We review the work of Bryan (1889) on the normal modes of a Maclaurin
spheroid, carrying out numerical calculations of the frequencies and spatial
forms of these modes that have not been previously published. We study all
modes of degree lβ€4, which includes both inertial modes and surface
gravity modes, with the aim of better understanding the effect of rapid
rotation on tidal interactions. The inclusion of these higher degree modes
greatly increases the number of frequencies at which tidal resonances may
occur. We derive an expression for the decay rates of these modes to first
order in viscosity and explicitly plot these for modes. We see that the
equatorial bulge of the spheroid has a significant effect on the decay rates
(changing some of these by a factor of 2 between an eccentricity of e=0 and
0.5), and a more modest effect on the mode frequencies. This suggests that
models of tidal interaction between rapidly rotating stars and giant planets
that model the Coriolis force while neglecting the centrifugal distortion of
the body may be in error by an order unity factor. In a subsequent paper we
shall examine the case of a forced flow in this spheroid, and complete the
model by considering how the tides raised by the orbiting companion change the
orbital elements.Comment: 27 pages, 39 figures, 1 table, accepted for publication in MNRA