3.1. F-$\varTheta$ dependence

The imposed magnetic field in RFPs is unstable for large values of $\varTheta$ and $S$, and it will form a dynamic helical structure with a certain amount of chaotic or turbulent motion superimposed.

The modification of the magnetic field can be quantified by the field reversal parameter $F$, which represents the normalized toroidal field at the boundary,

(3.1)\begin{equation} F = \frac{\overline{B_z}}{\langle B_z\rangle}. \end{equation}

As the current increases, the kink instability increases, which leads to the decrease of the toroidal magnetic field at the boundary, so that $F$ decreases as a function of $\varTheta$. This behaviour can be qualitatively predicted by Taylor's theory (Taylor Reference Taylor1974) and more sophisticated theories allow to improve this agreement (Reiman Reference Reiman1980, Reference Reiman1981; Taylor Reference Taylor1986). In studies by Paccagnella et al. (Reference Paccagnella, Bondeson and Lütjens1991) and Guo et al. (Reference Guo, Xu, Wang and Liu2013) based on two-dimensional equilibrium equations, it was shown that shaping does not alter the $F$-$\varTheta$ curve.

To be able to systematically assess the $F-\varTheta$ dependence, we carry out preliminary simulations for cylinders of small aspect ratio ${L_z}/{2{\rm \pi} b}\sim 2$, an ellipticity $a=1.6$ and moderate Lundquist number $S\sim 4200$. We compare with Taylor's prediction (Taylor Reference Taylor1974) and simulations with circular cross-section. Figure 2 shows the results of the field-reversal parameter $F$ versus $\varTheta$, which are in reasonable agreement with the two previous studies (Paccagnella et al. Reference Paccagnella, Bondeson and Lütjens1991; Guo et al. Reference Guo, Xu, Wang and Liu2013). In these references it was shown that shaping had a small destabilizing effect for large curvature, but the $F-\varTheta$ curve was unaffected. Indeed, the two geometries yield roughly the same behaviour.

Figure 2. Field-reversal parameter $F$ as a function of the pinch parameter $\varTheta$ for cylinders with circular and elliptic cross-section. Also shown is Taylor's prediction (Taylor Reference Taylor1974) for reference.

These moderate-Lundquist simulations allow a parametric investigation of the $F-\varTheta$ dependence for two different ellipticities. Carrying this out for higher values of $S$ would require substantially more computational resources. In the remainder of this investigation, we will consider, at a higher value of the Lundquist number, three different ellipticities but focus on only one $F-\varTheta$ value for each geometry. We have thereto performed simulations for values of the Lundquist number ranging up to $S \approx 2\times 10^4$ and a larger aspect ratio $L_z/2{\rm \pi} b = 4$. These higher values of the Lundquist number are still several orders of magnitude smaller than those in experimental RFPs, which are currently out of reach using precise numerical schemes. We consider three shapes, i.e. cylindrical devices with different cross-sections: a circle with radius $a=1$, an ellipse with $a=1.2$ and $b=0.83$ and an ellipse with $a=1.4$ and $b=0.714$. In the statistically steady states considered at this higher Lundquist number, both the $F$ and $\varTheta$ parameter fluctuate around their steady value, which is also plotted in figure 2. It is observed that these results are in the $F$–$\varTheta$ plane situated closer to Taylor's prediction, which is obtained using statistical mechanics of ideal MHD (Taylor Reference Taylor1974). We do not know whether higher values of $S$ will allow to approach this theoretical prediction even closer. The fluctuations are for $\varTheta$ inferior to $0.5\,\%$. The absolute value of the fluctuations of the $F$-parameter are of the order of $0.01$ (for instance, fluctuating between $F=-0.07$ and $F=-0.082$ for the circular geometry).

We observe thus that the effect of shaping is small on the global behaviour of the magnetic field, as characterized by $F$ and $\varTheta$. The present results indicate that the influence of the Lundquist number is significantly larger than that of shaping in the considered range of parameters. The reversal parameter is a global parameter and does not give insight into the fine structure of the dynamics. It is this fine structure, constituted by the nonlinear interplay of a large number of modes, which will determine the confinement quality of a reactor. The modification of the fine structure is in the following assessed by evaluating pressure gradients and the modal behaviour of the flow.

3.2. The role of the pressure: $\beta$ and gradient-$\beta$

Clearly, in fusion research, the plasma pressure plays a major role. In the present investigation, we do not focus on the confinement quality of the system, but rather on the dynamics and the velocity field. Therefore, we will assess the influence of shaping on the pressure and its influence on the dynamics. In figure 3, we show visualizations of the pressure fluctuations in cross-sections of the domains at the highest considered Lundquist number. Spatial fluctuations of the pressure are observed in all geometries. In particular, in the elliptic cases, the strongest values of the absolute pressure are situated in the centre of the domain and lower pressure values are observed close to the wall. In the circular case, the value of the pressure is smaller than for the elliptic cases. We will now focus on different measures of the influence of pressure on the dynamics.

Figure 3. Instantaneous visualizations of the pressure field for $S \approx 2 \times 10^4$ in cross-sections through the cylinders for the three different ellipticities. From (a–c), $a=1$, $a=1.2$, $a=1.4$.

Classically, to assess the influence of the pressure on the plasma dynamics, the quantity $\beta$ is evaluated, which measures the importance of the plasma pressure compared with the magnetic pressure,

(3.2)\begin{equation} \beta=2\frac{\langle P\rangle}{\langle B^2\rangle}, \end{equation}

where the brackets indicate a volume average over the plasma volume of interest. Other definitions are also used (Wesson & Campbell Reference Wesson and Campbell2011) based on the value of $\boldsymbol {B}$ at the wall of the plasma, for instance. Clearly, ${\beta }$ is an important parameter because the plasma pressure is a key parameter in the analysis of the confinement quality. However, if we are interested in the turbulence properties and the velocity field in general, it is the pressure gradients that are important. It is known from turbulence research that estimating simply the order of magnitude of pressure gradients by ${O}(\boldsymbol {\nabla } P) \sim P/L$, with $L$ a macroscopic lengthscale, can seriously underestimate their magnitude because the pressure gradients are, in general, dominated by small-scale contributions $l\ll L$. Therefore, we recently introduced the alternative, gradient-based $\beta$s (Chahine & Bos Reference Chahine and Bos2018),

(3.3a,b)\begin{equation} \beta_{\boldsymbol{\nabla}}\equiv2\frac{\langle\| \boldsymbol{\nabla} P\|\rangle}{\langle\|\boldsymbol{\nabla} B^2 \|\rangle} \quad \textrm{and}\quad {\beta'_{\boldsymbol{\nabla}}}\equiv\frac{\langle \| \boldsymbol{\nabla} P \|\rangle}{\langle \| \boldsymbol{J}\times \boldsymbol{B}\|\rangle}. \end{equation}

It is only when ${\beta '_{\boldsymbol {\nabla }}}$ is small compared to unity that the influence of the pressure term might be negligible in the dynamics, compared with the influence of the other terms in the velocity equation. This is not necessarily the case when $\beta$ is small.

In figure 4, we show the behaviour of the different versions of $\beta$. The size of the temporal fluctuations of the different quantities around the time-averaged value are indicated by error bars. In agreement with the results of Chahine & Bos (Reference Chahine and Bos2018), the value of $\beta$ is decreasing for increasing values of the Lundquist number. We see that for the highest values of $S$, $\beta$ has dropped most importantly in the circular geometry. This is not inconsistent with figure 3, which shows that the normalized pressure is weaker in this geometry than in the other two. Indeed, focusing only on this observation, one might be tempted to extrapolate and assume that the pressure plays no dominant role in the dynamics at high values of $S$. However, the gradient based $\beta '_{\boldsymbol {\nabla }}$ remains approximately constant when the Lundquist number is increased. This shows that the influence of the pressure gradients does not become less important for larger $S$.

Figure 4. Lundquist-number dependence of three alternative definitions of $\beta$ in cylinders with ellipticity (a) $a=1$, (b) $a=1.2$, (c) $a=1.4$. Whereas the pressure based $\beta$ is strongly decreasing with $S$, the gradient-based value $\beta '_{\boldsymbol {\nabla }}$ is roughly constant.

These observations are quite robust and qualitatively independent on the shape of the cylinder. What changes however, is the $S$-dependence of ${\beta }_{\boldsymbol {\nabla }}$. This indicates that $\langle \|\boldsymbol {\nabla } B^2 \|\rangle / \langle \| \boldsymbol {J}\times \boldsymbol {B}\|\rangle$ is influenced by the ellipticity of the cylinder cross-section. From the vector identity $\boldsymbol {J}\times \boldsymbol {B}=-\boldsymbol {\nabla } B^2/2+\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {B}$, this shows that the different behaviour is associated with the curvature term $\boldsymbol {B}\boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {B}$, which seems to be affected by the shaping. Shaping does therefore change the magnetic structure of the flow. We will further investigate this in the following, considering spectral considerations. Before that, we will visualize the magnetic structure of the simulated plasmas.

3.3. Characterization of the velocity and magnetic field

In figure 5, we show magnetic flux lines in cross-sections through the three cylindrical geometries. The lines correspond to iso-values of the axial component of the magnetic vector potential. This potential is computed from the induced field, i.e. the imposed field is subtracted from the total magnetic field.

Figure 5. Flux lines in a cross-section through the differently shaped geometries. From left to right, the cross-sections are a circle with radius $a=1$, an ellipse with major semi-axis $a=1.2$ and one with $a=1.4$, for $S \approx 2\times 10^4$. Visualized is the induced field, i.e. the magnetic field without the imposed contribution.

The most dramatic difference is observed for the $a=1.4$ geometry. Indeed, for the circular cross-section, the iso-lines are mostly concentric. Changing the shape, for $a=1.2$, a separatrix appears at the edges of the ellipse and for $a=1.4$, the central flux surfaces are destroyed and magnetic islands appear. Clearly, the shape of the magnetic field is dramatically altered by the shaping in this latter geometry, compared with the circular cross-section.

In figure 6, axial magnetic fluctuations are shown in various poloidal cross-sections. The fluctuations of the axial magnetic field are obtained by showing the magnetic field without the axially invariant ($k_z=0$) magnetic contribution. Only in a few sections is a clear poloidal mode structure observed. From these observations, it seems relatively clear that no single-helicity state is present in our simulations. A large number of modes seems to be evolving simultaneously and instantaneous local visualizations are not the best tool to illustrate this.

Figure 6. Poloidal cross-sections of the cylinder, illustrating the axial magnetic fluctuations at $t=9.8\times 10^3 \tau _A$ for $S \approx 2\times 10^4$. The geometries are a circle with radius $a=1$, an ellipse with major semi-axis $a=1.2$ and one with $a=1.4$.

To analyse the presence of different modes, we visualize now the effect of shaping on these helical modes by evaluating the axially Fourier-transformed magnetic field. In figure 7, we observe that all considered cases show strong fluctuations, which illustrates that we are in a highly nonlinear regime, far away from quasi-static equilibria.

Figure 7. Axial spectra of kinetic (a) and magnetic (b) energy, normalized respectively by the total kinetic and magnetic energy. Three shapes are considered, respectively from top to bottom, a circle with radius $a=1$, an ellipse with major semi-axis $a=1.2$ and one with $a=1.4$, for $S \approx 2\times 10^4$.

Figure 7 shows the predominance of a magnetic mode with toroidal mode number $n=7$ in the circular case, which is consistent with what has been observed in the RFX-mod device (Lorenzini et al. Reference Lorenzini, Martines, Piovesan, Terranova, Zanca, Zuin, Alfier, Bonfiglio, Bonomo and Canton2009; Martin et al. Reference Martin, Apolloni, Puiatti, Adamek, Agostini, Alfier, Annibaldi, Antoni, Auriemma and Barana2009). In the elliptical case ($a=1.2$), a tendency for mode $n=14$ to dominate is observed, while the magnetic modes $n=3$ and $n=4$ contain most of the magnetic energy for $a=1.4$. This last case seems to be closer to a multiple-helicity state, where it is not a single mode which contains most of the energy. Similar spectral differences are observed in the kinetic spectra, where $n=0$ and $n=1$ are the dominating kinetic modes in the circular case, $n=14$ in the $a=1.2$ elliptical case and $n=8$ in the $a=1.4$ elliptical case. A different representation of the modal dynamics is shown in the appendix, where the modal spectra are plotted using the double-logarithmic scale for the different cases.

We have not carried out a detailed poloidal mode composition, which is in particular less convenient in the elliptical geometries, and we have therefore no further interpretation of the possible underlying instabilities, which could be external tearing modes or other MHD instabilities. We think that a linear instability analysis of the present system would allow further insights in the origin of the magnetic structures. However, because the modal spectra indicate the large range of active modes, the outcome of such an analysis is of limited use in the here considered fully turbulent state.

It is clear from these observations that shaping significantly influences the topology of the velocity and magnetic fields, both qualitatively and quantitatively. All three geometries do however show the presence of a large range of active scales.

3.4. Poloidal angular momentum

The presented results show a clear influence of the shape of the cross-section on the dynamics. It would be beneficial if we could understand these differences in the light of a clear, large-scale dynamical feature which changes through the shaping. We have identified one such feature which behaves quite differently in the differently shaped geometries. This quantity is the poloidal angular momentum, a quantity which has received considerable attention in two-dimensional flows.

In fact, an interesting difference between two-dimensional flows in circular or elliptical domains is the change in dynamics associated with a form of symmetry breaking of the large-scale flow patterns. Indeed in 2-D turbulence, changing the flow geometry from circular to elliptical leads to the generation of angular momentum (Keetels, Clercx & van Heijst Reference Keetels, Clercx and van Heijst2008). This effect was shown to persist in 2-D MHD turbulence (Bos, Neffaa & Schneider Reference Bos, Neffaa and Schneider2008, Reference Bos, Neffaa and Schneider2010) and its investigation is considered to be interesting in the context of confinement studies, because large-scale poloidal motion could enhance radial transport barriers and stabilize MHD instabilities (Shan & Montgomery Reference Shan and Montgomery1994).

An analysis of a similar possibility in the present geometry is shown in figure 8, where for a given time instant, the angular momentum associated with the poloidal flow is computed for each cross-section. Its definition is

(3.4)\begin{equation} L_u(z)=\int_A \boldsymbol{e}_z\boldsymbol{\cdot}(\boldsymbol{r} \times \boldsymbol{u}) \,\mathrm{d}A, \end{equation}

where $\boldsymbol {r}$ is the radial vector, pointing from the centreline of the geometry outwards and $A$ is the surface of the poloidal cross-section. In a periodic cylinder with circular cross-section, global angular momentum can only be created by viscous effects. However, locally, at a given axial position, the angular momentum can be strong. This is exactly what is observed for the case of a circular geometry, where locally large values of the poloidal angular momentum exist. Through the associated rotating motion, confinement could be improved, because rotation suppresses instabilities (Shan & Montgomery Reference Shan and Montgomery1994). Note that this is one particular time instant, but qualitatively the same effect is observed at different time instants, with local fluctuations of $L_u$, which are largest in amplitude in the circular geometry. Possibly, this structure is associated with the $n=1$ velocity mode observed in the temporal spectrum of the kinetic energy in figure 7(a).

Figure 8. The $z$-dependence of the instantaneous poloidal angular momentum for the three geometries at $t=11\times 10^3\tau _A$.

This observation is somewhat the opposite of what is observed in investigations of spontaneous spin-up in 2-D systems. There, it is seen that the generation of angular moment is absent in circular 2-D domains and becomes more important in elliptical domains (Keetels et al. Reference Keetels, Clercx and van Heijst2008).

The important amount of circular movement characterized here by the poloidal angular momentum is perhaps simply owing to the fact that a circular shape is more compatible with circular movement, compared with non-circular-shaped cross-sections. However, such a poloidal flow could be beneficial for confinement, which shows here that with respect to that effect, circular domains might be favourable for good confinement. This is also what was concluded, for different reasons, by Guo et al. (Reference Guo, Xu, Wang and Liu2013). This observation is clearly of speculative nature and further research is required to investigate the link between poloidal angular momentum, shaping and the turbulent dynamics of the RFP.