vortex stretching and vorticity advection from momentum diagnostics?

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vortex stretching and vorticity advection from momentum diagnostics?

#1 Post by wtorres »

Hi there,

I am studying the vorticity equation in natural coordinates for idealized 2D barotropic toy problem, and was running into a little bit of trouble decomposing its constituent terms. Before getting to vorticity, here are the the depth-averaged continuity and momentum equations oriented in natural coordinates (s,n) where V is the flow speed, α is the direction, and k = dα/ds the curvature. The bottom stress is quadratic and depth-dependent specified in ana_drag.h, while the viscous and unsteady terms are negligible in this application. The names of the terms correspond to the diagnostic output by ROMS.

dη/dt + (V/H)*(dH/ds) + VH(dα/dn) = 0

Streamwise direction)
VdV/ds + Cd V^2 /H + gdη/ds = 0
HADV_s + DRAG_s + PRSGRD_s = 0

Normal direction
kV^2 + fV + gdη/dn = 0
HADV_n + COR_n + PRSGRD_n = 0

After taking the curl of the momentum equations to derive the barotropic vorticity equation on an f-plane in natural coordinates (see Signell + Geyer 1991 JGR for the coordinate agnostic form)

Vdω/ds + ωdV/ds + fdV/ds+ Cd (dH/dn)/H^2 - Cd(dV/dn)/H + Cd ω/H = 0
VRT_hadv + VRT_cor + VRT_drag = 0

Here I would like to separate the contributions of vorticity advection and vortex stretching towards the nonlinear term: VRT_hadv = ∇ x HADV = u•∇ω + ω (∇•u), but cannot get the terms to close; that is summing back to ∇ x HADV (the vorticity budget as a whole closes to machine precision thankfully). I was able to reconstruct the drag term by computing the individual components from ROMS average file output (Cd constant, V, ω, H = zeta + h provided), however the same approach does not work with the nonlinear term. Naively, it seems that simply differencing the terms that were successful in recovering the drag term (i.e. dω/ds, dV/ds) should also recover the nonlinear term, but this is not the case. I have tried instead of computing the divergence dV/ds by differencing V, directly computing the terms in the continuity equation, inferring it by dividing HADV_s by V, inferring by dividing VRT_cor by f, but all to no avail. As a side note, directly differencing dV/ds is equivalent to HADV_s/V, but is not equal to the divergence inferred by the Coriolis term in the vorticity equation and I am not sure why. Any ideas on how to make progress on this? Thanks in advance for the help!

I am happy to provide more details about the model configuration if warranted, but I think this is a conceptual issue rather than an implementation one. I have attached plots showing the bathymetry, transport (although the units should be m^2/s, not m/s^2 as pictured), and relative vorticity components with the jet center streamline overlaid. The only forcing is an inflow jet boundary condition and the domain is periodic on the east and west boundaries.
depth.png (48.53 KiB) Viewed 337 times
transport.png (47.56 KiB) Viewed 337 times
curvature.png (48.39 KiB) Viewed 337 times
shear.png (43.97 KiB) Viewed 337 times

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