  Long-term sunspot observations and solar activity reconstructions reveal that
the Sun occasionally slips into quiescent phases known as solar grand minima,
the dynamics during which is not well understood. We use a flux transport
dynamo model with stochastic fluctuations in the mean-field and
Babcock-Leighton poloidal field source terms to simulate solar cycle
variability. Our long-term simulations detect a gradual decay of the polar
field during solar grand minima episodes. Although regular active region
emergence stops, compromising the Babcock-Leighton mechanism, weak magnetic
activity continues during minima phases sustained by a mean-field
$\alpha$-effect; surprisingly, periodic polar field amplitude modulation
persist during these phases. A spectral analysis of the simulated polar flux
time series shows that the 11-year cycle becomes less prominent while high
frequency periods and periods around 22 years manifest during grand minima
episodes. Analysis of long-term solar open flux observations appears to be
consistent with this finding. Through numerical experimentation we demonstrate
that the persistence of periodic amplitude modulation in the polar field and
the dominant frequencies during grand minima episodes are governed by the speed
of the meridional plasma flow -- which appears to act as a clock.
