The nature of collisionless reconnection in a three-species plasma composed of a heavy species, protons, and electrons is examined. Besides the usual two length scales present in two-species reconnection, there are two additional larger length scales in the system: one associated with a "heavy whistler" which produces a large scale quadrupolar out-of-plane magnetic field, and one associated with the "heavy Alfven" wave which can slow the outflow speed and thus the reconnection rate. The consequences for reconnection in the magnetotail with an O+ population present are discussed.
Many in-situ spacecraft observations have demonstrated that magnetic reconnection in the Earth's magnetotail is largely controlled by the pre-reconnection current sheet configuration. One of the most important thin current sheet characteristics is the preponderance of electron currents driven by strong polarized electric fields, which are commonly observed in the Earth's magnetotail well before the reconnection. We use particle-in-cell simulations to investigate magnetic reconnection in the 2D magnetotail current sheet with a finite magnetic field component normal to the current sheet and with…
Magnetic field line reconnection is a universal plasma process responsible for the magnetic field topology change and magnetic field energy dissipation into charged particle heating and acceleration. In many systems, the conditions leading to the magnetic reconnection are determined by the pre-reconnection configuration of a thin layer with intense currents -- otherwise known as the thin current sheet. In this study we investigate two such systems: Earth's magnetotail and helmet streamers in the solar corona. The pre-reconnection current sheet evolution has been intensely studied in the magnet…
Among the fundamental and most challenging problems of laboratory, space, and astrophysical plasma physics is to understand the relaxation processes of nearly collisionless plasmas toward quasi-stationary states; and the resultant states of electromagnetic plasma turbulence. Recently, it has been argued that solar wind plasma $β$ and temperature anisotropy observations may be regulated by kinetic instabilities such as the ion-cyclotron, mirror, electron-cyclotron, and firehose instabilities; and that magnetic fluctuation observations are consistent with the predictions of the Fluctuation-Dissi…
The magnetotail current sheet configuration determines magnetic reconnection properties that control the substorm onset, one of the most energetic phenomena in the Earth's magnetosphere. The quiet-time current sheet is often approximated as a two-dimensional (2D) magnetic field configuration balanced by isotropic plasma pressure gradients. However, reconnection onset is preceded by the current sheet thinning and the formation of a nearly one-dimensional (1D) magnetic field configuration. In this study, using particle-in-cell simulations, we investigate the force balance of such thin current sh…
Magnetic reconnection is one of the most important processes in astrophysical, space and laboratory plasmas. Identifying the structure around the point at which the magnetic field lines break and subsequently reform, known as the magnetic null point, is crucial to improving our understanding reconnection. But owing to the inherently three-dimensional nature of this process, magnetic nulls are only detectable through measurements obtained simultaneously from at least four points in space. Using data collected by the four spacecraft of the Cluster constellation as they traversed a diffusion regi…
Earth's magnetotail region provides a unique environment to study plasma turbulence. We investigate the turbulence developed in an exhaust produced by magnetic reconnection at the terrestrial magnetotail region. Magnetic and velocity spectra show broad-band fluctuations corresponding to the inertial range, with Kolmorogov $-5/3$ scaling, indicative of a well developed turbulent cascade. We examine the mixed, third-order structure functions, and obtain a linear scaling in the inertial range. This linear scaling of the third-order structure functions implies a scale-invariant cascade of energy t…
The release of plasma in the jovian magnetotail is observed in the form of plasmoids, travelling compression regions, field-aligned particle beams and flux-rope like events. We demonstrate that electrons propagate along the magnetic field lines in the plasma sheet boundary layer (PSBL), while close to the current sheet center the electron distribution is isotropic. The evidences of the counterstreaming electron beams in the PSBLs are also presented. Most of the field-aligned energetic ion beams are associated with the field-aligned electron beams and about half of them have the bipolar fluctua…
We present the first statistical study of magnetic structures and associated energy dissipation observed during a single period of turbulent magnetic reconnection, by using the \textit{in-situ} measurements of the Magnetospheric Multiscale mission in the Earth's magnetotail on July 26, 2017. The structures are selected by identifying a bipolar signature in the magnetic field and categorized as plasmoids or current sheets via an automated algorithm which examines current density and plasma flow. The size of the plasmoids forms a decaying exponential distribution ranging from sub-electron up to …
Signatures of the dissipation region of collisionless magnetic reconnection are investigated by the Geotail spacecraft for the 15 May 2003 event. The energy dissipation in the rest frame of the electron's bulk flow is considered in an approximate form D*_e, which is validated by a particle-in-cell simulation. The dissipation measure is directly evaluated from the {plasma moments}, the electric field, and the magnetic field. Using D*_e, a compact dissipation region is successfully detected in the vicinity of the possible X-point in Geotail data. The dissipation rate is 45 pWm**{-3}. The length …
Models for collisionless magnetic reconnection in near-Earth space are distinctly characterized as 2D or 3D. In 2D kinetic models, the frozen-in law for the electron fluid is usually broken by laminar dynamics involving structures set by the electron orbit size, while in 3D models the width of the electron diffusion region is broadened by turbulent effects. We present an analysis of in situ spacecraft observations from the Earth's magnetotail of a fortuitous encounter with an active reconnection region, mapping the observations onto a 2D spatial domain. While the event likely was perturbed by …
Ion Temperature Anisotropy Limits from Magnetic Curvature Scattering in Magnetotail Reconnection Jets
Swedish Institute of Space Physics（瑞典空间物理研究所）；Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles（加州大学洛杉矶分校地球、行星与空间科学系）；Department of Physics and Engineering, University of Bergen（卑尔根大学物理与工程系）；Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文系）
We discuss electron acceleration and heating during collisionless magnetic reconnection by using the results of implicit kinetic simulations of Harris current sheets. We consider and compare electron dynamics in plasmas with different βvalues and perform simulations up to the physical mass ratio. We analyze the typical trajectory of electrons passing through the reconnection region, we study the electron velocity, focusing on the out-of-plane velocity, and we discuss the electron heating along the in-plane and out-of-plane directions.
We use the results of a three-dimensional kinetic simulation of an Harris current sheet to propose an explanation and to reproduce the ISEE-1/2, Geotail, and Cluster observations of the magnetotail current sheet structure. Current sheet flapping, current density bifurcation, and reconnection are explained as the results of the kink and tearing instabilities, which dominate the current sheet evolution.
I propose a new paradigm for solar coronal heating viewed as a self-regulating process keeping the plasma marginally collisionless. The mechanism is based on the coupling between two effects. First, coronal density controls the plasma collisionality and hence the transition between the slow collisional Sweet-Parker and the fast collisionless reconnection regimes. In turn, coronal energy release leads to chromospheric evaporation, increasing the density and thus inhibiting subsequent reconnection of the newly-reconnected loops. As a result, statistically, the density fluctuates around some crit…
Based on the first principles (i.e. (i) by balancing the magnetic field advection with the term containing electron pressure tensor non-gyrotropic components in the generalised Ohm's law; (ii) using the conservation of mass; and (iii) assuming that the weak magnetic field region width, where electron meandering motion supports electron pressure tensor off-diagonal (non-gyrotropic) components, is of the order of electron Larmor radius) a simple model of magnetic reconnection in a collisionless regime is formulated. The model is general, resembling its collisional Sweet-Parker analogue in that i…
I propose that solar coronal heating is a self-regulating process that keeps the coronal plasma roughly marginally collisionless. The self-regulating mechanism is based on the interplay of two effects. First, plasma density controls coronal energy release via the transition between the slow collisional Sweet-Parker regime and the fast collisionless reconnection regime. This transition takes place when the Sweet--Parker layer becomes thinner than the characteristic collisionless reconnection scale. I present a simple criterion for this transition in terms of the upstream plasma density (n_e), t…
The underlying concepts of collisionless magnetic field reconnection in the vicinity of the diffusion regions are developed qualitatively from first principles. One conclusion of this analysis is that energetic ions and electrons observed in the terrestrial magnetosphere or from the sun are not accelerated to the observed energies in the diffusion regions.
We propose an analytic approach to the problem of collisionless magnetic reconnection formulated as a process of Alfven eigenmodes' generation and dissipation. Alfven eigenmodes are confined by the current sheet in the same way that quantum mechanical waves are confined by the tanh^2 potential. The dynamical time scale of reconnection is the system scale divided by the eigenvalue propagation velocity of the n=1 mode. The prediction of the n=1 mode shows good agreement with the in situ measurement of the reconnection-associated Hall fields.
We point out that a conventional construction placed upon observations of accreting black holes, in which their nonthermal X-ray spectra are produced by inverse comptonization in a coronal plasma, suggests that the plasma is marginally collisionless. Recent developments in plasma physics indicate that fast reconnection takes place only in collisionless plasmas. As has recently been suggested for the Sun's corona, such marginal states may result from a combination of energy balance and the requirements of fast magnetic reconnection.
The results of kinetic simulations of magnetic reconnection in Harris current sheets are analyzed. A range of guide fields is considered to study reconnection in plasmas characterized by different $β$ values, $β> m_e/m_i$. Both an implicit Particle-in-Cell (PIC) simulation method and a parallel explicit PIC code are used. Simulations with mass ratios up to the physical value are performed. The simulations show that the reconnection rate decreases with the guide field and depends weakly on the mass ratio. The off-diagonal components of the electron pressure tensor break the frozen-in condition,…
Particle in cell (PIC) simulations of collisionless magnetic reconnection are presented that demonstrate that the electron dissipation region develops a distinct two-scale structure along the outflow direction. The length of the electron current layer is found to decrease with decreasing electron mass, approaching the ion inertial length for a proton-electron plasma. A surprise, however, is that the electrons form a high-velocity outflow jet that remains decoupled from the magnetic field and extends large distances downstream from the x-line. The rate of reconnection remains fast in very large…
Recently, magnetic reconnection during collisionless, stressed, X-point collapse was studied using kinetic, 2.5D, fully electromagnetic, relativistic Particle-in-Cell numerical code [D. Tsiklauri and T. Haruki, Phys. Plasmas, 14, 112905 (2007)]. Here we finalise the investigation of this topic by addressing key outstanding physical questions. It has been established here that: (i) reconnection out-of-plane electric field at the magnetic null is generated by the electron pressure tensor off-diagonal terms, resembling to the case of tearing unstable Harris current sheet studied by the GEM reconn…
We study the role of the Hall current and electron inertia in collisionless magnetic reconnection within the framework of full two-fluid MHD. At spatial scales smaller than the electron inertial length, a topological change of magnetic field lines exclusively due to electron inertia becomes possible. Assuming stationary conditions, we derive a theoretical scaling for the reconnection rate, which is simply proportional to the Hall parameter. Using a pseudo-spectral code with no dissipative effects, our numerical results confirm this theoretical scaling. In particular, for a sequence of differen…
A new measure to identify a small-scale dissipation region in collisionless magnetic reconnection is proposed. The energy transfer from the electromagnetic field to plasmas in the electron's rest frame is formulated as a Lorentz-invariant scalar quantity. The measure is tested by two-dimensional particle-in-cell simulations in typical configurations: symmetric and asymmetric reconnection, with and without the guide field. The innermost region surrounding the reconnection site is accurately located in all cases. We further discuss implications for nonideal MHD dissipation.
Spectral Properties and Energy Injection in Mercury's Magnetotail Current Sheet
Boston University（波士顿大学）；Laboratoire de Physique des Plasmas (LPP), CNRS, Observatoire de Paris, Sorbonne Université, Université Paris Saclay, Ecole polytechnique, Institut Polytechnique de Paris（等离子体物理实验室，法国国家科学研究中心，巴黎天文台，索邦大学，巴黎萨克雷大学，巴黎综合理工学院，巴黎理工学院）；Department of Climate and Space Sciences and Engineering, University of Michigan（密歇根大学气候与空间科学与工程系）；INAF-Institute for Astrophysics and Space Planetology（意大利国家天体物理学研究所空间与行星物理学研究所）
This paper presents two-fluid simulations of forced magnetic reconnection with finite electron inertia in a collisionless three-dimensional (3D) cube with periodic boundaries in all three directions. Comparisons are made to analogous two-dimensional (2D) simulations. Reconnection in this system is driven by a spatially localized forcing function that is added to the ion momentum equation inside the computational domain. Consistent with previous results in similar, but larger forced 2D simulations [B. Sullivan, B. N. Rogers, and M. A. Shay, Phys. Plasmas 12, 122312 (2005)], for sufficiently str…
A model of global magnetic reconnection rate in relativistic collisionless plasmas is developed and validated by the fully kinetic simulation. Through considering the force balance at the upstream and downstream of the diffusion region, we show that the global rate is bounded by a value $\sim 0.3$ even when the local rate goes up to $\sim O(1)$ and the local inflow speed approaches the speed of light in strongly magnetized plasmas. The derived model is general and can be applied to magnetic reconnection under widely different circumstances.
Large-scale resistive Hall-magnetohydrodynamic (Hall-MHD) simulations of the transition from Sweet-Parker (collisional) to Hall (collisionless) magnetic reconnection are presented, the first to separate effects of secondary islands from collisionless effects. Three main results are described. There exists a regime in which secondary islands occur without collisionless effects when the thickness of the dissipation regions exceed ion gyroscales. The reconnection rate with secondary islands is faster than Sweet-Parker but significantly slower than Hall reconnection. This implies that secondary is…
Particle-in-Cell simulations of collisionless magnetic reconnection with a guide field reveal for the first time the three dimensional features of the low density regions along the magnetic reconnection separatrices, the so-called "cavities". It is found that structures with further lower density develop within the cavities. Because their appearance is similar to the rib shape, these formations are here called "low density ribs". Their location remains approximately fixed in time and their density progressively decreases, as electron currents along the cavities evacuate them. They develop alon…
A stationary, two-dimensional, and non-driven antiparallel magnetic field configuration consisting of a central current film, flanked by two non-magnetic domains of electron inertial scale, resembling the Josephson model. Landau-Ginsburg conditions apply at the boundaries to the oppositely directed external magnetic fields, whose sources are located at infinity. On the microscopic level sufficiently large magnetic islands and X points form from exchange of small numbers of elementary magnetic fluxes without reference to any plasma instabilities which on these scales are purely electrostatic. T…
A prediction of the steady-state reconnection electric field in asymmetric reconnection is obtained by maximizing the reconnection rate as a function of the opening angle made by the upstream magnetic field on the weak magnetic field (magnetosheath) side. The prediction is within a factor of two of the widely examined asymmetric reconnection model [Cassak and Shay, Phys. Plasmas 14, 102114, 2007] in the collisionless limit, and they scale the same over a wide parameter regime. The previous model had the effective aspect ratio of the diffusion region as a free parameter, which simulations and o…
In this paper we develop a 3D Lattice-Boltzmann model that recovers in the continuous limit the two-fluids theory for plasmas, and consecuently includes the generalizated Ohm's law. The model reproduces the magnetic reconnection process just by given the right initial equlibrium conditions in the magnetotail, without any assumption on the resistivity in the diffusive region. In this model, the plasma is handled like two fluids with an interaction term, each one with distribution functions associated to a cubic lattice with 19 velocities (D3Q19). The electromagnetic fields are considered like a…
Magnetic reconnection (MR) in collisionless plasma is often attributed to the off-diagonal electron Reynolds stress, which can give rise to a large induction electric field in the reconnection region. However, in magneto-hydro-dynamics (MHD) simulations of MR, it is difficult to implement the full Reynolds stress, which is kinetic in nature. In this paper, we propose a theoretical model of effective resistivity from the first principle of particle dynamics. The derived theoretical formulation of the effective resistivity is verified by full particle-in-cell (PIC) simulations, and the correspon…
Although turbulence has been conjectured to be important for magnetic reconnection, still very little is known about its role in collisionless plasmas. Previous attempts to quantify the effect of turbulence on reconnection usually prescribed Alfvénic or other low-frequency fluctuations or investigated collisionless kinetic effects in just two-dimensional configurations and antiparallel magnetic fields. In view of this, we analyzed the kinetic turbulence self-generated by three-dimensional guide-field reconnection through force-free current sheets in frequency and wavenumber spaces, utilizing 3…
During magnetically dominated relativistic reconnection, inflowing plasma depletes the initial relativistic pressure at the x-line and collisionless plasma heating inside the diffusion region is insufficient to overcome this loss. The resulting pressure drop causes a collapse at the x-line, essentially a localization mechanism of the diffusion region necessary for fast reconnection. The extension of this low-pressure region further explains the bursty nature of anti-parallel reconnection because a once opened outflow exhaust can also collapse, which repeatedly triggers secondary tearing island…
We present a study of collisionless magnetic reconnection within the framework of full two-fluid MHD for a completely ionized hydrogen plasma, retaining the effects of the Hall current, electron pressure and electron inertia. We performed 2.5D simulations using a pseudo-spectral code with no dissipative effects. We check that the ideal invariants of the problem are conserved down to round-off errors. Our results show that the change in the topology of the magnetic field lines is exclusively due to the presence of electron inertia. The computed reconnection rates remain a fair fraction of the A…
System Size Dependence of Collisionless Reconnection Rate
University of Maryland, College Park（马里兰大学帕克分校）；National Aeronautics and Space Administration Goddard Space Flight Center（美国国家航空航天局戈达德太空飞行中心）；Princeton University（普林斯顿大学）
We report a newly found two-stage mechanism of electron acceleration near X-lines of 3D collisionless guide-field magnetic reconnection in the non-relativistic regime typical, e.g., for stellar coronae. We found that after electrons are first pre-accelerated during the linear growth of reconnection, they become additionally accelerated in the course of the nonlinear stage of 3D guide-field magnetic reconnection. This additional acceleration is due to the filamentation of electric and magnetic fields caused by streaming instabilities. In addition to enhanced parallel electric fields, the filame…
Non-Maxwellian electron velocity space distribution functions (EVDF) are useful signatures of plasma conditions and non-local consequences of collisionless magnetic reconnection. In the past, EVDFs were obtained mainly for antiparallel reconnection and under the influence of weak guide-fields in the direction perpendicular to the reconnection plane. EVDFs are, however, not well known, yet, for oblique (or component-) reconnection in dependence on stronger guide-magnetic fields and for the exhaust (outflow) region of reconnection away from the diffusion region. In view of the multi-spacecraft M…
Magnetic reconnection in curved spacetime is studied by adopting a general relativistic magnetohydrodynamic model that retains collisionless effects for both electron-ion and pair plasmas. A simple generalization of the standard Sweet-Parker model allows us to obtain the first order effects of the gravitational field of a rotating black hole. It is shown that the black hole rotation acts as to increase the length of azimuthal reconnection layers, per se leading to a decrease of the reconnection rate. However, when coupled to collisionless thermal-inertial effects, the net reconnection rate is …
Two cases of weakly and strongly stressed X-point collapse were considered. Here descriptors weakly and strongly refer to 20 % and 124 % unidirectional spatial compression of the X-point, respectively. In the weakly stressed case, the reconnection rate, defined as the out-of-plane electric field in the X-point (the magnetic null) normalised by the product of external magnetic field and Alfvén speeds, peaks at 0.11, with its average over 1.25 Alfvén times being 0.04. Electron energy distribution in the current sheet, at the high energy end of the spectrum, shows a power law distribution with th…
The kinetic features of plasmoid chain formation and evolution are investigated by two dimensional Particle-in-Cell simulations. Magnetic reconnection is initiated in multiple X points by the tearing instability. Plasmoids form and grow in size by continuously coalescing. Each chain plasmoid exhibits a strong out-of plane core magnetic field and an out-of-plane electron current that drives the coalescing process. The disappearance of the X points in the coalescence process are due to anti-reconnection, a magnetic reconnection where the plasma inflow and outflow are reversed with respect to the…
The nonlinear dynamics of collisionless reconnecting modes is investigated, in the framework of a three-dimensional gyrofluid model. This is the relevant regime of high-temperature plasmas, where reconnection is made possible by electron inertia and has higher growth rates than resistive reconnection. The presence of a strong guide field is assumed, in a background slab model, with Dirichlet boundary conditions in the direction of nonuniformity. Values of ion sound gyro-radius and electron collisionless skin depth much smaller than the current layer width are considered. Strong acceleration of…
Magnetic reconnection plays an important role in the release of magnetic energy and consequent energization of particles in collisionless plasmas. Energy transfer in collisionless magnetic reconnection is inherently a two-step process: reversible, collisionless energization of particles by the electric field, followed by collisional thermalization of that energy, leading to irreversible plasma heating. Gyrokinetic numerical simulations are used to explore the first step of electron energization, and we generate the first examples of field-particle correlation (FPC) signatures of electron energ…
Particle dynamics in the electron current layer in collisionless magnetic reconnection is investigated by using a particle-in-cell simulation. Electron motion and velocity distribution functions are studied by tracking self-consistent trajectories. New classes of electron orbits are discovered: figure-eight-shaped regular orbits inside the electron jet, noncrossing regular orbits on the jet flanks, noncrossing Speiser orbits, and nongyrotropic electrons in the downstream of the jet termination region. Properties of a super-Alfvénic outflow jet are attributed to an ensemble of electrons traveli…
Magnetic reconnection, breaking and reorganization of magnetic field topology, is a fundamental process for rapid release of magnetic energy into plasma particles that occurs pervasively throughout the universe. In most natural circumstances, the plasma properties on either side of the reconnection layer are asymmetric, in particular for the collision rates that are associated with a combination of density and temperature and critically determine the reconnection mechanism. To date, all laboratory experiments on magnetic reconnections have been limited to purely collisional or collisionless re…
Numerical studies implementing different versions of the collisionless Ohm's law have shown a reconnection rate insensitive to the nature of the non-ideal mechanism occuring at the X line, as soon as the Hall effect is operating. Consequently, the dissipation mechanism occurring in the vicinity of the reconnection site in collisionless systems is usually thought not to have a dynamical role beyond the violation of the frozen-in condition. The interpretation of recent studies have however led to the opposite conclusion that the electron scale dissipative processes play an important dynamical ro…
The kinetic features of secondary magnetic reconnection in a single flux rope undergoing internal kink instability are studied by means of three-dimensional Particle-in-Cell simulations. Several signatures of secondary magnetic reconnection are identified in the plane perpendicular to the flux rope: a quadrupolar electron and ion density structure and a bipolar Hall magnetic field develop in proximity of the reconnection region. The most intense electric fields form perpendicularly to the local magnetic field, and a reconnection electric field is identified in the plane perpendicular to the fl…
Using fully kinetic simulations, we study the suppression of asymmetric reconnection in the limit where the diamagnetic drift speed >> Alfven speed and the magnetic shear angle is moderate. We demonstrate that the slippage between electrons and the magnetic flux facilitates reconnection, and can even result in fast reconnection that lacks one of the outflow jets. Through comparing a case where the diamagnetic drift is supported by the temperature gradient with a companion case that has a density gradient instead, we identify a robust suppression mechanism. The drift of the x-line is slowed dow…
The propagation of reconnection signatures and their associated energy are examined using kinetic particle-in-cell simulations and Cluster satellite observations. It is found that the quadrupolar out-of-plane magnetic field near the separatrices is associated with a kinetic Alfvén wave. For magnetotail parameters, the parallel propagation of this wave is super-Alfvénic (V_parallel ~ 1500 - 5500 km/s) and generates substantial Poynting flux (S ~ 10^-5 - 10^-4 W/m^2) consistent with Cluster observations of magnetic reconnection. This Poynting flux substantially exceeds that due to frozen-in ion …
We introduce an extensible multi-fluid moment model in the context of collisionless magnetic reconnection. This model evolves full Maxwell equations, and simultaneously moments of the Vlasov-Maxwell equation for each species in the plasma. Effects like electron inertia and pressure gradient are self-consistently embedded in the resulting multi-fluid moment equations, without the need to explicitly solving a generalized Ohms's law. Two limits of the multi-fluid moment model are discussed, namely, the five-moment limit that evolves a scalar pressures for each species, and the ten-moment limit th…
It was recently proposed that the electron-frame dissipation measure, the energy transfer from the electromagnetic field to plasmas in the electron's rest frame, identifies the dissipation region of collisionless magnetic reconnection [Zenitani et al. Phys. Rev. Lett. 106, 195003 (2011)]. The measure is further applied to the electron-scale structures of antiparallel reconnection, by using two-dimensional particle-in-cell (PIC) simulations. The size of the central dissipation region is controlled by the electron-ion mass ratio, suggesting that electron physics is essential. A narrow electron j…
A mechanism for fast magnetic reconnection in collisionless plasma is studied for understanding sawtooth collapse in tokamak discharges. Nonlinear growth of the tearing mode driven by electron inertia is analytically estimated by invoking the energy principle for the first time. Decrease of potential energy in the nonlinear regime (where the island width exceeds the electron skin depth) is found to be steeper than in the linear regime, resulting in acceleration of the reconnection. Release of free energy by such ideal fluid motion leads to unsteady and strong convective flow, which theoretical…
