Rotating shallow water equations with bottom drag: bifurcations and growth due to kinetic energy backscatter

The candidate solution reproduces the paper’s Lyapunov–Schmidt (LS) reductions for both stationary Rossby-type and travelling inertia–gravity (IG) waves in the isotropic backscatter case with Q ≠ 0, and it obtains the same leading-order amplitude equations and coefficients. For Rossby waves, their bifurcation equation 0 = A1(α − 2bκ² − (16Qkc)/(3πH0)|A1| + Rns) coincides with the paper’s equation (4.13), including the explicit 16/(3π) coefficient and the stated remainder structure Rns ((Q+f̃)O(|A1|(|A1|+|α|+|κ|)) + Qf̃O(|A1|²) + O(|κ|³)) . For IG waves, they recover the two solvability conditions 0 = A1(−s + κ f²/ωc + O(|A1|² + |μ|²)) and 0 = A1(α − 2(Qkc/H0)(I1 + (k_c²gH0)/(2f² + k_c²gH0) I2)|A1| + O(|A1|² + |μ|²)), exactly as in Theorem 5.2 (equations (5.9a,b)) including the definitions of I1 and I2 and the leading-order wave shape . Minor differences are purely presentational: the model informally refers to a “first-order periodic ODE system” in ζ where the paper’s operator Lc contains up to fourth derivatives in ζ , and it does not spell out the H5-based inner-product and projection equivalence used in the paper’s LS setup for Rossby waves (but this equivalence is explicitly justified in the paper) . All key linear scalings (kc² = b/(2d), Cc/H0 = b²/(4d)) and the geostrophic plane-wave reduction are used consistently with the paper’s derivations . Overall, the model’s proof strategy and results match the paper’s, including supercriticality of both branches.