# Understanding Atmospheric Behaviour in Terms of Entropy: A Review of Applications of the Second Law of Thermodynamics to Meteorology

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Major Concepts of Entropy Used Frequently in Meteorology

#### 2.1. Thermodynamic Entropy

#### 2.2. The Entropy of Statistical Mechanics

_{i}is the probability of a microstate with k

_{B}as the Boltzmann constant.

#### 2.3. Information Entropy

_{1}, ..., x

_{n}} is:

## 3. On Entropy Balance

_{p}lnө with C

_{p}being the specific heat at constant pressure and ө the potential temperature that relates temperature T by:

_{0}reference pressure that is usually taken as 1000 hPa. In meteorological applications, not only is the total entropy for the whole climate system regarded as being an invariant but also entropy is assumed to be unchanged in many cases.

## 4. Entropy Production

**Figure 1.**A schematic of energy transport processes in the planetary system of the Earth, the Sun and space. The Earth receives the shortwave radiation from the hot Sun and emits longwave radiation into space. The atmosphere and oceans act as a fluid system that transports heat from the hot region to cold regions via general circulation.

**Figure 2.**Latitudinal distributions of (a) mean air temperature, (b) cloud cover, and (c) meridional heat transport in the Earth. Solid line curves indicate those predicted with the constraint of maximum entropy production (equation (9)), and dashed lines indicate those observed. (Reproduced with minor changes based on [4])

^{2}s, where s is specific entropy with δ

^{2}as the second order variational operator. Liu and Tao suggested a new stability criterion based on the Liapunov second method with the excess entropy production δ

^{2}s as a generalized Liapunov function by which the stability characteristics of a series of weather conditions are revealed [145]. On the other hand, Pujol and Llebot used a second differential of the entropy as a criterion for the stability in low-dimensional climate models [146].

## 5. Entropy Flow

_{1}, heated from its bottom at a higher temperature T

_{2}(see Figure 3), and noticed that a rather regular cellular pattern of hexagonal convective cells was abruptly organised when the temperature difference (T

_{2}− T

_{1}) reaching the value of the threshold of primary instability [30,44,45,149], suggesting that every system that obtains heat at a higher temperature but loses heat at a lower temperature will experience net negative entropy flow since in this case the entropy exchange ($\delta {s}_{e}$) of the system with its environment or entropy flow entering into the system through its boundaries is as follows:

_{1}and Q

_{2}are the heat fluxes through the top and bottom of the layer of fluid, respectively, and Q is the flux when the system becomes steady so that Q = Q

_{1}= Q

_{2}. Later, in the 1940s, Schrödinger stated in his monograph “What Is Life?” that life’s existence depends on its continuous gain of ‘negentropy’ from its surroundings [105]. This implies that negative entropy flow is something very significant for a system, whether it is living or nonliving [30,44,45,47,150,151]. That is, negative entropy flow will cause a system initially at equilibrium or even at rest to be organized, or lead a system already at non-equilibrium to a state further from equilibrium (i.e., cause it to strengthen).

**Figure 5.**The evolution of total entropy flow (TEF, in 10

^{6}J K

^{−1}s

^{−1}) for Hurricane Katrina at 6-hourly intervals, with changes in the maximum sustained wind speed (V

_{max}

_{,}in kt) as a reference.

**Figure 6.**The entropy flow fields at 850 hPa at 18:00 UTC 26 (left panel) and at 18:00 UTC 29 (right one) August 2005. The red spot stands for the corresponding position of the hurricane centre’s to the respective time. Green lines are for contours at 850 hPa and black ones are for isopleths of entropy flow with the areas of negative value shaded.

**Figure 8.**The numbers of thunderstorms, high winds with speed of at least 75 knots and hails with size of at least 2.5 inches in Wisconsin, between 1950 and 2003.

## 6. Applications of the Principle of the Second Law

_{e}the Renolds number. In the case of the initial condition taken as the linear distribution: u ＝ −x at t ＝ 0, −∞ < x < ∞ with the boundary condition as:

_{e}/2)x], which is a steady solution containing no time.

**Figure 9.**Comparison of analytic solution of the one-dimensional viscous Burger’s equation that runs to a steady flow with the numerical solutions between the original and physics-based diffusion schemes used.

**Figure 10.**Comparison of the surface maximum sustained wind velocity between the different schemes with observations.

## 7. Discussion and Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Liu, Y.; Liu, C.; Wang, D.
Understanding Atmospheric Behaviour in Terms of Entropy: A Review of Applications of the Second Law of Thermodynamics to Meteorology. *Entropy* **2011**, *13*, 211-240.
https://doi.org/10.3390/e13010211

**AMA Style**

Liu Y, Liu C, Wang D.
Understanding Atmospheric Behaviour in Terms of Entropy: A Review of Applications of the Second Law of Thermodynamics to Meteorology. *Entropy*. 2011; 13(1):211-240.
https://doi.org/10.3390/e13010211

**Chicago/Turabian Style**

Liu, Ying, Chongjian Liu, and Donghai Wang.
2011. "Understanding Atmospheric Behaviour in Terms of Entropy: A Review of Applications of the Second Law of Thermodynamics to Meteorology" *Entropy* 13, no. 1: 211-240.
https://doi.org/10.3390/e13010211