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Measurements and Modeling of Biased Electrode Discharges in HSX S.P. Gerhardt, D.T. Anderson, J.M. Canik, W.A. Guttenfelder, and J.N. Talmadge HSX Plasma Laboratory, U. of Wisconsin, Madison Key Points 4. Neoclassical Modeling of Plasma Flows •IV Characteristics of the Biased Electrode. •Evidence of Two Time Scale Damping in HSX. •Neoclassical Modeling of Plasma Flows. •Experimental Verification of Reduced Viscous Damping with Quasi-Symmetry •Computational Study of Different Configurations 1. Structure of the Experiments 3. Two Time Scales Observed in Flow Damping General Structure of Experiments Simple Flow Damping Example • Solve the Momentum Equations on a Flux Surface • mi N i • • Mach Probes in HSX • • • 6 tip mach probes measure plasma flow speed and direction on a magnetic surface. 2 similar probes are used to simultaneously measure the flow at high and low field locations, both on the outboard side of the torus. Data is analyzed using the unmagnetized model by Hutchinson. Looking To The Magnetic Surface • • As the damping is reduced, the flow rises more slowly, but to a higher value. Full problem involves two momentum equations on a flux surface2 time scales & 2 directions. U1,exp t X 2 t cos X 3 t • Fit flows to models U (t ) C 1 exp t / • s 2, fit f f f s s s Model Fits Flow Rise Well 3 This allows the calculation of the radial electric field evolution: U 1 e t /1 S1e S3e 1 e t / 2 S2e S4e • • • “Forced Er” Plasma Response Rate is Between the Slow and Fast Rates. Decay Time: =33x10-6 seconds Impedance: R=V/I=36 C= /R=8.9x10-7F Accumulated Charge from “Charging Current”: Q=2.33 Coulombs Voltage: V=310 Volts C=Q/V=7.5x10-7F Original Formulation of Capacitance/Impedance modeling published in J.G. Gorman, L.H. Rietjens, Phys. Fluids 9, 2504 (1966) Impedance is Smaller in the Mirror Configuration • • Current peaks at the calculated separatrix. Electrode current profile does not follow the density profileElectrode is not simply drawing electron saturation current. • Mirror • QHS Predicted Separatrix Position Impedance1/n consistent with radial conductivity scaling like n. Consistent with both neoclassical modeling by Coronado and Talmadge or anomalous modeling by, for instance, Rozhansky and Tendler. • • 1/n • Increasing the N4 Mirror Percentage is Efficient at Increasing the Damping 10% 7.5% 5% Mirror, slow rate QHS, slow rate 40 Capacitance is Linear in the Density •Plasma dielectric constant proportional to density. •Low frequency dielectric constant: 1 2 1500 8.854 1012 2 1.2 1F ln 17.5 11 N4 Mirror (Standard) + + + - - - - - N12 Mirror + - - Broad N Mirror + - + - + + - - + + - - + - - + - + - + Hill + + + + + + + + + + + + + - Percentage indicates the ratio of amp-turns in the two coil sets. Large Change in Surface Shape for Deep Hill Configuration QHS Other Mirror Configurations Result in Less Damping + + + 10% N4 Mirror 10% N12 Mirror Hamada Spectrum 10% N12 Mirror 10% Broad N QHS 16.66% Hill Hamada Spectrum B2 4 10 7 5 10 11 1.67 10 27 9 1016 1500 .52 C 10% N4 (Standard) Mirror Hamada Spectrum Deep Hill Mode Leads to a Slight Damping Increase o c 2 •Expected capacitance: 2.5% Aux. Coil Current Direction (+ for adding to the toroidal field of the mail coils) QHS, F Positive Part of I-V Curve is Linear Linear I-V relationship Consistent with linear viscosity assumption. Very little current drawn when collecting ions collect electrons in all experiments in this poster. •H signal is linear in the densityneutral density approximately constant. •See Poster by J. Canik for more details on the H system and interpretation. •Flow decay time becomes longer as density increases. 13% Two time scales/two direction flow evolution. Radial Conductivity Has a 1/n Scaling QHS Damping Does Not Appear to Scale with the Neutrals Scan of N4 Mirror Percentage 2. I-V Characteristics of Electrode Indicate Transport Limits the Current Bias Waveforms Indicate a “Capacitance” and an Impedance QHS Mirror, F t 0 Many Different Configurations are Accessible with the Auxiliary Coils Fast Rates, QHS and Mirror ExB flows and compensating Pfirsch-Schlueter flow will grow on the same time scale as the electric field. Parallel flow grows with a time constant F determined by viscosity and ion-neutral friction. Note: Floating Potential Rises Much Faster Than the Flows. Mirror Neutrals Only 6. Computational Study: Viscous Damping in Different Configurations of HSX t 0 t / U t U E 1 e e BQ1 E 1 1 Q2 e F t Q2 e t / Total Conductivity Viscosity Only QHS Damps Less Than Mirror; Some Physics Besides Neoclassical and Neutral Damping Appears to be Necessary to Explain the QHS Data. Assume that the electric field, d/d,is turned on quickly Er 0 Er 0 E 1 e t / QHS Mirror Formulation #2: The Electric Field is Quickly Turned On. • 1/in QHS 4 • Coronado and Talmadge Model Overestimates the Rise Times 1/in <B>=0 in net current free stellarator, but not a tokamak. <B>= Boozer g= 2*1e-7*48*14*5361 =.7205 • Measurements Combination of neutral friction and viscosity determines radial conductivity. Mirror agreement is somewhat better. t / 2 • • Define the radial conductivity as J R d d 2 t / 1 2 SS Similar model 2 time scale / 2 direction fit is used to fit the flow decay. B / Bo 1 H cos4 M cos4 Mirror : 1 d t d t 0 F1 1 et /1 F2 1 et / 2 d d t C 1 exp t / sin C 1 exp t / sin U U1, fit t C f 1 exp t / f cos f Cs 1 exp t / s cos s U1SS U Symmetry Can be Intentionally Broken with Auxiliary Coils fˆ C 1 exp t / sˆ Tokamak Basis Vectors Can Differ from those in Net Current Free Stellarator. t / 2 S1…S4, 1 (slow rate), and 2 (fast rate) are flux surface quantities related to the geometry. Break the flow into parts damped on each time scale: • Convert flow magnitude and angle into flow in two directions: Probe measures Vf with a proud pin. QHS : B / Bo 1 H cos4 U Flow Analysis Method Predicted formf of flow rise f from modeling: s S 1 e S S 1 e S 1 e U 1 e QHS Mirror The “Forced Er” Model Underestimates the QHS time Original calculation by Coronado and Talmadge After solving the coupled ODEs, the contravariant components of the flow are given by: t / 1 QHS Mirror Use Hamada coordinates, using linear neoclassical viscosities. No perpendicular viscosity included. U 2,exp t X 2 t sin X 3 t X I sat X 1 exp 2 .641 cos X 3 .71 cos X 3 2 • • • 0 t0 U jB 1 exp t / nm t 0 • •QHS Flow Rises and Damps More Slowly QHS Modeled Radial Conductivity agrees to a Factor of 3-4 QHS Flow Damps Slower, Goes Faster For Less Drive. •Flow Goes Faster For Per Unit Drive Formulation #1: The External Radial Current is Quickly Turned On. 0 t0 dU F U , F jB t 0 Has solution dt Need quantities like <e·e >, <e·e >, <e·e >,<||>, <| · |>. Previous calculation used large aspect ratio tokamak approximations. Method involves calculating the lab frame components of the contravariant basis vectors along a field line, similar to Nemov. Solve these with Ampere’s Law • • We Have Developed a Method to Calculate the Hamada Basis Vectors t B U B mi N iin B U t 4 J plasma J ext Take a simple 1D damping problem: • t c BP U J BP mi N iin BP U g B B mi N i mn • Two time scales/directions come from the coupled momentum equations on a surface. • • • 5. Comparisons Between QHS and Mirror Configurations of HSX and with Modeling QHS Scan of Hill Percentage 16.67% Hill 3.33% N4 Hill 13.33% Hill QHS