Coarse-grained local available potential energy

This paper develops a coarse-graining framework to derive multi-scale evolution equations for local available potential energy, including cross-scale flux terms, thereby enabling a complete analysis of energy cycles across spatial scales and reservoirs in stratified flows.

The Gist

Imagine a giant, invisible ocean where water layers of different densities (like oil and water, but mixed) are constantly swirling, mixing, and churning. Scientists have long known that this fluid has two main types of "fuel" or energy: Kinetic Energy (the energy of motion, like a current flowing) and Potential Energy (stored energy waiting to be released, like a heavy rock sitting on a hill).

This paper introduces a new way to watch how this stored energy moves around, breaks apart, and changes form. Here is the breakdown in simple terms:

1. The Problem: Looking at the "Big Picture" vs. the "Fine Print"

Traditionally, scientists looked at the total energy of the whole ocean at once. It's like looking at a forest from a helicopter and seeing just a green blob. You know energy is moving, but you can't see where a specific tree is falling or how the wind is pushing a specific leaf.

Other methods tried to split the forest into "average trees" and "wobbly leaves," but they often lost track of exactly where in the forest these things were happening.

2. The Solution: The "Coarse-Grained" Lens

The authors developed a new mathematical "lens" (called coarse-graining). Imagine taking a photo of the ocean and then blurring it slightly.

• The Blurry Photo (Large Scale): This shows the big, slow-moving currents and waves.
• The Sharp Details (Small Scale): This is the difference between the sharp, original photo and your blurry one. It shows the tiny eddies, the swirls, and the chaotic mixing.

The paper's main achievement is creating a set of rules (equations) that track how energy flows between these "blurry" big currents and the "sharp" tiny swirls.

3. The Energy Cycle: A Financial Analogy

Think of the ocean's energy like a bank account with two types of currency:

• Kinetic Energy (KE): Cash in your pocket (ready to spend/move).
• Available Potential Energy (APE): Money in a savings account (stored value that can be converted to cash).

The paper maps out a complete "energy cycle" with three main transactions:

1. Converting Savings to Cash: Sometimes, the stored energy (APE) turns into motion (KE). Imagine a heavy rock falling off a hill and starting a landslide.
2. The "Tax" (Dissipation): As the water mixes, some energy is lost forever as heat (irreversible mixing). This is like a transaction fee that disappears from the system.
3. Cross-Scale Transfers (The Big Innovation): This is the paper's big discovery. Energy doesn't just stay in the "Big" or "Small" category.
   • Downscale: Big waves can break apart and dump their energy into tiny swirls (like a large wave crashing into foam).
   • Upscale: Sometimes, tiny swirls can organize themselves to push a bigger current (like many small birds flying in formation to create a larger gust).

The authors derived a specific formula to measure exactly how much "Potential Energy" is moving from the big scale to the small scale (and vice versa) at any given moment.

4. The Test Drive: The "Rolling Wave" Experiment

To prove their new lens works, the authors ran a computer simulation of a specific phenomenon called Kelvin-Helmholtz instability.

• The Metaphor: Imagine two layers of water moving at different speeds, like a fast river flowing over a slow one. Eventually, they start to curl up into giant, rolling waves (like the clouds you see in the sky before a storm).
• What they found:
  • They watched these giant rolls form and break.
  • They saw that the "stored energy" (APE) rushed from the big rolls into the tiny, chaotic strands (braids) connecting the rolls.
  • Once in the tiny strands, that energy was converted into motion (KE) and then lost to heat (mixing).
  • They also noticed a "wobble" in the giant rolls later in the simulation, which caused energy to bounce back and forth between big and small scales, like a pendulum swinging.

5. Why This Matters

Before this paper, scientists had a great map for how motion (Kinetic Energy) moves between big and small scales. Now, they have the matching map for stored energy (Potential Energy).

By putting these two maps together, scientists can finally see the complete energy cycle of the ocean. They can now pinpoint exactly where energy is being stored, where it is turning into motion, and where it is being wasted as heat, all while keeping track of whether it is happening in a giant current or a tiny swirl.

In short: The authors built a new set of glasses that let us see how the ocean's "battery" (stored energy) charges, discharges, and moves between the big waves and the tiny ripples, all in real-time.

Technical Summary

Problem Statement
Available Potential Energy (APE) is a fundamental quantity for understanding the energetics and dynamics of stratified fluids, quantifying the portion of potential energy available for adiabatic conversion to kinetic energy (KE) and serving as a diagnostic for irreversible diapycnal mixing. While APE is often treated as a globally integrated metric, spatially local definitions exist. However, existing methods for analyzing the scale-dependent evolution of local APE—specifically spectral analysis and Reynolds averaging—have limitations. Spectral methods lose spatial locality, while Reynolds averaging does not offer independent control over the spatial length scale of decomposition. Consequently, there is a need for a framework that can decompose APE by spatial scale while retaining information about the physical location of energy transfer mechanisms, similar to the established coarse-graining (spatial filtering) approaches used for kinetic energy.

Methodology
The authors develop a theoretical framework for the coarse-grained evolution of local APE density using a spatial filtering approach.

• Definition: The local APE density ($E_A$) is defined based on the difference between the fluid's state and its adiabatically resorted minimum potential energy state.
• Decomposition: The APE is decomposed into large-scale ($E_A^l$) and small-scale ($E_A^s$) components via a spatial filter with kernel $G$. Crucially, the large-scale APE is defined using the filtered density field but retains the reference state calculated from the total unfiltered density field to maintain physical consistency with standard APE definitions.
• Derivation: Evolution equations for both large-scale and small-scale APE are derived by taking the material derivative of the large-scale APE and filtering the total APE equation. This involves expanding advective operators into large and small-scale velocities and utilizing the Leonard stress tensor $\tau(f, g) = \overline{fg} - \overline{f}\overline{g}$.
• Integration: These new APE equations are paired with existing coarse-grained KE equations to construct a complete, multi-scale energy cycle.
• Application: The framework is applied to a two-dimensional numerical simulation of Kelvin-Helmholtz (KH) instability using the finite-volume code Oceananigans.jl. The simulation uses a Reynolds number ($Re$) of 2097, a minimum Richardson number ($Ri$) of 0.1, and a Prandtl number ($Pr$) of 1.

Key Contributions

1. Derivation of Cross-Scale APE Flux: The primary theoretical result is the derivation of the cross-scale APE flux term, $\Pi_A = -\tau(u_i, \rho)\partial_i \Upsilon^l$. This Galilean-invariant term quantifies the transfer of APE between scales, serving as the APE counterpart to the well-studied cross-scale KE flux ($\Pi_K$).
2. Complete Energy Cycle: The paper establishes a complete energy budget that accounts for transfers between spatial scales (large to small and vice versa) and between energy reservoirs (APE and KE). This includes terms for:
   • Reversible conversions between APE and KE ($w b_r$ and $\tau(w, b_r)$).
   • Cross-scale fluxes ($\Pi_A$ and $\Pi_K$).
   • Transport terms.
   • Dissipation terms ($\varepsilon^l_A, \varepsilon^s_A$), distinguishing between large-scale dissipation (which includes internal-to-potential energy conversion) and small-scale dissipation (which represents irreversible diabatic mixing).
   • Terms related to the time-evolution of the reference density profile ($R$), which redistributes APE between scales.
3. Theoretical Distinction: The authors highlight a conceptual distinction between coarse-grained KE and APE: while large-scale KE depends only on filtered fields, large-scale APE implicitly depends on small-scale fields because the reference state is constructed from the total unfiltered density.

Results from the KH Instability Simulation

• Scale-Dependent Fluxes: The cross-scale KE flux ($\Pi_K$) shows a forward transfer at large scales (instability growth) but switches to an upscale transfer at scales around half the billow wavelength. Conversely, the cross-scale APE flux ($\Pi_A$) exhibits upscale transfers at larger scales and downscale transfers at smaller scales.
• Budget Dynamics:
  • At a filter scale of $l=7$ (half the initial instability wavelength), small-scale APE increases via conversion from KE, followed by a transfer to large-scale APE.
  • At a smaller scale ($l=1$), small-scale APE increases primarily via downscale transfer, partially offset by conversion to KE. This small-scale KE is then transferred inversely to larger scales, consistent with 2D turbulence expectations where direct KE cascade is inefficient.
  • An oscillatory exchange of APE across scales is observed, associated with the nutation of the vortex.
• Spatial Distribution: At the peak of cross-scale APE transfer, fluxes are most intense along the "braids" (regions of high strain) and the periphery of the billow.
• Dissipation: Small-scale APE dissipation (a proxy for diabatic mixing) is intensified along the braids and the junction of the braids with the vortex core. Notably, diabatic mixing remains enhanced at the periphery even in late-time evolution, despite the presence of statically unstable regions in the billow core.

Significance
The paper claims that this coarse-graining framework provides a powerful tool for investigating the multi-scale evolution of local APE. By offering control over the spatial length scale while retaining physical locality, the approach allows researchers to:

• Examine how potential energy is transferred across scales.
• Analyze scale-dependent exchanges between APE and KE.
• Connect cross-scale energy transfers to diabatic processes.

The authors position this work as a necessary extension to existing KE frameworks, enabling a more complete understanding of the full energy cycle in stratified flows, ranging from three-dimensional turbulence to planetary-scale circulations. They note that the implicit dependence of large-scale APE on small-scale fields via the reference state is a unique feature requiring further consideration.