Moist  air  thermodynamics  (Reading  Text  3.5-­‐3.7,  p79-­‐101)  

Topics:  •  Variables  that  descript  moist  air  •  staDc  stability  for  moist  convecDon,  saturated  and  pseudo-­‐  adiabaDc  lapse  rates,  equivalent  potenDal  temperature  

•  StaDc  stability  •  The  second  law  of  thermodynamics  for  moist  air,  the  Clausius-­‐Clapeyron  equaDon  

Variables  •  Mixing  raDo,  w,  and  specific  humidity,  q:  

–  W=mv/md,      mass  of  vapor  vs.  mass  of  dry  air      –  q=mv/(mv+md)=w/(w+1)    mass  of  vapor  vs.  total  air  mass  

•  Vapor  pressure,  e=[w/(w+ε)]p    –  Where  recall  ε=Mv/Md=Rd/Rv=0.622,  p:  air  pressure  Because:  

€

e =nv

nv + ndp =

mv

Mvmd

Md

+mv

Mv

p ×

Mv

mdMv

md

=

mv

mdMv

Md

+mv

md

p =w

ε + wp

•  nv,  nd:  numbers  of  mole  of  vapor  and  dry  air  molecules,  respecDvely.  mv/md,      mass  of  vapor  v  

•  Mv  and  Md,  molecular  weight  of  vapor  and  dry  air      

•  Virtual  temperature  Tv:  –  Recall  from  Sect.  4.1,  that  Tv=T/[1-­‐e/p(1-­‐ε)]    (Eq.  3.16  in  the  text),  we  can  express  Tv  ≈  T  (1+0.61W)    (see  below)  

 

€

e =w

ε + wp ⇒ e

p=

wε + w

Tv = T 1

1− ep

(1−ε)= T 1

1− wε + w

(1−ε)= T ε + w

ε + w − w + wε

Tv= T1+ w /ε(1+ w)

= T 1+ ( 10.622

−1) w1+ w

%

& '

(

) * = T 1+ 0.61 w

1+ w%

& '

(

) *

Tv≈ T 1+ 0.61w( ) because w << 1, w +1 ≈w

QuesDon:  

 If  the  atmospheric  consists  10  g  per  1  kg  of  dry  air  and  temperature  is  27C  at  the  sea-­‐level,  what  is  the  mixing  raDo  (w),  specific  humidity  (q),  vapor  pressure  (e)  and  virtual  temperature  of  this  atmosphere?  

 

•  W=10g/1  kg=0.01  •  q=(10g)/(1000g+10g)=0.0099=9.9g/kg  •  e=wP/(ε+w)=0.01X1000hPa/(0.622+0.01)=15.8Pha  

•  Tv≈T(1+0.61w)=(273+27)K(1+0.61X0.01)=301.8K=28.8C  

•  SaturaDon  vapor  pressure,  es:  the  maximum  vapor  pressure  with  respect  to  a  plane  surface  of  pure  water  at  T.    It  is  a  funcDon  of  T.  

•  Below  0°C,  the  saturaDon  vapor  pressure  with  respect  to  ice,  esi  for  a  plane  ice  surface  is  lower  than  that  with  respect  of    of  water.  

•  Thus,  atmosphere  can  be  saturated  for  ice  but  not  for  water.    Supper  saturaDon  e>esi  can  occur  in  real  atmosphere  (e.g.,  near  tropopause).    

•  SaturaDon  mixing  raDo,  ws=mvs/md=0.622es/(p-­‐es)≈0.622es/p  –  SaturaDon  mixing  raDo  is  more  commonly  used  in  meteorology  and  climate  than  es.  

–  See  p.  82  of  the  text  for  derivaDon.  

•  RelaDve  humidity,  RH=100w/ws=100e/es  –  RelaDve  humidity  is  derived  from  dew  point  or  frost  point  temperature,  Td,  and  air  temperature,  T,  which  can  be  measured  readily.  

•  Td:  the  temperature  at  which  dew  starts  to  form,  when    ws(Td)=w  of  the  air,  thus      RH=100ws(Td)/ws(T)      Thus,  RH  is  determined  by  saturaDon  mixing  raDo  of  dew/frost  point  temperature  vs.  that  of  air  temperature  for  the  same  pressure  value.  

Example:  

•  Meteorological  measurements  show  that  a)  air  T=30°C  and  Td=25°C;  b)  The  next  day,  T  remains  the  same,  but  Td  reduces  to  20°C.    Calculate  RH  for  both  cases  use  the  figure  below:  

a)  es(T=30C)=40  hPa,      es(Td=25C)=30  hPa,    RHa=100*es(Td)/es(T)=75%  

 b)  es(Td=20C)=25  hPa  RHb=100*es(Td)/es(T)  =25hPa/40hPa=62.5%  

 RelaDve  humidity  dropped    about  12.5%  in  case  b)  

•  Lii  condensaDon  level  (LCL):    –  T  in  a  rising  air  parcel  would  decrease  follow  the  dry  adiabaDc  lapse  

rate  (9.8C/km).    At  the  height  where  T  reduces  to    the  same  value  as  Td,  condensaDon  occurs.    This  height  is  referred  to  as  the  liiing  condensaDon  level  (LCL).    It  is  also  cloud  base.  

T  

height  

Td  

LCL  

•  LCL  in  a  skew  T-­‐ln  p  chart:  

Figure  3.10  The  liiing  condensaDon  level  of  a  parcel  of  air  at  A,  with  pressure  p,  temperature  T  and  dew  point  Td,  is  at  C  on  the  skew  T  −  ln  p  chart.  

Td   T  

•  T  in  a  rising  air  parcel  would  follow  constant  θ  line  (orange,  dry  adiabaDc  line).    The  green  isothermal  lines  indicate  T  and  Td  of  the  surface  air  (1000  hPa).    The  pressure  level  at  which  constant  θ  line  cross  the  isothermal  line  for  Td  is  the  LCL  for  this  air.  

Saturation moist adiabatic and pseudoadiabatic lapse rates:

•  As  an  rising  air  parcel  above  LCL,  condensaDon  occurs  and  the  temperature  change  inside  of  the  air  parcel  is  determined  by  combined  dry  adiabaDc  cooling  and  latent  heaDng  due  to  condensaDon  (dT/dz=(αdp/dz-­‐Ldw)/Cp,  vs.  dry  adiabaDc  dT/dz=(αdp/(Cpdz)).    Thus,  the  rate  of  temperature  decrease  with  height,  i.e.,  the  moist  adiabaDc  lapse  rate  (Γm),  is  less  than  that  of  dry  adiabaDc  lapse  rate.  

•  Unlike  the  dry  adiabaDc  lapse  rate  (Γd=R/Cp=9.8K/km),  Γm  is  not  a  constant.  Γm  depends  on  net  amount  of  condensaDon.  

•  Two  assumpDons  about  Γm:  –  SaturaDon  moist  adiabaDc  lapse  rate:  All  condensed  water  is  retained  in  

the  rising  air  parcel,  and  can  be  re-­‐evaporated  if  T  increases.    It  is  a  reversible  process.  

–  PseudoadiabaDc  lapse  rate:  All  condensed  water  falls  out  the  rising  air  parcel.    Thus,  re-­‐evaporated  is  not  possible.    It  is  a  irreversible  process.  

Saturation moist adiabatic vs. pseudoadiabatic lapse rates: •  Saturated moist adiabatic lapse rate is greater than pseudoadiabatic

lapse rate (T decreases faster with height), because –  In case of the former, condensed water can be re-evaporated which absorbs

latent heat. Thus, air parcel tends to be less buoyant and Tv (virtual temperature) decreases faster with height.

–  Under pseudoadiabatic case, no re-evaporation of condensed water. The rising air parcel is more buoyant and Tv decreases slower with height.

•  Moist lapse rate varies. Typically, Γm  is  about  4  K/km  in  the  warm  and  humid  lower  troposphere,  and  ~  6-­‐7  K/km  in  the  mid-­‐troposphere,  close  to  Γd  in  the  upper  troposphere.    Why?  

 

Example:  

 Meteorological  measurements  show  that  air  T=30°C  and  Td=25°C,  T=-­‐5°C  at  500  hPa.  

a)  Determine  the  LCL  of  an  rising  air  parcel  from  the  surface  

b)  Determine  the  lapse  rate  between  the  LCL  and  500  hPa  assuming  i)  all  condensed  liquid  water  falls  out  immediately;  ii)  only  a  half  of  the  liquid  fall  out,  the  other  half  rising  with  the  air  parcel  and  re-­‐evaporate  immediately.    Ignore  the  heat  absorbed  by  the  condensed  water.  

c)  Height  of  500  hPa  is  5.5  km,  950hPa  is  0.5  km.    Latent  heat  of  vaporizaDon,  Lv=2.5X106  J  kg-­‐1.    Surface  pressure  is  1000  hPa.  

SoluDon:  

€

a. From the surface to LCL, ∂T∂z

= Γd = 9.80K/km, T −Td =∂T∂z

zLCL

zLCL =T −TdΓd

=(30− 25)K9.8K /km

≈ 0.51km

b. At LCL, es,LCL = es (25C) = 30hPa, ws = 0.622 es

p= 0.622 30hPa

950hPa= 0.020 = 20g /kg

At 500 hPa, es,500hPa(-5C) = 5 hPa, based on es - T relation shownin the figure.

ws,500hPa = 0.622 esp

= 0.622 5hPa500hPa

= 6.2×10−3 = 6.2g /kg

i) The total condensed water between LCL and 500 hPa, δes = (20− 6.2)g /kg = 13.8g /kg

dTdz

=1Cp

αdpdz

=αρg=g

− L dwdz

*

+

, , , ,

-

.

/ / / /

=gCp=Γd

−LCp

dwdz

= 9.8K /km − 2.5X106.Jkg−1

1004JK −1kg−1⋅

13.8×10−3

5km ×103m /km)

= (9.8− 6.9)K /km ≈ 2.9K /Kmii) If a half of the condensed water re- evaporates, the net condensed water is 6.9g/kg.dTdz

= (9.8− 3.45) /1004K /m ≈ 6.35K /Km Thus, the value of moist lapse rate depends on the amount of net condensation in the rising air parcel.

•  Wet-­‐bulb  temperature,  Tw:  –  In  surface  meteorological  staDon,  web-­‐bulb  temperature  represents  T  measured  by  a  glass  bulb  thermometer  wrapped  by  wet  clothe  over  which  ambient  air  is  drawn.      

– Because  evaporaDon  of  wet  clothe  also  cool  ambient  T  and  increase  w,  thus  Tw  >  Td  (cool  dry  adiabaDcally  unDl  condensaDon  occurs)  <T,  usually  is  the  arithmeDc  mean  of  T  and  Td.  

Equivalent  potenDal  temperature,  θe:  

•  When  condensaDon  occurs  in  a  rising  air  parcel,  the  latent  heat  released  by  condensaDon,  Lvdws,  would  warm  the  air  parcel,    thus  against  the  dry  adiabaDc  cooling,  as  shown  by  the  2nd  law  of  thermodynamics  Lvdws=CpdT-­‐αdp.  

•  The  equivalent  potenDal  temperature,  θe,  is  defined  as  the  temperature  would  be  if  all  the  water  vapor  were  condensed  and  rain  out  immediately  (pseudoadiabaDc  moist  process),  and  all  the  potenDal  energy  were  used  to  heat  the  air.    Thus,  it  is  higher  than  the  potenDal  temperature  θ (only include internal and potential energy, not latent energy).  

•  θe  is  derived  from  the  2nd  law  of  thermodynamics      -­‐Lvdws=CpdT-­‐αdp  assuming  all  the  water  vapor  is  condensed  and  rain  out  immediately.  

 

€

because θ = T pop

#

$ %

&

' (

R / c p, lnθ = lnT - R

Cpln p+

RCp

ln pocons tan t

Cpdθθ

= CpdTT

− R dpp

, because dqT

= CpdTT

− R dpp

dqT

= Cpdθθ

⇒ Lvdws

T= Cp

dθθ

⇒

dθθ

= −Lvdws

CpT≈ −d Lvws

CpT

#

$ % %

&

' ( ( beause 1

T2 <<1T

dθθθ e

θ

∫ = − d Lvws

CpT

#

$ % %

&

' ( ( 0

ws∫

θe ≈θ ⋅ exp Lvws

CpT

#

$ % %

&

' ( (

θe is defined as the equivalent potential temperature, represents the potential tempearture (θ ) would be if all the vapor were condensedand its latent heat were used to increase θ .

•  Equivalent  potenDal  temperature,  θe  is  a  conserved  variable  for  an  pseudoadiabaDc  adiabaDc  process,  i.e.,  there  is  no  heat  exchange  between  the  system  and  the  ambient,  and  100%  latent  heat  release  due  to  condensaDon  of  water  vapor  is  used  to  increase  T.  

•  The  moist  adiabaDc  process  is  not  a  true  adiabaDc  process,  because  of  latent  heat  release.    However,  we  can  treat  it  as  it  were  a  adiabaDc  process  if  we  use  θe  concept.  

•  Θe  is  similar  to  θ  for  dry  air.  

Normand’s  rule-­‐finding  Td  and  Tw  using  θ  and  θe  lines  

•  Find  LCL:    –  Find  the  θ  line  that  intersects  with  

isothermal  line  that  represents    surface  T.  

–  Find  the  ws  line  that  intersects  with  Td  at  1000  hPa.  

–  The  intersecDon  between  θ  and  ws  represents  LCL.  

•  Finding  Tw  in  Skew  T-­‐lnp  chart:  –   From  the  LCL  follow  the  constant  θe  

line  down  to  the  sea  level  (1000  hPa),  you  will  find  the  wet  bulb  temperature,  Tw  at  the  surface.  

•  Q:  Why  does  LCL  is  determined  by  the  Td    follow  the  Ws  instead  of  isothermal  line?  

Effect  of  ascent  followed  and  descent  on  air  temperature  and  humidity:  

•  Once  a  convecDve  air  reaches  LCL  (point  2),  it  would  ascent  follow  constant  θe  line  unDl  it  reaches  cloud  top  (point  3)  and  begin  descending.    

•  The  temperature  of  the  descending  air  depends  on  the  type  of  moist  process  during  ascending  process:    –  If  it  follows  pseudoadiabaDc  process,  all  the  

condensed  water  falls  out,  no  evaporaDon  would  occur  during  the  descending,  T    follows  dry  adiabaDc  process  (constant  θ  line)  back  to  surface  (point  4,  900  hPa  in  this  case)  with  T2  (see  exercise  3.10  for  quanDtaDve  descripDon)  

–  If  it  follows  saturated  moist  process,  condensed  water  retained  in  the  air  parcel  would  re-­‐evaporate,  T  during  descending  follows  θe  to  LCL  (point  2),  then  follow  θ  back  to  surface  (point  1).    Thus,  T  is  reversal.  

T2  

LCL  

T   T2  1  

1  

2  

2  

3  

3  

4  

4  

In-­‐class  discussion-­‐summary:  

•  Specific  humidity  is  more  commonly  used  in  meteorological  applicaDon.  Do  you  expect  the  value  of  specific  humidity  more  or  less  than  that  of  mixing  raDo?  Why?  

•  How  is  RH  determined  by  meteorological  measurement?  •  What  is  the  difference  between  the  saturaDon  moist  adiabaDc  

and  pseudoadiabaDc  moist  assumpDons?    Which  assumpDon  would  lead  to  overesDmate  buoyancy  of  the  convecDng  air  (or  strength  of  convecDon)  and  which  assumpDon  would  underesDmate  the  buoyancy  of  convecDve  air?    Why?  

•  What  determine  value  of  the  moist  adiabaDc  lapse  rate?  •  What  type  of  lapse  rate  does  equivalent  potenDal  

temperature  represent?