Session JI3: Rotation and Flows

Validation of Kinetic-Turbulent-Neoclassical Theory for Edge Intrinsic Rotation in DIII-D Plasmas

Recent experiments on DIII-D with low-torque neutral beam injection (NBI) have provided a validation of a new model of momentum generation in a wide range of conditions spanning L- and H-mode with direct ion and electron heating. A challenge in predicting the bulk rotation profile for ITER has been to capture the physics of momentum transport near the separatrix and steep gradient region. A recent theory has presented a model for edge momentum transport which predicts the \textit{value and direction} of the main-ion intrinsic velocity at the pedestal-top, generated by the passing orbits in the inhomogeneous turbulent field [T. Stoltzfus-Dueck, PRL 108, 065002 (2012)]. In this study, this model-predicted velocity is tested on DIII-D for a database of 44 low-torque NBI discharges comprised of \textit{both }L- and H-mode plasmas. For moderate NBI powers (P$_{\mathrm{NBI}}$\textless 4 MW), model prediction agrees well with the experiments for both L- and H-mode. At higher NBI power the experimental rotation is observed to saturate and even degrade compared to theory. TRANSP-NUBEAM simulations performed for the database show that for discharges with nominally balanced - but high powered - NBI, the net injected torque through the edge can exceed 1 N.m in the counter-current direction. The theory model has been extended to compute the rotation degradation from this counter-current NBI torque by solving a reduced momentum evolution equation for the edge and found the revised velocity prediction to be in agreement with experiment. Projecting to the ITER baseline scenario, this model predicts a value for the pedestal-top rotation ($\rho \sim $ 0.9) comparable to 4 kRad/s. Using the theory modeled - and now tested - velocity to predict the bulk plasma rotation opens up a path to more confidently projecting the confinement and stability in ITER.   

Characterization of the core poloidal flow at ASDEX Upgrade

An essential result from neoclassical (NC) theory is that the fluid poloidal rotation ($u_{\mathrm{pol}}$) of the main ions is strongly damped by magnetic pumping and, therefore, expected to be small ($<2\,\mathrm{km/s}$). Despite many previous investigations, the nature of the core $u_{\mathrm{pol}}$ remains an open question: studies at DIII-D show that at low collisionalities, $u_{\mathrm{pol}}$ is significantly higher in the plasma core than expected. At higher collisionalities, however, a rather good agreement between experiment and theory has been found at both DIII-D and TCV. This is qualitatively consistent with the edge results from both Alcator C-Mod and ASDEX Upgrade (AUG). At AUG thanks to an upgrade of the core charge exchange recombination spectroscopy (CXRS) diagnostics, the core $u_{\mathrm{pol}}$ can be evaluated through the inboard-outboard asymmetry of the toroidal rotation with an accuracy of $0.5$-$1\,\mathrm{km/s}$. This measurement also provides the missing ingredient to evaluate the core $\vec{E}\times\vec{B}$ velocity ($u_{\vec{E}\times\vec{B}}$) via the radial force balance equation. At AUG the core $u_{\mathrm{pol}}$ ($0.35<\rho_{\mathrm{tor}}<0.65$) is found to be ion-diamagnetic directed in contradiction to NC predictions. However, the edge rotation is always found to be electron-directed and in good quantitative agreement with NC codes. Additionally, the intrinsic rotation has been measured in Ohmic L-mode plasmas. From the observed data, it is clear that the gradient of the toroidal rotation is flat to slightly negative at the critical density defining the transition from the linear to the saturated Ohmic confinement regime. Furthermore, the non-neoclassical $u_{\mathrm{pol}}$ observed in these plasma leads to a good agreement between the $u_{\vec{E}\times\vec{B}}$ determined from CXRS and the perpendicular velocity measured from turbulence propagation. The difference between these two quantities is the turbulent phase velocity. The gathered dataset indicates that the transition in the turbulence regime occurs after the saturation of the energy confinement time.   

Measurements and modeling of viscosity in a stochastic magnetic field

Controlled perturbation of the momentum in MST RFP plasmas has allowed the first comprehensive test of a theoretical model [Finn et al., Phys. Fluids B (1992)], originally derived for the tokamak, for rotation damping in a stochastic plasma. Both a resonant magnetic perturbation (RMP) and an inserted biased probe were applied, separately, to a wide variety of spontaneously rotating Ohmic plasmas with a 10-fold span in normalized magnetic fluctuation amplitude, b/B. These control techniques provide measurements of the perpendicular kinematic viscosity, which is found to increase as (b/B)\textasciicircum 2 and which agrees well with predictions from the model. The dominant magnetic fluctuations in MST are linearly unstable m$=$1 tearing modes resonant at multiple locations in the core. The islands associated with these modes commonly overlap, producing stochasticity. The applied RMP also has m$=$1, causing deceleration of the co-rotating core plasma and m$=$1 modes. The biased probe initially spins up the core, but when bias is turned off, the core decelerates. The viscosity is derived from the deceleration curves in both cases and reaches 50 m\textasciicircum 2/s, roughly 100 times the classical prediction in the absence of stochasticity. Applying both techniques to the same plasma conditions provides a valuable cross check. The theoretical model, targeting the tokamak edge with an applied magnetic perturbation, is based on stochastic field line diffusion, which increases as (b/B)\textasciicircum 2 [Rosenbluth et al., Nucl. Fusion (1966)]. Rotation damping in the Finn model occurs as the local radial electric field is shorted out, and this damping can be characterized by an effective perpendicular viscosity. The results described here are relevant to any magnetically confined plasma, such as the tokamak and RFP, where rotation is important, and magnetic stochasticity is either intrinsic or externally imposed. Work supported by USDoE.   

Parasitic momentum flux in the tokamak core

Tokamak plasmas rotate spontaneously without applied torque. This intrinsic rotation is important for future low-torque devices such as ITER, since rotation stabilizes certain instabilities. In the mid-radius 'gradient region,' which reaches from the sawtooth inversion radius out to the pedestal top, intrinsic rotation profiles may be either flat or hollow, and can transition suddenly between these two states, an unexplained phenomenon referred to as rotation reversal. Theoretical efforts to explain the mid-radius rotation shear have largely focused on quasilinear models, in which the phase relationships of some selected instability result in a nondiffusive momentum flux ("residual stress"). In contrast, the present work demonstrates the existence of a robust, fully nonlinear symmetry-breaking momentum flux that follows from the free-energy flow in phase space and does not depend on any assumed linear eigenmode structure. The physical origin is an often-neglected portion of the radial ExB drift, which is shown to drive a symmetry-breaking outward flux of co-current momentum whenever free energy is transferred from the electrostatic potential to ion parallel flows [1]. The fully nonlinear derivation relies only on conservation properties and symmetry, thus retaining the important contribution of damped modes. The resulting rotation peaking is counter-current and scales as temperature over plasma current. As first demonstrated by Landau [2], this free-energy transfer (thus also the corresponding residual stress) becomes inactive when frequencies are much higher than the ion transit frequency, which allows sudden transitions between hollow and flat profiles. Simple estimates suggest that this mechanism may be consistent with experimental observations. [1] T. Stoltzfus-Dueck, Phys. Plasmas 24, 030702 (2017). [2] L. Landau, J. Phys. (U.S.S.R.) 10, 25 (1946).   

Coupling of Shear Flows in a Cylindrical Plasma Device

Spontaneous generation of parallel flows has been observed in a cylindrical plasma device. The mean parallel velocity shearing rate, $V_z^\prime$, increases as the radial gradient of the plasma density, $\nabla_r n_e$, exceeds a critical value. Correspondingly, when critical density gradient is exceeded, the parallel Reynolds power, $\mathcal{P}^{Re}_z = -\langle v_z \rangle \nabla_r \langle \tilde{v}_r \tilde{v}_z \rangle$, increases substantially, indicating the mean parallel flow gains more energy from ambient turbulence. Meanwhile, the shearing rate of the mean azimuthal flow, $V_\theta^\prime$, increases with the density gradient, but soon saturates when critical density gradient is exceeded. Also, the azimuthal Reynolds power, $ \mathcal{P}^{Re}_\theta = -\langle v_\theta \rangle \nabla_r \langle \tilde{v}_r \tilde{v}_\theta \rangle$, drops at higher density gradient, implying that the mean azimuthal flow gains less energy from ambient turbulence. These results suggest that the energy of azimuthal flows may be coupled to that of parallel flows through ambient turbulence. A 4-field model is employed to explain the coupling between the azimuthal flow and the parallel flow.   

Velocity Space Degrees of Freedom of Plasma Fluctuations

Small scale wave modes are becoming more important in plasma physics. Examples include turbulent cascades in the solar wind[1], the energetics of fusion plasma electrostatic turbulence and transport[2][3], and low temperature basic plasma physics experiments[4]. In order to improve our understanding of these modes, I present an advance in experimental plasma diagnostics and use it to show the first measurement of a plasma ion velocity-space cross-correlation matrix. From this matrix I determine the eigenmodes of fluctuations on the ion distribution function as a function of frequency. I also determine the relative strengths of these modes - these are the velocity space degrees of freedom of plasma fluctuations. This measurement can detect the aforementioned smaller scale modes in plasmas through a localized measurement. The locality of this measurement means that it may be applied to plasmas in which a single - point velocity sensitive diagnostic is available and multipoint measurements may be difficult. Examples include \emph{in situ} measurements of space plasmas, fusion plasmas, trapped plasmas, and laser cooled plasmas. This fact, combined with the new perspective it can give on small scale plasma fluctuations, means it may be used to further research on the above cited subjects. Much work remains on fully understanding this measurement. This measurement opens a velocity space interpretation of small scale plasma wave modes, and understanding this perspective from theory requires the application or invention of new mathematical tools. I discuss open problems to follow up on, which include questions from experimental, theoretical, and instrumentation perspectives. [1] Howes et al, Phys. Rev. Lett. 107, 035004, 2011. [2] Terry et al, Physics of Plasmas 13, 022307 (2006). [3] Hatch et al, Phys. Rev. Lett. 106, 115003, 2011. [4] Ng et al, Phys. Rev. Lett. 92, 065002, 2004.