the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Oct 2025
Is the Lorenz reference state global or local and observable?
Abstract. Introduced over 70 years ago by Lorenz, the theory of available potential energy (APE) remains central to atmospheric and oceanic energetics. Yet the precise nature of its reference state is still debated and often misinterpreted. Because it is usually constructed from an energy-minimising adiabatic rearrangement of mass, the Lorenz reference state is commonly regarded as a global property of the fluid, requiring interactions between distant parcels. We argue instead that, analogously to the gravitational field, it should be viewed as a local and observable property of the environment. Gravity, though global in origin, functions in practice as a local property measurable from its effect on falling bodies. Likewise, the Lorenz reference density and pressure profiles, ρ0(z) and p0(z), can be inferred from observations of buoyancy oscillations near equilibrium. Farther from equilibrium, a structural analysis of the governing equations shows that only deviations from the reference state affect motion, regardless of amplitude, thereby recovering Lorenz's APE separation as a structural property of fluid mechanics. The Lorenz reference state is therefore best understood not as an arbitrary mathematical construct, but as an environmental constraint manifesting locally, reinforcing both the foundations of APE and its role in theories of ocean circulation and mixing.
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Status: open (until 14 Nov 2025)
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CC1: 'Comment on egusphere-2025-4595', Rainer Feistel, 14 Oct 2025
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AC2: 'Reply on CC1', Remi Tailleux, 19 Oct 2025
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The community comment by Prof. Feistel appears to be identical to that posted as a referee comment. For a response to this comment, see my response to the referee comments, also by Prof. Feistel.
Citation: https://doi.org/10.5194/egusphere-2025-4595-AC2
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AC2: 'Reply on CC1', Remi Tailleux, 19 Oct 2025
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RC1: 'Comment on egusphere-2025-4595', Rainer Feistel, 18 Oct 2025
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please refer to the pdf attached
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AC1: 'Reply on RC1', Remi Tailleux, 19 Oct 2025
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See my responses in the attached pdf
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AC1: 'Reply on RC1', Remi Tailleux, 19 Oct 2025
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RC2: 'Comment on egusphere-2025-4595', Andy Hogg, 01 Nov 2025
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This paper examines the definition of Available Potential Energy as per the Lorenz Reference State. The use of the Lorenz Reference State for oceanography has been debated over the last decade and a half, without a clear resolution. This paper looks to provide clarifying remarks on the nature of the Lorenz Reference State, with a view to providing that clarity.
One significant issue with the Lorenz Reference State is that it is defined relative to the global density field. This means that, in principle, altering the density in the Arctic Ocean may alter the energetics in the Southern Ocean, even though it’s somewhat unclear how that information could be communicated across the planet.
A second issue is that, since the Lorenz Reference State is regarded as the zero APE state, some people have interpreted that state as being somehow real. The problem with this view is that there is no practical way that the Lorenz Reference State can actually be attained in any real fluid with a complex density structure. It follows the Lorenz Reference state is in most cases just a hypothetical state; this leads to the question of what is gained by measuring Potential Energy relative to an unattainable state.
This paper aims to show that the Lorenz Reference State can be regarded as a local quantity, thereby resolving the first of the above two issues. It also links the local nature of the Lorenz Reference State with the governing equations, implying that Available Potential Energy can be directly recovered from the equations in this way.
To be honest, no-one would be happier than me if the arguments in this paper held water — it would allow us to overcome one of the last remaining obstacles to the widespread adoption of the Available Potential Energy framework in oceanography. Unfortunately, however, I remain unconvinced. I have two major objections which I will list below. But I don’t think this paper is ready to publish in its current form, and I can’t see a path to publication without major additional improvements.
- I like the analogy with gravity, but it is imperfect. In particular, the value of g = 9.81 m/s^2 is an approximation, and gravity does vary spatially. That is fine — it can still be measured locally. I also agree that the buoyancy frequency can also be measured locally. This measurement does give a local approximation to Available Potential Energy, and it’s well-known that, when considering linear internal waves, that local approximation is sufficiently accurate to close the wave energy budget. But that doesn’t mean that approximation suits all purposes. In particular, as first shown by Hughes, Hogg & Griffiths (2009) (curiously not cited here), when considering the large-scale overturning circulation and associated mixing, the local linear approximation is insufficient to yield understanding of the ocean energy budget. In other words, I don’t think it has been shown that the arguments made in section 3 apply beyond the linear range. More work is needed here.
- I’m not convinced that section 4 shows what the author claims to show. I accept the framework as it is laid out, and the argument that Available Potential Energy can be written from the dynamical component of static energy. However, I would point out that Equation (8) relies on the Lorenz definition of the Reference State, which remains global. In other words, to evaluate that statement, one still needs a globally defined reference state. So, even if the static energies themselves can be separated in this way, that doesn’t imply that there is an “external constraint” controlling the energy — because the separation itself depends on having this global knowledge. In other word, even though section 4 has some nice features, I am not convinced that it proves what it sets out to prove.
The issue at hand here is that there actually IS a non-local available potential energy effect. As a thought experiment — if I could take the current ocean, and magically make all water in the northern hemisphere 10 kg/m^3 denser, then most of it would sink below the southern hemisphere water. This would happen even though I haven’t made any changes to water the southern hemisphere — its energy and level of neutral buoyancy has been directly altered by a remote and magical effect.
In summary, I remain to be convinced that the author has demonstrated the Lorenz Reference State can be considered a local quantity. Instead, I would posit that the solution to the issue of nonlocal energetics is simply that physics is more complicated than we would like it to be!! And that potential energy in a fluid is more complicated than kinetic energy. In particular, the nonlocal generation of Available Potential Energy is not such an issue if one does not argue for the Lorenz Reference State as being either real or attainable. Instead, we get the most value from Available Potential Energy theory by considering the fluxes of energy from one reservoir to another, rather than the total amount of energy. It follows that a hierarchy of approximations to the exact Lorenz Reference State (from the local linear example proposed here to semi-local approximations) will help us to extract a full understanding of the energetics of the ocean.
Citation: https://doi.org/10.5194/egusphere-2025-4595-RC2
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Publisher’s note: this comment is a copy of RC1 and its content was therefore removed on 20 October 2025.