Naraine Persaud, Entry Code ME-11 1

Transcription

1 Naraine Persaud, Entry Code ME-11 1 Persaud, N Adiabatic cooling. In: Water Encyclopedia Volume 4: Oceanography; Meteorology; Physics and Chemistry; Water Law; and Water History, Art, and Culture. J. H. Lehr and J. Keeley Eds., John Wiley and Sons, New York, pp Adiabatic cooling: When any closed gaseous system expands against an external pressure with no compensatory input of heat from its surroundings to offset the pressure-volume work done by the system, the expansion results in adiabatic cooling of the gas. The viceversa process results in adiabatic heating. The term adiabatic (from the Greek word meaning "not passing through") implies an idealized process and a perfectly insulated system. Real, imperfectly insulated systems would permit some compensatory heat flux. However, if the effect of such fluxes on the system temperature is small relative to the cooling effect, the expansion can be considered as adiabatic for practical purposes. Adiabatic cooling and heating is the principal mechanism underlying the continual turbulent vertical mixing (auto-convection) in the earth s troposphere i.e. the lowest 9 km (polar regions) to 18 km (tropics) thick atmospheric layer comprising 80% of the atmospheric mass where all weather processes occur. It is especially important in the planetary boundary layer i.e. the lowest 1 to 2 km thick layer where surface effects on vertical motions are more manifest. Cloud formation and precipitation, observed temperature decrease with height, energy and mass transfers in the troposphere all depend on adiabatic cooling. This thermo-physical process is therefore sine qua non to the continuity of the earth s hydrological cycle. This article focuses on the physics of adiabatic cooling in the troposphere and the resulting temperature decreases with elevation. A global average of close to 30% of incoming solar short-wave radiation is absorbed at the earth's surface layers. The resulting increase in the temperature of these layers and heat transfer to the air is the most important driver for turbulent vertical convection and associated adiabatic cooling in the troposphere. The earth's atmosphere is a quasi-uniform thermo-physical system, meaning that pressure gradients rather than the pressure of the atmosphere as a whole can be defined. To facilitate the analysis of tropospheric adiabatic cooling, the concept of an air parcel is introduced. A parcel (or less commonly an eddy) is a discrete closed volume of tropospheric air with uniform properties that is small enough so that the pressure at any point is the same. This implies that, although each parcel has its own individual structure and evolution in time, the observed state and dynamics of tropospheric air is the net result of the motion of these parcels. Because of the many forces involved in such motions, applying Newton's second law to describe the hydrodynamics of these parcels in the earth s atmosphere would result in a complex set of three dimensional vector differential equations in spherical coordinates. However, only the relatively simpler vertical component of the vector equations is required for analysis of adiabatic cooling. This component is dominated by the vertical pressure and gravitational forces (Brown, 1990). In order to understand the interplay of these forces, consider a prizm-shaped, initially motionless parcel in a dry atmosphere air above the earth's surface as shown in Figure 1.

2 Naraine Persaud, Entry Code ME-11 2 Figure 1 P(z+Δz) Δz P(z) Δx Δy It is assumed that tropospheric air is continuously stratified meaning pressure (P) and density (ρ d ) are continuous and decreasing functions of vertical height (z) above the surface. The subscript d refers to dry air. The volume of the prizm (V) is (ΔxΔyΔz). The net force on the prizm is V ρ d g (downward weight) + (ΔxΔy) (dp/dz) Δx (upward net pressure buoyancy force). Since the prizm is assumed static, the net force per unit mass (acceleration) = 0 and therefore (1/ρ d ) dp/dz + g = 0. This equation (termed as the hydrostatic or aerostatic equation) states the necessary condition for vertical hydrostatic equilibrium in the troposphere namely, that gravity must be balanced by the vertical component of the atmospheric pressure gradient force. Any disturbance of this hydrostatic equilibrium would result in vertical motion of the parcel. In particular, if the density of the parcel (ρ p ) is different from that of the surrounding air (ρ o ), the parcel would accelerate. The buoyancy acceleration is g [(ρ o /ρ p ) -1] and is positive (upward) if ρ < ρ o and vice versa (assuming the vertical pressure gradient force is fixed). Substituting g = 9.81 m s -2, a 10% decrease in the density of the parcel would result in an acceleration of close to 1 m s -2. Such buoyancy accelerations are the direct result of temperature changes caused by heating (from surface or otherwise) of air parcels. Assuming dry air behaves as an ideal gas ρ d = P/R d T [where R d is the specific gas constant for dry air = J mol -1 o K -1 / kg (mol dry air) -1 and T is the absolute temperature ( o K)] the buoyancy acceleration can be written as g [(T/T o ) -1] where T and T o are the temperatures of the parcel and surrounding air at a given elevation.

3 Naraine Persaud, Entry Code ME-11 3 For purposes of analysis of adiabatic cooling an air parcel is considered as a perfectly insulated bubble of air submersed in the atmosphere, although in reality it could exchange heat with its surroundings. The thermal conductivity of dry air (k da in W m -1 o K -1 or cal s -1 cm -1 o K -1 ) at 1 atm pressure is linearly related to temperature over the range 263 to 473 o K. It increases from W m -1 o K -1 ( cal s -1 cm -1 o K -1 ) at o K by W m -1 o K -1 ( cal s -1 cm -1 o K -1 ) for each o K (or o C) change in temperature. The conductivity of moist air is always less than or equal to k da, the difference widening in non-linear manner with increasing temperature and increasing humidity. Upward acceleration and accompanying expansion of the parcel occur relatively quickly, and because of the low thermal conductivity values of moist air, any heat transfer to the parcel is negligible. This assumption makes possible analysis of the thermal physics of an ascending air parcel as an adiabatic cooling process. From the integral form of the first law of thermodynamics, a change in internal energy (ΔU) of a parcel = absorption (gain) or release (loss) of heat (q) - mechanical work done by parcel (loss) or work done on the parcel (gain). Assuming expansion/compression takes place reversibly (i.e. ideally-balanced quasi-static equilibrium is always maintained during process), the mechanical pressure-volume (PV) work done by the parcel against the external pressure (P) is given as P (ΔV) where (ΔV) is the volume change (positive for expansion). Thus in general q = ΔU + P(ΔV). The ideal gas equation of state for unit mass of dry air is PV= R d T. Assuming reversible adiabatic expansion (i.e. process is isentropic), q = 0 and ΔU = -P(ΔV) = -R(ΔT) for reversible adiabatic cooling of an ideal gas. For dry air (a homogeneous gaseous mixture) U in general is a function of absolute temperature (T) and P. However, for an ideal gas, U is a function of T only as demonstrated by James Prescott Joule ( ). Also, U is a state function implying that ΔU is the same for a given temperature change (ΔT) of the parcel, regardless of how ΔT was achieved. Two possibilities are discussed below namely: changing T at constant volume or at constant pressure. The specific heat of any substance is the heat input required to cause unit temperature change in unit mass of the substance. In general, specific heat is a function of temperature. The specific heat for gases at constant volume and constant pressure are defined as C v (T) = (dq/dt) v and C p (T) = dq/dt p. From the first law, for dry air C v < C p since "extra" heat is required at constant pressure for expansion work done against the surrounding pressure. Over tropospheric temperature ranges of practical importance, the specific heats for dry air (denoted as C vd and C pd ) can be considered as constants. Several important results follow from applying the first law to the analysis of adiabatic reversible cooling of a dry air parcel in a quasi-uniform atmosphere assuming ideal behavior and constant C vd and C pd. Applying ΔU = q P(ΔV) to the heating of unit mass dry air through ΔT at constant P, implies that C pd (ΔT) - ΔU = p(δv) = R d (ΔT). But ideal behavior implies that ΔU can be taken as C vd (ΔT), and therefore C pd (ΔT) - C vd (ΔT) = R(ΔT) which simplifies to C pd - C vd = R d.

4 Naraine Persaud, Entry Code ME-11 4 Taking the total differential of the equation of state for unit mass of dry air assuming ideal behavior (i.e. PV = R d T) gives VdP + PdV = R d dt or PdV = R d dt VdP. Substituting this, together with R d = C pd - C vd, and du = C vd dt into the differential form of the first law (i.e. δq = du PdV) gives δq = C vd dt VdP + (C pd - C vd ) dt = C pd dt VdP. The result δq = C pd dt VdP is in effect a restatement of the differential form of the first law for dry air. For an adiabatic ideal gas process δq = 0 and C pd dt = VdP = [(R d T)/P] dp. Separating variables gives d(ln T) = (R d /C pd ) d(ln P). Integrating between P = P o at T = T o and P = P RdCpd at T = T, and rearranging gives T/T o = [ P / Po ]. This result is one of the three adiabatic ideal gas process P-V-T relationships known as Poisson equations (Siméon Denis Poisson, ). It describes the quasi-static reversible adiabatic expansion for an ideal gas with constant specific heats from an initial temperature T o and pressure P o to a new temperature T and pressure P. Usually the ratio γ d = C pd /C vd is defined. Dividing C pd C vd = R d by C pd and rearranging gives C pd = (R d γ d ) /(γ d -1) = R d / κ d with κ d = (γ d -1)/ γ d. For dry air R d = J mol -1 o K -1 / kg (mol dry air) -1 = J kg -1 o K -1, C pd = 1004 J kg -1 o K -1 and κ d = d The result T/T o = [ P / P ] κ o can be solved for To d = T [ P / P ] κ o = θ d. This definition of θ d with P o = 1 bar (1000 mb) is termed as the potential temperature. It is the temperature that the parcel initially at T, P would assume when changed adiabatically to 1000 mb. From its definition, ln θ = ln T + κ d [ln P o ln P] and d(ln θ) = d(ln T) - κ d d(lnp). Dividing δq = C pd dt [(RT)/P]dP by C pd T gives δq/c pd T = d(ln T) - κ d d(ln P) and therefore d(ln θ d ) = δq/c pd T. Since δq = 0 for an adiabatic process, d(ln θ d ) = 0 implying the potential temperature is a parameter that remains invariant during the adiabatic ascent of a parcel. Since T = (θ d /P o ) P plots of T versus P for different θ d result in a family of straight lines. Using δq = C pd dt VdP with δq = 0 for adiabatic cooling, gives dt/dp = V/C pd = 1/(ρ d C pd ) since V per unit mass (specific volume) is the reciprocal of the density (ρ d ). As shown above, for hydrostatic equilibrium dp = -ρ d g dz. Substituting this for dp in the previous equation gives -dt/(ρ d gdz) = 1/ (ρ d C pd ) or dt/dz = -g /C pd = - (g κ d ) /R d. Writing this result as C pd dt = -gdz shows that adiabatic cooling can be interpreted physically as the dry air parcel trading its internal energy (C pd dt) to bootstrap its ascent in the troposphere to acquire gravitational potential energy (-gdz). This interpretation is appropriate since as already discussed the atmosphere is a quasi-uniform thermo-physical system. The temperature of an adiabatically cooled parcel of dry (ideally behaved) air ascending in a tropospheric air layer that is in vertical hydrostatic equilibrium therefore decreases linearly with altitude with slope = -(gκ d )/R d. This important theoretical result is known as the dry adiabatic lapse rate (DALR). For γ d = 1.4, κ d = 0.286, and together with R d = J kg -1 o K -1, g = 9.81 m s -2, gives dt/dz = o K m -1. Assuming a layer of dry air in the troposphere is rapidly and perfectly mixed by vertical convection of a large number of air parcels rising and falling adiabatically, the temperature would be expected to decrease

5 Naraine Persaud, Entry Code ME-11 5 linearly by approximately 1 o K (same as 1 o C) per 100 m change in elevation. The layer is isentropic (constant entropy with elevation) and therefore the potential temperature(θ d ) is constant. The DALR is a theoretical value achievable only in a turbulent well-mixed layer of dry tropospheric air. However, real tropospheric air is always moist because varying amounts of water vapor is always present. To facilitate including this factor in analysis of the behavior of rising moist parcels, the water vapor and dry air are treated as distinct components of a (one-phase) ideal gas mixture. In this case three independent variables are needed to define an equation of state for moist air namely: two (usually P,T) of the P- V-T variables together with a (humidity) variable that measures the mass ratio of water vapor in the air. Among several possible measures of this ratio the specific humidity (r q ) is the one traditionally used. It is defined as the mass of water vapor per unit mass of moist air. Consider a volume (V) of moist air of mass (m m ) = mass water vapor (m v ) + mass of dry air (m d ). The subscripts m, v, d denote properties associated with the moist mixture, vapor, and dry air respectively. The density ρ m = m m /V = m v /V + m d /V = ρ v + ρ d (i.e. the sum of the partial densities), and therefore specific humidity (r q ) = ρ v /ρ m. The total pressure P of the volume V of moist air at temperature T ( o K) would be equal to P v + P d (i.e. the sum of the partial pressures). P v is usually denoted as e, implying P d = P-e. Using ideal gas relations, e = ρ v R v T and P-e = ρ d R d T where R v = R/M v, R d = R/M d denote the specific gas constants for water vapor and dry air, R = J mol -1 o K -1 is the universal gas constant, and M is molecular weight (M v = kg mol -1 and M d = kg mol -1 ). This gives ρ v = e/(r v T) = (e M wv ) / (RT) and ρ d = (P e) / (R d T) Dividing numerator and denominator of the RHS for ρ v by M d gives ρ v = (0.622 e) / (R d T) where = M v / M d (usually denoted as ε), and R d is the specific gas constant for dry air ( = J kg -1 o K -1 ). Summing ρ v and ρ d, rearranging algebraically, and simplifying gives ρ m = [P/(R d T)] {1 [e (1-ε) / P]}. This shows that density of moist air is slightly less than the density of dry air [= P/(R d T)] at a given pressure and temperature, a fact first noted by Sir Isaac Newton in 1717 in his book entitled "Optiks". The value of the expression in curly brackets increases with decreasing P for a given e. Taking a maximum possible value of e = 4500 Pa at mean sea level with P = 1 atm = x 10 5 Pa, and ε = gives 1 [e (1-ε) / P] = 0.96 and implies ρ m ρ d. Using these results gives after substituting and simplifying, r q = ρ v /ρ m = (εe) / {P - [(1-ε) e] εe/p. The mixing ratio r w defined as mass vapor per unit mass dry air in the mixture = ρ v /ρ d = (εe) /(P e) r q. Eliminating e between ρ v and ρ d, rearranging algebraically, and simplifying gives the equation of state for moist air as P(T, r q ) = ρ m R d T( r q ). This is equivalent to the ideal gas equation for dry air with T(r q ) = T( r q ) or with R m = R d ( r q ). In the former case, T(r q ) is termed as the virtual temperature (denoted as T v ) and the moist air equation of state is written as P = ρ m R d T v. T v is the temperature dry air would assume in order to have the same density as moist air with specified values of P, T, and r q.

6 Naraine Persaud, Entry Code ME-11 6 The specific heats C pm, C vm for moist air would also be functions of r q. Partitioning the heat for a temperature change (ΔT) in mass (m m ) of moist air at a given r q expanding at constant pressure gives m m C pm (ΔT) = m v C pv (ΔT) + (m d C pd (ΔT). Dividing by m m (ΔT) gives C pm = r q C pv + (1 r q ) C pd since r q is by definition m v /m m and m d = m m m v. Similar reasoning gives C vm = r q C vv + (1 r q ) C vd. Values given in the Smithsonian Meteorological Tables (List, 1971) are (in J kg -1 o K -1 ) : C pv = 1846, C pd = 1005, C vv = 1386, and C vd = 716. Substituting these values and simplifying gives C pm = C pd ( r q ) and C vm = C vd ( r q ). From these γ m = C pm / C vm = γ [( r q )/ ( r q )] where γ = C pd /C vd = 1.4 is the ratio of the specific heats for dry air as described previously. Multiplying numerator and denominator of the term in square brackets by ( r q ) and neglecting r q raised to powers > 1, gives γ m γ(1 0.1 r q ). Similarly, κ m = R m /C pm = [R d ( r q )] /[C pd ( r q )]. Multiplying numerator and denominator of the term in square brackets by ( r q ) and neglecting r q raised to power > 1 gives κ m κ d ( r q ) where κ d = R d /C pd. γ m is lower for moist air than dry air but does not affect the DALR appreciably as long as the air does not become saturated. For example, using the foregoing results give the unsaturated moist adiabatic lapse rate (MALR) as dt/dz = - (g κ m ) / R m where κ m = (γ m -1)/ γ m and R m = R d ( r q ). At 20 o C and 50 % saturation the vapor pressure 1170 Pa. Using the equations for ρ v and ρ m with R d = J kg -1 o K -1 and P = x 10 5 Pa, give r q = ρ v / ρ m = kg vapor (kg moist air) -1, γ m = 1.4 x = 1.399, κ m = 0.285, R m = J kg -1 o K -1, and the MALR = o K m -1 compared to o K m -1 for the DALR. As was done for dry air, the potential temperature invariant for moist unsaturated air (θ m ) can be defined as θ m = T R d /C pd. [P / P o ] κm where as discussed above, κ m κ d ( r q ) with κ d = If a rising moist air parcel becomes cooled to the dew point, condensation occurs (if condensation nuclei are present). The elevation at which this occurs is called the lifting condensation level (LCL). The dew point temperature at the LCL (T dl ) is different than at the surface since, although r q ( εe/p) remains the same, the vapor pressure decreases as the ascending parcel expands. The dew point (T d0 ) at the surface depends on the observed ambient temperature (T) and relative humidity = (e/e s ) T of the air. The relationship can be obtained using the Clausius-Clapeyron equation de s /e s = (LdT)/(R v T 2 ) for the slope of the equilibrium saturation vapor pressure (e s ) versus temperature curve, where L is the latent heat of vaporization and R v (= J kg -1 o K -1 ) is the specific gas constant for water vapor (Salby, 1996). L (J kg -1 ) is usually a weak function of temperature approximated as L = x T o K. L is usually taken as constant value = 2.5 x 10 6 J kg -1 corresponding to T = 273 o K. Integrating the Clausius Clapeyron equation between e s = e at T to e s = e s at T d gives (after substituting for L and R v ), T T d0 = x 10-4 T T d0 ln [(e/e s ) T ]. When required, reasonable estimates of e s (T) can be obtained by integrating the Clausius- Clapeyron equation on the half-open interval using the triple point of water (e s = 611 Pa at T = o K) as the lower bound. After substituting numerical values for L and R v, rearranging, and simplifying the result is ln e s (in Pa) = ln [19.83 (5417/T)]

7 Naraine Persaud, Entry Code ME-11 7 The foregoing implies T dl depends on the surface T d0 and on the dew point lapse rate (DPLR). Logarithmic differentiation of r qs = εe s /P gives dr qs /r qs = de s /e s dp/p = 0 and therefore de s /e s = dp/p. Substituting the Clausius-Clapeyron equation for de s /e s and the aerostatic equation dp/p gives L/[R v (T d ) 2 ] dt d = (-g/r d T)dz where T d is the dew point of the ascending parcel elevation z and T is the corresponding air temperature, and R d R m. Substituting numerical values L = 2.5 x 10 6 K kg -1, R v = J kg -1 o K -1, R d = J kg - 1 o K -1, and g = 9.81 m s -1 gives dt d /dz (in o K m -1 ) = -(T d ) 2 /(0.158 x 10 6 )T. The LCL occurs when T by cooling at the DALR = T d by cooling at the DPLR. Therefore the LCL is obtained by solving T LCL = T d0 [(T d ) 2 /(0.158 x 10 6 T)] LCL where T 0 is the surface air temperature. This gives LCL = (T 0 -T d0 ) / { [(T d ) 2 /(0.158 x 10 6 T)]}. T d and T are functions of elevation, but in practice are considered equal to T d0 and T 0 respectively in estimating the LCL. For example, given T 0 = 20 o C (293 o K) and T d0 = 1 o C (274 o K) the LCL = 19 / { [(274) 2 /(0.158 x 10 6 x 293)]} = km. Alternatively, it has been shown that [(T d ) 2 /(0.158 x 10 6 T)] ranges from about 1.7 to 1.9 o K km -1 (Tsonis, 2002). Using an average value of 1.8 o K km -1, the LCL can be estimated as LCL (km) = (T 0 T d0 ) / 8. Using this alternate formula gives in the above example gives LCL = 19/8 = 2.38 km. For ascent above the LCL, Latent heat of condensation is released into the parcel, but the parcel remains saturated at the given temperature. A change d(r qs ) in r qs (where the subscript s denotes saturation), would result in latent heat released into the parcel = -L d(r qs ) where L denotes the latent heat of vaporization. As was done previously, substitution of the hydrostatic equation dp = -ρ s gdz into the first law as δq = C ps dt VdP gives (for unit mass) δq = C ps dt + gdz. Since no water is lost from the parcel ρ s = ρ m = P/R m T. Substituting -L d(r qs ) for δq gives C ps dt + gdz = -L d(r q ). Assuming almost all of the heat released by condensation goes to heating the dry air component in the parcel would imply C ps C pd. Making the substitution, dividing throughout by C pd dz, and rearranging gives dt/dz = - (L/C pd ) (dr qs /dz) - g/c pd. This is the saturated adiabatic lapse rate (SALR). Since r qs is a function of T and P, dividing its total differential (dr qs ) by dz gives dr qs /dz = [( r qs / P)] T (dp/dz) + [( r qs / T)] P (dt/dz). Substituting this result with dp/dz = -ρ m g into the SALR expression, rearranging, and simplifying gives: SALR = g C pd [1 ρml( r { L [1 + ( r C pd qs qs / P) / T) T P ] } ] Differentiating r qs εe s /P where e s (T) is the saturation vapor pressure gives [( r qs / P)] T = - (εe s ) /P 2 and [( r qs / T)] P = (ε/p) (de s /dt). Assuming constant L and ideal gas behavior, the Clausius-Clapeyron equation (Salby, 1996) for the slope of the saturation vapor pressure versus temperature curve is de s / dt = Le s / (R v T 2 ) where R v is the specific gas constant for water vapor. Substitution of these results together with ρ m ρ d = P/R d T into the equation for the SALR, and simplifying gives a more practical form as (Tsonis, 2002):

8 Naraine Persaud, Entry Code ME-11 8 SALR = L rqs [1 + ] g R dt { 2 C pd L rqs [1 + C R T pd v 2 } ] The value of (Lr qs / R d T) is usually small. In many applications this term is neglected resulting in the simpler expression usually encountered in atmospheric science literature. The SALR represents the saturated lapse rate above the LCL. It is assumed that the system remains closed i.e. r q remains the same (all the water and heat from condensation remains within the ascending parcel), and the ascent continues as a reversible saturated-adiabatic process. However, the system would no longer remain closed if (as may be expected) some or all of the water and heat from condensation at a given elevation is immediately released from the parcel. This would imply an irreversible and non-adiabatic process. However, the heat lost is small relative to the heat content of the parcel and the ascent in this case is termed as pseudo-adiabatic (meaning "as if it were adiabatic"). The SALR given above is considered equal to the pseudo-adiabatic rate for all practical purposes. However, the SALR is considerably lower than the DALR MALR, and is a function of temperature. The SALR is about 4 o K km -1 for humid air masses near the surface. It increases to about 9 o K km -1 as the humidity decreases with increasing elevation. An average value is about 5.4 o K km -1. Like its counterparts θ d and θ m for dry and unsaturated air, the potential temperature invariant for saturated adiabatic ascent (θ e termed as the equivalent potential temperature) is defined as d(ln θ e ) = δq/(c pd T) = (-Ldr qs ) /(C pd T) = - d[(lr qs ) /(C pd T)]. Integrating (on the half-open interval with lower bound θ e = θ d at r qs = 0) [L rqs ]/[ Cpd T] and exponentiating gives θ e = θ d e. Physically θ e is the temperature attained if moist air at temperature T is expanded pseudo-adiabatically until all the water has condensed out and removed from the parcel, and then compressed adiabatically to the standard pressure of 1000 mb. The theoretical lapse rates developed in the foregoing discussion serve as reference values for inferring tropospheric conditions from the observed environmental lapse rate (ELR) at a given location. Departures of the ELR from these reference lapse rates provide a measure of the stability (i.e. the auto-convective propensity or the tendency for vertical motions induced by density stratification) in a given layer of tropospheric air. Below the LCL, when the ELR = DALR MALR (lapse rate is adiabatic) a rising parcel will always be at the same temperature as its surroundings. Therefore it would have the same density and would neither tend to return or continue its displacement. The stability of a tropospheric layer with condition prevails is termed as neutral. If the ELR < DALR (lapse rate is sub-adiabatic) an ascending parcel will quickly become colder (more dense) than the surrounding air and will return to its original position. The layer is termed as absolutely stable. Stable air resists vertical movement. If the reverse is true and the ELR > DALR (lapse rate is super-adiabatic) an ascending parcel will become warmer(less dense) than the surrounding air and will continue to rise. Except in the heated layer next to the surface, such

9 Naraine Persaud, Entry Code ME-11 9 instabilities do not persist since they are rapidly counteracted by vigorous auto-convection and vertical mixing as fast they develop. In terms of θ d, these stability conditions correspond stable (θ d < 0), neutral (θ d = 0), and unstable (θ d > 0). In a similar manner saturated air parcels will be stable, neutral, or unstable with respect to vertical displacements depending on whether the ELR is <, =, or > the SALR or θ e <, =, or > 0. The tropospheric lapse rate for the Standard Atmosphere (NOAA, 1976) is -6.5 o C km -1. Parameters for the standard atmosphere have been specified by the International Civil Aeronautical Organization (ICAO) and are needed for standardization in aviation, aeronautics, and meterology. Using subscript sa to denote the standard atmosphere, solving o K m -1 = - [g(γ sa -1)] /(R d γ sa )] for γ sa with g = 9.81 m s -2 and R d =286.7 J kg - 1 o K -1, and gives γ sa = 1.23 compared to the value of γ d = 1.4 for dry air. Other parameters for the standard atmosphere are surface temperature = o K, pressure = 101,325 Pa, density = kg m -3, and mean molecular weight = g mol -1. The standard lapse rate implies T(z) = T o -βz where T(z) is the temperature ( o K) at elevation z (m), T o is the surface temperature, and β= o K m -1. Assuming ideal gas behavior the hydrostatic equation becomes Substituting T = T o - βz into the aerostatic equation dp/dz = (-Pg/RT)dz, separating variables, and integrating on a half-open interval with lower bound P = P o at T = T o gives after some rearranging (βz) / T o = 1 (P/P o ) (Rβ)/g. These relationships can be used to calculate P, T of tropospheric air for the Standard Atmosphere at any given elevation. As an example, for z = 10 km and T o = o K = T = o K ( o K m -1 x 10 4 m) = o K. Substituting T o = o K, P o = x 10 5 Pa, R = J kg -1 o K -1, β = o K m -1, gives P at 10 km and P = 0.26 atm with T = 223 o K. Literature Cited Brown, R. A Fluid mechanics of the atmosphere. New York, Academic Press, 490 pp. List, R. J Smithsonian Meteorological Tables, 6 th edition. Washington, D.C., Smithsonian Institution Press, 527 pp. NOAA The U.S. standard atmosphere. U.S. Government Printing Office, Washington, D.C., (available from National Technical Information Office, Springfield, Virginia, Product number: ADA ). Salby, M. L Fundamentals of atmospheric physics. San Diego, Academic Press, 627 pp. Tsonis, A. A An introduction to atmospheric thermodynamics. Cambridge, U.K., Cambridge University Press, 171 pp.