2.1 Physical interpretation of the wave kinetic equation

Figure 1 offers a way of understanding (2.11). The horizontal colour bars in this figure represent the spectra of outward-propagating and inward-propagating AWs, with red representing longer-wavelength waves and violet representing shorter-wavelength waves. AWs propagating in the $-\boldsymbol{B}_{0}$ direction at $|k_{z}|=k_{z3}$ decay into slow waves propagating anti-parallel to  $\boldsymbol{B}_{0}$ at $|k_{z}|\simeq 2k_{z3}$ and AWs propagating parallel to  $\boldsymbol{B}_{0}$ at $|k_{z}|=k_{z2}$ . AWs propagating parallel to  $\boldsymbol{B}_{0}$ at $|k_{z}|=k_{z2}$ decay into slow waves propagating parallel to $\boldsymbol{B}_{0}$ at $|k_{z}|\simeq 2k_{z2}$ and AWs propagating anti-parallel to $\boldsymbol{B}_{0}$ at $|k_{z}|=k_{z1}$ . Equation (2.11) is approximately equivalent to the statement that the rate at which $E_{2}^{+}=E^{+}(k_{z2})$ increases via the decay of AWs at $|k_{z}|=k_{z3}$ is

(2.13) $$\begin{eqnarray}R_{3\rightarrow 2}\sim \frac{k_{z2}E_{2}^{+}k_{z3}E_{3}^{-}}{\unicode[STIX]{x1D6FD}^{1/2}v_{\text{A}}},\end{eqnarray}$$

where $E_{3}^{-}=E^{-}(k_{z3})$ , while the rate at which $E_{2}^{+}$ decreases via the decay of AWs at $|k_{z}|=k_{z2}$ is

(2.14) $$\begin{eqnarray}R_{2\rightarrow 1}\sim \frac{k_{z2}E_{2}^{+}k_{z1}E_{1}^{-}}{\unicode[STIX]{x1D6FD}^{1/2}v_{\text{A}}},\end{eqnarray}$$

where $E_{1}^{-}=E^{-}(k_{z1})$ . The time derivative of $E_{2}^{+}$ is $R_{3\rightarrow 2}-R_{2\rightarrow 1}$ , or

(2.15) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}E_{2}^{+}}{\unicode[STIX]{x2202}t}\sim \frac{k_{z2}E_{2}^{+}(k_{z3}E_{3}^{-}-k_{z1}E_{1}^{-})}{\unicode[STIX]{x1D6FD}^{1/2}v_{\text{A}}}.\end{eqnarray}$$

Equation (2.8) implies that $k_{z3}-k_{z1}\sim k_{z2}c_{\text{s}}/v_{\text{A}}\sim \unicode[STIX]{x1D6FD}^{1/2}k_{z2}$ . A Taylor expansion of $k_{z3}E_{3}^{-}$ and $k_{z1}E_{1}^{-}$ about $k_{z2}$ in (2.15) thus allows this equation to be rewritten as

(2.16) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}E_{2}^{+}}{\unicode[STIX]{x2202}t}\sim \frac{k_{z2}^{2}E_{2}^{+}}{v_{\text{A}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k_{z}}(k_{z}E^{-})\right|_{k_{z}=k_{z2}},\end{eqnarray}$$

which is the same as (2.11) to within a factor of order unity.

To be clear, no independent derivation is being presented for (2.13) and (2.14). The foregoing discussion merely points out that (2.13) and (2.14) are equivalent (up to a factor of order unity) to (2.11), which is derived on the basis of weak-turbulence theory. It is worth pointing out, however, that several features of (2.13) and (2.14) make sense on a qualitative level. If either $E^{+}=0$ or $E^{-}=0$ , then $R_{3\rightarrow 2}=R_{2\rightarrow 1}=0$ , because the parametric instability is a stimulated decay, which ceases if initially all the AWs travel in the same direction. For fixed $E^{+}$ and  $E^{-}$ , $R_{3\rightarrow 2}$ and $R_{2\rightarrow 1}$ vanish as $v_{\text{A}}\rightarrow \infty$ , since the fractional nonlinearities vanish in this limit. Also, $R_{3\rightarrow 2}$ and $R_{2\rightarrow 1}$ are proportional to  $\unicode[STIX]{x1D6FD}^{-1/2}$ (when $S_{k}^{\pm }$ is negligibly small, as assumed) because the parametric-decay contribution to $\unicode[STIX]{x2202}A_{k}^{\pm }/\unicode[STIX]{x2202}t$ is an integral (over $\boldsymbol{p}$ and  $\boldsymbol{q}$ ) of third-order correlation functions such as $\langle \unicode[STIX]{x1D6FF}\boldsymbol{v}_{k}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{B}_{q}\unicode[STIX]{x1D6FF}n_{p}\rangle$ , where $\unicode[STIX]{x1D6FF}\boldsymbol{v}_{k}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{q}$ are the velocity and magnetic-field fluctuations associated with AWs at wave vectors  $\boldsymbol{k}$ and  $\boldsymbol{q}$ , and $\unicode[STIX]{x1D6FF}n_{p}$ is the density fluctuation associated with the slow waves at wave vector  $\boldsymbol{p}$ that are driven by the beating of the AWs at wave vectors  $\boldsymbol{k}$ and  $\boldsymbol{q}$ . For fixed AW amplitudes and fixed  $B_{0}$ and  $v_{\text{A}}$ , this driven density fluctuation is proportional to $\unicode[STIX]{x1D6FD}^{-1/2}$ , because as $\unicode[STIX]{x1D6FD}$ decreases the thermal pressure is less able to resist the compression along  $\boldsymbol{B}_{0}$ resulting from the Lorentz force that arises from the beating of the AWs.

5.1 Decaying, balanced turbulence

One family of exact solutions to (2.11) follows from setting

(5.1) $$\begin{eqnarray}E^{\pm }(k_{z},t)=f^{\pm }(k_{z},t)H(k_{z}-b(t))\end{eqnarray}$$

in (2.11), where

(5.2) $$\begin{eqnarray}H(x)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }x<0,\\ 1\quad & \text{if }x\geqslant 0,\end{array}\right.\end{eqnarray}$$

is the Heaviside function. When (5.1) is substituted into (2.11), each side of (2.11) becomes the sum of terms proportional to  $\unicode[STIX]{x1D6FF}(k_{z}-b)$ and terms that contain no delta function. By separately equating the two groups of terms, one can show that (5.1) is a solution to (2.11) if

(5.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}f^{\pm }(k_{z},t)=\frac{\unicode[STIX]{x03C0}}{v_{\text{A}}}k_{z}^{2}f^{\pm }(k_{z},t)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k_{z}}[k_{z}f^{\mp }(k_{z},t)],\end{eqnarray}$$

and

(5.4) $$\begin{eqnarray}\frac{1}{b^{3}}\frac{\text{d}b}{\text{d}t}=-\frac{\unicode[STIX]{x03C0}f^{+}(b,t)}{2v_{\text{A}}}=-\frac{\unicode[STIX]{x03C0}f^{-}(b,t)}{2v_{\text{A}}}.\end{eqnarray}$$

Equation (5.4) makes use of the relation $[H(x)]^{2}=H(x)$ and its derivative, $2H(x)\unicode[STIX]{x1D6FF}(x)=\unicode[STIX]{x1D6FF}(x)$ . In appendix A it is shown that (5.4) can be recovered by adding a small amount of nonlinear diffusion to (2.11) and replacing the discontinuous jump in the spectrum at $k_{z}=b(t)$ with a boundary layer. Equation (5.4) implies that, for solutions of the form given in (5.1), the mean-square amplitudes of forward- and backward-propagating AWs must be equal just above the break wavenumber  $b$ . An exact solution to (5.3) and (5.4) corresponding to decaying turbulence is

(5.5) $$\begin{eqnarray}f^{+}(k_{z},t)=f^{-}(k_{z},t)=\frac{a(t)}{k_{z}^{2}},\end{eqnarray}$$

(5.6) $$\begin{eqnarray}a(t)=a_{0}\left(1+\frac{\unicode[STIX]{x03C0}a_{0}t}{v_{\text{A}}}\right)^{-1},\end{eqnarray}$$

and

(5.7) $$\begin{eqnarray}b(t)=b_{0}\left(1+\frac{\unicode[STIX]{x03C0}a_{0}t}{v_{\text{A}}}\right)^{-1/2},\end{eqnarray}$$

where $a_{0}$ and $b_{0}$ are the values of $a$ and  $b$ at $t=0$ .

This solution can be further truncated at large  $k_{z}$ by setting

(5.8) $$\begin{eqnarray}E^{+}(k_{z},t)=E^{-}(k_{z},t)=\frac{a(t)H(k_{z}-b(t))H(q(t)-k_{z})}{k_{z}^{2}}\end{eqnarray}$$

with

(5.9) $$\begin{eqnarray}q(t)=q_{0}\left(1+\frac{\unicode[STIX]{x03C0}a_{0}t}{v_{\text{A}}}\right)^{-1/2},\end{eqnarray}$$

where $q_{0}$ is the value of  $q$ at $t=0$ , which is taken to exceed  $b_{0}$ . Equations (5.5) through (5.9) can be recovered numerically by solving (2.11) for freely decaying AWs. Whether the spectra satisfy (5.1) and (5.5) through (5.7) or, alternatively, equations (5.6) through (5.9), the number of wave quanta $N_{\text{tot}}$ defined in (4.6) is finite and independent of time.

5.2 Forced, balanced turbulence

An exact solution to (5.3) and (5.4) corresponding to forced turbulence is

(5.10) $$\begin{eqnarray}f^{+}(k_{z},t)=f^{-}(k_{z},t)=\frac{c}{k_{z}},\end{eqnarray}$$

and

(5.11) $$\begin{eqnarray}b(t)=\left(\frac{\unicode[STIX]{x03C0}ct}{2v_{\text{A}}}+\frac{1}{b_{0}}\right)^{-1},\end{eqnarray}$$

where $c$ is a constant and $b_{0}$ is the value of  $b$ at  $t=0$ . In this solution, the number of wave quanta  $N_{\text{tot}}$ is not constant, because there is a non-zero influx of wave quanta from infinity. A version of this solution can be realized in a numerical solution of (2.11) by holding $E^{\pm }$ fixed at some wavenumber $k_{\text{f}}$ , which mimics the effects of energy input from external forcing. In this case, the numerical solution at $k_{z}<k_{\text{f}}$ is described by (5.1), (5.10) and (5.11), with $b(t)<k_{\text{f}}$ .

The solution in (5.10) and (5.11) can be truncated at large  $k_{z}$ in a manner analogous to (5.8), but with $q=[(\unicode[STIX]{x03C0}ct/2v_{\text{A}})+(1/q_{0})]^{-1}$ , where $q_{0}$ is the value of  $q$ at $t=0$ . In this solution, $N_{\text{tot}}$ is independent of time. Numerical solutions of (2.11) show, however, that this solution is unstable. If the spectra initially satisfy $E^{\pm }=(c/k_{z})H(k_{z}-b)H(q-k_{z})$ , then they evolve towards the solution described by (5.5) through (5.9).

5.3 Exact solutions extending over all  $k_{z}$

In addition to the truncated solutions described in §§ 5.1 and 5.2, equation (2.11) possesses several exact solutions that extend over all  $k_{z}$ . These solutions are unphysical, because they correspond to infinite AW energy and neglect dissipation (which becomes important at sufficiently large  $k_{z}$ ) and finite system size (which becomes important at sufficiently small  $k_{z}$ ). However, they illustrate several features of the nonlinear evolution of the parametric instability, which are summarized at the end of this section.

The simplest solution to (2.11) spanning all  $k_{z}$ is

(5.12) $$\begin{eqnarray}E^{\pm }(k_{z},t)=\frac{c^{\pm }}{k_{z}},\end{eqnarray}$$

where $c^{\pm }$ is a constant. It follows from (4.5) that (5.12) corresponds to a constant flux of AW quanta to smaller  $k_{z}$ . In contrast to the truncated $E^{\pm }\propto k_{z}^{-1}$ forced-turbulence solution in § 5.2, $E^{+}$ and $E^{-}$ need not be equal in (5.12).

A second, non-truncated, exact solution to (2.11) is given by

(5.13) $$\begin{eqnarray}E^{\pm }(k_{z},t)=\frac{a^{\pm }(t)}{k_{z}^{2}},\end{eqnarray}$$

and

(5.14) $$\begin{eqnarray}a^{\pm }(t)=\frac{a_{0}^{\pm }(a_{0}^{\pm }-a_{0}^{\mp })}{a_{0}^{\pm }-a_{0}^{\mp }\text{e}^{-\unicode[STIX]{x03C0}(a_{0}^{\pm }-a_{0}^{\mp })t/v_{\text{A}}}},\end{eqnarray}$$

where $a_{0}^{+}$ and $a_{0}^{-}$ are the initial values of $a^{+}$ and  $a^{-}$ . In this solution,

(5.15) $$\begin{eqnarray}a^{+}(t)-a^{-}(t)=a_{0}^{+}-a_{0}^{-}.\end{eqnarray}$$

If $a_{0}^{+}>a_{0}^{-}$ , then $E^{-}$ decays faster than $E^{+}$ , and, after a long time has passed, $E^{-}$ decays to zero while $a^{+}$ decays to the value $a_{0}^{+}-a_{0}^{-}$ . Conversely, if $a_{0}^{-}>a_{0}^{+}$ , then $E^{+}$ decays faster than  $E^{-}$ , and the turbulence decays to a state in which $E^{+}=0$ . In the limit that $a_{0}^{+}\rightarrow a_{0}^{-}$ ,

(5.16) $$\begin{eqnarray}a^{\pm }(t)\rightarrow a_{0}\left(1+\frac{\unicode[STIX]{x03C0}a_{0}t}{v_{\text{A}}}\right)^{-1},\end{eqnarray}$$

where $a_{0}=a_{0}^{+}=a_{0}^{-}$ . Equations (5.13) and (5.16) are a non-truncated version of the decaying-turbulence solution presented in § 5.1.

Equations (5.12) and (5.13) can be combined into a more general class of solution,

(5.17) $$\begin{eqnarray}E^{\pm }(k_{z},t)=a^{\pm }(t)\left(\frac{1}{k_{z}^{2}}+\frac{d^{\pm }}{k_{z}}\right),\end{eqnarray}$$

where $d^{+}$ and $d^{-}$ are constants and $a^{\pm }(t)$ is given by (5.14). Another type of solution combining $k_{z}^{-1}$ and $k_{z}^{-2}$ scalings is

(5.18) $$\begin{eqnarray}\displaystyle & \displaystyle E^{+}(k_{z},t)=\frac{c_{0}\text{e}^{-\unicode[STIX]{x03C0}c_{2}t/v_{\text{A}}}}{k_{z}}, & \displaystyle\end{eqnarray}$$

(5.19) $$\begin{eqnarray}\displaystyle & \displaystyle E^{-}(k_{z},t)=\frac{c_{1}}{k_{z}}+\frac{c_{2}}{k_{z}^{2}}, & \displaystyle\end{eqnarray}$$

where $c_{0}$ , $c_{1}$ , and  $c_{2}$ are constants.

The exact solutions presented in this section illustrate three properties of the nonlinear evolution of the parametric instability at low  $\unicode[STIX]{x1D6FD}$ when slow waves are strongly damped. First, when $E^{\pm }\propto k_{z}^{-1}$ , $\unicode[STIX]{x2202}E^{\mp }/\unicode[STIX]{x2202}t$  vanishes. Second, if $E^{\pm }\propto k_{z}^{-2}$ , then $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t)\ln E^{\mp }$ is negative and independent of  $k_{z}$ , and $E^{\mp }(k_{z},t)$ can be written as the product of a function of $k_{z}$ and a (decreasing) function of time. (More general principles describing the evolution of  $E^{\pm }$ are summarized in figure 3 and (6.11).) Third, the parametric instability does not necessarily saturate with $E^{+}=E^{-}$ . For example, in (5.13) and (5.14), when $a_{0}^{+}\neq a_{0}^{-}$ , the AWs decay to a maximally aligned state reminiscent of the final state of decaying cross-helical incompressible MHD turbulence (Dobrowolny, Mangeney & Veltri Reference Dobrowolny, Mangeney and Veltri1980).

6.1 Heuristic explanation of the $e^{+}\propto f^{-1}$ and $e^{-}\propto f^{-2}$ scalings

In order to understand the time evolution illustrated in figure 2, it is instructive to first consider the case in which

(6.10) $$\begin{eqnarray}E^{\pm }=c^{\pm }k_{z}^{\unicode[STIX]{x1D6FC}^{\pm }}\end{eqnarray}$$

within some interval $(k_{z1},k_{z2})$ , where $c^{\pm }$ and $\unicode[STIX]{x1D6FC}^{\pm }$ are constants. Equation (2.11) implies that, within this interval,

(6.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\ln E^{\pm }=\frac{\unicode[STIX]{x03C0}c^{\mp }}{v_{\text{A}}}(1+\unicode[STIX]{x1D6FC}^{\mp })k_{z}^{\unicode[STIX]{x1D6FC}^{\mp }+2}.\end{eqnarray}$$

If $\unicode[STIX]{x1D6FC}^{\mp }>-1$ , then $\ln E^{\pm }$ grows at a rate that increases with  $k_{z}$ , causing $E^{\pm }$ to increase and ‘harden’, in the sense that the best-fit value of $\unicode[STIX]{x1D6FC}^{\pm }$ within the interval $(k_{z1},k_{z2})$ increases. If $\unicode[STIX]{x1D6FC}^{\mp }=-1$ , then $E^{\pm }$ does not change. If $-2<\unicode[STIX]{x1D6FC}^{\mp }<-1$ , then $\ln E^{\pm }$ decreases at a rate that increases with  $k_{z}$ , which causes the best-fit value of  $\unicode[STIX]{x1D6FC}^{\pm }$ within the interval $(k_{z1},k_{z2})$ to decrease. If $\unicode[STIX]{x1D6FC}^{\mp }=-2$ , then $\ln E^{\pm }$ decreases at the same rate at all  $k_{z}$ , and $\unicode[STIX]{x1D6FC}^{\pm }$ remains unchanged. Finally, if $\unicode[STIX]{x1D6FC}^{\mp }<2$ , then $E^{\pm }$ decreases at a rate that decreases with  $k_{z}$ , which causes the best-fit value of  $\unicode[STIX]{x1D6FC}^{\pm }$ in the interval $(k_{z1},k_{z2})$ to increase. These rules are summarized in figure 3 and apply to $e^{\pm }\propto f^{\unicode[STIX]{x1D6FC}^{\pm }}$ as well as $E^{\pm }\propto k_{z}^{\unicode[STIX]{x1D6FC}^{\pm }}$ .

Returning to figure 2, in the early stages of the numerical calculation, $e^{-}$ grows most rapidly at those frequencies at which $\unicode[STIX]{x1D6FE}^{-}$ in (3.1) is largest – namely, the high- $f$ end of the $f^{-0.5}$ range of $e^{+}$ . By the time $e^{-}$ reaches a sufficient amplitude that $e^{+}$ and $e^{-}$ evolve on the same time scale, $e^{-}$ develops a peaked frequency profile extending from some frequency $f=f_{\text{low}}$ to some larger frequency $f=f_{\text{high}}$ , as illustrated in figure 2(b). Near $f_{\text{low}}$ , $\text{d}e^{-}/\text{d}f>0$ (i.e. $\unicode[STIX]{x1D6FC}^{-}>0$ ), which causes $e^{+}$ to grow.Footnote ³ Near $f_{\text{high}}$ , $\unicode[STIX]{x1D6FC}^{-}<-1$ , which causes $e^{+}$ to decrease. Thus, $e^{+}$ steepens across the interval $(f_{\text{low}},f_{\text{high}})$ until it attains a $1/f$ scaling, at which point $e^{-}$ stops growing between $f_{\text{low}}$ and $f_{\text{high}}$ . However, at frequencies just below $f_{\text{low}}$ , $e^{+}$ and $e^{-}$ both continue to grow, causing $f_{\text{low}}$ to decrease. At the same time, $e^{+}$ continues to decrease at larger $f$ where $\unicode[STIX]{x1D6FC}^{-}<-1$ . Together, the growth of $e^{+}$ just below $f_{\text{low}}$ and the damping of $e^{+}$ at larger  $f$ cause the $f^{-1}$ range of $e^{+}$ to broaden in both directions, i.e. towards both smaller and larger frequencies.

The unique scaling of $e^{-}$ consistent with an $e^{+}$ spectrum $\propto f^{-1}$ that is a decreasing function of time is $e^{-}\propto f^{-2}$ . Moreover, the scalings $e^{+}\sim f^{-1}$ and $e^{-}\sim f^{-2}$ are, in a sense, stable, as can be inferred from figure 3. For example, if $\unicode[STIX]{x1D6FC}^{-}$ increases from $-2$ to a slightly larger value, then $e^{+}$ decreases at a rate that increases with  $f$ , causing $\unicode[STIX]{x1D6FC}^{+}$ to decrease to a value slightly below  $-1$ . This causes $e^{-}$ to decrease at a rate that increases with  $f$ , thereby causing $\unicode[STIX]{x1D6FC}^{-}$ to decrease back towards  $-2$ . A similar ‘spectral restoring force’ arises for any other small perturbation to the values $\unicode[STIX]{x1D6FC}^{+}=-1$ and $\unicode[STIX]{x1D6FC}^{-}=-2$ .

It is worth emphasizing, in this context, that the analytic solution presented in appendix B is approximate rather than exact. As the spectral break frequency decreases past some fixed frequency $f_{3}$ , the values of $e^{+}$ and $e^{-}$ at $f_{3}$ suddenly jump, but they do not jump to the precise values needed to extend the $e^{+}\sim f^{-1}$ and $e^{-}\sim f^{-2}$ scalings to smaller  $f$ . Instead, the spectra need further ‘correcting’ after the break frequency has swept past in order to maintain the scalings $e^{+}\sim f^{-1}$ and $e^{-}\sim f^{-2}$ in an approximate way. Also, the decrease in $e^{-}$ that occurs after $t=4~\text{h}$ is a consequence of the sub-dominant $f^{-2}$ component of  $e^{+}$ . This component of  $e^{+}$ becomes increasingly prominent near the break frequency as time progresses, leading to the pronounced curvature in the plot of $e^{+}$ near $f=10^{-4}~\text{Hz}$ in figure 4(c).

6.2 Comparison with Helios measurements

In the (average) plasma rest frame, the equations of incompressible MHD can be written in the form

(6.12) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{z}^{\pm }}{\unicode[STIX]{x2202}t}+(\boldsymbol{z}^{\mp }\pm \boldsymbol{v}_{\text{A}})\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{z}^{\pm }=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F1},\end{eqnarray}$$

where $\boldsymbol{z}^{\pm }=\unicode[STIX]{x1D6FF}\boldsymbol{v}\mp \unicode[STIX]{x1D6FF}\boldsymbol{B}/\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}$ are the Elsasser variables, $\unicode[STIX]{x1D6FF}\boldsymbol{v}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ are the velocity and magnetic-field fluctuations, $\unicode[STIX]{x1D70C}$ is the mass density, $\boldsymbol{v}_{\text{A}}=\boldsymbol{B}_{0}/\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}$ is the Alfvén velocity and $\unicode[STIX]{x1D6F1}$ is the total pressure divided by  $\unicode[STIX]{x1D70C}$ (Elsasser Reference Elsasser1950). Although the solar wind is compressible, equation (6.12) provides a reasonable approximation for the non-compressive, AW-like component of solar-wind turbulence. As (6.12) shows, the advection velocity of a $\boldsymbol{z}^{\pm }$ fluctuation is $\boldsymbol{z}^{\mp }\pm \boldsymbol{v}_{\text{A}}$ . This implies, as shown by Maron & Goldreich (Reference Maron and Goldreich2001), that $\boldsymbol{z}^{\pm }$ fluctuations propagate along magnetic field lines perturbed by $\boldsymbol{z}^{\mp }$ . As a consequence, in the solar wind, when the root-mean-square (rms) magnetic-field fluctuation $\unicode[STIX]{x1D6FF}B_{\text{in}}$ associated with inward-propagating AWs ( $\boldsymbol{z}^{-}$ ) is much smaller than the background magnetic field  $B_{0}$ , the outward-propagating AWs ( $\boldsymbol{z}^{+}$ ) propagate to a good approximation along the direction of  $\boldsymbol{B}_{0}$ . This is true even if the rms magnetic-field fluctuation  $\unicode[STIX]{x1D6FF}B_{\text{out}}$ associated with $\boldsymbol{z}^{+}$ is comparable to  $B_{0}$ . In the fast solar wind at $r<0.3~\text{AU}$ , the (fractional) cross-helicity is high (i.e. $E^{+}\gg E^{-}$ ), and $\unicode[STIX]{x1D6FF}B_{\text{in}}$ is indeed small compared to  $B_{0}$ (Bavassano, Pietropaolo & Bruno Reference Bavassano, Pietropaolo and Bruno2000; Cranmer & van Ballegooijen Reference Cranmer and van Ballegooijen2005). Moreover, the background magnetic field at $r=0.3~\text{AU}$ is nearly in the radial direction, because the Parker-spiral magnetic field begins to deviate appreciably from the radial direction only at larger  $r$ in the fast wind (Verscharen et al. Reference Verscharen, Chandran, Bourouaine and Hollweg2015). Hence, in high-cross-helicity fast-wind streams at $r=0.3~\text{AU}$ , the function $e^{+}$ defined by (6.3) and (6.4) corresponds to a good approximation to the frequency spectrum of outward-propagating AWs observed by a spacecraft in the solar wind. It is not clear, however, how well $e^{-}$ corresponds to the observed spectrum of inward-propagating AWs, because the inward-propagating AWs follow field lines perturbed by the outward-propagating AWs, which can be inclined relative to the radial direction by a substantial angle.

Figure 4 reproduces the $t=4~\text{h}$ and $t=8~\text{h}$ panels of figure 2, but with the same axis ranges as those in figure 2–2(c) of Tu & Marsch (Reference Tu and Marsch1995) to facilitate comparison. Figure 4 also includes a third panel that shows the spectra at $t=32~\text{h}$ . The $e^{+}$ spectrum in the $t=8~\text{h}$ panel of figure 2 shares a number of properties with the $e^{+}$ spectrum in figure 2–2(c) of Tu & Marsch (Reference Tu and Marsch1995), in addition to the $f^{-0.5}$ scaling at small  $f$ that was built in to the numerical calculation as an initial condition. In particular, $e^{+}\gg e^{-}$ at all frequencies, $e^{+}\sim f^{-1}$ at $f\gtrsim 3\times 10^{-4}~\text{Hz}$ , and there is a bump in the $e^{+}$ spectrum at the transition between the $f^{-0.5}$ and $f^{-1}$ scaling ranges of  $e^{+}$ .

Although this comparison is suggestive, it is not entirely clear how to map time in the numerical calculation to heliocentric distance in the solar wind, because the plasma parameters in the numerical calculation are independent of position and time, whereas they depend strongly upon heliocentric distance in the solar wind. For example, the turbulence is weaker (in the sense of smaller $\unicode[STIX]{x1D6FF}v_{0}/v_{\text{A}}$ ) the closer one gets to the Sun. (See also the discussion following (7.10).) Also, the choice of initial conditions in the numerical calculation artificially prolongs the linear stage of evolution, since in the solar wind there are sources of inward-propagating waves other than parametric instability, such as non-WKB reflection (Heinemann & Olbert Reference Heinemann and Olbert1980; Velli Reference Velli1993). Nevertheless, as a baseline for comparison, the travel time of an outward-propagating AW from the photosphere to $0.3~\text{AU}$ in the fast-solar-wind model developed by Chandran & Hollweg (Reference Chandran and Hollweg2009) is approximately 12 h.

7.1 The weak-turbulence approximation

A central assumption of the analysis is the weak-turbulence criterion in (2.1). Since $E^{+}$ and $E^{-}$ differ in the solar wind, equation (2.1) is really two conditions,

(7.1) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{nl}}^{\pm }\ll |k_{z}|v_{\text{A}},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\text{nl}}^{+}$ ( $\unicode[STIX]{x1D714}_{\text{nl}}^{-}$ ) is the inverse of the time scale on which nonlinear interactions modify outward-propagating (inward-propagating) AWs. The contribution to $\unicode[STIX]{x1D714}_{\text{nl}}^{\pm }$ from the parametric instability is

(7.2) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{\pm }\sim \frac{1}{E^{\pm }}\left|\frac{\unicode[STIX]{x2202}E^{\pm }}{\unicode[STIX]{x2202}t}\right|\sim \frac{k_{z}^{2}E^{\mp }}{v_{\text{A}}}.\end{eqnarray}$$

The contribution to $\unicode[STIX]{x1D714}_{\text{nl}}^{\pm }$ from one other type of nonlinear interaction is estimated in § 7.3. The estimate of $\unicode[STIX]{x2202}E^{\pm }/\unicode[STIX]{x2202}t$ in (7.2) follows from (2.11) and setting $E^{\mp }\sim |k_{z}|^{\unicode[STIX]{x1D6FC}^{\mp }}$ with $\unicode[STIX]{x1D6FC}^{\mp }$ not very close to  $-1$ . An approximate upper limit on $\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{\pm }$ results from replacing $k_{z}E^{\mp }$ in (7.2) with $(\unicode[STIX]{x1D6FF}v^{\mp })^{2}$ , where $(\unicode[STIX]{x1D6FF}v^{+})^{2}$ is the mean-square velocity fluctuation associated with outward-propagating AWs, and $(\unicode[STIX]{x1D6FF}v^{-})^{2}$ is the mean-square velocity fluctuation associated with inward-propagating AWs. This leads to an approximate upper limit on $\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{\pm }$ because $(\unicode[STIX]{x1D6FF}v^{\pm })^{2}$ includes contributions from all wavenumbers and is much larger than the value of $k_{z}E^{\pm }$ at some  $k_{z}$ . Equation (7.1), with $\unicode[STIX]{x1D714}_{\text{nl}}\sim \unicode[STIX]{x1D714}_{\text{nl},\text{PI}}$ , is thus satisfied provided

(7.3) $$\begin{eqnarray}(\unicode[STIX]{x1D6FF}v^{\mp })^{2}\ll v_{\text{A}}^{2}.\end{eqnarray}$$

Bavassano et al. (Reference Bavassano, Pietropaolo and Bruno2000) analysed Helios measurements of fluctuations in the fast solar wind at $r=0.4~\text{AU}$ and found that $(\unicode[STIX]{x1D6FF}v^{-})^{2}\ll (\unicode[STIX]{x1D6FF}v^{+})^{2}\simeq (60~\text{km}~\text{s}^{-1})^{2}$ . As mentioned above, the typical value of $v_{\text{A}}$ in the fast solar wind at $r=0.3~\text{AU}$ is ${\sim}150~\text{km}~\text{s}^{-1}$ (Marsch et al. Reference Marsch, Rosenbauer, Schwenn, Muehlhaeuser and Neubauer1982; Marsch & Tu Reference Marsch and Tu1990). Near $r=0.3~\text{AU}$ , $B_{0}\sim 1/r^{2}$ , $\unicode[STIX]{x1D70C}\sim 1/r^{2}$ and $v_{\text{A}}\sim 1/r$ , and so the typical value of $v_{\text{A}}$ in fast-solar-wind streams at $r=0.4~\text{AU}$ is ${\sim}112.5~\text{km}~\text{s}^{-1}$ . These measurements indicate that

(7.4) $$\begin{eqnarray}(\unicode[STIX]{x1D6FF}v^{-})^{2}\ll (\unicode[STIX]{x1D6FF}v^{+})^{2}\simeq 0.28v_{\text{A}}^{2}\end{eqnarray}$$

in the fast solar wind at $r=0.4~\text{AU}$ . Since $\unicode[STIX]{x1D6FF}v^{\pm }/v_{\text{A}}$ decreases as $r$ decreases below $0.4~\text{AU}$ (Cranmer & van Ballegooijen Reference Cranmer and van Ballegooijen2005; Chandran & Hollweg Reference Chandran and Hollweg2009), the condition $\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{+}\ll |k_{z}|v_{\text{A}}$ is well satisfied at $r<0.4~\text{AU}$ , and the condition $\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{-}\ll |k_{z}|v_{\text{A}}$ is at least marginally satisfied at $r<0.4~\text{AU}$ .

It is worth noting that weak-turbulence theory fails when applied to resonant interactions between three AWs, because such interactions occur only when one of the AWs has zero frequency, violating the weak-turbulence ordering (Schekochihin, Nazarenko & Yousef Reference Schekochihin, Nazarenko and Yousef2012; Meyrand, Kiyani & Galtier Reference Meyrand, Kiyani and Galtier2015). In contrast, the AW/slow-wave interactions in parametric decay do not involve a zero-frequency mode. Weak-turbulence theory is thus in principle a better approximation for the nonlinear evolution of the parametric instability than for incompressible MHD turbulence.

7.2 The low- $\unicode[STIX]{x1D6FD}$ assumption

The assumption that $\unicode[STIX]{x1D6FD}\ll 1$ is not satisfied at $r\gtrsim 0.3~\text{AU}\simeq 65R_{\odot }$ , where $\unicode[STIX]{x1D6FD}$ is typically ${\sim}1$ , but is reasonable at $r\lesssim 20R_{\odot }$ (Chandran et al. Reference Chandran, Dennis, Quataert and Bale2011). It is possible that the $\unicode[STIX]{x1D6FD}\ll 1$ theory presented here applies at least at a qualitative level provided $\unicode[STIX]{x1D6FD}$ is simply  ${\lesssim}1$ , and indeed this possibility motivates the comparison of the present model with Helios observations. However, further work is needed to investigate how the results of this paper are modified as $\unicode[STIX]{x1D6FD}$ increases to values ${\sim}1$ .

7.3 Neglect of other types of nonlinear interactions

Another approximation in §§ 2 through 6 is the neglect of all nonlinear interactions besides parametric decay. One of the neglected interactions is the shearing of inward-propagating AWs by outward-propagating AWs, which makes a contribution to $\unicode[STIX]{x1D714}_{\text{nl}}^{-}$ that depends on the perpendicular length scale of the AWs. At the perpendicular outer scale  $L_{\bot }$ (the overall correlation length of the AWs measured perpendicular to  $\boldsymbol{B}_{0}$ ), the contribution to $\unicode[STIX]{x1D714}_{\text{nl}}^{-}$ from shearing is approximately

(7.5) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{nl},\bot }^{-}\sim \frac{\unicode[STIX]{x1D712}\unicode[STIX]{x1D6FF}v^{+}}{L_{\bot }},\end{eqnarray}$$

where

(7.6) $$\begin{eqnarray}\unicode[STIX]{x1D712}=\frac{\unicode[STIX]{x1D6FF}v^{+}}{k_{z}L_{\bot }v_{\text{A}}}\end{eqnarray}$$

is the critical-balance parameter (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Ng & Bhattacharjee Reference Ng and Bhattacharjee1996; Lithwick, Goldreich & Sridhar Reference Lithwick, Goldreich and Sridhar2007). Equation (7.5) does not apply when $\unicode[STIX]{x1D712}$ is much larger than 1, but direct numerical simulations suggest that $\unicode[STIX]{x1D712}\lesssim 1$ at $r\gtrsim 10R_{\odot }$ for the bulk of the AW energy (J. Perez, private communication). Thus, at $r\gtrsim 10R_{\odot }$ ,

(7.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{-}}{\unicode[STIX]{x1D714}_{\text{nl},\bot }^{-}}\simeq (k_{z}L_{\bot })^{2}.\end{eqnarray}$$

As AWs propagate away from the Sun, they follow magnetic field lines, which leads to the approximate scaling $L_{\bot }\propto B_{0}^{-1/2}$ . In the WKB limit, the AW frequency in the Sun’s frame $k_{z}(U+v_{\text{A}})$ is independent of  $r$ . The scaling $k_{z}\sim 1/(U+v_{\text{A}})$ thus serves as a rough approximation for outward-propagating AWs in the turbulent solar wind. At the coronal base (just above the transition region), where $k_{z}$ and $L_{\bot }$ have the values $k_{z\text{b}}$ and $L_{\bot \text{b}}$ , the value of $k_{z\text{b}}L_{\bot \text{b}}$ for the energetically dominant AWs launched by the Sun can be estimated (in essence from the critical-balance condition) as $\unicode[STIX]{x1D6FF}v_{\text{b}}^{+}/v_{\text{Ab}}$ , where $\unicode[STIX]{x1D6FF}v_{\text{b}}^{+}$ and $v_{\text{Ab}}$ are the values of $\unicode[STIX]{x1D6FF}v^{+}$ and $v_{\text{A}}$ at the coronal base (Goldreich & Sridhar Reference Goldreich and Sridhar1995; van Ballegooijen & Asgari-Targhi Reference van Ballegooijen and Asgari-Targhi2016). Together, these scalings lead to the estimate

(7.8) $$\begin{eqnarray}k_{z}L_{\bot }\simeq \sqrt{\frac{B_{0\text{b}}}{B_{0}}}\left(\frac{U_{\text{b}}+v_{\text{Ab}}}{U+v_{\text{A}}}\right)\frac{\unicode[STIX]{x1D6FF}v_{\text{b}}^{+}}{v_{\text{Ab}}},\end{eqnarray}$$

where $B_{0\text{b}}$ and $U_{\text{b}}$ are the values of the background magnetic field and solar-wind outflow velocity at the coronal base. Between $r=10R_{\odot }$ and $r=60R_{\odot }$ , $B_{0\text{b}}/B_{0}\simeq f_{\max }(r/R_{\odot })^{2}$ , where $f_{\max }$ is the super-radial expansion factor (Kopp & Holzer Reference Kopp and Holzer1976). In the fast solar wind within this range of radii, $U+v_{\text{A}}\simeq 700{-}800~\text{km}~\text{s}^{-1}$ , which is comparable to $U_{\text{b}}+v_{\text{Ab}}\simeq v_{\text{Ab}}\simeq 10^{3}~\text{km}~\text{s}^{-1}$ . Equation (7.8) is thus approximately equivalent to

(7.9) $$\begin{eqnarray}k_{z}L_{\bot }\simeq \sqrt{f_{\max }}\,\left(\frac{r}{R_{\odot }}\right)\left(\frac{\unicode[STIX]{x1D6FF}v_{\text{b}}^{+}}{v_{\text{Ab}}}\right)\end{eqnarray}$$

for the energetically dominant fluctuations launched by the Sun. If we set $f_{\max }=9$ , $\unicode[STIX]{x1D6FF}v_{\text{b}}=30~\text{km}~\text{s}^{-1}$ , and $v_{\text{Ab}}=900~\text{km}~\text{s}^{-1}$ , then (7.9) becomes

(7.10) $$\begin{eqnarray}k_{z}L_{\bot }\sim \frac{r}{10R_{\odot }}\end{eqnarray}$$

for the energetically dominant AWs launched by the Sun. Equations (7.7) and (7.10) suggest that it is reasonable to neglect the shearing of inward-propagating AWs by outward-propagating AWs at $r\gtrsim 10R_{\odot }$ . On the other hand, at smaller radii, shearing could suppress the growth of inward-propagating AWs that would otherwise result from the parametric instability. Also, the requirement that $k_{z}L_{\bot }>1$ in order for  $\unicode[STIX]{x1D714}_{\text{nl},\text{PI}}^{-}$ to exceed  $\unicode[STIX]{x1D714}_{\text{nl},\bot }^{-}$ could prevent the $f^{-1}$ range from spreading to frequencies below some minimum ( $r$ -dependent) value.

The other nonlinearities in the weak-turbulence wave kinetic equations that are neglected in this paper include interactions involving fast magnetosonic waves, the turbulent mixing of slow waves by AWs, phase mixing of AWs by slow waves and the shearing of outward-propagating AWs by inward-propagating AWs (Chandran Reference Chandran2008). In situ measurements indicate that fast waves account for only a small fraction of the energy in compressive fluctuations at 1 AU (Yao et al. Reference Yao, He, Marsch, Tu, Pedersen, Rème and Trotignon2011; Howes et al. Reference Howes, Bale, Klein, Chen, Salem and TenBarge2012; Klein et al. Reference Klein, Howes, TenBarge, Bale, Chen and Salem2012). Also, fast waves propagating away from the Sun undergo almost complete reflection before they can escape into the corona (Hollweg Reference Hollweg1978). These findings suggest that nonlinear interactions involving fast waves have little effect upon the conclusions of this paper. The turbulent mixing of slow waves by AWs acts as an additional slow-wave damping mechanism and is thus unlikely to change the conclusions of this paper, which already assume strong slow-wave damping. Phase mixing of AWs by slow waves transports AW energy to larger  $k_{\bot }$ at a rate that increases with  $|k_{z}|$  (Chandran Reference Chandran2008). Although the fractional density fluctuations between $r=10R_{\odot }$ and $r=0.3~\text{AU}$ are fairly small (see, e.g. Tu & Marsch Reference Tu and Marsch1995; Hollweg et al. Reference Hollweg, Cranmer and Chandran2010), phase mixing could affect the parallel AW power spectra, and further work is needed to investigate this possibility. The shearing of outward-propagating AWs by inward-propagating AWs is enhanced by non-WKB reflection, which makes this shearing more coherent in time (Velli et al. Reference Velli, Grappin and Mangeney1989). The resulting nonlinear time scale for outward-propagating AWs is approximately $r/(U+v_{\text{A}})$ (Chandran & Hollweg Reference Chandran and Hollweg2009), where $U$ is the solar-wind outflow velocity. This time scale is comparable to the AW propagation time from the Sun to heliocentric distance  $r$ , and hence to the parametric-decay time scale at the small- $f$ end of the $1/f$ range of  $e^{+}$ . How this shearing modifies  $E^{+}(k_{z})$ , however, is not clear. For example, shearing by inward-propagating AWs may transport outward-propagating-AW energy to larger  $k_{\bot }=\sqrt{k_{x}^{2}+k_{y}^{2}}$ at a rate that is independent of  $|k_{z}|$ , in which case this shearing would reduce $E^{+}(k_{z})$ by approximately the same factor at all  $k_{z}$ , leaving the functional form of  $E^{+}(k_{z})$ unchanged.

7.4 Neglect of spatial inhomogeneity

In this paper, it is assumed that the background plasma is uniform and stationary. In the solar wind, however, as an AW propagates from the low corona to 0.3 AU, the properties of the ambient plasma seen by the AW change dramatically, with $\unicode[STIX]{x1D6FD}$ increasing from  ${\sim}10^{-2}$ to  ${\sim}1$ and $\unicode[STIX]{x1D6FF}v_{\text{rms}}/v_{\text{A}}$ increasing from ${\sim}0.02$ to  ${\sim}0.5$ (Bavassano et al. Reference Bavassano, Pietropaolo and Bruno2000; Cranmer & van Ballegooijen Reference Cranmer and van Ballegooijen2005; Chandran et al. Reference Chandran, Dennis, Quataert and Bale2011). Further work is needed to determine how this spatial inhomogeneity affects the nonlinear evolution of the parametric instability.

7.5 Approximate treatment of slow-wave damping

A key assumption in §§ 2 through 6 is that slow waves are strongly damped, and this damping is implemented by neglecting terms in the wave kinetic equations that are proportional to the slow-wave power spectrum  $S_{k}^{\pm }$ . There are two sources of error in this approach. First, damping could modify the polarization properties of slow waves, thereby altering the wave kinetic equations. Second, even if $S_{k}^{\pm }$ is much smaller than the AW power spectrum  $A_{k}^{\pm }$ , the neglected parametric-decay terms in the wave kinetic equations for AWs that are proportional to  $S_{k}^{\pm }$ could still be important, because they contain a factor of $\unicode[STIX]{x1D6FD}^{-1}$ , which is absent in the terms that are retained. This factor arises from the fact that the fractional density fluctuation of a slow wave is ${\sim}\unicode[STIX]{x1D6FD}^{-1/2}$ times larger than the fractional magnetic-field fluctuation of an AW with equal energy. These neglected terms act to equalize the 3-D AW power spectra $A^{+}$ and  $A^{-}$ , and hence to equalize $E^{+}$ and  $E^{-}$ . If these neglected terms were in fact important, they could invalidate the solutions presented in § 6, in which  $E^{+}\gg E^{-}$ . However, in situ observations indicate that $E^{+}\gg E^{-}$ in the fast solar wind at $r=0.3~\text{AU}$ (Marsch & Tu Reference Marsch and Tu1990; Tu & Marsch Reference Tu and Marsch1995), which suggests that the neglect of these terms is reasonable. Further work is needed to investigate these issues more carefully.