39 Problem set 7 solutions

Q1. Non-dimensionalisation of the Stommel box model

Consider the Stommel box model given in (Equation 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 \(R\).

Solution

The non-dimensionalisation procedure was done in the lecture notes and lectures.

The final parameters were established by \[ \delta = \frac{d}{c}, \qquad \lambda = \frac{c}{2\alpha k \Delta T^*}, \qquad R = \frac{\beta \Delta S^*}{\alpha \Delta T^*}. \] Their interpretations are as follows: \(\delta\) measures the relative relaxation temporal rates of the salt-basin exchange and the temperature-basin exchange. Since thermal energy turns out to exchange on a faster time scale than salinity, then we are typically interested in \(\delta \ll 1\).

The parameter \(\lambda\) is a measure of the strength of the THC. For example, with \(k\) large, this corresponds to significant flow through the pipes connecting the boxes, and so \(\lambda \to 0\) corresponds to strong THC.

Finally \(R\) allows us to compare the relative effects of temperature and salinity differences between the external basins. Salinity differences dominate if \(R > 1\) and temperature differences dominate if \(R < 1\).

Q2. Investigations for \(R\)

Using the script available on Noteable investigate the behaviour of the Stommel box model 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?

Solution

Remember that the steady states are found by examining the intersections of the curves \(\lambda f = \phi\). From (Equation 24.5), we have \[ \phi(f^*) = \frac{\delta R}{\delta + |f^*|} - \frac{1}{1 + |f^*|}. \] so essentially the effect of changing \(R\) is to multiply the curve \(\phi\) by a multiplier. Using the code on the JupyterHub we obtain the following graph for different values of \(R\).

Figure 39.1: Steady state intersections for Stommel’s box model

For the smallest values of \(R\), note that only the lowermost steady state will be present. As \(R\) crosses the critical value of \(R = 1\), two intersections are borne at \(\phi = 0\). Eventually \(R\) crosses another critical value, and the two leftmost intersections merge then disappear, leaving only the largest steady-state.

Because the stability of the solution is essentially given by the ordering of the curves (if the curve \(\lambda f\) is larger than the curve \(\phi\) on the left of the steady-states), then we can see that, as typical for stability in this course, the first and third steady states are stable, and the middle one unstable.

We can now plot a bifurcation diagram using the above information.

Figure 39.2: Bifurcation diagram for \(R\) vs \(f^*\)

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 Q1. 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.

Solution

Putting in the scalings, \[ \begin{aligned} \frac{[x]}{[t]}\frac{\mathrm{d}s}{\mathrm{d}t'} &= \delta(1 - [x]s) - [x]|f(x, y)|s, \\ \frac{[y]}{[t]}\frac{\mathrm{d}\theta}{\mathrm{d}t'} &= 1 - [y]\theta - |f(x, y)|\theta. \end{aligned} \]

Comparing the equations, we see that we should choose \[ [x] = \epsilon, \qquad \frac{\delta [t]}{[x]} = 1, \qquad [t] = \mu, \qquad [y] = [t] = \mu. \] Under this choice, \[ \begin{aligned} \frac{\mathrm{d}s}{\mathrm{d}t'} &= 1 - \epsilon s - [t]|f(x, y)|s, \\ \frac{\mathrm{d}\theta}{\mathrm{d}t'} &= 1 - \mu\theta - [t]|f(x, y)|\theta. \end{aligned} \] The latter quantity is \[ \mu f = \frac{\mu[y]}{\lambda}\left(-\theta + \frac{R[x]}{[y]}s\right). \] We there write \[ \kappa = \frac{\mu[y]}{\lambda}, \qquad \tilde{R} = \frac{R[x]}{[y]} \]

Thus we have produced \[ \begin{align} \frac{\mathrm{d}s}{\mathrm{d}t'} &= 1 - \epsilon s - |q|s, \\ \frac{\mathrm{d}\theta}{\mathrm{d}t'} &= 1 - \mu\theta - |q|\theta. \end{align} \] where \[ q = \kappa(-\theta + \tilde{R}s). \]

Q4. Box models for flooding

Solution

We walked through this problem in Lecture 33 so see the visualiser notes there.