Using global MHD simulations of the solar wind-magnetosphere-ionosphere system, we investigate the dependence of the contribution from the Earth's bow shock (I_1bs) to ionospheric region 1 field aligned current (FAC) (I₁). It is found that I_1bs increases with increasing southward interplanetary magnetic field (IMF) strength B_s, if the Alfvén Mach number M_A of the solar wind exceeds 2, a similar result as obtained by previous authors. However, if M_A becomes close to or falls below 2, I_1bs will decrease with B_s in both magnitude and percentage (i.e., I_1bs}/I₁) because of the resultant reduction of the bow shock strength. Both the surface current density J_bs at the nose of the bow shock and the total bow shock current I_bs share nearly the same relationship with M_A, and vary non-monotonically with M_A or B_s. The maximum point is found to be located at M_A = 2.7. Three conclusions are then made as follows: (1) The surface current density at the nose, which is much easier to be evaluated, may be used to largely describe the behaviour of the bow shock instead of the total bow shock current. (2) The peak of the total bow shock current is reached at about M_A = 2.7 when only B_s is adjusted. (3) The non-monotonic variation of the bow shock current with M_A causes a similar variation of its contribution to region 1 FAC. The turning point for such contribution is found to be nearly M_A= 2. The implication of these conclusions to the saturation of the ionospheric transpolar potential is briefly discussed.