Bridging the Prandtl number gap: 3D simulations of thermohaline convection in astrophysical regimes

This study bridges the long-standing Prandtl number gap in 3D simulations of thermohaline convection by demonstrating that the chemical mixing model of Brown, Garaud, & Stellmach (2013) remains valid across stellar regimes down to $\Pr = 10^{-6}$, thereby ruling out the Prandtl number discrepancy as the cause of tensions between the model and astronomical observations.

The Gist

The Big Picture: Why Stars are "Sticky" and "Sugary"

Imagine a star not as a giant ball of fire, but as a giant pot of soup. Sometimes, this soup has layers. In some layers, the heat is stable, and nothing moves up or down. But in other layers, something weird happens: the "ingredients" (chemicals) get mixed up in a way that creates tiny, finger-like currents. This is called thermohaline convection.

Think of it like this: Imagine you have a glass of water. If you carefully pour a layer of heavy syrup on top, it should stay there, right? But if the syrup is slightly warmer than the water, the heat escapes quickly (like steam), but the sugar stays put. The syrup becomes heavier than the water below it, and it starts to sink in little fingers, mixing the sugar into the water.

In stars, this process mixes chemicals (like lithium or carbon) from the deep interior up to the surface. Astronomers look at the surface of stars to see what's there, and they use this mixing to understand how stars age and die.

The Problem: The "Computer Gap"

For decades, scientists have tried to simulate this "syrup mixing" on supercomputers to predict how stars behave. But there was a huge problem, known as the Prandtl Number Gap.

• In Real Stars: The fluid is so thin and the heat moves so fast that the "stickiness" (viscosity) is almost zero compared to how fast heat spreads. It's like trying to mix honey with a jet of water; the water moves instantly, but the honey barely moves.
• In Computers: Our computers are powerful, but they can't handle that extreme difference. To make the math work, scientists had to slow down the heat or speed up the stickiness to make the numbers manageable. They were simulating a "thick syrup" scenario instead of the "thin water" scenario of real stars.

The Analogy: Imagine you are trying to predict how a feather falls in a hurricane.

• Real Life: The feather is light, the wind is fast.
• The Simulation: Because the computer can't handle the speed, you simulate a bowling ball falling in a gentle breeze.
• The Result: You get a result, but you don't know if it applies to the feather.

Because of this gap, whenever a computer simulation didn't match what astronomers saw in real stars, scientists would say, "Oh, that's just because our computer model was too 'thick' (wrong Prandtl number). Let's ignore the simulation and guess a different answer."

The Breakthrough: Finally Simulating the "Feather"

This paper, by Adrian Fraser, says: "Stop guessing. We finally have the computer power to simulate the real thing."

The author ran a new suite of 3D simulations that finally bridged the gap. They managed to simulate the extreme conditions found in real stars (where the "stickiness" is incredibly low) for the first time.

The Metaphor: They finally built a wind tunnel powerful enough to test the feather in the hurricane, rather than just the bowling ball in the breeze.

The Surprising Result: The Old Rules Still Work

Here is the twist. Everyone expected that because the physics changed so much (from "thick syrup" to "thin water"), the mixing would behave totally differently. They thought the old computer models were wrong because of the "gap."

But the paper found that the old models were actually right all along!

The author compared the new, ultra-realistic simulations with an existing model (called the BGS13 model).

• The Finding: The model predicted the mixing perfectly, even in the extreme "thin water" conditions of real stars.
• The Implication: The gap in computer power was not the reason for the disagreement between theory and observation. The model works.

So, Why Don't the Stars Match the Model?

If the model is right, and the stars show more mixing than the model predicts, then something else is missing from our understanding.

The paper suggests we need to look for other forces. The author points to magnetic fields as the likely culprit.

• The Analogy: Imagine you are trying to mix the soup with a spoon (the model). You expect a certain amount of mixing. But when you look at the pot, the soup is way more mixed than the spoon could do.
• The Conclusion: You wouldn't say, "The spoon is broken because I couldn't simulate the spoon perfectly." You would say, "There must be a hidden blender (magnetic fields) inside the pot that is doing extra mixing."

Why This Matters

1. Stop Blaming the Computer: We can no longer use "our computers aren't good enough" as an excuse for why our theories don't match the stars.
2. New Physics Needed: The mismatch between what we see in stars (like Red Giants and White Dwarfs) and what we calculate means there is hidden physics at play—likely magnetic fields—that we need to study.
3. Better Star Maps: By understanding this mixing correctly, we can build better models of how stars live, evolve, and eventually die.

In short: We finally built the perfect simulation, and it told us that our old theories were actually good. The problem isn't the math; it's that we forgot to include the "magnetic blender" in our recipes.

Technical Summary

1. Problem Statement

Thermohaline convection (also known as fingering convection) is a double-diffusive instability occurring in stellar radiation zones where the mean molecular weight ($\mu$) increases with radius. It is a critical mechanism for radial chemical mixing in stars, particularly in Red Giant Branch (RGB) stars (explaining the "luminosity bump" abundance anomalies) and polluted White Dwarfs (WDs).

The Core Conflict:

• Observations vs. Theory: Observations of RGB stars and polluted WDs suggest chemical mixing rates that are significantly faster than those predicted by the standard hydrodynamic thermohaline convection model (Brown et al. 2013, hereafter BGS13).
• The "Prandtl Number Gap": A major reason for dismissing the discrepancy between the BGS13 model and observations has been the inability to simulate astrophysical conditions.
  • In stars, the Prandtl number ($Pr = \nu/\kappa_T$, ratio of kinematic viscosity to thermal diffusivity) and diffusivity ratio ($\tau = \kappa_C/\kappa_T$) are extremely small, typically $Pr, \tau \sim 10^{-6}$ or lower.
  • Previous 3D Direct Numerical Simulations (DNS) were restricted to $Pr, \tau \gtrsim 10^{-2}$ due to computational limitations.
• The Hypothesis: It was widely suspected that the dynamics of thermohaline convection change fundamentally between the numerically accessible regime ($Pr \sim 10^{-2}$) and the astrophysical regime ($Pr \sim 10^{-6}$), rendering the BGS13 model (calibrated on high-$Pr$ data) unreliable for stars.

2. Methodology

The author performed a suite of 3D hydrodynamic simulations to bridge this gap, extending the range of $Pr$ and $\tau$ down to astrophysically relevant values.

Governing Equations & Setup:

• Physics: The simulations solve the Boussinesq equations for velocity ($\mathbf{u}$), temperature ($T$), and composition ($C$) under the assumption of fixed background gradients. Magnetic fields and rotation were neglected to isolate the hydrodynamic effect.
• Non-dimensionalization: The study utilized a specific non-dimensionalization scheme (following Xie et al. 2017; Fraser et al. 2025) where the unit of length is the characteristic finger width $d = (\kappa_T \nu / N_T^2)^{1/4}$. This choice ensures that dynamical fields remain $O(1)$ even as $\tau \to 0$, preventing numerical instability during initialization.
• Numerical Scheme:
  • Solver: Pseudospectral method using Dedalus v2 with periodic boundary conditions.
  • Time-stepping: A semi-implicit, 2nd-order Adams-Bashforth/backwards-difference scheme. Crucially, nonlinearities are treated explicitly, while linear terms (including buoyancy and advection) are treated implicitly. This allows for larger timesteps by avoiding the strict constraint imposed by the buoyancy frequency, which would otherwise be prohibitive at low $\tau$.
  • Resolution: Resolution was increased for higher Rayleigh ratios ($R$), ranging from 8 to 64 Fourier modes per characteristic wavelength.
• Parameter Space:
  • Fixed Schmidt number $Sc = Pr/\tau = 2$.
  • Varied $Pr$ from $10^{-1}$down to$10^{-6}$.
  • Varied $\tau = Pr/2$.
  • Varied the Rayleigh ratio $R$ (related to the reduced density ratio $r$) to explore different levels of supercriticality.

3. Key Contributions

1. First 3D Simulations at Astrophysical $Pr$: The paper presents the first 3D DNS of thermohaline convection reaching $Pr = 10^{-6}$ and $\tau = 5 \times 10^{-7}$, directly matching the regime of stellar interiors.
2. Validation of the BGS13 Model: The study demonstrates that the chemical mixing model proposed by Brown et al. (2013), which was previously only validated for $Pr \gtrsim 10^{-2}$, remains accurate and consistent with 3D simulations even at $Pr = 10^{-6}$.
3. Refutation of the "Gap" Excuse: The results prove that the discrepancy between stellar evolution models (using BGS13) and observations is not an artifact of the Prandtl number gap. The dynamics do not change qualitatively between the simulated regimes.
4. Efficiency of the $Pr=0$ Limit: The paper shows that simulations with $Pr=0$ (where thermal diffusion is instantaneous) yield results nearly identical to $Pr=10^{-6}$. This offers a computationally cheaper pathway for future studies in these regimes, provided the Péclet number remains low.

4. Results

• Flux Consistency: The turbulent vertical compositional flux ($|\tilde{F}_C|$) measured in the simulations collapses onto a single curve when plotted against the supercriticality parameter $\epsilon = R - 1$, regardless of whether $Pr$ is $10^{-1}$or$10^{-6}$.
• Model Agreement: The BGS13 model predictions (curves in Fig. 3) align perfectly with the simulation data points across the entire range of $Pr$. There is no evidence of a "break" in physics or a change in mixing efficiency as $Pr$ decreases.
• Flow Structure: Snapshots of vertical velocity show that as the Rayleigh ratio $R$ increases, flow amplitudes increase and structures become finer, but the fundamental nature of the fingering instability remains consistent across $Pr$ values.
• Control Parameters: The data suggests that the Rayleigh ratio $R$ (or supercriticality $\epsilon$) is a more practical control parameter than the reduced density ratio $r$ when studying convection at small $Pr$ and $\tau$.

5. Significance and Implications

• Closing the Gap: The "Prandtl number gap" is no longer a valid excuse for ignoring the predictions of the BGS13 model. The tension between the model and observations (e.g., rapid mixing in RGB stars, high accretion rates in polluted WDs) is real and must be resolved through other physical mechanisms.
• Need for Additional Physics: Since hydrodynamic thermohaline convection alone (as described by BGS13) cannot explain the observed mixing rates, the paper argues that additional physics must be included in stellar models.
  • Magnetic Fields: The author highlights that previous work (Harrington & Garaud 2019; Fraser et al. 2024) suggests that even weak magnetic fields ($\sim 100$ G) can significantly enhance mixing, potentially resolving the RGB tension.
  • Future Directions: Future research should focus on the interplay between thermohaline convection, magnetic fields, rotation, and internal gravity waves, rather than assuming the hydrodynamic limit is sufficient.

In summary, this paper establishes that current 3D simulations are capable of reaching astrophysical regimes and confirms that the standard hydrodynamic theory of thermohaline convection is robust. Consequently, the failure of this theory to match stellar observations points to missing physics (likely magnetic) rather than numerical limitations.