Discrete variance decay analysis of spurious mixing

This paper introduces a discrete variance decay framework to quantify spurious mixing in numerical advection-diffusion schemes, revealing the non-uniqueness of implied second-moment fluxes, the necessity of temporal averaging for high-order methods, and the correlation of spurious mixing with eddy kinetic energy distributions.

The Gist

The Big Picture: The "Ghost Mixing" Problem

Imagine you are a chef trying to keep a pot of soup perfectly separated into layers: hot broth at the bottom, cool cream at the top. You want to stir it gently so the flavors mix just enough to taste good, but not so much that the layers disappear entirely.

Now, imagine you are using a robot chef to stir this soup. The robot is very fast and follows complex instructions, but it has a glitch. Every time it stirs, it accidentally spills a little bit of the hot broth into the cool cream and vice versa, even when it's trying to be careful. This isn't a real physical process; it's a mistake made by the robot's code.

In the world of ocean modeling, this "glitch" is called Spurious Mixing. Ocean models use computers to simulate how water moves. Because computers can't handle infinite detail, they have to approximate movements. These approximations sometimes create "ghost mixing"—fake mixing that doesn't exist in the real ocean but ruins the computer simulation's accuracy.

This paper is about building a better stethoscope to listen to the computer and figure out exactly where and why this ghost mixing is happening.

---

The Old vs. The New Stethoscope

The Old Way (The "Guessing Game"):
Previously, scientists tried to measure this ghost mixing by looking at the "second moment" (a fancy math term for how spread out the temperature or salt is). To do this, they had to guess what the "flux" (the flow of that spread) looked like at the edges of their computer grid cells.

• The Problem: It was like trying to measure the speed of a car by guessing how fast the wheels were turning without looking at the road. Different scientists made different guesses, and the results were often ambiguous or contradictory, especially with high-speed, complex stirring methods.

The New Way (The "Direct Measurement"):
The authors (Banerjee, Danilov, and Klingbeil) decided to stop guessing. Instead of looking at the "spread" directly, they looked at the movement itself as it crosses the boundaries of the computer's grid cells.

• The Analogy: Imagine you are counting how many people walk through a doorway. Instead of trying to guess how crowded the room is based on noise, you just count the people stepping in and out.
• The Result: They derived a new, direct formula (Closed-Form Expression) to calculate exactly how much "variance" (mixing) is lost at every single step of the simulation.

The "Blurry Photo" Problem (Ambiguity)

Here is the catch: Even with their new, direct method, if you look at the mixing happening in a single tiny cell at a single moment in time, the picture is still a bit blurry.

• The Metaphor: Imagine taking a photo of a hummingbird's wings. If you snap a picture too fast, the wings look like a blur. You can't tell exactly where the wings are, only that they are moving.
• In the Paper: The authors found that for complex, high-speed stirring schemes (high-order advection), the "ghost mixing" signal is contaminated by mathematical "noise" (flux divergence). It's like the hummingbird's blur.
• The Solution: You can't fix the blur by looking at one frame. You have to take a video and average it over time or look at a wider area. The paper proves that if you average the data over time and space, the "blur" disappears, and you get a clear picture of where the real ghost mixing is happening.

What Did They Discover?

Once they cleaned up the data (by averaging), they found some surprising things about the ocean models:

1. The Mixing Follows the Energy: The ghost mixing isn't random. It happens exactly where the ocean has the most energy (Eddy Kinetic Energy).

   • Analogy: It's like dust motes dancing in a sunbeam. The dust (mixing) only appears where the light (energy) is brightest. Where the ocean is turbulent and swirling, the computer makes the most mistakes.

2. Up vs. Down: Most of the fake mixing happens horizontally (side-to-side), following the swirling currents. The vertical mixing (up and down) is much smaller and follows a different pattern, related to how buoyancy (floating/sinking) moves.

3. The "Too Good to Be True" Schemes: The scientists tested different "stirring recipes" (advection schemes). They found that even the most advanced, high-precision recipes still create ghost mixing.

   • The Shock: In some places, the computer's fake mixing was stronger than the real physical mixing (like the natural diffusion of heat in the ocean). This means the computer is sometimes "stirring" the ocean more than nature actually does, which could lead to wrong predictions about climate and currents.

Why Does This Matter?

Ocean models are used to predict climate change, sea-level rise, and weather patterns. If the computer is accidentally mixing water layers that should stay separate, the whole simulation can drift off course over time.

This paper gives scientists a new, more reliable tool to:

1. Diagnose exactly how much "fake mixing" their specific computer model is creating.
2. Compare different mathematical recipes to see which one creates the least amount of ghost mixing.
3. Fix the models by adjusting the "stirring" to be more accurate, ensuring that the digital ocean behaves more like the real one.

Summary in One Sentence

The authors invented a new, more direct way to measure the "mathematical errors" in ocean simulations, proving that these errors aren't random noise but follow the energy of the ocean, and that we must average our data over time to see the true picture clearly.

Technical Summary

1. Problem Statement

Ocean circulation models rely on high-order upwind schemes for horizontal advection to transport tracers (e.g., temperature, salinity). While necessary for stability, these schemes introduce Spurious Mixing (SM), a purely numerical artifact that alters watermass properties without physical justification.

• The Challenge: Diagnosing SM locally is difficult. Existing methods often rely on Reference Potential Energy (RPE), which provides a global measure but lacks local resolution.
• The Gap: Discrete Variance Decay (DVD) approaches offer local estimates but suffer from significant uncertainty. Previous attempts (e.g., Klingbeil et al., 2018) to derive local DVD rates required defining "second-moment fluxes" (fluxes of $T^2$). However, the discrete evolution of the first moment ($T$) does not uniquely define the flux of the second moment ($T^2$), leading to ambiguities in local estimates, particularly for high-order advection schemes where dispersive errors are prominent.

2. Methodology

The authors propose a new framework to derive closed-form expressions for local DVD rates directly from discrete tracer equations, avoiding arbitrary assumptions about second-moment fluxes.

• Derivation Approach:
  • Instead of discretizing the second-moment equation directly, the authors start with the discrete first-moment equation (tracer transport).
  • They multiply the discrete equation by $2T^*$ (where $T^* = T^{n+1} + T^n$) to generate a discrete time derivative of the second moment.
  • By regrouping terms, they separate the sink (variance decay) from the flux divergence.
  • This approach shifts the perspective from cell-centered values to cell faces, allowing the derivation of DVD rates without explicitly defining a unique second-moment flux initially.
• Decomposition: The total DVD rate ($\chi$) is decomposed into:
  • Physical Mixing: Vertical diffusion ($\chi_{dv}$).
  • Numerical Mixing (Spurious): Horizontal diffusion ($\chi_{dh}$) and Advection ($\chi_a$), further split into horizontal ($\chi_{ah}$) and vertical ($\chi_{av}$) components.
• Handling Ambiguity: The study acknowledges that for high-order schemes, the local DVD rate contains residual flux divergence (dispersive errors). The authors demonstrate that spatial and temporal averaging (coarse-graining) is necessary to filter out these non-physical oscillations and reveal the true dissipative nature of the mixing.
• Experimental Setup:
  • Model: Finite-volumE Sea ice–Ocean Model (FESOM2).
  • Test Case: A channel setup (Soufflet et al., 2013) with periodic boundaries, using triangular meshes (5 km and 10 km resolution).
  • Schemes Tested: Various advection schemes including MUSCL-type (GE), Quadratic Reconstruction (QR), Central Second-order (C2), and Compact (CO) schemes, with varying upwinding parameters ($n$).

3. Key Contributions

1. New Theoretical Framework: Derivation of closed-form DVD rates directly from first-moment equations, revealing that implied second-moment fluxes differ from previous literature (Klingbeil et al.) but share the same dissipative core.
2. Identification of Non-Uniqueness: The paper proves that for high-order advection schemes, the local DVD rate is ambiguous due to incomplete elimination of flux divergence (dispersive errors). It establishes that simple local estimates are unreliable without averaging.
3. Decomposition without Operator Splitting: The method allows for the separation of horizontal and vertical contributions to spurious mixing without requiring operator splitting, which is often computationally expensive or complex.
4. Quantification of SM vs. Physical Mixing: The study provides a rigorous comparison showing that numerical mixing can locally exceed physical background mixing.

4. Key Results

• Correlation with Eddy Kinetic Energy (EKE):
  • Spurious mixing (SM) is strongly correlated with the distribution of Eddy Kinetic Energy.
  • Horizontal advection and horizontal diffusion contributions to SM follow EKE patterns.
• Vertical Advection:
  • The contribution of vertical advection to SM is relatively small but correlates with buoyancy fluxes.
  • Peaks in vertical SM occur at depths where the conversion of Available Potential Energy (APE) to EKE is highest.
• Magnitude of Spurious Mixing:
  • In the tested configurations, Spurious Diapycnal Mixing (SDM) often exceeds Physical Diapycnal Mixing (PDM).
  • For third-order upwind schemes (or high upwinding parameters), SDM can be twice as large as the background physical mixing.
  • Even fourth-order schemes, when stabilized with biharmonic diffusion, produce significant SM.
• Resolution Dependence:
  • Refining the mesh (from 10 km to 5 km) reduces SDM, but the reduction is not linear.
  • On coarse meshes, the specific choice of advection scheme matters less for dissipation levels; on fine meshes, higher-order schemes (like QR) are demonstrably less dissipative.
• Time Stepping:
  • Temporal interpolation (Adams-Bashforth) introduces additional dissipation, but its contribution to total SM is generally weak compared to spatial discretization errors in this setup.

5. Significance and Implications

• Diagnostic Tool: The proposed DVD framework serves as a robust tool for diagnosing numerical mixing in ocean models, provided that sufficient spatial/temporal averaging is applied to filter out dispersive noise.
• Model Configuration: The results suggest that standard operational settings in global ocean models (often using upwinding parameters $n \le 0.5$) may be introducing significant artificial mixing that rivals or exceeds physical mixing.
• Future Directions:
  • The study highlights the need for coarse-graining in local DVD analysis.
  • It opens the door for optimizing the combination of viscosity, upwinding, and diffusion to minimize SM while preserving eddy dynamics.
  • It suggests that future model development should focus on reducing the "unknown" of numerical mixing, potentially through alternative vertical coordinates ($\tilde{z}$) or isoneutral diffusion operators.

In summary, this paper provides a rigorous mathematical foundation for quantifying numerical mixing in ocean models, revealing that spurious mixing is a dominant, spatially variable process driven by eddy activity, and that current high-order schemes often introduce more mixing than the physical processes they aim to simulate.