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Green-Function Analysis of Localized Rossby Wave Packets in β-Plane Flows

This paper employs multiscale asymptotic analysis and Green-function methods to derive an analytically tractable model demonstrating how vertical eddy viscosity and vortex radius govern the propagation direction, amplitude, and energy redistribution of localized Rossby wave packets in weakly diffusive $\beta$ -plane flows.

Original authors: Albertus Sulaiman

Published 2026-09-07

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CC BY 4.0

Original authors: Albertus Sulaiman

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The ocean and the atmosphere are not static sheets of water or air; they are churning, rotating fluids where energy constantly shifts between vast, slow currents and smaller, swirling eddies. In these rotating systems, the Earth's spin plays a crucial role. As you move away from the equator, the strength of this spin changes, creating a gradient that acts like a restoring force for waves. These are known as Rossby waves, massive ripples that travel through the ocean and atmosphere, carrying heat, salt, and nutrients across the globe. Often, these waves do not travel alone but are organized into packets, grouped together within coherent swirls of water called vortices. Understanding how these packets form, how they move, and how they interact with the surrounding water is essential for predicting climate patterns and ocean circulation. However, the real ocean is not perfectly smooth; it is turbulent, with friction and mixing occurring at small scales that can disrupt these large structures. Scientists have long sought to understand how this turbulent friction, or "eddy viscosity," influences the life and movement of these swirling wave packets.

In a recent study, a researcher at the National Research and Innovation Agency in Indonesia developed a new mathematical framework to explore exactly this interaction. The work focuses on localized packets of Rossby waves moving through a fluid layer where vertical mixing is present but relatively weak. By using a method that separates the problem into different time and space scales, the researcher was able to derive a clear description of how a vortex behaves when it is subjected to the Earth's changing rotation and vertical turbulence. The study begins by modeling a coherent, swirling mass of water that resembles a Gaussian shape, a smooth, bell-curve-like profile often seen in real ocean eddies. This model allows the scientist to track how the vortex traps water in its center while simultaneously acting as a carrier for Rossby waves.

The analysis reveals a surprising and specific relationship between the size of the vortex and the direction in which the wave packet travels. In traditional models, these waves are known to move westward. However, this new framework shows that the radius of the vortex itself sets a critical threshold. If the vortex is smaller than a certain size, the wave packet moves westward, following the classic pattern. But if the vortex is larger than that critical radius, the packet reverses course and begins to move eastward. This finding suggests that the physical size of the eddy is a primary control knob for the direction of energy transport in the ocean. Furthermore, the study demonstrates that the strength of the vertical mixing, represented by an eddy-viscosity coefficient, does not merely dampen the motion but actively reshapes the internal structure of the wave. Stronger mixing increases the amplitude of the disturbances and causes the vortex to lose its grip on the water it carries, allowing energy to leak out from the core into the surrounding flow.

To understand how these packets evolve over time, the researcher used a technique involving "Green's functions," a mathematical tool that acts like a map of how a system responds to a small push. By applying this method, the study showed that while the main vortex remains stable for a time, small disturbances at its edges eventually generate secondary swirls and ripples. These first-order corrections indicate that the energy of the primary vortex is not locked away forever; instead, it gradually scatters laterally, creating a cascade of smaller structures around the main packet. This process explains how coherent eddies can break down and redistribute their energy into the broader ocean environment. The simulations confirm that while the wave number determines the pattern and speed of the packet, the vertical mixing coefficient controls how much the packet grows or decays.

The results provide a theoretical explanation for observations of long-lived subsurface eddies in the eastern equatorial Indian Ocean, where warm, salty, and oxygen-rich water masses have been seen traveling westward for months. The study suggests that these eddies are not just passive blobs of water but are dynamic systems where the interplay between the vortex size, the Earth's rotation, and vertical turbulence dictates their fate. When the conditions are right, these structures can trap water efficiently, but as vertical mixing increases, they begin to shed their contents, releasing heat and nutrients into the surrounding currents. This work offers a clear, analytical link between the invisible friction of turbulent water and the visible movement of large-scale ocean waves, providing a foundation for better understanding how energy moves through the planet's fluid systems.

Technical Summary: Green-Function Analysis of Localized Rossby Wave Packets in β-Plane Flows

Problem Statement
The study addresses the dynamics of localized eddies and Rossby wave packets within weakly diffusive $\beta$ -plane flows, a common scenario in rotating geophysical systems like the ocean and atmosphere. While numerical and theoretical methods have previously explored vortex structures and Rossby wave evolution, there remains a scarcity of analytic descriptions for local wave packets that explicitly incorporate momentum diffusion. The paper seeks to resolve two primary questions: (1) how vortex strength influences vertical eddy viscosity intensity and vice versa, and (2) the specific relationship between Rossby wavenumber and eddy radius. The research is motivated by observations of energetic equatorial eddies and the need to understand how unresolved turbulent motion, parameterized via eddy-viscosity closures, alters the structure, propagation, and stability of localized Rossby wave packets.

Methodology
The author develops an analytical framework utilizing multiscale asymptotic expansion and Green's function techniques. The derivation begins with the incompressible momentum equations on a $\beta$ -plane, incorporating a Mellor-Yamada eddy-viscosity closure to represent turbulent kinetic energy and fluxes.

Model Reduction: By assuming a thin-layer regime ( $\epsilon = H/L \ll 1$ ) and weak baroclinicity, the author derives a reduced streamfunction equation. Vertical turbulent diffusion is shown to influence horizontal dynamics through a vertical eigenvalue problem, where the eigenvalues depend explicitly on the eddy-viscosity coefficient ( $\kappa_m$ ).
Separation of Variables: The streamfunction is separated into horizontal ( $\eta$ ) and vertical ( $\phi$ ) components. The vertical component yields a generalized eigenvalue problem solved via WKB methods, while the horizontal component is governed by a linearized equation involving the Jacobian and $\beta$ -effect.
Perturbation Analysis: A multiple-scale expansion is employed, distinguishing between fast time scales (vortex dynamics) and slow time scales (diffusive evolution). The solution is expanded in powers of a small parameter $\epsilon$ $ϵ$ .
- Zeroth Order: Solved as a localized Gaussian vortex.
- First and Second Order: Higher-order corrections are derived analytically using Green's function techniques to account for nonlinear perturbations and lateral energy scattering.
Validation: Theoretical results are supplemented with Lagrangian passive tracer simulations to visualize particle trapping, dispersion, and the deformation of water masses.

Key Contributions and Results

Radius-Dependent Dispersion Relation: The analysis yields a dispersion relation for vortex Gaussian Rossby wave packets: $\omega = -\beta k (k^2 + 4/R^2)^{-1}$ . Crucially, this relation reveals a critical threshold determined by the vortex radius ( $R$ ). Unlike the classical Rossby wave relation which predicts purely westward propagation, this formulation indicates that the vortex radius sets a boundary separating eastward and westward packet propagation.
Role of Eddy Viscosity: The vertical eddy-viscosity coefficient ( $\kappa_m$ ) modifies the vertical eigenstructure ( $\lambda_n$ ), which in turn controls the amplitude and evolution of the wave packet. Numerical experiments demonstrate that as $\kappa_m$ increases, the dispersion index of tracer particles grows exponentially, indicating that stronger vertical diffusion accelerates the loss of the vortex core's resilience and enlarges the effective eddy radius.
Green-Function Corrections and Energy Redistribution: The first-order perturbations, solved via Green's functions, generate secondary vorticity structures and lateral energy scattering around the primary vortex. This mechanism describes how energy is redistributed from coherent eddies to surrounding flow structures. The results show that while the eddy coefficient determines the magnitude of the wave packet amplitude, the wavenumber and vortex radius dictate the propagation pattern and speed.
Equatorial Applicability: The study confirms that localized Rossby wave packets (LRWPs), previously noted at high latitudes, can exist in equatorial $\beta$ -plane flows. The analytical framework provides a dynamical explanation for how coherent equatorial eddies (such as subsurface anticyclonic eddies observed in the Indian Ocean) can release energy into surrounding Rossby-wave disturbances, facilitating the transport of heat, salinity, and oxygen.

Significance
The paper claims to provide an analytically tractable description of the coupling between eddy viscosity, vortex dynamics, and localized Rossby-wave propagation. By linking vertical turbulent diffusion to horizontal flow evolution through a vertical eigenvalue problem, the framework offers a diagnostic tool for assessing how diffusivity shapes large-scale currents. The findings suggest that the interaction between external disturbances and Rossby wave dynamics is critical for understanding tropical variability and the redistribution of tracers in geophysical fluids. The study posits that while linear models may overlook subtle nonlinear interactions, they remain a useful tool for diagnosing large-scale circulation patterns and initializing more detailed numerical models, particularly in regions like the equatorial thermocline where stratification supports organized eddy formation.

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