Revisiting Batchelor's theory of two-dimensional turbulence

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X-2/3 k(-1) enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re -> infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(-1) with (omega(2))k(-1)(In Re)(-1).Publisher PDFPeer reviewe

Structure and superconductivity in the binary Re1−x_{1-x}Mox_x alloys

The binary Re$_{1-x}$Mo$_x$ alloys, known to cover the full range of solid solutions, were successfully synthesized and their crystal structures and physical properties investigated via powder x-ray diffraction, electrical resistivity, magnetic susceptibility, and heat capacity. By varying the Re/Mo ratio we explore the full Re$_{1-x}$Mo$_x$ binary phase diagram, in all its four different solid phases: hcp-Mg ($P6_3/mmc$), $\alpha$-Mn ($I\overline{4}3m$), $\beta$-CrFe ($P4_2/mnm$), and bcc-W ($Im\overline{3}m$), of which the second is non-centrosymmetric with the rest being centrosymmetric. All Re$_{1-x}$Mo$_x$ alloys are superconductors, whose critical temperatures exhibit a peculiar phase diagram, characterized by three different superconducting regions. In most alloys the $T_c$ is almost an order of magnitude higher than in pure Re and Mo. Low-temperature electronic specific-heat data evidence a fully-gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling across the series. Considering that several $\alpha$-Mn-type Re$T$ alloys ($T$ = transition metal) show time-reversal symmetry breaking (TRSB) in the superconducting state, while TRS is preserved in the isostructural Mg$_{10}$Ir$_{19}$B$_{16}$ or Nb$_{0.5}$Os$_{0.5}$, the Re$_{1-x}$Mo$_x$ alloys represent another suitable system for studying the interplay of space-inversion, gauge, and time-reversal symmetries in future experiments expected to probe TRSB in the Re$T$ family.Comment: 8 pages, 7 figures, accepted for publication on Physical Review Material