The absolute seawater entropy: Part I. Definition

This paper defines the absolute entropy of seawater by refining the TEOS10 formulation with updated thermodynamic tables and reference entropies, documenting the resulting differences from previous Millero-based calculations, and setting the stage for a subsequent analysis of observed oceanic profiles.

The Gist

The Big Idea: Giving Seawater a 'True Zero for entropy'

Imagine you are trying to measure the height of mountains. For a long time, oceanographers and meteorologists have been measuring the height of mountains relative to sea level. If you say a mountain is "5,000 meters high," you are really saying it is "5,000 meters above sea level." This works fine for most things, like hiking or flying planes.

However, in physics, there is a concept called Entropy. Think of entropy as a measure of "disorder" or "messiness" in a system. The more mixed up the molecules are, the higher the entropy.

For decades, scientists calculated the entropy of seawater using a "relative" scale. They arbitrarily decided that at a specific temperature (0°C) and a specific saltiness (35 grams of salt per kilogram), the entropy is zero. It's like saying, "Let's pretend this specific mountain is sea level, and measure everything else from there."

The Problem: This "relative zero" is a made-up number. It doesn't reflect the true physical reality of the universe. Just as a mountain has a true height from the center of the Earth, seawater has a "true" entropy based on the laws of quantum mechanics and the Third Law of Thermodynamics.

The Solution: Dr. Marquet's paper is about calculating the "Absolute Entropy" of seawater. He wants to stop measuring from "Sea Level" and start measuring from the "Center of the Earth" (Absolute Zero).

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The Analogy: The Bank Account vs. The Wallet

To understand why this matters, let's use a financial analogy.

• The Old Way (Relative Entropy): Imagine you and your friend are comparing your bank balances. You both agree to ignore the first $1,000 in your accounts and just count what you have above that.

  • You have $1,500. Your "relative" balance is $500.
  • Your friend has $2,000. Their "relative" balance is $1,000.
  • This works fine if you are just comparing who has more right now. But if you want to know the total wealth of the universe, or if you want to calculate interest based on the total principal, ignoring that first $1,000 creates a mathematical error.

• The New Way (Absolute Entropy): Dr. Marquet says, "Stop ignoring the first $1,000! Let's count every single penny from zero."

  • He takes one of the standard formulas used by oceanographers (called TEOS-10) and adds a "correction term." This term accounts for the "missing money" (the absolute reference values of pure water and sea salts) that was previously ignored.

Why Did They Ignore the "True Zero" Before?

For a long time, scientists thought, "Does it really matter?"

• The Argument: Most things we measure in the ocean (like how dense the water is, how fast sound travels, or how much heat it holds) don't change if you shift the starting point of your entropy scale. It's like saying the difference in height between two mountains is the same whether you measure from sea level or from the center of the Earth.
• The Flaw: Dr. Marquet argues that while the differences might look similar, the absolute values matter when you are doing complex calculations involving mixing, turbulence, and chemical reactions. By ignoring the true starting point, the old formulas (specifically those by Millero from the 1970s) were actually "broken" in subtle ways. They were calculating a "relative" messiness that didn't match the laws of physics.

The "Recipe" for the New Calculation

Dr. Marquet didn't throw away the old recipe (TEOS-10); he just added a missing ingredient.

1. The Base: He uses the modern, highly accurate TEOS-10 software, which is the current gold standard for ocean physics.
2. The Correction: He adds a specific "saltiness adjustment."
   • He calculated the true "messiness" (entropy) of pure water at 0°C.
   • He calculated the true "messiness" of sea salts at 0°C.
   • He found that the difference between these two values is huge.
3. The Result: When you mix water and salt, the total entropy isn't just a simple average. Because the "true" entropy of salt is much lower than water, adding salt actually lowers the average entropy per kilogram of the mixture in a way the old formulas missed or underestimated.

Why Should You Care? (The "So What?")

You might ask, "I'm not a physicist; why does this matter?"

Dr. Marquet suggests that this isn't just about math; it changes how we understand the ocean's behavior:

• The Ocean's "Weather": Just as the atmosphere has "isentropic" lines (lines of equal entropy that air follows), the ocean has them too. By using the absolute entropy, these lines shift. This means the "paths" ocean water takes as it moves, mixes, and circulates might be different than we thought.
• Turbulence: When water swirls and mixes (turbulence), according to Richardson it follows the rules of entropy. If we use the wrong "zero point," our models of how heat and salt mix in the ocean are slightly off. This could affect climate models.
• Chemistry: The "messiness" of water affects how chemicals dissolve and react. If the entropy baseline is wrong, our understanding of how carbon dioxide or other chemicals behave in the ocean could be inaccurate.

The "Aha!" Moment

The paper includes a fascinating comparison. The old formulas (Millero's) showed that as you add salt, the entropy goes up in a weird, increasing curve. Dr. Marquet's new calculation shows that as you add salt, the entropy actually goes down (because salt is "more ordered" than liquid water).

It's like realizing that if you add a perfectly organized deck of cards (salt) to a pile of scattered cards (water), the pile actually becomes more organized on average, not less. The old math missed this (or underestimated it) because it was looking at the wrong starting point.

Summary

Dr. Marquet's paper is a call to upgrade the "ruler" we use to measure the ocean's entropy.

• Old Ruler: Starts at an arbitrary point (Sea Level). Good for simple comparisons, but physically incomplete.
• New Ruler: Starts at the absolute bottom (Absolute Zero). It aligns with the fundamental laws of the universe (Quantum Mechanics and the Third Law of Thermodynamics).

By switching to this "Absolute Entropy," scientists can build more accurate models of how the ocean moves, mixes, and interacts with the atmosphere, potentially leading to better climate predictions and a deeper understanding of our planet's chemistry.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in oceanographic thermodynamics: the lack of a rigorously defined absolute entropy for seawater.

• Current Standard (TEOS-10): The current standard, TEOS-10 (Thermodynamic Equation of Seawater - 2010), calculates seawater entropy based on a Gibbs function fitted to observational data. However, it employs arbitrary reference values for the entropy of pure liquid water and sea salts. Specifically, TEOS-10 sets the entropy of standard seawater to zero at $0^\circ\text{C}$ and standard salinity, effectively ignoring the Third Law of Thermodynamics.
• Historical Context: Previous attempts by Millero (1976, 1983) to define absolute entropy were based on older data and "relative" formulations that contained numerical inconsistencies and did not fully align with modern thermodynamic tables.
• Consequence: Because the reference entropy values are arbitrary in standard models, the absolute value of the entropy variable is undetermined. This prevents the calculation of absolute entropy gradients and fluxes, which are theoretically required for a complete application of the Second Law of Thermodynamics in turbulent mixing and climate modeling.

2. Methodology

The author proposes a methodology to reconcile the modern, high-accuracy TEOS-10 formulation with the Third Law of Thermodynamics (which stipulates that the entropy of a perfect crystal is zero at absolute zero, $0\text{ K}$).

• Reconciliation Strategy: The paper retains the complex, non-linear dependence of entropy on pressure, temperature, and salinity provided by the TEOS-10 Gibbs function but modifies the reference constants ($\eta_{w0}$ and $\eta_{s0}$) to match absolute thermodynamic values.
• Derivation of Reference Values:
  • Pure Water ($\eta_{w0}$): The author adopts the absolute entropy of pure liquid water at $0^\circ\text{C}$ ($\approx 3513.4 \pm 1.7 \, \text{J K}^{-1} \text{kg}^{-1}$), derived from calorimetric data and statistical mechanics (including Pauling-Nagle residual entropy).
  • Sea Salts ($\eta_{s0}$): The author updates the mean absolute entropy of sea salts using modern thermodynamic tables (NEA-TDB, NIST-JANAF) rather than the older Millero values. The value at $0^\circ\text{C}$ is calculated as $\approx 1633.3 \pm 15 \, \text{J K}^{-1} \text{kg}^{-1}$.
• Correction Term ($\Delta\eta_s$): A linear correction term is derived to convert the standard TEOS-10 entropy ($\eta_{std}$) to the absolute entropy ($\eta_{abs}$):
  $$\eta_{abs} = \eta_{std} + \Delta\eta_s$$
  Where $\Delta\eta_s \approx (-1880 \pm 17) \times \frac{S_A - S_{SO}}{1000}$ (in $\text{J K}^{-1} \text{kg}^{-1}$). This term depends on the difference between the absolute reference entropies of sea salts and pure water.
• Validation: The author validates the Third Law approach by comparing statistical-quantum calculations (Sackur-Tetrode-Planck) with calorimetric calculations for water vapor and ice, showing they agree within $0.02\%$ when residual entropy is included.

3. Key Contributions

• Definition of Absolute Seawater Entropy: The paper provides the first rigorous definition of absolute seawater entropy compatible with the TEOS-10 framework and the Third Law of Thermodynamics.
• Correction of Millero's Formulation: The author demonstrates that the "relative" entropy formulations in Millero (1976, 1983) were mathematically inconsistent and did not actually represent absolute entropy. The new formulation corrects these historical errors.
• Introduction of $\theta_\eta$ (Absolute Potential Temperature): The paper proposes a new conservative variable, $\theta_\eta$, analogous to potential temperature but derived from absolute entropy. This variable has dimensions of Kelvin and is designed to be the natural variable for turbulent mixing schemes in ocean models.
• Thermodynamic Consistency: The study proves that absolute entropy values are not merely arbitrary constants; they impact physical relationships, such as the link between latent heat of sublimation and saturation vapor pressure.

4. Results

• Magnitude of Differences: The difference between standard TEOS-10 entropy and the new absolute entropy is significant and non-linear.
  • The standard TEOS-10 entropy is nearly constant with respect to salinity in certain ranges.
  • The absolute entropy decreases as salinity increases (because the reference entropy of sea salts is lower than that of pure water).
  • At high salinities, the difference can reach tens of $\text{J K}^{-1} \text{kg}^{-1}$.
• Impact on Isentropic Processes: In Temperature-Salinity ($t-S_A$) diagrams, the isentropic lines (lines of constant entropy) for the absolute version differ significantly from standard TEOS-10 lines.
  • For a process changing salinity from 10 to $35 \, \text{g kg}^{-1}$, the temperature change required to maintain constant absolute entropy is approximately $4^\circ\text{C}$ greater than predicted by the standard TEOS-10 model.
• Physical Implications: The paper shows that the absolute entropy gradient drives turbulent fluxes. Using the standard (arbitrary) entropy in turbulence parameterizations may lead to incorrect representations of mixing, particularly in regions with strong salinity gradients (e.g., surface layers, polar oceans).

5. Significance

• Theoretical Rigor: The work establishes that seawater entropy should be treated as an absolute state variable, consistent with the laws of thermodynamics, rather than a relative quantity defined by convention.
• Climate and Ocean Modeling: By introducing the variable $\theta_\eta$, the paper offers a path to improve ocean general circulation models (OGCMs). Turbulent mixing schemes based on absolute entropy could better resolve "double diffusion" phenomena and the vertical structure of temperature and salinity.
• Chemical Equilibrium: The study highlights that absolute entropies influence chemical equilibrium constants (e.g., for bicarbonate ionization in seawater), suggesting that absolute entropy definitions are crucial for accurate ocean chemistry modeling.
• Paradigm Shift: The paper challenges the prevailing oceanographic view that reference entropy values are irrelevant conventions. It argues that for non-stationary states (where $dT \neq 0$ and $dS_A \neq 0$), the choice of reference values fundamentally alters the entropy budget and the resulting physical predictions.

In summary, Part I of this series redefines the theoretical foundation of seawater entropy, moving from arbitrary conventions to a physically grounded, absolute scale, and sets the stage for Part II, which will apply these concepts to real-world observational data and surface datasets.