Magnetic reconnection is a fundamental process in magnetically confined plasmas, often responsible for sudden relaxation phenomena during which a significant amount of magnetic energy is converted into heat and ordered kinetic energy in a relatively short time. Where a local violation of the ideal ‘frozen’ magnetic flux condition is possible, the magnetic field lines can be torn and reconnected into a new topological configuration forming closed secondary structures called magnetic islands. In fusion experimental devices, they may have detrimental consequences for energy confinement and, in their nonlinear stage, are often precursors of disruptions of the confined plasma configuration (Waelbroeck Reference Waelbroeck2009).

The ordering of the basic space scales $\unicode[STIX]{x1D70C}_{e}<d_{e}<\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70F}}<d_{i}<\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D702}}$ (i.e. electron gyroradius, electron skin depth, sound Larmor radius, ion skin depth and resistive layer width) identifies the physical mechanisms associated with the reconnection process: electron pressure tensor and electron inertial effects, ion sound and ion inertial effects, viscosity and drift kinetic effects of deeply trapped particles and resistive diffusion of the perturbed magnetic field (Aydemir Reference Aydemir1992; Ottaviani & Porcelli Reference Ottaviani and Porcelli1993; Cafaro et al. Reference Cafaro, Grasso, Pegoraro, Porcelli and Saluzzi1998; Fitzpatrick & Porcelli Reference Fitzpatrick and Porcelli2004; Lazzaro, Comisso & Valdettaro Reference Lazzaro, Comisso and Valdettaro2010; Liu, Hastie & Hender Reference Liu, Hastie and Hender2012; Comisso et al. Reference Comisso, Grasso, Waelbroeck and Borgogno2013, Reference Comisso, Lingam, Huang and Bhattacharjee2016). The onset of magnetic reconnection in fusion devices depends on many factors such as the type of equilibrium current profile, the collisionality and viscosity regime, the $\unicode[STIX]{x1D6FD}$ effects (bootstrap and polarization currents), the plasma non-uniform equilibrium rotation and external drives (error fields with helical component resonant with a $q=m/n$ rational magnetic surface, sudden radiative cooling, coupling effects with other magnetic perturbations). Moreover, in the neoclassical regime, the tearing modes (NTMs) (Callen et al. Reference Callen, Qu, Siebert, Carreras, Shaing and Spong1987; Smolyakov et al. Reference Smolyakov, Hirose, Lazzaro, Re and Callen1995) present onset thresholds.

It is important to notice that the linear growth of the tearing instabilities depends on the different physical mechanisms driving the decoupling of the magnetic field motion from the plasma one: resistivity, electron inertia, agyrotropy of the electron pressure tensor. In particular, for the relevant case of reconnection processes in the low collisionality regime, the linear growth rate depends on the comparison between $d_{e}$ and $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70F}}$ (Grasso et al. Reference Grasso, Califano, Pegoraro and Porcelli2000). This regime may be characterized by a sudden increase in the time derivative of the reconnection rate (Grasso et al. Reference Grasso, Califano, Pegoraro and Porcelli2000; Comisso et al. Reference Comisso, Grasso, Waelbroeck and Borgogno2013) as nonlinearity takes over the linear phase. For a wide range of $d_{e}$ values, the peak nonlinear growth is approximately twice the value of the linear one, but it can increase up to one order of magnitude more for very low $d_{e}$ values (Comisso et al. Reference Comisso, Grasso, Waelbroeck and Borgogno2013). Besides the cases in which the instability grows spontaneously, it is important to note that in fusion devices the conditions (like error fields or resonant magnetic perturbations) that lead to forced reconnection events may frequently occur. In these cases also the plasmoid instability may develop and its relevant fast time scales must be considered (Loureiro, Schekochihin & Cowley Reference Loureiro, Schekochihin and Cowley2007; Comisso et al. Reference Comisso, Lingam, Huang and Bhattacharjee2016; Comisso & Grasso Reference Comisso and Grasso2016). We recall that the relevance of the physical parameters in determining the growth rate of the reconnection process depends also on the magnetic energy available to the system. The smaller this energy the larger the reconnection time scale is. For the spontaneous reconnection process a measure of this energy is given by the so called $\unicode[STIX]{x1D6E5}^{\prime }$ parameter (Furth, Killeen & Rosenbluth Reference Furth, Killeen and Rosenbluth1963), while for driven reconnection it depends on the intensity of the forcing (Comisso, Grasso & Waelbroeck Reference Comisso, Grasso and Waelbroeck2015).

The magnetic reconnection is then a multiscale phenomenon affecting the evolution of the continuum perturbed magnetic flux $\unicode[STIX]{x1D713}(\boldsymbol{r},t)$ , which can be described in its nonlinear stage with a coarse graining procedure. This procedure, first introduced by Rutherford (Rutherford Reference Rutherford1973), consists in evaluating the dynamics of a moment of the $(m,n)$ Fourier component of the field $\unicode[STIX]{x1D713}(\boldsymbol{r},t)$ , normally the island width $W(t)$ corresponding to the amplitude of the observable magnetic perturbation and its phase. Typically, coherent islands correspond to helical magnetic perturbations with low poloidal and toroidal mode numbers, such as classical and neoclassical tearing modes and $m=1$ resistive kinks (Ara et al. Reference Ara, Basu, Coppi, Laval, Rosenbluth and Waddel1978). The control of the dynamics of magnetic islands is a major challenge for present and future fusion devices, and its key issue is to prevent modes growth to dangerous amplitudes for the whole duration of a tokamak discharge. Starting from the 1980s, many efforts have been devoted to one of the most promising methods of controlling magnetic islands based on the attempt to drive a current directly into the magnetic island by absorption of radio-frequency (rf) waves at the lower hybrid frequency (Reiman Reference Reiman1983) or at the electron cyclotron frequency (ECCD) (Fisch & Boozer Reference Fisch and Boozer1980), with a $(m,n)$ helical component counteracting the destabilizing current perturbation. Successful ECCD experiments of NTM control, with different methods, have been carried out in ASDEX Upgrade (Gantenbein et al. Reference Gantenbein, Zohm, Giruzzi, Günter, Leuterer, Maraschek, Meskat and Yu2000), JT60U (Isayama et al. Reference Isayama, Kamada, Ide, Hamamatsu, Oikawa, Suzuki, Neyatani, Ozeki, Kajiwara and Ikeda2000), FTU (Berrino et al. Reference Berrino, Cirant, Gandini, Granucci, Lazzaro, Jannone, Smeulders and D’Antona2006), DIII-D (Petty et al. Reference Petty, La Haye, Luce, Humphreys, Hyatt, Lohr, Prater, Strait and Wade2004) and EAST (Wang, Ma & Zhang Reference Wang, Ma and Zhang2016). The main task of a theoretical study of rf control of these tearing instabilities, in the observable nonlinear phase, is the estimate of the necessary driven current, e.g. the rf power necessary to reduce the ‘state variable’ $W(t)$ and to design real time strategies for the rf launching for an effective power deposition and possible tracking of the moving island. A basic question for the design of a control concept is the order of the fastest and slowest time scales of the processes to be controlled (fast reconnection, slow nonlinear growth and saturation, island rotation period), and the associated space scales suggesting how sharp the focusing, radial and/or angular, should be. Moreover, in addition to all the physical scale lengths mentioned above, one has to consider also that the rf-driven current ‘forces’ another typical scale length, the absorption depth $d_{CD}$ . This length depends only on the wave beam launching and propagation conditions and plasma equilibrium quantities, such as density and temperature.

A generalized equation for the evolution of the island width $W=4\sqrt{\unicode[STIX]{x1D713}_{s}L_{s}/B}$ , which includes the effect of the ECCD, can be obtained from the rate of flux reconnection with surface averaging of the Faraday equation and currently goes under the name of generalized Rutherford equations (GRE) (see, e.g. Lazzaro & Ramponi Reference Lazzaro and Ramponi1996; Giruzzi et al. Reference Giruzzi, Zabiégo, Gianakon, Garbet, Cardinali and Bernabei1999):

(0.1) $$\begin{eqnarray}\displaystyle g_{1}\frac{\unicode[STIX]{x1D70F}_{R}}{r_{s}}\frac{\text{d}W}{\text{d}t} & = & \displaystyle r_{s}\left[\unicode[STIX]{x1D6E5}_{0}^{\prime }+\frac{(\unicode[STIX]{x1D6E5}_{bs0}^{\prime }-\unicode[STIX]{x1D6E5}_{GGJ0}^{\prime })W}{W^{2}+W_{c}^{2}}+\frac{\unicode[STIX]{x1D6E5}_{pol0}^{\prime }(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{E})(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{E}-\unicode[STIX]{x1D714}_{\ast i})}{\unicode[STIX]{x1D714}_{\ast e}^{2}}\frac{\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D703}i}^{2}}{W^{3}}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.\frac{I_{CD}}{I_{p}}\frac{L_{q}}{d_{cd}^{2}}\unicode[STIX]{x1D702}_{CD}\left(\frac{W}{d_{cd}}\right)\right].\end{eqnarray}$$

Here, $\unicode[STIX]{x1D702}_{CD}=(\int \text{d}\unicode[STIX]{x1D6FA}\int \text{d}\unicode[STIX]{x1D6FC}J_{CD}\cos (m\unicode[STIX]{x1D6FC})/\int \text{d}\unicode[STIX]{x1D6FA}\int \text{d}\unicode[STIX]{x1D6FC}J_{CD})$ is the expression of the ECCD efficiency of the $(m,n)$ helical current (Hegna & Callen Reference Hegna and Callen1997; Lazzaro & Ramponi Reference Lazzaro and Ramponi1996; Giruzzi et al. Reference Giruzzi, Zabiégo, Gianakon, Garbet, Cardinali and Bernabei1999), with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D703}-(n/m)\unicode[STIX]{x1D719}$ and $J_{CD}$ indicating the driven current density. This efficiency can be asymptotically estimated to give its dependence on $(W/d_{CD})$ (Sauter Reference Sauter2004). The quantity $W_{c}$ in the second (bootstrap current) term represents a lower limit of the island width related to the finite ratio of heat conductivity along and across the $B$ field ( $\unicode[STIX]{x1D712}_{\Vert }/\unicode[STIX]{x1D712}_{\bot }$ ), and governing the incomplete flattening of the temperature profile within the island separatrix (Fitzpatrick & Hender Reference Fitzpatrick and Hender1994; Fitzpatrick Reference Fitzpatrick1995). It is important to recall that the Rutherford equation is derived under the assumption that the linear condition of tearing modes growing slowly with respect to the diffusion time is valid also nonlinerly. For this reason, the Rutherford equation looses its validity as soon as highly nonlinear effects, such as mode–mode coupling, intervene in the evolution of the reconnection process.

In the neoclassical collisional regime, at low $\unicode[STIX]{x1D6FD}_{p}$ , a NTM is linearly and nonlinearly stable. At higher $\unicode[STIX]{x1D6FD}_{p}$ , in the phase plane $(\text{d}W/\text{d}t,W)$ there is an unstable region with $\text{d}W/\text{d}t>0$ . The plasma is metastable without an island, until a ‘seed’ perturbation (presumably of the same helicity) triggers the growth, proportional to $\unicode[STIX]{x1D6FD}_{p}$ (Callen et al. Reference Callen, Qu, Siebert, Carreras, Shaing and Spong1987; La Haye Reference La Haye2006). Often the seed island for NTM onset is associated with a sawtooth fast reconnection, providing appropriate resonant harmonics. The third term represents the polarization current due to the inertial ions effect. It depends on the mode rotation frequency and therefore can be stabilizing or destabilizing (Mikhailovskii Reference Mikhailovskii2003; Wilson et al. Reference Wilson, Alexander, Connor, Edwards, Gates, Grüber, Hastie, Hegna, Hender and La Haye1996). Finally, the last term is the ECCD coercive term, that should make negative or null the whole right-hand side of the GRE, leading to a chosen equilibrium value of $W$ , possibly, but not necessarily, zero.

A control problem could envisage either a prompt intervention to suppress the island of width $W$ as soon as the instability is detected, or a continuous control of a finite island within chosen bounds (Zohm et al. Reference Zohm1999; Pletzer & Perkins Reference Pletzer and Perkins1999; Gantenbein et al. Reference Gantenbein, Zohm, Giruzzi, Günter, Leuterer, Maraschek, Meskat and Yu2000), or alternatively a time optimal control policy to suppress an island with the maximum power as the mode reaches a target amplitude (LaSalle Reference LaSalle1959). Finally, a variant of the latter concept is provided by the strategy of sweeping the ECCD across the $q$ -rational surface (Fevrier et al. Reference Fevrier, Maget, Lütjens, Luciani, Decker, Giruzzi, Reich, Beyer, Lazzaro and Nowak2016) to mitigate the consequences of misalignment, which may lead to driven instability. Here we stress again that the instability time and space scales mentioned earlier constrain the choice of the best strategy. It is important to have in mind that the injection of a helical electron cyclotron current is accompanied both by a local intense (electron) heating and an axisymmetric current component, which may contribute to the evolution/control of the instability through the modification of the local magnetic shear and of the parameter $\unicode[STIX]{x1D6E5}_{0}^{\prime }$ (Bertelli & Westerhof Reference Bertelli and Westerhof2009). However, it has long been known (DeLuca, Jacchia & Lazzaro Reference DeLuca, Jacchia and Lazzaro1986) that the direct effect of EC heating and electron cyclotron (EC) current drive on tearing modes is destabilizing if the rf heating is applied on the inner (high field) side of the $q$ -rational surface. Therefore, the radial focusing (relative to the mode elliptic point) of the rf deposition is a control issue (Comisso & Lazzaro Reference Comisso and Lazzaro2010; Lazzaro et al. Reference Lazzaro, Comisso and Valdettaro2010). In the following of this paper, we will highlight some of the open physics problems which come from the evolution of the magnetic field topology and require special consideration in view of the realization of working control schemes.

The first aspect that must be underlined is that the equation obtained for $W(t)$ (and the related equation for the island phase (Smolyakov et al. Reference Smolyakov, Hirose, Lazzaro, Re and Callen1995; Lazzaro & Ramponi Reference Lazzaro and Ramponi1996)) necessarily masks several effects on the scales smaller than the island separatrix width. The injected EC current may form a new magnetic topology structure which is not completely symmetric (Comisso & Lazzaro Reference Comisso and Lazzaro2010; Lazzaro et al. Reference Lazzaro, Comisso and Valdettaro2010; Sun, Dong & Duan Reference Sun, Dong and Duan2015). The optimal localization and the duration of the ECCD pulse depend on the space scale and growth rates of both the fastest reconnection process and the strong nonlinear effect at the later phase. The amplitude of the total required ECCD is expected to be just of the order of the local current perturbation (e.g. ${\sim}1.2$ times the local bootstrap current density for NTMs (Prater et al. Reference Prater, La Haye, Lohr, Luce, Petty, Ferron, Humphreys, Strait, Perkins and Harvey2003)), but the local current density is distributed on the flux tubes intercepted by the beam ‘spot’ and therefore follows the island nonlinear shape and size evolution. A number of studies (Borgogno et al. Reference Borgogno, Comisso, Lazzaro and Grasso2014; Sun et al. Reference Sun, Dong and Duan2015; Wang et al. Reference Wang, Ma and Zhang2016) have established that the stabilizing effect is inversely proportional to the depostion width, $d_{CD}$ , for the same total driven current.

A second aspect that must be considered is related to new instabilities that may develop during the control itself. These can be for instance generated by an intense driven current, when its deposition depth $d_{CD}$ is narrower than the magnetic island separarix (like triple tearing modes (Wang et al. Reference Wang, Ma and Zhang2016)), while wider $d_{CD}$ leads to improved stabilization, albeit at the cost of reduced rf efficiency. Moreover, the appearance of secondary structures (plasmoids) destroys the concept of ‘island phase’ to be used in tracking an evolving island in a standard real time feedback control scheme (Volpe et al. Reference Volpe, Austin, La Haye, Lohr, Prater, Strait and Welander2009; van den Brand et al. Reference van den Brand, de Baar, Lopes Cardozo and Westerhof2012). Recently, simulations carried out for National Spherical Torus Experiment (NSTX) together with analysis of camera data directly from the experiment have proved the formation for plasmoids also in presence of a large guide field (Ebrahimi & Raman Reference Ebrahimi and Raman2015). Other studies in this direction (Ishii, Azumi & Smolyakov Reference Ishii, Azumi and Smolyakov2007) have shown that the rapid growth of the magnetic island is associated with its deformation that leads to the secondary magnetic island formation around the original X-points.

Furthermore, the control efficiency of the ECCD well focused on the island $O$ -point may be significantly reduced due to a magnetic topology transition occurring when the primary island is squeezed below the relevant scale of the reconnection layer. Under these circumstances, an unstable ideal MagnetoHydroDynamic (MHD) state is restored in which the $X$ - and $O$ -points have switched their positions and the field lines do reconnect, in a bifurcated configuration equivalent to a phase flip of $\unicode[STIX]{x03C0}$ on an inertial time scale (Borgogno et al. Reference Borgogno, Comisso, Lazzaro and Grasso2014; Fevrier et al. Reference Fevrier, Maget, Lütjens, Luciani, Decker, Giruzzi, Reich, Beyer, Lazzaro and Nowak2016), with total loss of control.

Also the injection of a current drive into a nonlinear island can be compromised by topological modifications. Indeed, the island nonlinear response to the ECCD leads to thin, bar-shaped velocity layers distributed along the axes, where the higher-order harmonics drive secondary Kelvin–Helmholtz instability, which affects the strong sheared plasma flows that form at the initial stages of the control process (Borgogno et al. Reference Borgogno, Comisso, Lazzaro and Grasso2014).

Finally, we highlight another aspect, related to the second term in (0.1), that would require a deeper insight. Indeed, in the low collisionality regime, neoclassical viscosity affects the stability of magnetic islands by providing an additional source of the transverse electric current contributing to the net longitudinal current in the magnetic island. It has been shown (Smolyakov & Lazzaro Reference Smolyakov and Lazzaro2004) that the additional current is associated with variations of plasma pressure (temperature) within the perturbed magnetic surfaces. These effects may provide a significant stabilizing influence that is comparable to the ion polarization current and finite $\unicode[STIX]{x1D712}_{\Vert }/\unicode[STIX]{x1D712}_{\bot }$ effects considered previously. Such variations can be expected from the ion polarization drift across the magnetic surface and by the finite parallel heat conductivity.

In conclusion, the description of the magnetic islands evolution by a GRE or a set of coupled GREs is a robust tool for the design of control algorithms, but the choice of control strategies requires consideration of finer, but strong, nonlinear effects related to the fragile topology of the reconnection layer. Further development of basic theory appears crucial in providing guidance toward a reliable system.