When solar observers attempt to explain the dynamics of solar eruptions, they often rely on the concept of magnetic pressure, e.g., there is a force associated with the gradient of the magnetic pressure pointed from the region of high pressure to the region low pressure. In contrast, laboratory plasma experimentalists prefer to use currents and the Lorentz force to analyze plasma dynamics. The choice between studying magnetic elds and studying currents appears to be one of convenience.
Solar physicists are unable to probe the sun to measure the currents so they rely on magnetic data through spectroscopic eects like the Zeeman eect. Experimentalists have direct control over many of the currents associated with the system and use those currents to recreate the necessary magnetic boundary conditions.
In theory, both the J×B approach and magnetic pressure (B2) approach are equivalent. In practice, force calculations by way of magnetic pressure are indirect and the user must take care to properly assess the geometry. Consider the magnetic conguration shown in Fig. B.2 (a). The magnetic eld is along the xˆdirection and is described to be zero fory <0and nite and positive fory >0. Through Ampere's law,
J×B = 1
µ0(∇ ×B)×B = 1 µ0
−∂Bx
∂y
ˆz×Bxxˆ =−µ0 2
∂
∂y Bx2 ˆ y
The force associated with the pressure clearly points in the −ˆy direction, but there appears to be no obvious currents, and thus no J×B force. In reality, there is no such thing as an innite slope like the one shown at y= 0 in Fig. B.2 (a), and if one zooms into a small region of size ε around y = 0 as shown in Fig. B.2 (b), one nds a nite slope corresponding to a nite −∂Bx/∂y which represents a thin layer of current owing along the zˆ direction. From the J×B point of view, the force associated with the magnetic conguration is determined by currents owing in the thin region around y= 0, thus reducing the study of the magnetic conguration into the study a localized region. Similarly, if the magnetic eld were inverted like shown in Fig. B.2 (c), both the ∂Bx/∂y and Bx terms would ip polarity; theJ×B approach and the magnetic pressure approach continue to give the same force.
The essential physics is contained in the behavior of the localized region over which currents ow, so both approaches are identical when given perfect information.
In practice, scientists have a dicult time measuring magnetic eld in the corona and solar models extrapolate the coronal magnetic eld from photospheric magnetic measurements [32, 58]. The imprecise spatial resolution and uncertainty about mag- netic measurements means that important local regions capturing key dynamics may be overlooked. Even scientists who advocate strongly for thinking about magnetic pressure recognize that the dynamics of huge phenomena such as CMEs may be controlled by detailed plasma processes that occur in relatively tiny regions [9].
Appendix C CME models
Reviews of CME models can be found in the works of Low [63], Forbes et al. [112], and Chen [29]. Certain textbooks by Aschwanden [74] and Crooker et al. [113] are also excellent resources.
C.1 CSHKP are model
The Carmichael-Sturrock-Hirayama-Kopp-Pneuman (CSHKP) are model is a 2D reconnection model which describes the evolution of a are along a vertical plane.
While no single model describes all possible ares, the CSHKP model ts most of the observations and is accepted as the standard model of ares. In particular, it explains the power source of the ares, the coronal streamer structure, the rising prominence, the brightening of the chromospheric footpoints, the are signatures in X-ray, EUV, and Hα, and the increased height and footpoints separation of the mag- netic structure. The standard model presents the physical mechanisms behind these observations, which include the solar wind energy source, magnetic reconnection, syn- chrotron radiation, shock acceleration, and Joule heating. A representative summary from the four papers is shown in Fig. C.1.
Carmichael [68] introduced the mechanical energy of the solar wind as the power source of solar ares: he suggested that the solar wind deposits magnetic energy into the are. This deposited energy is stored in the form of twisted eld lines (see Fig.
C.1 (a). Flares and the events that accompany ares are the manifestation of the
LaTnCarmichael-n1964 LbTnSturrock-n1966
LcTnHirayama-n1974 LdTnKoppnNnPneuman-n1976
CSHKPnFStandardFnFlarenModel
CORONALnSTREAMER
YUTYPEnNEUTRALnPOINT
Figure C.1: CSHKP phenomenological models for ares.
occasional release of this stored energy.
Sturrock [69] stressed the importance of coronal streamers (aka helmet streamers) containing ux of opposite polarity. The pinching of the current sheet above the Y-type neutral point results in a tearing-mode instability which reconnects magnetic eld lines. This reconnection leads to the development of strong electric elds which accelerate both electrons and protons. Plasma1 is sling-shot into the upper portion of the streamer, resulting in higher energy energy particles and shock waves, while some electrons (which are caught in the closed eld lines) accelerate towards the surface of the sun. The shocks from the ejected plasmas are expected to produce Type II solar radio bursts by kilovolt electron acceleration at the shock front2. The deection of sun-directed electrons by a magnetic mirror mechanism produces Type IV radio burst through synchrotron radiation. Fig. C.1 (b) is a summary of this process.
Hirayama [70] proposed that the pre-are process can be attributed to a rising prominence which leaves a magnetic cavity in its wake. The compression of this cav- ity from both sides (see Fig. C.1 (c) Maximum phase) near the X point generates Joule-heated downward ow towards the chromosphere, where the ows brighten the chromospheric footpoints and evaporate. These evaporated protons ll the newly reconnected eld lines with dense, heated plasma capable of producing soft X-ray- emitting are loops. These X-ray are loops cool through thermal conduction and radiation until they are detectable as EUV ares and eventually as two-ribbon Hα ares.
Kopp and Pneuman [71] proposed that the post-are loop prominence systems are the closing of magnetic eld lines which were torn open by are outbursts. They examined the cooling process of material supplied to the prominence region by en- hanced solar wind outow along open eld lines. They found threes stages of cooling (See Fig. C.1 (d): Condensation and infall of cool material) which correspond to loop structures in X-ray, EUV, and Hα. Their mechanism predicts the continuous rise of
1The plasma contains almost all the protons and the majority of electrons.
2This is similar to electrons accelerated in the Earth's bow shock.
the Y-type neutral point, which means that newly connected eld lines have larger height and wider footpoint separation.
While the CSHKP model can explain what happens during and after a are, it does not discuss what causes the system to go unstable, i.e., it does not discuss why a prominence arises in the rst place. The answer to that question is intimately related to the study of CMEs. Modern models have adapted CSHKP to t observations from new satellites (e.g., Yohkoh, SOHO, SDO) and advances in magnetic reconnection physics.