3. Consider the one-box model of the climate system, dT Q'=-XT's+c dt (1) where Q' is the extra radiative forcing (W m2), A is the climate radiative feedback parameter (W m2 K-), Tg is the surface temperature anomaly with respect to control conditions (K), c is the effective heat capacity per unit area (J m2 K-1) and t is time. Imagine the atmospheric concentration of CO2(t) increases at a rate of 1% per year from an initial concentration of CO2(0) at time t = 0 (This is not a bad approximation to what is actually happening to the atmospheric concentration of CO2 in recent decades). Total marks [23] (a) Write down the concentration of atmospheric CO2(t) at time t as a function of t (t is in units of years) and CO(0). [3] (b) How many years does it take for the concentration of atmospheric CO to double if it increases at a rate of 1% per year? [2] (c) The radiative forcing from atmospheric CO2 is logarithmic to a good approximation i.e. Q'(t) Q2xCO log 2 CO2(t) log CO(0) where Q2xCo = 3.71 W m is the extra radiative forcing from a doubling of CO2. Show that the expression for the forcing Q'(t) as a function of time if atmospheric CO2 increases at a rate of 1% per year is linear in time (use your answer for (a)). (d) Show that T's is given by kc T.(t) = (et-1) - kt [2] by solving the one-box model with the forcing Q'(t) explicit form for this constant was found in the last part). You can assume there is no temperature anomaly at t=0 i.e. initial condition T's (0) = 0. = kt, where k is a constant (an [8] (e) What is T's when the concentration of atmospheric CO2 has doubled? This is an impor- tant measure of warming in climate science called the transient climate response (TCR). (Hint: substitute for the value of t found in part (b).) (f) What is the equilibrium temperature if atmospheric CO2 is instantaneously doubled and held constant with time? This is another important measure of warming in climate science called the equilibrium climate sensitivity (ECS). (Hint: this is the t limit of T's when Q'(t) is a step function with height Q2xco. Alternatively, it is the energy balance temperature from perturbation Q = Q2xc02, see topic 2 notes.) (g) Is TCR or ECS a bigger warming? Why is one bigger than the other? [2] [2] [4]