Abstract
Recent work investigating the interaction of magnetic islands with microturbulence has uncovered the striking observation of large scale vortexmodes forming within the island structure [W. A. Hornsby et al., Phys. Plasmas17, 092301 (2010)]. These electrostaticvortices are found to be the size of the island and are oscillatory. It is this oscillatory behaviour and the presence of turbulence that leads us to believe that the dynamics are related to the geodesic acoustic mode (GAM), and it is this link that is investigated in this paper. Here, we derive an equation for the GAM in the MHD limit, in the presence of a magnetic island modified threedimensional axisymmetric geometry. The eigenvalues and eigenfunctions are calculated numerically and then utilised to analyse the dynamics of oscillatory largescale electrostatic potential structures seen in both linear and nonlinear gyrokinetic simulations.
This work used resources on the HECToR supercomputer that were provided by the Engineering and Physical Sciences Research Council [Grant No. EP/H002081/1].
I. INTRODUCTION
II. MATHEMATICAL MODEL
III. EIGENVALUE CALCULATION
IV. EIGENFUNCTION ANALYSIS
V. GYROKINETIC FRAMEWORK
VI. RESULTS AND COMPARISON
VII. CONCLUSIONS
Key Topics
 Magnetic islands
 50.0
 Rotating flows
 16.0
 Eigenvalues
 15.0
 Geodesic acoustic mode
 14.0
 Magnetic fields
 14.0
Figures
Time trace of the amplitude of the smallest radial wavevector (sideband) as a function of time for an island with poloidal wavevector, . There are two frequencies present, the fast, quickly damped geodesic acoustic mode and the slower, island geodesic mode, which is very slowly damped and of larger amplitude. The amplitudes of the modes have been adjusted so that a direct comparison can be made. Without the presence of turbulence the mode is initialised. When turbulence is present, the vortex mode is generated.
Time trace of the amplitude of the smallest radial wavevector (sideband) as a function of time for an island with poloidal wavevector, . There are two frequencies present, the fast, quickly damped geodesic acoustic mode and the slower, island geodesic mode, which is very slowly damped and of larger amplitude. The amplitudes of the modes have been adjusted so that a direct comparison can be made. Without the presence of turbulence the mode is initialised. When turbulence is present, the vortex mode is generated.
(Color online) Normalized electrostatic potential () in the plane perpendicular to the magnetic field (outboard midplane). Black lines represent the perturbed flux surfaces calculated from the total parallel vector potential. The presence of the island embedded in the turbulence not only generates flows around the island structure but also large scale electrostatic potential structures within the island which fluctuate in amplitude and sign. The top panel shows a vortex with a positive sign, while the lower panel shows that the vortex has flipped sign at a point later in the simulation.
(Color online) Normalized electrostatic potential () in the plane perpendicular to the magnetic field (outboard midplane). Black lines represent the perturbed flux surfaces calculated from the total parallel vector potential. The presence of the island embedded in the turbulence not only generates flows around the island structure but also large scale electrostatic potential structures within the island which fluctuate in amplitude and sign. The top panel shows a vortex with a positive sign, while the lower panel shows that the vortex has flipped sign at a point later in the simulation.
The nontrivial density eigenfunctions in the direction of the helical angle, at . Shown in inlay is the eigenfunction in the poloidal angle. Plotted here is one half of the island eigenfunction (i.e., positive or negative radial direction). The curves are numbered according to increasing eigenvalue.
The nontrivial density eigenfunctions in the direction of the helical angle, at . Shown in inlay is the eigenfunction in the poloidal angle. Plotted here is one half of the island eigenfunction (i.e., positive or negative radial direction). The curves are numbered according to increasing eigenvalue.
Cartoon representing two configurations of density perturbation around magnetic flux surfaces in a magnetic island. (left) reflected, symmetric solution between inner and outer island regions, producing an updown density asymmetry and a degree of compression, (right) zero compression perturbation where the outer solution is simply a copy of the inner solution, producing a radially asymmetric density perturbation.
Cartoon representing two configurations of density perturbation around magnetic flux surfaces in a magnetic island. (left) reflected, symmetric solution between inner and outer island regions, producing an updown density asymmetry and a degree of compression, (right) zero compression perturbation where the outer solution is simply a copy of the inner solution, producing a radially asymmetric density perturbation.
(Top) The square of the oscillation frequency for individual code runs where the value of , and hence the toroidal size of the island, is varied, plotted against the square of the wavevector. The dashed line represents a linear fit. This shows an exactly linear dependence. (Bottom) The calculated ratio of the eigenvalue and as a function of the perturbed flux label from the Opoint to the separatrix, . The agreement between the calculated gradient from the (top) figure (represented by the dashed line, here ) compares well with the value at approximately the island separatrix, . The code was run with 50 points in both the and directions.
(Top) The square of the oscillation frequency for individual code runs where the value of , and hence the toroidal size of the island, is varied, plotted against the square of the wavevector. The dashed line represents a linear fit. This shows an exactly linear dependence. (Bottom) The calculated ratio of the eigenvalue and as a function of the perturbed flux label from the Opoint to the separatrix, . The agreement between the calculated gradient from the (top) figure (represented by the dashed line, here ) compares well with the value at approximately the island separatrix, . The code was run with 50 points in both the and directions.
(Color online) Normalized electrostatic potential () in the plane perpendicular to the magnetic field (outboard midplane). Black lines represent the perturbed flux surfaces calculated from the total parallel vector potential. The top panel shows a vortex with a positive sign, while the lower panel shows that the vortex has flipped sign at a point later in the simulation. Data from gyrokinetic simulation without turbulence and initialised with a density perturbation which produces an electrostatic vortex structure similar to those seen in turbulent gyrokinetic simulations (see Fig. 2.)
(Color online) Normalized electrostatic potential () in the plane perpendicular to the magnetic field (outboard midplane). Black lines represent the perturbed flux surfaces calculated from the total parallel vector potential. The top panel shows a vortex with a positive sign, while the lower panel shows that the vortex has flipped sign at a point later in the simulation. Data from gyrokinetic simulation without turbulence and initialised with a density perturbation which produces an electrostatic vortex structure similar to those seen in turbulent gyrokinetic simulations (see Fig. 2.)
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