My current work, in collaboration with Aad van Ballegooijen, has produced some potentially useful plots of "generic" 1D plasma properties of the high-speed solar wind (corresponding to polar coronal holes at solar minimum), including detailed non-WKB models of damped Alfven-wave velocity amplitudes versus distance.
The main paper containing the results of this work was published by Ap. J. Suppl. in early 2005; see Cranmer and van Ballegooijen (2005). An earlier version of this work (without the non-WKB effects and damping) was published in the proceedings of the SOHO-13 Workshop (see preprint at astro-ph/0309676). Progress reports were given at the Fall 2003 AGU meeting (abstract) and the Spring 2004 AAS/SPD meeting (abstract).
Here we give a few summary figures. First is a summary "cartoon" of the magnetic field structure assumed in the models. Note the successive merging of flux tubes on granular and supergranular scales, which is important to include in modeling the physics of waves:
(a) Intergranular lanes host G-band bright points (i.e., kiloGauss-strength thin flux tubes) that are shaken transversely to generate kink-mode waves.
(b) Above the height where individual flux tubes merge (about 600 km), the supergranular network field is mainly open (with a funnel/canopy structure), and kink-mode waves are transformed into Alfven waves.
(c) Non-WKB waves in the solar wind propagate and reflect depending on their frequencies, and MHD turbulent cascade can occur where outward and inward waves interact nonlinearly. (Inverted solar image from EIT/SOHO).
Next we give some steady-state plasma speeds along the central axis of an empirically constrained superradial flux tube, from the photosphere to 4 AU. For reference, the transition region in the plot below is at a dimensionless height of 0.003. The orbit of the Earth (1 AU) is at a dimensionless height of 214.
The magnetic field B was computed below 1.02 solar radii with a 2.5D magnetostatic model of expanding granular and supergranular flux tubes. Above 1.02 solar radii, the magnetic field was adopted from the solar minimum model of Banaszkiewicz et al. (1998). The number density n was specified empirically from VAL/FAL models (at low heights) and white-light polarization brightness measurements (at large heights). We use B and n to derive the Alfven speed and the solar wind outflow speed (using mass flux conservation for the latter, normalized to Ulysses mass flux measurements). The sound speed above comes from the VAL/FAL models (at low heights) and an empirical fit to UVCS/SOHO empirical models and in-situ proton temperatures (at large heights). The modeled Alfven wave amplitude dV is described below.
Note that the solar wind goes supersonic at a heliocentric distance of about 2 solar radii, and goes super-Alfvenic at about 10 solar radii.
Next we show more details about the modeled Alfven wave velocity amplitude as a function of distance:
There are a large number of observational "data points" plotted here for comparison:
The solid lines show the transverse amplitudes of modeled non-WKB Alfven waves (black: velocity amplitude, red: magnetic field amplitude in velocity units). Frequency-dependent linear wave reflection is included self-consistently, and the amplitudes shown here are integrated over a full radially-dependent power spectrum. Note that in non-WKB models the kinetic and magnetic energy in Alfvenic fluctuations is not in equipartition, as can be seen most strongly around the transition region. The agreement between the red dB curve and the on-disk SUMER data points may be coincidental, but it may contain empirical constraints on possible mode coupling between transverse and longitudinal waves in the transition region.
At the photosphere, the power normalization is performed more or less empirically from an observed power spectrum of G-band bright point motions (e.g., van Ballegooijen et al. 1998; Nisenson et al. 2003). (One free parameter relating to occasional flux-tube merging/fragmenting allows us to adjust the overall normalization slightly, but not arbitrarily!)
Nonlinear turbulent damping is included in the corona using the dissipation rate given by, e.g., Dmitruk et al. (2002), and there is one additional free parameter in how the correlation length scale (i.e., the outer "stirring" scale length) is normalized. The value that fits the in-situ data best corresponds to a transverse length scale of about 1000 km in the middle chromosphere, which scales with radius as the square root of the magnetic field strength (e.g., Hollweg 1986).
The above references should be easy to find, but all of them are given by Cranmer and van Ballegooijen (2005). For further information, see other recent papers.
If there are any questions, just send me email at: scranmer @ cfa.harvard.edu (remove spaces to use address; an attempt at anti-spam)
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