  We consider the large time behavior of the solution to the $3$D Navier-Stokes
equation with horizontal viscosity $\Delta_{\rm h} u=\partial_1^2
u+\partial_2^2 u$ and show that the $L^p$ decay rate of the horizontal
components of the velocity field coinsides to that of the $2$D heat kernel,
while the vertical component decays like the $3$D heat kernel. Moreover, we
consider the asymptotic expansion of the solution and find that a portion of
the nonlinear term affect the leading term of the horizontal components of the
velocity field, whereas the leading term of the vertical component is given by
only the linear solution.
