# The wet bulb temperature is a thermodynamic property

## 3.1 Summary of some thermodynamic terms

Transcript

1 Chapter 3 The Analysis of Vertical Probes The thermodynamic structure of the atmosphere can be examined with vertical probes. They are mostly carried out with radiosondes (balloon ascents), in special cases also with so-called drop probes that are dropped from an airplane. These soundings can provide information about various atmospheric structures, e.g. Inversions, stable and unstable layers, fronts, the tropopause and cloud layers. Here some important thermodynamic terms are first summarized and then the basics of interpreting vertical probes are discussed. 3 Summary of some thermodynamic terms [a] The vapor pressure e denotes the partial pressure of water vapor. Saturation vapor pressure e s (T) means the vapor pressure in the case of saturation, i.e. the maximum vapor pressure at a certain temperature. 1

2 Fig. Solid line: saturation vapor pressure over a planar water layer as a function of temperature. Dashed line: difference between the saturation vapor pressure over a planar ice surface and that over a planar water layer [taken from Atmospheric Science, An Introductory Survey, J.M. Wallace and P.V Hobbs]. Experimentally, the following was found: e s = 6078exp 19.8 t 273 + t 6exp (0.073 t), where t here denotes the temperature in C and e s is calculated in hpa. Task: The saturation vapor pressure above water is greater than that above ice. What does this mean for the competing growth of water droplets and ice crystals? [b] The mixing ratio w = e / (p e) is the ratio of the mass of water vapor to the mass of dry air. The constant corresponds to the ratio of the gas constants for dry air and water vapor R / R v. As before, one defines a saturation mixing ratio w s = e s / (p e s). This is the ratio of the mass of water vapor to the mass of dry air in the case of saturation. Specific humidity q = e / p means the mass fraction of water vapor in the total air (unit kg / kg or g / kg). Usually the difference between w and q is negligible, i.e. w q. [c] The relative humidity r = e / e s w / w s, is usually given in%. It is an important measure to assess the dryness of the air. The following figure shows some of the variables discussed so far in a vertical section from west to east: 2

3 Fig. 2 Left: Vertical section from 100W to 0E along the 40N circle of latitude. The specific humidity in g / kg (shaded), the temperature in degrees Celsius (thin lines) and the dynamic tropopause (thick line) are shown. Right: Cross-section as on the left, but now with relative humidity in percent (shaded), vertical wind in Pa / s (thin lines) and dynamic tropopause (thick line). Task: The figure shows the vertical wind in Pa / s. What is the vertical speed in m / s? Use the basic hydrostatic equation for the conversion. Another important measure for assessing the dryness of the air is the dew point T d: This is the temperature at which saturation occurs for the observed vapor pressure e (i.e. temperature to which an air parcel has to be cooled down in order to achieve saturation (and possibly condensation) occurs. Formally: td (e) = 273 lne ln (lne ln6) 13.7ln (e / 6), here e is inhpa and t d in C. The dew point is the most frequently measured humidity variable. The other parameters (vapor pressure, relative and specific humidity) can then be approximately calculated as follows: e = 6exp (0.073 td) q = 3.794exp (0.073td / p) r = exp [0.073 (tdt)], where t and td are in C and p in hpa. The dew point depression TT d is a measure of the dryness of the air. The greater the dew point difference, the drier the air. 3

4 Task: In the figure above, calculate the dew point for one point in the troposphere and one in the stratosphere. [d] If the water vapor condenses in an air package, latent heat is released. If, on the other hand, condensed water evaporates (in the form of droplets), evaporation heat is required. In both cases there is a change in temperature of the air parcel. This is described by two temperature measures: The wet-bulb temperature T w is the minimum temperature that could be set in an air packet when water evaporates from an external reservoir (e.g. humidity thermometer). T w is generally smaller than: in the dry air, a lot of water can still evaporate from the external reservoir, and the evaporation cold results in a great cooling of the air. In saturated air, T w = T. Exercise: Find out what the wet bulb temperature is used for? Tip: It is a matter of measuring an important atmospheric quantity. As an additional task: Make it very clear to yourself how this measurement works. The equivalent temperature T e is the temperature that would be set if all the moisture present in the air package were to condense. T e is generally greater than T: the more humid the air, the more the temperature could rise due to the heat of condensation released. In completely dry air, T e = T. Task: How much does the temperature of an air package change, which originally has a temperature of 10 C and in which 5 g of water vapor condenses. [e] The potential temperature is an important variable: Θ = T (p 0 / p) κ. In the case of an adiabatic change of state (no heat exchange with the environment), the first law of thermodynamics can be written as follows: c p d (logt) R d (logp) = 0 where ic p denotes the specific heat at constant pressure and R the gas constant. If one integrates this equation from a state (T, p) to a state with pressure p 0 and temperature θ, the result is Possion's equation θ = T (p 0 / p) R / cp Pressure p 0 gives the potential temperature. Typically, in atmospheric physics, a reference pressure p 0 = 1000hPa is chosen. Clearly, θ is the temperature of an air parcel that is adiabatically compressed to 1000 hpa. The following diagram shows the temperature and the potential temperature in a north / south section: 4

5 Fig. 3 Temperature and potential temperature (both in K) in a north / south section along the 50W meridian. If you have an air parcel of temperature T (in K) at a height p (in hpa) and shift this air parcel vertically to the height p 0 = 1000hPa, its temperature will change (heating by compression, analogous to the air in a bicycle pump) . The resulting temperature at 1000 hpa is the potential temperature Θ. Analogous to the definition of the potential temperature, the equivalent potential temperature Θ e = T e (p 0 / p) κ can be defined. 1 Both temperatures have important conservation properties. If an air parcel originally had a potential temperature Θ, it will keep this potential temperature if no diabatic processes take place (heating or cooling by radiation, turbulent mixing with neighboring air parcels, condensation and evaporation). An analogous conservation property applies: 1 The calculation of the equivalent-potential temperature is quite complicated. A frequently used approximation formula can be found, for example, in the article The computation of equivalent potential temperature by D. Bolton, Monthly Weather Review, Volume 108, pages,

6 for the equivalent potential temperature, with condensation and evaporation now allowed. The following figure shows the potential temperature as it occurs during a northern foehn in a vertical north / south cross-section over the Alps. Fig. 4 Potential temperature in a vertical north / south cross-section over the Alps. As a first approximation, the isentropes (isolines of the potential temperature) can be viewed as streamlines. So there is a drop in the air over the Alps. The following figure shows the potential and the equivalent-potential temperature in a north / south section: 6

7 Fig.5 Potential and equivalent potential temperature (both in K) in a north / south section along the 50 W meridian. Task: The potential and equivalent-potential temperature agree very well in higher layers. On the other hand, there are major deviations in tropical regions. Why? Note that the potential temperature in the entire cross-section increases with height. We will see later that this has to be the case, otherwise turbulent mixing would take place immediately (convective instability). The equivalent-potential temperature exhibits negative vertical gradients, especially in the vicinity of the equator. This indicates possible convective instabilities in this area. The following figures show a front in the North Atlantic that has formed in connection with a low pressure vortex. Note how there is not only a (potential) temperature gradient across the front, but also a gradient in the specific humidity. Behind the front you will find dry, cold air, while in front of the front you will find warm, moist air. The moisture gradient has the effect that the gradient in the equivalent potential temperature is stronger than that of the potential temperature. Therefore θ e is often used to determine fronts in numerical weather forecasts. 7th

8 GM GM GM GM Fig. 6 Low pressure eddy over the Atlantic on October 14, 2005, 12 UTC. Shown from top left to bottom right at 850 hpa: (a) temperature (in degrees Celsius) and wind vectors, (b) potential temperature (in K), (c) specific humidity (in g / kg), and (d ) equivalent potential temperature (in K). [f] If you lift an air parcel, the pressure will decrease. As a rule, the decrease in pressure is accompanied by a decrease in temperature. This can lead to condensation of water vapor. This is taken into account in several important parameters: The LCL (lifting condensation level) is understood to mean the level at which saturation (condensation) would occur if an air package were lifted vertically. T LCL and p LCL denote temperature and pressure at this level (where T LCL corresponds to the dew point temperature t d. The following figure shows a vertical sounding near Flagstaff, Arizona (1201 MST, + August 1961). The solid line corresponds to Temperature, the dashed dew point. The sounding took place in the vicinity of an orographically induced thunderstorm. On the left side the LCL are given as a function of the altitude, 8

9 as they result from the left sounding. Example: An air parcel that is located in the sounding at the height of 5km ASL would therefore have to be raised by z = 500m in order to reach its LCL. Such an increase in the air package can occur, for example, when the air flows over a mountain range. So if an air parcel is located far upstream from the mountains at 5 km ASL and is raised by 500 m when the mountain flows over it, condensation sets in and a cloud forms. Fig. 7 Vertical probing of the temperature (solid line) and the dew point (dashed lines) as well as of LCL and LFC as a function of altitude. [taken from The Role of Mountain Flows in Making Clouds, R.M. Banta, Met. Monographs, Vol.23, June 1990, No. 45, Atmospheric Processes over complex terrain] The LFC (level of free convection) corresponds to the level to which an air parcel would have to be lifted so that it would then be lifted by itself would continue to rise (because it is warmer than the ambient air). The LFC is a sensible concept in the event of the possibility of frontal triggered thunderstorms. In order to reach the LFC, condensation must occur. Therefore, in the above figure, z of the LFC is always greater than that of the LCL. Task: Think about why free convection does not automatically set in with the onset of condensation in the LCL, so why LFC and LCL do not coincide. When expecting thermal tripping, the CCL (convective condensation level) is more often considered. The CCL describes the level up to which a well-mixed adiabatic layer would have to form due to solar radiation in order to achieve saturation with the CCL. The minimum required floor temperature is referred to as the trigger temperature CT (convective temperature). 9

10 Task: Think about why there is a well-mixed layer near the floor. How do you visualize the achievement of the CCL, ie. what will the daily course of the temperature profile look like near the ground on a hot summer day? For this, interpret the following figure, Fig. 8, illustration of the formation of the convective condensation level (CCL) between 1000 and 850 hpa. T 0, T 1 and T 3 indicate soil temperatures, the solid lines that emanate from these indicate the associated temperature profiles. The dashed line corresponds to the dew point temperature, assuming a constant specific humidity. Two temperature isolines are also drawn, 0 o C and 10 o C (from left and right to top right) [taken from A short course in cloud physics by R.Ṙ. Rogers and M.K.Yau]. 3.2 The skew T log p diagram The skew T log p diagram is the most frequently used thermodynamic map in which data from a vertical sounding can be plotted. The coordinates in this diagram are the temperature and the logarithm of the pressure, with the T-axis drawn skewed. In addition to the isotherms (T = const.), Dry and wet adiabats (i.e. isolines of Θ and Θ e) as well as isolines of the saturation-mixing ratio w s are shown. The following figure shows an example of how a number of other thermodynamic variables are graphically determined in the skew T logp diagram based on the measured values temperature T and dew point T d (here at 900 hpa) 10

11 can be. The LCL results as the intersection of the line with constant saturation mixing ratio through T d with the dry adiabats through T. If you follow the wet adiabats downwards from the LCL, you get the wet-bulb temperature T w at the level of the measurements and at 1000 hPa the potential wet-bulb temperature Θ w. If one follows the same wet adiabat upwards until the wet adiabat runs parallel to a dry adiabat and then back down along these dry adiabats, one obtains the equivalent temperature T e at the level of the measurements and at 1000hPa the equivalent potential temperature Θ e. Fig. 9 Determination of thermodynamic variables in a skew T log p diagram. The observed temperature and pressure values are shown in point A (T = 15 C, p = 900hPa and q = 6g / kg); the other circled points correspond to derived quantities. 11

12 3.3 Vertical soundings and hydrostatic stability [a] In a vertical sounding one can differentiate between different layers. This division is the first step in analyzing a vertical sounding. The following figure shows an example: 12

13 Fig0: An example of the vertical course of the temperature T a (p) and the dew point T d (p), shown in a skew T logp diagram. Several different layers are characterized by the different slopes of the curves. The vertical course of T and T d shows different layers, the 13

14 are characterized by a different inclination of the curves: Isothermal layers: layers of constant temperature (in the figure between 800 and 840hPa). Adiabatic layers: layers in which the temperature profile follows a dry adiabatic (in the figure between 430 and 470 hpa). Adiabatic layers are often characterized by strong turbulence (e.g. in the well-mixed atmospheric boundary layer, see the considerations on the CCL). The following figure shows a well-mixed planetary boundary layer over eastern Canada. Note in particular how the potential temperature in the lowest 1.7 km is almost independent of the altitude. It is also impressive how the well-mixed layer is limited towards the top by a temperature inversion to a height of approx. 2 km. Fig1 Vertical profile of temperature, potential temperature, dew point and ozone from an airplane measurement in the early afternoon in August over eastern Canada [taken from Introduction to Atmospheric Chemistry, Daniel J. Jacob]. Adiabatic layers can also arise in the free troposphere. The following figure shows approximately adiabatic layers in connection with a wind storm: 14

15 Fig2 Observation of the potential temperature in a vertical cross-section over the Rocky Mountains in Colorado (January 11, 1972). The dashed lines show the path of the research aircraft. Areas with strong turbulence are marked with plus signs along the paths. Note that the areas of high turbulence with near vertical isentropes, ie. adiabatic profiles coincide [taken from Atmosphere-Ocean Dynamics, A. Gill]. Inversions: Layers in which the temperature rises with increasing altitude (in the figure between 550 and 580 or between 420 and 430 hpa). Inversions are a sign of different atmospheric phenomena: cold fronts can be accompanied by underlying inversions, the tropopause is marked as a distinctive inversion in the middle latitudes and the formation of high fog at high pressure is often a sign of strong inversions close to the ground.Cloud layers (moist layers): layers in which the dew point is only slightly lower than the temperature (in the figure between 660 and 700 hpa). [b] The slope of the vertical temperature profile (i.e. the lapse rate Γ = T z) is also a measure of the hydrostatic stability of the atmosphere. According to the parcel method, a stratification is stable (unstable) if an air parcel, which is raised adiabatically as a thought experiment, is afterwards colder (warmer) than its surroundings, because it then sinks again (continues to rise) due to its greater (smaller) density. The basic idea of the parcel method is shown in the following figure. The left shows the change in pressure, temperature and density that results in the air package when it is raised slightly. The corresponding changes that prevail in the surrounding air are plotted on the right. 15th

16 Fig3 Basic idea of the parcel method. What is essential is the change in the density of the raised air parcel (left) compared to the density of the ambient air (right). If the density of the air parcel is greater than that of the surroundings, the air parcel returns to its rest position. Otherwise it will continue to rise and thus move further away from its rest position [taken from The Physics of Stars, Phillips]. There are four categories of hydrostatic stability (see following figure). In the case of conditional instability, the atmosphere is dry-stable and damp-unstable at the same time, i.e. the stratification is stable as long as no condensation occurs (but then immediately becomes unstable). This is a typical situation on summer thunderstorm days. 16

17 Fig4 The four main categories of hydrostatic stability, shown in a skew T log p diagram. The thick line shows a possible measured course of the temperature T (z); the other lines denote isotherms and dry or wet adiabats. Task: The following figure shows four profiles of temperature and dew point temperature in skew T-log p diagrams. Discuss these profiles with regard to stability: 24 Aug 2006, 06 UTC / Lat, Lon = 5.4 N, 0 E Aug 2006, 06 UTC / Lat, Lon = 28.8 N, 0 E km km km 10km km 303 8km p (hpa) 400 p (hpa) km km 6km 5km km 4km km km 800 2km 800 2km 900 1km 900 1km Aug 2006, 06 UTC / Lat, Lon = 54 N, 0 E Aug 2006, 06 UTC / Lat, Lon = 81 N, 0 E km km km 10km km 303 8km p (hpa) 400 p (hpa) km km 6km 5km km 4km km km 800 2km 800 2km 900 1km 900 1km Fig5 Four skewt-logp diagrams for August 24, 2006, 6UTC according to the operational Analysis of the ECMWF. The temperature (black) and the dew point temperature (red) are shown. Also note the geographical latitude at which the profiles were taken. [c] If a dry air parcel is moved a vertical distance z away from its rest position, it obeys the following equation of motion for small distances z. The Brunt-Väisälla frequency is defined by N 2 = g / θ Θ / z, where θ represents the potential temperature of the ambient air. D 2 z Dt 2 + N 2 z = 0 17

18 Exercise: What does the movement of the air parcel look like for positive and negative N 2? What conclusion can be drawn from this for the vertical course of the potential temperature? What is the connection with the above figure? How does the movement of the air parcel in the stratosphere differ from that in the troposphere? Use the following illustration to do this. How can the above equation of motion be generalized for air parcels where condensation occurs? Fig6 Potential temperature (in K), squared Brunt-Vaisälä frequency (in 10 4 s 1) and dynamic tropopause in a north / south section (50 W, October 14, 2005, 12 UTC). In the following, the above equation of motion is to be derived as an example. We follow the derivation as given in the book An Introduction to Dynamic Meteorology by James R. Holton. The equation of motion of an air parcel in the vertical direction can be written as Dw Dt = D2 Dt 2 (z) = g 1 ρ Here, g denotes the acceleration due to gravity and the second term on the right corresponds to the pressure gradient force. Note that this equation applies to the hydrostatic 18 p z

19 table approximation is reduced if the acceleration Dw / Dt on the left side is set equal to zero. In the packet method, it is assumed that the pressure of an air packet p adapts instantaneously to the ambient pressure p 0, ie. in the above equation, p may be replaced by p 0. We assume that the ambient pressure follows the hydrostatic equilibrium. The following should apply to the ambient pressure p 0 and the ambient density ρ 0: p 0 z = ρ 0g With this, the above equation of motion can be rewritten as Dw Dt = D2 Dt 2 (z) = g ρ0 ρ ρ With the definition of the ideal gas equation p = ρrt and the definition of the potential temperature θ = T (ps / p) κ, the buoyancy term can be written somewhat more simply on the right-hand side of the equation: g ρ0 ρ ρ = g θ θ 0 θ 0 where θ 0 is the potential temperature of the ambient air and θ θ 0 is the difference between the potential temperature of the air parcel and the environment. Now it is a matter of seeing how the potential temperature changes with a small displacement of the air parcel by the distance z. The potential temperature of the environment at the new position z can be determined by a Taylor expansion: θ 0 (z) = θ 0 (0) + dθ 0 (0) z + ... dz For the potential temperature θ of the air parcel we assume that it is an adiabatic process. Then the potential temperature θ of the air parcel is retained. So did it have the same potential temperature as the ambient air in its rest position, ie. equals θ 0 (0), it will also have this at the new position z: θ (z) = θ 0 (0) If these two terms are inserted into the equation of motion, the result is D 2 z Dt 2 + N 2 z = 0 where the squared Brunt-Väisälla frequency N is defined by: N 2 = g dln (θ 0) dz As a direct consequence we see that in a stable atmosphere (N> 0) the potential temperature must increase with altitude. 19th

20 [d] The following figure shows two examples of vertical soundings, one in the evening and the other in the early morning. In both cases the tropopause is clearly visible, once lower and once higher than 200hPa. High relative humidity occurs in the probe (a) around 650 hpa, in the probe (b) in a thicker layer between 550 and 700 hPa. Note also the very stable layer close to the ground in the probe (b) and the two small inversions at 500 hpa in the probe (a). These diagrams also show the LCL and LFC in (a) and the CCL and the tripping temperature CT in (b). The LCL results from the intersection of the line with constant saturation vapor pressure (through the measured value for T d) with the dry adiabatic (through the measured value for T). The LFC corresponds to the intersection of the moisture adiabats through the LCL with the measured temperature profile. In diagram (b) one finds the CCL as the intersection of the line with constant saturation vapor pressure (through the measured value for T d) with the measured temperature profile and finally the trigger temperature CT as the intersection of the dry adiabats through the CCL with the soil pressure. In other places it is considered more sensible when determining the CCL not to consider the soil value for w s, but rather a value averaged between the soil and around 800 hpa. (from 20

21 Fig7 Two examples of vertical soundings in the USA for (a) 00 UTC 7 May 1983 and (b) 12 UTC 14 June The observed vertical profiles of temperature (T a) and dew point (T d, dashed line), the Course of the lifting temperature (T l, dotted-dashed line), as well as in (a) the lifting condensation level (LCL) and the level of free convection (LFC) or in (b) the convective condensation level (CCL). [e] An interesting quantity is the energy that is possible when an air parcel is lifted and when condensation occurs. The lifting temperature T l is required to calculate this quantity. This temperature corresponds to the temperature curve for an air parcel close to the ground, which you keep increasing in a thought experiment. Up to the LCL, T l follows a dry adiabat, then the air packet is saturated and T l follows a wet adiabat. In a skew T log p diagram, the area between the measured temperature profile and the lifting temperature T 1 corresponds to the energy released in the event of a lifting of the air parcel. This quantity is called CAPE (convective available potential energy) and is an important parameter e.g. in the analysis and prediction of strong thunderstorms: CAPE = R pe p LFC (T (p) T a (p)) dlnp Here, R denotes the gas constant for dry air, p LFC and pe are the pressures at the level of free convection and am Equilibrium Level and T (p) and T a (p) indicate the temperature of the raised air parcel and the ambient air. CAPE values below 1000m 2 s 2 indicate a small probability of strong convection. In both diagrams in the figure, the CAPE is marked with + symbols. It is important that even if a large CAPE is present, a triggering mechanism is usually still required to set the strong convection in motion. Propagating fronts, orographic uplift (e.g. with NW currents in the foothills of the Alps) or wind systems in connection with existing thunderclouds come into question. 21

22 Fig8 (top left): CAPE calculates with the weather forecast model almo from MeteoSchweiz / DWD. The prediction is for August 12, 2004, 12UTC. The CAPE is given in J / m 2. (top right) Precipitation forecast for the same point in time. (bottom left) Geopotential and temperature at 850 hpa. The figure shows that CAPE reflects convective precipitation well in some areas, but in other areas the high CAPE values hardly coincide with precipitation (for example in the Mediterranean region). This illustrates very clearly that CAPE alone is not a good indicator of convective precipitation. As I said above, there is still a need for a mechanism that actually triggers convection. Only when this has been triggered can CAPE determine the energy released (and to some extent the precipitation). For this purpose, a further variable CIN (convective inhibition) is introduced, which specifies the energy that must be expended so that convection starts spontaneously: CIN = R plfc p 0 (T (p) T a (p)) dlnp Here denotes analogously to CAPE R is the gas constant for dry air, p LFC and p 0 are the pressures at the level of free convection and on the ground (mostly between 500m and 1000m above ground level) and T (p) and T a (p) indicate the temperature of the raised air parcel and the ambient air again. The conditions are analogous to a chemical reaction in which an activation energy (analogous to CIN) is also required until the exothermic reaction (analogous to CAPE) 22

23 expires. CAPE and CIN are used in some numerical weather forecast models to parameterize the convective precipitation that is not explicitly resolved. 3.4 Alternative vertical coordinates So far we have used either the geometric height z or the hydrostatic pressure p as vertical coordinates. Both quantities are suitable as vertical coordinates because they show a monotonous progression with height: increase for z, decrease for p. It is clear that any other size is suitable as a vertical coordinate that exhibits such a monotony. It was stated earlier that this essentially applies to the potential temperature Θ. Outside of unstable and turbulent layers, this increases monotonically with height. The following figure shows an example of what a picture looks like in the ß-Raumïm Θ -raum: 23

24 Fig9 (above) Vertical cross-section through a frontal zone. The solid lines indicate the potential temperature, the dashed lines indicate the wind speed. The tropopause is marked with a thick line. (below) The same shown with the potential temperature as a vertical coordinate. [Taken from Atmospheric Science, An Introductory Survey by J.M. Wallace and P.V. Hobbs.] Task: Think about the advantages of the different vertical coordinates. 3.5 References 1. A good overview of the various thermodynamic parameters (e.g. LCL, LFC) in connection with mountain overflows can be found in: The Role of Mountain Flows in Making Clouds, R.M. Banta, Met. Monographs, Vol.23, June 1990, No. 45, Atmospheric Processes over complex terrain] 2. The Weather Analysis book by Dusan Djuric has a good chapter on vertical soundings. Some of the diagrams in this chapter are taken from this book. 3. If you want to deal with the topic of convection in great detail, you should read the comprehensive book Atmospheric Convection by Kerry A. Emanuel

25 look. 4. There is also very specialized literature on atmospheric thermodynamics. Among others are Atmospheric Thermodynamics by Bohren and Albrecht and An Introduction to Atmospheric Thermodynamics by Tsonis. 25th

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