  We define various notions of Lagrangian solution in physical space for 3-d
incompressible geostrophic system with free upper boudary under different
conditions for initial data,then prove their existence via the minimization
with respect to a geostrophic functional.As a byproduct of our proof,we obtain
the existence of measure-valued dual space solutions when the initial measure
$\nu_0\in\mathcal{P}_2(\mathbf{R}^3)$ and is supported on
$\{-\frac{1}{\delta}\leq x_3\leq-\delta\}$
