E.J. Caramana

Semi-implicit magnetohydrodynamic calculations

Abstract A semi-implicit algorithm for the solution of the nonlinear, three-dimensional, resistive MHD equations in cylindrical geometry is presented. The specific model assumes uniform density and pressure, although this is not a restriction of the method. The spatial approximation employs finite differences in the radial coordinate, and the pseudo-spectral algorithm in the periodic poloidal and axial coordinates. A leapfrog algorithm is used to advance wave-like terms; advective terms are treated with a simple predictor-corrector method. The semi-implicit term is introduced as a simple modification to the momentum equation. Dissipation is treated implicitly. The resulting algorithm is unconditionally stable with respect to normal modes. A general discussion of the semi-implicit method is given, and specific forms of the semi-implicit operator are compared in physically relevant test cases. Long-time simulations are presented.

A pseudospectral algorithm for three-dimensional magnetohydrodynamic simulation

Abstract An algorithm for the solution of the three-dimensional resistive magnetohydrodynamic equations in toroidal geometry is presented. The algorithm employs the pseudospectral method for approximation in the two periodic coordinates, and finite differences in the radial direction. Efficient Fast Fourier Transforms are used to communicate between configuration and Fourier space. Leapfrog time advancement is used for advective terms. Diffusion terms are treated implicitly to avoid severe time step restrictions. Sample cases are presented, and a comparison of the method with standard finite difference techniques is presented and discussed.

The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics

This work explores the somewhat subtle meaning and consequences of the salient properties of the discrete, compatible formulation of Lagrangian hydrodynamics. In particular, since this formulation preserves total energy to roundoff error, the amount of error in the conservation of total energy cannot be used to gauge the internal consistency of calculations, as is often done with the older forms of this algorithm. However, the compatible formulation utilizes two definitions of zone volume: the first is the usual definition whereby the volume of a zone is defined as some prescribed function of the coordinates of the points that define it; the second is given as the integration in time of the continuity equation for zone volume as expressed in Lagrangian form. It is the use of this latter volume in the specific internal energy equation that enables total energy to be exactly conserved. These two volume definitions are generally not precisely equal. It is the analysis of this difference that forms the first part of this study. It is shown that this difference in zone volumes can be used to construct a practical internal consistency measure that not only takes the place of the lack of total energy conservation of the older forms of Lagrangian hydrodynamics, but is more general in that it can be defined on a single zone basis. It can also be used to ascertain the underlying spatial and temporal order of accuracy of any given set of calculations. The difference in these two definitions of zone volume may be interpreted as a type of entropy error. However, this entropy error is found to be significant only when a given calculation becomes numerically unstable, otherwise it remains at or far beneath truncation error levels. In fact, it can be utilized to provide an upper bound on the size of the spatial truncation error for a stable computation. It is also shown how this volume difference can be used as an indicator of numerical difficulties, since exact local conservation of total energy does not guarantee numerical stability or the quality of any numerical calculation. The discrete, compatible formulation of Lagrangian hydrodynamics utilizes a two level predictor/corrector-type of time integration scheme; a stability analysis, both analytical and numerical, is given. This analysis reveals a novel stability diagram that has not been heretofore published, and gives definitive information as to how the stabilizing corrector step should be centered in time.

The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics

This study presents the force and mass discretization of exceptional points in the compatible formulation of Lagrangian hydrodynamics. It concludes a series of papers that develop various aspects of the theoretical exposition and the operational implementation of this numerical algorithm. Exceptional points are grid points at the termination of lines internal to the computational domain, and where boundary conditions are therefore not applied. These points occur naturally in most applications in order to ameliorate spatial grid anisotropy, and the consequent timestep reduction, that will otherwise arise for grids with highly tapered regions or a center of convergence. They have their velocity enslaved to that of neighboring points in order to prevent large excursions of the numerical solution about them. How this problem is treated is given herein for the aforementioned numerical algorithm such that its salient conservation properties are retained. In doing so the subtle aspects of this algorithm that are due to the interleaving of spatial contours that occur with the use of a spatially-staggered-grid mesh are illuminated. These contours are utilized to define both forces and the work done by them, and are the central construct of this type of finite-volume differencing. Additionally, difficulties that occur due to uncertainties in the specification of the artificial viscosity are explored, and point to the need for further research in this area.

Curl- q : a vorticity damping artificial viscosity for essentially irrotational Lagrangian hydrodynamics calculations

The bane of Lagrangian hydrodynamics calculations in multi-dimensions is the appearance of vorticity that causes tangling of the mesh and consequent run termination. This vorticity may be numerical or physical in origin, and is in addition to the spurious ‘‘hourglass’’ modes associated with quadrilateral or hexahedral zones that in pure form have both zero curl and divergence associated with their velocity field. The purpose of this note is to introduce a form of vorticity damping, based on a previously published edgecentered artificial viscosity [1], that extends the runtime and range of calculations over which a pure Lagrangian code can compute. Since the explicit inclusion of an artificial viscosity into the fluid equations is often referred to as the ‘‘q term’’, we denote this new term as the ‘‘curl-q’’, because it is a function of the curl of the velocity field in a zone. It is formulated in the context of the ‘‘discrete, compatible formulation of Lagrangian hydrodynamics’’ [2,3]. This employs a staggered placement of variables in space (velocity and position at nodes, with density and stresses in zones), but a predictor/corrector time integration scheme so that all variables are known at the same time level, allowing total energy to be exactly conserved [2]. This new ‘‘curl-q’’ does not resolve shock waves and is always to be utilized with an artificial viscosity that performs this task. In order to set the stage for the introduction of the new curl-q force, the edge-centered artificial viscosity given in [1] is briefly reviewed in a slightly simplified form; after this the curl-q force is formulated as an analogy to this edge-centered artificial viscosity. Numerical results are given in both 2D and 3D that display its effectiveness. In particular, results are contrasted between this new term and a recently published tensor artificial viscosity [4]. It is shown that these two forms give quite similar results in 2D. We end with a brief discussion concerning the validity of the use of this type of numerical device. In Fig. 1 is shown a quadrilateral zone with its defining points, i = 1–4, and associated median mesh vectors ~Si. These vectors point in the indicated direction and have a magnitude of the surface area that lies between their defining points in 2D or in 3D [2,5]. In terms of the median mesh vector~S1, the force exerted by the edgecentered artificial viscosity between points ‘‘1’’ and ‘‘2’’ from zone ‘‘z’’ is given by

Theoretical and experimental studies of CO2 laser evaporation of clouds

A study of the effects of laser radiation on cloud drops and of the possibility of producing a clear optical channel in a cloud is presented. In order to produce a model that is appropriate to a realistic cloud with a distribution of drop sizes it is first necessary to study what happens to a single water drop subjected to laser radiation of different intensities. Various heating regimes are mapped out as a function of laser flux and fluence at the 10.6 μm wavelength. It is found that typical cloud drops can superheat until they become unstable and explode from the center. For a long laser pulse the boundary for this to occur is found to be 50(5/r)2 kW/cm2, where r is the drop radius in microns. Using these results a model that is spatially one‐dimensional through the cloud is constructed for a distribution of drop sizes. Laser beam intensity as the light penetrates a cloud is calculated from Mie scattering and absorption cross sections for a beam diameter that is small in the sense that light scattered o...

Cloud hole boring with long pulse CO 2 lasers: theory and experiment

Chemically generated CO(2) laser pulses at 10.6 microm have been used to clear a 5-cm diameter hole through a stratus-like cloud in a laboratory cloud chamber. The results show that 100% clearing can be achieved. The mechanism is shown to be droplet shattering followed by evaporation. In the experimental conditions, the channel closure is effected by turbulent mixing and droplet recondensation.