Nonlinear Evolution of Double Tearing Mode Mediated by Parallel Electron Viscosity

HE Zhixiong; DONG Jiaqi; JIANG Haibin

Introduction By adding parallel electron viscosity into the normal equation of resistance tearing mode, the nonlinear behavior of double tearing mode (DTM) mediated by parallel electron viscosity is numerically investigated considering the magnetohydrodynamics in a periodic cylinder. Method The evolution of magnetic islands and magnetic flux during the nonlinear behavior stage of double tearing mode mediated by parallel electron viscosity were analyzed, to study the changes of the magnetic field topology in different phases and associated kinetic characteristics for different distances between the two resonant rational flux surfaces. In addition, the nonlinear driving for each case was discussed. Result The results show that the distance between the two resonant rational flux surfaces has a significant impact on the nonlinear behavior of double tearing mode. Rapid magnetic reconnection would occur at intermediate distance, resulting in the fastest and most severe damage to the configuration. Conclusion The results of this study provide a reference for the configuration design and operational control of tokamaks.

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0. 引言

先进托克马克装置的一个重要特性是具有非单调安全因子剖面，能够提供负磁剪切^[1-2]。负磁剪切位形存在多个${q_s} = m/n$的共振有理面，这样的有理面上能够引起双撕裂模不稳定性^[3]。因此双撕裂模的研究，尤其是其非线性发展的特性的研究^[4-7]，对于探索和解决先进托克马克装置诸如内部输运垒的形成机制的一些问题，会有积极的帮助。由电阻引起的双撕裂模的非线性行为^[8-11]已经被很多学者研究过。王正汹等人用数值方法，给出了平板位形下两个有理面距离为中等距离的电阻双撕裂模的非线性演化^[12-14]，认为双撕裂模非线性演化大概可以分为4个阶段。KONOVALOV在DⅢ-D中的实验^[15]指出，由于准稳小幅度电磁扰动，在托卡马克等离子体中会存在反常电子粘滞性。这时候，作为耗散效应，反常电子粘滞性也能引起撕裂模不稳定性。由反常电子粘滞性引起的撕裂模被认为是引起破裂不稳定性的可能原因之一^[16]。电子粘滞撕裂模也被作为在实验上观察到的快锯齿破裂的可能原因^[17]而被研究。董家齐^[18]用解析的方法，研究了平板位形等离子体中由反常电子粘滞性引起的双撕裂模的线性行为。何志雄^[19]用数值模拟的方法，在直圆柱位形下，研究了由反常电子粘滞性引起的双撕裂模的线性发展。

1. 物理模型

（1）撕裂模MHD方程的推导

考虑圆柱位形的简化磁流体动力学（RMHD）方程，用大环径比近似和低-β展开，可以用两个标势，磁通量函数Ψ（G·cm²）和流函数Φ（cm³/s）来表示磁场和等离子体速度：$ \bar B = {B_0} - \nabla \varPsi \times \bar \varphi $，$ \bar \upsilon = {\upsilon _0} + \nabla \varPhi \times \bar \varphi $，包含电子粘滞项的欧姆定律：

$$ \vec E + \frac{{\text{1}}}{{{c}}}\vec \upsilon \times \vec B = \eta \vec j - \frac{{{m_{{e}}}{\mu _{{e}}}}}{{{n_{{e}}}{{\text{e}}^{\text{2}}}}}\nabla _ \bot ^2\vec j $$

(1)

运动方程可以写成：

$$ \rho \left[ {\frac{{\partial \vec \upsilon }}{{\partial t}} + (\vec \upsilon \cdot \nabla )\vec \upsilon } \right] = - \nabla P + \frac{{\text{1}}}{{{c}}}\vec j \times \vec B + {\mu _i}{\nabla ^2}\vec \upsilon $$

(2)

式中：

$ \vec B $ ——磁场强度（G）；

$ \vec E $ ——电场强度（G）；

$ \vec \upsilon $ ——磁流体元速度（cm/s）；

$ \vec j $ ——电流密度（emu/cm²）；

$e$ ——电子电荷（esu）；

$ {m_{\mathrm{e}}} $——电子质量（g）；

$ \eta $ ——等离子体电阻（Ω）；

$ {\mu _{\mathrm{e}}} $——平行电子粘滞性[emu·s/(g·cm)]；

$ \rho $——等离子体质量密度（g/cm³）；

$ {\mu _i} $——离子粘滞性[g/(cm·s)]。

简化方程式（1）和式（2）并归一化（无量纲）后得到撕裂模MHD方程：

$$ \frac{{\partial \psi }}{{\partial t}} = - [\psi ,\phi ] + {S^{ - 1}}\hat \nabla _ \bot ^2\psi - {R^{ - 1}}\hat \nabla _ \bot ^2(\hat \nabla _ \bot ^2\psi ) + {\hat E_{{\mathrm{wz}}}} - \frac{{\partial \phi }}{{\partial \zeta }}, $$

(3)

$$ \frac{\partial }{{\partial t}}(\hat \nabla _ \bot ^2\phi ) = [\hat \nabla _ \bot ^2\psi ,\psi ] - \frac{\partial }{{\partial \zeta }}(\hat \nabla _ \bot ^2\psi ) - [\hat \nabla _ \bot ^2\phi ,\phi ] + R_\mu ^{ - 1}\hat \nabla _ \bot ^2(\hat \nabla _ \bot ^2\phi ), $$

(4)

这里，归一化参量为（$ a,{\tau _{\mathrm{h}}},{B_0} $），并定义极向角$ \zeta \equiv z/{R_0}q(a) $，于是有归一化的磁通量函数$ \psi = \varPsi /a{B_0} $，流函数$ \phi = \varPhi /({a^2}/{\tau _{\mathrm{h}}}) $，电阻磁雷诺数$ S \equiv {\tau _{\text{η}}}/{\tau _{\mathrm{h}}} $，电子磁雷诺数$ R \equiv {\tau _{\text{υ}} }/{\tau _{\mathrm{h}}} $，离子磁雷诺数$ {R_{\text{μ}} } \equiv {\tau _{\text{μ}} }/{\tau _{\mathrm{h}}} $，$ {\tau _{\text{η}} } = {4{\text{π}}}{a^2}/{\text{η}} {c^2} $，$ {\tau _{\text{υ}} } = {4{\text{π}}}{a^4}{n_e}{e^2}/({m_e}{\mu _e}{c^2}) $，$ {\tau _{\text{μ}}} = \rho {a^2}/{\text{μ}} $。

Poisson括号定义为$ [a,b] = \dfrac{1}{r}\left( {\dfrac{{\partial a}}{{\partial r}}\dfrac{{\partial b}}{{\partial \theta }} - \dfrac{{\partial b}}{{\partial r}}\dfrac{{\partial a}}{{\partial \theta }}} \right) $。

式中：

$ a $ ——托卡马克小半径（cm）

$ {B_0} $——$ {R_0} $处的磁场大小（G）；

$ {\tau _h} $ ——宽为$a$的等离子体片的Alfven时间（s）；

将函数在$ \theta ,\varphi $方向做Fourier展开：

$$ \psi = {\psi _0}(r) + \sum\limits_{n = 0}^N {} \psi _{np,n}^c(r)\cos [n(p\theta + \varphi )] + \psi _{m,n}^s(r)\sin [n(p\theta + \varphi )] $$

(5)

$$ \phi = \sum\limits_{n = 0}^N {} \phi _{np,n}^c(r)\cos [n(p\theta + \varphi )] + \phi _{m,n}^s(r)\sin [n(p\theta + \varphi )] $$

(6)

这里，$p = m/n = {\mathrm{const}}$，$ {\psi _0} $是当$ \dfrac{{\partial \psi }}{{\partial t}} = 0 $和$ \dfrac{{\partial \phi }}{{\partial t}} = 0 $时的平衡量，其他的是扰动量，$ M,N $是$ \theta ,\varphi $方向所取的最大谐波数。将式（5）和式（6）应用到式（3）和式（4），利用Fourier级数的正交关系，得到一组关于扰动量的偏微分方程组。简记为算子形式：

$$ \frac{{\partial Y}}{{\partial t}} = NL(Y) + L(Y) $$

(7)

（2）方程数值化

对非线性部分有关时间的差分过程用二阶Adams-Bashforth显示格式，另外用半隐式格式于线性算子：

$$ {Y^{t + \Delta t}} - {Y^t} = \frac{{\Delta t}}{2}[L({Y^{t + \Delta t}}) + L({Y^t})] + \frac{{\Delta t}}{2}[3NL({Y^t}) - NL({Y^{t - \Delta t}})] $$

(8)

因此得到下面代数方程：

$$ \begin{split} &Y\frac{{t + \Delta t}}{p} - \frac{{\Delta t}}{2}L({Y^{t + \Delta t}}) = Y\frac{t}{p} + \frac{{\Delta t}}{2}L({Y^t}) +\\& \dfrac{{\Delta t}}{2}[3NL({Y^t}) - NL({Y^{t - \Delta t}})] \end{split} $$

(9)

极向平衡通量$ {\psi _0}(r) $可由安全因子$ q(r) $给出，其计算关系：

$$ \frac{{{\mathrm{d}}{\psi _0}(r)}}{{{\mathrm{d}}r}} = - \frac{r}{{q(r)}} $$

(10)

而安全因子剖面则参考文献[6]给出：

$$ q(r) = {q_{\mathrm{c}}}{\{ 1 + {\left( {\dfrac{r}{{{r_0}}}} \right)^{2\lambda }}\} ^{1/\lambda }}[1 + A\exp \{ - {\left( {\dfrac{{r - {r_\delta }}}{\delta }} \right)^2}\} ] $$

(11)

根据方程（9），采用差分方法，编写了一个数值代码《RDTM》，进行了网格无关乎验证，即当前网格数下再增加网格点数，数值结果在我们要求的精度内无差别。利用此代码进行数值模拟。

参量固定如下：$ \lambda = 1 $，$ {r_0} = 0.412 $，$ \delta = 0.273 $，$ {r_\delta } = 0 $，$ A = 3 $。用$ {q_{\mathrm{c}}} $来做调控参数，对于具体的$ {q_s} = m/n $，控制初始条件两个有理面的间距$ {D_{12}} $。

螺旋通量$ \psi _p^ * $和极向通量$ \psi $有如下关系^[20]：

$$ \begin{split} &\psi _p^ * = - p[({\psi _0} + \psi _{0,0}^c + \sum\limits_\ell ^M {} \psi _{\ell p,\ell }^c\cos (\ell (p\theta + \varphi )) + \\&\psi _{\ell p,\ell }^s\sin (\ell (p\theta + \varphi ))] - \frac{1}{2}({r^2} - 1) \end{split} $$

(12)

假设初始扰动有如下形式：

$$ \tilde \psi (t = 0) = \frac{1}{2}\sum\limits_m {{\psi _{0m}}r(r - 1){e^{i(m{\vartheta _*} + {\vartheta _{0m}})}}} + c.c. $$

(13)

这里，${\psi _{0m}}$是扰动振幅，${\vartheta _*} \equiv \vartheta - q_s^{ - 1}\zeta $是环向角坐标，${\vartheta _{0m}}$是初始相位，在数值计算里，取${\psi _{0m}} = {10^{ - 7}}$，${\vartheta _{0m}} = 0{\text{ }}$。

3. 结论

用圆柱位形的磁流体动力学，对由反常电子粘滞性引起的双撕裂模的非线性发展做了数值研究。研究表明，两个有理面的间距的大小，对双撕裂模非线性发展有很大影响。

1) 当两个有理面的间距比较大的时候，由于非线性耦合的驱动较弱，双撕裂模经过较长时间的线性发展，最后在两个有理面上两个磁岛各自达到饱和状态并形成稳态。

2) 当两个有理面距离为中等距离的时候，非线性发展可以分为4个阶段：第一个阶段是线性阶段，低极向模为支配模式，并且其模式增长率为线性增长率；第二阶段是非线性驱动的阶段，两个有理面上的低模的耦合驱动平衡模的发展；第一二阶段同两个有理面间距大的时候是一样的。由于两个有理面距离相对较近，两个有理面上的磁岛增长未达饱和之前，他们就交换了径向位置，并分别与两侧的开放磁力线进行快速的反向重联，即磁岛的湮灭，这就是第三阶段快速磁重联阶段；由于快速磁重联对磁场拓扑结构的破坏，模式最后进入衰减阶段。

3) 当两个有理面距离很小的时候，高极向模数的模式会发展起来并成为支配模式。各种极向模数的模式会经过短暂的第一阶段的线性增长，然后发展起来的高极向模数的模式会耦合驱动平衡模和低极向模数进入非线性驱动的第二阶段。由于有理面间距很小，两个有理面之间的磁岛很快会发生二次磁重联，但由于高极向模式的相位特点使得二次重联时只有部分磁岛发生了径向位置的交换，因此并没有快速重联过程，双撕裂模模式在较缓慢的磁重联中衰减直至消失。

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Nonlinear Evolution of Double Tearing Mode Mediated by Parallel Electron Viscosity

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References

通讯作者: 陈斌, bchen63@163.com

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