3. Wave Analyses

Wave analyses are used to determine the properties and characteristics of waves, such as propagation direction, wave frequency, wave amplitude, polarization, and mode. With measurements at multiple locations one can determine the phase speed, wave number, and etc. The waves of concern here are at frequencies near or lower than the ion gyrofrequency, or so-called ULF (Ultra Low Frequency) waves, measured in the spacecraft frame. Waves of higher frequencies will be discussed briefly in section 3.6. We will focus on electromagnetic waves with emphasis on MHD waves. Methods for wave analysis based solely on the magnetic field measurements have been well developed and will be discussed as routine procedures in section 3.2. From measurements at a single site these analyses can provide only the propagation direction (with an ambiguity in sign) and wave parameters in the spacecraft frame. To resolve the Doppler shift and to determine the direction of propagation, one needs to either use more than one satellite and plasma measurements or to make assumptions on the mode of the wave and use the dispersion relation calculated from theory as an accurate description of the wave. We note that dispersion relations developed for small amplitude linear waves may not be accurate for the large amplitude waves encountered in space. Also dispersion relations that ignore the interaction with the gyro and thermal motions, such as in the Hall-MHD treatment, may be inaccurate in the moderate and high beta collision-free plasmas in space. Related issues are discussed in section 3.3. To identify the mode of a perturbation is crucial to understand the underlying physics of the processes that generate the wave. Several schemes to make the identification have been developed in recent years and will be discussed in section 3.4.

3.1 Background.

The most powerful tool in general use for wave analyses is the Fourier analysis. Its principles can be found in most time-series data-analysis textbooks. However, in space physics, we deal with Fourier analysis of vectors. For a wave the perturbations in different components of a vector are correlated. This relationship defines a covariance matrix in the frequency domain, or the so-called spectral matrix. The Principal Axis Analysis can be used to analyze the covariance matrix in much the same way as it is used above in the time domain analysis [McPherron et al., 1972]. Fourier analysis, the resultant spectral matrix, and the Principal Axis Analysis form the main ingredients of the most popular wave analysis technique for waves in space plasmas.

3.1.1 Approximations.

Most of the existing wave analysis techniques in the time domain are based on the assumption that there is one dominant wave being analyzed. Most time series arise from a combination of many waves from various sources. If the coupling among different coexisting waves is weak, the analysis can be performed in the frequency domain. Spectral analysis transforms the time series into a simple superposition of waves of different frequencies. Usually when we analyze a wave we treat a finite band of frequencies, i.e. several successive Fourier amplitudes are grouped together. At each of the individual frequencies there is an amplitude and a phase, or a cosine (in phase) and a sine (quadrature phase) amplitudes. If there is a single source of all the wave amplitudes seen in this band, then the ratios of the amplitudes seen in two different components of the magnetic field over the band will be fixed (but not necessarily the amplitudes themselves) this quality is measured by the parameter called coherence which varies from zero to unity. In practice a random signal has a non-zero coherence whose value is determined by the number of Fourier estimates in the band analyzed.

Many magnetospheric waves appear with clear sinusoidal patterns, however, they may not be propagating but standing, in the sense they are the sum of two waves propagating in opposite directions with nodes in the ionosphere. For these waves, the Fourier analysis to be discussed in the following is applicable, but the matrix analysis is not. We note that there is another type of standing wave, one in which the wave propagation velocity is exactly balanced by the plasma bulk velocity so that the wave train stays fixed relative to its source. Such waves can be analyzed with the matrix method if the motion of the spacecraft or the source carries the observer through the wave train. We note further that the special property of the whistler mode that the group velocity exceeds the phase velocity allows energy to be pumped into a phase standing wave.

If the amplitude of the fluctuations is large, the harmonic generation associated with the nonlinear effects will affect the analysis. We discuss this issue in section 3.5.

3.1.2 An ideal wave.

The magnetic field vector can usually be measured relatively easily with adequate accuracy and time resolution. Moreover, it carries most of the Poynting flux in electromagnetic waves in plasmas of interest to us. (Here we recall that the ratio of the electric to magnetic energies for an electromagnetic wave equals the ratio of the phase velocity to the speed of light.) Thus it is often used to model wave fluctuations. The magnetic vector for a monochromatic plane wave propagating in the z direction is

B_x & = B_{x0} + \delta B_x \ {\rm exp} \left[ -i ...
 ...k \cdot r}
+ \phi) \right] \cr
B_z & = B_{z0} \cr
\eqno (3.1)\end{displaymath}

where subscript 0 and prefix $\delta$ denote the average and deviation from the average, respectively, and $\omega, {\bf k}, {\bf r}$ and $\phi$ are the frequency, wavenumber, spatial vector and the phase, respectively. Since $\grad \cdot {\bf B} = 0$, ${\bf k} \cdot \delta {\bf B} = 0$ for plane waves and therefore $\delta B_z = 0$ . This is a most important feature in wave analyses. The power spectral matrix is

\buildrel \leftrightarrow \over P(\omega) = 
\left( \matrix{...
 ...phi} & \delta B_y^2 & 0 \cr
 0 & 0 & 0 \cr}
\eqno (3.2)\end{displaymath}

where $P_{ij} (\omega) = B_{i} (\omega) B_j^* (\omega); i, j, = x, y, z$; and $B_{i}(\omega)$ is the Fourier transform of Bi(t). The asterisk denotes the corresponding complex conjugate.

In general, the off-diagonal elements can be written as a real part plus an imaginary part. The real (imaginary) part corresponds to the component of which $\delta
B_i$ and $\delta B_j$ are in or 180$^{\circ}$ out of phase (of phase shift of $\pm 90^{\circ})$.

The intensity of the wave is defined as

I (\omega) = Tr \buildrel \leftrightarrow \over P(\omega) = \delta B_x^2 + \delta B_y^2
\eqno (3.3a)\end{displaymath}

and the ellipticity [Rankin and Kurtz, 1970]

\epsilon = \tan \psi
\eqno (3.4a)\end{displaymath}


\sin 2 \psi = {2 I_m (P_{xy}) \over \left[ (Tr \buildrel \le...
 ...\vert\buildrel \leftrightarrow \over P_{xy}\vert

and $\buildrel \leftrightarrow \over P_{xy}$ is the 2x2 subtensor of $\buildrel \leftrightarrow \over P$.For a linearly polarized wave, $\phi = 0$ or $\pi , \ \ \epsilon = 0$.For a circularly polarized wave, $\phi = \pm {\pi \over 2}$, $\delta B_x
= \delta B_y $ and $ \epsilon = \pm 1$, where the plus sign is for right-hand polarization and the minus sign is for left-hand polarization.

3.1.3 Spectral analysis.

For a given interval, the spectral matrix $\buildrel \leftrightarrow \over P(\omega)$ can be evaluated for a selected frequency range, $\Delta \omega$.In general the matrix can be written as

\buildrel \leftrightarrow \over P= Re (P_{ij}) + i \ Im (P_{ij})
\eqno (3.5)\end{displaymath}

Its real part is symmetric, and its imaginary part is antisymmetric and consists of only off-diagonal elements because the covariance matrix is Hermitian. The real part can be diagonalized using the Principal Axis Analysis (see section 1.4). In principal axis coordinates, the matrix is

\buildrel \leftrightarrow \over P^{\prime} = 
\left( \matr...
 ...xz}^{\prime} & - P_{yz}^{\prime} & 0 \cr
} \right) 
\eqno (3.6)\end{displaymath}

where $\lambda_1 \gt \lambda_2 \gt \lambda_3$ are real.The matrix for isotropic noise is

\buildrel \leftrightarrow \over P_{noise} = \alpha 
 1 & & \cr
 & 1 & \cr
 & & 1 \cr}
\eqno (3.7)\end{displaymath}

Comparing the above expressions with equation (3.2), one finds that $\lambda_3$ corresponds to the noise in the ${\bf k}$ direction. The wave intensity should then be defined, if the noise is isotropic, as,

I = \lambda_1 + \lambda_2 -2 \lambda_3
\eqno (3.3b)\end{displaymath}

As we discussed in section 3.1.2, each imaginary element gives the correlation of the fractions of the two corresponding components that have $\pm 90^{\circ}$ phase shift. $Im (P_{ij})^{\prime}$ should be equal to or less than $\sqrt {\lambda_i
\lambda_j}$.If ${\bf k}$ is along $\hat z^{\prime}, Im P_{xz}^{\prime}$ and $Im P_{yz}^{\prime}$ should be very small. For a purely elliptically polarized wave with isotropic noise, $Im
P_{xy}^{\prime} = \sqrt {(\lambda_1 - \lambda_3) (\lambda_2 - \lambda_3)}$,and $\sqrt {\lambda_1 - \lambda_3}$ and $\sqrt {\lambda_2 - \lambda_3}$ are the lengths of the major and minor axes of the polarization ellipse, respectively. For a nearly linearly polarized wave, $\lambda_2 \approx \lambda_3$.The ellipticity is

\vert \epsilon \vert = \sqrt {{\lambda_2 - \lambda_3 \over \lambda_1 - \lambda_3}}
\eqno (3.4b)\end{displaymath}

and its sign is the same as of $Im P_{xy}^{\prime}$.The sense of the polarization is determined by the sign of $Im P_{xy}^{\prime}$ (right-hand for plus and left-hand for minus). The amplitude of the wave can then be defined as

a = \sqrt{{ I \over 1 + \epsilon^2}}
\eqno (3.8a)\end{displaymath}

A linearly polarized wave can be decomposed as two circularly, but with opposite senses, polarized waves. An elliptical polarized wave can also be decomposed into two circularly polarized waves with opposite senses but different amplitudes. Therefore, every wave has left-hand and right-hand components. One can show that the amplitudes of the right- and left-hand components are [Kodera et al., 1977]

a_{{R \atop L}} = {(\lambda_1 - \lambda_3) \pm I_m P_{xy}^{\...
 ...{ \lambda_1 - \lambda_3} \over 2} (1
\pm \epsilon)
\eqno (3.8b)\end{displaymath}

3.2 Routine Wave Analysis.

In practice a time series of data usually contains some flags and gaps. Before any wave analysis, one has to remove these flags and fill the gaps. Sometimes, spikes in the data can be considered as flags if their time scale is much much smaller than the period of the wave being studied. In general, deflagging reduces the wave power at higher frequencies and degapping adds wave power to low frequencies. Some instruments are operated periodically leaving periodic gaps in the data. The gaps will create false wave power in the corresponding periods. After the preparation of the data, one is ready for the routine wave analysis.

3.2.1 Coordinate system.

A wave can be analyzed more easily by choosing a proper coordinate system. Since one way to characterize a wave is to see if the wave is compressional or transverse, we suggest the use of field-aligned coordinates as discussed in section 1.2.2. In this coordinate system, if the perturbations are mainly along the background magnetic field direction, the wave is compressional, otherwise it is transverse.

Assume in magnetic field coordinates the field direction as $\hat z$.In general the spectral matrix is not diagonal, Pzz is the compressional power and Pxx + Pyy is the power transverse to the field. The real part of an off-diagonal element $Re P_{ij} / \sqrt{P_{ii} P_{jj}}$ gives the portion of which the two components are in phase (when positive) or 180$^{\circ}$ out of phase (when negative). Note here ij satisfies the right-hand rule, otherwise a minus sign needs to be added. Similarly, $Im P_{ij} / \sqrt {P_{ii} P_{jj}}$ gives the portion of the signals that are $\pm 90^{\circ}$ out of phase. The properties of a wave can be decided by careful examination of each element of the spectral matrix as will be discussed in sections 3.2.3 to 3.2.6.

3.2.2 Detrending.

Most wave analysis methods essentially perform Fourier analysis on a segment of a time series data. A critical but implicit assumption of Fourier analysis is that the time series is periodic. Thus a slow trend in the signal is transformed into a sawtooth variation by the Fourier analysis technique. Fig. 3.1 shows the effects of various detrending methods on subsequent Fourier analysis. The top line shows the Fourier spectra of linear, quadratic or cubic trends of the field increasing from zero to one nanotesla. The spectra for the three trends are almost identical. The slope is about -2. This slope will have a profound effect on studying the slopes of spectra as discussed in section 3.2.3. The corresponding amplitude can be as large as 10 percent in the low frequency range and decreases one order for every two-order increase in frequency. A non-zero average of a constant signal has only minimal effects on the spectral analysis as shown in the lowest line in Fig. 3.1. A linear detrending can remove a linear trend completely and lower the effects down to the level of the non-zero average trace shown.

Fig. 3.1. Effects of the trends in the background field on the Fourier analysis. The magnitude of the trend is one nanotesla over an interval of 2000 sec. The lowest line show the spectrum of a constant background. The noise (10 orders smaller than the trend) is associated with the finite digitization of the calculation. The top line shows the spectra of linear, quadratic and cubic trends. The middle three lines show the spectra of each specified trend after detrending.

Similarly quadratic detrending will remove the power due to the quadratic trend. A linear detrending is effective to reduce the error in higher frequencies for a quadratic trend, but not as effective at lower frequencies as shown in the second line from the bottom. The effects of a cubic trend are difficult to remove using either linear or quadratic detrending. The effects of higher order trends are especially important for magnetospheric wave studies since the measured background magnetospheric field compared with wave signals consists of strong higher order trends when a spacecraft moves radially through the Earth's field. A popular method, in addition to linear and quadratic detrending, is so-called prewhitening techniques. An example of such is the differencing method that analyzes the difference between two neighboring measurements. In principle, the differencing is similar to linear detrending between every two data points. Trends are removed unevenly throughout the interval being studied. It works very well in regions without discontinuities. In the differencing process, the power spectrum has been increased by a factor of $\omega^2$. The additional factor is slightly smaller than $\omega^2$ at the low frequency end [Takahashi et al., 1990].

3.2.3 Power spectrum.

After detrending the data, one can examine the power spectra of the waves. To make the spectra, one selects a time interval which includes the wave activity of interest. Observed waves often are not perfectly sinusoidal for a long period. How many wave cycles should be included for an analysis? The longest period in a Fourier spectrum is determined by the length of the selected interval. If only one wave cycle is selected, one would not obtain a meaningful peak in the spectrum. Furthermore, any residual trend in the background will strongly affect the analysis. It is common practice to select an interval containing at least five to six wave cycles in order to derive a statistically significant result. The more wave cycles are included, the better are its statistics, but the more chance has one to include waves of different sources or effects of source variation.

The spectrum can be spiky. To enhance the robustness of the result, one can average several Fourier estimates. The number of the total Fourier estimates in the average is called "bandwidth". Increasing the bandwidth improves statistical significance of the spectrum, reduces the resolution of frequency, and increases the lower-cutoff frequency of the spectrum.

To examine power spectra, firstly one may compare the power in the frequency range of interest for the different components and the strength of the field. Usually, the power for one or two components is much higher than that for the others. To determine whether a wave is compressional or transverse one can compare the power in the strength of the field Pt (the compressional component) with Px + Py + Pz - Pt (the transverse component) where Px, Py, and Pz are powers in Bx, By, and Bz, respectively.

Secondly, one may look for peaks in the spectra. A sharp peak indicates that the wave is nearly monochromatic and a resonance of some sort may occur in the process being studied. However, if a sharp peak occurs exactly at the frequency of the spin of the spacecraft or at some multiple, it may be caused by an imperfect despinning process. Usually the spin tone occurs only in the two transverse components to the spin axis and usually little in the field strength. A broad peak may indicate that many wave modes contribute to the wave. If the plasma is not in a rapid motion, or the Doppler shift is small (see detailed discussions in section 3.3), one may compare the peak frequency with the characteristic frequencies of the plasma, such as the gyrofrequencies. If the wave frequency is much smaller than the ion gyrofrequency, the wave is usually referred to, rightly or wrongly, as an MHD wave. If the wave frequency is near the ion gyrofrequency, the wave may be associated with ion gyromotion. Sometimes the trough between two peaks may be also of interest since it may be associated with resonant absorption by particles [Young et al., 1981; Anderson et al., 1991]. Thus a trough could occur at a local characteristic frequency for certain species.

Thirdly, one may examine the slope of the spectrum [LaBelle and Treumann, 1988]. The slope may provide some information about cascading processes in the frequency domain. Cascading processes describe the evolution of wavepower in wavenumber. A wave cycle may break into two wave cycles with smaller wavelengths or two wave cycles may coalesce to form one wave cycle with a longer wavelength. If this process continues, a single wave mode may evolve into a spectrum with many wavenumbers. However, we should point out that to compare the measured slopes with those given by theory is not as straight forward as many people thought. The cascading processes are described in the wavenumber domain in almost all theoretical work. They may appear differently in the frequency domain. In interplanetary space, since the solar wind velocity is much greater than the phase velocity for MHD waves, the measured spectrum is essentially due to the Doppler shift and hence is proportional to that of the wavenumber. Thus, a power spectrum in the frequency domain can be converted linearly into the wavenumber domain if the waves at different frequencies propagate in the same direction (which needs to be demonstrated), and is easily used in theoretical investigations. In other cases, the conversion from the frequency domain to the wavenumber domain may not be linear due to the dispersion. The slope of the power spectrum in the frequency domain may not be purely due to cascading. We should point out that an imperfect detrending process will strongly affect the analysis of the slope of a spectrum. One way to check this is to examine whether the amplitude in the lower frequency end is significantly smaller than a few percent of the trend in the background field.

3.2.4 Coherence.

In any region in space several wave modes may coexist. Each mode has a particular polarization at a particular frequency. When we observe these waves we measure the sum of the wave modes and the noise of the observing system. One way to separate these wave modes is to check the cross correlations between different components as a function of frequency. Coherence analysis is a particular way of using the cross-correlations i.e. the off-diagonal elements in equation (3.5). The coherence is defined as the square-root of |Pij|2 / (Pii Pjj). (see also section 4.2.) In the coordinate system of the waves, for a purely compressional wave, the field aligned component should have little coherence with the two transverse components. On the other hand, for a transverse wave, the two transverse components may have high coherence. Note, however, if one is not performing the analysis in the coordinate system of the waves so that any wave might excite all three sensors or directions of analysis, then high coherence could result in high coherence in all these possible pairs of signals. Combining the information gained from the coherence analysis with that from the power spectra, one may find sometimes that different peaks in the power spectra correspond to different wave modes. In coherence analysis, the phase difference between two components may also be calculated, $\phi_{ij} = {Im P_{ij} \over \vert Im P_{ij}\vert} \tan^{-1} (Im
P_{ij} / Re P_{ij})$. From the phase difference between the two transverse components, one may be able to determine the polarization of a transverse wave. The phase difference can also be used for mode identification as discussed in section 3.4.

3.2.5 Propagation direction.

In principle, there are two methods to determine the propagation direction ${\bf k}$.Comparing equation (3.6) with (3.2), in analogy of the minimum variance analysis in the discontinuity analysis, one can take ${\bf k}$ either parallel or antiparallel to the minimum eigenvector. This method is referred to as Born-Wolf method or principal axis analysis [Born and Wolf, 1965]. This method is based on the information contained in the real part of the covariance matrix, sometimes called the cospectrum. Its applicability is limited by the ratio of $\lambda_2 / \lambda_3$ in the same manner as the minimum variance analysis. An alternative method is based on the information contained in the imaginary part of the covariance matrix, sometimes called the quaspectrum. Means [1972] shows that the unit vector ${\bf k}^o$ can be given by

k_x^o & = \pm Im P_{yz}^{\prime} / IP \cr
k_y^o & ...
 ... IP \cr
k_z^o & = \pm Im P_{xy}^{\prime} / IP \cr
\eqno (3.9)\end{displaymath}

where $IP = (Im P_{xy}^{\prime 2} + Im P_{xz}^{\prime 2} + Im
P_{yz}^{\prime 2})^{1/2}$, the plus (minus) sign is for right (left)-handed polarization.

The differences between the two methods were studied by Arthur et al. [1976], who suggested that the Born-Wolf method may be better for linear polarization and the Means method for circular polarization. One advantage of the Means method is that it provides the sense of the polarization.

For a linearly polarized wave, the propagation direction cannot be well determined from the minimum variance analysis. If the waves are mainly compressional, their propagation direction can be determined according to the coplanarity theorem assuming that they are either fast or slow modes (see more discussion on wave modes in section 3.3). The coplanarity theorem requires that the background field, the perturbed field and the wave vector be coplanar for the fast and slow mode waves. Since the perturbed field is perpendicular to the wave vector, the propagation direction is [Russell et al., 1987]

{\bf k} \vert\vert (\Delta {\bf B} \times {\bf B}_0) \times \Delta {\bf B}
\eqno (3.10)\end{displaymath}

where the direction of the perturbed field, $\Delta \bf B,$ is in the direction of the maximum variance.

The propagation direction cannot be well determined for nearly linearly polarized intermediate mode waves.

3.2.6 Wave properties.

From principal axis analysis of the covariance matrix, one can derive properties of the wave such as its amplitude, ellipticity and compressibility using the definitions given in section 3.1. For a linearly polarized wave the intermediate eigenvalue and minimum eigenvalue may be close to each other, and hence the propagation direction not well determined. In this case, the direction of the perturbed field which is the direction of maximum eigenvector is well determined. If the linearly polarized wave is compressional (fast or slow mode), the propagation direction can be determined according to the coplanarity theorem. On the other hand, for a nearly circularly polarized wave, the maximum eigenvalue is close to the intermediate eigenvalue. In this case, the propagation direction is well determined. If all three eigenvalues are similar, say the differences are less than an order of magnitude, the fluctuations are turbulent. Neither the propagation direction nor polarization is well determined. Note that the wave properties are derived for only a selected frequency band. When one integrates over the entire frequency range, the results give the average properties over the whole spectrum and should be identical to the results of principal axis time series analysis. Wave analysis needs the background magnetic field for referencing the wave polarization. Thus if the wave properties are obtained after detrending or filtering, one must make sure that the background field be made available in the analysis.

3.2.7 Filtering.

Filtering the time series data can be used to examine the behavior, amplitude for example, of the wave changing with time or location in space. The interesting issues here are generation or damping of the waves, the nodes of the field line resonance and wave packets. Filtering may be performed in the time domain or the frequency domain. In the time domain filters may be symmetric about the point of interest with identical weights multiplying the data on either side of the central point. Such filters have no phase lag, but have the disadvantage for real time use that they look forward in time. Filters can also be recursive using only data previous to the point in question. Such filters have phase variations with frequency but can be used in real time situations. Filtering can also be done with weighting the powers in the frequency domain. Such a filtering process is accomplished by multiplying a filter function by the Fourier spectrum to be filtered, and then inverting the resultant spectrum back to the time domain. An ideal filter function removes the variations in the unwanted frequency range while keeping the rest unchanged. However, in reality, this process may either introduce artificial fluctuations into the resulting time series or leave some residual power in the frequencies to be filtered.

Fig. 3.2 shows examples of commonly used filter functions using time domain weighting. One selects a filter according to the requirements of his/her analysis. There are three major concerns in choosing a filter: how close to unity its pass-band is, how close to zero its stop band is and how sharp the cutoff is. In general, each filter is good in one or two of the three aspects. For example, the rectangular window, Figure 3.2a, has a sharp cutoff, but its stop band is not clean and it creates artificial waves in the pass band. The Kaiser window, Figure 3.2c, has the broadest cutoff transition but with the lowest stop band response. The Hanning window, Figure 3.2b, is a compromise between the rectangular and Kaiser windows and is widely used in data analysis. Hamming window, not shown, is similar to the Hanning window and is also popularly used.

Fig. 3.2. Response functions of three low-pass filter windows. (a) Rectangular window, (b) Hanning window, and (c) Kaiser window. The left panels show the linear response and right panels show decibel response. The Nyquist frequency is 2 Hz and the high-cutoff frequency is 1 Hz.

3.2.8 Hodogram.

The polarization and the decay or growth of a wave can be most clearly presented in form of hodograms [e.g., Russell et al., 1971, Le et al., 1989], see Fig 3.3 for example. Hodograms can be made either before or after filtering the data. In particular, hodograms should be able to illustrate clearly the properties derived from the wave analysis. If not, one should check over his/her analysis carefully.

Fig. 3.3. Hodograms of the wave packets upstream of the bow shock of comet Giacobini-Zinner. The upper frame shows the magnetic field measurements in minimum variance coordinates. The lower two frames show the hodograms with the average field removed.

3.2.9 Dynamic spectra.

A particular useful display of spectra is called the dynamic spectrum a time series of power spectra. It contains the information discussed above in sections 3.2.3 and 3.2.7. One may also examine the time evolution of the coherence and ellipticity of the waves. To understand the properties of a wave, one still has to perform the principal axis analysis, either explicitly or implicitly.

To make a dynamic spectrum, one needs to determine the number of data points, or the length of the window, for each individual spectrum and the overlap of the time series for successive spectra. Although a broad frequency range is presented in a dynamic spectrum, the window length should be determined according to the wave of most interest and the guidelines discussed in section 3.2.3. The lower frequency portion is in general under-represented and the higher frequency portion may be less coherent. If a wave is present only over a finite time interval, in the dynamic spectrum, it may ``propagate'' outside of the finite interval because of the length of the moving window. The presence of a discontinuity or a single spike in the time series can cause much serious problems in a dynamic spectrum analysis. The Fourier spectrum of a discontinuity or spike is broad-banded. It usually appears as a strip across all frequencies. Because an automated procedure to make a dynamic spectrum does not recognize a discontinuity, a strip with enhanced power will extend to a width of the window length on each side of the discontinuity (from the start to the end when the window includes the discontinuity). This could be potentially confused with a broadband wave of a finite region (unfortunately, this is not a hypothetical problem.) Therefore, it is extremely important to examine the original time series data and not to simply rely on a computerized automated dynamic spectral analysis, for example, in a statistical study.

3.3 Mode Identification.

With the routine wave analysis discussed in section 3.2, we have determined the frequency range of the wave of interest and overall properties of the wave, such as the propagation direction, polarization, and frequency. However, since these results are obtained in the spacecraft frame, the frequency in the plasma frame, wavelength, and phase velocity of the wave in general remain unknown. The magnetic field measurements from a single spacecraft alone in principle cannot determine these wave parameters. To further understand a wave, one needs measurements either from separated spacecraft or from plasma instruments.

Two major parameters of interest are the phase velocity and the frequency in the plasma frame, (the wavenumber then can be calculated). In this section we will discuss how to identify a wave mode which gives the range of the phase velocity and physical functions of a wave. In the next section, we will resolve the Doppler shift and determine the frequency in the plasma frame.

3.3.1 MHD modes in homogeneous plasmas.

Waves at frequencies much lower than the ion gyrofrequency can be treated as MHD waves in low beta plasma $(\beta \le 0.1)$. We note that it is common to refer to these low frequency waves as MHD waves even when the wave dispersion cannot be derived by the MHD assumption. In the recent years, there has been an increasing awareness that the dispersion and other properties of a wave described by the MHD theory are different from that by the kinetic theory and may not be appropriate for moderate and high beta plasmas, with which we are often dealing in space. Nevertheless, we base our discussion on MHD theory and point out where caution should be taken.

In MHD linear theory for isotropic uniform plasmas, there are four modes, the fast, intermediate, slow and entropy modes [e.g. Kantrowitz and Petschek, 1966; Kivelson and Russell, 1995]. There is an ongoing debate on the existence of the MHD slow mode. In linear kinetic theory, the slow mode is strongly Landau damped in high $\beta$ plasmas when the electron temperature is smaller than the ion temperature, as it is in the magnetosheath, plasma sheet and most of the magnetosphere, and its phase velocity is greater than the intermediate mode velocity [Gary, 1992]. However in data analyses, an observation should not eliminate the slow mode from consideration solely on the basis of theory, because conditions may exist that maintain this mode in the face of damping. In the following discussion, we will refer to the intermediate mode as the Alfven mode to avoid the implication of the ordering in the phase speeds.

The entropy mode is a nonpropagating perturbation with a zero phase velocity. The phase velocities for the other three modes are derived from linear MHD theory, to be,

v^2_{phase (f, s)} = {1 \over 2} \left[ (C_S^2 + C_A^2) \pm ...
C_A^2)^2 - 4 C_S^2 C_A^2 \cos^2 \theta} \right]
\eqno (3.11a)\end{displaymath}

v_{phase (a)} = C_A \cos \theta
\eqno (3.11b)\end{displaymath}

where CS, CA and $\theta$ are the sound speed, Alfven speed and the angle between the wavevector and the background field. The subscripts f and s stand for the fast and slow modes and correspond to the plus and minus signs on the right-hand-side of equation (3.11a). The subscript a stands for the Alfven mode. Their dependence of $\theta$ and $\beta = 2 C_s^2 / C_A^2 \gamma$ is given in Fig. 3.4.

Fig. 3.4. MHD phase velocities as functions of $\beta$ and field direction. The field direction is along y. Ratio CS 2 / CA 2 equals $\gamma \beta/2$, quantities CM, CA and CS are the phase velocities of the magnetosonic, intermediate and slow modes, respectively. The phase velocity for the entropy mode is zero, i.e., at the origin of each frame.

Characteristic features of each mode can be obtained from the perturbation relations. The Alfven mode is incompressible, and thus the perturbations should be transverse. Both fast and slow modes are compressible and hence contain perturbations in both field pressure and plasma pressure. For the fast (slow) mode, the two pressures vary in (180$^{\circ}$ out of) phase. These characteristics can be easily differentiated in the power spectra, (see section 3.2.3,) and coherence analysis, (see section 3.2.4). In inhomogeneous plasmas, the Alfven mode may contain variations in density and the field strength. However, it requires that the variations in the total pressure, the sum of the thermal and magnetic pressures, be zero. Here we can see how important to have an accurate intercalibration between the plasma and field measurements. Otherwise, an inhomogeneous Alfven wave may be misidentified as a homogeneous slow wave, and vice versa. Under this situation, careful examination of the perturbation and propagation directions is extremely important.

The fast mode propagates more isotropically than the other two propagating modes. Since the other two modes do not propagate perpendicular to the magnetic field, the fast mode most efficiently transmits the pressure perpendicular to the field. The Alfven mode bends and twists the magnetic field and the plasma motion. The function of the slow mode is more interesting. As shown in Fig. 3.5, if one applies a pressure perturbation perpendicular to a flux tube, the field strength will increase to conserve the flux while the cross-section of the tube decreases. If this is done slowly then, the thermal pressure will decrease because the pressure has a chance to equilibrate. Then the thermal pressure is anticorrelated with the magnetic field pressure, $B^2 / 2 \mu_0$.This is accomplished by the plasma moving away from the compressed region. Thus the role of the slow mode is to convert the perpendicular pressure perturbations to parallel pressure perturbations. With these physical pictures in mind, one can more readily understand why a particular mode exists in a certain region.

Fig. 3.5. Physical functions of the three propagating MHD modes.

3.3.2 Mirror mode.

The mirror mode instability can be derived from the MHD slow mode branch with inclusion of a finite temperature anisotropy, $ T_{\bot} / T_{\vert\vert} $.However, Southwood and Kivelson [1993] showed that the instability is a result of kinetic effects. The unstable condition is

{T_{\bot} \over T_{\vert\vert}} \geq 1+{1 \over \beta_{\bot}}
\eqno (3.12)\end{displaymath}

where $\beta_{\bot}$ is the plasma beta evaluated using the perpendicular temperature. The mirror mode is a purely growing mode with zero frequency in the frame of plasma. It has a maximum growth rate when ${\bf k}$ is about $70^{\circ}$ from the field direction [Gary, 1992]. The perturbations resulting from the instability convect with the plasma flow and are observed as oscillating waves. Krauss-Varban et al. [1994] showed that these perturbations correspond to the entropy mode in MHD. The mirror mode is expected to exist downstream of the bow shock [Crooker and Siscoe, 1977; Lee et al., 1988] and in outer magnetosphere where the plasma condition usually meets the mirror instability criterion. The perturbations associated with the mirror mode are similar to that of the perpendicular propagating slow modes. A major difference between the two is that the phase velocity is zero for the former but non-zero for the latter. However, when the slow mode phase velocity is small and the flow velocity is dominant, the two modes are difficult to distinguish as discussed in section 3.4. The schemes discussed in the next subsection are designed to differentiate the slow and mirror modes.

3.3.3 Transport ratios.

One way to identify a wave mode is to measure the phase velocity and compare it with the expected dispersion relation [Hoppe et al., 1981]. However, as will be discussed in section 3.4, to measure the phase velocity needs at least two spacecraft and 3-D plasma measurements. A different approach is to determine the mode according to the ratios among perturbations of different quantities. These ratios are referred to as transport ratios [e.g., Gary and Winske, 1992]. Because of the difference in the roles of waves, each mode has a particular set of values of the transport ratios which can be calculated from theory.

Transverse ratio. The transverse ratio is defined as the ratio of the transverse component of magnetic wave power to the compressional component power, or

T_R = \delta {\bf B}_{\bot} \cdot \delta {\bf B}_{\bot} / \delta {\bf B}_{\vert\vert}^2
\eqno (3.13)\end{displaymath}

where $\delta {\bf B}_{\bot} \cdot \delta {\bf B}_{\bot} = \delta {\bf B}
\cdot \delta {\bf B} - \delta B_{\vert\vert}^2$, and $\delta B_{\vert\vert}$ is the amplitude in the magnetic field strength. When $T_R \gg 1$, the wave is transversely polarized and when $T_R \ll 1$,compressionally polarized.

Compressional ratio. The compressional ratio is defined as the ratio of the compression in the plasma to the magnetic field perturbation, or

C_R = {\delta N^2 \over N_0^2} {B_0^2 \over \delta {\bf B} \cdot \delta {\bf
\eqno (3.14)\end{displaymath}

It represents the partition of the wave power between the plasma (density) and magnetic field. Since usually the error in the thermal pressure measurements is smaller than that in the density, as discussed in section 1.3.3, it is recommended to use $\delta P/P_0$ to replace $\delta N/N_0$.

Phase ratio. It has been defined as

P_R = {\delta P_i \over P_{i0}} {P_{B0} \over \delta P_B}
\eqno (3.15a)\end{displaymath}

and also by

P_R = {\delta N \over N_0} {B_0 \over \delta B_{\vert\vert}}
\eqno (3.15b)\end{displaymath}

Since only the sign of the ratio concerns us, the two definitions are essentially the same. In practice, the noise is lower for the first definition. This ratio determines whether the compression in the magnetic field and plasma is in phase or out of phase. It is useful only for compressional waves. For incompressible waves, the two perturbations are dominated by noise.

Alfven ratio. The Alfven ratio is one of the earliest recognized transport ratios, and is defined as

A_R = {\delta {\bf v} \cdot \delta {\bf v} \over C_A^2} {B_0^2 \over \delta
{\bf B} \cdot \delta {\bf B}}
\eqno (3.16)\end{displaymath}

It is particularly useful for theoretical investigations.

Doppler ratio. The Doppler ratio is defined as

D_R = {\delta {\bf v} \cdot \delta {\bf v} \over v_0^2} {B_0^2 \over \delta
{\bf B} \cdot \delta {\bf B}}
\eqno (3.17)\end{displaymath}

It is different from the Alfven ratio by a factor of MA2 where MA is the Alfven Mach number. This ratio is important because it contains not only the information on the velocity fluctuations but also on the background flow itself, which is not included in the Alfven ratio. The background velocity becomes crucial when one is to determine whether the phase velocity is greater or less than, or is significant at all compared with the flow velocity. Under some approximations, the Doppler ratio is related to the ratio of the frequency in the rest frame of the plasma to that in the spacecraft frame.

An important feature of the above transport ratios is that they are independent of frequency. Therefore, they can be generalized in the frequency domain by assuming a spectrum is a simple superposition of waves of different frequencies. They are also independent of the propagation direction, and hence isolated from the errors and uncertainty that may occur when determining the propagation direction.

When comparing the observed transport ratios with those calculated from theory, one could find that none of the modes in theory completely match the observations. In order to characterize a fluctuation with a mode, one has to choose a mode that is ``most likely" to represent the wave. Different schemes will evaluate the ``likelihood" from different angles. Song et al. [1994] first introduced a hierarchical scheme; Denton et al. [1995] proposed a parallel scheme. Recently, Omidi and Winske [1995] systematically investigated the mode identification problem using data from computer simulations.

3.3.4 Hierarchical scheme.

The scheme proposed by Song et al. [1994] is a qualitative and deterministic scheme. One follows the chart shown in Fig. 3.6 and makes a yes or no decision at each point. The levels of the boxes have been determined according to the accuracies of the measurements.

Fig. 3.6. A hierarchical scheme of wave mode identification [Song et al., 1994]. A given fluctuation can be distinguished among four different modes. At each step, the user makes a yes-no decision. The level of a box is determined according to the accuracy of the measurements. The scheme can be implemented in the frequency domain assuming that the waves are linear superpositions.

In the solar wind, the Alfven velocity is much less than the flow velocity and the lowest branch of this scheme does not apply.

3.3.5 Parallel scheme.

Denton et al.[1995] proposed a parallel scheme. In this scheme, all observed transport ratios are treated to be equally accurate. Each observed ratio is then compared with the theoretical values for all modes with different possible propagation angles. A mode is identified as the one with the smallest sum of the differences between theoretical and observational values of a select set of ratios.

3.4 Frequency and Phase Velocity.

If the frequency of a wave in the plasma frame is $\omega$ and the plasma flows with a velocity ${\bf V}$ relative to an observer, the frequency measured by the observer is

\omega^{\prime} = \omega + {\bf k} \cdot {\bf V}
\eqno (3.18)\end{displaymath}

This relationship can be derived from equation (1.1) with a Fourier transformation. The second term on the right is the Doppler shift. Dividing both sides of equation (17) by ${\bf k}$, we have

v^{\prime} = v_{phase} + V \cos \eta
\eqno (3.19)\end{displaymath}

where $v^{\prime} = \omega^{\prime}/ \vert {\bf k} \vert$, vphase and $\eta$ are the apparent velocity of the wave to the spacecraft, the phase velocity of the wave and the angle between the flow velocity and the propagation direction. The direction of the apparent velocity is along the propagation direction.

With plasma measurements and the routine wave analysis discussed in the last section, $V \cos \eta$, the Doppler velocity can be determined. The apparent velocity can be measured if there are two separated spacecraft,

v^{\prime} = {{{\bf L \cdot k}^{o}} \over {\Delta t}}
\eqno (3.20)\end{displaymath}

where ${\bf L}$ and $\Delta t$ are defined the same as equation (2.20) and will be discussed further in section 4. This method fails when the separation of the two space spacecraft is along the wave front, or ${\bf L} \cdot {\bf k}^{o} =0$. If the uncertainty in the determination of propagation direction which affects both $\eta$ and v' is large, the result has an extremely large uncertainty.

There have been debates about whether the apparent velocity defined by equation (3.20) corresponds to the phase velocity or group velocity. Because the phase velocity is the propagation velocity of the oscillations and the group velocity is the propagation velocity of the envelop of a group of oscillations, the group velocity corresponds to a time scale significantly longer than that of oscillations. Therefore, if the timing delay is measured from the frequency range of the oscillations, the apparent velocity corresponds to the phase velocity.

With measured frequency from the spectral analysis, the wave number, phase velocity and the frequency in the plasma frame can then in principle be determined. Without both 3-D plasma velocity and the timing difference measurements, to resolve the phase velocity, one has to make assumptions.

In the solar wind, the phase velocity of fluctuations is usually much smaller than the Doppler velocity. The phase velocity for entropy modes is zero in theory. In these two cases, the apparent velocity is similar to the Doppler velocity. The wavelength can be derived with only one spacecraft. It is important to point out however that to assume a fluctuation to be an entropy mode is to make a substantial physical assumption. For example, some oscillations in the magnetosheath which have been assumed to be mirror modes actually have a significant phase velocity [Song et al., 1992].

For quasi-standing waves, the apparent velocity is much smaller than the phase velocity. In this case, the Doppler velocity and the phase velocity are similar in magnitude but opposite in direction.

In summary, wave properties can be determined by combining plasma and field measurements, if a wave is nearly purely one of the four MHD modes. With more than one spacecraft, the apparent velocity of the wave to the spacecraft and hence the wavelength can be determined. The Doppler shift can be determined from 3-D plasma measurements and the minimum variance analysis. Therefore, the properties of the wave can be quantitatively determined with two of the above three determinations. The uncertainties for the three determinations are different. The determination of the apparent velocity has the least uncertainty but requires two spacecraft. The determination of the Doppler shift has the next to least uncertainty. However, if the flow velocity measurements are two dimensional, an additional assumption is needed and hence the uncertainty becomes larger. The phase velocity calculated from MHD dispersion relations has a relatively large uncertainty since the wave may not be purely a single mode and may not be in the linear stage, and may not be quantitatively described by MHD at all.

3.5 Nonlinear Effects.

In the previous sections, we have discussed how to analyze waves of small amplitudes. If the amplitude is large, nonlinear effects will become important.

3.5.1. Steepened wave.

If nonlinear effects involve the temperature, a wave will steepen into saw-tooth type profile. Physically, the steepening process is a harmonics generation process: the power at harmonics of the fundamental wave frequency is enhanced. The Fourier spectrum of a saw-tooth wave, Fig. 3.7, shows significant power in all harmonics. Therefore peaks at high frequencies in a spectrum may not necessarily indicate a set of waves, instead they could be simply the result of a single steepened wave. A visual inspection of time series data should effectively prove or disprove the possibility.

Fig. 3.7. The Fourier spectrum of a steepened, or saw-tooth, wave. The frequency of the primary wave is 1x10-3 Hz and the amplitude of the perturbation equals the background field strength.

3.5.2. Large amplitude transverse waves.

It is possible that nonlinearity occurs only in the magnetic field in particular when the plasma beta is high. The nonlinearity in the magnetic field alone may not lead to wave steepening. Both the positive and negative variations in the transverse components increase the magnitude of the magnetic field. As the result, there are new issues in analyzing linearly polarized perturbations.

One of the most important issues in treating large amplitude field fluctuations is $<\vert\bf B \vert \gt \neq \vert < \bf B \gt \vert$ where <> denotes averaging in time domain. The difference is significant in evaluation of the Alfven velocity, plasma $\beta$and relative wave amplitude of the field. Given a linearly polarized field perturbation ${\bf B} = (B_{x0} + B_1 \cos \omega t, B_{y0}, 0), 

the average field is $< {\bf B} \gt = {\bf B}_0 = (B_{x0},B_{y0}, 0) $ and the magnitude of the average field is

\vert< {\bf B} \gt\vert = \sqrt {B_{x0} ^2 + B_{y0} ^2 }
\eqno (3.21)\end{displaymath}

The magnitude of the field $B=\vert{\bf B}\vert = \sqrt {(B_{x0}+B_1 \cos \omega t) ^2 +
B_{y0} ^2 }$. The average field strength is

<\vert{\bf B}\vert\gt = {1 \over T}\int_{-T/2} ^{T/2} \vert{\bf B}\vert dt
\eqno (3.22)\end{displaymath}

The difference between $<\vert{\bf B} \vert\gt$ and $\vert< {\bf B} \gt\vert$ depends on the propagation angle. In general, $<\vert{\bf B} \vert\gt$ is greater than $\vert< {\bf B} \gt\vert$.They become the same for perpendicular propagation when Bx0 > B1. When comparing observations with theory, one has to decide which definition of the field strength corresponds to the value in theory. Most theories of waves assume that a wave is superposed on an average field. In these cases, one should use the definition given by equation (3.21). Namely, one should use the strength of the average field, instead of the average of the field strength.

A most important phenomenon when analyzing large amplitude field fluctuations is the appearance of the higher harmonics in the field strength in particular the second harmonic because when a component of the field varies between positive and negative, both positive and negative perturbations contribute positively to the field strength. The amplitudes of the harmonics in the field strength are

\vert B\vert _m = {2 \over T} \int _{-T/2} ^{T/2} B \cos m \omega t~ dt~~~~ m= 1,2,3,....
\eqno (3.23)\end{displaymath}

The harmonics and their amplitudes in the field strength can be significantly different from those in the components. There is no general simple analytical expression for the integral in equation (3.23). The first harmonic in the field strength would be dominant if Bx0 > B1. The second harmonic would be important if Bx0 < B1. Therefore, one may expect a significant presence of the second harmonic in the field strength for parallel propagation but not for perpendicular propagation. Figure 3.8 shows the Fourier spectra of the field strength that result from monochromatic large amplitude field fluctuations in one component ($B_0=\vert{\bf B}_1\vert=1$).

Fig. 3.8. Power spectrum of the field strength for large amplitude field fluctuations. The frequency of the primary wave is 1x10-3 Hz and the amplitude of the perturbation equals the background field strength. The angles given in the legend are the angle between the background field and the perturbed field. Peaks appear at the double frequency and higher harmonics of the primary wave when the wave is not purely compressional.

The spectrum in the field strength depends strongly on the angle between the background field and the perturbed field, namely the angle between ${\bf B}_0$ and ${\bf B}_1$. The power in the first harmonic decreases and the power in the second harmonic increases with the angle. Figure 3.9 shows an example in which wave power does not show in the field strength in the fundamental frequency but in the second harmonic. The dashed lines indicate spectra of white noise the energy density of which is independent of frequency. The most significant peak at the low frequency end of each component, as marked as F1, F2 and F3 has no significant corresponding peak in the spectrum of the field strength. This is consistent with nearly parallel propagation large amplitude waves as shown by the thin solid line in Figure 3.8. Peaks in the spectrum of the field strength clearly appear at the frequencies twice of the frequencies at the peaks in the components, as marked as 2F1, 2F2 and 2F3. At least one of them, 2F2, has no corresponding peak in the spectra of the components. Therefore, the second harmonic in the field strength is a major indicator of the nonlinear effects and creates problems in using wave analysis scheme based on linear theory, such as in Figure 3.6. If one examined only the spectrum of the field strength, this second harmonic peak could be mistakenly identified as a compressional wave. Therefore, it is important to check the spectra of components: for a compressional wave, enhanced power should appear in at least one of the components.

Fig. 3.9. Fourier spectra of large amplitude ($\sim$ average field) magnetic field perturbations. The scales for the three components have been shifted upward by one decade consecutively. The dashed lines indicate spectra of a slope of -1. The most significant peak at the low frequency end of each component, as marked as F1, F2 and F3 has no corresponding peak in the spectrum of the field strength. But there are peaks at the frequencies twice of these frequencies in the field strength, as marked as 2F1, 2F2 and 2F3.

3.6 Plasma Wave Analyses.

Electric field measurements are often recorded with a dipole antenna mounted perpendicular to the spacecraft spin axis. During the spacecraft rotation, the electric potential differences among the sensors are measured. The electric field variations with scales longer than or comparable to the length of the antenna can be derived by examining these potential differences with respect to the spin modulation and antenna's orientation. These signals are often Fourier analyzed on-board and the Fourier components are transmitted to the ground. The magnetic field fluctuations are usually measured by a search coil magnetometer in the same instrument package. If all six components of the electric and magnetic fields are available, one is able to determine the Poynting flux of each wave. Analyzing plasma wave measurements largely relies on the dynamic spectral analysis as discussed in sections 3.2.3 and 3.2.9.

Here are some general guidelines to reading the spectra. If the enhancements appear in both the electric and magnetic fields, the waves are eletromagnetic. From the Faraday's law, loosely speaking, the ratio between the power in the two components is the square of the phase velocity. For example, if the two spectra have a similar slope in a frequency range, the dispersion is weak in this range. If the strength of the electric component increases faster with frequency than the magnetic component, the phase velocity increases, and vice versa. If the enhancements occur only in the electric component, the waves are electrostatic. In this case, from Faraday's law, the wavevector is parallel to the electric perturbations and the phase velocity cannot be derived by using the ratio between the electric and (zero) magnetic amplitudes. Observationally, the latter may be largely determined by noise. Polarizations of the wave and the propagation direction (of electromagnetic waves) may be derived by examining the spin modulation of the signals [Scarf and Russell, 1988; Strangeway, 1991; Song et al., 1998].

To interpret a plasma wave is generally more difficult than low frequency waves. The first difficulty one encounters is whether the wave is generated where it is observed or not. Near the source region, a wave does not need to satisfy any particular dispersion because of the combination of the wave growth and possible strong spatial damping. One cannot assume a region of greater wave amplitude is the source region because a greater amplitude does not necessarily mean a greater wave energy flux. The phase velocity of a plasma wave is sensitive to the local plasma conditions. Strong waves usually occur in the region of rapid changes in plasma parameters. Many wave modes may be reflected at these boundaries which complicates the possible interpretations. One has to examine carefully the plasma parameters associated with the wave and estimate the unstable conditions for each of possible modes.

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