# Nonlinear dynamics of small-scale Alfvén waves

Alfred Mallet<EMAIL_ADDRESS>Space Sciences Laboratory,
University of California, Berkeley, CA 94720, USA  Seth Dorfman  Space Science
Institute, Boulder, CO 80301, USA  University of California Los Angeles, Los
Angeles, California 90095, USA  Mel Abler  Space Science Institute, Boulder,
CO 80301, USA  Trevor Bowen  Space Sciences Laboratory, University of
California, Berkeley, CA 94720, USA  Christopher H.K. Chen  Department of
Physics and Astronomy, Queen Mary University of London, UK

###### Abstract

We study the nonlinear evolution of very oblique small-scale Alfvén waves with
$k_{\perp}d_{i}\gtrsim 1$. At these scales, the waves become significantly
compressive, unlike in MHD, due to the Hall term in the equations. We
demonstrate that when frequencies are small compared to the ion gyrofrequency
and amplitudes small compared to unity, no new nonlinear interaction appears
due to the Hall term alone at the lowest non-trivial order, even when
$k_{\perp}d_{i}\sim 1$. However, at the second non-trivial order, we discover
that the Hall physics leads to a slow but resonant nonlinear interaction
between co-propagating Alfvén waves, an inherently 3D effect. Including the
effects of finite temperature, finite frequency, and electron inertia, the
two-fluid Alfvén wave also becomes dispersive once one or more of
$k_{\perp}\rho_{s}$, $k_{\perp}d_{e}$, or $k_{\parallel}d_{i}$ becomes
significant: for oblique waves at low $\beta$ as studied here, this can be at
a much smaller scale than $d_{i}$. We show that the timescale for one-
dimensional steepening of two-fluid Alfven waves is only significant at these
smaller dispersive scales, and also derive an expression for the amplitude of
driven harmonics of a primary wave. Importantly, both new effects are absent
in gyrokinetics and other commonly used reduced two-fluid models. Our
calculations have relevance for the interpretation of laboratory Alfvén wave
experiments, as well as shedding light on the physics of turbulence in the
solar corona and inner solar wind, where the dominant nonlinear interaction
between counter-propagating waves is suppressed, allowing these new effects to
become important.

##  I Introduction

Large-amplitude electromagnetic fluctuations are present in a wide range of
astrophysical and space plasma physics settings; in the latter case, we have
many increasingly precise measurements of these fluctuations from _in situ_
spacecraft in the Earth’s magnetosphere (Torbert  _et al._ , 2016), the solar
wind (Chen, 2016), and recently even a spacecraft (Parker Solar Probe,
henceforth PSP) in the solar corona (Kasper  _et al._ , 2021). In the solar
wind and especially in the corona, these fluctuations are often polarized (on
scales much larger than the ion gyroradius) quite precisely like _Alfvén
waves_ (AW), propagating along the magnetic field in the direction away from
the Sun (Bale  _et al._ , 2019; Bowen  _et al._ , 2021). PSP observations
especially have revealed patches of extremely large-amplitude ($\delta
B/B\gtrsim 1$) AW, referred to as "switchbacks" since they can even reverse
the direction of the background magnetic field. These switchbacks often have
remarkably steep boundaries (Kasper  _et al._ , 2019), suggesting that
nonlinear steepening may be an important process in their evolution and/or
generation.

Section | $\omega/\Omega_{i}$ | $k_{\perp}d_{i}$ | $k_{\perp}\rho_{s}$ | $k_{\perp}d_{e}$ | $\beta$ | $k_{\parallel}/k_{\perp}$ | $\delta B/B_{0}$
---|---|---|---|---|---|---|---
II | $\epsilon$ | $1$ | $0$ | $0$ | $0$ | $\epsilon$ | $\epsilon$
III | $1$ | $\epsilon^{-1}$ | $1$ | $1$ | $\epsilon^{2}$ | $\epsilon$ | $\epsilon$
Appendix B | $\epsilon$ | $\epsilon^{-1}$ | $1$ | $1$ | $\epsilon^{2}$ | $\epsilon^{2}$ | $\epsilon^{2}$
Table 1: Summary of the physical regimes studied in different sections of the
paper. In each case $\epsilon\ll 1$ is a small parameter in which the two-
fluid equations are expanded.

AW may perform a key role in coronal heating (Chandran and Hollweg, 2009;
Chandran, 2021), due to turbulence driven by reflection from large-scale
inhomogeneity in the Alfvén velocity $v_{\mathrm{A}}=B_{0}/\sqrt{4\pi
n_{0i}m_{i}}$. In the standard model of MHD turbulence, the nonlinear
interaction that permits a turbulent Alfvénic cascade is between
counterpropagating AW $\boldsymbol{z}^{\pm}=\boldsymbol{u}\pm\boldsymbol{b}$,
where $\boldsymbol{z}^{\pm}$ are the Elsasser variables and $\boldsymbol{u}$,
$\boldsymbol{b}$ are the velocity and magnetic field fluctuations in velocity
units(Schekochihin, 2022; Howes, 2015). However, the PSP observations reveal
(Bowen  _et al._ , 2021) that the flux of inward-travelling waves is extremely
small compared to that of the dominant outward-travelling AW. The dynamics of
this _imbalanced_ turbulence may be fundamentally different to balanced
turbulence with a comparable flux of AW in both directions, allowing different
physics to dominate the dynamics of the system. For example, in an important
theoretical development, it has recently been shown that the conservation of
generalized helicity causes a barrier to turbulent energy flux to appear at
the ion scales in imbalanced turbulence (Meyrand  _et al._ , 2021), leading to
remarkably large-amplitude fluctuations and enabling high-frequency ion-
cyclotron heating of the plasma (Squire  _et al._ , 2021), which allows for
the necessary perpendicular ion heating to power the fast solar wind. This
provides some motivation for revisiting the physics of small-scale AW,
including nonlinear steepening and co-propagating nonlinear interactions.

Alfvén waves are also routinely launched in laboratory experiments at the
LArge Plasma device (LAPD), which is a cylindrical device that produces a
$16.5\mathrm{m}$-long column of quiescent, magnetized plasma (Gekelman  _et
al._ , 1991). Waves with various parameters can be launched using antennae
placed at the ends of the column; there have been extensive experiments
performed using LAPD on the linear (Leneman, Gekelman, and Maggs, 1999;
Gekelman  _et al._ , 2000; Vincena, Gekelman, and Maggs, 2004; Palmer,
Gekelman, and Vincena, 2005) and nonlinear (Carter  _et al._ , 2006; Auerbach
_et al._ , 2010; Howes  _et al._ , 2012; Dorfman and Carter, 2013, 2015, 2016)
properties of AW. The plasma regime in which these waves are launched may be
similar to small-scale waves present in the solar corona and Earth’s aurora:
$k_{\perp}d_{i}\gg 1$ while $k_{\parallel}d_{i}\lesssim 1$, but
$k_{\perp}\rho_{\tau}\sim k_{\perp}d_{e}\lesssim 1$, implying
$k_{\parallel}/k_{\perp}\ll 1$ and $\beta\sim Zm_{e}/m_{i}\ll 1$, where
$k_{\parallel}$ ($k_{\perp}$) are the components of the wavenumber $k$
parallel (perpendicular) to the mean magnetic field ,
$d_{i}=v_{\mathrm{A}}/\Omega_{i}$ the ion inertial length,
$\rho_{\tau}=c_{s}/\Omega_{i}$ the ion sound radius,
$d_{e}=\sqrt{Zm_{e}/m_{i}}d_{i}$ the electron inertial length, and
$\beta=c_{s}^{2}/v_{\mathrm{A}}^{2}$, with the Alfvén velocity
$v_{\mathrm{A}}=B_{0}/\sqrt{4\pi n_{0i}m_{i}}$, sound velocity
$c_{s}=\sqrt{(T_{i}+ZT_{e})/m_{i}}$, and ion gyrofrequency
$\Omega_{i}=ZeB_{0}/m_{i}c$. Unlike the corona, the electron-ion collision
frequency in the device can be quite large, $\nu_{ei}\sim\omega$ where
$\omega\sim k_{\parallel}v_{\mathrm{A}}$ is the frequency of the waves; thus,
for this setting it is important to retain the Ohmic resistivity
$\eta=\nu_{ei}d_{e}^{2}$. Interestingly, harmonics are routinely observed in
AW experiments on LAPD(Brugman, 2007; Dorfman and Carter, 2016; Abler, 2023),
even without a counterpropagating wave, suggesting that improved modelling of
AW nonlinear behaviour is needed to understand the experimental results, as
well as the aforementioned coronal turbulence.

Within the framework of compressible magnetohydrodynamics (MHD), the Alfvén
wave is an exact nonlinear solution, for which velocity and magnetic field
fluctuations obey
$\delta\boldsymbol{u}=\pm\delta\boldsymbol{B}/\sqrt{4\pi\rho}$, and
additionally the magnetic-field strength, density and pressure are space-time
constants. This situation is known as spherical polarization since the
magnetic-field vector moves on the surface of a sphere: it should be noted
that this means that the large-amplitude AW is explicitly not monochromatic
(Goldstein, Klimas, and Barish, 1974; Barnes and Hollweg, 1974). Such a
configuration propagates at $v_{\mathrm{A}}$ without steepening of the
perpendicular fluctuations, regardless of amplitude or three-dimensional
structure(Goldstein, Klimas, and Barish, 1974). There are parallel
fluctuations $\delta B_{\parallel}=-\delta B_{\perp}^{2}/2B_{0}$, required to
enforce the space-time constancy of $|\boldsymbol{B}|$, appearing as harmonics
of the perpendicular fluctuations(Barnes and Hollweg, 1974; Vasquez and
Hollweg, 1998a): however, they do not depend on time in this solution and
their amplitude is much smaller than the harmonic amplitudes we will derive in
this paper. For MHD Alfvén waves, non-trivial nonlinear interaction only
occurs due to the presence of counter-propagating waves.

Even at large amplitude, the MHD Alfvén wave is precisely incompressible.
Going beyond MHD, including the Hall effect in Ohm’s law (see Appendix A for a
brief derivation of relevant equations), very oblique ($k_{\perp}\gg
k_{\parallel}$) AW develop density fluctuations(Hollweg, 1999),

$\frac{\delta n_{i}}{n_{0i}}\sim-ik_{\perp}d_{i}\frac{\delta B}{B_{0}},$ (1)

scaling with $k_{\perp}d_{i}$, along with a corresponding compressive flow
(the ion polarization drift). The wave only becomes _dispersive_ when one of
$k_{\perp}\rho_{\tau},k_{\perp}d_{e},k_{\parallel}d_{i}$ becomes significant:
the linear two-fluid dispersion relation in an appropriate limit, including
the effects of resistive damping, is (see Appendix A and Hollweg 1999Hollweg
(1999))

$(1+k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2})\omega^{2}+i\eta k^{2}\omega-
k_{\parallel}^{2}v_{\mathrm{A}}^{2}(1+k^{2}\rho_{\tau}^{2})=0,$ (2)

with solution

$\omega=-i\gamma_{0}\pm\sqrt{\omega_{0}^{2}-\gamma_{0}^{2}},$ (3)

where

$\omega_{0}=k_{\parallel}v_{\mathrm{A}}\sqrt{\frac{1+k^{2}\rho_{\tau}^{2}}{1+k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2}}},\quad\gamma_{0}=\frac{1}{2}\frac{\eta
k^{2}}{1+k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2}}.$ (4)

While not always accurate (e.g. when kinetic effects become important, see
Appendix A for more discussion), the two-fluid system thus provides a minimal
model for studying dispersive AW. For oblique waves at low $\beta$, the
dispersive terms only become relevant at much smaller scales than
$d_{i}$(Hollweg, 1999), so that if (for example) $k_{\perp}\rho_{\tau}$ is the
largest of these dispersive parameters, between $\rho_{\tau}\ll
1/k_{\perp}\lesssim d_{i}$ the wave is compressive but not significantly
dispersive (this conclusion holds for a system that is not too collisional,
$\nu_{ei}\lesssim\omega$, as the resistive damping then only enters when
$k_{\perp}d_{e}\sim 1$). Due to the density fluctuations, once
$k_{\perp}d_{i}\sim 1$ the local Alfven velocity $v_{A}^{\prime}=B/\sqrt{4\pi
n_{i}m_{i}}$ now varies with the wave’s phase: at sufficiently large
amplitude, it might be natural to assume that the Hall term might drive new
nonlinear interactions and steepening, even between co-propagating waves.
These new interactions are the subject of this paper.

First, in Section II, we derive a set of equations describing three-
dimensional dynamics of very oblique AW for $k_{\perp}d_{i}\sim 1$, but
assuming that $\rho_{s}=d_{e}=0$ and $\omega/\Omega_{i}\ll 1$, so that the
linear dispersion relation (in this oblique limit) is just $\omega=\pm
k_{\parallel}v_{\mathrm{A}}$. This results in an reduced expansion of the Hall
MHD equations. At the lowest non-trivial order, we reproduce the classic
equations of reduced MHD (RMHD), in which nonlinearity only occurs between
counter-propagating AW, despite the significant density fluctuations: a
previously known result (see, e.g. Zocco & Schekochihin 2011(Zocco and
Schekochihin, 2011), or Appendix B): thus, at this order, the Hall effect
makes no difference to the dynamics. However, at the next order in our
asymptotic expansion, a three-dimensional nonlinear interaction does exist
between co-propagating waves, despite no change in the linear dispersion
relation at this order either. Because this effect only appears at next order,
the timescale associated with the interaction is slow compared to the Alfvén
timescale $\tau_{A}=(k_{\parallel}v_{\mathrm{A}})^{-1}$. Moreover, this
interaction vanishes if the perpendicular wavevectors of two interacting waves
are aligned, e.g. in a one-dimensional system.

Once the AW becomes dispersive (e.g. when $k_{\perp}\rho_{\tau}$ or
$k_{\perp}d_{e}$ can no longer be neglected), it is already known nonlinear
interactions between co-propagating waves are possible, even at small
amplitude and low frequency (Seyler and Xu, 2003; Schekochihin  _et al._ ,
2009; Zocco and Schekochihin, 2011): waves of different $k_{\perp}$ "catch up"
with each other and interact. This well-known nonlinearity is not the main
focus of this paper, but we provide a systematic derivation of relevant
equations in Appendix B. Again, this nonlinear interaction vanishes if the
perpendicular wavevectors of the parent waves are aligned.

Both of the aforementioned nonlinear interactions vanish in one dimension. We
show in Sec. III that one-dimensional steepening of two-fluid AW is rather
unusual: it requires both large density fluctuations and significant
dispersive terms to be active. If all the dispersive terms and resistivity are
negligible, even with large density fluctuations ($k_{\perp}d_{i}\gg 1$), in
one dimension the compressive flow exactly cancels the modification to the
local Alfvén velocity, resulting in no net steepening. We derive an expression
for the harmonic of the primary AW driven by dispersive steepening, which can
be compared to harmonics observed in laboratory experiments on LAPD (Abler,
2023).

In a situation with equal fluxes of counterpropagating AW, the new nonlinear
interactions discovered in Secs. II and III would be subdominant for
$k_{\perp}d_{i}\sim 1$. The timescales associated with both processes scale
with the amplitude of the density fluctuations (Eq. 1). However, the
counterpropagating interaction is suppressed in the highly imbalanced, regime
in the solar corona, where moreover $\beta\ll 1$: this may allow these effects
to become important in the transition between the inertial-range and kinetic-
range turbulence, via a competition between the level of imbalance
$R_{E}=\delta z^{-}/\delta z^{+}$ and the smallness of the density
fluctuations. Moreover, the existence of these interactions means that care
must be taken interpreting laboratory experiments (Howes  _et al._ , 2012;
Abler, 2023) studying nonlinear AW interactions on LAPD, where
$k_{\perp}d_{i}\gg 1$, shortening the nonlinear timescales associated with
these new processes.

The different physical regimes studied in the paper are summarized in Table 1.
The three ordering schemes here are designed to isolate the three different
types of co-propagating nonlinear interaction: three-dimensional Hall-driven
interaction (Sec. II), one-dimensional steepening, which requires both density
fluctuations and the waves to be dispersive (Sec. III, and purely
dispersively-driven nonlinearity (Appendix B, already known e.g. from previous
reduced fluidSeyler and Xu (2003) and/or gyrokinetic modelsSchekochihin  _et
al._ (2009); Zocco and Schekochihin (2011)). Taken together, our results shed
new light on the physics of Alfvén waves in laboratory and space plasmas, and
may have relevance to both existing experiments and on the evolution of
extremely imbalanced turbulence and switchbacks in the corona and solar wind.

##  II Reduced Hall MHD to next order: 3D nonlinear interaction

As pointed out in the introduction, for $\beta\ll 1$,
$\nu_{ei}\lesssim\omega$, and $k_{\parallel}/k_{\perp}\ll 1$, AW become
compressive due to the Hall term at $k_{\perp}d_{i}\sim 1$, a much larger
scale than those at which the linear dispersion relation (2) differs from
ideal MHD. Are the nonlinear properties similarly unaffected? This question is
the subject of this section.

The essential new physics will appear even though from the outset we
completely neglect the kinetic and inertial AW dispersive terms and the
resistivity (in Eqs. 137–140, take $c_{s}/v_{A}\to 0$, $m_{e}/m_{i}\to 0$, and
$\eta\to 0$), leaving just the equations of compressible Hall MHD:

$\displaystyle\partial_{t}n$ $\displaystyle=-\nabla\cdot(n\boldsymbol{u}),$
(5) $\displaystyle\mathrm{d}_{t}\boldsymbol{u}$
$\displaystyle=\frac{1}{n}(\nabla\times\boldsymbol{b})\times\boldsymbol{b},$
(6) $\displaystyle\partial_{t}\boldsymbol{b}$
$\displaystyle=\nabla\times(\boldsymbol{u}\times\boldsymbol{b})-d_{i}\nabla\times\mathrm{d}_{t}\boldsymbol{u},$
(7) $\displaystyle\nabla\cdot\boldsymbol{b}$ $\displaystyle=0,$ (8)

where $\boldsymbol{u}$ is the ion velocity, $n=n_{i}/n_{0i}$, and the magnetic
field in Alfvén units is $\boldsymbol{b}=\boldsymbol{B}/\sqrt{4\pi
n_{0i}m_{i}}$. We write

$\boldsymbol{b}=\boldsymbol{b}_{\perp}+\hat{\boldsymbol{z}}b_{z}+\hat{\boldsymbol{z}}v_{\mathrm{A}},\quad
n=1+\tilde{n},\quad\boldsymbol{u}=\boldsymbol{u}_{\perp}+\hat{\boldsymbol{z}}u_{z}.$
(9)

where the Alfvén speed $v_{\mathrm{A}}=B_{0}/\sqrt{4\pi n_{0i}m_{i}}$ is the
magnitude of the (straight) mean magnetic field, taken to be in the
$\hat{\boldsymbol{z}}$ direction, and $\boldsymbol{b}_{\perp}$, $b_{z}$,
$\tilde{n}$, $\boldsymbol{u}_{\perp}$, and $u_{z}$ are fluctuations. We assume
that the waves are highly oblique, and use as our expansion parameter

$\epsilon\sim\frac{\partial_{z}}{\nabla_{\perp}}\ll 1.$ (10)

We also assume that the linear timescales of our fluctuations are Alfvénic,
and comparable to the nonlinear timescales,

$\partial_{t}\sim
v_{\mathrm{A}}\partial_{z}\sim\boldsymbol{u}_{\perp}\cdot\nabla_{\perp}\sim\boldsymbol{b}_{\perp}\cdot\nabla_{\perp},$
(11)

whence

$\frac{u_{\perp}}{v_{\mathrm{A}}}\sim\frac{b_{\perp}}{v_{\mathrm{A}}}\sim\epsilon.$
(12)

This means that the critical balance parameter(Goldreich and Sridhar, 1995)

$\chi=\frac{k_{\perp}b_{\perp}}{k_{\parallel}v_{\mathrm{A}}}\sim 1$ (13)

is permitted to be of order unity. Finally, we also take the perpendicular
scale of the fluctuations to be comparable to $d_{i}$

$d_{i}\nabla_{\perp}\sim 1,$ (14)

implying that the frequencies are small compared to $\Omega_{i}$,

$d_{i}\partial_{z}\sim\frac{1}{\Omega_{i}}\partial_{t}\sim\epsilon.$ (15)

Some of these orderings may be somewhat familiar from reduced MHD (Strauss,
1976; Schekochihin  _et al._ , 2009). For the other fluctuations, we write

$\tilde{n}\sim\epsilon^{\alpha_{n}},\quad\frac{\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}}{\Omega_{i}}\sim\epsilon^{\alpha_{\xi}},\quad\frac{b_{z}}{v_{\mathrm{A}}}\sim\epsilon^{\alpha_{b}},\quad\frac{u_{z}}{v_{\mathrm{A}}}\sim\epsilon^{\alpha_{u}},$
(16)

where we will deduce the values of
$\alpha_{n},\alpha_{\xi},\alpha_{b},\alpha_{u}$ in the following analysis. We
will keep terms at the lowest _two_ non-trivial orders in each equation. Note
that the dispersive correction to the linear AW frequency from finite
$\omega/\Omega_{i}$ (equivalently finite $k_{\parallel}d_{i}$) is
$\sim(\omega/\Omega_{i})^{2}k_{\parallel}v_{\mathrm{A}}\sim\epsilon^{3}\Omega_{i}$
(see Eq.2), and so will be neglected, meaning that the linear dispersion
relation in our system up to the lowest two orders is just $\omega=\pm
k_{\parallel}v_{\mathrm{A}}$.

We now order the terms in our equations; the orderings shown below the
equations are obtained by normalizing times by $\Omega_{i}^{-1}$, velocities
and magnetic field in Alfvén units by $v_{\mathrm{A}}$, and lengths by
$d_{i}=v_{\mathrm{A}}/\Omega_{i}$.

The perpendicular components of (6) are

$\displaystyle\underbrace{\partial_{t}\boldsymbol{u}_{\perp}}_{\epsilon^{2}}$
$\displaystyle+\underbrace{\boldsymbol{u}_{\perp}\cdot\nabla_{\perp}\boldsymbol{u}_{\perp}}_{\epsilon^{2}}+\cancel{\underbrace{u_{z}\partial_{z}\boldsymbol{u}_{\perp}}_{\epsilon^{2+\alpha_{u}}}}$
$\displaystyle=\frac{1}{n}$
$\displaystyle\left[\underbrace{v_{\mathrm{A}}\partial_{z}\boldsymbol{b}_{\perp}}_{\epsilon^{2}}+\underbrace{\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}\boldsymbol{b}_{\perp}}_{\epsilon^{2}}+\cancel{\underbrace{b_{z}\partial_{z}\boldsymbol{b}_{\perp}}_{\epsilon^{2+\alpha_{b}}}}\right.$
$\displaystyle\left.-\nabla_{\perp}\left(\underbrace{\frac{1}{2}b_{\perp}^{2}}_{\epsilon^{2}}+\underbrace{b_{z}v_{\mathrm{A}}}_{\epsilon^{\alpha_{b}}}+\cancel{\underbrace{b_{z}^{2}}_{\epsilon^{2\alpha_{b}}}}\right)\right],$
(17)

where, according to the normalizations in the previous paragraph, the
orderings shown below each term are relative to $\Omega_{i}v_{\mathrm{A}}$.
For the $\nabla_{\perp}b_{z}v_{\mathrm{A}}$ term to not be larger than all
others, $\alpha_{b}\geq 2$, so we take

$\frac{b_{z}}{v_{\mathrm{A}}}\sim\epsilon^{2},$ (18)

which allows us to neglect some of the pre-emptively crossed-out terms in
(17). The $\hat{\boldsymbol{z}}$-component of (6) is

$\displaystyle\underbrace{\partial_{t}u_{z}}_{\epsilon^{1+\alpha_{u}}}$
$\displaystyle+\underbrace{\boldsymbol{u}_{\perp}\cdot\nabla_{\perp}u_{z}}_{\epsilon^{1+\alpha_{u}}}+\cancel{\underbrace{u_{z}\partial_{z}u_{z}}_{\epsilon^{1+2\alpha_{u}}}}$
$\displaystyle=\frac{1}{n}\left[\underbrace{\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}b_{z}}_{\epsilon^{3}}+\cancel{\underbrace{b_{z}\partial_{z}b_{z}}_{\epsilon^{5}}}-\underbrace{\partial_{z}\left(\frac{1}{2}b_{\perp}^{2}\right)}_{\epsilon^{3}}\right].$
(19)

This implies $\alpha_{u}\geq 2$, and so we take

$\frac{u_{z}}{v_{\mathrm{A}}}\sim\epsilon^{2},$ (20)

allowing us to neglect some more of the crossed-out terms in (17) and (19).
The $\hat{\boldsymbol{z}}$-component of (7) gives

$\displaystyle\underbrace{\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}}_{\epsilon^{\alpha_{\xi}}}=$
$\displaystyle-\underbrace{\frac{1}{\Omega_{i}}\nabla_{\perp}\times\mathrm{d}_{t}\boldsymbol{u}_{\perp}}_{\epsilon^{2}}$
$\displaystyle+\underbrace{\frac{1}{v_{\mathrm{A}}}\nabla_{\perp}\cdot(u_{z}\boldsymbol{b}_{\perp}-b_{z}\boldsymbol{u}_{\perp})}_{\epsilon^{3}}-\frac{1}{v_{\mathrm{A}}}\underbrace{\partial_{t}b_{z}}_{\epsilon^{3}},$
(21)

We can deduce that $\alpha_{\xi}\geq 2$, and so we take

$\frac{\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}}{\Omega_{i}}\sim\epsilon^{2}.$
(22)

The different terms in (5) are then ordered as

$\underbrace{\partial_{t}\tilde{n}}_{\epsilon^{1+\alpha_{n}}}+\underbrace{\boldsymbol{u}_{\perp}\cdot\nabla_{\perp}\tilde{n}}_{{}_{\epsilon^{1+\alpha_{n}}}}+\cancel{\underbrace{u_{z}\partial_{z}\tilde{n}}_{\epsilon^{3+\alpha_{n}}}}+n(\underbrace{\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}}_{\epsilon^{2}}+\underbrace{\partial_{z}u_{z}}_{\epsilon^{3}})=0,$
(23)

so that $\alpha_{n}\geq 1$ and we take

$\tilde{n}\sim\epsilon.$ (24)

We may then drop the crossed-out term in (23) since it is two orders below the
leading terms.

Now, note that (8) gives

$\underbrace{\nabla_{\perp}\cdot\boldsymbol{b}_{\perp}}_{\epsilon}+\underbrace{\partial_{z}b_{z}}_{\epsilon^{3}}=0.$
(25)

Since the second term is two orders down from the first, we may write

$\boldsymbol{b}_{\perp}=\hat{\boldsymbol{z}}\times\nabla_{\perp}\psi+O(\epsilon^{3}v_{\mathrm{A}}),$
(26)

as in (165), so that at the lowest two orders in our expansion

$\nabla_{\perp}\cdot\boldsymbol{b}_{\perp}=0,\quad\nabla_{\perp}\times\boldsymbol{b}_{\perp}=\nabla_{\perp}^{2}\psi.$
(27)

We will also find it convenient to decompose the two independent components of
$\boldsymbol{u}_{\perp}$ as

$\boldsymbol{u}_{\perp}=\hat{\boldsymbol{z}}\times\nabla_{\perp}\phi+\nabla_{\perp}\xi,$
(28)

so that

$\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}=\nabla_{\perp}^{2}\xi,\quad\nabla_{\perp}\times\boldsymbol{u}_{\perp}=\nabla_{\perp}^{2}\phi.$
(29)

In terms of these scalar variables, up to the second non-trivial order we have

$\displaystyle\mathrm{d}_{t}f$
$\displaystyle=\underbrace{\partial_{t}f+\\{\phi,f\\}}_{\epsilon}+\underbrace{\nabla_{\perp}\xi\cdot\nabla_{\perp}f}_{\epsilon^{2}},$
(30) $\displaystyle\boldsymbol{b}\cdot\nabla f$
$\displaystyle=\underbrace{v_{\mathrm{A}}\partial_{z}f+\\{\psi,f\\}}_{\epsilon},$
(31)

where $\\{f,g\\}=\nabla_{\perp}f\times\nabla_{\perp}g$, $f$ and $g$ being
arbitrary functions, and the orderings are relative to $\Omega_{i}f$.

Operating on (17) with $\nabla_{\perp}\times$, we find

$\displaystyle\underbrace{\partial_{t}\nabla_{\perp}^{2}\phi+\\{\phi,\nabla_{\perp}^{2}\phi\\}}_{\epsilon^{2}}+\underbrace{\nabla_{\perp}\cdot(\nabla_{\perp}^{2}\phi\nabla_{\perp}\xi)}_{\epsilon^{3}}$
$\displaystyle=\underbrace{v_{\mathrm{A}}\nabla_{\perp}\cdot\left(\frac{1}{n}\partial_{z}\nabla_{\perp}\psi\right)+\left\\{\psi,\frac{1}{n}\nabla_{\perp}^{2}\psi\right\\}}_{\epsilon^{2}}+\underbrace{\left\\{v_{\mathrm{A}}b_{z},\frac{1}{n}\right\\}}_{\epsilon^{3}}.$
(32)

Operating on (17) instead with $\nabla_{\perp}\cdot$ yields

$\displaystyle\underbrace{\partial_{t}\nabla_{\perp}^{2}\xi}_{\epsilon^{3}}+\frac{1}{2}\nabla_{\perp}^{2}\left(\underbrace{|\nabla_{\perp}\phi|^{2}}_{\epsilon^{2}}+\cancel{\underbrace{|\nabla_{\perp}\xi|^{2}}_{\epsilon^{4}}}\right)$
$\displaystyle-\underbrace{\nabla_{\perp}\cdot\left(\nabla_{\perp}^{2}\phi\nabla_{\perp}\phi\right)}_{\epsilon^{2}}+\underbrace{\nabla_{\perp}^{2}\\{\phi,\xi\\}+\\{\xi,\nabla_{\perp}^{2}\phi\\}}_{\epsilon^{3}}$
$\displaystyle=\underbrace{v_{\mathrm{A}}\left\\{\partial_{z}\psi,\frac{1}{n}\right\\}}_{\epsilon^{3}}-\underbrace{\nabla_{\perp}\cdot\left(\frac{1}{n}\nabla_{\perp}^{2}\psi\nabla_{\perp}\psi+\frac{1}{n}\nabla_{\perp}b_{z}v_{\mathrm{A}}\right)}_{\epsilon^{2}}.$
(33)

Operating on the perpendicular components of (7) with $\nabla_{\perp}\times$,
we obtain

$\nabla_{\perp}^{2}\mathrm{d}_{t}\psi=v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\phi+d_{i}[\nabla_{\perp}^{2}\partial_{t}u_{z}-\partial_{z}\nabla_{\perp}\cdot\mathrm{d}_{t}\boldsymbol{u}_{\perp}].$
(34)

Using the lowest orders of (33) and (19), we can write this as

$\displaystyle\underbrace{\nabla_{\perp}^{2}\mathrm{d}_{t}\psi}_{\epsilon^{2}}=$
$\displaystyle\underbrace{v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\phi}_{\epsilon^{2}}$
$\displaystyle+\underbrace{d_{i}\nabla_{\perp}^{2}\left(\\{\psi,b_{z}\\}-\frac{1}{2}\partial_{z}\left(|\nabla_{\perp}\psi|^{2}\right)\right)}_{\epsilon^{3}}$
$\displaystyle+\underbrace{d_{i}\partial_{z}\nabla_{\perp}\cdot\left(\nabla_{\perp}^{2}\psi\nabla_{\perp}\psi+\nabla_{\perp}b_{z}v_{\mathrm{A}}\right)}_{\epsilon^{3}},$
(35)

where we have neglected all higher-order terms. (23), (21), (19), (33), (32),
and (35) constitute a closed set of equations for our six variables
$\phi,\psi,\xi,b_{z},u_{z},\tilde{n}$. We will now proceed to expand
systematically in $\epsilon$, writing
$\phi=\epsilon\phi_{1}+\epsilon^{2}\phi_{2}$, and so on for the other
variables, taking into account their overall ordering, e.g.
$b_{z}=\epsilon^{2}b_{z1}+\epsilon^{3}b_{z2}$.

###  II.1 Lowest order: RMHD$+$

If we keep only the lowest-order terms, Eqs. (32) and (35) reduce to the
equations of RMHD,

$\displaystyle\partial_{t}\nabla_{\perp}^{2}\phi_{1}+\\{\phi_{1},\nabla_{\perp}^{2}\phi_{1}\\}$
$\displaystyle=v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\psi_{1}+\left\\{\psi_{1},\nabla_{\perp}^{2}\psi_{1}\right\\},$
(36) $\displaystyle\partial_{t}\psi_{1}+\\{\phi_{1},\psi_{1}\\}$
$\displaystyle=v_{\mathrm{A}}\partial_{z}\phi_{1},$ (37)

i.e. the dynamics of the Alfvén wave is at this order unaffected by the fact
that $k_{\perp}d_{i}\sim 1$. The lowest orders of the other equations yield

$\displaystyle\tilde{n}_{1}$
$\displaystyle=\frac{1}{\Omega_{i}}\nabla_{\perp}^{2}\phi_{1},$ (38)
$\displaystyle\nabla_{\perp}^{2}\xi_{1}$
$\displaystyle=-\frac{1}{\Omega_{i}}\mathrm{d}_{t}\nabla_{\perp}^{2}\phi_{1},$
(39) $\displaystyle\nabla_{\perp}^{2}b_{z1}v_{\mathrm{A}}$
$\displaystyle=\nabla_{\perp}\cdot\left(\nabla_{\perp}^{2}\phi_{1}\nabla_{\perp}\phi_{1}-\nabla_{\perp}^{2}\psi_{1}\nabla_{\perp}\psi_{1}\right)$
$\displaystyle\quad-\frac{1}{2}\nabla_{\perp}^{2}|\nabla_{\perp}\phi_{1}|^{2},$
(40) $\displaystyle\mathrm{d}_{t}u_{z1}$
$\displaystyle=\\{\psi_{1},b_{z1}\\}-\frac{1}{2}\partial_{z}|\nabla_{\perp}\psi_{1}|^{2},$
(41)

It is worth noticing that

$\phi_{1}=\phi_{1}(x,y,z\pm v_{\mathrm{A}}t)=\pm\psi_{1}$ (42)

constitutes an exact nonlinear solution at this order, an Alfvén wave with
arbitrary perpendicular structure propagating antiparallel ($+$) or parallel
($-$) to the background magnetic field at group velocity $v_{\mathrm{A}}$. For
this solution, we obtain from (40) and (41)

$b_{z1}=-\frac{b_{\perp}^{2}}{2v_{\mathrm{A}}}=\pm u_{z1},$ (43)

and the total magnetic-field-strength-squared is constant to lowest order, a
generalization of the MHD large-amplitude AW(Barnes and Hollweg, 1974; Vasquez
and Hollweg, 1998a, b).

###  II.2 Next order

To obtain the equations to the next order, we must remember to take into
account the fact that $n=1+\epsilon\tilde{n}_{1}+O(\epsilon^{2})$, the second-
order pieces of the time derivative $\mathrm{d}_{t}f$,

$\left[\partial_{t}f_{2}+\\{\phi_{2},f_{1}\\}+\\{\phi_{1},f_{2}\\}+\nabla_{\perp}\xi_{1}\cdot\nabla_{\perp}f_{1}\right],$
(44)

and all the next-order terms that are already written explicitly in the
preceding equations. Solving for $n_{2},\xi_{2},b_{z2}$, and $u_{z2}$ will not
be necessary. The final two equations, (32) and (35), are at this order

$\displaystyle\partial_{t}\nabla_{\perp}^{2}\phi_{2}-v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\psi_{2}$
$\displaystyle+\\{\phi_{2},\nabla_{\perp}^{2}\phi_{1}\\}+\\{\phi_{1},\nabla_{\perp}^{2}\phi_{2}\\}-\\{\psi_{2},\nabla_{\perp}^{2}\psi_{1}\\}-\\{\psi_{1},\nabla_{\perp}^{2}\psi_{2}\\}$
$\displaystyle=\underbrace{-\nabla_{\perp}\cdot\left[\nabla_{\perp}^{2}\phi_{1}\nabla_{\perp}\xi_{1}+\tilde{n}_{1}v_{\mathrm{A}}\partial_{z}\nabla_{\perp}\psi_{1}\right]}_{A}$
$\displaystyle\quad\underbrace{-\\{\psi_{1},\tilde{n}_{1}\nabla_{\perp}^{2}\psi_{1}\\}+v_{\mathrm{A}}\\{\tilde{n}_{1},b_{z1}\\}}_{B},$
(45)
$\displaystyle\partial_{t}\nabla_{\perp}^{2}\psi_{2}-v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\phi_{2}+\nabla_{\perp}^{2}(\\{\phi_{1},\psi_{2}\\}+\\{\phi_{2},\psi_{1}\\})$
$\displaystyle=\underbrace{d_{i}\nabla_{\perp}^{2}\left\\{\psi_{1},b_{z1}\right\\}}_{C}\underbrace{-\nabla_{\perp}^{2}\left[\frac{1}{2}d_{i}\partial_{z}|\nabla_{\perp}\psi_{1}|^{2}+\nabla_{\perp}\xi_{1}\cdot\nabla_{\perp}\psi_{1}\right]}_{D}$
$\displaystyle\quad+\underbrace{d_{i}\left[\partial_{z}\nabla_{\perp}\cdot\left(\nabla_{\perp}^{2}\psi_{1}\nabla_{\perp}\psi_{1}+\nabla_{\perp}b_{z1}v_{\mathrm{A}}\right)\right]}_{E}.$
(46)

The evolution of $\phi_{2}$ and $\psi_{2}$ depend only on themselves and the
lowest-order variables. The labelled terms that comprise the RHS of each
equation only depend on the lowest-order fluctuations, i.e. they drive
$\phi_{2}$ and $\psi_{2}$. Using (39) combined with (36) to substitute for
$\xi_{1}$, and (38) to substitute for $\tilde{n}_{1}$,

$A=\frac{1}{\Omega_{i}}\nabla_{\perp}\cdot\left[\nabla_{\perp}^{2}\phi\nabla_{\perp}\nabla_{\perp}^{-2}\\{\psi_{1},\nabla_{\perp}^{2}\psi_{1}\\}\right],$
(47)

where $\nabla_{\perp}^{-2}$ is the inverse operator to $\nabla_{\perp}^{2}$,
and we cancel some terms. Next,

$\displaystyle B=$
$\displaystyle-\frac{1}{\Omega_{i}}\left[\nabla_{\perp}^{2}\phi_{1}\left\\{\psi_{1},\nabla_{\perp}^{2}\psi_{1}\right\\}+\nabla_{\perp}^{2}\psi_{1}\left\\{\psi_{1},\nabla_{\perp}^{2}\phi_{1}\right\\}\right.$
$\displaystyle\phantom{\frac{1}{\Omega_{i}}}\left.-\left\\{\nabla_{\perp}^{2}\phi_{1},b_{z1}v_{\mathrm{A}}\right\\}\right],$
(48)

$C$ does not benefit from any manipulation. Using (39) combined with (36) to
substitute for $\xi_{1}$, and cancelling two terms, we find that

$D=\frac{1}{\Omega_{i}}\nabla_{\perp}^{2}\left[\nabla_{\perp}\psi_{1}\cdot\nabla_{\perp}\nabla_{\perp}^{-2}\left\\{\psi_{1},\nabla_{\perp}^{2}\psi_{1}\right\\}\right]$
(49)

Finally, using (40), we find

$\displaystyle E$
$\displaystyle=d_{i}\partial_{z}\nabla_{\perp}\cdot\left[\nabla_{\perp}^{2}\phi_{1}\nabla_{\perp}\phi_{1}-\frac{1}{2}\nabla_{\perp}|\nabla_{\perp}\phi_{1}|^{2}\right]$
$\displaystyle=d_{i}\partial_{z}\nabla_{\perp}\cdot\left[(\hat{\boldsymbol{z}}\times\nabla_{\perp}\phi_{1})\cdot\nabla_{\perp}(\hat{\boldsymbol{z}}\times\nabla_{\perp}\phi_{1})\right].$
(50)

Notably, all of $A$-$E$ vanish if all perpendicular gradients in the system
are aligned. $A$-$D$ involve three powers of the lowest-order AW amplitude
(remembering that $b_{z1}$ is already nonlinear, given by Eq. 40), while $E$
involves only two; thus, $E$ would dominate if we performed a subsidiary
expansion in $\chi\ll 1$ (Eq. 13); for $\chi\sim 1$, all terms are comparable.
Defining Elsasser 1950 variables

$\zeta^{\pm}=\phi\pm\psi,$ (51)

and taking the sum and difference of (45) and (46), our equations may be
written

$\displaystyle\partial_{t}\nabla_{\perp}^{2}\zeta_{2}^{\pm}\mp
v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\zeta_{2}^{\pm}$
$\displaystyle=-\frac{1}{2}\left[\left\\{\zeta_{2}^{-},\nabla_{\perp}^{2}\zeta_{1}^{+}\right\\}+\left\\{\zeta_{2}^{+},\nabla_{\perp}^{2}\zeta_{1}^{-}\right\\}+\left\\{\zeta_{1}^{-},\nabla_{\perp}^{2}\zeta_{2}^{+}\right\\}+\left\\{\zeta_{1}^{+},\nabla_{\perp}^{2}\zeta_{2}^{-}\right\\}\mp\nabla_{\perp}^{2}\left(\left\\{\zeta_{1}^{+},\zeta_{2}^{-}\right\\}+\left\\{\zeta_{2}^{+},\zeta_{1}^{-}\right\\}\right)\right]$
$\displaystyle\quad+\frac{1}{\Omega_{i}}\left[\nabla_{\perp}\nabla_{\perp}^{2}\phi_{1}\cdot\nabla_{\perp}\nabla_{\perp}^{-2}\left\\{\psi_{1},\nabla_{\perp}^{2}\psi_{1}\right\\}-\nabla_{\perp}^{2}\psi_{1}\left\\{\psi_{1},\nabla_{\perp}^{2}\phi_{1}\right\\}+\left\\{\nabla_{\perp}^{2}\phi_{1},b_{z1}v_{\mathrm{A}}\right\\}\right]$
$\displaystyle\quad\pm\frac{1}{\Omega_{i}}\left[\nabla_{\perp}^{2}\left\\{\psi_{1},b_{z1}v_{\mathrm{A}}\right\\}-\nabla_{\perp}^{2}\left(\nabla_{\perp}\psi_{1}\cdot\nabla_{\perp}\nabla_{\perp}^{-2}\left\\{\psi_{1},\nabla_{\perp}^{2}\psi_{1}\right\\}\right)+v_{\mathrm{A}}\partial_{z}\nabla_{\perp}\cdot\left((\hat{\boldsymbol{z}}\times\nabla_{\perp}\phi_{1})\cdot\nabla_{\perp}(\hat{\boldsymbol{z}}\times\nabla_{\perp}\phi_{1})\right)\right]$
(52)

The first line of the RHS are just the standard RMHD nonlinearities between
$\zeta_{2}^{\pm}$ and $\zeta_{1}^{\mp}$. The second and third lines of the RHS
describe a nonlinear drive for $\zeta_{2}^{\pm}$. Most of these terms are
cubic in the fluctuation amplitude, but the final term is quadratic, and thus
dominates when $\chi\ll 1$.

The result of this calculation may be summarized as follows: at lowest
nontrivial order in the RMHD expansion arbitrary 3D structure is permitted for
Alfvén waves travelling in only one direction, even when
$d_{i}\nabla_{\perp}\sim 1$. However, at next order in the expansion, new
terms appear which break this nonlinear solution: taking $\zeta_{1}^{-}=0$,
these new nonlinear terms drive both $\zeta_{2}^{+}$ and $\zeta_{2}^{-}$.

###  II.3 Co-propagating three-wave interaction, $\chi\ll 1$

Let us suppose that

$\displaystyle\zeta_{1}^{-}=0,\quad\zeta_{1}^{+}=\zeta_{A}^{+}+\zeta_{B}^{+},$
$\displaystyle\zeta^{+}_{A,B}=s_{A,B}e^{i(\boldsymbol{k}_{A,B}\cdot\boldsymbol{r}+k_{zA,B}v_{\mathrm{A}}t)}+\text{c.c.},$
(53)

which is an exact nonlinear solution at lowest order (i.e. in RMHD), with

$|s_{A,B}|=\frac{\chi_{A,B}v_{\mathrm{A}}k_{z}}{k_{\perp}^{2}},$ (54)

and $\chi_{A,B}\ll 1$. At second order in this weakly nonlinear case, all but
the last term on the RHS of (52) may be neglected since they are smaller by a
factor $\chi$, and since we have taken $\zeta^{-}_{1}=0$ we are left with

$\displaystyle\partial_{t}\nabla_{\perp}^{2}\zeta_{2}^{\pm}\mp
v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\zeta_{2}^{\pm}$
$\displaystyle=\pm\frac{1}{4}d_{i}\partial_{z}\nabla_{\perp}\cdot\left((\hat{\boldsymbol{z}}\times\nabla_{\perp}\zeta_{1}^{+})\cdot\nabla_{\perp}(\hat{\boldsymbol{z}}\times\nabla_{\perp}\zeta_{1}^{+})\right).$
(55)

Note that if $\zeta^{-}_{1}\neq 0$, the term will also have contributions from
counter-propagating waves; however, in the case where there are
counterpropagating waves this interaction would be subdominant to the RMHD
nonlinearity at lower order. The RHS has nonzero components with two different
phase functions

$(\boldsymbol{k}_{A}\pm\boldsymbol{k}_{B})\cdot\boldsymbol{r}+(k_{zA}\pm
k_{zB})v_{\mathrm{A}}t,$ (56)

and we may write the resonant solution
$\zeta_{2}^{\pm}=\zeta_{A+B}^{\pm}+\zeta_{A-B}^{\pm}$ where

$\displaystyle\zeta_{A+B}^{\pm}$
$\displaystyle=s_{A+B}^{\pm}e^{i((\boldsymbol{k}_{A}+\boldsymbol{k}_{B})\cdot\boldsymbol{r}+(k_{zA}+k_{zB})v_{\mathrm{A}}t)}+\text{c.c.},$
(57) $\displaystyle\zeta_{A-B}^{\pm}$
$\displaystyle=s_{A-B}^{\pm}e^{i((\boldsymbol{k}_{A}-\boldsymbol{k}_{B})\cdot\boldsymbol{r}+(k_{zA}-k_{zB})v_{\mathrm{A}}t)}+\text{c.c.}$
(58)

It is not necessary to solve for both of these, so we choose $A+B$.
Substituting in for $\zeta_{1}^{+}$ on the RHS and matching the spatial
dependence of both sides,

$\displaystyle-k_{\perp+}^{2}\left[\partial_{t}\zeta_{A+B}^{\pm}\mp
ik_{z+}v_{\mathrm{A}}\zeta_{A+B}^{\pm}\right]$
$\displaystyle=\pm\frac{1}{2}ik_{z+}d_{i}|\boldsymbol{k}_{\perp
A}\times\boldsymbol{k}_{\perp B}|^{2}s_{A}s_{B}e^{i(\boldsymbol{k}_{+}\cdot
r+k_{z+}v_{\mathrm{A}}t)}+\text{c.c.},$ (59)

where $\boldsymbol{k}_{+}=\boldsymbol{k}_{A}+\boldsymbol{k}_{B}$. The solution
is a non-resonant $\zeta^{-}$ mode with amplitude

$s_{A+B}^{-}=\frac{s_{A}s_{B}}{4\Omega_{i}}\frac{|\boldsymbol{k}_{\perp
A}\times\boldsymbol{k}_{\perp B}|^{2}}{|\boldsymbol{k}_{\perp
A}+\boldsymbol{k}_{\perp B}|^{2}},$ (60)

and a resonant $\zeta^{+}$ mode,

$s^{+}_{A+B}=-\left[\frac{1}{2}i(k_{zA}+k_{zB})d_{i}\frac{|\boldsymbol{k}_{\perp
A}\times\boldsymbol{k}_{\perp B}|^{2}}{|\boldsymbol{k}_{\perp
A}+\boldsymbol{k}_{\perp B}|^{2}}s_{A}s_{B}\right]t,$ (61)

with the solution only valid at sufficiently early times that
$s_{A+B}^{+}/s_{A,B}\ll 1$. Writing these in terms of the magnitudes of vector
Elsasser variables
$\boldsymbol{z}^{\pm}=\boldsymbol{z}\times\nabla_{\perp}\zeta^{\pm}$,

$\displaystyle\frac{z^{-}_{A+B}}{v_{\mathrm{A}}}$
$\displaystyle=\frac{1}{4}\left(\frac{k_{\perp A}k_{\perp
B}}{|\boldsymbol{k}_{\perp A}+\boldsymbol{k}_{\perp
B}|}\right)d_{i}\sin^{2}\alpha\frac{z^{+}_{A}z^{+}_{B}}{v_{\mathrm{A}}^{2}},$
(62) $\displaystyle\frac{z^{+}_{A+B}}{v_{\mathrm{A}}}$
$\displaystyle=-\frac{1}{2}\left(\frac{k_{\perp A}k_{\perp
B}}{|\boldsymbol{k}_{\perp A}+\boldsymbol{k}_{\perp
B}|}\right)d_{i}\sin^{2}\alpha\frac{z^{+}_{A}z^{+}_{B}}{v_{\mathrm{A}}^{2}}(k_{zA}+k_{zB})v_{\mathrm{A}}t,$
(63)

where $\alpha$ is the angle between $k_{\perp A}$ and $k_{\perp B}$. Supposing
that the two primary waves have roughly similar scales and amplitudes, i.e.
$k_{\perp A}\sim k_{\perp B}\sim k_{\perp+}$, $k_{zA}\sim k_{zB}\sim k_{z+}$,
and $z^{+}_{A}\sim z^{+}_{B}\sim z^{+}_{1}$, in terms of scalings

$\displaystyle\frac{z^{-}_{2}}{v_{\mathrm{A}}}$ $\displaystyle\sim
k_{\perp}d_{i}\sin^{2}\alpha\left(\frac{z^{+}_{1}}{v_{\mathrm{A}}}\right)^{2},$
(64) $\displaystyle\frac{z^{+}_{2}}{v_{\mathrm{A}}}$ $\displaystyle\sim
k_{\perp}d_{i}\sin^{2}\alpha\left(\frac{z^{+}_{1}}{v_{\mathrm{A}}}\right)^{2}k_{z}v_{\mathrm{A}}t.$
(65)

The calculation breaks down once $z^{+}_{2}\sim z^{+}_{1}$, on a timescale

$\tau_{H2}\sim\tau_{A}(k_{\perp}d_{i}\sin^{2}\alpha
z^{+}_{1}/v_{\mathrm{A}})^{-1},$ (66)

where $\tau_{A}=1/k_{z}v_{\mathrm{A}}$; as expected, $\tau_{H2}\gg\tau_{A}$ by
a factor of $1/\epsilon$. More specifically, it is slower by a factor $\sim
n_{1}$, the density fluctuations (see Eq. 1): we will study a case where the
density fluctuations are permitted to become large in Sec. III.

###  II.4 Strong nonlinearity, $\chi\gtrsim 1$

When $\chi\gtrsim 1$ (as is likely the case in turbulence in the corona), the
cubic nonlinear terms in (52) are significant; moreover, the driven wave is
large enough that the RMHD-like terms on the first line of the RHS of (52) are
also important: resonant $n$-wave interactions are comparable for all $n\geq
3$. Another nonlinear timescale may be defined by comparing the cubic
nonlinear terms with the time-derivative term,

$\tau_{H3}\sim\left(k_{\perp}d_{i}|\sin\alpha|k_{\perp}(z^{+})^{2}/{v_{\mathrm{A}}}\right)^{-1}.$
(67)

Note that (ignoring factors of $\sin\alpha$)

$\frac{\tau_{H2}}{\tau_{H3}}\sim\chi.$ (68)

In this case with $\chi\gtrsim 1$, this process dominates until the RMHD-like
terms on the RHS of Eq. 52 are comparable to the cubic nonlinear terms, which
occurs at

$\frac{z^{+}_{2}}{v_{\mathrm{A}}}\sim\frac{z^{-}_{2}}{v_{\mathrm{A}}}\sim
k_{\perp}d_{i}\frac{{z^{+}_{1}}^{2}}{v_{\mathrm{A}}^{2}}.$ (69)

###  II.5 Coronal turbulence and laboratory experiments

Since this nonlinear interaction is a factor of $\epsilon$ slower than the
primary interaction between roughly-equal-amplitude counterpropagating waves,
clearly it can only be important when $\zeta^{-}/\zeta^{+}\lesssim\epsilon$,
i.e. the turbulence is rather imbalanced and the “standard" RMHD nonlinearity
is suppressed. While dispersive effects (e.g. at $\rho_{s}\nabla_{\perp}\sim
1$) also allow for co-propagating interactions (see Sec. III and Appendix B),
the mechanism in this section works even though linearizing the closed system
comprised of Eqs. (36),(37) and (52) results in non-dispersive AW, with
$\omega=\pm k_{\parallel}v_{\mathrm{A}}$.

In the corona, $\beta\ll 1$, justifying the use of Hall MHD, and moreover
fluctuations are observed to be oblique and very imbalanced, the perfect
situation for this interaction to become important. Moreover, the recently-
discovered helicity barrier (Meyrand  _et al._ , 2021; Squire  _et al._ ,
2021) may cause the amplitude of the turbulence (and therefore $\epsilon$) to
become rather large, helping the nonlinear effect here to become more
important.

Another situation where this calculation might be relevant is in laboratory
experiments on LAPD, where $k_{\perp}d_{i}\gg 1$ and so the density
fluctuations are relatively large. This decreases $\tau_{H2}$ (and
$\tau_{H3}$) relative to $\tau_{A}$, and so this nonlinear interaction could
be important when co-propagating waves with unaligned polarizations are
launched in the device.

##  III One-dimensional steepening of dispersive Alfvén waves

The reason the co-propagating nonlinearity was a higher-order effect in Sec.
II was that the density fluctuations (compare Eqs. 1 and 66) are still
$O(\epsilon)$: in this section, we will study the case where we allow them to
reach $O(1)$, by ordering $k_{\perp}d_{i}\gg 1$. Surprisingly we will find
that large density fluctuations alone is not enough to allow AW steepening:
even with $O(1)$ density fluctuations, a non-trivial one-dimensional nonlinear
interaction of a single AW only occurs if one or more of the parameters
$k_{\perp}^{2}\rho_{\tau}^{2}$, $k_{\perp}^{2}d_{e}^{2}$,
$k_{\parallel}^{2}d_{i}^{2}$, or $\eta k_{\perp}^{2}/\omega$ are non-
negligible: these are precisely those that cause the wave to be dispersive,
see Eq. 2.

###  III.1 Basic equations

To study nonlinear steepening, we may retreat to the comforting simplicity of
one-dimensional dynamics. Unlike in Section II, we assume that all variation
is in the $x$-direction, and that, without loss of generality, the background
magnetic field lies in the $x-z$ plane, i.e. the full magnetic field vector is

$\boldsymbol{b}=(v_{\mathrm{A}}\cos\theta,b_{y},b_{z}+v_{\mathrm{A}}\sin\theta),$
(70)

which defines the angle $\theta$ and the fluctuating components of the
magnetic field $b_{y}$ and $b_{z}$ (the $x$-component of the magnetic field is
a constant by magnetic Gauss’ law). Similarly, we write the ion velocity
fluctuations in components $u_{x},u_{y},u_{z}$, and again write

$n=1+\tilde{n},$ (71)

so that $\tilde{n}$ is the normalized density fluctuation. The two-fluid
equations (137-140) restricted to one dimension read

$\displaystyle\partial_{t}n$ $\displaystyle=-\partial_{x}(nu_{x}),$ (72)
$\displaystyle\mathrm{d}_{t}u_{x}$
$\displaystyle=-\frac{1}{n}\partial_{x}\left(c_{s}^{2}n+\frac{1}{2}b_{y}^{2}+\frac{1}{2}b_{z}^{2}+b_{z}v_{\mathrm{A}}\sin\theta\right)-\frac{Zm_{e}}{m_{i}}\mathrm{d}_{t}u_{x},$
(73) $\displaystyle\mathrm{d}_{t}u_{y}$
$\displaystyle=\frac{1}{n}v_{\mathrm{A}}\cos\theta\partial_{x}b_{y}-\frac{Zm_{e}}{m_{i}}\mathrm{d}_{t}u_{y}-\frac{Zm_{e}}{m_{i}}d_{i}\mathrm{d}_{t}\frac{\partial_{x}b_{z}}{n},$
(74) $\displaystyle\mathrm{d}_{t}u_{z}$
$\displaystyle=\frac{1}{n}v_{\mathrm{A}}\cos\theta\partial_{x}b_{z}-\frac{Zm_{e}}{m_{i}}\mathrm{d}_{t}u_{z}+\frac{Zm_{e}}{m_{i}}d_{i}\mathrm{d}_{t}\frac{\partial_{x}b_{y}}{n},$
(75) $\displaystyle\mathrm{d}_{t}b_{y}$
$\displaystyle=v_{\mathrm{A}}\cos\theta\partial_{x}u_{y}-b_{y}\partial_{x}u_{x}+d_{i}\partial_{x}\mathrm{d}_{t}u_{z}+\eta\partial_{x}\left(\frac{\partial_{x}b_{y}}{n}\right),$
(76) $\displaystyle\mathrm{d}_{t}b_{z}$
$\displaystyle=v_{\mathrm{A}}\cos\theta\partial_{x}u_{z}-b_{z}\partial_{x}u_{x}-v_{\mathrm{A}}\sin\theta\partial_{x}u_{x}-d_{i}\partial_{x}\mathrm{d}_{t}u_{y}$
$\displaystyle\quad+\eta\partial_{x}\left(\frac{\partial_{x}b_{z}}{n}\right),$
(77)

where $\mathrm{d}_{t}=\partial_{t}+u_{x}\partial_{x}$: because of the one-
dimensional propagation, the only velocity driving nonlinear interaction is
now the compressive velocity $u_{x}$.

To derive a system that describes nonlinear dispersive Alfvénic fluctuations,
we begin as in Sec. II, assuming the waves are highly oblique and using as our
expansion parameter

$\epsilon\sim\cos\theta.$ (78)

To keep the kinetic-Alfvén and inertial-Alfvén dispersive terms111Once again,
keeping both of these simultaneously ignores strong Landau damping, and our
results are really only valid when at least one of $d_{e}$ or $\rho_{\tau}$ is
small. as well as resistivity, we will also order

$\frac{c_{s}}{v_{\mathrm{A}}}\sim\sqrt{\frac{Zm_{e}}{m_{i}}}\sim\nu_{ei}\sim\epsilon.$
(79)

To keep the finite-frequency effects, we order

$\partial_{t}\sim v_{\mathrm{A}}\cos\theta\partial_{x}\sim\Omega_{i},$ (80)

which combined with (78) gives

$d_{i}\partial_{x}\sim\epsilon^{-1},$ (81)

the ion inertial scale is large compared to the typical lengthscale of the
fluctuations we are interested in. Combining the above equations, we have

$d_{i}\cos\theta\partial_{x}\sim\rho_{\tau}\partial_{x}\sim
d_{e}\partial_{x}\sim\eta\partial_{x}^{2}\sim 1,$ (82)

so that we are keeping all dispersive effects appearing in (2). For the system
to be strongly nonlinear, we require

$\partial_{t}\sim u_{x}\partial_{x}(\sim\Omega_{i}),$ (83)

whence

$\frac{u_{x}}{v_{\mathrm{A}}}\sim\epsilon.$ (84)

We will assume from the outset also that

$\frac{u_{y}}{v_{\mathrm{A}}}\sim\frac{b_{y}}{v_{\mathrm{A}}}\sim\epsilon,$
(85)

showing that this is consistent later in the calculation.

The ordering of the density fluctuation $\tilde{n}$ follows from (72);
ordering the different terms,

$\tilde{n}\sim 1,$ (86)

and the density fluctuations can be large. The ordering for the final two
variables, $b_{z}$ and $u_{z}$, may be deduced (similarly to how we obtained
their size in Sec. II) as follows. At lowest order in this expansion, (73)
becomes

$\mathrm{d}_{t}u_{x}=-\frac{1}{n}\partial_{x}\left(c_{s}^{2}n+\frac{1}{2}b_{y}^{2}+b_{z}v_{\mathrm{A}}\right),$
(87)

whence $b_{z}/v_{\mathrm{A}}\sim\epsilon^{2}$ to be consistent with the other
orderings.222We have preemptively used this to neglect the $b_{z}^{2}$ term;
this also puts some weak restrictions on the ranges of $c_{s}/v_{\mathrm{A}}$,
$b_{y}/v_{\mathrm{A}}$, and $\partial_{t}/\Omega_{i}$ allowed, which we will
not discuss. Note that if $\mathrm{d}_{t}u_{x}$ and
$(1/n)c_{s}^{2}\nabla_{\perp}\tilde{n}$ may be neglected, which are associated
with dispersion at finite $k_{\parallel}d_{i}$ and $k\rho_{\tau}$
respectively, all that is left is the MHD expression
$b_{z}=-b_{y}^{2}/2v_{\mathrm{A}}$ (Barnes and Hollweg, 1974; Vasquez and
Hollweg, 1996). Technically, this is already steepening, in that $b_{z}$
appears as a harmonic of $b_{y}$, but, when dispersive effects are relevant,
the steepening we will derive shortly also occurs in $b_{y}$, and drives much
larger-amplitude harmonics.

At lowest order, (75) is

$\mathrm{d}_{t}u_{z}=\frac{1}{n}v_{\mathrm{A}}\cos\theta\partial_{x}b_{z}+\frac{Zm_{e}}{m_{i}}d_{i}\mathrm{d}_{t}\frac{\partial_{x}b_{y}}{n},$
(88)

and, to ensure consistency with the other orderings,
$u_{z}/v_{\mathrm{A}}\sim\epsilon^{2}$.

These orderings mean that at lowest order, (77) simply reads

$v_{\mathrm{A}}\partial_{x}u_{x}=-d_{i}\partial_{x}\mathrm{d}_{t}u_{y}.$ (89)

where we have used $\sin\theta=1-O(\epsilon^{2})$.333This should be compared
with the similar equation (21) in the previous section. Now, integrate once;
the integration constant may be set to zero with either periodic or decaying
boundary conditions, and we obtain

$u_{x}=-\frac{1}{\Omega_{i}}\mathrm{d}_{t}u_{y},$ (90)

showing that our ordering of $u_{x}\sim u_{y}$ (85) is consistent. Note that
(90) shows that the compressive velocity results from the polarization drift
(Hollweg, 1999). In (74), all we need to keep are

$\mathrm{d}_{t}u_{y}=\frac{1}{n}v_{\mathrm{A}}\cos\theta\partial_{x}b_{y},$
(91)

showing that $u_{y}\sim b_{y}$, as anticipated by (85). To make the equations
more amenable to analysis, we first use (90) in (91) to obtain

$u_{x}=-\frac{1}{n}d_{i}\cos\theta\partial_{x}b_{y},$ (92)

which makes the way in which the dispersive parameter
$d_{i}\cos\theta\partial_{x}$ (i.e., $k_{\parallel}d_{i}$) controls the size
of $u_{x}$ slightly more transparent. Using this in (72), we obtain an
equation for the density

$\partial_{t}n=d_{i}\cos\theta\partial_{x}^{2}b_{y}.$ (93)

Use (87) in (88) to eliminate $b_{z}$,

$\displaystyle\mathrm{d}_{t}u_{z}=$
$\displaystyle-\frac{1}{n}c_{s}^{2}\cos\theta\partial_{x}n-\frac{1}{2n}\cos\theta\partial_{x}b_{y}^{2}$
$\displaystyle-\cos\theta\mathrm{d}_{t}u_{x}+\frac{Zm_{e}}{m_{i}}d_{i}\mathrm{d}_{t}\frac{\partial_{x}b_{y}}{n},$
(94)

and then use (94) to eliminate $u_{z}$ from (76),

$\displaystyle\partial_{t}b_{y}=$ $\displaystyle
v_{\mathrm{A}}\cos\theta\partial_{x}u_{y}-\partial_{x}(u_{x}b_{y})$
$\displaystyle-
d_{i}\cos\theta\partial_{x}\left(\frac{1}{n}c_{s}^{2}\partial_{x}n+\frac{1}{2n}\partial_{x}b_{y}^{2}+\mathrm{d}_{t}u_{x}\right),$
$\displaystyle+d_{e}^{2}\partial_{x}\mathrm{d}_{t}\frac{\partial_{x}b_{y}}{n}+\eta\partial_{x}\frac{\partial_{x}b_{y}}{n}.$
(95)

Using (92),

$\frac{1}{2n}d_{i}\cos\theta\partial_{x}b_{y}^{2}=-u_{x}b_{y},$ (96)

and using this in (95) two of the nonlinear terms cancel, resulting in

$\displaystyle\partial_{t}b_{y}=$ $\displaystyle
v_{\mathrm{A}}\cos\theta\partial_{x}u_{y}-d_{i}\cos\theta\partial_{x}\left(\frac{1}{n}c_{s}^{2}\partial_{x}n+\mathrm{d}_{t}u_{x}\right)$
$\displaystyle+d_{e}^{2}\partial_{x}\mathrm{d}_{t}\frac{\partial_{x}b_{y}}{n}+\eta\partial_{x}\frac{\partial_{x}b_{y}}{n}.$
(97)

This last cancellation eliminates the usual terms that give rise to the DNLS
(derivative nonlinear Schrödinger) type nonlinearity and associated solitons;
the remaining nonlinearity is intrinsically dispersive, as we will show later.

In summary, we have derived a closed set of model equations with which to
study the physics of one-dimensional small-amplitude, strongly nonlinear,
oblique, low-$\beta$ dispersive Alfvén waves; this set is

$\displaystyle u_{x}$
$\displaystyle=-\frac{1}{n}d_{i}\cos\theta\partial_{x}b_{y},$ (98)
$\displaystyle\partial_{t}n$
$\displaystyle=d_{i}\cos\theta\partial_{x}^{2}b_{y},$ (99)
$\displaystyle\mathrm{d}_{t}u_{y}$
$\displaystyle=\frac{1}{n}v_{\mathrm{A}}\cos\theta\partial_{x}b_{y},$ (100)
$\displaystyle\partial_{t}b_{y}$
$\displaystyle=v_{\mathrm{A}}\cos\theta\partial_{x}u_{y}-d_{i}\cos\theta\partial_{x}\left(\frac{1}{n}c_{s}^{2}\partial_{x}n\right)$
$\displaystyle\quad-
d_{i}\cos\theta\partial_{x}\mathrm{d}_{t}u_{x}+d_{e}^{2}\partial_{x}\mathrm{d}_{t}\frac{\partial_{x}b_{y}}{n}+\eta\partial_{x}\frac{\partial_{x}b_{y}}{n}.$
(101)

Equations equivalent to (98-101) written in different variables were
previously derived by Seyler & Lysak (1999)Seyler and Lysak (1999). In the
variables chosen here, the terms giving rise to dispersive waves (which we
will call "dispersive terms" in this section) are all collected on the RHS of
(101): from left to right, the terms on the RHS correspond to: (non-
dispersive) Alfvénic propagation, dispersion due to finite $k\rho_{\tau}$,
$k_{\parallel}d_{i}$, $kd_{e}$, and resistive damping, respectively. However,
each of these introduces not only linear dispersive terms but also additional
nonlinearity.

As an important caveat, this fluid model completely neglects many kinetic
phenomena: for example, wave-particle interactions like Landau and cyclotron
damping, and a realistic model of the ion response once $k\rho_{i}\sim 1$. It
should therefore be used with caution and skepticism. For example, if
$kd_{e}\sim k\rho_{\tau}\sim 1$, the real system exhibits strong Landau
damping(Zocco and Schekochihin, 2011), and even the real part of the linear
frequency predicted by our equations is incorrect. It is therefore only
reasonable to use these equations when at least one of $k_{\perp}\rho_{\tau}$
or $k_{\perp}d_{e}$ is small. Likewise, the model ceases to be reasonable if
$k_{\parallel}d_{i}$ becomes too large, in which case the real plasma exhibits
strong ion cyclotron damping.

###  III.2 Non-dispersive limit

First, we will briefly examine the equations in the strongly nonlinear but
non-dispersive limit, i.e. setting the last four terms of (101) to zero. Let
us try an Alfvén-wave solution, i.e. variables are functions of
$x-tv_{\mathrm{A}}\cos\theta$ (the opposite sign could be chosen if desired);
then, $\partial_{t}=-v_{\mathrm{A}}\cos\theta\partial_{x}$. Then, from (99),

$\tilde{n}=-\frac{\partial_{x}b_{y}}{\Omega_{i}}.$ (102)

(101) without the dispersive terms gives

$b_{y}=-u_{y}.$ (103)

Inserting (98) into (100), using
$\partial_{t}=-v_{\mathrm{A}}\cos\theta\partial_{x}$ and rearranging,

$-v_{\mathrm{A}}\cos\theta\partial_{x}u_{y}=\frac{1}{n}v_{\mathrm{A}}\cos\theta\left(\frac{\partial_{x}b_{y}}{\Omega_{i}}\partial_{x}u_{y}+\partial_{x}b_{y}\right)$
(104)

Inserting (102) and (103), we find that the nonlinear terms in this equation
cancel, showing that the Alfvén wave is in fact a valid nonlinear solution no
matter its (one-dimensional) spatial structure. Thus, if the dispersive terms
are sufficiently small, to lowest order444If we were to formally continue this
expansion to higher order in the dispersive parameters, we would find that the
wave steepens at higher order. the solution is a nonlinear Alfvén wave, with
strongly nonlinear compressive flow and density fluctuations. The fact that
the Alfvén wave survives so well in this regime agrees with the results
already obtained in Sec. II and in previous studies(Zocco and Schekochihin,
2011). This new calculation extends that result to larger $k_{\perp}d_{i}$
(larger density fluctuations), with the important restriction to one
dimension. In Mallet _et al._ 2023(Mallet, 2023), we show that in fact this
extension can continue all the way to $b_{y}/v_{\mathrm{A}}\sim 1$.

The main subject of this calculation, of course, is to to study the nonlinear
behaviour of these equations _without_ neglecting the dispersion. We will show
that once the dispersion is significant, a monochromatic $b_{y}$ (for example)
is no longer an exact solution, implying that the Alfvén wave is steepened due
to the dispersive nonlinearity.

###  III.3 Dispersive steepening

Let us study the weakly nonlinear but strongly dispersive case, taking the
overall amplitude of the fluctuations to be a factor $\delta\ll 1$ smaller
than in the original expansion in $\epsilon$ (Eq. 85)555These two expansions
commute, but if any of the dispersive parameters becomes small in $\delta$
they must be dropped in the following analysis, while keeping intact the
ordering of $kd_{i}\sim\epsilon^{-1}$, as well as the order-unity dispersive
parameters (Eq. 82). We then expand in powers of $\delta$:

$\displaystyle u_{x}$ $\displaystyle=u_{1x}+u_{2x}+\ldots,$ $\displaystyle
u_{y}$ $\displaystyle=u_{1y}+u_{2y}+\ldots,$ $\displaystyle b_{y}$
$\displaystyle=b_{1y}+b_{2y}+\ldots,$ $\displaystyle n$
$\displaystyle=1+n_{1}+n_{2}+\ldots,$ (105)

where

$\epsilon
v_{\mathrm{A}}\sim\frac{u_{1x}}{\delta}\sim\frac{u_{2x}}{\delta^{2}},$ (106)

and so on for the other variables. Qualitatively, one can already see the
basic result we will obtain as follows. The ratio between each of the linear
($\delta$) and leading-order ($\delta^{2}$) nonlinear dispersive terms on the
RHS of (101) is

$n_{1}\sim kd_{i}\frac{b_{1y}}{v_{\mathrm{A}}}\sim\delta,$ (107)

which suggests that the harmonic amplitude will scale with $kd_{i}$, which is
large. This will be confirmed by our mathematical analysis (see Eq. 116
later), and may explain why the harmonics observed on LAPD are quite
substantial(Abler, 2023); much larger than the MHD prediction (Barnes and
Hollweg, 1974; Vasquez and Hollweg, 1996).

####  III.3.1 $O(\delta)$

At first order, the equations are linear. Let us therefore assume sinusoidal
variation, with fluctuations $\propto\exp(i(kx-\omega t))$. From (98-101),

$\displaystyle u_{1x}$ $\displaystyle=-ik_{\parallel}d_{i}b_{1y},\quad
n_{1}=-\frac{ikk_{\parallel}d_{i}}{\omega}b_{1y},\quad
u_{1y}=-\frac{k_{\parallel}v_{\mathrm{A}}}{\omega}b_{1y},$ (108)
$\displaystyle\omega b_{1y}$
$\displaystyle=-k_{\parallel}v_{\mathrm{A}}u_{1y}+ikk_{\parallel}c_{s}^{2}d_{i}n_{1}-i\omega
k_{\parallel}d_{i}u_{1x}-k^{2}d_{e}^{2}\omega b_{1y}$
$\displaystyle\quad-i\eta k^{2}b_{1y},$ (109)

where

$k_{\parallel}=k\cos\theta\sim\epsilon k\ll k.$ (110)

Substituting the linear relationships (108) into (109), we obtain the
(expected) dispersion relation (2).

####  III.3.2 $O(\delta^{2})$

At second order, we must take into account the quadratic nonlinearities of the
first-order quantities. It is obvious that the second-order fluctuations must
therefore be proportional to $\exp(2i(kx-\omega t))$, and therefore do not in
general lie on the linear DAW dispersion relation.666If more than a single
sinusoid is considered at $O(\delta)$, one can also solve for the beat mode
resulting in the difference of the phase functions. In our case with a single
$O(\delta)$ wave, one just obtains $0=0$. We therefore do not consider this in
the present analysis. From (98-100) we obtain

$\displaystyle u_{2x}$
$\displaystyle=-2ik_{\parallel}d_{i}b_{2y}+ik_{\parallel}d_{i}n_{1}b_{1y},$
(111) $\displaystyle n_{2}$
$\displaystyle=-\frac{2ikk_{\parallel}d_{i}}{\omega}b_{2y},$ (112)
$\displaystyle u_{2y}$
$\displaystyle=-\frac{k_{\parallel}v_{\mathrm{A}}}{\omega}b_{2y}+\frac{1}{2}\frac{k}{\omega}u_{1x}u_{1y}+\frac{1}{2}\frac{k_{\parallel}v_{\mathrm{A}}}{\omega}n_{1}b_{1y}.$
(113)

Substituting the first-order relations (108) into the above equations, we
write the right hand sides of the above equations in terms of $b_{2y}$ and
$b_{1y}^{2}$ only,

$\displaystyle u_{2x}$
$\displaystyle=-2ik_{\parallel}d_{i}b_{2y}+\frac{kk_{\parallel}^{2}d_{i}^{2}}{\omega}b_{1y}^{2},\quad
n_{2}=-\frac{2ikk_{\parallel}d_{i}}{\omega}b_{2y},$ $\displaystyle u_{2y}$
$\displaystyle=-\frac{k_{\parallel}v_{\mathrm{A}}}{\omega}b_{2y},$ (114)

where in the last equation the nonlinearity coming from the convective
acceleration and the density inhomogeneities have cancelled. Equation (101) at
second order reads

$\displaystyle-2i\omega b_{2y}=$ $\displaystyle
2ik_{\parallel}v_{\mathrm{A}}u_{2y}+4kk_{\parallel}d_{i}c_{s}^{2}n_{2}-2kk_{\parallel}d_{i}c_{s}^{2}n_{1}^{2}$
$\displaystyle-4\omega
k_{\parallel}d_{i}u_{2x}+2kk_{\parallel}d_{i}u_{1x}^{2}$
$\displaystyle+8i\omega k^{2}d_{e}^{2}b_{2y}-4i\omega
k^{2}d_{e}^{2}n_{1}b_{1y}-2ik^{3}d_{e}^{2}u_{1x}b_{1y}$ $\displaystyle-4\eta
k^{2}b_{2y}+2\eta k^{2}n_{1}b_{1y},$ (115)

We use the first-order (108) and second-order (114) results to write the
equation solely in terms of $b_{2y}$ and $b_{1y}^{2}$. Solving for $b_{2y}$
and using (2) to tidy up the expression, we obtain

$\frac{b_{2y}}{v_{\mathrm{A}}}=ikd_{i}\frac{k_{\parallel}v_{\mathrm{A}}}{\omega}\left(\frac{k^{2}\rho_{\tau}^{2}-3k_{\parallel}^{2}d_{i}^{2}-3k^{2}d_{e}^{2}-\frac{i\eta
k^{2}}{\omega}-2k^{2}\rho_{\tau}^{2}(k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2})}{3(k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2}-k^{2}\rho_{\tau}^{2})+\frac{i\eta
k^{2}}{\omega}-2k^{2}\rho_{\tau}^{2}\frac{i\eta
k^{2}}{\omega}}\right)\frac{b_{1y}^{2}}{v_{\mathrm{A}}^{2}}$ (116)

This equation has some interesting properties. First, the driven harmonic
amplitude $b_{2y}$ is of course proportional to $b_{1y}^{2}$, but it also
scales with $kd_{i}$: this parameter is large in the ordering scheme (81), and
so this could explain the somewhat significant harmonics observed in AW
experiments on LAPD(Abler, 2023).

Second, the complicated fraction in between large brackets depends on the
dispersion and resistive damping via the parameters
$k_{\parallel}^{2}d_{i}^{2}$, $k^{2}d_{e}^{2}$, $k^{2}\rho_{\tau}^{2}$, and
$\eta k^{2}/\omega$. In (98-101), all these parameters are considered to be
order unity. However, (116) shows that even if they are all somewhat small
(provided that the $\delta$\- and $\epsilon$\- expansions both remain valid),
the numerator and denominator will be dominated by the largest of these. Then,
the dependence on top and bottom cancels out, giving a saturated harmonic
amplitude that is independent of the value of the dispersive terms even in the
weakly dispersive limit.

One serious limitation of this model is that it is one-dimensional, so that
all wave modes must have the same propagation direction $\mathbf{k}/k$. This
means that there are no RMHD-like interaction terms, for example, even if
counterpropagating waves are put into the main equations (99-101), their
wavevectors are perfectly aligned so the nonlinearity vanishes. It also means
that the second harmonic is not in general a normal mode of the system (as we
pointed out earlier).

Finally, note that without the resistive damping, there would be a resonance
at the point where the Alfvén wave dispersion vanishes (i.e. the positive and
negative dispersive terms cancel). For this special value of the parameters,
the coefficient of $b_{2y}$ vanishes: this may cause rather large harmonics
for some values of the parameters. Probing this resonance would be a useful
test of the model; however, as previously mentioned, care should be taken
since if $k_{\perp}d_{e}\sim k_{\perp}\rho_{\tau}\sim 1$ the dispersion
relation (2) is inaccurate, and the real system exhibits strong Landau
damping.

The timescale for the nonlinear steepening to occur may be estimated in
comparison with the non-dispersive timescale of the wave, i.e. we compare the
timescale of linear Alfvénic propagation (from the first term on the RHS of
Eq. 101), $\tau_{A}^{-1}\sim k_{\parallel}v_{\mathrm{A}}$ with the nonlinear
dispersive terms (the four final terms on the RHS of Eq. 101). A different
timescale may be estimated for each of the dispersive parameters
$k^{2}\rho_{\tau}^{2},k^{2}d_{e}^{2},k_{\parallel}^{2}d_{i}^{2}$, and the
resistivity $\eta k^{2}/\omega$: the relevant nonlinear dispersive term in Eq.
101 leads to the estimates

$\tau_{s}^{-1}\sim
n_{1}k_{\parallel}v_{\mathrm{A}}\times\max(k^{2}\rho_{\tau}^{2},k^{2}d_{e}^{2},k_{\parallel}^{2}d_{i}^{2},\eta
k^{2}/\omega),$ (117)

and so, using the estimate (107),

$\frac{\tau_{A}}{\tau_{s}}\sim
kd_{i}\frac{b_{y}}{v_{\mathrm{A}}}\times\max(k^{2}\rho_{\tau}^{2},k^{2}d_{e}^{2},k_{\parallel}^{2}d_{i}^{2},\eta
k^{2}/\omega),$ (118)

where the largest dispersive parameter is chosen in the equations above and
sets the steepening timescale. Thus, if the wave is strongly dispersive, i.e.
the largest of these parameters is $O(1)$, the steepening is slower than the
linear timescale $\tau_{A}$ by a factor $\delta$ in this weakly nonlinear
case. However, if all the dispersive parameters are also small, the steepening
is slower by a factor of $\delta$ times the largest dispersive parameter.
Steepening can then only be significant if the system is allowed to evolve on
this slow timescale; i.e. if other competing effects like the standard MHD-
like counter-propagating nonlinearity or the three-dimensional interactions
identified in Section II and Appendix B are suppressed. We re-emphasize that,
according to Eq. (116), the saturated harmonic amplitude tends to a finite
limit as the dispersive parameters become small, scaling only with $kd_{i}$:
however, this will only be attained if the steepening is the fastest
evolutionary process for the wave.

Our result shares some similarities with Eq. (3.136) of Brugman 2007Brugman
(2007), who studied the nonlinear interaction of two co-propagating DAW with
aligned polarizations using somewhat different ordering assumptions: their
beat wave amplitude also scales with $kd_{i}$, and has a resonance at the
point where the Alfven wave dispersion vanishes. Unlike our result, in which
this resonance only appears with cancellation of the positive and negative
dispersive terms, the Brugman expression incorrectly predicts this resonance
will also appear in the non-dispersive limit. This is because they did not
systematically keep all the relevant nonlinear terms: the cancellation of
terms in Section III.2 did not occur in this previous work, which would lead
to an incorrect prediction of nonlinear steepening even in the non-dispersive
limit.

###  III.4 Coronal turbulence and laboratory experiments: comparison with Sec.
II

As in Sec. II, in the turbulent corona, this effect may become important
because of the extreme imbalance. In laboratory experiments on LAPD, this
effect could be important even when the dispersive corrections to the real
frequency are small: the resistive damping can still be rather large for some
choices of parameters(Bose  _et al._ , 2019). Whether dispersive steepening
(this section) or 3D nonlinear interaction (Sec. II) dominates depends on how
aligned or dispersive the waves are: for example, comparing (117) with (66),

$\frac{\tau_{s}}{\tau_{H2}}\sim\frac{\sin^{2}\alpha}{\max(k^{2}\rho_{\tau}^{2},k^{2}d_{e}^{2},k_{\parallel}^{2}d_{i}^{2},\eta
k^{2}/\omega)},$ (119)

Thus, the scaling of turbulence dominated by these nonlinearities is likely to
be highly non-universal, with the fastest timescale for co-propagating
nonlinear interaction being one of $\tau_{H2}$, $\tau_{H3}$, or $\tau_{s}$
(with any of the dispersive terms), depending sensitively on parameters. In
general, the scale at which these interactions will "switch on" scales with
$k_{\perp}d_{i}$, which may explain why a spectral break at this scale appears
in the solar wind data at low $\beta$(Chen  _et al._ , 2014), although further
work is needed to develop a theory for turbulence dominated by these effects.

###  III.5 Historical note: No small-amplitude AW solitons.

If resistivity is negligible, one could attempt to find exact solutions to the
original equations (98–101) by looking for a solution with all variables
travelling as a wave at speed $v$, i.e. $f=f(x-vt)$. Then,
$\partial_{t}=-v\partial_{x}$. There are two potential classes of exact
solutions; periodic waves (which at small amplitude behave as in the analysis
above) and solitary waves, whose fluctuations tend to zero at infinity.
Historically, there has been considerable debate as to the existence or non-
existence of these small-amplitude AW solitons, with early analyses suggesting
that kinetic and inertial AW solitons were possible at small
amplitude(Hasegawa and Mima, 1976; Shukla, Rahman, and Sharma, 1982; Wu, Wang,
and Fälthammar, 1995). The subject of Seyler & Lysak (1999)(Seyler and Lysak,
1999) (who, as mentioned previously, used without a detailed derivation our
Eqs. 98–101) was in fact to show that all previous works neglected an
important nonlinear term (the nonlinear polarization drift), and upon
including this term, only singular solitary waves remain, necessarily
containing a pair of density discontinuities. Our systematic derivation will
hopefully convince an interested reader that the inclusion of this term is
correct. In Mallet _et al._ 2023(Mallet, 2023), we will treat the case of
large-amplitude, slightly dispersive waves, and show that we recover both the
small-amplitude singular solitons, but also new large-amplitude regular
solitons.

Section | Timescale symbol $\tau$ | $\tau_{A}/\tau$ | saturation amplitude $b_{2y}/v_{\mathrm{A}}$
---|---|---|---
Sec. II.3 | $\tau_{H2}$ |  $(k_{\perp}d_{i}b_{1y}/v_{\mathrm{A}})\sin^{2}\alpha$ [Eq. 66] | $b_{1y}/v_{\mathrm{A}}$
Sec. II.4 | $\tau_{H3}$ |  $(k_{\perp}d_{i}b_{1y}/v_{\mathrm{A}})(k_{\perp}b_{1y}/k_{\parallel}v_{\mathrm{A}})\sin\alpha$ [Eq. 67] |  $k_{\perp}d_{i}b_{1y}^{2}/v_{\mathrm{A}}^{2}$ [Eq. 69]
Sec. III | $\tau_{s}$ |  $k_{\perp}d_{i}{b_{1y}}/{v_{\mathrm{A}}}\times\max(k_{\perp}^{2}\rho_{\tau}^{2},k_{\perp}^{2}d_{e}^{2},k_{\parallel}^{2}d_{i}^{2},\eta k_{\perp}^{2}/\omega)$ [Eq. 117] |  $k_{\perp}d_{i}b_{1y}^{2}/v_{\mathrm{A}}^{2}$ [Eq. 116]
Table 2: The approximate nonlinear timescales and approximate saturated
fluctuation amplitudes associated with different effects studied in the paper.
Each nonlinear timescale is compared to the Alfvén time
$\tau_{A}=(k_{\parallel}v_{\mathrm{A}})^{-1}$, while the saturation amplitude
normalized to the Alfvén velocity $b_{2y}/v_{\mathrm{A}}$ is expressed in
terms of the amplitude of the primary wave(s) $b_{1y}$. The interaction in
Section II.3 is resonant, and only saturates once it reaches an amplitude
comparable to the primary wave, at which point the ordering assumptions break
down. The saturation amplitude in Sec. III also technically depends on the
values of the dispersive parameters (see Eq. 116); roughly speaking, this
amounts to a prefactor of order unity correcting the entry in this table.
$\alpha$ is a typical angle between perpendicular wavevectors of interacting
waves (see Sec. II).

##  IV Discussion

In this paper, we have analysed the nonlinear properties of small-scale two-
fluid Alfvén waves. Linearly, highly oblique two-fluid AW have two main
differences from MHD. First, when $k_{\perp}d_{i}\sim 1$, they develop a
significant density fluctuation(Hollweg, 1999) due to the Hall effect,

$\frac{\delta n_{i}}{n_{0i}}\sim-ik_{\perp}d_{i}\frac{\delta B}{B_{0}},$ (120)

Second, they become dispersive, with the dispersion controlled by the
parameters $k_{\perp}^{2}\rho_{\tau}^{2}$, $k_{\perp}^{2}d_{e}^{2}$ and
$k_{\parallel}^{2}d_{i}^{2}$ (see Eq. 2). Our nonlinear analysis uncovers two
new and interesting phenomena that stem from these new effects.

In Sec. II, we study a regime where $k_{\perp}d_{i}\sim 1$ in three
dimensions, but we consider non-dispersive AW (i.e, we take $\rho_{\tau}\to 0$
and $d_{e}\to 0$, and $\omega/\Omega_{i}\sim\epsilon$), leaving the equations
of compressible Hall MHD. Using a small-amplitude, anisotropic, low-frequency
expansion of these equations, we show that at the lowest non-trivial order we
recover reduced MHD, for which co-propagating AW with arbitrary spatial
structure are an exact, non-dispersive nonlinear solution. However, at the
second non-trivial order we show that there is a resonant nonlinear
interaction between co-propagating waves with non-aligned wavevectors: even
though the linear waves still obey the MHD linear dispersion relation
$\omega=\pm k_{\parallel}v_{\mathrm{A}}$. Importantly, this nonlinearity
vanishes if the perpendicular wavevectors in the system are aligned, and so a
single sinusoidal wave does not self-interact via this process. We estimate
the timescale for this process (66 and 67). From these timescales, it is clear
that the important parameter controlling the co-propagating interaction
timescale is the size of the AW’s density fluctuations (Eq. 120). Numerical
simulations of the decay of 2D large-amplitude Alfvén wavepackets have
recently been performed by Tenerani _et al._ 2023 Tenerani  _et al._ (2023),
with observed timescale for the interaction $\tau^{*}\sim\tau_{A}(l/d_{i})$:
these simulations were performed for fixed $\delta B/B_{0}\sim
l_{\perp}/l_{\parallel}\sim 1$. Inserting these fixed factors into the
expressions for our timescales, one finds that, indeed,
$\tau^{*}\sim\tau_{H2}\sim\tau_{H3}$, so the interaction derived analytically
in this paper may extend to large-amplitude, less oblique waves.

In Section III, we use a different ordering scheme, allowing these density
fluctuations to be large, $\delta n/n_{0}\sim 1$, and also retain the terms
that give rise to dispersive waves at lowest order (Eq.2) in a two-fluid
model. We show that, even with such large density fluctuations, due to a
fortuitous cancellation in the equations nonlinear steepening in one dimension
is absent if the terms giving rise to dispersion are neglected, only becoming
significant once at least one of the terms is non-negligible, as evidenced by
an expression for the timescale of this nonlinear steepening process (Eq.
117), which is inversely proportional to the dispersive parameters. We also
derive an explicit formula (116) for the amplitude of the harmonic of a
primary wave that would be produced by this mechanism; unlike the timescale,
this is only mildly dependent on the dispersion: weakly dispersive waves take
longer to evolve but have comparable saturated harmonic amplitudes. Both the
harmonic amplitude and the inverse of the timescale are additionally
proportional to $k_{\perp}d_{i}$ since each nonlinearity is driven by the
density fluctuations (Eq. 120). A related model which can deal with very-
large-amplitude waves with $\delta B/B_{0}\sim 1$ and solitons is developed in
Mallet _et al._ 2023(Mallet, 2023).

We summarize the main physical implications of the new nonlinear interactions
identified in in Sections II and III, in terms of their fully-saturated
amplitudes and dynamical timescales, in Table 2. The saturated amplitudes can
be quite large: in Section II the interaction is resonant, and continues until
the ordering assumptions break down, and in Section III the harmonic amplitude
scales with $k_{\perp}d_{i}$, which can be large. Conversely, the timescales
associated with both interactions are typically slow compared to the timescale
associated with the interaction of two roughly equal-amplitude counter-
propagating AW, even at $k_{\perp}d_{i}\sim 1$. However, there are two
important applications for which the new nonlinearities are likely to be
highly relevant.

First, the strength of the nonlinear interactions identified here scales with
$k_{\perp}d_{i}$ (equivalently, with the density fluctuations). This means
that care must be taken when interpreting the results of laboratory AW
experiments: typically on LAPD, $k_{\perp}d_{i}\gg 1$. Our results may provide
a way to understand the generation of wave harmonics on LAPD, as well as
understanding interactions between multiple co-propagating AW.

Second, the fluctuations in the solar corona and inner solar wind, currently
being explored by Parker Solar Probe and Solar Orbiter, are dominated by
large-amplitude, outward-travelling Alfvén waves, with only a relatively small
inward-travelling component. At large scales, this means that the usual
counter-propagating nonlinear interaction is suppressed, and the nonlinear
physics that we have studied here may be important, especially at sharp
switchback boundaries(Krasnoselskikh  _et al._ , 2020). Moreover, a recent
exciting theoretical result shows that the dominant nonlinear cascade is
further suppressed due to a "helicity barrier"(Meyrand  _et al._ , 2021;
Squire  _et al._ , 2021), which causes the turbulence amplitude to increase
until the usual small-amplitude, anisotropic approximation breaks down,
allowing the effects here to become even more significant. Further work will
study the complex interplay of the three-dimensional interaction of Sec. II
and the one-dimensional dispersive steepening of Secs. III, and their
potential application to turbulence in the corona. One important point is
that, unlike in the MHD counter-propagating interaction, these nonlinearities
couple waves to larger $k_{\parallel}$, meaning that they may help the
turbulence access ion cyclotron heating entering at $k_{\parallel}d_{i}\sim
1$, thought to be important for the ion heating necessary to accelerate the
solar wind(Hollweg and Isenberg, 2002).

###### Acknowledgements.

AM is grateful to C. Chaston, J. Bonnell, and S. Boldyrev for useful
discussions. A.M. was supported by NASA grant 80NSSC21K0462 and NASA contract
NNN06AA01C. S.D. was supported by DOE grant DE-SC0021237 and NASA grant
80NSSC18K1235. M.A. was supported by DOE grants DE-SC0021291 and DE-SC0023326.
T.B. was supported by NASA grant 80NSSC21K1771. C.H.K.C. was supported by UKRI
Future Leaders Fellowship MR/W007657/1 and STFC Consolidated Grant
ST/T00018X/1.

##  Appendix A Two-fluid model

Here, we discuss the basic set of model two-fluid equations that are the basis
for the calculations in the main body of the paper. We begin with isothermal
two-fluid equations for electrons and ions,

$\displaystyle\partial_{t}n_{i}+\nabla\cdot(n_{i}\boldsymbol{u})$
$\displaystyle=0,$ (121) $\displaystyle
Zm_{e}n_{i}(\partial_{t}\boldsymbol{u}_{e}+\boldsymbol{u}_{e}\cdot\nabla\boldsymbol{u}_{e})$
$\displaystyle=-Zen_{i}\left(\boldsymbol{E}+\frac{\boldsymbol{u}_{e}\times\boldsymbol{B}}{c}\right)$
$\displaystyle\phantom{=}\,-ZT_{e}\nabla
n_{i}-\nu_{ei}Zn_{i}m_{e}(\boldsymbol{u}_{e}-\boldsymbol{u}),$ (122)
$\displaystyle
m_{i}n_{i}(\partial_{t}\boldsymbol{u}+\boldsymbol{u}\cdot\nabla\boldsymbol{u})$
$\displaystyle={Zen_{i}}\left(E+\frac{\boldsymbol{u}\times\boldsymbol{B}}{c}\right)$
$\displaystyle\phantom{=}-{T_{i}}\nabla
n_{i}-\nu_{ie}n_{i}m_{i}(\boldsymbol{u}-\boldsymbol{u}_{e}),$ (123)
$\displaystyle\nabla\times\boldsymbol{B}$
$\displaystyle=\frac{4\pi}{c}\boldsymbol{J},$ (124)
$\displaystyle\boldsymbol{J}$
$\displaystyle=Zen_{i}(\boldsymbol{u}-\boldsymbol{u}_{e}),$ (125)
$\displaystyle\partial_{t}\boldsymbol{B}$
$\displaystyle=-c\nabla\times\boldsymbol{E},$ (126)
$\displaystyle\nabla\cdot\boldsymbol{B}$ $\displaystyle=0,$ (127)

where notations are standard except that we have denoted the ion velocity as
simply $\boldsymbol{u}$, and we are assuming quasineutrality, $n_{e}=Zn_{i}$.
We first use (122) to obtain

$\displaystyle\boldsymbol{E}=$
$\displaystyle-\frac{\boldsymbol{u}_{e}\times\boldsymbol{B}}{c}-\frac{m_{e}}{e}\left(\partial_{t}\boldsymbol{u}_{e}+\boldsymbol{u}_{e}\cdot\nabla\boldsymbol{u}_{e}\right)$
$\displaystyle-\frac{T_{e}}{e}\nabla\ln(n_{i}/n_{0i})+\frac{m_{e}}{Ze^{2}n_{i}}\nu_{ei}\boldsymbol{J}.$
(128)

In this equation, we then use (125) to eliminate $\boldsymbol{u}_{e}$, and
then Ampère’s law (124) to eliminate $\boldsymbol{J}$ in favour of
$\boldsymbol{B}$, obtaining

$\displaystyle\boldsymbol{E}=$
$\displaystyle-\frac{\boldsymbol{u}\times\boldsymbol{B}}{c}+\frac{1}{4\pi
Ze}\frac{(\nabla\times\boldsymbol{B})\times\boldsymbol{B}}{n_{i}}-\frac{m_{e}}{e}\mathrm{d}_{t}\boldsymbol{u}$
$\displaystyle+\frac{cm_{e}}{4\pi
Ze^{2}}\mathrm{d}_{t}\left(\frac{\nabla\times\boldsymbol{B}}{n_{i}}\right)+\frac{cm_{e}}{4\pi
Ze^{2}}\nu_{ei}\frac{\nabla\times\boldsymbol{B}}{n_{i}}$
$\displaystyle-\frac{T_{e}}{e}\nabla\ln(n_{i}/n_{0i})$
$\displaystyle+\frac{cm_{e}}{4\pi
Ze^{2}}\frac{\nabla\times\boldsymbol{B}}{n_{i}}\cdot\nabla\left(\boldsymbol{u}-\frac{c}{4\pi
Ze}\frac{\nabla\times\boldsymbol{B}}{n_{i}}\right),$ (129)

where we have denoted $\mathrm{d}_{t}=\partial_{t}+\boldsymbol{u}\cdot\nabla$.
We insert (129) into the ion momentum equation (123), noting that the
collisional drag term vanishes by symmetry, and thus obtain

$\displaystyle\mathrm{d}_{t}\boldsymbol{u}$ $\displaystyle=\frac{1}{4\pi
m_{i}}\frac{(\nabla\times\boldsymbol{B})\times\boldsymbol{B}}{n_{i}}-\frac{Zm_{e}}{m_{i}}\mathrm{d}_{t}\boldsymbol{u}$
$\displaystyle+\frac{cm_{e}}{4\pi
em_{i}}\mathrm{d}_{t}\left(\frac{\nabla\times\boldsymbol{B}}{n_{i}}\right)-c_{s}^{2}\nabla\ln(n_{i}/n_{0i})$
$\displaystyle+\frac{cm_{e}}{4\pi
em_{i}}\frac{\nabla\times\boldsymbol{B}}{n_{i}}\cdot\nabla\left(\boldsymbol{u}-\frac{c}{4\pi
Ze}\frac{\nabla\times\boldsymbol{B}}{n_{i}}\right),$ (130)

where

$c_{s}^{2}=\frac{T_{i}+ZT_{e}}{m_{i}}.$ (131)

Another expression for $\boldsymbol{E}$ may be found from (123), again using
(125) and (124) to eliminate $\boldsymbol{u}_{e}$,

$\boldsymbol{E}=\frac{m_{i}}{Ze}\mathrm{d}_{t}u-\frac{\boldsymbol{u}\times\boldsymbol{B}}{c}+\frac{T_{i}}{Ze}\nabla\ln(n_{i}/n_{0i})+\frac{\nu_{ie}m_{i}c}{4\pi
Ze^{2}}\frac{\nabla\times\boldsymbol{B}}{n_{i}}.$ (132)

Using (132) in (126),

$\partial_{t}\boldsymbol{B}=\nabla\times(\boldsymbol{u}\times\boldsymbol{B})-\frac{m_{i}c}{Ze}\nabla\times\mathrm{d}_{t}\boldsymbol{u}-\frac{\nu_{ie}m_{i}c^{2}}{4\pi
Z^{2}e^{2}}\nabla\times\left(\frac{\nabla\times\boldsymbol{B}}{n_{i}}\right).$
(133)

Equations (121),(130), (133) along with (127) are the complete set required to
determine the evolution; to make them more transparent, we normalize the
density and magnetic field according to

$n=\frac{n_{i}}{n_{0i}},\quad\boldsymbol{b}=\frac{\boldsymbol{B}}{\sqrt{4\pi
n_{0i}m_{i}}}.$ (134)

Additionally employing the definitions of the ion and electron inertial scales

$d_{i}=\frac{c\sqrt{m_{i}}}{\sqrt{4\pi Z^{2}e^{2}n_{0i}}},\quad
d_{e}=\frac{c\sqrt{m_{e}}}{\sqrt{4\pi
e^{2}n_{0e}}}=\sqrt{\frac{Zm_{e}}{m_{i}}}d_{i},$ (135)

and the Ohmic resistivity

$\eta=\nu_{ei}d_{e}^{2}=\nu_{ie}d_{i}^{2},$ (136)

the equations may be written

$\displaystyle\partial_{t}n$ $\displaystyle=-\nabla\cdot(n\boldsymbol{u}),$
(137) $\displaystyle\mathrm{d}_{t}\boldsymbol{u}$
$\displaystyle=\frac{1}{n}\boldsymbol{b}\cdot\nabla\boldsymbol{b}-\frac{1}{n}\nabla\left(c_{s}^{2}n+\frac{1}{2}b^{2}\right)+\frac{Zm_{e}}{m_{i}}d_{i}\mathrm{d}_{t}\frac{\nabla\times\boldsymbol{b}}{n}$
$\displaystyle\quad-\frac{Zm_{e}}{m_{i}}\mathrm{d}_{t}\boldsymbol{u}+\frac{Zm_{e}}{m_{i}}d_{i}\frac{\nabla\times\boldsymbol{b}}{n}\cdot\nabla\left(\boldsymbol{u}-d_{i}\frac{\nabla\times\boldsymbol{b}}{n}\right),$
(138) $\displaystyle\mathrm{d}_{t}\boldsymbol{b}$
$\displaystyle=\boldsymbol{b}\cdot\nabla\boldsymbol{u}-\boldsymbol{b}\nabla\cdot\boldsymbol{u}-d_{i}\nabla\times\mathrm{d}_{t}\boldsymbol{u}-\eta\nabla\times\left(\frac{\nabla\times\boldsymbol{b}}{n}\right),$
(139) $\displaystyle\nabla\cdot\boldsymbol{b}$ $\displaystyle=0.$ (140)

These equations, in different limits, are analysed in the main sections of
this paper. This fluid model obviously neglects many important plasma
processes, for example, wave-particle interaction resulting in
Landau/cyclotron damping. This means that our model is not always appropriate:
for example, when $k_{\parallel}d_{i}\sim 1$ or when $k_{\perp}\rho_{i}\sim
1$: future work will incorporate these effects.

We focus on the Alfvénic modes, and take

$\partial_{t}\sim v_{\mathrm{A}}\partial_{z}\lesssim\Omega_{i},$ (141)

which orders out the fast mode, and $c_{s}/v_{\mathrm{A}}=\sqrt{\beta}\ll 1$,
which removes the slow mode. We also assume oblique waves,
$\partial_{z}/\nabla_{\perp}\ll 1$. With these assumptions, one obtains the
dispersion relation(Hollweg, 1999)

$(1+k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2})\omega^{2}+i\eta k^{2}\omega-
k_{\parallel}^{2}v_{\mathrm{A}}^{2}(1+k^{2}\rho_{\tau}^{2})=0,$ (142)

with solution

$\omega=-i\gamma_{0}\pm\sqrt{\omega_{0}^{2}-\gamma_{0}^{2}},$ (143)

where

$\omega_{0}=k_{\parallel}v_{\mathrm{A}}\sqrt{\frac{1+k^{2}\rho_{\tau}^{2}}{1+k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2}}},\quad\gamma_{0}=\frac{1}{2}\frac{\eta
k^{2}}{1+k_{\parallel}^{2}d_{i}^{2}+k^{2}d_{e}^{2}}.$ (144)

As it results from a fluid model, this dispersion relation is a poor
approximation to reality in a number of important situations. For example,
when both $k\rho_{\tau}\sim 1$ and $kd_{e}\sim 1$, real AW undergo strong
Landau damping, completely ignored in this model; even the real frequency in
this regime also differs significantly from this two-fluid model(Thuecks  _et
al._ , 2009). For $k\rho_{s}\sim 1$ but $kd_{e}\ll 1$ and $k\rho_{i}\ll 1$
(fluid kinetic AW) and $kd_{e}\sim 1$ but $k\rho_{\tau}\ll 1$ (fluid inertial
AW), the collisionless damping is relatively weak, and (142) is somewhat
acceptable.

##  Appendix B Reduced dispersive AW equations

Evidence suggests that the small-scale turbulence in the solar wind(Chen  _et
al._ , 2013) consists of kinetic AW; in the aurora, both inertial and kinetic
AW exist in different regions (Stasiewicz  _et al._ , 2000). Modelling of this
turbulent cascade is often done assuming that typical linear and nonlinear
frequencies are small compared to $\Omega_{i}$(Seyler and Xu, 2003;
Schekochihin  _et al._ , 2009; Zocco and Schekochihin, 2011). To make contact
with these existing theories, we use the following ordering scheme. First, we
write the magnetic field $b$ as

$\boldsymbol{b}=v_{\mathrm{A}}+\boldsymbol{b}_{\perp}+\hat{\boldsymbol{z}}b_{z},$
(145)

where $v_{\mathrm{A}}$ is the mean Alfvén velocity and
$\boldsymbol{b}_{\perp}$ and $b_{z}$ are the (small-amplitude) magnetic
fluctuations in velocity units in the perpendicular and $z$ directions.
Similarly, we write

$n=1+\tilde{n},$ (146)

where $\tilde{n}$ is the normalized density fluctuation.

We assume that the perpendicular lengthscales of our fluctuations are small
compared to their parallel lengthscales, and expand in $\epsilon\ll 1$,
defined via

$\frac{\partial_{z}}{\nabla_{\perp}}\sim\epsilon^{2}\ll 1.$ (147)

We also assume that the linear timescales are Alfvénic, and comparable to the
nonlinear timescales and the electron-ion collision frequency, i.e.

$\partial_{t}\sim
v_{\mathrm{A}}\partial_{z}\sim\boldsymbol{u}_{\perp}\cdot\nabla_{\perp}\sim\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}\sim\nu_{ei},$
(148)

which implies that the fluctuation amplitudes are ordered as

$\frac{u_{\perp}}{v_{\mathrm{A}}}\sim\frac{b_{\perp}}{v_{\mathrm{A}}}\sim\epsilon^{2}\ll
1.$ (149)

We also restrict ourselves to low $\beta$ and small mass ratio, taking

$\beta=\frac{c_{s}^{2}}{v_{\mathrm{A}}^{2}}\sim\frac{Zm_{e}}{m_{i}}\epsilon^{2}.$
(150)

We wish to include the kinetic and inertial Alfvén dispersion, so we must
order777Again (cf. discussion in Appendix A), keeping both terms
simultaneously is a poor approximation, but we will here do it anyway to
compare with previous work.

$\rho_{\tau}\nabla_{\perp}\sim d_{e}\nabla_{\perp}\sim 1,$ (151)

implying that the resistive term is also kept: since $\eta=\nu_{ei}d_{e}^{2}$,

$\eta\nabla_{\perp}^{2}\sim\partial_{t}.$ (152)

Eq. (151) also means that $d_{i}$ is large compared to the perpendicular
scales,

$d_{i}\nabla_{\perp}\sim\epsilon^{-1},$ (153)

but small compared to the parallel scales,

$d_{i}\partial_{z}\sim\epsilon,$ (154)

a regime typical for auroral fluctuations and in some laboratory experiments.
This also means that the typical timescales are small compared to
$\Omega_{i}$,

$\frac{1}{\Omega_{i}}\partial_{t}\sim\epsilon.$ (155)

We will assume from the outset (and show that this is consistent) that $b_{z}$
and $u_{z}$ are at least as small as the perpendicular fluctuations
$\boldsymbol{u}_{\perp}$ and $\boldsymbol{b}_{\perp}$; we will deduce the
ordering for $\tilde{n}$ from the equations, but assume it is small compared
to unity. The $z$-component of the induction equation (139) at lowest order
yields

$\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}=-\frac{1}{\Omega_{i}}\nabla_{\perp}\times\mathrm{d}_{t}\boldsymbol{u}_{\perp}\sim\epsilon^{2}\Omega_{i},$
(156)

meaning that the perpendicular velocity is only slightly compressive: its
divergence is one order smaller than its curl. This can be inserted into the
continuity equation (137), which then gives at lowest order

$\mathrm{d}_{t}\tilde{n}=\frac{1}{\Omega_{i}}\nabla_{\perp}\times\mathrm{d}_{t}\boldsymbol{u}_{\perp},$
(157)

from which

$\tilde{n}\sim\epsilon.$ (158)

Turning now to the perpendicular momentum equation (138), at lowest order we
just obtain the pressure balance

$c_{s}^{2}\tilde{n}=-b_{z}v_{\mathrm{A}},$ (159)

whence in fact, we see that

$\frac{b_{z}}{v_{\mathrm{A}}}\sim\epsilon^{3}.$ (160)

The lowest order terms of the $z$-component of the momentum equation (138) are

$\displaystyle\mathrm{d}_{t}u_{z}=$ $\displaystyle
v_{\mathrm{A}}\partial_{z}b_{z}+\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}b_{z}+\frac{Zm_{e}}{m_{i}}d_{i}\mathrm{d}_{t}\nabla_{\perp}\times\boldsymbol{b}_{\perp}$
$\displaystyle+d_{e}^{2}[\hat{\boldsymbol{z}}\times\nabla_{\perp}b_{z}]\cdot\nabla_{\perp}\nabla_{\perp}\times\boldsymbol{b}_{\perp},$
(161)

from which we can deduce that888We have used this to preemptively drop the
term involving $u_{z}$ on the RHS of (138) when writing (161).

$\frac{u_{z}}{v_{\mathrm{A}}}\sim\epsilon^{3}.$ (162)

At lowest order, the perpendicular induction equation reads

$\mathrm{d}_{t}\boldsymbol{b}_{\perp}=v_{\mathrm{A}}\partial_{z}\boldsymbol{u}_{\perp}+\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}\boldsymbol{u}_{\perp}+d_{i}\hat{\boldsymbol{z}}\times\nabla_{\perp}\mathrm{d}_{t}u_{z}+\eta\nabla_{\perp}^{2}\boldsymbol{b}_{\perp},$
(163)

into which we insert (161) to eliminate $u_{z}$ and (159) to eliminate $b_{z}$
in favor of $\tilde{n}$, yielding

$\displaystyle\partial_{t}\boldsymbol{b}_{\perp}=$ $\displaystyle
v_{\mathrm{A}}\partial_{z}\boldsymbol{u}_{\perp}+\nabla_{\perp}\times(\boldsymbol{u}_{\perp}\times\boldsymbol{b}_{\perp})$
$\displaystyle-\frac{c_{s}^{2}}{\Omega_{i}}\hat{\boldsymbol{z}}\times\nabla_{\perp}\left[v_{\mathrm{A}}\partial_{z}\tilde{n}+\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}\tilde{n}\right]$
$\displaystyle-
d_{e}^{2}\frac{c_{s}^{2}}{\Omega_{i}}\hat{\boldsymbol{z}}\times\nabla_{\perp}[\hat{\boldsymbol{z}}\times\nabla_{\perp}\tilde{n}\cdot\nabla_{\perp}\nabla_{\perp}\times\boldsymbol{b}_{\perp}]$
$\displaystyle+d_{e}^{2}\hat{\boldsymbol{z}}\times\nabla_{\perp}\mathrm{d}_{t}\nabla_{\perp}\times\boldsymbol{b}_{\perp}+\eta\nabla_{\perp}^{2}\boldsymbol{b}_{\perp}.$
(164)

To make further progress, note that since (140) at lowest order is just
$\nabla_{\perp}\cdot\boldsymbol{b}_{\perp}=0+O(\epsilon^{3})$, we may write
$\boldsymbol{b}_{\perp}$ in terms of a flux function

$\boldsymbol{b}_{\perp}=\hat{\boldsymbol{z}}\times\nabla_{\perp}\psi.$ (165)

Similarly, because $\nabla_{\perp}\cdot\boldsymbol{u}_{\perp}$ is small (cf.
156), at lowest order we can write the perpendicular velocity in terms of a
stream function999We have not completely neglected the compressive part of the
velocity; it sets the density fluctuations via (156): in Secs. II and III, it
plays a more active role in the dynamics and we must include it explicitly.

$\boldsymbol{u}_{\perp}=\hat{\boldsymbol{z}}\times\nabla_{\perp}\phi.$ (166)

Nonlinear terms are expressed in terms of these scalar functions as

$\boldsymbol{u}_{\perp}\cdot\nabla_{\perp}f=\left\\{\phi,f\right\\},\quad\boldsymbol{b}_{\perp}\cdot\nabla_{\perp}f=\left\\{\psi,f\right\\}$
(167)

where the Poisson bracket
$\left\\{f,g\right\\}=\nabla_{\perp}f\times\nabla_{\perp}g$. To lowest order,

$\mathrm{d}_{t}f=\partial_{t}f+\left\\{\phi,f,.\right\\}$ (168)

Writing (157) in terms of $\phi$ results in

$\tilde{n}=\frac{1}{\Omega_{i}}\nabla_{\perp}^{2}\phi,$ (169)

the density fluctuations are proportional to the vorticity of the flow.
Writing (164) and the curl of the perpendicular momentum equation (138) in
terms of $\phi$ and $\psi$ only, we obtain a closed set of nonlinear equations
for the DAW,

$\displaystyle\partial_{t}\nabla_{\perp}^{2}\phi+\\{\phi,\nabla_{\perp}^{2}\phi\\}=$
$\displaystyle
v_{\mathrm{A}}\partial_{z}\nabla_{\perp}^{2}\psi+\\{\psi,\nabla_{\perp}^{2}\psi\\},$
(170) $\displaystyle\partial_{t}[\psi-d_{e}^{2}\nabla_{\perp}^{2}\psi]=$
$\displaystyle
v_{\mathrm{A}}\partial_{z}[\phi-\rho_{\tau}^{2}\nabla_{\perp}^{2}\phi]$
$\displaystyle+\\{\psi-
d_{e}^{2}\nabla_{\perp}^{2}\psi,\phi-\rho_{\tau}^{2}\nabla_{\perp}^{2}\phi\\}$
$\displaystyle+\eta\nabla_{\perp}^{2}\psi.$ (171)

These "reduced dispersive Alfvén wave" (RDAW) equations are identical to those
used in Seyler & Xu (2003)(Seyler and Xu, 2003) to study auroral fluctuations;
however, those authors did not provide a systematic derivation101010To the
author’s knowledge, one does not appear in press.. The equations are also
similar to the reduced gyrokinetic equations in Zocco & Schekochihin
2011(Zocco and Schekochihin, 2011) (a.k.a. "KREHM"): however, that model is
significantly more sophisticated, incorporating a rigorous treatment of the
ion and electron kinetics.

Here, we will just make two important points about the above equations. First,
the Poisson brackets all vanish if gradients are all in one direction: so any
one-dimensional configuration of fields is an exact solution; i.e., there is
no one-dimensional wave steepening retained in this system. Second, if both
$\rho_{\tau}\nabla_{\perp}\ll 1$ and $d_{e}\nabla_{\perp}\ll 1$, the RVDAW
equations become reduced magnetohydrodynamics (RMHD), for which $\phi=\pm\psi$
are exact nonlinear solutions, travelling in opposite directions along
$\hat{\boldsymbol{z}}$, regardless of three-dimensional structure. In this
non-dispersive limit, non-trivial nonlinear interaction only occurs between
counterpropagating waves. This is despite the density fluctuations (169) and
despite $d_{i}\nabla_{\perp}\gg 1$: to this order, the compressive nature of
the wave makes no difference to the nonlinear dynamics. Upon including the
dispersive effects, nonlinear interaction does occur between co-propagating
waves (provided their perpendicular wavevectors also point in different
directions) with different perpendicular scales, because smaller-scale waves
"catch up" to larger-scale ones.

## References

  * Torbert  _et al._ (2016) R. Torbert, C. Russell, W. Magnes, R. Ergun, P.-A. Lindqvist, O. LeContel, H. Vaith, J. Macri, S. Myers, D. Rau, _et al._ , Space Science Reviews 199, 105 (2016).
  * Chen (2016) C. H. K. Chen, J. Plasma Phys. 82, 535820602 (2016).
  * Kasper  _et al._ (2021) J. Kasper, K. Klein, E. Lichko, J. Huang, C. Chen, S. Badman, J. Bonnell, P. Whittlesey, R. Livi, D. Larson, _et al._ , Physical review letters 127, 255101 (2021).
  * Bale  _et al._ (2019) S. Bale, S. Badman, J. Bonnell, T. Bowen, D. Burgess, A. Case, C. Cattell, B. Chandran, C. Chaston, C. Chen, _et al._ , Nature 576, 237 (2019).
  * Bowen  _et al._ (2021) T. A. Bowen, S. T. Badman, S. D. Bale, T. D. de Wit, T. S. Horbury, K. G. Klein, D. Larson, A. Mallet, L. Matteini, M. D. McManus, _et al._ , arXiv preprint arXiv:2110.11454 (2021).
  * Kasper  _et al._ (2019) J. Kasper, S. Bale, J. W. Belcher, M. Berthomier, A. Case, B. Chandran, D. Curtis, D. Gallagher, S. Gary, L. Golub, _et al._ , Nature 576, 228 (2019).
  * Chandran and Hollweg (2009) B. D. Chandran and J. V. Hollweg, Astrophys. J. 707, 1659 (2009).
  * Chandran (2021) B. D. Chandran, Journal of Plasma Physics 87 (2021).
  * Schekochihin (2022) A. A. Schekochihin, Journal of Plasma Physics 88, 155880501 (2022).
  * Howes (2015) G. G. Howes, Phil. Trans. R. Soc. Lond. A 373, 20140145 (2015).
  * Meyrand  _et al._ (2021) R. Meyrand, J. Squire, A. A. Schekochihin, and W. Dorland, Journal of Plasma Physics 87 (2021).
  * Squire  _et al._ (2021) J. Squire, R. Meyrand, M. W. Kunz, L. Arzamasskiy, A. A. Schekochihin, and E. Quataert, arXiv preprint arXiv:2109.03255 (2021).
  * Gekelman  _et al._ (1991) W. Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs, Review of Scientific Instruments 62, 2875 (1991).
  * Leneman, Gekelman, and Maggs (1999) D. Leneman, W. Gekelman, and J. Maggs, Phys. Rev. Lett. 82, 2673 (1999).
  * Gekelman  _et al._ (2000) W. Gekelman, S. Vincena, N. Palmer, P. Pribyl, D. Leneman, C. Mitchell, and J. Maggs, Plasma Physics and Controlled Fusion 42, B15 (2000).
  * Vincena, Gekelman, and Maggs (2004) S. Vincena, W. Gekelman, and J. Maggs, Phys. Rev. Lett. 93, 105003 (2004).
  * Palmer, Gekelman, and Vincena (2005) N. Palmer, W. Gekelman, and S. Vincena, Physics of Plasmas 12, 072102 (2005), https://doi.org/10.1063/1.1930796 .
  * Carter  _et al._ (2006) T. A. Carter, B. Brugman, P. Pribyl, and W. Lybarger, Phys. Rev. Lett. 96, 155001 (2006).
  * Auerbach  _et al._ (2010) D. W. Auerbach, T. A. Carter, S. Vincena, and P. Popovich, Phys. Rev. Lett. 105, 135005 (2010).
  * Howes  _et al._ (2012) G. G. Howes, D. J. Drake, K. D. Nielson, T. A. Carter, C. A. Kletzing, and F. Skiff, Phys. Rev. Lett. 109, 255001 (2012).
  * Dorfman and Carter (2013) S. Dorfman and T. A. Carter, Phys. Rev. Lett. 110, 195001 (2013).
  * Dorfman and Carter (2015) S. Dorfman and T. A. Carter, Physics of Plasmas 22, 055706 (2015), https://aip.scitation.org/doi/pdf/10.1063/1.4919275 .
  * Dorfman and Carter (2016) S. Dorfman and T. Carter, Physical review letters 116, 195002 (2016).
  * Brugman (2007) B. T. Brugman,  _An investigation of the nonlinear interaction of Alfven waves_ , Ph.D. thesis, UCLA (2007).
  * Abler (2023) M. Abler, in preparation (2023).
  * Goldstein, Klimas, and Barish (1974) M. L. Goldstein, A. Klimas, and F. Barish, Solar Wind III , 385 (1974).
  * Barnes and Hollweg (1974) A. Barnes and J. V. Hollweg, J. Geophys. Res. 79, 2302 (1974).
  * Vasquez and Hollweg (1998a) B. J. Vasquez and J. V. Hollweg, J. Geophys. Res. Space Phys. 103, 349 (1998a).
  * Hollweg (1999) J. V. Hollweg, Journal of Geophysical Research: Space Physics 104, 14811 (1999).
  * Zocco and Schekochihin (2011) A. Zocco and A. A. Schekochihin, Phys. Plasmas 18, 102309 (2011).
  * Seyler and Xu (2003) C. Seyler and T. Xu, Geophysical research letters 30 (2003).
  * Schekochihin  _et al._ (2009) A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert, and T. Tatsuno, Astrophys. J. Supp. 182, 310 (2009).
  * Goldreich and Sridhar (1995) P. Goldreich and S. Sridhar, Astrophys. J. 438, 763 (1995).
  * Strauss (1976) H. R. Strauss, Phys. Fluids 19, 134 (1976).
  * Vasquez and Hollweg (1998b) B. J. Vasquez and J. V. Hollweg, J. Geophys. Res. Space Phys. 103, 335 (1998b).
  * Elsasser (1950) W. M. Elsasser, Phys. Rev. 79, 183 (1950).
  * Note (1) Once again, keeping both of these simultaneously ignores strong Landau damping, and our results are really only valid when at least one of $d_{e}$ or $\rho_{\tau}$ is small.
  * Note (2) We have preemptively used this to neglect the $b_{z}^{2}$ term; this also puts some weak restrictions on the ranges of $c_{s}/v_{\mathrm{A}}$, $b_{y}/v_{\mathrm{A}}$, and $\partial_{t}/\Omega_{i}$ allowed, which we will not discuss.
  * Vasquez and Hollweg (1996) B. J. Vasquez and J. V. Hollweg, J. Geophys. Res. Space Phys. 101, 13527 (1996).
  * Note (3) This should be compared with the similar equation (21) in the previous section.
  * Seyler and Lysak (1999) C. Seyler and R. Lysak, Physics of Plasmas 6, 4778 (1999).
  * Note (4) If we were to formally continue this expansion to higher order in the dispersive parameters, we would find that the wave steepens at higher order.
  * Mallet (2023) A. Mallet, in preparation (2023).
  * Note (5) These two expansions commute, but if any of the dispersive parameters becomes small in $\delta$ they must be dropped in the following analysis.
  * Note (6) If more than a single sinusoid is considered at $O(\delta)$, one can also solve for the beat mode resulting in the difference of the phase functions. In our case with a single $O(\delta)$ wave, one just obtains $0=0$. We therefore do not consider this in the present analysis.
  * Bose  _et al._ (2019) S. Bose, T. Carter, M. Hahn, S. Tripathi, S. Vincena, and D. W. Savin, The Astrophysical Journal 882, 183 (2019).
  * Chen  _et al._ (2014) C. Chen, L. Leung, S. Boldyrev, B. Maruca, and S. Bale, Geophysical research letters 41, 8081 (2014).
  * Hasegawa and Mima (1976) A. Hasegawa and K. Mima, Physical Review Letters 37, 690 (1976).
  * Shukla, Rahman, and Sharma (1982) P. Shukla, H. Rahman, and R. Sharma, Journal of Plasma Physics 28, 125 (1982).
  * Wu, Wang, and Fälthammar (1995) D.-J. Wu, D.-Y. Wang, and C.-G. Fälthammar, Physics of Plasmas 2, 4476 (1995).
  * Tenerani  _et al._ (2023) A. Tenerani, C. González, N. Sioulas, C. Shi, and M. Velli, Physics of Plasmas 30, 032101 (2023).
  * Krasnoselskikh  _et al._ (2020) V. Krasnoselskikh, A. Larosa, O. Agapitov, T. D. de Wit, M. Moncuquet, F. Mozer, M. Stevens, S. Bale, J. Bonnell, C. Froment, _et al._ , Astrophys. J. 893, 93 (2020).
  * Hollweg and Isenberg (2002) J. V. Hollweg and P. A. Isenberg, Journal of Geophysical Research: Space Physics 107, SSH (2002).
  * Thuecks  _et al._ (2009) D. Thuecks, C. Kletzing, F. Skiff, S. Bounds, and S. Vincena, Physics of Plasmas 16, 052110 (2009).
  * Chen  _et al._ (2013) C. H. K. Chen, S. Boldyrev, Q. Xia, and J. C. Perez, Phys. Rev. Lett. 110, 225002 (2013).
  * Stasiewicz  _et al._ (2000) K. Stasiewicz, P. Bellan, C. Chaston, C. Kletzing, R. Lysak, J. Maggs, O. Pokhotelov, C. Seyler, P. Shukla, L. Stenflo, _et al._ , Space Science Reviews 92, 423 (2000).
  * Note (7) Again (cf. discussion in Appendix A), keeping both terms simultaneously is a poor approximation, but we will here do it anyway to compare with previous work.
  * Note (8) We have used this to preemptively drop the term involving $u_{z}$ on the RHS of (138) when writing (161).
  * Note (9) We have not completely neglected the compressive part of the velocity; it sets the density fluctuations via (156): in Secs. II and III, it plays a more active role in the dynamics and we must include it explicitly.
  * Note (10) To the author’s knowledge, one does not appear in press.