The Disappearance of Gapless Kelvin and Yanai Modes in Shallow-Water Waves on a Rotating Sphere

Core Concepts

While the Matsuno spectrum predicts gapless Kelvin and Yanai modes in shallow-water waves on an unbounded β-plane, these modes disappear on a rotating sphere due to the curved metric inducing degeneracy points in the wave operator's Weyl symbol, effectively closing the frequency gap between Rossby and inertia-gravity waves as the rotation rate increases.

Abstract

This research paper investigates the topological properties of shallow-water waves on a rotating sphere, focusing on the disappearance of gapless Kelvin and Yanai modes observed in the Matsuno spectrum for an unbounded β-plane.

Research Objective: The study aims to explain the absence of gapless Kelvin and Yanai modes in the shallow-water wave spectrum on a rotating sphere, a phenomenon not predicted by the β-plane approximation.

Methodology: The authors employ a combination of numerical simulations using the Dedalus spectral solver and theoretical analysis based on the concept of modal flow and topological invariants like Chern numbers. They analyze the zeros of the meridional velocity of wave functions to track the transition of modes between Rossby and inertia-gravity wavebands.

Key Findings: The study reveals that the curved metric of the sphere introduces degeneracy points in the wave operator's Weyl symbol, which are absent in the β-plane approximation. These points carry non-zero Chern numbers, indicating a change in the topological properties of the system. As the rotation rate increases, the wave functions spread across the sphere, revealing additional zeros in the meridional velocity at non-zero latitudes. This alters the modal flow, leading to the disappearance of the spectral flow observed in the Matsuno spectrum and the closure of the frequency gap between wavebands.

Main Conclusions: The research demonstrates that the absence of gapless Kelvin and Yanai modes on the rotating sphere is a consequence of the curved metric, which induces topological changes in the wave operator. This highlights the limitations of the β-plane approximation in describing global-scale wave dynamics.

Significance: The findings provide a deeper understanding of the behavior of shallow-water waves on rotating spheres, with implications for modeling atmospheric and oceanic circulation patterns on Earth and other planets.

Limitations and Future Research: The study focuses on the linear shallow-water model. Further research could explore the impact of nonlinear effects and more complex geometries on the topological properties of the system.

Stats

For the first baroclinic mode in the equatorial ocean on Earth, the equatorial radius of deformation is approximately 300 kilometers.
In the terrestrial atmosphere, barotropic modes propagate at a phase speed of roughly 300 m/s, corresponding to an ϵ value of approximately 0.65.
Eastward-propagating motions strongly localized at the equator on Jupiter, consistent with an estimated ϵ of 1.4 x 10^-4, suggest the presence of Kelvin waves.

Quotes

Key Insights Distilled From

Topology of shallow-water waves on the rotating sphere

by Nicolas Pere... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2404.07655.pdf

Deeper Inquiries

How do the findings of this study impact our understanding of atmospheric and oceanic phenomena like the Madden-Julian Oscillation or the El Niño Southern Oscillation, which are influenced by equatorial waves?

This study deepens our understanding of equatorial waves on a rotating sphere, particularly by highlighting the limitations of the β-plane approximation in describing global-scale oscillations. While the β-plane approximation, which assumes a locally linear variation of the Coriolis parameter, is useful for studying equatorially trapped waves like those involved in the Madden-Julian Oscillation (MJO) and the El Niño Southern Oscillation (ENSO), it fails to capture the full complexity of wave behavior on a sphere.
Here's how the study's findings impact our understanding of these phenomena:

Modal Flow and Global Oscillations: The study demonstrates that the concept of "spectral flow," where wave modes transition between frequency bands as the zonal wavenumber changes, needs to be refined for a spherical geometry. Instead, the study introduces the concept of "modal flow," which accounts for the discrete nature of the azimuthal wavenumber on a sphere. This distinction is crucial for understanding the global behavior of equatorial waves, especially those with larger wavelengths that are not confined to the equatorial region.
Breakdown of β-plane Approximation: As the study shows, the β-plane approximation breaks down for waves with larger wavelengths or lower frequencies, which are not strictly confined to the equatorial waveguide. This implies that models based solely on the β-plane approximation might not accurately represent the global propagation and interaction of equatorial waves, particularly for phenomena like ENSO that involve basin-scale dynamics.
Importance of Spherical Geometry: The study emphasizes the importance of considering the complete spherical geometry and the latitudinal variation of the Coriolis parameter when studying large-scale atmospheric and oceanic phenomena. This is particularly relevant for understanding the interaction of equatorial waves with higher latitudes and the potential for energy exchange between these regions.
In the context of MJO and ENSO:

MJO: While the MJO is primarily an equatorially trapped phenomenon, its interaction with off-equatorial Rossby waves and the potential for global-scale teleconnections could be better understood by considering the insights from this study.
ENSO: ENSO involves basin-scale interactions between the ocean and atmosphere across the equatorial Pacific. The study's findings suggest that incorporating the complete spherical geometry and the accurate Coriolis parameter variation is crucial for accurately simulating and predicting ENSO events, especially their global impacts.
In summary, this study encourages a more nuanced understanding of equatorial wave dynamics by moving beyond the simplified β-plane approximation. This is essential for improving our ability to model and predict complex atmospheric and oceanic phenomena like the MJO and ENSO, which have far-reaching global consequences.

Could the presence of topography or other realistic features on a rotating sphere potentially alter the topological properties of the wave operator and lead to the reemergence of gapless modes under specific conditions?

Yes, the presence of topography or other realistic features on a rotating sphere can significantly alter the topological properties of the wave operator and potentially lead to the reemergence of gapless modes under specific conditions. Here's why:

Breaking of Symmetries: The study's findings rely on the idealized case of a perfectly smooth, rotating sphere. Introducing topography, like mountains and ocean basins, or other realistic features like spatially varying background flows or stratification, breaks the spherical symmetry of the system. This symmetry breaking can have profound effects on the wave operator's topological properties.
Modification of Wave Dispersion: Topography and other features can modify the dispersion relation of waves, potentially closing the frequency gap between different wave bands. For instance, the presence of a topographic waveguide, like a mid-latitude mountain range or an ocean basin, can support trapped waves with frequencies within the gap, effectively bridging the gap and allowing for mode transitions.
Emergence of Edge States: Topological properties often manifest as the presence of robust, localized states at the boundaries or interfaces of different regions. In the context of a rotating sphere with topography, the edges of topographic features can act as such interfaces. These edge states can have frequencies within the gap, leading to the reemergence of gapless modes.
Resonances and Mode Coupling: Topography can induce resonances and mode coupling between different wave types, such as between Rossby waves and inertia-gravity waves. This coupling can facilitate energy transfer across frequency bands and potentially lead to the emergence of modes within the previously forbidden gap.
Specific Examples:

Equatorial Waves and Island Chains: The presence of island chains in the equatorial Pacific can significantly modify the propagation and evolution of equatorial Kelvin and Rossby waves, potentially leading to wave trapping and the emergence of new wave modes.
Coastal Kelvin Waves: Coastal topography supports the existence of coastal Kelvin waves, which are trapped along coastlines and have frequencies within the gap between Rossby and inertia-gravity waves.
Research Directions:

Numerical Simulations: High-resolution numerical simulations of the shallow-water equations on a rotating sphere with realistic topography and other features are crucial for exploring the potential reemergence of gapless modes and their implications for atmospheric and oceanic dynamics.
Topological Analysis: Extending the topological analysis of the wave operator to include the effects of topography and other realistic features is essential for understanding the underlying mechanisms responsible for the emergence of gapless modes.
In conclusion, while the study's findings provide valuable insights into the idealized case of a smooth sphere, incorporating realistic features like topography significantly enriches the problem and opens up exciting avenues for exploring the interplay between topology, wave dynamics, and atmospheric and oceanic phenomena.

How can the insights gained from analyzing the topological properties of classical waves be applied to other physical systems, such as quantum fluids or electromagnetic waves in curved spacetime?

The insights gained from analyzing the topological properties of classical waves, such as those described in the study of shallow-water waves on a rotating sphere, have far-reaching implications and can be applied to a wide range of physical systems beyond classical fluid dynamics. Here are some key examples:
1. Quantum Fluids:

Topological Superfluids and Superconductors: Topological concepts, like Chern numbers and edge states, are central to understanding the behavior of topological superfluids and superconductors. These exotic states of matter exhibit quantized vortices, dissipationless edge currents, and other remarkable properties that arise from the topological properties of their wavefunctions.
Quantum Hall Effect: The quantum Hall effect, where the Hall conductivity of a two-dimensional electron gas is quantized in the presence of a magnetic field, is a prime example of a topological phenomenon in condensed matter physics. The topological invariants associated with the electronic band structure determine the quantized values of the Hall conductivity.
2. Electromagnetic Waves in Curved Spacetime:

Gravitational Lensing and Wavefront Singularities: The study of light propagation in curved spacetime, as described by Einstein's theory of general relativity, benefits from topological tools. Gravitational lensing, where massive objects bend the path of light, can be understood in terms of the topological properties of the wavefronts. Singularities in the wavefronts, like caustics, can be characterized using topological indices.
Hawking Radiation and Black Hole Thermodynamics: The event horizon of a black hole, a region of spacetime from which nothing can escape, possesses intriguing topological properties. Hawking radiation, the thermal radiation emitted by black holes, can be understood as a consequence of the topological properties of the event horizon and the quantum vacuum.
3. Other Systems:

Elastic Waves in Solids: Topological concepts are increasingly being applied to study the propagation of elastic waves in solids, leading to the discovery of topological insulators for sound and other mechanical waves. These materials exhibit robust, unidirectional wave propagation along their edges, immune to backscattering.
Photonic Crystals and Metamaterials: Photonic crystals and metamaterials, artificial structures with engineered optical properties, can be designed to exhibit topological properties, leading to novel optical devices with unusual waveguiding and light-manipulation capabilities.
Key Concepts and Tools:

Topological Invariants: Quantities like Chern numbers, winding numbers, and Berry phases are used to characterize the topological properties of wavefunctions, band structures, or other relevant mathematical objects.
Bulk-Edge Correspondence: This principle connects the topological properties of the bulk of a system to the existence of robust, localized states at its boundaries or interfaces.
Symmetry and Topology: The interplay between symmetry and topology plays a crucial role in determining the topological properties of physical systems.
In summary, the insights gained from analyzing the topological properties of classical waves, as exemplified by the study of shallow-water waves on a rotating sphere, provide a powerful framework for understanding a wide range of physical phenomena in diverse systems, from quantum fluids to electromagnetic waves in curved spacetime. These topological concepts are leading to new discoveries and technological advancements in various fields.