growth and decay factors of vegetation and are proxies of the plant biomechanical properties in relation to the specific hydromorphological context and K is the normalized carrying capacity (see SI Appendix, Section S4); θ(⋅) is the Heaviside function (i.e., the decay is active only at submerged sites, when the water depth D is positive). We emphasize that, since the water velocity U = (U,V) and depth D account for the presence of the bar, the coefficients of Eq. 4 are space- and time-dependent. During ordinary flows, it is likely that sediment transport vanishes in most of the submerged sites, so that the bars can be approximately regarded as stable, until the occurrence of the next formative event. Moreover, the different time scales for bar saturation, order of days, and vegetation growth, order of years, allow us to separate mathematically the process and treat it analytically.

In this description, vegetation develops as a secondary instability (36) over the finite-amplitude bed topography. Encompassing bar topography in space and flow variability in time, a secondary instability is therefore performed by linearizing Eq. 4 about a new basic state (labeled with tilde), namelyD̃=Du(t)+AsD^1(k)f1(n)eiks.[5]In Eq. 5, the first term r.h.s. refers to the uniform solution, which is dependent on time through the CPP (Du(t) is related to Q(t) through Manning’s equation). The second term accounts for the presence of the finite amplitude bars (see Methods for more details), in which s and n are the longitudinal and transversal coordinates and D^1 and f1 are the linear perturbation of the water depth and its transversal structure. Similar forms are considered for the other variables. To make the computation analytically accessible, we substitute the discharge stochastic time series with a statistically equivalent periodic one, constituted by a repetition of the typical mean hydrograph. The latter is obtained by the following compatibility conditions (see red curve in Fig. 2A): (i) The duration of the event equals the average interval between subsequent shots, T=τcv2; (ii) the discharge mean value between formative events is preserved; and (iii) the peak value is such that the coefficient of variation cv of the stochastic series is preserved. In this manner, the time dependency in Eq. 4 becomes periodic and Floquet’s theorem can be applied. Thus, linearizing Eq. 4 around Eq. 5, the solution for the vegetation is ψ∼P(t)eα(s,n)t, where P(t) is a periodic function and α(s,n)∈R is the so-called Floquet exponent, which readsα=1T∫0T(νgK̃−νdθ(D̃)D̃|Ũ|2)dt,[6]where K̃ = K(D̃). We remind the reader that α is spatially distributed and its sign provides the asymptotic behavior of the secondary instability: Wherever α>0 vegetation patterns develop. The value of α is more easily evaluated numerically, as the analytical solution of the integral in Eq. 6 is particularly cumbersome. Although the above theory has been described for river systems, it is quite general, and the same steps could be repeated for other ecogeomorphologic systems driven by unsteady flows.