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UDC 533.9
PHYSICS
L. S. SOLOV'EV, V. D. SHAFRANOV
ON THE THEORY OF PLASMA EQUILIBRIUM
IN TOROIDAL MAGNETIC TRAPS
(Presented by Academician M. A. Leontovich, 8 XII 1965)
The magnetic differential equation. The problem of plasma equilibrium in a magnetic field having toroidal magnetic surfaces leads to the “magnetic differential equation” \((^{1,2})\)
\[ \mathbf{B}\nabla r=s, \tag{1} \]
where \(\mathbf{B}\) is the magnetic field, \(s\) is known, and \(r\) is the sought single-valued function. The requirement of single-valuedness imposes certain conditions on \(r\). Indeed, integrating (1) over the volume between two magnetic surfaces, we obtain (the Kruskal–Kulsrud condition \((^{1})\))
\[ \int s\,dV=0. \tag{2} \]
If equation (1) is integrated along a field line, it is seen that on “rational surfaces,” where the magnetic field lines are closed, one must have (the Newcomb condition \((^{2})\))
\[ \oint \frac{s\,dl}{B}=0. \tag{3} \]
The meaning of these conditions is easy to clarify if one introduces curvilinear coordinates \(x_1=\Phi'/2\pi,\ x_2=\theta,\ x_3=\zeta\), associated with the magnetic surfaces \(\Phi=\mathrm{const}\), where \(\Phi(r)\) is the longitudinal flux of the magnetic field, and \(\theta\) and \(\zeta\) are angular coordinates on the surface, changing by \(2\pi\) upon a complete circuit of the torus. The coordinates \(\theta\) and \(\zeta\) can be chosen so that the contravariant components of the vector \(\mathbf{B}\) have the form \((^{1-5})\)
\[ B^i=\frac{1}{\sqrt{g}}\{0,\mu,1\}. \tag{4} \]
Here \(g\) is the determinant of the metric tensor \(g_{ik}\); \(\mu=\mu(\Phi)\) is the derivative of the transverse flux \(\chi(\Phi)\) with respect to the longitudinal flux,
\[ \mu(\Phi)=d\chi/d\Phi. \tag{5} \]
Since \(\sqrt{g}s\) is a periodic function of \(\theta\) and \(\zeta\), it can be expanded in a Fourier series
\[ \sqrt{g}s=\operatorname{Re}\sum_{m,n} a_{mn}(\Phi)e^{i(m\theta-n\zeta)}. \tag{6} \]
Taking into account that \(dV=\frac{1}{2\pi}\sqrt{g}\,d\Phi\,d\theta\,d\zeta\), we obtain condition (2) in the form
\[ \int s\,dV=\frac{d\Phi}{2\pi}\operatorname{Re}\sum_{m,n}a_{mn}(\Phi) \int_0^{2\pi}\int_0^{2\pi} e^{i(m\theta-n\zeta)}\,d\theta\,d\zeta =2\pi\,d\Phi\,a_{00}(\Phi)=0. \tag{7} \]
Since the field line on the surface \(\Phi=\mathrm{const}\), according to (4), is determined by the equation \(\theta=\mu\zeta\), condition (3) can be written in the form
\[ \oint \frac{s\,d\zeta}{B^3} =\oint \sqrt{g}s\,d\zeta =\operatorname{Re}\sum_{m,n}a_{mn}(\Phi)\oint e^{i(m\mu-n)\zeta}\,d\zeta =0. \tag{8} \]
Thus, condition (2) means the absence in the expansion (6) of the zero harmonic \(a_{00}=0\). Condition (3), according to (8), means the absence of “resonant” harmonics \(a_{mn}\) on the “rational” surfaces \(\Phi=\Phi_{mn}\), where \(\mu(\Phi_{mn})=n/m\):
\[ a_{mn}(\Phi_{mn})=0. \tag{9} \]
Equation (1) in the chosen coordinates has the form
\[ (\mu \partial/\partial \theta+\partial/\partial \zeta)r=\sqrt{g}\,s. \tag{10} \]
Expanding \(r\) in a series analogous to (6), we obtain for the coefficients of the series \(r_{mn}\)
\[ r_{mn}=-i\,\frac{a_{mn}(\Phi)}{m\mu(\Phi)-n}. \tag{11} \]
When the Newcomb condition is satisfied, the numerator vanishes simultaneously with the denominator, and, consequently, there is a finite periodic solution of equation (1).
It is of interest to clarify how requirements (2), (3) appear in concrete problems. Such problems are, for example, the problem of perturbations of toroidal magnetic surfaces and the problem of equilibrium toroidal plasma configurations. In these problems the Kruskal–Kulsrud condition (2) causes no difficulty. The Newcomb condition (3) at first sight appears to be very restrictive. However, as is shown below, it can be satisfied by renormalizing (appropriately choosing) the unperturbed solution.
Perturbations of magnetic surfaces. The equation for determining toroidal magnetic surfaces \(\psi(\mathbf r)=\mathrm{const}\) has the form
\[ \mathbf B\nabla\psi=0. \tag{12} \]
Let the true magnetic field \(\mathbf B\) differ from the field of the zeroth approximation \(\mathbf B_0\) by a vector \(\mathbf B_1\) of first order of smallness. Setting \(\psi=\psi_0+\psi_1\), where \(\psi_0\) is a known function satisfying the equation \(\mathbf B_0\nabla\psi_0=0\), we obtain in the linear approximation the magnetic differential equation for \(\psi_1\):
\[ \mathbf B_0\nabla\psi_1=-\mathbf B_1\nabla\psi_0. \tag{13} \]
Condition (2) is satisfied by virtue of the equality \(\operatorname{div}\mathbf B_1=0\). In the coordinates introduced above, equation (13) has the form (10), where \(\sqrt{g}s=-2\pi B^1\sqrt{g}\psi_0'(\Phi)\). Expanding \(\tilde B^1\sqrt{g}\) in a Fourier series
\[ \sqrt{g}B^1=\sum_{m,n} b_{mn}e^{i(m\theta-n\zeta)}, \]
we have
\[ \psi_{1mn}=2\pi i\,\frac{b_{mn}(\Phi)\psi_0'(\Phi)}{m\mu(\Phi)-n}. \tag{14} \]
The possibility of satisfying the solvability conditions (9) in the presence of resonant harmonics of the perturbing field \(b_{mn}(\Phi_{mn})\ne0\) is connected with two facts: 1) real perturbations have practically a finite number of resonant harmonics with appreciable amplitude; 2) the initial function \(\psi_0(\Phi)\) allows considerable freedom in its choice. The simplest way of choosing \(\psi_0\) consists in identifying it with some physical function which is, like \(\psi_0\), a “surface quantity” \((^1)\), for example, with the limiting flux \(\Phi\). This function usually increases monotonically from the axis. However, as \(\psi_0\) one may also take an arbitrary function \(\psi_0(\Phi)\), and it can be chosen so that on the resonant surfaces \(\Phi=\Phi_{mn}\) the derivative \(\psi_0'(\Phi)\) vanishes, for example, in the form of an integral with respect to \(\Phi\) of the product of the resonant denominators
\[ \psi_0(\Phi)=\int \prod_{m,n}[m\mu(\Phi)-n]\,d\Phi. \tag{15} \]
In this case the solution has the form
\[ \psi=\int \prod_{m,n}[m\mu(\Phi)-n]\,d\Phi +2\pi \operatorname{Re}\sum_{k,l} i b_{kl}(\Phi) \prod_{\substack{m\ne k\\ n\ne l}}[m\mu(\Phi)-n]\,e^{i(k\theta-l\zeta)} . \tag{16} \]
The solution obtained shows that the presence of resonant harmonics leads to the destruction of magnetic surfaces when \(\mu(\Phi)=\mathrm{const}\). However, if \(\mu'(\Phi_{mn})\ne 0\), then resonant perturbations lead only to the splitting of resonant magnetic surfaces and to the formation of an island structure \((^6,^7)\).
The form of the perturbed magnetic surfaces in the neighborhood of an individual resonant surface \(\Phi=\Phi_{MN}\) can be obtained by expanding (16) in \(\Phi-\Phi_{MN}\):
\[ \psi=\left\{ m\mu'(\Phi_{MN})\,\frac{(\Phi-\Phi_{MN})^2}{2} +\operatorname{Re}\,2\pi i b_{MN}e^{i(M\theta-N\zeta)} \right\} \prod_{\substack{m\ne M\\ n\ne N}}[m\mu(\Phi)-n]. \tag{17} \]
It is clear from this formula that only one (resonant) harmonic of the perturbing field affects the perturbation of magnetic surfaces near resonance.
Let us note that if all the other harmonics of the perturbation are discarded, then the solution obtained, for the specified particular choice of the renormalization (15), is exact in the linear approximation. It remains exact also in the more general case of perturbations having the same pitch
\(m/n=\mathrm{const}\). If \(\sqrt{g}B^1=\sum_N b_N(\Phi)e^{iN\Theta}\), where \(\Theta=m\theta-n\zeta\), then the function
\[ \psi=\int_{0}^{\Phi}[m\mu(\Phi)-n]\,d\Phi +\operatorname{Re}\sum_N 2\pi i\,\frac{b_N(\Phi)}{N}\,e^{iN\Theta} \tag{18} \]
gives an exact solution of the equation \(\mathbf{B}\nabla\psi=0\), as is easily verified by using the equality \(\operatorname{div}\mathbf{B}_1=0\).
The solution (16) was obtained above in a special coordinate system. However, the renormalization prescription (15) has an invariant form and can be used in solving problems in any particular coordinate system.
On the equilibrium of a plasma with finite conductivity. The problem of the theory of plasma equilibrium in toroidal magnetic traps is to determine the currents in the plasma and the perturbations of the magnetic surfaces associated with them. This problem can be solved by the perturbation method described by Spitzer \((^8)\) and by Kruskal and Kulsrud \((^1)\).
In the zeroth approximation there are vacuum toroidal magnetic surfaces \(\Phi_0=\mathrm{const}\). The plasma pressure in the first approximation is assumed known, \(p=p(\Phi_0)\). Decomposing the current-density vector into components transverse and longitudinal with respect to the vacuum magnetic field, \(\mathbf{j}=\mathbf{j}_{\perp}+h\mathbf{B}_0\), where \(\mathbf{j}_{\perp}=c[\mathbf{B}_0\nabla p]/B_0^2\), we obtain from the condition \(\operatorname{div}\mathbf{j}=0\) the magnetic differential equation for \(h\) \((^1)\):
\[ \mathbf{B}_0\nabla h=\mathbf{j}_{\perp}\nabla B_0^2/B_0^2 =c[\mathbf{B}_0\nabla p_0]\nabla B_0^2/B_0^4 . \tag{19} \]
The projection of Ohm’s law onto the magnetic field \(\mathbf{B}_0\) also leads to an analogous equation for the scalar potential of the electric field (which supports the longitudinal current that removes the charge separation caused by the toroidal effect) \((^1)\):
\[ \mathbf{B}_0\nabla\varphi=-\,\frac{(\mathbf{j}\mathbf{B}_0)}{\sigma_{\parallel}} =-\,\frac{h}{\sigma_{\parallel}}\,B_0^2 . \tag{20} \]
Here \(\sigma_{\parallel}\) is the longitudinal conductivity, which can usually be regarded as a “surface function,” \(\sigma_{\parallel}=\sigma_{\parallel}(\Phi_0)\).
The right-hand side of equation (19) satisfies the Kruskal–Kulsrud condition (2) automatically, since it is \(\operatorname{div} j_\perp\). In the absence of a vortical electric field, the right-hand side of equation (20) also satisfies condition (2) in the absence of the total longitudinal current (1). As for Newcomb’s conditions (3), in the general case they cannot be satisfied automatically for an arbitrary pressure distribution taken as the zero approximation. If the right-hand sides of these equations have harmonics resonant with respect to the vacuum magnetic surfaces, then the solvability condition for equation (20) requires that the function \(h\) have zeros on the surfaces \(\Phi_0 = \Phi_{0mn}\), and consequently \(\nabla p\) must have second-order zeros on the same surfaces:
\[ p(\Phi_0)=\int p_0(\Phi_0)\prod_{m,n}[m\mu(\Phi_0)-n]^2\,d\Phi_0; \tag{21} \]
\(p_0(\Phi_0)\) is an analytic function having no singularities at the points \(m\mu(\Phi_0)-n=0\).
It may therefore be concluded that the surfaces \(p=\mathrm{const}\), in the presence of resonant perturbations, must have a fibrous structure. To find the effects associated with the presence of “fibers” caused by external perturbations, it is necessary to solve the complete system of equations describing the behavior of the plasma, including the generalized Ohm’s law.
Summary. The solution of problems on equilibrium toroidal magnetic configurations by the perturbation method leads to an equation of type (10). The formal solution (11) of this equation contains “small denominators,” analogous to the solutions of equations for perturbations in problems of classical mechanics \((^{9,10})\). In the case under consideration, the difficulties associated with the presence of small denominators can be bypassed by renormalizing (appropriately choosing) the unperturbed solution, on which the form of the right-hand side of equation (10) depends. A concrete prescription for such a renormalization is proposed in the present work.
Received
29 X 1965
CITED LITERATURE
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