Smaller Scales: Five Physical ProcessesWaves and Turbulence

Consider the surface of a stream or river. We will often see surface waves propagating and experiencing convection, typically generated by some form of disturbance; mixed in are non-propagating fluctuations (a-b).

The solar wind is very similar, with an admixture of waves and non-propagating fluctuations, all of which interact in a highly non-linear fashion. Fluctuations in the solar wind are present on all measurable time scales, from the solar rotation period (27 days) to millisecond electrostatic noise (c).

c.
(Image - 104k)
The solar wind admits fluctuations on almost all conceivable scales. This series of plots illustrates the point by showing solar wind magnetic field data that has been averaged using four different time intervals.

The top left plot shows ACE data at 1 AU using 8-hour averaging. Large fluctuations are always present even at these scales for the full year of 2000.

The bottom left shows smaller scales using a 1-hour averaging and one sees as much structure as at the larger time scale.

The top right plot now uses 10-minute averaging and high frequency fluctuations are still present.

Finally, in the bottom right plot, going to the highest frequencies using 1-minute averaging, we see that fluctuations persist on these very small scales. Note too that the amplitudes of the fluctuations do not become smaller as the scales become finer.

Let us consider only periods less than one hour, so that we neglect stream structure. Most of the fluctuation energy in the solar wind appears at frequencies below the proton cyclotron frequency.

Proton Cyclotron Frequency:
The proton cyclotron frequency is the number of times per second that a proton orbits a magnetic field line. The frequency is completely determined by the strength of the field and the proton's charge-to-mass ratio.

We can measure energy in the fluctuations as a function of frequency or wavelength, or inverse wavelength.

Fluctuations in the solar wind are essentially incompressible. This means that the density of fluctuations in the solar wind does not change much in response to fluctuations in other variables, such as magnetic field. Thus, at these scales, the solar wind is nearly incompressible.

d.
(Image - 60k)
From observations of the solar wind density, we can plot the ratio dN/N of the amplitude of the density fluctuations dN to the mean or average density of the solar wind. By considering numerous 45-minute intervals, we can plot a histogram of dN/N. Similarly, we can construct a histogram of the Mach number for fluctuations in the solar wind. The results are shown in these figures. The "typical" solar wind fluctuation is less than 10% of the mean density, which indicates that the solar wind is nearly incompressible or only weakly compressible. The Mach number of the corresponding fluctuations is typically 1/10 of the background sound speed. Thus, fluctuations, in addition to being nearly incompressible, are also highly subsonic. Although somewhat surprising at first sight, we find that the solar wind can behave very much like an incompressible fluid.

Fluctuations observed in the solar wind by spacecraft are dominated by convection. As a wave is swept past a spacecraft, its frequency can be measured. However, because the wave is being convected in the highly supersonic solar wind, the frequency is dominated by the convection speed, rather than the wave speed. This means that information about the waves themselves, including their direction of propagation, is very difficult to extract.

Because of this, we need a well-developed theory of waves in the solar wind to understand the nature of fluctuations observed in the heliosphere.

Let's look at low frequency fluctuations in the solar wind, that is, from 10-3 to 10-1 Hz. In this case we can treat the solar wind as a tenuous fluid threaded by a magnetic field, which we call a magnetohydrodynamic, or MHD, description of the solar wind.

e.
(Image - 48k)
The spectral density of the power in the interplanetary magnetic field magnitude (B_mag) and in the trace of the magnetic field as a function of frequency at 1 AU. Fluctuations are clearly present on all frequency scales, ranging from very low to very high. Plotted in a log-log format, parts of the spectrum lie on a straight line indicating a power law dependence of the spectral density with frequency. Figure courtesy of C.W. Smith.

MHD yields three types of waves and two types of non-propagating structures.

Non-propagating structures can have constant solar wind plasma pressure and/or magnetic field pressure. Of the waves, the most important in the solar wind is the Alfvén wave (f), in which the velocity and magnetic field are correlated, while all other variables remain constant.

f.
(Image - 28k)
The propagation of Alfvén waves, illustrating the correlation of magnetic field and velocity fluctuations

The two other waves are versions of the usual sound wave in hydrodynamics, but now are influenced by the magnetic field. These are called fast and slow mode magnetoacoustic waves (g), because they are compressive and they travel faster or slower than the Alfvén speed. Fast and slow modes are easily damped and disappear.

g.
(Image - 40k)
Fast and slow magnetosonic waves, illustrating the relationship between P, plasma pressure perturbations, and B, magnetic pressure perturbations.

Most wave-like fluctuations in the solar wind are Alfvénic because correlations between velocity and magnetic field components are often seen, together with a constant density and field magnitude (h).

h.
(Image - 80k)
Alfven waves in the solar wind [Belcher and Davis, 1971]. The top three panels each show a different orthogonal component of the solar wind magnetic field and the associated (fluid) velocity component of the plasma. The bottom panel displays the magnitudes of the total magnetic field and flow speed. Notice the extraordinarily close correspondence between magnetic field and velocity fluctuations.

Where do Alfvén waves originate? In the inner heliosphere, waves are often seen propagating outward, but not inward, which implies that the Sun's atmosphere is the source of the Alfvén waves. The random direction of the waves below the Alfvén critical point implies that only outward propagating waves survive; in the super-Alfvénic region, all waves are convected out.

Although the Alfvén wave picture is attractive and tractable, it fails to explain many of the features of low-frequency fluctuations observed in the solar wind. For one, Alfvén theory predicts that the wave vectors will tend toward the radial direction but, in practice, this is not observed. Second, Kolmogorov-like spectra are a pervasive feature of low frequency fluctuations, which simply cannot be explained in the context of non-interacting Alfvén waves. Third, the solar wind is heated non-adiabatically, which cannot be explained by non-dissipative Alfvén waves.

Many of these issues are better understood in the context of turbulence. Non-linear stresses act to couple fluctuations so that energy is transported from large-scales to small-scales, generated initially by "stirring." In the solar wind, "stirring of magnetic fluid" can result from the interaction of streams, the propagation of shocks through the solar wind, or even the relaxation of beams or the pickup of interstellar or cometary neutral atoms.

i.
(Image - 48k)
A convenient way of plotting fluctuations is as a power spectrum. In this case, we plot fluctuations as a function of either the frequency or the wave number, k. Density, velocity, temperature, and magnetic field fluctuations are often plotted as a spectrum. Typically, the spectrum, especially that of the solar wind magnetic fluctuations, exhibits a broken power law shape, with a flatter spectrum at low k and a steeper spectrum at higher k. The flatter part of the spectrum is called the "energy containing" range and the steeper section the "inertial range." The inertial range frequently is Kolmogorov-like, in having a -5/3 power law index, which is what we expect of a turbulent incompressible fluid. These figures show the close similarity between the density and magnetic field fluctuation spectra-a consequence of nearly incompressible MHD in which weakly compressible fluctuations are enslaved to or generated by incompressible fluctuations associated with the magnetic field.

Turbulence is important for several reasons, including heating via dissipation of the solar wind, cosmic ray scattering and transport, and energy content. The current picture of magnetic fluctuations in the solar wind is that the fluctuations are a superposition of nearly incompressible slab fluctuations and 2D fluctuations.

j.
(QuickTime - 1.19M)
A cartoon animation of the current paradigm for turbulence in the solar wind. Based on the theory of nearly incompressible MHD, turbulent fluctuations in the solar wind are a superposition of "slab" turbulence (where Alfvén waves propagate parallel and anti-parallel to the interplanetary magnetic field and where wave number is parallel to the large-scale magnetic field) and "2D" turbulence (2D fluctuations which reside and interact in planes orthogonal to the interplanetary magnetic field and where wave number is perpendicular to the large-scale magnetic field). We find, theoretically and observationally, that about 80% of the solar wind fluctuations are 2D and the remainder slab turbulence. Riding parasitically on top of the incompressible superposition of 2D and slab turbulence are magnetosonic waves generated by the turbulence (analogous to the generation of sound by wind when it howls around a house).

The small-scale picture of the magnetic field is not nice and smooth, but instead very complicated.

 k. (Image - 36k) l. (Image - 52k)
The left image shows magnetic field lines shredding and diffusing in the presence of turbulence. The right image shows magnetic field lines when only Alfvén waves are present. Note the coherent character of all "magnetic flux surfaces."

Surface plot movies showing the evolution of (a) the vorticity--a measure of the local rate of rotation of the plasma, and (b) the electric current density. The movies were created using results from a computer simulation of the 2D MHD turbulence with Reynolds numbers of 800. MHD is the marriage of the equations of fluid dynamics with those of electrodynamics, and provides a good approximation to the behavior of various parts of the heliosphere. Important dynamical features of MHD include waves, turbulence, plasma heating, and particle acceleration. Note how vortices with the same sense of rotation tend to "collide'' and merge, leading to a smoothing of the vorticity. The merger process often involves a larger vortex shredding a smaller one that is wrapped around the larger one and eventually absorbed into it.

In the current density movie the dominant structures are seen to be much more sheet-like than in the case of the vorticity. In the heliosphere, similar (but more intense) ``current sheets'' are sites where particles are accelerated to relativistic energies and intense heating of the plasma can occur.

 m. (QuickTime - 52k) n. (QuickTime - 52k) o. (QuickTime - 52k) p. (QuickTime - 52k)