VOYAGER OBSERVATIONS OF MAGNETIC WAVES DUE TO NEWBORN INTERSTELLAR PICKUP IONS: 2–6 au

The initial distributions of newborn pickup ions resemble gyrating beams of finite pitch angle that depend on the orientation of the interplanetary magnetic field (IMF) relative to the radial direction. For quasi-radial IMF directions the beam excites resonant waves of the same wavelength as the distance along the IMF traveled by the ion in one orbit. The resultant waves have the same polarization (right-handed in the plasma frame) as the sense of ion-cyclotron motion about the mean magnetic field and propagate sunward in the same sense as the particles streaming through the ambient plasma. The particle speed in the solar wind essentially matches the solar wind speed because the speed of the neutral atom relative to the Sun is small at ∼23 km s⁻¹ (Bzowski et al. 2012; McComas et al. 2012; Möbius et al. 2012). Doppler shifting the waves into the spacecraft frame results in a left-hand polarized signal measured at the ion-cyclotron frequency. As the particles scatter to greater pitch angles, they excite waves of smaller wavelength, which means higher frequency in the spacecraft frame. Excitation of frequencies lower than the ion-cyclotron frequency would require particle energization, which generally is not seen in these observations. The result is a wave enhancement that extends from the ion gyrofrequency to higher frequencies in the spacecraft frame.

We apply the analyses used by Cannon et al. (2014a, 2014b), who studied waves due to newborn interstellar pickup protons (H⁺) seen by the Ulysses spacecraft. As with the observation of Joyce et al. (2010), we find intervals of wave excitation by pickup He⁺ in addition to examples of waves due to pickup H⁺. Events were identified by looking for spectral features near the ion-cyclotron frequency. The power spectra are generally enhanced. The observed spectral enhancements, including polarization, are limited to spacecraft-frame frequencies less than ∼5× the associated cyclotron frequency.

Lee & Ip (1987) computed the time-asymptotic power spectra resulting from fully scattered pickup ions to be

$$\begin{eqnarray}&&{I}_{\pm }(k,\infty )=\displaystyle \frac{1}{2}{\{{[C(k)]}^{2}+4{I}_{+}(k,0){I}_{-}(k,0)\}}^{1/2}\pm \displaystyle \frac{1}{2}C(k).\end{eqnarray} \tag{ 1 }$$

where ${I}_{+}(k,0)$ (${I}_{-}(k,0)$) are the background spectra for the antisunward (sunward) propagating fluctuations at wavenumber k. All k are assumed to be parallel to the mean magnetic field. For sunward-propagating waves, the wavevectors $k\lt 0$ ($k\gt 0$) denote the right-hand polarized fast-mode (left-hand polarized Alfvén) waves. The wave enhancement is described by

$$\begin{eqnarray}C(k) & = & {I}_{+}(k,0)-{I}_{-}(k,0)+2\pi {m}_{{\rm{i}}}{N}_{{\rm{i}}}{V}_{{\rm{A}}}| {{\rm{\Omega }}}_{{\rm{i}},{\rm{c}}}| {k}^{-2}\\ & & \times [{{\rm{\Omega }}}_{{\rm{i}},{\rm{c}}}{k}^{-1}{v}_{0}^{-1}-({{\rm{\Omega }}}_{{\rm{i}},{\rm{c}}}{k}^{-1}{v}_{0}^{-1}-{\mu }_{0})\\ & & \times | {{\rm{\Omega }}}_{{\rm{i}},{\rm{c}}}{k}^{-1}{v}_{0}^{-1}-{\mu }_{0}{| }^{-1}]\end{eqnarray} \tag{ 2 }$$

where m_i is ion mass, N_i is the pickup ion number density, ${{\rm{\Omega }}}_{{\rm{i}},{\rm{c}}}={e}_{{\rm{i}}}B/({m}_{\mathrm{ic}})$ is the ion-cyclotron frequency and ${f}_{{\rm{i}},{\rm{c}}}={{\rm{\Omega }}}_{{\rm{i}},{\rm{c}}}/(2\pi )$, V_A is the Alfvén speed, v₀ is the pickup ion speed in the plasma frame ($\simeq {V}_{\mathrm{SW}}$, where V_SW is the solar wind speed), and ${\mu }_{0}$ is the pitch angle of the newborn ions. We use ${f}_{{\rm{p}},{\rm{c}}}$ to be the more common expression for the cyclotron frequency of H⁺. We define $I({f}_{{\rm{i}},{\rm{c}}})$ to be the total background intensity ${I}_{+}(k,0)+{I}_{-}(k,0)$ at the cyclotron frequency, evaluating $| k|$ at $2\pi {f}_{{\rm{i}},{\rm{c}}}/{V}_{\mathrm{SW}}$.

The pickup ion density is given by

$$\begin{eqnarray}&&{N}_{{\rm{i}}}={\beta }_{{\rm{i}}}{N}_{{\rm{A}}}{\tau }_{\mathrm{ac}}\end{eqnarray} \tag{ 3 }$$

where ${\beta }_{{\rm{i}}}$ is the rate of ionization per neutral atom. For He⁺ this is just the photoionization rate ($\sim {10}^{-7}$ s⁻¹ at 1 au) scaled as the inverse square of the distance from the Sun (Ruciński et al. 1996). For H⁺, the charge exchange rate must also be included and is given by $\sigma {N}_{\mathrm{SW}}{V}_{\mathrm{SW}}$, where σ is the charge-exchange cross section for hydrogen (2 × 10⁻¹⁵ cm²) and N_SW is the solar wind proton density. N_A is the local density of interstellar neutral atoms. We take the density of neutral He to be constant at 0.015 cm⁻³ (Möbius et al. 2004). We take the density of neutral H to be given by ${N}_{{\rm{H}}}={N}_{{\rm{H}},\infty }\mathrm{exp}(-\lambda /R)$, where the heliocentric distance R is measured in the upwind direction approximated by the trajectory of the Voyager spacecraft at this time. ${N}_{{\rm{H}},\infty }\sim 0.1$ cm⁻³ is the density of neutral H at the termination shock (Gloeckler et al. 1997; Bzowski et al. 2009), and $\lambda =5.6\;{\rm{AU}}$ is the scale of the neutral H ionization cavity in the upwind direction.

The theory of Lee & Ip (1987) is a time-asymptotic formalism that computes the wave power generated by an initial population of pickup ions as they scatter completely to isotropy. It uses the accumulated density of pickup ions and the associated accumulation of wave energy to predict a spectrum a given distance from the Sun. For our purposes, and following Joyce et al. (2010), we wish to use this formalism as a means of approximating the rate of wave energy accumulation. We define an accumulation time ${\tau }_{\mathrm{ac}}$ in association with the above-prescribed ionization rates, which determines the density of the pickup population at a point in space. This parameter is adjusted so that the energy of the predicted wave spectrum matches the peak power in the observed spectrum. The resultant timescale and energy deposition rate are then compared to the timescale and energy transport rate for the turbulent destruction of the waves. The resulting growth rates are given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{W}}{{dt}}={R}_{\mathrm{ac}}\cdot \left(\displaystyle \frac{{10}^{2}}{3.6}\right)\left(\displaystyle \frac{2\pi f}{{V}_{\mathrm{SW}}}\right)\left(\displaystyle \frac{{21.8}^{2}}{{N}_{\mathrm{SW}}}\right)\end{eqnarray} \tag{ 4 }$$

where the factor 21.8 accounts for the conversion of the magnetic field to Alfvén units, V_SW is measured in km s⁻¹, and thermal proton density N_SW is measured in cm⁻³. Terms for unit conversion are included in the expression. R_ac is the derivative of the total power spectrum with respect to the accumulation time. Because ${\tau }_{\mathrm{ac}}$ is tens of hours, we compute the difference in predicted wave intensity between ${\tau }_{\mathrm{ac}}=20$ and 40 hr:

$$\begin{eqnarray}&&{R}_{\mathrm{ac}}=\displaystyle \frac{{I}_{\mathrm{tot}}({\tau }_{\mathrm{ac}}=40\;\mathrm{hr})-{I}_{\mathrm{tot}}({\tau }_{\mathrm{ac}}=20\;\mathrm{hr})}{40\;\mathrm{hr}-20\;\mathrm{hr}}\end{eqnarray} \tag{ 5 }$$

where ${I}_{\mathrm{tot}}({\tau }_{\mathrm{ac}})$ is the sum of the four components of the power spectrum (${I}_{\mathrm{tot}}={I}_{+}(+k)+{I}_{+}(-k)+{I}_{-}(+k)+{I}_{-}(-k)$). We compute separate growth rates for waves excited by H⁺ and He⁺ by evaluating I_tot at the peaks of the respective wave enhancements. Because I_tot becomes approximately linear in ${\tau }_{\mathrm{ac}}$ at the wave peaks (where the background terms are negligible), the results are not sensitive to the accumulation times used to compute R_ac.

In our application of kinetic theory we assume that the spatial distribution of interstellar neutral atoms is stationary. The density of neutral H is spatially dependent. The density of neutral He is not. The ionization efficiency due to charge exchange varies with local solar wind conditions. The ionization efficiency due to photoionization varies only with distance from the Sun.

The above explains how we use a time-asymptotic theory for wave generation in the absence of the turbulent cascade to compute growth rates as a function of ionization rates and ambient conditions. The resulting timescales required to achieve the observed wave energy are relatively long. Cannon et al. (2014b) argued that these timescales are sufficient to justify the assumption of complete particle isotropization that is at the core of the time-asymptotic analysis. Theories for the heating of the ambient plasma by absorbing the wave energy into the turbulent cascade also assume an average asymptotic form.

Numerous authors have argued through theory and observations that magnetohydrodynamic (MHD) turbulence in general and solar wind turbulence in particular evolve toward a two-dimensional (2D) state where the wavevectors are perpendicular to the mean magnetic field and the resulting turbulent cascade is well described by a simple extension of the Kolmogorov (1941) scaling law (Fyfe et al. 1977; Montgomery & Turner 1981; Shebalin et al. 1983; Higdon 1984; Matthaeus et al. 1990, 1996; Oughton et al. 1994; Hossain et al. 1995; Bieber et al. 1996; Ghosh & Goldstein 1997; Ghosh et al. 1998; Leamon et al. 1998b, 1999; Biskamp & Müller 2000; Dasso et al. 2005; Mininni et al. 2005; Hamilton et al. 2008; MacBride et al. 2010). Experiments with liquid metals in an applied DC magnetic field show that MHD turbulence evolves to nearly 2D flow (Lielausis 1975; Tsinober 1975; Volish & Koliesnikov 1976). We assume that the waves, which are largely parallel-propagating, couple to the 2D turbulence, which is itself responsible for determining the rate at which the wave energy is consumed. The resulting expression for the turbulent cascade rate in terms of observables is

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{{f}^{5/2}{[E(f)]}^{3/2}\cdot {21.8}^{3}}{{V}_{\mathrm{SW}}{N}_{{\rm{SW}}}^{3/2}}\end{eqnarray} \tag{ 6 }$$

where E(f) is the measured magnetic field power spectral density in units of nT${}^{2}$ Hz⁻¹ and is assumed to vary as ${f}^{-5/3}$. Equipartition of magnetic and kinetic energy is assumed, and the factors of N_SW and 21.8 are part of the conversion of the magnetic field to Alfvén units. This expression disagrees with that used by Leamon et al. (1999) by a factor of $2\pi {[(5/3)(1+{R}_{{\rm{A}}})/{C}_{K}]}^{3/2}\sim 6\pi$, where we take R_A = 1 and C_K = 1.6, but it is more nearly in agreement with the observed heating rates at 1 au (Vasquez et al. 2007). As we will adopt a turbulence model where the parallel-propagating waves couple to the background 2D turbulence, we will use the inferred background power levels, E(f), that omit the wave energy when computing the turbulent cascade rate.

The power spectra studied here show background power laws that vary in intensity and index. They do not necessarily conform to ${f}^{-5/3}$. There can be a great many reasons for this, including the fact that we are forced to use relatively short data intervals owing to the brief duration of the wave events in the data. Such short samples will not consistently agree with ensemble averages derived from large volumes of data. Nevertheless, we feel that it is a reasonable starting point for the discussion on which future efforts might be built.

We combine several spectral analysis methods in this paper. Power and magnetic helicity spectra are computed using Blackman–Tukey techniques (Matthaeus & Smith 1981; Smith 1981; Matthaeus & Goldstein 1982; Matthaeus et al. 1982; Leamon et al. 1998b; Smith et al. 2006a, 2006c; Hamilton et al. 2008; Joyce et al. 2010). We show the normalized magnetic helicity $-1\leqslant {\sigma }_{M}\leqslant +1$ (Matthaeus & Smith 1981; Smith 1981; Matthaeus & Goldstein 1982; Matthaeus et al. 1982), which is the spatial analog of the polarization. Fast Fourier transform techniques are used to compute the polarization parameters (Fowler et al. 1967; Rankin & Kurtz 1970; Means 1972; Mish et al. 1982), as well as power and helicity spectra, which agree with the Blackman–Tukey analyses. Polarization parameters are computed in mean-field coordinates (Belcher & Davis 1971; Bieber et al. 1996). The polarization parameters that we will show are the degree of polarization $0\leqslant {D}_{\mathrm{pol}}\leqslant 1$, coherence $0\leqslant {C}_{\mathrm{oh}}\leqslant 1$, ellipticity $-1\leqslant {E}_{\mathrm{lip}}\leqslant +1$, and angle between the minimum variance direction ${\boldsymbol{k}}$ and mean magnetic field ${\boldsymbol{B}}$ given by $0^\circ \leqslant {{\rm{\Theta }}}_{{kB}}\leqslant 90^\circ$. We part with the convention of Mish et al. (1982) and place the sign of polarization on the ellipticity instead of the degree of polarization so that ${E}_{\mathrm{lip}}\lt 0$ (${E}_{\mathrm{lip}}\gt 0$) denotes left-hand (right-hand) polarization. For typical solar wind observations where kinetic instabilites are not evident, ${D}_{\mathrm{pol}}\simeq {C}_{\mathrm{oh}}\simeq {E}_{\mathrm{lip}}\simeq {\sigma }_{M}\simeq 0$, although many of the background examples shown here do show a marginally elevated level of D_pol and C_oh. Fluctuations at dissipation scales that are generally higher in frequency than the waves studied here show ${{\rm{\Theta }}}_{{kB}}\simeq 90^\circ$. For typical wave events at the frequencies modified by the kinetic instability, we find ${D}_{\mathrm{pol}}\simeq {C}_{\mathrm{oh}}\simeq 1$ while ${E}_{\mathrm{lip}}\simeq -1$ and ${{\rm{\Theta }}}_{{kB}}\simeq 0^\circ$. The normalized magnetic helicity, ${\sigma }_{M}$, will take on values of ±1 according to the direction of the mean magnetic field (Smith et al. 1984).

The time intervals and ambient plasma parameters for data intervals shown in this paper (both wave events and their associated control intervals) are listed in Table 1. The table lists times, spacecraft locations, and ambient plasma conditions along with statements (Yes/No) of whether the interval displays waves in association with newborn interstellar pickup He⁺ or H⁺. The two intervals of questionable wave signatures are listed as "??." Most of the data are from rarefaction intervals. The exception to this is V1 on day 51 of 1978, when the spacecraft seems to be within an ejecta region of low density and cold plasma.

Table 1.  Data Interval Parameters

s/c	He+ Waves?	H+ Waves?	Year	Time	R	${{\rm{\Theta }}}_{{BR}}$	VSW	NSW	VA	${f}_{{\rm{H}},{\rm{c}}}$
	Yes/No	Yes/No			(AU)	(deg)	(km s${}^{-1})$	(cm−3)	(km s−1)	(mHz)
V2	??	No	1978	15.38–15.54	2.04	46.2 ± 2.5	414.0 ± 2.1	0.720 ± 0.070	44.7	26.4
V2	Yes	No	1978	16.04–16.13	2.05	11.3	394.1	0.541	48.6	24.9
V2	No	No	1978	16.29–16.50	2.05	49.7 ± 4.8	387.2 ± 3.3	0.601 ± 0.064	51.0	27.6
V1	No	No	1978	14.25–14.50	2.05	68.7 ± 14.8	446.1 ± 8.9	1.13 ± 0.26	45.6	33.8
V1	No	No	1978	14.50–14.75	2.06	56.2 ± 1.4	451.1 ± 0.8	1.01 ± 0.11	32.7	22.9
V1	Yes	Yes	1978	15.50–15.75	2.07	48.5 ± 4.4	403.0 ± 1.2	0.529 ± 0.024	55.5	28.1
V1	Yes	No	1978	16.25–16.50	2.07	41.6 ± 3.0	381.9 ± 1.0	0.425 ± 0.012	57.8	26.3
V1	No	No	1978	18.75–19.00	2.10	45.9 ± 6.0	319.4 ± 2.0	1.44 ± 0.13	39.3	32.9
V2	Yes	No	1978	52.04–52.25	2.37	11.5 ± 3.1	469.1 ± 3.4	0.39 ± 0.04	69.2	30.0
V2	Yes	No	1978	52.48–52.63	2.38	34.0 ± 11.5	455.1 ± 4.1	0.22 ± 0.06	88.8	29.2
V2	No	No	1978	52.48–52.92	2.38	43.5 ± 9.2	453.5 ± 4.7	0.16 ± 0.05	114.	31.8
V2	No	No	1978	52.71–52.88	2.38	48.4 ± 5.0	452.1 ± 4.9	0.13 ± 0.01	131.	33.4
V2	Yes	Yes	1979	7.81–8.00	4.52	27.6 ± 1.9	480.4 ± 0.9	0.045 ± 0.001	47.5	7.0
V2	No	No	1979	8.83–9.00	4.52	55.8 ± 1.4	460.8 ± 1.2	0.054 ± 0.001	46.8	7.6
V2	No	No	1979	264.15–264.35	5.61	59.7 ± 10.7	711.5 ± 10.9	0.32 ± 0.04	91.4	36.1
V2	Yes	Yes	1979	265.33–265.52	5.62	22.8 ± 3.6	642.1 ± 5.8	0.12 ± 0.01	145.	35.3
V2	Yes	Yes	1979	265.31–265.67	5.62	21.3 ± 3.7	636.9 ± 9.7	0.12 ± 0.01	143.	34.6
V2	Yes	Yes	1979	266.15–266.54	5.62	18.7 ± 10.4	592.8 ± 15.1	0.05 ± 0.01	155.	23.4
V2	??	??	1979	267.02–267.17	5.62	37.0 ± 5.5	542.2 ± 1.4	0.040 ± 0.004	117.	16.2
V1	No	Yes	1979	265.48–265.75	6.22	10.6 ± 4.0	719.3 ± 2.5	0.07 ± 0.01	216.	41.1
V1	No	No	1979	266.00–266.19	6.22	32.7 ± 8.2	649.1 ± 15.9	0.12 ± 0.02	155.	37.4
V1	No	No	1979	266.75–266.96	6.23	27.3 ± 9.5	659.1	0.06	200.	35.5
V1	Yes	Yes	1979	267.15–267.50	6.23	30.6 ± 2.5	640.9 ± 10.5	0.06 ± 0.01	166.	27.6
V1	Yes	Yes	1979	268.50–268.75	6.24	25.4 ± 2.1	616.4 ± 5.5	0.025 ± 0.002	191.	20.9
V1	No	No	1979	269.50–269.75	6.24	49.6 ± 10.9	595.4 ± 9.9	0.026 ± 0.005	143.	16.2

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Figure 3 shows our analysis of six data intervals that characterize the different types of observations found. The spacecraft and times are given at the top of each stack of magnetic spectra. Each stack of six panels represents a single data interval. The spectra are (top to bottom) total power spectrum (trace of the power spectral density matrix) and the power spectrum of the magnitude $| {\boldsymbol{B}}|$ time series, degree of polarization, coherence, ellipticity, angle between the minimum variance direction and mean-field direction, and normalized magnetic helicity as defined above.

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The top left panel of Figure 3 shows a control interval without evidence of waves due to pickup ions. The data were recorded by V2 in 1978 on day 15 from 09:00 to 13:00 UT. The spacecraft is in the same rarefaction interval shown in Figure 1 with only a minimal time shift allowing for the transit time of the wind between V1 and V2. Both the total power and $| {\boldsymbol{B}}|$ spectra are power laws with a factor of ∼10 difference between the two, indicating that the fluctuations are transverse to the mean field and largely noncompressive in the sense that compressive fluctuations show larger fluctuations in $| {\boldsymbol{B}}|$. D_pol, C_oh, E_lip, ${{\rm{\Theta }}}_{{kB}}$, and ${\sigma }_{M}$ are all $\simeq 0$, which is typical of the background interplanetary field. This indicates fluctuations that do not maintain coherence over many wavelengths, do not favor one sense of rotation over another, and are transverse to the mean magnetic field. ${{\rm{\Theta }}}_{{kB}}\simeq 0^\circ$ is often taken from wave theory to indicate propagation along the mean magnetic field, and this is consistent with certain wave types; however, it is also consistent with 2D turbulence, where the fluctuations are confined to the perpendicular plane and the minimum variance direction is along the mean magnetic field. At frequencies sufficiently higher than ${f}_{{\rm{p}},{\rm{c}}}$, the value of ${{\rm{\Theta }}}_{{kB}}$ increases to $\simeq 60^\circ$. This appears to mark the onset of dissipation dynamics.

The top middle panel of Figure 3 shows a data interval with strong evidence of waves due to pickup He⁺. The data were recorded by V2 in 1978 from 01:00 to 03:00 UT on day 16, and the spacecraft is again in the same rarefaction interval. There is a peak in the power at $f\gt {f}_{\mathrm{He},{\rm{c}}}$ that extends to $f\gt {f}_{{\rm{H}},{\rm{c}}}$ with an ${f}^{-2}$ behavior in the range ${f}_{\mathrm{He},{\rm{c}}}\lt f\lt {f}_{{\rm{H}},{\rm{c}}}$ as predicted by theory (Lee & Ip 1987). The enhanced power extends to $\sim 2{f}_{{\rm{H}},{\rm{c}}}$, but the spectrum falls more steeply than ${f}^{-2}$ in the range $f\gt {f}_{{\rm{H}},{\rm{c}}}$. The background spectrum appears to be a power law and is given by the dashed line. The spectrum of $| {\boldsymbol{B}}|$ also shows enhanced power in the same range, but is a factor of ∼10 smaller than the total power, again indicating that the fluctuations are largely transverse. D_pol and ${C}_{\mathrm{oh}}\simeq 1$ while ${E}_{\mathrm{lip}}\simeq -1$, ${{\rm{\Theta }}}_{{kB}}\simeq 0^\circ$, and ${\sigma }_{M}\simeq 0.5$. All of these findings point to parallel-propagating, transverse magnetic waves with left-hand polarization in the spacecraft frame arising from newborn pickup He⁺. There is some question as to whether some of these waves are due to pickup H⁺. The wave signatures become less evident for frequencies marginally larger than the cyclotron frequency, and the spectrum follows the ${f}^{-2}$ form that is predicted for a single resonance. For this reason, we suspect that all of the waves are due to the He⁺ resonance. However, it is difficult to assert that the H⁺ resonance has not produced significant wave growth that is separable from the He⁺ signature, and our analyses below indicate that it has. We suggest that when He⁺-associated waves extend to frequencies greater than ${f}_{{\rm{p}},{\rm{c}}}$, these waves can, at times, scatter the pickup H⁺ toward isotropy without the associated production of H⁺ spectral signatures. Again, as in most instances, ${{\rm{\Theta }}}_{{kB}}\simeq 60^\circ$ at frequencies greater than ${f}_{{\rm{p}},{\rm{c}}}$, which appears to mark the onset of dissipation dynamics.

The top right panel of Figure 3 shows our analysis of V1 data from 1978, day 15, hours 12:00 to 18:00 UT. The spacecraft is in the rarefaction interval shown in Figure 1 at a time with a strong He⁺ instability and undeniable H⁺ wave signatures. Except for ${{\rm{\Theta }}}_{{kB}}$, the analysis shows a distinct bimodal signature consistent with growth due to both pickup ion populations and weakened wave characteristics between the two. Careful examination indicates that there is no evidence of wave growth at frequencies less than the corresponding cyclotron frequency.

While theory predicts that the wave-associated power spectrum behaves as ${f}^{-2}$ (Lee & Ip 1987), this is not always seen here. There can also be a rather abrupt high-frequency cutoff to the polarization signatures. This is seen in spectra above and below. It may suggest an incomplete scattering of the pickup ions, which leads to less than nominal wave growth at the high frequencies (Möbius et al. 1998).

The bottom left panel of Figure 3 shows spectra from V2 in 1978 from 11:30 to 15:00 UT on day 52. There is a weak wave power enhancement associated with the He⁺ resonance and no evidence of an H⁺ signature. At this time the spacecraft is not within a rarefaction interval, but the observation comes ∼14 hr after a data gap of ∼12 hr duration when the wind speed increases and the density decreases. There may be a shock at ∼14:00 UT on day 50. There is no evidence of waves between the shock or the data gap and this time period, making it unlikely that the waves originate with a possible shock within the data gap. The density continues to decrease until ∼3 hr after the wave observation, although the wind speed is largely constant. The spacecraft may be within a magnetic cloud (Burlaga et al. 1990; Gosling 1990). The enhanced wave power is low, but the polarization signatures are strong for ${f}_{\mathrm{He},{\rm{c}}}\lt f\lt 2{f}_{\mathrm{He},{\rm{c}}}$ and do not extend to ${f}_{{\rm{p}},{\rm{c}}}$. This is certainly an interval where the waves are excited only by He⁺ and not H⁺.

The bottom middle panel of Figure 3 shows spectra from V2 in 1979 from 08:00 to 12:30 UT on day 265. The spacecraft is again in a rarefaction interval lasting 8 days when the wind speed falls from 1000 to 500 km s⁻¹ and the density falls from nearly 1 p⁺ cm⁻³ to under 0.01 p⁺ cm⁻³. There is strong evidence of an H⁺ resonance with the wave signatures extending back to ${f}_{\mathrm{He},{\rm{c}}}$. For this reason we contend that there are waves present due to the He⁺ resonance and the H⁺ resonance.

The bottom right panel of Figure 3 shows spectra from V1 in 1979 from 12:00 to 18:00 UT on day 268. The spacecraft is within a rarefaction interval, and there may be a shock within a data gap 6 days earlier. There is no evidence of shock-associated waves between the shock passage and this wave interval. There is clear evidence of both an He⁺ and H⁺ resonance. The two signatures are separate and distinct. The form of the power spectrum does approximate the ${f}^{-2}$ predicted by Lee & Ip (1987) for each of the wave power enhancements. It is likely that the resolution of two separate enhancements is attributable to the low level of the enhanced power, which allows the He⁺ waves to fade into the background spectrum before the onset of the H⁺ resonance.

We average the polarization spectra D_pol, C_oh, E_lip, and ${{\rm{\Theta }}}_{{kB}}$ over the range ${f}_{\mathrm{He},{\rm{c}}}\lt f\lt 2{f}_{\mathrm{He},{\rm{c}}}$ for all time periods listed in Table 1. Figure 4 shows the result of that analysis, where red triangles represent intervals with enhanced wave activity due to He⁺ and black circles represent intervals without waves attributed to He⁺. Green squares mark the two intervals where the presence of waves due to He⁺ is questionable. Note that D_pol and C_oh values are consistently greater when waves are present than when they are not. Values of E_lip are consistently more negative for wave events, while non-wave events show values ${E}_{\mathrm{lip}}\simeq 0$. The two intervals where the wave-like attributes of the spectra are questionable appear to be much more like background events than wave events in these plots. The minimum variance direction given by ${{\rm{\Theta }}}_{{kB}}$ is a less effective determinant of wave activity, with both wave events and control intervals showing ${{\rm{\Theta }}}_{{kB}}\leqslant 40^\circ$.

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Figure 5 shows our analysis of the polarization associated with the H⁺ resonance, averaging the parameters over the range ${f}_{{\rm{H}},{\rm{c}}}\lt f\lt 2{f}_{{\rm{H}},{\rm{c}}}$. The conclusions, except for the polarization of the one event described below, are the same as with the He⁺ resonance. The waves tend to have D_pol and C_oh higher than events without waves. The waves have ${E}_{\mathrm{lip}}\lt 0$ except for the one event described in the next paragraph and ${{\rm{\Theta }}}_{{kB}}\lt 45^\circ$. Except for ${{\rm{\Theta }}}_{{kB}}$, the wave events show polarization parameters that are distinct from non-wave events. Again, the one event with questionable spectra appears more like a control interval than a wave interval.

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Figure 6 shows our analysis of V1 data for the one case that is exceptional owing to the measured polarization of the waves. This is a V1 observation in 1979 from day 265.48 to 265.75. There is no evidence of wave generation by pickup He⁺ in this event. The spectrum of $| {\boldsymbol{B}}|$ demonstrates an unfortunate aspect of the Blackman–Tukey method, where undersampled or poorly sampled frequencies can exhibit negative power and the offending frequencies are omitted from the plot. This is not a problem here since the anomalous behavior is limited to the lowest frequencies and the spectrum still resolves ${f}_{\mathrm{He},{\rm{c}}}$. The D_pol and C_oh spectra, along with ${{\rm{\Theta }}}_{{kB}}$, all point to wave activity for $f\gt {f}_{{\rm{p}},{\rm{c}}}$. The E_lip and ${\sigma }_{M}$ spectra are consistent with right-hand polarized waves in the spacecraft frame. This is not what is expected, but ∼10% of the events studied by Cannon et al. (2014a, 2014b) exhibited this same unexpected behavior. We include this event in our analysis even though it is not fully understood.

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Two of the spectra shown in Figure 3 have overlapping analyses listed in Table 1. The first such instance is seen in the bottom left panel of Figure 3 and corresponds to V2 observations in 1978 from day 52.48 to 52.63 (11:30–15:00 UT). The overlapping analysis interval (not shown) extends from day 52.48 to 52.92 (11:30–22:00). There is no clear evidence of waves in this second analysis, which demonstrates that the wave signal, which is presumably during the earlier part of the overlapping time interval, is diluted by the later data. Efforts to compute spectra for significantly shorter data intervals result in poor spectral estimates. We do not guarantee that we have found all of the waves due to newborn interstellar pickup ions in the years of data studied.

The second such instance is seen in the bottom middle panel of Figure 3 and corresponds to V2 observations in 1979 from day 265.33 to 265.52 (8:00–12:30 UT). The overlapping analysis interval extends from day 265.31 to 265.67 (7:30–16:00 UT). The wave signatures in this second interval are present, but again reduced, indicating how brief the wave intervals can be. In both instances, but more prominantly in the former, the wave power for $f\lt {f}_{{\rm{p}},{\rm{c}}}$ is consistent with the formation of the anticipated ${f}^{-1}$ spectrum associated with the He⁺ resonance. This feature appears to be absent from the H⁺ resonance.

Having established a list of events, we now apply the theories for wave excitation and turbulent cascade to the observations. The power levels for the spectral peaks are obtained from the analysis. The values for the background spectra at the corresponding cyclotron frequency are obtained by extrapolation of the power-law background at lower and higher frequencies. Mean plasma parameters (V_SW, V_A, N_SW, etc.) are obtained by averaging the hourly Voyager observations over the prescribed time. In some instances, there are too few data to obtain an uncertainty for some parameters.

Figure 7 shows He ionization parameters for the time periods of Table 1. The top panel shows ${\beta }_{\mathrm{He}}$, the ionization rate that depends solely on solar UV radiation. The second panel shows the He⁺ production rate, which is the product of ${\beta }_{\mathrm{He}}$ and the neutral He density. We take the neutral He density to be a constant at these radial distances. Both of these rates follow a simple ${R}^{-2}$ dependence, and there is no difference between wave and non-wave events. The dashed lines in the top two panels give the ${R}^{-2}$ behavior expected for the variables. The bottom panel shows the rate of wave energy excitation as computed from Equation (4). Contrary to the production of He⁺, the wave energy excitation rate, ${{dE}}_{W}/{dt}$, increases with distance from the Sun. There is no consistent or significant difference between wave energy excitation in wave and non-wave events. This suggests that the observability of waves does not depend on the rate of wave excitation alone.

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Figure 8 shows some of the key parameters involved in the excitation of wave energy by H⁺. Charge exchange is a component of H⁺ production, so we plot the flux of thermal protons in the solar wind, F_SW. The flux decreases nominally as N_SW, which varies as ${R}^{-2}$, but rarefaction regions represent lower densities and average wind speeds, making the thermal ion flux low in these intervals. Still, ${F}_{\mathrm{SW}}\sim {R}^{-2}$ for the intervals studied. The density of neutral H increases with distance from the Sun, while the ionization rate ${\beta }_{{\rm{H}}}$ decreases. The resulting H⁺ production rate $H{\beta }_{{\rm{H}}}$ is nearly constant, but decreases precipitously inside 2 au owing to the exponential reduction in neutral H. The resultant excitation rate for wave energy, ${{dE}}_{W}/{dt}$, increases with distance from the Sun. In the case of H⁺ resonance, the rate of wave energy excitation in wave events does appear to be marginally greater than in data intervals not exhibiting wave activity, although there are exceptions to that statement, as seen in the figure.

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Figure 9 compares the background power spectrum level, P_back, at ${f}_{\mathrm{He},{\rm{c}}}$ and ${f}_{{\rm{H}},{\rm{c}}}$ for wave and non-wave events. Low background power levels make it easier for a prescribed amount of wave energy to be observable. While there is a strong tendency for wave events to have lower background levels, there are exceptions. The background power spectrum, ${P}_{\mathrm{back}}({f}_{{\rm{i}},{\rm{c}}})$, for these short intervals is very much the result of local flow conditions. Also, we have already shown that the magnetic field is generally more radial than the Parker field model would predict (Parker 1958, 1963). This is consistent with the observation that the rarefaction regions are of the type described by Gosling & Skoug (2002) and Schwadron (2002).

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Figure 10 shows the time required for Equation (4) to achieve the observed wave power. For control events the plot shows the time required to achieve the background power level. However, there is no assertion that the wave power at these times is as great as the background level. Two computed times for wave growth due to He⁺ are off the scale and represented by arrows at the top of the panel. It is evident that the 20–40 hr estimates used to compute the rate of wave energy growth are generally a good approximation. There are intervals requiring more time to achieve the observed wave power, but the growth rate is linear so that rates obtained using the 20–40 hr estimates remain valid. Accumulation times like these represent solar wind transit times for 0.3–1 au, suggesting that the multiple au accumulation times of Lee & Ip (1987) are unlikely. Whether this is the result of the interruption of the instability at some point in time or the gradual erosion of wave energy by the turbulence while the instability is active, wave power levels are not consistent with the accumulation of wave energy over many au.

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Cannon et al. (2014b) demonstrated that the determining factor in the observability of waves due to newborn interstellar H⁺ is low turbulent cascade rates as computed from Equation (6) when compared to wave growth rates as computed from Equation (4). Figure 11 reproduces that analysis using the waves found in this study. Wave events consistently show growth rates that exceed the turbulent cascade rate. The non-wave observations are less consistent in that there are a minority of background events showing weak turbulence conditions that favor wave growth despite the fact that waves are not seen during these intervals. Recent studies of the turbulent cascade at 1 au have shown that the turbulent cascade can be highly variable in the sense of intermittency relative to the mean predicted by Equation (6) (Coburn et al. 2014, 2015).

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