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# Scale and Time Dependence of Alfvénicity in the Solar Wind as Observed by the Parker Solar Probe

Panisara Thephthong¹, Peera Pongkitiwanichakul¹,<fnref n="7" />, David Ruffolo², Rungployphan Kieokaew³, Riddhi Bandyopadhyay⁴, William H. Matthaeus⁵, and Tulasi N. Parashar⁶

¹Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand; peera-po@ku.th

²Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

³Institut de Recherche en Astrophysique et Planétologie, CNRS, UPS, CNES, 9 Ave. du Colonel Roche, F-31028 Toulouse, France

⁴Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

⁵Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

⁶School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6012, New Zealand

Received 2023 October 31; revised 2023 December 11; accepted 2023 December 12; published 2024 February 5

## Abstract

Alfvénicity is a well-known property, common in the solar wind, characterized by a high correlation between magnetic and velocity fluctuations. Data from the Parker Solar Probe (PSP) enable the study of this property closer to the Sun than ever before, as well as in the sub-Alfvénic solar wind. We consider scale-dependent measures of Alfvénicity based on second-order functions of the magnetic and velocity increments as a function of time lag, including the normalized cross helicity $\sigma_c$ and residual energy $\sigma_r$. Scale-dependent Alfvénicity is strongest for lags near the correlation scale and increases when moving closer to the Sun. We find that $\sigma_r$ typically remains close to the maximally negative value compatible with $\sigma_c$. We did not observe significant changes in measures of Alfvénicity between sub-Alfvénic and super-Alfvénic wind. During most times, the solar wind was highly Alfvénic; however, lower Alfvénicity was observed when PSP approached the heliospheric current sheet or other magnetic structures with sudden changes in the radial magnetic field, non-unidirectional strahl electron pitch angle distributions, and strong electron density contrasts. These results are consistent with a picture in which Alfvénic fluctuations generated near the photosphere transport outward, forming highly Alfvénic states in the young solar wind, and subsequent interactions with large-scale structures and gradients lead to weaker Alfvénicity, as commonly observed at larger heliocentric distances.

Unified Astronomy Thesaurus concepts: Alfvén waves (23); Interplanetary magnetic fields (824); Interplanetary turbulence (830); Solar wind (1534)

## 1. Introduction

The solar wind is strongly turbulent and often exhibits an obvious correlation between large fluctuations in velocity and magnetic field over a wide frequency range (e.g., Belcher & Davis 1971; Kasper et al. 2019). This property is often referred to as “Alfvénicity” in view of the resemblance of these fluctuations to large-amplitude Alfvén waves (Barnes 1979). Often Alfvénic wind also displays a low level of density fluctuations and a nearly constant magnitude of the magnetic field vector (Barnes 1981). The domains of nearly constant magnetic fields describe magnetic pressure balance, and these domains are separated by significant changes in magnetic field magnitude (Ruffolo et al. 2021). The degree of Alfvénicity in the solar wind can vary depending on the location and its source (D’Amicis et al. 2021). Alfvénicity tends to be higher at higher heliolatitude (McComas et al. 2000), in the higher-speed solar wind at low latitude (Bruno et al. 2003), and closer to the Sun (Chen et al. 2020).

These Alfvénic properties are features of large-amplitude and low-frequency Alfvén waves, with velocity fluctuations $v$ and magnetic field fluctuations $b$ connected through the Walén

relation (Walén 1944),

$$v = \pm \frac{b}{\sqrt{\mu_0 \rho}}, \qquad (1)$$

where $\mu_0$ is the vacuum permeability, and $\rho$ is the plasma density. Even with a large amplitude, these incompressive waves are solutions of the compressible magnetohydrodynamic (MHD) equations, provided that the magnitude of the total (background plus fluctuating) magnetic field is constant (Goldstein et al. 1974; Barnes 1979). Based on this reasoning, it is commonly assumed that an ensemble of large-amplitude Alfvén wave packets of this type comprise much of the turbulence observed in the solar wind. Moreover, the strong Alfvénicity observed in the solar wind is usually associated with large-amplitude wave packets that are also polarized in the sense of outward propagation (Belcher & Davis 1971).

The Parker Solar Probe (PSP; Fox et al. 2016) is a spacecraft launched in 2018 to study the Sun using a combination of in situ and remote-sensing measurements to study phenomena in the solar wind, as well as energetic particles from solar storms and other sources. PSP enables us to explore the Sun closer than any previous spacecraft, providing the opportunity to study the features of Alfvénic turbulence in its early stages, including the high correlation between velocity and magnetic field fluctuations, as well as the nearly constant magnitude of the magnetic field. During its eighth orbit (also called solar encounter 8, or E8) at a perihelion distance of around 0.074 au

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Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="7">⁷ Corresponding author.</footnote>
(16 $R_⊙$), PSP became the first spacecraft to provide in situ observations of sub-Alfvénic solar wind (Kasper et al. 2021), where the solar wind speed is lower than the Alfvén speed, $V_A$, given by the following equation:

$$V_A = \frac{|\mathbf{B}|}{\sqrt{\mu_0 m_p n_e}}, \qquad (2)$$

where $|\mathbf{B}|$ is the magnitude of the magnetic field, $m_p$ is the proton mass, and $n_e$ is the electron density of the plasma (assuming plasma neutrality and neglecting the effect of the different mass-to-charge ratios of minor ions). Adopting the term “Alfvén critical zone” to refer to the spatial region where $V \sim V_A$ (DeForest et al. 2018; Chhiber et al. 2022), which has also been called the Alfvén critical point or surface, PSP now makes it possible to directly examine Alfvénicity near and inside this zone. This can help us gain a better understanding of the origin and nature of solar wind turbulence and the mechanisms that accelerate the solar wind and heat the solar corona.

In this work, we analyze data from multiple orbits of the PSP mission down to a distance of 0.062 au from the Sun. We determine quantitative measures of Alfvénicity from increments of the velocity and magnetic field. We explain the relationship between second-order functions of vector field increments and the corresponding Fourier spectra. We then analyze the scale dependence of Alfvénicity and investigate the association between Alfvénicity and other parameters as a function of distance and time. We also examine the time periods during which PSP observed anomalously low Alfvénicity and discuss possible mechanisms that can reduce Alfvénicity in the solar wind.

## 2. Data

We used magnetic field ($\mathbf{B}$) data from the fluxgate magnetometer of the PSP/FIELDS instrument suite (Bale et al. 2016) and proton partial moment ($V$) and temperature ($T$) data from the Solar Probe ANalyzer for Ions instrument in the PSP/SWEAP instrument suite (Kasper et al. 2016). The electron density ($n_e$) is derived from the quasi-thermal noise spectrum measured by the PSP/FIELDS Radio Frequency Spectrometer (Moncuquet et al. 2020), and the electron strahl pitch angle distribution is measured by the Solar Probe ANalyzer for Electrons of PSP/SWEAP (Whittlesey et al. 2020). To enable a consistent comparison of the data, we resampled all FIELDS and SWEAP data to a cadence of 1 NYs (a “New York second,” equal to 0.874 s), the native cadence of SWEAP, interpolating the two sets of data to the time stamp of the SWEAP data.

In this work, we analyzed public data during PSP encounter 8 from 2021 April 24 00:00 UTC to May 4 18:00 UTC, encounter 9 from 2021 August 4 00:00 UTC to August 14 23:59 UTC, and encounter 10 from 2021 November 15 00:00 UTC to November 25 23:59 UTC. During these time intervals, PSP observed several time periods of sub-Alfvénic solar wind (Kasper et al. 2021), characterized by an Alfvén Mach number $M_A$ less than 1. The Alfvén Mach number is calculated as the ratio of the proton radial velocity $V_R$ to the local Alfvén speed $V_A$.

## 3. Analysis Techniques

### 3.1. Increments and Structure Functions

The scale-dependent measures used to quantify the Alfvénicity of the solar wind are similar to those described by Parashar et al. (2020). In order to include a sensitivity to scale, the following measures will be based on increments and structure functions (or other second-order functions of increments) rather than on ordinary fluctuations that are defined in terms of departures from a mean field (see, e.g., Matthaeus & Goldstein 1982). These measures include the normalized increment cross helicity, the increment Alfvén ratio, the normalized increment residual energy, and the alignment cosine between velocity and magnetic field increments. Following the definitions below, for the remainder of the paper, we will drop the modifier “increment” from these terms, with the understanding that we are employing increment-based measures.

The solar wind velocity and magnetic increments for time lag $\tau$ are described as

$$\Delta V = V(t + \tau) - V(t), \qquad (3)$$

$$\Delta \mathbf{B} = \mathbf{B}(t + \tau) - \mathbf{B}(t), \qquad (4)$$

where $\Delta \mathbf{B}$ is usually measured in Alfvén speed units with implied division by $\sqrt{\mu_0 m_p n_e}$.

In certain circumstances, it would be useful to instead define the increments in terms of fluctuating fields $v \equiv V - \langle V \rangle$ and $b \equiv \mathbf{B} - \langle \mathbf{B} \rangle$. Such increments would differ from those in Equations (3) and (4) when the mean fields are time-dependent, e.g., based on running averages, a difference that becomes more pronounced at larger $\tau$. In the present work, we use Equations (3) and (4) directly and avoid the need for averaging. Indeed, avoiding an averaging or detrending procedure has been considered a substantial advantage of using structure functions rather than Fourier spectra of turbulent fluctuations (Lindborg 1999). In any case, at large $\tau$, the spacecraft locations at times $t$ and $t+\tau$ are often in distinct plasma streams with different physical features, so Taylor’s frozen-in hypothesis (Taylor 1938) is a poor approximation, and this technique, or indeed any single-spacecraft technique, is no longer providing information about the spatial properties of local turbulence; rather, it informs us about temporal decorrelation as influenced by such stream crossings.

As a first example, we define the increment cross helicity as

$$H_c = \langle \Delta V \cdot \Delta \mathbf{B} \rangle. \qquad (5)$$

It is straightforward to see that, so defined, this quantity behaves qualitatively as the second-order structure functions that are familiar in hydrodynamics (e.g., Frisch 1995; Monin & Yaglom 1999). Note that the bracket $\langle \dots \rangle$ introduced in Equation (5) represents a suitable time average appropriate to local solar wind conditions, and in this work, we average for each value of $t$ used in Equations (3) and (4) with a cadence of 1 NYs over nonoverlapping time intervals, usually of 10 minutes. This quantity measures the correlation between velocity and magnetic increments and is therefore relevant to assessing Alfvénicity (see Equation (1)). As will be explained later, the sign of $\Delta V \cdot \Delta \mathbf{B}$ typically reverses with the sign of $B_R$, i.e., with magnetic sector crossings. To avoid including both signs within the averaging of Equation (5), in which case the result would depend strongly on the durations of magnetic
sectors within the averaging period, we sometimes use a rectified cross helicity defined by

$$ \tilde{H}_c = \langle |\Delta \mathbf{V} \cdot \Delta \mathbf{B}| \rangle. \qquad (6) $$

Similar to $H_c$, one may define a second-order structure function separately for the magnetic field vector increments as

$$ S_b = \langle \Delta \mathbf{B} \cdot \Delta \mathbf{B} \rangle \qquad (7) $$

and for the velocity field vector increments as

$$ S_v = \langle \Delta \mathbf{V} \cdot \Delta \mathbf{V} \rangle. \qquad (8) $$

Note that all of these second-order quantities depend on the time lag $\tau$ as an argument, which is suppressed here.

These quantities begin at zero lag at a value of 0 and reach an asymptotic value at large lag of twice the estimated value of the total $\langle \mathbf{v} \cdot \mathbf{b} \rangle$, $\langle \mathbf{b}^2 \rangle$, or $\langle \mathbf{v}^2 \rangle$ defined in terms of the fluctuating fields $\mathbf{v} \equiv \mathbf{V} - \langle \mathbf{V} \rangle$ and $\mathbf{b} \equiv \mathbf{B} - \langle \mathbf{B} \rangle$. A relevant physical interpretation is that each function represents the contribution to its respective asymptotic value (energy or cross helicity) due to all fluctuations at timescales less than the lag $\tau$ (e.g., Davidson 2004).

This of course corresponds to the behavior of standard second-order structure functions in homogeneous hydrodynamical turbulence. We also note that these functions are related to rugged invariants of incompressible MHD turbulence (see, e.g., Matthaeus & Goldstein 1982).

## 3.2. Relations to Spectra

Some previous work (e.g., Chasapis et al. 2017; Chhiber et al. 2018; Parashar et al. 2018) has considered a structure function, computed from increments, in a form called an “equivalent spectrum” that is directly comparable to but with less scatter than a Fourier spectrum. We can explain the relationship as follows. The Fourier spectrum $P(\omega)$ for, say, magnetic fluctuations about a mean value can be normalized to satisfy the relation

$$ \langle b^2 \rangle = 2 \int_0^\infty P(\omega) d\omega. \qquad (9) $$

In terms of the structure function $S_b(\tau)$, as $\tau \to \infty$, there is no correlation between $\mathbf{b}(t+\tau)$ and $\mathbf{b}(t)$, so

$$ \langle b^2 \rangle = \frac{S_b(\infty)}{2} = \int_0^\infty \frac{1}{2} \frac{dS_b}{d\tau} d\tau. \qquad (10) $$

Now we identify $\omega$ with $1/\tau$ and therefore express the integral in terms of $1/\tau$:

$$ \langle b^2 \rangle = 2 \int_0^\infty \frac{1}{4} \tau^2 \frac{dS_b}{d\tau} d\left(\frac{1}{\tau}\right). \qquad (11) $$

The above derivation also applies if $S_b$ is replaced by $S_v$ or $\tilde{H}_c$; in the following, we use $S$ to refer to any of these quantities. Comparing Equations (9) and (11), $(1/4)\tau^2 dS(\tau)/d\tau$ as a function of $1/\tau$ can be directly compared with and interpreted in the same way as the Fourier spectrum. As noted above, $S(\tau)$ expresses the cumulative contributions of fluctuations at all timescales up to $\tau$, so $dS/d\tau$ relates to the specific contribution from the timescale $\tau$. Therefore, we identify

$$ S_{\text{eq}}(\tau) \equiv \frac{\tau^2}{4} \frac{dS(\tau)}{d\tau} \qquad (12) $$

as an equivalent spectrum based on structure functions.

Alternatively, we can define a function $\tilde{S}(\omega) \equiv S(1/\tau)$, giving us another expression of the equivalent spectrum:

$$ P_{\text{eq}}(\omega) = -\frac{1}{4} \frac{d\tilde{S}(\omega)}{d\omega}. \qquad (13) $$

According to Figure 1, our second-order functions often have a power-law form $S \propto \tau^\alpha$, and we find that they can be described by the Kolmogorov index $\alpha = 2/3$, at least over certain ranges of $\tau$. Then we can approximate the equivalent spectrum as $(1/6)\tau S(\tau)$.

Figure 1 compares the approximate equivalent spectra (1/6) $\tau S(\tau)$ based on second-order functions of rectified cross helicity, magnetic field, and velocity (black, red, and blue lines, respectively) with the corresponding Fourier transforms (light blue). Each equivalent spectrum is similar to but much smoother than the corresponding Fourier transform.

## 3.3. Measures of Alfvénicity

At this point, we introduce normalized quantities that are formed as ratios involving the above elementary second-order functions of increments and that provide physical interpretations related to solar wind properties. See Bruno & Carbone (2013) for other applications of increments to solar wind studies.

The normalized cross helicity is defined as

$$ \sigma_c = \frac{2 \langle \Delta \mathbf{V} \cdot \Delta \mathbf{B} \rangle}{\langle |\Delta \mathbf{V}|^2 \rangle + \langle |\Delta \mathbf{B}|^2 \rangle}. \qquad (14) $$

Note that $\langle |\Delta \mathbf{V}|^2 \rangle + \langle |\Delta \mathbf{B}|^2 \rangle \pm 2\langle \Delta \mathbf{V} \cdot \Delta \mathbf{B} \rangle \ge 0$, so $-1 \le \sigma_c \le 1$. This measure allows us to assess the cumulative degree of correlation between the velocity and magnetic field fluctuations, a characteristic of Alfvénic fluctuations, as a function of scale. To be precise, recalling that a structure function or related quantity represents the cumulative contribution of fluctuations at scales below and up to the lag $\tau$, $\sigma_c$ is the ratio of the cumulative cross helicity to the cumulative energy up to the scale $\tau$. A value of $|\sigma_c|$ close to 1 is indicative of highly Alfvénic fluctuations.

The Alfvén ratio is the ratio of the velocity field structure function to the magnetic field structure function:

$$ r_A = \frac{\langle |\Delta \mathbf{V}|^2 \rangle}{\langle |\Delta \mathbf{B}|^2 \rangle}. \qquad (15) $$

It represents the ratio of cumulative kinetic energy density to cumulative magnetic fluctuation energy density up to the scale $\tau$. In a spectrum of normal unidirectionally propagating Alfvén waves, the value of $r_A$ is equal to 1; other values indicate deviation from ideal Alfvénicity.

Another measure that we use to characterize Alfvénicity in solar wind plasma is the normalized residual energy, which combines information on the Alfvén ratio and the fluctuation energy. It is normally defined as the difference in kinetic and magnetic energies normalized by their sum. In keeping with the current approach based on structure functions and related quantities, it is defined here as

$$ \sigma_r = \frac{\langle |\Delta \mathbf{V}|^2 \rangle - \langle |\Delta \mathbf{B}|^2 \rangle}{\langle |\Delta \mathbf{V}|^2 \rangle + \langle |\Delta \mathbf{B}|^2 \rangle}. \qquad (16) $$

For Alfvén waves and ideal Alfvénicity, there is equal energy in the velocity and magnetic fluctuations, and the value of $\sigma_r$ is
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Equivalent spectra computed from increments at time lag $\tau$ of solar wind velocity and magnetic field using PSP data from a sample interval from 10:00 to 16:00 UTC on 2021 April 29 at a distance of 0.075 au from the Sun. Panel (a) shows equivalent spectra of rectified cross helicity (black solid curve), magnetic field (red dashed curve), and velocity (blue dashed-dotted curve). Panels (b)–(d) display equivalent spectra for each individual second-order function, along with the corresponding Fourier spectrum (light blue curve). The equivalent spectra provide information similar to Fourier spectra, with less scatter. The observation that $S_b > S_v$ at each timescale $\tau$, indicating the dominance of magnetic energy over kinetic energy, is a well-known feature of solar wind turbulence, which can be equivalently expressed in terms of a low Alfvén ratio, $r_A < 1$, and negative residual energy, $\sigma_r < 0$.</figcaption></figure>

equal to 0. However, solar wind observations usually indicate a negative value (Matthaeus & Goldstein 1982; Bruno et al. 1985).

The final measurement is the global alignment cosine angle between the magnetic and velocity fluctuations. This measure is defined as

$$ \cos \Theta = \frac{\langle \Delta \mathbf{V} \cdot \Delta \mathbf{B} \rangle}{\sqrt{\langle |\Delta \mathbf{V}|^2 \rangle \langle |\Delta \mathbf{B}|^2 \rangle}}. \quad (17) $$

This indicates the degree of directional alignment between the velocity and magnetic field fluctuations. A value of 1 indicates that the two quantities are perfectly aligned, while a value of -1 indicates that they are perfectly antialigned. Either of these values is consistent with ideal Alfvénicity.

These measures of Alfvénicity are based on second-order functions of increments, and their contributions come from fluctuations at scales smaller than $\tau$. Power spectra in the solar wind are observed to rise with increasing $\tau$ (see Figure 1), so the main contribution to such a second-order function comes from fluctuations at scales close to $\tau$, and our measures of Alfvénicity are generally dominated by fluctuations of scale $\sim \tau$.

We also note that these quantities are not all independent. For example,

$$ \sigma_r = \frac{r_A - 1}{r_A + 1} \quad (18) $$

and

$$ \sigma_c = \cos \Theta \sqrt{1 - \sigma_r^2}, \quad (19) $$

which implies that

$$ \sigma_r^2 + \sigma_c^2 \le 1. \quad (20) $$

Equivalently (Matthaeus & Goldstein 1982),

$$ \sigma_c = 2 \cos \Theta \frac{\sqrt{r_A}}{1 + r_A}. \quad (21) $$

The Elsässer increments are defined as

$$ \Delta \mathbf{Z}^{\pm} = \Delta \mathbf{V} \pm \Delta \mathbf{B}, \quad (22) $$

where $\mathbf{Z}^{-}$ is the Elsässer field that propagates parallel to the mean magnetic field and $\mathbf{Z}^{+}$ has antiparallel propagation. In further analysis, the scalar notation $\Delta \mathbf{Z}^{\pm}$ refers to the rms value of this vector increment.

## 4. Results

In this section, we present the results of our analysis, where we utilize data processed in terms of increments to quantify Alfvénicity. We examine four main issues: its relation to distance from the Sun, scale-dependent Alfvénicity, the time dependence of Alfvénicity near perihelia, and the occurrence of time periods of low Alfvénicity.

### 4.1. Radial and Scale Dependence of Alfvénicity

Figure 2 shows scatter plots of the normalized cross helicity and residual energy during PSP solar encounters 8, 9, and 10
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Normalized cross helicity as a function of normalized residual energy for PSP solar encounters 8, 9, and 10 using velocity and magnetic field increments over a lag of $\tau = 87.4$ s. Each data point is based on averages over a nonoverlapping 10 minute time window. The color scale represents the heliocentric distance from the Sun in solar radii ($R_s$). The black circle indicates the mathematical constraint $\sigma_c^2 + \sigma_r^2 = 1$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Equivalent spectra of rectified cross helicity (black solid curves) and second-order structure functions of magnetic field increments (red dashed curves) and velocity increments (blue dashed-dotted curves) as functions of the reciprocal of the lag $\tau$. Each panel displays these for a different range of radial distance from the Sun. The dotted lines indicate spectral slopes with a power-law index of -1 (pink), -3/2 (orange), and -5/3 (green).</figcaption></figure>

using the time lag $\tau = 87.4$ s, which is smaller than the correlation time measured during E1 of 300–600 s (Parashar et al. 2020), in order to study Alfvénicity in the inertial range. Each 10 minute data point is colored according to the distance of PSP from the Sun. According to Equation (20), $\sigma_c$ and $\sigma_r$ are constrained to lie within the unit circle.

At a small radial distance from the Sun, $|\sigma_c|$ tends to approach 1, indicating the predominance of waves that propagate in one direction (either parallel or antiparallel to

the mean field), with $\sigma_r$ tending toward 0 according to the constraint of Equation (20). These properties are indicative of high Alfvénicity in the solar wind.

However, as the heliocentric distance increases to $r \gtrsim 0.35$ au, $|\sigma_c|$ decreases. $\sigma_r$ almost always exhibits a negative value. Indeed, $\sigma_r$ is mostly distributed near the minimum value allowed by the constraint, and in some time intervals with $\sigma_c$ near 0, $\sigma_r$ is as low as -0.8 to -0.9 (though this is far from typical). Our results are consistent with previous results
<figure><img src="image_6.png" /><figcaption>**Figure 4.** The magnitude of normalized cross helicity, normalized residual energy, and the magnitude of the global alignment cosine angle as a function of time lag over different ranges of distance from the Sun. Each point indicates the mean value of the corresponding binned data. The blue shaded region corresponds to the correlation scale uncertainty range (Cuesta et al. 2022).</figcaption></figure>

(Chen et al. 2020; Shi et al. 2021), which indicate a decrease in $|σ_c|$ and $σ_r$ with increasing heliocentric distance $r$.

Figure 3 shows the equivalent spectra of rectified cross helicity, $(1/6)τ\dot{H}_c$, and the second-order structure functions, $(1/6)τS_b$ and $(1/6)τS_v$, for PSP data from different ranges of radial distance from the Sun. These spectra represent the average values of the second-order functions during encounters 8–10 with the heliospheric current sheet (HCS) crossing events removed, plotted as functions of $1/τ$, which can be interpreted as the fluctuation frequency $f$. Because previous work has reported an inertial range magnetic frequency spectrum proportional to $f^{-5/3}$ at larger $r$ and $f^{-3/2}$ at smaller $r$ in PSP data (Chen et al. 2020), we indicate these spectral dependences with dotted lines, as well as the $f^{-1}$ spectral dependence that has been found at lower frequencies (Russell 1972; Goldstein et al. 1984).

All of the equivalent spectra exhibit qualitatively similar trends, gradually but clearly bending from a steeper frequency dependence at high frequency to a flatter dependence at low frequency. The frequency of transition from the inertial range to the energy-containing $f^{-1}$ range decreases with increasing heliocentric distance, which is consistent with the classic results from the Helios spacecraft (e.g., Bavassano et al. 1982; Bruno & Carbone 2013; Wu et al. 2021). In the upper frequency range, corresponding to the inertial range of turbulence, our analysis is truncated near 1 NYs, the native cadence of the solar wind data from PSP/SWEAP. Thus, we analyze only about 1 order of magnitude of the inertial range and cannot precisely determine the power-law index of the frequency dependence, which seems consistent with a power-law index of either -3/2 or -5/3. At lower frequency, all equivalent spectra for $r$ between 15 and 55 $R_s$ are consistent with a gradual transition at decreasing frequency toward an $f^{-1}$ spectrum.

Our results confirm that magnetic field energy dominates over kinetic energy in all heliospheric distance ranges and at all scales examined. Closer to the Sun, the difference between $S_b$ and $S_v$ magnitude is small; however, this difference increases greatly as the heliocentric distance increases. Interestingly, $\dot{H}_c$ is similar to $S_v$ at all distances at scales, but because $S_b$ becomes much greater than both of these with increasing $r$, the normalized cross-helicity magnitude decreases greatly.

Figure 4 shows that $|σ_c|$, $σ_r$, and $|\cos Θ|$, for which a higher value indicates stronger Alfvénicity, usually exhibit a dependence on the time lag, especially at distances close to the Sun. The blue shaded region represents the range of $τ$ corresponding to the correlation length, where we employ the value obtained from Cuesta et al. (2022). Our analysis of Alfvénicity does not reach ion kinetic scales.

We find that these measures of Alfvénicity have their highest values for lags on the order of the correlation scale. For $τ$ below the correlation scale, $|σ_c|$, $σ_r$, and $|\cos Θ|$ always decrease with decreasing $τ$, indicating decreasing Alfvénicity. An exception is $σ_r$ in the highest distance range of 45–55 $R_s$, which fluctuates around a constant value and does not exhibit systematic scale dependence. This trend of decreasing Alfvénicity as the lag decreases in the inertial range was previously reported by Parashar et al. (2018), who observed this phenomenon in both the magnetosheath and the solar wind using data from the Magnetospheric Multiscale spacecraft.

As the lag increases beyond the correlation length, the values of $|σ_c|$, $σ_r$, and $|\cos Θ|$ again decrease, with the same exception as noted above. With further increases in the lag, there is evidently weaker correlation between increments of the velocity and magnetic fields.

We also note that while all measures of Alfvénicity systematically decrease with increasing $r$, there is a sudden change in the scale-dependent cross-helicity magnitude $|σ_c|$ and
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Time dependence of measures of Alfvénicity near the 8th–10th PSP perihelia using $\tau = 100$ NYs (87.4 s). The top axes show the radial distance from the Sun in units of solar radii, with perihelion distance labeled in red. The graphs show mean values of the (a) second-order structure functions of the magnetic field and velocity; (b) normalized cross helicity, $\sigma_c$; (c) cosine of the global alignment angle between magnetic and velocity increments, $\cos \Theta$; (d) normalized residual energy, $\sigma_r$; (e) Alfvén ratio, $r_A$; (f) ratio of rms Elsässer amplitudes; (g) radial magnetic field component, $B_r$, and magnetic field magnitude, $|B|$; (h) Alfvén Mach number, $M_A$, with a dashed blue line at $M_A = 1$; and (i) ratio of parallel to perpendicular velocity increments. Each panel shows a 10 minute average plotted at the centroid of the averaging interval. The blue shading indicates the sub-Alfvénic intervals with $M_A < 1$, and red shading indicates the times of low Alfvénicity $|\sigma_c| < 0.5$ other than full or partial crossings of the HCS.</figcaption></figure>

the alignment measure $|\cos \Theta|$ from the distance range of [35, 45] $R_s$ to [45, 55] $R_s$. Furthermore, all measures of Alfvénicity shown in Figure 4 become nearly independent of scale at $r > 45 R_s$.

## 4.2. Time Dependence of Alfvénicity

Figure 5 illustrates the time dependence of Alfvénicity near perihelion during PSP solar encounters E8–E10. In the context of the dependence on the radial distance described in Section 4.1, note that all these time periods have $r < 25 R_s$, corresponding to the innermost range of $r$ considered in that section. Here every parameter was calculated over nonoverlapping 10 minute time windows.

Panel (a) displays the second-order structure functions from the velocity and magnetic increments. Throughout most of the time, we observed that $\langle \Delta B^2 \rangle$ was slightly higher than $\langle \Delta V^2 \rangle$, resulting in $\sigma_r < 0$ and $r_A < 1$.

Panels (b) and (c) show the normalized cross helicity and cosine of the global alignment angle, respectively, which exhibit very similar time series. It is generally observed that Alfvénic fluctuations propagate predominantly outward from the Sun along magnetic field lines. Because a wave in $Z^+ = V + B$ propagates antiparallel to the mean field, when $B_R < 0$ (see panel (g)), the strongest increments are in $\Delta Z^+$,

with $\Delta V \cdot \Delta B > 0$ and therefore $\sigma_c > 0$ and $\cos \Theta > 0$. The predominance of $\Delta Z^+$ can be seen from the generally high value of the ratio of rms values, $\Delta Z^+/\Delta Z^- \gg 1$, when $B_R < 0$ (see panel (f)). On the other hand, when $B_R > 0$, we generally observe stronger increments in $\Delta Z^-$ with $\sigma_c < 0$, $\cos \Theta < 0$, and $\Delta Z^+/\Delta Z^- \ll 1$. A notable exception to this rule is that during a magnetic switchback, which can be defined as a temporary reversal in the $B_R$, the cross helicity is observed to be unchanged, as the predominant propagation direction of Alfvén waves continues to follow the field lines and reverses together with $B_R$ (McManus et al. 2020). However, here we plot 10 minute averaged quantities, and switchbacks are too small to be seen on this scale (Raouafi et al. 2023).

While $\sigma_c$ and $\cos \Theta$ usually remain quite close to $\pm 1$, the deviation from ideal Alfvénicity is more apparent when examining the normalized residual energy $\sigma_r$ and Alfvén ratio $r_A$ (see panels (d) and (e), respectively). For ideal Alfvénicity, we would have $\sigma_r = 0$ and $r_A = 1$, but in these observations, we almost always find $\sigma_r < 0$ and $r_A < 1$.

Sub-Alfvénic regions (highlighted in blue) are identified by $M_A < 1$ (see panel (h)), where the Alfvénic Mach number is defined by $M_A \equiv V/V_A$. In the sub-Alfvénic solar wind, we typically observed high Alfvénicity and a low ratio of the mean squared parallel to perpendicular velocity increments (see panel
(i)), which is consistent with the variance anisotropy in the sub-Alfvénic solar wind reported in Bandyopadhyay et al. (2022). We observe no noticeable changes in Alfvénicity between sub-Alfvénic and super-Alfvénic solar wind over this range of heliospheric distance.

We observed upward spikes in $\sigma_r$ and $r_A$ at the edges of sub-Alfvénic regions at some but not all of the times when $\sigma_c$ and $\cos\Theta$ changed sign, usually in association with PSP crossing boundaries between magnetic structures with different polarities or at full or partial HCS crossings. Some of these events involved sharply higher electron density, resulting in lower $V_A$ and higher $M_A$, and if the wind outside the high-density structure was sub-Alfvénic, the sharply higher $M_A$ inside the structure caused the wind there to be super-Alfvénic. Hence, some of these decreases in $\sigma_c$ and $\cos\Theta$, in some cases with upward spikes in $\sigma_r$ and $r_A$, occurred at the boundaries of sub-Alfvénic time periods.

We consider the time intervals where $|\sigma_c| < 0.5$ for a time increment of $\tau = 87.4$ s to have low Alfvénicity. This definition allows us to identify and study the brief periods of time when the normalized cross helicity of the solar wind plasma is relatively low, indicating that the velocity and magnetic field fluctuations are not strongly correlated. By analyzing the characteristics of the solar wind during these periods, we can gain insights into the processes that are driving the anomalously low Alfvénicity, i.e., the factors that are contributing to the weaker correlation between the velocity and magnetic field fluctuations.

In contrast to the usual behavior that $\Delta Z^+/\Delta Z^- \gg 1$ when $B_R < 0$ and $\Delta Z^+/\Delta Z^- \ll 1$ when $B_R > 0$, during these periods of low Alfvénicity, we observed a mixture of inward- and outward-propagating waves, as indicated by $\Delta Z^+/\Delta Z^- \approx 1$. Low Alfvénicity is commonly observed near the HCS. The times when PSP crosses the HCS are marked by a change in the polarity of the radial magnetic field $B_r$ from one polarity to another, corresponding with the change in degree of the suprathermal electron pitch angle distribution from 0° to 180°. Within the plasma sheet that surrounds the HCS, the properties of solar wind turbulence are different from the normal solar wind. There is an increase in ion density, temperature, and flow speed, accompanied by a weakening of the magnetic field strength (Smith 2001).

A reduction in Alfvénicity is also observed during incomplete or partial crossings. In these partial crossings, PSP approaches the HCS without crossing to the other side, as indicated by $B_R$ approaching zero but not fully reversing to the opposite direction, and there is a smaller magnetic field rotation compared with a complete crossing (Phan et al. 2021). The plot from encounter 10 illustrates one such partial crossing occurring between two sub-Alfvénic regions as reported by Zhao et al. (2022). When PSP encounters such structures, we also observe that the velocity field is dominated by the parallel velocity component, unlike the highly Alfvénic solar wind.

Away from HCS crossings, we observed two additional time intervals near perihelia that exhibited low Alfvénicity, as indicated by the red shading in Figure 5. In Figure 6, we show expanded plots for those brief non-Alfvénic time periods, showing $\sigma_c$, $\sigma_r$, and various solar wind parameters. Note that vector components are expressed in radial–tangential–normal coordinates. Both intervals exhibited $\sigma_c$ approaching zero, which indicates counterpropagating waves during this interval, and negative $\sigma_r$.

During the first interval of interest (19:30–21:00 UTC on 2021 April 28), the magnitude of the magnetic field was not constant and was accompanied by fluctuations in $n_e$, $T$, and $\beta$. The variability of these parameters suggests incompatibility with the Alfvén wave equation. Additionally, we observed a bidirectional electron pitch angle distribution. In the second interval of low Alfvénicity (22:20–23:10 UTC on 2021 August 8), we observed sudden changes in all magnetic field components, $V_R$, $n_e$, $T$, and $\beta$.

# 5. Discussion and Summary

In this work, we study the scale-dependent Alfvénicity of the solar wind using recent data during PSP orbits 8–10 as it approached to within 14 $R_s$ of the Sun in both super- and sub-Alfvénic solar wind. The PSP data were analyzed using increment-based measures to investigate Alfvénic properties. This method yields Alfvénicity measures that exhibit smooth changes as a function of the increment time lag $\tau$.

A structure function or related second-order function of increments at the time lag $\tau$ contains energy from all timescales smaller than and up to $\tau$. The derivative of the function with respect to $\tau$ expresses the energy at each timescale or frequency, so we can use an equivalent power spectrum based on that function that serves a similar purpose to the Fourier power spectrum. The equivalent spectrum exhibits a power-law break at the frequency marking the start (outer scale) of the inertial range of turbulence. We confirm the classic result that the break frequency decreases with increasing distance from the Sun (e.g., Bavassano et al. 1982; Bruno & Carbone 2013; Wu et al. 2021).

The decreases in $|\sigma_c|$ and $|\cos\Theta|$ with increasing $r$ (see Figures 2 and 4) suggest that the magnitudes and directions of changes in the velocity $\Delta V$ and magnetic field $\Delta B$ differ more strongly as the solar wind evolves. Indeed, all of our measures based on second-order functions of increments indicate that Alfvénicity decreases with increasing heliocentric distance and at small scales, which is in agreement with previous studies using Fourier analysis (Chen et al. 2020; Parashar et al. 2020; Shi et al. 2021).

The Alfvén ratio $r_A$ and normalized residual energy $\sigma_r$ are additional indicators of Alfvénicity, where an ensemble of unidirectional Alfvén waves would have $r_A=1$ and $\sigma_r=0$. However, solar wind observations usually indicate that $r_A < 1$ and $\sigma_r < 0$. Indeed, our Figure 2, as well as many studies in the past (e.g., Bavassano et al. 1998; Bruno et al. 2007), indicate that $\sigma_r$ in the solar wind is found near the minimum value compatible with the constraint $\sigma_c^2 + \sigma_r^2 \ll 1$ (marked by the bounding circle in the figure). When the normalized cross helicity is $\sigma_c = \pm 1$, $\sigma_r$ is constrained to be zero, but otherwise it is usually found at lower (minimal) values. Negative values of $\sigma_r$ indicate that the cumulative kinetic energy of the solar wind at all scales less than the lag is lower than the magnetic fluctuation energy in the same range of scales. The predominance of magnetic fluctuation energy could be related to interference between inward- and outward-type fluctuations, where the inward type can be generated by reflection in the expanding solar wind (Zhou & Matthaeus 1989). (However, it is interesting to note that a mixture of waves with mixed propagation directions—mixed cross helicities—is expected to have $r_A=1$ and $\sigma_r=0$ when the oppositely propagating wave packets are uncorrelated. This property is known as the “Alfvén effect”; Kraichnan 1965; Fyfe et al. 1977.) In simulations,
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Zoom-in of the red shaded time intervals in Figure 5. From the top, the panels show the 10 minute averaged normalized cross helicity $\sigma_c$ and normalized residual energy $\sigma_r$ from increments over a lag of $\tau = 87.4$ s, magnetic field components and magnitude, solar wind velocity components, pitch angle distribution of 314 eV superthermal (strahl) electrons normalized by the maximum electron flux during the time of interest, electron density $n_e$ with ion temperature $T$, and plasma beta $\beta$.</figcaption></figure>

negative residual energy is generated through nonlinear interactions of counterpropagating fluctuations, i.e., $z^+$ and $z^-$ (Shi et al. 2023). One physical mechanism for this is that current sheets generated in turbulent reconnection contribute to an excess of magnetic energy in the inertial range (Matthaeus & Lamkin 1986).

The reader may note that in Figure 2, while the data points cluster near the circular boundary defined by the mathematical constraint $\sigma_c^2 + \sigma_r^2 \le 1$, there is a gap between the data points and that circle, with the exception of some data points at $\sigma_c = \pm 1$ and $\sigma_r = 0$. In other words, while the data cluster near the minimum $\sigma_r$ for a given $\sigma_c$ as noted above, they do not reach the actual minimum value (except when $\sigma_c = \pm 1$). To understand the gap between the data and the boundary circle, let us consider the requirements to lie on the boundary circle. In general, $\sigma_c = \cos \Theta \sqrt{1 - \sigma_r^2}$ (Equation (19)), so reaching the boundary circle at $|\sigma_c| = \sqrt{1 - \sigma_r^2}$ requires $|\cos \Theta| = 1$; i.e., for each $t$ during the 10 minute averaging time, there must be perfect directional alignment or antialignment, which implies that $\Delta V = \alpha(t)\Delta B$ for a scalar $\alpha(t)$. When this occurs, it is a corollary of the local “Beltrami” alignment of magnetic and velocity fluctuations (Ting et al. 1986; Stribling & Matthaeus 1991; Matthaeus et al. 2008; Servidio et al. 2008; Osman et al. 2011). Furthermore, $\alpha$ must take the same value at each $t$ within the interval. To see this, note that for any distribution of $\alpha$, the variance is given by

$$ \langle (\alpha - \langle \alpha \rangle)^2 \rangle = \langle \alpha^2 \rangle - \langle \alpha \rangle^2 \ge 0, \quad (23) $$

which implies that

$$ \langle \alpha \rangle^2 \le \langle \alpha^2 \rangle, \quad (24) $$

and this is an equality if and only if the variance of $\alpha$ is zero. Then we note that in this case of perfect directional alignment, we have

$$ \sigma_c = \frac{2\langle\alpha\rangle}{\langle\alpha^2\rangle + 1}, \quad \sigma_r = \frac{\langle\alpha^2\rangle - 1}{\langle\alpha^2\rangle + 1}, \quad (25) $$

$$ 1 - \sigma_c^2 - \sigma_r^2 = \frac{4\langle\alpha^2\rangle - 4\langle\alpha\rangle^2}{(\langle\alpha^2\rangle + 1)^2}. \quad (26) $$

This is nonnegative, and the data point lies on the boundary circle if and only if $\langle\alpha\rangle^2 = \langle\alpha^2\rangle$ and the variance of $\alpha(t)$ is zero within that interval, so $\alpha$ is constant and $r_A = 1/\alpha^2$. In other words, if we do not have ideal Alfvénicity ($|\sigma_c| \ne 1$), then $\Delta V$ deviates from $\pm\Delta B$, but the only way for a data point to remain on the boundary circle is to deviate to the same fixed fraction $\alpha$, i.e., $\Delta V = \alpha\Delta B$, at every $t$ value.

Thus, we have a physical picture that near the Sun there is often near-ideal Alfvénicity with $\sigma_c \approx 1$ and $\sigma_r \approx 0$, as we see from Figure 2, which is consistent with the fluctuations being mostly Alfvén waves generated by the Sun. As the solar wind evolves farther from the Sun, the processes (interactions) that cause a deviation from ideal Alfvénicity, so that $\sigma_c \ne \pm 1$, also cause sufficient variation that we do not obtain increments of the same direction and magnitude ($\Delta V = \alpha\Delta B$) at every $t$ value. This leads to the zone of avoidance at the boundary,
where $\sigma_c^2 + \sigma_r^2 = 1$, despite the tendency noted above for $\sigma_r$ to approach near-minimal values.

Our results for $|\sigma_c|$ and $\sigma_r$ during E8–E10 exhibit lower values compared with those reported in a study of E1 by Chen et al. (2020), even when comparing at the same radial distance. According to Shi et al. (2021), Alfvénicity is weaker in slower solar wind, so the difference between our results and those of Chen et al. (2020) could be attributed to the lower solar wind speeds observed during E8–E10 perihelia in comparison with E1; Shi et al. (2021) reported a similar difference when examining data from E5, which also exhibited a slower solar wind speed than E1.

Near the Alfvén critical zone, with the radial velocity exceeding the Alfvén speed, the flow cannot propagate backward to the Sun (DeForest et al. 2014). It is interesting to look for possible changes in Alfvénicity when PSP crosses this zone. However, we did not observe noticeable changes in Alfvénicity parameters between sub-Alfvénic and super-Alfvénic wind in PSP data, which is indeed consistent with the model results of Lou (1994).

We find that scale-dependent Alfvénicity peaks at $\tau$ comparable to the correlation time and usually decreases at other $\tau$ ranges significantly different from the correlation time. In turbulence, the smaller scales of the inertial range have a shorter timescale for nonlinear interactions that tend to disrupt the Alfvénic correlations and reduce Alfvénicity. Shear was also proposed by Parashar et al. (2020) to serve as a factor responsible for reducing $|\cos\Theta|$ (see also Roberts et al. 1987, 1992; Zank et al. 1996). Without shear, $|\cos\Theta|$ would be expected to increase at $\tau$ smaller than the correlation time (Boldyrev 2006), as verified by computer simulations (Mason et al. 2006).

As $\tau$ increases above the correlation scale, the magnetic field structure function rises due to the increment technique acting as a high-pass filter in the frequency $1/\tau$, accumulating the contribution of energy from fluctuations at scales smaller than $\tau$. As a result, the structure functions increase with increasing $\tau$. However, the relationship between magnetic and velocity fluctuations is more limited to the scales comparable to the correlation time. Beyond the correlation time ($\tau$), we find that Alfvénicity becomes weaker. For example, as noted earlier, the normalized cross helicity is the ratio between the cross helicity $H_c$ and the energy $E=(1/2)(S_v + S_b)$. Thus, the gradual decrease in $|\sigma_c|$ with increasing lag $\tau$ can be understood as $H_c$ increasing more slowly than $E$, as can be seen visually from Figures 1 and 3. We note that some previous studies defined increments in terms of fluctuating fields relative to a running average, whereas here we use the **V** and **B** fields directly. Subtracting a running average does tend to remove the effects of fluctuations longer than the averaging time and may lead to different behaviors at long $\tau$. The decrease in $|\sigma_c|$ that we observe as $\tau$ increases beyond the correlation length is pronounced at closer distances to the Sun, but as we move further from the Sun, the Alfvénicity trend at different $\tau$ values becomes flatter.

The heliospheric plasma sheet (HPS) is a broader region characterized by high density and high $\beta$ surrounding the HCS, which may have repeated structures of flux ropes and magnetic reconnection sites (Sanchez-Diaz et al. 2019). The values of $\sigma_c$ and $\cos\Theta$ approach or cross zero as PSP nears the HCS, presumably because the solar wind is affected by the HPS. This behavior indicates the presence of non-Alfvénic fluctuations

and a reduced degree of alignment (or antialignment) between velocity and magnetic field increments. The decrease of $|\sigma_c|$ is also associated with strong velocity shear that occurs close to the current sheet (Roberts et al. 1992). Around the HPS, the inward- and outward-propagating waves are strongly mixed. These observations can be attributed to the differences in the mean magnetic field across the HCS and the absence of correlation between fields from opposite sides of the sheet.

During times when Alfvénicity is observed to be particularly low, PSP was crossing a boundary between regions with different plasma properties, as found in the HPS, velocity shear structures, and magnetic flux tube boundaries, with scattered strahl electrons in a non-unidirectional distribution (see Figure 6). These structures can scatter Alfvén waves, causing interactions between oppositely propagating wave packets and disrupting the correlation between velocity and magnetic field fluctuations, resulting in lower Alfvénicity. More specifically, if the sum of the parallel wavenumbers of the interacting waves is near 0, the interactions tend to produce two-dimensional fluctuations with wavenumbers perpendicular to the mean field (Shebalin et al. 1983; Tu & Marsch 1993) with predominantly magnetic rather than kinetic energy, tending to make the residual energy more negative as well (Wang et al. 2011; Howes & Nielson 2013). The decrease in Alfvénicity during these times may also be attributed to the spacecraft crossing discontinuity such as the boundary of a flux tube, where a random value of $\sigma_c$ is encountered.

## Acknowledgments

This research is funded by Kasetsart University through the Graduate School Fellowship Program. It has been partially supported in Thailand by Thailand Science Research and Innovation (RTA6280002), the National Science and Technology Development Agency (NSTDA) and National Research Council of Thailand (NRCT): High-Potential Research Team Grant Program (N42A650868), the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (B37G660015), and the Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation (OPS MHESI) and Thailand Science Research and Innovation (TSRI; RGNS 63-045). It was also supported by the Parker Solar Probe mission under the ISOIS project (NASA contract NNN06AA01C) and a subcontract to University of Delaware from Princeton University. R.K. acknowledges the support of the Centre National de la Recherche Scientifique (CNRS), the University of Toulouse III (UPS), and the Centre National d'Études Spatiales (CNES), France.

## ORCID iDs

Panisara Theuthong https://orcid.org/0000-0001-9597-1448

Peera Pongkitiwanichakul https://orcid.org/0000-0002-6609-1422

David Ruffolo https://orcid.org/0000-0003-3414-9666

Rungployphan Kieokaew https://orcid.org/0000-0003-0937-2655

Riddhi Bandyopadhyay https://orcid.org/0000-0002-6962-0959

William H. Matthaeus https://orcid.org/0000-0001-7224-6024

Tulasi N. Parashar https://orcid.org/0000-0003-0602-8381
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