On the baroclinic instability of inviscid non-conducting Boussinesq equations with rotation in 3-D

1. Problem Statement

The paper investigates the nonlinear instability of a specific vertical shear flow in a three-dimensional, inviscid, non-conducting Boussinesq system with rotation. The governing equations describe large-scale geophysical flows (atmospheric and oceanic) where the Rossby number ($Ro$) is small but non-zero.

• Governing System: The 3-D rotating Boussinesq equations (System 1.1) involving velocity $(u, v, w)$, pressure $p$, and temperature $\theta$, coupled with the Coriolis force and buoyancy.
• Base State: The authors analyze a basic shear flow confined between two rigid horizontal planes ($z=0, 1$), defined as:
  $$U = (z, 0, 0), \quad \Theta = -y, \quad P = -yz$$
  This setup corresponds to the famous Eady model in atmospheric science.
• The Challenge:
  • When $Ro = 0$, the system reduces to the geostrophic limit model, where the instability (baroclinic instability) is well-understood and driven by boundary effects.
  • When $Ro > 0$ (small but non-zero), the system cannot be reduced to a single equation for pressure. The coupling between the fast inertial waves and the slow geostrophic modes makes the mathematical analysis of nonlinear instability significantly more complex.
  • The goal is to rigorously prove that the shear flow is nonlinearly unstable for small $Ro > 0$, consistent with the $Ro=0$ limit.

2. Methodology

The authors employ a novel approach that avoids searching for growing modes directly in the linearized system with $Ro \neq 0$. Instead, they utilize an asymptotic expansion combined with the geostrophic limit model.

A. Geostrophic Limit Analysis ($Ro = 0$)

• The authors first analyze the system in the limit $Ro \to 0$.
• They derive the nonlinear geostrophic limit model (a single equation for pressure $p$).
• Using normal mode analysis ($p = \hat{p}(z)e^{i\alpha(x-ct)}\sin(\beta y)$), they prove the existence of exponentially growing modes (instability) for specific wavenumbers and Burger numbers ($B$).
• Key Finding (Lemma 2.1): The growing modes found in the linear geostrophic limit also solve the nonlinear geostrophic limit equations. This establishes a "growing profile" that persists in the nonlinear regime when $Ro=0$.

B. Construction of Approximate Solutions ($Ro > 0$)

• To handle the case $Ro > 0$, the authors construct an approximate solution $(u_a, \theta_a, p_a)$ using a high-order asymptotic expansion in terms of $Ro$:
  $$u_a = u_N + Ro \cdot u_e^{(1)} + Ro^2 \cdot u_e^{(2)} + \dots$$
  where $u_N$ is the solution to the nonlinear geostrophic limit model (containing the growing profile), and the subsequent terms are corrections.
• They rigorously derive the equations for the coefficients of the expansion (up to order $Ro^3$) to ensure the remainder terms are controlled.

C. Error Analysis and Nonlinear Instability Proof

• Let $(u^*, \theta^*)$ be the true solution and $(u_a, \theta_a)$ be the approximate solution. The error is defined as $(u', \theta') = (u^* - u_a, \theta^* - \theta_a)$.
• Energy Estimates: The authors derive energy estimates for the error terms in Sobolev spaces ($H^3$). They prove that the error remains small (of order $O(Ro^k)$) over a specific time scale.
• Instability Mechanism: Since the approximate solution contains the exponentially growing profile from the geostrophic limit, and the error is small, the true solution must eventually deviate significantly from the base shear flow.
• Time Scale: The instability is proven to occur up to a time scale $T^\delta = O(\log(1/\delta))$, where $\delta$ is the size of the initial perturbation.

3. Key Contributions

1. Rigorous Nonlinear Instability Proof: The paper provides the first rigorous mathematical proof of nonlinear baroclinic instability for the 3-D inviscid rotating Boussinesq system with small, non-zero Rossby numbers.
2. Novel Methodological Approach: Unlike previous works (e.g., Guo-Strauss or Grenier methods) that rely on finding the fastest growing mode of the linearized operator for $Ro \neq 0$, this paper:
   • Identifies a growing mode in the nonlinear geostrophic limit ($Ro=0$).
   • Constructs an approximate solution for $Ro > 0$ based on this limit.
   • Proves that the instability of the limit model transfers to the full system via asymptotic estimates.
3. Boundary-Driven Instability: The analysis confirms that the instability is driven by the physical boundaries ($z=0, 1$), consistent with the Eady model mechanism, even in the presence of rotation.
4. Handling Multi-Scale Singularities: The paper addresses the multi-scale singular limit problem arising from the transition from the full Boussinesq system to the geostrophic limit, specifically managing the singularities in the $L^\infty$ norm (though the $L^\infty$ estimate remains an open problem noted in the remarks).

4. Main Results

• Theorem 1.1: For any small $Ro > 0$ and initial perturbation size $\delta$, there exists a time $T^\delta$ such that the $L^2$ norm of the perturbation grows from $\delta$ to a macroscopic scale $\eta$ (where $\eta > \delta$).
• Condition for Instability: The instability occurs if the Burger number and wavenumbers satisfy $q < q_c$ (where $q_c \approx 1.2$ is the root of a transcendental equation derived from the boundary conditions).
• Growth Rate: The growth rate is determined by the imaginary part of the phase speed $c$ from the geostrophic limit model ($\alpha \Im(c)$).

5. Significance

• Geophysical Relevance: The results validate the physical intuition that baroclinic instability (a primary driver of mid-latitude weather systems) persists in rotating fluids even when the Rossby number is small but finite, bridging the gap between idealized geostrophic models and realistic rotating flows.
• Mathematical Advancement: The paper overcomes the difficulty of analyzing the full rotating system by leveraging the solvability of the geostrophic limit. This "limit-based construction" strategy offers a new template for studying instabilities in other singularly perturbed fluid systems where direct spectral analysis of the full system is intractable.
• Foundation for Future Work: The authors identify the multi-scale singular limit in $L^\infty$ as an open problem, setting the stage for future research into higher-order regularity and more precise asymptotic behaviors.

In summary, Mao and Wang successfully demonstrate that the Eady-type shear flow is nonlinearly unstable in the 3-D rotating Boussinesq system, proving that the instability mechanism driven by boundaries in the geostrophic limit survives the introduction of small rotation effects.