Effective resolution: turbulent cascade and numerical dissipation
Researchers: P. Marchesiello, Y. Soufflet, L. Debreu, X. Capet., J. Jouanno
Project: ANR COMODO
The increase of spatial and temporal resolution naturally leads to the representation of a wider energy spectrum in the numerical solution. As a result, in recent years, understanding of oceanic submesoscale dynamics has significantly improved (Thomas et al., 2008; and Capet et al., 2008abc, Klein et al., 2008, Molemaker and al., 2010) and the ubiquity of upper ocean frontal dynamics is now acknowledged.
The increase of numerical resolution toward the kilometer scale (or improvement of observational data coverage and treatment; Le Traon et al., 2008) invariably leads to flatter surface kinetic energy (KE) spectra than those expected from quasi-geostrophic (QG) mesoscale dynamics: spectral slope is closer to -2 than -3 or steeper. KE injection initiated by mesoscale straining and reinforced by submesoscale instability is the underlying mechanism that produces shallower spectra. Advection then plays an important role by fluxing KE energy upward from mesoscale to large scale (inverse cascade) and downward to submesoscale (forward cascade). Consequently, part of the kinetic energy released from available potential energy (APE) leaks toward smaller scales, en route to dissipation.
The forward cascade at submesoscale implies that numerical closure can be made more consistent with physical closure. Nevertheless, dissipation in submesoscale models remains dominated by numerical constraints rather than physical ones. In Marchesiello et al. (2011), model convergence at submesoscale is controlled by numerical dissipation, which overpowers submesoscale energy production and transfer. Effective resolution can thus be defined as the dissipation wavelength marking the start of dissipation range, below which model dynamics become unphysical. This dissipation range is a function of the model numerical filters.
To evaluate the effective resolution associated with various model filters, Skamarock (2004) used an approach based on kinetic energy spectra, which are “a direct measure of dissipation in the model’s dynamics”. The figure on the right presents his definition of effective resolution, based on the behavior of KE spectral tail, i.e., the wavelength marking the start of exponential tail off due to dissipation. It is found that effective resolution in models using high-order schemes (e.g. WRF and ROMS) follows a linear function of grid spacing: ~7*dx (Marchesiello et al., 2011).
This is arguably close to the limit of resolving capabilities of finite-difference models. A strong argument in this sense is in the phase error of numerical schemes. The figure below (from Durran, 1999) shows that phase errors in 4th order advection schemes are significant at resolution of 5-10*dx while they appear for 2nd order schemes at larger scales around 50*dx. Since dispersive computational modes need to be ultimately filtered, the dispersion relation not only shows the achievable limits of effective resolution but also its strong dependence on the order of the discretization schemes.
Going beyond our present knowledge requires a better understanding of numerical dispersion/dissipation ranges and their connection to the submesoscale range in models. This is the objective of TASK 1 of the ANR-COMODO project.
We use a Baroclinic Jet test case, which provides a controllable test of a model capacity at resolving submesoscale dynamics. We first compare simulations from two models, ROMS and NEMO, at different mesh sizes (from 20 to 2 km). Through a spectral decomposition of kinetic energy and its budget terms, we identify the characteristics of numerical dissipation and effective resolution. It shows that numerical dissipation appears in different parts of a model, especially in spatial advection-diffusion schemes for momentum equations (KE dissipation) and tracer equations (APE dissipation) and in the time stepping algorithms. Dissipation does not always decay at large scale and we conclude that the definition of effective resolution is not always meaningful, depending on mesh size and numerical methods. Our results argue in favor of high-order filters (higher than fourth order) for the dissipation range to be always restricted to small scales. To avoid dispersive errors, advection schemes must be consistent and be of higher order as well.
Changes in spatial schemes must also be consistent with the temporal scheme. The combination of low-order temporal schemes and high-order spatial schemes generally show poor performances for dispersion/diffusion properties and for robustness with respect to the Courant number (Durran, 1991; Shchepetkin and McWilliams, 1998, 2009; Lemarie et al., 2015). The latter property is particularly relevant to ocean modeling. Any scheme can always be made to work its best in a particular range of resolution, but realistic applications have a large range of Courant number. The upgrade of time stepping algorithms (from filtered Leapfrog), a cumbersome task in a model, appears critical from our results, not just as a matter of model solution quality but also of computational efficiency (extended stability range of predictor-corrector schemes).
It is generally assumed that eddy-resolving models have only small spurious diapycnal mixing (Veronis effect) associated with numerical necessities. This argument has been opposed by Roberts and Marshall (1998) and is confirmed here. The energy spectrum shows a large sensitivity at submesoscale to the orientation of mixing. The effect of spurious diapycnal mixing is to suppress submesoscale KE injection by draining the supply of APE, which is needed to sustain frontal processes. It argues in favor of third-order rotated diffusion schemes such as RSUP3 (Marchesiello et al., 2009) or higher-order advection/diffusion methods.
- Soufflet Y., P. Marchesiello, J. Jouanno, X. Capet, L. Debreu, F. Lemarie: On effective resolution in ocean models. Ocean Modelling, 98, 36-40. doi:10.1016/j.ocemod.2015.12.004
- Marchesiello P., X. Capet, C. Menkes, and S.C. Kennan, 2011: Submesoscale dynamics in Tropical Instability Waves. Ocean Modelling, 39, 31-46.