Switchbacks—rapid, large deflections of the solar wind's magnetic field—have generated interest as possible signatures of the mechanisms that heat the corona and accelerate the solar wind. In this context, an important task for theories of switchback formation and evolution is to understand their observable distinguishing features, allowing them to be carefully assessed using spacecraft data. Here, we work toward this goal by studying how Alfvénic switchbacks evolve in the expanding solar wind beyond the Alfvén radius, when the background magnetic field also rotates due to the Parker spiral. Using simple analytic arguments based on the physics of one-dimensional spherically polarized (constant-field-magnitude) Alfvén waves, we find that, by controlling the wave's obliquity, a Parker spiral strongly impacts switchback properties. Surprisingly, parallel magnetic-field deflections (switchbacks) can grow faster in a Parker spiral than in a radial background field, even though normalized wave amplitudes grow more slowly. In addition, switchbacks become strongly asymmetric: large switchbacks preferentially involve magnetic-field rotations in the plane of the Parker spiral (tangential deflections) rather than perpendicular (normal) rotations, and such deflections are strongly “tangentially skewed,” meaning switchbacks always involve field rotations in the same direction (toward the positive-radial direction for an outward mean field). In a companion paper [Johnston *et al.,* Phys. Plasmas **29**, 072902 1346 (2022)], we show that these properties also occur in turbulent 3D fields with switchbacks, given various caveats. Given that these nontrivial asymmetries and correlations develop purely as a consequence of switchback propagation in the solar wind, our results show that *in situ* observed asymmetrical switchback features cannot be used straightforwardly to infer properties of sources in the low corona.

## I. INTRODUCTION

Early observations of switchbacks by Parker solar probe (PSP) provided a stark demonstration of the dynamic nature of the near-sun solar wind.^{1} These strong and sudden reversals of the magnetic field, which are known from electron measurements to have the topology of local folds in field lines (as opposed to global polarity reversals^{2}), contain significant energy content compared to the background plasma. This suggests they play a role in—or are least a helpful diagnostic of—the key processes that heat and accelerate the solar wind.^{3} As such, a range of different explanations have been put forth for their origin, each relating in some way to fundamental low-coronal or solar-wind physics, with switchbacks emerging as an observable consequence. These explanations include models relating to interchange reconnection near the solar surface,^{4–7} low-coronal jets or other motions,^{8,9} field-line folding due to asymmetries from interchange reconnection,^{10} instabilities between different wind streams,^{11} or the growth of Alfvénic fluctuations due to plasma expansion.^{12–14} Given these widely differing ideas, each of which provide different predictions for switchback properties and occurrence rates, a clear path forward to bettering our understanding of the solar wind presents itself: quantify various (hopefully) unique and observationally testable predictions for each model, then compare these to observations.

In this paper, we continue this process of better understanding model predictions for the scenario where switchbacks form due to the growth of Alfvénic fluctuations. This scenario, which is most naturally associated with wave-driven solar-wind acceleration theories,^{15,16} is in some sense the simplest of all those mentioned above: it requires Alfvén waves of some form to be released into the low corona;^{17,18} in growing to normalized amplitudes $\u22731$ due to expansion, they create switchbacks. Importantly, the scenario does not assume anything in particular about the process that creates the Alfvénic fluctuations in the first place, but concerns only the role of expansion and the assertion that (because they are Alfvénic) they should be “spherically polarized,” viz*.,* have a constant magnetic field strength (as is observed^{19}). Thus, its predictions relate exclusively to how fluctuations grow and change shape as they propagate. Accordingly, a particular correlation, structure, or feature that arises as a basic prediction of the scenario would arise naturally from almost any mechanism that generates Alfvénic fluctuations at low altitudes, purely as a consequence of their propagation to the radius at which they are observed. While this certainly does not eliminate the possibility that the original fluctuations result from an interesting mechanism (e.g., interchange reconnection), it does undermine the credibility of using switchbacks as evidence for the importance of this mechanism. Several examples of this mindset have already been discussed in the theory of Ref. 14 (hereafter M + 21), which argues, for example, that Alfvénic switchbacks are preferentially elongated along the background magnetic field as a simple consequence of being divergence-free as well as predicting various non-trivial compressive correlations that result from expansion.

Our results here concern how the Parker spiral—the azimuthal rotation of the mean field with increasing heliocentric radius^{20}—influences the evolution of Alfvénic switchbacks in an expanding plasma. Our method is to consider the idealized physics governing the evolution of large-amplitude, one-dimensional waves. We analyze their evolution in regions beyond the Alfvén radius $RA$ (where the solar-wind and Alfvén speeds are equal), imagining that they have arrived at $RA$ after propagating outward from some source in the low corona (quasi-Alfvénic perturbations are clearly observed at $RA$^{21}). As the waves continue propagating beyond $RA$, the plasma's expansion also causes the mean-field direction to rotate away from being nearly radial (the manifestation of the Parker spiral in the local parcel of plasma), thus changing the wave's propagation direction and causing the development of a number of asymmetrical features in the magnetic-field fluctuations. These are the main predictions of this paper. However, an important limitation of this analysis is the assumption of 1D structure, which cannot be rigorously justified. In particular, perturbations released from the low solar atmosphere presumably have multi-dimensional structure, which implies that predictions of wave properties that rely on them being 1D (for instance the variance of parallel field perturbations at a given amplitude) cannot be quantitatively correct. Perhaps more importantly, the treatment neglects the nonlinear interactions that lead to turbulence^{22} and/or parametric decay,^{23,24} which will cause enhanced dissipation and mode coupling, invalidating our results if sufficiently strong. Nonetheless, we do note that our main results are at least qualitatively confirmed by fully nonlinear 3D calculations, which contain all of the aforementioned effects. These are presented in our companion paper (Ref. 25) (hereafter J + 22), which is designed to be considered in conjunction with this work. J + 22 shares the same overarching goal of honing the predictions of the Alfvénic switchback model, but uses realistic 3D expanding magnetohydrodynamic (MHD) simulations that capture the full complexity of 3D structures, turbulence, and compressibility. Thus, despite the obvious limitations of treating 1D waves (some of which are quite severe), the approach seems to yield useful dividends in straightforwardly understanding a number of seemingly perplexing properties of switchbacks in 3D simulations. Our hope—backed by some preliminary evidence—is that such conclusions apply also to the real solar wind.

The main result of this work (likewise, a key result of J + 22) is that the Parker spiral causes switchbacks to become pronouncedly asymmetric in a number of ways. Defining the radial (R), tangential (T), and normal (N) directions in the usual way—the Parker spiral lies in the radial-tangential (R-T) plane with a component in the +T direction, while normal is perpendicular to the R-T plane (see Fig. 1)—we show, among other results, that: (i) switchbacks preferentially involve field deflections in the R-T plane, as opposed to the normal direction, (ii) the switchback prevalence and intensity (as measured, e.g., by the variance of parallel magnetic-field perturbations) is enhanced by the Parker spiral, and (iii) field deflections become strongly tangentially skewed, meaning they always deflect toward the positive radial direction in an outward mean field, as opposed to the opposite direction (more generally, they deflect toward the mean field's radial component). A corollary of point (iii) is that the most common magnetic-field direction (its mode) can be very different to its mean, because its probability density function is highly skewed. Per our discussion above, this implies that any fluctuation directional asymmetries in observed switchbacks do not necessarily signify that the original source of the switchbacks is asymmetric; rather, asymmetries (of our predicted form) arise organically due to plasma expansion as fluctuations propagate outwards in the super-Alfvénic solar wind.

The remainder of the paper is organized as follows: Sec. II provides the necessary background to our calculations, introducing key concepts such as the definition of a nonlinear Alfvén wave, and how the divergence-free and constant-magnetic-field-strength constraints determine its associated fluctuations parallel to the mean field. This section effectively summarizes some key results of M + 21, which forms the basis for our work. Section III presents results related to how the size of the parallel field perturbation, which is loosely interpreted as the switchback prevalence, evolves in 1D waves as they grow and decay due to expansion. We will see that the Parker spiral has a strong influence because of its effect on the evolution of the wave's obliquity. Section IV then considers the magnetic-field structure of switchbacks that evolve as discussed in Sec. III, illustrating their tangential skewness and providing a simple proof for why this occurs in 1D. Section V is then concerned with the thorny question of how our 1D results might apply to fully 3D fields, including the effects of 3D structure, turbulence, and parametric decay, with reference to many of the results from J + 22. We argue that while the results cannot be universally applicable, they can likely be reasonably applied to certain regimes, so long as various important caveats are kept in mind. We conclude in Sec. VI.

Three appendixes present tangential results. Appendix A provides a different, 3D, argument for the main results of Sec. III based on integrating over a spectrum of waves that has been influenced by plasma expansion. The fact that the same conclusions are reached through an unrelated argument (albeit one with its own severe assumptions) seems to suggest that aspects of our main results should be relatively robust. Appendix B provides a cursory comparison of idealized wave results based on M + 21 to two-dimensional MHD simulations with expansion, showing mostly good agreement (and allowing better understanding of a case where the method of M + 21 fails). Appendix C demonstrates that, independent of the mean-field direction, wave amplitudes scale with expansion in the same way.

## II. NONLINEAR ALFVÉN WAVES AND THE INFLUENCE OF EXPANSION

In this section, we discuss and derive some key properties of large-amplitude Alfvén waves, which will form the basis for our analysis of how such waves evolve in an expanding plasma with a Parker spiral. We wish to explore the creation of switchbacks, which, for simplicity, we define as a reversal of the background magnetic field for most of our calculations and discussion. Although other properties (e.g., sharp field rotations) have been used in some literature to define switchbacks (see, e.g., Ref. 26 and further discussion in Sec. IV), our definition is simple, unambiguous, and consistent with many previous works (e.g., Refs. 27 and 28). Similarly, for simplicity of language, we will loosely associate the “switchback prevalence” with the normalized amplitude of magnetic-field fluctuations along the mean-field direction (see below).

Our notation used below is as follows: *P*, *ρ*, ** u,** and

**are the plasma's thermal pressure, density, flow velocity, and magnetic field. Where appropriate, we will denote spatial averaging with a bar (e.g., $B\xaf$) and fluctuating quantities (i.e., the remainder) with a**

*B**δ*(e.g., $\delta B$), using periodic boundary conditions. In order to clarify notation, we will almost exclusively reference the mean field through the Alfvén speed, $vA\u2261B\xaf/4\pi \rho \xaf$, and reference magnetic field fluctuations (i.e., the waves) in velocity units $b\u2261\delta B/4\pi \rho \xaf$. We will work in the comoving frame in which $\delta u=u$ ($u\xaf=0$). The magnetic field strength is $B=|B|$ or $vA=|vA|$, and we will use $\xb7\xb7$ to denote unit vectors (e.g., $v\u0302A=vA/vA$). In the discussion of waves,

**denotes the wavevector, which makes an angle ϑ to $vA$ ($cos\u2009\u03d1=p\u0302\xb7v\u0302A$). In the discussion of expansion,**

*p**a*will denote the plasma's expansion factor (starting from some reference position with

*a*=

*1; see Sec. II C) and the coordinate system is Cartesian $x=(x,y,z)$, with*

*x*the radial direction (that of the mean flow in the solar wind),

*y*the tangential direction, and

*z*the normal direction. Important parameters are listed in Table I for quick reference.

Symbol . | Description and/or definition . |
---|---|

R, T, N | Radial, tangential, and normal direction |

x, y, z | Radial, tangential, normal: local coordinates |

a | EBM expansion parameter |

$g\xaf$ | Spatial average of some parameter g |

$g0$ | Initial value of parameter g at a = 1 |

ρ, P | Density, pressure (locally spatially constant) |

, BB | Total magnetic field, magnetic-field strength |

$vA,\u2009v\u0302A,\u2009vA$ | Background Alfvén velocity, direction, magnitude |

b | Magnetic-field perturbation in Alfvén units |

A | Perturbation amplitude ($|b|/vA$) |

$b||$ | Parallel magnetic-field perturbation ($v\u0302A\xb7b$) |

, $p\u0302$ p | Wavevector (direction of perturbation's variation) |

λ | Spatial coordinate along the $p\u0302$ direction |

$n\u0302,\u2009m\u0302$ | Orthogonal basis perpendicular to $p\u0302$ |

ϑ | Wave obliquity: angle between $v\u0302A$ and $p\u0302$ |

θ, $\theta p0$ _{p} | Angle between $p\u0302$ and the radial ($x\u0302$) direction |

$\theta \u22a50$ | $90\xb0\u2212\theta p0$ |

$\Phi ,\u2009\Phi 0$ | Parker spiral angle ($cos\u2009\Phi =vAx/vA$) |

$\phi $ | Projected angle of $p\u0302$ on the tangential-normal plane |

Symbol . | Description and/or definition . |
---|---|

R, T, N | Radial, tangential, and normal direction |

x, y, z | Radial, tangential, normal: local coordinates |

a | EBM expansion parameter |

$g\xaf$ | Spatial average of some parameter g |

$g0$ | Initial value of parameter g at a = 1 |

ρ, P | Density, pressure (locally spatially constant) |

, BB | Total magnetic field, magnetic-field strength |

$vA,\u2009v\u0302A,\u2009vA$ | Background Alfvén velocity, direction, magnitude |

b | Magnetic-field perturbation in Alfvén units |

A | Perturbation amplitude ($|b|/vA$) |

$b||$ | Parallel magnetic-field perturbation ($v\u0302A\xb7b$) |

, $p\u0302$ p | Wavevector (direction of perturbation's variation) |

λ | Spatial coordinate along the $p\u0302$ direction |

$n\u0302,\u2009m\u0302$ | Orthogonal basis perpendicular to $p\u0302$ |

ϑ | Wave obliquity: angle between $v\u0302A$ and $p\u0302$ |

θ, $\theta p0$ _{p} | Angle between $p\u0302$ and the radial ($x\u0302$) direction |

$\theta \u22a50$ | $90\xb0\u2212\theta p0$ |

$\Phi ,\u2009\Phi 0$ | Parker spiral angle ($cos\u2009\Phi =vAx/vA$) |

$\phi $ | Projected angle of $p\u0302$ on the tangential-normal plane |

The basis for all of our discussion is the realization^{29} that

is a nonlinear solution to the compressible MHD equations, which propagates along the mean field at the Alfvén speed. More generally, the solution (1) is valid even for the Kinetic MHD equations that describe large-scale collisionless dynamics,^{30,31} if *P* is replaced with separate constraints on the perpendicular and parallel pressure and $vA$ is modified to account for any mean pressure anisotropy (in fact, it is valid under even more general conditions than this; see Ref. 32). Here, ** b** can be of arbitrary amplitude compared to $vA$, but $B=|B\xaf+\delta B|$ (or $|vA+b|=const.$) must include both the mean-field and fluctuation contributions. Such solutions are the natural generalization of Alfvénic fluctuations to nonlinear amplitudes

^{29}and are often referred to as

*spherically polarized*waves.

Throughout the remainder of the text, we will ignore density and pressure fluctuations to explore Alfvénic, spherically polarized solutions. This implies $\rho =\rho \xaf$ and $P=P\xaf$, and we leave off the bar for simplicity of notation.

### A. One-dimensional nonlinear Alfvén waves and field reversals

Although Eq. (1) is valid for general three-dimensional ** b**, we now specialize to 1D solutions that vary only along the direction $p\u0302$, so that $b(x)=b(p\xb7x)=b(\lambda )$. The functional form $b(\lambda )$ is arbitrary, providing that it satisfies the constraints $\u2207\xb7b=0,\u2009b\xaf=0$, and $B=const.$ Using $\u2207\xb7b=p\xb7db/d\lambda =0$, which implies $p\u0302\xb7b=0$ via $b\xaf=0$, we see that

**has two independent nonzero components. We choose these to be in the $n\u0302=(p\u0302\xd7v\u0302A)/|p\u0302\xd7v\u0302A|$ direction (the perturbation direction of a linear Alfvén wave) and the $m\u0302=(p\u0302\xd7n\u0302)/|p\u0302\xd7n\u0302|$ direction. Defining $bn=n\u0302\xb7b$ and $bm=m\u0302\xb7b$, we see that the parallel field perturbation—that which can lead to switchbacks—is**

*b*We are interested in relating $b||/vA$, the relative parallel-field perturbation, to the wave's amplitude

A simple argument from M + 21 (see also Refs. 29 and 33) goes as follows. First, we note that $n\u0302\xb7vA=0$ and define $vAp=p\u0302\xb7vA$ and $vAm=m\u0302\xb7vA$. Along with $p\u0302\xb7b=0$, this yields the constant-*B* condition

which implies $A2/\u2009sin2\u03d1+1+2bm/vAm=const.$ on dividing by $vAm2$. Taking the spatial average and multiplying by $\u2009sin2\u03d1$ fixes the constant, thus allowing one to solve for *b _{m}* to find $b||/vA=(A2\xaf\u2212A2)/2$. This shows that for $A\u226a1,\u2009b||/vA$ scales with the wave amplitude squared. On the other hand, the amplitude definition (3) shows that $bm/vA\u2272A$, so that for $A\u22731,\u2009b||/vA\u2272A\u2009sin\u2009\u03d1$, with approximate equality for $bm\u2248bn$. Combined, these constraints give

which we find is very well satisfied by constant-*B* 1D solutions. Note that in using Eq. (5), *A* and $b||$ are simply numbers rather than functions of *λ* (i.e., *A* is loosely equated with its spatial average, $A\u223cA\xaf$), a distinction that should be clear from context in the discussion below. We see that large-amplitude waves, with $A\u2273\u2009sin\u2009\u03d1$, preferentially form switchbacks ($b||\u2273vA$) when they are oblique ($sin\u2009\u03d1\u22481$). Unsurprisingly, waves with $A\u226a\u2009sin\u2009\u03d1$ can never form switchbacks, because, given that $sin\u2009\u03d1<1$, the *A*^{2} scaling only ever applies for $A\u22721$ (implying $b||\u2272vA$).

The result (5) can be simply explained intuitively as follows: field perturbations are confined to be perpendicular to ** p** by $\u2207\xb7B=0$, so $b||$ perturbations are necessarily small when

**and $vA$ are nearly parallel; additionally, in large-amplitude waves,**

*p**b*causes large variation in

_{n}*B*, which must be compensated by a similar-magnitude

*b*component. Thus, large switchbacks result from large-amplitude, oblique nonlinear Alfvén waves. Importantly, as discussed by M + 21, this provides a simple explanation for the observed radial elongation of switchbacks:

_{m}^{28,34}in a random magnetic field with power spread across a wide range of wavenumbers, only preferentially perpendicular (radially elongated) structures generate significant $b||$, even for $A\u22731$.

### B. Constant-*B* wave solutions

A question that naturally arises, and one that will be important for our analysis later in this manuscript, is how to form constant-*B* solutions. In general, this is simple in one dimension for small-amplitude perturbations, but not for larger amplitudes, and not in two or three dimensions (see, e.g., Refs. 35–38 and J + 22). In one dimension, one can simply arbitrarily specify the functional form of $bn(\lambda )$, then solve Eq. (4) for *b _{m}*,

In doing so, one must enforce $bm\xaf=0$ (otherwise, *b _{m}* would contribute to $vA$), which is used as a constraint to solve for the unknown constant field magnitude

*B*

^{2}. Moreover, Eq. (6) does not always have real solutions for an arbitrary choice of

*b*, which leads to an effective amplitude limit on constructing such a wave for the chosen functional form of

_{n}*b*. For example,

_{n}^{29}if we choose a sinusoidal perturbation, $bn/vA=An\u2009sin\u2009(k\lambda )$, with $vA=vAx\u0302$ and $p\u0302=cos\u2009\u03d1x\u0302+sin\u2009\u03d1y\u0302,$ the spherically polarized wave is $b=bm\u2009sin\u2009\u03d1x\u0302\u2212bm\u2009cos\u2009\u03d1y\u0302+AnvA\u2009sin\u2009(k\lambda )z\u0302$ with

*b*given by Eq. (6). This can be straightforwardly verified to satisfy $|vA+b|2=(vA+bx)2+by2+bz2=constant$ and $\u2207\xb7b=0$. However, there remains the constraint $\u222bd\lambda \u2009bm=0$, which must be solved for

_{m}*B*

^{2}from $CE(An2/C2)=(\pi /2)\u2009sin\u2009\u03d1$, where $C2=B2/B\xaf2\u2212cos2\u03d1$ and

*E*is the complete elliptic integral of the second kind; this constraint only has a real solution for $C>An$, which implies $An<(\pi /2)\u2009sin\u2009\u03d1$. Thus, it is not possible to form spherically polarized solutions of arbitrary amplitude with sinusoidal

*b*.

_{n}However, this amplitude limit is artificial: it arises because the constant-*B* requirement constrains the functional form of both *b _{n}* and

*b*simultaneously, meaning

_{m}*b*cannot be chosen arbitrarily. If

_{n}*b*is chosen correctly, it is perfectly possible to form smooth constant-

_{n}*B*waves of arbitrary amplitude for arbitrary ϑ. Below, we discuss how to do this by starting a with lower-amplitude wave, for which one can solve Eq. (6), then growing it via expansion using the asymptotic theory of M + 21 [see Eq. (8) and Ref. 38]. Such a process is also physical: Refs. 29 and 39 showed that, no matter how large it becomes, the average amplitude $A\xaf$ of a spherically polarized 1D wave grows due to plasma expansion in exactly the same way as does a linear ($A\u226a1$) Alfvén wave. Thus, starting from small amplitudes in the lower solar atmosphere, and neglecting the influence of turbulence and parametric decay (see Sec. V), waves can in principle grow to $A\u22731$, so long as both

*b*and

_{n}*b*change shape in order to allow a consistent solution with constant

_{m}*B*. For the same reasons, in the WKB limit and absent parametric instability, an initially 1D wave must remain 1D, irrespective of its amplitude (indeed, large-amplitude wave characteristics and other properties are essentially identical to those of linear Alfvén waves; see appendix of Ref. 29). Of course, neglect of the parametric instability cannot be rigorously justified (see Sec. V C), and moreover, there is no reason to expect waves released at low altitudes to be one-dimensional; however, the 1D assumption is at least self-consistent, in that an initially 1D nonlinear solution can in principle remain 1D in an expanding plasma.

### C. Wave evolution and growth with expansion

M + 21 consider 1D wave evolution in the MHD expanding-box model (EBM),^{40} which describes the evolution of a small patch of plasma in a spherically expanding wind with constant velocity *U*. Note that, by assuming a constant *U*, the EBM applies only to radii where $U\u226bvA$, outside the Alfvén radius $RA$; inner regions with $R\u2272RA$ exhibit some important differences as concerns switchback formation (see, e.g., Sec. II D of J + 22 for further discussion). They also have an almost radial mean field^{41} so are less interesting from the perspective of the Parker spiral anyway. We are thus imagining perturbations arriving at $RA$ from lower altitudes, and studying only their subsequent evolution. We do not reproduce the full EBM equations here (see J + 22), but just note that the model is parameterized by the expansion parameter *a*, which starts at *a *=* *1 at some reference radius *R*_{0} and evolves as $a=1+a\u0307t$, where $a\u0307/a=U/R=U/(R0+Ut)$ is the expansion rate ($a\u0307=U/R0$ is constant). The perpendicular dimensions of a plasma parcel scale $\u221da$, due to the spherical expansion, while the parallel dimension remains constant. Thus, the wavevector scales as $p=(px,py,pz)=(px0,a\u22121py0,a\u22121pz0)$, where $p0=p(a=1)$ (with components $pi0$), because it remains constant in the expanding frame. In contrast, the background Alfvén speed evolves with expansion according to $vA=(vAx,vAy,vAz)=(a\u22121vAx0,vAy0,vAz0)$, where $vA0=vA(a=1)$. This scaling is a consequence of mass and flux conservation:^{40} the volume of the expanding box increases $\u221da2$, implying $\rho \u221da\u22122$; the area of a plane normal to the radial increases $\u221da2$, implying $Bx\u221da\u22122$ via flux conservation, while planes perpendicular to this increase in area $\u221da$, implying $By\u221dBz\u221da\u22121$. These different scalings of the radial and perpendicular mean-field components cause its direction to rotate with *a* and is the manifestation of the Parker spiral within this simplified model.

Our analysis is based on the dynamics of nonlinear Alfvén waves with frequency $\omega A\u226ba\u0307/a$ (termed the WKB regime). This regime is applicable to all but the largest-scale fluctuations in the solar wind: in particular, $\omega A\u226ba\u0307/a$ is satisfied for scales $k||\u22121\u226avA0R0/U\u223c7.0\xd7106km(R/35R\u2299)(vA0/100kms\u22121)(U/350kms\u22121)\u22121$, or via Taylor's hypothesis, to frequencies

as measured by a nearly stationary spacecraft. Due to expansion, waves in the WKB regime change in amplitude according to $|b|\xaf\u221da\u22121/2$, independently of the wave propagation direction (see Appendix C). Combined with the scaling of $vA$, this leads to the well-known result that $A\xaf\u221da1/2$ in a radial background field, which, as mentioned above, applies for both linear and nonlinear waves ($A\u22731$).

M + 21 derives a simple equation that captures the slow growth and evolution of $b(\lambda )$ due to constant expansion. The method involves using an asymptotic expansion in $\epsilon =(a\u0307/a)/(p\xb7vA)$, which is the ratio of the expansion rate to the wave frequency $\omega A=p\xb7vA$; $\epsilon \u226a1$ implies the waves are in the WKB regime (note that *ε* remains constant during expansion because $\omega A\u221da\u22121$). Averaging over the fast wave motion, they obtain

where $vAT\u2261vA\u2212p\u0302\u2009p\u0302\xb7vA$ is the transverse part of the mean field, and $p\xb7u1$ is the is the (higher-order) compressive flow that is responsible for changing the shape of ** b** to maintain constant

*B*, given by

Wave amplitude growth is contained in the second term in Eq. (8) (coupled with the scaling of $vA$), which implies $|b|\xaf\u221da\u22121/2$ as expected. In addition, M + 21 derive equations for the higher-order density and *B*^{2} fluctuations that are driven as part of this process, but these will not feature in our discussion.

Later in this work, we examine the properties of solutions to Eqs. (8) and (9) by means of its numerical solution. For this, we use a standard Fourier pseudospectral method with fourth- and fifth-order Runga–Kutta timestepping and 1024 grid points in *λ*. We also provide some cursory comparisons of the solutions Eqs. (8) and (9) with true MHD solutions in Appendix B, finding mostly good agreement except for a specific case where Eq. (8) seems to break down (it fails to maintain constant *B*).

Finally, it is worth noting that Eqs. (8) and (9) provide a convenient and practical method to construct large-amplitude Alfvénic solutions when the method described above (Sec. II B) fails at large *A*. One simply starts with a chosen form of $b(\lambda )$ at smaller *A*, then evolves it according to Eqs. (8) and (9) to reach any desired amplitude. This process demonstrates that in one dimension at least, the apparent limits on *A* for some chosen form of *b _{n}* are artificial; in all choices for

*b*that we have tried, waves grow to arbitrary amplitude without forming particularly sharp gradients or discontinuities in

_{n}**. This general idea has been extended to construct multi-dimensional spherically polarized solutions in Ref. 38, which seem to form sharp gradients much more readily than in 1D.**

*b*## III. SWITCHBACK FORMATION DUE TO EXPANSION

In this section, we study the evolution of 1D Alfvén waves in an expanding plasma using the simple analytic scalings of $b||$ and *A* outlined above (Sec. II C). Throughout, we will heuristically equate $b||/vA$ with the “prevalence” of switchbacks, estimating how this evolves due to expansion using Eq. (5). Correspondingly, we will regard *A* as a number (evolving with *a*) rather than a function of *λ* throughout this section. We will consider the wave to exhibit strong switchbacks once $b||/vA\u22731$, which (as we will see below) is actually a rather conservative estimate in some regimes because the volume filling fraction of large-$b||$ deviations can evolve to be quite small.

While we will find that the influence of the Parker spiral on switchbacks can be quite dramatic, it is worth cautioning the detailed results of this section [e.g., the exact form of $b||(a)$] may be of limited applicability to a real plasma. In particular, they apply only in the absence of turbulence, which both causes energy decay (thus changing the amplitude scalings) and acts to populate different wavenumbers ** p** (thus invalidating the assumed wave angle ϑ evolution). Nonetheless, we will be able to draw some more general conclusions, which seem to match key results from 3D nonlinear EBM simulations (J + 22). These issues will be appraised in detail in Sec. V; for now, we simply take as given the applicability of 1D wave scalings and examine their consequences.

### A. Switchback growth in a radial field

Before considering the evolution of waves in a Parker spiral, it is helpful to examine the radial-field case to better understand its important features. In this case, $vA=a\u22121vAx0x\u0302$, so $A=A0a1/2$, where *A*_{0} is the initial amplitude at *a *=* *1. We take $p(a)=p0(cos\u2009\theta p0,0,a\u22121\u2009sin\u2009\theta p0)$, where $\theta p0$ is the initial angle between the radial direction and the wave, which we have arbitrarily taken to lie in the *x*, *z* plane. We also define $\theta p\u2261\u2009tan\u22121(pz/px)=tan\u22121(a\u22121\u2009tan\u2009\theta p0)$ as the *a*-dependent evolution of this angle. To obtain simple, physically intuitive results, we imagine starting with a nearly perpendicular wave $\theta \u22a50=\pi /2\u2212\theta p0\u226a1$ and treat $\theta \u22a50\u226a1$ as a small expansion parameter (see Fig. 1).

We proceed by noting that $\u03d1=\theta p$ for a radial field, so that

where we expanded in $\theta \u22a50\u226a1$ to obtain the final expression. This shows that $sin\u2009\u03d1\u22431$ for $a\u226a1/\theta \u22a50$ ($\theta p\u22731$; mostly oblique propagation) and $sin\u2009\u03d1\u221d1/a$ for $a\u226b1/\theta \u22a50$ ($\theta p\u22721$; mostly parallel propagation). Thus, combined with the continual increase in $A\u221da1/2$, from Eq. (5) we see that for $a\u226a1/\theta \u22a50,\u2009b||/vA$ grows either as $b||/vA\u221da$ (for $b||/vA\u22721$) or as $b||/vA\u221da1/2$ (for $b||/vA\u22731$). Then, it reaches a maximum at $a\u22481/\theta \u22a50$, before decreasing as $b||/vA\u221da\u22121/2$ for $a\u226b1/\theta \u22a50$ even though *A* continues to increase. The cause of the transition between these regimes is simply the transition from oblique ($\u03d1=\theta p\u22731$) to parallel propagation ($\u03d1\u22721$), because a parallel-propagating wave always involves small $b||$ perturbations. The decrease in $b||/vA$ once $\u03d1\u22731$ also implies that *A*_{0} must satisfy $A02\u2273\theta \u22a50$ in order to form switchbacks at all—that is, for $A02\u2272\theta \u22a50,\u2009b||/vA$ reaches a maximum value $b||/vA\u22721$ before decreasing again.

An example $b||/vA$ evolution, which involves each of the regimes discussed above, is illustrated in Fig. 2 with the thick black lines.

### B. Switchback growth in the Parker spiral

In the presence of a perpendicular component to the mean field (the Parker spiral), the scalings described above become more complex and interesting because of non-monotonic behavior of $sin\u2009\u03d1$ and *A*. First, let us consider the amplitude evolution. We take the Parker spiral to lie in the *x*, *y* plane, $vA0=vA0(cos\u2009\Phi 0,\u2009sin\u2009\Phi 0,0)$, which implies that the radial (R), tangential (T), and normal (N) directions correspond to *x*, *y*, and *z*, respectively (see Fig. 1). We define $0<\Phi 0\u226a1$ as the initial Parker spiral angle, which (like $\theta \u22a50$) will be considered a small parameter and used to simplify the results. We also define the *a*-dependent Parker spiral angle $\Phi \u2261\u2009tan\u22121(vAy/vAx)=tan\u22121(a\u2009tan\u2009\Phi 0)$. Using $vA=vA0(a\u22121\u2009cos\u2009\Phi 0,\u2009sin\u2009\Phi 0,0)$ and $|b|\xaf\u221da\u22121/2$ gives

showing that $A$ grows like the radial-field case, $A\u221da1/2$, for $\Phi \u22721$ ($a\u22721/\Phi 0$), but starts decreasing $A\u221da\u22121/2$ once $\Phi \u22731$ ($a\u22731/\Phi 0$) because the mean field decays more slowly than the wave-like perturbations.

We must now allow ** p** to have components in all three directions to capture the full range of possible behaviors of $sin\u2009\u03d1$. We thus parameterize it with $p(a)=p0(cos\u2009\theta p0,a\u22121\u2009sin\u2009\theta p0\u2009cos\u2009\phi ,a\u22121\u2009sin\u2009\theta p0\u2009sin\u2009\phi )$, so that $\phi =\xb1\pi /2$ corresponds to

**lying in the plane perpendicular to the Parker spiral mean field, and $\phi =0$ or**

*p**π*corresponds to

**lying in the same plane as the Parker spiral. We will sometimes describe these cases as $p\u0302(Z)$ and $p\u0302(Y)$, respectively, as illustrated with the green and red arrows in Fig. 1. We again imagine starting from a highly oblique wave ($\theta \u22a50=\pi /2\u2212\theta p0\u226a1$, also with $|\u03d1|<\pi /2$) and compute**

*p*as a function of *a*. The result is that $sin\u2009\u03d1$ decreases in the same way as radial-$vA$ case initially, but its evolution starts to differ markedly as *a* approaches

at which point $sin\u2009\u03d1$ reaches a local minimum

From this point, unlike the radial case, $sin\u2009\u03d1$ starts increasing again back toward oblique propagation, because $vA$ rotates toward the perpendicular direction.

We illustrate the effect of these features on the evolution of Eq. (5) in Fig. 2, using Eqs. (11) and (12) for *A* and $sin\u2009\u03d1$. All curves have the same initial $\u03d1=80\xb0$, and the different colors show a variety of initial Parker spiral angles $\Phi 0$. Note that $\theta p0$ must be adjusted to keep fixed initial ϑ while varying $\Phi 0$, which implies $\theta p0=cos\u22121(cos\u2009\u03d1/\u2009cos\u2009\Phi 0)$ for $\phi =\xb1\pi /2$, or $\theta p0=\u03d1\u2009cos\u2009\phi +\Phi 0$ for $\phi =0$ or *π*. We plot the waves with $\phi =\xb1\pi /2$ ($p\u0302(Z)$) and with $\phi =0,\pi $ ($p\u0302(Y)$) separately in the two panels, because their evolution differs significantly and this will suggest important conclusions about switchback properties. In order to understand the illustrated behavior, let us first consider the importance of the two angles $\Phi 0$ and $\theta \u22a50$ (controlling the Parker spiral and the wave's obliquity, respectively), then consider the wave's orientation $\phi $.

For $\theta \u22a50\u2273\Phi 0$—that is, when the Parker Spiral makes a smaller angle to the radial than the wavevector makes to the perpendicular—$a\u03d1min<1/\Phi 0$. This means $sin\u2009\u03d1$ reaches its minimum and starts increasing again before *A* starts decreasing at $\Phi =45\xb0$ [$a\u223c1/\Phi 0$; see Eq. (11)], but after the maximum of $b||/vA$ for the radial-$vA$ waves at $a\u223c1/\theta \u22a50$. This explains the inflection points seen in the green curves in Fig. 2: $sin\u2009\u03d1$ decreases significantly below unity by $a\u223c1/\theta \u22a50$ when $\theta p\u22721$, then reaches its minimum at $a\u223c1/\theta \u22a50\Phi 0$ causing $b||/vA$ to start increasing again, but then *A* itself starts decreasing at $a\u223c1/\Phi 0$ causing $b||/vA$ to decrease. Thus, in this regime of modestly oblique waves $\theta \u22a50\u2273\Phi 0$, switchbacks usually form significantly more efficiently than in the radial-$vA$ case, because the waves evolve to become more oblique after $p\u0302$ rotates to be mostly radial, but before the Parker spiral rotates past $\u224345\xb0$. In contrast, in the opposite regime of a large Parker spiral with $\theta \u22a50\u2272\Phi 0$ (yellow curves in Fig. 2), *A* starts decreasing, at $a\u22481/\Phi 0\u2272a\u03d1min$, before the minimum in $sin\u2009\u03d1$. Because $sin\u2009\u03d1min\u22481$ in this regime unless $cos\u2009\phi \u22481$ [see Eq. (14)], $b||/vA\u223cmin(A,A2)$, which simply peaks when $\Phi \u223c1$ then decreases again, making for inefficient switchback formation even though the wave remains oblique at all times. In both regimes, the decrease $A\u221da\u22121/2$ always wins out and causes $b||/vA$ to decrease as $b||/vA\u221da\u22121$ once $\Phi \u22731$ and $A\u22721$, which is faster than in the radial-$vA$ case.

As seen by the comparison between panels (a) and (b) of Fig. 2, the wave's direction $\phi $ is also key in determining its evolution. For $cos\u2009\phi \u22601$, as applies to $p\u0302(Z)$ waves or when $p\u0302$ and $vA$ lie in the same plane but different quadrants ($\phi =\pi $), the evolution occurs broadly as described above, with $\phi =\pi $ having a modestly larger $sin\u2009\u03d1min$ and thus slightly larger $b||/vA$ for intermediate times [cf. green-yellow curves in panels (a) and (b)]. However, for $\phi =0,\u2009p\u0302$ and $vA$ pass through each other at $a\u03d1min$, viz., the wave becomes perfectly parallel with $\u03d1=0$ (this effect can be seen clearly by following the red dotted and blue dotted lines in Fig. 1). At this point, $b||/vA$ must go to zero, and ϑ flips sign. Although $sin\u2009\u03d1$ then increases rather rapidly in the opposite direction, we see that for most of these wave's evolution, they produce only small $b||/vA$ fluctuations. Thus, the Parker spiral can create strong differences between switchback properties, depending on the direction of fastest variation ($p\u0302$) of the wave or structure in question.

Finally, we note that in order for any physics related to $sin\u2009\u03d1$ to be relevant, the maximum of $b||/vA$ should occur when $A2\u2273A\u2009sin\u2009\u03d1$ [see Eq. (5)]. Because the maximum of $b||/vA$ occurs together with the maximum of *A*, which is at $a\u223c1/\Phi 0$, we see that for initial wave amplitudes $A02\u2272\Phi 0,\u2009b||/vA\u223cA2$ for all *a* and there are no significant switchbacks ($b||/vA\u22721$).

#### 1. Comparison to wave solutions

The above arguments and Fig. 2 are based purely on Eq. (5). How well do these estimates compare to true expanding wave solutions? To test this, we solve Eqs. (8) and (9) with periodic boundary conditions, starting from random initial conditions constructed from the first 10 Fourier modes with $\u03d1=80\xb0$ and $\phi =\pi /2$, and varying $\Phi 0$ in order to match the computations Fig. 2(a). Results are shown in Fig. 3. The definition of $b||/vA$ is only valid as a scaling in Eq. (5), so to make a reasonable comparison we measure the standard deviation of $b||/vA$ across the domain, with $b||(\lambda )=b\xb7v\u0302A$. The agreement, both in the general form and the predicted scalings with *a* (black dotted lines), is extremely good, including in features such as the inflection point, which one might have expected to be an artifact of the idealized nature of Eq. (5). The spatial form of the solutions themselves is shown in Figs. 4 for $\Phi 0=2\xb0$ and 5 for $\Phi 0=0$ (radial $vA$) and will discussed in detail below. In addition, we show in Appendix B (see Fig. 8) that the solutions of Eq. (8) match true 2D expanding MHD solutions very well.

In Fig. 3, we do not consider waves with $\phi =0$ or *π*, for which $b||$ estimates are shown in Fig. 2(b). The reason for this is that Eqs. (8) and (9) fail to produce constant-*B* solutions when $\phi =0$ or *π*. Although the cause for this behavior remains unclear (we speculate that it results from the more rapid evolution of $sin\u2009\u03d1$ in this geometry), it is important to note that Eq. (8) was derived assuming constant *B*, so if this is not satisfied we should not trust its solutions. Thus, it is not worthwhile to compare to the predictions of Eq. (5) and Fig. 2(b) in detail. The consequences of this discrepancy are discussed below (Sec. III C 1) and in Appendix B (see Fig. 9 for MHD solutions of this geometry).

### C. Consequences for the solar wind

Although we will delay detailed discussion of turbulence and 3D fields until Sec. V, it is helpful to briefly outline some possible observable consequences of these wave properties for switchbacks in the solar wind. The most obvious property from Fig. 2 is that a modest Parker spiral with $\Phi \u22721$ ($vAx>vAy$) can significantly enhance switchback formation (as measured by $b||/vA$). This is because when $\theta \u22a50\u226b\Phi 0$ (with $\theta \u22a50=\pi /2\u2212\theta p0\u2243\pi /2\u2212\u03d10$ in most regimes), the simultaneous rotation of $vA$ and ** p** causes the wave obliquity to increase even when $\Phi \u226a1$. This seems to be the more relevant regime for the solar wind, since we measure $\Phi $ to be rather small near $RA$ (where the EBM becomes applicable), and there are a wider range of wavenumbers with $\theta \u22a50\u2273\Phi 0$ than with $\theta \u22a50\u2272\Phi 0$ if $\Phi 0\u226a1$. Thus, we predict more robust growth of switchbacks due to expansion in the presence of a sub-$45\xb0$ Parker spiral than not, an observationally testable prediction that is also seen in the simulations of J + 22.

Another interesting conclusion we can draw concerns the directions of switchback deflections. All else being equal, waves with ** p** perpendicular to the plane of the Parker spiral ($p\u0302(Z)$) generate more switchbacks than waves with

**in the plane of the Parker spiral ($p\u0302(Y)$): half of the $p\u0302(Y)$ waves (those with $\phi =0$) evolve to become purely parallel and cause only small $b||/vA$ over a wide portion of their evolution. Because these are Alfvénic fluctuations, the strongest**

*p***fluctuation lies in the $n\u0302=(p\u0302\xd7v\u0302A)/|p\u0302\xd7v\u0302A|$ direction, which means that $p\u0302(Z)$ waves cause large**

*b**b*fluctuations, and $p\u0302(Y)$ large

_{y}*b*fluctuations. Thus, we expect switchbacks to preferentially involve rotations of the field in the plane of the Parker spiral (the tangential direction), rather than the normal direction. This seems to be observed, at least partially, in the simulations of J + 22 (see their Fig. 7) and more clearly in PSP observations.

_{z}^{19,28}In essence, this argument is nothing more than the statement that for a distribution of waves with wavevectors that are biased toward the radial direction (as caused by expansion), wavevectors that lie in the plane perpendicular to the mean field are more oblique, on average, than those in the plane of the mean field. This interpretation is explored in more detail in Appendix A to provide another argument for this general effect.

#### 1. The assumption of constant B for $p\u0302(Y)$ waves

We noted above that when $\phi =0$ or *π* ($p\u0302(Y)$), Eq. (8) fails to maintain a constant-*B* solution as the wave grows. This raises the obvious question of whether Eq. (5) is valid for such waves, and, if it is not, what will be the consequences. In Appendix B, we address this question by directly comparing solutions of Eq. (8) to 2D expanding MHD solutions, finding that indeed Eq. (8) overpredicts the variation in *B* compared to MHD for this geometry, although the general form of the solutions is similar (see Fig. 9). However, we do still see tentative evidence that, even in MHD, larger variation in *B* occurs compared to a radial $vA$ or $\phi =\xb1\pi /2$, particularly for $\phi =\pi $ (which, recall, generates larger $b||$ than $\phi =0$). While a more careful study is needed, if this result holds, it only strengthens our main results from this section, implying that not only do $p\u0302(Y)$ waves generate relatively smaller $b||$ because of the geometry of $sin\u2009\u03d1$, they also generate larger *B* fluctuations that will then be more prone to dissipation by other means (e.g., kinetic damping or shocks). This would only act to enhance the dominance of *b _{y}* (tangential) over

*b*(normal) rotations in switchbacks, strengthening our second conclusion above.

_{z}## IV. THE STRUCTURE OF SWITCHBACKS IN THE TANGENTIAL PLANE

Above, we saw that the addition of a small Parker spiral can significantly enhance switchback generation (increase $b||/vA$), particularly when ** p** lies in the

*x*,

*z*plane and

**is dominated by its**

*b**y*component. In this section, we explore the structure of the wave solutions that develop under these conditions, governed by the constant-

*B*constraint and the rotating mean field and wavevectors. We will show that in the regime where the Parker spiral significantly modifies switchbacks—that is, as

*a*approaches and exceeds $a\u03d1min$ [Eq. (13)]—field rotations are generically tangentially skewed: the deflection of

**always causes**

*b**b*to increase, rather than decrease. This implies that through the switchback, the field deflects around toward the positive radial ($vAx$) direction (then past it), as opposed to the negative radial direction. We provide a simple proof for why this must occur based on certain assumptions about the form of such switchbacks and the $\u2207\xb7b=0$ and constant-

_{x}*B*constraints. For simplicity of notation, we will assume $vAx$ points in the $+x\u0302$ direction and $vAy$ in the $+y\u0302$ direction; in the case with $vAx$ in the $\u2212x\u0302$ direction,

**deflects toward the negative radial direction instead (i.e., still $vAx$), but is otherwise identical. We will also assume the wave starts with $\theta \u22a50\u2273\Phi 0$, because the opposite (large-Parker-spiral) limit with $\theta \u22a50\u2272\Phi 0$ was shown above to be ineffective at generating switchbacks (it also requires extremely perpendicular waves for small $\Phi 0$). Through this section, we do not consider waves with $\phi =0$ or**

*b**π*($p\u0302(Y)$); this is both because constant-

*B*solutions for such waves are not so easily understood (see Sec. III C 1), and because field rotations will be more symmetric in this case anyway (since they are predominantly in the normal direction; see Fig. 9).

The evolution of a representative 1D wave solution with a Parker spiral is shown in Fig. 4, starting from $\u03d1=80\xb0,\u2009\Phi 0=2\xb0$, and $\phi =\pi /2$, to match the parameters of the dark green solutions in Figs. 2(a) and 3. The equivalent solution without a Parker spiral, matching the black lines in Figs. 2(a) and 3, in is shown in Fig. 5 to provide a reference for comparison (see also M + 21 Fig. 4). Similar to Fig. 3, the initial conditions are constructed using Eq. (6) with random amplitudes in the first 5 Fourier modes for *b _{n}* =

*b*. We then evolve these according to Eqs. (8) and (9) with periodic boundary conditions, capturing the change in shape of

_{y}**needed to keep**

*b**B*constant as the wave grows due to expansion. This initial condition, as expected, involves predominantly

*b*perturbations (red line). Strong switchbacks (thick purple lines) develop at later times as expected from Fig. 2(a). They are dominated by sudden changes in the direction of

_{y}*b*and always involve an increase in

_{y}*b*(blue lines) through the sharp change in $b||$ (gray shaded regions). This implies that the magnetic-field lines always rotate toward the positive radial direction during the field reversal, viz., they are tangentially skewed. Switchbacks do not grow nearly as effectively in the radial-$vA$ case (Fig. 5).

_{x}### A. A simple proof of switchback tangential skewness

To understand why this behavior occurs, let us first consider the regime in which it occurs. Because $b||=bxv\u0302Ax+byv\u0302Ay$ must decrease below $\u2243\u2212vA$ to form a switchback, it must involve either a large-negative *b _{x}* or large-negative

*b*(or both). The former case is simply a standard radial-$vA$ switchback as explored in M + 21 and should not be expected to involve preferential deflections. This situation will apply even with a Parker spiral when $a\u22721/\theta \u22a50$ (i.e., when $\theta p\u227345\xb0$, which also implies $\Phi \u226a45\xb0$), because such waves behave like the radial-$vA$ case anyway (see Sec. III B). However, the latter case, with a large

_{y}*b*perturbation, is different. As we now show, it takes over before or around $a\u223ca\u03d1min$, which is well before $vAy\u2248vAx$ [$\Phi \u224845\xb0$; see Eq. (13)]. To understand why, we first note that $by\u2248bn$ and $bm2\u2248bx2+\u2009bz2$ because $vA$ remains nearly radial, while $p\xb7b=0$ implies

_{y}(this can be clearly observed in the blue and yellow lines in Fig. 4). For $a\u2248a\u03d1min\u22481/\theta \u22a50\Phi 0$ and $\theta \u22a50\u226b\Phi 0$, we thus see $bx\u226abz\u2248bm$ implying $bx\u2248bm\Phi 0/\theta \u22a50$. This shows that at $a\u2248a\u03d1min$,

where we have used the fact that $v\u0302Ax\u22481$ and $v\u0302Ay\u2248a\Phi 0\u2248\Phi 0/\theta \u22a50$ (because $\Phi \u226a1$ since $a\u03d1min\u226a1/\Phi 0$). Thus, since $bn\u2273bm$ (with near equality holding once $A\u22731$; see Sec. II A), $byv\u0302Ay$ dominates for $a\u2273a\u03d1min$, meaning switchbacks result from large *b _{y}* fluctuations, rather than large

*b*fluctuations, even though $v\u0302Ax>v\u0302Ay$ for a wide range of

_{x}*a*after this point (until $a\u22721/\Phi 0$). This behavior can be seen in the second (

*a*=

*10) panel of Fig. 4, which is pictured slightly before $a\u03d1min\u224812.7$ for these parameters: at this*

*a*, some $b||$ minima are dominated by

*b*fluctuations (e.g., around $\lambda \u22480.1$), some are dominated by

_{y}*b*fluctuations (e.g., around $\lambda \u22480.45$), while the largest $b||$ perturbations involve both (e.g., around $\lambda \u22480.7$). In contrast, by later times (e.g., the next panel where $\Phi \u224830\xb0$), $b||$ fluctuations are nearly completely determined by large changes in

_{x}*b*to

_{y}*b*< 0.

_{y}With this piece of information in hand—that the enhanced $b||/vA$ from the Parker spiral involves switchbacks that are dominated by *b _{y}* perturbations—it is straightforward to demonstrate that

*b*must increase through a switchback. First, we form the spatial constant

_{x}which is the difference between the total and mean-field magnitude [we use Eq. (15) for *b _{z}*]. Solving for

*b*gives

_{x}Here, $\Delta b2+vAy2$ is simply a positive constant, while $by+vAy$ is the total *y*-directed field. The key insight from Eq. (18) is that *b _{x}* is a monotonic function of $(by+vAy)2$ and is maximized when $(by+vAy)2=0$. Then, we recall that $b||$ changes are driven by $byv\u0302Ay$, while $v\u0302Ay$ is rather small, implying that any large change to $b||/vA$ must involve

*b*and thus $by+vAy$, passing through zero (since

_{y}*b*must also clearly remain less than the total field magnitude). Thus, any large change to $b||$ must occur around the same location as

_{y}*b*being maximized, which implies that the magnetic-field vector rotates through the positive-radial direction during the switchback [note that

_{x}*b*can subsequently decrease as

_{x}*b*becomes large and negative and $(by+vAy)2$ increases]. This feature is highlighted by the gray shading in Fig. 4, which show the regions where $|db||/d\lambda |$ is near its maximum in each panel; clearly, such regions generically line up with positive peaks in

_{y}*b*. Physically, all that Eq. (18) is saying is that the constant-

_{x}*B*constraint implies that if $by+vAy$ passes through zero,

*b*must increase to compensate, even though if it decreased instead it could in principle help to form switchbacks. It is also worth mentioning that Eq. (18) remains valid even for radial $vA$ or for $a\u22721/\theta \u22a50$, when $b||$ is instead dominated by the contribution from $bxv\u0302Ax$, but in this regime, it does not provide any obviously useful constraint.

_{x}Also of interest is that Eq. (18) excludes the possibility that the field rotates beyond $90\xb0$ in the $+y\u0302$ direction (i.e., the +T direction) at all. Such a rotation would involve $(by+vAy)2$ reaching a maximum value then decreasing, while *b _{x}* would have to continuously decrease, which is impossible due to the monotonic dependence of

*b*on $(by+vAy)2$. Put together with the discussion of the previous paragraph, this provides an alternate way to consider the switchback skewness: a field rotation from $v\u0302A$ in the +T ($+y\u0302$ direction) is limited to be less than $90\xb0\u2212\Phi $ when projected onto the

_{x}*x*,

*y*plane, but a field rotation in the −T direction can rotate the field by $90\xb0+\Phi $. This leads to a strongly asymmetrical distribution of field rotations. This feature is clearly seen in the 3D simulations of J + 22 (see their Figs. 7–9), despite the fields therein obviously not satisfying the 1D approximation used to derive Eq. (18).

### B. Switchback sharpness, irregularity, and the direction of the mean field

The most obvious feature of the solutions shown in Fig. 4 is how sharp and irregular the switchbacks become, viz., solutions feature wide quiet sections interspersed by sudden and rapid switchbacks as *b _{y}* changes sign. That these solutions evolve to become significantly sharper and more sporadic than those with radial $vA$ is clear from a quick by-eye comparison with Fig. 5, which shows wave evolution in a radial mean field with the same initial

*b*. We also show in Fig. 6 the “steepening factor” $Q\u2261|\u2202\lambda b|2\xaf/|b|2\xaf$ (M + 21), for the same set of 1D waves as shown in Fig. 3, which demonstrates the same idea more quantitatively and shows how the effect is even stronger for larger $\Phi $. Promisingly, we see evidence that this effect of the Parker spiral enhancing switchback sharpness carries over to fully 3D solutions (see Fig. 5 of J + 22), which suggests it may be observable in the solar wind.

_{n}To understand why the effect occurs, we must simply couple the fact that $by\xaf=0$ to the conclusion of the previous paragraph that ±T field rotations are limited to angles $<90\xb0\u2213\Phi $ from the mean field (projected onto the radial-tangential plane). As the Parker spiral rotates further, the difference between these two directions increases further, but $by\xaf$ remains zero, implying that the field must spend more volume rotated in the +T direction than the −T direction to compensate for the limited size of the +T rotations. The consequence is clearly seen in the bottom half of Fig. 4 (for $a\u227315$ or so, once $\Phi $ becomes significant): the solutions spend wide regions with modest *b _{y}* > 0, then suddenly rotate to

*b*< 0 over short distances. This leads to both sharper waveforms and a more irregular, intermittent distribution of $b||$ fluctuations.

_{y}A corollary of the previous paragraph's discussion relates to measuring the Parker spiral direction. In particular, the mode of the field direction (its most-common direction) becomes quite different to the mean-field direction $v\u0302A$, which is the direction in which the waves actually propagate. This is particularly clear from, for example, the *a *=* *30 panel in Fig. 4, which has $\Phi \u224845\xb0$, but, discounting the switchbacks, one would conclude $3vAx\u2248vAy$, or an angle of $\u224870\xb0$. This feature is also clearly seen in the simulations of J + 22 (see their Fig. 7), suggesting that in measuring the Parker spiral angle in the solar wind, one must be careful to resolve the distinction between statistical mean and mode.

Finally, we note that the comparison between Figs. 3, 4, and 6 reveals an interesting consequence of the intermittent switchback rotations. Once $\Phi $ rotates beyond $45\xb0$ and *A* starts decreasing, fluctuations in $b||/vA$ also decrease, if measured by their root mean square deviation [see Fig. 3, which matches the prediction Eq. (5)]. However, we see from Fig. 4 that this decrease does not involve individual switchbacks becoming smaller, but rather a decrease in their volume-filling fraction. This is clear from the fact that the relative size of individual $b||/vA$ fluctuations increases from *a *=* *50 to *a *=* *200 in Fig. 4, and that *Q* (the waveform steepness) continues to grow rapidly at large *a* (Fig. 6). This trend appears to continue up until arbitrary large *a* for waves with $\theta \u22a50\u226b\Phi 0$, with nominally small-*A* solutions containing extreme and sudden changes in field direction. Nonetheless, although interesting, we do not expect this property to be particularly important to the solar wind: by such large *a* with $\Phi \u227345\xb0$, there has likely been significant reflection of forward into backward propagating waves and turbulence, which, in general, seems to destroy the constant-*B* requirement needed to form such solutions [see Sec. V B below and Fig. 3(b) of J + 22].

## V. DISCUSSION: HEURISTIC APPLICATION TO 3D FIELDS

In this section, we provide some commentary on how our results can be applied to fully 3D fields, as needed for application to the solar wind. In Appendix A, we also provide a different—more generic but less informative—argument for the results from Sec. III B that switchbacks are enhanced by the Parker spiral and form preferentially with field rotations that lie in its plane ($p\u0302(Z)$ in the out-of-plane direction). A key idea in this argument, as well as for the qualitative discussion, is that expansion generically tends to expand structures in the perpendicular plane (i.e., make them pancake shaped), or equivalently, to rotate the wave vectors to become more radial. In the presence of turbulence—which, to the contrary, tends to elongate structures along the background magnetic field—the competition with expansion will presumably enhance the power in radially aligned (as opposed to field perpendicular) wavevectors, compared to turbulence without expansion. Indeed, this seems to be observed in EBM simulations,^{42} and likely also in the solar wind.^{43,44}

In order to appraise the application to 3D fields in more detail, let us start by pointing out that there are (at least) three important differences in 3D that are lacking from the 1D results above. The first and most obvious is structure: simply the fact that fields vary in all three directions, not just in ** p**. The second is reflection-driven turbulence, which cannot affect 1D fields because the nonlinear term vanishes [indeed, it is this feature that enables the derivation of Eq. (8); M + 21], but will in general cause the destruction of the pure Alfvénic state, the decay of fields compared to the WKB expectations, and the (re-)population of power across a wide range of wavenumbers. The third is parametric decay

^{24,45,46}—that is, instability of the nonlinear Alfvénic solutions—which can destroy the wave if its perturbations grow sufficiently large. While this can occur in 1D also, it is convenient to discuss it here because it is not captured by our analysis or by Eqs. (8) and (9). Let us address each of these in turn.

### A. Structure

Realistic solar-wind fields presumably involve power across a wide range of wavenumber directions, distributed in such a way as to ensure constant *B*^{2}. We suggest that because the key physical ingredients needed for most of our results above are relatively simple—$\u2207\xb7b=0,\u2009B2=const.$, and the driving of ** p** toward the radial direction—these results can also apply in 3D with important caveats. In this application, the

**direction should correspond to the direction of fastest variation across some particular substructure of the 3D field. The simplest example of this viewpoint is from Eq. (5), which shows that near-perpendicular wavenumbers generate large $b||$ perturbations because of the constant-**

*p**B*constraint. As discussed in M + 21, the application to 3D fields is straightforward, implying that structures that vary rapidly in the nearly perpendicular direction—in other words, those that are elongated along the mean magnetic field—drive larger $b||$. This feature has been seen in both simulations

^{47}and observations.

^{34}

A similar application is to the conclusions of Sec. III B, that a Parker spiral should enhance switchback formation (i.e., increase $b||/vA$), and that switchback field deflections occur preferentially in the tangential plane. As demonstrated in more detail in Appendix A, when structures are compressed to perpendicular pancakes by expansion ($px\u2273py,\u2009pz$ on average), wavenumbers are on average more perpendicular to a mean field with a Parker spiral than a radial mean field, causing enhanced $b||/vA$ (more switchbacks) by Eq. (5). Similarly, if $vA$ involves a Parker spiral and the wavenumber distribution is biased toward the radial direction, wavenumbers that lie perpendicular to the plane of Parker spiral ($p\u0302(Z)$ waves) are on average more perpendicular to $vA$ than those in the plane of the Parker spiral. Even in 3D, Alfvénically polarized fluctuations involve larger field perturbations perpendicular to their direction of fastest variation, implying that the structures that generate larger switchbacks are more likely to involve tangential rotations of ** b** in the plane of the Parker spiral (i.e.,

*b*). Indeed, both of these features are seen in the simulations of J + 22, with their Figure 6 showing the stronger switchback growth with a Parker spiral, and their Figs. 7–9 showing various measures of deflections becoming tangentially asymmetric. Tangential asymmetry of deflections is also seen quite clearly in PSP data.

_{y}^{19}Nonetheless, turbulence provides an important caveat, particularly to the conclusion about switchback growth (see below).

More complex are the conclusions of Sec. IV, where we showed that waves become strongly tangentially skewed. As well as $\u2207\xb7b=0$ and constant *B*^{2}, this conclusion relies on the idea that the $b||$ of a switchback becomes dominated by the *b _{y}* contribution, rather than the radial (

*b*) fluctuation. This in turn relied on the wavevector becoming predominantly radial as $vA$ rotates away from the radial [see Eq. (15)]. If these conditions are satisfied, Eq. (17) suggests that structures that vary fastest in the near-radial direction ($\theta p\u227245\xb0$), but nonetheless remain somewhat perpendicular to $vA$ ($\u03d1\u227345\xb0$), must increase their radial-field perturbation through a switchback in order to maintain constant

_{x}*B*. This makes them tangentially skewed, therefore causing a significant difference between the statistical mean and mode of the magnetic-field direction. This condition—that quasi-radial $\theta p\u227245\xb0$ structures start dominating switchbacks for modest $\Phi \u227245\xb0$—does not seem unreasonable, so long as the turbulence is not continually repopulating modes along $vA$ as fast as they are being rotated radially by expansion (see below). Indeed, such tangential skewness is undeniably obvious in the Parker spiral simulation of J + 22 [see their Figs. 7(c) and 8], which provides at least a basic confirmation of the above scenario. The feature is less clear in a recent analysis of PSP deflections,

^{19}although it seems to be present in some cases (most prominently, encounter E6) and will be influenced by the Parker spiral angle, the fluctuation's amplitude, and the analysis method (e.g., the deflection angles that are counted as switchbacks). The enhanced switchback sharpness (Sec. IV B) seems to result from $sin\u2009\u03d1$ being an increasing function of time, so presumably applies under similar circumstances with similar caveats. This feature is also observed in Fig. 5 of J + 22, with the Parker spiral simulation exhibiting more sharp field rotations.

### B. Turbulence

Turbulence in the solar wind is thought to be caused by (among other possibilities) the reflection of outward- to inward-propagating perturbations,^{22} whose amplitudes we will term *z*^{+} and *z*^{−}, respectively. If this happens sufficiently rapidly so as to cause the ratio $z\u2212/z+$ (the imbalance) to grow continuously, the process will destroy the nonlinear solution Eq. (1), eventually breaking the constant-*B* condition and invalidating all of our arguments above. Indeed, the start of this process can be observed at late times in most simulations of J + 22, with $CB2$ (a measure of the relative spherical polarization) increasing at late times as the imbalance decreases [see their Figs. 2(b) and 3(b)]. In the solar wind, such a process is at least only partially complete by $\u223c1AU$, where turbulence is still observed to be relatively imbalanced and spherically polarized;^{48} nonetheless, all of our results for $\Phi \u227345\xb0$ (e.g., the bottom two panels of Fig. 4) are clearly suspect and will likely be invalidated by this effect. However, even well before $z\u2212\u223cz+$, turbulence causes two other effects that invalidate our arguments if they are sufficiently strong: the first is the turbulent decay of the magnetic field; the second is the re-population of wavenumbers through nonlinear interactions. Turbulent decay will decrease the growth of wave amplitude below $A\u221da1/2$, as used in our estimates (for $\Phi \u226a45\xb0$); clearly if waves stop growing there will likewise be no growth of switchbacks (unless perhaps if $A\u22731$ already and $sin\u2009\u03d1$ increases). Similarly, if the interaction between different modes ** p** is stronger than the effect of expansion, the scaling of

**used in the arguments above will be incorrect (although there will presumably be some expansion-driven bias toward the radial direction). While this does not necessarily hinder switchback formation—indeed, stopping the decrease in $sin\u2009\u03d1$ would be helpful—it would at least invalidate our scalings.**

*p*J + 22 argued (see their Sec. II C), based on previous work,^{49–51} that the importance of the effects discussed above should be determined by the parameter $\chi \u2248k\u22a5z+/k||vA\u2248Ak\u22a5/k||$, which is a measure of the relative size of nonlinear effects ($k\u22a5z+$) and wave propagation ($k||vA$) for the *z*^{−} waves (here $k\u22a5$ and $k||$ should be interpreted as average inverse correlation lengths of the energetically dominant *z*^{+} structures, which source *z*^{−} through reflection). For $\chi \u22731$, the phenomenology suggests turbulent decay balances expansion-induced growth such that $A\u221da0$ and waves do not grow at all [this is tentatively supported by the results of a $\chi >1$ simulation in J + 22; see their Fig. 2(a)]. In this case, our results likely do not apply. In contrast, for $\chi \u22721$, turbulent effects are weaker, and many of our results for 1D waves seem to apply relatively well to 3D, as evidenced by the multiple items of agreement discussed above. This conclusion—that the applicability our results in 3D is determined by *χ*—requires further study and is currently quite poorly understood. For instance, the phenomenology seems to predict faster decay than seen in simulations and observations,^{50–52} and kinetic effects could also play a dominant role if they halt turbulent decay.^{53}

Finally, it is worth reiterating that these conclusions about turbulence, and indeed all of our results, apply only to regions of super-Alfvénic wind beyond the Alfvén surface because they use the expanding box model. In the sub-Alfvénic wind, fluctuation amplitudes almost certainly grow robustly even in the presence of turbulence,^{54,55} while without turbulence waves grow much more rapidly than Eq. (11). In addition, if we interpret *a* as the cross-sectional area of a flux tube (see J + 22, Sec. II D), the scaling of ** p** and $\Phi $ with

*a*is different in sub-Alfvénic regions,

^{41,56}and of course $\Phi \u226a1$ in such regions anyway. Thus, for all our results, it is imagined that fluctuations have arrived already with a relatively large amplitude, and perhaps switchbacks, at the Alfvén point. It does not seem possible to grow large switchbacks from very-small-amplitude fluctuations in the EBM because of the turbulent decay (see Sec. II C of J + 22).

### C. Parametric decay

A third physical aspect that is missed out by our analysis is parametric decay, viz., instability of the nonlinear Alfvénic solution (1).^{24} This can afflict even 1D waves; however, it is not captured by the reduced equations (8)–(9) on which we base our analysis. In general, it can cause breakup of the wave if the instability grows to saturate at large amplitudes. Its growth rate, which will determine the time before saturation, generally increases with wave amplitude and at lower plasma *β*. For both general parallel fluctuations^{57} and oblique large-amplitude waves of the form discussed in Sec. II B,^{23} parametric instability has been found to be quite virulent, leading to saturation with $z+\u223cz\u2212$. However, Ref. 58 found that expansion strongly decreased the growth rate of the instability in the EBM, which is likely due to the dynamics slowing down as the plasma expands, suggesting waves with frequencies $\omega A$ not too far above $a\u0307/a$ could propagate undisrupted over a relatively wide range of *a* (in contrast, Ref. 59 find parallel waves are rapidly destroyed in the inner heliosphere, so more work is needed to better understand the influence of expansion). Perhaps more importantly, there are hints that the instability saturates at much lower levels in 2D or 3D fields,^{36,60} maintaining the nonlinear Alfvénic state (1) nearly unchanged with $z\u2212\u226az+$ even after saturation (although the instability may still play a key role in seeding turbulence^{47}). In addition, parametric decay is at least partially stabilized by damping of compressive fluctuations (which should be strong in a collisionless plasma), and random structure in the background Alfvénic state.^{46} Overall, more study is needed to better understand the relevance of parametric instability compared to reflection-driven turbulence, but if it either grows too slowly or saturates at low levels, it will not significantly change our results.

## VI. CONCLUSIONS

In this paper, we have explored the influence of the Parker spiral on the evolution of Alfvénic switchbacks in an expanding plasma. Using simple, geometric arguments based on spherically polarized 1D waves, we find a surprisingly large effect. This highlights the interesting, nonintuitive physics of nonlinear Alfvénic perturbations, underscoring that particular care must be taken before attributing any observed asymmetrical (or otherwise unexpected) characteristics of switchbacks to properties of their source. The key differences compared to the case with a radial mean field all result from the nontrivial (non-monotonic) evolution of the wave's obliquity $sin\u2009\u03d1$, which is brought by the simultaneous rotation of the mean field $vA$ and mode wavevector ** p** in different directions. Surprisingly, despite the normalized amplitude of waves in a Parker spiral growing more slowly than with a radial mean field, the formation of switchbacks, as measured by the growth of fluctuations parallel to the background magnetic field, is strongly enhanced in the most relevant regimes (cf. black and green lines in Fig. 2). This conclusion may be testable in the solar wind by comparing streams with different mean-field directions but similar fluctuation amplitudes. Our other main conclusions are as follows:

Switchbacks preferentially involve field rotations in the tangential direction, viz., a rotation in the plane of the Parker spiral. This is because wavevectors perpendicular to the plane of the Parker spiral ($p\u0302(Z)$ or $cos\u2009\phi =0$) are more effective at generating parallel field perturbations than those in the plane of the Parker spiral ($p\u0302(Y)$ or $cos\u2009\phi =\xb11$). This conclusion is based on both the evolution of 1D waves (Sec. III B) and a simple argument based on the average obliquity of wavevectors in a spectrum biased by radial expansion ( Appendix A).

Tangential switchbacks [those discussed in conclusion (i)] become strongly tangentially skewed, meaning the sharp field rotations of the switchback preferentially occur in one direction (toward the radial component of the mean field). This is a consequence of the divergence-free constraint in a spherically polarized wave, which forces the radial-field fluctuation to increase, rather than decrease, as the tangential field passes through zero (Sec. IV A). A similar constraint is that, projected on to the radial-tangential plane, field rotations in the ±T direction are limited to angles $\u226490\xb0\u2213\Phi $, where $\Phi $ is the Parker-spiral angle; thus, −T field rotations can be significantly larger, causing a highly asymmetrical rotation distribution.

As a consequence of conclusion (ii), and in order to maintain mean-zero fluctuations, the field-direction mode (its most common direction) is strongly skewed toward the +T direction compared to the mean field, viz., it usually rotated to a larger angle than $\Phi $. This suggests that care must be taken in measuring the Parker spiral, which should be the mean-field direction (that being the direction in which Alfvénic perturbations propagate).

As another consequence of conclusions (ii) and (iii), switchback fluctuations in a Parker spiral become more intermittent and sharper than those in a radial mean field. There are long quiet periods in which the field is rotated beyond $\Phi $, interspersed with short and sudden large rotations (see Fig. 4).

Although these conclusions are clearly limited by our reliance on 1D wave physics, we provide an extended commentary in Sec. V about the general applicability to 3D fields with turbulence. This suggests that there can be significant caveats, sometimes to the point of nullifying most of our results (e.g., in regimes where turbulence nonlinearities completely dominate over expansion), but also regimes where we might expect our results to apply qualitatively, even in 3D. More importantly, in our companion paper J+22, we see evidence for all five of the above conclusions [enhanced switchbacks in the Parker spiral, plus each of points (i)–(iv) above] in full 3D compressible expanding-box MHD simulations. Conclusions (ii)–(iii), on the skewness of tangential switchbacks, are particularly clear in field-deflection distributions (see Figs. 7–9 of J+22).

There may also be tentative evidence for observations of these features in PSP and other spacecraft data: Refs. 19, 28, and 61, see enhanced numbers of switchbacks with tangential deflections per point (i) above, although the Parker spiral is quite small in these works. Reference 10 reports that switchbacks preferentially deflect to one side, which is consistent with point (ii) above (this feature is also plausibly present in the analysis of Ref. 19). References 28 and 61 also report on the tendency of switchbacks to deflect in the same direction over long periods, which may relate to the same physics, but this requires further understanding (in particular, we predict this tendency only for –T deflections as is the case in Fig. 8 of Ref. 61; if this also occurs for other deflection directions, it is likely a result of other processes). Finally, Ref. 62 has shown that, consistent with our assumptions, for large Parker spiral angles (near $1AU$), there still exist Alfvénic fluctuations with properties very similar to switchbacks, although their observational appearance can be quite different due to the rotated background field. While they do not directly study properties related to points (i)–(iv) above, they may also see tentative evidence for a skewed field-direction distribution in agreement with points (ii) and (iii) [see their Fig. 3(a)]. Our other predictions are also potentially observable—for instance, one could compare switchbacks between streams with different Parker spiral angles to investigate point (iv) or the overall prevalence of switchbacks—but this requires further work.

## ACKNOWLEDGMENTS

The authors thank R. Laker, T. Horbury, S. Bale, N. Fargette, and C. Hill for interesting discussions in the course of this work. Support for J.S. was provided by Rutherford Discovery Fellowship Grant No. RDF-U001804, which is managed through the Royal Society Te Apārangi. Z.J. was supported by a postgraduate scholarship publishing bursary from the University of Otago. A.M. acknowledges the support of NASA through Grant No. 80NSSC21K0462. R.M. was supported by Marsden fund Grant No. MFP-U0020, managed through the Royal Society Te Apārangi.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jonathan Squire:** Conceptualization (lead); Formal analysis (lead); Investigation (lead); Writing – original draft (lead). **Zade Johnston:** Conceptualization (supporting); Methodology (equal); Validation (equal); Writing – review & editing (equal). **Alfred Mallet:** Conceptualization (equal); Formal analysis (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **Romain Meyrand:** Conceptualization (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that supports the findings of this study are available within the article.

### APPENDIX A: PREFERENTIAL TANGENTIAL DEFLECTIONS IN 3D FIELDS DUE TO EXPANSION

In this appendix, we provide an alternate argument for two of the main results of Sec. III: (i) that the Parker spiral tends to enhance switchback formation (increase $b||/vA$), and (ii) that switchbacks in a Parker spiral tend to involve tangential, rather than normal, deflections. Our method is simply to posit a simple Gaussian form for a spectrum of waves with its axis aligned along the radial direction, then compute the average $\u2009sin2\u03d1$ formed by such a spectrum when the mean field lies at angle $\Phi $ to the radial. This method is unrelated to the 1D wave arguments put forth in the manuscript, allowing for general 3D structure, but does of course rely on its own unjustified assumptions related to the form of the spectrum. In reality, of course, the spectrum should be neither Gaussian nor aligned perfectly along the radial: its alignment will presumably result from a competition between the expansion, which would tend to create spectral contours that are elongated along the radial direction, and turbulence, which tends to create spectral contours that are pancake shaped about the mean field. Nonetheless, the argument is relatively simple, does not have strong dependence on the chosen functional form of the spectrum, and can be trivially extended to account for nonradial alignment of the spectrum by simply redefining $\Phi $ as the angle between $vA$ and the spectrum's axis of symmetry (if this exists).

With this idea in mind, the assumed geometry is shown in Fig. 7(a). We take the Gaussian spectrum

where $p\u22a52=py2+pz2$ implies we assume symmetry about the radial (*x*) axis. As in the main text, we take $vA=vA(cos\u2009\Phi ,\u2009sin\u2009\Phi ,0)$, where $\Phi $ is the Parker spiral angle. The spectral properties are then specified by the anisotropy $\xi p\u2261px0/p\u22a50$, with $\xi p>1$, implying that outer-scale eddies are perpendicularly extended pancakes (elongated in *p _{x}*), and $\xi p<1$, implying the opposite.

^{63}Equation (5) says that for large-amplitude ($A\u2273\u2009sin\u2009\u03d1$) waves, a measure of the switchback prevalence is $(b||/vA)2\u223cA2\u2009sin2\u03d1$. Equating $E(px,p\u22a5)$ with

*A*

^{2}of a given mode at $px,p\u22a5$, we see that the relative switchback prevalence for a given amplitude, in the large-amplitude regime of Eq. (5), is $(b||2/vA2)/A2\u223c\u27e8\u2009sin2\u03d1\u27e9=E0\u22121\u222bdp\u2009E(px,p\u22a5)\u2009sin2\u03d1$. We can evaluate this integral using $\u2009sin2\u03d1=1\u2212(p\xb7v\u0302A)2/p2$ by writing $py=p\u22a5\u2009cos\u2009\phi ,\u2009pz=p\u22a5\u2009sin\u2009\phi $ and integrating over $p\u22a5$ and

*p*. This gives

_{x}where $\xi \u0303p2=\xi p2\u22121$ and (despite appearances) the expression is valid for both $\xi p>1$ and $\xi p<1$ (the $\u2009tan\u22121\xi \u0303p$ becomes $itanh\u22121|\xi \u0303p|$ for $\xi p<1$).

If we first consider carrying out the $\phi $ integral in Eq. (A2), this allows the comparison of the total relative prevalence of switchbacks at different $\Phi $. The result is plotted in Fig. 7(b) as a function of $\xi p=px0/p\u22a50$. As expected and intuitive from the geometry, for $\xi p>1$ the Parker spiral (green to yellow curves) increases $\u27e8\u2009sin2\u03d1\u27e9$, while the opposite occurs for $\xi p<1$. This demonstrates that if eddies become expanded into perpendicular pancake structures by expansion, the Parker spiral increases the average obliquity of the spectrum thus enhancing switchbacks for the same amplitude. This is effectively the same physics as discussed in Sec. III B, which showed that nonzero $\Phi $ increases $b||/vA$ significantly for $a\u2273a\u03d1min$, which can only occur once $\theta p>45\xb0$ in the relevant regime.

The second conclusion from Sec. III B was that switchback deflections should be primarily tangential (in *b _{y}*), because many of the wavevectors in the Parker-spiral plane (which create normal field deflections) become highly parallel [Fig. 2(b)]. To assess this conclusion for the spectrum (A1), we imagine considering the contribution of each $\phi $ in Eq. (A2) separately. Wavevectors with $\phi =0$ or

*π*will cause larger

*b*perturbations (because an Alfvénic perturbation has polarization $\u223cp\u0302\xd7vA$), while those with $\phi =\xb1\pi /2$ will cause large

_{z}*b*perturbations. We plot twice the integrand of Eq. (A2) in Fig. 7(c), which captures the contribution to $\u27e8\u2009sin2\u03d1\u27e9$ from the

_{y}*p*-space plane angled at $\phi $, as illustrated in Fig. 7(a) [this normalization is such that

*π*multiplied by the black Φ = 0 curve in Fig. 7(c) yields the same curve in Fig. 7(b)]. We see that when $\Phi =30\xb0$ the contribution from $\phi =0$ or

*π*(orange curve) is quite small compared to that from $\phi =\xb1\pi /2$ (blue curves), for any value of

*χ*. This is not surprising and is indeed rather obvious by inspection of Fig. 7(a), but it demonstrates mathematically that in a random collection of waves with a random series of Alfvénic deflections, field deflections in the tangential direction (those with $|\phi |\u2248\pi /2$) will cause larger parallel-field perturbations than deflections in the normal direction.

_{p}### APPENDIX B: COMPARISON TO MHD SOLUTIONS

In this appendix, we directly compare the predictions of Eqs. (8) and (9), which has been used in the main text to understand nonlinear wave evolution, to nonlinear isothermal expanding-box MHD simulations with the Athena++ code. The purpose of this comparison is twofold. First, it is simply interesting to better understand the accuracy and applicability of Eqs. (8) and (9), given it was derived through an asymptotic expansion in a slow expansion rate. On this aspect, the comparison is extremely positive. Second, we noted in Sec. IV that when ** p** and $vA$ lie in the same plane (the $p\u0302(Y)$ case with $\phi =0$ or

*π*), Eqs. (8) and (9) usually fail to produce solutions with constant

*B*. Importantly, because Eqs. (8) and (9) are derived by assuming that

**maintains constant**

*b**B*, if it does not, we cannot trust their results. It is thus interesting to see whether this production of non-constant

*B*is truly physical—that is, whether it also occurs in true MHD evolution—or whether it results for another reason related to the approximations used to derive Eqs. (8) and (9). Although the detailed cause of this behavior remains unclear, we speculate that it relates to $sin\u2009\u03d1$ changing particularly rapidly in this geometry, with the shape of

**not able to adjust fast enough to maintain constant**

*b**B*.

To generate the MHD solutions, we use the MHD code Athena++, with the modifications to capture plasma expansion detailed in J + 22. We set up each wave in a 2D domain of dimensions $Lx=4Ly$ at *a *=* *1, by initializing the sinusoidal $px=2\pi /Lx,\u2009py=2\pi /Ly$ mode in *b _{n}* (the component out of the plane) with amplitude 1, such that the wave obliquity is $\theta p0=tan\u22121(Lx/Ly)\u224876\xb0$. The other

**components are constructed as described in Eq. (6) to ensure constant**

*b**B*, with an initial Parker spiral angle of $5\xb0$ (if this is included). We choose the expansion rate to be $a\u0307/a=0.5\omega A$ (i.e., $\epsilon =0.5$) and use 128 grid points in each direction. In order to compare the solutions to Eq. (8), we perform a “wavefront average” at each output step, meaning we rotate the coordinate system to align with the wave (accounting for the periodicity of the domain), then spatially average in the direction perpendicular to

**.**

*p*Results for radial $vA$ and $\phi =\pi /2$—viz., the situations in which Eqs. (8) and (9) successfully maintain constant *B*, are shown in Fig. 8. Left and right subpanels compare wavefront-averaged solutions to the MHD equations to those of Eq. (8) at the same *a* and other parameters. We see excellent agreement between the general shape of the waveforms, discounting the phase of the wave, which evolves in the MHD case but not in Eq. (8). The MHD solutions do involve small fluctuations that are not present in Eq. (8), which are most clearly observable in the field magnitude profile; these result in part from the small compressive components neglected in Eq. (8) (see M + 21), and in part from the parametric instability (these fluctuations vary across the wavefront direction so are averaged in Fig. 8). The parametric instability fluctuations slowly grow and eventually overwhelm the wave, but are not our primary interest here (see Ref. 58).

Results for $\phi =0$ and $\phi =\pi $, when ** p** and the Parker spiral lie in the same plane, are shown in Fig. 9. In these cases, we can clearly see in the right-hand subpanels that the solutions of Eqs. (8) and (9) do not maintain constant

*B*, calling the validity of these solutions into question. Indeed, we see that

*B*remains much more spatially constant in the MHD solutions, and the spatial form of the individual components differs more significantly than those shown in Fig. 8, although they clearly maintain similar structures. Particularly for the $\phi =\pi $ case, we see tentative evidence that the MHD solutions exhibit larger variation in

*B*than the cases in Fig. 8, suggesting that at least some of the failure of Eq. (8) to maintain constant

*B*is physical. In the MHD solutions, these variations propagate around the box, steepening and reducing in size due to compressive processes that are not captured by Eqs. (8) and (9) (see, e.g., Ref. 64). While the situation in the $\phi =0$ solution is less clear, we clearly see smaller switchbacks in MHD in both cases, which presumably results from the combination of MHD more robustly maintaining a constant-

*B*Alfvénic solution, and MHD dissipating or smoothing the wave energy that is converted into compressive structures. This latter effect would not be captured by the arguments in Sec. III, suggesting that the $b||/vA$ estimated therein could be an overestimate. Also of note is that we do not see any obvious singular behavior (e.g., mode conversion) when $sin\u2009\u03d1$ passes through zero in the $\phi =0$ case (around $a\u22486$, as seen by the nearly flat $b||$ at this time).

Overall, we see tentative evidence that $p\u0302(Y)$ waves have a tendency to generate larger compressive variations than $p\u0302(Z)$ waves. This could enable other dissipation mechanisms in a real plasma, thereby reducing the switchbacks generated by such waves. Thus, this strengthens our conclusions from Sec. III, implying that, as well as naturally generating smaller $b||$ due to geometry, switchbacks that involve field rotations in the normal direction also are likely to dissipate more strongly, thus enhancing the dominance of tangential switchbacks.

### APPENDIX C: THE SCALING OF WKB WAVES IN A STRONG PARKER SPIRAL

In deriving the scaling of wave amplitudes with expansion in the presence of a Parker spiral $vAy\u223cvAx$, we used the same scaling of the wave amplitude with expansion, $|b|\xaf\u221da\u22121/2$, as in the radial-background-field case. In this appendix, we confirm that this is indeed correct, viz., that the scaling of the unnormalized amplitude of WKB Alfvén waves with *a* is independent of their direction of propagation. This property has been shown in a number of previous works for small amplitude waves^{65,66} and is also contained in the large-amplitude results of Refs. 29 and 39 and M + 21 [the second term $(a\u0307/2a)b$ in Eq. (8) is independent of $v\u0302A$]. Nonetheless, we feel that the EBM derivation below is helpful both for its generality (it does not assume 1D waves or low amplitude) and its simplicity, which helps to illustrate the physical cause of the $v\u0302A$-independence of the amplitude scaling.

Starting from the EBM equations (see, e.g., J + 22), we assume constant *ρ* and incompressible motions, then form the equations for $z\xb1=u\u2009\xb1\u2009B/4\pi \rho =u\u2009\xb1\u2009b\u2009\xb1\u2009vA$ (where, as above, ** b** is $\delta B/4\pi \rho $). This gives

where $T=diag(0,1,1),\u2009\u2207\u0303$ is the $\u2207$ operator in the expanding frame, and $p\u0303$ is chosen to constrain $\u2207\u0303\xb7z\xb1=0$. The final two terms arise due to the differing influence of expansion on ** b** (the second-to-last term) and

**(the last term). To understand the scaling in the WKB limit neglecting nonlinear interactions, we can simply set $z\u2212$ to zero. This is justified because when the $vA\xb7\u2207\u0303$ term dominates the others, the reflection terms (those involving $a\u0307/a\u2009z+$), cannot cause $z\u2212$ grow because it quickly moves out of phase with the source $z+$ wave. Thus, in this limit, $(\u2202t+vA\xb7\u2207\u0303)z+\u2248\u2212(a\u0307/2a)z+$ because both the third and fourth terms on the right-hand side of Eq. (C1) involve the same factor $\u2212a\u0307/2a$. This implies that $|z+|$ decays as $|z+|\u221da\u22121/2$, no matter its direction or the direction of the mean field.**

*u*## References

*p*and

_{z}*p*widths. Nonetheless, given that we lack a good model of how this competition between turbulence and expansion manifests, while numerical integrations suggest that this does not make a significant difference to our qualitative conclusions anyway, it does not seem worthy of detailed exploration.

_{y}