modeling internal tides in a canyon: solution not converged

[schen]

Hi all,
I am simulating a regional sea off Taiwan where tidal flows over a submarine canyon generates internal tides (~ 50m isotherm displacement). The canyon is about 10km wide and is around 300-400 meter deeper than its rim. The tidal forcing is from OSU TPXO, and the model-derived sea-level agrees well with observation (as it should be).

Problem: As I change the barotropic time step from 0.5 to 0.1 sec (DT=2 is fixed, NDTFAST = 4,8,10,20), I find that the numerical solutions of 3D u and v in the canyon do not converge. The surface tides do not get changed, but the vertical structures of u and v (which is sensitive to stratification) look quite different between cases with different NDTFAST.

Questions: Does anyone know the cause of this? I certainly expect to see convergence of numerical solutions.

Here is my setup:
Initial condition: observed climatological TS profile that applied everywhere in the model
BC:
FSCHAPMAN + M2FLATHER
M3RADIATION
TRADIATION + nudging to climatological TS profile (nudging time scale is 2-day on the boundary and decreases linearly inward to 20 days over 6 grid points)

Tides:
SSH_TIDES
UV_TIDES
RAMP_TIDES

The maximum barotropic Courant number ranges from 0.3 to 0.06.

Any suggestion is appreciated!! thanks ---- Shih-Nan

[rduran]

Possibility:

by reducing your time step you are able to resolve different types of waves possibly related to spin up.

Convergence also depends on spatial grid refinement, what is your spatial resolution?

[schen]

thanks for the reply!
The spatial resolution is variable. The highly resolved canyon region is 200m x 200m. But the grid coarsens to 4km near the boundaries.

Maybe the spin-up period of 30days is not long enough. I will try spatial grid refinement as well.

[rduran]

I would continue asking before spending too much time creating grid files with different resolutions.

Other possibilities are sensitivity to boundary conditions.

What is your vertical resolution? In general, the finer the resolution the longer the spin up time (allows for (more) higher mode waves, specially where topography is expected to interact with lower order waves).

What are your maximum grid stiffness ratios?

[arango]

The resolving of all the tidal harmonics that you may using for forcing will take long time because of the nonlinear interactions. As you increase the tidal constituents, the time that it takes to resolve such frequencies increases. It takes around 30 days to separate M2 from S2. We have solutions that take about a year to resolve 8 tidal constituents: 4 semi-diurnal (K2, S2, M2, N2) and 4 diurnal (K1, P1, O1, Q1) periods.

You just need to think more in terms of tidal harmonics and beat frequencies...

[schen]

To rduran: The max water depth is around 2000m, and I tried 20, 30, and 40 layers. The max. stiffness ratio is 0.4 (a bit high).

I understand the period needed to resolve different harmonics. I just thought that the same dominant wave modes for adjustment would be resolved in the cases with different barotropic time steps (as the model is stable). So I expect that the leading-order features of 3D u and v in the canyon would be similar between cases. Looks like I have to think harder.....

[ezaron]

Hi Shih-Nan

This can be a difficult problem. So it is good you are checking out convergence.

From my understanding of Shchepektin's descriptions of the numerics, the energetics of ROMS should be pretty accurate, but I think DiLorenzo is the only one to systematically verify the convergence of internal tide fields and energetics (and in an idealized 2-D setting). You might look for a paper by MacCready et al who applied ROMS to the tides in the Columbia River, estuary, and plume. I don't recall accuracy of their energy budgets, but it ought to give you some indication of resolution and accuracy. [Assuming lack of closure to volume-integrated energy budget is indicative of model convergence and truncation error.]

Anyway, there are truncation errors in the vertical mode-splitting that can influence barotropic to baroclinic tidal conversion, so I'd suggest being more careful in your study of convergence. To eliminate one class of nonlinear interactions, you might just check convergence with a single constituent. Look for convergence as dt->0 for constant mode-splitting ratio, and then see how things change as you alter mode-splitting ratio. Be aware too that the higher vertical modes propagate slowly, so it can take a long time to fully "spin up" (it iss easier to look at convergence by comparing solutions at specific time, before complete spin-up).

If the boundaries are too close and sponge layer isn't working as expected, the faster low modes can reflect back into the domain and make it very complicated to diagnose convergence. Glenn Carter and colleagues's papers on Monterrey Canyon might give you a target for necessary resolution and domain size.

All the best,
Ed

[schen]

Hi Ed, thanks for the suggestions. I am testing the convergence with idealized M2 tidal forcing. Will report back!