8 A. Finally, a similar integral can be performed to com- pute the total flow velocity V tot s . This has the form n tot s V tot s ≡ ∫ d vv ̂ f$_{∗}$$_{s}$ ≃ n$_{s}$ [ V$_{s}$ + Δ U$_{s}$ ] , where by def- inition V$_{s}$ = Ω$_{s}$( ψ ) R e$_{ϕ}$ and Δ U$_{s}$ represents the self- consistent FLR velocity corrections given by: to ε ) as the characteristic scale-lengths associated with the species pseudo-densities η$_{s}$ , the temperatures T$_{‖}$$_{s}$ and T$_{⊥}$$_{s}$ , and the toroidal rotational frequencies Ω$_{s}$. To conclude this section we point out that the very existence of the present asymptotic kinetic equilibrium solution and the realizability of the kinetic constraints implied by it, must be checked for consistency also with the constraints imposed by the Maxwell equations, as discussed in Sec. VII. where Δ$_{ϕs}$ ≡ Δ$_{n}$$_{s}$Ω$_{s}$ R + Δ$_{2}$$_{s}$ + Δ$_{3}$$_{s}$ I $_{RB}$. Note that in Eq.(24) the terms proportional to Δ$_{n}$$_{s}$, Δ$_{2}$$_{s}$ and Δ$_{3}$$_{s}$ come from the asymptotic expansion of the KDF for strongly magnetized plasmas and are of O ( ε ) with re- spect to the toroidal velocity Ω$_{s}$ R . The full expressions for Δ$_{n}$$_{s}$, Δ$_{2}$$_{s}$ and Δ$_{3}$$_{s}$ are given in Appendix A. The first-order term Δ U$_{s}$ provides corrections to the zero- order toroidal flow velocity with components in all of the three space directions and so we can conclude that, al- though the dominant fluid velocity is mainly toroidal, there is also a poloidal component of order ε , associated with the term Δ $^{3}$s B ∇ ψ × ∇ ϕ . However, this is not nec- essarily an accretion velocity, especially under the hy- pothesis of closed nested magnetic surfaces which de- fine a local domain in which the disc plasma is confined. Moreover, note that the ratio between the toroidal and poloidal velocities depends also on δ$_{Ts}$ , in the sense that | Δ $^{3}$s B ∇ ψ ×∇ ϕ | | V $_{s}$| ∼ O ( ε ) O ( δ$_{Ts}$ ). The magnitude of the tem- perature anisotropy can therefore be relevant in further decreasing the poloidal velocity in comparison with the toroidal one, which on the contrary is not affected by δ$_{Ts}$ . However, the real importance of this result in connection with the astrophysics of collisionless AD plasmas is, in- stead, the fact that this poloidal velocity is a primary source for a poloidal current density which in turn can generate a finite toroidal magnetic field (see the section on the Maxwell equations). This means that, even with- out any net accretion of disc material (which would re- quire at least a redistribution of the angular momentum), the kinetic equilibrium solution provides a mechanism for the generation of a toroidal magnetic field, with serious implications for the stability analysis of these equilib- ria. The physical mechanism responsible for this poloidal drift is purely kinetic and is essentially due to the conser- vation of the canonical toroidal momentum and the FLR effects associated with the temperature anisotropy. As a last point, consider the species non-isotropic pressure tensor, which is defined by the following moment of the KDF: Π$_{s}$ = ∫ d v M$_{s}$ ( v − V tot s )( v − V tot s ) ̂ f$_{∗}$$_{s}$. Then, the overall pressure tensor of the system is obtained by sum- ming the single species pressure tensors: Π = ∑ s = $_{i,e}$Π$_{s}$ . A direct analytical calculation retaining only the zero- order terms (with respect to all of the small dimension- less parameters) in the Taylor expansion of the KDF ̂ f$_{∗}$$_{s}$ (whose leading-order expression coincides with the bi- Maxwellian KDF), shows that the corresponding species tensor pressure is non-isotropic to leading order and in this approximation is given by: It is well known that, given a distribution function, it is always possible to compute the fluid moments as- sociated with it, which are defined through integrals of the distribution over the velocity space. Although an exact calculation of the fluid moments could be carried out (e.g., numerically) for prescribed kinetic closures, in this section we want to take advantage of the asymptotic expansion of the KDF in the limit of strongly magne- tized plasmas to evaluate them analytically, thanks to the properties of the bi-Maxwellian KDF. In the follow- ing, we provide approximate expressions for the number density and the flow velocity, which allow one to write the Poisson and Ampere equations for the EM fields in a closed form, and for the non-isotropic species pressure tensor. Since these fluid fields are then known (in terms of suitable kinetic flux functions and with a prescribed accuracy), the closure problem characteristic of the fluid theories is then naturally solved as well. The main feature of this calculation is that the number density and flow velocity are computed by performing a transformation of all of the guiding-center quantities ap- pearing in the asymptotic equilibrium KDF to the actual particle position, to leading order in ε$_{M}$ (according to the order of accuracy of the adiabatic invariant used), and they are then determined up to first order in ε , in agree- ment with the order of expansion previously set for the KDF. Terms of higher order, i.e. O ( ε n $_{M}$) , with n ≥ 1, and O ( ε k ) , with k ≥ 2, as well as mixed terms of or- der O ([ εε$_{M}$ ] $^{n}$) with n ≥ 1, are therefore neglected in the present calculation. This approximation clearly holds if ε$_{M}$ ε ≪ 1, which is consistent with the present assump- tions. In fact, from the definitions given for these two small dimensionless parameters it follows that To first order in ε , the total number density n tot s is given by n tot s ≡ ∫ d v ̂ f$_{∗}$$_{s}$ ≃ n$_{s}$ [1 + Δ$_{n}$$_{s}$]. Note here that the resulting number density has two distinct con- tributions: n$_{s}$ is the zero order term given in the pre- vious section, while Δ$_{n}$$_{s}$ represents the term of O ( ε ) which carries all of the corrections due to the asymp- totic expansion of the KDF for strongly magnetized plas- mas. The full expression for Δ$_{n}$$_{s}$ is given in Appendix Π$_{s}$ = p$_{⊥}$$_{s}$ I + ( p$_{‖}$$_{s}$ − p$_{⊥}$$_{s}$ ) bb , (25) ε$_{M}$ ε ∼ O ( δ ) ≪ 1 . (23) Δ U$_{s}$ ≡ Δ$_{ϕs}$ e$_{ϕ}$ + Δ$_{3}$$_{s}$ B ∇ ψ × ∇ ϕ, (24) V. MOMENTS OF THE KDF