A Kinetic Alfvén wave cascade subject to collisionless damping cannot reach electron scales in the solar wind at 1 AU
Turbulence in the solar wind is believed to generate an energy cascade that is supported primarily by Alfvén waves or Alfvénic fluctuations at MHD scales and by kinetic Alfvén waves (KAWs) at kinetic scales . Linear Landau damping of KAWs increases with increasing wavenumber and at some point the damping becomes so strong that the energy cascade is completely dissipated. A model of the energy cascade process that includes the effects of linear collisionless damping of KAWs and the associated compounding of this damping throughout the cascade process is used to determine the wavenumber where the energy cascade terminates. It is found that this wavenumber occurs approximately when , where and are, respectively, the real frequency and damping rate of KAWs and the ratio is evaluated in the limit as the propagation angle approaches 90 degrees relative to the direction of the mean magnetic field. For plasma parameters typical of high-speed solar wind streams at 1 AU, the model suggests that the KAW cascade in the solar wind is almost completely dissipated before reaching the wavenumber . Consequently, an energy cascade consisting solely of KAWs cannot reach scales on the order of the electron gyro-radius, . This conclusion has important ramifications for the interpretation of solar wind magnetic field measurements. It implies that power-law spectra in the regime of electron scales must be supported by wave modes other than the KAW.
Subject headings:Solar wind — turbulence, magnetohydrodynamics, kinetic theory
Recent papers by Schekochihin et al. (2009), Howes et al. (2008a), and Schekochihin et al. (2008) have described a scenario for turbulence in collisionless magnetized plasmas, such as the solar wind, that can be briefly described as follows. The turbulence at large scales, scales larger than the ion inertial length and the thermal ion gyro-radius , consists of an energetically dominant Alfvén wave cascade that transfers energy from large to small scales. At scales on the order of the proton gyro-radius, , the wavevector spectrum of the turbulence is highly anisotropic with energy concentrated in wavevectors nearly perpendicular to the mean magnetic field so that . At , two things happen. On the one hand, there may be some nonlinear effects that are not well understood. On the other hand, a significant fraction of the energy in the Alfvén wave cascade excites a kinetic Alfvén wave cascade (KAW cascade) that carries the energy down to scales on the order of the thermal electron gyro-radius where the turbulence is finally dissipated by collisionless Landau damping.
Sahraoui et al. (2009) used the above scenario to interpret spacecraft measurements of solar wind turbulence in the kinetic range of scales . These authors used the linear KAW dispersion relation and damping rates to argue that the KAW cascade should reach electron scales before being strongly damped, noting that the relative damping rate does not become of order unity until , where is the thermal electron gyroradius. The purpose of the present paper is to point out that this argument is incomplete because it does not take into account the compounding of the damping throughout the course of the cascade process. KAWs are damped by collisionless Landau and transit-time damping throughout the entire wavenumber range of their existence from to . If the energy cascade process is thought of as taking place in a sequence of discrete steps, then there is dissipation at each step in the sequence and the effects of damping are compounded with each step, analogous to the way compound interest works. When this compounding is taken into account it is found that for typical solar wind plasma parameters near 1 AU the KAW cascade is dissipated before reaching electron scales. This conclusion has important ramifications for the interpretation of solar wind power spectra in the kinetic regime.
In astrophysical plasmas, the effects of collisionless damping on the turbulence spectrum have been studied primarily using quasilinear theory (see, for example, Melrose, 1994) and phenomenological models in which the energy cascade process is modeled as a diffusion process in wavenumber space (for example, Li et al., 2001; Stawicki et al., 2001; Cranmer & van Ballegooijen, 2003; Jiang et al., 2009; Matthaeus et al., 2009). Other approaches include weak turbulence theory (Yoon, 2006, 2007; Galtier, 2006; Chandran, 2008) and gyro-kinetic theory (Howes et al., 2008b; Schekochihin et al., 2009). Here we adopt a somewhat simpler approach that, like the diffusion models, is based on an equation expressing the conservation of energy in wavenumber space. Generally speaking, the objective of any of these models is to describe the essential physics as accurately and economically as possible.
In this study, the effects of damping on the KAW cascade are modeled using two complimentary approaches. The first approach is heuristic and shows how this damping takes place through a sequence of steps in wavenumber space whereby energy is damped at each step before being transferred to higher wavenumbers. The second approach is based on an equation for the conservation of energy in wavenumber space and two different expressions for the energy cascade rate that are similar to Kolmogorov’s relation . This approach has much in common with the cascade model developed by Howes et al. (2008a) and allows both the energy cascade rate and the energy spectrum to be computed as functions of wavenumber throughout the inertial range and dissipation range. In this study, the main objective is to determine the point in wavenumber space where the KAW cascade terminates and, therefore, the energy cascade rate is the quantity of primary interest. Assuming that the propagation angles of the waves are nearly perpendicular to , analytic solutions for the energy cascade rate may be derived that are convenient for purposes of analysis and prediction. These analytic solutions are new and are presented here for the first time.
One of the principal assumptions of this work is that the damping rates of linear wave theory are applicable at kinetic scales, even though nonlinear interactions are still strong. Although this same assumption has been made before by many investigators (Leamon et al., 1998; Quataert, 1998; Gary, 1999; Leamon et al., 1999; Quataert & Gruzinov, 1999; Li et al., 2001; Stawicki et al., 2001; Cranmer & van Ballegooijen, 2003; Howes et al., 2008a), there is no well established criteria for its validity. This is, perhaps, the greatest source of uncertainty for the theory presented here. It is also easy to envision a scenario in which turbulence at MHD scales is dissipated via collisionless magnetic reconnection involving structures (reconnection sites) that span lengthscales from the ion inertial length to the electron inertial length (Birn & Priest, 2007, Chapter 3). To some extent, this is a possible alternative to the wave cascade and damping scenario described by Schekochihin et al. (2009) and others.
For the moment, adopting the KAW cascade and linear damping scenario and assuming that the wave damping rates of the linear Vlasov-Maxwell theory are valid in the kinetic regime, then for typical solar wind conditions at 1 AU the models derived here predict that the KAW cascade terminates at wavenumbers less than or equal to which implies that the KAW cascade cannot reach electron scales in the solar wind at 1 AU. These results agree with and are supported by those obtained previously using the cascade model of Howes et al. (2008a) which show a transition to an exponentially decaying magnetic energy spectrum that decays rapidly at around the same wavenumber. On the contrary, the magnetic energy spectra in the gyro-kinetic simulations of Howes et al. (2008b) do not appear to show any deviations from power-law behavior in the range as there should be if kinetic damping is having a significant effect. However, because these scales are near the smallest spatial scales in the simulation where an artificial electron hypercollisionality takes effect, the simulation may not accurately describe the physics when .
One note on terminology. The term “energy cascade” or “cascade,” for short, refers to a nonlinear energy transfer process that transfers energy from large to small scales or vice versa. This term usually refers to the inertial range where the effects of dissipation are negligible. Here, this term is also used to describe the nonlinear energy transfer in the dissipation range. This slight abuse of notation should cause no confusion.
2. Simple heuristic model
We begin with some general considerations that apply to the entire paper. Consider a homogeneous proton-electron plasma with a constant uniform background magnetic field. The equilibrium state is charge neutral, free of macroscopic electric fields, free of electric currents, and characterized by isotropic Maxwell distribution functions for both particle species. It is assumed that the wave amplitudes at kinetic scales are small enough that linear wave theory adequately describes the collisionless damping process. For a given wavenumber (real-valued), the real and imaginary parts of the wave frequency may be computed numerically using the hot plasma dispersion relation (Stix, 1992, chap 10). For , the KAW is uniquely identified by means of its asymptotic dispersion relation in the limit as with or the angle of propagation held constant.
In general, the damping is minimized for all wavenumbers in the range from to when the angle of propagation is close to , that is, near perpendicular to . It is important to note that for all angles in a sufficiently small neighborhood of , the ratio of the damping rate to the real frequency, , is approximately independent of the angle so that all propagation angles in this range yield the same damping per wave period. In practice, this means that the damping per wave period cannot be reduced further by going from to , for example. For simplicity, it will be assumed in this section that is close, but not equal to . This range of near-perpendicular angles is physically relevant because as a consequence of three-wave interactions the energy cascade process creates a spectrum in which the inequality is increasingly well satisfied as the cascade progresses to higher wavenumbers.
Investigation of the damping of the KAW cascade begins with the plot in Figure-1
which shows the attenuation of the wave amplitude in one wave period, that is, versus for a fixed angle of propagation close to . As just mentioned, the curve for in Figure-1 applies for all angles sufficiently close to . Now recall the critical balance hypothesis of Goldreich & Sridhar (1995, 1997) which states that in the inertial range the cascade time is equal to the Alfvén wave period. Using this as a guide, we assume that the cascade time in the kinetic regime is equal to the wave period of the KAWs. Consider the sequence of wavenumbers , where and . In one cascade time, one wave period, the energy at scale cascades to scale but in the process that energy is damped by a factor , where is the attenuation of the wave amplitude in one wave period. If is the energy cascade rate at wavenumber , that is, the rate at which energy reaches scale , then the rate at which energy reaches the next scale is
where . This is the change in the energy cascade rate caused by linear damping after just one step in the cascade process. Starting from , the energy cascade rate after steps is equal to
This describes how the energy of the waves is damped as it cascades through the kinetic range of scales. To compute the wavenumber dependence of for the KAW cascade it is only necessary to compute the relative damping rate from the hot plasma dispersion relation and then use equation (2).
By way of illustration, consider the damping rates for shown in Figure-1. For each wavenumber it is straightforward to read off the values of from the plot in Figure-1 and then compute the cascade rate from equation (2). This yields the results in Table-1 which show that the energy cascade rate is reduced to less than 1% of its original value by the time the cascade reaches . The plasma parameters used to compute the damping rates in Figure-1 are typical of high-speed solar wind streams at 1 AU. For these parameters, the electron gyro-radius occurs when or, equivalently, , and the electron inertial scale occurs when or, equivalently, . This simple calculation indicates that the KAW cascade is likely to be dissipated before reaching electron scales. As we show below, a more detailed model yields results that are in good agreement with those in Table-1.
3. Model based on conservation of energy
The energy spectrum is defined such that is the energy per unit mass contained in the fluctuations for all wavenumbers between and . The total energy contained in the entire spectrum is
The cascade rate is the average energy per unit mass that passes through wavenumber per unit time. The energy flows from low to high wavenumbers. Consider a small interval from to in wavenumber space as illustrated in Figure 2.
Energy is flowing into the boundary from the left and out of the boundary from the right. Conservation of energy requires that the energy flowing out per unit time is equal to the energy flowing in per unit time minus the rate at which energy is dissipated in the layer. Hence,
where is the linear damping rate obtained from the KAW dispersion relation. This yields
which expresses the conservation of energy in the cascade process. Note that if , then const and the energy cascade rate is independent of .
Equation (5) gives one relation between and . A second is Kolmogorov’s relation
where is the energy cascade time. Komlogorov’s relation assumes there is no dissipation of energy during the cascade process so that all the energy at scale cascades to higher wavenumbers. When dissipation is present, a fraction of the total energy at scale cascades to higher wavenumbers and a fraction is dissipated. The energy that is dissipated does not cascade to higher wavenumbers. In this case, the relation (6) is replaced by
where is the energy attenuation factor in time .
It is clear from physical considerations that in the dissipation range the energy cascade rate must depend on the dissipation rate. Hence, Kolmogorov’s relation (6) that was initially postulated for the inertial range needs to be modified. What is not clear is what the precise functional form should be. Justification for the form (7) can be obtained by considering a simple shell model. Let denote a sequence of shells in wavenumber space, where is an integer and is an arbitrary wavenumber. The energy contained in the interval between and is . In the absence of dissipation, this energy cascades to scale in one cascade time so that the energy cascade rate is
When there is no dissipation, const and the energy cascade rate is independent of scale. When dissipation is present, the energy at scale is partly dissipated before it can cascade to the next level. If the energy is attenuated by a factor in time , then the energy cascade rate becomes
where may depend on . This is another way to derive equation (7).
In general, equations (6) and (7) should also contain a constant coefficient which is dimensionless. In hydrodynamic turbulence it follows from dimensional analysis that this coefficient is of order unity. In plasma turbulence there are other parameters in the problem which may cause this coefficient to differ from unity. For example, in the inertial range, one such parameter is the normalized cross-helicity . Thus, equation (7) takes the final form
where in many cases of practical interest the dimensionless constant is of order unity.
3.1. Complete model
Assuming that critical balance holds for the KAW cascade, then the energy cascade time is equal to the wave period, , and the energy cascade rate (10) has the form
where is the attenuation factor for the wave amplitudes in one wave period and is the attenuation factor for the energy. When equation (11) is substituted into the conservation law (5), it follows that
When the exponential factor on the right-hand side is omitted, this equation is almost identical to equation (8) in Howes et al. (2008a) with the source term in Howes et al. (2008a) set equal to zero. The one minor difference is that the damping rates and in (13) are obtained from the hot plasma dispersion relation (Stix, 1992) whereas Howes et al. (2008a) employ the damping rates obtained from gyro-kinetic theory.
Because and , the exponential factor in (13) causes to decrease more rapidly than it would if the exponential factor were omitted. When , the exponential drives toward zero at a faster than exponential rate as can be seen in the solutions presented below. At this point, the dissipation becomes so effective that the energy cascade is abruptly terminated. However, even if the exponential factor is omitted from equation (13) which is equivalent to using equation (6) in place of (7), the wavenumber where the cascade rate goes to zero is still roughly of the same order of magnitude. Therefore, the presence of this exponential factor is not crucial for the conclusions reached in this study.
If and are known from the hot plasma dispersion relation, then equation (13) may be solved to find how depends on . Once the solution for is known, the energy spectrum is obtained from equation (11). In general, the ratio depends on the angle of propagation of the waves which needs to be taken into account. However, when or, equivalently, when the angle of propagation is sufficiently close to , then the ratio becomes approximately independent of angle. This property is used to derive an approximate analytic solution below. In cases where the angle is not sufficiently close to the angle dependence may be taken into account as follows. Assume that the energy in the wavevector spectrum is concentrated near the critical balance curve so that the angle of propagation at a given scale is determined by the critical balance relation
where is the electron velocity perturbation of the wave in the plane perpendicular to and is the parallel phase speed of the KAWs. Note that the parallel phase speed can be much larger than the Alfvén speed in the kinetic regime and this needs to be included in the critical balance relation (14).
According to the critical balance hypothesis, the longitudinal crossing time of two wavepackets is equal to the nonlinear eddy turnover time , where is the parallel group speed. The parallel phase and group velocities are approximately equal as can be seen from the approximate dispersion relation (see, for example, Hollweg, 1999; Cranmer & van Ballegooijen, 2003). In addition to equation (14), the self consistent calculation of the propagation angle requires the relation
where the magnetic field perturbation is measured in velocity units. For the plasma parameters of interest here, the energy density of KAWs is dominated by magnetic and thermal energy (i.e., density fluctuations) with an approximate equipartition between magnetic and thermal energy as discussed, for example, by Terry et al. (2001). These two contributions have been lumped together into a single term in (15). Critical balance tightly couples the magnitude of the energy spectrum to the propagation angle through equations (14) and (15). For KAWs, the ratio may be determined from the hot plasma dispersion relation using the relations , where is the electron current density and is the electron susceptibility (Stix, 1992). This completes the specification of the model which consists of equations (11), (13), (14), and (15).
3.2. Analytic model
The above model can be significantly simplified when so that the ratio becomes approximately independent of the propagation angle . Simulations of incompressible MHD turbulence have shown that the inequality is usually well satisfied at the smallest inertial range scales and, therefore, the assumption is justified in the kinetic regime. It is useful to adopt the approximate expressions for the ratio derived by Howes et al. (2006) using gyrokinetic theory. Using equations (62) and (63) in Howes et al. (2006) one obtains
where and is a constant given by
Here, is the electron mass, is the proton mass, is the equilibrium electron temperature, is the equilibrium proton temperature, , and , where is Boltzmann’s constant, is the equilibrium particle number density, is the equilibrium magnetic field, and is the permeability of free space (SI units). Equation (16) is valid when and . Using typical plasma parameters for the solar wind at 1 AU, equation (16) was compared to the ratio obtained from the hot plasma dispersion relation and found to be an excellent approximation except when where (16) underestimated the damping although it remained accurate to within a factor of 2 or so.
where . The solution is given by
which has the solution
Equations (19) and (21) describe how the energy cascade rate changes as a result of collisionless damping. The associated energy spectrum can be obtained from equation (11) if the propagation angle is known since the propagation angle is necessary to compute the wave frequency . Therefore, the energy spectrum can only be obtained by solving the complete model consisting of equations (11), (13), (14), and (15). This is not attempted here because the energy spectrum is not needed for the purpose of this study.
The theoretical model developed here may also be adapted to study hydrodynamic turbulence. In hydrodynamics, the damping rate is , where is the kinematic viscosity, and the cascade time is the nonlinear eddy turnover time , where . The model is based on equation (5) and either (6) or (7) with . An analytic solution is only possible when (6) is used. Calculations of the energy cascade rate versus wavenumber obtained using these hydrodynamic models show that the energy cascade rate appoaches zero near the Kolmogorov scale . Moreover, the model calculations in the range are in reasonable agreement with the cascade rate obtained using the model spectrum in Pope (2000). In fact, the wavenumber where the energy cascade terminates in Pope’s solution lies between the cutoff wavenumbers and for the two solutions obtained using either (6) or (7).
Results are now presented for typical high-speed solar wind in the ecliptic
plane near 1 AU with the plasma parameters km/s, , , and . This choice of parameters is based on some of our own work, both published (Podesta, 2009) and unpublished, and results published in the literature (Feldman et al., 1977; Newbury et al., 1998; Schwenn, 2006). For these parameters and for highly oblique angles of propagation (), the damping of KAWs is negligible for small wavenumbers, . KAW damping starts to become significant around . The initial wavenumber in the model calculations is . The results for the two different theoretical models, equations (19) and (21), are shown in Figures 3 and 4.
The energy cascade rate shown in Figures 3 and 4 decreases to zero around and , respectively. When the energy cascade terminates. Even though the two models yield noticeably different results, the wavenumber where the cascade rate becomes negligibly small, , is of the same order of magnitude in both Figures 3 and 4. Thus, the two models are roughly consistent with each other and show that the KAW energy cascade cannot continue beyond the wavenumber because the energy flux is almost completely damped at that point. This is the central conclusion of this study. The behavior of the solutions when the parameter is varied are also shown in Figures 3 and 4. The model (19) that includes the factor in Kolmogorov’s relation (7) is less sensitive to variations in the parameter than the model (21) that is based on Kolmogorov’s relation (6). Taken together, the results in Figures 3 and 4 suggest that the KAW cascade will be completely damped by collisionless Landau damping before the energy can reach the scale of the electron gyro-radius; the electron gyro-radius occurs at for the plasma parameters used here.
The wavenumber where the KAW cascade terminates can be estimated from equations (19) and (21). The wavenumber where reaches some small value, say with , can be determined as follows. From equation (19), assuming , the wavenumber where is given by
For and , this yields . These formulas are very convenient for estimating the wavenumber where the KAW cascade terminates.
The termination point can also be expressed in terms of the ratio . Using equation (16), the condition (22) with and is equivalent to . Similarly, the condition (23) with and is equivalent to . The application of theoretical models like those in Section 3 to hydrodynamic turbulence suggests that the actual termination point is somewhere between these two model predictions, and . Hence, it is reasonable to conclude that the cutoff for the KAW cascade occurs approximately when . Note that for the heuristic model in section 2 the cascade terminates as soon as which implies . Thus, the cutoffs for the KAW cascade predicted by the heuristic model in section 2 and the more elaborate models presented in section 3 agree to within a factor of 2.
In this study, two different methods were employed to calculate the collisionless damping of the KAW cascade. The first method demonstrates the effect of compounding on wave dissipation during the energy cascade process as illustrated by the calculation in Table-1. For nearly perpendicular propagating KAWs under typical solar wind conditions at 1 AU, the ratio is usually small from to so that . Nevertheless, because the wave damping is compounded at each stage of the energy cascade process the energy cascade is damped more rapidly than might be expected from an examination of the wavenumber dependence of . The simple heuristic model used to quantify this compounding effect assumes that all the energy is carried by KAWs, that the cascade time is equal to the wave period, and that damping of these waves is governed by the linear Vlasov-Maxwell dispersion relation. The second method based on equation (13) yields quantitatively similar results eventhough the precise form of Kolmogorov’s relation is unknown. Thus, both methods support the main conclusion of this work, that a cascade consisting solely of KAWs cannot reach the electron gyro-scale in the solar wind at 1 AU. This conclusion is also supported by the somewhat different cascade model developed by Howes et al. (2008a).
The conclusions are, of course, subject to some uncertainty. For example, if the cascade time is reduced from one wave period to half a wave period, then the wavenumber where the KAW cascade terminates is roughly twice as large. And, if the cascade time is reduced to one quarter of a wave period, then the wavenumber where the KAW cascade terminates is roughly four times as large. Consequently, the results are sensitive to the cascade time which is not known precisely.
A simple expression for the wavenumber where the KAW cascade terminates has been obtained from the analytical solutions (19) and (21). For , the cutoff occurs approximately when . Hence, one must be careful when using damping rates obtained from the linear Vlasov-Maxwell wave theory to estimate the stopping point of the cascade. As a consequence of the effects of compounding, the wavenumber where the cascade terminates occurs not when but approximately when .
The conclusion reached in the study of Sahraoui et al. (2009) should be reexamined in light of this result. Sahraoui et al. (2009) have suggested that high-frequency magnetic field spectra in the solar wind may be caused by a KAW cascade from proton to electron scales. To support this idea Sahraoui et al. (2009) compute dispersion curves from the hot plasma dispersion relation using the solar wind parameters in their measurements and show that the ratio increases to values of order , indicating strong damping, when the wavenumber reaches the scale of the thermal electron gyro-radius . The physical interpretation suggested by Sahraoui et al. (2009) appears to be untenable based on the model calculations presented here. Using the dispersion curves in Figure 5 of the paper by Sahraoui et al. (2009), the condition for the termination of the KAW cascade occurs when . Hence, the KAW cascade cannot reach the electron gyro-scale at and cannot by itself account for the high frequency solar wind spectra reported by Sahraoui et al. (2009). This means that at least part of the high-frequency power-law spectrum measured in the solar wind by Sahraoui et al. (2009), Kiyani et al. (2009), and Alexandrova et al. (2009) must be supported by some other kind of wave modes.
The theoretical model developed here depends crucially on the wave damping rates computed from the linear dispersion relation and on the hypothesis of critical balance. Particle distribution functions in the solar wind are usually anisotropic with high energy tails and sometimes other features which can deviate significantly from the isotropic Maxwell distributions considered here. The effects of these features on the wave damping rates need to be taken into account in a more thorough analysis. If the hypothesis of critical balance fails to hold or is modified somehow in the dissipation range, then the theory presented here would be incorrect and would have to be revised. A separate effect that needs to be taken into account for the interpretation of solar wind measurements is the breakdown of Taylor’s hypothesis caused by the increasing phase velocities of the waves for . This should reduce the magnitude of the spectral exponent somewhat by systematically shifting power to higher wavenumbers, thereby creating a shallower spectrum.
It is of interest to apply the model developed here to study the damping of MHD turbulence in the solar corona. Because the plasma parameters vary significantly between the low and high corona and the particle distribution functions are nonthermal, a survey of parameter space is required. This is an important avenue of investigation for future work. It is also of some practical importance since NASA plans to launch a spacecraft into the high corona in the coming decade (McComas et al., 2007). The Solar Probe will be the first spacecraft to reach perihelion in the ecliptic plane at a heliocentric distance of approximately 10 solar radii where the plasma beta is around .
For the representative coronal parameters and K the prediction for an isothermal electron-proton plasma is that the KAW cascade terminates when . For and K the theory predicts that the KAW cascade terminates when . It should be noted that for these plasma parameters the approximation (16) overestimates the damping rate at by a factor of and, consequently, these preliminary estimates could be too small by a factor of 2 or so. Nevertheless, these preliminary calculations suggest that the KAW cascade cannot reach the electron gyro-scale in the solar corona.
The cascade terminates more rapidly at low because for a fixed value of the ratio increases as decreases. This is shown in Figure-5 where the ratio obtained from the hot plasma dispersion relation is compared to the approximation (16) used in the theoretical model. Note from equations (16) and (17) that when and , if is held fixed (), then even though for a fixed value of the ratio is independent of .
This work was supported by the NASA Solar and Heliospheric Physics Program, the NASA Heliospheric Guest-Investigator Program, the NSF SHINE Program, and by the Los Alamos National Laboratory LDRD Program.
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