Expected in situ velocities from a hierarchical model for expanding interplanetary coronal mass ejections

In situ data provide only a one-dimensional sample of the plasma velocity along the spacecraft trajectory crossing an interplanetary coronal mass ejection (ICME). Then, to understand the dynamics of ICMEs it is necessary to consider some models to describe it. We derive a series of equations in a hierarchical order, from more general to more specific cases, to provide a general theoretical basis for the interpretation of in situ observations, extending and generalizing previous studies. The main hypothesis is a self-similar expansion, but with the freedom of possible different expansion rates in three orthogonal directions. The most detailed application of the equations is though for a subset of ICMEs, magnetic clouds (MCs), where a magnetic flux rope can be identified. The main conclusions are the following ones. First, we obtain theoretical expressions showing that the observed velocity gradient within an ICME is not a direct characteristic of its expansion, but that it depends also on other physical quantities such as its global velocity and acceleration. The derived equations quantify these dependencies for the three components of the velocity. Second, using three different types of data we show that the global acceleration of ICMEs has, at most, a small contribution to the in situ measurements of the velocity. This eliminates practically one contribution to the observed velocity gradient within ICMEs. Third, we provide a method to quantify the expansion rate from velocity data. We apply it to a set of 26 MCs observed by Wind or ACE spacecrafts. They are typical MCs, and their main physical parameters cover the typical range observed in MCs in previous statistical studies. Though the velocity difference between their front and back includes a broad range of values, we find a narrow range for the determined dimensionless expansion rate. This implies that MCs are expanding at a comparable rate, independently of their size or field strength, despite very different magnitudes in their velocity profiles. Furthermore, the equations derived provide a base to further analyze the dynamics of MCs/ICMEs.