6.5 Sea level

In the model, two processes impact the sea level:

• Thermal expansion, which depends on the ocean temperature $T_{oce}$.
• The ice sheet melting, which depends on the ice sheet extent $phi_{g}$.

We note by $N(t)$ the sea level anomaly with respect to the present-day level: $N(t)=H_{mer}(t)-H_{mer,actuel}$, where $H_{mer}$ is the average sea depth

The average sea depth is calculated as:

$$H_{mer}=alpha(T_{oce})cdotfrac{M_{mer}}{S_{mer}}$$

where $alpha(T_{oce})$ is the volumetric mass of water at temperature $T_{oce}$, $T_{oce}$ is the global-mean ocean temperature, which is supposed to be an average of the global surface temperatures over the previous 100 years , $M_{mer}$ the total sea water mass and $S_{mer}$ the surface of ocean basins.

• We calculate $alpha(T_{oce})$ assuming a linear relationship as a function of $T_{oce}$, given the thermal expansion coefficient $c=2.6cdot10^{-4}/{^circ}C$:

$$alphaleft(T_{oce}right)=alphaleft(T_{oce,actuel}right)left(1+ccdot(T_{oce}-T_{oce,actuel}right)$$

• We calculate $frac{M_{mer}}{S_{mer}}$ by a mass balance: let $M_{tot}$ be the total mass of the water in the system {ice sheets + ocean}, and $f(phi_{g})$ the fraction of this water trapped in ice sheets. We have:

$$M_{mer}=M_{tot}cdotleft(1-f(phi_{g})right)$$

Assuming that the surface of the ocean basins is constant, we get:

$$frac{M_{mer}}{S_{mer}}=H_{tot}cdotleft(1-f(phi_{g})right)$$

where $H_{tot}$ id the average sea depth if all ice sheets had melted. We take $H_{tot}$ =3.8km ([Herring and Clarke, 1971]).

Therefore:

$$H_{mer}=left(1+ccdot(T_{oce}-T_{oce,actuel})right)cdot H_{tot}cdotleft(1-f(phi_{g})right)$$

The fraction $f(phi_{g})$ is proportional to the area covered between the latitude $phi_{g}$ and the poles: $1-cos(90-\phi_{g})$ . We alaso assume that ice thickness increases as the latitude of the ice caps approaches the equator, according to a function in $\left(1-\phi_{g}/90\right)^{n_{mer}}$ with $n_{mer}$ a parameter adjusted to meet the constraints summarized in section 2.1. So:

$$f(\phi_{g})=\left(1-cos(90-\phi_{g})\right)\cdot\left(1-\phi_{g}/90\right)^{n_{mer}}$$