4 does not correspond to a fluid equilibrium and there are non-stationary fluid fields). In the following we shall take the AD plasma to be: a) non-relativistic , in the sense that it has non–relativistic species flow velocities, that the gravitational field can be treated within the classical Newtonian theory, and that the non-relativistic Vlasov kinetic equation is used as the dynamical equation for the KDF; b) collisionless , so that the mean free path of the plasma particles is much longer than the largest characteristic scale length of the plasma; c) axi-symmetric , so that the relevant dynamical variables characterizing the plasma (e.g., the fluid fields) are independent of the toroidal angle ϕ, when referred to a set of cylindrical coordinates ( R, ϕ, z ); d) acted on by both gravitational and EM fields. fined as δ ≡ r max R 0 , where R$_{0}$ is the radial distance from the vertical axis to the center of the nested magnetic surfaces and r$_{max}$ is the average cross-sectional poloidal radius of the largest closed toroidal magnetic surface; see Fig.1 for a schematic view of the configuration geometry and the meaning of the notation introduced here. Then, we impose the requirement 0 < δ ≪ 1, which is referred to as “small inverse aspect ratio ordering”. The main motivation for introducing this ordering is that we are discussing only local solutions where this asymptotic con- dition holds; this property also follows from the results presented in [12, 13], and has aready been used in other previous work on the subject [14, 15]. The requirement δ ≪ 1 is also needed in order to satisfy the constraint con- dition imposed by Ampere’s law, as discussed in Sec. VII. We stress that the δ − ordering here introduced is consis- tent with the assumption of nested and closed magnetic surfaces that are assumed to be localized in space. For definiteness, we shall consider here a plasma con- sisting of at least two species of charged particles: one species of ions and one of electrons. We also introduce some convenient dimensionless pa- rameters which will be used in constructing asymptotic orderings for the relevant quantities of the theory. The first one, which enters into the construction of the gy- rokinetic theory, is defined as ε$_{M}$ ≡ max { r$_{Ls}$ L , s = i, e } , where r$_{Ls}$ = v$_{⊥}$$_{ths}$/ Ω$_{cs}$ is the species average Larmor radius, with v$_{⊥}$$_{ths}$ = { T$_{⊥}$$_{s}$/M$_{s}$ } 1 / 2 denoting the species thermal velocity perpendicular to the magnetic field di- rection and Ω$_{cs}$ = Z$_{s}$eB/M$_{s}$c denoting the species Lar- mor frequency. Here L is the characteristic length- scale of the inhomogeneities of the EM field, defined as L ∼ L$_{B}$ ∼ L$_{E}$ , where L$_{B}$ and L$_{E}$ are the char- acteristic lengths of the gradients of the absolute val- ues of the magnetic field B ( r , t ) and the electric field E ( r , t ), defined as 1 L$_{B}$ ≡ max {∣ ∣ ∂ ∂r$_{i}$ ln B ∣ , i = 1 , 3 } and 1 L$_{E}$ ≡ max {∣ ∣ ∂ ∂r$_{i}$ ln E ∣ , i = 1 , 3 } . For typical tempera- tures and magnetic fields in AD plasmas, 0 < ε$_{M}$ ≪ 1. The second parameter is the inverse aspect ratio de- where B self and B ext denote the self-generated magnetic field produced by the AD plasma and a non-vanishing ex- ternal magnetic field produced by the central object. We also impose the following relative ordering between the Finally we introduce a parameter δ$_{Ts}$ which measures the magnitude of the species temperature anisotropy and is defined as δ$_{Ts}$ ≡ T$_{‖}$$_{s}$ − T$_{⊥}$$_{s}$ T$_{‖}$$_{s}$ , where T$_{‖}$$_{s}$ and T$_{⊥}$$_{s}$ denote the parallel and perpendicular temperatures, as mea- sured with respect to the magnetic field direction. Note that, in the following, we will use a prime “ ′ ” to denote a dynamical variable defined at the guiding-center position. In the following, we shall assume that the magnetic field is of the form B ≡ ∇ × A = B $^{self}$( r , t ) + B $^{ext}$( r , t ) , (1) A. The treatment of EM and gravitational fields FIG. 1: Schematic view of the configuration geometry. Also, we will focus on the situation where the equilib- rium magnetic field B admits, at least locally, a family of nested axi-symmetric closed toroidal magnetic surfaces { ψ ( r ) } ≡ { ψ ( r ) = const. } , where ψ denotes the poloidal magnetic flux of B (see [12, 13] for a proof of the pos- sible existence of such configurations in the context of astrophysical accretion discs; see also [14, 15] for further discussions in this regard and Fig.1 for a schematic view of such a configuration). In this situation, a set of mag- netic coordinates ( ψ, ϕ, ϑ ) can be defined locally, where ϑ is a curvilinear angle-like coordinate on the magnetic sur- faces ψ ( r ) = const. Each relevant physical quantity A ( r ) can then be expressed as a function of these magnetic coordinates, i.e. A ( r ) = A ( ψ, ϑ ) , where the ϕ depen- dence has been suppressed due to the axi-symmetry. It follows that it is always possible to write the following de- composition: A = A ∼ + 〈 A 〉 , where the oscillatory part A ∼ ≡ A − 〈 A 〉 contains the ϑ -dependencies and 〈 A 〉 is the ψ − surface average of the function A ( r ) defined on a flux surface ψ ( r ) = const. as 〈 A 〉 = ξ − 1 ∮ dϑA ( r ) / | B · ∇ ϑ | , with ξ denoting ξ ≡ ∮ dϑ/ | B · ∇ ϑ | . For definiteness, we shall consider here a plasma con-