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My professor posed this question, without any relevant material. I suppose the currents last until the density gradient is neutralized, but I can't find any literature nor articles on this, regarding the time scale. Any suggestions? Thanks!

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    $\begingroup$ Are you talking about the theoretical/lecture perspective without friction etc. or a real application? In the case where only the pressure gradient balances the coriolis force, those flows can last indefinetly, because the flow is circular and therefore induces no mass-transport that would eliminate pressure gradients. $\endgroup$ Aug 27, 2021 at 13:29
  • $\begingroup$ Thanks for the reply! The question is, as far as I know, applied to real situations. I kept digging around and found that, for the Coriolis force to take non-neglectable effect, the spatial scale should be at least 100 km, then, using the space-time scale in oceanography and combining with average wind speeds, the duration of an average geostrophic currents is somewhere from a week to several months. Is this an acceptable estimate? $\endgroup$
    – a Ljeonjko
    Aug 27, 2021 at 14:20

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The geostrophic equilibrium is indeed without any time evolution, and even the quasi-geostrophic version only introduces a weak time dependence in the momentum equation (but in $O(Rossby)$). Geostrophic equilibrium is then purely diagnostic, but not prognostic. Density gradient can only be neutralized by adding some dissipation (either through mixing or friction).

As a quite rough first guess, adding a dissipation coefficient $K$ and having geostrophic density gradient scaling with the Rossby deformation radius $R_d$, I would dimensionnally answer that the typical timescale would be $ \tau \sim R_d^2/K$

However numerical application with ocean value of horizontal mixing $K \approx 1 \,m^2/s$ and $R_d \approx 50 \,km$ leads to decay in decades (lol).

Some mesoscale eddies can last for as much as 4-5 years, but it probably means that in the real ocean other forcings (atmospheric in particular) can destroy geostrophic structure over shorter timescales.

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