2. The inertial range of reflection-driven Alfvénic turbulence

We view Alfvénic turbulence as a collection of localized, non-compressive fluctuations in the Elsasser variables $\boldsymbol{z}^\pm$ defined in (1.1) and take these fluctuations to be characterized by length scales $\lambda$ and $l_\lambda ^\pm$ perpendicular and parallel to the magnetic field, respectively. We define the Elsasser increment

(2.1) \begin{equation} \Delta \boldsymbol{z}^\pm _\lambda \left (\boldsymbol{x}, \hat{\boldsymbol{s}}, t\right )= \boldsymbol{z}^\pm \left (\boldsymbol{x} + 0.5 \lambda \hat{\boldsymbol{s}}, t \right ) - \boldsymbol{z}^\pm \left (\boldsymbol{x} - 0.5 \lambda \hat{\boldsymbol{s}}, t \right ), \end{equation}

where $\hat{\boldsymbol{s}}$ is a unit vector perpendicular to $\boldsymbol{B}$ , and we take

(2.2) \begin{equation} \delta z^\pm _\lambda (\boldsymbol{x},t) = \frac {1}{2\pi } \int _0^{2\pi } {\rm d}\theta \, \left |\Delta \boldsymbol{z}^\pm _\lambda \left (\boldsymbol{x}, \hat{\boldsymbol{s}}, t\right )\right | \end{equation}

to be the characteristic amplitude of the $\boldsymbol{z}^\pm$ structure of perpendicular scale $\lambda$ at position $\boldsymbol{x}$ and time $t$ , where the angle $\theta$ specifies the direction of $\hat{\boldsymbol{s}}$ within the plane perpendicular to $\boldsymbol{B}$ .

In intermittent turbulence, the tail of the distribution of fluctuation amplitudes becomes increasingly prominent as $\lambda$ decreases. To account for this, we follow an approach that has been used in two previous studies of balanced, intermittent reduced MHD (RMHD) turbulence: Chandran, Schekochihin & Mallet (Reference Chandran, Schekochihin and Mallet2015) and Mallet & Schekochihin (Reference Mallet and Schekochihin2017), which we henceforth refer to as CSM15 and MS17, respectively. In particular, we parameterize the scale-dependence of the fluctuation-amplitude distribution by treating $\delta z^\pm _\lambda$ as a random variable given by the equation

(2.3) \begin{equation} \delta z_\lambda ^\pm = \overline { z}^\pm \beta ^q, \end{equation}

where $\overline { z}^\pm$ is a scale-independent random number, $\beta$ is a constant satisfying $0\lt \beta \lt 1$ that we will determine in the analysis to follow, and $q$ is a random integer that is uncorrelated with $\overline { z}^\pm$ and that has a Poisson distribution with scale-dependent mean $\mu$ ,Footnote ³

(2.4) \begin{equation} P(q) = \frac {e^{-\mu } \mu ^{q}}{q!}. \end{equation}

For simplicity, we have taken $\mu$ and $\beta$ to be the same for $\delta z^+_\lambda$ and $\delta z^-_\lambda$ , but we assume that

(2.5) \begin{equation} \left \langle \overline { z}^+ \right \rangle \gg \left \langle \overline { z}^- \right \rangle, \end{equation}

as we consider imbalanced turbulence, where $\langle \ldots \rangle$ denotes an ensemble average.

Given (2.3) and (2.4), the volume filling factor of the most intense fluctuations is $P(0) = e^{-\mu }$ . As in several previous studies of intermittent MHD turbulence (Grauer, Krug & Marliani Reference Grauer, Krug and Marliani1994; Politano & Pouquet Reference Politano and Pouquet1995; Chandran et al. Reference Chandran, Schekochihin and Mallet2015; Mallet & Schekochihin Reference Mallet and Schekochihin2017), we take this filling factor to be $\propto \lambda$ and therefore set

(2.6) \begin{equation} \mu = A + \ln \left (\frac {L_\perp }{\lambda }\right ), \end{equation}

where $A$ is a constant that characterizes the breadth of the distribution of fluctuation amplitudes at the perpendicular outer scale $L_\perp$ . We discuss the possible physical origins of this scaling in § 4.1.

Equation (2.6) implies that $\mu \gg 1$ within the inertial range, in which $\lambda \ll L_\perp$ . It follows from (2.4) that both the most common value of $q$ and the median value of $q$ are $\simeq \mu$ (Choi Reference Choi1994), and hence the most common fluctuation amplitude at scale $\lambda$ and the median fluctuation amplitude at scale $\lambda$ are approximately

(2.7) \begin{equation} w^\pm _\lambda = \left \langle \overline { z}^\pm \right \rangle \beta ^\mu \propto e^{\mu \ln \beta } \propto \lambda ^{-\ln \beta }. \end{equation}

Fluctuations with $q\ll \mu$ satisfy $\delta z^\pm _\lambda \gg w^\pm _\lambda$ and form the tail of the distribution, which becomes broader as $\lambda$ decreases and $\mu$ increases.

The functional form of $P(q)$ makes it straightforward to compute the $n$ th-order structure function of $\delta z^\pm _\lambda$ ,

(2.8) \begin{equation} \left \langle \left (\delta z^\pm _\lambda \right )^n \right \rangle = \left \langle \left ( \overline { z}^\pm \right )^n \right \rangle e^{-\mu }\sum _{q=0}^\infty \frac {\left (\mu \beta ^n\right )^q}{q!} = \left \langle \left ( \overline { z}^\pm \right )^n \right \rangle e^{-\mu + \mu \beta ^n}, \end{equation}

where we have utilized the Taylor expansion $\sum _{q=0}^\infty x^q/q! = e^x$ . The scaling exponent $\zeta _n$ of the $n$ th-order structure function is defined by the proportionality relation

(2.9) \begin{equation} \left \langle \left (\delta z^\pm _\lambda \right )^n \right \rangle \propto \lambda ^{\zeta _n}. \end{equation}

Upon substituting (2.6) into (2.8), we obtain (Chandran et al. Reference Chandran, Schekochihin and Mallet2015; Mallet & Schekochihin Reference Mallet and Schekochihin2017)

(2.10) \begin{equation} \zeta _n = 1 - \beta ^n. \end{equation}

As $q$ increases from 0, the summand in (2.8) increases until $q$ reaches a value $\simeq \mu \beta ^n$ , after which the summand decreases with increasing $q$ . The amplitudes of the fluctuations that make the largest contribution to $\langle (\delta z^\pm _\lambda )^n \rangle$ are thus approximately

(2.11) \begin{equation} \delta z^\pm _{(n),\lambda } = \overline { z}^\pm \beta ^{\mu \beta ^n} \propto \lambda ^{-\beta ^n \ln \beta }. \end{equation}

It follows from (2.7) and (2.11) that $w^\pm _{\lambda } = \delta z^\pm _{(0),\lambda } \lt \delta z^\pm _{(1),\lambda } \lt \delta z^\pm _{(2), \lambda } \lt \ldots$ . As $n$ increases, $\mu \beta ^n$ decreases, and $\delta z^\pm _{(n),\lambda }$ characterizes the amplitudes of fluctuations that are farther out in the tail of the distribution.

We restrict our attention to the strong-turbulence regime, in which the time $\tau _{{\rm nl,} \lambda }^-\sim \lambda /\delta z^+_\lambda$ required for a $\delta z^+_\lambda$ fluctuation to shear a $\delta z^-_\lambda$ fluctuation is comparable to or smaller than the time $\tau _{{\rm lin,} \lambda }^- \sim l_\lambda ^+/(2v_{\textrm {A}})$ required for a point moving with the $\delta z^-_\lambda$ fluctuation at velocity $ \boldsymbol{v}_{\textrm {A}}$ to pass through the (counter-propagating) $\delta z^+_\lambda$ fluctuation. In other words, we assume that

(2.12) \begin{equation} \chi ^+_\lambda \equiv \frac {\delta z^+_\lambda l_\lambda ^+}{\lambda v_{\textrm {A}}}\gtrsim 1. \end{equation}

In this limit, each cross-section (in the plane perpendicular to $\boldsymbol{B}$ ) of a propagating $\delta z^-_\lambda$ structure is strongly deformed after a time $\sim \lambda /\delta z^+_\lambda$ , causing the parallel correlation length of the $\delta z^-_\lambda$ fluctuation to satisfy (Lithwick et al. Reference Lithwick, Goldreich and Sridhar2007)

(2.13) \begin{equation} l^-_\lambda \sim \frac {v_{\textrm {A}} \lambda }{ \delta z^+_\lambda }. \end{equation}

Two locations within a $\delta z^+_\lambda$ structure separated by a distance $l^-_\lambda$ along the magnetic field are deformed by $\delta z^-_\lambda$ structures in uncorrelated ways, and hence (Lithwick et al. Reference Lithwick, Goldreich and Sridhar2007)

(2.14) \begin{equation} l^+_\lambda \simeq l^-_\lambda . \end{equation}

Henceforth (except in § 4), we will drop the distinction between $l^+_\lambda$ and $l^-_\lambda$ and use simply $l_\lambda$ . Equations (2.13) and (2.14) imply that the inequality in (2.12) is replaced by the critical-balance relation

(2.15) \begin{equation} \chi ^+_\lambda \sim 1, \end{equation}

which states that $\tau ^-_{{\rm nl,}\lambda } \sim \tau _{{\rm lin,}\lambda }^-$ at all scales (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Lithwick et al. Reference Lithwick, Goldreich and Sridhar2007) and within structures of differing amplitudes at the same scale (Mallet, Schekochihin & Chandran Reference Mallet, Schekochihin and Chandran2015).

We define

(2.16) \begin{equation} \epsilon _\lambda ^+= \frac {\left (\delta z^+_\lambda \right )^2}{\tau _{\textrm {nl,} \lambda }^+}, \end{equation}

which is the rate at which a localized $\delta z^+_\lambda$ fluctuation’s energy cascades to smaller scales, where $\tau _{{\rm nl,} \lambda }^+$ is the energy cascade time scale of the $\delta z^+_\lambda$ fluctuation. In reflection-driven MHD turbulence in the solar wind, the fluctuations propagating towards the Sun in the plasma frame (the $\boldsymbol{z}^-$ fluctuations) are produced by the reflection of the outward-propagating $\boldsymbol{z}^+$ fluctuations (Heinemann & Olbert Reference Heinemann and Olbert1980; Velli Reference Velli1993; Hollweg & Isenberg Reference Hollweg and Isenberg2007). The $\boldsymbol{z}^-$ fluctuations are also subsequently sheared and cascaded by nonlinear interactions with $\boldsymbol{z}^+$ fluctuations (Velli et al. Reference Velli, Grappin and Mangeney1989). As a consequence, in a reference frame that propagates with the $\boldsymbol{z}^+$ fluctuations away from the Sun, the $\boldsymbol{z}^-$ fluctuations at scale $\lambda$ remain coherent until the $\boldsymbol{z}^+$ fluctuations at scale $\lambda$ evolve due to nonlinear interactions (Lithwick et al. (Reference Lithwick, Goldreich and Sridhar2007); see also § 5 of Chandran & Perez (Reference Chandran and Perez2019)). This ‘anomalous coherence’ implies that the $\boldsymbol{z}^+$ fluctuations at scale $\lambda$ have a cascade time scale

(2.17) \begin{equation} \tau _{{\rm nl,}\lambda }^+ \sim \frac {\lambda }{\delta z^-_\lambda }, \end{equation}

even though this time exceeds the time $\sim l_\lambda /2v_{\textrm {A}}$ required for $\boldsymbol{z}^+$ and $\boldsymbol{z}^-$ structures at perpendicular scale $\lambda$ to propagate through each other at the Alfvén speed.

For values of $\lambda$ in the inertial range, we assume that the average cascade flux is independent of $\lambda$ :

(2.18) \begin{equation} \left \langle \epsilon _\lambda ^+ \right \rangle \propto \lambda ^0. \end{equation}

Upon substituting (2.16) and (2.17) into (2.18), we obtain

(2.19) \begin{equation} \left \langle \frac {(\delta z^+_\lambda )^2 \delta z^-_\lambda }{\lambda } \right \rangle \propto \lambda ^0. \end{equation}

We cannot evaluate the left-hand side of (2.19) by separately averaging $(\delta z^+_\lambda )^2$ and $\delta z^-_\lambda$ using (2.3) and (2.4), because $\boldsymbol{z}^+$ and $\boldsymbol{z}^-$ fluctuations interact with each other, causing $\delta z^+_\lambda$ and $\delta z^-_\lambda$ to become correlated. To model how this correlation affects $\langle \epsilon ^+_\lambda \rangle$ , we assume that $\langle \epsilon ^+_\lambda \rangle$ is dominated by $\delta z^+_\lambda$ fluctuations in the tail of the distribution, which fill some volume $\Omega$ . We further assume that, in and around volume $\Omega$ , the $\boldsymbol{z}^-$ fluctuations at scales somewhat larger than $\lambda$ are ordinary, with amplitudes comparable to the median value (2.7) for their scale. These larger-scale fluctuations provide the $\boldsymbol{z}^-$ energy that is fed into fluctuations of scale $\lambda$ within volume $\Omega$ , and we make the simplifying approximation that $\delta z^-_\lambda$ is driven or injected at the same rate throughout volume $\Omega$ . We also assume that nonlinear interactions at scale $\lambda$ in effect damp $\delta z^-_\lambda$ on the time scale $\lambda /\delta z^+_\lambda$ , causing $\delta z^-_\lambda$ to be $\propto 1/\delta z^+_\lambda$ in volume $\Omega$ . Choosing the proportionality factor so that $\delta z^-_\lambda \rightarrow w^-_\lambda$ as $\delta z^+_\lambda \rightarrow w^+_\lambda$ , we find that

(2.20) \begin{equation} \delta z^-_\lambda \sim \frac {w^-_\lambda w^+_\lambda }{\delta z^+_\lambda } \end{equation}

in volume $\Omega$ , which corresponds to the tail of the $\delta z^+_\lambda$ distribution. (Our estimate of $\delta z^-_\lambda$ in (2.20) differs from the estimate of $\delta z^-_\lambda$ in CSM15 for reasons that we discuss in § 4.)

Upon substituting (2.20) into (2.19), averaging, and making use of (2.7), (2.9) and (2.10), we obtain

(2.21) \begin{equation} \left \langle \epsilon _\lambda ^+\right \rangle \sim \frac {\left \langle \delta z^+_\lambda \right \rangle w^+_\lambda w^-_\lambda }{\lambda } \propto \lambda ^{-\beta -2\ln \beta } . \end{equation}

Equation (2.18) then implies that

(2.22) \begin{equation} \beta = - 2 \ln \beta . \end{equation}

The solution to (2.22) is

(2.23) \begin{equation} \beta = 2 W_0(1/2) = 0.7035, \end{equation}

where $W_0$ is the Lambert $W$ function. We note that the $\delta z^+_\lambda$ fluctuations that make the largest contribution to $\langle \epsilon ^+_\lambda \rangle$ in (2.21) have amplitudes $\simeq \delta z^+_{(1),\lambda } \propto \lambda ^{-\beta \ln \beta } = \lambda ^{0.247}$ , where $\delta z^+_{(1),\lambda }$ is defined in (2.11). Deep in the inertial range, $\delta z^+_{(1),\lambda } \gg w^+_\lambda \propto \lambda ^{-\ln \beta } = \lambda ^{0.352}$ , consistent with our assumption that $\langle \epsilon ^+_{\lambda }\rangle$ is dominated by the tail of the $\delta z^+_\lambda$ distribution.

One might expect that, in reflection-driven MHD turbulence, the spatial distribution of $\delta z^-_\lambda$ should mirror the spatial distribution of $\delta z^+_\lambda$ , so that $\delta z^+_\lambda$ and $\delta z^-_\lambda$ are highly correlated. However, the reflection of inertial-range $ \boldsymbol{z}^+$ fluctuations produces inertial-range $\boldsymbol{z}^-$ fluctuations with a very steep spectrum (Velli et al. Reference Velli, Grappin and Mangeney1989). As a consequence, the inertial-range $\delta z^-_\lambda$ fluctuations are primarily the result of $\boldsymbol{z}^-$ energy that originates from the reflection of outer-scale $\boldsymbol{z}^+$ fluctuations and that subsequently cascades to smaller scales. We do not expect the spatial distribution of such ‘cascaded’ $\delta z^-_\lambda$ fluctuations to approximate the spatial distribution of $\delta z^+_\lambda$ fluctuations in the inertial range, and indeed we have argued in (2.20) that $\delta z^+_\lambda$ and $\delta z^-_\lambda$ are anticorrelated.

From (2.11), the fluctuations that make the largest contribution to the second-order structure function have amplitude

(2.24) \begin{equation} \delta z^\pm _{(2),\lambda } \propto \lambda ^{-\beta ^2 \ln \beta } = \lambda ^{0.174}, \end{equation}

whereas the root-mean-square fluctuation amplitude scales as

(2.25) \begin{equation} \delta z^\pm _{{(\rm rms),}\lambda } \equiv \left \langle \left (\delta z^\pm _{\lambda }\right )^2 \right \rangle ^{1/2} \propto \lambda ^{(1-\beta ^2)/2} = \lambda ^{0.253}. \end{equation}

The scaling in (2.24) is shallower than in (2.25) because the second-order structure function is dominated by a fraction $f_\lambda$ of the volume that decreases as $\lambda$ decreases, within which the $\boldsymbol{z}^\pm$ fluctuations are unusually strong. This volume filling factor $f_\lambda$ could be defined in different ways, one being the relation $[\delta z^\pm _{(\rm rms),\lambda }]^2 \sim f_\lambda [\delta z^\pm _{(2),\lambda }]^2$ . The parallel length scale $l_\lambda$ that can be inferred from an analysis of the second-order $\boldsymbol{z}^+$ structure function (see, e.g. Sioulas et al. Reference Sioulas2024) is approximately the value of $l_\lambda$ within the fraction $f_\lambda$ of the volume that makes the dominant contribution to $\langle (\delta z^+_\lambda )^2\rangle$ , which is approximately

(2.26) \begin{equation} l_{(2),\lambda } \equiv \frac {v_{\textrm {A}} \lambda }{ \delta z^+_{(2),\lambda }} \propto \lambda ^{1+\beta ^2 \ln \beta } = \lambda ^{0.826}. \end{equation}

To connect our results to the Elsasser power spectra, we take $E^\pm (f)$ to be the $z^\pm$ power spectrum measured by a spacecraft in the solar wind, where $f$ is frequency in the spacecraft frame. Setting $f \sim U/\lambda$ , where $U$ is the plasma velocity in the spacecraft frame, and taking $fE^\pm (f)$ to be $\propto \langle (\delta z^\pm _\lambda )^2 \rangle$ , we obtain

(2.27) \begin{equation} E^\pm (f) \propto f^{-1 - \zeta _2} = f^{-1.51}. \end{equation}

3. Comparison with observations

We compare our model with the analysis by Sioulas et al. (Reference Sioulas2024) of magnetic-field fluctuations during the first perihelion encounter (E1) of the Parker Solar Probe (PSP), from 1 November, 2018, to 11 November, 2018, when PSP’s heliocentric distance $r$ ranged between $0.166$ and $0.244$ astronomical units (au). Sioulas et al. (Reference Sioulas2024) analyzed the merged SCaM data product (Bowen et al. Reference Bowen2020), which combines measurements from the flux-gate magnetometers and search-coil magnetometer of the PSP FIELDS instrument suite (Bale et al. Reference Bale2016). Sioulas et al. (Reference Sioulas2024) subdivided the data into 12 hr intervals, with 6 hrs of overlap between consecutive intervals, and restricted their analysis to intervals in which the absolute value of the fractional cross-helicity

(3.1) \begin{equation} \sigma _{\textrm {c}} = \frac {\left \langle (\delta z^+_\lambda )^2 - (\delta z^-_\lambda )^2 \right \rangle }{ \left \langle (\delta z^+_\lambda )^2 + (\delta z^-_\lambda )^2 \right \rangle } \end{equation}

exceeded 0.75 at $\lambda = 10^4 {\rm d}_{{i}}$ , where ${\rm d}_{{i}}$ is the proton inertial length. Sioulas et al. (Reference Sioulas2024) then computed five-point structure functions of the magnetic field in a coordinate system in which one axis (the $l$ axis) is aligned with the local background magnetic field $\boldsymbol{B}_{{\rm local}}$ , a second axis (the $\xi$ axis) is aligned with the component of the magnetic-field fluctuation perpendicular to $\boldsymbol{B}_{{\rm local}}$ , and the third axis (the $\lambda$ axis) is perpendicular to the first two. Each increment $\Delta \boldsymbol{B}$ computed from the data corresponds to a spatial separation $\Delta \boldsymbol{x}$ parallel to the instantaneous value of the relative velocity between the plasma and spacecraft, which in general has a non-zero component along each of the $l$ , $\xi$ and $\lambda$ directions. Sioulas et al. (Reference Sioulas2024) considered three different subsets of increments, one in which $\Delta \boldsymbol{x}$ is within $5^\circ$ of the $l$ direction, one in which $\Delta \boldsymbol{x}$ is within $5^\circ$ of the $\xi$ direction and one in which $\Delta \boldsymbol{x}$ is within $5^\circ$ of the $\lambda$ direction. It is this last subset that we compare with our model results in this section.

To convert time intervals into spatial intervals, Sioulas et al. (Reference Sioulas2024) applied Taylor’s frozen-flow hypothesis based on the average plasma velocity $U$ in the spacecraft frame. Plasma measurements from the SWEAP instrument suite (Kasper et al. Reference Kasper2016) show that the average value of $U$ during the selected intervals was $277 \mbox { km} \mbox { s}^{-1}$ . Quasithermal-noise electron-density measurements from FIELDS data (Moncuquet et al. Reference Moncuquet2020) and quasineutrality imply that the average value of $d_{\textrm {i}}$ during the selected intervals was $16.3 \mbox { km}$ .

Figure 1. (a) The scaling exponent $\zeta _n$ of the $n$ th-order $\boldsymbol{z}^+$ structure function from (2.10) and (2.23), and the scaling exponent of the $n$ th-order magnetic-field structure function obtained by Sioulas et al. (Reference Sioulas2024) from measurements during PSP’s first perihelion encounter. (b) The power spectrum $E_B(f)$ of the magnetic field in PSP encounter-1 data as a function of spacecraft-frame frequency $f$ , as well as the $\boldsymbol{z}^+$ power-spectrum scaling from (2.27). The shaded grey rectangle shows the frequency interval that corresponds to the scale range $200 {\rm d}_{{i}} \lt \lambda \lt 6000 {\rm d}_{{i}}$ based on the correspondence $f = U/(2 \lambda )$ suggested by figure 1 of Huang et al. (Reference Huang2023), with $U = 277 \mbox { km} \mbox { s}^{-1}$ and ${\rm d}_{{i}} = 16.3 \mbox { km}$ .

Within the large- $\sigma _{\textrm {c}}$ intervals analysed by Sioulas et al. (Reference Sioulas2024), $\delta z^-_\lambda \ll \delta z^+_\lambda$ , and hence $\delta z^+_\lambda \simeq 2 \delta B_\lambda /\sqrt {4 \pi \rho }$ , where $\delta B_\lambda$ is defined by analogy to (2.2). We thus expect the structure functions of $\boldsymbol{B}$ in these intervals to scale with $\lambda$ in the same way as the $\boldsymbol{z}^+$ structures functions. In figure 1(a), we plot the magnetic-field structure-function scaling exponents obtained by Sioulas et al. (Reference Sioulas2024) over the range $200 {\rm d}_{{i}} \lt \lambda \lt 6000 {\rm d}_{{i}}$ , as well as the $\boldsymbol{z}^+$ structure-function scaling exponents from (2.10) and (2.23). The error bars on the scaling exponents are the errors associated with the power-law fits to the structure functions and do not account for errors arising from the sensitivity of high-order structure functions to rare events in the tail of the distribution (see, e.g. Dudok de Wit et al. Reference Dudok de Wit, Alexandrova, Furno, Sorriso-Valvo and Zimbardo2013). Further work is needed to quantify such errors; here we simply note that the observationally inferred $\zeta _n$ values at large $n$ in figure 1 should be viewed with caution.

Using the subset of E1 magnetic-field increments obtained by Sioulas et al. (Reference Sioulas2024) in which $\Delta \boldsymbol{x}$ is within $5^\circ$ of the $\lambda$ direction, we compute the conditional power spectral density of the magnetic field, $E_B(f)$ , obtained via the maximum overlap discrete wavelet transform (Percival & Walden Reference Percival and Walden2000), where $f$ is frequency in the spacecraft frame. We plot $E_B(f)$ in figure 1(b), as well as the Elsasser power-spectrum scaling from (2.27).

In figure 2 we plot the result of Sioulas et al. (Reference Sioulas2024) for the scale-dependent parallel correlation length $l_\lambda$ , which they obtained by comparing the parallel second-order structure function $\mbox {SF}_{2\parallel }(l)$ in which $\Delta \boldsymbol{x}$ is within $5^\circ$ of the $l$ direction and the perpendicular second-order structure function $\mbox {SF}_{2\perp }(\lambda )$ obtained from increments in which $\Delta \boldsymbol{x}$ is within $5^\circ$ of the $\lambda$ direction. They defined $l_\lambda$ to be that value of $l$ for which $\mbox {SF}_{2\perp }(\lambda ) = \mbox { SF}_{2\parallel }(l)$ . We also plot in figure 2 the scaling $l_{(2),\lambda } \propto \lambda ^{1+\beta ^2 \ln \beta } = \lambda ^{0.826}$ from (2.26) for the fluctuations that make the largest contribution to the second-order structure function of $\boldsymbol{z}^+$ .

Figure 2. The parallel correlation length $l_\lambda$ inferred from PSP magnetic-field measurements (Sioulas et al. Reference Sioulas2024), and the $l_{(2),\lambda }$ scaling from (2.26). The shaded rectangle corresponds to the scale range that was used to calculate the PSP E1 structure-function scaling exponents in figure 1.

Our model agrees quite well with the power spectrum and $\zeta _n$ values inferred by Sioulas et al. (Reference Sioulas2024) from PSP E1 data for the scale range $200 {\rm d}_{{i}} \lt \lambda \lt 6000 {\rm d}_{{i}}$ . We note, however, that the $\zeta _n$ values that Sioulas et al. (Reference Sioulas2024) found for the scale range $8 {\rm d}_{{i}} \lt \lambda \lt 100 {\rm d}_{{i}}$ are larger than those shown in figure 1 and do not agree with our model. On the other hand, the scaling of $l_{(2), \lambda }$ in (2.26) agrees reasonably well with the $l_\lambda$ values inferred by Sioulas et al. (Reference Sioulas2024) over the broader scale range $8 {\rm d}_{{i}} \lt \lambda \lt 3 \times 10^4 {\rm d}_{{i}}$ .

4. Comparison with models of balanced, intermittent, homogeneous RMHD turbulence

Our model shares several features with the previous models of CSM15 and MS17, which were developed for balanced (i.e. zero-cross-helicity), intermittent, homogeneous RMHD turbulence. In particular, we adopt the same procedure in (2.3) and (2.4) for determining the scale dependence of the fluctuation-amplitude distribution and the same volume filling factor in (2.4) and (2.6) for the most intense $\delta z^+_\lambda$ fluctuations. As a consequence, we obtain the same formula, (2.10), as these previous authors for the scaling exponent $\zeta _n$ of the $n$ th-order structure function of $\delta z^+_\lambda$ . These shared assumptions receive considerable support from the results of Sioulas et al. (Reference Sioulas2024) on the structure-function scaling exponents of magnetic fluctuations in the scale range $200 {\rm d}_{{i}} \lt \lambda \lt 6000 {\rm d}_{{i}}$ in the near-Sun solar wind, which are shown in figure 1. In particular, the finding of Sioulas et al. (Reference Sioulas2024) that $\zeta _n$ asymptotes to a constant value at large $n$ implies that the amplitude of the most intense fluctuations at scale $\lambda$ is independent of $\lambda$ , consistent with (2.3). Moreover, the finding of Sioulas et al. (Reference Sioulas2024) that $\zeta _n$ asymptotes to $\simeq 1$ at large $n$ implies that the volume filling factor of the most intense fluctuations is $\propto \lambda$ , consistent with (2.4) and (2.6).

On the other hand, we depart from CSM15 and MS17 by taking large-amplitude fluctuations to be tube-like. In contrast, CSM15 and MS17 took such fluctuations to be sheet-like, in agreement with numerical simulations of homogeneous MHD turbulence (e.g. Maron & Goldreich Reference Maron and Goldreich2001). Our assumption of tube-like fluctuations is motivated by the results of Sioulas et al. (Reference Sioulas2024) on the three-dimensional geometry of the second-order structure function of the magnetic field in the scale range $200 {\rm d}_{{i}} \lt \lambda \lt 6000 {\rm d}_{{i}}$ in PSP data.

The assumption of CSM15 that fluctuations in the tail of the distribution are sheet-like led them to the conclusion that $l^+_\lambda$ is an increasing function of $\delta z^+_\lambda$ . To explain this, we briefly review CSM15’s analysis of the interaction between large-amplitude, sheet-like $\boldsymbol{z}^+$ structures and much weaker, counter-propagating $\boldsymbol{z}^-$ fluctuations. CSM15 characterized sheet-like $\boldsymbol{z}^+$ structures in the tail of the distribution as having dimensions $l^+_\lambda$ in the $\boldsymbol{B}$ direction, $\xi _\lambda$ in the $\Delta \boldsymbol{z}^+_\lambda$ direction, and $\lambda$ in the $\boldsymbol{B} \times \Delta \boldsymbol{z}^+_\lambda$ direction, with $\xi _\lambda \gg \lambda$ and $\xi _\lambda \propto \delta z^\pm _\lambda$ . To determine the rate of deformation of a sheet-like $\delta z^+_\lambda$ structure, CSM15 defined a trial-volume sphere of radius $\sim \lambda$ located somewhere in the middle of the sheet. Noting that the $\boldsymbol{z}^-$ fluctuations propagating at velocity $\boldsymbol{v}_{\textrm {A}}$ within that trial volume have been preprocessed by the part of the $\boldsymbol{z}^+$ sheet upstream of the trial volume, CSM15 defined the ‘source region’ for the $\boldsymbol{z}^-$ fluctuations in the trial volume to be the region upstream of the trial volume closest to the trial volume within which the $\boldsymbol{z}^-$ fluctuations have not yet interacted with the $\boldsymbol{z}^+$ sheet. CSM15 made the approximation that the $\boldsymbol{z}^-$ fluctuations in this source region are median-amplitude $\boldsymbol{z}^-$ fluctuations, which are tube-like in the CSM15 model. CSM15 used the notation $l^\ast _\lambda$ to denote the parallel correlation length of such median-amplitude fluctuations at perpendicular scale $\lambda$ . CSM15 then showed that the $\boldsymbol{z}^-$ fluctuations in the trial volume that are most effective at shearing the $\boldsymbol{z}^+$ structure are the $\boldsymbol{z}^-$ fluctuations that had a perpendicular scale $\xi _\lambda$ in the source region. As these $\boldsymbol{z}^-$ fluctuations propagate from the source region to the trial volume, shearing by the $\boldsymbol{z}^+$ sheet reduces their correlation length in the $\boldsymbol{B} \times \Delta \boldsymbol{z}^+_\lambda$ direction from $\xi _\lambda$ to $\lambda$ and rotates them into alignment with the $\boldsymbol{z}^+$ sheet, so that the angle $\theta ^\pm _\lambda$ between $\Delta \boldsymbol{z}^+_\lambda$ and $\Delta z^-_\lambda$ within the trial volume is small, which decreases the rate at which the $\boldsymbol{z}^-$ fluctuations shear the $\boldsymbol{z}^+$ structure.

CSM15 estimated $l^+_\lambda$ using two different arguments. First, CSM15 set $l^+_\lambda \sim v_{\textrm {A}} \tau _\lambda ^+$ , taking the parallel correlation length of a $\boldsymbol{z}^+$ fluctuation to be the distance it can propagate along the magnetic field before its energy cascades to smaller scales. We call this the survival-time argument. Second, CSM15 took $l^+_\lambda$ for a $\boldsymbol{z}^+$ structure to be the parallel correlation length of the $\boldsymbol{z}^-$ fluctuations that dominate the shearing of the $\boldsymbol{z}^+$ structure and evaluating the parallel correlation length of those $\boldsymbol{z}^-$ fluctuations before they started interacting with the $\boldsymbol{z}^+$ structure – i.e. in the source region described in the previous paragraph. This led to the estimate $l^+_\lambda \sim l^\ast _{\xi _\lambda }$ . We call this second argument the agent-of-shearing argument. CSM15 showed that these two arguments lead to the same estimate for $l^+_\lambda$ , i.e. that $v_{\textrm {A}} \tau ^+_\lambda \sim l^\ast _{\xi _\lambda }$ .

These two arguments were the basis for CSM15’s aforementioned conclusion that $l^+_\lambda$ is an increasing function of $\delta z^+_\lambda$ . For the agent-of-shearing argument, this is because $\xi _\lambda$ is an increasing function of $\delta z^+_\lambda$ , and hence so is $l^\ast _{\xi _\lambda }$ . For the survival-time argument, this is because as $\delta z^+_\lambda$ increases, $\theta ^\pm _\lambda$ decreases, the cascade time scale $\tau ^+_\lambda$ increases, and the $\boldsymbol{z}^+$ fluctuation propagates farther before cascading.Footnote ⁴

In our model, large-amplitude fluctuations are tube-like and therefore their nonlinear interactions are not described by the analysis of CSM15. In particular, there is no preprocessing of $\boldsymbol{z}^-$ fluctuations as they propagate from a preinteraction source region through an extended sheet-like $\boldsymbol{z}^+$ structure, and nonlinear interactions do not cause $\boldsymbol{z}^+$ and $\boldsymbol{z}^-$ fluctuations to become progressively more aligned as $\delta z^+_\lambda$ increases. We estimate $l^-_\lambda \sim v_{\textrm {A}} \lambda / \delta z^+_\lambda$ in (2.13) using the survival-time argument. We then use the agent-of-shearing argument to estimate $l^+_\lambda$ in (2.14), setting $l^+_\lambda \sim l^-_\lambda$ . This latter estimate makes our survival-time estimate of $l^-_\lambda$ consistent with the agent-of-shearing estimate of $l^-_\lambda$ . However, our agent-of-shearing estimate of $l^+_\lambda$ differs from the survival-time estimate of $l^+_\lambda$ , because $v_{\textrm {A}} \tau ^+_\lambda \sim v_{\textrm {A}} \lambda / \delta z^-_\lambda \gg l^-_\lambda \sim v_{\textrm {A}} \lambda / \delta z^+_\lambda$ . Thus, although CSM15 did not need to choose between the agent-of-shearing argument and the survival-time argument to determine $l^+_\lambda$ , we do need to choose, and we choose the agent-of-shearing argument.

Whereas CSM15 found that $l^+_\lambda$ is an increasing function of $\delta z^+_\lambda$ in balanced RMHD turbulence, we find in (2.13) and (2.14) that $l^+_\lambda$ is a decreasing function of $\delta z^+_\lambda$ in imbalanced, reflection-driven, Alfvénic turbulence. Because $\delta z^+_{\lambda }$ exceeds $\delta z^+_{{\rm rms,}\lambda }$ within the small fraction of the volume that dominates the second-order $\boldsymbol{z}^+$ structure function (see (2.24) and (2.25)), $l^+_\lambda$ is smaller in this small fraction of the volume than it would be throughout the plasma as a whole in a turbulence model that neglects intermittency. Intermittency in reflection-driven Alfvénic turbulence thus increases the effective parallel wavenumbers $k_\parallel \sim 1/l^+_\lambda$ of the energetically dominant fluctuations at small $\lambda$ , making these fluctuations more isotropic.

Our assumption that large-amplitude $\boldsymbol{z}^+$ fluctuations are tube-like also explains why our estimate of $\delta z^-_\lambda$ in (2.20) differs from the estimate of CSM15. CSM15 showed that when a large-amplitude, sheet-like $\boldsymbol{z}^+$ structure interacts with a smaller-amplitude $\boldsymbol{z}^-$ fluctuation, the amplitude of the $\boldsymbol{z}^-$ fluctuation is not altered. In contrast, in our analysis, large-amplitude, tube-like $\boldsymbol{z}^+$ fluctuations can alter, and in particular decrease, the amplitudes of the $\boldsymbol{z}^-$ fluctuations with which they interact.