Deriving deterministic parameters for a climate change model
- 2 minsHello!
I realized that many of the models and techniques used in climate change modeling are extremely deterministic (like (6) in the previous post), so accurate parametrization is crucial. I will elaborate on my attempt at a simple deterministic parameter climate change model, global average temperature. Deterministic and stochastic parameters coexist within a climate change model where the effect is approximated both by a nonzero $ \varepsilon $ and a deterministic expression.
We define $ T_i(t) $ as the temperature of the Earth with uniform heat distribution at time $ t $, and the heat capacity as $ C_E $. Thus,
$$ C_E T_i (t) = E_E $$
where $ E_E $ is the energy needed to heat up Earth from 0K to $ T_i (t) $.
Derivative on both sides:
$$ C_E \frac{d T_i (t)}{dt} = \frac{dE}{dt} $$
Dividing by Earth's surface area:
$$ \frac{C_E}{A_E} \frac{d T_i (t)}{dt} = \frac{1}{A} \frac{dE}{dt} $$
And we define $ \frac{C_E}{A_E}$ as $ c $, heat capacity per unit area.
The RHS has a unit of W/m^2, thus can be modified for radiative flux:
$$ C_E \frac{d T_i (t)}{dt} = I(t) - O(t) $$
Applying the Stefan-Boltzmann Law on the RHS (net radiative output):
$$ C_E \frac{d T_i (t)}{dt} = I(t) - \varepsilon \sigma T_i (t)^4 $$
where $ \varepsilon $ is the emissivity of the Earth and $ \sigma $ is the Stefan-Boltzmann constant.
In order to model change in temperature, we define
$$ T_i (t) = T_{eq} + T(t) $$
where $ T_{eq} $ is a constant equilibrium temperature, and $ T(t) $ is net change in temperature. We modify the equation so that:
$$ C_E \frac{d (T_{eq} + T(t))}{dt} = I(t) - \varepsilon \sigma (T_{eq} + T(t))^4 $$
and
$$ C_E \frac{d (T_{eq} + T(t))}{dt} = I(t) - \varepsilon \sigma T_{eq}^4 (1 + \frac{T(t)}{T_{eq}})^4 $$
Since we know that Earth's initial absolute temperature is 275K and we're looking for fluctuations around 3K, it is safe to assume that $ \frac{T(t)}{T_{eq}} \ll 1 $. So we contract the binomial:
$$ = I(t) - \varepsilon \sigma T_{eq}^4 (1 + 4 \frac{T(t)}{T_{eq}}) $$
and
$$ = I(t) - \varepsilon \sigma T_{eq}^4 - 4 \varepsilon \sigma T_{eq}^3 T_(t) $$
Modify using net radiative output:
$$ = I(t) - O(t) - 4 \varepsilon \sigma T_{eq}^3 T_(t) $$
$$ \frac{dT}{dt} = \frac{N(t) - 4 \varepsilon \sigma T_{eq}^3 T_(t)}{c} $$
The approximate change given previous conditions of time $ t' $, as well as the equilibrium temperature $ T_{eq} $ can be approximately accounted for my the model.
In the future I think I will spend some time on possible ways to apply heat distribution (diffusion equation via discrete 2D laplacian) to the climate model. Goodbye!
İzge Bayyurt
Art / Science / Design