M. L. Goldstein

Cluster - Science and Mission Overview

The European Space Agencys Cluster programme is designed to study the small-scale spatial and temporal characteristics of the magnetospheric and near-Earth solar wind plasma. The programme is composed of four identical spacecraft which will be able to make physical measurements in three dimensions. The relative distance between the four spacecraft will be varied between 200 and 18000 km during the course of the mission. This paper provides a general overview of the scientific objectives, the configuration and the orbit of the four spacecraft and the relation of Cluster to other missions.

Properties of the fluctuating magnetic helicity in the inertial and dissipation ranges of solar wind turbulence

We investigated the inertial and dissipation ranges of the reduced magnetic helicity spectrum of solar wind fluctuations and have found that this spectrum appears insensitive to solar cycle variations and changes in solar wind flow parameters. In the inertial range of the spectrum, the reduced helicity is large but random and independent of heliocentric distance between 0.3 and 10 AU. At small scales, in the dissipation range of the spectrum, a correlation appears to exist between the average value of the normalized reduced magnetic helicity and the polarity of magnetic sectors, suggesting that these fluctuations, if outward propagating, are predominantly right-hand polarized. In the inertial range the statistical properties of the normalized magnetic helicity are well approximated by a simple model of the magnetic field in which the total magnetic field vector randomly walks with only small variations in magnitude. The behavior of the inertial range spectrum is very similar to that seen in three- and two-and-a-half-dimensional simulations of the incompressible and compressible equations describing magnetohydrodynamic turbulence, consistent with the paradigm that the solar wind is a turbulent magnetofluid.

Spectral Exponents of Kinetic and Magnetic Energy Spectra in Solar Wind Turbulence

Kinetic and magnetic energy spectra in the ecliptic plane near 1 AU are found to exhibit different power-law behaviors in the inertial range, with the magnetic spectrum often having a power-law exponent near 5/3 and the kinetic energy spectrum often having a power-law exponent near 3/2 (the inertial range extends from approximately 5 × 10-4 to 10-1 Hz). The total energy, kinetic plus magnetic, has a power-law exponent that lies between 3/2 and 5/3, with a value near 1.6. The Alfven ratio, the ratio of kinetic to magnetic energy, is found to be a slowly increasing function of frequency in the inertial range, increasing from roughly 0.5 to 0.9 in the frequency range from 10-3 to 10-1 Hz. These conclusions are based on the analysis of four distinct time intervals of solar wind magnetic field and plasma data obtained by the Wind spacecraft near the end of solar cycle 22 and at different times throughout solar cycle 23. Three 54 day intervals and one 81 day interval are used to compute power spectra in the range from 10-5 to 1.7 × 10-1 Hz. Power-law exponents are estimated from linear least-squares fits to the logarithm of the power spectral density versus the logarithm of the frequency over the frequency interval from 10-3 to 10-2 Hz. To prevent errors due to spectral aliasing, the last decade of the spectrum is omitted from the calculation of the power-law exponents. The results show that a measurable difference exists between the power-law exponents of velocity and magnetic field fluctuations and that this difference persists throughout the solar cycle.

Simulation of high-frequency solar wind power spectra using Hall magnetohydrodynamics

Solar wind frequency spectra show a distinct steepening of the ƒ−5/3 power law inertial range spectrum at frequencies above the Doppler-shifted ion cyclotron frequency. This is commonly attributed to dissipation due to wave-particle interactions. We consider the extent to which this steepening can be described, using a magnetohydrodynamic formulation that includes the Hall term. An important characteristic of Hall MHD is that although the ion cyclotron resonance is included, there is no wave-particle dissipation of energy. In this study we use a compressible Hall MHD code with a constant magnetic field and a polytropic equation of state. Artificial dissipation in the form of a bi-Laplacian operator is used to suppress numerical instabilities, allowing for a clear separation of the dissipative scales from the ion cyclotron scales. A distinct steepening appears in the simulation power spectra near the cyclotron resonance for certain types of initial conditions. This steepening is associated with the appearance of right circularly polarized fluctuations at frequencies above the ion cyclotron resonance. Similar steepenings and polarization enhancements are observed in solar wind magnetic field data.

A numerical study of the nonlinear cascade of energy in magnetohydrodynamic turbulence

Power spectra of solar wind magnetic field and velocity fluctuations more closely resemble those of turbulent fluids (spectral index of −5/3) than they do predictions for magnetofluid turbulence (a −3/2 index). Furthermore, the amount the solar wind is heated by turbulence is uncertain. To aid in the study of both of these issues, we report numerically derived energy cascade rates in magnetohydrodynamic (MHD) turbulence and compare them with predictions of MHD turbulence phenomenologies. Either of the commonly predicted spectral indices of 5/3 and 3/2 are consistent with the simulations. Explicit calculation of inertial range energy cascade rates in the simulations show that for unequal levels of fluctuations propagating parallel and antiparallel to the magnetic field, the majority species always cascades faster than does the minority species, and the cascade rates are in better agreement with a Kolmogoroff-like MHD turbulence phenomenology than with a generalized Kraichnan phenomenology even in situations where the fluctuations are much smaller than the mean magnetic field. The “Kolmogoroff constant” for MHD turbulence for small normalized cross helicity is roughly 6.7 in two dimensions and 3.6 for one calculation in three dimensions. For large normalized cross helicity, however, none of the existing models can account for the numerical results, although the Kolmogoroff-like case still works somewhat better than the Kraichnan-like. In particular, the applied magnetic field has much less influence than expected, and Alfvenicity is more important than predicted. These results imply the need for better phenomenological models to make clear predictions about the solar wind.

Waves, structures, and the appearance of two-component turbulence in the solar wind

Spacecraft observations of magnetic field fluctuations in the solar wind reveals a “Maltese Cross” pattern in the two-dimensional correlation function measurements of solar wind fluctuations [Matthaeus et al., 1990]. This pattern suggests the presence of two components: fluctuations with their (Fourier) wave vector approximately parallel to the ambient magnetic field (e.g., slab turbulence) and fluctuations with their (Fourier) wave vector approximately perpendicular to the ambient magnetic field (e.g., quasi two-dimensional turbulence). To date, the appearance of such a pattern has never been reproduced from numerical simulation studies. Here we present results of several MHD simulations that address this issue using both two-and-one-half dimensional and three-dimensional compressible models and a wide variety of initial states and plasma parameters. Slab turbulence and quasi two-dimensional turbulence appear in various runs; however, their simultaneous appearance is difficult to achieve and seems to rely upon their separate existence in the initial data. In contrast, the presence of transverse pressure-balanced magnetic structures causes slab turbulence to evolve in such a manner that a two-component correlation function emerges through time averaging. We suggest that the Maltese Cross and similar observations may be a consequence of either the initial data or of averaging over different parcels of solar wind.

Numerical simulation of Alfvénic turbulence in the solar wind

Low-frequency fluctuations in the solar wind magnetic field and plasma velocity are often highly correlated, so much so that the fluctuations can be thought of as nearly perfect Alfven waves. Evidence from the Helios and Ulysses spacecraft suggest strongly that these fluctuations emanate from the solar corona with high correlation and flat power spectra (∼f−1). These fluctuations constitute a source of free energy for a turbulent cascade of magnetic and kinetic energy to high wave numbers, a cascade that evolves most rapidly in the vicinity of velocity shears and the heliospheric current sheet. Numerical solutions of both the compressible and incompressible equations of magnetohydrodynamics (MHD) in Cartesian geometry showed that sharp gradients in velocity would decrease substantially the Alfvenicity of initially pure Alfvenic fluctuations; however, the effects of solar wind expansion on this turbulent evolution is, as yet, undetermined. We demonstrate that as was the case in Cartesian geometry, in an expanding volume, velocity shears and pressure-balanced flux tubes still reduce the Alfvenicity of parallel propagating wave packets. These three-dimensional spherically expanding simulations include velocity shears separating fast and slow flows, pressure-balanced flux tubes, and a central current sheet which is the site of magnetic reconnection. Two-dimensional spectra constructed in the r – θ plane resemble closely those resulting from similar initial conditions in Cartesian geometry.

The evolution of slab fluctuations in the presence of pressure-balanced magnetic structures and velocity shears

The traditional view that solar wind fluctuations are well-described as a spectrum of parallel-propagating Alfven waves has been challenged many times but is still a frequently encountered perspective. Here we examine whether it remains consistent to view most of the fluctuation energy as resident in parallel-propagating Alfven waves in situations in which there are also present either transverse pressure-balanced (PB) magnetic structures or transverse velocity shears. We address these questions through direct simulation of compressible magnetohydrodynamics, with expansion effects neglected. We show that parallel-propagating Alfven waves are redirected to large oblique angles after refractive interactions with PB structures or advective interactions with velocity shears, reflecting the nonequilibrium nature of the initial spectral distribution. The timescale for these processes ranges from 2–8 eddy-turnover times or characteristic nonlinear times. Relatively small amounts of PB structure and/or shear energy can redirect initially parallel-propagating Alfven waves to highly oblique angles. Velocity microstreams appear to be particularly efficient at creating highly oblique waves. Even though the excited wave vectors are eventually primarily oblique, the magnetic variance ratios show a minimum variance in the mean magnetic field direction.

Time development of field‐aligned currents, potential drops, and plasma associated with an auroral poleward boundary intensification

[1] We present a detailed case study of the plasma and fields measured by the Cluster spacecraft fleet at the high-altitude auroral zone (-3.5 R E ) across the plasma sheet boundary layer and into the polar cap. This event, which occurred during quiet geomagnetic conditions (Kp = 1 + , AE = 50 nT), is of particular interest in that Cluster provides measurements at key instances during the time development of a new large-scale auroral arc system. Central to the formation of the arc system is the depletion of ionospheric plasma through a region of small-scale, field-aligned currents having the properties of Alfven waves. This depletion occurred prior to the growth of and ultimately bounded a well-defined equatorward moving, upward and downward current sheet pair. In association with the transverse scales approaching the electron inertial scale, the Alfvenic currents have amplitudes that appear to be attenuated subsequent to the formation of the cavity. Potential structures essentially time invariant over particle transit times (quasi-static) associated with the current pair are identified and observed to drive a poleward boundary intensification (PBI) identified in coincident IMAGE satellite far ultraviolet measurements. The PBI formed in association with a local thickening of the plasma sheet via the injection of new magnetospheric plasma, which may be the result of a bursty, patchy reconnection process. Estimates of the ionospheric equatorward velocity and thickness of the PBI are consistent with their ionospheric mapped cavity counterparts, suggesting that the motion and thickness are controlled by the plasma and electrodynamic features at or above the altitude sampled by Cluster. The magnitude of the upward and downward current region parallel potentials is correlated with the temperature of the newly injected electrons suggesting that the electron temperature is an important controlling factor. These novel observations indicate that quasi-static systems of field-aligned currents do form out of the highly dynamic Alfvenic region at the plasma sheet boundary layer, and perhaps suggest that the Alfvenic region can be the initial stage in the development of quasi-static systems. The observed time sequence of the currents is qualitatively similar to the expectations of transient response models of magnetospheric-ionospheric coupling, however, such models may need to be modified to account for the attenuation of electron inertial scale currents/Alfven waves.

Alfvén wave phase mixing driven by velocity shear in two-dimensional open magnetic configurations

Phase mixing of torsional Alfven waves in axisymmetric equilibrium magnetic configurations with purely poloidal magnetic field and stationary flow along the field lines in resistive viscous plasmas is studied. The characteristic wavelength along the magnetic field lines is assumed to be much smaller than the characteristic scale of inhomogeneity in the magnetic field direction, and the WKB method is used to obtain an analytic solution describing phase mixing. The general solution is applied to a particular configuration with the radial magnetic field and flow under the assumptions that the magnetic field and density are independent of the polar angle in the spherical coordinates and the flow velocity is independent of the radial coordinate. The only source of phase mixing in this configuration is velocity shear. The analytical solution is compared with a numerical simulation of the fully nonlinear resistive MHD equations. The numerical and analytical results are in good agreement. Consequences for wave energy deposition into the solar corona and solar wind and for the evolution of the Alfven wave energy spectrum are discussed.