  We study the tidal excitation of gravity modes (g-modes) in compact white
dwarf binary systems with periods ranging from minutes to hours. As the orbit
of the system decays via gravitational radiation, the orbital frequency
increases and sweeps through a series of resonances with the g-modes of the
white dwarf. At each resonance, the tidal force excites the g-mode to a
relatively large amplitude, transferring the orbital energy to the stellar
oscillation. We calculate the eigenfrequencies of g-modes and their coupling
coefficients with the tidal field for realistic non-rotating white dwarf
models. Using these mode properties, we numerically compute the excited mode
amplitude in the linear approximation as the orbit passes though the resonance,
including the backreaction of the mode on the orbit. We also derive analytical
estimates for the mode amplitude and the duration of the resonance, which
accurately reproduce our numerical results for most binary parameters. We find
that the g-modes can be excited to a dimensionless (mass-weighted) amplitude up
to 0.1, with the mode energy approaching $10^{-3}$ of the gravitational binding
energy of the star. This suggests that thousands of years prior to the binary
merger, the white dwarf may be heated up significantly by tidal interactions.
However, more study is needed since the physical amplitudes of the excited
oscillation modes become highly nonlinear in the outer layer of the star, which
can reduce the mode amplitude attained by tidal excitation.
