Limitations of Hall MHD as a model for turbulence in weakly collisional plasmas

Two complications arise when making a detailed comparison of standard Hall MHD with Vlasov-Maxwell kinetic theory. The first is the use of an isotropic equation of state (4) in standard Hall MHD. Condition (9) suggests that the electron response is so fast that, if the ion temperature is ignorable, the pressure response may be correctly treated as isothermal and isotropic (Hirose et al., 2004). But for finite ion temperatures, the ion response is anisotropic and appears to be better described with a double polytropic equation of state,

For example, a choice of $\gamma_{\perp}=2$ and $\gamma_{\parallel}=3$ corresponds to the double adiabatic, CGL closure for the ions (Chew et al., 1956; Abraham-Shrauner, 1973; Ballai et al., 2002). Even this more complicated closure, however, fails to yield a dispersion relation that is a rigorous limit of kinetic theory for a weakly collisional plasma with a finite ion temperature (Hirose et al., 2004). But, it appears that the slow wave branch is the most strongly affected by this complication; for example, the one-dimensional collisionless dynamics of ions along the magnetic field leads to a parallel sound speed $C_{s}^{2}=(T_{e}+3T_{i})/m_{i}$ (Hirose et al., 2004), and this produces unusual behavior, even at MHD scales $k\rho_{i}\ll 1$, such as a slow wave with a faster phase speed than the Alfvén wave (Ballai et al., 2002). The closest connection between kinetic theory and the standard Hall MHD dispersion relation appears to occur by defining an isotropic sound speed $v_{s}^{2}=(T_{e}+T_{i})/m_{i}$ (Hirose et al., 2004), so this is the definition taken here. Although this choice is not entirely satisfactory, we will see that the slow wave is strongly damped via wave-particle interactions when warm ions are present, so the differences arising from the details of the fluid closure are often irrelevant.

The second complication in comparing Hall MHD with kinetics is that the linear fluid modes of Hall MHD cannot be assigned a one-to-one correspondence with the linear kinetic modes over all parameter space (Orlowski et al., 1994; Krauss-Varban et al., 1994; Yoon and Fang, 2008). For example, in a study of low-frequency waves in the solar wind upstream of Venus, Orlowski et al. (1994) found a correspondence to the fast Hall MHD mode at low $\beta$ and to the Alfvén Hall MHD mode at high $\beta$, while the results were uniformly consistent with the fast mode of kinetic theory. A subsequent study comparing Hall MHD with kinetic theory found this mixing of modes to be commonplace (Krauss-Varban et al., 1994), perhaps accounting for the surprising changes in the linear wave properties of Hall MHD at small scales noted in a recent paper (Hameiri et al., 2005).

This mixing of solutions to the linear dispersion relations can lead to confusion about the connections between wave modes in different regimes of parameter space. For a chosen set of plasma parameters $\beta_{i}$, $T_{i}/T_{e}$, and $v_{ti}/c$, each solution to the dispersion relation describes a manifold in $(\mbox{Re}[\omega/\Omega_{i}],k_{\perp}d_{i},k_{\parallel}d_{i})$ space, termed a “dispersion surface” (Andre, 1985). A single dispersion surface may describe distinct wave modes (with different physical properties) in different regimes of wave vector space. For example, the Alfvén dispersion surface describes the MHD Alfvén wave at large scale ($k_{\perp}d_{i}\ll 1$ and $k_{\parallel}d_{i}\ll 1$), the ion cyclotron wave at small parallel scales ($k_{\perp}d_{i}\ll 1$ and $k_{\parallel}d_{i}\gtrsim 1$), and the kinetic Alfvén wave at small perpendicular scales ($k_{\perp}d_{i}\gtrsim 1/\sqrt{\beta_{i}}$ and $k_{\parallel}d_{i}\ll 1$) (Yoon and Fang, 2008). In this paper we avoid confusion by naming the solution branch of the dispersion relation according to the MHD mode to which it is topologically connected along the dispersion surface. Thus, we will investigate the properties of the slow (S), Alfvén (A), and fast (F) branch solutions of both the linear Hall MHD and Vlasov-Maxwell systems. Since the solutions of these branches in the two systems do not exhibit a one-to-one correspondence, we take the position that we want to compare the Hall MHD wave modes with the most similar kinetic mode at each point in parameter space, and will remark upon which modes were chosen when relevant.

To connect quantitatively Hall MHD to kinetic theory, we must specify the relation between $\beta_{0}$ in the Hall MHD dispersion relation and $\beta_{i}$ in the Vlasov-Maxwell dispersion relation. For a finite ion temperature, we choose to define $\beta_{0}=v_{s}^{2}/v_{A}^{2}$, where we take an isotropic sound speed $v_{s}^{2}=(T_{e}+T_{i})/m_{i}$. Thus, the necessary relation is given by $\beta_{0}=\beta_{i}(1+T_{i}/T_{e})/(2T_{i}/T_{e})$. Note that, in the limit $T_{i}\ll T_{e}$ in which Hall MHD is rigorously valid, the ion pressure is ignorable and so $v_{s}\rightarrow c_{s}=\sqrt{T_{e}/m_{i}}$, the ion acoustic speed; for turbulence in the solar wind, the ion and electron temperatures are comparable $T_{i}\sim T_{e}$, but standard Hall MHD has no means to account for distinct ion and electron temperatures, so we merely quote the value of $T_{i}/T_{e}$ used for the Vlasov-Maxwell calculation.

Hall MHD is a fluid theory of the behavior in a magnetized plasma, and therefore does not account for the effect of collisionless damping; the eigenfrequencies of the Hall MHD dispersion relation (5) are real, $\omega_{h}$. Therefore, in comparing with the complex eigenfrequencies of the Vlasov-Maxwell dispersion relation $\omega_{v}-i\gamma_{v}$, we must separately account for the error due to the differences in the real frequencies and the error due to the absence of damping in Hall MHD. Discrepancies in the real mode frequencies may yield Hall MHD results for a turbulent cascade that are quantitatively incorrect but still qualitatively correct with respect to a kinetic solution. But without capturing the often strong kinetic damping (via wave-particle interactions) occurring in a weakly collisional plasma, the Hall MHD model may give results for turbulence in the plasma that are both quantitatively and qualitatively incorrect.

We therefore will present two measures of comparison: first, the normalized difference between the Hall MHD frequency and the real part of the complex kinetic frequency $\Delta\omega\equiv|\omega_{h}-\omega_{v}|/\omega_{v}$; and, second, the normalized collisionless damping rate $\gamma_{v}/\omega_{v}$ in linear kinetic theory. In general , a normalized damping rate of $\gamma/\omega=1/(2\pi)\simeq 0.16$ is sufficient to reduce the amplitude of a wave mode by $e^{-1}\simeq 0.37$ over a single wave period. The position is taken in this work that any damping rate $\gamma/\omega\geq 0.1$ is sufficient to cause qualitative differences in the dynamics.

First, we wish to verify that standard Hall MHD and Vlasov-Maxwell kinetic theory do agree in the limit $T_{i}\ll T_{e}$. We choose the plasma beta $\beta_{0}=1$ and take $T_{i}/T_{e}=10^{-3}$, yielding an ion plasma beta in the kinetic theory $\beta_{i}\simeq 2\times 10^{-3}$. Figure 1 presents real frequencies (dashed line for Hall MHD, dotted line for Vlasov-Maxwell) in the top panel and damping rates (solid line for Vlasov-Maxwell) in the bottom panel for parallel wavenumbers $10^{-2}\leq k_{\parallel}d_{i}\leq 10$ and perpendicular wavenumber $k_{\perp}d_{i}=10^{-2}$. The solution labeled F is the fast magnetosonic wave at $k_{\parallel}d_{i}\ll 1$ and the parallel whistler wave at $k_{\parallel}d_{i}\gtrsim 1$. At $k_{\parallel}d_{i}\ll 1$, the solution A is the shear Alfvén wave and the solution S is the slow wave, often termed the ion acoustic wave in this cold ion limit. In Figure 1, the inset in the top panel shows an expanded view of the region around $k_{\parallel}d_{i}\sim 0.1$, demonstrating that neither the Hall MHD nor the kinetic solutions cross. An examination of the linear kinetic eigenfunctions (not shown) shows that the A and S solution branches undergo a mode conversion (Swanson, 1989; Stix, 1992) at $k_{\parallel}d_{i}\simeq 0.09$, exchanging physical characteristics with each other. Thus, at $k_{\parallel}d_{i}\gtrsim 0.1$, the A branch corresponds to the slow wave and the S branch corresponds to the ion cyclotron wave. This mixing of the A and S branches occurs commonly in Hall MHD (Krauss-Varban et al., 1994; Yoon and Fang, 2008). In this limit $T_{i}\ll T_{e}$, the real frequencies for all three modes are in excellent agreement; damping is weak for all modes over this range except for the S branch at $k_{\parallel}d_{i}>3$, due to cyclotron damping of the corresponding ion cyclotron waves.

Figure 2 presents real frequencies and damping rates for perpendicular wavenumbers $10^{-2}\leq k_{\perp}d_{i}\leq 10$ and parallel wavenumber $k_{\parallel}d_{i}=10^{-2}$. The Hall MHD slow wave solution S agrees well with the weakly damped kinetic solution. The Alfvén wave mode A also shows good agreement, but after it transitions to the kinetic Alfvén wave at $k_{\perp}d_{i}\sim 1$, damping via the Landau resonance becomes significant at $k_{\perp}d_{i}>4$. The fast wave solution F from kinetic theory, on the other hand, deviates from the Hall MHD solution at $k_{\perp}d_{i}>0.5$; here the numerical solver used is unable to follow the fast wave root as it undergoes a mode conversion into an ion Bernstein mode. Bernstein waves arise due to finite Larmor radius effects that have no counterpart in cold plasma theory and that are not described by Hall MHD (Swanson, 1989); the impact of the ion Bernstein waves is discussed at length in §4.3.

Despite some of these difficulties with ion Bernstein waves and non-negligible kinetic damping, Figures 1 and 2 bear out the general finding (Hirose et al., 2004; Ito et al., 2004) that Hall MHD is a valid limit of kinetic theory in the cold ion approximation, $T_{i}\ll T_{e}$.

Next, we explore the limit of warm ions relevant to most space and astrophysical plasmas. We take $\beta_{0}=1$ and $T_{i}/T_{e}=1$, giving an ion plasma beta in the kinetic theory $\beta_{i}=1$. Figure 3 presents real frequencies (top) and damping rates (bottom) for parallel wavenumbers $10^{-2}\leq k_{\parallel}d_{i}\leq 10$ and perpendicular wavenumber $k_{\perp}d_{i}=10^{-2}$. The frequency of the kinetic fast wave F (dotted) is well reproduced by Hall MHD solution F’ (dashed), and its damping is weak except for a small region around $k_{\parallel}d_{i}\sim 2\times 10^{-2}$. Comparison of the slow and Alfvén wave solutions demonstrates some of the complications arising in a comparison of Hall MHD to kinetic theory as described in §3.1. The kinetic slow wave solution S (dotted) has a phase speed that is faster than that of the Alfvén wave solution A (dotted) of kinetic theory (this is perhaps more easily seen Figure 4); the anisotropic ion pressure response in the kinetic theory that leads to this curious result (Ballai et al., 2002) is not captured by the isotropic equation of state used in the standard Hall MHD model. But this error is overshadowed by a far more serious discrepancy between Hall MHD and kinetic theory: at finite ion temperature the kinetic slow wave S (dotted)—the ion acoustic wave—is heavily damped with $\gamma_{v}/\omega_{v}\sim 1$, yet a fluid theory such as Hall MHD does not account for this damping. Therefore, at finite ion temperature the Hall MHD slow wave represents an unphysical, spurious wave that does not exist in a weakly collisional plasma.

The second complication apparent in Figure 3 is the mixing of the modes between the solution branches of kinetic theory and those of Hall MHD (Krauss-Varban et al., 1994; Yoon and Fang, 2008). The inset in the top panel of Figure 3 shows that the Hall MHD solutions do not cross, however the correspondence between kinetic A and S branches and Hall MHD A’ and S’ branches changes with increasing $k_{\parallel}d_{i}$: at $k_{\parallel}d_{i}\ll 1$, the solution A (dotted) of kinetic theory agrees closely with the Hall MHD solution A’ (dashed); but at $k_{\parallel}d_{i}>0.1$, the solution A (dotted) from kinetic theory (the ion cyclotron wave) appears to correspond to the Hall MHD solution S’ (dashed) and the kinetic solution S (dotted) appears to correspond most closely with the Hall MHD solution A’. This mixing up of fluid and kinetic modes complicates any attempted comparison of the two models, but a careful matching of the most similar wave modes will yield a fair evaluation. Again, however, the major discrepancy between the models is that Hall MHD does not capture the strong kinetic damping of the kinetic S branch at all scales and of the solution A from kinetic theory (the ion cyclotron wave) at $k_{\parallel}d_{i}>0.4$ .

Figure 4 presents real frequencies and damping rates for perpendicular wavenumbers $10^{-2}\leq k_{\perp}d_{i}\leq 10$ and parallel wavenumber $k_{\parallel}d_{i}=10^{-2}$ for the warm ion case with $\beta_{0}=1$ and $T_{i}/T_{e}=1$. The kinetic branch S is once again strongly damped by the warm ions, so Hall MHD supports a spurious, undamped slow wave mode S’. The kinetic branch A, representing MHD Alfvén waves and kinetic Alfvén waves, is quite well represented over the entire wavenumber range, with some discrepancy near $k_{\perp}d_{i}\sim 1$. The kinetic fast wave solution F is well matched by the Hall MHD fast wave solution F’, with only a slight deviation, for $k_{\perp}d_{i}\lesssim 0.4$. At $k_{\perp}d_{i}=0.4$, however, the kinetic branch F goes through a mode conversion to become an ion Bernstein wave, a situation discussed at length in §4.3.

In summary, at finite ion temperature, Hall MHD does a good job modeling the kinetic fast wave branch F for nearly parallel wave vectors and the kinetic Alfvén wave branch for nearly perpendicular wave vectors. But, it fails to capture the strong kinetic damping of the slow mode branch in all cases and of the Alfvén branch in the nearly parallel case. The correspondence of the Hall MHD fast wave branch F’ at nearly perpendicular wave vectors to a series of kinetic ion Bernstein waves is discussed next.

The fast solution F from kinetic theory in Figure 4 deviates greatly from the Hall MHD fast solution F’ at $k_{\perp}d_{i}\gtrsim 0.4$. The cause of this deviation is the conversion, in kinetic theory, from an MHD fast wave to an $n=1$ ion Bernstein wave (with $\omega<\Omega_{i}$) on the dispersion surface of solution branch F. Not shown in Figure 4 is the conversion of an $n=1$ ion Bernstein wave (with $\omega>\Omega_{i}$) to a continuation of the fast wave at $\omega>\Omega_{i}$; this wave lies on a different dispersion surface from branch F. Mode conversion from fast wave solutions to ion Bernstein waves at nearly perpendicular propagation is a well known physical phenomenon (Swanson, 1989; Stix, 1992; Li and Habbal, 2001). Here we aim to evaluate the accuracy of the Hall MHD description of the kinetic plasma behavior in this regime.

In Figure 5 are presented the real frequencies (top panel) and damping rates (bottom panel) for the Hall MHD fast branch F’ (dashed), the kinetic fast branch F (dotted), and the kinetic $n=1$ through $n=9$ ion Bernstein solutions for perpendicular wavenumbers $10^{-2}\leq k_{\perp}d_{i}\leq 10$ and parallel wavenumber $k_{\parallel}d_{i}=5\times 10^{-2}$ for $\beta_{0}=1$ and $T_{i}/T_{e}=1$. Note that each of the ion Bernstein solutions lie on distinct dispersion surfaces. In the top panel, the relation between the Hall MHD branch F’ and the series of ion Bernstein wave solutions is clear, yet the summed effect of these kinetic wave modes differs significantly in a number of ways from the Hall MHD fast branch solution F’. First, between each of the ion Bernstein solutions lies a frequency gap in which no solution exists. Second, the real frequency of the ion Bernstein waves (and thus the perpendicular phase velocity $\omega/k_{\perp}$) is only coincident with the frequency given by Hall MHD at a single point for each of the the $n\geq 2$ Bernstein modes. Finally, although the perpendicular group velocity (the slope of the plot $\partial\omega/\partial k_{\perp}$) of the Hall MHD F’ solution is always positive and increasing with $k_{\perp}d_{i}$, each of the ion Bernstein waves have a positive but smaller group velocity at lower $k_{\perp}d_{i}$, and a negative group velocity at higher $k_{\perp}d_{i}$, corresponding to a backward wave (Stix, 1992) at these scales. The lower panel of Figure 5 shows that the kinetic fast branch F and $n=1$ through $n=9$ ion Bernstein waves are not strongly damped. Our conclusion is that the kinetic dynamics of the fast branch for nearly perpendicular wave vectors in a weakly collisional plasma is not well represented by Hall MHD.

The axisymmetric nature of any uniform magnetized plasma about the direction of a uniform, straight magnetic field $\mathbf{B}_{0}$ reduces the dependence on the wave vector $\mathbf{k}$ to its two components perpendicular and parallel to the magnetic field, $(k_{\perp},k_{\parallel})$; this is clearly seen in the dispersion relations of both the Hall MHD and the Vlasov-Maxwell systems with their dependence on the parameters $k_{\perp}d_{i}$ and $k_{\parallel}d_{i}$. Therefore, plots of the real frequency difference $\Delta\omega$ and of the kinetic damping rate $\gamma_{v}/\omega_{v}$ on the plane $(k_{\perp},k_{\parallel})$ provide excellent quantitative measures of the fidelity of Hall MHD to the kinetic physics in a weakly collisional plasma over a wide range of scales.

In panel (a) of Figure 6, the normalized real frequency difference between Hall MHD and Vlasov-Maxwell kinetics, $\Delta\omega\equiv|\omega_{h}-\omega_{v}|/\omega_{v}$, is plotted for the Alfvén solution over $10^{-2}\leq k_{\perp}d_{i}\leq 10$ and $10^{-2}\leq k_{\parallel}d_{i}\leq 10$; in panel (b) is plotted the normalized linear kinetic damping rate, $\gamma_{v}/\omega_{v}$. Plasma parameters relevant to the near-earth solar wind, $\beta_{i}=1$ and $T_{i}/T_{e}=1$, are chosen. Areas of panel (a) with $\Delta\omega>0.1$ are shaded to highlight the regions of parameter space over which the frequency using Hall MHD deviates significantly from the kinetic value; as evident, the Alfvén solution from Hall MHD²²2Because of the tendency for the kinetic modes to correspond to different fluid modes as parameters vary, as discussed in §3.1, in this paper the calculation of $\Delta\omega$ at each point uses the Hall MHD mode that the yields the minimum value of this difference. deviates substantially from the Vlasov-Maxwell frequency at $k_{\parallel}d_{i}\gtrsim 1$ and moderately at $k_{\perp}d_{i}\sim 1$. Areas of panel (b) with $\gamma_{v}/\omega_{v}>0.1$ are shaded to denote regions of parameter space over which the linear collisionless damping of this mode in Vlasov-Maxwell kinetic theory is strong. It is evident that for $k_{\parallel}d_{i}\gtrsim 1$, the regime corresponding to ion cyclotron waves, strong damping occurs via the ion cyclotron resonance; it seems likely that this strong damping, due to wave-particle interactions not captured by Hall MHD, is also responsible for the poor agreement in real frequency. Also evident in panel (b) is the increasingly strong damping of the Alfvén solution at $k_{\perp}d_{i}\gg 1$ and $k_{\perp}\gg k_{\parallel}$; this is the regime of kinetic Alfvén waves in which damping occurs via Landau resonance with the electrons.

The analogous calculation for the fast wave branch is presented in Figure 7, with the frequency difference $\Delta\omega$ plotted in panel (a) and the kinetic damping rate $\gamma_{v}/\omega_{v}$ in panel (b), again for plasma parameters $\beta_{i}=1$ and $T_{i}/T_{e}=1$. Note that this calculation employs only the kinetic solutions on the dispersion surface of the F branch and does not consider the $n\geq 1$ ion Bernstein wave dispersion surfaces. The real frequency derived from Hall MHD differs from the kinetic value in two regimes: a moderate difference occurs at $k_{\parallel}d_{i}\ll 1$ and $0.05\lesssim k_{\perp}d_{i}\lesssim 0.5$; and the difference becomes more dramatic at $k_{\perp}d_{i}\gtrsim 1$ and $k_{\parallel}d_{i}\lesssim 1$ due to the mode conversion in kinetic theory of the F branch from the fast wave to an ion Bernstein wave in this regime. The kinetic damping, shown in panel (b), is relatively weak except for the regime of the ion Bernstein waves, $k_{\perp}d_{i}\gtrsim 1$ and $k_{\parallel}d_{i}\lesssim 1$; these waves are strongly damped unless $k_{\parallel}d_{i}\ll 1$ (Swanson, 1989).

The slow wave branch suffers very strong linear collisionless damping in a plasma with finite ion temperature, such as the solar wind, over the entire range of scales considered in Figures 6 and 7. A comparison of the real frequency in this case would be meaningless since a wave with a damping rate $\gamma/\omega\sim 1$ is so heavily damped that it can scarcely be considered a wave.

A very important aspect of the behavior of weakly collisional plasmas is made clear by the damping rate plots in panel (b) of both Figures 6 and 7: at small length scales, with wavenumbers $kd_{i}\gtrsim 1$, there exist only four regimes in which undamped wave modes occur. These are:

1. 1.

   Kinetic Alfvén Waves: The Alfvén branch solution of Vlasov-Maxwell kinetic theory in the nearly perpendicular regime $k_{\parallel}d_{i}\ll 1$ and $k_{\perp}d_{i}\gtrsim 1$ corresponds to the kinetic Alfvén wave mode (KAW) .

2. 2.

   Parallel Whistler Waves: The fast branch solution in the regime $k_{\parallel}d_{i}\gtrsim 1$ and $k_{\perp}d_{i}\ll 1$ corresponds to the whistler wave with a nearly parallel wave vector (PW).

3. 3.

   Oblique Whistler Waves: The fast branch solution in the regime $k_{\parallel}d_{i}\gtrsim 1$ and $k_{\perp}d_{i}\gtrsim 1$ corresponds to the whistler wave with an oblique wave vector (OW).

4. 4.

   Ion Bernstein Waves: The fast branch solution in the nearly perpendicular regime $k_{\parallel}d_{i}\ll 1$ and $k_{\perp}d_{i}\gtrsim 1$ corresponds to an ion Bernstein wave (IBW).

Since the plasma turbulence at small length scales—for example, the dissipation range in the solar wind—is likely to consist of one, or a mixture, of these four undamped modes, we would like to explore the ability of Hall MHD to reproduce their linear wave frequencies accurately (relative to linear Vlasov-Maxwell kinetic theory). We note, however, that the ion Bernstein wave is an electrostatic wave mode (Swanson, 1989; Stix, 1992), and therefore has no corresponding magnetic field fluctuation; since the solar wind dissipation range is observed to have magnetic fluctuations (Goldstein et al., 1995; Leamon et al., 1998; Smith et al., 2006; Hamilton et al., 2008), we do not further pursue the ion Bernstein waves here, but rather we focus solely on the first three modes, all of which are electromagnetic.

For a kinetic Alfvén wave at $k_{\perp}d_{i}=1$ and $k_{\parallel}d_{i}=0.1$, denoted by the triangle (KAW) in panel (b) of Figure 6, we plot the frequency difference $\Delta\omega$ and kinetic damping rate $\gamma_{v}/\omega_{v}$ over $0.01\leq\beta_{i}\leq 100$ and $0.01\leq T_{i}/T_{e}\leq 100$ in panels (a) and (b) of Figure 8. The kinetic Alfvén wave appears to be well reproduced by Hall MHD for $\beta_{i}<1$, but is less accurately determined for $\beta_{i}\gtrsim 1$. The linear kinetic damping of the mode appears to become strong only for $\beta_{i}\gtrsim 1$ and $T_{i}/T_{e}\lesssim 1$.

For a parallel whistler wave (on the fast wave branch) at $k_{\perp}d_{i}=0.1$ and $k_{\parallel}d_{i}=1$, denoted by the triangle (PW) in panel (b) of Figure 7, we plot the frequency difference and kinetic damping rate on the $(\beta_{i},T_{i}/T_{e})$ plane in panels (a) and (b) of Figure 9. This parallel whistler wave appears to be well reproduced by Hall MHD over all parameter space $(\beta_{i},T_{i}/T_{e})$ except for $\beta_{i}\gtrsim 3$ and $T_{i}/T_{e}\gtrsim 0.1$; in this problematic regime, the whistler wave mode converts to an ion Bernstein wave and is therefore both strongly damped and poorly described by Hall MHD.

For an oblique whistler wave (on the fast wave branch) at $k_{\perp}d_{i}=1$ and $k_{\parallel}d_{i}=1$, denoted by the triangle (OW) in panel (b) of Figure 7, we plot the frequency difference and kinetic damping rate on the $(\beta_{i},T_{i}/T_{e})$ plane in Figure 10. The results are very similar to those for the parallel whistler wave, with strong damping, and correspondingly poor agreement in linear real frequency, occurring over the regime $\beta_{i}\gtrsim 1$ and $T_{i}/T_{e}\gtrsim 0.1$.

With the exception of the regime $\beta_{i}\gtrsim 1$ and $T_{i}/T_{e}\gtrsim 1$ for the kinetic Alfvén wave, where Hall MHD predicts a quantitatively different real frequency from kinetic theory, all of the differences between Hall MHD and kinetic theory occur when the damping becomes strong, with $\gamma_{v}/\omega_{v}>0.1$. That the weakly damped wave modes should be well represented by Hall MHD is precisely what is expected—the fluid limit of the kinetic plasma physics is valid only when kinetic effects, such as collisionless damping by wave-particle interactions, are negligible (Ballai et al., 2002).

By exploiting the highly super-Alfvénic speed of the solar wind flow, the temporal fluctuations measured by in situ spacecraft can be mapped, according to the Taylor hypothesis (Taylor, 1938), to spatial fluctuations. Thus, the break in the solar wind magnetic energy spectrum, typically observed at around $f\sim 0.4$ Hz, is thought to correspond to some characteristic length scale in the plasma, perhaps the ion inertial length $d_{i}$ or the ion Larmor radius $\rho_{i}$ (Leamon et al., 1998; Howes et al., 2008a; Hamilton et al., 2008). At frequencies above this spectral break, corresponding to yet smaller spatial scales, the magnetic energy spectrum appears steeper (Smith et al., 2006)—this regime is denoted the “dissipation range.”

The kinetic theory results presented in §4.4 clearly demonstrate that, for a weakly collisional plasma with a finite ion temperature—such as the solar wind—all slow branch modes, the ion cyclotron wave of the Alfvén branch, and the fast branch ion Bernstein wave at $k_{\parallel}d_{i}\gtrsim 0.1$ are strongly damped. Only three weakly damped electromagnetic wave modes exist at scales $kd_{i}\gtrsim 1$: the kinetic Alfvén wave, the parallel whistler wave, and the oblique whistler wave. It seems unlikely, even in a nonlinearly turbulent plasma, that wave modes with strong linear kinetic damping rates could be responsible for the dissipation range fluctuations. The linear wave properties are quite likely to play a large role in the turbulent dynamics, in particular at the small length scales $kd_{i}\gtrsim 1$ where the amplitude of the magnetic fluctuations is small compared to the magnitude of the local interplanetary magnetic field (Howes et al., 2008a). This line of reasoning suggests that the fluctuations in the dissipation range must consist of one, or perhaps a mixture, of the three undamped electromagnetic wave modes. The finding of §4.5 suggests that, for typical solar wind parameters with $\beta_{i}\lesssim 1$, Hall MHD generally does an adequate job of describing these undamped wave modes.

How effective are linear collisionless damping mechanisms in a turbulent plasma where nonlinear interactions are continually transferring energy from one wave mode to another? This question essentially weighs the importance of linear kinetic wave-particle interactions versus nonlinear fluid wave-wave interactions in turbulent, weakly collisional plasma. According to models for turbulent fluids, the energy in a particular wave mode will be transferred to other wave modes through nonlinear interactions on some characteristic nonlinear time scale. The theory for strong incompressible MHD turbulence takes this nonlinear timescale to be the eddy-turnover time in the perpendicular direction (Goldreich and Sridhar, 1995); the central conjecture of this theory is that, as the turbulent energy cascades to smaller scales, this nonlinear timescale remains in an approximate critical balance with the linear Alfvén wave period, and consequently predicts an anisotropic cascade to high perpendicular wavenumbers that is supported by numerical evidence (Cho and Vishniac, 2000; Maron and Goldreich, 2001) and is consistent with a careful analysis of solar wind turbulence observations (Horbury et al., 2008).

The key question in weakly collisional plasmas is whether or not this nonlinear transfer suppresses the linear collisionless damping via wave-particle interactions. Conversely, it is possible that nonlinear wave-wave interactions can transfer energy into a strongly damped mode, ultimately leading to energy loss from the turbulent fluctuations through linear wave-particle interactions of that strongly damped mode. But it is conceivable that nonlinear couplings involving strongly damped modes are inhibited by an impedance mismatch, thus preventing energy loss in such a manner. Ultimately, of course, the turbulence must be dissipated via kinetic damping mechanisms (Howes, 2008); in this case, an evaluation of both the rate of energy transfer by the turbulent cascade and the rate of kinetic damping is necessary to determine the dissipation as a function of wavenumber, as in the cascade model of Howes et al. (2008a).

These important questions will most likely only be answered through detailed nonlinear kinetic numerical simulations. On this front, turbulence simulations using undamped fluid models, such as Hall MHD, will provide valuable comparison points for analysis and interpretation of the kinetic simulation results.

Until such work can provide further guidance, it seems a reasonable strategy to assume the applicability of the linear kinetic damping rates to turbulent plasmas. In fact, the first such nonlinear kinetic simulation of turbulence at the scale of the ion Larmor radius has been well fit by a model assuming linear damping rates (Howes et al., 2008c). The plasma parameters chosen for this study, however, predicted rather weak linear collisionless damping over the range of scales simulated, so we must await further simulations in more strongly damped parameter regimes to test more thoroughly the effectiveness of the kinetic damping in a turbulent plasma. The study did show, however, that the damping via kinetic mechanisms in the turbulent plasma was not stronger than the linear prediction, a result that is not obvious a priori.

Despite the many unanswered questions about damping in kinetic turbulence, there is no question that the dissipation of turbulence in a weakly collisional plasma cannot be studied by Hall MHD, or any standard fluid model. The viscous and resistive dissipation terms used in such models are merely ad hoc fluid closures that do not accurately represent the underlying kinetic mechanisms. Only a much more complicated model, such as a Landau-fluid model (Snyder et al., 1997; Passot and Sulem, 2007), is capable of adequately representing the kinetic dissipation. The steeper spectral index of the magnetic energy spectrum in the dissipation range of solar wind turbulence has been variously attributed to either kinetic dissipation (Coleman, 1968; Leamon et al., 1999; Gary, 1999; Howes et al., 2008a) or to wave dispersion (Stawicki et al., 2001; Krishan and Mahajan, 2004; Galtier, 2006). If dissipation does play a significant role, further investigation requires a kinetic model. It has even been argued that, without a kinetic model to determine that kinetic dissipation is negligible for a specific case, the use of a fluid model such as Hall MHD is unjustified (Howes et al., 2008b).

Hall MHD has proven to be a valuable framework for the study of a number of plasma phenomena, in particular magnetic reconnection. For example, the finding that whistler waves mediated a faster reconnection rate (Mandt et al., 1994) paved the way in identifying the importance of the Hall term in magnetic reconnection. Such applications depend on the accurate description of the linear properties of a particular wave mode. But turbulence is consequence of nonlinear wave couplings; to provide a useful framework for kinetic turbulence, a model must accurately describe the behavior not of just one mode, but of all modes.³³3The nonlinear interactions in a turbulent plasma will depend not only on the complex eigenfrequencies of the normal modes but on their eigenfunctions as well. Although this paper restricts its focus to the frequencies, a detailed comparison of all mode properties at $kd_{i}=0.1$ found that the kinetic mode properties are not always well represented by Hall MHD (Krauss-Varban et al., 1994).

Because the turbulent dynamics and evolution will depend on the nonlinear couplings between all possible wave modes, the existence of certain undamped wave modes in Hall MHD—modes that are strongly damped in a weakly collisional plasma according to kinetic theory—is troubling. These spurious wave modes effectively provide additional degrees of freedom to the turbulence that would otherwise be strongly impeded in a kinetic plasma. Consider, for example, waves in the parameter regime $k_{\parallel}d_{i}\gtrsim 1$ in a plasma with $\beta_{i}=1$ and $T_{i}/T_{e}=1$ as presented in Figure 3 of §4.2. In Vlasov-Maxwell kinetic theory, both the Alfvén and slow branch modes are heavily damped in this regime, leaving only fast branch waves available. But the Hall MHD model supports undamped Alfvén and slow waves. It seems unlikely that the nonlinear dynamics would be the same in the presence of three undamped modes as when only the fast mode is available, but just how the presence of the unphysical modes will alter the nonlinear wave-wave couplings is unclear. Here, once again, a promising avenue for progress is a detailed study contrasting nonlinear couplings in the presence and absence of these spurious modes. A suite of nonlinear kinetic simulations compared to Hall MHD simulations in the same regime will provide valuable insight, identifying the regimes of validity for which Hall MHD provides an adequate description.

Two arguments exist that may serve to diminish the impact of the spurious undamped waves in Hall MHD. First, as shown in Figure 4, in the limit that $k_{\perp}\gg k_{\parallel}$, the fast wave frequency is much higher than the Alfvén wave frequency; since the strength of nonlinear interactions typically diminishes rapidly as the wave mode frequencies become widely separated, the presence of a spurious wave mode may negligibly impact the turbulent couplings in such a case. Of course, in the opposite limit $k_{\perp}\ll k_{\parallel}$, seen in Figure 3, all three wave modes have similar frequencies for $k_{\parallel}d_{i}\lesssim 1$. A second argument, derived in the gyrokinetic limit⁴⁴4The fast wave branch is ordered out of the system in the gyrokinetic approximation. $k_{\perp}\gg k_{\parallel}$ and $\omega\ll\Omega_{i}$, shows that the turbulent cascade of Alfvén waves does not exchange energy with the slow wave cascade except in the regime where the perpendicular scale is near the ion Larmor radius $k_{\perp}\rho_{i}\sim 1$ (Schekochihin et al., 2007). Thus, the existence of a spuriously undamped slow wave may not influence the Alfvén wave cascade. To explore the exchange of energy between the separate cascades at the scale of the ion Larmor radius $k_{\perp}\rho_{i}\sim 1$, however, will certainly require nonlinear kinetic simulations.

As a model that extends beyond the limits of MHD, Hall MHD has seen increasing use in recent years as a framework for describing turbulence in weakly collisional plasmas, such as the near-earth solar wind. Its applicability to turbulence in kinetic systems has been called into question (Krauss-Varban et al., 1994; Howes et al., 2008b), so a thorough evaluation of the limitations of Hall MHD in this context is desirable. This paper takes the first step in this process by quantitatively comparing the real linear eigenfrequencies of standard Hall MHD with the complex linear eigenfrequencies of Vlasov-Maxwell kinetic theory.

Previous work has shown that Hall MHD is a rigorous limit of kinetic theory only in the cold ion limit satisfying $T_{i}\ll T_{e}$, and $k_{\parallel}v_{ti}\ll\omega\ll k_{\parallel}v_{te}$ (Ito et al., 2004; Hirose et al., 2004); the fluid description is only valid when wave-particle interactions and finite-Larmor-radius effects are negligible (Ballai et al., 2002). The quantitative comparisons with kinetic theory in §4 bear out the general finding that Hall MHD is a valid limit of Hall MHD in the cold ion limit $T_{i}\ll T_{e}$; for finite ion temperature $T_{i}\sim T_{e}$, however, the lack of collisionless damping in Hall MHD leads to undamped wave modes that do not exist in a weakly collisional plasma.

Three key issues are identified regarding the use of Hall MHD to describe turbulence in kinetic plasmas: (1) what are the wave modes comprising the turbulence at scales $kd_{i}\gtrsim 1$?; (2) are the collisionless damping rates from linear kinetic theory applicable in a nonlinearly turbulent plasma?; and (3) are the nonlinear wave-wave mode couplings inherent in turbulence altered by the presence of spurious, undamped wave modes in Hall MHD? In a weakly collisional plasma, the only three undamped electromagnetic wave modes that exist at scales $kd_{i}\gtrsim 1$ are the kinetic Alfvén wave, the parallel whistler wave, and the oblique whistler wave; each of these waves is generally well-described by Hall MHD for $\beta_{i}\lesssim 1$. To determine the effective collisionless damping in a turbulent plasma, a research program using nonlinear numerical simulations to contrast the predictions of a kinetic approach with those of a fluid approach is the most promising path forward. For studies focusing on the dissipation of turbulence and the thermalization of the turbulent energy, a kinetic description is certainly required. Both fluid Hall MHD and kinetic nonlinear numerical simulations will be instrumental in shedding light on the question of whether the presence of the spuriously undamped waves in Hall MHD alters the nonlinear couplings of the available wave modes, thus changing the dynamics and evolution of the turbulence due to unphysical effects. A study focused on the turbulent dynamics at the characteristic length scales in the plasma, such as the ion Larmor radius or the ion inertial length, is the natural starting point for such a numerical investigation. In conclusion, although nonlinear kinetic simulations will be indispensable for the study of turbulence at kinetic scales in weakly collisional plasmas, Hall MHD will certainly continue to provide useful insights and valuable points for comparison in the study of kinetic turbulence.