Originally Posted by Mike Gilson

Mine does the occasional buzz sometimes almost a scream when the ignition is turned on but goes away completely after the engine has started. Mine comes from the RH side of the dash as my car is RHD. I'm convinced the noise comes from the small electric water pump inside the bulkhead behind the engine, I think it feeds the heater but is always running. The water may drain out on standing and it runs dry until the engine water pump gives it a charge.

You can check the small fan supplying air to the temp sensor in the interior light cluster by turning on the ignition and putting you ear to the glove box, you can just perceive a faint hum if all is working as it shoulb be - you can't hear it when the engine is on.