Here's what Walt Beineke, retired professor of forestry at Purdue University and black walnut grower, has to say on the subject.
"Mowing is not a weed control method and is mostly a cosmetic effect.
"The root systems of the weeds are still intact, and the weeds resprout using additional moisture and nutrients. In fact, in numerous plantations, mowing has done much more harm than good. The tendency is to mow too close to the trees and cause a serious disease known as tractor or mower blight. By bumping or scraping the bark, pockets of decay develop beneath the bark, even if the bark remains intact. These areas provide ideal points of infection for various canker and wood rotting diseases which can lead to wind breakage at that point or severely degrade the log at harvest.
"If you must mow, at least use herbicide or hand pull weeds from around the base of the trees before you mow so that you won't be tempted to mow too closely."