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Atmospheric Thermodynamics TMD July-Aug 2008 L03 Atmospheric Motion Dynamics Newton’s second law m F x Mass × Acceleration = Force d2x m 2 F dt Thermodynamics Concerned with changes in the internal energy and state of moist air. The Atmosphere as an ideal gas Atmosphere as an ideal gas. Volume V, pressure p, p TV Temperature T, mass m Specific volume = volume of 1 kg a = V/m. Density r = mass per unit volume = 1/a Equation of state for dry air pa = RdT Pressure – Partial pressure The molecules of a gas are in constant random motion In an ideal gas each molecule has kinetic energy Occasionally molecules collide with each other or with the walls of the containing vessel. The nature of pressure v Momentum change due to a perfectly elastic collision = m(-v) - m(v) = -2mv v Unit area The momentum change by many collisions averaged over unit time and unit area represents a force. The force per unit area is the pressure. The unit is the Pascal. 1 Pa = 1 Nm-2. Partial pressure O2 N2 Unit area The O2 molecules exert a partial pressure p1 and the N2 molecules a partial pressure p2 on the unit area. The pressure is thee sum of the partial pressures. Air motion V The molecules of a fluid (gas or liquid) are in a constant state of motion. The mean motion of all molecules in a fluid parcel is the macroscopic velocity of the parcel, V. The (internal) energy residing in the random motion is characterized by the absolute temperature of the fluid, T. The nature of temperature According to the kinetic theory of gasses, the absolute temperature of a gas is proportional to the mean kinetic energy of the molecules (including rotational energy). This energy is called the internal energy. The state of unsaturated moist air The state of moist air is characterised by: • pressure, p • absolute temperature, T • density, r (or specific volume a = 1/r), and • Some measure of the moisture, for example. • The water vapour mixing ratio, r, defined as the mass of water vapour per unit mass of dry air. Equation of state for moist, unsaturated air V Water vapour mv Dry air md Now p d V md R d T and eV m v R v T pV (pd e)V (md R d mv R v )T Divide by m mv R v 1 m R (md R d m v R v ) d d pa T Rd mv md m v 1 md T Equation of state for moist, unsaturated air mv R v 1 m R 1 r / (m d R d m v R v ) d d pa T Rd T Rd T mv md m v 1 r 1 md Let = Rd/Rv = 0.622 r = mv/md is the water vapour mixing ratio (typically << 1, max 0.04) pa R d Tv Tv 1 0.61rT is the virtual temperature The density of a sample of moist air is characterized by its pressure and its virtual temperature, i.e. p r RTv Moist air (r > 0) has a larger virtual temperature than dry air (r = 0) => the presence of moisture decreases the density of air --- important when considering the buoyancy of an air parcel! For cloudy air Tr T 1 r / 1 rT p r RTr is the density temperature is the total water mixing ratio The virtual temperature a R dTv / p Tv T(1 0.61r) Dry air with the virtual temperature Tv has the same specific volume as moist air with temperature T at the same pressure. To determine Tv one must convert r into kg/kg and convert T into Kelvin. The hydrostatic equation Except for motion on small scales, e.g. thunderstorms, the atmosphere is to a very good approximation in hydrostatic balance. The hydrostatic equation 2 Hydrostatic balance Mass = rAdz p(z)A - p(z + dz)A = grAdz As dz 0 Cross section A dp g (z) dz The minus sign is because the The hydrostatic equation pressure decreases with height The hydrostatic equation 3 dp g (z) dz When r(z) is known, we can integrate with respect to z : p(z) z g r(z)dz z Note that p(z) 0 as z . The pressure at height z is just the weight of a column of air with unit cross section. The hydrostatic equation 4 Mean sea level pressure: p(z) g r(z)dz 0 0 The product of the mean sea level pressure (= 105 Pa) times the area of the Earth‘s surface (= 5 1014 m2) gives approximately the mass of the atmosphere (5 1019 kg). The hydrostatic equation 5 The vertical density profile r(z) is difficult to measure: p and T are easier to measure. r(z) can be obtained from the ideal gas equation p = rRdTv : dp -g r(z) dz 1 dp g p dz R dTv (z) ln p(z) - ln p(0) z 0 p(z) p(0) exp z 0 g dz R dTv (z) g dz R dTv (z) Other moisture variables The partial pressure of water vapour, e = rp/( + r) The relative humidity, RH = 100 e/e*(T). • e* = e*(T) is the saturation vapour pressure, which is the maximum amount of water vapour, that an air parcel can hold, without condensation occurring. The specific humidity, q = r/(1 + r), is the mass of water vapour per unit mass of moist air. Saturation vapour pressure e*(T). A more accurate empirical formula is (see E94, p117): ln e* 53.67957 6743.769 / T 4.8451 ln T e* in mb and T in K A corresponding expression for ice-vapour equilibrium is: ln e* 23.33086 611172784 . / T 015215 . ln T These formulae are used to calculate the water vapour content of a sample of air. If the air sample is unsaturated, the dew point temperature (or ice point temperature) must be used. Water vapour the relationship between e, r and p Water vapour mv Dry air md Now p d V md R d T and eV m v R v T Divide the equations of state => pd V m d R d T eV m v R v T p-e e r e r p-e More moisture variables The dew point temperature, Td, is the temperature at which an air parcel first becomes saturated as it is cooled isobarically. The wet-bulb temperature, Tw, is the temperature at which an air parcel becomes saturated when it is cooled isobarically by evaporating water into it. The latent heat of evaporation is extracted from the air parcel. Vertical distribution of r and RH from a radiosonde sounding on a humid summer day in central Europe. Aerological (or thermodynamic) diagrams T = constant p ln p . . (p, a) (p, T) T = constant a pa R dT T Aerological diagram with plotted sounding T = constant ln p . . (p, Td) (p, T) The first law of thermodynamics The increase in the internal energy of a system is equal to the amount of energy added by heating the system, minus the amount lost as a result of the work done by the system on its surroundings. The internal energy of a system includes the kinetic and potential energy of the molecules or atoms. When the kinetic part of the internal energy increases (i.e. the molecules move faster on average), the temperature of the gas increases Materials. The potential energy of the molecules is determined by their position relative to neighbouring molecules. James Joule (1818-1889). The mechanical equivalent of heat: Joule was a very enthusiastic experimenter. During his honeymoon in Switzerland he tried to determine the temperature change of water at a waterfall. Waterfall Water gains kinetic Energy from potential Energy. The kinetic Energy of the Water is in converted first to turbulence and later to heat. Der erste Hauptsatz der Thermodynamik 9 Joule showed that for a thermally isolated system, dU = dQ + dW • dU = the increase of the internal energy • dQ = the heat input • dW = the work done on the gas. U dQ, dW U + dU dU = dQ + dW dQ = dU - dW Not all the heat is available to increase the internal energy When the gas expands (i.e. dV > 0), it does work on its surroundings. dQ U = U + dU - dW A thought experiment cylinder gas Pressure p Temperature T Piston Volume V Graphical representation of state changes The thermodynamic state of dry air can be represented by a point in a pV- or pa-diagram. p p A (a,p) B a The change in state can be represented by a curve in such a diagram. a Ein Gedankenexperiment 2 Area A dx Pressure force = pA Pressure p Work done pA dx = pdV Volume change dV = Adx Work done pdV/(unit mass) = pda per unit mass p A Work done dW = pdV p V1 dV B V2 Total work = V -W V2 V1 pdV The first law for 1 kg of an ideal gas dq = du - dw = du + pda Heat input The work done by the gas Change in internal energy Temperature increase For a sample of moist air du cv dT Then or dq cv dT pda dq cp dT - adp d(pa) pda adp d(R T) cv cvd (1 0.94r) c vd cp cpd (1 0.85r) cpd Adiabatic processes An adiabatic process is one in which there is zero heat input dq = 0 d ln T R / cp d ln p RT 0 cp dT dp p dq c p dT - adp d ln T d ln p where ln T ln p ln A R d / c p 0.2865 a constant Define A such that, when p equals some standard pressure, po, usually taken to be 1000 mb, T = . The quantity is called the potential temperature The potential temperature The potential temperature and is given by: po T p We define the virtual potential temperature, v by F p I T G J Hp K o v v take the value of for dry air. Enthalpy The first law of thermodynamics can be expressed as dq d(u pa) - adp = dk - adp k = u + pa is called the specific enthalpy. The enthalpy is a measure of heat content at constant pressure. For an ideal gas, k = cpT. Entropy An excellent reference is Chapter 4 of the book: C. F. Bohren & B. A. Albrecht Atmospheric Thermodynamics Oxford University Press Specific entropy dq ds c pd ln T s c p ln cons tan t The equivalent potential temperature dq c p dT - adp dq dT dp ds cp - Rp cp d ln T T p Suppose dq results from latent heat release, i.e. dq = -Lvdr Lv c p d ln dr 0 T Lv r Lv r ln ln e d ln 0 cp T c T p Lv r e exp c T p a constant e is called the equivalent potential temperature Some notes e, L, and Lv are conserved in reversible adiabatic processes involving changes in state of unsaturated or cloudy air. e, L, and Lv are not functions of state - they depend on p, T, r and rL Curves representing reversible, adiabatic processes cannot be plotted in an aerological diagram In a saturated process, r = r*(p, T) The pseudo-equivalent potential temperature The formula e* exp (Lvr*/cpdT) is an approximation for the pseudo-equivalent potential temperature ep . A more accurate formula is: p I F TGJ Hp K 0.2854 / (1 - 0.28 r ) ep o L O F I 3376 exp M r(1 0.81 r ) G - 2.54J P T H K N Q LCL Temperature at the LCL TLCL is given (within 0.1°C) by the empirical formula: L ln(T / T ) O 1 M 56 P T - 56 800 Q N -1 TLCL K d d TK and Td in Kelvin The reversible equivalent potential temperature The reversible equivalent potential temperature is defined by po er T pd R d /(cpd rT cL ) (RH) - rR v /(cpd rT cL ) Lv r exp (cpd rT cL )T It is based on the assumption that all water vapour is carried with an air parcel. Lines of constant er cannot be plotted on an aerological diagram. If an air parcel is lifted pseudo-adiabatically to the high atmosphere until all the water vapour has condensed out, ep = . The pseudo-equivalent potential temperature is the potential temperature that an air parcel would attain if raised pseudo-adiabatically to a level at which all the water vapour were condensed out. The isopleths of ep can be plotted on an aerological diagram. These are sometimes labelled by their temperature at 1000 mb which is called the wet-bulb potential temperature, w. The adiabatic lapse rate Lift a parcel of unsaturated air adiabatically it expands and cools, conserving its v its T decreases with height at the dry adiabatic lapse rate, d : g 1 r dT d - dz dq 0 c pd 1 r(c pv / c pd ) Note that r is conserved, but r* decreases because e*(T) decreases more rapidly than p. Saturation occurs at the lifting condensation level (LCL) when T = TLCL and r = r*(TLCL, p). Above the LCL, the rate of which its temperature falls, m, is less than d because condensation releases latent heat. For reversible ascent: g 1 rT dT m - dz s cpd 1 r cpv cpd Lv r 1 R dT L2v (1 r / )r cL 1 rL 2 c rc R T (c rc ) pd pv v pd pv When rT is small, the ratio m/d is only slightly less than unity, but when the atmosphere is very moist, it may be appreciably less than unity. The moist static energy and related quantities The first law gives dq = dk - addp where dq is expressed per unit mass of dry air. Adiabatic process (dq = 0) dk - addp = 0 ad = a(1 + rT) For a hydrostatic pressure change, adp = -gdz. Under these conditions: dh (c pd rT c L )dT d(L v r) (1 rT )gdz 0 Some notes If rT is conserved, we can integrate h (c pd rTc L )T L v r (1 rT )gz cons tan t. The quantity h is called the moist static energy. h is conserved for adiabatic, saturated or unsaturated transformations in which mass is conserved and in which the pressure change is strictly hydrostatic. h is a measure of the total energy: (internal + latent + potential) The dry static energy Define the dry static energy, hd. Put rT = r => h d (c pd rc L )T (1 r)gz. This is conserved in hydrostatic unsaturated transformations. h and hd are very closely related to e and . Vertical profiles of dry conserved variables Dry static energy z (km) z (km) , v v deg K hda hd hd = (cpd+ rcL)T + (1 + r)gz 105 J/kg hda = cpdT +gz Vertical profiles of moist conserved variables Moist static energy z (km) z (km) , pseudo e e ep esp ha h hs epa epa exp (Lvr*/cpdT) deg K 105 J/kg ha = cpT + Lvr + gz The stability of the atmosphere Consider the vertical displacement of an air parcel from its equilibrium position Calculate the buoyancy force at its new position Consider first an infinitesimal displacement ; later we consider finite-amplitude displacements Parcel motion is governed by the vertical momentum equation d 2 b 2 dt r p - ra b( ) -g rp a p - aa g a a is the buoyancy force per unit mass Newton's law for an air parcel: d 2 buoyancy force 2 dt unit mass buoyancy force b : b() unit mass z z 0 d 2 b 0 2 dt z z 0 The motion equation for small displacements is: d 2 2 N 0 2 dt where b N z 2 The motion equation for small displacements is: d 2 2 N 0 2 dt b N z 2 where For an unsaturated displacement, vp is conserved and we can write Tvp - Tva vp - va b() g g Tva va Since vp = constant va, vp va b g va N - g 2 z va z va z 2 Stability criteria Parcel displacement is: • stable if va/z > 0 • unstable if va/z < 0 • neutrally-stable if va/z = 0 A layer of air is stable, unstable, or neutrally-stable if these criteria are satisfied in the layer. In a saturated (cloudy) layer of air, the appropriate conserved quantity is the moist entropy s (or the equivalent potential temperature, e, or L) Must use the density temperature to calculate b. Replace ap in b by the moist entropy, s. In this case 1 N 1 rT 2 ap - aa b() g a a rT s m z - cLm ln T g z A layer of cloudy air is stable to infinitesimal parcel displacements if s (or e) increases upwards and the total water (rT) decreases upwards. It is unstable if e decreases upwards and rT increases upwards. Some notes The stability criterion does not tell us anything about the finite-amplitude instability of an unsaturated layer of air that leads to clouds. Parcel method okay, but must consider finite displacements of parcels originating from the unsaturated layer. Potential Instability A layer of air may be stable if it remains dry, but unstable if lifted sufficiently to become saturated. Such a layer is referred to as potentially unstable. The criterion for instability is that de/dz < 0. unstable lift stable unsaturated saturated/cloudy Conditional Instability The typical situation is that in which a displacement is stable provided the parcel remains unsaturated, but which ultimately becomes unstable if saturation occurs. This situation is referred to as conditional instability. To check for conditional instability, we examine the buoyancy of an initially-unsaturated parcel as a function of height as the parcel is lifted through the troposphere, assuming some thermodynamic process (e.g. reversible moist adiabatic ascent, or pseudo-adiabatic ascent). If there is some height at which the buoyancy is positive, we say that the displacement is conditionally-unstable. If some parcels in an unsaturated atmosphere are conditionally-unstable, we say that the atmosphere is conditionally-unstable. Conditional instability is the mechanism responsible for the formation of deep cumulus clouds. Whether or not the instability is released depends on whether or not the parcel is lifted high enough. Put another way, the release of conditional instability requires a finite-amplitude trigger. The conventional way to investigate the presence of conditional instability is through the use of an aerological diagram. 200 LNB 300 dry adiabat pseudoadiabat 500 10 g/kg 700 850 LFC LCL 1000 20 oC 30 oC Positive and Negative Area Convective Inhibition (CIN) The positive area (PA) PA 1 u2 2 LNB - 1 u2 2 LFC p z cT LFC p LNB vp h - Tva R d d ln p The negative area (NA) or convective inhibition (CIN) NA CIN p parcel p LFC T vp - Tva R d d ln p Convective Available Potential Energy - CAPE The convective available potential energy or CAPE is the net amount of energy that can be released by lifting the parcel from its original level to its LNB. CAPE = PA - NA We can define also the downdraught convective available potential energy (DCAPE) DCAPE i z R (T po pi d ra - Trp )d ln p The integrated CAPE (ICAPE) is the vertical mass-weighted integral of CAPE for all parcels with CAPE in a column. Reversible e z (km) z (km) Pseudo e b m s-2 b m s-2 Liquid water z (km) z (km) Buoyancy zL km zL km Height (km) reversible with ice reversible pseudo-adiabatic Buoyancy (oC) Downdraught convective available potential energy (DCAPE) DCAPE i Td z R (T po pi d ra - Trp )d ln p qw = 20oC T 700 LCL Tw 800 mb, T = 12.3oC Td 850 r* = 6 g/kg 1000 Trp DCAPE Tra The End Summary: Various forms of the equation of state Für m kg pV = mRT Für 1 kg pa = RT Für eine beliebige Menge Für ein Kmole pV = MRT oder pV = R*T Für n Kmole pV = nR*T Für 1 kg feuchte Luft p = rRT pa = RdTv Example p = 990 mb T = 26 C w = 8 g/kg 99000 Pa 299 K 0,008 kg/kg pa R dTv p rR dTv Tv = 299 (1 + 0,61 0,008) = 300,46 K r = p/RdTv = 99000/(287 300,46) = 1,15 kg/m3 a = 1/r = 1/1.15 = 0,87 m3/kg Der erste Hauptsatz der Thermodynamik 6 Wasser Rührwerk Gewicht Joule schloß daraus Mechanische Energie Wärme The equation of state for cloudy air Consider cloudy air as a single, heterogeneous system specific volume = (total volume)/(total mass) a Va Vl Vi / M d M v M l M i Divide by Md total mixing ratio of water substance a a d 1 rl (a l / a d ) ri (a i / a d ) / 1 rT RdT 1 R d T pd e 1 RdT 1 r / a p d 1 rT p p d 1 rT p 1 rT = Rd/Rv = 0.622 defines the density temperature for cloudy air: Tr = T(1 + r/)/(1 + rT) specific heat of water vapour The reversible equivalent potential temperature po e T pd R d /(cpd rT cL ) (RH) =p - rR v /(cpd rT cL ) =1 Lv r exp (cpd rT cL )T =1 For dry air (r = 0, rL = 0, rT = 0), e reduces to . Note that e is not a state variable isopleths of constant e cannot be plotted on an aerological diagram. The (virtual) liquid water static energy Define two forms of static energy related to L and Lv. These are the liquid water static energy: h w (c pd rT c pv )T L v rL (1 rT )gz and the virtual liquid water static energy h Lv rT Lv rL cpd gz Tr 1 rT rT - rL hLv is almost precisely conserved following slow adiabatic displacements. If rL = 0, hLv = cpdTv + gz (just as L reduces to ). Buoyancy and e z Lifted parcel b -g Tvp rp Environment (rp - ro ) ro To(z), ro(z), p(z), ro(z) or bg (Tvp - Tvo ) To Lifted parcel bg Trp rp Environment (Trp - Tvo ) To To(z), ro(z), p(z), ro(z) Below the LCL (Trp= Tvp) sgn (b) = sgn { Tp(1 + rp) - To(1 + ro) = Tp - To + [Tprp - Toro]} At the LCL (Trp= Tvp) = 0.61 sgn (b) = sgn [Tp(1 + r*(p,Tp)) - To(1 + ro)] = sgn [Tp(1 + r*(p,Tp) - To(1 + r*(p,To)) + To(r*(p,To) - ro)] sgn (b) = sgn [Tp(1 + r*(p,Tp) - To(1 + r*(p,To)) + To(r*(p,To) - ro)] Since Tv is a monotonic function of e, b (*ep - *eo ) small z parcel saturated eo *eo LFC LCL