  We apply the coadjoint orbit technique to the group of area preserving
diffeomorphisms (APD) of a 2D manifold, particularly to the APD of the
semi-infinite cylinder which is identified with $w_{\infty}$. The geometrical
action obtained is relevant to both $w$ gravity and 2D turbulence. For the
latter we describe the hamiltonian, which appears to be given by the Schwinger
mass term, and discuss some possible developments within our approach. Next we
show that the set of highest weight orbits of $w_{\infty}$ splits into subsets,
each of which consists of highest weight orbits of $\bar{w}_N$ for a given N.
We specify the general APD geometric action to an orbit of $\bar{w}_N$ and
describe an appropriate set of observables, thus getting an action and
observables for $\bar{w}_N$ gravity. We compute also the Ricci form on the
$\bar{w}_N$ orbits, what gives us the critical central charge of the
$\bar{w}_N$ string, which appears to be the same as the one of the $W_N$
string.
