Numerical Weather Prediction Moist Thermodynamics

cemeterymarylandMechanics

Oct 27, 2013 (3 years and 7 months ago)

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Numerical Weather Prediction



Moist Thermodynamics

Peter Bechtold and Adrian Tompkins


Can we represent clouds in a GCM

ie
. Moisture transport and phase changes ?


e.g. T799 36h forecast from 20080525


Meteosat 9 versus forecasted satellite image

What is the annual global mean cloud cover ?

The same for GOES12 versus IFS
!


Tropical Cyclone (Gustaf),

Tropical continental convection

Stratocumulus

,
Cirrus

Overview of Clouds/Convection


Introduction


Moist thermodynamics


Parametrization of moist
convection

(4 lectures, Peter),


Theory of moist convection


Common approaches to parametrization including the ECMWF scheme


Cloud Resolving Models

(1 lecture
-

Richard)


Their development and use as parametrization tools


Parametrization of

clouds

(4 lectures
-

Richard)


Basic microphysics of clouds


The ECMWF cloud scheme and problem of representing cloud cover


Issues concerning validation


Planetary boundary
-
layer

(
4 lectures


Martin+Anton
)


--

Surface fluxes, turbulence, mixing and clouds


Exercise Classes

(
1 afternoon, Peter and Richard)

Moist Thermodynamics


The great Belgian tradition:

“Thermodynamique de l’atmosphere”

Dufour and v. Mieghem (1975)

Most recent:


Maarten Ambaum (2010) “Thermal physics of the
atmosphere”



For simplified Overview:

Rogers and Yau (1989) “
A short course in cloud physics



Thermodynamics and Kinematics:

K. A. Emanuel (1994) “
Atmospheric Convection


Textbooks

The First and Second Law


-----


The First Law of Thermodynamics:

Heat is work and work is heat.

Heat is work and work is heat

Very good! The Second Law of Thermodynamics:

Heat cannot of itself pass from one body to a hotter body,

Heat cannot of itself pass from one body to a hotter body


Heat won't pass from a cooler to a hotter,

Heat won't pass from a cooler to a hotter

You can try it if you like but you'd far better notter,

You can try it if you like but you'd far better notter

'Cos the cold in the cooler will get hotter as a ruler,

'Cos the cold in the cooler will get hotter as a ruler

'Cos the hotter body's heat will pass to the cooler,

'Cos the hotter body's heat will pass to the cooler


Good, First Law:

Heat is work and work is heat and work is heat and heat is work

Heat will pass by conduction,

Heat will pass by conduction

And heat will pass by convection,

Heat will pass by convection

And heat will pass by radiation,

Heat will pass by radiation

And that's a physical law.


Heat is work and work's a curse,

And all the heat in the universe,

Is gonna cooooool down

'Cos it can't increase,

Then there'll be no more work

And there'll be perfect peace

Really?

Yeah, that's entropy, maan!





And its all because of the Second Law of Thermodynamics, which lays
down:


That you can't pass heat from a cooler to a hotter,

Try it if you like but you far better notter,

'Cos the cold in the cooler will get hotter as a ruler,

'Cos the hotter body's heat will pass to the cooler.

Oh you can't pass heat, cooler to a hotter,

Try it if you like but you'll only look a fooler

'Cos the cold in the cooler will get hotter as a ruler

And that's a physical Law!


Oh, I'm hot!

Hot? That's because you've been working!

Oh, Beatles
-

nothing!

That's the First and Second Law of Thermodynamics!



The First and Second Law

Authors: M. Flanders (1922
-
1975) & D. Swann (1923
-
1994)

From "At the Drop of Another Hat“

Moist Thermodynamics

Ideal Gas Law


Assume that “moist air” can be treated as mixture of two
ideal gases: “dry air” + vapour

Dry air equation
of state:

Gas Constant for

dry air = 287 J Kg
-
1

K
-
1

Pressure

Density

Water Vapour
equation of
state:

Vapour

pressure

vapour

density

Gas constant for

Vapour = 461 J Kg
-
1

K
-
1

0.622

Temperature

What is
definition of
ideal gas?

Pressure, partial pressures and gas
law for moist air


Partial pressures add if
both gases occupy same
volume V. N
x
are the mol
masses

First law of thermodynamics

Heat supplied by
diabatic process

Change in internal
Energy

Work done

by Gas

Energy conservation and Heat

All quantities are per unit mass (specific)

dQ is not a perfect differential, but ds
(change in entropie is !)

Can write as

Specific Heat at

constant volume

First law of thermodynamics:
Enthalpy
and Legendre transformation

Changing variables

Special processes: “Adiabatic Process


dQ=0 or better ds=0

Special significance since many atmospheric
motions

can be approximated as adiabatic

Enthalpy and flow process

In isentropic (adiabatic) flow

velocity

Summary :Potentials and Maxwell
relations

Internal Energy

Enthalpy

Helmholtz free Energy

Gibbs free Energy

Conserved Variables

Using enthalpy
equation and
integrating, obtain
Poisson’s
equation

Setting reference pressure to 1000hPa
gives the definition of potential
temperature for dry air

Conserved in dry
adiabatic motions, e.g.
boundary layer
turbulence

What is the
speed of
sound?


4. Mixing ratio

Humidity variables

There are a number of common ways to describe vapour
content etc

1. Vapour Pressure

2. Absolute humidity


3. Specific humidity

Mass of water vapour per unit moist air


5. Relative humidity

Mass of water vapour per unit dry air


6. Specific liquid water content


7. Total water content

e

Humidity variables

How to define “moist quantities” and how to switch from mixing
ratio to specific humidity.

For any intensive quantity we have

Virtual temperature

T
v

Another way to describe the
vapour content is the
virtual
temperature

, an artificial
temperature.

By extension, we define the
virtual potential temperature
,
which is a conserved variable
in unsaturated ascent, and
related to density


It describes the temperature required for
dry air, in order to have at the same
pressure the same density as a sample of
moist air


Definition:

The Clausius
-
Clapeyron equation


For the
phase change between water and water
vapour

the equilibrium pressure (often called
saturation water vapour pressure) is a function of
temperature only

water

air+water vapour

Consider this closed system in equilibrium:

T equal for water & air, no net evaporation
or condensation

Air is said to be saturated


with

v

>>

w
,
and the ideal gas law


v
=R
v
T/e
s

The Clausius
-
Clapeyron equation
-

Integration


The problem of integrating
the Clausius
-
Clapeyron
equation lies in the
temperature dependence
of
L
v
.


Fortunately this
dependence is only weak,
so that approximate
formulae can be derived.

Nonlinearity has
consequences for
mixing in convection

e
s0

= 6.11 hPa at T
0
=273 K

Meteorological energy diagrams

Total heat
added in
cyclic process:

Thus diagram with ordinates T versus
ln


will have the properties of

“equal areas”=“equal energy”

Called a
TEPHIGRAM

rotate to have pressure (almost) horizontal

Dry
adiabatic
motion

Pressure



T



T

Tephigram (II)

pressure

r

Saturation
specific humidity

Saturation
mixing ratio

Function of temperature and
pressure only


tephigrams
have isopleths of
r
s

Using a Tephigram

At a pressure of

950 hPa


Measure

T=20
o
C

r=10 gkg
-
1

plot a atmospheric
sounding

Enthalpy and phase change

Have we been a bit negligeant ? Yes, more
precisely

Divide by
m
d

(or
m
d
+m
v
) and assume adiabatic process

Ways of reaching saturation


Several ways to reach saturation:




Diabatic Cooling (e.g. Radiation)



Evaporation (e.g. of precipitation)



Expansion (e.g. ascent/descent)

Cooling:

Dew point temperature

T
d

Temperature to which air must be
cooled to reach saturation, with p
and r held constant

All of these are
important cloud
processes!!!

Evaporation:

Wet
-
bulb temperature
T
w

Temperature to which air
may be cooled by
evaporation of water into
it until saturation is
reached, at constant p

Will show how to determine graphically from tephigram

Ways of reaching saturation:

Expansion: (Pseudo) Adiabatic Processes

As (unsaturated) moist air expands (e.g.
through vertical motion), cools
adiabatically conserving

.


Eventually saturation pressure is reached,
T,p are known as the “
isentropic
condensation temperature and pressure
”,
respectively. The level is also known as
the “Lifting Condensation Level”.


If expansion continues, condensation will
occur (assuming that liquid water
condenses efficiently and no super
saturation can persist), thus the
temperature will decrease at a slower rate.

Ways of reaching saturation:

Expansion: (Pseudo) Adiabatic Processes

Have to make a decision concerning the
condensed water.




Does it falls out instantly or does it remain in the
parcel? If it remains, the heat capacity should be
accounted for, and it will have an effect on parcel
buoyancy





Once the freezing point is reached, are ice
processes taken into account? (complex)


These are issues concerning
microphysics
, and
dynamics
. The air parcel history will depend on
the situation. We take the simplest case: all
condensate instantly lost as precipitation, known
as “
Pseudo adiabatic process


Pseudo

adiabatic

process



T

Tephigram (III)

pressure

r

Pseudoadiabat

(or moist adiabat)

Remember: Involves
an arbitrary “cloud
model”

Cooling:
(Isobaric
process)
gives dew
point
temperature

parcel mixing ratio=5g/kg

Expansion,
(adiabatic
process) gives
condensation
temperature

Wet Bulb

Temperature

Moist Adiabat

Parcel at 850 hPa,
T=12.5
o
C

r=6 g/kg

T
e

Raise parcel
pseudoadiabatically
until all humidity
condenses and then
descend dry
adiabatically to
reference pressure


e

(=315K)

Equivalent Potential

temperature

conserved in adiabatic motions

Summary: Conserved Variables

Dry adiabatic processes

Moist adiabatic processes

potential temperature

equivalent potential temperature

moist static energy

total water specific humidity

In HYDROSTATIC ATMOSPHERE:
dry static energy

Specific humidity

Last not least: how to compute
numerically saturation (adjustment)

given
T, q

check if q > q
s
(T) then

solve for adjusted T
*
,q
*
so that

q
*
= q
s
(T
*
)

q
l
= q
-
q
s
(T
*
)

using c
p
dT =
-
L
v
dq
s

either numerically through iteration or with the aid
of a linearisation of


q
s
(T*) (see Excercises !!)

T,q
v

T*,q
vs
(
T*)

q
v

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-
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-
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