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Euler equation on a fast rotating ellipsoid

Haoran Wu

TL;DR

This work extends the fast-rotating Euler dynamics from the sphere to a biaxial ellipsoid by adapting the Cheng–Mahalov PDE–geometric averaging framework to ellipsoidal geometry. It introduces a latitude-dependent Coriolis operator $\mathcal{L}$ and an ellipsoidal-harmonic spectral setup to obtain $\omega$-independent, local-in-time $H^k$ bounds and a precise time-averaging estimate. In the fast-rotation limit, the time-averaged flow converges to a longitude-independent zonal state $\nabla^{\perp} f(z)$, with an explicit latitudinal dependence encoded by $f$, bridging spherical and ellipsoidal theories of geophysical flows. The results provide a rigorous foundation for zonalization on rotating planets with flattening and demonstrate how ellipsoidal geometry modifies the leading-order fast-rotation dynamics.

Abstract

This paper extends the analytical study of the incompressible Euler equations from the classical spherical setting to the more realistic geometry of a biaxial ellipsoid. Motivated by the work of Cheng and Mahalov on fast rotating spheres and Xu on Rossby-Haurwitz solutions on ellipsoids, we adapt their framework to establish a parallel result for Euler flows on a rotating ellipsoidal surface. In the regime of rapid rotation, we prove that the time-averaged velocity field remains uniformly bounded in Sobolev norms independent of the rotation rate and converges to a longitude-independent zonal flow depending only on latitude. This shows that the zonalization phenomenon discovered by Cheng and Mahalov on the sphere persists on biaxial ellipsoids, thereby bridging the gap between spherical and ellipsoidal theories of fast rotating Euler dynamics.

Euler equation on a fast rotating ellipsoid

TL;DR

This work extends the fast-rotating Euler dynamics from the sphere to a biaxial ellipsoid by adapting the Cheng–Mahalov PDE–geometric averaging framework to ellipsoidal geometry. It introduces a latitude-dependent Coriolis operator and an ellipsoidal-harmonic spectral setup to obtain -independent, local-in-time bounds and a precise time-averaging estimate. In the fast-rotation limit, the time-averaged flow converges to a longitude-independent zonal state , with an explicit latitudinal dependence encoded by , bridging spherical and ellipsoidal theories of geophysical flows. The results provide a rigorous foundation for zonalization on rotating planets with flattening and demonstrate how ellipsoidal geometry modifies the leading-order fast-rotation dynamics.

Abstract

This paper extends the analytical study of the incompressible Euler equations from the classical spherical setting to the more realistic geometry of a biaxial ellipsoid. Motivated by the work of Cheng and Mahalov on fast rotating spheres and Xu on Rossby-Haurwitz solutions on ellipsoids, we adapt their framework to establish a parallel result for Euler flows on a rotating ellipsoidal surface. In the regime of rapid rotation, we prove that the time-averaged velocity field remains uniformly bounded in Sobolev norms independent of the rotation rate and converges to a longitude-independent zonal flow depending only on latitude. This shows that the zonalization phenomenon discovered by Cheng and Mahalov on the sphere persists on biaxial ellipsoids, thereby bridging the gap between spherical and ellipsoidal theories of fast rotating Euler dynamics.

Paper Structure

This paper contains 6 sections, 11 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hodge decomposition
  3. Null space of L and associated $L^2$-orthogonal projection
  4. Key estimates
  5. Uniform estimates independent of $\omega$
  6. Proof of the main theorem

Key Result

Theorem 1.1

Consider the incompressible Euler equation 1.1 on $\mathbb{E}^2$ with initial data $u_0\in H^k(\mathbb{E}^2)$ for $k\geq3$. Define the time-averaged flow $\overline{{u}}$ as in 1.2. Then, there exists a function $f(\cdot):[-1,1]\mapsto\mathbb{R}$ and constants $C_0,T_0$ independent of $\omega$ and $ with $M_0:=\|{u}_0\|_{H^k}$.In elliptic coordinates, the approximation $\nabla^\perp f(z)$ is whic

Figures (1)

  • Figure 1: A biaxial ellipsoid rotating with angular velocity $\omega$Xu2024

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 1.1
  • Remark 1.1
  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 4.1
  • Definition 4.1
  • Definition 4.2
  • Remark 4.2
  • ...and 7 more