  It was recently revealed that a rotating compact body responds dynamically
when it is subjected to a gravitomagnetic tidal field, even when this field is
idealized as time-independent. The dynamical response is characterized by
time-changing internal currents, and it was suspected to originate from
zero-frequency g-modes and r-modes driven by the tidal forces. In this paper we
provide additional insights into the phenomenon by examining the tidal response
of a rotating body within the framework of post-Newtonian gravity. This
approach allows us to develop an intuitive picture for the phenomenon, which
relies on the close analogy between post-Newtonian gravity and Maxwell's theory
of electromagnetism. In this picture, the coupling between the gravitomagnetic
tidal field and the body's rotational velocity is naturally expected to produce
an unbalanced Lorentz-like force within the body, and it is this force that is
responsible for the tidal currents. The simplicity of the fluid equations in
the post-Newtonian setting allows us to provide a complete description of the
zero-frequency modes and demonstrate their precise role in the establishment of
the tidal currents. We estimate the amplitude of these currents, and find that
for neutron-star binaries of relevance to LIGO, the scale of the velocity
perturbation is measured in kilometers per second when the rotation period is
comparable to 100 milliseconds. This estimate indicates that the tidal currents
may have a significant impact on the physics of neutron stars near merger.
