On the Description of Turbulent Transport in Magnetic Confinement Systems

Abstract

We show how a source-aware fluid closure framework for turbulent transport performs well on the confinement timescale in magnetically confined plasmas. A central result is that whether a source is resonant with the turbulence determines which fluid moments must be retained. Using a nonlinear current formulation, we show that resonance broadening—the dominant kinetic nonlinearity—cancels linear resonances and thereby justifies a quasilinear fluid closure already on the turbulence timescale. We derive a practical negative-energy criterion and identify parameter regimes satisfied by ion-temperature-gradient (ITG) modes (slab and toroidal), with parallel ion compressibility and magnetic curvature controlling the sign. The framework clarifies when velocity-space dynamics must be retained in the kinetic Fokker–Planck equation (for example, for fast-particle instabilities at frequencies about 10² higher than drift-wave frequencies). The present study provides additional support for our model by predicting transport that increases with radius and by showing—consistent with nonlinear kinetic simulations—that the diamagnetic flow dominates the Reynolds stress. Altogether, the results obtained provide a consistent, reduced-cost path to fluid closures that retain the essential kinetic physics while remaining tractable on confinement timescales.

1. Introduction

Ever since fusion research started in the beginning of the 1950’s, there has been strong focus on understanding turbulent transport [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125], which typically has a magnitude of about 100 times that of neoclassical transport [2]. A major problem has been that the instabilities responsible for the main transport have a phase velocity close to the thermal particle velocity [2]. Because of this, it appears that a kinetic theory, including wave–particle resonances, seem to be necessary. Several pioneering studies have also been devoted to kinetic instabilities such as Refs. [3,4]. These examples, like many others, either consider special cases or ignore toroidal effects.

Furthermore, we need to include strongly nonlinear effects in order to calculate transport. Since toroidicity represents the third direction in which the magnetic field does not confine a plasma, it is highly fundamental and, for example, leads to transport coefficients which grow with radius [71]. It is possible to include all these effects in nonlinear kinetic codes. However, because of the considerably long computation times needed for nonlinear kinetic codes, it was not possible to run these on the confinement timescale. Because of this, advanced fluid models with different attempts to include wave–particle resonances have been suggested [11,14,15]. First, it is important to recognize that many different timescales are involved in this description. The two closest related to the use of fluid models are the confinement timescale and the turbulence timescale, which is the timescale of the growth and saturation of the turbulence. Since it is known that the plasma gets cold if not heated, it was realized that sources will play a role on the confinement timescale. Thus, we started with the rule that we include all moments with sources in the experiment [10,11]. However, our transport model also performed considerably well on the turbulence timescale [31,55,57]. This is due to resonance broadening, which is a strongly nonlinear kinetic phenomenon [1,2,5,6]. It removes linear kinetic resonances and justifies our fluid closure.

Resonance broadening was originally discovered by Thomas Dupree by adding up phase-independent, intensity-type terms in a general turbulent expansion [1]. This leads to a Fokker–Planck equation for turbulent collisions that includes a nonlinear frequency shift. The nonlinear frequency shift changes the phase velocity of turbulent waves in such a way that their phase velocity moves out of resonance with the particles. It can be considered that the wave–particle interaction tends to flatten the velocity distribution, but all interactions have to be mutual, so the phase velocity of waves is also changed. Since a perturbation is changing, this is a highly nonlinear effect; thus, our fluid closure is due to resonance broadening in the kinetic description, leading to a fluid closure where one can use quasilinear theory [11,16,45].

The reason why a strongly nonlinear description is needed in the kinetic case is that the kinetic velocity is so large that square terms in the velocity become comparably large while the fluid velocity is a few orders of magnitude smaller, so that one may neglect terms that change the frequency but still to keep terms that cause quasilinear transport [16]. Such effects have a tendency to flatten the gradients in the velocity space, thus reducing the effect of linear wave–particle resonances [2,16]. However, it has been argued that since drift waves have such a wide spectrum, an unrealistically wide portion of the distribution function would have to be flattened in order to cancel all wave–particle resonances [17], and it is instead due to resonance broadening. This effect is large enough to, in its coherent limit, explain the difference between advanced fluid models [14,15] and kinetic simulations for parameters of the Cyclone simulations [31]. Here, our model was close to the kinetic simulations [31], above the experimental gradient. However, nonlinear upshift physics was not included in this version of the model. This was included later with quite good results [33,34,35,36,37]. These were mainly due to the diamagnetic drive of the Reynolds stress.

We note that, recently, research on fast-particle instabilities has also involved resonance broadening [18,19,20]. It is observed that resonance broadening can occur also in the coherent limit [7,8]. This led to a reduction in transport of approximately the magnitude needed to explain the Cyclone results [31]. Also, Ref. [9] discusses the effects of nonlinear frequency shifts, although they are not directly related to drift waves. It is, however, striking to observe the similarity of solutions between Refs. [8,9], as pointed out in Ref. [8]. Returning now briefly to our argument of moments with sources in the experiment, these can be both external and internal. Internal sources can be due to off-diagonal components of a full transport matrix or resistive equilibration effects from another species.

Here, it is worthy to mention the simulation of the heat pinch on DIII-D [47,48]. Here, the external sources were the Electron Cyclotron Resonance Heating (ECRH) heating of electrons at half radius, given by the experiment, and a particle source at the edge, which was unknown. The main source of the electron heat pinch was the density gradient. Thus, an artificial source of density at the edge was introduced [48]. This source was varied until it reproduces the experimental electron heat pinch inside the ECRH source. Then, approximately the right electron temperature, ion temperature, and density profiles had been reached. Here, the source of the ion temperature was the resistive equilibration term from electrons.

At the final state, there was no particle flux (no internal source), while inside the ECRH source, there was an inward electron heat flux and an outward ion heat flux. These were combined to give a net outward energy flux. This was a low beta shot, so a rather early and simple version of our model could be used. However, this simulation included the most fundamental aspects of the fluid closure. We later added electromagnetic effects, zonal flows at the correlation length, and ion viscosity effects in the closure. Electromagnetic effects were included in our simulation of the particle pinch on Tore Supra [28]. This was quite close to the corresponding simulation using QualiKiz [27]. We note that although named “quasilinear”, QualiKiz has been fitted to a nonlinear kinetic code and therefore includes resonance broadening. This is necessary in order to obtain a particle pinch.

The above-mentioned extensions of our model have led to successful simulations or explanations for the fast-particle instabilities at higher frequencies [38,39,40], internal transport barriers [35], the low-to-high confinement mode (L-H) transition [36], the Dimits nonlinear upshift [33,34], isotope scaling [13,56], the density limit [56], and the fast particle instabilities at frequencies higher than drift wave frequencies [37]. Already, our first transport code, including only the absorbing boundary for long wavelengths, has led to several citations of our model, as well as by the International Thermonuclear Experimental Reactor (ITER) central team [57]. However, the latest results from comparisons with experiments are even more impressive [55,56,57,59,60,61]. These also involve local transport barriers generated by the Reynolds stress, where diamagnetic flow is the main drive [35,125]. We again note the importance of sources. In order to include comparably fast particle instabilities, one needs to include a resonant source in the Fokker–Planck equation [37]. This occurs only at frequencies about two orders of magnitude above drift wave frequencies.

Finally, let us note that we focused on the nonlinearly (up to quadratic terms) unstable case in Ref. [61]. Here, the sign of the energy changes at the stabilization amplitude so that the sign of both linear and nonlinear growth changes. This is a process that repeats itself so that linear Landau growth and damping is averaged out [61]. This is the reason why we have shown here that this applies to both slab and toroidal ion-temperature-gradient (ITG) modes. Part of this study compares the results with experimental ones in detail. Such a comparison has its foundation in several of our references, starting with Ref. [10] and continuing with several papers by the central ITER team [55,57,94,95,96,97]. Finally, let us to stress that the results of Ref. [5] are not a model but a first-principles derivation, as long as we consider stationary turbulence. This is true for all cases studied here, including [31,44]. We furthermore stress that we have not used any rigged setups in our simulations of transport barriers. This signifies that the radial step length in the codes has always been independent of radius.

Section 2 develops the nonlinear current framework, deriving the dielectric and dispersion relation, the ion density response with parallel dynamics, and a practical wave energy/negative energy criterion. We then apply fluid closure to the Cyclone and DIII-D heat pinch cases and benchmark against kinetic simulations to isolate resonance broadening and the nonlinear upshift. Section 3 incorporates zonal flows explicitly at the correlation length and examines their role in transport barriers. Section 4 discusses the importance of the real part of the linear eigenfrequency, and Section 5 summarizes the implications for source-driven fluid closures and resonant forcing in the Fokker–Plank equation.

2. Nonlinear Current and Wave Energy

A rather common way of describing nonlinear effects in plasmas is by the use of a nonlinear current [9].

$$D\left(\omega ,k\right)E_{\omega }=i{J}^{\left(2\right)},$$

(1)

where J⁽²⁾ is a nonlinear current, $E_{\omega }$ is the Fourier component of the electric field at frequency $\omega$, k is the wavevector (with $k=\left|\mathbf{k}\right|$). For low-frequency electrostatic modes, we take the (dimensionless) dielectric $\left(D(\omega ,k)\right)$ and Equation (1) can be regarded as a nonlinear dispersion relation. For low frequencies,

$$D\left(\omega ,k\right)=ϵ{ϵ}_0={\displaystyle \frac{1}{k^2{\lambda }_{De}^2}}\left[{\displaystyle \frac{\delta n_e}{n_e}}-{\displaystyle \frac{\delta n_i}{n_e}}\right],$$

(2)

where k is the wavenumber magnitude, $ϵ$ and ${ϵ}_0$ are, respectively, the absolute and vacuum permittivities, λ_De is the electron Debye length, and δn_e, δn_i and n_e, n_i are the perturbed and equilibrium electron/ion densities, respectively.

Thus, the linear dispersion relation is written as

$$D\left(\omega ,k\right)=0.$$

(3)

We can now obtain a condition for linear instability as γ > 0, where [2,9]

$$\omega ={\omega }_r+i \gamma \mathrm{with} \gamma ={\displaystyle \frac{i\mathrm{m}D}{\partial D/\partial \omega }} ,$$

(4)

with ${\omega }_r$ the real part of the frequency. Now, to calculate the wave energy, we need the ion density perturbation. We will also include here the parallel ion motion [42], as per Equation (6.165) in Ref. [42]:

$$\begin{gathered} {\displaystyle \frac{\delta n_i}{n_i}} ={\displaystyle \frac{A\left(\omega \right)}{N\left(\omega \right)}}, \\ A\left(\omega \right)=\omega \left({\omega }_{\ast e}-{\omega }_{De}\right)+\mathrm{EIH} {\omega }_{\ast e}{\omega }_{Di}+{\displaystyle \frac{k_{\parallel }^2{c}_s^2}{\omega }} \left[\omega -{\displaystyle \frac{5}{3}}{\omega }_{Di}-{\omega }_{\ast i}\left({\eta }_i-{\displaystyle \frac{2}{3}}\right)\right]-\mathrm{F}\mathrm{L}, \\ \mathrm{F}\mathrm{L}=k^2{\rho }_s^2\left(\omega -{\omega }_{\ast ip}\right)\left(\omega -{\displaystyle \frac{5}{3}}{\omega }_{Di}\right), \\ A\left(\omega \right)=\omega \left({\omega }_{\ast e}-{\omega }_{De}\right)+\mathrm{EIH} {\omega }_{\ast e}{\omega }_{Di}+{\displaystyle \frac{k_{\parallel }^2{c}_s^2}{\omega }} \left[\omega -{\displaystyle \frac{5}{3}}{\omega }_{Di}-{\omega }_{\ast i}\left({\eta }_i-{\displaystyle \frac{2}{3}}\right)\right]-\mathrm{F}\mathrm{L}, \\ \mathrm{F}\mathrm{L}=k^2{\rho }_s^2\left(\omega -{\omega }_{\ast ip}\right)\left(\omega -{\displaystyle \frac{5}{3}}{\omega }_{Di}\right), \\ N(\omega )={\omega }^2-{\displaystyle \frac{10}{3}}\omega {\omega }_{Di}+{\displaystyle \frac{5}{3}}{\omega }_{Di}^2-{\displaystyle \frac{5}{3\tau }}(k_{\parallel }{c}_s{)}^2\left(1-{\displaystyle \frac{{\omega }_{D_i}}{\omega }}\right), \\ \mathrm{EIH}={\eta }_i-{\displaystyle \frac{7}{3}}+{\displaystyle \frac{5}{3}}{\epsilon }_n. \end{gathered}$$

(5)

Here, ${\omega }_{\ast e}$ and ${\omega }_{\ast i }$are the electron and ion diamagnetic drift frequencies; ${\omega }_{\ast ip}$ is the ion pressure-gradient diamagnetic frequency (as used in FL); ω_De and ω_Di are the electron/ion magnetic-curvature (drift) frequencies; $k_{\parallel }$ is the longitudinal wavenumber; τ = T_i/T_e; where T_e and T_i are the electron and ion temperatures; $c_s=\sqrt{T_e/m_i}$ is the ion-sound speed with $m_i$ the ion mass; ${\rho }_s=c_s/{\Omega }_{ci}$ is the ion-sound Larmor radius with ${\Omega }_{ci} =eB/m_i$ the ion cyclotron frequency, where e is the elementary charge and B is the magnetic field magnitude; ${\eta }_i=L_n/L_{Ti}$ with $L_n=-{\left(d\mathrm{l}\mathrm{n}{n}_i/dr\right)}^{-1}$ and $L_{Ti}=-{\left(d\mathrm{l}\mathrm{n}{T}_i/dr\right)}^{-1}$with r the local small radius; ε_n is the normalized density-gradient parameter, and we introduced N(ω) as the denominator in Equation (5).

As it turns out, the wave energy can be expressed as [9]

$$W\propto {\displaystyle \frac{\partial D}{\partial \omega }}.$$

Thus, Equation (5) needs to be differentiated with respect to ω:

$${\displaystyle \frac{\partial }{\partial \omega }}\left({\displaystyle \frac{\delta n_i}{n_i}}\right)={\displaystyle \frac{A’N\left(\omega \right)-A\left(\omega \right)N’\left(\omega \right)}{{\left|N\left(\omega \right)\right|}^2,}}$$

(6)

with

$$\begin{gathered} A’\left(\omega \right)=\left\{{\omega }_{\ast e}-{\omega }_{De}+{k_{\parallel }}^2{c_s}^2\left[{\displaystyle \frac{5}{3}}{\displaystyle \frac{{\omega }_{\ast i}}{{\omega }^2}}+\left({\eta }_i-{\displaystyle \frac{2}{3}}\right){\displaystyle \frac{{\omega }_{\ast i}}{{\omega }^2}}\right]\right\}-k^2{{\rho }_s}^2\left[2\omega -\left({\omega }_{\ast ip}+{\displaystyle \frac{5}{3}}{\omega }_{Di}\right)\right], \\ N’(\omega )=2\omega -{\displaystyle \frac{10}{3}}{\omega }_{Di}-{\displaystyle \frac{5}{3}}{\displaystyle \frac{k_{\parallel }^2{c_s}^2}{\tau }}{\displaystyle \frac{{\omega }_{Di}}{{\omega }^2}}, \end{gathered}$$

$${\displaystyle \frac{\partial \mathrm{Re}D}{\partial \omega }}=-{\displaystyle \frac{1}{{k_{\perp }}^2{{\lambda }_{De}}^2}}{\displaystyle \frac{\partial }{\partial \omega }}\left({\displaystyle \frac{\delta n_i}{n}}\right)={\displaystyle \frac{1}{{k_{\perp }}^2{{\lambda }_{De}}^2}}{\displaystyle \frac{A^{’}N\left(\omega \right)-A\left(\omega \right)N´\left(\omega \right)}{{\left|N\left(\omega \right)\right|}^2}},$$

Thus, the criterion for negative energy is

$${\displaystyle \frac{\partial \mathrm{Re}D}{\partial \omega }}=<0\iff A’N(\omega )-A(\omega )N’(\omega )>0.$$

(7)

Physical Interpretation

We note the significance of the factor

$$S=A’N\left(\omega \right)-A\left(\omega \right)N’\left(\omega \right).$$

From Equation (6), one finds that parallel compressibility contributes:

$$A´\left(\omega \right)\propto {\displaystyle \frac{{k_{\parallel }}^2{c_s}^2}{{\omega }^2}}\left({\displaystyle \frac{5}{3}}{\omega }_{Di}+{\omega }_{\ast i}\left({\eta }_i-{\displaystyle \frac{2}{3}}\right)\right)\iff N’(\omega )=-{\displaystyle \frac{5}{3}}{\displaystyle \frac{{k_{\parallel }}^2{c_s}^2}{{\omega }^2}}{\displaystyle \frac{{\omega }_{Di}}{{\omega }^2}},$$

(8)

where we introduce N(ω) as the denominator in Equation (5), which reinforces the tendency towards negative energy when ω < 0.5 (${\omega }_{\ast i\mathrm{p}}$+${\omega }_{Di})$ and gives a trend towards positive energy at higher ω. In the flute limit$k_{\parallel }=0$, the parallel contribution vanishes, making sign reversal less likely.

Modes with negative wave energy satisfy (7), implying that small dissipative losses (for example, weak collisions or Landau damping) increase the mode amplitude, i.e., a dissipation-induced instability; c.f. Equation (6). In the present model, sign reversal is promoted primarily by compressibility in A′ and magnetic curvature though ${\omega }_{Di }$with finite Larmor radius (FLR) corrections shifting the boundary between positive and negative energy regimes.

We here note that we have possibilities for negative energy, both due to parallel motion and due to magnetic curvature. However, magnetic curvature is typically the dominating source of instability.

3. Zonal Flows

A phenomenon often mentioned in connection with plasma transport is zonal flows.

In early versions of our model, zonal flows were only included and implicitly represented by an absorbing boundary for long wavelengths [11] (this includes also the Cyclone paper [31]).

However, we later included zonal flows, also explicitly at the correlation length, using the Reynolds stress, as implemented using our fluid model and subtracting the rotation shearing rate from the growth rate. This led us to produce different types of transport barriers. These turn out to depend strongly on the fluid closure through the strength of the temperature perturbation. The latter, in the absence of ion viscosity, is written as [10,11]

$${\displaystyle \frac{\delta T_i}{T_i}}={\displaystyle \frac{\omega }{\omega -\frac{5}{3}{\omega }_{Di}}}\left[{\displaystyle \frac{2}{3}}{\displaystyle \frac{\delta n_i}{n_i}}-{\displaystyle \frac{{\omega }_{\ast e}}{\omega }}\left({\displaystyle \frac{2}{3}}-{\eta }_i\right){\displaystyle \frac{e\varphi }{T_e}} \right],$$

(9)

$${\Gamma }_p=⟨V_{Er}{V}_{\theta }⟩=-{\displaystyle \frac{1}{2}}{D}_B^2{k}_r{k}_{\theta } {\hat{\varphi }}^{\ast }\left[\hat{\varphi }+{\displaystyle \frac{1}{\tau }}{\hat{P}}_i\right]+\mathrm{c}.\mathrm{c}.,$$

(10)

where $\varphi$ is the electrostatic potential, $V_{Er}$and $V_{\theta }$ are, respectively, the radial and poloidal components of the velocities, $k_r$and $k_{\theta }$are the corresponding wavenumbers, $D_B$ represents the Bohm diffusivity, the asterisk denotes complex conjugation (c.c.), and $⟨\cdot ⟩$ denotes a flux-surface average. We normalize

$$\hat{\varphi }={\displaystyle \frac{e\varphi }{T_e}}, \widehat{P_i}={\displaystyle \frac{\delta p_i}{n_i{T}_i}}.$$

Here, the temperature perturbation from Equation (9) is the most significant part of Equation (10).

When ion viscosity ${\nu }_{ii }$ is included according to Ref. [2], the temperature perturbation changes to [57]

$${\displaystyle \frac{\delta T_i}{T_i}}={\displaystyle \frac{\omega }{\omega -\frac{5}{3}{\omega }_{Di}+i {\nu }_{ii }}}\left[{\displaystyle \frac{2}{3}}{\displaystyle \frac{\delta n_i}{n_i}}-{\displaystyle \frac{{\omega }_{\ast e}}{\omega }}\left({\displaystyle \frac{2}{3}}-{\eta }_i\right){\displaystyle \frac{e\varphi }{T_e}} \right].$$

(11)

Note that the viscosity term enters at the fluid resonance in the ion energy equation. This term plays a central role in setting the system response. Although rotation is only included implicitly in our earlier studies, it enters through the absorbing boundary for long wavelengths, thus strongly influencing the turbulence level. The temperature perturbation controls the whole system through the Reynolds stress, and thus the fluid closure is crucial. The ion viscosity in Equation (11) increases the turbulence level by reducing the rotation. It has the ion mass in the denominator and the density in the numerator. However, in ordinary discharges, the ion viscosity is quite small, thus introducing the favorable isotope scaling, as studied in [13,56], and it does not change our previous results where the ion viscosity was ignored. However, if one increases the density strongly, quite a severe reduction is obtained in the confinement, associated with the density limit [56]. It is only for our reactive fluid closure that it is the diamagnetic part which dominates in the Reynolds stress. This has recently been confirmed by nonlinear gyrokinetic simulations [124]. The dominance of the diamagnetic drive of the Reynolds stress is in our model due to a fluid resonance in Equation (11) and was, as far as we know, the reason for the first successful simulation of the spin-up of poloidal rotation in internal transport barriers [35].

4. The Importance of the Linear Eigenfrequency

As we have mentioned in Section 1, our model is considerably good in reproducing various transport barriers, including the nonlinear Dimits upshift. The reason can be found in Equations (10) and (11). As it turns out, the seal eigenfrequency is typically close to 5/3 ω_Di so that one is quite close to resonance in Equation (9) to Equation (11) for a typical small ion viscosity. This can be found in most of our publications, such as Refs. [10,11].

However, the strongest support comes from the exceptionally good agreement for the real part of our eigenfrequency with that from Rewoldt’s code in [64]. The linear part of the real eigenfrequency is stabilizing in the same way as the lowest-order FLR effect, ω_¤I for magnetohydrodynamics (MHD) ballooning modes. One can see this from Equations (6.144)–(6.146) of Ref. [42]:

$${\omega }_r={\displaystyle \frac{1}{2}}{\omega }_{\ast e}\left[1-\left(1+{\displaystyle \frac{10}{3\tau }}\right){\epsilon }_n+\mathrm{F}\mathrm{L}{\mathrm{R}}_1\right],$$

(12)

$$\gamma ={\displaystyle \frac{{\omega }_{\ast e}\sqrt{{\epsilon }_n/\tau }}{1+k^2{\rho }^2}}\sqrt{{\eta }_i-{\eta }_{ith}},$$

(13)

$${\eta }_{ith}={\displaystyle \frac{2}{3}}-{\displaystyle \frac{\tau }{2}}+{\epsilon }_n\left({\displaystyle \frac{\tau }{4}}+{\displaystyle \frac{10}{9\tau }}\right)+{\displaystyle \frac{\tau }{4{\epsilon }_n}}+\mathrm{F}\mathrm{L}{\mathrm{R}}_2,$$

(14)

where ε_n = 2 L_n/R (R is the main radius). We start by noting that we have used Ref. [69] in deriving Equations (9) and (11). This signifies that our closure term, the diamagnetic heat flow, q_×, contributes only with its toroidal (curvature) parts.

This term contributes the major part of the term proportional to ε_n in Equations (12)–(14). Since Equation (12) is in exceptionally good agreement with [64], we can conclude that our closure term works at least for the real eigenfrequency. We also note that, in our model, the threshold increases towards the axis, where ε_n is large. In the fluid case, we know that quasilinear theory studies [11,45]. This means that we can be confident that we remain close to the fluid resonance in Equations (9) and (11). Accordingly, then the diamagnetic flux dominates in Equation (10). We also note that our threshold is largest near the axis. This means that the linear growth rate and, accordingly, the transport will be growing with radius. This is the experimental trend, and it was also found in nonlinear kinetic simulations in Ref. [71]. We note also that the linear kinetic growth rate in Ref. [64] would counteract the radial growth of the transport. This is because the kinetic growth rate is larger than the fluid growth rate, indicating inverse Landau damping or drift resonance. Since the added gyro-Landau resonance in Ref. [15] here is evidently destabilizing and since the magnetic drift part in general dominates, the destabilizing effect would increase towards the axis. This would be against experimental trends and the simulation results in Ref. [71]. Further support for our closure is that the diamagnetic part of the Reynolds stress dominates also in a nonlinear gyrokinetic code [124]. We also note that we obtained the right trend for the radial variation in the ion thermal conductivity as shown in Figure (6.13) in Ref. [42]. This result was obtained with an interpretative code. Predictive codes are more accurate and often give a stronger radial growth of the ion thermal conductivity.

5. The Multi-Mode Model

The Multi-Mode Model (MMM), which incorporates the Weiland model for ion temperature gradient (ITG), trapped electron mode (TEM), kinetic ballooning mode (KBM), and peeling–ballooning (PB) instabilities, along with high-mode-number MHD and other transport models, provides a comprehensive framework for predicting turbulent transport in magnetically confined plasmas. Implemented within the TRANSP integrated modeling framework [125], the MMM captures the key transport characteristics of National Spherical Torus Experiment (NSTX) discharges, exhibiting neoclassical-like ion heat transport due to strong plasma shaping, high normalized beta, and substantial equilibrium flow shear that suppress microturbulence—consistent with experimental observations. In contrast, electron heat transport remains anomalous, also in agreement with experimental findings. The simulated ion and electron temperature profiles are in quite good agreement with the experimental measurements, demonstrating the model’s capability to reproduce the overall confinement characteristics of spherical tokamaks [60].

MMM-based simulations have also been carried out for a range of conventional tokamak plasmas, including the Joint European Torus (JET), DIII-D, Experimental Advanced Superconducting Tokamak (EAST), and Korea Superconducting Tokamak Advanced Research (KSTAR). Across these devices, the model consistently reproduces the observed transport trends and temperature profile shapes, validating its predictive capability over a broad spectrum of plasma conditions [56]. Recent benchmarking against CGYRO gyrokinetic simulations further supports the fidelity of the fluid closure approach used in MMM, confirming its ability to capture key turbulence-driven transport trends across multiple regimes [59]. These results underscore the robustness of MMM in bridging gyrokinetic-level physics and integrated modeling applications, supporting its use for both interpretive and predictive simulations of present and next-generation tokamak plasmas.

6. Discussion

We have discussed the most fundamental parts of the physics of magnetic confinement for nuclear fusion. We note that our description has been derived from first principles. The analytical solution by Subrahmanyan Chandrasekhar [53] is exactly valid only for stationary turbulence, where the coefficients in the Fokker–Planck equation for turbulent collisions have constant coefficients. However, this is a state which is always approached [31,44], and this was studied for quite general conditions in Ref. [44]. Thus, our derivation in Ref. [5] is always fulfilled. The derivation for the explosively unstable case in Ref. [61] is thus just an example of this fluid closure.

Recently, there has been experimental confirmation [63] of the role of Reynolds stress (2) for the L-H transition [36] and density limit [56] through space correlations in a way predicted by our fluid model. The efficiency of Equation (2) for giving strong rotation is due to our reactive fluid closure. This comes from the $E\times B$ (electric by magnetic fields) convection of the perturbed diamagnetic drift, which is typically close to the fluid resonance in the energy equation. This resonance occurs because the real part of frequency ω is typically close to 5/3 ω_Di, where fluid and kinetic frequencies are in general close [61]. While the effect of ion viscosity in Equation (3) is typically quite small for normal density, it becomes considerably large when the density limit is approaching. Thus, there, a quite large ion mass can significantly increase the density limit. An essential point is that the linear real eigenfrequency in our model was found to agree reasonably well with linear kinetic theory [64]. This is particularly interesting since the linear real eigenfrequency in our model depends strongly enough on our closure term. The linear eigenfrequency has a stabilizing influence, thus giving agreement with the experimental trend, also seen in nonlinear kinetic simulations [71], that transport grows with radius. The linear kinetic growth rate, containing effects of gyro-Landau resonances, however, counteracts this. An obvious conclusion here is that resonance broadening nonlinearly removes the gyro-Landau resonances.