A new Energy Equation Derivation for the Shallow Water Linearized Moment Equations

This paper presents a new systematic derivation of the energy equation for the Shallow Water Linearized Moment Equations (SWLME) by extending the standard Shallow Water Equations approach to include skew-symmetric formulations, thereby facilitating the extension to other SWME variants and improving their numerical solutions.

The Gist

Imagine you are trying to predict how a wave moves down a river or how a mudslide flows down a hill. For a long time, scientists have used a standard set of rules called the Shallow Water Equations (SWE). Think of these rules like a "flat map" of the water. They assume that if you look at the water from the bottom to the surface, everyone is moving at the exact same speed. It's like assuming a whole crowd of people walking down a hallway are all marching in perfect lockstep.

The Problem: The River Isn't Flat
In reality, water doesn't move in lockstep. The water near the bottom might be slow due to friction, while the water near the surface is fast. The old "flat map" rules miss this vertical difference. To fix this, scientists created a more advanced model called Shallow Water Moment Equations (SWME).

Think of the SWME as upgrading from a flat map to a 3D hologram. Instead of just one speed for the whole depth, it breaks the water's speed into layers, like a stack of pancakes, where each layer can have its own speed. This gives a much more accurate picture of how the water actually behaves.

The Specific Model: SWLME
The paper focuses on a specific, simplified version of this 3D hologram called the Shallow Water Linearized Moment Equations (SWLME). It's a streamlined version that keeps the 3D accuracy but removes some of the messy, complicated math to make it easier to solve on a computer.

The Big Discovery: The Energy Equation
The main goal of this paper was to write down a new "Energy Equation" for this specific model.

Here is the best way to understand what that means:
Imagine you are balancing a checkbook. You have money coming in (energy) and money going out (energy flux). For a physical system like water flowing, the total energy (kinetic energy from movement + potential energy from height) must be conserved. It can't just disappear or appear out of nowhere.

• The Old Way: Previously, scientists had written down the energy rule for this SWLME model, but they did it quickly, skipping many of the steps. It was like showing someone the final math answer on a test without showing the work.
• The New Way: This paper provides a step-by-step, systematic derivation. The author, Julian Koellermeier, rebuilt the energy equation from the ground up, starting with the basic rules of the simpler "flat map" model and carefully adding the 3D layers one by one.

Why This Step-by-Step Approach Matters
The author didn't just get the right answer; he found a special "secret sauce" along the way called the skew-symmetric form.

Think of the equations as a machine with gears. If the gears aren't aligned perfectly, the machine might grind and break when you try to simulate it on a computer. The "skew-symmetric form" is like a perfectly balanced gear system. It ensures that the math is stable and won't crash when you run complex simulations.

The Takeaway
The paper proves that:

1. We can now calculate the total energy of these complex, 3D water flows with a clear, verified method.
2. The method used to get there (the step-by-step derivation) is so clear that other scientists can use it to build energy rules for even more complex water models in the future.
3. The "balanced gear" (skew-symmetric) structure found during the process will help engineers build better, more stable computer programs to simulate floods, tsunamis, and avalanches.

In short, the paper didn't invent a new type of water, but it provided a better, clearer instruction manual for calculating how energy moves through complex water flows, ensuring our computer simulations are accurate and stable.

Technical Summary

Technical Summary: A New Energy Equation Derivation for the Shallow Water Linearized Moment Equations

Problem Statement
Shallow Water Equations (SWE) are the standard model for simulating free-surface flows, such as river waves and avalanches. However, the SWE rely on a depth-averaged velocity profile, which assumes a uniform vertical velocity distribution. This assumption limits the model's accuracy in scenarios where vertical velocity variations are significant. To address this, Shallow Water Moment Equations (SWME) were developed, extending the SWE by modeling the vertical velocity profile as a polynomial expansion. While the SWME offer improved physical fidelity, deriving a consistent energy equation for these extended systems is non-trivial. Specifically, for the Shallow Water Linearized Moment Equations (SWLME)—a simplified variant of SWME where certain non-linear coupling coefficients are set to zero—a previous energy equation derivation was provided in [2] but was presented only as a side remark with omitted intermediate steps. This lack of a systematic derivation hinders the generalization of energy-stable methods to other SWME variants and their numerical solutions.

Methodology
The paper presents a new, systematic derivation of the energy equation for the SWLME system. The methodology extends the step-by-step derivation technique used for standard SWE found in [5]. The process involves the following steps:

1. System Definition: The SWLME system is defined by $N+2$ equations: a continuity equation for water height $h$, a momentum balance for depth-averaged velocity $u_m$, and $N$ moment equations for polynomial coefficients $u_i$ (representing vertical velocity variations). The system assumes a hydrostatic pressure, frictionless flow, and a time-independent bottom topography.
2. Potential Energy Derivation: The potential energy equation is derived by multiplying the continuity equation by $g(h+b)$, following standard SWE procedures.
3. Kinetic Energy Derivation (Momentum): The momentum balance is manipulated to isolate the acceleration terms. By computing the difference between the momentum equation and the continuity equation multiplied by $u_m$, and then averaging this result with the original momentum equation, the authors derive a skew-symmetric form. Multiplying this skew-symmetric form by $u_m$ yields the evolution equation for the kinetic energy associated with the mean flow.
4. Kinetic Energy Derivation (Moments): A similar procedure is applied to the moment equations. An alternative form of the moment equation is derived, and a skew-symmetric form is obtained by averaging it with the original. Multiplying by the respective coefficient $u_i$ yields the evolution equation for the "partial kinetic energy" associated with each moment coefficient.
5. Total Energy Synthesis: The total kinetic energy equation is constructed by summing the mean flow kinetic energy and the partial kinetic energies of all moments. This is then added to the potential energy equation. Through careful algebraic manipulation of the flux terms, the cross-terms cancel or combine to form a conservative flux.

Key Contributions and Results
The primary result of this work is the rigorous derivation of the total energy equation for the SWLME system (Equation 1.29). The derived equation confirms that the total energy $e$, defined as the sum of the kinetic energy of the mean flow, the kinetic energy of the higher-order moments, and the potential energy, is a conserved quantity in the absence of friction and source terms.

Key findings include:

• Conservation Law: The total energy $e = \frac{h u_m^2}{2} + \frac{h}{2}\sum_{i=1}^N \frac{u_i^2}{2i+1} + \frac{gh^2}{2} + ghb$ satisfies a conservation law $\partial_t e + \partial_x f = 0$, with a specific entropy flux $f$.
• Skew-Symmetric Formulation: A significant byproduct of the derivation is the explicit identification of skew-symmetric forms for both the momentum equation (Equation 1.17) and the moment equations (Equation 1.23).
• Entropy Variables: The paper explicitly computes the entropy variables ($q_1, q_2, q_{u_i}$) associated with the total energy, demonstrating that the system admits an entropy pair.
• Recovery of Previous Results: The derived energy equation matches the one previously presented in [2], validating the new systematic approach.

Significance and Claims
The paper claims that this new derivation is beneficial for the broader field of shallow water modeling in several ways:

• Generalizability: The systematic, step-by-step nature of the derivation serves as a template for extending energy equation derivations to more complex SWME variants (where non-linear coefficients $A_{ijk}$ and $B_{ijk}$ are non-zero) and other dispersive models.
• Numerical Stability: The identification of the skew-symmetric formulation and the entropy variables provides the necessary theoretical foundation for developing entropy-conservative numerical schemes. The authors suggest these formulations can be used to construct numerical methods that preserve energy discretely, similar to approaches used for standard SWE.
• Transparency: By filling in the intermediate steps omitted in [2], the work clarifies the mathematical structure of the SWLME, making the connection between the polynomial velocity profile and the resulting energy conservation properties explicit.

The work does not propose new experimental applications or future numerical implementations but focuses strictly on the theoretical derivation required to enable such developments.