Sharp Growth Rate of Linear Rayleigh–Taylor Instability of Primitive Equations

We investigate the linear Rayleigh–Taylor instability within the framework of two-dimensional viscous primitive equations (VPE) under hydrostatic approximation, which govern large-scale oceanic and atmospheric flows. For steady quiescent background states with arbitrary density profiles satisfying the Rayleigh–Taylor instability criterion (i.e., heavier fluid overlying lighter fluid), we conduct a rigorous spectral analysis of the linearized perturbation system. By eliminating the vertical velocity and pressure fields, we derive a nonlocal eigenvalue problem governing the horizontal velocity perturbations. To address the intrinsic non-self-adjoint nature of the system, we introduce an auxiliary variational problem with a frozen parameter, which enables us to establish the existence and uniqueness of a positive eigenvalue λk for each admissible horizontal wavenumber k. We further prove that λk is bounded above by a wavenumber-dependent critical value sk, and verify the sharpness of the maximal growth rate defined as Λ:=supk≠0λk. In particular, we show that no small perturbation can grow faster than the exponential rate eΛt, and construct explicit normal-mode solutions that achieve growth rates arbitrarily close to Λ. We also derive complete energy estimates for the linearized system, establishing uniform temporal control of solutions via the bound e2Λt. Additionally, we demonstrate that positive viscosity dissipation uniformly bounds all unstable eigenvalues, whereas the zero-dissipation limit gives rise to an unbounded unstable spectrum where eigenvalues grow without bound as the horizontal wavenumber increases. This spectral behavior stands in stark contrast to that of the classical Euler equations, for which all unstable eigenvalues remain uniformly bounded. This fundamental discrepancy reveals that the hydrostatic approximation essentially distorts the intrinsic instability properties of the original fluid system, thereby invalidating its physical validity for numerical simulations of Rayleigh–Taylor instability.