I have a similar issue and in my case the whine is related to how much current my alternator is producing. After a start, the alternator generates more current than normal getting the battery charged back up. I have an ammeter installed that measures current going into and out of my battery, and the whine volume is proportional to the amount of charge going into the battery. Also, if I turn on additional loads, landing lights, etc. right after a start I notice an increase in the whine volume. As my battery got older it took more current and longer to charge back up after a start and the whine became more noticeable. Because my battery was 7 years old and getting a bit weak I decided to replace it before it let me down. Now with the new battery the whine is much less noticeable and only lasts a couple minutes after a start.