2 Alfvénic turbulence model

In the theory of intermittent Alfvénic turbulence of Mallet & Schekochihin (Reference Mallet and Schekochihin2017), the turbulence is modelled as an ensemble of structures, each of which is characterised by an Elsasser amplitude $\unicode[STIX]{x1D6FF}z$ and three characteristic scales: $l_{\Vert }$ (parallel), $\unicode[STIX]{x1D706}$ (perpendicular) and $\unicode[STIX]{x1D709}$ (fluctuation-direction). We normalise these variables by their values at the outer scale:

(2.1a-d ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}=\frac{\unicode[STIX]{x1D6FF}z}{\overline{\unicode[STIX]{x1D6FF}z}},\quad \hat{\unicode[STIX]{x1D706}}=\frac{\unicode[STIX]{x1D706}}{L_{\bot }},\quad \hat{l}_{\Vert }=\frac{l_{\Vert }}{L_{\Vert }},\quad \hat{\unicode[STIX]{x1D709}}=\frac{\unicode[STIX]{x1D709}}{L_{\bot }},\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D6FF}z}$ is the outer-scale fluctuation amplitude, and $L_{\bot }$ and $L_{\Vert }$ are the perpendicular and parallel outer scales. In the following, we will treat $\hat{\unicode[STIX]{x1D706}}$ as a parameter (i.e. we are conditioning on $\hat{\unicode[STIX]{x1D706}}$ ): the distribution of $\unicode[STIX]{x1D6FF}\hat{z}$ depends on $\hat{\unicode[STIX]{x1D706}}$ , and $\hat{\unicode[STIX]{x1D709}}$ and $\hat{l}_{\Vert }$ are calculated from $\unicode[STIX]{x1D6FF}\hat{z}$ and $\hat{\unicode[STIX]{x1D706}}$ . It is assumed that the turbulence is critically balanced already at the outer scale. The normalised amplitude is given by

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}\sim \unicode[STIX]{x1D6EC}^{q},\end{eqnarray}$$

where $q$ is a Poisson-distributed random variable,

(2.3) $$\begin{eqnarray}P(q)=\frac{\unicode[STIX]{x1D707}^{q}}{q!}\text{e}^{-\unicode[STIX]{x1D707}},\end{eqnarray}$$

with mean $\unicode[STIX]{x1D707}=-\ln \hat{\unicode[STIX]{x1D706}}$ ,Footnote ² and $\unicode[STIX]{x1D6EC}=1/\sqrt{2}$ is a dimensionless constant (which Mallet & Schekochihin Reference Mallet and Schekochihin2017 called $\unicode[STIX]{x1D6FD}$ , but which we here rename to avoid confusion with $\unicode[STIX]{x1D6FD}_{e}$ ). The scalings of perpendicular structure functions are then given by

(2.4) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6FF}\hat{z}^{n}\rangle \sim \hat{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D701}_{n}^{\bot }},\quad \unicode[STIX]{x1D701}_{n}^{\bot }=1-\unicode[STIX]{x1D6EC}^{n}.\end{eqnarray}$$

The fluctuation-direction scale $\hat{\unicode[STIX]{x1D709}}$ is related to the amplitude via

(2.5) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D709}}\sim \hat{\unicode[STIX]{x1D706}}^{1/2}\unicode[STIX]{x1D6EC}^{q},\end{eqnarray}$$

while the parallel scale depends only on $\hat{\unicode[STIX]{x1D706}}$ :

(2.6) $$\begin{eqnarray}\hat{l}_{\Vert }\sim \hat{\unicode[STIX]{x1D706}}^{1/2}.\end{eqnarray}$$

Following Mallet et al. (Reference Mallet, Schekochihin and Chandran2017), we define the ‘effective amplitude’ of structures that dominate the $n$ th-order perpendicular structure function:

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}[n]\equiv \langle \unicode[STIX]{x1D6FF}\hat{z}^{n}\rangle ^{1/n}\sim \hat{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D701}_{n}^{\bot }/n}.\end{eqnarray}$$

The effective amplitude $\unicode[STIX]{x1D6FF}\hat{z}[n]$ is a strictly increasing function of $n$ , and so $n$ may be used as a convenient proxy for the amplitude of the structures at a given scale. The scalings for three interesting cases can be immediately obtained from (2.7): first,

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}[\infty ]\sim 1\end{eqnarray}$$

describes the ‘most intense’ structures, whose amplitude is independent of scale; secondly,

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}[2]\sim \hat{\unicode[STIX]{x1D706}}^{1/4}\end{eqnarray}$$

describes the fluctuations that dominate the second-order structure function and the energy spectrum, and thus determine the spectral index; finally, the ‘bulk’ fluctuations are described by $n\rightarrow 0$ , and their amplitudes scale as

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}[n\rightarrow 0]\sim \hat{\unicode[STIX]{x1D706}}^{-\ln \unicode[STIX]{x1D6EC}}.\end{eqnarray}$$

We will also need an expression for the (effective) fluctuation-direction scale for the $n$ th-order fluctuations, given by

(2.11) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D709}}[n]\sim \hat{\unicode[STIX]{x1D706}}^{1/2}\unicode[STIX]{x1D6FF}\hat{z}[n]\sim \hat{\unicode[STIX]{x1D706}}^{1/2+\unicode[STIX]{x1D701}_{n}^{\bot }/n},\end{eqnarray}$$

and for the cascade time,

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{C}\sim \frac{\unicode[STIX]{x1D709}}{\unicode[STIX]{x1D6FF}z}\sim \frac{L_{\bot }}{\overline{\unicode[STIX]{x1D6FF}z}}\hat{\unicode[STIX]{x1D706}}^{1/2}.\end{eqnarray}$$

One can easily see from (2.11) that the structures are anisotropic in the perpendicular plane, with $\unicode[STIX]{x1D709}\gg \unicode[STIX]{x1D706}$ , and that the higher-amplitude structures are more anisotropic, consistent with numerical evidence (Mallet et al. Reference Mallet, Schekochihin and Chandran2015, Reference Mallet, Schekochihin, Chandran, Chen, Horbury, Wicks and Greenan2016).

We now have all the information needed about the turbulent structures to determine whether they can be disrupted by tearing.

3 Collisionless tearing mode

Scalings for the low- $\unicode[STIX]{x1D6FD}_{e}$ collisionless tearing mode are reviewed in appendix B.3 of Zocco & Schekochihin (Reference Zocco and Schekochihin2011). Our sheet-like turbulent structures have a width $\unicode[STIX]{x1D706}$ and a length $\unicode[STIX]{x1D709}$ in the perpendicular plane. We will assume that the perturbed magnetic field reverses across the structure $\unicode[STIX]{x1D6FF}b\sim \unicode[STIX]{x1D6FF}z$ . There is also a velocity perturbation $\unicode[STIX]{x1D6FF}u$ associated with the $\unicode[STIX]{x1D6FF}z$ . If the situation were that $\unicode[STIX]{x1D6FF}u>\unicode[STIX]{x1D6FF}b$ , the Kelvin–Helmholtz instability would disrupt the sheets much faster than the tearing mode. However, this situation does not typically occur, because the vortex stretching terms for the different Elsasser fields $\boldsymbol{z}_{\bot }^{\pm }$ have opposite sign (Zhdankin, Boldyrev & Uzdensky Reference Zhdankin, Boldyrev and Uzdensky2016), meaning that ‘current sheets’ are more common than ‘shear layers’ in RMHD turbulence, i.e. $\unicode[STIX]{x1D6FF}u<\unicode[STIX]{x1D6FF}b$ (this is also true in the solar wind; see, e.g. Chen Reference Chen2016, Wicks et al. Reference Wicks, Mallet, Horbury, Chen, Schekochihin and Mitchell2013), and the Kelvin–Helmholtz instability is naturally stabilised (Chandrasekhar Reference Chandrasekhar1961). For simplicity, we assume that the velocity fluctuations $\unicode[STIX]{x1D6FF}u\lesssim \unicode[STIX]{x1D6FF}b$ present in the sheet-like structures do not significantly affect the dynamics that we will describe in this paper.

The structure of the collisionless tearing mode involves three scales: the perpendicular scale $\unicode[STIX]{x1D706}$ of the turbulent structure, and a nested inner layer, where two-fluid effects become important at the ion sound scale $\unicode[STIX]{x1D70C}_{s}$ , while flux unfreezing happens due to electron inertia in a thinner layer controlled by the electron inertial scale $d_{e}=c/\unicode[STIX]{x1D714}_{pe}\ll \unicode[STIX]{x1D70C}_{s}$ , where $\unicode[STIX]{x1D714}_{pe}=\sqrt{4\unicode[STIX]{x03C0}n_{e}e^{2}/m_{e}}$ is the electron plasma frequency. The tearing instability’s growth rates will, therefore, involve all of these scales. The scalings that we use here will cease to apply if $\unicode[STIX]{x1D70C}_{s}\lesssim d_{e}$ (i.e. $\unicode[STIX]{x1D6FD}_{e}\lesssim m_{e}/m_{i}$ ), at which point the ion scale becomes unimportant, and if $\unicode[STIX]{x1D6FD}_{e}\gtrsim 1$ , when the flux unfreezing happens at the electron gyroradius $\unicode[STIX]{x1D70C}_{e}=d_{e}\sqrt{\unicode[STIX]{x1D6FD}_{e}}$ , rather than at $d_{e}$ . This means that we are restricting ourselves to ‘moderately’ small beta, $1\gg \unicode[STIX]{x1D6FD}_{e}\gg m_{e}/m_{i}$ .

We will assume a Harris-sheet-like equilibrium (Harris Reference Harris1962)Footnote ³ . For long-wavelength modes ( $k\ll 1/\unicode[STIX]{x1D706}$ ), the instability parameter $\unicode[STIX]{x1D6E5}^{\prime }$ is given by

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{\prime }\unicode[STIX]{x1D706}\approx \frac{1}{k\unicode[STIX]{x1D706}}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6E5}^{\prime }\unicode[STIX]{x1D6FF}_{\text{in}}\ll 1$ , where $\unicode[STIX]{x1D6FF}_{\text{in}}$ is the width of the inner layer, the linear growth rate and the inner-layer width are

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{{>}}\sim k\unicode[STIX]{x1D6FF}z\frac{d_{e}\unicode[STIX]{x1D70C}_{s}\unicode[STIX]{x1D6E5}^{\prime }}{\unicode[STIX]{x1D706}}\sim \unicode[STIX]{x1D6FF}z\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x1D706}^{3}},\quad \unicode[STIX]{x1D6FF}_{\text{in}}\sim d_{e}\unicode[STIX]{x1D70C}_{s}^{1/2}{\unicode[STIX]{x1D6E5}^{\prime }}^{1/2}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6E5}^{\prime }\unicode[STIX]{x1D6FF}_{\text{in}}\sim 1$ , they are

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{{<}}\sim k\unicode[STIX]{x1D6FF}z\frac{d_{e}^{1/3}\unicode[STIX]{x1D70C}_{s}^{2/3}}{\unicode[STIX]{x1D706}},\quad \unicode[STIX]{x1D6FF}_{\text{in}}\sim d_{e}^{2/3}\unicode[STIX]{x1D70C}_{s}^{1/3}.\end{eqnarray}$$

The wavenumber $k_{\text{tr}}$ of the transition between these two regimes can be found by balancing the two expressions for the growth rate, giving

(3.4) $$\begin{eqnarray}k_{\text{tr}}\sim \frac{d_{e}^{2/3}\unicode[STIX]{x1D70C}_{s}^{1/3}}{\unicode[STIX]{x1D706}^{2}}.\end{eqnarray}$$

For $k<k_{\text{tr}}$ , the growth rate is $\unicode[STIX]{x1D6FE}_{{<}}$ , while for $k>k_{\text{tr}}$ , it is $\unicode[STIX]{x1D6FE}_{{>}}$ , which is independent of wavenumber. This breaks down when $k\sim 1/\unicode[STIX]{x1D706}$ , because (3.1) ceases to apply – but, since $\unicode[STIX]{x1D709}\gg \unicode[STIX]{x1D706}$ , this only happens for a very large number of islands. Therefore, the maximum growth rate is attained for all $k>k_{\text{tr}}$ , and is given simply by $\unicode[STIX]{x1D6FE}_{{>}}$ . Thus there is always a mode with the maximum growth rate and a large enough wavenumber to fit into a sheet of any length $\unicode[STIX]{x1D709}>\unicode[STIX]{x1D706}$ . This is somewhat different from the resistive-RMHD case studied by Mallet et al. (Reference Mallet, Schekochihin and Chandran2017), in which $\unicode[STIX]{x1D6FE}_{{>}}^{\text{RMHD}}\propto k^{-2/5}$ , while $\unicode[STIX]{x1D6FE}_{{<}}^{\text{RMHD}}\propto k^{2/3}$ , and the maximum growth rate is attained at the transitional wavenumber.

The linear growth stage of tearing ends when the width of the islands reaches $\unicode[STIX]{x1D6FF}_{\text{in}}$ , which decreases with increasing $k$ for $k>k_{\text{tr}}$ . Thus at the end of the linear stage, the largest islands are produced by the mode with $k\sim k_{\text{tr}}$ , and so, despite the independence of the linear growth rate on $k$ , we can assume that this transitional mode dominates the nonlinear dynamics. We assume that the $X$ -points between the islands then collapse quickly (i.e. on a time scale at most comparable to $\unicode[STIX]{x1D6FE}_{{>}}^{-1}$ ), circularising the islands and forming a set of flux ropes of width $\unicode[STIX]{x1D706}$ , as appears to be consistent with numerical evidence (Loureiro, Schekochihin & Zocco Reference Loureiro, Schekochihin and Zocco2013).Footnote ⁴ Since these structures are as wide as the original sheet, the latter should at this point be disrupted and broken up – being effectively replaced by a set of flux ropes. The scale of these ropes parallel to the (exact) magnetic field is set as usual by critical balance. Since we assume that the $X$ -point collapse is at least as fast as the linear tearing stage, we estimate the disruption time using the linear growth rate (3.2) of the tearing mode:

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\text{D}}\sim \unicode[STIX]{x1D6FE}_{{>}}^{-1}.\end{eqnarray}$$

It is important to point out that the restriction to low $\unicode[STIX]{x1D6FD}_{e}$ limits the applicability of our conclusions in the solar wind, where, more often than not, $\unicode[STIX]{x1D6FD}_{e}\sim 1$ , but our results will be more relevant to the turbulence closer to the Sun and in the corona: indeed, at the perihelion (approximately 10 solar radii) of the upcoming Parker Solar Probe mission, $\unicode[STIX]{x1D6FD}_{e}\approx 0.01$ , at least in fast solar-wind streams (Chandran et al. Reference Chandran, Dennis, Quataert and Bale2011). Moreover, the growth rate scaling (3.2) appears to be quite robust even at moderately large $\unicode[STIX]{x1D6FD}_{e}$ : Numata & Loureiro (Reference Numata and Loureiro2015) showed that, keeping all other parameters fixed, $\unicode[STIX]{x1D6FE}_{{>}}\propto \unicode[STIX]{x1D6FD}_{e}^{-1/2}\propto d_{e}$ up to at least $\unicode[STIX]{x1D6FD}_{e}=10$ , in agreement with (3.2), and despite the width of the reconnecting layer being set by $\unicode[STIX]{x1D70C}_{e}$ rather than $d_{e}$ . Therefore, we expect our conclusions to be at least qualitatively relevant at $\unicode[STIX]{x1D6FD}_{e}\sim 1$ .

4 Disruption scale

A sheet-like structure will be disrupted if its nonlinear cascade time (2.12) is longer than its disruption time (3.5). The disruption scale $\hat{\unicode[STIX]{x1D706}}_{\text{D}}$ is then determined by demanding

(4.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70F}_{C}}{\unicode[STIX]{x1D70F}_{\text{D}}}\sim \unicode[STIX]{x1D709}\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x1D706}^{3}}\gtrsim 1.\end{eqnarray}$$

Using (2.11) and (2.7), we find that, for $n$ th-order aligned structures, this inequality is satisfied for

(4.2) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D706}}\lesssim \hat{\unicode[STIX]{x1D706}}_{\text{D}}[n]\sim \left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{(2/5)(1-2\unicode[STIX]{x1D701}_{n}^{\bot }/5n)^{-1}}.\end{eqnarray}$$

The scale $\hat{\unicode[STIX]{x1D706}}_{\text{D}}$ is an increasing function of $n$ . It is largest for the most intense structures, with $n\rightarrow \infty$ , for which

(4.3) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D706}}_{\text{D}}[\infty ]\sim \left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{2/5}.\end{eqnarray}$$

The scale at which the $n=2$ structures, which determine the scaling of the second-order structure function and the energy spectrum, are disrupted isFootnote ⁵

(4.4) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D706}}_{\text{D}}[2]\sim \left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{4/9},\end{eqnarray}$$

and, finally, the bulk fluctuations ( $n\rightarrow 0$ ) are disrupted at

(4.5) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D706}}_{\text{D}}[0]\sim \left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{0.46}.\end{eqnarray}$$

The disruption may effectively be thought of as taking place over a narrow range of scales between $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[\infty ]$ and $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[0]$ , with $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[2]$ as a good representative. The disruption will only be relevant if any of these scales is larger than the scale at which the waves become dispersive, i.e.

(4.6) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D706}_{\text{D}}[n]}{\unicode[STIX]{x1D70C}_{s}} & {\sim} & \displaystyle \left(\frac{d_{e}}{\unicode[STIX]{x1D70C}_{s}}\right)^{(2/5)(1-2\unicode[STIX]{x1D701}_{n}^{\bot }/5n)^{-1}}\left(\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}\right)^{1-(4/5)(1-2\unicode[STIX]{x1D701}_{n}^{\bot }/5n)^{-1}},\nonumber\\ \displaystyle & {\sim} & \displaystyle \left(\frac{m_{i}}{m_{e}}\frac{\unicode[STIX]{x1D6FD}_{e}}{Z}\right)^{-(1/5)(1-2\unicode[STIX]{x1D701}_{n}^{\bot }/5n)^{-1}}\left(\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}\right)^{1-(4/5)(1-2\unicode[STIX]{x1D701}_{n}^{\bot }/5n)^{-1}}\gtrsim 1.\end{eqnarray}$$

This gives us a critical $\unicode[STIX]{x1D6FD}_{e}$ for structures at any given $n$ to be disrupted:

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{e}\lesssim \unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[n]\sim Z\frac{m_{e}}{m_{i}}\left(\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}\right)^{1-2\unicode[STIX]{x1D701}_{n}^{\bot }/n}.\end{eqnarray}$$

For the $n=2$ structures,

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[2]\sim Z\frac{m_{e}}{m_{i}}\left(\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}\right)^{1/2},\end{eqnarray}$$

while for the most intense fluctuations ( $n\rightarrow \infty$ ),

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[\infty ]\sim Z\frac{m_{e}}{m_{i}}\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}.\end{eqnarray}$$

It is interesting to note that, despite the fact that the dependence of $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[n]$ on $n$ does not appear to be very strong [the exponents in (4.3), (4.4) and (4.5) are close together, at $0.40$ , $0.44$ and $0.46$ respectively], the dependence of $\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}$ on $L_{\bot }/\unicode[STIX]{x1D70C}_{s}$ is a strong function of $n$ .

In the solar wind, typically $L_{\bot }/\unicode[STIX]{x1D70C}_{s}\approx 10^{3}$ (Chen Reference Chen2016), and so $\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[2]\sim 10^{-2}$ , which is rare at 1 AU but should be rather common closer to the Sun in the region to be explored by the Parker Solar Probe (Fox et al. Reference Fox, Velli, Bale, Decker, Driesman, Howard, Kasper, Kinnison, Kusterer and Lario2016; Chandran et al. Reference Chandran, Dennis, Quataert and Bale2011). On the other hand, $\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[\infty ]\sim 1$ , and so one might expect the most intense sheet-like structures to become unstable to the onset of reconnection even at moderate $\unicode[STIX]{x1D6FD}_{e}$ . Flux-rope-like ‘Alfvén vortex’ structures extended in the parallel direction were indeed observed at ion scales in the solar wind by Perrone et al. (Reference Perrone, Alexandrova, Mangeney, Maksimovic, Lacombe, Rakoto, Kasper and Jovanovic2016) and Lion et al. (Reference Lion, Alexandrova and Zaslavsky2016). It is tempting to identify the structures produced by the disruption due to tearing with these observationsFootnote ⁶ . We will study the structures produced by the disruption process in the next section, quantifying the relationship between their amplitude and scale, and further examining their importance as a function of $\unicode[STIX]{x1D6FD}_{e}$ and of $\unicode[STIX]{x1D70C}_{s}/L_{\bot }$ .

Finally, let us set aside for a moment the precise values of $\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}$ for which we predict that disruption happens, and focus instead on the scaling of the break in the energy spectrum, $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[2]$ , with physical parameters, i.e. the dependence of $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[2]$ on $\unicode[STIX]{x1D6FD}$ . Chen et al. (Reference Chen, Leung, Boldyrev, Maruca and Bale2014) observed that at low $\unicode[STIX]{x1D6FD}_{i}$ , the break scale of solar-wind turbulence appeared to scale as $d_{i}\propto \unicode[STIX]{x1D70C}_{i}/\sqrt{\unicode[STIX]{x1D6FD}_{i}}$ , in contradiction with expectations based on the linear physics of low- $\unicode[STIX]{x1D6FD}$ plasmas (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Here we predict (ignoring the factor of $(L_{\bot }/\unicode[STIX]{x1D70C}_{s})^{1/9}$ , which barely changes with the relevant physical parameters)

(4.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D706}_{\text{D}}[2]}{\unicode[STIX]{x1D70C}_{s}}\propto \unicode[STIX]{x1D6FD}_{e}^{-2/9}\quad \Rightarrow \quad \frac{\unicode[STIX]{x1D706}_{\text{D}}[2]}{d_{i}}\propto \left(\unicode[STIX]{x1D6FD}_{i}\frac{T_{e}}{T_{i}}\right)^{5/18}.\end{eqnarray}$$

For disruption due to reconnection to explain the anomalous break scale observed by Chen et al. (Reference Chen, Leung, Boldyrev, Maruca and Bale2014), there would therefore have to be correlations between $\unicode[STIX]{x1D6FD}_{i}$ and $T_{e}/T_{i}$ in their chosen intervals (namely, $T_{i}/T_{e}\propto \unicode[STIX]{x1D6FD}_{i}$ to match precisely). Encouragingly, in their data it does appear that the lower- $\unicode[STIX]{x1D6FD}_{i}$ intervals are associated with markedly higher $T_{e}/T_{i}$ .

5 Statistical properties of flux ropes

The dependence (4.2) of $\hat{\unicode[STIX]{x1D706}}_{\text{D}}$ on $n$ is one way of quantifying the scales at which the disrupted structures appear. In this section, we recast our calculation, treating the amplitude of the fluctuation as a random variable, i.e. we return to (2.2), and determine what fraction of the aligned structures remain undisrupted at any given scale, in terms of $q$ (we remind the reader that this is an integer distributed as a Poisson random variable with mean $\langle q\rangle =\unicode[STIX]{x1D707}=-\ln \hat{\unicode[STIX]{x1D706}}$ ). For a structure to be disrupted, we again demand (4.1) and, using (2.5), find that

(5.1) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D706}}^{-5/2}\unicode[STIX]{x1D6EC}^{q}\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\gtrsim 1.\end{eqnarray}$$

This is satisfied for

(5.2) $$\begin{eqnarray}\displaystyle \displaystyle q\lesssim q_{\text{D}} & = & \displaystyle \frac{(5/2)\ln \hat{\unicode[STIX]{x1D706}}-\ln \left(d_{e}\unicode[STIX]{x1D70C}_{s}/L_{\bot }^{2}\right)}{\ln \unicode[STIX]{x1D6EC}}\nonumber\\ \displaystyle \displaystyle & = & \displaystyle \frac{(5/2)\ln \hat{\unicode[STIX]{x1D706}}-2\ln \left(\unicode[STIX]{x1D70C}_{s}/L_{\bot }\right)+(1/2)\ln \left(\unicode[STIX]{x1D6FD}_{e}m_{i}/Zm_{e}\right)}{\ln \unicode[STIX]{x1D6EC}}.\end{eqnarray}$$

5.1 Filling factor of aligned turbulence

At any given scale, the filling factor of sheet-like, aligned structures that have not been affected by the disruption process (i.e. the probability of encountering them) is given by

(5.3) $$\begin{eqnarray}f_{0}(\hat{\unicode[STIX]{x1D706}})=P(q>q_{D})=1-\mathop{\sum }_{q=0}^{\lfloor q_{\text{D}}\rfloor }P(q),\end{eqnarray}$$

where the distribution of $q$ is given by (2.3). Similarly, the disruption causes a fractional reduction of energy contained in aligned sheet-like structures that is, using (2.4) and (2.2),

(5.4) $$\begin{eqnarray}\displaystyle f_{2}(\hat{\unicode[STIX]{x1D706}})=1-\frac{1}{\langle \unicode[STIX]{x1D6FF}\hat{z}^{2}\rangle }\mathop{\sum }_{q=0}^{\lfloor q_{\text{D}}\rfloor }\unicode[STIX]{x1D6FF}\hat{z}^{2}P(q)=1-\frac{1}{\hat{\unicode[STIX]{x1D706}}^{1/2}}\mathop{\sum }_{q=0}^{\lfloor q_{\text{D}}\rfloor }2^{-q}P(q). & & \displaystyle\end{eqnarray}$$

Obviously, (5.3) and (5.4) can only be considered quantitatively good estimates if $f_{0}$ and $f_{2}$ are close to unity, i.e. if the overall ‘RMHD ensemble’ (described in § 2) is not significantly altered.

Using (5.2), both $f_{0}$ and $f_{2}$ may be calculated numerically, as functions of $\hat{\unicode[STIX]{x1D706}}$ , $\unicode[STIX]{x1D70C}_{s}/L_{\bot }$ , and $\unicode[STIX]{x1D6FD}_{e}$ . A particularly interesting case is $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D70C}_{s}$ , since this quantifies the cumulative effect of reconnection on the turbulence at the ion scale. Figure 1 shows the dependence of $f_{0}(\unicode[STIX]{x1D70C}_{s}/L_{\bot })$ and $f_{2}(\unicode[STIX]{x1D70C}_{s}/L_{\bot })$ on $\unicode[STIX]{x1D6FD}_{e}$ . The effect of disruption becomes more important at smaller $\unicode[STIX]{x1D6FD}_{e}$ . Note that the amount of energy in the undisrupted structures at the ion scale is significantly reduced for values of $\unicode[STIX]{x1D6FD}_{e}$ somewhat larger than $\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[2]$ given by (4.8). This suggests that, in practice, the turbulence is significantly affected by the disruption at only moderately small $\unicode[STIX]{x1D6FD}_{e}$ : e.g. for $\unicode[STIX]{x1D6FD}_{e}\sim 0.1$ , only around half of the energy that would be in sheets without disruption actually makes it to the ion scale, despite only approximately $10\,\%$ by volume of the turbulence being disrupted.

Figure 1.

The filling factor of the aligned (i.e. undisrupted) structures at the ion sound scale $f_{0}(\unicode[STIX]{x1D70C}_{s}/L_{\bot })$ (blue) and fraction $f_{2}(\unicode[STIX]{x1D70C}_{s}/L_{\bot })$ of energy in them (red), plotted as a function of $\unicode[STIX]{x1D6FD}_{e}$ . We have taken $\unicode[STIX]{x1D70C}_{s}/L_{\bot }=10^{-3}$ , a reasonable value for the solar wind. Since $q$ is an integer but $q_{\text{D}}$ is not, the sums (5.3) and (5.4) are performed up to $\lfloor q_{\text{D}}\rfloor$ , resulting in the discontinuities shown in the plot. In reality, of course, $f_{0}$ and $f_{2}$ will be smooth.

5.2 Amplitude of flux ropes

We have proposed that the sheet-like structures with $q\sim q_{\text{D}}$ disrupted by tearing at the scale $\unicode[STIX]{x1D706}$ are converted into circular flux ropes, with perpendicular scale $\unicode[STIX]{x1D706}$ . We will assume that, just after they are created, they have the same amplitude as the sheet-like structure that produced them:

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{fr}\sim \unicode[STIX]{x1D6EC}^{q_{\text{D}}}.\end{eqnarray}$$

The flux ropes will not stay around for long: they will interact with each other and the remaining sheet-like structures, cascade, align, and form smaller, more sheet-like structures: this process will be studied in the next section. However, we can predict the relationship between the amplitude $\unicode[STIX]{x1D6FF}z_{\text{fr}}$ and radius $\unicode[STIX]{x1D706}$ of the newly created flux ropes: upon inserting (5.2) into (5.5), we get

(5.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{\text{fr}}\sim \hat{\unicode[STIX]{x1D706}}^{5/2}\left(\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}\right)^{2}\left(\frac{\unicode[STIX]{x1D6FD}_{e}}{Z}\frac{m_{i}}{m_{e}}\right)^{1/2}.\end{eqnarray}$$

It is important to note that this is not a prediction for the scaling of any structure function or the spectrum of the disrupted turbulence (which will be worked out in the next section); rather, this is a relationship describing individual flux ropes upon their formation within an aligned structure.

Thus, provided that $\unicode[STIX]{x1D6FD}_{e}$ and $\unicode[STIX]{x1D70C}_{s}/L_{\bot }$ are small enough that $q_{\text{D}}>0$ before the cascade reaches $\unicode[STIX]{x1D70C}_{s}$ , i.e. $\unicode[STIX]{x1D6FD}_{e}<\unicode[STIX]{x1D6FD}_{e}^{\text{crit}}[\infty ]$ given by (4.9), there should be a strong relationship between the amplitude and radius of the structures at scales between $\hat{\unicode[STIX]{x1D706}}_{\text{D}}[\infty ]$ , given by (4.3) and $\unicode[STIX]{x1D70C}_{s}$ : indeed, the scaling (5.6) is very steep with $\unicode[STIX]{x1D706}$ and the flux ropes with the largest radius $\unicode[STIX]{x1D706}_{\text{D}}[\infty ]$ also have the largest amplitude, $\unicode[STIX]{x1D6FF}z_{\text{fr}}\sim \overline{\unicode[STIX]{x1D6FF}z}$ . The scaling (5.6) could in principle be tested against observations such as those reported by Perrone et al. (Reference Perrone, Alexandrova, Mangeney, Maksimovic, Lacombe, Rakoto, Kasper and Jovanovic2016), who observed 12 ‘Alfvén vortices’ with diameters between $5\unicode[STIX]{x1D70C}_{i}$ and $17\unicode[STIX]{x1D70C}_{i}$ (as part of a sample of over 100 coherent structures of different types).

6 Disruption range turbulence

In a realistic situation relevant to coronal or solar-wind turbulence, the separation between $\unicode[STIX]{x1D706}_{\text{D}}$ and $\unicode[STIX]{x1D70C}_{s}$ is so small that it would be challenging to establish a robust distinction between these two scales. It is nonetheless interesting to speculate on the nature of the turbulence in the interval $\unicode[STIX]{x1D706}_{\text{D}}\gg \unicode[STIX]{x1D706}\gg \unicode[STIX]{x1D70C}_{s}$ in the asymptotic case where $L_{\bot }\rightarrow \infty$ .

As described in § 3, we expect the disruption process to convert the sheet-like structures just above $\unicode[STIX]{x1D706}_{\text{D}}$ into flux-rope-like structures just below $\unicode[STIX]{x1D706}_{\text{D}}$ . These are roughly circular in the perpendicular plane, with radius $\unicode[STIX]{x1D706}_{\text{D}}$ , but extended in the parallel direction due to critical balance. In order to treat this ‘disruption range’ properly, we would need to account for the intermittency of both sheets and flux ropes. We do not attempt such a treatment in this paper. Instead, we develop a simpler model, in which we take the sheets and flux ropes to be effectively volume filling, and the sheets to have the properties of the $n=2$ structures of § 2 (since we would like to explore the scaling of the energy spectrum below $\unicode[STIX]{x1D706}_{\text{D}}$ ). For simplicity of notation, we will drop the ‘ $[n=2]$ ’ argument of all relevant quantities. A ‘characteristic fluctuation amplitude’ $\unicode[STIX]{x1D6FF}z_{1,-}$ for the turbulence just below $\unicode[STIX]{x1D706}_{\text{D}}$ may be defined by assuming that there is negligible dissipation during the disruption process, so that the energy flux stays constant across the disruption scaleFootnote ⁷ . Further assuming (radically) that the turbulence just below $\unicode[STIX]{x1D706}_{\text{D}}$ loses all its dynamic alignment, the scale $\unicode[STIX]{x1D709}$ is no longer relevant and the only perpendicular scale in the problem is $\unicode[STIX]{x1D706}_{\text{D}}$ :

(6.1) $$\begin{eqnarray}\unicode[STIX]{x1D716}\sim \frac{\overline{\unicode[STIX]{x1D6FF}z}^{3}}{L_{\bot }}\sim \frac{\unicode[STIX]{x1D6FF}z_{1,-}^{3}}{\unicode[STIX]{x1D706}_{\text{D}}},\end{eqnarray}$$

giving

(6.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{1,-}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D}}^{1/3}.\end{eqnarray}$$

Note that this is smaller than the amplitude of the aligned turbulence at the same scale:

(6.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{1,+}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D}}^{1/4}.\end{eqnarray}$$

The definition (6.2) might appear to be in contradiction with our earlier assumption (5.5) that the flux ropes should be formed with the same amplitude as their ‘mother sheet’. Indeed, if one took (5.5), (6.2) and (6.3) together, they imply that the flux ropes created by a disrupting sheet fill only a fraction of the mother sheet’s volume, contradicting our supposition above. However, (6.2) is not meant to be the amplitude of any individual structure, but rather an effective estimate that would on average give (6.1). In this sense, there is no difference between (6.2) or (6.3) and the usual ‘twiddle’ relations in Kolmogorov-style turbulence phenomenologies relying on constant energy flux and ignoring intermittency and local imbalance: the amplitude (6.2) is an estimate that effectively absorbs within itself the filling fraction (probability of occurrence) of energetic structures that contribute to averages, as well as the (unknown) details of how precisely the nonlinear interactions within or between flux ropes (and between flux ropes and ambient turbulence) actually occur. In the case of flux ropes, it is clear that to make a connection between (5.5) and any average quantity, one would need to take into account the fact that they are clearly less volume filling than their mother sheets, have shorter lifetimes, and might mainly cascade due to interactions between different types of structures. We leapfrog these issues with the aid of the requirement (6.1) that energy flux should stay the same. This will allow us to make progress and develop a simple model in this section: (6.2) will define the ‘outer scale amplitude’ for the Alfvénic cascade below $\unicode[STIX]{x1D706}_{\text{D}}$ . Intermittency is known to be of crucial importance in Alfvénic turbulence (Chandran et al. Reference Chandran, Schekochihin and Mallet2015; Mallet & Schekochihin Reference Mallet and Schekochihin2017), and so a more rigorous model incorporating intermittency should be the subject of future work.

Following Mallet et al. (Reference Mallet, Schekochihin and Chandran2017), the turbulence just below $\unicode[STIX]{x1D706}_{\text{D}}$ should behave just like the usual Alfvénic turbulence described in § 2 (since $\unicode[STIX]{x1D706}_{\text{D}}\gg \unicode[STIX]{x1D70C}_{s}$ ): the flux ropes will interact with each other and the rest of the turbulence, causing a cascade to smaller scalesFootnote ⁸ . In the course of this secondary cascade, the turbulence will again start to dynamically align and form sheet-like structures, and may eventually be disrupted by the onset of reconnection at a secondary disruption scale $\unicode[STIX]{x1D706}_{\text{D},2}$ , at which the whole process repeats – provided that $\unicode[STIX]{x1D706}_{\text{D},2}>\unicode[STIX]{x1D70C}_{s}$ . Therefore, the turbulence between the first disruption scale $\unicode[STIX]{x1D706}_{\text{D}}$ (which we will now rechristen $\unicode[STIX]{x1D706}_{\text{D},1}$ ) and $\unicode[STIX]{x1D70C}_{s}$ is characterised by a sequence of disruptions, between which the turbulence realigns. We will now show that this recursive disruption process is unlikely to be fully realised, and usually terminates after only one disruption.

6.1 Recursive disruption?

The sequence of disruptions described above may be understood in terms of a recursion relation. After the $(i-1)$ st disruption, the turbulence undergoes the $i$ th ‘mini-cascade’, with ‘outer-scale’ values of the relevant quantities given by the values just below the $(i-1)$ st disruption scale $\unicode[STIX]{x1D706}_{\text{D},i-1}$ :

(6.4) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6FF}z}\rightarrow \unicode[STIX]{x1D6FF}z_{i-1,-},\quad L_{\bot }\rightarrow \unicode[STIX]{x1D706}_{\text{D},i-1}.\end{eqnarray}$$

Substituting this into (4.4) and normalising by $L_{\bot }$ , we obtain the $i$ th disruption scale

(6.5) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D706}}_{\text{D},i}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}^{1/9}\left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{4/9}\sim \left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{(1/2)(1-\frac{1}{9^{i}})}.\end{eqnarray}$$

Unlike in the resistive case (Mallet et al. Reference Mallet, Schekochihin and Chandran2017), this sequence will always terminate after a finite number of disruptions, because $\unicode[STIX]{x1D70C}_{s}>d_{e}$ and so eventually $\unicode[STIX]{x1D706}_{\text{D},i}<\unicode[STIX]{x1D70C}_{s}$ . The number of disruptions is given by the greatest $i=i_{\text{max}}$ for which

(6.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D706}_{\text{D},i}}{\unicode[STIX]{x1D70C}_{s}}\sim \left(\frac{d_{e}}{\unicode[STIX]{x1D70C}_{s}}\right)^{1/2(1-(1/9^{i}))}\left(\frac{L_{\bot }}{\unicode[STIX]{x1D70C}_{s}}\right)^{1/9^{i}}>1.\end{eqnarray}$$

It is obvious from the exponent of $L_{\bot }/\unicode[STIX]{x1D70C}_{s}$ in (6.6) that, for there to be more than one disruption, $L_{\bot }/\unicode[STIX]{x1D70C}_{s}$ must be unrealistically large. Namely, (6.6) may be solved for $i_{\text{max}}$ , showing an extremely weak dependence on $L_{\bot }/\unicode[STIX]{x1D70C}_{s}$ :

(6.7) $$\begin{eqnarray}i_{\text{max}}=\frac{\ln \left[1+{\displaystyle \frac{2\ln (L_{\bot }/\unicode[STIX]{x1D70C}_{s})}{\ln (\unicode[STIX]{x1D70C}_{s}/d_{e})}}\right]}{2\ln 3}.\end{eqnarray}$$

With $d_{e}/\unicode[STIX]{x1D70C}_{s}$ kept constant as $L_{\bot }/\unicode[STIX]{x1D70C}_{s}\rightarrow \infty$ , the number of disruptions grows extremely slowly, so, in a moderately low- $\unicode[STIX]{x1D6FD}_{e}$ situation where $d_{e}/\unicode[STIX]{x1D70C}_{s}<1$ , it is very unlikely that more than one disruption will occur.

6.2 Effective spectral index below $\unicode[STIX]{x1D706}_{\text{D}}$ : many disruptions

Nevertheless, in the spirit of asymptotic fantasising, let us determine the effective spectral index in a scale range featuring many disruptions. With the substitutions (6.4), the characteristic fluctuation amplitude just below each disruption scale is [cf. (6.2)]

(6.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{i,-}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D},i}^{1/3}.\end{eqnarray}$$

Therefore, the fluctuation amplitude associated with the $i$ th aligning ‘mini-cascade’ between disruption scales $\hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}$ and $\hat{\unicode[STIX]{x1D706}}_{\text{D},i}$ is

(6.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{i}\sim \unicode[STIX]{x1D6FF}\hat{z}_{-,i-1}\left(\frac{\hat{\unicode[STIX]{x1D706}}}{\hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}}\right)^{1/4}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}^{1/3}\left(\frac{\hat{\unicode[STIX]{x1D706}}}{\hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}}\right)^{1/4}.\end{eqnarray}$$

The characteristic aspect ratio of the turbulence (equivalently, the inverse alignment angle) between disruptions is then

(6.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D709}_{i}}{\unicode[STIX]{x1D706}}\sim \left(\frac{\hat{\unicode[STIX]{x1D706}}}{\hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}}\right)^{-1/4},\end{eqnarray}$$

which is much smaller than the value ( $\hat{\unicode[STIX]{x1D706}}^{-1/4}$ ) that would have been attained without the disruptions. The ‘coarse-grained’ fluctuation amplitude calculated at the scale just above the $i$ th disruption is [cf. (6.3)]

(6.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{i,+}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}^{1/3}\left(\frac{\hat{\unicode[STIX]{x1D706}}_{\text{D},i}}{\hat{\unicode[STIX]{x1D706}}_{\text{D},i-1}}\right)^{1/4}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D},i}\left(\frac{d_{e}\unicode[STIX]{x1D70C}_{s}}{L_{\bot }^{2}}\right)^{-1/3},\end{eqnarray}$$

where we have used (6.5) to obtain the second expression. Since (6.11) is larger than (6.8), our model spectrum looks like a sloping staircase, with (6.11) providing an upper envelope for the true scaling of the fluctuation amplitude (the true spectrum and structure function will not, of course, have discontinuities). Thus, $\unicode[STIX]{x1D6FF}z_{+,i}\propto \unicode[STIX]{x1D706}_{\text{D},i}$ , and so the effective scaling exponent of the fluctuation amplitudes is $\unicode[STIX]{x1D6FC}_{\text{eff}}=1$ . This implies a spectral index of $-1-2\unicode[STIX]{x1D6FC}_{\text{eff}}=-3$ , slightly steeper than the observed spectral index ${\approx}-2.8$ below the ion scales in strong kinetic Alfvén wave turbulence in the solar wind (Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Chen et al. Reference Chen, Horbury, Schekochihin, Wicks, Alexandrova and Mitchell2010; Sahraoui et al. Reference Sahraoui, Goldstein, Belmont, Canu and Rezeau2010).

Exactly the same scaling is (of course) found by observing that $\unicode[STIX]{x1D70F}_{\text{C}}\sim \unicode[STIX]{x1D6FE}_{{>}}^{-1}$ on the coarse-grained points $\hat{\unicode[STIX]{x1D706}}_{\text{D},i}$ : then, constancy of energy flux through all scales implies that (cf. Boldyrev & Loureiro Reference Boldyrev and Loureiro2017)

(6.12) $$\begin{eqnarray}\unicode[STIX]{x1D716}\sim \frac{\unicode[STIX]{x1D6FF}z_{i,+}^{2}}{\unicode[STIX]{x1D70F}_{\text{C}}}\sim \unicode[STIX]{x1D6FE}_{{>}}\unicode[STIX]{x1D6FF}z_{i,+}^{2}\propto \frac{\unicode[STIX]{x1D6FF}z_{i,+}^{3}}{\unicode[STIX]{x1D706}_{\text{D},i}^{3}}\sim \text{const.}\quad \Rightarrow \quad \unicode[STIX]{x1D6FF}z_{+,i}\propto \unicode[STIX]{x1D706}_{\text{D},i}.\end{eqnarray}$$

6.3 Effective spectral index below $\unicode[STIX]{x1D706}_{\text{D}}$ : one disruption

For realistic values of $L_{\bot }$ and $\unicode[STIX]{x1D70C}_{s}$ , there is only one disruption, i.e. $i_{\text{max}}=1$ . The aligning cascade below $\unicode[STIX]{x1D706}_{\text{D}}$ then gives an amplitude at $\unicode[STIX]{x1D70C}_{s}$ of

(6.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{z}_{\unicode[STIX]{x1D70C}_{s}}\sim \hat{\unicode[STIX]{x1D706}}_{\text{D}}^{1/3}\left(\frac{\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x1D706}_{\text{D}}}\right)^{1/4},\end{eqnarray}$$

using (6.9). The characteristic aspect ratio of the turbulence at $\unicode[STIX]{x1D70C}_{s}$ is, using (6.10),

(6.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x1D70C}_{s}}\sim \left(\frac{\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x1D706}_{\text{D}}}\right)^{-1/4}\sim \left(\frac{m_{i}}{m_{e}}\frac{\unicode[STIX]{x1D6FD}_{e}}{Z}\right)^{-1/18}\left(\frac{\unicode[STIX]{x1D70C}_{s}}{L_{\bot }}\right)^{-1/36},\end{eqnarray}$$

which is approximately unity for any realistic set of parameters. Therefore, the turbulence at the ion scale, i.e. at the largest scales in the kinetic Alfvén wave cascade, will be very different depending on the presence or absence of the disruption: namely, it will either be nearly isotropic (in the perpendicular plane), as per (6.10), or highly anisotropic (aligned) with aspect ratio $(\unicode[STIX]{x1D70C}_{s}/L_{\bot })^{-1/4}$ , respectively. In reality, there may be a mixture of both types of structures, as is suggested by the discussion in § 5.1 – so the reduction in the alignment may not be as drastic as suggested by the extreme estimate (6.14).

The effective scaling of the fluctuation amplitudes between $\unicode[STIX]{x1D706}_{\text{D}}$ and $\unicode[STIX]{x1D70C}_{s}$ will be steeper than $\unicode[STIX]{x1D6FC}_{\text{eff}}=1$ derived in § 6.2, because the recursive disruptions are cut off by the presence of the ion scale $\unicode[STIX]{x1D70C}_{s}$ . The effective scaling in this case is given by

(6.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\text{eff}}=\frac{\log (\unicode[STIX]{x1D6FF}\hat{z}_{1,+}/\unicode[STIX]{x1D6FF}\hat{z}_{\unicode[STIX]{x1D70C}_{s}})}{\log (\unicode[STIX]{x1D706}_{\text{D}}/\unicode[STIX]{x1D70C}_{s})}.\end{eqnarray}$$

This ranges between $1\leqslant \unicode[STIX]{x1D6FC}_{\text{eff}}<\infty$ , increasing the closer $\unicode[STIX]{x1D70C}_{s}$ gets to $\unicode[STIX]{x1D706}_{\text{D}}$ . Thus the effective spectral index in a ‘realistic’ short range of scales between $\unicode[STIX]{x1D706}_{\text{D}}$ and $\unicode[STIX]{x1D70C}_{s}$ , with only one disruption, may be somewhat steeper than $-3$ , and may depend on $\unicode[STIX]{x1D6FD}_{e}$ (i.e. it is not universal). Here again, the caveat that disruption does not in fact occur at a single scale or in every aligned structure implies that the spectrum should not be as dramatically steepened as (6.15) suggests. A spectrum slightly steeper than $-3$ is probably a reasonable expectation.

Short intervals of steep spectra are indeed sometimes observed near the ion scales in the solar wind: Sahraoui et al. (Reference Sahraoui, Goldstein, Belmont, Canu and Rezeau2010) and Lion et al. (Reference Lion, Alexandrova and Zaslavsky2016) report a spectral index close to $-4$ in a small ‘transition range’ of scales near the ion gyroradius. Lion et al. (Reference Lion, Alexandrova and Zaslavsky2016) attribute this spectral index to Alfvén vortices (Alexandrova Reference Alexandrova2008) with scales a few times the ion gyroradius.