**2. Geometry and basic magnetohydrodynamic equations**

The simplest model of spicules is a straight vertical cylinder (see Fig. 4) with radius *a*

Fig. 4. Geometry of a spicule flux tube containing flowing plasma with velocity **U**.

filled with ideal compressible plasma of density *<sup>ρ</sup>*<sup>i</sup> <sup>∼</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>13</sup> g cm−<sup>3</sup> (Sterling, 2000) and immersed in a constant magnetic field **B**<sup>i</sup> directed along the *z* axis. Such a cylinder is usually termed *magnetic flux tube* or simply 'flux tube.' The most natural discontinuity, which occurs at the surface binding the cylinder, is the tangential one because it is the discontinuity that ensures an equilibrium total pressure balance. Moreover, it is worth noting that the jet is non-rotating and without twist – otherwise the centrifugal and the magnetic tension forces should be taken into account. Due to the specific form of the real flux tube which models a spicule, that part of the whole flux tube having a constant radius actually starts at the height of 2 Mm from the tube footpoint. The flow velocity, **U**i, like the ambient magnetic field, is directed along the *z* axis. The mass density of the environment, *ρ*e, is much, say 50–100 times, less than that of the spicule, while the magnetic field induction *B*e might be of the order or less than *B*<sup>i</sup> ∼ 10–15 G. Both the magnetic field, **B**e, and flow velocity, **U**<sup>e</sup> (if any), are also in the *z*ˆ-direction. We note that while the parameters of classical Type I spicules are well-documented (Beckers, 1968; 1972) those of Type II spicules are generally disputed; Centeno et al. (Centeno et al., 2010), for example, on using a novel inversion code for Stokes profiles caused by the joint action of atomic level polarization and the Hanle and Zeeman effects to interpret the observations, claim that magnetic fields as strong as ∼50 G were detected in a very localized area of the slit, which might represent a lower field strength of organized network spicules.

The flux tube modelling of the X-ray jets is actually the same as that for spicules, however, with different magnitudes of the mass densities, flow velocities, and background magnetic fields. When studying waves' propagation and their stability/instability status for a given solar structure (spicule or X-ray jet), the values of the basic parameters will be additionally specified. Now let us see what are the basic magnetohydrodynamic equations governing the motions in a flowing solar plasma.
