Thd science of twinkling stars is a huge field in astronomy and its instruments and optical transmission in general. They twinkle because our retinas recognize fluctuation of (…luminance, primarily?) of the light as it passes through our atmosphere.
So our star’s gotta be twinkling away.
Have I just never noticed, or are our human optical receptor–primarily the speed of gain adjustment at the retina–or visual cortex not up to it?
In case you never noticed - planets don’t twinkle, either
It takes a point source to twinkle.
So, our sun will twinkle to some observer on a distant planet (out of our solar system).
I don’t know if it’s “twinkling”, but I can certainly perceive fluctuations in the brightness of the “rim” of the sun if I look at it. it almost looks as though there’s a bright circle randomly spinning around the edge of the sun.
I assume it’s because it’s too close to Earth.
Stars twinkle because of fluctuations in the refractive index of the atmosphere (caused mainly by temperature fluctuations, although there are other contributing factors). These flucuations act kinda like small prisms, and looking at the stars through them, they sometimes appear to move or to blink on and off as the light that would otherwise come to your eye in a straight line through the atmosphere gets buffeted around.
Stars are essentially points of light as seen by your eye. The apparent diameter is the result of the limitations of your visual system. You see an Airy spot of light, and it’s tiny. Displace it by a little bit, and it disappears.
The sun is big and close. It subtends half a degree in the sky. If the randomly refracting cells in the atmosphere bump a bit of it off-couirse, theres plenty of other light nearby to fill in the gap. The sun’s just too big in your vision to be switched on and off by index fluctuations. It doesn’t twinkle.
Besides, it’s BRIGHT! Persistence of vision from that awesomely bright object takes longer for you to recover from than the “twinklle” would last. Even if the Sun were small enough to twinkle, you wouldn’t see the effect.
You certainly can onserve the effects of atmospheric index changfes, if you look at the setting sun (where it doesn’t dazzle you), or at films of the csun – the edge looks all wavery. This isn’t solar prominences, for the most part, it’s the atmosphere.
Has twinkling been observed during a total solar eclipse, especially during the “Bailey’s Beads” phase?
Is that what you see just before your retinas burn out? 
The sun is still not a point source during an eclipse. Would you expect your desk lamp to twinkle if you held your hand in front of it?
Probably a persistence-of-vision thing since your gaze is probably wobbling by a hairsbreadth - I know exactly what phenomenon you mean.
Please don’t look at the Sun directly unless it is on, or nearly on, the horizon.
The Sun would twinkle if seen from an Earth-like planet near Alpha Centauri or other nearby star.
The brightness of the day isn’t only coming from the sun’s image, its coming from reflection and Rayleigh scattering from all over the sky…
So variations in the jet stream at one spot don’t have a noticable effect brightness of the day… the star twinkles because the jet stream has gusts in it, patches of dense and less dense air… causing the star to , apparently, shift …
I’m not taking about staring directly at it. But sometimes it is there in your field of view unavoidably.
That’s why I mentioned Bailey’s Beads, containing various irregular points of light. Possibly also some annular eclipses.
Just an observation: Sometimes, you can find planets do appear to twinkle if the atmosphere between the object and the observer is significantly turbulent.