Neoclassical tearing mode control using vertical shifts on MAST

Triggered vertical shifts of the MAST spherical tokamak plasma have been found to stabilize 2/1 neoclassical tearing modes (NTMs) for a number of MAST shots, without impacting on core confinement. This stabilization is a result of favourable modifications of the density, temperature and pressure profiles at the location of an NTM by means of a brief transition from high (H) to low (L) confinement mode. Using this method, the high confinement phase can typically be recovered, and the NTM removed, within 20 ms of onset.


Introduction
A major performance limitation for future magnetically confined fusion (MCF) devices is predicted to be the onset of neoclassical tearing modes (NTMs) [1][2][3][4], which can give rise to a soft beta limit. If NTMs grow to a saturated size they can cause plasma terminating disruptions and unacceptable damage to the first wall of the device. Spherical tokamaks (STs) are being investigated as more compact potential MCF devices and have a number of advantages over the conventional tokamak scheme [5]. However, STs are overdense to conventional electron cyclotron waves and therefore cannot use the electron cyclotron current drive (ECCD) that is conventionally used to stabilize NTMs [6][7][8]. An alternative stabilization method has recently been demonstrated on MAST and is reported here. It involves brief transitions from high (H) to low (L) confinement modes, which are triggered using controlled vertical shifts of the plasma magnetic axis. Such transitions are possible because on MAST both H-mode access and edge pedestal height are sensitive to the position of the magnetic axis [9].
Triggered H-L transitions are shown to stabilize m/n = 2/1 NTMs (where m is the poloidal mode number and n the toroidal mode number) and prevent locked mode disruptions for several double null discharges on MAST, typically doubling the H-mode duration [10], with no significant lasting decrease in electron temperature (T e ), density (n e ) or ion temperature (T i ) measured in the plasma core (figure 1). The H-mode phase is typically recovered, and the NTM removed, within 20 ms of onset using this method. In this paper, the mechanisms by which triggered H-L transitions stabilize 2/1 NTMs are explored using MAST data.

Description of Experiments performed
The MAST discharges reported here are double null, 800 kA plasma current (I p ) discharges each with a toroidal field (B T ) of 0.5 T at the magnetic axis and beam heating of 2-4 MW. This scenario is characterized by weakly reversed shear q profiles and high performance, typically reaching β n of 3. Here, q is the safety factor, which represents the number times a magnetic field line travels around a torus toroidally (n) for each time around poloidally (m) and β n is the ratio of β T (β T = P /(B 2 T /2µ 0 )) to the Troyon limit (I p /aB T ), where P is the volume averaged total pressure, B 2 T /2µ 0 is the magnetic energy density and a is the minor radius [12].
The shots typically transition into H-mode at ∼0.2 s and a 2/1 NTM occurs a short time (20- [11]. The NTM disappears at both 0.265 s and 0.33 s. programmable gate array system was developed to trigger the MAST vertical control system, upon detection of a 2/1 NTM by the MAST Mirnov coil array [10]. Vertical shifts of the order of 1-2 cm in these scenarios result in an H-L back transition on MAST. The triggering system is designed to hold the vertical position down until the Mirnov coils no longer detect the 2/1 NTM. At this point, the vertical position returns to the optimum position for H-mode access and H-mode is rapidly restored. Figure 1 shows the evolution of the NTM island width when using this approach. In this shot two 2/1 NTMs occur and each is successfully removed using a vertical shift. For the MAST discharges considered here, pressure profiles are characterized by three distinct regions, a steep pressure gradient in the pedestal region (R = 1.38-1.4 m), a low gradient region that arises due to impurity accumulation and the resulting n e 'ears' (R = 1.28-1.38 m) and, finally, a steep gradient region that results from reduced ion transport and fast ion pressure (R = 1.1-1.28 m). This region of enhanced core confinement is found to be largely unaffected by brief H-L back transitions and the core values of density, temperature and pressure are maintained (figures 1, 2 and 3). In the current flat top phase, the q = 2 surface is typically located inside the low pressure gradient region. As the majority of terms which determine NTM growth are dependent on the pressure gradient, it is critical to measure the pressure profile. For example, equilibrium reconstructions using the equilibrium fitting code (EFIT) [13] with 7th order splines were found to overestimate the pressure gradient by an order of magnitude. In order to accurately describe the pressure and current in this region, the equilibrium boundary is first determined using EFIT code. The fixed boundary code CHEASE [14] is then used to recalculate the equilibrium using high resolution measurements of pressure, current and poloidal flux from the MAST motional Stark effect (MSE) magnetic field internal pitch angle profile diagnostic, using a method previously described by Petty et al [15] and DeBock et al [16].
The ff and P terms required for the equilibrium solution are calculated directly from MSE data, where ff is the product of the poloidal current flux function (f ) and its derivative with respect to poloidal flux (f ) and P is the derivative of the plasma pressure with respect to the poloidal flux. The P term at q = 2 agrees well with the total pressure profile estimated from Thomson scattering (TS) and charge exchange (CXR) data (figure 3). The fast ion contribution, which was not directly measured on these discharges, agrees qualitatively with estimates from transport codes. The bootstrap current terms are then calculated using TS, CXR and visible bremsstrahlung (ZEBRA) profile measurements of the T e , n e , T i and Z eff profiles, where Z eff is the effective charge of the plasma as defined in Wesson et al [12].

Effects of H-L transitions on NTM stability
The modified Rutherford equation (MRE) details the different contributions to NTM stability as a function of island full radial width (W ). It can be written as the sum of the classical stability term ( CL ), the bootstrap drive term ( BS ), the magnetic field curvature term ( GGJ ) and the ion polarization term ( POL ). These terms can be calculated using equilibrium codes and written in terms of measurable parameters, and the form used here is very similar to that used in previous works [11,[18][19][20].
Here, r s is the minor radius of the rational surface, t r the resistive time, P e the electron pressure, P the total pressure, W d the finite island transport diffusion width, W d,Te the finite island heat transport diffusion width, W b,i the ion banana width, D r the resistive interchange term, B θ the flux average poloidal magnetic field, L q the safety factor gradient scale length (L q = q(dq/dr) −1 ), L p the pressure gradient scale length (L p = p(dp/dr) −1 ) and β p is the ratio of kinetic pressure to poloidal magnetic pressure (β p = 2µ 0 P / B θ 2 ). In equation (1), the ion profile contribution to the bootstrap term has been neglected because of a high trapped fraction (∼80%) at the location of the q = 2 rational surface. The standard forms of BS,βp and POL,βp are shown in equations (3) and (5) respectively. These expressions are typically used in experiments where β p scans are performed [11,18,20,21] and assume a constant L p during the island evolution. As transient H-L-H transitions principally modify L p , these stability terms has been rewritten in terms of the P and dP /dr, by substituting L p and β p into equations (3) and (5). W d,Te and the coefficients a NL , a 1 , a 2 and a 3 are kept fixed during the time evolution of the mode and have been determined from fitting a heat transport model and previous beta scan experiments respectively on similar MAST discharges [11]. The values used for these coefficients are shown in table 1 and, as discussed by Snape et al [11], are close to the theoretically expected sizes. In conventional aspect ratio tokamaks ECCD is the favoured scheme for NTM control and mitigation. ECCD modifies NTM stability by the addition of a localized current at the position of the NTM, which makes CL more stabilizing and replaces the missing bootstrap current. However, transient H-L-H mode transitions are found to influence both the pressure and current at the NTM location and, as a result, modify all stability terms in equation (1). The effects of these transitions on each on the stability terms will now be considered in turn.

Changes to the resistivity
An H-L transition reduces T e and Z eff ( figure 3) and results in an increase in plasma resistivity (η) at the q = 2 surface. The size of the resistive time (t r = 1.22µ 0 l 2 /η) determines the timescale over which the current profile can evolve on a given length scale (l). The value of t r is important in determining the how quickly W can change (equation (1)). t r is reduced during an H-L transition and as dW/dt is proportional to 1/t r , this in turn increases the magnitude of dW/dt. As an H-L transition is stabilizing (dW/dt is negative) this enhances NTM stability ( figure 3(b)). If the length scale of interest is assumed to be ∼15 cm, the t r in these discharges (1 < t r < 5 ms) matches the typical time scale of island decay.

Changes to the bootstrap stability term
The bootstrap drive is typically regarded as the principal destabilizing term which drives NTMs and is represented by in the MRE. This effect of the bootstrap current on tearing mode stability was first predicted by Rutherford et al [22] and NTMs were first identified experimentally in TFTR [1]. Experimental measurements showed there to be a threshold island width above which NTMs grow and the two principal mechanisms proposed to explain this threshold are the effects of the finite island width and the ion polarization current. The effect of the ion polarization current is discussed further in the next section.
As an NTM grows, dPe dr is locally reduced at the island position and this creates a hole in the bootstrap current, which then further drives NTM growth. The size of this reduction depends on the degree of flattening of the n e and T e profiles, which is described by the finite island width (W d ) for each [23]. The finite island width used in this paper is W d,Te and has been determined from fitting a heat transport model from Fitzpatrick et al [23] to TS electron temperature profiles of similar discharges as discussed in Snape et al [11].
Generally W d,Te W d,ne and therefore dTe dr shows a much greater reduction than dne dr during NTM growth. Thus, the size of the bootstrap hole is predominantly determined by dTe dr n e . A triggered H-L transition causes a reduction of n e at the island location, but usually little change in dTe dr . The reduction in n e reduces the dominant dTe dr n e term and H-L transitions are therefore predicted to decrease the size of a perturbed bootstrap current hole. The bootstrap drive term is therefore represented by 1.  figure 4(c).

Changes to the curvature and polarization current stability term
GGJ is the stabilizing contribution of the magnetic field curvature and is typically an order of magnitude larger in  (6)) and evolution of δ BS calculated from the hole in < J BS > (equation ( (7))). (c). The n e dT e /dr (red, equation (3)) and T e dn e /dr (blue) contributions to the total BS . (d). The evolution of the GGJ (equation (4)) and POL (equation (5)) terms. (e). The island growth rate with and without the r 2 s /t r term and using δ or δ BS in the MRE.
STs than in conventional aspect ratio tokamaks, as a result of the high level of plasma shaping in these machines. Glasser, Greene and Johnson derived the mathematics to describe this stabilizing contribution [24], which is referred to as the Glasser Green Johnson (GGJ) effect. The dependence of the GGJ term on W was modified by Lutjens et al [25] to take into account the finite island width. The size of GGJ is dependent on the resistive interchange parameter D r calculated from the CHEASE code. D r is proportional to dP dr and this can be seen in its low aspect ratio approximation [11], given by (q 2 −1)Lq qR0 dP dr , where R 0 is the major radius at the magnetic axis.
The size of the ion polarization term [26,27] is dependent on how ions and electrons respond to the island rotation (ω) in the rest frame of the plasma, given by the difference in island frequency measured in the lab frame (ω M ) and the frequency where the electric field is zero (ω Er=0 ) [28]. This complex dependency is represented by g( , ν i , ω) in equation (5), where ν i is the ion collisional frequency. During vertical shifts the ω M is reduced by a factor of two in 100 µs, a similar decrease is found in the ω Er=0 . However, the minimum time resolution of this measurement is 500 µs. The value of POL plotted in figure 4(d) therefore assumes that g( , ν i , ω) does not vary as a result of the H-L transition. A large uncertainty exists in the theoretical description of g( , ν i , ω) and in the absence of a complete theory, the unknown contribution is absorbed into the dimensionless fitting parameter a 3 and therefore the size of POL , and thus measurements of this term, are mostly qualitative. In order to limit the effect of the ion polarization term below the ion banana width we have adopted the heuristic model of Sauter et al [29].
During a triggered H-L shift the dP dr at the q = 2 rational surface is found to increase and the tokamak curvature ( GGJ ) and ion polarization ( POL ) terms, in equations (4) and (5) respectively, scale with dP dr and ( dP dr ) 2 P . A triggered vertical shift results in a reduction in the destabilizing bootstrap drive ( BS ) and an increase in the stabilizing GGJ and POL terms; the magnitudes of these terms are shown in figure 4(d). GGJ and POL both have larger influences at smaller W, which indicates that the triggered H-L mode transitions are more effective at smaller W. The size of the GGJ term may have different dependencies on T e , T i and n e profiles similar to the BS , but no complete theory exists to describe these dependencies.

Changes to the classical tearing stability term
The classical tearing stability term [30,31] is generally assumed to be stabilizing, as if it were not, tearing modes would be present in all discharges. However, a growing body of work [11,32] suggests that this term can be modified to be weakly stabilizing or destabilizing, depending on the evolution of plasma parameters. In the discharges used in this work, the onset of the NTM occurs without a clear trigger, which suggests that CL evolves towards a point of being weakly stabilizing or destabilizing. EFIT reconstructions and estimates of the current profile using the MSE diagnostic show a hole develops in the current profile ( J hole ) inside the H-mode pedestal, near r s ( figure 3). This may make CL destabilizing and it is likely that triggered H-L transitions remove this hole, by modifying the bootstrap current ( J bs ) and increasing the inductive current density inside the pedestal via a reduction in Z eff (figure 1). A full calculation of CL in a realistic tokamak geometry remains a challenging problem. Here, a simple model has been developed in order to qualitatively estimate the effects of the 'current hole' on CL and uses an expression describing the modification of CL by the addition of a Gaussian current profile. This expression was previously developed to incorporate the ECCD current drive into the MRE framework [8,33] and here is simply adapted by reversing the current drive sign to represent a current hole, where the shape parameter (a s ) is taken as 4 (low aspect ratio value), δ hole is the full-width half maximum of the Gaussian hole and x = |r s − r hole |/δ hole . F (x) depends on the alignment of the current hole with r s . F (x) is destabilizing when x < 0.6 and stabilizing when 0.6 < x < 1. δ BS is determined from the hole in the derived J bs profile and δ from the estimated J profile. Typically, as the q profile evolves, then r s becomes more closely aligned with the current hole and this term is more destabilizing. The evolution of these terms is shown in figure 4(b). In the case of the δ BS the current hole is predicted to be initially stabilizing, but to become destabilizing as the shot evolves, whilst δ is found to be destabilizing for the entire shot. In both cases, the triggered H-L transition is found to remove, or significantly reduce, the current hole and both δ BS and δ are reduced to approximately zero.
A narrowing of the current channel indicative of a current hole has also being observed on 'triggerless' NTMs seen in ITER like discharges in a similar experiment on DIII-D [32]. Here it is proposed that 'current holes' are responsible for positive CL and triggerless tearing modes in the shots examined.

Merits of this NTM stabilization scheme
This approach has a number of advantages which may permit further application on ST. The most unstable tearing modes are expected in the high edge magnetic shear which is typical in close proximity to the pedestal region, coupling the evolution of the H-mode pedestal profiles to those at the location of the mode. Evidence of this coupling has also been observed on the conventional aspect ratio tokamak DIII-D [34]. The drop in density associated with an H-L transition reduces the bootstrap drive term. The increase in the overall pressure gradient drives the stabilizing tokamak curvature and ion polarization terms. On STs, the curvature term is large enough to rival the bootstrap drive and an order of magnitude greater than on conventional tokamaks [11,35].
The theory of the ion polarization term remains incomplete, but comparisons with current theories are encouraging. Current hole formation as a result of pedestal bootstrap current and impurity accumulation are suggested as a means of reducing CL . Brief H-L transitions can modify J , removing current holes centred on the r s .
Brief H-L transitions are found to have little effect on the core temperature, density and pressure obtained in MAST. This may be a result of the core performance on MAST being dominated by suppression of the ion temperature gradient (ITG) turbulence [36] rather than by the edge pedestal performance. In a number of other tokamaks a linear scaling has been obtained between the edge pressure and the core confinement and therefore more work is needed to characterize the effect of brief transitions on other devices.
On MAST, the sensitivity of H-mode access on the vertical positions is used to trigger H-L transitions when the signature of 2/1 NTMs is detected by the magnetic coils. This mechanism is sufficient to trigger an H-L transition at maximum input power on MAST. A range of methods exist to trigger similar transitions on existing conventional and spherical tokamaks, as discussed by Meyer et al [9]. One of the possible disadvantages of this scheme is a high divertor heat flux (8 MW m −2 ) associated with the H-L transition, which is of the order of a typical type I edge localized mode (ELM) [12] on MAST. ELMs are instabilities which occur in H-mode plasmas and cause heat loads on the tokamak's divertor. A number of new divertors designed [37,38] to handle high ELM heat fluxes are currently being built [39] and tested and it is anticipated that successful operation of these schemes will mitigate this problem.

Conclusion and future work
Triggered, transient H-L-H mode transitions, induced by small vertical plasma displacements, have been demonstrated as a method for removing 2/1 NTMs in high performance discharges on MAST, extending the H-mode duration by a factor of two. A detailed analysis of changes to the terms in the MRE suggests that the stabilization mechanism is likely to be via a reduction in the destabilizing terms BS and CL , as well as an increase in the stabilizing GGJ and POL terms. The modification of these terms is a result of the favourable modification of the kinetic and current profiles during the brief H-L-H transitions. Further work will focus on optimizing this scheme on a range of high performance ST scenarios, in particular those with greater bootstrap drives. It will also look to improve the triggering hardware, in order to permit triggering of H-L transitions at smaller island widths, where this methodology is expected to be more efficient.