## The physics of turbulence localised to the tokamak divertor volume

### Experimental database

This paper focusses on results from the Mega Ampere Spherical Tokamak device, MAST^{41}, during it’s final experimental campaign in 2013. During these experiments a visible light camera capable of recording in excess of 120,000 frames per second was placed on the divertor with a tangential view into the vessel (see Fig. 1) for several hundred individual plasma discharges. Rather than base this study on individual plasma discharges within this set, a database has been drawn together that covers the widest available parameter range of the plasmas viewed by the camera. Plasma parameters from the database are given in Table 1. The database is constructed of discharges mainly configured in the lower single null (LSN, where only the lower X-point is active) configuration (pictured in Fig. 1) where the data quality is highest, but also considers the impact of resonant magnetic perturbations (used to control violent edge instabilities) and High confinement (H-) mode. The strategy employed in this paper is to compare simulation and experiment, across a wide-ranging experimental database, with robust measurements to draw high-level conclusions around the characteristics of the turbulence, and importantly to validate these aspects of the simulations. With the simulations validated, the flexibility of the code will be leveraged to diagnose the fundamental physics drivers of the turbulence.

### Imaging analysis

The method developed and deployed in this article for the tomographic inversion of camera images, described and rigorously tested by ref. ^{30,42}, provides a mapping between the complex image recorded by a high speed camera and a two-dimensional plane in the divertor, taken here as the poloidal (radial-vertical) plane around the inner and outer divertor legs. It assumes that the 3D structures being imaged by the camera align to the background magnetic field (an assumption that is confirmed in simulation). This allows for the formation of a basis on which to perform a tomographic inversion using standard minimisation routines. During the pre-processing stage, subtraction of the pixel-wise minimum of a given frame with its 19 predecessors^{43,44} is applied to isolate fluctuations from the slowly-varying background component of the light. Figure 8 a), b) and d) show an example of a typical camera frame with important features of the plasma indicated. The effect of background subtraction on that frame is shown in panel (b), and the inversion of the background subtracted image onto the inner and outer divertor legs is shown in panel (d). The inversion domain is chosen to isolate the PFR and near SOL region of both divertor legs, avoiding the X-point and core plasma. The light emission contained in the camera images is dominated by Balmer 3 ⇒ 2 emission and is a complex nonlinear function of plasma quantities—density, temperature and neutral density. Without a multi-measurement comparison, which is extremely challenging for turbulent structures and was not practicable for MAST, the direct experimental inference of these thermodynamic quantities and (more importantly) their fluctuations utilising the diagnostic camera images could not be carried out, though previous studies indicate consistency between camera and probe fluctuation measurements^{26,28}. Instead, this study utilises the turbulence code (STORM) for predictions of the plasma turbulent solution to forward model the Balmer 3 ⇒ 2 light emission observed in synthetic camera image measurements. This provides like-for-like comparison of experiment and simulation, ensuring that any systematic uncertainties are respected in both datasets and allowing high-level comparisons and conclusions to be drawn with confidence.

### Computational modelling and synthetic imaging

The STORM model is based on a 3D drift-reduced two-fluid plasma model, with the electron density, *n*, electron temperature, *T*, parallel ion velocity *U*, parallel electron velocity *V*, and parallel scalar vorticity Ω as dynamic variables. The plasma potential, *ϕ*, is derived through an inversion of the parallel vorticity, \({{\Omega }}=\nabla \cdot \left({B}^{-2}\nabla \phi \right)\). The set of equations is solved in a field-aligned coordinate system on a grid with toroidal symmetry, with geometric factors derived directly from an equilibrium reconstruction^{45} of the experimental plasma discharge under study, and Bohm sheath boundary conditions are applied at upper and lower divertor boundaries. The grid and geometric properties of the system are not evolved during the simulation. The simulation evolves the full fields (ie no specification of a background profile) and is driven by a particle source centered on the last closed flux surface to mimic neutral ionisation, and an energy source in the core region of the simulation. These sources are scaled until *n* and *T* within the simulation match experiment at the outer midplane separatrix, and do not evolve within the simulation. The model makes the cold-ion, Boussinesq, and electrostatic assumptions to make the system tractable in the complex geometry employed for the simulation. The latter is justified by the high resistivity of the SOL and divertor plasma in MAST, however the former two assumptions may impact the detailed characteristics of turbulence in the simulation. Nevertheless, detailed experimental validation has demonstrated that the STORM model captures the main aspects of SOL turbulence well^{35,36}, and without a more detailed simulation available, is a good basis for a first detailed study of divertor turbulence within this manuscript.

This paper employs synthetic images of the divertor turbulence derived from simulations conducted by ref. ^{36}. Data from the the simulation is interpolated onto a grid identical to that used in the experimental analysis, which is then projected along the path of the magnetic field to produce a camera image accounting for line-integration effects and occlusion by machine structures. The emissivity in the poloidal plane is a complex function of thermodynamic quantities of the plasma and neutral gas, and atomic physics, and is forward-modelled in this paper using the OpenADAS database^{46} for the Balmer 3 → 2 transition, employing a neutral particle distribution from a complementary laminar simulation including plasma-neutral interactions. This complementary simulation was conducted with the SOLPS-ITER (Scrape-Off Later Plasma Simulation – ITER) code^{47}, with Monte-Carlo neutral transport and diffusive cross-field plasma transport. The frames are then processed in the same manner as the experimental data. A synthetic camera frame is shown in Fig. 8c). By design the image does not account for any emission from the X-point, core plasma or outer-SOL regions to capture only the salient features of the divertor legs allowing for robust comparison between simulation and experiment.

The use of a fixed, axisymmetric neutral distribution from a auxiliary laminar simulation to generate the *D*_{α} emissivity for the synthetic simulation images was the best estimate available, but means that experimental and synthetic images cannot be considered entirely alike. The thermodynamic fluctuations in the plasma may induce fluctuations in the ionisation of neutrals, which cannot be captured in the synthetic images used here since there is no interaction between the turbulence and the neutral gas in the simulation. For this reason the magnitude of the emission is not compared between experiment and simulation, only the geometric positional and geometric properties are compared. It is important to recognise that plasma-neutral interactions are neglected in the turbulent simulation, but are present in the experimental situation, meaning that such an approach to comparison should be limited to leading order turbulent characteristics.

The STORM simulation analysed is in the slightly different lower disconnected double null (LDN) configuration’, where both X-points are active, but the lower is still the primary X-point. In the shot studied by Riva et al. the gap between primary and secondary separatrix is between 2 mm and 5 mm. In such a configuration between 5% and 30% of the total power entering the SOL is measured on the lower inner divertor^{48}. Reference ^{48} also shows that an LSN plasma may have up to twice the power to the lower inner target compared to an LDN, however a wide range of input powers in the LSN configuration has been studied here, with no clear leading order variation in fluctuation properties. As such, this potential variation in power between the LSN and LDN configurations is not considered likely to impact the features of the turbulence studied. Therefore, from the perspective of the PFR of the lower divertor which is the area of study in this paper, the STORM simulation in the LDN configuration is considered sufficiently comparable to an LSN plasma to justify the comparison.

### Shape, distribution, and spectra of turbulent structures in the divertor

Turbulence is complex and difficult to diagnose with acceptable uncertainty. In order to draw robust conclusions, this article focusses on simple and robust measurements that can be readily compared between divertor legs, and between experiment and simulation. The first such set of measurements forms an assessment of the shape and distribution of turbulence structures across the database by calculating a quasi toroidal mode-number (the number of structures in 2*π* radians toroidally around the device), calculated by counting peaks in the emission along the projection of a magnetic field line in the R–Z plane, and the poloidal structure width calculated as the full-width half maximum of these identified peaks. A useful radial coordinate is the ‘poloidal magnetic flux’ normalised using values at the magnetic axis *ψ*_{ax} and separatrix *ψ*_{sep}, such that *ψ*_{N} = (*ψ* − *ψ*_{ax})/(*ψ*_{ax} − *ψ*_{sep}). The analysis is carried out on the flux surface at *ψ*_{N} = 0.99 which is sufficiently far into the PFR to avoid questions of magnetic field reconstruction misalignment, but sufficiently close to the separatrix that the flux of turbulent structures across the surface is significant. A systematic offset of the experimental flux-surfaces is present which results in a radial shift of measurements by Δ*ψ*_{N} = 0.005, though this has little impact on the conclusions of this study.

In Fig. 8 (d) the embedded white lines show the trajectory of the *ψ*_{N} = 0.99 surface in the R–Z plane in the inner and outer divertor legs, and in (e) the emissivity along the surface is shown in an example discharge. This is cast onto the toroidal angle subtended by the analysed section of the magnetic field line simply by mapping the projection of the magnetic field. By casting this data onto the toroidal angle it is possible to directly compare the features of the inner and outer legs.

### Turbulent flow in the inner divertor leg

Since the tomographic inversion employed in this paper produces 2D time-histories in the R–Z plane, flow velocities can be derived by mapping the trajectory of turbulent structures. Velocimetry based on two-point time-delayed cross-correlations has been used here to map the average flow of structures in the inner divertor leg. No clear directive flow was reliably measurable in the outer divertor leg, as demonstrated by the symmetric *k*_{θ} spectra in Fig. 4.

### Turbulence drives in inner and outer divertor legs

To determine the driving mechanisms for turbulence in the divertor a simulation study has been carried out in the manner of refs. ^{49,50} by eliminating terms from the vorticity equation, which determines the electrostatic potential and therefore regulates turbulence, that are known to drive certain classes of turbulent transport. The vorticity equation in STORM is [Eq. 1]^{36}

$$\frac{\partial {{\Omega }}}{\partial t}+U{{{{{{{\bf{b}}}}}}}}\cdot \nabla {{\Omega }}=-\frac{1}{B}{{{{{{{\bf{b}}}}}}}}\times \nabla \phi \cdot \nabla {{\Omega }}+\frac{1}{n}\nabla \times \left(\frac{{{{{{{{\bf{b}}}}}}}}}{B}\right)\cdot \nabla P+\frac{1}{n}\nabla \cdot \left({{{{{{{\bf{b}}}}}}}}{J}_{\parallel }\right)+{\mu }_{{{{\Omega }}}_{0}}{\nabla }_{\perp }^{2}{{\Omega }}$$

(1)

where *ϕ* is the plasma potential, Ω = ∇ ⋅ (*B*^{−2}∇_{⊥}*ϕ*) the scalar vorticity, *B* the magnetic field strength, *P* = *n**T* the electron pressure, *n* and *T* the electron density and temperature, *J*_{∥} = *n*(*U* − *V*) the parallel current with *U* and *V* the ion and electron velocities parallel to the magnetic field, **b** the magnetic field unit vector and *μ*_{Ω} the (small) collisional perpendicular viscosity. This equation has three terms that drive different classes of turbulence. The term \(\frac{1}{n}\nabla \times \left(\frac{{{{{{{{\bf{b}}}}}}}}}{B}\right)\cdot \nabla P\) drives interchange turbulence^{51}, which is analogous to Rayleigh–Taylor turbulence, and is driven by thermodynamic gradients in regions where the curvature of the magnetic field has a destabilising effect. The term \(\frac{1}{B}{{{{{{{\bf{b}}}}}}}}\times \nabla \phi \cdot \nabla {{\Omega }}\) drives Kelvin–Helmholtz turbulence via sheared flows^{52}, whilst the term \(\frac{1}{n}\nabla \cdot \left({{{{{{{\bf{b}}}}}}}}{J}_{\parallel }\right)\) term mediates drift-wave turbulence driven ubiquitously by cross-field thermodynamic gradients in a resistive plasma. To test the effect of these three different mechanisms, three simulations were performed beginning from the baseline simulation presented in this paper thus far, with the three turbulent drive terms removed in turn. To remove interchange turbulence from the simulation, \(\nabla \times \left(\frac{{{{{{{{\bf{b}}}}}}}}}{B}\right)\to 0\) was set in the lower divertor. To remove Kelvin–Helmholtz turbulence, **b** × ∇*ϕ* ⋅ ∇Ω → < **b** × ∇ *ϕ* ⋅ ∇Ω > _{Φ} in the vorticity equation, whilst to remove drift-waves the substitution \(\frac{1}{n}{\nabla }_{\parallel }P\to < \frac{1}{n}{\nabla }_{\parallel }P \,{{ > }}_{{{\Phi }}}\) is made in parallel Ohm’s law (equation 4 from ref. ^{36}) which blocks energy transfer into resistive drift-waves. < > _{Φ} indicates a toroidal average in the divertor volume.