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Generation of magnetic holes in fully kinetic simulations of collisionless turbulence Vadim Roytershteyn1,2 , Homa Karimabadi1 and
Research
Aaron Roberts3
Cite this article: Roytershteyn V, Karimabadi H, Roberts A. 2015 Generation of magnetic holes in fully kinetic simulations of collisionless turbulence. Phil. Trans. R. Soc. A 373: 20140151. http://dx.doi.org/10.1098/rsta.2014.0151
1 SciberQuest, Inc., Del Mar, CA 92014, USA
Accepted: 9 February 2015 One contribution of 11 to a theme issue ‘Dissipation and heating in solar wind turbulence’. Subject Areas: plasma physics, astrophysics Keywords: plasma turbulence, magnetic holes, fully kinetic simulations Author for correspondence: Vadim Roytershteyn e-mail: [email protected]
2 Space Science Institute, Boulder, CO 80301, USA 3 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
The results of three-dimensional fully kinetic simulations of decaying turbulence with the amplitude of the fluctuating magnetic field comparable to that of the mean field are presented. Coherent structures in the form of localized depressions in the magnitude of the magnetic field are observed to form self-consistently in the simulations. These depressions bear considerable resemblance to the so-called magnetic holes frequently reported in spacecraft observations. The structures are pressurebalanced and tend to be aligned with the local magnetic field. In the smallest structures observed, the decrease in the magnetic field strength is compensated by an increase in the electron perpendicular pressure, such that the transverse size of these structures is comparable to the electron gyroradius inside the depression. It is suggested that the structures evolve self-consistently out of the depressions in the fluctuating magnetic field, rather than being the consequence of instability growth and saturation. This is confirmed by additional, small-scale simulations, including those with realistic mass ratio between protons and electrons.
1. Introduction
Electronic supplementary material is available at http://dx.doi.org/10.1098/rsta.2014.0151 or via http://rsta.royalsocietypublishing.org.
Localized depressions in the magnitude of the magnetic field, frequently referred to as magnetic holes (e.g. [1,2], see also review in [3]), have attracted considerable interest since their initial discovery. Despite their significant history, a number of important questions remain unanswered. For example, limited information is
2015 The Author(s) Published by the Royal Society. All rights reserved.
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The simulations described in the paper were performed using the general-purpose plasma simulation code VPIC [11], which solves the relativistic Vlasov–Maxwell system of equations using a particle-in-cell (PIC) algorithm [12]. Turbulence was seeded by imposing an initial perturbation on uniform magnetized plasma with density n0 . The unperturbed distribution for both electrons and ions is a Maxwellian with temperature T0 . The ratio of total plasma thermal energy to the reference magnetic energy is β0 = 16π n0 T0 /B20 = 0.5, where B0 is the strength of the uniform magnetic field applied in the z-direction, B = B0 ez . The simulations were conducted in a fully periodic three-dimensional domain of size L3 with the resolution of 20483 cells, where L ≈ 42di , ds = c/ωps is the inertial length for species s with mass ms , and ωps = (4π n0 e2 /ms )1/2 is the corresponding plasma frequency. The length of the domain is chosen in such a way that the maximum wavelength in each direction is kmin ρi = 0.075, where ρs = (2Ts /ms )1/2 /Ωs is the gyroradius and Ωs = eB0 /(ms c) is the gyrofrequency of species s. The ion-to-electron mass ratio is mi /me = 50 and the ratio ωpe /Ωce = 2. The average number of particles per cell per species is 150, corresponding to approximately 2.6 × 1012 total simulation particles. The time step is tωpe ≈ 0.08.
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available from observations on the three-dimensional shape of the structures or the plasma distributions. More significantly, the physical mechanisms leading to the generation of the magnetic holes are not entirely understood. These structures are often observed in high-β plasmas (where β is the ratio of plasma thermal energy to the magnetic energy density) and are frequently associated with proton anisotropy [4–6]. This has led to their interpretation as remnants of mirror instability [4]. Alternatively, magnetic holes have been proposed to result from steepening of large-amplitude Alfvén waves [7]. Setting aside the dynamics of magnetic hole formation, a number of soliton-like solutions with properties resembling magnetic holes have been introduced [8]. The majority of the observations reported magnetic holes of scales larger than the proton gyroradius, but smaller-scale structures have also been reported [9]. In this paper, we present the results of three-dimensional fully kinetic simulations of decaying plasma turbulence where long-lived coherent structures bearing many characteristics of magnetic holes are generated. The structures form at relatively early stages of the simulations and some of them persist for several Alfvén transit times. The simulations evolve a full Vlasov– Maxwell system of equations for both electrons and ions, thus providing a self-consistent model corresponding to the most complete description of collisionless plasmas. Many of the observed structures have scales comparable to the electron gyroradius within the depression, such that the electron kinetic physics plays an important role. The fully kinetic formalism provides an accurate description of such magnetic holes and enables their internal structure to be studied in detail. Additional small-scale three-dimensional simulations, including those with realistic proton-toelectron mass ratio, demonstrate that the structures can evolve self-consistently out of localized depressions in the magnetic field associated with turbulent fluctuations. We note that the short-wavelength range of collisionless plasma turbulence has attracted considerable interest owing to its role in the dissipation of cascading energy (see e.g. the review by Howes [10]). Several of the various dissipation mechanisms discussed in the literature are associated with coherent structures, such as current sheets. In this context, the results presented in this paper are of note because they demonstrate the possibility of formation of another class of coherent structures, distinct from the planar quasi-two-dimensional current sheets that are most commonly considered. The paper is organized as follows. The simulation parameters and the initial conditions are introduced in §2. The main results are presented in §3, including the general properties of the decaying turbulence (§3a) and the structure of the magnetic holes (§3b). The conclusions of the paper are summarized in §4.
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Figure 1. (a) A schematic of the initial conditions. The perturbations of magnetic field δB and of velocity δU are initialized inside a cube |kx | ≤ kmax , |ky | ≤ kmax , |kz | ≤ kmax√in k-space. The phase differences between U and B fluctuations are zero for all modes except those inside the sphere |k| ≤ 3kmin , where the phase difference is randomized. (b–d) Properties of magnetic field at t = 0. Shown are the profile of magnetic field amplitude |B|/B0 along a randomly chosen one-dimensional cut through the simulation domain (b); profiles of the magnetic field components Bx,y,z (red, green and blue curves, respectively) along the same cut (c); and the power spectrum estimate of the magnetic field along the same cut (d). (Online version in colour.)
x ) cos(k y + φ y ) × The initial perturbation is of the form δB = p=1,2 k δBp,k cos(kx x + φp,k y p,k z cos(kz z + φp,k ). Here, p denotes two orthogonal polarizations chosen such that δB1,k · B0 = 0,
k · δBp,k = 0 and δB1,k · δB2,k = 0. The amplitudes |δBp,k |2 ∝ k−3 are chosen to yield an overall power spectrum decaying as k−1 for a range of wavenumbers kmin ≤ |k| ≤ kmax ∼ d−1 i with equal power in both polarizations and the root-mean-squared amplitude |δB|2 = B20 , where . . . is x,y,z the average over the domain. The phase angles φp,k are random, but are chosen to satisfy the above-listed constraints on δB. In addition to the perturbation of the magnetic field, a velocity perturbation is loaded at time t = 0 by off-setting the Maxwellian particle distributions by velocity δU that has the same form as the magnetic perturbation. The relative phase angle between the velocity and magnetic field is equal to zero for all but the six modes corresponding to the lowest k in each direction (figure 1a). For the latter modes, the velocity and magnetic field perturbations are randomly phased. This mixture of Alfvénic and randomly phased perturbations yields the initial value of normalized cross-helicity σc ≈ 0.44. These initial conditions bear some resemblance to the large-scale perturbations observed in the solar wind, but also differ from them in several important ways. First, the k−1 portion of the spectrum extends up to k ∼ d−1 i , much further than in the solar wind. Second, the solar wind magnetic field is typically characterized by the condition |B0 + δB| ≈ const. The initial conditions used in the simulations do not enforce a similar constraint. We note that methods for constructing such magnetic fields have been proposed [13]. The structure of the magnetic field at t = 0 is illustrated in figure 1b–d, which show profiles and spectrum of B along a randomly, chosen one-dimensional cut through the simulation. It is immediately apparent that the imposed large perturbation creates multiple regions where |B/B0 | is rather small. As we discuss below, these imposed depressions in the magnetic field amplitude may play a role in the formation of the magnetic holes observed in the simulations.
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Figure 2. Evolution of the parallel (a) and perpendicular (b) spectra of the magnetic field. (c) The shape of the spectrum at t ≈ 4τA . Here, ‘t.a.’ refers to the time-averaged field and ‘s.f.’ is the spatially filtered field (see text for details). (Online version in colour.)
3. Results (a) Evolution of the initial perturbation After the simulation is initialized, the subsequent evolution is entirely self-consistent, with no applied external drive. Because the initial perturbation is a generic strong perturbation of the equilibrium that does not correspond to a solution of the Vlasov–Maxwell system, it quickly decays in time, generating fluctuations across all scales. The evolution of the spectrum of magnetic field fluctuations is illustrated in figure 2a,b. The parallel (with respect to B0 ) ˆ x , ky , k )|2 , where B(k ˆ x , ky , kz ) is the three-dimensional spectrum is defined as EB (k ) = kx ,ky |B(k fast Fourier transform (FFT) of B(x, y, z). The perpendicular spectrum is defined by EB⊥ (k⊥ ) = 2 2 2 1/2 = k . The ˆ ⊥ kx ,ky ,kz |B(kx , ky , kz )| , where the sum is restricted to the modes with (kx + ky ) shape of the perpendicular spectra remains mostly stationary after a relatively short transient √ interval t ≈ 0.2τA , where τA = L/VA = L 4π n0 mi /B0 . The parallel spectrum S undergoes a more pronounced evolution, becoming somewhat steeper in time. This is not surprising given that spectra in magnetized turbulence are typically anisotropic, whereas the spectrum of the imposed initial perturbation was isotropic. The shape of the spectra is further illustrated in figure 2c, which demonstrates the parallel and perpendicular spectra at time t ≈ 4τA . The spectra obtained from instantaneous magnetic field B(x, y, z, t) are characterized by an upturn at high k, which is a cumulative effect of the discrete particle noise inherent in the PIC algorithm. Several techniques for reducing the effects of the noise have been proposed, including spatial filtering and time averaging. Throughout this paper, we use the fields smoothed by both methods. The timeaveraged fields were obtained by performing in situ averaging over 100 steps, corresponding to a time interval of approximately 0.65 times the electron gyroperiod in the reference magnetic field B0 . This procedure averages out high-frequency oscillations and PIC sampling noise, but should have a minimal effect on the relatively low-frequency phenomena that are of interest here. The time-averaged fields are rather expensive to compute and are available only at later times in the simulation. In order to perform uniform analysis, some of the quantities below (e.g. kurtosis) were computed from the fields that were spatially filtered using a low-pass filter with a cut-off at [14]. The spectra of the smoothed magnetic fields obtained using both methods k = 3d−1 e are included in figure 2c for comparison. It is interesting to note that filtering high-k noise dramatically improves the spectrum at lower k, suggesting that the dominant pollution of parallel spectra is associated with sampling PIC noise at very high perpendicular wavenumbers. Both the parallel and the perpendicular spectra are well approximated by the expression S ∝
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k−α exp(−k/k0 ), with α = 2.8 for S and α ≈ 2.1 for S⊥ . The respective values of k0 are k0 de ≈ 0.3 and k0 de ≈ 0.5. In general, spectra of this shape are expected in a strongly dissipative system. Given the relatively low mass ratio employed by the present simulations, it is not surprising that almost the entire available range of scales could be fitted with a single function S ∝ k−α exp(−k/k0 ). Solar wind observations, in general, support the idea of dissipation at electron scales, although the details remain controversial [15–19]. Figure 3 demonstrates the global energy balance in the simulations by tracking in time the changes of the magnetic energy EB = (1/8π ) B2 d3 x, flow energy in each species EU,s = 2 3 (ms /2) Us d x, and the ‘random’ particle energy EP,s = 12 ( i Pii,s ) d3 x, where Pij,s is the pressure tensor. We note that ions receive the largest portion of the dissipated energy. This is in contrast to what is observed in a similar simulation with perturbation of lower energy and probably indicates that the partition of the dissipated energy depends on the strength of the initial perturbation, as first observed in two-dimensional simulations [20]. In figure 3, an estimate of the corresponding numerical heating was subtracted from all of the quantities. This estimate was obtained by performing a control simulation with parameters identical to the case presented here, but of smaller size and without the initial perturbation. We note that the estimate of total energy ET = EB + EU,i + EU,e + EP,i + EP,e remains constant in time with a good accuracy, indicating the validity of the approach. The turbulence that develops during the decay of the initial perturbation is characterized by the presence of coherent volume-filling current structures at multiple scales. This is illustrated in figure 4, which shows volume rendering of the current density in the simulation domain at four different times. An animation showing full time evolution is available in the electronic supplementary material. At early times, when the dissipation is relatively strong, the current density distribution is characterized by the presence of intense two-dimensional current sheets. At later times, the current density structures tend to be quasi-one-dimensional elongated strands. The intermittent nature of the magnetic field is further illustrated by the typical values of the kurtosis of magnetic field elements χ (s) = [B(x + se) · e]4 /[B(x + se) · e]2 2 . As illustrated in figure 5, the kurtosis peaks at small scales in the manner typical of both observations and other simulations. The peak value of kurtosis is observed at early times in the simulations, but only after the fluctuations develop sufficiently. In order to compute the kurtosis, the spatially filtered magnetic field was sampled along multiple trajectories in the simulation domain oriented in the y-direction. The filtering is crucial to recover correct behaviour across scales [14].
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(b) Magnetic holes An intriguing result of the simulations described here is the existence of long-lived isolated current structures (see figure 4 and the electronic supplementary material, movie S1). These structures are associated with strong depressions in the magnetic field and bear considerable
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Figure 7. Profiles of various quantities across the cut shown in panel (a) of figure 6: (a) magnetic field |B|/B0 and the angle of magnetic field rotation θ ; (b) electron parallel (red) and perpendicular (blue) temperatures; (c) ion parallel (red) and perpendicular (blue) temperatures; (d) electron (red) and ion (blue) densities; (e) x (red), y (blue) and z (green) components of the electron flow speed; (f ) the three components of the current density; (g) pressure balance, i.e. changes in magnetic pressure B2 /(8π B20 ) (black), ion perpendicular pressure (blue), electron perpendicular pressure (red) and −δP (green), where δP⊥ is the total perpendicular plasma pressure; (h) total perpendicular (blue) and parallel (red) plasma β together with the MHD mirror instability threshold g = β⊥ /β − 1/β⊥ multiplied by 20 to fit on the same scale (black). The profiles are taken along a cut in the y-direction indicated by the solid line in panel (a) of figure 6. All quantities shown are time-averaged over 100 steps as described in the main text. (Online version in colour.)
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Figure 8. Profiles of various quantities across a magnetic hole: (a) magnetic field |B|/B0 and the angle of magnetic field rotation θ ; (b) electron parallel (red) and perpendicular (blue) temperatures; (c) ion parallel (red) and perpendicular (blue) temperatures; (d) electron (red) and ion (blue) densities; (e) x (red), y (blue) and z (green) components of the electron flow speed; (f ) the three components of the current density; (g) pressure balance, i.e. changes in magnetic pressure B2 /(8π B20 ) (black), ion perpendicular pressure (blue), electron perpendicular pressure (red) and −δP (green), where δP⊥ is the total perpendicular plasma pressure; (h) total perpendicular (blue) and parallel (red) plasma β together with the MHD mirror instability threshold g = β⊥ /β − 1/β⊥ multiplied by 20 to fit on the same scale (black). The profiles are taken along a cut in the y-direction indicated by the solid line in panel (b) of figure 6. All quantities shown are time-averaged over 100 steps as described in the main text. (Online version in colour.)
resemblance to the magnetic holes frequently reported in spacecraft observations. Figure 6 demonstrates the geometrical shape and the distribution of magnetic field depressions at the later stage of the simulation t/τA ≈ 3.8. All of the depressions are field-aligned. Depressions with relatively small perpendicular size tend to be cylindrical, whereas the larger depressions have more complicated shapes (e.g. structures marked A and B in figure 6a). The profiles of various quantities along the cut passing through the two largest structures are shown in figure 7. The magnetic field does not appreciably change its direction through the structure A, whereas the angle of rotation through the structure B is θ ≈ 20◦ . Here, cos θ(l) = B(x0 + le) · B(x0 )/(|B(x0 + le)||B(x0 )|) is the angle of rotation of the magnetic field along the trajectory passing through a reference point x0 and parametrized by unit vector e. Similar to the magnetic holes observed in space, the structures are pressure-balanced, with both ion and electron perpendicular temperatures rising to compensate the depression in B2 . Owing to the rapid increase in plasma β, the anisotropy T⊥ /T inside the structures is close to the magnetohydrodynamic (MHD) threshold of mirror instability. This is highly surprising, given that the transverse size of the smallest structures is comparable to the electron gyroradius inside the structure, so that they are clearly outside of the regime where MHD considerations should apply. In what follows, we focus on the smaller structures, which apart from their size are distinguished by nearly symmetric shapes and relatively long lifetimes. A typical example of
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a structure with perpendicular size of approximately one ion inertial length is shown in figure 6b and figure 8. Most of the profiles show qualitative behaviour similar to those observed in larger structures, with the magnetic field rotation angle θ < 5◦ . However, the decrease of the magnetic field is mostly balanced by the electron perpendicular pressure in this case, whereas the ion temperature remains constant through the structure. The total anisotropy is still near marginal stability for the mirror threshold. We note that the perpendicular size of the structure corresponds to approximately one electron gyroradius defined with the peak temperature and the minimum magnetic field. This is in contrast to the larger structures described above, where the size is several electron gyroradii, but still below the ion gyroradius. In order to gain further insights into the structure of the magnetic holes observed in the simulations, we have conducted several additional three-dimensional simulations with smaller domain sizes, including those with realistic proton-to-electron mass ratio mi /me = 1836. The electron-scale depressions were also observed to form in these simulations. Here, we discuss one such simulation with the size of the spatial domain reduced by a factor of 4 in each direction compared to the simulation described in §2, but other parameters (including mi /me = 50) remaining the same. The much smaller size of the reduced simulation made it possible to obtain field and full particle data with a much higher temporal resolution, thus allowing detailed analysis of the distribution function inside the magnetic holes. Figure 9 demonstrates some of the salient features of the electron distribution. Figure 9a,d show the x–y and y–z plane cuts through a magnetic hole. The distribution function was measured at two locations indicated by white squares in panel (a). Figure 9b,c show the electron distribution function at the edge of the hole (location 1 in panel (a)). The distribution is very well approximated by a Maxwellian at vy < 0 (which corresponds to the negative angular direction in a cylindrical coordinate system located at the hole centre). In contrast, the distribution deviates from the
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Figure 10. (a) Profiles of magnetic field, vector potential Aθ and quasi-potential Φ along a line cut in the x-direction originating at the centre of the hole shown in figure 9. (b) Effective potential Ueff (r) for particles with several initial values of vθ at r = 4 (see the main text for details). The dashed horizontal lines correspond to the considered values of the perpendicular energy ⊥ . The particles move in the region defined by Ueff (r) ≤ ⊥ . (Online version in colour.) Maxwellian at positive vy , so that the azimuthal current is predominantly carried by particles with v0 vy 3v0 , where v0 = 2Te0 /me is the initial electron thermal velocity. In order to interpret the observed features of the distribution, consider a static axisymmetric magnetic field, which in cylindrical coordinates can be written as B = Bz ez . The field can be described by a single component of the vector potential Aθ (r), so that Bz = (1/r) d(rAθ )/dr. The Hamiltonian for a particle of species s is 2 1 qs 1 pθ − rAθ + qs Ψ , (p2r + p2z ) + (3.1) H= 2 2ms c 2ms r where pr = ms vr , pz = ms vz , pθ = ms r2 θ˙ + (qs /c)rAθ and Ψ (r) is the electrostatic potential. Because H, pz and pθ are constants of motion, the problem of finding the particle orbits reduces to solving the one-dimensional equation for radial motion in an effective potential Ueff , ms dr 2 + Ueff (r) = ⊥ (3.2) 2 dt and Ueff (r) =
2 1 qs rA p − + qs Ψ . θ θ c 2mr2
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The angular motion is then determined by ms r2 θ˙ = pθ − (qs /c)rAθ . Figure 10a shows the profile of r the magnetic field Bz , the vector potential Aθ and a quasi-potential Φ = − 0 Er dr along the cut in the x-direction originating at the centre of the hole. Because the magnetic field inside the structure r is mostly in the z-direction, Aθ (r) ≈ (1/r) 0 (rBz ) dr. We also assume that the electric field is mostly electrostatic, such that Φ ≈ Ψ . Because the electric field does not appear to play an important role in the radial confinement of particles inside the hole, the latter assumption is not essential. Figure 10b shows the effective potential Ueff for particles located at r ≈ 4de with vr = 0 and several values of vθ = vy . It is clear that the particles with vθ ≈ 2v0 have pθ ≈ 0, and execute relatively wide orbits (note that, ignoring the electrostatic potential, only particles with pθ ≈ 0 can reach r = 0). Moreover, it is easy to see that particles with pθ > 0 never change sign of the azimuthal ˙ which indicates that their orbits encircle the centre of the hole. We conclude that the velocity θ, diamagnetic current supporting the magnetic hole is carried predominantly by particles executing wide orbits inside the hole with characteristic size comparable to the transverse size of the hole (the characteristic length scale for the magnetic field gradient). The mechanism of the longitudinal confinement of particles in the hole is considerably more difficult to analyse, because this is essentially a two-dimensional problem. The small-scale
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Figure 11. Ion distribution functions at the centre of the hole (location ‘2’ in figure 9). (Online version in colour.)
magnetic holes have a mirror-like configuration of the magnetic field. If the magnetic moment 2 /(2B2 ) was conserved for all electrons, the distribution function at mid-plane would μe = me v⊥ √ be expected to show characteristic loss cones defined by |v /v⊥ | > Bmax /Bmin − 1. However, no obvious loss cones are observed in the distribution functions. This is not entirely surprising, given that a significant fraction of the electrons move on wide orbits of complicated shapes such that μe is not conserved. We leave detailed analysis of the longitudinal structure of the magnetic holes observed in our simulation to a future exploration, but point out that the ion distribution function f (vz ) shows a distinct two-beam structure throughout the hole (figure 11). The peaks appear to be related to the presence of an axial electric field E . In principle, such a distribution can become unstable if the peaks are sufficiently separated.
4. Discussion and conclusions In this paper, we reported on the formation of magnetic holes in fully kinetic three-dimensional simulations of decaying turbulence. The main findings are as follows: — The simulation demonstrates that long-lived coherent structures associated with strong local depressions of the magnetic field can evolve self-consistently in a turbulent environment. — The structures have many characteristics similar to the magnetic holes reported in spacecraft observations. They are pressure-balanced, field-aligned significant depressions of the magnitude of the magnetic field. — The smallest structures observed in the simulations are distinguished by nearly cylindrical shapes and relatively long lifetimes. They correspond to electron-scale toroidal current sheets where the decrease of the magnetic field is supported predominantly by the increase in the electron perpendicular pressure. The transverse size of these structures is comparable to the size of electron orbits inside the structure. — Owing to the increase in perpendicular pressure, the central regions of the depressions are slightly anisotropic. — The ion distribution function f (v ) inside the small-scale structures is found to have a distinct two-peak shape resulting from the presence of axial electric fields. — The magnetic holes in the simulation evolve out of slightly anisotropic ambient plasma with T > T⊥ for both electrons and ions, suggesting that mirror instability is not the origin of the observed structures. Instead, the structures evolve self-consistently out of magnetic field depressions associated with the strongly fluctuating magnetic field.
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is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the State of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. Additional simulations were performed on the Pleiades supercomputer provided by the NASA HEC program. Author contributions. H.K. proposed the concept of the simulation; A.R. proposed the initial conditions; V.R. executed the simulation and wrote the paper; A.R., V.R. and H.K. extensively discussed all aspects of the project and of the paper. Conflict of interests. We have no competing interests.
References 1. Turner J, Burlaga L, Ness N, Lemaire J. 1977 Magnetic holes in solar wind. J. Geophys. Res. Space Phys. 82, 1921–1924. (doi:10.1029/JA082i013p01921) 2. Burlaga L, Lemaire J. 1978 Interplanetary magnetic holes: theory. J. Geophys. Res. Space Phys. 83, 5157–5160. (doi:10.1029/JA083iA11p05157) 3. Tsurutani BT, Lakhina GS, Verkhoglyadova OP, Echer E, Guarnieri FL, Narita Y, Constantinescu DO. 2011 Magnetosheath and heliosheath mirror mode structures, interplanetary magnetic decreases, and linear magnetic decreases: differences and distinguishing features. J. Geophys. Res. Space Phys. 116, A02103. (doi:10.1029/2010JA015913) 4. Winterhalter D, Neugebauer M, Goldstein B, Smith E, Bame S, Balogh A. 1994 Ulysses field and plasma observations of magnetic holes in the solar-wind and their relation to mirrormode structures. J. Geophys. Res. Space Phys. 99, 23 371–23 381. (doi:10.1029/94JA01977) 5. Franz M, Burgess D, Horbury T. 2000 Magnetic field depressions in the solar wind. J. Geophys. Res. Space Phys. 105, 12 725–12 732. (doi:10.1029/2000JA900026) 6. Neugebauer M, Goldstein B, Winterhalter D, Smith E, MacDowall R, Gary S. 2001 Ion distributions in large magnetic holes in the fast solar wind. J. Geophys. Res. Space Phys. 106, 5635–5648. (doi:10.1029/2000JA000331) 7. Buti B, Tsurutani B, Neugebauer M, Goldstein B. 2001 Generation mechanism for magnetic holes in the solar wind. Geophys. Res. Lett. 28, 1355–1358. (doi:10.1029/2000GL012592) 8. Baumgärtel K. 1999 Soliton approach to magnetic holes. J. Geophys. Res. Space Phys. 104, 28 295– 28 308. (doi:10.1029/1999JA900393) 9. Sun WJ et al. 2012 Cluster and TC-1 observation of magnetic holes in the plasma sheet. Ann. Geophys. 30, 583–595. (doi:10.5194/angeo-30-583-2012) 10. Howes GG. 2015 A dynamical model of plasma turbulence in the solar wind. Proc. R. Soc. A 373, 20140145. (doi:10.1098/rsta.2014.0145)
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Data accessibility. The full simulation data are available upon request. Acknowledgements. We gratefully acknowledge support from NASA grant NNX14AI63G at SSI. This research
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We note that the simulations used in this work solve a Vlasov equation for each plasma species coupled to a full set of Maxwell’s equations, thus providing a self-consistent model with a physically accurate description of various kinetic effects. At the same time, the large-scale simulations necessarily use reduced parameters, such as rather small mass ratio mi /me = 50 and a modest value of ωpe /Ωce = 2. The presented simulations are close to the limits of current computing capabilities, and the full investigation of how the results change with the above parameters will only be possible with the advent of the next generation of supercomputers and/or advanced algorithms that can circumvent the limitations of the fully explicit PIC simulations presented here. Because the formation of the electron-scale structures can be clearly traced to diamagnetic effects associated with wide electron orbits inside the hole (see §3b), we expect that the main results concerning the existence and the structure of electron-scale magnetic holes will persist at realistic values of mi /me and ωpe /Ωce . Indeed, our small-scale three-dimensional simulations with realistic values of mi /me demonstrated the formation of electron-scale holes with nearly identical structure. While this paper was under review, we also became aware of recent two-dimensional simulations with realistic mass ratios demonstrating the formation of small-scale magnetic holes [21] in a manner quite similar to that discussed here.
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11. Bowers KJ, Albright BJ, Yin L, Bergen B, Kwan TJT. 2008 Ultrahigh performance threedimensional electromagnetic relativistic kinetic plasma simulation. Phys. Plasmas 15, 055703. (doi:10.1063/1.2840133) 12. Birdsall CK, Langdon AB 2004 Plasma physics via computer simulation. New York, NY: CRC Press. 13. Roberts DA. 2012 Construction of solar-wind-like magnetic fields. Phys. Rev. Lett. 109, 231102. (doi:10.1103/PhysRevLett.109.231102) 14. Wan M, Matthaeus WH, Karimabadi H, Roytershteyn V, Shay M, Wu P, Daughton W, Loring B, Chapman SC. 2012 Intermittent dissipation at kinetic scales in collisionless plasma turbulence. Phys. Rev. Lett. 109, 195001. (doi:10.1103/PhysRevLett.109.195001) 15. Sahraoui F, Goldstein ML, Robert P, Khotyaintsev YV. 2009 Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys. Rev. Lett. 102, 231102. (doi:10.1103/PhysRevLett.102.231102) 16. Alexandrova O, Lacombe C, Mangeney A, Grappin R, Maksimovic M. 2012 Solar wind turbulent spectrum at plasma kinetic scales. Astrophys. J. 760, 121. (doi:10.1088/0004-637X/760/2/121) 17. Sahraoui F, Huang SY, Belmont G, Goldstein ML, Rétino A, Robert P, De Patoul J. 2013 Scaling of the electron dissipation range of solar wind turbulence. Astrophys. J. 777, 15. (doi:10.1088/0004-637X/777/1/15) 18. Alexandrova O, Bale SD, Lacombe C. 2013 Comment on ‘Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale’. Phys. Rev. Lett. 111, 149001. (doi:10.1103/PhysRevLett.111.149001) 19. Sahraoui F, Robert P, Goldstein ML, Khotyaintsev YV. 2013 Sahraoui et al. reply. Phys. Rev. Lett. 111, 149002. (doi:10.1103/PhysRevLett.111.149002) 20. Wu P, Wan M, Matthaeus WH, Shay MA, Swisdak M. 2013 von Kármán energy decay and heating of protons and electrons in a kinetic turbulent plasma. Phys. Rev. Lett. 111, 121105. (doi:10.1103/PhysRevLett.111.121105) 21. Haynes CT, Burgess D, Camporeale E, Sundberg T 2015 Electron vortex magnetic holes: a nonlinear coherent plasma structure. Phys. Plasmas 22, 012309. (doi:10.1063/1.4906356)
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Generation of magnetic holes in fully kinetic simulations of collisionless turbulence Vadim Roytershteyn1,2 , Homa Karimabadi1 and
Research
Aaron Roberts3
Cite this article: Roytershteyn V, Karimabadi H, Roberts A. 2015 Generation of magnetic holes in fully kinetic simulations of collisionless turbulence. Phil. Trans. R. Soc. A 373: 20140151. http://dx.doi.org/10.1098/rsta.2014.0151
1 SciberQuest, Inc., Del Mar, CA 92014, USA
Accepted: 9 February 2015 One contribution of 11 to a theme issue ‘Dissipation and heating in solar wind turbulence’. Subject Areas: plasma physics, astrophysics Keywords: plasma turbulence, magnetic holes, fully kinetic simulations Author for correspondence: Vadim Roytershteyn e-mail: [email protected]
2 Space Science Institute, Boulder, CO 80301, USA 3 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
The results of three-dimensional fully kinetic simulations of decaying turbulence with the amplitude of the fluctuating magnetic field comparable to that of the mean field are presented. Coherent structures in the form of localized depressions in the magnitude of the magnetic field are observed to form self-consistently in the simulations. These depressions bear considerable resemblance to the so-called magnetic holes frequently reported in spacecraft observations. The structures are pressurebalanced and tend to be aligned with the local magnetic field. In the smallest structures observed, the decrease in the magnetic field strength is compensated by an increase in the electron perpendicular pressure, such that the transverse size of these structures is comparable to the electron gyroradius inside the depression. It is suggested that the structures evolve self-consistently out of the depressions in the fluctuating magnetic field, rather than being the consequence of instability growth and saturation. This is confirmed by additional, small-scale simulations, including those with realistic mass ratio between protons and electrons.
1. Introduction
Electronic supplementary material is available at http://dx.doi.org/10.1098/rsta.2014.0151 or via http://rsta.royalsocietypublishing.org.
Localized depressions in the magnitude of the magnetic field, frequently referred to as magnetic holes (e.g. [1,2], see also review in [3]), have attracted considerable interest since their initial discovery. Despite their significant history, a number of important questions remain unanswered. For example, limited information is
2015 The Author(s) Published by the Royal Society. All rights reserved.
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The simulations described in the paper were performed using the general-purpose plasma simulation code VPIC [11], which solves the relativistic Vlasov–Maxwell system of equations using a particle-in-cell (PIC) algorithm [12]. Turbulence was seeded by imposing an initial perturbation on uniform magnetized plasma with density n0 . The unperturbed distribution for both electrons and ions is a Maxwellian with temperature T0 . The ratio of total plasma thermal energy to the reference magnetic energy is β0 = 16π n0 T0 /B20 = 0.5, where B0 is the strength of the uniform magnetic field applied in the z-direction, B = B0 ez . The simulations were conducted in a fully periodic three-dimensional domain of size L3 with the resolution of 20483 cells, where L ≈ 42di , ds = c/ωps is the inertial length for species s with mass ms , and ωps = (4π n0 e2 /ms )1/2 is the corresponding plasma frequency. The length of the domain is chosen in such a way that the maximum wavelength in each direction is kmin ρi = 0.075, where ρs = (2Ts /ms )1/2 /Ωs is the gyroradius and Ωs = eB0 /(ms c) is the gyrofrequency of species s. The ion-to-electron mass ratio is mi /me = 50 and the ratio ωpe /Ωce = 2. The average number of particles per cell per species is 150, corresponding to approximately 2.6 × 1012 total simulation particles. The time step is tωpe ≈ 0.08.
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available from observations on the three-dimensional shape of the structures or the plasma distributions. More significantly, the physical mechanisms leading to the generation of the magnetic holes are not entirely understood. These structures are often observed in high-β plasmas (where β is the ratio of plasma thermal energy to the magnetic energy density) and are frequently associated with proton anisotropy [4–6]. This has led to their interpretation as remnants of mirror instability [4]. Alternatively, magnetic holes have been proposed to result from steepening of large-amplitude Alfvén waves [7]. Setting aside the dynamics of magnetic hole formation, a number of soliton-like solutions with properties resembling magnetic holes have been introduced [8]. The majority of the observations reported magnetic holes of scales larger than the proton gyroradius, but smaller-scale structures have also been reported [9]. In this paper, we present the results of three-dimensional fully kinetic simulations of decaying plasma turbulence where long-lived coherent structures bearing many characteristics of magnetic holes are generated. The structures form at relatively early stages of the simulations and some of them persist for several Alfvén transit times. The simulations evolve a full Vlasov– Maxwell system of equations for both electrons and ions, thus providing a self-consistent model corresponding to the most complete description of collisionless plasmas. Many of the observed structures have scales comparable to the electron gyroradius within the depression, such that the electron kinetic physics plays an important role. The fully kinetic formalism provides an accurate description of such magnetic holes and enables their internal structure to be studied in detail. Additional small-scale three-dimensional simulations, including those with realistic proton-toelectron mass ratio, demonstrate that the structures can evolve self-consistently out of localized depressions in the magnetic field associated with turbulent fluctuations. We note that the short-wavelength range of collisionless plasma turbulence has attracted considerable interest owing to its role in the dissipation of cascading energy (see e.g. the review by Howes [10]). Several of the various dissipation mechanisms discussed in the literature are associated with coherent structures, such as current sheets. In this context, the results presented in this paper are of note because they demonstrate the possibility of formation of another class of coherent structures, distinct from the planar quasi-two-dimensional current sheets that are most commonly considered. The paper is organized as follows. The simulation parameters and the initial conditions are introduced in §2. The main results are presented in §3, including the general properties of the decaying turbulence (§3a) and the structure of the magnetic holes (§3b). The conclusions of the paper are summarized in §4.
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Figure 1. (a) A schematic of the initial conditions. The perturbations of magnetic field δB and of velocity δU are initialized inside a cube |kx | ≤ kmax , |ky | ≤ kmax , |kz | ≤ kmax√in k-space. The phase differences between U and B fluctuations are zero for all modes except those inside the sphere |k| ≤ 3kmin , where the phase difference is randomized. (b–d) Properties of magnetic field at t = 0. Shown are the profile of magnetic field amplitude |B|/B0 along a randomly chosen one-dimensional cut through the simulation domain (b); profiles of the magnetic field components Bx,y,z (red, green and blue curves, respectively) along the same cut (c); and the power spectrum estimate of the magnetic field along the same cut (d). (Online version in colour.)
x ) cos(k y + φ y ) × The initial perturbation is of the form δB = p=1,2 k δBp,k cos(kx x + φp,k y p,k z cos(kz z + φp,k ). Here, p denotes two orthogonal polarizations chosen such that δB1,k · B0 = 0,
k · δBp,k = 0 and δB1,k · δB2,k = 0. The amplitudes |δBp,k |2 ∝ k−3 are chosen to yield an overall power spectrum decaying as k−1 for a range of wavenumbers kmin ≤ |k| ≤ kmax ∼ d−1 i with equal power in both polarizations and the root-mean-squared amplitude |δB|2 = B20 , where . . . is x,y,z the average over the domain. The phase angles φp,k are random, but are chosen to satisfy the above-listed constraints on δB. In addition to the perturbation of the magnetic field, a velocity perturbation is loaded at time t = 0 by off-setting the Maxwellian particle distributions by velocity δU that has the same form as the magnetic perturbation. The relative phase angle between the velocity and magnetic field is equal to zero for all but the six modes corresponding to the lowest k in each direction (figure 1a). For the latter modes, the velocity and magnetic field perturbations are randomly phased. This mixture of Alfvénic and randomly phased perturbations yields the initial value of normalized cross-helicity σc ≈ 0.44. These initial conditions bear some resemblance to the large-scale perturbations observed in the solar wind, but also differ from them in several important ways. First, the k−1 portion of the spectrum extends up to k ∼ d−1 i , much further than in the solar wind. Second, the solar wind magnetic field is typically characterized by the condition |B0 + δB| ≈ const. The initial conditions used in the simulations do not enforce a similar constraint. We note that methods for constructing such magnetic fields have been proposed [13]. The structure of the magnetic field at t = 0 is illustrated in figure 1b–d, which show profiles and spectrum of B along a randomly, chosen one-dimensional cut through the simulation. It is immediately apparent that the imposed large perturbation creates multiple regions where |B/B0 | is rather small. As we discuss below, these imposed depressions in the magnetic field amplitude may play a role in the formation of the magnetic holes observed in the simulations.
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3. Results (a) Evolution of the initial perturbation After the simulation is initialized, the subsequent evolution is entirely self-consistent, with no applied external drive. Because the initial perturbation is a generic strong perturbation of the equilibrium that does not correspond to a solution of the Vlasov–Maxwell system, it quickly decays in time, generating fluctuations across all scales. The evolution of the spectrum of magnetic field fluctuations is illustrated in figure 2a,b. The parallel (with respect to B0 ) ˆ x , ky , k )|2 , where B(k ˆ x , ky , kz ) is the three-dimensional spectrum is defined as EB (k ) = kx ,ky |B(k fast Fourier transform (FFT) of B(x, y, z). The perpendicular spectrum is defined by EB⊥ (k⊥ ) = 2 2 2 1/2 = k . The ˆ ⊥ kx ,ky ,kz |B(kx , ky , kz )| , where the sum is restricted to the modes with (kx + ky ) shape of the perpendicular spectra remains mostly stationary after a relatively short transient √ interval t ≈ 0.2τA , where τA = L/VA = L 4π n0 mi /B0 . The parallel spectrum S undergoes a more pronounced evolution, becoming somewhat steeper in time. This is not surprising given that spectra in magnetized turbulence are typically anisotropic, whereas the spectrum of the imposed initial perturbation was isotropic. The shape of the spectra is further illustrated in figure 2c, which demonstrates the parallel and perpendicular spectra at time t ≈ 4τA . The spectra obtained from instantaneous magnetic field B(x, y, z, t) are characterized by an upturn at high k, which is a cumulative effect of the discrete particle noise inherent in the PIC algorithm. Several techniques for reducing the effects of the noise have been proposed, including spatial filtering and time averaging. Throughout this paper, we use the fields smoothed by both methods. The timeaveraged fields were obtained by performing in situ averaging over 100 steps, corresponding to a time interval of approximately 0.65 times the electron gyroperiod in the reference magnetic field B0 . This procedure averages out high-frequency oscillations and PIC sampling noise, but should have a minimal effect on the relatively low-frequency phenomena that are of interest here. The time-averaged fields are rather expensive to compute and are available only at later times in the simulation. In order to perform uniform analysis, some of the quantities below (e.g. kurtosis) were computed from the fields that were spatially filtered using a low-pass filter with a cut-off at [14]. The spectra of the smoothed magnetic fields obtained using both methods k = 3d−1 e are included in figure 2c for comparison. It is interesting to note that filtering high-k noise dramatically improves the spectrum at lower k, suggesting that the dominant pollution of parallel spectra is associated with sampling PIC noise at very high perpendicular wavenumbers. Both the parallel and the perpendicular spectra are well approximated by the expression S ∝
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k−α exp(−k/k0 ), with α = 2.8 for S and α ≈ 2.1 for S⊥ . The respective values of k0 are k0 de ≈ 0.3 and k0 de ≈ 0.5. In general, spectra of this shape are expected in a strongly dissipative system. Given the relatively low mass ratio employed by the present simulations, it is not surprising that almost the entire available range of scales could be fitted with a single function S ∝ k−α exp(−k/k0 ). Solar wind observations, in general, support the idea of dissipation at electron scales, although the details remain controversial [15–19]. Figure 3 demonstrates the global energy balance in the simulations by tracking in time the changes of the magnetic energy EB = (1/8π ) B2 d3 x, flow energy in each species EU,s = 2 3 (ms /2) Us d x, and the ‘random’ particle energy EP,s = 12 ( i Pii,s ) d3 x, where Pij,s is the pressure tensor. We note that ions receive the largest portion of the dissipated energy. This is in contrast to what is observed in a similar simulation with perturbation of lower energy and probably indicates that the partition of the dissipated energy depends on the strength of the initial perturbation, as first observed in two-dimensional simulations [20]. In figure 3, an estimate of the corresponding numerical heating was subtracted from all of the quantities. This estimate was obtained by performing a control simulation with parameters identical to the case presented here, but of smaller size and without the initial perturbation. We note that the estimate of total energy ET = EB + EU,i + EU,e + EP,i + EP,e remains constant in time with a good accuracy, indicating the validity of the approach. The turbulence that develops during the decay of the initial perturbation is characterized by the presence of coherent volume-filling current structures at multiple scales. This is illustrated in figure 4, which shows volume rendering of the current density in the simulation domain at four different times. An animation showing full time evolution is available in the electronic supplementary material. At early times, when the dissipation is relatively strong, the current density distribution is characterized by the presence of intense two-dimensional current sheets. At later times, the current density structures tend to be quasi-one-dimensional elongated strands. The intermittent nature of the magnetic field is further illustrated by the typical values of the kurtosis of magnetic field elements χ (s) = [B(x + se) · e]4 /[B(x + se) · e]2 2 . As illustrated in figure 5, the kurtosis peaks at small scales in the manner typical of both observations and other simulations. The peak value of kurtosis is observed at early times in the simulations, but only after the fluctuations develop sufficiently. In order to compute the kurtosis, the spatially filtered magnetic field was sampled along multiple trajectories in the simulation domain oriented in the y-direction. The filtering is crucial to recover correct behaviour across scales [14].
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(b) Magnetic holes An intriguing result of the simulations described here is the existence of long-lived isolated current structures (see figure 4 and the electronic supplementary material, movie S1). These structures are associated with strong depressions in the magnetic field and bear considerable
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Figure 7. Profiles of various quantities across the cut shown in panel (a) of figure 6: (a) magnetic field |B|/B0 and the angle of magnetic field rotation θ ; (b) electron parallel (red) and perpendicular (blue) temperatures; (c) ion parallel (red) and perpendicular (blue) temperatures; (d) electron (red) and ion (blue) densities; (e) x (red), y (blue) and z (green) components of the electron flow speed; (f ) the three components of the current density; (g) pressure balance, i.e. changes in magnetic pressure B2 /(8π B20 ) (black), ion perpendicular pressure (blue), electron perpendicular pressure (red) and −δP (green), where δP⊥ is the total perpendicular plasma pressure; (h) total perpendicular (blue) and parallel (red) plasma β together with the MHD mirror instability threshold g = β⊥ /β − 1/β⊥ multiplied by 20 to fit on the same scale (black). The profiles are taken along a cut in the y-direction indicated by the solid line in panel (a) of figure 6. All quantities shown are time-averaged over 100 steps as described in the main text. (Online version in colour.)
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Figure 8. Profiles of various quantities across a magnetic hole: (a) magnetic field |B|/B0 and the angle of magnetic field rotation θ ; (b) electron parallel (red) and perpendicular (blue) temperatures; (c) ion parallel (red) and perpendicular (blue) temperatures; (d) electron (red) and ion (blue) densities; (e) x (red), y (blue) and z (green) components of the electron flow speed; (f ) the three components of the current density; (g) pressure balance, i.e. changes in magnetic pressure B2 /(8π B20 ) (black), ion perpendicular pressure (blue), electron perpendicular pressure (red) and −δP (green), where δP⊥ is the total perpendicular plasma pressure; (h) total perpendicular (blue) and parallel (red) plasma β together with the MHD mirror instability threshold g = β⊥ /β − 1/β⊥ multiplied by 20 to fit on the same scale (black). The profiles are taken along a cut in the y-direction indicated by the solid line in panel (b) of figure 6. All quantities shown are time-averaged over 100 steps as described in the main text. (Online version in colour.)
resemblance to the magnetic holes frequently reported in spacecraft observations. Figure 6 demonstrates the geometrical shape and the distribution of magnetic field depressions at the later stage of the simulation t/τA ≈ 3.8. All of the depressions are field-aligned. Depressions with relatively small perpendicular size tend to be cylindrical, whereas the larger depressions have more complicated shapes (e.g. structures marked A and B in figure 6a). The profiles of various quantities along the cut passing through the two largest structures are shown in figure 7. The magnetic field does not appreciably change its direction through the structure A, whereas the angle of rotation through the structure B is θ ≈ 20◦ . Here, cos θ(l) = B(x0 + le) · B(x0 )/(|B(x0 + le)||B(x0 )|) is the angle of rotation of the magnetic field along the trajectory passing through a reference point x0 and parametrized by unit vector e. Similar to the magnetic holes observed in space, the structures are pressure-balanced, with both ion and electron perpendicular temperatures rising to compensate the depression in B2 . Owing to the rapid increase in plasma β, the anisotropy T⊥ /T inside the structures is close to the magnetohydrodynamic (MHD) threshold of mirror instability. This is highly surprising, given that the transverse size of the smallest structures is comparable to the electron gyroradius inside the structure, so that they are clearly outside of the regime where MHD considerations should apply. In what follows, we focus on the smaller structures, which apart from their size are distinguished by nearly symmetric shapes and relatively long lifetimes. A typical example of
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a structure with perpendicular size of approximately one ion inertial length is shown in figure 6b and figure 8. Most of the profiles show qualitative behaviour similar to those observed in larger structures, with the magnetic field rotation angle θ < 5◦ . However, the decrease of the magnetic field is mostly balanced by the electron perpendicular pressure in this case, whereas the ion temperature remains constant through the structure. The total anisotropy is still near marginal stability for the mirror threshold. We note that the perpendicular size of the structure corresponds to approximately one electron gyroradius defined with the peak temperature and the minimum magnetic field. This is in contrast to the larger structures described above, where the size is several electron gyroradii, but still below the ion gyroradius. In order to gain further insights into the structure of the magnetic holes observed in the simulations, we have conducted several additional three-dimensional simulations with smaller domain sizes, including those with realistic proton-to-electron mass ratio mi /me = 1836. The electron-scale depressions were also observed to form in these simulations. Here, we discuss one such simulation with the size of the spatial domain reduced by a factor of 4 in each direction compared to the simulation described in §2, but other parameters (including mi /me = 50) remaining the same. The much smaller size of the reduced simulation made it possible to obtain field and full particle data with a much higher temporal resolution, thus allowing detailed analysis of the distribution function inside the magnetic holes. Figure 9 demonstrates some of the salient features of the electron distribution. Figure 9a,d show the x–y and y–z plane cuts through a magnetic hole. The distribution function was measured at two locations indicated by white squares in panel (a). Figure 9b,c show the electron distribution function at the edge of the hole (location 1 in panel (a)). The distribution is very well approximated by a Maxwellian at vy < 0 (which corresponds to the negative angular direction in a cylindrical coordinate system located at the hole centre). In contrast, the distribution deviates from the
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Figure 10. (a) Profiles of magnetic field, vector potential Aθ and quasi-potential Φ along a line cut in the x-direction originating at the centre of the hole shown in figure 9. (b) Effective potential Ueff (r) for particles with several initial values of vθ at r = 4 (see the main text for details). The dashed horizontal lines correspond to the considered values of the perpendicular energy ⊥ . The particles move in the region defined by Ueff (r) ≤ ⊥ . (Online version in colour.) Maxwellian at positive vy , so that the azimuthal current is predominantly carried by particles with v0 vy 3v0 , where v0 = 2Te0 /me is the initial electron thermal velocity. In order to interpret the observed features of the distribution, consider a static axisymmetric magnetic field, which in cylindrical coordinates can be written as B = Bz ez . The field can be described by a single component of the vector potential Aθ (r), so that Bz = (1/r) d(rAθ )/dr. The Hamiltonian for a particle of species s is 2 1 qs 1 pθ − rAθ + qs Ψ , (p2r + p2z ) + (3.1) H= 2 2ms c 2ms r where pr = ms vr , pz = ms vz , pθ = ms r2 θ˙ + (qs /c)rAθ and Ψ (r) is the electrostatic potential. Because H, pz and pθ are constants of motion, the problem of finding the particle orbits reduces to solving the one-dimensional equation for radial motion in an effective potential Ueff , ms dr 2 + Ueff (r) = ⊥ (3.2) 2 dt and Ueff (r) =
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The angular motion is then determined by ms r2 θ˙ = pθ − (qs /c)rAθ . Figure 10a shows the profile of r the magnetic field Bz , the vector potential Aθ and a quasi-potential Φ = − 0 Er dr along the cut in the x-direction originating at the centre of the hole. Because the magnetic field inside the structure r is mostly in the z-direction, Aθ (r) ≈ (1/r) 0 (rBz ) dr. We also assume that the electric field is mostly electrostatic, such that Φ ≈ Ψ . Because the electric field does not appear to play an important role in the radial confinement of particles inside the hole, the latter assumption is not essential. Figure 10b shows the effective potential Ueff for particles located at r ≈ 4de with vr = 0 and several values of vθ = vy . It is clear that the particles with vθ ≈ 2v0 have pθ ≈ 0, and execute relatively wide orbits (note that, ignoring the electrostatic potential, only particles with pθ ≈ 0 can reach r = 0). Moreover, it is easy to see that particles with pθ > 0 never change sign of the azimuthal ˙ which indicates that their orbits encircle the centre of the hole. We conclude that the velocity θ, diamagnetic current supporting the magnetic hole is carried predominantly by particles executing wide orbits inside the hole with characteristic size comparable to the transverse size of the hole (the characteristic length scale for the magnetic field gradient). The mechanism of the longitudinal confinement of particles in the hole is considerably more difficult to analyse, because this is essentially a two-dimensional problem. The small-scale
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magnetic holes have a mirror-like configuration of the magnetic field. If the magnetic moment 2 /(2B2 ) was conserved for all electrons, the distribution function at mid-plane would μe = me v⊥ √ be expected to show characteristic loss cones defined by |v /v⊥ | > Bmax /Bmin − 1. However, no obvious loss cones are observed in the distribution functions. This is not entirely surprising, given that a significant fraction of the electrons move on wide orbits of complicated shapes such that μe is not conserved. We leave detailed analysis of the longitudinal structure of the magnetic holes observed in our simulation to a future exploration, but point out that the ion distribution function f (vz ) shows a distinct two-beam structure throughout the hole (figure 11). The peaks appear to be related to the presence of an axial electric field E . In principle, such a distribution can become unstable if the peaks are sufficiently separated.
4. Discussion and conclusions In this paper, we reported on the formation of magnetic holes in fully kinetic three-dimensional simulations of decaying turbulence. The main findings are as follows: — The simulation demonstrates that long-lived coherent structures associated with strong local depressions of the magnetic field can evolve self-consistently in a turbulent environment. — The structures have many characteristics similar to the magnetic holes reported in spacecraft observations. They are pressure-balanced, field-aligned significant depressions of the magnitude of the magnetic field. — The smallest structures observed in the simulations are distinguished by nearly cylindrical shapes and relatively long lifetimes. They correspond to electron-scale toroidal current sheets where the decrease of the magnetic field is supported predominantly by the increase in the electron perpendicular pressure. The transverse size of these structures is comparable to the size of electron orbits inside the structure. — Owing to the increase in perpendicular pressure, the central regions of the depressions are slightly anisotropic. — The ion distribution function f (v ) inside the small-scale structures is found to have a distinct two-peak shape resulting from the presence of axial electric fields. — The magnetic holes in the simulation evolve out of slightly anisotropic ambient plasma with T > T⊥ for both electrons and ions, suggesting that mirror instability is not the origin of the observed structures. Instead, the structures evolve self-consistently out of magnetic field depressions associated with the strongly fluctuating magnetic field.
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is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the State of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. Additional simulations were performed on the Pleiades supercomputer provided by the NASA HEC program. Author contributions. H.K. proposed the concept of the simulation; A.R. proposed the initial conditions; V.R. executed the simulation and wrote the paper; A.R., V.R. and H.K. extensively discussed all aspects of the project and of the paper. Conflict of interests. We have no competing interests.
References 1. Turner J, Burlaga L, Ness N, Lemaire J. 1977 Magnetic holes in solar wind. J. Geophys. Res. Space Phys. 82, 1921–1924. (doi:10.1029/JA082i013p01921) 2. Burlaga L, Lemaire J. 1978 Interplanetary magnetic holes: theory. J. Geophys. Res. Space Phys. 83, 5157–5160. (doi:10.1029/JA083iA11p05157) 3. Tsurutani BT, Lakhina GS, Verkhoglyadova OP, Echer E, Guarnieri FL, Narita Y, Constantinescu DO. 2011 Magnetosheath and heliosheath mirror mode structures, interplanetary magnetic decreases, and linear magnetic decreases: differences and distinguishing features. J. Geophys. Res. Space Phys. 116, A02103. (doi:10.1029/2010JA015913) 4. Winterhalter D, Neugebauer M, Goldstein B, Smith E, Bame S, Balogh A. 1994 Ulysses field and plasma observations of magnetic holes in the solar-wind and their relation to mirrormode structures. J. Geophys. Res. Space Phys. 99, 23 371–23 381. (doi:10.1029/94JA01977) 5. Franz M, Burgess D, Horbury T. 2000 Magnetic field depressions in the solar wind. J. Geophys. Res. Space Phys. 105, 12 725–12 732. (doi:10.1029/2000JA900026) 6. Neugebauer M, Goldstein B, Winterhalter D, Smith E, MacDowall R, Gary S. 2001 Ion distributions in large magnetic holes in the fast solar wind. J. Geophys. Res. Space Phys. 106, 5635–5648. (doi:10.1029/2000JA000331) 7. Buti B, Tsurutani B, Neugebauer M, Goldstein B. 2001 Generation mechanism for magnetic holes in the solar wind. Geophys. Res. Lett. 28, 1355–1358. (doi:10.1029/2000GL012592) 8. Baumgärtel K. 1999 Soliton approach to magnetic holes. J. Geophys. Res. Space Phys. 104, 28 295– 28 308. (doi:10.1029/1999JA900393) 9. Sun WJ et al. 2012 Cluster and TC-1 observation of magnetic holes in the plasma sheet. Ann. Geophys. 30, 583–595. (doi:10.5194/angeo-30-583-2012) 10. Howes GG. 2015 A dynamical model of plasma turbulence in the solar wind. Proc. R. Soc. A 373, 20140145. (doi:10.1098/rsta.2014.0145)
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Data accessibility. The full simulation data are available upon request. Acknowledgements. We gratefully acknowledge support from NASA grant NNX14AI63G at SSI. This research
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We note that the simulations used in this work solve a Vlasov equation for each plasma species coupled to a full set of Maxwell’s equations, thus providing a self-consistent model with a physically accurate description of various kinetic effects. At the same time, the large-scale simulations necessarily use reduced parameters, such as rather small mass ratio mi /me = 50 and a modest value of ωpe /Ωce = 2. The presented simulations are close to the limits of current computing capabilities, and the full investigation of how the results change with the above parameters will only be possible with the advent of the next generation of supercomputers and/or advanced algorithms that can circumvent the limitations of the fully explicit PIC simulations presented here. Because the formation of the electron-scale structures can be clearly traced to diamagnetic effects associated with wide electron orbits inside the hole (see §3b), we expect that the main results concerning the existence and the structure of electron-scale magnetic holes will persist at realistic values of mi /me and ωpe /Ωce . Indeed, our small-scale three-dimensional simulations with realistic values of mi /me demonstrated the formation of electron-scale holes with nearly identical structure. While this paper was under review, we also became aware of recent two-dimensional simulations with realistic mass ratios demonstrating the formation of small-scale magnetic holes [21] in a manner quite similar to that discussed here.
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11. Bowers KJ, Albright BJ, Yin L, Bergen B, Kwan TJT. 2008 Ultrahigh performance threedimensional electromagnetic relativistic kinetic plasma simulation. Phys. Plasmas 15, 055703. (doi:10.1063/1.2840133) 12. Birdsall CK, Langdon AB 2004 Plasma physics via computer simulation. New York, NY: CRC Press. 13. Roberts DA. 2012 Construction of solar-wind-like magnetic fields. Phys. Rev. Lett. 109, 231102. (doi:10.1103/PhysRevLett.109.231102) 14. Wan M, Matthaeus WH, Karimabadi H, Roytershteyn V, Shay M, Wu P, Daughton W, Loring B, Chapman SC. 2012 Intermittent dissipation at kinetic scales in collisionless plasma turbulence. Phys. Rev. Lett. 109, 195001. (doi:10.1103/PhysRevLett.109.195001) 15. Sahraoui F, Goldstein ML, Robert P, Khotyaintsev YV. 2009 Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys. Rev. Lett. 102, 231102. (doi:10.1103/PhysRevLett.102.231102) 16. Alexandrova O, Lacombe C, Mangeney A, Grappin R, Maksimovic M. 2012 Solar wind turbulent spectrum at plasma kinetic scales. Astrophys. J. 760, 121. (doi:10.1088/0004-637X/760/2/121) 17. Sahraoui F, Huang SY, Belmont G, Goldstein ML, Rétino A, Robert P, De Patoul J. 2013 Scaling of the electron dissipation range of solar wind turbulence. Astrophys. J. 777, 15. (doi:10.1088/0004-637X/777/1/15) 18. Alexandrova O, Bale SD, Lacombe C. 2013 Comment on ‘Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale’. Phys. Rev. Lett. 111, 149001. (doi:10.1103/PhysRevLett.111.149001) 19. Sahraoui F, Robert P, Goldstein ML, Khotyaintsev YV. 2013 Sahraoui et al. reply. Phys. Rev. Lett. 111, 149002. (doi:10.1103/PhysRevLett.111.149002) 20. Wu P, Wan M, Matthaeus WH, Shay MA, Swisdak M. 2013 von Kármán energy decay and heating of protons and electrons in a kinetic turbulent plasma. Phys. Rev. Lett. 111, 121105. (doi:10.1103/PhysRevLett.111.121105) 21. Haynes CT, Burgess D, Camporeale E, Sundberg T 2015 Electron vortex magnetic holes: a nonlinear coherent plasma structure. Phys. Plasmas 22, 012309. (doi:10.1063/1.4906356)
