  The hydrostatic equilibrium state is the consequence of the exact hydrostatic
balance between hydrostatic pressure and external force. Standard finite volume
or finite difference schemes cannot keep this balance exactly due to their
unbalanced truncation errors. In this study, we introduce an auxiliary variable
which becomes constant at isothermal hydrostatic equilibrium state and propose
a well-balanced gas kinetic scheme for the Navier-Stokes equations with a
global reconstruction. Through reformulating the convection term and the force
term via the auxiliary variable, zero numerical flux and zero numerical source
term are enforced at the hydrostatic equilibrium state instead of the balance
between hydrostatic pressure and external force. Several problems are tested
numerically to demonstrate the accuracy and the stability of the new scheme,
and the results confirm that, the new scheme can preserve the exact hydrostatic
solution. The small perturbation riding on hydrostatic equilibria can be
calculated accurately. The viscous effect is also illustrated through the
propagation of small perturbation and the Rayleigh-Taylor instability. More
importantly, the new scheme is capable of simulating the process of converging
towards hydrostatic equilibrium state from a highly non-balanced initial
condition. The ultimate state of zero velocity and constant temperature is
achieved up to machine accuracy. As demonstrated by the numerical experiments,
the current scheme is very suitable for small amplitude perturbation and long
time running under gravitational potential.
