# Nonlinear Interactions in Spherically Polarized Alfvénic Turbulence

Trevor A. Bowen<EMAIL_ADDRESS>Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  Samuel T. Badman  Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Stuart D.
Bale  Space Sciences Laboratory, University of California, Berkeley, CA
94720-7450, USA  Physics Department, University of California, Berkeley, CA
94720-7300, USA  Thierry Dudok de Wit  LPC2E, CNRS, CNES, and University of
Orléans, Orléans, France  Timothy S. Horbury  The Blackett Laboratory,
Imperial College London, London, SW7 2AZ, UK  Kristopher G. Klein  Department
of Planetary Sciences & Lunar and Planetary Laboratory, University of Arizona,
Tucson, AZ 85721  Davin Larson  Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  Alfred Mallet  Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Lorenzo
Matteini  The Blackett Laboratory, Imperial College London, London, SW7 2AZ,
UK  Michael D. McManus  Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA  Physics Department, University of California,
Berkeley, CA 94720-7300, USA  Jonathan Squire  Department of Physics,
University of Otago, 730 Cumberland St., Dunedin 9016, New Zealand

###### Abstract

Turbulent magnetic field fluctuations observed in the solar wind often
maintain a constant magnitude condition accompanied by spherically polarized
velocity fluctuations; these signatures are characteristic of large-amplitude
Alfvén waves. Nonlinear energy transfer in Alfvénic turbulence is typically
considered in the small-amplitude limit where the constant magnitude condition
may be neglected; in contrast, nonlinear energy transfer in the large-
amplitude limit remains relatively unstudied. We develop a method to analyze
finite-amplitude turbulence through studying fluctuations as constant
magnitude rotations in the stationary wave (de Hoffmann-Teller) frame, which
reveals that signatures of finite-amplitude effects exist deep into the MHD
range. While the dominant fluctuations are consistent with spherically-
polarized large-amplitude Alfvén waves, the subdominant mode is relatively
compressible. Signatures of nonlinear interaction between the finite-amplitude
spherically polarized mode with the subdominant population reveal highly
aligned transverse components. In theoretical models of Alfvénic turbulence,
alignment is thought to reduce nonlinearity; our observations require that
alignment is sufficient to either reduce shear nonlinearity such that non-
Alfvénic interactions may be responsible for energy transfer in spherically
polarized states, or that counter-propagating fluctuations maintain anomalous
coherence, which is a predicted signature of reflection-driven turbuence.

## Introduction

Spacecraft observations of the solar wind provide much of the context for our
understanding of nonlinear interactions occurring in plasma environments Bruno
and Carbone [2013]. Although the solar wind is collisionless, many of its
dynamics can be understood using fluid approximations, such as
magnetohydrodynamics (MHD) Kulsrud [1983]. To understand nonlinear
interactions in MHD, the velocity $\mathbf{v}$ and magnetic field $\mathbf{B}$
are cast into the Elsasser variables ${\bf{z}}^{\pm}={\bf{v}}\pm{{\bf{b}}}$
with ${\bf{b}}={{\bf{B}}}/{\sqrt{\mu_{0}\rho}}$, and mass density $\rho$
Elsasser [1950]. Assuming incompressbility, which is motivated by our
observations,

$\displaystyle\partial_{t}{\bf{z}}^{\pm}=-{\bf{z}}^{\mp}\cdot\nabla{\bf{z}}^{\pm}-\frac{1}{\rho}\nabla(p_{th}+\frac{\rho}{2}b^{2}).$
(1)

The Elsasser fluctuations can be identified as travelling parallel
($\mathbf{z}^{-}$) and anti-parallel ($\mathbf{z}^{+}$) to the mean magnetic
field $\mathbf{b}_{0}$Alfvén [1942], Elsasser [1950]. Fluctuations in the
solar wind, defined as $\delta{\bf{x}}={\bf{x}}-{\bf{x}}_{0}$, are dominated
by fluctuations perpendicular to $\mathbf{b}_{0}$ with
$\delta\mathbf{v_{\perp}}\approx\pm\delta\mathbf{b_{\perp}}$ Belcher and Davis
[1971]. In the small-amplitude limit, Alfvén waves are approximated by
transverse linear plane-waves polarized perpendicular to both wave-vector
$\mathbf{k}$ and the background field $\mathbf{b}_{0}$. This limit is well
described by Reduced MHD (RMHD), where $\mathbf{z}^{\pm}$ perturbations are
perpendicular, $\mathbf{z}^{\pm}=\mathbf{z}^{\pm}_{\perp}$, polarized
perpendicular to their wave vector ${\bf k}$, and can be written using the
Elsasser potentials $\zeta^{\pm}$ as $\mathbf{z}^{\pm}=\hat{\bf
b}_{0}\times\nabla_{\perp}\zeta^{\pm}$ Strauss [1976], Schekochihin et al.
[2009]. Observations in the solar wind are often dominated by one of
$\delta\mathbf{z}^{\pm}_{\perp}$Belcher and Davis [1971]–a condition that is
commonly known as imbalance Lithwick et al. [2007]. In the solar wind,
imbalance favors the anti-sunward propagating Alfvén wave population Roberts
et al. [1987a, b]. In turn, the subdominant Elsasser variable corresponds to a
sunward propagating mode that interacts nonlinearly with the dominant mode via
the ${\delta\bf{z}}^{\mp}\cdot\nabla{\delta\bf{z}}^{\pm}$ term in Equation 1.
This mutual shearing of counter-propagating fluctuations in MHD is thought to
generate a cascade, similar to neutral-fluid turbulence, that results in
inertial range energy transfer Kolmogorov [1941], Kraichnan [1965], Coleman
[1968], Dobrowolny et al. [1980], Grappin et al. [1982], Goldreich and Sridhar
[1995], Boldyrev [2005, 2006], Mason et al. [2006], Schekochihin et al.
[2009], Chandran et al. [2015], Howes [2015].

Theoretical consideration of magnetized turbulence is often performed in a
small-amplitude limit Kolmogorov [1941], Kraichnan [1965], Coleman [1968],
Strauss [1976], Grappin et al. [1982], Dobrowolny et al. [1980], Goldreich and
Sridhar [1995], Boldyrev [2006], Schekochihin et al. [2009], Boldyrev [2005],
Beresnyak and Lazarian [2006], Mason et al. [2006], Podesta et al. [2009],
Chandran et al. [2015], Mallet et al. [2015], Howes [2015], Mallet et al.
[2016]; however, the solar wind is subject to large-amplitude
($|\delta\mathbf{b}|/|\mathbf{b}_{0}|\sim 1$) fluctuations that maintain

$|\mathbf{b}|=|\mathbf{b}_{0}+\delta\mathbf{b}|=const,$

and thus appear spherically-polarized Goldstein et al. [1974], Lichtenstein
and Sonett [1980], Riley et al. [1996]. This condition is characteristic of
large-amplitude Alfvén waves, which are not entirely perpendicular to
$\mathbf{b}_{0}$, but can acquire a component parallel to $\mathbf{b}_{0}$ in
order to maintain constant magnitude of the total magnetic field Barnes and
Hollweg [1974], Goldstein et al. [1974]. The small-amplitude limit neglects
the higher-order corrections that maintain constant total magnitude.

Observations from Parker Solar Probe (PSP) reveal that both the constant-
magnitude condition and high-Alfvénicity are pronounced in the inner-
heliosphere Bale et al. [2019], Dudok de Wit et al. [2020], Horbury et al.
[2020], Chen et al. [2020], McManus et al. [2020], Chaston et al. [2020],
Bourouaine et al. [2020], Chen et al. [2021], Martinović et al. [2021], and
are consistent with the large-amplitude Alfvén mode; however, these studies
mostly omit discussion of observed finite-amplitude signatures in turbulence.
Both magnetic field and velocity fluctuations show finite-amplitude signatures
of spherical polarization Wang et al. [2012], Matteini et al. [2015]
consistent with the transverse Alfvén mode Barnes and Hollweg [1974]. While
spherical polarization of the velocity fluctuations is independent of
(Galilean) reference frame, constant-magnitude of the total velocity vector is
only maintained in the frame at the center of spherical polarization Barnes
and Hollweg [1974], Matteini et al. [2015], associated with the de Hoffmann-
Teller frame (dHTf) de Hoffmann and Teller [1950]. Analysis in the dHTf
enables fluctuations to be characterized in terms of constant magnitude
_rotations_ corresponding to large-amplitude Alfvén waves Barnes and Hollweg
[1974], Hollweg [1974]. Formally, the dHTf minimizes the convected electric
field ${\bf{E}}=-{\bf{v}}\times{\bf{B}}$, which is ideally zero in a
stationary frame of transverse electromagnetic waves. The existence of an
empirically measurable dHTf indicates significant alignment between magnetic
and velocity fluctuations, which constrains turbulent energy transfer.

In this Letter, we discuss the nonlinear interactions observed in spherically
polarized turbulence, which we measure to be highly aligned in nature. The
relation between alignment and nonlinearity is clear when RMHD is written
using the Elsasser potentials

$\partial_{t}\nabla_{\perp}^{2}\zeta^{\pm}\propto\left\\{\zeta^{+},\nabla_{\perp}^{2}\zeta^{-}\right\\}+\left\\{\zeta^{-},\nabla_{\perp}^{2}\zeta^{+}\right\\}\mp\nabla_{\perp}^{2}\left\\{\zeta^{+},\zeta^{-}\right\\},$
(2)

with
$\\{A,B\\}=\left(\nabla_{\perp}A\times\nabla_{\perp}B\right)\cdot\hat{{\mathbf{b}}}_{0}.$
Because $\\{A,B\\}$ vanishes if $\nabla_{\perp}A$ and $\nabla_{\perp}B$ are
parallel, the final term vanishes for aligned $\mathbf{z}_{\perp}^{\pm}$. The
other nonlinear terms are significantly reduced if contours of
$\nabla_{\perp}^{2}\zeta^{\pm}$ are approximately aligned with $\zeta^{\mp}$,
a condition that is satisfied for perturbations that look locally like sheets
or tubes. In the theory of dynamic alignment, cascading turbulent structures
are sheet-like and $\delta\mathbf{z}^{+}$ and $\delta\mathbf{z}^{-}$ shear
themselves into alignment (i.e., becoming more parallel at smaller scales),
such that
$\delta\mathbf{z}_{\perp}^{\mp}\cdot\nabla\delta\mathbf{z}_{\perp}^{\pm}$ is
reduced by a factor $\sim\\!\sin\phi_{\bf k}$, where $\phi_{\bf k}$ is the
alignment angle between $\delta\mathbf{z}_{\perp}^{+}$ and
$\delta\mathbf{z}_{\perp}^{-}$ at scale ${\bf k}$. This depletes the
nonlinearity towards smaller scales, flattening $\sim k_{\perp}^{-5/3}$,
energy spectrum to $\sim k_{\perp}^{-3/2}$, which is often observed in the
solar wind Podesta et al. [2009], Chen et al. [2013], Bowen et al. [2018a],
Chen et al. [2020].

The dynamic alignment argument neglects the fact that perfectly aligned
fluctuations may have significant nonlinearity. Because
$\delta\mathbf{z}^{\pm}\cdot\nabla\delta\mathbf{z}^{\mp}=\delta\mathbf{z}^{\mp}\cdot\nabla\delta\mathbf{z}^{\pm}-\nabla\times(\delta\mathbf{z}^{\pm}\times\delta\mathbf{z}^{\mp})$
it is possible to have both
$\mathbf{z}_{\perp}^{+}\propto\mathbf{z}_{\perp}^{-}$ and significant
nonlinearity as long as
$\delta\mathbf{z}^{\mp}\cdot\nabla\delta\mathbf{z}^{\pm}\approx\delta\mathbf{z}^{\pm}\cdot\nabla\delta\mathbf{z}^{\mp}$.
In essence, it is possible that
$\mathbf{z}_{\perp}^{+}\propto\mathbf{z}_{\perp}^{-}$
($\zeta^{+}\propto\zeta^{-}$), while not satisfying a specific topology, in
which case aligned fluctuations have significant nonlinearity. Here, we show
that measurements of highly aligned $\mathbf{z}^{\pm}$ suggests either (i)
that large-amplitude fluctuations are sheet or tube-like structures and
nonlinearity is significantly reduced to an extent that non-Alfvénic modes
contribute significantly to nonlinear energy transfer, or (ii)
$\mathbf{z}^{-}$ and $\mathbf{z}^{+}$ are “anomalously coherent” (nearly
proportional), but that alignment does not affect nonlinear energy transport.

Figure 1: (a) Velocity measurements in dHTf (green), magnetic field
fluctuations in Alfvén units (black). (b) Elsasser variables
$\mathbf{z}^{\prime}\pm$ constructed in the dHTf (black +, blue -).

## Data

PSP provides measurements of the inner heliosphere using the electromagnetic
FIELDS [Bale et al., 2016] and Solar Wind Electron Alpha and Proton (SWEAP,
Kasper et al. [2016]) instrument suites. We study a stream from PSP perihelion
4 from 2020-01-28/09:56-21:43. The mean magnetic field was directed sunward,
such that ${\bf{z}^{+}}$ is the outward propagating mode. Magnetic field data
is obtained from PSP/FIELDS Bale et al. [2019], Bowen et al. [2020]. Nonlinear
bi-Maxwellian fits to proton beam and core distributions measured by the
PSP/SWEAP Solar Probe ANalyzer (SPANi) provide estimates of ${\bf{v}}$,
$T_{\parallel,\perp}$ Kasper et al. [2016], Klein et al. [2021]. The proton
core and beam are well resolved. Density measurements are from quasi-thermal
noise (QTN) from FIELDSMalaspina et al. [2016], Pulupa et al. [2017].
Uncertainty in the density is order 10% due to the resolution of FIELDS/RFS
Pulupa et al. [2017].

## Turbulent Signatures in the dHTf

Fig. 1(a) shows measurements of ${\bf{b}}$ and
${{\bf{v}}^{\prime}={\bf{v}}-{\bf{v}}_{dHT}}$, with projections on each 2D
plane. ${\bf{v}}_{dHT}$ is found through minimizing

$\displaystyle
E^{2}=\sum_{i}\left(({\bf{v}}_{dHT}-{\bf{v}}_{i})\times{\bf{B}}_{i}\right)^{2}$
(3)

with respect to ${\bf{v}}_{dHT}$. Fig. 1(b) shows the dHTf Elsasser variables
defined as ${{\bf{z}^{\prime}}^{\pm}}=\mathbf{v}^{\prime}\pm{\bf{b}}$, with
${\bf{v}}_{{dHT}}=[-71,23,-342]$ km/s
$\langle\mathbf{v}\rangle={\bf{v}}_{sw}=[-99,26,-265]$ km/s. The Alfvén speed
is ${\bf{v}}_{A}=\langle{\bf{b}}\rangle=[19,10,97]$ km/s. The mean solar wind
speed is $\langle|{\bf{v}}_{sw}|\rangle=295$ km/s, with central half of values
$265<|{\bf{v}}_{sw}|<310$ km/s. Fig. 1(b) shows that ${{\bf{z}}^{\prime+}}$ is
well approximated by a spherical surface, with constant magnitude
$|{{\bf{z}}^{\prime+}}|$=221 km/s. In contrast, ${{\bf{z}}^{\prime-}}$ has
significant compressibility. The magnetic field is taken as frame invariant
and no substantial effects were found when introducing kinetic Alfvén speed
normalizations Barnes [1979], Chen et al. [2013].

A local, scale-dependent, dHTf, ${\bf{\tilde{v}}}_{t,t+\tau}$, is constructed
using two point increments and averages

$\displaystyle{\Delta{\bf{x}}=\bf{x}}(t+\tau)-{\bf{x}}(t)$ (4)
$\displaystyle\bar{\bf{x}}=\frac{{\bf{x}}(t)+{\bf{x}}(t+\tau)}{2}$ (5)

in $\mathbf{v}$ and $\mathbf{b}$ for every increment pair Khrabrov and
Sonnerup [1998],

$\displaystyle
E_{t,t+\tau}^{2}=\left[({\bf{\tilde{v}}}-{\Delta\bf{v}})\times({\bar{{\bf{b}}}+\Delta{\bf{b}}})\right]^{2}$
(6) $\displaystyle\frac{\partial{E^{2}}}{\partial{\bf{\tilde{v}}}}=0.$ (7)

Figure 2: (a-b) Structure functions $S_{2}$ for finite-amplitude increments
($\Delta{\bf{x}}$, $\Delta{\bf{x}}_{R}$, $\Delta{\bf{x}}_{C}.$) for ${\bf{b}}$
(red), ${\bf{v}}^{\prime}$ (green) and ${\bf{z}}^{+}$ (black) normalized to
$|\mathbf{z}^{\prime+}|$; total $\Delta{\bf{z}}^{-}$ is in blue. Power-law
fits to structure functions of $\Delta{\bf{z}}^{+}$ are in black lines. (c)
Small amplitude $\Delta{\bf{z}}^{\prime+}$ (black), and
$\Delta{\bf{z}}^{\prime-}$ (blue) increments. Perpendicular $\perp$ and
parallel $\parallel$ increments are shown respectively as $+$ and $\times$. As
described in the text, a bound on instrumental noise determined from unaligned
increments in $\mathbf{v}$ is shown in orange. Panels (d) and (e) show the
ratio of $S_{2}^{\perp}/S_{2}$ and $S_{2}^{R}/S_{2}$ for
$\Delta{\bf{z}}^{\prime+}$, $\Delta{\bf{v}}^{\prime}$ and $\Delta{\bf{b}}$.

## Rotational Increments

In the small-amplitude limit, perpendicular and parallel increments discern
between Alfvénic and compressible fluctuations Mason et al. [2006],
Schekochihin et al. [2009], Podesta et al. [2009], Wicks et al. [2013a, b]

$\displaystyle\Delta{\bf{x}}_{\parallel}(t,\tau)=$
$\displaystyle(\Delta{\bf{x}}(t,\tau)\cdot\hat{\bar{\mathbf{b}}})\hat{\bar{\mathbf{b}}},$
(8) $\displaystyle\Delta{\bf{x}}_{\perp}(t,\tau)=$
$\displaystyle\Delta{\bf{x}}(t,\tau)-\Delta{\bf{x}}_{\parallel}(t,\tau),$ (9)
$\displaystyle\hat{{\bf{x}}}=$ $\displaystyle\frac{{{\bf{x}}}}{|{\bf{x}}|}.$
(10)

In contrast, the finite-amplitude Alfvén mode is associated with a constant
magnitude rotation, we thus introduce a _rotational increment_ :

$\displaystyle\Delta{\bf{x}}_{R}(t,\tau)=$
$\displaystyle{\bf{x}}(t)-{\bf{R}}_{t,t+\tau}{\bf{x}}(t),$ (11)
$\displaystyle{\bf{R}}_{t,t+\tau}=$
$\displaystyle\hat{{\bf{x}}}(t+\tau)\hat{{\bf{x}}}(t)^{T}$ (12)

which is the part of $\Delta\mathbf{x}$ involving only a rotation of
$\mathbf{x}$. The rotational increment is only sensible to define for
$\mathbf{v}$ and $\mathbf{z}^{+}$ fluctuations in the dHTf, where the velocity
fluctuations have (approximately) constant magnitude. Outside of the dHTf, a
constant magnitude rotation in velocity includes contribution from the offset-
center of the spherical surface into the estimate of $\Delta\mathbf{v}_{R}$.
Each rotational increment for $\mathbf{v}^{\prime}$ and $\mathbf{z}^{\prime+}$
is defined in its local, scale dependent, ${\bf{\tilde{v}}}_{t,t+\tau}$ frame.
The non-rotational portion of the increment $\Delta{\bf{x}}_{C}$ is

$\displaystyle\Delta{\bf{x}}_{C}=\Delta{\bf{x}}-\Delta{\bf{x}}_{R}.$ (13)

Fig. 2(a) shows the square-root of the trace of the second order structure
functions $S_{2}=\langle\Delta{\bf{x}}^{2}\rangle$ for finite-amplitude
increments in ${\bf{b}}$ and ${\bf{v}}^{\prime}$. The amplitudes are
normalized to the magnitude $|\mathbf{z}^{\prime+}|=221$ km/s. The non-
rotational portion of the velocity fluctuations $\Delta{\bf{v}}_{C}$ is
significantly larger than $\Delta{\bf{b}}_{C}$. Fig. 2(b) shows total,
rotational, and non-rotational increments in ${\bf{z}}^{\prime\pm}$ normalized
to $|\mathbf{z}^{\prime+}|$. Alfvénic fluctuations in ${\bf{z}}^{\prime-}$ are
not expected to be pure rotations in the empirically measured dHT, and thus
$\Delta{\bf{z}}_{C}^{-}$ and $\Delta{\bf{z}}_{R}^{-}$ are omitted. The total
$\Delta{\bf{z}}^{+}$ increments are almost precisely equal to the rotational
$\Delta{\bf{z}}_{R}^{+}$ increments, indicating that the fluctuations are
dominated by the finite-amplitude Alfvén wave. $\Delta{\bf{z}}_{C}^{+}$ is
dominated by non-rotational $\Delta{\bf{v}}_{C}$ fluctuations.

Power-law scalings are fit to the $\Delta{\bf{z}}^{\prime+}$ increments with
$S^{z+}_{2}\sim\tau^{\alpha}$. An outer range index is $\alpha=0.1$, while a
steeper MHD range scaling, commonly associated with an inertial range, is
found to be $\alpha=0.7$. A spectral break occurs at approximately 100 s or,
assuming Taylor’s hypothesis, length-scales of $L\sim 30000$ km.

Fig. 2(c) shows the application of small amplitude increments (Equations 8 &
9) to the Elsasser variables. Measurable differences between the small
amplitude and rotational increments in Fig. 2(d,e), which are admittedly
small, are a signature of the finite-amplitude nature of the fluctuations.
Fig. 2(e) shows the persistence of the constant total magnitude condition well
below outer-scales, suggesting that the constant magnitude condition is
dynamically relevant to nonlinear energy transfer across MHD scales Matteini
et al. [2018, 2019].

## Nonlinearities

Nonlinear turbulent interactions of finite-amplitude Alfvén waves are poorly
understood, yet key to understanding how these fluctuations evolve and
dissipate. We study the strength of nonlinear interactions using cross-terms
$\langle|\Delta\mathbf{x}||\Delta\mathbf{y}|\rangle$Mason et al. [2006],
Beresnyak and Lazarian [2006]; this assumes that nonlinear interactions are
local in scale. The strength of the nonlinearity associated with transverse
shear Alfvén wave interactions is given as
$\langle|\Delta\mathbf{z}_{R}^{+}||\Delta\mathbf{z}^{-}_{\perp}|\rangle$.
Similarly, we estimate shear interactions not associated with Alfvén modes
using
$\langle|\Delta\mathbf{z}^{+}_{R}||\Delta\mathbf{z}^{-}_{\parallel}|\rangle$.
Fig. 3 shows the mean magnitude quantities of nonlinear terms. The transverse
nonlinearity
$\langle|\Delta\mathbf{z}^{+}_{R}||\Delta\mathbf{z}^{-}_{\perp}|\rangle$
dominates across MHD scales. Fig. 3 additionally shows nonlinearity in the
magnetic pressure $1/2\langle{\Delta b^{2}}\rangle$

$\displaystyle\Delta b^{2}={{b}^{2}}(t+\tau)-{{b}^{2}}(t),$ (14)

and thermal pressure gradients assuming $p_{th}=c_{s}^{2}\rho$

$\displaystyle c_{s}^{2}/b_{0}^{2}=\beta$
$\displaystyle=\frac{2\mu_{0}n_{p}}{B^{2}}(2/3T_{\perp_{p}}+1/3T_{\parallel
p}),$ (15) $\displaystyle\Delta p_{th}=$
$\displaystyle\beta\bar{b_{0}^{2}}\frac{\Delta\rho}{\bar{\rho}},$ (16)

where $T_{\perp,\parallel}$ is defined using proton core and beam populations
Klein et al. [2021]. The median $\beta=0.66$ with the central half of values
between $0.48<\beta<0.75$. Though the isothermal equation of state is likely
not well satisfied, and full compressible treatment requires additional terms
in Equation 1, Fig. 3 shows the dominance of shear nonlinearities, justifying
an incompressible approximation.

We study the effect of alignment on nonlinearity using the terms
$\langle|\Delta\mathbf{z}_{R}^{+}\times\Delta\mathbf{z}^{-}_{\perp}|\rangle$
and
$\langle|\Delta\mathbf{z}^{+}_{R}\times\Delta\mathbf{z}^{-}_{\parallel}|\rangle$
Beresnyak and Lazarian [2006], Mason et al. [2006], Servidio et al. [2008],
Podesta et al. [2009]. Fig. 3 shows that the alignment strongly affects the
shear Alfvén-wave nonlinearity. This effect is observed across scales, such
that if the shear nonlinearity is depleted according to
${\delta\bf{z}}^{\mp}\cdot\nabla{\delta\bf{z}}^{\pm}\sim\nabla\times({\delta\bf{z}}^{\pm}\times{\delta\bf{z}}^{\mp})$,
then the non-transverse terms become as strong as the Alfvénic nonlinearity.
Reduction in the Alfvénic nonlinearity indicates that the rotational
$\Delta\mathbf{z}_{R}^{+}$ is highly aligned with
$\Delta\mathbf{z}_{\perp}^{-}$. In contrast, including alignment has little
effect on the nonlinearity associated with $\mathbf{z}^{-}_{\parallel}$.

The sensitivity of our results to instrumental noise is tested by computing
unaligned fluctuations in $\Delta\mathbf{v}_{\perp}$ as
$\epsilon_{v}=\Delta\mathbf{v}_{\perp}\times\hat{\Delta\mathbf{b}_{\perp}}$,
as a bound on measurement uncertainty (i.e. assuming any unaligned
$\Delta\mathbf{v}_{\perp}$ is entirely noise). The rms value of $\epsilon_{v}$
at smallest scales ($\tau_{min}$=7s) is 5.4 km/s, which is an upper-bound on
the error and corresponds to $\sqrt{\epsilon_{v}^{2}}\sim
0.02|\mathbf{z}^{\prime+}|$; $\Delta{\mathbf{z}}_{\parallel}\sim
0.08|\mathbf{z}^{\prime+}|$, at smallest scales. The trace noise is
$\sqrt{3\epsilon_{v}^{2}}\sim 9$ km/s or $\sqrt{3\epsilon_{v}^{2}}\sim
0.05|\mathbf{z}^{\prime+}|$. Smallest $\Delta\mathbf{z}_{\perp}^{-}$
fluctuations are at $\sim|0.1\mathbf{z}^{\prime+}|$, Fig. 2(c). Sensitivity in
the nonlinear cross-term structure functions is bounded assuming
$\Delta\mathbf{z^{-}}$ at the smallest scale is purely noise $\epsilon_{\Delta
z^{-}}=\Delta\mathbf{z}_{\tau_{min}}^{-}$; the scale dependent noise is
computed as $\epsilon_{\Delta z^{-}}\times\Delta{\mathbf{z}}_{R}$.

Figure 3: Cross-term structure functions
$\langle|\Delta\mathbf{z}^{+}_{R}||\Delta\mathbf{z}^{-}_{\perp}|\rangle$ and
$\langle|\Delta\mathbf{z}^{+}_{R}||\Delta\mathbf{z}^{-}_{\parallel}|\rangle$(black,
blue +). Cross-term structure functions under consideration of alignment
$\langle|\Delta\mathbf{z}^{+}_{R}\times\Delta\mathbf{z}^{-}_{\perp}|\rangle$
and
$\langle|\Delta\mathbf{z}^{+}_{R}\times\Delta\mathbf{z}^{-}_{\parallel}|\rangle$
(black, blue,*).Magnetic $1/2{\Delta b^{2}}$ and thermal pressure $\Delta
p_{th}$ terms are shown in green + and $\diamond$. An error estimate
$\epsilon_{\Delta z^{-}}\times\Delta{\mathbf{z}}_{R}$ described in the text is
shown in orange.

## Conclusions

Analysis of MHD fluctuations in the dHTf enables the study of Alfvénic
turbulence in a large-amplitude limit, in which the dominant mode corresponds
to constant magnitude rotations Barnes and Hollweg [1974], Goldstein et al.
[1974]. The Alfvénic constant $|B|$ state is a nonlinear equilibrium solution
satisfied in physical space that affects all scales contributing significantly
to $\delta B$. Our development of rotational increments shows that the
constant magnitude condition, and large amplitude fluctuations $\delta
B$/$B>0.3$, can be maintained deep into scales traditionally associated with
an “inertial range” (Fig. 2d-e) Kraichnan [1965], Goldreich and Sridhar
[1995]. Our observations suggest that fluctuations across all MHD scales may
maintain correlations to keep $|B|$ constant Matteini et al. [2018].

While incompressible shearing between counter-propagating Alfvén waves is
often invoked in describing magnetized Alfvénic turbulence, theories are
typically developed in limits that omit correlations that maintain the
constant magnitude conditionKraichnan [1965], Coleman [1968], Strauss [1976],
Grappin et al. [1982], Dobrowolny et al. [1980], Goldreich and Sridhar [1995],
Boldyrev [2006], Beresnyak and Lazarian [2006], Schekochihin et al. [2009],
Boldyrev [2005], Mason et al. [2006], Podesta et al. [2009], Chandran et al.
[2015], Mallet et al. [2015], Howes [2015], Mallet et al. [2016]. The presence
of correlations between that maintain constant magnitude throughout MHD-range
fluctuations, suggest that finite-amplitude dynamics Derby [1978], Viñas and
Goldstein [1991], Del Zanna [2001], Tenerani and Velli [2013], Squire et al.
[2017], Bowen et al. [2018b], Chandran [2018], González et al. [2020],
Tenerani et al. [2020] are likely important nonlinear processes. Finite-
amplitude effects may contribute to the growth and evolution of MHD-scale
compressible fluctuations, as well as kinetic processes Matteini et al.
[2010], González et al. [2021].

Our observations show that large-amplitude fluctuations in the solar wind are
highly aligned and that measured nonlinearities are not in agreement with
common understandings of alignment on Alfvénic turbulence Boldyrev [2006].
Across all scales, the Alfvénic (rotational) increments of ${\bf z}^{+}$ are
aligned with Alfvénic (perpendicular) increments of ${\bf z}^{-}$, such that
$\langle|\Delta\mathbf{z}_{R}^{+}\times\Delta\mathbf{z}_{\perp}^{-}|\rangle\ll\langle|\Delta\mathbf{z}_{R}^{+}||\Delta\mathbf{z}_{\perp}^{-}|\rangle$.
Strong alignment only affects the Alfvénic part of ${\bf z}^{-}$: the parallel
${\bf z}^{-}$ increment is not strongly aligned
($\langle|\Delta\mathbf{z}_{R}^{+}\times\Delta\mathbf{z}_{\|}^{-}|\rangle\approx\langle|\Delta\mathbf{z}_{R}^{+}||\Delta\mathbf{z}_{\|}^{-}|\rangle$),
such that
$\langle|\Delta\mathbf{z}_{R}^{+}\times\Delta\mathbf{z}_{\perp}^{-}|\rangle\approx\langle|\Delta\mathbf{z}_{R}^{+}\times\Delta\mathbf{z}_{\|}^{-}|\rangle$
even though
$\langle|\Delta\mathbf{z}_{\|}^{-}|^{2}\rangle<\langle|\Delta\mathbf{z}_{\perp}^{-}|^{2}\rangle$.
The nonlinearity associated with the magnetic pressure is weak relative to the
shear interactions, suggesting that the waves relax to the constant magnitude
state efficiently Cohen and Kulsrud [1974]. Recent-work shows that small
perturbations in magnetic pressure may be an effect of expanding large-
amplitude Alfvén waves with oblique wave-vectors Mallet et al. [2021].

Interpreting these results hinges on the relevance of alignment to strength of
nonlinear interactions. If the Alfvénic nonlinearity is depleted by
$\sim\\!\sin\phi_{\bf k}$, Boldyrev [2006], then our results imply that the
non-Alfvénic nonlinearity (involving ${\bf z}^{-}_{\|}$) is approximately
equal to the Alfvénic shear-nonlinearity, challenging the basis for RMHD
turbulence phenomenologies. If alignment is not directly related to the
nonlinearity, which occurs if
$\delta\mathbf{z}^{\mp}\cdot\nabla\delta\mathbf{z}^{\pm}\approx\delta\mathbf{z}^{\pm}\cdot\nabla\delta\mathbf{z}^{\mp}$,
then our results imply that ${\bf z}^{+}$ and ${\bf z}^{-}$ are anomalously
coherent, despite propagating in opposite directions. Either way, our results
do not support a dynamically aligned cascade phenomenology often invoked to
explain observed spectra Boldyrev [2005, 2006], Mason et al. [2006]. Our
results further suggest that alignment has little bearing on the development
of inertial range turbulence from outer scales Wicks et al. [2013a, b]. The
possibility that nonlinearity is unrelated to alignment, and that the Elsasser
variables are anomalously coherent, resembles theories of reflection-driven
turbulence Perez and Chandran [2013], Velli et al. [1989], Chandran and Perez
[2019], where ${\bf z}^{-}$ is strongly aligned with ${\bf z}^{+}$ because it
is driven directly by wave reflection from large-scale gradients. In any case,
our results question the relevance of RMHD turbulence theories to the inner-
heliosphere and provide constraints on dynamics of finite-amplitude
turbulence.

###### Acknowledgements.

The authors thank Prof. Benjamin Chandran, Dr. Christopher H.K Chen, Dr.
Romain Meyrand and Dr. Julia Stawarz for useful discussions surrounding this
manuscript. TAB is suppored through NASA Grant NNN06AA01C KGK is supported by
NASA Grants NNN06AA01C and 80NSSC19K0829. Support for JS was provided by
Rutherford Discovery Fellowship RDF-U001804 and Marsden Fund grant UOO1727,
which are managed through the Royal Society Te Apārangi.

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