2 Instability thresholds

The description of the resonant destabilisation of the toroidal ITG is based on the following eigenvalue equation (Terry et al. Reference Terry, Anderson and Horton1982; Biglari et al. Reference Biglari, Diamond and Rosenbluth1989; Romanelli Reference Romanelli1989)

where $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ , $\unicode[STIX]{x1D714}$ is the mode complex frequency, $\boldsymbol{k}_{\bot }$ is the wave vector perpendicular to the equilibrium magnetic field, $\hat{v}^{2}=(v_{\Vert }^{2}+v_{\bot }^{2})/v_{\text{thi}}^{2}$ , with $v_{\text{thi}}=\sqrt{2T_{i}/m_{i}}$ the ion thermal speed and $m_{i}$ and $n_{0}$ are the ion mass and density, respectively. We are effectively considering a local limit, in the sense that the eigenmode structure along the field-following coordinate is neglected, and we restrict our attention to a particular location along the equilibrium magnetic field. This is equivalent to completely neglecting the slab branch discussed in the introduction, and taking $k_{\Vert }v_{\text{thi}}\ll \unicode[STIX]{x1D714}$ , where $k_{\Vert }$ is the parallel wavenumber. The equilibrium distribution function is taken to be Maxwellian,

Two characteristic frequencies are present in (2.1)

with $\unicode[STIX]{x1D714}_{\ast i}=0.5k_{y}\unicode[STIX]{x1D70C}_{i}v_{\text{thi}}/L_{n}$ , $\unicode[STIX]{x1D702}_{i}=L_{n}/L_{T}$ , and

where $\unicode[STIX]{x1D70C}_{i}=v_{\text{thi}}/\unicode[STIX]{x1D6FA}_{ci}$ is the ion Larmor radius, $\unicode[STIX]{x1D6FA}_{ci}=eB/(m_{i}c)$ the ion cyclotron frequency, $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}=0.5k_{y}\unicode[STIX]{x1D70C}_{i}v_{\text{thi}}/a$ and $\unicode[STIX]{x1D714}_{B}=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}-\text{d}\unicode[STIX]{x1D6FD}/\text{d}x$ the magnetic curvature and $\unicode[STIX]{x1D735}B$ -drift frequencies, while $L_{n}^{-1}=-n_{0}^{-1}\text{d}n_{0}/\text{d}x$ and $L_{T}^{-1}=-T_{i}^{-1}\text{d}T_{i}/\text{d}x$ define the characteristic density and temperature-gradient scales, and $\unicode[STIX]{x1D6FD}=8\unicode[STIX]{x03C0}n_{0}(T_{e}+T_{i})/B^{2}$ is the ratio of kinetic to magnetic pressure (which will shortly be set to zero), $x$ is the radial coordinate and $y$ is a coordinate associated with the field-line labels on a magnetic surface. Notice that we are assuming $k_{x}=0$ for simplicity. The Bessel function squared originates, as usual, from a Fourier transform of the gyroaveraged potential followed by a $\text{d}^{3}\boldsymbol{v}$ integral at constant $\boldsymbol{r}$ , the particle position. In (2.1) the eigenvalue $\unicode[STIX]{x1D714}$ acquires non-zero imaginary values that regularise the, otherwise, singular integrand. The analytical form of the integral $I_{i}$ for arbitrary Larmor radii is given in appendix A. The drift-kinetic limit is sufficient for the following discussion. In this case $\boldsymbol{k}_{\bot }\unicode[STIX]{x1D70C}_{i}\rightarrow 0$ , $J_{0}^{2}\rightarrow 1$ , and it is easy to show that

which corresponds to $D_{0}$ in (3) of BDR (Biglari et al. Reference Biglari, Diamond and Rosenbluth1989). In the appendices we prove this result in two different ways; both are alternatives to the derivation of Biglari et al. (Reference Biglari, Diamond and Rosenbluth1989). The real frequency at marginality is evaluated by solving $\text{Im}[D(\unicode[STIX]{x1D714}_{r})]\equiv D_{i}(\unicode[STIX]{x1D714}_{r})=0$ , where one sets $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{r}+\text{i}\unicode[STIX]{x1D6FE}$ , and takes $\unicode[STIX]{x1D6FE}\rightarrow 0$ . Using (2.5) this means

Once $\unicode[STIX]{x1D714}_{r}$ is found, the condition that determines destabilisation is $\text{Re}[D(\unicode[STIX]{x1D714}_{r})]>0$ , which can be determined only numerically. However, Romanelli (Romanelli Reference Romanelli1989) has evaluated the integral $I_{i}$ by replacing $v_{\bot }^{2}/2+v_{\Vert }^{2}\rightarrow 4/3(v_{\bot }^{2}+v_{\Vert }^{2})$ in the definition of the particle magnetic drifts (see (2.4)) in order to fit the numerical solution of (2.1) at $R/L_{n}\ll 1$ , for $k_{y}\unicode[STIX]{x1D70C}_{s}=\sqrt{0.1}\approx 0.32$ , where $\unicode[STIX]{x1D70C}_{s}=\sqrt{T_{i}/m_{i}}/\unicode[STIX]{x1D6FA}_{i}=\unicode[STIX]{x1D70C}_{i}/\sqrt{2}$ . In his case, the condition $\text{Re}[D(\unicode[STIX]{x1D714}_{r})]>0$ yields a critical gradient $\left.R/L_{T}\right|_{\text{crit}}=(1+\unicode[STIX]{x1D70F})(4/3)$ whose $\unicode[STIX]{x1D70F}$ dependence comes from the obvious fact that $\text{Re}[D(\unicode[STIX]{x1D714}_{r})]>0$ implies $\text{Re}[I_{i}(\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{r})]>1+\unicode[STIX]{x1D70F}$ . Obtaining the correct mode frequency at marginality is therefore crucial. From direct numerical simulations we see that this quantity shows a dependence on the ion temperature that is not captured by Romanelli’s treatment, nor by any other analytical theory known to us (see figure 1).

Figure 1. Real frequency from the GENE code as a function of the temperature gradient for different $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ . The $\unicode[STIX]{x1D70F}$ dependence is evident. Cyclone base case parameters are used (see text for details).

In our numerical tests, we consider adiabatic electrons and typical cyclone base case (CBC) parameters for ${\hat{s}}-\unicode[STIX]{x1D6FC}$ geometry: ${\hat{s}}=0.786$ , $q=\unicode[STIX]{x1D704}^{-1}=1.4$ , $\unicode[STIX]{x1D716}=r/R=0.18$ , $R/L_{n}=2$ , $k_{y}\unicode[STIX]{x1D70C}_{s}=0.3$ , where $R$ is the device major radius. From the imaginary part of the eigenvalue, we extract the critical gradient (see figure 2). Figure 3 shows the resulting critical gradient compared to several theories and one fitting formula. The linear fit of the numerical data givesFootnote ³ $\left.R/L_{T}\right|_{\text{crit}}=(2.45\pm 0.07)\unicode[STIX]{x1D70F}+2.07\pm 0.06$ , with a reduced $\unicode[STIX]{x1D712}^{2}=6.5\times 10^{-4}$ . The fitting formula of Jenko et al. (Reference Jenko, Dorland and Hammett2001) (hereafter denoted by JDH) is $\left.R/L_{T}\right|_{\text{crit}}=(1+\unicode[STIX]{x1D70F})(4/3+1.91\,{\hat{s}}/q)$ . The curve derived by Romanelli (Reference Romanelli1989), $R/L_{T}=(1+\unicode[STIX]{x1D70F})\left(4/3\right)$ , which should work better for flat densities, $R/L_{n}\rightarrow 0$ , is not expected to match the numerical results for $R/L_{n}=2$ , $R/L_{T}=O(1)$ , even if $q\rightarrow \infty$ (in which case the streaming term effect measured by the factor ${\hat{s}}/q$ in the JDH formula should be negligible, as it is neglected in (2.1)). The BDR condition for destabilisation (Biglari et al. Reference Biglari, Diamond and Rosenbluth1989) would give $R/L_{T}=2.7$ , with virtually no dependence on $\unicode[STIX]{x1D70F}$ . The full ion Larmor radius resonant theory developed in the present article predicts higher critical gradients than the long-wavelength resonant theory. Like results from numerical simulations, it gives a critical gradient that depends linearly on $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ , but it underestimates the numerical results. As already mentioned, this discrepancy can be attributed to the fact that analytical theories predict a real frequency at marginality that does not depend on $\unicode[STIX]{x1D70F}$ , as opposed to the numerical findings. This is true for all the analytical theories revisited and derived here, and seems to be a consequence of the neglect in (2.1) of the streaming term. The problem is, of course, that parallel streaming has been neglected in the derivation of (2.1), which can only be satisfied at a single (or a pair of) poloidal location on the flux surface. Taken literally, this equation thus implies that the entire mode structure is concentrated at this point, which, in the analytical theories quoted, is the location where the field curvature is most unfavourable. However, in all practical circumstances, that is not what the mode structure is, and the streaming term (with its $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}$ derivatives, where $\unicode[STIX]{x1D703}$ is the ballooning angle) determines the rate at which the eigenmode decays along the field line.

Figure 2. Growth rate from the GENE code as a function of the temperature gradient for different $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ . The $\unicode[STIX]{x1D70F}$ dependence is evident. CBC parameters are used (see text for details).

Figure 3. Critical threshold from GENE data, linear fit to GENE data, fitting formula of Jenko Dorland and Hammett and different local theories. The GENE data at $q=100$ should be compared to the local theories.

The results from GENE simulations are somewhat consistent with the fitting formula of Jenko et al. (Reference Jenko, Dorland and Hammett2001) (hereafter JDH) but do not seem to relate to the theory of Hahm & Tang (Reference Hahm and Tang1989), since we find a stabilising contribution from finite shear. This is in line with a statement present in the work of Biglari et al. (Reference Biglari, Diamond and Rosenbluth1989).Footnote ⁴ The remaining two curves are derived from the numerical solution of the dispersion relation with full Larmor orbits, equation (A 1), for small ( $\boldsymbol{k}_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ ) and for arbitrary Larmor radii. Strictly speaking, all our numerical results are valid for one specific wavelength: $k_{y}\unicode[STIX]{x1D70C}_{s}=0.3$ . However, a dependence of the numerical results on $k_{y}$ cannot be excluded. A sensitivity scan was performed for the reference value $\unicode[STIX]{x1D70F}=1$ . If we vary $0.2\leqslant k_{y}\unicode[STIX]{x1D70C}_{s}\leqslant 0.4$ , we find a critical gradient in the range $4.35\leqslant \left.R/L_{T}\right|_{\text{crit}}\leqslant 4.8$ This indeterminacy is small enough to discriminate the analytical results of figure 3, and large enough to let us consider the JDH formula quite reliable.

This preliminary analysis is in fact more general that it would seem. Equations (2.1) and (A 1) were derived by neglecting the contribution associated with particle streaming, but the magnetic drift frequencies $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}$ and $\unicode[STIX]{x1D714}_{B}$ are essentially arbitrary. They must therefore be applicable to the stellarator case as well. There is a key factor that now brings tokamaks and stellarators together. A low-bootstrap-current stellarator, such as Wendelstein 7-X, possesses a low global shear. This condition is, incidentally, the same condition that is required to neglect particle streaming in the kinetic equation, and the slab branch of the ITG instability (Jenko et al. Reference Jenko, Dorland and Hammett2001) for the tokamak case. Thus, for low shear, the resonant destabilisation of the ITG mode in tokamaks and stellarators could show some common features. In the following sections we will discover what these common aspects are.

Figure 4. Imaginary part of eigenvalue from the GENE code as a function of the temperature gradient for different $\unicode[STIX]{x1D70F}$ . Parameters as in figure 1, but ${\hat{s}}=0.1$ , $\unicode[STIX]{x1D716}=0.03$ . The ‘background’ modes at marginality are evident.

Figure 5. Real part of eigenvalue from the GENE code as a function of the temperature gradient for different $\unicode[STIX]{x1D70F}$ . Parameters as in figure 1, but ${\hat{s}}=0.1$ , $\unicode[STIX]{x1D716}=0.03$ . The ‘background’ modes at marginality are evident.

Figure 6. Ion distribution function in phase space. $\unicode[STIX]{x1D70F}=1$ , $R/L_{T}=3$ , for the ‘background’ mode that belongs to the ‘foot’ of figure 5. No clear resonance at finite $v_{\Vert }$ . The velocity-space resolution in the $v_{\Vert }$ -direction is $5.86\times 10^{-3}$ . Other parameters as in figure 5.

Figure 7. Ion distribution function in phase space. $\unicode[STIX]{x1D70F}=1$ , $R/L_{T}=5$ , for the toroidal branch mode. Resonance occurs outside the $\unicode[STIX]{x1D707}\pm \hat{v}_{\Vert }^{2}\equiv \text{const.}$ boundary. The velocity-space resolution in the $v_{\Vert }$ -direction is $5.86\times 10^{-3}$ . Other parameters as in figure 5.

Figure 8. Eigenfunctions as a function of the ballooning angle for the ${\hat{s}}-\unicode[STIX]{x1D6FC}$ tokamak equilibrium from the GENE code (flux tube). Here $\unicode[STIX]{x1D70F}=1$ , $k_{y}\unicode[STIX]{x1D70C}_{s}=0.3$ , $a/L_{T}=3$ (Floquet), $a/L_{T}=5$ (toroidal branch). The velocity-space resolution in the $v_{\Vert }$ direction is $5.86\times 10^{-3}$ . Other parameters as in figure 4.

Figure 9. A ‘zoom’ of figure 8. Eigenfunctions as a function of the ballooning angle for the ${\hat{s}}-\unicode[STIX]{x1D6FC}$ tokamak equilibrium from the GENE code (flux tube). Here $\unicode[STIX]{x1D70F}=1$ , $k_{y}\unicode[STIX]{x1D70C}_{s}=0.3$ , $a/L_{T}=3$ (Floquet), $a/L_{T}=5$ (toroidal branch). The velocity-space resolution in the $v_{\Vert }$ direction is $5.86\times 10^{-3}$ .

3 Temperature dependence of critical thresholds

In the previous section we have conjectured that the resonant destabilisation of the ITG mode in low-shear tokamaks and stellarators possesses common features. Numerical simulations corroborate this hypothesis. We refrain from proposing some universal formula for critical gradients at low shear, nonetheless we find it instructive to explore the temperature dependence of the critical gradient, since this puts the new results in the same context as previous analytical studies.

In the asymptotic limit of large inverse aspect ratio, $\unicode[STIX]{x1D716}=0.03$ , the ${\hat{s}}-\unicode[STIX]{x1D6FC}$ tokamak case gives a critical inverse gradient that increases with $\unicode[STIX]{x1D70F}$ , for $\unicode[STIX]{x1D70F}\gtrsim 1$ . For $\unicode[STIX]{x1D70F}\lesssim 1$ , the instability threshold is fixed by the density gradient, e.g. $(R/L_{T})_{\text{crit}}\approx R/L_{n}$ (see figure 15). In accordance with the JDH formula, a decreasing shear and an increasing $L_{n}/L_{T}$ have a destabilising effect (see figure 16). The results for Wendelstein 7-X (figure 17) were all derived in the flat density limit $R/L_{n}=0$ . The critical temperature gradient is a growing function of $\unicode[STIX]{x1D70F}$ . We observe that $(a/L_{T})_{\text{crit}}\approx 1$ for sufficiently hot ions, $\unicode[STIX]{x1D70F}\gtrsim 2$ .

Figure 15. Critical threshold from GENE data of figure 5 for several density gradients,  $R/L_{n}.$

Figure 16. Critical threshold from GENE. CBC parameters are used (see text for details), however ${\hat{s}}=0.1$ for $R/L_{n}=0$ and ${\hat{s}}=0.786$ for $R/L_{n}=2$ .

Figure 17. Critical gradient for Wendelstein $7-X$ as a function of the temperature ratio $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ .

4 Floquet solutions

What are the modes that emerge below the critical threshold of the toroidal branch at low shear? In the classical nomenclature of ITG modes in tokamaks, for ${\hat{s}}/q\approx \unicode[STIX]{x1D704}{\hat{s}}\ll 1$ , the slab branch of the instability tends to be negligible. Below marginality the toroidal branch is stable. We saw that the background of modes at marginality is not eliminated at large inverse aspect ratios in tokamaks. High-mirror, as well as low-mirror, Wendelstein 7-X configurations also exhibit this persistent feature, and we recall that density gradients are stabilising since the electrons are taken to be adiabatic. To shed some light on the nature of these modes, we are compelled to abandon for a moment the regime of marginality and consider these modes at their maximum growth rate. We first verify where this maximum is. We select two representative cases for $\unicode[STIX]{x1D716}=0.1$ , and $\unicode[STIX]{x1D70F}=1:$ $R/L_{T}=3$ (background) and $R/L_{T}=4$ (strong toroidal branch). The spectra of the two instabilities are shown in figure 18. We first notice that both the background and toroidal branch modes occur approximately at the same ion scale, $k_{y}\unicode[STIX]{x1D70C}_{s}\approx 0.3$ . This does not necessarily imply that the two modes would be destabilised for the same $R/L_{T}$ . Let us now consider the non-resonant limit. We take the fluid limit of the ITG equation (see (19) of Zocco et al. (Reference Zocco, Plunk, Xanthopoulos and Helander2016), or the Fourier transform of (12) of Connor & Taylor (Reference Connor and Taylor1987), or (B1) of Candy, Waltz & Rosenbluth (Reference Candy, Waltz and Rosenbluth2004)), neglect altogether the shear, i.e. ${\hat{s}}\equiv 0$ , and restore the streaming term, to obtain the Mathieu equation

with $q(\unicode[STIX]{x1D714})=8\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}^{(0)}/(v_{\text{thi}}^{2}/\ell _{\Vert }^{2})$ , $a(\unicode[STIX]{x1D714})=-8(\unicode[STIX]{x1D714}^{3}/\unicode[STIX]{x1D714}_{T}v_{\text{thi}}^{2}/\ell _{\Vert }^{2})[\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D714}_{T}k_{y}^{2}\unicode[STIX]{x1D70C}_{i}^{2}/(2\unicode[STIX]{x1D714})]$ and $z=l/2$ , where $l$ is the field-following coordinate and $\ell _{\Vert }$ is the parallel connection length. Basically, when the system approaches marginality, the toroidal driving term (proportional to $q$ , not to be confused with the safety factor), becomes smaller and the second-order derivative associated with the streaming term cannot be neglected. The absence of secular terms due to very low shear, however, makes this non-resonant analysis still valid. The resonant theory of such modes does not seem straightforward.

Figure 18. Spectra of instability for an ${\hat{s}}-\unicode[STIX]{x1D6FC}$ tokamak, $\unicode[STIX]{x1D716}=0.1$ , $\unicode[STIX]{x1D70F}=1$ , $R/L_{T}=3$ (background) and $R/L_{T}=4$ (strong toroidal branch).

Equation (4.1) has been studied in the context of formation of internal transport barriers in ITG turbulence by Candy et al. (Reference Candy, Waltz and Rosenbluth2004) and by Connor & Hastie (Reference Connor and Hastie2004). A more general form was investigated by Taylor & Wilson (Reference Taylor and Wilson1996). In stellarator research, ‘weakly localised’ Mathieu solutions were first identified by Bhattacharjee et al. (Reference Bhattacharjee, Sedlak, Similon, Rosenbluth and Ross1983).

The most general solution of (4.1) is

where $\unicode[STIX]{x1D6F7}$ is periodic with period $\unicode[STIX]{x03C0}k$ , with $k$ integer. The evaluation of $\unicode[STIX]{x1D707}$ was first published in 1886 by Hill (Reference Hill1886), and it will now be applied to the ITG case. One introduces the Fourier series for the $\unicode[STIX]{x03C0}$ -period solution

to obtain

For a $2\unicode[STIX]{x03C0}$ -period solution, $\unicode[STIX]{x1D711}_{2\unicode[STIX]{x03C0}}(z)=\text{e}^{\unicode[STIX]{x1D707}z}\sum _{n=-\infty }^{\infty }b_{2n}^{\prime }\exp [\text{i}nz]$ , we have

where $c_{n}^{\prime }\equiv b_{2n}^{\prime }$ . Equation (4.5) is, for $\unicode[STIX]{x1D707}=0$ , see (21) of Connor & Taylor (Reference Connor and Taylor1987). It is clear that $\unicode[STIX]{x1D707}=0$ is the eigenvalue condition, because twisting boundary conditions require $\unicode[STIX]{x1D711}^{\prime }(0)=\unicode[STIX]{x1D707}\unicode[STIX]{x1D6F7}(0)+\unicode[STIX]{x1D6F7}^{\prime }(0)=\unicode[STIX]{x1D707}\unicode[STIX]{x1D6F7}(0)=0$ , which, indeed, implies $\unicode[STIX]{x1D707}=0$ . Similarly, we introduce $c_{n}\equiv b_{2n}$ and write (4.4) in matrix form, $\unicode[STIX]{x1D608}_{nm}c_{n}$ , where (for $\unicode[STIX]{x03C0}$ -period solutions), after setting $\unicode[STIX]{x1D707}=0$ ,

is an infinite tridiagonal matrix. The eigenvalue, $\hat{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}/(v_{\text{thi}}/a)$ , is therefore determined from

Notice that for $q\sim a$ , the off-diagonal elements decay like $m^{-2}$ , so the determinant of the matrix is convergent for $m\rightarrow \infty$ . For a given large $M$ , $-M\leqslant (m,n)\leqslant M$ , one can use the iterative formula for determinants of tridiagonal matrices, to show that (4.7) can be written in terms of the truncated continued fraction

where the recursive formula is iterated from $m=M$ , to $m=-M$ , with $\det A_{-M-1}\equiv 1$ . Then

is the eigenvalue equation for the Floquet ITG mode. Candy et al. (Reference Candy, Waltz and Rosenbluth2004) report an eigenvalue condition which reads

and is valid only for $q\gg 1$ . Connor & Taylor (Reference Connor and Taylor1987) derived, for large $q$ ,

after expanding a finite ${\hat{s}}$ version of (4.1) in Fourier harmonics, and treating the Fourier index as a continuous variable. Following Connor and Taylor, we can consider slowly varying $c_{m}$ coefficients, for the $\unicode[STIX]{x03C0}$ -period solution, and obtain

By using the ansatz $c_{m}=\exp [-\unicode[STIX]{x1D706}m^{2}]$ , the eigenvalue condition,

is derived. This corresponds to equation (B9) of Candy et al. (Reference Candy, Waltz and Rosenbluth2004) for $q\rightarrow -q$ ((4.10) in this work). In fact, both eigenvalue equations (4.5) and (4.13) are approximate forms of the exact one, for a different periodicity of $2\unicode[STIX]{x03C0}$ or $\unicode[STIX]{x03C0}$ , respectively. For each periodicity, we have two eigenmodes (and eigenvalues) depending on the argument of $q$ . For instance, the $2\unicode[STIX]{x03C0}$ -period solutions have maxima at the poloidal location $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ , equation (4.11) and $\unicode[STIX]{x1D703}=0$ ((4.11) with $q\rightarrow -q$ ).

Figure 19. Numerical solution of (4.7) and (4.13) derived for $\unicode[STIX]{x03C0}$ -periodic solutions with maximum at $\unicode[STIX]{x1D703}\propto z=0$ . The eigenvalue of the local toroidal branch is given by the solution of $a=2q$ , where $a$ and $q$ are defined in the text.

The numerical solution of (4.7), $\hat{\unicode[STIX]{x1D6FE}}=\text{Im}[\hat{\unicode[STIX]{x1D714}}]$ , is plotted in figure 19 as a function of the matrix dimension for $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}^{(0)}/(v_{\text{thi}}/a)=0.45$ , $\unicode[STIX]{x1D714}_{T}/(v_{\text{thi}}/a)=0.6$ , $v_{\text{thi}}/\ell _{\Vert }=0.5(v_{\text{thi}}/a)$ and $k_{y}\unicode[STIX]{x1D70C}_{i}=0.3$ (here $a$ is a normalising length). The result is compared to the solution of (4.13). In this case $q=10$ . The agreement is excellent. For $q\sim 1$ , which is the relevant regime, since (4.1) is derived for $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D705}}/\unicode[STIX]{x1D714}\sim v_{\text{thi}}^{2}/\ell _{\Vert }^{2}\ll 1$ , equation (4.7) is the correct eigenvalue equation. It is worth mentioning that Taylor & Hastie (Reference Taylor and Hastie1968), in their seminal work where linear gyrokinetics was derived for the first time for electrostatic perturbations in a torus, report an eigenvalue equation of the form of our (4.7). Their eigenvalue equation (50) reduces to a determinant of a tridiagonal matrix (derived by Coppi et al. Reference Coppi, Laval, Pellat and Rosenbluth1968), when the plane slab ITG is modified by a sinusoidal periodic gravity term. Coppi et al. recognised the role of the Hill determinant in the evaluation of the true eigenvalue given by the solution of (4.7), but the approximated eigenvalue of (4.11) was derived only $19$ years later! We now see that the resonant destabilisation of Floquet modes in stellarators can possibly be compared with the final result derived by Coppi et al. (Reference Coppi, Laval, Pellat and Rosenbluth1968) in appendix B of their work, if temperature-gradient effects are included.

5 Discussion and conclusions

The aim of this work was to elucidate what sets the linear threshold for destabilisation of the electrostatic ion-temperature-gradient driven mode in toroidal magnetic fusion devices. We study both axisymmetric (tokamaks) and non-axisymmetric (stellartors) confinement concepts.

In regimes of interest for the stellarator Wendelstein 7-X, a background of unstable modes exists below the threshold of destabilisation of the toroidal branch of the ITG mode. We proposed the hypothesis that such modes are ITG modes of the Floquet type. The eigenfunctions resulting from GENE numerical simulations of Wendelstein 7-X show a much slower decay along the field line for these modes than for modes of the toroidal branch (see figure 14). This behaviour is evocative of the prediction of weakly localised drift-wave modes in stellarators, described in terms of Mathieu functions by Bhattacharjee et al. (Reference Bhattacharjee, Sedlak, Similon, Rosenbluth and Ross1983). In the non-resonant limit, a complete eigenvalue equation is then proposed and studied in relation to previously known results.

The importance of our findings relies on the fact that such modes result from having a very low global shear, ${\hat{s}}$ , and feature similar qualitative behaviour in tokamaks and stellarators. In the former case, one could imagine them being relevant for advanced confinement scenarios with very flat safety factor profiles. For finite density gradients, the insensitivity of the derived temperature gradient thresholds to the temperature ratio $\unicode[STIX]{x1D70F}=T_{i}/T_{e}$ for $\unicode[STIX]{x1D70F}<1$ seems a rather clear effect, distinguishing these modes from the more conventional branches. For stellarators, our findings could have a non-negligible impact on confinement if Floquet solutions play any role nonlinearly. If stellarator electrostatic turbulence is affected by the nonlinear activity of these modes, there is an open question regarding what is really the achievable temperature gradient for a stellarator with low global shear, as opposed to known tokamak cases, where a finite turbulent transport is observed above a finite critical temperature gradient, generally larger than the one discussed in this work (Dimits et al. Reference Dimits, Bateman, Beer, Cohen, Dorland, Hammett, Kim, Kinsey, Kotschenreuther and Kritz2000). Additional stabilising effects can of course suppress these modes. Our results are perhaps also relevant for trapped-electron mode driven turbulence, where similar eigenfunctions and ‘background’ modes can be observed (Proll, Xanthopoulos & Helander Reference Proll, Xanthopoulos and Helander2013; Faber et al. Reference Faber, Pueschel, Proll, Xanthopoulos, Terry, Hegna, Weir, Likin and Talmadge2015).