The Astrophysical Journal, 866:106 (8pp), 2018 October 20 https://doi.org/10.3847/1538-4357/aade04 © 2018. The American Astronomical Society. All rights reserved. Incompressive Energy Transfer in the Earth’s Magnetosheath: Magnetospheric Multiscale Observations Riddhi Bandyopadhyay1 , A. Chasapis1 , R. Chhiber1 , T. N. Parashar1 , W. H. Matthaeus1 , M. A. Shay1 , B. A. Maruca1 , J. L. Burch2, T. E. Moore3, C. J. Pollock4, B. L. Giles3, W. R. Paterson3, J. Dorelli3, D. J. Gershman3 , R. B. Torbert5, C. T. Russell6 , and R. J. Strangeway6 1 Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA; [email protected], [email protected] 2 Southwest Research Institute, San Antonio, TX, USA 3 NASA Goddard Space Flight Center, Greenbelt, MD, USA 4 Denali Scientific, Fairbanks, AK, USA 5 University of New Hampshire, Durham, NH, USA 6 University of California, Los Angeles, CA, USA Received 2018 July 10; revised 2018 August 24; accepted 2018 August 28; published 2018 October 17 Abstract Using observational data from the Magnetospheric Multiscale mission in the Earth’s magnetosheath, we estimate the energy cascade rate at three ranges of length scale, employing a single data interval, using different techniques within the framework of incompressible magnetohydrodynamic (MHD) turbulence. At the energy-containing scale, the energy budget is controlled by the von Kármán decay law. Inertial range cascade is estimated by fitting a linear scaling to the mixed third-order structure function. Finally, we use a multi-spacecraft technique to estimate the Kolmogorov–Yaglom-like cascade rate in the kinetic range, well below the ion inertial length scale, where we expect a reduction due to involvement of other channels of transfer. The computed inertial range cascade rate is almost equal to the von Kármán–MHD law at the energy-containing scale, while the incompressive cascade rate evaluated at the kinetic scale is somewhat lower, as anticipated in theory. In agreement with a recent study, we find that the incompressive cascade rate in the Earth’s magnetosheath is about 1000 times larger than the cascade rate in the pristine solar wind. Key words: magnetohydrodynamics (MHD) – planets and satellites: magnetic fields – plasmas – solar wind – turbulence 1. Introduction Taylor (1935) suggested, based on heuristic arguments, that the decay rate in a neutral fluid is controlled by the energy- One of the long-standing mysteries of space physics is the containing eddies. Later, de Kármán & Howarth (1938) derived anomalous heating of the solar wind. Assuming adiabatic Taylor’s results more rigorously, assuming that the shape of the expansion, the temperature profile of the solar wind is expected − / two-point correlation function remains unchanged during the to scale as T(r)∼r 4 3, where r is the radial distance from decay of a turbulent fluid at high Reynolds number. In one of his the Sun. Yet, the best fittoVoyager temperature observa- three famous 1941 papers, Kolmogorov (1941) derived an exact tion(Richardson et al. 1995) results in a radial profile — ( )∼ −1/2 expression for the inertial range cascade rate the so-called T r r . Turbulence provides a natural explanation, third-order law for homogeneous, incompressible, isotropic supplying internal energy through a cascade process that neutral fluids—from the Navier–Stokes equation, based on channels available energy, in the form of electromagnetic ( ) fl only a few general assumptions. Hossain et al. 1995 attempted uctuations and velocity shear at large scales, to smaller scales to extend the von Kármán phenomenology for magnetized and ultimately into dissipation and heating. In collisionless fluids based on dimensional arguments. Politano & Pouquet plasmas, such as the solar wind or the magnetosheath, the (1998a, 1998b;hereafterPP98) extended Kolmogorov’s third- situation is more complicated due to kinetic effects. Never- order law to homogeneous, incompressible MHD turbulence ( ) theless, magnetohydrodynamics MHD , which models the using Elsasser variables. plasma as a single fluid, has proven to be a very successful Following these theoretical advances, the third-order law has theoretical framework in describing even weakly collisional been used in several studies to estimate the energy cascade rate plasmas, such as the solar wind, provided one focuses on large- in the solar wind(Sorriso-Valvo et al. 2007; MacBride scale features and processes. Over the last few decades, there et al. 2008; Coburn et al. 2012, 2014, 2015; Marino et al. have been extensive studies related to energy cascade rate and 2012). Density fluctuations in the solar wind are usually low dissipation channels in collisionless plasmas (e.g., Sorriso- enough that incompressible MHD works well. Carbone et al. Valvo et al. 2007; MacBride et al. 2008; Osman et al. 2011; (2009) and Carbone (2012) first made an attempt, based on Coburn et al. 2015), largely based on ideas originating in MHD heuristic reasoning, to include density fluctuations for estimat- studies (Politano & Pouquet 1998a). Recently, there has been ing compressible transfer rate using the “third-order” law. an effort to understand the more complex pathways of energy Recently, Banerjee & Galtier (2013; hereafter BG13) worked cascade in plasmas(Howes et al. 2008; Servidio et al. 2017; out an exact transfer rate for a compressible medium. Hadid Yang et al. 2017a, 2017b; Hellinger et al. 2018). Prior to et al. (2015, 2018) compared the cascade rates derived presenting new observational results on this timely subject, to from BG13 and PP98 in weakly and highly compressive provide context we now briefly digress on this history. media in various planetary magnetosheaths and solar wind 1 The Astrophysical Journal, 866:106 (8pp), 2018 October 20 Bandyopadhyay et al. (Banerjee et al. 2016). It was found that the fluxes derived from the applicability of an incompressible MHD approach for the the two theories lie close to each other for weakly compressive selected interval. Contrary to the pristine solar wind, the ratio media and the deviation starts to become significant as the of root mean squared (rms) fluctuations to the mean is high for plasma compressibility becomes higher, as expected. the magnetic field, indicating isotropy, thus making this Parallel to observational works in space plasma systems, on interval suitable for this study since we only consider the theoretical side there have been numerous efforts to refine incompressible, isotropic MHD here. the “third-order” law derived for incompressible homogeneous The separation of the four MMS spacecraft was ≈8 km, which MHD, by including more kinetic physics, like two-fluid corresponds to about half the ion inertial length (di ≈ 16 km). MHD(Andrés et al. 2016), Hall MHD(Andrés et al. 2018), We used MMS burst resolution data which provides magnetic electron MHD(Galtier 2008), by incorporating the effect field measurements (FGM) at 128 Hz(Russell et al. 2016; of large-scale shear, slowly varying mean field(Wan et al. Torbert et al. 2016), and ion density, temperature and velocity 2009, 2010), etc. One would expect, as kinetic effects become (FPI) at 33Hz(Pollock et al. 2016). The small separation, important in a plasma, such corrections would need to be taken combined with the high time resolution of the measurements of into account. In this work, we consider only incompressible, the ion moments, allows us to use a multi-spacecraft approach homogeneous MHD phenomenologies. similar to that used by Osman et al. (2011) at the very small In this study, we focus on the Earth’s magnetosheath. While scales of the turbulent dissipation range. similar to the turbulence observed in the pristine solar wind, the shocked solar wind plasma in the magnetosheath, downstream 3. Energy-containing Scale: ò1 of Earth’s bow-shock, provides a unique laboratory for the It is reasonable to expect that the global decay is controlled, study of turbulent dissipation under a wide range of conditions, to a suitable level of approximation, by the von Karman decay like plasma beta, particle velocities, compressibility, etc. Past law, generalized to MHD(Hossain et al. 1995; Politano & studies, both numerical(Karimabadi et al. 2014) and observa- Pouquet 1998a, 1998b; Wan et al. 2012), tional(Sundkvist et al. 2007; Huang et al. 2014; Chasapis et al. 2015, 2017; Breuillard et al. 2016; Yordanova et al. 2016) have dZ() 22 ()ZZ probed the properties of turbulent dissipation in the magne- 1 =- =a ,1() tosheath. Such studies have established the contribution of dt L intermittent structures, such as current sheets to turbulent α ± fl dissipation in the kinetic range. However, the properties of where ± are positive constants and Z are the rms uctuation turbulence at kinetic scales and quantitative treatments of values of the Elsasser variables, defined as energy dissipation at those scales remain scarce(Huang B()t et al. 2017; Breuillard et al. 2018; Gershman et al. 2018; ZV()tt= () ,2() ) Hadid et al. 2018 . m0mnpi() t Here, we investigate the energy transfer at several scales, including kinetic scales, using the state-of-the-art capabilities of where the local mean values have been subtracted from V and the Magnetospheric Multiscale (MMS) mission(Burch et al. B, respectively the plasma (ion) velocity and the magnetic field 2016). The combination of high time resolution plasma data vector. Here, μ0 is the magnetic permeability of vacuum, with multi-spacecraft observations at very small separations m ?m are proton and electron mass, respectively and n is allows us to carry out estimation of the cascade rate using p e i the number density of protons. several strategies employing a single data interval. For steady The similarity length scales L± appearing in Equation (1) are conditions, we expect that the decay rate of the energy- related to the characteristic scales of the “energy-containing” containing eddies at large scales, ò and the cascade rate 1 eddies. Usually, a natural choice for the similarity scales is the measured in the inertial range, ò , will be comparable.