Drift kinetic equation orderings

I would like to start by saying that I really appreciate this question. I have struggled to find a clear explanation of the ordering used to derive "The Drift-Kinetic Equation". I too tried to work through 6.5 in Helander and Sigmar but was disappointed by their hasty explanation of the ordering.

In the business of “ordering” it’s hard to be very rigorous unless you know exactly what to do (and I do not). So with that caveat, here is my admittedly very heuristic answer to your question:

For completeness this is the starting point,

$$frac{partial f}{partial t}+dot{mathbf{R}}cdotnabla f+dot{mathscr{E}}frac{partial f}{partialmathscr{E}}+dot{mu}frac{partial f}{partialmu}+dot{vartheta}frac{partial f}{partialvartheta}=C(f)tag{1}$$

Energy Term

The third term on the LHS of eq. (1) (term proportional to $dot{mathscr{E}}$) can be ordered using the assumption that time-variations of quantities are driven by diffusive processes. This amounts to substituting:

$$frac{partial}{partial t} rightarrow nudelta^2$$

(see eq. 8.2 in H&S and the surrounding text.)

One might also be able to justify this ordering using equipartition of energy to relate energy to temperature ($T$) by writing $mathscr{E} sim T$.

Then, using material from Chapter 1 of Helander and Sigmar (H&S),

$$mathbf{q} = -kappanabla T hskip{1cm}text{with}hskip{1cm} kappa propto nu_{ii}rho_i^2$$

and from Eq. 2.17 in H&S,

$$frac{dT}{dt} propto -nabla cdot mathbf{q}$$

Therefore,

$$frac{dmathscr{E}}{dt} sim frac{dT}{dt} sim nabla cdot(nurho^2nabla T) sim nurho^2/L^2 = nudelta^2.$$

This is what H&S essentially claims. I will admit, this is not exactly correct since I simply replace the gradient with $1/L$ instead of finding a factor of $1/L_B = |nabla ln B|$ as per H&S's definition of $delta = rho/L_B$. But H&S is actually inconsistent in its use of $L$ vs. $L_B$ (see for example Ch. 8) so I believe they are interchangeable.

Drift Term

To order the "drift term" in eq. (1), i.e. $dot{mathbf{R}}cdot nabla f$. I think we need an additional piece of information which is not given to the reader explicitly. I believe we are meant to take the guiding center velocity as approximately equal to the thermal speed, $dot{mathbf{R}} sim v_T$.

I do expect this to hold in the parallel direction given the approximation: $dot{mathbf{R}}_parallel approx v_parallel$ which is given to the reader, but I do not expect this to be true in the perpendicular direction. In fact, the g.c. velocity in the perpendicular direction we have an additional factor of $delta$ as per the "transport ordering" discussed in H&S Ch. 8 i.e. that macro-flows are suppressed by a factor of $delta$ when compared to the thermal speed, $V sim delta v_T.$

With all this, the desired ordering follows quickly,

$$dot{mathbf{R}}cdot nabla f sim v_T L^{-1}f sim Omega(rho_s/L)f sim Omega delta f$$

where I have introduced the "sound gyro-radius": $rho_s = v_T/Omega$ and extended the definition of $delta$ s.t. $delta = rho_s/L$.

We then recover the ordering claimed by H&S, i.e. that the drift term is an order of $delta$ smaller than the $partial f/partial vartheta$ term.

Conclusion

My genuine hope is that someone comes along and provides a more rigorous answer to this question. But this is what I can offer at this juncture.

Regarding the apparent energy non-conservation: I recommend you take a look at Section 5 of this famous paper: Baños A., Jr. (1967) Plasma Phys. 1, 305.. Essentially, this guiding-centre framework is an approximation, and non-constant, externally-modulated fields are doing work on the particle hence the gain in energy.

Also, you should be careful with your final statement that $f_0$ is the "averaged" distribution function. I read this part of H&S as saying that once you order eq. (1), you look at terms of order $mathcal{O}(delta^0, Delta^0)$ and you find that the largest term is simply,

$$frac{partial f_0}{partial vartheta} = 0$$

implying - to this order - $f_0$ is independent of the gyrophase $vartheta$. To solve for $f_0$ one must advance to the next order and retain the other terms in eq. (1), which is subsequently gyro-averaged to make the whole equation independent of $vartheta$.