Q: At idle, on an intermittent basis, the rpm repeatedly surges and returns to normal

There a few things that can cause this situation on your car. The most common have to do with the throttle body and the idle air control system.

The throttle body is the where your your throttle valve is housed. The throttle valve is connected to the gas pedal via a cable. When the throttle body gets gummed up with carbon, it can cause a variety of issues. It should be thoroughly clean with some carburetor cleaner and a rag. Use the carburetor cleaner sparingly. The build up of carbon can have adverse affects on idle quality.

The most likely cause of your problem is the idle air control valve (IAC). The IAC is the most common part that fails on your particular car and is often gummed up with carbon as well. It has a couple mounting locations that I can’t be exact about unless the car were in front of me. Most times, it is round cylinder with two hoses attached to it and an electrical connector. It is either by the throttle body or mounted somewhere on the intake manifold. One hose connects to the intake manifold and the other will be connected to the hose that connects your air cleaner to the motor. Often times, these valves can be cleaned, but most often they will need to be replaced.

The other possible problem is the idle speed control unit that is mounted in the dash above the glove box. This doesn’t fail as often as the IAC, but does fail at times. This component unique to BMW. Most vehicles use the PCM (Powertrain Control Module) to control idle.

There may be other issues causing this, such as intake manifold leaks that can occur in many places. I do believe your particular problem, based on your detailed description, has something to do with the IAC system.

If you want to have the faulty component identified in person, consider YourMechanic. They will be able to properly diagnose your rough engine idle and determine exactly what needs repair.