The dependence of tokamak L-mode confinement on magnetic field and plasma size, from a magnetic field scan experiment at ASDEX Upgrade to full-radius integrated modelling and fusion reactor predictions

The dependence of the confinement of a tokamak plasma in L-mode on the magnetic field is explored with a set of dedicated experiments in ASDEX Upgrade and with a theory-based full-radius modelling approach, based on the ASTRA transport code and the TGLF-SAT2 transport model and only using engineering parameters in input, like those adopted in scaling laws for the confinement time. The experimental results confirm the weak dependence of the global confinement on the magnetic field, consistent with the scaling laws for L-mode plasmas and in agreement with the full-radius TGLF-SAT2 predictions. The modelling approach is then extended to numerically investigate the confinement dependence on magnetic field, plasma current and plasma size. The weak dependence of the L-mode confinement on the magnetic field at constant plasma current and plasma size is shown to be produced by a balance between the decrease of confinement mainly produced by the reduction of the E×B shearing rate and the increase of confinement provided by the reduced gyro-Bohm factor, when the magnetic field is increased. The ASTRA/TGLF-SAT2 predicted increase of confinement with increasing plasma size is investigated in comparison with the predictions of the global confinement scaling laws for L-mode plasmas and the Bohm and gyro-Bohm dependencies of confinement, highlighting interesting similarities and important differences. Full-radius TGLF-SAT2 simulations with increasing plasma size are then extended to dimensions which are compatible with reactor relevant fusion power production, using ITER and the European DEMO as references. ASTRA/TGLF-SAT2 predictions of fusion power and confinement of an L-mode fusion reactor are presented at both 5.7 T and 10 T of magnetic field on the magnetic axis.


Introduction
Together with the plasma current and the plasma size, for given aspect ratio and isomorphic plasma shape, the magnetic field is the third critical parameter which enters basic aspects in the design of a tokamak. These three parameters are directly connected in determining the safety factor, which limits the amount of current which can be 'safely' carried by the tokamak plasma. An analytical expression for the edge safety factor has the form where a and R are the plasma minor and major radius respectively, B T the magnetic field at the plasma centre, I p the total plasma current, and k the plasma elongation. Current, field and size also critically enter the expression of the Troyon β limit, which is proportional to I p /(aB T ), where β is the ratio of the volume averaged plasma pressure to the magnetic pressure. This property is regularly expressed through the normalized parameter β N = β/(I p /(aB T )). In general, an increase of plasma current has to be compensated by an increase of magnetic field or of plasma size, that also determines the achievable maximum density in a tokamak, through the Greenwald limit I p /π a 2 [1]. Within these general arguments, the observation that the impacts of plasma current and magnetic field on the plasma confinement are strongly different is of critical interest. In both L-mode and H-mode regimes, the confinement time is found to increase with increasing plasma current, whereas it has a weak dependence on magnetic field [2]. This dependence can even be slightly negative in most recent scaling laws for H-mode plasmas [3][4][5], where, comparing different regressions, usually the sums of the exponents of plasma current and magnetic field are observed to feature smaller variations than the current and field exponents separately. Specifically, in L-mode plasmas, the magnetic field dependence was found to be practically negligible in the scaling law IPB98 L-mode [2,6,7] that we report here below for convenience of the reader where I p is the plasma current in MA, B T the toroidal magnetic field in T at the geometrical major radius R in meters, P is the power loss in MW, n is the line averaged density in 10 19 m −3 , M the average ion mass in AMU, ϵ = a/R is the inverse aspect ratio and k is the elongation. The resulting confinement time is in s. The almost linear dependence on plasma current strongly contrasts with the virtually inexistent dependence on magnetic field. A similar feature was obtained in the ITER89-P scaling law [8].
In contrast to the H-mode plasmas in which edge localized modes are present (ELMy H-modes) and where the confinement properties are determined by a combination of effects coming from the transport and the MHD stability of the pedestal, the global confinement properties of L-mode plasmas can be considered to directly reflect the properties of the turbulent transport, which determines the plasma temperature and density profiles in response to the heat and particle sources over the entire cross section. Considering that turbulent transport is generally expected to scale proportional to the Bohm or the gyro-Bohm diffusivities, proportional to B −1 T and B −2 T respectively, it can come as a surprise the experimental result that global confinement in L-mode has a practically negligible dependence on magnetic field. Recent work has been dedicated to the consistency between full-radius predictions from a local quasi-linear turbulent transport description of the plasma and the dependencies of confinement on various regression variables of global confinement scaling laws, such as the plasma current, the heating power and the plasma density, which can be considered as engineering parameters, and are regularly adopted as regression variables. The results from the full-radius transport modelling with the TGLF-SAT2 transport model [9][10][11][12] and the ASTRA transport code [13,14] have been found to reproduce the dependencies of global confinement observed in dedicated experiments in ASDEX Upgrade (AUG) reasonably well. The experimental observations were also found to be broadly consistent with the dependencies of the scaling laws for L-mode confinement [15]. In particular, the dependence on plasma current was confirmed in AUG experiments, but also observed to be weaker than in the scaling laws. The application of the TGLF-SAT2 quasi-linear turbulent transport model showed a relatively good consistency with the experimental results, although the predicted increase of confinement with increasing plasma current was slightly weaker than that found in the experiment in the electron heated cases. Considering that the variation of plasma current is performed keeping fixed all the other engineering parameters, in particular the magnetic field and the plasma shape, and considering that the simulations are performed only using engineering parameters in input, from the standpoint of local transport, the variation in plasma current translates into a variation of the safety factor profile. From an experimental standpoint, a current scan at fixed density and fixed auxiliary power leads to an increase of the total heating power through an increase of the Ohmic power, which can be non-negligible in L-mode plasmas. The variation of the safety factor impacts the local turbulent transport, as well as the, usually smaller, neoclassical transport, and therefore also the predicted stored energy of the plasma. As already mentioned, fullradius modelling with ASTRA and TGLF-SAT2 of L-mode current scans in AUG have been compared to the experimental results in [15]. However, from the perspective of theory validation, the equally critical dependence on magnetic field was not considered in that work and it is subject of this specific study, in combination with a comparison with the dependencies on current and size predicted by the model in dedicated numerical scans. In section 2, the experimental results from a magnetic field scan at AUG and the related full-radius modelling with ASTRA and TGLF-SAT2 are described. In section 3, the results of different sets of simulations representing numerical magnetic field and plasma current scans assess the role of different transport ingredients in determining the (absence of) magnetic field dependence obtained in the experiments and in the related modelling, in contrast to the dependence of confinement on the plasma current. In section 4 the dependence on plasma size produced by the full-radius modelling approach is explored. In section 5 the dependence on plasma size is extended to reactor type plasmas, which can be relevant for nuclear fusion production, with the inclusion of deuterium-triutium fusion reactions. Finally in section 6 conclusions are drawn and an outlook for future verification and validation activities is briefly presented, which could be motivated by the present work.

Experimental results and related TGLF-SAT2 modelling
As a consequence of the strong dependence of the threshold in the heating power for the transition from L-to H-mode on the magnetic field [16], the L-mode operational window at AUG becomes small in the usual AUG magnetic field configuration with low single null, with the low divertor and the ∇B drift pointing to the X-point, more favourable to access H-mode. The motivation provided by the recent results on full-radius modelling of L-mode plasmas [15] has suggested to directly perform some experiments in AUG during the last campaign, in which only operation in such usual magnetic field configuration was planned. This limited the L-mode operational window, particularly with respect to the total heating power, but still allowed a significant variation of magnetic field values which gave results largely consistent with the global confinement scaling laws, that is, a very weak dependence of the stored energy on the magnetic field. A variation from 2.38 T to 2.98 T at two low NBI heating power levels has been possible in L-mode, whereas a broader variation also including 1.79 T and 1.99 T has been performed, but in this case the plasma directly moved to a I-phase [17] already at the lowest power level, featuring a small, although still not completely negligible, increase of stored energy at the transition. While these experiments cannot be considered appropriate for a very accurate determination of the magnetic field dependence of L-mode confinement, nevertheless, when the stored energy of all of these plasmas is examined, very limited variations with increasing magnetic field are observed, in complete consistency with the scaling law [2]. This observation allows us to state that present AUG results, although somewhat limited in the parameter variations, confirm the absence of a strong magnetic field dependence of the L-mode confinement time. These results are shown in figures 1 and 2 for two NBI power levels, corresponding to 0.38 MW and 0.57 MW. These very low NBI power levels have been made possible by operating the NBI system at AUG at reduced voltage, in particular the NBI box 1 at 31 kV and the NBI box 2 at 30 kV, profiting of the recently installed variable gap on the first source of box 2 [18]. The discharges were performed with a plasma current of 0.83 MA and a line averaged density around 3.6 10 19 m −3 . The four values of magnetic field allowed a relatively large variation of the safety factor q 95 , with values of 3.67, 4.10, 4.85 and 6.04, respectively corresponding to the magnetic fields on axis of 1.79 T, 1.99 T, 2.38 T, 2.98 T. The reduced voltages used with the NBI also allowed us to produce a dominant auxiliary ion heating in the plasma, with about 62%-65% of the auxiliary heating delivered to the ions. The Ohmic power is about 0.5 MW, the effective charge is reconstructed to be very low by integrated data analysis (IDA) (around 1.1) [19] and the tungsten concentrations, measured by grazing incidence spectrometry [20], are also observed to be very low, between 1.5-2.0 10 −5 , leading to total radiated powers inside the last closed flux surface of about 0.1 MW. These heating power levels provide a total heat flux on the electrons of about 0.6 MW combining Ohmic power, NBI power and radiation, and a total heat flux on the ions from 0.25 to 0.5 MW, for the low and high NBI power levels respectively, therefore still producing slightly more total electron heating than ion heating even in the high NBI power phases. Profile measurements have been challenged by the change in magnetic field for the electron cyclotron emission radiometry, and by the low NBI voltage for active charge exchange recombination spectroscopy (CXRS), which, particularly in the lowest NBI heating phase, has made use of a short blip at that reduced voltage. Despite these limitations and the related uncertainties, these AUG experimental results appear to confirm the general expectation which can be derived from the IPB98-L mode scaling law of a practically negligible dependence of the confinement time and the plasma stored energy on the magnetic field.
The electron and ion temperatures, as well as the electron density measurements and the related profile fits used for comparison with the predictive transport modelling results, are illustrated in figures 1(a)-(c) and 2(a)-(c) respectively, with figure 1 presenting the results during the low NBI power phase and figure 2 those during the higher NBI power phase. The corresponding thermal stored energies on electrons, ions and total are plotted in figures 1(d) and 2(d). The electron temperature and density profiles are obtained by reconstructions with the IDA [21] combining different diagnostics, the ion temperature Experimental profiles as a function of the normalized toroidal minor radius ρ ϕ of the electron temperature reconstructed by IDA (a), ion temperature from CXRS measurements (b) and electron density from IDA (c) as well as the corresponding total, electron and ion thermal stored energies (d) as a function of the magnetic field at the geometrical axis for the four plasma phases at 0.83 MA and 0.38 MW of NBI heating. The solid line in (d) shows the predicted dependence of the IPB98-L scaling law, with the numerical factor in front which has been renormalized to match the mean value of the AUG points.
is obtained by CXRS measurements [22]. While experimental uncertainties do not allow a precise determination of the exponent, also due to the limitations provided by the fact that the two points at low magnetic field directly featured a I-phase behaviour, it is clear that the dependence on magnetic field is very small and consistent with the ITER98 L-mode scaling, which is also plotted in figures 1(d) and 2(d), rescaled by a factor 1.45 to match the average value of the stored energies of these AUG plasmas (the overprediction on average by the IPB98 L-mode scaling law of the AUG L-mode confinement was already found in [15]).
Profiles of the safety factor from magnetic equilibrium reconstruction obtained with the ASTRA code in interpretive mode are shown in figure 3. The corresponding profiles of the ratio of the magnetic shear to the safety factor s/q, relevant for the threshold of microinstabilities [23], are also shown.
The corresponding results obtained with full-radius integrated modelling with the ASTRA code and the transport model TGLF-SAT2 [11,12] are presented in figures 4 and 5 respectively for the two NBI power levels, demonstrating a relatively good agreement, considering that all of these simulations have been performed with the same modelling approach described in [15] and therefore do not use any input from the temperature and density profile measurements. Here 161 radial grid points are used from the magnetic axis to the separatrix, and the TGLF-SAT2 transport model is called at each radial grid point. Input parameters are the engineering variables and plasma shape, the NBI heating source, computed with the RABBIT code [24], and the volume averaged density, which is used to obtain the required density in the simulations by a feedback procedure on the gas puff level. The relatively small toroidal rotation, with Mach numbers below 0.15, is not predicted but is taken from experimental measurements and included in the calculation of the radial electric field.
The general result is that, consistently with the experimental observations and the IPB98 L-mode scaling law, TGLF-SAT2 predicts an almost negligible dependence on magnetic field. This overall agreement should not be considered a trivial result. The theoretical model contains several elements which critically depend on the magnetic field, in particular the gyro-Bohm factor, with which all of the turbulent fluxes scale Experimental profiles as a function of the normalized toroidal minor radius ρ ϕ of the electron temperature reconstructed by IDA (a), ion temperature from CXRS measurements (b) and electron density from IDA (c) as well as the corresponding total, electron and ion thermal stored energies (d) as a function of the magnetic field at the geometrical axis for the four plasma phases at 0.83 MA and 0.57 MW of NBI heating. The solid line in (d) shows the predicted dependence of the IPB98-L scaling law, with the numerical factor in front which has been renormalized to match the mean value of the AUG points.
proportionally. Thereby it could be considered even surprising that the theoretical model predicts such a weak dependence in agreement with the experimental results. These considerations motivate an investigation of the reasons behind the absence of a strong dependence in magnetic field, also in comparison with the dependence of confinement on the plasma current. To this end, in the next section simulations of artificially created numerical scans in magnetic field and plasma current are performed to single out the impact of different ingredients in the transport.
Finally, it is worth to notice that figures 4(b) and 5(b) show that the predicted ion temperature profiles can feature a small peripheral layer with very high transport, and therefore locally very flat ion temperature profiles, at high values of the magnetic field. This trend suggests a role of the E×B shearing rate at the edge, which becomes weaker at high magnetic field, as it will be more explicitly presented in the next section. When the E×B shearing rate becomes too weak, it cannot compensate the turbulence drive, which is particularly produced by the strength of the density gradient. A modification of the boundary conditions can mitigate this effect. We also observe that the general boundary condition which is applied for the density profile predictions, that is n e,sep = ⟨n e ⟩ Vol , sets the separatrix density values slightly higher than those of the experimental IDA density profiles. However it is important to mention that usually these features do not strongly affect the total stored energy, since predicted profiles re-adjust in such a way that they largely overlay in the core, although boundary conditions are strongly modified (figures 4 and 6 of [15] show examples of this effect). Moreover, some of the experimental profile shapes, particularly of the ion temperature, are not correctly reproduced by the TGLF-SAT2 modelling. The theoretical predictions show pretty similar, typically linear, radial profiles at all magnetic field values, as one can expect from a theoretical model. Such theoretically predicted profile shapes have been found consistent with the majority of ion temperature profile measurements at AUG, as shown in 14. In contrast, some differences in the core profile shapes are obtained in the ion temperature measurements obtained in this work. As we have already mentioned, the conditions of these experiments, which adopted NBI at very low voltage to allow low heating power and keep the plasma in L-mode, have been challenging for CXRS measurements. Therefore no general conclusion can be drawn on the relevance of this disagreement.

Magnetic field and plasma current dependencies in numerical scans
First, a broader three points numerical magnetic field scan is performed keeping fixed all of the plasma parameters except the magnetic field and the corresponding variation of the safety factor profile. This numerical scan explores values of the magnetic field of 1.66 T, 2.48 T and 4.92 T, therefore beyond those which are possible on AUG (AUG has maximum magnetic field of 3.1 T, and is usually operated around 2.5 T to allow central electron heating with the 140 GHz gyrotrons in ×2 mode). With a current of 0.83 MA, this implies the corresponding values of 3.40, 5.05 and 10.0 for the safety factor q 95 . Such a broader scan in magnetic field can be expected to more clearly reveal the trends predicted by TGLF-SAT2. In the simulations, we artificially consider conditions in which 1.5 MW of auxiliary heating are provided to the electrons or to the ions only, with a localized central deposition of the heat. This procedure allows the investigation of the impact of these two different heat flux conditions, which also impact the consequent turbulent state of the plasma. While in the electron heated case, this can be considered to be a realistic heating profile, as produced by electron cyclotron resonance heating, in the ion heated case such a centrally localized ion heating profile corresponds to an idealization. However this allows us to numerically study electron and ion heated conditions in a completely symmetrical way. In order to ensure such a difference in the heat flux, the density is kept small, volume averaged density 1.7 10 19 m −3 , in such a way to minimize the collisional thermal exchange between electrons and ions. Ohmic heating is consistently included in the simulations, and the same 0.32 MW of radiated power are also included in all of the simulations. The toroidal rotation profile is imposed equal to zero, but the consistent effect of the radial electric field produced by the combination of main ion pressure gradient and neoclassical poloidal rotation is kept. The poloidal rotation is computed with NCLASS [25], like all the other neoclassical transport coefficients which are required.
The thermal stored energies predicted by these simulations are presented in figure 6 and feature a very weak increase of the stored energy with increasing magnetic field in the ion heated cases, and a somewhat stronger increase in the electron heated cases, with in general a stronger increase of the electron energy confinement as compared to the ion energy confinement. The exponents of the dependencies are reported in table 1. Additional simulations with mixed electron and ion heating fractions have been performed and are not shown here, since they feature a dependence of the stored energy on the magnetic field which is intermediate between these two extreme cases of electron and ion heating only. We can conclude that overall these simulations show a reasonable consistency with the scaling law results as well as with the specific AUG experiments and related dedicated modelling presented in the previous section.
The corresponding predicted electron and ion temperatures, electron density and the self-consistently computed E×B shearing rate are shown in figures 7 and 8 for the electron and ion heated cases respectively. The strength of the E×B shearing rate γ E is computed through the Waltz-Miller formula [26], A small manipulation shows that from which we observe that the E×B shearing rate is inversely proportional to the toroidal magnetic field, but also that it has a component that increases in absolute value with increasing magnetic shearŝ. In the absence of toroidal rotation, and assuming neoclassical poloidal rotation, like in the simulations performed in this section, the radial electric field can be expressed in the form where Z i is the main ion charge, R/L ni and R/L Ti are the logarithmic main ion density and temperature radial gradients, normalized to the major radius R, and K 1 is the neoclassical flow coefficient, which is function of both trapped particle fraction and collisionality. As a consequence of these dependencies, in figures 7(d) and 8(d) the strength of the E×B shearing rate at the edge increases with decreasing magnetic field, particularly from the high field case to the intermediate field case, leading to an increase of plasma pressure in the peripheral region, more significant in the ion heated cases. Being mainly connected with the ion pressure gradient, the strength of the E×B is significantly stronger in the ion heated cases (figure 8) than in the electron heated cases (figure 7). As an additional comment, we observe that density profiles exhibit an increasing peaking with increasing magnetic field in the ion heated magnetic field scan, with a particularly stronger peaking at 4.92 T, as shown in figure 8(c). Such a variation of density profile shape is not observed in the AUG experimental results, which however cover a more limited range of variation of the magnetic field. The strong reduction of the E×B shearing rate at this high field increases the peripheral transport and reduces the impact of the particle source in that region, leading to a weaker density gradient at the periphery. At the same time, the predicted stronger electron temperature gradient, connected with reduced gyro-Bohm factor, drives a stronger particle convection in the core in these conditions of dominant ion temperature gradient (ITG) turbulent transport [27,28]. The stronger density peaking in the confinement region also allows the condition of matching the particle content to be obtained at lower gas puff rate, which is also consistent with a reduction of the density at the edge. At lower field, with a lower electron temperature gradient and a reduced inward electron convection, the electron density profile is less peaked in the core, what requires a higher edge value. This is produced by an increase gas puff rate in combination with a stronger E×B shearing rate. This explains the different density profile shapes, particularly of the case at 4.92 T. We observe that the cases at 1.7 T and 2.5 T, which are in the accessible AUG range, show more limited variation of the electron density profile, more consistent with the experimental results, which were also obtained at lower heating power, as compared to the present numerical   Values of the exponent of the magnetic field in the dependence of the total, electron and main ion stored energies in the magnetic field scan from the TGLF-SAT2 predictions with and without the inclusion of the E×B shearing rate from the consistent radial electric field. The last column reports the corresponding figure in the paper. scans. A similar trend of increased electron density peaking at increased magnetic field is also present in the electron heated magnetic field scan, figure 7(c). However, in this case the more dominant trapped electron mode (TEM) turbulence implies that these effects on the predicted electron density profile are weaker, as also weaker are the effects at the edge, given the overall lower values of E×B shearing rate.
Since the E×B shearing rate introduces an explicit dependence on magnetic field, the impact of this physics ingredient is assessed by reproducing the same set of simulations while imposing the entire profile of the radial electric field equal to zero in the TGLF input and therefore removing the effect of the E×B shearing rate (we recall that in the previous set of simulations E r was consistently computed from the combination of the diamagnetic and neoclassical poloidal rotation terms, assuming zero toroidal rotation). The results are presented in figure 9. The corresponding regression exponents are also given in table 1. The removal of the E×B shearing effect has a strong impact on the resulting reduction of ion energy confinement, particularly at low field, where the E×B shearing rate is stronger, while in contrast the predictions at high field are almost independent of the inclusion of the E×B shearing effect. The combination of the strong impact of the E×B shearing and its dependence on the magnetic field can be also connected to the magnetic field dependence of the power threshold for the transition from L-to H-mode confinement [16]. The electron thermal confinement is found to be less sensitive to the value of the E×B shearing in these ASTRA/TGLF-SAT2 simulations, leading to the result that in conditions of dominant electron heating the resulting dependence of confinement on the magnetic field is almost negligibly modified by the removal of the shearing rate. This is in clear contrast to what happens to the ion thermal confinement and to the stored energy in the ion heated cases, as reported in table 1 in terms of exponents of the predicted dependencies. Since in these simulations the radial electric field is produced by the combination of ion density and ITGs, it is also weaker in the electron heated cases, as already shown in the comparison of figure 7 with figure 8. Recently, the impact of the E×B shearing in TGLF-SAT2 on the edge transport has been also found to reasonably well reproduce nonlinear gyrokinetic results of the GENE code for AUG edge parameters approaching the L-H transition, although with a weaker stabilization at high shearing rates (much higher rates than those obtained in the simulations presented here) [29]. In this recent study, it has been also found to lead to the possibility of self-consistently producing pedestal formations and L to H type of transitions in full-radius transport simulations [29].
A companion set of simulations is now performed removing another explicit dependence of transport on the magnetic field, that is, the dependence which comes from the gyro-Bohm factor, T/eB (ρ s /a) ∝ B −2 . The gyro-Bohm scaling of turbulent transport is directly produced by the dimensionless form of the gyro-kinetic model, and it represents the natural scaling of turbulent transport in the local limit [30]. Following these considerations, a corresponding set of simulations is performed by setting a constant magnetic field in the gyro-Bohm factor of the TGLF-SAT2 model, and equal to 2.48 T, that is the middle value of the present B T scan. Therefore, the middle B T point of this new set of simulations provides exactly the same  result, whereas the points at low and high magnetic fields are modified through the replacement of the actual magnetic field values (respectively 1.66 T and 4.92 T) by 2.48 T, leading to reduction of transport by a factor 2.25 and an increase of transport by a factor 4, at low and high field respectively. Figures 10  and 11. show that the resulting magnetic field dependencies of confinement, respectively obtained with and without the inclusion of the E×B shear from the consistent radial electric field, are strongly affected. A strong decrease of confinement with increasing magnetic field is particularly obtained in the ion heated cases with the inclusion of the E×B shearing rate ( figure 10(b)), whereas, in agreement with the previous results, the removal of the impact of the radial electric field removes both dominant effects, the E×B shearing and the magnetic field dependence in the gyro-Bohm factor, leading to a weaker dependence, in both electron and ion heated cases (figure 11). In general the effects are found to be significantly stronger in the ion heated conditions as compared to the electron heated cases. The specific impact of a change of the safety factor value in the plasma periphery on the transport is usually weak. This can be understood through the fact that effects on transport due to an increase of q saturate at large values of q, and therefore a change of the safety factor at the edge, where the safety factor is in any case large, usually has limited impact.
The exponents of the magnetic field which describe the predicted dependencies of the confinement as obtained in this new set of simulations are reported in table 2. By comparison with table 1, these numbers quantify the differences provided by the inclusion of the E×B shearing rate and the removal of the magnetic field dependence of the gyro-Bohm factor.
The last element that we consider is connected with the modification of the safety factor profile. This is a common feature which equally affects a magnetic field and a plasma current scan, although in opposite directions. Moreover, while the plasma current does not enter the gyro-Bohm factor, the modifications of the safety factor profile can play a role on the local impact of the E×B shearing, leading to the expectation that the E×B shearing rate can also affect the current scan. Therefore, a corresponding set of simulations has been performed with and without the inclusion of the E×B shearing rate from the consistent radial electric field, with fixed magnetic field at 2.48 T, and with plasma current which varies from 0.42 MA to 1.24 MA. In terms of safety factor, 0.42 MA and 1.24 MA correspond to the previous magnetic field cases at 4.92 T and at 1.66 T respectively. The results with and without the inclusion of the radial electric field are presented in figures 12 and 13 respectively. The related exponents which quantify the dependence of confinement on the plasma current for these scans are reported in table 3.
We note that there is an additional difference when the safety factor profile is changed by modifying the magnetic field or by modifying the plasma current. This is connected with the variation of the Ohmic heating. While the increase of the magnetic field at fixed plasma current produces a very limited, practically negligible, variation of the Ohmic power, the increase of plasma current at fixed magnetic field is accompanied by a corresponding non-negligible increase of the Ohmic power. Within the simulation strategy that we have applied so far, the Ohmic power has been consistently computed inside ASTRA. For all of the magnetic field scans presented so far in figures 6-11 the variation of the Ohmic power, and correspondingly of the total heating power, has been extremely limited, so that these results can be considered to be representative of pure magnetic field dependencies only, at constant total heating power. For instance, the results presented in figure 6 have Ohmic power around 0.21 MW for the electron heated cases and around 0.44 MW for the ion heated cases, corresponding to total heating powers of 1.38 MW and 1.63 MW, with variation of the total heating powers below 3% in both the electron and the ion heated cases respectively, and a relatively small impact of the Ohmic heating on the total heating power, which also has 1.5 MW of auxiliary heating and 0.32 MW of radiated power for all of the cases. In contrast, when the plasma current is increased, the Ohmic heating power also increases when this is consistently included in the transport simulations, leading to an increase of the total heating power with increasing current. As an example, in the plasma current scans presented in figure 12, which include the E×B shearing rate, the increase of the Ohmic power is  Values of the exponent of the magnetic field in the dependence of the total, electron and main ion stored energies in the magnetic field scan from the TGLF-SAT2 predictions, removing the magnetic field dependence from the gyro-Bohm factor, with and without the inclusion of the E×B shearing rate from the consistent radial electric field. The last column reports the corresponding figure in the paper. The variation of the total electron heating produced by the variation in Ohmic heating power has small impact on the auxiliary electron heated cases, which have small fraction of Ohmic heating, and more significant impact on the auxiliary ion heated cases. Interestingly, the increased electron temperatures, produced by the increased total electron heating when the Ohmic power increases with increasing plasma current, produce a visible variation in the stored ion energy, through modifications of the transport. This modification of the stored ion energy produces an increased dependence of the stored energy on the plasma current in the case in which the total heating power is kept constant, open symbols in figure 12(b), as compared to the case in which the total heating power increases with increasing current, full symbols in figure 12(b). In figure 13, analogous results of the same current scans are presented, this time obtained neglecting the impact of the E×B shearing rate in TGLF-SAT2. A significantly weaker dependence on the plasma current is predicted, this time specifically reflecting the impact of the modification of the safety factor profile only in the TGLF-SAT2 predictions when the E×B shearing rate is neglected. As it should be expected, the open symbols in figures 12 and 13, obtained with constant heating power, practically mirror the results obtained in figures 10 and 11 respectively. From the transport modelling standpoint, once the magnetic field is kept fixed in the gyro-Bohm factor in the magnetic field scan and the Ohmic heating power is rescaled to a constant value in the current scan, the simulations become largely equivalent. In fact, there would be one additional potentially critical parameter which is still differently modified, and which we have been not explicitly mentioned so far. This is the β parameter, ratio of plasma pressure to the magnetic pressure. At constant plasma pressure, this parameter is proportional to B −2 T . However, this parameter, while it has been consistently included as input in all  the TGLF-SAT2 calculations, is not discussed in detail here, since it cannot be expected to play a particularly important role in these L-mode conditions, particularly in the context of the present simulations which adopt a quasi-linear transport model. While the variation of electron β e is significant (in the numerical magnetic field scans β e varies from 0.45% to 0.06% with electron heating and from 0.27% to 0.04% with ion heating), the β e values remain sufficiently limited not to produce a strong effect. Moreover, in the electron heated cases, where β e reaches higher values, turbulent transport is mostly produced by TEMs, which are weakly affected by β [31]. Last point which is worth mentioning, when the impact of the E×B shearing rate is neglected in TGLF, the dependencies on magnetic field in the electron heated and ion heated conditions become closer. The same happens for the corresponding dependencies on the plasma current. Finally we mention that the impact of sawteeth has been only included through an artificially applied additional transport in the central region of the plasma, inside ρ ϕ = 0.2, in the same way in all cases, without considering the broader q = 1 region obtained with decreasing q 95 . The case with the lowest q 95 has the q = 1 location at ρ ϕ = 0.29. Even in the extreme case that the predicted profiles are cut completely flat inside that radial location of q = 1, the reduction in total stored energy is below 8%. We consider therefore that the accurate treatment of the impact of sawteeth on confinement is not significant for the type of analysis that is performed in this work.
In conclusion, by singling out the effects connected with the modification of the safety factor profile, the role of the radial electric field, and the magnetic field dependence in the gyro-Bohm factor, we obtain that only a relatively small part of the TGLF predicted dependencies are connected with the modification of the safety factor profile alone. An important effect is clearly given by the inclusion of the E×B shearing rate. The combination of these effects mainly determines the predicted dependence of the stored energy on the plasma current in the plasma current scans. In the magnetic field scans, the impact of these dependencies is offset by the magnetic field dependence of the gyro-Bohm factor, leading to a predicted weaker dependence of the stored energy on the magnetic field as compared to the dependence on the plasma current, at least in the ion heated cases. The electron heated cases feature comparable weak, but still significant, positive dependencies of the predicted stored energies as a function of both the magnetic field and the plasma current. The magnetic field dependence of the gyro-Bohm factor yields a positive dependence of the stored energy on the magnetic field, in spite of a positive dependence on the plasma current.
The weak, but still non-negligible, increase of stored energy with increasing magnetic field in the electron heated cases could appear inconsistent with the experimental results presented in section 2. However, it has to be underlined that these AUG experiments explored conditions of dominant ion heating, which are indeed predicted to have an almost negligible dependence of stored energy on magnetic field. At the same time, a magnetic field scan with auxiliary central electron heating would be difficult to perform in experiments, due to the dependence of the resonance location of electron cyclotron resonance heating with magnetic field.
As a final remark, in an attempt of more clearly identifying specific magnetic field effects on confinement, an additional set of simulations has been performed at 9.8 T. These were motivated by the idea that the effect of the strongly reduced gyro-Bohm factor at high field, and therefore strongly reduced transport stiffness, can become more visible in the overall dependence of confinement on the magnetic field, when the corresponding compensation from reduced E×B shearing rate becomes negligible. However, the results only show small deviations from the trends identified with the three point B T scans presented in these section. This can be understood from the fact that the gyro-Bohm factor also has a sort of asymptotic behaviour at very large magnetic field, due to the B −2 T dependence. Therefore, even in such an extreme conditions, we have not observed clear signatures of a strong deviation from the trends determined by the lower values of the magnetic field, nor from the weak dependence on the magnetic field of the Lmode confinement scaling law. Moreover, within the strategy of the present numerical approach, with constant plasma current (0.83 MA) and heating power (1.5 MW), a plasma with 9.8 T (q 95 = 18.7) is largely outside the usual domain of operation of existing and future tokamaks, and therefore also of the datasets over which scaling laws are developed. In particular, from equations (4) and (5), γ E becomes extremely low at high magnetic field and low plasma pressure, that is, at low heating power, whereas normal operation at that magnetic field would be obtained at much higher heating power, and therefore much higher plasma pressure, also leading to more significant E×B shearing rates. These arguments suggest that, although one can enter pretty extreme regimes of parameters where the compensation between the B T dependence of the gyro-Bohm factor and the E×B shearing rate does not occur any longer, these regimes are at the limits of these corresponding dependencies, where asymptotic behaviours occur, and outside the usual domain of operation within which tokamaks operate (simply because one would not waste a very high magnetic field with concomitant very low plasma current and very low heating power).

Predicted dependence on plasma size from an L-mode numerical scan
We consider now the impact of an increase in plasma size, adopting the same modelling procedure, which allows us to fix a set of engineering parameters, in particular the total heating power and the volume averaged electron density, as well as to keep fixed the plasma aspect ratio R/a and keep isomorphic the plasma shape, also leading to the same elongation and triangularity. Once all these parameters are fixed, the major radius (and proportionally the minor radius and the plasma boundary) can be increased in three basically different ways. The first is by also keeping fixed the magnetic field and the safety factor profile. In this case the plasma current increases proportional to the major radius I p ∝ R maj . The second case is by also keeping fixed the plasma current and the safety factor, and in this case the magnetic field has to decrease inversely proportional to the major radius B T ∝ 1/R maj . Finally, the third possibility is that the plasma current and the magnetic field are kept fixed, then in this case the safety factor profile has to change, with q 95 ∝ R maj . This last approach corresponds to that usually adopted by scaling laws for global confinement, which include magnetic field and plasma current as independent regression variables in the multivariate regression. In the present specific application, we consider D plasmas with 3 MW of total heating power (which is kept constant, by also rescaling the OH power), and auxiliary heating divided in 1.5 MW to electrons and 1.5 MW to ions, with localized central deposition. A volume averaged density of 3 10 19 m −3 is also imposed by peripheral gas puff feedback and the toroidal rotation is assumed to be equal to zero over the entire profile. The plasma shape is taken from an AUG equilibrium reconstruction. The plasma current is increased over the three values 1.24, 2.48 and 3.72 MA, allowing an increase of the major radius from 1.60 to 3.20 and 4.81 m. The same scan in major radius is obtained by keeping the current fixed at 1.24 MA, and decreasing the magnetic field from 2.49 T to 1.21 T and 0.82 T. Finally, three compatible safety factor profiles from AUG equilibrium reconstructions, and shown in figure 14, have been adopted, corresponding to q 95 = 3.86, 5.17 and 9.56, concomitantly allowing the major radius to increase from 1.60 m to 2.31 m and 4.31 m respectively. The corresponding profiles of the ratio of magnetic shear to safety factor s/q are also shown. We observe that, analogously to the safety factor profiles of the experimental cases shown in figure 3, they exhibit a considerable variation at the plasma periphery. The case with the lowest safety factor has the highest value of s/q. A summary of the global confinement results of the full-radius ASTRA/TGLF-SAT2 modelling solutions for these three cases is provided Figure 14. Profiles of the safety factor (a) and corresponding profiles of the ratio magnetic shear to safety factor s/q (b) used in the major radius scan at constant magnetic field and plasma current. by figure 15. The ASTRA/TGLF-SAT2 predicted confinement times, plotted with stars, are shown to increase with increasing plasma size in figure 15(a), with exponents of the major radius which are 2.77, 2.55 and 2.59 respectively with increasing current, decreasing magnetic field and increasing safety factor. The corresponding logarithmic-linear fitting curves are presented with solid lines. As it could have been expected, a stronger increase of confinement time with size is obtained when the plasma current is proportionally increased, whereas the weakest increase is obtained when the magnetic field is decreased with increasing plasma size, keeping constant safety factor and current. In the case of B T ∝ 1/R maj , the decrease of magnetic field increases the gyro-Bohm factor of the transport, proportional to B 2 T . A very similar dependence is obtained when the safety factor is increased with increasing major radius, keeping fixed magnetic field and current. It remains that these three ways of increasing plasma size do not exhibit very strong differences in the resulting major radius exponent. The dash-dotted lines in figure 15(a) illustrate the results obtained with the IPB98-L mode scaling law [2], with the same colour code as in the legend. We observe that the TGLF-SAT2 results are more optimistic with respect to the scaling law regression, but at the same time consistent with the scaling law predictions in the case plasma size and plasma current are increased proportionally. In particular it is interesting to note that in the case the plasma current is increased proportionally to the plasma size, then the TGLF-SAT2 predictions are perfectly matched by the IPB98-L mode regression, with the sum of the current and major radius exponents, 0.93 and 1.83 respectively, which perfectly match the logarithmic fit of the TGLF-SAT2 results (with exponent 2.77), as also shown in figure 15(a). Such a perfect match should be considered a coincidence. In contrast, in the case the magnetic field and the plasma current are kept fixed, which rigorously corresponds to the multivariate regression approach of the scaling law in which magnetic field, plasma current and major radius are independent regression variables, the major radius exponent of 2.59 is significantly larger than those of the ITER-89 P scaling law [8], which features the exponent 1.5, and of the IPB98-L mode scaling law [2], which features the exponent 1.83.
We also observe that both electron and ion thermal stored energies increase almost exactly in the same way in the simulated conditions of 50% electron and ion heating fractions, figure 15(b), although the electron stored energy is in general higher than the ion stored energy, due to the predicted higher heat transport in the ions than in the electrons. As a consequence of this, there is no significant change of the T e /T i ratio with increasing plasma size in these conditions. In figures 15(c) and (d), the confinement times are plotted normalized to the Bohm and the gyro-Bohm confinement times, obtained from the ratio of the minor radius squared divided by the corresponding Bohm and gyro-Bohm heat conductivities, computed with volume averaged temperatures. We have defined the gyro-Bohm conductivity as χ gB = ρ s c s (ρ i /a) where ρ i is the ion Larmor radius and ρ s is the Larmor radius computed with the sound speed, and c s is the sound speed. The Bohm heat conductivity, proportional to T e /ZeB, has been expressed as χ B ∝ ρ s c s and, for plotting purposes, the proportionality factor has been chosen in such a way that the corresponding value of τ th /τ B be the same as that of τ th /τ gB at the lowest magnetic field value in each R maj scan. We observe that all of the three different ways of increasing the plasma size lead to TGLF-SAT2 predictions which behave more favourably than Bohm, figure 15(c). In the comparison with the gyro-Bohm normalization, the case, in which the plasma size is increased with B T ∝ 1/R maj , behaves more favourably than gyro-Bohm, figure 15(d). We notice that in this case the gyro-Bohm confinement time is reduced by the reduction of the magnetic field with increasing plasma size (τ gB ∝ B 2 ). In contrast, the case in which plasma size is increased with increasing current has a behaviour which is less favourable than gyro-Bohm, and even less favourable is the case in which the plasma size is increased with increasing safety factor. The corresponding dependencies of the IPB98-L mode scaling law are plotted with dash-dotted lines. In dimensionless parameters the IPB98-L scaling law reads, Definitions of these physical quantities can be found in [2], where in particular τ B is the Bohm confinement time. By noticing the positive exponent of ρ * in equation (6), we observe that the L-mode scaling law behaves even worse than Bohm. This is also demonstrated in figure 15(c), where the corresponding predictions of the IPB98-L mode scaling law are plotted normalized to the Bohm confinement time. As already mentioned, appropriate direct comparison with the scaling law can be made when current and magnetic field are kept constant, which is the approach used in the multivariate regression. In this case we observe that the IPB98-L mode scaling predictions are indeed worse than Bohm, consistent with the dimensionless form of this scaling law [2], reported in equation (6). The property of a confinement time normalized to Bohm which increases with increasing ρ * is clearly opposite to what the gyro-Bohm limit would prescribe, that is, the ratio of the confinement time to the Bohm time is proportional to ρ −1 * (this limit is approached by the IPB98(y,2) scaling law, with a ρ * exponent of −0.7 when the scaling law confinement time is normalized to Bohm, whereas the new ITPA20-IL [3] is extremely close to Bohm, that is, the ρ * exponent of the confinement time normalized to Bohm is very close to zero, 0.03). It can be speculated that the worse-than-Bohm confinement of the L-mode scaling law could reflect the fact that this scaling law contains data from the late eighties and early nineties [6,7] which have an important fraction of observations from relatively small devices, which are characterized by a more global and Bohm-like type of confinement behaviour. A new L-mode scaling law based on larger devices might give a more favourable size scaling, this is what is at least suggested by the present integrated modelling results, although the remarkable difference in the ρ * exponents of IPB98(y,2) and ITPA20-IL scaling laws shows how sensitive this parameter is to the database properties.
A local turbulent transport model like TGLF-SAT2 has to be expected to provide a major radius dependence of the confinement time which is closer to the gyro-Bohm limit, although several physics ingredients, in particular connected with the non-adiabatic electron response and the E×B shearing rate, can significantly modify this dependence. In particular, the role of the E×B shearing rate belongs to a class of gyro-Bohm breaking mechanisms which are connected with profile shearing effects [32][33][34], and therefore, in practice, with second derivatives of the radial profiles. The E×B shearing rate can be considered the only component of these profile shearing effects which is included in the usual local description. The E×B shearing rate also has an important impact in altering the isotope dependence of confinement [35].
For comparison, the major radius scan performed with constant plasma current and safety factor, and therefore with concomitant decrease of the magnetic field, has been repeated neglecting the impact of the E×B shearing rate. The results are plotted with open symbols in figures 15(a), (c) and (d) for the predicted confinement time in seconds and normalized to the corresponding Bohm and gyro-Bohm confinement times respectively. In this case the size scaling is significantly reduced, as one can expect from the increase of the E×B shearing rate with decreasing magnetic field, and Figure 16. ASTRA/TGLF-SAT2 full-radius predictions of the electron temperature (a), ion temperature (b) and electron density (c), as well as corresponding electron and ion heat conductivities normalized to the gyro-Bohm heat conductivity (d) for a major radius scan in which the plasma current is increased proportional to the major radius, keeping magnetic field and safety factor constant. therefore with increasing plasma size, when the effect of γ E is included. The major radius exponent for the confinement time decreases from 2.55 to 2.29, becoming closer to a gyro-Bohm like behaviour, which would result in a horizontal line in figure 15(d). The corresponding predicted profiles for the cases with I p ∝ R maj and B T ∝ 1/R maj are shown in figures 16 and 17 respectively. The close-to-gyro-Bohm behaviour of the electron and ion transport coefficients in the confinement region figure 16(d) translates into temperature profiles which exhibit very limited change when plotted over the normalized toroidal minor radius ρ ϕ , as shown in figures 16(a) and (b), despite the strong increase in plasma volume, with the same volume averaged density and the same total heating power. In contrast, in figure 17, a clear reduction of the temperature profiles is obtained with increasing plasma size, leading to a weaker increase of confinement with increasing size, despite the partial compensation provided by the stronger reduction of the heat conductivities at the edge, as a consequence of the stronger E×B shearing rate at low magnetic field and large major radius. Figure 18 shows the same type of results as figure 15, this time obtained with 3 MW of electron heating and no auxiliary heating to the ions. Therefore, in these cases, the ions are only heated by the electron-ion thermal exchange. The results show a significantly stronger size dependence, with exponents 3.15, 2.79 and 2.88 respectively for the three situations I p ∝ R maj , B T ∝ 1/R maj and q 95 ∝ R maj , figure 18(a). Such a stronger increase of confinement with increasing major radius is clearly reflected also in the scaling normalized to the Bohm and gyro-Bohm confinement times, figures 18(c) and Figure 17. ASTRA/TGLF-SAT2 full-radius predictions of the electron temperature (a), ion temperature (b) and electron density (c), as well as corresponding electron and ion heat conductivities normalized to the gyro-Bohm heat conductivity (d) for a major radius scan in which the magnetic field is decreased inversely proportional to the major radius, keeping plasma current and safety factor constant.
(d) respectively. We observe in particular that, at least over the range of R maj which has been considered in this work, all of the three cases practically perform at least as well as gyro-Bohm, although the cases with I p ∝ R maj and q 95 ∝ R maj exhibit a clear non-monotonic trend. The reason behind this more favourable scaling can be understood from figure 18(b) and is better explained by figure 19. Because the ion heating only comes from thermal equipartition from electrons to ions, the stored energy in the ions is very low at small plasma size and strongly increases with increasing size, figure 18(b). This is the result of stronger coupling between electrons and ions allowed by the strong decrease of the ratio between the thermal exchange time τ ei and the confinement time τ th , as demonstrated in figure 19(a). As a result of this, we also observe that the central ion to electron temperature ratio, which is very small at small plasma size, strongly increases with increasing major radius, figure 19(b). This also has a consequence on the TGLF-SAT2 predicted heat turbulent transport, which decreases with increasing T i /T e , as shown in figure 19(c), leading to a strong increase of confinement with increasing plasma size. When T i approaches T e , the strength of this effect is reduced, and the same trends, as those obtained with mixed electron and ion heating and shown in figure 15, are progressively recovered. These elements provide an explanation of why non-monotonic dependencies are found in figure 18(d). From these considerations, we can also conclude that the variation of local plasma parameters produced by the change in plasma size, in this specific case the electron to ion temperature ratio, can significantly affect the resulting global confinement scaling.

Size scan at reactor conditions
Starting from the results of section 4, we extend the modelling to conditions which can potentially become relevant for fusion power production. We consider a magnetic field of 5.7 T, consistent with the current design of the European DEMO [36] and we scale the plasma size proportional to the plasma current keeping fixed the plasma shape parameters (aspect ratio, elongation and triangularity) to the current values of the European DEMO (and practically of ITER), that is, R/a = 3.1, k a = 1.75 and δ a = 0.45, where k a and δ a are elongation and triangularity of the last closed flux surface. We consider a current of 19 MA and a major radius of R maj = 8.940 m, minor radius a = 2.884 m as reference plasma, with q 95 = 3.65 and, keeping fixed the auxiliary heating power to 50 MW, we increase the plasma size from a corresponding ITER size device (R maj = 6.2 m, I p = 13.5 MA) up to the achievement of fusion power levels which can be considered in a reactor range, that is with P fus > 1.5 GW and a fusion power multiplication factor Q > 30. The 50 MW of auxiliary heating power are deposited in the plasma centre ρ ϕ = 0.1 with a Gaussian profile and are assumed to provide electron heating only. Figure 19. Ratio of the electron-ion thermal exchange time to the confinement time (a) and central ion to electron temperature ratio (b) as a function of the major radius for the three major radius scans as indicated in the legend with 3 MW of auxiliary electron heating only. Corresponding ion heat conductivities from TGLF-SAT2 averaged in the radial window ρ ϕ ⩾ 0.35 are plotted with the same colour code as a function of the ion to electron temperature ratio, averaged over the same radial window.
As a consequence of the increasingly long confinement times with increasing size and current of the plasma, as well as of the requirement of sufficiently stationary levels of fusion power, simulations have been run over considerably long times in order to reach sufficiently stationary conditions, usually from 20 s up to 50 s of actual simulation time of the transport code. The time step of these simulations in ASTRA is kept constant at 50 ms. Considered the explorative (and mainly curiosity driven) nature of these modelling activity, the ASTRA radial grid is kept to 161 points, but TGLF-SAT2 is only called every second point, that is over 80 radial grid points. An interpolation from the reduced grid on which TGLF-SAT2 is called to the ASTRA grid is then performed. The boundary conditions for the electron and ion temperature have been kept fixed at T i,sep = T e,sep = 170 eV for all the cases, the electron density has a separatrix value fixed at 0.3 the volume averaged density, and the peripheral gas puff is applied in feedback in the simulations in order to allow the density profile to go below the Greenwald density limit [1] at ρ ϕ = 0.85. While this condition provides density profiles which eventually can have line averaged densities which are above the Greenwald density limit, it is nowadays established that the density limit is a limit for the edge density, and that densities can exceed the Greenwald limit thanks to central density peaking, while keeping the edge values below the limit. Typical examples in present devices are provided by experiments with sufficiently large inward convection [37][38][39], or with sufficiently central pellet deposition, in H-mode [40] as well as in L-mode [41]. In a reactor central density peaking, potentially enabling operation above the Greenwald limit, is predicted to be provided by turbulence through the pinch of trapped electrons at sufficiently low collisionality [27,42,43]. In the simulations, equal fractions of tritium and deuterium are assumed, and a single main ion species is included in TGLF-SAT2, with atomic mass 2.5. Helium has been included as only impurity and included in the TGLF-SAT2 calculations, although the helium density profile is not evolved in the transport code. A helium density profile proportional to the electron density profile has been assumed, with a concentration which is provided by the sum of fusion ash and of the presence of an additional impurity. The concentration of He ash has been assumed to increase with increasing size of the device, and therefore with increasing fusion power, from 3% to 7.5% moving from ITER size to a 15 m major radius plasma. This dependence has been introduced to more realistically include the effect of increasing fusion power, and therefore of increasing He ash production, with increasing reactor size, with respect to a more simplified treatment, which simply considers a constant He ash concentration. A consistent treatment of the He ash, which computes the time evolution of the He ash density, as for instance performed in [44], has been judged outside the scope of the present work, but could be considered in case of future applications. The charge concentration of an additional light impurity providing a contribution of the effective charge of ∆Z eff = 0.3 has been included through an additional helium content, which, assuming a reference charge of 6 for the light impurity, is equivalent to a helium concentration of 0.03, that is, a charge concentration of 0.06. In this way, a sufficient impurity content in terms of charge concentration is included in order to take into account the dilution effect, while keeping a single impurity in the TGLF-SAT2 calculations.
Simulations with and without the inclusion of the E×B shearing rate have been performed, with the radial electric field self-consistently computed with the combination of the main ion pressure and neoclassical poloidal rotation, while the toroidal rotation has been assumed to be equal to zero.
The results are presented in figures 20 and 22. Figure   Given the increasing interest on the possibility of building a tokamak reactor at very high magnetic field, as potentially allowed by high temperature superconductors [46][47][48][49], an additional set of three simulations has been performed with a magnetic field on axis of 10 T. Identical modelling procedure has been applied, with a three point scan at major radius of 5.1 m, 6.0 m and 7.0 m, with corresponding plasma currents of 19.0 MA, 22.4 MA and 26.1MA. The smallest of these plasmas has the same current of the reference DEMO case. The corresponding temperature and density profiles of these three cases are shown in figure 21, where also the corresponding profiles of the lowest and highest current cases at 5.7 T are plotted with black dashed lines for comparison. Only simulations with the consistent inclusion of the E×B shearing rate have been performed in the 10 T cases. The corresponding global results are also included in figure 22 with diamond symbols, for direct comparison with the results at 5.7 T.
The first preliminary consideration that can be made is that L-mode conditions have been obtained in all the predicted profiles, as shown by the absence of a pedestal in the temperature profile shapes at the edge in figures 20(a), (b), (d), (e) and 21(a), (b), (d), (e). However, as illustrated in figure 22(c), the L-H power threshold for deuterium plasmas from the Martin's scaling [16], plotted with a dash-dotted line, is exceeded by the total heating power at around 12 m in the 5.7 T cases and is exceeded by all cases at 10 T. Although it has been shown that strong edge E×B shearing rates in TGLF-SAT2 can develop pedestal like structures at the edge in TGLF-SAT2 simulations [29], these conditions are not achieved in the present modelling approach, which can still be reliably considered to represent L-mode conditions. In a very hypothetical L-mode reactor, if L-mode conditions had to be ensured, then ways to prevent access to H-mode should be adopted, like, for instance, an H-mode unfavourable grad-B drift configuration, usually doubling the power threshold. Occurrence of a transition to I-mode [45] could of course take place at some power level, but also this transition cannot be predicted within the present ASTRA/TGLF-SAT2 modelling approach.
The second preliminary consideration is connected to the predicted central values of electron to ion temperature ratio, which decrease from around 1.5 for the ITER size plasma, to below 1.2 for the 15 m major radius plasma, in connection with the common strong increase of both the electron and the ion central temperatures with increasing plasma size and fusion power. In the conditions of the very high density plasmas at 10 T, this ratio is around 1.2 for all of the three cases. This demonstrates the increasing strength of the thermal coupling in these reactor conditions, consistent with the results presented in figure 19. The fraction of ion heating from the fusion reactions only increases from 0.24 to 0.37 in correspondence to the increase from an ITER size plasma to the 15 m major radius plasma. This result confirms that, provided the confinement time is large enough, an increase of ion temperature with increasing plasma size and ion temperatures close to the electron temperatures can be also achieved in L-mode reactor conditions. This property is consistently predicted with a transport model like TGLF-SAT2, which, at the same time, has been shown to also predict the central clamping of the ion temperature [15,50], which is observed in present devices when the electron heating is increased [51]. While this central clamping of the ion temperature is mainly determined by the inverse electron temperature dependence of the thermal coupling, we notice that TGLF is also characterized by a particularly strong increase of the ion heat transport with increasing T e /T i [50], which even leads to the unfavourable (and not observed) prediction that in low density AUG plasmas the central ion temperature slightly decreases with increasing central electron heating.
We move now to the examination of the temperature profile shapes. We observe that, while they do not feature a pedestal structure at the edge, figures 20(a), (b) and 21(a), (b), consistent with L-mode confinement, they are characterized by a confinement region (approximately 0.3 ⩽ ρ ϕ ⩽ 0.8) in which the profiles are largely parallel in logarithmic scale, demonstrating that they have very similar logarithmic gradients, particularly in the absence of E×B shearing rate, figures 20(d) and (e). The impact of the E×B shearing rate is particularly strong in the centre in these simulations. This can be understood from the fact that, within the present modelling approach, the dominant component of the radial electric field in the plasma core is produced by the ITG, as described in equation (5). Therefore, the inclusion of the effect of the E×B shearing rate provides two concomitant positive feedback loops connected with a central ion temperature increase, which causes both an increase of the heating power, through the ion temperature dependence of the fusion reaction cross-section, and a reduction of the heat transport due to an increase of the E×B shearing rate. We remark that this somewhat synergistic effect is not present in conditions which do not have dominant heating produced by fusion reactions. This also explains why the difference in the temperature profiles with and without the inclusion of the E×B shearing rate is almost negligible in the ITER size device, which only has Q = 2.2, whereas becomes strong in the 15 m device, with Q = 49.0. In contrast to the general self-similar behaviour in the core, the temperature profiles strongly differ at the edge. In the periphery, steeper temperature profiles are obtained with increasing plasma size, also when the the E×B shearing rate is neglected. We recall that the same separatrix temperature is applied to all cases, however not the same density, since the limit provided by the Greenwald density is taken into account. The density profiles at 5.7 T, figure 20(c) and (f ), exhibit an increase of peaking in the confinement region with increasing plasma size, concomitant with the decrease of collisionality. The normalized logarithmic density gradient R/L n in the confinement region increases from 1.0 to 2.7, partly compensating the strong reduction of plasma density at the edge determined by the density limit. In the cases at 10 T, the variation of the density peaking with increasing plasma size is more limited, as more limited is the corresponding change in collisionality, but it is still clearly visible around ρ ϕ = 0.7, figures 21(c) and (f ). The TGLF-SAT2 predicted density peaking allows the line averaged density to exceed the Greenwald limit. The ratio of the line averaged density to the Greenwald density n e,lin /n GW increases from 1.14 to 1.31 from the ITER size to the 15 m major radius plasma the 5.7 T plasmas. When the pressure profiles are considered, it is found that the four different plasma sizes are characterized by very similar pressure profiles, and therefore by similar values of volume averaged plasma β. This is shown in figure 22(b), which exhibits variations which slightly exceed 20% in the simulations with E×B included (only 11% when the E×B is neglected). We remark that, despite the higher magnetic field, the plasmas at 10 T develop significantly higher β values, also higher with respect to the corresponding IPB98 L-mode scaling law projections.
At 5.7 T the total heating power, shown in figure 22(c), increases from 74 MW at 6.2 m to 542 MW at 15 m, corresponding to an increase of the total stored energy from 158 MJ to 2740 MJ (the plasma volume increases from 826 m 3 to 11 700 m 3 ). In figure 22(b), where the volume averaged thermal plasma β is plotted, we observe therefore that a factor 17.3 increase of the stored energy corresponds to a factor 7.3 increase of the heating power, with an increase of the stored energy which is stronger than the factor 14.2 increase of the plasma volume (when the impact of the E×B shearing rate is neglected, the increase of stored energy is only by a factor 15.8, therefore closer to the increase of plasma volume). In comparison, the IPB98 L-mode scaling law only foresees an increase of the thermal stored energy from 131 MJ to 1929 MJ, mainly proportional to the increase in plasma volume, as shown by the almost horizontal dash-dotted line of the parabolic fit in figure 22(b). Simulations at 10 T show an even stronger improvement with respect to the IPB98 L-mode scaling law, comparable in magnitude to the case at 5.7 T with the largest major radius, and therefore also with the highest heating power. The better confinement properties of TGLF-SAT2 with respect to the scaling law should not be considered in contradiction with the results in section 4. Figure 15 has shown that the TGLF-SAT2 predictions for an increase of major radius with proportional increase of the plasma current, at fixed magnetic field and safety factor (that is, the situation of the present reactor size scan), were perfectly reproduced by the IPB98 L-mode scaling law, with the scaling law having a stronger dependence on current and weaker dependence on major radius, whereas TGLF-SAT2 has stronger dependence on size and weaker on current, but the two were practically providing the same sum of the corresponding exponents. The results of figure 15 were obtained at constant power, which is not the case of the present simulations, where the confinement degradation produced by an increase of heating power is an additional critical ingredient when projecting to a reactor [52]. In [15] an AUG L-mode power scan was specifically modelled and it was found that the IPB98 L-mode scaling law had stronger power degradation (exponent −0.73) than that obtained by the TGLF-SAT2 simulations and that observed in the experiment. The strong power degradation of IPB98 Lmode can be identified as the main source of disagreement with the TGLF-SAT2 results. By modifying the scaling law and introducing a weaker power degradation provides results which closely follow those of TGLF-SAT2. Largely consistent with the results of table 2 in [15], by replacing the power exponent −0.73 of the scaling law with −0.65, we obtain a very good match of the TGLF-SAT2 predicted dependencies of these reactor size scan (both stored energy and fusion power) at both 5.7 T and 10 T. It is interesting to notice that the observed increase of the stored energy as a function of heating power in AUG was not only stronger than that of the IPB98 L-mode scaling law, but also of that predicted by the TGLF-SAT2 simulations. The impact of this exponent on the reactor predictions is considerable.
We finally examine the predicted fusion power, plotted in figure 22(a). The TGLF-SAT2 results show that at 5.7 T a fusion power of 2.5 GW (Q = 50) is reached by the 15 m (31.9 MA) plasma. Neglecting the impact of the E×B shearing rate, this plasma almost reaches 2.0 GW (Q = 40). The ITER size L-mode plasma is predicted to develop a fusion power slightly above 110 MW, thereby only a Q = 2.2, whereas the DEMO plasma only 466 MW, that is, slightly below Q = 10.
The dash-dotted line shows the projection of fusion power from the IPB98 L-mode, which is expected to only provide 1.21 GW at 15 m, consistent with the stronger power degradation of the scaling law. For additional comparison, the dashed oblique line shows the projections with the IPB98(y,2) scaling law for H-mode plasmas, which qualifies the ITER parameters for a Q = 10, with 500 MW of fusion power, and the DEMO parameters for a fusion power of 2 GW (Q = 40). According to the predictions of TGLF-SAT2, this highly desirable target of 2 GW should be achievable by a reactor in L-mode with major radius of 14.3 m, and the (somewhat 'symbolic') current of 30 MA (we recall, this always assuming 5.7 T of magnetic field and isomorphic plasma shape with the ITER aspect ratio of 3.1). This 2 GW target is not expected to be reached by the IPB98 L-mode scaling law, not even at 15 m, and we identify the strong power degradation of this scaling law as the main cause of this more pessimistic prediction in comparison to the TGLF-SAT2 results.
Coming now to the results at 10 T, we observe that they show a much higher fusion power at the same plasma volume, and a much faster increase with increasing size of the plasma. They also show a much higher performance as compared to the IPB98 L-mode scaling law, which, due to the very strong power degradation, is very unfavourable for the prediction of devices with very high fusion power densities. As we have just discussed, there are evidences that suggest that the power degradation of the IPB98 L-mode scaling law is too strong. Clearly the higher fusion powers at 10 T are connected with the higher densities that are allowed by the higher Greenwald density limits, in a device which allows the same current at smaller minor radius (we recall that in figure 22, the green diamond at 10 T and the red square at 5.7 T have the same DEMO reference current of 19 MA). In contrast, and as one can expect, the thermal energy confinement times of the 10 T plasmas are significantly shorter as compared to the 5.7 T plasmas. Given the smaller size and much larger power densities at 10 T, the obtained confinement times increase only from 1.15 s to 1.35 s, whereas at 5.7 T the confinement times increase from 2.15 s to 5.05 s, from the smallest to the biggest major radii which have been respectively simulated. At 10 T, the reference target of a 2 GW fusion power reactor, with a fusion multiplication factor Q = 40, would be already achieved by a 21 MA plasma, in a device with major radius of only 5.7 m.
As a conclusive remark, we underline that this study has been only dedicated to the investigation of the plasma confinement and the related fusion power production from the confined plasma side, but does not consider any aspect related to the technological feasibility, nor to the compatibility with the plasma control or with heat and particle exhaust solutions as well as with consequent plasma-wall interactions.

Conclusions
Experiments in AUG have been performed to investigate the impact of a variation in magnetic field on the plasma confinement in the L-mode confinement regime. Despite the limitations connected with the difficulty of performing these experiments, a result largely consistent with the scaling law is obtained, which is also reproduced by specific full-radius ASTRA/TGLF-SAT2 simulations, with a weak dependence of confinement on the magnetic field. Numerical magnetic field scans are performed in order to better investigate the dependencies and single out the effects which are responsible of the weak dependence of confinement with magnetic field and the stronger dependence on plasma current. The ASTRA/TGLF-SAT2 simulations show that the increase of confinement produced by the decrease of safety factor and magnetic field is in large part connected with the increased role of the E×B shear and that this effect is compensated by the magnetic field dependence of the gyro-Bohm factor of turbulent transport in the magnetic field scan. The gyro-Bohm factor, in contrast, is not modified by a change in current. The combination of these effects explains, at least from the standpoint of this modelling approach, the reason why the confinement time in L-mode increases with increasing current, while does not increase with decreasing magnetic field, in spite of the opposite role that current and field have on the safety factor. This study therefore allows the identification of the dominant physics ingredients which determine the dependence of L-mode confinement of the ASTRA/TGLF-SAT2 results on magnetic field and plasma current. By identifying those physical ingredients which determine these fundamental confinement properties in the modelling, these results can also motivate a closer verification of these TGLF-SAT2 predicted dependencies against nonlinear gyrokinetic simulations. Moreover, a set of numerical scans on the plasma size has shown that these ASTRA/TGLF-SAT2 simulations predict an increase of confinement time with increasing major radius which is stronger than that expected by L-mode scaling laws, when, consistently with the multivariate regression approach, the plasma current and the magnetic field are kept constant with increasing plasma size. Depending on whether the auxiliary heating is provided in equal fractions to the ions and the electrons, or to the electrons only, different increases of confinement with increasing plasma size are obtained. With mixed heating, the increase in size accompanied by a proportional increase in current leads to a scaling which is moderately less faborable then gyro-Bohm. In contrast, when the auxiliary heating is given to the electrons only, the increase of confinement is more favourable than gyro-Bohm, at least in the range of parameters in which the electron heating condition allows a concomitant significant increase of the ion to electron temperature ratio with increasing plasma size, caused by the electron to ion thermal coupling. Full-radius ASTRA/TGLF-SAT2 simulations with increasing plasma size have been also performed in a reactor relevant range of plasma dimensions, with 50 MW of auxiliary central electron heating power and including deuterium-triutium reactions at both 5.7 T and 10 T magnetic fields on the magnetic axis. At 5.7 T, starting from the ITER size, which in L-mode is predicted to only slightly exceed Q = 2, the plasma has been increased up to the production 2.5 GW (Q = 50) of fusion power in deuterium-tritium Lmode plasmas, which is obtained at 15 m and 31.9 MA. These simulations have again underlined the important role of the E×B. At 10 T, with the same aspect ratio and the same auxiliary heating, a fusion power of 2.5 GW (Q = 50) is already obtained with a major radius of 6 m and a current of 22.5 MA. The reduced strength of the degradation of the confinement time with increasing heating power found in the TGLF-SAT2 predictions in comparison to the IPB98 L-mode scaling law, as shown in [15], explains the result that higher fusion powers in comparison to the scaling law are obtained with TGLF-SAT2.
These set of modelling results could motivate further investigation of the properties of the predicted E×B stabilization of turbulent transport in TGLF-SAT2 in comparison with gyrokinetic simulations, particularly in edge turbulence, given the important role that these not only play in determining the current and magnetic field dependencies of L-mode confinement, the impact on the scaling as a function of plasma size, but also potentially in the possibility of accessing an L-H confinement transition in fullradius simulations, on which promising results have been recently obtained [29]. From an experimental standpoint, further investigations of the L-mode confinement dependence on plasma current and magnetic field under dominant electron and ion heating conditions would allow a more complete validation of the TGLF-SAT2 results presented in this work. Regarding the dependence of the L-mode confinement time on the plasma size, some information could already be obtained by examining the data already available in the L-mode ITPA confinement database and in examining the level of agreement of full-radius TGLF-SAT2 predictions on plasmas from present tokamaks of different size, possibly in sufficiently similar conditions.