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Simulations of the onset and dynamical evolution of inertial waves in solar/stellar interior

This study utilizes global numerical simulations to demonstrate that inertial modes in solar and stellar interiors are spontaneously excited via baroclinic instability driven by differential rotation, leading to the formation of high-latitude retrograde polar vortices and equatorial Rossby waves that can subsequently modify the internal rotation profile through Reynolds stresses.

Original authors: Mariane D. Souza-Gomes, Conrado S. Finotti, Gustavo Guerrero, Santiago A. Triana, Mausumi Dikpati, Piotr K. Smolarkiewicz, Eric S. Botelho

Published 2026-03-24

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CC BY 4.0

Original authors: Mariane D. Souza-Gomes, Conrado S. Finotti, Gustavo Guerrero, Santiago A. Triana, Mausumi Dikpati, Piotr K. Smolarkiewicz, Eric S. Botelho

Original paper licensed under CC BY 4.0 (http://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

Imagine the Sun not as a static, burning ball of gas, but as a giant, swirling ocean of plasma. Just like water in a bathtub, this solar "ocean" spins, but it doesn't spin perfectly like a solid top. The equator spins faster than the poles, and different layers spin at different speeds. This creates friction and shear, much like when you stir a cup of coffee and the liquid creates little whirlpools.

This paper is about a team of scientists using a super-computer to simulate what happens inside this solar ocean. They are trying to understand a specific type of "wave" called inertial waves (specifically Rossby waves), which were recently detected on the Sun's surface but are still a bit of a mystery.

Here is the story of their discovery, broken down into simple concepts:

1. The Setup: A Cosmic Bathtub

The scientists built a digital model of the Sun's interior. Think of it as a giant, transparent, rotating sphere filled with gas.

The Test Run: First, they spun the sphere perfectly evenly (like a solid top) and gave it a little nudge. They wanted to see if their computer model could recreate the known "Rossby waves."
The Result: It worked! The waves behaved exactly as physics predicts. They found that when you poke the system, it doesn't just make one wave; it creates a ripple effect where energy jumps to other waves, some bigger and some smaller, creating a complex dance of energy.

2. The Real Challenge: The Solar "Stirring"

The real Sun isn't a solid top; it's a messy, differential rotator. The equator spins faster than the poles. To simulate this, the scientists added a "force" to their model to make the equator spin faster than the rest, creating shear (friction between layers moving at different speeds).

They asked: If we spin the Sun like this, do these mysterious waves appear on their own?

3. The Discovery: The "Baroclinic" Spark

The answer was a resounding yes, but with a twist.

The Spark: The waves didn't start randomly. They were ignited by something called baroclinic instability.
The Analogy: Imagine a pot of soup where the bottom is hot and the top is cold. If you stir it, the heat and the spinning create a chaotic mix. In the Sun, the difference in rotation speed creates a temperature difference (entropy gradient). This mismatch acts like a spring that gets squeezed and then snaps, launching powerful waves.
The Result: These waves manifested as giant polar vortices (swirling storms) near the Sun's poles. They spin in the opposite direction of the Sun's rotation (retrograde), just like the waves observed by telescopes.

4. The "Traffic Jam" of Energy

The scientists watched how these waves behaved over time:

The Cascade: When the "stirring" (shear) was weak, the waves were small and quiet. But as they increased the speed difference between the equator and poles, the waves grew stronger.
The Polar Vortex: At high speeds, the energy concentrated into massive, organized storms at the poles. These storms are so powerful they can actually push the Sun's rotation around, speeding up the poles slightly. It's like a group of dancers (the waves) suddenly grabbing the lead dancer (the Sun's rotation) and pulling them into a faster spin.
The Equator: Interestingly, the waves near the equator (Rossby waves) didn't appear spontaneously. They needed a little "push" (an external nudge) to get started. This suggests that on the real Sun, something else—perhaps the churning of convection or magnetic fields—is needed to wake them up.

5. Why This Matters

This study is a breakthrough because it connects the dots between how the Sun spins and how it creates waves.

The Tachocline: The waves seem to be born deep inside the Sun, at a boundary layer called the tachocline (where the Sun's rotation changes from solid-body to differential).
The Missing Piece: The simulations showed that these waves can transport angular momentum (spin energy) from the equator to the poles. This helps explain how the Sun maintains its shape and rotation profile.
What's Still Missing: The model didn't perfectly reproduce every type of wave seen in real life (specifically the very high-frequency ones). The scientists suspect that magnetic fields (which they didn't include in this specific run) are the missing ingredient that fine-tunes these waves.

The Bottom Line

Think of the Sun as a giant, spinning, friction-filled machine. This paper shows that when you spin the machine fast enough, the friction creates giant, organized storms at the poles. These storms aren't just noise; they are active participants that help regulate the Sun's spin.

The scientists have successfully built a digital "Sun" that proves these waves are real, natural consequences of the Sun's rotation, driven by the friction between its spinning layers. It's a major step toward understanding the hidden heartbeat of our star.

1. Problem Statement

Inertial modes, including equatorial Rossby waves and high-latitude polar vortices, have been recently detected in the Sun via helioseismology. However, their excitation mechanisms, nonlinear evolution, and specific roles in the dynamics of the solar plasma and magnetic field remain poorly understood. While linear eigenvalue studies have identified these modes, there is a lack of an integrated framework that addresses the conditions for their excitation in the solar interior and their subsequent nonlinear evolution, particularly regarding their contribution to angular momentum transport and the maintenance of differential rotation.

2. Methodology

The authors employ global numerical simulations using the hydrodynamic branch of the EULAG-MHD code, which solves the Lipps-Hemler anelastic system of equations.

Domain and Stratification: The simulations cover a spherical shell representing the solar interior ( $0.60 R_\odot$ $0.60 R_{⊙}$ to $0.95 R_\odot$ $0.95 R_{⊙}$ ). The thermodynamic structure includes:
- A bottom layer with stiff stratification and solid-body rotation.
- A middle layer with stiff stratification and differential rotation.
- An upper layer that is either adiabatic or slightly sub-adiabatic (to allow for baroclinicity) with differential rotation.
- The stratification is designed to match solar models (Christensen-Dalsgaard et al., 1996), with a tachocline located in the stable radiative zone.
Forcing Mechanism: To impose differential rotation (DR), a physically motivated body force term ( $F$ ) is added to the momentum equation. This force relaxes the longitudinal velocity toward a target solar-like DR profile (including radial shear at the tachocline and latitudinal shear) over a timescale of $\sim 5$ Carrington rotations.
Simulation Cases:
1. Rigid Rotation Validation: Simulations with solid-body rotation where initial perturbations are imposed as sectoral and tesseral spherical harmonics to validate the dispersion relations.
2. Differential Rotation Evolution: Simulations with varying shear amplitudes (relative shear $\Omega_2/\Omega_0$ ranging from 5% to 40%) starting from zero perturbations to observe spontaneous instability onset.
Analysis Tools: The study utilizes Hovmöller diagrams, power spectra of radial vorticity, and spectral analysis of velocity components to track mode frequencies, wavenumbers, and energy cascades.

3. Key Contributions

Validation of Inertial Mode Excitation: The study confirms that perturbations in rigidly rotating spheres excite neighboring modes via direct and inverse energy cascades, strictly adhering to the Rossby wave dispersion relation ( $\omega = -2m\Omega_0 / l(l+1)$ ).
Identification of Baroclinic Instability as the Driver: The paper establishes that in differentially rotating, stratified fluids, the spontaneous onset of inertial modes is primarily driven by baroclinic instability arising from latitudinal gradients of potential temperature (entropy) induced by thermal wind balance.
Characterization of Polar Vortices: The simulations reproduce high-latitude inertial modes in the form of retrograde polar vortices. The study analyzes their morphology, frequency, and dependence on shear strength.
Angular Momentum Transport: The work quantifies the role of Reynolds stresses generated by these inertial modes in transporting angular momentum, specifically their ability to accelerate polar rotation against the imposed forcing.

4. Key Results

A. Rigid Rotation (Validation)

Imposed sectoral and tesseral perturbations evolve into Rossby waves propagating retrograde.
Nonlinear interactions lead to energy cascades (both direct and inverse) across different wavenumbers ( $m$ ) and latitudinal degrees ( $l$ ), while maintaining frequencies consistent with the theoretical dispersion relation.
Tesseral modes exhibit more complex evolution, including temporary prograde propagation and energy exchange between layers.

B. Differential Rotation and Instability Onset

Threshold: Instability onset occurs when relative shear exceeds a threshold between 5% and 10%.
Growth Rate: The growth rate ( $\gamma$ ) of the instability increases with shear amplitude, saturating at shear values $>35\%$ .
Mechanism: The instability is confirmed to be baroclinic. The radial shear at the tachocline creates a latitudinal gradient in potential temperature ( $\Theta'$ ). When the upper layer is slightly sub-adiabatic, this gradient persists, fueling the instability.
Taylor-Proudman Theorem: In strictly adiabatic layers, the flow obeys the Taylor-Proudman theorem (contours aligned with the rotation axis). A slightly sub-adiabatic stratification breaks this symmetry, allowing tilted rotation contours and the development of baroclinic instability.

C. Morphology and Dynamics of Modes

High-Latitude Modes: Strong shear (e.g., 20–40%) generates retrograde polar vortices. In the 20% shear case (solar analogue), the dominant frequencies ( $m=1, 2, 3$ $m = 1, 2, 3$ ) match observational data from helioseismology ( $\sim -78$ $\sim - 78$ to $-164$ nHz).
- Note on Orientation: The raw simulation shows clockwise vortices in the Northern Hemisphere, whereas observations show counter-clockwise. The authors suggest this discrepancy may be due to missing physics (magnetic fields, specific viscosity values) or that the observed modes are hidden within higher-order energetic modes in the simulation.
Equatorial Rossby Modes: These do not arise spontaneously from the shear instability. They require external perturbations (e.g., convection) to be excited. Once excited, they participate in energy cascades similar to the rigid rotation case.
Critical Latitude Modes: Modes propagating at the local rotation speed (zero frequency in the co-rotating frame) are observed at mid-latitudes but not at the equator.

D. Angular Momentum Transport

Meridional Circulation (MC): Dominates the transport of angular momentum in weak shear regimes, balancing the external forcing.
Reynolds Stresses (RS): In high-shear regimes (e.g., 40%), RS generated by inertial modes become significant. They transport angular momentum poleward, effectively accelerating the poles and modifying the imposed differential rotation profile. This suggests a feedback loop where inertial modes can regulate solar differential rotation.

5. Significance and Conclusion

This study provides a comprehensive global hydrodynamic framework for understanding solar inertial modes.

Origin of Polar Vortices: It supports the hypothesis that high-latitude inertial modes and polar vortices originate from baroclinic instability driven by the tachocline's radial shear and thermal stratification.
Dynamical Impact: The results demonstrate that these modes are not merely passive oscillations but active agents in angular momentum transport, capable of modifying the solar rotation profile in high-shear environments.
Future Directions: The discrepancy in vortex orientation and the absence of high-frequency modes in the simulations suggest that magnetic fields and more realistic turbulent viscosity are critical missing ingredients. The authors conclude that while the hydrodynamic mechanism is sound, a full reproduction of the solar inertial spectrum requires coupling with magnetohydrodynamics (MHD).

In summary, the paper bridges the gap between linear theoretical predictions and nonlinear global dynamics, confirming that baroclinic instability in a differentially rotating, stratified shell is a robust mechanism for generating the inertial modes observed in the Sun.