# Scale-Dependent Dynamic Alignment in MHD Turbulence: Insights into
Intermittency, Compressibility, and Imbalance Effects

Nikos Sioulas  Department of Earth, Planetary, and Space Sciences, University
of California, Los Angeles, CA, USA  Marco Velli Department of Earth,
Planetary, and Space Sciences, University of California, Los Angeles, CA, USA
Alfred Mallet Space Sciences Laboratory, University of California, Berkeley,
CA 94720-7450, USA  Trevor A. Bowen Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  B. D. G. Chandran Space Science
Center and Department of Physics, University of New Hampshire, Durham, NH
03824, USA  Chen Shi (时辰) Department of Earth, Planetary, and Space Sciences,
University of California, Los Angeles, CA, USA  S. S. Cerri Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de
l’Observatoire, CS 34229, 06304 Nice5cedex 4, France  Ioannis Liodis
Department of Physics, Aristotle University of Thessaloniki
GR-52124 Thessaloniki, Greece  Tamar Ervin Department of Physics, University
of California, Berkeley, Berkeley, CA 94720-7300, USA  Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Davin E.
Larson Space Sciences Laboratory, University of California, Berkeley, CA
94720-7450, USA

###### Abstract

Scale-Dependent Dynamic Alignment (SDDA) in the polarization of Elsässer field
fluctuations is theorized to suppress nonlinearities and modulate the energy
spectrum. Despite its significance, empirical evidence for SDDA within the
inertial range of solar wind turbulence has been conspicuously absent. We
analyze a large dataset of homogeneous intervals from the WIND mission to
assess the effects of compressibility, intermittency, and imbalance on SDDA.
Our key findings are as follows: SDDA is consistently evident at energy-
containing scales, but a trend towards misalignment is, on average, observed
at inertial scales. While compressible fluctuations do not exhibit any
increasing alignment, their impact on the average behavior of SDDA is
negligible. Therefore their mixing with incompressible fluctuations cannot
explain the trend of misalignment at inertial scales. The alignment angles
exhibit an inverse correlation with the intensity of field gradients. This may
stem from “anomalous” and/or “counterpropagating” wave packet interactions.
Regardless, this observation indicates that the physical origin of alignment
arises from the mutual shearing of the Elsässer fields during imbalanced
($\delta\bm{z}^{\pm}\gg\delta\bm{z}^{\mp}$) wave-packet interactions.
Stringent thresholding on field gradient intensity reveals SDDA signatures
within a significant portion of the inertial range. The scaling of the
alignment angle in the Elsässer increments, $\Theta^{z}$, becomes steeper with
increasing global Alfvénic imbalance, while the angle between
magnetic/velocity field increments. $\Theta^{ub}$, becomes shallower. Only
$\Theta^{ub}$ is correlated with global Elsässer imbalance, becoming steeper
as the imbalance increases. Additionally, increasing alignment in
$\Theta^{ub}$ extends deep into the inertial range of balanced intervals but
breaks down at large scales for imbalanced ones. We present a simplified
theoretical analysis and model the effects of high-frequency, low-amplitude
noise in the velocity field on SDDA measurements. Our analysis indicates that
noise can significantly affect alignment angle measurements even at very low
frequencies, with the impact growing as global imbalance increases.

Solar wind; Interplanetary turbulence; Magnetohydrodynamics; Space plasmas

††software: Python (Van Rossum & Drake Jr, 1995), SciPy (Virtanen et al.,
2020), Pandas (McKinney et al., 2010), Matplotlib (Hunter, 2007), PySpedas
(Angelopoulos et al., 2019), MHDTurbPy (Sioulas, 2023)

##  1 Introduction

In the framework of reduced magnetohydrodynamics (RMHD), applied to highly
conducting, magnetized plasmas threaded by a large-scale background magnetic
field, $\bm{B}_{0}$, on scales larger than the ion gyroradius, $\rho_{i}$, the
Alfvén (1942) polarized fluctuations decouple from the compressive cascade and
and obey the equation (Kadomtsev & Pogutse, 1967; Schekochihin et al., 2009)

$\partial_{t}\bm{z}_{\perp}^{\mp}\pm
V_{A}\partial_{z}\bm{z}_{\perp}^{\mp}+\bm{z}_{\perp}^{\pm}\cdot\nabla_{\perp}\bm{z}_{\perp}^{\mp}=-\nabla_{\perp}p,$
(1)

where $\bm{V}_{a}=\bm{B}_{0}/\sqrt{4\pi\rho}$ is the Alfvén speed,
$\bm{z}_{\perp}^{\pm}=\bm{u}_{\perp}\mp\bm{b}_{\perp}/\sqrt{4\pi\rho}$ are the
Elsasser (1950) variables, $\bm{u}_{\perp}$ and $\bm{b}_{\perp}$ represent
transverse (to $\bm{B}_{0}$), small-amplitude
($|\bm{b}_{\perp}|/|\bm{B}_{0}|\ll 1$) velocity and magnetic-field
perturbations, respectively, and the total pressure, $p$, can be determined by
taking the (perpendicular) divergence of Equation (1) and imposing the
condition $\nabla_{\perp}\cdot\bm{z}_{\perp}^{\pm}=0$. In this framework,
nonlinearity arises from the collisions of counterpropagating wave packets,
resulting in their distortion and fragmentation (Iroshnikov, 1963; Kraichnan,
1965). The effectiveness of the nonlinear interactions hinges on the relative
importance of the linear and nonlinear terms in Equation 1, which can be
quantified by the nonlinearity parameter,
$\chi^{\pm}=(\ell^{\pm}_{||,\lambda}/\lambda)/(\delta
z_{\lambda}^{\pm}/V_{a})$ which compares the linear wave propagation time
$\tau^{\pm}_{A}=\ell^{\pm}_{||}/V_{A}$ and non-linear decorrelation time
$\tau^{\mp}_{nl}\sim\lambda/\delta\bm{z}^{\pm}_{\lambda}$, where,
($\ell_{||}\sim 1/k_{||}$), ($\lambda\sim 1/k_{\perp}$) the coherence lengths
parallel and perpendicular to $\bm{B}_{0}$.

The interaction of Alfvénic wavepackets, described by the dispersion relation
$\omega=k_{||}v_{A}$, where $k_{||}$ is the wavevector parallel to $B_{0}$,
gives rise to a distinctive feature of MHD turbulence wherein the flux of
energy is primarily directed towards smaller scales perpendicular to the
magnetic field . The cascade is strongly anisotropic, and even more so at
smaller scales, resulting in structures satisfying $\ell_{||}\gg\lambda$
(Robinson & Rusbridge, 1971; Shebalin et al., 1983; Higdon, 1984; Oughton et
al., 1994). The anisotropy amplifies the non-linearity of the interactions
(Galtier et al., 2000a). Consequently, even in the case where the system was
forced at large scales such that $\chi\ll 1$ (referred to as the “weak
turbulence” regime), it inevitably transitions, at sufficiently small scales,
to a state where within a single collision, the wave packet undergoes
deformation of a magnitude comparable to its own, $\chi\sim 1$, i.e., the
cascade time $\tau_{c}$ and dynamical $\tau_{d}=1/wk_{\perp}$ timescales are
similar. Since the perturbation frequency $\omega$ has a lower bound due to an
uncertainty relation $\omega\tau_{c}>1$, the cascade is forced to remain in
the $\chi\sim 1$ regime. This realization forms the basis of what is formally
known as the “critical balance” (CB) conjecture (Goldreich & Sridhar, 1995),
denoted hereafter as GS95. In the framework of _balanced_ turbulence, i.e.,
assuming equal energy fluxes, $\epsilon^{\pm}$, in counterpropagating
wavepackets, CB implies an anisotropic scaling relationship between the
parallel and perpendicular wavevectors, specifically $k{||}\propto
k_{\perp}^{2/3}$, leading to expected field-perpendicular and field-parallel
energy spectra of the form $E(k_{\perp})\propto k_{\perp}^{-5/3}$ and
$E(k_{||})\propto k_{||}^{-2}$, resepectively.

While in situ observations of the solar wind appeared consistent with the
phenomenology proposed by GS95 (Horbury et al., 2008; Wicks et al., 2010),
numerical simulations of homogeneous, incompressible MHD turbulence revealed a
significant discrepancy. Specifically, the observed spectral index
perpendicular to the locally defined background magnetic field,
$\bm{B}_{\ell}$, was closer to $-3/2$ (Maron & Goldreich, 2001; Müller et al.,
2003; Müller & Grappin, 2005). These findings prompted refinements to the GS95
model (Boldyrev, 2005; Galtier et al., 2005; Beresnyak & Lazarian, 2006a;
Gogoberidze, 2007).

One approach to addressing this discrepancy was the incorporation of scale-
dependent dynamic alignment (SDDA) into the GS95 framework. Boldyrev (2006),
henceforth B06, proposed that a self-consistent mechanism that results in
increasing alignment would modify the spectral slope of the field-
perpendicular inertial range energy spectrum from the -5/3 Kolmogorov slope to
the numerically observed -3/2 slope. Specifically, Boldyrev (2006) suggests
that the non-linearity in MHD turbulence is reduced due to the increasing
alignment between $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$ as
$\lambda$ decreases, with $\theta^{ub}_{\lambda}\sim\delta
b/v_{A}\propto\lambda^{1/4}$. At small scales, this leads to the emergence of
three-dimensional anisotropic eddies characterized by
$\ell_{||}\gg\xi\gg\lambda$, where $\xi$ represents the coherence length in
the direction of $\delta\bm{b}$, and $\lambda$ is perpendicular to both
$\bm{B}_{\ell}$ and $\delta\bm{b}$.

A substantial body of numerical studies on homogeneous MHD turbulence has
provided evidence supporting the scale-dependence of certain alignment
measures across a sizable portion of the inertial range (Mason et al., 2006;
Perez et al., 2012, 2014; Mallet et al., 2015; Chandran et al., 2015; Cerri et
al., 2022). However, concerns have been raised suggesting that the observed
alignment may be a finite-range effect intrinsically linked to dynamics
occurring at the outer scale (Beresnyak, 2012). For instance, Beresnyak (2012)
interpret these signatures based on the idea that MHD turbulence is much less
local in k-space compared to hydrodynamic turbulence (see, e.g., Beresnyak,
2011; Schekochihin, 2022). Consequently, the driving mechanism does not fully
replicate the properties of the inertial range, and the transition to
asymptotic statistics is broad, causing many quantities to appear scale-
dependent as they adjust to the asymptotic regime. Moreover, the B06 model has
faced criticism for violating the rescaling symmetry of the RMHD equations
(Beresnyak, 2012). As a result, an ongoing debate persists regarding whether
this numerical evidence accurately reflects the scale-dependent dynamic
alignment angle in the asymptotic state of the inertial range (Mason et al.,
2011; Beresnyak, 2012; Perez et al., 2014).

The theory of SDDA has been revisited by Chandran et al. (2015, hereafter,
CSM15), refining it in a manner that aligns SDDA with the rescaling symmetry
of RMHD. Their model distinguishes between two archetypal types of alfvénic
interactions: imbalanced ($\delta\bm{z}^{\pm}\gg\delta\bm{z}^{\mp}$) and
balanced ($\delta\bm{z}^{\pm}\sim\delta\bm{z}^{\mp}$). They develop an
approximate theory which demonstrates that in imbalanced interactions, the
mutual shearing of the interacting wave packets leads to the rapid cascading
of the subdominant field to smaller scales. This occurs as it rotates into
alignment with the dominant field, notably without distorting their
amplitudes. Consequently, at any given scale, wave packets subjected to the
smallest number of balanced collisions—those that alter both amplitude and
coherence lengths—roughly retain their outer scale amplitudes, resembling
three-dimensional anisotropic current sheet structures (see also Howes, 2015;
Mallet & Schekochihin, 2017). This indicates that as the cascade progresses
toward smaller scales, the fluctuating energy becomes confined within an
increasingly smaller volume fraction, thus establishing a link between SDDA
and intermittency (Oboukhov, 1962; Kolmogorov, 1962).

The framework established by CSM15 has garnered significant support from the
studies conducted by Mallet et al. (2015, 2016), who employed RMHD numerical
simulations to demonstrate that the degree of alignment at any given scale
increases with the fluctuation amplitude. This finding indicates that
alignment angles exhibit intermittency rather than scale invariance (see also
Beresnyak & Lazarian, 2006b). Moreover, scaling predictions derived from these
phenomenological models, particularly those related to higher-order moments
and alignment angles, show strong agreement with numerical simulations of
forced, homogeneous, and isotropic MHD turbulence Chandran et al. (2015);
Mallet et al. (2016); Mallet & Schekochihin (2017); Shi et al. (2023); Dong et
al. (2022).

However, in-situ solar wind observations reveal certain deviations from the
model predictions. For example, while inertial range scalings of the
perpendicular components for highly Alfvénic intervals tend towards the
predicted -3/2 value, for both magnetic field and velocity fluctuations, the
former steepen towards -5/3 in balanced streams (Chen et al., 2013a; Sioulas
et al., 2022; McIntyre et al., 2023). Moreover, emerging evidence indicates
the presence of two sub-regimes within the traditionally defined inertial
range, particularly prominent in strongly imbalanced streams and when
considering anisotropic spectra in locally defined frames (Wicks et al., 2011;
Sioulas et al., 2022; Wu et al., 2022; Sioulas et al., 2023). Although 3D
anisotropic eddies are evident at the smaller scale end of the inertial range,
anisotropy qualitatively changes across scales (Chen et al., 2012; Verdini et
al., 2018; Sioulas et al., 2024).

Finally, in-situ observations of SDDA in the solar wind remain inconclusive.
While SDDA is evident at large, energy-containing scales in the 1/f range, a
shift occurs in the inertial range ($\lambda\approx 10^{4}d_{i}$), where
increasing misalignment of fluctuations is observed towards smaller scales
Podesta (2009); Hnat et al. (2011); Wicks et al. (2013); Parashar et al.
(2018, 2020). The increasing misalignment at inertial scales persists even in
intervals selected to minimize solar wind expansion effects Verdini et al.
(2018), although in this case, the 3D anisotropic inertial range spectral
scaling and topology of eddies align with proposed models Boldyrev (2006);
Chandran et al. (2015); Mallet & Schekochihin (2017).

These observations could suggest that additional physical mechanisms beyond
standard homogeneous MHD are necessary to fully explain the observed
properties of solar wind turbulence. Such mechanisms include, but are not
limited to, the breaking of local anisotropy due to the additional radial
symmetry axis introduced by expansion (Dong et al., 2014; Verdini & Grappin,
2015; Verdini et al., 2018), the influence of non-Alfvénic interactions in the
turbulent cascade (see, e.g., Chapman & Hnat, 2007; Bowen et al., 2021), the
spherically polarized nature of Alfvénic fluctuations (Matteini et al., 2014;
Mallet et al., 2021; Matteini et al., 2024), and the imbalance in the fluxes
of counterpropagating wave packets (Lithwick et al., 2007; Chandran, 2008;
Perez & Boldyrev, 2009; Schekochihin, 2022).

In recent years, considerable efforts have been made to gauge the impact of
imbalance, compressibility, intermittency, and expansion on the statistical
properties of MHD turbulence (Lithwick & Goldreich, 2001; Hnat et al., 2005;
Salem et al., 2009; Lithwick et al., 2007; Chandran, 2008; Podesta & Borovsky,
2010; Podesta & Bhattacharjee, 2010; Li et al., 2011; Chen et al., 2013b;
Wicks et al., 2013; Matteini et al., 2014; Bowen et al., 2018a, c; Shoda et
al., 2019; Vech & Chen, 2016; Meyrand et al., 2021; Sioulas et al., 2022,
2022; Dunn et al., 2023; Shi et al., 2023; Wang et al., 2023). However,
previous investigations, particularly those focusing on in-situ observations,
have not thoroughly explored these effects on the statistical properties of
SDDA. This work aims to bridge this gap.

Figure 1:  Joint probability distribution of $V_{\text{sw}}$ and (a)
$\sigma_{c}$, (b) $\sigma_{r}$ for the homogeneous intervals selected for our
analysis. $\sigma_{c}$ and $\sigma_{r}$ are computed based on fluctuations at
the scale of $2\cdot 10^{4}d_{i}$. In Panel (c), a histogram displays interval
counts and cumulative durations relative to $V_{\text{sw}}$.

To this end, we conduct a statistical analysis of carefully selected solar
wind intervals spanning a 28-year period from WIND observations. We utilize
scale and time-dependent proxies to isolate the properties of interest, namely
compressibility, intermittency, and imbalance, enabling us to quantify their
effects on SDDA measurements.

The remainder of this paper is structured as follows: Section 2 details the
selection and processing of the data. Section 3 provides background on the
methods applied for this analysis. The results of this study are presented in
Section 4, followed by a discussion of these results and the conclusions in
Section 6.

##  2 Data Selection

We utilize data collected by instruments aboard the WIND spacecraft,
positioned at Earth’s L1 Lagrange point, approximately 1 AU from the Sun.

This study primarily analyzes magnetic field and ion measurements obtained
from the MFI instrument (Lepping et al., 1995) and the 3DP/PESA-L instrument
(Lin et al., 1995), which provide data with resolutions of 0.1 seconds and 3
seconds, respectively. It should be noted that, while a 3-second dataset is
available for the magnetic field timeseries, an issue has been identified with
this specific dataset, as detailed in .5. Consequently, the full-resolution
MFI data were downsampled—following the application of a low-pass Butterworth
filter to mitigate aliasing—to match the temporal resolution of the ion moment
data.

In the preliminary phase of our analysis, we conducted a meticulous visual
inspection of the WIND timeseries data spanning from January 1, 1995, to
August 1, 2023. Based on this examination, we classified the data into three
distinct categories:

  1. 1.

Slow Alfvénic wind: $V_{sw}\leq 450$ km/s, $\sigma_{c}\geq 0.85$

  2. 2.

Slow non-Alfvénic wind: $V_{sw}\leq 450$ km/s, $\sigma_{c}\leq 0.85$

  3. 3.

Fast Alfvénic wind: $V_{sw}\geq 500$ km/s, $\sigma_{c}\geq 0.85$

Throughout the visual inspection, strict adherence was maintained to several
selection criteria:

  1. 1.

A minimum interval duration of 6 hours was mandated,

  2. 2.

The stability of plasma parameters ($V_{sw}$, $n_{p}$, $\beta$, $\Theta_{VB}$,
$\sigma_{c}$, $\sigma_{r}$, $\text{sign}(B_{x})$) was maintained, ensuring
that these parameters did not undergo significant variations throughout the
selected interval,

  3. 3.

Intervals showing clear indications of transient events, like Coronal Mass
Ejections or Heliospheric Current Sheet crossings, were omitted,

  4. 4.

Intervals were restricted to those with a maximum of 2% missing magnetic field
data and 5% missing plasma data.

The selection process resulted in a total of 2335 intervals. The distribution
of interval characteristics within our dataset is visualized in Fig. 1. Panels
(a) to (c) depict 2D histograms illustrating the solar wind speed in
correlation with normalized cross helicity ($\sigma_{c}$), normalized residual
energy ($\sigma_{r}$), both evaluated at a scale of $\ell\approx 2\cdot
10^{4}d_{i}$. Finally, column (c) provides a histogram portraying the number
of selected intervals (solid line) and their cumulative duration associated
with different $V_{sw}$ values.

Furthermore, to facilitate a more direct comparison with (Podesta, 2009)
(hereinafter referred to as P09), we examined four intervals spanning: (a)
January 1, 1995, to July 29, 1995, (b) May 15, 1996, to August 16, 1996, (c)
January 8, 1997, to June 9, 1997, (d) August 23, 2000, to February 15, 2001.
For an exhaustive discussion regarding the properties of the chosen intervals,
readers are directed to P09. An in-depth comparison between our results and
those of P09 is provided in Appendix .1.

##  3 Data Analysis

###  3.1 Estimating 5-point increments

Assuming the validity of Taylor (1938) hypothesis the 2-point increments for a
field $\bm{\phi}$ at a specific spatial lag, where
$\bm{\ell}=\tau\cdot\bm{V}_{sw}$ and $\bm{V}_{sw}$ represents the solar wind
speed, can be estimated as:

$\delta\bm{\phi}=\bm{\phi}(\bm{r}+\bm{\ell})-\bm{\phi}(\bm{r}),$ (2)

However, in the context of estimating five-point increments, Equation 2
necessitates a redefinition of $\delta\bm{\phi}$ to:

$\displaystyle\delta\bm{\phi}={}$
$\displaystyle[\bm{\phi}(\bm{r}-2\bm{\ell})-4\bm{\phi}(\bm{r}-\bm{\ell})+6\bm{\phi}(\bm{r})$
(3)
$\displaystyle\qquad-4\bm{\phi}(\bm{r}+\bm{\ell})+\bm{\phi}(\bm{r}+2\bm{\ell})]/\sqrt{35}.$

Using the five-point method, we estimate the scale-dependent increments of the
magnetic field—in velocity units—as:

$\delta\bm{b}=\frac{\delta\bm{B}}{\sqrt{\mu_{0}\rho}},$ (4)

where $\rho=1.16m_{p}n_{p}$, $\mu_{0}$ is the permeability of free space, and
$m_{p}$ and $n_{p}$ represent the proton mass and a 1-minute rolling average
of the number density, respectively. The factor 1.16 accounts for alpha
particles, assuming $n_{a}/n_{p}=0.04$ (Podesta et al., 2009). It is
noteworthy that similar results were obtained even when this analysis was
performed without the prefactor.

The perpendicular components of the increments are defined by

$\delta\bm{\phi}_{\perp}=\delta\bm{\phi}-(\delta\bm{\phi}\cdot\hat{\bm{z}})\hat{\bm{z}},$
(5)

where, $\hat{\bm{z}}=\bm{B_{\ell}}/|\bm{B_{\ell}}|$, is a unit vector in the
direction of the localy-defined scale-dependent background magentic field
estimated as

$\displaystyle\bm{B}_{\ell}={}$
$\displaystyle[\bm{B}(\bm{r}-2\bm{\ell})+4\bm{B}(\bm{r}-\bm{\ell})+6\bm{B}(\bm{r})$
(6)
$\displaystyle\qquad+4\bm{B}(\bm{r}+\bm{\ell})+\bm{B}(\bm{r}+2\bm{\ell})]/16.$

For our analysis, we evaluated both the 2-point and 5-point increment methods.
Across the range of scales considered, both methods yielded similar results.
However, the 2-point method has been shown to produce erroneous results at
scales where the turbulent cascade leads to steep scalings (Sioulas et al.,
2024). Therefore, for consistency with our previous and forthcoming studies,
we have chosen to proceed with the 5-point increments approach.

Figure 2: (a) $\sin(\theta_{\perp}^{\bm{z}})$, (b)
$\sin(\theta_{\perp}^{\bm{ub}})$. Each subplot categorizes alignment angles
into 50 bins based on $\mathcal{I}_{\bm{z^{o}}}$, with the bins ranging from
the lowest PVI values (black) to the highest PVI values (red), as shown in
panel (c). Reference lines depict the scaling parameter
$\theta\propto\lambda^{\alpha}$, facilitating comparative analysis.

###  3.2 Alignment Angles

Conventionally, the alignment angle between the perpendicular components of
field increments, $\delta\bm{\phi}_{\perp}$ and $\delta\bm{\psi}_{\perp}$, is
estimated using:

$\sin(\theta^{\phi\psi}_{\perp})=\left\langle\frac{|\delta\bm{\phi}_{\perp}\times\delta\bm{\psi}_{\perp}|}{|\delta\bm{\phi}_{\perp}||\delta\bm{\psi}_{\perp}|}\right\rangle.$
(7)

This definition can be applied to estimate the angle between
$\delta\bm{u}_{\perp}$ and $\delta\bm{b}_{\perp}$, denoted as
$\theta_{\perp}^{ub}$, as well as the angle $\theta_{\perp}^{z}$, which is
between the perpendicular components of the increments in the outwardly and
inwardly propagating modes:

$\delta\bm{z}_{\perp}^{o,i}=\delta\bm{v}_{\perp}\mp\text{sign}(\bm{B}^{r}_{\ell})\delta\bm{b}_{\perp},$
(8)

where, $\bm{B}^{r}_{\ell}$ is the radial component of $\bm{B}_{\ell}$, in RTN
coordinates (Franz & Harper, 2002). The local average of $B^{r}$ assists in
determining the polarity of the radial magnetic field (Shi et al., 2021). It
is worth noting that we consider data in the GSE coordinate system; therefore,
we use $B^{r}=-B^{x}$.

An alternative definition of these angles is obtained by separately averaging
the numerator and denominator:

$\sin(\tilde{\theta}^{\phi\psi}_{\perp})=\frac{\langle|\delta\bm{\phi}_{\perp}\times\delta\bm{\psi}_{\perp}|\rangle}{\langle|\delta\bm{\phi}_{\perp}||\delta\bm{\psi}_{\perp}|\rangle}.$
(9)

This approach allows for the identification of “dynamically relevant”
fluctuations, as the averaging procedure considers that, at a given $\lambda$,
fluctuations with amplitudes near the RMS value are the primary contributors
to turbulent dynamics (Mason et al., 2006).

The CSM15 model predicts $\tilde{\theta}_{\perp}^{z}\propto\lambda^{0.10}$,
and $\tilde{\theta}_{\perp}^{ub}\propto\lambda^{0.21}$. For simplicity, in
subsequent discussions, when referring to either definition (Equations 7 and
9), $\Theta^{ub}$ will represent magnetic-velocity alignment, and $\Theta^{z}$
will represent Elsässer alignment angles.

###  3.3 Quantifying the Influence of Intermittency and Compressibility on
SDDA

To assess the influence of intermittency and compressibility on alignment
angle measurements, we utilize scale- and time-dependent proxies for these
properties. Employing a scale-dependent conditional averaging technique
enables us to isolate specific types of fluctuations, thus focusing on
particular properties of interest.

To identify coherent structures, we employ the Partial Variance of Increments
(PVI) method, which is effective for detecting sharp gradients within a
turbulent field (Greco et al., 2008; Servidio et al., 2012). The scale-
dependent PVI time series can be estimated as follows:

$\mathcal{I}_{\bm{q}}\ =\
\frac{|\delta\bm{q}(t,\ell)|}{\sqrt{\langle|\delta\bm{q}(t,\ell)|^{2}\rangle}_{\tau}},$
(10)

where $\delta\bm{q}$ is defined by Equation 3, and $\langle...\rangle_{\tau}$
denotes the average over a $\tau=60~{}\text{min}$ window (Greco et al., 2018).

To quantify the compressibility of fluctuations, we utilize

$n_{\bm{q}}\
=\left[\frac{\delta|\bm{q}(t,~{}\ell)|}{|\delta\bm{q}(t,~{}\ell)|}\right]^{2}.$
(11)

To conduct our analysis, we obtain $\delta\bm{b}_{\perp}$ and
$\delta\bm{v}_{\perp}$ using the method detailed in Section 3.1. Additionally,
we compute $\mathcal{I}_{\bm{q}}$ and $n_{\bm{q}}$ using equations 10 and 11,
respectively, for a range of $\ell$ values, specifically $\ell=1.2^{j}d_{i}$,
where $j=15,...,80$.

Here, $d_{i}$ represents the ion inertial length, defined as
$d_{i}={V_{A}}/{\Omega_{i}}$, where $\Omega_{i}={e|B|}/{m_{p}}$, is the proton
gyrofrequency, $e$ is the elementary charge, $|B|$ is the magnetic field
magnitude.

Subsequently, we employ a scale-dependent conditional binning technique. This
process entails partitioning $\mathcal{I}_{\bm{q}}$, $n_{\bm{q}}$, and their
corresponding alignment angle values into $N=50$ consecutive bins for each
scale $\ell$. More precisely, the $i$-th bin at scale $\ell$ is defined as the
interval $[2(i-1),~{}2i)$, where $i=1,...,50$. The results of this analysis
are presented in Section 4.1.

###  3.4 Quantifying Imbalance Effects on SDDA

The total energy $E_{t}=E^{+}+E^{-}$ and cross-helicity $H_{c}=E^{+}-E^{-}$,
expressed in terms of the energy associated with fluctuations in
$\bm{z}^{\pm}$, $E^{\pm}=\langle|\delta\bm{z}^{\pm}|^{2}\rangle/4$, are ideal
(i.e., with zero viscosity and resistivity) invariants of the RMHD equations.
Elsässer imbalance can be quantified by the normalized cross-helicity,
$\sigma_{c}=H_{c}/E_{t}$, which measures the relative fluxes of
counterpropagating wavepackets in the system. In the context of solar wind
turbulence, Elsässer imbalance is assessed by examining the relative
magnitudes of inwardly and outwardly propagating Alfvén waves (Velli et al.,
1991; Velli, 1993).

$\sigma_{c}=\frac{|\delta\bm{z}^{o}_{\perp}(t,~{}\ell)|^{2}-|\delta\bm{z}^{i}_{\perp}(t,~{}\ell)|^{2}}{|\delta\bm{z}^{o}_{\perp}(t,~{}\ell)|^{2}+|\delta\bm{z}^{i}_{\perp}(t,~{}\ell)|^{2}}.$
(12)

In addition, we consider the normalized residual energy, $\sigma_{r}$, to
investigate the effects of Alfvénic imbalance. This metric evaluates the
relative energy in kinetic and magnetic fluctuations:

$\sigma_{r}=\frac{|\delta\bm{v}_{\perp}(t,~{}\ell)|^{2}-|\delta\bm{b}_{\perp}(t,~{}\ell)|^{2}}{|\delta\bm{v}_{\perp}(t,~{}\ell)|^{2}+|\delta\bm{b}_{\perp}(t,~{}\ell)|^{2}}.$
(13)

Figure 3: (a) $\sin(\theta_{\perp}^{\bm{z}})$, (b)
$\sin(\theta_{\perp}^{\bm{ub}})$. Each subplot categorizes alignment angles
into 50 bins based on magnetic compressibility, $n_{\bm{B}}$, with the bins
ranging from the lowest magnetic compressibility (black) to the highest
magnetic compressibility (red).

##  4 Results

To understand the influence of imbalance, we opted for an approach different
from the one presented in Section 3.3. This decision was based on the results
presented Appendix .2. More specifically, among the variables $\sigma_{c}$,
$\sigma_{r}$, $\theta^{ub}_{\perp}$, and $\theta^{z}_{\perp}$, only two are
independent. Thus, using the previously described binning method would merely
replicate the relationships shown in Figure 13. However, a more pressing
question remains: how does global-scale imbalance affect SDDA measurements?

To address global imbalance, we adhered to the following methodology: We
computed the average values of $\sigma_{c}(\ell^{\ast})$ and
$\sigma_{r}(\ell^{\ast})$ for each interval identified by visual inspection,
where $\ell^{\ast}=10^{4}~{}d_{i}$.

Following this, we segregate our intervals into 10 linearly spaced bins,
ranging from [0, 1] for Elsässer imbalance and [-1, 0] for Alfvénic imbalance,
based on the values of $|\sigma_{c}(\ell_{\ast})|$ and
$\sigma_{r}(\ell^{\ast})$, respectively. Within each bin, we calculate a
scale-dependent weighted mean of the alignment angle, using the number of
samples in each interval at that scale as the weighting factor. The results of
this analysis are discussed in Section 4.3.

In this section, our aim is to assess the impact of intermittency,
compressibility, and global imbalance on SDDA measurements. However, before
presenting our findings, it is essential to address a key aspect of our
analysis.

We first investigated the scaling of SDDA for each stream type (slow Alfvénic,
fast Alfvénic, slow non-Alfvénic) separately. Then, we aggregated all
homogeneous intervals and applied the conditional averaging method outlined in
Section 3.3. Interestingly, the trends observed when analyzing stream types
with similar characteristics separately persisted when considering a mix of
various interval types. This observation could be attributed to the inherently
diverse nature of fluctuations across all wind streams. For instance,
intervals primarily characterized as incompressible may still contain a
fraction of compressible fluctuations. Similarly, wind streams classified as
generally balanced may exhibit localized patches of imbalance (Matthaeus et
al., 2008a; Perez & Boldyrev, 2009; Chen et al., 2013b). The use of
conditional averaging enables us to concentrate on specific types of
fluctuations, facilitating an investigation into particular properties of
interest, such as segregating fluctuations based on scale-dependent proxies
for compressibility and intermittency.

Finally, it is important to point out that the conditional averaging analysis
utilized here only considers normalized quantities like $\theta_{\perp}$,
$I_{\bm{\xi}}$, and $n_{\bm{\xi}}$. Therefore, at this stage, we did not
consider $\tilde{\theta}_{\perp}$, Equation 9. This is because it entails
averaging non-normalized quantities, and extreme values among different
intervals, owing to significant variations in the root mean square of the
fluctuations, could potentially dominate the mean. We will explore the scaling
of the alignment angle as defined in Equation 9 in Section 4.3, where weighted
averages of the alignment angles among homogeneous intervals are considered,
ensuring that the previously mentioned concern does not hinder our analysis.

Figure 4: (a) $\sigma_{r}$. (b) $\sigma_{c}$. (c) $n_{B}$ as a function of
$\lambda$ and $\mathcal{I}_{\bm{B}}$. In panel (b), contours indicate the
count levels of the distribution. Additionally, panel (b) displays
$\mathcal{F}(\mathcal{I}_{\bm{B}}^{4})^{1/4}$, where $\mathcal{F}$ represents
either the mean (black line) or median operator (red line).

###  4.1 Influence of Intermittency & Compressibility on measurements of SDDA

In this section, we delve into the effects of intermittency and
compressibility on SDDA using the conditional averaging method described in
Section 3.3.

Figure 2 presents the results for intervals of mixed streams. The first two
columns show $\theta_{\perp}^{\bm{z}}$ and $\theta_{\perp}^{\bm{ub}}$,
conditioned on $\mathcal{I}_{\xi}$ with $\xi=\bm{z}^{o}$, plotted against
scales normalized in $d_{i}$ units. Although analyses for
$\xi=\bm{z}^{i},\bm{B},\bm{V}$ were also conducted, they are not shown here;
key findings are discussed below.

Figure 2 clearly illustrates that the scaling of SDDA varies significantly
depending on the percentile bin considered. At any given scale, an inverse
relationship exists between alignment angles and the intensity of field
gradients. More specifically, curves derived from averaging alignment angles
for fluctuations with weak field gradients (black lines) tend to exhibit flat
profiles, indicating negligible or weak SDDA at smaller scales. In contrast,
curves for fluctuations with the strongest $\mathcal{I}_{\bm{z}^{o}}$ indices
show steeper scaling laws, suggesting increasing alignment down to
$\lambda\approx 8\cdot 10^{2}d_{i}$.

While the anticorrelation holds in both cases, it is apparent that alignment
in $\theta_{\perp}^{ub}$ is consistently tighter compared to
$\theta_{\perp}^{z}$. This is likely because the intervals considered here are
slightly skewed towards the globally Alfvénic side, as shown in Figure 1,
primarily because such intervals were prioritized during the selection
process. This was the only notable difference observed between the analyses of
“pure” and “mixed” intervals. This effect will become more pronounced in
Section 4.3, where we demonstrate that $\theta_{\perp}^{ub}$ and
$\theta_{\perp}^{z}$ exhibit very similar behavior in balanced intervals, but
they tend to diverge, with the former showing tighter alignment as the
imbalance increases. Nonetheless, this does not affect the conclusions of this
analysis. The primary objective here is to illustrate the relationship between
alignment angles and field gradients, which, as explained earlier, shows
consistent behavior in all cases.

For a more straightforward comparison with the numerical simulations by Mallet
et al. (2015), we investigated the relationship between the alignment angle
and the normalized Elsässer fluctuation magnitudes,
$|\delta\bm{z^{o}}|/V_{A}$. The results, not shown here, reveal a trend
qualitatively similar to those in Figure 2, as expected since the PVI
diagnostics provide an estimate of fluctuation gradient intensity normalized
by the standard deviation of local gradients.

We proceeded to segregate fluctuations into percentile bins based on the
compressibility diagnostic, $n_{\xi}$, as detailed in Section 3. Our findings,
illustrated in Figure 3, reveal a strong correlation between alignment and
magnetic compressibility, $n_{\bm{B}}$. Specifically, at the outer scale,
strongly compressible fluctuations exhibit no signs of increasing dynamic
alignment; the scaling of SDDA becomes progressively steeper as $n_{B}$
decreases. Conditioning on $n_{\bm{V}}$ reveals only a weak and statistically
insignificant correlation with SDDA, not shown here. It is worth noting,
however, that the compressibility in the velocity field is considerably
stronger than that in the magnetic field, with $n_{\bm{V}}(\ell)\gg
n_{\bm{B}}(\ell)$ for the majority of intervals considered.

In general, curves derived from magnetically incompressible fluctuations
exhibit behaviors at outer scales qualitatively similar to those associated
with the most intense coherent structures, suggesting a strong correlation
between compressibility and intermittency (see also Vasko et al., 2022;
Lotekar et al., 2022). However, unlike Figure 2, increasing misalignment is
observed in most cases at $\lambda\lesssim 10^{4}d_{i}$, which roughly
coincides with the large-scale break separating the energy-injection range
from the inertial range of the magnetic field power spectrum, usually
associated with $f^{-1}$ and $f^{\alpha}$, $\alpha\in[-5/3,-3/2]$ scalings,
respectively (Bruno & Carbone, 2013, and references therein).

To quantify our findings, we applied a power-law fit to the curves
representing the top 5% of fluctuations measured by $|\delta\bm{z}^{\pm}|$ and
$\mathcal{I}_{\xi}$, with $\xi=\bm{z}^{o},\bm{z}^{i},\bm{B},\bm{V}$, and to
the bottom 5% in terms of $n_{\bm{\xi}}$ for $\xi=\bm{B},\bm{V}$. The derived
scalings, evaluated over the range $4\times 10^{4}-4\times 10^{5}d_{i}$, are
summarized in Table 1.

###  4.2 Nature of fluctuations

In this section, we aim to better understand the nature of fluctuations at
energy-injection and inertial scales. Figure 4 illustrates (a) $\sigma_{r}$,
(b) $\sigma_{c}$, and (c) $n_{B}$ as functions of $I_{\bm{B}}$ and $\lambda$.
Additionally, panel (b) includes a contour plot of the $I_{\bm{B}}$.

The contours of $I_{\bm{B}}$ show a slight downward trend at inertial scales,
indicating a decrease in the median PVI index towards smaller scales. This may
seem counterintuitive given in-situ observations of non-Gaussian increment
distributions at smaller scales (see e.g., Sorriso-Valvo et al., 1999).
However, examining $\mathcal{F}(\mathcal{I}_{\bm{B}}^{n})^{1/n}$ for
$n=2,\ldots,6$, where $\mathcal{F}$ represents either the mean (black line) or
median operator (red line), reveals that while the median curve trends
downward, the mean curve trends upward at scales $\lambda\leq 10^{4}d_{i}$.
The steepness of the mean curve increases with $n$, reflecting intermittency
in the magnetic field time series. For $n=1$, the average is influenced by
median-amplitude eddies, while for larger $n$ values, rare large-amplitude
eddies dominate. The scale-dependent behavior of both the mean and median
suggests a trend towards distributions exhibiting progressively heavier tails
at smaller scales, consistent with previous findings (Sorriso-Valvo et al.,
1999; Chhiber et al., 2021).

Table 1: Power-law scaling of alignment angles within $4\times 10^{4}-4\times
10^{5}\,d_{i}$ using top 5% fluctuations in $|\delta\bm{z}^{\pm}|$ and
$\mathcal{I}_{\xi}$, where $\xi=\\{\bm{B},\bm{V},\bm{z}^{o},\bm{z}^{i}\\}$,
and bottom 5% in $n_{\bm{\xi}}$ for $\xi=\\{\bm{B},\bm{V}\\}$.

| $\theta^{z}_{\perp}$ | $\tilde{\theta}_{\perp}^{z}$ | $\theta^{ub}_{\perp}$ | $\tilde{\theta}_{\perp}^{ub}$
---|---|---|---|---
$\mathcal{I}_{\bm{z}^{o}}$ | $0.3\pm 0.02$ | $0.32\pm 0.02$ | $0.29\pm 0.02$ | $0.33\pm 0.01$
$\mathcal{I}_{\bm{z}^{i}}$ | $0.3\pm 0.02$ | $0.28\pm 0.03$ | $0.30\pm 0.02$ | $0.35\pm 0.04$
$\mathcal{I}_{\bm{B}}$ | $0.30\pm 0.01$ | $0.30\pm 0.02$ | $0.27\pm 0.01$ | $0.31\pm 0.01$
$\mathcal{I}_{\bm{V}}$ | $0.23\pm 0.01$ | $0.25\pm 0.02$ | $0.29\pm 0.02$ | $0.30\pm 0.01$
$n_{\bm{B}}$ | $0.19\pm 0.01$ | $0.26\pm 0.02$ | $0.25\pm 0.03$ | $0.32\pm 0.03$
$n_{\bm{V}}$ | $0.17\pm 0.03$ | $0.25\pm 0.03$ | $0.33\pm 0.04$ | $0.38\pm 0.02$

Conversely, the roughly scale-independent behavior observed in both the mean
and median curves at larger scales reflects the well-established phenomenon of
Gaussian distributions of fluctuations at energy injection scales (Sorriso-
Valvo et al., 1999; Bruno et al., 2003).

Interestingly, the transition from intermittent to Gaussian regimes appears to
coincide with the scale at which we typically observe the switch from
increasing alignment to increasing misalignment, as shown in Figure 3.
However, the impact of this regime change on the scaling of SDDA remains
unclear.

Furthermore, we observe that at any given scale, fluctuations with the highest
PVI index exhibit the lowest $n_{B}$ and highest $\sigma_{c}$ values, along
with strongly negative residual energy indices. This implies an
anticorrelation between $\sigma_{r}$ and $I_{\bm{B}}$, consistent with the
findings of Bowen et al. (2018c), who noted that intervals with large scale-
dependent kurtosis, $K(\ell)$, predominantly exhibit negative $\sigma_{r}$.
Our analysis confirms the persistence of this relationship across the observed
spectrum, substantiating the claim that intermittency and residual energy are
most likely interconnected.

Finally, we turn our attention to the nature of the compressible fluctuations.
To this end, we consider the PVI diagnostic applied to the magnetic field
magnitude and proton density time series, $I_{|\bm{B}|}$ and $I_{n_{p}}$,
respectively. Figure 5 illustrates $I_{n_{p}}$ as a function of $I_{|B|}$ and
scale $\lambda$. This method of visualization has not been previously
implemented. Earlier works have compared the magnetic and thermal pressure
(Burlaga & Ogilvie, 1970; Thieme et al., 1990), used wavelet cross-coherence
analysis (Kellogg & Horbury, 2005; Yao et al., 2011), or utilized the zero-lag
cross-correlation between $|B|$ and $n_{p}$ (Howes et al., 2012; Bowen et al.,
2018a).

A well-established result is confirmed: fluctuations in $|B|$ and $n_{p}$ are
anticorrelated at inertial scales, similar to observations by Thieme et al.
(1990); Yao et al. (2011); Howes et al. (2012); Bowen et al. (2018a),
attributed to either the presence of the kinetic slow-mode (Howes et al.,
2012; Klein et al., 2012) or the MHD slow-mode (Verscharen et al., 2017).
Focusing on the strongly compressible fluctuations, we observe that the
anticorrelation persists at the outer scale, $\lambda\gtrsim 10^{4}d_{i}$,
with a tendency for both $I_{n_{p}}$ and $I_{|B|}$ to predominantly obtain
negative values, indicating a preference for magnetic field magnitude
depressions accompanied by a decrease in proton density. Finally, at even
larger scales, $\lambda\geq 5\times 10^{5}d_{i}$, a rough correlation is
observed, consistent with the results of Burlaga & Ogilvie (1970).

It is not clear whether this observation can provide any insights into the
effects of compressible fluctuations on SDDA. Nevertheless, we present this
finding for completeness without further discussion.

Figure 5: $I_{n_{p}}$, as a function of $I_{|B|}$ and scale $\lambda$
(normalized to the ion inertial length $d_{i}$). The plot utilizes a color
gradient to represent the mean values of $I_{n_{p}}$ within each bin. Black
and red dots show the mean and median values of $I_{|B|}$ as a function of
scale. Figure 6: Weighted averages of (a)
$\sin(\tilde{\theta}_{\perp}^{\bm{z}})$, (b)
$\sin(\tilde{\theta}_{\perp}^{\bm{ub}})$, and (c) $\sigma_{c}$ for the
homogeneous intervals identified through visual inspection. After estimating
these quantities as a function of scale for each identified interval, the
curves are divided into $N=10$ bins based on $\sigma_{c}(\ell^{\ast})$, where
$\ell^{\ast}=2\cdot 10^{4}d_{i}$, and a scale-dependent weighted average of
the curves is computed.

###  4.3 Influence of Imbalance on SDDA

In this section, we seek to understand the influence of global Alfvénic and
Elsässer imbalance on alignment angle scaling using homogeneous intervals
identified through visual inspection (see Section 2). To this end we employ
the conditional averaging technique described in Section 3.4, and segregate
intervals based on the outer scale values of either the cross-helicity or
normalized residual energy $\sigma_{c}(\ell^{\ast})$ and
$\sigma_{r}(\ell^{\ast})$, where $\ell^{\ast}=10^{4}d_{i}$.

We first consider alignment curves conditioned on $\sigma_{c}(\ell^{\ast})$.
The results are shown in Figure 6. For $\sigma_{c}(\ell^{\ast})<0.7$, there is
no clear trend in the scaling of $\tilde{\theta}_{\perp}^{z}$, as curves from
all bins overlap. However, for $\sigma_{c}(\ell^{\ast})>0.7$, the scaling of
$\tilde{\theta}_{\perp}^{z}$ becomes progressively shallower, especially at
larger scales. In contrast, large-scale Elsässer imbalance organizes the
$\tilde{\theta}_{\perp}^{ub}$ curves more clearly. As
$\sigma_{c}(\ell^{\ast})$ increases, (a) $\tilde{\theta}_{\perp}^{ub}$ shows
steeper scaling, and (b) the rollover of the curves, i.e., the range at which
increasing misalignment is observed, shifts towards larger scales. The local
scaling index, $\alpha(\lambda)$, illustrated in the inset of Figure 6b, shows
that $\alpha(\lambda)\approx 0.4$ for $\sigma_{c}(\ell^{\ast})>0.90$, while a
shallower scaling of $\alpha(\lambda)\approx 0.2$ is observed for
$\sigma_{c}(\ell^{\ast})<0.10$. The range of scales over which the slope
remains positive increases as $\sigma_{c}(\ell^{\ast})\rightarrow 0$.

It is worth noting that while alignment in $\tilde{\theta}_{\perp}^{ub}$ is
considerably tighter compared to $\tilde{\theta}_{\perp}^{z}$ for strongly
imbalanced intervals, the two definitions yield qualitatively similar results
at $\sigma_{c}(\ell^{\ast})\rightarrow 0$.

When intervals are segregated based on $\sigma_{r}(\ell^{\ast})$, a similar
trend is observed for $\tilde{\theta}_{\perp}^{ub}$. Additionally, as Alfvénic
imbalance decreases ($\sigma_{r}(\ell^{\ast})\rightarrow 0$), SDDA signatures
in $\tilde{\theta}_{\perp}^{z}$ diminish, resulting in flatter curves between
$10^{2}-10^{6}d_{i}$. Conversely, with increasing magnetic energy dominance
over kinetic energy, SDDA signatures become more pronounced, exhibiting steep
power-law behavior, particularly at larger scales. In practice, this implies
that equipartitioned intervals ($E_{u}\approx E_{b}$) show no signs of
increasing alignment in the Elsässer field fluctuations, but at least at large
scales still show increasing alignment in the polarizations of
magnetic/velocity field fluctuations.

To summarize: (a) Large-scale Alfvénic imbalance is a good predictor for the
expected behavior of both $\tilde{\theta}_{\perp}^{ub}$ and
$\tilde{\theta}_{\perp}^{z}$. As $\sigma_{r}(\ell^{\ast})$ approaches zero,
the scalings of $\tilde{\theta}_{\perp}^{ub}$ become steeper, while the
scalings of $\tilde{\theta}_{\perp}^{z}$ become shallower, extending to a
narrower and wider range of scales, respectively. (b) Elsässer imbalance
mostly correlates with the $\tilde{\theta}_{\perp}^{ub}$ scaling. SDDA
signatures persist to smaller scales for globally balanced streams compared to
their imbalanced counterparts, but with the scaling becoming progressively
shallower. Is it worth noting that the persistence of SDDA signatures to
smaller scales is consistent with noise being responsible for misalignment at
small scales. That is because the globally imbalanced streams reach small
alignment angles at larger scales and so become affected by noise at larger
scales.

Figure 7: Weighted averages of (a) $\sin(\tilde{\theta}_{\perp}^{\bm{z}})$,
(b) $\sin(\tilde{\theta}_{\perp}^{\bm{ub}})$, and (c) $\sigma_{r}$ for the
homogeneous intervals identified through visual inspection. After estimating
these quantities as a function of scale for each identified interval, the
curves are divided into $N=10$ bins based on $\sigma_{r}(\ell^{\ast})$, where
$\ell^{\ast}=2\cdot 10^{4}d_{i}$, and a scale-dependent weighted average of
the curves is computed.

###  4.4 High-Frequency Instrumental Noise

In this section, we investigate the extent to which high-frequency, small-
amplitude noise in the velocity field measurements can affect our ability to
reliably estimate alignment angles in the Elsässer field fluctuations,
$\theta_{\perp}^{z}$.

Let $\delta\bm{b}$ denote the true value of the magnetic field fluctuation
vector in velocity units. We assume that the error in its measurement is
negligible. Although a more realistic assessment would incorporate an error
term, especially since $\delta\bm{b}$ measurements may be contaminated by
errors in proton density measurements, we assume for highly Alfvénic intervals
that $\delta\rho/\rho$ is sufficiently small. This assumption is supported by
observations (Shi et al., 2021), allowing rolling averages of $\rho$ to be
used for the Alfvénic normalization.

We define the real value of the velocity field as $\delta\bm{u}$ and introduce
an error term $\bm{\Delta}$, where $\bm{\Delta}$ is a normally distributed
random vector with mean $\bm{0}$ and standard deviation $\sigma$, represented
by $\bm{\Delta}\sim\mathcal{N}(\bm{0},\sigma)$. We assume that the magnitude
of $\bm{\Delta}$ is significantly smaller than the magnitudes of
$\delta\bm{b}$ and $\delta\bm{u}$ (i.e.,
$|\bm{\Delta}|\ll|\delta\bm{b}|,|\delta\bm{u}|$). For simplicity, we assume
that the real vectors are perfectly aligned,
$\delta\bm{u}=\alpha\delta\bm{b}$, and that
$\delta\bm{z}_{i}=\epsilon\delta\bm{z}_{o}$.

We can then write:

$\delta\bm{z}^{\prime}_{o}=\delta\bm{z}_{o}+\bm{\Delta}$ (14)

$\delta\bm{z}_{i}^{\prime}=\epsilon\delta\bm{z}_{o}+\bm{\Delta}.$ (15)

The quantity controlling the error in the measurements of $\delta\bm{z}_{i}$
can be represented by the dimensionless parameter $Q$:

$Q=\frac{\epsilon\delta z_{o}}{\Delta}$ (16)

We can estimate the alignment angles as follows:

$\displaystyle\cos(\theta^{\prime})$
$\displaystyle=\frac{\delta\bm{z}_{o}^{\prime}\cdot\delta\bm{z}_{i}^{\prime}}{|\delta
z_{o}^{\prime}||\delta z_{i}^{\prime}|}$ $\displaystyle=\frac{\epsilon\delta
z_{o}^{2}+(1+\epsilon)\delta\bm{z}_{o}\cdot\bm{\Delta}+\Delta^{2}}{\epsilon\delta
z_{o}^{2}\sqrt{1+\frac{2\delta\bm{z}_{o}\cdot\bm{\Delta}}{\delta
z_{o}^{2}}+\frac{\Delta^{2}}{z_{o}^{2}}}\sqrt{1+\frac{2\delta\bm{z}_{o}\cdot\bm{\Delta}}{\epsilon\delta
z_{o}^{2}}+\frac{\Delta^{2}}{{\epsilon}^{2}\delta z_{o}^{2}}}}$

Considering the case where $Q\gg 1$, we can estimate:

$\cos(\theta^{\prime})\approx 1-O(Q^{-1})$ (17)

On the other hand, when $Q\ll 1$:

$\cos(\theta^{\prime})\approx\delta\bm{z}^{o}\cdot\bm{\Delta}/(\delta
z^{o}\Delta).$ (18)

Therefore, the accuracy of the angle measurement depends on the errors in the
velocity compared to the true size of the $\delta z_{i}$ signal. Specifically,
when $Q\gg 1$, the alignment angle can be estimated reliably. Conversely, for
$Q\ll 1$, a significant deviation is expected between the true and estimated
values of the alignment angles.

###  4.5 Modeling High Frequency Instrumental Noise: Perfectly aligned fields.

To further investigate the impact of high-frequency noise on alignment angle
measurements, we simulate the previous analysis by generating synthetic
magnetic and velocity field data, subsequently polluted by artificial high-
frequency white noise of specified amplitude. The synthetic time series, power
spectra, and alignment angle estimates are obtained following the method
described in Appendix .3.

Figure 8: Synthetic power-spectra of (a) the velocity field ($P_{\bm{v}}$) and
(b) the minor Elsässer field ($P_{\bm{z}_{i}}$) for different levels of
imbalance. Dotted lines indicate the original time series, while solid lines
represent the time series after adding white noise. Panel (c) shows the
alignment angle of the Elsässer/Alfvén fields for the original time series
(dotted lines), which remain constant at $\theta^{z}=180^{\circ}$,
$\theta^{ub}=0^{\circ}$, respectively, independent of scale, and for the
noise-affected time series (solid lines). The insets in panels (a), (b)
illustrate $\sigma_{c}$, $\sigma_{r}$ for the different runs, respectively.

The results of this analysis are illustrated in Figure 8. Panels (a)-(c)
display the power-spectral density of the velocity field $P_{\bm{v}}$, the
ingoing Elsässer fields $P_{z_{i}}$, and the alignment angles
$\theta_{\perp}^{z}$ and $\theta_{\perp}^{ub}$, respectively. The insets in
panels (a) and (b) show $\sigma_{r}$ and $\sigma_{c}$. Different colors
represent varying levels of Elsässer imbalance, with red lines indicating
imbalanced intervals and black curves showing balanced intervals. Dotted lines
represent the expected behavior, while solid lines show the impact of
introducing white noise into the velocity-field time series.

Introducing noise causes $P_{\bm{v}}$ to flatten as $kd_{i}$ increases beyond
a critical value $k_{n}d_{i}\approx 10^{-2}$ for the lowest cross-helicity
intervals, as shown in Figure 8(a). As $\sigma_{c}\rightarrow 1$,
$\bm{V}_{RMS}$ increases and becomes comparable to $\bm{B}_{RMS}$. This
results in a decrease in $\bm{z}_{i,\text{RMS}}$, leading to an artificial
flattening of the power spectrum of $\bm{z}_{i}$ at progressively smaller
scales. Consequently, in the highest $\sigma_{c}$ bin, $P_{z_{i}}$ flattens at
a $kd_{i}\ll k_{n}d_{i}$.

When $P_{\bm{v}}$ reaches the noise floor at $k_{n}d_{i}$, the power spectrum
of $\bm{z}_{i}$ becomes unmeasurable for $P_{z_{i}}(kd_{i}>k^{\ast}d_{i})$,
where $P_{z_{i}}(k^{\ast}d_{i})=P_{u}(k_{n}d_{i})$. For the balanced intervals
(black curves), although $P_{\bm{v}}$ is flattened at smaller scales, the
impact on $P_{z_{i}}$ is subtle and mainly evident at large scales.

Although our assumption that the noise floor is independent of imbalance might
be an oversimplification, our analysis demonstrates that small amplitude high-
frequency noise can significantly affect $P_{z_{i}}$, thereby reducing the
accuracy of $P_{z_{i}}$ estimates as the Elsässer imbalance increases.

Noise has an even greater impact on alignment angle estimates. It distorts the
phases of the velocity field measurements, causing them to decorrelate from
the magnetic field phases without significantly altering their amplitude. The
extent and scale at which these effects become significant depend on the level
of imbalance: $\theta_{\perp}^{z}$ is more affected during imbalanced
intervals, while $\theta_{\perp}^{ub}$ is more affected during balanced
intervals. This can be attributed to the fact that the RMS of the velocity
field fluctuations is considerably lower for balanced intervals (see, e.g.,
Figure 14), and thus the noise floor is reached at smaller scales.

It is important to note that by imposing perfect alignment between magnetic
and velocity field fluctuations, simulating different levels of imbalance
requires the RMS of the velocity field fluctuations to decrease faster than
that of the magnetic field fluctuations for $\sigma_{c}$ to decrease. The
opposite scenario could also be considered, but that would result in positive
$\sigma_{r}$, which is not usually observed in the solar wind (Bruno &
Carbone, 2013, and references therein). With this assumption, we have
effectively restricted our analysis to the periphery of the circle plot shown
in Figure 13. However, this is not always the case; in principle, we could
maintain the RMS of both fields constant and adjust the alignment angles or
use a combination of these factors. Given that the ratio of the RMS in the
synthetic time series is informed by our in-situ observations, while the
correlation in the fluctuations of the fields is not, our approach does not
allow $\sigma_{c}$ at large scales to decrease below $\sigma_{c}\lesssim 0.45$
due to our assumption of perfect alignment.

###  4.6 Modeling High Frequency Instrumental Noise: Correlation coefficient
informed by in-situ observations.

To address the simplifications made in Section 4.5, we generate time series
for the magnetic and velocity fields that satisfy three simultaneous
conditions based on our in-situ observations: (1) the fluctuations are
correlated with a specified correlation coefficient, (2) their inertial-range
spectral scaling matches the observed scaling, and (3) the RMS values of these
time series align with those observed in-situ. A detailed description of the
steps followed to generate the synthetic data is presentd in Appendix .4.

The results of this analysis are presented in Figure 9, with a setup identical
to Figure 8. In addition the inset of panel (c) illustrates the normalized
diffrence of the alignemnt angle estimates of the clean and noise-affected
timeseries. While the exact spectral scaling of the different fields does not
make a big difference in the observed behavior, not shown here, allowing the
correlation coefficient, to vary in a manner consitent with in-situ data,
enables for a more realistic comparison between the synthetic and in-situ
data.

To further clarify this aspect, we estimate the band-pass RMS value of the
different fields within a specified $k^{\ast}=kd_{i}$, range from the power
spectral density through:

$\mathcal{J}_{\bm{\phi}}=\sqrt{\int_{k^{\ast}_{j}}^{k^{\ast}_{j+1}}P_{\bm{\phi}}(k^{\ast})\,dk^{\ast}}$
(19)

where, $k^{\ast}_{j}$ and $k^{\ast}_{j+1}$ denote the lower and upper bounds
of the $j$-th bin, respectively, with $k^{\ast}_{j}\in[10^{-4},4\times
10^{-1}]$. The bins are linearly spaced in logarithmic space.

The results of this analysis are presented in Figure 10. Panels (a), (b), and
(c) illustrate the variations of $\mathcal{J}_{\bm{z}_{o}}$,
$\mathcal{J}_{\bm{z}_{i}}$, and
$\mathcal{J}_{\bm{v}}/\mathcal{J}_{\bm{z}_{i}}$ as functions of $kd_{i}$ and
$\sigma_{c}$. These values are initially calculated for individual intervals
in our dataset. Afterward, the weighted mean values for each $kd_{i}$ and
$\sigma_{c}$ bin are determined, with the weights being the interval duration.

Figure 9: The setup is similar to that in Figure 8, but with the time series
generated using the empirical relationships obtained for $\rho$,
$\bm{B}_{\text{RMS}}$, $\bm{V}_{\text{RMS}}$, $a_{b}$, and $a_{u}$.
Additionally, the inset in panel (c) features the normalized difference
$D=100(\theta-\theta_{c})/\theta_{c}$, where $\theta$ and $\theta_{c}$ are the
noise-affected and clean estimates of the alignment angles, respectively.

Panel (c) additionally includes estimates of the same quantities derived from
the synthetic dataset, obtained by integrating the power-spectra shown in
Figure 9. These synthetic estimates are represented as solid lines overlaid on
the dots from the in-situ data.

The good agreement between the synthetic and in-situ data, at all scales
considered, suggests that the noise level introduced in the synthetic dataset
is roughly consistent with the quantization noise observed in the in-situ
data.

Overall, this analysis indicates that small amplitude high-frequency noise can
cause significant deviations in observed alignment angles from their true
values, even at low frequencies (large scales). However, reliable estimates of
$\sigma_{c}$ and $\sigma_{r}$ can still be obtained at higher frequencies, as
the distortion in amplitude is minimal, but the phase effects are significant.

##  5 Discussion

In recent years, the inertial-range scaling behavior of the alignment angle
has ignited intense discussions and debate. SDDA, if related to the reduction
of nonlinearities111A dedicated discussion on why this may not be the case can
be found in Bowen et al. (2021), holds the potential to flatten the inertial
range spectrum from $E(k_{\perp})\propto k_{\perp}^{-5/3}$ to
$E(k_{\perp})\propto k_{\perp}^{-3/2}$. Although the spectral exponents of
-5/3 and -3/2 are numerically close, they signify fundamentally different
physical mechanisms underlying the turbulent energy cascade. Therefore, a
thorough understanding of the inertial range behavior of SDDA is crucial,
particularly when applying MHD turbulence models to astrophysical systems with
extensive inertial ranges. For example, predictions about the turbulent
heating rate can deviate significantly between models that ingore and
incorporate SDDA (Chandran et al., 2010; Chandran & Perez, 2019).

Figure 10: (a) $\mathcal{J}_{\bm{z}_{o}}$, (b) $\mathcal{J}_{\bm{z}_{i}}$, and
(c) $\mathcal{J}_{\bm{v}}/\mathcal{J}_{\bm{z}_{i}}$ as functions of
$\sigma_{c}$. Here, $\mathcal{J}_{\bm{\phi}}$ represents the band-pass RMS of
the field $\bm{\phi}$, as defined in Equation 19. These values are initially
estimated for individual intervals in our dataset. Subsequently, the weighted
mean values at a given $kd_{i}$ and $\sigma_{c}$ bin are calculated, with the
weights being the interval duration. Panel (c) also includes the estimates of
the same quantities for the synthetic dataset obtained by integrating the
power-spectra presented in Figure 9. These are shown as solid lines overlaid
on the dots from the in-situ data.

Nonetheless, in-situ observations have cast doubt on our comprehension of the
role of SDDA in Alfvénic turbulence. While several aspects of model
incorporating SDDA (Boldyrev, 2006; Chandran et al., 2015; Mallet &
Schekochihin, 2017) are consistent with in-situ observations (Chen et al.,
2012; Verdini et al., 2018; Shi et al., 2023; Sioulas et al., 2024),
signatures of increasing alignment usually fade within the inertial range
(Podesta, 2009; Verdini et al., 2018; Parashar et al., 2019, 2020; Sioulas et
al., 2024).

It is imperative to recognize, however, that the B06 and CSM15 models omit the
potential effects of compressibility, imbalance, solar wind expansion, and
various instabilities on field alignment. These factors have been deliberately
neglected either because they were deemed negligible, given that these models
focus on homogeneous alfvénic turbulence, or to simplify computations, e.g.,
by assuming negligible cross-helicity. Nonetheless, these effects often become
significant in the solar wind, raising questions about the applicability of
these models to such conditions (Verdini et al., 2018; Bowen et al., 2021).

In this work, we have cast our analysis within a framework that allows us to
isolate and study the influence of effects such as compressibility,
intermittency, and imbalance, which have not been adequately addressed in
previous works.

Before drawing conclusions in Section 6, we review related work and discuss
the characteristics of solar wind turbulence that diverge from conventional
models of homogeneous MHD turbulence. Additionally, we evaluate how
instrumental noise might obscure the precise estimation of alignment angles in
solar wind studies.

###  5.1 Wavepackets, shearing, alignment & residual energy

We have shown that the quenching of the alignment angle typically observed at
inertial scales shifts toward smaller scales when fluctuations with strong
gradients are isolated from residual fluctuations, Figure 2a. It is well known
that the breadth of the inertial range narrows with Elsässer imbalance (Bruno
& Carbone, 2013, and references therein). Nevertheless, at 1 AU, the inertial
range begins, on average, at $\ell\approx 10^{4}d_{i}$. Combining this
observation with the results presented in Figure 2, it is clear that by
carefully thresholding based on the field gradients, SDDA signatures can still
be detected across a significant portion of the inertial range.

This latter observation is crucial because it is not the typical or median
amplitude eddy that is expected—or required for the purposes of the CSM15
model—to be strongly aligned at a given scale. Such wave packets have likely
undergone several balanced collisions, which could impede increasing
alignment. Instead, it is the “atypical,” yet “dynamically relevant,” eddies
residing at the tails of the PDFs of increments that are expected to show such
behavior for the model to work. This is indeed the case, at least down to the
point where instrumental effects likely dominate the statistics (see analysis
in Section 4.6). These wave packets are characterized by strong field
gradients, remain strongly incompressible, and have highly correlated
velocity/magnetic field fluctuations, as evidenced by the low $n_{B}$ and high
$\sigma_{c}$ indices, respectively. In this sense, our results are consistent
with the CSM15 model, although it should be noted that CSM15 did not account
for the residual energy that we and others find to be widespread in solar-wind
turbulence.222It is perhaps worth noting that although the average cross
helicity was zero in the CSM15 model, the local cross helicity was high within
the volume occupied by the large-amplitude fluctuations in the tail of the
distribution.

Taking all this into account, our observations support the following picture:
The nonlinear, intermittent dynamics perpendicular to $\bm{B}_{0}$ produce
highly aligned, strongly turbulent, current sheet structures that retain
certain properties associated with the linear-wave response, given their high
cross-helicity. Kinetic-magnetic energy equipartition is spontaneously broken
due to imbalanced wave packet collisions, even if it is present initially at
large scales, where $\delta\bm{u}\approx\pm\delta\bm{b}$. This process
culminates in the formation of incompressible current sheet structures at
smaller scales, residing at the periphery of the circle plot illustrated in
Appendix .2, a result also recovered in homogeneous and EBM simulations (see
e.g., Wang et al., 2011; Dong et al., 2014; Shi et al., 2023; Meyrand et al.,
2023). As such, our results provide observational support for the in-situ
generation of coherent structures as a by-product of the turbulent cascade. At
the same time, however, our results cannot definitively rule out the presence
of advected flux tube structures originating from the inner heliosphere
(Borovsky, 2008).

###  5.2 Dependence of SDDA measurements to Elsässer and Alfvénic imbalance

In Section 4.3, we demonstrated that the scaling of $\Theta^{z}$ and
$\Theta^{ub}$ is strongly correlated with the global Alfvénic and Elsässer
imbalance, but with some nuances.

For instance, the large-scale value of normalized cross helicity,
$\sigma_{c}(\ell^{\ast})$, with $\ell^{\ast}=2\times 10^{4}d_{i}$, shows a
strong correlation with $\Theta^{ub}$. As $\sigma_{c}(\ell^{\ast})\rightarrow
1$, the scaling of $\Theta^{ub}$ becomes steeper and is observed over a
progressively narrower range of scales. However, the scaling of $\Theta^{z}$
does not appear to change significantly with global Elsässer imbalance, at
least for $\sigma_{c}(\ell^{\ast})<0.7$. This finding is in qualitative
agreement with the incompressible, homogeneous MHD simulations by Beresnyak &
Lazarian (2009), who found that the scaling of $\Theta^{z}$ remains unaffected
by the global Elsässer imbalance.

Conversely, large-scale Alfvénic imbalance correlates with both $\Theta^{z}$
and $\Theta^{ub}$. As $\sigma_{r}(\ell^{\ast})\rightarrow 0$, the scaling of
$\Theta^{ub}$ becomes steeper and narrows in scale range; that is, the
rollover of the curves shifts to smaller scales as
$\sigma_{r}(\ell^{\ast})\rightarrow-1$ or $\sigma_{c}(\ell^{\ast})\rightarrow
0$. For $\Theta^{z}$, the scaling becomes steeper as
$\sigma_{r}(\ell^{\ast})\rightarrow-1$, but there is no clear trend in the
extent of $\Theta^{z}$ with $\sigma_{r}(\ell^{\ast})$.

While our analysis clearly indicates that the scaling of SDDA depends strongly
on the degree of large-scale imbalance in the system, the observation that the
extent over which these signatures persist might also depend on imbalance is
less convincing. It is tempting to assume that for globally imbalanced
intervals, the fields are already tightly aligned at large scales, resulting
in a strong depletion of nonlinearities. This would effectively halt the
mechanism that results in SDDA and prevent the average value of the different
alignment angles from becoming any tighter towards smaller scales. This could
lead to a saturation of the alignment angles at some minimum value, or allow
for other mechanisms that result in misalignment of the fields to become
significant, as discussed in Section 5.6.

Another possible explanation could be instrumental effects. As discussed in
Section 4.6, small uncertainties in high-frequency velocity field measurements
can result in significant uncertainty in alignment angle measurements, with
the effect becoming increasingly important as Elsässer imbalance increases.
For example, it is clear from the inset in Figure 9c that as
$\sigma_{c}\rightarrow 1$, the effects of high-frequency noise become
increasingly important at larger scales for $\Theta^{ub}$, consistent with the
results presented in Figure 7b. Thus, the dependence of alignment angle
contamination due to instrumental effects on the imbalance could provide an
alternative explanation for the observed trend.

In any case, the extent to which SDDA persists deeper into the inertial range
depending on the global imbalance in the system is an interesting area for
future research. Further investigation with better quality particle
measurements or high-resolution numerical simulations is necessary to
conclusively address this matter.

###  5.3 Sensitivity of SDDA measurements to instrumental noise

In recent years, concerns have been raised regarding the extent to which our
ability to estimate statistical quantities that depend on the sub-dominant
Elsässer mode is hampered by uncertainty in velocity field measurements due to
instrument characteristics (Podesta, 2009; Chen et al., 2012; Podesta et al.,
2009; Gogoberidze et al., 2012; Bowen et al., 2021; Sioulas et al., 2024;
Ervin et al., 2024).

For example, the velocity data from the Wind mission are quantized before
transmission back to Earth, a process that involves rounding real, rational
numbers to the nearest integer. A close inspection of the velocity field time
series shows that almost every data point has a time-discontinuity. While
these discontinuities are small—corresponding to the bit size of the data—they
induce an artificial $\propto f^{-2}$ spectrum. This high-frequency
quantization noise leads to a decoupling of velocity and magnetic field
fluctuations near the Nyquist frequency, identified as the primary reason for
the observed decrease in $\sigma_{c}$ at high frequencies (Podesta & Borovsky,
2010). Other sources of uncertainty, including aliasing and Poisson noise, are
further discussed by Chen et al. (2013a).

Since good-quality, high-frequency velocity field measurements are not
available at the moment, we have tried to quantify the uncertainty in our
measurements to assess the degree of confidence we can place in our
observations and to attribute them to physical mechanisms underlying the
turbulent cascade.

In Section 4.4, we presented a simplified theoretical analysis along with an
effort to model the effects of noise on SDDA measurements. Our analysis
indicates that despite the fact that other quantities, e.g., $P_{\bm{v}}$,
$\sigma_{c}$, or even $P_{z_{i}}$, may still be measurable for the largest
portion of the inertial range, the effects of noise on alignment angle
measurements can be dramatic even at very low frequencies. The strong
dependence on the level of Elsässer imbalance can render such measurements
impossible even at scales as large as $5-10\times 10^{3}d_{i}$. Therefore,
while we discuss other alternative mechanisms of physical origin that could
result in misalignment of the fields in subsequent sections, we believe, based
on the analysis presented in Section 4.6, that the increasing misalignment of
the fields usually observed at inertial scales is of instrumental origin.

###  5.4 Effects of compressibility in SDDA

As the cascade progresses toward smaller spatial scales, the topology and
characteristics of turbulent fluctuations can change dramatically. Various
types of coherent structures, both incompressible (such as current sheets and
Alfvén vortices) and compressible (including magnetic holes and solitons),
have been observed in different space-plasma environments (Alexandrova et al.,
2006; Rees et al., 2006; Perrone et al., 2016; Vasko et al., 2022; Vinogradov
et al., 2023). In many cases, compressive fluctuations have been shown to
significantly impact the dynamics of the turbulent solar wind (Klein et al.,
2012; Howes et al., 2012; Verscharen et al., 2017; Bowen et al., 2018b;
Chandran, 2018; Shoda et al., 2019).

To bridge the gap between in-situ observations, theoretical predictions, and
numerical evidence, we sought to quantify the effects of compressibility on
SDDA. This involved a distinct examination of the dynamics of compressible and
incompressible fluctuations to gain insights into the persistent observation
of increasing misalignment at the inertial scale of solar wind turbulence.

As anticipated, our results indicated that strongly compressible fluctuations
generally exhibit very weak to negligible signatures of increasing alignment,
as shown in Figure 3. These fluctuations can, to some extent, affect the
average scaling of SDDA when mixed with Alfvénic fluctuations. However,
compressible fluctuations are usually associated with weak field gradients and
have considerably lower amplitudes than incompressible fluctuations, Figure 4.
Therefore, their impact on the average scaling behavior is minimal, especially
when amplitude-weighted definitions of the alignment angle are considered (see
Equation 9).

In fact, though not shown here, the average behavior (i.e., without
segregating compressible from incompressible fluctuations) of the alignment
angles almost perfectly overlaps with the black (i.e., incompressible) lines
in Figure 3, indicating that the effects of compressibility on SDDA are
negligible and can thus not explain the trend of misalignment at inertial
scales.

###  5.5  Alignment at the Outer Scale: Assessing Consistency with
Homogeneous, Incompressible MHD Phenomenologies

Our findings confirm a well-established result: increasing signatures of SDDA
are observable for the majority of the intervals considered, irrespective of
associated plasma parameters, at the outer scale. However, before concluding
that the observed behavior aligns with the phenomenological models of
homogeneous MHD turbulence discussed above, another aspect, which has not been
extensively explored in the literature, warrants further discussion.

In homogeneous MHD, the turbulent cascade advances through uncorrelated
collisions between counterpropagating wave packets. However, in the stratified
solar wind, dynamics become more intricate due to linear couplings of outgoing
waves with large-scale inhomogeneities, leading to non-WKB (Wentzel-Kramers-
Brillouin) reflections. In this scenario, the Elsässer fields can be
decomposed into primary and secondary components (Velli et al., 1989, 1990;
Hollweg & Isenberg, 2007). The primary component, known as the “classical”
component $z^{i}_{c}$, travels at the characteristic phase speed
$V_{sw}-V_{a}$. The secondary, or “anomalous” component $z^{i}_{a}$, maintains
the same phase function and thus propagating properties as the forcing,
$z^{o}$. Consequently, $z^{i}_{a}$ maintains a strong correlation with
$z^{o}$, $\bm{z}^{o}\propto-\bm{z}^{i}_{a}$. Numerical demonstrations of this
anomalous coherence effect in inhomogeneous MHD turbulence have been explored
by Verdini et al. (2009) and Meyrand et al. (2023).

The nonlinear interactions between $z^{o}$ and $z^{i}_{c}$ are uncorrelated
and transient, limited to the duration of their encounters. Conversely, in the
frame of the outgoing wave, $z^{i}_{a}$ appears stationary, and the shearing
between $z^{o}$ and $z^{i}_{a}$ remains coherent over time. It is intuitive to
anticipate that as the imbalance in the fluxes of counterpropagating
wavepackets increases, the efficiency of $z^{o}$ and $z^{i}_{c}$ decreases—it
becomes more difficult for a $z^{i}_{c}$ fluctuation to locate and interact
with the dominant $z^{o}$ fluctuations. Consequently, the influence of the
anomalous coherence effect would intensify with increasing Elsässer imbalance.

The anomalous coherence of wave packets in the expanding solar wind enhances
nonlinear interactions compared to the homogeneous scenario, thereby altering
the phenomenology of the energy cascade. This results in $P_{z_{o}}\propto
k^{-1}$ and $P_{z_{i}}\propto k^{-3/2}$ outer range scalings for inwardly and
outwardly propagating modes, respectively (Velli et al., 1989; Perez &
Chandran, 2013; Meyrand et al., 2023). By selecting the most strongly
imbalanced ($\sigma_{c}(\ell^{\ast})>0.95$) fast wind intervals—having similar
levels of RMS—and calculating an average second-order moment, we demonstrate
that this is indeed the case, as shown in Figure 11. Therefore, the coherent
nature of the interactions could potentially affect the scale-dependence of
the alignment angle. For example, the fact that the anomalous fluctuations are
highly aligned, and that the fraction of $z^{i}_{c}$ becomes larger as the
cascade proceeds to smaller scales, could provide an alternative explanation
for the weakening or even reversal of SDDA.

This discussion underscores the need for a comprehensive theoretical framework
to understand the nature of SDDA in the solar wind, taking into account the
relative contributions of $z^{i}_{a}$ and $z^{i}_{c}$ to the shearing of
$z^{o}$ across various scales.

Figure 11: Averaged trace $SF_{5}^{2}$ of $\bm{z}_{o}$ (black) and
$\bm{z}_{i}$ (red) for the fast, strongly Alfvénic ($\sigma_{c}>0.95$) solar
wind. At inertial scales, indicated by the pink shading, both fields follow a
$-3/2$ spectral scaling. Two outer scale regimes may be observed: at scales
$\ell\in[2\times 10^{4}-2\times 10^{5}]d_{i}$, spectral scalings for both
fields, $P_{z_{o}}\propto k^{-1}$ and $P_{z_{i}}\propto k^{-3/2}$, are
consistent with models based on “anomalous coherence” effects (Velli et al.,
1989; Perez & Chandran, 2013; Meyrand et al., 2023). At even larger scales, a
range with $P_{z_{o}}\propto k^{-1}$ and $P_{z_{i}}\propto k^{-2}$ scalings is
observed, consistent with the model by Chandran (2018) based on PDI.

###  5.6 Instabilities

An important question arising from our study concerns the regime change
observed in the polarization alignment, which tends to plateau, at best, as
inertial scales are approached. An additional explanation for the observed
increasing misalignment at smaller spatial scales involves the idea that
dynamically aligned structures of a particular amplitude become unstable to
disruption by tearing instabilities and the onset of magnetic reconnection
(Furth et al., 1963). Particularly, when the maximum growth rate of the Coppi
et al. (1976) mode, $\gamma_{t}$, becomes comparable to the non-linear cascade
time $\tau_{nl}$, $\gamma_{t}\tau_{nl}\gtrsim 1$, the current sheets can no
longer remain stable (Pucci & Velli, 2014; Uzdensky & Loureiro, 2016). The
disruption of the current sheets, at a certain scale $\lambda_{D}$, interrupts
dynamic alignment, accelerating the turbulent cascade and leading to a
pronounced steepening of the power spectrum (Mallet et al., 2017a; Loureiro &
Boldyrev, 2017). In the tearing-dominated regime, the alignment angle is
expected to increase with decreasing scale, $\theta^{ub}\sim\lambda^{-4/5}$
(Mallet et al., 2017b; Boldyrev & Loureiro, 2017; Comisso et al., 2018).

Using three-dimensional gyrokinetic simulations, Cerri et al. (2022) explored
the dynamics of collisions between Alfvén waves (AW), revealing distinct
behaviors in SDDA depending on whether the wave packets are colliding or well
separated. When colliding, alignment demonstrates scale-dependent behavior
likely induced by the packets shearing each other, whereas when well
separated, a reconnection-mediated cascade dominates, leading to misalignment.
This differentiation, as depicted in Figure 4 of Cerri et al. (2022), is
crucial; without it, the average curve lacks specific scaling and remains
nearly flat. This averaging effect could significantly impact in-situ
measurements, especially when alignment is measured over extended periods,
exacerbating as the measurements move towards smaller scales where turbulence
evolution timescales become shorter.

In many astrophysical scenarios, MHD turbulence is typically driven by
localized sources (e.g., shear, instabilities), leading to non-balanced
states, i.e., non-negligible cross helicity. Indeed, localized regions of
imbalance have been prominently observed even in globally imbalanced
simulations of MHD turbulence (Matthaeus et al., 2008b; Perez & Boldyrev,
2009). Patches of positive and negative cross-helicity are also evident in
globally balanced solar wind streams (Chen et al., 2011; Wicks et al., 2013).
Considering these observations and based on their simulations Cerri et al.
(2022) propose that aligning thin, long-lived current sheets are generated by
the turbulent cascade, then misaligning through tearing, in a patchy fashion
in space and time, rather than stepwise in k-space. Taking these observations
into account, we can conclude that while the tearing mediated regime is highly
unlikely to occur at scales as large as $10^{3}-10^{4}d_{i}$, the impact of
tearing of the current sheets on the alignment angle at inertial scales
warrants further investigation.

In addition, other types of ideal MHD instabilities, such as the Kelvin-
Helmholtz instability (Malagoli et al., 1996), could potentially manifest in
the solar wind. For example, it has been demonstrated that despite the highly
Alfvénic nature of the fluctuations at large scales, the signatures of
Alfvénicity (namely, $\sigma_{c},~{}\sigma_{r},~{}\theta^{ub}$) diminish at
considerably larger scales than those observed at 1 AU (Podesta & Borovsky,
2010).

Parashar et al. (2020) interpret these observations as indicative of
substantial energy found in velocity shears, which disrupt an initial spectrum
of high cross helicity by injecting equal amounts of the two Elsässer energies
(Roberts et al., 1987; Goldstein et al., 1989; Roberts et al., 1992) within
the inner heliosphere (Ruffolo et al., 2020). Since both $\sigma_{c}$ and
$\sigma_{r}$ are interdependent with the alignment angles, such effects could
potentially influence the SDDA scalings. While it is not possible to rule out
such effects, especially given recent in-situ observations (Paouris et al.,
2024), it’s essential to note that, as demonstrated in Section 5.3, high-
frequency instrumental noise in velocity field measurements can contaminate
the power spectrum of the ingoing Elsässer field even at lower frequencies.
This contamination becomes progressively more significant as the Elsässër
imbalance increases, see 8b. Therefore, another plausible explanation for such
observations may be contamination by high-frequency velocity field noise.
Additionally, as pointed out in (Schekochihin, 2022), since the vortex
stretching terms for the different Elsässer fields have opposite signs
(Zhdankin et al., 2016), there will generally be more “current sheets” than
“shear layers”. In such sheets, at least in the homogeneous case, the
Kelvin–Helmholtz instability will be suppressed (Chandrasekhar, 1961).

###  5.7 Parametric Decay Instability & “Anomalous coherence”: Two Possible
Outer Scale Regimes?

The coupling of large amplitude outwardly propagating waves to slow
magnetosonic waves through the parametric decay instability (PDI; Galeev &
Oraevskii, 1963a; Goldstein, 1978; Pruneti & Velli, 1997; Réville et al.,
2018; Malara et al., 2022) results in two daughter waves: an antisunward
propagating slow wave and a sunward propagating Alfvén wave of frequency
slightly lower than the mother wave, could be another mechanism resulting in
the misalignment of the fields. In the limit of low $\beta$, PDI is known to
have a growth rate given by $\gamma/\omega_{0}\sim\delta B/B_{0}\beta^{-1/4}$
(Galeev & Oraevskii, 1963b; Goldstein et al., 1989), and thus its role could
become important at the outer scale.

Considering the limit of strongly imbalanced, weak, compressible turbulence,
Chandran (2018) has demonstrated that PDI can lead to an inverse cascade of
Alfvén wave quanta, resulting in outer-range scalings of $P_{z_{o}}\propto
k^{-1}$ and $P_{z_{i}}\propto k^{-2}$. For this mechanism to become dominant
over either the inward-outward interactions or the anomalous shearing
discussed in Section 5.5, the condition
$\omega^{\pm}_{nl,~{}PI}>\omega^{\pm}_{nl,\perp}$ must hold, where
$\omega_{nl}$ represents the inverse of the respective dynamical timescales.
Given the dependence of $\omega^{\pm}_{nl,~{}PI}$ on $\delta B/B_{0}$, it is
possible that two outer scale regimes may be realizable in the solar wind.

At large scales, $\lambda\gtrsim 10^{5}d_{i}$, where the weak-turbulence
regime allows PDI effects to become dominant, increasing misalignment of the
fields and thus lower $\sigma_{c}$ is observed towards larger scales, see
e.g., Figure 6. It is noteworthy that magnetic compressibility monotonically
increases with scale, see e.g., Figure 3, providing further evidence
supporting the idea that PDI might be dominant over this scale-range.

When considering the averaged trace $SF_{5}^{2}$ of $\bm{z}_{o}$ (black) and
$\bm{z}_{i}$ (red) for the fast, strongly Alfvénic ($\sigma_{c}>0.95$) solar
wind, two outer scale regimes may be observed. At scales $\ell\in[2\times
10^{4}-2\times 10^{5}]d_{i}$, the spectral scalings for both fields are
consistent with models based on “anomalous coherence” effects (Velli et al.,
1989; Perez & Chandran, 2013; Meyrand et al., 2023). At even larger scales, a
range with scalings consistent with the Chandran (2018) model is observed.
This indicates that both mechanisms may be active in the solar wind, each
becoming dominant at different scales.

Recent works considering Parker Solar Probe data (PSP; Fox et al., 2016) have
provided evidence for the dynamic formation of the $P_{z_{o}}\propto k^{-1}$
range of the power spectrum (Davis et al., 2023; Sioulas et al., 2023; Huang
et al., 2023). While the PDI has been suggested as the plausible formation
mechanism, the scaling of $P_{z_{i}}$ has not been considered, leaving open
questions for future work. Additionally, it was shown that for $R<0.3$ au and
$\lambda\gtrsim 10^{4}d_{i}$, which roughly corresponds to the scale at which
the $P_{z_{o}}\propto k^{-1}$ scaling was observed, turbulence is weak with
$\chi^{\pm}\ll 1$ (Sioulas et al., 2024). However, as the Alfvénic wavepackets
propagate through the heliosphere, the decreasing Alfvén speed results in the
growth of normalized magnetic-field amplitudes to nonlinear magnitudes,
$\delta B/B_{0}\sim 1$ (Squire et al., 2020; Chen et al., 2020; Mallet et al.,
2021; Tenerani et al., 2021), indicating that the regime of strong turbulence,
$\chi^{-}\gtrsim 1$, might be extending to progressively larger scales with
increasing heliocentric distance. This could allow for nonlinear shearing
mechanisms to dominate over PDI, thus modifying the scaling of $P_{z_{i}}$.

Investigating the weak-to-strong turbulence transition (Galtier et al., 2000b;
Verdini et al., 2012; Meyrand et al., 2016), the degree of compressibility,
along with the scaling of both $P_{z_{i}}$ and $P_{z_{o}}$ in a range of
different imbalance regimes, could provide a deeper understanding of the
origin of the outer scale of solar-wind turbulence and the effects of the
underlying physical mechanisms on SDDA. This will be the subject of future
work.

##  6 Summary & Conclusions

Using a large dataset of carefully selected homogeneous intervals from the
WIND mission, we explored the impact of compressibility, intermittency, and
imbalance on the statistical signatures of Scale-Dependent Dynamic Alignment
(SDDA) in MHD turbulence within the solar wind (Boldyrev, 2006; Chandran et
al., 2015).

Below, we summarize the key findings of our analysis. For clarity, we refer to
the alignment angles of the Elsässer variables as ($\Theta^{z}$) and the
alignment between velocity and magnetic field fluctuations as ($\Theta^{ub}$).

(1) SDDA in both $\Theta^{z}$ and $\Theta^{ub}$ is consistently evident at
energy-containing scales, $\lambda\gtrsim 10^{4}d_{i}$; over the same scale
range, $|\sigma_{c}|$ increases and $\sigma_{r}$ becomes more negative.

SDDA in both $\Theta^{z}$ and $\Theta^{ub}$ is consistently evident at energy-
containing scales, $\lambda\gtrsim 10^{4}d_{i}$. The alignment increases as
$|\sigma_{c}|$ increases and $\sigma_{r}$ becomes more negative.

(2) $\Theta^{z}$ and $\Theta^{ub}$ exhibit an inverse correlation with the
intensity of field gradients. This may stem from “anomalous” and/or
“counterpropagating” wave packet interactions. Nevertheless, this observation
indicates that the physical origin of alignment arises from the mutual
shearing of the Elsässer fields during imbalanced wave-packet interactions
(Chandran et al., 2015).

(2) Compressible fluctuations do not exhibit any signs of SDDA. However, their
effects on the average behavior of SDDA are negligible due to their relatively
low amplitude and thus cannot explain the trend of misalignment at inertial
scales.

(3) Stringent thresholding on proxies for intermittency reveals SDDA
signatures within a portion of the inertial range. Regardless of the approach
we follow increasing misalignment is consistently observed for scales
$\lambda\leq 8\times 10^{2}d_{i}$. Based on analytical arguments and modeling,
we believe this misalignment is not a physical phenomenon but rather an
artifact caused by high-frequency noise in the velocity field measurements.

(4) The scaling of $\Theta^{z}$/$\Theta^{ub}$ becomes steeper/shallower with
increasing global Alfvénic imbalance ($\sigma_{r}(\ell^{\ast})\rightarrow-1$).
However, only $\Theta^{ub}$ is correlated with global Elsásser imbalance,
becoming steeper with ($\sigma_{c}(\ell^{\ast})\rightarrow 1$).

(5) Signatures of increasing alignment in $\Theta^{ub}$ extend deeper into the
inertial range of balanced intervals. This could be due to two factors: first,
in globally imbalanced intervals, the fields are tightly aligned at large
scales, depleting nonlinearities and halting the SDDA mechanism, causing
alignment angles to saturate at a minimum value. Alternatively, instrumental
noise may have a greater impact on alignment angle measurements as imbalance
increases, leading to significant measurement errors.

(6) Two outer scale regimes in the solar wind may be realizable: one dominated
by parametric decay instability (PDI) (Chandran, 2018) at larger scales where
turbulence is weak ($\chi^{\pm}\ll 1$), and another by anomalous coherence
effects (Velli et al., 1989) at intermediate scales where nonlinear Alfvénic
interactions strengthen.

While we do not disregard the need for theoretical advancements in existing
turbulence models—especially to address the effects of imbalance and the
expansion of the solar wind—it is more plausible that our results underscore
the current limitations in precisely estimating alignment angles due to
instrumental constraints, particularly in the context of highly Alfvénic,
strongly imbalanced turbulence. Unfortunately, the most intriguing cases are
also the most difficult to investigate.

To further investigate the scaling of alignment angles, it is essential to
consider alternative methodologies. For example, employing electric field
measurements could facilitate the estimation of the perpendicular component of
magnetic field fluctuations, given by
$\delta\mathbf{V}_{\perp}(\mathbf{\ell},t)=\delta\mathbf{E}_{\perp}(\mathbf{\ell},t)\times\mathbf{B}_{0}/c$,
where $\delta\mathbf{E}_{\perp}(\mathbf{\ell},t)$ represents the perpendicular
component of the electric field fluctuations and $c$ is the speed of light.
This approach is planned for future research.

Moreover, 3D expanding box simulations of compressible MHD, which include both
balanced and imbalanced turbulence, will offer deeper insights and facilitate
more precise quantification of these effects on alignment measurements (Shi et
al., in prep).

NS acknowledges insightful discussions with Dr. Lynn B. Wilson on the use of
WIND data and with Dr. Michael Terres on MHD turbulence in the solar wind.

This research was funded in part by the FIELDS experiment on the Parker Solar
Probe spacecraft, designed and developed under NASA contract NNN06AA01C; the
NASA Parker Solar Probe Observatory Scientist grant NNX15AF34G and the HERMES
DRIVE NASA Science Center grant No. 80NSSC20K0604.

NS and MV were supported by the International Space Science Institute (ISSI)
in Bern, through ISSI International Team project. # 550, Solar Sources and
Evolution of the Alfvénic Slow Wind. T.A.B. acknowledges NASA Grant No.
80NSSC24K0272. BC was supported in part by NASA grant 80NSSC24K0171.

Figure 12: Alignment angle measurements as defined by Equations 7 and 9 in top
and bottom panels, respectively, across the three (out of four) intervals
studied in (Podesta et al., 2009).Black curves represent the alignment between
Elss̈ser fields, while red curves illustrate the alignment between velocity
and magnetic field fluctuations.

###  .1 Revisiting Podesta et al. (2009)

To verify the effectiveness of our modified alignment angle estimation method
in comparison with P09, we revisited the intervals analyzed in their study.
Their investigation primarily focused on the alignment between
$\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$, employing Equation 9. In
addition they introduced a weighted average angle, defined in Equation 8 of
P09. Our analysis indicates that the latter definintion results in scalings
that are in most cases identical to Equation 9.

Figure 12 showcases our findings for $\tilde{\theta}_{\perp}^{ub(z)}$, with
each column representing one of the three (out of four) intervals analyzed by
P09. The Elsässer fields are denoted by black lines, while the angles between
magnetic and velocity fields are depicted in red. In all instances, the
scaling obtained aligns with those reported in P09, with both the 2-pt and
5-pt increment methods yielding very similar results, as detailed in Section
3. Thus, we can confidently proceed with our analysis utilizing the 5-pt
increment method.

###  .2 Geometrical Constrains and Types of Alignment

To provide a clear illustration of the interdependence between $\sigma_{c}$,
$\sigma_{r}$, $\theta^{ub}_{\perp}$, and $\theta^{z}_{\perp}$ we provide a
graphical representation of the equations:

$\cos(\theta^{z}_{\perp})=\frac{\sigma_{r}}{\sqrt{1-\sigma_{c}^{2}}},~{}~{}\cos(\theta^{ub}_{\perp})=\frac{\sigma_{c}}{\sqrt{1-\sigma_{r}^{2}}},$
(1)

To generate random values for $\sigma_{c}$ and $\sigma_{r}$ within a unit
circle on the Cartesian plane, constrained to the range of [-1, 1], we
employed a random sampling approach. We independently selected values for
$\sigma_{c}$ and $\sigma_{r}$ from a continuous uniform distribution spanning
[-1, 1], while ensuring that the condition $\sigma_{c}^{2}+\sigma_{r}^{2}\leq
1$ was met to keep them within the unit circle. Subsequently, for each pair,
we estimated the alignment angles according to Equation 1. The results are
illustrated in Figure 13, colored by $\theta^{z}_{\perp}$ in panel (a), and
$\theta^{ub}_{\perp}$ in panel (b).

Figure 13: Graphical representation of Equations 1 illustrated in panels (a)
and (b), respectively.

###  .3 Modeling High Frequency Instrumental Noise: Perfectly aligned fields

We generate two sets of three-component Gaussian white noise time series for
the magnetic field ($\mathbf{B}$) and the velocity field ($\mathbf{V}$), with
the corresponding components being perfectly correlated ($\rho=1$). The
duration of the time series is set to 48 hours ($d=48\text{ hours}$), with a
cadence of 3 seconds ($dt=3\text{ s}$), mimicking the typical duration and
cadence of our identified intervals. The base time series are transformed to
the frequency domain using the Fast Fourier Transform (FFT). To shape the
power-spectral density (PSD) of the time series, we apply filters in the
frequency domain. For a given desired spectral exponent $\alpha$, the filter
response $H(f)$ is defined as:

$H(f)=|f|^{\alpha/2},$ (2)

where $f$ denotes the frequency.

The spectral shaping filters are applied by multiplying the Fourier-
transformed series by the filter response $H(f)$. The filtered series are then
transformed back to the time domain using the inverse FFT. At this point, two
time series with the desired correlation and spectral properties have been
constructed.

Next, we need to properly normalize the two time series to model different
levels of Elsässer imbalance. This requires input from in-situ observations.
It is well known that the RMS of the fluctuations in both velocity and
magnetic field increases with $\sigma_{c}$ (Borovsky et al., 2019; Pi et al.,
2020; Sioulas et al., 2023). To quantify this dependency, we estimate the RMS
value of the fluctuations in the different fields for each of our selected
intervals. For any given field $\bm{\Phi}$, fluctuations are estimated as:

$\Delta\bm{\Phi}=\bm{\Phi}-\bm{\Phi}_{0},$ (3)

where $\bm{\Phi}=V_{a},V,Z_{o},Z_{i}$ and
$\bm{\Phi}_{0}=\langle\bm{\Phi}\rangle_{D}$ is a moving average with a window
of duration $D=8\text{ hours}$. Subsequently, the RMS of each field is
estimated using $d=1\text{ minute}$ moving averages of the fluctuations:

$\bm{\Phi}_{RMS}=\sqrt{\langle\Delta\bm{\Phi}^{2}\rangle_{d}}.$ (4)

Using the fluctuations estimated through Eq. 3, 1-minute moving averages of
$\sigma_{c}$ are then also estimated. This process yielded a dataset of size
$N\approx 7.5\times 10^{7}$ total measurements. $\bm{\Phi}_{RMS}$ are then
plotted against $\sigma_{c}$, and an empirical fit is extracted by finding the
best 12th-degree polynomial fit. The fit obtained for the different fields is
plotted against $\sigma_{c}$ in Figure 14 with the legends showing only the
first three terms.

Using the empirical relationship for the RMS of the fields as a function of
$\sigma_{c}$, the time series are normalized such that the RMS of the
synthetic data matches that of the real data at different levels of imbalance
while retaining the original correlation and desired spectral scalings. To
simplify our analysis, we assume—though this is strictly true only for the
velocity field (Chen et al., 2012; Sioulas et al., 2023; McIntyre et al.,
2023)—that the inertial-range trace power-spectral scalings of the two fields
are independent of $\sigma_{c}$ and scale as $a_{b}=a_{v}=-3/2$.

Figure 14: The empirical relationship for the RMS of the fluctuations is shown
for (a) $\bm{B}$ (black line), (b) $\bm{V}$ (red line), (c) $\bm{z}_{o}$ (blue
line), and (d) $\bm{z}_{i}$ (gray line), plotted against $\sigma_{c}$. Using
the fluctuations estimated through Eq. 3, 1-minute moving averages of
$\sigma_{c}$ were also estimated, yielding a dataset of size $N\approx
7.5\times 10^{7}$. The RMS quantities of the fields were plotted against
$\sigma_{c}$, and an empirical fit was extracted by finding the best 12th-
degree polynomial fit. The fits obtained for the different fields are plotted
against $\sigma_{c}$, with the legends showing only the first three terms of
the empirical fit.

Small amplitude, high-frequency noise is then added to the $\bm{V}(t)$ series.
We assume, though this is not strictly true (see, e.g., Ervin et al., 2024),
that the noise floor in the velocity field measurements is uniform across all
intervals considered and independent of $\sigma_{c}$. Given that the RMS of
the fluctuations peaks at $\sigma_{c}\approx 1$, we define the noise level as
$\Delta=\alpha\bm{V}_{\text{RMS}}^{\ast}(f_{\text{Nyq}})\mathcal{N}(0,1)$,
where $\bm{V}_{\text{RMS}}^{\ast}(f_{\text{Nyq}})$ is the RMS value of the
velocity field at the Nyquist frequency for $\sigma_{c}=1$, and
$\mathcal{N}(0,1)$ represents a normal distribution with mean, $\mu=0$ and
standard deviation $\sigma=1$. Different values of $\alpha$ have been
considered. In the following, we examine the case that was found to provide
the most realistic results, with $\alpha=0.1$, corresponding to an error of
$\Delta\approx 1\,\text{km/s}$.

For each simulated interval, the Fourier trace power spectral density $F(f)$
was calculated, smoothed by averaging over a sliding window with a factor of
2, and then transformed into a wavenumber spectrum expressed in physical units
$P(kd_{i})$:

$P(kd_{i})=\frac{V_{sw}}{2\pi d_{i}}F(f),$ (5)

where $kd_{i}=(2\pi fd_{i})/V_{sw}$ (Sioulas et al., 2023). To perform this
normalization, the ratio $V_{sw}/d_{i}$ was estimated for all intervals in our
dataset, resulting in a mean value of $3.95\pm 1.27s^{-1}$. The mean value was
used to normalize the synthetic spectra to enable a more meaningful comparison
with the in-situ derived power spectra and spectral scaling indices.

###  .4 Modeling High Frequency Instrumental Noise: Correlation coefficient
informed by in-situ observations.

The previous approach, by assuming perfect alignment between magnetic and
velocity field fluctuations, restricts our analysis and does not allow
$\sigma_{c}$ at large scales to decrease below $\sigma_{c}\lesssim 0.45$, thus
preventing a fully consistent and realistic simulation. This limitation arises
from the imposed correlation, which is not always observed in the solar wind.

We therefore need to follow a more rigorous approach. To this end, we generate
time series for the magnetic and velocity fields that meet three simultaneous
conditions informed by our in-situ observations: (1) the fluctuations are
correlated with a specified correlation coefficient, (2) their inertial-range
spectral scaling, and (3) the RMS values of these time series align with those
observed in-situ. Therefore, in addition to considering the RMS values of the
fields, the inertial-range spectral-scalings, $\alpha_{\phi}$, where
$\phi=\bm{B},\bm{V},~{}\bm{Z}_{i},~{}\bm{Z}_{o}$ were estimated by finding the
best-fit linear gradient in log–log space over the range $kd_{i}\in[5\times
10^{-4},10^{-2}]$. In addition, the correlation coefficient

Figure 15: (a) The inertial-range power-spectral scalings for $\bm{B}$ (black
line), $\bm{V}$ (red line), $\bm{z}_{o}$ (blue line), and $\bm{z}_{i}$ (gray
line) plotted against $\sigma_{c}$, estimated by determining the best-fit
linear gradient in log–log space over the range $kd_{i}\in[5\times
10^{4},10^{-2}]$ for each selected interval in our dataset. (b) The
correlation coefficient, $C_{\phi}$, as defined in Equation 6, for the
velocity/magnetic fields (black) and Elsässer fields (red) plotted against
$\sigma_{c}$. In contrast to the scaling index in panel (a), $C_{\phi}$ is
estimated using the 1-minute moving average, similar to Figure 14, from a
dataset of size $N\approx 7.5\times 10^{7}$.

$C_{\phi}=\frac{\langle\Delta\bm{\phi}\cdot\Delta\bm{\psi}\rangle}{\sqrt{\langle\Delta\bm{\phi}^{2}\rangle\langle\Delta\bm{\psi}^{2}\rangle}},$
(6)

between either magnetic-velocity or Elsässer field fluctuations, was estimated
and an empirical relationship was obtained for these quantities as a function
of $\sigma_{c}$. The obtained functional form for the spectral scalings and
the correlation coefficients between the magnetic-velocity fields,
$\rho_{ub}$, and Elsässer fields, $\rho_{z}$, are illustrated in Figure 15.

While the third condition is straightforward to enforce and satisfying either
of the first two conditions independently is also manageable, applying
spectral shaping filters as described in Equation 2 to achieve the desired
spectral scaling can disrupt the specified correlation coefficient, and vice-
versa. Thus, imposing both conditions simultaneously in a rigorous manner is
challenging.

To achieve this, we follow an iterative approach. Specifically, we generate
time series for the magnetic field components $B_{i}$ with the desired
spectral scaling of $a_{B}=-1.74+0.26\cdot\sigma_{c}$. Each component $B_{i}$
of the magnetic field is paired with an initially uncorrelated component
$V_{i}$ of the velocity field. The desired scaling index for the velocity
field is $-1.5$, independent of $\sigma_{c}$. However, the iterative process
to achieve the desired correlation coefficient modifies the imposed scaling.

We have empirically found that if the initial scaling of the generated
velocity time series is given by $a_{v}=-1.51+0.18\cdot\sigma_{c}$, we can
obtain both the desired scaling and correlation coefficient simultaneously
through iterative adjustment. This process aims to match the desired
correlation between the increments of $B_{i}$ and $V_{i}$. The input velocity
field component is given by

$V_{i}=\rho\cdot B_{i}+\sqrt{1-\rho^{2}}\cdot\eta_{i},$ (7)

where $\rho$ is the desired correlation coefficient, and $\eta_{i}$ is the
initially generated velocity time series with the imposed scaling. This
provides an initial guess for $V_{i}$.

For each iteration, the current correlation coefficient $r$ between $B_{i}$
and $V_{i}$ increments over specified intervals is computed. If the absolute
difference between the current and desired correlation coefficients is less
than the specified tolerance, the iteration stops. Otherwise, an adjustment is
applied to $V_{i}$ based on the correlation error, defined as

$\delta=\gamma\times(\rho-r)\times B_{i},$ (8)

where $\gamma$ is the learning rate and $r$ is the current correlation
coefficient. This process is repeated until the desired correlation
coefficient is achieved.

Once the desired correlation is achieved, we normalize the RMS values of the
magnetic and velocity fields to ensure they meet the specified conditions.

By carefully balancing the spectral properties and correlation through this
iterative process, we successfully produced time series for the magnetic and
velocity fields that simultaneously satisfied the specified conditions and
adhered to the desired spectral scalings and correlation coefficients.

Figure 16: Trace power spectrum of the magnetic field time series for three
randomly sampled 1-day intervals, analyzed using CDAWeb data at varying
cadences. Displayed are the full resolution ($\delta\tau=0.1s$, black line),
downsampled data ($\delta\tau=3s$, blue line; $\delta\tau=24s$, cyan line),
and full resolution data downsampled to the velocity field cadence after
processed with a 10th order (Butterworth, 1930) low-pass filter (red line).

###  .5 A problem with downsampled WIND data provided in CDAWeb

In digital signal processing, downsampling is commonly used to reduce the size
of data sets, decrease computational demands , and align timeseries sampled at
different cadence. However, to maintain the integrity of the signal, it is
crucial to apply low-pass filtering before downsampling. According to the
Nyquist-Shannon sampling theorem, to reconstruct a signal without loss of
information, the sampling frequency, $f_{s}$, must be at least twice the
highest frequency component, $f_{max}$, in the signal:

$f_{s}\geq 2f_{max}$ (9)

Failing to adhere to this principle results in aliasing, where frequency
components higher than $\frac{f_{s}}{2}$—the Nyquist frequency—appear as
lower, incorrect frequencies in the sampled signal. Therefore, not applying a
low pass filter before downsampling would result in an artificial flattening
of the power-spectrum. Prior to downsampling, it is necessary to apply a low-
pass filter that eliminates frequencies above the new Nyquist frequency. The
cutoff frequency of this filter, $f_{c}$, should ideally be at or below
$f_{s_{new}}/2$, where $f_{s_{new}}$ is the new, reduced sampling rate:

$f_{c}\leq\frac{f_{s_{new}}}{2}$ (10)

The design of the low-pass filter, including its cutoff frequency and the
steepness of its roll-off, significantly impacts the resulting signal. A
filter with a steep roll-off efficiently attenuates frequencies near $f_{c}$,
minimizing aliasing but possibly altering the natural characteristics of the
signal. Conversely, a filter with a gentle roll-off may inadequately suppress
high frequencies, resulting in a flatter than expected power spectral density
(PSD). A well-designed filter ensures the PSD is shaped appropriately:

$PSD(f)\approx\left\\{\begin{array}[]{ll}S(f)&\text{if }f<f_{c}\\\ 0&\text{if
}f>f_{c}\end{array}\right.$ (11)

where $S(f)$ represents the original power spectral density of the signal. For
example, if a signal sampled at 1 kHz is to be downsampled to 250 Hz, the
original Nyquist frequency is 500 Hz, and the new Nyquist frequency post-
downsampling is 125 Hz. Thus, a suitable low-pass filter should have a cutoff
frequency at or below 125 Hz to prevent aliasing. Proper implementation of
these filtering and downsampling steps preserves the fidelity of the signal,
thereby ensuring the accuracy of any subsequent analysis.

While analyzing WIND data, we encountered issues with the downsampled magnetic
field datasets available on CDWeb. Figure 16 compares the trace power-spectral
densities: the properly downsampled signal using a Butterworth low-pass filter
of order $N=10$ is shown against the full-resolution dataset
($\delta\tau=0.1s$, black line), the downsampled $\delta\tau=3s$ dataset (blue
line), and the downsampled $\delta\tau=24s$ dataset (cyan line). As
illustrated, both downsampled signals deviate from the original and
appropriately downsampled power spectra. Although the deviations are less
critical for the $\delta\tau=3s$ dataset, they are considerably pronounced for
the $\delta\tau=24s$ dataset, potentially leading to erroneous interpretations
of the turbulent cascade across the frequency range of $10^{-3}-10^{-2}$ Hz.
We therefore recommend caution when utilizing these datasets and advise
adopting a manual downsampling methodology if aligning the cadence of the
magnetic and velocity field time series is required.

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