In the two-dimensional radial geometry, a DBM aggregate has a circular envelope of radius with radially radiating branches. We have discussed this geometry in detail in reference  and include an overview here for completeness and to contrast with results for other geometries. A circular interface preserves its shape under Eqs. (1)-(7) and advances with a radial velocity
The conductivity anisotropy does not appear in Eq. (11) because there are no tangential currents in this radially symmetric solution. The linear stability of the interface is determined by studying the evolution of an m-fold sinusoidal perturbation of infinitesimal amplitude
As we previously reported , the relative growth rate of perturbations in this model is
This result was found to be in good qualitative and fair quantitative agreement with both numerical simulations and quasi-two-dimensional electrochemical deposition experiments.
For sufficiently large internal dissipation (values of larger than zero) and sufficiently strong anisotropy (small values of ), the relative growth rate is negative for small mode numbers; the circular envelope is stable against long wavelength perturbations under these conditions. Both dissipation and current confinement are required to stabilize the smooth envelope of densely branched structures. In contrast to Erlbacher et al.'s suggestion  that the dense branching morphology is stabilized in quasi-two dimensional experiments by three-dimensional effects, our result indicates that the DBM can be stable in purely two-dimensional systems.
Earlier one-sided models which attempted to account for dissipation in the the growth channels through corrections to the interfacial boundary condition failed to account for stable dense branching structures in electrochemical deposition when reasonable estimates for were used [4, 9, 11]. Even with a conductivity contrast as small as , the present model can account for the stability of the DBM provided the confinement of currents to the branches is sufficiently strong. Comparable values for have been estimated from measurements in quasi-two-dimensional electrochemical deposition experiments under conditions which formed the DBM . Strong confinement of currents to the branches in these experiments is reasonable since the electrolyte between the branches is known to be largely depleted of metal ions .