 Ridge-push force in a plate model

The ridge-push force in a plate model is the depth integral of the horizontal pressure difference between hot and cold compensated columns. To leading order, the old plate's linear excess density gives $F=g{\int }_0^{a}s\delta \rho (s)ds={\rho }_{m}g\alpha \Delta T{a}^2/6$. Here $s$ is depth below the old seafloor. Retaining the geometrical contribution from its lowered surface adds $g({\rho }_m-{\rho }_w)h^2/2$ within the linear-density model. Force per ridge length has units $\mathrm{N}{\mathrm{m}}^{-1}$, distinct from stress.