3.6: Frameworks for Understanding the Atmosphere
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)All atmospheric science concepts are shared with other disciplines like physics and chemistry, but they often use a more specific set of equations or variables as shown above with the first law of thermodynamics. This helps to simplify things because general physics equations can be applied to nearly any form of matter, while in atmospheric science we primarily deal with gases that are not constrained to a specific volume.
In the two sections that follow, two important frameworks are described that will help you to view the atmosphere in ways that will simplify concepts in the sections and chapters to come.
Lagrangian vs. Eulerian
When we look at or try to solve a problem in the atmosphere, there are two different lenses or frameworks through which we can view the problem. An Eulerian framework is a fixed framework, relative to a single point on the Earth’s surface. When a weather forecast is done for a given location on Earth or when you look at a dataset from one weather station, you are viewing the atmosphere from an Eulerian perspective — that is, how the wind and air travels past a fixed point. With a Eulerian framework, we need to be concerned about things like temperature and moisture advection, properties that travel with and are carried by the wind.
Another framework is Lagrangian, which is a framework that is constantly moving and travels with the air. When we are looking at motions within the atmosphere, such as rising or sinking air, it is useful to use this framework as a way to see how properties within the rising plume of air are changing. Both frameworks will be used in the coming sections and chapters.
Air Parcels
At this point, we should discuss a naming convention used in atmospheric sciences. Many times it is useful to think about a mass of air instead of individual molecules. We generally call a mass of air an “air parcel”. This is especially useful for distinguishing processes happening within the air, versus processes happening within the environment. An air parcel is often thought of as an amorphous bubble or blob of air, roughly the scale of a party balloon or a hot air balloon that contains uniform properties (temperature, density, pressure) throughout. Air parcels are simplified theoretical constructions used as a way to think about and examine motions and instability in the atmosphere.
For example, let’s revisit the important topic of latent heating. We discussed how condensation of water vapor to liquid water is a warming process. This is only clear if we separate the processes happening within a parcel of air and within the rest of the environment. When water vapor condenses within an air parcel, it gives off energy, which acts to warm the environment. We will often refer to air parcels when we want to distinguish changes within a piece of air with respect to the rest of the atmosphere.
In reality, as air parcels move through the atmosphere, there will be some mixing between the air parcel and the environment, and heat can be exchanged with the environment or added via radiation. However, with the concept of air parcels, we simplify the situation and imagine that radiative effects are small, and that mixing only occurs on the outer edges of the parcel, with a protected inner core.
Defining Changes in the Atmosphere
In order to study changes in temperature, momentum, and moisture in an air parcel, a Lagrangian framework is always used. With a Lagrangian framework, we can see changes from the parcel’s perspective as it moves, not from a fixed point on the ground.
Applying the hydrostatic equation from Chapter 1 to the first law of thermodynamics, we can get the Lagrangian first law of thermodynamics equation for a moving air parcel.
\[\Delta T=-\left(\frac{|g|}{C_p}\right) * \Delta z+\frac{\Delta q}{C_p} \nonumber \]
\(\Delta q\), or heat transfer, can be caused by various processes, outlined in the following figure. The figure shows an air parcel moving both horizontally (through advection) and vertically (through convection), as well as the various processes both inside and outside that are causing heat exchange. The temperature inside the air parcel is conserved unless heat is transferred to or from the environment, or if it loses or gains heat by rising or sinking, which will cause it to expand or contract, respectively.
Lapse Rates
The atmospheric lapse rate, \(\Gamma\), denoted by an upper-case gamma, is defined as the change in temperature with altitude, specifically a reduction in temperature with altitude.
\[\Gamma=-\frac{T_2-T_1}{z_2-z_1}=-\frac{\Delta T}{\Delta z} \nonumber \]
A positive lapse rate indicates that the temperature is decreasing with increasing height, while a negative lapse rate indicates a “temperature inversion” meaning that the temperature is increasing with increasing height. Lapse rates are typically defined for:
- The environment, \(\Gamma_e\)
- A dry air parcel, \(\Gamma_d\)
- A saturated air parcel (moist), \(\Gamma_m\)
We can observe the environmental lapse rate by using atmospheric sensors attached to weather balloons. As a weather balloon rises through the atmosphere, it measures temperature and other properties on the way. The environmental lapse rate varies depending on time of day, altitude, latitude, land surface properties, heat fluxes, and air movement. A typical \(\Gamma_e\) is around 6.5 K km-1 but this will be discussed in later chapters. You will have experienced this decrease in temperature with altitude if you have hiked in the mountains or seen an outdoor thermometer reading on a commercial flight.
Dry Adiabatic Lapse Rate
In atmospheric science, you will often hear the word “adiabatic”, typically accompanied by the words “cooling”, “warming”, or “lapse rate”. An Adiabatic process just means that there is no heat transfer taking place (∆q = 0) during the process. For an air parcel, this means that no thermal energy is entering or leaving the air parcel from the outside. However, internal processes are allowed, such as the ones shown in the figure above (most notably adiabatic expansion and latent heating). An adiabatic lapse rate indicates that air is cooling or warming with altitude without any external heat exchange. An air parcel that contains no liquid water or ice (none of the moisture in the parcel has condensed into liquid, no saturation or latent heat release), will cool at the dry adiabatic lapse rate.
\[\Gamma_d=9.8 \frac{K}{km}=9.8 \frac{{ }^{\circ} C}{km} \nonumber \]
Pro Tip: A positive lapse rate means that the air temperature is decreasing with height.
The reason that an air parcel expands adiabatically as it rises is due to the fact that the environmental air pressure decreases with height. The air parcel’s pressure will adjust to the lower pressure of its environment, but the parcel must expand in order to do so (in order for the air molecules inside the parcel to exert a smaller force). The parcel uses some of its own internal energy to do the work of expansion, and its temperature decreases as a result. The opposite is true as an air parcel sinks, the environmental pressure rises, and the parcel’s pressure adjusts in order to maintain pressure equilibrium, and the parcel must shrink for its pressure to increase. As it shrinks, work is done on it, and the temperature rises.
In later chapters we’ll define the moist or wet adiabatic lapse rate (Chapter 4), as well as the environmental lapse rate (Chapter 5) and their significance.
Potential Temperature
The potential temperature, \(\theta\), of a parcel completely ignores the temperature change of the parcel due to it having done work or been worked on (expanding or contracting). As a result, potential temperature is constant for an adiabatic process and \(\theta\) does not change when \(\Delta q=0\). Potential temperature is proportional to the sensible heat contained in a parcel and can increase or decrease when sensible heat is added or removed through diabatic (non-adiabatic) processes (\(\Delta q \neq 0\)). Examples of diabatic heating include turbulent mixing, condensation (latent heat), and radiative heating. Potential temperature has units of K or °C, and can be found if you know the air temperature, \(T\), at pressure-level, \(P\), using the following equation
\[\theta=T *{\frac{P_0}{P}}^{R_d / C_p} \nonumber \]
where \(R_d / C_p\)is a constant equal to 0.28571, and has no units, and \(P_0\) is a reference pressure, typically 1,000 hPa (100,000 Pa), or the local surface pressure.