Laplacian networks: Growth, local symmetry, and shape optimization

Abstract

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry, and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.

Distributary Channel Networks as Moving Boundaries: Causes and Morphodynamic Effects

“…Preliminary model runs were dominated by many narrow channels that scaled with the segment length chosen in the method, which we deemed unreasonable. This is a common behavior in models controlled by Laplace's equation, where all wavelengths are unstable and grow, sometimes called the “ultraviolet crisis” (Devauchelle et al, ; Pecelerowicz & Szymczak, ). It is also important to note that for a given channel depth as modeled here, channel widths cannot be arbitrarily narrow.…”

Finger Growth and Selection in a Poisson Field

“…where d 1,2 are coefficients determined by the global numerical solution of the Poisson problem, r and θ are local polar coordinates such that θ = ±π coincides with the finger near its tip at r = 0. The principle of local symmetry requires that the fingers grows in a direction such that d 2 = 0 [3,7]. As noted in [4] while the path selection and growth mechanism of fingers is similar at a local level near finger tips for both harmonic (Laplacian) and Poisson fields, the trajectories will differ since the coefficients in (1) depend on the global field.…”

“…In the Laplacian case the principle of local symmetry states that the coefficient of δ vanishes in the expansion (8) [3]. Effectively this guarantees that the path taken by the finger is such that it maintains local symmetry in the potential field about the tip and is equivalent to maximising the flux into the tip [4,7]. However in the Poisson case this must be modified owing to the contribution of the term (2) that has been added to ψ needed to satisfy the right hand side of Poisson's equation.…”

“…Laplacian growth models have been successful in describing many features of these networks: a prominent example of this being an explanation of the remarkable observation that the angle at which streams bifurcate is close to 2π/5 e.g. [6,7,15]. However, the groundwater flow field φ is more appropriately modelled by Poisson's equation, in non-dimensional form, Δφ = −1 where the constant right-hand side represents a constant source (precipitation).…”

Symmetric rearrangement of groundwater-fed streams

“…If streams advance at an angle narrower than 72 • , the streamlines entering the two springs will bend away from each other; if streams advance at an angle wider than 72 • , the streamlines entering the springs will bend towards each other [10,11]. The 72 • confluence therefore can be considered a stable fixed point for stream advance [13]. An illustration of such a confluence and its streamlines is shown in figure 1.…”