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beta plane corrections to nonlinear atmospheric flow patterns application to jupiters great red spot (GRS) drift dynamics

Oladiran Johnson Abimbola

TL;DR

The paper extends the f-plane description of Jupiter’s Great Red Spot to a β-plane framework by incorporating the meridional gradient of planetary vorticity, characterized through the Rossby parameter $\beta$ and the deformation parameter $\gamma$. Using a thin-shell, far-from-equilibrium perturbation approach with a Lagrangian formulation, it derives an analytic westward drift velocity $c_R\approx-3.7$ to $-3.9\mathrm{m\,s^{-1}}$ in close agreement with observations, and explains a 90-day oscillation as a beat between vortex rotation and Rossby waves. The β-plane model also reveals meridional asymmetry in winds, a quadratic pressure correction, and a latitudinally varying PV field, while demonstrating broad universality across Saturn, Neptune, and Earth without free parameters. The study discusses the regime of validity ($\gamma\lesssim0.5$), potential time-dependent extensions, and applications to exoplanets and other planetary vortices, highlighting the critical role of planetary vorticity gradients in long-lived atmospheric vortex dynamics.

Abstract

The Great Red Spot (GRS) of Jupiter has been observed for over a century, with researchers studying its characteristics and dynamics, including its size, depth, movement, and interactions with its environment. Recently, the f-plane thin-shell asymptotic analysis was used to explain some of the GRS features, but the method failed to capture the observed westward drift of the GRS. In this study, the f-plane theory was extended by including the Rossby parameter in the β-plane approximation and using the dimensionless Rossby deformation parameter γ, to systematically apply perturbation theory. The westward drift velocity of 3.7 m/s was analytically predicted, which is 95% in agreement with the observed 3.9 m/s. The observed 90-day oscillation in drift rate was explained. Also explained is the north-south asymmetry in circulation patterns. The universality of the \b{eta}-plane theory was demonstrated by its application to the vortices on Saturn, Neptune and Earth, without free parameters. It was demonstrated in this study that for the understanding of long-lived atmospheric vortex dynamics, the planetary vorticity gradient is very critical.

beta plane corrections to nonlinear atmospheric flow patterns application to jupiters great red spot (GRS) drift dynamics

TL;DR

The paper extends the f-plane description of Jupiter’s Great Red Spot to a β-plane framework by incorporating the meridional gradient of planetary vorticity, characterized through the Rossby parameter and the deformation parameter . Using a thin-shell, far-from-equilibrium perturbation approach with a Lagrangian formulation, it derives an analytic westward drift velocity to in close agreement with observations, and explains a 90-day oscillation as a beat between vortex rotation and Rossby waves. The β-plane model also reveals meridional asymmetry in winds, a quadratic pressure correction, and a latitudinally varying PV field, while demonstrating broad universality across Saturn, Neptune, and Earth without free parameters. The study discusses the regime of validity (), potential time-dependent extensions, and applications to exoplanets and other planetary vortices, highlighting the critical role of planetary vorticity gradients in long-lived atmospheric vortex dynamics.

Abstract

The Great Red Spot (GRS) of Jupiter has been observed for over a century, with researchers studying its characteristics and dynamics, including its size, depth, movement, and interactions with its environment. Recently, the f-plane thin-shell asymptotic analysis was used to explain some of the GRS features, but the method failed to capture the observed westward drift of the GRS. In this study, the f-plane theory was extended by including the Rossby parameter in the β-plane approximation and using the dimensionless Rossby deformation parameter γ, to systematically apply perturbation theory. The westward drift velocity of 3.7 m/s was analytically predicted, which is 95% in agreement with the observed 3.9 m/s. The observed 90-day oscillation in drift rate was explained. Also explained is the north-south asymmetry in circulation patterns. The universality of the \b{eta}-plane theory was demonstrated by its application to the vortices on Saturn, Neptune and Earth, without free parameters. It was demonstrated in this study that for the understanding of long-lived atmospheric vortex dynamics, the planetary vorticity gradient is very critical.
Paper Structure (28 sections, 85 equations, 9 figures, 1 table)

This paper contains 28 sections, 85 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing Equations with Latitudinally Varying Coriolis Parameter
  3. The $\beta$-Plane approximation and the modified Navier-Stoke’s equation
  4. Nondimensionalization
  5. Thin-Shell Asymptotic Reduction with $\beta$-Effect
  6. Leading-Order System
  7. Vorticity Dynamics and Rossby Wave Generation
  8. Scale-Separation Analysis and Consistency Conditions
  9. $\beta$-plane system with standard coordinates
  10. Pertubation Solution for Jupiter's GRS
  11. Lagrangian Framework
  12. Perturbation Expansion
  13. $f$-Plane Base Solution (Order $\gamma^0$)
  14. $\beta$-Plane Corrections (Order $\gamma$)
  15. Rossby Wave Drift Velocity
  16. ...and 13 more sections

Figures (9)

  • Figure 1: Jupiter and the Great Red Spot (GRS). [https://science.nasa.gov/asset/hubble/jupiter-and-the-great-red-spot/]
  • Figure 2: Jupiter’s Great Red Spot (GRS) Westward Drift Velocity (Inset: Agreement between theory and observation, as indicated by recent data from 2015 to 2024).
  • Figure 3: Particle trajectories in (a) $f$-plane and (b) $\beta$-plane.
  • Figure 4: Temporal evolution and 90-day oscillation in GRS position (a) 4-year long-term drift, (b) Detrended longitude anomaly, (c) Fourier power spectrum, (d) Phase portrait.
  • Figure 5: North-South asymmetry meridional structure induced by $\beta-$plane effect (a) Azimuthal wind speed, (b) pressure anomaly, (c) temperature anomaly, (d) vertical temperature profiles.
  • ...and 4 more figures