2.1. Statistical mechanics of quasi-2-D nearly inviscid fluid and plasma flows

Fjortoft's (cf. Lim & Assad Reference Lim and Assad2005) and later Kraichnan's study (Kraichnan Reference Kraichnan1975) of energy inverse cascades in nearly inviscid quasi-2-D turbulence – a non-equilibrium result – renewed interest in Onsager's approach (Onsager Reference Onsager1949) to 2-D turbulence, which is based on equilibrium statistical mechanics (Lundgren & Pointin Reference Lundgren and Pointin1977). Onsager introduced the notion of negative temperatures to vortical flows in the 1940s with his seminal paper (Onsager Reference Onsager1949). His approach is based on the first of the above discretizations, namely a particle method leading to a 2-D vortex gas. The mean field sinh-Poisson equations arising from the Lagrangian vortex gas methodology have yielded notable results on the emergence of domain-scale flows consisting of one or several large coherent vortices (Onsager Reference Onsager1949; Lundgren & Pointin Reference Lundgren and Pointin1977). This particle approach was also used to study the anomalous diffusion in quasi-2-D plasmas.

For the discussion in this paper, the second and third methods of discretizations are emphasized in an equilibrium statistical mechanics approach that is further based on fixing certain physical quantities of the plasma. The use of equilibrium statistical mechanics has to be justified in a setting with dissipation, albeit small, as in nearly inviscid and perfectly conducting plasmas. Partial justification can be based on selective decay, which is a consequence of the inverse cascade of the spectrum of a physical quantity superposed on the usual forward cascades of others. In the quasi-2-D fluid dynamics context where it first arose, it is also known as the constrained minimum enstrophy theory; enstrophy is the square norm of vorticity. The key physical quantity here is the ratio or quotient of enstrophy to mechanical energy of the fluid. Under small viscosity, this ratio in quasi-2-D flows tends over time to a minimum for a large class of initial fluid flows, because the mechanical energy spectrum undergoes an inverse cascade to low wavenumbers while the enstrophy spectrum has a forward cascade to large wavenumbers. Because viscous dissipation is larger at large wavenumbers, the enstrophy decays faster versus that of energy. This then justifies the 2-D mechanical energy as a robust invariant versus the enstrophy, the variational principle of minimum enstrophy for fixed energy and also its dual variational principle of maximum energy for fixed enstrophy. Its equilibrium statistical mechanical analogues arise as Kac's spherical models for energy-enstrophy theories, where the energy of a lattice Hamiltonian interacts with a heat bath while its enstrophy is fixed by a hard constraint.

Mapped to the quasi-2-D HM plasma in this paper, we expect that its generalized energy spectrum cascades to low wavenumbers, while the potential enstrophy spectrum cascades to large wavenumbers. That means, for quasi-2-D plasmas, the generalized energy is the robust invariant. In the equilibrium statistical mechanics approach here, we will focus on constructing and solving a version of Kac's spherical model for the HM system, with hard constraint on the potential enstrophy of the plasma.

In the fully three-dimensional (3-D) toroidal plasmas in Taylor's relaxation theory, the total helicity dissipates at a slower rate than the magnetic energy to highest order and is taken to be the robust invariant. Instead of minimum enstrophy, we now have minimum energy configurations in Taylor's relaxation theory for 3-D plasmas. These will not be included here.

Equilibrium statistical mechanics is further justified in the small dissipation quasi-2-D plasmas, by either postulating that: (I) the plasma is a damped-driven system; or (II) the unforced slightly damped plasma has a relatively short mixing or equilibration time compared with the dissipation time scale. In case (I), the forcing can be external, like stirring by mechanical and electromagnetic means, or internal, where in the Fourier analysis of the problem, a UV cutoff is postulated after which the smaller length scales are NOT modelled but are taken as part of the heat bath. This heat bath comprises both mechanical and electromagnetic (EM) random effects.

In all of the above cases involving only the two physical quantities, $E$ and $Q$, the statistical mechanics approach can be exhaustively classified in one of the following ways: (a) canonical in $E$, canonical in $Q$; (b) canonical in $E$, microcanonical (hard) in $Q$; (c) microcanonical in $E$, canonical in $Q$; and (d) microcanonical in $E$ and in $Q$. Formulations (a), (c) and (d) will be discussed in other papers. In this paper, we will analyse completely case (b).

A brief argument for case (b) to be in the category of spherical models is that the hard constraint on potential enstrophy is on the square-norm of the local spins in a spin-lattice setting.

Like many geophysical flow models with Coriolis effect, such as the $\beta$-plane model and the barotropic vorticity equation (BVE) on a rotating massive sphere, the Hasegawa–Mima generalized energy should have, at first sight, an inhomogeneity arising from the interaction of the potential vorticity $q$ with a term that is linear and odd in the $x$-coordinate. One may associate the latitude coordinate in the BVE model with the $x$-coordinate in the HM model. The longitude in the geophysical flow models then corresponds to the $y$-coordinate in the domain:

(2.1)\begin{equation} D=\{(x,y)\in R\times \lbrack -{\rm \pi} ,{\rm \pi} )\} \end{equation}

on which, the HM model takes $y$-periodic boundary conditions. To see how this domain $D$ fits into the tokamak geometry for plasmas, $D$ should be viewed as the generic cross-sectional plane normal to the $\theta$-direction around the tokamak. The HM model is invariant in the $\theta$-direction, hence its 2-D nature.

Owing to the doubly canonical form of its Gibbs ensemble or partition function, the classical energy-enstrophy theories are exactly solvable Gaussian models which are not well defined at low numerical values of statistical temperatures (Lim Reference Lim2005). However, most of the interesting physics of transitions to domain-scale flows is expected to occur at sufficiently low numerical values of the relevant temperature. Note the important fact that the critical temperatures could be either positive or negative in the equilibrium statistical mechanics of many quasi-2-D flows. In the latter case, the ordered phase usually coincides with negative temperatures that have numerical values less than the critical, which corresponds to flow states with extremely high kinetic energy.

2.2. Canonical energy constraint

For the canonical constraint on generalized energy in the statistical mechanics formulation of the HM model, it should be emphasized that the largest scales (near the domain length scale) in the plasma flow are excluded by design from this exchange of energy with the reservoir – these scales comprise the long-range order from the phase transition and are treated in our application here of the large $N$ saddle point method as non-ergodic scales separately from smaller so-called ergodic ones.

2.3. Model equations

The Hasegawa–Miwa equation for 2-D drift turbulence in a plasma at fast time scales states that the evolution of generalized (electric) potential $\varphi$ is given by

(2.2)\begin{equation} \partial_{t}(\varphi -\Delta \varphi )+\boldsymbol{n}\times \boldsymbol{\nabla} \varphi \boldsymbol{\cdot}\boldsymbol{\nabla} (-\Delta \varphi )+\nu \partial_{y}\varphi =0, \end{equation}

where $\boldsymbol {n}$ is the unit normal to the domain $D$ above. This equation has conserved quantities: (i) generalized energy, $E$; and (ii) potential vorticity, $q$

(2.3)\begin{gather} E =\frac{1}{2}\int\int_{D}\,\textrm{d}x\,\textrm{d}y\left( \varphi ^{2}+(\boldsymbol{\nabla} \varphi )^{2}\right), \end{gather}

(2.4)\begin{gather}q=(\varphi -\Delta \varphi )-\nu x, \end{gather}

where $-\nu x$ is the inhomogeneous term which makes the HM model similar to the $\beta \text {-}plane$ model of geophysical flows. All moments, $Q_{n}=\int \int \textrm {d}x\,\textrm {d}y$ $q^{n}$ of $q$ are conserved by the inviscid Hasegawa–Miwa dynamics: the domain of integration is $D=R\times \lbrack -{\rm \pi},{\rm \pi} )$.

2.4. Physical quantities of the HM model

Equilibrium statistical mechanics for 2-D plasma flows are based fundamentally on the inverse cascade of energy to large scales, as in Kraichnan (Reference Kraichnan1975), Miller (Reference Miller1990), Robert & Sommeria (Reference Robert and Sommeria1991). In the energy-enstrophy theory here, the key object is the partition function, i.e. the path or functional integral, and the action $S$,

(2.5)\begin{gather} Z(\beta ;q) =\int D[q]\textrm{e}^{{-}S[q]}\delta \lbrack Q_{2}-Q^{{\ast} }], \end{gather}

(2.6)\begin{gather}S[q] =\frac{\beta }{2}\int_{D}\int \textrm{d}x\,\textrm{d}y\left( \varphi ^{2}+(\boldsymbol{\nabla} \varphi)^{2}\right), \end{gather}

where $\beta$ is the inverse temperature of the plasma's generalized energy, and the potential enstrophy $Q_{2}$ is constrained by the (hard) delta function.

In most other statistical mechanics formulations of 2-D flows, following the Miller– Robert (MR) approach, the (Shannon) entropy is used but this requires a strong assumption on the independence of neighbouring parcels of potential vorticity distributions. This is equivalent to a mean field theory. Unlike the MR approach, we will ignore all moments of $q$ except the second, known as the potential enstrophy, $Q_{2}=\int \int \textrm {d}x\,\textrm {d}y\,q^{2}$ and view the resulting statistical mechanics as an instance of a 2-D lattice theory of spherical model-type. Similar to the 2-D Euler equations (and related 2-D flows on a variety of geometries) which conserve the energy, $E=\frac {1}{2} \int \int \textrm {d}x\,\textrm {d}y\,(\boldsymbol {\nabla } \varPsi )^{2}=-\frac {1}{2}\int \int \textrm {d}x\,\textrm {d}y\,\varPsi \Delta \varPsi$ in terms of stream function, $\varPsi$, the generalized energy for the HM model is given in terms of the generalized electric potential $\varphi$ by,

(2.7)\begin{align} E & =\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\left( \varphi ^{2}+(\boldsymbol{\nabla} \varphi)^{2}\right) \end{align}

(2.8)\begin{align} & =\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\,\varphi^{2}-\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\,\varphi \Delta \varphi \end{align}

(2.9)\begin{align} & =\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\,\varphi (\varphi -\Delta \varphi )=\frac{ 1}{2}\int \int \textrm{d}x\,\textrm{d}y\, \varphi (I-\varDelta )\varphi \end{align}

(2.10)\begin{align} & =\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\,\varphi (q+\nu x). \end{align}

We will show next by symmetry considerations that the inhomogeneous term $\nu x$ has no effect on the energy $E$. In view of $(q+\nu x)=(I-\varDelta )\varphi$, we solve the first (homogeneous) part by the Green's function for the problem:

(2.11)\begin{equation} (I-\varDelta )G((x,y),(x^{\prime },y^{\prime }))=\delta ((x,y)-(x^{\prime },y^{\prime })) \end{equation}

with the following boundary conditions in the domain $D$. The corresponding eigenvalue problem in domain $D$, is

(2.12)\begin{gather} (I-\Delta )\varphi (x,y) =\mu \varphi (x,y) \end{gather}

(2.13)\begin{gather}\Delta \varphi_{k} =(1-\mu (k))\varphi_{k}=\lambda (k)\varphi_{k} \end{gather}

with vanishing boundary conditions for large $|x|$ and periodic boundary conditions on $y=\pm {\rm \pi}$. For the unbounded $D=R\times \lbrack -{\rm \pi},{\rm \pi} ),$ these eigenfunctions are $\varphi _{k}(x,y)=\textrm {e}^{\textrm {i}ky}\exp ( -{ |x|}/{\lambda _{k}})$; in real and imaginary parts,

(2.14a,b)\begin{gather} \varphi_{k}(x,y) =\cos \left( ky\right) \exp \left( -\frac{|x|}{\lambda _{k}}\right) , \quad \phi_{k}(x,y)=\sin \left( ky\right) \exp \left( - \frac{|x|}{\lambda_{k}}\right) \end{gather}

(2.15)\begin{gather}\lambda_{k}^{2} =(\lambda (k)+k^{2})^{{-}1} \end{gather}

where $k$ are integers. For $a\gg 1$ very large in the corresponding compact domain $D=[-a,a]\times \lbrack -{\rm \pi},{\rm \pi} )$, there is a complete orthonormal (o.n.)  set of eigenfunctions for $(I-\varDelta )$, which are well approximated by the normalized versions of $\varphi _{k}$, $\phi _{k}$. The potential vorticity has the following expansion in terms of these o.n. set:

(2.16)\begin{equation} q(x,y)=\sum_{k\in Z}\alpha_{k}\varphi_{k}(x,y)+\sum_{k}b_{k}\phi _{k}(x,y), \end{equation}

where $\alpha _{k}$ and $b_{k}$ are real amplitudes. In the case of a neutral plasma where total charge vanishes, $\alpha _{0}=b_{0}=0$. We will discuss both cases below.

This Green's function $G$ for the above boundary conditions on $\partial D$ can be expressed in terms of the following expansion in terms of these eigenfunctions (using $z=(x,y))$,

(2.17)\begin{equation} G(z,z^{\prime })=\sum_{k}\frac{1}{\lambda (k)}\left\{ \varphi_{k}(z)\varphi_{k}^{{\ast} }(z^{\prime })\right\}. \end{equation}

Fixing $z$, and taking $z^{\prime }\rightarrow \bar {z}^{\prime }=(-x^{\prime },y)$, it follows from the evenness of $\exp ( -{|x| }/{\lambda _{k}})$ in $x$ that $G(z,\bar {z}^{\prime })=G(z,z^{\prime })$, that is, the Green's function $G$ is even in $x^{\prime }$. Similarly, by self-adjointness of $G$, we get $G(\bar {z} ,z^{\prime })=G(z,z^{\prime })$, where $\bar {z}=(-x,y)$.

The inhomogeneous part of the problem, i.e. $(I-\varDelta )\varphi =\nu x$ on the domain $D$ – which cannot be solved using the above Green's function $G$ – has the following solution,

(2.18)\begin{equation} (I-\varDelta )^{{-}1}(\nu x)=\nu x, \end{equation}

which clearly does not vanish for $x$ large in $D$. It follows that the generalized (electric) potential in the HM model,

(2.19)\begin{equation} \varphi(x,y)=\nu x+\int \textrm{d}z\,G(z,z^{\prime }) q(z), \end{equation}

where $z=(x,y)\in D$.

Because the o.n. set of eigenfunctions $\varphi _{k},\phi _{k}$ have the same $x\text {-dependence}$, $\exp ( -{|x|}/{\lambda _{k}})$ which is even,

(2.20)\begin{equation} \frac{\upsilon }{2}\int_{{-}a}^{a}{\textrm{d} x}\int_{-{\rm \pi} }^{\rm \pi}{\textrm{d} y}\,x q(x,y)=0. \end{equation}

From this it follows that the HM model's generalized energy,

(2.21)\begin{align} E & =\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\,(q+\nu x)\varphi (x,y) \end{align}

(2.22)\begin{align} & =\frac{1}{2}\int \int \textrm{d}x\,\textrm{d}y\, q(x,y)\int \int {\textrm{d} x}^{\prime}\, {\textrm{d} y}^{\prime }G((x,y),(x^{\prime },y^{\prime }))q(x^{\prime },y^{\prime }) \end{align}

after dropping the (inactive) constant term $E_{0}=({\upsilon ^{2}}/{2}) \int _{-a}^{a}{\textrm {d} x}\int _{-{\rm \pi} }^{{\rm \pi} }{\textrm {d} y}\,x^{2}$. We have shown that the inhomogeneous term $\upsilon x$ does not contribute to $E$, unlike the $\beta \text {-plane}$ model where the Coriolis force is energetically active.

We will use the action $S[q]$ in the energy-enstrophy theory here to be $S[q]=\beta E$. In standard statistical mechanics (Lim & Nebus Reference Lim and Nebus2006), $S[q]$ is not the free energy and only after evaluating the partition function $Z(q)$ is the free energy $F[q^{\ast }]$ derived as the effective exponent in the expression for $Z$. Moreover, the free energy is to be calculated at the most probable distribution of potential vorticity called $q^{\ast }$.

3.1. Solution of the spherical model

The partition function of the above spherical models is calculated using Laplace's integral form for path integrals,

(3.22)\begin{align} Z_{N} & \propto \int D[\boldsymbol{q}]\exp \left( -\beta H_{N}[ \boldsymbol{q}]\right) \delta \left( N-\frac{A}{Q^{{\ast} }} \sum_{j=1}^{N}q_{j}^{2}\right) \delta \left( \sum_{j=1}^{N}q_{j}-\varGamma\right) \nonumber\\ & =\int_{\delta \left( \sum_{j=1}^{N}q_{j}-\varGamma \right) }D[\boldsymbol{ q}]\exp \left( -\beta H_{N}[\boldsymbol{q}]\right) \left( \frac{1}{2{\rm \pi} \textrm{i} }\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\textrm{d}\eta \exp \left( \eta \left( N-\frac{A}{ Q^{{\ast}}} \sum_{j=1}^{N}q_{j}^{2}\right) \right) \right)\nonumber\\ & =\int_{\delta \left( \sum_{j=1}^{N}q_{j}-\varGamma \right) }D[\boldsymbol{ q}]\exp \left( \frac{\beta }{2}\sum_{j\neq k}^{N}J_{jk}q_{j}q_{k}\right) \left( \frac{1}{2{\rm \pi} \textrm{i}}\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\textrm{d}\eta \exp \left( \eta \left( N-\frac{A}{Q^{{\ast} }}\sum_{j=1}^{N}q_{j}^{2}\right) \right) \right), \end{align}

where $a>0$ is chosen large enough for the calculation below. The total charge constraint is given by delta functions $\delta ( \sum _{j=1}^{N}q_{j}-\varGamma )$, with $\varGamma =0$ in the neutral case. As shown above, this last constraint can be implemented by taking the amplitude $\alpha _{0}=0$ in the neutral case. We focus on the neutral case because the total charge is conserved in the HM model and $\alpha _{0}\neq 0$ is a constant of motion and uninteresting in the statistical mechanics formulation.

We have replaced the microcanonical enstrophy constraint, $\delta ( Q^{\ast }-({A}/{N})\sum _{j=1}^{N}q_{j}^{2})$ by the mathematically equivalent $\delta ( N-({A}/{Q^{\ast }})\sum _{j=1}^{N}q_{j}^{2} )$, and then substituted it by the mathematically equivalent Laplace integral $({1}/{2{\rm \pi} i})\int _{a-\textrm {i}\infty }^{a+\textrm {i}\infty }\textrm {d}\eta \exp ( \eta ( N-({A}/{Q^{\ast }})\sum _{j=1}^{N}q_{j}^{2}) )$. Thus, we get

(3.23)\begin{align} Z_{N} & =\int_{\delta \left( \sum_{j=1}^{N}q_{j}-\varGamma \right) }D[ \boldsymbol{q}]\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\frac{\textrm{d}\eta }{2{\rm \pi} \textrm{i}}\exp \left( N\left( \begin{array}{c} \eta -\dfrac{A}{Q^{{\ast} }N}\eta \sum\limits_{j=1}^{N}q_{j}^{2} \\ +\dfrac{\beta }{2N}\sum\limits_{j\neq k}^{N}J_{jk}q_{j}q_{k} \end{array}\right) \right) \nonumber\\ & =\int_{\delta \left( \sum_{j=1}^{N}q_{j}-\varGamma \right) }D[\boldsymbol{q}]\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\frac{\textrm{d}\eta }{2{\rm \pi} \textrm{i}}\exp \left( N\left( \eta -\frac{1}{N}\sum\limits_{j,k}^{N}K_{jk}(Q^{{\ast} },\beta ,\eta ^{\prime })q_{j}q_{k}\right) \right), \end{align}

where

(3.24)\begin{equation} K_{jk}(Q^{{\ast} },\beta ,\eta ;z_{j},z_{k})=\left\{\begin{array}{@{}cc@{}} \dfrac{A}{Q^{{\ast} }}\eta & j=k \\ -\dfrac{\beta }{2}J_{jk}(z_{j},z_{k}) & j\neq k \end{array} \right\}. \end{equation}

To evaluate the Gaussian integrals in $Z_{N}$, in the case of a compact domain $D$, we expand the potential vorticity in terms of the o.n. complete system of eigenfunctions (associated with $G)$. Solution of the Gaussian integrals requires diagonalizing the interaction $J_{jk}$ in $H_{N}$ in terms of the above normalized orthogonal and complete eigenfunctions,

(3.25)\begin{equation} \frac{1}{N}\sum_{j\neq k}^{N}K_{jk}(Q^{{\ast} },\beta ,\eta )q_{j}q_{k}=\sum_{k\neq 0}^{\infty }\left( \frac{\beta }{2N}\lambda_{k}^{{-}1}+\frac{\eta }{Q^{{\ast} }}\right) (\alpha_{k}^{2}+b_{k}^{2}) \end{equation}

in the neutral case; for non-zero total charge $\varGamma$, the sum is over all integer $k$. Thus, in the general case of total charge,

(3.26)\begin{align} Z_{N} & =\int \textrm{d}\alpha_{0}\,\textrm{d}\alpha_{1}\,\textrm{d}b_{1}\int D_{k>1}(\alpha ,b) \nonumber\\ & \quad \int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\frac{\textrm{d}\eta }{2{\rm \pi} \textrm{i}}\exp \left\{ N\left[ \begin{array}{c} \eta -\sum\limits_{k=2}^{\infty }\left( \dfrac{\beta }{2N}\lambda_{k}^{{-}1}+\dfrac{ \eta }{Q}\right) (\alpha_{k}^{2}+b_{k}^{2}) \\ -\left( \dfrac{\beta }{2N}\lambda_{1}^{{-}1}+\dfrac{\eta }{Q}\right) (\alpha _{1}^{2}+b_{1}^{2}) \end{array} \right] \right\}, \end{align}

where we have split off the integral over the ground state harmonics for $k=1$ in the neutral case and for $k=0,1$ in the non-zero total charge cases. They represent the so-called non-ergodic modes which motivates this split.

By choosing real part of $\eta =a>0$ large enough, we interchange the order of integration of the term $\exp ( -N\sum _{k=2}^{\infty }( ({ \beta }/{2N})\lambda _{k}^{-1}+{\eta }/{Q}) (\alpha _{k}^{2}+b_{k}^{2}))$ to obtain

(3.27)\begin{align} Z_{N}& \propto \int \textrm{d}\alpha_{1}\,\textrm{d}b_{1}\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\frac{ \textrm{d}\eta }{2{\rm \pi} i}\exp \left\{ N\left[ \eta -\left( \frac{\beta }{2N}\lambda_{1}^{{-}1}+\frac{\eta }{Q}\right) (\alpha_{1}^{2}+b_{1}^{2})\right] \right\}\nonumber\\ & \quad \int D_{k\geqslant 2}(\alpha ,b)\exp \left({-}N\sum_{k=2}^{\infty }\left( \frac{ \beta }{2N}\lambda_{k}^{{-}1}+\frac{\eta }{Q}\right) (\alpha_{k}^{2}+b_{k}^{2})\right). \end{align}

3.2. Restricted partition function and non-ergodic modes

Next we write the problem in terms of the restricted partition function $Z_{N}(\alpha _{1},b_{1};\beta,Q)$, that is,

(3.28)\begin{align} Z_{N}(\beta ,Q) & \propto \int \textrm{d}\alpha_{1}\,\textrm{d}b_{1} Z_{N}(\alpha_{1},b_{1};\beta ,Q) \nonumber\\ & =\int \textrm{d}\alpha_{1}\,\textrm{d}b_{1}\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\frac{\textrm{d}\eta }{2{\rm \pi} \textrm{i}} \exp \left\{ N\left[ \eta -\left( \frac{\beta }{2N}\lambda_{1}^{{-}1}+ \frac{\eta }{Q}\right) (\alpha_{1}^{2}+b_{1}^{2})\right]\right\}\nonumber\\ & \quad\int D_{k\geqslant 2}(\alpha ,b)\exp \left({-}N\sum_{k=2}^{\infty }\left( \frac{ \beta }{2N}\lambda_{k}^{{-}1}+\frac{\eta }{Q}\right) (\alpha_{k}^{2}+b_{k}^{2})\right). \end{align}

Owing to non-ergodicity of the ordered domain-scale nature of modes, we do not integrate over these ordered modes whose amplitudes are denoted by $\alpha _{1},b_{1}$. We also note that all the higher harmonics will turn out to have zero amplitudes in the ordered phase of this problem. The statistics of the problem are therefore completely determined by the restricted partition function $Z_{N}(\alpha _{1},b_{1};\beta,Q)$. Amplitude of the ordered modes appear as parameters in this restricted partition function and will have to be evaluated separately.

Standard Gaussian integration is used to evaluate the last integral, which yields, after using the non-extensive thermodynamic limit scaling, $\beta ^{\prime }N=\beta$,

(3.29)\begin{equation} \int D_{k\geqslant 2}(\alpha ,b)\exp \left({-}N\sum_{k=2}^{\infty }\left( \frac{ \beta }{2N}\lambda_{k}^{{-}1}+\frac{\eta }{Q}\right) (\alpha_{k}^{2}+b_{k}^{2})\right) .=\prod_{k=2}^{\sqrt{N}}\left( \frac{{\rm \pi} }{\dfrac{ N\eta }{Q}+\dfrac{\beta ^{\prime }N}{2}\lambda_{k}^{{-}1}}\right) ^{1/2}, \end{equation}

provided the physically significant Gaussian conditions hold: for $k\geqslant 2$,

(3.30)\begin{equation} \frac{\beta ^{\prime }\lambda_{k}^{{-}1}}{2}+\frac{\eta }{Q}>0. \end{equation}

Substituting these Gaussian expressions in the partition function gives

(3.31)\begin{equation} Z_{N}(\alpha_{1},b_{1};\beta ,Q)\propto \frac{1}{2{\rm \pi} \textrm{i}}\int_{a-\textrm{i}\infty }^{a+\textrm{i}\infty }\textrm{d}\eta \exp \left\{ N\left[\begin{array}{c} \eta -\left( \dfrac{\beta ^{\prime }}{2}\lambda_{1}^{{-}1}+\dfrac{\eta }{Q} \right) (\alpha_{1}^{2}+b_{1}^{2}) \\ -\dfrac{1}{2N}\sum\limits_{k=2}\ln \left( \dfrac{N\eta }{Q}+\dfrac{\beta ^{\prime }N}{2 }\lambda _{k}^{{-}1}\right) \end{array} \right] \right\} , \end{equation}

where the partition function is now in the form where the saddle point method can be applied for $N$ large. Doing that yields the free energy per site, $f$, evaluated at the most probable macrostate, to be given by

(3.32)\begin{align} f(\beta ^{\prime },Q) & =U-TS={-}\frac{1}{\beta ^{\prime }}\ln Z_{N}(\alpha_{1},b_{1};\beta ^{\prime },Q) \end{align}

(3.33)\begin{align} & ={-}\frac{1}{\beta ^{\prime }}F(\eta (\beta ^{\prime }),Q,\beta ^{\prime }) \end{align}

with

(3.34)\begin{align} F(\eta (\beta ^{\prime }),Q,\beta ^{\prime }) & =\eta ^{\prime }(\beta^{\prime })\left[ 1-\frac{1}{Q}(\alpha_{1}^{2}+b_{1}^{2})\right] -\frac{ \beta ^{\prime }}{2}\lambda_{1}^{{-}1}(\alpha_{1}^{2}+b_{1}^{2})\nonumber\\ & \quad -\frac{1}{2N}\sum_{k=2}\ln \left( \frac{N\eta (\beta ^{\prime })}{Q}+\frac{ \beta ^{\prime }N}{2}\lambda_{k}^{{-}1}\right). \end{align}

Note that ‘the free energy per site evaluated exactly at the most probable macrostate’, is precisely, extracting the dominant term, as $N\rightarrow \infty$, in the integral $Z_{N}(\alpha _{1},b_{1};\beta,Q)$, to be that given by the saddle point, $\eta =\eta (\beta ^{\prime })$. In other words,

(3.35)\begin{equation} \left( Z_{N}(\alpha_{1},b_{1};\beta ,Q)\right) ^{1/N}\rightarrow _{N\rightarrow \infty }\exp \left[ F(\eta (\beta ^{\prime }),Q,\beta ^{\prime })\right]. \end{equation}

3.3. Saddle points and energy threshold for phase transitions

Provided that the saddle point $\eta (\beta ^{\prime })$ can be determined at given inverse temperature $\beta ^{\prime }$, the stable (most probable) macrostate is given by the extremum of the expression $F(\eta (\beta ^{\prime }),Q,\beta ^{\prime })$. The saddle point condition gives one equation for the determination of three variables $\eta,\alpha _{1},b_{1}$ in terms of inverse temperature $\beta ^{\prime }$ and enstrophy $Q$,

(3.36)\begin{equation} 0=\frac{\partial F}{\partial \eta ^{\prime }}=\left( 1-\frac{1}{Q}(\alpha _{1}^{2}+b_{1}^{2})\right) -\frac{1}{2NQ}\sum_{k=2}\left( \frac{\eta (\beta ^{\prime })}{Q}+\frac{\beta ^{\prime }}{2}\lambda_{k}^{{-}1}\right) ^{{-}1} , \end{equation}

where $\eta =\eta (\beta ^{\prime })$ is taken to be the value of the saddle point, with $\alpha _{0}=0$ in the neutral case.

We need more conditions to determine the amplitudes $\alpha _{1},b_{1}$ and the saddle point $\eta (\beta ^{\prime })>0$. They are provided by the following equations of state representing the variational principle for free energy at stable states:

(3.37)\begin{gather} 0 =\frac{\partial F}{\partial \alpha_{1}}={-}\left( \frac{2\eta (\beta ^{\prime })}{Q}+\frac{\beta ^{\prime }}{1}\lambda_{1}^{{-}1}\right) \alpha_{1} \end{gather}

(3.38)\begin{gather}0 =\frac{\partial F}{\partial b_{1}}={-}\left( \frac{2\eta (\beta ^{\prime }) }{Q}+\frac{\beta ^{\prime }}{1}\lambda_{1}^{{-}1}\right) b_{1}. \end{gather}

Thus, a coupled system of three algebraic equations (3.36), (3.38) determines three unknowns in terms of the given potential enstrophy $Q>0$ and the scaled inverse temperature $\beta ^{\prime }$. The equations of state implies that

(3.39)\begin{equation} \alpha_{1}=0=b_{1} \left( \frac{2\eta (\beta ^{\prime })}{Q}+ \frac{\beta ^{\prime }}{1}\lambda_{1}^{{-}1}\right) =0. \end{equation}

Because the saddle point is on the line with real part, $a>0$, we deduce $\eta (\beta ^{\prime })>0$ and with enstrophy $Q>0$, we obtain

(3.40)\begin{equation} \beta ^{\prime }={-}\frac{2\eta }{Q}\lambda_{1}<0 \end{equation}

when $\alpha _{1}\neq 0$ or $b_{1}\neq 0$. This proves that to have positive energy in the condensed blob-hole modes, the HM model's generalized energy must be sufficiently high to make $T^{\prime }$ not only negative but numerically small (and inverse temperature $\beta ^{\prime }<0$ and numerically large). Thus, we conclude that the critical temperature $T_{c}^{{\prime }}<0$.

In addition, the Gaussian conditions imply that for $k>1$,

(3.41)\begin{equation} \frac{\beta ^{\prime }}{2}\lambda_{k}^{{-}1}+\frac{\eta (\beta ^{\prime })}{Q} >0, \end{equation}

which implies that the large $N$ limit on the right-hand side of the saddle point condition is well defined and finite:

(3.42)\begin{align} \left( 1-\frac{1}{Q}(\alpha_{1}^{2}+b_{1}^{2})\right) & =\lim_{N\rightarrow \infty }\frac{1}{2NQ}\sum_{k=2}\left( \frac{\eta (\beta ^{\prime })}{Q}+ \frac{\beta ^{\prime }}{2}\lambda_{k}^{{-}1}\right) ^{{-}1} \end{align}

(3.43)\begin{align} & =\lim_{N\rightarrow \infty }\frac{1}{2NQ}\sum_{k=2}\left( \frac{\eta (\beta ^{\prime })}{Q}+\frac{\beta ^{\prime }}{2}\lambda_{k}^{{-}1}\right) ^{{-}1}. \end{align}

When either $T^{\prime }>0$ or $T^{\prime }<0$ and numerically large, there is no energy in the condensed modes with $l=2$, that is,

(3.44)\begin{equation} \left( 1-\frac{1}{Q}(\alpha_{1}^{2}+b_{1}^{2})\right) =1. \end{equation}

However, for sufficiently high energy or hot enough negative temperatures, given by a threshold $T_{c}^{\prime }< T^{\prime }<0$, the energy in these modes increases until they contain all of the fixed enstrophy $Q$ at absolute zero $T^{\prime }=0^{-}$:

(3.45)\begin{equation} \left( 1-\frac{1}{Q}(\alpha_{1}^{2}+b_{1}^{2})\right) \searrow 0. \end{equation}

So to calculate $T_{c}^{\prime }(Q)<0$, we need to find the most negative value of inverse temperature, denoted by $\beta _{c}^{\prime }$, for which

(3.46)\begin{equation} 1=\lim_{N\rightarrow \infty }\frac{1}{2NQ}\sum_{k=2}^{\sqrt{N}}\left( \frac{ \eta (\beta ^{\prime })}{Q}+\frac{\beta ^{\prime }}{2}\lambda_{k}^{{-}1}\right) ^{{-}1} \end{equation}

and after which, for values of $\beta ^{\prime }<\beta _{c}^{\prime }<0$, $\lim _{N\rightarrow \infty }({1}/{2NQ})\sum _{k=2}^{\sqrt {N}}({ \eta (\beta ^{\prime })}/{Q}+({\beta ^{\prime }}/{2})\lambda _{k}^{-1}) ^{-1}<1$.

Inserting $\eta =-({\beta ^{\prime }Q}/{2})\lambda _{1}^{-1}$ into the right-hand side of the above equation, we calculate $\beta _{c}^{\prime }$ to be given by

(3.47)\begin{equation} {-}\infty <\beta_{c}^{\prime }=\lim_{N\rightarrow \infty }\frac{1}{QN} \sum_{k=2}^{\sqrt{N}}\left( \lambda_{k}^{{-}1}-\lambda_{1}^{{-}1}\right) ^{{-}1}<0\end{equation}

and check that for $\beta ^{\prime }<\beta _{c}^{\prime }(Q)<0$,

(3.48)\begin{equation} \lim_{N\rightarrow \infty }\frac{1}{\beta ^{\prime }NQ}\sum_{k=2}^{\sqrt{N} }\left( \lambda_{k}^{{-}1}-\lambda_{1}^{{-}1}\right) ^{{-}1}<1. \end{equation}

Hence the importance of the spectral gap $( \lambda _{k}-\lambda _{1})$. We note the significance of the critical temperature's linear dependence on the given potential enstrophy $Q$ of the flow.

In other words, the extreme saddle point

(3.49)\begin{equation} \eta ^{{\ast} }={-}\frac{\beta ^{\prime }Q}{2}\lambda_{1}^{{-}1} \end{equation}

is no longer adequate to solve (3.46) for $\beta ^{\prime }<\beta _{c}^{\prime }(Q)<0$, which is the so-called sticking of the saddle point at the critical point $\beta _{c}^{\prime }$.

Moreover, the saddle point equation gives us a way to compute the equilibrium amplitudes of the blob-hole modes for temperatures hotter than the negative critical temperature $T_{c}$. For temperatures $T$ so that $T_{c}< T< T_{b}<0$,

(3.50)\begin{equation} (\alpha_{1}^{2}+b_{1}^{2})(T)=Q\left( 1-\frac{T}{T_{c}}\right). \end{equation}

Here $T_{b}<0$ is a possible second critical temperature hotter than $T_{c}$, thus, $T_{c}$ $< T_{b}<0$, at which higher modes condense. A significant point to make here is that, true to a variational theory based on symmetries such as the statistical mechanics formulation here, the actual values of the amplitudes, $\alpha _{1},b_{1}$, are not directly obtainable from this analysis, only the sum of squares $(\alpha _{1}^{2}+b_{1}^{2})(T)$ as a function of the temperature $T$. It is interesting, however, to note that we can make the further case that either $\alpha _{1}\neq 0$ or $b_{1}\neq 0,$or $|\alpha _{1}|=$ $|b_{1}|$. Otherwise, the two modes $\varphi _{1}$ and $\phi _{1}$ interact nonlinearly through the HM equation and the amplitudes $\alpha _{1}(t),b_{1}(t)$ are dynamic quantities. This violates the premise of an equilibrium statistical mechanics.

For $b_{1}\neq 0$, inspection of the spatial structure of mode $\phi _{1}$, in particular its oddness in $y$, implies that this mode is responsible for the so-called blob-hole coherent structure in plasmas.