Diabatic Processes and Entrainment

Diabatic Processes
• Diabatic processes are non-adiabatic
processes such as
• precipitation fall-out
• entrainment and mixing
• radiative heating or cooling
Parcel Model
dθ
dt
dw
dt
dl
dt
dr
dt
=
L
(C − Er ) + Dθ
cp π̄
= −(C − Er ) + Dw
= C − Ar + Dl
= Pr + Ar − Er + Dr
π̄ = (p̄/p0 )R/cp , C is the net condensation rate, Er is the
rain evaporation rate, Ar is the cloud-to-rain water
conversion rate, Pr is the convergence of rain water flux,
and Di represents the effects of entrainment and mixing.
Microphysics
on
ati
or
ap
ev
(Er)
cloud
droplets
co
nd
co ens
lle ati
cti on
on
condensation
evaporation
(C)
water
vapor
(Ar)
rain
drops
(Pr)
fall out
(precipitation)
ned bythan
the with
saturation
adjustment
algorithm.
The
remaining
proces
ssure
how they
change
with
time. The
equations
that
go
dp
dp
of
cloudwith
waterpressure
to rain (A
entrainment
(Dθ , Dw ,properties
Dl ), are diaba
change
of
parcel’s
thermodynamic
are
ra) and
Diabatic
Processes dw
dw
ally
-(3) represented
by dividing explicitly.
by −dp/dt:
−
= −Ĉ + D̂w −
=
−
Ĉ
+
D̂
w
dp
1)-(3), the process
ratesrates
are per
interval.
dp For example,
Process
perunit
unittime
time
interval:
dθ
dl
� �
�
�D̂θ
dl
−
=
γ
Ĉ
+
− dp= Ĉ − Â
r + D̂l −
dl
dr
=
Ĉ
−
Â
+
D̂
r
l
dp
dp
Ar ≡
=
.
dt conversion to rain
dt conversion from cloud water
-(20), the process
rates arethe
per
unit decrease
in per
pressure.
dwprocess
In (18)-(20),
rates are
unit decrease in p
−
=
−
Ĉ
+
D̂
w
before,
the
net
condensation
rate,
Ĉ,
in
(18)-(20),
is
implicitly
dete
Process
rates
per
unit
pressure
interval:
As
before,
the
net
condensation
rate,
Ĉ,
in
(18)-(20
often more interested in howdpthe thermodynamic properties of a parc
saturation
adjustment
algorithm.
The
diabatic
processes,
conversion
o
by
the
saturation
adjustment
algorithm.
The
diabatic
pr
essure than with how theydlchange with time. The equations that go
o rain (Âr ) and
turbulent
mixing
(
D̂
,
D̂
,
D̂
),
remain
to
be
specified
θ
w
l
=
Ĉ
−
Â
+
D̂
−
water
to
rain
(
Â
)
and
turbulent
mixing
(
D̂
,
D̂
r
l
r
θ
w , D̂l ),
change with pressure of dp
a parcel’s
thermodynamic
properties
arere
ery simple formulation
ofsimple
the conversion
rateofofthe
cloud
water to
rainofisc
A
very
formulation
conversion
rate
)-(3) by dividing by −dp/dt:
� per�unit decrease�in pressure.
20), the process rates are
�
dl
dl
efore, the net condensation
rate,
Ĉ,
in
(18)-(20),
is implicitly =de−
dθ
−Âr ≡ −
=
−Cl,
−
Â
≡
−
r D̂θ
−
=
γ
Ĉ
+
dp
dp conversionconversion
to rain
aturation adjustment algorithm.
The diabatic
processes,
dp conversion
to rain
rain
(
Â
(
D̂
to
be
specifie
−2
−1
r ) and turbulent mixing
θ , D̂w , D̂l ), remain
−2
−1
dt < 0 only, with
C = 2<×010
mb
.
for
dp/dt
only,
with
C
=
2
×
10
mb
.
dw
ry simple formulation of the
conversion
rate
of
cloud
water
to
rain
−
= −Ĉ + D̂w
dp�
�
dl
ˆ
dl
Entrainment
Entrainment is the
incorporation of
environmental air into
a parcel or cloud.
Evidence for Entrainment in Cu
Liquid water content (g m–3)
2.5
2.0
Adiabatic LWC
1.5
1.0
0.5
0.0
0.0 0.1 0.2 0.3
0.4 0.5
0.6 0.7
Distance (km)
0.8 0.9 1.0
Evidence for Entrainment in Cu
Height above cloud base (m)
2500
Adiabatic LWC
2000
1500
1000
500
0
0
1
2
3
4
Liquid water content, LWC (g m–3)
5
parcel
mixin
parcel
ancy
severa
substa
as pen
in par
cumul
bution
drople
partia
when
Ove
clouds
invers
the to
strato
Entrainment in Stratocumulus
e
b
s
F
h
b
b
f
p
d
E
c
p
plays an important role in the marine stratocumulustopped boundary layer. The rate at which this
Entrainment in Stratocumulus
Altitude (m)
Altitude (m)
e)
b)
f)
c)
g)
d)
h)
1250
1000
1500
Altitude (m)
a)
1250
1000
1500
1250
1000
1500
Altitude (m)
Entrainment in a
3D high-resolution
simulation of Sc.
1500
dQ1 # cp(T" $ T
1250
1000
1500
2000
2500
Horizontal distance
3000 1500
2000
2500
Horizontal distance
3000
Entrainment: Kelvin-Helmholtz Instability
Entrainment
into a
turbulent jet
droplet evaporation
molecular diffusion
turbulent deformation
entrainment
saturated parcel
Fractional Rate of Entrainment
8.33 g/kg (10 × 100 + 0 × 20)/(100 + 20) = 8.33
120 kg
mixing
8.33 g/kg
10 g/kg
mixing
ratio
10 g/kg
0
g/kg
120 kg
entrainment
20 kg
10 g/kg
100 kg
mass
Entrainment
The fractional rate of entrainment of a parcel of
mass m that entrains a blob of mass dm while the
pressure changes by −dp (due to ascent) is
1 dm
λ̂ ≡ −
.
m dp
The rate of change of a scalar φ due to entrainment
is
�
�
dφ
D̂φ ≡ −
= −λ̂(φ − φe ),
dp entrainment
where φe is the value of φ in the entrained air.
Entrainment
We can derive this from
�
�
dφ
φafter
−
= lim
dp entrainment ∆p→0
using
φbefore
and
φafter
Substitution gives
�
�
dφ
−
dp entrainment
ent
ent
− φbefore
−∆p
ent
ent
=φ
mφ + ∆m φe
=
.
m + ∆m
1
∆m
= lim
(φ − φe )
∆p→0 m + ∆m ∆p
1 dm
=
(φ − φe ) = −λ̂(φ − φe ).
m dp
1 dm
Entrainment
=
(φ − φe )
m dp
= −λ(φ − φe ).
ing (21) to θ, w, and l, we obtain
D̂θ = −λ(θ − θe ),
D̂w = −λ(w − we ),
D̂l = −λ(l − le ) = −λl.
us clouds, the fractional rate of entrainment, λ, ranges fro
−1
km . Cloud-top height is largely determined by λ: dee
ith small values, and shallow clouds with large values. It ha
udies that λ ∼ 0.2/R, where R is the cloud radius.
Entrainment
• In cumulus clouds, the fractional rate of entrainment, λ ≡ (1/m) dm/dz, ranges from
−1
−1
about 0.1 km to 2 km .
• Cloud-top height is largely determined by λ:
deep clouds are associated with small values,
and shallow clouds with large values.
• Field studies suggest that λ ∼ 0.2/R, where
R is the cloud radius.
Entrainment
pressure (mb)
800
lem50
lem100
lem200
entrainment only
−1
850
λ = 1.5 km
900
950
0.0
0.2
0.4
0.6
0.8
Fraction of unmixed cloud base air
1.0