🔭 astrophysics

Effects of the radiative interior on solar inertial modes

This study utilizes the Dedalus code to demonstrate that while the Sun's radiative interior only slightly alters the frequencies of most inertial modes, it significantly influences high-frequency columnar modes and supports mixed Rossby-inertial modes that, despite their complex structure, are difficult to excite stochastically due to their high mass in the radiative zone.

Original authors: Suprabha Mukhopadhyay, Yuto Bekki, Xiaojue Zhu, Laurent Gizon

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Suprabha Mukhopadhyay, Yuto Bekki, Xiaojue Zhu, Laurent Gizon

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 ball of fire, but as a giant, swirling, rotating fluid. Deep inside, it has two main neighborhoods: a turbulent, churning outer layer called the Convection Zone (where the weather happens), and a calm, stable, deep interior called the Radiative Zone.

For a long time, scientists studying the "music" of the Sun (its oscillations) mostly focused on the outer neighborhood. They thought the deep interior was just a solid floor that the waves bounced off of, having no real say in the melody.

This new paper asks a simple but profound question: What if the deep interior isn't just a floor, but actually an instrument that can play along?

Here is the breakdown of their findings using everyday analogies:

1. The "Dance" of the Sun

The Sun spins, but not like a rigid top. The equator spins faster than the poles. This differential rotation creates "inertial modes"—think of them as giant, global waves or ripples moving through the Sun's fluid, held together by the Coriolis force (the same force that makes hurricanes spin).

Scientists have been observing these waves at the Sun's surface for years. They wanted to know: Does the deep, quiet interior change how these waves dance?

2. The "Glass Wall" Experiment

To find out, the researchers built a super-accurate computer simulation.

The Old Setup: They simulated just the outer churning layer, treating the bottom as a solid, impenetrable wall.
The New Setup: They extended the simulation deep down into the Radiative Zone, allowing the waves to potentially leak into the deep interior.

The Result: For most of the Sun's "songs," the deep interior is like a glass wall. The waves hit it and bounce back. The deep interior barely changes the pitch (frequency) or the shape of the waves we see at the surface. If you are standing on the surface, you wouldn't notice the difference. The outer layer is mostly doing its own thing.

3. The "Deep Echo" (Rossby Modes)

However, the deep interior does have its own music. Because the deep zone is stable and rotates at a constant speed, it can support its own special type of wave called Rossby modes.

Think of the Convection Zone as a busy, noisy dance floor where people are bumping into each other. The Radiative Zone is a quiet, empty ballroom below.

In the noisy dance floor, only certain dance moves (waves) are possible.
In the quiet ballroom, a wider variety of dance moves can happen, including some that are impossible on the noisy floor.

The paper found that these deep "ballroom dances" (Rossby modes) exist and are very stable. They follow their own strict rules based on how fast the deep interior spins.

4. The "Rare Duet" (Mixed Modes)

Here is the most exciting part. Sometimes, a wave from the noisy dance floor and a wave from the quiet ballroom happen to have the exact same rhythm (frequency).

When this happens, they don't just bounce off each other; they sync up. They form a "duet" or a mixed mode.

The Catch: For this to happen, the two waves must be incredibly close in pitch (within a tiny fraction of a hertz) and they must have the same "handedness" (symmetry).
The Problem: Even when they do sync up, the resulting wave spends most of its energy deep in the quiet ballroom. It's like a singer trying to sing a duet with a ghost; the ghost is doing most of the work. Because so much energy is trapped deep inside, it is very hard for the Sun's surface turbulence to "kickstart" or excite these duets. They are likely too quiet to be heard from Earth.

5. The "Leaky Floor" (Overshooting)

There is one more detail. While the deep interior is mostly a glass wall, there is a thin "transition zone" right at the bottom of the churning layer (called the overshooting layer).

Some waves, especially those that move up and down vigorously, do dip their toes into this transition zone.
This acts like a friction pad. It doesn't change the song's pitch much, but it does make the waves lose energy faster (damping). It's like running through shallow water; you don't change direction, but you get tired faster.

The Big Picture Takeaway

Good News: Scientists can keep using their simpler models (ignoring the deep interior) to interpret most of what they see at the Sun's surface. The deep interior doesn't mess up the data much.
New Discovery: The deep interior does have its own hidden orchestra of waves (Rossby modes) that we haven't fully mapped yet.
The Mystery: Occasionally, the surface waves and deep waves might try to sing together. If they do, the song is likely too deep and quiet for us to hear easily, but finding them would be a huge breakthrough in understanding how the Sun's deep interior works.

In short: The Sun's deep interior is mostly a silent observer, but occasionally, it whispers a secret that changes the song just enough to be interesting, even if we can't hear it clearly yet.

1. Problem Statement

Solar inertial modes are global oscillations in the Sun's differentially rotating convection zone (CZ) restored by the Coriolis force. While these modes are well-observed and utilized for diagnosing solar differential rotation and internal dynamics, previous theoretical studies have largely treated the CZ as an isolated shell with impenetrable lower boundaries.
The central problem addressed in this study is the coupling between solar inertial modes in the convection zone and the underlying radiative interior (RZ). Specifically, the authors investigate:

How the sub-adiabatic, uniformly rotating RZ affects the frequencies and eigenfunctions of CZ inertial modes.
Whether the RZ supports its own distinct oscillation modes (Rossby modes) and if these can couple with CZ modes to form "mixed modes."
Whether these interactions lead to measurable changes at the solar surface.

2. Methodology

The authors employed a linear eigenmode analysis using the Dedalus spectral code to solve the linearized compressible fluid dynamical equations.

Domain: The computational domain spans from $0.5 R_\odot$ (deep in the radiative interior) to $0.985 R_\odot$ (just below the photosphere), encompassing the Radiative Zone (RZ), the Overshoot (OS) layer, and the Convection Zone (CZ).
Physics:
- Equations: Fully compressible equations including momentum, entropy, and continuity, coupled with a quasi-adiabatic equation of state.
- Background State: Based on the standard solar Model S. The model includes a simplified analytic profile for differential rotation (with a tachocline) and observed profiles for high-latitude modes.
- Stratification: The RZ is modeled as strongly sub-adiabatic (stable), while the CZ is near-adiabatic. A smooth transition (overshoot layer) is defined between them.
- Diffusivities: Radially varying turbulent viscosity ( $\nu$ ) and thermal diffusivity ( $\kappa$ ) are applied, with lower values in the RZ and higher values in the CZ.
Boundary Conditions: Unlike previous studies using impenetrable boundaries, this study imposes free-surface boundary conditions at both the top and bottom of the domain. This allows for radial motion and is crucial for correctly modeling $g$ -modes and gravito-inertial modes in the RZ.
Comparison: The authors compared results from the extended model (CZ + RZ) against a traditional "CZ-only" model (lower boundary at $0.71 R_\odot$ with impenetrable conditions).

3. Key Contributions

First Linear Eigenmode Analysis of the Coupled System: This is the first study to compute linear inertial modes across the entire CZ and upper RZ simultaneously, moving beyond the isolated shell approximation.
Characterization of RZ Rossby Modes: The study identifies a spectrum of quasi-toroidal Rossby modes trapped entirely within the RZ, which exist due to the sub-adiabatic stratification allowing for non-sectoral modes ( $\ell \neq m$ ).
Discovery of CZ–RZ Mixed Modes: The authors define and characterize a specific class of "mixed modes" that arise when RZ Rossby modes and CZ inertial modes have nearly identical frequencies and matching north–south symmetry.
Quantification of Mode Mass: The study calculates the mode mass for these mixed modes, revealing they are heavily dominated by the radiative interior.

4. Key Results

A. Impact on Standard CZ Inertial Modes

Frequencies: For most inertial modes (e.g., equatorial Rossby, high-latitude modes), including the RZ modifies the frequencies by less than 10 nHz.
Eigenfunctions: The surface eigenfunctions are nearly identical to the CZ-only models (correlation coefficient $C \gtrsim 0.9$ ).
Exceptions: Significant frequency shifts ( $>20$ nHz) occur for High-Frequency Retrograde (HFR) modes ( $m \ge 12$ ) and prograde columnar modes ( $m \le 6$ ). These modes have significant radial motion and critical layers that penetrate the overshoot layer, altering their effective rotation rate.
Damping: The inclusion of the RZ increases the damping rates (reduces growth rates) for most modes due to viscous dissipation in the overshoot layer where modes exhibit finite motion before decaying in the stable RZ.

B. RZ Rossby Modes

Dispersion: The RZ supports Rossby modes that follow the classical dispersion relation $\omega_R = -2m\Omega_{RZ}/[\ell(\ell+1)] + m(\Omega_{RZ} - \Omega_0)$ .
Non-Sectoral Nature: Unlike the adiabatic CZ where only sectoral modes ( $\ell=m$ ) exist, the sub-adiabatic RZ supports non-sectoral modes ( $\ell \neq m$ ). This is possible because buoyancy forces help balance the radial Coriolis force.
Radial Nodes: These modes can have any number of radial nodes ( $n_{RZ}$ ) within the RZ without changing their frequency, though higher $n_{RZ}$ leads to higher dissipation.

C. CZ–RZ Mixed Modes

Formation Condition: Mixed modes form only when an RZ Rossby mode and a CZ inertial mode satisfy two conditions:
1. Frequency separation $|\Delta \omega| \lesssim 10$ nHz.
2. Identical north–south symmetry.
Characteristics: These modes exhibit significant motion in both zones. While their surface signatures resemble pure CZ modes, their internal structure involves substantial flow in the RZ.
Mode Mass: Crucially, mixed modes possess a very high mode mass (dominated by the RZ contribution), often orders of magnitude larger than pure CZ modes (e.g., $163.5 \times 10^{-3} M_\odot$ vs $5.9 \times 10^{-3} M_\odot$ for a specific $m=1$ mode).

5. Significance and Implications

Observational Interpretation: For most solar inertial modes, the "CZ-only" approximation remains valid for interpreting surface observations (frequencies and surface eigenfunctions). The RZ introduces negligible corrections except for specific high- $m$ retrograde and prograde modes.
Excitation Mechanism: The high mode mass of the CZ–RZ mixed modes suggests they are energetically difficult to excite stochastically via convection in the CZ. This implies that even if these modes exist, they may not reach observable amplitudes unless excited by other mechanisms (e.g., tachocline shear instability, though none were found in this linear study).
Tachocline Dynamics: The study confirms that inertial modes have significant horizontal velocities in the overshoot layer (up to $\sim 3$ m/s for $m=1$ ). This motion could contribute to horizontal turbulent diffusion and the confinement of the solar tachocline via local dynamo effects.
Future Diagnostics: The identification of mixed modes provides a theoretical framework for interpreting observed modes that lie close to RZ Rossby frequencies. If such modes are detected with high amplitudes, it would imply the existence of excitation mechanisms capable of overcoming the high mode mass barrier.

In conclusion, while the radiative interior does not drastically alter the global properties of most solar inertial modes, it hosts a rich spectrum of Rossby modes that can couple with the convection zone under specific resonant conditions, creating high-mass mixed modes with potential implications for solar interior dynamics.