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Recent improvements to MACH2 and MACH3 for fast Z-pinch modeling
Abstract
The U.S. Air Force Research Laboratory's 2 1/2 and 3-dimensional magnetohydrodynamic simulation codes, MACH2 and MACH3, are used by many laboratories to simulate fast Z-pinch experiments. In this talk we will describe numerous improvements made recently in those codes specifically related to modeling those experiments, and show the improvements that result in the simulations.
... Previously published simulations considered a r ∼ 5 -20 mm, Xe plasma liner → DT target plasma pinch [47], however the present configuration provides a far greater fusion-energy yield. MACH2 is a 2-1/2D MHD code, which contains the essential-physics for the SZP [48,49]. MACH2 was also modified to include an inductive-electric field, produced when the magnetic fields are compressed. ...
This paper is dedicated to Norman Rostoker, our (FJW and HUR) mentor and long-term collaborator, who will always be remembered for the incredible inspiration that he has provided us. Norman’s illustrious career dealt with a broad range of fundamental-physics problems and we were fortunate to have worked with him on many important topics: intense-charged-particle beams, field-reversed configurations, and Z-pinches. Rostoker ’s group at the University of CA, Irvine was well known for having implemented many refinements to the Z-pinch, that make it more stable, scalable, and efficient, including the development of: the gas-puff Z-pinch [1], which provides for the use of an expanded range of pinch-load materials; the gas-mixture Z-pinch [2], which enhances the pinch stability and increases its radiation efficiency; e-beam pre-ionization [3], which enhances the uniformity of the initial-breakdown process in a gas pinch; magnetic-flux-compression [4, 5], which allows for the amplification of an axial-magnetic field B z ; the Z-θ pinch [6], which predicts fusion in a pinch-on-fiber configuration; the Staged Z-pinch (SZP) [7], which allows for the amplification of the pinch self-magnetic field, B θ , in addition to a B z , and leads to a stable implosion and high-gain fusion [8, 9, 10].
This paper describes the physical basis for a magneto-inertial compression in a liner-on-target SZP [11]. Initially a high-atomic-number liner implodes under the action of the J⃗ ×B⃗
, Lorentz Force. As the implosion becomes super Alfvénic, magnetosonic waves form, transporting current and magnetic field through the liner toward the interface of the low-atomic-number target. The target implosion remains subsonic with its surface bounded by a stable-shock front. Shock waves that pass into the target provide a source of target plasma pre-heat. At peak compression the assembly is compressed by liner inertia, with flux compression producing an intense-magnetic field near the target. Instability develops at the interface, as the plasma decelerates, which promotes the formation of target-hot spots. Early experiments provide evidence for the magneto-inertial implosion [8, 9, 10]. Studies underway are designed to verify these predictions on the National Terawatt Facility, Zebra Generator, located at the University of Nevada, Reno. Simulations for an unmagnetized, silver-plasma liner imploding onto a deuterium-tritium plasma target, driven by a 200 TW generator, predict fusion beyond break-even, with a 200 MJ yield in an ignited plasma, with an engineering gain factor of, G = E fusion /E stored ∼20.
... II. SIMULATION The 2-1/2-D, MACH2 MHD code [10], [11] is used to simulate the Z -pinch implosion. It solves the continuity, momentum, energy, and magnetic-field equations, using a finite-volume-differencing technique on an R-Z grid of quadrilateral cells. ...
... II. SIMULATION The 2-1/2-D, MACH2 MHD code [10], [11] is used to simulate the Z -pinch implosion. It solves the continuity, momentum, energy, and magnetic-field equations, using a finite-volume-differencing technique on an R-Z grid of quadrilateral cells. ...
The implosion of a liner-on-target Z-pinch is
simulated for fusion. The simulation code is the 2-1/2-D, MACH2
MHD code and the driver parameters are τ1/4 ∼ 130 ns,
Ipeak ∼ 22 MA, and Estored ∼ 22 MJ. Simulations are
run for a Staged Z-pinch, configured as an unmagnetized,
silver-plasma liner imploding onto a deuterium-tritium plasma
target. Magnetosonic-shocks play a decisive role in the attainment
of fusion: preheating the target plasma, producing a stagnation-shock
front at the liner–target interface, and transporting current
and magnetic field that is flux-compressed by liner inertia. The
target implosion is magneto-inertial and stable, up to the last
of couple of nanoseconds when the interface decelerates and
target-hot spots form, leading to ignition, with a 100-MJ yield,
∼5 × Estored. A simulation is also provided for MagLIF,
configured as a magnetized Beryllium liner → Deuterium
target implosion; these results compare favorably with recent
measurements.
The implosion of a Staged Z-pinch is simulated for parameters that are characteristic of the ZR accelerator, at Sandia National Laboratories. The Staged Z-pinch configuration is comprised of a cylindrical, Silver (Ag) plasma shell, 3-mm outer radius, 0.01-cm thick, imploding onto a uniform-mixture fill (target) of Deuterium-Tritium (DT); the Z-R parameters are: 130 ns, 27 MA, 22 MJ; and the simulation code is MACH2, a 2-1/2 D, radiation-MHD code. During implosion magnetosonic shock waves are generated which propagate radially inward at different speeds in the liner and target plasmas, producing a shock front at the interface between the liner and target plasmas, and a conduction channel ahead of the liner that facilitates target preheating. The Ag/DT interface remains stable throughout the compression, even as the outer surface of the liner becomes RT unstable. At peak compression a string of hot spots forms in the target plasma, that triggers ignition and a fusion-energy yield in excess of 200 MJ; net-energy gain approaching a factor of 10, based on the total stored energy.
Typical MHD models do not include the effects of a finite-electric field, finite gyro-radius, and gyro-period. The magnetohydrodynamic code, MACH2, is modified in one dimension to account for two-fluid behavior, in order to include these effects during the formation of a “driven”, field-reversed configuration (FRC). The simulation is run for a period of 150 μs, during which time an azimuthal ion current accelerates, the FRC forms, compresses radially and axially, and then begins to decay. Once the FRC is formed, an electron current develops, which sharpens the magnetic-field profile outside the null-field region. The equilibrium that is formed is characteristic of a Rigid Rotor.[1] The simulations also agree with prior experiments,[2] specifically the r-z shape of the FRC and the magnitude of the total current, including the ion and electron flows.
A Z-pinch liner, imploding onto a target plasma, evolves in a step-wise manner, producing a stable,
magneto-inertial, high-energy-density plasma compression. The typical configuration is a cylindrical,
high-atomic-number liner imploding onto a low-atomic-number target. The parameters for a
terawatt-class machine (e.g., Zebra at the University of Nevada, Reno, Nevada Terawatt Facility)
have been simulated. The 2-1/2 D MHD code, MACH2, was used to study this configuration. The
requirements are for an initial radius of a few mm for stable implosion; the material densities
properly distributed, so that the target is effectively heated initially by shock heating and finally by
adiabatic compression; and the liner’s thickness adjusted to promote radial current transport and
subsequent current amplification in the target. Since the shock velocity is smaller in the liner, than in
the target, a stable-shock forms at the interface, allowing the central load to accelerate magnetically
and inertially, producing a magneto-inertial implosion and high-energy density plasma. Comparing
the implosion dynamics of a low-Z target with those of a high-Z target demonstrates the role of
shock waves in terms of compression and heating. In the case of a high-Z target, the shock wave does
not play a significant heating role. The shock waves carry current and transport the magnetic field,
producing a high density on-axis, at relatively low temperature. Whereas, in the case of a low-Z
target, the fast moving shock wave preheats the target during the initial implosion phase, and the later
adiabatic compression further heats the target to very high energy density. As a result, the
compression ratio required for heating the low-Z plasma to very high energy densities is greatly
reduced.
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