Elsasser Variables Research Articles

Decay of magnetohydrodynamic turbulence in the expanding solar wind: WIND observations

We have studied the decay of turbulence in the solar wind. Fluctuations carried by the expanding wind are naturally damped because of flux conservation, slowing down the development of a turbulent cascade. The latter also damps fluctuations but results in plasma heating. We analyzed time series of the velocity and magnetic field ($ v $ and $ B $, respectively) obtained by the WIND spacecraft at 1 au. Fluctuations were recast in terms of the Elsasser variables, $ z v B rho $, with rho being the average density, and their second- and third-order structure functions were used to evaluate the Politano-Pouquet relation, modified to account for the effect of expansion. We find that expansion plays a major role in the Alfv\'enic stream, those for which $z_+ z_-$. In such a stream, expansion damping and turbulence damping act, respectively, on large and small scales for $z_+$, and also balance each other. Instead, $z_-$ is only subject to a weak turbulent damping because expansion is a negligible loss at large scales and a weak source at inertial range scales. These properties are in qualitative agreement with the observed evolution of energy spectra that is described by a double power law separated by a break that sweeps toward lower frequencies for increasing heliocentric distances. However, the data at 1 au indicate that injection by sweeping is not enough to sustain the turbulent cascade. We derived approximate decay laws of energy with distance that suggest possible solutions for the inconsistency: in our analysis, we either overestimated the cascade of $z_ or missed an additional injection mechanism; for example, velocity shear among streams.

Alternative formulation of weak magnetohydrodynamic turbulence theory

In a recent paper [P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021)], the weak turbulence theory for incompressible magnetohydrodynamics is formulated by employing the method customarily applied in the context of kinetic weak plasma turbulence theory. Such an approach simplified certain mathematical procedures including achieving the closure relationship. The formulation in the above-cited paper starts from the equations of incompressible magnetohydrodynamic (MHD) theory expressed via Elsasser variables. The derivation of nonlinear wave kinetic equation therein is obtained via a truncated solution at the second-order of iteration following the standard practice. In the present paper, the weak MHD turbulence theory is alternatively formulated by employing the pristine form of incompressible MHD equation rather than that expressed in terms of Elsasser fields. The perturbative expansion of the nonlinear momentum equation is carried out up to the third-order iteration rather than imposing the truncation at the second order. It is found that while the resulting wave kinetic equation is identical to that obtained in the previous paper cited above, the third-order nonlinear correction plays an essential role for properly calculating derived quantities such as the total and residual energies.

Direct numerical simulatuion of homogeneous isotropic cross-helical mhd turbulence

Magnetohydrodynamic (MHD) turbulence with cross-helicity excited by a large-scale source is considered. The source is a random external force contributing energy with a controlled level of cross-helicity. The proposed approach allows one to form and maintain high levels of cross-helicity without using an external constant magnetic field. The force is implemented in the software package TARANG. A series of numerical experiments with a constant energy flow and various levels of injection of cross-helicity have been carried out. The ratio of the cross-helicity to the total energy varied in the range from 0 to 0.6. The calculations were carried out at kinetic Reynolds number R = 2094 and at magnetic Prandtl number Prm= 1 on the grid 5123. An equal amount of energy per unit of time was introduced into velocity and magnetic induction fields. Regardless of the level of cross-helicity, the rate of dissipation of magnetic energy has always been higher than the intensity of injection. Spectral energy fluxes indicated a positive energy flux from the velocity field to the magnetic induction field. The high levels of cross helicity led to energy accumulation on a large scale and to changes in energy distribution over the scale. The spectral energy densities in the Elsasser variables z±showed a significant difference in the slopes of the z+and z-spectra. On the spectral flux z-, a region close to the inertia interval was detected. Qualitative agreement with the results obtained earlier using cascade models is shown. Varying the ratio of codirectional change of fields did not affect the integral and spectral characteristics of the flow. The results obtained allow us to use the proposed force for a detailed study of the evolution of homogeneous isotropic turbulence with a high level of cross-helicity and its dissipation.

A least squares finite element method using Elsasser variables for magnetohydrodynamic equations

There are various different forms of magnetohydrodynamic(MHD) equations and they have been studied for years due to its complicated coupling between variables. This paper proposes to use an equivalently transformed MHD equations with Elsasser variables and the least squares finite element method to find the approximation to them. Introducing new variables by combining fluid velocity and magnetic field yields a Navier–Stokes like system. Then the first-order system least squares method using displacement recasts the transformed MHD equations into a system of first order partial differential equations and the Newton’s algorithm linearizes the problem. An L 2 -residual functional is defined to minimize and the unique existence of corresponding weak solution is shown. Finally, the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory.

Partitioned second order method for magnetohydrodynamics in Elsässer variables

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes equations coupled with Maxwell equations via Lorentz force and Ohm's law. Monolithic methods, which solve fully coupled MHD systems, are computationally expensive. Partitioned methods, on the other hand, decouple the full system and solve subproblems in parallel, and thus reduce the computational cost. This paper is devoted to the design and analysis of a partitioned method for the MHD system in the Elsasser variables. The stability analysis shows that for magnetic Prandtl number of order unity, the method is unconditionally stable. We prove the error estimates and present computational tests that support the theory.

Reflection of Alfven waves and plasma turbulization in solar corona

The continuous reflection of Alfven waves in the coronae of the Sun and stars is considered. Based on the WKB approximation, the solution to the linear wave equation in the case of a stratified isothermal atmosphere has been obtained. A critical analysis of results obtained by Ferraro and Plumpton (1958) and Hollweg (1972), as well as the relations in the Elsasser variables has been carried out. It has been shown that Alfven disturbances do not undergo a continuous reflection within the accepted model and are transformed into intermediate-type modes that possess the properties of vibrations and travelling waves. The problem of the turbulization of corona plasma is discussed. The origin of the Alfven waves that propagate toward the Sun is related to the development of parametric instability.

Anisotropy of Imbalanced Alfvénic Turbulence in Fast Solar Wind

We present the first measurement of the scale-dependent power anisotropy of Elsasser variables in imbalanced fast solar wind turbulence. The dominant Elsasser mode is isotropic at lower spacecraft frequencies but becomes increasingly anisotropic at higher frequencies. The subdominant mode is anisotropic throughout. There are two distinct subranges exhibiting different scalings within what is normally considered the inertial range. The low Alfvén ratio and the different scaling of the Elsasser modes suggests an interpretation of the observed discrepancy between the velocity and magnetic field scalings, the total energy is dominated by the latter. These results do not appear to be fully explained by any of the current theories of incompressible imbalanced MHD turbulence.

The Energy Cascade in Solar Wind MHD Turbulence

Direct evidence for the presence of an inertial energy cascade, the most characteristic signature of hydromagnetic turbulence (MHD), is observed in the solar wind by the Ulysses spacecraft. A linear relation is indeed observed for the scaling of mixed third order structure functions involving Elsasser variables. This experimental result, confirming the prescription stemming from a theorem for MHD turbulence, firmly establishes the turbulent character of low-frequency velocity and magnetic field fluctuations in the solar wind plasma.

The Turbulent Cascade at 1 AU: Energy Transfer and the Third‐Order Scaling for MHD

We perform a test of MHD turbulent cascade theory in the solar wind and directly evaluate the contribution of local turbulence to heating the solar wind at 1 AU. We look at turbulent fluctuations in the solar wind velocity V and magnetic field B, using the vector Elsasser variables Z ± ≡ V ± B/(4πρ)1/2 as measured at the ACE spacecraft stationed at the Earth's L1 point. We combine the fluctuations δZ± over time lags in the inertial range, from 64 s to several hours, to form components of the mixed vector third moments, and we adopt the work of Politano & Pouquet, who derive an exact scaling law, similar to the Kolmogorov 4/5 law, but valid in anisotropic MHD turbulence, for these components. We demonstrate that the scaling is reasonably linear, as is expected for the inertial range. The total turbulent energy injection/dissipation rate that we derive this way agrees with the in situ heating of the solar wind that is inferred from the temperature gradient, whereas methods using the power spectra only seldom agree with the heating rates derived from gradients of the thermal proton distribution. We derive expressions of the third-order moments that are applicable to the spectral cascades parallel and perpendicular to the mean magnetic field. We apply these expressions to fast- and slow-wind subsets of the data, with additional subsetting for mean field direction. We find that both the fast wind and the slow wind exhibit an active energy cascade over the inertial range scales. Furthermore, we find that the energy flux in the parallel cascade is consistently smaller than in the perpendicular cascade.

Simulations of nonhelical hydromagnetic turbulence.

Nonhelical hydromagnetic forced turbulence is investigated using large scale simulations on up to 256 processors and 1024(3) mesh points. The magnetic Prandtl number is varied between 1/8 and 30, although in most cases it is unity. When the magnetic Reynolds number is based on the inverse forcing wave number, the critical value for dynamo action is shown to be around 35 for magnetic Prandtl number of unity. For small magnetic Prandtl numbers we find the critical magnetic Reynolds number to increase with decreasing magnetic Prandtl number. The Kazantsev k(3/2) spectrum for magnetic energy is confirmed for the kinematic regime, i.e., when nonlinear effects are still unimportant and when the magnetic Prandtl number is unity. In the nonlinear regime, the energy budget converges for large Reynolds numbers (around 1000) such that for our parameters about 70% is in kinetic energy and about 30% is in magnetic energy. The energy dissipation rates are converged to 30% viscous dissipation and 70% resistive dissipation. Second-order structure functions of the Elsasser variables give evidence for a k(-5/3) spectrum. Nevertheless, the three-dimensional spectrum is close to k(-3/2), but we argue that this is due to the bottleneck effect. The bottleneck effect is shown to be equally strong both for magnetic and nonmagnetic turbulence, but it is far weaker in one-dimensional spectra that are normally studied in laboratory turbulence. Structure function exponents for other orders are well described by the She-Leveque formula, but the velocity field is significantly less intermittent and the magnetic field is more intermittent than the Elsasser variables.

A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equations

By considering the three-dimensional incompressible Euler equations, a 4-vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω·∇ u=S ω . The evolution equation for ζ can then be cast naturally into a quaternionic Riccati equation. This is easily transformed into a quaternionic zero-eigenvalue Schrödinger equation whose potential depends on the Hessian matrix of the pressure. With minor modifications, this system can alternatively be written in complex notation. An infinite set of solutions of scalar zero-eigenvalue Schrödinger equations has been found by Adler and Moser, which are discussed in the context of the present problem. Similarly, when the equations for ideal magneto-hydrodynamics (MHD) are written in Elsasser variables, a pair of 4-vectors ζ ± are governed by coupled quaternionic Riccati equations.

Initial condition sensitivity of global quantities in magnetohydrodynamic turbulence

The effect of subtle changes in initial conditions on the evolution of global quantities in two-dimensional magnetohydrodynamic (MHD) turbulence is studied. It is found that a change in the initial phases of complex Fourier modes of the Elsässer variables, while keeping the initial values of total energy, cross helicity, and Alfvén ratio unchanged, has a significant effect on the evolution of cross helicity. On the contrary, the total energy and Alfvén ratio are insensitive to the initial phases. The simulations are based on direct numerical simulation using the pseudospectral method.

Large‐Amplitude Magnetohydrodynamic Waves in Nonhomogeneous Plasmas

We consider the propagation of large-amplitude magnetohydrodynamic (MHD) waves in an anisotropic nonhomogeneous plasma. A classical MHD formulation is adopted, and, by means of a sequence of transformations and use of Elsasser's variables, we derive an exact nonlinear system of equations in compact form that describes such waves. The system incorporates compressibility of the plasma and an arbitrary polarization for the waves. Accordingly, the system provides for the analysis of parallel-propagating, nonlinear Alfven waves and magnetosonic waves in a general space plasma context, including the solar wind. We obtain a new large-amplitude incompressible Alfven wave solution to the system, and also show how to construct Riemann solutions that are valid under broad assumptions. Brief consideration is given to Alfven wave propagation in a stationary plasma, and we show that the equation of propagation reduces to classical second-order partial differential equations in two special cases. For the special case in which the Alfven speed depends linearly on the spatial coordinate, a simple, exact Riemann solution is presented.

Pick‐up ions and associated wave energy transport at comet P/Halley: A case study

During the flyby of the spacecraft Giotto at comet p/Halley Poynting vectors and Elsässer variables have been determined to study wave propagation directions. The observed wave properties are compared with the theoretically predicted RH−, LH+ and LH− wave modes. Between 10:36 and 19:11 SCET on March 13, 1986 the predicted dominating energy propagation direction mostly corresponds with the observed one. Only between 12:11 and 13:44 SCET there is no agreement. In this region the excitation of waves is dominated by pick‐up ions which are not locally implanted ions in the solar wind but have been picked‐up outside the plasma regime studied, and whose free energy is not already used up.

Discrete vortex representation of magnetohydrodynamics.

We present an alternative approach to statistical analysis of an intermittent ideal magnetohydrodynamics fluid in two dimensions, based on the hydrodynamic discrete vortex model applied to the Elsasser variables. The model contains negative temperature states which predict the formation of magnetic islands, but also includes a natural limit under which the equilibrium states revert to the familiar twin-vortex states predicted by hydrodynamic turbulence theories. Numerical dynamical calculations yield equilibrium spectra in agreement with the theoretical predictions.

Self-Similar Statistics in MHD Turbulence

Abstract The fully developed decaying turbulence of 3-d resistive, viscous, incompressible magnetohydrodynamics is investigated using Elsasser variables and Hopf equation for probability distributions. The method is an extension of a previous work for Navier Stokes equations done by Foias et al. based on a suggestion by Hopf. It uses essentially self-similar properties of the statistics, which "almost" determine the turbulence spectrum up to a mild assumption on an unknown function. This spectrum is the well known Kolmogorov spectrum