Estimating Sub-Grid Scale Processes Using Oceanographic Data

William A. Gough

University of Toronto at Scarborough

Ontario, Canada

Gough discussed oceanic measurements and their current and potential uses for coarse resolution ocean and climate modeling. He reviewed ocean measurements and climatologies, discussed how these measurements are used, and then focused on the use of inverse methods to estimate the oceanic flow field and mixing coefficients that are of potential use to modelers. He then assessed the efficacy of such approaches and the complicating problems of the ocean data. The following questions were examined: What comprises ocean data? How is it currently used? What are inverse methods? Can large scale mixing coefficients be estimated by inverse methods? Where do we go from here?

What comprises ocean data?

Humankind has a long history of observing the sea. Traditionally, temperature and salinity dominate the recorded observations, although in more recent years these data have been complemented by an increasing array of geo- and biochemical tracers such as dissolved oxygen, tritium, nitrates, phosphates, and CFCs. Direct current measurements have also been made. In the last decade, satellite technology has greatly enhanced our ability to assess the ocean surface. There are, however, strong biases in the ocean data set, i. e., for practical reasons there are many more surface observations than measurements of the deep ocean. The North Atlantic Ocean is relatively data rich compared to other parts of the world ocean.

The nature of the ocean data set in some respects has lead to a specific oceanographic culture or paradigm. Traditional views of ocean circulation in many ways may still haunt our current approach to collecting and using ocean data. Upscaling of site-specific measurements is a case where our perceptions of what processes are taking place may influence the use of the data.

How are ocean data used?

The question that has faced oceanographers is how to use this spatially and temporally incoherent data which is neither synoptic (snapshot at one time) nor truly climatological. With the heavy weighting of surface data, the question has been refined to: How much can be inferred about the ocean circulation, both surface and deep, from surface observations?

Upscaling of site-specific measurements is a case where our perceptions of what processes are taking place may influence the use of the data.

Two distinct camps have evolved to try to answer this question. The first is composed of prognostic modelers who have developed models of the ocean circulation based on the fundamental physics of fluid flow. Oceanic data are used by this group in two ways, estimating (parameterizing subgrid scale processes using data to tune the model) and for validating the model.

The other camp uses ocean data to infer what is occurring in the ocean. Olbers (1989) quotes Fofonoff, "Given the answer, what was the question?" This has been the traditional approach in descriptive oceanography. Following temperature and salinity distributions has lead to the characterization of water masses in the oceans, an extremely useful representation of the world ocean circulation, although potentially misleading (Wunsch, 1996). The view that the world ocean is a simple, large scale flow prevails, often in subtle ways, in much of oceanic thought. However, the MODE experiment of the 1970s and current meter measurements showed the presence of mesoscale eddies (tens of km scale) which locally swamp the large scale signal in the measurements. This approach has evolved rapidly in the last twenty years through the use of powerful mathematical tools usually referred to as inverse methods. These tools, discussed below, use hydrographic (temperature and salinity) and tracer data to calculate velocity fields and in some cases mixing coefficients.

What are inverse methods?

As defined above, inverse methods seek to determine characteristics of the ocean flow from the available data. To do this, a theoretical framework that can be exploited needs to be developed. In essence, given that we have some oceanographic measurements, we can backtrack and deduce the oceanic dynamics (flow and diffusion) necessary to produce the measured distribution of ocean properties such as temperature, salinity and other tracers. Refer to Wunsch (1996) for details on this methodology.

Can large scale mixing coefficients be determined from inverse methods?

Smaller scale processes, such as mesoscale eddies, are modeled as diffusive processes. This modeling is referred to as parameterization, the representation of subgrid scale processes using resolvable model variables. In this instance, the mesoscale eddies are assumed to behave diffusively. This is a presumption of an emergent property. Holloway (1989) says that this parameterization lacks, "... systematic derivation from some averaging procedure over sub grid scale motion,"but that it is "... understandable at an intuitive level and is relatively straightforward to implement into models."

There is a high degree of anisotropy with oceanic diffusion; horizontal diffusion is typically represented as seven orders of magnitude larger than vertical diffusion. It has been found that ocean models are highly sensitive to the magnitude of diffusion constants. Bryan (1987) found that the strength of meridional flow depended strongly on the value chosen for vertical diffusivity.

Mesoscale eddies (tens of km scale) locally swamp the large scale signal in the measurements.

A further subtlety is introduced by considering that mixing occurs predominantly along isopycnals (surfaces of constant density) rather than horizontally. It is therefore of interest to obtain estimates of mixing coefficients from observations and to investigate if mixing preferentially occurs along isopycnals in this data.

Obtaining mixing coefficients from observations

Mixing coefficients have been estimated using inverse methods in a number of studies using both climatological and "synoptic" data for both large and regional scale circulations. The results have been mixed and raise a number of issues about this approach and the data used. Olbers et al. (1985) and Olbers and Wenzel (1989) examined climatological data in the North Atlantic and Southern Ocean respectively. Mixing is explicitly represented in the tracer conservation equation along and across isopycnals. The Levitus (1982) data set is used for the North Atlantic analysis. Using inverse methods flow velocities and mixing coefficients are estimated. The surface flow produced in this work appears quite reasonable. Values for the isopycnal mixing coefficient vary from 1.0 to 3.0 x 103 m2/s for the upper 800 m of the ocean and 0 - 10.0 x 102 m2/s for the deeper ocean. These estimates are similar to that typically used in ocean general circulation models (103 m2/s). The diapycnal diffusivities ranged from 0 - 3.0 m -4/s in the upper ocean and 0 - 1.0 m-4/s in the deep ocean. Peak values for both diffusivities occurred in Gulf Stream and North Atlantic Drift regions.

Olbers et al. (1985) called into question the suitability of using the Levitus data set for this type of calculation and its impact on his results. The temperature and salinity data were collected from hydrographic stations and bathythermograph surveys. The data was averaged and gridded onto 1° square resolution. This is a smoothing, filtering, and interpolating process that aims to reduce data noise arising from both measurement and unresolvable processes. If this is not done there is the potential of aliasing high frequency oscillations into lower frequencies. In this process the upscaling of site specific measurements has been done by eliminating the variability. This resulted in the western boundary flow (Gulf Stream) being less intense than expected. They also assert that much of the diffusive structure implied in the isopycnal and diapycnal coefficients may be due to the climatological averaging.

It has been found that ocean models are highly sensitive to the magnitude of diffusion constants.

In Olbers and Wenzel (1989) the Southern Ocean was examined using a different climatological data set. Mixing coefficients (along with reference velocities) were calculated. The mixing parameterization was tested in two ways by orienting the anisotropic mixing first, isopycnally, and then, horizontally. In this way, isopycnal and diapycnal mixing coefficients and vertical and lateral coefficients can be compared. It was found that there was not a significant difference between the two parameterizations. In a zonal average diapycnal diffusivity ranged from 3.0 x 10 -5 m2/s to 3.0 x 10-4 m2/s for the upper ocean and an order of magnitude larger for the deeper ocean. Spatially the peak value is coincident with the strong Antarctic Circumpolar Current, a result consistent with the North Atlantic analysis. The isopycnal diffusivity ranges from 102 m2/s to 103 m2/s for the upper ocean with a reduction by a factor of two for the deeper ocean levels. Once again the peak value tended to coincide with the Antarctic Circumpolar Current. As before, diapycnal/vertical diffusivity values fall within the range of values currently used in ocean modeling. However, the vertical structure of diapycnal diffusivity is different for the two locations, decreasing with depth in the North Atlantic and increasing with depth in the Southern Ocean. It is possible that the increasing diffusivity for the Southern Ocean is a result of periodic convection which would tend to result in an increase in diapycnal diffusivity.

Because of the reputed inconsistencies in the original Levitus data and its gridding procedure (Wunsch, 1996), it is of interest to examine the calculation of mixing coefficients using a temporally and spatially coherent data set. This had been attempted by several researchers. Tziperman (1988) concluded that the inverse model could not fully resolve the mixing coefficients, i. e., they were not significantly different from zero and were not needed to produce a reasonable inverse.

Where do we go from here?

The upscaling of site specific oceanographic data has been reviewed. Specifically the focus was on obtaining mixing coefficients from existing oceanographic data sets suitable for use in coarse resolution models. Usable coefficients have been obtained by employing methods on climatological data. This was based on the presumption of an emergent property when upscaling, i. e., that sub-grid scale processes acted diffusively. This is further hampered by the necessary use of smoothing, filtering and interpolating in order to produce an apparently cohesive climatology of ocean observations. How useful or meaningful are the coefficients generated? Are they an artifact of a presumptive emergent property or massaged data and perhaps are overestimates as suggested by Wunsch (1996)?

There is also the possibility that oceanographic community has fallen into a tautological pitfall. Olbers et al. (1985) used the Levitus data set. This data set is also commonly used by ocean general circulation modelers. In the model development, tunable parameters, such as the mixing coefficients, are adjusted to achieve a "reasonable" flow as measured by the overturning strength, thermocline depth, and bottom temperature. Is it surprising that the inverse of the Levitus data set, which uses a simple conceptual framework for the ocean physics, produces mixing coefficients similar to those that are used in the models? In some respects both groups may have fallen victim to the traditional view of smooth oceanic flow.

Tunable parameters, such as the mixing coefficients, are adjusted to achieve a "reasonable" flow as measured by the overturning strength, thermocline depth, and bottom temperature.

References

Bryan, F., 1987: Parameter sensitivity of primitive equation ocean general circulation models. J. Phys. Oc., 17:970-985.

Holloway, G., 1989: Parameterizing sub-grid scale processes. In Ocean Circulation Models: Combining Data with Dynamics . D. Anderson and J. Willebrand (eds.). Kuwer, Dordrecht.

Olbers, D., Wenzel, J. and Willebrand, J., 1985: The inference of North Atlantic circulation patterns from climatological hydrographic data. Revs. Geophys., 23:313-356.

Olbers, D., 1989: A geometrical interpretation of inverse problems. In Ocean Circulation Models: Combining Data with Dynamics , pp. 65 -94, D. Anderson and J. Willebrand (eds.), Kuwer, Dordrecht.

Olbers, D. and Wenzel, J., 1989: Determining diffusivities from hydrographic data using inverse methods with application to the Circumpolar Current. In Ocean Circulation Models: Combining Data with Dynamics , pp. 95-140, D. Anderson and J. Willebrand (eds.). Kluwer, Dordrecht.

Tziperman, E., 1988: Calculating the time-mean oceanic general circulation and mixing coefficients from hydrographic data. J. Phys. Oc. , 18:519-525.

Wunsch, C., 1996: The Ocean Circulation Inverse Problem . Cambridge University Press. Cambridge. 442 p.

In some respects both groups may have fallen victim to the traditional view of smooth oceanic circulation.
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