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Coupled EBM Project Update 7/29

Overview of Current Model

Change in Population

To model rise in population, we begin with a basic logistic growth equation, which models rise in any population with a carrying capacity, by having exponential rise slow as the population size approaches its maximum.

\frac{dN}{dt}=\alpha (T,E)N(1-\frac{N}{N_{max}})
\alpha (T,E) = \alpha_{birth} (T,E) - \alpha_{death} (T,E)=E(B_{0}e^{-(\frac{T-T_{max}}{\Delta t})^{2}}-D_{0}e^{(\frac{T-T_{max}}{\Delta t})^{2}})

\alpha (T,E) = relative growth rate
- T = temperature
- E = technological efficiency (ie: measurement of how much technology a civilization has)
N = current population
N_{max} = maximum population \approx 20\ billion\ ppl
T_{max} = global temperature resulting in maximum growth rates \approx 75^{o}F \approx 297K
B_{0} = initial growth rate \approx 1/20
D_{0} = initial death rate \approx 1/70
- assuming a 70 year avg lifespan
\Delta t = width of gaussian (or exponential ramping distance) \approx 10

Change in Temperature

In order to model rise in temperature, we use an energy balance model, which balances the incoming solar radiation with the outgoing longwave radiation, in order to give an approximate planetary temperature.

C\frac{\partial T}{\partial t}= S(1-a)-IR+\frac{1}{cos(L)}\frac{\partial}{\partial L}(Dcos(L)\frac{\partial T}{\partial L})

S(1-a) = amount of solar flux being absorbed by the planet
- S = diurnally averaged solar flux
- a = albedo of the planet (1 for perfect reflector)
IR = outgoing infrared flux, modeled with a higher-order polynomial parameterization of outgoing longwave radiation
C = effective heat capacity of surface and atmosphere = 2.1*10^{8} \frac{J}{m^{2}K}
D = diffusive parameter describing efficiency of energy transport = 0.38 \frac{W}{m^{2}K}
L = latitude (18 bands total)

Change in CO2

We model changes in CO2 using a constant, annual, per-capita CO2 increase, assuming constant technology (ie: constant E). We can eventually progress this model by having E progress with time, as it would for any realistic technological civilization. If I were to leave the model here, we would inevitable end up with extinction. Thus, the extra term that is subtracted represents the constant reduction of CO2 by trees

\frac{dP_{co2}}{dt}=C_{o}EN-T_{r}

C_{o} = annual average pco2 contributions from 1 million people \approx 3.459*10^{-4} \frac{ppm\ from\ million\ ppl}{yr}
T_{r} = constant reduction of CO2 by trees = 1 \frac{ppm}{yr}

Results (E=1)

E acts to quantify a civilizations technological capacities, and affects our model in two ways. Higher E results in…

a higher peak growth rate
- the more technology you have, the lower your infantile death rates becomes, which raises the relative growth rate
higher per capita CO2 contributions
- the more technologically advanced a civilization, the more evenly distributed the technology becomes. (ex: cars were invented in the late 1700's, yet weren't commonplace until the 1900's)

Next Steps

Put a dTdt dependence on the death rate as a way to quantify acclimation
Put a time dependence on E as a way to quantify the technological advancements of energy-harvesting civilizations
Include more complete models for the environmental impacts of various resources, so that given a set of planetary/climate parameters, we could calculate climate sensitivity for the use of various resources
Put more accurate temperature dependence onto the relative growth rate

astrobear