Adiabatic Lapse Rate

8/16/2019 Adiabatic Lapse Rate

1/13

ATMO 551a Fall 2010

Dry Adiabatic Temperature Lapse Rate

As we discussed earlier in this class, a key feature of thick atmospheres (where thickmeans atmospheres with pressures reater than 100!200 m"# is temperature decreases with

increasin altitude at hiher pressures definin the troposphere of these planets$ %e want to

understand why tropospheric temperatures systematically decrease with altitude and what therate of decrease is$ The first order e&planation is the dry adia"atic lapse rate$ An adia"atic

process means no heat is e&chaned in the process$ For this to "e the case, the process must "e

'fast so that no heat is e&chaned with the en)ironment$ *o in the first law of thermodynamics,we can anticipate that we will set the dQ term e+ual to ero$

To et at the rate at which temperature decreases with altitude in the troposphere, we

need to introduce some atmospheric relation that defines a dependence on altitude$ This relation

is the hydrostatic relation we ha)e discussed pre)iously$ -n summary, the adia"atic lapse ratewill emere from com"inin the hydrostatic relation and the first law of thermodynamics with

the heat transfer term, dQ, set to ero$

The gravity sideThe hydrostatic relation is

dP = −g ρdz (1#

which relates pressure to altitude$ %e rearrane this as

−g dz = dP

ρ=αdP (2#

where α . 1/ ρ is known as the specific )olume which is the )olume per unit mass$ This is the

e+uation we will use in a moment in deri)in the adia"atic lapse rate$

otice that

g dz 1 d 2 = F g

mdz = dW

g

m= dE

g

m(3#

where Φ is known as the eopotential , F g is the force of ra)ity and dW g is the work done "y

ra)ity$ *o g dz is the ra)itational work per unit mass that is the work that must "e done aainst

the 4arths ra)itational field to raise a mass of 1 k from some reference heiht (nominally sea

le)el# to that heiht$ 6om"inin the last two e+uations, we ha)e

d Φ= gdz = −αdP (7#

-nteratin this to heiht, z , yields

Φ z( ) = g dz

o

z

8 ' (5#%e can also define somethin called the eopotential heiht, 9:

Z 1 2 z( )

g0

=1

g0

g dzo

z

8 (;#

1


2/13

ATMO 551a Fall 2010

where 0 is the lo"ally a)eraed acceleration due to ra)ity at the 4arth surface normally taken

as =$>1 m/s2$ Z is often used as a )ertical coordinate either independent so that P ( Z # or

dependent, Z ( P #$

The Thermodynamics side%e return to the first law of thermodynamics

dQ ? dW . dU ? dE B (@#

%hen we chane the altitude of an air parcel, there are two enery transfers in)ol)in work,

• One associated with doin work to raise the air parcel in 4arths ra)ity field which

chanes the "ulk potential enery$

• the work associated with the air parcel decompression as the air parcel is raised to hiher

altitude and lower pressure$

-n the first type of work, remem"er that the )ertical force on an air parcel that is not mo)in

)ertically is ero and in hydrostatic "alance$ To mo)e the parcel )ertically means the )ertical

force "alance of ra)ity and pressure is no loner e&actly ero such that the air parcel accelerates)ertically$ -n this case, there is some work transferrin enery to the "ulk enery of the air

parcel$ %hile this is necessary to mo)e the air parcel )ertically, this is not the enery transfer weare interested in riht now$

%e are interested in the second type of work associated with the chane in pressure as the air

parcel is lifted (or lowered#$ %e assume the liftin is fast enouh that there is no heat transfer toor from the air parcel$ -n this case,

dW . dU (>#

*u"stitutin for "oth the work and the internal enery terms, we et

-P dV . m C v dT (=#

where m is the mass of the air parcel$ sin d ( PV # . P dV ? V dP , we et

V dP - d(PV) . m C v dT (10#

From the ideal as law, PV . nR*T $ Therefore

V dP - d(nR*T) . V dP - nR*dT . m C v dT (11#

-n this case, n is the num"er of moles in the air parcel with )olume, V , and the mass of the air

parcel is i)en "y m . n µ where µ is the mean molecular mass per mole of the as molecules in

the parcel$ Bi)idin throuh "y the mass of the air parcel yields

V

m

dP − n

m

R * dT = C v dT

1

ρdP −

n

n µ R * dT =αdP −

R *

µdT =C vdT (12#

2


3/13

ATMO 551a Fall 2010

α dP = R *

µ

+Cv

⎝

⎜

⎠

⎟dT =C pdT (13#

6om"inin (13# with either e+uation (2# or (7# from the ra)ity and eopotential deri)ationa"o)e, yields

−g dz = C p dT (17#

*o

dT

dz= −

g

C p(15#

This is eneral, true for any planetary atmosphere$ For 4arth, the heat capacity at

constant pressure of air is appro&imately 1000 C/k/< (you ha)e a homework assinment onthis#$ The a)erae ra)itational acceleration at 4arths surface is =$>1 m/s2$ *o the dry adia"atic

)ertical temperature radient is a"out !=$>


4/13

ATMO 551a Fall 2010

This fiure a"o)e is a 'skew T plot of the Tucson radiosonde profile taken on *ept 22 at 00

T6 in 200> which is *ept 21, 5 EM local time$ Elotted on the fiure are the temperature profile

(the thick solid white line#, the dew point temperature profile (the thick reen dashed line# which

pro)ides a measure of the )ertical profile of water )apor and the wind profile which is these+uence of arrows alon the riht hand ede of the fiure$

The thin solid white lines tilted to the riht are lines of constant temperature$ This tilt is to

compensate for the fact that temperature enerally decreases with altitude in the troposphere$The solid yellow lines that tilt to the left represent dry adia"ats, which we are discussin at

present$ The dashed yellow lines that start out somewhat )ertical near the surface and then tilt to

the left at hiher altitude (lower pressure# are moist adia"ats$ The moist adia"at includes the

effect of the latent heat release as water condenses out$The temperature profile shows the inter)al "etween the surface and a"out ;00 m" follows

a dry adia"at or adia"atic temperature profile almost perfectly$ This is an indicator of a deep, dry

con)ecti)e "oundary layer in which the air is rapidly o)erturnin, in this case due to heatinfrom the )ery hot surface "elow in our semi!arid reion$

Dry Adiabatic Temperature vs. Pressure and Potential Temperature

7


5/13

ATMO 551a Fall 2010

-n e+uation (15# a"o)e, we determined the dry adia"atic lapse rate as a function of

altitude$ %e will now deri)e the temperature dependence on pressure for a dry adia"atic liftin

or sinkin process$ *tartin with (12# from a"o)e

1

ρ

dP − n

n µ

R * dT =αdP − R *

µ

dT =C vdT (12#

1

ρdP =

R*

µ+ Cv

⎝ ⎜

⎠ ⎟dT =CPdT (1;#

From the ideal as law,

P = ρ R * T

µ, so

ρ = µ P

R * T (1@#

*u"stitutin this e&pression for density into (1;# yields

dP

ρ=

dP

P

R * T

µ= C p dT

dP

P= µC p

R*

dT

T (1>#

d lnP = µC p

R*d lnT = C

p '

R*d lnT (1=#

%e interate this

d lnPP0

P1

8= ln P

1( ) − ln P

0( ) = ln

P1

P0

⎝ ⎜

⎠ ⎟=

C p '

R *d lnT

T 0

T 1

8=C p '

R *d lnT

T 0

T 1

8=C p '

R*ln T 1

T 0

⎝ ⎜

⎠ ⎟Takin e!p(# of "oth sides and usin the fact that e!p(a ln !G# . !a, yields

5


6/13

ATMO 551a Fall 2010

P1

P0

= T

1

T 0

⎛

⎝ ⎜

⎞

⎠ ⎟

C p '

R*

(20#

This shows there is a power law relation "etween pressure and temperature under dry adia"atic

conditions$ ote - ha)e chosen to write the e+uations in terms of C p"R* (the molar form# ratherthan C p R, where R. RH/µ, "ecause the relation is more o")ious "ecause C p" in 4arths

atmosphere is a"out @/2 RH$ Therefore C p"R* is +uite close to @/2$ ow we rearrane (20# to

o"tain the dependence of temperature on pressure which is what we really want$

T 1

P1( ) = T 0 P0( )

P1

P0

⎡

⎣ ⎢

⎤

⎦ ⎥

R

C p

= T 0

P0( )

P1

P0

⎡

⎣ ⎢

⎤

⎦ ⎥

R*

C p '

(21#

- ha)e written T # as T # ( P ## to make it clear that T # is the atmospheric temperature at pressure .

P #$ ote that R*/Cp" is +uite close to 1/(@/2# . 2/@$

Alternati)ely we can rewrite (21# as

θ = T 0 = T 1P

0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p

= T 1P

0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R*

C p '

(22#

where θ is known as the potential temperature$ (22# tells us that i)en air with a temperature of

T #, at pressure, P #, the temperature of that air parcel would "e θ if the air parcel were mo)ed

adia"atically to another pressure, E0$ On 4arth, potential temperature is defined with P $ taken to

"e the a)erae sea le)el pressure, 1013 m"$ ote that a powerful feature of the potential temperature of an air parcel is that it is

conser)ed under adia"atic liftin or sinkin motion$ As a result, θ is )ery useful for determinin

the )ertical sta"ility of the air, that is its likelihood of o)erturnin$ Eotential temperature can "e

used as a )ertical coordinate as lon as the air is somewhat 'sta"le, that is temperature does notdecrease with altitude as fast as the dry adia"atic lapse rate$

Stable vs. neutral vs. unstable

The potential temperature pro)ides a way of determinin atmospheric sta"ility, that iswhether or not hot air rises and cold air sinks$ The adia"atic lapse rate defines how the

temperature of an air parcel will chane when it is displaced$ %hat is important is the )ertical

temperature structure of the en)ironment relati)e to an adia"atic lapse rate$ -f the en)ironmental

temperature decreases more slowly with increasin altitude*ta"le (unsta"le# means the potential temperature of the en)ironmental air column

increases (decreases# with altitude$ eutral sta"ility conditions occur when the potential

temperature is constant with altitude which is usually the sin of acti)e dry con)ection

dθ/d I 0 sta"le

dθ/d . 0 neutral

dθ/d J 0 unsta"le

;


7/13

ATMO 551a Fall 2010

%ell draw some fiures in class to o o)er this$

The oscillation frequency of displaced air parcels

6onsider the force on a displaced air parcel$ -ts pressure is the same as the rest of the air

at that altitude "ecause pressure adKusts at the speed of sound which is much faster than the

)ertical )elocity of the air parcels displacement$ Lowe)er, the )ertically displaced air parcelstemperature differs from that of the surroundin air$ Therefore its density differs from that of the

surroundin air$ Therefore the ra)itational force on the air parcel differs from that on thesurroundin air$ Therefore the air parcel is no loner in hydrostatic "alance and there is a net

)ertical force on it and it will "ein to accelerate )ertically$

The net )ertical force on the air parcel is the difference "etween the ra)itational force on

it and the surroundin air (which is in hydrostatic e+uili"rium#$ Therefore

F vert . g ( ρ par%el − ρ env)V . g ∆ρ V (23#

where V is the parcel )olume$ The density difference is

∆ ρ = ρ parcel − ρenv = µP R*

1T parcel

− 1T env

⎝ ⎜ ⎜

⎠ ⎟ ⎟= µP R*

T env −

T parcelT parcelT env

⎝ ⎜ ⎜

⎠ ⎟ ⎟

= µ P

R*T parcel

T env − T parcel

T env

⎝ ⎜

⎠ ⎟= ρ parcel

T env − T parcel

T env

⎝ ⎜

⎠ ⎟ (27#

The mass of the air parcel is ρ par%el V so the )ertical acceleration of the air parcel is

a = F

m=

g∆ ρV

ρ parcel V =

g∆ ρ

ρ parcel(25#

The fractional density difference "etween the air parcel and the en)ironment is

∆ ρ

ρ parcel= ρ parcel − ρenv

ρ parcel

= − T parcel − T env

T env

⎝ ⎜

⎠ ⎟ (2;#

The temperature of the parcel rapidly displaced )ertically "y a small amount from z $ to z is i)en

"y

T parcel z( ) = T z0( ) + z − z0( )∂T

∂ z adiabatic(2@#

The temperature of the en)ironment is i)en "y

T env z( ) = T z0( ) + z − z0( )∂T ∂ z env

(2>#

*o the temperature difference "etween the displaced air parcel and the surroundin

en)ironmental air is

T parcel z( ) − T env z( ) = z − z0( )∂T

∂ z adiabatic−∂T

∂ z env

⎝ ⎜

⎠ ⎟

@


8/13

ATMO 551a Fall 2010

∆ ρ

ρ parcel= −

T parcel − T env

T env

⎝ ⎜

⎠ ⎟= −

z − z0( )

T env

∂T

∂ z adiabatic−

∂T

∂ z env

⎛ ⎝ ⎜

⎞ ⎠ ⎟ (2=#

*o the )ertical acceleration of the air parcel is

g∆ ρ

ρ parcel = −

g

T env

∂T

∂ z adiabatic −

∂T

∂ z env

⎝ ⎜

⎠ ⎟ z − z0( ) (30#

This can "e e&pressed more compactly usin the potential temperature$

θ = T 0 = T 1P

0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p

= T 1P

0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R*

C p '

(22#

∂θ

∂ z=

∂ T 1P0

P1

⎛

⎝

⎜ ⎞

⎠

⎟

R

C p

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

∂ z= P0

P1

⎛ ⎝ ⎜ ⎞

⎠ ⎟

R

C p ∂T 1

∂ z+ T 1

∂ P0

P1

⎛

⎝

⎜ ⎞

⎠

⎟

R

C p

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

∂ z= P0

P1

⎛ ⎝ ⎜ ⎞

⎠ ⎟

R

C p ∂T 1

∂ z+ T 1 P0

RC p[ ]

∂ P1

−

R

C p

⎡ ⎣ ⎢

⎤ ⎦ ⎥

∂ z

∂θ

∂ z=

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p ∂T 1

∂ z+ T 1P0

RC p −

R

C p

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟P1

−

R

C p−1 ∂ P1[ ]

∂ z=

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p ∂T 1

∂ z− T 1

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p R

C p

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟1

P1

∂ P1[ ]

∂ z

"ut d ln( P #/dz . !1/ & . g / RT so

∂θ

∂ z=

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p ∂T 1

∂ z− T

1

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p R

C p

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ −

g

RT

⎛ ⎝ ⎜

⎞ ⎠ ⎟=

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p ∂T 1

∂ z+

g

C p

⎡

⎣ ⎢

⎤

⎦ ⎥ (31#

%e di)ide throuh "y θ to et

∂θ

θ ∂ z=

P0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p ∂T 1

∂ z+

g

C p

⎡

⎣ ⎢

⎤

⎦ ⎥

T 1P

0

P1

⎛

⎝ ⎜

⎞

⎠ ⎟

R

C p

=

∂T 1+

∂ z

g

C p

⎡

⎣ ⎢

⎤

⎦ ⎥

T 1

= −

−

g

C p−

∂T 1

∂ z

⎡

⎣ ⎢

⎤

⎦ ⎥

T 1

= −

∂T adiabatic

∂ z−

∂T 1

∂ z

⎡ ⎣ ⎢

⎤ ⎦ ⎥

T 1

(32#

6om"inin this with the e+uation a"o)e yields

g∆ ρ

ρ parcel= −

g

T env

∂T

∂ z adiabatic−∂T

∂ z env

⎝ ⎜

⎠ ⎟ z − z0( ) =

g

θ

∂θ

∂ z z − z0( ) (33#

otice the parcels )ertical acceleration has the form of the simple harmonic oscillator (*LO#, Fm . a .!'m ! where

k

m=

g

θ

∂θ

∂ z(37#

The fre+uency of a *LO is deri)ed as follows

>


9/13

ATMO 551a Fall 2010

F = ma = m d 2 x

dt 2= −kx (35#

a = d 2 x

dt 2= −

k

m x (3;#

Takin the solution to "e sinusoidal: !(t # . sin(ω t ? φ # then

dx

dt =ω X cos ωt +φ( )

d 2 x

dt 2=−ω

2 X sinωt +φ( ) =−ω

2 x =−

k

m x (3@#

Therefore the oscillator fre+uency in radians per second is ('m#1/2$ *o the fre+uency of theoscillation of a displaced air parcel is

ω = N = g

θ

∂θ

∂ z

⎛

⎝

⎜ ⎞

⎠

⎟1/ 2

(3>#

This is known as the runt!Naisala fre+uency$ The more sta"le the air is, that is the larer d θ dz ,

the stron the restorin force to displaced air parcels and the faster they oscillate$ otice also

that the acceleration is proportional to $

=


10/13

ATMO 551a Fall 2010

Temperature related climatology figures from Peixoto and ort !"##"$

10


11/13


12/13

ATMO 551a Fall 2010

12


13/13

ATMO 551a Fall 2010

Diurnal cycle on %ars

13

Adiabatic Lapse Rate

Documents

Transcript of Adiabatic Lapse Rate