Strong vorticity fluctuations and antiferromagnetic correlations in axisymmetric fluid equilibria

The macroscale structure and microscale fluctuation statistics of late-time asymptotic steady-state flows in cylindrical geometries is studied using the methods of equilibrium statistical mechanics. The axisymmetric assumption permits an effective two-dimensional (2D) description in terms of the (toroidal) flow field $\sigma$ about the cylinder axis and the vorticity field $\xi$ that generates mixing within the (poloidal) planes of fixed azimuth. As for a number of other 2D fluid systems, extending the classic 2D Euler equation, the flow is constrained by an infinite number of conservation laws, beyond the usual kinetic energy and angular momentum. All must be accounted for in a consistent equilibrium description. It is shown that the most directly observable impact of the conservation laws is on $\sigma$, which displays interesting large-scale radius-dependent flow structure. However, unlike in some previous treatments, we find that the thermodynamic temperature is always positive. As a consequence, except for an infinitesimal boundary layer that maintains the correct (conserved) value of the overall poloidal circulation, the impact on $\xi$ resides in the statistics of the strongly fluctuating, fine-scale mixing, where it is sensitive to “antiferromagnetic” microscale correlations that help maintain the analog of local charge neutrality. The poloidal flow is macroscopically featureless, displaying no large-scale circulating jet- or eddylike features (which typically emerge as negative temperature states in analogous Euler and quasigeostrophic equilibria).

Figure 1

Axisymmetric flow geometry. The pattern of flows is assumed to be invariant under rotation about the cylinder axis, and hence may be fully specified by the toroidal flow field $\sigma$ about the axis [Eq. (7)], and poloidal vorticity field $\xi$ [Eq. (9)] within any 2D radial plane $D$.

Figure 2

Schematic level surfaces of the toroidal field $\sigma$, illustrating the conserved vorticity integrals in an approximation in which $\sigma$ takes only a finite number of discrete values ($...>s_2>s_1>0>-s_1>...$). As formally defined by Eq. (40), the total area $\gamma \left[\sigma \right]\left(s\right)$ of each level set $\left\{\sigma \right(\mathit{\rho })=s\}$ is conserved by the flow, as is the total integral $\tilde{\gamma }[\sigma ,\xi ]\left(s\right)$ of $\xi$ over each such set. In general each set is the union of some number of (in general, multiply connected) pieces. For clarity, only a few of these are explicitly labeled in the figure. Since the exact value of $\sigma$, but only the mean value of $\xi$, is specified on each set, the former is far more strongly constrained. Arbitrary fluctuations of the latter about the mean are permitted, and this greatly impacts the predicted equilibrium states.

Figure 3

Example equilibrium results for the two-level model described by Eqs. (59, 60, 61, 62, 63, 64) using $s_1=0$ and $s_2=1$, scaled temperature $\overline{T}=0.1$, and cylinder boundaries $r_{\mathrm{in}}=1$, $r_{\mathrm{out}}=2$ yielding $y_{\mathrm{in}}=0.5$, $y_{\mathrm{out}}=2$. Left: Probability distribution $p_{\sigma }(s_2,y)$ for a range of chemical potential values $0.2\le {\mu }_{\mathrm{\Delta }}\le 1.7$ in steps of 0.1. The interface moves left (larger region occupied by $s_2$) for increasing ${\mu }_{\mathrm{\Delta }}$. For the chosen $s_1,s_2$ values, the curves coincide with the mean toroidal flow profile $\langle \sigma \left(\mathit{\rho }\right)\rangle$ [third equality in Eq. (61)]. Right: Fractional area $p_2\left({\mu }_{\mathrm{\Delta }}\right)$, defined by Eqs. (59) and (64), occupied by $\sigma =s_2$. This illustrates the 1–1 correspondence between the Lagrange multiplier ${\mu }_{\mathrm{\Delta }}$ and the conserved integrals $g\left(s\right)$.

Figure 4

Example poloidal field equilibrium results for the same two-level model and parameters described in Fig. 3. Left: Poloidal spatial distribution ${\tilde{G}}_0\left(y\right)$ defined by Eq. (131) for a range of chemical potential values $0.2\le {\mu }_{\mathrm{\Delta }}\le 1.7$ in steps of 0.1. The peak moves to left for increasing ${\mu }_{\mathrm{\Delta }}$. Right: Normalized area integral ${\tilde{g}}_0\left({\mu }_{\mathrm{\Delta }}\right)$ defined by Eq. (133).