A Mathematical Analysis of the Intermediate Behaviour of the Energy Cascades of Quantum Turbulence

We propose a mathematical interpolation between several regimes of energy cascade in quantum turbulence in He II. On the basis of a physical interpretation of such mathematical expression we discuss in which conditions it is expected to appear an intermediate k2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k^{2}$\end{document} regime (equipartition regime) in the transition region between the hydrodynamic regime and the Kelvin wave regime (namely, between the k−5/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k^{-5/3}$\end{document} and k−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k^{-1}$\end{document} regions in coflow situations and between the k−3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k^{-3}$\end{document} and k−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k^{-1}$\end{document} regions in counterflow situations). It is seen that if the energy rate transfer from the hydrodynamic region to the Kelvin wave region is sufficiently slow, such equipartition region will be present, but for higher values of such energy rate transfer it will disappear. For high rates of the energy rate transfer, the transition regime between the hydrodynamic and the Kelvin wave regimes will be monotonous, characterized by a negative exponent of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k$\end{document} between −5/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-5/3$\end{document} and −1 (or between −3 and −1), instead of the positive 2 exponent of the equipartition regime.