# The nonlinear dynamo effect of tearing modes

Luo Yuhang<EMAIL_ADDRESS>Gao Zhe  Department of Engineering
Physics, Tsinghua University, Beijing, 100084

###### Abstract

The nonlinear dynamo effect of tearing modes is derived with the resistive MHD
equations. The dynamo effect is divided into two parts, parallel and
perpendicular to the magnetic field. Firstly, the force-free plasma is
considered. It is found that the parallel dynamo effect drives opposite
current densities at the different sides of the rational surface, making the
$\lambda=\bm{j}\cdot\bm{B}/|\bm{B}\rvert^{2}$ profile completely flattened
near the rational surface. There are many rational surfaces for the turbulent
plasma, which means the plasma is tending to relax into the Taylor state. In
contrast, a bit far from the rational surface, the parallel dynamo effect is
much smaller, and the nonlinear dynamo form approximates the quasilinear form.
Secondly, the pressure gradient is included. It is found that rather than the
$\lambda$ profile, the $\bm{j}\cdot\bm{B}$ profile is flattened by the
parallel dynamo effect. Besides, the perpendicular dynamo effect of tearing
modes is found to eliminate the pressure gradient near the rational surface.
In addition, our result also provides another basis for the assumption that
current density is flat in the magnetic island for the tearing modes theory.

##  I Introduction

The dynamo effectMoffatt (1978); Sokoloff, Stepanov, and Frick (2014) is an
average electric motive force produced by velocity and magnetic field
fluctuations. In astrophysical plasmas, the dynamo effect is widely discussed
as a mechanism of magnetic flux amplification.Yoshizawa (1990); Brandenburg
(2001) In fusion plasmas, the dynamo effect has been invoked in explaining the
relaxation process of self-organized systems like Reversed-Field Pinch
(RFP)Hokin  _et al._ (1991) and SpheromakAl-Karkhy  _et al._ (1993); Jarboe
_et al._ (2012).

A typical relaxation process in RFP and Spheromak contains four stages.
Firstly, the current density profile is modified by the external current drive
method. Generally, the driven current concentrates at the edge of plasma when
using non-inductive procedures like helicity injection and wave injection.
Secondly, the modified current density triggers instabilities, which are
mainly considered as tearing modes. Thirdly, the overlap of the growing
instabilities leads to turbulent plasma. Fourthly, the turbulent plasma
relaxes into a state with a stable current density profile. The first three
stages are intuitive, while the last stage is hard to understand.

At the fourth stage, TaylorTaylor (1986) shows that plasma will relax into the
state where the magnetic energy is minimized while the magnetic helicity is
conserved relative to the magnetic energy. This state is called the Taylor
state. According to Taylor’s theory, $\bm{j}=\lambda\bm{B}$ holds in the
relaxation area, and the parameter,
$\lambda=\bm{j}\cdot\bm{B}/|\bm{B}\rvert^{2}$, is constant. This means the
pressure gradient goes to zero; thus, the confinement is not well. In reality,
the plasma would be stable before fully relaxing into the Taylor state, which
makes plasma with a finite pressure gradient can be obtained. Taylor specifies
to what state the turbulent plasma will relax, but how this relaxation occurs
remains unsolved.

The dynamo effect has been widely discussed these years to explain the
relaxation process’s detail, both theoreticallyStrauss (1985); Bhattacharjee
and Hameiri (1986); Ji and Prager (2002) and numericallyCappello, Bonfiglio,
and Escande (2006); Mirnov, Hegna, and Prager (2004); Bonfiglio, Cappello, and
Escande (2005). A lot of experimentsJi  _et al._ (1996); Den Hartog  _et al._
(1999); Fontana  _et al._ (2000); Brower  _et al._ (2002) have been developed
to observe and measure the dynamo effect.

In the most widely studied MHD dynamo model, the dynamo effect is expressed as
$\langle\widetilde{\bm{v}}\times\widetilde{\bm{B}}\rangle$,
$\widetilde{\bm{v}}$ and $\widetilde{\bm{B}}$ correspond to the fluctuating
velocity and magnetic field respectively, $\langle~{}\rangle$ indicates
average over poloidal and toroidal direction.

The dynamo effect can be divided into two components. One component is
parallel to the mean magnetic field,
$\varepsilon_{\parallel}=\langle\widetilde{\bm{v}}\times\widetilde{\bm{B}}\rangle\cdot\overline{\bm{B}}/|\overline{\bm{B}}\rvert$,
which is also called the $\alpha$ effectSeehafer (1996); Ji and Prager (2002).
This component is important in driving parallel current.Ji  _et al._ (1995);
Brower  _et al._ (2002) The other component is perpendicular to the mean
magnetic field and radial direction,
$\varepsilon_{\perp}=\langle\widetilde{\bm{v}}\times\widetilde{\bm{B}}\rangle\cdot(\overline{\bm{B}}\times\nabla
r)/|\overline{\bm{B}}\rvert$, which plays a crucial role in driving
perpendicular current density and change the pressure gradient.

The parallel dynamo effect induced by the tearing fluctuations is first
obtained by StraussStrauss (1985). The turbulent plasma is decomposed into
many tearing modes in Strauss’s theory. Each tearing mode is considered
separately, which means the interactions of the tearing modes are ignored
here. Bhattacharjee et al. Bhattacharjee and Hameiri (1986) also found a
similar parallel dynamo effect form of the tearing modes. The velocity and
magnetic fluctuations are derived using linear tearing equations in their
theories. Therefore the form of the parallel dynamo effect is quasilinear. The
quasilinear form of parallel dynamo effect clearly shows a flattening effect
of $\lambda$ profile. This makes the theory popular in the explanation of
Taylor’s relaxation process.

However, the quasilinear approach is only suitable in the linear growth stage
of the tearing modes. When the tearing modes evaluate the nonlinear growth
stage and the final saturated state, the quasilinear theory is not applicable.
To our best knowledge, the nonlinear theory of the parallel dynamo effect has
not been developed yet.

The perpendicular dynamo effect is omitted in StraussStrauss (1985)’s theory
since the force-free plasma is considered. Bhattacharjee et al. Bhattacharjee
and Hameiri (1986) used the interchange mode to give the nonlinear form of the
perpendicular dynamo effect. The tearing mode induced perpendicular dynamo
effect is also not discussed yet.

The tearing mode has been widely discussed. The linear theory of a single
tearing mode is first derived by Furth et al. Furth, Killeen, and Rosenbluth
(1963). RutherfordRutherford (1973) proposed the nonlinear theory in which the
quasilinear modified current was considered. The convincing theory of multiple
tearing modes interactions has not been developed. So that we also treat the
turbulent plasma as multiple individual tearing modes in this paper.
Furthermore, the nonlinear treatment of a single tearing mode is used to give
the nonlinear form of the dynamo effect.

Our treatment of the tearing mode is a little different from most tearing mode
theories did. We do not perform the flux average procedure, which is extensive
used in most tearing mode theories. As we mentioned earlier, the plasma we
study here is turbulent. Thus the fast transport along with the magnetic flux
no longer exists.

This paper gives the nonlinear result of both parallel and perpendicular
dynamo effect in the nonlinear tearing mode theory frame without using flux
average. We find the driven current by the nonlinear parallel dynamo effect in
the force-free plasma totally flattens $\lambda$ near the rational surface. In
terms of the turbulent plasma, the dynamo effect should be significant because
there are many rational surfaces. This makes the $\lambda$ gradient vanished
in the turbulent area, which means relaxing into the Taylor state.

When the distance goes far from the rational surface, the parallel dynamo
effect becomes smaller. Our nonlinear form approximates well with the
quasilinear form as the distance goes really far from the rational surface.
From a quantitative view, we can ignore the parallel dynamo effect in the
region far from the rational surface. This makes the quasilinear form of the
dynamo effect less critical in explaining the relaxation process.

Furthermore, the apparent singularity in the parallel dynamo effect’s
quasilinear form makes the quasilinear theory not suitable near the rational
surface of the tearing mode. Although Strauss declared that the inclusion of
inertia would remove the singularity, we will demonstrate that as soon as
performing the nonlinear tearing mode procedure, the singularity is no longer
exists when using the MHD model.

When the pressure gradient is included, we find the perpendicular dynamo
effect will drive an opposite current density to cancel the mean perpendicular
current density. We also find the nonlinear form of the parallel dynamo effect
is different from that in the force-free plasma. The parallel dynamo effect
flattens $\bm{j}\cdot\bm{B}$ instead of $\lambda$. The constant $\lambda$ is
required by the Taylor state in the force-free plasma as mentioned above. The
requirement of $\bm{j}\cdot\bm{B}$ may give an idea of extending Taylor’s
theory into the plasma with the pressure gradient.

Because the tearing modes’ interactions are not concerned in our theory, our
result is also suitable for a single tearing mode. For most single tearing
mode theories, the flattened current density is a premise hypothesis when
performing flux average. Our result gives another perspective of how the
current density flattens near the rational surface for a single tearing mode.

This paper is organized as follows. The resistive MHD model is presented in
Sec. II. The force-free plasma is considered in Sec. III, where the nonlinear
form of parallel dynamo effect is obtained and compared with the quasilinear
form. In Sec. IV, the nonlinear forms of both the parallel and perpendicular
dynamo effect are derived with the pressure gradient existed. Finally, the
conclusions and discussion are included in Sec. V.

##  II  Physical Model

This section presents the definition of the dynamo effect under the frame of
the resistive MHD model. Consider the full resistive MHD equations:

$\displaystyle\rho\left(\frac{\partial\bm{v}}{\partial
t}+\bm{v}\cdot\nabla\bm{v}\right)=\bm{j}\times\bm{B}-\nabla p,$ (1)
$\displaystyle\bm{E}+\bm{v}\times\bm{B}=\eta\bm{j},$ (2)
$\displaystyle\frac{\partial\bm{B}}{\partial t}=-\nabla\times\bm{E},$ (3)
$\displaystyle\bm{j}=\nabla\times\bm{B}.$ (4)

Gaussian units is used here, light speed $c=1$ and $4\pi$ is absorbed for
convenience. Equation (1) is the momentum or force balance equation where
$\rho$ is the mass density, $\bm{v}$ is the plasma velocity, $\bm{j}$ is the
current density, and $p$ is the pressure. Eq. (2) is the ohm’s law where
$\bm{E}$ is the electric field, and $\eta$ is the resistivity which is assumed
to be constant. Eq. (3) is Faraday’s law, and Eq. (4) is Ampère’s circuital
law, and the displacement current is neglected.

The cylindrical coordinate system$(r,\theta,\phi)$ is employed, $r$, $\theta$
are the radial and poloidal direction, respectively, and $\phi=z/R$ indicates
the toroidal direction with $R$ the major radius and $z$ the axial direction.
The large aspect ratio limit is not implied in this paper, making our theory
more suitable for small aspect radio systems. At the same time, toroidal
effects are not considered here.

All the physical quantities are separeted into mean and fluctuating parts, for
example, the magnetic field

$\displaystyle\bm{B}=\overline{\bm{B}}(r,t)+\widetilde{\bm{B}}(r,\theta,\phi,t),$
(5)
$\displaystyle\overline{\bm{B}}=\frac{1}{4\uppi^{2}}\oint\bm{B}\operatorname{d\\!}\theta\operatorname{d\\!}\phi.$
(6)

The perturbation part $\widetilde{\bm{B}}$ takes the form of
$\mathrm{e}^{\mathrm{i}(m\theta-n\phi)}$ for a tearing mode with the polodial
number $m$ and torodial number $n$. The mean magnetic field satisfies the
resonant condition $\bm{k}\cdot\overline{\bm{B}}=0$ at rational surface
$r=r_{\mathrm{s}}$ , where $\bm{k}=m\nabla\theta-n\nabla\phi$ is the
constructed mode number vector.

In the following derivation, we introduce a vector potential $\bm{A}$ and an
electrostatic potential $U$. Here the Coulomb gauge $\nabla\cdot\bm{A}=0$ is
used. Therefore,

$\displaystyle\bm{B}=\nabla\times\bm{A},$ (7) $\displaystyle\bm{E}=-\nabla
U-\frac{\partial\bm{A}}{\partial t}.$ (8)

The Ohm’s law Eq. (2) becomes

$\frac{\partial\bm{A}}{\partial t}=\bm{v}\times\bm{B}-\nabla U-\eta\bm{j}.$
(9)

If we averages Eq. (9) over $\theta$ and $\phi$ and remain the
$\langle\widetilde{\bm{v}}\times\widetilde{\bm{B}}\rangle$ term, the dynamo
induced current density can be expressed as

$\delta\overline{\bm{j}}=\frac{1}{\eta}\left<\widetilde{\bm{v}}\times\widetilde{\bm{B}}\right>.$
(10)

To calculate the form of dynamo effect, the velocity perturbation,
$\widetilde{\bm{v}}$, should be represented as a function of magnetic
perturbation, $\tilde{\bm{B}}$. Take linearization of Eq. (9) and cross with
$\overline{\bm{B}}/|\overline{\bm{B}}\rvert^{2}$, we get

$\widetilde{\bm{v}}_{\perp}=-\nabla\tilde{U}\times\frac{\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}-\frac{\eta\widetilde{\bm{j}}\times\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}-\frac{\partial\widetilde{\bm{A}}}{\partial
t}\times\frac{\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}.$ (11)

The parallel dynamo effect is

$\begin{split}\varepsilon_{\parallel}&=\left<\widetilde{\bm{v}}\times\widetilde{\bm{B}}\right>\cdot\frac{\overline{\bm{B}}}{|\overline{\bm{B}}\rvert}=\\\
&\qquad\qquad-\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\nabla\tilde{U}}{2|\overline{\bm{B}}\rvert}-\frac{\widetilde{\bm{B}}_{\perp}^{*}}{2|\overline{\bm{B}}\rvert}\cdot\left(\eta\widetilde{\bm{j}}+\frac{\partial\widetilde{\bm{A}}}{\partial
t}\right).\end{split}$ (12)

The perpendicular dynamo effect is

$\varepsilon_{\perp}=\left<\widetilde{\bm{v}}\times\widetilde{\bm{B}}\right>\cdot\frac{\overline{\bm{B}}\times\nabla
r}{|\overline{\bm{B}}\rvert}=\frac{\tilde{B}_{r}^{*}\widetilde{\bm{v}}\cdot\overline{\bm{B}}-\widetilde{\bm{B}}^{*}\cdot\overline{\bm{B}}\tilde{\upsilon}_{r}}{2|\overline{\bm{B}}\rvert}.$
(13)

The perpendicular dynamo effect does not occur in the force-free plasma
because the current density is always parallel to the magnetic field, and the
perpendicular current can not be driven.

Dynamo effect equals zero in the perfect conducting plasma because of the
“frozen-in” restriction. The resistivity must be included in the analysis of
the dynamo effect.

##  III  the force-free plasma

In this section, we will give the nonlinear form of the dynamo effect in the
force-free plasma. Both the steady state and the growth stage are considered.
We will also discuss the difference between the quasilinear form and our
nonlinear form.

In force-free plasma, $\bm{j}\times\bm{B}=0$ always holds, and all the
pressure terms are neglected. Taking $\bm{B}/|\bm{B}\rvert^{2}$ component of
Eq. (9) to eliminate $\bm{v}$, we get the relation of $\tilde{U}$ and
$\tilde{\lambda}$

$\eta\tilde{\lambda}=-\frac{\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}\cdot\nabla\tilde{U}-\left(\frac{\partial\bm{A}}{\partial
t}\cdot\frac{\bm{B}}{|\bm{B}\rvert^{2}}\right)_{1}.$ (14)

where $()_{1}$ indicates the fluctuating part, and

$\tilde{\lambda}=\left(\frac{\bm{j}\cdot\bm{B}}{|\bm{B}\rvert^{2}}\right)_{1}.$
(15)

Crossing Eq. (1) with $\bm{B}/|\bm{B}\rvert^{2}$, taking divergence and then
linearizing gives

$\nabla\cdot\left(\frac{\overline{\rho}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\widetilde{\bm{v}}}{\partial
t}\times\overline{\bm{B}}\right)=\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}+\overline{\bm{B}}\cdot\nabla\tilde{\lambda},$
(16)

where
$\overline{\lambda}=\overline{\bm{j}}\cdot\overline{\bm{B}}/|\overline{\bm{B}}\rvert^{2}$
is the mean part of $\lambda$, and $\nabla\cdot\bm{j}=0$ is used.

Substituting Eq. (14) and linearized Eq. (9) into Eq. (16) leads to

$\begin{split}&\frac{\partial}{\partial
t}\nabla\cdot\left(\frac{\overline{\rho}}{|\overline{\bm{B}}\rvert^{2}}\nabla\tilde{U}\right)-\frac{(\bm{k}\cdot\overline{\bm{B}})^{2}}{\eta|\overline{\bm{B}}\rvert^{2}}\tilde{U}=\\\
&\qquad\qquad\qquad\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}-\frac{\mathrm{i}\bm{k}\cdot\overline{\bm{B}}}{\eta}\left(\frac{\partial\bm{A}}{\partial
t}\cdot\frac{\bm{B}}{|\bm{B}\rvert^{2}}\right)_{1}.\end{split}$ (17)

The time derivatives of $\overline{\bm{B}}$ and $\overline{\rho}$ are assumed
to be neglected, and $\overline{\rho}/|\overline{\bm{B}}\rvert^{2}$ is assumed
constant. Eq. (LABEL:lineqn) is obtained using linear theory, and the
nonlinear term will be added in the following discussion.

We replace $\overline{\lambda}$ by
$\overline{\lambda}+\delta\overline{\lambda}$ in Eq. (LABEL:lineqn), where

$\delta\overline{\lambda}=\frac{\delta\overline{\bm{j}}\cdot\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}=-\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\nabla\tilde{U}}{2\eta|\overline{\bm{B}}\rvert^{2}}-\frac{\widetilde{\bm{B}}_{\perp}^{*}}{2\eta|\overline{\bm{B}}\rvert^{2}}\cdot\left(\eta\widetilde{\bm{j}}+\frac{\partial\widetilde{\bm{A}}}{\partial
t}\right).$ (18)

This idea is first proposed by RutherfordRutherford (1973). The adding of the
quasilinear term $\delta\overline{\lambda}$ into the linear theory Eq.
(LABEL:lineqn) leads to the nonlinear theory equation,

$\begin{split}&\frac{\partial}{\partial
t}\nabla\cdot\left(\frac{\overline{\rho}}{|\overline{\bm{B}}\rvert^{2}}\nabla\tilde{U}\right)-\frac{(\bm{k}\cdot\overline{\bm{B}})^{2}}{\eta|\overline{\bm{B}}\rvert^{2}}\tilde{U}=\\\
&\qquad\qquad\widetilde{\bm{B}}\cdot\nabla\left(\overline{\lambda}+\delta\overline{\lambda}\right)-\frac{\mathrm{i}\bm{k}\cdot\overline{\bm{B}}}{\eta}\left(\frac{\partial\bm{A}}{\partial
t}\cdot\frac{\bm{B}}{|\bm{B}\rvert^{2}}\right)_{1}.\end{split}$ (19)

Note that this is the equation govern the plasma behavior in the resistive
layer. Usually, most of the tearing mode theories perform the asymptotic
matching of the inner resistive and the outer ideal region to get the growth
rate. However, we will just focus on the resistive layer. Calculate
$\tilde{U}$ from Eq. (LABEL:nlineqn) in terms of other quantities, then obtain
the form of dynamo effect using Eq. (12).

Using Eqs. (12) and (18) to substitute $\delta\overline{\lambda}$ in Eq.
(LABEL:nlineqn) gives

$\begin{split}\frac{\partial}{\partial
t}\nabla\cdot\left(\frac{\overline{\rho}\nabla\tilde{U}}{|\overline{\bm{B}}\rvert^{2}}\right)+\widetilde{\bm{B}}\cdot\nabla\left(\frac{\widetilde{\bm{B}}_{\perp}^{*}}{2\eta|\overline{\bm{B}}\rvert^{2}}\cdot\frac{\partial\widetilde{\bm{A}}}{\partial
t}\right)+\frac{\mathrm{i}\bm{k}\cdot\overline{\bm{B}}}{\eta}\left(\frac{\partial\bm{A}}{\partial
t}\cdot\frac{\bm{B}}{|\bm{B}\rvert^{2}}\right)_{1}=\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}-\widetilde{\bm{B}}\cdot\nabla\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot(\nabla\tilde{U}+\eta\widetilde{\bm{j}})}{2\eta|\overline{\bm{B}}\rvert^{2}}+\frac{(\bm{k}\cdot\overline{\bm{B}})^{2}}{\eta|\overline{\bm{B}}\rvert^{2}}\tilde{U}.\end{split}$
(20)

All the time derivative terms are on the left hand side of the equation. The
$\overline{\lambda}$ gradient term is reserved, because the flux average
assumption can not be applied for the turbulent plasma. LiLi (1995) did not
use flux average either in his tearing mode theory, but he did not include
this $\lambda$ gradient term in his model. So that our solutions of the
velocity and magnetic field fluctuations are different from those Li obtained.

The quasilinear theoriesStrauss (1985); Bhattacharjee and Hameiri (1986) did
not include the nonlinear term,
$\widetilde{\bm{B}}\cdot\nabla\delta\overline{\lambda}$, which is of great
importance in the nonlinear growth stage and steady state of a tearing mode.
As we can see, $\delta\overline{\lambda}$ is actually driven by the parallel
dynamo effect.

###  III.1 Steady state

Firstly, the steady state is considered, which means the time derivative terms
are omitted. Eq. (20) becomes

$\widetilde{\bm{B}}\cdot\nabla\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\nabla\tilde{U}}{2\eta|\overline{\bm{B}}\rvert^{2}}-\frac{(\bm{k}\cdot\overline{\bm{B}})^{2}}{\eta|\overline{\bm{B}}\rvert^{2}}\tilde{U}=\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}-\widetilde{\bm{B}}\cdot\nabla\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\widetilde{\bm{j}}}{2|\overline{\bm{B}}\rvert^{2}}.$
(21)

Some simplifications will be made in order to analyse Eq. (21).
$|\overline{\bm{B}}\rvert^{2}=\text{constant}$, because
$p+|\overline{\bm{B}}\rvert^{2}/2=\text{constant}$ in the cylindrical geometry
and $\nabla p$ is neglected in a force-free plasma. The “constant-$\psi$”
approximationFurth, Killeen, and Rosenbluth (1963) is also applied as the
resistive layer is assumed narrow, limiting the treatment only suitable for
$m\geq 2$ modes. Therefore, $\tilde{B}_{r}$ is assumed constant and the
another component of $\widetilde{\bm{B}}_{\perp}$,
$\overline{\bm{B}}\times\nabla r\cdot\widetilde{\bm{B}}_{\perp}$, is assumed
zero.

Taylor expansion of $\overline{\lambda}$ around the rational surface gives

$\overline{\lambda}=\overline{\lambda}_{0}+\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}s+\cdots,\quad\text{where}~{}s=r-r_{s}.$ (22)

The form of $\bm{k}\cdot\overline{\bm{B}}$ is determined by
$\overline{\lambda}$, and the leading order is

$\bm{k}\cdot\overline{\bm{B}}\approx|\bm{k}||\overline{\bm{B}}|\overline{\lambda}_{0}s,$
(23)

where the orders greater than $\mathcal{O}(s)$ are neglected.

The $\widetilde{\bm{A}}$ is assumed containing two components,

$\widetilde{\bm{A}}=\frac{\widetilde{\bm{A}}\cdot\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}\overline{\bm{B}}+\tilde{A}_{r}\nabla
r.$ (24)

The $\overline{\bm{B}}$ component of $\widetilde{\bm{A}}$ can be expressed as
$\tilde{B}_{r}$, and we only keep the leading term,

$\frac{\widetilde{\bm{A}}\cdot\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}=\frac{\tilde{B}_{r}}{\mathrm{i}\bm{k}\times\overline{\bm{B}}\cdot\nabla
r}\approx\frac{\tilde{B}_{r}}{\mathrm{i}|\bm{k}||\overline{\bm{B}}|}.$ (25)

And using Coulomb gauge, $\nabla\cdot\widetilde{\bm{A}}=0$, $\tilde{A}_{r}$
can be expressed as

$\frac{\partial\tilde{A}_{r}}{\partial
r}=-\mathrm{i}\bm{k}\cdot\overline{\bm{B}}\frac{\widetilde{\bm{A}}\cdot\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}\approx-\overline{\lambda}_{0}\tilde{B}_{r}s,$
(26)

which makes

$\tilde{j}_{r}=\nabla^{2}\tilde{A}_{r}\approx-\overline{\lambda}_{0}\tilde{B}_{r}.$
(27)

Therefore, the second term of Eq. (21)’s right hand side(RHS) becomes

$\widetilde{\bm{B}}\cdot\nabla\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\widetilde{\bm{j}}}{2|\overline{\bm{B}}\rvert^{2}}=\widetilde{\bm{B}}\cdot\nabla\frac{\overline{\lambda}_{0}|\tilde{B}_{r}\rvert^{2}}{2|\overline{\bm{B}}\rvert^{2}}$
(28)

At most fusion plasma, the fluctuating magnetic field is much smaller than
mean magnetic field, so this term can be neglected when compared to the first
term of Eq. (21)’s RHS.

Comparison of the two terms of Eq. (21)’s left hand side(LHS) shows that

$\widetilde{\bm{B}}\cdot\nabla\biggl{(}\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\nabla\tilde{U}}{2\eta|\overline{\bm{B}}\rvert^{2}}\biggr{)}\bigg{/}\frac{(\bm{k}\cdot\overline{\bm{B}})^{2}}{\eta|\overline{\bm{B}}\rvert^{2}}\tilde{U}\approx\frac{|\tilde{B}_{r}\rvert^{2}}{2(\bm{k}\cdot\overline{\bm{B}})^{2}\tilde{U}}\frac{\partial^{2}\tilde{U}}{\partial
r^{2}}.$ (29)

Although, the structure of magnetic island doesn’t exist in the turbulent
plasma, we also use the “island width”, $w$, as a characteristic length of the
magnetic perturbation, which can be written as

$w=4\left(\frac{|\tilde{B}_{r}|}{|\bm{k}||\overline{\bm{B}}|\overline{\lambda}_{0}}\right)^{1/2}.$
(30)

The radial derivative can be written as radial characteristic length $L$. The
electrostatic potential fluctuation, $\tilde{U}$, is corresponding to
fluctuating velocity field from Eq. (11). Therefore, $L$ is also the space
scale of velocity fluctuation.

From the flow structure of nonlinear tearing modeRutherford (1973), we can
assume $L\approx w/4$. Therefore, the ratio, Eq. (29), becomes

$\frac{|\tilde{B}_{r}\rvert^{2}}{2(\bm{k}\cdot\overline{\bm{B}})^{2}\tilde{U}}\frac{\partial^{2}\tilde{U}}{\partial
r^{2}}=\frac{w^{4}}{2^{9}L^{2}s^{2}}\approx\frac{w^{2}}{2^{5}s^{2}}.$ (31)

From the ratio, we can see the first term here is dominate at small $|s|$ and
can be ignored as $|s|$ becomes big. In the following, we will give the form
of the parallel dynamo effect at different regions.

####  III.1.1  The region $|s|\ll 2^{-5/2}w$

In this region, the ratio given by Eq. (31) is large. Therefore, we can
neglect the second term of Eq. (21)’s LHS, which gives

$\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}-\widetilde{\bm{B}}\cdot\nabla\left(\frac{\widetilde{\bm{B}}_{\perp}^{*}\cdot\nabla\tilde{U}}{2\eta|\overline{\bm{B}}\rvert^{2}}\right)=0.$
(32)

As we mentioned before, the first term involves the dynamo driven parallel
current density, $\delta\overline{\lambda}$. If we substitute
$\delta\overline{\lambda}$ back into the above equation, we can find

$\widetilde{\bm{B}}\cdot\nabla\left(\overline{\lambda}+\delta\overline{\lambda}\right)=0,$
(33)

which means the modified parallel current density,
$\overline{\lambda}+\delta\overline{\lambda}$, is constant at this region.

Eq. (33) shows that no matter what the original $\overline{\lambda}$ profile
is, the steady state parallel current density will be adjusted into constant
by the dynamo driven $\delta\overline{\lambda}$.

####  III.1.2  The region $|s|\gg 2^{-5/2}w$

In this region, which is far from the rational surface, the first term of Eq.
(21)’s LHS should be neglected. We can get

$\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}+\frac{(\bm{k}\cdot\overline{\bm{B}})^{2}}{\eta|\overline{\bm{B}}\rvert^{2}}\tilde{U}=0.$
(34)

Then, $\tilde{U}$ can be easily found as

$\tilde{U}=-\frac{\eta|\overline{\bm{B}}\rvert^{2}}{(\bm{k}\cdot\overline{\bm{B}})^{2}}\widetilde{\bm{B}}\cdot\nabla\overline{\lambda}.$
(35)

Recalling the expression of the parallel dynamo effect, Eq. (12), gives

$\varepsilon_{\parallel}=\frac{1}{2r|\overline{\bm{B}}\rvert}\frac{\partial}{\partial
r}\left[\frac{\eta
r|\overline{\bm{B}}\rvert^{2}|{\tilde{B}_{r}}|^{2}}{(\bm{k}\cdot\overline{\bm{B}})^{2}}\frac{\partial\overline{\lambda}}{\partial
r}\right],$ (36)

This is actually the same form as StraussStrauss (1985) found. It is not
surprising because the first term’s exclusion makes the treatment degenerate
into the linear tearing mode theory. And Strauss got the result using the
linear tearing mode theory.

####  III.1.3 The whole region

The analytical solution of $\tilde{U}$ exists when the two terms are both
considered simultaneously. Rewriting Eq. (21) gives

$\frac{\partial^{2}\tilde{U}}{\partial
r^{2}}-\frac{2(\bm{k}\cdot\overline{\bm{B}})^{2}}{\lvert\tilde{B}_{r}\rvert^{2}}\tilde{U}=\frac{2\eta|\overline{\bm{B}}\rvert^{2}}{\lvert\tilde{B}_{r}\rvert}\frac{\partial\overline{\lambda}}{\partial
r}.$ (37)

where we only keep the second order radial derivatives of $\tilde{U}$ in the
first term of Eq. (21)’s LHS.

To illustrate the main effect and simplify the equation, we only take the
first two terms of the $\overline{\lambda}$’s expansion, Eq. (22), into
consideration. The form of $\bm{k}\cdot\overline{\bm{B}}$ is taken as
$|\bm{k}||\overline{\bm{B}}|\overline{\lambda}_{0}s$ as mentioned earlier.
Then, Eq. (37) is written in a more convenient form by introducing the
characteristic length $d_{1}$ and new variables $X_{1}$ and $Y_{1}$ defined by

$\begin{split}&d_{1}=\left(\frac{|\tilde{B}_{r}|^{2}}{2|\bm{k}|^{2}|\overline{\bm{B}}|^{2}\overline{\lambda}_{0}^{2}}\right)^{1/4}=2^{-9/4}w,\\\
&X_{1}=\frac{s}{d_{1}}\quad\text{and}\quad
Y_{1}=\left(\frac{2\eta|\overline{\bm{B}}\rvert^{2}d_{1}^{2}}{\lvert\tilde{B}_{r}\rvert}\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}\right)^{-1}\tilde{U}.\end{split}$ (38)

Thus, substituting into Eq. (37) gives

$\frac{\operatorname{d\\!}^{2}Y_{1}}{\operatorname{d\\!}X_{1}^{2}}-X_{1}^{2}Y_{1}=1$
(39)

According to Appendix A, the solution of the above second-order differential
equation is

$Y_{1}=-\frac{1}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-3/4}\exp{\left(-\frac{1}{2}\mu
X_{1}^{2}\right)}.$ (40)

Actually, the solution is determined by the boundary condition. We assume the
domain of $X_{1}$ is $(-\infty,\infty)$ and $Y_{1}=0$ at infinity for
simplicity.

So that we can get the form of dynamo effect from Eqs. (12), (LABEL:chalen)
and (40),

$\varepsilon_{\parallel}=-\eta|\overline{\bm{B}}\rvert
d_{1}\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}\frac{\operatorname{d\\!}Y_{1}}{\operatorname{d\\!}X_{1}}.$
(41)

Obviously, $Y_{1}$ is an even function hence $\varepsilon_{\parallel}$ is an
odd function. Therefore it drives opposite current density at different sides
of the rational surface.

The modification of $\overline{\lambda}$ induced by the parallel dynamo
effect,

$\delta\overline{\lambda}=-\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}s\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-3/4}\frac{\mu}{2}\exp{\left(-\frac{\mu
X_{1}^{2}}{2}\right)}.$ (42)

The integration here approximately equal to 1 near $X_{1}=0$, which makes
$\overline{\lambda}+\delta\overline{\lambda}\approx 0$ near the rational
surface.

Figure 1:  The $\delta\overline{\lambda}$ both in nonlinear and quasilinear
form.The $\overline{\lambda}$ profile before and after modification by the
nonlinear dynamo effect.

Fig. 1 shows the nonlinear form of $\delta\overline{\lambda}$. The quasilinear
$\delta\overline{\lambda}$ from Eq. (36) is also illustrated as a comparison.
The original $\overline{\lambda}$ profile and modified $\overline{\lambda}$
profile are also plotted. The $\overline{\lambda}_{0}$ is deducted so the
values equals 0 at $s=0$. The “saturated island width”, $w_{\mathrm{s}}$, and
the separatrix, $|s|=2^{-5/2}w_{\mathrm{s}}$, of the two regions Section
(III.1.1) and (III.1.2) are also displayed.

The original $\overline{\lambda}$ profile is assumed as a linear function of
$s$. The modified $\overline{\lambda}$ profile is obtained by adding the
nonlinear $\delta\overline{\lambda}$ into the original $\overline{\lambda}$.
The original $\overline{\lambda}$ profile is almost entirely flattened by
$\delta\overline{\lambda}$ near the rational surface. This is the current
redistribution effect driven by the parallel dynamo effect. As for the
external driven current concentrate at the edge, the dynamo effect may be
useful in explaining the current transport from the edge to the core.

As the distance goes far from the rational surface, the nonlinear
$\delta\overline{\lambda}$ becomes smaller and close to the quasilinear
$\delta\overline{\lambda}$. The reason has been explained in Section
(III.1.2).

The above discussion shows the parallel dynamo effect’s analytical expression
under the simple linear distributed parallel current density profile. For an
arbitrary parallel current density profile, the conclusions of Section
(III.1.1) and (III.1.2) are still valid, but the solution of the whole region
is more complicated. A practical procedure is to perform Taylor expansion of
$\overline{\lambda}$ and treat each term separately. However, the first two
terms of Taylor expansion are enough because of the narrowness of the region
where the dynamo effect is essential.

Figure 2: The parallel dynamo effect generated by magnetic and velocity
perturbation.

Fig. 2 demonstrates the direction of the parallel dynamo effect at different
sides of the rational surface. The parallel dynamo effect,
$\bm{\varepsilon}_{\parallel}$, is generated by $\widetilde{\bm{B}}_{r}$ and
$\widetilde{\bm{v}}\cdot\overline{\bm{B}}_{0}\times\nabla r$, where
$\overline{\bm{B}}_{0}$ is the mean magnetic field at the rational surface.

The mean current density is along the direction of $\overline{\bm{B}}_{0}$ at
the whole region. At the $r>r_{s}$ side, $\bm{\varepsilon}_{\parallel}$ is
opposite to $\overline{\bm{B}}_{0}$ and reduces the mean current density. At
the $r<r_{s}$ side, $\bm{\varepsilon}_{\parallel}$ changes direction because
of the reversal $\widetilde{\bm{v}}\cdot\overline{\bm{B}}_{0}\times\nabla r$,
making the mean current density increase. This clearly shows how the current
redistribution occurs near the rational surface.

###  III.2 Growth stage

The time derivative terms are included in this section. Some assumptions have
to be made to simplify Eq. (20). Since
$\widetilde{\bm{B}}_{\perp}\approx\tilde{B}_{r}\nabla r$ and $\tilde{B}_{r}$
is constant, the second term of Eq. (20)’s LHS becomes

$\widetilde{\bm{B}}\cdot\nabla\left(\frac{\widetilde{\bm{B}}_{\perp}^{*}}{2\eta|\overline{\bm{B}}\rvert^{2}}\cdot\frac{\partial\widetilde{\bm{A}}}{\partial
t}\right)=\frac{|\tilde{B}_{r}|^{2}}{2|\overline{\bm{B}}\rvert^{2}}\frac{\mathrm{i}\bm{k}\cdot\overline{\bm{B}}}{\eta}\frac{\partial}{\partial
t}\frac{\widetilde{\bm{A}}\cdot\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}},$
(43)

where Eq. (26) is used to replace $\tilde{A}_{r}$. Then, if the time
derivative of $\overline{\bm{A}}$ is neglected, comparing the second and third
terms of Eq. (20)’s LHS, we can easily find the ratio is
$|\tilde{B}_{r}|^{2}/2|\overline{\bm{B}}\rvert^{2}$. Therefore, the second
term can be ignored when compared to the third term of Eq. (20)’s LHS.

Introducing the growth rate, $\gamma$, to represent $\partial/\partial t$, the
simplified Eq. (20) can be written as

$\frac{2\eta\rho\gamma+|\tilde{B}_{r}\rvert^{2}}{2\eta|\overline{\bm{B}}\rvert^{2}}\frac{\partial^{2}\tilde{U}}{\partial
s^{2}}-\frac{|\bm{k}|^{2}\overline{\lambda}_{0}^{2}}{\eta}s^{2}\tilde{U}=\tilde{B}_{r}\frac{\partial\overline{\lambda}}{\partial
s}-\frac{\tilde{B}_{r}\overline{\lambda}_{0}\gamma}{\eta}s.$ (44)

This equation is the same as Rutherford’s except that we keep the
$\overline{\lambda}$ gradient term. Again, we only consider the first two
terms of the $\overline{\lambda}$’s expansion. Then, introducing another
characteristic length $d_{2}$ and new variables $X_{2}$ and $Y_{2}$ defined by

$\begin{split}&d_{2}=\left(\frac{2\eta\rho\gamma+|\tilde{B}_{r}|^{2}}{2|\bm{k}|^{2}|\overline{\bm{B}}\rvert^{2}\overline{\lambda}_{0}^{2}}\right)^{1/4},\\\
&X_{2}=\frac{s}{d_{2}},\quad\quad
Y_{2}=\left(\frac{2\eta|\overline{\bm{B}}\rvert^{2}|\tilde{B}_{r}|d_{2}^{2}}{2\eta\rho\gamma+|\tilde{B}_{r}|^{2}}\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}\right)^{-1}\tilde{U},\\\ &\text{and}\quad C=\frac{\gamma
d_{2}\overline{\lambda}_{0}}{\eta}\left(\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}\right)^{-1}.\end{split}$ (45)

Eq. (44) can be written in a more convenient form,

$\frac{\operatorname{d\\!}^{2}Y_{2}}{\operatorname{d\\!}X_{2}^{2}}-X_{2}^{2}Y_{2}=1-CX_{2}.$
(46)

According to Appendix A, the solution is

$\begin{split}Y_{2}&=-\frac{1}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-3/4}\exp{\left(-\frac{1}{2}\mu
X_{2}^{2}\right)}\\\
&+\frac{CX_{2}}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-1/4}\exp{\left(-\frac{1}{2}\mu
X_{2}^{2}\right)}.\end{split}$ (47)

The first term of Eq. (47)’s RHS is the same as Eq. (40) and plays the role of
flattening the current density. The second term of Eq. (47)’s RHS is the eddy
current to slow down the growth of tearing mode as in nonlinear tearing mode
theoryRutherford (1973). The first term is an even part, and it does not
affect the growth rate. The second term actually changes the growth rate from
the linear stage to the nonlinear stage.

If we include the higher order of $\overline{\lambda}$’s expansion, the second
derivative of $\overline{\lambda}$ will contribute to the second term. The
growth rate may have a slight correction by this effect.

From Eqs. (12), (LABEL:chalen2) and (47), the form of dynamo effect is

$\varepsilon_{\parallel}=-\frac{|\tilde{B}_{r}\rvert^{2}}{2\eta\rho\gamma+|\tilde{B}_{r}\rvert^{2}}\eta|\overline{\bm{B}}\rvert
d_{2}\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}\frac{\operatorname{d\\!}Y_{2}}{\operatorname{d\\!}X_{2}},$
(48)

Where the second term of Eq. (12) is omitted because of Eqs. (28) and (43).
The ratio of the two components in the denominator of Eq. (48) is

$\frac{2\eta\rho\gamma}{|\tilde{B}_{r}|^{2}}=\frac{2^{9}}{w^{4}|\bm{k}|^{2}\overline{\lambda}_{0}^{2}}\frac{\tau_{\mathrm{A}}^{2}\gamma}{\tau_{\mathrm{R}}},$
(49)

where $\tau_{\mathrm{A}}=\rho^{1/2}a/|\overline{\bm{B}}|$ is the Alfvén
transit time, $\tau_{\mathrm{R}}=a^{2}/\eta$ is the resistive diffusion time,
and $a$ is the minor radius. If this radio is small, the dynamo effect is
essential.

At linear growth stage, the growth rateWesson (2011)

$\gamma=0.55\left[(|\bm{k}|\overline{\lambda}_{0})^{1/2}a^{2}\Delta^{\prime}\right]^{4/5}\tau_{\mathrm{R}}^{-3/5}\tau_{\mathrm{A}}^{-2/5},$
(50)

where $\Delta^{\prime}$ is an important parameter in tearing mode theory and
is determined by outer region. The ratio becomes

$\frac{2\eta\rho\gamma}{|\tilde{B}_{r}|^{2}}=282\left(\frac{a^{2}\Delta^{\prime}}{|\bm{k}|^{2}\overline{\lambda}_{0}^{2}w^{5}}\right)^{4/5}\left(\frac{\tau_{\mathrm{R}}}{\tau_{\mathrm{A}}}\right)^{-8/5}.$
(51)

As the “island width” is small at the beginning, the ratio is large. However,
the magnetic Reynolds number, $\tau_{\mathrm{R}}/\tau_{\mathrm{A}}$, is
prominent at most fusion plasmas. For a typical $m=2$ mode, if we take
$\tau_{\mathrm{R}}/\tau_{\mathrm{A}}\approx 10^{6}$, $a\Delta^{\prime}\approx
10$, $|\bm{k}|\approx m/a$ and $\overline{\lambda}_{0}\approx 1/a$. The
condition for the ratio, Eq. (49), being unity gives $w/a\approx 0.026$. It is
about a quarter of the normally saturated “island width”,
$w_{\mathrm{s}}/a=0.1$. The unity ratio means the dynamo effect is half of
that in the steady state. We should expect the dynamo effect is essential even
in the linear growth stage.

At the nonlinear growth stage, the growth rate becomes smaller as “island
width” grows bigger. This makes the ratio, Eq. (49), decrease much faster. The
dynamo effect becomes significant at this stage.

##  IV  The pressure gradient

The situation that includes the pressure gradient is considered in this
section. We give both the perpendicular and parallel dynamo effect of tearing
modes. The perpendicular dynamo effect is first derived for the tearing modes.
The parallel dynamo effect has some difference from that in the force-free
plasma.

Taking curl of the momentum equation, Eq. (1), to eliminate the $\nabla p$ and
linearizing gives

$\overline{\rho}\frac{\partial}{\partial
t}\nabla\times\widetilde{\bm{v}}=\overline{\bm{B}}\cdot\nabla\widetilde{\bm{j}}+\widetilde{\bm{B}}\cdot\nabla\overline{\bm{j}}-\overline{\bm{j}}\cdot\nabla\widetilde{\bm{B}}-\widetilde{\bm{j}}\cdot\nabla\overline{\bm{B}}.$
(52)

The elimination of $\nabla p$ avoids dealing with the $\nabla\tilde{p}$ term.
The mean current density can be separated into two parts, parallel and
perpendicular to the mean magnetic field respectively,

$\overline{\bm{j}}=\overline{\lambda}\overline{\bm{B}}+\frac{\overline{j}_{\perp}}{|\overline{\bm{B}}\rvert}\overline{\bm{B}}\times\nabla
r,$ (53)

The perpendicular current density, $\overline{j}_{\perp}$, can be represented
in term of $\nabla\overline{p}$. Using the mean part of force balance
equation, Eq. (1), we get

$\overline{j}_{\perp}=\overline{\bm{j}}\cdot(\overline{\bm{B}}\times\nabla
r)=\frac{\partial\overline{p}}{\partial r}.$ (54)

###  IV.1 The perpendicular dynamo effect

The $\nabla r$ component of Eq. (52) is

$\overline{\rho}\frac{\partial}{\partial t}(\nabla
r\cdot\nabla\times\widetilde{\bm{v}})=\mathrm{i}\bm{k}\cdot\overline{\bm{B}}\tilde{j}_{r}-\mathrm{i}\bm{k}\cdot\overline{\bm{j}}\tilde{B}_{r}.$
(55)

Substituting Eq. (53) into above equation, and replacing
$\overline{j}_{\perp}$ with $\overline{j}_{\perp}+\delta\overline{j}_{\perp}$.
Then Eq. (55) becomes

$\begin{split}&\overline{\rho}\frac{\partial}{\partial
t}\left[\mathrm{i}\overline{\bm{B}}\cdot(\nabla
r\times\bm{k})\frac{\widetilde{\bm{v}}\cdot\overline{\bm{B}}}{|\overline{\bm{B}}\rvert^{2}}-\mathrm{i}\bm{k}\cdot\overline{\bm{B}}\frac{\widetilde{\bm{v}}\cdot(\overline{\bm{B}}\times\nabla
r)}{|\overline{\bm{B}}\rvert^{2}}\right]=\\\
&\qquad\quad-\mathrm{i}\tilde{B}_{r}\bm{k}\cdot\overline{\bm{B}}(\overline{\lambda}-\overline{\lambda}_{0})+\mathrm{i}\tilde{B}_{r}\overline{\bm{B}}\times\nabla
r\cdot\bm{k}\frac{\overline{j}_{\perp}-\delta\overline{j}_{\perp}}{|\overline{\bm{B}}\rvert},\end{split}$
(56)

where $\tilde{j}_{r}$ is replaced using Eq. (27). From Eq. (13), the dynamo
induced perpendicular current density, $\delta\overline{j}_{\perp}$, is

$\delta\overline{j}_{\perp}=\frac{\tilde{B}_{r}^{*}\widetilde{\bm{v}}\cdot\overline{\bm{B}}}{2\eta|\overline{\bm{B}}\rvert}-\frac{\widetilde{\bm{B}}^{*}\cdot\overline{\bm{B}}\tilde{\upsilon}_{r}}{2\eta|\overline{\bm{B}}\rvert}.$
(57)

If looking at the steady state, the form of $\delta\overline{j}_{\perp}$ is
directly obtained from Eq. (LABEL:rpnlineqn),

$\delta\overline{j}_{\perp}=-\overline{j}_{\perp}-\overline{\lambda}_{0}|\overline{\bm{B}}\rvert\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}s^{2},$ (58)

where $\overline{\lambda}$ has been Taylor expanded. The approximations, Eq.
(23) and $\overline{\bm{B}}\times\nabla
r\cdot\bm{k}\approx|{\bm{k}}\rvert|\overline{\bm{B}}\rvert$, are adopted here.

This form clearly shows that the current driven by the perpendicular dynamo
effect offsets the mean perpendicular current density into zero near the
rational surface. From Eq. (54), the pressure gradient is eliminated near the
rational surface. Again, the flux average is not used here.

If the growth state is considered, we need to know the relation of fluctuating
velocity’s three components to solve Eq. (LABEL:rpnlineqn). For
simplification, we neglect the second term of Eq. (57)’s RHS, because
$\tilde{B}_{r}$ is usually much greater than
$\widetilde{\bm{B}}\cdot\overline{\bm{B}}/|\overline{\bm{B}}\rvert$. We also
ignore the second term of Eq. (LABEL:rpnlineqn )’s LHS, as
$\bm{k}\cdot\overline{\bm{B}}\ll\overline{\bm{B}}\cdot(\nabla r\times\bm{k})$.
Then, Eq. (LABEL:rpnlineqn) only involves
$\widetilde{\bm{v}}\cdot\overline{\bm{B}}$, and becomes

$\frac{2\overline{\rho}\gamma\eta+|\tilde{B}_{r}\rvert^{2}}{2\eta|\overline{\bm{B}}\rvert^{2}}\widetilde{\bm{v}}\cdot\overline{\bm{B}}=-\tilde{B}_{r}\overline{\lambda}_{0}\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}s^{2}-\tilde{B}_{r}\frac{\overline{j}_{\perp}}{|\overline{\bm{B}}\rvert}.$
(59)

So that $\widetilde{\bm{v}}\cdot\overline{\bm{B}}$ can be easily calculated as

$\widetilde{\bm{v}}\cdot\overline{\bm{B}}=-\frac{2\eta|\overline{\bm{B}}\rvert^{2}\tilde{B}_{r}}{2\overline{\rho}\gamma\eta+|\tilde{B}_{r}\rvert^{2}}\left(\frac{\overline{j}_{\perp}}{|\overline{\bm{B}}\rvert}+\overline{\lambda}_{0}\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}s^{2}\right),$ (60)

The dynamo induced perpendicular current density is

$\delta\overline{j}_{\perp}=-\frac{|\tilde{B}_{r}\rvert^{2}}{2\overline{\rho}\gamma\eta+|\tilde{B}_{r}\rvert^{2}}\left(\overline{j}_{\perp}+\overline{\lambda}_{0}|\overline{\bm{B}}\rvert\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}s^{2}\right).$ (61)

When tearing mode grows, the magnetic perturbation becomes bigger while the
growth rate becomes smaller. $\delta\overline{j}_{\perp}$ becomes more
approaches the steady state and the pressure more flatten. The coefficient
here is the same as that in Eq. (48).

Figure 3: The perpendicular dynamo effect generated by magnetic and velocity
perturbation.

Fig. 3 demonstrates that the direction of the perpendicular dynamo is opposite
to $\overline{\bm{j}}_{\perp}$. The generation of $\bm{\varepsilon}_{\perp}$
requires $\widetilde{\bm{B}}_{r}$ and
$\widetilde{\bm{v}}\cdot\overline{\bm{B}}$. It is different with the force-
free situation where $\widetilde{\bm{v}}\cdot\overline{\bm{B}}$ is not needed.
This means the velocity perturbation will develop a 3D structure when the
pressure gradient is included.

###  IV.2 The parallel dynamo effect

Taking $\overline{\bm{B}}$ component of Eq. (52) leads to

$\begin{split}\overline{\rho}\frac{\partial}{\partial
t}(\overline{\bm{B}}\cdot\nabla\times\widetilde{\bm{v}})&=\mathrm{i}\bm{k}\cdot\overline{\bm{B}}(\widetilde{\bm{j}}\cdot\overline{\bm{B}})-\mathrm{i}\bm{k}\cdot\overline{\bm{j}}(\widetilde{\bm{B}}\cdot\overline{\bm{B}})\\\
&+\tilde{B}_{r}\left(\overline{\bm{B}}\cdot\frac{\partial\overline{\bm{j}}}{\partial
r}\right)-\tilde{j}_{r}\left(\overline{\bm{B}}\cdot\frac{\partial\overline{\bm{B}}}{\partial
r}\right).\end{split}$ (62)

The divergence of linearized Eq. (9) gives

$\overline{\bm{B}}\cdot\nabla\times\widetilde{\bm{v}}=\nabla^{2}\tilde{U}-\widetilde{\bm{v}}\cdot\overline{\bm{j}}\approx\frac{\partial^{2}\tilde{U}}{\partial
r^{2}},$ (63)

where we only keep the dominant second radial derivative term. Because
$\widetilde{\bm{v}}\sim\nabla\tilde{U}\times\overline{\bm{B}}/|\overline{\bm{B}}\rvert$
and the resistive layer is narrow.

The $\overline{\bm{B}}$ component of linearized Eq. (9) gives

$\widetilde{\bm{j}}\cdot\overline{\bm{B}}=-\frac{1}{\eta}\left(\mathrm{i}\bm{k}\cdot\overline{\bm{B}}\tilde{U}+\frac{\partial\widetilde{\bm{A}}\cdot\overline{\bm{B}}}{\partial
t}\right),$ (64)

where the time derivative of $\overline{\bm{B}}$ is ignored. Substituting Eqs.
(63) and (64) to Eq. (62). Utilizing the approximations, Eqs. (23), (27) and
$\overline{\bm{B}}\times\nabla
r\cdot\bm{k}\approx|{\bm{k}}\rvert|\overline{\bm{B}}\rvert$, we get the
equation for $\tilde{U}$ after some algebra,

$\begin{split}&\overline{\rho}\gamma\frac{\partial^{2}\tilde{U}}{\partial
s^{2}}-\frac{|\bm{k}\rvert^{2}|\overline{\bm{B}}\rvert^{2}\overline{\lambda}_{0}^{2}}{\eta}s^{2}\tilde{U}=\\\
&\qquad\qquad\tilde{B}_{r}|\overline{\bm{B}}\rvert^{2}\left(\frac{\partial\overline{\lambda}}{\partial
r}-2\frac{\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
r}\right)-\frac{\gamma\tilde{B}_{r}|\overline{\bm{B}}\rvert^{2}\overline{\lambda}_{0}}{\eta}s,\end{split}$
(65)

where the time derivative is represented as $\gamma$. Again, to include the
nonlinear term, we replace $\overline{\lambda}$ with
$\overline{\lambda}+\delta\overline{\lambda}$. Finally we can get

$\begin{split}&\frac{2\eta\rho\gamma+|\tilde{B}_{r}\rvert^{2}}{2\eta|\overline{\bm{B}}\rvert^{2}}\frac{\partial^{2}\tilde{U}}{\partial
s^{2}}-\frac{|\bm{k}|^{2}\overline{\lambda}_{0}^{2}}{\eta}s^{2}\tilde{U}=\\\
&\qquad\qquad\qquad\quad\tilde{B}_{r}\left(\frac{\partial\overline{\lambda}}{\partial
s}-2\frac{\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
s}\right)\Bigg{|}_{s=0}-\frac{\tilde{B}_{r}\overline{\lambda}_{0}\gamma}{\eta}s,\end{split}$
(66)

where $\overline{\lambda}$ and $\overline{p}$ are Taylor expanded, and we only
keep the leading terms. Except for the first term, this form is the same as
Eq. (44). If the pressure gradient is neglected as in force-free plasma, this
equation will degenerate into Eq. (44). Therefore, the solution can be
obtained conveniently by the replacement,

$\frac{\partial\overline{\lambda}}{\partial
s}\Big{|}_{s=0}\rightarrow\left(\frac{\partial\overline{\lambda}}{\partial
s}-2\frac{\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
s}\right)\Bigg{|}_{s=0},$ (67)

in $Y_{2}$ and $C$. We introduce $Y_{2}^{\prime}$ and $C^{\prime}$,

$\begin{split}&Y_{2}^{\prime}=\left[\frac{2\eta|\overline{\bm{B}}\rvert^{2}|\tilde{B}_{r}|d_{2}^{2}}{2\eta\rho\gamma+|\tilde{B}_{r}\rvert^{2}}\left(\frac{\partial\overline{\lambda}}{\partial
s}-\frac{2\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
s}\right)\Bigg{|}_{s=0}\right]^{-1}\tilde{U},\\\ &C^{\prime}=\frac{\gamma
d_{2}\overline{\lambda}_{0}}{\eta}\left[\left(\frac{\partial\overline{\lambda}}{\partial
s}-\frac{2\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
s}\right)\Bigg{|}_{s=0}\right]^{-1}.\end{split}$ (68)

So that, the solution of $Y_{2}^{\prime}$ is the same as $Y_{2}$ when
replacing $C$ by $C^{\prime}$ in Eq. (47).

The parallel dynamo effect is calculated as

$\varepsilon_{\parallel}=-\frac{\eta|\overline{\bm{B}}\rvert|\tilde{B}_{r}\rvert^{2}d_{2}}{2\eta\rho\gamma+|\tilde{B}_{r}\rvert^{2}}\left(\frac{\partial\overline{\lambda}}{\partial
s}-\frac{2\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
s}\right)\Bigg{|}_{s=0}\frac{\operatorname{d\\!}Y_{2}^{\prime}}{\operatorname{d\\!}X_{2}}.$
(69)

If the relation, $p+|\overline{\bm{B}}\rvert^{2}/2=\text{constant}$, is used,

$\frac{\partial\overline{\lambda}}{\partial
s}-2\frac{\overline{\lambda}_{0}}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{p}}{\partial
s}\approx\frac{1}{|\overline{\bm{B}}\rvert^{2}}\frac{\partial\overline{\bm{j}}\cdot\overline{\bm{B}}}{\partial
s}.$ (70)

This seems that the parallel dynamo effect will flatten
$\overline{\bm{j}}\cdot\overline{\bm{B}}$ profile instead of
$\overline{\lambda}$ profile. This is contradicting with Taylor’s theory,
where the tendency of flattening $\overline{\lambda}$ is required. Taylor’s
theory is suitable in the force-free plasma, while our result is derived with
the pressure gradient included. This may give an idea of extending Taylor’s
theory into the plasma with a pressure gradient.

##  V  conclusions and discussion

In this paper, the nonlinear dynamo effect of the tearing modes is derived in
the frame of the resistive MHD model. The dynamo effect parallel to
$\overline{\bm{B}}$ was considered both in the force-free plasma and the
plasma with pressure gradient included. The dynamo effect perpendicular to
$\overline{\bm{B}}$ was first derived in the plasma with pressure gradient.

In the force-free plasma, it was found that the parallel dynamo effect would
drive opposite current densities at different sides of the rational surface.
Thus the $\overline{\lambda}$ profile is flattened by the dynamo effect near
the rational surface. The fast transport along the magnetic flux surface is
not used here. For the turbulent plasma, there are many rational surfaces, so
the $\overline{\lambda}$ profile is flattened in the entire turbulent region.
This provided a reasonable explanation of the Taylor relaxation process
through the dynamo effect.

The quasilinear theory also indicated that the parallel dynamo effect produced
the flattening effect of the current density. However, it was only suitable
for the region that far from the rational surface, and the flattening effect
was minimal. This paper’s nonlinear theory gave the proper form of the
parallel dynamo effect in the entire region. It was found that the flattening
effect mainly influences the region near the rational surface. At the region
far from the rational surface, the parallel dynamo effect is relatively small.
Moreover, our nonlinear form of the parallel dynamo effect is the same as the
quasilinear form under the approximation that the distance is far from the
rational surface. The reason is the nonlinear term can be ignored at the
region far from the rational surface.

In the plasma with the pressure gradient, it was found that the perpendicular
dynamo effect would drive a current density opposite to the mean perpendicular
current density. This eliminates the mean perpendicular current density, which
indicates the elimination of the pressure gradient.

Besides, with the pressure gradient included, the parallel dynamo effect is
found different from that in force-free plasma. The parallel dynamo effect
flattens $\overline{\bm{j}}\cdot\overline{\bm{B}}$ instead of
$\overline{\lambda}$ when considering the pressure gradient. This may give
inspiration for extending the Taylor relaxation theory with the pressure
gradient included.

Most tearing mode theories had assumed the flatten current density using the
flux average in the magnetic island. Some tearing mode theories naturally
ignored the current density gradient in the inner region like LiLi (1995) did.
Our result gave another explanation of how the current density flattened in
the magnetic island without using flux average. This indicates that the
current density flattening assumption has a solid foundation in the tearing
mode theory. In addition, the tearing mode theories considering the current
density gradient perpendicular to the magnetic flux may take this dynamo-
induced flattening effect into consideration.

The turbulent plasma was decomposed into many tearing modes in this paper.
Although we treated the tearing modes separately, the interactions of the
tearing modes, especially the adjacent ones, should be considered in future
work.

###### Acknowledgements.

We wish to acknowledge Dr. Zhuping’s team’s support with the using of NIMROD
and offering suggestions. We also thanks Dr. Huang Wenlong for the discussion
about tearing mode. This work was supported by National Natural Science
Foundation of China (Nos. 11827810 and 11875177), International Atomic Energy
Agency Research Contract No. 22733.

##  Appendix A The solution of
$\frac{\operatorname{d\\!}^{2}y}{\operatorname{d\\!}x^{2}}-x^{2}y=a+bx$

Note that $a$ and $b$ are arbitrary real numbers here, do not confuse with
minor radius mentioned above. This is a linear equation and can be separated
into two equations:

$\displaystyle\frac{\operatorname{d\\!}^{2}y}{\operatorname{d\\!}x^{2}}-x^{2}y=1,$
(71)
$\displaystyle\frac{\operatorname{d\\!}^{2}y}{\operatorname{d\\!}x^{2}}-x^{2}y=x.$
(72)

The solutions of above two equations are even and odd functions respectively.
Therefore, we can deal with $x\geq 0$ first and get another part using
symmetricity. Taking substitutions with

$z=\frac{1}{2x^{2}},\quad u=z^{-1/4}y.$ (73)

Then, the equations becomes

$\displaystyle
z^{2}\frac{\operatorname{d\\!}^{2}u}{\operatorname{d\\!}z^{2}}+z\frac{\operatorname{d\\!}u}{\operatorname{d\\!}z}-\left(z^{2}+\frac{1}{16}\right)u=\frac{1}{2}z^{3/4},$
(74) $\displaystyle
z^{2}\frac{\operatorname{d\\!}^{2}u}{\operatorname{d\\!}z^{2}}+z\frac{\operatorname{d\\!}u}{\operatorname{d\\!}z}-\left(z^{2}+\frac{1}{16}\right)u=\frac{\sqrt{2}}{2}z^{5/4}.$
(75)

These are two special cases of the non-homogeneous Bessel’s differential
equation, and the solutions are the second-kind version of modified Struve
functionsAbramowitz and Stegun (1972). The two modified Struve functions here
can be represented in term of the Poisson’s integral:

$\displaystyle
u(z)=-\frac{1}{2}z^{-1/4}\int_{0}^{1}(1-t^{2})^{-3/4}\exp{(-zt)}\operatorname{d\\!}t,$
(76) $\displaystyle
u(z)=-\frac{\sqrt{2}}{2}z^{1/4}\int_{0}^{1}(1-t^{2})^{-1/4}\exp{(-zt)}\operatorname{d\\!}t.$
(77)

Substituting back with Eq. (73) and using the symmetricity, the solutions of
Eqs. (71) and (72) are given as

$\displaystyle
y(x)=-\frac{1}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-3/4}\exp{\left(-\frac{1}{2}\mu
x^{2}\right)},$ (78) $\displaystyle
y(x)=-\frac{x}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-1/4}\exp{\left(-\frac{1}{2}\mu
x^{2}\right)}.$ (79)

Combine the above two solutions, we can get the solution of the original
equation:

$\begin{split}y(x)&=-\frac{a}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-3/4}\exp{\left(-\frac{1}{2}\mu
x^{2}\right)}\\\
&-\frac{bx}{2}\int_{0}^{1}\operatorname{d\\!}\mu(1-\mu^{2})^{-1/4}\exp{\left(-\frac{1}{2}\mu
x^{2}\right)}.\end{split}$ (80)

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