User:Galassig/examples

General issues
The Sea Level Equation is the linear, integral equation that describes the sea– level variations associated with the Glacial Isostatic Adjustement (GIA). The basic idea of the SLE dates back to 1888, when Woodward published his pio- neering work on the form and position of mean sea level, and only later has been refined by Platzman and Farrell in the context of the study of the ocean tides. In the words of Wu and Peltier, the solution of the SLE yields the space– and time–dependent change of ocean bathymetry which is required to keep the gravitational potential of the sea surface constant for a specific deglaciation chronology and viscoelastic earth model. The SLE theory was then developed by other authors as Mitrovica & Peltier, Mitrovica et al. and Spada & Stocchi.

In is simpliest form, the SLE reads


 * $$ S=N-U, $$

where S is the sea–level change, N is the sea surface variation as seen from Earth's center of mass, and U the vertical displacement. In a more complete form the SLE can be written as follow:


 * $$S (\omega, t) = \frac{\rho_i}{\gamma} G_s \otimes_i I + \frac{\rho_w}{\gamma} G_s \otimes_o S + S^E - \frac{\rho_i}{\gamma}\overline{G_s \otimes_i I } - \frac{\rho_w}{\gamma}\overline{G_o \otimes_o S } $$

where $$\theta$$ is colatitude and $$\lambda$$ is longitude, $$t$$ is time, $$\rho_i$$ and $$\rho_w$$ are the densities of ice and water, respectively, $$\gamma$$ is the reference surface gravity, $$G_s=G_s(h,k)$$ is the sea–level Green’s function (dependent upon the h and k viscoelastic load–deformation coefficients – LDCs), $$ I= I(\theta, \lambda, t)$$ is the ice thickness variation, $$S^E=S^E(t)$$ represents the eustatic term (i.e. the ocean–averaged value of S),  $$\otimes_i$$ and $$\otimes_o$$ denote spatio–temporal convolutions over the ice– and ocean–covered regions, and the overbar indicates an average over the surface of the oceans that ensures mass conservation.

This Equation represents the gravitationally selfconsistent form of the SLE, first presented by Mitrovica & Peltier and then refined by Spada & Stocchi.