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Characteristics of drift effects in the quasi-geostrophic equation arising from nonlinear symmetry

Masakazu Yamamoto

TL;DR

This work analyzes the large-time behavior of two 2D diffusion-type models in meteorology: the quasi-geostrophic equation $\partial_t \theta + \mathcal{R}^\bot \theta \cdot \nabla \theta = \Delta \theta$ and the convection-diffusion equation $\partial_t \rho - \Delta \rho = a \cdot \nabla (|\rho|\rho)$. Using the Escobedo–Zuazua (EZ) expansion with renormalization, the authors decompose solutions into a Gaussian core and gradient corrections to isolate nonlinear contributions. They find a stark contrast: the convection-diffusion model exhibits a logarithmic drift term and a symmetry-distortion in the asymptotics, while the quasi-geostrophic model shows no nonlinear drift due to the odd-type symmetry and skew-adjointness of the Riesz transform, effectively mitigating the logarithmic growth. This reveals that nonlinear symmetry strongly governs the large-time profiles, aligning meteorological intuition that Coriolis effects preserve the radially symmetric component without altering the mean potential temperature. $\,$

Abstract

This paper compares two similar diffusion equations that appear in meteorology. One is the quasi-geostrophic equation, and the other is the convection-diffusion equation. Both are two-dimensional bilinear equations, and the order of differentiation is the same. Naturally, their scales also coincide. However, the direction in which the nonlinear effects act differs: one acts along the isothermal surface, while the other acts along the temperature gradient in a specified direction. The main assertion quantifies this difference through the large-time behavior of their solutions. In particular, the nonlinear distortions in the asymptotic profiles of both equations are compared. In this context, the spatial symmetry of the first approximation plays a crucial role, but the solutions require no symmetry.

Characteristics of drift effects in the quasi-geostrophic equation arising from nonlinear symmetry

TL;DR

This work analyzes the large-time behavior of two 2D diffusion-type models in meteorology: the quasi-geostrophic equation and the convection-diffusion equation . Using the Escobedo–Zuazua (EZ) expansion with renormalization, the authors decompose solutions into a Gaussian core and gradient corrections to isolate nonlinear contributions. They find a stark contrast: the convection-diffusion model exhibits a logarithmic drift term and a symmetry-distortion in the asymptotics, while the quasi-geostrophic model shows no nonlinear drift due to the odd-type symmetry and skew-adjointness of the Riesz transform, effectively mitigating the logarithmic growth. This reveals that nonlinear symmetry strongly governs the large-time profiles, aligning meteorological intuition that Coriolis effects preserve the radially symmetric component without altering the mean potential temperature.

Abstract

This paper compares two similar diffusion equations that appear in meteorology. One is the quasi-geostrophic equation, and the other is the convection-diffusion equation. Both are two-dimensional bilinear equations, and the order of differentiation is the same. Naturally, their scales also coincide. However, the direction in which the nonlinear effects act differs: one acts along the isothermal surface, while the other acts along the temperature gradient in a specified direction. The main assertion quantifies this difference through the large-time behavior of their solutions. In particular, the nonlinear distortions in the asymptotic profiles of both equations are compared. In this context, the spatial symmetry of the first approximation plays a crucial role, but the solutions require no symmetry.
Paper Structure (5 sections, 2 theorems, 28 equations)

This paper contains 5 sections, 2 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. The EZ expansion with the renormalization
  4. The case of convection-diffusion equation
  5. The case of quasi-geostrophic equation

Key Result

Proposition 1.1

Let $\rho_0 \in L^1 (\mathbb{R}^2) \cap L^\infty (\mathbb{R}^2)$ and $x\rho_0 \in L^1 (\mathbb{R}^2)$. Then the solution $\rho$ of bg fulfills that as $t \to +\infty$ for $1 \le q \le \infty$ and $\gamma_q = 1-\frac{1}{q}$, where $M_0 = \int_{\mathbb{R}^2} \rho_0 (x) dx \in \mathbb{R}$ and $M_1 = - \int_{\mathbb{R}^2} x\rho_0 (x)dx + a \int_0^\infty \int_{\mathbb{R}^2} ((|\rho|\rho)(s,y) - |M_0|

Theorems & Definitions (2)

  • Proposition 1.1
  • Theorem 1.2