Numerical studies on the regularity properties of the fluid dynamical equations

为了研究磁流体动力学中的快速重联现象，我们在Eulerian-Lagrangian方法的基础上进行了数值实验，并在2004年将Navier-Stokes方程的Eulerian-Lagrangian方法推广到磁流体动力学方程。有两种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形，我们证明了一个连接张量就足以重构磁流体动力学系统。利用该方法对二维Orszag-Tang模型进行了直接数值模拟，发现扩散标记A与空间位置x之间的对应关系变得不可逆（重置现象）.我们发现它与磁重联有关。在此基础上，采用三维广义O-T涡和正交偏置磁通管作为初始条件，对三维非线性流场进行了数值模拟。在2005年，对扭曲磁通管进行了更实际的数值模拟。平行/反平行通量管和连接通量环被用作初始条件。证实了磁重联与复位现象有关。由重置间隔定义的时间尺度小于由全局特征估计的时间尺度，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。我们还通过可视化发现重联磁场与重联现象之间存在空间对应关系，并给出了理想磁流体动力学方程的一个正则性判据，该判据可以用涡度场和电流密度场来表示。关于螺旋度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，从而表明上述判据还有改进的空间。

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.