PHYSICAL REVIEW LETTERS
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Anomalous k−8=3 Spectrum in Electron Magnetohydrodynamic Turbulence ⊥ Romain Meyrand and Sébastien Galtier Laboratoire de Physique des Plasmas, Ecole Polytechnique, F-91128 Palaiseau Cedex, France (Received 30 May 2013; published 26 December 2013) Electron magnetohydrodynamic turbulence is investigated under the presence of a relatively strong external magnetic field b0 e∥ and through three-dimensional direct numerical simulations. Our study reveals −8=3 D D the emergence of a k⊥ scaling for the magnetic energy spectrum at scales kD ∥ ≤ k⊥ ≤ k⊥ , where k∥ and D k⊥ are, respectively, the typical largest dissipative scales along and transverse to the b0 direction. Unlike standard magnetohydrodynamic, this turbulence regime is characterized by filaments of electric currents parallel to b0 . The anomalous scaling is in agreement with a heuristic model in which the transfer in the parallel direction is negligible. Implications for solar wind turbulence are discussed. DOI: 10.1103/PhysRevLett.111.264501
PACS numbers: 47.27.ek, 47.27.Jv, 47.65.-d, 52.30.Cv
Introduction.—Plasma is the natural state of matter in the Universe. Since the beginning of space exploration in the 1960s, our knowledge of space plasma physics, and in particular about the solar wind, has greatly evolved [1]. Nowadays, the solar wind is seen as a highly turbulent medium with velocity and magnetic fluctuations characterized by extended power laws observed in the frequency range 10−5 ≤ f ≤ 102 Hz [2–4]. At variance with neutral fluids, the interplanetary plasma is not a scale-free medium. There exist several spatial and temporal characteristic scales, such as the ion and electron inertial lengths, Larmor radii, and cyclotron frequencies. As expected, when one probes the solar wind plasma towards high frequencies, the physical properties evolve and several breaks in the magnetic field fluctuation spectrum are detected [5–9]. At 1 AU, the first break detected at the frequency f 1 ∼ 0.5 Hz may be attributed to the decoupling between ions and electrons and defines, therefore, the scale at which one has to abandon the standard magnetohydrodynamic (MHD) model. The precise mechanism that drives the physics is still unclear: basically, the frequency range f 1 ≤ f ≤ 100 Hz is seen as either a dissipative range or a new turbulence regime. The difficulty resides, in particular, in the collisionless nature of the plasma and also its anisotropy. Both kinetic and fluid models are used to investigate plasma turbulence at subproton scales (see, e.g., Refs. [10–14]). In the latter case, we have to abandon the MHD approximation and consider a two-fluid description. The electron momentum equation, with all electron inertia terms neglected, gives the generalized ideal Ohm’s law (in SI units):
E þ ue × b þ
∇pe ¼ 0; ne
(1)
where E is the electric field, ue the electron velocity, b the magnetic field, n the electron density, e the magnitude of the electron charge, and pe the electron pressure. 0031-9007=13=111(26)=264501(5)
The introduction of Eq. (1) into the Maxwell-Faraday law, with the MHD momentum equation and, for example, the polytropic closure, leads to the so-called Hall MHD system in which the Hall effect becomes dominant at length scales smaller than the ion inertial length di (di ≡ c=ωpi , with c the speed of light and ωpi the ion plasma frequency) and time scales of the order, or shorter than, the ion cyclotron period ω−1 ci . Note that in Hall MHD, the electron pressure pe is assumed to be a scalar (this can be justified in the collisional limit or in the isothermal electron fluid approximation [15]). Hall MHD is, in fact, widely used to investigate several questions, such as the origin of the fast magnetic reconnection [16–18]. Three-dimensional Hall MHD turbulence is much more difficult to investigate numerically than pure MHD because the Hall effect brings a new kind of nonlinear term with a second-order derivative [19] which sometimes forces us to use lower-dimensional models [20,21]. Because of this difficulty, it is interesting to investigate first the limit for which the Hall effect is asymptotically large, i.e., the so-called electron MHD (EMHD) regime. In this limit, the ions can be considered as a motionless neutralizing background such that the electron flow determines entirely the electric current. Since the ions are static, electron compressibility corresponds to a violation of quasineutrality and reciprocally. Therefore, EMHD is only valid for large enough βe [22]. Direct numerical simulations (DNS) of isotropic EMHD showed without ambiguity that the magnetic energy spectrum scales like k−7=3 [23,24]. This scaling was explained by a heuristic model in the manner of Kolmogorov which turns out to be compatible dimensionally with an exact relation derived for third-order correlation functions [25]. It is widely believed that Hall MHD should exhibit the same (magnetic) spectrum. However, in the framework of 3D incompressible Hall MHD, a recent study has revealed the strong influence of the (left or right) polarity on the (magnetic) energy spectra [26] with the possibility to get different power laws at different scales, a tendency already observed with a low-dimensional
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model [21]. The case of a plasma embedded in an external magnetic field b0 is even more difficult to investigate because numerically it brings a strong constraint on the time step and physically it leads to anisotropy. Nevertheless, this situation has been investigated numerically in strong turbulence [10,19,27] and analytically in weak turbulence [28,29]. In the EMHD limit, 3D DNS reveal that a state of critical balance may be reached [30,31] when the external magnetic field b0 is of the order of the fluctuations. In this Letter, EMHD turbulence is studied through 3D DNS and under the influence of a relatively strong uniform b0 . A new turbulence regime is found for which the magnetic energy spectrum is compatible with k−8=3 . This result ⊥ might be interpreted with a heuristic model in which the transfer in the parallel direction is negligible. EMHD turbulence.—Our numerical simulation is based on the 3D incompressible EMHD equations: ∂ t b þ b0 ∂ ∥ ð∇ × bÞ ¼ −∇ × ½ð∇ × bÞ × b þ η3 ∇6 b; (2) ∇ · b ¼ 0;
(3)
where b is magnetic field normalized to a velocity pffiffiffiffiffiffiffiffiffiffiffiffiffi (b → μ0 nmi b, with mi the ion mass) and ∥ is the direction along the external magnetic field b0 ¼ b0 e∥ . (Note that we have fixed the dimensionless parameter di ¼ 1.) An inertial range can be clearly defined, in particular, if the dissipative term implies derivatives of higher degrees than that present in the nonlinear term, hence the hyperdiffusivity η3. From a numerical point of view, it is also a condition to avoid the emergence of numerical instabilities. From Eq. (2) we may define two time scales: i.e., the Alfvén-whistler wave time τw ∼ 1=ðk∥ kb0 Þ and the nonlinear time τnl ∼ 1=ðk2 bÞ. In strong turbulence, it is generally assumed that such a system reaches a critical balance regime [30,31] for which the relation τw ∼ τnl holds in the inertial range. Because of the anisotropy, we may assume k⊥ ≫ k∥ and rewrite the critical balance relation as k∥ b0 ∼ k⊥ b:
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direction as the external magnetic field. However, in our situation where δb=b0 ∼ 0.1, with δb the magnetic fluctuation, we may assume that the local and global directions are roughly the same [32]. Clearly, the approximation is much better for the perpendicular direction. Note that anisotropy of the local structure function is not a measure of anisotropy of the energy spectrum [33] which renders any local definition of a mean field nonuniversal and therefore not well adapted for statistical studies of turbulence. Numerical results.—We use a modified version of the TURBO code [26,34] in which we have implemented the Hall term. Equation (2) is solved using a pseudospectral solver with periodic boundary conditions in all three directions. We use 5122 × 128 collocation points, the lower resolution being along the b0 direction for which the transfer is reduced. The initial state consists of magnetic field fluctuations with random phases such that the energy is equal to 1=2 and localized at the largest scales of the system (see Fig. 1). The solver uses a fast Fourier transform algorithm for the space discretization and a third-order Williamson-Runge-Kutta method for the time advancement. The time step is computed automatically to be consistent with the Courant-Friedrich-Levy criterion; we have fixed the Courant-Friedrich-Levy coefficient equal to 0.4. The nonlinear terms are partially dealiased using a phase-shift method [35]. We fix η3 ¼ 10−11 and b0 ¼ 15 (a limit never explored before in previous similar DNS). External forcing is not included. Our analysis is made at a time where the mean dissipation rate reaches its first maximum. The perpendicular and parallel magnetic energy spectra are defined as 1 Eðk⊥ Þ ¼ hjbðkÞj2 iC ; 2
1 Eðk∥ Þ ¼ hjbðkÞj2 iD ; 2
(7)
where hi means an integration over a cylinder (C) or a disk (D) whose axis of symmetry is b0 . These spectra are reported in Fig. 1. Note that they have the same form at
(4)
A heuristic approach in the manner of Kolmogorov leads to the (reduced) spectral prediction: Eðk⊥ Þ ∼ ε2=3 k−7=3 ; ⊥
(5)
where ε is the mean energy transfer rate per unit mass. A combination of Eqs. (4) and (5) leads to k∥ ∼ k1=3 ⊥ , and finally to Eðk∥ Þ ∼ ε2=3 k−5 ∥ :
(6)
It is important to recall that in the critical balance phenomenology the symbol ∥ means the direction parallel to the local mean magnetic field which is not necessarily the same
FIG. 1. Parallel (dashed line) Eðk∥ Þ and perpendicular (solid line) Eðk⊥ Þ magnetic energy spectra compared, respectively, with −8=3 . The initial spectra are displayed in gray. Inset: The k−5 ∥ and k⊥ perpendicular spectrum compensated by k8=3 ⊥ is flat, whereas a −7=3 scaling should follow the power law k1=3 k⊥ ⊥ .
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the largest scales. It is not in contradiction with anisotropy whose presence is detected by the unequal extension of the inertial ranges which is the consequence of the direct cascade and the reduction of energy transfer towards the small parallel length scales. As expected for a critical balance regime, we find approximately Eðk∥ Þ ∼ k−5 ∥ in its inertial range, i.e., for k0∥ ≤ k∥ ≤ kD , with, respectively, k0∥ ∼ 4 ∥ D and k∥ ∼ 14 the typical integral and largest dissipative length scales along b0 . The situation is different for the perpendicular spectra for which a clear inertial range emerges only for approximately k⊥ ≥ kD ∥ . It is believed that it is due to the initial condition which fixes the spectrum for k⊥ ∈ ½1; 5: in this freely decaying simulation, an inertial range can only be developed beyond these wave numbers. We find Eðk⊥ Þ ∼ k−8=3 , which does not correspond to the ⊥ expected scaling for a critical balance regime. Plots of the bidimensional axisymmetric energy spectra Eðk⊥ ; k∥ Þ—which is linked to the magnetic energy E of the system by the double integral E ¼ ∬ Eðk⊥ ; k∥ Þdk⊥ dk∥ —at fixed k∥ are given in Fig. 2. A −8=3 scaling is found when k∥ ¼ 1–10, or when Eðk⊥ ; k∥ > 0Þ is considered, whereas the spectrum for the slow mode, Eðk⊥ ; k∥ ¼ 0Þ, behaves quite differently. The isocontours of the spectral energy are also given to demonstrate the anisotropic character of our simulation with a strong elongation of the isocontours in the perpendicular direction. The critical balance relation is also overlaid [36,37].
An important question that we may address in DNS is about the form of the structures produced in anisotropic EMHD turbulence. It is well known that MHD turbulence behaves differently from hydrodynamics with, in particular, the presence of current (and vorticity) sheets in the former case [38] whereas vorticity filaments are observed in the latter case. Surprisingly, this question has not received the same attention in 3D EMHD. In Fig. 3, we show the norm of the electric current fluctuations on our numerical box. Unlike the standard MHD picture, the plot reveal the presence of filamentary (or hairlike) structures which are mainly parallel to b0 . Heuristic model.—The scaling previously reported cannot be understood in the framework of the critical balance. However, as we will see, it may be seen as a particular solution of a more general formulation. Below, we summarize this idea which was mainly proposed in the past for MHD turbulence [39]. Since the problem is anisotropic, one needs to introduce the bidimensional axisymmetric magnetic energy spectrum: −β Eðk⊥ ; k∥ Þ ∼ k−α ⊥ k∥ :
We shall derive an equation for the coefficients α and β which will lead to a family of solutions for our problem. Note that the spectral form (8) is quite general: for example, it is used in weak anisotropic turbulence when we want to derive (with the Kuznetsov-Zakharov transform) exact power law solutions [29]. If we assume that the nonlinear transfer is mainly driven by local interactions, then the contribution of counterpropagating waves dominates [40,41] and we may define the rate of energy transfer per unit mass as −2αþ5 −2βþ1 k∥
b2 b4 k3⊥ k⊥ ε∼ 2 ∼ ∼ τnl =τw k∥ b0
FIG. 2. Top: Compensated bidimensional axisymmetric energy spectra at fixed k∥ . Spectra for k∥ ¼ 1–10 are in solid lines (respectively, from top to bottom), for k∥ ¼ 0 (dashed line), whereas Eðk⊥ ; k∥ > 0Þ is the dash-dotted line (it is shifted to the top to see it better). Bottom: Isocontours of ln½Eðk⊥ ; k∥ Þ with the critical balance relation k∥ ∝ k1=3 (dot-dashed line). ⊥ The dashed lines delimit the inertial ranges.
(8)
b0
;
(9)
FIG. 3 (color online). Norm of the electric current fluctuations. Unlike standard MHD, the plot reveals the presence of filamentary (or hairlike) structures along the external magnetic field.
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The comparison between Eqs. (10) and (13) leads eventually to the relation
which can be written as ð−2αþ5Þ=ð2β−1Þ
k∥ ∼ ðεb0 Þ1=ð−2βþ1Þ k⊥
:
(10)
A second relation is necessary to conclude; it is given by the balance relation χ¼
τw k⊥ b ∼ ∼ const; τnl k∥ b0
(11)
where the constant has to be understood as an order of magnitude which is not necessarily of order one; i.e., it can be much smaller or much larger than one. Although the value of a constant does not change a scaling relation, the order of magnitude does influence the result because it determines the phenomenology used. As we will see, in this way we can recover both the strong and the weak turbulence predictions. Figure 4 confirms that in the inertial range both εðk⊥ ; k∥ Þ and χðk⊥ ; k∥ Þ are reasonably constant at fixed k∥ and that χ is much less than unity everywhere. These two properties are at the basis of our heuristic interpretation. The introduction of expression (8) into (11) gives χ∼
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1 ð−αþ3Þ=2 −ðβþ1Þ=2 k k∥ ; b0 ⊥
(12)
which can also be written as ð−αþ3Þ=ðβþ1Þ
k∥ ∼ ðχb0 Þ−2=ðβþ1Þ k⊥
:
(13)
FIG. 4. Isocontours of ln½χðk⊥ ; k∥ Þ (top) and ln½εðk⊥ ; k∥ Þ (bottom). The dashed lines delimit the inertial ranges.
3α þ β ¼ 8;
(14)
which gives a family of solutions (α, β) for EMHD turbulence satisfying the relation k∥ ∼ k1=3 ⊥ . In particular, we see that relation (14) contains the strong (7=3, 1) [23] and the weak (5=2, 1=2) [41] turbulence predictions. It is believed that this general phenomenology may also describe the transition between weak and strong anisotropic EMHD turbulence as well as anomalous scalings like the one found in our DNS. Precisely, if we fix β ¼ 0, one obtains α ¼ 8=3, which corresponds apparently to our DNS. What is the meaning of β ¼ 0? By taking this value, we remove the spectral dependence of expression (8) in k∥ which may be interpreted as a way to consider the nonlinear transfer along the b0 direction negligible and therefore to suppress the presence of an inertial magnetic energy range in that direction. In other words, a k−8=3 ⊥ spectrum might be seen as a signature of quasibidimensional dynamics in 3D EMHD turbulence. In our case, the −8=3 D D scaling emerges at scales kD ∥ ≤ k⊥ ≤ k⊥ , with k⊥ ∼ 60 the typical largest dissipative scale transverse to the b0 direction. Discussion.—Our numerical study reveals the emergence of a new turbulence regime in EMHD which appears under the application of a relatively strong external magnetic field b0 . The intensity of b0 is such that (i) the inertial range in the parallel direction is localized at large scales (i.e., for k∥ < kD ∥ ) and (ii) the inertial range in the perpendicular direction extends beyond kD ∥ whose main characteristic is a departure from the classical k−7=3 energy spectrum ⊥ −8=3 to a steeper power law in k⊥ . This regime is also characterized by hairlike structures parallel to the external magnetic field b0 . A first understanding of our findings may be reached through the introduction of a heuristic model in which the parallel transfer is negligible. Our results are relevant for space plasma physics where subion scales are excited in the presence of a strong guide field such as for example, the magnetosphere of Saturn [42], the geomagnetic tail [43], or the solar wind, where a debate exists in the community (see, e.g., Ref. [44]) about the physical origin of anomalous magnetic spectra measured beyond the first spectral break at f 1 ∼ 0.1 Hz (at 1 AU). According to some recent observations, the data seem to converge towards a scaling in −8=3 [45]. We may think that kinetic effects could explain the observations. For example, the Landau damping is cited to explain power law indices greater (in absolute value) than 7=3 although no clear numerical demonstration has been made in a realistic situation mainly because of the difficulty to simulate a nonlinear kinetic system in the fully turbulence regime. For that reason, the numerical study of a model equation of kinetic-Alfvén turbulence has been done recently [46]. The anomalous scaling k−8=3 was also reported, but ⊥ it was associated with sheetlike structures. In this context
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the authors proposed a fractal model to explain their results and concluded that the −8=3 spectrum observed in solar wind turbulence may be the consequence of a pure nonlinear dynamics which can be described at the fluid level. The new results presented here reinforce this idea and, because of the discrepancy between the structures (sheets versus filaments), it suggests the possibility to make a distinction between the two models which will help further understanding of solar wind turbulence. More generally, our findings can be used to investigate other systems like MHD for which it is believed that the anisotropic turbulence regime falls either in the critical balance description [30] (see also [36] for a discussion) or in the weak turbulence one [47]. Previous numerical studies [32] have already shown the marginal presence of an anomalous scaling in k−7=3 , which is precisely the predic⊥ tion given by the heuristic model applied for MHD [39]. In light of this Letter, we think that it is important now to reinvestigate this question. This work was granted access to the HPC resources of CCRT/CINES/IDRIS under allocation 2012 [x2012046736] made by GENCI.
[1] W. Baumjohann and R. A. Treumann, Basic Space Plasma Physics (Imperial College Press, London, 1997). [2] V. Carbone, Space Sci. Rev. 172, 343 (2012). [3] M. L. Goldstein and D. A. Roberts, Phys. Plasmas 6, 4154 (1999). [4] S. Galtier, J. Low Temp. Phys. 145, 59 (2006). [5] C. W. Smith, K. Hamilton, B. Vasquez, and R. J. Leamon, Astrophys. J. 645, L85 (2006). [6] S. A. Markovskii, B. J. Vasquez, and C. W. Smith, Astrophys. J. 675, 1576 (2008). [7] O. Alexandrova, J. Saur, C. Lacombe, A. Mangeney, J. Mitchell, S. J. Schwartz, and P. Robert, Phys. Rev. Lett. 103, 165003 (2009). [8] F. Sahraoui, M. L. Goldstein, G. Belmont, P. Canu, and L. Rezeau, Phys. Rev. Lett. 105, 131101 (2010). [9] S. Bourouaine, O. Alexandrova, E. Marsch, and M. Maksimovic, Astrophys. J. 749, 102 (2012). [10] S. Ghosh and M. L. Goldstein, J. Plasma Phys. 57, 129 (1997). [11] Ö. D. Gürcan, X. Garbet, P. Hennequin, P. H. Diamond, A. Casati, and G. L. Falchetto, Phys. Rev. Lett. 102, 255002 (2009). [12] B. D. G Chandran, B. Li, B. N. Rogers, E. Quataert, and K. Germaschewski, Astrophys. J. 720, 503 (2010). [13] R. Meyrand and S. Galtier, Astrophys. J. 721, 1421 (2010). [14] L. Rudakov, M. Mithaiwala, G. Ganguli, and C. Crabtree, Phys. Plasmas 18, 012307 (2011). [15] A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert, and T. Tatsuno, Astrophys. J. Suppl. Ser. 182, 310 (2009). [16] A. Bhattacharjee, Annu. Rev. Astron. Astrophys. 42, 365 (2004).
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[17] Y. Ren, M. Yamada, S. Gerhardt, H. Ji, R. Kulsrud, and A. Kuritsyn, Phys. Rev. Lett. 95, 055003 (2005). [18] L. S. Shepherd and P. A. Cassak, Phys. Rev. Lett. 105, 015004 (2010). [19] P. D. Mininni, A. Alexakis, and A. Pouquet, J. Plasma Phys. 73, 377 (2007). [20] S. Ghosh, E. Siregar, D. A. Roberts, and M. L. Goldstein, J. Geophys. Res. 101, 2493 (1996). [21] S. Galtier and É. Buchlin, Astrophys. J. 656, 560 (2007). [22] D. Biskamp, Magnetic Reconnection in Plasmas (Cambridge University Press, Cambridge, England, 2000). [23] D. Biskamp, E. Schwarz, and J. F. Drake, Phys. Rev. Lett. 76, 1264 (1996). [24] C. Ng, A. Bhattacharjee, K. Germaschewski, and S. Galtier, Phys. Plasmas 10 1954 (2003). [25] S. Galtier, Phys. Rev. E 77, 015302(R) (2008). [26] R. Meyrand and S. Galtier, Phys. Rev. Lett. 109, 194501 (2012). [27] D. Gómez, S. M. Mahajan, and P. Dmitruk, Phys. Plasmas 15, 102303 (2008). [28] S. Galtier, J. Plasma Phys. 72, 721 (2006). [29] S. Galtier, Nonlinear Proc. Geophys. 16, 83 (2009). [30] P. Goldreich and S. Sridhar, Astrophys. J. 438, 763 (1995). [31] J. Cho and A. Lazarian, Astrophys. J. 615, L41 (2004). [32] B. Bigot and S. Galtier, Phys. Rev. E 83, 026405 (2011); B. Bigot, S. Galtier, and H. Politano, Phys. Rev. E 78, 066301 (2008). [33] W. H. Matthaeus, S. Servidio, P. Dimitruk, V. Carbone, S. Oughton, M. Wan, and K. T. Osman, Astrophys. J. 750, 103 (2012). [34] B. Teaca, M. K. Verma, B. Knaepen, and D. Carati, Phys. Rev. E 79, 046312 (2009). [35] G. S. Patterson and S. A. Orszag, Phys. Fluids 14, 2538 (1971). [36] R. Grappin and W. C. Muller, Phys. Rev. E 82, 026406 (2010). [37] A. Verdini and R. Grappin, Phys. Rev. Lett. 109, 025004 (2012). [38] P. D. Mininni, A. G. Pouquet, and D. C. Montgomery, Phys. Rev. Lett. 97, 244503 (2006). [39] S. Galtier, A. Pouquet, and A. Mangeney, Phys. Plasmas 12, 092310 (2005). [40] S. Dastgeer, A. Das, P. Kaw, and P. H. Diamond, Phys. Plasmas 7, 571 (2000). [41] S. Galtier and A. Bhattacharjee, Phys. Plasmas 10, 3065 (2003). [42] F. Akalin, D. A. Gurnett, T. F. Averkamp, A. M. Persoon, O. Santolik, W. S. Kurth, and G. B. Hospodarsky, Geophys. Res. Lett. 33, L20107 (2006). [43] A. Tenerani, O. Le Contel, F. Califano, P. Robert, D. Fontaine, N. Cornilleau-Wehrlin, and J. A. Sauvaud, J. Geophys. Res. 118, 6072 (2013). [44] W. H. Matthaeus, S. Servidio, and P. Dmitruk, Phys. Rev. Lett. 101, 149501 (2008). [45] O. Alexandrova, C. Lacombe, A. Mangeney, R. Grappin, and M. Maksimovic, Astrophys. J. 760, 121 (2012). [46] S. Boldyrev and J.-C. Perez, Astrophys. J. 758, L44 (2012). [47] S. Galtier, S. V. Nazarenko, A. C. Newell, and A. Pouquet, J. Plasma Phys. 63, 447 (2000).
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Anomalous k−8=3 Spectrum in Electron Magnetohydrodynamic Turbulence ⊥ Romain Meyrand and Sébastien Galtier Laboratoire de Physique des Plasmas, Ecole Polytechnique, F-91128 Palaiseau Cedex, France (Received 30 May 2013; published 26 December 2013) Electron magnetohydrodynamic turbulence is investigated under the presence of a relatively strong external magnetic field b0 e∥ and through three-dimensional direct numerical simulations. Our study reveals −8=3 D D the emergence of a k⊥ scaling for the magnetic energy spectrum at scales kD ∥ ≤ k⊥ ≤ k⊥ , where k∥ and D k⊥ are, respectively, the typical largest dissipative scales along and transverse to the b0 direction. Unlike standard magnetohydrodynamic, this turbulence regime is characterized by filaments of electric currents parallel to b0 . The anomalous scaling is in agreement with a heuristic model in which the transfer in the parallel direction is negligible. Implications for solar wind turbulence are discussed. DOI: 10.1103/PhysRevLett.111.264501
PACS numbers: 47.27.ek, 47.27.Jv, 47.65.-d, 52.30.Cv
Introduction.—Plasma is the natural state of matter in the Universe. Since the beginning of space exploration in the 1960s, our knowledge of space plasma physics, and in particular about the solar wind, has greatly evolved [1]. Nowadays, the solar wind is seen as a highly turbulent medium with velocity and magnetic fluctuations characterized by extended power laws observed in the frequency range 10−5 ≤ f ≤ 102 Hz [2–4]. At variance with neutral fluids, the interplanetary plasma is not a scale-free medium. There exist several spatial and temporal characteristic scales, such as the ion and electron inertial lengths, Larmor radii, and cyclotron frequencies. As expected, when one probes the solar wind plasma towards high frequencies, the physical properties evolve and several breaks in the magnetic field fluctuation spectrum are detected [5–9]. At 1 AU, the first break detected at the frequency f 1 ∼ 0.5 Hz may be attributed to the decoupling between ions and electrons and defines, therefore, the scale at which one has to abandon the standard magnetohydrodynamic (MHD) model. The precise mechanism that drives the physics is still unclear: basically, the frequency range f 1 ≤ f ≤ 100 Hz is seen as either a dissipative range or a new turbulence regime. The difficulty resides, in particular, in the collisionless nature of the plasma and also its anisotropy. Both kinetic and fluid models are used to investigate plasma turbulence at subproton scales (see, e.g., Refs. [10–14]). In the latter case, we have to abandon the MHD approximation and consider a two-fluid description. The electron momentum equation, with all electron inertia terms neglected, gives the generalized ideal Ohm’s law (in SI units):
E þ ue × b þ
∇pe ¼ 0; ne
(1)
where E is the electric field, ue the electron velocity, b the magnetic field, n the electron density, e the magnitude of the electron charge, and pe the electron pressure. 0031-9007=13=111(26)=264501(5)
The introduction of Eq. (1) into the Maxwell-Faraday law, with the MHD momentum equation and, for example, the polytropic closure, leads to the so-called Hall MHD system in which the Hall effect becomes dominant at length scales smaller than the ion inertial length di (di ≡ c=ωpi , with c the speed of light and ωpi the ion plasma frequency) and time scales of the order, or shorter than, the ion cyclotron period ω−1 ci . Note that in Hall MHD, the electron pressure pe is assumed to be a scalar (this can be justified in the collisional limit or in the isothermal electron fluid approximation [15]). Hall MHD is, in fact, widely used to investigate several questions, such as the origin of the fast magnetic reconnection [16–18]. Three-dimensional Hall MHD turbulence is much more difficult to investigate numerically than pure MHD because the Hall effect brings a new kind of nonlinear term with a second-order derivative [19] which sometimes forces us to use lower-dimensional models [20,21]. Because of this difficulty, it is interesting to investigate first the limit for which the Hall effect is asymptotically large, i.e., the so-called electron MHD (EMHD) regime. In this limit, the ions can be considered as a motionless neutralizing background such that the electron flow determines entirely the electric current. Since the ions are static, electron compressibility corresponds to a violation of quasineutrality and reciprocally. Therefore, EMHD is only valid for large enough βe [22]. Direct numerical simulations (DNS) of isotropic EMHD showed without ambiguity that the magnetic energy spectrum scales like k−7=3 [23,24]. This scaling was explained by a heuristic model in the manner of Kolmogorov which turns out to be compatible dimensionally with an exact relation derived for third-order correlation functions [25]. It is widely believed that Hall MHD should exhibit the same (magnetic) spectrum. However, in the framework of 3D incompressible Hall MHD, a recent study has revealed the strong influence of the (left or right) polarity on the (magnetic) energy spectra [26] with the possibility to get different power laws at different scales, a tendency already observed with a low-dimensional
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© 2013 American Physical Society
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model [21]. The case of a plasma embedded in an external magnetic field b0 is even more difficult to investigate because numerically it brings a strong constraint on the time step and physically it leads to anisotropy. Nevertheless, this situation has been investigated numerically in strong turbulence [10,19,27] and analytically in weak turbulence [28,29]. In the EMHD limit, 3D DNS reveal that a state of critical balance may be reached [30,31] when the external magnetic field b0 is of the order of the fluctuations. In this Letter, EMHD turbulence is studied through 3D DNS and under the influence of a relatively strong uniform b0 . A new turbulence regime is found for which the magnetic energy spectrum is compatible with k−8=3 . This result ⊥ might be interpreted with a heuristic model in which the transfer in the parallel direction is negligible. EMHD turbulence.—Our numerical simulation is based on the 3D incompressible EMHD equations: ∂ t b þ b0 ∂ ∥ ð∇ × bÞ ¼ −∇ × ½ð∇ × bÞ × b þ η3 ∇6 b; (2) ∇ · b ¼ 0;
(3)
where b is magnetic field normalized to a velocity pffiffiffiffiffiffiffiffiffiffiffiffiffi (b → μ0 nmi b, with mi the ion mass) and ∥ is the direction along the external magnetic field b0 ¼ b0 e∥ . (Note that we have fixed the dimensionless parameter di ¼ 1.) An inertial range can be clearly defined, in particular, if the dissipative term implies derivatives of higher degrees than that present in the nonlinear term, hence the hyperdiffusivity η3. From a numerical point of view, it is also a condition to avoid the emergence of numerical instabilities. From Eq. (2) we may define two time scales: i.e., the Alfvén-whistler wave time τw ∼ 1=ðk∥ kb0 Þ and the nonlinear time τnl ∼ 1=ðk2 bÞ. In strong turbulence, it is generally assumed that such a system reaches a critical balance regime [30,31] for which the relation τw ∼ τnl holds in the inertial range. Because of the anisotropy, we may assume k⊥ ≫ k∥ and rewrite the critical balance relation as k∥ b0 ∼ k⊥ b:
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direction as the external magnetic field. However, in our situation where δb=b0 ∼ 0.1, with δb the magnetic fluctuation, we may assume that the local and global directions are roughly the same [32]. Clearly, the approximation is much better for the perpendicular direction. Note that anisotropy of the local structure function is not a measure of anisotropy of the energy spectrum [33] which renders any local definition of a mean field nonuniversal and therefore not well adapted for statistical studies of turbulence. Numerical results.—We use a modified version of the TURBO code [26,34] in which we have implemented the Hall term. Equation (2) is solved using a pseudospectral solver with periodic boundary conditions in all three directions. We use 5122 × 128 collocation points, the lower resolution being along the b0 direction for which the transfer is reduced. The initial state consists of magnetic field fluctuations with random phases such that the energy is equal to 1=2 and localized at the largest scales of the system (see Fig. 1). The solver uses a fast Fourier transform algorithm for the space discretization and a third-order Williamson-Runge-Kutta method for the time advancement. The time step is computed automatically to be consistent with the Courant-Friedrich-Levy criterion; we have fixed the Courant-Friedrich-Levy coefficient equal to 0.4. The nonlinear terms are partially dealiased using a phase-shift method [35]. We fix η3 ¼ 10−11 and b0 ¼ 15 (a limit never explored before in previous similar DNS). External forcing is not included. Our analysis is made at a time where the mean dissipation rate reaches its first maximum. The perpendicular and parallel magnetic energy spectra are defined as 1 Eðk⊥ Þ ¼ hjbðkÞj2 iC ; 2
1 Eðk∥ Þ ¼ hjbðkÞj2 iD ; 2
(7)
where hi means an integration over a cylinder (C) or a disk (D) whose axis of symmetry is b0 . These spectra are reported in Fig. 1. Note that they have the same form at
(4)
A heuristic approach in the manner of Kolmogorov leads to the (reduced) spectral prediction: Eðk⊥ Þ ∼ ε2=3 k−7=3 ; ⊥
(5)
where ε is the mean energy transfer rate per unit mass. A combination of Eqs. (4) and (5) leads to k∥ ∼ k1=3 ⊥ , and finally to Eðk∥ Þ ∼ ε2=3 k−5 ∥ :
(6)
It is important to recall that in the critical balance phenomenology the symbol ∥ means the direction parallel to the local mean magnetic field which is not necessarily the same
FIG. 1. Parallel (dashed line) Eðk∥ Þ and perpendicular (solid line) Eðk⊥ Þ magnetic energy spectra compared, respectively, with −8=3 . The initial spectra are displayed in gray. Inset: The k−5 ∥ and k⊥ perpendicular spectrum compensated by k8=3 ⊥ is flat, whereas a −7=3 scaling should follow the power law k1=3 k⊥ ⊥ .
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the largest scales. It is not in contradiction with anisotropy whose presence is detected by the unequal extension of the inertial ranges which is the consequence of the direct cascade and the reduction of energy transfer towards the small parallel length scales. As expected for a critical balance regime, we find approximately Eðk∥ Þ ∼ k−5 ∥ in its inertial range, i.e., for k0∥ ≤ k∥ ≤ kD , with, respectively, k0∥ ∼ 4 ∥ D and k∥ ∼ 14 the typical integral and largest dissipative length scales along b0 . The situation is different for the perpendicular spectra for which a clear inertial range emerges only for approximately k⊥ ≥ kD ∥ . It is believed that it is due to the initial condition which fixes the spectrum for k⊥ ∈ ½1; 5: in this freely decaying simulation, an inertial range can only be developed beyond these wave numbers. We find Eðk⊥ Þ ∼ k−8=3 , which does not correspond to the ⊥ expected scaling for a critical balance regime. Plots of the bidimensional axisymmetric energy spectra Eðk⊥ ; k∥ Þ—which is linked to the magnetic energy E of the system by the double integral E ¼ ∬ Eðk⊥ ; k∥ Þdk⊥ dk∥ —at fixed k∥ are given in Fig. 2. A −8=3 scaling is found when k∥ ¼ 1–10, or when Eðk⊥ ; k∥ > 0Þ is considered, whereas the spectrum for the slow mode, Eðk⊥ ; k∥ ¼ 0Þ, behaves quite differently. The isocontours of the spectral energy are also given to demonstrate the anisotropic character of our simulation with a strong elongation of the isocontours in the perpendicular direction. The critical balance relation is also overlaid [36,37].
An important question that we may address in DNS is about the form of the structures produced in anisotropic EMHD turbulence. It is well known that MHD turbulence behaves differently from hydrodynamics with, in particular, the presence of current (and vorticity) sheets in the former case [38] whereas vorticity filaments are observed in the latter case. Surprisingly, this question has not received the same attention in 3D EMHD. In Fig. 3, we show the norm of the electric current fluctuations on our numerical box. Unlike the standard MHD picture, the plot reveal the presence of filamentary (or hairlike) structures which are mainly parallel to b0 . Heuristic model.—The scaling previously reported cannot be understood in the framework of the critical balance. However, as we will see, it may be seen as a particular solution of a more general formulation. Below, we summarize this idea which was mainly proposed in the past for MHD turbulence [39]. Since the problem is anisotropic, one needs to introduce the bidimensional axisymmetric magnetic energy spectrum: −β Eðk⊥ ; k∥ Þ ∼ k−α ⊥ k∥ :
We shall derive an equation for the coefficients α and β which will lead to a family of solutions for our problem. Note that the spectral form (8) is quite general: for example, it is used in weak anisotropic turbulence when we want to derive (with the Kuznetsov-Zakharov transform) exact power law solutions [29]. If we assume that the nonlinear transfer is mainly driven by local interactions, then the contribution of counterpropagating waves dominates [40,41] and we may define the rate of energy transfer per unit mass as −2αþ5 −2βþ1 k∥
b2 b4 k3⊥ k⊥ ε∼ 2 ∼ ∼ τnl =τw k∥ b0
FIG. 2. Top: Compensated bidimensional axisymmetric energy spectra at fixed k∥ . Spectra for k∥ ¼ 1–10 are in solid lines (respectively, from top to bottom), for k∥ ¼ 0 (dashed line), whereas Eðk⊥ ; k∥ > 0Þ is the dash-dotted line (it is shifted to the top to see it better). Bottom: Isocontours of ln½Eðk⊥ ; k∥ Þ with the critical balance relation k∥ ∝ k1=3 (dot-dashed line). ⊥ The dashed lines delimit the inertial ranges.
(8)
b0
;
(9)
FIG. 3 (color online). Norm of the electric current fluctuations. Unlike standard MHD, the plot reveals the presence of filamentary (or hairlike) structures along the external magnetic field.
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The comparison between Eqs. (10) and (13) leads eventually to the relation
which can be written as ð−2αþ5Þ=ð2β−1Þ
k∥ ∼ ðεb0 Þ1=ð−2βþ1Þ k⊥
:
(10)
A second relation is necessary to conclude; it is given by the balance relation χ¼
τw k⊥ b ∼ ∼ const; τnl k∥ b0
(11)
where the constant has to be understood as an order of magnitude which is not necessarily of order one; i.e., it can be much smaller or much larger than one. Although the value of a constant does not change a scaling relation, the order of magnitude does influence the result because it determines the phenomenology used. As we will see, in this way we can recover both the strong and the weak turbulence predictions. Figure 4 confirms that in the inertial range both εðk⊥ ; k∥ Þ and χðk⊥ ; k∥ Þ are reasonably constant at fixed k∥ and that χ is much less than unity everywhere. These two properties are at the basis of our heuristic interpretation. The introduction of expression (8) into (11) gives χ∼
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1 ð−αþ3Þ=2 −ðβþ1Þ=2 k k∥ ; b0 ⊥
(12)
which can also be written as ð−αþ3Þ=ðβþ1Þ
k∥ ∼ ðχb0 Þ−2=ðβþ1Þ k⊥
:
(13)
FIG. 4. Isocontours of ln½χðk⊥ ; k∥ Þ (top) and ln½εðk⊥ ; k∥ Þ (bottom). The dashed lines delimit the inertial ranges.
3α þ β ¼ 8;
(14)
which gives a family of solutions (α, β) for EMHD turbulence satisfying the relation k∥ ∼ k1=3 ⊥ . In particular, we see that relation (14) contains the strong (7=3, 1) [23] and the weak (5=2, 1=2) [41] turbulence predictions. It is believed that this general phenomenology may also describe the transition between weak and strong anisotropic EMHD turbulence as well as anomalous scalings like the one found in our DNS. Precisely, if we fix β ¼ 0, one obtains α ¼ 8=3, which corresponds apparently to our DNS. What is the meaning of β ¼ 0? By taking this value, we remove the spectral dependence of expression (8) in k∥ which may be interpreted as a way to consider the nonlinear transfer along the b0 direction negligible and therefore to suppress the presence of an inertial magnetic energy range in that direction. In other words, a k−8=3 ⊥ spectrum might be seen as a signature of quasibidimensional dynamics in 3D EMHD turbulence. In our case, the −8=3 D D scaling emerges at scales kD ∥ ≤ k⊥ ≤ k⊥ , with k⊥ ∼ 60 the typical largest dissipative scale transverse to the b0 direction. Discussion.—Our numerical study reveals the emergence of a new turbulence regime in EMHD which appears under the application of a relatively strong external magnetic field b0 . The intensity of b0 is such that (i) the inertial range in the parallel direction is localized at large scales (i.e., for k∥ < kD ∥ ) and (ii) the inertial range in the perpendicular direction extends beyond kD ∥ whose main characteristic is a departure from the classical k−7=3 energy spectrum ⊥ −8=3 to a steeper power law in k⊥ . This regime is also characterized by hairlike structures parallel to the external magnetic field b0 . A first understanding of our findings may be reached through the introduction of a heuristic model in which the parallel transfer is negligible. Our results are relevant for space plasma physics where subion scales are excited in the presence of a strong guide field such as for example, the magnetosphere of Saturn [42], the geomagnetic tail [43], or the solar wind, where a debate exists in the community (see, e.g., Ref. [44]) about the physical origin of anomalous magnetic spectra measured beyond the first spectral break at f 1 ∼ 0.1 Hz (at 1 AU). According to some recent observations, the data seem to converge towards a scaling in −8=3 [45]. We may think that kinetic effects could explain the observations. For example, the Landau damping is cited to explain power law indices greater (in absolute value) than 7=3 although no clear numerical demonstration has been made in a realistic situation mainly because of the difficulty to simulate a nonlinear kinetic system in the fully turbulence regime. For that reason, the numerical study of a model equation of kinetic-Alfvén turbulence has been done recently [46]. The anomalous scaling k−8=3 was also reported, but ⊥ it was associated with sheetlike structures. In this context
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the authors proposed a fractal model to explain their results and concluded that the −8=3 spectrum observed in solar wind turbulence may be the consequence of a pure nonlinear dynamics which can be described at the fluid level. The new results presented here reinforce this idea and, because of the discrepancy between the structures (sheets versus filaments), it suggests the possibility to make a distinction between the two models which will help further understanding of solar wind turbulence. More generally, our findings can be used to investigate other systems like MHD for which it is believed that the anisotropic turbulence regime falls either in the critical balance description [30] (see also [36] for a discussion) or in the weak turbulence one [47]. Previous numerical studies [32] have already shown the marginal presence of an anomalous scaling in k−7=3 , which is precisely the predic⊥ tion given by the heuristic model applied for MHD [39]. In light of this Letter, we think that it is important now to reinvestigate this question. This work was granted access to the HPC resources of CCRT/CINES/IDRIS under allocation 2012 [x2012046736] made by GENCI.
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