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I saw a clip of a recent (Nov. 5th, 2021) JAXA Epsilon launch on Instagram (link). Shown differently here in a YouTube video @~0:23s:

It's an entirely solid fueled vehicle so turbine exhaust can (I believe) be ruled out. Wikipedia says the first stage is a modified H-IIA side booster which seen here does not emit any black smoke:

My understanding and (hobby) experience with solid motors is that they light kind of like a gas barbeque (i.e., wholly and suddenly), yet in the video the vehicle seems to sit emitting the smoke for ~10 seconds before 'fully' igniting.

What causes the black smoke?

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It's the Solid Motor Side Jet (SMSJ) system used for roll control - and more!

When you look at a picture of the launch, you will see black smoke being discharged from the SMSJ motors. They are ignited 10 seconds before launch, and keep burning until the separation of the first stage. Their role is to control the rocket’s rotation around its central axis, in order to control its attitude.

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After the first stage burnout, it takes about 40 seconds for it to separate. During that time, the two SMSJ systems continue to perform three-axis attitude control – that is, controlling roll, pitch and yaw. The SMSJ has jet nozzles placed at perpendicular angles in three different directions. When attitude control is not needed, combustion gas is exhausted equally from opposing nozzles to counteract the thrust of each. And when attitude needs to be controlled, a hot gas valve is opened to discharge more combustion gas from a selected nozzle.

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Source: https://global.jaxa.jp/article/special/epsilon/tokudome_e.html

See also How does a single SRB control attitude? for other examples

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