In [Zhang, Zujin; Yao, Zheng-an. 3D axisymmetric MHD system with regularity in the swirl component of the vorticity. Comput. Math. Appl. 73 (2017), no. 12, 2573--2580], we have obtained the following fine property of the convective terms of axisymmetric MHD system Let $u,v,w$ be smooth axisymmetric $bR^3$-valued functions. Then $$eelabel{lem:me:equal} ea &quadsum_{i,j,k=1}^3p_ku_jcdot p_jv_icdot p_kw_i\ &=frac{u^r}{r}cdot frac{v^r}{r}cdot frac{w^r}{r} +frac{u^r}{r}cdot frac{v^ t}{r}cdot frac{w^ t}{r}\ &quad+frac{u^ t}{r}cdot p_rv^rcdot frac{w^ t}{r} -frac{u^ t}{r}cdot p_rv^ tcdot frac{w^r}{r}\ &quad+ p_ru^ tcdot frac{v^r}{r}cdotp_rw^ t +p_zu^ t cdot frac{v^r}{r}cdot p_zw^ t -p_ru^ tcdot frac{v^ t}{r}cdot p_rw^r -p_zu^ tcdot frac{v^ t}{r}cdot p_zw^r \ &quad +p_ru^rcdot p_rv^rcdot p_rw^r +p_ru^rcdot p_rv^ tcdot p_rw^ t +p_ru^rcdot p_rv^zcdot p_rw^z\ &quad +p_ru^zcdot p_zv^rcdot p_rw^r +p_ru^zcdot p_zv^ tcdot p_rw^ t +p_ru^zcdot p_zv^zcdot p_rw^z\ &quad +p_zu^rcdot p_rv^rcdot p_zw^r +p_zu^rcdot p_rv^ tcdot p_zw^ t +p_zu^rcdot p_rv^zcdot p_zw^z\ &quad +p_zu^zcdot p_zv^rcdot p_zw^r +p_zu^zcdot p_zv^ tcdot p_zw^ t +p_zu^zcdot p_zv^zcdot p_zw^z. eea eee$$ With this above fine property, we could be able to find a regularity criterion in terms of $om^ t$ and $j^ t$. Moreover, using the governing equations of $j^ t$: $$eelabel{j_tt} ea &p_t j^ t +u^rp_rj^ t+u^zp_zj^ t -sex{p_r^2+p_z^2+frac{1}{r}p_r-frac{1}{r^2}}j^ t\ &=b^rp_rom^ t +b^zp_zom^ t +(p_ru^r-p_zu^z)(p_zb^r+p_rb^z) -(p_zu^r+p_ru^z) (p_rb^r-p_zb^z), eea eee$$ we could show that if $$eelabel{thm:me:om^tt} om^ tin L^p(0,T;L^q(bR^3)),quadfrac{2}{p} +frac{3}{q}=2,quad 2leq qleq 3, eee$$ then the solution is smooth on $(0,T)$.