  This is the second in a series of papers aimed at developing a practical
time-domain method for self-force calculations in Kerr spacetime. The key
elements of the method are (i) removal of a singular part of the perturbation
field with a suitable analytic "puncture" based on the Detweiler--Whiting
decomposition, (ii) decomposition of the perturbation equations in azimuthal
($m$-)modes, taking advantage of the axial symmetry of the Kerr background,
(iii) numerical evolution of the individual $m$-modes in 2+1-dimensions with a
finite difference scheme, and (iv) reconstruction of the physical self-force
from the mode sum. Here we report an implementation of the method to compute
the scalar-field self-force along circular equatorial geodesic orbits around a
Kerr black hole. This constitutes a first time-domain computation of the self
force in Kerr geometry. Our time-domain code reproduces the results of a recent
frequency-domain calculation by Warburton and Barack, but has the added
advantage of being readily adaptable to include the back-reaction from the self
force in a self-consistent manner. In a forthcoming paper---the third in the
series---we apply our method to the gravitational self-force (in the Lorenz
gauge).

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena