Experimental validation of momentum transport theory in the core of H-mode plasmas in the ASDEX Upgrade tokamak

This study employs the established momentum transport analysis at ASDEX Upgrade [Zimmermann et al. , Nucl. Fusion 63 , 124003 (2023)] to investigate the parametric variations of the momentum transport coefficients in the core of H-mode plasmas. These experimental results are compared to a comprehensive database of gyrokinetic calculations. Generally, good agreement between predicted and measured diffusive and convective transport coefficients is found. The predicted and measured Prandtl numbers correlate most dominantly with the magnetically trapped particle fraction. The experimentally inferred pinch numbers strongly depend on the logarithmic density gradient and magnetic shear, consistent with the theoretical predictions of the Coriolis pinch. The intrinsic torque from residual stress in the inner core is small, scales with the local logarithmic density gradient, and the data indicate a possible sign reversal. In the outer periphery of the core, the intrinsic torque is always co-current-directed and scales with the pressure gradient. This is consistent with prior experimental findings and global, non-linear gyrokinetic predictions. It suggests that profile shearing effects generate the intrinsic torque in the inner core. Toward the outer core, most likely, effects from E (cid:1) B -shearing become more influential. These results offer the first comprehensive picture of this transport channel in the core plasma and contribute to validating the corresponding theoretical understanding. The derived scaling laws are used to construct a reduced momentum transport model, which has been validated against an additional dataset. This demonstrates that the model captures the essential contributions to momentum transport in the core of H-mode plasmas.

Experimental validation of momentum transport theory in the core of H-mode plasmas in the ASDEX Upgrade tokamak
9][20][21][22][23][24][25] However, experimental measurements of momentum transport and corresponding validation of these theoretical concepts is still ongoing work, see, for example, Refs.26-36.This is because of the complicated nature of momentum transport, encompassing diffusive and convective mechanisms and a transport component not directly proportional to the rotation velocity or its gradient.This third component is usually referred to as residual stress. 23,25,37,38This phenomenon gives rise to an intrinsic torque [39][40][41][42][43][44][45][46][47] and is a significant uncertainty for the extrapolation of the toroidal rotation profiles for future devices.
Accurate prediction of rotation profiles requires a comprehensive understanding of momentum sinks, sources, and transport.In future tokamaks, such as ITER, the direct torque from neutral beams may be too small due to the large size and inertia of the plasma to drive a significant rotation.Alternative mechanisms, such as inward convection of momentum or intrinsic torque, may provide the desired level of rotation for stable operation, e.g., to avoid locked modes.Therefore, validated momentum transport theory, particularly for the residual stress, is vital for understanding rotation dynamics in present-day devices and optimizing operational conditions in future reactors.
This study utilizes the well-established momentum transport analysis developed at ASDEX Upgrade (AUG) and presented in prior works. 48,49The reader is referred to those papers for more information on the methodology and the used models.The methodology relies on the analysis of NBI modulation experiments and determines the contributions of diffusion, convection, and residual stress to momentum transport in the fusion plasma core.The time-averaged steady-state, the modulation amplitude, and phase profiles of the toroidal plasma rotation are central to the modeling.The experimental ion temperature and rotation data are measured via the charge exchange recombination spectroscopy (CXRS), which assumes the main impurity data to be representative of the main ions. 50,51Modeling is based on the toroidal momentum conservation equation as formulated in Ref. 52.An experimental boundary condition of the toroidal rotation at the normalized toroidal flux coordinate q u ¼ 0:8 is given to concentrate on analyzing the transport in the plasma core.Varying the precise position only has a minor influence on the results as long as it is chosen to avoid the impact of edge localized modes.The flux-surface-averaged radial flux of toroidal momentum can be written as Here, m is the main ion mass, n is the main ion density, R is the fluxsurface-averaged local major radius, and v u is the momentum diffusivity.v u ¼ hRX u i is the toroidal velocity with hÁi denoting the flux surface average and X u is the toroidal component of the angular velocity, which is constant on flux surfaces for sufficiently small poloidal rotation as in the core plasma of such discharges. 53If the poloidal rotation were not small enough, the measured toroidal rotation would need to be corrected to be compared with the parallel velocity in the modeling.Furthermore, @=@r is the derivative with respect to the minor radius r, V c is the convective velocity, and P int is the residual stress flux, which gives rise to an intrinsic torque s int ¼ À@V=@r P int with V the enclosed plasma volume.
In the modeling, the momentum diffusivity v u is scaled with the experimental ion heat diffusivity v i through a linear function of the normalized toroidal flux radius q u .5][56][57][58][59] The ion heat diffusivity is calculated via power balance from experimental data.The scaling ensures a correct response of the turbulent momentum transport to the possible changes in turbulence amplitude due to the modulation of the neutral beam heating.Convection and residual stress flux profiles are prescribed as radial cubic polynomials, ensuring continuity at q u ¼ 0. The ratio of convective velocity to momentum diffusivity, ÀR V c =v u , is usually called pinch number.The momentum conservation equation is solved using the ASTRA transport code. 60,61The externally applied torque from the beams is calculated via the Monte Carlo code NUBEAM, 62 which is part of the TRANSP code suite 63,64 and consistently included in the momentum conservation equation.
In a previous Letter, 65 the methodology was applied to a small set of discharges.This showed parametric dependences of the pinch number in the inner (q u % 0:3) and of the intrinsic torque in the outer plasma core (q u % 0:7).This work aims to expand on this to obtain a more detailed understanding of the main parameter dependences of momentum transport and to more thoroughly validate theory models with experimental data.This sets the basis for coherent, physics-based, and validated predictions of momentum transport for a prospective reactor scenario.The paper is structured as follows: Sec.II presents detailed theoretical predictions on diffusive and convective momentum transport.Section III provides experimental results on parameter dependences of transport mechanisms, with Sec.IV discussing the experimental findings in the context of prior research.Section V combines the findings to propose a reduced momentum transport model for the plasma core.The paper concludes with a summary and an outlook in Sec.VI.

II. GYROKINETIC PREDICTIONS
The gyrokinetic code GKW 66 was utilized to simulate turbulent momentum transport.The code solves the gyrokinetic equations [67][68][69][70] and includes kinetic electrons, electromagnetic effects, collisions, and a realistic reconstruction of the plasma equilibrium.Moreover, it takes into account the influence of plasma rotation, making it well-suited for investigating the Coriolis momentum pinch. 66All gyrokinetic calculations in this work are local and linear.Such calculations result in predictions for the Prandtl and the pinch numbers.They can be compared with experimentally inferred transport coefficients, as done in Sec.III.For the prediction of the dominant residual stress fluxes in the core, global, non-linear simulations are required, 71 which are computationally costly and unsuitable for analyzing databases, as done in this study.
Here, the code is used with a time integration scheme to identify the micro-instability with the highest growth rate.The resulting fluxes are then weight-averaged across a spectrum of five binormal wavenumbers in the range of 0.2 < k y q i < 0.9.This ensures the inclusion of contributions from modes with different wavelengths.As the different tested wavenumbers exhibit distinct growth rates c k , their results are spectrally averaged.

A. Isolated parameter scans
In previous theoretical works, Peeters et al. 19,24,72 investigated the parametric dependences of the pinch number.It was found that the pinch number is expected to scale most strongly with the logarithmic density gradient R=L n (Ref.24, Fig. 7).The strong dependence on the logarithmic density gradient results from the coupling of the density, temperature, and parallel velocity perturbation via the Coriolis force and can be derived in an analytical fluid model even when neglecting momentum flux from the development of a finite parallel wavenumber k k . 19When the parallel dynamics is included, see, for example, Ref. 73, one finds that the deformation of the mode eigenfunction along the fieldlines also contributes to the momentum flux.The parallel dynamics weakens the scaling of the pinch number on the density gradient 72,74 and causes a dependence on the magnetic shear.Furthermore, it is expected that the pinch vanishes when the trapped particle fraction approaches zero due to a compensation effect from the passing particles via the self-consistent parallel wave number. 72,74With a similar approach, the Prandtl number was studied by Strintzi et al. 21,24 and found to be rather constant under variations of the gyrokinetic input parameters with values of the order of unity.
Most of those previous works have used gyrokinetic calculations to perform isolated parameter scans of the momentum transport coefficients, i.e., by varying one gyrokinetic input parameter and keeping the others fixed to the value of a standard case.This has the advantage that parameter dependences can be separated, whereas for experimentbased input data, often many variables are cross-correlated.The Waltz or GA standard case is usually chosen as the starting point for such investigations. 75It is defined by a logarithmic density gradient of R=L n ¼ 3, a logarithmic temperature gradient of R=L T ¼ 9, T e ¼ T i (if the temperature ratio is not varied), a safety factor of q ¼ 2, a magnetic shear of s ¼ 1, and an inverse aspect ratio of ¼ 0:16.Furthermore, the mass ratio of D is used, m e =m p ¼ 0.000 27 (kinetic electrons).A concentric circular equilibrium is assumed.This brings the advantage of decoupling the effects of safety factor, shear, and inverse aspect ratio, which is not straightforward to do with an experimental equilibrium reconstruction that combines those values consistently.Furthermore, electromagnetic effects are neglected (ergo electrostatic) and collisions between the species are neglected.The standard case focuses solely on one mode with k y q i % 0:42 ( ffiffi ffi 2 p included in the thermal velocity in the definition of q i ), which usually denotes the fastest-growing mode of instabilities in the plasma core [ion temperature gradient modes (ITGs)].In this work, additionally, an adapted standard case with input parameters more representative of the studied experimental dataset is created.This adapted standard case has the following input quantities: R=L n ¼ 1.2 (instead of 3), R=L T ¼ 5.7 (9), q ¼ 2.27 (2), and s ¼ 0.8 (1).Other than that, all settings were kept fixed to those of the Waltz standard case.
As already noted, the Waltz and the adapted standard cases are constrained and simplified by using an analytical equilibrium, neglecting collisions and electromagnetic effects, and assuming equal temperatures and gradients.Although one cannot separate the dependences on q, s, and based on a more realistic experimental equilibrium, it is possible to test the effect of the simplifications for the scans of the gradients and temperature ratio.Therefore, in the following, a more realistic standard case is introduced, including all of the effects neglected for the Waltz and the adapted case.Experimental profiles were used as input to enhance realism, specifically those from the reference discharge #40 076, from 2.0 to 4.2 s at q u ¼ 0.7.This radial position was selected due to its strong temperature gradients, balancing potential stabilizing effects throughout the scans.The precise input parameters are Z eff ¼ 1.26, m e =m p ¼ 0.000 27, n e ¼ 0.6 Â10 19 m À3 , R=L Ti ¼ 7.42, R=L Te ¼ 6.9, R=L ne ¼ 0.19, T e =T i ¼ 0.94, T e ¼ 0.654 keV, ¼ 0.22, s ¼ 1.54, and q ¼ 3.04.This parameter set is close to the adapted standard case and representative of the studied experimental dataset in Sec.III.Various k y q i values were tested, as mentioned earlier.For comparability to the simplified scans, only the values corresponding to k y q i % 0:43 are discussed in the following.Nonetheless, the results for spectrum-averaged Prandtl and pinch numbers closely align with those from these scans, suggesting that the chosen k y q i represents a valid test case, and the derived scalings can be later compared to spectrumaveraged outcomes.
The parameter scans performed by Peeters et al. and Strintzi et al. are repeated starting from the Waltz, the adapted, and the more realistic standard case.The differences between the Waltz and the adapted standard cases were minor, mainly resulting from the smaller gradients.This is shown in Fig. 1 with the Waltz standard case shown with blue crosses and the adapted standard case with orange circles.The range of variation of the varied parameters FIG. 1. Gyrokinetic parameter scans, based on the Waltz standard case and its adaption.Furthermore, parameter scans with more realistic gyrokinetic models are shown, in which and s cannot be decoupled due to the use of an experimental numerical equilibrium.Linear fits are shown with lines, and their fitting coefficients are given in the corresponding plots.For clarity, only scalings that are later compared quantitatively are noted.Panel (a) and (b) show the Prandtl and pinch numbers for a scan in R=L T , (c) and (d) for a scan in , (e) and (f) for a scan in R=L n , (g) and (h) for a scan in T e =T i , and (i) and (j) for a scan in s.
was adapted to the values seen in the experimental data to understand their relative influence.In the same figure, the more realistic scans are shown with black triangle symbols.As one can see, the more realistic scans do not agree in their dependences on every varied parameter with the Waltz and adapted standard case, indicating that the simplifications and the different starting points in those scans affect the observed trends.
As expected from earlier work, the Prandtl number is constant around unity and shows a variety of parametric dependences.It tends to decrease with increasing R=L Ti for the Waltz and the adapted standard case [panel (a)].In the more realistic standard case, the variation of the ion temperature gradient does not show a significant influence.The Prandtl number clearly increases for [panel (c)].It shows a slight decrease with increasing R=L ne for the Waltz and the adapted standard case [panel (e)].A similar trend is found for the more realistic case.The temperature ratio does not strongly modify the Prandtl number for the Waltz and the adapted case, a slight decrease with increasing T e =T i is found, as shown in panel (g).Contrary to the simplified scans, the Prandtl number positively correlates with the temperature ratio for the more realistic case, most likely due to the different temperature and density gradients assumed.Analyzing the dominant mode frequencies shows that the more realistic case is characterized by larger frequency in the ion drift direction (and, thus, deeper in the ITG turbulence regime) than the Waltz and the adapted standard case, and this modifies the reaction of the turbulence to the temperature ratio, resulting in the different trends observed.Interestingly, there is a non-monotonic behavior with the shear, see panel (i).
For the pinch number, one finds increasing values for increasing R=L Ti [panel (b)].However, for the realistic case, the overall variation is small.As expected from earlier works, the pinch number scales clearly with the inverse aspect ratio as a proxy for the trapped particle fraction [panel (d)].The fundamental and robust scaling of the pinch number on R=L ne is consistently found in all three scans [panel (f)].For the simplified Waltz and adapted standard cases, the pinch number decreases for increasing T e =T i [panel (h)].In contrast, the pinch number correlation with the temperature ratio differs for the more realistic case, and the overall variation is much smaller.The differences between the simplified and the realistic cases observed in both the Prandtl and pinch numbers for the temperature ratio scan could be associated with stronger trapped electron modes (TEMs) emerging with the larger density gradient of the Waltz and the adapted standard case.Panel (j) shows a non-monotonic trend for s, likely due to a stabilizing effect of the shear on the growth of the modes.
A similar plot can be made for the mode propagation frequency and growth rates of the fastest-growing modes.However, there is no clear pattern of how the growth rate and the mode propagation frequency exactly propagate into the calculation of the Prandtl and pinch number.This is not shown for the sake of brevity.As a summary for these parameter scans, the Prandtl number is expected to exhibit the most clear scaling with s and .The pinch number shows the strongest and clearest dependence on the density gradient.Notably, elevated values of s can potentially reduce both the Prandtl and the pinch numbers.These calculations show that scalings and trends derived from simplified reference cases reflect certain general behaviors, but these predictions are modified when more complete models and specific parameters, which are obtained in experimental conditions, are considered.Therefore, whenever possible, the most realistic and detailed gyrokinetic model should be employed and a representative starting point of the scan should be chosen.For the following comparisons, the more realistic scalings of R=L Ti ; R=L ne , and T e =T i are used while maintaining the simplified ones for , f Tr , and s, as those cannot be repeated with an experimental equilibrium.In principle, future theoretical work could involve conducting more detailed scans to gauge the single effects of various simplifications on the predictions of the Prandtl and pinch numbers.

B. Gyrokinetic database
In the following, the parameter dependences of the gyrokinetically predicted Prandtl and pinch numbers will be examined by fitting scaling laws to a database of gyrokinetic simulations, which were calculated based on realistic experimental input data from AUG H-mode plasmas.This approach includes the more realistic physics effects previously neglected for the simplified standard case.All following gyrokinetic scaling laws result from the regression of this database with realistic experimental input.
The experimental data for this database are derived from 29 discharges, and the list of these discharges is provided in Table II in the Appendix.Gyrokinetic calculations were conducted for each discharge at radial positions of q u ¼ [0.3, 0.4, 0.5, 0.6, 0.7].The parameter space these gyrokinetic predictions cover is shown in column (a) of Table I.The median values of this database correspond exactly to the parameters used for the previously examined adapted standard case, allowing for comparison of the following database regression to the isolated scaling laws shown in Fig. 1.

TABLE I.
Parameter space and underlying engineering parameters covered by the used databases.Column (a) is for the experimental input data for the gyrokinetic database.Column (b) is for the data points for the experimental analysis.Column (c) is for the additional 85 phases from 49 discharges employed as a test bed and validation dataset for the reduced model.Local quantities sampled for q u ¼ 0.3-0.7.The quality of the regression of scaling laws can be quantified via the normalized root mean square error

Quantity
where y is the mean of the observed data, n is the number of data points, Y i is the regressed data points, and y i is the observed data.A similar definition holds for the v red , but instead of normalizing with the averaged observed values y, the individual measured uncertainties r i are used.
Returning to the isolated parameter scans, one would expect dependences of Pr in this database on , s, T e =T i , and ÀR=L ne .Employing all these parameters in a linear scaling law regression results in good agreement between the gyrokinetic prediction and the regression with RMSE ¼ 17.4%.The regressed scaling law reads as A comparison of the coefficients with the linear fits presented in Figs.1(c), 1(e), 1(g), and 1(i) reveals quantitative agreement between the database regression and the isolated parameter scans.An uncertainty calculation for the scaling coefficients has also been conducted, relying on mapping out the fitting solutions up to a variation of 1.5 of the underlying v 2 cost function.The uncertainties obtained are considerably larger than the absolute values of the fitted coefficients due to the significant cross correlation among the selected fitting parameters and the limited dataset.Reducing the number of parameters in the regression can help reduce these uncertainties.
Hence, in the subsequent analysis, individual parameters were assessed as potential ordering parameters for regressing the Prandtl number.It can be shown that removing the scaling with T e =T i or R=L ne in Eq. ( 2) does not severely lower the regression quality.In fact, the calculation of the Pearson correlation coefficients suggests that the predicted Prandtl numbers exhibit the strongest correlation with either s or .It is worth noting that while the isolated parameter scans indicated a decrease in Pr for s > 1 (around q u ¼ 0.5-0.6 for most of the studied discharges herein), see Fig. 1(i), this trend is not observed when examining the predictions across entire radial profiles in the gyrokinetic database.This can be seen, for example, from the gyrokinetic predictions of the Prandtl number presented later to the experimental findings, see column (d) of Fig. 4. In contrast, those simulations show Prandtl numbers that consistently increase monotonically with radius.This raises questions about the precise physics mechanisms underlying the scaling with shear, as the observed correlation (Pr reg ¼ 0:3 s þ 0:7 with RMSE % 17.8%) is likely a result of cross correlation rather than an accurate representation of a specific physics mechanism.
A more coherent picture emerges when regressing Pr with .In the isolated parameter scans, Pr consistently increased monotonically with in both the standard cases and the scans involving more realistic gyrokinetic calculations, as illustrated in Fig. 1(c).The regression of a scaling law to the database results in Pr reg ¼ ð4:862:9Þ þ ð0:260:5Þ with RMSE % 18.8%.The scaling with is shown in Fig. 2. The scaling law, Eq. ( 3), can be compared to the isolated parameter scans, see Fig. 1(c).The slope of the scaling in the database regression (4.8 6 2.9) is steeper than the linear fit to the isolated parameter scan (%1.23), but within uncertainties.Overall, the results of the isolated parameter scans were qualitatively reproduced using the database approach.It is shown that is the most physical and robust single quantity for a scaling of Pr.It is speculated that such a general dependence of Pr on could be explained by a similar compensational effect of passing particles modifying the parallel mode structure as found previously for the pinch.
In the following, the results for the pinch number are analyzed.Based on the isolated scans, a robust scaling of the pinch number with R=L ne is expected, along with potential scalings involving R=L Ti , s, and T e =T i .While all these quantities can be incorporated into a scaling law, it is observed that the impact of R=L Ti is minimal.Cross correlations likely make the inclusion of R=L Ti unnecessary, and its contribution can be omitted without compromising the quality, The primary term contributing to the variation is measured to be R=L ne (causing 59% of the variation), with s (26%) and T e =T i (15%) playing a secondary role.This suggests that R=L ne is the dominant ordering parameter.However, the quality of this regression is significantly poorer compared to those achieved for the Prandtl number, with an RMSE % 46.7%.The uncertainties associated with the fitting coefficients are substantial.A comparison with the linear fits from the isolated parameter scans (refer to Fig. 1) reveals similar values.For the fits to the parameter scans of T e =T i and s, as shown in Figs.1(h) and 1(j), only the linear increase was considered, as the majority of the input data falls within this parameter range.
To reduce uncertainties, components are systematically eliminated and it is found that neglecting T e =T i causes a minor loss in regression quality compared to neglecting s.It is feasible to rely solely on s and R=L ne , ÀRV c =v ureg ¼ ð0:560:2Þ R=L ne þ ð0:4460:4Þ s; (5) achieving acceptable RMSE % 47.6%.The regression coefficients can be compared to the linear fits obtained in the isolated parameter scans (see Fig. 1), and similar trends are found.As s and are strongly cross-FIG.2. Regressed Prandtl numbers from Eq. ( 3), a scaling with (x-axis) vs the gyrokinetic prediction (y-axis).Unity is shown as a black line.
correlated in this dataset, one can replace the shear scaling in Eq. ( 5) with the inverse aspect ratio without loss of quality.This, however, contradicts the concept of the compensational effect of the passing particles, see previous work by Peeters et al., 72 which requires vanishing pinch number for !0, which cannot be enforced by using as a term in the sum.Enforcing a fundamental scaling for the pinch number on or the trapped particle fraction does not improve the quality of the regression laws, ÀRV c =v ureg ¼ f Tr ð1:07 R=L ne þ 0:58 sÞ gives RMSE ¼ 51.6%.Therefore, enforcing such a scaling was not included in the assessment of gyrokinetic scaling laws.The authors speculate that the effect of the passing particles is small in the radial range considered here, and the shear dependence compensates for this.The scaling combining f Tr and s is still provided as it could be tested on plasma with different aspect ratios.
Reducing to one scaling parameter (together with a constant), clearly, a scaling with R=L ne is most accurate and results in ÀRV c =v ureg ¼ ð0:3460:3Þ R=L ne þ ð0:760:6Þ with an RMSE ¼ 49.6%, which is close to the regression together with s before [cf.Eq. ( 5)].The regression utilizing R=L ne demonstrates a good agreement, considering the uncertainties, with the linear fit derived from the isolated parameter scan [y ¼ 0:44x þ 1:04, see Fig. 1(b)].The agreement between the regression results and gyrokinetic values is visualized in Fig. 3.These findings indicate that the density gradient and shear are predicted to primarily influence the pinch number.Furthermore, the density gradient emerges as the only parameter capable of serving as a robust and single ordering parameter, with consistently stable scaling coefficients of % 0.3-0.5 observed across all examined regressions.Compared to prior studies, these results reproduce the dominant role of the density gradient in influencing the pinch number, as reported in works such as Refs.19 and 24.Those studies noted a nonmonotonic impact of shear and a relatively small influence of R=L Ti .A similar investigation conducted by Weisen et al. 76 explored the gyrokinetically computed pinch number dependences using a database of experimental data from the Joint European Torus (JET).In that study, a linear scaling law was regressed to the pinch number.Remarkably, the coefficients obtained for the influence of the density gradient on the pinch number in that publication align quantitatively with the coefficients derived from the isolated parameter scans and the database regression conducted in this study, thereby strengthening these findings.However, also in that work, gyrokinetic calculations were missing the residual stress.

III. EXPERIMENTAL RESULTS
This section presents the outcomes of the experimental analysis conducted on a smaller subset of H-mode discharges, comparing the parameter dependences of the experimental transport coefficients with the theoretical predictions of Sec.II.

A. Analyzed dataset
This study compares eight experimental discharge phases, as listed in Table III in the Appendix.This subset enables an examination of the main experimental ordering parameters compared to theoretical predictions.Since diffusive and convective coefficients are considered as local quantities, their values can be sampled at various radial positions, such as q u ¼ [0.3, 0.4, 0.5, 0.6, 0.7].This results in 40 data points, covering the parameter space outlined in column (b) of Table I.The range of parameters mirrors that explored in Sec.II, except for slightly lower values of R=L ne ; T e =T i , and b e due to the limitation of P ECRH < 1 MW in this dataset.Despite this limitation, the experimental results should be comparable to theoretical predictions and align if no fundamental deviations exist between prediction and measurement.

B. Modeling results
Figure 4 presents a comparison of the experimental (brown) and modeled (green) profiles for the toroidal rotation's steady-state (column a), amplitude (column b), and phase (column c) across the eight plasma phases examined in this section.The transport coefficients, depicted in columns (d)-(g), are compared with gyrokinetic predictions, shown as black points, which were calculated specifically for these cases.
The first three columns demonstrate that the model accurately reproduces the experimental data with high fidelity and within acceptable uncertainties.Regarding the steady-state profiles (column a), there is observable variation in peak values, particularly in discharges (1), (3), and (6).For discharge (1), the high rotation is attributed to the substantial NBI power and the resulting torque, see Table III.A relatively low experimental diffusivity in discharge (3) contributes to the peaked steady-state profile.Conversely, discharge (6) exhibits remarkably low NBI power, but the positive intrinsic torque over the entire FIG. 3. The plots show regressed pinch numbers on the x-axis against gyrokinetic predictions on the y-axis.In panel (a), the regression from Eq. ( 5) is depicted, a scaling with R=L ne and s.In panel (b), the regression from Eq. ( 6) shows a linear scaling with R=L ne .The black line represents unity.
analysis domain leads to a peaked profile.Notably, the steady-state rotation profiles exhibit different curvatures, with most slightly concave.Discharges (3) and ( 6) feature slightly convex shapes.As in those discharges no ECRH was applied, this finding indicates that the heating mixture can have an effect on the formation of the rotation profiles, e.g., via changes to the transport.The observation of hollow or negative rotation profiles together with ECRH, as known from earlier studies, will be further discussed later.
The amplitude and phase profiles are predominantly observed to vary depending on the utilized NBI modulation frequency f mod , a previously documented effect discussed in Ref. 48.In column (b), the amplitudes are higher when lower modulation frequencies are used, see, for example, discharge (8) with f mod ¼ 2 Hz.This trend is accompanied by flat phase profiles, as illustrated in column (c) for the same discharge.Conversely, despite discharge (7) being conducted with f mod ¼ 3 Hz, the amplitude remains relatively small, and the phase shift between the inner and outer edges of the fitting domain aligns with the order of magnitude observed in other discharges conducted with f mod ¼ 5 Hz.This suggests the presence of different transport characteristics in discharge (7), as indicated by the flat experimental Prandtl number profile.
Regarding the Prandtl numbers (column d), it is evident that most of the discharges exhibit an upward trend in Prandtl numbers with increasing radius.Except for discharges (3) and ( 7), the gyrokinetic predictions generally align with the experimental analysis across the measured radius.For discharges (3) and ( 7), there may be transport effects that the applied experimental or gyrokinetic models do not entirely capture.For the remaining discharges, prediction and measurement display approximately a variation of two between the inner and outer core.The resulting diffusion profiles are illustrated in column (e), exhibiting a monotonic increase with radius.The lowest values are observed for discharge (3), characterized by the lowest overall heating power of approximately 2.75 MW and, consequently, smaller values for the power balance v i $ Q i .
The gyrokinetic predictions closely follow the experimentally inferred pinch numbers (column f).Noteworthy variations are observed, such as the peaked profile in the inner core of discharge (1)  or the remarkably flat pinch number predicted for the innermost radial position in discharge (6).
Column (g) displays the intrinsic torque profiles, which remain flat in the inner core and steepen toward the outer core, being co-FIG.4. Steady-state, amplitude, and phase profiles of the toroidal rotation are depicted in the left three columns, with experimental data in brown and modeled data in green.The columns on the right compare the experimental transport coefficients (green) and the gyrokinetic predictions (black).
current directed toward the edge.While some discharges, like (5), feature highly peaked profiles toward the outer core, others, such as (3) and ( 8), maintain relatively flat profiles or even tend to roll over.However, it is essential to note that the error bars are often substantial, especially for the outermost radial position at q u ¼ 0:8 due to the imposed boundary condition in the modeling.
In summary, this initial analysis highlights that the formation of the steady-state, amplitude, and phase profiles of the toroidal rotation results from a complex interaction involving the modulation of the externally applied torque and various transport mechanisms.

C. Diffusion
The subsequent analysis tests the experimental dependences of different transport mechanisms and contrasts them with theoretical outcomes of Sec.II.
Beginning with the Prandtl number, theoretical predictions pointed to the local inverse aspect ratio, serving as a proxy for the trapped particle fraction as the clearest ordering parameter.Similarly, the inverse aspect ratio emerges as the most effective ordering parameter for the Prandtl number in the experimental assessment.Linear fitting leads to Pr reg ¼ ð3:462:9Þ þ ð0:460:5Þ with RMSE ¼ 20% and v 2 red % 0.85.The uncertainty associated with the scaling parameters represents a 1.5 variation in the cost function value.No weighting was applied with respect to experimental uncertainties during the fitting process, as it could not adequately capture the asymmetry of the error bars.In Fig. 5, the experimentally determined Prandtl numbers are plotted against the inverse aspect ratio.The green dashed line represents the experimental fit from Eq. (7).For clarity, uncertainties are not plotted for all data points.Instead, a representative value corresponding to the dataset's average is shown for one data point.The linear fits generally align within uncertainties for most points.Notably, the scaling law derived from the gyrokinetic database [see Eq. ( 3)], depicted as a solid black line, exhibits agreement in both trend and magnitude with the experimental results.Attempts to significantly reduce scatter by incorporating additional quantities such as q, q 95 , or T e =T i into the scaling proved unsuccessful, and including further regression parameters would likely result in overfitting, as indicated by the shown error bar and the v 2 red .The lower points mainly originate from discharges (3) and ( 7) with a flat Prandtl number profile.
Comparable outcomes are achieved by employing f Tr as an ordering parameter, given that is a proxy for the trapped particle fraction.In this experimental examination, R=L Ti ; Ã , and s also serve as ordering parameters, albeit with higher, though worse, RMSE values.Given the insights gained from Sec. II, it is assumed that experimental cross correlations likely cause the ordering characteristic of these quantities, as these quantities generally exhibit monotonic increases over the radius and do not represent a physical dependency.
Moving forward, the resulting momentum diffusivity is discussed.Mutual cross correlation exists due to its dependence on the Prandtl number and ion heat diffusivity, e.g., with or with quantities such as Q i , rT i , or R=L Ti .The logarithmic ion temperature gradient holds particular significance as a main parameter determining the growth rates of the ITG modes.
As the fluxes from linear gyrokinetic calculations are constantly growing with the potential fluctuation, they cannot be compared to experimental values, but plotting the experimental diffusion coefficients against the growth rates c from gyrokinetic calculations yields additional insights.As depicted in Fig. 6, the experimentally determined momentum diffusion exhibits a scaling with the growth rate of the fastest-growing mode and, thus, its associated instability.These findings align with the fundamental understanding of turbulent transport, where D $ c=k 2 ?links the growth rate to the turbulent diffusion coefficient.The clarity of the comparison between theoretically predicted turbulence properties and experimentally inferred diffusion values is noteworthy.

D. Convection
The theoretical predictions in Sec.II proposed a variety of potential parameter dependences for the pinch number.From the regression of the gyrokinetic database, key candidates were found to be T e =T i ; R=L ne , and s.Regressing the experimentally measured pinch numbers, the logarithmic density gradient emerged as the most effective single-order parameter, aligning with the theoretical predictions.
FIG. 5.The experimentally determined Prandtl numbers plotted against the inverse aspect ratio.The linear fit to the experimental data points, depicted by a dashed green line [refer to Eq. ( 7)], is presented alongside the regression derived from the gyrokinetic database, illustrated by a black solid line [refer to Eq. ( 3)].The error bars displayed correspond to the average error bar of the dataset.FIG. 6.The experimentally determined momentum diffusion plotted against the growth rates from gyrokinetic calculations, with a linear fit for visual guidance.The displayed error bar represents the average uncertainty across the dataset.
T e =T i or R=L Ti were not found to significantly improve the quality of the regression of the measured pinch numbers.
The most significant variation in the density gradient within the analyzed dataset is present at q u ¼ 0.35.For this radial position, the predicted and measured pinch numbers are plotted against R=L ne in Fig. 7.The graph illustrates the increase in the pinch number with the density gradient, suggesting that the density gradient can indeed locally order the pinch number.Linear fits are presented as dashed lines (experimental data) and solid lines (gyrokinetic prediction) for visual guidance.Both trends exhibit qualitative agreement.Additional parameters do not notably reduce the scatter.This is shown in the figure, where different symbols represent I p ¼ 0.6 MA (triangles) and I p ¼ 0.8 MA (squares), with no apparent grouping.
Encouraged by this scaling behavior, the existing dataset was used to derive simple scaling laws.The most effective regression of the experimental pinch number with R=L ne and s is given by ÀRV c =v ureg ¼ ð0:760:3Þ R=L ne þ ð0:4460:5Þ s with RMSE % 63% and v 2 red % 1:2.The variability in the regressed value is primarily attributed to the changes in the density gradient.Figure 8 depicts the experimental values plotted against the regressed ones from Eq. ( 8).The solid black line represents unity.Despite the worse RMSE values, compared to the scaling laws in Sec.II, the representative error bar in this plot shows the regression is reasonable given the uncertainties, and introducing a larger number of regression parameters is not recommended to prevent over-fitting.
The values obtained from this scaling law align within uncertainties with the gyrokinetic database regression [see Eq. ( 5), ð0:560:2Þ R=L ne þ ð0:4460:4Þ s].However, the experimental dependence on density gradients appears slightly stronger.The inclusion of shear does not contradict the earlier finding that the local ordering at q u ¼ 0.35 cannot be enhanced by incorporating q or s, as both peak toward the edge and, thus, have a minor impact at smaller radii.Additionally, there is minimal variation in q and s values at this radial position within the studied dataset.

E. Intrinsic torque
Deriving parameter dependences for the local intrinsic torque in the inner plasma core (q u ¼ 0.3-0.6) is challenging due to the limited variation in values.The experimentally inferred range from À0.5 to 1.0 Nm at mid-radius, and they are comparable to the NBI torque, which falls between 1.2 and 1.9 Nm at mid-radius for the investigated discharges.Notably, the case with co-current intrinsic torque in the core, #39 015, features the lowest R=L ne values at that location.Conversely, #41 551, characterized by the steepest core density gradient, displays the strongest countercurrent intrinsic torque.This is illustrated in Fig. 9, with the density gradient in panel (a) and the intrinsic torque values in panel (b).
Due to the simple cubic shape assumed for the residual stress, the intrinsic torque values mainly increase between q u ¼ 0.3-0.6 for all analyzed cases.This leads to strong correlations with variables that exhibit a monotonic shape in this radial region.The propagation frequency of the fastest-growing mode does not exhibit a monotonic behavior across the radius.Remarkably, the measured co-and countercurrent core intrinsic torque and the density gradient can be effectively ordered by the fastest-growing propagation frequency, as illustrated in Fig. 10.This is interpreted as two sign reversals in the local residual stress flux: one occurring during the transition from TEM to a mixed regime and another as ITGs become more dominant.The comparison of panels (a) and (b) reveals that the co-current intrinsic torque occurs with flat density profiles.In contrast, the transition regime, with peaked density gradients, is associated with countercurrent intrinsic torque.To draw a conclusive picture, it would be necessary to include stronger TEM-dominated cases, which were not available in this study.A regression of the core intrinsic torque values with the logarithmic density gradient results in a reasonable v 2 red value of approximately 0.57.The relationship between the density gradient and the dominant mode propagation frequency was previously motivated by Angioni et al. 77 and explained by Fable et al. 78 In addition to the scaling with R=L ne , various parameters were tested for their ordering characteristics.It was evident that the TEMdominated data points exhibited small R=L Ti and large R=L Te values.While for ITG-dominated and mixed cases, R=L Ti successfully ordered the core intrinsic torque, it failed to order the TEM-dominated cases, yielding v 2 red % 0.76, which is worse than the ordering achieved by the logarithmic density gradient.
0][81][82][83] Moreover, the intrinsic torque driven by equilibrium up-down asymmetry, as discussed in Refs.37 and 84, is negligible in the studied scenarios.6][87] Note that the observed sign reversal may not coincide with a sign reversal of the propagation frequency or density gradient.The propagation frequency is assessed in local, linear calculations, which only identify the most unstable mode, and do not compute the entire spectrum of multiple unstable co-existing modes.In the experiment, in the presence of multiple unstable modes which interact in the turbulent state, non-linear mode mixing is a relevant mechanism for how the produced radial wave vectors tilt the turbulent eddies and induce residual stress.In addition, not all residual stress mechanisms scale with the propagation frequency and the profile shearing.Therefore, no linear or direct relationship between the signs of those quantities can be expected.
The intrinsic torque at the edge of the fitting domain (q u % 0.7) does not exhibit a clear correlation with x r or R=L ne .In the investigation of the scaling of the intrinsic torque toward the edge of the fitting domain, excluding discharge #39 015 was necessary due to its significant uncertainties at this radial position, making its contribution less meaningful.As a starting point, the well-known Rice scaling [88][89][90] was tested, which connects the intrinsic rotation to the plasma stored energy and current, expressed as Dv u $ W p =I p .As previously noted, the variability in I p within this dataset is minimal.The intrinsic torque at the edge of the fitting domain was ordered by W p with a RMSE % 40%.Also rT i at q u % 0.7 ordered the intrinsic torque with a RMSE % 47%.Slightly worse ordering was observed with rT i in the steepest region with RMSE % 50%.No ordering was observed with the density gradient.It is important to note that this comparison is somewhat limited, as the Rice scaling concerns an effective intrinsic rotation, while this study focuses on the extracted intrinsic torque.
Pearson cross correlation matrices were computed to explore potential scalings.The local ion pressure gradient Àrp i emerged as the most effective ordering parameter, as shown in Fig. 11(a).The linear fit (dashed line) aligns well with experimental uncertainties, resulting in a RMSE value of 23%.In panel (b), it is evident that the ion pressure gradient in the steepest region also serves as a suitable ordering parameter.This, in the first place, only shows that the pressure gradients at those radial positions are correlated but could also guide future work on the localization of the intrinsic torque-generating mechanisms in the plasma edge.The corresponding linear fit remains within error bars, though with a slightly higher RMSE ¼ 28%.Those regression values are much better than the ordering with W p or rT i , but, of course, in the discharges studied here, the ion temperature gradient is the dominant contribution to the pressure gradient.
The conclusion drawn from these plots is that the experimental values can be effectively ordered by both the local and steepest ion pressure gradients, which exhibit cross correlation within this dataset.Given the relatively small variation compared to the size of the error bars, it is not possible to decisively favor one over the other.This result can be interpreted such as rp i may serve as a proxy for the radial electric field E r and, consequently, as a measure for E Â B-shearing.Therefore, the effective ordering of the edge intrinsic torque with the pressure gradient aligns with the concept of E Â B-driven residual stress 79,80,82,83 as a dominant mechanism for creating co-current intrinsic torque at this radial position.Furthermore, other intrinsic torque generating mechanisms scaling with local or edge gradients could contribute, such as ion orbit losses. 25,42,91

IV. DISCUSSION OF THE EXPERIMENTAL RESULTS
This section discusses the experimental findings in the context of prior experimental and theoretical investigations.][94][95][96][97] Although its radial variation has been rarely resolved in experimental analyses, studies at JET by Tala et al. 29 and Mantica et al. 31 indicate an increasing trend over radius, in line with this study.However, the experimentally determined Prandtl numbers were often larger than the corresponding gyrokinetic prediction.Experimental work at JET by Tala et al. 29 revealed a constant Pr during parameter scans of q, R=L ne , and Ã , in agreement with this work.A study at JT-60U 98 arrived at similar conclusions, but identified a dependence of Pr on b N that has not been recovered.
The pinch scaling with density gradient agrees with earlier experimental results, e.g., from the JET tokamak, such as those by Tala et al. 29 This observation further agrees with inter-machine comparisons by Yoshida et al. 33 and Tala et al. 97 In the latter, the inferred pinch numbers were larger than the theory predictions.These discrepancies, found for the Prandtl and pinch numbers, likely arise from a distortion of transport coefficients in that analysis, attributed to the exclusion of intrinsic torque and its time dependence.The q dependence of the pinch 29,30,97 was not evaluated in this work due to limited current variation in the dataset.However, scaling with s could be considered a proxy for the dependence on q.In Refs.29, 43, and 99, no or only marginal dependence of the pinch on collisionality was identified, aligning with the results in this work.The scaling of the pinch on R=L ne and q was also noted in a JET database study. 76hen investigating experimental steady-state toroidal rotation profiles in this work, discharges lacking ECRH exhibited a more convex shape.Specifically, a discharge without ECRH was the only studied plasma with distinctly co-current intrinsic torque in the inner core.Other discharges featuring low ECRH power displayed slightly concave profiles.Similar effects of ECRH on rotation profile shape were previously observed in DIII-D, 100 JT-60U, 98 and AUG. 101,102In this study, the absolute effect of the ECRH on the intrinsic torque appeared weak, likely due to the relatively small P ECRH .No analysis of discharges with sign reversal of the rotation gradient was performed.The author speculates that such reversals occur in scenarios with strong countercurrent intrinsic torque in the core coupled with low external torque, inward momentum pinch, and edge intrinsic torque.However, firm conclusions necessitate the study of more strongly TEM-dominated plasmas with clearly hollow rotation profiles.
Previous works from AUG investigating the LOC-SOC transition 86,102 demonstrated a clear dependence of residual stress on the density gradient R=L n .The presented work made a consistent observation by the two discharges with the lowest/highest density gradients, displaying the most co/countercurrent core intrinsic torque.In global, non-linear gyrokinetic calculations by Hornsby et al., the effects of profile shearing of density profiles were crucial in determining residual stress. 71Other global, non-linear gyrokinetic calculations by Grierson et al., 103 exploring the transition from flat to hollow toroidal rotation with increased ECRH power, attributed the changes most strongly to the modified ion temperature gradient.Those results also, to some extent, align with the findings of the presented work.
Solomon et al. 42 have highlighted the decoupling of intrinsic torque mechanisms in the inner and outer core in agreement with the observations in this work.However, those studies 42,43 struggled to draw firm conclusions about the parameter dependences of the core intrinsic torque.Consistent with this work, core intrinsic torque was found to be small and became more strongly countercurrent with increased ECRH. 42or edge intrinsic torque, various empirical models and scaling laws have been explored in earlier works, including the Rice scaling.In a subsequent publication, 104 Rice et al. found intrinsic rotation ordering by the edge ion temperature gradient.Similar results were obtained in other fusion experiments, such as NSTX, 105 LHD, 106 and DIII-D, 91 identifying strong correlations of intrinsic torque with the ion temperature gradient.Both the Rice scaling and dependence on the edge ion temperature gradient were confirmed in this work.This is expected, given the common cross correlation of stored energy, ion temperature gradient, and pressure gradients at the edge.However, in the present work, these parameters showed inferior ordering characteristics compared to the pressure gradient.Importantly, these studies often focused on net intrinsic rotation, encompassing the interacting effects of all transport mechanisms.
Experiments at the DIII-D tokamak using balanced beams resulted in zeroed rotation profiles. 41,42In those works, diffusive and convective fluxes are neglected, and the effective intrinsic torque is calculated based on the net applied beam torque.Those measurements of intrinsic torque, which exhibit correlation at the plasma edge with the pressure gradient, align with the findings presented in this study.The pressure gradient, especially in the absence of toroidal and poloidal terms of the E r force balance, was identified as a proxy for 11.The experimentally measured intrinsic torque at q u ¼ 0:7 plotted against the pressure gradient, at q u ¼ 0:7 [panel (a)], and in the steepest region of the pedestal [panel (b)].Linear fits are added to guide the eye.
E Â B-shearing, generating residual stress.A similar scaling of intrinsic toroidal rotation with pressure gradient was observed at JT-60U by Yoshida et al. 39 This work refrains from presenting dimensional scaling laws for intrinsic torque, recognizing the limitations of universal applicability.Previous scalings utilized dimensional parameters, such that they cannot be reliably applied to other machines or parameter ranges.The proper normalization for such extrapolation, discussed in Refs.30, 99, and 107, still needs to be solved.An open question not addressed in this work is the dependence of intrinsic torque on q Ã .As the variation of q Ã is relatively limited within the studied dataset, disentangling it from other parameter dependences requires further investigation.Given a tokamak's restricted variation of q Ã , inter-machine comparisons are deemed essential.

V. REDUCED MOMENTUM TRANSPORT MODEL
The scaling laws investigated in this study can be employed to develop a simplified model for predicting rotation in the plasma core of H-modes when an appropriate boundary condition is available.The scalings for the Prandtl and pinch numbers were derived from the regressions of the gyrokinetic database (based on experimental input data, considering all available realistic gyrokinetic effects), which covered a broad parameter space, and were comparable to the experimental findings.Hence, these scalings are adopted as an initial basis for a reduced model: and Through experimental examinations of the intrinsic torque, it is evident that the intrinsic torque in the core exhibits a distinct correlation with the logarithmic density gradient, showing the most pronounced variation in the analyzed dataset around q u % 0.35.Additionally, at q u % 0.7, the intrinsic torque demonstrates a scaling with the local pressure gradient.In the applied modeling, the residual stress was modeled as with the ion sound speed c s and other basic plasma quantities for normalizing to the correct dimensions.As discussed in previous publications, 49,65 the scaling with the momentum diffusivity serves as a proxy to compensate for the change of the turbulence amplitude in the plasma.The dimensionless function values gðq u Þ is for now approximated as a quadratic polynomial without a constant part (for continuity in zero), as only a two-point model is envisioned corresponding to the two main scalings observed in this work.If, in future work, additional independent scaling mechanisms are discovered, they can included easily via a different radial shape.The values of the chosen quadratic polynomial in q u ¼ 0:35 and q u ¼ 0:7 can be fitted to the experimental findings, such that gð0:35Þ ¼ 0:068 Á R=L ne À 0:048 (11)   and gð0:7Þ % 0:0057 ðm=kPaÞ Á rp i þ 0:024: Equation (11) deliberately can lead to a sign change in the intrinsic torque, occurring for values below R=L ne % 0.7.The constant component in the second equation is not meant to contribute to the sign change, but rather represents a constant offset from effects that do not scale with the pressure gradient at this radial position.According to Eq. ( 12), a sign change would occur at Àrp i < 5 kPa/m, a value considerably lower than the experimental values observed at this radial position.Therefore, it is assumed that the contribution of the intrinsic torque at this radial position is consistently co-current.The values of gðq u Þ are interpolated due to the used quadratic polynomial.As with the entire modeling in this work, the torque from the NBI is provided through TRANSP.Additionally, the experimental boundary condition for the rotation is set at q u % 0.8.This specific form of the model is limited to approximate experimental data at AUG within the studied parameter space.While the gyrokinetic scaling laws, Eqs. ( 9) and (10), are not specific to AUG and, in principle, applicable to any other machine in a similar parameter range and turbulence regime, a proper normalization of the intrinsic torque scaling remains a subject for future investigation.The need for normalizing the extrapolation of the intrinsic torque was discussed in Sec.IV.
As an initial validation step, the performance of this reduced model is assessed using the discharge #40 076 (2.0-4.2 s), which served as a reference discharge in previous works. 49,65The results, depicted in Fig. 12, reveal a satisfactory agreement.Panels (a)-(c) display the steady-state, amplitude, and phase profiles, respectively, where the reduced model aligns closely with the error bars of the analysis solution of the reference discharge.Notably, the Prandtl number [panel (d)] and intrinsic torque [panel (f)] exhibit precise matches, while the pinch number is closely aligned, almost within the error bars.Subsequent validation of the reduced model, however, will prioritize evaluating the reconstruction accuracy of the steady-state profiles over transient dynamics.
As a subsequent validation step, the reduced model is applied to the discharges from which the gyrokinetic database was constructed, as discussed in Sec.II, and which served as the basis for the Prandtl and pinch number regression of the reduced model.This specifically allows the simple intrinsic torque model derived from experimental values rather than from gyrokinetic calculations to be examined.In Fig. 13(a), the reduced model solution (y-axis) is compared with the measured toroidal rotation (x-axis), with various radial positions represented by different colors and symbols.As expected, the accuracy of the reconstruction diminishes as predictions move away from the boundary (q u % 0.8).Some points are slightly underpredicted, which could be linked to the observed trend of marginally smaller Prandtl numbers in the experimental regression [refer to Eq. ( 7)] or slightly higher values for the pinch number in the experimental regression [refer to Eq. ( 8)].Overall, the RMSE is approximately 7%, indicating the model's ability to capture the essential features in the core effectively.
Following this successful validation, the reduced model is now employed on a distinct database of H-mode discharges, which was not previously considered in this study.This dataset encompasses experiments conducted in helium, scans involving current and heating variations, and a scan in q Ã .Discharges exhibiting substantial core MHD activities are excluded.In total, this dataset comprises 85 plasma phases from 49 discharges.The parametric variation within this test dataset surpasses that of the experimental dataset from which the scalings were derived, not only in terms of engineering parameters, but also in local quantities, as indicated in column (c) of Table I.Consequently, this dataset is an excellent test bed to evaluate whether the identified scaling laws effectively capture the most crucial mechanisms across a broader range of parameters.This is illustrated in Fig. 13(b).Most of the modeled values cluster around the unity line in black.Aside from the effect of the NBI modulation, the modeled rotation profiles are found to be temporally stable.Therefore, only the steady-state values are presented in this depiction.The RMSE % 9.6% is comparable to that obtained for the gyrokinetic database, and data points are evenly distributed around the unity line.It is observed that the deviation between modeled and measured values increases with distance from the boundary, with RMSE % 10.5% for q u ¼ 0.3, RMSE % 9.0% for q u ¼ 0.4, and RMSE % 6.7% for q u ¼ 0.5.The boundary condition within this dataset varies between 15 and 40 km/s (at q u ¼ 0.8), while the modeling reaches values between 10 and 125 km/s in the inner core (at q u ¼ 0.2).This indicates that the reproduced variation within the dataset is not solely attributed to the boundary condition, but, of course, it plays an important role.Overall, the simple model successfully reproduces the fundamental transport for all three mechanisms in AUG over a wide range of background parameters and can provide a meaningful estimate for the steady-state rotation profiles provided a reliable boundary condition at the edge is given.
In the following, the AUG H-mode discharge #29 216 (5.5-7.0 s) is studied.This NBI modulation discharge features hollow rotation profiles.Analyzing this case has been challenging due to the large uncertainties associated with the experimental data.This discharge involved dominant ECRH with approximately P ECRH % 3.4 MW and NBI at P NBI % 3.0 MW, aiming to investigate the potential sign reversal of the intrinsic torque.The reduced model, which scales the intrinsic torque with the core density gradient and the pressure gradient at the edge, successfully reproduces the sign reversal and the hollow rotation profiles.Figure 14 I.That dataset did not contribute to the assessment of the scaling laws of the reduced model.A unity reference is shown in both plots as a black dotted line.
within experimental uncertainties for most radial positions.The experimental logarithmic density gradient profile in panel (b) reveals high values in the inner core, while the intrinsic torque in panel (c), modeled using Eq. ( 11), exhibits a strong countercurrent behavior in the inner core.Gyrokinetic calculations suggest a mixed turbulence regime in the inner core for this discharge, connecting this sign reversal to the high-density-gradient branch in Fig. 10.The reproduction of the hollow profiles was similarly observed for the comparable discharge #29 217.

VI. SUMMARY AND OUTLOOK
This work used the momentum transport analysis framework established at AUG to study the parametric dependences of diffusion, convection, and intrinsic torque to momentum transport within the core of H-mode plasmas.
Gyrokinetic parameter scans were performed using, among others, the Waltz standard case 75 to compare the following experimental results to theoretical predictions.Scans employing simplified gyrokinetic models are found to be modified when taking more complete, realistic gyrokinetic models into account, such as collisions, electromagnetic effects, or an experimental equilibrium.In addition, the precise results of such scans depend on the chosen starting point in parameter space.A database of realistic gyrokinetic calculations based on experimental data was constructed.It was demonstrated that the Prandtl number exhibits dependence on the inverse aspect ratio, likely as a result of an analogous compensation effect from the self-consistent parallel wave number, as it occurs for the pinch number. 72,74Concerning the pinch number, a strong dependence on the logarithmic density gradient is observed, aligning with the understanding of the Coriolis momentum pinch. 19Additionally, magnetic shear contributes to the scaling of the pinch number.Overall, the scaling coefficients derived from the database of gyrokinetic calculations performed with experimental profiles as inputs agreed qualitatively with fits from the isolated parameter scans, strengthening the findings.
Subsequently, the experimental results are used to validate these theoretical predictions.The analyzed parameter dependences exhibit consistent trends and magnitudes with the theoretical predictions for Prandtl and pinch numbers.The Prandtl number exhibits a monotonic increase across the radius in most discharges in the experimental analysis.The local inverse aspect ratio is the most reliable ordering parameter for the Prandtl number in the experimental analysis.A linear fit provides values for PrðÞ comparable to those predicted by theory, thus confirming this prediction.In the case of the pinch number, the most effective ordering parameter is found to be the logarithmic density gradient.This outcome is consistent with corresponding predictions.A regression analysis involving the density gradient and shear produces trends similar to those derived from regressing the gyrokinetic database.
The intrinsic torque is found to be small and slightly countercurrent directed in the inner core in the experimental analysis.Furthermore, it is observed to become strongly co-current toward the plasma edge.Most likely, different mechanisms drive the intrinsic torque in the inner and outer core regions.
The findings for the intrinsic torque values in the inner core indicate a correlation with the density gradient.Examination of the dominant turbulence regimes reveals two sign reversals: one from dominant TEM with flat density profiles to a mixed regime with peaked density profiles and another from the mixed regime to strong ITG modes with flat density profiles.A correlation with the applied ECRH power is identified.This aligns with previous research on the LOC-SOC transition, 53,86,102 the impact of ECRH on the rotation profile, 98,[100][101][102] and investigations of the effects of profile shearing on the generation of intrinsic torque. 71,86,102,103It is speculated that the intrinsic torque and, in particular, its sign reversal in this radial domain results from profile shearing effects, which would result from terms of higher order in q Ã in the gyrokinetic equations.Furthermore, intrinsic torque could be generated from E Â B-shearing or turbulence intensity gradient effects.
The pressure gradient best orders the experimental data at the outer edge of the fitting domain.The intrinsic torque is co-current directed and does not indicate a sign reversal.This suggests that terms that are linear in q Ã in the gyrokinetic equations, associated with E Â B-shearing, might be accountable, as those effects cannot lead to a sign reversal of the intrinsic torque in this particular scenario.Other effects, such as ion orbit losses, could also contribute.Overall, the observed trends of the intrinsic torque align with expectations from prior experimental 39,41,42 and theoretical work. 80,81,83However, nonlinear global gyrokinetic calculations would be necessary for thorough validation, 71 as they would facilitate interpreting these results by disentangling the different effects.
A discussion of these results in the context of earlier theoretical 19,21,24,72,76 and experimental work 29,31,33,76,97 demonstrates overall agreement regarding the parameter dependences of the Prandtl and pinch numbers.The study confirms previously suggested parameter scalings regarding intrinsic torque, but a conclusive unified picture across multiple tokamak devices is still under development.
Finally, the gyrokinetic scaling laws for the Prandtl and pinch numbers and experimental scalings for the intrinsic torque were utilized to construct a reduced momentum transport model for the core of H-modes in the AUG tokamak.The scalings for the Prandtl and pinch numbers can be expected to have relatively general validity in the parameter range over which they were constructed.They should be applicable also for non-AUG plasmas, as they are dimensionless transport parameters derived from dimensionless gyrokinetic input parameters.One of the limitations of the spanned parameter space is the absence of a thorough scan of the safety factor.In contrast, the intrinsic torque scaling is dimensional, and the regression coefficients could be AUG-specific.However, one can optimistically expect that this scaling could be generalized to other tokamaks with appropriate scaling factors.The model successfully reproduced experimental rotation data for numerous discharges with extensive parametric variations, including those not analyzed in this study, such as discharges with hollow rotation profiles.This success is notable for such a simplified transport model and shows that the model captures the most important contributions to momentum transport in the core plasma in the studied parameter regime.
The next step in enhancing the understanding of this transport channel involves analyzing and validating discharges at AUG with stronger ECRH to include more experiments with mixed or even TEM-dominated turbulence regimes.Furthermore, the validation on other machines includes investigating the q Ã dependence of the intrinsic torque mechanisms.This also addresses the question of normalizing the intrinsic torque terms for extrapolation.Additionally, the boundary condition, set at q u ¼ 0.8 in this work, requires replacement, prompting substantial future work on both theoretical and experimental fronts.
A potential starting point for understanding momentum transport in the outer core or pedestal top could involve applying the presented methodology in small-ELM scenarios or L-mode, where the experimental boundary condition can be set further outside, offering insights into the core-edge coupling of this transport channel.Future work could focus on inferring scaling laws of the pedestal top rotation depending on the applied external torque and local and global machine parameters.This should be done based on a dataset that includes different-sized tokamak devices.
The reduced model has potential for integrated modeling approaches and real-time control.To improve its flexibility, a viable option is to explore alternatives to the time-intensive TRANSP calculations, such as the RABBIT code, 108 which has been integrated into the ASTRA suite.The reduced model could find application in the FENIX flight simulator at AUG, 109 offering potential benefits for scenario development and real-time control.Due to the relatively more straightforward CXRS analysis of edge rotation compared to the core, there is a possibility of providing real-time boundary values for edge rotation to the control system while approximating core values using the reduced model.Additionally, the integration of this model into integrated modeling suites, like IMEP, [110][111][112] stands out as a logical next step.
From the results of this work, a future machine such as ITER, SPARC, or DTT could achieve elevated rotation profiles through steep gradients that are beneficial for significant inward pinch or an edge-localized, co-current intrinsic torque.Whether the overall magnitude of these effects is sufficient to peak the rotation profiles is an open question.The validation of momentum transport theory and the proposed reduced model in this work provide a starting point for a meaningful extrapolation of the rotation profiles of future machines.

Analyzed discharges for experimental analysis
List of discharges analyzed in Sec.III.

FIG. 7 .FIG. 8 .FIG. 9 .
FIG. 7.Measured and predicted values of the pinch number are plotted against the logarithmic density gradient at q u ¼ 0.35.Linear, unweighted fits are included to provide visual guidance.Upward triangles represent discharges with I p ¼ 0.6 MA, while squares denote those with I p ¼ 0.8 MA.

FIG. 10 .
FIG.10.The local core intrinsic torque [panel (a)] and the experimental logarithmic density gradient [panel (b)] plotted against the frequency of the fastestgrowing mode, derived from the gyrokinetic calculations.Fitted lines are included to aid visualization.The error bars shown correspond to the average error bar of the dataset.The sign convention for x r assigns a positive sign to mode propagation in the ion diamagnetic direction (e.g., ITG) and a negative sign for a mode in the electron diamagnetic direction (e.g., TEM).

FIG. 12 .
FIG.12.Evaluation of the optimized modeling results for the reference discharge (#40 076, 2.0-4.2s) in comparison with transport coefficients derived from the reduced model.
FIG.13.Comparison between the experimentally measured toroidal rotation (on the x-axis) and the outcomes from the reduced model (on the y-axis) at various radial positions.Panel (a) depicts the modeling results of the discharges which were used to construct the gyrokinetic database and which are the basis for scaling law assessments of the Prandtl and pinch numbers.Panel (b) shows the results for a distinct dataset of H-mode discharges with the parameter variations outlined in column (c) of TableI.That dataset did not contribute to the assessment of the scaling laws of the reduced model.A unity reference is shown in both plots as a black dotted line.

FIG. 14 .
FIG. 14.(a) Comparison between the experimentally measured toroidal rotation with a hollow profile (brown) and the results from reduced modeling (black, dashed) for discharge #29 216 (5.5-7.0 s, P ECRH ¼ 3.4 MW).In panel (b), it is evident that a significant logarithmic density gradient leads to a strong countercurrent intrinsic torque in the inner core [panel (c)].