copyright © 2013 r. r. dickerson & z.q. li 1 hydrostatic equilibrium chapt 3, page 28 g :...

Copyright © 2013 R. R. Dickerson & Z.Q. Li

1

Hydrostatic EquilibriumChapt 3, page 28

forcegradient pressure:z

p

g : gravity

p-dp

p+dp z-dz

z+dz

gdz

dp mequilibriuin

(one of the best approximations in meteorology)


2

Remember pressure is force per unit area so pressure per unit height is weight (mg) per unit volume (Nm-2 m-1 = kg m s-2 m-3). In this class we will usually ignore horizontal variations in thermodynamic variables (T* is virtual temp) and write

dgdzp

dpRT

dzRT

g

p

dpor

RTpbut

gdz

dp

z

p

*

*

*/


3

p

dpRTSo Increment of geopotential energy.

Thus the thermodynamic coordinates -RlnP, Tyield areas proportional to energy (emagram)

Consider the case for g constant (good assumption).Integrate the hydrostatic equation from p = po to p and zo to z.

)(exp)(

* oo zzRT

gpzp


4

To find the mean virtual temperature take the weighted average:

o

o

p

p

p

p

pdp

pdp

TT

*

*

p

po

T*

-Rlnp

*T

Thus thermodynamic diagrams can be used to determine the geopotential thickness between pressure levels.


5

g has a slight variation with latitude and altitude which can usually be ignored; in AOSC652 you will integrate for g = f(z). So we can define an increment of geopotential thickness, or height.

d dgo; go = 9.8 m/s2

Geopotential

Define an increment of geopotential

p

dpRTgdz *

ss)(energy/ma gdzd

)ln( 12

*

12

*

ppg

TR

p

dp

g

TRd

o

o


6

Moisture effects on Geopotential

)6.0()(

)ln(

:includednot weremoisture If

)ln()6.01(

)ln(

12

12

12

*

w

ppg

TR

ppwg

TR

ppg

TR

o

oo

o

o


7

Pressure Variation with z for “Special” Atmospheres

1. Constant Density o (Homogenous Atmosphere)

g

RT

g

RT

g

PH

dzgdP

gdzgdzdP

o

o

o

o

o

H

o

p

o

o

0

0

H → “Scale Height” = height of const. T (288K), homogenous atmosphere ~ 8 km.


8

2. Constant Lapse Rate Atmosphere

Define Lapse Rate z

T


9

3. Isothermal Atmosphere : = 0

Hzo

o

ePzP

H

dzdz

RT

g

P

dP

/)(

The isothermal approximation is good to about 30% for the troposphere. Problem for students: What fraction of the mass of atmosphere lies between 800 and 700 hPa?


10

Stability Criteria

Air parcelproperties

P, T, w

Environment properties

P’, T’, ’,’w’

Assume 1. Parcel and environment are in instantaneous dynamic equilibrium: P = P’2. Atmosphere is in hydrostatic equilibrium.3. Parcel and environment do not mix.4. No compensating motion by environment as an air parcel moves.


11

Consider a dry adiabatic displacement by an air parcel:


12

Consider a Saturated Adiabatic displacement (pseudo adiabatic):

dz

dP

dz

dTc

dz

dwLor

dPdTcdwLqd

po

v

pov

Note that wo= f(T,P)

We want to derive dT/dz for this process using the hydrostatic equation and assuming ’ ~ 1.

so gdz

dTc

dz

dwL p

ov


13

s

p

sv

sv

p

s

v

sv

s

s

s

s

ss

RTc

wL

RT

wL

c

g

dz

dT

RT

gw

dz

dT

TR

wL

RT

g

TR

g

P

g

dz

Pd

Pdz

dP

Pbut

dz

dP

Pdz

dT

dT

de

edz

dw

w

P

ewbut

T

2

2

2s

s

1

1

dz

dw

:EquationClapeyron Clausius the toapplied becan above The

11

111

:by w dividing and atingdifferentiby So

)(


14

Saturated Adiabatic Lapse Rate

The table below lists values of s in oK/km

1000 mb 700 mb 500 mb

-30 C 9.2 9.0 8.7

0 C 6.5 5.8 5.1

20 C 4.3 3.7 3.3

pressure

tem

pera

ture

Note that s < d


15

Buoyant Force on an Air Parcel

The environment is in hydrostatic equilibrium (no acceleration)

z

Pg

dt

zdso

0

2

2

In general, an air parcel WILL be subject to an acceleration due to density differences with the environment, so for the parcel:

z

Pgz


16

*

**

*

)(

)(then

since

T

TTgzonacceleratiso

P

RTbut

gz

z

P

z

P


17

Stability Criteria

We are interested in small displacements of a parcel from its original location.

If with a small displacement we

find thatThe environment is

said to be

Unstable

Neutral

Stable0

0

0

z

z

z

zo


18

For convenience, consider the parcel location zo to be zero. The temperature of the environment may be written as:

...22

*2

21

**

0*

z

dz

Tdz

dz

TdTT

If the displacement z is sufficiently small,

rate" lapse talenvironmen" the*

*0

*

**

0*

dz

Td

zTTor

zdz

TdTT


19

For the parcel we may write:

zTTor

zdz

dTTT

p

*0

*

**

0*

p = parcel lapse rate.

zT

zgz

T

TTgz

p

*0

*

**

)(

)( remember


20

zT

zgz p

*

0

)(

Thus if…

For dry displacements, use p= d

For saturated displacements, use p= s


21

Since s< d, we must also consider conditions between the two stability criteria for dry and saturated.

unstable Absolutely

neutralDry

unstablelly Conditiona

neutral Saturated

stable Absolutely

d

d

ds

s

s

= parcel lapse rate = environment lapse rate


22

DIURNAL CYCLE OF SURFACE HEATING/COOLING:

z

T0

1 km

MIDDAY

NIGHT

MORNING

Mixingdepth

Subsidenceinversion

NIGHT MORNING AFTERNOON


23

*** Emagram ***


24

Convective Instabilitydry air


25

Creterion for Convective Stability


26

Criterion for Convective Instability

unstablez

Neutralz

Stablez

w

w

w

,0

,0

,0

DIURNAL CYCLE OF SURFACE HEATING/COOLING:

z

T0

1 km

MIDDAY

NIGHT

MORNING

Mixingdepth

Subsidenceinversion

NIGHT MORNING AFTERNOON


28

ATMOSPHERIC LAPSE RATE AND STABILITYATMOSPHERIC LAPSE RATE AND STABILITY

T

z

= 9.8 K km-1

Consider an air parcel at z lifted to z+dz and released.It cools upon lifting (expansion). Assuming lifting to be adiabatic, the cooling follows the adiabatic lapse rate :

z

“Lapse rate” = -dT/dz

-1/ 9.8 K kmp

gdT dz

C

ATM(observed)

What happens following release depends on the local lapse rate –dTATM/dz:• -dTATM/dz > upward buoyancy amplifies initial perturbation: atmosphere is unstable• -dTATM/dz = zero buoyancy does not alter perturbation: atmosphere is neutral• -dTATM/dz < downward buoyancy relaxes initial perturbation: atmosphere is stable• dTATM/dz > 0 (“inversion”): very stable

unstable

inversion

unstable

stable

The stability of the atmosphere against vertical mixing is solely determined by its lapse rate


29

EFFECT OF STABILITY ON VERTICAL STRUCTUREEFFECT OF STABILITY ON VERTICAL STRUCTURE


30

WHAT DETERMINES THE LAPSE RATE OF THE WHAT DETERMINES THE LAPSE RATE OF THE ATMOSPHERE?ATMOSPHERE?

• An atmosphere left to evolve adiabatically from an initial state would eventually tend to neutral conditions (-dT/dz = at equilibrium

• Solar heating of surface disrupts that equilibrium and produces an unstable atmosphere:

Initial equilibriumstate: - dT/dz =

z

T

z

T

Solar heating ofsurface: unstableatmosphere

ATM

ATM

z

Tinitial

final

buoyant motions relaxunstable atmosphere to –dT/dz =

• Fast vertical mixing in an unstable atmosphere maintains the lapse rate to Observation of -dT/dz = is sure indicator of an unstable atmosphere.


Plume looping, Baltimore ~2pm.


Plume Lofting, Beijing in Winter ~7am.


35


36

Did you report the calibrated T?


37


38

Mean (±σ) oC

24.7 (2.2)


39

Results

• Calibrating thermometers reduced σ from 2.2 to 1.5 oC.

• Eliminating siting differences and operator bias further reduced σ to 0.74 oC, a major improvement in precision, even without applying the calibration coefficients.

• To identify mesoscale differences in temperature, we need precision. We can identify a T of 1.5 oC with 95% confidence.

copyright © 2013 r. r. dickerson & z.q. li 1 hydrostatic equilibrium chapt 3, page 28 g :...

Documents

meteorology slide

lapse rate slide

s d slide

p popo t

d d g o g o

w environment properties

air parcel moves

gravity pdp p dp zdz