Statistical description of coalescing magnetic islands via magnetic reconnection

The physical picture of interacting magnetic islands provides a useful paradigm for certain plasma dynamics in a variety of physical environments, such as the solar corona, the heliosheath and the Earth’s magnetosphere. In this work, we derive an island kinetic equation to describe the evolution of the island distribution function (in area and in flux of islands) subject to a collisional integral designed to account for the role of magnetic reconnection during island mergers. This equation is used to study the inverse transfer of magnetic energy through the coalescence of magnetic islands in two dimensions. We solve our island kinetic equation numerically for three different types of initial distribution: Dirac delta, Gaussian and power-law distributions. The time evolution of several key quantities is found to agree well with our analytical predictions: magnetic energy decays as t ̃−1, the number of islands decreases as t ̃−1 and the averaged area of islands grows as t ̃, where t ̃ is the time normalised to the characteristic reconnection time scale of islands. General properties of the distribution function and the magnetic energy spectrum are also studied. Finally, we discuss the underlying connection of our island-merger models to the (self-similar) decay of magnetohydrodynamic turbulence.相互作用磁岛的物理图景为多种物理环境中的特定等离子体动力学提供了实用范式，例如日冕、太阳鞘以及地球磁层。在本研究中，我们推导了磁岛动力学方程，用以描述磁岛分布函数（磁岛面积与磁通量）的演化，该方程采用碰撞积分项来刻画磁岛合并过程中的磁重联（magnetic reconnection）作用。本方程被用于研究二维空间中磁岛合并引发的磁能逆输运过程。我们针对三类初始分布——狄拉克δ分布（Dirac delta distribution）、高斯分布（Gaussian distribution）以及幂律分布（power-law distribution）——对磁岛动力学方程进行了数值求解。研究发现，若干关键物理量的时间演化与我们的解析预测吻合良好：磁能随归一化时间t̃以t̃⁻¹衰减，磁岛数量随t̃⁻¹减少，而磁岛平均面积随t̃增长，其中t̃为归一化至磁岛特征重联时间尺度的时间。我们还对分布函数与磁能谱的一般特性展开了研究。最后，我们探讨了本研究的磁岛合并模型与磁流体动力学湍流（magnetohydrodynamic turbulence）的（自相似）衰减过程之间的内在关联。