The climate system and the second law of thermodynamics

The second law of thermodynamics implies a relationship between the net entropy export by Earth and its internal irreversible entropy production. The application of this constraint for the purpose of understanding Earth’s climate is reviewed. Both radiative processes and material processes are responsible for irreversible entropy production in the climate system. With a focus on material processes, an entropy budget for the climate system is derived that accounts for the multiphase nature of the hydrological cycle. The entropy budget facilitates a heat-engine perspective of atmospheric circulations that has been used to propose theories for convective updraft velocities, tropical cyclone intensity, and the atmospheric meridional heat transport. Such theories can be successful, however, only if they properly account for the irreversible entropy production associated with water in all its phases in the atmosphere. Irreversibility associated with such moist processes is particularly important in the context of global climate change, for which the concentration of water vapor in the atmosphere is expected to increase, and recent developments toward understanding the response of the atmospheric heat engine to climate change are discussed. Finally, the application of variational approaches to the climate and geophysical flows is reviewed, including the use of equilibrium statistical mechanics to predict the behavior of long-lived coherent structures and the controversial maximum entropy production principle.

Figure 1

Schematic of different definitions of the climate system and its boundary in the application of the second law of thermodynamics. The planetary system is defined as a control volume including all particles (matter and radiation) encompassed by a fictitious surface denoted “top of the atmosphere” (TOA). The material system includes only matter (circles) and excludes all radiation (arrows) as part of the surroundings. The transfer system ([99]) includes internal radiative transfer (photons that are emitted and absorbed by matter within the system; solid black arrows), but excludes external radiative transfer (photons that are incident from the Sun or emitted directly to outer space; dashed black arrows).

Figure 2

A simple model for Earth’s climate and the associated vertical energy and entropy fluxes. The climate system is assumed to be horizontally homogeneous, and it includes a surface with temperature $T_s$ and a single-layer atmosphere with temperature $T_a$. Arrows show energy fluxes $F$ and their corresponding entropy fluxes $J$ expressed per unit area. The subscripts LW and SW denote shortwave and longwave radiant fluxes, respectively, and the subscripts $s$ and $a$ refer to the surface and atmosphere, respectively. Shortwave radiation is divided into its upward and downward components. The turbulent flux of enthalpy from the surface to the atmosphere is made up of a sensible heat flux $F_{\mathrm{SH}}$ and a latent heat flux $F_{\mathrm{LH}}$. Other terms in the equations are described in the text.

Figure 3

Schematic of (a) dry and (c) moist radiative-convective equilibrium. Horizontal black lines denote the tropopause in each case and vertical arrows show the mean sensible (gray) and latent (black) heat fluxes into the atmosphere in $\mathrm{W}{\mathrm{m}}^{-2}$. (b) Mean temperature profiles in the dry (gray) and moist (black) cases, and temperature of a parcel initialized with the mean temperature and humidity of the lowest model level of the moist simulation and lifted adiabatically while assuming all condensate is immediately removed from the parcel by precipitation (dotted line). Mean quantities are calculated as horizontal averages over the domain and over days 90–100 in each simulation.

Figure 4

Snapshot of vertical velocity at a height of 4 km at day 90 in simulations of (a) dry and (b) moist radiative-convective equilibrium. The black contour in (b) shows regions within clouds (cloud water content greater than $0.01\mathrm{g}{\mathrm{kg}}^{-1}$).

Figure 5

Empirical probability distribution functions (PDFs) of the vertical velocity at a height of 4 km in simulations of dry (gray) and moist (black) RCE. PDFs are constructed from hourly snapshots from days 90–100 of each simulation.

Figure 6

Snapshots of column relative humidity (colors) and contour of column cloud water content of $0.5\mathrm{kg}{\mathrm{m}}^{-2}$ (black) in a simulation of moist RCE with interactive radiation at (a) day 30 and (b) day 230. Simulation is run following the configuration of [331], but on a $288\times 288\mathrm{km}$ horizontal domain and with a horizontal grid spacing of 2 km. Lower boundary condition is an ocean surface held fixed at 301.5 K, and simulation is initialized from a state of rest.

Figure 7

Schematic of two idealized tropical cyclone overturning circulations (radial cross section). Gray arrows indicate dry air mass fluxes. Black arrows indicate boundary fluxes of heat $Q$, entropy $S$, water content $r_T$, and momentum $M$ into and out of the domain, and arrows indicate the characteristic direction of the fluxes, in particular, regions. (a) Closed circulation bounded by an outer wall or doubly periodic boundary conditions, with the idealized Carnot cycle legs indicated. (b) TC in lateral contact with the rest of the atmosphere.

Figure 8

Structure and overturning circulation of an axisymmetric tropical cyclone simulation at steady state [data from a 30-day window of a TC in a 6000 km radius domain at 40° latitude as described by 245]. (a) Equivalent potential temperature [${\theta }_e$ (K); darker shades for higher values] and absolute angular momentum (dashed lines; contour value increases with radius). (b) Eulerian mass overturning streamfunction $\mathrm{\Psi }$ ($\mathrm{kg}{\mathrm{s}}^{-1}$) integrated radially outward to 800 km; each contour represents 10% of the mass flux. (c) Isentropic mass overturning streamfunction ${\mathrm{\Psi }}_e$ ($\mathrm{kg}{\mathrm{s}}^{-1}$) integrated radially outward to 800 km. Each contour represents 10% of the mass flux. (d) Temperature-entropy ($T\text{−}s$) diagram corresponding to parcel paths in (c) using the MAFALDA technique ([274]), where data gaps present in the contouring procedure in (c) have been filled via linear interpolation. Numbers in (b) and (c) give maximum streamfunction magnitudes in units of ${10}^9\mathrm{kg}{\mathrm{s}}^{-1}$.

Figure 9

Mean meridional mass overturning streamfunction estimated from the ERA-Interim reanalysis ([51]) using six-hourly snapshots for the years 1981–2000 and calculated using (a) pressure, (b) potential temperature, and (c) equivalent potential temperature as a vertical coordinate using the method described by [270]. Black contours represent clockwise motion and gray contours represent counterclockwise motion. The contour interval is $10\times {10}^9\mathrm{kg}{\mathrm{s}}^{-1}$, with the zero contour omitted. Numbers give the maximum streamfunction magnitude in each hemisphere in units of ${10}^9\mathrm{kg}{\mathrm{s}}^{-1}$. Hadley, Ferrel, and polar cells are labeled in (a). Thick maroon curves show mean (b) potential temperature and (c) equivalent potential temperature near the surface (at the pressure level $p=0.9988p_s$, where $p_s$ is the surface pressure).

Figure 10

Entropy budget as a function of sea-surface temperature (SST) in simulations of radiative-convective equilibrium plotted with log scale. Material entropy export $⟨{\dot{S}}_e^{\mathrm{mat}}⟩$, black circles; total irreversible entropy production $⟨{\dot{S}}_i^{\mathrm{mat}}⟩$, black line; irreversible entropy production owing to frictional dissipation $⟨{\dot{S}}_i^{\mathrm{fric}}⟩$, red triangles; precipitation sedimentation $⟨{\dot{S}}_i^{\mathrm{sed}}⟩$, blue squares; irreversible phase change and mixing $⟨{\dot{S}}_i^{\mathrm{mem}}⟩$, green pluses). Dotted gray lines show Clausius-Clapeyron scaling, which increases in proportion to the saturation vapor pressure at the sea surface. Adapted from [331].

Figure 11

Schematic of release of available potential energy. (a) Motionless initial water glass with a nonzero interface slope between stably stratified water masses. (b) Motionless final state after available potential energy $A$ was released, converted to kinetic energy $K$, and then converted to internal energy $IE$ and nonavailable potential energy through frictional heating and the associated thermal expansion.

Figure 12

Total energy $E$ vs total entropy $S$. The solid black curve $S(E)$ is the entropy of an isolated system of total energy $E$ in thermodynamic equilibrium. For an isolated system, the initial state $a$ and final equilibrium state $e$ have the same total energy. The reference state $c$ maximizes exergy $B_{\max }$ if it can be reached isentropically ([150]). If $T_r$ is chosen such that the reference state cannot be reached isentropically, the exergy available for mechanical work $B<B_{\max }$. Adapted from [170], and [129].

Figure 13

Examples of geophysical flow phenomena that have been described using equilibrium statistical mechanics. (a) Streaklines of ocean surface flow exhibiting oceanic rings from a numerical model. From NASA/Goddard Space Flight Center Image Visualization Studio. (b) Southern Hemisphere ozone hole on October 12, 2018, indicating the location of the stratospheric polar vortex. From NASA Earth Observatory, NASA/NOAA Suomi NPP satellite. (c) Submesoscale vortices mixing the eyewall of Hurricane Isabel on September 12, 2003, at 1315 UTC as seen by GOES-12. From NASA/NOAA. (d) Jupiter and the Great Red Spot as seen by the Hubble Space Telescope on August 25, 2020. From NASA, ESA, STScI, A. Simon (Goddard Space Flight Center), M. H. Wong (University of California, Berkeley), and the OPAL team.

Figure 14

Jupiter’s (a) north and (b) south polar cyclones imaged by the NASA Juno mission in infrared. From [2]. (c) Vortex crystal experimental equilibria. From [81].