# How do we measure (or empirically calculate) the amount of entropy in the atmosphere and oceans?

Here's a paper on the entropy budget1, where moist entropy is defined in equation (8) as

$$s = (1-q_t)(C_{pd} \ln T - R_d \ln p_d) + q_t C_l \ln T + \frac{q_v L_v}{T} - q_v R_v \ln \mathcal{H}$$

In this expression, $q_v$ is the speciﬁc humidity for water vapor, $C_{pd}$ is the speciﬁc heat at constant pressure of dry air, $C_l$ is the speciﬁc heat of liquid water, $T$ is the temperature of moist air, $R_d$ and $R_v$ are the gas constants of dry air and water vapor, $p_d$ is the partial pressure of dry air, $\mathcal{H} = e/e_s$ is the relative humidity, $e$ is the water vapor pressure, and $e_s$ is the saturation vapor pressure.

I'm just curious: is entropy directly measured, or indirectly measured from other sources? If we measured it, what would be the spatial-temporal resolution of such a measurement?

1 Pauluis, Olivier, and Isaac M. Held. "Entropy Budget of an Atmosphere in Radiative–Convective Equilibrium. Part I: Maximum Work and Frictional Dissipation." Journal of the Atmospheric Sciences 59: 125-139.

• Is the answer not in the paper or its references?
– gerrit
Apr 15 '14 at 22:07
• It's a very theoretical paper, so it doesn't discuss measurement much. It's also back from 1991, so I suspect that there are more ways to measure it now. But it still doesn't seem to be a very well-researched topic. Apr 15 '14 at 22:11
• I don't know, but I strongly expect it's not directly measured.
– gerrit
Apr 15 '14 at 22:12
• I don't really understand why that paper is linked from the question. Is there anything it says that adds to the question? If so, could you perhaps quote that bit, instead of just linking to the whole paper? Apr 16 '14 at 11:19

There is a convention for moist air entropy built from dry air entropy being 0 at T = 273.15K and P = 101325Pa, and liquid water entropy being 0 at its triple point. See the page numbered 47 of this NIST document:

Entropy is not measured directly.

Entropy $S$ is defined as $dS \equiv dQ_{rev}/T$

Therefore, entropy can measured by measuring heat transfer and temperature.

See Calculation of Entropy Change in Some Basic Processes for more information.

It is possible to compute the relevant moist air entropy by using the Third law values. This has been done first by Hauf and Höller (1987). A moist-air potential temperature has been defined by Marquet (2011, ..., 2017). This moist-air potential temperature is truly synonymous with the moist-air entropy and is different from the previous values published by Emanuel or Pauluis, because the Third law values generate a new term which has large physical impacts.

Hauf, T., and H. Höller, 1987: Entropy and potential temperature. J. Atmos. Sci., 44 (20), 2887–2901

Marquet, P., 2011: Definition of a moist entropy potential temperature: application to FIRE-I data flights. Quart. J. Roy. Meteorol. Soc., 137 (9), 768–791

Marquet, P., 2017: A third-law isentropic analysis of a simulated hurricane. J. Atmos. Sci., 74 (10), 3451–3471