Plasma Rotation and Transport in the MAST Spherical Tokamak

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Plasma Rotation and Transport in the MAST Spherical Tokamak
1A. R. Field, 1C. Michael, 1R. J. Akers, 2J. Candy, 1, 3G. Colyer, 4W. Guttenfelder, 3Y.-c. Ghim(Kim), 1C. M. Roach, 1S. Saarelma and the MAST team.
1EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxon, UK. 2General Atomics, P.O. Box 85608, San Diego, CA 92186-5608, USA. 3Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK. 4Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ, USA.
E-mail: [email protected]
Abstract: The formation of internal transport barriers (ITBs) is investigated in MAST spherical tokamak (ST)
plasmas. The roles of E×B flow shear, q-profile (magnetic shear) and MHD activity in their formation and
evolution are studied using data from high-resolution kinetic- and q-profile diagnostics. In L-mode plasmas, with co-current directed NBI heating, ITBs in the momentum and ion thermal channels form in the negative shear region just inside qmin. In the ITB region the anomalous ion thermal transport is suppressed, with ion thermal transport close to the neo-classical level, although the electron transport remains anomalous. Linear stability analysis with the gyro-kinetic code GS2 shows that all electrostatic micro-instabilities are stable in the negative magnetic shear region in the core, both with and without flow shear. Outside the ITB, in the region of positive magnetic shear and relatively weak flow shear, electrostatic micro-instabilities become unstable over a wide range of wave-numbers. At ITG length scales, flow shear reduces linear growth rates and narrows the spectrum of unstable modes, but flow shear suppression of ITG modes is incomplete. Flow shear has little impact on growth rates at ETG scales. This is consistent with the observed anomalous electron and ion transport in this
region. With counter-NBI ITBs of greater radial extent form outside qmin due to the broader profile of E×B flow
shear produced by the greater prompt fast-ion loss torque.
1. Introduction
Shear-flow suppression of anomalous turbulent transport plays an important role in determining the confinement properties of spherical tokamak plasmas. In MAST the strong toroidal rotation, with typical Mach number Mφ = Rωφ vth,i ≤ 0.5 (where ωφ is the rotation rate and vth,i the ion thermal velocity), driven by tangential neutral beam injection (NBI) heating produces E×B shearing rates sufficient to exceed the growth rate of low-k ion-scale turbulence. In H-mode plasmas ion heat transport is within a factor 1-3 of the ion neoclassical level over most of the radius, while the electron transport remains highly anomalous [1, 2]. In L-mode plasmas however the ion transport can significantly exceed neo-classical levels in the outer regions but is suppressed by flow shear at mid-radius and, under favorable conditions can exhibit an internal transport barrier (ITB) with ion transport at the neo-classical level. Studies of transport in such regimes in MAST are facilitated by the availability of advanced diagnostics, including a high-resolution, multi-pulse NdYAG Thomson scattering system with resolution (~ 1 cm) matching that of the CXRS system and a multi-channel Motional Stark Effect (MSE) diagnostic for q-profile measurements. The integrated analysis chain (MC3) used to prepare data for transport analysis using TRANSP [3] now incorporates the MSE constrained EFIT equilibrium reconstruction, allowing local transport properties to be related to details of the q-profile evolution.
An Internal Transport Barrier (ITB) can be defined as a region of reduced anomalous transport in the plasma core, associated with strong E×B flow shear, low or negative magnetic shear sˆ = r q ⋅ (dq dr) and a suppression of turbulence. In several tokamaks ITB formation is observed to be linked to low-order rational surfaces, where the transport improvement is initiated or tied to locations with rational q-values. In many cases the ITB is initiated when the location of zero magnetic shear ( qmin ) passes through an integer value. Two mechanisms

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proposed for ITB formation are related to coherent MHD modes and to the density of rational surfaces respectively. In the first [4, 5] tearing modes or fast-particle driven ‘fishbone’ modes are thought to modify the radial electric field (Er) profile and hence trigger the ITB through increased E×B flow shear suppression of the turbulence. The second relies on the rarefaction of low-order rational surfaces in the vicinity of integer q-values [6, 7], which could lead to decreased coupling between micro-instabilities. In regions of negative magnetic shear, growth rates of micro-instabilities are also reduced due to a reduction in the interchange drive [8, 9].
The phenomenology of MAST discharges exhibiting ITBs, heated with both co- and counter NBI heating, is presented in Sect. 2, including results of local transport analysis. The location and time of ITB formation is shown to be related to the q-profile evolution, and the role of MHD activity in limiting the sustainment and in the termination of the ITB is discussed. In Sect. 3 results of micro-stability analysis of the co-NBI ITB discharge are presented both from the linear gyro-kinetic flux-tube code GS2, which incorporates flow shear [10], and also with the global, particle in cell (PIC) code ORB5 [11]. These analyses reveal the importance of flow shear and magnetic shear to micro-stability and hence to the level of anomalous transport.

2. ITB discharges
In DIII-D [12], discharges exhibiting transport changes at rational q values are low-density, L-mode discharges with NBI heating power marginal for ITB formation applied early to produce negative magnetic shear in the plasma core. Steepening in gradients of Ti and toroidal rotation ωφ is observed as qmin passes through integer values, with a simultaneous reduction in low-k turbulence observed with the BES diagnostic. On JET [4], ITBs have also been linked to integer q values, forming when qmin reaches an integer value and then bifurcating, propagating both inwards and outwards with foot points following the two integer q surfaces. In NSTX [13] ITBs in the ion thermal and momentum channels are observed to form in the vicinity of maximum E×B shear, consistent with the E×B shear stabilization of ITG turbulence, while electron ITBs are more closely correlated with the region of most negative magnetic shear.
The ITB scenario used on MAST is similar to that of other tokamaks [14, 15], with early NBI heating of a low-density Lmode discharge applied during the current ramp to slow current penetration, resulting in strong toroidal rotation and reversed

Fig. 1 Evolution of (a) magnetic shear sˆ , (b)

normalised ion temperature ρ L and (c)

s

Ti

toroidal rotation ρs Lω gradients, (d) rate of

change of toroidal rotation ω&φ and (e) MHD in

co-NBI ITB discharge #24600. Locations of

rational surfaces (green, labled below plots) and

qmin (cyan) are also shown.

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magnetic shear in the core. One aim is to investigate the relative importance of rotation shear compared to magnetic shear by comparing discharges with co- and counter-NBI heating in which the ratio of power to torque is different.

Co-NBI discharge:

The evolution of such an ITB discharge with ~ 3 MW of co-NBI heating ( I p ~ 850 kA, Bt ~

0.53 T) is shown in Fig. 1. The ITB in the ion thermal channel forms at about 0.1 s when the

ion thermal diffusivity χi locally falls to the ion neo-classical value. At this time the core Ti

is only about 300 eV, although the normalised ion temperature gradient ρs LT (where ρs is i

the ion Larmor radius at the sound speed and L = (T ' T )−1 ) already reaches 0.1. The ITB

Ti

i

i

forms in the negative shear region just inside qmin about the time at which qmin reaches 3. An

ITB in the momentum channel forms before this, at the onset of negative magnetic shear at

~ 0.07 s, with a peak in ρs Lω localised to the region of most negative magnetic shear a few

cm inside qmin . This is stronger than the ion thermal ITB with the maximum of ρs Lω ~ 0.2

located inside the peak of ρs LT . There is no sign of ITB formation in the electron channel, i

as the Te profile is broad with the gradient distributed over the full radial region outside qmin .

The formation of the ITB is better illustrated in Fig. 2, which shows the evolution of the

maxima of ρs LT and ρs Lω and their radial locations, which are close to that of qmin . The

i

φ

times when qmin reaches rational values are also indicated. These times show some correlation

with temporary increases in ρs Lωφ and temporary reductions in ρs LTi , which increases in the periods between the integer-q crossings. Such behaviour has also been observed on DIII-D

[12].

Fig. 2 Evolution of: (a) the maximum values

of ρ L (black), ρ L (blue) and (b)

s

Ti

s

ωφ

their radial locations along with that of qmin

(dashed) and (c) the ratio χi χi,NC at the

location of (ρs LTi )max (red) and at r/a = 0.7 (black) for the discharge shown in Fig. 1.

Times when qmin crosses rational values are

also shown in (a).

Thermal transport analysis requires knowledge of the heating profiles. The deposition of the dominant NBI heating is determined in TRANSP [3] using the NUBEAM MonteCarlo model, which assumes classical behaviour of the fast-ions. Several factors indicate that, under some conditions, the loss rate of fast-ions must exceed that due to classical diffusion, primarily due fast-ion driven MHD activity. Firstly, the plasma energy WMEHFDIT estimated from EFIT is much lower than that calculated by TRANSP. In discharge #24600 in the period with two beams, TRANSP overestimates WMHD by a factor ≤ 1.7 compared to EFIT. Overestimating the total pressure results a predicted Safranov shift ∆RSh that is ≤ 5 cm larger than the value obtained from EFIT. A further consequence is that the predicted D-D neutron rate RN is a factor ≤ 2 too high compared to that measured.
Anomalous fast-ion losses can be represented in TRANSP using a diffusion coefficient Dfast which is isotropic in pitch angle, with prescribed radial and energy dependencies. It is necessary to assume anomalous losses in discharge #24600 only after 0.21 s when the NBI power is increased from ~ 1.8 MW to

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~ 3.2 MW, indicating a strong power dependence of the losses. In this phase a value of Dfast ~ 3 m2s-1 is required and after 0.27 s, following the onset of an n = 1 internal kink mode in the core, an even higher value ~ 5 m2s-1 has to be assumed. Anomalous diffusion is applied only
to fast-ions with energies > 30 keV, which is necessary to reduce RN by a larger factor than WMHD , as the D-D fusion cross-section is strongly weighted to higher energies. The enhanced fast-ion losses result in the absorbed power being reduced by factors of ~ 0.4 and ~ 0.5 in
these two latter phases respectively.

With these assumptions on the fast-ion losses, as shown in Fig. 2 (c), the inferred level of ion
thermal transport at the ITB location is close to the neo-classical value with χi χi,NC ~ 1, whereas χi considerably exceeds the neo-classical value in the positive shear region outside qmin , e.g. by an order of magnitude at r/a ~ 0.7. In the latter phase (> 0.21s) with two beams, there is a period (0.24-0.27 s) exhibiting negative values of qi and hence χi (indicated by the shaded region). This occurs because the magnitude of qi is relatively small and hence sensitive to systematic uncertainties in the power deposition profile, which is dependent on the crude ad-hoc fast-ion redistribution model, and to rapid changes in the kinetic energy W&kin at this time.

During the period from 0.07-0.16 s, during which the ITB forms, there is strong ‘fishbone’
MHD activity driven by the fast-ion pressure gradient, localised around qmin . At this time the rotation exhibits an inverted (positive) gradient just outside qmin indicating the presence of a localised negative torque there. Such a torque would arise from an outward redistribution of
the fast-ions by the fishbone MHD activity, which would be balanced by an inward radial
return current resulting in a jr × Bθ torque counter to I p . Although this braking reduces ρs Lω at qmin , it enhances the rotation gradient just inside qmin in the region of favourable negative magnetic shear, hence locally increasing the E×B shearing rate γ E , which is proportional to L−ω1 . This process may therefore facilitate formation of the ITB and perhaps also help localise it to the region inside qmin .

Once the ITB is established Ti and ωφ both increase rapidly inside qmin , which is located at

about r a ~ 0.5, with T increasing to 2.5 keV and ω to 2×105 s-1 on axis, while T reaches

i

φ

e

1.5 keV. This rotation rate corresponds to a Mach number Mφ ~ 0.5. The stored energy Wpl

reaches 100 kJ, including that of the fast-ions, which is comparable to the thermal energy. The

strong pressure gradient at the ITB causes growth of NTM instabilities. Such a mode is

observed in the discharge in Fig. 1 at about 0.16 s, which causes a localised braking of

rotation inside qmin and an acceleration further outside, due to coupling with another mode of

different mode number. This weakens the gradients, causing the mode to die away and the

ITB to be sustained. Another tearing mode occurs later in the discharge at 0.28 s, which

causes a similar localised braking of the core rotation. An example of such coupling of n = 3

and n = 2 tearing modes at q = 4/3

and 3/2 surfaces respectively is

shown in Fig. 3, where the three

independent measurements are all

consistent. Although the ITB

terminates around this time the

tearing mode is not the sole cause.

Once the central q0 approaches unity

at about 0.25 s, the plasma core

becomes unstable to an internal n = 1, m = 1 internal kink mode, localised to the region where q < 1, which subsequently grows in

Fig. 3 Example of coupling of n= 3 mode at q = 4/3
surface to n= 2 mode at q = 3/2 showing: (a) n×ωφ and MHD spectrum, (b) torque ∝ dωφ/dt, (c) q-profile.

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amplitude and saturates [16]. The resulting axially asymmetric perturbation causes braking of the core rotation due to neo-classical toroidal viscocity (NTV) [17]. This flattens the rotation profile inside the q = 1 surface and leads to termination of the ITB within about 30 ms. Such modes, which are referred to as Long-Lived Modes (LLM) usually occur in MAST plasmas once q0 falls below ~ 1.3.

Counter-NBI discharge:

Heating by counter-current directed NBI is less efficient due to higher level of first-orbit
losses of the fast ions. The beam torque however is of similar magnitude for a given injected
power Pinj because of the increased co-beam directed jr × B torque produced by the return current which balances the loss of fast-ions. Typically, the fraction of Pinj absorbed by the plasma is ~ 0.5 compared to over ~ 0.8 for co-injection, while the beam torque ~ 1 Nm is
comparable. The rotation rate achieved is therefore similar to that with co-NBI. Perhaps
surprisingly, the peak stored energy WMHD ~ 80 kJ is not greatly reduced compared to the similar co-NBI discharge (shown in Fig. 1), hence the confinement with counter-NBI is
improved. The temperatures in the counter-discharges are however lower with Ti ≤ 800 eV and Te ≤ 600 eV and the profiles broader. Although the fuelling rates are comparable increased particle confinement with counter-NBI leads to a factor ~ 2 higher density. As in
the co-discharge, the magnetic shear is weakly negative in the core inside r / a ~ 0.4.

Fig. 4 Evolution of a counter-NBI ITB discharge #22543 with plots as defined in Fig. 1.

An ITB forms in the momentum channel
at about 0.07 s (see Fig. 4). This can be
seen as an acceleration of the core plasma
inside qmin and a braking outside. The ITB forms just outside qmin at about r / a ~ 0.5 in the region of sˆ > 0 and the
rotation gradient then broadens across the
whole outer region of the plasma. This is
consistent with the broad profile of
jr × B torque from the beams. The Ti gradient also increases at the same time
at the same location outside qmin , although the ITB is weaker with ρs LT ~
i
0.1. After qmin falls below 5/2, the ion ITB appears to bifurcate with a second
ITB at larger radius following the
approximate location of the q = 5/2
surface. Across most of the plasma radius χi is within a factor ~ 2 of the ion neoclassical value while χe is several times larger. The assumption of anomalous
fast-ion losses is not required in these
counter-NBI discharges to match WMHD and RN.

3. Simulations
It is well known that radially sheared equilibrium flows V can suppress turbulence if the E×B shearing rate γ E is larger than the maximum linear growth rate γ max [18], i.e. γ E = dV⊥ dr > γ max . ITG turbulence can be affected when γ E ~ O(vth,i L) , where L is an equilibrium scale length. Equilibrium flows which approach sonic speeds are toroidal as the

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poloidal component generated by Er is cancelled by the parallel neo-classical flow. Sheared

toroidal equilibrium flow V = Rω(ψ ) eφ has been implemented in the local, flux-tube

geometry, gyro-kinetic code GS2 [19] in the intermediate flow ordering, where

vth,i >> V >> (ρi L)vth,i [10]. Sheared flows introduce two additional physics terms into the

gyro-kinetic equation. The perpendicular component γ E tears apart radially extended eddies

and is generally stabilising, while the parallel component ~ (B B )γ drives the parallel

φ

θE

velocity gradient driven instability [20] and can enhance the growth rate of ITG modes [21].

The ratio of turbulence suppression to additional drive is proportional to (B B ) ~ r (Rq) , i.e.

θ

φ

suppression is favoured at low q and at large r R , which arise by definition in an ST.

Nonlinear simulations for the conventional aspect ratio Cyclone base case equilibrium have demonstrated that the sheared parallel component of the flow can rekindle turbulence at large sheared flow [10, 22, 23]. This analysis has recently been extended to study turbulent toroidal momentum transport, and to assess the detailed sensitivity of turbulent fluxes to ρs LT and
i
γ E [24, 25]. At high values of sheared flow or low magnetic shear, the sheared parallel flow transiently drives instability growth, which can support sub-critical turbulence observed in recent simulations [24, 25, 26]. Furthermore, a possible physics mechanism, through which plasma equilibria may bifurcate to ITB profiles with steeper gradients, has been found to be more effective when the magnetic shear is zero rather than finite [25]. These results may be relevant to ITB formation in MAST.

A procedure to determine effective linear growth rates in local linear gyro-kinetic simulations with flow shear was outlined in [10]. In previous work [10, 27], local linear electrostatic analyses were performed at close to mid-radius ( ρn (= Φ1N/2 ) = 0.4 , where ΦN is the normalised toroidal flux) for the MAST H-mode plasma #6252. The trapped particle drives for low-k modes ( ky ρi < O(1) ) were shown to be weak owing to collisions, and the flow shear was found to be sufficient to stabilise all linear, low-k electrostatic instabilities in this region [10].

A similar linear analysis has been performed for the co-NBI ITB discharge #22807 (which is similar to #24600 shown in Fig. 1) at the time of peak ITB strength (0.25 s) at three radial locations of ρN = 0.3, 0.52 and 0.7, which are located inside the ITB region, just outside qmin and in the outer region of the plasma. At the innermost location where sˆ < 0, the linear electrostatic analysis shows that all modes at both ion and electron scales are stable with or without flow shear. Including magnetic perturbations without sheared flow did not change this result. Further calculations are underway to determine whether the core region is also fully stable at earlier times prior to the ITB formation. Fig. 5 (a, b) shows that at mid-radius, just outside qmin where sˆ is slightly positive, it is important to include kinetic electron

Fig. 5: Effective linear growth rate γ * vs. perpendicular wave-number ky ρi with (triangles) and without sheared equilibrium toroidal flow (circles) for co-NBI ITB discharge #22807 (0.25 s) at (a, b) mid-radius and (c) in the outer region. Simulations were carried out with: (a) kinetic ions and adiabatic electrons and (b, c) with kinetic electrons and ions. The horizontal dashed lines show the local level of flow shear γ E and symbols at the axis minimum denote stable modes.

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physics because there is an appreciable trapped electron drive for TEM modes ( ky ρi < O(1) ) there. It is important to note that the collisionality is lower and the density gradient higher
than in the H-mode equilibrium discussed in Refs. 10 and 27.

Including flow shear and kinetic electrons results in substantial but incomplete suppression of these low-k instabilities (Fig. 5 (b)), but without including a kinetic drive from trapped electrons, the ITG modes would be fully stabilised by the flow shear (Fig. 5(a)). On the outermost surface, a stable gap appears in the spectral range between the ITG and ETG regions: i.e. the TEM modes are stable because the plasma is more collisional. Strongly growing ITG modes ( ky ρi < O(1) ) are not fully stabilized by the weak flow shear in this outer region. Anomalous thermal and momentum transport is hence to be expected in this region, which is consistent with observations (see Fig. 2 (c)). Doubling the flow shear would be sufficient to fully suppress the ITG modes here. In summary, these calculations indicate that flow shear is not required to sustain the ITB in the negative shear core region but that it is acting to reduce the level of transport due to low-k TEM modes at mid-radius. The outer region is, however, unstable to ITG modes at the prevailing level of flow shear, which would result in anomalous ion heat transport.

The linear stability to ITG modes of discharge #22807 at 0.25s when the ITB is strongest has also been analysed using the global electrostatic code ORB5 [28, 11] and the local version of GYRO [29]at the most unstable location ( ρN = 0.52). The results of these calculations are consistent with the GS2 analysis.

In the event where low-k, ion scale micro-instabilities are present, global, non-linear

simulations are required to capture equilibrium variation over the large domain required for

the simulation. This has motivated modelling of

the ITG turbulence in this ITB discharge using

ORB5 [11]. Simplified, non-linear ITG

simulations without flow shear and with

adiabatic electrons, show that the turbulence

spreads inwards from the linearly unstable outer

region to the stable inner region. With the

experimental Ti gradient, the ion heat flux qi

saturates at just below the neo-classical level. As

shown in Fig. 6, modest increases in the gradient

Fig. 6 Ion heat flux qi from non-linear ORB5 calculations with adiabatic electrons for cases with R L at the experimental
Ti
value and increased by a factor 1.15 and 1.3. The ion heat flux from NCLASS and the linear flux are also shown.

result in considerably increased transport indicating that ITG turbulence may be clamping the Ti profiles, i.e. that the ion transport is stiff in the region outside the ITB. Global simulations with both sheared equilibrium flow and kinetic electrons are clearly required to capture fully the dynamics of the ion scale turbulence.

4. Summary
Internal transport barriers in the ion thermal and momentum channels form in MAST L-mode discharges in the vicinity of qmin . With co-NBI heating, the momentum ITB forms first in a region of negative magnetic shear in the core plasma. Some correlation is found with the strength of the ITB and the passing of qmin through rational values. The strength of the ITB is limited by coupling of MHD modes which reduce the rotation gradient and ultimately, an internal kink mode removes the core flow shear destroying the ITB when q0 approaches unity. Micro-stability analysis indicates that the negative magnetic shear core region is stable, and flow shear is not required to form the ITB there, while in the outer regions the flow shear is

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too weak to stabilize low-k ITG modes resulting in anomalous ion transport. At mid-radius shear flow is sufficient to partially stabilise low-k TEM modes, leading to incomplete turbulence suppression. With counter-NBI an ion thermal ITB tracks the location of qmin and later bifurcates, an outer ITB following the location of a low order rational surface. The rotation profile is broader than with co-NBI and the thermal transport close to the ion neoclassical level across most of the radius. Measurements of the low-k density turbulence in such plasmas using a BES imaging system newly installed on MAST [30] will in future allow direct comparison of results from global turbulence simulations with observations.
This work was funded by the RCUK Energy Programme under grant EP/G003955 and the European Communities under the contract of Association between EURATOM and CCFE. The views and opinions
expressed herein do not necessarily reflect those of the European Commission.

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