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 ? 1 Case 1: no condensation
Specific humidity: Mixing Ratio since m1 q m1 m2 q2 w q m1 w m1 m2 So vapor pressure m2 q1 m1 m2
e m2 w1 m1 m2 w2 m1 e1 m2 e2 m1 m2 m1 m2 Copyright 2013 R. R. Dickerson & Z.Q. Li 2 Heat lost by warm sample = heat gained by cold sample
m1 (c p w1ccp )(T1 T ) m2 (c p w2ccp )(T T2 ) or T since m1 m2 T1 T2 m1 m2 m1 m2 w1 w2 1 Copyright 2013 R. R. Dickerson & Z.Q. Li 3
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 4 Clausius-Clapeyron e es(T) e2 es
ef e1 T1 Tf T2 T es > ef so isobaric mixing in this case does NOT result in condensation. Copyright 2013 R. R. Dickerson & Z.Q. Li 5 e es(T) e2 ef
es e1 T1 Tf T2 T Isobaric mixing in this case will result in condensation because es < ef Copyright 2013 R. R. Dickerson & Z.Q. Li 6 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 7 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 8 Latent Heat released during condensation: Isobaric Process: dq = -Lvdw dq = cpdT Since w ~ e/p - Lvdw = Lv de/p = cpdT Or pc p de dT
Lv Copyright 2013 R. R. Dickerson & Z.Q. Li the equation of a line! 9 e Final uncondensed state es(T) (e2 ,T2) ef True final state e
Isobaric condensation line (e1 ,T1) Tf T Copyright 2013 R. R. Dickerson & Z.Q. Li T 10 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: es (T ') de e pc p Lv T' dT so es (T ) e T
Copyright 2013 R. R. Dickerson & Z.Q. Li pc p Lv (T ' T ) 11 The Clausius Clapeyron Equation Lv es (T ' ) es (T ) exp Rv 1 1 T T '
Simplifies for T ~ T to Lv es (T ' ) es (T ) 1 Rv 1 1 T T' Copyright 2013 R. R. Dickerson & Z.Q. Li 12 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 dont allow you To simplify the equations? Copyright 2013 R. R. Dickerson & Z.Q. Li 13 Consider Two functions of x: f(x) and g(x) Assume both are continuous and have continuous derivatives. f g Find x0 such that f(x0) = g(x0)
xo Copyright 2013 R. R. Dickerson & Z.Q. Li 14 Since we can not find xo analytically, how do we proceed? Expand f and g in a Taylors 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. Isnt this what we did for the CC equation? Copyright 2013 R. R. Dickerson & Z.Q. Li 15 f(x) = f(x*) + f(x*)(x- x*) = g(x*) + g(x*)(x- x*)
or * * g(x ) f (x ) x x * * g'(x ) f '(x ) * or * * x j 1 x j
* g(x j ) f (x j ) * * g'(x j ) f '(x j ) Newton Raphson iteration. Copyright 2013 R. R. Dickerson & Z.Q. Li 16 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 17 Specific Humidity of a Mixed Parcel z2 qmixed 1 qdz m z1 Where mass of air per unit area
z2 m dz z1 Using the hydrostatic equation we can show p1 1 qmixed qdp Dp p2 Copyright 2013 R. R. Dickerson & Z.Q. Li 18 p1
Likewise, 1 q mixed q dp Dp p2 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. d Copyright 2013 R. R. Dickerson & Z.Q. Li 19 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 20 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 21
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 22 Fair Weather Cumulus Fair weather cumulus 1 pm EST July 7, 2007, a smoggy day Copyright 2013 R. R. Dickerson & Z.Q. Li
23 Boundary Layer Venting Through Fair Weather Cumulus (Cumulus Humilis) H2SO4 Cumulus H2SO4 Cumulus Inversion SO2 SO2 Copyright 2013 R. R. Dickerson & Z.Q. Li 24
Two Reservoir Model (Taubman et al., JAS, 2004) H2SO4 cloud cloud clou clou dd SO2 Copyright 2013 R. R. Dickerson & Z.Q. Li 25