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Evolution of the bounded magnetized jet and comparison with Helimak experiments
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10.1063/1.3166598
/content/aip/journal/pop/16/7/10.1063/1.3166598
http://aip.metastore.ingenta.com/content/aip/journal/pop/16/7/10.1063/1.3166598

Figures

Image of FIG. 1.
FIG. 1.

Vertical flow profiles of a Helimak plasma at bias voltage values (a) , (b) , and (c) . The error bars correspond to the statistical uncertainty in spectroscopic Doppler shift measurements of spectral atomic emission lines from a working gas of singly ionized argon.

Image of FIG. 2.
FIG. 2.

Fluctuation amplitudes of the electric potential and the electron density measured with Langmuir probes in a 10 eV argon Helimak plasma.

Image of FIG. 3.
FIG. 3.

A schematic drawing of the Helimak illustrating the coordinate system used in this paper.

Image of FIG. 4.
FIG. 4.

Plot of the basic flow and magnetic field profiles for the calculations in this paper in the laboratory frame of reference. In the laboratory frame of reference corresponds to the radial direction; corresponds to the toroidal direction; and corresponds to the axial direction (Since most of our research in performed in a rotated frame of reference, we use primes to denote the laboratory frame of reference). Note that the maximum flow speed is one-tenth of the maximum Alfvén speed.

Image of FIG. 5.
FIG. 5.

Plot of the basic flow and magnetic field profiles for the calculations in this paper in the rotated frame of reference. The fields shown in Fig. 4 are rotated through an angle to move to the frame of reference. This is the frame of reference used for the calculations described in this paper.

Image of FIG. 6.
FIG. 6.

The complete eigenmode spectrum for a typical axisymmetric vertical mode for the reference flow profile given in Eqs. (5)–(8). Equations (12) and (13) are solved to obtain this solution.

Image of FIG. 7.
FIG. 7.

The unstable eigenfunction for the bounded magnetized jet . This mode resembles the ideal sinuous mode found in the unbounded magnetized jet. For Helimak parameters the Alfvén time is 0.6 ms so the eigenmode has an angular frequency of and a growth rate of . It propagates perpendicular to the helical magnetic field at with . This is well into the resistive MHD regime with .

Image of FIG. 8.
FIG. 8.

Dispersion relations for the 2D case, parametrized by the spanwise wavenumber . For these calculations .

Image of FIG. 9.
FIG. 9.

Variation of growth rate with respect to spanwise wavenumber, parametrized by the streamwise wavenumber . For these calculations .

Image of FIG. 10.
FIG. 10.

Variation of growth rate with respect to Lundquist number , parametrized by the viscous Lundquist number . For these calculations and . Here corresponding to for the parameters in Table I.

Image of FIG. 11.
FIG. 11.

Variation of growth rate with respect to viscous Lundquist number , parametrized by the Lundquist number . For these calculations and . Here .

Image of FIG. 12.
FIG. 12.

Linear stresses for the model Helimak for the unstable mode shown in Fig. 7.The Reynolds and Maxwell stresses dominate the other terms.

Image of FIG. 13.
FIG. 13.

Variation of perturbation energy balance terms with respect to streamwise wavenumber. Here the -axis is the growth rate in units of and, , , , , and are the normalized values of the fluctuation energy contributions due to the Reynolds stress, Maxwell stress, cross stress, viscous dissipation, and Ohmic dissipation, respectively. For these calculations and .

Image of FIG. 14.
FIG. 14.

Variation of perturbation energy balance terms with respect to viscous Lundquist number . Here is the growth rate and, , , , , and represent normalized versions of the perturbed energy contributions due to the Reynolds stress, Maxwell stress, cross stress, viscous dissipation, and Ohmic dissipation, respectively. For these calculations , and .

Image of FIG. 15.
FIG. 15.

Variation of perturbation energy balance terms with respect to Lundquist number . Here is the growth rate and , , , , and represent normalized versions of the perturbed energy contributions due to the Reynolds stress, Maxwell stress, cross stress, viscous dissipation, and Ohmic dissipation, respectively. For these calculations , and .

Image of FIG. 16.
FIG. 16.

Fluctuating kinetic energy and fluctuating magnetic energy for a simulation with the same parameters as those used in Fig. 7. The results shown here are computed using a 3D, nonlinear Chebyshev collocation-Fourier pseudospectral code that solves Eqs. (9) and (10) in a channel geometry. An exponential phase of growth is seen to occur in both and after about 200 characteristic times.

Image of FIG. 17.
FIG. 17.

Fluctuating kinetic energy growth rate as a function of time computed from the result shown in Fig. 16. This case uses the same parameters as are used in Fig. 7. Note that the growth rate computed from the linearized equations [Eqs. (12) and (13)] is 0.008 52.

Image of FIG. 18.
FIG. 18.

Perturbation energy balance stress components as functions of time as given by Eq. (37). This case uses the same parameters as are used in Fig. 7. Here is the Reynolds stress term, is the Maxwell stress term, is the cross-stress term, is the viscous dissipation term, and is the Ohmic dissipation term. The growth rate determined from addition of these components is . For comparison, direct computation from Eq. (37) gives the results: , , , , and . Adding these up, we compute that . Note as well that the growth rate computed from the linearized equations [Eqs. (12) and (13)] is 0.008 52.

Tables

Generic image for table
Table I.

Helimak viscoresistive MHD parameters.

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/content/aip/journal/pop/16/7/10.1063/1.3166598
2009-07-15
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Evolution of the bounded magnetized jet and comparison with Helimak experiments
http://aip.metastore.ingenta.com/content/aip/journal/pop/16/7/10.1063/1.3166598
10.1063/1.3166598
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