  Ring seismology has recently revealed the presence of internal gravity waves
inside Saturn that extend up to 60% of Saturn's radius starting from the
center, in what is recognized today as Saturn's stably-stratified dilute core.
Similarly, gravity measurements on Jupiter suggest the existence of a dilute
core of still poorly constrained radial extent. These cores are likely in a
double-diffusive regime, which prompt the question of their long-term
stability. Indeed, previous DNS (Direct Numerical Simulations) studies in
triply-periodic domains have shown that, in some regimes, double-diffusive
convection tends to spontaneously form shallow convective layers, which coarsen
until the region becomes fully convective. In this letter, we study the
conditions for layering in double-diffusive convection using different boundary
conditions, in which temperature and composition fluxes are fixed at the domain
boundaries. We run a suite of DNS varying microscopic diffusivities of the
fluid and the strength of the initial stratification. We find that convective
layers still form as a result of the previously discovered gamma-instability
which takes place whenever the local stratification drops below a critical
threshold that only depends on the fluid diffusivities. We also find that the
layers grow once formed, eventually occupying the entire domain. Our work thus
recovers the results of previous studies, despite the new boundary conditions,
suggesting that this behavior is universal. The existence of Saturn's
stably-stratified core, today, therefore suggests that this threshold has never
been reached, which places a new constraint on scenarios for the planet's
formation and evolution.
