Nonlinear dynamics of foreshock structures: Application of nonlinear autoregressive moving average with exogenous inputs model to Cluster data

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007ja012493, 2008 Nonlinear dynamics of foreshock structures: Application of nonlinear autoregressive moving average with exogenous inputs model to Cluster data D. Zhu, 1 M. A. Balikhin, 1 M. Gedalin, 2 H. Alleyne, 1 S. A. Billings, 1 Y. Hobara, 1 V. Krasnosel skikh, 3 M. W. Dunlop, 4 and M. Ruderman 5 Received 25 April 2007; revised 7 August 2007; accepted 8 October 2007; published 23 April [1] Nonlinear processes identification techniques based on multi-input nonlinear autoregressive moving average with exogenous inputs model has been applied to four-point Cluster measurements in order to study nonlinear processes that take place in the terrestrial foreshock. It is shown that both quadratic and cubic processes are involved in the evolution of shocklets in particular in the steepening of their leading edge and generation of whistler precursor. Nonlinear processes do not play an essential role in the dynamics and propagation of small-amplitude whistler packets. However, for large-amplitude wave packets, cubic processes lead to the considerable modification of apparent propagation velocity. Citation: Zhu, D., M. A. Balikhin, M. Gedalin, H. Alleyne, S. A. Billings, Y. Hobara, V. Krasnosel skikh, M. W. Dunlop, and M. Ruderman (2008), Nonlinear dynamics of foreshock structures: Application of nonlinear autoregressive moving average with exogenous inputs model to Cluster data, J. Geophys. Res., 113,, doi: /2007ja Introduction [2] The ultimate goal of the experimental investigation of waves in space plasma is to deduce from the measurements a comprehensive physical model that accounts for distribution of turbulence energy among plasma modes and provides quantitative description of physical processes involved in the dynamics of turbulence. In the case of the developed turbulence that are observed in various regions of solar-terrestrial plasma such as foreshock, magnetosheath, cusp, etc., the latter include both linear and nonlinear processes. Linear processes are processes in which there is no energy exchange between various scales of plasma turbulence. Therefore they are limited to the energy exchange between plasma and turbulence and involve growth of plasma waves due to some plasma instability or damping of waves and transfer of wave energy into the plasma population. Nonlinear processes involve coupling between various scales of plasma turbulence. Obvious examples for quadratic and cubic process are 3-wave decay instability and 4-wave modulational instability correspondingly. A comprehensive model of plasma turbulence derived from data should include such quantitative characteristics of linear processes as growth (damping) rates and kernels of 1 Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UK. 2 Department of Physics, Ben Gurion University, Beer-Sheva, Israel. 3 Le Laboratoire de Physique et Chimie de l Environnement, Centre National de Recherche Scientifique, Orleans, France. 4 Space Science and Technology Department, Rutherford Appleton Laboratory, Didcot, UK. 5 Department of Applied Mathematics, University of Sheffield, Sheffield, UK. Copyright 2008 by the American Geophysical Union /08/2007JA nonlinear quadratic, cubic, and higher-order nonlinearities to be able to assess separately each contribution to the evolution of observed waves. For such a thorough model to be deduced directly from measurements, a process identification approach, in which plasma is treated as a black box input-output system, is used in the present study. Two spacecraft measurements should be considered first for simplicity. The wave field is measured first by one of these two spacecraft and can be considered as the input into the plasma system. The wave then propagates between two spacecraft. During this propagation the wave field undergoes evolution due to the linear and nonlinear processes before being registered by the second spacecraft. The wave field as measured by the second spacecraft represent an output of the plasma system. Such an approach to plasma field measurements has been proposed for study of turbulence in laboratory plasma [Ritz and Powers, 1986]. In space plasma AMPTE UKS AMPTE IRM pair have been used to identify nonlinear processes in plasma turbulence [e.g., Coca et al., 2001; McCaffrey et al., 2000; Balikhin et al., 2001]. However, in order to use these in processes identification two-point measurements, some conditions should be satisfied. The same plasma structures should be measured by both spacecraft. This can only take place either if the separation between spacecraft is very small or when the direction of the wave motion is very close to the spacecraft separation line. Generally, in the plasma rest frame the velocity of waves is omnidirectional, therefore the second condition can be satisfied only if plasma convection velocity is much higher than the plasma rest frame wave velocities and is directed almost as the spacecraft separation vector. Such conditions are not often satisfied for two point spacecraft measurements. This is the reason that a number of studies concentrated on a single case [Dudok de Wit et al., 1999; Coca et al., 2001] of AMPTE UKS AMPTE IRM measurements in the terrestrial foreshock. When four or 1of10

2 more points of measurements are available (eg. CLUSTER, THEMIS) the full three-dimensional (3-D) separation between the spatial and temporal variations can be achieved and the Multi-Input Single Output (MISO) modeling can be used to identify the dynamic processes in plasma turbulence even if the condition of collinearity between the plasma velocity and the spacecraft separation vector is not satisfied. In such an approach the data gained by three spacecraft are considered as three inputs into the system and the measurements of the fourth spacecraft as the system s output. In the present paper, MISO modeling is used to identify the dynamics of nonlinear structures and to quantitatively assess the separate contributions of linear and nonlinear processes to the evolution of shocklets in the terrestrial foreshock. The technique has been applied to Cluster observations of the terrestrial foreshock that took place on 18 February The reason that the foreshock is chosen for the first application of the MISO modeling is because the foreshock is saturated with nonlinear processes such as wave steepening for example, for which comprehensive analytical models have been developed. Therefore data based identification of the dynamic processes in the foreshock turbulence can be compared with the existing theoretical models. 2. NARMAX MISO Modeling [3] A linear dynamical system with the input u(t) and the output y(t) can be described as a convolution of the input u(t) and the impulse response h(t) yt ðþ¼ Z 1 0 h n ðtþut ð tþdt ð1þ In the discrete-time case the system can be represented as yk ð Þ ¼ X1 hi ðþuk ð iþ ð2þ i¼1 where h(k) is the discrete-time impulse response. [4] For a nonlinear dynamical system, the Volterra series model takes the form of Z yt ðþ¼ X1 1 n¼1 0 Z 1 0 Z 1 0 h n ðt 1 ; t 2 ;;t n Þ Y1 ut ð t i Þdt i ð3þ i¼1 The discrete-time form of the Volterra series model is yk ð Þ ¼ X1 n¼1 X 1 g 1 ¼1 X1 g n ¼1 h n ðg 1 ; g 2 ;;g n Þ Y1 uk ð g i Þ ð4þ i¼1 The functions h n (t 1, t 2,..., t n ), h n (g 1, g 2,...,g n ) are referred to as the Volterra kernels in the continuoustime and discrete-time cases, respectively. These series were introduced by Volterra in his studies of integral equations and functionals. The application of Volterra series to the system identification is based on a proof that a continuous real function could be approximated to any degree of accuracy by truncated Voltera series [Frechet, 1910]. Application of Volterra series to the identification of dynamical systems from input-output data sets was first proposed by Wiener [1942, 1958]. Since Wiener s pioneering works comprehensive and rigorous mathematical apparatus has been developed to solve the main problem to identify kernels of Volterra decomposition from the measurements of the input output data series and to identify Volterra model of the unknown dynamical system. [5] Most importantly, the measurement of the Volterra kernels is required in the identification of nonlinear systems based on the Volterra series model. The Volterra series representation is not only an explicit nonlinear representation of the system response in terms of the input but also affords insight into the system operation. However, the Volterra series model has its limitations and for systems described by nonanalytic models a convergent Volterra series does not exist and systems with sub-harmonics can not be modeled by a Volterra series [Boaghe and Billings, 2003]. In practice, the Volterra series model must be truncated to a finite number of terms so that it can only represent well the class of fading memory systems. The nonlinear autoregressive moving average with exogenous inputs (NARMAX) model is more general than the Volterra series model and can represent a larger class of nonlinear systems, including the Volterra, Hammerstein, and Wiener models as special cases. The NARMAX model has no restriction to systems with properties of fading memory. [6] The NARMAX model takes the form of a set of nonlinear difference equations, one for every output of the system. The general form of the NARMAX model for a multi-input multi-output (MIMO) system with m outputs and r inputs is yk ðþ¼f yðk 1Þ; ; yk n y ; uk ð 1Þ; ; uk ð nu Þ; where ek ð 1 Þ; ; ek ð n e 2 3 y 1 ðþ k 6 yk ð Þ ¼. 4 y m ðþ k 7 5; uk ðþ¼ ÞŠþek ðþ 2 3 u 1 ðþ k 6. 4 u r ðþ k 7 5; ek ðþ¼ 2 3 e 1 ðþ k 6 4. e m ðþ k 7 5 are the system outputs, inputs, and noises, respectively; n y, n u, and n e are the maximum lags in the outputs, inputs, and noises; e(k) is assumed to be independent zero-mean sequences; and F[] is some vector-valued nonlinear function [Billings et al., 1989]. [7] In our work, the underlying system is considered as a multi-input and single-output (MISO) nonlinear system, which is a special case of MIMO nonlinear systems. In this case, there are m = 1 and r = 3. Therefore this MISO system can be represented by yk ðþ¼f ðk 1Þ; ; yk n y ; u1 ðk 1Þ; ; u 1 ðk n u1 Þ; u 2 ðk 1Þ; ; u 2 ðk n u2 Þ; u 3 ðk 1Þ; ; u 3 ðk n u3 Þ; ek ð 1 Þ; ; ek ð n e ÞŠþek ðþ Since the nonlinear function F[] is generally unknown, it is convenient to take a linear in the parameters model structure to represent this function [Korenberg et al., 1988]. The generic form of a linear-in-parameters model can be expressed as yk ð Þ ¼ XM I¼1 ð5þ ð6þ q i f i ðxk ðþþþxðþ; k k ¼ 1; ; N ð7þ 2of10

3 Figure 1. Magnitude and GSE components of the magnetic field measured by Cluster 1 during the foreshock crossing on 18 February Time in hours after 1120:00 UT. where N is the data length, x(k) are monomials of degrees 0 to l, each consisting of delayed output, input, and/or noise terms (degree 0 corresponding to a constant term), f i () canbe chosen to be some global or local basis functions such as polynomials of degree l or wavelets, M is the model size (N M), x(k) is the modelling error, and q i are the unknown parameters to be estimated. In the present study polynomial basis functions are used. [8] The determination of the model structure and the unknown parameters play crucial roles in the NARMAX methodology. To determine the model structure involves the selection of significant terms from all possible terms in the model. Usually, the algorithms perform structure detection as well as parameter estimation simultaneously. For the linear in the parameters models, the forward orthogonal least squares (OLS) algorithm with the error reduction ratio (ERR) [Billings et al., 1988] has proven to be very efficient to perform structure detection and parameter estimation simultaneously both for SISO (single-input and single-output) and MIMO NARMAX models [Billings et al., 1989]. The forward OLS algorithm involves a stepwise orthogonalization of the regressors and a forward selection of the corresponding terms based on the error reduction ratio criterion. The algorithm also provides the unbiased parameter estimation for the corresponding parameters q i. Therefore the forward OLS algorithm is employed to iden-tify the MISO NARMAX models for the underlying system. [9] The identified model can be validated based on correlation tests. A MIMO nonlinear model validity test procedure [Billings and Zhu, 1995] is employed to form a global-to-local hierarchical validation diagnosis of the identified multivariable nonlinear model. This statistical validation consisting of global tests and local tests is based on higher order correlation functions. For global tests the identified model is valid if the following conditions hold: f xx ðtþ ¼ E½xðÞx k ðk þ tþš ¼ rdðtþ g f #h ðtþ ¼ E ½# ðþh k ðk þ tþš ¼ 0 8t where 0 < r < 1 and x(k), h(k), #(k) are given by x(k) = e 2 (k), h(k) =y(k) e(k), and #(k) =u 1 2 (k) +u 2 2 (k) +u 3 2 (k). [10] For local tests the submodel is correct if the following conditions hold: f e 2 hðtþ ¼ rdðtþ ð8þ ð9þ 2 3 f u 2 1 hðtþ f u 2 hðtþ ¼ f u 2 2 hðtþ ¼ 0 ð10þ f u 2 3 hðtþ 3. Foreshock Crossing on 18 February 2003 [11] The data for this study were collected by FGM instruments [Balogh et al., 2001] on board four Cluster Spacecraft at 1120: :00 UT on 18 February During this time interval all four spacecraft were in the foreshock and for 3of10

4 Figure 2. Magnitude and GSE components of the magnetic field measured by Cluster 1 during the foreshock crossing on 18 February Time in hours after 1134:00 UT. Figure 3. Magnitude and GSE components of the magnetic field measured by Cluster 1 during the time period 1120: : UT on 18 February of10

5 Figure 4. (a) Bz GSE component of the magnetic field measured by Cluster 1 (blue) and the model predicted output of the identified NARMAX model (11) (Black). Time in seconds after 1121:00 UT. (b) Bz GSE component of the magnetic field measured by Cluster 1 (blue) superimposed with the model predicted output of the identified NARMAX model (11) (black), the output corresponding to the linear part of the model (11) (red), the output corresponding to the quadtratic part of the model (11) (green), and the output corresponding to the cubic part of the model (11) (cyan). Time in seconds after 1121:00 UT. (c) By GSE component of the magnetic field measured by Cluster 1 (blue) superimposed with the model (12) (red), the output corresponding to the quadratic part of the model (12) (green), and the output corresponding to the cubic part of the model (12) (cyan). Time in seconds after 1134:00 UT. all four the FGM sampling rate was ms. Intersatellite distance was ranging from 4000 to 9000 km. The three components and the absolute value of the magnetic field as measured by the Cluster-1 spacecraft is displayed in Figure 1. The strong level of turbulence with sb B / 1 is evident from this picture. Waves with such large amplitudes are usually observed in the foreshock. Plasma population that is a combination of solar wind plasma and ions reflected from the shock front is the source of plasma instabilities that lead to the generation of the foreshock turbulence. Ions reflected from the perpendicular and quasi-perpendicular part of the terrestrial bow shock are turned back to the shock forming a rather narrow foot region. The width of the foot for a perpendicular shock in the case of specular reflection is about 0.68 of the ion convected gyroradius and is defined through the upstream bulk velocity and ion cyclotron frequency [Woods, 1969]. Modification of this relation in case of nonspecular reflection can be found in the work of Gedalin [1997]. As the angle between the normal to the shock front and the upstream magnetic field q Bn deviates from the perpendicular, the width of the foot increases. However if this angle is greater than 45, that is, the shock is quasi-perpendicular the width of the foot is still rather small and comparable with that of a purely perpendicular shock. Propagation of the beam of reflected ions in the foot leads to a number of possible plasma instabilities. However experimental observations point out that high amplitude low frequency wave packets observed in the foot are not generated by plasma instabilities but are the result of shock front dynamics [e.g., Walker et al., 1999; Balikhin et al., 2005]. This can be explained by the high solar wind velocity and narrow width of the unstable region in the foot. In case of quasi-parallel shocks, that is, if q Bn <45,the magnetic field is not able to prevent the escape of some reflected ions to infinity. The region upstream of the shock front where the motion of the bulk of reflected ions takes place is called ion foreshock. Foreshock plasma that is a combination of solar wind and reflected ion is a source of free energy for a number of plasma instabilities. Therefore foreshock region is filled with various high-amplitude plasma waves. These waves are involved both in processes of energy exchange with plasma and nonlinear processes of energy transfer between different scales of plasma turbulence. In addition to the plasma instabilities the interaction of plasma and foreshock waves leads to the scattering of reflected ions, 5of10

6 which is one of key processes in cosmic ray acceleration by collisionless shocks. Dynamics of foreshock waves has a fundamental importance for the acceleration of cosmic rays because these are the waves that provide containment of cosmic rays that undergo Fermi type of acceleration at the collisionless shocks associated with supernova remnants and other remote astrophysical objects. Some of the most remarkable wave structures observed in the foreshock are shocklets [Russell, 1988]. Shocklets are large-amplitude compressional nonlinear structures that propagated upstream in the plasma rest frame but are convected toward the shock by the solar wind flow. Discrete whistler wave packets that exhibit circular left-handed polarization in the spacecraft reference frame are often observed at the upstream edge of a shocklet. These whistler wave packets are reminiscent of wave precursors at the front of subcritical dispersive shocks [Kennel et al., 1985]. Figure 2 shows two examples of shocklets and discrete wave packets observed by Cluster-1 during the foreshock crossing on 18 February A dualsatellite measurement technique [e.g., Balikhin et al., 1997; Hobara et al., 2007] has been applied to discrete wave packets to determine the corresponding plasma wave mode in the plasma rest frame. Both examples posses similar wave characteristics. In the plasma rest frame waves propagate in upstream direction with phase velocity about 45 km/s, almost along the magnetic field q B0,k 20. The wave frequency in the plasma rest frame is close to the local proton cyclotron frequency 0.08 Hz. Since the wave polarization, which should be reversed in the plasma rest frame due to the Doppler effects, is intrinsically a right-handed polarization it can be concluded that these waves propagate in the whistler mode. Moreover, the experimentally derived rest frame phase velocity of the waves is in good agreement with that from the theoretical whistler wave dispersion relation under the quasi-longitudinal approximation (35 km/s). [12] In case of MISO methodology described above the shocklet orientation relative to 3 Cluster spacecraft from which measurements are used as inputs differs for different shocklets. Therefore the identification of common model for all shocklets is not feasible. Instead, individual models are derived for particular shocklets. However, the generic dynamics of shocklets should be common for all particular models. For the general case of 3-D turbulence a recently developed modification of NARMAX, which extends its identification capabilities to incorporate 3-D spatiotemporal systems, should be applied. However, a variance analysis of the waves investigated in the current paper indicates that their dynamics can be considered to be planar, and so a simplified version of the NARMAX MISO modeling methodology may be implemented. The results of the MISO modeling for a number of individual shocklets are presented in the following sections Shocklet 1 at 1121 UT [13] The magnitude and three components of the magnetic field in the GSE coordinate frame measured by Cluster-1 spacecraft during the time interval 1120: : UT are plotted in Figure 3. Shocklet with the adjacent whistler wave packet can be seen in this figure at 1121: : UT. The B z component is dominant for the main nonlinear part of the shocklet. Comparison of B x, B y,andb z indicates that the polarization plane of the discrete wave packet is close to y-z plane. It is the B z component that has been used for identification of MISO model. The identified polynomial NARMAX model is the following: yk ðþ¼ 7:5124e 01u 2 ðk 1Þ 6:3669e 01u 2 ðk 6Þ2:1407e þ 00u 2 ðk 10Þ 2:8297e 01u 1 ðk 1Þ2:5841e 01u 3 ðk 6Þu 3 ðk 6Þ 1:7580e 01u 1 ðk 1Þu 1 ðk 10Þ 8:9509e 02u 1 ðk 8Þu 2 ðk 10Þ 1:4263e 01u 3 ðk 1Þu 3 ðk 1Þ 1:7514e 01u 1 ðk 2Þu 3 ðk 10Þ 4:7180e 01u 3 ðk 2Þu 3 ðk 7Þ 1:0896e þ 00u 2 ðk 10Þu 3 ðk 1Þ 3:0514e 01u 1 ðk 1Þu 2 ðk 8Þ 2:2304e 01u 1 ðk 10Þu 2 ðk 1Þ 3:0054e 01u 2 ðk 10Þu 3 ðk 10Þ 9:7857e 01u 2 ðk 10Þu 3 ðk 3Þ 3:8638e 01u 1 ðk 10Þu 2 ðk 10Þ 2:1107e 01u 1 ðk 9Þu 3 ðk 4Þ 1:9067e 01u 1 ðk 3Þu 2 ðk 9Þ 9:9080e 02u 1 ðk 1Þu 1 ðk 10Þu 2 ðk 7Þ 8:1782e 02u 2 ðk 10Þu 3 ðk 10Þu 3 ðk 10Þ 2:9559e 01u 2 ðk 1Þu 3 ðk 1Þu 3 ðk 10Þ 5:4656e 02u 1 ðk 10Þu 2 ðk 10Þu 3 ðk 10Þ 4:0114e 02u 1 ðk 1Þu 1 ðk 1Þu 3 ðk 1Þ 2:1764e 01u 2 ðk 2Þu 3 ðk 1Þu 3 ðk 10Þ 7:1171e 02u 1 ðk 10Þu 1 ðk 10Þu 2 ðk 10Þ 1:5856e 02u 1 ðk 6Þu 1 ðk 10Þu 2 ðk 1Þ 3:6498e 03u 2 ðk 9Þu 3 ðk 1Þu 3 ðk 2Þ 2:9398e 01u 1 ðk 1Þu 2 ðk 1Þu 3 ðk 10Þ 3:0420e 01u 1 ðk 4Þu 2 ðk 10Þu 3 ðk 2Þ 4:7804e 02u 1 ðk 5Þu 1 ðk 9Þu 2 ðk 1Þ 3:2607e 01u 1 ðk 1Þu 2 ðk 2Þu 3 ðk 9Þ 1:1452e 01u 1 ðk 3Þu 1 ðk 3Þu 2 ðk 10Þ 1:1951e 01u 1 ðk 5Þu 1 ðk 5Þu 2 ðk 3Þ 9:0876e 02u 1 ðk 4Þu 2 ðk 1u 3 ðk 9Þ 5:2836e 02u 1 ðk 6Þu 1 ðk 6Þu 2 ðk 9Þ 1:2264e 01u 1 ðk 10Þu 1 ðk 10Þu 3 ðk 7Þ 1:2920e 01u 1 ðk 3Þu 2 ðk 10Þu 3 ðk 4Þ 9:2773e 02u 1 ðk 8Þu 1 ðk 10Þu 3 ðk 7Þ 1:1903e 01u 2 ðk 10Þu 3 ðk 1Þu 3 ðk 4Þ 2:6639e 01u 1 ðk 4Þu 2 ðk 10Þu 3 ðk 1Þ ð11þ It is worth noting that individual terms in the above equation do not represent orthogonal functions. However the right-hand side of (11) represents a sum of auxiliary orthogonal functions used in the application of OLS algorithm [Billings et al., 1988; Korenberg et al., 1988] to Cluster data. Each of these auxiliary orthogonal functions represents linear combination of the individual terms of equation (11). [14] Data sets measured by Cluster 2, 3, and 4 have been considered as inputs and data of Cluster-1 as output for this 6of10

7 Figure 5. Magnitude and GSE components of the magnetic field measured by Cluster 1 during the time period 1134: : UT on 18 February and all other models discussed in the present paper. Both the output of the MISO NARMAX model based on equation (11) (black) and the real measurements of the B z component as measured by the Cluster-1 (blue) during the time interval 1121: : are shown in Figure 4a. Very good correspondence between the output of the NARMAX model and the real measurements for the nonlinear structure of the shocklet and for adjacent wave packet can be used as an extra validation for the identified model in addition to nonlinear correlations. One of the advantages of NARMAX modelling based on structure detection is that it leads to physically interpretable models. It makes NARMAX methodology a more comprehensive tool in the study of unknown dynamical systems in comparison with other techniques such neural networks, for example, that often lead to noninterpretable models. Explicit NARMAX polynomial model can be decomposed in linear, quadratic, cubic, and higher-order nonlinear components. In the particular case of NARMAX model (11) the highest-order nonlinearity contribution is the cubic one. Nonlinear correlations used in the process of validation ensure that the relative contribution of these components to the output of the model reflects the importance of corresponding processes in the dynamics of the object under investigation. Linear (red), quadratic (green), and cubic (cyan) contributions in the integral model output (black) are plotted in Figure 4b. In addition to the main shocklet with adjacent whistler wave train a small-amplitude whistler wave packet (db 1 nt) can be seen in this Figure during time period 1121: : UT. For this small-amplitude wave packet the linear contribution accounts for almost the whole model output and matches the Cluster-1 measurements, while both quadratic and cubic contributions are negligible. It should be emphasized that linear contribution models not only the magnitude but also the phase of this wave packet. These results can be expected for a small-amplitude waves. For the main nonlinear structure of the shocklet observed during the time interval of 1121: : UT, the contribution of nonlinear terms is comparable to the linear contribution in the initial part of shocklet with quadratic term dominating in the nonlinear contribution. However, the nonlinear contribution becomes more significant toward the steepened edge of the shocklet. The increase of cubic component is more substantial and its contribution exceeds the quadratic one at the very edge of the steepened part of the shocklet structures. The dominance of the cubic contribution at the steepened edge of shocklets were reported previously as a result of AMPTE data analysis [Coca et al., 2001]. The interplay between linear a nonlinear contributions in the dynamics of the wave packet adjacent to the shocklet is much more complex. Figure 4a shows very close correspondence between the model and the real measurements of this wave packet in magnitude and in the phase. At the same time it can be seen in Figure 4b that the linear contribution to the identified model results in the wave processes that has the same frequency, higher magnitude, and considerable phase shift in comparison with real observations by Cluster-1. In the higher-amplitude part of these wave packet that are close to the shocklet-wave packet interface the phase shift between the linear contribution and the real data is so big that it seems that they are in antiphase with each other. However, as the wave magnitude decreases this phase shift changes and for a relatively small amplitude oscillation observed around 1121: UT it seems that the phase difference between the linear contribution and the real waves packet is close to zero. As it was mentioned above, the output of the whole model is in phase with real measurements, which is only possible if the nonlinear contribution will affect the mishap of phase difference produced by the linear part of the model. Indeed it can be 7of10

8 seen in Figure 4b that it is the cubic nonlinearity that corrects the phase of the model output Shocklet 2 at 1134 UT [15] The magnetic filed as measured by Cluster-1 spacecraft during the time interval 1134: : UT are plotted in Figure 5. The shocklet with the attached discrete wave packet is observed during time interval 1134: : UT. The B y component is dominant for the main nonlinear part of the shocklet and whistler wave packet polarization plane is close to the y-z plane. Therefore it is the B y component that has been used for identification of MISO NARMAX polynomial model: yk ðþ ¼5:4974e 01u 2 ðk 1Þ 3:0814e 02u 1 ðk 8Þ 9:1199e 01u 2 ðk 7Þ4:6364e 01u 1 ðk 9Þ 3:4956e 01u 1 ðk 4Þ 5:7026e 01u 2 ðk 10Þ 3:3731e 01u 1 ðk 6Þ4:1238e 01u 2 ðk 6Þ 3:0650e 02u 2 ðk 1Þu 2 ðk 1Þ 5:1960e 02u 2 ðk 6Þu 2 ðk 7Þ 1:0450e 01u 3 ðk 9Þu 3 ðk 9Þ 8:8730e 02u 1 ðk 8Þu 2 ðk 10Þ 6:4654e 02u 2 ðk 10Þu 2 ðk 10Þ 1:5470e 01u 1 ðk 3Þu 1 ðk 9Þu 1 ðk 9Þ 1:4526e 02u 3 ðk 9Þu 3 ðk 9Þu 3 ðk 9Þ 1:0380e 01u 1 ðk 9Þu 1 ðk 9Þu 2 ðk 1Þ 1:6611e 01u 1 ðk 3Þu 1 ðk 3Þu 1 ðk 9Þ 1:3526e 01u 1 ðk 3Þu 1 ðk 8Þu 2 ðk 1Þ 2:6467e 01u 1 ðk 6Þu 1 ðk 9Þu 2 ðk 6Þ 6:3286e 02u 1 ðk 6Þu 2 ðk 6Þu 2 ðk 10Þ 7:9848e 02u 1 ðk 4Þu 2 ðk 10Þu 2 ðk 10Þ 1:1780e 01u 1 ðk 4Þu 1 ðk 6Þu 2 ðk 6Þ 1:2979e 01u 1 ðk 3Þu 1 ðk 3Þu 2 ðk 10Þ 5:7834e 02u 1 ðk 8Þu 1 ðk 9Þu 1 ðk 9Þ 7:4359e 02u 1 ðk 4Þu 1 ðk 4Þu 1 ðk 6Þ 1:8648e 01u 1 ðk 6Þu 1 ðk 9Þu 1 ðk 9Þ 6:4989e 02u 1 ðk 8Þu 2 ðk 6Þu 2 ðk 7Þ 6:1767e 02u 1 ðk 4Þu 2 ðk 1Þu 2 ðk 10Þ 8:4160e 02u 2 ðk 1Þu 2 ðk 7Þu 2 ðk 7Þ 1:3418e 01u 1 ðk 6Þu 2 ðk 1Þu 2 ðk 7Þ 1:8785e 01u 1 ðk 3Þu 2 ðk 6Þu 2 ðk 10Þ 2:0939e 01u 1 ðk 3Þu 2 ðk 6Þu 2 ðk 7Þ 4:8670e 02u 2 ðk 6Þu 2 ðk 7Þu 2 ðk 7Þ 2:4331e 01u 1 ðk 6Þu 1 ðk 9Þu 2 ðk 1Þ 1:3268e 01u 1 ðk 6Þu 1 ðk 6Þu 1 ðk 8Þ 1:1696e 01u 1 ðk 6Þu 1 ðk 6Þu 2 ðk 10Þ 1:1822e 01u 1 ðk 3Þu 1 ðk 9Þu 2 ðk 6Þ 1:1494e 01u 1 ðk 6Þu 1 ðk 9Þu 2 ðk 10Þ 1:1557e 01u 1 ðk 3Þu 2 ðk 1Þu 2 ðk 7Þ 6:2012e 02u 1 ðk 6Þu 2 ðk 1Þu 2 ðk 10Þ ð12þ The B y component as measured by the Cluster-1 spacecraft is plotted in Figure 4c (black) together with the output of the identified model (12) (blue). As in the previous model (11) there is a very close similarity between the output of the model and real measurements. Resemblance provides additional validation for the model (12). Decomposition of linear (red), quadratic (green), and cubic (cyan) component of the model (12) are plotted in the same Figure 4c. The dynamic of the shocklet nonlinear structure (1134: : UT) is almost linear as the linear contribution is the dominant one and exceeds both quadratic and cubic one. The relation between relative contributions of quadratic and cubic terms varies for different parts of the shocklet. For the main part of the shocklet 1134: : UT the cubic contribution is negligible in comparison to the quadratic. However, the gradients in the B y observed around 1134: UT coincides with the decrease in the quadratic and the increase in cubic components. From that moment the cubic component dominates the nonlinear contribution. The steepened leading edge is the region of strongest gradient in the shocklet structure. It is also an interface between the main structure of the shocklet and the adjacent wave packet. At this edge the cubic contribution exceed the linear contribution. It stays the dominant for the almost the whole discrete wave packet, except the very low amplitude part of it where the magnitude of oscillations is around 1 nt. For these last couple of oscillations observed 1134: : UT the linear contribution again becomes the dominant one. Similar to the Shocklet 1 the model (12) represents the dynamic of the discrete wave packet very well without exhibiting any apparent phase shift between the model output and Cluster 1 measurements, at the same time the essential phase shift between the linear contribution and the measurements can be seen in Figure 4. This phase shift is corrected by the cubic contribution, while the effect of quadratic terms is negligible in comparison to the cubic and linear terms. Therefore the relative relation for linear, quadratic, and cubic contributions to the shocklet and wave packet dynamics are almost the same as for the shocklet Discussion [16] MISO NARMAX modelling of other shocklets in the foreshock crossing on 18 February leads to similar interplay of the linear, quadratic, and cubic contributions as was unveiled for three shocklets analyzed above. Previously, frequency domain modeling [Dudok de Wit et al., 1999] technique has been applied to shocklets in the foreshock (in the work of Dudok de Wit et al. [1999] shocklets are referred to as SLAMS). As frequency domain methods are severely limited by the length of data sets, even combination of all three magnetic field components in one surrogate data sets did not allow authors to include the cubic component (or four wave processes in their modelling). The authors [Dudok de Wit et al., 1999] assumed that cubic processes can be disregarded for the foreshock turbulence. This assumption was not supported by rigorous physical arguments but was absolutely necessary in order to implement frequency domain modelling advocated by Dudok de Wit et al. [1999] for a data set of limited length. It should be noted that many analytical models for waves similar to 8of10

9 those observed in the foreshock involve essential contribution of third-order nonlinearities. One of the examples is derivative nonlinear Schroedinger equation (DNLS) [Shukla and Stenflo, 1985; Pokhotelov et al., 1996] which describes evolution of low-frequency (Alfven) circularly polarized waves. Exclusion of third order nonlinearity from frequency domain modelling in the work of Dudok de Wit et al. [1999] makes impossible the comparison of their results with the present paper. [17] Probably one of the most well understood and simple nonlinear processes that occurs in ordinary hydrodynamic and in plasma waves of finite amplitude is wave steepening. In case of MHD waves the wave velocity depends on the absolute value of the magnetic field jbj and upon density that is related to the magnetic field for MHD waves. For the typical conditions in the terrestrial foreshock fast magnetosonic waves will always steepen [Kennel et al., 1985; Hada and Kennel, 1985]. It is customary to relate steepening in hydrodynamic waves to the advection þ ¼ 0; ð13þ where n is some parameter (e.g., magnetic field). The solution is n(x, t) =n(x V(n)t). If the wave mode and medium are such that dv dn > 1 the leading edge will steepen. For MHD waves the magnetic pressure leads to the quadratic nonlinearity of the which also contributes to the wave steepening. For moderate amplitudes the velocity can be Taylor expanded, V = V 0 +(dv/ db)db, where db is the magnetic field perturbation, so that the quadratic nonlinearity is the most important. Since dv/ db > 0, the leading edge should steepen. Such steepening of the leading edge for shocklets have been reported in many experimental studies. Figure 4 represent experimental evidence that it is the advection type quadratic processes that dominates the nonlinear part of the shocklet dynamics. Steepening can be also considered in the spectral domain. It has been shown many times [e.g., Sagdeev and Galeev, 1969] that in the spectral domain steepening corresponds to the transfer of energy to shorter scales and higher frequencies. As amplitudes of these short-wavelength components grow the nonlinearity starts to play a role in their dynamics as well, leading to the energy transfer to even shorter scales and so on. This cascade of energy transfer can be interrupted either by dissipation or dispersion. If this cascade reaches the characteristic scale of some dissipative processes (e.g., dissipation related to ion-sound anomalous resistivity), energy supplied to this spectral range by steepening will dissipate. In the other scenario steepening will terminate when the dispersive scale is reached. In the long-wavelength limit, MHD approximation can be used to describe dynamics of waves in the foreshock. Infinitesimal amplitude MHD waves are nondispersive, as their velocity does not depend upon spatial coordinate. As steepening process transfers energy to shorter scales at some point validity of classical MHD approximation will be reached, and two-fluid model should be used instead [e.g., Kennel et al., 1985]. [18] Infinitesimal amplitude waves in the frame of two fluid model are dispersive. The dispersion of short-wavelength waves generated by steepening limits as the speed of short-wavelength waves differs from the speed of the main shocklet they are carrying energy away from the steepening edge. This is the same process that leads to the formation of subcritical dispersive shocks [Kennel et al., 1985]. Evolution of nonlinear Alfven waves is determined by cubic nonlinearity [see, e.g., Shukla and Stenflo, 1985; Pokhotelov et al., 1996] appearing in the nonlinear derivative Sroedinger equation for weakly nonstationary one-dimensional waves. If shocklet steepening reaches the characteristic scale of whistler mode, as happens in our case [Hobara et al., 2007], effective pumping of energy to the whistler waves takes place that results in the formation of circular polarized wave precursor. The comprehensive model of such a precursor should account for both pumping of energy from the main shocklet structure, nonlinear dynamics of wave packet, and dissipation of energy to the plasma. Such a comprehensive theoretical model is not going to be used in the present paper. Instead, we shall discuss one-dimensional quasi-stationary models of nonlinear waves with whistler-like dispersion [see, e.g., Gedalin, 1993, and references therein]. Dynamics of a whistler wave packet: B y þ ib z ðt; x Þ ¼ B0 bt; ð x ð Þ ð14þ Þe ik0x w0t where B 0 is background magnetic field and the relative complex amplitude b(t, X) = sb B 0 slowly depends on x and time t, is described by the nonlinear Schroedinger W i 2 jbj2 b þ v 2 b w 2 ¼ 0; ð15þ where w pi is the ion plasma frequency. The wave number k 0 and the frequency w 0 are central wave number and frequency of the whistler wave packet. The ratio w 0 k 0 corresponds to the phase velocity of linear whistler waves. Substitution of b / e idkx idw t into above equation leads to the frequency shift for the central wave number k 0 of the wave packet: Dw ¼ W i 2 jbj2 ð16þ Thus the frequency shift and related change in apparent phase velocity are determined by the cubic nonlinearity. [19] The linear contribution can be used to describe propagation of very low amplitude waves. Indeed in Figure 4 linear component does not have essential phase shift with low amplitude part of wave packet. However, if the magnitude of waves is high, the linear component misrepresents the wave phase and the apparent phase velocity. It is the cubic nonlinearity that leads to the difference between the apparent phase velocity of oscillations and phase velocity of infinitesimal waves. Both in accordance with experimentally determined features that it is the cubic nonlinearity that corrects the phase speed in Figure 4. [20] It is worth noting that the strict application of the above theoretical model equations is limited because they were derived based upon a number of assumptions that are violated within the foreshock turbulence (for example it was 9of10

10 assumed that the waves possessed a small amplitude). The comparison of our experimental results with those of the theoretical models above has been performed solely as an illustration of the effects observed in the dynamics of the foreshock turbulence. 5. Conclusions [21] For the first time MISO (Multi Input-Single Output) NARMAX modelling has been successfully applied for the identification of nonlinear dynamics of space plasma turbulence observed in multispacecraft data. [22] The linear part of the identified model can be used to describe propagation of very low amplitude waves. It is shown that for the main shocklet structure effects of these second order processes leads to the steepening of the leading edge of the shocklet and generation of a whistler wave train. [23] Third-order (cubic) nonlinear processes prevail in the dynamics of discrete wave packets adjacent to the shocklets. Cubic nonlinearity leads to the the dependence of phase upon the amplitude in the observed wave packets. [24] These experimentally identified dynamical features are in very good agreement with the theoretical results. [25] Acknowledgments. The authors acknowledge that this research was supported by PPARC and EPSRC. Authors are grateful to Cluster FGM team (PI E. Lucek) for providing high quality magnetic field data. [26] Wolfgang Baumjohann thanks Gary P. Zank and another reviewer for their assistance in evaluating this paper. References Balikhin, M. A., L. J. C. Woolliscroft, H. S. Alleyne, M. Dunlop, and M. A. Gedalin (1997), Determination of wave dispersion, on the basis of cospectral characteristics of turbulence: Application to the study of plasma waves in the downstream of quasi-perpendicular shock, Ann. Geophys., 15, Balikhin, M. A., I. Bates, and S. Walker (2001), Identification of linear and nonlinear processes in space plasma turbulence data, Adv. Space Res., 28, Balikhin, M., S. Walker, R. Treumann, H. Alleyne, V. Krasnoselskikh, M. Gedalin, M. Andre, M. Dunlop, and A. Fazakerley (2005), Ion sound wave packets at the quasiperpendicular shock front, Geophys. Res. Lett., 32, L24106, doi: /2005gl Balogh, A., et al. (2001), The cluster magnetic field investigation: Overview of in-flight performance and initial results, Ann. Geophys., 19, Billings, S. A., and Q. M. Zhu (1995), Model validation tests for multivariable nonlinear models including neural networks, Int. J. Control, 62(4), Billings, S. A., M. J. Korenberg, and S. Chen (1988), Identification of nonlinear output-affine systems using an orthogonal least-squares algorithm, Int. J. Syst. Sci., 19, Billings, S. A., S. Chen, and M. J. Korenberg (1989), Identification of MIMO non-linear systems using a foward-regression orthogonal estimator, Int. J. Control, 49, Boaghe, O. M., and S. A. Billings (2003), Subharmonic oscillation modelling and MISO Volterra series, IEEE Trans. Circuits Syst. I, 50(7), Coca, D., M. A. Balikhin, S. A. Billings, H. S. K. Alleyne, and M. Dunlop (2001), Time domain analysis of plasma turbulence observed upstream of a quasi-parallel shock, J. Geophys. Res., 106, 25,005 25,022. Dudok de Wit, T., V. V. Krasnosel skikh, M. Dunlop, and H. Lühr (1999), Identifying nonlinear wave interactions in plasmas using two-point measurements: A case study of short large amplitude magnetic structures (slams), J. Geophys. Res., 104, 17,079 17,090. Frechet, M. (1910), Sur les Fonctionnelles Continues, Ann. Ecole Normale Suppl., 27, 3rd ser. Gedalin, M. (1993), Nonlinear waves in two-fluid hydrodynamics, Phys. Fluids, B5, Gedalin, M. (1997), Ion dynamics and distribution at the quasiperpendicular collisionless shock front, Surv. Geophys., 18, Hada, A. T., and C. F. Kennel (1985), Nonlinear evolution of slow waves in the solar wind, J. Geophys. Res., 90, 531. Hobara, Y., S. N. Walker, M. A. Balikhin, O. A. Pokhotelov, M. Dunlop, H. Nilsson, and H. Réme (2007), Characteristics of terrestrial foreshock ulf waves: Cluster observations, J. Geophys. Res., 112, A07202, doi: /2006ja Kennel, C. F., J. P. Edmiston, and T. Hada (1985), A quarter century of collisionless shock research, in Collisionless Shocks in the Heliosphere: A Tutorial Review, Geophys. Monogr. Ser., vol. 34, edited by R. G. Stone and B. T. Tsurutani, pp. 1 36, AGU, Washington, D. C. Korenberg, M., S. A. Billings, Y. P. Liu, and P. J. McIlroy (1988), Orthogonal parameter estimation algorithm for non-linear stochastic systems, Int. J. Control, 48, 193. McCaffrey, D., I. Bates, M. A. Balikhin, H. S. K. Alleyne, M. Dunlop, and W. Baumjohann (2000), Experimental method for identification of dispersive three-wave coupling in space plasma, Adv. Space Res., 25, Pokhotelov, O. A., D. O. Pokhotelov, M. B. Gokhberg, F. Z. Feygin, L. Stenflo, and P. K. Shukla (1996), Alfven solitons in the Earth s ionosphere and magnetosphere, J. Geophys. Res., 101, Ritz, C. P., and E. J. Powers (1986), Estimation of nonlinear transfer functions for fully developed turbulence, Phys. Scr., 20D, 320. Russell, C. T. (1988), Multipoint measurements of upstream waves, Adv. Space Res., 8, Sagdeev, R. Z., and A. A. Galeev (1969), Nonlinear Plasma Theory, Benjamin, White Plains, N. Y. Shukla, P. K., and L. Stenflo (1985), Nonlinear propagation of electromagnetic ion-cyclotron Alfven waves, Phys. Fluids, 28, Walker,S.N.,M.A.Balikhin,H.S.K.Alleyne,W.Baumjohann,and M. Dunlop (1999), Observations of a very thin shock, Adv. Space Res., 24, Wiener, N. (1942), Reponse of a Nonlinear Device to Noise, MIT Press, Cambridge, Mass. Wiener, N. (1958), Nonlinear Problems in Random Theory, MIT Press, Cambridge, Mass. Woods, L. C. (1969), On the structure of collisionless magneto-plasma shock waves at supercritical alfven-mach numbers, Plasma Phys., 3, 435. H. Alleyne, M. A. Balikhin, S. A. Billings, M. Gedalin, Y. Hobara, V. Krasnosel skikh, M. Ruderman, and D. Zhu, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK. (h.alleyne@sheffield.ac.uk; m.balikhin@sheffield.ac.uk; s.billings@sheffield.ac.uk; m.gedalin@bgu. ac.il; y.hobara@sheffield.ac.uk; m.ruderman@sheffield.ac.uk; d.zhu@ sheffield.ac.uk) M. Dunlop, Space Science and Technology Department, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK. (m.w.dunlop@ rl.ac.uk) 10 of 10