Energy stability of the Charney–DeVore quasi-geostrophic equation for atmospheric blocking

The paper proves asymptotic H^1-stability of the parallel flow U0(y)=C sin y + C' for the barotropic β-plane QG model with Rayleigh friction and specially chosen forcing/topography by deriving an exact Lyapunov identity d/dt ∫∫[ζ'^2+(η−1)(u^2+v^2)] = −2k ∫∫[ζ'^2+(η−1)(u^2+v^2)] when F′(y)=kC(sin y−2/π) and h(y)=Cη cos y (equations (45)–(53)), and then combining it with a sharp static inequality that controls kinetic energy by enstrophy plus two scalar functionals to obtain the H^1-decay estimate (69) with rate e^{-kt} (Theorem 3.6). The constant-Uave case (Theorem 3.7) follows under the same structure. The candidate solution reproduces this route almost step-by-step: basic state via (46); disturbance/vorticity equations (47)–(52); the same Lyapunov functional and exact identity (53); exponential decay of the mean and a boundary trace (the paper’s (65), (66)); the same streamfunction decomposition and sharp 1D trace/mean inequality (paper’s (68)); and the same convexity step yielding (69). The constants match exactly, and the case with prescribed Uave is treated identically, noting C2≡0 in that formulation. Overall, the model’s solution is a faithful, technically sound restatement of the paper’s proof with equivalent cancellations and bounds. Key supporting equations/results in the paper: model and well-posedness discussion (37)–(44) ; choice of F′ and basic state (45)–(46) ; disturbance energy/enstrophy identities leading to (53) ; the sharp static inequality and final estimate (68), (69) ; and the constant-Uave extension (Theorem 3.7) .