Shell models of magnetohydrodynamic turbulence

Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though shell models cannot account for the spatial complexity of MHD turbulence, their dynamics are not over simplified and do reflect those of real MHD turbulence including intermittency or chaotic reversals of large-scale modes. Furthermore, these models use realistic values for dimensionless parameters (high kinetic and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing extended inertial range and accurate dissipation rate. Using modern computers it is difficult to attain an inertial range of three decades with direct numerical simulations, whereas eight are possible using shell models.     

Magnetic discontinuities in magnetohydrodynamic turbulence and in the solar wind

Recent measurements of solar wind turbulence report the presence of intermittent, exponentially distributed angular discontinuities in the magnetic field. In this Letter, we study whether such discontinuities can be produced by magnetohydrodynamic (MHD) turbulence. We detect the discontinuities by measuring the fluctuations of the magnetic field direction, Δθ, across fixed spatial increments Δx in direct numerical simulations of MHD turbulence with an imposed uniform guide field B(0). A large region of the probability density function (pdf) for Δθ is found to follow an exponential decay, proportional to exp(-Δθ/θ(*)), with characteristic angle θ(*)≈(14°)(b(rms)/B(0))(0.65) for a broad range of guide-field strengths. We find that discontinuities observed in the solar wind can be reproduced by MHD turbulence with reasonable ratios of b(rms)/B(0). We also observe an excess of small angular discontinuities when Δx becomes small, possibly indicating an increasing statistical significance of dissipation-scale structures. The structure of the pdf in this case closely resembles the two-population pdf seen in the solar wind. We thus propose that strong discontinuities are associated with inertial-range MHD turbulence, while weak discontinuities emerge from dissipation-range turbulence. In addition, we find that the structure functions of the magnetic field direction exhibit anomalous scaling exponents, which indicates the existence of intermittent structures.     

Characterization of the three-dimensional supersonic flow for the MHD generator

A numerical procedure based on a five-wave MHD model associated with non-ideal, low magnetic Reynolds number MHD flows was developed in the present study for analyzing the flow fields in the MHD generator of a MHD bypass scramjet. The numerical procedure is composed of an entropy conditioned scheme for solving the non-homogeneous Navier-Stokes equations, in conjunction with an SOR method for solving the elliptic equation governing the electrical potential. It was found that a separation would take place near the downstream edge of the second electrode, where the local adverse pressure gradient is large, and the core of the flow field is characterized as a 2-D flow due to the Hartmann effects along the direction of the magnetic field. The electric current lines would be increasingly distorted as the magnetic interactive parameter increases, and even induce an eddy current. Induced eddy current was also found in the different cross-sections along the axial direction, all of these would definitely deteriorate the performance of the MHD generator. The cross-sectional M-shape velocity profile found along the axial direction between the insulating walls is responsible for the formation of the vortex flow at the corner of the insulator cross-section, which, in turn, induces the corner eddy current at the corner. A numerical parametric study was also performed, and the computed performance parameters for the MHD generator suggest that, in order to enhance the performance of MHD generator, the magnetic interaction parameter should be elevated.     

Nonaxisymmetric anisotropy of solar wind turbulence as a direct test for models of magnetohydrodynamic turbulence

Single point spacecraft observations of the turbulent solar wind flow exhibit a characteristic nonaxisymmetric anisotropy that depends sensitively on the perpendicular power spectral exponent. We use this nonaxisymmetric anisotropy as a function of wave vector direction to test models of MHD turbulence. Using Ulysses magnetic field observations in the fast, quiet polar solar wind we find that the Goldreich-Sridhar model of MHD turbulence is not consistent with the observed anisotropy, whereas the observations are well reproduced by the "slab+2D" model. The Goldreich-Sridhar model alone cannot account for the observations unless an additional component is also present.     

Direct numerical simulation of the turbulent MHD channel flow at low magnetic Reynolds number for electric correlation characteristics

Direct numerical simulation (DNS) of incompressible magnetohydrodynamic (MHD) turbulent channel flow has been performed under the low magnetic Reynolds number assumption.The velocity-electric field and electric-electric field correlations were studied in the present work for different magnetic field orientations.The Kenjeres-Hanjalic (K-H) model was validated with the DNS data in a term by term manner.The numerical results showed that the K-H model makes good predictions for most components of the velocity-electric field correlations.The mechanisms of turbulence suppression were also analyzed for different magnetic field orientations utilizing the DNS data and the K-H model.The results revealed that the dissipative MHD source term is responsible for the turbulence suppression for the case of streamwise and spanwise magnetic orientation,while the Lorentz force which speeds up the near-wall fluid and decreases the production term is responsible for the turbulence suppression for the case of the wall normal magnetic orientation.     

Turbulent diffusion of magnetic fields in two-dimensional magnetohydrodynamic turbulence with stable stratification

We calculate the correction, due to nonlinear wave-wave interactions, to the Zel'dovich estimate for the turbulent diffusivity of magnetic fields in a model of two-dimensional magnetohydrodynamic turbulence in the presence of stable stratification. Such a model has some relevance to hydromagnetic turbulence in stellar interiors. The significance of this correction is that, unlike the lowest-order Zel'dovich balance, it is independent of the molecular resistivity eta and so will not vanish in the limit of a large magnetic Reynolds number, although the correction is O(sigma;{4}), where sigma is the wave slope, which necessarily is small. Thus, we are led to the counterintuitive result that the presence of stable stratification can actually increase the vertical flux of magnetic fields relative to that in 2D MHD turbulence without stratification.     

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采用FLUENT软件分别对外加均匀横向磁场的等截面三维充分发展液态金属管流的层流模型和低雷诺数湍流Lam/Bremhost(LB)模型进行了数值模拟,分析了外加磁场对普通方管LB模型速度分布和压降的影响。比较在相同哈特曼数下,层流和湍流模型方管截面上速度分布和管道中MHD压降。其中,对电流的计算采用磁感应方程来求得。数值模拟结果证明了用低雷诺数LB湍流模型解决方管磁流体流动的可行性。通过层流模型和湍流模型的对比可知,层流模型有较短的入口长度,但管内流体的压降却很大;而湍流模型管内速度更加平均化,管内压降较小,但管内入口长度较长。     

Dynamos at extreme magnetic Prandtl numbers: insights from shell models

We present a magnetohydrodynamic (MHD) shell model suitable for computation of various energy fluxes of MHD turbulence for very small and very large magnetic Prandtl numbers Pm; such computations are inaccessible to direct numerical simulations. For small Pm, we observe that both kinetic and magnetic energy spectra scale as k^?5/3 in the inertial range, but the dissipative magnetic energy scales as k^?11/3exp?(? k/k_η). Here the kinetic energy at large length scale feeds the large-scale magnetic field that cascades to small-scale magnetic field, which gets dissipated by Joule heating. The large-Pm dynamo has a similar behaviour except that the dissipative kinetic energy scales as k^?13/3. For this case, the large-scale velocity field transfers energy to the large-scale magnetic field, which gets transferred to small-scale velocity and magnetic fields; the energy of the small-scale magnetic field also gets transferred to the small-scale velocity field, and the energy thus accumulated is dissipated by the viscous force.     

Mean magnetic field and noise cross-correlation in magnetohydrodynamic turbulence: results from a one-dimensional model

We show that a recently proposed [J. Fleischer, P.H. Diamond, Phys. Rev. E 58, R2709 (1998)] one-dimensional Burgers-like model for magnetohydrodynamics (MHD) is in effect identical to existing models for drifting lines and sedimenting lattices. We use the model to demonstrate, contrary to claims in the literature, that the energy spectrum of MHD turbulence should be independent of mean magnetic field and that cross-correlations between the noise sources for the velocity and magnetic fields cannot change the structure of the equations under renormalisation. We comment on the scaling and the multiscaling properties of the stochastically forced version of the model. Received 29 October 1998 and Received in final form 8 December 1998     

Energy decay laws in strongly anisotropic magnetohydrodynamic turbulence

We investigate the influence of a uniform magnetic field B(0)=B(0)e( parallel) on energy decay laws in incompressible magnetohydrodynamic (MHD) turbulence. The nonlinear transfer reduction along B(0) is included in a model that distinguishes parallel and perpendicular directions, following a phenomenology of Kraichnan. We predict a slowing down of the energy decay due to anisotropy in the limit of strong B(0), with distinct power laws for energy decay of shear- and pseudo-Alfvén waves. Numerical results from the kinetic equations of Alfvén wave turbulence recover these predictions, and MHD numerical results clearly tend to follow them in the lowest perpendicular planes.     

Generation and amplification of magnetic islands by drift interchange turbulence

We investigate the multiscale nonlinear dynamics of a linearly stable or unstable tearing mode with small-scale interchange turbulence using 2D MHD numerical simulations. For a stable tearing mode, the nonlinear beating of the fastest growing small-scale interchange modes drives a magnetic island with an enhanced growth rate to a saturated size that is proportional to the turbulence generated anomalous diffusion. For a linearly unstable tearing mode the island saturation size scales inversely as one-fourth power of the linear tearing growth rate in accordance with weak turbulence theory predictions. Turbulence is also seen to introduce significant modifications in the flow patterns surrounding the magnetic island.     

Decaying magnetohydrodynamics: effects of initial conditions

We study the effects of homogenous and isotropic initial conditions on decaying magnetohydrodynamics (MHD). We show that for an initial distribution of velocity and magnetic-field fluctuations, appropriately defined structure functions decay as a power law in time. We also show that for a suitable choice of initial cross correlations between velocity and magnetic fields even-order structure functions acquire anomalous scaling in time where as scaling exponents of the odd-order structure functions remain unchanged. We discuss our results in the context of fully developed MHD turbulence.     

聚变堆液态包层准二维磁流体湍流模型研究

为研究聚变堆氚增殖包层中液态金属湍流磁流体动力学(MHD)效应,开发了一种新的准二维单方程 MHD 湍流模型,并进行了相关数值模拟程序的编制及验证。对于矩形管道中的准二维 MHD 湍流流动,三维流 动主要发生在哈德曼层中,中心的主流区呈现出二维流动。为了反映这种特殊的流动特征,新湍流模型在标准 k-ε 模型的基础之上去掉了传统的耗散项,代之以电磁耗散项来模拟湍流 MHD 效应。同时,采用 Bradshaw 假设来对 湍流涡粘系数进行模化。为验证该湍流模型是否合理,编制了相关数值模拟程序,并利用直接数值模拟(DNS)结 果对该程序进行了校正,数值模拟结果与 DNS 结果吻合较好。计算结果表明,该湍流模型可应用于聚变堆液态 包层 MHD 湍流流动的数值模拟。     

Piecewise parabolic method on a local stencil for magnetized supersonic turbulence simulation

Stable, accurate, divergence-free simulation of magnetized supersonic turbulence is a severe test of numerical MHD schemes and has been surprisingly difficult to achieve due to the range of flow conditions present. Here we present a new, higher order-accurate, low dissipation numerical method which requires no additional dissipation or local “fixes” for stable execution. We describe PPML, a local stencil variant of the popular PPM algorithm for solving the equations of compressible ideal magnetohydrodynamics. The principal difference between PPML and PPM is that cell interface states are evolved rather that reconstructed at every timestep, resulting in a compact stencil. Interface states are evolved using Riemann invariants containing all transverse derivative information. The conservation laws are updated in an unsplit fashion, making the scheme fully multidimensional. Divergence-free evolution of the magnetic field is maintained using the higher order-accurate constrained transport technique of Gardiner and Stone. The accuracy and stability of the scheme is documented against a bank of standard test problems drawn from the literature. The method is applied to numerical simulation of supersonic MHD turbulence, which is important for many problems in astrophysics, including star formation in dark molecular clouds. PPML accurately reproduces in three-dimensions a transition to turbulence in highly compressible isothermal gas in a molecular cloud model. The low dissipation and wide spectral bandwidth of this method make it an ideal candidate for direct turbulence simulations.     

Nanoflares and MHD turbulence in coronal loops: a hybrid shell model

A model to describe injection, due to footpoint motions, storage, and dissipation of MHD turbulence in coronal loops, is presented. The model is based on the use of the shell technique in the wave vector space applied to the set of reduced MHD equations. Numerical simulation showed that the energy injected is efficiently stored in the loop where a significant level of magnetic and velocity fluctuations is obtained. Nonlinear interactions among these fluctuations give rise to an energy cascade towards smaller scales where energy is dissipated in an intermittent fashion. The statistical analysis performed on the intermittent dissipative events compares well with all observed properties of nanoflare emission statistics.     

高速飞行器磁控阻力特性

采用低磁雷诺数磁流体数学模型,对外加磁场下的高超声速半球体流场进行数值模拟.选取三种简单理想磁场(轴向、径向、周向均布磁场),分析了不同磁场类型对流场结构、气动阻力与洛伦兹阻力的影响及作用机理.研究发现,轴向磁场径向"挤压"效应使得激波外形凸出,且壁面静压存在"饱和现象";径向磁场存在轴向"外推"效应,较大的磁场强度会导致肩部形成高温区;周向磁场下感应电场的存在导致增阻效果很差.进而对比了两种相同驻点磁感应强度特殊分布磁场(偶极子磁场、螺线管磁场)下的流场,发现了不同于理想磁场的径向"扩张"效应.按增阻效果从大到小依次为径向磁场、螺线管磁场、轴向磁场、偶极子磁场、周向磁场.     

Self-similar signature of the active solar corona within the inertial range of solar-wind turbulence

We quantify the scaling of magnetic energy density in the inertial range of solar-wind turbulence seen in situ at 1 AU with respect to solar activity. At solar maximum, when the coronal magnetic field is dynamic and topologically complex, we find self-similar scaling in the solar wind, whereas at solar minimum, when the coronal fields are more ordered, we find multifractality. This quantifies the solar-wind signature that is of direct coronal origin and distinguishes it from that of local MHD turbulence, with quantitative implications for coronal heating of the solar wind.     

Nonaxisymmetric anisotropy of solar wind turbulence

A key prediction of turbulence theories is frame-invariance, and in magnetohydrodynamic (MHD) turbulence, axisymmetry of fluctuations with respect to the background magnetic field. Paradoxically the power in fluctuations in the turbulent solar wind are observed to be ordered with respect to the bulk macroscopic flow as well as the background magnetic field. Here, nonaxisymmetry across the inertial and dissipation ranges is quantified using in situ observations from Cluster. The observed inertial range nonaxisymmetry is reproduced by a "fly through" sampling of a direct numerical simulation of MHD turbulence. Furthermore, fly through sampling of a linear superposition of transverse waves with axisymmetric fluctuations generates the trend in nonaxisymmetry with power spectral exponent. The observed nonaxisymmetric anisotropy may thus simply arise as a sampling effect related to Taylor's hypothesis and is not related to the plasma dynamics itself.     

A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability,implementation and tests

The ideal MHD equations are a central model in astrophysics, and their solution relies upon stable numerical schemes. We present an implementation of a new method, which possesses excellent stability properties. Numerical tests demonstrate that the theoretical stability properties are valid in practice with negligible compromises to accuracy. The result is a highly robust scheme with state-of-the-art efficiency. The scheme’s robustness is due to entropy stability, positivity and properly discretised Powell terms. The implementation takes the form of a modification of the MHD module in the FLASH code, an adaptive mesh refinement code. We compare the new scheme with the standard FLASH implementation for MHD. Results show comparable accuracy to standard FLASH with the Roe solver, but highly improved efficiency and stability, particularly for high Mach number flows and low plasma β. The tests include 1D shock tubes, 2D instabilities and highly supersonic, 3D turbulence. We consider turbulent flows with RMS sonic Mach numbers up to 10, typical of gas flows in the interstellar medium. We investigate both strong initial magnetic fields and magnetic field amplification by the turbulent dynamo from extremely high plasma β. The energy spectra show a reasonable decrease in dissipation with grid refinement, and at a resolution of 512³ grid cells we identify a narrow inertial range with the expected power law scaling. The turbulent dynamo exhibits exponential growth of magnetic pressure, with the growth rate higher from solenoidal forcing than from compressive forcing. Two versions of the new scheme are presented, using relaxation-based 3-wave and 5-wave approximate Riemann solvers, respectively. The 5-wave solver is more accurate in some cases, and its computational cost is close to the 3-wave solver.