Introduction to Tephigrams:
Introduction to Tephigrams
Recommended Texts:
** J T Houghton, 'The Physics of the Atmospheres', 2nd ed. Cambridge University Press, 1986 (Background for dynamics, radiation transfer and thermodynamics) *** D. H. McIntosh and A. S. Thom , „Essentials of Meteorology‟, Wykham (A good introductory text, especially section on thermodynamics and tephigrams) ** R. R. Rogers and Yau , „A Short Course in Cloud Physics‟ Pergamon (Good introductory text on thermodynamics of dry and moist (incl. Tephigrams), Cloud formation and growth, cold clouds, precipitation). ** J.S.A.Green, 'Atmospheric Dynamics', C.U.P. 1999 (A novel and interesting approach with good Physics for Dynamics) * J R Holton, 'An Introduction to Dynamic Meteorology', 2nd ed. Academic Press, 1979 (Dynamics of the atmosphere) A good reference text on microphysics * H. R. Pruppacher and J. D. Klett, „Microphysics of Clouds and Precipitation‟ Reidel mathematical treatment of cloud microphysics. A very complex but complete text)
Detailed
Useful symbols, units and values p T R* pressure Nm-2 or in hectapascals, hp, or millibars, mb. Standard pressure p0 = 1013.25mb . 1 mb / hp = 102 Nm-2 density , kg m-3 temperature in K or oC . Tw and Td are Wet bulb and Dew-point temperatures.
Gas constant (per Kmol) 8.314*103 J kmol-1 K-1 ( Specific gas constant R = R* / M ). M is the molecular weight kg kmol-1 For dry air , R = 287 J kg-1 K-1 Md and Mm are the molecular weights of dry and moist air. the mixing ration r is defined as mass of water vapour to that of the mass of dry air, or r = v / d often g kg-1 or kg kg-1 relative humidity U = e / es(T) where e is the vapour pressure of water and es is the saturation vapour pressure, a function of T only, ignoring curvature effects etc. potential temperature , K or
o
r
U B k Lv LI
C
static stability, 1/*d/dz , km-1 Stefan‟s constant , von Karman‟s 5.70*10-8 W m-2 K-4 ~ 0.4
Latent heat of fusion of ice , 3.35*105 J kg-1 Latent heat of vaporisation of water , 2.47*106 J kg-1
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Cp g S G
Specific hear of air at constant pressure , 1.01* 103 J kg-1 K-1 . Cpv ~ 2*Cp gravitational acceleration , 9.81m s-2 dry adiabatic lapse rate , 9.8 K km-1 solar constant , 1.39 kW m-2 gravitational constant 6,67 * 10-11 N m2 kg-2
Planet Earth Mars Venus
Mass 59.8*1023 kg 6.4*1023 kg 48.7*1023kg
Radius 6.37*103 km 3.39*103 km 6.05*103 km
Solar Distance 1.49*108 km 2.28*108 km 1.08*108 km
Angular Velocity 7.3*10-5 rad s-1 7.8*10-5 rad s-1 3.0*10-7 rad s-1
Orbital Inclin. 23.50 25.20 <30
Orbital Ecc. 0.017 0.093 0.007
g 9.81ms-2 3.76ms-2 8.80ms-2
Albedo 0.33 0.17 0.71
Solar Constant 1.39 kWm-2 0.6 kWm-2 2.7 kWm-2
Adiabatic lapse rate 9.8 0C km-1 4.5 0C km-1 10.7 0C km-1
0.
Introduction:
Meteorology is the study of the physical state of the atmosphere. The atmosphere is a heat engine transporting energy from the warm ground to cooler locations, both vertically and horizontally. The driving force is the solar radiation. The whole atmospheric system transforms and transports energy. The heat engine's source of energy is the sun. Short wave radiation is absorbed primarily at the bottom. The working fluid is the atmosphere, which distributes heat by motion systems of all time and space scales. The heat sink is space with long-wave radiation being emitted. A few atmospheric motion systems are listed below, with typical time and space scales. Their dependence on radiation is given. However, the examples given are often found with vastly different sizes and time-scales.
Type of motion Dust Devils Vertical Convection (cumulus, cumulonimbus) Sea Breeze Tornado system Hurricane Depression / cyclone General Circulation
Size Scale 1 -100 m 1 - 10 km 10 - 100 km ~ < 100..300 km A few hours 10 - 100 km < ~ a few weeks ~ < 1000 km 1 - 10 days 1-3 103 km Years 40,000 km
Time scale 1 - 10 mins 20 mins to a few hours A few hours
Radiation's importance Yes ( solar ) Sometimes (solar) Yes ( solar) No No No Yes (solar + longwave )
The atmosphere is full of waves of different wavelengths , from 40,000 km to eddies of ~ 1 cm. Cloud droplets and Cloud Condensation Nuclei are of the size of micros, and growth rates for these cloud microphysical processes can be of the order of milliseconds, compared with days for large scale systems. There are many feedback processes e.g.
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between dynamics, radiation and clouds and it is this range of science that makes the study of meteorology and atmospheric physics so exciting. 1. Physical properties of the atmosphere. The atmospheric composition is uniform in the lowest 80km. The atmosphere is a thin veil above the surface (equivalent to “ a month‟s” dust on a model globe) Gas Nitrogen Oxygen Argon Carbon Dioxide composition 78.09% 20.95% 0.93% 0.04% Molecular weight (g mol-1) 28.013g 32g 39.948g 44.010g
N2 O2 A CO2
Mean molecular weight, i.e. the weighted average, for air < 80 km , is 28.966 kg kmol-1 There are many other components, some of which can be critically important. e.g. Ne 0.002%, CH4 0.0002%, O3 10-6 % , H2 10-5 % and water (variable amounts depending on depth) to mention but the largest. For example water vapour is a much more powerful “green-house gas” , see radiation and heat transfer $2. The composition of the atmosphere depends on outgassing from the surface, chemical reactions with surface materials and capture from solar nebula, solar wind meteorites and comets. Losses include thermal evaporation, chemical reactions with the surface, rotational effects and any sweeping action of the solar wind Above 400km most atoms are dissociated. The light atoms, e.g. Hydrogen, have a lower mass, and can have enough energy to reach the escape velocity, ve and thus can overcome the Earth‟s gravitational field. In the exosphere, above say 600km, if a particle has enough velocity it can escape into space. If the root mean square velocity of the atoms is a small fraction of the escape velocity, then the loss will be negligible. However, doing rough calculations to calculate the escape velocities , at T = 1500 K hydrogen will escape over a prolonged period of time. Molecule H He N,O N2
1/2
Molecular Weight 1 4 14 , 16 28
c / ve 0.57 0.28 0.15 0.11
Half life time constant, years ~ 104 ~ 1010 ~ 1035 ~ 1060
Table 1.0 Planetary half-lives of exosphere constituents at T = 1500 K , c = (3 R* T / M ) Given root mean square velocity c = (2? R* T / M ) 1/2 and ve = (2 G {mass of planet} / r)1/2 and r is the radial distance to the exosphere then for Mars where r ~ 3500km, the ratio of c / ve is much larger ( how much?? and what are the consequences?). An alternative method is to use a Boltzman distribution which produces the following:. v/v0 fraction 0 1.0 1 0.7 2 0.1 3 5*10-4 4 10-7 5 10-20 10 10-60
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Table 1.0.1 The "approximate" fraction of atmospheric molecules with speeds greater than v to the most probable velocity v0 1.1 Temperature structure of the Earth‟s Atmosphere.
The earth‟s atmosphere temperature structure depends critically on the radiative balances and dynamical motions of the atmosphere (e.g. on average, for the Earth, 90% of the surface radiation is absorbed by the atmosphere, for Mars 90% is transmitted directly to space).
Figure 1.1 Typical mid-latitude values of pressure, temperature, and density for the lowest 100km of the earth‟s atmosphere.
Figure 1.1.1 Average temperature (0C) in the troposhere and lower stratosphere. 1.2 Pressure variation in the vertical
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At sea level, the surface pressure is about 105 N m-2 . Thus since pressure is force per unit area, m = p/g = 1.02*104kg or about 10 tonnes per square metre. If air was incompressible, then since m = g h , then h would be about 8.4km. In reality air is compressible, but as a first approximation, it is often assumed to be incompressible in the lowest few kilometres. At the surface, one cubic metre of air weighs about 1kg. (1 m3 of water weighs 103 kg) Governing equations: Standard Gas equation Across a volume of gas Thus and
p= R T
δp = - g δz ( change in Force per unit area )
δp / δz = - g .
δz = R T / g δp/p.
This called the hydrostatic equation, or dp / p = - g dz / (R T) Integrating gives
p(z) = p(z0) exp(-gz/RT)
We can define RT/g as a scale height = km (height at which the pressure reduces by -1 e ) and the decimal scale height = km. 1.3 Water vapour in the atmosphere.
In the Earth‟s atmosphere, water vapour has important effects. The amount of water vapour in a volume, is defined by its mixing ratio, r . In a unit volume of moist air, r = v / d or mass of water in the volume compared with the mass of the dry air. Moist air p = R*/Mm T , dry air pd = d R*/Md T , for water vapour e = v R*/Mv T p – e = pd = d R*/Md T . Thus r = Mv / Md {e / (p-e ) } or r ~ 5 / 8
{e / (p-e ) }
e is the water vapour pressure. The saturated vapour pressure, es(T) is the vapour pressure of water above a flat surface of pure water, super cooled water and ice. Ignoring all curvature terms the s.v.p. is a function of temperature only, and tables of data are available to define es(T) .
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Figure 1.3 Saturation vapour pressure over ice eI(T) and water ew(T). For water, es(T) = ew(T) Relative humidity is defined e / es(T) . The dewpoint is that temperature at which cooling the air at constant pressure will produce saturated air where e = es(Td) . For the measurement of humidity , dew point and wet bulb temperatures are important, see Figure 1.3. Recent electronic methods are available to calculate humidity and mixing ratio. Prior to this, a thermodynamic diagram (e.g. Figure 1.3 or a “Tephigram” ) were required, or a special “whirling hygrometer” with tabulated values could be used to calculate the wet bulb depression and hence the humidity. The basic principle is that if you surround a thermometer bulb by a wick soaked in water, latent heat will be required to evaporate the water. Thus depending on the ventilation and the efficiency of the evaporation, the thermometer bulb will be cooled to a point, somewhere between points A to C. If the sensible heat from the parcel at constant pressure, the point A will be reached, the dewpoint. If the bulb stays at the same temperature, then the wet bulb will not cool, and effectively point C will be reached. In reality, we assume point B will be reached when we bring air to point B adiabatically. When water is evaporated in the sample volume, the temperature will decrease to provide energy for this process. i.e Td < Tw < T. How can this be used to measure the vapour pressure e and hence the humidity U and mixing ratio r? Consider the latent heats of water Lv and the specific heats Cp and Cpv or dry air and vapour and the change in energy caused by condensation. The saturated mixing ratio is rs or rw . However, the process is iterative, because as you change the temperature of the air, so es(T) changes. It is easier to write the energy equation. Latent heat release Lv ( rs - r ) r Cpv ( T –Tw) = change in heat capacity of the dry air plus the vapour = Cp ( T –Tw)
+
We often ignore r Cpv as it is much smaller than Cp. Hence r = rs -Cp / Lv ( T -Tw ) and e = es(T) - 8/5 Cp / Lv p ( T -Tw )
The atmospheric lapse rate is ~ 6.5 K km-1 and thus at the tropopause, es(T) ~ 1% of the surface value, whereas pressure is ~ 25% of the surface. The ability of the atmosphere to hold water rapidly decreases with height; 75% of the water vapour lies below 2.5km
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4.0
Thermodynamic visualisation of wet and dry processes in the atmosphere.
A thermodynamic diagram ( T / Ф ) Tephigram is used to enable thermodynamic comparison of different air masses. There are others; the US use almost identical "Skew T / log P" diagrams. Entropy, Ф or S , is related to the potential temperature θ m is one of the axes of the diagram d Ф = dQ / T = ( cpdT - v dp ) / T Hence = cpdT/T -R dp/p .
Ф = cp ln T - R ln p + constant ln θ = ln T + R/cp (ln (1000) - ln p)
θ / T = ( 1000/p ) R/cp and since Thus Ф = cp lnθ + constant
Thus the ordinate of the diagram Ф is a loge of the potential temperature (K) As ln θ = ln T + R/cp (ln (1000) - ln p) cp d (ln θ ) = cp dT / T - R dp / p cp T d(ln θ ) = cp dT - R T dp / p
and
Integrating around a closed loop, cp ∫ T d(ln θ ) = cp = = = Hence the area dA ∫ dT - R ∫ T dp/p 0 ∫v dp ∫ p dv ∫ R dT dW + 0
= the work done dW
The isobars are curved. Given a temperature T and potential temperature θ then pressure values can be calculated using θ / T = ( 1000/p ) R/cp The lines of saturated mixing ratio , rs or rw can be obtained from rs = 5/8 es / p . es can be obtained from tables or formulae. Pseudo saturated adiabats are the loci of points that a saturated volume would follow. As the parcel rises, it cools and water condenses, produce more heat. These curves will asymptote to the dry adiabats. The heat released is cp ΔT = L Δ dr. However in calculating the new temperature, the value of es(T) changes. The curves are called pseudo wet adiabats as no account is taken of the change in heat content of the condensed water. However, these can be calculated as before by
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ΔQ = 0 = cpdT + Ldrs - dp/ρ = cp dT + ε Lv d (es / p ) Since
1/T d (es / p ) ~ d (es / pT) then
log θs = log T - R / cp log (p /p0 ) + ε L es (T) / pT - ε L es(θs)/ ( p0θs ) The saturated vapour pressure, es(T) can be taken from tables or calculated from formulae such as Lowe & Ficke, 1974 es(T) = a0 + T(a1 + T(a2 + T(a3 + T(a5 + a6 T))))) where for water a0 a1 a2 a3 a4 a5 a6 valid for : 6.107799961 4.436518521 10-1 1.428945805 10-2 2.650648471 10-4 3.031240396 10-6 2.034080948 10-8 6.136820929 10-11 -500 to + 500 C for ice 6.109177956 5.034698970 10-1 1.886013408 10-2 4.176223716 10-4 5.824720280 10-6 4.838826904 10-8 1.838826904 10-10 and -500 to 00 C
An alternative formula is
e s (T ) 6.112 exp (
17.67 T ) T 243.5
Where es(T) is in mb (hecta pascals) and temperature T is in degrees Celsius. This is another empirical formula and fits within 0.1% over the range -300 C < T < 350 C . The Latent heats vary only slowly with temperature and are displayed in Figure 4.0.1 T0 Celsius -40 -20 0 20 40 es Pascals 19.05 125.63 611.21 2338.54 7381.27 eI Pascals 12.85 103.28 L Joules/g 2603 2549 2501 2453 2406 Ls Joules/g 2839 2838
Fig. 4.0.1 Saturation Vapour Pressure over water and ice and Latent Heats of Condensation and Sublimation
4.1
Use of the Tephigram.
A “radiosonde” is used to measure the atmospheric temperature, humidity and wind velocity profiles. In plotting a sounding, the “dry bulb” temperatures and the “dewpoint” temperatures are drawn. The dewpoint is the temperature at which the water would
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condense out when an air parcel is cooled at constant pressure. This defines the mixing ratio (grams/Kilogram) of that parcel of air. When the two curves meet, the radiosonde passes through a cloud. Using a tephigram, you can estimate the height of the cloud base. The heights, measured at the side of the Tephigram, result from using the hydrostatic approximation. Given the dewpoint temperature, which defines the mixing ratio line , and the dry bulb temperature, which defines the potential temperature isopleth, the “lifting condensation level” is where the two meet. If a profile is available, the mixing condensation level is where the average adiabat and the average mixing ratio line for the layer meet. The amount of precipitable water in a column can be estimated. In a column from z1 to z2 the mass of water mw = ∫ ρw dz = ∫ ρw / (ρg) dp { hydrostatic equation) or
mw = raver (p1 – p2 ) / g
{ kg m-2 }
If you can calculate the vertical velocity, then the precipitation rate can also be estimated. The instability and the height of convection can be calculated using a tephigram. From measuring the areas of the cloud, an estimate can be made to calculate the height of the cloud top. Below is a sample set of data as collected from a radiosonde. Complete the table using the supplied formulae. Tephigrams, with associated wind speed and direction observations are used throughout the world as a way of recording and examining the vertical profile and structure of the atmosphere.
An example for use with a Tephigram.
Height (m) (%) 0
Pressure(hp)
Temp. (C)
Dewpoint Temp. (C)
r (g/kg)
es(T)
Humidity
1000 950 900 850 800 750 700 600
20 16.5 12.5 10.5 7.5 4 2 -1
14 12.5 10 6 2 -3 -6 -16
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At what height is the mixing condensation level? At what height is the lifting condensation level? How much water is there available between 1000 - 700 hp? What height is the convective cloud likely to reach?
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