Excitation of Inertial Modes in 3D Simulations of Rotating Convection in Planets and Stars

Using 3D simulations of rotating convection in spherical shells, the study demonstrates that inertial modes naturally emerge in rotationally constrained turbulence when the convective Rossby number falls below approximately one-half, suggesting that such modes are likely driven by differential rotation instabilities in the interiors of stars and giant planets.

The Gist

Imagine a giant, spinning ball of hot gas or liquid, like the inside of a planet or a star. Deep inside these celestial bodies, heat rises and falls, creating a chaotic, churning soup known as convection. Usually, we think of this churning as just random turbulence, like boiling water in a pot. But this paper asks: What happens when you spin that pot really fast?

The authors, using powerful computer simulations, discovered that when you spin a rotating fluid fast enough, that chaotic boiling doesn't just stay messy. Instead, it organizes itself into distinct, rhythmic "songs" or inertial modes.

Here is a breakdown of their findings using everyday analogies:

1. The "Spin" Threshold

Think of the rotation speed as a volume knob.

• Slow Spin (High Rossby Number): If you spin the pot slowly, the heat just bubbles up randomly. It's like a crowd of people milling about in a room; everyone is moving, but there's no pattern. The paper found that in this state, no distinct "songs" emerge.
• Fast Spin (Low Rossby Number): Once the spin gets fast enough (specifically, when the rotation period is less than half the time it takes for a bubble of heat to rise), the chaos suddenly snaps into order. It's like a crowd of people suddenly starting to march in a synchronized parade. The paper found that these organized "marches" (inertial modes) only appear when the rotation dominates the heat.

2. What are these "Songs"?

These inertial modes are waves that are held together by the Coriolis force—the same invisible force that makes hurricanes spin and laundry dryers spin clothes to the side.

• The Analogy: Imagine a spinning top. If you poke it, it wobbles in a specific, predictable way. In the planet's interior, the "poke" comes from the churning heat, and the "wobble" is the inertial mode.
• The Direction: Most of these waves travel "backward" relative to the spin of the planet (retrograde), like a runner jogging against the direction of a moving walkway.
• The Location: They don't happen everywhere. They are mostly confined to the "middle and high latitudes" (the mid-to-polar regions), avoiding the equator, much like how certain weather patterns only happen in specific bands on Earth.

3. The Secret Ingredient: Viscosity and "Sticky" Fluids

The paper tested what happens if the fluid is "thinner" or "stickier" (changing the Prandtl number, which relates to how easily heat moves compared to how easily the fluid flows).

• Thicker Fluid (Pr = 1): The waves were there, but quiet and few.
• Thinner Fluid (Pr = 0.1): When they simulated a fluid that behaves more like the actual hot, thin gases found in stars and giant planets, the "music" became much louder and more complex. Suddenly, many more different "notes" (modes) appeared, and they were much stronger. It's as if switching from a heavy wool blanket to a silk sheet allowed the wind to create a much richer, more complex sound.

4. How Do They Start? (The Mystery)

The paper notes that these waves didn't need an outside hand to start them (like a drummer tapping a rhythm). They started naturally because of the shear (the difference in speed between layers of the fluid).

• The Mechanism: The heat creates different rotation speeds in different parts of the planet (differential rotation). The authors suggest that the waves are likely triggered by instabilities in these speed differences, rather than just being random bumps from the heat. It's like a river flowing over rocks; the water doesn't just splash randomly; it forms specific, repeating ripples where the current changes speed.

5. Can We Hear Them?

The authors conclude that while these waves almost certainly exist inside giant planets (like Jupiter and Saturn) and stars, they are very hard to detect.

• The Problem: They are very low-frequency waves. If you were to listen to Jupiter, these waves would be like a deep, slow hum that takes days to complete a cycle.
• The Detection: Current tools might miss them because they are too slow or too quiet. However, the paper mentions that we might have already seen hints of them in Saturn's rings (where the rings act like a seismograph for the planet), but we haven't seen them in stars yet.

Summary

In short, this paper shows that if you spin a hot, churning fluid fast enough, the chaos organizes into specific, rhythmic waves. These waves are a natural consequence of the planet spinning and the heat moving, and they become much more active and numerous if the fluid is "thinner" (like real planetary gases). While they are likely singing inside our solar system's giants right now, they are singing so quietly and slowly that we haven't quite learned how to hear them yet.

Technical Summary

Technical Summary: Excitation of Inertial Modes in 3D Simulations of Rotating Convection in Planets and Stars

Problem Statement
Thermal convection in rotating stars and planets drives anisotropic turbulence and differential rotation, both of which can feed energy into global oscillations. While inertial modes (oscillations restored by the Coriolis force) are well-observed in various geophysical and astrophysical systems, their excitation mechanisms in realistic, self-consistent environments remain poorly understood. Previous studies have largely focused on spherical Couette flows, where differential rotation is imposed via boundary conditions. In contrast, astrophysical differential rotation emerges self-consistently from rotating convection. The specific conditions under which inertial modes are excited by this self-generated shear, and the nature of their properties, require investigation in 3D rotating convection simulations.

Methodology
The authors performed a suite of 3D numerical simulations of thermal convection within a rotating spherical shell using the Dedalus pseudospectral solver. The simulations employed the Boussinesq approximation with stress-free, isothermal boundaries.

• Parameters: The study fixed the Rayleigh number ($Ra = 5 \times 10^6$) and Prandtl number ($Pr = 1$) for the primary suite, varying the Ekman number ($Ek$) to explore a range of convective Rossby numbers ($Ro_c \approx 0.17$ to $\infty$). A single additional simulation was conducted at a lower Prandtl number ($Pr = 0.1$) with $Ro_c \approx 0.17$ to mimic astrophysical diffusivities.
• Analysis: The researchers analyzed the spatial and frequency spectra of the resulting turbulence. They decomposed the velocity field using spherical harmonics to compute power spectra as a function of spherical harmonic degree ($\ell$), azimuthal order ($m$), and frequency ($\omega$). They also examined the morphology of specific modes by filtering velocity components in space and time.

Key Results

1. Excitation Threshold: Coherent inertial modes emerge naturally in the simulations only when the convective Rossby number falls below approximately $Ro_c \approx 0.5$. Below this threshold, rotation dominates the dynamics, and the system transitions from isotropic turbulence to rotationally constrained flow with strong differential rotation.
2. Mode Properties:
   • Frequency: The excited modes have discrete frequencies well below twice the background rotation rate ($|\omega| < 2\Omega_0$).
   • Symmetry and Direction: Most modes are equatorially symmetric and retrograde in the rotating frame. They are confined primarily to mid and high latitudes.
   • Structure: The modes exhibit complex structures with internal shear layers resembling wave attractors, particularly visible at high latitudes.
   • Azimuthal Order: The simulations excited modes with azimuthal orders ranging from $m=1$ to $m=8$.
3. Dependence on Prandtl Number: The simulation with $Pr = 0.1$ (at the same $Ro_c \approx 0.17$ as the $Pr=1$ case) exhibited a significantly richer and more vigorous spectrum of inertial modes. The excitation efficiency was higher, with larger amplitudes and a broader range of excited modes compared to the $Pr=1$ case.
4. Differential Rotation: The simulations produced strong zonal flows, including prograde equatorial jets at low $Ro_c$ and retrograde jets at higher latitudes. The power spectrum of these zonal flows followed an $\ell^{-5}$ scaling, consistent with zonostrophic turbulence observed in giant planets.
5. Excitation Mechanism: The authors observed that the excited modes are discrete and phase-coherent, rather than a broad stochastic background. This suggests the modes are driven by instabilities related to differential rotation (likely shear instabilities) rather than stochastic forcing by convection. The presence of "critical layers" (where the mode drift frequency matches the local flow rotation) was noted, though the persistence of modes in these layers suggests complex interactions beyond simple absorption.

Significance and Claims
The paper concludes that inertial modes are likely to exist in the interiors of giant planets and stars, driven by the differential rotation that naturally arises from convection. The study highlights that:

• Self-Consistent Excitation: Inertial modes can be excited without external forcing (such as tides or libration), solely through the interaction of convection and rotation.
• Detection Challenges: While these modes likely exist, their low frequencies (in the inertial frame, periods of hours to days for giant planets) and small amplitudes make them difficult to detect via current ground-based methods like radial velocity monitoring.
• Astrophysical Relevance: The findings offer a potential explanation for observed phenomena, such as the $m=3$ waves detected in Saturn's rings via ring seismology, which may correspond to unstable inertial modes in the planet's convective interior.
• Role of Prandtl Number: The results suggest that the Prandtl number plays a critical role in mode excitation efficiency, implying that simulations with astrophysically relevant low Prandtl numbers ($Pr \ll 1$) are necessary to fully understand the wave spectrum in planetary and stellar interiors.

The authors emphasize that while the precise excitation pathway remains uncertain, the emergence of these modes in self-consistent simulations supports the hypothesis that shear instabilities in differentially rotating convection zones are a viable mechanism for generating inertial modes in real astrophysical objects.