In this paper, weak-turbulence theory is used to investigate the nonlinear evolution of the parametric instability in three-dimensional low-$\unicode[STIX]{x1D6FD}$ plasmas at wavelengths much greater than the ion inertial length under the assumption that slow magnetosonic waves are strongly damped. It is shown analytically that the parametric instability leads to an inverse cascade of Alfvén wave quanta, and several exact solutions to the wave kinetic equations are presented. The main results of the paper concern the parametric decay of Alfvén waves that initially satisfy $e^{+}\gg e^{-}$, where $e^{+}$ and $e^{-}$ are the frequency ($f$) spectra of Alfvén waves propagating in opposite directions along the magnetic field lines. If $e^{+}$ initially has a peak frequency $f_{0}$ (at which $fe^{+}$ is maximized) and an ‘infrared’ scaling $f^{p}$ at smaller $f$ with $-1<p<1$, then $e^{+}$ acquires an $f^{-1}$ scaling throughout a range of frequencies that spreads out in both directions from $f_{0}$. At the same time, $e^{-}$ acquires an $f^{-2}$ scaling within this same frequency range. If the plasma parameters and infrared $e^{+}$ spectrum are chosen to match conditions in the fast solar wind at a heliocentric distance of 0.3 astronomical units (AU), then the nonlinear evolution of the parametric instability leads to an $e^{+}$ spectrum that matches fast-wind measurements from the Helios spacecraft at 0.3 AU, including the observed $f^{-1}$ scaling at $f\gtrsim 3\times 10^{-4}~\text{Hz}$. The results of this paper suggest that the $f^{-1}$ spectrum seen by Helios in the fast solar wind at $f\gtrsim 3\times 10^{-4}~\text{Hz}$ is produced in situ by parametric decay and that the $f^{-1}$ range of $e^{+}$ extends over an increasingly narrow range of frequencies as $r$ decreases below 0.3 AU. This prediction will be tested by measurements from the Parker Solar Probe.

中文翻译：

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参数不稳定性、逆级联和太阳风湍流的范围


本文利用弱湍流理论研究三维低维参数不稳定性的非线性演化。 $\unicode[STIX]{x1D6FD}$假设慢磁声波受到强烈阻尼，等离子体的波长远大于离子惯性长度。分析表明，参数不稳定性会导致阿尔文波量子的逆级联，并给出了波动力学方程的几个精确解。论文的主要结果涉及阿尔文波的参数衰减，最初满足$e^{+}\gg e^{-}$ ， 在哪里$e^{+}$和$e^{-}$是频率（ $f$ ）沿磁场线以相反方向传播的阿尔芬波的频谱。 如果$e^{+}$最初有一个峰值频率$f_{0}$ （此时$fe^{+}$最大化）和“红外”缩放$f^{p}$较小时$f$和$-1<p<1$ ， 然后$e^{+}$获得一个$f^{-1}$在整个频率范围内进行缩放，该频率范围从$f_{0}$ 。同时， $e^{-}$获得一个$f^{-2}$在同一频率范围内进行缩放。如果等离子参数和红外$e^{+}$选择光谱来匹配日心距离为 0.3 天文单位 (AU) 的快速太阳风条件，然后参数不稳定性的非线性演化导致$e^{+}$与快风测量相匹配的频谱赫利俄斯航天器在 0.3 AU，包括观察到的$f^{-1}$缩放至$f\gtrsim 3\times 10^{-4}~\text{Hz}$ 。本文的结果表明$f^{-1}$看到的光谱赫利俄斯在快速的太阳风中$f\gtrsim 3\times 10^{-4}~\text{Hz}$已生产就地通过参数衰减并且$f^{-1}$的范围$e^{+}$延伸到越来越窄的频率范围$r$降至 0.3 AU 以下。这一预测将通过帕克太阳探测器的测量得到检验。