Theory of Magnetic Switchbacks Fully Supported by Parker Solar Probe Observations

Magnetic switchbacks are rapid high-amplitude reversals of the radial magnetic field in the solar wind that do not involve a heliospheric current sheet crossing. First seen sporadically in the 1970s in Mariner and Helios data, switchbacks were later observed by the Ulysses spacecraft beyond 1 au and have been recently discovered to be a typical component of solar wind fluctuations in the inner heliosphere by the Parker Solar Probe spacecraft. While switchbacks are now well understood to be spherically polarized Alfvén waves thanks to Parker Solar Probe observations, their formation has been an intriguing and unsolved puzzle. Here we provide a simple yet predictive theory for the formation of these magnetic reversals: the switchbacks are produced by the distortion and twisting of circularly polarized Alfvén waves by a transversely varying radial wave propagation velocity. We provide an analytic expression for the magnetic field variation, establish the necessary and sufficient conditions for the formation of switchbacks, and show that the proposed mechanism works in a realistic solar wind scenario. We also show that the theoretical predictions are in excellent agreement with observations, and the high-amplitude radial oscillations are strongly correlated with the shear of the wave propagation speed. The correlation coefficient is around 0.3–0.5 for both encounter 1 and encounter 12. The probability of this being a lucky coincidence is essentially zero with p-values below 0.1%.


Introduction
The solar wind, to a good approximation, can be described by the equations of ideal magnetohydrodynamics (MHD).Parker Solar Probe (PSP) observations of switchbacks (Bale et al. 2019;Kasper et al. 2019) show a tight correlation of magnetic and velocity perturbations that are characteristic of Alfvén waves (Kasper et al. 2019).Alfvén waves are typically thought of as transverse oscillations of the velocity and magnetic fields around a guide field B r , which, in the case of the solar wind, points approximately in the radial direction within Mercury's orbit.The magnitudes of transverse velocity and magnetic perturbations, u ⊥ and B ⊥ , are related as m r = ^û B 0 , where ρ is the mass density of the solar wind, and μ 0 is the magnetic permeability of vacuum.Circularly polarized Alfvén waves are, in fact, exact solutions of the MHD equations, even when their amplitude B ⊥ is large.The most puzzling property of the observed switchbacks is that the presumed guide field B r changes sign with frequent largeamplitude oscillations as shown in the top middle and bottom middle panels of Figure 1.
It is now well established that PSP observes spherically polarized Alfvén waves (Barnes & Hollweg 1974).For these waves, both the magnetic field vector B and velocity vector u oscillate in arbitrary directions, and m r = u B 0  in the coordinate frame moving with the wave, where the minus/plus signs determine whether the wave propagates parallel/ antiparallel to the background magnetic field direction.This is fully consistent with PSP measurements (Kasper et al. 2019).
There is another requirement for nonlinear spherical Alfvén waves to be an exact solution: the magnetic pressure p B = B 2 /(2μ 0 ) must be constant, a nontrivial constraint, because a nonconstant divergence-free magnetic field typically has a spatially varying amplitude, with the exception of circularly polarized Alfvén waves (Marris & Wang 1970;Vasquez & Hollweg 1998).The observed spherically polarized Alfvén waves are only approximately stationary because the magnetic pressure is only approximately constant, but they can travel large distances in the solar wind without significant dissipation.
An important feature of spherically polarized Alfvén waves is that they must originate from genuinely multidimensional perturbations.A one-dimensional MHD wave in plane-parallel or spherical geometry cannot perturb the normal component of the background field (B r in the case of spherical symmetry) because it would violate the ∇ • B = 0 condition.This means that spherically polarized Alfvén waves, unlike circularly polarized waves, require a multidimensional perturbation.In particular, observations suggest that switchbacks come in patches that are on the transverse scale of supergranulation when projected back to the Sun and may be associated with both the supergranular spatial gradient scale and a temporal modulation associated with emerging flux (Fargette et al. 2021;Shi et al. 2022).In any case, a perturbation in the transverse direction with the characteristic length scale of supergranulation is present, potentially with smaller-scale gradients at granular scales as well.
There have been several ideas put forward to explain how switchbacks form, including magnetic reconnection (Zank et al. 2020;Drake et al. 2021), Kelvin-Helmholtz instability (KHI; Mozer et al. 2020;Ruffolo et al. 2020) velocity shears and jets (Landi et al. 2006;Schwadron & McComas 2021), but none of these provide a fully selfconsistent explanation for all observed properties.In particular, mechanisms unrelated to Alfvénic turbulence (reconnection, KHI, or sub-and super-Parker radial velocity) cannot easily explain the Alfvénic nature of the observed magnetic and velocity perturbations.Compressible turbulence theory on its own, on the other hand, lacks the coherent multidimensional perturbation that can organize the switchbacks into the observed large-scale structures (patches).Landi et al. (2006) studied the interaction of velocity shears and jets with Alfvénic turbulence, which has similar ingredients as our proposed theory, but the small spatial length scale of the shear assumed by Landi et al. is not suitable to produce switchbacks in the solar wind observed by PSP.

Large-scale Shearing of Circularly Polarized Alfvén Waves
We propose a new explanation for the formation of switchbacks and provide supporting observational, theoretical, and numerical evidence.Switchbacks are produced by circularly polarized Alfvén waves distorted and twisted by a large-scale transverse shear of the radial wave speed.The radial speed of an outward-traveling Alfvén wave is A 0 and u r are the Alfvén and solar wind speeds in the radial direction, respectively.The wave velocity can vary for three reasons: variations of u r , B r , or ρ.Our numerical tests confirm that any of these can produce switchbacks.The transverse shear combined with the variation associated with the radial wavelength of circularly polarized waves provides a genuinely multidimensional perturbation.This process of generalized phase mixing of Alfvén waves has been considered before in the context of coronal heating (Hasegawa & Chen 1974;Heyvaerts & Priest 1983;Ofman & Davila 1995).
Consider a sinusoidally sheared radial wave velocity profile ( ) , where λ y is the wavelength in the ydirection that is perpendicular to the radial direction, and z completes the coordinate system.The wave velocity shear impacts an initially circularly polarized sinusoidal Alfvén wave with radial wavelength λ r .The magnetic field lines of the wave oscillate within a width w = (B ⊥ /B r )λ r /π.A long-wavelength velocity perturbation, λ y ?w, will shear the circularly polarized waves and rotate the field in the r-y plane across several waves.The left panels of Figure 2 show numerical simulation results for this case.When λ y ∼ w, a much more complex solution emerges, as shown in the middle panels.Finally, for λ y = w, the velocity shear can bend the transverse field lines, as found by Landi et al. (2005Landi et al. ( , 2006)), but it does not create switchbacks unless the magnetic field is weak, which is not the case near PSP.
The long-wavelength case can be regarded locally as a constant shear of the radial wave velocity, = dv dy const, which can be studied analytically., so the wavelength is λ r = 2π.After a time t, the field lines at a distance y from the center of the wave will be pushed from position r to ¢ = + r r sy, where s = t(dv/dy) is the shear at time t.To a first-order approximation, the shear will simply shift B y and B z in the radial direction, ( ) The evolution of the shear in a radially expanding flow is different than in the plane-parallel case examined so far.The transverse gradient of the average radial velocity can be characterized as dv/dα, where dv is the difference in the radial velocities, and dα is the angle between two streamlines in radians, as depicted at the bottom of Figure 3.The shear is s = dr/dy = (tdv)/(rdα) = (t/r)(dv/dα).We can estimate the time as t = D/v, where D is the distance from where the shearing starts.For small distances D = r, the shear grows linearly with time t and D similar to the plane-parallel case.For radial distances where D/r ≈ 1 and the average wave velocity v is not changing rapidly, the shear s = (1/v)(dv/dα) becomes roughly constant.This is in contrast with the plane-parallel case, where s = t(dv/dy) is growing proportionally with time t and the x-coordinate indefinitely, at least in the linear approximation.
A switchback occurs when ¢ B r changes sign.This happens when dB r = sB ⊥ = (B ⊥ /v)(dv/dα) exceeds B r .This does not require a very large transverse gradient in the wave velocity.For example, during solar minimum, the slow solar wind is about 400 km s −1 near the equator, the fast wind is about 800 km s −1 near the poles separated by 90°= π/2 rad, and the Alfvén speed is much smaller for distances at encounter 1 or larger.Assuming that the wave velocity changes linearly with latitude, (1/v)(dv/dα) = 2/π ≈ 0.6 at the equator.Given that B ⊥ and B r have comparable magnitudes during encounter 1, it is clear that even this minimal velocity shear can produce a dB r comparable to B r .The wave velocity gradients of the hourly values in Figure 1 are much larger and can easily produce switchbacks, as discussed in Section 4. The observations also show strong correlations between the oscillations of B r and the perpendicular components at most times.This confirms that the oscillations are the radial and perpendicular components of a sheared oscillation.The direction of the shear determines if dB R is proportional to B T or B N (or some linear combination of them), and the sign of s determines if there is a positive correlation or an anticorrelation.
The first-order approximation satisfies the divergence-free property, but the magnetic pressure 0 is no longer constant.The magnetic pressure gradient will compress the plasma and modify ¢ B r and ¢ B y while maintaining the ( ) relationship so that the field remains divergence-free.The plasma will move toward the small magnetic pressure region where B r is small and the switchbacks form.This explains why the observed switchbacks are narrow peaks, while the regions with a normal B r direction are wide and flat (see Figure 1).In addition, the top left panel of Figure 1 demonstrates that low magnetic pressure, a small value of B, is highly correlated with enhanced density, which is in full agreement with the notion of compression of the plasma toward the switchbacks.
An additional requirement for a switchback to occur is that the shear velocity has sufficient energy to distort the original wave.A simple estimate is that the average energy density of the shear motion is comparable to or larger than the magnetic energy density of the transverse magnetic field, rv 1 2  m For the 12th encounter, the 24 hr period prior 19:00 UT May 31 is shown, as there are no public plasma data after that.The wave speed varies along the PSP orbit in both cases.The panels on the right show the high-cadence observations of the three components of the magnetic field for half-hour periods.The background is removed by subtracting a 7.4 minute sliding average.During the first encounter, all three components vary with similar amplitudes.For the second encounter, dBR is multiplied by 4 to make its variation similar to the perpendicular oscillations.The correlation coefficients r(dBR, dBT) and r(dBR, dBN) are calculated over a 2 minute sliding window.The light red rectangles highlight where dBR is highly correlated with dBT or dBN.The yellow rectangles highlight anticorrelation, where the signs are opposite.The green rectangles indicate times when B r is approximately constant.The bottom left panel shows one of these times, when dBT and dBN vary with similar amplitudes consistent with roughly circularly polarized Alfvén waves.The top left panel shows BR, the magnetic field magnitude B, the density n (all smoothed over 9.6 s), and the correlation coefficient r(B, n) for a selected short time period with many switchbacks.
energy density of the shear of the wave speed rv 2 1 2 gets reduced.The hourly averaged PSP plasma data during the first encounter (see Figure 1) suggest that this is indeed happening.The U r , B r , and m r 0 vary by 5%, 29%, and 6%, respectively, which would result in ≈8% variation in the wave speed v if these were independent of each other.But the observed wave speed only varies by 5.5%, which means that the velocity, magnetic field, and density variations contributing to the wave speed partially cancel each other out.This cancellation is caused by the distortion of the field reducing the energy of the shear as the system tries to find an approximate equilibrium solution with a constant wave speed.

Numerical Experiments
The basic dynamics of shearing a circularly polarized Alfvén wave can be captured in a two-dimensional (2D) MHD simulation with three vector components for velocity and magnetic field.The simulation domain is a double periodic rectangle.The r-direction corresponds to the radial direction in the solar wind.The frame of reference is chosen such that the initial circularly polarized wave, without the perturbation of the wave speed, is at rest.The setup is normalized by setting the units of distance, time, and mass, so that λ r = 4, B ⊥ = u ⊥ = 1, and m r = 1 0 , where r is the unperturbed density.The initial magnetic and velocity fields are .We can perturb either u r , B r , or ρ to change the wave speed.The size of the domain in the r-direction is λ r , while in the y-direction, it is a multiple of λ y .
The simulations are performed with the BATS-R-US code (Powell et al. 1999;Tóth et al. 2012) on a fine grid (cell size Δr = Δy = 0.04 = λ r /100) with a fifth-order accurate scheme (Chen et al. 2016) to minimize numerical errors.The left panels of Figure 2 show the solution for the long-wavelength case with the perturbation applied to B r in the part of the domain where the shear is near maximal.The result is a distorted wave, similar to the analytic description, with large switchbacks (bottom left panel) that look remarkably similar to the observations in Figure 1.The middle panels show the solution for a case where the wavelength of the perturbation λ y is comparable to w.The solution has complex structures that do not resemble a circularly polarized wave; still, the Alfvénic relations, −B y ≈ u y , −B z ≈ u z , and , hold (subtracting the initial perturbation from u r removes the background variation).In this case, the y = 0 cuts show more complicated switchback structures.The right panels show a solution for λ y that is much smaller than w.In this case, there are no switchbacks.
Figure 4 shows four simulations with various perturbations in the radial velocity, radial field, and density.The first three columns show simulations where the wave velocity v has a shear that results in switchbacks.The fourth column is a control experiment: both u r and B r have shear, but their effects on v cancel each other out, and no switchbacks are formed.These numerical experiments confirm that the crucial quantity producing switchbacks is the shear of v no matter what causes it.
Finally, we show that the mechanism also works in the radially expanding solar wind.We use physical units for easier comparison with observations.The 2D computational domain is a spherical wedge extending from r = 25 to 40 R s , and the azimuthal angle goes from −5°to 5°.The 2D computational grid consists of 4000 × 1600 cells.The boundaries are periodic in the azimuthal direction, and the outflow condition is applied at r = 40 R s .The circularly polarized Alfvén waves enter at r = 25 R s with amplitude B ⊥ = 80 nT and wavelength λ r = 0.1 R s .The number density, radial velocity, and temperature are 800 cm −3 , 300 km s −1 , and Figure 5 shows the solution at t = 30 hr, which is more than enough for the solar wind to propagate from 25 to 40 R s with 300 km s −1 speed.The figure shows that switchbacks develop with their characteristic asymmetric shapes, and the Alfvénic relationships between magnetic and velocity fields are satisfied.The anticorrelation between magnetic field magnitude and density is also similar to the observations near the switchbacks.

Theoretical Predictions Tested against Observations
This section validates our theory against observations.We use the RTN coordinate system, where R is the radial direction, the RN plane contains the rotation axis of the Sun, and T completes the system.To indicate observed variables, we use the usual notation: n for number density; B R , B T , and B N for the magnetic field components; and V R , V T , and V N for the velocity components.The corresponding quantities in the theoretical description were ρ, B r , B y , B z , u r , u y , and u z , respectively.The top panels show the variation of the radial velocity u r , radial field B r , and density n and the resulting variation in the wave velocity v = u r + V A along y at t = 0.The first three columns have perturbations of the wave velocity due to perturbations in u r , B r , and n, respectively.The fourth column has perturbations in both u r and B r that cancel each other out, so v is constant.The bottom panels show B r (colors) with white field lines in the region with maximum shear at various times.The simulations in the first three panels produce switchbacks, B r < 0 (red stripes), but the fourth does not because there is no shear in v.

Radial Oscillations and Wave Velocity Shear
Our theory states that the radial magnetic field perturbations dB R are created from the perpendicular field B ⊥ by the transverse angular shear of the wave speed, where the 〈.〉 indicates averaging over the period of the oscillations.On the right-hand side, D is the distance of PSP from where the transversely polarized Alfvén waves start shearing, and v is the average wave velocity over this distance.
The average velocity can be estimated as A , which should approximately hold within a constant factor that cancels out as v occurs in both the numerator and the denominator of the equation.
The PSP measures in situ all quantities needed for the wave speed and the radial and perpendicular magnetic field perturbations as a function of time.To calculate the derivative with respect to the angular α coordinate, we need to divide by the transverse velocity of PSP relative to the radially propagating plasma in the heliographic inertial coordinate frame: The waves can, of course, be sheared in the N (or latitude θ) direction as well, but we have no information about the gradient with respect to N because PSP moves in the T direction.Nevertheless, we can distinguish waves sheared in the T and N directions, respectively, by comparing the correlation coefficients r(dB R , B T ) and r(dB R , B N ).We can, for example, restrict our analysis The bottom left panels show the magnetic field and velocity components, as well as the total pressure p + p B , magnetic pressure p B , density n, and correlation coefficient r(B, n) between B and n over a sliding window of 0.3 R s along the trajectory.The virtual satellite is moving +5°longitude per day, similar to PSP near the perihelion of encounter 1.All quantities are comparable to PSP observations during the first encounter.The gradients of the total pressure are small but not zero.Density variations are also substantial and show a strong anticorrelation with the magnetic field magnitude near the switchbacks.The bottom right panels compare the magnetic (black solid lines) and velocity (red dotted lines) perturbations around a switchback.The magnetic field components are converted to Alfvén velocity components: For the radial components, the background variation is removed with a smoothing over 100 grid cells (0.34 R s ).All components satisfy the Alfvénic relationship to a high accuracy similar to PSP observations (Kasper et al. 2019).
to the time period where these correlation coefficients exceed some threshold.
For encounter 12, we use observations from 2022 May 30 to June 4 with a 3.5 s cadence.Encounter 12 contains a current sheet crossing at around 17 UT June 2.To simplify the analysis, we flip the sign of the magnetic field beyond this time, ¢ = -B B R R for time t > 89 hr measured from 00 UT May 30, and do not include the 1 hr period surrounding the crossing into the analysis.Next, we calculate RS s , where the 〈.〉 indicates a boxcar smoothing over S = 3500 s (or 1000 data points), and s = S/10.These smoothing widths are significantly longer than the periods of the rapid oscillations (we checked that varying the smoothing lengths has a small impact on the results).The perpendicular component is obtained as . The outward-going wave speed is obtained as . The plus sign is used because the background B R > 0 everywhere after flipping the field beyond the current sheet.Due to the Aflvénicity, the high-frequency oscillations in V R and V A mostly cancel out.We apply an additional smoothing over s (or 100 data points) to eliminate any remaining residuals of the high-frequency oscillations, so that v and ( ) dv dt V T PSP are accurate estimates of the wave speed and its gradient.
We also study a 24 hr period (2018 November 5) of encounter 1.The analysis is the same as for encounter 12 except that there is no current sheet crossing and the background B r is negative everywhere, so the wave speed is . Since the time period is 1 day long instead of 5 days and the cadence is 1 s instead of 3.5 s, the smoothing lengths are reduced to S = 1000 s (corresponding to 1000 data points) and s = 100 s.This is longer than the typical periods of the rapid oscillations of about 30 s.
First, we calculate crude estimates for the two sides of Equation (1) to obtain an approximate value for D. The lefthand side is observed to be ≈0.24 for encounter 12. Based on the hourly average of the wave speed shown in Figure 1, the derivative on the right-hand side can be roughly estimated as a change from ≈670 to ≈570 km s −1 over a 24 hr period, while the average radial distance is R ≈ 25 R s , and the distance traveled in the T direction is about 10 R s .This gives for the right-hand side = .Taking D = R − 1 = 24 R s , the right-hand side becomes 0.39, about a factor of 1.6 larger than the left-hand side, 0.24.For 2018 November 5 of encounter 1, the left-hand side is ≈0.28, while on the righthand side, the velocity changes from 300 to 400 km s −1 .The spacecraft is at R ≈ 36.5 R s radial distance, and it travels about 12 R s in 24 hr, giving = .Substituting D = R − 1 = 35.5 R s , the right-hand side becomes 0.84, which is three times larger than the left-hand side.Given all of the various approximations, the agreement is quite reasonable.The shorter-timescale velocity gradients dv/dt are much larger than the gradient over a 24 hr period.This means that the large dB R fluctuations can be easily generated by the velocity shear.For encounter 1, the ratio of the average B R and B ⊥ magnitudes is around 1.2, while for encounter 12, the ratio is around 2, roughly inversely proportional to the radial distance.Since the shear s = (1/v)(dv/dα) = dB R /B ⊥ does not vary much with radial distance, we expect more switchbacks during encounter 1 than encounter 12, which is indeed observed.
We now investigate if large local wave velocity shear and fluctuations of B R are indeed correlated in a statistically significant manner.We calculate the left-and right-hand sides of Equation (1) over temporal bins containing 100 consecutive data points.This corresponds to a bin size of b = 350 and 100 s for the two encounters, respectively.This bin size is significantly longer than the high-frequency oscillations and significantly (10 times) shorter than the smoothing length S. Figure 6 shows the left-and right-hand sides of Equation (1) as black and red lines.The correlation between the two curves is extremely strong in all cases given the various approximations and the fact that the spacecraft only observes the transverse variation of the wave speed along its trajectory.
We use a χ 2 test appropriate for time series to calculate the statistical significance of the correlation.First, both time series are normalized to have zero mean and unity standard deviation.Next, we calculate the variance s 2 of the difference between the two normalized time series.If the two normalized time series were uncorrelated (our null hypothesis), the variance of the difference would be σ 2 = 2. Assuming that the difference has a Gaussian distribution, the probability of having a variance s 2 or less is 2 , where f = N − 1 is the degrees of freedom, and N is the number of bins in the time series.All p-values shown in Figure 6 are very small, proving that the correlation between the radial field perturbation and the transverse velocity shear is statistically significant.

Correlations of the Magnetic Field Components
According to our theory, the switchbacks are produced from circularly polarized Alfvén waves, so we expect the three components of the magnetic field to be highly correlated and the magnitudes of the two transverse components to be comparable.For encounter 12, we indeed find that the Pearson's and Spearman's correlation coefficients are r(〈|dB R |〉 s , 〈B ⊥ 〉 s ) = 0.59, r s = 0.75 and r(〈|B T |〉 s , 〈|B N |〉 s ) = 0.56, r s = 0.59.The boxcar smoothing is applied over 100 data points corresponding to an s = 350 s time interval.The average magnitudes |dB R |, |B T |, and |B N | over the whole 6 day interval from 2022 May 30 to June 4 are 24.5, 71.2, and 67.4 nT, respectively, so the two transverse components are not only correlated but have very similar magnitudes, while dB R is highly correlated but has a smaller amplitude, around one-third.
For encounter 1, the 6 day time period from 2018 November 2 to November 7 is studied for the sake of good statistics, although the results are essentially the same for the single day of November 5.The time cadence is 1 s, so we increase the boxcar smoothing to 350 data points.The correlation coefficients are r(〈|dB R |〉 s , 〈B ⊥ 〉 s ) = 0.57, r s = 0.62 and r(〈|B T |〉 s , 〈|B N |〉 s ) = 0.12, r s = 0.17.Clearly, the correlation between the two transverse components is much smaller than for encounter 12, but it still has a p-value of less than 10 −3 .The average magnitudes |dB R |, |B T |, and |B N | are 12.9, 28, and 21.9 nT, respectively.Again, the two transverse components have comparable magnitudes, while the magnitude of the radial component is about half.
Comparing the two encounters suggests that as the solar wind moves from around 26 R s to around 40 R s , the relative magnitude of the radial perturbation is roughly a constant fraction (one-third to half) of the transverse oscillations, which is consistent with the prediction of our theory for radial outflow.In addition, the correlation between the two transverse components is reduced, which can be understood in terms of the circularly polarized waves getting distorted by the nonlinear shearing process, which makes the waves less circular and the two transverse components less correlated with each other.

Anticorrelation of Density and Magnetic Field Magnitude
Finally, we check that the anticorrelation of the density n and magnitude of the magnetic field B illustrated for a short time period in the top left panel of Figure 1 are indeed general and significant features of the observations.The fluctuations are defined as = -á ñ dn n n S and dB = B − 〈B〉 S , where the boxcar smoothing is done over 1000 data points.For 2018 November 5 of encounter 1, the Pearson's and Spearman's correlation coefficients for the full 24 hr period are ( )=r dB dn , 0 . 2 2 and r s = −0.23,respectively, and the p-value is zero.If we restrict the analysis to the time period when B R > −50 nT, the correlation coefficients become r ≈ r s ≈ −0.31 with a zero pvalue.For 2022 May 30 to June 4 of encounter 12, the anticorrelation is even stronger.The background is removed with the same boxcar smoothing over 1000 data points corresponding to 3500 s in time.For the full time period, r = −0.73 and r s = −0.29,and the p-value is zero.If we restrict the data to |B R | < 200 nT (the absolute value accounts for the sign change over the current sheet crossing), the correlations become r = −0.8 and r s = −0.3 with a zero p-value.

Conclusions
Our simulations were set up to demonstrate the formation of switchbacks in an idealized solar wind.In the real solar wind, Alfvénic fluctuations are fully 3D and turbulent, presenting well-developed power laws and a total energy decay dominated by the spherical expansion (Velli 1993;Dong et al. 2014).According to previous theoretical and numerical studies (Squire et al. 2020;Mallet et al. 2021), the turbulence will preserve the spherically polarized Alfvén waves and further enhance their amplitudes.Our theory states that switchbacks form in Alfvénic turbulence due to the large-scale transverse shear of the radial wave velocity, just like in the idealized simulations.Indeed, the observed features of the turbulent solar wind are remarkably similar to the idealized simulations, but the turbulence produces switchbacks over a range of spatial scales, while the simulations produce them at the wavelength imposed by the boundary conditions.
While the simulations are idealized, the theory provides predictions that are directly supported by observations.Most importantly, as discussed in the previous section in detail, the observations show a clear correlation between the magnitude of the radial field oscillations and the shear of the wave velocity.While correlation cannot prove causality, this strong and statistically significant correlation is a clear validation of the theory that the radial magnetic field perturbations are produced from the transverse perturbations due to the shear of the wave speed.We also show that the global (over 1 day) velocity shear is sufficient to produce the observed magnitudes of dB R .On a shorter timescale, the shear is in fact much stronger than the minimum requirement.This suggests that the nonlinear evolution is limiting the growth of the radial oscillations, and the local wave shear may result from saturated nonlinear processes.
The sheared circularly polarized waves result in highly correlated oscillations in the radial and perpendicular components of the magnetic field, as confirmed by the right panels of Figure 1.The theory predicts that the shear tends toward a maximum value with increasing radial distance.Indeed, the shear estimated from dB r /B ⊥ is comparable for encounter 1 at 35 R s and encounter 12 at around 14 R s .On the other hand, B r /B ⊥ decreases with increasing radial distance, which makes switchbacks more likely to occur during encounter 1 than encounter 12.The initially symmetric perturbations in B r become asymmetric due to the gradient of the magnetic pressure that pushes the plasma and the frozen-in magnetic field toward the switchbacks where the magnetic field is weaker.This prediction is confirmed both by the asymmetric shape of the B r peaks (narrow, sharp crests in the switchback direction and wide, flat troughs in the background field direction) and by the anticorrelation of the magnetic field magnitude with the plasma density near the switchbacks.The latter prediction is, in fact, the opposite of what the general notion of a frozen-in magnetic field would imply (the plasma and magnetic field move together).The top left panel in Figure 1 shows another clear validation of our theory, which is also confirmed by the analysis of the 6 days of data during encounters 1 and 12. Finally, we predict that the energy associated with the wave velocity shear is gradually depleted, as it is distorting the magnetic field.Indeed, we find that the variation of the wave velocity is smaller than expected from the variations of u r , B r , and density in the hourly average plots.Overall, the theory and observations are in excellent agreement, suggesting that we have identified the crucial mechanism that results in the formation of switchbacks.
There can be several reasons for a transverse gradient of the wave velocity.We showed that even the transition from slow to fast wind from the equator to the pole can create substantial shear.On a smaller scale, the wave speed may vary spatially, for example, due to the presence of active regions, current sheets, or interchange reconnection (Bale et al. 2021) or temporally, for example, due to the breathing of supergranulation (Fargette et al. 2021;Shi et al. 2022).Both spatial and temporal variations will create transverse shear of the wave speed, which will tilt the Alfvén waves and create switchbacks if B r and B ⊥ are comparable.
This paper focuses on explaining the puzzling observations by PSP.At a more fundamental level, the interaction of wave velocity shear with circularly polarized Alfvén waves can play an important role in the physics of the solar wind.The interaction can create mode conversion from Alfvén turbulence to compressive turbulence that can heat and accelerate the solar wind (Akhavan-Tafti et al. 2022).
, compressible turbulence (Squire et al. 2020; Mallet et al. 2021), and radial Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
in Figure 3. On the other hand, the originally constant B r will change to ( ) ¢ B r y , sB y (r − sy) an amplitude dB r = sB ⊥ .

ĈB 2 0Figure 1 .
Figure1.The PSP observations of switchbacks.The panels in the top half are from 2018 November 5, during the first encounter at about 35 R s from the Sun, and the rest are from 2022 May 30 to June 1 during the 12th encounter between 30 and 13.3 R s distance.The top and bottom middle panels show the radial field B r with 0.22 s time cadence containing switchbacks where the black line is inside the blue areas.The panels in the middle show hourly averages of radial velocity u r , magnitude of B r , number density, radial wave speed v = u r + v A , and the Alfvénic Mach number M A = u r /v A .For the 12th encounter, the 24 hr period prior 19:00 UT May 31 is shown, as there are no public plasma data after that.The wave speed varies along the PSP orbit in both cases.The panels on the right show the high-cadence observations of the three components of the magnetic field for half-hour periods.The background is removed by subtracting a 7.4 minute sliding average.During the first encounter, all three components vary with similar amplitudes.For the second encounter, dBR is multiplied by 4 to make its variation similar to the perpendicular oscillations.The correlation coefficients r(dBR, dBT) and r(dBR, dBN) are calculated over a 2 minute sliding window.The light red rectangles highlight where dBR is highly correlated with dBT or dBN.The yellow rectangles highlight anticorrelation, where the signs are opposite.The green rectangles indicate times when B r is approximately constant.The bottom left panel shows one of these times, when dBT and dBN vary with similar amplitudes consistent with roughly circularly polarized Alfvén waves.The top left panel shows BR, the magnetic field magnitude B, the density n (all smoothed over 9.6 s), and the correlation coefficient r(B, n) for a selected short time period with many switchbacks.
, which correspond to the Alfvén wave propagating in the +R direction relative to the plasma.There are only four free dimensionless parameters: the relative strength of the unperturbed guide field ¯B B r (which also determines ¯¯= -= - u v B r r Ato make the wave standing), the plasma beta ¯b = p p B that defines the pressure p, and the two parameters, v 1 /u ⊥ and λ y /λ r , for the wave velocity perturbation (

Figure 2 .
Figure 2. Numerical solutions of sheared circularly polarized Alfvén waves in a double periodic box.Colors show the components of the magnetic field and velocity.Black lines are field lines and streamlines.In the top left panels, the long-wavelength λ y /λ r = 10 shear produces a highly distorted Alfvén wave.Only a part of the domain is shown where the shear is maximal.The parameters are B r = 1, p = 1, v 1 = 0.5 (by perturbing B r ), and time t = 20.In the top middle panels, the comparable wavelength shear λ y /λ r = 1.25 results in a complex spherically polarized Alfvén wave solution.The parameters are B r = 0.5, p = 1, v 1 = 0.5 (by perturbing u r ), and t = 6.6.In the top right panels, the small wavelength shear λ y /λ r = 0.25 results in distorted field lines but no switchbacks.The parameters are B r = 0.5, p = 1, v 1 = 0.2 (by perturbing u r ), and t = 40.Bottom row: cuts along r = 0 in the full domain for the three simulations.Solid lines show the −B r magnetic field component, while the dotted lines are the u r velocity components in the coordinate system moving with the wave.

Figure 3 .
Figure 3. Illustration of how switchbacks form.The top part of each panel shows the evolution of three magnetic field lines (solid black curves with arrows) projected to the r-y plane.The dashed lines follow r = 3π + sy, indicating the amount of shear s = 1.5t that grows linearly with time t.At time t = 0, the original circularly polarized Alfvén wave is shown with wavelength λ r = 2π and width w = (B ⊥ /B r )λ r /π = 0.8 At this time, B r is uniform, and the magnitude of the perpendicular field is B ⊥ = 0.4B r .The wave velocity v = 1.5y shown by the red arrows changes linearly with y.At time t = 1, the field lines are mildly distorted.By time t = 2, the field line folds over and B r changes sign, creating a switchback around the positions where the dashed line intersects the field lines.At t = 3, there is a substantial radial field reversal.The bottom part of each panel shows B r (t) = B r (t = 0) + sB y (solid lines) and the magnetic pressure ( )( ) m = + p B B 2 B r2 2 0 (dashed lines) at the four time instances.Switchbacks occur when the line enters the blue regions.The gray arrows show the gradient of the magnetic pressure that compresses the plasma and magnetic field.The red lines show numerical simulation results for comparable conditions.The switchbacks are narrow peaks between flatter background field regions.The sketch at the bottom of the figure depicts the evolution of shear in a radial outflow.The two streamlines are separated by an angle dα, and the wave velocities are v and v + dv, respectively.The dotted and solid curves show the field line without and with shear, respectively.The shear s grows fast near the starting point from s 0 = 0 to s 1 and s 2 , but it eventually reaches an asymptotic maximum value of (1/v)(dv/dα).Still, due to B ⊥ /B r ∝ r, switchbacks can form at larger radii, as shown by the radially inward-pointing black arrow on the right.

Figure 4 .
Figure4.Formation of switchbacks due to different causes of the wave velocity shear.The top panels show the variation of the radial velocity u r , radial field B r , and density n and the resulting variation in the wave velocity v = u r + V A along y at t = 0.The first three columns have perturbations of the wave velocity due to perturbations in u r , B r , and n, respectively.The fourth column has perturbations in both u r and B r that cancel each other out, so v is constant.The bottom panels show B r (colors) with white field lines in the region with maximum shear at various times.The simulations in the first three panels produce switchbacks, B r < 0 (red stripes), but the fourth does not because there is no shear in v.

Figure 5 .
Figure 5. Formation and propagation of spherically polarized Alfvén waves in the spherically expanding solar wind.The top panels show the three components of the magnetic field and its magnitude in part of the 2D computational domain.The circularly polarized Alfvén waves enter through the left boundary at R = 25 R s .The incoming radial field is perturbed along the Y-direction, which causes a shear in the Alfvén wave speed and the development of spherical polarization.Switchbacks with B r > 0 form at r > 27.5 R s .The total magnetic field (top right) is relatively smooth.The black curve indicates a possible PSP trajectory at r = 31.4R s .The satellite symbol in the top left panel indicates a spot along the trajectory where switchbacks occur.The bottom left panels show the magnetic field and velocity components, as well as the total pressure p + p B , magnetic pressure p B , density n, and correlation coefficient r(B, n) between B and n over a sliding window of 0.3 R s along the trajectory.The virtual satellite is moving +5°longitude per day, similar to PSP near the perihelion of encounter 1.All quantities are comparable to PSP observations during the first encounter.The gradients of the total pressure are small but not zero.Density variations are also substantial and show a strong anticorrelation with the magnetic field magnitude near the switchbacks.The bottom right panels compare the magnetic (black solid lines) and velocity (red dotted lines) perturbations around a switchback.The magnetic field components are converted to Alfvén velocity components:rm = V B

Figure 6 .
Figure 6.Analysis of PSP observations to check the satisfaction of Equation (1).The magnitude of dB R /B ⊥ (black line) is compared with the transverse gradient of the wave speed dv/(rdα) multiplied by the time t = D/v it takes to reach PSP (red line), where D is set to match the averages of the two curves.The left column shows the results when the full time period is used.The right column shows only the bins where the magnitude of the correlation coefficient between dB R and B y is |r(dB R , B y )| > 0.5.The top and bottom rows are for encounters 1 and 12, respectively.The number of bins N, Pearson's correlation r, Spearman's rank correlation r s , and statistical significance (p-value) p are shown in the top left corner of each plot.