11affiliationtext: Center for Soft Matter Research, Department of Physics, New
York University 726 Broadway Avenue, New York, NY 10003, U.S.A.
22affiliationtext: Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, Maryland 21218, U.S.A. 33affiliationtext: School of
Physics and Astronomy, and the Center for Physics and Chemistry of Living
Systems, Tel Aviv University, Tel Aviv 6997801, Israel

### Figures

Figure 1: Experimental setup to drive micro-rotors asynchronously. a Optical
setup for introducing a broad ($D\approx 440\;\mu\rm{m}$) circularly polarized
beam into a microscope sample. b Schematic and polarized microscopy image of
birefringent vaterite particles rotating while moving freely in the
illuminated region. The transmitted light intensities of two particles (blue
and green) are tracked over the duration of the experiment and are shown in c
and the inset of d. c One-half of the particles’ (blue and green) blinking
cycle, demonstrating that their optical axes are asynchronous. The incident
electric field – whose direction is set by the orientation of the polarizer
(P) – is de-polarized whenever the optical axis of the rotating particles is
aligned with neither the polarizer nor analyzer (A). d Computing the magnitude
of the Fourier transform ($\sqrt{\mathcal{F}\mathcal{F}^{*}}$) of the blinking
patterns (inset) of the two particles in b shows that the frequencies at which
the particles de-polarize the incident L.E.D. light are centered around
$0.5\;\rm{Hz}$, corresponding to a rotation frequency of 0.125 Hz. The
magnitude of the sum of transforms, $\left|\sum_{i}\mathcal{F}_{i}\right|^{2}$
(solid line), decays, confirming that the particles’ orientations are out of
phase. Scale bar: $5\;\mu\rm{m}$; Figure 2: Synthesis of silica-coated
vaterite particles. a Synthesis procedure starting with (1) mixing of equal
volumes of buffered $\rm{CaCl_{2}}$ and $\rm{Na_{2}C.O._{3}}$ for 10s with a
magnetic stir bar, followed by a two-step coating procedure starting with (2)
APTMS The $\rm{CaCl_{2}}$-A.T.P.M.S. solution was placed in a shaker for
approximately 2.5 hours. Particles were then coated with (3) TEOS and placed
in a shaker for 5.5 hours. The two-step coating procedure was then repeated.
Figure 3:  The translational and rotational diffusion of an individual
particle is described by a sphere near a non-slip wall in Stokes flow. a
Snapshots (and time-colored trajectory) of a birefringent vaterite particle
viewed through crossed-polarizers, showing Brownian translation and rotation.
b Near a no-slip wall (solid line), particles have lower translational
diffusion $D_{t}$ relative to particles in bulk, consistent across a measured
size range of $d=2-6\;\mu\rm{m}$. Inset shows the translational mean-squared
displacement (MSDs) for particles with sizes $d=1.92\pm 0.11\mu$m and $5.92\pm
0.35\mu$m. The translational diffusion constants $D_{\rm{t}}$ obtained from
these two MSDs are indicated by grey arrows in the main panel. c Rotational
diffusion perpendicular to the wall, (spinning) $D_{\rm{r},\perp}$, measured
using depolarization intensity decorrelation, $g_{\rm{PA}}$.
$D_{\rm{r},\perp}$ is largely unchanged by the presence of a no-slip wall
(solid line). The blue arrow indicates $D_{\rm{r},\perp}$ obtained from
fitting $g_{\rm{PA}}$ for a $d=2.31\pm 0.14\mu$m particle (inset). Scale bar:
$5\;\mu\rm{m}$. Figure 4: Measurement of rotational and translational dynamics
in a uniform optical torque field. Individual rotors freely diffuse and spin
in a force-free optical torque field. a The ensemble-averaged mean square
displacement $\langle\Delta r^{2}\rangle$ is linear (diffusive) for different
photonic fluxes $J$. A representative trajectory of a single particle for
different photonic fluxes is shown in the inset. b Translational diffusion
increases with $J$ as the gravitational height $h_{g}$ increases (solid line)
until it approaches the bulk value (dashed line) c The effective birefringence
of a particle, $\Delta n$, can be measured by monitoring the minimal
ellipticity, $\phi$, where rotation begins for different particle sizes
(inset). d Measured spin rates for different fluxes and different particle
sizes (inset) as predicted by Eq. S4 (solid lines), are consistent with
measured birefringence. Figure 5: Flow field of a single rotor spinning near a
wall. Geometry of a rotating sphere and its corresponding image charge
generated near a solid no-slip wall while rotating about an axis perpendicular
to the wall. Figure 6: Pairs of free rotors couple hydrodynamically through
rotation and translation. a Diagram of an orbiting pair near a wall with
corresponding image charges. b Snapshot of two mutually advecting particles. c
Angular speed $\omega$ of two orbiting spheres (diameters $6.7\;\mu$m and
$5.1\;\mu$m) at different separations, along with the dependence of the
spinning rate $\Omega$ on the normalized separation (inset). Curves show the
predicted spinning rates for each particle at different separations – derived
from Faxen’s laws (Eqs. 6 and 7 in the main text) – given the particles’
asymptotic spinning rate, $\Omega_{i,j}^{0}$, measured at large separations
(colors correspond to panel b). d The change in spin $\Delta\Omega$, and the
orbital frequency $\omega$ for rotors of varying size, separation, and optical
flux follows a geometric relation given by Eq. LABEL:eqSpinOrbit, where
$\alpha=4$ (solid line). The scatter relative to the trend line originates
from thermal fluctuations in transient orbits of freely diffusing Brownian
rotors (see Supplementary Information). Scale bar: $5\;\mu\rm{m}$.

Hydrodynamic spin-orbit coupling in asynchronous optically driven micro-rotors
Supplementary Information

## Optical setup

Our instrument is designed to apply a uniform optical torque field by
delivering a collimated beam to the sample plane. A schematic diagram of the
setup can be seen in Supplementary Fig. S1a. The experimental system consists
of a 3mm infrared (IR) laser, $\lambda=1064$nm (IPG Photonics, $M^{2}=1$),
propagating first through a half-wave plate ($\lambda/2$) followed by a
Galilean beam contractor. The telescope can shrink the beam by a factor of 30,
while preserving the initial collimation of the laser source. Once the beam
diameter has been reduced, the laser propagates through a polarizing beam-
splitter (PBS) cube where the S-component of light is reflected upwards. The
handedness of the linearly polarized light can be altered via a quarter wave
plate ($\lambda/4$) fixed immediately after the PBS atop a precision manual
rotation mount.

We image the distribution of the incident near-infrared beam using an infrared
viewing card. The pixel intensities of the image are then fit to a Gaussian
surface of revolution to obtain an intensity distribution of the beam flux at
the sample plane. The flux distribution $J\left(\bm{r}\right)$ can be written
as

$J\left(\bm{r}\right)=J_{0}\exp\left[-2\frac{\textbf{r}^{2}}{\sigma^{2}}\right]$
(S1)

The prefactor corresponds to the flux at the center of the beam and is
obtained by integrating the 2-dimensional Gaussian over the area of the beam.
Supplementary Fig. S1b shows a typical $J\left(\bm{r}\right)$ for
$\sigma=372.8\pm 5.6\mu$m (FWHM = $438.9\pm 6.6\mu$m). We can adjust the flux
at the sample by changing the size of the incoming beam, modulating the power
at the laser head, or tuning the orientation of the $\lambda/2$\- plate.

We illuminate the sample using a 505 nm LED (Thorlabs) to image our photonic
rotors. The light emitted by the LED passes through a N-BK7 ground glass
diffuser (600 grit) and is then pseudo-collimated with an N-BK7 plano-convex
lens (f=25.4mm) that acts as a primary collector lens. After the plano-convex
lens, a secondary achromatic doublet lens (f=76.2mm) forms an image of the
filament at the position of a field diaphragm. A condenser lens focuses the
light at the objective’s back focal plane. Particles are imaged using an
infinity-corrected Leica HCX PL APO 40x objective (NA=0.85). High-pass index
matched IR filters mounted before the camera are used to prevent interference
of IR light with the 3.1MP monochrome camera (Imaging Source).

Figure S1: Optical setup used to create a torque field to drive birefringent
micro-rotors. a Detailed schematic of the optical elements used to couple the
imaging and laser systems to deliver a defocused beam to the sample plane. b
Broad field image of the laser beam (contours show relative power given by
Gaussian fit). Experiments were performed in a region $\sim 165\mu$m x
$125\mu$m, where gradients in the flux are negligible.

## Measuring rotational diffusion using transmission decorrelation through
crossed-polarizers

Rotational diffusion is measured by calculating the auto-correlation of the
depolarization of the light intensity transmitted through the particles, as
measured using cross-polarizers on our custom-built microscope (see
Supplementary Fig. 1(a)). Following previous work 36, 37, we define the
intensity auto-correlation function as $g_{PA}(\tau)=\langle
I_{PA}^{*}(0)I_{PA}(\tau)\rangle/\langle|I_{PA}|^{2}\rangle$ where $I_{PA}$ is
the time-averaged scattered intensity field detected by the camera, and $\tau$
is the lag time. To leading order, the translational and rotational dynamics
are uncoupled, thus enabling direct measurement of the diffusion matrix
$\mathbf{D}_{r}$ by fitting $g_{PA}=\exp(-6\mathbf{D}_{r}\tau)$ (Fig. 3 in the
main text). In the lubrication limit, diffusive rolling, $D_{r,||}$ (rotation
on an axis parallel to the wall) is much slower than diffusive spinning
(rotation perpendicular to the wall $D_{r,\perp}$) which dominates the
diffusion matrix, $\mathbf{D}_{r}\approx D_{r,\perp}$ 37.

## Measuring the translational diffusion constant of rotors

We record videos of freely-diffusing vaterite micro-spheres and track their
instantaneous positions of particles using the Python package TrackPy. TrackPy
was downloaded and used without further modification 30. The package
implements the widely used Crocker-Grier particle tracking algorithm 31, which
identifies local brightness maxima within an image as candidate particle
locations. We track particles diffusing throughout the movie and extract their
diffusion constants by computing their mean squared displacements,
$\langle\Delta r^{2}\rangle=4D_{t}\tau$ for different lag-times $\tau$. The
y-intercept of a linear fit in log directly measures $D_{t}$.

## Measuring the reflection coefficient of a vaterite rotor

### Quantifying the back-scattered radiation by a photonic rotor

Back-scattered light from a photonic rotor generates a radiation pressure that
counteracts the gravitational force $F_{g}=\pi\Delta\rho gd{{}^{3}}/6$
(Supplementary Fig. S2). For a given flux $J$, the average force on a particle
with diameter $d$ is given by $\langle F_{rad}\rangle=\frac{RJ\pi d^{2}}{4c}.$
Here $c$ is the speed of light, and $R$ is the reflection coefficient. For
vaterite micro-spheres, absorption of IR light is negligible 16. Micro-spheres
travel at a constant velocity through the $100\mu$m capillary when $\langle
F_{rad}\rangle>F_{g}$. This upward motion is balanced by the viscous drag
force $F_{D}=3\pi\eta dv$. Here $v$ is the average rise velocity of the
particle as it travels through a $100\mu$m tall capillary. The procedure for
measuring $v$ can be found in the subsection below.

Balancing the forces acting on the particle gives,

$\frac{RJ_{0}}{12c\eta}-\frac{1}{18}\frac{\Delta\rho g}{\eta}d=\frac{v}{d},$
(S2)

where the buoyant density $\Delta\rho$ = 1.43 g/cc for vaterite particles
suspended in $\rm{D_{2}O}$. The velocities for different particle sizes and
fluxes are shown in Supplementary Fig. S2. $R$ is extracted by linear fit of
Eq. S2 to the data. We find that $R=0.22\pm 0.01$ – to our knowledge, this is
the first direct measurement of the reflection coefficient of vaterite micro-
spheres and is in good agreement with spherical particles of similar indices
of refraction 18.

### Measuring the average rise velocity $v$ of a particle in the presence of
an optical flux

When $\langle F_{rad}\rangle>F_{g}$, particles steadily rise. To monitor the
out-of-plane motion of the particles, we first focus on the particles as they
freely diffuse in the plane in the presence of no optical flux. Using a
translation stage, we vertically shift the imaging focal plane by $100\mu$m,
corresponding to the capillary’s height where the particles are suspended. We
time how long it takes for the particles to re-appear in focus at the top of
the capillary. The depth of field of our 40x objective is $\approx 1\mu$m –
less than the diameter of the particles – ensuring good accuracy of our
measurements.

Figure S2: Linear fits to the ratio between the average rise velocity and
particle size $v/d$ allows for direct measurement of the reflection
coefficient R for vaterite microspheres. Across three different fluxes, we
compute an average $R=0.22\pm 0.01$. Buoyant densities obtained from the slope
of the lines vary by $<10\%$ from the expected value.

## Spinning angular frequency $\Omega$ of a birefringent particle

In the low-Reynolds number limit, a sphere spinning in a fluid at angular
frequency $\Omega$ experienced a viscous torque $\tau_{\rm{v}}=\pi\eta
d^{3}\Omega$, dependant on the viscosity of the fluid $\eta$ and the
particle’s diameter, $d$. $\tau_{\rm{v}}$ is balanced by the optical torque,
$\tau_{\rm{p}}$, that is responsible for generating the spinning motion. The
magnitude of $\tau_{\rm{p}}$ depends on the incident light’s polarization
ellipticity $\phi$ and wavelength $\lambda$ according to 55,

$\tau_{\rm{p}}=\frac{\lambda PT}{2\pi c}\left[\left(1-\cos\frac{2\pi\Delta
nt}{\lambda}\right)\right]\sin 2\phi-\sin\left(\frac{2\pi\Delta
nt}{\lambda}\right)\cos 2\phi\sin 2\theta.$ (S3)

Here, $P$ represents the power incident on the particle, and $c$ is the speed
of light. The first term of Eq. S3 corresponds to the ”spinning torque”; the
argument of $\cos$ can be thought of as the degree to which the particle acts
as a phase-retarder, providing the maximum change in the transmitted light’s
spin-angular momentum when the particle is exactly the thickness of a half-
wave plate. Therefore, it must depend on the physical properties of the
particle, such as its birefringence $\Delta n$, transmissivity $T$, and
thickness $t$. The second term in Eq. S3 is a torque that seeks to align the
fast axis of the particle with the plane of polarization. To derive an
equation for the spinning rate in the presence of a circularly polarized
incident beam $(\phi=\pi/4)$, we approximate the spherical particles as
cylinders so that their diameter $d\approx t$ 40. In this limit, the alignment
torque vanishes, and the optical torque is proportional to $\Delta\sigma\equiv
1-\cos\left(\frac{2\pi\Delta nd}{\lambda}\right)$. Rewriting $P$ in terms of
the power flux $J$, $P=J\pi d^{2}/4$, the optical torque becomes
$\tau_{\rm{p}}=\Delta\sigma TJ\lambda d^{2}/8c$. Equating $\tau_{\rm{p}}$ and
$\tau_{\rm{v}}$ and solving for $\Omega$ gives,

$\Omega=\frac{\rm{T}J\lambda}{8\pi c\eta d}\left[1-\cos\left(\frac{2\pi\Delta
nd}{\lambda}\right)\right].$ (S4)

Figure S3: Characterization of synthesized vaterite micro-spheres. a SEM
micrograph of vaterite microspheres. Inset shows the poly-crystalline
structure of the particle’s surface. b Size distribution of vaterite
microspheres.

## Measuring the frequency and phase of a rotating particle

When spinning, the angle $\theta$ a vaterite particle’s optical axis makes
with incident linearly polarized light varies periodically so that a
particle’s rotation frequency $f\propto\dot{\theta}$. Throughout one period,
vaterite particles depolarize the transmitted light from the polarized LED
source four times. The observed depolarizations, or ”blinks”, occur whenever
the optical axis of the particle coincides with neither the axis of the
polarizer nor the analyzer (Fig. 1c). Tracking the transmitted light
intensities of individual particles as a function of time,
$I_{i}\left(t\right)=I_{0}\sin^{2}\left(2f_{i}t+\Phi_{i}\right)$, allows for a
direct measure of each particles’ rotation frequency and phase, $\Phi_{i}$. We
normalized all signals so that $I_{0}=1$.

To measure $\Phi_{i}$ and $f_{i}$ of rotating particles, we compute the
Fourier transform of their transmitted light intensities

$\displaystyle\mathcal{F}_{i}=\mathcal{F}_{i}\left[I_{i}\left(t\right)\right]\left(\omega\right)$
$\displaystyle\equiv\int dte^{-\mathrm{i}\omega t}I_{i}\left(t\right)$ (S5)
$\displaystyle\propto-
I_{0}e^{2\mathrm{i}\Phi_{i}}\delta\left(\omega-4f\right),$

where we have assumed that $\omega,\Phi_{i},f$ are positive quantities. Eq. S5
allows us to compare the phases of multiple particles globally. For individual
particles, the magnitude of the Fourier transform
$\sqrt{\mathcal{F}_{i}\mathcal{F}_{i}^{*}}\propto\delta\left(\omega-4f\right)$.
However, the sum of N-individual Fourier transforms,
$F_{N}=\left|\frac{1}{N}\sum_{i}^{N}\mathcal{F}_{i}\right|^{2}$, decays with
the number of particles. Explicitly,

$\displaystyle F_{N}$
$\displaystyle=\left|\frac{1}{N}\sum_{i}^{N}\mathcal{F}_{i}\left[I_{i}\left(t\right)\right]\left(\omega\right)\right|^{2}$
(S6) $\displaystyle=\left|\frac{1}{N}\sum_{i}^{N}\int dte^{-\mathrm{i}\omega
t}I_{i}\left(t\right)\right|^{2}$
$\displaystyle\propto\frac{I_{0}^{2}}{N^{2}}\delta^{2}\left(\omega-4f\right)\left[N+\sum_{i}^{N}\sum_{k}^{N}e^{2\mathrm{i}\left(\Phi_{i}-\Phi_{k}\right)}\right].$

As an example, let us consider the case of $N=2$ particles, Eq. S6 becomes,

$F_{N}=\frac{I_{0}^{2}}{4}\delta^{2}\left(\omega-4f\right)\left(2+e^{2i\left(\Phi_{1}-\Phi_{2}\right)}+e^{2i\left(\Phi_{2}-\Phi_{1}\right)}\right)$
(S7)

When particles have the same global phases, $\Phi_{1}=\Phi_{2}$,
$\sqrt{F_{2}F_{2}{{}^{\star}}}=I_{0}=1$. When the optical axes of the
particles are perfectly out of phase $\Phi_{1}-\Phi_{2}=\pi/2$,
$\sqrt{F_{2}F_{2}{{}^{\star}}}=0$. For all other intermediate relative phases,
as – is the case in Fig. 1(c,d) – the asynchronous phases of the light
intensities do not add up constructively, and the amplitude of the sum of the
individual Fourier transforms is reduced.

## Flow generated by a rotating sphere near a wall in the Stokes-flow regime

Consider an isolated sphere with radius $a$ subjected to a constant torque
$\bm{\tau^{0}}=8\pi\eta a^{3}\mathbf{\Omega}_{j}^{0}$, where $\eta$ is the
fluid viscosity, and $\Omega^{0}_{j}$ is the initial angular speed of the
rotor in the absence of any neighboring rotating particles. The Green’s
function for this rotlet is 42

$\mathbf{G_{ij}}=\frac{\epsilon_{ijk}r_{k}}{r^{3}}$

and the corresponding free space flow is

$\mathbf{u_{j}\left(r\right)}=\epsilon_{ijk}\Omega_{j}^{0}a^{3}\frac{r_{k}}{r^{3}}.$

Here $\epsilon_{ijk}$ is the Levi-Cevita symbol with indices representing the
$x,y,z$ directions in the Cartesian coordinate system, and $r$ is a three-
dimensional (3D) vector, $|r|\equiv\sqrt{x^{2}+y^{2}+z^{2}}$, from the source
of the disturbance to a point $\left(x,y,z\right)$ in space.

In our experiments, rotors spin near the wall of a capillary. We consider a
rotating sphere spinning about an axis perpendicular to a nearby plane located
at $z=0$. The sphere’s center is at a height $\delta=h+a$ (see Fig. 5 in the
main text). We assume that the plane has a no-slip boundary condition,
$\left(\bm{u_{j}}(z=0)=0\right)$, so that we may approximate the far-field
fluid flow as the superposition of two rotlets with equal and opposite
torques,$\bm{\tau}=\pm\bm{\tau^{0}},$ located at $z=\pm\delta$, respectively
(Fig.5). The resulting fluid flow is,

$\bm{u}_{j}\left(r\right)=\Omega_{j}^{0}a^{3}\left[\left(\frac{1}{|R_{+}|^{3}}-\frac{1}{|R_{-}|^{3}}\right)\left(-y\mathbf{\hat{x}}+x\mathbf{\hat{y}}\right)\right]$
(S8)

Here,
$|R_{\pm}|\equiv\left(x^{2}+y^{2}+\left(z\mp\delta\right)^{2}\right)^{\frac{1}{2}}$
representing the distance to a point $\left(x,y,z\right)$ in space from the
source and image charges, respectively. Next, we consider specifically the
far-field limit $\left(\delta\ll r\right)$ where
$|R_{\pm}|=\left[(r^{2}\left(1\mp\frac{2\delta
z}{r^{2}}\right)\right]^{\frac{1}{2}}$ and
$|R_{\pm}|^{-3}\approx\frac{1}{r^{3}}\left(1\pm\frac{3\delta^{2}}{r^{2}}\right)$.
In the final equality, we make the approximation that the flow is strictly
two-dimensional (2D) and confined to the special symmetry plane located
approximately one particle radius from the wall at $z=\delta$. Eq.S8 reduces
to:

$\bm{u}_{j}\left(r\right)=\frac{6\Omega_{j}^{0}a^{3}\delta^{2}}{r^{4}}\bm{\hat{\theta}}.$
(S9)

Note that the $r^{-4}$ scaling is a result of the conversion from Cartesian
(with basis vector $\bm{\hat{x}},\bm{\hat{y}}$) to polar coordinates (with
basis vectors $\bm{\hat{r}},\bm{\hat{\theta}}$). The polar component of the
flow, $\bm{\hat{\theta}}$, has been rewritten using
$-r\sin\theta\bm{\hat{x}}+r\cos\theta\bm{\hat{y}}=-y\bm{\hat{x}}+x\bm{\hat{y}}=r\bm{\hat{\theta}}$.
In Eq. S9, $r$ is now the distance in the 2D plane.

Faxen’s first law 56, $\bm{F}_{i}=6\pi\eta
a\left\\{\left[\bm{u}_{j}\left(\bm{r}\right)+\frac{1}{6}a^{2}\nabla^{2}\bm{u}_{j}\left(\bm{r}\right)\right]_{\bm{r}=\bm{r}_{i}}-\bm{v}_{i}\right\\}$,
in the presence of no external forces ($\bm{F}_{i}=0$) reduces to

$\mathbf{v}_{i}=\frac{d\mathbf{r}_{i}}{dt}=\bm{u}_{j}\left(\bm{r}\right)+\mathcal{O}\left(\frac{1}{r^{6}}\right)$
(S10)

Eq. S10 describes the tangential velocity of particle $i$ resulting from the
advective flow field generated by particle $j$. The modification of particle
$i\rm{{}^{\prime}s}$ angular velocity (spinning rate) due to the vorticity of
$j\rm{{}^{\prime}s}$ flow is found by application of Faxen’s second law56,

$\bm{\Delta\Omega}_{i}\equiv\bm{\Omega}_{i}-\bm{\Omega}^{0}_{i}=\frac{1}{2}\nabla\times\bm{u}_{j}\left(\bm{\bm{r}=\bm{r}_{i}}\right).$
(S11)

Rewriting Eq.S9 in terms of a general flow power law, $\alpha$ and prefactor,
$A$,
$\mathbf{u}_{j}\left(r\right)=Ar^{-\alpha}{\Omega}^{0}_{j}\bm{\hat{\theta}}$.
We compute the curl as
$\left(\nabla\times\mathbf{u}_{j}\right)=\frac{1}{r}\left(\frac{\partial}{\partial
r}\left[Ar\cdot r^{\alpha}\Omega^{0}_{j}\right]\right)$
$=\mathbf{u}_{j}\left(r\right)\left(1-\alpha\right)/r$. Considering the case
where rotors are of equal size, we arrive at

$\Delta\mathbf{\Omega}=\frac{1-\alpha}{4}\bm{\omega},$ (S12)

where we have rewritten the expression in terms of the orbital frequency of
the rotating pair about their common center by using
$\bm{\omega}=2\mathbf{\omega}_{i}=2\mathbf{u}_{j}\left(r\right)/r$. For the
case of a particle near a wall, $\alpha=4$ and Eq.S12 reduces to
$\Delta\bm{\Omega}=-3\bm{\omega}/4$.

### Measuring the change in the instantaneous spinning rate $\Delta\Omega$ of
a rotor

We record videos of hydrodynamically-coupled rotors and track their
trajectories and transmitted light intensities $I_{i}\left(t\right)$ using
TrackPy 30. For each rotor pair, local maxima and minima in
$I_{i}\left(t\right)$ are found based on a topographic prominence algorithm
distributed through the SciPy library. We compute the difference between
neighboring minima and maxima in $I_{i}\left(t\right)$, proportional to $1/8$
of the particle rotation frequency. For a given window, the instantaneous
$\Omega$ is found from the mean and standard error of the minima-maxima
difference. The window is then advanced by a $1/8$ of a rotation period, and
this procedure is repeated until the entire signal is evaluated.

### Expected fluctuations in the extracted spin rate for measuring the spin-
orbit coupling

The accuracy of measuring the instantaneous rotation rate of a Brownian rotor
is limited by the time over which the orientation rate is tracked. An active
rotor that follows the overdamped Langevin equation
$\dot{\theta}=\Omega_{0}+\eta\left(t\right)$ (where $\Omega_{0}$ is the
nominal spinning rate, and $\eta\left(t\right)$ is the thermal noise term),
follows the average dynamic analogs to a Brownian particle undergoing thermal
diffusion in 1D:

$\begin{split}&\langle\Delta\theta\rangle=\Omega_{0}t\\\
&\langle\Delta\theta^{2}\rangle=2D_{r}t.\end{split}$ (S13)

The mean evolution of the orientation
($\Delta\theta\equiv\theta\left(t\right)-\theta\left(0\right)$) has a drift-
like term
($\langle\Delta\theta\rangle^{2}\propto\left(\Omega_{0}t\right)^{2}$). When
tracking the change in orientation over some finite time, $\tau$, the
instantaneous spinning rate can be approximated by the mean
$\langle\Omega_{0}\rangle\approx\langle\Delta\theta\rangle/\tau$. Measurement
over a finite time will also be subjected to thermal fluctuations, and hence
the measured instantaneous spinning rate will have a finite uncertainty:
$\langle\Delta\Omega_{0}\rangle\approx\sqrt{\langle\Delta\theta^{2}\rangle}/\tau\approx\sqrt{2D_{r}/\tau}$.
Therefor there is an intrinsic uncertainty for the measured spinning rate when
evaluated over a finite period of time as done in an experiment. The relative
error decreases with time as:
${\bf\rm{RE}}=\frac{\langle\Delta\Omega_{0}\rangle}{\langle\Omega_{0}\rangle}=\frac{\sqrt{2D_{r}}}{{\Omega_{0}\sqrt{\tau}}}$.

When two optical rotors interact, they make a transient orbit, until they
diffuse apart. We evaluated the instantaneous spinning rate over the duration
$\tau_{B}$ — the time between two blinking events (see Fig. 1c in the main
text). We can now compute the uncertainty in the spinning rate for a typical
$4\;\mu\rm{m}$ particle using the empirically found parameters: $D_{r}\approx
0.02\;\rm{rad}^{2}/\rm{s}$, $\Omega_{0}\approx 1\;\rm{rad}/\rm{s}$,
$\tau_{B}\approx 2\;\rm{s}$ (see Fig. 1c, Fig 3c, and Fig. 4d in the main
text). This gives relative error of ${\bf\rm{RE}}\approx 0.14$. Compared with
the nominal spinning rate, the relative error is not very large. But when
compared with $\Delta\Omega$ – that is the change in the spinning rate
(originating from the hydrodynamic spin-orbit coupling) – this thermal
fluctuation is more significant. We measure a relative change in spinning rate
$\Delta\Omega/\Omega_{0}\lessapprox 0.3$. This value is consistent with
previously theoretically predicted value 22. As can be seen in Fig. 6d, the
thermal fluctuations are small (but compatible) to the measured spin-orbit
coupling.

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# Dispersive and kinetic effects on kinked Alfvén wave packets: a comparative
study with fluid and hybrid models

Anna Tenerani<EMAIL_ADDRESS>Department of Physics,
University of Texas at Austin, TX 78712  Carlos González  Department of
Physics, University of Texas at Austin, TX 78712
<EMAIL_ADDRESS>Nikos Sioulas  Department of Earth, Planetary
and Space Sciences, University of California, Los Angeles, CA 90095  Chen Shi
Department of Earth, Planetary and Space Sciences, University of California,
Los Angeles, CA 90095  Marco Velli  Department of Earth, Planetary and Space
Sciences, University of California, Los Angeles, CA 90095

###### Abstract

We investigate dispersive and kinetic effects on the evolution of a two-
dimensional kinked Alfvén wave packet by comparing results from MHD, Hall-MHD
and hybrid simulations of a low-$\beta$ plasma. We find that the Hall term
determines the overall evolution of the wave packet over a characteristic time
$\tau^{*}=\tau_{a}\ell/d_{i}$ in both fluid and hybrid models. Dispersion of
the wave packet leads to the conversion of the wave energy into internal
plasma energy. When kinetic protons are considered, the proton internal energy
increase has contributions from both plasma compressions and phase space
mixing. The latter occurs in the direction parallel to the guiding mean
magnetic field, due to protons resonating at the Alfvén speed with a
compressible mode forced by the wave packet. Implications of our results for
switchbacks observations and solar wind energetics are discussed.

##  I introduction

After E. Parker put forward the first theory of a supersonic solar wind
(Parker, 1958), it has become clear that a thermally driven wind cannot
reproduce the highest speeds measured in-situ, reaching values up to $750-800$
km/ (Shi et al., 2022; Halejas et al, 2022), and that an additional source of
energy is required to explain both the heating of the solar corona and the
acceleration of high speed streams. Since the first observations in
interplanetary space (Coleman, 1966), turbulence and waves have been proposed
as a mechanism to heat and accelerate the solar wind plasma (Cranmer et al.,
2015). Alfvénic fluctuations indeed represent the dominant contribution to
solar wind turbulence, especially, but not limited to, the fastest streams
(D’amicis et al., 2020). Switchbacks are part of this turbulent flux
continually emitted by the sun and correspond to large amplitude Alfvénic
kinks of the magnetic field that lead to a local magnetic field polarity
reversal. Parker Solar Probe (PSP) observations have shown that switchbacks
are a common feature of the Alfvénic wind, regardless of its speed (Horbury et
al., 2020; Dudok de Wit et al., 2020).

What is the origin of switchbacks and what is their role in solar wind
dynamics and turbulence still remain important open questions. On the one
hand, it has been proposed that switchbacks are an intrinsic part of the
evolving turbulence and that they form in-situ dynamically, driven by solar
wind expansion or large scale shear flows (Landi et al., 2006; Squire et al.,
2020; Ruffolo et al., 2020). Alternatively, it has been argued that
switchbacks are kinked magnetic field lines resulting from interchange
reconnection in the corona. In the latter scenarios, switchbacks are
considered as the outcome of processes that may generate the wind itself
(Schwadron and McComas, 2021; Zank et al., 2020; Drake et al., 2020). An
analysis of the occurrence rate of switchbacks as a function of radial
distance has shown that the probability to observe longer duration switchbacks
increases with radial distance, while the probability to observe shorter
duration switchbacks, shorter than a few tens of minutes, decreases. These
observations suggest that the dynamics of switchbacks is complex and scale
dependent, and that different types of switchbacks may coexists — those formed
closer to the corona and that gradually decay or degrade as they propagate
out, and those formed in-situ as the turbulence evolves with radial distance
(Tenerani et al., 2021).

An important property of Alfvénic fluctuations in the solar wind, including
switchbacks, is that they are characterized by a nearly constant magnetic
field amplitude (Matteini et a., 2015). Large amplitude monochromatic or
broadband magnetic and velocity fluctuations correlated like Alfvén waves are
an exact solution — although unstable — to the nonlinear compressible
Magnetohydrodynamic (MHD) model, provided the total magnetic field magnitude
is also constant (Derby, 1978; Malara and Velli, 1996; Tenerani and Velli,
2013). In previous work, we showed via MHD simulations that the parametric
instability leads to the disruption of an Alfvén wave packet similar to a
switchback over a timescale that can reach up to a few hundreds of Alfvén
times (Tenerani et al., 2020). Those results support the idea that some of the
observed switchbacks may be formed back in the corona and then propagate out
to distances of a few tens of solar radii, before eventually decaying. On the
other hand, switchbacks occur over a wide range of scales. They can be as long
as several hours (MHD scales), and as short as a few seconds, approaching the
proton cyclotron period. Switchbacks can therefore be affected also by
dispersion and other kinetic effects faster than expansion and MHD processes,
like the parametric decay instability mentioned above. Wave activity at the
scale of a few seconds at switchback boundaries and degraded switchbacks with
signatures of magnetic holes have been reported during the first encounter of
PSP (Farrell et al., 2021). Emission of dispersive waves can thus provide
another channel for switchbacks’ evolution and disruption. The goal of this
work is to determine how and on what timescales dispersion and wave-particle
interactions affect the dynamics of Alfvénic wave packets such as switchbacks.

It is known that broadband Alfvénic fluctuations are no longer an exact
solution to the Hall-MHD model due to dispersion. Exact nonlinear solutions to
the Hall-MHD system still exist, but in the form of monochromatic right or
left-handed circularly polarized waves (Sakai and Sonnerup, 1983; Hollweg,
1994). Nonlinear Alfvénic solutions to reduced equations, namely, the DNLS
(derivative nonlinear Schrödinger) equation, have also been found in the form
of one-dimensional solitary wave-packets with a modulated envelope (Mjholus
and Wyller, 1986; Spangler, 1989, 1990). The effect of dispersion on large
amplitude Alfvénic fluctuations in plane geometry has been investigated
extensively in the past (Spangler, 1985; Hollweg, 1994; Champeaux et al.,
1999; Buti et al., 2000; Araneda et al., 2008; Matteini et a., 2010; González
et al., 2021) but, to the best of our knowledge, never for a two-dimensional
wave packet such as the one considered here.

In this work we consider a two-dimensional Alfvén wave packet with constant
magnetic and thermal pressure in quasi-parallel propagation and investigate
the role of dispersive and kinetic effects on its dynamics by comparing
results from MHD, Hall-MHD and hybrid simulations. In section II we describe
the numerical models used and the initial conditions for our simulations.
Results are reported in Section III. We discuss the implications of our
results on switchbacks lifetime and observations in Section IV and we provide
a summary in Section V.

Table 1: Summary of the simulation runs, where $\ell_{x}$ is the longitudinal length of the wave packet, $d_{i}$ the proton inertial length, $\tau_{a}=B_{0x}/\sqrt{4\pi\rho_{0}}$ and $\tau^{*}=\tau_{a}\ell_{x}/d_{i}$.  | $(L_{x}\times L_{y})/\ell_{x}$ | $\ell_{x}/d_{i}$ | $\beta$ | $\tau_{*}/\tau_{a}$
---|---|---|---|---
run 0 (MHD) | $33.5\times 8.37$ | $\infty$ | 0.5 | $\infty$
run 3 (Hall) | $150$ | 150
run 1 (Hall) | $30$ | 30
run 2a (Hall) | $6$ | 6
run 2b (hybrid) | $34.13\times 8.53$ | $6$ | $\beta_{p,e}=0.25$ | 6

##  II Numerical models

We make use of a 2.5D numerical code (two-dimensional domain and three-
dimensional vectors) that integrates the full set of compressible Hall-MHD
equations in conservative form, where an adiabatic closure is assumed (Shi et
al., 2019). The code adopts periodic boundary conditions and derivatives are
calculated via the Fast Fourier Transform. An explicit third-order Runge-Kutta
method is used for the time integration and the Courant–Friedrichs–Lew
condition is used to determine the appropriate time step. We do not impose
explicit resistivity and use instead a pseudo-spectral filter to avoid energy
accumulation at the grid-scale.

Simulations are initialized with a two-dimensional analytical model for a
switchback as discussed in Landi et al. (2005) and Tenerani et al. (2020). The
magnetic field is defined starting from the two-dimensional magnetic scalar
potential $\psi(x,y)$,

${\bf
B}={\bm{\nabla}}\times\psi(x,y){\bf\hat{z}}+B_{z}(x,y){\bf\hat{z}}+B_{x0}{\bf\hat{x}},$
(1)

where

$\psi(x,y)=-\psi_{0}\left(e^{-r_{1}^{2}}-e^{-r_{2}^{2}}\right),$ (2)
$r_{1,2}^{2}=\left(\frac{x-x_{1,2}}{\ell_{x}}\right)^{2}+\left(\frac{y-y_{1,2}}{\ell_{y}}\right)^{2}.$
(3)

The component $B_{z}$ of the magnetic field is then determined by imposing a
constant total magnetic field strength $B$,

$B_{z}(x,y)^{2}=B^{2}-[B_{x}(x,y)^{2}+B_{y}(x,y)^{2}].$ (4)

The velocity fluctuation ${\delta\bf u}$ follows directly from the Alfvénicity
condition, ${\delta\bf u}=-{\bf\delta B}/\sqrt{4\pi\rho_{0}}$, ${\bf\delta B}$
being the fluctuating magnetic field (Primavera et al., 2019; Tenerani et al.,
2020).

The setup described above corresponds to a wave-packet similar to a switchback
localized in the $(x,y)$ plane, with initial constant magnetic field strength,
that propagates in a homogeneous plasma with density $\rho_{0}$, pressure
$p_{0}$, and guiding mean field ${\bf B}_{0x}$ that we take in the
${\bf\hat{x}}$ direction. Lengths are normalized to a reference length $L$,
the magnetic field to $B_{0x}$, density to the background density $\rho_{0}$,
speed to the corresponding Alfvén speed. In these units, the wave packet has
length $\ell_{x}=1.5$, width $\ell_{y}=1$, and we set
$|x_{1}-x_{2}|=|y_{1}-y_{2}|=2$. We have considered three Hall-MHD cases, each
corresponding to a different value of the normalized proton inertial length,
$d_{i}=0.25,0.05,0.01$ (where $d_{i}=v_{a}/\Omega_{ci}$, with
$\Omega_{ci}=eB_{0x}/(m_{i}c)$ and $v_{a}=B_{x0}/\sqrt{4\pi\rho_{0}}$), and
one MHD case ($d_{i}=0$). Decreasing $d_{i}$ allows us to increase the scale
separation between the typical length of the wave packet and dispersive
scales. In all of the runs the plasma beta $\beta=8\pi p_{0}/B_{0x}^{2}$ is
set to $\beta=0.5$ and the mesh resolution is $\Delta x=\Delta y=0.1$ with a
domain size of $L_{x}\times L_{y}=(8\times 2)2\pi$. The mesh resolution has
been chosen to well resolve the dynamical evolution of system. A summary of
the numerical parameters used is reported in Table 1.

The analytical form of the magnetic field given in eq. (1)-(4) includes a
finite mean magnetic field ${\bf B}_{0z}$ in the $z$-direction, which is
required to maintain a constant magnetic field strength and whose magnitude
depends on the amplitude of the fluctuation $\psi_{0}$. The mean field ${\bf
B}_{0z}$ mimics self-consistently the Parker spiral magnetic field (at a fixed
angle). We therefore consider a small but finite amplitude fluctuation,
$\psi_{0}=0.25$, which allows us to simulate an Alfvénic wave-packet
propagating at an angle $\theta=\arctan(B_{z0}/B_{x0})=0.61$ ($\theta\simeq
35^{\circ}$). The corresponding minimum value of the longitudinal field
$B_{x}$ is $B_{x}^{(min)}=0.76$. A larger value for $\psi_{0}$ leading to a
longitudinal field reversal with $B_{x}\lesssim 0$ would require a larger
propagation angle, which strongly affects dispersion properties and would make
a comparison with observations difficult. The amplitude of the fluctuations
may also affect parametric instabilities. Here we have chosen the parameters
so that the Alfvén wave-packet is in quasi-parallel propagation and stable in
the MHD limit. This allows us to focus solely on dispersive effects.

To investigate proton kinetic effects, we performed 2.5D (2D 3V for particles)
simulations with the hybrid code CAMELIA (see, e.g., Franci et al. (2018)) in
which electrons are described as a massless isothermal fluid. The hybrid code
is periodic, and it uses the current advance method Matthews (1994) and Boris
scheme for the particle pusher. Explicit resistivity ($\eta=0.001$ in units of
$4\pi/cv_{a}\omega_{pi}^{-1}$, where $\omega_{pi}=\sqrt{4\pi ne^{2}/m_{i}}$)
has been added to improve conservation of energy by avoiding the formation of
magnetic fluctuations at the grid scale. In the hybrid code, lengths are
normalized to $d_{i}$ and time to the inverse of the proton gyrofrequency
$\Omega_{ci}$. We impose the same initial condition as in the Hall-MHD
simulations, an initial Maxwellian proton distribution and
$\ell_{x}/d_{i}\simeq 6$, so that the hybrid simulation can be compared with
run 2a. We use 2000 particles-per-cell, a mesh resolution of $\Delta x=\Delta
y=0.2d_{i}$ and a domain size of $L_{x}\times L_{y}=(204.8\times 51.2)d_{i}$.
In the hybrid simulation energy is conserved within 0.016% and in the fluid
code within $6\times 10^{-7}\%$. A summary of the numerical parameters is
reported in Table 1. Both fluid and hybrid simulations are performed in a
frame moving with the wave-packet at the Alfvén speed $v_{a}$.

##  III Results

###  III.1 Fluid model

Figure 1: MHD and Hall-MHD runs. Variance of magnetic and velocity fields (top
and second panel), rms of density fluctuations (third panel) and internal
energy (bottom panel) as a function of time in units of Alfvén time.

In Fig. 1 and Fig. 2 we report the overview of the evolution of the system for
the Hall-MHD and MHD simulations. Figure 1 displays the variances of the
magnetic and velocity fields (top and second panels, respectively), the root-
mean-square (rms) of density fluctuations (third panel) and the evolution of
the internal energy (bottom panel) as a function of time in units of the
Alfvén time $\tau_{a}=\ell_{x}/v_{a}$. The MHD simulation ($d_{i}=0$, red
color) remains nearly stationary. The slight decrease in the variances of
magnetic and velocity fields is due to a continuous, slow growth of
compressible fluctuations. Such a compressible fluctuations are expected to
grow at a slow rate due to parametric instabilities, however, they remain
negligible over the time interval considered, reaching a maximum value of
$\delta\rho_{rms}/\rho_{0}=0.003$. The wave packet is thus not affected by the
parametric instability in the MHD limit, at least until time $t=600\tau_{a}$.

When dispersion is included, a decrease in the kinetic and magnetic energy of
the fluctuations is observed, together with an increase of the internal energy
and of density fluctuations. In particular, the internal energy gain matches
the loss in kinetic and magnetic energy of the wave packet, that is,

$\Delta(<1/2\rho|\delta{\bf u}|^{2}+1/2|\delta{\bf B}|^{2}>)=-\Delta e_{T},$
(5)

where $\Delta g=g(t)-g(0)$ and the internal energy is $e_{T}=3/2<p>$.
Conversion of magnetic and bulk kinetic energy into internal energy is due to
the coupling of the Alfvén wave packet to compressible modes, which becomes
stronger as $\ell_{x}/d_{i}$ decreases. As a consequence, the largest internal
energy gain (for the time intervals considered) occurs in run 2a, where the
internal energy increases by $\Delta e_{T}/e_{T}(0)\simeq 0.2\%$ when the wave
packet has undergone complete dispersion and has disrupted, with
$\Delta<|\delta{\bf B}|^{2}>/<|\delta{\bf B}(0)|^{2}>\simeq-40\%$ and
$\Delta<\rho|\delta{\bf u}|^{2}>/<\rho|\delta{\bf u}(0)|^{2}>\simeq-33\%$.
Also the amplitude of density fluctuations in run 2a are larger than in the
less dispersive cases, with a value of $\delta\rho_{rms}/\rho_{0}\simeq 0.04$.
Although compressible fluctuations are generated in all of our simulations,
the wave packet does not display signatures of instabilities, such as
modulational instability, even when dispersion is included. As it will be
discussed later, the evolution of the wave packet is determined primarily by
the Hall term in Ohm’s law.

Figure 2: Hall-MHD runs. Fluctuations’ magnetic and kinetic energy as a
function of time in units of the inverse proton gyrofrequency $\Omega_{ci}$
for run 3 (top panel), run 1 (middle panel) and run 2a (bottom panel).

Figure 3: Characteristic curves of the magnetic field in the $(t,x)$ plane at
$y=L_{y}/2$. Top panels: contour plot of $B_{x}$. Bottom panels: contour plot
of $B_{y}$. From left to right, top to bottom, results for run 3, run 1, run
2a and run 2b are shown. Time is normalized to the characteristic time
$\tau^{*}$ and length to $\ell_{x}$. Only a portion of the spatial domain is
shown.

Figure 2 shows the evolution of the fluctuations’ magnetic and kinetic energy
density (blue and orange, respectively) for the Hall-MHD simulations. We show,
from top to bottom, results for run 3, run 1 and run 2a as a function of time
in units of $\Omega_{ci}^{-1}$. For very small but finite dispersion (run 3
and run 1) we find that magnetic and kinetic energy remain in equipartition to
a very good approximation, while an excess of kinetic energy is observed in
run 2a. In Fig. 3 we show the characteristic curves of the magnetic field
components $B_{x}$ and $B_{y}$ in the $(t,x)$ plane at $y=L_{y}/2$, for run 3
(top left panels), run 1 (top right panels) and run 2a (bottom left panels).
This set of simulations shows that the evolution of the wave packet is
characterized by two stages marked by a characteristic time
$\tau^{*}=\tau_{a}\ell_{x}/d_{i}$. The origin of the time $\tau^{*}$ will be
discussed in section IV.1.

During the first stage $t<\tau^{*}$, dispersive waves that propagate ahead and
behind the wave packet itself are emitted. Dispersive waves are found in all
the simulations with dispersion, but their amplitude is negligible for weak
dispersion (run 1 and run 3). We analize these emitted waves in Fig. 4, top
panel, where we show the same characteristic curves as in Fig. 3 for run 2a,
but for a shorter time interval. As a reference, we also show the $(t,x)$
characteristic curves of fast, slow and intermediate modes that match the
emitted waves for this run. We have identified those modes by calculating the
phase speed from the dispersion relation of the Hall-MHD system for
$kd_{i}\lesssim 1$. Although the system does not evolve through a nonlinear
turbulent cascade, the magnetic energy spectrum (not reported here) shows that
energy is transferred to that range of scales during the time interval
$t\lesssim 2.7\tau^{*}$ (or $t\Omega_{ci}\lesssim 100$). In particular, the
plotted curves correspond to phase speeds $v_{f}=\pm 1.65$, $v_{s}=-0.46$ and
$v_{i}=-0.8$ at $kd_{i}=0.87$ in the plasma rest frame. In addition to
dispersive waves, two slow-mode compressive wave packets localized in the
$(x,y)$ plane are emitted in all of the simulations with finite dispersion.
Such a compressible wave packets correspond to localized structures, of about
the size of the initial Alfvén wave packet, comprising magnetic pressure
depletions and compressions anti-correlated with density fluctuations,
propagating along the guiding field ($x$ direction) at nearly the slow mode
speed (in the MHD limit the slow mode speed for our parameters is
$v_{s}=0.5$). The two slow mode structures can be seen in Fig. 4, top panel,
in the form of a depletion and a bump of $B_{y}$, respectively, moving at a
speed of $v_{1}\simeq 0.5$ (blue dashed line) and $v_{2}\simeq 0.36$ (black
dashed line) in the plasma rest frame. For the sake of illustration, Fig. 4,
bottom panel, shows the contour plot of $B_{y}$ in the $(x,y)$ plane at
$t=1.4\tau^{*}$. In this plot, the dispersive waves are visible in the $(x,y)$
plane on the right of the wave-packet (which lies at the center); the slow
modes correspond to the localized “blobs” lying on the left of the wave
packet. Slow mode wave packets propagate at similar speeds also in run 1 and
run 3, for which we estimated $v_{1}\simeq 0.5$ and $v_{2}\simeq 0.45$ (not
shown here).

Figure 4: Hall-MHD run 2a. Top panel: contour plot of $B_{y}$ in the $(t,x)$
plane at $y=L_{y}/2$. Dashed lines represent fast forward and backward (f+ and
f$-$) modes, the backward intermediate (I-) mode and backward slow mode (s-).
Two compressible wave packets of the slow mode type are also emitted and
denoted as s1 and s2. Time is normalized to the characteristic time $\tau^{*}$
and length to $\ell_{x}$. Bottom panel: contour plot of $B_{y}$ in the $(x,y)$
plane at $t=1.4\tau^{*}$

The second stage starts at $t\simeq\tau^{*}$, when the initial Alfvén wave
packet disperses by spreading out. During the dispersion stage the kinetic and
magnetic fluctuations remain in quasi energy partition, particularly for run 1
and run 3, as can be seen from Fig. 2. Because of dispersion, large amplitude
waves appear to propagate from the trailing and leading edge of the wave
packet along the guiding field $B_{x0}$, as well as in the transverse ($y$)
direction. Dispersion leads to a significant reduction of the longitudinal
magnetic field $\delta B_{x}$ on a timescale that ranges between $10\tau^{*}$
and $40\tau^{*}$. We find that in the higher dispersion case (run 2a) the
longitudinal magnetic field has decreased by about 20% with respect to its
initial value at time $t=10\tau^{*}$, after which it remains stationary. Not
surprisingly, when dispersion is weak (run 1 and run 3) and energy is not lost
into many wave modes, the same 20% decrease occurs later, at time
$t=40\tau^{*}$, and the longitudinal fluctuation continues to slowly decrease
making the wave packet shallower with time.

In Section IV.1 we discuss a simple model to interpret the evolution of the
magnetic field over long time scales when compressibility is negligible.

###  III.2 Hybrid model

In Fig. 5 we show the overview of the time evolution of fluctuation’s kinetic
and magnetic energy density, of the density rms, and of the mean pressures for
the hybrid run. The system evolves in a way which is consistent with the Hall-
MHD runs shown in Fig. 1 and Fig. 2. In Fig. 3, bottom right panel, we show
the characteristic curves of the magnetic field in the $(t,x)$ plane at
$y=L_{y}/2$ for a comparison with the Hall-MHD model.

Although the hybrid simulation is qualitatively similar to the Hall-MHD
simulations, there are a few differences. In the hybrid model we do not
observe slow-mode wave packets emitted at early times $t<\tau^{*}$, and
density fluctuations are overall weaker, with $\delta\rho_{rms}/\rho_{0}\simeq
0.018$. Fast modes with positively correlated density and magnetic pressure
perturbations are instead emitted within a few tens of gyroperiods. Such a
fast mode perturbations are generated within the Alfvén wave packet, forced by
magnetic pressure imbalance induced by dispersive effects. Since the
compressible mode is forced, a portion of it remains stuck to the Alfvén wave
packet propagating at the Alfvén speed, and a portion of it propagates ahead
and behind the wave packet at a velocity of about $v_{f}\simeq\pm 1.35v_{a}$
(in the plasma rest frame). The fast modes can be seen in the top two panels
of Fig. 6, where we show the contour plots of $<\rho(x,t)>_{y}$ and $<|{\bf
B}(x,t)|>_{y}$, averaged over the $y$ coordinate to reduce noise in the
density. The forward fast mode is clearly visible in both density and magnetic
field magnitude contour plots. The backward fast mode is of smaller amplitude
and is not visible in the density contour. The fast mode perturbations and the
corresponding signature in the electric field are shown in the third panel of
Fig. 6, where we plot the fluctuation of density and magnetic field strength,
$<\delta\rho(x)>_{y}$ and $<\delta|{\bf B}(x)|>_{y}$, respectively, and of the
electric field $<e_{x}(x)>_{y}$ as a function of $x$ at time $t\Omega_{ci}=30$
($t=0.83\tau^{*}$).

Figure 5: Hybrid run 2b: time evolution of the magnetic and kinetic energy
density of fluctuations (top panel), of the density rms (middle panel) and the
diagonal mean pressures (bottom panel).  Figure 6: Hybrid run. Top two panels:
characteristic curves of $<\rho(x,t)>_{y}$ and $<|{\bf B}(x,t)|>_{y}$,
averaged over $y$. Dashed lines indicate the forward and backward fast modes.
Bottom panel: fluctuation of the averaged density and magnetic field strength,
$<\delta\rho(x)>_{y}$ and $<\delta|{\bf B}(x)|>_{y}$, respectively, and of the
electric field $<e_{x}(x)>_{y}$ at time $t\Omega_{ci}=30$ ($t=0.83\tau^{*}$).

After time $t=\tau^{*}$, the evolution is consistent with the Hall-MHD
simulations. However, in the hybrid model $\delta B_{x}$ persists longer than
in its Hall-MHD counterpart (cfr. Fig. 3, bottom right panels, which
correspond to the hybrid simulation, and bottom left panels, which correspond
to the Hall-MHD run 2a). We ascribe the persistence of the wave packet over
longer timescales to the fact the in the hybrid model less energy is lost
initially into different types of dispersive waves.

The field-aligned mean proton pressures, $<p_{xx}>$ and $<p_{zz}>$ shown in
the bottom panel of Fig. 5, tend to increase as the wave packet disperses,
just like in the fluid simulations. The increase in proton internal energy is
determined by $<p_{xx}>$, which changes by $\Delta p_{xx}/p_{xx}(0)=0.2\%$ in
a time interval $\Delta t=2000\Omega_{ci}^{-1}$, comparable to the relative
change of internal energy in run 2a in the same time interval (which is
$\Delta e_{T}/e_{T}(0)\simeq 0.14\%$). While in the fluid models the gain in
internal energy is due to compressions of the plasma, in the hybrid model
contributions from both compressions and phase space mixing determine the
changes in internal energy of protons. The corresponding signature of proton
heating in phase space is reported in Fig. 7. The top panel shows the
variation $\delta f$ of the spatially averaged proton distribution function
$<\delta f({\bf x},v_{\parallel},v_{\perp},t)>_{\bf x}$ at $t\Omega_{ci}=2000$
($t=55\tau^{*}$). The bottom panel shows the averaged distribution function
$<f({\bf x},{\bf v},t)>_{{\bf x},v_{y},v_{z}}$ as a function of $v_{x}$ at
$t=0$ and at $t\Omega_{ci}=2000$. It is interesting to see that $<\delta f>$
displays a clear structure in velocity space at the Alfvén speed, resulting in
the small shoulder in $<f(v_{x})>$ at $v_{x}>v_{a}$. Such a signature starts
to emerge in phase space at time $t\Omega_{ci}=50$ ($t=1.38\tau^{*}$). At that
time $<p_{xx}>$ has reached its highest increase rate, and density rms and
electric field their highest decrease rate. This suggests that a wave-particle
resonance exists with the fast mode forced initially and discussed above.
Proton resonance is however not sufficient to damp entirely density
fluctuations. In Sec. IV we provide a rough estimate to compare the increase
of internal energy found in our simulations with the one inferred from
observations.

Figure 7: Hybrid run. Top panel: contour plot of $<\delta f({\bf
x},v_{\parallel},v_{\perp})>_{\bf x}$ at $t\Omega_{ci}=2000$. Bottom panel:
averaged distribution function $<f({\bf x},{\bf v},t)>_{{\bf x},v_{y},v_{z}}$
as a function of $v_{x}$ at $t=0$ and $t\Omega_{ci}=2000$ ($t=55\tau^{*}$).

##  IV Discussion

###  IV.1 The Hall effect on Alfvén wave packets

In order to interpret our simulation results, let us assume that Alfvénicity
($\delta{\bf u}=\pm\delta{\bf B}/\sqrt{4\pi\rho_{0}}$) is conserved and that
compressibility (thermal and magnetic) remains negligible for weak dispersion,
two conditions that are satisfied in run 1 and run 3. In this case, the
motional electric field contributes to the propagation of the wave packet at
the Alfvén speed, and departures from the MHD exact solution are determined
only by the Hall term in Ohm’s law. By neglecting the ${\bf u}\times{\bf B}$
term while retaining the Hall term, the induction equation in a two-
dimensional system can be cast into the following set of equations for the
magnetic potential $\psi$ and the out-of-plane component $B_{z}$ of the
magnetic field:

$\frac{\partial\psi}{\partial t}=-d_{i}{\bf B}\cdot\bm{\nabla}B_{z}$ (6)
$\frac{\partial B_{z}}{\partial t}=d_{i}{\bf
B}\cdot\bm{\nabla}\nabla^{2}\psi,$ (7)

where we have normalized the magnetic field to $B_{x0}$, lengths to an
arbitrary length $L$, density to a reference $\rho_{0}$, and we have
approximated $\rho\simeq\rho_{0}$. Equations (6)-(7) introduce a
characteristic time $\tau^{*}$. If the wave packet is nearly isotropic, as in
the case considered here, $\tau^{*}\sim\tau_{a}\ell_{x}/d_{i}$. If the wave-
packet is highly anisotropic then the timescale will be determined by the
shortest time $\tau_{x,y}^{*}\sim\tau_{a}\ell_{x,y}/d_{i}$. When equations
(6)-(7) are linearized, they can be recast into an harmonic oscillator
equation in Fourier space for the Fourier components $\tilde{\psi}(t,{\bf
k})$,

$\frac{\partial^{2}}{\partial t^{2}}\tilde{\psi}(t,{\bf
k})=-\omega_{h}^{2}\tilde{\psi}(t,{\bf k}),$ (8)

with $\omega_{h}^{2}=d_{i}^{2}|k_{x}|^{2}|{\bf k}|^{2}$ (in our
normalization). The amplitude of each Fourier mode is a periodic function of
time, with a periodicity that depends on ${\bf k}$ and $d_{i}$. In general,
the solution for a non-monochromatic wave packet can be found for given
initial conditions $\psi(x,y,0)$ and $\partial\psi(x,y,0)/\partial t$, and by
transforming back from Fourier to real space. We have solved eq. (6)-(7) with
a 3rd order Runge Kutta scheme and pseudo-spectral method and eq. (8) by using
eq. (2)-(4) as initial conditions, and we did not find significant
quantitative differences. In Fig. 8 we show the nonlinear solution to eq.
(6)-(7) for $d_{i}=0.25$. Note that changing $d_{i}$ introduces a rescaling of
time included into $\tau^{*}$, so that the plot in Fig. 8 can be compared with
run 1 and run 3 as well (Fig. 3). As can be seen, the model reproduces quite
well the observed evolution for run 1 and run 3, although the full Hall-MHD
system evolves somewhat slower, by about a factor of two. Nevertheless, the
main features of the long-term evolution, in particular wave dispersion along
and across the guiding field that characterizes all of our simulations, can be
recognized in this simple model and can therefore be ascribed to the Hall
effect. This model does not capture entirely the evolution of run 2a and run
2b, where departures from equipartition of magnetic and kinetic energy are
larger, and where compressible effects are in general more important.

In summary, the role of the Hall effect is twofold. When dispersion is weak,
$\ell/d_{i}\gg 1$, the Hall term determines a slow dispersion of the initial
wave packet that starts to affect its dynamics after a characteristic time
$\tau^{*}$. When dispersion becomes stronger, $\ell/d_{i}\gtrsim 1$,
departures from MHD are larger and thus the initial wave packet couples with
compressible and other dispersive modes at times $t<\tau^{*}$, before
dispersing at about $t\simeq\tau^{*}$. Such a coupling with compressible
modes, mediated by dispersion, leads to an increase of internal energy.

Figure 8: Solution to eq. (6)-(7). Top panel: contour plot of
$B_{x}(x,L_{y}/2,t)$. Bottom panel: contour plot of $B_{y}(x,L_{y}/2,t)$. Only
a portion of the spatial domain is shown.

Our results are remarkably different from Hall-MHD and hybrid simulations of
1D Alfvén waves (broadband or monochromatic) and solitons. Previous work has
shown that dispersion can cause modulational instabilities and wave steepening
and collapse when a plane geometry is adopted for the fluctuation (Spangler,
1985; Buti et al., 2000; Matteini et a., 2010; González et al., 2021). Our
work instead shows that the evolution of a 2D (non-plane) wave packet, in
quasi-parallel propagation, differs significantly from its 1D counterpart (at
least without initial strong amplitude modulations). First, just like without
dispersion, a localized wave packet tends to be more stable than an equivalent
large amplitude plane wave. Second, the two-dimensional dynamics allows for
additional channels for wave evolution, inhibiting strong field aligned wave
steepening. As a result, the overall evolution appears to be determined mainly
by the Hall electric field leading to dispersion along and across the mean
field. A field aligned beam, a signature of steepened Alfvén wave fronts
(Matteini et a., 2010; González et al., 2021), does not form in the case
considered here. Instead, a wave-particle resonance is triggered by forced
compressible modes, leading to parallel heating.

We conclude by comparing the rate of change of the internal energy (either
parallel or mean internal energy) resulting from the coupling with
compressible modes with the rates of internal energy change estimated from
observations (Hellinger et al., 2011). In particular, observations show that
parallel pressure increases with radial distance after $R\gtrsim 0.6$ au. We
therefore provide an order of magnitude estimate of the heating rates by
taking typical values of density and temperature at $R\simeq 0.6$ au. By using
results from run 2b we can estimate $\Delta p_{xx}/\Delta t\simeq
p_{xx}(0)\times 0.002/(2000\Omega_{ci}^{-1})$. With a number density of
$n\simeq 7.7\times 10^{6}\,m^{-3}$ at $R=0.6$ au, parallel temperature
$T_{\parallel}\simeq 355764\,K$, and $2\pi/\Omega_{ci}\simeq 1\,s$, we obtain
$\Delta p_{xx}/\Delta t\simeq 2\times 10^{-16}\,Wm^{-3}$. Such a value is of
the same order of magnitude of the parallel heating rate extrapolated from in-
situ data at $R\geq 0.6-0.7$ au (Hellinger et al., 2011). A similar estimate
for the total internal energy increase rate from run 2a, with $T\simeq
364845\,K$ at $R=0.6$ au, gives $\Delta e_{T}/\Delta t\simeq 4\times
10^{-16}\,Wm^{-3}$, still of the same order of the observational
extrapolation. This estimate provides a lower limit, in the sense that a
larger amplitude wave packet can transfer more of its energy to internal
energy. While this comparison is encouraging and suggests that dispersion of
large amplitude wave packets can provide the required heating rates,
perpendicular heating, which represents the dominant contribution to the solar
wind heating rate, in particular for $R<0.6$ au (Hellinger et al., 2011), is
not observed in our simulations. Dispersion in the transverse direction of the
wave packet however may impact the turbulent cascade and thus, indirectly,
perpendicular heating when a fully turbulent plasma is considered.

###  IV.2 Implications for switchbacks observations

Even if switchbacks are generally within the inertial range of scales,
dispersive effects can still be non negligible for switchbacks at smaller
scales ($\ell_{x,y}/d_{i}\gtrsim 1$), or, as our results show, affect the
evolution of those at larger scales ($\ell_{x,y}d_{i}\gg 1$) over sufficiently
long times. To provide context, we show in Fig. 9 the probability distribution
of the length of switchbacks $p(\ell)$, where $\ell$ is expressed in units of
km (top panel) and of the ion inertial length (bottom panel), for distances
from the sun $R=0.06-1$ au. Details on the datasets and methods can be found
in the Appendix. Our simulations cover the first decade of the distribution
shown in the bottom panel of Fig. 9.

Figure 9: Probability distribution of the length of switchbacks at radial
distances $0.06<R<1$ au in units of km (top panel) or ion inertial length
(bottom panel). The inset plots show the scatterplot of $\ell$ and
$\ell/d_{i}$, respectively, as a function of $R$ (in gray), and the mean value
for each radial bin is shown in red.

In prior work we determined that, in MHD, parametric decay can take a time of
up to several hundreds of Alfvén times before destroying the switchbacks, if
the wind is “quiet” (large system size) (Tenerani et al., 2020). Here we have
shown that if a wave packet is stable with respect to parametric
instabilities, then dispersive effects determine the time evolution of the
wave packet. For example, if a switchback is stable over a few hundreds of
Alfvén times, we expect the dispersive timescale $\tau^{*}$ to be shorter than
that of parametric instabilities for those switchbacks of size
$\ell/d_{i}\lesssim 100$. Switchbacks of duration $\delta t$ in the range
$\delta t\simeq 10-100$ s in a wind with speed $V_{sw}=500$ km/s have an
approximate size of $\ell\simeq(0.5-5)\times 10^{4}$ km and
$\ell/d_{i}\simeq(0.5-5)\times 100$ at a radial distance of $R\simeq 0.1-0.2$
au (cfr. Fig. 9). With a mean Alfvén speed of about $v_{a}\simeq 50$ km/s we
obtain $\tau^{*}\simeq(5-500)\times 10^{3}$ s, independent from radial
distance. Dispersive effects should start to degrade those switchbacks over a
distance of $\Delta R\simeq V_{sw}\tau^{*}$. By taking $V_{sw}\simeq 500$
km/s, we obtain a corresponding range of $\Delta R\simeq 0.017-1.67$ au. Note
that the large variation in $\Delta R$ is due to the fact that
$\tau^{*}\propto\ell^{2}$. Nevertheless, this estimate indicates that there
should be a subset of switchbacks undergoing dispersion within 1 au.

Our kinetic and weakly dispersive fluid simulations show that while waves are
emitted from the leading and trailing edge of the wave packet, causing a fast
dispersion of the transverse components (i.e., tangential and normal in RTN
coordinates), the longitudinal (i.e., radial) perturbation persists for
several tens of $\tau^{*}$. Thus, dispersive effects on switchbacks could
result in a defined field reversal in the radial direction with wave activity
at its boundaries, including strong perturbations in magnetic pressure, and
associated with large amplitude transverse waves. We conclude by noting that
it may be interesting to investigate in-situ signatures of wave-particle
resonances such as those reported in our hybrid simulation via the field-
particle correlation technique (Klein and Howes, 2016).

##  V Summary

We have considered dispersive and kinetic effects on a 2D Alfvén wave packet
with constant magnetic field pressure similar to a switchback in a low-$\beta$
plasma. This complements our previous work where the parametric decay of a 2D
switchback was investigated in MHD. Our results can be summarized as follows:

  * •

Dispersion due to the Hall term introduces a characteristic time
$\tau^{*}=\tau_{a}\ell/d_{i}$, where $\ell$ is the wave packet’s smallest
scale (parallel, $\ell_{x}$, or transverse, $\ell_{y}$, to the guiding field),
$\tau_{a}=\ell_{x}/v_{a}$ and $d_{i}$ the proton inertial length.

  * •

If $\ell/d_{i}\gg 1$, dispersion of the initial wave packet starts to affect
its dynamics after a time $t\simeq\tau^{*}$.

  * •

If $\ell/d_{i}\gtrsim 1$ the wave packet couples with compressible and other
dispersive modes at early times $t<\tau^{*}$. The wave packet then starts to
disperse and disrupt at time $t\simeq\tau^{*}$.

  * •

In the Hall-MHD model, coupling with compressible modes leads to the gradual
conversion of the wave packet’s kinetic and magnetic energy into internal
energy.

  * •

When proton kinetic effects are included, the generation of dispersive waves
and slow-modes is inhibited. Compressible fast modes are emitted by the wave
packet, which undergo Landau resonance at the Alfvén speed leading to parallel
heating.

  * •

The resulting heating rates (parallel and total) are estimated to be of about
the same order of the heating rates estimated from fast solar wind
observations at $R\simeq 0.6$ au.

  * •

Observationally, we expect that the shortest switchbacks, of duration $\delta
t\lesssim 100$ s, display signatures of fast-modes and dispersive waves
propagating from their leading and trailing edge, both along the radial and
transverse directions.

##  VI Appendix

The plots shown in Fig. 9 were obtained by combining PSP data at radial
distances $0.06<R<0.5$ au (E1-E12) with data from Solar Orbiter at distances
$0.5<R<1$ au (between June 1, 2018 to March 1, 2022). We have used Level 2
magnetic field measurements from the Flux Gate Magnetometer (FGM) (Bale et
al., 2016) onboard PSP, as well as Level 3 plasma moment data from the Solar
Probe Cup (SPC) for E1-E8, and Solar Probe Analyzer (SPAN) part of the Solar
Wind Electron, Alpha and Proton (SWEAP) suite for E9-E12 (Kasper et al.,
2016). The plasma data are comprised of moments of the distribution function
including the proton velocity vector $\bm{V}_{p}$, number density $n_{p}$, and
temperature $T_{p}$. When available, electron number density data derived from
the quasi-thermal noise from the FIELDS instrument (Moncuquet et al., 2020),
have been used for estimating proton number density. In order to get the
proton density from the electron density, one must consider charge neutrality,
and consequently a $\approx 4\%$ abundance of alpha particles. Accordingly,
electron density from QTN has been divided by 1.08. For Solar Orbiter data, we
used magnetic field measurements from the Magnetometer (MAG) instrument
(Horbury et al., 2020) and particle moments from the Proton and Alpha Particle
Sensor (SWA-PAS) onboard the Solar Wind Analyser (SWA) suite of instruments
(Owen et al., 2020).

Starting from this dataset, we have then identified and removed heliospheric
current sheet crossings and magnetic field data were resampled at 1 second
resolution. We have then determined the duration $\delta t$ of field
reversals. These are defined as those time intervals in which

$\theta_{sb}\equiv\arccos({\bf B\cdot<B>}/(|{\bf B}||<{\bf B}>|))>\pi/2,$ (9)

where here we have calculated the mean magnetic field $<{\bf B}>$ over a time
interval $\Delta t=8$ h. We defer the reader to Tenerani et al. (2021) for
details on the identification process of switchbacks. For each identified
switchback, we have determined the field reversal duration $\delta t$, the
mean solar wind radial velocity $V_{r}$, $v_{a}$ and $d_{i}$. Assuming that
switchbacks are advected at the solar wind speed in the radial direction, we
have then defined $\delta\ell=\delta t|V_{r}+v_{a}-V_{sc}|$ with $V_{sc}$ the
spacecraft speed.

###### Acknowledgements.

This research was supported by NASA grant #80NSSC18K1211. We acknowledge the
Texas Advanced Computing Center (TACC) at The University of Texas at Austin
for providing HPC resources that have contributed to the research results
reported within this paper. URL: http://www.tacc.utexas.edu.

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