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

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?

• Be careful to verify that Chen is not implying this drift is ignored for "one-fluid" approximations, not "two-fluid" approximations. The difference is that the typical MHD approximation is just a single particle species, magnetized fluid. The grad-B drift has a charge dependence and so requires very special treatment if you ignore multiple species (e.g., could make the kludgy approximation of being in one particle species rest frame but even that has serious problems here). – honeste_vivere Feb 10 at 17:51

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:

• Thanks so much for the answer and book suggestions! I'll take a look and likely switch. – Eublepharis Feb 12 at 16:59