3.7: Introduction to Thermodynamic Diagrams

Potential temperature is also very useful in thermodynamic diagrams, which will be briefly introduced here, but covered in more detail in Chapter 5. Thermodynamic diagrams are useful in diagnosing the state of the atmosphere and the buoyancy of air parcels by comparing the temperature difference \(\Delta T\) between the parcel and its environment. Parcels that are warmer than their environment will tend to rise due to lower density, and the change of the parcel’s temperature with height can be anticipated based on the parcel’s moisture content. In the thermodynamic diagrams you will use, dry adiabat lines will be plotted to show the dry adiabatic lapse rate, as parcels will cool at this rate until condensation occurs within the parcel. Dry adiabats are labelled with \(\theta\) because \(\theta\) is constant along these lines. A simple example is provided here.

Sample thermodynamic diagram (CC BY-NC-SA 4.0). Horizontal lines show isobars, lines of constant pressure; vertical lines show isotherms, lines of constant temperature; and orange lines show dry adiabats, lines of constant potential temperature.

Heat Budget at Earth’s Surface

Before moving on from thermodynamics, let’s add another layer of complication to our understanding of Earth’s surface heat budget. In Chapter 2 we discussed how the Earth’s heat budget could be defined by the incoming shortwave radiation (\(K\downarrow\)), reflected shortwave radiation (\(K\uparrow\)), longwave radiation emitted by the Earth (\(I\uparrow\)), and the downwelling longwave radiation emitted from the atmosphere (\(I\downarrow\)) received by the Earth’s surface.

\[F^*=K \uparrow+K \downarrow+I \uparrow+I \downarrow \nonumber \]

Earth’s surface is considered to be infinitesimally thin with no volume, and no heat can be stored, so the sum of all incoming and outgoing heat fluxes at the surface must balance. The net heat flux at the surface must be zero. In addition to the incoming and outgoing shortwave and longwave radiation, there are three other fluxes that must be considered. But first, let’s discuss what is meant by a “flux” of heat.

Heat Flux

Let’s say you have a cube of air, somewhere fixed relative to the ground. From an Eulerian framework (fixed-location), pressure changes can be neglected in the First Law equation, as they will be small and slow.

\[\Delta q=C_p \times \Delta T-\frac{\Delta P}{\rho}=C_p \times \Delta T-\frac{0}{\rho}=C_p \times \Delta T \nonumber \]

Leftover, you have an equation that states that the heat you transfer to the cube (per unit mass) causes the temperature change:

\[\Delta T=\frac{\Delta q}{C_p} \nonumber \]

If you divide this by a time interval ∆t (note: lower-case t is used for time, \(T\) is temperature), gives an equation for temperature change with time:

\[\frac{\Delta T}{\Delta t}=\left(\frac{1}{C_p}\right) \frac{\Delta q}{\Delta t} \nonumber \]

The temperature of the air cube could be increased if there were a heat transfer into it. Heat flux is the rate of heat transfer through a surface over time, as illustrated in the below figure. The units of heat flux can be given as J·m^-2·s^-1, or W·m^-2, because W = J·s^-1. It is heat moving through an area over time.

This figure represents a flux of some material or energy through a surface or area (CC BY-SA 3.0).

Temperature could be increased by heat flux into the cube of air, and could also be decreased by heat flux out of the cube. For example, if there is a net heat flux into a cube of air there will be a net heating effect because heat will be transferred into the cube more quickly than it will leave the cube.

Earth’s Surface Budget

Fluxes are defined as positive for heat moving upward. In addition to the net radiation (F*) from shortwave and longwave radiation, the fluxes include:

\(F^*\) = the net radiation between the surface and atmosphere, defined above;
\(F_H\) = effective surface turbulent heat flux (sensible heat flux, SH);
\(F_E\) = effective surface latent heat flux, caused by evaporation or condensation (latent heat flux, LH); and
\(F_G\) = molecular heat conduction to/from deeper below the surface, basically heat being conducted from nearby molecules.

All of these fluxes have to balance.

\[0=F^*+F_H+F_E-F_G \nonumber \]

Note

In the class we will likely use SH or SHF to represent the sensible heat flux, and LH or LHF to represent the latent heat flux. Remember, SHF is a dry heat flux from convection and LHF is heat transfer from moisture.

Over the course of a day, the relative contributions from the different terms in the surface heat budget vary. The below figure shows the four terms as they vary over a moist surface during one average day. Notice the yellow line first. There is negative \(F^*\) through most of the day as the surface gains heat from the excess shortwave radiation. At night, \(F^*\) is positive as the surface radiates infrared radiation upward away from the surface. This is similar to \(F_G\)_, where heat is transferred downward into the ground during the day and upward at night. However \(F_H\) and \(F_E\) have opposite signs from \(F^*\) and \(F_G\). The surface sensible heat flux, \(F_E\) and surface latent heat flux, \(F_H\) are positive during the day as convection and evaporation draws heat upward away from the surface.

This figure shows the surface heat balance throughout the day over a moist surface (CC BY-NC-SA 4.0).

Now let’s look at how the fluxes vary instantaneously over a moist (a, b) or dry (c, d) surface during the day (a, c) and night (b, d). The size of each arrow corresponds to the strength of each type of flux.

This diagram is a representation of Earth’s surface fluxes with different sized arrows showing the magnitude of each under different conditions (CC BY-NC-SA 4.0).

In the above image, note the difference between (a) and (c). Both have a large incoming radiation \(F^*\), but the moist surface (a) has a larger latent heat flux \(F_E\) as compared to the dry desert in (c) with a larger sensible heat flux \(F_H\). This shows that heat is transferred from the ground to the atmosphere through evaporation when the surface is moist, but through convection when the surface is dry.

The Bowen ratio helps to distinguish various types of surfaces. The Bowen ratio is defined as

\[B R=\frac{F_H}{F_E}=\frac{S H F}{L H F}, \nonumber \]

the ratio between the sensible heat flux and the latent heat flux. Moist surfaces have a small Bowen ratio because latent heating dominates over sensible heating while dry surfaces have a large Bowen ratio because sensible heating dominates over latent heating.

What other surface heat flux changes do you notice with day or night? How does energy partitioning change depending on the available moisture?

Remember our learning goals for this chapter:

1. Define and describe four methods of energy transfer
2. Describe the change in energy associated with changes in water state
3. Define and apply the first law of thermodynamics
4. Differentiate Eulerian and Lagrangian frameworks
5. Describe the importance of the dry adiabatic lapse rate, and recall what sets its constant value in the atmosphere
6. Compute potential temperature and apply the conserved variable approach
7. Draw a diagram of surface heat fluxes and Earth’s radiation budget
8. Compute the Bowen ratio, and define latent and sensible heat flux

Do you feel comfortable with all of the goals?