Local available potential energy: what is it and why we need it

By: Prof. Remi Tailleux 

As is well known, atmospheric winds and ocean currents ultimately derive their energy from the Sun. In general, this involves a two-step process, whereby the solar energy is first transformed into potential energy (PE) before finding its way into kinetic energy (KE). Some aspects of this transformation remain mysterious and poorly understood, however, which are to do with the so-called `cooling paradox’. In the oceans, cooling at high latitudes creates dense waters that sink and give rise to the Atlantic meridional overturning circulation (AMOC). The puzzle here is that the cooling decreases the PE of the oceans. How then can it be converted into KE?  

The cooling paradox has been a recurring source of confusion and controversies in energetic studies of the atmosphere and oceans, so much so that one of the most famous textbooks on buoyancy-driven flows (Turner, 1973) stays away from energetics altogether. It is only with the work of Lorenz (1955) that a resolution of the cooling paradox started to emerge. Lorenz’s key contribution was to remark that some configurations, such as the equilibrium resting state depicted in the right panel of Fig. 1, possesses lots of potential energy, yet none of it is readily available for conversions into KE.

Figure 1. Schematics depiction of the potential temperature field in the atmosphere. Left panel shows that in the actual state, potential temperature varies with latitude, with cold air at in the surface layers in the polar regions (indicated in purple). Right panel shows the reference or global equilibrium state of the atmosphere, obtained by allowing each layer to relax towards their equilibrium state while retaining their mass and potential temperature. The equilibrium state is horizontally uniform and therefore a function of height or pressure only. Reproduced from Tailleux (2013).

Another way to look at the problem is by considering the two fluid configurations illustrated in Fig. 2. Both configurations have identical amounts of potential energy, but only the one in the right panel is expected to develop motions. Lorenz therefore posited that potential energy must come into two distinct flavours: Available potential energy (APE) and background potential energy (BPE): 

                                                           PE = APE + BPE 

Figure 2. Schematics illustrating two idealised stratifications having the same potential energy (PE), but a different partition APE/BPE. Intuitively, the fluid on the left is purely static and has no APE. No motion is expected to develop. The fluid on the right, however, is clearly unstable, as intuitively, one expects the fluid on the top right to start sinking owing to being denser than the other parcels. The fluid on the right has APE, while the fluid on the left has none, despite the two fluids having the same PE. Reproduced from Hughes et al. (2009).

Lorenz APE theory resolved the cooling paradox because although high latitudes cooling decreases both the PE and BPE of the oceans, it increases the APE, which is what matters, as this is the part of the PE that can be converted into KE.  

While the Lorenz discovery was revolutionary and dramatically altered energetics studies in the oceans and atmosphere, this is not to say that everybody was happy with Lorenz formulation of APE theory. Indeed, unlike kinetic energy, which can be defined for individual fluid parcels, Lorenz APE could only be defined for the fluid as whole. This was a major issue, which prompted the quest for local principle. Understanding how to do this took over two decades, as the first satisfactory local APE theory only appeared in 1981 as reviewed in Tailleux (2013). These early formulations, however, were still relatively obscure and complicated. It took over two decades for the local APE theory to be digested and reformulated before it started to be used in practical applications. One key process that can only be discussed with the local APE theory is the advection or transport of APE between different regions. As it turns out, advection of APE was recently established to be of key importance for understanding the energetics of atmospheric storm tracks (Novak and Tailleux, 2008) and of tropical cyclone intensification (Harris et al, 2022) by two PhD students in the meteorology department, thus highlighting the usefulness and importance of the local APE framework.   

Another advantage of the locally defined APE is that it can be partitioned into `mean’ and `eddy’ components, similarly as what is commonly done with kinetic energy. In the atmosphere, the mean APE and kinetic energy (KE) can be seen in Fig. 3 to characterise the large-scale circulation of the atmosphere, whereas the eddy component characterise the storm tracks, that is, the regions dominated by low pressure systems. In the oceans, one may similarly diagnose the eddy APE, which similarly characterise the storm tracks of the oceans, as depicted in Fig. 4.  

Figure 3. Mean and eddy components of Available Potential Energy and Kinetic Energy in the Northern Hemisphere. (a) Mean APE, (b) Mean KE, (c) Eddy APE, (d) Eddy KE. Reproduced from Fig. 5.8 of Novak (2016).

So far, the power and usefulness of the local APE framework has only been used in a few studies and therefore remain under-exploited. The local theory of APE is not completely settled yet and continue to evolve (Tailleux, 2018). Nevertheless, it is now clear that it represents a much better framework than Lorenz global APE theory, but more work is needed to unlock its full potential.    

Figure 4. Eddy APE at 175 m estimated from the monthly Armor3D data. Regions of intense meso-scale eddy activity appear in red. Notable regions are the Gulf Stream and Kuroshio regions and the Southern Ocean.

References and Further Reading: 

Harris, B.L., R. Tailleux, C. E. Holloway, and P.L. Vidale, 2022. A moist available potential energy budget for an axisymmetric tropical cyclone. J. Atm. Sci., 79, 2493—2513 

Hughes, G., O, A. M. Hogg, and R. W. Griffiths, 2009: Available potential energy and irreversble mixing in the meridional overturning circulation. J. Phys. Oceanogr., 39, 31300—3146. 

Lorenz, E. N., 1955. Available potential energy and the maintenance of the general circulation. Tellus, 7, 157—167. 

Novak, L., 2016. The lifecycle of storm tracks. PhD Thesis. University of Reading.  

Novak, L, and R. Tailleux, 2008: On the local view of atmospheric potential energy. J. Atm. Sci., 75, 1891—1907.  

Tailleux, R., 2013. Available potential energy and exergy in stratified fluids. Annual Review of Fluid Mechanics. 45, 35—58.  

Tailleux, R., 2018. Local available energetics of multicomponent compressible stratified fluids. J. Fluid Mec., 842, R1.  

Turner, JS, 1973. Buoyancy effects in fluids. Cambridge University Press.  

About sdriscoll

https://twitter.com/SimonDriscoll_ Researching machine learning and thermodynamics of Arctic sea ice. Part of SASIP (2021-present) @UniofReading (Schmidt Futures). Previously DPhil Physics @UniofOxford (climate/volcanoes/geoengineering). Also nuclear war/winter + X-risk.
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