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tokens: strength of the magnetic field (Gehman, Adams Watkins 1996b ). The purpose of this paper is to add the effect of the ambipolar diffusion in the gravitational instability of filamentary MCs with magnetic field using non-ideal MHD. The outline of this article is as follows: in Section 2 , we present the basic MHD equations, which include the induction equation with an ambipolar diffusion term. We also describe the physical assumptions that are applied to the system. The unperturbed state of the system is introduced in Section 3 . In Section 4 , we perform linear perturbation analysis and then describe briefly the boundary conditions (BCs) and numerical method which are used to solve the perturbed equations in Sections 5 and 6 , respectively. We express the results in Section 7 . In Section 8 , we discuss our findings and finally, we summarize our conclusions in Section 9 . 2 MHD EQUATIONS WITH AD If the degree of ionization in a medium is very low, the force associated with the pressure gradient of the ions and their gravitational force could be neglected in comparison with the Lorentz and ion-neutral drag force. Accordingly, the Lorentz force and the drag force exerted by the neutrals on the ions could virtually balance each other. The equations of MHD in this so-called strong coupling approximation (Shu 1983 ) with self-gravity and AD term are as follows: where is the neutral gas density, is the fluid velocity, p is the pressure, is the gravitational potential and is the magnetic field strength, whereas A is the AD coefficient indicated by (Shu, Adams Lizano 1970 ), where i and are the ions density and the drag coefficient, respectively. Draine, Roberge Dalgarno ( 1983 ) proposed the value of 3.510 13 cm 3 g 1 s 1 . Due to the cosmic radiation, MC are partially ionized. Elmegreen ( 1979 ) showed that one could approximate the relation between ions and neutrals density as where C is a constant equals to C 310 16 cm 3/2 g 1/2 . So the equation ( 5 ) can be recast as where is a substitution for C and equals to 10.510 3 cm 3/2 g 1/2 s 1 . These assumptions lead to a fractional ionization of n i / n n 10 7 for n n 10 4 cm 3 , that is completely centred within 10 6 to 10 8 of observation of MC cores. By making the equations dimensionless, and A become 11.465 and 0.007 3/2 , respectively (see Section A ). The aforementioned fluid equations, are closed by addition of one equation of state, which is considered here to be isothermal, i.e. and, where c s is the isothermal sound speed which is about 0.2kms 1 for an MC with temperature of 10K and mean molecular weight of 2.36. The above values for the number density and sound speed, will give the value of 0.7 for a typical MC threaded by a magnetic field of 10G. 3 UNPERTURBED STATE The unperturbed fluid is supposed to be in the hydrostatic equilibrium state ( = 0). We make the assumption that the initial magnetic field does not have any role in stability of the filament. Therefore the density and