Tests of Profile Stiffness Using Modulated Electron Cyclotron Heating

Transcription

1 Tests of Profile Stiffness Using Modulated Electron Cyclotron Heating by T.C. Luce in collaboration with J.C. DeBoo, C.C. Petty, J. Pino, J.M. Nelson, and J.C.M. dehaas Presented at 9th EU-US Transport Task Force Workshop Cordoba, Spain September 1, 2

2 CORRELATION OF EXPERIMENTAL AND THEORETICAL SCALE LENGTHS IS OFTEN TAKEN AS PROOF OF STIFFNESS R/L Te 15 1 Experiment (L mode, 4 MW NBI) Critical gradient for ETG modes Two complementary tests of this hypothesis are needed to validate it: Demonstrate correlation persists as sensitive parameters in the theory are varied Demonstrate stiffness (causality) by perturbative techniques (subject of this presentation) ρ

3 MODULATION ANALYSIS IS BASED ON A LINEARIZED ELECTRON ENERGY CONSERVATION EQUATION Fundamental assumptions Perturbation is small enough that nonlinearities can be linearized Heat flux is of the general form: q nχ (T, T, x) T + nu (T, x) T + ξ(x) Simplifying assumptions Only T e perturbed Particle convection terms can be ignored Periodic perturbations ( e iωt ) The resulting equation is (for slab geometry): χ dt D χ + ( T) dx D d2 T + V dt ( i 3 ωτ ) 2 T = S dx 2 dx τ where V χ dn χ χ dt dn dt n ( T) dt x ( T) ( T) dx 2 + U + U T n dx x T dx 1 χ 2 χ χ d 2 T dx dx dx T 1 τ 1 dn χ dt 2 χ dt x T χ n dx T dx dx T + 3Q OH + Q ie (3T i T) 2nT 2nT T i T d 2 T dx n dn U + U + 1 dn U n T + 2 U x T T + U dt dx x dx T T dx

4 ROUGH ESTIMATES OF THE PERTURBED FLUX CAN BE USED AS AN INDICATION OF STIFFNESS Solutions to the linearized equations in source-free regions with constant diffusion coefficient yield simple relations for the Infinite Slab Solution amplitude and phase variations in space: χ φ 3ω ( dφ ) 2 dx = χ 4 χ T 3ω ( 1 dt 1 ) 2 T1 = χ 4 dx where T = T 1 e iφ T 1 φ x Source In the presence of convective-like and damping terms χ φ and χ T are modified and only give the diffusivity when ω SAN DIEGO Note that χ φ is always increased and is independent of the direction of the convective term χ φ,t /χ χ φ Damping important χ T (pulse moving with convection) χ T (pulse moving against convection) ω (rad/s)

5 1 SLAB ESTIMATORS FOR χ GIVE POSSIBLE INDICATIONS OF STIFFNESS L mode, NBI (4 MW), no sawteeth 1 χ T = 5.8 m 2 /s 8 χ φ = 137 m 2 /s Amplitude (ev) 1 χ T = 5.4 m 2 /s χ T = 9.6 m 2 /s Phase (degree) 6 4 χ φ = 6.8 m 2 /s χ φ = 45.3 m 2 /s 25 Hz Modulation 25 Hz Modulation ρ ρ

6 FREQUENCY DEPENDENCE OF THE SLAB ESTIMATORS INDICATES NO STIFFNESS IN THE CORE AND SIGNIFICANT DAMPING AT THE EDGE 1 L mode, NBI (4 MW), ρ EC =.76, no sawteeth χ φ, χ T (m 2 /s) 1 χ PB χ φ (ρ =.5.75) χ φ (ρ =.75.9) χ T (ρ =.5.75) χ T (ρ =.75.9) f mod (Hz)

7 MINIMUM PHASE LAG IN THE DEPOSITION REGION ALSO INDICATES NO STIFFNESS 1 L mode, NBI (4 MW), ρ EC =.76, no sawteeth 8 φ min (deg) 6 4 χ = 1 m 2 /s f mod (Hz)

8 ANALYTIC SOLUTIONS FOR THE MINIMUM PHASE ARE DEPENDENT ON THE SOURCE FUNCTION AND BOUNDARY CONDITIONS φ min (deg) Bounded Solutions With Fixed Boundary Gradient χ = 1 m 2 /s Source at ρ =.75 Box Source (asymptotes to 9 ) δ Function Source Extended source in a bounded domain with fixed edge is the only solution consistent with the data Bounded Solutions With Fixed Boundary Temperature f (Hz)

9 MINIMUM PHASE LAG IN H MODE IS CONSISTENT WITH IMPROVED CONFINEMENT AND NO STIFFNESS 1 H mode, NBI (4 MW), ρ EC =.6, no sawteeth L mode, NBI (4 MW), ρ EC =.76, no sawteeth 8 φ min (deg) 6 4 χ = 1 m 2 /s f mod (Hz)

10 SIGNIFICANT ION TEMPERATURE PERTURBATIONS ARE INDUCED BY THE ECH PULSE Amplitude (ev) Phase (deg) ρ SAN DIEGO Electrons ECH Location ECH Location Amplitude (ev) Phase (deg) Ions ρ DeBoo, et al., Nucl. Fusion (1999) Anticipated from power balance studies (Petty, et al.) and seen in ITG simulations (Kinsey, et al.) Precedent exists in JET sawtooth analysis of density and electron temperature (dehaas et al.) Frequency space formalism exists for coupled fluid analysis (Hogeweij, et al.)

11 QUESTIONS FOR DISCUSSION Are there any unique signatures of a threshold in the perturbative response? (I don t think so) Can progress be made using only a one-fluid model? (I hope so, but probably not) What are the appropriate boundary conditions to use for analytic and numerical perturbation models? (I don t know)

12 CONCLUSIONS Proof of threshold phenomena as the origin of profile consistency requires demonstration that the plasma strongly resists increases in gradient. L mode data is not consistent with this mechanism in the electron temperature profile Electron energy spreading in L mode plasmas can be much faster than expected on the basis of purely diffusive heat flux at power balance values. However, analysis indicates that this is not due to large incremental diffusivity Preliminary analysis of H mode plasmas indicates no strong incremental diffusivity Measurements of the perturbed ion temperature response to a purely electron energy pulse indicates a direct strong dependence of the ion heat flux on the electron temperature, as shown in power balance studies and ITG simulations Boundary conditions and use of generalized functions for source models can have profound (unintended) consequences on the perturbation analysis. Only extended source, fixed boundary solutions are consistent with the data