Numerical studies on the regularity properties of the fluid dynamical equations

为了阐明磁流体力学中的快速重联现象，我们在欧拉-拉格朗日形式的基础上进行了数值实验。2004年，我们将Navier-Stokes方程的欧拉-拉格朗日形式推广到磁流体动力学方程。有两个韦伯变换对应于两种螺旋度的守恒。对于单位普朗特数的情况，我们已经证明了一个连接张量足以重新表述磁流体动力系统。利用它对二维Orszag-Tang进行了直接数值模拟，发现扩散标签A与空间位置x之间的对应变得不可逆（重置现象）。我们发现它与磁重联有关。在三维广义O-T涡旋和正交偏置磁通管初始条件下进行了数值模拟。在这些情况下也会发生重置现象。2005年，用扭流式磁通管进行了更实际的数值模拟。以平行/反平行磁通管和连通磁通环为初始条件。结果表明，磁重联与复位现象有关。由重置间隔定义的时间尺度比由全局特征估计的时间尺度更小，更接近于快速重联的时间尺度。从这个意义上说，我们证明了这种方法可以量化快速重连接。我们还通过可视化发现，重新连接的磁场和重置现象之间存在空间对应关系。一个关于螺旋不变量的简单论证表明，如果磁场是光滑的，那么速度场也是光滑的，从而表明上述准则还有改进的余地。

In order to clarify fast reconnection phenomena in magnetohydrodynamics, we performed numerical experiments on the basis of the Eulerian-Lagrangian formalism.In 2004, we extended the Eulerian-Lagrangian formalism for the Navier-Stokes equations to magnetohydrodynamical equations. There are two Weber transforms corresponding to conservation of two kinds of helicities. For the case of unit Prandtl number, we have shown that one connection tensor is sufficient to reformulate the magnetohydrodynamical system. By using it, direct numerical simulations of 2D Orszag-Tang were done and it was found that correspondence between diffusive labels A and spatial positions x becomes non-invertible (resetting phenomena). We showed that it is related with magnetic reconnection. Furthermore, numerical simulations were done with initial conditions of 3D generalized O-T vortices and orthogonally offset magnetic flux tubes. Resetting phenomena also take place for these cases.In 2005, more practical numerical simulations with twisted magnetic flux tubes were performed. Parallel/anti-parallel flux tubes and linked flux rings were used as initial conditions. It was confirmed that magnetic reconnections are associated with resetting phenomena. The time scales defined by the resetting intervals are smaller that those estimated by global characteristics and are closer to time-scales of fast reconnections. In this sense we showed that this method can quantify fast reconnections. We also found by visualizations that a spatial correspondence between reconnecting magnetic fields and the resetting phenomena.A regularity criterion for ideal magnetohydrodynamical equations is known to be given in terms of the vorticity and the current density fields A simple argument respecting helicity invariants shows that if the magnetic field is smooth, then so is the velocity field, thereby suggesting some room for improving the above criterion.