research

My research deals with the representation of turbulence in an isopycnal model (HIM) of ocean dynamics. A little more detail is provided towards the end of this section, while the following few paragraphs are meant as a general introduction to this area of science and its main unanswered questions (with, as readers, mainly those rare existing and hypothetical individuals in mind who would, in an actual or hypothetical situation, ask me: “So, what is it exactly that you do?”)

General Overview

Climate Models

Climate research’s most important tool are computer-based models which simulate the dynamics and thermodynamics of the atmosphere and oceans. As we learn more and more about the earth and its climate, the importance of ever more additional physical, chemical and biological processes in determining the behavior of the whole, integrated system is recognized, and once these processes are well-enough understood, they are incorporated into the latest models. (As an obvious example, the amount of absorption of sunlight throughout the atmosphere, and therewith the overall energy supply to the earth, is dependent on the abundance and distribution of many gases – in particular carbon dioxide. The abundance of carbon dioxide, in turn, depends not only on how much of it we humans pump into the atmosphere, but also on how much is taken up by biological and other processes in the oceans and on land.)

These computer models are based on differential equations that describe the motions of fluids (in their most comprehensive form, the Navier-Stokes equations). Computers can, however, represent real world entities only by means of a finite collection of numbers – the 10 million pixels of your digital photograph, for example, or the 44.100 times per second that a sound signal is sampled to produce a CD. Equations, too, can only be solved at a finite number of points by a computer, and how many of them depends on how powerful your computer is. Climate models usually represent the earth as a sphere covered by a regular grid of discrete points; at the moment, a typical resolution of such grids is about one point per degree latitude and longitude. More becomes difficult to handle for all but the most powerful supercomputers.

These lattices of points on the sphere correspond to physical variables (such as wind velocities or water temperatures) which in the real world occur in continuous fields. In solving the equations, the computer will spew out a number at every point, for which two straightforward but different interpretations spring to mind: a point may represent the value of the physical variable at precisely this point; or it might represent an average over a certain area of this variable, where all such individual areas (called grid cells) taken together cover the whole globe exhaustively – much like tiles cover the bathroom floor. Which or whether any of these interpretations is the correct one depends solely on the way the model has been formulated; but let us, for the remainder, assume we are dealing with the latter one: with grid cell averages.

“Subgridscale Processes”

The important point is that in the real world, a lot is going on within one of these bathroom tiles (all of it being referred to as subgridscale processes). What exactly is going on, however, is impossible to tell within the framework of the computer simulation – unless, of course, we reduce the size of our tiles (or in proper terminology, increase the resolution of our grid). Apart from the limitations set by the available computer power, this game resembles a snake that swallows its own tail, since there will always be something happening on a level smaller than what we can resolve. Since our model points represent averages, this would not be a problem if our equations were linear, that is, if they did not contain products of some of the physical variables in our equations. Why? Because the product of two averages is not, in general, the same as the average of a product. For example, if our equations contain the product of the water’s velocity and density, computing this product everywhere within our grid cell (which is, of course, not possible) and then taking the average gives a different result than using the averages of velocity and temperature and then computing the product (which is the only the the computer can do). Yet the former would be the correct thing to do. It turns out that the nonlinear terms in the equations that cause this dilemma are precisely the reason for the existence of turbulence; in a way, subgridscale processes are simply that part of the turbulence that cannot any longer be resolved by the computer simulation. While turbulence itself is a process that often persists over a large range of spatial scales (it is said to be scale-invariant), it turns out that in the ocean, energy accumulates in eddies with a typical diameter of some tens of kilometers (so called mesoscale eddies). These eddies are just small enough to escape the grid resolution of current global climate models (to slip through the cracks between the bathroom tiles, so to speak) – they just don’t occur in them. (In the atmosphere, on the other hand, turbulent energy accumulates on the much larger scale of cyclones and anticyclones, which can be adequately modeled.)

Hence a discrepancy arises between what should and what can be done, and ever since the dawn of computational fluid dynamics have scientist tried to come up with ways to remedy the situation. Parameterizations are attempts to come up with the `missing’ subgridscale terms (the difference between the average of the product and the product of the averages, if you will) from a somewhat different direction, namely by assuming that they can in some way be related to the larger-scale average fields (which the computer can and does calculate). It is not known to the present day whether this approach is actually justified; while theory seems to say, “not really” (but this is also not sure), there have been encouraging successes in practice, even though the ultimate goal of a perfect parameterization remains (and probably always will) elusive.

As I final note, I should point out that the problems with turbulence are by no means confined to computer simulations. In fact, without the help of computers, we don’t know how to solve the equations in the first place, and this, too, is mainly due to their nonlinear nature.

My Research

My own research deals with eddy parameterizations in the context of a specific abstract oceanographic situation: the dynamics of a wind-driven reentrant two-layer channel with topography (based on the model set-up in Hallberg and Gnanadesikan, 2001). Channel models are frequently used as highly idealized representations of the Antarctic Circumpolar Current (ACC), the world’s strongest ocean current, and, unsurprisingly, also its most turbulent one (e.g., MacCready and Rhines, 2001; Wang and Huang, 1995; Hallberg and Gnanadesikan, 2001).

Besides studying the dynamics of the ACC for their own sake, the topic carries implications for the large-scale ocean circulation and with it for global climate in general. All oceanic motion can be divided into two distinct regimes: wind-driven ocean currents and the thermohaline circulation. Wind-driven currents comprise the Gulf Stream and the Kuroshio, both of which owe their existence to the lateral confinement of the (Atlantic and Pacific) oceans into their respective basins, at the western edge of which they occur (hence the terminology western boundary currents). The ACC is also wind-driven, but occurs at a latitude where there is no north-south blockage through continents, which allows it to encircle the globe freely. Whereas the wind-driven currents (at least those other than the ACC) are for the most part confined to the ocean surface, the thermohaline circulation is the deep-reaching counterpart that is driven by differences in water density (caused, in turn, but differences in temperature and salinity). Often also referred to as the Meridional Overturning Circulation (MOC), it connects the northern with the southern hemisphere and, interacting with the ACC, also the different ocean basins amongst each other.

On of the most prominent (and most talked-about) features of the MOC is the sinking of dense water in the vicinity of Greenland in the North Atlantic. This sinking is intimately connected to the formation of sea ice, because the freezing water rejects salt and makes the remaining liquid water more saline and hence more dense. While the media has often talked about the “slowdown of the Gulf Stream” by 30% due a reduction in sea ice formation, it is actually the MOC which is affected (Bryden et al. 2005).

Interestingly, it has been shown that the ACC can have an effect on the magnitude of North Atlantic sinking and the strength of the MOC (Toggweiler and Samuels, 1995). This has to do with the fact that the water that sinks in the North Atlantic needs to well up somewhere else, and it has been suggested that it does so to a certain (and debated) extent in the ACC (cf. Gnanadesikan, 1999). In course resolution ocean models such as the one used by Toggweiler and Samuels (1995), any change in upwelling in the ACC (potentially due to changes in the strength of the winds in the region) translates directly into a change of the strength of the MOC. However – and we finally come full circle – the presence of eddies (only possible in higher resolution models) changes this picture, since eddy-induced flows can lead to a short-cut of the overturning circulation in the southern hemisphere, reducing the influence of the ACC on the global MOC (Gnanadesikan, 1999; Hallberg and Gnanadesikan, 2001). How this affects the integrity of global climate simulations remains to be studied in more detail; however, it is clear that as long as such simulations cannot resolve ocean eddies, they will always benefit from improved eddy parameterizations.

(As a second final note, the Meridional Overturning Circulation is often quoted as an important mechanism for poleward heat transport by the climate system. It seems more and more clear, however, that compared to the atmosphere and wind-driven currents, the MOC is of relatively minor importance in absolute terms; cf. Trenberth and Caron, 2001. Nonetheless, it may well be a major player as far as variability of the transport is concerned.)

References