Exploring Experimental Isotope Scaling and Density Limit in Tokamak Transport

1. Introduction

The general problem of tokamak transport has been one of the main issues in fusion research for a long time [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,51]. However, the specific mechanism behind isotope scaling remains undetermined [1,2,3,4]. Similarly, the underlying reason for the density limit has not been clearly understood [11,36,51]. Several other papers [38,39,40,41,42,43] address the density limit. It is commonly associated with increased turbulent transport and collisions, which is also true in our model. In some cases (Gates et al. Ref. [40]), the coupling to magnetic perturbations is emphasized. Of course, magnetic field perturbations increase with the turbulence level in our model as well, but we do not see a causal relationship here. However, our model for the L-H transition [28], without ion viscosity, is electromagnetic and agrees very well with the model in Ref. [31]. The density limit in Ref. [31] is associated with collisions (as in our model) and was also discussed by Giacomin et al. Since we obtain the density limit by including ion viscosity, we are consistent with Ref. [31]. In the paper by Giacomin et al., it is also mentioned that the pressure gradient can become a considerable fraction of the minor radius, as in the case of MARFEs.

In this study, we link both phenomena to fluid closure, a mechanism that must also be present in kinetic formulations [10], though it is not always prominently highlighted. One aspect that appears to be important is the effects of classical dissipation [1,3]. As shown numerically in Ref. [6] and analytically in Ref. [7], using the reductive perturbation method [8], the Landau fluid resonance [14] changes the nonlinear Dimits upshift [10] strongly. Our fluid model results, in agreement with the kinetic Dimits shift [6,7], show that we have the correct strength of zonal flows. This means that the inverse turbulent cascade [16] is damped out so that there is no pileup of waves at the system size. This also verifies the choice of absorbing boundary for long wavelengths, as used in Ref. [17]. It is interesting to compare it with the resistive drift wave fluid model [9], which in Ref. [18] gave a kernel transport coefficient of the same form as ours. With kernel, we mean that we keep only diagonal elements of the transport matrix. This requires an outgoing turbulence boundary without pileup at the system size. In this work, both analytical and numerical methods are used. However, ion viscosity is typically at least an order of magnitude smaller than the Landau fluid resonance [14] and can thus be incorporated into our general approach to fluid closure. As it turns out, usually edge data give a stronger isotope effect (about 20% difference in transport between hydrogen and tritium). Thus, we also have an interest in the L-H transition [28,31,32,33] which is also due to zonal flows. Actually, our fluid model did very well in comparison with local transport simulations as made in Ref. [31]. Although our simulations were originally set up for EAST [28], we were able to investigate the same parameter regimes used in Refs. [31,32] for Alcator C-Mod. This allowed us to construct a stability diagram in the magnetohydrodynamic (MHD) parameter alpha and a corresponding diamagnetic resistive parameter alpha, showing good agreement. As found in Refs. [18,29], a quasilinear approach is quite adequate for studying drift wave turbulence, and in Ref. [29], a fluid approach was found to be accurate. Our scaling of temperature at the separatrix with the magnetic field and the power threshold were also consistent with those reported in Ref. [33]. Thus, we have secured the validity of our simulation of the L-H transition [28,44]. All our new findings are related to the excitation of zonal flows and the associated fluid closure aspects. This connects to our previous work on fast particles [12], where resonance broadening was shown to be important for both fast particle instabilities and drift waves, eliminating wave-particle resonances in the latter case. The foundation of this research lies in our drift wave fluid model [17] (usually referred to as the Weiland model), which still serves as a valid limit of the current model. The motivation behind implementing fluid closure was to incorporate all moments with sources observed in the experiment. This transport model still has the same normalization as in Ref. [17]. It has never been fitted to any other theoretical model or experiment, yet it consistently yields good agreement with conventional (JET, EAST, KSTAR, and DIII-D) and low aspect ratio tokamak (NSTX) plasma profiles [45,46,47].

As is clear from this introduction, zonal flows play an important role in tokamak transport. We here mention three more papers in this field [20,21,22]. However, none of these include resonance broadening [23,24], which is now the main argument for our fluid closure [12]. Contrary to flattening, which means that particles move out of resonance with waves, resonance broadening means that waves move out of resonance with particles.

The manuscript is structured as follows: Section 2 discusses the inclusion of all moments with sources in the experimental closure approach and the impact of resonance broadening on the fluid model. It concludes that resonance broadening stabilizes fast particle instabilities and validates the reactive closure for drift waves, allowing the use of Braginskii’s highest moment for successful modeling. Section 3 outlines the original motivation behind a fluid closure, driven by the aim to encompass all moments with sources and justified by a nonlinear Fokker-Planck equation. It further discusses the implications of the Fokker-Planck equation in understanding turbulence and resonance broadening effects in wave-particle interactions. In Section 4, we explore the incorporation of ion viscosity, emphasize the pivotal role of zonal flows generated by Reynolds stress—specifically, in the form of off-diagonal poloidal momentum flux—and stress the critical consideration of averaged resonances for accurate predictions. Section 5 highlights the relevance of edge data for understanding the H-mode barrier, presenting ion thermal diffusivity against normalized temperature gradients. The observed isotope mass effect leads to a confinement time scaling in agreement with experimental results, along with a global scaling involving heating power. Section 6 explores the strong dependence on system size and the density limit, highlighting the impact of gyro-Landau resonances on rotation dynamics. The interplay with $\mathbf{E}\times \mathbf{B}$ convection is elucidated, emphasizing the need for strong zonal flows to avoid wave pileup and reflections, particularly in finite radial systems. The discussion encompasses stabilizing nonlinearity, showcasing its critical role in preventing an inverse cascade towards longer wavelengths. The importance of such considerations is underscored in reactive systems, as demonstrated in previous works. Section 7 delves into successful applications of the transport model, focusing on isotope scaling and the density limit, while acknowledging challenges in the latter. The discussion underlines the importance of ion temperature perturbation resonance in turbulence-rotation balance and tokamak transport dynamics.

2. Fluid Model

The original approach to our fluid closure was to include all moments with sources in the experiment [17], the reason being that moments without sources would die out on the transport time scale. An important mechanism turned out to be resonance broadening [19]. This was applied to our model, with the conclusion that we would get a reactive closure with a diagonal part of the same type as in our fluid model [23]. We found that turbulent collisions have the same effect on a fluid description as classical collisions. A generalization to include fast particles verified both the stabilizing trend of resonance broadening on fast particle instabilities [12] and our previous result that our closure is valid for drift waves. Since resonance broadening only shows that we have a reactive closure, we again employed our original rule for the exact point of closure [52]. This closure means that we can take the highest moment at the Braginskii $q_{*}$, and this has been extremely successful in previous studies in cases where the ion viscosity could be ignored. Resonance broadening means that nonlinear frequency shifts change the phase velocity of waves in such a way that waves move out of resonance with particles when we average over turbulent fluctuations. Thus, dissipative wave-particle resonances like Landau damping and magnetic drift resonances are averaged out. Of course, this will be true for all processes, so we can also use the Braginskii derivation for ion viscosity.

3. Formulation

The original motivation for our fluid closure was to include all moments with sources in the experiment. The elimination of higher moments was then motivated by a nonlinear Fokker-Planck equation. The Fokker-Planck equation for turbulent collisions [12,23] can be written as

$$\left({\displaystyle \frac{\partial }{\partial t}}+v·{\displaystyle \frac{\partial }{\partial r}}\right)W(X,X^{\prime },t,t^{\prime })={\displaystyle \frac{\partial }{\partial v}}\left[\beta v+C_i+D^{\nu }{\displaystyle \frac{\partial }{\partial v}}\right]W(X,X^{\prime },t,t^{\prime })+S_{\mathrm{v}}.$$

(1)

$$\begin{array}{cc}\hfill \beta & =\sum _j{\beta }_j{\left|{\varphi }_j\right|}^2\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill D^{\nu }& =\sum _j{d}_j{\left|{\varphi }_j\right|}^2.\hfill \end{array}$$

(2b)

We are here interested in drift waves, so we take $S_{\mathrm{v}}=\langle S_{\mathrm{v}}\rangle =0$. In the given context, W represents the transition probability between states X and $X^{\prime }$ over the time interval $t-t^{\prime }$. The terms $\beta$, $C_i$, $D^{\nu }$, and $\varphi$ denote the nonlinear turbulent friction, classical friction term due to ion-ion collisions, nonlinear diffusion coefficient in velocity space, and electrostatic potential, respectively. The Fokker-Planck equation leads to the mean square of velocity deviation (velocity dispersion) developing as [23]

$$\langle ▵v^2\rangle ={\displaystyle \frac{D^{\nu }}{\beta }}(1-e^{-\beta t}).$$

(3)

This quantity measures the average deviation of particle velocities from their mean due to turbulent collisions over time. It helps in understanding the intensity and impact of turbulence on plasma behavior and illustrates how particles spread over time. A simplified derivation of Equation 3 was given in Ref. [34] Chapter 9, where the Chandrasekhar solution [53] of Equation 1 without viscosity was used as a weight function. Now, the classical collision term just gives the usual classical viscosity, so we will recover Equation 4 since Equation 3 means that we have a reactive closure as discussed in Section 4.

The Fokker-Planck equation can also be applied to fast particles if we add a fast particle source, and then we can recover equations for both fast particles and drift waves within appropriate limits [12]. In both cases, resonance broadening [19,23] reduces the wave-particle interaction. As it turns out, the resonance broadening is due to nonlinear frequency shifts, which would also remain in the coherent limit [26,27].

A solution for coherent explosive instability with a nonlinear frequency shift (Figure 1) was shown already in our book [24]. Similar to our recent work [48], we can explore the coherent three-wave interaction regime as also studied in Ref. [13] and Ref. [54] for drift waves [26,27].

The coherent, unstable three-wave system is stabilized by nonlinear frequency shifts. These work so as if the sign of the wave energy is shifted, thus we have alternatively Landau growth and damping. This is the way the effect of waves-particles [23] interaction is averaged out, leading to the absence of energy transfer between waves and particles, as shown by Equation 3 in the turbulent case.

In summary, this section motivates the use of a fluid closure approach, which utilizes a nonlinear Fokker-Planck equation to include all experimental moments and eliminate higher moments. The equation accounts for nonlinear turbulent friction, classical collision friction, and nonlinear diffusion, and can be extended to fast particles and drift waves, leading to resonance broadening and balanced wave-particle interactions. This approach provides a comprehensive framework for understanding turbulent collision dynamics and wave-particle interactions. By incorporating nonlinear effects, it offers a more accurate and robust description of plasma behavior, enabling better predictions and insights in various plasma physics applications.

4. Exploring Ion Viscosity Effects in Drift Wave Turbulence

In order to study isotope scaling, we add ion viscosity to our usual derivation.

$${\displaystyle \frac{3}{2}}{n}_{\mathrm{i}}\left({\displaystyle \frac{\partial }{\partial t}}+v_{\mathrm{i}}·\nabla \right)T_{\mathrm{i}}+P_{\mathrm{i}}\nabla ·v_{\mathrm{i}}=-\nabla ·q_{*\mathrm{i}}+i{\nu }_{\mathrm{ii}},$$

(4)

As discussed after Equation 3, the model can use ion viscosity from Braginskii’s equations directly. Turbulent viscosity behaves like nonlinear friction (see Ref. [23]) and can be included without changing the model’s final results, where ion viscosity

$${\nu }_{\mathrm{ii}}={\nu }_{\mathrm{ee}}{\left({\displaystyle \frac{T_{\mathrm{e}}}{T_{\mathrm{i}}}}\right)}^{3/2}{\left({\displaystyle \frac{m_e}{m_i}}\right)}^{0.5}{\displaystyle \frac{1}{A^{0.5}}},$$

(5)

where A is the isotope mass number and ${\nu }_{\mathrm{ee}}$ is the electron electron collision frequency. Our viscosity, however, has been taken from Ref. [13], Equation (1.8). This gives the scaling $T_i^{-3/2}n/A^{0.5}$ for the viscosity. Thus, we see that viscosity is smaller for heavier isotopes (favorable scaling) and larger for higher density (unfavorable scaling). Equation 5 leads to the ion temperature perturbation

$${\displaystyle \frac{\delta T_{\mathrm{i}}}{T_{\mathrm{i}}}}={\displaystyle \frac{\omega }{\omega -5{\omega }_{\mathrm{Di}}/3+i{\nu }_{\mathrm{ii}}}}\left[{\displaystyle \frac{2}{3}}{\displaystyle \frac{\delta n_{\mathrm{i}}}{\mathrm{n}}}+{\displaystyle \frac{{\omega }_{*\mathrm{e}}}{\omega }}({\eta }_{\mathrm{i}}-{\displaystyle \frac{2}{3}}){\displaystyle \frac{e\varphi }{T_{\mathrm{e}}}}\right].$$

(6)

Note that the viscosity term enters at the fluid resonance in the ion energy equation, the most sensitive point in our fluid modeling. This sensitivity is the main improvement in our model. Here $n_i$ is the ion density, $v_i$ is the ion flow velocity, $q_{*i}$ is the diamagnetic ion heat flow, $T_e\left(T_i\right)$ is the electron (ion) temperature, e is the electron charge, $m_i$ is the ion mass, $m_e$ is the electron mass, ${ϵ}_0$ is the permittivity of free space, $\omega ={\omega }_{\mathrm{r}}+i\gamma$, where ${\omega }_{\mathrm{r}}$ is the real frequency and $\gamma$ is the mode’s growth rate, ${\omega }_{\mathrm{Di}}$ is the magnetic drift frequency, ${\omega }_{*\mathrm{e}}$ is the electron diamagnetic drift frequency, and ${\eta }_i$ is the ratio of ion temperature gradient to ion density gradient. It is important to recall that in our usual reactive limit, the linear threshold of Ion Temperature Gradient (ITG) modes appears exactly at the resonance in Equation 6. As it turns out, zonal flows play important roles in both Dimits shifts, the L-H transition, isotope scaling, and finally the density limit. Zonal flows are generated by the Reynolds stress.

We note that Equations (4), (5),(6) are a generalization of our fluid closure, where the principle is to omit dissipative kinetic resonances, keeping only moments with sources in the experiment. This can be motivated by resonance broadening, as explained in Ref. [12]. Resonance broadening and profile flattening are parallel phenomena that both remove wave particle resonances, although resonance broadening is strongly nonlinear. If you have a particle source that injects particles into the resonant region in phase space, it can balance the flattening so that the resonance survives. The same thing can happen with resonance broadening. Here we have the source of fast particles, which can balance the resonance broadening so that the resonance remains. In this case, it is the question of waves moving out of resonance with particles. However, as we clarify in our paper on fast particles [12], drift waves lack a source because their frequency is approximately two orders of magnitude lower than the source’s, leading to the averaging of the source. Then only resonance broadening remains, and the wave particle resonance is cancelled. We have shown this in Ref. [23], and our paper on fast particles [12] just uses this.

$${\Gamma }_p=\langle V_{Er}{V}_{\theta }\rangle =-{\displaystyle \frac{1}{2}}{D}_B^2{k}_r{k}_{\theta }{\widehat{\varphi }}^{*}\left[\widehat{\varphi }+{\displaystyle \frac{1}{\tau }}{\widehat{P}}_i\right]+c.c,$$

(7)

In the expression, ${\Gamma }_p$ represents the off-diagonal poloidal momentum flux, $V_{Er}$ denotes the radial component of the $\mathbf{E}\times \mathbf{B}$ drift, $V_{\theta }$ signifies the poloidal flow velocity, $D_B$ is defined as ${\rho }_s{c}_s$ where $c_s$ represents the sound speed, $k_{\theta }$ stands for the poloidal wavenumber, $\widehat{\varphi }$ is defined as $e\varphi /T_e$, and normalized ion pressure, ${\widehat{P}}_i$ is calculated as $\delta P_i/P_i$. In Equation 7, it is noteworthy to observe the dependence on ion temperature through ion pressure ($P_i$). This dependence is highly sensitive to the fluid closure, as indicated by Equation 6. Here, the last part, the $\mathbf{E}\times \mathbf{B}$ convection of the diamagnetic flow, is often ignored but is, in fact, the most important part.

In the absence of viscosity, marginal stability for the reactive system enters at the resonance in Equation 6. Because of the resonance, we get a particularly strong drive of rotation here (see Figure 2). Although ion viscosity is usually small, it can still be of importance here, and it turns out to generate the isotope scaling. We recall from Ref. [7] that the resonance in Equation 6 is completely smeared out by gyro-Landau fluid resonances, giving a significantly smaller rotation. We note that gyro-Landau resonances are averaged out by resonance broadening and are, without this averaging, typically an order of magnitude larger than ion viscosity but added in the same place. We may here add that nonlinear flattening means that particles move out of resonance with waves, while resonance broadening means that waves are moving out of resonance with particles. Thus, there is no way that we could obtain the isotope scaling if we kept the unaveraged gyro-Landau resonances.

In summary, this section examines the impact of ion viscosity on drift wave turbulence and isotope scaling, utilizing Braginskii’s formulation for ion viscosity and noting that turbulent viscosity behaves similarly to nonlinear friction. The derived equations illustrate the effect of ion viscosity on ion temperature perturbations and validate the fluid closure approach. The study highlights the importance of zonal flows and resonance broadening in achieving isotope scaling and maintaining plasma stability.

5. Isotope Effects on Transport and Confinement: Edge Considerations and Predictive Comparison

As pointed out above, edge data are relevant since the transport flux has to pass the edge. Since the H-mode barrier depends on the same type of zonal flows as the Dimits shift, ion thermal diffusivity as a function of the normalized temperature gradient is shown for the parameters relevant to the plasma edge in Figure 3. Figure 3 directly continues from Figure 2. We are extending the results of Ref. [7] by including ion viscosity.

It is known that the pedestal can be close to force balance, but the L-H transition process involves the creation of zonal flows via poloidal rotation. This allows the use of analytic results from Ref. [7] to determine transport calculations. When switching from the use of hydrogen to tritium, a 6% drop in transport is noted in typical edge data.

Using data from our local code and from Figure 3, the scaling leads to the confinement time (${\tau }_E$) scaling with isotope mass (A),

$${\tau }_E=A^{0.2}.$$

(8)

This result agrees with the experimental result in Ref. [36]. We also recall our result for global scaling with heating power (P) [37]. Additionally, it is important to note the strong FLR stabilization for large gradients (see, e.g., Ref. [34], Equation 6.161).

In summary, these results may be stated as follows:

$${\tau }_E=P^{-0.67}{A}^{0.2}.$$

(9)

Now, using our predictive code for ITER simulations of hydrogen and tritium, we instead obtain:

$${\tau }_E=A^{0.5}.$$

(10)

This result was obtained by comparing runs with different isotopes at different temperatures assuming that ${\tau }_E$ scales as $T^{-2/3}$. The ITER simulations, focusing on ELMY H-mode discharges and employing the Weiland [17,49] and NCLASS [50] neoclassical models for predictive simulation, are shown in Figure 4. The simulations initiate with prescribed sources and an assumed L-mode profile, progressing to the temperature of the L–H transition and pedestal. We do not make any assumptions about the occurrence of an L–H transition or the specific locations of barriers to temperature [44]. We note that these calculations are rather sensitive since we are close to a pole in the fluid equations. Thus, we expect that the difference in these energy confinement scalings (c.f. Ref. [4]) can be due to impurities in the ITER simulations, where $Z_{\mathrm{eff}}=1.65$. We did not include impurities in the local code.

In summary, this section examines isotope effects on transport and confinement at the plasma edge. It highlights the importance of edge data, as transport flux must pass through the edge, and shows that the H-mode barrier depends on zonal flows. Ion thermal diffusivity decreases with higher isotope mass. The L-H transition, driven by poloidal rotation, enables the use of analytic transport calculations. A switch from hydrogen to tritium results in a 6% reduction in transport, with confinement time (${\tau }_E$) scaling as $A^{0.2}$, consistent with experimental data. However, ITER simulations predict a different scaling of ${\tau }_E=A^{0.5}$, possibly due to impurities not accounted for in local code simulations. The sensitivity of these calculations is noted, particularly near poles in the fluid equations.

6. Density Limit and System Size Dependence: Insights from Gyro-Landau Resonances and $\mathbf{E}\times \mathbf{B}$ Convection

We now recall from Ref. [7] that there is hardly any rotation if we include the unaveraged gyro-Landau resonances. Then we get much weaker zonal flow, which could hardly absorb the inverse turbulent cascade. Then we get a pileup of waves at the system size, leading to very strong transport, which is in accordance with the density limit Equation (11). Now we know that the gyro-Landau resonances are typically about an order of magnitude larger than the ion viscosity. However, ion viscosity will increase with density, so if we increase density by about a factor 10, we reach a density limit [38].

$$n_G={\displaystyle \frac{I_p}{\pi a^2}},$$

(11)

where $I_p$ the plasma current, and a the minor radius. Our first observation is the strong dependence on the system size, meaning a strong potential dependence on perturbations with radial scale length approaching the system size. Now, the stabilization of an instability due to $\mathbf{E}\times \mathbf{B}$ convection is written:

$$\gamma \delta T={\mathbf{v}}_{\mathrm{E}}·\nabla \delta T.$$

(12)

leading to

$${\displaystyle \frac{e\delta \varphi }{T_{\mathrm{e}}}}={\displaystyle \frac{\gamma }{{\omega }_{*\mathrm{e}}}}{\displaystyle \frac{1}{k_x{L}_{\mathrm{n}}}},$$

(13)

where $k_x$ is the radial wavenumber and $L_n$ is the density gradient scale length. Note that in Equation 13, the diamagnetic drift accounts only for the density gradient. Thus, the density gradient is canceled (see, e.g., Ref. [34]). The saturation level in Equation (13) is in agreement with typical experiments and used both by us and in other works. A very critical point is here whether the nonlinearity in Equation (12) is entirely stabilizing. This is only the case if we are looking at the correlation length (corresponding to the mode number with the largest growth rate as normalized by the drift frequency) and if there are no reflections in k-space. This may be critical for the inverse cascade, leading to cascade towards longer wavelengths, i.e., towards the system size. Clearly, it is very important to take into account that we are here looking at systems with a finite radial size. To avoid getting a pileup of waves at the system size with reflections, we need strong zonal flows that absorb the inverse cascade. This is usually only available in reactive systems, as seen in Ref. [18] for only density transport and in Ref. [17] using our reactive fluid closure. We note that in both cases, we obtain a transport kernel of the type

$$D={\displaystyle \frac{{\gamma }^3/k_x^2}{{\omega }_r^2+{\gamma }^2}}.$$

(14)

In Ref. [18], we have an exact fluid closure by taking a zero-ion temperature, while in our case, we have no kinetic dissipation because of resonance broadening [7,13]. In our case Equation (12) is applied to ion temperature. In our derivation, we also included pinch terms from a full quasilinear procedure [17]. The quasilinear approach was found in Ref. [29] to be valid within a few percent in the fluid case. We now recall that if the ion density is increased by about a factor 100, the ion viscosity in Equation 6 becomes comparable to an unnormalized gyro-Landau resonance that was, in fact, displayed in Ref. [7]. The result for the rotation was similar to that in Figure 5 below, where the density was only increased by a factor 10 above the Cyclone case. The result of the Cyclone work was an increase of up to a factor 3 in thermal conductivity for an unnormalized gyro-Landau fluid model. However, recent gyro-Landau fluid models have been normalized to the nonlinear gyrokinetic code [11].

Consequently, if the density is increased by approximately ten times compared to the density investigated in the Cyclone work [10], it is expected that there will be an increase in thermal transport, a decrease in flow shear, and the disappearance of the Dimits shift. The increase in transport will be accompanied by the breakdown of the H-mode barrier and the appearance of significant turbulence structures, known as MARFES (for multifaceted asymmetric radiation from the edge), similar to those observed in experimental studies at the density limit. Due to the normally larger sizes of MHD modes compared to drift waves, it is anticipated that MHD ballooning, kinetic ballooning modes, resistive ballooning modes, and peeling modes will also be present near the edge.

This section explores the density limit and system size dependence in plasma transport, focusing on gyro-Landau resonances and $\mathbf{E}\times \mathbf{B}$ convection. Including unaveraged gyro-Landau resonances results in weaker zonal flows, leading to increased transport and aligning with the density limit equation. Gyro-Landau resonances are significantly larger than ion viscosity, but ion viscosity increases with density, reaching the density limit when density is increased by about a factor of ten. Stabilization of instabilities due to $\mathbf{E}\times \mathbf{B}$ convection is crucial, with saturation levels matching experimental observations. Strong zonal flows are necessary to prevent wave pileup at system size, which is typically available in reactive systems. Increasing ion density significantly raises thermal transport, potentially breaking the H-mode barrier and causing substantial turbulence structures, similar to those seen in experiments at the density limit.

7. Summary

In this work, we have achieved two additional successful applications of our transport model: isotope scaling and the density limit. Regarding isotope scaling, we believe it fits perfectly. However, for the density limit [38,39], the evidence is less accurate. We found that increasing the density in our model to the gyro-Landau limit, giving viscosity comparable to the gyro-Landau resonance, would require a density far exceeding the density limit. We also discuss the implications of encountering MARFEs and ballooning instability in this scenario. MARFEs are very large-scale structures and can therefore be torn apart by zonal flows.

We recall the derivation and development of our fluid model, first established in 1988 using theory and simulations with an absorbing boundary for long wavelengths. The original model remains valid within the appropriate limit. The motivation for the absorbing boundary for long wavelengths is now supported by the damping of zonal flows. This first-principles model is not tailored to any other theoretical model or experiment. A notable extension of the model includes the incorporation of fast particles, applying resonance broadening for fast particle modes—an aspect utilized by other research groups.

From a broader perspective, transport in tokamaks is observed to be governed by a delicate balance between turbulence and rotation. In our model, similar to any fluid model, both turbulence and rotation (resulting from turbulence) originate at the resonance in the ion temperature perturbation. This resonance defines the linear threshold of the ITG and acts as the primary source of rotation. We assert that this interplay fundamentally controls transport. Given that both our isotope scaling and density limit rely predominantly on this resonance, we consider them fundamental. While acknowledging the existence of several other mechanisms contributing to both isotope scaling and density limits, we maintain confidence in the fundamentality of our proposed mechanisms. This confidence arises from their close relation to our fluid closure, previously shown to give excellent agreement with experiment and from their integral role in overall transport control, leading us to expect their dominance.