CN117396985A

CN117396985A - Electrode configuration for extended plasma confinement

Info

Publication number: CN117396985A
Application number: CN202280038132.6A
Authority: CN
Inventors: E·迈耶; B·A·纳尔逊; U·舒姆拉克
Original assignee: Zap Energy Co
Current assignee: Zap Energy Co
Priority date: 2021-05-28
Filing date: 2022-05-27
Publication date: 2024-01-12
Also published as: CN117441215A

Abstract

Methods and systems for plasma confinement with various electrode and valve configurations are provided. In one example, an apparatus includes: a first electrode positioned to define an outer boundary of an acceleration volume; a second electrode coaxially arranged with respect to the first electrode and positioned to define an inner boundary of the acceleration volume; at least one power supply to drive a current along a Z pinch plasma column between the first electrode and the second electrode; and a set of valves that provide gas to the acceleration volume to fuel the Z pinch plasma column, wherein electrons of the current flow in a first direction from the second electrode to the first electrode. In an additional or alternative example, a shaping portion is conductively connected to the second electrode to cause gas breakdown of the gas in the presence of the gas to produce a shear flow rate profile.

Description

Electrode configuration for extended plasma confinement

Cross reference to related applications

The present application claims priority from each of U.S. provisional application No. 63/194,866, titled "APPARATUS AND METHOD FOR EXTENDED PLASMA CONFINEMENT", filed 5/28 a year 2021, and U.S. provisional application No. 63/194,877, titled "ELECTRODE CONFIGURATION FOR EXTENDED PLASMA CONFINEMENT", filed 5/28 a year 2021. The entire contents of each of the above applications are incorporated herein by reference for all purposes.

Statement regarding federally sponsored research or development

The present invention has been completed, at least in part, with government support under grant numbers DE-AR001010 and DE-AR001260 awarded by the United states department of energy. The government has certain rights in this invention.

Background

Unless otherwise indicated herein, the statements disclosed in this section are not to be considered as prior art to the claims in this application and are not admitted to be prior art by inclusion in this section.

Nuclear fusion is the process by which two nuclei combine. If the atomic number is less than 26[ i.e., the atomic number is lower than iron (Fe)]The two nuclei of the element of (a) merge and then release energy. The release of energy is due to a slight mass difference between the reactants and the products of the fusion reaction (e.g., in a high Wen Jubian plasma reactor), as expressed by the expression e=mc ² Controlled by the controller.

Nuclear fusion is expected to provide unlimited energy and more manageable waste products that are more efficient than some existing energy sources.

In the case where the plasma reaction continues for a long time, controlled nuclear fusion in fusion plasma may be hindered by rapidly growing plasma instabilities. By studying different plasma confinement methods, a viable approach to such controlled nuclear fusion (hereinafter also referred to as "controlled fusion" or simply "fusion" as a noun or adjective indicating nuclear fusion related features and/or properties) continues to be sought. These methods have distinct advantages at different levels of scientific maturity.

Drawings

The foregoing and other embodiments, features and aspects of the invention are considered in more detail in connection with the following description of the embodiments illustrated in the accompanying drawings, in which:

FIG. 1 illustrates modeling results of Z pinching without shear velocity flow in accordance with at least one embodiment;

FIG. 2 illustrates modeling results of shear velocity flow stabilization Z-pinch in accordance with at least one embodiment;

FIG. 3 illustrates modeling results of shear velocity flow stabilization Z-pinch in accordance with at least one embodiment;

FIG. 4 illustrates modeling results of shear velocity flow stabilization Z-pinch in accordance with at least one embodiment;

FIG. 5 illustrates the result of integrating Z-pinch quality in accordance with at least one embodiment;

FIG. 6A illustrates an isometric view of an apparatus to generate and maintain azimuthally symmetric shear ion velocity streams in accordance with at least one embodiment;

FIG. 6B shows a cross-sectional view of the device shown in FIG. 6A;

FIG. 7 illustrates a shaped portion in accordance with at least one embodiment;

FIG. 8 schematically illustrates a process of initiating and driving azimuthally symmetric shear flow to stabilize a Z pinch discharge in accordance with at least one embodiment;

FIG. 9 illustrates a schematic diagram showing an apparatus for generating and maintaining azimuthally symmetric shear ion velocity flows in accordance with at least one embodiment;

FIGS. 10A through 10F schematically illustrate a process of initiating and driving azimuthally symmetric shear flows to stabilize Z pinch discharge of different anode/cathode configurations in accordance with at least one embodiment;

FIG. 11 illustrates a normalized radial profile of magnetic field density and temperature at present NetTet equilibrium (Bennett equilibrium) in accordance with at least one embodiment;

FIG. 12A illustrates an example time trace in accordance with at least one embodiment;

FIG. 12B illustrates an example time trace of integrated radial ion kinetic energy normalized by initial magnetic energy in accordance with at least one embodiment;

FIG. 13A illustrates an ideal five-moment dual fluid (5M 2F) result in accordance with at least one embodiment;

FIG. 13B illustrates an example time trace of integrated radial ion kinetic energy normalized by initial magnetic energy in accordance with at least one embodiment;

FIG. 14 illustrates ideal 5M2F results with several disturbance wave numbers in accordance with at least one embodiment;

FIG. 15 illustrates an ideal 5M2F mode structure in accordance with at least one embodiment;

FIG. 16A illustrates an ideal 5M2F result with linear flow stability in accordance with at least one embodiment;

FIG. 16B illustrates ideal 5M2F results with parabolic shear flow stability in accordance with at least one embodiment;

FIG. 17 illustrates a pattern structure in ideal 5M2F modeling with shear flow in accordance with at least one embodiment;

18A-18D illustrate pattern growth behavior in a 5M2F model with and without an initial phase shift in the perturbation, in accordance with at least one embodiment;

FIG. 19 illustrates momentum diffusivity in a FuZE-like equilibrium in accordance with at least one embodiment;

FIG. 20 illustrates corrected and uncorrected ion thermal diffusivity in accordance with at least one embodiment;

FIG. 21 illustrates an instability growth rate in accordance with at least one embodiment;

FIG. 22 illustrates an instability growth rate in accordance with at least one embodiment;

FIG. 23 illustrates a non-linear simulated ion density profile from FuZE-like equilibrium in accordance with at least one embodiment; and is also provided with

Fig. 24 illustrates normalized ion inventory and thermal energy in accordance with at least one embodiment.

Detailed Description

Embodiments of the present disclosure may be better understood by reference to the following description, which should be read in connection with the accompanying drawings of certain exemplary embodiments. This description of illustrated examples set forth below to enable a person to make and use embodiments of the invention is not intended to limit the invention but is used as a particular example of the invention. Those skilled in the art should appreciate that they may readily use the conception and the specific embodiment disclosed as a basis for modifying or designing other methods and systems for carrying out the same purposes of the present disclosure. It should also be realized by those skilled in the art that such equivalent components do not depart from the spirit and scope of the disclosure in its broadest form.

In some examples, typical power supply devices and methods arranged to form and maintain an axial Z-pinch current may not be suitable or capable of generating and maintaining sufficient shear rate axial flow that can be used to stabilize the Z-pinch plasma at all. Accordingly, embodiments of apparatus and processes for generating and maintaining sufficient shear velocity flow in a transition medium associated with a boundary region of a Z-pinch discharge are described herein.

Fusion devices based on Z-pinch [ e.g., U.Shumlak, B.A.Nelson, E.L.Claveau, E.G.Forbes, R.P.Golingo, M.C.Hughes, R.J.Oberto, M.P.Ross and t.r.weber, "Increasing plasma parameters using sheared flow stabilization of a Z-pinch," plasma physics "24,055702 (2017); "Shomlak' 17"; incorporated herein by reference may be attractive due to its simple geometry, compactness in nature, and relatively low cost. Some of the latest publications [ e.g., U.S. Shomlak, "Z-pinch fusion," application Physics journal (J.appl.Phys.) 127,200901 (2020); on-line publication: 2020, 5 months, 27 days; "Shomlak' 20"; also incorporated by reference in further detail is Shear Flow Stability (SFS) to create a balanced Z pinch that can maintain a compressed plasma state for a duration significantly longer than other plasma time scales.

One area of shear flow optimization involves enhanced control of both neutral gas and ionized gas fed into the evacuated volume of the acceleration volume of the Z-pinch apparatus. An example of a pre-existing device can be found in figure 3 of the shimlak' 20, showing a schematic cross-sectional view of the vacuum vessel of a fusion Z pinch test (FuZE) SFS Z pinch test device. In fig. 3, an internal gas injection valve is arranged substantially at an intermediate axial position of the accelerating volume to provide a portion of the selected neutral fill gas via an "inner electrode" (which, when the term is used herein, means that the recited features, parameters or values need not be precisely achieved, but deviations or variations in the appearance of the effects that the feature is intended to provide may not be precluded, including, for example, tolerances, measurement errors, measurement accuracy limits, and other factors known to those skilled in the art). The additional (external) gas injection valve is shown substantially at the same axial position radially opposite the internal gas injection valve and is arranged to provide a separate portion of the fill gas through an opening in the "outer electrode" of the FuZE SFS Z pinch test setup.

The arrangement shown in fig. 3 of the shimlak' 20 depends on the diffusion of the neutral gas from the position of the gas injection valve into the surrounding evacuated volume to produce a substantially axisymmetric neutral gas density profile having a maximum substantially at the axial position of the injection valve. Such profiles may provide sufficient fuel gas to drive the shear rate flow for a duration commensurate with the duration of the Z pinch discharge. After the neutral gas inventory is depleted (e.g., due to driven outflow around the Z pinch plasma column and/or gas diffusion into other regions of the enclosed volume), the Z pinch current may decay due to instability even though at least a portion of the energy from the high voltage power supply is still available. To enhance and improve the shear flow profile created by neutral gas injection, a plasma injector, a plasma gun, or an ion source may be used in combination to inject the pre-ionized gas.

Thus, in at least one embodiment, one or more valves (e.g., one or more gas injection valves and one or more plasma injectors) may be fluidly coupled to the fuel gas supply and configured to direct sufficient fuel gas (e.g., neutral gas and/or pre-ionized gas) derived from the fuel gas supply to drive the shear velocity plasma flow for the duration of each of the Z pinch discharges. In particular, in one such embodiment, sufficient neutral gas may be directed to support the local breakdown path between the inner electrode and the outer electrode and establish a shear rate plasma flow. In an additional or alternative embodiment, sufficient pre-ionized gas may be directed to maintain a shear velocity plasma stream (e.g., make-up neutral gas).

In connection with the methods and apparatus described herein, shear rate flow stability is supported by at least the following modeling results. Axisymmetric plasma configurations represented by the FuZE SFS Z pinch experimental setup have been simulated using the WARPXM computer code based on a nonlinear five-moment two-fluid (5M 2F) plasma model [ U.Shumlak, R.Lilly, N.Reddell, E.Sousa and b.srinivasan, "Advanced physics calculations using a multi-fluid plasma model," computer physical communications (comp. Phys. Comm.) 182,1767 (2011) ]. The model contains viscosity and heat conduction effects based on the boson endosik (braginsky).

Some selected simulation results are depicted in fig. 1-5. The results of the shear-free velocity flow are depicted in fig. 1 for reference. The results of three cases with three different parabolic shear flow profiles are shown in figures 2 to 4. The situation can be distinguished by different shear flow rate values "vsfa" at r=a (where a is the nominal radius of the Z pinch plasma), which are defined by the characteristic Alfv n velocity v at the pinch edge (r=a) _A Normalization. The initial condition (t=0) for the case shown includes a peak density of 4×10 ²⁴ m ^-3 Ion and electron temperatures of 1.27keV, and peak magnetic fields of 33.0T. Effective pinch radius [ ]a) Is 0.91mm.

The simulation uses a specific normalized diffusivity limit (difflim=32m ² ·s ^-1 ) And a minimum diffusivity is applied (diffmin=3.2m ² ·s ^-1 ). Electrons are given a viscosity with a diffusivity level equal to the minimum diffusivity. In each case, the perturbation is used to trigger a mode with a wavelength equal to the axial length of the domain.

In fig. 1, vsfa=0.0 indicates a reference case without a shear rate flow. The grey scale is plotted at time points t=0.000 (corresponding to the establishment of an undisturbed Z pinch plasma column) and t=8.000 (normalized to radial alveol time τ), respectively _A ) A snapshot of two-dimensional (r/z) cross-sections 100 and 110 of normalized ion density at that location. In addition, the axial velocity (v _z ) Profiles 120 and 130 (defined by characteristic Alvin velocity v _A Normalized) has substantially no shear, as in v of FIG. 1 _z /v _A And r/a graph.

The situation represented in fig. 1 supports the understanding of: the Z-pinch without stability exhibits a rapidly increasing instability, for example indicated by density perturbation 140, which exhibits a significant plasma ion loss that increases in a relatively small number of alveoli time scale units measured by the time required for a magnetized plasma perturbation to propagate from the axis (r=0) to the edge (r=a) of an undisturbed plasma column.

Three additional modeling results for shear velocity flow stabilization Z-pinch are depicted in fig. 2-4. The situation shown in fig. 2 is characterized by vsfa=0.25. The grey scale is plotted at time points t=0.000 (corresponding to the establishment of an undisturbed Z pinch plasma column) and t=14.000 (normalized to τ), respectively _A ) A snapshot of two-dimensional (r/z) cross-sections 200 and 210 of normalized ion density. The initial (t=0) parabolic shear velocity profile 220 develops into a profile 230 at the (normalized) time t=14, yet exhibits significant shear outside the initial plasma column boundary r=a. Although detectable, the ion density perturbation 250 is located predominantly at a radius commensurate with the initial r=a radius of the undisturbed Z pinch plasma column.

The case featuring vsfa=0.5 in fig. 3 shows a stronger stabilizing effect relative to the case shown in fig. 1 and 2. The grey scale is plotted at time points t=0.000 (corresponding to the establishment of an undisturbed Z pinch plasma column) and t= 26.400 (normalized to τ _A ) A snapshot of two-dimensional (r/z) cross-sections 300 and 310 of normalized ion density at that location. The initial (t=0) parabolic shear velocity profile 320 develops into a perturbed ion axial velocity profile 330 at a (normalized) time t=26.4, exhibiting a near parabolic radial dependence and significant shear outside the initial plasma column boundary r/a=1. The ion density perturbation 350 is located primarily inside the plasma column (r<a) Is a volume of (c).

The case shown in fig. 4 is characterized by vsfa=0.75. The results in fig. 4 show a stronger shear flow stabilization effect relative to the cases shown in fig. 1 to 3. The grey scale is plotted at time points t=0.000 (corresponding to the establishment of an undisturbed Z pinch plasma column) and t= 37.000 (normalized to τ _A ) A snapshot of two-dimensional (r/z) cross-sections 400 and 410 of normalized ion density at that location. The initial (t=0) parabolic ion axial shear velocity profile 420 develops into a (slightly) perturbed ion axial velocity profile 430 at a (normalized) time t=37, exhibiting a substantially parabolic radial dependence. The ion density perturbation 450 is located primarily inside the plasma column (r <a) Is a volume of (c).

Some results of the WARPXM computer code are depicted in FIG. 5. The graph depicted in FIG. 5 shows the plot at t/τ _A The time dependence of the Z-pinch quality is integrated (normalized time corresponding to normalized time t in fig. 1 to 4). The dependence in fig. 5 emphasizes the effect of shear flow rate values characterized by "vsfa" values. It can be observed that for the case without stability (vsfa=0.0), while the confinement of the Z pinch plasma starts to drop after t=5 (marked by the initial decay of the normalized mass 500) and shows significant losses before the nominal time t=8.0, the corresponding mass ratios 525, 550 and 575 (vsfa=0.25, 0.5 and 0.75, respectively) show increasingly stronger plasma confinement, indicating that sufficient axial plasma current is provided only and sufficient shear, square around the plasma column is generated and maintainedThe stable Z pinch plasma can be maintained by the azimuthally symmetric ion velocity flow.

At least in the foregoing context, embodiments of an apparatus and method for generating and maintaining azimuthally symmetric shear ion velocity streams in accordance with the present disclosure are described below. Some components of a particular embodiment of a plasma (confinement) apparatus for stabilizing a Z-pinch are depicted in isometric view 600 (fig. 6A) and cross-sectional view 610 (fig. 6B) in fig. 6A and 6B.

In general, a Z-pinch plasma device having a vacuum vessel as shown in fig. 6A and 6B (with associated connections omitted for clarity, such as cables and tubing, vacuum pumps and conduits, diagnostic guide holes, optical windows, etc.) may be enlarged relative to certain other Z-pinch plasma devices, at least except for neutral gas feed valves (discussed in detail below) associated with improving the process of shear azimuth velocity flow generation and maintenance according to certain embodiments provided in the present disclosure.

More specifically, in at least one embodiment, the acceleration volume 620 may be increased relative to the acceleration volume of some other Z-pinch plasma device and arranged to be filled with a gas mixture (e.g., a neutral working gas mixture) substantially along the central axis of the acceleration volume 620 via, for example, at least one gas injection valve (providing neutral gas to the acceleration volume 620) and/or at least one internal valve 630 of the plasma injector 630 (providing pre-ionized gas to the acceleration volume 620). Additionally or alternatively, a plurality of external valves, such as a plurality of gas injection valves (providing neutral gas to the acceleration volume 620) and/or plasma injectors 640 (providing pre-ionized gas to the acceleration volume 620), may be installed as a regular array on an external vacuum boundary, which may be arranged as an external or external electrode 650.

Depending on the particular embodiment, the gas injection valve and/or plasma injector 630, 640 may be electronically triggered to deliver a "burst" of fill neutral and/or pre-ionized gas starting at a start time programmable to a fraction of a millisecond and having a duration of at most hundreds of microseconds (e.g., at most 1 millisecond). The fill gas (also referred to herein as "fuel gas") that is delivered (e.g., in units of "strands")Body ") may also be controlled by adjusting the fill gas pressure supplied to the gas injection valves and/or the plasma injectors 630, 640, alone or as a subset of the selected valves (wherein the subset of valves may include only some or all of the valves and/or injectors 630, 640). In addition, the different valves and/or injectors 630, 640 (or different combinations of the plurality of valves and/or injectors 630, 640) may be fed by different fill gas mixtures having, for example, different fill gas element ratios and/or different isotope ratios (e.g., adjustable D) ₂ /T ₂ Molecular ratio). In at least one embodiment, the various gas injection valves and/or plasma injectors may be uniform (e.g., all of the same type/size and all of the same operational settings if configurable), but in other embodiments, different valves may be used in different positions. In additional or alternative embodiments, one or more gas jets or other gas valves and/or plasma injectors may control the flow of gas into the acceleration volume 620 via a manifold containing a plurality of ports providing access to the acceleration volume 620. In such embodiments, the ports of the manifold may be uniform or may vary in configuration (e.g., to deliver different amounts of gas to different locations of the acceleration volume 620 when the respective valves are open).

Similar to neutral gas injection via gas injection valves, combinations of plasma injectors or manifolds at different locations may be used to inject ionized gas or plasma. The plasma formed from the gas mixture may also be generated and injected in a manner similar to neutral gas injection. Plasma injection may provide finer control over the final axial plasma distribution and its shear flow profile, which in turn may allow for higher fidelity control over plasma stability and lifetime. Since the plasma particles are charged particles, additional control over the plasma injection may be provided, which may be accelerated by an electric field generated by a variable bias (or voltage) on the injection electrode. Thus, the velocity of the injected plasma may be finely controlled to allow tuning and optimization of the breakdown of any neutral gas present (e.g., in the acceleration volume 620). Furthermore, the injected plasma may travel faster than the injected neutral gas, which may travel in an almost static manner (relative to the injected plasma) during the Z pinch discharge pulse. Thus, the plasma injection may provide pre-ionized fuel "on demand" (e.g., more immediately) relative to neutral gas injection, for example, to replenish fuel gas during the Z pinch discharge pulse.

In some embodiments, the plasma to be injected into the acceleration volume 620 may be generated by pre-ionizing a neutral gas with a spark plug or via inductive ionization. More broadly, the gas injection valve and/or the plasma injectors 630, 640 may comprise one or more electrode plasma injectors and/or one or more electrodeless plasma injectors. In examples that include one or more electrode plasma injectors, the plasma to be injected into the acceleration volume 620 may be generated at least in part by an electrode discharge. In additional or alternative examples including one or more electrodeless plasma injectors, the plasma to be injected into the acceleration volume 620 may be generated at least in part by an inductive discharge generated by an external coil window (e.g., a radio frequency antenna operating at 400kHz, 13.56MHz, 2.45GHz, and/or other frequencies permitted for use in a given local jurisdiction, such as within a frequency range permitted by the federal communications commission). In some embodiments, the neutral gas used for pre-ionization may be limited by the configuration of the neutral gas reservoir (not shown in fig. 6A and 6B) and/or the neutral gas may be conducted to a selected plasma injector configuration.

In some embodiments, the axial distribution of the injected plasma may be ensured via an axisymmetric plasma injector configuration. In at least one embodiment, eight plasma injectors 640 may be positioned at eight equally spaced ports of the manifold, respectively. The eight ports may each be configured at an oblique angle (e.g., between 5 ° and 90 ° relative to a central axis of the acceleration volume 620) relative to the housing of the acceleration volume 620. In one example, the tilt angle may be 45 ° with respect to the central axis of the acceleration volume 620. In some embodiments, eight ports may be configured at a single axial location along the central axis of the acceleration volume 620 (i.e., eight ports may be equally spaced around the circumference or other perimeter of the acceleration volume 620 at an axial location). In other embodiments, the ports may comprise multiple sets of eight ports, with each set of eight ports equally spaced about different axial locations along the central axis of the acceleration volume 620. In an example embodiment, the set of eight ports may be configured as interleaved pairs of sets, wherein a first set of eight ports may be positioned at a first axial position and a second set of eight ports may be positioned at a second, different axial position and rotated relative to the first set such that each port in the second set is positioned between a pair of ports in the first set relative to the circumference of the acceleration volume 620. Specifically, in such an embodiment, each port of the first set of eight ports may be spaced every 45 ° around the circumference of the acceleration volume 620, and each port of the second set of eight ports may be spaced every 45 ° offset (rotated) from the first set of ports by 22.5 ° around the circumference of the acceleration volume 620, such that one port of the first and second sets is disposed every 22.5 ° around the circumference of the acceleration volume 620. In additional or alternative embodiments, the plasma injection may be performed, for example, along an chordal azimuth angle perpendicular to the central axis of the acceleration volume 620, so as to generate an azimuthal flow within the acceleration volume 620. In additional or alternative embodiments, the valve may be configured with other variations in different ways (e.g., asymmetrically placed angular distribution and/or having different angular distributions) to achieve a substantially equivalent profile by compensating for the effects of the variations.

In some embodiments, the injection of the acceleration volume 620 with the pre-ionized gas may generate a plasma having a plasma temperature in the range of 1 to 10 eV. Furthermore, and as mentioned above, since the injection rate of the pre-ionized gas may be significantly greater than the injection rate of the neutral gas, the velocity of the plasma within the acceleration volume 620 may be as high as 50 x 10 ³ m/s. In some embodiments, the injection of pre-ionized gas may provide flexibility in the amount of particles injected. In particular, in an example embodiment, 1/5 of the time for injecting the same amount of neutral gas particles may be0, a certain amount of pre-ionized gas particles are injected. For example, the amount of time for injecting 10 Torr-L neutral gas particles (where 1 Torr-L and 2.5X10 ¹⁹ The molecules are proportional at 273K) may be the amount of time for injecting 500 torr-L of pre-ionized gas particles. Similarly, in some embodiments, the injection rate (or mass flow rate) of the pre-ionized gas may vary depending on the supply current and voltage (i.e., the waveform of the injection pulse). For example, increasing the supply voltage (e.g., to between 100V and 500V) may increase the injection rate at the same time. As another example, increasing the supply current (e.g., to between 1A and 500A) may increase the injection rate simultaneously.

In particular, the injection of neutral gas can be achieved by a purge valve or by releasing hydrogen from a metal hydride, such as titanium deuteride (TiD ₂ ) Or other metal hydrides based on scandium, vanadium, or other metals. In some embodiments, the injection valve may be a solenoid-driven injection valve (although other configurations may be implemented and are within the scope of the disclosure).

As discussed above, at least one internal gas injection valve and/or plasma injector 630 and a plurality of external gas injection valves and/or plasma injectors 640 may be activated individually or in groups. The initial gas load inside the acceleration volume 620 with the desired axial and azimuthal profiles can be achieved by timing individual valves and/or groups of valves. Such valves (or groups thereof) may be timed in a manner to align the arrival of neutral and/or pre-ionized gases and/or mixtures thereof with a desired initial profile, such as the embodiments discussed in detail below and shown in fig. 8 and 10A-10F. A power supply (not shown in fig. 6A and 6B) may be timed to achieve ionization at the desired axial location and to generate and maintain a shear flow with an initial gas load.

Various combinations of (neutral gas) gas injection valves and plasma injectors may be enabled to further adjust (e.g., optimize) such parameters, for example, to achieve a desired power output level. In addition, the plasma may be injected into the acceleration volume 620 much faster (e.g., -100 times) than the injected neutral gas. The combination of such different implantation speeds allowed by acceleration of the plasma implantation with neutral gas implantation provides even greater parameter space for optimization. In addition, plasma injectors can be used to inject mass and carefully control the location of neutral gas ionization.

The embodiment shown in fig. 6A and 6B incorporates a pre-formed acceleration volume 620 to incorporate connectors or other coupling elements for at least one internal gas injection valve and/or plasma injector 630 extending from within the internal or inner electrode 660. For example, the at least one internal gas injection valve and/or plasma injector 630[ and the corresponding coupling element(s) ] may comprise eight valves 630 (having 45 ° angular separation) distributed azimuthally symmetrically at z= -50cm prescription (with a z=0 position at the unsupported end 665 of the inner electrode 660, wherein the z axis coincides with the central axis of the acceleration volume 620, and wherein the negative direction of the z axis extends along the central axis of the inner electrode 660 from the unsupported end 665 and the positive direction of the z axis extends through the acceleration volume 620 from the unsupported end 665 in a direction opposite to said negative direction), the eight valves 630 being similarly distributed at z= -75cm, and the eight valves 630 being similarly distributed at z= -100 cm. The illustrated embodiment can be easily updated with additional valves to allow for more fuel gas injection (e.g., for a more durable pinch discharge) and control the axial pressure profile of the neutral-filled gas in the acceleration volume 620 (e.g., for additional enhanced shear rate shear flow duration). In additional or alternative embodiments, the valve may be configured with other variations in different ways (e.g., asymmetrically placed angular distribution and/or having different angular distributions) to achieve a substantially equivalent profile by compensating for the effects of the variations. Such considerations may apply equally to plasma injectors.

The gas injection valve of the illustrated embodiment incorporates prismatic structural elements that may allow a tool actuator (e.g., a force transmitting element of a hand tool or a coupling insert of a power tool) to be used therewith to directly transfer torque and other associated stresses to the stronger primary structural element while avoiding various inserts, connectors, contacts, vacuum or pressure bars, potting and/or welded joints.

In embodiments, the gas injection valve of the present disclosure may be designed to incorporate an orifice diameter of not less than 0.075in and at least 1cm ³ Is provided. Additionally, one feature of the gas injection valve according to embodiments of the present disclosure is the ability to close (and remain closed) at a preprogrammed time before and during the Z pinch discharge.

As stated above (including documents incorporated by reference), preserving the azimuthal symmetry of the plasma and associated shear velocity flow is one advantage of embodiments of the present disclosure. Thus, at least some embodiments of the present disclosure may enable reproducible formation and shaping of an initial azimuthally symmetric plasma structure at a predetermined axial location in the acceleration volume 620. In various embodiments, various "plasma forming" devices and methods may be used. Such devices and methods may include, but are not limited to: dedicated systems for plasma generation [ some of which may be configured to take into account the complexity associated with specific power supplies and piping and/or complexity (pre-ionization subsystem) ], plasma injectors and tuning algorithms and other methods of operation.

The plasma confinement device of fig. 6A and 6B may include a controller or other computing device (not shown) that may include a non-transitory memory on which executable instructions may be stored. The executable instructions may be executed by one or more processors of the controller to perform various functions of the plasma confinement arrangement. Accordingly, the executable instructions may include various routines for operation, maintenance, and testing of the plasma confinement device. The controller may also include a user interface at which an operator of the plasma confinement device may enter commands or otherwise modify operation of the plasma confinement device. The user interface may include various components for facilitating operator use of the plasma confinement device and for receiving operator input (e.g., a request to generate a plasma for thermonuclear fusion, etc.), such as one or more displays, input devices (e.g., a keyboard, a touch screen, a computer mouse, depressible buttons, mechanical switches, other mechanical actuators, etc.), lights, and the like. The controller may be communicatively coupled to various components of the plasma confinement device (e.g., valve, power supply, etc.) to command actuation and use thereof (wired and/or wireless communication paths between the controller and the various components are omitted from fig. 6A and 6B for clarity).

Aspects of the plasma initiation and subsequent shaping portions associated with embodiments of the present disclosure are schematically depicted in fig. 7. These shaped portions may include sharp points (e.g., tips formed at the local concave elements 725; see below) that enhance the local electric field and facilitate plasma breakdown. Various configurations of the "passive" (i.e., no dedicated power or gas supply for active drive field emission) shaping portion 700 may be arranged in the form of ring electrodes in one or more recesses in the inner electrode 660 at one or more negative z-axial positions, typically proximate to one or more inner gas injection valves 630 (inner electrode 660 and gas injection valve 630 are not shown in fig. 7; see fig. 6B and 8). In at least one embodiment, one significant function of such a portion is to initiate and maintain a multi-channel breakdown of the surrounding neutral gas (starting from various substantially independent azimuthally distributed radial discharge currents) that helps to generate and maintain substantially equal currents in all radial directions. It should be noted that in various embodiments, the shaping portion 700 may be placed at the location of the gas valve or downstream thereof (e.g., on the right side of fig. 6B and 8-10F). Although a sharp shaped segment is described with reference to the inner electrode 660, such features may be at the inner electrode 660, the outer electrode 650 (not shown in fig. 7; see fig. 6A, 6B, and 8), or both. In some embodiments, it is beneficial to enhance the electric field near such sharp points at the cathode (from which electrons are emitted), which may be the inner electrode 660 or the outer electrode 650. Sharp points on the anode surface may also be included so that a breakdown path may be selectively established between the shaped portions on the cathode and the shaped portions on the anode (which may be, for example, inner electrode 660 and outer electrode 650, respectively).

The shaped portion 700 shown in fig. 7 incorporates a conductive ring 710, the conductive ring 710 arranged to include at least one contact surface 720, the at least one contact surface 720 forming a low contact resistance surface contact with a cylindrical or conical or otherwise tapered outer surface of the inner electrode 660 (e.g., to cause a voltage drop of less than 100V between the at least one contact surface 720 and the inner electrode 660). In some embodiments, the conductive ring 710 may be formed of one or more conductive materials that may be fully or at least partially chemically and/or thermo-mechanically compatible with the conductors of the inner electrode 660 (e.g., the heat and stress experienced during operation does not significantly affect the life usage of the conductive ring 710). In addition, the plasma-facing portion 715 of the shaped portion 700 may be formed of a conductor that resists chemical and physical damage caused by supporting the discharge. In many embodiments, one or more refractory metals (e.g., one or more of W, ta, nb, mo or Re; additionally or alternatively comprising one or more of Ti, V, cr, mn, zr, tc, ru, rh, hf, os or Ir) and/or alloys or combinations thereof may be used for at least relatively low chemical reactivity, relatively high melting point, and relatively high plasma ablation and sputtering resistance.

In additional or alternative embodiments, the plasma-facing material may be based on carbon in an electrically conductive form, including graphite, sintered or pressed carbon powder, carbon fiber matrix, and/or carbon nanotube bonding structures and compositions. In addition to being relatively insensitive to plasma effects, degradation, and damage, carbon-based structures (particularly carbon nanotubes) may also exhibit desirable electron multiplication properties during the plasma generation phase.

In other embodiments, the plasma-facing portion 715 may be textured to incorporate a plurality of local concave elements 725 so as to form a structured array. In certain embodiments, such elements may enhance the local electric field and facilitate electron field emission from solid (and liquid) surfaces. The elements 725 may be formed by mechanical action (including cutting, scraping, sanding, sandblasting, grooving, embossing, stuccoing, knurling, etc.). Different chemical and/or thermal processes (e.g., etching, chemical deposition, spraying, sputtering, ion and neutral implantation, epitaxial growth, etc.) may also be involved. In certain embodiments, multiple elements 725 having relatively small characteristic dimensions (e.g., as compared to the size of the plasma-facing portion 715) may be created and maintained to avoid significant changes in geometry when/during any individual element 725 is damaged or deformed (e.g., by arcing or localized overheating). For example, in some embodiments, the elements 725 may have an average height that is 1-10% of the height of the plasma-facing portions 715 (excluding the elements 725).

In at least one embodiment, the shaped portion 700 surrounding the inner electrode 660 (whether with or without the element 725) may be configured as a substantially uniform annular ring, e.g., varying where the ports of the inner gas injection valve 630 traverse the cross-section of the plasma-facing portion 715 and the at least one contact surface 720. However, in some embodiments (e.g., when the shaping portion 700 is configured as a substantially uniform annular ring), the shaping portion 700 may have a varying cross-section around and/or along the inner electrode 660. Other variations, such as multiple discrete shaped portions 700 where a single loop is not formed, are placed around the inner electrode 660, are also within the scope of the present disclosure.

In addition, as discussed in detail below with reference to fig. 8, a shaped portion similar to shaped portion 700 shown in fig. 7 may form a ring along the inner surface of outer electrode 650. This shaped portion and corresponding contact surface (similar to the at least one contact surface 720) may be configured as shown in fig. 7 (or according to variations discussed herein), except that the cross-section of the at least one contact surface 720 and the plasma-facing portion 715 is rotated 180 ° to accommodate being located on the inner surface of the outer electrode 650 instead of the outer surface thereof. The slope of at least one contact surface 720 may also be different from that shown in fig. 7 to accommodate for the taper or other surface variations or lack of surface variations of the outer electrode 650 (see "tapered electrode surface" discussed below with reference to fig. 9).

Other methods of assisting/controlling/directing plasma formation may be used in addition to the shaping portion 700 shown in fig. 7, alone or in combination. One possibility includes the use of radioactive material(s) embedded in the outer electrode 650 and/or the inner electrode 660. In particular, energetic particles or photons emitted from the radioactive decay process may result in pre-ionization near the embedded radioactive material(s), resulting in regions of increased plasma breakdown relative to regions not containing the embedded radioactive material(s). For example, a beta emitter and/or a gamma emitter may be selected for the embedded radioactive material(s).

Additional or alternative possibilities include irradiating the outer electrode 650 and/or the inner electrode 660 with incident laser light in the region where plasma pre-ionization is desired. Within such regions, the electrode surfaces (e.g., of the outer electrode 650 and/or the inner electrode 660) may comprise materials specifically selected to emit X-rays or other forms of ionizing radiation when subjected to incident laser light. Additional or alternative possibilities include directly ionizing the gas using a laser (e.g., via direct interaction of the laser with neutral gas particles). In such embodiments, the laser may pass through a neutral gas and deposit energy throughout the laser path, resulting in pre-ionization and directional channels for plasma breakdown (e.g., channels with greater plasma breakdown than the surrounding volume).

Other methods to assist/control/direct plasma formation may utilize various forms of cathodes, such as field emitters or thermionic emitters, located on the electrode surfaces (e.g., the electrode surfaces of the outer electrode 650 and/or the inner electrode 660) where breakdown is desired. A field emitter may use a relatively high electric field to emit electrons from small sharp features. Examples of such emitters may include nanostructures, such as carbon nanotubes, graphene emitters, nanowire emitters, schottky (Schottky) emitters, and the like. Additionally or alternatively, a thermionic emitter may be used to cause plasma breakdown. Examples of such emitters may include heated tungsten filaments that emit electrons at relatively high temperatures. Schottky emitters can be considered field enhanced thermionic emitters.

One embodiment of a process for initiating and driving azimuthally symmetric shear flow for stabilizing a Z pinch discharge in a plasma confinement device such as the plasma confinement device described in detail above with reference to fig. 6A-7 is schematically depicted in fig. 8. In this example, the process may be characterized by a schematic diagram of steps or stages 810, 820, 830, 840, 850, 860, 870, and 880 of substantially unequal duration. In certain embodiments, the process may include performing steps 810, 820, 830, 840, 850, 860, 870, and 880 in sequence.

In some embodiments, the process, or a portion thereof, may be implemented as executable instructions stored in a non-transitory memory of a computing device, such as a controller communicatively coupled to a plasma confinement device. Furthermore, in certain embodiments, additional or alternative sequences of steps may be implemented as executable instructions on such a computing device, where various steps discussed with reference to the processes may be added, removed, substituted, modified, or interchanged.

The process begins at step 810, which step 810 may include applying a high voltage between electrodes 650 and 660 to generate a radial electric field (not shown), and sequentially activating one or more of inner valve 630 and outer valve 640 (possibly in combination with plasma injector 640). Valves 630 and 640 may be arranged to locally introduce the initially measured and predetermined concentration 812 of fill gas. In certain embodiments, it may be desirable to initiate a jet and/or plasma injection during the initialization phase and continue to deliver a sufficient initial concentration 812 of fill gas near the shaped portion 700 to provide additional protection against premature and/or asymmetric gas breakdown.

During step 820, the initial concentration 812 may spontaneously evolve through a neutral gas diffusion process to form a continuous (e.g., uninterrupted) axisymmetric volume of neutral fill gas formation 822 that occupies a majority (e.g., a major portion) of the acceleration volume 620. In some embodiments, the volume of neutral fill gas formation 822 exhibits a neutral gas molecular number density gradient in an axial direction along the central axis of acceleration volume 620 (e.g., toward unsupported end 665 of inner electrode 660) while substantially maintaining azimuthal symmetry that facilitates a substantially symmetric distribution of discharge current during initial breakdown of the fill gas. In certain embodiments, the neutral gas molecular number density gradient may be such that pam Shen Jichuan (Paschen breakdown) occurs at the shaped portion 700. In additional or alternative embodiments, pre-ionization may be injected at the shaping portion 700 to facilitate the formation of an ionization wave that moves into the neutral gas that has been injected upstream (e.g., toward the supported end of the inner electrode 660 opposite the unsupported end 665).

During step 830, the proximal electric field structure shaped by the geometry and material properties of the shaping portion 700 may promote neutral gas breakdown, forming an axisymmetric plasma structure 835, e.g., located axially near the shaping portion 700, supporting a current 837 between the inner electrode 660 and the (surrounding) outer electrode 650. The current 837, supported by energy from a power source (e.g., a capacitor bank or similar device), may form a continuous (e.g., uninterrupted) current loop (from the outer electrode 650, through the plasma structure 835, and into and through the inner electrode 660) that may generate a substantially azimuthal magnetic field 838 (as indicated by azimuthal magnetic field lines). In additional or alternative embodiments, to form the plasma structure 835, pre-ionized gas may be injected from the plasma injector 640 toward the unsupported end 665 of the inner electrode 660.

The Lorentz force (Lorentz force) interaction between the current 837 and the magnetic field 838 may cause the current 837 to migrate from the shaping portion 700 in the direction of the unsupported end 665, as shown in the schematic representation of step 840. In addition, the lorentz force interaction may induce a current 837 along the surface of the outer electrode 650.

During step 850, the current 837 may continue to progress toward the unsupported end 665 and reach the unsupported end 665. At the same time, the magnetic pressure driven by the magnetic field 838 enclosed by the current 837 may displace the plasma structure 835 being formed in the direction of the opposite portion 655 of the outer electrode 650 arranged to face the unsupported end 665. In existing embodiments, the ionization wave moving into the neutral gas may be controlled by injecting different amounts of pre-ionized gas, for example, at the internal valve 630 and/or the external valve 640. In at least one embodiment, the substantial azimuthal symmetry of the plasma structure 835 supporting the current 837 can significantly contribute to the efficiency of the process, as any significant disturbance in the current 837 can lead to instability, electrode damage, and/or the introduction of metallic impurities into the discharge being formed.

In some embodiments, steps 830, 840, and 850 in the formation of the discharge may last from a fraction of a microsecond to a few microseconds, e.g., significantly shorter than step 810 (corresponding to filling acceleration volume 620 with neutral gas) or steps 860, 870, and 880 (corresponding to a Z pinch discharge). Thus, the neutral fill gas formation 822 is depicted as stationary because the neutral fill gas formation 822 may be formed (e.g., only) during a time interval that is significantly longer than the duration of steps 830, 840, and 850.

In certain embodiments, plasma implantation may occur between steps 810 and 820. In additional or alternative embodiments, plasma implantation may occur rapidly and on the same scale as steps 830, 840 and 850, and may be used to control the formation/initialization and kinetics of these steps.

Step 860 corresponds to an initial Z-pinch operation step including forming a support Z-pinch current I generated _pinch Is a Z pinch plasma column 865. In addition, a residual plasma structure 866 may be formed to support a residual (radial) current 867 flowing through the neutral fill gas formation 822 in the acceleration volume 620. Furthermore, in various embodiments, the propagation of the plasma structure 835 may drive an initial shear velocity plasma stream 868 that surrounds (and stabilizes) the Z-pinch plasma column 865. In some embodiments, residual plasma structure 866 may be initiated near shaping portion 700, characterized by a (local) highest number density of neutral gas components (molecules and/or atoms).

As discussed above, in at least one embodiment, during discharge maintenance in step 870, the Z-pinch plasma column 865 may be maintained and stabilized by a continuous plasma flow from the acceleration volume 620. The ionization front 872 may continuously generate plasma that is accelerated from the acceleration volume 620 by the residual current 867 to drive the shear velocity plasma stream 868.

During step 880, in at least one embodiment, the ionization front 872 may move toward the trailing end of the acceleration volume 620 to ionize the remaining neutral fuel gas in a continuous (e.g., uninterrupted) manner until all or substantially all of the fuel gas available in the acceleration volume 620 is ionized, resulting in the disappearance of the ionization front 872 and a Z pinch currentI _pinch For example, a current through the Z pinch plasma column 865 and the inner electrode 660, such as current 950 shown in fig. 9, as described below. When steps 810, 820, 830, 840, 850, 860, 870, and 880 are completed, the plasma confinement system may be flushed to remove fusion byproducts and the above process may be repeated for another pulse. In some embodiments, the process and repetition thereof may be automated and controlled by a software application, for example, implemented by a controller communicatively coupled to the plasma confinement system.

Another embodiment of a plasma confinement device, a Z-pinch plasma device 900, is schematically illustrated in fig. 9. The Z-pinch plasma device 900 may generate a plasma within an assembled volume 635 of the plasma confinement chamber 615, the plasma being confined, compressed, and maintained by an axially symmetric magnetic field. The axially symmetric magnetic field may be stabilized by a shear ion velocity flow driven by an electrical discharge between a pair of electrodes that interface with the plasma confinement chamber 615.

Devices belonging to the illustrated class of plasma confinement devices are generally related to the previous embodiments discussed above and shown in fig. 6A-8, and have similar features except for the additional or alternative subsystems and functions below. The description provided above with reference to fig. 6A-8 may be additionally applied to the embodiment depicted in fig. 9, except for certain assembly and operational aspects that may be caused by such differences. In certain embodiments, additional subsystems and/or functionality may also be included in the Z-pinch plasma device 900, which are not described in detail above with reference to fig. 6A-8 and may be additionally applied to the embodiments depicted in fig. 6A-8.

In an example embodiment, the Z-pinch plasma device 900 may include an outer electrode 650 that is physically and functionally separated from an outer vacuum boundary 910, which outer vacuum boundary 910, together with portions of the inner electrode 660, form a vacuum vessel 645 as a low pressure vessel that includes the plasma confinement chamber 615. The intermediate electrode 920 may be positioned so as to have a radius between the radius of the inner electrode 660 and the radius of the outer electrode 650. In particular, the intermediate electrode 920 may substantially surround the inner electrode 660, and the outer electrode 650 may substantially surround the intermediate electrode 920. For example, the inner electrode 660 may include one end 665 at least partially surrounded by the intermediate electrode 920, and the intermediate electrode 920 may include one end 965 at least partially surrounded by the outer electrode 650.

The Z-pinch plasma device 900 may incorporate at least two functionally separate power sources, e.g., arranged and controlled primarily to drive a Z-pinch (discharge) current 950 (I _pinch ) At least one main power supply 930, and at least one additional power supply 940 arranged and controlled primarily to drive a residual current 867. In some embodiments, the at least one primary power source 930 may be a separate power source device(s) from the at least one additional power source 940. In other embodiments, the at least one main power supply 930 and the at least one additional power supply 940 may be components of the same power supply apparatus.

For example, in at least one embodiment, a single power supply device may have multiple outputs that individually provide an amount of power to achieve performance of the respective function (e.g., drive Z pinch current 950, drive residual current 867, etc.). This arrangement may be based on at least two power sources (e.g., one main power source 930 and one additional power source 940), and may allow additional control of the Z-pinch current 950 and its shear flow stability. In principle, the at least two power sources may be scaled, charged and controlled such that the Z pinch current 950 and its stability may be maintained for a commensurate period of time before either of the at least two power sources prematurely depletes or runs out of stored energy.

In certain embodiments, the Z-pinch plasma device 900 may incorporate a "tapered electrode" configuration characterized by widening the gap between the inner electrode 660 and the intermediate electrode 920 by tapering the end 965 of the intermediate electrode 920 outwardly along the central axis of the acceleration volume 620, e.g., in the direction of the (unsupported) ends 665 and 965, to increase the volume of at least a portion of the acceleration volume 620. In one example, the taper may be between 0 ° and 15 ° from the central axis of the acceleration volume 620. This arrangement may facilitate transfer of momentum, e.g., along the central axis, from the plasma heated by residual current 867 to the neutral gas, thereby creating and maintaining shear flow stability. Momentum transfer may be described and modeled using a method suitable for design/optimization of a "de Laval nozzle" as is known in the jet propulsion art.

While the techniques described herein are discussed in connection with thermonuclear fusion and utilize energy generated thereby, for example, the techniques described herein may be used for other purposes such as heat generation (e.g., for manufacturing with relatively high temperatures) and propulsion. For example, the embodiments of fig. 6A-8 or 9 may be modified by at least removing the vacuum chamber 338 or the outer vacuum boundary 910, respectively, and introducing an opening in one end of the outer electrode 650 to allow fusion products to escape (e.g., parallel to the central axis of the acceleration volume 620). In one embodiment, a magnetic nozzle (not shown in fig. 9) is positioned downstream of the outer electrode 650, e.g., to the right of the outer electrode 650 in the plane of fig. 9, to collimate the plasma to reduce any exhaust plume emissions.

The Z-pinch plasma device 900 may include a controller or other computing device 948, which may include a non-transitory memory on which executable instructions may be stored. The executable instructions may be executed by one or more processors of controller 948 to perform various functions of Z-pinch plasma device 900. Accordingly, the executable instructions may contain various routines for operation, maintenance, and testing of the Z-pinch plasma device 900. The controller 948 may also contain a user interface at which an operator of the Z-pinch plasma device 900 may enter commands or otherwise modify the operation of the Z-pinch plasma device 900. The user interface may include various components that facilitate operator use of the Z-pinch plasma device 900 and for receiving operator input (e.g., a request to generate a plasma for thermonuclear fusion, etc.), such as one or more displays, input devices (e.g., a keyboard, touch screen, computer mouse, depressible buttons, mechanical switches or other mechanical actuators, etc.), lights, and the like. The controller 948 may be communicatively coupled to various components (e.g., valves, power supplies, etc.) of the Z-pinch plasma device 900 to command actuation and use thereof (wired and/or wireless communication paths between the controller 948 and the various components are omitted from fig. 9 for clarity).

Fig. 10A to 10F schematically illustrate an embodiment of a process for initiating and driving azimuthally symmetric shear flow for stabilizing Z pinch discharge in a plasma confinement device. Fig. 10A to 10F show the series of two configurations, a first configuration in which the inner electrode is the cathode and the outer electrode is the anode (on the left side of each of fig. 10A to 10F when fig. 10A to 10F are oriented such that the alphanumeric characters depicted therein are oriented in a standard manner), and a second configuration in which the inner electrode is the anode and the outer electrode is the cathode (on the right side of each of fig. 10A to 10F when fig. 10A to 10F are oriented such that the alphanumeric characters depicted therein are oriented in a standard manner). Some Z-pinch plasma confining means may correspond to a first configuration that may be more simply constructed and successfully operated. However, as shown in more detail in the discussion below with respect to fig. 11-24, the second configuration may produce advantageous and unexpected results in accordance with various embodiments described herein. Some of the non-alphanumeric symbols (e.g., current arrows, gas valves, gas clouds, magnetic field symbols, gas flow arrows, etc.) used in fig. 10A-10F are the same as those used in fig. 8. It should be noted that the portions labeled with "cathode" and "anode" may be electrically connected to portions having the same name ("cathode" or "anode"). It should be noted that while fig. 10A-10F illustrate a set of gas valves in physical contact with or in close proximity to an outer electrode, other configurations are within the scope of the present disclosure, such as the configuration of valves shown and described in detail with reference to fig. 8, which may or may not include one or more shaping portions, as described in detail above with reference to fig. 7.

In at least one embodiment, greater stability of the Z-pinch plasma may be achieved by a plasma confinement system as disclosed herein, wherein the outer electrode is a cathode and the inner electrode is an anode. Specifically, and as discussed in more detail below with reference to fig. 11-24, a nonlinear ideal 5M2F model and extensions of the model (including buson inner stokes heat and momentum transfer) were used to study Z pinch m=0 instability and its radial shear axial flowStability. The linear growth rate results were compared to previous work using the ideal 5M2F model, using magneto-hydrogen dynamics (MHD) and Hall MHD (Hall MHD). The consistency with hall MHD is excellent with and without radial shear axial flow, indicating that in the dual fluid item, the hall item dominates. The results are also consistent with MHD, subject to the constraint of small ion inertial lengths. Comparison with intracellular Particle (PIC) modeling without shear m=0 stability focused on the plasma scenario based on recent experimental results. In the sweep of the mode wavenumber, the ideal 5M2F result is similar in nature to PIC: the growth rate rises to a peak at moderate wavenumbers and falls at large wavenumbers compared to MHD results, which show that the long rate saturates with increasing wavenumbers rather than falling. Peak normalized 5M2F growth rate is γτ _A =1.5, where τ _A Is the alvanii transmission time spanning the pinch. The peak occurs at normalized wavenumber ka=10, where a is the effective pinch radius. In contrast, PIC results have a peak increase γτ at ka=5 _A =0.77. The businessk-based closure comprising the 5M2F model does not change in nature the ideal outcome in this particular case. With pinch edge shear flow velocity equal to half the alveol velocity, nonlinear 5M2F modeling was performed using dissipative buson-stokes closure. The non-linear mixing caused by the shear flow produces a saturated quasi-steady state in which pinch ion inventory and pinch heat energy are moderately lost, respectively: about 30% and 10%. The 5M2F modeling captures the fundamental physical principle of m=0 instability and provides a computationally tractable approach to high-fidelity modeling of 3D Z pinch behavior (including m=1 instability).

Experimental evidence from SFS Z pinch studies and open numerical stability analysis indicate that radial shear axial flow can achieve the observed long plasma lifetime; static Z-pinch is typically terminated by m=0 (cylindrical) and m=1 (kinked) instabilities with a growth rate close to the radial alveoli transmission frequency.

The insight of computational modeling is expected to be an important component of ongoing SFS Z pinch development. As plasma parameters increase in future experiments, high-fidelity modeling may be used to explore various processes involved in plasma formation, assembly, and confinement. Processes of great concern include, for example, deflagration modes associated with maintenance of the effective resistivity of the shear flow due to electron drift micro-turbulence, and the dynamics of the flow Z pinch itself.

Five-moment multiple fluid models are excellent candidates for accurate capture of the physical principles of interest. A five-moment two-fluid (5M 2F) plasma model, where the two fluids represent ions and electrons, has previously been applied to model Z pinch instability, capturing the actual m=0 growth rate and electron drift instability of interest when the electron cross-field drift velocity exceeds the ion thermal velocity. The 5M2F model may allow an efficient tradeoff between the fidelity of the kinetic model (but excessive computational expense) and the computationally easy handling (but limited fidelity) of the MHD-based approach.

The study presented herein applied a 5M2F model to explore m=0 instability with and without shear flow. The 5M2F model contains the finite inertial length correction of ions and electrons, as well as the effect of the finite speed of light. Under the constraint that the electronic inertia is negligible and the speed of light is infinite, the result should match the hall MHD result. In some examples, the linearized hall MHD model is applied to the present equilibrium and parabolic shear flow profile (v _sf ∝r ² ) M=0 stability. The same setup has been used, but taking into account the linearity (v _sf Oc r) and parabolic shear flow profiles, additional work was done using a nonlinear MHD model. Hereinafter, the 5M2F model is based on these MHD and hall MHD results.

To evaluate the physical fidelity of the 5M2F model with complete buson inner stokes closure, comparisons were made by complete kinetics of m=0z pinch stability (i.e., PIC modeling that was not gyrokinetic or otherwise reduced), including coulomb collisions (Coulomb collision). In this work, the PIC model was applied to study a FuZE-like bennett profile with a linear shear flow profile. Scans of ka without shear flow show a similar growth rate to MHD results up to ka=5, when PIC growth rate reaches a maximum. For larger k, the PIC results show a decrease in growth rate, as opposed to a constant at k in generalOr an increased growth rate. Simulation at ka=5 with shear flow shows that this would be true forM=0 stability of (c). Stability analysis performed under reduced collisional reactor-like conditions produced similar results.

The five moment fluid equation for a given species is derived by taking the moment of the associated boltzmann equation (Boltzmann equation). The first three moments of the boltzmann equation for species α produce the evolution equation of five independent variables, as described by boson inner stokes: number density (n) _α ) Three momentum components (m _α n _α v _α Wherein m is _α And v _α Is species mass and velocity) and scalar pressure (p _α ). Five moment multiple fluid model and its reduction to 5M2F are summarized below, and the boson inner stark closure details are presented below. Implementation of the model in the WARPXM framework is also discussed below.

The moment of the boltzmann equation for species α yields the following fluid equation:

wherein the method comprises the steps ofIs the total fluid energy density, and q _α Is the species charge. The identity matrix is represented by->In one example, an insulation system is usedThe number Γ=5/3. Species temperature is determined by relationship p _α ＝n _α k _B T _α Determining, where k _B Is the boltzmann constant. The non-ideal term to be discussed below is the stress tensor (pi _α ) Heat flux (h) _α ) And Source item->Andwhich represents a collision source of particles, momentum and energy from reactions and interactions between species. In this example, there are five equations in total, two scalar equations and one vector equation, resulting in a "five moment" name. All expressions are expressed in SI units unless otherwise indicated.

Fluid equations are coupled to Maxwell's equations for magnetic (B) fields and electric (E) fields,

here, μ ₀ Sum epsilon ₀ The permeability and permittivity of free space, respectively. Current density j= Σq _α n _α v _α And a charge density ρ _c ＝∑q _α n _α (the sum exceeds α) provides coupling to the fluid equation. If the divergence constraint equations (6) and (7) are satisfied in the initial value problem, itThey will remain mathematically satisfied; equations (4) and (5) then fully describe the evolution of E and B. This strict mathematical guarantee is broken by the presence of numerical errors or domain boundaries, thereby stimulating formulas that explicitly preserve constraints. In the results presented herein, the divergence constraint is fully satisfied and no such special formulation is used.

To obtain the 5M2F model, the species is limited to ions and electrons, α=i, e. The collision source term occurs only due to coulomb scattering between ions and electrons. Specifically, the sources obtained are respectivelyAnd->Wherein the method comprises the steps ofAnd->Is a frictional exchange of momentum and energy. The physical and digital aspects of the 5M2F model are described in detail in earlier work.

The model was closed after businessman using a Chapman-enstock type closure (Chapman-Enskog-type closure). Stress tensor (pi) _α ) And heat flux (h _α ) Allows arbitrary magnetization, x, according to the Brissen Neisseria formulation specification _α ＝ω _cα τ _α Wherein omega _cα Is the cyclotron frequency, and τ _α Is the collision time of species alpha.

Momentum and heat exchange item And->) The reason for this is as follows. The friction momentum is exchanged on a time scale tau _exch ≈(m _i /m _e )τ _e Upper modifier plasma momentum, whichMiddle tau _e Is the electron impact relaxation time. The exchange of thermal energy occurs on the same timescale. If τ _exch /τ _dyn > 1, where τ _dyn Is a dynamic time scale of interest, then friction and heat exchange terms can be omitted without losing accuracy. For example, the condition τ is satisfied in a FuZE-like plasma considered hereinafter _exch /τ _dyn > 1. Although the term does not present a particular computational challenge, the term is omitted to allow presentation and analysis focusing on viscosity and heat flux terms that are more important for m=0 unstable behavior.

The heat flux is

Wherein the non-magnetic heat flux term (related to) And sign is taken. The magnetic field direction is b=b/B, where b= |b|. Relates to->Has fallen because b is in the symmetrical direction. Vertical thermal conductivity is

And the non-magnetic heat flux coefficient is

Wherein the method comprises the steps ofFor electrons, the constant is

(γ′ _0e ,γ′ _1e ,γ″ _0e ,γ″ _1e ,δ _0e ,δ _1e )

＝(11.92,4.664,21.67,2.5,3.7703,14.79)，

And for ions

(γ′ _0i ,γ′ _1i ,γ″ _0i ,γ″ _1i ,δ _0i ,δ _1i )

＝(2.645,2.0,4.65,2.5,0.677,2.7)。

Magnetization is calculated as x _α ＝ω _cα τ _α Wherein the cyclotron frequency is ω _cα ＝eB/m _α . Assuming hydrogen ions, the species collision frequency is

And

in these expressions, lnΛ is the coulomb logarithm, which is assumed herein to be equal to 10, and the temperature is in eV. It should be noted that a portion of the heat flux associated with ion-electron friction is omitted here; in this axisymmetric formulation, the moiety will be Where u=v _e -v _i . Assume a larger ω on the body of the Z-pinch _ce τ _e In the case of (2), this term falls.

The stress tensor is defined by the strain rate tensor

And five viscosity coefficients, eta ₀ 、η ₁ 、η ₂ 、η ₃ And eta ₄ The composition is formed. The uniform viscosity coefficient is

η ₀ ＝0.96n _i k _B T _i τ _i ， (14)

Wherein the method comprises the steps ofBy using 2 omega _ci Replacement of omega _ci From eta ₂ And eta ₄ Obtaining odd coefficient eta ₁ And eta ₃ The method comprises the steps of carrying out a first treatment on the surface of the I.e. eta ₁ ＝η ₂ (2x _i ) And eta ₃ ＝η ₄ (2x _i ). In this context, a cylindrical coordinate system is used with radial, azimuthal, and axial coordinates r, θ, and z, respectively. The magnetic field is strictly acquired in the azimuthal direction. With no change in azimuth direction and assumption of zero azimuth velocity, the component of the stress tensor is

Π _zθ ＝Π _θz ＝0， (21e)

Π _rθ ＝Π _θr ＝0。 (22f)

Such as Busen NeisseDiscussed and eta ₀ The proportional term corresponds to the stress associated with the compression or expansion of the plasma. And eta ₁ The proportional term is associated with the diffusion of the cross magnetic field and the step frequency set by the collisions, with the step equal to the Larmor radius (larmorradius). Coefficient eta ₃ Associated with a gyrovisconsity, which is a non-magnetic flux of momentum. And eta ₂ And eta ₄ The proportional term has disappeared. In this embodiment the electron viscosity (except for the isotropic viscosity applied for numerical purposes) is omitted, which is based on the electron viscosity coefficient for similar ionic and electron temperatures And->Respectively smaller than the ion counterparts thereof (m _e /m _i ) ^1/2 、m _e /m _i And (m) _e /m _i ) ^3/2 And further assuming that the electron and ion velocity gradients are comparable.

Three types of corrections to the businessman transmission coefficients are contemplated herein. The first is related to the assumption that the time scale is derived by a long coefficient compared to the particle impact time. Stress in the r-z plane caused by plasma compression or expansion is due to the unmagnetized viscosity η ₀ ＝0.96p _i τ _i And (5) adjusting. As described by the bescens-like,(compression) increasing stress(expansion) reduces stress. The magnitude of this stress is +.about +.>The physical mechanism is as follows. The continuity equation shows(ignoring the gradient of n); for dynamic time scale, the +.>Then, suppose that flux freezes into the fluid, +.> Assuming that the ion lamo rail size is small compared to the size of the compression or expansion zone, the increased magnetic field gives an increased lateral velocity and associated lateral energy and stress. This effect is manifested as temperature anisotropy, as observed in continuous kinetic simulations. The increased energy is set as tau _i Is divided between the transverse direction and the parallel direction within the time of (a); this process is called gyroreflex relaxation (gyroreflex). For τ _i ≈τ _dyn The lateral stress is about pδB/B. For example, for δb/b=1, the magnitude of the stress is similar to isotropic pressure. When τ is _i <τ _dyn At this time, the stress is reduced due to the rapid halving. However, when τ _i >τ _dyn The derivative effect is not physically strong when it is, giving a stress for δb/b=1 that is greater than the isotropic pressure. In the model implemented herein, the correction factor,

applied to eta ₀ Coefficients. Here the number of the elements is the number,is a representative ion collision time. The form used for this correction is similar to the parallel heat flux correction commonly employed in the modeling based on businessman in tokamak. By using a representative strike time, this correction is a global constant, rather than a global constant that varies depending on local plasma parameters. This isMethod requires tau _dyn Is defined in the specification. For modeling m=0z pinch instability τ _dyn Set as characteristic Alvin time τ _A Defined as the characteristic pinch radius divided by the alveoli velocity (see full definition herein).

The second correction and the third correction are related to breakdown of the boson inner stark model when the larmor radius is large compared to the length scale of interest. (there are no analogs of these corrections in the tokamak modeling community that normally assume small larmor radii.) one is a global correction,

wherein->And

otherwise, (24)

Based on representative lamo radius And the length scale of interest +.>Correction of linear dependence r even if the lamo radius is larger than the feature size _L Strong transmissions will also be allowed. The nonlinear dependence of equation (24) on the lamo radius is physically more reasonable. For->Correction is 1/2 and for +.>The correction becomes stronger (e.g. for +.>)。This correction applies to the rotational viscosity (. Eta.) ₃ ) And non-magnetic heat flux->And is also applicable to eta ₀ The associated transverse (in the r-z plane) stress, which is due to the scale length +.>Is related to the gradient induced change in lamo rail size. As for τ _dyn A priori specifies->In order to model the linear increase of modes with known wavenumber k, a judicious choice is +.>For non-linear modeling, the judicious choice is +.>Wherein k is _max Is the wavenumber at which linear growth is greatest. Because for k>k _max The linear growth rate tends to decrease so the choice ensures that the transmission correction is applied to the fast growth mode.

Another correction of the large lamo radius involves a special case around r=0, where the magnetic field is close to zero and the lamo track is no longer a simple spiral. Moving radially from r=0, the lamo orbit becomes finite and eventually it is possible to reach a value matching the radius. Using critical radius r _crit To approximate this position, r-correlation correction,

wherein r is<r _crit A kind of electronic device

Otherwise, (25)

For use inTransmission around r=0 is reduced, where the ion trajectory is no longer a simple helix. This correction applies only to η ₃ Andas mentioned in the discussion of equation (24) above, +.>The non-linear dependence of (c) gives a strong cut-off in case the local lamo radius exceeds the distance from the cylinder axis. It is worth mentioning that small magnetic field regions around r=0 are also expected to affect the sum η ₀ The associated transverse stress. However, instead of approximating zero at r=0, η is at some level representing the random walk process within the non-spiral track region ₀ Should match the vertical transport η ₁ . No associated correction is attempted here.

The combination of these three correction factors is sufficient to explore the fundamental impact of the businessman transmission on Z pinch stability. As described further herein, examples of correction factors are presented in the context of FuZE-like balancing.

The WARPXM modeling framework is used to solve a 5M2F model on an unstructured mesh of triangles using the Dragon-Kutta (Runge-Kutta) discontinuous Gaverken (Runge-Kutta discontinuous Galerkin, RKDG) technique. WARPXM uses MPI parallelization and the extensibility of RKDG is suitable for problems with larger dimensions.

Since explicit time stepping is employed, the fastest time scale must be resolved. For the hyperbolic phenomenon, the time step (Δt _hyp ) According to uDeltat _hyp /h _eff C is limited, where u is the wave velocity, h _eff Is the effective grid resolution, and C is Ke Lang number (Courant number) which depends on the particular longgrid-kuta scheme selected, but is typically +.1. In 5M2F applications, the speed of light c=1/(μ) ₀ ∈ ₀ ) ^1/2 Typically the fastest speed. To allow for a larger Δt _hyp The E can be increased manually ₀ To reduce c. To ensure system stability, c must exceed electron thermal velocity v _Te ＝(2k _B T _e /m _e ) ^1/2 . Reducing v by artificially enhancing electron mass _Te Δt can be further increased _hyp . For c and m _e The changes in (c) should be adjusted to maintain physical accuracy. The total Δt allowed is limited not only by the hyperbolic physics, but also by the oscillation and diffusion behavior. Oscillation with angular frequency ωFor diffusion behavior, the time step must be satisfied +.>Wherein D is the relative diffusivity, and C _D Is the number of diffusions Ke Lang in order units. For Deltat _diff The limitations of (c) may be severe and it is sometimes useful to artificially reduce the diffusivity when physically reasonable. Taking all these limitations into account, at each time step Δt=min (Δt _hyp ,Δt _osc ,Δt _diff )。

Three additional details of WARPXM need to be explained. First, special care needs to be taken to accurately calculate the required gradient in flux associated with the boson inner stark closure, as the dispersion of the primary variables is discontinuous. To solve this problem, the Brazilian Rainbow (Bassi-Rebay) method is used. Second, to accommodate the cylindrical coordinate system, vector calculus operations are written based on linear derivatives and source terms. For example, the divergence of vector A is typically written as

In certain embodiments of the axisymmetric system described herein, the azimuthal derivative is zero. The term involving the radial derivative can be rewritten to give

Next, the basis of WARPXMThe structure may naturally process the first two items and the last item is included as the source item. The convolution, tensor divergence, and vector gradient operations are also written in terms of straight-line derivatives plus cylindrical source terms. The third detail is related to the spatial integration of these cylindrical source items. The basic DG method implemented in WARPXM integrates source terms using quadrature (quadrature) based on values at Legendre Gao Siluo barton (LGL) nodes. The triangle has LGL nodes on the edges, and thus some nodes are at r=0. Orthogonalization is performed, followed by requesting computation of the source item at r=0. Some cylindrical source terms involve the first derivative of the primary variable divided by r. Applying the lobida rule rule) will require knowledge of the second derivative. To avoid the need for a second derivative, the LGL quadrature is replaced by a symmetrical gaussian quadrature that does not have a positive intersection on the triangle edge and thus avoids dividing by 0 at r=0.

Diffusion of the bennett Z pinch balance is the focus of modeling presented below. This balance can be achieved by pinching the radius (a) and the plasma current (I _p ) And (5) parameterizing. Azimuthal magnetic field (B) _θ ) Axial current density (j) _z ) And the total plasma pressure (p) is then given by:

FIG. 11 shows the magnetic field (B) _θ ) Normalized radial profile of density (n) and temperature (T).

Total pressure is contributed equally by ion and electron pressureComposition is prepared. Assuming a uniform temperature (T) such that the density (n for no shear equilibrium _i ＝n _e =n) and pressure (p/2=nk _B T) is proportional. Exactly half the plasma mass and current is contained within r=a. The total pressure at r=a is equal to the magnetic pressure at r=aThis balanced magnetic field, density and temperature profile is normalized by its corresponding peak value, shown in fig. 11.

Moment balance requirement in 5M2F model

And

except for the case of shear flow (see below), equal and opposite ionic and electronic axial velocities are assumed so as to balance E _r Zero. Thus v _iz ＝-v _ez ＝j _z /(2 en), the embodiments herein assume that there is a charge q _i ＝-q _e In the case of singly charged ions of =e, where e is the fundamental charge. Because j is _z And n have the same radial dependence, so these speeds are radially uniform. The total current satisfiesAnd the lorentz force balances the radial pressure gradient of each species. To establish with v _iz Balance=0, finite E, as is commonly assumed in static MHD balance _r It would also be desirable to perform charge separation to satisfy gaussian law. Some early work uses and v using 5M2F model _iz =0 and E _r Balance of =0; in this case there is an oscillation around the equilibrium state, but they may be benign from the point of view of growth rate determination.

Shear flow was added to the equilibrium as follows. The ion axial velocity is v _iz ＝j _z /(2en)+v _sf . Shear flow rates with a linear or parabolic radius are considered herein, i.eOr->Wherein->Is the shear flow rate at r=a. The required electric field is determined by equation (29), where n _i =n, remain unchanged without shear balancing. Gauss law is used to derive n _e . The axial current remains unchanged without shear balancing and v is set accordingly _ez . The electron pressure is then determined by equation (30). V for practical interest _sf The electron density and pressure are only slightly different from the no shear value (< 1%).

The perturbations used herein are radially located using known methods and contain an optional phase shift. By including a phase shift, the perturbation mode shape can closely match the final characteristic mode structure, depending on the applied shear flow profile and other plasma parameters. Disturbance density and current density are n+δn and j _z +δj _z Wherein

And

where k is the disturbance wave number. Phase shift is determined by the parameter phi ₀ Sum factor determination r ^ζ The method comprises the steps of carrying out a first treatment on the surface of the ζ=1 and 2 were used for simulations with linear and parabolic shear flows, respectively. Radial positioning uses the parameter b=a/3. The disturbance added to the ion and electron velocity is δv _iz ＝-δv _ez ＝δj _z /(2 en). The equilibrium magnetic field is unchanged, so Faraday's Law is unbalanced, so that the electric field immediately begins to evolve in response to the disturbance.

The computational domain is rectangular in the r-z plane. The axial length being adapted to the disturbance wavelength, i.e. L _z =2pi/k. The radial extent of the domain is 4a. This domain setting matches the previous work on m=0 instability analysis. The axial boundaries are periodic. At r=0, the standard axisymmetric boundary condition is used: the radial and azimuthal components of the vector quantity are zero, while the scalar and axial components of the vector quantity have no radial variation. At r=4a, a perfect sliding conductive wall boundary condition is applied. That is, the radial velocity, radial magnetic field, and axial electric field are zero. There is no radial change in density, pressure, axial velocity, radial electric field, and axial magnetic field.

Several characteristic quantities are used in the analysis and discussion below. Characteristic Alvin speed v _A ＝B _θ,pk /(n _pk m _i μ ₀ ) ^1/2 Wherein B is _θ,pk And n _pk Is the peak magnetic field and number density. The characteristic time is defined as τ _A ＝a/v _A . The ion heat velocity is v _Ti ＝(2k _B T/m _i ) ^1/2 . Ion lamo radius r _Li ＝m _i v _Ti /(eB _θ,pk ). The heat rate is related to the Alvin rate, which is Hydrogen ions are assumed.

For a given selection of a and I _p The pressure was determined, but the density and temperature were not specified. The density and temperature are established by specifying the ratio of pinch size to ionic lamo radius, a/r _Li . Using v _Ti Definition of (1) given for T _i Solution to (2)

Ion and electron density follow the relationship nk _B T＝p/2。

In an ideal 5M2F model, the kinetics are normalized, e.g. normalized to the instability growth rate γτ _A Depending on a/r _Li But a and I _p The particular choice of (c) is not critical. However, the boson endo-stokes transport depends on the plasma properties (density, temperature and magnetic field), and in the application of 5M2F with boson endo-stokes closure discussed below, a, I _p And a/r _Li All specified.

Sotnikov et al [ V.I.Sotnikov, I.Paraschiv, V.Makhin, B.S.Bauer, J.N.Leboeuf and J.M. Dawson, "Linear analysis of sheared flow stabilization of global magnetohydrodynamic instabilities based on the Hall fluid model," plasma physics 9,913 (2002) ]The linearized hall MHD model was applied to study the increase in m=0 instability in the present inner-ter equilibrium, using various hall parameter intensities. Hall effect usage e _Sot. ＝c/(ω _pi R) parameterization, wherein ω _pi ＝[n _i0 e ² /(∈ ₀ m _i )] ^1/2 Is the ion plasma frequency, and R is the radius of the modeled domain, and n _i0 Is the ion density at r=0. In order to E _Sot. And parameter a/r _Li In relation, note that Sotnikov et al use r=3a. Using r previously introduced _Li And v _Ti Obtaining the relationshipThe situation E of Sotnikov et al _Sot. =0.1 and 0.01 as reference cases; these cases correspond to +.>And 23.57. It should be noted that the characteristic time τ used by Sotnikov et al _Sot. ＝R/v _Ti And Ala (Ala)The Erwining time is related to->For E without shear flow _Sot. =0.1, the rate of increase at ka=10/3 is γτ _A =1.27, and at e _Sot. In the case of =0.01, the growth rate is γτ _A =0.80. In E shaped _Sot. When=0, the hall MHD model is reduced to the ideal MHD, γτ _A ＝0.73。/>

In an ideal 5M2F simulation without shear flow, close agreement with MHD and hall MHD results can be seen. The linear growth rate of m=0 instability is determined by the linear growth phase in the nonlinear simulation. In the case of normalized axial wave number ka=10/3, a/r in the range of 4 to 200 is shown in fig. 12A _Li Is a growth rate of (2). For these simulations, the mass ratio was m _e /m _i =1/100, disturbance level e=10 ^-3 . At the slave time t ₀ To t ₁ During an exponential growth period of (2), the growth rate is obtained by taking into account the volumetric integrated radial kinetic energy of the ionic fluidWherein v is _ir Is the radial velocity of the ions.

The growth rate is calculated as

In FIG. 12B, ln (KE _i,rad. /ME ₀ ) Is an example time trace of (1), where ME ₀ Is the initial magnetic energy.

Fig. 12A and 12B show ideal 5M2F results at ka=10/3. FIG. 12A shows a/r in the range of 4 to 200 _Li Is a growth rate of (2). At a large a/r _Li The 5M2F growth rate is consistent with that obtained by Sotnikov et al for pure MHD (dashed line). For smaller a/r _Li The 5M2F results were close to the corresponding Sotnikov et al results. FIG. 12B depicts the product normalized by the initial magnetic energyAn example time trace of radial ion kinetic energy is divided. The vertical dashed line covers the period of measuring a linear increase, and the dashed line shows an exponential increase of the measurement.

Again for ka=10/3, the results of the relatively large lamo radius like scheme are shown in fig. 13A. For the followingUsing m _e /m _i The result of =1/100 is similar to that with a true mass ratio m _e /m _i As a result of=1/1836, the deviation is a/r _Li Up to about 30% at about 1.2. The rapid increase occurs with +. >Is defined in the above. This rapid increase is due to electron drift instability, as explored further below. For a/r _Li Radial ion kinetic energy increase is plotted in fig. 13 b=1.2. At the end of the simulation, at t/τ _A A rapid energy increase was observed at ≡ 2. Examination of the solution shows that the rapid growth is due to the development of high-k modes of the wavelength of the computational grid setup. Optimizing the grid results in a faster, higher k-pattern and is thus contrary to the goal of identifying the growth rate of the perturbed ka=10/3 pattern. For->In the simulation under the circumstance, the linear growth period of the disturbance mode is very short, so that it is difficult to accurately determine the simulation growth rate; for a/r _Li The increase rate obtained for the case of < 1.3 has an error of about 10%.

FIGS. 13A and 13B show ideal 5M2F results at ka=10/3, focusing on small a/r _Li . FIG. 13A shows a/r for two electron ion mass ratios in the range of 0.75 to 4 _Li Is a growth rate of (2). For a/r between 1.5 and 4 _Li The growth rate is slowly changing. For a/r less than about 1.1 _Li A faster increase was observed. Fig. 13B depicts an example time trace of integrated radial ion kinetic energy normalized by initial magnetic energy.

A/r is performed at ka=20/3, 40/3 and 80/3 _Li Is included in the scan pattern. All of these simulations use m _e /m _i =1/1836. The results are shown in fig. 14. Regions with rapid growth due to electron drift instability remain in the regionIs a kind of medium. No growing region appears, wherein->And expands as the number of disturbance waves increases. For a given ka, with r _Li Near the mode wavelength, the growth becomes zero. In the case of ka, a/r is lower than the expected damping increase _Li Can be approximated as (a/r) _Li ) _damp And (2 pi). At ka=40/3, for example, (a/r) _Li ) _damp And ≡ 2.11, consistent with the results seen in figure 14. Zero growth region slave (a/r) _Li ) _damp Extends to a/r _Li And ≡ 1.8, in which electron drift instability occurs.

Fig. 14 shows ideal 5M2F results with several disturbance wave numbers. The fast growth mode is continuously inBut as the number of disturbance waves increases, the region without growth appears and widens.

For m _e /m _i =1/100 and ka=10/3 for a/r _Li The density structure of the linear growth pattern is shown in fig. 15, =4, 10, and 50. At large r _Li A slight drift of the mode structure in the +z direction is visible below; at a/r _Li At=4, the total shift is about 1/8 of the axial wavelength. At large r _Li The tilt is seen below, where the structure at the larger radius drifts more in the +z direction than at the smaller radius. The spatial resolution of the result in fig. 15 is a 64×16 (radial×axial) region with third-order spatial accuracy.

FIG. 15 shows the ratio at a/r _Li Ideal 5M2F mode structure at=4, 10 and 50. Plotting the equilibrium density relative to the peak in the r-z planeThe logarithm of the density variation of the degree normalization is based on 10. At a/r _Li At=50 (MHD-like), a single lobe (lobe) indicated by a dashed oval dominates the radial structure. At a smaller a/r _Li There are two lobes, with a/r _Li The case of=4 is indicated by a dashed oval: the smaller lobes exist near the axis and are not aligned with the main lobe.

The rate of increase value shown in FIG. 12A hasIs a function of the error of (a). Three known error sources are considered to draw this conclusion:

spatial resolution: the scan uses a base spatial resolution 40 x 10 (radial x axial) region with third order spatial accuracy. At a/r _Li At=4, 10, and 50, convergence behavior with respect to spatial resolution was studied. In each case, the error at the base resolution is<0.1%. As seen in fig. 15, the feature size is half of the axial domain and is represented by five regions. By a third order representation (using a quadratic polynomial) the accuracy is very good. The radial representation of the pattern is as good as or better than the axial representation.

Mass ratio: using enhanced electron mass m in scanning _e /m _i =1/100. Using the true mass ratio (m _e /m _i =1/1836) is simulated at a/r _Li Run =4, 10 and 50 and used to determine the error associated with enhanced quality. At a/r _Li At=4, 10 and 50, the errors were 1.0%, 0.6% and 0.8%, respectively. At m _e /m _i In the case of =1/100, the combined ionic electron fluid density in the 5M2F model is 1% greater than the density of the MHD fluid, so an error of about 1% is expected. With a strong dual fluid effect, the enhanced electron mass can cause larger errors, as can be seen in fig. 13A.

Light velocity: the light speed is set to 3v in scanning _Te . At a/r _Li At=4, 10 and 50, the simulations were run at double and re-double speeds of light in each case, up to 12v _Te . Within this range, the measured rate of increase varies at a/r _Li All three of (3)At a value of<0.5% indicating that the reduced electromagnetic wave velocity has only a slight effect on the modeling instability.

The ideal 5M2F simulation with ka=10/3 mode of linear and parabolic shear flow profiles gives similar results to previous operation with MHD and hall MHD. Figures 16A and 16B show the m=0 increase rate in the shear flow intensity scan,will come from having a/r _Li The growth rate of the 5M2F model=50 is compared with the MHD result and will have a/r _Li 5M2 f=2.357 is compared with hall MHD results using equivalent hall parameters. As shown, at a/r _Li The MHD-like 5M2F modeling at=50 provides similar results in nature as the ideal MHD modeling with linear and parabolic shear flow profiles. At pinch edge speed +.>Complete stability occurs in the case of (a). In the presence of a/r _Li In plasma=2.357, 5M2F simulation with linear shear flow indicates stability requirement +.>(note that a similar simulation of a linear shear flow with hall MHD is not readily available.) in the case of a parabolic shear flow, 5M2F modeling shows that at +.>The lower part is provided with a/r _Li Plasma stability=2.357, similar to the linear shear flow case. The 5M2F modeling is very consistent with MHD and hall MHD results.

FIGS. 16A and 16B illustrate the ideal 5M2F results for linear and parabolic shear flow stability, respectively. All simulations have ka=10/3. The growth rate obtained using MHD and hall MHD modeling is shown for comparison. The scan shown in FIG. 16A comprises a scan with reverse shear flow, labeled "negative"; all scans in FIG. 16A have positive +.>

Previous work with hall MHD (and PIC results discussed herein) uses positive shear flow, i.e Positive values of (2). It is expected that the dual fluid effect will depend on the shear flow direction, scanning with minus +.>I.e., reverse shear flow is complete, as shown in fig. 16A. When the flow is reversed, the stabilizing effect of the shear flow is reduced, and +.>Is necessary for stability. With a/r _Li The rate of increase of the shear flow, positive or negative, will approach MHD results independent of the direction of the shear flow.

FIG. 17 shows the pattern structure in an ideal 5M2F modeling with shear flow. Shows the results of linear (top row) and parabolic (bottom row) shear flows, as indicated for MHD-like (a/r _Li =50) and large lamo radius (a/r _Li =2.357) of the plasma. All cases use positive shear flow (in the sense discussed in the text), except for the case where one negative linear shear flow is labeled "negative". In all cases, the shear flow velocity at the pinch edge was

The shear flow axially stretches the exponential growth mode. FIG. 17 shows a flow rate for a flow with shear flowIs a simulated pattern structure of (a). Compare a given a/r _Li Linear result and parabolic junction of (2)As a result, parabolic shear flow limits the mode structure to smaller radii. For having a/r _Li Linear shear flow=2.357, showing the results of both positive and negative shear flow. In the case of negative shear flow, alignment of the paraxial lobes of the structure with the main lobes appears to promote the observed larger growth rate.

In order to accurately capture the steady exponential growth in the presence of such stretching, the resolution and perturbation size and shape are carefully selected. The complete development of the mode structure is about 5 tau depending on the plasma parameters and shear flow details _A Or longer. If the disturbance level is e=10 ^-3 As used in the no-shear simulation, then nonlinearities may affect the mode growth prior to full-scale development. Therefore, in these shear flow simulations, less disturbance is used. In these cases, higher resolution is also used, both to limit noise, which may obscure the growth of small perturbations, and to minimize accidental seeding of short wavelength modes, which may grow rapidly, and to interfere with development at the axial wavenumber of the perturbations. At a/r _Li In the case of=2.357, the problem of the short wavelength mode is small because the strong double fluid effect suppresses the high-k growth. To further alleviate the difficulties of high-k growth and nonlinearity, phase shifts are included as discussed elsewhere herein. By phi ₀ Perform preliminary simulation to obtain phase shift parameter phi =0 ₀ Is observed for pattern stretching in the late stages of the simulation before the growth is interrupted by a nonlinear or high-k pattern growth. For the second and final simulations, phi is selected ₀ Such that the shape of the initial conditions approximately matches the stretch pattern seen in the preliminary simulation. Fig. 18A to 18D show examples of mode growth with and without phase shift initial conditions. Without phase shift, mode stretching results in a steady decrease in growth rate before the simulation is eventually destroyed by high-k growth, at t≡18τ _A The following can be observed. By including a phase shift, the simulation quickly enters a linearly increasing period. The line in FIGS. 18A and 18C is denoted by τ _A Data calculated at intervals of/10. In fig. 18B and 18D, γτ at time t is calculated using equation (34) _A Wherein t is a value of ₁ T and t ₀ ＝t-2τ _A 。

Class MHD 5M2F simulation (a/r _Li =50) all uses with five-order elements and e=1×10 ^-5 Is a 64 x 16 (radial x axial) region of (c). Large r _Li Simulation (a/r) _Li =2.357) using a four-order element with a sum of e=1×10 ^-7 Is a 64 x 16 (radial x axial) region of (c). The phase shift is given in table I.

Table I: ideal 5M2F simulated setup for shear flow. Giving a phase shift parameter phi for each run with shear flow ₀ 。

For linear shear flow, the total phase shift is φ ₀ r, and for parabolic shear flow it is phi ₀ r ² The method comprises the steps of carrying out a first treatment on the surface of the See equations (31) and (32).

Deviations from the exact linear growth rate in these simulations are estimated to be no more than 10% and typically no more than a few percent or less. This level of error is higher than the no-shear result, mainly due to the challenges associated with avoiding both high-k modes and nonlinearities.

Fig. 18A to 18D show the pattern growth behavior in a 5M2F model with and without an initial phase shift (effectively tilting) in the perturbation. Results simulation for MHD-like scheme (a/r _Li =50), including having pinch edge speedIs a linear shear flow of (c). Fig. 18A shows the evolution of radial ion kinetic energy normalized by the initial magnetic energy. FIG. 18B shows the derivation at 0.1 τ _A Normalized growth rate γτ of kinetic energy of interval growth _A . The method for deriving each plotted value (x) is discussed in the main text. The red line shows 1- τ for a single value _A Moving average. FIGS. 18C and 18D classSimilar to fig. 18A and 18B, but for a phase shift phi with perturbation ₀ r, wherein phi ₀ ＝2π。

In a recent study using PIC modeling, m=0 instability was considered in the FuZE-like balance. In the discussion that follows, a balance is set forth, and details regarding the implementation of the boson inner-stokes transmission in this balance are provided below. Also below, 5M2F modeling with boson inner stokes transmission was compared to PIC results.

This Butt balance is used to represent a typical FuZE plasma. As discussed in detail above, the selection of I may be made _p A and a/r _Li To give the dimensions of the normalized cross-section in fig. 11. After previous work, fuZE-like equilibrium has a/r _Li =5.825, a=0.91 mm and I _p =300 kA. These choices give B _θ,pk ＝33.0T、n _pk ＝4.25×10 ²⁴ m ^-3 And t=1.27 keV. The associated characteristic time and alveol velocity is τ _A =2.61 ns and v _A ＝3.49×10 ⁵ m/s。

Presented herein are closed-form models of businessman including kinetic corrections. Here, the transmission coefficients are specifically considered in the FuZE-like balance before and after correction. Fig. 19 shows corrected and uncorrected viscous diffusivity. Correction is performed according to equations (23), (24) and (25). In the equation (23) for the case of the vehicle,for τ calculated as peak density _i And τ _dyn ＝τ _A . In equation (24), the ∈ ->Is r _Li The characteristic ion lamor radius discussed herein, and l is selected to be 0.2a, corresponding to k ^-1 Wherein ka=5. In equation (25), r _crit =0.5a, which corresponds to position r=0.31a, where the ion lamo radius matches the radius itself. The diffusivity (ρ=m) is obtained by dividing the viscosity coefficient by the mass density _i n _i ). Discussed belowThe bussinesike heat transfer coefficient of (c) is also expressed as diffusivity to facilitate comparison. Correction factor->Is found to be->Also performs a diffusivity limit D _lim Wherein at r=a, D _lim Equal to->The diffusivity limit is imposed purely for numerical reasons: and (3) withThe associated diffusion typically sets the maximum time step available for simulation. By applying D _lim The dynamics of interest remain, but the time step can be significantly increased. Correction factor->Is found to be->Correction factor->Is found to be->

Fig. 19 shows the momentum diffusivity in the FuZE-like balance. Uncorrected diffusivity (without ^* Coefficients of (c) have non-physically and/or numerically challenging characteristics. As described in the main text, the corrected diffusivity (with ^* Coefficients of (c) to solve these problems. Correction is performed according to equations (23), (24) and (25), where l=0.2a and r _crit ＝0.5a。

Fig. 20 shows corrected and uncorrected ion thermal diffusivity. For momentum diffusivity, useAnd r _crit Correction was performed=0.5a. A diffusivity limit for momentum diffusivity is also imposed, although it affects +.>Electron thermal diffusivity is not shown; in the businessman electron transport term, only the non-magnetic heat flux term is retained and the associated coefficient +.about.except for very close to r=0>Almost identical to its ionic counterpart. In the zone, < >>Radial variation ratio +.>More extreme. Does not apply +.>Because of the correction factor of->The characteristic electron lamor radius is smaller than the others. The position where the electronic lamo radius matches the radius itself is 0.09a, so r is used _crit It will be reasonable to=0.2a. Using such a small r according to equation (25) _crit And use->Produce->Which peak at r=0.15 is close to 300m ² ·s ^–1 . Select r _crit =0.5a to get +.>Is easier to handle numerically, wherein the peak at r=0.4 is close to 120m ² ·s ^–1 . After r=0.4 and +.>Almost the same uncorrected +.>And consistent.

Fig. 20 shows the thermal diffusivity of ions in the FuZE-like equilibrium. Uncorrected diffusivity (without ^* Coefficients of (c) have non-physically and/or numerically challenging characteristics. As described in the main text, the corrected diffusivity (with ^* Coefficients of (c) to solve these problems. Correction is performed according to equations (24) and (25), whereinAnd r is _crit ＝0.5a。

Simulation indicates that in all the Brissen Nernst transmission terms, the transmission term is represented by eta ₀ The adjusted transverse stress has the strongest effect on the m=0 unstable growth. To investigate η compared to cyclotron viscosity and non-magnetic heat flux ₀ Three series of simulations were performed at a fixed wavenumber ka=5. The results are presented in fig. 21. To scan eta ₀ For intensity purposes by a single multiplier f _η0 Replacing the product of the two global correction factors, i.e.The multiplier is used to calculate the modified unmagnetized viscosity: In the first series, the complete boson-inner-stark model mentioned above is applied, wherein r is based on _crit The correction of the large larmor radius is applied to the gyratory viscosity and the non-magnetic heat flux coefficient, but the correction of the large larmor radius is not applied to these coefficients. In the first placeIn the two series, the gyratory viscosity was omitted, but the non-magnetic heat flux was retained. In the third series, the non-magnetic heat flux was omitted, but the gyratory viscosity was retained. Using a complete boson inner stark model, at f _η0 At approximately 0.1, the modeled growth rate approximates the PIC result. At a small f _η0 In the case of (a), the gyratory viscosity and the non-magnetic heat flux contributions each decrease the growth rate moderately (10%) from the ideal 5M2F value. When f _η0 Approximately 0.1, the effect of the gyratory viscosity is reduced (i.e., the "no gyratory viscosity" result approximates the "complete buson's" result in fig. 21), but the effect of the non-magnetic heat flux remains moderate (i.e., the "no non-magnetic heat flux" result remains approximately 10% higher than the "complete buson's" result in fig. 21). The 5M2F result shown in fig. 21 uses the true electron mass (M _i /m _e 1836), perturbation size e=10 ^-3 And a 40×10 (radial×axial) region having a third order.

Fig. 21 shows the increase rate of 5m2f m=0 instability with the magnetization free viscosity coefficient η at ka=5 ₀ Is the total multiplier f of (2) _η0 But vary. The results of the complete boson inner stokes model, the complete model minus the cyclotron viscosity term, and the complete model minus the non-magnetic thermal flux term are shown. Results at ka=5 from PIC modeling by tuneel et al [ K.Tummel, D.P.Higginson, A.J.Link, A.E.W.Schmidt, D.T.Offermann, D.R.Welch, R.E.Clark, U.Shumlak, B.A.Nelson, R.P.Golingo and h.s.mclean, "Kinetic simulations of sheared flow stabilization in high-temperature Z-pin plasma," physical plasma 26,062506 (2019)]For comparison.

Examining the dependence of growth rate on k shows that applying the 5M2F model with corrected buson's transmission can result in a result that shares some features of the PIC result. As shown in fig. 21, has f _η0 The complete boson inner stark model =0.06 yields a growth rate similar to PIC. (note that the overall factorDiffusivity for correction as shown in figure 19) To fix f _η0 Scan ka, γτ=0.06 _A Ka rises and shows a symbol that does not reach a peak even at ka=23, as shown in fig. 22. At f _η0 In this scan of=0.06, correction of the large larmor radius is not used. At large ka γτ _A The cause of the rise is not clear. Additional simulations indicate that the rise at high k persists even though the gyratory viscosity and non-magnetic heat flux are omitted. The results show that a large eta is used ₀ (not physically large, as discussed above) is not a suitable method of reproducing PIC results over a broad range of ka. The results of a scan employing the complete size-based correction model discussed above are also shown. In the scan, +_>Wherein->And->Applied to eta ₃ And->With the size-based model, the reduction of growth at high k is restored. However, it is worth noting that the results are similar to the ideal 5M2F results, which are shown for comparison. Finally, scanning is accomplished by designing a transmission correction for nonlinear modeling; see the basic principle discussion herein. In a non-linear setting, < > a->And minimum diffusivity D _min ＝0.01av _A Applied to eta ₁ And->Also comprises a D-based _min Is a polymer having an isotropic electronic viscosity. According to equations (24) and (25), including correctionWherein r is _crit =0.5a and fixed feature size +.>With this nonlinear arrangement, the growth rate is reduced, resulting in a higher than if the true electron mass and D were used _min The result is more similar to the PIC result when=0. The mass ratios used for the simulation in fig. 22 are shown in the legend. The disturbance is again e=10 ^-3 And the resolution is a 40×10 area with a third order.

Fig. 22 shows the 5m2f m=0 instability increase rate at the mode wave numbers ranging from ka=5 to 23. Results from PIC modeling by tunel et al are included for comparison. Using a constant viscosity correction factor f _η0 =0.06, the growth rate does not physically increase at high k. Using size-based modelsCalculating f _η0 The peak growth rate is seen at ka≡12 and decreases in the ideal case as k increases. The results of the corrected transmission model for nonlinear modeling are also shown: enhanced electron mass (m _i /m _e =1/100), based on a fixed size +.>F of (2) _η0 And a diffusivity minimum D _min ＝0.01av _A . The nonlinear setup provides a growth rate most similar to the PIC result.

The above model is more strictly nonlinear, but the results presented so far only consider the linear approach of an unstable growth, i.e. growth results in a small deviation from the equilibrium state. In this section, consider a simulation that tracks the plasma's progression to a nonlinear regime. To achieve this nonlinear modeling, a combination of high resolution and artificial large vertical transport is used. The numerical method readily available via WARPXM does not itself provide strong numerical dissipation in the event that the gradient becomes steep, and the polynomial representation is susceptible to non-physical oscillations (similar to Gibbs) phenomenon) that may terminate the simulation. The artificial vertical viscosity and thermal conductivity help mitigate sharp gradients, and the high resolution helps solve these problems.

In the simulations presented herein, the minimum diffusivity, D _min ＝0.01av _A ≈3.2m ² ·s ^–1 Applied to eta ₁ Andand for isotropic electronic viscosity:Wherein ρ is _e ＝m _e n _e . As seen in fig. 19 and 20, η ₁ Rho and->Approximately 0.04 and 0.27m at r=a, respectively ² ·s ^–1 . These correspond to Reynolds numbers, respectivelyAnd Peclet number->The artificial minimum diffusivity reduces it to +.>At characteristic time and feature size set to τ _dyn ＝τ _A And->The dynamic correction according to equations (24) and (25) is performed. Additional corrections are specifically made to this nonlinear modeling to prevent false numerical behavior near the outer radial boundary associated with the nonmagnetic heat and momentum flux term. Will be a factor f _trunc Cutoff η applied in the outer region of a plasma ₃ And->In particular, the method comprises the steps of,

wherein r is>r _trunc A kind of electronic device

f _trunc Otherwise, =1, (35)

Wherein r is _max =4a and r _trunc =2a. The difficulty of non-magnetic flux is associated with boundary conditions on the derivative of the 5M2F variable. In principle, good boundary conditions are possible; however, the truncation described above enables accurate simulation of the region of interestIs a non-linear plasma dynamics in (a).

The same initial conditions were used as for PIC comparison at ka=5, but in the case of parabolic shear flow application, the simulation was performed with resolution increased to 96×24 (radial×axial) regions with fourth order elements. No phase shift is used in the disturbance. A series of parabolic shear flow intensities were studied, in which =0, 0.25, 0.5 and 0.75. For the followingThe evolution of the two-dimensional density profile is depicted in fig. 1 and 3 for the case of =0 and 0.5, respectively. As the shear flow velocity increases, the radial injection of the plasma is limited. This is further illustrated in fig. 23, where the normalized pinch ion inventory and total thermal energy are plotted as a function of time. Pinch ion inventory and thermal energy are defined as

And

the normalized amounts are N (t)/N (t=0) and W (t)/W (t=0). In the absence of shear, the pinch has lost more than 50% of its initial inventory at the end of the simulation, while at the endIn the case of ≡ 30% of the initial inventory is lost. The heat energy loss without shearing is ≡20% and at +.>In this case ≡10%.

Fig. 23 shows a non-linear simulated ion density profile starting from a FuZE-like equilibrium with seed pattern ka=5. The profile is axially repeated three times and reflected across r=0. In the case of zero shear flow (top row), by t=10τ _A Radial jet-like structures have been developed. At the position ofIn the case of (bottom row), the growth delays, the unstable structure is sheared and shear limited in the radial range, and the relaxed near equilibrium state passes t=35τ _A And (5) establishing.

FIG. 24 depicts a schematic representation of a system havingFuZE-like nonlinear simulations of =0, 0.25, 0.5, and 0.75 normalized ion inventory and thermal energy within r=a by initial inventory and energy normalization. In all cases, ka=5. The inclusion of shear flow improves particle and energy confinement.

The results presented here are shown at a small a/r _Li Lower rapidly growing instability. As can also be seen in the previous 5M2F modeling, instability is identified as an electron drift pattern. There are two important electron drift instabilities (sometimes referred to as micro-instabilities), where the electron cross-field drift velocity (v _dr ＝|v _e -v _i I) approximates or exceeds v _Ti : lower mixing drift instabilitySex (LHDI) and ion acoustic instability. In the 5M2F balance used herein, the radially uniform electron drift velocity is v _dr ＝j _z /(en). Using the balance expression for j _z And n, thereby giving

Equation (26) where r=a is evaluated for using peak B _θ Replacement I _p And apply r _Li Is defined in (a). Thus, in having a/r _Li <2 such that v _dr >v _Ti . This expectation and atThe existence of the fast-growing mode seen below in fig. 13A through 14 is consistent. Most likely, the observed instability is not LHDI, as LHDI occurs at less than r _Le Is at a wavelength of (c). At a/r _Li At =1.8, r _Le A.apprxeq.0.013 (where m _i /m _e =1836). The shortest disturbance wavelength considered has a ka=80/3 and a ratio r _Le The associated wavelength is approximately 18 times larger (λ/a=2pi/ka approximately 0.24). Instead of LHDI, the instability may be ionoacoustic instability. As discussed, at T only _e ＞＞T _i Such instability is expected in single ionization plasmas. For T _e ＝T _i As in the plasma considered herein, the speed of sound is comparable to the ionic thermal speed, and the sound waves are Landmark damped. Since the 5M2F model does not capture langerhans damping, it may be allowed to have T _i ≈T _e Ion acoustic instability in the scheme of (a). In future work, the ion acoustic mode and LHDI behavior in the 5M2F model can be considered in more detail.

In the presence of positive shear flow, a/r is utilized _Li The 5M2F model at=2.357 models linear growth, yielding results almost identical to hall MHD modeling. Whether linear or parabolic shear flow profile, is expected forComplete stability will occur. This result is consistent with the PIC result of shear flow stability, which indicates that FuZE-like plasma is at +.>And (3) stabilizing. At negative shear flow (i.e., ion flow is zero at r=0 and for r >0, opposite to the current direction), the stabilizing effect is weaker and is expected for +.>Stability occurs. In SFS Z pinch experiments, the shear flow velocity observed at the pinch edge during steady plasma operation is typically +.>Sometimes a velocity gradient near the pinch edge is observed even steeper than a parabolic profile, so this concentrated gradient may play a role in the observed stability. Yet another possibility is that the experimental shear flow does not stabilize the bennett profile; in contrast, the observed m=0 stability may occur because the profile relaxes, making it (m=0) stable with a lower shear level.

The 5M2F model with true ion-to-electron mass ratio, with or without corrected boson inner stokes transport, gives different results than the PIC results, as shown in fig. 22. In particular, the 5M2F growth rate peak is two times higher and the peaks appear at ka≡10 and ka≡5 in the PIC results. The 5M2F results do reproduce the growth rate flip at high k and have a similarity in nature to the PIC results. Understanding the specific causes of peak rise and drift in 5M2F modeling requires further investigation, but the kinetic effects appear to be functioning. Cyclotron kinetic modeling shows close agreement with PIC even with capturing some but not all large r _Li The same is true of electrostatic models of effects and of simplified cyclotron kinetic Poisson (Poisson) equations (e.g., not containing cyclotron orbit averaging). These results are combined with Hall MHD, ideally 5It is interesting to compare M2F with Kelvin-Helmholtz instability (KHI) analysis of continuous kinetic modeling. In this work, the KHI increase rate of hall MHD is higher than the kinetic result. The 5M2F growth rate is higher than the kinetics, but is closer than the hall MHD results.

The nonlinear modeling results shown in fig. 3 indicate that after initial instability, the flow was measured at moderate shear (withParabolic) the plasma establishes a quasi-stable configuration with more than half of the original pinch mass and more than 80% of the initial pinch energy. Further investigation is necessary to see if such self-organizing behavior might resemble actual experimental behavior.

Previous MHD modeling has indicated that shear flow stabilizes the short wavelength m=0 mode more effectively than long wavelengths. The same setup as the above-described ka=5 nonlinear simulation was used to run a shear flow with ka=5/3 and a moderately parabolic curveTo explore the nonlinear constrained behavior of longer wavelength instabilities. Instability grows faster and mass and energy losses are earlier, but the total losses are comparable. In addition, to represent a more realistic noisy Z-pinch plasma, the "multimode" case is operated with ka.ltoreq.25. That is, the domain length is such that L _z A=6pi/5 (capture ka=5/3 of one wavelength); until each of the available modes (for which 15 wavelengths are captured) of ka=25 is perturbed. For each mode, the phases in equations (31) and (32) are chosen randomly ₀ Set, and no radial dependence (ζ=0). The result is an increase in mass and energy loss, which may be due to mode coupling. In all cases, however, the quality is lost<50% and energy loss<20%. The general case is the same as the case of ka=5: moderate shear flow appears to promote the development of a quasi-stable plasma with limited losses.

The modeling presented here focuses on the Z-pinch balance using a bennett profile with a uniform initial temperature. Studies considering other profiles including those matching the best available experimental data may provide different m=0 stability behavior, including stability effects due to pilot center drift, even in the case of zero body flow. It is also well known that profiles meeting the card-east Mu Saifu standard (Kadomtsev criterion) are MHD stable. The stability of the MHD stability profile in a non-MHD model (e.g. a 5M2F model) may be considered in future work. Another consideration is that even a profile stabilized by the carton Mu Saifu may be unstable for so-called entropy modes, but for gases with actual adiabatic coefficients (for real gases with three or more degrees of freedom Γ+.5/3), entropy mode instability only occurs when the temperature profile is non-uniform. In the most recent modeling, the entropy mode is considered in the special case of Γ >2, with a uniform temperature in the present Nernst section. A 5M2F modeling of entropy mode behavior in MHD stability profiles with non-uniform temperature and Γ+.5/3 would be interesting.

The 5M2F model is presented and the ideal model is extended to contain a boson-stokes based closure. The model was used to study m=0z pinch instability, focusing on initial conditions based on bunte equilibrium.

The ideal 5M2F modeling results are based on previous MHD and hall MHD results. The growth rate closely matches the previous results with and without shear flow. At the position ofComplete stability occurs at the edge flow velocity of (c). In addition to the previous work, reverse linear shear flow was studied, revealing that the growth rate decreased relatively slowly with increasing flow velocity, and was expected to be at ≡1.5v _A Is completely stable at the edge flow velocity. Another interesting feature of the 5M2F results is that, inThe electron drift instability is seen and temporarily identified as an ion acoustic mode.

The 5M2F model also comprisesThe latest PIC modeling initialized with this inter-body equilibrium of class FuZE is used as a benchmark. The peak increase in the 5M2F result is about twice faster than in the PIC (γτ _A Approximately 1.5 and 0.77) and the peak at k is approximately twice larger (ka approximately 10 and 5). The 5M2F result is similar to the PIC result, such that the growth rate peak occurs at moderate k, and then drops to a higher k. The application of the boson-inner-stark based transport model does not change this behavior. More generally, exploring businessman-based transport in a 5M2F framework provides insight into relevant physics, including cyclotron viscosity, non-magnetic heat flux, and ion cyclotron relaxation effects. Electron cyclotron relaxation is marked as a potentially important effect for future consideration. In addition to the 5M2F model, even with boson inner stokes transmission, dynamics appear to be responsible for the m=0 stability differences observed in PIC modeling.

Nonlinear modeling of m=0 instabilities in the FuZE-like plasma was done using a buson-based 5M2F model. By 0.5v _A The simulation of the edge shear flow velocity of (c) shows instability followed by non-linear mixing due to shear and relaxation to a quasi-steady state. The pinch ion inventory and pinch heat loss are limited to ≡ 30% and ≡ 10%, respectively.

The 5M2F model provides a convincing platform for high fidelity computing Z-pinch studies. In future work, various equilibrium profiles may be considered in addition to the bunte profile; a particularly interesting profile is the MHD stabilized karst Mu Saifu profile, the temperature gradient of which enables the entropy mode. Simulations of linear and nonlinear 3D Z pinch dynamics, including m=1 instability, are also plausible. Finally, the ability of 5M2F modeling to capture electron drift instability should be studied; accurate and efficient modeling of the associated micro-turbulence can yield important insights about current profiles and axial heat transfer in the reactor-level Z-pinch plasma.

For the balance of the shear flow, a true two-fluid balance is obtained. The total ion axial velocity is determined as v _iz ＝v _iz0 +v _sf Wherein(Linear) or->(parabolic). Here, v _iz0 ＝j _z /(2en _i ) Is the shear-free ion velocity which provides half of the equilibrium current. The magnetic field and current profile remains unchanged at the shear-free equilibrium profile of equations (26) and (27). The ion pressure and density also remain unchanged. Specifically, the ion pressure is half of the total pressure given in equation (28). The ion temperature is uniform and is a free parameter; herein, T _i Is according to a/r _Li And is set as shown in equation (33). Next, n _i ＝p _i /(k _B T _i ). The radial electric field of the ion momentum balance is determined according to equation (29). The result of linear and parabolic shear flows is

Gauss's law, equation (7) is used to determine n _e Thereby giving

The electron pressure is then determined according to equation (30). The result is

These pressures are combined with the shear-free pressure p at r=0 _e Match, and deviate slightly elsewhere. The electron temperature is determined as p _e /(n _e k _B ). By meeting j _z ＝v _ez n _e q _e +v _iz n _i q _i To determine the electron axial velocity.

Embodiments of the present disclosure may be described in view of the following clauses:

1. an apparatus, comprising:

a first electrode positioned to define an outer boundary of an acceleration volume;

a second electrode positioned to define an inner boundary of the acceleration volume;

at least one power supply to drive a current along a Z pinch plasma column between the first electrode and the second electrode;

A set of valves including at least one gas injection valve to provide neutral gas to the acceleration volume to fuel the Z pinch plasma column; and

a shaping portion conductively connected to the second electrode to cause gas breakdown of the neutral gas provided by the at least one gas injection valve in the presence of the neutral gas to produce a shear flow rate profile.

2. The device of clause 1, wherein the electron flow of the current is from the second electrode to the first electrode.

3. The device of any of clauses 1 and 2, wherein the shaping portion incorporates at least one conductive ring comprising at least one contact surface electrically connected to an outer surface of the second electrode.

4. The device of clause 3, wherein the at least one conductive ring incorporates a conductive material that is chemically and thermo-mechanically compatible with the conductor of the second electrode, and a plasma-facing portion of the at least one shaping portion incorporates at least one refractory metal.

5. The apparatus of clause 4, wherein the at least one refractory metal comprises one or more of W, ta, nb, mo, re, ti, V, cr, mn, zr, tc, ru, rh, hf, os, ir or an alloy of any one or more of the foregoing metals.

6. The apparatus of any one of clauses 4 and 5, wherein the plasma-facing portion incorporates at least one conductive form of carbon comprising one or more of graphite, sintered carbon powder, pressed carbon powder, carbon fibers, or carbon nanotube bonding structures.

7. The device of any of clauses 4-6, wherein the plasma-facing portion comprises at least one textured surface formed to incorporate a plurality of local concave elements forming a structured array to enhance local electric fields and promote electron field emission.

8. The device of clause 7, wherein the at least one textured surface has been formed by a mechanical treatment comprising one or more of cutting, scraping, sanding, sandblasting, grooving, embossing, stuccoing, embossing, or knurling.

9. The apparatus of any of clauses 7 and 8, wherein the at least one textured surface has been formed by a chemical process comprising one or more of etching, chemical deposition, spraying, sputtering, ion and neutral implantation, or epitaxial growth.

10. The apparatus of any of clauses 1-9, wherein the set of valves further comprises at least one plasma injector to provide ionized gas to the acceleration volume to further fuel the Z-pinch plasma column.

11. The apparatus of any of clauses 1-10, wherein the second electrode incorporates a conical electrode surface arranged to enhance momentum transfer to ions and neutral particles in an axial direction of the first and second electrodes.

12. The device of any one of clauses 1 to 11, further comprising a third electrode arranged between and coaxially with respect to the first and second electrodes, wherein the third electrode exhibits a conical electrode configuration and incorporates a conical electrode surface arranged to enhance momentum transfer to ions and neutral particles in an axial direction of the first, second and third electrodes.

13. A method, comprising:

activating one or more gas injection valves to introduce an axisymmetric volume of neutral gas into the acceleration volume;

Generating a radial electric field to support a first current by promoting breakdown of the neutral gas, the first current flowing between an inner electrode and an outer electrode via the introduced neutral gas; and

forming a Z pinch plasma column from the introduced neutral gas to support a second current flowing between the inner electrode and the outer electrode,

wherein the Z pinch plasma column is surrounded and stabilized by a shear velocity plasma stream formed at least in part by the neutral gas.

14. The method of clause 13, activating one or more plasma injectors to introduce an axisymmetric volume of ionized gas into the acceleration volume.

15. The method of clause 14, wherein the axialiy volume of the ionized gas is introduced after forming the Z pinch plasma column to replenish the axialiy volume of the neutral gas.

16. The method of any of clauses 13-15, wherein the inner electrode is an anode and the outer electrode is a cathode.

17. A plasma confinement system, comprising:

an external electrode;

an inner electrode;

at least one power source conductively coupled to each of the inner electrode and the outer electrode, a terminal of the at least one power source configured to generate a potential difference between the inner electrode and the outer electrode; and

One or more first valves fluidly coupled to a fuel gas supply and configured to direct sufficient neutral gas from the fuel gas supply to support a local breakdown path between the inner electrode and the outer electrode and establish a shear velocity plasma flow between the inner electrode and the outer electrode for the duration of a Z pinch discharge.

18. The plasma confinement system of clause 17, wherein the inner electrode and the outer electrode delimit an acceleration volume to which the neutral gas is directed by the one or more first valves.

19. The plasma confinement system of clause 17, further comprising an intermediate electrode,

wherein the inner electrode and the intermediate electrode delimit an acceleration volume to which the neutral gas is directed by the one or more first valves.

20. The plasma confinement system of any of clauses 17-19, further comprising one or more second valves fluidly coupled to the fuel gas supply and configured to direct sufficient ionized gas derived from the fuel gas supply to maintain the shear velocity plasma flow during the duration of the Z-pinch discharge.

21. An apparatus, comprising:

a second electrode coaxially arranged with respect to the first electrode and positioned to define an inner boundary of the acceleration volume;

at least one power supply to drive a current along a Z pinch plasma column between the first electrode and the second electrode; and

a set of valves providing gas to the acceleration volume to fuel the Z-pinch plasma column,

wherein electrons of the current flow in a first direction from the second electrode to the first electrode.

22. The apparatus of clause 21, wherein the gas comprises a neutral gas, and

wherein the device further comprises a shaping portion conductively connected to the second electrode to cause gas breakdown of the neutral gas in the presence of the neutral gas provided by the set of valves to produce a shear flow rate profile in a second direction opposite the first direction.

23. The device of clause 22, wherein the shaping portion incorporates at least one conductive ring comprising at least one contact surface electrically connected to an outer surface of the second electrode.

24. The device of clause 23, wherein the at least one conductive ring incorporates a conductive material that is chemically and thermo-mechanically compatible with the conductor of the second electrode, and a plasma-facing portion of the at least one shaped portion incorporates at least one refractory metal.

25. The apparatus of clause 24, wherein the at least one refractory metal comprises one or more of W, ta, nb, mo, re, ti, V, cr, mn, zr, tc, ru, rh, hf, os, ir or an alloy of any one or more of the foregoing metals.

26. The apparatus of any one of clauses 24 and 25, wherein the plasma-facing portion incorporates at least one conductive form of carbon comprising one or more of graphite, sintered carbon powder, pressed carbon powder, carbon fibers, or carbon nanotube bonding structures.

27. The device of any one of clauses 24 to 26, wherein the plasma-facing portion comprises at least one textured surface formed to incorporate a plurality of local concave elements forming a structured array to enhance local electric fields and promote electron field emission.

28. The apparatus of clause 27, wherein the at least one textured surface has been formed by a mechanical treatment comprising one or more of cutting, scraping, sanding, sandblasting, grooving, embossing, stuccoing, embossing, or knurling.

29. The apparatus of any of clauses 27 and 28, wherein the at least one textured surface has been formed by a chemical process comprising one or more of etching, chemical deposition, spraying, sputtering, ion and neutral implantation, or epitaxial growth.

30. The apparatus of clause 21, wherein the gas is provided to the acceleration volume as an ionized gas.

31. The apparatus of any of clauses 21-30, wherein the second electrode incorporates a conical electrode surface arranged to enhance momentum transfer to ions and neutral particles in an axial direction of the first and second electrodes.

32. The device of any one of clauses 21 to 31, further comprising a third electrode arranged between and coaxially with respect to the first and second electrodes, wherein the third electrode exhibits a conical electrode configuration and incorporates a conical electrode surface arranged to enhance momentum transfer to ions and neutral particles in an axial direction of the first, second and third electrodes.

33. A method, comprising:

activating one or more valves to introduce an axisymmetric volume of fuel gas into the acceleration volume; and

forming a Z pinch plasma column from the introduced fuel gas to support a Z pinch current flowing between the inner anode and the outer anode surrounding the unsupported end of the inner anode,

wherein the Z pinch plasma column is surrounded and stabilized by a shear velocity plasma stream formed from a fuel gas.

34. The method of clause 33, further comprising, prior to forming the Z pinch plasma column, generating a radial electric field to support an initial current flowing between the inner anode and the outer cathode via the introduced fuel gas.

35. The method of clause 34, wherein the fuel gas comprises a neutral gas, and

wherein the radial electric field supports the initial current at least by promoting breakdown of the neutral gas.

36. The method of any one of clauses 33 to 35, whereby introducing a fuel gas into the acceleration volume, the fuel gas comprising an ionized gas.

37. A plasma confinement system, comprising:

an external electrode;

an inner electrode concentrically positioned within the outer electrode;

At least one power source conductively coupled to each of the inner and outer electrodes, terminals of the at least one power source oriented to generate a potential difference between the inner and outer electrodes to drive electrons from the inner electrode to the outer electrode; and

one or more valves fluidly coupled to a fuel gas supply and configured to direct sufficient fuel gas from the fuel gas supply to drive a shear velocity plasma stream for the duration of a Z-pinch discharge between the inner and outer electrodes.

38. The plasma confinement system of clause 37, wherein the inner electrode and the outer electrode delimit an acceleration volume into which the fuel gas is directed by the one or more valves.

39. The plasma confinement system of clause 37, further comprising an intermediate electrode concentrically positioned between the inner electrode and the outer electrode,

wherein the inner electrode and the intermediate electrode delimit an acceleration volume into which the fuel gas is directed by the one or more valves.

40. The plasma confinement system of any of clauses 37-39, wherein the fuel gas comprises one or both of a neutral gas and an ionized gas.

Although specific values, relationships, materials, and components have been set forth for purposes of describing the concepts of the invention, those skilled in the art will understand that many variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the basic concepts and principles of operation of the invention as broadly described. It will be appreciated that those skilled in the art can modify those details in light of the above teachings without departing from the invention taught herein. For example, the numerical ranges recited herein are exemplary and can be modified based on the mode of operation of a given plasma confinement system or based on modifications to the size, function, configuration, etc. of a given plasma confinement system. For example, if the size of a given plasma confinement system increases, such a range may scale proportionally (e.g., linearly, exponentially, etc.).

Having now fully set forth embodiments of the inventive concepts and certain modifications, various other embodiments, as well as certain variations and modifications of the embodiments shown and described herein, will occur to persons skilled in the art upon becoming familiar with such underlying concepts. It is intended to include all such modifications, alternatives, and other embodiments as fall within the scope of the appended claims or equivalents thereof. It is, therefore, to be understood that this invention may be practiced otherwise than as specifically set forth herein. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the subject matter as set forth in the claims.

Other variations are within the spirit of the present disclosure. Thus, while the disclosed technology is susceptible to various modifications and alternative arrangements, certain illustrated embodiments thereof are shown in the drawings and have been described above in detail. It should be understood, however, that there is no intention to limit the subject matter recited in the claims to the specific forms disclosed, but on the contrary, the intention is to cover all modifications, alternative arrangements, and equivalents falling within the spirit and scope of the disclosure as defined in the appended claims.

The use of the terms "a" and "an" and "the" and similar referents in the context of describing the disclosed embodiments (especially in the context of the following claims) are to be construed to cover both the singular and the plural, unless otherwise indicated herein or clearly contradicted by context. Similarly, the use of the term "or" should be interpreted to mean "and/or" unless clearly or contextually contradicted. Unless otherwise indicated, the terms "comprising," "having," "including," and "containing" are to be construed as open-ended terms (i.e., meaning "including, but not limited to"). The term "connected" when unmodified and referring to a physical connection should be interpreted as partially or completely contained within, attached to, or joined together, even if there is something intervening. Recitation of ranges of values herein are merely intended to serve as a shorthand method of referring individually to each separate value falling within the range, unless otherwise indicated herein, and each separate value is incorporated into the specification as if it were individually recited herein. Unless otherwise indicated or contradicted by context, use of the term "set" (e.g., "a set of items") or "subset" should be understood to include a non-empty set of one or more members. Furthermore, unless indicated otherwise or contextually contradictory, the term "subset" of a corresponding set does not necessarily denote the proper subset of the corresponding set, but the subset and the corresponding set may be equal. Unless explicitly stated otherwise or clear from the context, the use of the phrase "based on" means "based at least in part on" and is not limited to "based only on".

The phrase "at least one of the forms" A, B, and C "or" at least one of A, B and C "(i.e., the same phrase with or without oxford comma) is otherwise understood within the context of normal use to mean an item, etc., which may be any non-empty subset of a or B or C, a set of a and B and C, or any set containing at least one a, at least one B, or at least one C that is not inconsistent or otherwise excluded from the context, unless the context clearly contradicts otherwise. For example, in the illustrative example of a set of three members, the joint phrase "A, B, and at least one of C" and "A, B, and at least one of C" refers to any one of the following sets: { A }, { B }, { C }, { A, B }, { A, C }, { B, C }, { A, B, C }, and if there is no explicit or contextual conflict, any set has { A }, { B }, and/or { C } as a subset (e.g., a set with multiple "A"). Thus, such a joint language is not generally intended to imply that certain embodiments require at least one of A, at least one of B, and at least one of C to be present. Similarly, phrases such as "A, B, or at least one of C" and "A, B, or at least one of C" refer to the same as "A, B, and at least one of C," A, B, and at least one of C "refer to any one of the following sets: { A }, { B }, { C }, { A, B }, { A, C }, { B, C }, { A, B, C }, unless otherwise explicitly stated or clear from the context. In addition, unless indicated otherwise or contextually contradicted, the term "plurality" indicates a plurality of states (e.g., the term "plurality of items" indicates a plurality of items). The number of the plurality of items is at least two, but may be more when explicitly or as indicated by the context.

The operations of the processes described herein may be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. In an embodiment, processes such as those described herein (or variations and/or combinations thereof) are performed under control of one or more computer systems configured with executable instructions and are implemented by hardware or combinations thereof as code (e.g., executable instructions, one or more computer programs, or one or more application programs) executing jointly on one or more processors. In an embodiment, the code is stored on a computer readable storage medium, for example, in the form of a computer program comprising a plurality of instructions executable by one or more processors. In an embodiment, the computer-readable storage medium is a non-transitory computer-readable storage medium that does not contain transitory signals (e.g., propagating transitory electrical or electromagnetic transmissions) but contains non-transitory data storage circuitry (e.g., buffers, caches, and queues) within a transceiver of the transitory signals. In an embodiment, code (e.g., executable code or source code) is stored on a set of one or more non-transitory computer-readable storage media having stored thereon executable instructions that, when executed (i.e., as a result of being executed) by one or more processors of a computer system, cause the computer system to perform operations described herein. In one embodiment, the set of non-transitory computer-readable storage media includes a plurality of non-transitory computer-readable storage media, and one or more of the individual non-transitory storage media in the plurality of non-transitory computer-readable storage media lacks all code, and the plurality of non-transitory computer-readable storage media collectively store all code. In embodiments, the executable instructions are executed such that different instructions are executed by different processors, e.g., in embodiments, a non-transitory computer-readable storage medium stores instructions, and a host CPU executes some of the instructions while a graphics processor unit executes other instructions. In another embodiment, different components of the computer system have separate processors, and different processors execute different subsets of the instructions.

Thus, in embodiments, a computer system is configured to implement one or more services that individually or collectively perform the operations of the processes described herein, and such computer system is configured with the applicable hardware and/or software that enables the performance of the operations. Further, in an embodiment of the present disclosure, the computer system is a single device, and in another embodiment, a distributed computer system that includes multiple devices that operate in different ways, such that the distributed computer system performs the operations described herein, and such that a single device does not perform all of the operations.

The use of any and all examples, or exemplary language (e.g., "such as") provided herein, is intended merely to better illuminate the various embodiments and does not pose a limitation on the scope of the claims unless otherwise claimed. No language in the specification should be construed as indicating any non-claimed element as essential to the practice of the subject matter of the invention disclosed herein.

Embodiments of the present disclosure are described herein, including the best mode known to the inventors for carrying out the inventive concepts described herein. Variations of those embodiments may become apparent to those of ordinary skill in the art upon reading the foregoing description. The inventors expect skilled artisans to employ such variations as appropriate, and the inventors intend for the embodiments of the disclosure to be practiced otherwise than as specifically described herein. Accordingly, the scope of the present disclosure includes all modifications and equivalents of the subject matter recited in the claims appended hereto as permitted by applicable law. Moreover, any combination of the above-described elements in all possible variations thereof is encompassed by the disclosure unless otherwise indicated herein or otherwise clearly contradicted by context.

All references, including publications, patent applications, and patents cited herein are hereby incorporated by reference to the same extent as if each reference were individually and specifically indicated to be incorporated by reference and were set forth in its entirety herein.

Claims

1. An apparatus, comprising:

2. The device of claim 1, wherein the gas comprises a neutral gas, and

3. The device of claim 2, wherein the shaping portion incorporates at least one conductive ring comprising at least one contact surface electrically connected to an outer surface of the second electrode.

4. The device of claim 3, wherein the at least one conductive ring incorporates a conductive material that is chemically and thermo-mechanically compatible with a conductor of the second electrode, and a plasma-facing portion of the at least one shaping portion incorporates at least one refractory metal.

5. The apparatus of claim 4, wherein the at least one refractory metal comprises one or more of W, ta, nb, mo, re, ti, V, cr, mn, zr, tc, ru, rh, hf, os, ir or an alloy of any one or more of the foregoing metals.

6. The apparatus of any one of claims 4 and 5, wherein the plasma-facing portion incorporates at least one conductive form of carbon comprising one or more of graphite, sintered carbon powder, pressed carbon powder, carbon fibers, or carbon nanotube bonding structures.

7. The device of any one of claims 4-6, wherein the plasma-facing portion contains at least one textured surface formed to incorporate a plurality of local concave elements forming a structured array to enhance local electric fields and promote electron field emission.

8. The device of claim 7, wherein the at least one textured surface has been formed by a mechanical treatment comprising one or more of cutting, scraping, sanding, sandblasting, grooving, embossing, stuccoing, embossing, or knurling.

9. The apparatus of any one of claims 7 and 8, wherein the at least one textured surface has been formed by a chemical process comprising one or more of etching, chemical deposition, spraying, sputtering, ion and neutral implantation, or epitaxial growth.

10. The apparatus of any one of the preceding claims, wherein the gas is provided to the acceleration volume as an ionized gas.

11. The apparatus of any one of the preceding claims, wherein the second electrode incorporates a conical electrode surface arranged to enhance momentum transfer to ions and neutral particles in the axial direction of the first and second electrodes.

12. The device of any one of the preceding claims, further comprising a third electrode arranged between and coaxially with respect to the first and second electrodes, wherein the third electrode exhibits a conical electrode configuration and incorporates conical electrode surfaces arranged to enhance momentum transfer to ions and neutral particles in the axial direction of the first, second and third electrodes.

13. A method, comprising:

forming a Z pinch plasma column from the introduced fuel gas to support a Z pinch current flowing between an inner anode and an outer cathode surrounding an unsupported end of the inner anode,

wherein the Z pinch plasma column is surrounded and stabilized by a shear velocity plasma stream formed from the fuel gas.

14. The method of claim 13, further comprising generating a radial electric field to support an initial current flowing between the inner anode and the outer anode via the introduced fuel gas prior to forming the Z pinch plasma column.

15. The method of claim 14, wherein the fuel gas comprises a neutral gas, and

16. A method according to any one of claims 13 to 15, whereby the fuel gas is introduced into the acceleration volume, the fuel gas comprising an ionised gas.

17. A plasma confinement system, comprising:

an external electrode;

An inner electrode concentrically positioned within the outer electrode;

one or more valves fluidly coupled to a fuel gas supply and configured to direct sufficient fuel gas from the fuel gas supply to drive a shear velocity plasma flow for the duration of a Z-pinch discharge between the inner electrode and the outer electrode.

18. The plasma confinement system of claim 17, wherein the inner electrode and the outer electrode delimit an acceleration volume into which the fuel gas is directed by the one or more valves.

19. The plasma confinement system of claim 17, further comprising an intermediate electrode concentrically positioned between the inner electrode and the outer electrode,

20. The plasma confinement system according to any one of claims 17 to 19, wherein the fuel gas comprises one or both of a neutral gas and an ionized gas.