PHYSICAL REVIEW FLUIDS 4, 104611 (2019) Modeling Ekman and quasi-static magnetohydrodynamic turbulence using Pao’s hypothesis Mohammad Anas * Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India Mahendra K. Verma † Department of Physics, Indian Institute of Technology, Kanpur 208016, India (Received 10 July 2019; published 28 October 2019) Two-dimensional (2D) turbulence with Ekman friction and quasi-static magnetohydro- dynamic (QS MHD) turbulence are complex flows. In this paper, we present models for these flows by extending Pao’s hypothesis [Phys. Fluids 8, 1063 (1965)] for hydrodynamic turbulence to them. For 2D Ekman turbulence, the energy spectrum predicted by the model is steeper than its hydrodynamic counterpart due to Ekman friction. The model predictions are consistent with earlier theoretical predictions and experimental and numerical results. Similarly, the model for QS MHD turbulence predicts a steeper energy spectrum due to Joule dissipation; the model predictions fit with earlier numerical results quite well. DOI: 10.1103/PhysRevFluids.4.104611 I. INTRODUCTION To date, turbulence remains a poorly understood phenomenon. One of the important known results of three-dimensional (3D) turbulence is due to Kolmogorov [1,2], according to which a constant energy flux cascades from large scales to small scales, leading to energy-spectrum scaling as E u (k ) = K Ko ǫ 2/3 u k −5/3 . (1) Here, ǫ u is the energy flux, and K Ko is Kolmogorov’s constant. The above energy spectrum and flux have been observed in three-dimensional homogeneous isotropic turbulence at high Reynolds number [3–6] but is modified for different forcing. For example, inclusion of rotation, magnetic field, and buoyancy alter turbulence properties significantly [5,7–11]. In some flows, the presence of an external dissipation (for example, Ekman friction [12–14] and Joule dissipation [10,15]) dampens the energy of multiscale flow structures more than that predicted by Kolmogorov’s theory. This leads to steepening in the energy flux and spectrum. Modeling such complex flows is quite difficult. Fortunately, an extension of Pao’s hypothesis for hydrodynamic turbulence offers a set of models for two-dimensional (2D) Ekman turbulence and quasi-static magnetohydrodynamic (QS MHD) turbulence; this is the topic of the present paper. Friction at the bottom of the container in a shallow layer [16], and the friction of the surrounding air in soap film [17] induce drag which is commonly termed Ekman friction. Such flows are typically modeled as 2D turbulence because of the shallow nature of the flow, and they have been widely studied by researchers using experiments and numerical simulations. A common theme in * anas@iitk.ac.in † mkv@iitk.ac.in 2469-990X/2019/4(10)/104611(11) 104611-1 ©2019 American Physical Society