Constrained Resonance in Magnetic Self-Organization

The helicity-conserving Taylor relaxation of a plasma in a toroidal chamber to a force-free configuration, which means j=(j‖∕B)B with j‖∕B independent of position, can be generalized to include the external injection of magnetic helicity. When this is done, j‖∕B has resonant values, which can be understood using the eigenmodes of Taylor-relaxed plasmas enclosed by a perfectly conducting toroidal shell. These eigenmodes include a toroidal generalization of those found by Chandrasekhar and Kendall (CK) [Astrophys. J. 126, 457 (1957)] for a spherical chamber, which has no externally produced magnetic flux. It is shown that the CK modes in an axisymmetric torus are of three types: (1) helical modes as well as axisymmetric modes that have (2) and have no (3) net toroidal flux. Yoshida and Giga (YG) [Math. Z. 204, 235 (1990)] published a fourth class of modes: axisymmetric modes that have no net toroidal flux in the chamber due to toroidal flux produced by a net poloidal current in the shell canceling the net toroidal flux from the plasma currents. Jensen and Chu [Phys. Fluids 27, 2881 (1984)], as well as Taylor [Rev. Mod. Phys. 58, 741 (1986)], considered modes in which the vector potential was zero on the axisymmetric toroidal chamber. It is shown that these Jensen-Chu-Taylor modes include only the CK helical modes and the CK axisymmetric modes without net toroidal flux. If the toroidal chamber is perfectly conducting except for a cut that prevents a net poloidal current from flowing, resonances in j‖∕B occur at the eigenvalues of the axisymmetric CK modes. Jensen and Chu studied this type of resonance. Without the cut, so a poloidal current flows to conserve the net toroidal flux, it is shown that j‖∕B resonances occur at the eigenvalues of the CK modes that have no net toroidal flux and at the eigenvalues of the YG modes, which are upshifted from the eigenvalues of the axisymmetric CK modes that carry net toroidal flux.

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“…͑29͒, as we have done earlier, 28 the most mathematically convenient choice is the Yoshida-Giga modes. ͑26͒.…”

Section: Yoshida-giga Modes and Tang-boozer Resonancesmentioning

confidence: 97%

Chandrasekhar-Kendall modes and Taylor relaxation in an axisymmetric torus

Tang

Boozer

2005

Physics of Plasmas

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“…(3), where 0 < < 1 denotes a partially relaxed plasma that has a higher edge k than its core values. This Letter focuses on the simply connected geometry (e.g., spheromak and radio lobes), with the understanding that the results can be generalized to a torus (field reversed pinch and spherical tokamak under coaxial helicity injection) where the constrained resonance [12] appears in place of the unconstrained Jensen-Chu resonance [10] due to toroidal flux conservation. Since k ÿdG=d, one finds that in a simply connected geometry,…”

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confidence: 98%

Driven Resonance in Partially Relaxed Plasmas

Tang

2007

Phys. Rev. Lett.

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A Taylor-relaxed plasma (j kB with k a constant) under external magnetic helicity injection encounters resonances in spatial frequencies of its force-free eigenmodes. Such driven resonance underlies the physics of magnetic self-organization and determines the flux amplification in laboratory helicity injection applications. Here we show that for partially relaxed plasmas where the deviation from the fully relaxed Taylor state, for example, a flux-dependent k, is a function of the normalized flux = a with a the poloidal flux at the magnetic axis, a modified driven resonance persists even if k has an order-unity variation across the flux surfaces.Magnetic relaxation was first suggested by Taylor [1] as the cause of the spontaneous formation of a quiescent reversed field pinch configuration in the laboratory toroidal pinch experiments. An important and robust laboratory application of magnetic relaxation [2] has been to form plasma confining magnetic fields such as those of spheromak [3] and spherical tokamak [4] by external magnetic helicity injection. This is carried out by electrically biasing open magnetic field lines that intercept the electrodes and hence inducing a primarily parallel current flowing along the open field lines with a nominal j k =B inj [5]. The open field line kinks driven by the open field line current [6] set off a helical instability cascade in space [7] that tends to flatten j k =B throughout the plasma [8]. Since the helical fluctuations that facilitate global relaxation usually have a much smaller energy content in comparison with the axisymmetric component of the magnetic field [7,9], it is often useful to examine the toroidally averaged magnetic field alone and speak of a j k =B int inside the separatrix of the mean field, i.e., the field lines of the mean field that do not connect to the electrodes. In the limit of Taylor relaxation [1,2], j k =B inj j k =B int k is a global constant.The reason that the injected magnetic energy and helicity are expected to self-organize onto the device-scale spheromak and spherical tokamak magnetic fields is the result of a resonance phenomenon when k approaches the eigenvalues of the intrinsic Chandrasekhar-Kendall and Yoshida-Giga modes of the discharge chamber [10 -12]. For a given magnetic helicity, the relaxed state of k 2 < k 2 1 , compared with that of k 2 > k 2 1 , always has the lower magnetic energy [10]. For laboratory spheromak applications, k 1 is the eigenvalue of the first axisymmetric Chandrasekhar-Kendall mode that carries a net toroidal flux [13], so the minimum magnetic energy state bounded by k 2 < k 2 1 has a magnetic field on the spatial scale of the entire discharge chamber (1=k 1 ). From an operational perspective, as the injector and plasma currents are ramped up, the initial increase in k j k =B inj because of higher j k , saturates to the limit of k 1 because of a diverging B that is proportional to the plasma current. This is the plasma response to a resonant barrier at k k 1 that a Taylorrelaxed spheromak plasma cannot be ...

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“…(11) and (12) to depend on the eigenmode structure. Although helical CK modes is strongly indicated, it is cautioned that the limited range of h (a factor of ten) in the convergence studies prevents a conclusive proof that lim h!0 Err b n;m u ¼ 0. are plotted as functions of (R, Z) for the first four n = 1 CK modes.…”

Section: Discussionmentioning

confidence: 99%

“…(11) and ( 12), we need to verify the subset that is the eigensolution to the Taylor state. In other words, whether the solutions satisfy Eq.…”

Section: Verify the Helical Ck Modes In The Divergence-free Formulationmentioning

confidence: 99%

“…These force-free eigensolutions are uniquely determined by the chamber geometry, and play an essential role in determining the relaxed states of a driven plasma [4,7], in which the spatial overlap of the CK modes and a vacuum field generated by external current provides the coupling between the external helicity source and the driven plasma [8]. In fact, resonant coupling [9][10][11][12] is the physical mechanism underlying the self-organization of system-scale magnetic fields by magnetic relaxation in the laboratory formation of spherical tokamak [13,14], spheromak [15], and reversed field pinch by helicity injection. It is also thought to be a competing paradigm for the generation and sustainment of large scale magnetic fields in astrophysical radio lobes [16].…”

Section: Introductionmentioning

confidence: 99%

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