The Kelvin Force and Loop-Top Concentration

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Number: 307
1st Author: Kiyoto Shibasaki
2nd Author:
Published: 18 September 2017
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The Lorentz force causess charged particles in a magnetized plasma to gyrate in a plane perpendicular to the direction of the field. This consequence is easy to understand if the Larmor frequency is greater than the collision frequency; the particle will move in a circle in 2D, or in a helical path in 3D. In the absence of strong gradients this gyrational motion is the first of the three adiabatic invariants of charged particles in a plasma, an extremely convenient concept for understanding the Van Allen Belts, for example.

But what happens when the collision frequency increases and the plasma becomes a fluid, in the MHD approximation to plasma kinetic physics? Does the Lorentz force cease to be important? This question underlies the important Bohr-Van Leeuwen theorem, first reported in the independent PhD theses of Niels Bohr and Hendrika Van Leeuwen. According to this theorem, the bulk magnetism of a solid cannot be explained by classical physics and must have a quantum-mechanical explanation.

The theorem, based on statistical mechanics and classical physics, asserts that the net magnetism of a body in thermal equilibrium simply vanishes. Thus, quite surprisingly, any magnetic effect on an equilbrium solid body (diamagnetism, paramagnetism, ferromagnetism) requires an explanation based on quantum physics. How does this result apply to an inhomgenous plasma? It would be safe to say that these concepts do not often get addressed in practical research affairs in solar physics. But what if they were decisively important? In a magnetic fluid the a fluid approximation (MHD), this is called the Kelvin force. In a plasma this represents the bulk force resulting from the magnetic moment associated with the gyrations of the individual particles. In this Nugget we describe a possible application of the Kelvin force to solve a perplexing problem that arose largely from the soft X-ray observations of the Yohkoh SXT soft X-ray telescope: what supports loop-top concentrations of plasma in solar flares?

The Observations

The loop-top concentrations in solar flares can be seen in a number of ways: soft X-rays (commonly), microwaves (Ref. [1]), and white light. Figure 1 provides an illustration that is little known or appreciated: a "white light prominence" (Ref. [2]). This rare kind of event is hard to observe, but probably also very common; typically one sees loop-top concentrations better at wavelengths with high image contrasts. White-light views such as this one are hampered by the glare of the photosphere, but the emission mechanism is probably the same as for a coronagraphic white-light view: Thomson scattering, and therefore a direct measure of column density. What we see seems like a large mass elevated mysteriously to coronal heights by the flare. Such a concentration is inconsistent with hydrostatic equilibrium.

Figure 1: A "white-light prominence," (right) in which Thomson scattering provides direct evidence for plasma concentration at the tops of coronal loops (Ref. [2]). Much later in the flare development, a typical Hα loop prominence system develops, but the process of overdense loop formation is continuous and may continue for hours at higher and higher altitudes.

The Mathematics

We do not present a detailed derivation here, but note that the Larmor motion of the constituent particles produces a magnetic moment M and results in a net magnetic field B = μ0(H + M), where B is the magnetic induction and H the magnetic field strength as spelled out in Maxwell's equations. The M term is not present in classical MHD theory and, in the real world, must be included. This leads to the Kelvin force


and then to a revised formula for the MHD equation of motion:


where the terms have their usual meanings; α is the frictional force and F represents any other forces in the system. To put this in perspective we can compare it with the gravitational force very simply: if the vertical gradient of the magnetic field exceeds the gradient of gravity, it is upward and tends to levitate the plasma. Because of this it provides a simple explanation for the observed loop-top concentrations, noted recently by Ref. [4] with the comment "The origin of these brightenings and their dynamics is not well understood to date." The sketch in Figure 2 illustrates the situation.

Based on this, we have simulated a simple dipole flux tube, as shown in Figure 2, and find that with the inclusion of the Kelvin force the excess mass concentration at loop top results naturally. Note that we assume a low [plasma beta] so that the additional force does not alter the shape of the dipole flux tube. The result is consistent with the many observations of loop-top structures in soft X-ray images, for example the Yohkoh results in Ref. [3].

There are of course many other possible applications of this revision to the MHD equation of motion.

Figure 2: Toy model of a dipole magnetic field (left), with one flux tube identified; on the right, the plasma concentration resulting from adding the Kelvin force. Without this the flux tube would be in hydrostatic equilibrium. In this simulation the hydrostatic scale length is set to 10.


The Kelvin force represents the mirror action of a gradient in the magnetic field of plasma volume. It has long been neglected for subtle reasons, including the influential but frequently misunderstood Bohr-van Leeuwen theorem. The magnetic moment is one of the most important properties of a plasma. It influences plasma dynamics not only along the magnetic field by the Kelvin force, but also perpendicular to the magnetic field by the Lorentz force. A magnetic moment does exist even in a highly collisional and thermally relaxed plasma. We have described one application, but there are others.

The Kelvin force dominates the gravitational force wherever the vertical gradient of B exceeds the gradient of gravity; this is almost everywhere in the corona, since gravity originates at Sun center and the coronal magnetic field mainly via a multipole structure originating near the photosphere.


[1] "Loop-Top Nonthermal Microwave Source in Extended Solar Flaring Loops"

[2] "White-light flare observed at the solar limb"

[3] "The bright knots at the tops of soft X-ray flare loops: Quantitative results from Yohkoh"

[4] "On the Bright Loop Top Emission in Post-eruption Arcades"

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