Timescale to propagate a perturbation through a system?

• tonydoss
In summary, the conversation discusses the concept of a river catchment with a mass of material and a flux in and out of the box. The individual is seeking help in understanding how a change in the input flux would affect the output over time. They mention a potential solution involving a non-trivial volume and a transition time. The conversation ends with a suggestion to refer to a PowerPoint presentation for further information on a similar problem.
tonydoss
Hi,

I'm a geologist, and I really suck in physics so I would need some help, please! This is not a homework question, I'm an academic! This is for my research...

Let's say I have a "box" (a river catchment) with a mass, M, of material (sediments) in this box, which resides in the box for a time, T. There's a flux in the box, Fin, balanced by a flux out of the box, Fout, and equal to M/T (see attached PDF file)

Let's say I change Fin to a new value, Fin_new. I have the gut feeling it would take the time T for this perturbation to propagate through the box and for Fout to reach a new value (Fout_new = Fin_new)... but I have no idea how to put that in equations (for example, express Fout as a function of T and the old and new values of Fin).

Attachments

• catchment box.pdf
234.6 KB · Views: 221
If the catchment had zero volume, then surely Fout=Fin regardless of how Fin might vary in the future (eg: Fin_new). However, the purpose of the catchment (which has a non trivial volume) is to allow the absorption of excess mass (Fin_new - Fin_old) over an initial transition time (T'). Once the catchment has filled up, then Fin_new = Fout_new.

So, Fout_new gradually increases above Fout_old until T' has elapsed, thereafter Tout_new stabilizes and equals Tin_new. During that transition phase, the catchment was acquiring the excess Mass (delta_M).

I have seen this exact problem treated somewhere (I think it in the context of maintaining a fluid level), but I can't recall where- it's a "system with lag", and I think it's nonlinear.

If the change in input is a simple step (from one constant to another), you can readily determine the new equilibrium, but maybe not the detailed approach to equilibrium. If the input is allowed to vary continuously, the system becomes much more difficult to analyze.

say the change in input is a step function, how do I write Fout_new as a function of Fin_new, t (the time as a variable) and T (the residence time in the box)?

Like I said, I don't recall exactly where I saw it. Here's a PPT set of slides that treats a similar problem:

hotohke.ou.edu/~astriolo/ProcessDynamics&Control/Class08.ppt

The first 10-12 slides are great but then it gets a little rough. Hopefully it's enough to get you started.

1. How does the size of a system affect the timescale to propagate a perturbation?

The larger the system, the longer it takes for a perturbation to propagate through it. This is because there are more components and interactions within a larger system, which can slow down the spread of the perturbation.

2. Can the complexity of a system affect the timescale to propagate a perturbation through it?

Yes, the complexity of a system can also affect the timescale to propagate a perturbation. A more complex system with numerous interconnected components and feedback loops can take longer for a perturbation to spread through compared to a simpler system.

3. How do the properties of the perturbation impact the timescale for propagation?

The properties of the perturbation, such as its magnitude and direction, can greatly impact the timescale for propagation. A larger and more powerful perturbation will likely spread faster through a system compared to a smaller and weaker one.

4. Is the timescale for propagation the same for all types of systems?

No, the timescale for propagation can vary depending on the type of system. For example, the timescale for a perturbation to spread through a physical system like a body of water may be different from the timescale for it to spread through a biological system like a population of animals.

5. What are some factors that can affect the timescale for propagation of a perturbation through a system?

Some factors that can affect the timescale for propagation include the size and complexity of the system, the properties and magnitude of the perturbation, and any external influences or constraints on the system. Additionally, the type of interactions and feedback within the system can also impact the timescale for propagation.

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