In-­‐class  exercise:  

When  westerly  wind  from  Pacific  arrive  California  coast,  the  air  would  be  li>ed  by  Sierra  Nevada.    Assume  that  air  temperature  is  T=25°C  and  Td=20°C,  the  mountain  top  is  1500  m,  use  the  Skew  T-­‐lnp  chart  to  determine:  

a)  Determine  the  cloud  base  (LCL)  as  air  rise  on  windward  side.  b)  Determine  air  temperature  at  the  top  of  the  mountain.  c)  Determine  temperature  of  the  air  at  the  based  of  leeside  for  

pseudoadiabaPc  process  and  saturated  moist  adiabaPc  process,  respecPvely.    Assume  that  the  base  of  the  windside  is  0  m  above  sea  level,  the  base  of  the  leeside  is  500  m  or  950  hPa  above  sea  level.  Latent  heat  of  vaporizaPon,  Lv=2.5X106  J  kg-­‐1.      

a.  Find  LCL:  follw  θ  line  unPl  it  intersects  with  20C  isothermal  at  950  hPa  b.  Above  LCL,  T  change  follows  θe  line.    T  at  mountain  top  is  determined  by  intersects  

between  θe  at  1500  m  or  850  hPa  and  isothermal  line,  which  is  15C,  c.  If  the  air  followed  pseudoadiabaPc  process  when  it  rises,  T  would  follows  θ  line  that  

intersects  with  θe  at  850  hPa  as  it  flows  downhill  to  the  boZom  of  the  leeside  because  there  is  no  condensaPon.    T  at  1000  hPa  is  30C,  5C  warmer  than  it’s  iniPal  T  at  boZom  of  windward  side  of  the  mountain.  

d.  If  the  air  followed  saturated  moist  adiabaPc  process,  T  would  follow  θe  line  back  to  LCL  and  then  follow  the  θ  line  back  to  1000  hPa.    T=25C,  the  same  as  T  at  the  boZom  of  windward  side.  The  process  is  reversible.  

25C  

LCL=950hPa  

Top=1500m=850hPa  Ttop=12C  

Tlee,pseudo=30C  

Tlee,sat=25C  

θ θε

T   ws  

4.6  StaPc  stability:  

Sta$c  stability  of  unsaturated  air  –  Determines  whether  a  verPcally  displayed  air  parcel  would  oscillated  around  its  original  

locaPon  or  would  convect.    For  example,  –  When  the  ambient  lapse  rate,  Γ<  Γd,  an  air  parcel  would  be  colder  and  less  buoyant  

than  the  ambient  air  when  it  is  displayed  upward,  thus    it  would  conPnue  to  move  upward  without  any  forcing.    Likewise,  it  would  be  cooler  and  heavier  than  the  ambient  air  when  it  is  displayed  downward.    Thus,  it  would  conPnue  to  sink  downward  without  any  external  forcing.    The  ambient  atmosphere  is  stable  for  convecPon.  

–  When  ,  Γ>  Γd,  an  air  parcel  would  be  warmer  and  lighter  than  the  ambient  air  when  it  is  displayed  upward,  thus    it  would  sink  with  external  forcing,.    Likewise,  it  would  be  warmer  and  lighter  than  the  ambient  air  when  it  is  displayed  downward,  consequently,  rise.    Thus,  this  verPcally  displaced  air  parcel  would  oscillate  around  its  original  locaPon.    The  ambient  atmosphere  is  un  stable  for  convecPon.      

How do we quantify the influence of Γ on static instability? •  Stability  of  unsaturated  air  can  be  quanPfied  by  

–  T’-­‐T:  temperature  difference  between  the  convecPng  parcel  and  ambient  atmosphere    T’-­‐T>0,  unstable,  T’-­‐T<0,  stable,  T’-­‐T=0,  neutral  

 –  Γd-­‐Γ:    lapse  rate  difference  between  dry  adiabaPc  process  and  ambient  

atmosphere    Γd-­‐Γ<0 or Γ>Γd:  unstable,  only  occurs  near  surface  in  summer  desert    Γ<Γd:  stable,  occurs  most  of  Pme  in  reality,  Γ=Γd:  neutral  

 –  dθ/dz:  verPcal  slope  of  potenPal  temperature  

 dθ/dz>0:  stable,  occurs  most  of  Pme  in  reality,      dθ/dz<0:  unstable,  only  occurs  near  surface  in  summer  desert    dθ/dz=0:  neutral    

   

Why?

ρ 'w = FB where w ≡dzdt

is the veritcal velocity, FB is buoyancy force as

determined by density difference between the air parcel and the ambient air, i.e., ρ'-ρ, where ρ' is density of the rising/sinking air parcel, ρ is density of the ambient air

dwdt

=FBρ '= −g ρ'-ρ

ρ '= −g

P/R 1T'−

1T

#

$%

&

'(

P/R 1T'#

$%

&

'(

= gT '−TT

For a rising air parcel: If T'-T>0, dz/dt>0, unstable, If T'-T<0, dz/dt<0, stable

Or dwdt

= gT '−TT

= gTo −Γdz '− (To −Γ z ')T

=gz 'T

Γ−Γd( )

where z'=z-zo zo : original height of the air parcel

In-­‐class  exercise:  show    

€

1θ∂θ∂z

=1T(Γd −Γ)

The  following  derivaPon  shows  that  air  rises,  i.e.,  convects,  when  Γ>Γd.  

€

Thus, ∂θ∂z

> 0 means Γd − Γ > 0 or Γ < Γd stable

€

because θ = T po

p#

$ %

&

' (

R / c p

, lnθ = lnT - RCp

ln p +RCp

ln poconstant

Cpdθθ

= CpdTT− R

dpp

= CpdTT

+ Rρgdzp

= CpdTT

+gdzT

ideal gas law

1θdθdz

=1T

dTdz

+gCP

#

$ %

&

' ( =

1TΓd −Γ( )

How  does  dry  stability  affect  weather  and  climate?    

•  StaPc  stability  affect  gravity  wave  generated  by  mountains  or  weather  disturbances.    The  wave  length  (or  frequency)  depends  on  stability,  or  buoyancy  oscillaPon.      

•  Waves  would  develop  if  the  dynamic  wave  length  is  comparable  to  that  corresponding  to  buoyancy  frequency.  

•  The  buoyancy  wave  frequency  is  determined  by  

Stable  atmosphere  

dwdt

=dwdt

= −gTΓd −Γ( ) z ' where z'=z-zo

d2z 'dt2 + N 2z '=0 wave equation

where N2 ≡gTΓd −Γ( )

referred to as the Brunt-Vaisala frequencyor buoyancy frequency

Gravity  wave  induced  clouds  under  stable  condiPon:  

What  determines  the  wavelength  or  the  space  between  cloud  rolls?  

Example:  

•  Meteorological  measurements  show  that  the  lapse  rate  of  the  boundary  layer  is  5  C/km,  the  temperature  is  27C  and  wind  speed  is  10  m/s  over  coast  of  California.    What  would  be  distance  between  rolls  of  boundary  layer  clouds?  

•  The  buoyancy  frequency  would  be  N2=10  m/s/(273+27)K[10K/km-­‐5K/km]/1000m,  N=1.29X10-­‐2  s-­‐1.  

•  The  gravity  waves  that  form  rolls  of  clouds  would  have  frequency  similar  to  N,  thus  

Τ=L/U=2π/Ν L=2πU/Ν=2X3.14X10m/s/1.29X10-­‐2  s=4.87X103m=4.87  km The distance between rolls of clouds would be 4.87 km. What would happen to the distance between roll of clouds a.  if wind speed increases to 20 m/s? 9.74 km b.  Lapse rate increases to 7C/km and wind remains 10m/s? 6.28 km  

Stability  of  saturated  air:  

•  Γ<Γd: the atmosphere would be stable for dry convection, however, if –  Γ>Γm, or dθe/dz>0, the atmosphere would be unstable for saturated air,

referred to as conditional unstable; –  Γ<Γm, or dθe/dz<0, the atmosphere would be stable for saturated air, in this

case, referred to absolute stable

Γd

Γm

LFC: the level of free convection, referred to the height above which the convecting air would be warmer than the ambient atmosphere, therefore, would rise spontaneously.  

Γ

LCL  

LFC  

Convective Available Potential Energy •  Vertical integrated buoyancy from the level of free convection to the

limit of convection

Γd: dry adiabatic

Γm: moisture adiabatic

Γ: measured by radiosondes

zi

LCL

LFC

LOC

CAPE

T

B

unstable for dryconvection

unstable for moistconvection

CINE

positvie buoyant

negative buoyant

Updraft velocity: If CAPE were 100% converted to kinetic energy and without frictional drag, then the updraft velocity at the LOC would be:

w = 2 • CAPE Trigger mechanism: A lift of surface air to the LFC, such as fronts, dry lines, sea-breeze

fronts, gust fronts from other thunderstorms, atmospheric buoyancy waves, mountains and localized regions of excess surface heating.

How strong the trigger, i.e., the initial lifting, is required depending on the magnitude of negative buoyancy a convective parcel has to overcome between the surface and the LFC. This negative buoyancy is referred to as the Convective Inhibition Energy (CINE).

Γd: dry adiabatic

Γm: moisture adiabatic

Γ: measured by radiosondes

zi

LCL

LFC

LOC

CAPE

T

B

unstable for dryconvection

unstable for moistconvection

CINE

positvie buoyant

negative buoyant€

CINE = −g (Δz •i= sfc

LFC

∑ Tc −TeTe

)i

Discussion:  •  Why  does  occurrence  of  convecPon  strongly  depends  on  surface  air  humidity  in  the  tropics?