tokens: and the quasi-parallel case for the acceleration of the electrons could better reproduce the observed distribution of the Stokes parameter Q . For another remnant G1.90.1, which also has two opposite protrusions on the X-ray morphology, assuming it has evolved inside an elliptical planetary nebula with ears and clumps, Tsebrenko Soker ( 2015 ) investigated the formation of the shape with two ear-like protrusions using 3D HD simulations, and the interaction of the ejecta with the clumps produces the double-shock structure. SNRs evolved in inhomogeneous mediums have morphologies with interesting signatures. In this paper, we intend to study the formation mechanism of the peculiar structures illustrated by the multiband observations for SN 1006. By introducing an ambient medium with a density discontinuity, we investigate how the substructures of it are produced via numerical simulations. Different from Schneiter etal. ( 2010 ), the HD simulations in this paper are performed in 3D, and the NE and SW protrusions and the NW filament can be reproduced using the model. The model and numerical set-up for SN 1006 are shown in Section 2 . The results are presented in Section 3 . Finally, the discussion and some conclusions are given in Section 4 . 2 THE MODEL AND NUMERICAL SET-UP To investigate the dynamical evolution of the remnant SN 1006 to reproduce the peculiar morphology, we assume the ejecta have evolved in the non-uniform medium with a density discontinuity. As illustrated in Fig. 1 , the plane of the density discontinuity is perpendicular to the z -axis, and the number density of the ambient medium is assumed to be where is the unit vector along the z -axis, n H is the hydrogen number density, 1.4 is the mean atomic mass for a gas of 10:1 H:He ratio, the ratio x 3 is adopted in this paper, which is consistent with that derived from the observations with XMM Newton (Acero, Ballet Decourchelle 2007 ). Figure 1. Scheme of the simulation model. The explosion occurs at the centre, and there is a density discontinuity at the plane . The LoS is in the xz plane with an angle of to the z -axis. After the supernova explosion, the spherical ejecta with a radius of R ej , a mass of M ej , and a kinetic energy of E ej is set at the centre of the simulation. Initially, the material in the ejecta has a velocity of , where v 0 is the velocity at the boundary of the ejecta (Jun Norman 1996 ; Truelove McKee 1999 ). The inner part of the ejecta within a radius of r c is homogenous with a density of c , whereas the outer part has a mass of M ej and a density with a power-law dependence on r with an index of s (Jun Norman 1996 ), i.e. where 0 is the density at r R ej . s 7 and 3/7 are used for the Type Ia SNR in this paper. The other details of the initial condition for the ejecta were shown in the equations(1)(3) in Jun Norman ( 1996 ). Without including the effects of the radiative cooling and the particle acceleration, the dynamical properties of the
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