Timescale to propagate a perturbation through a system?

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Discussion Overview

The discussion revolves around the timescale required for a perturbation in flux within a river catchment system to propagate and reach a new equilibrium. Participants explore the relationship between input and output fluxes, the concept of residence time, and the dynamics of the system in response to changes in input flux.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a river catchment as a system where a mass of sediments is balanced by an input flux (Fin) and an output flux (Fout), suggesting that a change in Fin would take time T to propagate and affect Fout.
  • Another participant argues that if the catchment has zero volume, Fout would equal Fin regardless of future variations, but with a non-zero volume, there is a transition time (T') during which Fout gradually increases until it stabilizes at the new value.
  • A different participant mentions that the problem may be treated as a "system with lag" and notes the complexity of analyzing continuous variations in input compared to a simple step change.
  • A follow-up question seeks to express Fout_new as a function of Fin_new, time, and residence time, indicating a desire for a mathematical formulation of the problem.
  • One participant references a PowerPoint presentation that discusses a similar problem, suggesting it may provide useful insights for the discussion.

Areas of Agreement / Disagreement

Participants express differing views on how the system responds to changes in input flux, particularly regarding the effects of volume and the nature of the input change (step vs. continuous). The discussion remains unresolved with multiple competing perspectives on the dynamics of the system.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the system's volume and the nature of the input changes. The mathematical relationships and specific conditions under which the dynamics operate are not fully established.

tonydoss
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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).
Any help, please?

Thanks in advance
 

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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.
 

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