Stability of densely branched growth in dissipative diffusion-controlled systems

The dense branching morphology appears in a number of pattern-forming systems. Neither ordered nor fractal, this pattern is characterized by a large number of branches advancing at constant areal density behind a smooth envelope. We propose a two-sided model which accounts for the stability of the dense branching morphology (DBM) through dissipative and anisotropic current transport in the evolving pattern. Confinement of currents to slightly resistive branches suffices to stabilize radially symmetric DBM growth in two and three dimensions. Stability of the planar DBM, on the other hand, is found to require, in addition, the introduction of a characteristic length scale, such as a short diffusion length.