Diagnostics of diapycnal diffusion in z-level ocean models. Part II: 3-Dimensional OGCM

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Abstract

We present a robust method for diagnosing total diapycnal diffusivities, i.e. explicitly applied plus numerically induced diffusivities, from tracer release experiments in numerical z-level models. To this extent, numerical experiments differing only in the advection scheme used (CTRD using 2nd order centred differences, UPWIND using the upwind/upstream advection scheme, QUICK using the quicker advection scheme after Farrow and Stevens (1995) and FCT after Gerdes et al. (1991)) are analysed and compared. To obtain regionally resolved estimates of diapycnal diffusivities, individual inert dye tracers are released in dynamically different regions of a North Atlantic model, namely (i) in the interior of the subtropical gyre and (ii) in the western boundary current. Diagnosed diffusivities are robust with respect to changes in temporal and spatial sampling of the simulated dye tracer for both advection schemes and for both regions. The numerically induced diffusivity is generally positive, but can become negative for centred differences advection numerics after several months of simulated tracer dispersion.

Highlights

► Calculating total diapycnal diffusivities in a full 3-dimensional z-level model. ► Our approach follows real-ocean tracer release experiments. ► Two experiments differing only in the advection scheme used are compared. ► Individual inert dye tracers are analysed for two dynamically completely different regions.

Introduction

Diapycnal diffusion is a key process in the ocean, responsible for water mass transformation and the conversion of turbulent kinetic energy into mean flow potential energy. Despite its widely assumed importance in controlling ocean dynamics, diapycnal diffusion is difficult to quantify both in the real ocean and in ocean models. Here we apply the method described in a one-dimensional context (Getzlaff et al., 2010,) to a three-dimensional general ocean circulation model (OGCM) based on vertical z-level grid, arguably the most common vertical grid architecture of current ocean general circulation models.

In general, z-level models have difficulties in adequately representing along-isopycnal processes that are hard to formulate in a fully adiabatic way on a fixed geopotential model grid. This is because diffusion is not only determined by the explicit diffusion term, but also by numerically induced diffusion, an effect arising from discretisation errors. Present numerical advection schemes are either diffusive (e.g. upstream), dispersive (e.g. 2nd order centred differences in space and in time) or a mixture of both (e.g. Modified Split Quick (MSQ) (Webb et al., 1998), second order moment (SOM) (Prather, 1986), MPDATA (Smolarkiewicz, 1982, Smolarkiewicz, 1983)). While modellers are interested in developing advection schemes that represent the oceanic process best, there is no general method available to determine total (explicitly applied plus numerically induced) diffusivities, needed to quantify the numerically induced diapycnal diffusion in typical ocean model applications in along- and cross-isopycnal direction.

Morales Maqueda and Holloway (2006) considered the variance field of a dye tracer and estimated total diffusivities from the tracer variance decay and the tracer gradient. Thereby, they could determine the numerically induced diffusivity for each grid box, but this method was initially restricted to models that used a linear second-order moment (SOM) advection scheme (Prather, 1986). Burchard et al. (2008) generalised these diagnostics so that they could be applied to any model and any advection scheme. However, this approach cannot discriminate between diapycnal and isopycnal components of the diffusive mixing. Griffies et al. (2000) analysed the basin-wide average of the diapycnal diffusivity by equating the temporal change in the volume of density classes with the diapycnal flux of density. As long as diabatic forcings, such as surface buoyancy fluxes, are precisely known, this approach can also be used for any model configuration to isolate the diapycnal component of the diffusive mixing. Still, this method is limited to basin-wide averages and does not allow for estimates of local diapycnal diffusion.

In order to develop an approach for estimating total diapycnal diffusivities regionally, we here simulate tracer release experiments such as those performed in the real ocean as e.g. described by Ledwell et al., 1993, Ledwell et al., 1998 (North Atlantic Tracer Release Experiment). An inert dye tracer is injected into the model and total diapycnal diffusivities are inferred from the temporal evolution of the dye tracer as a result of the ocean dynamics (advection and diffusion). Similar to the observational studies, the results are restricted to spatial mean values in relation to the spatial distribution of the tracer patch, which, insofar as this patch occupies a volume much smaller than en entire ocean basin, can be considered local estimates.

In part I of this study (GNO10), we isolated the difficulties involved in following the diapycnal movement of such a tracer on a z-level grid, and investigated consequences for estimating diffusivities from simulated tracer release experiments. We tested three different methods for diagnosing total diffusivities from the temporal evolution of the tracer field. It turned out that only the tracer flux method based on the work of Griffies et al. (2000) gave robust results and therefore became our method of choice. Here, diapycnal diffusivities are inferred from the flux balance across isopycnal surfaces, with the temporal change of the total amount of a conservative tracer above an isopycnal equaling the diapycnal flux through the isopycnal.

The current study is organised as follows: In Section 2 the ocean circulation model and the model simulations are described. Section 3 covers a short theoretical background and introduces the method for estimating total diapycnal diffusivities. More detailed information about the theoretical background is given in GNO10. Section 4 presents the results for two dynamically very different regions, first the interior ocean and second the western boundary current. Section 5 contains a short discussion and conclusions.

Section snippets

Model

To test and illustrate our method for diagnosing total diapycnal diffusivities, we employ a three-dimensional circulation model of the North Atlantic. The numerical code is based on modular ocean model version 2, MOM2 (Pacanowski, 1995) and the model configuration used spans the Atlantic Ocean from 20°S to 70°N at a horizontal resolution of 4/3° × cos Φ (Φ denoting latitude), i. e. it is a “non-eddy-resolving” model. The vertical model grid consists of 45 non-equidistant z-levels, with spacing of

Modelling strategy and diagnostics

For the in situ experiment described by Ledwell et al. (1998), an inert dye tracer was deployed at about 300 m depth in the eastern North Atlantic. Its isopycnal and diapycnal distribution was repeatedly measured by ship surveys at subsequent times to estimate isopycnal and diapycnal diffusivities in the thermocline. We mimic this experiment in the model by placing a passive tracer (T1) at the location where the in situ experiment by Ledwell et al. (1998) was started. Additionally, we also

Results: diagnosed diapycnal diffusivities from various regions

In the following we show results for diagnosed diffusivities for two different regions. A first dye tracer is released in the interior Atlantic at 300 m depth at the same position used by Ledwell et al. (1998) in the real ocean. A second dye is release in the model’s western boundary current.

Summary and Conclusion

We presented a robust method to diagnose total diffusivities from tracer release experiments in numerical models. Although the method is limited for the analysis of spatial mean values of a dye tracer patch, the results can be considered regional estimates as the patch occupies a volume much smaller than the ocean basin.

Results for the mean values of total diffusivities for a local tracer patch were estimated for two dynamically different ocean regions in a coarse resolution model. In the

Acknowledgements

The present work received financial support from the Deutsche Forschungsgemeinschaft (DFG). We would like to thank Stephen Griffies and three anonymous reviewers for their constructive comments.

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