Does my turkey need to be completely covered by the brine? Is it OK if the top part is not covered?
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If you don't have a container big enough (or not enough brine), it should be okay if you're rotating the bird every now and then. Ideal is completely covered, but this is a decent workaround!
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Covering it completely is a good idea, but if you don't want to dilute the brine (adding a cup or two of water is probably not going to make that much difference), just flip the bird over a few times and whatever else you do, please don't worry about it. ;o)