vortex stretching and vorticity advection from momentum diagnostics?

[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.

Continuity
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.