# Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline

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## Abstract

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## 1. Introduction

#### Outline

## 2. Two-Dimensional Euler Equation

#### 2.1. Vorticity and Stream Function

#### 2.2. Conservation Laws

## 3. Statistical Equilibrium Concepts

#### 3.1. Phase Space Measure and the Liouville Theorem

#### Liouville Theorem for the Euler Equation

#### 3.2. Choice of Statistical Ensemble

#### 3.2.1. Microcanonical Ensemble

#### 3.2.2. Grand Canonical Ensemble

#### 3.2.3. Thermodynamic Free Energy

#### 3.2.4. Grand Canonical Formulation of the Euler Equation

## 4. Thermodynamics of the Euler Equation: Exact Solution

#### 4.1. Mean Field Approach

#### 4.2. Microscale Entropy

#### 4.3. Rotating Fluids and Generalization to the Beta Plane

#### More General Curvilinear Domains

#### 4.4. Simplified Model Examples

## 5. Some Limitations of the Statistical Equilibrium Hypothesis

#### 5.1. Metastable Steady States

#### 5.2. Viscosity Effects

#### 5.3. Strongly Fluctuating Long-Lived States

## 6. General Statistical Theory of Single-Field Systems

#### 6.1. Statistical Mechanics

#### 6.2. Quasi-Geostrophic Flow and Nonlinear Rossby Waves

#### 6.3. Adiabatic Conservation Laws and Slow Equilibration

#### 6.4. Generalized Surface Quasigeostrophic Equations

## 7. 3D Axisymmetric Flow

#### 7.1. Axisymmetric Equation of Motion

#### 7.2. Conservation Laws

#### 7.3. Axisymmetric Equilibria

#### 7.4. Pure Poloidal Flows

#### 7.5. Axisymmetric Flow Equilibration Issues

## 8. Shallow Water Dynamics and Wave-EDDY Interactions

#### 8.1. Conservation Laws

#### 8.2. Liouville Theorem and Statistical Measures

#### 8.3. Shallow Water Equilibria

#### 8.4. Quasi-Hydrostatic Shallow Water Equilibria

## 9. 2D Magnetohydrodynamics and the Solar Tachocline

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## References

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**Figure 1.**Strip and annular (or disc if ${R}_{2}=0$) geometries for which, respectively, a conserved linear momentum (10) or angular momentum (11) exists. The strip has periodic boundary conditions along x and Dirichlet boundary conditions on the lower and upper boundaries ${\Gamma}_{1,2}$. The annulus has Dirichlet boundary conditions on both boundaries. The latter lead to two independent circulation integrals (8) for each domain (which are seen to actually have the same topology).

**Figure 2.**Notional illustration of the equilibrium state of an overall neutral set of point-like vortices or charges as a function of internal energy [5]. The low energy state (1) on the left corresponds to a molecular dipole state with strongly bound charges. The middle state (2) corresponds to a higher energy plasma-like state with unbounded charges but that continue to obey local charge neutrality. The state (3) on the right exhibits large scale structure obtained by increasing the energy even further, forcing the charges to segregate into separate non-neutral regions. This negative temperature state is accessible in fluid dynamics because the charges are not conventional momentum and kinetic energy carrying particles. In the vortex field description, charges carry only potential energy of interaction.

**Figure 3.**Highly schematic illustration of the turbulent mixing process that begins here with a well defined though irregular region of finite, fixed vorticity $\omega ={q}_{0}$, surrounded by a vorticity free (potential flow) region, $\omega =0$. Over time the vortex region stretches and folds to give rise as $t\to \infty $ to a fully mixed smoothly varying macroscale steady state. However, the macro-view obscures the continuing microscale dynamics (illustrated in Figure 4) where restriction to values $\omega =0,q$ is preserved, consistent with the Casimir constraints.

**Figure 4.**Illustration of separation of scales entering the exact thermodynamic solution. The vortex self-advection is dominated by the large-scale flow, while the small scale fluctuations asymptotically obey a simple a-cell permutation rule generating the microscale entropy (65) characterizing each intermediate scale l-cell. Within each l-cell one may define the local vorticity distribution ${n}_{0}({\mathbf{r}}_{l},\sigma )$ which has a well defined continuum limit $a,l\to 0$ but in such a way that $l/a\to \infty $. Its first moment defines the equilibrium vorticity (58) and its area integral is constrained by the Casimir function (59). This illustrates the formal limiting process by which, e.g., a discrete set of (a-scale) vorticity levels controlled by the Casimirs produces a smooth (l-scale) average.

**Figure 5.**Example equilibrium vorticity profiles ${\omega}_{0}\left(\mathbf{r}\right)$ for the two level system on the unit disk for a sequence of inverse temperatures $-\infty \le \beta \le \infty $, obtained by numerically solving the nonlinear Laplace Equation (74). Vorticity level $q=1$ occupies fractional area $\alpha =0.2$, hence total vorticity ${\Omega}_{0}=\pi \alpha $. For each temperature, the Lagrange multiplier ${\mu}_{q}\left(\beta \right)$ must be determined iteratively to satisfy this constraint. As seen, the $\beta =-\infty $ ($T={0}^{-}$) maximum energy solution gathers all vorticity near the disc center, while the $\beta =+\infty $ ($T={0}^{+}$) minimum energy solution compacts all vorticity against the disc boundary. The $\beta =0$ ($T=\pm \infty $) maximum entropy solution distributes the vorticity uniformly (center panel).

**Figure 6.**Schematic illustration of the entropy function $S\left(E\right)$ associated with the two level system (72), and also of the point vortex system pictured in Figure 2. As described in the text, the Casimir constraints on the vorticity allow for both positive and negative temperatures, and corresponding entropy limited to a finite energy interval, vanishing with infinite slope at both ends. This general picture will hold for any $g\left(\sigma \right)$ with bounded support. The dashed line corresponds to conventional particle systems in which the momentum degree of freedom can absorb unbounded energy.

**Figure 7.**Axisymmetric flow geometry confined to a cylinder of height H and inner and outer radii ${R}_{1}<{R}_{2}$. The pattern of flows is taken to be invariant under rotation about the cylinder axis, and is therefore specified by a toroidal flow field s about the axis, and a poloidal vorticity field q within any 2D radial planar section D.

**Figure 9.**Example numerically generated long-time (near-equilibrium) behavior of freely decaying 2D magnetohydrodynamics on the sphere. The zonal velocity field (above) and zonal magnetic field (below) undergo coupled dynamics according to (134), reducing to (135) and (136) in the 2D solar tachocline model [34,44,45]. In the $(\psi ,A)$ representation (141) of the statistical functional, where $\mathbf{v}$ is defined by the level curves of $\psi $ and $\mathbf{B}$ is defined by the level curves of A, the model is that of two gradient-coupled membranes in an external potential which, among other things, tends to preferentially align the two vector fields.

**Table 1.**Summary of the (classical) field theoretic formulations of the statistical mechanics of the fluid systems discussed in this article. Statistical averages are governed by a phase space equilibrium probability density ${\rho}_{\mathrm{eq}}={Z}_{\mathrm{eq}}^{-1}{e}^{-\beta \mathcal{K}\left[{\Phi}\right]}$ in which $\beta =1/T$ is an inverse temperature variable (quite distinct from the physical temperature) conjugate to the system entropy. The partition function ${Z}_{\mathrm{eq}}$ normalizes ${\rho}_{\mathrm{eq}}$ to a probability. The statistical functional $\mathcal{K}\left[{\Phi}\right]$ is a certain conserved integral of the system dynamics, generalizing the energy (or Hamiltonian) familiar from conventional statistical mechanics. Its argument is a vector field ${\Phi}\left(\mathbf{r}\right)$ whose components are the fluid degrees of freedom, and where $\mathbf{r}$ is a 2D physical spatial coordinate (sometimes scaled or transformed) ranging over a finite domain $\mathcal{D}$. The allowed configurations of ${\Phi}$ define the thermodynamic phase space. The table summarizes the forms of $\mathcal{K}$ and ${\Phi}$ for the systems treated in the referenced sections of the paper. The Coriolis parameter $f\left(\mathbf{r}\right)$, through which rotation enters, has been left out of these forms for simplicity, but will be restored in the later sections.

2D Euler [Section 2, Section 3, Section 4 and Section 5; vorticity $\omega $, planar coordinate $\mathbf{r}=(x,y)$] | $\mathcal{K}\left[\omega \right]=\frac{1}{2}\int d\mathbf{r}\int d{\mathbf{r}}^{\prime}\omega \left(\mathbf{r}\right)G(\mathbf{r},{\mathbf{r}}^{\prime})\omega \left({\mathbf{r}}^{\prime}\right)$ | |

and QG flow [Section 6; potential vorticity $\omega $] | ||

Continuous spin, long range interacting Ising-type model, with spin | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-\int d\mathbf{r}\left\{{\mu}_{P}\alpha \left(\mathbf{r}\right)\omega \left(\mathbf{r}\right)+\mu \left[\omega \left(\mathbf{r}\right)\right]\right\}$ | |

weighting function $\mu \left(\omega \right)$, inhomogeneous magnetic field ${\mu}_{P}\alpha \left(\mathbf{r}\right)$ | ||

3D Axisymmetric flow [Section 7; poloidal vorticity q, toroidal | $\mathcal{K}[q,s]=\pi \int d\mathbf{\rho}\int d{\mathbf{\rho}}^{\prime}q\left(\mathbf{r}\right)G(\mathbf{\rho},{\mathbf{\rho}}^{\prime})q\left({\mathbf{r}}^{\prime}\right)$ | |

circulation s, radial-vertical “cylinder slice” coordinate | ||

$\mathbf{\rho}=({\rho}_{1},{\rho}_{2})=({r}^{2}/2,z)$] | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+\int d\mathbf{\rho}\left\{\frac{\pi}{2{\rho}_{1}}s{\left(\mathbf{\rho}\right)}^{2}-\mu \left[s\left(\mathbf{\rho}\right)\right]-\tilde{\mu}\left[s\left(\mathbf{\rho}\right)\right]q\left(\mathbf{\rho}\right)\right\}$ | |

Continuous Ising-type spin s, weighting function $\mu \left(s\right)$, | ||

mediated by Gaussian “charge” field q, coupling strength $\tilde{\mu}\left(s\right)$ | ||

Shallow water equations [Section 8; vorticity $\omega $, potential vorticity $\omega /h$, | $\mathcal{K}[\omega ,q,h]=\frac{1}{2}\int d\mathbf{r}\int d{\mathbf{r}}^{\prime}{\left[\begin{array}{c}\omega \left(\mathbf{r}\right)\\ q\left(\mathbf{r}\right)\end{array}\right]}^{T}{\mathcal{G}}_{h}(\mathbf{r},{\mathbf{r}}^{\prime})\left[\begin{array}{c}\omega \left({\mathbf{r}}^{\prime}\right)\\ q\left({\mathbf{r}}^{\prime}\right)\end{array}\right]$ | |

compression field q, surface height h, planar coordinate $\mathbf{r}=(x,y)$] | ||

Continuous spin, long-range interacting Ising-type field $\omega $, with spin | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+\int d\mathbf{r}\left\{\frac{1}{2}gh{\left(\mathbf{r}\right)}^{2}-h\left(\mathbf{r}\right)\mu [\omega \left(\mathbf{r}\right)/h\left(\mathbf{r}\right)]\right\}$ | |

weighting $\mu (\omega /h)$, tensor-coupled nonlinearly to Gaussian fields $q,h$ | ||

2D magnetohydrodynamics [Section 9; stream function $\psi $, | $\mathcal{K}[A,\psi ]=\int d\mathbf{r}\{\frac{1}{2}{|\nabla A\left(\mathbf{r}\right)|}^{2}+\frac{1}{2}{|\nabla \psi \left(\mathbf{r}\right)|}^{2}$ | |

magnetic vector potential A, planar section $\mathbf{r}=(x,y)$ | ||

orthogonal to electric current density $J=-{\nabla}^{2}A$ along $\widehat{\mathbf{z}}$] | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-\phantom{\rule{4pt}{0ex}}{\tilde{\mu}}^{\prime}\left[A\left(\mathbf{r}\right)\right]\nabla A\left(\mathbf{r}\right)\xb7\nabla \psi \left(\mathbf{r}\right)+\mu \left[A\left(\mathbf{r}\right)\right]\}$ | |

Model is equivalent to that of a pair of gradient-coupled elastic | ||

membranes, external confining and coupling potentials $\mu \left(A\right),\tilde{\mu}\left(A\right)$ |

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**MDPI and ACS Style**

Weichman, P.B.; Marston, J.B. Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline. *Entropy* **2022**, *24*, 1389.
https://doi.org/10.3390/e24101389

**AMA Style**

Weichman PB, Marston JB. Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline. *Entropy*. 2022; 24(10):1389.
https://doi.org/10.3390/e24101389

**Chicago/Turabian Style**

Weichman, Peter B., and John Bradley Marston. 2022. "Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline" *Entropy* 24, no. 10: 1389.
https://doi.org/10.3390/e24101389