First Law of Thermodyamics U = Q + W where U is the increase in internal energy of the system, Q is the heat supplied to the system and W is the work done on the system. We can rewrite this by introducing two physical concepts, entropy and enthalpy. Entropy, S, is a measure of disorder in the system. Enthalpy, H, is the quantity U + pv, as this combination of terms frequently appears in thermodynamic equations. so we can rewrite the rst law in terms of entropy and enthalpy by considering conditions of constant pressure or constant volume: U = TS,pV H = TS +Vp For a simplyfying assumption we consider only reversible changes so that Q = TS;W = pv Specic Heats At constant volume:
which becomes c v =( dq dt ) c v =( du dt )! U = c vt for an ideal gas (pv = RT). Therefore, H = c v T + RT = T(c v + R) =c p T!H = c p T where c p is the specic heat at constant pressure. We can then solve for TS TS = H, Vp eliminating V yields TS = c p T, ( RT p )p Dividing by T yields a convenient form: S = c p ( T T ), R(p p ) and integrating over all the 's yields (A) S = c p (lnt),r(lnp)+s o! S = c p (ln(tp, ))+S o where R cp
For a diatomic gas, is approximately 2/7. Now let's consider an adiabatic process, one in which there is no heat transfer. This means Q = 0 which, by our denition means S =0. To fullll this condition means c p ( T T )=R(p p ) if the system goes from T to a nal temperature T f and from p to p o then when we integrate the lefthand side from T to T f and the righthand side from p to p o we get or c p (ln( T f T )) = R(ln(p o p )) T f = T ( p o p) The quantity T f is usually referred to as and is known as the potential temperature. For reference, p o is usually taken to be 1000 millibars. Parcels of Air and Lapse Rates We assume that a parcel of air is like our idealized cylinder. The parcel is in pressure equilibrium with respect to the environment but may have diering temperature and density. Of course, there is no such thing as a parcel as real air mixes fast.
For an adiabatic parcel, the potential temperature and entropy are constant as heigh changes so we can write: d dz ] parcel =[ds dz ] =0 parcel or from equation A c p T dz ], R dp parcel p dz ] =0 parcel or rearranging terms dz ]=RT dp c p p dz ] (B) Now we remember hydrostatic equilibrium to yield dp dz =,gp RT dz ]=,g c p or more conveniently, dz ]= g c p This species the rate of temperature decrease with height (z) that would be expected for an adiabatic parcel of air. So this is called the adiabatic
lapse rate. In its simplest form when applied to a mass of dry air its value is 9.8 K/km. The introduction of this concept now allows for us to physically dene a stability criterion. We consider a parcel of air with parameters T,p, and z which adiabatically rises to a height z. At this new height z 1 = z + z. The new temparature will be: or T 1 = T + dz ] parcel z T 1 = T, g c p z The environmetal temperature, at height z 1 will be dierent than T 1 put the parcel is assumed to be in pressure equilibrium so that p 1 = p + dp dz ] env z Using the ideal gas law p 1 parcel = ; env = p 1 RT parcel RT env stability occurs whenever parcel is greater than env ; the displaced parcel can't continue to rise and will fall back down, Instability occurs if the rising
parcel nds its density to be less than the density of the environment. Hence the concept of buoyancy now comes into play. The buoyancy force occurs because the eective mass is dierent at height z than it is at z 1 as we assume constant volume for the parcel of air (there are no pressure dierences to cause expansion or contraction). The mass dierence is V 1 ( e, pl ) where e refers to environment and pl refers to parcel. Now we apply newtons second law: F = mx Now the notation gets a little cumbersome but: Force= gv 1 ( e, pl )= pl V 1 d2 (z) dt 2 d 2 (z) = g[ e, 1]) = g[ T pl, 1] dt 2 pl T e now we use our previous expressions for T e and T pl and dene to get, a = dt ;,= dt dz parcel dz env d 2 (z) dt 2 = g[ (T,, az) (T,,z), 1)
or d 2 (z) dt 2 =, g T (, a,,)z This reduces to an equation of the form: where N 2 d 2 (z) dt 2 + N 2 (z)=0 = g T (, a,,) = g T ( dz + g c p ) where T in this case is environmental temperature. If N 2 > 0 then this is the equation of motion for a harmonic oscillator where N is the frequency (called buoyancy frequency). We can return to our previous equation: or =T( p o p ) ln =lnt, lnp dierentiating yields d = T, dp p now we diereniate both sidez by dz
1 d dz = 1 T remember that dp =,g dz so now since = R cp 1 d dz = 1 T dp dz, p dz dz + g p and the ideal gas law we get 1 d dz + dz = 1 T so that N 2 = g d dz as the stability criteria. g c p T