A straight (screw or z-)pinch (tokamak surrogate) is diagnosed using
magnetic flux ψ (per unit length along z in Webers) and magnetic field
(in Tesla) measurements at four positions on the unit circle (lengths
in meters). Taking the longitudinal coordinate to be z, along which
everything is invariant, and x and y to be the coordinates in the
plane of variation, the positions are
i
x(i)
y(i)
0
1.00
0.00
1
-1.00
0.00
2
0.00
1.00
3
0.00
-1.00
where i is the index of the measurement position.
You must devise, explain, and implement solution techniques that allow
you to reconstruct the magnetic configuration and answer the questions
posed for the following two situations.
(a)
The interior of the unit circle can be considered to be a
current-free region, prior to the appearance of the plasma. But there
are external shaping fields applied by relatively distant coils. The
measurements are
i
ψ(i)
By(i)
−Bx(i)
0
0.2851
0.4800
0.0480
1
0.1251
-0.3200
0.0480
2
-0.1469
0.0800
-0.3520
3
-0.2429
0.0800
0.4480
and you must reconstruct the fields inside
the circle to determine:
(i)
What is the magnetic field (Bx,By) at the center
(x,y)=(0,0)?
(ii)
Is there a place in the circle where the total
transverse field is zero Bx=By=0, and if so where is it?
(b)
The plasma is now present and can be considered to have
current which is only in the z-direction, concentrated in a rather
narrow filament, whose spatial extent is negligible as far as this
measurement is concerned. The measurement values are:
i
ψ(i)
By(i)
−Bx(i)
0
0.7037
2.9077
0.0385
1
1.3038
-3.1072
0.0588
2
-0.9383
-0.3096
-0.8589
3
-1.0392
-0.2899
1.1562
(i)
What is the value of the plasma current in Amps?
(ii)
What is the position of the plasma current (centroid)?
(iii)
Do the flux-surfaces inside the circle have any
x-points? If so, where are they?
[Hint. In regions where there is no current, the flux
satisfies ∇2 ψ = 0. The problem calls for an approximate
solution under appropriate assumptions. One way is to fit a small
number of Fourier harmonics. You probably should write a little code
to do the fitting. If you do, submit a print-out of your code with your
solutions. If it is well commented, then it presumably serves the
requirement of explanation as well as implementation.]
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