Semigeostrophic equations in physical space with free upper boundary

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 $\{-\frac{1}{\delta}\leq x_3\leq-\delta\}$