Copyright © 2013 R. R. Dickerson1 Mixing and Convection Chapt 4 page 44. Isobaric Mixing (p...

Copyright © 2013 R. R. Dickerson 1

Mixing and ConvectionChapt 4 page 44.

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 & Z.Q. Li

2

Case 1: no condensation

Specific humidity: 221

21

21

1 qqqmm

m

mm

m

Mixing Ratio – since qw

221

21

21

1 wwwmm

m

mm

m

So vapor pressure 2

21

21

21

1 eeemm

m

mm

m


3

Heat lost by warm sample = heat gained by cold sample

))(())(( 222111 TTcwcmTTcwcm cppcpp

or

221

21

21

1 TTTmm

m

mm

m

since121 ww


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!


5

e

TT1 T2

e1

e2

es

Tf

ef

es > ef so isobaric mixing in this case does NOT result in condensation.

es(T)

Clausius-Clapeyron


6

e

TT1 T2

e1

e2

es

Tf

ef

es(T)

Isobaric mixing in this case will result in condensation because es < ef


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 es(T):

if e < es(T) - no condensation

if e > es(T) - condensation will occur.


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.


9

Latent Heat released during condensation: dq = -Lvdw

Isobaric Process: dq = cpdT

Since w ~ e/p

- Lvdw = Lv de/p = cpdT

Or vp

L

pc

dT

de the equation

of a line!


10

e

T

(e1 ,T1)

es(T)

(e2 ,T2)

Tf

ef

Final uncondensedstate

e’

T’

True final state

Isobaric condensation line


11

To Determine the Final e & T:Find the intersection of the isobaric condensation equationwith the Clausius-Clapeyron equation using e &T as “initial conditions”.

The isobaric condensation equation must be integrated to arriveat an algebraic form:

')'( T

Tv

pTe

e

dTL

pcde

s

so )'()( TT

L

pceTe

v

ps


12

The Clausius Clapeyron Equation

'

11exp)()'(

TTR

LTeTe

v

vss

Simplifies for T ~ T’ to

'

111)()'(

TTR

LTeTe

v

vss


13

The simplified form of the Clausius-Clapeyronequation can be combined with the isobariccondensation equation to find the final values of e and T.

But what if conditions don’t allow youTo simplify the equations……?


14

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)


15

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?


16

f(x) = f(x*) + f’(x*)(x- x*) = g(x*) + g’(x*)(x- x*)

or

)(')('

)()(**

***

xfxg

xfxgxx

or

)(')('

)()(**

***

1

jj

jjjj

xfxg

xfxgxx

Newton – Raphson iteration.


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.


18

Specific Humidity of a Mixed Parcel

2

1

1z

z

mixed dzqm

q

Where mass of air per unit area 2

1

z

z

dzm

Using the hydrostatic equation we can show


19

Likewise,

Thus for a well mixed layer, q, w and areconstant throughout. With no condensation, this must mean that the lapse rate correspondsto dry adiabatic.

d


20

Convective Condensation Level

CCL – Pressure and temperature at whichcondensation occurs in/at top of a well mixedlayer. It can be found by the intersection ofthe dry adiabat for the layer with the mixingratio isopleth for the layer.


21

Lifting Condensation Level

LCL – level at which condensation will occur if a parcel is lifted from the surface in a dryadiabatic process with constant w until justsaturated.

Note: LCL = CCL if the layer is well mixed.

Copyright © 2013 R. R. Dickerson22

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.


23

Fair Weather CumulusFair weather cumulus

1 pm EST July 7, 2007,

a smoggy day


24

Boundary Layer Venting Through Fair Weather Cumulus (Cumulus Humilis)

SO2

H2SO4

SO2

H2SO4

Inversion

CumulusCumulus


25

Two Reservoir Model (Taubman et al., JAS, 2004)

cloud

cloud

cloudcloud

SO2

H2SO4

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