SH31A-1645
Intermittent structures and magnetic discontinuities in MHD turbulence and solar wind
In this work we re-examined the statistics of rapid spatial variations of the magnetic field in simulations of Hall magnetohydrodynamic (HMHD) turbulence, using analysis of intermittency properties of the turbulence, and also using methods often employed to identify discontinuities in the solar wind (as in the earlier work of Tsurutani&Smith 1979). The hypothesis is that the statistics of intermittent events might be related to the statistics of classical MHD discontinuities. Indeed, those methods give similar distributions of events, often identifying the same structures. This suggests that observed discontinuities might not be static solutions to the MHD equations, but instead may be related to flux tube boundaries and intermittent structures that appear spontaneously in MHD turbulence. Then, we further examine the link between intermittency and MHD discontinuities, directly comparing statistical analysis from solar wind data and 3D and 2D simulations of MHD turbulence. The comparison between ACE solar wind data and simulations of magnetohydrodynamic turbulence shows a good agreement in the Waiting-Time analysis of magnetic field discontinuities. This result adds to evidence that solar wind magnetic structures may emerge fast and locally from nonlinear dynamics that can be properly described in the framework of MHD theory. Finally, probability distribution functions of increments in ACE data and in simulations reveal a robust structure consisting of small random currents, current cores, and intermittent current sheets. This classification provides a real-space picture of the nature of intermittent MHD turbulence.
SH31A-1646
Numerical simulations of the turbulence dissipation in the solar wind and corona
The physical mechanism responsible for the turbulence dissipation in the solar wind and corona is not completely understood. Analysis of the observational data shows that the damping rate of the turbulent fluctuations has to depend on their amplitude. This suggests a nonlinear mechanism of dissipation. However, existing theoretical mechanisms of nonlinear wave damping require fluctuation amplitudes that are too high compared to the actual values. We will report the results of hybrid simulations designed to mimic quasi-parallel and quasi-perpendicular turbulent fluctuations in a cold and warm plasma. We will analyze how nonlinearity contributes to the damping of the waves when their amplitudes conform to the limitations imposed by observations. In the in situ solar, the amplitudes are estimated directly wind from the measured magnetic spectra. In the corona, the turbulent fluctuations can become compressional due to certain nonlinear and kinetic effects. This allows us to tie the fluctuation amplitudes to the density spectra inferred from the interplanetary scintillation observations.
SH31A-1647
Dynamics of the Auroral Electrojet Index Time Series
We analyze the Auroral Electrojet (AE) index data for periods of Solar minimum and maximum, with respect to predictability and intermittency. Neural networks are employed to the AE- data for different intermittent as well as smooth intervals, and it is shown that the intermittent bursts can be predicted from the smooth sequences of the the AE-data, whereas the opposite does not hold true. This implies that the smooth dynamics in the data could cause intermittent bursts, but that it is difficult to find a causal link between the bursty dynamics and the subsequent smooth signal. It is, however, possible to establish a causal link between the smooth signals before and after the intermittent burst. We also compute the multifractal singularity spectrum as additional evidence for the existence of intermittency, and show that this spectrum is independent of the Solar activity.
SH31A-1648
A Fractional Differential Kinetic Equation and Applications to Modelling Bursts in Turbulent Nonlinear Space Plasmas
Since the 1960s Mandelbrot has advocated the use of
fractals for the description of the non-Euclidean
geometry of many aspects of nature. In particular he
proposed two kinds of model to capture persistence in
time (his Joseph effect, common in hydrology and with
fractional Brownian motion as the prototype) and/or prone
to heavy tailed jumps (the Noah effect, typical of
economic indices, for which he proposed Lévy flights
as an exemplar). Both effects are now well demonstrated
in space plasmas, notably in the turbulent solar wind.
Models have, however, typically emphasised one
of the Noah and Joseph parameters (the Lévy exponent
μ and the temporal exponent β) at the other's
expense. I will describe recent work in which we
studied a simple self-affine stable model-linear
fractional stable motion, LFSM, which unifies both
effects and present a recently-derived diffusion equation for LFSM. This replaces the second order spatial
derivative in the equation of fBm with a fractional
derivative of order μ, but retains a diffusion
coefficient with a power law time dependence rather than
a fractional derivative in time.
I will also show work in progress using an LFSM model and simple analytic scaling arguments to study the
problem of the area between an LFSM curve and a threshold. This problem relates to the burst size measure
introduced by Takalo and Consolini into solar-terrestrial physics and further studied by Freeman et al [PRE,
2000] on solar wind Poynting flux near L1. We test how expressions derived by other authors generalise to
the non-Gaussian, constant threshold problem. Ongoing work on extension of these LFSM results to
multifractals will also be discussed.
http://arxiv.org/abs/0807.1053
SH31A-1649
On Velocity Gradient Tensor in Space Plasma Turbulent Flows: A Case Study
The studies on space plasma turbulence mainly focused on the spectral and statistical features of magnetic and plasma parameter longitudinal fluctuations. This description is, however, incomplete being the longitudinal velocity and/or magnetic field difference only one of the locally independent components of the gradient tensors governing the global dynamics of the plasma. In the last decade significant advances have been found in the characterization and in the experimental techniques to measure and model all the components of the velocity gradient tensor in the framework of fluid turbulence [Chertkov et al., Phys. Fluids, 11, 2394, 1999]. Here, we attempt an evaluation of the small scale velocity gradient tensor (at the edge of the MHD domain) for a case study of space plasma turbulence. The statistics P(R, Q) of the velocity gradient (topological) invariants is similar to what is found in the case of the low end of the inertial range for hydrodynamic turbulence, suggesting the occurrence of vortex stretching (Vieillefosse tail) [Viellefosse, Physica 125A, 150, 1984].
SH31A-1650
Field-Line Dispersal and the Death of Lyapunov Exponents
Turbulent magnetic field lines have long been thought to be diverging from each other (or converging towards each other) at exponential rates known as Lyapunov exponents. It is now shown that in a turbulent magnetized plasma, subexponential divergence (convergence) and diffusive twist better characterize the dispersal of magnetic field lines than do the usual Lyapunov exponents or exponentiation rates. Pairs of nearby magnetic field lines diverge (converge) sub-exponentially rather than exponentially, and as soon as they diverge (converge) by a significant amount, they also experience substantial twist or rotation relative to each other. More distant magnetic field lines follow the same dynamics of twist and sub-exponential divergence (convergence), though at a slower rate. It is also found that on a very broad range of separation length scales, the statistics of the field-line separations are log-normal rather than Gaussian. Most importantly, the field-line dispersal can now be evaluated quantitatively and accurately. These results will be presented and some implications for the dispersal and mixing of solar wind magnetic field lines and particles will be discussed.
SH31A-1651
Aniostropic kinetic dissipation in collisionless turbulent plasmas.
The nature of the collisionless dissipation at small scales in solar wind turbulence is a problem of critical importance. To gain some insight into the nature of the dissipation, we simulate the Orszag-Tang vortex using collisionless hybrid simulations. In magnetohydrodynamics this configuration leads rapidly to broadband turbulence. At small scales, differences from magnetohydrodynamics arise, as energy dissipates into heat almost exclusively through the magnetic field. A key result is that protons are heated preferentially in the plane perpendicular to the mean magnetic field, creating a proton temperature anisotropy as is observed in the corona and solar wind. In order to gain insight into the heating mechanism, the scaling properties of the dissipation are examined, as well as length and time spectra.
SH31A-1652
Scaling relations for compressible turbulence in the solar wind
Incompressible and isotropic magnetohydrodynamic turbulence in plasmas can be described by an exact relation for the energy flux through the scales. This Yaglom-like scaling law has been recently observed in some samples of data collected by the Ulysses spacecraft in the solar wind above solar poles, where turbulence is Alfvenic. An analogous scaling law, suitably modified to take into account compressible fluctuations, can be observed in a more extended fraction of the same dataset. Large scale density fluctuations, despite their low amplitude, play thus a crucial role in the basic scaling properties of turbulence. The compressive turbulent cascade, moreover, can supply the energy needed to account for the local heating of the non-adiabatic solar wind.
SH31A-1653
Magnetic turbulence in space plasmas: scale dependent effects of anisotropy
The presence of a background magnetic field induces anisotropy in magnetic turbulence. This paper presents a systematic analysis of anisotropy in three different regions of the heliosphere, namely in the solar wind, and in the Earth's foreshock and magnetosheath behind a quasi parallel bow shock. A strong anisotropy is found in all cases, with very interesting cross-scale effects at the ion cyclotron frequency. In particular: i) the eigenvalues of the variance matrix have a strong intermittent behaviour, with very high localized fluctuations at small scales. As a consequence the Probability Distribution Functions are almost Gaussians at large scales and become power laws at small scales; ii) the minimum variance direction is almost parallel to the large scale magnetic field at large scales, while it become perpendicular at small scales. Differences among the three regions are discussed.
SH31A-1654
Intermittency analysis of magnetic field disturbances in the fast solar wind
The intermittency properties of the magnetic field fluctuations in the fast solar wind (wind speed greater than 550 km/s) are investigated here at various heliocentric distances based on the measurements by the Helios spacecraft during 1974-1981. Seven quantities of normalized PDF (Probability Distribution Function) associated with the magnetic field and its disturbances and the magnetic energy are utilized to characterize the degree of intermittency by fitting the PDF with a Castaing distribution and by using the idea of ¡§Flatness¡¨. The fluctuating magnetic fields in a number of fast solar wind events from 0.29 AU to 1.0 AU are analyzed separately to figure out the dependence of intermittency on the heliocentric distances at different time scales. It is apparent that the magnetic field fluctuations are more intermittent at farther distances from the sun. The increase in the degree of intermittency with decreasing time scale is more obvious farther from the sun. Moreover, for a given quantity, the intermittency decreases with increasing time scales and in particular, the PDF of the perpendicular component of perturbed magnetic field is found to approach a Gaussian distribution much more quickly than others at a large time scale. Due to the large value and scattered distribution of ¡§Flatness¡¨ in the measured magnetic fields, we suppose that the quantities associated with the perturbed fields is more apparent to represent the spatial variations of intermittent characteristics than those associated with the measured fields.
SH31A-1655
Two-Dimensional Nonlinear Evolution of the Electron-Beam-Plasma Instability
Recently, numerical and simulation efforts on the weak beam plasma interactions in one-dimensional space showed that the collisionallity of a plasma does a crucial role for electron acceleration and suprathermal tail generation. In this situation the collisionallity related to nonlinear wave-particle effects brings the Langmuir inverse cascade and this makes heating. Another effort, beam plasma interaction in weakly turbulent regime is numerically investigated including 2nd order nonlinear effect by Ziebell et al in the 2D space. This shows that, inverse cascade of 1D situation is numerical projection effect, and thus inverse cascade is not generated. Evolution of a two-dimensional electron-beam-plasma system is studied using a particle-in-cell (PIC) simulation. At the same time, the set of the quasilinear equations is numerically solved allowing direct comparisons of obtained electron and Langmuir wave distributions to the simulation results. From the comparison, it is found that the spectrum of the primary Langmuir wave obtained from PIC simulation is attributed to the linear wave-particle interaction. Furthermore we run simulations until the nonlinear interaction regime to examine the Langmuir wave inverse cascade and superthermal tail generation. We will present the nonlinear evolution of the electron distribution and Langmuir wave spectrum in a 2D simulation space.
SH31A-1656
Alignment of Velocity and Magnetic Fields in Fast and Slow Solar Wind
The earliest measurements of velocity and magnetic fields in the solar wind showed that the turbulence can be strongly Alfvenic in the fast wind. This property is very important as the cross-helicity, that is the correlation between velocity and magnetic field is an invariant of the ideal MHD equations and can inhibit the turbulent energy cascade if it is completely realized. It is however very important to study the local values of the cross-helicity, and in particular its probability distribution function, to determine the statistics of the local turbulent energy transfer. Indeed, even if there is no global Alfvenicity, the fluctuations of the local correlation can be very strong and, as they are related to the cosine of the angle between the velocity and the magnetic field, their distribution will depend on the dimensionality of these vector fluctuations. We have determined the probability distribution of the angle and its cosine between random vectors for different values of the cross-correlation, and we have compared these distributions with those observed at different distances by the Ulysses spacecraft in high-latitude fast solar wind, and those measured by the Helios satellite in fast and slow wind in the ecliptic. The distributions are found to always differ from the casual ones, but to be closer to two-dimensional distributions in the fast solar wind, suggesting a possible link between Alfvenicity and dimensionality of the fluctuations.
SH31A-1657
Gyrokinetic Particle Simulation of Kinetic Alfven Wave
Random magnetic fluctuations resembling Alfven waves are ubiquitously observed in laboratory, space and astrophysical plasmas. The issue of spectral cascade and plasma heating in Alfvenic turbulence is a major unsolved problem in space plasma physics. Gyrokinetic particle simulation is applied in this work to study the cascade and heating in Alfvenic turbulence with fully self-consistent nonlinear kinetic effects. A particle-in-cell code with gyrokinetic ions and fluid-kinetic hybrid electrons is being developed to study the coupling between shear Alfven wave and ion acoustic wave, which lead to energy exchange between waves and particles.
SH31A-1658
Propagation of a shock-wave in a strongly inhomogeneous background
In this poster, we present the numerical solution of the evolution of a shock-wave which propagates through a large-scale rapidly changing inhomogeneous background. By using the publicly-available Pluto code, which is a modular Godunov-type code intended mainly for astrophysical applications, we investigate the radial evolution of a spherically expanding shock-wave for a variety of large-scale density profiles. Implications for shock-wave propagation in the Heliosphere and Solar corona will be discussed.
SH31A-1659
Electromagnetic Fluctuations in the Dissipation Range of Solar Wind MHD Turbulence: Kinetic Alfven Waves or Whistlers?
Electromagnetic fluctuations in the inertial range of solar wind MHD turbulence and beyond (up to frequencies of 10Hz) have recently been studied for the first time using both magnetic field and electric field measurements on Cluster [Bale et al., PRL, 2005]. It has been shown that at frequencies above the spectral breakpoint at ~0.4Hz, in the so-called dissipation range, the wave modes become dispersive and are consistent with Kinetic Alfven Waves (KAW). This interpretation is based on the simple assumption that the measured frequency spectrum is actually a Doppler shifted wave number spectrum (ω ≈ k Vsw), commonly used in the solar wind and known as Taylor's hypothesis. While Taylor's hypothesis is valid in the inertial range of solar wind turbulence, it may break down in the dissipation range where temporal fluctuations can become important. In this work, we analyze the effect of Doppler shift on KAW as well as compressional proton whistler waves, and revisit Cluster solar wind data using this approach. We focus our analysis on low-beta (β < 1) ambient solar wind intervals. We first determine, both analytically and numerically, the dispersive properties of the KAW and the whistler wave modes and estimate the electric to magnetic field (E⊥/B⊥) ratio in the plasma and the spacecraft frame. Finally, we compare those estimates with the data directly in the spacecraft frame.
SH31A-1660
Solar Cycle Dependence of Spatial Correlation Lengthscales in the Solar Wind- in situ Turbulence and Coronal Signatures
The spatial correlation lengthscale is a key observable for quantitative modelling of fluctuations in the solar wind, in the context of either MHD turbulence or of propagating coherent structures. We present direct measurements of the spatial correlation lengthscale of solar wind magnetic field and ion density, obtained from simultaneous in-situ observations by multiple spacecraft. We focus on comparisons between different parameters and different phases of the solar cycle, which taken together clarify the respective roles of in-situ evolving turbulence and structure of more direct coronal origin. We calculate the spatial correlation properties of the solar wind in the ecliptic at 1AU using simultaneous in-situ observations by the ACE and WIND spacecraft. We present the first direct study of the spatial correlation lengthscale λ of solar wind ion density fluctuations, and find it to be smaller than that of the magnetic field. We find that there is the same statistically significant increase in λ, by a factor ≈ 2 from solar minimum to solar maximum for both density and magnetic field magnitude. In contrast, the λ of the individual components of the magnetic field, shows no discernible solar cycle variation. This behaviour is present both in the inertial range of turbulence and on larger scales. Our results suggest that long range correlation in ion density and |B| is of direct coronal origin, in contrast to that found in the B components, which is more strongly dominated by in situ evolving turbulence. The distinct correlation lengths of the density and magnetic field, together with their solar cycle variation, thus provide new quantitative insights into the mapping of coronal processes out into the solar wind. The lack of variation in the correlation length of the components of the magnetic field implies the continual presence of shear Alfvénic turbulence throughout the solar cycle. The difference between the correlation lengths of the magnetic field magnitude and the magnetic field components could indicate the relative scales of compressive versus shear Alfvénic fluctuations or, at larger temporal scales, different aspects of propagating coherent structures of coronal origin. These results contribute to our understanding of the interplay between in-situ turbulence and fluctuations of coronal origin, and also provide quantitative input for models of cosmic ray propagation within the heliosphere. We acknowledge the ACE and WIND magnetometer and SWE teams for data provision and the EPSRC and UKAEA for support.
SH31A-1661
Fluctuations in the solar wind that show scaling- MHD turbulence and coronal origin.
In- situ spacecraft observations of plasma parameters are at minute (or below) resolution for intervals spanning the solar cycle and provide a large number of samples for statistical studies. These observations reveal that the power spectrum of the components of magnetic field typically has two characteristic features, an inertial range of turbulence over several orders of magnitude with approximately Kolmogorov power law and at lower frequencies, an approximately '1/f' energy containing range believed to be of direct coronal origin. On the other hand, the (much lower energy density) magnetic field magnitude power spectrum typically shows a single scaling range that spans these timescales. This is consistent with the idea that the power seen in the components, but not necessarily the magnitude, of magnetic field is dominated by Alfvenic turbulence in the evolving solar wind. Here, we use quantitative statistical techniques to explore the idea that the solar wind exhibits fluctuations over a broad range of timescales characteristic of magnetohydrodynamic (MHD) turbulence evolving in the presence of structures of direct coronal origin. We find a strong correlation between the solar cycle variation in the scaling properties of magnetic energy density fluctuations and the magnetic complexity of the coronal magnetic fields. At solar maximum in the ecliptic, the magnetic energy density as seen by WIND and ACE shows a fractal signature, whereas at minimum it is multifractal. This is corroborated by ULLYSES polar observations at solar minimum in quiet, fast solar wind where again, multifractal scaling is found. High magnetic complexity in the corona then corresponds to fractal, rather than multifractal scaling in magnetic energy density seen at 1AU; remarkably, this fractal signature dominates the full dynamic range of observations, extending across timescales typically identified with both the '1/f' and 'inertial range'. Intervals when WIND and ACE simultaneously sample the solar wind also provide direct observations of the correlation lengthscale of these fluctuations; these show (i) distinct correlation lengthscales for magnetic field magnitude and components and (ii) the correlation lengthscale for magnetic field magnitude tracks the solar cycle whereas that of the components is insensitive to it. An important question which we will also address is then whether the observation of multifractal scaling per –se uniquely maps onto in- situ turbulence, or whether multifractal scaling, seen in the magnitude of magnetic field rather than components, is of direct coronal origin, with implications for our understanding of the heating of the solar wind.
SH31A-1662
Statistical Analysis of Solar Wind Turbulence at Solar Maximum and Solar Minimum
We have applied statistical methods to a study of turbulence in the solar wind using data from the SWE experiment on NASA's WIND mission during solar minimum (1996 and 2006) and solar maximum (2001). We examined the probability distribution functions (PDFs) for solar wind velocity differences at lag times from 1 to 105 minutes. The statistical features of these PDFs reveal information regarding the cascade of momentum from large to small scale structures as well as intermittency within the solar wind. We have analyzed the velocity differences using three component velocity data, allowing us to determine turbulent anisotropies and examine the PDFs for velocity differences parallel and perpendicular to the local magnetic field. At small time differences, the PDFs of bulk velocity differences have the form of a double exponential function. For increasing time differences, there is a shift in the mean of the PDF to negative velocity difference with a noticeable asymmetry as seen previously by Burlaga et al. (J. Geophys. Res., 107, 1403, 2002). However, the shift and asymmetry are absent when examining velocities parallel to the magnetic field suggesting anisotropy in the turbulence is found in the component perpendicular to the magnetic field. The second statistical moment (variance) of the PDFs are found to have different behavior in time scales less and greater than about 103 minutes. These two regimes each follow a separate power law scaling with the time difference. The regimes appear to be related to the Kolmogorov 2/3 and 4/5 laws that define the power law scaling for inertial and Gaussian regimes. The third and fourth statistical moments (skew and kurtosis) also follow power laws. Comparison between solar minimum and solar maximum shows general similarity in the results. However, at solar maximum only a single regime exists for the kurtosis, while at solar minimum two regimes are present, similar to the regimes found for the variance.
SH31A-1663
Solar Wind MHD Turbulence for Different Phases of the Solar Cycle
Recent results have shown that solar wind MHD turbulence can be described not only as a mixture of inward and outward stochastic Alfvénic fluctuations but includes also advected coherent structures, partially dominated by an excess of magnetic energy. The present study analyzes the role played by these two contributions for different phases of the solar cycle, comparing also different solar cycles. In addition, we show a study on the radial evolution of solar wind turbulence between 1 and 1.4 AU, obtained from a recent alignment between Earth and Ulysses occurred at the end of August 2007. This unusual event gave us the opportunity to analyze the same plasma sample at different observational points.
SH31A-1664
Modeling of the turbulent cross-helicity dissipation rate: Comparison using the solar-wind observations
The turbulent cross helicity (velocity--magnetic-field correlation of turbulence) W ≡ 〈u' · b'〉, as well as the turbulent magnetohydrodynamic (MHD) energy K ≡ 〈 u'2 + b'2 〉 / 2, is a quantity of primary importance which represents statistical properties of turbulence. The presence of the cross helicity in turbulence may alter the transport properties of turbulence, then it affects the magnitude and configuration of large-scale fields much. A typical example is the turbulent dynamo. If the cross helicity exists in turbulence accompanied by the large-scale vortical motions, electromotive force parallel to the vorticity is induced. This may counterbalance a huge magnetic diffusivity due to turbulence, and work for the magnetic-field generation. Although spacecraft observations of solar-wind turbulence have provided precious information on the turbulent cross helicity, their results have not been fully utilized in the studies of the MHD turbulence modeling. As for the dissipation rate of the turbulent cross helicity, εW, very little is known. This is in marked contrast with the dissipation rate of the turbulent energy, ε, whose model equation has long been discussed. We propose a few models for the turbulent cross-helicity dissipation rate εW: an algebraic model, a model equation for εW evolution, etc. Using comparison with the large-scale behavior of the cross helicity obtained by several solar-wind observations, we evaluate these models. The detailed observations by Roberts et al. (1987) inferred that in the absence of flow shear the turbulent cross helicity W remains to be relatively large value as the heliocentric distance increases. We will show that a turbulence model simulation with the algebraic model of εW can reproduce this W behavior with a reasonable model constant. Further discussions including the model equation for the εW evolution will be also presented.
SH31A-1665
Turbulent Cascade Rates and Geometries in the Solar Wind at 1 AU
Recent efforts that focus on the MHD extensions to Kolmogorov's 4/5 Law have shown that the interplanetary spectrum at 1 AU results from the intrinsically turbulent dynamic of the interplanetary medium. Cascade rates for energy flowing from large to small scales are in agreement with the rate of local heating as determined by the radial gradient of the thermal proton temperature. We continue this effort to place greater restrictions on solar wind type that refine our comparison with observations and we show that the transport of energy through the multi-dimensional spectrum leads to a 2D state that populates wave vectors perpendicular to the mean magnetic field. However, we also resolve a reduced cascade rate associated with moving energy to larger parallel wave vectors. The balance between these two cascades is examined in detail for fast and slow wind observations.
SH31A-1666
How to relate the 3D wavevector spectrum of Alfvenic fluctuations to the frequency spectrum observed by a single spacecraft when Taylor's hypothesis is not valid
A longstanding problem is to discover the nature of the three-dimensional (3D) wavevector spectrum of Alfvenic fluctuations in the solar wind. Although some progress has been made using structure function analysis and also the wave telescope technique, we currently have very little knowledge of the scale dependent anisotropy of the fluctuations in wavevector space, a quantity that is central to existing phenomenological theories of MHD turbulence. To make progress in this area, a simple method has been developed that allows the frequency spectrum in the spacecraft frame to be computed for any 3D wavevector spectrum in Fourier space. The technique is based on the well known formula for the doppler shift in a moving medium together with the random phase approximation of turbulence theory. Because the method does not rely on Taylor's hypothesis, it also applies when the Alfven speed is large compared to the solar wind speed, a circumstance that occurs close to the sun at heliocentric distances less than 20 solar radii or so (0.1 AU). Different model wavevector spectra are used to investigate the effects of wavevector anisotropy on single spacecraft measurements. It is shown, for example, that for typical solar wind and Alfven speeds at 1 AU, wavevector spectra that are anisotropic power laws with Goldreich-Sridhar-like scaling can produce spectral exponents in the spacecraft frame that appear unrelated to the power law behaviors in k-space. In particular, a wavevector spectrum with an Iroshnikov-Kraichnan-like spectral index in the perpendicular direction could be seen by a spacecraft observer as a 5/3 spectrum.