Resistive Skin Depth Research Articles

A theory of contact resistance under AC conditions

Electrical contact is an important issue to high power microwave sources, pulsed power systems, field emitters, thin film devices and integrated circuits, interconnects, etc. Contact resistance and the enhanced ohmic heating that results have been treated mostly under steady state (DC) condition. In this paper, we consider the AC contact resistance for a simple geometry, namely, that of two semi-infinite slab conductors of different thicknesses joined at z = 0, with current flowing in the z-direction. The conductivity of the two planar slabs may assume different values. We propose a procedure to accurately calculate the normalized contact resistance under the assumption σ≫ωϵ, where ω is the frequency, σ is the electrical conductivity, and ϵ is the dielectric constant of the material in either channel. We found that in the low frequency limit, the normalized AC contact resistance reduces to the DC case, which was solved exactly by Zhang and Lau. At very high frequency, we found that the normalized contact resistance is proportional to ω, in which case the resistive skin depth becomes the effective channel width, and the physical origin of the contact resistance is identified. The transition between the high and low frequency limits was explored, where, in some cases, the normalized contact resistance may become negative, meaning that the total resistance is less than the total bulk resistance expected from the two current channels. In other cases, the numerical data suggest that the normalized contact resistance is proportional to ω in the transition region. Other issues are addressed.

Validation of single-fluid and two-fluid magnetohydrodynamic models of the helicity injected torus spheromak experiment with the NIMROD code

We present a comparison study of 3-D pressureless resistive MHD (rMHD) and 3-D presureless two-fluid MHD models of the Helicity Injected Torus with Steady Inductive helicity injection (HIT-SI). HIT-SI is a current drive experiment that uses two geometrically asymmetric helicity injectors to generate and sustain toroidal plasmas. The comparable size of the collisionless ion skin depth di to the resistive skin depth predicates the importance of the Hall term for HIT-SI. The simulations are run with NIMROD, an initial-value, 3-D extended MHD code. The modeled plasma density and temperature are assumed uniform and constant. The helicity injectors are modeled as oscillating normal magnetic and parallel electric field boundary conditions. The simulations use parameters that closely match those of the experiment. The simulation output is compared to the formation time, plasma current, and internal and surface magnetic fields. Results of the study indicate 2fl-MHD shows quantitative agreement with the experiment while rMHD only captures the qualitative features. The validity of each model is assessed based on how accurately it reproduces the global quantities as well as the temporal and spatial dependence of the measured magnetic fields. 2fl-MHD produces the current amplification ItorIinj and formation time τf demonstrated by HIT-SI with similar internal magnetic fields. rMHD underestimates ItorIinj and exhibits much a longer τf. Biorthogonal decomposition (BD), a powerful mathematical tool for reducing large data sets, is employed to quantify how well the simulations reproduce the measured surface magnetic fields without resorting to a probe-by-probe comparison. BD shows that 2fl-MHD captures the dominant surface magnetic structures and the temporal behavior of these features better than rMHD.

Parallel resistivity and ohmic heating of laboratory dipole plasmas

The parallel resistivity is calculated in the long-mean-free-path regime for the dipole plasma geometry; this is shown to be a neoclassical transport problem in the limit of a small number of circulating electrons. In this regime, the resistivity is substantially higher than the Spitzer resistivity due to the magnetic trapping of a majority of the electrons. This suggests that heating the outer flux surfaces of the plasma with low-frequency parallel electric fields can be substantially more efficient than might be naively estimated. Such a skin-current heating scheme is analyzed by deriving an equation for diffusion of skin currents into the plasma, from which quantities such as the resistive skin-depth, lumped-circuit impedance, and power deposited in the plasma can be estimated. Numerical estimates indicate that this may be a simple and efficient way to couple power into experiments in this geometry.

Hall magnetohydrodynamic reconnection in the plasmoid unstable regime

A set of reduced Hall magnetohydrodynamic (MHD) equations are used to evaluate the stability of large aspect ratio current sheets to the formation of plasmoids (secondary islands). Reconnection is driven by resistivity in this analysis, which occurs at the resistive skin depth dη≡SL-1/2LνA/γ, where SL is the Lundquist number, L, the length of the current sheet, νA, the Alfvén speed, and γ, the growth rate. Modifications to a recent resistive MHD analysis [N. F. Loureiro et al., Phys. Plasmas 14, 100703 (2007)] arise when collisions are sufficiently weak that dη is shorter than the ion skin depth di ≡ c/ωpi. Secondary islands grow faster in this Hall MHD regime: the maximum growth rate scales as (di/L)6/13SL7/13νA/L and the number of plasmoids as (di/L)1/13SL11/26, compared to SL1/4νA/L and S3/8, respectively, in resistive MHD.

Reconnection in semicollisional, low-β plasmas

Reconnection of semicollisional, low-β plasmas is studied numerically for two model problems using a two-field description of the plasma including electron pressure effects (and hence kinetic Alfvén-wave dynamics). The tearing unstable Harris sheet, with the global parameters of the Geospace Environment Modeling-challenge case, shows a linear growth of the peak reconnection rate with the drift parameter ρs when this scale is significantly larger than the resistive skin depth, and the island is smaller than the Harris sheet current layer width. As exemplary for a driven, rather than a spontaneous reconnection situation we study as second model system two coalescing islands, starting from a nonequilibrium situation. The peak reconnection rate again increases initially linearly with ρs but saturates and becomes ρs independent for larger values. In this saturated regime, no flux pileup occurs, and the reconnection is limited by the rate of approach of the two coalescing islands. The qualitative differences between spontaneous and driven reconnection cases and their scaling behavior are best understood by considering the reconnection rate as a triple product of outflow Mach number, outflow to inflow channel width ratio, and magnetic energy density at a height ρs above the X point.

The Ohmic heating of particulates in a lossless medium

This paper provides a general theory on the Ohmic dissipation of electromagnetic energy by a spherical particulate that is embedded in a lossless medium. The particulate may possess an arbitrary electrical conductivity, and both the medium and the particulate may assume general values of permittivity and permeability. Under the assumption that the wavelength of the electromagnetic field in the medium is large compared with the particulate size, we provide an accurate account of the degree of Ohmic heating by the radio frequency (rf) electric field and by the rf magnetic field of the electromagnetic field. It is found that, in general, heating by the rf magnetic field is dominant whenever δ<a, where δ is the resistive skin depth and a is the radius of the particulate. Analytic scaling laws in the various regimes are derived, from the static case to very high frequency, and for ratios of δ∕a ranging from zero to infinity. The calculation is extended to a transient electromagnetic pulse. Also constructed is the loss tangent of the medium, resulting from a distribution of particulates.

Heating of a particulate by radio-frequency electric and magnetic fields

With the use of a highly symmetrical model, the heating of a spherical particulate by a predominantly radio-frequency electric field and by a predominantly rf magnetic field is solved exactly using the Maxwell equations. It is found that, in general, heating by the rf magnetic field is dominant whenever δ<a, where δ is the resistive skin depth and a is the radius of the particulate, which may either be nonmagnetic or magnetic. The known analytic scaling laws in the various regimes are recovered, from the static case to very high frequency, subject to λ⪢a, where λ is the free space wavelength of the rf field. The analysis may form a theoretical basis in the heating phenomenology of particulates.

Microwave absorption on a thin film

With the use of a simple model, it is shown that a thin film of contaminant on a microwave window may absorb up to 50% of the incident power, even if the film thickness is only a small fraction of its resistive skin depth. This unexpectedly large amount of absorption is conjectured to have played a significant role in window failure. The temperature rise in a thin film is estimated.

Low mass recyclable transmission lines for Z-pinch driven inertial fusion

Recyclable transmission lines (RTLs) are being studied as a means to repetitively drive Z pinches. Minimizing the mass of the RTL should also minimize the reprocessing costs. Low mass RTLs could also help reduce the cost of a single shot facility such as the proposed X-1 accelerator and make Z-pinch driven nuclear space propulsion feasible. Calculations are presented to determine the minimum electrode mass to provide sufficient inertia against the magnetic pressure produced by the large currents needed to drive the Z pinches. The results indicate an electrode thickness which is much smaller than the initial resistive skin depth. This suggests that the minimum electrode thickness may be not be solely determined by inertial effects, but also by the ability of the electrode to efficiently carry the current. A series of experiments have been performed to determine the ability of the electrodes to carry current as a function of the electrode thickness. The results indicate that electrodes much thinner than the initial resistive skin depth can efficiently carry large currents presumably due to the formation of a highly conducting plasma. This result implies that a transmission line with only a few tens of kilograms of material can carry the large Z-pinch currents needed for inertial fusion.

Small scale magnetic flux-averaged magnetohydrodynamics

By relaxing exact magnetic flux conservation below a scale λ a system of flux-averaged magnetohydrodynamic equations are derived from Hamilton’s principle with modified constraints. An energy principle can be derived from the linearized averaged system because the total system energy is conserved. This energy principle is employed to treat the resistive tearing instability and the exact growth rate is recovered when λ is identified with the resistive skin depth. A necessary and sufficient stability criteria of the tearing instability with line tying at the ends for solar coronal loops is also obtained. The method is extended to both spatial and temporal averaging in Hamilton’s principle. The resulting system of equations not only allows flux reconnection but introduces irreversibility for appropriate choice of the averaging function. Except for boundary contributions which are modified by the time averaging process total energy and momentum are conserved over times much longer than the averaging time τ but not for less than τ. These modified boundary contributions correspond to the existence, also, of damped waves and shock waves in this theory. Time and space averaging is applied to electron magnetohydrodynamics and in one-dimensional geometry predicts solitons and shocks in different limits.

Use of diffusing inductive fields of a relativistic beam-plasma system to determine plasma conductivity

As an intense relativistic electron beam passes through neutral gas it produces a plasma channel. A technique to measure the conductivity profile of this plasma is discussed. The analysis shows that diagnosing the radial and temporal variation of inductive field strengths may be used to determine both the plasma conductivity and channel radius. The essence of the method is to measure magnetic diffusion time as a function of radial position, which fixes the resistive skin depth.

Plasma fluid equations in the high collision-frequency limit

The nature of the fluid approximation for plasmas in the high collision-frequency limit is examined in an effort to understand some of the limits of validity of the commonly used ideal magnetohydrodynamic equations. The plane wave dispersion relation for a homogeneous medium is evaluated. The dependence of the damping on the various dissipative terms is examined for each type of wave predicted by the dispersion relation. The effects of these terms are then compared, using the standard values for transport quantities with collision frequency larger than gyro-frequency. When the total dissipation is considered as a function of collision frequency, it is found that there is a minimum in the region where the collision and gyro-frequencies are about equal. This is the régime for which ideal magnetohydrodynamics is a good approximation. As collision frequency is increased, dissipation reaches a maximum when the wave is in resonance with the resistive skin depth, and then decreases into a régime where ordinary sound and light waves propagate. Lower collision frequencies, where thermal conduction and equilibration are important, are considered.