Distance from the Sun is not a major driver for vorticity of gas giants in our Solar System
According to Yadav et al., Jupiter indeed has a higher vorticity than Saturn:
There is a stark contrast in the number of vortices between the two
planets: about 200 with 1000 km or larger diameter on Jupiter, while
only 10 to 50 on Saturn. Note that while both planets also have
smaller vortices, the disparity in the numbers still persists.
Vortices are also present at or very close to the rotational poles:
Saturn has a cyclonic polar vortex at each pole , while Jupiter has a
cluster of cyclones at each pole
The authors claim the difference in the vorticity is mostly due to the differences in atmospheric make-up of the two planets (and their magnetic fields) and don't mention distance from the Sun as a factor. The below image is from their paper:
The difference in vorticity between Saturn and Neptune is less clear. Vortices may be less visible on Neptune as there is less color contrast in the atmosphere. Neptune certainly has many vortices including its own Great Dark Spots. It also has the highest winds observed in the Solar System.
The vortexes on Neptune are also modeled based on what we know about the atmosphere. Hadland et al. model methane cloud formation in their paper: EPIC Simulations of Neptune’s Dark Spots Using an Active Cloud Microphysical Model. Here is a picture of the Great Dark Spot (center) as taken by Voyager 2 in 1989.
You can contrast that image against this more recent image of Saturn from Cassini:
It certainly isn't clear to me which planet is more turbulent.
Reading the aforementioned papers, the vorticity of Jupiter, Saturn, and Neptune are due to:
- The atmospheric makeup
- The quantity of heat transfer in convective currents
- The Coriolis Effects on the atmosphere due to the spin rate of the planets
Note that the ratio for power emitted to power absorbed by the Sun are 2.5 for Jupiter, 2.4 for Saturn, and 2.7 for Neptune, which means the convection currents are mostly driven by geothermal energy loss, rather than solar heating (as opposed to Earth's atmospherics).
Hot Jovian exo-planets are indeed expected to have "extremely swirly, complex weather patterns!"
Though the Sun doesn't play the primary role for our distantly gas giants, it probably plays an enormous role in the weather patterns of Hot Jupiters! Most of the Hot Jupiters will be tidally locked to their stars since the time to tidal lock is proportional to $a^6$, where $a$ is the semi-major axis of the orbit of the planet. Planets with a small $a$ tidal lock very quickly.
Hot Jupiters in tidal lock with their star have a much slower rotation rate (on the order of days) than our Jupiter's rotation rate (~10 hours). That means the contribution to vorticity due to Coriolis Forces will be proportionally less. However, heat transfer in the upper atmosphere will be much more dramatic since much of the heat absorbed on the "day side" of the planet will be emitted from the "night-side" of the planet. Skinner and Cho (2021) perform numerical simulations in their paper Numerical Convergence of Hot-Jupiter Atmospheric Flow Solutions and highlight that their solution:
features a dynamic, zonally (east-west) asymmetric jet with a copious
amount of small-scale vortices and gravity waves.
Here is one of the beautiful simulation images from their paper:
