Experiments such as those depicted in Fig. 1 have three-dimensional analogs which have been studied [13, 14, 15] almost as extensively as their more easily interpreted two-dimensional variants. Recent advances in admittance spectroscopy [14] and image analysis [15] make it possible to analyze the shapes of evolving three-dimensional patterns. High speed magnetic resonance imaging also has been applied to the study of fluid flow in porous media. Three dimensional radial pattern formation in which branches grow outward from the end of a conduit constitute another class of systems to which our growth model should pertain.

In the full three-dimensional radial geometry model, the unperturbed spherical interface advances with velocity

An infinitesimal perturbation to the growth front with a spherical harmonic

grows at the relative rate

where .
Figure 2 shows a plot of the growth rate as a function of the
perturbation harmonic for a range of dissipation values.
The growth rate is negative for values of *l* less than a
critical mode number which depends on the degree of
dissipation in the growth channels.
Such long wavelength perturbations shrink as the aggregate grows.
The greater , therefore,
the more nearly spherical the evolving pattern appears.
In Fig. 3 this marginally stable harmonic number is
plotted as a function of the amount of dissipation ( ) and
anisotropy ( ) in the system. As seen from the plot,
dissipation and current confinement act in concert to
stabilize the dense branching morphology in the
three dimensional radial geometry.

In the limit , we can solve for
the critical aggregate radius beyond which the envelope
will be stable against perturbations of harmonic number *l*:

In the anisotropic limit ( ), the expression simplifies to

For the DBM is stable against perturbations
of harmonic number greater than *l*.
Even for ,
an initially disordered core pattern
can cross over into a regime where densely branched growth
will be stable, provided the enveloping equipotential still is
reasonably smooth by the time the interface has advanced
to a mean position .
The size at which the envelope of the growing pattern becomes stable
scales with the size of the inner boundary, , and depends
strongly on the amount of dissipation in the system.
The position at which such a crossover to stable dense radial growth
might occur for
and
appears in Fig. 4 as a rapid increase in with .
Under certain conditions, decreases again for very
near *R* and a second crossover from stable
DBM to unstable growth is possible.

Mon May 20 13:07:47 CDT 1996