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