Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 1 Derivation of dynamo current drive in a closed current volume and stable current sustainment in the HIT SI experiment A C Hossack, D A Sutherland and T R Jarboe University of Washington, Seattle, WA 98195, USA E mail: saracen@uw.edu A derivation is given showing that the current inside a closed current volume can be sustained against resistive dissipation by appropriately phased magnetic perturbations. Imposed dynamo current drive (IDCD) theory is used to predict the toroidal current evolution in the HIT SI experiment as a function of magnetic fluctuations at the edge. Analysis of magnetic fields from a HIT SI discharge shows that the injector imposed fluctuations are sufficient to sustain the measured toroidal current without instabilities whereas the small, plasma generated magnetic fluctuations are not sufficiently large to sustain the current. It was previously thought that the sustainment of any spheromak required flux surface breaking relaxation activity which would prevent confinement sufficient for a fusion reactor[1]. The discovery of imposed dynamo current drive (IDCD)[2] on the helicity injected torus with steady inductive helicity injection (HIT SI) experiment shows that dynamo current drive can be performed without gross, currentdriven kink instabilities. The injector linking flux is externally sustained with / higher than that of the spheromak. By applying appropriately phased perturbations, current is transported to inner flux surfaces by imposed dynamo action, maintaining the stable IDCD profile. This method if sustainment has been demonstrated on the HIT SI experiment and is theoretically steady state. The IDCD model predicts the evolution of toroidal current, the helicity injector impedance scaling, and internal magnetic field profile of driven spheromaks. Further analysis in reference [3] demonstrates that both helicity injection and current drive across closed flux surfaces is possible, but the analysis neglected ion fluid motion for simplicity. Analysis of surface magnetic probe data in reference [4] further supports the claim of sustainment without kink instabilities. Through biorthogonal decomposition (BD), injector correlated signals are subtracted from the data set revealing that almost all of the energy present in the HIT SI experiment is injector imposed rather than plasma generated. The relative effect of the injector and plasma modes were not quantified, however. Demonstrating that flux surface breaking relaxation activity is not fundamental to sustainment enables the spheromak as a confinement device. In this letter, we report three advances in the understanding of IDCD theory and supporting experimental evidence. First, the derivation showing that current sustainment along a current path within a closed current volume is possible has been generalized to include the ion fluid velocity. Second, the IDCD model has been reformulated to use the measured magnetic fluctuations as the dynamo field rather than an approximation based on injector currents. Third, a better upper bound on the level of plasma generated fluctuations is found by excluding injector imposed perturbations and equilibrium asymmetries from magnetic field measurements. This model, evaluated with either injector imposed fluctuations or plasma generated fluctuations, confirms that the plasma current in HIT SI is sustained by imposed dynamo action and cannot be sustained by instabilities. For this calculation, we assume the electric field is well described by Generalized MHD Ohm s Law of the form
Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 2 # 1 Where the electron pressure terms have been neglected since they are likely small compared to the other included terms. Next, assume a perturbation is imposed on an equilibrium faster than the current diffusion time and slower than the inverse plasma frequency. A perturbative component of relevant quantities is assumed as follows, # 2 # 3 # 4 The above assumption implies that the imposed perturbation is frozen into the electron fluid, and we assume that the electron fluid is exclusively carrying the equilibrium current for mathematical convenience. The ion fluid is assumed at rest in the lab frame. In the perturbation frame of reference, in which subsequent calculations will be made, the ion fluid is moving at the drift speed, such that Using these assumptions, we perform the dot product of the electric field with a unit vector in the direction of the total current as follows # 5 Performing the dot product, cancellations of terms occur, and all resulting non zero terms that are higher than third order in are ignored. We find the following result, 1 # 6 In reference [3] we assumed for simplicity, and the above result reduces to the previously published one, including the resistive contribution, as follows, E # 7 For sustainment, the loop voltage on any current path within the closed current volume must be zero, such that. We have shown more generally with this calculation that the dynamo terms in Generalized MHD Ohm s Law can sustain current against resistive dissipation so long as the current driving terms expressed in equation (6), and in a simplified form, equation (7), are nonzero and and can balance along all current paths within a closed current volume of interest in a magnetic fusion device With appropriately applied perturbations of relevant quantities with appropriate phasing, the electromagnetic energy on all current paths within the closed current volume can be maintained in steady state. Should these magnetic perturbations be externally imposed instead of requiring gross, current driven (kink) instabilities, steady state sustainment with sufficient confinement may be possible in an eventual imposed dynamo driven reactor[5]..
Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 3 Plasma current evolution within a particular flux surface of interest can be predicted from the currentdriving fluctuations on the surface using mean dynamo analysis. The following derivation is reproduced from Jarboe et al.[2]. The two fluid, parallel, generalized MHD Ohm s law for a turbulent plasma is.# 8 The electron fluid is frozen to the magnetic field and perturbations in this fluid create distortions in B, so the above equation can be written as.# 9 To estimate the fluctuations required to drive the current inside a mean flux surface consider a simplified geometry: a cylinder with arbitrary axial magnetic field and current, and. Integrating over the volume and calculating the Maxwell stress due to the fluctuations yields # 1 The maximum strength of the current drive is limited to the amplitude of the imposed perturbations that become distorted. This occurs when the contribution of is produced by the bending of causing the kernel to be proportional to. Assuming saturation at 2, the above can be evaluated for a toroidal geometry: 2 2 2 2 # 11 where and are the major and minor radii, respectively. The left hand side is the maximum force that can be transmitted to the inside of the flux surface and the right hand side is the force required to sustain the current inside the flux surface. The terms are equal when slippage occurs; the inequality applies when no slippage occurs and the force from the perturbations is saturated. Spheromaks have similar amounts of toroidal and poloidal current, so the current that must be sustained inside a surface is 2. Assuming a separatrix at 23 cm,, 4, and 4, the above equation can be evaluated solving for the current evolution: 4.# 12 The L/R time / is calculated using helicity balance, which has been validated on HIT SI[6]. The L/R time as a function of helicity is 2 # 13 where 3.5 1 5. 1, 5. 1, 4. 1,,, where the coefficients have been numerically computed from Taylor state equilibria. The helicity injection rate is 2. Magnetic fields at the wall of the HIT SI experiment are measured with 2 axis dot probes arranged in four poloidal arrays and two toroidal arrays of 16 probes each[7]. The poloidal arrays are located at toroidal angles, 45, 18 and 225 degrees. The reported toroidal current is the average of the currents through the four arrays. The two toroidal arrays are located on either side of the midplane diagnostic
Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 4 gap at 3 cm and enable toroidal Fourier mode resolution up to. Injector currents are measured with Rogowski coils. BD analysis is used to separate the equilibrium field from the injector imposed and plasma generated perturbations. Following the method in Victor et al.[4], the data from all of the surface probe arrays are arranged in a matrix such that each row is the time evolution of a particular probe signal. The data matrix is then decomposed using singular value decomposition into orthonormal spatial eigenvectors, orthonormal temporal eigenvectors, and eigenvalues (weights). Thus, the data is separated into empirical modes ordered by the amplitudes of the spatio temporal, coherent modes. Initial analysis of HIT SI data shows that the largest mode is the spheromak equilibrium and the next two modes are wellcorrelated with the injector currents. Further modes, not correlated with the spheromak or injectors, are plasma generated perturbations. To enhance the injector correlated signal, the injector current signals are multiplied by an arbitrary constant and appended to the magnetic probe data array before the decomposition. After the decomposition, any signals in the magnetic probe array correlated with the injector currents are grouped together in the two injector correlated modes. In Figure 1, data is plotted from a discharge with an injector driving frequency of 68.5 khz. Plot (a) gives the injector and toroidal currents for comparison. The injector circuits begin ramping up at (not shown), breakdown occurs at.7 ms, and the spheromak forms at 1 ms. The driving circuits are turned off at 1.85 ms because a lack of pumping leads to excess density in the spheromak. Plot (b) shows the toroidal Fourier modes before BD analysis. Plot (c) shows the toroidal modes after the injector correlated signals have been subtracted from the probe data. This plot shows that the injectors have mostly structure since the previously dominant drops below the level of the. Plot (d), where any nonaxisymmetric Fourier modes associated with the equilibrium have also been subtracted, further reduces the amplitude, showing that the spheromak equilibrium contains an distortion. The plasma generated fluctuations (Figure 1d) are added in quadrature and their magnitude plotted as in Figure 2b. Additionally, the nonaxisymmetric Fourier modes from the injector correlated BD modes are added in quadrature and plotted as in Figure 2b. The electron density, shown in Figure 2a, is required to solve the IDCD current evolution equation. Starting at 1.25 ms, the current evolution is computed in Figure 2c for the two cases of using the experimentally measured /. In Figure 3 the same analysis as in Figure 1 is shown for a different shot, 129191. Electron density data is not available for shot 129191 but Figure 3(d) shows significantly lower activity than Figure 1(d). By subtracting the magnetic signal components correlated with the equilibrium and injectors, the plasma generated mode activity is clearly shown. Figure 1d shows the quiescent period during sustainment. The plasma generated fluctuation levels during sustainment of <5% are very low for sustained, cold spheromaks and are insufficient to sustain the toroidal and poloidal current, as shown in Figure 2c. Further, the remaining, transient activity shown in Figure 1(d) is not fundamental for sustainment as shown by the more quiescent shot in Figure 3(d). Thus, the injector imposed fluctuations must be sustaining the current by imposed dynamo action. We have shown theoretically that appropriately phased magnetic fluctuations can sustain the current along a current path within a closed current volume of interest against resistive decay. The evolution of toroidal current predicted by the model further demonstrates that the imposed, fluctuating fields from the injectors are indeed at an appropriate level to drive the experimentally measured current. The plasma generated fluctuations, however, are not of sufficient amplitude to sustain the measured
Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 5 toroidal current. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award Number DE FG2 96ER54361. Figure 1. (a) Injector currents, their quadrature sum, and toroidal current for shot 129175, (b) toroidal Fourier modes from the toroidal probe arrays normalized to the average magnetic field at the wall during sustainment from t = 1.25 to 1.85 ms, (c) toroidal Fourier modes after subtracting the injectorcorrelated BD modes, normalized to the instantaneous, and (d) Fourier modes after also subtracting the equilibrium correlated BD mode.
Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 6 Figure 2. (a) Chord averaged electron density as measured by far infrared interferometry, (b) quadrature sum of injector correlated and plasma generated magnetic fluctuations and (c) toroidal currents measured and predicted by the IDCD model for the two δb cases.
Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 7 Figure 3. (a) Injector currents, their quadrature sum, and toroidal current for shot 129191, (b) toroidal Fourier modes from the toroidal probe arrays normalized to the average magnetic field at the wall during sustainment from t = 1.25 to 1.85 ms, (c) toroidal Fourier modes after subtracting the injectorcorrelated BD modes, normalized to the instantaneous, and (d) Fourier modes after also subtracting the equilibrium correlated BD mode. [1] Hooper E B 211 Nucl. Fusion 53 858 [2] Jarboe T R, Victor B S, Nelson B A, Hansen C J, Akcay C, Ennis D A, Hicks N K, Hossack A C, Marklin G J and Smith R J 212 Nucl. Fusion 52 8317 [3] Jarboe T R, Nelson B A and Sutherland D A 215 Phys. Plasmas 22 7253 [4] Victor B S, Jarboe T R, Hansen C J, Akcay C, Morgan K D, Hossack A C and Nelson B A 214 Phys. Plasmas 21 8254 [5] Sutherland D A, Jarboe T R, Morgan K D, Pfaff M, Lavine E S, Kamikawa Y, Hughes M, Andrist P, Marklin G and Nelson B A 214 Fusion Eng. Des. 89 412 425 [6] O Neill R G, Marklin G J, Jarboe T R, Akcay C, Hamp W T, Nelson B A, Redd A J, Smith R J, Stewart B T and Wrobel J S 27 Phys. Plasmas 14 11234 [7] Wrobel J S, Hansen C J, Jarboe T R, Smith R J, Hossack A C, Nelson B A, Marklin G J, Ennis D A, Akcay C and Victor B S 213 Phys. Plasmas 2 1253
3 2 (a) shot 129175 I INJ, X I INJ, Y [ka] 1 I INJ, QUAD I TOR δb / <B> δb / B δb / B -1.3.2.1.15.1.5.15.1.5 (b) (c) (d) Toroidal Modes Injector-subtracted Injector & Equilibriumsubtracted 1 1.5 2 2.5 time [ms]
n e 1 19 [m -3 ] δ B [mt] 1.5 1.5 6 4 2 (a) (b) δb PLASMA δb INJ I TOR [ka] 3 2 1 (c) shot 129175 Measured Model, δb PLASMA Model, δb INJ 1 1.5 2 2.5 time [ms]
3 2 (a) shot 129191 I INJ, X I INJ, Y [ka] 1 I INJ, QUAD I TOR δb / <B> δb / B δb / B -1.3.2.1.15.1.5.15.1.5 (b) (c) (d) Toroidal Modes Injector-subtracted Injector & Equilibriumsubtracted 1 1.5 2 2.5 time [ms]