1. Introduction
Magnetic reconnection is a fundamental plasma physics process, involving a topological rearrangement of the magnetic field, and accompanied by the conversion of magnetic energy into bulk plasma flow and heat. Reconnection occurs in a wide range of contexts, for example solar flares (Yan et al. Reference Yan2022), at planetary magnetopauses (Paschmann, Øieroset & Phan Reference Paschmann, Øieroset and Phan2013), in magnetic confinement fusion devices (Kadomtsev Reference Kadomtsev1975; Zanini et al. Reference Zanini2020) and other laboratory plasmas (Ji et al. Reference Ji2023) and in current sheets ubiquitous in many space and astrophysical environments (Eriksson et al. Reference Eriksson, Swisdak, Weygand, Mallet, Newman, Lapenta, Wilson III, Turner and Larsen2022, Reference Eriksson, Swisdak, Mallet, Kruparova, Livi, Romeo, Bale, Kasper, Larson and Pulupa2024).
Here, we are specifically motivated by observations of current sheets in the near-Sun solar wind, currently being explored in situ for the first time by NASA’s Parker Solar Probe (PSP) spacecraft. While such sheets are observed essentially ubiquitously (Vasko et al. Reference Vasko, Alimov, Phan, Bale, Mozer and Artemyev2022; Lotekar et al. Reference Lotekar, Vasko, Phan, Bale, Bowen, Halekas, Artemyev, Khotyaintsev and Mozer2022), only a small fraction actually reconnect (Eriksson et al. Reference Eriksson, Swisdak, Weygand, Mallet, Newman, Lapenta, Wilson III, Turner and Larsen2022). In the Alfvénic solar wind (D’Amicis & Bruno Reference D’Amicis and Bruno2015; D’Amicis et al. Reference D’Amicis, Alielden, Perrone, Bruno, Telloni, Raines, Lepri and Zhao2021), the fluctuations are close to Alfvén waves propagating away from the Sun, with velocity fluctuations
$\delta \boldsymbol {u} \approx \pm \delta \boldsymbol {b}$
,
$\delta \boldsymbol {b}=\delta \boldsymbol {B}/\sqrt {4\pi n_{i} m_i}$
being the magnetic field fluctuation in velocity units. An even more pronounced absence of reconnection events has been observed in this type of solar wind (Phan et al. Reference Phan2020; Fargette et al. Reference Fargette2023; Eriksson et al. Reference Eriksson, Swisdak, Mallet, Kruparova, Livi, Romeo, Bale, Kasper, Larson and Pulupa2024), suggesting that the quasi-Alfvénic velocity shear accompanying the magnetic shear across current sheets suppresses the onset of reconnection, as observed in steady-state nonlinear reconnection simulations (Cassak Reference Cassak2011). Here, we develop a linear theory to describe the suppression of reconnection onset by such a flow shear.
Many numerical studies of reconnection begin with a kinetic-scale current sheet: for example, the GEM reconnection challenge problem (Birn et al. Reference Birn2001) has a current sheet with a width comparable to
${\rm d}_i=c/\omega _{pi}$
, where
$\omega _{pi}=(4\pi n_i Z^2 e^2/m_i)^{1/2}$
is the ion plasma frequency, with
$Z=q_i/e$
. With this set-up, reconnection proceeds rapidly from the outset. However, in many of the natural systems mentioned above, reconnection occurs as part of a bursty, two-time-scale process: first a long, quiescent phase in which magnetic energy builds up in thinning current sheets, followed by an extremely rapid disruption as reconnection occurs.
Recently, new models of reconnection onset that explain this property have been developed (Pucci & Velli Reference Pucci and Velli2014; Uzdensky & Loureiro Reference Uzdensky and Loureiro2016; Tolman, Loureiro & Uzdensky Reference Tolman, Loureiro and Uzdensky2018). On large scales, ideal dynamics leads to the progressive ‘thinning’ of the current sheet width
$a$
, for example via the Chapman–Kendall collapse (Chapman & Kendall Reference Chapman and Kendall1963), on a characteristic (ideal) time scale
$T(a)$
. This can be compared with the growth rate of the tearing mode (the linear stage of reconnection),
$\gamma (a)$
: only when
$\gamma (a) T(a) \gtrsim 1$
will reconnection onset occur: at which point, the ideal dynamics is disrupted and the current sheet is usually destroyed. This model has also been applied to turbulence, where
$T$
can be identified as the nonlinear time scale
$T_{{nl}}$
, whose scaling depends on the details of the turbulence (Mallet et al. Reference Mallet, Schekochihin and Chandran2017a
,Reference Mallet, Schekochihin and Chandran
b
; Loureiro & Boldyrev Reference Loureiro and Boldyrev2017a
,Reference Loureiro and Boldyrev
b
; Comisso et al. Reference Comisso, Huang, Lingam, Hirvijoki and Bhattacharjee2018).
To apply this model to the highly sheared current sheets observed by PSP, we need to understand the scalings of both
$\gamma$
and
$T$
in the presence of strong flow shear, neither of which are currently well understood. Here, we focus on the former problem, and develop a new analytic theory for the collisionless tearing mode with significant flow shear,
$\delta u \sim \delta b = \delta B/\sqrt {4\pi n_{i} m_i}$
, determining the growth rate
$\gamma$
as a function of the physical parameters. We find that flow shear strongly suppresses the tearing instability, with the growth rate proportional to
$1-\alpha ^2$
, where
$\alpha = \delta u/\delta b$
: as
$\alpha \to 1$
, corresponding to Alfvénic flow shear, the growth rate of the tearing mode vanishes (for
$\alpha \gt 1$
, we would have instead the ideal Kelvin–Helmholtz instability, whose growth rate is proportional to
$\sqrt {\alpha ^2-1}$
: we will not comment further on this instability here). Finally, we discuss how our results might apply to the PSP observations.
1.1. Tearing mode without shear flow
To compare with the new results we will find in our calculation, we first present an overview of the collisionless tearing mode scalings without equilibrium flow shear.Footnote
1
We assume that the current sheet normal is in the
$\hat {\boldsymbol {x}}$
direction, with reconnecting field
$v_{\mathrm {A}y}=\delta b=\delta B / \sqrt {4\pi n_{0i} m_i}$
, the wavenumber
$k$
of the tearing mode both in the
$\hat {\boldsymbol {y}}$
direction and a strong constant guide field
$B_0\gg \delta B$
pointing in the
$\hat {\boldsymbol {z}}$
direction. The growth rate is given by
\begin{align} \frac {\gamma _0 a}{v_{\mathrm {A}y}} \sim \begin{cases} k\Delta ' {\rm d}_e \rho _s\sqrt {1+\tau /Z}, &\Delta '\delta _{in}\ll 1 \\ kd_e^{1/3}\rho _s^{2/3}\sqrt {1+\tau /Z}, &\Delta '\delta _{in}\gg 1, \end{cases} \end{align}
where
$\rho _s = \sqrt {ZT_{0e}/m_i}/\Omega _i$
is the ion sound radius and
${\rm d}_e=c/\omega _{pe}$
is the electron inertial length, with
$Z=q_i/e$
,
$\tau =T_{0i}/T_{0e}$
,
$\omega _{pe}=(4\pi n_{0e}e^2/m_e)^{1/2}$
the electron plasma frequency and
$\Omega _i=ZeB_0/m_i c$
the ion gyrofrequency (Zocco & Schekochihin Reference Zocco and Schekochihin2011). Here,
$\Delta '$
is a measure of the ideal discontinuity (see § 3 for details), with
$\Delta '\gt 0$
required for instability, while
$\delta _{in}\ll a$
is the inner layer width over which the microphysics becomes important. The explicit form of
$\delta _{in}$
depends on the regime, but is not needed here. We refer the reader to Zocco & Schekochihin (Reference Zocco and Schekochihin2011) for further details. The parameter
$\Delta '$
depends on
$k$
in a way that depends on the specific equilibrium profile (Boldyrev & Loureiro Reference Boldyrev and Loureiro2018): for
$ka\ll 1$
, we have
$\Delta 'a \propto (k a)^{-n}$
with
$n=1$
for a Harris-type equilibrium (Harris Reference Harris1962) with
$f(x/a) = \tanh (x/a)$
and
$n=2$
for
$f(x/a) = \sin (x/a)$
. For
$\Delta '\delta _{in}\ll 1$
, the growth rate
$\gamma _0$
decreases with
$k$
, while at
$\Delta '\delta _{in}\gg 1$
,
$\gamma _0$
increases with
$k$
. The maximum growth rate and wavenumber at which it is attained may be found by equating the two expressions in (1.1)
\begin{equation} \frac {\gamma _{0{tr}}a}{v_{\mathrm {A}y}} \sim \frac {\displaystyle {\rm d}_e^{1/3+2/(3n)} \rho _s^{2/3+1/(3n)}\sqrt {1+\tau /Z}}{\displaystyle a^{1+1/n}}, \quad k_{0{tr}} a \sim \frac {\displaystyle {\rm d}_e^{2/(3n)}\rho _s^{1/(3n)}}{\displaystyle a^{1/n}}. \end{equation}
For
$n=1$
the growth rate in the
$\Delta '\delta _{in}\ll 1$
case does not depend on
$k$
, and so the growth rate is the same for all
$k\gt k_{0{tr}}$
: for
$n=2$
this is not the case. In the calculation that follows, we will find that these scalings are strongly affected by the presence of even relatively modest flow shear.
2. Equations
We will use the kinetic reduced electron heating model (KREHM) equations (Zocco & Schekochihin Reference Zocco and Schekochihin2011), in the collisionless limit. These equations are derived from gyrokinetics (and thus assume low-frequency fluctuations
$\omega \ll \Omega _i$
and a strong guide field,
$\delta B/B_0\ll 1$
) in the limit of small
$\beta _e = 8\pi n_{0e}T_{0e}/B_0^2$
, and, assuming no variations in the
$\hat {\boldsymbol {z}}$
direction, may be written
\begin{align} \frac {d}{{\rm d}t}(1-\hat {\Gamma }_0)\varPhi &=-\frac {1}{2}\rho _i^2\left \{{\varPsi },{\nabla _\perp ^2\varPsi }\right \}\!, \end{align}
\begin{align} \frac {d}{{\rm d}t}(\varPsi - d_e^2 \nabla _\perp ^2\varPsi ) &= \left \{{\varPsi },{\frac {Z}{\tau }(1-\hat {\Gamma }_0)\varPhi - \frac {c}{eB_0}\delta T_{\parallel e}}\right \} \!,\end{align}
\begin{align} \frac {{\rm d}{g_e}}{{\rm d}t} + \frac {v_\parallel }{v_A}\left \{{\varPsi },{g_e - \frac {\delta T_{\parallel e}}{T_{0e}}F_{0e}}\right \} &= -\frac {1}{\Omega _i}\left (1-\frac {2v_\parallel ^2}{v_{{th}e}^2}\right )F_{0e}\left \{{\varPsi },{\nabla _\perp ^2\varPsi }\right \}\!. \\[8pt]\nonumber\end{align}
where the Poisson bracket is
$\{f,g\}=\hat {\boldsymbol {z}} \cdot (\nabla _\perp \,f \times \nabla _\perp g)$
, and the ion gyroradius is
$\rho _i= v_{{th}i}/\Omega _i$
, with
$v_{{th}i}=\sqrt {2T_{0i}/m_i}$
the ion thermal speed . Here,
$B_0$
is again the (strong) guide magnetic field in the out-of-plane
$\hat {\boldsymbol {z}}$
direction,
$F_{0e}$
is the equilibrium Maxwellian electron distribution function, and
\begin{equation} \frac {\rm d}{{\rm d}t} = \frac {\partial }{\partial t} + \left \{{\varPhi },{\ldots }\right \}. \end{equation}
The equations have been written in different variables to Zocco & Schekochihin (Reference Zocco and Schekochihin2011)
\begin{equation} \varPhi = \frac {c}{B_0}\phi, \quad \varPsi = -\frac {A_\parallel }{\sqrt {4\pi m_i n_{0i}}}, \end{equation}
chosen to be notationally similar to reduced magnetohydrodynamics (RMHD):
$\phi$
is the electric potential and
$A_\parallel$
is the parallel magnetic vector potential, while the perpendicular (to
$\hat {\boldsymbol {z}})$
magnetic field (in velocity units) is
$\boldsymbol {b}_\perp = \hat {\boldsymbol {z}}\times \nabla _\perp \varPsi$
and the
$\boldsymbol {E}\times \boldsymbol {B}$
velocity is
$\boldsymbol {u}_{E\times B} = \hat {\boldsymbol {z}}\times \nabla _\perp \varPhi$
. The operator
$\hat {\Gamma }_0$
is the inverse Fourier transform of
\begin{equation} \Gamma _0(k_\perp ^2\rho _i^2/2) = I_0(k_\perp ^2\rho _i^2/2)e^{-k_\perp ^2\rho _i^2/2}, \end{equation}
where
$I_0$
is the modified Bessel function: at large and small scales, we have
\begin{equation} 1-\hat {\Gamma }_0 \approx \begin{cases} -\frac {\rho _i^2}{2}\nabla _\perp ^2, \quad &\rho _i^2\nabla _\perp ^2\ll 1\\[3pt] 1, \quad &\rho _i^2\nabla _\perp ^2\gg 1. \end{cases} \end{equation}
In (2.1) we have already substituted
\begin{equation} \frac {\delta n_e}{n_{0e}} = - \frac {2}{\rho _i^2\Omega _i}(1-\hat {\Gamma }_0)\varPhi = \frac {\delta n_i}{n_{0i}},\end{equation}
for the density fluctuations. Equation (2.3) evolves the reduced parallel electron distribution function
$g_e$
, defined in terms of the perturbed parallel electron distribution function
$\delta f_e$
as
\begin{equation} g_e = \delta f_e - \left [\frac {\delta n_e}{n_{0e}} + \frac {2v_\parallel u_{\parallel e}}{v_{{th}e}^2}\right ]F_{0e}. \end{equation}
The parallel electron temperature fluctuation is given by the second moment of
$g_e$
\begin{equation} \frac {\delta T_{\parallel e}}{T_{0e}} = \frac {1}{n_{0e}}\int d^3\boldsymbol {v} \frac {2v_\parallel ^2}{v_{{th}e}^2}g_e. \end{equation}
The KREHM is designed to be a set of equations appropriate for studying reconnection: the equations contain a rigorous treatment of electron heating via the electron kinetic equation (2.3), the dispersion at the ion scales (
$\rho _i$
and
$\rho _s = \rho _i\sqrt {Z/2\tau }$
), important for achieving fast reconnection (Shay et al. Reference Shay, Drake, Rogers and Denton2001) and also the flux unfreezing at the electron inertial-scale
${\rm d}_e$
. This model cannot be expected to apply to all the current sheets observed in the solar wind: in particular, the assumptions that
$\beta \ll 1$
and
$\delta B/B_0\ll 1$
are often not satisfied, nor do we attempt to model sheared parallel flow or density gradients across the current sheet. However, it will allow us to make progress in understanding reconnection onset in the near-Sun solar wind.
2.1. Equilibrium and linearised equations
Our chosen equilibrium is
\begin{equation} \varPhi _0 = \alpha \varPsi _0, \quad b_{0y} = v_{{A}y} f(x/a) = \partial _x \varPsi _0, \end{equation}
where we will be especially interested in the case where the shear flow in the
$\hat {\boldsymbol {y}}$
direction is comparable to the reconnecting magnetic field,
$\alpha \sim 1$
. As in § 1.1, we have the current sheet normal in the
$\hat {\boldsymbol {x}}$
direction and both the reconnecting field and wavenumber in the
$\hat {\boldsymbol {y}}$
direction. We also assume that the equilibrium length scale is large compared with ion and electron scales,
$a\gg \rho _i\sim \rho _s\gg {\rm d}_e$
. We have also implicitly assumed that
$u_{0y},b_{0y}\ll v_A$
, so we are limited to guide-field reconnection. Relaxing this assumption is not possible with the KREHM equations, which assume that there are no flows on the same level as the background (guide) magnetic field
$B_0$
. Linearising (2.1)–(2.3) and assuming fluctuations of the form
\begin{align} \delta \varPhi = \varPhi (x) \exp (iky + \gamma t),\quad \delta \varPsi = \varPsi (x) \exp (iky + \gamma t), \end{align}
and that
$k^{-1}\gg \rho _i, {\rm d}_e$
, we obtain
\begin{align} &(\gamma +i\alpha k v_{{A}y} f)(1-\hat {\Gamma }_0)\varPhi + \frac {1}{2}i\alpha k v_{{A}y} \rho _i^2f''\varPhi = - \frac {1}{2}ikv_{{A}y} f\rho _i^2 \left [\varPsi '' - k^2\varPsi - \frac {f''}{f}\varPsi \right ], \end{align}
\begin{align} &(\gamma + i \alpha k v_{{A}y} f)(\varPsi -{\rm d}_e^2\varPsi '')= ikv_{{A}y} f\left [\left (1+\frac {Z}{\tau }(1-\hat {\Gamma }_0)\right )\varPhi - \frac {c}{eB_0}\delta T_{\parallel e}\right ], \end{align}
\begin{align} &(\gamma + i\alpha k v_{{A}y} f)g_e + ikv_{{A}y} f \frac {v_\parallel }{v_A}\left (g_e - \frac {\delta T_{\parallel e}}{T_{0e}}F_{0e}\right )=\nonumber \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad -\frac {1}{\Omega _i}ikv_{{A}y} f\left (1-\frac {2v_\parallel ^2}{v_{{th}e}^2}\right )F_{0e}\left [\varPsi '' - k^2\varPsi - \frac {f''}{f}\varPsi \right ]. \\[8pt]\nonumber\end{align}
We first solve the linearised kinetic equation (2.15) for
$g_e$
, integrating according to (2.10) to find
\begin{equation} \frac {\delta T_{\parallel e}}{T_{0 e}} = -\frac {2}{\Omega _i}\frac {v_{{A}}}{v_{{th}e}}\frac {Z(\zeta )+\zeta Z'(\zeta )}{Z'(\zeta )}\left (\varPsi '' - k^2\varPsi - \frac {f''}{f}\varPsi \right ), \end{equation}
where
\begin{equation} \zeta = \left [\frac {i\gamma }{kv_{{A}y} f} -\alpha \right ]\frac {v_{{A}}}{v_{{th}e}}, \end{equation}
and
$Z(\zeta )$
is the plasma dispersion function, with
$Z'(\zeta )=-2(1+\zeta Z(\zeta ))$
. Using (2.13),
\begin{equation} \frac {c}{eB_0}\delta T_{\parallel e} = -(G-1)\frac {Z}{\tau } (1-\hat {\Gamma }_0)\varPhi, \end{equation}
where we have dropped a term since
$\rho _s^2 f''/f\ll 1$
, and
\begin{equation} G=2\left (\zeta ^2 -\frac {1}{Z'(\zeta )}\right ). \end{equation}
For
$\zeta \to \infty$
,
$G\to 3$
(adiabatic electrons), while as
$\zeta \to 0$
,
$G\to 1$
(isothermal electrons). Inserting (2.18) into (2.14), we obtain
\begin{equation} (\gamma + i \alpha k v_{{A}y} f)(\varPsi -d_e^2\varPsi '')= ikv_{{A}y} f\left [\left (1+G\frac {Z}{\tau }(1-\hat {\Gamma }_0)\right )\varPhi \right ]. \end{equation}
We will solve (2.13) and (2.20) in the ‘outer region’
$x\sim a$
and in the ‘inner region’ close to
$x=0$
. Compared with the resistive magnetohydrodynamics (MHD) case (Chen & Morrison Reference Chen and Morrison1989), the main differences are, first, that we have the electron inertia term involving
${\rm d}_e$
allowing reconnection instead of resistivity, and second, the terms involving
$1-\hat {\Gamma }_0$
which encode the ion-scale behaviour. Since there are two microscales in the problem,
$\rho _s$
(or
$\rho _i$
) and
${\rm d}_e\ll \rho _s$
, there will be nested ion and electron boundary layers. Because
$\gamma$
is real, the real and imaginary parts of the eigenmodes will be even and odd respectively around
$x=0$
, and it will turn out that the imaginary part is small compared with the real part.
3. Outer region
Here,
$x\sim a \sim (f'/f)^{-1} \sim k^{-1} \gg \rho _s, {\rm d}_e$
. On these scales
\begin{equation} 1-\hat {\Gamma }_0 \approx -\frac {1}{2}\rho _i^2 \nabla _\perp ^2 \ll 1, \end{equation}
so it may be neglected in (2.20) (but not in 2.13, where all the terms are at least this small). Assuming
$\gamma \ll kv_{\mathrm {A}y}$
, we may also neglect the growth terms. (2.20) becomes
\begin{equation} \varPhi =\alpha \varPsi, \end{equation}
and inserting this into (2.13), we obtain
\begin{equation} (1-\alpha ^2)f [\varPsi '' - k^2\varPsi - (\,f''/f)\varPsi ]=0, \end{equation}
so that the outer solution for
$\varPsi$
is the same as in the MHD tearing mode. As
$x\to 0$
,
$\partial _x \gg k$
and
$f\approx x/a$
, and of the outer equation all we are left with is
$\varPsi ''=0$
, whence the real and imaginary parts are
\begin{align}varPsi _R &\to \varPsi _{\infty R}\left(1+ \frac {1}{2}\Delta '|x|\right), \quad x\to 0,\nonumber \\ \varPsi _I &\to \varPsi _{\infty I}\left(\frac {1}{2}\Delta ' x \pm 1\right), \quad x\to 0, \\[8pt]\nonumber\end{align}
defining
$\Delta ' = {[\varPsi ']_{-0}^{+0}}/{\varPsi (0)}$
, the discontinuity in the outer solution’s magnetic field.
4. Inner region
In the inner region, of width
$\delta _{in}\ll a$
, the ion- and electron-scale effects become important. Here,
$x\ll a$
,
$f\approx x/a$
and
$\partial ^2/\partial x^2 \gg k^2, f''/f$
. Defining
\begin{equation} \delta = \frac {\gamma a}{kv_{{A}y}}, \end{equation}
we obtain the inner region equations
\begin{align} \left (\delta + i\alpha x\right )(1-\hat {\Gamma }_0)\varPhi &= - \frac {1}{2}ix\rho _i^2\varPsi '', \end{align}
\begin{align} \left (\delta + i\alpha x\right )(\varPsi -{\rm d}_e^2\varPsi '')&=ix\left [\varPhi + G\frac {Z}{\tau }(1-\hat {\Gamma }_0)\varPhi \right ]. \\[8pt]\nonumber\end{align}
We need to deal with the non-local operator
$1-\hat {\Gamma }_0$
: substituting (4.2) into (4.3) we obtain
\begin{equation} \left (\delta + i\alpha x\right )(\varPsi -{\rm d}_e^2\varPsi '')=ix\varPhi + x^2 G\rho _s^2\frac {\varPsi ''}{\delta + i\alpha x}, \end{equation}
but we still need to calculate
$1-\hat {\Gamma }_0$
to solve (4.2) itself. One simplification is to take cold ions: as
$\rho _i^2\partial _x^2\to 0$
,
$1-\hat {\Gamma }_0\to -(1/2)\rho _i^2\partial _x^2$
and we recover a MHD-like version of (4.2). Instead, following Pegoraro & Schep (Reference Pegoraro and Schep1986) and more recently Zocco & Schekochihin (Reference Zocco and Schekochihin2011), we incorporate the hot-ion response (non-rigorously) by using the Padé approximant
\begin{equation} 1-\hat {\Gamma }_0 \approx \frac {-(1/2) \rho _i^2\partial _x^2}{1-(1/2) \rho _i^2\partial _x^2}. \end{equation}
Then, (4.2) becomes
\begin{equation} \varPhi '' = \frac {ix\varPsi ''}{\delta + i\alpha x} - \frac {1}{2}i\rho _i^2\left (\frac {x\varPsi ''}{\delta + i\alpha x}\right )''. \end{equation}
We now rescale our equations by the inner length scale
$\delta _{in}$
, to be determined later: writing
\begin{equation} \xi = \frac {x}{\delta _{in}}, \quad \lambda = \frac {\delta }{\delta _{in}}, \quad \epsilon = \frac {d_e}{\rho _s}\ll 1, \end{equation}
and with
$'$
now denoting differentiation by
$\xi$
, (4.6) and (4.4) become
\begin{align} \varPhi '' = \frac {i\xi \varPsi ''}{\lambda + i\alpha \xi } - \frac {1}{2}i\frac {\rho _i^2}{\delta _{in}^2}\left (\frac {\xi \varPsi ''}{\lambda + i\alpha \xi }\right )'', \end{align}
\begin{align} (\lambda + i\alpha \xi )\varPsi - i \xi \varPhi = \frac {\rho _s^2}{\delta _{in}^2}\left [G\xi ^2 + (\lambda + i\alpha \xi )^2\epsilon ^2\right ]\frac {\varPsi ''}{\lambda + i\alpha \xi }, \\[8pt]\nonumber\end{align}
where the argument of
$G(\zeta )$
(see equation (2.19)) is
\begin{equation} \zeta = \frac {1}{\sqrt {2}}\epsilon \left (\frac {i\lambda }{\xi } - \alpha \right ). \end{equation}
The electron inertia term in (4.9) becomes important when
$\xi \sim \lambda \epsilon$
(
$x\sim \delta \epsilon$
): this is also the scale below which the electrons are no longer isothermal, i.e. when
$\zeta \sim 1$
so that
$G(\zeta )$
starts to differ from
$1$
. We are free to choose
$\delta _{in}$
to be of the same order as the ion scale,
$\delta _{in}\sim \rho _s$
, and anticipate
$\lambda \sim \epsilon \ll 1$
.
4.1. Ion layer
We first solve the equations on the ion scales,
$\xi \sim 1$
. Since
$\lambda \ll \alpha \sim 1$
, to lowest order the equations are
\begin{align} \varPhi '' &= \frac {1}{\alpha }\left (\varPsi '' - \frac {1}{2}\frac {\rho _i^2}{\delta _{in}^2}\varPsi ''''\right ), \end{align}
\begin{align} \alpha \varPsi - \varPhi &= -\frac {1}{\alpha }\frac {\rho _s^2}{\delta _{in}^2}\varPsi ''. \\[8pt]\nonumber\end{align}
The solution that matches onto the outer layer solution (3.4) is
\begin{align} \varPsi _{i} &= \varPsi _\infty \left (1+\frac {1}{2}\Delta '\delta _{in}\xi \right ) + C_i e^{-\xi }, \end{align}
\begin{align} \varPhi _i &= \alpha \varPsi _\infty \left (1+\frac {1}{2}\Delta '\delta _{in}\xi \right ) + \frac {1}{\alpha }C_i e^{-\xi }\left [1-\frac {1-\alpha ^2}{1+Z/\tau }\right ], \\[8pt]\nonumber\end{align}
where we have set
\begin{equation} \delta _{in} = \frac {\rho _s\sqrt {1+\tau /Z}}{\sqrt {1-\alpha ^2}}. \end{equation}
It is worth noticing that the solution for
$\varPhi _i$
is very different from the tearing mode without shear as
$\xi \to \infty$
, to match the outer solution (3.4), which is also very different from its equivalent with
$\alpha =0$
, since the flow shear dominates over the growth rate on the large equilibrium scales. Moreover, it is clear that (4.11) depends strongly on the ratio of
$\rho _i$
to
$\rho _s$
, i.e. on
$\tau$
. These two facts will conspire to change the
$\tau$
-dependence of the growth rate from the case with no shear flow (1.1). As
$\xi \to 0$
, we have
\begin{align} \varPsi _i &\to \varPsi _\infty + C_i + \left (\frac {1}{2}\Delta '\delta _{in}\varPsi _\infty -C_i\right )\xi, \end{align}
\begin{align} \varPhi _i &\to \alpha \varPsi _\infty +\frac {1}{\alpha }C_i \left [1-\frac {1-\alpha ^2}{1+Z/\tau }\right ] + \left (\frac {1}{2}\alpha \Delta '\delta _{in}\varPsi _\infty -\frac {1}{\alpha }C_i\left [1-\frac {1-\alpha ^2}{1+Z/\tau }\right ]\right )\xi .\\[8pt]\nonumber \end{align}
4.2. Electron layer
We rescale the equations again to find the solution on electron scales, defining
\begin{equation} y = \frac {\xi }{\lambda \epsilon } = \frac {x}{\delta \epsilon }. \end{equation}
The equations are
\begin{align} \varPhi '' &= \frac {i\epsilon y \varPsi ''}{1+i\alpha \epsilon y} - \frac {1}{\lambda ^2\epsilon ^2}\frac {1-\alpha ^2}{1+Z/\tau } \left (\frac {i\epsilon y \varPsi ''}{1+i\alpha \epsilon y}\right )'', \end{align}
\begin{align} \lambda ^2(1+i\alpha \epsilon y)\varPsi -i\lambda ^2\epsilon y \varPhi &= \frac {1-\alpha ^2}{1+\tau /Z}\left [(1+i\alpha \epsilon y)^2+Gy^2\right ]\frac {\varPsi ''}{1+i\alpha \epsilon y}. \\[8pt]\nonumber\end{align}
On these scales, the shear terms are small compared with the growth terms. Dividing (4.20) by
$y$
, differentiating twice and substituting for
$\varPhi ''$
using (4.19), we obtain an equation for
$\varPsi$
\begin{align} \lambda ^2\left [\left ({1}/{y}+i\alpha \epsilon \right )\varPsi \right ]'' &+ \lambda ^2\epsilon ^2\frac {\varPsi ''}{1/y+i\alpha \epsilon }\nonumber \\&= \frac {1-\alpha ^2}{1+\tau /Z}\left [\left (G+\frac {\tau }{Z}+\frac {1}{y^2}\left (1+i\alpha \epsilon y\right )^2\right )\frac {\varPsi ''}{1/y+i\alpha \epsilon }\right ]''. \end{align}
Anticipating
$\lambda \sim \epsilon$
, we can now solve order by order. At lowest order, using (4.20), we have
$\varPsi _0''=0$
, and so from (4.19) we also have
$\varPhi _0''=0$
. To match the solution at larger scales, the lowest-order solution must be real and even, and we have
\begin{equation} \varPsi _0=\varPsi _{0e}, \quad \varPhi _0 = \varPhi _{0e}, \end{equation}
both constants. At first order,
$\varPsi _1''=0$
, and we take
$\varPsi _1=0$
: we can absorb any even constant piece into the zeroth-order solution, and since the solution is even and real to lowest order overall, any term linear in
$y$
must appear at order
$\lambda \epsilon$
or higher. At second order, we obtain
\begin{equation} \varPsi _2'' = \frac {\lambda ^2(1+\tau /Z)}{1-\alpha ^2}\varPsi _{0e} \frac {1}{(G+\tau /Z)y^2+1}, \end{equation}
so that
\begin{equation} \varPsi _2 = \frac {\lambda ^2\sqrt {1+\tau /Z}}{1-\alpha ^2}\varPsi _{0e} \int _0^y {\rm d}z \int _0^z\frac {{\rm d}u\sqrt {1+\tau /Z}}{(G+\tau /Z)u^2+1}. \end{equation}
As
$y\to \infty$
, we then have (in terms of
$\xi$
)
\begin{equation} \varPsi _e\to \varPsi _{0e}\left [1 + \frac {\lambda \sqrt {1+\tau /Z}}{\epsilon (1-\alpha ^2)}I_G\xi - \frac {\lambda ^2\sqrt {1+\tau /Z}}{1-\alpha ^2}\log (\xi )\right ], \end{equation}
where
\begin{equation} I_G = \int _0^\infty \frac {{\rm d}y\sqrt {1+\tau /Z}}{(G+\tau /Z)y^2+1}. \end{equation}
If we had assumed isothermal electrons (
$G=1$
),
$I_G=\pi /2$
. The solution for
$\varPhi$
can be found by integrating (4.19) twice
\begin{equation} \varPhi = \varPhi _{0e}- \frac {1}{\lambda ^2\epsilon ^2}\frac {1-\alpha ^2}{1+Z/\tau } \left (\frac {i\epsilon y \varPsi ''}{1+i\alpha \epsilon y}\right )+i\epsilon \int _0^y {\rm d}z \int _0^z {\rm d}u \frac { u \varPsi ''}{1+i\alpha \epsilon u}. \end{equation}
The second term on the right-hand side, resulting from the small-scale asymptotic of
$1-\hat {\Gamma }_0$
, is dominant for
$y\sim 1$
, but always decays for
$y\gg 1$
, since
$\varPsi ''$
(e.g.
$\varPsi _2''$
) is strongly peaked on the electron scales. The final term on the right-hand side does produce terms in the solution for
$\varPhi$
that do not decay as
$y\to \infty$
: however, note that they are still a factor
$\epsilon$
smaller than the corresponding terms in
$\varPsi$
. Inserting
$\varPsi _2''$
into (4.27), the lowest-order piece is, as
$y\to \infty$
and in terms of
$\xi$
,
\begin{align} \varPhi _e \to\, \varPhi _{0e}&+i\epsilon \int _0^y {\rm d}z \int _0^z {\rm d}u { u \varPsi _2''} \to \varPhi _{0e}+\frac {i\lambda (1+\tau /Z)}{1-\alpha ^2}\varPsi _{0e} \left [\xi \log (\xi )-\xi + \text {const.}\right ] \nonumber\\&+ \frac {i\lambda \varPsi _{0e}}{1+Z/\tau }\frac {1}{\xi }, \end{align}
so the non-constant piece of
$\varPhi _e$
is
$O(\lambda )$
for
$\xi \sim 1$
.
Finally, if we were to solve to third order for
$\varPsi$
, we could determine the odd part of the eigenmode in terms of the even one, and thus determine the small imaginary part of the solution: because this appears at third order, the linear term that appears in the asymptotic solution as
$y\to \infty$
is
$\sim \lambda \xi$
, showing that the imaginary part of the eigenmode is a factor of around
$\lambda$
smaller than the real part. However, this is not necessary to obtain the growth rate.
5. Matching and dispersion relation
We can now match the ion and electron solutions. We match the constant and linear terms from
$\xi \to 0$
from the ion solution (4.16 and 4.17) and
$y\to \infty$
in the electron solution (4.25 and 4.28), assuming
$\Delta '\delta _{in}\sim 1$
. From the constant terms of
$\varPsi$
, we obtain
\begin{equation} \varPsi _{0e}= \varPsi _\infty + C_i. \end{equation}
As we found above, at lowest order the leading non-constant terms of
$\varPhi _e$
are
$O(\lambda )$
, and so to lowest order, the linear term in (4.17) must be zero
\begin{equation} \frac {1}{2}\alpha \Delta '\delta _{in}\varPsi _\infty -\frac {1}{\alpha }C_i\left [1-\frac {1-\alpha ^2}{1+Z/\tau }\right ]=0, \end{equation}
whence
\begin{equation} C_i = \frac {\frac {1}{2}\alpha ^2\Delta '\delta _{in}\varPsi _\infty (1+Z/\tau )}{Z/\tau + \alpha ^2}. \end{equation}
The
$O(\lambda )$
terms of
$\varPhi _e$
in (4.28) can in principle be matched with the next-order solution to
$\varPhi _i$
(which we have not calculated), providing a small correction to our results, but we do not attempt this here. Finally, matching the linear terms for
$\varPsi _i$
as
$\xi \to 0$
(4.16) and
$\varPsi _e$
as
$y\to \infty$
(4.25)
\begin{equation} \frac {1}{2}\Delta '\delta _{in}\varPsi _\infty -C_i = \frac {\lambda \sqrt {1+\tau /Z}}{\epsilon (1-\alpha ^2)}I_G\varPsi _{0e}. \end{equation}
Inserting (5.1) and (5.3), we obtain the dispersion relation
\begin{equation} \lambda = \frac {\frac {1}{2}\Delta '\delta _{in}\epsilon (1-\alpha ^2)^2Z/\tau }{Z/\tau + \alpha ^2 +\frac {1}{2}\alpha ^2\Delta '\delta _{in}(1+Z/\tau )}\frac {1}{I_G\sqrt {1+\tau /Z}}. \end{equation}
5.1. Growth rate
Using the definitions of
$\lambda$
(4.7) and
$\delta$
(4.1), the growth rate is
\begin{equation} \frac {\gamma a}{v_{{A}y}} = \frac {k\Delta '\delta _{in}^2\epsilon (1-\alpha ^2)^2}{2 I_G \sqrt {1+\tau /Z}(1+\alpha ^2\tau /Z+\frac {1}{2}\alpha ^2\Delta '\delta _{in}(1+\tau /Z))} .\end{equation}
Inserting
$\delta _{in}$
(4.15), we find that
\begin{equation} \frac {\gamma a}{v_{{A}y}}\sim \begin{cases} k\Delta ' \rho _s {\rm d}_e \frac {\displaystyle (1-\alpha ^2)\sqrt {1+\tau /Z})}{\displaystyle 2I_G(1 + \alpha ^2\tau /Z)}, &\Delta '\delta _{in} \ll 1\\[10pt] k {\rm d}_e \frac {\displaystyle (1-\alpha ^2)^{3/2}}{\displaystyle \alpha ^2 I_G(1+\tau /Z)}, &\Delta '\delta _{in} \gg 1. \end{cases} \end{equation}
The growth rate for
$\Delta '\delta _{in}\ll 1$
has the same scaling with
$d_e$
and
$\rho _s$
as the no-flow scaling (1.1), but for
$\Delta '\delta _{in}\gg 1$
, we have obtained the surprising result that the growth rate is independent of
$\rho _s$
: in the no-flow case, we had
$\gamma _{0} \propto \rho _s^{2/3}{\rm d}_e^{1/3}$
. Note that the growth rate for
$\Delta '\delta _{in}\gg 1$
does not match smoothly onto the no-flow scaling (1.1): this is because the expansion in the ion region is invalidated for
$\alpha \ll \lambda$
(see § 5.2 below).
The transition between these scalings occurs where they match. Inserting
$\Delta 'a \propto 1/(ka)^n$
(see § 1.1), the transition occurs roughly at
\begin{equation} k_{{tr}} a \sim (\rho _s/a)^{1/n} \frac {\alpha ^{2/n}}{(1-\alpha ^2)^{1/2n}}\left (\frac {(1+\tau /Z)^{3/2}}{1+\alpha ^2\tau /Z}\right )^{1/n}, \end{equation}
for which the growth rate is of order
\begin{equation} \frac {\gamma _{{tr}} a}{v_{{A}y}} \sim \frac {\rho _s^{1/n}d_e}{a^{1+1/n}}\frac {(1-\alpha ^2)^{3/2-1/2n}}{\alpha ^{2-2/n}}\frac {(1+\tau /Z)^{3/2n-1}}{(1+\alpha ^2\tau /Z)^{1/n}}. \end{equation}
For
$n=2$
, the growth rate increases with
$k$
for
$\Delta '\delta _{in} \gg 1$
and decreases with
$k$
for
$\Delta '\delta _{in}\ll 1$
, so that
$\gamma _{{tr}}$
is also the maximum growth rate, occuring uniquely at
$k_{{tr}}$
. For the
$n=1$
case,
$\gamma$
is independent of
$k$
for
$\Delta '\delta _{in}\ll 1$
, and equal to
$\gamma _{{tr}}$
for all
$k\gt k_{{tr}}$
.
This differs in several ways from the growth rate for
$\alpha =0$
(1.1). First, the growth rate scales strongly with
$1-\alpha ^2$
: as
$\alpha \to 1$
, the growth rate vanishes, as is the case for the resistive MHD tearing mode with shear (Hofman Reference Hofman1975; Chen & Morrison Reference Chen and Morrison1989; Boldyrev & Loureiro Reference Boldyrev and Loureiro2018; Shi et al. Reference Shi, Artemyev, Velli and Tenerani2021a
), where
$\gamma _{{MHD}}\propto (1-\alpha ^2)^{1/2}$
: thus the growth rate of the tearing mode is more strongly suppressed in a collisionless plasma than in resistive MHD.Footnote
2
Second, the growth rate depends differently on
$\tau$
. For
$\tau \ll 1$
, the growth rate becomes independent of
$\tau$
, similarly to the tearing mode without flow shear. In contrast, for
$\tau \gg 1$
, the maximum growth rate is proportional to
$\tau ^{(1/2n)-1}$
; the opposite dependency to the case without flow shear, for which
$\gamma _{0{tr}}\propto \tau ^{1/2}$
for
$\tau \gg 1$
. Finally, in general the growth rate also depends differently on the scales
$\rho _s$
and
$d_e$
than in the no-flow case: without shear flow,
$\gamma _{0{tr}} \propto {\rm d}_e^{1/3+2/3n}\rho _s^{2/3+1/3n}$
, but with shear flow,
$\gamma _{{tr}}\propto {\rm d}_e \rho _s^{1/n}$
. For
$n=1$
, these scalings coincide, but for
$n=2$
,
$\gamma _{{tr}}/\gamma _{0{tr}}\propto \epsilon ^{1/3}$
, where
$\epsilon ={\rm d}_e/\rho _s\ll 1$
.
5.2. Validity and the small shear flow limit
For the expansion in the ion region to be valid, we must have
$\lambda \ll \alpha$
. For
$\Delta '\delta _{in} \ll 1$
, this is easily satisfied, and indeed, the growth rate in this case smoothly joins onto the no-flow growth rate (1.1) as
$\alpha \to 0$
. For
$\Delta '\delta _{in}\gg 1$
, the inequality is
\begin{equation} \frac {\epsilon (1-\alpha ^2)^2}{\alpha ^2 (1+\tau /Z)^{3/2}} \ll \alpha, \end{equation}
or
$\alpha$
above a critical
$\alpha _c$
defined by
\begin{equation} \frac {\alpha _c}{(1-\alpha _c^2)^{2/3}} = \epsilon ^{1/3}(1+\tau /Z)^{-1/2}. \end{equation}
For
$\epsilon \ll 1$
,
$\alpha _c \sim \epsilon ^{1/3}$
. Taking
$\beta _e\approx 0.1$
in the low-
$\beta$
solar wind and corona,
$\epsilon \approx 0.1$
and
$\alpha _c \approx 0.5$
. Thus, even a modest shear flow (far enough from
$\alpha =1$
that
$1-\alpha ^2$
is not small) can affect the growth rate scalings with
${\rm d}_e$
and
$\rho _s$
for
$\Delta '\delta _{in}\gg 1$
. This is because the growth rate of the mode is slow compared with the shearing rate on the ion scales.
6. Numerical tests
We use an eigenvalue code to solve (2.13–2.14) assuming cold ions and isothermal electrons (and thus we do not need to solve (2.15)): these assumptions (while unjustified physically for the case of the solar wind) make the equations much simpler to solve numerically. We use the profile (Loureiro et al. Reference Loureiro, Cowley, Dorland, Haines and Schekochihin2005)
\begin{align} f(x/a) &= -2\tanh (x/a)\textrm {sech}^2(x/a). \end{align}
For
$ka\ll 1$
,
$\Delta 'a \sim 15/(ka)^2$
; i.e.
$n=2$
. For this profile, the maximum growth rate is attained uniquely at the transitional wavenumber, rather than for all
$k\gt k_{{tr}}$
as would be the case for the more usual
$f(x/a)=\tanh (x/a)$
profile, for which
$n=1$
.
Growth rate as a function of
$ka$
for
$\alpha =0,0.05,\ldots, 0.95$
, red to blue lines. We set
$\rho =0.01a$
and
${\rm d}_e=0.001a$
.
Growth rates as a function of
$k$
for different
$\alpha$
are shown in figure 1, showing the expected scalings with
$k$
for both small and large
$k$
(large and small
$\Delta '$
). Because the transition moves to larger
$k$
with increasing shear flow, the interval in
$k$
over which the small-
$\Delta '$
scalings are relevant gets narrower as
$\alpha$
increases towards
$1$
.
We plot growth rates at fixed
$k$
as a function of
$1-\alpha ^2$
in figure 2. For both small and large
$\Delta '$
(top left and right panels, respectively), the scalings agree with (5.7). In the top right panel, we have also marked on the x-axis the position at which
$\alpha =\epsilon ^{1/3}$
: for smaller
$\alpha$
(larger
$1-\alpha ^2$
), we do not expect our scaling to apply – and indeed, the behaviour changes at around this point. The maximum/transitional growth rate (5.9) and corresponding transitional wavenumber (5.8) are shown in the bottom panels, and also agree quite well with the predicted scalings, shown as black lines.
Top left:
$\gamma$
for
$ka = 1.12$
, at which wavenumber
$\Delta '\delta _{in}\ll 1$
. Top right:
$\gamma$
for
$ka=10^{-3}$
(
$\Delta '\delta _{in}\gg 1$
). Bottom left: maximum growth rate
$\gamma _{max}$
. Bottom right: the wavenumber
$k_{{tr}}$
at which
$\gamma _{max}$
is attained. The vertical dashed lines on the top right and bottom left panels mark
$\alpha =\epsilon ^{1/3}$
.
We also check the dependence of the growth rate on both
$\rho _s$
and
${\rm d}_e$
. As a reminder, for
$\alpha =0$
the dependencies are given by (1.1) with
$\tau =0$
: our code reproduces these scalings (not shown). In figure 3 we plot the scalings for
$\alpha =0.9$
. For large
$\Delta '$
, the growth rate depends linearly on
${\rm d}_e$
, and does not depend on
$\rho _s$
: both as predicted (5.7), and very different to the
$\alpha =0$
case. For small
$\Delta '$
, at a wavenumber of
$ka=1.12$
, the growth rate depends on the product
${\rm d}_e \rho _s$
, as in the no-flow case, also in agreement with our predictions.
In all panels
$\alpha =0.9$
. Top left: growth rate vs
$k$
, different lines correspond to different
$\rho _s$
. For large
$\Delta '\delta _{in}$
(smaller wavenumbers), the growth rate is independent of
$\rho _s$
. Top right: growth rate as a function of
$\rho _s$
for
$ka=1.12$
, i.e. for
$\Delta '\delta _{in}\ll 1$
. Bottom left: growth rate as a function of
${\rm d}_e$
for
$ka=10^{-3}$
, i.e.
$\Delta '\delta _{in}\gg 1$
. Bottom right: growth rate as a function of
${\rm d}_e$
for
$ka=1.12$
.
We can also check the behaviour of the width of the perturbed current profile: this corresponds to the scale at which the electron terms become important,
$x\sim \delta \epsilon$
. We have checked that this agrees with the scalings found numerically for the growth rate.
7. Conclusions
We have studied the collisionless tearing mode in the presence of significant shear flows, as parametrised by
$\alpha = \delta u / \delta b$
, where
$\delta u$
and
$\delta b$
are the amplitudes of the background velocity and magnetic field fluctuations, pointing in the
$y$
(or L) direction and varying in the
$x$
(or N) direction across the current sheet. We find that the growth rates depend strongly on
$\alpha$
, with a maximum growth rate given by (5.9),
\begin{equation} \frac {\gamma _{{tr}} a}{v_{{A}y}} \sim \frac {\rho _s^{1/n}{\rm d}_e}{a^{1+1/n}}\frac {(1-\alpha ^2)^{3/2-1/2n}}{\alpha ^{2-2/n}}\frac {(1+\tau /Z)^{3/2n-1}}{(1+\alpha ^2\tau /Z)^{1/n}}, \end{equation}
where
$n=1$
(e.g. for a Harris-type profile) or
$n=2$
(e.g. for a sinusoidal type profile). As
$\alpha \to 1$
(an exactly Alfvénic flow),
$\gamma \to 0$
. Moreover, for large ion-to-electron temperature ratios,
$\tau =T_{0i}/T_{0e}$
, the growth rate decreases with
$\tau$
,
$\gamma _{{tr}}\propto \tau ^{1/2n-1}$
. With
$n=1$
for simplicity, relative to the maximum tearing mode growth rate with
$\alpha =0$
,
$\gamma _{0{tr}}$
(see (1.2)), we have
\begin{equation} \frac {\gamma _{{tr}}}{\gamma _{0{tr}}} = \frac {1-\alpha ^2}{1+\alpha ^2 \tau /Z}. \end{equation}
As described in the introduction, a remarkable absence of reconnection was reported in the near-Sun solar wind observed during PSP’s first perihelion (Phan et al. Reference Phan2020). This has been confirmed in a comprehensive recent study of the PSP data by Eriksson et al. (Reference Eriksson, Swisdak, Mallet, Kruparova, Livi, Romeo, Bale, Kasper, Larson and Pulupa2024), who found that reconnection was extremely rare more specifically in faster, high-ion-temperature solar wind emerging from coronal holes: this is also the wind that is typically highly Alfvénic (D’Amicis & Bruno Reference D’Amicis and Bruno2015; D’Amicis et al. Reference D’Amicis, Alielden, Perrone, Bruno, Telloni, Raines, Lepri and Zhao2021), with highly correlated velocity and magnetic field fluctuations,
$\delta \boldsymbol {u} \sim \pm \delta \boldsymbol {b}$
or
$1-\alpha ^2\ll 1$
(Ervin et al. Reference Ervin, Bale, Badman, Rivera, Romeo, Huang, Riley, Bowen, Lepri and Dewey2024). Typically, this wind also has large
$\tau =T_{0i}/T_{0e}$
(Shi et al. Reference Shi2023), probably due to higher ion heating in imbalanced turbulence due to the recently discovered helicity barrier (Meyrand et al. Reference Meyrand, Squire, Schekochihin and Dorland2021; Squire et al. Reference Squire, Meyrand, Kunz, Arzamasskiy, Schekochihin and Quataert2022). Similar clustering was observed at larger heliocentric distances by Fargette et al. (Reference Fargette2023), possibly with the same underlying cause.
We have shown that both these parameters typical of the Alfvénic wind,
$1-\alpha ^2\ll 1$
and
$\tau \gtrsim 1$
, suppress the tearing mode growth rate, and thus increase the reconnection onset time, comparable to
$\gamma ^{-1}$
. We can roughly estimate a typical reconnection time from our expression for the growth rate: taking
$n=1$
and typical values for the PSP Alfvénic wind,
$\alpha \approx 0.9$
and
$\tau \approx 2$
(Chen et al. Reference Chen2020; Shi et al. Reference Shi2021b
, Reference Shi2023; Ervin et al. Reference Ervin, Bale, Badman, Rivera, Romeo, Huang, Riley, Bowen, Lepri and Dewey2024), we find
$\gamma _{{tr}}/\gamma _{0{tr}} \approx 0.07$
. On the other hand, the eddy turnover time
$\tau _{nl}\sim \lambda / \delta u(1-\alpha ^2)$
in imbalanced turbulence is increased by the same factor
$(1-\alpha ^2)^{-1}$
(Schekochihin Reference Schekochihin2022), so if all the current sheets are produced by turbulence, the product
$\gamma _{{tr}}\tau _{nl}$
is left relatively unchanged. If the development of current sheets is controlled by the eddy turnover time, then the overall prevalence of reconnection would not be suppressed by this mechanism. However, it is far from clear that this is the only time scale available. For example, sharp switchback boundaries may evolve instead due to the large-scale inhomogeneity in the system, on the solar wind expansion time of order
$\tau _{exp} \sim R/U_{sw}$
, where
$R$
is the radial distance from the Sun and
$U_{sw}$
is the solar wind velocity (Mallet et al. Reference Mallet, Squire, Chandran, Bowen and Bale2021).
Our work could be extended in several important ways to better describe the solar wind current sheets. First, KREHM cannot handle large equilibrium flows, and we are therefore limited to guide-field reconnection. Second, we have not allowed an equilibrium parallel flow shear, which is also often observed in the solar wind (Eriksson et al. Reference Eriksson, Swisdak, Mallet, Kruparova, Livi, Romeo, Bale, Kasper, Larson and Pulupa2024). Third, KREHM is formally limited to small
$\beta$
. This is perhaps reasonable for the solar wind close to the Sun, but further out in the heliosphere
$\beta$
is typically of order unity. Finally, we have also not allowed an equilibrium density gradient (again, often observed in the data), which can suppress reconnection (Swisdak et al. Reference Swisdak, Rogers, Drake and Shay2003) and also requires a more careful treatment of the matching between the ion and outer layers (Connor et al. Reference Connor, Ham, Hastie and Zocco2019). Nevertheless, our theory may help to explain the patchy suppression of reconnection in the PSP data, but further work is needed to generalise the equilibrium and to elucidate the formation time scale of the observed sheet-like structures.
Acknowledgements
The authors would like to thank the anonymous reviewers for helpful comments that significantly improved the paper. A.M., S.E. and M.S. were supported by NASA grant 80NSSC20K1284. J.J. was funded by the U.S. Department of Energy under Contract No. DE-AC 02–09CH1146 via an LDRD grant.
Editor Nuno Loureiro thanks the referees for their advice in evaluating this article.
Declaration of interests
The authors report no conflict of interest.
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