# Generalized Clebsch Variables for Compressible Ideal Fluids: Initial Conditions and Approximations of the Hamiltonian

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## Abstract

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## 1. Introduction

## 2. Canonical Equations for Compressible Nonhomogeneous Fluids

#### 2.1. Clebsch Variables for a State Equation $P({V}_{r}/V)$

#### 2.2. Clebsch Variables for More General State Equations

#### 2.3. Computing Clebsch Variables from the Initial Conditions: One Auxiliary Pair of Clebsch Variables

#### 2.4. Two Auxiliary Pairs of Clebsch Variables

## 3. Expansions of the Hamiltonian

#### 3.1. Secular Growth of Clebsch Variables

#### 3.2. Canonical Approximations of the Linear Dynamics

#### 3.3. Nonlinear Canonical Approximations

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Small fluid parcel with surfaces of constant $\varphi $, $\beta $, and $\sigma $. (

**b**) The gradients $\nabla \varphi $, $\nabla \sigma $, and $\nabla \beta $ are adjusted by scalars $\nu $, $\lambda $, and $\alpha $ to yield the local momentum density $\mathbf{p}=\nu \nabla \varphi +\lambda \nabla \sigma +\alpha \nabla \beta $.

**Figure 2.**(

**a**) Stationary state of a stratified fluid: the growing variables $\varphi =\widehat{t}f$ and $\lambda =\widehat{t}l$ yield a zero momentum field $\mathbf{p}={\nu}_{eq}\widehat{t}\nabla f+\widehat{t}l\nabla {\sigma}_{eq}=\mathbf{0}$. (

**b**) Nonstationary state: the terms $\widehat{t}f$ and $\widehat{t}l$ are separated from the canonical variables as $\varphi =\widehat{t}f+\widehat{\varphi}$, $\lambda =\widehat{t}l+\widehat{\lambda}$, the momentum is $\mathbf{p}=\nu \nabla (\widehat{t}f+\widehat{\varphi})+(\widehat{t}l+\widehat{\lambda})\nabla \sigma $.

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**MDPI and ACS Style**

Rumpf, B.; Lvov, Y.V.
Generalized Clebsch Variables for Compressible Ideal Fluids: Initial Conditions and Approximations of the Hamiltonian. *Fluids* **2022**, *7*, 122.
https://doi.org/10.3390/fluids7040122

**AMA Style**

Rumpf B, Lvov YV.
Generalized Clebsch Variables for Compressible Ideal Fluids: Initial Conditions and Approximations of the Hamiltonian. *Fluids*. 2022; 7(4):122.
https://doi.org/10.3390/fluids7040122

**Chicago/Turabian Style**

Rumpf, Benno, and Yuri V. Lvov.
2022. "Generalized Clebsch Variables for Compressible Ideal Fluids: Initial Conditions and Approximations of the Hamiltonian" *Fluids* 7, no. 4: 122.
https://doi.org/10.3390/fluids7040122