  Nacre, commonly known as mother-of-pearl, is a remarkable biomineral that in
red abalone consists of layers of 400-nm thick aragonite crystalline tablets
confined by organic matrix sheets, with the $(001)$ crystal axes of the
aragonite tablets oriented to within $\pm$ 12 degrees from the normal to the
layer planes. Recent experiments demonstrate that this orientational order
develops over a distance of tens of layers from the prismatic boundary at which
nacre formation begins.
  Our previous simulations of a model in which the order develops because of
differential tablet growth rates (oriented tablets growing faster than
misoriented ones) yield patterns of tablets that agree qualitatively and
quantitatively with the experimental measurements. This paper presents an
analytical treatment of this model, focusing on how the dynamical development
and eventual degree of order depend on model parameters. Dynamical equations
for the probability distributions governing tablet orientations are introduced
whose form can be determined from symmetry considerations and for which
substantial analytic progress can be made. Numerical simulations are performed
to relate the parameters used in the analytic theory to those in the
microscopic growth model. The analytic theory demonstrates that the dynamical
mechanism is able to achieve a much higher degree of order than naive estimates
would indicate.
