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E2. IntroductionThe ideal MHD equationsThe Grad-Shafranov equation and its solutionThe equilibrium fieldEquilibrium by external currentsPlasma ellipticity and triangularityThe ITER equilibrium configurationConclusions. Summary. E3. . Introduction. . The forces determining plasma equi
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1. E1 International Doctorate in Fusion Science and Engineering
Basic Course
Padova, 4-15 May 2009
Axisymmetric Equilibrium and Stability
of Toroidal Plasmas
Francesco Gnesotto, Universitŕ di Padova
7 and 8 May 2009
2. E2 Introduction
The ideal MHD equations
The Grad-Shafranov equation and its solution
The equilibrium field
Equilibrium by external currents
Plasma ellipticity and triangularity
The ITER equilibrium configuration
Conclusions
Summary
3. E3
Introduction
4. E4
Maxwell equations under quasi-stationary conditions:
The Ohm’s law
Thermo-fluidodynamics equations, single fluid
mass conservation
momentum conservation
adiabatic process
Ideal MHD ?? = ?
E + v x B = 0
5. E5 Under stationary conditions:
J x B = ?p
thus
B ? ?p = 0: the magnetic surfaces are isobaric surfaces as well
J ? ?p = 0: the current field lines lie on isobaric surfaces, i.e. on magnetic surfaces
NB: from here onwards, we will only consider an axisymmetric geometry (2D)
The equilibrium equations
6. E6 Cylindrical coordinates (R, ?, z)
def.: poloidal flux function : ?
? is the magnetic flux linked with the circumference passing across point P(R,z) and having its center on the z axis
More frequently we use
Since the magnetic surfaces are isobaric, p = p(?)
Poloidal field components:
The poloidal flux function
7. E7 Cylindrical coordinates (R, ?, z)
def.: poloidal current function (IPOL):
IPOL is the current linked with the circumference passing through point P(R, z) and having its center on the z axis
Ampčre’s law
More frequently we use:
The current density components:
The poloidal current function
8. E8 The Grad-Shafranov Equation Usually it is solved by numerical methods
An analytical solution is possible under particularly simple geometrical conditions
circular plasma
large aspect ratio
We will now use a quasi-cylindrical
coordinate system
9. E9 Analytical solution The flux surfaces ?(r, ?) = const
are circumferences whose center moves outwards when their radius decreases
This corresponds to a condition where B? is higher on the outside of the torus, to counteract the tendency of the plasma to expand
At the equilibrium, the poloidal field on plasma surface is:
10. E10 Equilibrium vertical field To maintain the plasma horizontal equilibrium,
the following magnetic field is necessary:
Produced only by the external field sources
Proportional to plasma current
Increasing with ?
Uniform under the assumed hypotheses
11. E11 The Virial’s theorem “It’s impossible to sustain any MHD equilibrium without currents external to the plasma”
where
12. E12 Equilibrium currents - 1 Ideal case
Currents distributed in a thin layer on a circumference of radius b, concentric to the plasma
To decouple these currents from the transformer:
i0=0
Let’s consider the first harmonic i1
And let’s impose the boundary
conditions on radius b
We obtain a vertical field
13. E13 Equilibrium currents - 2 Semi-ideal case
Currents concentrated in a finite number
of filamentary conductors
Distribution 1: 4 conductors localized at:
? = ?/4, 3?/4, -3?/4, -?/4
and carrying the currents:
This distribution results in a dipole field
Fourier analysis
The plasma cross-section remains circular
14. E14 Distribution 2: 4 conductors localized at:
? = 0, ?/2, ?, 3 ?/2
And carrying the currents:
This ristribution results in a quadrupole field
Fourier analysis
When i? is present, the plasma cross-section becomes ellyptical
?=f (i ? /i1)
Plasma ellipticity - 1
15. E15 Plasma ellipticity - 2 i??0
i?<0
Triangularity
16. E16 Vertical stability The quadrupole field bends the field lines
For instance, if i??0
“spindle” deformation, the equilibrium is unstable
Vice-versa, if i??0
“barrel” deformation, the equilibrium is stable
17. E17 Stability analysis Def.: field decay index (adimensional)
Rigid vertical displacement dz
Associated radial field variation dBR
Vertical force dFz?-2?R0 I dBR
The equilibrium is stable if:
i.e. if
18. E18 Dynamic phenomena
The stability of the plasma column is influenced by eddy currents flowing along passive conducting structures
Eddy currents tend to counteract dB/dt, i.e. changes in current density or current position
A passive shell, made by a high conductivity material (copper), can be provided as close as possible to the plasma
The vacuum vessel is indeed a shell, made by a medium conductivity material (Ni alloy), but continuous and very close to the plasma
The vessel allows to significantly decrease the growth rate of vertical instabilities, so easing the design of active control systems
19. E19 ITER
20. E20 ITER poloidal field
21. E21 The plasma toroidal current
22. E22 The ITER equilibrium windings
23. E23 The ITER poloidal field
24. E24 ITER control issues
25. E25 Conclusions The equilibrium of a Tokamak plasma column is maintained by a complex system of poloidal field inductors
The equilibrium of highly elongated plasmas, like in ITER, is vertically unstable
Fast feedback control systems are needed to counteract vertical instabilities (VDE)
On shorter timescales an important role is played by the passive structures
A reliable diagnostic (magnetics), data handling and power amplifier system is needed to stabilize the plasma column
The modern electronics and IT allowed to drastically improve the active control capabilities in the last years
26. E26 Further reading….
