A Dynamical Model of Plasma Turbulence in the Solar Wind
A dynamical approach, rather than the usual statistical approach, is taken to explore the physical mechanisms underlying the nonlinear transfer of energy, the damping of the turbulent fluctuations, and the development of coherent structures in kinetic plasma turbulence. It is argued that the linear and nonlinear dynamics of Alfvén waves are responsible, at a very fundamental level, for some of the key qualitative features of plasma turbulence that distinguish it from hydrodynamic turbulence, including the anisotropic cascade of energy and the development of current sheets at small scales. The first dynamical model of kinetic turbulence in the weakly collisional solar wind plasma that combines self-consistently the physics of Alfvén waves with the development of small-scale current sheets is presented and its physical implications are discussed. This model leads to a simplified perspective on the nature of turbulence in a weakly collisional plasma: the nonlinear interactions responsible for the turbulent cascade of energy and the formation of current sheets are essentially fluid in nature, while the collisionless damping of the turbulent fluctuations and the energy injection by kinetic instabilities are essentially kinetic in nature.
Plasma Physics, Astrophysics, Solar System
Gregory G. Howes
Five centuries ago, Leonardo da Vinci first marveled at the intricate flow patterns arising in a turbulent cascade of water. But the first quantitative understanding of the physics of hydrodynamic turbulence had to wait until the seminal work by Kolmogorov in the 1940s, and humankind’s grasp of hydrodynamic turbulence remains yet incomplete. To advance the frontiers of our knowledge beyond the limits of our terrestrial environment, we strive to understand the impact of turbulence on the astrophysical plasmas that comprise most of the visible matter in the universe.
On the face of it, it would appear that unraveling the details of plasma turbulence, with electromagnetic forces in addition to the gas pressure force that occurs
in hydrodynamics, would be a much more challenging task than the study of turbulence in water and air. But the study of plasma turbulence is facilitated by one fundamental difference between a hydrodynamic fluid and a plasma: in a magnetized plasma, magnetic tension provides a restoring force that is absent in hydrodynamics. The importance of this distinction cannot be overstated, because magnetic tension supports the propagation of a fundamental wave mode, the Alfvén wave, that plays a governing role in plasma turbulence.
The linear Alfvénic response, due to magnetic tension, of the plasma
to applied perturbations provides a critical theoretical foothold in
the quest to understand the dynamics and energetics of plasma
turbulence. Specifically, in the limit of weak
Lacking this foothold of linear physics, hydrodynamics ruthlessly forces the physicist to resort almost immediately to a statistical approach to describe the turbulent evolution, with the typical focus on the power spectra, the development of intermittency, and the scaling of higher-order statistics. In the weakly collisional solar wind, researchers at the forefront strive to illuminate the physics of the nonlinear energy transfer to small scales, the kinetic mechanisms of dissipation of the turbulent fluctuations, and the resulting plasma heating. To illuminate these mechanisms, I contend that we must step beyond the usual statistical treatments, adopting instead a dynamical approach, to determine definitively the dominant physical processes at play in the evolution of plasma turbulence. Only with a dynamical understanding of both the nonlinear wave-wave interactions responsible for the turbulent cascade and the collisionless wave-particle interactions responsible for the dissipation will it be possible to construct a predictive theory of plasma turbulence and its effect on energy transport and plasma heating.
A significant fraction of the heliospheric turbulence research community appears to believe, likely based on the analogy with hydrodynamics, that linear physics properties are not relevant to strong plasma turbulence. On the contrary, there are numerous counterexamples in which linear physics properties have been shown to be relevant to strong plasma turbulence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and in some cases significant insights into the nature of magnetized plasma turbulence have been achieved by the exploitation of intuition from the linear physics properties [12, 13, 14, 15, 16, 17, 18, 19]. One particularly significant example is that the qualitative picture of the nonlinear energy transfer in the weak turbulence limit persists in the limit of strong turbulence . In fact, there is a long history of linear wave properties being explicitly or implicitly used to analyze spacecraft measurements and numerical simulations of plasma turbulence [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 5, 38, 39, 40, 41, 42, 43, 7, 44, 45].
In fact, as will be explained below, the linear and nonlinear dynamics of Alfvén waves are responsible, at a very fundamental level, for some of the key qualitative features of plasma turbulence that distinguish it from hydrodynamic turbulence, including the anisotropic cascade of energy and the development of current sheets at small scales. In the opinion of this author, to attempt a purely statistical investigation of plasma turbulence, analogous to the common approach in hydrodynamic turbulence, without exploiting the plasma physics of the turbulent fluctuations is bound to meet with moderate success at best.
The current frontier of research on plasma turbulence in the weakly collisional solar wind is framed by three fundamental questions:
What is the physical mechanism underlying the nonlinear transfer of energy from large to small scales? This question must be answered both for the MHD regime of the inertial range and the kinetic regime of the dissipation range.
What are the dominant physical mechanisms responsible for damping the turbulent electromagnetic fluctuations? Note that which mechanisms dominate may depend on both the plasma parameters and the characteristics of the turbulence, such as the scale and amplitude of the energy injection. These mechanisms determine the partitioning of dissipated turbulent energy into heat, or other energization, of the protons, electrons, and minor ions.
How do coherent structures arise from and/or affect both the nonlinear energy transfer and the dissipation mechanism? The concentration of dissipation in coherent structures, specifically current sheets, is well established by numerical simulations of plasma turbulence and is inferred from spacecraft measurements in the solar wind.
The ultimate goal of the study of space and astrophysical plasma turbulence is to develop a sufficiently detailed understanding of the physics to enable the construction of predictive models of the nonlinear energy transfer, the damping of the turbulent fluctuations, and the resulting heating of the plasma species, for example the heating of the solar corona . In addition to establishing a more thorough knowledge of the physics of the heliosphere—our home in the universe—it will provide a crucial foundation for an improved understanding of complex astrophysical phenomena in remote regions of the cosmos.
Below I present the first dynamical model of kinetic turbulence in the weakly collisional solar wind plasma that combines self-consistently the physics of Alfvén waves with the development of small-scale current sheets. Based on this model, I will address each of the questions above, as well as numerous subsidiary issues, as we follow the turbulent cascade of energy from large to small scales in the solar wind.
2 The Dynamics of the Turbulent Cascade
Incompressible MHD is one of the most simple, self-consistent descriptions of turbulent plasma dynamics. Although incompressible MHD lacks much of the rich physical behavior possible in the weakly collisional plasma conditions relevant to the solar wind, I argue here that incompressible MHD systems appear to contain the minimum number of physical ingredients necessary to yield the key qualitative features of plasma turbulence that distinguish it from hydrodynamic turbulence, specifically the anisotropic cascade of energy and dissipation dominantly occurring within current sheet structures. These key features persist as more physically complete plasma descriptions are adopted, yet do not occur in hydrodynamic turbulence.
Early research on incompressible MHD turbulence in the 1960s [48, 49] suggested that nonlinear interactions between counterpropagating Alfvén waves—or Alfvén wave collisions—support the turbulent cascade of energy from large to small scales. The incompressible MHD equations, expressed here in the symmetrized Elsasser form , are
and . Here is the Alfvén velocity due to the local mean magnetic field where the total magnetic field is . Note that the direction of the local mean magnetic field plays a key role in the physics of incompressible MHD turbulence , and here the coordinate system is chosen so that the unit vector direction is aligned with the mean field direction. is total pressure (thermal plus magnetic), is mass density, and are the Elsasser fields which represent waves that propagate up or down the mean magnetic field. The term governs the nonlinear interactions between counterpropagating Alfvén waves, denoted Alfvén wave collisions.
The strength of the nonlinearity in incompressible MHD turbulence may be characterized by the nonlinearity parameter, , the ratio of the magnitude of the nonlinear term to that of the linear term in \eqrefeq:elsasserpm. Weak incompressible MHD turbulence corresponds to the limit , while a state of strong incompressible MHD turbulence is characterized by [52, 53].
2.1 How is Energy Transferred to Small Scales?
Following significant previous studies on weak incompressible MHD turbulence [52, 54, 55, 56], the nonlinear energy transfer in Alfvén wave collisions has recently been solved analytically in the weakly nonlinear limit, . The key physics is illustrated in Fig. 1 for the interaction between two perpendicularly polarized, counterpropagating Alfvén waves with wavevectors and , denoted by the red circles in panel (a). In this section, and are positive constants, and the components of a particular wavevector are denoted , , and . The asymptotic solution is ordered by the small parameter . The lowest order nonlinear interaction creates an inherently nonlinear, purely magnetic mode with wavevector (green triangle). The nonlinear evolution is presented in panel (b) of Fig. 1, where Elsasser field amplitudes are represented by Elssaser potentials, , given by . The Elsasser potential is normalized by and time is normalized by the primary Alfvén wave frequency . In panel (b), the secondary mode (green) has and frequency , but does not grow secularly in time. At next order, the primary modes then interact with this secondary mode to transfer energy secularly (with an amplitude that grows linearly in time) to two nonlinearly generated Alfvén waves and (blue squares), where transfers energy to an Alfvén wave with , and transfers energy to . This process is the fundamental mechanism by which turbulence transfers energy from large to small scales .
This fundamental energy transfer mechanism in the weakly nonlinear limit, derived in the limit of incompressible MHD, has been confirmed numerically with gyrokinetic numerical simulations in the MHD regime  and verified experimentally in the laboratory [57, 58, 59], establishing Alfvén wave collisions as the fundamental building block of astrophysical plasma turbulence. This nonlinear mechanism is likely to be the physics leading to a non-zero third-moment of turbulent magnetic field measurements, a statistical measure often used to estimate the turbulent energy cascade rate .
The validation of the incompressible MHD analytical solution for the nonlinear evolution of Alfvén wave collisions with a kinetic numerical simulation has two important implications. First, the properties of incompressible MHD solution persist even under the weakly collisional plasma conditions relevant to realistic space and astrophysical plasmas. Second, the dynamical behavior of the turbulent cascade in the MHD regime, , in particular the nonlinear energy transfer, is adequately described by the simplified framework of incompressible MHD. This second point highlights that, even in a kinetic plasma, the nonlinear energy transfer is essentially fluid in nature, and can be modeled as a nonlinear wave-wave interaction. On the other hand, the physical mechanisms responsible for damping of the turbulent fluctuations under weakly collisional plasma conditions, as will be argued below, must be essentially kinetic in nature, dominated by wave-particle interactions.
I also note that the physics of Alfvén wave collisions highlights the fact that plasma turbulence is inherently three-dimensional [9, 17, 61]. The linear term in \eqrefeq:elsasserpm that governs the propagation of the Alfvén waves along the equilibrium magnetic field and is nonzero only when the parallel wavenumber , requiring the inclusion of the field-parallel dimension. And the vector nature of the nonlinear term requires variation in both of the directions perpendicular to the magnetic field for the nonlinearity to be nonzero. The manifestly three-dimensional nature of plasma turbulence not only applies to incompressible MHD plasmas, but persists as a general characteristic of the turbulence for more complex plasmas, such as compressible MHD plasmas or kinetic plasmas .
2.2 What Causes the Anisotropic Cascade in Turbulent Plasmas?
The anisotropic nature of the nonlinear energy transfer in Alfvénic plasma turbulence has long been recognized in laboratory plasmas [62, 63, 64] and in the solar wind , as well as in the results of numerical simulations [65, 66]. It is observed that energy is preferentially transferred to small scales perpendicular to the local magnetic field direction, leading to anisotropic, small-scale turbulent fluctuations that are elongated along the local magnetic field, characterized by the wavevector anisotropy .
Why is magnetized plasma turbulence anisotropic? This question can be answered rather simply, in the context of incompressible MHD, through physical intuition derived from the picture of nonlinear energy transfer presented above. Basically, the anisotropy arises as a direct consequence of two facts: (i) only counterpropagating Alfvén waves interact nonlinearly, and (ii) the nonlinear energy transfer is maximized for interactions between perpendicularly polarized Alfvén waves.
Consider the nonlinear interaction between two plane Alfvén waves with arbitrary wavevectors and . We adopt the convention that the wave frequency must be non-negative, , so that the propagation direction of the wave along the mean magnetic field is given by the sign of the parallel component of the wavevector. The nonlinear interaction between these two modes is nonzero only for counterpropagating Alfvén waves [48, 49, 17], so the parallel components and must be opposite in sign. A mode receiving energy via the nonlinear interaction between these two modes must satisfy , so its parallel component must have and . The nonlinear term in \eqrefeq:elsasserpm has the form . The magnitude of this nonlinear term is maximized when the perpendicular components of the wavevectors of the interacting Alfvén waves are orthogonal, , so we expect the nonlinear energy transfer to be dominated by interactions in which the colliding Alfvén waves are perpendicularly polarized . For this case of orthogonal perpendicular wavevector components, , the resulting magnitude of the perpendicular component of will satisfy and . In summary, the parallel wavevector component of Alfvénic fluctuations decreases or remains constant due to the fact that only counterpropagating Alfvén waves interact nonlinearly, while the perpendicular wavevector component of the fluctuations generally increases due to the dominance of nonlinear interactions between perpendicularly polarized Alfvén waves. Thus, the resulting nonlinear energy transfer is expected to be anisotropic, with energy preferentially flowing towards smaller scales in the perpendicular direction than in the parallel direction.
The argument presented above is similar to a previous explanation of anisotropic energy transfer [65, 66, 52, 54, 55, 56] which relies on the wavevector and frequency matching conditions in the weak turbulence limit for resonant interactions between Alfvén waves with wavevectors and . Retaining the convention of non-negative frequencies, the linear Alfvén wave frequency is given by . Therefore, the parallel component of the wavevector matching condition and the frequency matching condition lead to the equations and . Since only counterpropagating Alfvén waves interact nonlinearly, and thus and must be opposite in sign (or zero), these equations have a solution only if either or . This argument establishes the necessity of modes with in resonant nonlinear interactions in the weak turbulence limit, but elucidates neither the origin nor the nature of such modes. In summary, the parallel wavenumber does not increase through nonlinear interactions in weak incompressible MHD turbulence, while the perpendicular wavenumber generally does increase, yielding an anisotropic transfer of energy to modes with .
The analytical solution for Alfvén wave collisions in the weakly nonlinear limit presented in §2.1 can be used to illustrate this previous explanation for the anisotropic energy transfer and to clarify the origin and role of the modes. Using the wavevector notation from §2.1, where represents the wavevector component parallel to the local mean magnetic field, the nonlinear interaction between counterpropagating plane Alfvén waves given by wavevectors and may lead to energy transfer to two other Fourier modes and . The detailed analytical solution  demonstrates that the resonant interactions that lead to the secular transfer of energy to smaller scales are mediated by a nonlinearly generated mode with , so this result singles out as the key mode mediating the nonlinear energy transfer. For this particular problem, the primary counterpropagating Alfvén waves and each then interact nonlinearly with this self-consistently generated mode to transfer energy resonantly to Alfvén waves with wavevectors and . It is these two subsequent interactions that correspond directly to the previous explanation [65, 66, 52, 54, 55, 56] of anisotropic energy transfer in the weak turbulence limit. In the presence of pre-existing energy in modes, this energy transfer corresponds to a dominant three-wave interaction. In the absence of such pre-existing modes, it is the self-consistently generated mode that mediates the energy transfer. In this case, there is no energy transfer via three-wave interactions, but rather the combined interactions and , together equivalent to a four-wave interaction, constitute the dominant mechanism of nonlinear energy transfer. Note that the mode has no associated velocity fluctuation, , but does include a non-zero perpendicular magnetic field fluctuation, , so this mode has both and , and therefore interacts with Alfvén waves propagating in either direction along the magnetic field. Physically, the mode can be interpreted as a shear in the magnetic field that the Alfvén waves traveling in either direction attempt to follow.
These arguments demonstrate that, in the weak turbulence limit, , no parallel cascade occurs, yielding a maximally anisotropic
energy transfer only to smaller perpendicular scales. Does the
physical picture of energy transfer mediated by self-consistently
generated modes persist in the strong turbulence limit, ? Let us consider an Alfvén wave collision problem equivalent
to the weakly nonlinear case presented in §2.1, but
with the initial Alfvén wave amplitudes increased to yield a strongly
nonlinear case with . Simulations of Alfvén wave collisions
in this strong turbulence limit  demonstrate that
the energy transfer in perpendicular Fourier space flows primarily to
modes along the three diagonal lines represented in panel (a) of
Fig. 1. Due to the wavevector matching conditions for
this particular problem, the parallel wavenumber
In summary, the anisotropic cascade of energy in incompressible MHD turbulence is a consequence of the two facts that only counterpropagating Alfvén waves interact nonlinearly and that the nonlinear term is greatest for interactions between perpendicularly polarized Alfvén waves. Note this explanation of the anisotropy in terms of the linear and nonlinear properties of the turbulent fluctuations differs substantially from a statistical argument in this issue  relying on the coupling of third-order to second-order correlations in the turbulence. It is an open question why the property of anisotropic energy transfer appears to persist under less restrictive plasma conditions than incompressible MHD (in particular for the dispersive modes at scales below the ion Larmor radius) which allow for copropagating modes to interact nonlinearly. An analytical solution of the nonlinear energy transfer in the dispersive regime of kinetic Alfvén waves may help to answer this open question.
2.3 What is Low-Frequency Turbulence in a Weakly Collisional Plasma?
As a relevant aside, it is worthwhile to consider the nature of
A Boltzmann equation for each plasma species together with Maxwell’s equations form the closed set of Maxwell-Boltzmann equations governing the dynamics of a kinetic plasma. Since Maxwell’s equations are linear, the only nonlinearities appearing in the Maxwell-Boltzmann equations occur in the electromagnetic Lorentz force term and the collisional term in \eqrefeq:boltzmann. Under the weakly collisional conditions relevant to the solar wind, the collisional term is subdominant, and thus the physics of solar wind turbulence is controlled by the electromagnetic Lorentz force on the particles.
The charge density and current density give rise to the electromagnetic fields through Maxwell’s equations, and therefore the fields are entirely determined by the two lowest-order moments of the ion and electron distribution functions. Thus, the forces in weakly collisional plasma turbulence are essentially fluid in nature, since they depend on only the low-order moments. For the non-relativistic conditions of solar wind turbulence, the plasma fluctuations are quasineutral, with typical charge density fluctuations, , normalized by the total ion charge density, , having an order of magnitude . Therefore, with negligible charge density fluctuations, the electromagnetic fields mediating the low-frequency turbulence in the solar wind are determined entirely by the current density, given by the first moments of the ion and electron distribution functions.
The energy in a volume of plasma is given by
the sum of the electromagnetic field energy and kinetic energy of the plasma particles . Note that the kinetic energy density of a plasma species is given by , and we may define a “kinetic” temperature (expressed in units of energy) for a monatomic plasma species in three spatial dimensions by
where is the bulk fluid velocity of species , given by the first moment of the distribution function. Therefore, the energy in a kinetic plasma may be expressed by
Note that, if the distribution function for species is Maxwellian (corresponding to local thermodynamic equilibrium), then the kinetic temperature is equal to the thermodynamic temperature of that species, . In this case, the last term in \eqrefeq:vlasov_energy2 corresponds entirely to the thermal energy. Energy cannot be extracted from a thermal distribution without lowering the entropy, so this energy is thermodynamically inaccessible. But, if the distribution function deviates from a Maxwellian, as will generally be the case for a turbulent weakly collisional plasma, then the last term contains not only thermal energy but also non-thermal free energy associated with the deviations from a Maxwellian distribution. These deviations generally have the form of small-scale structure in velocity space. In summary, the first term in \eqrefeq:vlasov_energy2 represents the electromagnetic fluctuation energy, the second term represents the kinetic energy of the bulk fluid velocity, and the third term includes both non-thermal free energy in the distribution function as well as thermal energy.
The energy given in \eqrefeq:vlasov_energy2 is conserved by the Maxwell-Boltzmann equations independent of collisionality. It is worthwhile considering the various physical mechanisms that can lead to the transfer of energy among the different terms in \eqrefeq:vlasov_energy2. The low-frequency linear wave response of the kinetic plasma leads to the continual exchange of energy between the electromagnetic energy and bulk kinetic energy—the magnetic tension in Alfvén waves, for example, leads to the periodic transfer of energy between the magnetic energy and the kinetic energy of the bulk plasma motion, an exchange that is purely fluid in nature. Collisionless wave-particle interactions, such as Landau damping, lead to the transfer of energy from the electromagnetic fields into non-thermal free energy in the distribution function. Conversely, unstable distribution functions can lead to the transfer of energy from the non-thermal free energy in the velocity distributions to electromagnetic fluctuations via collisionless wave-particle interactions, such as kinetic temperature anisotropy instabilities . Both of these physical mechanisms of energy transfer by wave-particle interactions are essentially kinetic in nature. In addition to conserving energy, all of the processes mentioned above also conserve entropy since the collisionless Boltzmann equation (or the Vlasov equation) conserves entropy. Collisions are the final physical mechanism that converts non-thermal free energy in the distribution function to thermal energy, a process that increases the entropy of the plasma, thereby realizing irreversible thermodynamic heating of the plasma. Collisions, of course, do not transfer energy between terms in \eqrefeq:vlasov_energy2, but merely convert non-thermal to thermal energy within the last term in the equation.
Based on these considerations, here I suggest a practical definition for what constitutes the turbulence in a kinetic plasma. The turbulence is represented by the first moments of the distribution functions, specifically the electromagnetic fluctuations (determined by the current density) and the bulk fluid velocities of the plasma species, corresponding to the first two terms in \eqrefeq:vlasov_energy2. The nonlinear interactions underlying the turbulent cascade of energy from large to small scales are mediated by the electromagnetic fields, and since these fields depend only on the first moments of the ion and electron distribution functions, the nonlinear dynamics of the turbulence is therefore fluid in nature. Particle motions contributing to the third term, containing both non-thermal and thermal energy in the distribution function, are not part of the turbulence by this definition. Collisionless wave-particle interactions transfer energy from the turbulent electromagnetic and bulk velocity fluctuations (the first two terms in \eqrefeq:vlasov_energy2) into non-thermal free energy in the plasma species distribution functions (the third term). This process damps the turbulence, increasing the “kinetic” temperature, but does not lead to thermodynamic heating of the plasma (an increase of the thermal temperature). The physical mechanisms that damp the turbulent fluctuations are therefore kinetic in nature. Collisions then act to smooth out the non-thermal fluctuations in the velocity distribution, irreversibly converting the non-thermal free energy in the distribution function to thermal energy. Thus, the dissipation of turbulence in a weakly collisional plasma is necessarily a two-step process, whereby first entropy-conserving collisionless wave-particle interactions damp the turbulent electromagnetic and bulk velocity fluctuations, and then entropy-increasing collisions thermalize the energy removed from the turbulent fluctuations. Other forms of particle energization, such as the development of beams or high-energy tails in the velocity distribution, may also occur via collisionless wave-particle interactions, and correspond to the transfer of energy from the electromagnetic fields to non-thermal free energy in the distribution functions, a transfer that is also essentially kinetic in nature.
2.4 Why are Parametric Instabilities Subdominant?
The inherently anisotropic nature of plasma turbulence, with turbulent fluctuations that satisfy , also leads to the prediction that the nonlinearity associated with Alfvén wave collisions dominates over other potential nonlinear mechanisms driven by gradients parallel to the magnetic field, in particular parametric instabilities, such as the decay,[71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83] modulational,[84, 85, 86, 77, 87, 81, 88] and beat[87, 81] instabilities. For example, even for oblique Alfvén wave modes, numerical studies using both MHD [89, 90] and kinetic ion/fluid electron hybrid  simulations find parametric instability growth rates that are proportional to , where is the angle between the wavevector and the magnetic field. The root of this property is that the nonlinearity associated with parametric instabilities is proportional to , whereas the nonlinearity associated with Alfvén wave collisions is proportional to . Therefore, the relative magnitude of the effect of parametric instabilities to that of Alfvén wave collisions is , so parametric instabilities are expected to be subdominant in anisotropic plasma turbulence . A more thorough investigation is warranted to confirm this prediction.
2.5 Why is Critical Balance Natural for Turbulent Plasmas?
The nonlinearity parameter, , measures the strength of incompressible MHD turbulence using the ratio of the magnitude of the nonlinear term to that of the linear term in \eqrefeq:elsasserpm. In the weak turbulence limit, , the dynamics may be rigorously calculated analytically using perturbation theory [55, 56, 92, 17], with the results of such a calculation described above in §2.1. Strong incompressible MHD turbulence is conjectured to maintain a state of critical balance  in which the nonlinear and linear terms are of equal magnitude, or . Although the concept of critical balance is not universally accepted, a significant number of studies employing various techniques to estimate the variation along the local mean magnetic field direction support critical balance [51, 1, 93, 94, 95, 96, 97, 98], while contradictory studies have uniformly used a global mean field [99, 100, 101], an approach that has been shown to give misleading results . Since the linear terms are equal in magnitude to the nonlinear terms in critically balanced strong turbulence, the linear plasma physics continues to have an important impact on the dynamics in magnetized plasma turbulence, unlike hydrodynamic turbulence. An important consequence of this fact is that the turbulent fluctuations may exhibit some linear mode properties, properties which may be exploited to illuminate the nature of plasma turbulence [14, 102]. The relevance of linear physics to plasma turbulence is responsible for the numerical finding that the picture of energy transfer to smaller scales presented in §2.1, derived rigorously in the weakly nonlinear limit , remains qualitatively correct, even in the limit of strong, critically balanced turbulence, well beyond the formal regime of applicability of the solution .
It is worthwhile to explain physically why critical balance—equivalently described by a state in which the characteristic linear timescale and nonlinear timescale are in balance—naturally arises through the dynamics of a turbulent plasma. In these variables, . Consider an MHD plasma, initially at rest, that is shaken transverse to the magnetic field at one position with a sinusoidal variation of velocity of amplitude over a distance . The timescale associated with this perturbation of the plasma (and of the frozen-in magnetic field) is . Magnetic tension will lead to the propagation of Alfvén waves up and down the magnetic field, driven by the applied perturbation. One may ask, for the period of forcing , what is the parallel wavelength of the resulting waves? The parallel wavelength will be given by , so this leads to the relation , or . If the transverse perturbation of the plasma is due to the nonlinear terms in \eqrefeq:elsasserpm such that , then this leads to the relation between the linear and nonlinear timescales, . Alternatively, critical balance can be expressed as a balance between frequencies , or .
One may wonder, why is the limit not considered? In this case, the nonlinear term is much larger than the linear term, which occurs when , or , the two-dimensional limit where fluctuations perpendicular to the magnetic field are correlated for long distances along the magnetic field. But, transverse fluctuations in a turbulent plasma will only remain correlated along a magnetic field if they an exchange information along the field. Since information propagates along the magnetic field at the Alfvén speed, for a transverse oscillation with a period , the distance along the magnetic field over which the transverse motions can remain correlated must have . Therefore, one obtains the constraint , or in terms of the nonlinearity parameter, . Therefore, fluctuations with will decorrelate and evolve toward a state with . Thus, the turbulence cannot be any more two dimensional than allowed by the critical balance because fluctuations in any two planes perpendicular to the mean field can only remain correlated if an Alfvén wave can propagate between them in less than their perpendicular decorrelation time [1, 39].
In closing, it is worthwhile pointing out the common misconception that plasma turbulence in the solar wind must be weak because the turbulent magnetic field fluctuations are small compared to the mean magnetic field magnitude , or . In fact, the strength of the turbulence, estimated by the nonlinearity parameter as the ratio of the nonlinear to the linear term in \eqrefeq:elsasserpm, depends not only on the normalized amplitude of the fluctuations, , but also on the anisotropy of the fluctuations, . Direct multi-spacecraft measurements of turbulent magnetic field fluctuations at the small-scale end of the inertial range find [7, 103, 104], so even the small amplitude magnetic field fluctuations measured at that scale, typically of magnitude , lead to strong nonlinear interactions.
2.6 What is the Nature of the Compressible Turbulent Fluctuations?
In addition to the incompressible dynamics of Alfvén waves that
dominate turbulence in the inertial range of magnetized plasmas,
plasma turbulence also contains a small fraction (typically 10% or
less) of energy in compressible fluctuations .
Recent high-frequency measurements of the density fluctuations are
presented elsewhere in this issue . Exploiting
the properties of compressible fluctuations in the inertial range
using linear kinetic theory, the nature of the compressible
fluctuations may be identified by analyzing the correlation between
the density and parallel magnetic field fluctuations, . A recent
comparison of between 10 years of WIND spacecraft data at 1 AU
and theoretical predictions using the synthetic spacecraft data method
 has shown that there is negligible energy in the
kinetic counterpart of the fast magnetosonic mode, and that the
observed spectrum of compressible energy is composed entirely of
kinetic slow mode
2.7 How Does Temperature Anisotropy Arise from or Affect Turbulence?
Recent numerical  and theoretical  results suggest that the compressible fluctuations in turbulence may control the observed spread of proton temperature anisotropy in the solar wind plasma. The observations of in the solar wind are distributed across the broad range of values between the kinetic instability boundaries due to the mirror instability at and the oblique firehose instability at [109, 110, 111, 112, 113, 114, 115]. Although a wide range of physical mechanisms affect —including collisions, ion parallel heat flux, spherical expansion, radial compression and expansion, turbulent heating, turbulent compression, and kinetic instabilities—theoretical considerations show that kinetic slow mode fluctuations may be responsible for the observed spread of values , and recent kinetic numerical simulations are consistent with this explanation .
In addition to the proton temperature anisotropy being affected by the
turbulence, when exceeds certain
threshold values, kinetic temperature anisotropy instabilities can
There exist four ion temperature anisotropy instabilities that may serve to inject energy into the turbulent plasma: the parallel (or whistler) firehose instability [116, 117], the Alfvén (or oblique) firehose instability , the mirror instability [119, 120, 121], and the proton cyclotron instability . For each of these instabilities, unstable wave growth peaks at , where is the ion inertial length. For the parallel firehose and proton cyclotron instabilities, these waves have , and for the Alfvén firehose and mirror instabilities, they have . Therefore, these instabilities inhabit a combined region with and , with a typical wavevector anisotropy of . In contrast, at the small scales around the ion inertial length, turbulent fluctuations in a magnetic plasma have a typical wavevector anisotropy [53, 122, 5, 39]. Therefore, the instability-driven fluctuations do not occupy the same region of wavevector space as the fluctuations of the turbulent cascade from large scales.
Note that kinetic instabilities of the electrons can also inject fluctuation energy into the solar wind at yet smaller scales, and recent work has indeed identified the signature of localized whistler wavepackets at scales which appear to be driven by the whistler heat flux instability . These whistler modes have quasi-parallel wavevectors with , and therefore also inhabit a region of wavevector space distinct from the anisotropic cascade of energy from large scales.
Discriminating fluctuations with from those with , however, is not generally possible using single-point spacecraft measurements. Such measurements only provide the projection of the wavevector along the solar wind velocity, so there is insufficient information to separate the parallel from the perpendicular component of the wavevector. Only when the magnetic field direction is parallel to the solar wind flow, a relatively rare occurrence in the near-Earth solar wind, can one uniquely determine the value of . Measurements of the fluctuating magnetic helicity as a function of the angle between the solar wind velocity and indeed have discovered two distinct signatures in regions perpendicular and parallel to at frequencies Hz [124, 125], corresponding to . It has been suggested that the perpendicular signature arises from kinetic Alfvén waves with , while the parallel signature arises from either ion cyclotron waves propagating away from the sun or whistler waves propagating toward to the sun with [125, 124, 126]. An analysis comparing predictions from the synthetic spacecraft data method  to observations confirms that such a two-component model—consisting primarily of kinetic Alfvén waves with with a small amount (5% in energy) of either ion cyclotron or whistler waves with —indeed reproduces the observations, supporting the hypothesis that kinetic-instability-driven waves persist in the solar wind alongside the Alfvénic turbulent fluctuations that mediate the nonlinear transfer of energy from large to small scales .
Are these instability-driven parallel waves turbulent themselves (do they transfer energy nonlinearly to small scales?), or do they merely persist in the turbulent environment caused by the anisotropic Alfvénic cascade of energy from large to small scales? That the parallel waves have small amplitude and are propagating unidirectionally suggests they do not interact nonlinearly among themselves . In addition, since the Alfvénic wave frequencies are linearly proportional to the parallel wavenumber, , the instability-driven waves with have much higher frequencies than the anisotropic fluctuations of the turbulent cascade that have . Here I suggest that this frequency mismatch impedes significant nonlinear coupling of the parallel waves to the turbulent fluctuations, so the parallel waves are not turbulent but merely persist alongside the anisotropic Alfvénic turbulence. Therefore, the kinetic instabilities and their resulting fluctuations, distinguished observationally by the parallel magnetic helicity signature, operate separately from the turbulent cascade of energy from large to small scales.
2.8 How Are Coherent Structures Generated in Plasma Turbulence?
Numerical simulations of plasma turbulence demonstrate the ubiquitous development of coherent structures, specifically current sheets, at small scales [127, 128, 129, 130, 1, 131], and dissipation is largely concentrated in these current sheets [132, 133, 134, 135, 136]. There has been a flurry of activity recently seeking evidence for such localized heating through statistical analyses of solar wind observations [137, 138, 139, 140, 141, 142, 135, 143]. But one fundamental question remains unanswered: what mechanism governs the generation of current sheets in plasma turbulence? New research suggests that current sheets arise naturally as a consequence of Alfvén wave collisions in the strong turbulence limit .
Consider first the physics of the Alfvén wave collision problem described in §2.1 in the weak turbulence limit, . For initial Alfvén waves that have equal amplitudes given by the constants , the nonlinearity parameter is . An asymptotic expansion of the nonlinear evolution in the weakly nonlinearly limit can be performed, ordered by the small expansion parameter . In this limit, the physics of the nonlinear energy transfer is dominated by just five modes as depicted in panel (a) of Fig. 1: the primary Alfvén waves at order (red circles), the secondary mode at order (green triangles), and the tertiary Alfvén waves at order (blue squares). The secular transfer of energy to small scales is accomplished by just three nonlinear interactions, the interaction at order and the two interactions at order . These nonlinear interactions, governed by the form of the nonlinear terms in \eqrefeq:elsasserpm, determine the phase and amplitude relationships among these five relevant Fourier modes. For example, for the particular problem described in §2.1, the nonlinearly generated Alfvén waves are phase-shifted from the primary Alfvén waves by [17, 58]. In fact, recent analytical and numerical work argues for the importance of intermittency, or phase synchronization, in weak incompressible MHD turbulence .
As the initial Alfvén wave amplitudes increase to the strongly nonlinear regime, , the asymptotic expansion of the equations of evolution ceases to be well-ordered, and higher-order terms—terms that can safely be neglected in the weakly nonlinearly limit—begin to contribute significantly. The nonlinear interactions associated with these higher-order terms mediate significant energy transfer to additional Fourier modes in the perpendicular plane, as depicted in Fig. 1. Yet, even in the strong turbulence limit, the phase and amplitude relationships among all of these modes remain completely specified by the nonlinear terms in the equations of evolution.
Nonlinear gyrokinetic simulations using the Astrophysical Gyrokinetics Code, AstroGK , have demonstrated the development of a current sheet as a natural consequence of the nonlinear evolution of an Alfvén wave collision in the strong turbulence limit, . Through these simulations, the qualitative understanding of the nonlinear energy transfer gleaned from the analytical solution in the weakly nonlinear limit  is found to persist as one approaches and reaches the strongly nonlinear limit, . In particular, the nonlinearly generated modes continue to play a key role in the energy transfer. It is found that the energy transfer to higher perpendicular wavenumbers flows predominantly along the three diagonal lines in panel (a) of Fig. 1, largely confined to these modes with , , and . Generally, it appears that the higher-order modes are also purely magnetic, inherently nonlinear modes, similar to . And, the energy in the higher-order modes with appears to primarily consist of counterpropagating Alfvén waves, where the perpendicular wavenumber of these Alfvén waves increases, given by the distance of the mode from the origin of the plot. Further analytical work is necessary is necessary to confirm the nature of these higher-order, nonlinearly generated modes. The modes along these three diagonals represent a strikingly small fraction of the total possible perpendicular Fourier modes, typically a few percent. It is constructive interference among just this small number of modes that results in the appearance of a coherent structure, in this case a current sheet.
All of the modes shown in panel (a) of Fig. 1 can, in principle, be computed analytically by extending the asymptotic calculation to higher order, but the such a calculation involves a rapidly growing number of terms. However, the confinement of the turbulent energy to a few modes at each order can provide a guide to limit the effort of detailed analytical calculations to just the nonlinear modes directly involved in the dominant flow of energy to small scales along the diagonal lines in Fig. 1. Another complication is that, as the nonlinearity parameter increases to the strong turbulence limit, , the convergence of the asymptotic series becomes poor, and the calculation must be taken to increasingly higher powers of . Nevertheless, the onset of current sheet development in a moderately nonlinear numerical calculation with is well reproduced by an analytical calculation employing only the five modes up to and including order . This result suggests that the physical mechanism of nonlinear energy transfer in Alfvén wave collisions, outlined in §2.1, is indeed ultimately responsible for the development of current sheets in plasma turbulence, and that such coherent structure development can be computed analytically.
Note that any coherent structure can be assembled through constructive interference among spatial Fourier modes with particular amplitudes and phases. For example, a square wave can be expressed as the sum of sinusoidal modes . In the Alfvén wave collision problem, it is constructive interference among wave modes in the perpendicular plane—specifically the Fourier modes depicted in Fig. 1, where the amplitudes and phases of these modes are determined by the form of the nonlinear terms and can be computed analytically—that results in the development of a coherent structure. The observation of coherent structures in plasma turbulence arising in the form of current sheets is often invoked as evidence that their presence is inconsistent with the strongly interacting Alfvén wave interpretation of plasma turbulence. The results reviewed here suggest not only that Alfvén waves are consistent with, but also that they are directly responsible for, the development of current sheets. The development of a current sheet is a natural consequence of the nonlinear interaction between two counterpropagating Alfvén waves in the strong turbulence limit, a result that is not a priori obvious in any way. This key finding resolves the apparent contradiction between investigations that view plasma turbulence as a sea of nonlinearly interacting Alfvén waves and those that focus on the development of coherent structures and their role as sites of enhanced dissipation. Ultimately, this investigation of the nonlinear dynamics of Alfvén waves enables a first-principles prediction of current sheet development in plasma turbulence through the linear superposition of the nonlinearly generated modes, and illuminates the nature of these current sheets as constructive interference between counterpropagating Alfvén waves.
Along with the picture of nonlinear energy transfer mediated by counterpropagating Alfvén waves, this discovery of Alfvén-wave-driven current sheet development completes the foundation necessary to construct the first dynamical model of plasma turbulence that self-consistently combines the physics of Alfvén waves and small-scale current sheets, as presented below. This dynamical understanding of the origin and nature of current sheets provides important constraints on how such coherent structures may be dissipated, to be discussed in next section.
Before ending our discussion of current sheets, it is important to point out that there are other mechanisms, such as particular flow and magnetic field geometries, that can produce current sheets. The Orzsag-Tang vortex is one such example . Do such alternative mechanisms play any role in the development of current sheets in solar wind turbulence, or are strong Alfvén wave collisions sufficient to account for all of the current sheets observed in plasma turbulence? To answer this open question, the properties of current sheets generated by Alfvén wave collisions must be carefully compared to those arising in numerical simulations and inferred from solar wind observations.
3 The Damping of Turbulent Fluctuations and Plasma Heating
At the forefront of plasma turbulence research stands the effort to identify and characterize the physical mechanisms responsible for the dissipation of the turbulent fluctuations and the conversion of their energy to heat, or other energization, of the protons, electrons, and minor ions. As discussed above, for a weakly collisional plasma, the dissipation of turbulence necessarily includes two stages: first, the entropy-conserving, collisionless damping of the turbulent fluctuations; then, the ultimate conversion of the damped non-thermal free energy through entropy-increasing collisions into plasma heat.
There are presently three leading candidates for the damping of the
turbulent fluctuations: (i) coherent wave-particle interactions, in
particular Landau damping
3.1 A Model of Anisotropic Alfvénic Turbulence
The observed one-dimensional magnetic energy frequency spectrum in the solar wind demonstrates a power law with spectral index at low frequencies , a break at around Hz, and a steeper spectrum at higher frequencies with a spectral index of approximately [38, 162, 163, 41, 7]. The low-frequency range is denoted the inertial range of solar wind turbulence, the break is believed to be associated with an ion kinetic length scale, and the high-frequency range is commonly denoted the dissipation range, although whether dissipation is actively occurring at all scales in this range remains an open question. In addition, multi-spacecraft observations show that the turbulent fluctuations at the small scales near the break are highly anisotropic, with [7, 103, 104].
The following model of critically balanced, strong Alfvénic turbulence has been proposed to explain these observed features and to
provide a foundation upon which predictive models of solar wind
turbulence, its dissipation, and the resulting plasma heating may be
[5, 164, 39, 165, 17, 20].
In this model, the inertial range occurs at scales and consists of counterpropagating Alfvén waves that transfer
energy to successively smaller scales via strong Alfvén wave
collisions. At the scale , the linear wave
physics transitions from that of nondispersive MHD Alfvén waves to
dispersive kinetic Alfvén waves, leading to the steepening of the
energy spectrum. At smaller scales, I suggest here that the nonlinear
energy transfer continues to be mediated by counterpropagating wave
collisions, only between kinetic Alfvén waves rather than MHD Alfvén waves.
Although the collisionless damping of the Alfvén waves is weak in the inertial range at , ion Landau damping peaks at and is expected to transfer energy from the turbulent electromagnetic fluctuations into non-thermal free energy in velocity space through collisionless wave-particle interactions. The resulting small-scale structure in the ion velocity distribution, through a nonlinear phase mixing process, may undergo a dual-cascade to smaller scales in both configuration space and velocity space, termed the ion entropy cascade [39, 167, 168, 169, 170]. The entropy cascade enables efficient collisional thermalization of the non-thermal free energy associated with fluctuations in the ion distribution function even under conditions of arbitrarily weak collisionality. In addition to ion Landau damping, electron Landau damping becomes increasingly strong as the perpendicular wavenumber increases within the dissipation range, . Ultimately, the electron Landau damping becomes sufficiently strong to dominate over the nonlinear energy transfer, terminating the cascade with an exponential roll-off of the magnetic energy spectrum around the scale of the electron Larmor radius, [171, 98].
Based on such a dynamical understanding of Alfvénic turbulence and its collisionless damping, it is possible to construct predictive models for the turbulent cascade of energy and its dissipation [5, 172, 43, 165, 173]. Turbulent cascade models have been used to predict: the density fluctuations in the solar corona and solar wind [174, 11]; the ratio of ion to electron heating as a function of and ; and the proton-to-total turbulent heating ratio in the solar wind . As well, such cascade models have proven invaluable in the interpretation of kinetic numerical simulations of weakly collisional plasma turbulence [4, 165, 98].
3.2 Relation to Stochastic Heating
Although it may not be immediately obvious, the model of stochastic
heating is entirely compatible with the dynamical model of anisotropic
Alfvénic turbulence described above. Recent work has resulted in a
very well-developed model of stochastic ion heating in heliospheric
[40, 152, 153, 154]. A
key component of this model is a turbulent spectrum of Alfvén and
kinetic Alfvén waves, such as the Alfvénic cascade described above.
If a particular ion has a Larmor radius
In principle, stochastic heating could be incorporated into the cascade models mentioned in §3.1, but to attempt such an implementation raises several unanswered questions: (i) what fraction of the large-scale turbulent cascade energy is stochastically lost to the ions? (ii) how is this energy loss distributed over different scales of the turbulent Alfvénic fluctuations?, and (iii) does this mechanism impact the nature of the fluctuations at smaller scales that receive the remaining turbulent energy not lost stochastically to the ions? Indeed, a recent observational study finding an amplitude dependent drop in the magnetic energy spectrum at the ion kinetic scale  may be evidence of turbulent cascade energy lost stochastically to ions.
3.3 Dissipation Associated with Current Sheets
Although the property that dissipation in plasma turbulence occurs dominantly in small-scale current sheets is well established [127, 128, 129, 130, 1, 131, 132, 133, 134, 135, 136], the kinetic physical mechanism by which dissipation occurs in current sheets has not been elucidated. Indeed, stochastic ion heating [176, 156] and Landau damping [177, 134, 178] have both been suggested as the physical process underlying the kinetic damping of current sheets. In addition, parallel electric fields [179, 180, 181, 182, 183, 184] and Fermi acceleration , both arising through the process of magnetic reconnection, have also been proposed. It should be emphasized, however, that even though magnetic reconnection often springs to mind when current sheets are found, it has not yet been firmly established that magnetic reconnection plays a significant role in the dissipation of energy in three-dimensional plasma turbulence in the solar wind. The topics of the connection between magnetic reconnection and turbulence and the role played by current sheets in the dissipation of turbulence are central to several reviews in this issue [160, 186, 187].
One important question is whether the current sheets inferred from solar wind observations are all dynamically generated by the turbulence itself [188, 189], or whether some fraction of the current sheets merely represent advected flux tube boundaries [190, 191, 192, 193]. I have highlighted in §2.8 recent research that explains, from first-principles, the development of current sheets in plasma turbulence as a natural consequence of the nonlinear energy transfer caused by strong Alfvén wave collisions . If this mechanism operates in the solar wind, then at least some of the observed current sheets consist of a constructively interfering sum of counterpropagating Alfvén waves. This finding provides new insight into the underlying nature of dynamically generated current sheets in plasma turbulence, and the damping of the associated turbulent electromagnetic fluctuations may indeed be dominated by collisionless wave-particle interactions via the Landau resonance, as previously suggested by a recent analysis of gyrokinetic numerical simulations . If these conjectures are correct, then the fact that the heating is concentrated in current sheets is merely a consequence of the turbulent nonlinear dynamics but does not determine the mechanism of dissipation. This idea does not conflict with results reported elsewhere in this issue suggesting that intermittency develops identically in both ideal and resistive MHD simulations .
A statement that is likely to prove very controversial is that intermittency, although a well established characteristic of plasma turbulence, may not be critically important for the prediction of the plasma heating due to the dissipation of turbulent fluctuations, particularly if coherent (or possibly even stochastic) wave-particle interactions dominate the dissipation. This directly contradicts the conclusion of Matthaeus et al.  that “recent studies of kinetic dissipation of the turbulent cascade suggest that coherent structures and associated nonuniform dissipation play a very important and possibly dominant role in the termination of the cascade and the effectively irreversible conversion of fluid macroscopic energy into microscopic random motions.” In short, the viewpoint proposed here is that current sheets are not a cause of turbulent dissipation, but are merely a consequence of the nonlinear dynamics underlying the turbulent energy cascade. If this contentious statement is correct, it represents good news for the endeavor to develop predictive models of plasma heating arising from the dissipation of weakly collisional plasma turbulence. Simple cascade models would remain a valid means of predicting the differential heating of the plasma species since the collisionless damping would depend only on the energy content in different wavevectors, while the phases that lead to intermittency would not come into play.
4 Dynamical Model of Plasma Turbulence
Here I present a brief summary of the first dynamical model of plasma turbulence that combines the physics of Alfvén waves with the self-consistent development of current sheets, illustrated by the diagrams of the magnetic energy spectrum and wavevector anisotropy in Fig. 2.
Isotropic fluctuations at the outer scale of the turbulent inertial range in the near-Earth solar wind, typically at , consist primarily of incompressible Alfvén waves with a small admixture of kinetic slow wave fluctuations (not depicted). Nonlinear energy transfer from large to small scales through the inertial range is governed by the physics of Alfvén wave collisions. The theory of the critically balanced Alfvénic turbulence  predicts a magnetic energy spectrum , where the turbulent fluctuations become increasingly more anisotropic as the cascade progresses to smaller scales, following a scale-dependent anisotropy in the inertial range. Note that a refined version of critically balanced turbulence that accounts for the dynamic alignment of the turbulent fluctuations  predicts slightly different scalings, and , and appears to be more consistent with the high resolution numerical simulations of incompressible MHD turbulence .
At the perpendicular scale of the ion Larmor radius, , the turbulent cascade transitions from the inertial range (), in which the nonlinear energy transfer is mediated by Alfvén wave collisions, to the dissipation range (), in which the nonlinear energy transfer is mediated by kinetic Alfvén wave collisions. The dispersive nature of kinetic Alfvén waves leads to a steepening of the magnetic energy spectrum , and an extension of the concept of critical balance to this regime  predicts an anisotropy scaling as .
The nonlinear energy transfer is governed by Alfvén and kinetic Alfvén wave collisions, which transfer energy to Alfvénic waves at smaller scales. The particular phases and amplitudes of the nonlinearly generated waves, determined by the mathematical form of the nonlinear terms in the dynamical equations, leads to the development of coherent structures through constructive interference . In plasma turbulence, these coherent structures take the form of current sheets, and dissipation is found to be concentrated within these current sheets. But, since the current sheets themselves are primarily made up of counterpropagating Alfvén and kinetic Alfvén waves, these turbulent electromagnetic fluctuations will damp by collisionless wave-particle interactions via the Landau resonance. Ion Landau damping peaks at scales , and electron Landau damping begins to be significant at and becomes increasingly strong as the wavenumber increases. At the electron scales , electron Landau damping overwhelms the nonlinear energy transfer and the cascade is terminated, producing an exponential roll-off in the magnetic energy spectrum. Ultimately, the energy removed from the turbulent fluctuations by collisionless wave-particle interactions will ultimately be converted to plasma heat by collisions. This process is facilitated by an entropy cascade (not shown) that mediates the transfer of nonthermal structure to sufficiently small scales in velocity space that weak collisions can thermalize the energy .
In addition to the physics of the Alfvénic turbulent cascade of
energy from large scales that is ultimately terminated at electron
scales, kinetic instabilities can inject energy into fluctuations at
An improved understanding of turbulence in the weakly collisional solar wind will impact our ability not only to predict physical processes within the heliosphere but also to illuminate complex astrophysical phenomena in remote regions of the universe. The linear response due to magnetic tension in a magnetized plasma, a physical property absent in turbulent hydrodynamical flows, provides an important foothold to understand the dynamics of plasma turbulence. To unravel the physics of the nonlinear energy transfer to small scales, the kinetic mechanisms of dissipation of the turbulent fluctuations, and the resulting plasma heating in kinetic plasma turbulence, I contend that we must step beyond the usual statistical treatments of turbulence, adopting instead a dynamical approach. It is the linear and nonlinear dynamics of Alfvén waves that are responsible, at a very fundamental level, for some of the key qualitative features of plasma turbulence that distinguish it from hydrodynamic turbulence, including the anisotropic cascade of energy and the development of current sheets at small scales. The ultimate goal is to create a predictive theory of plasma turbulence. Only a thorough understanding of the turbulent plasma dynamics will enable the development of a simplified theoretical framework upon which predictive models of plasma turbulence and its effect on energy transport and plasma heating can be constructed.
The arguments presented here support a simplified perspective on the nature of turbulence in a weakly collisional plasma. The nonlinear interactions responsible for the turbulent cascade of energy and the formation of current sheets are essentially fluid in nature, and can be understood in terms of nonlinear wave-wave interactions. On the other hand, the collisionless damping of the turbulent fluctuations and the energy injection by kinetic instabilities are essentially kinetic in nature, and can be understood in terms of linear collisionless wave-particle interactions.
From this perspective, it is easy to understand why reduced models, such as incompressible MHD and reduced MHD, are valuable tools in the study of plasma turbulence. The simplicity of such reduced models is that they satisfy a number of exact constraints that facilitate the development of an intuitive understanding of the turbulent dynamics. Although these constraints do not strictly hold under more general plasma conditions, often the behavior observed in the simplified system persists, at least to lowest order, in the more general system. For example, we employ here our understanding of the nonlinear energy transfer between counterpropagating Alfvén waves in incompressible MHD to argue that the anisotropic cascade of energy ubiquitously observed in magnetized plasma turbulence is due to the facts that only counterpropagating Alfvén waves interact nonlinearly and that the nonlinear term is greatest for interactions between perpendicularly polarized Alfvén waves. Although the constraint that only counterpropagating waves interact ceases to hold strictly when compressibility or kinetic effects (dispersion due to finite Larmor radius effects) are introduced, the anisotropy of the turbulent cascade persists.
The separation of essentially fluid versus essentially kinetic properties of kinetic plasma turbulence enables further simplifications. Although non-Maxwellian particle velocity distributions are widely found in the solar wind and in kinetic numerical simulations, the departures from a Maxwellian equilibrium (bi-Maxwellian, kappa, or multi-component with core/halo structure) do not affect the electromagnetic dynamics of the turbulence very much . Since the nonlinear energy transfer and development of current sheets in plasma turbulence are due to wave-wave interactions, they depend only on the first moments of the distribution functions, and are therefore not very sensitive to the particular form of the distribution functions. The physical mechanisms of dissipation and the injection of energy by kinetic instabilities, on the other hand, which depend on growth or damping rates due to collisionless wave-particle interactions, may depend sensitively on the form of the distribution functions. For example, a kappa distribution, having a much larger population of particles at high energy with respect to a Maxwellian, may experience significantly enhanced rates of collisionless damping of waves that are resonant with these high energy particles. This viewpoint proposed here strongly contradicts the recent suggestion from a kinetic study of plasma turbulence  that “it seems increasingly clear that significant kinetic effects including heating have strong association with coherent structures and with the turbulent cascade that produces intermittency.”
Looking toward the future, to identify definitively the kinetic mechanism responsible for the dissipation of solar wind turbulence, it is likely that it will be necessary to step beyond the analysis of the turbulent electromagnetic fields alone, delving much more deeply into the dynamic effect of turbulent fluctuations and their dissipation on the particle velocity distributions. These velocity distributions are already measured directly by instruments on existing spacecraft missions. And, on upcoming missions, such as MMS and Solar Probe Plus, with a combination of higher cadence and better resolution than previous missions, the distribution function data represent an important untapped potential source for discovery science. A new frontier will be the exploration of the dynamics of the perturbations to the distribution functions associated with the turbulent fluctuations and their kinetic dissipation, where kinetic plasma turbulence simulations will be essential to exploit fully the potential of MMS and Solar Probe Plus.
I would like to thank Jason TenBarge and Kris Klein for their invaluable feedback and contributions. Financial support has been provided by NSF grant PHY-10033446, NSF CAREER Award AGS-1054061, and NASA grant NNX10AC91G.
- Weak nonlinearity means that the magnitude of the nonlinear terms in the equations of evolution are small relative to the magnitude of the linear term that supports the propagation of Alfvén waves.
- I denote here the MHD regime as and the kinetic regime as , although compressible fluctuations in the weakly collisional solar wind require a kinetic treatment, even at the large scales of the inertial range [39, 46, 14]. Here refers to the component of the wavevector perpendicular to the local mean magnetic field and is the thermal ion Larmor radius, where the ion thermal velocity is defined by (with temperature expressed in units of energy) and the ion cyclotron frequency is given by .
- Recall that, in this notation, is a positive constant describing the initial counterpropagating Alfvén waves.
- Low frequency here refers to frequencies much smaller than the electron plasma frequency, .
- Future work to confirm this finding should address the issue of whether modes driven by the kinetic mirror instability are also consistent with the WIND observations, but the ubiquity of the compressible fluctuations measured suggest that they are not restricted to plasma intervals with the temperature anisotropy necessary for the mirror instability to generate fluctuations.
- In addition to temperature anisotropy instabilities, other kinetic instabilities, such as those driven by beams or the relative drift between different plasma species, can also give rise to electromagnetic fluctuations.
- I include in my use of the term “Landau damping” all damping associated with the Landau resonance, thereby also including transit-time damping, the magnetic analogue of Landau damping.
- The growing body of observational evidence in support of kinetic Alfvén waves as the dominant contributor to the dissipation range of solar wind turbulence has recently been reviewed .
- Note that this Larmor radius is not the thermal ion Larmor radius , but rather is the gyroradius for an ion with a particular perpendicular velocity .
- Note that , so for the typical value in the near-Earth solar wind, .
- J. Maron and P. Goldreich. Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J., 554:1175–1196, 2001.
- J. Cho and A. Lazarian. Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications. Mon. Not. Roy. Astron. Soc., 345:325–339, October 2003.
- O. Alexandrova, V. Carbone, P. Veltri, and L. Sorriso-Valvo. Small-Scale Energy Cascade of the Solar Wind Turbulence. Astrophys. J., 674:1153–1157, February 2008.
- G. G. Howes, W. Dorland, S. C. Cowley, G. W. Hammett, E. Quataert, A. A. Schekochihin, and T. Tatsuno. Kinetic Simulations of Magnetized Turbulence in Astrophysical Plasmas. Phys. Rev. Lett., 100(6):065004, February 2008.
- G. G. Howes, S. C. Cowley, W. Dorland, G. W. Hammett, E. Quataert, and A. A. Schekochihin. A model of turbulence in magnetized plasmas: Implications for the dissipation range in the solar wind. J. Geophys. Res., 113(A12):A05103, May 2008.
- V. A. Svidzinski, H. Li, H. A. Rose, B. J. Albright, and K. J. Bowers. Particle in cell simulations of fast magnetosonic wave turbulence in the ion cyclotron frequency range. Phys. Plasmas, 16(12):122310–+, December 2009.
- F. Sahraoui, M. L. Goldstein, G. Belmont, P. Canu, and L. Rezeau. Three Dimensional Anisotropic k Spectra of Turbulence at Subproton Scales in the Solar Wind. Phys. Rev. Lett., 105(13):131101–+, September 2010.
- P. Hunana, D. Laveder, T. Passot, P. L. Sulem, and D. Borgogno. Reduction of Compressibility and Parallel Transfer by Landau Damping in Turbulent Magnetized Plasmas. Astrophys. J., 743:128, December 2011.
- G. G. Howes, J. M. TenBarge, W. Dorland, E. Quataert, A. A. Schekochihin, R. Numata, and T. Tatsuno. Gyrokinetic simulations of solar wind turbulence from ion to electron scales. Phys. Rev. Lett., 107:035004, 2011.
- J. M. TenBarge, J. J. Podesta, K. G. Klein, and G. G. Howes. Interpreting Magnetic Variance Anisotropy Measurements in the Solar Wind. Astrophys. J., 753:107, July 2012.
- C. H. K. Chen, G. G. Howes, J. W. Bonnell, F. S. Mozer, K. G. Klein, and S. D. Bale. Kinetic scale density fluctuations in the solar wind. In G. P. Zank, J. Borovsky, R. Bruno, J. Cirtain, S. Cranmer, H. Elliott, J. Giacalone, W. Gonzalez, G. Li, E. Marsch, E. Moebius, N. Pogorelov, J. Spann, and O. Verkhoglyadova, editors, American Institute of Physics Conference Series, volume 1539 of American Institute of Physics Conference Series, pages 143–146, June 2013.
- G. G. Howes. Prediction of the Proton-to-total Turbulent Heating in the Solar Wind. Astrophys. J., 738:40, 2011.
- G. G. Howes, S. D. Bale, K. G. Klein, C. H. K. Chen, C. S. Salem, and J. M. TenBarge. The Slow-mode Nature of Compressible Wave Power in Solar Wind Turbulence. Astrophys. J. Lett., 753:L19, 2012.
- K. G. Klein, G. G. Howes, J. M. TenBarge, S. D. Bale, C. H. K. Chen, and C. S. Salem. Using Synthetic Spacecraft Data to Interpret Compressible Fluctuations in Solar Wind Turbulence. Astrophys. J., 755:159, 2012.
- C. S. Salem, G. G. Howes, D. Sundkvist, S. D. Bale, C. C. Chaston, C. H. K. Chen, and F. S. Mozer. Identification of Kinetic Alfvén Wave Turbulence in the Solar Wind. Astrophys. J. Lett., 745:L9, 2012.
- C. H. K. Chen, S. Boldyrev, Q. Xia, and J. C. Perez. Nature of Subproton Scale Turbulence in the Solar Wind. Phys. Rev. Lett., 110(22):225002, May 2013.
- G. G. Howes and K. D. Nielson. Alfvén wave collisions, the fundamental building block of plasma turbulence. I. Asymptotic solution. Phys. Plasmas, 20(7):072302, July 2013.
- K. D. Nielson, G. G. Howes, and W. Dorland. Alfvén wave collisions, the fundamental building block of plasma turbulence. II. Numerical solution. Physics of Plasmas, 20(7):072303, July 2013.
- K. G. Klein, G. G. Howes, J. M. TenBarge, and J. J. Podesta. Physical Interpretation of the Angle-dependent Magnetic Helicity Spectrum in the Solar Wind: The Nature of Turbulent Fluctuations near the Proton Gyroradius Scale. Astrophys. J., 785:138, April 2014.
- G. G. Howes. The Dynamical Generation of Current Sheets in Astrophysical Plasma Turbulence. Science, 2015. in preparation.
- P. J. Coleman, Jr. Turbulence, Viscosity, and Dissipation in the Solar-Wind Plasma. Astrophys. J., 153:371–388, August 1968.
- J. W. Belcher and L. Davis. Large-Amplitude Alfvén Waves in the Interplanetary Medium, 2. J. Geophys. Res., 76:3534–3563, 1971.
- C.-Y. Tu, Z.-Y. Pu, and F.-S. Wei. The power spectrum of interplanetary Alfvenic fluctuations Derivation of the governing equation and its solution. J. Geophys. Res., 89:9695–9702, November 1984.
- W. H. Matthaeus, M. L. Goldstein, and D. A. Roberts. Evidence for the presence of quasi-two-dimensional nearly incompressible fluctuations in the solar wind. J. Geophys. Res., 95:20673–20683, December 1990.
- W. H. Matthaeus, Y. Zhou, G. P. Zank, and S. Oughton. Transport theory and the WKB approximation for interplanetary MHD fluctuations. J. Geophys. Res., 99:23421–+, December 1994.
- C.-Y. Tu and E. Marsch. On the nature of compressive fluctuations in the solar wind. J. Geophys. Res., 99:21481–+, November 1994.
- M. K. Verma, D. A. Roberts, and M. L. Goldstein. Turbulent heating and temperature evolution in the solar wind plasma. J. Geophys. Res., 100:19839–19850, October 1995.
- R. J. Leamon, W. H. Matthaeus, C. W. Smith, and H. K. Wong. Contribution of cyclotron-resonant damping to kinetic dissipation of interplanetary turbulence. Astrophys. J., 507:L181–L184, 1998.
- E. Quataert. Particle Heating by Alfvénic Turbulence in Hot Accretion Flows. Astrophys. J., 500:978–991, June 1998.
- R. J. Leamon, C. W. Smith, N. F. Ness, and H. K. Wong. Dissipation range dynamics: Kinetic alfvén waves and the importance of . J. Geophys. Res., 104:22331–22344, 1999.
- E. Quataert and A. Gruzinov. Turbulence and Particle Heating in Advection-dominated Accretion Flows. Astrophys. J., 520:248–255, July 1999.
- R. J. Leamon, W. H. Matthaeus, C. W. Smith, G. P. Zank, D. J. Mullan, and S. Oughton. MHD-driven Kinetic Dissipation in the Solar Wind and Corona. Astrophys. J., 537:1054–1062, July 2000.
- O. Stawicki, S. P. Gary, and H. Li. Solar wind magnetic fluctuation spectra: Dispersion versus damping. J. Geophys. Res., 106:8273–8282, May 2001.
- M. K. Verma. Statistical theory of magnetohydrodynamic turbulence: recent results. Phys. Rep., 401:229–380, November 2004.
- S. D. Bale, P. J. Kellogg, F. S. Mozer, T. S. Horbury, and H. Reme. Measurement of the Electric Fluctuation Spectrum of Magnetohydrodynamic Turbulence. Phys. Rev. Lett., 94(21):215002–+, June 2005.
- S. A. Markovskii, B. J. Vasquez, C. W. Smith, and J. V. Hollweg. Dissipation of the Perpendicular Turbulent Cascade in the Solar Wind. Astrophys. J., 639:1177–1185, March 2006.
- K. Hamilton, C. W. Smith, B. J. Vasquez, and R. J. Leamon. Anisotropies and helicities in the solar wind inertial and dissipation ranges at 1 AU. J. Geophys. Res., 113(A12):A01106, January 2008.
- F. Sahraoui, M. L. Goldstein, P. Robert, and Y. V. Khotyaintsev. Evidence of a Cascade and Dissipation of Solar-Wind Turbulence at the Electron Gyroscale. Phys. Rev. Lett., 102(23):231102–+, June 2009.
- A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert, and T. Tatsuno. Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas. Astrophys. J. Supp., 182:310–377, May 2009.
- B. D. G. Chandran, B. Li, B. N. Rogers, E. Quataert, and K. Germaschewski. Perpendicular Ion Heating by Low-frequency Alfvén-wave Turbulence in the Solar Wind. Astrophys. J., 720:503–515, September 2010.
- C. H. K. Chen, T. S. Horbury, A. A. Schekochihin, R. T. Wicks, O. Alexandrova, and J. Mitchell. Anisotropy of Solar Wind Turbulence between Ion and Electron Scales. Phys. Rev. Lett., 104(25):255002–+, June 2010.
- G. G. Howes. A prescription for the turbulent heating of astrophysical plasmas. Mon. Not. Roy. Astron. Soc., 409:L104–L108, 2010.
- J. J. Podesta, J. E. Borovsky, and S. P. Gary. A Kinetic Alfvén Wave Cascade Subject to Collisionless Damping Cannot Reach Electron Scales in the Solar Wind at 1 AU. Astrophys. J., 712:685–691, March 2010.
- S. Saito, S. P. Gary, and Y. Narita. Wavenumber spectrum of whistler turbulence: Particle-in-cell simulation. Phys. Plasmas, 17(12):122316–+, December 2010.
- L. Rudakov, C. Crabtree, G. Ganguli, and M. Mithaiwala. Quasilinear evolution of plasma distribution functions and consequences on wave spectrum and perpendicular ion heating in the turbulent solar wind. Physics of Plasmas, 19(4):042704, April 2012.
- G. G. Howes. Limitations of hall mhd as a model for turbulence in weakly collisional plasmas. Nonlin. Proc. Geophys., 16:219–232, 2009.
- Steven R. Cranmer, Mahboubeh Asgari-Targhi, John C. Raymond Mari Paz Miralles and, Leonard Strachan, Hui Tian, and Lauren N. Woolsey. The Role of Turbulence in Coronal Heating and Solar Wind Expansion. Phil. Trans. Roy. Soc. A, 2015. this issue.
- R. S. Iroshnikov. The turbulence of a conducting fluid in a strong magnetic field. Astron. Zh., 40:742, 1963. English Translation: Sov. Astron., 7 566 (1964).
- R. H. Kraichnan. Inertial range spectrum of hyromagnetic turbulence. Phys. Fluids, 8:1385–1387, 1965.
- W. M. Elsasser. The Hydromagnetic Equations. Physical Review, 79:183, July 1950.
- J. Cho and E. T. Vishniac. The Anisotropy of Magnetohydrodynamic Alfvénic Turbulence. Astrophys. J., 539:273–282, 2000.
- S. Sridhar and P. Goldreich. Toward a theory of interstellar turbulence. 1: Weak Alfvenic turbulence. Astrophys. J., 432:612–621, September 1994.
- P. Goldreich and S. Sridhar. Toward a Theery of Interstellar Turbulence II. Strong Alfvénic Turbulence. Astrophys. J., 438:763–775, 1995.
- D. Montgomery and W. H. Matthaeus. Anisotropic Modal Energy Transfer in Interstellar Turbulence. Astrophys. J., 447:706–+, July 1995.
- C. S. Ng and A. Bhattacharjee. Interaction of Shear-Alfven Wave Packets: Implication for Weak Magnetohydrodynamic Turbulence in Astrophysical Plasmas. Astrophys. J., 465:845–+, July 1996.
- S. Galtier, S. V. Nazarenko, A. C. Newell, and A. Pouquet. A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys., 63:447–488, June 2000.
- G. G. Howes, D. J. Drake, K. D. Nielson, T. A. Carter, C. A. Kletzing, and F. Skiff. Toward Astrophysical Turbulence in the Laboratory. Phys. Rev. Lett., 109(25):255001, December 2012.
- G. G. Howes, K. D. Nielson, D. J. Drake, J. W. R. Schroeder, F. Skiff, C. A. Kletzing, and T. A. Carter. Alfvén wave collisions, the fundamental building block of plasma turbulence. III. Theory for experimental design. Physics of Plasmas, 20(7):072304, July 2013.
- D. J. Drake, J. W. R. Schroeder, G. G. Howes, C. A. Kletzing, F. Skiff, T. A. Carter, and D. W. Auerbach. Alfvén wave collisions, the fundamental building block of plasma turbulence. IV. Laboratory experiment. Physics of Plasmas, 20(7):072901, July 2013.
- Jesse T. Coburn, Miriam A. Forman, Charles W. Smith, Bernard J. Vasquez, and Julia E. Stawarz. Third-Moment Descriptions of the Interplanetary Turbulent Cascade, Intermittency, and Back Transfer. Phil. Trans. Roy. Soc. A, 2015. this issue.
- G. G. Howes. The inherently three-dimensional nature of magnetized plasma turbulence. Journal of Plasma Physics, FirstView:1–19, 12 2014.
- D. C. Robinson and M. G. Rusbridge. Structure of turbulence in the zeta plasma. Phys. Fluids, 14:2499–2511, 1971.
- S. J. Zweben, C. R. Menyuk, and R. J. Taylor. Small-scale magnetic fluctuations inside the macrotor tokamak. Phys. Rev. Lett., 42:1270–1274, 1979.
- D. Montgomery and L. Turner. Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids, 24:825–831, 1981.
- J. V. Shebalin, W. H. Matthaeus, and D. Montgomery. Anisotropy in mhd turbulence due to a mean magnetic field. J. Plasma Phys., 29:525–547, 1983.
- S. Oughton, E. R. Priest, and W. H. Matthaeus. The influence of a mean magnetic field on three-dimensional magnetohydrodynamic turbulence. Journal of Fluid Mechanics, 280:95–117, 1994.
- S. Oughton, W. H. Matthaeus, M. Wan, and K. T. Osman. Anisotropy in Solar Wind Plasma Turbulence. Phil. Trans. Roy. Soc. A, 2015. this issue.
- G. G. Howes, S. C. Cowley, W. Dorland, G. W. Hammett, E. Quataert, and A. A. Schekochihin. Astrophysical Gyrokinetics: Basic Equations and Linear Theory. Astrophys. J., 651:590–614, November 2006.
- N. A. Krall and A. W. Trivelpiece. Principles of plasma physics. International Student Edition - International Series in Pure and Applied Physics, Tokyo: McGraw-Hill Kogakusha, 1973, 1973.
- S. Peter Gary. Short-Wavelength Plasma Turbulence and Temperature Anisotropy Instabilities: Recent Computational Progress. Phil. Trans. Roy. Soc. A, 2015. this issue.
- A. A. Galeev and V. N. Oraevskii. The Stability of Alfvén Waves. Soviet Physics Doklady, 7:988, May 1963.
- R. Z. Sagdeev and A. A. Galeev. Nonlinear Plasma Theory. 1969.
- A. Hasegawa. Kinetic theory of MHD instabilities in a nonuniform plasma. Sol. Phys., 47:325–330, March 1976.
- N. F. Derby, Jr. Modulational instability of finite-amplitude, circularly polarized Alfven waves. Astrophys. J., 224:1013–1016, September 1978.
- M. L. Goldstein. An instability of finite amplitude circularly polarized Alfven waves. Astrophys. J., 219:700–704, January 1978.
- S. R. Spangler and J. P. Sheerin. Properties of Alfven solitons in a finite-beta plasma. J. Plasma Phys., 27:193–198, April 1982.
- J.-I. Sakai and U. O. Sonnerup. Modulational instability of finite amplitude dispersive Alfven waves. J. Geophys. Res., 88:9069–9079, November 1983.
- S. R. Spangler. The evolution of nonlinear Alfven waves subject to growth and damping. Phys. Fluids, 29:2535–2547, August 1986.
- T. Terasawa, M. Hoshino, J.-I. Sakai, and T. Hada. Decay instability of finite-amplitude circularly polarized Alfven waves - A numerical simulation of stimulated Brillouin scattering. J. Geophys. Res., 91:4171–4187, April 1986.
- V. Jayanti and J. V. Hollweg. On the dispersion relations for parametric instabilities of parallel-progagating Alfvén waves. J. Geophys. Res., 98:13247–13252, August 1993.
- J. V. Hollweg. Beat, modulational, and decay instabilities of a circularly polarized Alfven wave. J. Geophys. Res., 99:23431–+, December 1994.
- V. I. Shevchenko, R. Z. Sagdeev, V. L. Galinsky, and M. V. Medvedev. The DNLS equation and parametric decay instability. Plasma Phys. Rep., 29:545–549, July 2003.
- Y. M. Voitenko and M. Goossens. Nonlinear coupling of Alfvén waves with widely different cross-field wavelengths in space plasmas. J. Geophys. Res., 110:10, July 2005.
- C. N. Lashmore-Davies. Modulational instability of a finite amplitude Alfven wave. Phys. Fluids, 19:587–589, April 1976.
- E. Mjolhus. On the modulational instability of hydromagnetic waves parallel to the magnetic field. J. Plasma Phys., 16:321–334, December 1976.
- K. Mio, T. Ogino, S. Takeda, and K. Minami. Modulational instability and envelope-solitons for nonlinear Alfven waves propagating along the magnetic field in plasmas. Journal of the Physical Society of Japan, 41:667–673, August 1976.
- H. K. Wong and M. L. Goldstein. Parametric instabilities of the circularly polarized Alfven waves including dispersion. J. Geophys. Res., 91:5617–5628, May 1986.
- P. K. Shukla, N. Shukla, and L. Stenflo. Kinetic modulational instability of broadband dispersive Alfvén waves in plasmas. J. Plasma Phys., 73:153–157, April 2007.
- L. Del Zanna. Parametric decay of oblique arc-polarized Alfvén waves. Geophys. Res. Lett., 28:2585–2588, 2001.
- L. Del Zanna, L. Matteini, S. Landi1, A. Verdini1, and M. Velli. Parametric decay of parallel and oblique Alfvén waves in the expanding solar wind. J. Plasma Phys., 2015.
- L. Matteini, S. Landi, L. Del Zanna, M. Velli, and P. Hellinger. Parametric decay of linearly polarized shear Alfvén waves in oblique propagation: One and two-dimensional hybrid simulations. Geophys. Res. Lett., 37:20101, October 2010.
- Y. Lithwick and P. Goldreich. Imbalanced weak magnetohydrodynamic turbulence. Astrophys. J., 582:1220–1240, 2003.
- T. S. Horbury, M. Forman, and S. Oughton. Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett., 101:175005, Oct 2008.
- J. J. Podesta. Dependence of Solar-Wind Power Spectra on the Direction of the Local Mean Magnetic Field. Astrophys. J., 698:986–999, June 2009.
- R. T. Wicks, T. S. Horbury, C. H. K. Chen, and A. A. Schekochihin. Power and spectral index anisotropy of the entire inertial range of turbulence in the fast solar wind. Mon. Not. Roy. Astron. Soc., 407:L31–L35, September 2010.
- M. A. Forman, R. T. Wicks, and T. S. Horbury. Detailed Fit of ”Critical Balance” Theory to Solar Wind Turbulence Measurements. Astrophys. J., 733:76–+, June 2011.
- J. M. TenBarge and G. G. Howes. Evidence of critical balance in kinetic Alfvén wave turbulence simulations. Phys. Plasmas, 19(5):055901, May 2012.
- J. M. TenBarge, G. G. Howes, and W. Dorland. Collisionless Damping at Electron Scales in Solar Wind Turbulence. Astrophys. J., 774:139, September 2013.
- W. H. Matthaeus, S. Oughton, S. Ghosh, and M. Hossain. Scaling of Anisotropy in Hydromagnetic Turbulence. Phys. Rev. Lett., 81:2056–2059, September 1998.
- J. A. Tessein, C. W. Smith, B. T. MacBride, W. H. Matthaeus, M. A. Forman, and J. E. Borovsky. Spectral Indices for Multi-Dimensional Interplanetary Turbulence at 1 AU. Astrophys. J., 692:684–693, February 2009.
- R. Grappin and W.-C. Müller. Scaling and anisotropy in magnetohydrodynamic turbulence in a strong mean magnetic field. Phys. Rev. E, 82(2):026406–+, August 2010.
- G. G. Howes, K. G. Klein, and J. M. TenBarge. The Quasilinear Premise for the Modeling of Plasma Turbulence. ArXiv e-prints, April 2014. 1404.2913.
- Y. Narita, S. P. Gary, S. Saito, K.-H. Glassmeier, and U. Motschmann. Dispersion relation analysis of solar wind turbulence. Geophys. Res. Lett., 38:L05101, March 2011.
- O. W. Roberts, X. Li, and B. Li. Kinetic Plasma Turbulence in the Fast Solar Wind Measured by Cluster. Astrophys. J., 769:58, May 2013.
- R. Bruno and V. Carbone. The Solar Wind as a Turbulence Laboratory. Living Reviews in Solar Physics, 2:4, September 2005.
- M. O. Riazantseva, V. P. Budaev, G. N. Zastenker, L. M. Zeleniy, G. P. Pavlos, J. Safrankova, Z. Nemecek, and L. Prech. Dynamic properties of small scale solar wind plasma fluctuations. Phil. Trans. Roy. Soc. A, 2015. this issue.
- S. Servidio, K. T. Osman, F. Valentini, D. Perrone, F. Califano, S. Chapman, W. H. Matthaeus, and P. Veltri. Proton Kinetic Effects in Vlasov and Solar Wind Turbulence. Astrophys. J. Lett., 781:L27, February 2014.
- J. M. TenBarge, R. T. Wicks, and G. G. Howes. Radial Evolution of the Solar Wind Temperature Anisotropy Dependence on Magnetic Field Angle. Astrophys. J., 2014. in preparation.
- S. P. Gary and M. A. Lee. The ion cyclotron anisotropy instability and the inverse correlation between proton anisotropy and proton beta. J. Geophys. Res., 99:11297–11302, June 1994.
- S. P. Gary, R. M. Skoug, J. T. Steinberg, and C. W. Smith. Proton temperature anisotropy constraint in the solar wind: ACE observations. Geophys. Res. Lett., 28:2759–2762, July 2001.
- J. C. Kasper, A. J. Lazarus, and S. P. Gary. Wind/SWE observations of firehose constraint on solar wind proton temperature anisotropy. Geophys. Res. Lett., 29:20–1, September 2002.
- E. Marsch, X.-Z. Ao, and C.-Y. Tu. On the temperature anisotropy of the core part of the proton velocity distribution function in the solar wind. J. Geophys. Res., 109:4102, April 2004.
- P. Hellinger, P. Trávníček, J. C. Kasper, and A. J. Lazarus. Solar wind proton temperature anisotropy: Linear theory and WIND/SWE observations. Geophys. Res. Lett., 33:9101–+, May 2006.
- L. Matteini, S. Landi, P. Hellinger, F. Pantellini, M. Maksimovic, M. Velli, B. E. Goldstein, and E. Marsch. Evolution of the solar wind proton temperature anisotropy from 0.3 to 2.5 AU. Geophys. Res. Lett., 34:20105, October 2007.
- S. D. Bale, J. C. Kasper, G. G. Howes, E. Quataert, C. Salem, and D. Sundkvist. Magnetic Fluctuation Power Near Proton Temperature Anisotropy Instability Thresholds in the Solar Wind. Phys. Rev. Lett., 103:211101, 2009.
- C. F. Kennel and H. E. Petscheck. Limit on stably trapped particle fluxes. J. Geophys. Res., 71:1–28, 1966.
- S. P. Gary, M. D. Montgomery, W. C. Feldman, and D. W. Forslund. Proton temperature anisotropy instabilities in the solar wind. J. Geophys. Res., 81:1241–1246, March 1976.
- P. Hellinger and H. Matsumoto. New kinetic instability: Oblique Alfvén fire hose. J. Geophys. Res., 105:10519–10526, May 2000.
- A. A. Vedenov and R. Z. Sagdeev. Some properties of a plasma with an anisotropic ion velocity distribution in a magnetic field. In M. A. Leontovich, editor, Plasma Physics and the Problem of Controlled Thermonuclear Reactions, volume 3, page 332. Pergamon, New York, 1958.
- M. Tajiri. Propagation of Hydromagnetic Waves in Collisionless Plasma. II. Kinetic Approach. Journal of the Physical Society of Japan, 22:1482, June 1967.
- D. J. Southwood and M. G. Kivelson. Mirror instability. I - Physical mechanism of linear instability. J. Geophys. Res., 98:9181–9187, June 1993.
- S. Boldyrev. Spectrum of Magnetohydrodynamic Turbulence. Phys. Rev. Lett., 96(11):115002–+, March 2006.
- C. Lacombe, O. Alexandrova, L. Matteini, O. Santolík, N. Cornilleau-Wehrlin, A. Mangeney, Y. de Conchy, and M. Maksimovic. Whistler Mode Waves and the Electron Heat Flux in the Solar Wind: Cluster Observations. Astrophys. J., 796:5, November 2014.
- J. J. Podesta and S. P. Gary. Magnetic Helicity Spectrum of Solar Wind Fluctuations as a Function of the Angle with Respect to the Local Mean Magnetic Field. Astrophys. J., 734:15–+, June 2011.
- J. He, E. Marsch, C. Tu, S. Yao, and H. Tian. Possible Evidence of Alfvén-cyclotron Waves in the Angle Distribution of Magnetic Helicity of Solar Wind Turbulence. Astrophys. J., 731:85, April 2011.
- J. J. Podesta and S. P. Gary. Effect of Differential Flow of Alpha Particles on Proton Pressure Anisotropy Instabilities in the Solar Wind. Astrophys. J., 742:41, November 2011.
- D. Biskamp and H. Welter. Dynamics of decaying two-dimensional magnetohydrodynamic turbulence. Phys. Fluids B, 1:1964–1979, October 1989.
- S. R. Spangler. Magnetohydrodynamic turbulence and enhanced atomic processes in astrophysical plasmas. Phys. Plasmas, 5:3006–3020, August 1998.
- S. R. Spangler. Two-dimensional Magnetohydrodynamics and Interstellar Plasma Turbulence. Astrophys. J., 522:879–896, September 1999.
- D. Biskamp and W.-C. Müller. Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence. Phys. Plasmas, 7:4889–4900, December 2000.
- J. A. Merrifield, W.-C. Müller, S. C. Chapman, and R. O. Dendy. The scaling properties of dissipation in incompressible isotropic three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas, 12(2):022301, February 2005.
- M. Wan, W. H. Matthaeus, H. Karimabadi, V. Roytershteyn, M. Shay, P. Wu, W. Daughton, B. Loring, and S. C. Chapman. Intermittent Dissipation at Kinetic Scales in Collisionless Plasma Turbulence. Phys. Rev. Lett., 109(19):195001, November 2012.
- H. Karimabadi, V. Roytershteyn, M. Wan, W. H. Matthaeus, W. Daughton, P. Wu, M. Shay, B. Loring, J. Borovsky, E. Leonardis, S. C. Chapman, and T. K. M. Nakamura. Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. Phys. Plasmas, 20(1):012303, January 2013.
- J. M. TenBarge and G. G. Howes. Current Sheets and Collisionless Damping in Kinetic Plasma Turbulence. Astrophys. J. Lett., 771:L27, July 2013.
- P. Wu, S. Perri, K. Osman, M. Wan, W. H. Matthaeus, M. A. Shay, M. L. Goldstein, H. Karimabadi, and S. Chapman. Intermittent Heating in Solar Wind and Kinetic Simulations. Astrophys. J. Lett., 763:L30, February 2013.
- V. Zhdankin, D. A. Uzdensky, J. C. Perez, and S. Boldyrev. Statistical Analysis of Current Sheets in Three-dimensional Magnetohydrodynamic Turbulence. Astrophys. J., 771:124, July 2013.
- K. T. Osman, W. H. Matthaeus, A. Greco, and S. Servidio. Evidence for Inhomogeneous Heating in the Solar Wind. Astrophys. J. Lett., 727:L11+, January 2011.
- J. E. Borovsky and M. H. Denton. No Evidence for Heating of the Solar Wind at Strong Current Sheets. Astrophys. J. Lett., 739:L61, October 2011.
- K. T. Osman, W. H. Matthaeus, M. Wan, and A. F. Rappazzo. Intermittency and Local Heating in the Solar Wind. Phys. Rev. Lett., 108(26):261102, June 2012.
- K. T. Osman, W. H. Matthaeus, B. Hnat, and S. C. Chapman. Kinetic Signatures and Intermittent Turbulence in the Solar Wind Plasma. Phys. Rev. Lett., 108(26):261103, June 2012.
- S. Perri, M. L. Goldstein, J. C. Dorelli, and F. Sahraoui. Detection of Small-Scale Structures in the Dissipation Regime of Solar-Wind Turbulence. Phys. Rev. Lett., 109(19):191101, November 2012.
- X. Wang, C. Tu, J. He, E. Marsch, and L. Wang. On Intermittent Turbulence Heating of the Solar Wind: Differences between Tangential and Rotational Discontinuities. Astrophys. J. Lett., 772:L14, August 2013.
- K. T. Osman, W. H. Matthaeus, J. T. Gosling, A. Greco, S. Servidio, B. Hnat, S. C. Chapman, and T. D. Phan. Magnetic Reconnection and Intermittent Turbulence in the Solar Wind. Phys. Rev. Lett., 112(21):215002, May 2014.
- R. Meyrand, K. H. Kiyani, and S. Galtier. Intermittency in Weak Magnetohydrodynamic Turbulence. ArXiv e-prints, September 2014.
- R. Numata, G. G. Howes, T. Tatsuno, M. Barnes, and W. Dorland. AstroGK: Astrophysical gyrokinetics code. J. Comp. Phys., 229:9347, 2010.
- S. A. Orszag and C.-M. Tang. Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech., 90:129–143, January 1979.
- J. R. Johnson and C. Z. Cheng. Stochastic ion heating at the magnetopause due to kinetic Alfvén waves. Geophys. Res. Lett., 28:4421–4424, December 2001.
- L. Chen, Z. Lin, and R. White. On resonant heating below the cyclotron frequency. Phys. Plasmas, 8:4713–4716, November 2001.
- R. White, L. Chen, and Z. Lin. Resonant plasma heating below the cyclotron frequency. Phys. Plasmas, 9:1890–1897, May 2002.
- Y. Voitenko and M. Goossens. Excitation of kinetic Alfvén turbulence by MHD waves and energization of space plasmas. Nonlin. Proc. Geophys., 11:535–543, November 2004.
- S. Bourouaine, E. Marsch, and C. Vocks. On the Efficiency of Nonresonant Ion Heating by Coronal Alfvén Waves. Astrophys. J. Lett., 684:L119–L122, September 2008.
- B. D. G. Chandran. Alfvén-wave Turbulence and Perpendicular Ion Temperatures in Coronal Holes. Astrophys. J., 720:548–554, September 2010.
- B. D. G. Chandran, T. J. Dennis, E. Quataert, and S. D. Bale. Incorporating Kinetic Physics into a Two-fluid Solar-wind Model with Temperature Anisotropy and Low-frequency Alfvén-wave Turbulence. Astrophys. J., 743:197, December 2011.
- S. Bourouaine and B. D. G. Chandran. Observational Test of Stochastic Heating in Low- Fast-solar-wind Streams. Astrophys. J., 774:96, September 2013.
- P. Dmitruk, W. H. Matthaeus, and N. Seenu. Test Particle Energization by Current Sheets and Nonuniform Fields in Magnetohydrodynamic Turbulence. Astrophys. J., 617:667–679, December 2004.
- S. A. Markovskii and B. J. Vasquez. A Short-timescale Channel of Dissipation of the Strong Solar Wind Turbulence. Astrophys. J., 739:22, September 2011.
- W. H. Matthaeus and M. Velli. Who Needs Turbulence?. A Review of Turbulence Effects in the Heliosphere and on the Fundamental Process of Reconnection. Space Sci. Rev., 160:145–168, October 2011.
- S. Servidio, A. Greco, W. H. Matthaeus, K. T. Osman, and P. Dmitruk. Statistical association of discontinuities and reconnection in magnetohydrodynamic turbulence. J. Geophys. Res., 116:9102, September 2011.
- K. T. Osman, K. H. Kiyani, S. C. Chapman, and B. Hnat. Anisotropic Intermittency of Magnetohydrodynamic Turbulence. Astrophys. J. Lett., 783:L27, March 2014.
- M. L. Goldstein, R. T. Wicks, S. Perri, , and F. Sahraoui. Kinetic scale turbulence and dissipation in the solar wind: Key observational results and future outlook. Phil. Trans. Roy. Soc. A, 2015. this issue.
- M. L. Goldstein, D. A. Roberts, and W. H. Matthaeus. Magnetohydrodynamic Turbulence In The Solar Wind. Ann. Rev. Astron. Astrophys., 33:283–326, 1995.
- K. H. Kiyani, S. C. Chapman, Y. V. Khotyaintsev, M. W. Dunlop, and F. Sahraoui. Global scale-invariant dissipation in collisionless plasma turbulence. Phys. Rev. Lett., 103:075006, 2009.
- O. Alexandrova, J. Saur, C. Lacombe, A. Mangeney, J. Mitchell, S. J. Schwartz, and P. Robert. Universality of Solar-Wind Turbulent Spectrum from MHD to Electron Scales. Phys. Rev. Lett., 103(16):165003–+, October 2009.
- G. G. Howes. Inertial range turbulence in kinetic plasmas. Phys. Plasmas, 15(5):055904, May 2008.
- G. G. Howes, J. M. Tenbarge, and W. Dorland. A weakened cascade model for turbulence in astrophysical plasmas. Phys. Plasmas, 18:102305, 2011.
- J. J. Podesta. Evidence of Kinetic Alfvén Waves in the Solar Wind at 1 AU. Sol. Phys., 286:529–548, September 2013.
- T. Tatsuno, A. A. Schekochihin, W. Dorland, G. Plunk, M. A. Barnes, S. C. Cowley, and G. G. Howes. Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence. Phys. Rev. Lett., 103(1):015003, 2009.
- G. G. Plunk, S. C. Cowley, A. A. Schekochihin, and T. Tatsuno. Two-dimensional gyrokinetic turbulence. Journal of Fluid Mechanics, 664:407–435, December 2010.
- G. G. Plunk and T. Tatsuno. Energy Transfer and Dual Cascade in Kinetic Magnetized Plasma Turbulence. Phys. Rev. Lett., 106(16):165003–+, April 2011.
- E. Kawamori. Experimental Verification of Entropy Cascade in Two-Dimensional Electrostatic Turbulence in Magnetized Plasma. Phys. Rev. Lett., 110(9):095001, March 2013.
- O. Alexandrova, C. Lacombe, A. Mangeney, R. Grappin, and M. Maksimovic. Solar Wind Turbulent Spectrum at Plasma Kinetic Scales. Astrophys. J., 760:121, December 2012.
- Y. W. Jiang, S. Liu, and V. Petrosian. Cascade and Damping of Alfvén-Cyclotron Fluctuations: Application to Solar Wind Turbulence. Astrophys. J., 698:163–183, June 2009.
- A. Schreiner and J. Saur. Turbulence in the Solar Wind - a Model for Dissipation by Kinetic Alfvén-waves. In EGU General Assembly Conference Abstracts, volume 16 of EGU General Assembly Conference Abstracts, page 1778, May 2014.
- B. D. G. Chandran, E. Quataert, G. G. Howes, Q. Xia, and P. Pongkitiwanichakul. Constraining Low-Frequency Alfvénic Turbulence in the Solar Wind Using Density-Fluctuation Measurements. Astrophys. J., 707:1668–1675, December 2009.
- R. Bruno, L. Trenchi, and D. Telloni. Spectral Slope Variation at Proton Scales from Fast to Slow Solar Wind. Astrophys. J. Lett., 793:L15, September 2014.
- T. N. Parashar, M. A. Shay, P. A. Cassak, and W. H. Matthaeus. Kinetic dissipation and anisotropic heating in a turbulent collisionless plasma. Phys. Plasmas, 16(3):032310–+, March 2009.
- N. F. Loureiro, A. A. Schekochihin, and A. Zocco. Fast Collisionless Reconnection and Electron Heating in Strongly Magnetized Plasmas. Phys. Rev. Lett., 111(2):025002, July 2013.
- R. Numata and N. F. Loureiro. Ion and electron heating during magnetic reconnection in weakly collisional plasmas. ArXiv e-prints, June 2014.
- J. F. Drake, M. Swisdak, C. Cattell, M. A. Shay, B. N. Rogers, and A. Zeiler. Formation of Electron Holes and Particle Energization During Magnetic Reconnection. Science, 299:873–877, February 2003.
- P. L. Pritchett and F. V. Coroniti. Three-dimensional collisionless magnetic reconnection in the presence of a guide field. J. Geophys. Res., 109:1220, January 2004.
- J. Egedal, W. Fox, N. Katz, M. Porkolab, M. ØIeroset, R. P. Lin, W. Daughton, and J. F. Drake. Evidence and theory for trapped electrons in guide field magnetotail reconnection. J. Geophys. Res., 113:12207, December 2008.
- J. Egedal, W. Daughton, J. F. Drake, N. Katz, and A. Lê. Formation of a localized acceleration potential during magnetic reconnection with a guide field. Phys. Plasmas, 16(5):050701, May 2009.
- J. Egedal, A. Lê, Y. Zhu, W. Daughton, M. Øieroset, T. Phan, R. P. Lin, and J. P. Eastwood. Cause of super-thermal electron heating during magnetotail reconnection. Geophys. Res. Lett., 37:10102, May 2010.
- J. Egedal, W. Daughton, and A. Le. Large-scale electron acceleration by parallel electric fields during magnetic reconnection. Nature Physics, 8:321–324, April 2012.
- J. F. Drake, M. Swisdak, H. Che, and M. A. Shay. Electron acceleration from contracting magnetic islands during reconnection. Nature, 443:553–556, October 2006.
- A. Lazarian, G. Eyink, E. Vishniac, and G. Kowa. Turbulent Reconnection and its Implications. Phil. Trans. Roy. Soc. A, 2015. this issue.
- W. H. Matthaeus, Minping Wan, S. Servidio, A. Greco, K. T. Osman, S. Oughton, and P. Dmitruk. Intermittency, Nonlinear Dynamics, and Dissipation in the Solar Wind and Astrophysical Plasmas. Phil. Trans. Roy. Soc. A, 2015. this issue.
- S. Boldyrev, J. C. Perez, J. E. Borovsky, and J. J. Podesta. Spectral Scaling Laws in Magnetohydrodynamic Turbulence Simulations and in the Solar Wind. Astrophys. J. Lett., 741:L19, November 2011.
- V. Zhdankin, S. Boldyrev, J. Mason, and J. C. Perez. Magnetic Discontinuities in Magnetohydrodynamic Turbulence and in the Solar Wind. Phys. Rev. Lett., 108(17):175004, April 2012.
- J. E. Borovsky. Flux tube texture of the solar wind: Strands of the magnetic carpet at 1 AU? J. Geophys. Res., 113:8110, August 2008.
- J. E. Borovsky. Contribution of Strong Discontinuities to the Power Spectrum of the Solar Wind. Phys. Rev. Lett., 105(11):111102, September 2010.
- B. Miao, B. Peng, and G. Li. Current sheets from Ulysses observation. Ann. Geophys., 29:237–249, February 2011.
- L. Arnold, G. Li, X. Li, and Y. Yan. Observation of Flux-tube Crossings in the Solar Wind. Astrophys. J., 766:2, March 2013.
- J. C. Perez, J. Mason, S. Boldyrev, and F. Cattaneo. On the Energy Spectrum of Strong Magnetohydrodynamic Turbulence. Phys. Rev.X, 2(4):041005, October 2012.
- K. G. Klein and G. G. Howes. The Impact of Proton Temperature Anisotropy on Solar Wind Turbulence. Phys. Plasmas, 2015. submitted.
- S. Servidio, F. Valentini, D. Perrone, A. Greco, F. Califano, W. H. Matthaeus, and P. Veltri. A kinetic model of plasma turbulence. Journal of Plasma Physics, FirstView:1–35, 12 2014.