30 Problem set 7
Q1. Non-dimensionalisation of the Stommel box model
Consider the Stommel box model given in (Equation 22.1). By following the procedure outlined in Section 22.1 non-dimensionalise the model in order to produce the set of equations given in (Equation 22.2), repeated here: \[ \begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}t} &= \delta(1 - x) - |f(x, y)|x, \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= 1 - y - |f(x, y)|y, \end{aligned} \] where we have introduced the function, \[ f(x, y; R, \lambda) = \frac{1}{\lambda}(Rx - y). \]
Provide a brief description of the physical interpretations of the parameters \(\lambda\), \(\delta\), and \(R\).
Q2. Investigations for \(R\)
Using the script available on Noteable, investigate the behaviour of the system in the above question under changing values of \(R\) and fixed values of the other parameters. Based on your intuition, what do you expect to be the features of the bifurcation diagram, as shown in the \((R, f)\)-plane?
Q3. Alternative scalings of the Stommel box model
It is possible to scale the problem differently, and this may allow for simpler analysis. Consider the set of equations in Q2. Write instead \[ x = [x] s, \qquad y = [y] \theta, \qquad t = [t] t', \] Choose the correct scalings, \([x]\), \([y]\), and \([t]\), so that we obtain a re-scaled version: \[ \begin{aligned} \frac{\mathrm{d}s}{\mathrm{d}t'} &= 1 - (\epsilon + |q|) s, \\ \frac{\mathrm{d}\theta}{\mathrm{d}t'} &= 1 - (\mu + |q|)\theta, \end{aligned} \]
where now the flow term is given by \[ q = \kappa(-\theta + \tilde{R}s). \] In addition to the choice of scalings, you will need to describe how the new parameters, \(\epsilon\), \(\mu\), \(\kappa\), and \(\tilde{R}\), are related to the older parameters.
Q4. Box models for flooding 1
In Chapter 23, two models for flooding were proposed. These models are conceptual models (or so-called box models), like the ones used previously in our study of the ocean. Create a diagram of the two box models, using arrows to indicate inputs and outputs. Label annotations related to precipitation, evapotranspiration, and runoff.
The analysis of the flood models contains numerous interesting complexities that seem very different than what you may have observed in your undergraduate courses on dynamical systems and differential equations (it is a shame this comes nearly at the last possible week in your studies!). Examining the form of the associated differential equations, name several reasons why the analysis of such equations is difficult and beyond what you may have studied in Year 2 differential equations (and beyond).
Q5. Box models for flooding 2
The intention of this question is to show you that, despite its apparent simplicity, the ODE models used here have some oddities because of the non-smooth nature of the Heaviside function.
Consider the basic flooding model given in (Equation 23.1), and repeated below: \[ \frac{\mathrm{d}S}{\mathrm{d}t} = P(t) - P(t)H(S - S_{\text{max}}) - E_p(t) \frac{S}{S_{\text{max}}}. \] Consider the initial condition of \(S(0) = 0\), and a constant rainfall \(P(t) = P\) and constant potential evapotranspiration \(E_p(t) = E_p\).
Let’s use the power of scaling and non-dimensionalisation. Show that the problem can be re-scaled to: \[ x(T) = 1 - H(x - x_m) - x. \] Explain how to choose \(x\) and \(T\).
We drop all tildes below. Consider firstly the evolution from the initial condition at \(t = 0\). Solve the problem analytically. Distinguish between the two cases of \(x_m > 1\) and \(x_m < 1\).
The situation for \(x_m < 1\) is unusually tricky. Attempt to develop an analytical solution. What do you think happens?