2.1.1. Magnetic flux balance

One can consider two times $t_{1}$ and $t_{2}$ during which the plasma-fluid element displacement is $\boldsymbol {u}(t_{2}-t_{1})=\boldsymbol {u}\Delta t$ (figure 2). At time $t_{1}$, the magnetic flux $\varPhi _{1}$ through the surface $S_{1}$ bounded by the contour $C_{1}$ is

(2.1)\begin{equation} \varPhi_{1}=\int_{S_{1}}\boldsymbol{B}\left(\boldsymbol{x},t\right)\boldsymbol{\cdot} \boldsymbol{dS_{1}}, \end{equation}

where $\boldsymbol {d_{S_{1}}}$ is the oriented surface element related to the surface $S_{1}$.

Figure 2. Magnetic flux balance.

At time $t_{2}=t+\Delta t$, the plasma-fluid element has moved by a distance $\boldsymbol {u}\Delta t$ and crosses a new surface $S_{2}$ (deformed with respect to $S_{1}$) and delimited by the contour $C_{2}$ (also deformed with respect to $C_{1}$). The magnetic flux $\varPhi _{2}$ through this new surface is $\varPhi _{2}=\int _{S_{2}}\boldsymbol {B}(\boldsymbol {x}, t)\boldsymbol {\cdot } \boldsymbol {dS_{2}}$ (with $\boldsymbol {dS_{2}}$ being the oriented surface element related to $S_{2}$).

A first Taylor expansion, in time, gives $\varPhi _{2}\sim \int _{S_{2}}\boldsymbol {B}(\boldsymbol {x}+\boldsymbol {u}\Delta t, t)\boldsymbol {\cdot } \boldsymbol {dS_{2}}+\Delta t\int _{S_{2}}\partial _{t}\boldsymbol {B}(\boldsymbol {x}+\boldsymbol {u}\Delta t, t)\boldsymbol {\cdot } \boldsymbol {dS_{2}}$. Then, a second Taylor expansion in $\boldsymbol {x}$ (with $\boldsymbol {dx}=\boldsymbol {u}\Delta t$) gives $\boldsymbol {B}(\boldsymbol {x}+\boldsymbol {u}\Delta t, t)\sim \boldsymbol {B}(\boldsymbol {x}, t)+\boldsymbol {u}\Delta t({\partial \boldsymbol {B}(\boldsymbol {x},t)}/{\partial \boldsymbol {x}})$, where the position $\boldsymbol {x}$ is related to the surface $S_{1}$ and the position $\boldsymbol {x}+\boldsymbol {u}\Delta t$ is related to the surface $S_{2}$. Finally, keeping only the order $1$ terms (with $\Delta t \ll \Delta t^{2}$), the linearized $\varPhi _{2}$ can be written as

(2.2)\begin{equation} \varPhi_{2}\sim\int_{S_{2}}\boldsymbol{B}\left(\boldsymbol{x}+\boldsymbol{u}\Delta t, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{2}}+\Delta t\int_{S_{1}}\partial_{t}\boldsymbol{B}\left(\boldsymbol{x}, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{1}}. \end{equation}

Applying the Ostrogradski theorem to the local Maxwell–Thomson equation, it is well known that at a given time, $\int _{V}\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {B}\,{\rm d}V=\int _{S}\boldsymbol {B}\boldsymbol {\cdot } \boldsymbol {dS}=0$, where $S=S_{1}+S_{2}+S_{L}$ (with $\boldsymbol {dS}$ its associated oriented surface element) is the surface enveloping the volume $V$ crossed by the plasma-fluid element. At a time $t$, the plasma-fluid element ‘enters’ by $S_{1}$ and ‘leaves’ the volume $V$ through $S_{2}$ and $S_{l}$. This simple balance of flows gives

(2.3)\begin{equation} \int_{S}\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{dS}=0={-}\int_{S_{1}}\boldsymbol{B}\left(\boldsymbol{x}, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{1}}+\int_{S_{2}}\boldsymbol{B}\left(\boldsymbol{x}+\boldsymbol{u}\Delta t, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{2}}+\int_{S_{l}}\boldsymbol{B}\left(\boldsymbol{x}, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{l}}. \end{equation}

Using the expressions of the fluxes $\varPhi _{1}$ (2.1) and $\varPhi _{2}$ (2.2), one can write

(2.4)\begin{equation} \varPhi_{1}-\varPhi_{2}+\Delta t\int_{S_{1}}\partial_{t}\boldsymbol{B}\left(\boldsymbol{x}, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{1}}-\int_{S_{l}}\boldsymbol{B}\left(\boldsymbol{x}, t\right)\boldsymbol{\cdot} \boldsymbol{dS_{l}}=0. \end{equation}

Then (figure 2), on the lateral surface $S_{l}$, $\boldsymbol {dS_{l}}=\boldsymbol {dl}\times \boldsymbol {u}\Delta t$, where $\boldsymbol {dl}$ is the oriented displacement element along the contour $C_{l}=C_{1}$ which is the common boundary of the surfaces $S_{1}$ and $S_{l}$. Using the Stokes theorem, one can get $\int _{S_{l}}\boldsymbol {B}\boldsymbol {\cdot } \boldsymbol {dS_{l}}=\int _{C_{l}}\boldsymbol {B}\boldsymbol {\cdot }(\boldsymbol {dl}\times \boldsymbol {u}\Delta t)=\Delta t\int (\boldsymbol {u}\times \boldsymbol {B})\boldsymbol {\cdot } \boldsymbol {dl}=\Delta t\int _{S_{1}}\boldsymbol {\nabla }\times (\boldsymbol {u}\times \boldsymbol {B})\boldsymbol {\cdot } \boldsymbol {dS_{1}}$.

Finally, the flux balance (2.4) can be re-written using the Maxwell–Faraday equation and representing the conservation (or the non-conservation) of the magnetic flux during the plasma motion:

(2.5)\begin{equation} \frac{{\rm d}\varPhi}{{\rm d}t}=\frac{\varPhi_{2}-\varPhi_{1}}{\Delta t}=\int_{S_{1}}\left[-\boldsymbol{\nabla}\times\left(\boldsymbol{E}+\boldsymbol{u}\times \boldsymbol{B}\right)\right]\boldsymbol{\cdot} \boldsymbol{dS_{1}}. \end{equation}

2.1.2. Magnetic connectivity

At a given time $t$, $M_{1}$ is a plasma-fluid element of the magnetic field line moving with $\boldsymbol {u_{1}}\equiv \boldsymbol {u}$ and $M_{2}$ is another plasma-fluid element (moving with $\boldsymbol {v_{2}}$) belonging to the same line and very close to $M_{1}$: $M_{1}$ and $M_{2}$ are separated by the infinitesimal length $\boldsymbol {\Delta } \boldsymbol {l}$ and $\Delta \boldsymbol {u}=\boldsymbol {u_{2}}-\boldsymbol {u}$ (figure 3). By definition, magnetic field lines are parallel at any point of the magnetic field $\boldsymbol {B}$ leading to $\boldsymbol {\Delta } \boldsymbol {l}\times \boldsymbol {B}=\boldsymbol {0}$, and it is said that $M_{1}$ and $M_{2}$ are connected by the same magnetic field line. The notion of magnetic reconnection is therefore naturally linked to the non-conservation (between two moments $t$ and $t+dt$) of this magnetic connectivity which should be investigated by computing $({{\rm d}}/{{\rm d}t})(\boldsymbol {\Delta } \boldsymbol {l}\times \boldsymbol {B})=\underset {I}{\underbrace {({{\rm d}\boldsymbol {\Delta } \boldsymbol {l}}/{{\rm d}t})\times \boldsymbol {B}}}+\underset {II}{\underbrace {\boldsymbol {\Delta } \boldsymbol {l}\times {{\rm d}\boldsymbol {B}}/{{\rm d}t}}}$.

Figure 3. Magnetic connectivity concept. At $t$, $M_1$ and $M_2$ belong to the same magnetic field line: they are connected and $\boldsymbol {\Delta } \boldsymbol {l}\times \boldsymbol {B} = \boldsymbol {0}$. If the connectivity is conserved, at $t + dt$, $M_1'$ and $M_2'$ still belong to the same magnetic field line and $(\boldsymbol {\Delta } \boldsymbol {l}+d (\boldsymbol {\Delta }\boldsymbol {l}))\times \boldsymbol {B}=\boldsymbol {0}$.

To that purpose, figure 3 is plotted: $M_{1}^{'}$ and $M_{2}^{'}$ are respectively $M_{1}$ and $M_{2}$ after $dt$. An evaluation of $I$Footnote ¹ gives ${{\rm d}\boldsymbol {\Delta }\boldsymbol {l}}/{{\rm d}t}\times \boldsymbol {B}=[(\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {u}]\times \boldsymbol {\Delta } \boldsymbol {l}$. Additionally, an evaluation of $II$Footnote ² gives $\boldsymbol {\Delta } \boldsymbol {l}\times {{\rm d}\boldsymbol {B}}/{{\rm d}t}=[\boldsymbol {\nabla }\times \boldsymbol {E}-(\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {B}]\times \boldsymbol {\Delta }\boldsymbol {l}$. As a consequence, one can write that $({{\rm d}}/{{\rm d}t})(\boldsymbol {\Delta l\times B})=I+II=[(\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {u}+\boldsymbol {\nabla }\times \boldsymbol {E}-(\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {B}]\times \boldsymbol {\Delta } \boldsymbol {l}$. Finally, thanks to vector identities, one can write the conservation (or the non-conservation) of the magnetic connectivity:

(2.6)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\boldsymbol{\Delta} \boldsymbol{l}\times \boldsymbol{B}\right)=\left[\boldsymbol{\nabla}\times\left(\boldsymbol{E}+ \boldsymbol{u}\times \boldsymbol{B}\right)\right]\times\boldsymbol{\Delta} \boldsymbol{l}. \end{equation}

2.1.3. Frozen-in law and general definition of magnetic reconnection

It is very interesting to highlight that, so far, the fluid character of the plasma has not been used, in particular, to obtain (2.5) and (2.6). $M_1$ and $M_2$ have been considered as a plasma-fluid volume element $\delta V = {{\rm d}\kern0.7pt x}^{3}_{{\rm fluid}}$ (see figure 1) attached to the magnetic field line as well as to a single point that belongs to this line. However, at this stage, it is useful to use the MHD framework where $M_{1}$ and $M_{2}$ are plasma-fluid volume elements moving with the fluid velocity $\boldsymbol {u}$ which satisfies the well-known generalized Ohm's law (Biskamp Reference Biskamp2000):

(2.7)\begin{equation} \boldsymbol{E}= \boldsymbol{E_{ind}} +\boldsymbol{E_{Ohm}}+ \boldsymbol{E_{Hall}} + \boldsymbol{E_{iner}} +\boldsymbol{E_{therm}}. \end{equation}

In (2.7):

1. (i) $\boldsymbol {E_{ind}}=-\boldsymbol {u}\times \boldsymbol {B}$ is the ideal induction or dynamo field representing the electric field created by the global motion $\boldsymbol {u}$ of the plasma within the magnetic field $\boldsymbol {B}$. It plays at large fluid scale $L$ (see figure 1);

2. (ii) $\boldsymbol {E_{Ohm}}=\eta \boldsymbol {j}$ is the non-ideal ohmic field, the plasma being an electric conductor of resistivity $\eta$ and of current $\boldsymbol {j}$. This term becomes important in thin resistive layer $L_{\eta } \ll L$ (see further for an evaluation of $L_{\eta }$ for typical tokamaks);

3. (iii) $\boldsymbol {E_{Hall}}=({1}/{n_{e}e})\boldsymbol {j}\times \boldsymbol {B}$ is the non-ideal Hall field corresponding to the Lorentz force $\boldsymbol {j}\times \boldsymbol {B}$ ($n_e$ being the fluid density of the electrons and $e$ being the elementary charge). This term becomes important at scales of the order of the ionic Larmor radius $\rho _i \ll L_{\eta } \ll L$;

4. (iv) $\boldsymbol {E_{iner}}=({m_e}/{n_{e}e^{2}})[\partial _{t}\boldsymbol {j}+\boldsymbol {\nabla }\boldsymbol {\cdot }(\boldsymbol {\overline {\overline {uj}}} +\boldsymbol {\overline {\overline {ju}}}-({1}/{n_{e}e})\boldsymbol {\overline {jj}})]$ (where $\bar {\bar {\bullet }}$ represents a tensor) is the non-ideal inertial term which plays a role at scales of the order of the electron skin depth $d_{e} \ll \rho _{i} \ll L_{\eta } \ll L$ ($m_e$ being the electron mass);

5. (v) $\boldsymbol {E_{therm}}=-({1}/{n_{e}e})\boldsymbol {\nabla }\boldsymbol {\cdot }\overline {\overline {P_{e}}}$ is the non-ideal thermal electromotive field playing a role at ion sound Larmor radius scale being smaller than $L$ and with $\boldsymbol {\overline {P_e}}$ being the electron's pressure tensor.

Thus, at large scale, non-ideal effects can be neglected and the ideal Ohm's law is simply $\boldsymbol {E}=\boldsymbol {E_{ind}}=-\boldsymbol {u}\times \boldsymbol {B}$ leading to a conservation of the magnetic flux ((2.5) becomes ${{\rm d}\varPhi }/{{\rm d}t}=0$) and the magnetic connectivity ((2.6) becomes $({{\rm d}}/{{\rm d}t})(\boldsymbol {\Delta } \boldsymbol {l}\times \boldsymbol {B})=0$). This is the well-known frozen-in law, where at large scale, the plasma and the magnetic field $\boldsymbol {B}$ are strongly coupled in the same dynamic and where magnetic reconnection cannot occur.

However at smaller scales, non-ideal effects can originate from an electric field leading to a non-conservation of the magnetic connectivity with (2.6) becoming

(2.8)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\boldsymbol{\Delta} \boldsymbol{l}\times \boldsymbol{B}\right)=\left[\boldsymbol{\nabla}\times \boldsymbol{E^{NI}}\right]\times\boldsymbol{\Delta} \boldsymbol{l}\neq0. \end{equation}

To go further by investigating the required form of $\boldsymbol {E^{NI}}$ leading to a non-conservation of magnetic connectivity (2.8), a general definition of magnetic reconnection can be drawn. Indeed, magnetic reconnection is due to a particular dynamics of the magnetic field and as a consequence of the Faraday law due to the existence of non-zero rotational electric field. However, not all rotational electric fields can lead to magnetic reconnection. More precisely, having $\boldsymbol {E^{NI}}\neq \boldsymbol {0}$ is not sufficient to allow magnetic reconnection. Additionally, to get a general definition of magnetic reconnection, the impact of the form of $\boldsymbol {E^{NI}}$ has to be investigated on (2.5) and (2.8). Let us set $\boldsymbol {E^{NI}}=\boldsymbol {E_{\parallel }^{NI}}+\boldsymbol {E_{\perp }^{NI}}$, where the subscripts $_{\parallel }$ and $_{\perp }$ refer respectively to the parallel direction to $\boldsymbol {B}=B\boldsymbol {b}$ with $\boldsymbol {E^{NI}_{\parallel }}=(\boldsymbol {b}\boldsymbol {\cdot } \boldsymbol {E^{NI}})\boldsymbol {b}$ and to the orthogonal direction with $\boldsymbol {E^{NI}_{\perp }}=\boldsymbol {b}\times (\boldsymbol {E^{NI}}\times \boldsymbol {b})$. Thus, $\boldsymbol {E^{NI}_{\perp }}$ can be written as $\boldsymbol {E^{NI}_{\perp }}= \boldsymbol {v}\times \boldsymbol {B}$, where the physical meaning of the velocity $\boldsymbol {v}$ will be understood just after. As a consequence, $\boldsymbol {E^{NI}}=\boldsymbol {E^{NI}_{\parallel }}+\boldsymbol {v}\times \boldsymbol {B}$. The expression of the generalized Ohm's law written as $\boldsymbol {E}=-\boldsymbol {u}\times \boldsymbol {B}+\boldsymbol {v}\times \boldsymbol {B}+\boldsymbol {E^{NI}_{\parallel }}$ leads to the following expression for the conservation of the magnetic flux (2.5):

(2.9)\begin{equation} \frac{{\rm d}\varPhi}{{\rm d}t}=\int_{S_{1}}-\left[\boldsymbol{\nabla}\times \boldsymbol{E^{NI}}_{{\parallel}}\right]\boldsymbol{\cdot}\boldsymbol{dS_{1}} \end{equation}

and for the conservation of the magnetic connectivity (2.8):

(2.10)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\boldsymbol{\Delta} \boldsymbol{l}\times \boldsymbol{B}\right)=\left[\boldsymbol{\nabla}\times \boldsymbol{E_{{\parallel}}^{NI}}\right]\times\boldsymbol{\Delta} \boldsymbol{l} \end{equation}

in the moving frame of velocity $\boldsymbol {w} = \boldsymbol {u}-\boldsymbol {v}$ with ${{\rm d}}/{{\rm d}t}=\partial _{t}+\boldsymbol {w}\boldsymbol {\cdot }\boldsymbol {\nabla }$. One can note that the perpendicular non-ideal electric field $\boldsymbol {E^{NI}_{\perp }}$ does not contribute to the magnetic field connectivity conservation (2.10). From (2.9) and (2.10), one can conclude that the perpendicular component of the non-ideal terms in generalized Ohm's law (2.7) written as $\boldsymbol {E^{NI}_{\perp }}=\boldsymbol {v}\times \boldsymbol {B}$ leads only to a non-conservation of the magnetic flux (2.9) in the plasma flow frame of velocity $\boldsymbol {u}$.

So having $\boldsymbol {E^{NI}}=\boldsymbol {E^{NI}_{\perp }}$ in a thin non-ideal layer does not originate in magnetic reconnection. However, this term leads to a slippage of the magnetic field line moving at a velocity $\boldsymbol {v}$ different to the plasma velocity $\boldsymbol {u}$. (It is interesting to note that in the generalized Ohm's law (2.7), the Hall term $\boldsymbol {E_{Hall}}$ induces such slippage of magnetic field lines and can accelerate (without originating it) magnetic reconnection.) Finally, following (2.9) and (2.10), only the parallel component of the non-ideal electric field $\boldsymbol {E^{NI}_{\parallel }}$ in the generalized Ohm's law (2.7) leads to a non-conservation of magnetic connectivity in a thin non-ideal layer and as a consequence leads to magnetic reconnection.

In the past (Vasyliunas Reference Vasyliunas1975; Axford Reference Axford1984; Sonnerup Reference Sonnerup1984; Schindler & Hesse Reference Schindler and Hesse1988; Hornig & Rastätter Reference Hornig and Rastätter1997, Reference Hornig and Rastätter1998; Priest & Forbes Reference Priest and Forbes1998), from a general form of $\boldsymbol {E^{NI}}$, attempts have been made to provide a general definition of magnetic reconnection and some questions are still open today. Here, associating the magnetic reconnection to a non-conservation of magnetic connectivity given by (2.10) and following the point of view of Schindler & Hesse (Reference Schindler and Hesse1988), a general definition of magnetic reconnection is given: having $\boldsymbol {E^{N}_{\parallel }}\neq \boldsymbol {0}$ in a extremely thin region is a necessary and sufficient condition for general magnetic reconnection in two-dimensional (2-D) as well as in three-dimensional (3-D) space. Magnetic reconnection is the result of a local non-conservation of magnetic connectivity between two times due to non-ideal effects leading to the presence of $\boldsymbol {E^{NI}_{\parallel }}$ in the thin non-ideal region. The 2-D process can be illustrated by figure 4 where magnetic field lines are not conserved during the magnetic reconnection process: two plasma-fluid elements $P_{1}$, $P_{2}$ (regardless of the distance separating them), connected by the same magnetic field line at a given time, are no longer so after the magnetic field lines have been disconnected and then reconnected. Thus, magnetic reconnection is by nature a multi-scale process. Most of the time, the frozen-in law is satisfied, the plasma and the magnetic field being strongly attached together. However, there exist extremely thin regions where non-ideal effects become important to break the frozen-in law. In these non-ideal regions (also known as diffusive regions), magnetic field lines can experience a local tearing and then a local reconnection leading to a global change of magnetic field topology and to the formation of magnetic island(s) (for a 2-D picture, see figure 4). It is said that the magnetic field diffuses through the plasma. As mentioned in the introduction, although the historical point of view emphasizes large-scale topology changes, the resulting structures (with an island shape) can develop on both small and large scales.

Figure 4. Simple picture of 2-D magnetic reconnection process. Before magnetic reconnection, $P_1$ and $P_2$ are connected by the same magnetic field line. The plasma-fluid element $P_3$ belongs to another magnetic field line. During the reconnection process, a narrow non-ideal region (in orange) forms between the oppositely directed field lines: magnetic reconnection is a non-ideal local process. After the reconnection, $P_2$ is no longer connected to $P_1$, but is now connected to $P_3$.

2.2.1. Sweet–Parker model

Sweet and Parker were the first to propose a physical mechanism of magnetic reconnection (Parker Reference Parker1957; Sweet Reference Sweet1958a,Reference Sweetb). The so-called Sweet–Parker mechanism is a 2-D model of steady-state magnetic reconnection of a resistive current sheet in a plasma flow of velocity $\boldsymbol {u}$. To illustrate this model, let us consider a simple magnetic configuration in a 3-D slab geometry such as $\boldsymbol {B}=B_{z}\boldsymbol {e_{z}}+B_{y}x\boldsymbol {e_{y}}$ with $B_{z} \gg B_{y}$. Here, $B_{z}\boldsymbol {e_{z}}$ is called the guide field, where $\boldsymbol {e_{z}}$ is the parallel direction, and the reconnection process takes place in the 2-D perpendicular plane $(x,y)$. For such configuration, in $(x,y)$, along the line $x=0$, the magnetic field is null and is of opposite sign on either side of the null points line. Resistivity is supposed to play a role only in a thin layer called the resistive or the diffusive layer of length $2L$ and width $2l$ and follows the null points line. In this layer, following the well-known Ampère law, the parallel current $\boldsymbol {j}=\mu _{0}^{-1}B_{y}\boldsymbol {e_{z}}$ induces a non-zero $\boldsymbol {E^{NI}_{\parallel }}=\eta \boldsymbol {j}$ (where $\mu _{0}$ is the magnetic constant) leading to magnetic reconnection. From the generalized Ohm's law (2.7) in the resistive layer, $\boldsymbol {E}\sim \eta \boldsymbol {j}$, combined with the Faraday law, one can write the induction equation for the magnetic field: $\partial _{t}\boldsymbol {B}\sim \eta \boldsymbol {\Delta } \boldsymbol {B}$. It is said that the magnetic field diffuses in the resistive layer. Outside the resistive layer, the system is considered as ideal where the coupling between the generalized Ohm's law (2.7), $\boldsymbol {E}\sim -\boldsymbol {u}\times \boldsymbol {B}$, and the well-known Faraday law gives the ideal induction equation for the magnetic field, $\partial _{t}B\sim \boldsymbol {\nabla }\times (\boldsymbol {u}\times \boldsymbol {B})$. In the $(x, y)$ plane, the two components of the plasma flow $\boldsymbol {u}$ does not play the same role regarding the reconnection phenomena. The component along the $x$ direction, i.e. $\boldsymbol {u_{x}}=\boldsymbol {u_{in}}$, ‘pushes’ the flow inside the resistive layer where magnetic reconnection can take place by the resistive diffusion of the magnetic field. Thus, it is assumed that $\boldsymbol {u_{in}}$ is directed along increasing $x$ when $x<0$ and along decreasing $x$ when $x>0$. The component along the $y$ direction, i.e. $\boldsymbol {u_{y}}=\boldsymbol {u_{out}}$, is attached to the magnetic field and drives the flow outside the resistive layer. An illustration of the Sweet–Parker configuration is presented in figure 5.

Figure 5. Illustration of the Sweet–Parker mechanism.

First, the Sweet–Parker model proposes an elegant description of the physical mechanism at play at the resistive layer during a steady-state (i.e. $\partial _{t}$ going to $0$) magnetic reconnection process. As mentioned previously, the flow pushes the plasma inside the resistive layer. The counter-reaction to this flow pushing the magnetic field induces the existence of a Lorentz force opposing $\boldsymbol {u_{in}}$. In the MHD framework, the Lorentz force can be evaluated as $\boldsymbol {F_{L}}\sim \boldsymbol {j}\times \boldsymbol {B}$, where the current can be estimate taking the generalized Ohm's law (2.7) with only the inductive and resistive terms, i.e. $\boldsymbol {j}\sim ({\boldsymbol {u}\times \boldsymbol {B}})/{\eta }=({u_{in}B_{y}}/{\eta })\boldsymbol {e_{z}}$. It follows an estimation of the Lorentz force $\boldsymbol {F_{L}}\sim -({u_{in}B_{y}^{2}}/{\eta })\boldsymbol {e_{x}}$ . Thus, in the ideal region where $u_{in}B_{y}^{2}\neq 0$ and where $\eta \rightarrow 0$, magnetic reconnection is not allowed: the Lorentz force is infinite and prevents flow from breaking magnetic field lines that can only be distorted. However, in the resistive layer, $B_{y}(x=0)=0$ and since $\eta$ is very small indeed, but has a finite value, one can write that $\boldsymbol {F_{L}}=0$. There is no more counter-reaction to the incoming flow $\boldsymbol {u_{in}}$. The pressure applied by the incoming flow $\boldsymbol {u_{in}}$ on the magnetic field lines leads to the breaking of these field lines: magnetic reconnection occurs.

Then, the Sweet–Parker model gives an estimate of the magnetic reconnection time defined by Sweet and Parker as $\tau _{RM}=L/u_{in}$. To evaluate $\tau _{RM}$, $u_{in}$ has to be expressed as a function of the system parameters. To this end, one can start by using the incompressibility property of the flow (i.e. $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {u}=0$) to write that ${u_{in}}/{l}\sim {u_{out}}/{L}$. This leads to $u_{in}\sim ({l}/{L})u_{out}$. For the steady-state regime, the induction equation in the resistive layer (i.e. considering the magnetic field $B_{y}x\boldsymbol {e_{y}}$ in the reconnected plane $(x,y)$) gives $\boldsymbol {\nabla }\times (\boldsymbol {u}\times \boldsymbol {B})+\eta \boldsymbol {\Delta } \boldsymbol {B}\sim 0\Rightarrow ({u_{in}B_{y}}/{l})\sim \eta ({B_{y}}/{l^{2}})\Rightarrow u_{{\rm in}}\sim {\eta }/{l}$. From these evaluations, one can express $l$ as $l=({\eta L}/{u_{{\rm out}}})^{2}$. It results that $\tau _{RM}\sim \eta ^{-1/2}u_{{\rm out}}^{-1/2}L^{3/2}$. To express $u_{{\rm out}}$, the fluid momentum equation as well as the Ampère–Maxwell equation in the ideal region (meaning that the total magnetic field $\boldsymbol {B}=B_{z}\boldsymbol {e_{z}}+B_{y}x\boldsymbol {e_{y}}$ has to be considered) are used to give

(2.11)\begin{equation} \rho\frac{u_{{\rm out}}}{T}\sim\frac{B^{2}}{\mu_{0}L}\Rightarrow\rho\frac{u_{{\rm out}}^{2}}{L}\sim\frac{B^{2}}{\mu_{0}L}\Rightarrow u_{{\rm out}}\sim\frac{B}{\sqrt{\rho\mu_{0}}}=v_{A}, \end{equation}

where $v_{A}$ is the well-known Alfvén velocity. Thus, the flow is driven out of the resistive layer with the Aflvèn velocity. Finally, using the characteristic Alfvén time definition $\tau _{A}=L/\tau _{A}$ and the definition of the characteristic resistive time $\tau _{\eta }=L^{2}/\eta$, the characteristic reconnection time based on the Sweet–Parker model can be expressed as

(2.12)\begin{equation} \tau_{RM}\sim\sqrt{\tau_{\eta}\tau_{A}}. \end{equation}

2.2.2. Origin of magnetic reconnection in TCV, WEST and JET

In the past, evaluations of the reconnection time $\tau _{RM}$ (2.12) have been done taking $L$ as the ideal macroscopic length scale (i.e. corresponding to a fraction of the tokamak minor radius $\sim$ of a few centimetres) leading to the conclusion that resistive magnetic reconnection cannot occur in a fusion device: due to the high core temperature, the weak resistivity gives a resistive time $\tau _{\eta }=L^{2}/\eta$ and, as a consequence, a reconnection time $\tau _{RM}$ too long compared with experimental observations. Such evaluations of $\tau _{RM}$ for three modern tokamaks of various sizes (small size, TCV; medium size, WEST; large size, JET) can be found in table 1 (third column).

Table 1. Estimation of the resistive characteristic reconnection time in the centre of TCV, WEST and JET using $\tau _{MR}\sim \sqrt {\tau _{\eta }\tau _{A}}$ (column 3) and using $\tau _{MR}\sim \sqrt {\tau _{\eta }^{l}\tau _{A}}$ (column 4).

It is interesting to discuss these evaluations. Indeed, these are based on the Sweet–Parker model which is a resistive and steady-state model, where reconnection occurs in the resistive layer and where the drive of the reconnected structure is outside the resistive layer and is led by the ideal induction term $\boldsymbol {u}\times \boldsymbol {B}$. From that picture, the result obtained for the expression of the reconnection time (2.12) is suitable: $\tau _{RM}$ is naturally derived from the mechanism inside the resistive layer, i.e. $\tau _{\eta }$, and from the mechanism outside the resistive layer i.e. $\tau _{v_{A}}$. However, one can imagine that once the magnetic reconnection is initiated in the resistive layer of radial size $l$, other physical mechanisms than the induction field can drive the reconnection structures outside the layer (see § 3). In this new paradigm, the evaluation of the resistive time has to be done taking in consideration the resistive layer size $l$ instead of the macroscopic size $L$, i.e. $\tau _{\eta }^{l}\sim {l^{2}}/{\eta }$. Considering $\tau _{\eta }^{l}$, a new evaluation of $\tau _{RM}$ has been done for the three modern tokamaks in the fourth column of table 1. A better agreement with observations is found. Thus, one can argue that in modern tokamaks, resistive reconnection can occur with reasonable characteristic time in a thin non-ideal layer. The question is then to find relevant mechanism(s) that drive(s) the reconnected structures (i.e. magnetic islands) outside the layer. From this analysis, one can define a new general characteristic reconnection time as $\tau _{RM}\sim \sqrt {\tau _{{\rm origin}}^{l}\tau _{{\rm drive}}^{L}}$. Here, $\tau _{{\rm origin}}^{l}$ is the characteristic time of the physical process that originates from magnetic reconnection by breaking the frozen-in law on the typical length scale $l$. Additionally, $\tau _{{\rm drive}}^{L}$ is the characteristic time of the driving physical process that plays a role on the typical length scale $L$. In the case where resistivity is at the origin of magnetic reconnection, $\tau _{{\rm origin}}^{l}=\tau _{\eta }^{l}$.

In the same spirit, although the reconnection process due to electron inertia (see (2.7)) is faster (an evaluation of the characteristic reconnection time based on electronic inertia for modern tokamaks gives $\tau _{RM}^{m_{e}}\sim {L\tau _{A}}/{d_{e}}\sim 10^{-3}\,{\rm s}$), it was considered as non-relevant in fusion devices. Indeed, the resulting reconnected structures are too small (of the order of the electron skin depth $d_{e}$) compared with the centimetre magnetic islands observed. However, in light of the above, one can consider electronic inertia as a source of possible magnetic island(s) in modern fusion devices: after a initiating phase of magnetic reconnection at small scale due to electronic inertia, another mechanism can drive island(s) outside the non-ideal layer. In this case, one can define the characteristic reconnection time as $\tau _{RM}\sim \sqrt {\tau _{{\rm origin}}^{d_{e}}\tau _{{\rm drive}}^{L}}$.

3.1.1. Tearing mode, TM

The Sweet–Parker mechanism presents a physical picture of steady-state and resistive magnetic reconnection. The first linear analysis of unsteady reconnection can be found from Furth, Killeen & Rosenbluth (Reference Furth, Killeen and Rosenbluth1963), where the linear investigation of finite resistive instabilities in a current sheet pinch is proposed for the first time. In particular, the linear dispersion relation is derived for the so-called tearing mode. The tearing mode instability is the dynamical extension of the Sweet–Parker model where the resistivity originates in magnetic reconnection in the current layer of size $l$ and drives the growth of the island outside the layer of size $L$ thanks to the mechanism proposed by Sweet and Parker, i.e. thanks to the ideal incoming flow that ‘pushes’ the magnetic field line towards the null magnetic field line.

To investigate the linear tearing instability, one can consider a simple 3-D slab configuration of a current sheet as seen in § 3 (figure 5), where $\boldsymbol {B}=B_{z}\boldsymbol {e_{z}}+B_{y}\boldsymbol {e_{y}}+B_{x}\boldsymbol {e_{x}}$ with a strong guide field in the $\boldsymbol {e_{z}}$ direction having $B_{z} \gg B_{y}$ and $B_{z} \gg B_{x}$. In such a configuration, the presence of a resistive layer (where $\eta \neq 0$) induces the existence of a current sheet in the perpendicular (to the guide field $B_{z}\boldsymbol {e_{z}}$) plane $(x,y)$ due to the parallel current $j_{z}\boldsymbol {e_{z}}$ with $j\equiv j_{z}(x,y)$. Under particular conditions that will be developed just after, such a current sheet can be unstable from the tearing instability point of view: in the perpendicular plane, magnetic reconnection can take place and a 2-D magnetic island can grow with the linear tearing growth rate.

To derive this linear growth rate, the magnetohydrodynamic framework can be used to give a reduced (Strauss Reference Strauss1976) model of the dynamical behaviour of the plasma coupled with the magnetic field:

(3.1)\begin{gather} \partial_{t}\psi+\left[\phi,\psi\right]=\eta\nabla_{{\perp}}^{2}\psi, \end{gather}

(3.2)\begin{gather} \partial_{t}\nabla_{{\perp}}^{2}\phi+\left[\phi,\nabla_{{\perp}}^{2}\phi\right] =\left[\psi,\nabla_{{\perp}}^{2}\psi\right]+\mu\nabla_{{\perp}}^{4}\phi, \end{gather}

where $\mu$ is the plasma viscosity. Here, all the quantities have been normalized using the Alfvén velocity $v_{A}$, a characteristic macroscopic perpendicular length $L_{\perp }$ and the Alfvén time $\tau _{A}=L_{\perp }/v_{A}$. Equation (3.1) gives the time evolution of the magnetic flux function in the reconnected plane, i.e. $\psi (x,y)$ defined as $\boldsymbol {B}=B_{z}\boldsymbol {e_{z}}+\boldsymbol {\nabla }\times (\psi \boldsymbol {e_{z}})$ and represents a reduction of Ohm's law. In particular, $\boldsymbol {\nabla _{\perp }}\psi \equiv j(x,y)$, which is the current in the parallel direction (i.e. along $\boldsymbol {e_{z}}$) due to the resistivity that breaks the frozen-in law and allows magnetic reconnection. Equation (3.2) gives the time evolution of the electrostatic potential in the reconnected plane, i.e. $\phi (x,y)$ defined from the fluid velocity in the plane $(x,y)$ $\boldsymbol {u_{\perp }}=-\boldsymbol {\nabla }\times (\phi \boldsymbol {e_{z}})$ and represents the momentum equation. (One can emphasize that symbols $[f,g]$ refer to the nonlinearity of the system and are the well-known Poisson bracket with $[f,g]=\partial _{x}f\partial _{y}g-\partial _{y}f\partial _{x}g$ in such a configuration.)

The linear tearing mode instability of such a current sheet of size $(L_{x},L_{y})$ can be investigated by linearizing (3.1) and (3.2):

(3.3)\begin{gather} \psi\left(x,y,t\right)=\psi_{0}\left(x\right)+\psi_{1}\left(x,\right)\exp\left(\gamma t\right)\cos\left(k_{y}y\right), \end{gather}

(3.4)\begin{gather} \phi\left(x,y,t\right)=\phi_{1}\left(x\right)\exp\left(\gamma t\right)\sin\left(k_{y}y\right). \end{gather}

In (3.1) and (3.2), it has been assumed that the poloidal direction is a periodic direction with $k_{y}=2{\rm \pi} /L_{y}$ being the poloidal wave number, and that the fields $\psi$ and $\phi$ vanish at the boundaries of the radial direction. Here, $\psi _{0}$ represents the magnetic flux of the equilibrium current sheet. Additionally, the stability of the well-known Harris current sheet (Harris Reference Harris1962) is investigated, so $\psi _{0}(x)$ is such that the associated equilibrium magnetic field is $B_{0}(x)=\psi '_{0}(x)=\tanh (x/a)$, where $a$ represents the thickness of the magnetic field in the resistive layer. For such a current sheet, $B_{0}(x=0)=0$ and $B_{0}$ is going in opposite directions on either side of the $x=0$ line, which are necessary conditions to have tearing instability. For a linear unstable current sheet, the fluctuations $\psi _{1}$ ($\ll \psi _{0}$) and $\phi _{1}$ grow exponentially with a linear growth rate $\gamma$.

To derive the linear dispersion relation of the tearing mode instability (considering the non-viscous case where $\mu =0$), the domain has to be separated into an ideal region, where $\eta =0$, and a resistive region. The asymptotic matching between the two regions leads to this linear dispersion relation.

In the ideal region (labelled as the outside region) of characteristic radial size $L$, resistivity is neglected $\eta =0$. Using the decomposition given by (3.3) and (3.4), the linearization of Ohm's law (3.1) leads to the following coupling $\phi _{1}^{out}(x)=\psi _{1}^{out}(x)/B_{0}(x)$. Moreover, it is supposed that the linear growth rate of the tearing mode $\gamma$ is relatively weak compared with the Alfvén time leading to $1 \ll \gamma \ll \tau _{\eta }/\tau _{A}$ and to $\partial _{t}\nabla _{\perp }^{2}\phi \sim 0$ in (3.2). Thus, under this assumption and using (3.3) and (3.4), the linearization of (3.2) leads to

(3.5)\begin{gather} \left[\psi_0+\psi_{1}^{{\rm out}} \cos\left(k_{y}y\right),\nabla_{{\perp}}^{2}\left(\psi_0+\psi_{1}^{{\rm out}}\cos\left(k_{y}y\right)\right)\right]=0, \end{gather}

(3.6)\begin{gather}\psi''^{{\rm out}}_{1}(x)=\left(\frac{\psi'''^{{\rm out}}_{0}(x)}{B_{0}(x)}+k_{y}^{2}\right)\psi^{{\rm out}}_{1}(x), \end{gather}

which, for fields vanishing at the the radial boundaries and Harris current sheet, the solution is

(3.7)\begin{equation} \psi_{1}^{{\rm out}}(x) = \exp({-}k_{y}|x|)\left[1+\frac{\tanh\dfrac{|x|}{a}}{ak_{y}}\right]. \end{equation}

Due to the presence of the resonant surface where $\eta \neq 0$, this solution presents a discontinuity at $x=0$ from which the well-known index stability parameter $\varDelta '$ is defined:

(3.8)\begin{equation} \varDelta'=\lim_{x\rightarrow 0}2\psi'^{{\rm out}}_{1}(x). \end{equation}

Finally, for a Harris current sheet and assuming that the fields ($\psi$ and $\phi$) vanish at the radial boundaries, one can derive

(3.9)\begin{equation} \varDelta'=2\left(\frac{1}{a^{2}k_{y}}-k_{y}\right). \end{equation}

It is important to note that $\varDelta '$ is entirely defined from the outside solution $\psi ^{{\rm out}}_{1}$, which is entirely defined by the equilibrium magnetic field in the ideal region. Here, $\varDelta '$ does not depend at all on the mechanism that originates in the magnetic reconnection (i.e. resistivity or electronic inertia).

In the thin resistive layer (labelled as the inside region) of characteristic radial size $l$, the resistivity cannot be neglected any longer. Moreover, due to the large scale involved in the linear tearing mode, it is assumed that radial gradients are larger than poloidal gradients leading to $\nabla _{\perp }^{2}\sim \partial _{x}^{2}$. It is also assumed that in these non-ideal regions, $\psi _{1}(x)$ is constant having non-zero derivatives (this is the well-known ‘constant-$\psi$ approximation’ Furth et al. Reference Furth, Killeen and Rosenbluth1963). Then, a Taylor development around $x=0$ is done to derivate $\psi '_{0}(x)=B_{0}(x)\sim -x/a$ for a Harris current sheet. Following these assumptions, the linearization of (3.1) and (3.2) using (3.3) and (3.3) leads to a reduction of the system to an equation independent of the physical parameters:

(3.10)\begin{equation} z+z^{2}\chi(x)=\chi''(z), \end{equation}

with $x=(\eta \gamma )^{1/4}\sqrt {a/k_{y}}$z and $\phi _{1}^{{\rm in}}(z)=(\eta \gamma )^{-1/4}\sqrt {k_{y}a}\chi (z)$, and which is the linearized Ohm's law (3.1). The coupling between $\psi$ and $\phi$ is given by the linearized momentum equation (3.2):

(3.11)\begin{equation} \psi''^{in}_{1}(x)=\frac{\gamma^{2}a}{k_{y}^{2}x}\phi''^{{\rm in}}_{1}(x). \end{equation}

Finally, the linear dispersion relation is derived from the matching between the two regions and using the definition of $\varDelta '$ from the ideal region (3.8):

(3.12)\begin{equation} \varDelta'=\varDelta'(\psi_{1}^{{\rm out}})=\int_{{\rm in}}\frac{\psi''^{{\rm in}}_{1}(x)} {\psi_{1}(0)} \,{{\rm d}\kern0.7pt x}. \end{equation}

This matching condition can be solved by deriving a solution for $\psi ''^{{\rm in}}_{1}(x)$ from (3.10) and (3.11), and leads to the following expression of the tearing mode linear growth rate:

(3.13)\begin{equation} \gamma = 2.13\varDelta'^{5/4}\eta^{3/5}\left(\frac{a}{k_{y}}\right)^{{-}2/5}. \end{equation}

Thus, the Harris current sheet is tearing unstable for $\varDelta '>0$. It is said that $\varDelta '$ represents the magnetic energy available in the ideal region for instability. During the magnetic reconnection process, the magnetic energy stored in the equilibrium is converted into kinetic energy. As expected, (3.13) shows that the linear drive of the tearing magnetic island is resistive. Figure 6(b) (from Takeda et al. Reference Takeda, Agullo, Benkadda, Sen, Bian and Garbet2008) shows the scaling law in resistivity of a linear tearing mode for linear simulations of (3.1) and (3.2), and is in agreement with (3.13). It is interesting to note that the linear dispersion relation (3.13) has been obtained here for a resistive tearing mode by considering the asymptotic matching of the ideal region (which does not depend on the physics of the non-ideal layer) with a resistive layer. As a consequence, the linear growth rate $\gamma$ is a function of the resistivity $\eta$. For a collisionless tearing mode, the solution in the ideal region is unchanged, i.e. the expression for $\varDelta '$ given by (3.9) is still valid. However, the asymptotic matching with the non-ideal layer originating from the electron inertia gives a different linear dispersion relation for the tearing growth rate $\gamma$ that depends on the electron skin depth. A detailed derivation of the growth rate of the collisionless tearing mode can be found from Porcelli (Reference Porcelli1991) and Porcelli et al. (Reference Porcelli, Borgogno, Califano, Grasso, Ottaviani and Pegoraro2002).

Figure 6. (a) Scaling law of the linear resistive tearing mode, reproduced from Takeda et al. (Reference Takeda, Agullo, Benkadda, Sen, Bian and Garbet2008), with permission from AIP Publishing. (b) Structure of the fields $\psi$ and $\phi$ resulting from a tearing instability in the $(x,y)$ reconnected plane.

The linearization of (3.1) and (3.2) leads to a characteristic radial parity of the tearing mode and the associated flow. In figure 7(a), which shows the radial structure obtained numerically of the fields, the tearing mode $\psi _{1}(x)$ is even while $\phi _{1}(x)$ is odd. Figure 6(b) shows in the $(x,y)$ reconnected plane, the structure of the fields $\psi$ and $\phi$ resulting from a tearing instability. The dynamic of the flow (through $\phi$) and the magnetic field (through $\psi$) are strongly coupled. The characteristic quadrupole structure observed on $\phi$ is essential to understand the drive of the magnetic island observed on $\psi$.

Figure 7. (a) Radial structure of the fields for a mode (at large scale $m=1$) having a tearing parity. (b) Radial structure of the fields for a mode (at small scale $m*= 33$) having an interchange parity.

Except during the initial phase, where the current is increased to set the plasma discharge, the tearing instability is usually not observed in modern fusion devices since $\varDelta '<0$. Moreover, the drive of the tearing instability is led by the resistivity. As a consequence, even with an unstable magnetic equilibrium (with $\varDelta '>0$), the resulting growth of a magnetic island due to tearing instability would be too slow compared with the characteristic discharge time of a fusion device. Although the tearing instability is not relevant to explain the existence of magnetic island(s) in modern fusion devices, its study is meaningful and it is useful to understand the physical mechanisms at play during magnetic reconnection processes.

3.1.2. Neoclassical tearing mode, NTM

In fusion devices, neoclassical effects through the bootstrap current lead to a physical mechanism that can drive nonlinear large-scale magnetic islands that are deleterious for confinement and transport. Such ‘neoclassical tearing modes’ (NTMs) (Chang et al. Reference Chang, Callen, Fredrickson, Budny, Hegna, McGuire and Zarnstorff1995; La Haye Reference La Haye2006; Sauter et al. Reference Sauter, La Haye, Chang, Gates, Kamada, Zohm, Bondeson, Boucher, Callen and Chu2010) can become very large quickly and reach the tokamak wall bringing with it fast and hot particles which can damage the device. More precisely, the toroidal shape of the tokamak leads to a non-uniform magnetic field in the poloidal direction. Such a magnetic configuration exhibits two classes of trajectory for particles: passing particles that travel all around the torus and trapped particles that are confined in a smaller device region following their well-known ‘banana’ orbits. The bootstrap current results from the friction between trapped and passing particles. It has been empirically defined as the current carried by the passing electrons and proportional to the radial electronic pressure gradient (resulting from the collisions between the passing electrons and the trapped particles): $j_{BS}=C_{b}\partial _{x}p(x)$, where $C_{b}$ represents the strength of the bootstrap current (Militello, Ottaviani & Porcelli Reference Militello, Ottaviani and Porcelli2008). From that definition, the mechanism at play in the nonlinear drive of a large magnetic island by the bootstrap current can be drawn. A seed magnetic island (see later for the origin of such an island) initially flattens the pressure profile (Fitzpatrick Reference Fitzpatrick1995) leading to a perturbation of the bootstrap current. This perturbation amplifies the magnetic island that grows until a fully flattened pressure profile is obtained (at saturation of the island width). Such nonlinear dynamics is well described by the so-call Rutherford model that will be investigated in § 4. To investigate numerically the drive of a magnetic island by the bootstrap current, an heuristic simple model can be derived in the $(x,y)$ reconnected plane using the reduced-MHD framework and giving the following time evolution equations (Muraglia et al. Reference Muraglia, Agullo, Poyé, Benkadda, Dubuit, Garbet and Sen2017):

(3.14)\begin{gather} \partial_t\psi+\left[\phi-P,\psi\right]={-}v^{{\star}}\partial_y\psi+\eta\nabla_\perp^2\psi + {{\eta C_{b}\partial_{x}P}}, \end{gather}

(3.15)\begin{gather} \partial_{t} \nabla_{{\perp}}^{2}\phi+\left[\phi,\nabla_{{\perp}}^{2}\phi\right]=\left[\psi,\nabla_{{\perp}}^{2}\psi\right]+\mu\nabla_{{\perp}}^{4}\phi, \end{gather}

(3.16)\begin{gather} \partial_t P+\left[\phi,P\right] ={-}v^{{\star}}\partial_{y}\phi+\rho^{{\star}^{2}}\left[\psi,\nabla_{{\perp}}^{2}\psi\right]+\chi_{{\perp}}\nabla_{{\perp}}^{2}P. \end{gather}

Equation (3.16) gives the time evolution of the electronic pressure $P$ with $v^{\star }$ being the electronic diamagnetic velocity induced by the equilibrium electronic pressure (i.e. at the equilibrium $\partial _{x}P_{0}(x)=v^{\star }$). Here, $\rho ^{\star }$ is the normalized ionic Larmor radius and $\chi _{\perp }$ is the perpendicular conductivity. The model contains implicitly a parallel conductivity through the coupling between (3.14) and (3.16) given by $\chi _{\parallel }=\rho ^{\star ^{2}}/\eta$ (Agullo et al. Reference Agullo, Muraglia, Poyé, Benkadda, Yagi, Garbet and Sen2014). This term is required to initiate the pressure flattening inside the island when its radial width becomes larger than (Fitzpatrick Reference Fitzpatrick1995)

(3.17)\begin{equation} w_{c}=\sqrt{8}\left(\chi_{{\perp}}/\chi_{{\parallel}}\right)^{1/4}\sqrt{a/k_{y}}. \end{equation}

In fusion devices, the nonlinear drive of a magnetic island by the bootstrap current requires a seed island of radial width $w_{{\rm seed}}$ (which is not usually originated from the tearing instability since $\varDelta '<0)$. The NTM triggering is a threshold mechanism: for a given bootstrap current (of strength $C_{b}$), $w_{{\rm seed}}$ has to be large enough to initiate the NTM mechanism. This is well observed in figure 8(a) that shows the NTM stability diagram obtained thanks to nonlinear simulations of (3.14)–(3.16). In these simulations, $\varDelta '<0$ and the required seed island of radial width $w_{seed}$ is numerically implemented in the initial conditions of the simulations. In figure 8(b) taken from Agullo et al. (Reference Agullo, Muraglia, Poyé, Benkadda, Yagi, Garbet and Sen2014), the evolution of the pressure profile during the nonlinear NTM growth is shown. Once the island width reaches the critical width $w_d$, the pressure profile starts to become flat inside the island until the saturation phase where the island width saturates and the pressure profile is completely flattened.

Figure 8. (a) NTM stability diagram obtained thanks to nonlinear simulations of (3.14)–(3.16). (b) Pressure profile $P_{0}(x)$ is flattened while the NTM island grows (reproduced from Agullo et al. (Reference Agullo, Muraglia, Poyé, Benkadda, Yagi, Garbet and Sen2014), with permission from AIP Publishing).

3.1.3. Turbulence driven magnetic island, TDMI

NTM requires a seed magnetic island which is amplified by the bootstrap current. Some ‘NTM precursors’ have been identified in experiments. For example, sawtooth crashes, fishbone instabilities or edge-localized modes are some possible MHD events that can generate the required seed magnetic island. However, the basic physical mechanism at play in the seed island generation is still not understood. Moreover, in some discharges, in particular, in high-poloidal-beta discharges, NTMs are observed without any MHD precursor events (Isayama et al. Reference Isayama, Matsunaga and Hirano2013). Thus, many questions related to the NTMs origin are still open. In hot fusion plasmas, large-scale MHD structures (like magnetic islands) co-exist with small-scale turbulence. The recent work in the tokamak KSTAR (Choi et al. Reference Choi, Bardöczi, Kwon, Hahm, Park, Kim, Woo, Park, Yun and Yoon2021) shows the existence of a mutual interaction between turbulence and the magnetic island. Naturally, the small-scale turbulence could be one candidate to explain the origin of the seed island required for NTM. In this last decade, several theoretical works have addressed the question of the role of turbulence in the seed island generation. One of the main results is that a magnetic island can be driven by small-scale turbulence and such an island is called a turbulence driven magnetic island (TDMI) (Muraglia et al. Reference Muraglia, Agullo, Benkadda, Yagi, Garbet and Sen2011). Here, the physical mechanism at play in the drive of a TDMI is presented as well as a review of works devoted to TDMI.

Muraglia et al. (Reference Muraglia, Agullo, Benkadda, Garbet, Beyer and Sen2009a,Reference Muraglia, Agullo, Yagi, Benkadda, Beyer, Garbet, Itoh, Itoh and Senb) derived a reduced-MHD model for the multi-scale interaction of a large magnetic island and interchange small-scale turbulence. Typically, this model describes in the 2-D reconnected plane $(x, y)$ the nonlinear time evolution of the magnetic flux $\psi$, (3.18), of the electric potential $\phi$, (3.19), and of the electronic pressure $P$, (3.20):

(3.18)\begin{gather} \partial_t\psi+\left[\phi-P,\psi\right] ={-}v^\star\partial_y\psi+\eta\nabla_\perp^2\psi, \end{gather}

(3.19)\begin{gather} \partial_t \nabla_\perp^2\phi+[\phi,\nabla_\perp^2\phi]= [\psi,\nabla_\perp^2\psi]- \kappa_1 \partial_y P+\mu\nabla_\perp^4\phi, \end{gather}

(3.20)\begin{gather} \partial_t P+[\phi,P] ={-}v^\star((1- \kappa_2) \partial_y\phi+ \kappa_{2} \partial_y P)+\rho^{{\star}^{2}}[\psi,\nabla_\perp^2\psi]+\chi_\perp\nabla_\perp^2 P. \end{gather}

This model includes both the tearing physics through the magnetic equilibrium shaped with a Harris current sheet (i.e. having $\boldsymbol {B_{0}}(x)=\tanh (({x-L_{x}/2})/{a})\boldsymbol {e_{y}}$) and the interchange physics through $\kappa _1$ and $\kappa _2$ terms. Indeed the $\kappa _1$ represent the magnetic curvature and the $\kappa _2$ and $v^{\star }$ parameters represents the equilibrium pressure gradient. When the curvature is in opposition with this gradient, interchange instability develops at small scales.

Like for the linear tearing mode, in the box $([-L_{x}/2,L_{x}/2],[0,L_{y}])$, the fields are considered as periodic in the poloidal direction (i.e. in the $y$-direction) and vanish at the radial boundaries. Therefore, one can decomposed the fields ($\psi$, $\phi$ and $P$) using a Fourier decomposition with some of the poloidal modes $m$: $\psi (x, y, t)=\sum _{m\in \mathbb {Z}}\psi _{m}(x, t)\exp ({\rm i}k_{m}y)$ with $k_{m}=2{\rm \pi} m/L_{y}$. The parity, i.e. odd or even symmetry in the radial direction, of the eigenfunctions $\psi _{m}(x, t)$, $\phi _{m}(x, t)$, $P_{m}(x, t)$ provides a distinct marker of identification of a given mode $m$ and helps to pinpoint the instability mechanism generating it. Small-scale interchange modes have (odd, even, even) parities with respect to $x\in [-L_{x}/2, L_{x}/2]$ for ($\psi _{m}$, $\phi _{m}$, $P_{m}$), respectively. Usually, interchange modes are unstable at small scales, i.e. $m \gg 1$. Tearing mode has (even, odd, odd) parities with respect to $x\in [-L_{x}/2, L_{x}/2]$ for ($\psi _{m}$, $\phi _{m}$, $P_{m}$), respectively. Usually, the tearing mode is unstable at large scales, i.e. $m\sim 1$ (see figure 7).

To understand how unstable interchange modes at small scales can drive the growth of an $m=1$ magnetic island, one can consider the linear spectrum with $C_{b}=0$ (blue crosses) presented in figure 9(a). This spectrum is obtained by a linear simulation of (3.18), (3.19) and (3.20) using parameters such as at large scales, all the modes are stable and at small scales, the modes are unstable. In particular, at small scales, the unstable modes present an interchange parity and the most unstable mode is $m^{\star }=33$ with a linear growth rate $\gamma ^{\star }$. At large scale, the mode $m=1$ is stable and presents an interchange parity with $\varDelta '<0$. Thus, linearly, there is no magnetic island. Then, from this linear spectrum, nonlinear simulation is performed without bootstrap current (i.e. $C_{b}=0$) and the nonlinear growth of a $m=1$ magnetic island is observed whereas $\varDelta '<0$. Indeed, figure 9(b) gives the time evolution of the island for a nonlinear simulation with $\varDelta '<0$ and without bootstrap current (i.e. $C_{b}=0$ with the blue line). The contour plot of $\psi +\psi _{0}$ at the end of the simulation is shown and exhibits a magnetic island in figure 9(b). This magnetic island has been nonlinearly generated thanks to the nonlinear interaction between small scales and large scales (Muraglia et al. Reference Muraglia, Agullo, Benkadda, Yagi, Garbet and Sen2011).

Figure 9. (a) Linear spectrum of a typical TDMI simulation. Two cases are considered: without bootstrap current, i.e. $C_{b}=0$ (blue crosses) and with bootstrap current, i.e. $C_{b}=40\neq 0$ (purple circles). (b) Nonlinear TDMI simulations with $\varDelta '<0$ without (blue line) and with (magenta line) bootstrap current.

The time evolution of the kinetic energy of the large-scale modes $m=0$ and $m=1$ for the nonlinear simulation with $\varDelta '<0$ and $C_{b}=0$ is presented in figure 10(a). It clearly reveals that the beginning of the simulation can be decomposed into three phases.

Figure 10. (a) TDMI generation mechanism. Time evolution of the kinetic energy of the large-scale modes $m=1$ and $m=0$ at the beginning of the nonlinear simulation. (b) Hornsby et al. (Reference Hornsby, Migliano, Buchholz, Grosshauser, Weikl, Zarzoso, Casson, Poli and Peeters2016) observe TDMI in global gyrokinetic simulations including collisions. Ion temperature gradient (ITG) turbulence accelerates the growth of a resistive tearing mode following the rule $\gamma _{1}^{NL}\sim 2\gamma ^{\star }$.

First, for $t/\tau _{A}=[0, 200]$, there is the linear phase which is in agreement with the linear spectrum presented in figure 9(b): the modes $m=0$ and $m=1$ are stable and present an interchange parity. There is no magnetic island. Around the resonance surface $x=0$, small-scale modes grow.

Then, the second phase ($t/\tau _{A}=[200, 500]$ in figure 10a) is quasi-linear: a positive growth with $\gamma _{0}^{NL}\sim \gamma _{1}^{NL}\sim 2\gamma ^{\star }$ of the large-scale modes $m=0$ and $m=1$ is observed. During this phase, the parity of the large-scale modes $m=0$ and $m=1$ change into a tearing parity: the magnetic island is generated. Using the parity properties of the system, one can explain the physical mechanism at play during this phase where the TDMI is generated. Indeed, at small scales, the two most interchangeable unstable modes $m^{\star }$ and $m^{\star }+1$ with a comparable linear growth rate $\gamma ^{\star }$ (see figure 9a) beat together to drive nonlinearly a magnetic island with $\gamma _{1}^{NL}\sim 2\gamma ^{\star }$. This nonlinear beating is dominant in the simulation dynamics and leads to a change of parity of the modes $m=0$ and $m=1$. Indeed, the nonlinear properties of (3.18), (3.19) and (3.20) are such that if the system is linearly driven by small-scale (ss) interchange modes $I_{ss}$, their mutual nonlinear interaction can drive only a tearing parity large-scale (ls) fluctuation $T_{ls}$: $[I_{ss,}\:I_{ss}] \rightarrow T_{ls}$. More precisely, the projection of Ohm's law (3.18) on the mode $m=1$ illustrates the nonlinear beating at small scales of the interchange modes driving a tearing parity mode at large scale: $\partial _{t}\psi _{1}\sim [\psi _{m^{*}}, \phi _{m^{\star }+1}-p_{m^{\star }+1}] +\eta \boldsymbol {\nabla _{\perp }^{2}\psi _{1}}$ (Muraglia et al. Reference Muraglia, Agullo, Benkadda, Yagi, Garbet and Sen2011; Ishizawa et al. Reference Ishizawa, Kishimoto and Nakamura2019).

Finally, up to $t/\tau _{A}>500$ (figure 10a), the system enters into a fully nonlinear regime and the island width is saturated as presented in figure 9(a) with $C_{b}=0$.

Thus, one can conclude that the large-scale magnetic island generated by a nonlinear beating of small-scale unstable interchange modes is a TDMI. The drive of such an island is not governed by the resistivity. However, it is the resistivity that originates in the island by allowing magnetic reconnection. This important result, which was first presented by Muraglia et al. (Reference Muraglia, Agullo, Benkadda, Yagi, Garbet and Sen2011) in the MHD context, was recovered later with gyrokinetic simulations (Hornsby et al. Reference Hornsby, Migliano, Buchholz, Zarzoso, Casson, Poli and Peeters2015b, Reference Hornsby, Migliano, Buchholz, Grosshauser, Weikl, Zarzoso, Casson, Poli and Peeters2016) where the ITG turbulence accelerates the growth of a resistive tearing mode following the rule $\gamma _{1}^{NL}\sim 2\gamma ^{\star }$ (see figure 10b).

From this work, the question of the amplification of a TDMI by the bootstrap current to lead to a NTM becomes natural. To answer this question, linear and nonlinear simulations of (3.18), (3.19) and (3.20) including both the bootstrap current ($C_{b}\neq 0$) and turbulence (i.e. interchange small-scale unstable modes with $\kappa _{1}\neq 0$ and $\kappa _{2}\neq 0$) have been performed with $\varDelta '<0$. In figure 9(b), the linear spectrum of such simulations is presented with purple circles ($C_{b}=40$). The bootstrap current does not affect so much the linear spectrum which presents always unstable interchange modes at small scales and stable tearing mode at large scales. Linearly, there is no magnetic island with bootstrap current. Nonlinearly, as presented in blue in figure 9(a), without bootstrap current, a TDMI is generated. Additionally, the magenta line shows the time evolution of the island when the bootstrap current is included in the model: the TDMI is amplified by the bootstrap current to lead to an NTM. Thus, one can conclude that the TDMI generation mechanism can explain the origin of the required seed island for the growth of NTM in fusion devices (Muraglia et al. Reference Muraglia, Agullo, Poyé, Benkadda, Dubuit, Garbet and Sen2017).

In the first study by Muraglia et al. (Reference Muraglia, Agullo, Benkadda, Yagi, Garbet and Sen2011), the model used is by nature 2-D (in the reconnected plane) with a Cartesian geometry. Moreover, turbulence and island are located at the same resonant surface whereas in fusion devices, magnetic reconnection can take place at a different radial location than that of turbulence. As an example, a sawtooth crash can generate magnetic island in the core of the machine while interchange-like turbulence develops at the device pedestal where gradients are strong. In that context, Poyé et al. (Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015) investigated the resilience of the TDMI mechanism found by Muraglia et al. (Reference Muraglia, Agullo, Benkadda, Yagi, Garbet and Sen2011) using an RMHD framework in 3-D cylindrical geometry in the situation where interchange-like turbulence is located outside the $q=2$ resonant surface, which is the surface where a large $(2,1)$ magnetic island is usually observed. One can note that here and in the following, the bootstrap current is not included. In figure 11(a), the linear spectrum as a function of the radial coordinate $q$ used in the 3-D geometrical nonlinear simulations of Poyé et al. (Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015) is shown: the input parameters of the simulations are chosen to have stable tearing mode around the $q=2$ surface and unstable interchange modes in the region outside the $q=2$ surface. So linearly, there is no magnetic island. As shown in figure 11(b), nonlinearly, a $(2,1)$ magnetic island is observed on the magnetic flux $\psi.$ Here, after the linear phase, similar to 2-D, an efficient beating between turbulent modes is produced with large tearing radial structure $(5,2)$, $(7,3)$ and $(9,4)$ using the parity rule $[I_{ss,}\:I_{ss}] \rightarrow T_{ls}$. However, those modes are located in the region where the interchange modes are unstable and so outside the $q=2$ surface. In figflagfigure 12(a), the radial structure of these modes is presented and shows that these modes, due to their tearing structure, have an extended radial structure and reach the $q=2$ surface. Using the parity rule $[T_{ss,}\:T_{ss}] \rightarrow T_{ls}$, these modes beat together nonlinearly at the surface $q=2$ giving birth to the growth of a $(2:\,1)$ magnetic island with an interchange characteristic time. From this work (Poyé et al. Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015), one can give rules for a 3-D nonlinear coherent and non-local beating for TDMI growth: (1) the modes beat if they overlap; (2) the beating is efficient if the resulting mode is resonant at its birth location.

Figure 11. (a) Linear spectrum as a function of the radial coordinate $q$ used in the 3-D geometrical nonlinear simulations (reproduced from Poyé et al. (Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015), with permission from AIP Publishing). (b) Magnetic flux $\psi$ obtained during the saturation phase of 3-D cylindrical nonlinear simulations (reproduced from Poyé et al. (Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015), with permission from AIP Publishing).

Figure 12. (a) Radial structure of modes in a 3-D nonlinear simulation of Poyé et al. (Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015). (b) Reproduced from Dubuit et al. (Reference Dubuit, Agullo, Muraglia, Frank, Garbet and Maget2021), with permission from AIP Publishing, it is shown that the saturated width of the $(2, 1)$ TDMI generated in 3-D toroidal configuration is proportional to input power source, i.e. to the turbulence level. One can note that after a input power source $>4$, the island becomes too large compared with the size of the numerical box.

Dubuit et al. (Reference Dubuit, Agullo, Muraglia, Frank, Garbet and Maget2021) added toroidal effects in the 3-D RMHD model. Two paths towards TDMI are found. The first path, based on linear toroidal coupling, generates a turbulence cascade from small scales to large scales. At small scales, local (as in 2-D) and non-local (as in 3-D cylindrical) turbulence modes cascade together towards the large scale using first the $[I_{ss,}\:I_{ss}] \rightarrow T_{ls}$ rule and then the $[T_{ss,}\:T_{ss}] \rightarrow T_{ls}$ rule to produce a $(2,1)$ magnetic island. The second path is the nonlinear TDMI generation mechanism already investigated in 3-D cylindrical configuration by Poyé et al. (Reference Poyé, Agullo, Muraglia, Garbet, Benkadda, Sen and Dubuit2015). As shown in figure 12(a), the width at saturation of the generated $(2, 1)$ TDMI is proportional to the input power source, i.e. to the turbulence level. Moreover, the path chosen by the system to generate the $(2, 1)$ TDMI depends also on the input power. For weak turbulence and input power source, the toroidal linear interaction is dominant and the system follows the first path to generate a TDMI. When the input power source increases, the nonlinear interactions overcome the linear cascade and the second path is followed.

Recently, in a work just submitted, D. Villa et al. used a 2-D and six-fields RMHD model (Frank et al. Reference Frank, Agullo, Maget, Garbet, Dubuit and Muraglia2020) to find a new path towards TDMI involving coupling between kinetic ballooning modes and zonal flows at low magnetic shear. In figure 13(a), the nonlinear time evolution of the $m=2$ mode width is presented. The kinetic ballooning modes get tearing parity to drive TDMI at the beginning of the nonlinear phase. In the presence of zonal flow, a coalescence of modes towards the $m=1$ scale with a tearing structure is observed.

Figure 13. (a) Nonlinear time evolution of the $m=2$ mode width in simulations of 2-D and six-fieldsRMHD model (by courtesy of D. Villa et al.). (b) Time evolution of a $(2, 1)$ magnetic island in PIC and collisionless simulation, and in the presence of ITG turbulence (by courtesy of F. Fidmer et al.). Thanks to the electronic inertia, a tearing mode grows linearly at the beginning of the simulation. Then, an acceleration of its growth due to nonlinear beating of ITG modes is observed.

In a recent work of F. Widmer et al. (that is currently in the process for submission), the acceleration by ITG turbulence of the drive of a tearing mode originated by electronic inertia is observed in PIC and collisionless simulations (figure 13b).

Seto et al. (Reference Seto, Xu, Dudson and Yagi2024) used RMHD simulations to find that TDMIs (driven by drift ballooning modes) can be interpreted as ELM precursors and, by consequence, can play a role in the ELM triggering.

More generally, various examples exist in the literature where the magnetic island is driven/accelerated by a physical mechanism that is not the one that initially breaks the frozen-in law. Without being exhaustive, one can quote Ishii, Azumi & Kishimoto (Reference Ishii, Azumi and Kishimoto2002), where simulations show that the magnetic reconnection process is accelerated thanks to a nonlinear interaction between two unstable tearing modes. Moreover, in the ELM context, plasmoid-mediated reconnection is observed in MHD simulations of nonlinear peeling–ballooning edge-localized modes (Ebrahimi & Bhattacharjee Reference Ebrahimi and Bhattacharjee2023). Finally, Grasso et al. (Reference Grasso, Borgogno, Tassi and Perona2020) found by means of RMHD simulations that collisionless magnetic reconnection (originating from electron inertia) can be accelerated in the presence of Kelvin–Helmoltz instability.

Up to now, although the work of Choi et al. (Reference Choi, Bardöczi, Kwon, Hahm, Park, Kim, Woo, Park, Yun and Yoon2021) shows experimentally the existence of a mutual interaction between turbulence and magnetic island in KSTAR, there is no experimental evidence for the existence of TDMI in modern fusion devices and for the generation of NTM from TDMI. This work is in progress thanks to collaborations between European researchers.

3.2.1. Relevance of the microtearing mode instability in modern fusion devices

The microtearing mode (MTM) is another example where a magnetic island is originated from collisions, i.e. resistivity, that allow magnetic reconnection and another physical mechanism to drive the growth of the mode: for MTM, the energy source for the instability is the electron temperature gradient. Here, the mechanisms in the presence of collisions will be highlighted (although an electron temperature gradient can also destabilize MTM originating from electron inertia).

More precisely, microtearing instability corresponds to the destabilization at ion small scales of a collisional current layer by the electron temperature gradient (Hazeltine et al. Reference Hazeltine, Dobrott and Wang1975; Drake & Lee Reference Drake and Lee1977). Due to the characteristic length scale $l_{MTM}\sim r_{L_{i}}$ (where $r_{L_{i}}$ is the ionic Larmor radius) that is involved in the MTM destabilization mechanisms, the MHD framework is useless to investigate this instability. MTM analysis requires a kinetic or a gyrokinetic framework.

For many years, on the argument that collisions were very weak in a high temperature fusion plasma, MTMs have been the subject of few investigations with the assumption that they were stable in fusion devices. However, in this last decade, several works have found unstable MTMs in gyrokinetic simulations of fusion plasmas. In particular, they were found to be unstable in the pedestal region where the gradients (and of course the electron temperature gradient) are strong. Applegate et al. (Reference Applegate, Roach, Connor, Cowley, Dorland, Hastie and Joiner2007) and Dickinson et al. (Reference Dickinson, Roach, Saarelma, Scannell, Kirk and Wilson2013) found unstable MTMs in the pedestal of the spherical tokamak MAST. Doerk et al. (Reference Doerk, Jenko, Pueschel and Hatch2011) found unstable MTMs for standard tokamaks like ASDEX-Upgrade. Then, Predebon & Sattin (Reference Predebon and Sattin2013) found unstable MTMs in reversed field pinches. Later, Hatch et al. (Reference Hatch, Kotschenreuther, Mahajan, Valanju, Jenko, Told, Görler and Saarelma2016) observed that unstable MTM were found to play an important role in the pedestal of JET-ILW. Due to their possible impact on the electron heat transport in the pedestal, theses results are very important and lead to open questions. What exactly is the role played by collisions in the MTM destabilization mechanism? Are there other physical mechanisms leading to an increase of MTMs drive and growth rate? Here, a review of the possible physical mechanisms at play for the drive of the microtearing mode is presented.

3.2.2. Linear destabilization of a current sheet by MTM

To illustrate what could be an MTM, the slab configuration of figure 5 can be used. Here again, there is a strong magnetic field in the parallel direction ($\boldsymbol {e_{z}}$) and the reconnection process takes place in the $(x,y)$ plane. The linear theory of the MTM follows the same procedure as that for the large-scale tearing mode. First, the domain is decomposed into an ideal region and a collisional region.

In the ideal region, there is no current, i.e. $-\mu _{0}j_{\parallel {\rm out}}=\boldsymbol {\nabla _{\perp }^{2}}\psi _{{\rm out}}=0$. As for the large-scale tearing mode, this leads to $\psi _{{\rm out}}(x)\sim \exp ^{-k_{m}|x|}$, where it has been considered that the boundary conditions are far (even at infinity) from the current sheet (which is a good hypothesis for a small-scale MTM). In the collisional region, collisions lead to the formation of a current sheet, where $j_{\parallel in}\neq 0$ and the magnetic flux is assumed to be constant (the so-called ‘$\psi$-constant’ approximation already mentioned for the large-scale tearing mode) $\psi _{in}=cte$. Then, the two regions are matched given the expression of the $\Delta _{k_{m}}'^{MTM}$ parameter:

(3.21)\begin{equation} \Delta_{k_{m}}'^{MTM}=\lim_{L\rightarrow0}\frac{1}{\psi\left(0\right)}\left .\frac{{\rm d}\psi}{{\rm d}\kern0.7pt x}\right|_{{-}L}^{L}=\int_{-\infty}^{+\infty}\sigma\left(x\right)\,{{\rm d}\kern0.7pt x}={-}2\left|k_{m}\right|<0, \end{equation}

where $\sigma (x)={j_{\parallel }}/{\psi }$ is the kinetic conductivity. Thus, for an MTM, $\varDelta '^{MTM}$ is always negative. The destabilization is done by the electron temperature gradient which drives the mode.

The kinetic theoretical derivation of the MTM linear dispersion relation has been done by Hamed et al. (Reference Hamed, Muraglia, Camenen, Garbet and Agullo2019) taking into account collisions as the mechanism allowing for the breaking of the frozen-in law (i.e. originating from the magnetic reconnection process). Note that due to the scales involved, the resolution of the system in the non-ideal layer has to be done using a kinetic framework. Following the Sweet–Parker mechanism, where the coupling between the magnetic field and the flow is an essential key of the dynamics, the electric potential is taken into account in the MTM linear theory through the well-known Maxwell–Gauss equation. The magnetic drift is also added in the evaluation of the collisional Vlasov equation given the time evolution of the electron distribution function $f_{e}$. This leads to a derivation of a model that couples the magnetic flux and the electric potential in the reconnected plane. This model is the kinetic equivalent to the reduced-MHD model for the large-scale tearing mode given by (3.1), (3.2) and it is naturally called the kinetic reduced-MHD (KRMHD) model. In the 1-D-1V phase space $(\hat {x}, \hat {v})$, it can be expressed as (Hamed et al. Reference Hamed, Muraglia, Camenen, Garbet and Agullo2019)

(3.22)\begin{gather} \boldsymbol{\nabla}_{{\perp}}^{2}\hat{\psi}\left(\hat{x}\right)+\hat{\beta}\hat{\sigma}_{\omega_{d}}(\hat{x}, \hat{v})\left(\hat{\psi}\left(\hat{x}\right)-\frac{\hat{x}}{\hat{\omega}}\hat{\phi}\left(\hat{x}\right)\right)=0, \end{gather}

(3.23)\begin{gather} \boldsymbol{\nabla}_{{\perp}}^{2}\hat{\phi}\left(\hat{x}\right)+\hat{\mu}_{e}\left(\hat{\omega}\right)\frac{\hat{x}}{\hat{\omega}}\hat{\sigma}_{\omega_{d}}(\hat{x}, \hat{v})\left(\hat{\psi}\left(\hat{x}\right)-\frac{\hat{x}}{\hat{\omega}}\hat{\phi}\left(\hat{x}\right)\right)-\hat{C}_{int}\hat{\phi}\left(\hat{x}\right)=0, \end{gather}

where $\hat {\beta }$ is the well-known normalized beta parameter, $\hat {\sigma }_{\omega _{d}}$ is the normalized conductivity that depends on the magnetic drift $\omega _{d}$ (for more details, see (Hamed et al. Reference Hamed, Muraglia, Camenen, Garbet and Agullo2019)) and $\hat {\mu }_{e}$ is a normalized quantity linked to the mass ratio $\frac {m_{e}}{m_{i}}$, to the magnetic shear length $L_{s}$, to the electron temperature gradient length scale $L_{T_{e}}$ and to the normalized mode growth and frequency, i.e. $\hat {\omega }=\hat {\omega }_{r}+i\hat {\gamma }$. Finally, $\hat {C}_{int}$ is a normalized quantity linked to interchange terms. The resolution of the KRMHD model given by (3.22) and (3.23) gives the linear theory of MTM, i.e. the expression of $\hat {\omega }$, and the profile of $\hat {\psi }(\hat {x})$ and $\hat {\phi }(\hat {x})$.

3.2.3. Impact of the electric potential and of the magnetic curvature on the MTM drive

To investigate the mechanisms at play in the MTM drive, the KRMHD model given by (3.22) and (3.23) can be solved by using an eigenvalue code (here the SolveAP code) taking into account the boundary conditions.

figure 14 represents the theoretical linear growth rate $\gamma$ as a function of the collisions $\nu$ for different models. Whatever the models used, the microtearing mode is stable in the absence of collisions. As expected, collisions are the mechanism that originates in the reconnection process. Although collisions are weak in a fusion device, they are strong enough to allow magnetic reconnection (the vertical blue dash lines in figure 14 give the experimental range for the collisions in the JET-ILW pedestal). Then, another physical model drives the growth of the island. To investigate the possible mechanisms at play in the drive of MTM, various models have been tested in figure 14.

Figure 14. Linear destabilization of the microtearing mode: theoretical linear growth rate $\gamma$ of the mode as a function of collisions $\nu$ (reproduced from Hamed et al. (Reference Hamed, Muraglia, Camenen, Garbet and Agullo2019), with permission from AIP Publishing).

The most simple model is represented by the red crosses, where the electrostatic field and the magnetic drift have been removed from the model. In the absence of an electric potential, the required coupling between the magnetic field and the plasma flow is assured by the electronic pressure through the distribution function of the electrons (Muraglia et al. Reference Muraglia, Agullo, Benkadda, Garbet, Beyer and Sen2009a). In this model, the electron temperature gradient drives the MTM growth as in the past work of Hazeltine et al. (Reference Hazeltine, Dobrott and Wang1975) and Drake & Lee (Reference Drake and Lee1977). The impact of the electric potential and the magnetic drift on the MTM drive is investigated by Hamed et al. (Reference Hamed, Muraglia, Camenen, Garbet and Agullo2019), as shown in figure 14. The green circles show the linear MTM growth rate using a model without electric potential and including the magnetic drift. The magenta triangles show the growth rate for a model without the magnetic drift and with the electric potential. Finally, the blue diamonds show the result for a model including both. One can conclude that the electric potential and the magnetic drift do not destabilize MTM at the low collisional regime, and in the presence of collisions, they amplify the MTM drive by the electron temperature gradient and collisions.

It is interesting to highlight that here, the drive of MTM has been addressed only in a linear regime. However, nonlinear interactions with others (turbulent) modes can affect the nonlinear growth of MTMs. In particular, Maeyama, Watanabe & Ishizawa (Reference Maeyama, Watanabe and Ishizawa2017) showed that MTMs can be suppressed in the presence of electron temperature gradient (ETG) modes.

4.1.1. History of Rutherford-like models and magnetic island width definition

To limit the impact of a large-scale NTM island, a theory on the saturation mechanisms of tearing has emerged over the last decade giving a variety of Rutherford-like models. From Ohm's law, (3.1) or (3.14), these models give a zero-dimensinoal (0-D) time evolution equation of the island radial width from which the width at saturation $w_{sat}$ can be predicted (Rutherford Reference Rutherford1973). Naturally, these models have been largely used in experiments to control and reduce the width of a NTM(s) (Sauter et al. Reference Sauter, La Haye, Chang, Gates, Kamada, Zohm, Bondeson, Boucher, Callen and Chu2010; Kong et al. Reference Kong, Felici, Sauter, Galperti, Vu, Ham, Hender, Maraschek and Reich2022, Reference Kong, Sauter, Felici, Hogeweij, Merle and Nowak2020) (see figure 15b). However, up to now, systematic comparisons of Rutherford-like models with numerical simulations evolving NTM have failed. Militello et al. (Reference Militello, Ottaviani and Porcelli2008) performed one of the first works dedicated to validation of Rutherford-like model with simulations using the reduced-MHD model (similar to (3.14), (3.15) and (3.16)). As shown by figure 16(a), in this work, only the prediction of the saturated island width, and not the complete dynamics, has been investigated and good agreement is found with theory only in the case of the metastable magnetic island having a small saturated island width. Later, the work of Maget et al. (Reference Maget, Février, Garbet, Lütjens, Luciani and Marx2016) used an extended MHD model to investigate NTM computations to test the validity of the Rutherford-like model including bootstrap current (Carrera et al. Reference Carrera, Hazeltine and Kotschenreuther1986). Agreement is found only by adjusting empirically the theoretical model (by adjusting what will be called later ‘Rutherford parameters’) to numerical results instead of computing these parameters from first principles. In the same way, in the work of Westerhof, Blank & Pratt (Reference Westerhof, Blank and Pratt2016), based on reduced-MHD simulations, all NTM dynamics is not recovered numerically and only the contribution of the bootstrap current agrees with theory. In fact, to understand the discrepancy between simulations and Rutherford-like models in the prediction of NTM dynamics and saturation, some preliminary steps are required: first, a discussion on the island radial width definition is required, and then, a better understanding of the nonlinear dynamics and saturation of a resistive tearing mode without bootstrap current is required.

Figure 15. (a) Successful comparison between Rutherford-like model (using fitting empirically the ‘Rutherford parameters’) and experimental results in TCV of the time evolution of an NTM width (from Kong et al. Reference Kong, Felici, Sauter, Galperti, Vu, Ham, Hender, Maraschek and Reich2022). (b) ‘Rutherford plot’ (island width $w$ as $x$-coordinateand time derivative of the island width ${\rm d}w/{\rm d}t$ as $y$-coordinate) for a nonlinear evolution of $m=1$ tearing mode with $\varDelta '=2.5$ given by simulations of (3.1) and (3.2). In blue, the Rutherford behaviour of $w_{{\rm sep}}$ is plotted and compared with the Rutherford behaviour of $w_{1}$ in orange and the first Rutherford model (4.2) in green.

Figure 16. (a) Tearing mode magnetic island saturated width as a function of $\varDelta '$ for three different magnetic equilibria (profile $H$, profile $A$ and profile $C$)(reproduced from Poyé et al. (Reference Poyé, Agullo, Smolyakov, Benkadda and Garbet2013), with permission from AIP Publishing). (b) Agreement between Rutherford-like model and reduced-MHD simulations including bootstrap current for the saturated width prediction in the case of a metastable magnetic island having a small saturated width (reproduced from Militello et al. (Reference Militello, Ottaviani and Porcelli2008), with permission from AIP Publishing).

First of all, an intuitive definition of the island width can be given from the $\psi$ contour plot in the reconnected plane shown in figure 6(a). Indeed, the radial width of the magnetic island can be defined as the radial extension of the island envelop drawn from lines called separatrices. In the following, the island width defined from this method is labelled $w_{{\rm sep}}$ and can be computed directly from simulations.

Then, Biskamp (Reference Biskamp2000) gives a first theoretical definition of the radial island width associated to the width of the $m=1$ mode and labelled $w_{1}$:

(4.1)\begin{equation} w_{1}=4\sqrt{2a\psi_{1}^{{\rm res}}}, \end{equation}

which can be computed from the simulations by evaluating $\psi _{1}^{{\rm res}}$.

Finally, following the pioneering work of Rutherford (Reference Rutherford1973), one can consider that the saturation mechanism is mainly driven by the nonlinear interaction between the unstable equilibrium $\psi _{0}$ and the mode $m=1$. Thus, projecting this nonlinear interaction on Ohm's law given by (3.1) for the $m=1$ mode, one obtains the first Rutherford-like model given the nonlinear evolution of the $m=1$ island width, i.e. $w^{{\rm Ruth}}_{1}$:

(4.2)\begin{equation} \partial_{t}w^{{\rm Ruth}}_{1}=1.22\eta\varDelta'. \end{equation}

The comparison between the time evolution of $w_{{\rm sep}}$ and of $w_{1}$ (computed from simulations) to the first Rutherford model (4.2) is rich in teaching: in figure 15(a), a ‘Rutherford plot’ is presented (i.e. plot having island width as $x$-coordinate and the time derivative of the island width as $y$-coordinate) for a nonlinear evolution of $m=1$ tearing mode with $\varDelta '=2.5$ given by simulations of (3.1) and (3.2). In blue, the Rutherford behaviour of $w_{{\rm sep}}$ is plotted and compared with the Rutherford behaviour of $w_{1}$ in orange. One can conclude that the definition of the island width from the computation of separatrix positions $w_{{\rm sep}}$ is not equivalent to the definition given by Biskamp with (4.1). Indeed, in Biskamp's definition, only the $m=1$ mode is taken into account whereas in nonlinear simulation, the resulting magnetic island observed on the contour plot of $\psi$ results from nonlinear interactions of all the simulation modes. It then comes that, naturally, the evaluation of the island width from Biskamp's definition is underestimated since only one mode is taken into account. Then, the first model of Rutherford (4.2) is also plotted in green in figure 15(a). This first model gives the maximum growth of the $m=1$ tearing mode, but fails at predicting the saturation width. The model has to be improved.

4.2.1. Prediction of NTM dynamics and saturation with generalized Rutherford model

In fusion devices, the tearing mode is usually stable. However, bootstrap current can amplify a seed magnetic island and lead to the nonlinear growth of an NTM that can have dramatic consequences for the discharge and the device itself. As a consequence, the NTM predictability in terms of behaviour and saturation are of utmost importance for successful fusion by magnetic confinement. In that context, Rutherford-like models including bootstrap effects have been widely and successfully used in experiments to control and reduce NTM (La Haye Reference La Haye2006; Sauter et al. Reference Sauter, La Haye, Chang, Gates, Kamada, Zohm, Bondeson, Boucher, Callen and Chu2010; Kong et al. Reference Kong, Sauter, Felici, Hogeweij, Merle and Nowak2020, Reference Kong, Felici, Sauter, Galperti, Vu, Ham, Hender, Maraschek and Reich2022). NTM leads to a current deficit in the island, and control strategies consist in compensating for this deficit by depositing a highly localized power inside the island. More precisely, to reduce and even suppress the island, electron cyclotron resonant heating and electron cyclotron current drive are used to inject the required current. These techniques, which have met with great success in various tokamaks, are based on the NTM dynamics predictions from Rutherford-like model(s) including bootstrap current (Carrera et al. Reference Carrera, Hazeltine and Kotschenreuther1986; Sauter et al. Reference Samoylov, Igochine, Yu and Zohm1997):

(4.6)\begin{equation} \partial_{t}w_{1}=1.22\eta\varDelta'-1.22\eta\frac{0.41}{a^{2}}w_{1}+\alpha C_{b}v^{{\star}}\frac{w_{1}}{w_{1}^{2}+w_{d}^{2}}. \end{equation}

Here, $w_{d}$ is deduced from the critical island width $w_c$ (3.17) from which pressure flattening starts, i.e. $w_{d}=1.8w_c$ (Fitzpatrick Reference Fitzpatrick1995; Militello et al. Reference Militello, Ottaviani and Porcelli2008; Agullo et al. Reference Agullo, Muraglia, Poyé, Benkadda, Yagi, Garbet and Sen2014). The parameter $\alpha$ should be equal to one and is added to fit the model to experimental or numerical data. In figure 15(a) taken from Kong et al. (Reference Kong, Felici, Sauter, Galperti, Vu, Ham, Hender, Maraschek and Reich2022), a successful comparison (by empirical fitting) between Rutherford-like model and experimental NTM width evolution in TCV is presented. This is typically what is used to control NTM width in fusion devices. However, as expected by theoretical results on the tearing mode saturation prediction, the comparison of the Rutherford-like model including bootstrap current (4.6) and numerical simulations without fitting any parameters fails. Figure 18(a) taken from Muraglia et al. (Reference Muraglia, Poyé, Agullo, Dubuit and Garbet2021) shows disagreement between the evolution of $w_{1}$ from the model (4.6) with $\alpha =1$, and computing the exact value of $w_d$ (in purple) and the evolution of $w_1$ from nonlinear simulations of NTM using (3.14), (3.15) and (3.16) (with the dashed green line). The discrepancy can be reduced with the condition of fitting $\alpha$ and $w_d$: the empirical fitting with $\alpha =18$ and $w_d=1$ (in orange) shows a relatively good agreement with the numerical results (in green). One can note that the systematic comparisons between theoretical models and simulations (Muraglia et al. Reference Muraglia, Poyé, Agullo, Dubuit and Garbet2021) do not take into account the polarization current. However, in a recent work (Dudkovskaia et al. Reference Dudkovskaia, Connor, Dickinson, Hill, Imada, Leigh and Wilson2023), it has been shown that this current impacts the NTM threshold. In future work, it could be very interesting to investigate also its impact on NTM saturation.

Figure 18. From Muraglia et al. (Reference Muraglia, Poyé, Agullo, Dubuit and Garbet2021). (a) Impact of the seeding mechanism on the NTM dynamics. (b) NTM without turbulence and seeding by unstable tearing mode ($\varDelta '=0.98$), width evolution from numerical simulations of (3.14), (3.15), (3.16) in green, from empirical fitting of (4.6) and from (4.6) computing the exact value of $w_d$ and taking $\alpha =1$.

4.2.2. Impact of seed island mechanism on NTM dynamics and saturation

The Rutherford-like model including bootstrap effects (4.6) is a 0-D model taking into account for the seeding mechanism only the $\varDelta '$ parameter suggesting that the nonlinear evolution of the NTM does not depend on the physical nature of the seed. In figure 18(a) taken from Muraglia et al. (Reference Muraglia, Poyé, Agullo, Dubuit and Garbet2021), the NTM has been seeded by an unstable tearing mode with $\varDelta '=0.98$. However, in figure 18(b), numerical simulations are shown for NTM having $\varDelta '<0$ and seeding by turbulence. The two evolutions suggest that the seeding mechanism impacts the nonlinear evolution of the NTM and its saturated width. One can conclude that future improved Rutherford-like model(s) taking into account bootstrap current have to take into account also the seeding mechanism. One can also note that the threshold mechanism of the bootstrap current is well recovered for NTM seeding by turbulence: figure 18(b) shows that for weak bootstrap current (i.e. weak $C_{b}$), the TDMI is not amplified to an NTM. Here, $C_{b}$ has to be strong enough to drive NTM from TDMI as in the absence of turbulence.