MHD stability analysis of the SMall Aspect Ratio Tokamak (SMART)

University of Seville Interuniversity Master in Nuclear Physics MHD stability analysis of the SMall Aspect Ratio Tokamak (SMART) ` Alex Reyner Vi˜nolas Supervisors: Manuel Garc´ıa Mu˜noz Eleonora Viezzer Jes´us Jos´e Dom´ınguez-Palacios Dur´an Departamento de F´ısica At´omica, Molecular y Nuclear Facultad de F´ısica Plasma Science and Fusion Technology Stability Analysis of SMART Alex Reyner Vi˜nolas Abstract At the University of Seville, a significant leap forward is being pursued in the race towards achieving viable fusion energy. In this pursuit, the construction of the SMall Aspect Ratio Tokamak (SMART), a spherical tokamak, is currently underway. Having good confinement is essential for the proper function of a tokamak. Different operation regimes have been proposed. The advanced tokamak regime seems to be the more promising for nuclear fusion plants. The high pressures in this regime give rise to pressure driven instabilities that deteriorate the confinement and limit the operational space. A good understanding of the instabilities is key to operate the tokamak. The objective of this master thesis is to ascertain the influence of an ideal wall on the plasma stability. To accomplish this, CHEASE and MARS-F codes have been employed. The simulations are centered on the ideal magnetohydrodynamic (MHD) scenario, neglecting plasma rotation and kinetic effects, as well as plasma resistivity and viscosity. The aim of this simplified approach is to isolate and comprehend the fundamental impacts of the wall on plasma stability. MHD instabilites in the advanced tokamak regime for low-nmodes (n= 1, ..., 6), and the effects of the wall at varying distances, were studied. Taking advantage of SMART’s capability to operate at different triangularities, a study to determine the most favorable triangularity for achieving good confinement has been conducted. The highest positive triangularities have shown better MHD stability. Final Master Thesis 3 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Final Master Thesis 4 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Contents 1 Introduction 7 1.1 Nuclear fusion .................................. 7 1.2 The tokamak concept .............................. 8 1.3 SMART ..................................... 11 1.4 Structure of the work .............................. 11 2 Theoretical background 13 2.1 Magnetohydrodynamic model ......................... 13 2.2 Linear ideal MHD equations .......................... 14 2.3 MHD instabilities ................................ 15 2.3.1 Internal kink .............................. 15 2.3.2 External Kink .............................. 15 2.3.3 Infernal Mode .............................. 16 3 CHEASE and MARS-F 17 3.1 Equilibria definition ............................... 18 3.2 Normalization .................................. 20 4 Simulation results 21 4.1 Plasma βscan ................................. 21 4.2 Ideal wall inclusion ............................... 25 4.3 Plasma βand wall distance scan ....................... 28 4.4 AT regime scan ................................. 31 4.5 Triangularity scan ................................ 34 5 Summary and future work 37 A Data 41 B Acronyms and symbols 45 Final Master Thesis 5 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Final Master Thesis 6 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 1 Introduction 1.1 Nuclear fusion Plasma, usually called the fourth state of matter, is a highly ionized gas composed of positive and negative charges, that forms when matter is so hot and energetic that atoms are separated in ions (nucleus) and electrons. Stars, like the Sun, produce tremendous amounts of energy through nuclear fusion. While on a universal scale this process may not be eternal, we can consider it as an infinite source of energy. The plasma in their core, composed of hydrogen, helium, carbon, and other heavier nuclei, is subjected to extremely high temperatures and pressures due to the intense gravitational forces. This causes the light nuclei to fuse together, forming heavier elements and liberating energy. This process is called Nucleosynthesis. For Sun-like stars, the dominant fusion process is the proton-proton chain. Of course, we can not replicate the same processes that occur in the core of the stars here, in the Earth. With the objective of imitating the stars, the development of fusion power plants is of special interest for our future. The ultimate goal is the creation of a fusion reactor capable of achieving sustainable, clean, and virtually limitless energy generation. The most promising reaction is the deuterium-tritium1fusion, 2 1D + 3 1T−→ 4 2He + n + Q(1.1) Figure 1.1: Cross sections of nuclear fusion reactions. [1] produces Q= 17.6 MeV of energy. Although this reaction does not yield the largest release of energy, it is the most promising one in terms of the cross-sections for the achievable temperatures (Fig. 1.1). Tritium is a highly contaminating element and does not exist in large quantities on earth. However, this problem can be solved by utilizing the neutron from the fusion reaction to induce a reaction with lithium, n + 6 3Li −→ 4 2He + 3 1T (1.2) Thus, the problem of fuel acquisition is solved, as lithium is a relatively common element, and deuterium can be found in water. 1Deuterium and tritium are isotopes of hydrogen, 2H and 3H respectively. Final Master Thesis 7 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 1.2 The tokamak concept As we lack the gravitational forces of the stars to confine the plasmas, we had to search other methods for plasma confinement. Since plasma is composed of charged particles, we take advantage of the Lorentz force and magnetic fields to confine the plasma. Numerous fusion devices have been designed for this purpose. The tokamak shows the highest confinement rates so far, due to the strong decrease of MHD fluctuations at the plasma edge in the high-confinement mode (H-mode) [2]. The plasmas in tokamaks are much less denser than the ones in the stars. To compensate this and having nuclear fusion reactions we need really high temperatures (∼keV ). The high temperatures reached in fusion devices require avoiding physical contact with any material of the vessel, as the plasma temperature is well over the melting point of any material. In a purely toroidal confinement the toroidal component is not uniform (Bt∝1/R, being R the radius of the torus). This leads to the distinction of two regions. The high field side (HFS), located in the interior part of the torus, and low field side (LFS), in the outer most part. Therefore, a gradient directed towards the center of the torus is observed. Particles gyrating in such a non-uniform magnetic field will experience different magnetic fields during their gyration. This leads to drift forces that, eventually, will transport particles to the vessel walls. To counteract this effect, a poloidal magnetic field is added to the toroidal magnetic field. Then, the magnetic field lines take a helical shape. A schematic representation of the coils used in a tokamak, field lines, and trajectories can be seen in Fig. 1.2. Figure 1.2: Structure of a typical tokamak device. The coils, magnetic fields and plasma are represented. Credit: EUROfusion. Final Master Thesis 8 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas In order to have profitable energy production, it is necessary for these plants to operate at a constant energy output, and be as efficient as possible and produce an equivalent or larger amount of energy. The advanced tokamak regime (AT) [3,4] appears to be very promising for this goal. This regime is characterized by flat or reversed safety factor profiles (Fig. 1.3) and high bootstrap current, only possible for high pressures gradients. Most of the most harmful resonant surfaces are avoided due to the elevated safety factor. The normalized plasma beta [5], which is the ratio between the plasma pressure and magnetic field pressure, as it will be an important parameter in this work, is introduced: β=2µ0⟨p⟩ ⟨B2⟩−→ βN=β(%)(a(m)B0(T) Ip(MA)) (1.3) where ⟨p⟩is the average plasma pressure, B0the magnetic field, athe minor radius and Ip the plasma current. It is especially important to control the instabilities to maximise this parameter, as fusion power scales as ⟨p⟩2∼β2 N[6]. However, βNis limited by numerous magnetohydrodinamic (MHD) fluctuations that deteriorate the plasma confinement and, thus, reduce the plasma βN. The ideal external kink is the instability that limits the maximum plasma pressure achievable, which is rather low for AT tokamaks [7]. Therefore, the understanding of the physics underlying MHD instabilities, as well as of control techniques, is mandatory for future fusion reactors. The AT regime offers significant advantages only when the beta limit is significantly increased by the presence of a closely fitting wall [8]. Figure 1.3: Different safety factor profiles corresponding to different scenarios. [16] In a tokamak, the safety factor qis defined as the ratio of toroidal magnetic flux to the poloidal magnetic flux. It is expressed as, q=dΦ dΨ(1.4) Final Master Thesis 9 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas stabilize the modes caused by plasma current, EK modes induced by pressure are the most common in tokamaks. These modes typically have low toroidal mode numbers, usually n= 1−3. When pressure exceeds a certain threshold βN,no−wall, the mode becomes highly unstable. Past this limit the EK grows really fast, with a growth rate γ0∼1/τA, where τAis the Alfv´en time. The mode becomes impossible to control and plasma confinement is lost. However, the presence of an ideal conducting wall surrounding the plasma within a critical distance rchas shown to have a stability effect over the EK. Below a higher βN threshold called the “ideal-wall” limit, βN,ideal−wall, the external kink is stabilized by the wall [26,27]. This value depends on the shape and geometry of the wall, and increases the βNlimit by a factor of 1.2−2. If the plasma pressure surpasses this limit, the EK will also rapidly increase like in the no-wall case. 2.3.3 Infernal Mode The Infernal mode (IF) is a plasma current and pressure driven instability, but mainly pressure-driven. An IF is an EK with resonant surface inside the plasma, in a region where the magnetic shear is low and the pressure gradient is large around this resonant surface [17]. One of the main characteristics to identify an IF is that the main instability with (m0,n) is coupled to the neighboring modes (m0±1,n). The growth rate of the IF increases with smaller ratio between the mode and q-profile: δq =qmin −m0/n, and is highly dependent on the pressure [23]. The safety factors considered in AT regime are most unstable to IF. IF is most unstable for qmin just below a rational number [24]. Final Master Thesis 16 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 3 CHEASE and MARS-F The MARS-F code [28–30] solves the standard, linearized, ideal or resistive, MHD equations (2.4) in a toroidal geometry as an eigenvalue problem. The equilibrium input to MARS-F is produced with CHEASE [31]. CHEASE solves the Grad-Shafranov equation for a toroidal MHD equilibria. ∇· 1 R2 0∇ψ=jϕ R0 =−p′(ψ)−1 R2 0 TT′(ψ) (3.1) where jϕis the toroidal plasma current density, p′the plasma pressure gradient and T is the poloidal current flux function. To solve the equation, the free functions p′(ψ) and TT′(ψ) need to be determined. The coordinate system used is the flux coordinate system (s, χ, ϕ), where the radial coordinate is defined as s=sψ−ψaxis ψedge −ψaxis =pψN≡ρpol (3.2) where ψis the poloidal magnetic flux. χis a generalized poloidal angle, and ϕis the toroidal angle. Some important parameters mentioned in further explanations, for CHEASE are: •R0: normalisation factor for the length. •B0: normalisation for the magnetic field. •CFBAL: scaling factor of the pressure gradient. •NVC: radial resolution of final vacuum mesh, in terms of s. •REXT: vacuum radius normalized by the minor radius of the plasma, a. and for MARS-F are: •NVM: cell number specifying vacuum extension. •M1 & M2: denotes the lower and higher poloidal modes simulated. •NWALL: number of resistive walls considered in the simulation. The complete set of parameters and their explanations can be found in reference [32]. In MARS-F simulations without a wall are not possible. In our code, the boundary of the computational domain acts as an ideal conductive wall. However, if the boundary is far enough, it can be treated as if there were no wall, and the effect will be the same as an infinite vacuum. This will be seen in the next sections. The parameters values for the simulations are: CHEASE: NVC=160, NVEXP=0, REXT=3, R0= 0.526 m, B0=0.37 T MARSF: NVM=160, NWALL=0, M1=-9, M2=30 Final Master Thesis 17 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 3.1 Equilibria definition Equilibria with a reversed, or nearly reversed, q-profile according to the advanced tokamak regime, with qmin >1, have been produced. Due to the way the version used of the CHEASE code functions q-profiles can not be entered as an input. A discretization of the pressure (RPPF), the current profile (RFUN) and the plasma boundary, or last closed flux surface (LCFS) are used to determine the equilibrium. The starting point is an equilibrium generated by the FIESTA code. Modifications to the current profile and plasma shape have been made to alter the output of the q-profile. The pressure gradient is given by, RPPF = 1.54ρ2 pol −5.38 ×10−8ρpol −1.54 (3.3) Instead of the TT′quantity of equation (3.1), the surface averaged current profile has been used, I∥=HJ·BJdχ HB·∇ϕJdχ (3.4) The procedure followed was manually creating current profiles using analytical expressions, such as Gaussian and Fermi like distributions, as a basis. G(A, x, µ, σ) = A·e−(x−µ)2/σ ;F(x, µ, σ) = 1 1 + e(x−µ)/σ (3.5) Combining this expressions, current densities and q-profiles similar to those used in Refs [7, 24,36] are obtained. Once the shape of the profile is determined, it is normalized so that its maximum value is 1. Then, to adjust the current profile amplitude a multiplier, M, is used. RFUN = M·[G(A, ρpol, µ, 0.2) + F(ρpol,0.9,0.15)] ·F(ρpol,0.85,0.05) ([G(A, ρpol, µ, 0.2) + F(ρpol,0.9,0.15)] ·F(ρpol,0.85,0.05))max (3.6) The LCFS used is given by following expression: R=R0+acos (θ+δsin θ) ; Z=aκ sin θ(3.7) where R0and aare the major and minor radius respectively, κis the elongation and δ the triangularity of the plasma. The wall used in this study will have a shape conformal to the LCFS shape, meaning it is described by: R=R0+awcos (θ+δsin θ) ; Z=awκsin θ(3.8) where awis the wall radius. A section of the plasma surface and wall, the surface averaged current profile, and plasma pressure gradient can be seen in Fig. 3.1. Final Master Thesis 18 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas -2 -1 0 1 2 0 0.5 1 1.5 2 Z/R0 R/R0 Plasma section LCFS Wall 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 Normalized Current ρpol Surface averaged parallel current density EK6 -2 -1.5 -1 -0.5 0 0 0.2 0.4 0.6 0.8 1 ∂p/∂ψ ρpol Pressure gradient Figure 3.1: On the left, a representation of the normalized plasma LCFS and wall. In this case R0= 0.53 m, a= 0.27 m, κ= 1.98, δ= 0.5 and aw= 0.4 m. On the right, the surface averaged current profile. Below them, the ∇pinput. As mentioned, CHEASE produces the equilibrium input for MARS-F simulations from these expressions. The main characteristics of the equilibrium: total current, q-profile, plasma beta, etc.; are obtained with CHEASE. A problem encountered in our simulations is that CHEASE can not produce an equilibrium for every set of parameters, especially the high pressure equilibria (βN>6). Once the CHEASE simulation is done and the output equilibrium is produced, MARS-F is used to simulate the plasma stability. Among all the results obtained with MARS-F, the focus will be placed on the growth rate of the specific Final Master Thesis 19 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas toroidal mode n, and its mode structure. Since MARS-F solves the problem iteratively starting from an initial guess of the growth rate, it is possible that for some cases (mainly the most stable plasmas with lower growth rates) non-physical results are obtained. These types of results are characterized by having very narrow and tall peaks in the eigenstate structure (Fig. 3.2). Figure 3.2: Example of a result considered non-physical. 3.2 Normalization In this work, the normalized variables for the quantities computed are used. Table 3.1 lists the normalization used for each variable, the normalization used for each variable can be seen. Both CHEASE and the MARS-* codes share the same normalization scheme. Quantity SI unit Norm. factor length m R0 mag. field T B0 current A R0B0/µ0 current density A/m2B0/(µ0R0) pressure Pa B2 0/µo γ0sωA Table 3.1: Normalization factors of variables. The Alfv´en frequency, ωA, is computed as, ωA=νa/R0;νA=B0/√µ0mHn0(3.9) where mHis the Hydrogen mass and n0the plasma density at the magnetic axis. In this work, where B0= 0.4 T and n0= 3.72·1019 m−3, the Alfv´en frequency is ωA= 2.514 µs−1. Final Master Thesis 20 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 4 Simulation results Our work has been divided into four phases. Firstly, with a fixed current profile, how the increase in pressure affects our plasma has been examined. Secondly, how the ideal wall affects plasma stability. Then, a scan for different currents has been conducted. Finally, the triangularity of the plasma has been modified to produce a scan over positive and negative triangularities, using the current profile from the first part of the work. In this section the results obtained in the simulations over the different scans are presented. 4.1 Plasma βscan For this first part of the work, the averaged current profile (3.6) is fixed to: RFUN(M= 2.3, µ = 0.7, A = 1) (4.1) By changing the CFBAL parameter, the pressure gradient is modified and, thus, the plasma beta is modified. Tab. 4.1 shows the different parameters of the equilibria for each CFBAL value considered. A wide range of pressures has been covered to observe the effect plasma beta has on the GR of the modes. Tag CFBAL q0qmin q95 qedge β(%) Ip(kA)βN ek6v4.0 4.0 1.80 1.62 4.12 5.70 19.22 359.36 5.276 ek6v3.8 3.8 1.78 1.59 4.04 5.62 18.66 356.99 5.157 ek6v3.6 3.6 1.76 1.56 3.97 5.53 18.08 354.56 5.030 ek6v3.4 3.4 1.74 1.53 3.89 5.45 17.47 352.06 4.895 ek6v3.2 3.2 1.72 1.50 3.82 5.36 16.82 349.49 4.750 ek6v3.0 3.0 1.70 1.48 3.74 5.27 16.15 346.86 4.594 ek6v2.8 2.8 1.69 1.45 3.67 5.18 15.44 344.16 4.426 ek6v2.6 2.6 1.67 1.43 3.59 5.08 14.69 341.39 4.246 ek6v2.4 2.4 1.66 1.40 3.52 4.99 13.90 338.54 4.051 ek6v2.2 2.2 1.65 1.38 3.44 4.89 13.06 335.63 3.840 ek6v2.0 2.0 1.65 1.36 3.37 4.79 12.18 332.64 3.612 ek6v1.8 1.8 1.64 1.33 3.29 4.70 11.24 329.58 3.364 ek6v1.6 1.6 1.64 1.31 3.22 4.59 10.24 326.45 3.096 ek6v1.4 1.4 1.64 1.29 3.14 4.49 9.19 323.25 2.805 ek6v1.2 1.2 1.65 1.28 3.07 4.39 8.07 319.99 2.488 ek6v1.0 1.0 1.65 1.26 2.99 4.29 6.89 316.68 2.146 Table 4.1: q0,qmin,qedge, plasma beta, total plasma current and normalized plasma beta (1.3) for different CFBAL values. The tag allows us to reference each case. Fig. 4.1 illustrates the pressure and safety factor profiles for some of the cases listed in Tab.4.1. This equilibrium returns us a slightly reversed q-profile, with a qvalue at the edge close to 5. This is a typical profile for an AT scenario (see Fig. 1.3). Since CHEASE calculates the equilibrium for each CFBAL value self-consistently, this implies that not all the equilibirum parameters can be kept constant when changing one of them (in this case, Final Master Thesis 21 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas CFBAL). It can be observed that qdecreases with decreasing pressure. When changing the pressure is expected to have a slight change in the q-profile, specially in the edge zone. In our case this change is more global. From Tab. 4.1, it can be seen that qedge changes ∼28%, q95 ∼32%, qmin ∼25% and q0∼10%. The effects of a change like this can modify the results obtained for the GRs. There is also an important value change on the total current Ip, that varies ∼10%. Figure 4.1: Pressure and safety factor for some of the EK6 equilibria. The growth rates and eigenfunctions for modes n= 1, ..., 6 have been computed. Fig. 4.2 shows the normalized GR of these modes for some cases of the plasma beta scan. A clear tendency for an increase in the GR with plasma beta can be seen. For the lowest pressures, the GR has an oscillatory behaviour with the toroidal mode number n. For elevated pressures, this behaviour disappears and the the GR increases monotonically with n. 0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 γ/ωA n ek6v1.0 ek6v1.6 ek6v2.2 ek6v2.8 ek6v3.4 ek6v4.0 Figure 4.2: Normalized growth rates obtained for some of the simulations of the EK6 equilibria. Full data in Tab. A.1 in the appendix. Final Master Thesis 22 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas In Fig. 4.3 all the modes are plotted for the ek6v3.0 case. For n=m= 1, despite having q0>1, an IK with edge component can be seen. Plasmas with this kind of qprofile are unstable for the IK even for high qmin and β[33]. For lower βthe n= 1 mode is unstable for the m= 1, and as plasma pressure is increased the m= 2 contribution gains importance. The IF modes are also unstable in this regime. The GR of these IF has an oscillatory character over n, as observed in this work, that becomes stronger as qmin increases until a certain value where it disappears [33]. In this case, EK are mainly present in modes n= 1,2 and even 3. EK can be identified by their component at the boundary, and because they are globally defined. The higher nIF are also coupled to an EK. Final Master Thesis 23 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Figure 4.3: Eigenstate structure of the n= 1, ..., 6 modes for the ek6v3.0. The dominant poloidal modes m= 1, ..., 6 has been marked to differentiate them. Final Master Thesis 24 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 4.2 Ideal wall inclusion Our goal is to study the effect that an ideal wall has on the stability of the external kinks. Tthe EK6 equilibrium with CFBAL=3 (tag: ek6v3.0, Tab. 4.1) has been choosen, because this case has a normalized plasma beta value βN= 4.6, reasonable for the operation of a spherical tokamak [17]. This case will be named as the ICW6 equilibrium, in particular the icw6v3.0 case. The ideal wall effects are investigated changing the NVMparameter in MARS-F, i.e., by changing the radial location of the ideal wall. By setting NVM=NVC, the entire vacuum region is included, as in the previous section. Otherwise, the ideal wall condition at the plasma boundary is achieved by setting NVM=1. The relation between the wall radius and NMin an equidistant mesh is, rw a= 1 + (REXT −1)NVM−1 NVC (4.2) Fig. 4.4 shows the normalized linear GRof n= 1, ..., 6 modes for different ideal wall radial locations. For low nvalues, the component at the edge has more influence on the mode’s GR, making the effect of the wall more pronounced. As nincreases, the effect appears for shorter distances of the ideal wall respect of the LFCS. This suggests that for these nvalues, the dominant IF is not stabilized by the wall. If the wall very close enough to the plasma, these modes are partially stabilized as well. 0 0.1 0.2 0.3 0.4 1 1.25 1.5 1.75 2 γ/ωA rw/a n=1 n=2 n=3 n=4 n=5 n=6 Figure 4.4: Normalized growth rates versus the ideal wall radius. Modes with lower nare affected by the wall effect earlier. Data in Tab. A.2 in the appendix. The results obtained for the case with the closest wall are shown in Fig. 4.5. It is worth noting that while the outermost component becomes negligible in all cases, the modes still exhibit their IK and IF with the same dominant poloidal number. Having a wall prevents the displacement from expanding to the plasma edge. In Fig. 4.6, the evolution of the n= 1 in terms of the wall distance can be seen. The IK is not suppressed in any case and its contribution to the GR is really important. The internal component of the modes is barely affected by the ideal wall presence, and is probably the reason why the GR tending to zero at a certain radius is not observed, rc, as seen in the literature [34,35]. Final Master Thesis 25 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Simulations have been conducted for different current profiles with A= 1 and A= 2.5, over a broad set of pressures, for the n= 1 and n= 2 modes, with and without the ideal wall. The GR results have been filtered to determine which equilibria are stable. The GR threshold value below which the equilibrium has been considered stable is γT/ωA= 1×10−3. The mode profiles have been studied, and it has been found that this value is a good discriminator for ruling out where there are not IK, EK, or IF. Some simulations have converged to non-physical results (see Fig. 3.2). To identify these modes, a more in-depth study was conducted by identifying the peaks of each mode. If the width of the highest peak is lower than 0.03, the mode is considered to be non-physical, and if the value was lower than γ= 5 ×10−3, the mode has been considered stable. For cases where the current profile is very high, the qmin drops to values below 1. The results corresponding to this cases have been also filtered out. Fig. 4.12 shows the results for the A= 1 equilibria, and Fig. 4.13 shows the results for the A= 2.5 equilibria. The results show that SMART exhibits stability for βN<3 without a wall. When an ideal wall is considered, the region of stability increases to βN≈6. Similar results has been obtained in studies for other spherical tokamaks [37,38]. At high currents the stabilization thanks to the ideal wall is less effective. The βNlimit found presents a step like behaviour, found also in [39]. Comparing both figures, it can be seen that having a more reversed q-profile gives us access to higher normalized current operational range. In Fig. 4.14 the GR obtained for the RFUN(M= 1.90, µ = 0.7, A = 1) equilibrium are plotted. It can be seen that the βNlimits found change substantially from the ones found in Fig. 4.8. While the βN,no−wall limits for both n= 1 and 2 are almost the same, the βN,ideal−wall are significantly increased from βN,ideal−wall ≈3.3 to βN,ideal−wall ≈5. Figure 4.12: Operational range for equilibria with A= 1. βNvs. normalized current for equilibria found stable without the presence of an ideal wall (black) and with an ideal wall at the LCFS (red). The parameters of the equilibria are in Tab. A.4.B0= 0.4 T. Final Master Thesis 32 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Figure 4.13: Operational range for equilibria with A= 2.5. βNvs. normalized current for equilibria found stable without the presence of an ideal wall (black) and with an ideal wall at the LCFS (red). The parameters of the equilibria are in Tab. A.4.B0= 0.4 T. 10-4 10-3 10-2 10-1 100 2 2.5 3 3.5 4 4.5 5 5.5 6 γ/ωA βΝ no-wall (n=1) rw/a=1 (n=1) no-wall (n=2) rw/a=1 (n=2) Figure 4.14: Normalized growth rates of the n= 1 and n= 2, for the “no-wall” case and ideal wall at the LCFS, vs βN. Equilibrium: RFUN(M= 1.90, µ = 0.7, A = 1). Final Master Thesis 33 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 4.5 Triangularity scan -1 0 1 0 0.5 1 1.5 2 Z/R0 R/R0 Plasma section δ=0.5 δ=-0.5 Figure 4.15: LCFS for two different triangularities. As mentioned in Sec. 1.3, SMART has been designed to operate with positive and negative triangularities. In the following, the effects of the triangualrity on mode stability are discussed. Until now, the triangularity has been fixed to δ= 0.5. A scan has been conducted over trinagularities between 0.3 and -0.3 (Tab. A.5). CHEASE has been unable to obtain the equilibrium for δ=−0.5. In this scan, the pressure gradient and average parallel current density input are fixed to the EK6 equilibrium (Fig. 3.1). The shape of the LCFS and wall changes according to equation (3.7). In Fig. 4.16 it can be seen how changing δmodifies the q-profile. Due to the change in plasma shape [40], for decreasing triangularities the q-profile becomes more reversed, and the edge value is lower. The pressure profile also changes for different triangularities, but not as much as the safety factor. Triang. q0qmin q95 qedge β(%) Ip(kA)βN δ= 0.5 1.70 1.48 3.74 5.27 16.15 346.86 4.569 δ= 0.3 1.91 1.51 3.27 4.04 17.61 359.33 4.809 δ= 0.1 2.12 1.53 2.93 3.40 18.94 366.62 5.068 δ=−0.1 2.82 1.53 2.71 3.08 20.08 366.62 5.375 δ=−0.3 2.92 1.54 2.64 2.98 20.55 360.21 5.600 Table 4.2: q0,qmin,qedge, plasma beta, total plasma current and normalized plasma beta (1.3) for different triangularity simulations. Figure 4.16: Pressure and safety factor for the different triangularities. Final Master Thesis 34 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas In Fig. 4.17 the GRs without considering the ideal wall in the simulations are presented. It can be observed that the GR increases for decreasing triangularity. This shows that for our case of study PT are better than NT. Its interesting to see that δ= 0.3 gives us a lower GR than δ= 0.5, except for n= 1. 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 γ/ωA n δ=0.5 δ=0.3 δ=0.1 δ=-0.1 δ=-0.3 Figure 4.17: Normalized growth rates versus the toroidal mode number n, without considering an ideal wall, for different triangularities. Full data in Tab. A.5 in the appendix. In Fig. 4.18 the GRs with an ideal wall in the simulations are presented. As expected, the wall affects the plasma stability in both PT and NT. The main difference is how the wall affects the modes. For the n= 1 mode, for δ= 0.3, the decrease in GR is progressive with wall distance, while for δ=−0.3, the effect of the wall increases significantly when it is brought close to the LCFS. This behavior leads us to think that this mode has a greater component at the edge, or that the IK is less important than for δ= 0.3, since it has been observed that the effect of the wall on the IK is less noticeable than on the EK. In Fig. 4.19Ir can be see how, indeed, this mode has less edge component. This change in the profile is probably because of the change in the q-profile (higher q0and lower qedge). Final Master Thesis 35 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 0 0.1 0.2 0.3 0.4 0.5 0.6 1 1.25 1.5 1.75 2 γ/ωA rw/a δ=0.3 n=1 n=2 n=3 n=4 n=5 n=6 0 0.1 0.2 0.3 0.4 0.5 0.6 1 1.25 1.5 1.75 2 γ/ωA rw/a δ=-0.3 n=1 n=2 n=3 n=4 n=5 n=6 Figure 4.18: Effect of the ideal wall on the normalized growth rates of the δ= 0.3 and δ=−0.3 triangularities for modes n= 1, ..., 6. Data in Tab. A.6 and Tab. A.7, respectively, in the appendix. Figure 4.19: Eigenstate structure of the n= 1 mode for δ= 0.3 and δ=−0.3. Final Master Thesis 36 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas 5 Summary and future work In this thesis, the aim of this master thesis is to study the differences in plasma stability in the SMART tokamak when considering the presence or absence of the surrounding wall. Simulations over n= 1, ..., 6 have been carried out using the CHEASE and MARS-F codes. The first objective of the work was to generate an equilibrium according to our objective, and simulate and evaluate the instabilities within it. The q-profile chosen was one in the advanced tokamak regime. The plasma shape with positive triangularity, A= 1.97, B0= 0.37 T and a total current Ip<500 kA. It has been found that in our case, the most unstable modes are not the external kinks, but rather the infernal modes and the internal kink. This internal kink is driven by the high plasma pressure in the tokamak. For the second part of the project, the goal was to include the ideal wall and study its effect on plasma stability. As seen in the literature, the wall has a significant influence over the growth rate the closer it is to the plasma surface. However, due to the presence of these internal kink and infernal modes, the full stabilization of the external kink at a certain wall radius via the computation of the growth rate could not be seen. Instead, it can be seen that the edge component of all modes disappears when the wall is close enough to the plasma. The plasma pressure limits found for the initial case: βN,no−wall ≈2.4 and βN,ideal−wall ≈3.3, which implies an increase of ≈37% in the plasma pressure. To conclude this study, various AT regimes have been explored. It has been found that the plasma beta limit without the presence of a wall limit in SMART is βN= 3, but it extends to as much as βN= 6 when an ideal wall is introduced. This represents a significantly larger increase than anticipated, effectively doubling the beta value. Lastly, a scan over positive and negative triangulatities has been conducted to give completeness to the work done. The current and pressure gradient profiles are fixed, and only the triangularity of the plasma has been changed. The result obtained is that positive triangularities have a better MHD stability than negative triangularities.These conclusions must be approached with caution as the plasma pressure varies substantially over the different triangularities, being the negative triangularities the ones with higher βN, and plasmas at higher pressures are generally less stable. Nonetheless, positive triangularities has been found more stable than negative triangularities for low-nmodes (the ones studied in this work) in other studies [41]. To continue this work, the resistivity of the wall should be included in the simulations. This would allow us to simulate a more real case and the RWM. The wall does not necessarily need to have the same shape as the plasma. The CHEASE code allows us to input an arbitrary shape for the wall. In this work, all positive values of mare considered until m= 30, but the presence of the m= 1 internal kink disturbs our analysis. The work can be repeated by excluding the m= 1, and thus the internal kink, for a more focused study of the wall effects on the external kink. Additionally, our simulations are based on the ideal MHD case, without considering resistivity, plasma rotation, and the effects of free particles. This would significantly change the results, as these mechanisms help control modes such as the infernal mode. Final Master Thesis 37 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Final Master Thesis 38 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Acknowledgments First of all I want to express my gratitude to Manolo and Eli, my supervisors, for the trust they placed in me to carry out this work and allow me to collaborate in this project. I’ve had a warm welcome to the PSFT group, and I hope I can continue working closely with you. I want to give an special “thank you” to Jes´us, for his invaluable assistance through this year. For teaching me how the codes work, the Matlab routines, helping me understand the MHD and infinite patience in reviewing all the complications I’ve had. Finally, I would like to thank my family and friends, the true supports in my life, with whom I have shared countless moments and who have constantly motivated me to work hard towards achieving my goals. Final Master Thesis 39 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas Final Master Thesis 40 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas A Data Data tables from the growth rates computed. Tag n= 1 n= 2 n= 3 n= 4 n= 5 n= 6 ek6v4.0 3.34E-01 4.03E-01 4.27E-01 4.41E-01 4.49E-01 4.56E-01 ek6v3.8 3.18E-01 3.83E-01 4.05E-01 4.17E-01 4.25E-01 4.33E-01 ek6v3.6 3.02E-01 3.62E-01 3.81E-01 3.92E-01 4.01E-01 4.08E-01 ek6v3.4 2.84E-01 3.40E-01 3.56E-01 3.67E-01 3.76E-01 3.83E-01 ek6v3.2 2.65E-01 3.17E-01 3.30E-01 3.43E-01 3.50E-01 3.57E-01 ek6v3.0 2.45E-01 2.93E-01 3.04E-01 3.17E-01 3.22E-01 3.30E-01 ek6v2.8 2.23E-01 2.69E-01 2.78E-01 2.90E-01 2.95E-01 3.01E-01 ek6v2.6 2.00E-01 2.44E-01 2.51E-01 2.60E-01 2.68E-01 2.69E-01 ek6v2.4 1.76E-01 2.17E-01 2.24E-01 2.27E-01 2.41E-01 2.35E-01 ek6v2.2 1.51E-01 1.89E-01 1.98E-01 1.91E-01 2.13E-01 2.03E-01 ek6v2.0 1.24E-01 1.58E-01 1.74E-01 1.51E-01 1.80E-01 1.78E-01 ek6v1.8 9.58E-02 1.23E-01 1.51E-01 1.11E-01 1.39E-01 1.56E-01 ek6v1.6 6.57E-02 8.40E-02 1.27E-01 7.51E-02 8.50E-02 1.30E-01 ek6v1.4 3.08E-02 3.97E-02 9.95E-02 5.22E-02 2.32E-02 8.97E-02 ek6v1.2 5.24E-03 4.50E-03 6.77E-02 4.45E-02 4.21E-03 2.86E-02 ek6v1.0 2.40E-04 7.91E-04 2.98E-02 4.29E-02 2.77E-03 3.22E-03 Table A.1: Growth rates obtained with MARS-F code for the different pressures and modes n= 1, ..., 6 in the EK6 equilibrium. NVMrw/a n = 1 n= 2 n= 3 n= 4 n= 5 n= 6 80 1.99 2.27E-01 2.89E-01 3.03E-01 3.17E-01 3.22E-01 3.30E-01 72 1.89 2.21E-01 2.87E-01 3.02E-01 3.17E-01 3.22E-01 3.30E-01 64 1.79 2.14E-01 2.84E-01 3.01E-01 3.16E-01 3.22E-01 3.30E-01 56 1.69 2.04E-01 2.81E-01 3.00E-01 3.16E-01 3.22E-01 3.30E-01 48 1.59 1.92E-01 2.75E-01 2.97E-01 3.15E-01 3.21E-01 3.30E-01 40 1.49 1.76E-01 2.67E-01 2.93E-01 3.12E-01 3.20E-01 3.29E-01 32 1.39 1.57E-01 2.54E-01 2.85E-01 3.08E-01 3.18E-01 3.28E-01 24 1.29 1.35E-01 2.36E-01 2.71E-01 2.99E-01 3.12E-01 3.24E-01 16 1.19 1.13E-01 2.10E-01 2.47E-01 2.81E-01 2.98E-01 3.14E-01 8 1.09 9.12E-02 1.79E-01 2.11E-01 2.46E-01 2.66E-01 2.88E-01 1 1.00 7.15E-02 1.49E-01 1.67E-01 1.96E-01 2.10E-01 2.33E-01 Table A.2: Growth rates versus wall radius for n= 1, ..., 6 for the icw6v3.0 equilibrium. Final Master Thesis 41 Sevilla, June 2023 Stability Analysis of SMART Alex Reyner Vi˜nolas [24] H.A. Holties et al, 1996 Nucl. Fusion 36 973 [25] G. Q. Dong, Y. Q. Liu, S. Wang, et al, Phys. 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Fusion 61 116014 [41] Zheng L.J., Koschenreuther M.T., Waelbroeck F.L. et al, 2018 Negative triangularity effects on tokamak MHD stability IAEA-FEC,TH/P5-31 Final Master Thesis 48 Sevilla, June 2023