  Important insights into the dynamics of spherically symmetric AdS-scalar
field perturbations can be obtained by considering a simplified time-averaged
theory accurately describing perturbations of amplitude epsilon on time-scales
of order 1/epsilon^2. The coefficients of the time-averaged equations are
complicated expressions in terms of the AdS scalar field mode functions, which
are in turn related to the Jacobi polynomials. We analyze the behavior of these
coefficients for high frequency modes. The resulting asymptotics can be useful
for understanding the properties of the finite-time singularity in solutions of
the time-averaged theory recently reported in the literature. We highlight, in
particular, the gauge dependence of this asymptotics, with respect to the two
most commonly used gauges. The harsher growth of the coefficients at large
frequencies in higher-dimensional AdS suggests strengthening of turbulent
instabilities in higher dimensions. In the course of our derivations, we arrive
at recursive relations for the coefficients of the time-averaged theory that
are likely to be useful for evaluating them more efficiently in numerical
simulations.
