Mixing and Convection R&Y Chapt 4 page 44. Salby Chapt. 5&7. Isobaric Mixing (p constant) of two samples of moist air: m1, w1, P1, q1, T1 m2, w2, P2, q2, T2 m, w, P, q, T ? Copyright © 2013 R. R. Dickerson

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

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

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

Copyright © 2013 R. R. Dickerson & Z.Q. Li Clausius-Clapeyron e es(T) e2 Tf ef es e1 T1 T2 T es > ef so isobaric mixing in this case does NOT result in condensation. Copyright © 2013 R. R. Dickerson & Z.Q. Li

Copyright © 2013 R. R. Dickerson & Z.Q. Li es(T) e2 Tf ef es e1 T1 T2 T Isobaric mixing in this case will result in condensation because es < ef Copyright © 2013 R. R. Dickerson & Z.Q. Li

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 es(T): if e < es(T) - no condensation if e > es(T) - condensation will occur. Copyright © 2013 R. R. Dickerson & Z.Q. Li

If condensation occurs, what are 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

Copyright © 2013 R. R. Dickerson & Z.Q. Li Latent Heat released during condensation: dq = -Lvdw Isobaric Process: dq = cpdT Since w ~ ee/p - Lvdw = Lv e de/p = cpdT Or the equation of a line! Copyright © 2013 R. R. Dickerson & Z.Q. Li

Copyright © 2013 R. R. Dickerson & Z.Q. Li Tf ef Final uncondensed state es(T) (e2 ,T2) e’ T’ True final state Isobaric condensation line (e1 ,T1) T Copyright © 2013 R. R. Dickerson & Z.Q. Li

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

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

Copyright © 2013 R. R. Dickerson & Z.Q. Li 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

Solving Transcendental Equations Consider Two functions of x: f(x) and g(x) Assume both are continuous and have continuous derivatives. f g xo Find x0 such that f(x0) = g(x0) Copyright © 2013 R. R. Dickerson & Z.Q. Li

Copyright © 2013 R. R. Dickerson & Z.Q. Li Since we can not find xo 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

Copyright © 2013 R. R. Dickerson & Z.Q. Li 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

Copyright © 2013 R. R. Dickerson & Z.Q. Li 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

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

Copyright © 2013 R. R. Dickerson & Z.Q. Li Likewise, Thus for a well mixed layer, q, w and q 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

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

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 & Z.Q. Li

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

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

Boundary Layer Venting Through Fair Weather Cumulus (Cumulus Humilis) H2SO4 H2SO4 Cumulus Cumulus Inversion SO2 SO2 Copyright © 2013 R. R. Dickerson & Z.Q. Li

Two Reservoir Model (Taubman et al., JAS, 2004) H2SO4 cloud cloud SO2 Copyright © 2013 R. R. Dickerson & Z.Q. Li