Why should we keep working on theory and fluid dynamics in climate sciences?

By: Prof. David Ferreira

Recently, a colleague pointed out that, in the Northern Hemisphere subtropics, the winds blow eastward around 30-40N (the jet stream), and westward around 20-10N (the trade winds), and these winds make the ocean spin in the clockwise direction. This is akin to thinking of the ocean as a solid plate rotating around an axis: if one pushes the plate in one direction on one side, and the opposite direction on the other side, the plate will spin.

Fig. 1: The observed Mean Ocean topography estimated from a combination of satellite measurements and ocean drifters from Maximenko et al. (2009). Data, figure, and further details can be found here https://apdrc.soest.hawaii.edu/projects/DOT/. The pink arrows represent the dominant winds, while the blue arrows show the directions of the ocean current averaged over the full depths of the ocean.

Fig. 1 shows the Dynamical Ocean Topography (DOT). The currents tend to flow along the contours of DOT. Focusing on the North Pacific, the figure highlights the most striking example of a subtropical gyre, with a clockwise motion between about 10 and 40N (but these gyres are found in all basins). Applying the solid plate reasoning to the southern hemisphere suggests that the gyres spin counterclockwise (which they do! See Fig. 1).  

A “correlation” analysis supports the solid plate explanation. All subtropical gyres spin in the direction imparted by the winds, and indeed on some level, this is a decent explanation of subtropical gyres.  However, the remark of my colleague provides us with an opportunity to emphasize why we bother with fluid dynamics, equations and idealized modelling. 

The solid plate reasoning hides a complex dynamic. When the wind blows over the ocean, the first effect is not to make it spin. The winds first create horizontal currents confined to the upper 50-100 m of the oceans. Somewhat unintuitively, these currents, called Ekman currents, are at right angle of the wind and to their right (in the Northern Hemisphere). The meeting of the Ekman currents around 25N creates a bulge of the ocean surface between the Jet stream and the trade winds, but also a descending motion w_e (water cannot accumulate in the surface layer forever). This is where we use a version of the Sverdrup balance: 

βV = f w_e

where f is the Coriolis parameter (Earth is spinning on itself), V is the north–south flow in the deep ocean and β represents the variations of the Coriolis parameter with latitude.

Fundamentally, the parameter β represents a geometrical effect: Earth is a sphere. If Earth was flat, β would be zero. A direction that particularly matters for the dynamics is that along the axis of rotation. The sphericity of Earth implies that, even in a flat bottom ocean, columns of water aligning with the axis of rotation are smaller near the pole than near the Equator (Fig. 2 a).

Fig. 2: Schematic showing the variations of the height of water column in the direction of the rotation axis. Case a) corresponds to a flat-bottom ocean where the depth of the ocean is the same everywhere in the direction of gravity. Cases b) and c) illustrate what happens when the bottom of the ocean is tilted in some region of the planet.

We can now interpret the Sverdrup balance (1). The wind pushes water downward into the ocean interior (w_e <0) making water columns taller. As a result, water columns must move southward where they can accommodate their new height. Indeed, we see in the real ocean (Fig. 1) that the southward motions of the ocean spread over the full width of the basin and is only broken in a very narrow region on the western side of the basin (in the Kuroshio current, the Gulf Stream, the Agulhas current etc.).

Obviously, we cannot experiment with the real ocean to test our understanding. However, we can do it in numerical models that solve more complex equations than Eq. (1) (where nonetheless we can afford some simplifications to make things easier).

This is illustrated in Fig. 3 using a square basin with a flat bottom. The ocean is assumed to be of uniform density. It is driven by eastward wind to the north of the basin and westward wind to the south, mimicking the real-world wind pattern. Fig. 3a shows that the circulation is southward throughout the basin, except in a very narrow current on the western boundary.

Fig. 3: Simulations with a numerical ocean model (the MITgcm, mitgcm.org). The pink arrows in the top left panel show the direction of the winds. The blue arrows indicate the direction of the currents. The upper row has a flat bottom ocean with (a) and without (b) beta effect. In the bottom row, the sea floor is tilted, shallower in the northern part of the basin (corresponds to Fig. 2c)) and shallower in the southern part of the basin (corresponds to Fig. 2b)

What happens if we set β to zero, i.e. make a flat Earth? This is shown Fig. 3b (same winds as in Fig. 3a): the circulation becomes circular around the center of the basin. The flow is southward over half of the basin, and northward over the other half, nothing like the real ocean.

The geometrical interpretation of β suggests that the shape of the ocean bottom could play a role here. Imagine that, in some region of the ocean, the sea floor is getting deeper as one moves northward. Thus, seen from the perspective on the sphere, water columns in that region have similar heights (Fig. 2b). The tilted bottom floor makes Earth “flatter” in this region. This effect is called the topographic β-effect. In this case, it partially cancels effect. For a given wind, columns need to travel southward faster than before to accommodate for the pilling up of water by the wind. So, the gyre must be stronger.

Conversely, if the ocean floor is getting shallower moving northward in some region (Fig. 2c), the height of water columns varies much more, the Earth is more “spherical”. The topographic β-effect reinforces the normal β-effect, and we predict that the gyre should be weaker than for a flat bottom ocean.

Again, we can test our predictions through numerical experiments. Fig. 3c) and d) show the circulation in our simplified geometry, but now with tilted bottoms. As predicted, the gyre in Fig. 3c (corresponding to the tilt in Fig. 2b) is stronger than with a flat bottom while in Fig. 3d (corresponding to the tilt in Fig. 2c) it is weaker than with a flat bottom.

The β-effect is therefore critical to explain the characteristic pattern of the subtropical gyres. Something we could not get from the “correlation” analysis. One could do much more with the theory. It is possible to make more analytical advances, which reveals that β also controls the width of the narrow western boundary current (the smaller β, the wider the boundary current). Incidentally, the analytical solution can be used to test numerical models (and be happy that discretization, used in climate predictions, can be trusted). Retrospectively, it appears that the correlation analysis of wind patterns and gyres did not tell us much that was interesting.

The gyre dynamics discussed above was done decades ago and is pretty much textbook material nowadays, but the need to develop theory for new research remains. Recently, we explored the connection between Arctic sea ice decline and Ocean heat transport under climate change (Aylmer et al. 2024). Climates models that predict larger sea ice decreases in the future also have larger increases in ocean heat transport into the Arctic. This sounds intuitively OK, but is it what happens? Is it even quantitatively possible? That is, is the additional heat to the Arctic ocean large enough to explain the additional melting in these models? Less intuitive, models that predict more heat to the Arctic by the atmosphere tend to show relatively less sea ice retreat. These correlations have been noted for decades, but their interpretation has remained speculative. Aylmer et al. (2024) provides a theoretical framework that predicts, quantitatively, the behavior of the climate models, providing firm ground for interpretation.

Interestingly, this work started in 2019 with the development of an analytical model of climate by Dr Jake Aylmer during the first year of his PhD (published in Aylmer et al., 2020).

Yes, it takes time. As our field turns more and more toward statistical analysis, big data, and is pushed toward more applied, impactful research by funding agencies (with cascading effect on doctoral training programs), there is less and less room for this type of work.

As I am polishing this blog, I am listening to a seminar by David Karoly discussing Hoskins and Karoly (1981), a paper which had a massive impact in the field of meteorology and climate dynamics. This paper has been cited more than 3000 times and is still cited 100 times per year 40 years after publication! David Karoly pointed out that this work had major limitations and should be critically evaluated before being cited. Hoskins and Karoly (1981) is grounded in mathematics, geophysical fluid dynamics, and idealized experiments. One needs to be trained in such things to critically assess and carry on fundamental research in climate science. Unfortunately, the number of PhD students trained with these skills is gradually decreasing.

Correlation analysis is cool, but a bit of theory on top is cooler.

References:

Aylmer, J., D. Ferreira, and D. Feltham, 2020: Impacts of oceanic and atmospheric heat transports on sea-ice extent. J. Climate, 33, 7197–7215. https://doi.org/10.1175/JCLI-D-19-0761.1 

Aylmer, J., D. Ferreira, and D. Feltham, 2024: Ocean heat transport regulates Arctic sea ice loss. In press in Nature Communications.  https://doi.org/10.21203/rs.3.rs-3249087/v1 

Hoskins, B. J., and D. J. Karoly, 1981: The Steady Linear Response of a Spherical Atmosphere to Thermal and Orographic Forcing. J. Atmos. Sci., 38, 1179–1196, 2.0.CO;2″>https://doi.org/10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2. 

Maximenko, N., P. Niiler, M.-H. Rio, O. Melnichenko, L. Centurioni, D. Chambers, V. Zlotnicki, and B. Galperin, 2009: Mean dynamic topography of the ocean derived from satellite and drifting buoy data using three different techniques. J. Atmos. Oceanic Tech., 26 (9), 1910-1919. DOI: https://doi.org/10.1175/2009JTECHO672.1