Parasitic Momentum Flux in the Tokamak Core

Parasitic Momentum Flux in the Tokamak Core


A geometrical correction to the drift causes an outward flux of cocurrent momentum whenever electrostatic potential energy is transferred to ion parallel flows. The robust symmetry breaking follows from the free energy flow in phase space and does not depend on any assumed linear eigenmode structure, acting both for axisymmetric fluctuations (such as geodesic acoustic modes) as well as more general nonaxisymmetric fluctuations. The resulting rotation peaking is countercurrent and scales as electron temperature over plasma current. This peaking mechanism can only act when fluctuations are low-frequency enough to excite ion parallel flows, which may explain some recent experimental observations related to rotation reversals.

toroidal rotation, tokamak, transport, intrinsic rotation, rotation reversal
52.25.Dg, 52.25.Fi, 52.25.Xz, 52.30.Gz, 52.35.We, 52.55.Dy, 52.55.Fa

Tokamak plasmas without applied torque routinely rotate spontaneously in the toroidal (symmetry) direction, exhibiting nonzero, sheared toroidal rotation profiles. (deGrassie, 2009) This so-called “intrinsic” rotation is not only of fundamental interest: toroidal rotation helps suppress certain instabilities(Strait et al., 1995) and its shear may reduce turbulent heat transport.(Biglari, Diamond, and Terry, 1990) These advantages are important for future burning plasma devices such as ITER, in which the dominant -heating will not exert toroidal torque, unlike the neutral beam heating typical of present-day devices. (Doyle et al., 2007)

Although experimentally measured intrinsic rotation profiles are very diverse, many exhibit three distinct radial regions: an edge region with cocurrent rotation (toroidal rotation in the direction of the plasma current ), a mid-radius “gradient region” where rotation either becomes increasingly countercurrent with decreasing radius (countercurrent peaking) or stays relatively flat, and a flat or weakly cocurrent-peaked central region affected by sawtoothing.deGrassie et al. (2007); ?; ?; Stoltzfus-Dueck et al. (2015a); ?; Rice et al. (2011); Angioni et al. (2011); Sauter et al. (2010) Previous theoretical,Stoltzfus-Dueck (2012a); ? numerical,(Seo et al., 2014) and experimental(Stoltzfus-Dueck et al., 2015a, b) work suggests that the edge rotation is driven by the interaction of passing-ion drift orbit excursions with spatial variation of the turbulent fluctuations. The present work focuses on the “gradient region” at intermediate radius, where radial variation of plasma parameters is much slower, allowing other effects to compete with those of orbit excursions.

Over the last decade, intrinsic rotation at mid-radius has undergone intense theoretical and experimental investigation. Nonaxisymmetric magnetic fields can strongly affect the toroidal rotation.(Park et al., 2013) The present work will focus exclusively on the case of axisymmetric confining magnetic field, for which the conservation of toroidal angular momentumScott and Smirnov (2010); ? excludes the possibility of a self-generated torque. Intrinsic rotation must therefore result from a nondiffusive component to the momentum flux. Neoclassical (collisional) momentum fluxes are much too small to explain experimental observations, implying that turbulent transport is dominant.(deGrassie, 2009) A number of turbulent calculations suggest the presence of a momentum pinch, a component of momentum flux that is proportional to the toroidal rotation itself, rather than its gradient.Hahm et al. (2007); ? However, these models cannot explain the common observation of sheared velocity profiles passing through zero.(deGrassie et al., 2007; Eriksson et al., 2009; Sauter et al., 2010; Rice et al., 2011; Angioni et al., 2011; McDermott et al., 2011; Stoltzfus-Dueck et al., 2015b) Such measurements imply the presence of a “residual stress,” meaning a momentum flux contribution that is independent of both toroidal rotation and its radial gradient. For up-down symmetric geometries, often a good approximation for tokamak core plasmas, symmetry arguments restrict the leading-order momentum flux terms from driving residual stress.(Peeters et al., 2011) Theoretical work has accordingly focused on symmetry-breaking mechanisms(Peeters et al., 2011) such as shear,(Dominguez and Staebler, 1993) up-down-asymmetric geometry,(Camenen et al., 2009) and polarization effects.(McDevitt et al., 2009) Particularly challenging to theory are the experimental observations of rotation reversals in the “gradient region,” in which countercurrent rotation peaking suddenly flattens or switches to weak cocurrent peaking when plasma density or current cross threshold values.(Sauter et al., 2010; Angioni et al., 2011; Rice et al., 2011) The rapidity of these reversals suggests that the direction of peaking is not determined by neoclassical flows or other quantities that vary smoothly with plasma parameters, following instead from the properties of the turbulence itself, which may suddenly change character e.g. as an instability threshhold is crossed. In this letter, I identify a geometrical correction to the drift, neglected in all previous analytical work, that causes the free energy flows within the turbulence to drive a robust, fully nonlinear symmetry-breaking momentum flux. This flux causes counter-current core rotation peaking consistent with experimental measurements, and explains several observations related to rotation reversals.

To develop intuition, consider first a low-frequency axisymmetric density perturbation, as sketched in Fig. 1. At low frequencies and large scales, electron parallel force balance ensures that the nonzonal electrostatic potential is proportional to the nonzonal ion gyrocenter density . The pressure gradient and electric field then cause ions to flow out of the dense region along the magnetic field. The poloidal electric field also causes a radial drift that advects counter- (co-)current ion momentum inward (outward), regardless of the signs of and the toroidal magnetic field .

Key to this mechanism is a dual role for the weak electric field caused by the poloidal variation of the potential on length scales comparable to the minor radius . The nonvanishing parallel component of this electric field allows it to cause local ion acceleration, resulting in energy transfer between electrostatic potential () and parallel ion flow (). Because the background plasma gradients predominantly supply energy to even moments of the distribution function (such as density), while odd moments (such as ) are subjected to dissipation,(Scott, 2010; Stoltzfus-Dueck, ) steady-state energy balance often requires a net transfer of free energy from the potential (a function of even moments) to the ion parallel flows, causing a statistical symmetry breaking in the corresponding energy transfer term. Although toroidal angular momentum conservation does not allow the self-generated electric field to impart a net torque to the plasma, the weak radial drift due to the poloidally varying may transport toroidal angular momentum in the radial direction. The correlations between the ion parallel flows and the weak radial drift, resulting from the statistical symmetry breaking due to energy transfer, cause this part of the momentum flux to have a preferred sign, independent of plasma rotation and its radial gradient. In this letter, we will consider this residual stress in two separate cases: first a simpler special case with axisymmetric fluctuations, where the momentum flux occurs due to damping of geodesic acoustic modes (GAMs) via ion parallel flows, and later a more general case including nonaxisymmetric fluctuations, where the momentum flux can occur for any turbulent fluctuations in which energy transfer from potential to ion parallel flow is nonnegligible.

Both calculations use the simplest model capturing the relevant physics: the large-aspect-ratio limit of the electrostatic, isothermal gyrofluid equations in a radially thin geometry,(Scott, 2003, 2010) written in cgs units as


with species subscript meaning ions or electrons ; species charge state (-1 for electrons), mass , and (constant) temperature ; fluctuating and equilibrium species density (assuming ); drift ; gyroaveraged potential ; curvature operator capturing the magnetic drifts and divergence; parallel gradient ; parallel flow velocity ; and dissipation operator . The gyroaveraging operators and take the low- limits and , for the species gyroradius, with thermal speed and gyrofrequency . We take safety factor order unity, so the poloidal field and inverse aspect ratio are comparably small, , allowing explicit appearances of and to be replaced with representative constants and , and setting for magnetic and toroidal directions and . Since the toroidal component of is small in , Eqs. (1)–(3) conserve a simplified toroidal angular momentum involving only the zonal (flux-surface) average of , assuming :


with toroidal angular momentum density and flux , , with , , and radial (flux-surface) label . Eq. (4) shows that toroidal angular momentum is advected by the and magnetic drifts, without sources or sinks.

Figure 1: Poloidal cut of a low-frequency axisymmetric fluctuation, with axis of symmetry on the left. Darker shading shows larger , proportional to by the low- electron adiabatic response. Given enough time, ions flow out of the dense region along the magnetic field, causing counter-current (red) toroidal flow toward decreasing and co-current (blue) toward increasing . The poloidal variation of causes an flow that is inward for the countercurrent ion flux and outward for the cocurrent flux. Reversing the toroidal magnetic field switches the poloidal direction of counter- and cocurrent flow as well as the sign of the drift , leaving the momentum flux unchanged. The poloidal orientation of the density perturbation has no effect on the sign or magnitude of this momentum flux.

A nondiffusive momentum flux as in Fig. 1 may be driven by geodesic acoustic mode (GAM) damping, which we may treat in a shearless simple-circular geometry, for poloidal angle . Following Ref. Scott, 2005, we retain only one axisymmetric Fourier component from each of Eqs. (1)–(3), specifically , , and with . We neglect electron polarization and take low- gyroaveraging and quasineutrality , and electron adiabatic response , obtaining


in which and . We have taken for parallel flow damping rate . The , , and terms respectively capture the divergences of ion density, parallel/toroidal momentum, and /poloidal momentum fluxes due to unresolved Fourier components. The form follows from Eq. (3), with plus FLR corrections.(Scott and Smirnov, 2010) Linearizing Eqs. (5)–(7) and neglecting , , and yields a simple dispersion relation


For , Eq. (8) contains a pair of weakly damped high-frequency GAMs . However, for near 1, as is typical in tokamak core plasmas, the GAMs damp at a significant fraction of , as seen in more detailed kinetic calculations.Novakovskii et al. (1997); ?; ?

To evaluate and physically understand the resulting toroidal momentum flux, we examine the free energy balance for , , and :


in which , , and . Turbulence simulations show that the Reynolds stress () typically acts as a source for , the geodesic transfer term () moves free energy from to , and both parallel flow excitation () and the turbulent density flux sideband () move energy out of .(Miyato, Kishimoto, and Li, 2004; Scott, 2005) Note next that the electron adiabatic response combined with nonzero implies a radial drift . Recalling Eq. (4), this beats with to cause a contribution to the toroidal angular momentum flux . Since is directly proportional to the parallel flow excitation term in Eqs. (9) and (10), we conclude that energy transfer from the pressure sideband to the parallel ion flow necessarily implies an outflux of cocurrent toroidal angular momentum. Indeed, since the turbulent flux term will typically transfer energy out of , we may use the statistical average of Eq. (10), , to conservatively estimate the (signed) momentum flux as . Although simple ordering estimates suggest this flux may only drive toroidal ion thermal Mach numbers of order times the ratio of GAM kinetic energy over turbulent fluctuations’ kinetic energy, its fixed relation to the free-energy transfer guarantees robust symmetry breaking whenever there is strong GAM damping acting via ion parallel flows. Quantitative evaluation of its magnitude will require numerical simulation.

What is happening here physically? First, Reynolds stress excites a poloidal flow. Poloidal variation of causes a divergence in the velocity, resulting in up-down-asymmetric density fluctuations, like those sketched in Fig. 1. The resulting poloidal electric field (due to adiabatic electron response) and ion pressure gradient jointly excite an ion flow along . The net energy flow from Reynolds stress drive to damping via the ion parallel flow implies a positive correlation of the poloidal electric field and poloidal ion flow. The poloidal electric field also causes a weak radial drift. Due to the pitch of the magnetic field, the poloidal ion flow along the field corresponds to co- (counter-)current toroidal flow where the drift points radially outward (inward), which causes countercurrent rotation peaking.

Energy transfer from nonaxisymmetric potential fluctuations to ion parallel flows can drive an even stronger toroidal momentum flux, but in order to understand its origin we must first discuss the field-aligned magnetic coordinates used in most gyrokinetic formulations: Consider now an axisymmetric geometry with good nested flux surfaces, but otherwise arbitrary. Radial position is specified by a flux-surface label , which is axisymmetric and satisfies . Poloidal position is specified by a distended but axisymmetric poloidal angle label . The third coordinate is chosen so that , letting it label perpendicular position within the flux surface. These choices are not arbitrary: The definition of implies that so contains only slow variation. The use of an axisymmetric and implies that the partial is proportional to a simple toroidal derivative , since holding and fixed is equivalent to holding and vertical position fixed. This property has two important implications. First, appropriate choice of allows toroidal periodicity to imply simple periodicity in . Second, vanishes for any axisymmetric quantity, in particular for equilibrium plasma parameters and the magnetic geometry. These properties allow one to construct symmetry arguments that the dominant toroidal angular momentum flux, due to the portion of , must vanish in the statistical average for leading-order local gyrokinetic formulations with up-down symmetric magnetic geometry.(Peeters et al., 2011) In contrast, the portion of , neglected in all previous analytical works, is unrestricted by the symmetry arguments(Peeters et al., 2011) and indeed must break symmetry in the (common) case of net energy transfer from to ion parallel flows, as we will derive now.

We begin with the contribution of the higher-order part of the drift, in a simple, geometric way. Defining the radial and poloidal directions and , decompose . Since , the radial component of the drift is


The first term is the leading-order contribution, restricted by symmetry. The second term does not represent true parallel physics, it simply cancels the parallel gradient contribution that was included in the first term, leaving the true . Although nominally smaller than the first term by , it has symmetry-breaking properties, as we will identify in its contribution to :


For emphasis, does not represent any effect of parallel acceleration, it is simply the advection of the parallel portion of toroidal angular momentum by a small but robustly symmetry-breaking portion of the drift.

To understand the momentum flux caused by we must examine the free-energy balance for Eqs. (1)–(3), derived following Ref. Scott, 2010:


with fluctuating pressure and parallel flow free energies, energy including FLR corrections , and volume integral . Boundary terms have been assumed to vanish. The key point here is that the momentum flux term is directly proportional to the electrostatic acceleration of ion parallel flows, in Eq. (15). Although this term is sometimes referred to as ion Landau damping, it is in fact conservative, representing a transfer of energy from the potential to ion parallel flow . In cases with damping of turbulence via parallel ion flows, this term will tend to be positive, so that transports cocurrent momentum outward, corresponding to countercurrent rotation peaking,1 c. f. Fig. 2. This will especially occur when there are density fluctuations at low , due to low frequencies, electron adiabatic response and low- quasineutrality , which reduces . Since ion parallel flows are excited predominantly at low , we may use this with to estimate , in which for poloidal ion gyroradius , (signed) , and . Alternatively, if a fraction of turbulent free energy is dissipated via ion parallel flows, one may estimate the resulting volume-averaged momentum flux as , with pressure gradient scale lengths and turbulent radial electron and ion heat fluxes. Assuming comparable turbulent transport coefficients for heat and toroidal angular momentum, this corresponds to countercurrent velocity peaking with ion thermal Mach number of order . Although the isothermal model cannot distinguish between a particle and a heat flux, the energy balance for a six-moment non-isothermal gyrofluid model clearly shows that the necessary density fluctuations may be driven by an electron or ion heat flux, even in the absence of a particle flux.(Scott, 2010) Analogous manipulations in a gyrokinetic formulation also lead to the same result: energy transfer from to necessarily implies a corresponding exhaust of cocurrent momentum, with the same basic properties, magnitude, and scaling as derived here. (Stoltzfus-Dueck, )

Figure 2: Side view of a toroidally asymmetric low-frequency fluctuation, with darker shading again showing larger , proportional to by the electron adiabatic response. At frequencies , ions flow along out of the density hump. The symmetry-breaking portion of the radial drift, , again brings counter- (co-)current parallel momentum inwards (outwards). Although the drift due to (not shown) is now nonzero, its contribution to the momentum flux nearly vanishes by symmetry.

A few comments: is a residual stress, following from symmetry-breaking due to energy transfer from to , regardless of the background rotation profile. The symmetry breaking is statistical: it occurs simply because free energy flows through phase space from sources to sinks. In particular, is nonlinear, not quasilinear—it follows from the energy transfer term summed over all modes (including damped ones) and does not depend on the linear mode structure of any particular instability. However, it does require the presence of fluctuations (unstable or damped) at low enough frequency to excite ion parallel flows, . It survives in a radially local (fluxtube) limit, not requiring any radially global effects. Although results from a higher-order part of the drift (), which should have little direct impact on the leading-order turbulence, it is slaved to free energy fluxes that are determined by the leading-order physics. It can therefore be estimated even by simulations that neglect , simply by evaluating the relevant energy flux term a posteriori.

Although quantitative evaluation requires nonlinear simulation, we may qualitatively compare with experimental rotation observations. The general scaling for the countercurrent velocity peaking (roughly the toroidal velocity at the surface minus that at the pedestal top) is , which resembles Rice scaling () and has a magnitude comparable with experimental observations.(deGrassie et al., 2007; Eriksson et al., 2009; Rice et al., 2011; Angioni et al., 2011; McDermott et al., 2011; Stoltzfus-Dueck et al., 2015b) Also, in ASDEX-Upgrade (AUG), countercurrent momentum peaking has correlated strongly with density peaking across many discharge types.(Angioni et al., 2011) The relation may be more coincidental than causal: density peaking tends to occur due to electron precessional resonance for fluctuations with ,(Angioni et al., 2012) which (at core ) are the same modes that can excite ion parallel flows, thus driving countercurrent peaking. Interestingly, on Alcator C-mod, the presence of countercurrent peaking is correlated with the disappearance of broadband high- density fluctuations.(Rice et al., 2011) Viewed theoretically, dominant dissipation via low- ion parallel flows, which implies countercurrent rotation peaking in the present model, would also imply the reduction or elimination of a strong direct cascade of density fluctuations to high , consistent with C-mod measurements. Further qualitative and quantitative comparisons are needed.

In conclusion, a geometrical correction to the drift causes an outward flux of cocurrent momentum whenever electrostatic potential energy is transferred to ion parallel flows. The robust symmetry breaking follows from the free energy flow in phase space and does not depend on assumed linear eigenmode structure. The resulting rotation peaking is countercurrent and scales with . This peaking mechanism can only act when fluctuations are low-frequency enough to excite ion parallel flows, which may explain some recent experimental observations.(Angioni et al., 2011; Rice et al., 2011)

Helpful discussions with C. Angioni, G. Hammett, P. Helander, J. Krommes, and B. Scott, and funding by the Max-Planck/Princeton Center for Plasma Physics are gratefully acknowledged.


  1. For an atypical case with inverse ion Landau damping, meaning energy transfer from to , would reverse sign and cause cocurrent peaking.


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