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We revisit the global modes and instabilities of homogeneous rotatingellipsoidal fluid masses, which are the simplest global models of rotationallyand tidally deformed gaseous planets or stars. The tidal flow in a short-periodplanet may be unstable to the elliptical instability, a hydrodynamicinstability that can drive tidal evolution. We perform a global (and local WKB)analysis to study this instability using the elegant formalism of Lebovitz &Lifschitz. We survey the parameter space of global instabilities with harmonicorders $\ell\leq 5$, for planets with spins that are purely aligned (prograde)or anti-aligned (retrograde) with their orbits. In general, the instability hasa much larger growth rate if the planetary spin and orbit are anti-alignedrather than aligned. We have identified a violent instability for anti-alignedspins outside of the usual frequency range for the elliptical instability (when$\frac{n}{\Omega}\lesssim -1$, where $n$ and $\Omega$ are the orbital and spinangular frequencies, respectively) if the tidal amplitude is sufficientlylarge. We also explore the instability in a rigid ellipsoidal container, whichis found to be quantitatively similar to that with a realistic free surface.Finally, we study the effect of rotation and tidal deformation on modefrequencies. We find that larger rotation rates and larger tidal deformationsboth decrease the frequencies of the prograde sectoral surface gravity modes.This increases the prospect of their tidal excitation, potentially enhancingthe tidal response over expectations from linear theory. In a companion paper,we use our results to interpret global simulations of the ellipticalinstability.

Nonlinear tides in a homogeneous rotating planet or star: global modes and elliptical instability