32 Problem set 7
Q1. Non-dimensionalisation of the Stommel box model
Consider the Stommel box model given in (Equation 24.1). By following the procedure outlined in Section 24.1 non-dimensionalise the model in order to produce the set of equations given in (Equation 24.2), 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
Consider the basic flooding model given in (Equation 25.1), 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 \(S(0) = S_0 \geq 0\), and a constant rainfall \(P(t) = P\) and constant potential evapotranspiration \(E_p(t) = E_p\). Describe, using analysis, the evolution of the system.
Draw graphs of the storage, \(S\), and saturation excess runoff \(Q_{se}\) (the second term of the above equation).