Question

The earth's temperature over the shorter term, compared to centuries, might be modelled as τp​dT/dt=−(T−T[infinity]​(t)) where T[infinity]​(t)=Aexp(t/τw​)+B+Ccos(ωt)). The parameters are: (i) τp​ the planet's process time constant, (ii) τw​ the global warming process time constant, and (iii) A,(B,K) and are contributions of people to global warming and 'natural' processes which tend to be or less settled over a long time scale relative to τw​. Use the method of undetermined coefficients (MUD) to solve this problem subject to an initial temperature T(0)=T0​. Use the Auxiliary problem to find the response due to the input 1exp(αt) and assume that there is no duplication. Sketch the transfer function associated with the Auxiliary problem. Omit the 'Sketch' component. Why is no steady-state possible in this problem? Use the supplied Matlab code (or code your own version) to plot the solution for the case where T0​=1.0,τp​=1yr,τw​=5 years, A=0.1,B=1.0,C=0.1, ω=2π/yr. You should plot the solution over a range where it reaches about 400% more than T0​. Estimate how long the average solution stays within about 1.5T0​ and note that it approximately doubles again in the same amount of time to an average of about 3T0​. Assume that 3T0​ was fine, but we cannot withstand 4T0​. About how much time is left at 3T0​ before 4T0​ occurs?

Answer 1

To solve the given equation using the method of undetermined coefficients (MUD) and find the transfer function associated with the Auxiliary problem. Explain why no steady-state is possible in this problem. Plot the solution for the given parameters and estimate the times it takes for the temperature to reach certain levels. Estimate how much time is left at a certain temperature before it reaches another level.

Step-by-step explanation:

The equation given is τp​dT/dt=−(T−T[infinity]​(t)) where T[infinity]​(t)=Aexp(t/τw​)+B+Ccos(ωt)). To solve this equation using the method of undetermined coefficients (MUD) subject to the initial temperature T(0)=T0​, we need to assume a particular solution of the form Tp(t)=Dexp(αt). By substituting this particular solution into the equation, we can determine the value of α. The transfer function associated with the Auxiliary problem can be obtained by substituting Tp(t)=1exp(αt) into the equation and solving for α. In this problem, there is no steady-state possible because the temperature keeps increasing without reaching a stable point.

To plot the solution for the given values, we can use Matlab code or any other programming language to calculate the values of the temperature over time. By plotting the solution and estimating how long it takes for the average solution to reach about 1.5 times the initial temperature, we can also estimate the time it takes to double again to about 3 times the initial temperature. To estimate how much time is left at 3 times the initial temperature before it reaches 4 times the initial temperature, we can analyze the rate of increase and extrapolate the time period.