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The S3 already has a electric pump that circulates coolant around the turbo's jacket to help it cool down, after the engine has been shut off.
besides the fact leaving a car running unatteneded is illegal in the uk.
certainly on public roads/ land it is.
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Yes, the S3 does have this device to help cool the turbo after a drive, but it is COOLANT that is being circulated. It is still benefitial to cool the turbo down with mild driving - off boost - for the last 1-2 miles of your journey, as it is the circulating of the oil that keeps the turbo in good nick.