Math 5421 Spring 2025
Introduction to Climate Models

Assignment 6
due February 27, 2025

Budyko's Equation

For these exercises, consider Budyko's equation \[ R\frac{\partial T}{\partial t} = Qs(y)(1 - \alpha(y,\eta)) - (A + BT) + C(\bar T - T) \] with standard parameters \(Q = 343\), \(A = 202\), \(B = 1.90\), \(C = 3.04\), \[ \alpha(y,\eta) = \begin{cases} \alpha_1 = 0.32, &y\lt\eta, \\ \alpha_2 = 0.62, &y\gt\eta, \end{cases} \] and \(s(y) = 1 - 0.241(3y^2 - 1)\).

Exercise 1

Remove the heat transport in the model by replacing the parameter \(C\) with zero, and consider the ice line \(\eta\) to be a fixed constant. Compute the equilibrium temperature distribution \(T^\star(y)\) and graph the distribution for ice lines at these latitudes: \(23.5^\circ\),\(45^\circ\), and \(66.5^\circ\). Compare the graphs to those of the equilibrium solutions for Budyko's equation with standard parameters. Discuss the differences.

Exercise 2

Continuing with Budyko's equation with no heat transport (\(C=0\)), compute the value of \(\eta\) where the ice line condition is met. (The ice line condition is that the average temperature across the discontinuity at the ice line is \(-10^\circ\text{C}\).) Discuss the difference between the location of the ice line in your answer and the locations of where the ice line condition is met for the standard values of Budyko's equation.