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Vertical Structure of the Atmosphere

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Vertical Structure of the Atmosphere Chapter 5: Vertical structure of the atmosphere 5.1 Sounding of the atmosphere Fig.2.1 indicates that globally temperature decreases with altitude in the troposphere. However, already Fig.4.2 showed that locally the temperature can also stay constant with height (isothermal layer), or even increase (inversion layer). The actual stratification of the atmosphere is highly variable in space and time and is, thus, routinely probed by all national weather services. Generally, this probing consists of launching a radiosonde every 3 to 6 hours. Fig. 5.1: Photo sequence of the launching of a radiosonde on a ship and (on the right) the schematic structure of a radiosonde; La METEO de A à Z; Météo France ISBN/2.234.022096 The balloon is filled with hydrogen. Attached is a parachute to assure a soft landing, an aluminium foil to assure a high radar reflectivity and the actual sonde containing instruments for measuring pressure, temperature and humidity. For a description of the humidity sensor see chapter 3.4.1. Temperature, humidity and pressure are radiotransmitted to the station at the ground, while wind speed and wind direction are derived from telemetric data following the balloon by radar. The vertical profile of the atmosphere is then transferred on thermodynamic diagram papers. An eXample of such a representation can be found in Fig. 5.3. The diagram paper used here is a skew T –log p diagram paper which has a logarithmic aXis of pressure in the vertical and the isotherms are straight lines tilted 45° with respect to the horizontal. Other diagram papers are known also. More details can be found in the following sections. A.I. Flossmann 34 Fig.5.2 : schematic display of the data transmisson from the balloon sonde; La METEO de A à Z; Météo France ISBN/2.234.022096 Fig.5.3 summarizes the vertical information of the atmosphere and it is easy to detect humid (both curves are close) and dry (both curves are far apart) layers. Fig. 5.3: Example of a vertical temperature profile (solid line) and a humidity (dew point temperature) profile (dashed line) in skew T- log p paper 5.2 Thermodynamic diagram papers There eXist several different kinds of thermodynamic or aerological diagrams. It is a graph upon which observations of temperature, pressure and moisture content are plotted. Various lines are constructed from theoretical equations and drawn as a permanent support. The diagram looks complicated at first, busy due to the numerous additional lines. But actually, it is quite simple and will become obvious after the first use. A.I. Flossmann 35 All diagrams are quite similar, due to the identical basis used for their construction. Their preference in different countries and meteorological services is more historical, as most of them have their advantages as well as disadvantages. 5.2.1 the skew T – log p diagram The skew T – log p diagram is the most widely used in North America, and in many services with which the United States (various) weather services have had connections. Fig. 5.4: EXample of the skew T- log p diagram; highlighted are the sense of the isotherms, isobars, line of constant saturation vapour pressure and dry adiabats It is in fact a variation on the original emagram (see below). In 1947, N. Herlofson proposed a modification to the emagram which allows straight, horizontal isobars, and provides for a large angle between isotherms and dry adiabats, similar to that in the tephigram. The ordinate is ln (p/p0) and the isotherms are lines inclined 45° with respect to the abscissa. In France, the skew T-log p diagram is called “emagramme oblique à 45°” or “emagramme 761”. 5.2.2 The emagram The original emagram was first devised in 1884 by H. Hertz. It has orthogonal axes of temperature (T) and pressure (p). The dry adiabats form an angle of about 45 degrees with the isobars, and isopleths of saturation miXing ratio are almost straight and vertical. 5.2.3 The tephigram The tephigram was invented by Napier Shaw in 1915 and is used primarily in the United Kingdom and Canada. The name evolved from the original name "T-φ-gram" to describe the aXes of temperature (T) and entropy (φ) used to create the plot. Nowadays, entropy is given the symbol S with �� �� � �� = = − �� � � � and for the atmosphere as a perfect gas (compare chapter 4.1.3) one gets: A.I. Flossmann 36 �� � �� �� + ! = = � ln � � �! � �! using the definition of θ from eq (4.3). Thus, it comes evident that θ is also a measure of entropy. Fig.5.5: EXample of the original emagram, taken from Wikipedia highlighted are the sense of the isotherms, isobars, line of constant saturation vapour pressure and dry adiabats Consequently, the principal axes of a tephigram are temperature and potential temperature; these are straight and perpendicular to each other, but rotated through about 45° anticlockwise so that lines of constant temperature run from bottom left to top right on the diagram. This rotation makes lines of constant pressure almost horizontal, though gently curving down towards bottom left, so that altitude increases from bottom to top of the diagram. Fig.5.6: EXample of the tephigram, taken from Wikipedia; highlighted are the sense of the isotherms, isobars, line of constant saturation vapour pressure and dry adiabats A.I. Flossmann 37 5.2.4 The Stüve-diagram The Stüve diagram was developed circa 1927 by Georg Stüve (1888-1935) and quickly gained widespread acceptance. It is quite simple as it uses straight lines for the three primary variables: pressure, temperature and potential temperature. In doing so, however, it sacrifices the equal-area requirements (from the original Clausius-Clapeyron relation) that are satisfied in two of the other diagrams (Skew-T and Tephigram). Fig.5.7: EXample of the Stüve diagram, taken from Wikipedia; highlighted are the sense of the isotherms, isobars, line of constant saturation vapour pressure and dry adiabats In the following, the skewT - log p diagram paper will be eXplained in more detail. The construction details for the others can be found in the literature. 5.3 The emagramme 761 Several lines can be found in this diagram paper that will be detailed below: 5.3.1 Isotherms The isotherms are solid, continuous lines, inclined 45° from bottom left to top right. They are drawn every degree and labelled in degrees Celsius. Their spacing is constant, and, thus, their scale is linear. 5.3.2 Isobars The isobars are solid, continuous, horizontal lines. They are labelled in hPa and given every 10 hPa between 1050 and 200 hPa and every 5 hPa between 100 and 200 hPa. The values increase towards the bottom and their spacing is not constant. The scale of the isobars is logarithmic of the form: 1000 � = � log � At the right hand side of the diagram paper, approXimate altitudes in km are indicated. 5.3.3 Lines of constant saturation miXing ratio Saturation miXing-ratio lines are the straight, dashed, lines sloping from the lower left to the upper right. Saturation miXing-ratio lines are labelled in parts of water vapour per 1000 parts of dry air (g/kg). Values range between 40 g/kg to 0.1 g/kg, from right to left (compare eq 3.5): A.I. Flossmann 38 Fig. 5.8: Emagramme 761 (document Météo-France); source: Météo France A.I. Flossmann 39 !!"#,!(!) �!"#,! ≈ 0.622 (5.1) !!!!"#,! 5.3.4 Dry adiabats Dry adiabats are the slightly curved, solid lines that slope from the lower right to the upper left. They indicate the rate of temperature change in a parcel of dry air, which is rising or descending adiabatically when no phase change is occurring with water (compare eq. 4.3) ! !! � = � ! (5.2) ! 5.4 The saturation adiabats Saturation adiabats, also called pseudo adiabats are the slightly curved, dashed lines sloping from the lower right to the upper left, eXcept those on the eXtreme right. They indicate the temperature change eXperienced by a saturated parcel of air rising pseudo- adiabatically through the atmosphere. Pseudo-adiabatic means all the condensed water vapor is assumed to fall out (precipitate) immediately as the air rises. 5.4.1 The reversible moist adiabat The pseudo adiabatic equation drawn in the emagramme is closely related to the reversible moist adiabat, both being also derived from 1st principle, and allowing for phase changes. These phase transitions add a heat source to the system. In the reversible moist adiabat, the condensed liquid water is remaining in the system. Thus, eq. (4.1) now reads: �� � �� � − ! = − ! ��!"#,! � �! � �!� !"#,! !"#,! � − � �!� �!� !" ! !"#,! �� = −� − ! �� + � �!� �!� �! � here, the temperature dependence of lv gives that last term, according to chapter 3.2. Replacing dT by the Clausius-Clapeyron expression (3.3): � �� ! = � � ln �!"#,! �! ! yields �� � �� � �!"#,! � �!"#,! � − � �� − ! = −� ! − ! � ln �!"#,! + !" ! �!"#,! � �! � �!� �! �! � or with introducing k=Rh/cp and Rd/Rv=0.622 �� �� � �!"#,! � �!"#,! � − � �� − � = −� ! − ! � ln �!"#,! + !" ! �!"#,! � � �!� �!0.622 �! � the second last term of this equation can be rewritten, by using the definition of chapter 3.3: A.I. Flossmann 40 �!"#,! �!"#,! = 0.622 � − �!"#,! which yields also: �!"#,! �!"#,! = � 0.622 + �!"#,! A logarithmic differentiation gives: 0.622 � ln �!"#,! = � ln � + � ln 0.622 + �!"#,! �!"#,! Replacing in the reversible moist adiabatic equations yields: �� �� � �!"#,! �� − � + ! � � �!0.622 � � �!"#,! � � − � �� = −� ! − ! � ln 0.622 + �!"#,! + !" ! �!"#,! �!� �! �! � the factors for dT can be combined: �� �� � − � − � �!"#,! ! � !" ! � �� �� = � + � �!"#,! + � �!"#$"% − � − � �!"#,! !" !" ! � !" ! � �� �� = � + � �!!! ≈ � !" ! � !" � as well as the factors for dp: !"#,! !"#,! !"#,! −�!�� + �!� �� = −�! − �!� �� + �!� �� = −�!�� yielding finally when approXimating also in the two remaining terms cp≈cpd !"#,! (5.1) �� �� �!�! � !"#,! − �! = −� ! ! − �!� ln(0.622 + � ) � � �!� this is the expression for the reversible moist adiabat. It is possible to integrate this sat,w sat,w equation between a state of zero condensate (r =0; p0; T->0) and r yields the equation for the potential equivalent temperature θe: !! !"#,! !! !!!!! !"#,! �! = � ��� + �! ln 1 + 1.608� (5.2) ! !!! In a first order approXimation, the second term in the eXponential eXpression can be neglecting.
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