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Motivated by lopsided structures observed in some massive transition discs,we have carried out 2D numerical simulations to study vortex structure inmassive discs, including the effects of disc self-gravity and the indirectforce which is due to the displacement of the central star from the barycenterof the system by the lopsided structure. When only the indirect force isincluded, we confirm the finding by Mittal & Chiang (2015) that the vortexbecomes stronger and can be more than two pressure scale heights wide, as longas the disc-to-star mass ratio is >1%. Such wide vortices can excite strongdensity waves in the disc and therefore migrate inwards rapidly. However, whendisc self-gravity is also considered in simulations, self-gravity plays a moreprominent role on the vortex structure. We confirm that when the disc Toomre Qparameter is smaller than pi/(2h), where h is the disc's aspect ratio, thevortices are significantly weakened and their inward migration slows downdramatically. Most importantly, when the disc is massive enough (e.g. Q~3), wefind that the lopsided gas structure orbits around the star at a speedsignificantly slower than the local Keplerian speed. This sub-Keplerian patternspeed can lead to the concentration of dust particles at a radius beyond thelopsided gas structure (as shown in Paper II). Overall, disc self-gravityregulates the vortex structure in massive discs and the radial shift betweenthe gas and dust distributions in vortices within massive discs may be probedby future observations.

Gas and dust hydrodynamical simulations of massive lopsided transition discs - I. Gas distribution