\f3. Changes in the latitudinal temperature distribution in time Finally, let's look at how the latitudinal distribution of temperatures changes in time. We'll focus on Budyko's model. For a climate perturbation Q ! Q + Q, we can derive a latitude- and time-dependent version of equation (1), B @ T (t, y) = (1 @t ) Q(t)s(y) B T (t, y)/g + C( T (t) T (t, y)), (3) where T (t, y) is the perturbation to the atmospheric temperature at time t and latitude y. For simplicity, we'll approximate the albedo as a latitude-independent constant. For a time-dependent perturbation (1 ) Q(t) = bt, we showed in class that the response of the mean temperature obeyed the equation (1) and had solution T (t) = where (t) = g 1 exp t g g (1 B ) Q(t (t)), is the time-dependent time lag of the climate's response to the perturbation. a. (20 points) Let T (t, y) = T (t) + (t, y), where (t, y) is the deviation of the temperature at latitude y from the global mean temperature. Plug this expression and (1 ) Q(t) = bt into equation (3) and, using equation (1) to cancel out terms involving T , derive a dierential equation for (t, y). Solve this dierential equation using the initial condition (0, y) = 0. Because there are no y-derivatives, you can just treat y as a constant parameter. Show that your result can be written as 1 (t, y) = (s(y) 1) (1 ) Q(t (t)), B/g + C where (t) is a time-lag (dierent from the previously defined ) that you should determine an exact expression for. R1 R1 Note that because 0 dy s(y) = 1, 0 dy (t, y) = 0 for every time t. b. (15 points) Show that for a certain range of y the temperature change (t, y) is not only negative but growing increasingly negative as time goes on. i.e., at some latitudes the temperature is getting colder (relative to the increasing mean global temperature) in response to the increase in energy input Q! Last year, a certain United States senator infamously threw a snowball onto the floor of the senate to \"prove\" that global warming is not real because it was unseasonably cold outside, rather than unseasonably hot. Based on the calculation we just performed, explain why decreasing temperatures around the globe do not necessarily disprove global warming. 3