Action-Angle Transform of the Guiding-Center motion & Orbital Spectrum Analysis of Perturbation Modes
Y. Kominis, P. Zestanakis, G. Anastassiou, K. Hizanidis National Technical University of Athens
Motivation Need for phase space analysis and engineering in order to study and design synergetic effects between different types of non-axisymmetric and wave perturbations: MHD instabilities, RMPs, magnetic ripples, Alfvenic modes, turbulence, RF waves in terms of particle, momentum and energy transport. Crucial role of resonance mechanisms for particle redistribution.
Action-Angle transform and Orbital Spectrum Analysis The transformation to Action-Angle (AA) transformation for the GC motion enables the Orbital Spectrum Analysis (OSA) of all perturbations, that is: The simple expression of all resonance conditions and the pinpoint of the location of each resonance in the phase space. The a-priori knowledge of the effective strength of each perturbation in the Action space. The calculation of the AA transform can be carried out once per equilibrium and characterizes the equilibrium. Finite Orbit Effects are important as the Orbital Spectrum depends on the orbit width
Orbital Spectrum Analysis (OSA) • The nonlinear character of the AA transform that a monochromatic mode
= σ Am,n (ψ )exp i ( mχ + nτ − ωt )
gives an infinite series of modes in the AA space
= H1 ∑ H s ,m ( J , Pχ , µ )exp i ( mχ + sθ − ωt ) s
• The equation above yields the resonance condition in the simple form
mωχ ( J , Pχ , µ ) + sωθ ( J , Pχ , µ ) − ω = 0
Resonance Chart for the location of Resonances 2.5
x 10
3
Separatrix 2
Resonance Chart of the banana area of the phase space for resonance families with m=8, 10 and ω=0.
1.5
J
Different resonance ratios are depicted with different colors.
1
0.5
0 3
2.5
2
The perturbed motion is confined on the energy surfaces.
B
A
Wall
P
1.5
1
0.5 x 10
3
RESULTS: proof of concept The semi-analytically calculated resonance positions and widths are marked with solid red and dashed blue lines, respectively.
a
a
2
2
The chain of overlapping resonances in energy surface B can connect a deeply trapped area to the wall. b 2
Poincare cuts for two modes with m=8, 10 for subcritical (left) and critical (right) amplitude on the energy surface A.
b
2
Poincare cuts for two modes with m=8, 10 for subcritical (left) and critical (right) amplitude on the energy surface B. The wall is denoted with a thick dashed black line.
RESULTS: importance of Finite Orbit Width effects Toroidal over poloidal frequency ratio as a function of the Action J on the energy surface B. The resonances with the m=8 and m=10 are located at the intersections with the horizontal dashed lines. Solid blue curve: numerical calculation with Full Orbit Width (FOW) effects. Dashed-dotted red curve: analytic Zero Orbit Width (ZOW) approximation.
The two approaches predict entirely different resonance ratios and locations. ZOW approximation fails for energetic particles. 2.5
x 10
9
The corresponding (J,J) element of the quasilinear tensor, taking into account FOW effects.
2
1.5
The diffusion tensor D used in the Fokker-Plank equation is a smooth function that interpolates the singular diffusion tensor at the resonance points, calculated with FOW.
DJJ 1
0.5
0 1
1.1
1.2
J
1.3
1.4 x 10
3
