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On the theory of magnetic diagnostics in stellarators and tokamaks
Fusion Engineering and Design 34-35 (1997) 689-693 ELSEVIER
Fusion Engineering and Design
On the theory of magnetic diagnostics in stellarators and tokamaks V.D.
Pustovitov
1
National Institute for Fusion Science, Nagoya 464-01, Japan
Abstract The main problem of magnetic diagnostics is discussed here: which plasma characteristics can be determined from magnetic measurements in tokamaks and stellarators. The reason is elucidated why diamagnetic measurements are reliable and easily interpretable. Explanation is given of the paradox that the measured diamagnetic signal is always proportional to plasma energy, even in the case when magnetic flux is frozen into the plasma. Diagnostics based on the measurements of poloidal fields outside the plasma is also discussed. The integral nature of measurable signals is shown to be a serious obstacle for estimating plasma pressure and current profiles from magnetic measurements. © 1997 Elsevier Science S.A.
1. Introduction Two basic questions in the theory for magnetic diagnostics are what plasma characteristics can be found from magnetic measurements and with what accuracy. We imagine that the measurements themselves are perfect so that the discussion below is addressed to the interpretation of measurable magnetic signals. We must find relations between plasma parameters and magnetic fields created by the plasma current. The underlying physics is governed by the equations (la)
Vp = j x B
div B = 0,
j = rot B
(lb)
1Permanent address: Russian Research Centre "Kurchatov Institute", Moscow 123182, Russia.
where p is the plasma pressure, j is the current density and B = Bext + Bpl is the magnetic field, with Bext being the part produced by external currents. A distinctive feature of the problem is that we must get useful information from Eq. (1) without knowledge about the internal distributions of p and j. Some part of Bcxt may also be unknown because of eddy currents in the surrounding metal. Finally, magnetic measurements are performed by magnetic probes or coils outside the plasma. These factors are the main sources of uncertainties. However, there is one definitely positive point: the calculation of the magnetic field outside the plasma is a classical boundary value problem. Inaccuracy of its solution may be related only to drawbacks of models but not to mathematics. We discuss here related theoretical problems keeping in mind practical applications.
0920-3796/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0920-3796(96)00565-0
690
V.D. Pustovitov/Fusion Engineering and Design 34-35 (1997) 689-693
1 I~ F - F b
2. Theory of diamagnetic measurements
--7--
Diamagnetic measurements allow us to find the plasma energy content (fl). The following expression was used for more than 35 years to interpret their results: 2 * 0 - Bg -
(2)
It was derived by Braginskii and Shaffanov in the framework of a very simple model: a straight circular plasma cylinder. It is amazing that this formula was almost always good enough for real toroidal devices despite the evident weakness of the model. On one hand this observation engendered the first question of the theory, why Eq. (2) is so good. On the other hand it was found recently [1] that experimental, numerical and analytical results on the diamagnetic effect in a compact helical system (CHS) could not be satisfactorily fitted, which made urgent another question on how to improve or modify Eq. (2) in such a way as to account for the real geometry of the experiment. For conventional stellarators such an improvement has been achieved in Ref. [2]. In this case, to calculate Aqb, which is the small difference between the toroidal magnetic flux (I)p inside the plasma and the similar flux qbv of a vacuum field, A(I) = @p - @v = [
( B - Bv) .dS±
where F b is the vacuum value of F characterizing the external toroidal field. For a stellarator Eq. (6) is approximate, but for a tokamak it is exact. In the cylindrical limit, V( in Eq. (5) must be replaced by ec/R, where R = const., and r by R in Eq. (6). Toroidicity reveals itself in Eq. (6) first by the obvious presence of r in the denominator and second, implicitly, through the dependence F(~). In toroidal systems, when fl increases, magnetic surfaces ¢ = const, are shifted outwards. As a result the maximum of IF-Fb[ is shifted into the region of larger r and its contribution to the integral becomes smaller because it is divided by r > R, where R is the major radius of the device. Thus, with a natural outward shift of the magnetic axis, A@ should increase more slowly with increasing fl than is expected from Eq. (2). However, taking account of the toroidicity in Eq. (6) can lead at most to corrections of the order of the inverse aspect ratio b/R, which can be noticeable in A~ only for a strong relative shift of magnetic surfaces. This is why a cylindrical approach may give a reliable result for Aqb in many cases for toroidal systems too. The above comments about toroidicity can be illustrated by the formula [2] Oo
stellarator expansion can be used. In stellarators (4)
where B is the helical (fast oscillating) component of the magnetic field and /~ is the axisymmetric component of B. F o r / ~ the following representation is valid: v(¢, - ¢,d x v ( +
F
v(
(6)
(3)
.Is ±
B = B + B
dS±
(5)
where ( is the toroidal angle; below, r, ( and z are ordinary cylindrical coordinates. We use standard notation and known theoretical results described in detail in Ref. [3]. The difference between/~ and its vacuum value Bv is insignificant and can be disregarded. After substitution of Eq. (5) into Eq. (3) we get
2
3R
(7)
explicitly showing a weak dependence of A(I) on the Shafranov shift. It is valid for stellarators without shear that have circular shifted magnetic surfaces. Here qbo = rcb2Bo, flOq= p2b/R, b is the minor radius and /x is the rotational transform. Even for fl close to the equilibrium limit the last term in Eq. (7) cannot be larger than 0.7b/R. In the general case [2] a~b = - 1 -- dsFb- dh ~vI-" pp dV
(8)
where ds is a quantity of the order of b/R describing the dependence of Aqb on the relative shift of magnetic surfaces and fih is another small term due to the helical field. This formula is valid for a currentless plasma in conventional stellarators
V.D. Pustovitov / Fusion Engineering and Design 34-35 (1997) 689-693
with an arbitrary magnetic surface shape, aspect ratio and/z(a). It is surprising that after removing all geometrical restrictions of the model we obtained a result which looks almost as simple as Eq. (2) and is ready for immediate practical use. The integral over the plasma volume in Eq. (8) is the value of interest to be found from the measured Ago. Fortunately, the main dependence of Ago on the plasma shape is contained in this integral. However, the coefficient in front of the integral turned out to be practically independent of the plasma shape. The coefficient is approximately equal to - 1/Fb and all possible deviations from this constant are described by the small terms gs and gh. In present-day experiments fl is still far from the equilibrium limit fl~q. Then ~s 0 if there is a shift of the magnetic axis in a vacuum configuration. In this case 6s as well as csh can be calculated from the given vacuum field. Finally, diamagnetic measurements can be easily interpreted to find fl with high accuracy in currentless stellarators. There is another advantage to diamagnetic measurements: the value of Ago as determined by Eq. (6) does not depend on the shape of the transverse area S±, because only integration over the plasma cross-section gives a non-vanishing contribution. Thus the shape of the diamagnetic coils may be arbitrary. Of course, these coils should lie in a plane perpendicular to the geometrical axis of the device. The parameter AgO is often referred to as the change in the toroidal flux in the plasma. In some cases this is correct, but in general such an interpretation is wrong. By definition, Ago is the change in the magnetic flux through the area encircled by the diamagnetic coil. The plasma boundary divides this area into two parts: S j_ = Sin+ Sext. Correspondingly, the magnetic flux through S : consists of two parts, i.e. the flux goin through the transverse cross-section of the plasma, Sin, and the flux (I)ext of the external field through Se~t:
go = q~n + @~xt
(9)
During the discharge the plasma can move and change its shape. Then Sext and, as a result, (I)ext are also changed. Thus some part of the signal
691
measured by the diamagnetic coil can be attributed to the change in goext, sometimes up to 100%. Indeed, a hot plasma is a perfect conductor and during fast processes magnetic fluxes inside the plasma must be frozen irrespective of changes in ft. However, Ago -~ 0. There is no contradiction: in Ago, which is the difference of two fluxes, Eq. (3), only one characterizes the plasma, while the other is determined by the vacuum field Bv. If the vacuum toroidal field does not vary, Ago is equal to the difference of fluxes through the cross-section of the plasma; see Eqs. (3) and (6). However, this value shows how much the toroidal flux goinin the plasma differs from the flux of the vacuum field Bv through the same cross-section S=. Such an interpretation is valid always for any geometry and history of the discharge. We need only the assumption that the final state is described by the equilibrium Eq. (1). In the calculations above the toroidal magnetic field outside the plasma was assumed to be unchanged, cSBt = 0. Then, if the equilibrium is flux conserving, the flux go~xt~ BtSext can vary only owing to a change in Sin (~S~xt= -~Sin). The plasma size must increase with increasing t , because in this case Ago = Agoext < 0; see Eqs. (2) and (8). This was discussed in Refs. [2,3]. However, a similar discussion in Ref. [4] has been spoilt by an inconsistent analysis in which the contribution due to c~S,×t in ggo~xt= B t ~ S e x t -~ SextaNt was disregarded.
3. Self-field of equilibrium plasma currents outside plasma column The diamagnetic coil embracing the plasma column records the change in the field inside the plasma. The plasma current j creates a magnetic field Bpl outside the plasma also. This field can be calculated directly if its source j is known. However, we can find Bpl as a vacuum field by solving Eq. (lb) outside the plasma boundary Fp. If the boundary conditions are given at Fv, there is no need to know the current j and field B inside I~p, though they actually contribute to determining the shape of Fp and the magnetic field Br o n Fp.
692
V.D. Pustovitov/ Fusion Engineering and Design 34-35 (1997) 689-693
Independence of the external solution from the internal one for given boundary conditions means that two configurations, both with the same Fp and Br but one of them without a magnetic field inside Up, Bin = 0, must have the same magnetic field Bwo in the external region. To provide a jump in the magnetic field at the plasma boundary (Br outside and Bin = 0 inside in the latter case), a surface current must flow on I'p: (10)
ir = - - B r x n
where n is the unit external normal to Fp. In a configuration with such a surface current, B~+ Bext = Bv~c in the external region and B~+ e e x t : 0 in the inner region, where B~ is the field of the surface current (10) and B~xt is the field produced by the currents flowing in external conductors. However, by definition, Bv~c= Bpl + B,xt, so Bpl Bi =
-- Bex t
outisde Fp inside Fp
(11)
The last two equations show that two problems can be solved once the magnetic field on the plasma boundary is known: the plasma-generated magnetic field Bpl outside the plasma can be found and the external magnetic field B~xtwhich is needed for sustaining a given configuration in equilibrium can be calculated. This general result itself is well known, but its implications do not appear to be fully appreciated in some works on magnetic diagnostics, where discussions start from another level. However, a very strong conclusion follows from Eq. (11): the plasma-generated field Bp~ in the vacuum is determined only by the shape of the plasma boundary and the magnetic field Br on this boundary. Thus, by measuring the plasma-generated field Bpl outside the plasma, one can recover only that information about the plasma which is 'hidden' in Fp and Br. If Fp and neighbouring magnetic surfaces in a tokamak, for example, are shifted tori with a circular cross-section (r - R - iX)2+ z 2 = a 2, we get from Eq. (5) ~' n x e~ Br = 2~r 1 -- iX' cos 0
(12)
where 0 is the poloidal angle. There are only two parameters, ~' and A'. By ~' the amplitude of Br is described, which can be measured by a Rogowski coil. Any other measurements of Bpl can give nothing more than A' on the plasma boundary. In other words, only one more value can be found besides the net plasma current J. This fact follows from the most general principles and there is no need to solve equilibrium equations in order to answer the question [5] why pp and f~ cannot be separately measured in a nearcircular tokamak, because N = - (b/R)(~p + Ei/ 2). There is another interesting question on whether it is possible to determine the plasma current or pressure distribution by measuring Bp~ outside the plasma. Eq. (11) indicates that the answer is most probably negative, because information is hidden in Fp. For example, only the net current J can be found in a straight cylinder and only J and A' in a tokamak with a circular plasma cross-section. For non-circular Fp we can refer to the results of Ref. [6]: in four cases the plasma shape and poloidal field at Fp are approximately the same, but the current profiles are strongly different. According to Eqs. (10) and (11), all four configurations must have the same Bpl outside the plasma for the same F v and Br. It is then impossible to distinguish the difference in current (and pressure) profiles by measuring Bpl. " N o " seems to be a natural answer. However, there is a lot of optimism in some publications [7-10]. The example given by Eq. (12) above prompts the simplest way to reveal the real ground of positive answers: instead of profiles we must speak about numbers, which could be found by measuring Bpl. If it is clearly understood that there is a substitution "one number-one profile" in Refs. [8,10], it is logical to ask then what this number is. In the case described by Eq. (12) it is iX' and only three values, 3", tip and ~, can be found from magnetic measurements. Can we then expect high accuracy in evaluating the pressure profile by known fl and the current profile by Y~? The above comments about tokamaks are qualitatively applicable to stellarators also. Often it is said that stellarators are better for magnetic diagnostics. One reason is that they operate without a
V.D. Pustovitov / Fusion Engineering and Design 34-35 (1997) 689-693 net c u r r e n t a n d b o t h Aq) a n d Bpl m u s t give inf o r m a t i o n a b o u t the p l a s m a pressure. I n stellarators the m a g n i t u d e o f Bpl o n ]"p is given b y [3]
~b a 2 p ' ( a )
• oa
(13)
F o r an Y = 3 s t e l l a r a t o r with /~ oca 2 we o b t a i n
B/~ = Cflo. This result allows us to m a k e some guess a b o u t p(a), b e c a u s e two n u m b e r s can be f o u n d , average fl a n d flo, w h i c h is the local value o f fl at the m a g n e t i c axis; b u t only a guess, a n d this is the case o f the best resolution. I f / t = const., Eq. (13) gives B~ = Cfl. H e r e fl is the s a m e as in Eq. (2), so m e a s u r i n g Bpl does n o t give a n y i n f o r m a t i o n a d d i t i o n a l to d i a m a g netic m e a s u r e m e n t s . O r d i n a r y E = 2 s t e l l a r a t o r s are 'in b e t w e e n ' a n d b o t h Bpl a n d AcI) give inform a t i o n t h a t is less t h a n fl a n d flo. Profile e v a l u a t i o n s b y Bpl a n d A ~ are p o o r b e c a u s e these values are integral; see Eqs. (3) a n d (11). H o w e v e r , the integral n a t u r e o f Bpl a n d A(I) helps us to find fl a n d the p l a s m a shape a n d p o s i t i o n with g o o d accuracy. F o r stellarators, recent H e l i o t r o n E results [11] are g o o d examples.
693
References [1] H. Yamada, K. Ida, H. Iguchi, et al., Shafranov shift in a low aspect ratio heliotron/torsatron compact helical system, Nucl. Fus. 32 (1992) 25-32. [2] V.D. Pustovitov, Refined theory of diamagnetic effect in stellarators, J. Plasma Fus. Res. 69 (1993) 34-40. [3] V.D. Pustovitov, Fundamental stellarator MHD theory, J. Plasma Fus. Res. 70 (1994) 943-991. [4] E.D. Andryukhina, K.S. Dyabilin and O.I. Fedyanin, in Proc. IOFAN, Vol. 31, Stellarators, Nauka, Moscow, 1991, pp. 186-192 (in Russian). [5] J.P. Friedberg, et al., Why tip and li cannot be separately measured in a near circular tokamak, Plasma Phys. Controlled Fus. 35 (t993) 1641-1648. [6] L.L. Lao, et al., Separation of tip and li in tokamaks of non-circular cross-section, Nucl. Fus. 25 (1985) 1421- 1436. [7] Yu.K. Kuznetsov, V.N. Pyatov and I.V. Yasin, Possibility of determining equilibrium plasma current and pressure profile in a tokamak from magnetic measurements, Soy. J. Plasma Phys. 13 (1987) 75-80. [8] Yu.V. Gutarev, et al., Method for magnetic measurements of equilibrium plasma parameters in a stellarator, Soy. J. Plasma Phys. 14 (1988) 164-167. [9] V.K. Pashnev and V.V. Nemov, Use of magnetic diagnostics in stellarators, Nucl. Fus. 33 (1993) 435-447. [10] A.B. Kuznetsov, S.V. Shchepetov and D.Yu. Sychugov, Is it possible to extract information on the plasma pressure profile from magnetic measurements in stellarators?, Nucl. Fus. 34 (1994) 185-190. [11] S. Besshou, et al., Detection of the boundary plasma shift in a toroidal helical plasma on Heliotron-E, Nucl. Fus. 35 (1995) 173 182.
Fusion Engineering and Design
On the theory of magnetic diagnostics in stellarators and tokamaks V.D.
Pustovitov
1
National Institute for Fusion Science, Nagoya 464-01, Japan
Abstract The main problem of magnetic diagnostics is discussed here: which plasma characteristics can be determined from magnetic measurements in tokamaks and stellarators. The reason is elucidated why diamagnetic measurements are reliable and easily interpretable. Explanation is given of the paradox that the measured diamagnetic signal is always proportional to plasma energy, even in the case when magnetic flux is frozen into the plasma. Diagnostics based on the measurements of poloidal fields outside the plasma is also discussed. The integral nature of measurable signals is shown to be a serious obstacle for estimating plasma pressure and current profiles from magnetic measurements. © 1997 Elsevier Science S.A.
1. Introduction Two basic questions in the theory for magnetic diagnostics are what plasma characteristics can be found from magnetic measurements and with what accuracy. We imagine that the measurements themselves are perfect so that the discussion below is addressed to the interpretation of measurable magnetic signals. We must find relations between plasma parameters and magnetic fields created by the plasma current. The underlying physics is governed by the equations (la)
Vp = j x B
div B = 0,
j = rot B
(lb)
1Permanent address: Russian Research Centre "Kurchatov Institute", Moscow 123182, Russia.
where p is the plasma pressure, j is the current density and B = Bext + Bpl is the magnetic field, with Bext being the part produced by external currents. A distinctive feature of the problem is that we must get useful information from Eq. (1) without knowledge about the internal distributions of p and j. Some part of Bcxt may also be unknown because of eddy currents in the surrounding metal. Finally, magnetic measurements are performed by magnetic probes or coils outside the plasma. These factors are the main sources of uncertainties. However, there is one definitely positive point: the calculation of the magnetic field outside the plasma is a classical boundary value problem. Inaccuracy of its solution may be related only to drawbacks of models but not to mathematics. We discuss here related theoretical problems keeping in mind practical applications.
0920-3796/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0920-3796(96)00565-0
690
V.D. Pustovitov/Fusion Engineering and Design 34-35 (1997) 689-693
1 I~ F - F b
2. Theory of diamagnetic measurements
--7--
Diamagnetic measurements allow us to find the plasma energy content (fl). The following expression was used for more than 35 years to interpret their results: 2 * 0 - Bg -
(2)
It was derived by Braginskii and Shaffanov in the framework of a very simple model: a straight circular plasma cylinder. It is amazing that this formula was almost always good enough for real toroidal devices despite the evident weakness of the model. On one hand this observation engendered the first question of the theory, why Eq. (2) is so good. On the other hand it was found recently [1] that experimental, numerical and analytical results on the diamagnetic effect in a compact helical system (CHS) could not be satisfactorily fitted, which made urgent another question on how to improve or modify Eq. (2) in such a way as to account for the real geometry of the experiment. For conventional stellarators such an improvement has been achieved in Ref. [2]. In this case, to calculate Aqb, which is the small difference between the toroidal magnetic flux (I)p inside the plasma and the similar flux qbv of a vacuum field, A(I) = @p - @v = [
( B - Bv) .dS±
where F b is the vacuum value of F characterizing the external toroidal field. For a stellarator Eq. (6) is approximate, but for a tokamak it is exact. In the cylindrical limit, V( in Eq. (5) must be replaced by ec/R, where R = const., and r by R in Eq. (6). Toroidicity reveals itself in Eq. (6) first by the obvious presence of r in the denominator and second, implicitly, through the dependence F(~). In toroidal systems, when fl increases, magnetic surfaces ¢ = const, are shifted outwards. As a result the maximum of IF-Fb[ is shifted into the region of larger r and its contribution to the integral becomes smaller because it is divided by r > R, where R is the major radius of the device. Thus, with a natural outward shift of the magnetic axis, A@ should increase more slowly with increasing fl than is expected from Eq. (2). However, taking account of the toroidicity in Eq. (6) can lead at most to corrections of the order of the inverse aspect ratio b/R, which can be noticeable in A~ only for a strong relative shift of magnetic surfaces. This is why a cylindrical approach may give a reliable result for Aqb in many cases for toroidal systems too. The above comments about toroidicity can be illustrated by the formula [2] Oo
stellarator expansion can be used. In stellarators (4)
where B is the helical (fast oscillating) component of the magnetic field and /~ is the axisymmetric component of B. F o r / ~ the following representation is valid: v(¢, - ¢,d x v ( +
F
v(
(6)
(3)
.Is ±
B = B + B
dS±
(5)
where ( is the toroidal angle; below, r, ( and z are ordinary cylindrical coordinates. We use standard notation and known theoretical results described in detail in Ref. [3]. The difference between/~ and its vacuum value Bv is insignificant and can be disregarded. After substitution of Eq. (5) into Eq. (3) we get
2
3R
(7)
explicitly showing a weak dependence of A(I) on the Shafranov shift. It is valid for stellarators without shear that have circular shifted magnetic surfaces. Here qbo = rcb2Bo, flOq= p2b/R, b is the minor radius and /x is the rotational transform. Even for fl close to the equilibrium limit the last term in Eq. (7) cannot be larger than 0.7b/R. In the general case [2] a~b = - 1 -- dsFb- dh ~vI-" pp dV
(8)
where ds is a quantity of the order of b/R describing the dependence of Aqb on the relative shift of magnetic surfaces and fih is another small term due to the helical field. This formula is valid for a currentless plasma in conventional stellarators
V.D. Pustovitov / Fusion Engineering and Design 34-35 (1997) 689-693
with an arbitrary magnetic surface shape, aspect ratio and/z(a). It is surprising that after removing all geometrical restrictions of the model we obtained a result which looks almost as simple as Eq. (2) and is ready for immediate practical use. The integral over the plasma volume in Eq. (8) is the value of interest to be found from the measured Ago. Fortunately, the main dependence of Ago on the plasma shape is contained in this integral. However, the coefficient in front of the integral turned out to be practically independent of the plasma shape. The coefficient is approximately equal to - 1/Fb and all possible deviations from this constant are described by the small terms gs and gh. In present-day experiments fl is still far from the equilibrium limit fl~q. Then ~s 0 if there is a shift of the magnetic axis in a vacuum configuration. In this case 6s as well as csh can be calculated from the given vacuum field. Finally, diamagnetic measurements can be easily interpreted to find fl with high accuracy in currentless stellarators. There is another advantage to diamagnetic measurements: the value of Ago as determined by Eq. (6) does not depend on the shape of the transverse area S±, because only integration over the plasma cross-section gives a non-vanishing contribution. Thus the shape of the diamagnetic coils may be arbitrary. Of course, these coils should lie in a plane perpendicular to the geometrical axis of the device. The parameter AgO is often referred to as the change in the toroidal flux in the plasma. In some cases this is correct, but in general such an interpretation is wrong. By definition, Ago is the change in the magnetic flux through the area encircled by the diamagnetic coil. The plasma boundary divides this area into two parts: S j_ = Sin+ Sext. Correspondingly, the magnetic flux through S : consists of two parts, i.e. the flux goin through the transverse cross-section of the plasma, Sin, and the flux (I)ext of the external field through Se~t:
go = q~n + @~xt
(9)
During the discharge the plasma can move and change its shape. Then Sext and, as a result, (I)ext are also changed. Thus some part of the signal
691
measured by the diamagnetic coil can be attributed to the change in goext, sometimes up to 100%. Indeed, a hot plasma is a perfect conductor and during fast processes magnetic fluxes inside the plasma must be frozen irrespective of changes in ft. However, Ago -~ 0. There is no contradiction: in Ago, which is the difference of two fluxes, Eq. (3), only one characterizes the plasma, while the other is determined by the vacuum field Bv. If the vacuum toroidal field does not vary, Ago is equal to the difference of fluxes through the cross-section of the plasma; see Eqs. (3) and (6). However, this value shows how much the toroidal flux goinin the plasma differs from the flux of the vacuum field Bv through the same cross-section S=. Such an interpretation is valid always for any geometry and history of the discharge. We need only the assumption that the final state is described by the equilibrium Eq. (1). In the calculations above the toroidal magnetic field outside the plasma was assumed to be unchanged, cSBt = 0. Then, if the equilibrium is flux conserving, the flux go~xt~ BtSext can vary only owing to a change in Sin (~S~xt= -~Sin). The plasma size must increase with increasing t , because in this case Ago = Agoext < 0; see Eqs. (2) and (8). This was discussed in Refs. [2,3]. However, a similar discussion in Ref. [4] has been spoilt by an inconsistent analysis in which the contribution due to c~S,×t in ggo~xt= B t ~ S e x t -~ SextaNt was disregarded.
3. Self-field of equilibrium plasma currents outside plasma column The diamagnetic coil embracing the plasma column records the change in the field inside the plasma. The plasma current j creates a magnetic field Bpl outside the plasma also. This field can be calculated directly if its source j is known. However, we can find Bpl as a vacuum field by solving Eq. (lb) outside the plasma boundary Fp. If the boundary conditions are given at Fv, there is no need to know the current j and field B inside I~p, though they actually contribute to determining the shape of Fp and the magnetic field Br o n Fp.
692
V.D. Pustovitov/ Fusion Engineering and Design 34-35 (1997) 689-693
Independence of the external solution from the internal one for given boundary conditions means that two configurations, both with the same Fp and Br but one of them without a magnetic field inside Up, Bin = 0, must have the same magnetic field Bwo in the external region. To provide a jump in the magnetic field at the plasma boundary (Br outside and Bin = 0 inside in the latter case), a surface current must flow on I'p: (10)
ir = - - B r x n
where n is the unit external normal to Fp. In a configuration with such a surface current, B~+ Bext = Bv~c in the external region and B~+ e e x t : 0 in the inner region, where B~ is the field of the surface current (10) and B~xt is the field produced by the currents flowing in external conductors. However, by definition, Bv~c= Bpl + B,xt, so Bpl Bi =
-- Bex t
outisde Fp inside Fp
(11)
The last two equations show that two problems can be solved once the magnetic field on the plasma boundary is known: the plasma-generated magnetic field Bpl outside the plasma can be found and the external magnetic field B~xtwhich is needed for sustaining a given configuration in equilibrium can be calculated. This general result itself is well known, but its implications do not appear to be fully appreciated in some works on magnetic diagnostics, where discussions start from another level. However, a very strong conclusion follows from Eq. (11): the plasma-generated field Bp~ in the vacuum is determined only by the shape of the plasma boundary and the magnetic field Br on this boundary. Thus, by measuring the plasma-generated field Bpl outside the plasma, one can recover only that information about the plasma which is 'hidden' in Fp and Br. If Fp and neighbouring magnetic surfaces in a tokamak, for example, are shifted tori with a circular cross-section (r - R - iX)2+ z 2 = a 2, we get from Eq. (5) ~' n x e~ Br = 2~r 1 -- iX' cos 0
(12)
where 0 is the poloidal angle. There are only two parameters, ~' and A'. By ~' the amplitude of Br is described, which can be measured by a Rogowski coil. Any other measurements of Bpl can give nothing more than A' on the plasma boundary. In other words, only one more value can be found besides the net plasma current J. This fact follows from the most general principles and there is no need to solve equilibrium equations in order to answer the question [5] why pp and f~ cannot be separately measured in a nearcircular tokamak, because N = - (b/R)(~p + Ei/ 2). There is another interesting question on whether it is possible to determine the plasma current or pressure distribution by measuring Bp~ outside the plasma. Eq. (11) indicates that the answer is most probably negative, because information is hidden in Fp. For example, only the net current J can be found in a straight cylinder and only J and A' in a tokamak with a circular plasma cross-section. For non-circular Fp we can refer to the results of Ref. [6]: in four cases the plasma shape and poloidal field at Fp are approximately the same, but the current profiles are strongly different. According to Eqs. (10) and (11), all four configurations must have the same Bpl outside the plasma for the same F v and Br. It is then impossible to distinguish the difference in current (and pressure) profiles by measuring Bpl. " N o " seems to be a natural answer. However, there is a lot of optimism in some publications [7-10]. The example given by Eq. (12) above prompts the simplest way to reveal the real ground of positive answers: instead of profiles we must speak about numbers, which could be found by measuring Bpl. If it is clearly understood that there is a substitution "one number-one profile" in Refs. [8,10], it is logical to ask then what this number is. In the case described by Eq. (12) it is iX' and only three values, 3", tip and ~, can be found from magnetic measurements. Can we then expect high accuracy in evaluating the pressure profile by known fl and the current profile by Y~? The above comments about tokamaks are qualitatively applicable to stellarators also. Often it is said that stellarators are better for magnetic diagnostics. One reason is that they operate without a
V.D. Pustovitov / Fusion Engineering and Design 34-35 (1997) 689-693 net c u r r e n t a n d b o t h Aq) a n d Bpl m u s t give inf o r m a t i o n a b o u t the p l a s m a pressure. I n stellarators the m a g n i t u d e o f Bpl o n ]"p is given b y [3]
~b a 2 p ' ( a )
• oa
(13)
F o r an Y = 3 s t e l l a r a t o r with /~ oca 2 we o b t a i n
B/~ = Cflo. This result allows us to m a k e some guess a b o u t p(a), b e c a u s e two n u m b e r s can be f o u n d , average fl a n d flo, w h i c h is the local value o f fl at the m a g n e t i c axis; b u t only a guess, a n d this is the case o f the best resolution. I f / t = const., Eq. (13) gives B~ = Cfl. H e r e fl is the s a m e as in Eq. (2), so m e a s u r i n g Bpl does n o t give a n y i n f o r m a t i o n a d d i t i o n a l to d i a m a g netic m e a s u r e m e n t s . O r d i n a r y E = 2 s t e l l a r a t o r s are 'in b e t w e e n ' a n d b o t h Bpl a n d AcI) give inform a t i o n t h a t is less t h a n fl a n d flo. Profile e v a l u a t i o n s b y Bpl a n d A ~ are p o o r b e c a u s e these values are integral; see Eqs. (3) a n d (11). H o w e v e r , the integral n a t u r e o f Bpl a n d A(I) helps us to find fl a n d the p l a s m a shape a n d p o s i t i o n with g o o d accuracy. F o r stellarators, recent H e l i o t r o n E results [11] are g o o d examples.
693
References [1] H. Yamada, K. Ida, H. Iguchi, et al., Shafranov shift in a low aspect ratio heliotron/torsatron compact helical system, Nucl. Fus. 32 (1992) 25-32. [2] V.D. Pustovitov, Refined theory of diamagnetic effect in stellarators, J. Plasma Fus. Res. 69 (1993) 34-40. [3] V.D. Pustovitov, Fundamental stellarator MHD theory, J. Plasma Fus. Res. 70 (1994) 943-991. [4] E.D. Andryukhina, K.S. Dyabilin and O.I. Fedyanin, in Proc. IOFAN, Vol. 31, Stellarators, Nauka, Moscow, 1991, pp. 186-192 (in Russian). [5] J.P. Friedberg, et al., Why tip and li cannot be separately measured in a near circular tokamak, Plasma Phys. Controlled Fus. 35 (t993) 1641-1648. [6] L.L. Lao, et al., Separation of tip and li in tokamaks of non-circular cross-section, Nucl. Fus. 25 (1985) 1421- 1436. [7] Yu.K. Kuznetsov, V.N. Pyatov and I.V. Yasin, Possibility of determining equilibrium plasma current and pressure profile in a tokamak from magnetic measurements, Soy. J. Plasma Phys. 13 (1987) 75-80. [8] Yu.V. Gutarev, et al., Method for magnetic measurements of equilibrium plasma parameters in a stellarator, Soy. J. Plasma Phys. 14 (1988) 164-167. [9] V.K. Pashnev and V.V. Nemov, Use of magnetic diagnostics in stellarators, Nucl. Fus. 33 (1993) 435-447. [10] A.B. Kuznetsov, S.V. Shchepetov and D.Yu. Sychugov, Is it possible to extract information on the plasma pressure profile from magnetic measurements in stellarators?, Nucl. Fus. 34 (1994) 185-190. [11] S. Besshou, et al., Detection of the boundary plasma shift in a toroidal helical plasma on Heliotron-E, Nucl. Fus. 35 (1995) 173 182.