Copyright © 2013 R. R. Dickerson1 Mixing and Convection Chapt 4 page 44. Isobaric Mixing (p constant) of two samples of moist air: m 1, w 1, P 1, q 1, T 1 m 2, w 2, P 2, q 2, T 2 m, w, P, q, T ??

Copyright © 2013 R. R. Dickerson & Z.Q. Li 2 Case 1: no condensation Specific humidity: Mixing Ratio – since So vapor pressure

Copyright © 2013 R. R. Dickerson & Z.Q. Li 3 Heat lost by warm sample = heat gained by cold sample or since

Copyright © 2013 R. R. Dickerson & Z.Q. Li 4 Case 2: condensation and mixing Question: can condensation occur during the mixing of two unsaturated samples (isobaric mixing)? Yes, in the winter when you see your breath!

Copyright © 2013 R. R. Dickerson & Z.Q. Li 5 e T T1T1 T2T2 e1e1 e2e2 eses TfTf efef e s > e f so isobaric mixing in this case does NOT result in condensation. es(T)es(T) Clausius-Clapeyron

Copyright © 2013 R. R. Dickerson & Z.Q. Li 6 e T T1T1 T2T2 e1e1 e2e2 eses TfTf efef es(T)es(T) Isobaric mixing in this case will result in condensation because e s < e f

Copyright © 2013 R. R. Dickerson & Z.Q. Li 7 How does one determine if condensation will occur? 1. Determine T & e that would result if no condensation were to occur. 2. Compare e with e s (T): if e < e s (T) - no condensation if e > e s (T) - condensation will occur.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 8 If Condensation occurs, what is the final e & T? e must be less than that calculated assuming no condensation because vapor will be removed. T must be greater because latent heat has been released.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 9 Latent Heat released during condensation: dq = -L v dw Isobaric Process: dq = c p dT Since w ~  e/p - L v dw = L v  de/p = c p dT Or the equation of a line!

Copyright © 2013 R. R. Dickerson & Z.Q. Li 10 e T (e 1,T 1 ) es(T)es(T) (e 2,T 2 ) TfTf efef Final uncondensed state e’e’ T’T’ True final state Isobaric condensation line

Copyright © 2013 R. R. Dickerson & Z.Q. Li 11 To Determine the Final e & T: Find the intersection of the isobaric condensation equation with the Clausius-Clapeyron equation using e &T as “ initial conditions ”. The isobaric condensation equation must be integrated to arrive at an algebraic form: so

Copyright © 2013 R. R. Dickerson & Z.Q. Li 12 The Clausius Clapeyron Equation Simplifies for T ~ T ’ to

Copyright © 2013 R. R. Dickerson & Z.Q. Li 13 The simplified form of the Clausius-Clapeyron equation can be combined with the isobaric condensation equation to find the final values of e and T. But what if conditions don ’ t allow you To simplify the equations……?

Copyright © 2013 R. R. Dickerson & Z.Q. Li 14 Consider Two functions of x: f(x) and g(x) Assume both are continuous and have continuous derivatives. fg xoxo Find x 0 such that f(x 0 ) = g(x 0 )

Copyright © 2013 R. R. Dickerson & Z.Q. Li 15 Since we can not find x o analytically, how do we proceed? Expand f and g in a Taylor ’ s series: f(x) = f(x * ) + f ’ (x * )(x- x * ) + … g(x) = g(x * ) + g ’ (x * )(x- x * ) + … Neglect higher order terms and solve for x. Isn ’ t this what we did for the CC equation?

Copyright © 2013 R. R. Dickerson & Z.Q. Li 16 f(x) = f(x * ) + f ’ (x * )(x- x * ) = g(x * ) + g ’ (x * )(x- x * ) or Newton – Raphson iteration.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 17 Adiabatic Mixing Parcels from different pressure levels are mixed after being brought together adiabatically. The final state of the combined parcel can be calculated as shown previously. When a column of air is thoroughly mixed, the specific humidity becomes constant throughout.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 18 Specific Humidity of a Mixed Parcel Where mass of air per unit area Using the hydrostatic equation we can show

Copyright © 2013 R. R. Dickerson & Z.Q. Li 19 Likewise, Thus for a well mixed layer, q, w and  are constant throughout. With no condensation, this must mean that the lapse rate corresponds to dry adiabatic.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 20 Convective Condensation Level CCL – Pressure and temperature at which condensation occurs in/at top of a well mixed layer. It can be found by the intersection of the dry adiabat for the layer with the mixing ratio isopleth for the layer.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 21 Lifting Condensation Level LCL – level at which condensation will occur if a parcel is lifted from the surface in a dry adiabatic process with constant w until just saturated. Note: LCL = CCL if the layer is well mixed.

Copyright © 2013 R. R. Dickerson 22 Fair Weather Cumulus Clouds Fair weather cumulus are form atop buoyant bubbles of air (thermals) that rise from Earth's surface. As bubbles rise, the water vapor mixing ratio remains constant but the temperature falls and the relative humidity increases until it reaches the saturation vapor pressure, 100% RH. Here droplets condense and clouds form. This occurs at the Lifting Condensation Level, (LCL) where the flat cloud bases are seen.

Copyright © 2013 R. R. Dickerson & Z.Q. Li 23 Fair Weather Cumulus Fair weather cumulus 1 pm EST July 7, 2007, a smoggy day

Copyright © 2013 R. R. Dickerson & Z.Q. Li 24 Boundary Layer Venting Through Fair Weather Cumulus (Cumulus Humilis) SO 2 H 2 SO 4 SO 2 H 2 SO 4 Inversion Cumulus

Copyright © 2013 R. R. Dickerson & Z.Q. Li 25 Two Reservoir Model (Taubman et al., JAS, 2004) clou d SO 2 H 2 SO 4