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. The first part of the equation can be identified as the dry potential temperature, giving: !"#,! �!� �! = � exp �!"� often also called liquid water potential temperature.

A.I. Flossmann 41 Since in this equation the exponent is a small number, the function can be developed keeping only the 1st order term:

!"#,! ! !"#,! ! �!� �! ! �!� �! ! �! ≈ � 1 + = � + = �!,!"# �!"� � �!" �

The newly introduced temperature Te,sat is called equivalent temperature of saturated air. It is also applied to unsaturated air and is defined, when neglecting the contribution of liquid water to the heat capacity, as

!!! �! = � + (5.3) !!"

It is called equivalent temperature Te. This temperature is often used to measure the degree of human comfort since it involves temperature and humidity. High values of T and r represent marine tropical air masses.

5.4.2 The moist adiabat temperature decrease Similar as was done in (4.5) the adiabat equation can be used to estimate the temperature decrease with altitude in a water vapour saturated environment:

�� � �� � − ! = − ! ��!"#,! � �! � �!� which is also: � ln � � � ln � � ��!"#,! − ! = − ! �� �! �� �!� ��

Developping ��!"#,!/�� with respect to T and p gives:

� ln � � � ln � � ��!"#,! �� ��!"#,! �� − ! = − ! + �� � �� � � �� �� �� �� ! ! ! ! dp/dz=-gρ which gives:

�� � � ��!"#,! �� ��!"#,! − = − ! − �� �� � � �� �� �� ! ! ! ! using (5.1) and neglecting the saturation vapour pressure in the denominator yields: ��!"#,! �!"#,! = − �� � ! also deriving (5.1) with respect to T and inserting Clausius-Clapeyron eq.(3.3) yields:

��!"#,! 0.622 ��!"#,! 0.622 � �!"#,! � �!"#,! = = ! = ! �� � �� � � �! � �! ! ! ! ! inserting both equation above and solving for dT/dz yields:

A.I. Flossmann 42 ! !!"#,! !! ! !" ! !!! = − ! !"#,! (5.4) !" !"# !!" !.!"" !!! !! ! !!!!!

The temperature decrease in a saturated environment depends on T, but one can estimate that dT/dz≈ −0.6°�/100�, and consequently quite a bit smaller than the dry adiabatic temperature decrease. This can be explained with the additional heat released during condensation.

5.4.3 The irreversible moist adiabat The derivation of the irreversible moist adiabat follows in large parts the one for the reversible adiabat. Only, the condensed liquid water is not remaining in the considered air parcel, but falling out immediately after condensation. Thus, the element does not contain any liquid water at any time. Consequently, the total mixing ratio rH2O, used in 5.4.1 is not conserved. The temperature dependency now reads: �� �� � − � − � �!"#,! ! � !" ! � �� �� = � + � �!"#,! − � − � �!"#,! !" !" � !" ! � !"#,! �� �! !"#,! �� = �!" + �!� ≈ �!" 1 + � � �!" � The pseudo-adiabatic equation now reads:

� �� �� � � �!"#,! � ���,� ! ! !"#,! 1 + � − �! = −� − �!� ln 0.622 + � ��� � � �!� and integration yields the pseudo-potential temperature:

!! !"#,! !! !!!!! !"#,! !! ! !"#,! �!" = � ��� + �! ln 1 + 1.608� − � �� (5.5) ! !!! !!" !

This equation cannot be solved analytically any more. It requires a numeric and iterative technique. The reality is to be found somewhere between the concept of the reversible moist adiabat and the pseudo-potential adiabat. Their numerical values are, in any case, relatively close to each other.

All derivations have been made for a saturation state between water vapour and liquid water. When ice is present, the indices have to be changed accordingly, but the reasoning stays the same.

5.5 Application of the emagramme 761 (skew T – log p)

5.5.1 Determination of unreported meteorological data from observations Surface measurements and soundings have provided a table of data of the form:

A.I. Flossmann 43

Pressure (hPa) Temperature (°C) Dew point (°C) 1000 20 15 900 18 12 700 5 0 Tab.5.1 example values of pressure, temperature and dew point used to illustrate how to obtain unreported data from the emagramme

Other information can be obtained either by using the above presented formula, or by reading the values from the emagramme.

5.5.1.1 Mixing ratio At the intersection of the environmental dew-point temperature line and the desired pressure level, interpolate the mixing ratio value horizontally along the pressure level line between the values of the dashed brown saturation mixing-ratio lines (rsat) on either side of the intersection. On the table τ at 900 hPa is 12°C (red cross in Fig. 5.9) and the saturation mixing-ratio line passing through 12°C is 9.8 g/kg. Hence, the mixing ratio of the air at the 900-hPa level on the sounding is 9.8 g/kg.

5.5.1.2 Saturation mixing ratio At the intersection of the environmental temperature T and the pressure level p, interpolate the saturation mixing ratio value horizontally along the pressure level between the values of the dashed, brown saturation mixing ratio lines on either side of the intersection. In Fig. 5.9 , T at 900 hPa is 18°C (black cross); and the saturation mixing-ratio value at 900 hPa and 18°C is interpolated to be 14.5 g/kg. Hence the saturation mixing ratio (the amount of moisture the air could hold) at the 900 hPa level is 14.5 g/kg.

Fig. 5.9: part of an emagramme illustrating the determination of the mixing ratio, the saturation mixing ratio and relative humidity as explained in the text.

5.5.1.3 Relative humidity Following chapter 3.3.4.1

A.I. Flossmann 44 � �� = 100 �!"#(�)

As the mixing ratio and the saturation mixing ratio have been determined previously, for the example given, RH= 9.8/14.5 100= 68%. The alternate procedure for determining relative humidity is shown graphically in Fig.5.9 (blue lines), and described below: • At the pressure level for which relative humidity is desired, start at the dew point temperature value and follow parallel to the saturation mixing ratio line to the 1000-hPa pressure level. • From this intersection, draw a line upward parallel to the isotherm lines. • From the pressure level for which relative humidity is desired, start at the temperature value and draw a line parallel to the saturation mixing ratio line to the intersection with the line drawn in the previous step. The numerical value of the isobar through this last intersection divided by ten is the percent relative humidity (ex: 680/10=68%).

Fig. 5.10: part of an emagramme illustrating the determination of the vapour pressure, the saturation vapour pressure and potential temperature as explained in the text.

5.5.1.4 Vapour pressure Vapour pressure e is the partial pressure exerted by water vapour molecules in a given volume of air. As shown by Fig. 5.10 from the pressure level at which e is desired, start at the environmental dew-point temperature value (τ), and go parallel to the isotherm lines to the 622 hPa isobar. Read the saturation mixing ratio value at this point. This value is the vapour pressure in hPa for the original level, since:

� � � = 0.622

The example in Fig.5.10 shows the dew point temperature τ at 900 hPa to be approximately 12°C. Following upward, parallel to the isotherms to 622 hPa and

A.I. Flossmann 45 interpolating the saturation mixing ratio value gives a value of 14.5 hPa for the vapour pressure.

5.5.1.5 Saturation vapour pressure The saturation vapour pressure is the partial pressure exerted by water vapour in a given volume of air when the vapour is saturated at the current temperature. The procedure is the same as with determination of the vapour pressure except the starting point is the temperature value at the desired level. See Fig.5.10. From the pressure level at which esat,w is desired, start at the environmental temperature value, (T) and go parallel to the isotherms to the 622-hPa level. The saturation mixing ratio value at this point is 21.5 and the saturation vapour pressure is 21.5 hPa for the original level. The ratio e/ esat,w gives again a relative humidity of 68%

5.5.1.6 Potential temperature Potential temperature (θ) is the temperature that a parcel of dry air would have if it were brought dry-adiabatically from its initial state to a pressure of 1000 hPa (compare chapter 4.1.3.1). From the environmental temperature value T at the pressure level for which potential temperature is desired, follow parallel to the dry adiabats to the 1000 hPa level, as shown in Fig. 5.10. The value of the temperature at the intersection of the dry adiabat and the 1000 hPa isobar is the potential temperature (in the example θ=27°C). Since the dry adiabats are labelled with the value of the isotherm they intersect at the 1000-hPa surface, the potential temperature is also the value of the dry adiabat which passes through the environmental temperature value at the desired pressure level (in the example 900 hPa).

5.5.1.7 Wet bulb temperature The wet-bulb temperature is the lowest temperature to which a volume of air at constant pressure can be cooled by evaporating water into it. Physically, the wet-bulb temperature is the temperature of the wet-bulb thermometer (compare chapter 3.4.1). Fig. 5.11 illustrates the method of finding the wet-bulb temperature for a given pressure level. • From the dew-point at the level for which the wet-bulb temperature is desired, draw a line upward parallel to the saturation mixing-ratio lines. • From the temperature value at the desired pressure level, draw a line upward parallel to the dry adiabat lines to where it intersects the line drawn first. • From this intersection point, follow parallel to the saturation adiabats back to where it intersects the original pressure level. The temperature value at this last intersection is the wet-bulb temperature. For the example this gives a Tw of 14.3°C.

A.I. Flossmann 46

Fig. 5.11: part of an emagramme illustrating the determination of the wet bulb temperature and the wet bulb potential temperature as explained in the text.

5.5.1.8 Wet bulb potential temperature The wet-bulb potential temperature is the wet-bulb temperature a sample of air would have if it were brought saturation-adiabatically to a pressure of 1000 hPa. From the wet-bulb temperature follow down parallel to the saturation adiabat lines to the 1000-hPa level. The temperature value at this intersection is the potential wet-bulb temperature θw. In Fig. 5.11, the wet-bulb potential temperature is 18.5°C.

5.5.1.9 Equivalent temperature The equivalent temperature Te (compare section 5.4.1) is the temperature a parcel of air would have if all its moisture were condensed out by a pseudo-adiabatic process (i.e., with the latent heat of condensation being used to heat the air sample), and the sample then brought dry- adiabatically to its original pressure. This equivalent temperature is sometimes termed the "adiabatic equivalent temperature." Fig.5.12 illustrates the method of finding Te. • From the dew-point temperature at the pressure level for which the equivalent temperature is desired, (in the example, 900 hPa is used), draw a line upward paralleling the saturation mixing-ratio lines. • From the temperature at the desired pressure level (900 hPa in the example), draw a line upward paralleling the dry adiabat lines until it intersects the first This is the LCL for the 900 hPa level. • From this LCL point, draw a line upward following the saturation adiabat lines to a pressure level where both the saturation and dry adiabats are parallel; i.e., to a pressure where all moisture has been condensed out of the parcel. • From this pressure, follow the dry adiabat lines back to the original pressure (900 hPa in the example). The isotherm value at this point is equal to the equivalent temperature. In the example shown in Fig. 5.11, Te(900) =42.5°C (outside range of the emagramme).

A.I. Flossmann 47

Fig. 5.12: part of an emagramme illustrating the determination of the equivalent temperature and the equivalent potential temperature as explained in the text.

5.5.1.10 Equivalent Potential Temperature (θe) The equivalent potential temperature is the temperature a parcel of air would have if all its moisture were condensed out by a pseudo-adiabatic process and the sample were brought dry adiabatically to 1000 hPa. From the equivalent temperature, follow the dry adiabat lines to the 1000 hPa isobar. The isotherm value at this point is equal to the equivalent potential temperature. In the example shown in Fig.5.12, θe =56°C.

5.5.2 Determination of cloud base

5.5.2.1 Lifting Condensation Level (LCL) The height at which a parcel of air becomes saturated when it is lifted dry adiabatically is the Lifting Condensation Level. When a parcel of air is ascending upward, as by being forced upward across land, a mountain, or over a layer of colder air, the air cools dry adiabatically. This is called mechanical lifting. If the air is lifted high enough, and cools enough, the parcel is saturated and any further cooling will result in condensation of moisture. This is the Lifting Condensation Level. • From the dew-point temperature of the level for which the LCL is desired to be determined, draw a line upward parallel to the saturation mixing ratio lines. • From the temperature value of the level for which the LCL is desired, draw a line upward parallel to the dry adiabat lines. The level where these two lines intersect is the LCL. Using the observations of the Table 5.1 for 1000 hPa give the LCL at 930 hPa (compare Fig.5.12).

5.5.2.2 Convection Condensation Level (CCL) The convection condensation level is the height at which a parcel of air, if heated sufficiently from below, will rise dry adiabatically until it is just saturated. This is the height of the base of cumuliform clouds, which are, or would be, produced by thermal

A.I. Flossmann 48 convection from surface heating. Thus, the procedure always starts with temperature and dew-point temperature values of the surface air. From the surface dew-point temperature, draw a line up the saturation mixing ratio line to where it intersects the environmental temperature curve. This level is the CCL. The value read 790 hPa for the meteorological data of Tab. 5.1 (see Fig.5.11).

Fig. 5.13: part of an emagramme illustrating the determination of the LCL, the CCC and the convection temperature as explained in the text.

5.5.2.3 Convection Temperature (Tc) The Tc is the surface temperature that must be reached to start formation of convective clouds by heating of the surface air layer. When this temperature is reached, air can rise dry adiabatically to the convection condensation level. Start at the CCL and follow down the dry adiabat to the surface pressure isobar. The temperature at this intersection is the convection temperature. From Fig.5.12 Tc=31.5°C.

5.5.3 Determination of cloud dimensions

5.5.3.1 Level of Free Convection (LFC) The level of free convection is the height at which a parcel of air lifted dry adiabatically until saturated and then moist adiabatically thereafter, would first become warmer (less dense) than the surrounding environmental air. The parcel would then continue to rise freely above this level until it becomes colder (more dense) than the surrounding air. Note, a LFC may not be present for all atmospheres. From the LCL go upwards parallel to the saturation adiabats until you intersect the temperature curve. The level where the intersection occurs is the LFC (compare Fig.5.14). Note, that a parcel, which is forced upward, will first cool at the dry adiabatic lapse rate until it is saturated (follow the dry adiabat to the LCL). Thereafter as it is forced upward, it cools at the saturation adiabatic lapse rate, (moisture is condensing out of the parcel; follow the saturation adiabat from the LCL upward). The parcel must be forced upward as long as it is cooler than the surrounding air. Once the parcel becomes warmer than the surrounding air it rises by itself.

A.I. Flossmann 49

Fig. 5.14: part of an emagramme illustrating for mechanical lifting the determination of the LFC, the LC=EL and the positive and negative areas as explained in the text.

5.5.3.2 Level of Convection (LC) The level of convection, also called equilibrium level (EL), is the height where a buoyantly rising parcel, (rising freely because it is warmer than the surrounding air), again becomes equal to the temperature of the surrounding environmental air. Above this level, the parcel is cooler, (denser) than the surrounding air and will not rise freely. For mechanical lifting: From the LFC, continue drawing a line upward paralleling the saturation adiabat lines until the drawn line intersects the temperature curve (compare Fig. 5.13). For convective lifting: From the CCL, continue drawing a line upward paralleling the saturation adiabat lines until the drawn line intersects the temperature curve. This is the level of convection (compare Fig. 5.14). The actual cloud top will slightly exceed the level of convection due to an overshooting of the rising air parcel. The depth of the overshooting into the negatively buoyant zone will depend on the velocity of ascent.

Fig. 5.15: part of an emagramme illustrating for convective lifting the determination of the CCL,

A.I. Flossmann 50 the LC and the positive and negative areas as explained in the text.

5.5.3.3 Positive and Negative Areas On a thermodynamic diagram, such as the Skew-T, Log-P Chart, a given area can be considered proportional to a certain amount of energy of a vertically and adiabatically moving air parcel.

Positive Area: When a parcel can rise freely because it is in a layer where the adiabat it follows is warmer than the surrounding environment, the area between the adiabat and the environmental temperature curve is proportional to the Convective Available Potential Energy (CAPE), also called positive areas (compare chapter 5.7), which is available for conversion to kinetic energy of motion of the parcel. A parcel rising in these CAPE or positive areas finds itself warmer than the surrounding air and continues to rise freely. These are considered unstable areas and are regions where clouds of greater vertical extent can form (compare Fig.5.13 and 5.14).

Negative Area: When a parcel on a sounding lies in a negative area, energy has to be supplied to it to move it either up or down. The area between the path of such a parcel moving along an adiabat and the environmental temperature curve is proportional to the amount of energy that must be supplied to move the parcel. For this reason, this negative area is called a region of Convective Inhibition (CIN) (compare Figs.5.13 and 5.14). The negative and positive areas are not uniquely defined on any given sounding. They depend on the parcel chosen and on whether the movement of the chosen parcel is assumed to result from surface heating, (insolation), release of latent heat of condensation, or from forcible lifting, (i.e., convergence, orographic lifting, etc.).

5.6 Stability As already mentioned in chapter 4.1.3.4, a density difference between a rising air element and its environment can result in a vertical motion (eq 4.9):

!" ! !! ! !! !!! = � !"# = � ! !"#! ≈ � !"# (5.6) !" ! !!"#! !!"#

This motion can be ascending or descending depending on the difference on the right hand side of the equation. The environmental meteorological values are given by the sounding obtained in regular intervals. The temperature variation of the air parcel will follow a dry or a saturated adiabat, depending whether the parcel is already saturated or not. The dry adiabatic temperature decrease is given by the dry adiabatic lapse rate (eq 4.5):

�� � Γ! = − = �� ! �! while the saturated temperature decrease is described by the saturated adiabatic lapse rate (eq 5.3):

A.I. Flossmann 51 � �!"#,! 1 + ! �� � �!� Γ!"# = − = ! !"#,! �� !"# �!" 0.622 �!� 1 + ! �!�!�

Fig. 5.15 compares the adiabatic lapse rates with an environmental temperature decrease:

Fig.5.16: an adiabatic air parcel moves vertically in an unstable environment (left: dry; right: saturated)

The environmental temperature profile is indicated by the black line, and the initial position of the parcel by the blue circle. Initially, the parcel has the same temperature as the environment. Moving upwards along an adiabat leaves the parcel at any pressure level always warmer than the environment. It will consequently continue to rise. If, however, the parcel descends, then it will always become colder than the environment at any given pressure level. It will, thus, continue to descend. An atmosphere where any vertical motion of an air parcel is maintained is called unstable (γatmosphere>Γd, Γ sat).

In the case where the temperature of the environment decreases less with altitude, the result can change:

Fig.5.17: an adiabatic air parcel moves vertically in an stable environment (left: dry; right: saturated)

In the cases displayed in Fig.5.16 the air parcel when ascending gets colder than its environment, and will, thus, develop negative buoyancy. Left alone, it will redescend to its initial position. And when descending, it will become warmer than its environment and rise to regain its initial position. An atmosphere stratified in this manner, i.e. an air parcel will return to its starting position when perturbed is called a stable environment

A.I. Flossmann 52 (γatmosphere<Γd, Γsat). In particular, isothermal layers and inversions are stable, i.e. supressing vertical motion.

Between the stable and the unstable, there is the neutral, or indifferent atmosphere (γatmosphere=Γd or γatmosphere=Γ sat):

Fig.5.18: an adiabatic air parcel moves vertically in a neutral environment (left: dry; right: saturated)

Here, the air parcel will find exactly the same environmental temperature conditions than its adiabatic evolution will give. Consequently, it will experience zero buoyancy and stay at whatever pressure levels it will be put. Note, however, that a neutral stratification is different for a dry or a saturated parcel. Consequently, it needs to be specified if the neutral conditions pertain to a dry or a saturated adiabat.

Fig.5.19: an adiabatic air parcel moves vertically in a conditionally unstable environment

Furthermore, the different lapse rate can also result in a particular case, given in Fig. 5.18. If the atmospheric lapse rate takes values between the dry adiabatic and the saturated lapse rate, then the air parcel will experience the atmosphere as stable when it is dry and as unstable if it is saturated. Such an atmosphere is called conditionally unstable (Γd> γatmosphere >Γ sat) or latent unstable. Such an atmosphere is rather common: stable with respect to dry conditions, which supresses cloud formation from

A.I. Flossmann 53 each little vertical displacement of an air element. However, once condensation as occurred, the saturated element finds itself in an unstable environment, and deep convection can develop. These conditions are rather current e.g. during summer.

To summarize: �!"#$%&!!"! < Γ!"# unconditionally stable Γ! > �!"#$%&!!"! > Γ!"# conditionally unstable �!"#$%&!!"! > Γ! unconditionally unstable

The lapse rate of an unconditionally unstable atmosphere are also called superadiabatic lapse rate. Instead of using the environmental lapse rate to characterize stability of the atmosphere, the derivative of the potential temperature can be used (dry or pseudo potential temperature): �� > 0 ≫ � ↗ stable �� �� = 0 neutral �� �� < 0 ≫ � ↘ unstable ��

The Brunt- Vaïsälä frequency, as derived in chapter 4.1.3.5 from

�!∆� = − �!∆� ��! gives � � = Γ! − �!"#$%&!!"! �!"#! and shows that in a stable atmosphere where Γd>γatmosphere N is positive. This results in a pure oscillation after a displacement Δz:

∆� = � cos �� + � sin �� with ω=N. In the case of N2<0 (unstable atmosphere), an exponential solution results:

∆� = � exp �� + � exp −��

And the neutral atmosphere with N2=0 will provide a solution

∆� = �� + � in agreement with the discussion above on stability.

5.7 Convective available potential energy

Based on the stability concept, the positive areas identified in Fig. 5.13 and 5.14 of chapter 5.5 can be associated to unstable parts of the atmosphere, while the negative

A.I. Flossmann 54 areas comprise stable parts. The size of these two areas will determine the strength of the convection. Based on eq. (4.9) �� �! − �!"#! �� �� �� = � = = � �� �!"#! �� �� �� the total potential energy PE of an air parcel ascending in the atmosphere from the ground to the LC is equal to the kinetic energy gained during the ascent:

!!" !!" 1 �! − �!"#! �� = ��! = � �� 2 �!"#! ! !

This PE commonly is sub-divided into two separate parts: the one for the positive area and the one for the negative area.

!!" �! − �!"#! ���� = ! � �� (5.9) �!"#! ! !"#

The positive part is called Convective Available Potential Energy (CAPE): It is the amount of potential energy that a parcel can obtain from the unstable part of the environment. Mathematically, it is the area between the level of free convection and the equilibrium level (EL) or (LC= level of convection). In order to identify CAPE on a sounding, find the level of free convection and follow the moist adiabat through this level up to the equilibrium level. The area between this curve and the temperature curve is positive area, CAPE (see Fig.5.13).

!!"# �� − ����� ��� = − � �� ����� !

The negative part is called Convective Inhibition (CIN): Amount of energy that has to be provided for a parcel to reach the level of free convection. The CIN is proportional to the area between the temperature curve and a parcels ascent via both a dry and moist adiabatic lapse rates (see Fig.5.13). From this, �� = ���� − ���

Typical CAPE values are around 500 m2 s-2, however also much higher values of 2000- 3000 m2 s-2 are observed in severe storms. In those systems generally higher vertical velocities are observed as suggested by eq (5.6).

A.I. Flossmann 55