Scale-by-scale kinetic energy flux calculations in simulations of rotating convection

This paper evaluates spatial filtering and adapted Fourier-based techniques for calculating scale-by-scale kinetic energy fluxes in rotating Rayleigh-Bénard convection, revealing that while the bulk flow is dominated by a direct energy cascade, significant inverse cascading occurs near the boundaries due to Ekman plume vortex merging.

The Gist

Imagine a giant, spinning pot of soup. If you heat the bottom and cool the top, the soup starts to swirl and churn. This is what scientists call convection. Now, imagine you put a lid on that pot and spin the whole thing really fast. This creates a special kind of chaotic flow called rotating convection, which is a bit like how weather systems behave on Earth or how fluids move inside stars.

The big question this paper asks is: How does energy move through this swirling soup?

The Two Ways Energy Moves

In a normal, non-spinning turbulent flow (like a raging river), energy usually flows from big, slow swirls down to tiny, fast ripples until it disappears as heat. Scientists call this the direct cascade. Think of it like a waterfall: big drops break into smaller drops, then into mist.

But when you add rotation (like spinning the pot), something magical happens. Some of that energy decides to go upstream. Instead of breaking into tiny bits, the small swirls merge together to form giant, slow-moving vortices. This is called the inverse cascade. It's like if the mist in our waterfall suddenly decided to reassemble itself into a giant drop at the top.

The Problem: Measuring the Invisible

Scientists want to measure exactly how much energy is flowing "down" (direct) versus how much is flowing "up" (inverse). However, measuring this is tricky.

• The Ideal Lab: In a perfect computer simulation where the walls are invisible (periodic), it's easy to measure.
• The Real World: In real experiments or simulations with solid walls (like a real cylinder), the flow gets messy, bumpy, and uneven. The standard tools for measuring energy flow often break down or give confusing results in these messy environments.

The Solution: Two Different Rulers

The authors of this paper tested two different "rulers" to measure this energy flow in these messy, spinning systems to see if they agree.

1. The Fourier Method (The "Perfect Slices" Ruler): This method tries to cut the flow into perfect, mathematical slices based on size. It works great in ideal, repeating boxes but struggles when the flow hits a solid wall or isn't perfectly uniform.
2. The Spatial Filtering Method (The "Blurry Lens" Ruler): This method is like looking at the soup through a lens that blurs out the tiny details. By adjusting how blurry the lens is, they can see how energy moves between big and small scales. This method is more flexible and works well even in messy, real-world shapes.

What They Found

The researchers ran simulations of this spinning soup in two different containers:

1. A Box with Invisible Walls: A perfect, repeating environment.
2. A Solid Cylinder: A realistic container with solid walls all around.

The Results:

• The Rulers Agree: Surprisingly, even in the messy, solid-walled cylinder, both the "Perfect Slices" and the "Blurry Lens" methods gave very similar answers. This is great news because it means scientists can use the more flexible "Blurry Lens" method for real-world experiments where the "Perfect Slices" method might fail.
• Where the Magic Happens: They discovered that the "upstream" energy flow (the inverse cascade) mostly happens near the top and bottom lids of the container. It's like the tiny whirlpools near the floor and ceiling are merging together to build giant, slow-moving storms.
• The Middle is Different: In the middle of the container (the bulk), the energy mostly flows the "normal" way—breaking down from big swirls to tiny ripples (the direct cascade).

The Bottom Line

This paper proves that we have reliable tools to measure how energy moves in complex, spinning fluids, even when they are trapped in solid containers. They found that while the middle of the flow behaves like a normal waterfall (energy breaking down), the edges near the top and bottom act like a reverse waterfall, where tiny swirls merge to create giant structures. This helps us better understand how energy moves in nature, from our atmosphere to the cores of planets.

Technical Summary

Technical Summary: Scale-by-scale kinetic energy flux calculations in simulations of rotating convection

Problem Statement
Turbulent flows are characterized by non-zero net fluxes of kinetic energy between different scales, a process crucial for the formation of flow structures in nature. While the "direct cascade" (energy transfer to smaller scales) is well-understood in three-dimensional (3D) turbulence, systems constrained by rotation, stratification, or magnetic fields often exhibit "quasi-two-dimensional" behavior, leading to an "inverse cascade" (energy transfer to larger scales). Rotating Rayleigh-Bénard convection (RRBC) serves as a canonical model for studying these geophysical and astrophysical flows.

However, quantifying these scale-by-scale energy fluxes in practical settings presents significant challenges. Traditional methods rely on Fourier decompositions, which require homogeneous and periodic boundary conditions. Most experimental setups and many realistic simulations involve inhomogeneous, anisotropic, and aperiodic domains (e.g., bounded cylinders), rendering standard Fourier techniques difficult to apply directly. Furthermore, while spatial filtering methods (common in Large Eddy Simulations) offer an alternative for non-periodic domains, there is a lack of comparative studies validating their viability against Fourier-based methods in the specific context of rotating convection.

Methodology
The authors employ Direct Numerical Simulations (DNS) of RRBC in two distinct geometries:

1. Horizontally Periodic Domain: A box with solid top and bottom plates and periodic horizontal boundaries.
2. Cylindrically Bounded Domain: A fully confined cylinder with solid walls on all sides, mimicking experimental conditions.

The simulations utilize the Oberbeck-Boussinesq approximation with stress-free boundary conditions to promote large-scale flow formation. Two distinct techniques are developed and compared to calculate the scale-by-scale kinetic energy flux ($\Pi$):

• Fourier-based Method: Based on Frisch (1995), this approach splits the velocity field into low-pass and high-pass components using a sharp spectral cut-off in wavenumber space ($k_0$). The flux is calculated via the interaction term $\langle \mathbf{u}_{<k_0} \cdot (\mathbf{u} \cdot \nabla) \mathbf{u}_{>k_0} \rangle$. For the cylindrical domain, the velocity field is interpolated onto a Cartesian grid, windowed, and padded with zeros to enable Fourier transformation.
• Spatial Filtering Method: Based on the Large Eddy Simulation (LES) framework (Sagaut 2005; Pope 2000), this approach filters the velocity field in real space using a Gaussian kernel with a filter width $\Delta$. The flux is derived from the residual stress tensor and the filtered rate of strain tensor: $\Pi_\Delta \equiv \langle -\boldsymbol{\tau}_\Delta : \mathbf{S} \rangle$.

The study compares these methods across various Rayleigh ($Ra$) and Ekman ($Ek$) numbers, focusing on regimes where both direct and inverse cascades are expected.

Key Results

• Methodological Agreement: In both the periodic and cylindrical domains, the spatial filtering and Fourier-based methods yield largely consistent qualitative results regarding the directionality of the energy cascade. Both methods identify a transition from inverse to direct cascades at specific wavenumbers (e.g., $k \approx 37$ in the periodic case).
• Fluctuations and Smoothing: The Fourier-based method produces flux curves that are more erratic and exhibit a larger spread (higher variance) compared to the smoother results of the spatial filtering method. This is attributed to the discrete nature of Fourier shells in low-wavenumber bins and the integer-multiple constraints of the grid. The spatial filtering method allows for arbitrary filter widths, offering greater flexibility.
• Vertical Distribution of Flux:
  • Inverse Cascade: Significant inverse energy flux (negative $\Pi$) is observed predominantly near the top and bottom plates (within the thermal boundary layers). The authors attribute this to the merging of like-signed vortical Ekman plumes into larger structures.
  • Direct Cascade: The bulk of the flow (the vertical middle region) is dominated by the direct cascade (positive $\Pi$), where energy is transported to smaller scales.
  • Boundary Effects: In the thermal boundary layers, the Fourier-based method sometimes shows positive flux where the spatial filtering method does not, a discrepancy the authors note requires further research.
• Cylindrical Domain: Even in the fully bounded cylindrical domain, where flow inhomogeneity is highest, both methods successfully retrieve energy flux data. While the agreement is slightly less pronounced than in the periodic case due to necessary preprocessing (interpolation and windowing), the results confirm the presence of inverse cascades in rapidly rotating confined systems.

Significance and Claims
The paper claims to provide a validated methodology for retrieving scale-by-scale kinetic energy flux in anisotropic and aperiodic domains, which are common in experimental fluid dynamics but difficult to analyze with standard spectral tools.

• Versatility: The spatial filtering method is highlighted as more versatile because the filter width can be chosen arbitrarily, independent of grid resolution, whereas the Fourier method is constrained by the grid spacing.
• Robustness: The study demonstrates that despite the complexities of confined geometries and the need for data preprocessing, the spatial filtering approach yields trustworthy results comparable to the established Fourier method.
• Physical Insight: The results reinforce the physical understanding that in rotating convection, the inverse cascade is a boundary-layer phenomenon driven by plume merging, while the bulk flow remains dominated by the direct cascade.

The authors conclude that these findings enable the application of either method to a broader range of numerical and experimental works, particularly those involving complex geometries where periodicity cannot be assumed. They note that future work will focus on implementing these calculations for immediate run-time averaging to improve convergence.