Validation of the kinetic-turbulent-neoclassical theory for edge intrinsic rotation in DIII-D

Toroidal rotation is well documented as being beneficial to the performance of tokamaks and spherical tori. By contributing to the sheared radial electric field, rotation enhances the suppression of turbulent transport.¹ Another benefit of toroidal rotation is the stabilization of MHD instabilities such as resistive wall modes (RWM)^2,3 and neoclassical tearing modes (NTM).^4–6 In the absence of applied neutral beam torque, tokamak plasmas are observed to spontaneously rotate in the toroidal direction. This intrinsic rotation^7,8 is typically co-current in the edge and can reach magnitudes as high as fractions of the ion thermal speed v_ti.^9–11 In future devices, where there is little relative applied torque, rotation may be at intrinsic levels. Taking advantage of the positive impacts of this intrinsic rotation for optimizing the performance of future devices requires the understanding of its underlying physics.

A general behavior of the intrinsic rotation velocity (V_ζ) is that this velocity increases with the plasma stored energy and inversely with plasma current (⁠$Vζ∼Wp/Ip$⁠), which is known as the Rice scaling^12,13 in the magnetic fusion community. More recent observations show that edge intrinsic rotation might be independent of density and solely proportional to temperature or temperature gradients.^14,15

In recent theoretical work^16,17 the intrinsic rotation in the tokamak edge is explained using a simplified kinetic transport model. In that work, spontaneous generation of toroidal rotation results from the fact that co-current (co-I_p) and counter-current (ctr-I_p) passing ions interact differently with a spatially inhomogeneous turbulent viscosity. For a typical DIII-D configuration, with X-point on the high-field-side (HFS), ctr-I_p passing ions experience a stronger turbulent diffusion than co-I_p ions. As a result, ctr-I_p ions are depleted at a higher rate, resulting in a co-I_p intrinsic rotation of the edge plasma. In this model, momentum transport is assumed to be turbulence dominated (as supported by experimental evidence^18,19) and is characterized by a spatially dependent turbulent viscosity.

Previous comparisons of this theory to experiments have been successful for Ohmic L-mode.^20,21 However, comparison to the rotation at the pedestal top in H-mode plasma has not been done before. The H-mode is the planned operational state of ITER and future tokamaks, and therefore, it is essential to test the theory in this more relevant regime. In this work, we test the theory with both L-mode and H-mode data, for a wide range of control parameters such as electron cyclotron heating (ECH), beam heating, and plasma current. The study of intrinsic rotation in the H-mode on DIII-D showed that although intrinsic rotation showed a dependence on the pedestal pressure gradient, there was a mismatch between the plasma rotation and the Reynolds stress torque at the edge. This implies that additional mechanisms are necessary to explain the edge intrinsic rotation generation.²² 

References 16 and 17 make the following prediction for the flux-surface-averaged (FSA) toroidal velocity of main-ions at the boundary between the steep gradient edge and the core:

$R¯X$ is the normalized major radius of the X-point defined as

$R¯X$ varies from −1 for an X-point at the HFS of the last closed flux surface (LCFS), to +1 for an X-point on the low field side (LFS) of the LCFS, where R_out (R_in) is the major radius of outer (inner) separatrix at the midplane. The parameter d_c represents the poloidal variation of the turbulent viscosity, which ranges from −2 to +2 (see Refs. 20 and 21) q is a representative edge safety factor; here, we evaluate q at the pedestal top located by tanh-fit to the electron temperature profile, typically near ρ ∼ 0.9²³ (ρ is the square root of the normalized toroidal flux coordinate $ρ=ψN$⁠, where $ψN=ψ(r)/ψ(a)$⁠, a is the minor radius). B_T is the toroidal field for which we use the value on the magnetic axis; this choice is appropriate as a rough average of LFS and HFS for passing (circulating) ions which generate the intrinsic rotation in Eq. (1). L_ϕ is the e-folding decay length of the potential fluctuation intensity $⟨ϕ̃2⟩1/2$⁠. The predicted direction of the rotation is co-I_p for typical inboard X-point plasma. T_i is the ion temperature measured at the pedestal top. Using Eq. (1), the FSA toroidal angular velocity is given by

where R₀ is the major radius of the magnetic axis and is approximately the average major radial location of passing particles. As a short-hand, the plasma toroidal rotation angular velocity, which has units of rad/s, will be referred to as the plasma rotation.

A key prediction of Eq. (1) is the strong dependence of the rotation on the radial position of the X-point. This prediction was found to be in good agreement with a series of Ohmic L-mode discharges on TCV tokamak that spanned both positive and negative triangularity shapes.^20,21

The rest of this paper is organized as follows: In Sec. II, implementation of the model for DIII-D experimental data is discussed. In Sec. III, we present an overview of the results from the entire database, including not only the intrinsic rotation conditions, but also discharges with large neutral beam injection (NBI) torque, for which theory-experiment agreement is not expected. In Sec. IV, a detailed comparison of the theory prediction and the measured rotation is presented. We show that for the L-modes and H-modes at moderate β_N (i.e., $βN≲1.7$⁠), the theory prediction for edge intrinsic rotation has a reasonably good agreement with the experiments. In Sec. V, edge rotation degradation at high NBI powers is discussed and is explained by excess ctr-I_p torque deposited in the edge. In the final section (Sec. VI), we summarize our results and discuss our future works.

In this section, modification to the theory prediction, which is necessary for comparison with the experiments, is explained. In the second part (Sec. II B), the analytical method used to obtain the deuterium main-ion rotation velocities from carbon experimental rotation measurements is discussed. The inferred deuterium rotations are validated with direct main-ion charge-exchange recombination (CER) measurements in Sec. II C. Section II D provides the technical details on how the rotation database was assembled. In the last part, collisionalities of main-ions and impurities at the pedestal top are discussed.

In obtaining Eq. (1) in Refs. 16 and 17, the entire plasma population was assumed to be made up of passing particles. However, in DIII-D we find that at the pedestal top the main-ions are mostly in the banana regime. Therefore, considering the typical inverse aspect ratio ϵ ∼ 0.28 at the pedestal top in DIII-D, the majority of plasma population is trapped. To incorporate the effect of trapped particles, Eq. (1) is revised to

where the predicted rotation has been multiplied by the average fraction of passing particles at the pedestal top, (⁠$|pt$ symbol means that f_pass is evaluated at the pedestal-top); for a typical H-mode in the database at the pedestal-top $fpass≈30±2%$⁠.

A simple explanation for this follows the results in Ref. 17 which analytically estimates both a residual stress and a viscous momentum flux through the pedestal. Based on Ref. 17, in the absence of an applied torque, the intrinsic velocity is determined by the balance between the net flux due to residual stress and viscous diffusion

Here, the integrations are over the flux surface at the core-edge boundary, v_rig is the intrinsic rigid rotation velocity in the absence of a momentum source, and L_ped is the pedestal width. The term on the left hand side (LHS) of Eq. (5) is the net viscous diffusion flux, and $∫Πres dA$ is the net momentum flux due to residual stress. The viscous momentum flux results from radially outward diffusion of both trapped and passing particles. With intrinsic rotation typically increasing in the positive (co-I_p) direction, this term will have a positive sign. In contrast, the residual stress flux (i.e., $∫Πres dA$⁠) results from the deviation of particles' toroidal velocity away from rigid rotation, which is only significant for passing particles. Moreover, since the residual stress flux preferentially depletes (ctr-I_p) passing ions for a typical tokamak configuration, it has a negative sign. Therefore, the viscous momentum flux is proportional to the total plasma population (⁠$∫vrigχϕnmi/Lped dA∝Ntot$⁠), whereas the residual stress flux is proportional to the passing population only (⁠$∫Πres dA∝Npass$⁠). Solving Eq. (5) for the intrinsic rotation velocity gives

which explains the presence of this factor in Eq. (4).

The direct rotation velocity measurements used in this study are from charge-exchange recombination (CER) spectroscopy for carbon 6+ impurity ions only. In DIII-D, the deuterium main-ion CER diagnostics has recently started making measurements of the main-ions in the edge and data analysis is underway. Currently, a very limited amount of the direct main-ion data has been processed and analyzed.^24–31 Therefore, in this work main-ion rotation velocities are indirectly obtained from the carbon velocities using the analytical calculation of the neoclassical offset between the main-ions and impurities following the model by Kim, Diamond, and Groebner (KDG).³² In Ref. 29, toroidal rotation velocities calculated by this model showed reasonable agreement in the plasma core (⁠$ρ<0.75$⁠) with other more sophisticated neoclassical models, NCLASS, NEO, and GTC-NEO. KDG also has the advantage that it can be easily evaluated, as it is analytical and does not require numerical simulations, which are time consuming and inconvenient for applying to a large database of measurements. Toroidal angular velocity of species a can be expressed as (see the  Appendix)

where $Vθa$ is the poloidal rotation velocity, B_θ is the poloidal component of the magnetic field, and I = RB_ζ, where B_ζ is the magnetic field in the toroidal direction. The first term is the θ-independent, flux-surface-averaged, rigid rotation part of the toroidal velocity; the second term is poloidally asymmetric. Although the latter is small in the core region, near the edge its value can be non-negligible and must be treated with caution. The poloidally asymmetric part of the impurity rotation can often manifest as a localized dip in the edge velocity, near the steep pressure gradient in the outboard midplane.^33–35 Note that the derivation leading to Eq. (7) is valid when density is a flux function. In the case of in-out asymmetry in the impurity density, as measured C-Mod³⁶ and AUG,^35,37 correction must be made to Eq. (7).

The difference between the FSA toroidal rotation for main-ions (⁠$⟨Ωζ,D⟩$⁠) and the impurities (⁠$⟨Ωζ,C⟩$⁠) is given by the neoclassical offset [see Eq. (A23)]

$ρip=micvTi/ZieBp$ and K₂ is a function of viscous matrix elements and impurity density and temperature. Obtaining Eq. (8) in Ref. 32 was based on the assumption that the impurity viscous force is negligible compared to main-ion—impurity friction force. Although at the pedestal top the impurity density is not small, we find that impurity ions are predominantly in the Pfirsch-Schluter regime. Hence, the physics based assumption required for the validity for the use of Eq. (8) is satisfied.³² Hereafter, TSD refers to the theory prediction given by Eq. (4) (in reference to the author of Ref. 16) and KDG refers to the inferred experimental main-ion rotations obtained from Eq. (8).

The uncertainty in either the theoretical rotation (TSD) or inferred experimental rotation (KDG) results from the uncertainties of the fits; these are calculated by a Monte Carlo program using the random error of individual measurement points. For the theoretical prediction of Eq. (1), we find the average uncertainty of ∼1.3 krad/s (5%–10%). For the inferred experimental rotations, the uncertainty is on average ∼0.5 krad/s (2%–5%).

In this investigation, we use a unique DIII-D capability of directly measuring the main-ion (deuterium) toroidal velocity on the outboard midplane.^24,27 This capability has been used to compare main-ion and impurity toroidal rotation to neoclassical theory²⁶ in the deep core and shown that the agreement with neoclassical predictions has a collisionality dependence²⁹ where the strongest deviations from neoclassical appear at $ν*≲0.05$⁠, consistent with related studies of poloidal flow³⁸ at low collisionality. More recently, the diagnostic capability has been expanded to the plasma edge with first measurements from a four-channel diagnostic system reported in Ref. 39 and full 16-channel capability reported in Refs. 30 and 31.

An example of the impurity and main-ion rotation evolution during a representative discharge in this investigation is presented in Fig. 1(a), showing that the deuterium is rotating faster in the co-I_p direction than the dominant impurity carbon. The main-ion rotations inferred from impurity measurements using the analytical neoclassical method³² (KDG) is shown in Fig. 1(a) as well. The agreement between KDG rotation and the experimental main-ion rotations can be seen in Fig. 1(b), for which KDG rotations have been calculated for the main-ion CER measurement times. The measured main-ion rotation is within reasonable expectations from analytical neoclassical theory. Hence, the KDG method is exploited for producing a database of thousands of time-slices, because only a limited set of discharges have the direct main-ion measurements available.

A database of main-ion rotations at the pedestal top is created from nominally low injected torque DIII-D discharges to compare with the theory prediction given by Eq. (4). This database includes 43 discharges varying in pedestal density (⁠$ne∈[3e18, 6.8e19] m−3, nC∈[1.5e17, 5.7e18] m−3$⁠), temperature (⁠$Te∈[0.6,1.2]$ keV, $TC∈[0.15,2.1]$ keV, ECH power (maximum 3.6 MW), NBI power (maximum 7 MW), and positive and negative plasma current [I_p = ±1 MA except for the ITER baseline scenario (IBS) discharge for which I_p = 1.26 MA]. The database only includes discharges and time periods for which the magnitude of the time-averaged injected torque was less than 0.5 Nm. However as will be discussed in Sec. V, the torque applied to the edge plasma can be significantly larger in certain cases with conditions included in the database. All of the considered discharges are either upper or lower single null. Magnetic field is in the negative direction, which for DIII-D is in the clockwise direction when the torus is being looked at from above.

MHD instabilities such as NTMs and RWMs can lead to the slow collapse of density and rotation and eventual loss of confinement. Moreover, they create drag on the rotation and result in slowing the rotation down. This complex behavior and extra rotation sink are not addressed by the model. Therefore, any measurements which coincide with such MHD activities have been eliminated from the database. Following the same logic, all the measurements acquired during the edge-localized modes (ELM) crash events, and also during the L- to H-mode transition have also been removed from the database.

Evaluation of the theoretical rotation prediction [Eq. (4)] and also the neoclassical offset between main-ion and impurity [Eq. (8)] requires knowledge of the local values of the physical quantities at the pedestal top. For individual time-slices, these quantities are normally obtained by detailed profile fitting. However, due to the relatively large number of discharges in the database, automatic evaluation of the desired quantities is more practical than fitting the profiles for each discharge individually: the computer program goes discharge by discharge through the database, fits the desired profiles, and evaluates derived quantities. The pedestal top location is given by Tanh fit to electron temperature readily available and stored in MDSplus database. Moreover, further expansion of this database for use in future works is much more simplified and streamlined if it is automated. Within the OMFIT⁴⁰ framework, the database for pedestal top rotation velocities is built by automatically obtaining the raw experimental data, mapping to the CER measurement times, fitting curves to the edge profiles of the needed physical quantities (here only T_i and impurity poloidal velocity V_θ), and calculating the desired quantities. This automatic method, although it sacrifices some accuracy in fitting to profiles, is much faster for application to a large database of discharges.

Fluctuation measurements were not available in the edge region for the considered discharges. However, since the typical radial variation of $eϕ̃/Te$ is weaker than that of T_e itself,^41–45 the measured electron temperature gradient length $LTe$ is used as a substitute for L_ϕ. Previous reports of $Lϕ/LTe$ vary in the range from 0.5 to around 4, with most typical values in the range of 1 to 2.^41–49 A closer evaluation of L_ϕ could refine the predictions and can make for interesting future work. Here, we use a simple linear estimate to evaluate $LTe$ using the following prescription:

where “pt” and “sep,” respectively, refer to the pedestal top and separatrix radial location on the outboard midplane and R is their major radius. Electron density and temperature are obtained from the high spatial resolution Thomson scattering diagnostic, with a 230 Hz (edge measurements) repetition frequency. The pedestal top location (R_pt) is identified using the tanh-fit to the electron temperature pedestal, measured through Thomson scattering.²³ The pedestal width is taken to be twice the width of tanh-fit to T_e. Thomson scattering measurements are arranged in the vertical (z direction), which is not perpendicular to the flux surfaces. The obtained pedestal width is different from the pedestal width on the outboard midplane, due to the divergence of the flux surfaces, and the fact the measurement direction with the plasma surface is not perpendicular. Therefore, the pedestal width is mapped to the outboard midplane using the equilibrium reconstruction code EFIT;⁵⁰ then R_pt, the major radius at the pedestal top at the outboard midplane, is defined as

where $Δped$ is the pedestal width on the outboard midplane. Moreover, the radial location of the pedestal top is taken from the profile fit to the electron temperature.

The predicted rotation in Eq. (4) is for flux-surface-averaged deuterium rotation. However, the CER data used here are for carbon measurements only. We make the simplifying assumption that ions are thermalized at the pedestal top T_D = T_C which allows us to use the carbon temperature measurements for D⁺. The edge tangential CER chords (in the radial range $0.75≲ρ≲1$⁠) are used to find T_i, the ion temperature at the pedestal top. The temperature gradient length scale defined as $LTi−1=−d log Ti/dr$ is also evaluated on the outboard midplane using the temperature measurements made using these CER channels by fitting a modified (slanted) tanh to these temperature measurements. $LTi$ is used in the calculation of the neoclassical offset between Ω_D and Ω_C. Making the assumption that carbon is the dominant impurity, deuterium density is derived from the quasi-neutrality relation $nD=ne−6nC$⁠.

As seen in Eq. (7), poloidal velocities of impurity (⁠$VθC$⁠) are needed to extract the FSA rotation from the locally measured impurity toroidal rotations. Using the OMFIT⁴⁰ toolset, these poloidal velocities are obtained from the measurements made by the vertical CER diagnostics.

In a series of experiments, the DIII-D tokamak has been testing the practicality of ITER operation by producing the ITER shape scaled to DIII-D dimensions and operating with similar normalized current (I_N) and normalized β (β_N) as the ITER baseline scenario (IBS).⁵¹ IBS will use an inductively driven plasma current of 15MA which will access the H-mode that exhibits edge-localized modes (ELM).⁵² Here, $IN=Ip/aBT$ and $βN=β/(Ip/aBT)$⁠, where β is the volume-average ratio of the plasma pressure to the magnetic pressure, I_p is the plasma current (in MA), a is the minor radius (in m), and B_T the toroidal magnetic field (in T). The X-point location is on the HFS for all the present data regardless of the location of the single null; the value of the normalized X-point radius is in the close vicinity of $R¯X∼−0.72(±0.03)$ for the entire database, except for the discharge for which R_X ∼ –0.81 (discharge 164 988). This is a DIII-D IBS discharge that has been included in the database for comparison and has some physical parameters which are different from the rest of the database by design: higher plasma current, thus lower q at the edge, and higher density.

Close agreement between the rotations calculated automatically for the database to accurate (manual) profile fits using the OMFIT⁴⁰ toolset is shown in Fig. 2 for a discharge in the database. The latter profiles have been fitted to the experimental data over the whole radial range. In this discharge, the plasma current is in the positive direction (I_p > 0, positive direction is counter clockwise as looked from above), and the plasma configuration is upper single null (USN). The comparison is for the time between 2550 and 5500 (ms) for which the plasma is in the H-mode.

Prediction of the TSD model depends on the fraction of passing population, which in turn depends on the collisionality regime of main-ions. Moreover, the physics of the neoclassical rotation offset between the main-ion deuterium and impurity carbon depends on the collisionality of each species. Collisionality of the species a is defined as

where $νa/ε$ is the effective collision frequency, $vTa=2Ta/ma$ is the thermal velocity of a, and $Ωb=εvTa/qR$ is a lower bound approximation for the trapped particle bounce frequency. For ν^a in Eq. (11), collisions with all the present species including self-collisions are taken into consideration

where $νa/b$ is a basic collision rate of a test particle of type a in a background of type b particles⁵³ 

For a small inverse aspect ratio, the banana regime (plateau regime) is defined as $ν*≪1$(⁠$ν*≫1$⁠). In the banana regime, all ions—trapped and passing—are essentially collisionless. In the plateau regime, the majority of passing particles are collisionless, whereas trapped orbits cannot be completed due to interruption by collisions. In this DIII-D database, the inverse aspect ratio at the pedestal top is $ε≈0.28$ which is used for calculating $ν*$⁠. For this relatively large inverse aspect ratio, in terms of the neoclassical transport regimes, the distinction between the banana regime and the plateau regime may be less meaningful.

The discharges considered for the database have carbon as a dominant impurity. At the pedestal top, ion-impurity collisions are more frequent than ion-ion collisions $(νi/I∼3νi/i)$⁠. This is due to the fact that carbon impurities (C6+) are highly charged and have relatively high densities at the pedestal top $(Zeff≈2.5±0.7)$⁠.

In evaluating Eq. (11), for the main-ion collision frequency νⁱ collisions with the dominant impurity (C6+) and with main-ions themselves are taken into account. From Eq. (11), the main-ion collisionality is $ν*≈0.43±0.3$ (average and standard deviation). Hence, at the pedestal top main-ions are in the banana-plateau regime, for which the passing particles are collisionless. Use of the term banana-plateau for the collisionality here is due to the fact that making a distinction between banana or plateau regime with the relatively large aspect ratio of DIII-D may not be meaningful.

The fraction of passing particles is given by

The above equation can be expanded and approximated as³² 

The distribution of f_pass at the pedestal top for this DIII-D database, evaluated using Eq. (15), peaks at f_pass ∼ 0.29. As a result according to Eq. (4), the original prediction in Ref. 16 is diluted by a fraction of 0.29.

Highly charged impurities are more collisional than the main-ions. For impurities, the ratio of their collision frequency ν^I and the transit time, $Ωt=vTI/qR$⁠, gives a measure of collisionality. Impurities with $νI/Ωt≳1$ are expected to be in the high-collisionality or Pfirsch-Schluter regime. For ν,^I we use Eq. (12) and include impurity-ion and impurity-impurity collisions. For carbon impurities in this database, at the pedestal top $νI/Ωt∼1.9±1.1$⁠. Therefore, at the pedestal top, for the discharges included in the database, impurity is predominantly in the Pfirsch-Schluter regime.

In this section, we provide an overview for the entire database, including both intrinsic rotation discharges and also discharges with finite edge torque. Dependencies on a number of physical parameters of the system are examined, such as β_N, heating, plasma current, geometry, and density. Experimental measurements are compared with the theoretical prediction [Eq. (4)], which assumes that the injected torque is small enough for the experimental main-ion rotation to be considered “intrinsic.” However, in Sec. V we show that for the balanced beam configuration used at high beam power (P_NBI ∼ 6.5 MW) the torque absorbed in plasma edge can exceed 1 Nm in the ctr-I_p direction. This large excess torque can significantly reduce the measured rotation velocity from its intrinsic value. For data under these finite-torque conditions, all of which are included in Figs. 3–7, 10(b), and 11(b), the assumptions underlying Eq. (4) do not hold, so the intrinsic rotation theory-experiment agreement is not expected.

Comparison between the inferred deuterium experimental rotation in Fig. 3(a) with the carbon rotation in Fig. 3(c) shows that the former is more co-I_p than the latter in all β_N ranges. Even though some of the measured carbon rotation data for β_N > 1.7 are ctr-I_p, the inferred main-ion rotation in Fig. 3(a) for the same β_N range is co-I_p. These measured rotations have formed three populated clusters in low, medium, and high β_N ranges. The vertical dashed lines drawn in this figure [and also in Fig. 3(b)] do not represent any physical limits and are used to help separate the approximate β_N ranges of the data clusters. The cluster of data for the low β_N range (⁠$βN≲0.75$⁠) mostly consists of measurements of the plasma in L-mode. Transition from the L-mode to H-mode, for $0.75≲βN≲1.7$⁠, results in an increase in the value of Ω. For $βN≳1.7$⁠, average rotation velocity is lower than for medium range β_N data and is comparable in value to L-mode data.

Figure 3(b) shows the theoretical value for rotation versus β_N, obtained from Eq. (4), with a fixed ballooning parameter d_c = 1 for all data. The theoretical prediction for rotation monotonically increases over the entire β_N range. For high β_N (i.e., $βN≳1.7$⁠), the monotonic increase in the predicted rotation in Fig. 3(b) disagrees with the measured main-ion rotation in Fig. 3(a), for which the rotation decreases. This disagreement is to be expected because, as discussed in Sec. V, most of the discharges with β_N > 1.7 have strong effective ctr-I_p torque in the edge, violating the assumptions underlying Eq. (4)

As β_N crosses over to values larger than ∼0.75 and the L-to-H transition takes place, the theory predicted rotation increases in the co-I_p direction, similar to the experimental result in Fig. 3(a). This increase in the theory value with β_N, calculated from Eq. (4), comes from the decrease in the value of the electron temperature scale length L_Te [Fig. 4(a)] and increase in the ion temperature at the pedestal top T_i [Fig. 4(b)]. L_Te calculated from Eq. (9), and the measured pedestal top T_i are shown, respectively, in Figs. 4(a) and 4(b). To capture the trend of the data shown in these scatter-plots, mean of the y variable (vertical axis) is calculated for the data in equal finite spans of the x variable (horizontal axis), and shown in diamond shapes. Also, the standard deviation of the variation of y is shown with a vertical uncertainty bar. This mean-standard deviation calculation has also been performed for other scatter-plots in this paper. T_i linearly increases with β_N as the result of the L-to-H transition. However, L_Te drops (nonlinearly) from low to medium β_N and does not change much as β_N further increases. Thus, the theory prediction exhibits a jump in value at β_N ∼ 0.75 and a linear increase in value for β_N > 0.75.

In DIII-D, ECH by itself is not enough to reach high β_N values. Therefore, balanced-NBI is applied to augment or replace ECH for raising β_N to ITER values (i.e., β_N > 1.8). The dependence of the measured main-ion rotation on the injected neutral beam power (P_NBI) and electron cyclotron heating power (P_ECH) is shown in Figs. 5(a) and 5(c). P_ECH does not increase beyond ∼3 MW and is not responsible for reaching high β_N. Therefore, as expected, there is no correlation between the P_ECH and main-ion rotation degradation at high β_N.

In this database, the magnitude of P_NBI, and as a result the value of β_N, does not increase continuously [see Fig. 5(d)]. Instead, for H-modes the value of β_N is elevated from moderate values of average β_N ∼ 1.3 to high values of average β_N ∼ 2, as the P_NBI is raised from moderate values of average $PNBI∼3.5$ to high values of average P_NBI ∼ 6.5. Figure 5(c) shows that the high P_NBI data cluster (yellow to bright green), which is centered at $βN∼2$⁠, has lower rotation than the moderate P_NBI data cluster (dark green) centered at $βN∼1.3$⁠. The value of NBI power separating the low and high range is $PNBI∼4$ MW. This separation can also be seen in Fig. 5(d) showing the rotation dependence on NBI heating; H-modes with higher NBI heating (⁠$PNBI∼6.5$ MW) , which mostly are in $βN>1.7$(red), have average lower rotations than H-modes with $PNBI<4$ MW, which are in the medium β_N range (⁠$0.75<βN<1.7$ cyan).

Observation of this degradation of the measured toroidal rotation for D⁺ and C⁶⁺ at high β_N is consistent with the previous report in DIII-D in Ref. 9 for He²⁺. There, it was shown that the measured rotation velocity falls below Rice scaling, when the balanced NBI was employed to achieve high β_N. As discussed in Sec. V, much of the rotation decrease can be attributed to a strong effective ctr-I_p edge torque resulting from increased NBI prompt losses and change in the NBI duty cycle.

The data in Fig. 6 do not show a significant difference in this database between the measured main-ion rotation velocities for the lower single null (LSN), represented by blue points, versus the upper single null (USN) represented by red points. Note that the former (LSN) is the favorable $∇B$ geometry for standard DIII-D setting (⁠$Bζ<0$⁠) which was used for all discharges included in the database. This is in contrast with the previous report of an upshift in rotation from USN (unfavorable) to LSN (favorable.) in the L-mode.^21,54 This observation quantitatively agrees with the theoretical prediction, which is the same for USN and LSN geometry.

The dependence of measured rotation velocities on the density of main-ions at the pedestal top is shown in Fig. 7. Within each β_N range, the rotation is roughly independent of density, in qualitative agreement with the theoretical prediction Eq. (4). For the $βN<0.75$ range (blue), density varies from $n∼3e18 m−3$ to $n∼9e18 m−3$⁠. For $0.75<βN≲1.7$ (cyan), due to L-H transition rotation increases from 6 to 20 krad/s. The density of the H-modes in $0.75<βN≲1.7$ varies over a wide range of $n∼0.9e19 m−3$ to $n∼2.2e19 m−3$⁠; however, the average rotation does not change with density. For $βN>1.7$ (red), the density varies over the range $n∼0.9e19 m−3$ to $n∼2.2e19 m−3$⁠), with its average velocity 15 krad/s. IBS discharge and has higher density compared to the rest of the database and has not been included in this figure (respectively, $n∼3.6e19 m−3$⁠).

Theory prediction for main-ion rotation is compared to the measured rotation for different ranges of NBI power P_NBI: $PNBI<4$ MW in Fig. 8(a), and $PNBI>4$ MW in Fig. 8(b). The ballooning parameter in Eq. (4) has been chosen to be d_c = 1 for all the rotations calculated, which corresponds to a modest outboard ballooning. Using a universal d_c for rotations is strictly for convenience and as we discuss later, this parameter may be different for the L-mode and H-mode. There is general agreement between theory and the measured rotations for $PNBI<4$ MW [Fig. 8(a)]. However, there is a clear gap between theory and experiments for $PNBI>4$ MW, with theory being roughly 3 times larger than the experiment. These high NBI power data are discussed next, where we show that for $PNBI>4$ MW the NBI torque deposited in the edge is ctr-I_p and large (⁠$∼−1$ Nm). For $PNBI<4$ MW, we will show that although the deposited ctr-I_p torque in the edge is smaller, there is still a resulting finite ctr-I_p rotation shift.

An outlier among the $PNBI>4$ MW data is the IBS discharge [inverted triangles in Fig. 8(b)], for which the theory prediction is closer to the experimental value: $ΩTSD≈1.3 ΩKDG$ in the H-mode. For this discharge, high β_N of $≈2.1$ is achieved by employing balanced NBI. As shown in Sec. V, the effective edge torque and torque-driven rotation shift for discharge are smaller than for other high-β_N discharges. One reason for this is the high plasma current (for the IBS discharge I_p = 1.26, for the rest of the database I_p = ±1), which reduces prompt beam-ion losses and the associated edge torque. Also, unlike most other high-β_N discharges, IBS discharge has positive plasma current, which results in a smaller effective ctr-I_p edge torque (see next paragraph). For both of these reasons, the IBS discharge has a relatively small torque-driven rotation offset, comparable to that at moderate β_N, explaining why it also shows comparably good agreement with Eq. (3), which neglects applied torque.

In DIII-D, the edge CER viewing chords intersect with the NBI line which is directed in the positive toroidal direction (i.e., the 330 beam), and the timing of the CER measurement is synchronized with this beam. Therefore, the edge CER measurements of toroidal rotation are particularly sensitive to the torque exerted by this beam (see Sec. V). Therefore, if the plasma current is in the standard positive direction, the beam blips spin up the plasma in the co-I_p direction, which to an extent cancel some of the ctr-I_p prompt-loss torque. For negative-current discharges, this effect further adds a ctr-I_p torque to the prompt loss.

For $PNBI<4 MW$⁠, theory prediction is compared to experimental main-ion rotation velocity, in Fig. 9(a). Although there is a general agreement between theory and experiment, the choice of a constant d_c = 1 results in under-predicting the rotation velocity for $βN<0.75$ (L-mode, yellow) and over-predicting it for $βN>0.75$ (H-mode, blue). The ballooning parameter d_c currently cannot be measured and can be considered as a free parameter in the theory. The available free parameter of the theory d_c can be used to move the theory prediction closer to the experiments for the L-modes and the H-modes separately. This can be carried out by finding an optimal d_c which gives the least square of difference between theory prediction and the experimental measurement. For $βN≲0.75$ (L-modes), the optimal value is $dc=1.66$⁠, which corresponds to a moderately strong outboard ballooning [see Fig. 9(b)]. This value is consistent with the similarly calculated d_c values for the Ohmic L-modes in the TCV tokamak.²¹ 

For the moderate β_N H-modes in Fig. 9(a) (blue), we will show in Sec. V that finite ctr-I_p torque in the edge results in a ctr-I_p shift in rotation, which reduces the predicted rotation, from intrinsic rotation given by Eq. (4) to values closer to the experiment. Therefore, without taking the finite edge torque into consideration fitting for an optimum d_c will not give a significant result. In fact, for the moderate β_N H-mode, after taking the finite torque into consideration, keeping the initial choice of $dc∼1$ results in reasonably good agreement with experiments.

Current dependence of rotations is examined in Fig. 10, which compares the theory to experiment with the color of data points representing different plasma current values. Figure 10(a) shows general agreement for $PNBI<4$ MW [as was depicted in Fig. 9(a)], regardless of plasma current. Figure 10(b) shows theory-experiment discrepancy, as expected due to strong effective edge torque violating the low-torque assumption. However, the torque-driven rotation offset is reduced at larger plasma current and for positive plasma current (see Sec. V), resulting in better theory-experiment agreement for those cases. The IBS discharge [red “+” in in Fig. 10(b)] has the highest plasma current (⁠$Ip≈1.26$ MA) and is positive I_p. Discharge 157 910 [light red squares in Fig. 10(b)] is the only other positive-I_p with $βN>1.7$ and also shows a reduced discrepancy between rotation and (torque-neglecting) theory prediction.

Although, as mentioned earlier, the ELM crash measurements have been excluded from the database, measured rotation in H-mode during the ELM recovery phase may be reduced by the presence of ELMs, which are neglected entirely by the theory. In this database, the inter-ELM time period varies between 7 to 45 ms; discharges employing ECH have more frequent ELM occurrence. ELMs have been experimentally observed to significantly break edge toroidal rotation on DIII-D⁵⁵ and JET.^56,57 However, the region with large rotation loss due to ELMs is between the steep-gradient-region to the LCFS. In contrast, it has been shown that the effect of ELMs on the rotation at the pedestal top is relatively smaller.⁵⁸ Detailed investigation of the effects of ELMs on edge rotation in DIII-D will be presented in a future article. In Sec. V we show that taking the excess NBI torque into account can, to an extent, correct the over-predicted intrinsic rotation in the medium β_N range [see Fig. 16(a)].

In this section we show that during the time periods of beam pulses, the torque absorbed by plasma in the edge region is typically prompt and ctr-I_p, resulting in a torque-driven ctr-I_p offset of the edge rotation velocity. For $PNBI>4 MW$ the net injected torque can exceed 1 Nm in the ctr-I_p direction, which can result in a significant degradation of the edge rotation at $βN>1.7$⁠.

As was shown in Fig. 3(a), the experimental value for main-ion rotation is degraded in the high β_N range (⁠$βN>1.7$⁠). Moreover, it was shown in Fig. 5(c) that this high β_N range is reached when the applied NBI power is large (⁠$PNBI∼6.5 MW$⁠). Although the time averaged toroidal torque emitted by NBI sources is approximately balanced, prompt-losses cause a net ctr-I_p torque with a strongly radially dependent profile.

Newly ionized trapped fast ions generated by a co-I_p directed NBI will move radially inward, depositing their co-I_p momentum in the core region. The fast ions generated by a ctr-I_p NBI move radially outward and deposit their ctr-I_p momentum in the edge region. Therefore, when both direction of NBI are applied the radial torque profile is typically co-I_p in the core region and ctr-I_p in the edge region. Further, most of ctr-I_p fast ions cross the separatrix and are effectively lost due to charge exchange or hitting the vessel wall. These ions deliver a momentum to the plasma that is larger than the initial momentum leaving the beam. The effect of this prompt-loss is that although the calculated torque leaving the beam sources might be balanced, the total torque absorbed by the plasma is net ctr-I_p.⁵⁹ 

For implementing balanced beams for a low torque discharge, the beams are deployed such that the net torque from the beam lines has a periodic time variation as shown in Fig. 11. During one neutral beam duty cycle the net injected torque is at times co- or counter-current. As an example, in the duty cycle between t = 2750–2786 for discharge 157 906 (blue time trace), initially the net torque is positive with maximum torque $≈$2 Nm. Since the current is in the negative direction, this torque is deposited mostly in the edge region. In the remaining time of the duty cycle, the net torque is in the negative (counter-current) direction with maximum $≈$−2 Nm. However, on average over a duty cycle, the net injected torque is much smaller.

Volume integrated profiles of the absorbed torque by the plasma, calculated by NUBEAM are shown in Fig. 12, for the same two DIII-D discharges of Fig. 11, discharge 159 386 which to simplify we call G (for green), and 157 906 which we call B (for blue); the former is in the medium β_N range (⁠$βN≈$ 1.1) and the latter in high β_N range (⁠$βN≈2$⁠). In both discharges, plasma current is in the negative toroidal direction. For discharge G at the pedestal top $ne=2e19$ m⁻³, $Te=0.6$ keV, $Ti=9.8$ keV, $νi*=0.4$⁠; for discharge B: $ne=2.8e19$ m⁻³, $Te=0.6$ keV, $Ti=1.1$ keV, $νi*=0.5$⁠. Solid curves show the time averaged torque and the colored area around this curve represents the spread of the calculated absorbed torque over the chosen time window $Δt=±36$ ms around the time t = 2.9 s. This time window contains one NBI duty cycle for G and two cycles for B. This seemingly large spread results from the way co and counter beams are deployed: net injected torque varies in time periodically; in each direction(positive or negative) the net torque reaches up to 2 Nm, and over the time period of the duty cycle the net injected torque is small. The locally absorbed torque density, given by the slope of the curves, is co-I_p in the core (⁠$ρ≲0.8$⁠), and ctr-I_p in the edge (⁠$0.8≲ρ≤1$⁠). The total torque deposited in the plasma is $≈−1$ Nm for B, and $≈−0.22$ Nm for G. This larger net ctr-I_p torque for the former discharge is consistent with the observed degradation of rotation for this discharge.

In Fig. 13, the net torque calculated by NUBEAM is compared to the net torque leaving all the beam lines (tinj). In other words, tinj is the mechanical angular momentum of the neutrals per unit time, as they enter the vessel volume. tinj has been averaged over a 10 ms time window. The criterion for low torque measurements to be added to the intrinsic rotation database was $|tinj|<0.5$ Nm. For low to medium NBI power (⁠$PNBI<4 MW$⁠), the torque calculated by TRANSP is close to tinj, whereas at high NBI power $PNBI∼6.5 MW$⁠, the torque calculated for TRANSP is ctr-I_p and large, $∼−1$ Nm (vertical axis), whereas tinj is both co-I_p and ctr-I_p, and small in magnitude (⁠$|tinj|≲0.25$ Nm).

Absorption of the collisionless torque by the bulk plasma takes place over the averaged poloidal orbit transit time of the trapped fast ion (⁠$ωbi∼20$ krad/s at the pedestal top). This timescale is much smaller than the collision timescale $ν−1∼4$ ms, which in turn is smaller than the beam cycle period $tNBI∼20$ ms. Generally, in a turbulence dominated transport regime, momentum diffusivity (⁠$χϕ$⁠) and thermal diffusivity (χ_i) are observed to be comparable.^9,60 The Prandtl number, defined as $Pr=χϕ/χi$⁠, for the turbulent transport regime is thus near unity. Therefore, momentum transport timescale $tϕ$ is similar to the energy transport timescale, which is typically of the order of $∼100–200$ ms in the core and $∼5$ ms in the edge. Hence, we have the following hierarchy of timescales

Therefore, in the edge region, deposition of the ctr-I_p torque, which is predominantly collisionless, is effectively immediate in comparison to $tNBI$⁠. This prompt torque results in a prompt ctr-I_p momentum and rotation change in the edge.⁶⁰ 

In the core region, the co-I_p momentum change, resulting from the deposition of the co-I_p torque, is only transported to the edge on the slow $tϕcore$ timescale, which is longer than $tNBI$⁠. Since this transported co-I_p momentum is smoothed in time by slow core transport and by relatively large core inertia, its measurable effect corresponds roughly to the time average of the torque deposited in the core. As a result, the net torque in the edge at any time is

$τedgefast$ is the locally promptly absorbed torque in the edge and $τcoreslow$ is the torque deposited in the core and transported to the edge on the slow $tϕcore$ timescale. The time averaging window for the computed torque is chosen to be −10 ms for the edge, which is smaller than a typical beam blip time. The torque deposited in the core is time averaged over −200 ms, which is roughly equal to the momentum transport timescale. Since the core torque is averaged over a transport timescale, it does not show the prompt absorbed torque in the core; rather it represents the momentum transported to the edge from the core and smoothed out during this process.

The TRANSP code and its NUBEAM package are used for calculating NBI torque and the resulting ctr-I_p rotation shift for the entire database. Figure 14 shows the net torque in the edge, calculated from Eq. (17). For the high P_NBI measurements (yellow to light green), the edge torque is $τedge∼−1$ Nm. This relatively large ctr-I_p torque in the high β_N range reduces the generated co-I_p intrinsic rotation and appears to explain the observed rotation degradation for high β_N (i.e., β_N > 1.7) H-mode [see Fig. 3(a)]. For the moderate β_N H-mode (0.75 < β_N < 1.7), the edge torque is $τedge∼−0.3$ Nm. This smaller ctr-I_p torque results in a smaller rotation reduction for the moderate β_N H-mode.

To approximate the prompt rotation change in the edge, we use a reduced momentum evolution equation for the edge region

Here, ω_τ is the spatially averaged edge rotation change in response to the edge torque τ_edge given by Eq. (17). The Predicted rotation at the pedestal top is the sum of intrinsic rotation plus NBI-driven rotation shift

In Eq. (18), the first term on the right side is an approximation to the viscous momentum diffusion term (⁠$∂rχϕ∂rωτ∼−ωτ/tϕedge$⁠). I is the moment of inertia of the edge region

where R₀ is the major radius at the magnetic axis, a is the minor radius at the pedestal top, and $Lped$ is the width of the pedestal. The edge momentum transport time can be estimated as

where Pr $=χϕedge/χiedge$ is the edge Prandtl number, and $χiedge$ and $χϕedge$ are, respectively, the ion heat diffusivity and ion momentum diffusivity. $χiedge$ can be obtained through ion edge heat flux $Qiedge$ which is roughly defined as

Solving for $χiedge$ from Eq. (22) and replacing in Eq. (21) yields

Edge momentum transport time $tϕedge$⁠, given by Eq. (23), and assuming Pr = 0.634,¹⁷ is evaluated by using the TRANSP calculated $Qiedge$⁠. Integrating Eq. (18) in time gives $ωτ$ as

The change in toroidal rotation due to NBI torque is calculated using Eq. (24) for the database and shown in Fig. 15(a). Colors represent the magnitude of NBI power. ω_τ is negative, corresponding to ctr-I_p rotation degradation. For the high NBI power measurements (yellow), i.e., P_NBI > 4 MW, the rotation change is clearly larger in magnitude compared to the measurements with P_NBI < 4 MW (green).

Figure 15(b) shows the same rotation change, now with colors representing the plasma current magnitude. In the high β_N range, the two positive I_p discharges, (inverted triangles and diamonds) have the smallest change in rotation due to torque. As was explained in Sec. IV or positive I_p measurements, the co-I_p beam torque used for CER spectroscopy opposes the ctr-I_p prompt loss torque, resulting in less net excess torque. Thus, these two discharges have smaller rotation changes due to smaller excess torque. Moreover, the IBS discharge (diamonds) has the smaller ion banana widths due to the large plasma current, reducing prompt beam-ion losses and the corresponding ctr-I_p torque and rotation shift ω_τ. There is another outlier discharge shown with squares (β_N ∼ 2.4), which has the ω_τ amplitude smaller than the rest in the high β_N range. For this discharge, the width of the edge region, as determined from the L_Te profile, is wider than other high β_N measurements by a factor of ≈1.6. Therefore, for this discharge, moment of inertia of the edge is larger and the calculated ω_τ resulting from NBI torque is relatively smaller in amplitude.

Intrinsic rotation from Eq. (4), and rotation from Eq. (19) which includes the NBI torque effect are compared to the experimental rotation in Fig. 16. The former (i.e., intrinsic) are shown with blue, and the latter with red points. High NBI power (P_NBI > 4 MW) rotations are show in Fig. 16(b). The prediction for intrinsic rotation (blue) is degraded by a large ctr-I_p rotation shift (ω_τ ∼ 30 krad/s). As a result, the NBI-torque corrected rotation (red) is much closer to the experimental rotations. Figure 16(a) shows the H-mode with moderate NBI power (P_NBI < 4) and in the moderate β_N range. The NBI torque driven rotation shift is relatively modest ω_τ ∼ 5–10 krad/s. Nevertheless, this finite co-I_p rotation shift brings the theory prediction closer to the experimental rotations (dashed diagonal line).

The theoretical model of Refs. 16 and 17 is tested using a database of experimental main-ion rotation velocities in nominally low-injected-torque discharges on DIII-D. These discharges include different geometrical and physical parameter ranges, including positive and negative plasma current, lower and upper single null, wide range of ECH and NBI input power, and L-mode and H-mode plasmas.

For the comparison, the theory from Refs. 16 and 17 was modified to take into account the trapped particle population. Generation of intrinsic rotation in this model is due to the effect of neoclassical orbits of passing particles on inhomogeneous turbulent transport. Natural inclusion of the trapped particle population reduces the fraction of passing particles and thus, reduces the theory prediction for toroidal rotation at the core-edge boundary [see Eq. (4)]. We show that in a typical DIII-D plasma with main-ions at the pedestal top being in the banana regime, this dilution due to trapped particles reduces the theory prediction by a factor of ∼3. For NBI injected powers in the range P_NBI < 4, MW theory prediction for rotation velocities has reasonable agreement with the experimental measurements. In the L-mode range of β_N (i.e., $βN≲0.75$⁠), the theory prediction (assuming d_c = 1) slightly under-predicts the toroidal rotation and can be corrected by adjusting the ballooning parameter; performing this optimization for β_N < 0.75 data results in d_c ∼ 1.66. This value is consistent with the previous reports for Ohmic L-modes.^20,21 For the moderate β_N H-mode (0.75 < β_N < 1.7), the theory (assuming d_c = 1) slightly over predicts the rotation; taking the excess NBI torque into account can to an extent correct the over-predicted intrinsic rotation in the moderate β_N range. Another possible explanation for lower than expected rotations in the H-mode is the appearance of high frequency ELMs, which are neglected by the theory but are experimentally observed to reduce the pedestal-top rotation. Another possible reason is that the ballooning nature of turbulence could be diminished in H-mode, leading to reduced values for d_c.

An ITER baseline scenario discharge has been included in the database. Although the physical parameters of this discharge are different from the rest of the database, the model prediction for the rotation velocity agrees quite well with the experimental results (⁠$Ωpred≃ΩKDG$ for a typical d_c = 1).

Main-ion rotation decreases from medium (0.75 < β_N < 1.7) to high β_N values (β_N > 1.7). These high β_N plasmas are obtained by employing higher NBI injection powers, i.e., P_NBI > 4 MW, which require more perpendicular injection and a different NBI duty cycle, resulting in a strong edge torque. For this reason, the observed rotation behavior in this β_N range does not fit the model (as expected due to the strong edge torque here): the theory model predicts a monotonic increase in the main-ion rotation (see Fig. 8), due to the increase in the pedestal top temperature and decrease in L_Te.

The TRANSP code and NUBEAM package are used to compute the torque absorbed in the plasma edge for the database of discharges. These TRANSP simulations show that in high NBI power discharges (P_NBI > 4 MW), the net torque deposited in the edge has a value of ∼1 Nm in the ctr-I_p direction. This excess torque can significantly reduce the co-I_p rotation below its intrinsic value.

We derive an approximation for the torque driven rotation degradation [Eq. (24)] and show that this excess torque can—to a large extent—explain the rotation degradation at high β_N's and high balance NBI powers (previously reported in Ref. 9). The calculated ctr-I_p rotation correction is large for β_N > 1.7, (ω_τ ∼ 30 krad/s), which brings the intrinsic theory prediction much closer to the experimental rotations. Moreover, for medium β_N H-mode data, torque driven rotation degradation, although much smaller in value (ω_τ ∼ 5–10 krad/s), helps improve the theory prediction for the rotations by shifting the theory prediction closer to the experimental rotation values.

In future works, the effect of ELMs on edge rotation will be included in the model and evaluated over the database of experimental measurements.

We are grateful to Colin Chrystal, Jose Boedo, and John deGrassie for helpful discussions. This work was supported by the U.S. Department of Energy under DE-AC02-09CH11466 and DE-FC02-04ER54698.

In Ref. 32, neoclassical toroidal velocities for main-ions and impurities were derived neglecting the poloidal in-out asymmetry. There, derivations were made for the core region in which the in-out asymmetry effect is negligible. As we will show (and as was stated in Ref. 32) near the edge this in-out asymmetry is not necessarily negligible. In this appendix, we obtain the flux-surface-averaged rotation of impurities by subtracting the calculated in-out asymmetric Pfirsch-Schluter term. Main-ion toroidal rotation velocity is then obtained using the neoclassical offset between main-ions and impurities.

Axisymmetric magnetic field is expressed as

where $|∇ψ|=RBθ$ and $I=RBζ$⁠. R is the major radius of torus and ψ is the poloidal flux function. Toroidal unit vector given by $ζ̂=R∇ζ$⁠. Re-ordering Eq. (A1) gives

Inserting $∇ζ$ on the RHS of above gives

Adding $Bθ2Bζ2∇ζ$ to both sides of Eq. (A3) and multiplying by $I2/B2$ results in the identity

Flow velocity for species a can be decomposed in terms of perpendicular and parallel terms

The force balance relation (∂/∂ t = 0) gives the lowest order perpendicular flow as

The relation between parallel and poloidal flows is obtained by taking the product of $.∇θ/(B.∇θ)$ with Eq. (A5)

The second term on the RHS of the above equation is obtained from Eq. (A5)

where we defined the flux function

Substituting from Eq. (A8) in Eq. (A7) gives

Multiplying Eq. (A10) by B² and taking the flux surface average (FSA) yields

Now substituting from Eqs. (A6) and (A8) in Eq. (A5) results in the flow velocity expressed in terms of the sum of a toroidal and a parallel term

Note that $uθa$ is the related to the poloidal velocity as

where we took the dot product of $θ̂$ with Eq. (A12). To obtain the toroidal angular velocity, we take the product $.ζ̂/R=.∇ζ$ and Eq. (A12)

$Ωζa$ can be expressed in terms of a FSA part, plus a poloidally asymmetric part. Taking FSA from Eq. (A14) we find

$⟨Ωζa⟩$ is the rigid rotation angular velocity. FSA angular rotation can alternatively be defined by using the definition of angular moment $⟨L⟩=⟨R2Ωζa⟩$

$⟨ΩζI⟩$ and $Ω¯ζI$ are, respectively, obtained from Eqs. (A15) and (A16) as

For standard DIII-D sign convention, positive I_p (counter-clockwise) leads to positive B_θ on the LFS midplane (downward); thus $Bθ=sign(Ip)|Bθ|$⁠. Moreover for standard toroidal field in DIII-D points in the negative direction (clockwise) and $I=−|RBζ|=−|I|$⁠. Hence, for DIII-D parameters, the above equations can be written more explicitly as

Subtracting the main-ion (i) and Impurity (I) rigid rotations, and using Eq. (A11) we obtain

where we used $|∇ψ|=RBθ$ and $d log p/dr=1/Lp$⁠. From Eqs. (37) and (38) of Ref. 32 (KDG), we have the following relation

Using Eq. (A22) in Eq. (A21) and making the assumption T_i = T_I yields

Equivalently, for $Ω¯ζi$ we find

By making the assumption $⟨Bζ2⟩≈⟨B2⟩$ and $⟨Bθ2⟩/⟨B2⟩≪1$⁠, we simplify Eq. (A23) to obtain