The Luminous Blue Variable stars exhibit behavior ranging from light curve `microvariations on timescales of tens of days, to `outbursts accompanied by mass loss of up to 10e-03 solar masses per year, occurring decades apart, to `giant eruptions such as seen in Eta Carinae ejecting one or more solar masses and recurring on timescales of centuries. Here we review the work of the Los Alamos group since 1993 to investigate pulsations and instabilities in massive stars using linear pulsation models and non-linear hydrodynamic models. The models predict pulsational variability that may be associated with the microvariations. Using a nonlinear pulsation hydrodynamics code with a time-dependent convection treatment, we show that, in some circumstances, the Eddington limit is exceeded periodically in the pulsation driving region of the stellar envelope, accelerating the outer layers, and perhaps initiating mass loss or LBV outbursts. We discuss how pulsations and mass loss may be responsible for the location of the Humphreys-Davidson Limit in the H-R diagram. The `giant eruptions, however, must involve much deeper regions in the stellar core to cause such large amounts of mass to be ejected. We review and suggest some possible explanations, including mixing from gravity modes, secular instabilities, the epsilon mechanism, or the SASI instability as proposed for Type II supernovae. We outline future work and required stellar modeling capabilities to investigate these possibilities.
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