As indicated by its title, "Electron demagnetization and heating in quasi-perpendicular shocks, " the main point of our work has been to show that electrons are not magnetized while crossing the bow shock. While this conclusion is implied in earlier work that is referenced in our paper and in the Schwartz reply, the vast majority of work on electron heating at shocks has been based on the assumption that shock-crossing electrons are magnetized and satisfy the first adiabatic invariant. Thus, our work offers an alternative view of electron heating at the terrestrial bow shock.The main thrust of Schwartz's comment concerns adiabatic electron behavior, which is irrelevant to our main conclusion because the electrons are not magnetized. The comment provides other interesting and useful insights. In particular, Schwartz's discussion of the difference between the adiabatic behavior of a single electron and an entire distribution clarifies misinterpretations in ours and others discussions of adiabatic behavior (response to Schwartz points 1, 2, and 4).Schwartz points out that demagnetization would inhibit the electron drift along the shock and asks what would then limit the amount of energy gained by the cross-shock potential. The behavior of demagnetized electrons in turbulence is a complex problem. The strong electric field fluctuations we observe have frequencies around the electron gyrofrequency. Some aspects of the behavior of the electrons can perhaps be explained by electron cyclotron drift waves [Muschietti and Lembège, 2013], where stochastic demagnetized electron orbits in strong turbulence can play a major role [Dupree, 1966]. The effect of cyclotron waves for scattering of electrons has been extensively studied in the past [see, e.g., Horne and Thorne, 2000; Guest, 2009, and references therein]. In such strong turbulence, it is plausible that electrons are heated equally in perpendicular and parallel directions by a fraction of the cross-shock potential (response to Schwartz point 3).Schwartz discusses the deHoffman-Teller (dHT) and Normal Incidence (NIF) frames of reference, along with his theoretical preference for the dHT frame in which the perpendicular electric field is zero. It is worth noting that experimentally, the preference for one frame over the other is less evident. Figure 2 of our paper gives the electric field components (in Figures 2e-2g) in what is described as the NIF frame in the paper but is actually in the dHT frame (due to an error in the figure sent to the journal). The dHT frame of reference was found by transforming to the frame tied to the plasma and, in which, the perpendicular electric field in the solar wind was zero, as can be verified by the zero values of the field components of Figures 2e-2g after 20 s. While the theoretical expectation in this frame is that the perpendicular electric field is also zero in the shock ramp and downstream, the experimental data does not generally behave in this manner. As shown in Figure 2, the perpendicular field in the ramp was as large as 3...

A novel state of turbulent plasma characterized by small scale phase-space granulations called “clumps” is proposed. Clumps are produced when regions of different phase space density are mixed by the fluctuating electric field. They move along ballistic orbits and drive the turbulent field in a manner similar to that in which thermal fluctuations are driven by particle discreteness. In the coherent wave limit the clumps become the familiar trapped particle eddies of a Bernstein-Green-Kruskal mode. The turbulent state can exist in the absence of linear instability although it is more likely to occur in a linearly unstable plasma. The spectrum contains a ballistic portion as well as resonances at the wave (collective) frequencies. The discreteness of the clumps produces collision-like process. For example, the average distribution function satisfies a Fokker-Planck equation instead of a quasilinear diffusion equation.

The turbulent plasma state which develops from unstable, current-driven drift waves is analyzed. In the nonlinear theory, the wave growth predicted by the linear theory is ultimately suppressed by ion damping. Since the phase speeds of the unstable waves are much greater than the ion thermal velocity, the ions cannot absorb wave energy until they become trapped. The amplitude of the turbulent spectrum grows until trapping occurs, and a quasi-steady state is reached in which the directed electron energy is converted into ion thermal energy at a constant rate. In this state the perturbation of the density gradient due to the turbulence is equal to the mean gradient. Nonlinear limitation due to mode coupling does not suppress the unstable wave growth until much larger density perturbations have developed. Therefore, ion trapping is established as the controlling nonlinear mechanism. In the steady state, the ions diffuse across the magnetic field with a diffusion coefficient D⊥ ≈ (k⊥ −2γ)max, where γ is the growth rate predicted by the linear theory and k⊥ is a perpendicular wavenumber. Although the detailed treatment is for a specific instability, the general conclusions appear to apply to a variety of drift waves.

A Bernstein–Green–Kruskal mode consisting of a depression or ’’hole’’ in the phase-space density is shown to be a state of maximum entropy subject to constant mass, momentum, and energy. The parameter space of such holes is studied. The maximum entropy property is used to develop a simplified approximate analytic method as well as to infer the results of hole collisions including coalescing and decay. The maximum entropy property suggests that random, turbulent fluctuations tend to form into such self-trapped structures and this nonperturbative concept is related to the physics of ’’clumps’’ which occur in a renormalized perturbative theory of turbulence.

An electron drift driven by an applied constant electric field causes an ion acoustic instability. A steady state is proposed in which ballistic clumps of plasma behave like dressed test particles and collisionally scatter each other. The applied field is balanced by the dynamical friction force on the clumps. The resulting conductivity is a« 10oj pe /kX D .In a collisionless plasma the resistivity is generally believed to be due to the scattering of particles by waves driven unstable by the applied field. However, because of the formation of plasma clumps 1 ' 2 it appears likely that a rather different state develops in which the electron and ion clumps behave like macroparticles and collisionally scatter each other.We assume that the plasma contains a uniform electric field E 0 (parallel to a strong magnetic field) which causes the electrons to acquire an average velocity u which in turn drives an ionacoustic instability. We further assume that a quasisteady state develops in which macroscopic properties such as the average distribution function/(v) change slowly compared with the ionacoustic frequency oo s -kv s ^h{T e /m^) 112 . (e = electron, i = ion; q e> m e , n e , and T e are electron charge, mass, average number density, and temperature; w Pe 2 = 4Tfn e q e 2 /m ef T e = jm e v e 2 , ^D 2 -T e /4Trn e q e 2 .) This problem has been considered by a number of authors. 3 ' 5 There are two basic questions to be answered. First, what ultimately limits the wave growth? Second, what, if anything, inhibits the electrons from freely accelerating? Several processes can limit wave growth; however, we shall assume that for a sufficiently large field E 0 the required energy and momentum transfer to the ions can be achieved only by trapping a portion of the ions in the waves,, 6 To answer the second question one must consider the types of wave-particle interactions that are possible. The conventional collisionless theory for/(v) provides only for diffusion. However, diffusion can retard only the positive-slope portion of f e (v) and then only for v&v s . For large wave amplitudes the resonance can be broadened or one can include higher-order diffusion and thereby involve a greater portion of velocity space in the resonance. But the negative-slope portion of f e remains a problem if diffusion is the only operative process. The negative-slope portion must be retained if a quasisteady state is to be achieved. What is needed is a drag or dy-namical friction force equal to -E 0 which will retard electrons independent of 8/ e /8v. Such a force, which has previously been omitted from the theory, can occur in collisionless plasma.To understand this force consider the development of the instability. As the electrons feed energy to the wave at the linear growth rate y e ~hf 1/2 (u/v e )(A) s , the wave grows and the ion waveparticle resonance width increases. Eventually some ions become resonant (trapped) and absorb wave energy at a rate y i sufficient to make the total nonlinear growth rate y NL vanish, i.e., y NL = y i +y ...