  Luminous blue variables (LBVs) are hot, very luminous massive stars
displaying large quasi-periodic variations in brightness, radius,and
photospheric temperature, on timescales of years to decades. The physical
origin of this variability, called S Doradus cycle after its prototype, has
remained elusive. Here, we study the feedback of stellar wind mass-loss on the
envelope structure in stars near the Eddington limit. We perform a
time-dependent hydrodynamic stellar evolutionary calculation, applying a
stellar wind mass-loss prescription with a temperature-dependence inspired by
the predicted systematic increase in mass-loss rates below 25 kK. We find that
when the wind mass-loss rate crosses a well-defined threshold, a discontinuous
change in the wind base conditions leads to a restructuring of the stellar
envelope. The induced drastic radius and temperature changes, which occur on
the thermal timescale of the inflated envelope, impose in turn mass-loss
variations that reverse the initial changes, leading to a cycle that lacks a
stationary equilibrium configuration. Our proof-of-concept model broadly
reproduces the typical observational phenomenology of the S Doradus
variability. We identify three key physical ingredients needed to trigger the
instability: inflated envelopes in close proximity to the Eddington limit, a
temperature range where decreasing opacities do not lead to an accelerating
outflow, and a mass-loss rate that increases with decreasing temperature,
crossing a critical threshold value within this temperature range. Our scenario
and model provide testable predictions, and open the door for a consistent
theoretical treatment of the LBV phase in stellar evolution, with consequences
for their further evolution as single stars or in binary systems.
