Milestone 2 - Heat Capacity and Time Dependence

We are currently considering the simple algebraic equilibrium EBM,

As our next step, we want to include a temporal component, i.e., the possibility that temperature $T$ changes in time. Following the general derivations of the EBM chapter, we add the temporal term, which is proportional to the temporal derivative $\frac{\partial T}{\partial t} = T_t$. Hence, we add the temporal term with the constant $C$ to our model

to get

where analysis of the physical units reveal that the Earth's surface area normalized heat capacity $C$ has the units $[J/m^2/K]$.

In general, we need an estimate of the mass density $\rho$ with units $[kg/m^3]$, the specific heat capacity $C_p$ with units $[J/kg/K]$, and the estimated height $d$ in meters $[m]$ of the material column at a given location $x$.

For the atmosphere (air), we have $\rho_{atm} = 1.225$, $C_{p,atm}=1000$, and $d_{atm}=3850$ (here, we take into account that the density of the atmosphere reduces the higher we get and compute an corresponding integrated average height), which gives the value

For the ocean (mixed layers of water), we have $\rho_{mixed} = 1030$, $C_{p,mixed}=4000$, and $d_{mixed}=70$, which gives the value

For the sea ice, we have $\rho_{ice} = 917$, $C_{p,ice}=2000$, and $d_{ice}=1.5$, which gives the value

For the snow cover, we have $\rho_{snow} = 400$, $C_{p,snow}=880$, and $d_{snow}=0.5$, which gives the value

For the soil, we have $\rho_{soil} = 1350$, $C_{p,soil}=750$, and $d_{soil}=1$, which gives the value

With these estimates of the heat capacities for different materials, we are almost ready to define the heat capacity values for the geography map from milestone 1.

But first we introduce a normalization with respect to our time scale that we will need later on for our full EBM. The typical units in physics for time $t$ is seconds $[s]$. However out of convenience we will instead use the unit year $[yr]$ instead. This affects the time derivative of temperature $T_t$ that we need to scale properly with the number of seconds per year. We choose $sec\_per\_year = 3.15576\cdot 10^7$ following the paper by Zhuang et al. (2017), which corresponds to $365,25$ days in the year. To not carry around this scaling factor explicitly, out of convenience, we absorb this parameter into the heat capacity $C(x)$ as it gets multiplied by $T_t$ anyways.

We hence get, for the different geopgraphy types in the map:

Land/soil ($geo=1$): $C = (\widetilde{C}_{soil} + \widetilde{C}_{atm})/sec\_per\_year$.

Sea ice ($geo=2$): $C = (\widetilde{C}_{ice} + \widetilde{C}_{atm})/sec\_per\_year$.

Snow cover ($geo=3$): $C = (\widetilde{C}_{snow} + \widetilde{C}_{atm})/sec\_per\_year$.

Lakes/inland sea ($geo=4$): $C = (\widetilde{C}_{mixed}/3 + \widetilde{C}_{atm})/sec\_per\_year$.

Ocean ($geo=5$): $C = (\widetilde{C}_{mixed} + \widetilde{C}_{atm})/sec\_per\_year$.

In summary, we have now our first time dependent (still simplified) EBM