WASHINGTON — It has played host to civil rights marches, presidential inaugurations, fireworks displays and decades of picnics. But in recent years the National Mall, envisioned more than a century ago as a lush carpet of green connecting the nation’s most treasured monuments, has looked more like a West Texas dust bowl.

Now National Park Service officials are trying to teach the nearly 30 million people who gather here each year a difficult lesson: Tread softly or keep off the grass.

With the installation of exquisite, expensive new turf on a little less than half the mall, the Park Service has established strict rules that include limits on festivals and concerts, and even restrictions on when pickup softball and Frisbee games may be played. The requirements have created an identity crisis for the mall and set off a deeper debate.

Should the National Mall remain a utilitarian gathering place, rough and resilient and welcoming to all? Or should it be a more pristine landscape, a monument to the nation’s commitment to parks and preservation?