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NASA NTRS1985

Large-scale computations in fluid mechanics; Proceedings of the Fifteenth Summer Seminar on Applied Mathematics, University of California, La Jolla, CA, June 27-July 8, 1983. Parts 1 & 2

Papers are presented on such topics as the use of semi-Lagrangian advective schemes in meteorological modeling; computation with high-resolution upwind schemes for hyperbolic equations; dynamics of flame propagation in a turbulent field; a modified finite element method for solving the incompressible Navier-Stokes equations; computational fusion magnetohydrodynamics; and a nonoscillatory shock capturing scheme using flux-limited dissipation. Consideration is also given to the use of spectral techniques in numerical weather prediction; numerical methods for the incorporation of mountains in atmospheric models; techniques for the numerical simulation of large-scale eddies in geophysical fluid dynamics; high-resolution TVD schemes using flux limiters; upwind-difference methods for aerodynamic problems governed by the Euler equations; and an MHD model of the earth's magnetosphere.

Engquist, B. E.↗

NASA NTRS1978

Magnetohydrodynamic models of coronal transients in the meridional plane. I - The effect of the magnetic field

The propagation of coronal transients through the lower corona in the meridional plane is studied on the basis of numerical solutions of the time-dependent MHD equations of motion. The importance of the topology of the initial coronal magnetic field is demonstrated by considering two configurations: one which is essentially radial (open), and another which is essentially parallel to the solar surface (closed). The effect of coronal field magnitude is investigated by using values for the beta parameter of 1 and 0.1 at the coronal base. The consequences of including radiative losses and of varying the polytropic index are also considered. The solar event is simulated by a step-function increase in the pressure at the base of an initially hydrostatic atmosphere by a factor of 5, lasting for a period of 5 min.

Steinolfson, R. S.↗

NASA NTRS1994

Structure and Dynamics of the Solar Corona

Advanced computational techniques were used to study solar coronal heating and coronal mass ejections. A three dimensional, time dependent resistive magnetohydrodynamic code was used to study the dynamic response of a model corona to continuous, slow, random magnetic footpoint displacements in the photosphere. Three dimensional numerical simulations of the response of the corona to simple smooth braiding flows in the photosphere were calculated to illustrate and understand the spontaneous formation of current filaments. Two dimensional steady state helmet streamer configurations were obtained by determining the time asymptotic state of the interaction of an initially one dimensinal transponic solar wind with a spherical potential dipole field. The disruption of the steady state helmet streamer configuration was studied as a response to shearing of the magnetic footpoints of the closed field lines under the helmet.

Schnack, D. D.↗

NASA NTRS2012

Statistical Theory of the Ideal MHD Geodynamo

A statistical theory of geodynamo action is developed, using a mathematical model of the geodynamo as a rotating outer core containing an ideal (i.e., no dissipation), incompressible, turbulent, convecting magnetofluid. On the concentric inner and outer spherical bounding surfaces the normal components of the velocity, magnetic field, vorticity and electric current are zero, as is the temperature fluctuation. This allows the use of a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity, current and the temperature fluctuation. The resulting dynamical system, based on the Boussinesq form of the magnetohydrodynamic (MHD) equations, represents MHD turbulence in a spherical domain. These basic equations (minus the temperature equation) and boundary conditions have been used previously in numerical simulations of forced, decaying MHD turbulence inside a sphere [1,2]. Here, the ideal case is studied through statistical analysis and leads to a prediction that an ideal coherent structure will be found in the form of a large-scale quasistationary magnetic field that results from broken ergodicity, an effect that has been previously studied both analytically and numerically for homogeneous MHD turbulence [3,4]. The axial dipole component becomes prominent when there is a relatively large magnetic helicity (proportional to the global correlation of magnetic vector potential and magnetic field) and a stationary, nonzero cross helicity (proportional to the global correlation of velocity and magnetic field). The expected angle of the dipole moment vector with respect to the rotation axis is found to decrease to a minimum as the average cross helicity increases for a fixed value of magnetic helicity and then to increase again when average cross helicity approaches its maximum possible value. Only a relatively small value of cross helicity is needed to produce a dipole moment vector that is aligned at approx.10deg with the rotation axis.

Shebalin, J. V.↗

NASA NTRS1983

Magnetohydrodynamic modeling of coronal structure and expansion

The presence of a magnetic field in the corona adds structure to the solar wind and almost certainly plays an important role in the energetics of the flow. Analytical and numerical modeling of gas-magnetic field interactions as used to compute steady, global flow are discussed. The approach used in, and results from a recent global model (Steinolfson, Suess and Wu, 1982) are discussed. Ideas on the most effective ways to improve the physical content and numerical efficiency of these models are outlined. Solutions of the MHD equations are discussed only in order to find steady-state flows, even though this often entails solving time-dependent equations.

Suess, S. T.↗

NASA NTRS1985

Stellar fibril magnetic systems. II - Two-dimensional magnetohydrodynamic equations. III - Convective counterflow

The dynamics of magnetic fibrils in the convective zone of a star is investigated analytically, deriving mean-field equations for the two-dimensional transverse motion of an incompressible fluid containing numerous small widely spaced circular cylinders. The equations of Parker (1982) are extended to account for the inertial effects of local flow around the cylinders. The linear field equation for the stream function at the onset of convection is then rewritten, neglecting large-scale heat transport, and used to construct a model of convective counterflow. The Kelvin impulse and fluid momentum, convective motion initiated by a horizontal impulse, and the effects of a viscous boundary layer are considered in appendices.

Parker, E. N.↗

NASA NTRS1984

Ion viscous effects on magnetosonic shock

Profiles of magnetic field and plasma bulk velocity associated with perpendicular shocks are investigated theoretically by solving magnetohydrodynamic equations. The main dissipation mechanism is assumed to be due to ion viscosity. Shock structures with zero temperatures and finite pressure are examined separately. The numerical solutions indicate that ion viscosity tends to smooth profiles of magnetic field and bulk velocity for high Mach number shocks. For smaller ion viscosity and lower Mach number, a damped wave train would appear downstream. The results suggest that kinetic effects such as ion reflections and microturbulence are necessary in order to explain magnetic field overshoots as well as oscillatory magnetic-field structure in high Mach number shocks.

Lin, C. S.↗

NASA NTRS1991

Inverse energy cascades in three-dimensional turbulence

Fully three-dimensional magnetohydrodynamic (MHD) turbulence at large kinetic and low magnetic Reynolds numbers is considered in the presence of a strong uniform magnetic field. It is shown by numerical simulation of a model of MHD that the energy inverse cascades to longer length scales when the interaction parameter is large. While the steady-state dynamics of the driven problem is three-dimensional in character, the behavior has resemblance to two-dimensional hydrodynamics. These results have implications in turbulence theory, MHD power generator, planetary dynamos, and fusion reactor blanket design.

Hossain, Murshed↗

NASA NTRS1983

Proposed Short-Throat Supersonic Nozzles

Numerical procedure analyzes mach numbers along wall and in flow field, wall pressures, gas temperatures, and nozzle-throat discharge coefficients. Nozzles used in turbines, jet engines, magnetohydrodynamic systems, laser systems and other supersonic-flow devices.

Wagner, W. R.↗

NASA NTRS2016

The Efficiency of Magnetic Field Amplification at Shocks by Turbulence

Turbulent dynamo field amplification has often been invoked to explain the strong field strengths in thin rims in supernova shocks (approx.100 micrograms) and in radio relics in galaxy clusters (approx. micrograms). We present high-resolution magnetohydrodynamic simulations of the interaction between pre-shock turbulence, clumping and shocks, to quantify the conditions under which turbulent dynamo amplification can be significant. We demonstrate numerically converged field amplification which scales with Alfven Mach number, B/B0 varies as MA, up to MA approx.150.This implies that the post-shock field strength is relatively independent of the seed field. Amplification is dominated by compression at low MA, and stretching (turbulent amplification) at high MA. For high MA, the B-field grows exponentially and saturates at equipartition with turbulence, while the vorticity jumps sharply at the shock and subsequently decays; the resulting field is orientated predominately along the shock normal (an effect only apparent in 3D and not 2D). This agrees with the radial field bias seen in supernova remnants. By contrast, for low MA, field amplification is mostly compressional, relatively modest, and results in a predominantly perpendicular field. The latter is consistent with the polarization seen in radio relics. Our results are relatively robust to the assumed level of gas clumping. Our results imply that the turbulent dynamo may be important for supernovae, but is only consistent with the field strength, and not geometry, for cluster radio relics. For the latter, this implies strong pre-existing B-fields in the ambient cluster outskirts.

lementary particles â€“ magnetic fields â€“ radiat↗

NASA NTRS2004

Theory and Simulation of Real and Ideal Magnetohydrodynamic Turbulence

Incompressible, homogeneous magnetohydrodynamic (MHD) turbulence consists of fluctuating vorticity and magnetic fields, which are represented in terms of their Fourier coefficients. Here, a set of five Fourier spectral transform method numerical simulations of two-dimensional (2-D) MHD turbulence on a 512(sup 2) grid is described. Each simulation is a numerically realized dynamical system consisting of Fourier modes associated with wave vectors k, with integer components, such that k = |k| less than or equal to k(sub max). The simulation set consists of one ideal (non-dissipative) case and four real (dissipative) cases. All five runs had equivalent initial conditions. The dimensions of the dynamical systems associated with these cases are the numbers of independent real and imaginary parts of the Fourier modes. The ideal simulation has a dimension of 366104, while each real simulation has a dimension of 411712. The real runs vary in magnetic Prandtl number P(sub M), with P(sub M) is a member of {0.1, 0.25, 1, 4}. In the results presented here, all runs have been taken to a simulation time of t = 25. Although ideal and real Fourier spectra are quite different at high k, they are similar at low values of k. Their low k behavior indicates the existence of broken symmetry and coherent structure in real MHD turbulence, similar to what exists in ideal MHD turbulence. The value of PM strongly affects the ratio of kinetic to magnetic energy and energy dissipation (which is mostly ohmic). The relevance of these results to 3-D Navier-Stokes and MHD turbulence is discussed.

Shebalin, John V.↗

NASA NTRS1974

The dynamical responses of the thermosphere due to a geomagnetic storm

A theoretical model for the dynamic responses to geomagnetic storms in the thermosphere is derived from magnetohydrodynamic theory. The validity claim for this model is based on the assumption that the thermosphere behaves like an electrically conductive fluid. In order to test the proposed model, a numerical example is presented. Recommendations for improving the model are also offered.

Wu, S. T.↗

NASA NTRS1986

Filament formation due to photospheric shear

Recently, Wu et al. (1983, 1986) developed a three-dimensional, time-dependent axisymmetric ideal magnetohydrodynamic (MHD) model which has been used to study lower solar atmospheric response due to photospheric motion. They have shown that various observed solar features can be formed from these numerical simulation experiments. Some work currently in progress which has resulted from a specific numerical simulation experiment is reported. These results may elucidate the possible physical mechanism of formation of filament.

Wu, S. T.↗

NASA NTRS2023

Modeling the Formation and Eruption of Coronal Structures by Linking Data-Driven Magnetofrictional and MHD Simulations for AR 12673

Context .The data-driven and time-dependent modeling of coronal magnetic fields is crucial for understanding solar eruptions. These efforts are complicated by the challenges of finding a balance between physical realism and computing efficiency. One possible technique is to couple two modeling approaches. Aims . Our aim here is to showcase our progress in using time-dependent magnetofrictional model (TMFM) results as input to dynamical magnetohydrodynamic (MHD) simulations. However, due to the different evolution processes in these two models, using TMFM snapshots in an MHD simulation is nontrivial. We address these issues, both physically and numerically, discuss the incompatibility of the TMFM output to serve as the initial condition in MHD simulations, and show our methods of mitigating this. The evolution of the flux systems and the cause of the eruption are investigated. Methods . TMFM is a prevalent approach that has proven to be a very useful tool in the study of the formation of unstable structures in the solar corona. In particular, it is capable of incorporating observational data as initial and boundary conditions and requires shorter computational time compared to MHD simulations. To leverage the efficiency of data-driven TMFM and also to simulate eruptive events in the MHD framework, one can apply TMFM up to a certain time before the expected eruption(s) and then proceed with the simulation in the full or ideal MHD regime in order to more accurately capture the eruption process. Results . We show the results of a benchmark test case with a linked TMFM and MHD simulation to study the evolution of NOAA active region 12673. A rise of a twisted flux bundle through the MHD simulation domain is observed, but we find that the rate of the rise and the altitude reached depends on the time of the TMFM snapshot that was used to initialize the MHD simulation and the helicity injected into the system. The analysis suggested that torus instability and slip-running reconnection could play an important role in the eruption. Conclusions . The results show that the linkage of TMFM and zero-β MHD models can be successfully used to model the eruptive coronal magnetic fields.

F. Daei↗

NASA NTRS1981

Nonlinear magnetohydrodynamic stability

The computer code developed by Bauer et al. (1978) for the study of the magnetohydrodynamic equilibrium and stability of a plasma in toroidal geometry is extended so that the growth rates of instabilities may be estimated more accurately. The original code, which is based on the variational principle of ideal magnetohydrodynamics, is upgraded by the introduction of a nonlinear formula for the growth rate of an unstable mode which acts as a quantitative measure of instability that is important in estimating numerical errors. The revised code has been applied to the determination of the nonlinear saturation, ballooning modes and beta limits for tokamaks, stellarators and torsatrons.

Bauer, F.↗

NASA NTRS1983

Dynamic alignment and selective decay in MHD

Under some circumstances, incompressible magnetohydrodynamic turbulence will evolve toward a state in which the velocity fields and magnetic fields are aligned or anti-aligned. We propose a mechanism for this effect and illustrate with numerical computations. Under some other circumstances, the energy appears to decay selectively toward a minimum energy state in which the kinetic energy has disappeared. It has not been possible so far to identify a boundary in the phase space which divides the two regimes.

Matthaeus, W. H.↗

NASA NTRS1984

Dynamic alignment and selective decay in MHD

Matthaeus, W. H.↗

NASA NTRS2001

Fluid Aspects of Solar Wind Disturbances Driven by Coronal Mass Ejections

Transient disturbances in the solar wind initiated by coronal eruptions have been modeled for many years, beginning with the self-similar analytical models of Parker and Simon and Axford. The first numerical computer code (one-dimensional, gas dynamic) to study disturbance propagation in the solar wind was developed in the late 1960s, and a variety of other codes ranging from simple one-dimensional gas dynamic codes through three-dimensional gas dynamic and magnetohydrodynamic codes have been developed in subsequent years. For the most part, these codes have been applied to the problem of disturbances driven by fast CMEs propagating into a structureless solar wind. Pizzo provided an excellent summary of the level of understanding achieved from such simulation studies through about 1984, and other reviews have subsequently become available. More recently, some attention has been focused on disturbances generated by slow CMEs, on disturbances driven by CMEs having high internal pressures, and disturbance propagation effects associated with a structured ambient solar wind. Our purpose here is to provide a brief tutorial on fluid aspects of solar wind disturbances derived from numerical gas dynamic simulations. For the most part we illustrate disturbance evolution by propagating idealized perturbations, mimicking different types of CMEs, into a structureless solar wind using a simple one-dimensional, adiabatic (except at shocks), gas dynamic code. The simulations begin outside the critical point where the solar wind becomes supersonic and thus do not address questions of how the CMEs themselves are initiated. Limited to one dimension (the radial direction), the simulation code predicts too strong an interaction between newly ejected solar material and the ambient wind because it neglects azimuthal and meridional motions of the plasma that help relieve pressure stresses. Moreover, the code ignores magnetic forces and thus also underestimates the speed with which pressure disturbances propagate in the wind.

Gosling, J. T.↗