Apparently, proton model VDFs that obey the multi-moment fluid Equations (32)–(37) can be constructed on the basis of the 16-moment expansion (38) and (39), where in addition terms related to viscosity may be included. Demars and Schunk (1990) used this expansion to model such measured VDFs as shown in Figure 3 and Figure 4. By comparison and visual inspection of the measured and modelled velocity space contours they concluded that many features of the VDFs could be reproduced, even a possible secondary peak, as long as the beam relative density stayed between 0.1 and 0.4, which is to say massive beams cannot be described properly, and dilute beams with drift speed of several thermal speeds either. A double beam is better represented by a two-Maxwellian (double-humped) VDF in the first place. They found that the components of the stress tensor were not required, but the four parameters, , mostly were sufficient to characterise the shape of the distributions published by Marsch et al. (1982c).

However, spherical-harmonics or Legendre expansions (Dum et al., 1980) previously had indicated that the polynomial must often be of order up to ten, as to obtain credible representations of in the presence of an extended tail or a strong beam, for example, or a distinct thermal core anisotropy. A more recent comparison of proton model VDFs by Li (1999) and Olsen and Leer (1999) with the ones measured in the solar wind indicated that a lower-order moment expansion badly failed in reproducing the key features of those observed proton VDFs that were shown in previous figures. Some model results of Li (1999) for solar wind protons are reproduced in Figure 10, which clearly illustrates the problem with and inadequacy of the moment expansion for ions. The model includes perpendicular wave heating that leads to the exaggerated ion conic-like VDFs. The overall tendency for magnetic moment conservation in the magnetic field mirror is also obvious, which leads to pitch-angle focussing such that the distribution at reveals .

In parallel to the activities described in Subsection 4.4, Leblanc and Hubert (1997, 1998) and Leblanc et al. (2000) in a series of three papers also developed a generalised multi-moment fluid model, in which solar wind proton VDFs were constructed by expansions to higher-order moments. However, instead of using a zeroth-order background single Maxwellian or bi-Maxwellian, which are distinguished by the fact that they describe local thermal equilibrium, a non-thermal was chosen at the outset. It was constructed such as to be closer to the true (though unknown) solution of the non-uniform Vlasov–Boltzmann Equation (9), and thus to ensure more rapid convergence than the series (38) and (39).

The impasse to which all such expansions about an assumed background (weight) function leads is illustrated in Figure 11 taken from Leblanc and Hubert (1997), in which a VDF with a beam and its model representations (left a bi-Maxwellian and right a skewed weight function ) are displayed as one-dimensional cuts in the field direction of the normalised VDF (velocity component ), with crosses indicating the measurements from Marsch et al. (1982c) and lines the various fits. The general slowness of the convergence is obvious in either case. While only if is a higher-order polynomial (up to 7) the convergence is acceptable (see left bottom frame), it is worse for a skewed background VDF, that was intended to account for the beam in the first place (see right bottom frame of Figure 11). Also, the core density is much better fitted by a bi-Maxwellian.

For their generalised model Leblanc and Hubert (1998) derived the associated transport equation, which are substantially more involved than the set (32)–(37). The related collisional exchange terms were provided in the work of Leblanc et al. (2000). The background function has to be estimated well and calculated such that essential aspects (such as the electrostatic field or mirror force producing skewness) are incorporated in this zeroth-order solution. Formally, after (Leblanc and Hubert, 1997) we may write their VDF as:

where the new function is normalised to unity and of the same order than the perpendicular Gaussian. It essentially describes heat conduction and thus is a skewed distribution in the parallel velocity. The small function is an appropriate polynomial correction. To obtain already requires to solve a kinetic equation. The advantage is the intrinsic asymmetry, representing a possible suprathermal tail. Then the heat flux is not involved in the construction of the polynomial part , according to Leblanc and Hubert (1997).

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