Apparatus and method for extended plasma confinement

Abstract

Methods and systems for plasma confinement utilizing various electrode and valve configurations are provided. In one embodiment, the apparatus includes a first electrode positioned to define an outer boundary of an acceleration volume, a second electrode positioned coaxially relative to the first electrode and to define an inner boundary of the acceleration volume, at least one power source for driving a current along a Z-pinch plasma column between the first and second electrodes, and a set of valves for providing a gas to the acceleration volume to fuel the Z-pinch plasma column, the electronic flow of the current being in a first direction from the second electrode to the first electrode. In additional or alternative embodiments, the molded part is conductively connected to the second electrode to cause gas breakdown of the gas in the presence of the gas to generate a shear flow velocity profile.

Description

Technical Field

[0001] (Cross - Reference to Related Applications) This application claims priority to each of U.S. Provisional Patent Application No. 63 / 194,866, filed on May 28, 2021, entitled "APPARATUS AND METHOD FOR EXTENDED PLASMA CONFINEMENT", and U.S. Provisional Patent Application No. 63 / 194,877, filed on May 28, 2021, entitled "ELECTRODE CONFIGURATION FOR EXTENDED PLASMA CONFINEMENT". The entire contents of each of the above - mentioned applications are hereby incorporated by reference herein for all purposes.

[0002] (Description of Research and Development Sponsored by the Federal Government) This invention was made, at least in part, under Government support awarded by the United States Department of Energy under grant numbers DE - AR001010 and DE - AR001260. The Government has certain rights in this invention.

Background Art

[0003] Unless otherwise indicated in this specification, the descriptions disclosed in this section are not considered prior art with respect to the claims of this application and are not admitted to be prior art by including them in this section.

[0004] Nuclear fusion is the process of combining two atomic nuclei. When two atomic nuclei of elements having an atomic number less than 26 [i.e., having an atomic number lower than iron (Fe)] are fused, energy is released. The release of energy is due to a slight difference in mass between the reactants and products of the fusion reaction (e.g., in a thermonuclear fusion plasma reactor) as defined by the equation E = mc 2 as.

[0005] Regarding nuclear fusion, virtually limitless energy with waste that is easier to manage than some existing energy sources is expected.

[0006] Controlled nuclear fusion in a fusion plasma where the plasma reaction persists over a long period may be suppressed by rapidly growing plasma instabilities. Feasible approaches to such controlled nuclear fusion (hereinafter also described as "controlled fusion" or simply "fusion" as a noun or adjective indicating fusion-related features and / or characteristics) will continue to be pursued through research on different plasma confinement approaches. Such approaches offer clear advantages across various levels of scientific maturity.

Brief Description of the Drawings

[0007] The above and other embodiments, features, and aspects of the present invention will be considered in more detail in connection with the following description of the embodiments shown in the accompanying drawings.

[0008]

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[0009] Embodiments of the present disclosure can be better understood by reference to the following description, which should be read in conjunction with the accompanying drawings of specific exemplary embodiments. This description of the illustrated embodiments set forth below is not intended to limit the invention, but rather is intended to function as specific examples thereof. Those skilled in the art should understand that the concepts and specific embodiments disclosed can be readily utilized as a basis for modifying or designing other methods and systems for carrying out the same purposes of the present disclosure. Those skilled in the art should also understand that such equivalent assemblies do not depart from the spirit and scope of the present disclosure in its broadest form.

[0010] In some embodiments, typical power supply devices and power feeding methods arranged to form and maintain an axial Z-pinch current may not be suitable or may be completely impossible to generate and maintain a sufficient shear rate axial flow that can be used for the stabilization of the Z-pinch plasma. Therefore, embodiments of devices and processes for generating and maintaining a sufficient shear rate flow in the transition medium associated with the boundary region of the Z-pinch discharge are described herein.

[0011] Fusion devices based on Z-pinches [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," Phys. Plasmas 24, 055702 (2017); "Shumlak '17", incorporated herein by reference] can be attractive because they are geometrically simple, inherently compact, and relatively low cost. Some more recent publications [e.g., U. Shumlak, "Z-pinch fusion," J. Appl. Phys. 127, 200901 (2020); published online: May 27, 2020; "Shumlak '20", incorporated by reference] have further clarified sheared flow stabilization (SFS) to produce an equilibrium Z-pinch, which may maintain a compressed plasma state for a duration that is considerably longer than other plasma time scales.

[0012] One area of shear flow optimization relates to enhanced control of both neutral gas supply and ionized gas supply to the vacuum volume of the acceleration volume of a Z-pinch device. An example of an existing device can be found in FIG. 3 of Shumlak '20, which shows a schematic cross-section of the vacuum vessel of the Fusion Z-pinch Experiment (FuZE) SFS Z-pinch experimental device. In FIG. 3, one internal gas puff valve is disposed at a substantially central axial position of the acceleration volume to provide a portion of a selected neutral fill gas through an "inner electrode" (when the term "substantially" is used herein, the recited characteristics, parameters, values need not be achieved exactly, and deviations or variations including, for example, tolerances, measurement errors, limits of measurement accuracy, and other factors known to those of ordinary skill in the art may occur in an amount that does not prevent the intended effect given by the characteristic). An additional (external) gas puff valve is shown at a substantially the same axial position radially opposite the internal gas puff 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 experimental device.

[0013] The arrangement shown in FIG. 3 of Shumlak '20 relies on the diffusion of neutral gas from the position of the gas puff valve into the surrounding vacuum volume to create a substantially axially symmetric neutral gas density profile that generally has a maximum value at the axial position of the puff valve. Such a profile may provide sufficient fuel gas to drive a shear velocity flow for a duration comparable to the duration of the Z-pinch discharge. After depletion of the neutral gas inventory (e.g., due to driven outflow around the Z-pinch plasma column and / or diffusion of gas to other regions of the enclosed volume), the Z-pinch current may decay due to instability even if at least some of the energy from the high voltage power supply remains available. Injection of pre-ionized gas using a plasma injector, plasma gun, or ion source may also be employed in conjunction to enhance and improve the shear flow profile created by neutral gas injection.

[0014] Accordingly, in at least one embodiment, one or more valves (e.g., one or more gas puff valves and one or more plasma injectors) are fluidly coupled to a fuel gas source and configured to direct sufficient fuel gas (e.g., neutral gas and / or pre-ionized gas) supplied from the fuel gas source to drive a sheared velocity plasma flow during each duration of a Z-pinch discharge. Specifically, in such an embodiment, sufficient neutral gas may be directed to support a local disruption path between the inner and outer electrodes and establish a sheared velocity plasma flow. In additional or alternative embodiments, sufficient pre-ionized gas may be directed to maintain the sheared velocity plasma flow (e.g., to replenish neutral gas).

[0015] In connection with the methods and apparatuses described herein, the stabilization of the sheared velocity flow is supported by at least the following modeling results. An axisymmetric plasma configuration representing the FuZE SFS Z-pinch experiment device was simulated using the WARPXM computer code [U. Shumlak, R. PLUS, N. Reddell, E. Sousa, and B. Srinivasan, "Advanced physics calculations using a multi-fluid plasma model." Comput. Phys. Comm. 182, 1767 (2011)] based on a non-linear 5-moment 2-fluid (5M2F) plasma model. This model includes viscosity and heat conduction effects based on Braginskii.

[0016] Some selected simulation results are shown in FIGS. 1-5. Results without a sheared velocity flow are shown in FIG. 1 for reference. Results for three cases with three different parabolic sheared flow velocity profiles are shown in FIGS. 2-4. The cases may be distinguished by different sheared flow velocity values "vsfa" at r = a (where a is the nominal radius of the Z-pinch plasma) normalized by the characteristic Alfven velocity v A at the pinch edge (r = a). The initial conditions (t = 0) for the illustrated cases include 4×1024 m -3 includes a Bennett equilibrium with a peak density of, an ion temperature and an electron temperature of 1.27 keV, and a peak magnetic field of 33.0 T. The effective pinch radius (a) is 0.91 mm.

[0017] The simulations use a specific normalized diffusivity limit (difflim = 32 m 2 ·s -1 ) and impose a minimum diffusivity (diffmin = 3.2 m 2 ·s -1 ). The electrons are given a viscosity with a diffusivity level equal to the minimum diffusivity. Perturbations are used in each case to trigger modes with a wavelength equal to the axial length of the domain.

[0018] In FIG. 1, vsfa = 0.0 shows a reference case without a shear velocity flow. Schematic diagrams of two-dimensional (r / z) cross-sections 100 and 110 of the normalized ion density at time points t = 0.000 (corresponding to the establishment of the unperturbed Z-pinch plasma column) and t = 8.000 (normalized with respect to the radial Alfvén time τ A ) are shown in grayscale, respectively. Further, the axial velocity (v Z ) profiles 120 and 130 (normalized by the characteristic Alfvén velocity v A ) are substantially shear-free as shown in the r / a graph of FIG. 1 compared to v Z / v A .

[0019] The case represented in FIG. 1 shows that an unstable Z-pinch grows in a relatively small number of Alfvén time scale units, as measured by, for example, the time required for a magnetized plasma perturbation to propagate from the axis (r = 0) to the edge (r = a) of the unperturbed plasma column, indicating a substantial plasma ion loss due to density perturbations that exhibit rapidly growing instabilities. 150 This supports the understanding that it exhibits rapidly growing instabilities, as shown by.

[0020] Three additional modeling results of the shear velocity stabilized Z - pinch are shown in Figs. 2 - 4. The case illustrated in Fig. 2 is characterized by vsfa = 0.25. Schematic diagrams of the two - dimensional (r / z) cross - sections 200 and 210 of the normalized ion density at time points t = 0.000 (corresponding to the establishment of the unperturbed Z - pinch plasma column) and t = 14.000 (normalized with respect to τ A are shown in grayscale, respectively. The initial (t = 0) parabolic shear velocity profile 220 evolves into profile 230 at (normalized) time t = 14 and still shows substantial shear outside the initial plasma column boundary r = a. The ion density perturbation 250 is detectable but is mainly localized at a radius comparable to the initial r = a radius of the unperturbed Z - pinch plasma column.

[0021] The case illustrated in Fig. 3, which is characterized by vsfa = 0.5, shows a stronger stabilization effect compared to the cases illustrated in Figs. 1 and 2. Schematic diagrams of the two - dimensional (r / z) cross - sections 300 and 310 of the normalized ion density at time points t = 0.000 (corresponding to the establishment of the unperturbed Z - pinch plasma column) and t = 26.400 (normalized with respect to τ A are shown in grayscale, respectively. The initial (t = 0) parabolic shear velocity profile 320 evolves into the perturbed ion axial velocity profile 330 at (normalized) time t = 26.4 and shows a nearly parabolic radial dependence and substantial shear outside the initial plasma column boundary r / a = 1. The ion density perturbation 350 is mainly localized in the volume inside the plasma column (r < a).

[0022] The case illustrated in Fig. 4 is characterized by vsfa = 0.75. The results in Fig. 4 show a stronger shear flow stabilization effect compared to the cases illustrated by Figs. 1 - 3. At time points t = 0.000 (corresponding to the establishment of the unperturbed Z - pinch plasma column) and t = 37.000 (normalized with respect to τ ASchematic diagrams of the two-dimensional (r / z) cross-sections 400 and 410 of the normalized ion density in (normalized) are shown in grayscale, respectively. The initial (t = 0) parabolic ion axial shear velocity profile 420 evolves into an ion axial velocity profile 430 with (slight) perturbation in (normalized) time, showing a substantial parabolic radial dependence. The ion density perturbation 450 is mainly localized in the volume inside the plasma column (r < a).

[0023] Some results of the WARPXM computer code are shown in FIG. 5. The graphs shown in FIG. 5 show the time dependence of the integrated Z-pinch mass with respect to t / τ A (Normalized time corresponding to the normalized time t in FIGS. 1-4). The dependence in FIG. 5 emphasizes the effect of the shear flow velocity value, characterized by the "vsfa" value. In the absence of stabilization (vsfa = 0.0), the confinement of the Z-pinch plasma begins to decline after t = 5 (shown by the initial decay of the normalized mass 500), showing significant losses by the nominal time t = 8.0, while the corresponding mass ratios 525, 550, and 575 (vsfa = 0.25, 0.5, and 0.75, respectively) show a gradual increase in plasma confinement, indicating that a stabilized Z-pinch plasma can be sustained as long as sufficient axial plasma current is supplied. It is also observed that a sufficiently sheared azimuthally symmetric ion velocity flow surrounding the plasma column may be generated and maintained.

[0024] At least in the above context, embodiments of apparatuses and methods for generating and maintaining an azimuthally symmetric sheared ion velocity flow according to the present disclosure are listed below. Some components of a particular embodiment of a plasma (confinement) apparatus for a stabilized Z-pinch are shown in FIGS. 6A and 6B in an isometric view 600 (FIG. 6A) and a cross-sectional view 610 (FIG. 6B).

[0025] Generally, a Z-pinch plasma device having a vacuum vessel shown in FIGS. 6A and 6B (however, related systems such as electrical cables and conduits, vacuum pumps and ducts, diagnostic feedthroughs, optical windows, etc. are omitted for clarity) may be extended to other specific Z-pinch plasma devices, except for a neutral gas supply valve (discussed in detail below) related to an improved process for generating and maintaining a sheared azimuthal velocity flow according to certain specific embodiments provided by the present disclosure.

[0026] More specifically, in at least one embodiment, the acceleration volume 620 is increased compared to the acceleration volume of other specific Z-pinch plasma devices and is disposed substantially along the central axis of the acceleration volume 620. For example, at least one gas puff valve (to provide neutral gas to the acceleration volume 620) and / or a plasma injector 630 (to provide pre-ionized gas to the acceleration volume 620) may be disposed to be filled with a gas mixture (e.g., a neutral operating gas mixture) via at least one internal valve 630. Additionally, or alternatively, a plurality of external valves such as a plurality of gas puff valves (to provide neutral gas to the acceleration volume 620) and / or a plasma injector 640 (to provide pre-ionized gas to the acceleration volume 620) may be disposed as an external electrode or an outer electrode 650 or installed as a regular array on an external vacuum boundary.

[0027] Depending on the particular embodiment, the gas puff valves and / or plasma injectors 630, 640 may be electronically triggered to deliver a “puff” of fill gas and / or pre-ionized gas that starts at a programmable start time up to on the order of 1 millisecond and has a duration of up to several hundred microseconds (e.g., up to 1 millisecond). The amount of fill gas (also referred to herein as “fuel gas”) delivered (e.g., in a “puff”) may also be controlled by adjusting the fill gas pressure supplied to the gas puff valves and / or plasma injectors 630, 640, either individually or as a selected subset of the valves (the subset of valves may include only some, or all, of the valves and / or injectors 630, 640). Further, different valves and / or injectors 630, 640 (or different combinations of multiple valves and / or injectors 630, 640) may be supplied with different fill gas mixtures having, for example, different elemental ratios of fill gas and / or different isotope ratios (e.g., an adjustable D2 / T2 molecular ratio). In at least one embodiment, the various gas puff valves and / or plasma injectors may be uniform (e.g., all of the same type / size and having all of the same operating settings if so configured), but in other embodiments, different valves may be used for different positions. In additional or alternative embodiments, one or more gas puffs or other gas valves and / or plasma injectors may control the flow of gas into the acceleration volume 620 via a manifold that includes a plurality of ports providing passageways into 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 within the acceleration volume 620 when each respective valve is open).

[0028] Similar to neutral gas injection through a gas puff valve, ionized gas or plasma may be injected using a combination of plasma injectors or a manifold at various locations. Plasma formed from a gas mixture may also be generated and injected in a manner similar to neutral gas injection. Plasma injection may provide finer control of the final axial plasma distribution, as well as its shear flow profile, which may enable higher fidelity control of plasma stability and lifetime. Further control of plasma injection may be provided because plasma particles are charged particles that may be accelerated by an electric field generated by a variable electrical bias (or voltage) on the injection electrode. Thus, the velocity of the injected plasma may be finely controlled to enable fine tuning and optimization of the disruption of any neutral gas present (e.g., within the accelerator volume 620). Additionally, the injected plasma may move at a faster velocity than the injected neutral gas, which may move in a substantially static manner (compared to the injected plasma) during a Z-pinch discharge pulse. In this way, relative to neutral gas injection, plasma injection may provide “on demand” (e.g., more immediate) pre-ionized fuel to replenish fuel gas, for example, during a Z-pinch discharge pulse.

[0029] In some embodiments, the plasma injected into the acceleration volume 620 may be generated by pre-ionizing a neutral gas with a spark plug or via inductive ionization. More generally, the gas puff valve and / or plasma injectors 630, 640 may include one or more electrode plasma injectors and / or one or more electrodeless plasma injectors. In embodiments that include one or more electrode plasma injectors, the plasma injected into the acceleration volume 620 may be at least partially generated by an electrode discharge. In additional or alternative embodiments that include one or more electrodeless plasma injectors, the plasma injected into the acceleration volume 620 may be at least partially generated by an inductive discharge (e.g., a radio frequency antenna operating at 400 kHz, 13.56 MHz, 2.45 GHz, and / or other frequencies within a frequency range permitted for use in a given local jurisdiction, e.g., permitted by the Federal Communications Commission) generated by an external coil window. In some embodiments, the neutral gas for pre-ionization may be limited by a neutral gas reservoir (not shown in FIGS. 6A and 6B) and / or the conductance of the neutral gas to a selected plasma injector configuration.

[0030] In some embodiments, the axial distribution of the injected plasma may be ensured via an axially symmetric plasma injector configuration. In at least one embodiment, eight plasma injectors 640 may be respectively positioned at eight equally spaced ports of the manifold. Each of the eight ports may be configured at an inclination angle (e.g., 5 degrees to 90 degrees with respect to the central axis of the acceleration volume 620) with respect to the housing of the acceleration volume 620. In one example, the inclination angle may be 45 degrees with respect to the central axis of the acceleration volume 620. In some embodiments, the eight ports may be configured at a single axial position along the central axis of the acceleration volume 620 (i.e., the eight ports may be equally spaced around the circumference or other perimeter of the acceleration volume 620 at the axial position). In other embodiments, the ports may include multiple sets of eight ports, and each set of eight ports may be equally spaced around a different axial position along the central axis of the acceleration volume 620. In an exemplary embodiment, the set of eight ports may be configured as a pair of alternating sets, the first set of eight ports may be positioned at a first axial position, the second set of eight ports may be positioned at a second different axial position, and may be rotated with respect to the first set such that each port of the second set is positioned between a pair of ports of the first set around the perimeter of the acceleration volume 620. Specifically, in such embodiments, each port of the first set of eight ports may be spaced 45 degrees around the perimeter of the acceleration volume 620, and each port of the second set of eight ports may be spaced 22.5 degrees around the perimeter of the acceleration volume 620 for each 45-degree offset (rotation) from the ports of the first set, such that one port of the first set and one port of the second set are provided 22.5 degrees around the perimeter of the acceleration volume 620. In additional or alternative embodiments, the plasma injection may be performed azimuthally, e.g., along a chord 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 differently using other deformations to achieve substantially equivalent profiles by compensating for the effects of deformation (e.g., asymmetrically azimuthally distributed and / or having different angular distributions).

[0031] In some embodiments, injecting the acceleration volume 620 with a pre-ionized gas may result in a plasma having a plasma temperature in the range of 1 to 10 eV. Further, as described above, the injection rate of the pre-ionized gas may be significantly greater than the injection rate of the neutral gas, so the velocity of the plasma within the acceleration volume 620 may be up to 50×10 3 m / s. In some embodiments, the injection of the pre-ionized gas may provide flexibility in the amount of particles injected. Specifically, in an exemplary embodiment, a certain amount of pre-ionized gas particles may be injected in 1 / 50 of the time utilized to inject the same amount of neutral gas particles. For example, the time utilized to inject 10 Torr-L (1333.2 Pa-L) of neutral gas particles (1 Torr-L (133.32 Pa-L) is proportional to 2.5×10 19 molecules at 273 K) may be the same amount as the time utilized to inject 500 Torr-L (6666 Pa-L) of pre-ionized gas particles. Similarly, in some embodiments, the injection rate (or mass flow rate) of the pre-ionized gas may be varied according to the current and voltage of the power supply (i.e., the waveform of the injection pulse). As an example, increasing the power supply voltage (e.g., 100 V to 500 V) may simultaneously increase the injection rate. As another example, increasing the power supply current (e.g., 1 A to 500 A) may simultaneously increase the injection rate.

[0032] In particular, the injection of the neutral gas may be achieved through a puff valve or through the release of hydrogen gas from a metal hydride, such as titanium deuteride (TiD2) or other metal hydrides based on scandium, vanadium, or other metals. In some embodiments, the puff valve may be a solenoid-driven puff valve (however, other configurations may be implemented and are within the scope of the present disclosure).

[0033] As described above, at least one internal gas puff valve and / or plasma injector 630, and a plurality of external gas puff valves and / or plasma injectors 640 may be actuated individually or as a group. The initial gas load inside the acceleration volume 620 having the desired axial and azimuthal profiles may be achieved by timing the individual valves / or groups of valves. Such valves (or groups thereof) may be timed, for example, in a manner that matches the arrival of neutral gas and / or pre-ionized gas and / or mixtures thereof to the desired initial profile, such as the embodiments discussed in detail below and shown in FIGS. 8 and 10A - 10F. A power supply (not shown in FIGS. 6A and 6B) may be timed to achieve ionization at the desired axial position and to generate and maintain a shear flow using the initial gas load.

[0034] These parameters may be further adjusted (e.g., optimized) by actuating various combinations of (neutral gas) gas puff valves having plasma injectors to achieve, for example, a desired level of power output. Additionally, the plasma may be injected into the acceleration volume 620 significantly (e.g., about 100 times) faster than the puffed neutral gas. Such a combination of different injection speeds allowed by the acceleration of plasma injection with neutral gas injection provides a larger parameter space for optimization. Additionally, the plasma injector may serve to inject mass and carefully control the location of neutral gas ionization.

[0035] The embodiments shown in FIGS. 6A and 6B incorporate a pre-assembled acceleration volume 620 so as to incorporate connectors or other coupling elements for at least one internal gas puff valve and / or plasma injector 630 that extend from within the internal or inner electrode 660. For example, at least one internal gas puff valve and / or plasma injector 630 [and respective coupling elements] are (having a separation at an angle of 45°) at z = -50 cm (with respect to the z = 0 position of the unsupported end 665 of the inner electrode 660, where the z-axis coincides with the central axis of the acceleration volume 620, and the negative direction of the z-axis extends from the unsupported end 665 along the central axis of the inner electrode 660, and the positive direction of the z-axis extends from the unsupported end 665 through the acceleration volume 620 in a direction opposite to the negative direction) and may include eight valves 630 that are azimuthally symmetrically distributed (having an angular separation), the eight valves 630 are similarly distributed at z = -75 cm, and the eight valves 630 are similarly distributed at z = -100 cm. The illustrated embodiments may be readily updated with additional valves to allow for more fuel gas injection (e.g., for a longer sustained pinch discharge) and control of the axial pressure distribution of the fill neutral gas within the acceleration volume 620 (e.g., for further enhancement of the duration of the shear rate shear flow). In additional or alternative embodiments, the valves may be configured differently using other deformations (e.g., asymmetrically azimuthally distributed and / or having different angular distributions) to achieve substantially equivalent profiles by compensating for the effects of deformation. Such considerations may equally apply to plasma injectors.

[0036] The gas puff valves of the illustrated embodiments incorporate prism structural elements that can be used with a tool effector (e.g., a hand tool for electric appliances or a force transmission element of a coupling insert) to directly transmit torque and other related stresses to a more robust main structural element while avoiding various inserts, connectors, contacts, vacuum or pressure seals, potting, and / or solder joints.

[0037] In one embodiment, the gas puff valve of the present disclosure may be designed to incorporate an orifice diameter of 0.075 inches or greater and a plenum volume of at least 1 cm 3 . Additionally, one feature of the gas puff valve according to embodiments of the present disclosure is the ability to close (and remain closed) at a pre-programmed time before and during the Z-pinch discharge.

[0038] As described above (including the documents incorporated by reference), the preservation of the azimuthal symmetry of the plasma and the associated shear velocity flow is one advantage of the embodiments of the present disclosure. Thus, the reproducible formation and shaping of an initial azimuthally symmetric plasma structure at a given axial position of the acceleration volume 620 may be achieved by at least some of the embodiments of the present disclosure. In various embodiments, various "plasma formation" 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 account for the complexity associated with specific power supplies and conduits and / or complex (pre-)ionization subsystems], plasma injectors, as well as tuned operating algorithms and other methods.

[0039] The plasma confinement devices of FIGS. 6A and 6B may include a controller or other computing device (not shown), which may include a non-transitory memory in which executable instructions may be stored. The executable instructions may be executed by one or more processors of the controller, and various functions of the plasma confinement device may be performed. Thus, the executable instructions may include various routines for the operation, maintenance, and testing of the plasma confinement device. The controller may further include a user interface through which an operator of the plasma confinement device may enter commands or otherwise modify the operation of the plasma confinement device. The user interface may include various components for facilitating use by an operator of the plasma confinement device and for receiving operator inputs such as one or more displays, input devices (e.g., keyboards, touchscreens, computer mice, pushable buttons, mechanical switches, other mechanical actuators, etc.), and optical (e.g., a request to generate a plasma for thermonuclear fusion). The controller may be communicatively coupled to various components of the plasma confinement device (e.g., valves, power supplies, etc.) to command their operation and use (for clarity, the wired and / or wireless communication paths between the controller and the various components are omitted in FIGS. 6A and 6B).

[0040] Some aspects of plasma initiation and subsequent shaped parts related to embodiments of the present disclosure are schematically shown in FIG. 7. These shaped parts may include sharp points (e.g., tips formed by local concave elements 725. See below) that enhance the local electric field and assist in plasma breakdown. Various configurations of "passive" (i.e., without a dedicated power source or gas supply actively driving field emission) shaped part 700 may be disposed in the form of ring electrode fitting in one or more recesses of the inner electrode 660 at one or more negative z-axis positions generally proximate to one or more internal gas puff valves 630 (the inner electrode 660 and the gas puff valve 630 are not shown in FIG. 7. See FIGS. 6B and 8). One important function of such parts is to initiate and maintain multi-channel breakdown of the surrounding neutral gas in at least one embodiment, resulting in the generation and maintenance of a substantially equal current flow in all radial directions (initiated by various substantially independently azimuthally distributed radial discharge streamers). The shaped part 700 may be disposed at the position of the gas valve or downstream therefrom (e.g., FIGS. 6B and 8 to 10F) in various embodiments. The sharp shaped part is described with reference to the inner electrode 660, but such features may be used with either the inner electrode 660, the outer electrode 650 (not shown in FIG. 7. See FIGS. 6A, 6B, and 8), or both. In some embodiments, enhancing the electric field near such sharp points is beneficial at the cathode, which may be either the inner electrode 660 or the outer electrode 650 (from which electrons are emitted). Sharp points on the anode surface may also be included so that the breakdown path may be selectively established between the shaped part on the anode and the shaped part on the cathode (e.g., the inner electrode 660 and the outer electrode 650, respectively).

[0041] The molded part 700 shown in FIG. 7 incorporates a conductive ring 710 arranged to include at least one contact surface 720 that forms a low contact resistance surface contact with the cylindrical or conical or otherwise tapered outer surface of the inner electrode 660 (e.g., to cause a voltage drop between at least one contact surface 720 and the inner electrode 660 of less than 100 V). In some embodiments, the conductive ring 710 may be formed from one or more conductive materials that may be fully or at least partially chemically and / or thermomechanically compatible with the conductor of the inner electrode 660 (e.g., the heat and stress experienced during operation do not have a perceptible effect on the lifetime use of the conductive ring 710). Further, the plasma-facing portion 715 of the molded part 700 may be formed from a conductor that is resistant to chemical and physical damage by the supported discharge. In many embodiments, one or more refractory metals (e.g., one or more of W, Ta, Nb, Mo, or Re, and additionally or alternatively including one or more of Ti, V, Cr, Mn, Zr, Tc, Ru, Rh, Hf, Os, or Ir) and / or alloys or combinations thereof may be utilized due to at least relatively low chemical reactivity, relatively high melting point, and relatively high resistance to plasma ablation and sputtering.

[0042] In additional or alternative embodiments, the plasma-facing material may be based on conductive forms of carbon including graphite, sintered carbon powder or pressed carbon powder, carbon fiber matrix, and / or carbon nanotubes incorporating structures and compositions. In addition to relative insensitivity to plasma effects, decomposition, and damage, carbon-based structures (particularly carbon nanotubes) may exhibit desirable electron multiplication characteristics during the plasma generation stage.

[0043] In other embodiments, the portion 715 facing the plasma may be textured to incorporate a number 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 field emission from solid (and liquid) surfaces. The elements 725 may be formed by mechanical actions (including cutting, scratching, polishing, lapping, sandblasting, grooving, checking, punching, embossing, knurling, etc.). Also, different chemical and / or thermal processes (such as etching, chemical deposition, spraying, sputtering, ion implantation and neutral implantation, epitaxial growth, etc.) may be included. In certain embodiments, a number of elements 725 having relatively small characteristic dimensions (e.g., compared to the size of the portion 715 facing the plasma) may be generated and maintained to avoid significant changes in the geometric shape when any individual element 725 is damaged or deformed (e.g., by arc discharge or local overheating). For example, the elements 725 may have an average height that is 1-10% of the height of the portion 715 facing the plasma (excluding the elements 725) in some embodiments.

[0044] In at least one embodiment, the shaped part 700 (whether or not including the elements 725) around the inner electrode 660 may be configured as a generally uniform annular ring that varies, for example, when the port of the internal gas puff valve 630 traverses the cross-section of the portion 715 facing the plasma and at least one contact surface 720. However, in some embodiments (such as when the shaped part 700 is configured as a generally uniform annular ring), the shaped part 700 may have a varying cross-section around and / or along the inner electrode 660. Other variations, such as when a plurality of individual shaped parts 700 that do not form a single ring are arranged around the inner electrode 660, are also within the scope of the present disclosure.

[0045] Also, as will be discussed in detail below with reference to FIG. 8, a shaped component similar to the shaped component 700 shown in FIG. 7 may form a ring along the inner surface of the outer electrode 650. Such shaped components and corresponding contact surfaces (similar to at least one contact surface 720) may be configured as shown in FIG. 7 (or according to the variations discussed herein), except that the cross-section of at least one contact surface 720 and the portion 715 facing the plasma is rotated 180° so that it comes onto the inner surface of the outer electrode 650 instead of its outer surface. The inclination of at least one contact surface 720 may also differ from that illustrated in FIG. 7 (e.g., see FIG. 9 and refer to the following description of the "tapered electrode configuration") in order to adapt to the taper or other surface variations of the outer electrode 650, or the lack thereof.

[0046] In addition to the shaped component 700 illustrated in FIG. 7, other methods of assisting / controlling / directing plasma formation may be used, either individually or in combination. One possibility includes using radioactive substances embedded in the outer electrode 650 and / or the inner electrode 660. Specifically, the high-energy particles or photons emitted from the radioactive decay process may cause pre-ionization near the embedded radioactive substance, resulting in a region with increased plasma breakdown compared to regions without the embedded radioactive substance. For example, a beta emitter and / or a gamma emitter may be selected for the embedded radioactive substance.

[0047] As an additional or alternative possibility, in regions where pre-ionization of the plasma is desired, the outer electrode 650 and / or the inner electrode 660 may be irradiated with incident laser light. Within such regions, the electrode surface (e.g., of the outer electrode 650 and / or the inner electrode 660) may comprise a material specifically selected to emit X-rays or other forms of ionizing radiation when receiving the incident laser light. As an additional or alternative possibility, the laser light may be used to directly ionize the gas (e.g., via direct interaction of the laser light with neutral gas particles). In such embodiments, the laser light passes through the neutral gas and deposits energy throughout the laser path, which may result in a directed channel for pre-ionization and plasma breakdown (e.g., a channel having a larger plasma breakdown than the surrounding volume).

[0048] Other ways to assist / control / direct plasma formation may utilize various forms of cathodes, such as field emitters or thermionic emitters, located on the electrode surface (e.g., the outer electrode 650 and / or the inner electrode 660) where breakdown is desired. A field emitter may emit electrons from small sharp features using a relatively high electric field. Examples of such emitters may include nanostructures such as carbon nanotubes, graphene emitters, nanowire emitters, Schottky emitters, etc. Additionally or alternatively, a thermionic emitter may be used to cause plasma breakdown. Examples of such emitters may include a heated tungsten filament that emits electrons at a relatively high temperature. A Schottky emitter may be considered a field-enhanced thermionic emitter.

[0049] In a plasma confinement device such as the plasma confinement device described in detail above with reference to FIGS. 6A-7, one embodiment of the process of initiating and driving an azimuthally symmetric shearing flow for the stabilization of Z-pinch discharge is schematically illustrated in FIG. 8. The process may, in this example, be characterized by schematic diagrams of steps or stages 810, 820, 830, 840, 850, 860, 870, and 880 of generally unequal durations. In certain embodiments, the process may include performing steps 810, 820, 830, 840, 850, 860, 870, and 880 in sequence.

[0050] 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 the plasma confinement device. Further, in certain embodiments, an additional or alternative order of steps may be implemented as executable instructions on such a computing device, and the individual steps discussed with reference to the process may be added, removed, replaced, modified, or exchanged.

[0051] The process may begin at step 810, which may include the application of a high voltage generating a radial electric field (not shown) between electrodes 650 and 660, and the sequential activation of one or more internal valves 630 and external valves 640 (combinable with the plasma injector 640). Valves 630 and 640 may be arranged to locally introduce an initially measured and predetermined concentration of fill gas 812. In certain embodiments, it may be desirable to initiate gas puffing and / or plasma injection during the initialization phase and continue to deliver a sufficient initial concentration of fill gas 812 in the vicinity of the shaped part 700 for additional protection against premature and / or asymmetric gas breakdown.

[0052] During step 820, the initial concentration 812 may spontaneously evolve through a diffusion process of the neutral gas to form a continuous (e.g., uninterrupted) axially symmetric volume of neutral fill gas formation 822 that occupies a substantial portion (e.g., the majority) of the acceleration volume 620. In some embodiments, the volume of neutral fill gas formation 822 exhibits an axial neutral gas molecular number density gradient along the central axis of the acceleration volume 620 (e.g., toward the unsupported end 665 of the inner electrode 660), while substantially maintaining an azimuthal symmetry that is advantageous for a substantially symmetric distribution of discharge streamers during the initial breakdown of the fill gas. In certain embodiments, the neutral gas molecular number density gradient may be a gradient such that breakdown according to Paschen's law occurs in the shaped part 700. In additional or alternative embodiments, pre-ionized material may be injected in the shaped part 700 to facilitate the formation of an ionization wave that moves into the upstream-injected neutral gas (e.g., toward the supported end of the inner electrode 660 opposite the unsupported end 665).

[0053] During step 830, the proximal electric field structure formed by the geometric shape and material properties of the shaped part 700 may facilitate the neutral gas breakdown that forms an axially symmetric plasma structure 835 that supports a flow of current 837 between the inner electrode 660 and the (surrounding) outer electrode 650 that is axially localized near the shaped part 700. The flow of current 837 supported by energy from a power source (e.g., a capacitor bank or the like) may form a continuous (e.g., uninterrupted) current loop (from the outer electrode 650, through the plasma structure 835, into the inner electrode 660, and through the inner electrode 660) that may generate a substantially azimuthal magnetic field 838 (as indicated by the azimuthal magnetic field lines). In additional or alternative embodiments, a pre-ionized gas may be injected from the plasma injector 640 toward the unsupported end 665 of the inner electrode 660 to form the plasma structure 835.

[0054] The Lorentz force interaction between the current flow 837 and the magnetic field 838 may cause the movement of the current flow 837 in the direction of the unsupported end 665 of the shaped part 700, as shown in the schematic diagram of step 840. Further, the Lorentz force interaction may induce the current flow 837 along the surface of the outer electrode 650.

[0055] During step 850, the current flow 837 may continue to occur towards and up to the unsupported end 665. At the same time, the magnetic pressure driven by the magnetic field 838 surrounded by the current flow 837 may displace the plasma structure 835 in progress in the direction of the opposing part 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, for example, by injecting various amounts of pre-ionized gas with the internal valve 630 and / or the external valve 640. In at least one embodiment, the substantially azimuthal symmetry of the plasma structure 835 supporting the current flow 837 may significantly contribute to the efficiency of the process, since any significant interference with the current flow 837 may cause instability, electrode damage, and / or introduction of metal impurities into the discharge in progress.

[0056] In certain embodiments, the discharge generation steps 830, 840, and 850 may continue for a fraction of a microsecond to several microseconds, which is significantly shorter than, for example, either step 810 (corresponding to filling the acceleration volume 620 with neutral gas) or steps 860, 870, and 880 (corresponding to Z-pinch discharge). Thus, the neutral filling gas formation 822 is shown as stationary, since the neutral filling gas formation 822 may evolve over a time interval that is significantly longer than the duration of steps 830, 840, and 850 (e.g., only over).

[0057] In certain embodiments, plasma injection may occur between steps 810 and 820. In additional or alternative embodiments, plasma injection may occur rapidly on the same scale as steps 830, 840, and 850 and may be used to control the formation / initiation and dynamics of such steps.

[0058] Step 860 corresponds to an initial Z-pinch operation step that includes the formation of a Z-pinch plasma column 865 created to support a Z-pinch current I pinch Furthermore, a residual plasma structure 866 may be formed to support a residual (radial) current 867 flowing through the neutral fill gas formation 822 of the acceleration volume 620. Further, in various embodiments, the propagation of the plasma structure 835 may drive an initial shear rate plasma flow 868 that surrounds (and stabilizes) the Z-pinch plasma column 865. In some embodiments, the residual plasma structure 866 may be initiated in the vicinity of the shaped part 700 such that it is characterized by (locally) the highest number density of neutral gas components (molecules and / or atoms).

[0059] As described above, in at least one embodiment, during the discharge duration at step 870, the Z-pinch plasma column 865 may be sustained and stabilized by the continuation of the plasma flow from the acceleration volume 620. The ionization front 872 may continuously generate plasma accelerated from the acceleration volume 620 by the residual current 867 to drive the shear rate plasma flow 868.

[0060] During step 880, in at least one embodiment, the ionization front 872 may move towards the bleach end of the acceleration volume 620 until all or substantially all of the fuel gas available in the acceleration volume 620 is ionized, continuously (e.g., without interruption) ionizing the residual neutral fuel gas, resulting in the disappearance of the ionization front 872 and subsequent decay of the Z-pinch current I pinch(For example, a current (such as current 950 shown in FIG. 9 below) passing through the Z-pinch plasma column 865 and the inner electrode 660) is provided. When steps 810, 820, 830, 840, 850, 860, 870, and 880 are completed, the plasma confinement system may be cleaned to remove fusion by-products, and the above-described process may be repeated for another pulse. In certain embodiments, the process and its repetition may be automated and controlled by a software application implemented, for example, by a controller communicatively coupled to the plasma confinement system.)

[0061] Another embodiment of a plasma confinement device, a Z-pinch plasma device 900, is schematically shown in FIG. 9. The Z-pinch plasma device 900 may generate a plasma within the assembled volume 635 of the plasma confinement chamber 615, and the plasma is 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 a discharge between a pair of electrodes facing the plasma confinement chamber 615.)

[0062] Devices belonging to the exemplified class of plasma confinement devices are generally related to the previous embodiments discussed above and shown in FIGS. 6A - 8 and have similar features, except for the following additional or alternative subsystems and functions. Except for certain assemblies and modes of operation that may result from such differences, the description provided above with reference to FIGS. 6A - 8 may also be additionally applicable to the embodiment shown in FIG. 9. In certain embodiments, additional subsystems and / or functions that are also not described in detail above with reference to FIGS. 6A - 8 and that may be additionally applicable to the Z-pinch plasma device 900 shown in FIG. 9 may also be included.)

[0063] In an exemplary embodiment, the Z-pinch plasma device 900 may include an outer electrode 650 that is physically and functionally separated from an external vacuum boundary 910, and the external vacuum boundary 910 forms a vacuum vessel 645 as a low-pressure container that includes a plasma confinement chamber 615 together with a portion of the inner electrode 660. The intermediate electrode 920 may be positioned to have a radius between the radius of the inner electrode 660 and the radius of the outer electrode 650. Specifically, 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 that is at least partially surrounded by the intermediate electrode 920, and the intermediate electrode 920 may include one end 965 that is at least partially surrounded by the outer electrode 650.

[0064] The Z-pinch plasma device 900 incorporates at least two functionally distinct power supplies, for example, at least one primary power supply 930 that is primarily arranged and controlled to drive a Z-pinch (discharge) current 950 (I pinch ) and at least one additional power supply 940 that is primarily arranged and controlled to drive a residual current 867. In some embodiments, the at least one primary power supply 930 may be a separate power device from the at least one additional power supply 940. In other embodiments, the at least one primary power supply 930 and the at least one additional power supply 940 may be components of the same power device.

[0065] For example, in at least one embodiment, a single power supply device may have multiple outputs that individually provide power to enable the performance of each function (e.g., driving a Z-pinch current 950, driving a residual current 867, etc.). Such an arrangement may be based on at least two power supplies (e.g., one primary power supply 930 and one additional power supply 940), and may enable additional control of the Z-pinch current 950 and stabilization of its shear flow. In principle, at least two power supplies may be scaled, charged, and controlled such that the Z-pinch current 950 and its stabilization can be maintained for a corresponding period before the energy stored in any of the at least two power supplies runs out prematurely or is exhausted.

[0066] In a particular embodiment, 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, for example, by tapering the end 965 of the intermediate electrode 920 outward along the central axis of the acceleration volume 620 to increase the volume of at least a portion of the acceleration volume 620 in the direction of the (unsupported) ends 665 and 965. In one example, the taper may be at 0° and 15° from the central axis of the acceleration volume 620. Such an arrangement may, for example, facilitate the transfer of momentum from the plasma heated by the residual current 867 to the neutral gas along the central axis, thereby generating and maintaining shear flow stabilization. The momentum transfer may be explained and modeled using methodologies applicable to the design / optimization of "Laval nozzles" as known in the field of jet propulsion.

[0067] The technology described herein is discussed in relation to nuclear fusion and, for example, the utilization of energy generation therefrom, but the technology described herein can be used for other purposes such as heat generation (e.g., manufacturing utilizing relatively high temperatures) and propulsion. For example, the embodiments of FIGS. 6A - 8 or FIG. 9 may be modified by introducing an opening at one end of the outer electrode 650 so as to remove at least the vacuum chamber 338 or the external vacuum boundary 910 respectively and allow the fusion products to escape (e.g., parallel to the central axis of the accelerator 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 parallelize the plasma and reduce the divergence of any exhaust plume.

[0068] The Z-pinch plasma device 900 may include a controller or other computing device 948, which may include a non-transitory memory that may store executable instructions. The executable instructions may be executed by one or more processors of the controller 948 to perform various functions of the Z-pinch plasma device 900. Thus, the executable instructions may include various routines for the operation, maintenance, and testing of the Z-pinch plasma device 900. The controller 948 may further include a user interface through which an operator of the Z-pinch plasma device 900 can input commands or otherwise modify the operation of the Z-pinch plasma device 900. The user interface may include various components for facilitating the use by an operator of the Z-pinch plasma device 900 and for receiving operator inputs such as one or more displays, input devices (e.g., keyboard, touch screen, computer mouse, pushable buttons, mechanical switches, or other mechanical actuators, etc.), and light (e.g., a request to generate a plasma for thermonuclear fusion, etc.). The controller 948 may be communicatively coupled to various components of the Z-pinch plasma device 900 (e.g., valves, power supplies, etc.) to command their operation and use (for clarity, the wired and / or wireless communication paths between the controller 948 and the various components are omitted in FIG. 9).

[0069] Figures 10A to 10F schematically show an embodiment of the process of initiating and driving an azimuthally symmetric shearing flow for the stabilization of Z-pinch discharge in a plasma confinement device. A series of Figures 10A to 10F show two configurations: a first configuration where the inner electrode is the cathode and the outer electrode is the anode (the left side of Figures 10A to 10F when Figures 10A to 10F are oriented such that the alphanumeric characters shown therein are oriented in a standard manner), and a second configuration where the inner electrode is the anode and the outer electrode is the cathode (the right side of Figures 10A to 10F when Figures 10A to 10F are oriented such that the alphanumeric characters shown therein are oriented in a standard manner). A particular Z-pinch plasma confinement device may correspond to the first configuration which is easier to construct and can operate successfully. However, as will be discussed in more detail below with respect to Figures 11 to 24, the second configuration may result in advantageous and unexpected results according to various embodiments described herein. The particular non-alphanumeric symbols used in Figures 10A to 10F (for example, arrows indicating the direction of current flow, gas valves, gas clouds, magnetic field symbols, arrows indicating gas flow, etc.) are the same as those used in Figure 8. Note that the parts labeled "cathode" and "anode" can be electrically connected to parts of the same name ("cathode" or "anode"). Figures 10A to 10F show a set of gas valves that physically contact or are directly adjacent to the outer electrode, and other configurations such as the valve configuration shown and described in detail above with reference to Figure 8 are also within the scope of the present disclosure, which may or may not include one or more shaped parts as described in detail above with reference to Figure 7.

[0070] In at least one embodiment, greater stability of the Z-pinch plasma may be achieved using a plasma confinement system as disclosed herein, where the outer electrode is the cathode and the inner electrode is the anode. Specifically, as will be discussed in more detail below with reference to FIGS. 11-24, the Z-pinch m = 0 instability, and its stabilization by a radially sheared axial flow, are studied using the nonlinear ideal 5M2F model, and an extension of that model including heat and momentum transport by Braginskii. Using the ideal 5M2F model, the results of the linear growth rate are compared to previous studies using magnetohydrodynamics (MHD) and Hall MHD. In scenarios with or without a radially sheared axial flow, the agreement with Hall MHD is excellent, suggesting that the Hall term is dominant among the two-fluid terms. At the limit of small ion inertial length, the results also agree with MHD. Comparison with cell-in-particle (PIC) modeling of the m = 0 stability without shear focuses on plasma scenarios based on recent experimental results. In the scan of the mode wave number, the ideal 5M2F results are qualitatively similar to PIC, and in contrast to the MHD results showing saturation of the growth rate with increasing rather than decreasing wave number, the growth rate rises to a peak at intermediate wave numbers and decreases at large wave numbers. The peak normalized 5M2F growth rate is γτ A = 1.5, where τ A is the Alfvén transport time across the pinch. The peak occurs at a normalized wave number ka = 10, where a is the effective pinch radius. For comparison, the PIC results have a peak growth γτ AIt has η = 0.77. Including the Braginskii closure of the 5M2F model does not qualitatively change the ideal outcome in this particular case. Nonlinear 5M2F modeling using the diffusive Braginskii closure is performed for a pinch edge shear flow velocity equal to half of the Alfvén velocity. Nonlinear mixing by the shear flow results in a saturated quasi-steady state, with a small loss of pinch ion inventory and pinch thermal energy, approximately 30% and 10% respectively. The 5M2F modeling captures the essential physics of the m = 0 instability and provides a computationally tractable path to high-fidelity modeling of 3D Z-pinch behavior including the m = 1 instability.

[0071] Experimental evidence from SFS Z-pinch studies, along with a significant numerical stability analysis, suggests that axially sheared radial flows enable a long plasma lifetime where they are observed, and that static Z-pinches typically terminate due to m = 0 (sausage) and m = 1 (kink) instabilities with growth rates near the radial Alfvén transport frequency.

[0072] Insights from computational modeling are expected to be an important part of ongoing SFS Z-pinch generation. As plasma parameters increase in future experiments, high-fidelity modeling can be used to explore the various processes involved in plasma formation, assembly, and confinement. Particularly interesting processes include, for example, the deflagration mode associated with the sustenance of the effective resistivity of the shear flow by electron drift microscopic turbulence, and the dynamics of the flowing Z-pinch itself.

[0073] The five - moment multi - fluid model is an excellent candidate for accurately capturing the physics of interest. The five - moment two - fluid (5M2F) plasma model (having two fluids representing ions and electrons) has been previously applied for the modeling of the Z - pinch instability and captures realistic m = 0 growth rates and interesting electron drift instabilities when the electron cross - field drift velocity exceeds the ion thermal velocity. The 5M2F model can enable an effective compromise between the fidelity of kinetic models (however, exorbitant computational costs) and the computational tractability of MHD - based approaches (however, limited fidelity).

[0074] The research presented herein explores the m = 0 instability by applying the 5M2F model, with or without shear flow. The 5M2F model also includes finite - inertia - length corrections for ions and electrons, as well as the effects of finite light speed. In the limit where electron inertia can be ignored and the speed of light is infinite, the results should match those of Hall MHD. In some examples, a linearized Hall MHD model is applied to study the m = 0 stability of Bennett equilibria with a parabolic shear - flow profile (v sf ∝r 2 ). Using the same setup, but with a non - linear MHD model, additional work was done considering both linear (v sf ∝r) and parabolic shear - flow profiles. Below, the 5M2F model is used as a benchmark against these MHD and Hall MHD results.

[0075] To evaluate the physical fidelity of the 5M2F model with complete Braginskii closure, comparisons are made using fully dynamic (i.e., not gyrokinetic or reduced in some other way) PIC modeling of m = 0 Z - pinch stability that includes Coulomb collisions. In that work, the PIC model is applied to study a FuZE - like Bennett profile with a linear shear - flow profile. A scan of ka without shear flow shows growth rates similar to MHD results up to a maximum ka = 5, at which point the PIC growth rate reaches a maximum. For larger k, the PIC results typically show a decrease in growth rate, unlike MHD results which typically show a constant or increasing growth rate at large k. Simulations at ka = 5 with shear flow show

Number

[0076] The five - moment fluid equations for a given species are derived by taking moments of the relevant Boltzmann equation. As described by Braginskii, the first three moments of the Boltzmann equation for species α yield evolution equations for the five independent variables: the number density (n α ), the three momentum components (

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[0077] The moments of the Boltzmann equation for species α result in the following fluid equations.

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[0078] The fluid equations are coupled to the Maxwell equations for the magnetic field (

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[0079] To reach the 5M2F model, the species are limited to ions and electrons, α = i, e. The collisional source term arises only from Coulomb scattering between ions and electrons. Specifically, the resulting source is

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[0080] This model is closed following Braginskii using Chapman-Enskog type closures. The stress tensor (Π α ) and the heat flux (

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[0081] The terms for momentum and heat exchange ( [[ID='28']]

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[0082] The heat flux is given by the following equation

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[0083] Magnetization is calculated as x α = ω cα τ α wherein the cyclotron frequency is ω cα = eB / m α Assuming hydrogen ions, the various collision frequencies are as follows:

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[0084] [[ID=3�]]In these equations, lnΛ is the Coulomb logarithm assumed to be equal to 10 in this specification, and the temperature is in eV units. Note that a part of the heat flux related to ion-electron friction is omitted here. In this axisymmetric equation, that part is

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[0085] The stress tensor is constructed from the strain rate tensor,

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Equation

[0086] As considered by Braginskii, the term proportional to η0 corresponds to the stress associated with the compression or expansion of the plasma. The term proportional to η1 is associated with diffusion across the magnetic field, the step size is equal to the Larmor radius, and the step frequency is set by collisions. The coefficient η3 is related to the gyroviscosity, which is the diamagnetic flux of momentum. The terms proportional to η2 and η4 are omitted. The electron viscosity, for similar ion and electron temperatures, has electron viscosity coefficients η e 0, η e 3, and η e 1 that are approximately (m e / m i ) 1 / 2 , (m e / m i ), and (m e / m i ) 3 / 2 smaller than their ion counterparts, respectively, and are omitted in this implementation (except for the isotropic viscosity applied for numerical purposes), further assuming that the electron velocity gradient and the ion velocity gradient are equal.

[0087] Three types of corrections to the transport coefficients by Braginskii are considered in this specification. The first relates to the assumptions made when deriving coefficients with a long time scale compared to the particle collision time. The stress in the r-z plane due to plasma compression or expansion is non-magnetized viscosity η0 = 0.96p i τ i is regulated by. As explained by Braginskii, [Number] (Compression) increases the stress, while [Number] (Expansion) decreases the stress. The magnitude of this stress is [Number] . The physical mechanism is as follows. The continuity equation shows [Number] with respect to the dynamic time scale [Number] (ignoring the gradient of n). Next, assuming that the flux is frozen in the fluid, [Number] . Assuming that the ion Larmor orbit size is small compared to the size of the compression or expansion region, an increase in the magnetic field gives an increase in the transverse velocity and the associated transverse energy and stress. This effect appears as temperature anisotropy, as observed in continuous motion simulations. The increased energy is divided between the transverse and parallel directions over a time set by τ i , but this process is known as gyro relaxation. [Number] Then, the transverse stress is approximately pδB / B. For δB / B = 1, for example, the magnitude of the stress is similar to the isotropic pressure. τ i <τ dyn In the case of, the stress is reduced by fast equipartition. However, for τ i >τ dyn in the case of, the derivation time effect is unphysically strong and gives a stress larger than the isotropic pressure for δB / B = 1. In the model implemented here, the correction coefficient,

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[0088] The second and third corrections are related to the breakdown of the Braginskii model when the Larmor radius is large compared to the length scale of interest. (In the tokamak modeling community, where a small Larmor radius is usually assumed, there is no analog of these corrections). One is a global correction,

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[0089] Other corrections for large Larmor radii are relevant to special situations near r = 0 where the magnetic field approaches zero and the Larmor orbits are no longer simple helices. Moving radially away from r = 0, the Larmor orbits become finite and can eventually reach values that coincide with the radius. The critical radius, r crit is used to approximate this position with an r-dependent correction, [Number] in the case of [Number] and otherwise [Number] is applied to reduce transport near r = 0 where the ion orbits are no longer simple helices. This correction is only applied to η3 and [Number] As mentioned in the above considerations of Equation (24), the non-linear dependence of f corr r gives a strong cutoff where the local Larmor radius exceeds the distance from the cylindrical axis. It is worth noting that the region of small magnetic fields near r = 0 is also expected to affect the transverse stress related to η0. However, instead of approaching zero at r = 0, η0 should match the vertical transport η1 at some level representing a random walk process within the region of non-helical orbits. The related corrections are not attempted here.

[0090] The combination of these three correction factors is sufficient to explore the fundamental effects of transport according to Braginskii on Z-pinch stability. As further described herein, examples of correction factors are presented in the context of a FuZE-like equilibrium.

[0091] The WARPXM modeling framework uses techniques based on the Runge-Kutta Discontinuous Galerkin (RKDG) method to solve the 5M2F model on an unstructured grid of triangles. WARPXM uses MPI parallelization, and the scalability of RKDG is suitable for problems with large dimensionality.

[0092] For the explicit time advancement used, the fastest time scales must be resolved. For hyperbolic phenomena, the time step size (

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[0093] Three additional details of WARPXM are justified. First, because the discretization of the primary variables is discontinuous, special care is needed to accurately calculate the gradients required for the fluxes associated with the closures by Braginskii. To address this problem, the Bassi - Rebay approach is used. Second, to accommodate the cylindrical coordinate system, the vector analysis operations are described from the perspective of the straight derivative and the source term. For example, the divergence of a vector

Math

Math

[0094] The azimuthal derivative is zero in certain embodiments of the axisymmetric systems described herein. Terms involving the radial derivative can be rewritten as follows: [Number]

[0095] The WARPXM infrastructure can then naturally handle the first two terms, and the last term is included as a source term. Operations such as curl, divergence of tensors, and vector gradients are also described in terms of the linear derivatives and the cylindrical source terms. A third detail concerns the spatial integration of these cylindrical source terms. The basic DG method implemented in WARPXM integrates the source terms using a quadrature method based on the values at the Legendre-Gauss-Lobatto (LGL) nodes. The triangles have LGL nodes on their edges, so some nodes are at r = 0. Then, to perform the quadrature, the calculation of the source term at r = 0 is required. Part of the cylindrical source term involves dividing the first derivative of the primary variable by r. Knowledge of the second derivative is needed to apply L'Hopital's theorem. To avoid the need for the second derivative, the LGL quadrature is replaced by a symmetric Gaussian quadrature that does not have quadrature points on the edges of the triangles, and thus, division by zero at r = 0 is avoided.

[0096] The diffusive Bennett Z-pinch equilibrium is the focus of the modeling presented below. This equilibrium may be parameterized by the pinch radius (a) and the plasma current (I p ). The azimuthal magnetic field (B θ ), the axial current density (j z ), and the total plasma pressure (p) are then given by: [Number]

[0097] Figure 11 shows the normalized radial profiles of the magnetic field (B θ ), density (n), and temperature (T) in the Bennett equilibrium state.

[0098] The total pressure consists of equal contributions from the ion pressure and the electron pressure. A uniform temperature (T) is assumed to be proportional to the pressure (p / 2 = nk i = n e = n) in an equilibrium without shear where the density is B Half of the plasma mass and current is contained within r = a. The total pressure at r = a is equal to the magnetic pressure B θ 2 / (2μ0) at r = a. The magnetic field, density, and temperature profiles of this equilibrium, normalized by their respective peak values, are shown in Fig. 11.

[0099] The momentum balance in the 5M2F model requires the following:

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[0100] Except for the case of shear flow (see below), equal and opposite ion axial velocity and electron axial velocity are assumed so that the equilibrium E r is zero. Thus, v iz = -v ez = j z / (2en), and in the embodiments of this specification, monovalent ions with charge q i = -q e = e are assumed, where e is the elementary charge. Since j z and n have the same radial dependence, these velocities are uniform in the radial direction. The total current is

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[0101] Add the shear flow to the equilibrium as follows. The ion axial velocity is v iz = j z / (2en) + v sf . The shear flow velocity with a linear or parabolic radius is, in this specification, i.e., v sf (r) = v a sf r / a or v sf (r) = v a sf (r / a) 2 is considered, and v a sf is the shear flow velocity at r = a. The required electric field is determined by equation (29) with n i = n and does not change from the shear-free equilibrium. Gauss's law is used to find n e . The axial current does not change from the shear-free equilibrium, and v ez is set accordingly. Then, the electron pressure is determined by equation (30). For practical interests (

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[0102] + δj z and z , [Number] and [Number] where k is the perturbation frequency. The phase shift is determined by the parameters φ0 and the coefficient r ζ and ζ = 1 and 2 are used in simulations with linear and parabolic shear flows, respectively. Radial localization uses the parameter b = a / 3. The perturbation added to the ion and electron velocities is δv iz = -δv ez = δj z / (2en). Since the equilibrium magnetic field does not change, Faraday's law is unbalanced, and as a result, an electric field immediately begins to evolve in response to the perturbation.

[0103] The computational domain is rectangular in the r-z plane. The axial length matches the perturbation wavelength, i.e., Lz = 2π / k. The radial extent of the domain is 4a. This domain setup is consistent with previous studies on m = 0 instability analysis. The axial boundaries are periodic. At r = 0, standard axisymmetric boundary conditions are used. The radial and azimuthal components of the vector quantities are zero, while the scalar and axial components of the vector quantities have no radial variation. At r = 4a, a perfectly conducting wall boundary condition is applied. That is, the radial velocity, radial magnetic field, and axial electric field are zero. The density, pressure, axial velocity, radial electric field, and axial magnetic field have no radial variation.

[0104] In the following analysis and discussion, several characteristic quantities are used. The characteristic Alfvén velocity is v A = B θ,pk / (n pk m i μ0) 1 / 2 where B θ,pk and n pk are the peak magnetic field and number density, respectively. The characteristic time is defined as τ A = a / v A The ion thermal velocity is v Ti = (2kB T / m i ) 1 / 2 is. The ion Larmor radius is r Li =m i v Ti / eB θ,pk is. The thermal velocity is related to the Alfvén velocity as [Number] Hydrogen ions are assumed.

[0105] a and I p For a given choice of a and I, the pressure is determined, but the density and temperature are not yet specified. The density and temperature are established by specifying the pinch size: ion Larmor radius ratio a / r Li Using the definition of v Ti to solve for T i gives the following: [Number] The ion density and electron density follow from the relation nk B T = p / 2.

[0106] In the ideal 5M2F model, the normalized instability growth rate γτ A and other normalized dynamics such as depend on the choice of a / r Li but the specific choice of a and I p is not important. However, the transport by Braginskii depends on the plasma properties (density, temperature, and magnetic field), and for the applications discussed below for 5M2F with Braginskii closure, a, I p , and a / r Li are all specified.

[0107] 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 stabilize of global magnetohydrodynamic instabilities based the Hall fluid model," Phys. Plasmas 9, 913 (2002)] applied the linearized Hall MHD model to study the growth of m = 0 instabilities in a Bennett equilibrium state with various Hall parameter strengths. The Hall effect is parameterized using ε Sot. = c / (ω pi R), where ω pi = [n i0 e 2 / (ε0m i )] 1 / 2 is the ion plasma frequency, R is the radius of the modeled domain, and n i0 is the ion density at r = 0. Note that Sotnikov et al. used R = 3a to relate the parameter ε Sot. to a / r Li . Using the aforementioned r Li and v Ti , the relation is found to be

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[0108] In an ideal 5M2F simulation without shear flow, a close agreement with the MHD and Hall MHD results is seen. The linear growth rate of the m = 0 instability is determined from the linear growth phase of the nonlinear simulation. Using the normalized axial wavenumber ka = 10 / 3, the growth rate for a / r in the range of 4 - 200 is shown in Fig. 12A. In these simulations, the mass ratio is m Li / m e / m i = 1 / 100, and the perturbation level is ε = 10 -3 . The growth rate is found by considering the change in the volume-integrated radial kinetic energy of the ion fluid (

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[0109] An exemplary time trace of ln(KE i,rad. / ME0), where the initial magnetic energy is ME0, is shown in Fig. 12B.

[0110] Figs. 12A and 12B show the ideal 5M2F results at ka = 10 / 3. Fig. 12A shows the growth rate for a / r in the range of 4 - 200. a / r Li Li ​When it is large, the 5M2F growth rate is consistent with the results found by Sotnikov et al. for pure MHD (dashed line). For smaller a / r Li for, the 5M2F results are close to the corresponding Sotnikov et al. results. FIG. 12B shows an exemplary time trace of the integrated radial ion kinetic energy normalized by the initial magnetic energy. The vertical dotted line brackets the period during which the linear growth is measured, and the dashed line shows the measured exponential growth.

[0111] The results for the relatively large Larmor radius regime are again shown in FIG. 13A for ka = 10 / 3.

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[0112] Figures 13A and 13B show ideal 5M2F results at ka = 10 / 3 focused on small a / r Li . Figure 13A shows the growth rates for a / r in the range 0.75 - 4 for two electron-ion mass ratios. For a / r between 1.5 - 4 Li , the growth rate changes slowly. For a / r less than approximately 1.1 Li , faster growth is observed. Figure 13B shows an exemplary time trace of the integrated radial ion kinetic energy normalized by the initial magnetic energy. Li Additional scans of a / r

[0113] are performed at ka = 20 / 3, 40 / 3, and 80 / 3. All these simulations use m Li / m e / m i = 1 / 1836. The results are shown in Figure 14. The region with high growth due to the electron drift instability persists within the region

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[0114] Figure 14 shows ideal 5M2F results with several perturbation frequencies. The high-growth modes persist at

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[0115] m e / m i = 1 / 100 and ka = 10 / 3, the density structures of the linear growth modes are shown in Figure 15 for a / r Li = 4, 10, and 50. At large r Li a slight drift of the mode structure in the +z direction is visible, and at a / r Li = 4, the total shift is about 1 / 8 of the axial wavelength. A tilt is seen at large r Li and the structure at larger radii shifts more in the +z direction than the structure at smaller radii. The spatial resolution of the results in Figure 15 is 64 × 16 (radial × axial) cells with cubic spatial accuracy.

[0116] Figure 15 shows the ideal 5M2F mode structure at a / r Li = 4, 10, and 50. The decimal logarithm of the change in density normalized to the peak equilibrium density is plotted in the r-z plane. a / r Li = 50 (MHD-like), a single lobe shown by the dashed ellipse dominates the radiation structure. For smaller a / r Li values, two lobes exist, and for a / r Li = 4, it is shown by the dashed ellipse. The smaller lobe is near the axis and not aligned with the main lobe.

[0117] The growth rate values shown in Figure 12A have errors

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[0118] Spatial resolution: The scan uses a base spatial resolution of 40×10 (radial × axial) cells with three-dimensional spatial accuracy. For a / r Li = 4, 10, and 50, the convergence behavior with respect to spatial resolution is studied. In all cases, the error at the base resolution is <0.1%. As shown in Figure 15, the characteristic size is half of the axial domain and is represented by five cells. In the cubic representation (using a quadratic polynomial), the accuracy is very good. The radial representation of the mode is as good as or better than the axial representation.

[0119] Mass ratio: An enhanced electron mass m e / m i = 1 / 100 is used in the scan. Simulations using the actual mass ratio (m e / m i = 1 / 1836) are performed at a / r Li = 4, 10, and 50 and are used to determine the error associated with the enhanced mass. The errors are 1.0%, 0.6%, and 0.8% at a / r Li = 4, 10, and 50, respectively. m e / mi Using 1 / 100, the combined ion - electron fluid density in the 5M2F model is 1% greater than the ion - electron fluid density of the MHD fluid, so an error of about 1% is expected. In situations where the two - fluid effect is strong, the increase in electron mass can cause a larger error, as seen in Figure 13A.

[0120] Speed of light: The speed of light is set in the scan

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[0121] The growth rate from the 5M2F model with a / r a sf = 50 is compared to the MHD results, and the 5M2F with a / r Li = 2.357 is compared to the Hall MHD results using equivalent Hall parameters. As shown, the MHD - like 5M2F modeling with a / r Li = 50 yields results qualitatively similar to ideal MHD modeling with linear and parabolic shear - flow profiles. Complete stabilization occurs when the pinch - edge velocity is Li

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[0122] Figures 16A and 16B respectively illustrate ideal 5M2F results with stabilization by linear and parabolic shear flows. All simulations have ka = 10 / 3. The growth rates seen in MHD and Hall MHD modeling are shown for comparison. The scan shown in Figure 16A includes a scan by "neg (negative) v a sf " of reverse shear flow, and the other scans in Figure 16A have positive v a sf .

[0123] Previous work using Hall MHD (and PIC results discussed herein) used positive shear flow, i.e., positive values of v a sf . As shown in Figure 16A, anticipating that the two-fluid effect depends on the shear flow direction, the scan is performed with negative v a sf , i.e., reverse shear flow. The stabilizing effect of the shear flow is reduced when the flow is reversed, and v a sf > v A is required for stabilization. a / r Li As a / r increases, the growth rate seen in the shear flow, whether positive or negative, approaches the MHD results regardless of the shear flow direction.

[0124] Figure 17 shows the mode structure in an ideal 5M2F modeling with shear flow. As shown, for MHD-like (a / r Li = 50) plasma and large Larmor radius (a / r Li = 2.357) plasma, the results of linear (upper) and parabolic (lower) shear flows are shown. All cases use positive shear flow (in the sense discussed in the text), except for one case with a negative linear shear flow labeled "neg". In all cases, the shear flow velocity at the pinch edge is |v a sf | = 0.3v A .

[0125] Shear flow extends modes that grow exponentially in the axial direction. Figure 17 shows the mode structure for simulations with a shear flow velocity |v a sf | = 0.3v A . For a given a / r Li , comparing the linear and parabolic results, the parabolic shear flow confines the mode structure to a smaller radius. For the linear shear flow with a / r Li = 2.357, the results for both positive and negative shear flows are shown. In the case of negative shear flow, the alignment of the lobe near the axis of the structure with the main lobe seems to promote a larger observed growth rate.

[0126] To accurately capture the stable exponential growth in the presence of such an extension, the resolution and perturbation size and shape are carefully selected. The complete development of the mode structure takes approximately 5τ A or more, depending on the plasma parameters and the details of the shear flow. The perturbation level is ε = 10 -3If so, when used in a shear-free simulation, the non-linearity can affect the mode growth before complete development. Therefore, in these shear flow simulations, smaller perturbations are used. In these cases, higher resolution is also used to limit the noise that can obscure the growth of small perturbations and to minimize the unintentional seeding of short-wavelength modes that can grow rapidly at the perturbed axial wavenumber and disrupt the generation. Short-wavelength modes have fewer problems when a / r Li = 2.357. To further reduce the difficulties with high-k growth and non-linearity, as discussed elsewhere in this specification, a phase shift is included. The appropriate value φ0 of the phase shift parameter is found by performing a preliminary simulation with φ0 = 0 and observing the mode elongation in the later stages of the simulation before the growth is interrupted by non-linearity or high-k mode growth. For the second and final simulations, φ0 is selected such that the shape of the initial state closely matches the elongated mode seen in the preliminary simulation. FIGS. 18A - 18D show examples of mode growth with or without a phase-shifted initial state. Without a phase shift, the mode elongation steadily reduces the growth rate before the simulation is ultimately interrupted by high-k growth,

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[0127] All MHD-like 5M2F simulations (a / r Li = 50) use fifth-order elements and 64×16 (radial × axial) cells with ε = 10 -5 . For large r LiSimulation (a / r Li =2.357) is a fourth-order element and ε=10 -7 A 64x16 (radial x axial) cell with θ = 0.05 is used. The phase shifts are shown in Table I.

[0128] Table I: Settings used for idealized 5M2F simulations with shear flow. The phase shift parameter φ is given for each run with shear flow. For linear shear flow, the overall phase shift is φr, and for parabolic shear flow, φr 2 See equations (31) and (32). [Table 1]

[0129] The deviation from an exact linear growth rate in these simulations is estimated to be less than 10% and typically less than a few percent. This level of error is higher than in shear-free results, primarily due to the challenges associated with simultaneously avoiding high-k modes and nonlinearities.

[0130] 18A-18D show the mode growth behavior in the 5M2F model with and without an initial phase shift (effectively a tilt) in the perturbation. The results show that the pinch edge velocity v a sf =0.3v A MHD-like regimes (a / r Li = 50). Figure 18A shows the evolution of the radial ion kinetic energy normalized by the initial magnetic energy. Figure 18B shows the evolution of the radial ion kinetic energy normalized by the initial magnetic energy. A Normalized growth rate γτ derived from kinetic energy growth over the interval A The approach used to find each plotted value (×) is discussed in the text. The red lines represent the 1-τ A 18C and 18D are similar to FIGS. 18A and 18B, but for the case where the perturbation phase shift φr has φ=2π.

[0131] In recent studies using PIC modeling, the m = 0 instability at the FuZE-like equilibrium was considered. In the following discussion, an equilibrium is presented and details related to the implementation of transport by Braginskii at this equilibrium are provided below. Also, below, 5M2F modeling using transport by Braginskii is compared with PIC results.

[0132] The Bennett equilibrium is used to represent a typical FuZE plasma. As discussed in detail above, the normalized profiles in Figure 11 are dimensioned by choosing I p , a, and a / r Li . After previous work, the FuZE-like equilibrium has a / r Li = 5.825, a = 0.91 mm and I p = 300 kA. These choices give B θ,pk = 33.0 T, n pk = 4.25×10 24 m -3 , and T = 1.27 keV. The relevant characteristic times and Alfvén velocity are τ A = 2.61 ns and v A = 3.49×10 5 m / s.

[0133] A closed model by Braginskii including kinetic corrections is presented herein. Here, the transport coefficients before and after correction are considered to be particularly FuZE-like equilibria. Figure 19 shows the corrected viscous diffusivity and the uncorrected viscous diffusivity. The correction is made according to equations (23), (24), and (25). In equation (23),

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[0134] Figure 19 shows the momentum diffusion rate in the FuZE-like equilibrium. The uncorrected diffusion rate ( * coefficient without) has nonphysical and / or numerically difficult characteristics. The diffusion rate is corrected to address these issues as described in the main text ( * coefficient with). The correction is performed according to equations (23), (24), and (25) with l = 0.2a and r critIt is 0.5a.

[0135] Figure 20 shows the corrected ion thermal diffusivity and the uncorrected ion thermal diffusivity. The correction is performed with respect to the momentum diffusivity using l = 0.2a and r crit = 0.5a. A diffusivity limit is also imposed on the momentum diffusivity, but this only affects a small region near r = 0

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[0136] Figure 20 shows the ion thermal diffusivity in the FuZE-like equilibrium. The uncorrected diffusivity ( * coefficient without) has non-physical and / or numerically difficult features. The diffusivity is corrected to address these issues as described in the main text ( * coefficient with). The correction is performed according to equations (24) and (25), with l = 0.2a and r crit = 0.5a.

[0137] Simulations show that among all the Braginskii transport terms, the cross stress regulated by η0 has the strongest effect on the m = 0 instability growth. To study the effect of η0 compared to the gyroviscosity and diamagnetic heat flux, three series of simulations are performed at a fixed wave number ka = 5. The results are shown in Figure 21. For the purpose of scanning the intensity of η0, the product of the two global correction factors described above, i.e., f corr τ f corr L is replaced by a single multiplier f η0 which is used to calculate the modified non-magnetized viscosity η * 0 = f η0 η0. In the first series, the complete Braginskii model described above is applied, and the gyroviscosity coefficient and the diamagnetic heat flux coefficient are set to r critThe base correction is applied, but the correction for large Larmor radii is not used for those coefficients. In the second series, the gyroviscosity is omitted but the diamagnetic heat flux is retained. In the third series, the diamagnetic heat flux is omitted but the gyroviscosity is retained. Using the full Braginsky model, f η0 As f approaches 0.1, the modeled growth rate approaches the PIC result. η0 In the case of the gyroviscosity and diamagnetic heat flux, the contributions of the gyroviscosity and diamagnetic heat flux each vary from the ideal 5M2F value to a moderate (

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[0138] Figure 21 shows the overall multiplier f for the non-magnetizing viscosity coefficient η0. η0Shows the growth rate of the 5M2F m = 0 instability at ka = 5 as a function. Results are shown for the full Braginskii model, the full model with the gyroviscous term subtracted, and the full model with the diamagnetic heat flux term subtracted. Results at ka = 5 from PIC (Tummel 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-pinch plasmas," Phys. Plasmas 26, 062506 (2019)]) are included for comparison.

[0139] Testing the dependence of the growth rate on k shows that applying the 5M2F model with corrected Braginskii transport can yield results that share some features of the PIC results. As shown in Fig. 21, the full Braginskii model yields a growth rate f η0 = 0.06, similar to PIC. (Note that the overall coefficient f corr τ f corr L = 0.018 is used for the corrected diffusivity η * 0 = ρ shown in Fig. 19.) Scanning ka with a fixed f η0 = 0.06, γτ A increases with ka and does not show signs of reaching a peak even at ka = 23, as shown in Fig. 22. In this scan with f η0 = 0.06, the large Larmor radius correction is not used. Generally, γτ increases with ka AThe reason for the increase is unclear. Additional simulations show that the increase at high-k persists even when the gyro-viscosity and diamagnetic heat flux are omitted. The results indicate that using a large η0 (unphysically large as discussed above) is not a suitable approach to reproduce the PIC results over a wide range of ka. The results are also shown for the scan using the fully size-based correction model discussed above. In that scan, f η0 = f corr τ f corr L where l = k -1 and f corr L is η3 and

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[0140] Figure 22 shows the 5M2F m = 0 instability growth rate at mode wavenumbers in the range ka = 5 - 23. The results of PIC modeling by Tummel et al. are included for comparison. Using a constant viscosity correction factor f η0 = 0.06, the growth rate increases unphysically at high k. Based on a size-based model using l = k -1 to calculate f η0 , the peak growth rate is

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[0141] The models described above are more rigorously nonlinear, but the results presented so far have only considered the linear regime of instability growth, i.e., growth that leads to a small deviation from the equilibrium state. In this section, we consider simulations that track plasma generation into the nonlinear regime. To enable this nonlinear modeling, a combination of high resolution and artificially large perpendicular transport is used. The numerical methods readily available via WARPXM do not inherently provide strong numerical dissipation where the gradients become steep, and polynomial representations are susceptible to the effects of nonphysical oscillations (similar to the Gibbs phenomenon) that can terminate the simulation. Artificial perpendicular viscosity and thermal conductivity help to relax sharp gradients, and high resolution helps to resolve them.

[0142] In the simulations presented herein, the minimum diffusion rate

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[0143] Using ka = 5 and the same initial conditions as those used for comparison with PIC, but with a parabolic shear flow applied, the simulation is performed such that the resolution increases to 96×24 (radial×axial) cells by the quartic element. No phase shift is used in the perturbation. The range of the parabolic shear flow strength is studied using v a sf / v A =0, 0.25, 0.5 and 0.75. For the cases with v a sf / v A =0 and 0.5, the evolution of the two-dimensional density profile is shown in Figs. 1 and 3, respectively. As the shear flow velocity increases, the radial emission of the plasma is restricted. This is further shown in Fig. 23, where the normalized pinch ion inventory and the total thermal energy are plotted as functions of time. The pinch ion inventory and the thermal energy are defined as follows. [Number] and [Number] The normalized quantities are N(t) / N(t = 0) and W(t) / W(t = 0). In the case without shear, the pinch loses 50% of the initial inventory by the end of the simulation, while for va sf / v A In the case of / v = 0.5,

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[0144] Figure 23 shows the ion density profiles of a non - linear simulation starting from a FuZE - like equilibrium with a seed mode ka = 5. The profiles are repeated axially three times and reflected across r = 0. In the case where the shear flow is zero (upper panel), at t = 10τ A a radial jet - like structure is generated. v a sf / v A = 0.5 (lower panel), the growth is retarded, the instability structure is sheared, the radial extent is limited by the shear, and a relaxed near - equilibrium state is established at t = 35τ A by.

[0145] Figure 24 shows the normalized ion inventory and thermal energy within r = a, normalized by the initial inventory and energy, for a FuZE - like non - linear simulation, where v a sf / v A = 0, 0.25, 0.5, and 0.75. In all cases, ka = 5. By including the shear flow, the confinement of particles and energy is improved.

[0146] The results presented herein are for small a / r Lishows rapidly growing instabilities. The instabilities are identified as electron drift modes, as seen in previous 5M2F modeling. The electron cross-field drift velocity

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[0147] a / r Li =2.357, the modeling of the linear growth in the presence of a positive shear flow using the 5M2F model yields results almost identical to Hall MHD modeling. For either a linear or parabolic shear - flow profile, complete stabilization is

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[0148] The 5M2F model with the actual ion / electron mass ratio, with or without the corrected Braginskii transport, is shown in Fig. 22. In particular, the 5M2F growth rate peak is twice as high, and the peak is at

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[0149] / v a sf / vA For (v / v

[0150] Previous MHD modeling has shown that shear flow stabilizes shorter wavelength m = 0 modes more effectively than longer modes. Using the same setup as the ka = 5 non - linear simulation above, simulations were performed with ka = 5 / 3 and moderate parabolic shear flow (v a sf / v A = 0.5) to explore the non - linear confinement behavior of longer wavelength instabilities. The instability grows more rapidly and mass and energy are lost earlier, but the total losses are equivalent. Further, to represent a more realistic noisy Z - pinch plasma, a "multi - mode" case is [Number] performed. That is, the domain length is such that L z / a = 6π / 5 (capturing one wavelength for ka = 5 / 3), up to a maximum ka = 25 (capturing 15 wavelengths), with each available mode perturbed. For each mode, the phases of equations (31) and (32) are set such that φ0 is randomly chosen and there is no radial dependence (ζ = 0). The results are that mass and energy losses increase, perhaps due to mode coupling. However, in all cases, the mass loss is < 50% and the energy loss is < 20%. Overall, the picture remains the same as for the a = 5 case. Moderate shear flow appears to facilitate the generation of a quasi - stable plasma with limited losses.

[0151] The modeling presented here has focused on Z-pinch equilibria using Bennett profiles with a uniform initial temperature. Studies considering other profiles that include profiles matching the best available experimental data can give different m = 0 stability behaviors, including stabilization effects due to the induction of central drift even when the bulk flow is zero. It is also well known that profiles satisfying the Kadomtsev criterion are MHD stable. The stability of MHD stability profiles in non-MHD models, such as the 5M2F model, could be considered in future studies. Another consideration is that even Kadomtsev stability profiles can be unstable with respect to the so-called entropy mode, but for a gas with realistic adiabatic coefficients (for real gases with three or more degrees of freedom [Number] ) the instability of the entropy mode occurs only when the temperature profile is non-uniform. In recent modeling, the entropy mode is considered in a special case of Γ > 2 with a uniform temperature in the Bennett profile. When the temperature is non-uniform and [Number] 5M2F modeling of the entropy mode behavior in MHD stability profiles having will be of interest.

[0152] The 5M2F model is presented, extending the ideal model to include closures by Braginskii. This model is applied to study the instability of m = 0 Z-pinches, focusing on initial conditions based on Bennett equilibria.

[0153] The ideal 5M2F modeling results are benchmarked against previous MHD and Hall MHD results. The growth rates are in close agreement with the previous results, regardless of the presence or absence of shear flow. Complete stabilization is [Number] It occurs at the edge flow velocity. Beyond previous work, reverse linear shear flow has been studied, revealing a relatively slow decrease in growth rate with increasing flow rate, and it is predicted to fully stabilize at the edge flow velocity

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[0154] The 5M2F model is also benchmarked against recent PIC modeling initialized with a FuZE-like Bennett equilibrium. The peak growth of the 5M2F results is about twice as fast as that of PIC

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[0155] Nonlinear modeling of m = 0 instability in FuZE-like plasmas is performed using the 5M2F model by Braginskii. The edge shear flow velocity is 0.5v AThe simulation in showed instability, followed by non-linear mixing due to shear and relaxation to a quasi-steady state. The losses of pinch ion inventory and pinch thermal energy are limited to and respectively. [Number] and [Number] respectively.

[0156] The 5M2F model provides a compelling platform for high-fidelity computational Z-pinch research. In future studies, in addition to the Bennett profile, various equilibrium profiles may be considered, particularly the MHD stability Kadomtsev profile with a temperature gradient that can trigger the entropy mode. Simulations of linear and non-linear 3D Z-pinch dynamics, including m = 1 instability, are also within the scope. Finally, it is necessary to study the ability of the 5M2F model to capture electron drift instability. Accurate and efficient modeling of the associated microturbulence can provide important insights into the current profile and axial heat transport in reactor-grade Z-pinch plasmas.

[0157] For an equilibrium with shear flow, a true two-fluid equilibrium is derived. The total ion axial velocity is determined as v iz = v iz0 + v sf , where v sf (r) = v a sf r / a (linear) or v sf (r) = v a sf r 2 / a 2 (parabolic). Here, v iz0 = j z / (2en i ) iz =v iz0 +v sf and the equation is sf (r)=v a sf r / a (linear) or v sf (r)=v a sf r 2 / a 2 (parabolic). Here, v iz0 =j z / (2en i) is the ion velocity without shear and provides half of the equilibrium current. The profiles of the magnetic field and current do not change from the shearless equilibrium profiles of equations (26) and (27). The ion pressure and density also do not change. Specifically, the ion pressure is half of the total pressure given by equation (28). The ion temperature is uniform and a free parameter, and in this specification, T i is set to a / r as shown in the following equation (33): It does not change from the shearless equilibrium profile of (27). The ion pressure and density also do not change. Specifically, the ion pressure is equation (33). Next, n Li = p i = p i / (k B T i ). The radial electric field is determined from equation (29) for the ion momentum balance. The results for linear and parabolic shear flows are as follows,

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[0158] Next, the electron pressure is determined from equation (30). The results are as follows.

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[0159] These pressures match p e without shear at r = 0 and deviate slightly elsewhere. The electron temperature is determined as p e / (n e k B ). The electron axial velocity is such that its j z = v ez n e q e + v iz n i q iis determined by requiring that it be satisfied.

[0160] Embodiments of the present disclosure can be described in consideration of the following sections.

[0161] 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 source for driving 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 puff valve for providing a neutral gas to the acceleration volume to supply fuel to the Z - pinch plasma column; a shaped component conductively connected to the second electrode to cause gas breakdown of the neutral gas and generate a shear flow velocity profile in the presence of the neutral gas provided by the at least one gas puff valve.

[0162] 2. The apparatus according to item 1, wherein an electron flow of the current flows from the second electrode to the first electrode.

[0163] 3. The apparatus according to any one of items 1 and 2, wherein the shaped component incorporates at least one conductive ring including at least one contact surface electrically connected to an outer surface of the second electrode.

[0164] 4. The apparatus according to item 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 shaped component incorporates at least one refractory metal.

[0165] 5. The apparatus according to item 4, wherein the at least one refractory metal includes 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.

[0166] 6. The apparatus according to any one of claims 4 and 5, wherein the portion facing the plasma incorporates at least one conductive form of carbon including one or more of graphite, sintered carbon powder, pressed carbon powder, carbon fiber, or carbon nanotube.

[0167] 7. The apparatus according to any one of claims 4 to 6, wherein the portion facing the plasma includes at least one textured surface formed to incorporate a plurality of local concave elements that form a structured array to enhance the local electric field and promote field emission.

[0168] 8. The apparatus according to claim 7, wherein the at least one textured surface is formed by a mechanical treatment including one or more of cutting, scratching, polishing, sandblasting, grooving, checkering, punching, embossing, or knurling.

[0169] 9. The apparatus according to any one of claims 7 and 8, wherein the at least one textured surface is formed by a chemical treatment including one or more of etching, chemical deposition, spraying, sputtering, ion and neutral implantation, or epitaxial growth.

[0170] 10. The apparatus according to any one of claims 1 to 9, further comprising at least one plasma injector that provides an ionized gas to the acceleration volume to further supply fuel to the Z - pinch plasma column.

[0171] 11. The apparatus according to any one of claims 1 to 10, wherein the second electrode incorporates a conical electrode surface arranged to enhance the momentum transfer to ions and neutral particles in the axial direction of the first and second electrodes.

[0172] 12. The apparatus according to any one of items 1 to 11, further comprising a third electrode disposed coaxially with respect to the first electrode and the second electrode and between the first electrode and the second electrode, the third electrode showing a tapered electrode configuration and incorporating a conical electrode surface arranged to enhance momentum transfer to ions and neutral particles in the axial direction of the first, second, and third electrodes.

[0173] 13. A method comprising: activating one or more gas puff valves to introduce an axially symmetric volume of neutral gas into the acceleration volume; generating a radial electric field to support a first current by promoting destruction of the neutral gas, the first current flowing between the inner electrode and the outer electrode via the introduced neutral gas; 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. The method, wherein the Z-pinch plasma column is surrounded and stabilized by a sheared velocity plasma flow formed at least in part from the neutral gas.

[0174] 14. The method according to item 13, further comprising activating one or more plasma injectors to introduce an axially symmetric volume of ionized gas into the acceleration volume.

[0175] 15. The method according to item 14, wherein the axially symmetric volume of ionized gas is introduced to replenish the axially symmetric volume of neutral gas after formation of the Z-pinch plasma column.

[0176] 16. The method according to any one of items 13 to 15, wherein the inner electrode is an anode and the outer electrode is a cathode.

[0177] 17. A plasma confinement system comprising: an outer electrode; an inner electrode; At least one power source conductively coupled to each of the inner electrode and the outer electrode, wherein a terminal of the at least one power source is configured to generate a potential difference between the inner electrode and the outer electrode, the at least one power source; One or more first valves fluidly coupled to a fuel gas source and configured to direct sufficient neutral gas supplied from the fuel gas source to support a local disruption path between the inner electrode and the outer electrode and to establish a sheared velocity plasma flow during the duration of a Z-pinch discharge between the inner electrode and the outer electrode, a plasma confinement system comprising the same.

[0178] 18. The plasma confinement system according to item 17, wherein the inner electrode and the outer electrode define an acceleration volume in which the neutral gas is directed by one or more first valves.

[0179] 19. Further comprising an intermediate electrode, The plasma confinement system according to item 17, wherein the inner electrode and the intermediate electrode define an acceleration volume in which the neutral gas is directed by one or more first valves.

[0180] 20. The plasma confinement system according to any one of items 17 to 19, further comprising one or more second valves fluidly coupled to a fuel gas source and configured to direct sufficient ionized gas supplied from the fuel gas source to maintain a sheared velocity plasma flow during the duration of the Z-pinch discharge.

[0181] 21. An apparatus, A first electrode positioned to define an outer boundary of the acceleration volume, A second electrode coaxially disposed with respect to the first electrode and positioned to define an inner boundary of the acceleration volume, At least one power source for driving a current along a Z-pinch plasma column between the first electrode and the second electrode, A set of valves for providing gas to the acceleration volume to fuel the Z-pinch plasma column, comprising the same. An apparatus in which the electron flow of the current is in a first direction from a second electrode to a first electrode.

[0182] 22. The apparatus according to claim 21, wherein the gas contains a neutral gas, further comprising a shaped component conductively connected to the second electrode so as to cause gas breakdown of the neutral gas in the presence of the neutral gas provided by a set of valves and generate a shear flow velocity profile in a second direction opposite to the first direction.

[0183] 23. The apparatus according to claim 22, wherein the shaped component incorporates at least one conductive ring including at least one contact surface electrically connected to the outer surface of the second electrode.

[0184] 24. The apparatus according to claim 23, wherein at least one conductive ring incorporates a conductive material that is chemically and thermomechanically compatible with the conductor of the second electrode, and the plasma-facing portion of at least one shaped component incorporates at least one refractory metal.

[0185] 25. The apparatus according to claim 24, wherein at least one refractory metal includes 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.

[0186] <7001993>26. The apparatus according to any one of claims 24 and 25, wherein the plasma-facing portion incorporates at least one conductive form of carbon including one or more of graphite, sintered carbon powder, pressed carbon powder, carbon fiber, or carbon nanotube.

[0187] 27. The apparatus according to any one of claims 24 to 26, wherein the plasma-facing portion includes at least one textured surface formed to incorporate a plurality of locally concave elements forming a structured array to enhance a local electric field and promote field emission.

[0188] 28. The device according to claim 27, wherein at least one texturized surface is formed by a mechanical treatment including one or more of cutting, scratching, polishing, sandblasting, grooving, checkering, punching, embossing, or knurling.

[0189] 29. The device according to any one of claims 27 and 28, wherein at least one texturized surface is formed by a chemical treatment including one or more of etching, chemical deposition, spraying, sputtering, ion and neutral implantation, or epitaxial growth.

[0190] 30. The device according to claim 21, wherein the gas is provided to the acceleration volume as an ionized gas.

[0191] 31. The device according to any one of claims 21 to 30, wherein the second electrode incorporates a conical electrode surface arranged to enhance the momentum transfer to ions and neutral particles in the axial direction of the first and second electrodes.

[0192] 32. The device according to any one of claims 21 to 31, further comprising a third electrode arranged coaxially between and with respect to the first and second electrodes, the third electrode showing a tapered electrode configuration and incorporating a conical electrode surface arranged to enhance the momentum transfer to ions and neutral particles in the axial direction of the first, second, and third electrodes.

[0193] 33. A method comprising: activating one or more valves to introduce an axially symmetric volume of fuel gas into the acceleration volume; 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 the unsupported end of the inner anode; wherein the Z-pinch plasma column is surrounded and stabilized by a shear velocity plasma flow formed from the fuel gas.

[0194] 34. The method according to claim 33, further comprising generating a radial electric field to support an initial current flowing between the inner anode and the outer cathode through the introduced fuel gas before forming the Z-pinch plasma column.

[0195] 35. The fuel gas includes a neutral gas, The method according to claim 34, wherein the radial electric field supports the initial current by promoting at least the breakdown of the neutral gas.

[0196] 36. The method according to any one of claims 33 to 35, wherein when the fuel gas is introduced into the acceleration volume, the fuel gas includes an ionized gas.

[0197] 37. A plasma confinement system, an outer electrode, an inner electrode concentrically positioned within the outer electrode, at least one power source conductively coupled to each of the inner electrode and the outer electrode, wherein the terminals of the at least one power source are oriented to generate a potential difference between the inner electrode and the outer electrode and drive electrons from the inner electrode to the outer electrode, the at least one power source; one or more valves fluidly coupled to a fuel gas supply source and configured to drive a shear rate plasma flow with sufficient fuel gas supplied from the fuel gas supply source during the duration of a Z-pinch discharge between the inner electrode and the outer electrode.

[0198] 38. The plasma confinement system according to claim 37, wherein the inner electrode and the outer electrode define an acceleration volume in which the fuel gas is directed by one or more valves.

[0199] 39. Further comprising an intermediate electrode concentrically positioned between the inner electrode and the outer electrode, The plasma confinement system according to claim 37, wherein the inner electrode and the intermediate electrode define an acceleration volume in which the fuel gas is directed by one or more valves.

[0200] 40. The plasma confinement system according to any one of items 37 to 39, wherein the fuel gas contains one or both of a neutral gas and an ionized gas.

[0201] Certain values, relationships, materials, and components have been described for the purpose of describing the concept of the present invention. However, those skilled in the art will understand that numerous variations and / or modifications can be made to the present invention without departing from the spirit or scope of the basic concept and operating principle of the present invention, as shown in specific embodiments. In light of the above teachings, those skilled in the art should recognize that they can modify those details without departing from the present invention taught herein. For example, the numerical ranges listed herein are exemplary and may be modified based on the operating mode of a given plasma confinement system or 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 ranges may be expanded proportionally (e.g., linearly, exponentially, etc.).

[0202] Here, although embodiments and specific modifications of the underlying concept of the present invention have been fully described, various other embodiments, as well as specific variations and modifications of the embodiments shown and described herein, will occur to those skilled in the art who have become proficient in such underlying concepts. All such modifications, alternatives, and other embodiments are intended to be included as long as they are within the scope of the appended claims or their equivalents. Therefore, it should be understood that the present invention can be practiced in ways other than those specifically described herein. As a result, this embodiment should be considered exemplary in all respects and not restrictive.

[0203] Therefore, the present specification and drawings should be considered in an illustrative rather than a restrictive sense. However, it will be apparent that various modifications and changes can be made without departing from the broader spirit and scope of the subject matter recited in the claims.

[0204] Other variations are within the spirit of the disclosure. Thus, while the disclosed techniques are susceptible to various modifications and alternative constructions, the particular illustrated embodiments have been shown in the drawings and described in detail above. However, it should be understood that there is no intention to limit the claims of the subject matter recited by the claims to the particular forms disclosed, but on the contrary, it is intended to cover all modifications, alternative constructions, and equivalents within the spirit and scope of the disclosure as defined by the appended claims.

[0205] In the context of describing the disclosed embodiments, terms such as "a," "an," and "the," as well as the use of similar referents (particularly in the context of the following claims), should be construed to cover both the singular and plural forms unless otherwise indicated herein or clearly contradicted by the context. Similarly, the use of the term "or" should be construed to mean "and / or" unless explicitly stated otherwise or clearly contradicted by the context. Unless otherwise stated, terms such as "comprising," "having," "including," and "containing" should be construed as open-ended terms (i.e., meaning "including but not limited to"). When referring to a connection that has not been modified and is a physical connection, the term "connected" should be construed to mean that, even if there are intervening elements, it is contained within, attached to, or joined together with, either partially or completely. The recitation of a range of values herein is merely intended to serve as a concise way of referring individually to each separate value that falls within that range, and each separate value is incorporated herein as if it were individually recited herein. Unless otherwise stated or contradicted by the context, the use of the term "set" (e.g., "a set of items") or "subset" should be construed to mean a non-empty set containing one or more members. Further, unless otherwise stated or contradicted by the context, the term "subset" of a corresponding set does not necessarily denote a proper subset of the corresponding set, and the subset and the corresponding set may be equal. Unless otherwise explicitly stated in the context, the use of the phrase "based on" means "at least partially based on" and not "based solely on."

[0206] Connective phrases such as phrases in the form of "at least one of A, B, and C" or "at least one of A, B, and C" (i.e., the same phrase with or without the Oxford comma) are, unless otherwise clearly stated or clearly not in conflict with the context, within the context, items, terms, etc. are either any of A or B or C, any non-empty subset of the set of A, B, and C, or any set containing at least one A, at least one B, or at least one C, and are generally understood to be used to indicate any set that is not in conflict with the context or excluded in another way. For example, in an exemplary embodiment of a set having three members, the connective phrases "at least one of A, B, and C" and "at least one of A, B, and C" refer to any of the following sets: {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}, and, explicitly or not in conflict with the context, any set having {A}, {B}, and / or {C} as subsets (e.g., a set having multiple "A"s). Thus, such connective phrases are not generally intended to mean that a particular embodiment requires the presence of at least one of each of at least one A, at least one B, and at least one C. Similarly, phrases such as "at least one of A, B, or C" and "at least one of A, B or C" refer to the same thing as "at least one of A, B, and C", and "at least one of A, B, and C" refers to any of the following sets: {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}, provided that a different meaning is not explicitly stated or is not clear from the context. Further, unless otherwise stated or not in conflict with the context, the term "plural" indicates a plural state (e.g., "a plurality of items" indicates a plurality of items). The number of items in a plurality is at least two, but may be more if indicated explicitly or by the context.

[0207] The operations of the processes described in this specification can be performed in any suitable order, unless otherwise indicated herein or unless clearly inconsistent with the context. In one embodiment, a process such as the processes described herein (or variations and / or combinations thereof) is performed under the control of one or more computer systems configured with executable instructions and implemented as code (e.g., executable instructions, one or more computer programs, or one or more applications) that is executed collectively on one or more processors by hardware or combinations thereof. In one embodiment, the code is stored on a computer-readable storage medium in the form of a computer program that includes, for example, a plurality of instructions executable by one or more processors. In one embodiment, the computer-readable storage medium excludes transient signals (e.g., propagating transient power or electromagnetic transmissions) but includes non-transient data storage circuits (e.g., buffers, caches, and queues) within a transceiver of the transient signal, and is a non-transient computer-readable storage medium. In one embodiment, the code (e.g., executable code or source code) is stored on a set of one or more non-transient computer-readable storage media that stores executable instructions that cause a computer system to perform the operations described herein when executed (i.e., as a result of execution) by one or more processors of the computer system. In one embodiment, the set of non-transient computer-readable storage media includes a plurality of non-transient computer-readable storage media, and not all of the code is present on one or more of the individual non-transient storage media of the plurality of non-transient computer-readable storage media, while the plurality of non-transient computer-readable storage media collectively stores all of the code. In one embodiment, the executable instructions are executed such that different instructions are executed by different processors. For example, in one embodiment, the non-transient computer-readable storage medium stores the instructions, the main CPU executes a portion of the instructions, while the graphics processing 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.

[0208] Thus, in one embodiment, a computer system is configured to implement one or more services that singly or collectively perform the operations of the processes described herein, and such a computer system is composed of applicable hardware and / or software that enables the execution of the operations. Further, in embodiments of the present disclosure, the computer system is a single device, and in other embodiments, a distributed computer system includes a plurality of devices that perform different operations such that the distributed computer system performs the operations described herein and a single device does not perform all of the operations.

[0209] The use of any and all examples or exemplary language (e.g., "such as") provided herein is merely intended to better illustrate various embodiments and does not impose a limitation on the claims unless otherwise claimed. No language in this specification should be construed as indicating any non-claimed element as essential to the practice of the inventive subject matter disclosed herein.

[0210] 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 will be apparent to those skilled in the art upon reading the foregoing description. The inventors expect those skilled in the art to adopt such variations as appropriate, and the inventors intend for the embodiments of the present 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. Further, unless otherwise indicated herein or clearly contradicted by context, any combination of any of the above-described elements in all possible variations thereof is included within the scope of the present disclosure.

[0211] All references, including publications, patent applications, and patents, cited in this specification 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: 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 source for driving a current along a Z-pinch plasma column between the first and second electrodes; a set of valves including at least one gas puff valve for providing a neutral gas to the acceleration volume to supply fuel to the Z-pinch plasma column; a shaped component conductively connected to the second electrode to cause gas breakdown of the neutral gas in the presence of the neutral gas provided by the at least one gas puff valve and generate a shear flow velocity profile.

2. The apparatus of claim 1, wherein the current is directed from the second electrode to the first electrode.

3. The apparatus of claim 1, wherein the shaped component incorporates at least one conductive ring including at least one contact surface electrically connected to an outer surface of the second electrode.

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

5. The apparatus of claim 4, wherein the at least one refractory metal includes 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 claim 4, wherein the plasma-facing portion incorporates at least one conductive form of carbon including one or more of graphite, sintered carbon powder, pressed carbon powder, carbon fiber, or carbon nanotubes.

7. The apparatus of claim 4, wherein the plasma-facing portion includes at least one textured surface formed to incorporate a plurality of local concave elements forming a structured array to enhance a local electric field and promote field emission.

8. The apparatus of claim 7, wherein the at least one textured surface is formed by a mechanical process including one or more of cutting, scratching, polishing, sandblasting, grooving, checkering, punching, embossing, or knurling.

9. The apparatus according to claim 7, wherein the at least one surface subjected to texture processing is formed by a chemical treatment including one or more of etching, chemical deposition, spraying, sputtering, ion and neutral implantation, or epitaxial growth.

10. The apparatus according to claim 1, wherein the set of valves further comprises at least one plasma injector that provides an ionized gas to the acceleration volume to further supply fuel to the Z-pinch plasma column.

11. The apparatus according to any one of claims 1 to 10, 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 apparatus according to any one of claims 1 to 10, further comprising a third electrode disposed coaxially between and with respect to the first and second electrodes, the third electrode exhibiting a tapered electrode configuration and incorporating a conical electrode surface 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: activating one or more gas puff valves to introduce an axially symmetric volume of neutral gas into the acceleration volume; generating a radial electric field using a shaped component conductively connected to the inner electrode to promote breakdown of the introduced neutral gas and support a first current flowing between the inner and outer electrodes through the introduced neutral gas; forming a Z-pinch plasma column from the introduced neutral gas to support a second current flowing between the inner and outer electrodes; The method, wherein the Z-pinch plasma column is surrounded and stabilized by a shear velocity plasma flow formed at least in part from the neutral gas.

14. The method according to claim 13, wherein one or more plasma injectors are activated to introduce an axially symmetric volume of ionized gas into the acceleration volume.

15. The method according to claim 14, wherein the axially symmetric volume of the ionized gas is introduced to replenish the axially symmetric volume of the neutral gas after formation of the Z-pinch plasma column.

16. The method according to any one of claims 13 to 15, wherein the inner electrode is an anode and the outer electrode is a cathode.

17. A plasma confinement system comprising: an outer electrode, an inner electrode, at least one power source conductively coupled to each of the inner and outer electrodes, the at least one power source configured such that a terminal of the at least one power source generates a potential difference between the inner and outer electrodes, one or more first valves fluidly coupled to a fuel gas source and directing sufficient neutral gas supplied from the fuel gas source to support a local disruption path between the outer electrode and a shaped component conductively connected to the inner electrode, and configured to establish a sheared velocity plasma flow during a duration of a Z-pinch discharge between the inner and outer electrodes, a plasma confinement system comprising the same.

18. The plasma confinement system according to claim 17, wherein the inner and outer electrodes define an acceleration volume into which the neutral gas is directed by the one or more first valves.

19. further comprising an intermediate electrode, The plasma confinement system according to claim 17, wherein the inner and intermediate electrodes define an acceleration volume into which the neutral gas is directed by the one or more first valves.

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

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