Thermosteric sea level change ($\Delta {\eta }_{\mathrm{thermal}}=\int \alpha \Delta Tdz$)

Thermosteric sea level change is the thermal expansion part of steric sea level change. For a uniform layer of depth $H$, conservation of mass gives $\rho H=(\rho +\Delta \rho )(H+\Delta \eta )$, hence $\Delta \eta \simeq -H\Delta \rho /\rho =H\alpha \Delta T$.