Two-dimensional Cartesian Geometry Research Articles

Numerical simulations of a two-fluid jet at a magnetic null point in a solar arcade

ABSTRACT We study the formation and evolution of jets in the solar atmosphere using numerical simulations of partially ionized plasma. The two-fluid magnetohydrodynamic equations with ion+electron and neutral hydrogen components are used in two-dimensional Cartesian geometry. Numerical simulations show that a localized non-linear Gaussian pulse of ion and neutral pressures initially launched from the magnetic null point of a potential arcade located below the transition region quickly develops into a shock due to the decrease of density with height. The shock propagates upwards into the solar corona and lifts the cold and dense chromospheric plasma behind in the form of a collimated jet with an inverted-Y shape. The inverted-Y shape of jets is connected with the topology of a magnetic null point. The pulse also excites a non-linear wake in the chromosphere, which leads to quasi-periodic secondary shocks. The secondary shocks lift the chromospheric plasma upwards and create quasi-periodic jets in the lower corona. Ion and neutral fluids show generally similar behaviour, but their relative velocity is higher near the upper part of the jets, which leads to enhanced temperature or heating due to ion–neutral collisions. Simulations of jets with an inverted-Y shape and their heating may explain the properties of some jets observed in the solar atmosphere.

Time-explicit Darwin PIC algorithm

A new approach to Darwin particle-in-cell plasma simulation is described. Using a finite-element approach, the vector potential is assured to be exactly solenoidal. This allows writing the system of multi-species particles as an action-at-a-distance Hamiltonian system. Applying recently-developed explicit symplectic methods to this gives a time-explicit algorithm with all desired properties. Elliptic systems for the electrostatic and magneto-static portions of the problem are inverted efficiently by an algebraic multi-grid algorithm. The algorithm is implemented in a two-dimensional Cartesian-geometry code and tested by application to Weibel instability and to Alfvén wave dynamics. Results show the effectiveness of this approach. Extensions to arbitrary meshes and to a partially time-implicit scheme are briefly discussed.

Metodologia Espectronodal Determinística para Cálculos de Blindagem de Nêutrons usando a Equação de Equação de Transporte Multigrupo Considerando Problemas Bidimensionais na Formulação das Ordenadas Discretas (SN)

In this work, we present the most recent numerical results in a nodal approach, which resulted in the development of a new numerical spectral nodal method, based on the spectral analysis of the multigroup, isotropic scattering neutron transport equations in the discrete ordinates ($S_N$) formulation for fixed-source calculations in non-multiplying media (shielding problems). The numerical results refer to simulations of typical problems from the reactor physics field, in rectangular two-dimensional Cartesian geometry, $X, Y$ geometry, and are compared with the traditional Diamond Difference ($DD$) fine-mesh method results, used as a reference, and the spectral coarse-mesh method Green's function ($SGF$) results.

On the Regularity Order of the Pointwise Uncollided Angular Flux and Asymptotic Convergence of the Discrete Ordinates Approximation of the Scalar Flux

To determine the angular discretization error asymptotic convergence rate of the uncollided scalar flux computed with the discrete ordinates (S) method, a comprehensive theory of the regularity order with respect to the azimuthal angle of the exact pointwise SN uncollided angular flux is derived based on the integral form of the transport equation in two-dimensional Cartesian geometry. With this theory, the regularity order of the pointwise uncollided angular flux can be estimated for a given problem configuration. Our new theory inspired a novel Modified Simpson’s (MS) quadrature that converges the uncollided scalar flux faster than any of the traditional quadratures by avoiding integration across points of irregularity in the azimuthal angle. Numerical results successfully verify our new theory in four variants of a test configuration, and the angular discretization errors in the corresponding scalar flux computed with conventional angular quadrature types and with our new quadrature types are found to converge with different orders. The error convergence rates obtained with traditional quadrature types are limited by the regularity order of the exact angular flux and the quadrature’s integration intervals while our new MS quadrature types converge with order two to four times higher than traditional quadratures. A detailed study of oscillations observed in certain quadrature errors is provided by introducing the effective length of the irregular interval and the associated oscillating function.

Método Espectro Nodal de Grades Compostas com Aproximação Exponencial na Formulação de Ordenadas Discretas em Meios Homogêneos Não-multiplicativos

Neutron transport problems in non-multiplying media appear in several applications in science and engineering, with emphasis on the design of shielding structures.  In this work, we present the composite spatial grids spectral nodal method with an exponential approximation for neutron transport problems in homogeneous non-multiplying media.  We use as a mathematical model the discrete ordinates formulation in two-dimensional Cartesian geometry with one-speed approximation and isotropic scattering source. The discretization process is done in two steps like is as composite spatial grid methods. An exponential function is used to approximate the transverse leakage terms in the region. The multi-step discretization mechanism transforms the two-dimensional problem into two coupled one-dimensional problems, that are solved using the alternative directions iterative scheme. Numerical results are offered for a model problem, showing that the proposed method generates accurate results with a low computational cost.

An analytical discrete ordinates nodal solution to the two-dimensional adjoint transport problem

A solution to the one-group discrete ordinates approximation of the adjoint to the transport equation in two-dimensional Cartesian geometry is developed in this work. Starting from an integration process over a region (node) of the domain, the model is reduced to one-dimensional transverse-integrated equations in the x and y variables which are then solved by the Analytical Discrete Ordinates Method (ADO Method). Particular attention is given to the effect of non-homogeneous boundary conditions in the formulation. Explicit expressions are derived and used for evaluating, in a fast way, absorption rates of internal detectors from the adjoint formulation. Excellent agreement was found between numerical results obtained for the absorption rates evaluated from the forward and adjoint problems.

Iteration Methods with Multigrid in Energy for Eigenvalue Neutron Diffusion Problems

In this paper we present nonlinear multilevel methods with multiple grids in energy for solving the k-eigenvalue problem for multigroup neutron diffusion equations. We develop multigrid-in-energy algorithms based on a nonlinear projection operator and several advanced prolongation operators. The evaluation of the eigenvalue is performed in the space with smallest dimensionality by solving the effective one-group diffusion problem. We consider two-dimensional Cartesian geometry. The multilevel methods are formulated in discrete form for the second-order finite volume discretization of the diffusion equation. The homogenization in energy is based on a spatially consistent discretization of the group diffusion equations on coarse grids in energy. We present numerical results of model reactor-physics problems with 44 energy groups. They demonstrate performance and main properties of the proposed iterative methods with multigrid in energy.

An analytical model of an urban heat island circulation in calm conditions

We obtain an analytical model solution to an idealization of an urban heat island (UHI) circulation—the steady shallow convective flow of a viscous stably stratified fluid over a differentially heated lower boundary without system rotation (no Coriolis force) or background wind. The coupled linearized equations of motion and thermal energy are solved for flows in two-dimensional Cartesian (slab-symmetric) and axisymmetric geometries forced by laterally varying surface buoyancies considered as Gaussian and parabolic arch functions. The scaled problem is free of governing parameters, and the solutions are universal. In all cases the flow is dominated by an in-up-out circulation. The updraft is stronger in the axisymmetric case than in the slab-symmetric case, while the surrounding downdraft, and the inflow and outflow branches of the circulation are stronger in the slab-symmetric case. These differences are explained in terms of the response of the perturbation pressure to the thermal forcing in the different geometries. The scalings themselves provide insight into observations that daytime UHI circulations can be stronger than nighttime circulations despite the relative weakness of urban–rural temperature contrasts during the daytime.

A composite spatial grid spectral Green's function method for one speed discrete ordinates eigenvalue problems in two-dimensional Cartesian geometry

Spectral Green's function nodal methods (SGF) are well established as a class of coarse mesh methods. For this reason, they are widely used in the solution of neutron transport problems in discrete ordinates formulation (SN). When compared with fine mesh methods, SGF are considered efficient, as solutions are as accurate as, using a smaller number of spatial nodes, reducing floating point operations. However, the development of spectral-nodal methods for XY Cartesian geometries, has been limited due to (a) difficulties in implementing efficient computational algorithms and, (b) high algebraic and computational costs. This is because these methods need to use NBI-type (One-Node Block Inversion) sweep schemes. The composite spatial grid methods were developed to overcome these challenges. In this work, we describe a composite spatial grid spectral-nodal method to solve one-speed discrete ordinate eigenvalue problems in XY Cartesian geometry with isotropic scattering. The discretization is developed into two stages and two 1D problems coupled by transverse leakage terms in each domain region are obtained. In order to converge toward the numerical solution, we used an alternating-direction iterative technique and a modified source iteration sweep scheme. Also, we used the conventional power method to estimate the problem's dominant eigenvalue. Numerical results for benchmark problems are presented to illustrate the accuracy and performance of the developed method. This approach offers more accurate and efficient results for integral quantities if compared with others SGF methods.

Lie symmetry Analysis and Invariant Solutions for Multiregion Neutron Diffusion Equation

In this paper, an approach of determining analytical solutions of the mono-kinetic multiregion neutron diffusion equation from two-dimensional Cartesian geometry is presented. The technical approach is based on the Lie symmetry group for partial differential equation. The local symmetry groups to the one-parameter transformation are obtained. The invariant solutions spanned of an expansion of neutron fluxes with respect to the space, time and material regions are reported.

Non-linear growth of double tearing mode: Explosive reconnection, plasmoid formation, and particle acceleration

The nonlinear evolution of double tearing modes (DTMs) is investigated within the framework of resistive magnetohydrodynamic (MHD) simulations in a two-dimensional Cartesian geometry. We have explored the explosive reconnection phase associated with the growth of the secondary structure-driven instability for a range of resistivity values. The time scale of the explosive phase (that is of order of a few Alfvénic time scales) is shown to be quasi-independent of the resistivity, even when fast growing plasmoids develop for the highest enough Lundquist number cases. Test particle accelerations are performed using the MHD explosive simulations as input parameters. Our results show that reconnection DTM dynamics is able to provide an efficient process for accelerating charged particles far beyond characteristic thermal velocities within the reconnection layers. The main acceleration mechanism is attributed to the strong inductive electric field generated by the island structure-driven instability, with an additional smaller contribution due to the presence of plasmoids. Finally, our results are used to discuss some features of the accelerated particle spectra during flaring activity in the solar corona.

Two-phase convection in Ganymede’s high-pressure ice layer — Implications for its geological evolution

Ganymede, the largest moon in the solar system, has a fully differentiated interior with a layer of high-pressure (HP) ice between its deep ocean and silicate mantle. In this paper, we study the dynamics of this layer using a numerical model of two-phase ice–water mixture in two-dimensional Cartesian geometry. While focusing on the generation of water at the silicate/HP ice interface and its upward migration towards the ocean, we investigate the effect of bottom heat flux, the layer thickness, and the HP ice viscosity and permeability. Our results suggest that melt can be generated at the silicate/HP ice interface for small layer thickness ( ≲ 200 km) and high values of heat flux ( ≳ 20 mW m−2) and viscosity ( ≳ 1015 Pa s). Once generated, the water is transported through the layer by the upwelling plumes. Depending on the vigor of convection, it stays liquid or it may freeze before melting again as the plume reaches the temperate (partially molten) layer at the boundary with the ocean. The thickness of this layer as well as the amount of melt that is extracted from it is controlled by the permeability of the HP ice. This process constitutes a means of transporting volatiles and salts that might have dissolved into the melt present at the silicate/HP ice interface. As the moon cools down, the HP ice layer becomes less permeable because the heat flux from the silicates decreases and the HP ice layer thickens.

Explosive Magnetic Reconnection in Double-current Sheet Systems: Ideal versus Resistive Tearing Mode

Abstract Magnetic reconnection associated with the tearing instability occurring in double-current sheet systems is investigated within the framework of resistive magnetohydrodynamics (MHD) in a two-dimensional Cartesian geometry. A special emphasis on the existence of fast and explosive phases is taken. First, we extend the recent theory on the ideal tearing mode of a single-current sheet to a double-current layer configuration. A linear stability analysis shows that, in long and thin systems with (length to shear layer thickness) aspect ratios scaling as (S L being the Lundquist number based on the length scale L), tearing modes can develop on a fast Alfvénic timescale in the asymptotic limit . The linear results are confirmed by means of compressible resistive MHD simulations at relatively high S L values (up to ) for different current sheet separations. Moreover, the nonlinear evolution of the ideal double tearing mode (IDTM) exhibits a richer dynamical behavior than its single-tearing counterpart, as a nonlinear explosive growth violently ends up with a disruption when the two current layers interact trough the merging of plasmoids. The final outcome of the system is a relaxation toward a new state, free of magnetic field reversal. The IDTM dynamics is also compared to the resistive double tearing mode dynamics, which develops in similar systems with smaller aspect ratios, , and exhibits an explosive secondary reconnection, following an initial slow resistive growth phase. Finally, our results are used to discuss the flaring activity in astrophysical magnetically dominated plasmas, with a particular emphasis on pulsar systems.

Nonlinear Diffusion Acceleration Method with Multigrid in Energy for k-Eigenvalue Neutron Transport Problems

We present a multilevel method for solving multigroup neutron transport k-eigenvalue problems in two-dimensional Cartesian geometry. It is based on the nonlinear diffusion acceleration (NDA) method. The multigroup low-order NDA (LONDA) equations are formulated on a sequence of energy grids. Various multigrid cycles are applied to solve the hierarchy of multigrid LONDA equations. The algorithms developed accelerate transport iterations and are effective in solving the multigroup NDA low-order equations. We present numerical results for model reactor-physics problems with a large number of groups to demonstrate the performance of different iterative schemes.

Water generation and transport below Europa's strike-slip faults

Jupiter's moon Europa has a very young surface with the abundance of unique terrains that indicate recent endogenic activity. Morphological models as well as spectral observations suggest that it might possess shallow lenses of liquid water within its outer ice shell. Here we investigate the generation and possible accumulation of liquid water below the tidally activated strike-slip faults using a numerical model of two-phase ice-water mixture in two-dimensional Cartesian geometry. Our results suggest that generation of shallow partially molten regions underneath Europa's active strike-slip faults is possible, but their lifetime is constrained by the formation of Rayleigh-Taylor instabilities due to the negative buoyancy of the melt. Once formed, typically within a few million years, these instabilities efficiently transport the meltwater through the shell. Consequently, the maximum water content in the partially molten regions never exceeds 10% which challenges their possible detection by future exploration mission.

The mantle convection model with non-Newtonian rheology and phase transitions: The flow structure and stress fields

The mantle convection model with phase transitions, non-Newtonian viscosity, and internal heat sources is calculated for two-dimensional (2D) Cartesian geometry. The temperature dependence of viscosity is described by the Arrhenius law with a viscosity step of 50 at the boundary between the upper and lower mantle. The viscosity in the model ranges within 4.5 orders of magnitude. The use of the non-Newtonian rheology enabled us to model the processes of softening in the zone of bending and subduction of the oceanic plates. The yield stress in the model is assumed to be 50 MPa. Based on the obtained model, the structure of the mantle flows and the spatial fields of the stresses σ xz and σ xx in the Earth’s mantle are studied. The model demonstrates a stepwise migration of the subduction zones and reveals the sharp changes in the stress fields depending on the stage of the slab detachment. In contrast to the previous model (Bobrov and Baranov, 2014), the self-consistent appearance of the rigid moving lithospheric plates on the surface is observed. Here, the intense flows in the upper mantle cause the drift and bending of the top segments of the slabs and the displacement of the plumes. It is established that when the upwelling plume intersects the boundary between the lower and upper mantle, it assumes a characteristic two-level structure: in the upper mantle, the ascending jet of the mantle material gets thinner, whereas its velocity increases. This effect is caused by the jump in the viscosity at the boundary and is enhanced by the effect of the endothermic phase boundary which impedes the penetration of the plume material from the lower mantle to the upper mantle. The values and distribution of the shear stresses σ xz and superlithostatic horizontal stresses σ xx are calculated. In the model area of the subducting slabs the stresses are 60–80 MPa, which is by about an order of magnitude higher than in the other mantle regions. The character of the stress fields in the transition region of the phase boundaries and viscosity step by the plumes and slabs is analyzed. It is established that the viscosity step and endothermic phase boundary at a depth of 660 km induce heterogeneities in the stress fields at the upper/lower mantle boundary. With the assumed model parameters, the exothermic phase transition at 410 km barely affects the stress fields. The slab regions manifest themselves in the stress fields much stronger than the plume regions. This numerically demonstrates that it is the slabs, not the plumes that are the main drivers of the convection. The plumes partly drive the convection and are partly passively involved into the convection stirred by the sinking slabs.

Shear instabilities in a fully compressible polytropic atmosphere

Shear flows have an important impact on the dynamics in an assortment of different astrophysical objects including accreditation discs and stellar interiors. Investigating shear flow instabilities in a polytropic atmosphere provides a fundamental understanding of the motion in stellar interiors where turbulent motions, mixing processes, as well as magnetic field generation takes place. Here, a linear stability analysis for a fully compressible fluid in a two-dimensional Cartesian geometry is carried out. Our study focuses on determining the critical Richardson number for different Mach numbers and the destabilising effects of high thermal diffusion. We find that there is a deviation of the predicted stability threshold for moderate Mach number flows along with a significant effect on the growth rate of the linear instability for small P\'eclet numbers. We show that in addition to a Kelvin-Helmholtz instability a Holmboe instability can appear and we discuss the implication of this in stellar interiors.

A Compressible Nonhydrostatic Cell-Integrated Semi-Lagrangian Semi-Implicit Solver (CSLAM-NH) with Consistent and Conservative Transport

Abstract A cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic solver for the fully compressible moist Euler equations in two-dimensional Cartesian (x–z) geometry is presented. The semi-implicit CISL solver uses the inherently conservative semi-Lagrangian multitracer transport scheme (CSLAM) and a new flux-form semi-implicit formulation of the continuity equation that ensures numerically consistent transport. The flux-form semi-implicit formulation is based on a recent successful approach in a shallow-water equations (SWE) solver (CSLAM-SW). With the new approach, the CISL semi-implicit nonhydrostatic solver (CSLAM-NH) is able to ensure conservative and consistent transport by avoiding the need for a time-independent mean reference state. Like its SWE counterpart, the nonhydrostatic solver presented here is designed to be similar to typical semi-Lagrangian semi-implicit schemes, such that only a single linear Helmholtz equation solution and a single call to CSLAM are required per time step. To demonstrate its stability and accuracy, the solver is applied to a set of three idealized test cases: a density current (dry), a gravity wave (dry), and a squall line (moist). A fourth test case shows that shape preservation of passive tracers is ensured by coupling the semi-implicit CISL formulation with existing shape-preserving filters. Results show that CSLAM-NH solutions compare well with other existing solvers for the three test cases, and that it is shape preserving.

Nodal synthetic kernel (N-SKN) method for solving neutron transport equation in one- and two-dimensional X–Y geometries

In this study, a nodal method based on the synthetic kernel (SKN) approximation is presented for solving the neutron transport equation in one- and two-dimensional cartesian geometries. The two-dimensional neutron transport equation for a node is transformed to one-dimensional transport equation based on the face-averaged scalar flux and the current. At the node interfaces, DP1 expansion is employed to the surface angular fluxes in conjunction with isotropic angular dependence of the transverse leakage term. The one-dimensional integral transport equation is obtained in terms of the node-face-averaged incoming/outgoing neutron flux and the currents. The synthetic kernel approximation is employed to the transport kernels and nodal-face contributions. The resulting SKN equations are solved analytically. One-dimensional interface-coupling nodal SK1 and SK2 equations (incoming/outgoing flux and current) are derived for the small nodal-mesh limit. These equations have simple recursive forms which do not pose burden on either the memory or the computational time. The method was applied to one- and two-dimensional benchmark problems and compared with the solutions obtained with nodal integral method.

Explosive reconnection of double tearing modes in relativistic plasmas: application to the Crab flares

Abstract Magnetic reconnection associated to the double tearing mode is investigated by means of resistive relativistic magnetohydrodynamic simulations. A linearly unstable double current sheet system in two-dimensional Cartesian geometry is considered. For initial perturbations of large enough longitudinal wavelengths, a fast reconnection event is triggered by a secondary instability that is structurally driven by the non-linear evolution of the magnetic islands. The latter reconnection phase and time-scale appear to weakly depend on the plasma resistivity and magnetization parameter. We discuss the possible role of such explosive reconnection dynamics to explain the MeV flares observed in the Crab Pulsar nebula. Indeed, the time-scale and the critical minimum wavelength give constraints on the Lorentz factor of the striped wind and on the location of the emission region, respectively.