Do plasmas really not experience a $\nabla \vec B$ drift even though the individual particles do?

Question

I'm currently working through Chen's Introduction to Plasma Physics and Controlled Fusion, and I just got to the chapter on the fluid theory of plasmas. Chen claims that the $\nabla\vec B$ drift does not exist for fluids. I have trouble understanding this as each particle would experience this drift. Is this a limitation of the fluid theory of plasmas or do plasmas as a whole really not experience it, even though the particles individually do? If it is not a limitation of the theory, how does this make sense?

Answer 1

I finally read the part in Section 3.4 of Chen's book (Introduction to Plasma Physics and Controlled Fusion, Volume 1: Plasma Physics) to which you refer. I think he is implying the following though I must admit I do not like the hand-waviness of the argument. For reference, I have the 2nd Edition.

If you take a continuous, isotropic Maxwellian and evolve it with the Vlasov equation in the absence of electric fields and the magnetic field gradients are gradual (i.e., the gradient scale length is larger than the gyroradius of the particles), then the particle distribution will remain an isotropic Maxwellian. A $\nabla B$-drift will generate anisotropies in the distribution function (i.e., oblateness orthogonal the quasi-static magnetic field), thus changing the distribution function away from an isotropic Maxwellian.

Note that you must also neglect sources and losses in the evolution of the distribution function described above. It's a lengthy way of saying, rather obtusely, that stationary, static magnetic fields cannot do work on charged particles. Another part of this that is not shown for some reason is that in describing the dynamics, Chen does not explicitly delineate between kinetic and fluid. That is, he mentions the bulk fluid velocity (see the following for discussion of velocity moments https://physics.stackexchange.com/a/218643/59023) but does not really explain in detail why the fluid velocity would not show a $\nabla B$-drift. He provides a highly idealized, single-particle picture to explain the reason why the $\nabla B$-drift does not arise, but it seems like a highly selective choice of a physical region to perform the ensemble average over which to calculate the fluid moments.

The problem is that a flow of plasma incident on a magnetic field gradient will induce a $\nabla B$-drift, which illustrates one of the many major weaknesses in fluid models. Again, I think Chen is trying to state that if you calculate the fluid moments prior to examining any dynamics, then the $\nabla B$-drift will not arise.

One way to handle this if you are going to use a fluid approximation is to do something called gyroaveraging – ensemble time average over a gyroperiod for each species. This will give you guiding center motions and is a starting point for something called gyrokinetics. This method will elicit $\nabla B$-drifts.

In short, you are right to be confused by this discussion (assuming it has not improved in the 3rd Edition).

As an aside, I took a class that used this book once as well and found myself continually confused and frustrated by the numerous errors, typos, and opaque descriptions of processes. I would recommend the following books to supplement and provide, hopefully, better explanations and details:

• Plasma Confinement by Hazeltine and Meiss
• Introduction to Plasma Physics by Gurnett and Bhattacharjee
• Fundamentals of Plasma Physics by Bellan

Answer 2

Fluids do experience a $\nabla B$ drift, just like the single particle picture. This is shown in Section 4.5 of Hazeltine's textbook. Chen's hand-wavy arguments are misleading/incorrect in multiple instances (for example, I've heard people here at UW-Madison, and elsewhere, complain about the correctness of his "intuitive" explanation for Landau damping).

Hazeltine shows in that section that in the fluid case you get both $\nabla B$ and curvature drifts just like in the single particle picture; however, in the fluid case there is an additional drift which comes from the curl of the plasma magnetization for a given species divided by $n_sq_s$ (where $n_s$ is the species' number density and $q_s$ is the species' charge), $$1/(n_sq_s)\nabla\times \mathbf{M}_s\ .$$ You can think of the plasma (in the drift kinetic picture) as consisting of little spinning dipoles whose guiding centers are drifting. The spinning dipoles throughout the plasma result in a bulk magnetization of the plasma. Just like in the traditional E&M picture, the curl of the total magnetization will give a net current. Conveniently, when you add this extra flow in the fluid picture to that from the $\nabla B$ and curvature drifts, (and the parallel guiding center drift which Hazeltine derives has to be included as well), it exactly cancels just the right pieces to leave only the $\mathbf{E}\times\mathbf{B}$ flow and the diamagnetic flow.

The guiding center drifts are important in fluid theory whenever one is considering the divergence of the current density. The divergence annihilates the magnetization current (divergence of a curl is zero), leaving only the divergence of the current due to the guiding center drifts.