Curvature and grad-B drifts

[henry2004y]

From the fluid theory, the curvature drift exists while the grad-B drift does not. Are we able to demonstrate this with our test particle model?

[TCLiuu]

It is very interesting, I am not familiar with the fluid theory. But according to the kinetic theory, the effect of grad-B drift has an influence on the distribution function and we call it kinetic effect (include curvature drift, grad-B drift and FLR).

I have found a thread about this question.

[TCLiuu]

I found two papers relating to this question.
Paper 1: calculating the drift flow by gyrokinetic moments.
Paper 2: also gyrokinetic model.

[henry2004y]

It is very interesting, I am not familiar with the fluid theory. But according to the kinetic theory, the effect of grad-B drift has an influence on the distribution function and we call it kinetic effect (include curvature drift, grad-B drift and FLR).

I have found a thread about this question.

The linked thread is slightly biased, but I get his point. In the fluid theory, which I quoted from Chen's book, there is a strong assumption

If f(v) remains Maxwellian in a nonuniform B field, and there is no density gradient, then the net momentum
carried into any fixed fluid element is zero.

Obviously even if only $\nabla B$ exists, an initially Maxwellian distributed plasma will be distorted since

$$ \mathbf{v}_{\nabla B} = \pm \frac{1}{2}v_\perp r_L\frac{\nabla B\times\mathbf{B}}{B^2} \propto v_\perp^2 $$

However, I think this will NOT generate anisotropies in the distribution since it is symmetric with $|v_\perp|$? If there are still the same number of particles with the same $|v_\perp|$, then Chen's argument is still valid?

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I haven't read those two papers. Do you have a quick scan of what their conclusion is about our question @TCLiuu ?

[TCLiuu]

It is very interesting, I am not familiar with the fluid theory. But according to the kinetic theory, the effect of grad-B drift has an influence on the distribution function and we call it kinetic effect (include curvature drift, grad-B drift and FLR).
I have found a thread about this question.

The linked thread is slightly biased, but I get his point. In the fluid theory, which I quoted from Chen's book, there is a strong assumption

If f(v) remains Maxwellian in a nonuniform B field, and there is no density gradient, then the net momentum
carried into any fixed fluid element is zero.

Obviously even if only ∇B exists, an initially Maxwellian distributed plasma will be distorted since

v∇B=±12v⊥rL∇B×BB2∝v⊥2

However, I think this will NOT generate anisotropies in the distribution since it is symmetric with |v⊥|? If there are still the same number of particles with the same |v⊥|, then Chen's argument is still valid?

I haven't read those two papers. Do you have a quick scan of what their conclusion is about our question @TCLiuu ?

To summarize, from the viewpoint of gyrokinetics, the fluid perpendicular current in isotropic plasmas, $c\mathbf{b}\times\nabla p_{\perp}/B$, contains both the diamagnetic current and part of the current caused by the gradient drift. The other part of the current caused by the gradient drift cancels the current generated by the curvature drift for isotropic distribution functions. From the viewpoint of gyrokinetics, the diamagnetic current is $c\nabla\times (p_{\perp}\mathbf{b}/B)$ actually.

[TCLiuu]

We can calculate the perpendicular current by the method of moment integral. It can be write as:
$$\mathbf{J}_\perp =\mathbf{J}_d+\mathbf{J}_M= \int e(\dot{\mathbf{X}}+\dot{\mathbf{\rho}})\delta(\mathbf{X}+\mathbf{\rho}-\mathbf{r})F(\mathbf{Z})\mathrm{d}^6\mathbf{Z}$$
$$\mathbf{J}_M= \int e\dot{\mathbf{\rho}}\delta(\mathbf{X}+\mathbf{\rho}-\mathbf{r})F(\mathbf{Z})\mathrm{d}^6\mathbf{Z}$$
$$\mathbf{J}_d= \int e\dot{\mathbf{X}}\delta(\mathbf{X}+\mathbf{\rho}-\mathbf{r})F(\mathbf{Z})\mathrm{d}^6\mathbf{Z}$$

We can separate naturally the perpendicular current to two terms (the so-called diamagnetic drift term and the guiding center drift term). The part of diamagnetic drift term will cancel the effect of the grad-B drift and reduce to the result of the fluid theory.

[TCLiuu]

Recently, I have been preparing for this part of the content and therefore studied it again. I think ffchen's statement here is not very rigorous.

If considering anisotropic plasma, then the fluid velocity in the direction perpendicular to the magnetic field should be written as

where

And the actual requirements that need to be met is ignoring the parallel flow, that is $u_\parallel$≈0. If this item is not ignored, there will be a corresponding curvature drift.

After ignoring the parallel flow, according to my previous post, magnetic field gradient drift and curvature drift still exist, but all the gradient drift parts and some of the curvature drift parts and diamagnetic drift parts are cancelled out. In isotropic plasma, only the remaining diamagnetic drift part can be seen, while in anisotropic plasma, we can still see curvature drift. This does not mean that there is no gradient drift. From the perspective of kinetic theory, it is clear that diamagnetic drift is a result of FLR effect. As for FFChen's discussion on distribution, I don't think it's rigorous enough but it's acceptable in introductory courses.

I strongly recommend that you take a look at paper2 mentioned above, which clearly demonstrates the entire process of this issue from gyrokinetic theory to MHD.

I am curious about your current understanding of this issue.

[henry2004y]

My current understanding of this issue is based on Section 3.6 in Paul Bellan's Fundamentals of Plasma Physics: Relations of Drift Equations to the Double Adiabatic MHD equations. He showed that by taking the diamagnetic current, grad B current, curvature current, and polarization current into account, the drift equations for phenomena with characteristic frequencies $\omega << \omega_{ci}$ and the double adiabatic MHD equations are equivalent descriptions of plasma dynamics.

[henry2004y]

Updates from Gemini:

In the context of standard fluid theory (specifically single-fluid MHD), the statement is true regarding the explicit appearance of drift terms in the macroscopic momentum equation.In the single-particle picture, both curvature drift ($\mathbf{v}c$) and grad-B drift ($\mathbf{v}{\nabla B}$) are distinct, explicit drift velocities for a guiding center. However, when you derive the fluid equations by taking moments of the Vlasov equation (averaging over a distribution of particles):
Curvature Drift: This effect survives the averaging process and appears explicitly in the fluid momentum equation as part of the inertial term (specifically the convective derivative $\rho \mathbf{v} \cdot \nabla \mathbf{v}$). It corresponds to the macroscopic centrifugal force.
Grad-B Drift: This term cancels out or is "absorbed" when you sum over a Maxwellian distribution. The single-particle $\nabla B$ drift effects are mathematically accounted for by the magnetization current term ($\nabla \times \mathbf{M}$) and the perpendicular pressure gradient ($\nabla P_\perp$). Therefore, it does not appear as a distinct "drift velocity" term in the fluid center-of-mass velocity equation, unlike the $\mathbf{E} \times \mathbf{B}$ or diamagnetic drifts.

We can demonstrate this distinction using a test particle model, though it requires comparing single-particle trajectories with ensemble averages.Single Particle Level: You can use TestParticle.jl to trace individual charged particles in a vacuum field (e.g., a magnetic dipole or a toroidal field). The code will correctly reproduce both $$\mathbf{v}c$$ and $$\mathbf{v}{\nabla B}$$ in the particle trajectories. The existing examples in the documentation (specifically "Grad-B drift" and "Curl-B drift") already demonstrate these individual effects.
Fluid Level Demonstration: To demonstrate the fluid theory conclusion, you would need to initialize a Maxwellian distribution of particles (an ensemble). By computing the bulk velocity (center of mass velocity) of this ensemble, you would observe that the net macroscopic flow matches the fluid prediction (where explicit $\nabla B$ drift is absent/cancelled by magnetization effects), differing from what one might expect if they simply summed the guiding center drifts without correcting for magnetization current.