Pages 707-711

KINETIC THEORY MODE PROPERTIES: APPLICATION TO LOW-FREQUENCY WAVES IN THE ION FORESHOCK

X. Blanco-Cano1, S. J. Schwartz2

1Instituto de Geofisica, UNAM, Coyoacan, Mexico DF, 04510, Mexico, E-mail:   xbc@tonatiuh.igeofcu.unam.mx
2Astronomy Unit, Queen Mary and Westfield College, Mile End Road, London E1 4NS, U.K.

ABSTRACT

The majority of previous works studying ULF waves in the Earth's ion foreshock have been based only on the magnetic field fluctuations and most of them have based their mode identification on fluid models. To provide a more complete description of the waves, it is necessary to consider the fluctuations of the particle population and to use the values of wave identifiers derived from a kinetic model. We use the electromagnetic dispersion relation derived from linear Vlasov theory to evaluate various plasma-field correlations for the hot beam right-hand and left-hand resonant instabilities under conditions consistent with the foreshock. Comparison of the observed Alfvén ratio, parallel compressibility, cross-helicity, magnetic compression and polarization for the waves observed in 10 intervals by AMPTE-UKS with the theoretical values, allows us to identify regions with Alfvénic waves and regions with magnetosonic fluctuations.

INTRODUCTION

Ultra low frequency (ULF), transverse, large amplitude (~5 nT) waves with frequencies ~10-2 Hz propagate in the ion foreshock (Hoppe et al., 1981; Hoppe and Russell, 1983). ULF waves have been observed with left-hand and right-hand polarizations, that due to Doppler shift correspond respectively to right-hand and left-hand polarizations in the plasma frame. Comparison to cold plasma dispersion relations identifies these as magnetosonic and Alfvén waves (Hoppe and Russell, 1983). Identification of wave modes in the foreshock has relied only on magnetic field data and has been based on MHD models. However, since ULF waves also have plasma signatures a complete description of them requires considering the fluctuation of the proton velocity and density in addition to the magnetic fluctuations. Moreover, because the plasma in the foreshock is hot, and because kinetic effects such as the generation of ion instabilities are very important in this region, the use of a kinetic model to derive plasma wave identifiers is imperative.

In this work we study the waves' properties considering fluctuations in the proton density and velocity together with magnetic field data. We use AMPTE-UKS ion and magnetic data to evaluate various transport ratios (Alfvén ratio, cross-helicity, parallel compressibility, magnetic compression and magnetic transverse ratio) for the ULF waves and compare them with the values derived from linear kinetic theory for the left-hand resonant and the right-hand resonant instabilities. These modes can be generated in the foreshock by the interaction of diffuse ions with the solar wind. At small angles of propagation, the transport ratios for these two modes are very similar, so we also evaluate the waves polarization. There are limitations in comparing the results of a single linear mode theory with observations of a nonlinear mixed mode spectrum of waves. Thus, we investigate several examples in detail and consider several transport ratios.

THEORETICAL TRANSPORT RATIOS

We use the program WHAMP (Rönmark, 1982) to solve the dispersion relation from linear Vlasov theory for a three-species plasma (one electron and two proton populations) and evaluate the transport ratios (defined below) of the left-hand and the right-hand resonant ion beam instabilities for a set of hot beam ion distributions similar to the diffuse ions observed in the foreshock.

The magnetic polarization is P= iBs/BA ( indicates the amplitudes of fluctuating quantities). A and S are the typical directions of fluctuation for Alfvén and magnetosonic waves respectively. A is perpendicular to the plane defined by k, the wave vector and the unperturbed magnetic field Bo and S is in the plane k-Bo, perpendicular to k.

The magnetic compression and the noncoplanar ratio are defined as

 CB = (<B|| B|| >)/(<B B> ) ;

Rnc=( <BA BA >)/(<B B> )                                             (1)

B||=(B Bo)/|Bo|, and <ab> denotes the correlation between a and b for a pair of omega and k.

The Alfvén ratio, the cross-helicity and the parallel compressibility are

 rAj=(<vjvj>)/( <bb>)

cj=(2 <vj b>)/(<vj2> + <b2>)

C||  j={(<nj B || >)/ njBo}{ Bo2/(<B|| B ||  >) }                  (2)

where b=(Bo)/(4mp(np+nb))1/2 and the subscript j denotes the j-species in the plasma (p, e and b refer to the proton core, the electrons and the ion beam).

All quantities are evaluated for the core protons as functions of kBo, the angle between k and Bo, and of p, the core proton beta. The input plasma distributions are given in table 1. Case 1 shows a beam similar to the "typically observed" diffuse ions in the foreshock (Thomsen, 1985). Other cases show variations in the ion distributions. Our mode identification is done by comparing the observed wave properties with the values obtained for the wave identifiers considering these five plasma configurations.

  Table 1. Plasma input distributions. In all cases vA/c=10-4 and mp/me=1836.1. Case 1 corresponds to the "fundamental" hot beam distribution, the remaining cases give only the values of the parameters that change, while the rest remain as in case 1. The ratio Tb/Tp is given for p=1. Tb is constant but Tp varies as pdoes.

 

 

Protons

Electrons

Beam

1

nj/ne

0.99

1.0

0.01

 

Tj/Tp

1.0

1.0

100

 

voj/vA

-0.02

0.0

1.98

2

nj/ne

0.95

1.0

0.05

3

nj/ne

0.90

1.0

0.10

4

Tj/Tp

1.0

1.0

495

5

voj/vA

-0.05

0.0

4.95

6

nj/ne

0.95

1.0

0.05

 

Tj/Tp

1.0

1.0

1000

 

AMPTE-UKS DATA ANALYSIS

The AMPTE-UKS ion instrument (Coates et al., 1985) operated a solar wind mode in which high resolution in angle and energy measurements of an anti-sunward sector accumulated over 0.625 s were made. The solar wind distribution was measured every 10 s. The velocity, the temperature and the density of the solar wind protons were found as described in Blanco-Cano and Schwartz (1996). The magnetic field data are from the UKS magnetometer (Southwood et al., 1985). We use high resolution (8 vectors/s) magnetic data when available and low resolution spin averaged data (1 vector/5 s) for the other cases. The ion data were obtained with a cycle time (10 s) of the order of the wave period observed in previous studies, thereby precluding the ability to follow the variations within an individual wave cycle. Instead, we use a statistical approach to determine the transport ratios. All the analysis was done as a function of time, taking sliding averages over a period Ta=2 min every 0.5 min. To evaluate rAp, cp, C ||  p we interpolated the magnetic field corresponding to each ion measurement as follows; when we used high resolution magnetic field data, an average field vector was evaluated, representative of the time during which the ion measurements were taken (0.625 s). In the case of low resolution magnetic field data the values of the magnetic field corresponding to the ion data points were interpolated using a cubic spline. CB and Rnc are derived from the reduced magnetic field points. The magnitude of the polarization is evaluated as |PBokBs/BA (Bs and BA, are the variances along S and A). The sense of polarization is estimated with the method of Elaoufir et al. (1990).

RESULTS

We use AMPTE-UKS observations from day 30 October 1984, and selected ten intervals in which the waves' characteristics remained fairly constant for each interval. In all the cases, diffuse ion distributions are observed (Blanco-Cano and Schwartz, 1996). Figures 1 and 2 show the observed transport ratios and the values derived from linear kinetic theory for the two hot beam instabilities. The theoretical values are given for all the beam distributions in table 1 for a range in p that covers the observed solar wind betas. Observed average quantities are plotted with an error bar given by their associated standard deviation. In figure 1 diamonds show the properties of the waves that are right-handed in the spacecraft frame, and dots are used for waves with left-hand polarization. In all the events cp < 0 and we use this value to remove the 180o ambiguity in the direction given by the minimum variance analysis. The waves are moving upstream in the plasma rest frame and provided that their phase velocities are smaller than the solar wind velocity, they suffer a reversal in their sense of polarization. Thus, diamonds correspond to waves that are left-handed in the plasma frame, and dots to waves with intrinsic right-hand polarization. In figure 1 the value of PBok includes the sense of polarization in the plasma frame for the events with intrinsic left-hand polarization. For these waves, observed |PBok|  are always larger than the theoretical values. For most of the cases, observed C ||  p values are within the range of theoretical values derived for the left-hand mode, while the majority of the observed rAp values are smaller than the theoretical curves. The observed cp magnitudes are smaller than the theoretical value | cp|  ~1, with the larger discrepancies present for the intrinsically right-handed waves. Observed CB (Rnc) values are larger (smaller) than the theoretical curves.

Figure 1. Observed waves' properties and the values derived for the left-hand resonant instability.

Even if some discrepancies exist between the observed values of PBok, C|| p, rAp, cp, CB and Rnc and the values given by the theory, we believe that in 5 cases, the waves with intrinsic left-hand polarization (diamonds), can be identified as the left-hand resonant instability. For the left-handed waves with kBo= 18o, | cp|  is very small, in disagreement with the theoretical value for the Alfvén instability. Nevertheless, for this event the deviations between the rest of the observed transport ratios and the theoretical values are very similar to the differences found for the other intrinsically left-handed events. So these waves have also some Alfvénic properties. CB, Rnc and PBok values show that even in the events identified as Alfvénic, the magnetic field fluctuations presented a compressive component, and are not completely Alfvénic.

In figure 2 we compare the observed transport ratios with the properties of the hot beam right-hand resonant mode. Diamonds indicate intrinsically left-hand polarized waves, while dots indicate right-handed waves. Observed PBok and Rnc values are in very good agreement with the theory, for the waves that are right-handed in the plasma rest frame. For most of the events, there are some discrepancies between observed and theoretical values. Observed C ||  p and rAp values are in general smaller than the theoretical curves, while | cp|  for the intrinsically right-handed waves is smaller than the theoretical values. For the waves with kBo=15o, cp ~-0.55. In the other cases with right-hand polarization (dots) | cp|  is even smaller, in disagreement with the theoretical predictions for modes with a small thetakBo. Such small cp values may not be reliable to remove the 180o ambiguity in the minimum variance direction. For the waves with intrinsic right-hand polarization, observed CB values are larger than the values predicted by the theory. Thus, the observed waves are more compressive than the right-hand linear mode. The intrinsically left-handed waves with kBo=26o and kBo=31o, have cp and CB values that match the theoretical values of the right-hand resonant instability, indicating that these waves identified as Alfvénic are not 100% Alfvénic.

Fig. 2. Observed wave's properties and the valuesderived for the right-hand resonant instability.

From the values of the plasma-field correlations and the sense of polarization, we identify the waves with kBo=15o as the hot beam right-hand instability, and believe that the remaining right-handed waves may be identified as the hot beam right-hand resonant mode, but the small cp makes mode identification uncertain. The observed transport ratios for intrinsic left-handed and right-handed waves are comparable, indicating that their properties are very similar. The observed waves show a compressive component that is in contrast to the incompressive character predicted by linear theory for low frequency modes with small kBo.

DISCUSSIONS AND CONCLUSIONS

The discrepancies found between the observed transport ratios and the values given by the theory may be due to several factors that are different between the theory and the real foreshock. Linear theory considers small amplitudes whereas the studied events had normalized amplitudes ~0.5 - 1.20 so that nonlinear processes may be important. Linear theory is based on plane waves, but the minimum variance analysis reveals only a marginal determination of k in many instances. The theoretical transport ratios are calculated from an individual wave, in reality what is observed in the foreshock is the superposition of different modes with different frequencies. Thus, any mode identification is describing the main mode in each region. We evaluated the theoretical transport ratios by considering a isotropic beam while in reality the diffuse ions are not 100 % isotropic. Besides, absolute errors in the proton density (e.g. the under-estimation due to the solar wind coverage) can impact the value of p and the transport ratios.

The differences found between the observed and theoretical transport ratios values also show that the observed modes are distinct from the theoretical ones and that mode identification based only on magnetic polarization quantities does not give a complete picture of the waves' characteristics and can lead to mode identification of waves whose polarization may agree with theoretical predictions even when other properties can diverge from those of the theoretical modes. This accentuates the importance of using transport ratios in addition to the polarization to characterize and identify the observed fluctuations. A more accurate description of the waves also requires a theory in which nonlinear processes are taken into account.

REFERENCES

Blanco-Cano, X., and S. J. Schwartz, Ann. Geophys., in press, (1996).

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Elaoufir, J., A. Mangeney, T. Passot, C. C. Harvey, and C. T. Russell., Ann. Geophys., 8, 297, (1990).

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Southwood, D. J., W. A. C. Mier-Jedrzejowicz, and C. T. Russell, IEEE Trans. Geosci. Remote Sensing, GE-23(3), 301, (1985).

Thomsen, M. F., in Reviews of Current Research, Geophys. Monogr. Ser., 35, ed. B. T. Tsurutani, pp. 253, AGU, Washington, D. C., (1985).