Abstract
Magnetized jets are important features of many systems of physical interest. To date, most interest has focused on solar and space physics and astrophysical applications, and hence the unbounded magnetized jet, and its cousin, the unbounded magnetized wake, have received the most attention. This work presents calculations of a bounded, magnetized jet for a laboratory experiments in the Helimak device [K. W. Gentle and H. He, Plasma Sci. Technol.10, 284 (2008)]. The Helimak device has a toroidalmagnetic field with a controlled velocity flow that represents jets in bounded systems. Experimental and theoretical features include three spatial dimensions, the inclusion of resistivity and viscosity, and the presence of noslip walls. The results of the linearized model are computed with a Chebyshev algorithm. The bounding walls stabilize the ideal varicose mode found in unbounded magnetized jets. The ideal sinuous mode persists in the bounded system. A comparison theorem is proved showing that twodimensional modes are more unstable than the corresponding threedimensional modes for any given set of system parameters. This result is a generalization of the hydrodynamic Squires theorem. An energystress theorem indicates that the Maxwell stress is crucial for the growth of the instability. The results of the analysis are consistent with the observed plasma fluctuations with in the limits of using a simple model for the more complex measured jet velocity flow profile. The working gas is singly ionized argon and the jet velocity profile is accurately measured with Doppler shift spectroscopy.
The authors would like to thank T. A. Zang for helpful discussions and acknowledge useful discussion with Professor Ken Gentle on the physics of the Helimak device.
R.B.D. was supported by the Office of Naval Research. W.H., C.C., and J.P. were supported by the U.S. Department of Energy contract DEFG0204ER 54742 and NSF Grant ATM0539099. W.R. was supported by DENSF. The numerical computations were performed on the LCP&FD SGI Origin 3400.
I. INTRODUCTION
II. MAGNETIZED JET REVIEW: THE UNBOUNDED CASE
III. HELIMAK AND ITS RELATION TO THE MAGNETIZED JET
A. Description of the Helimak experiment and its parameters
B. The slab model for the Helimak
C. Rotation to fieldaligned coordinates
IV. LINEAR CALCULATIONS
A. Derivation of linearized equations
B. Numerical method
C. Linear results: Spectrum, unstable eigenfunction, and dispersion relations
D. A magnetohydrodynamic Squires theorem
E. Effects of viscous and resistive dissipation
F. Perturbation energy analysis
V. DISCUSSION
Key Topics
 Magnetic fields
 49.0
 Viscosity
 29.0
 Flow instabilities
 18.0
 Maxwell equations
 18.0
 Reynolds stress modeling
 15.0
Figures
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.
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.
Fluctuation amplitudes of the electric potential and the electron density measured with Langmuir probes in a 10 eV argon Helimak plasma.
Fluctuation amplitudes of the electric potential and the electron density measured with Langmuir probes in a 10 eV argon Helimak plasma.
A schematic drawing of the Helimak illustrating the coordinate system used in this paper.
A schematic drawing of the Helimak illustrating the coordinate system used in this paper.
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 onetenth of the maximum Alfvén speed.
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 onetenth of the maximum Alfvén speed.
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.
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.
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.
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.
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 .
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 .
Dispersion relations for the 2D case, parametrized by the spanwise wavenumber . For these calculations .
Dispersion relations for the 2D case, parametrized by the spanwise wavenumber . For these calculations .
Variation of growth rate with respect to spanwise wavenumber, parametrized by the streamwise wavenumber . For these calculations .
Variation of growth rate with respect to spanwise wavenumber, parametrized by the streamwise wavenumber . For these calculations .
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.
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.
Variation of growth rate with respect to viscous Lundquist number , parametrized by the Lundquist number . For these calculations and . Here .
Variation of growth rate with respect to viscous Lundquist number , parametrized by the Lundquist number . For these calculations and . Here .
Linear stresses for the model Helimak for the unstable mode shown in Fig. 7.The Reynolds and Maxwell stresses dominate the other terms.
Linear stresses for the model Helimak for the unstable mode shown in Fig. 7.The Reynolds and Maxwell stresses dominate the other terms.
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 .
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 .
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 .
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 .
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 .
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 .
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 collocationFourier 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.
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 collocationFourier 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.
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.
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.
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 crossstress 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.
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 crossstress 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
Helimak viscoresistive MHD parameters.
Helimak viscoresistive MHD parameters.
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