Reply to comment on “Identification of widespread turbulence of dispersive Alfv´ en waves” K. Stasiewicz1 and Y. Khotyaintsev Swedish Institute of Space Physics, SE-75121 Uppsala
We welcome the comment made by Lund [this issue] as this provides us an opportunity to clarify some important implications of our paper [Stasiewicz et al., 2000a], hereafter referred to as S2000a. In his comment, Lund claims that results of S2000a are questionable because: (1) the derivation of our expression for δE⊥ /δB⊥ is incorrect, and that (2) we do not prove the uniqueness of our solution. In this reply we argue that the conclusions of our paper hold and represent important new findings in space physics. 1. Derivation of the δE/δB relation We would like to emphasize that, generally, waves measured by a satellite are observed not at the true frequency ω but at an apparent frequency ω ω = |ω − k · v|
(1)
where v is the velocity of the medium (plasma) with respect to the satellite. At the topside ionosphere (e.g. 1500 km, at Freja altitudes) the convective plasma flows are generally smaller than the satellite speed vs and therefore v ≈ vs ∼ 7 km/s. On the other hand at the magnetospheric boundary layers the convective plasma flow speeds are much larger than the satellite speed and therefore v ≈ vE ∼ 100 km/s. In the Freja environment, typical characteristic spatial scales are: inertial electron length λe = c/ωpe ∼ 100 m, ion gyroradius ri ∼ 10 m, Debye length λD ∼ 0.1 m. Observed by a moving satellite, plasma wave structures at, say, ω /2π < 1 Hz, and with perpendicular scales λ⊥ on the order of λe , ri , and 10λD would be seen as wave structures at the apparent frequency of 70, 700, and 7000 Hz, respectively, if k is along the satellite velocity. Thus, “broadband waves” ∆ω can be produced by time domain waves ∆ω as well as by Doppler shifted spatial structures ∆k · v. By “spatial” we do not mean purely static structures (ω = 0), but waves that fulfill the condition ω < |k · v|, i.e. dominated by Doppler shift. Experimental evidence that ELF broadband space plasma waves may be produced by Doppler shifted zerofrequency spatial turbulence has existed for at least two 1
2 decades, as shown in several references cited in S2000a. In that paper we provide three arguments for the interpretation of BB-ELF as spatial turbulence: (i) bare eye evidence, (ii) evidence from cross-spectra, and (iii) indirect evidence from the δE/δB relation. By studying the density structures measured by two density probes one can observe a time delay related to the fact that a spatial structure is encountered first by one probe and later by the other probe (see Figure 3 of S2000a). The main discovery of S2000a is not that the structures are spatial, but that the experimental ratio of perpendicular components δEy /δBx fulfills the well-known dispersion relation for inertial Alfv´en waves δEy 2 2 (2) δBx ≈ vA 1 + ky λe , where vA is the Alfv´en velocity and the Oz axis is directed along the ambient magnetic field. For details of the derivation the reader is referred to a recent review by Stasiewicz et al. [2000b]. Using equation (1) in (2), one finds the approximation 2 δEy ≈ vA 1 + 2πλe fd2 + R(f ), (3) δBx v cos θ where fd (= k · v/2π) is an apparent Doppler frequency in the satellite frame, and R(f ) is an unknown contribution from true time domain waves. Here, θ is the angle between the k-vector and the velocity v, and brackets represent a spatial average. It is obvious that the transformation between the apparent Doppler frequency fd and the k⊥ involves a geometric factor that depends on the nature of turbulence and the satellite trajectory, as Lund quite correctly points out in his comment. We use here the electric field component that is close to the satellite velocity direction. However, the axis in Figure 2 of S2000a is perfectly correct as this is a simple transformation from the apparent frequency f to vs /f . We point out that the experimental data fit very well equation (3) with R(f ) ≈ 0 over wide range of spatial scales 50–10,000 m (or equivalent frequencies), and on this basis we draw a conclusion on the Alfv´enic nature of the investigated turbulence. We do not discuss scales shorter than the antenna length (< 20 m), where resonance fingerprints should be visible under certain geometries studied by Temerin [1978]. Obviously, the dispersion relations are usually derived for plane waves. While the experimental fact that the main dispersive properties of (2) and (3) are preserved in a general 2D Alfv´enic turbulence deserves a further theoretical study, this cannot be used against the proposed theory.
3 A careful reading of Lund’s comment reveals that his problems are related mainly to ion heating mechanisms which are not discussed in the commented S2000a but in another paper [Stasiewicz et al., 2000c]. Indeed, the comment appears to be a promotion of one particular ion heating mechanism: gyroresonance heating by broadband waves. This mechanism would not work if the broadband waves were in k domain and not in the frequency domain, which is implied by S2000a and many other earlier publications. We would like to clarify that we also regard ion cyclotron heating as a most direct and efficient way to energize ions. However, the presence of large wave powers at O+ or H + cyclotron frequencies in the satellite frame does not imply the presence of any ion cyclotron waves in space, not to mention left hand circularly polarized waves which are needed to heat the ions via gyroresonance. When ion cyclotron waves are present in space, they are clearly seen both in the power spectra and also in the δEy /δBx ratio (see e.g. Figure 3 in Stasiewicz et al., 1998). Thus, frequency domain spectral features (waves) are seen in the δEy /δBx plots as R(f ) deviations from the purely Doppler part of equation (3) for dispersive Alfv´en waves. In Doppler shifted structures ∆ω ≈ ∆k · v, a large spectral power at the ion gyrofrequency cannot be used to heat ions via gyroresonance, and other mechanisms must be invoked, as has been recently discussed by Stasiewicz et al. [2000c]. 2. Uniqueness of the δEy /δBx relation The second problem that Lund has with S2000a is that we allegedly failed to show that equation (2) for δEy /δBx is unique for dispersive Alfv´en waves. He speculates that perhaps the same properties could be attributed to some other phenomena such as (cited from Lund): electrostatic solitary structures, inhomogeneous energy-driven density instability, ion acoustic waves, electron acoustic waves, or “electrostatic” cyclotron waves. Again, we do not question that these waves may play important rˆ oles in auroral physics under certain circumstances. However, we believe that the fulfillment of a dispersion relation (equation 2) is a strong argument for the identification of a particular wave mode. Until our critic will show that any of his alternative phenomena would also fulfill equation (2) (or its equivalent (3) in the frequency domain), we will insist that waves which satisfy the dispersion relation for inertial Alfv´en waves should be called inertial (dispersive) Alfv´en waves.
4 Summary The objections raised by Lund (this issue) do not affect the results of S2000a. Direct and unambiguous identification of spatial nature of ELF fluctuations can be made with a bare eye when two density probe measurements are available on a single spacecraft and augmented with a cross-spectral analysis of these two signals. Identification of broadband ELF turbulence as dispersive Alfv´en waves on the basis of the dispersion relation (equation 4, S2000a) is an important result in space physics, which appears to hold not only at ionospheric altitudes but also at the magnetospheric boundary layer [Stasiewicz et al., 2000d]. This discovery, based on several case studies, would require additional statistical assessments.
References Lund, E. J., Comment on “Identification of widespread turbulence of dispersive Alfv´en waves”, Geophys. Res. Lett., this issue. Stasiewicz, K., G. Holmgren, and L. Zanetti, Density depletions and current singularities observed by Freja, J. Geophys. Res., 103, 4251–60, 1998. Stasiewicz, K., Y. Khotyaintsev M. Berthomier and J-E. Wahlund, Identification of widespread turbulence of dispersive Alfv´en waves, Geophys. Res. Lett., 27, 173-176, 2000a. Stasiewicz, K., P. Bellan, C. Chaston, C. Kletzing, R. Lysak, J. Maggs, O. Pokhotelov, C. Seyler, P. Shukla, L. Stenflo, A. Streltsov, and J-E. Wahlund., Small scale Alfv´enic structure in the aurora, Space Sci. Reviews, 92, 423–533, 2000b. Stasiewicz, K., R. Lundin, and G. Marklund, Stochastic ion heating by orbit chaotization on electrostatic waves and nonlinear structures, Phys. Scripta, T84, 60–65, 2000c. Stasiewicz, K., E. C. Seyler, F. Mozer, G. Gustafsson, J. Pickett and B. Popielawska, Magnetic bubbles and kinetic Alfven waves in the high-latitude magnetopause boundary, submitted to J. Geophys. Res., http://cluster.irfu.se/ks/res/publications.html, 2000d. Temerin, M., The polarization, frequency, and wavelengths of high-latitude turbulence, J. Geophys. Res., 83, 2609, 1978. Y. Khotyaintsev, K. Stasiewicz Swedish Institute of Space Physics, Box 537, SE-75121 Uppsala. (e-mail:
[email protected],
[email protected]) (Received October 31, 2000; accepted December 11, 2000.) 1 Also at Space Research Centre, Polish Academy of Sciences, Warsaw.
