The Grey Zone

The Grey Zone
Molecular
scale
Planetary
scale
Truncation
scale
The Earth System starts at the molecular scale and works up.
Modelers start at the planetary scale and work down.
Conventional Parameterizations
Global circulation
Cloud-scale
&mesoscale
processes
Radiation,
Microphysics,
Turbulence
Parameterized
Parameterize less at high resolution.
Global circulation
Cloud-scale
&mesoscale
processes
Radiation,
Microphysics,
Turbulence
Parameterized
Heating and drying
on coarse and fine meshes
Increasing
resolution
GCM
Parameterizations for lowresolution models are designed
to describe the collective effects
of ensembles of clouds.
CRM
Parameterizations for highresolution models are designed to
describe what happens inside
individual clouds.
Expected values --> Individual realizations
Scale-dependence of heating & drying
1 ∂
1
Q1 − QR = LC −
(ρ w′s′) − ∇ H ⋅ (ρ v H ′s′),
ρ ∂z
ρ
1 ∂
1
Q2 = −LC −
(ρ w′qv′ ) − ∇ H ⋅ (ρ v H ′qv′ ).
ρ ∂z
ρ
These quantities are defined in terms of spatial averages.
As the averaging length becomes smaller:
The vertical transport terms become less important. Later horizontal
averaging does not change this.
The horizontal transport terms become more important locally.
Horizontal averaging kills them, though.
The phase-change terms become dominant.
Typical vertical profiles of the apparent moist static energy source
due to convective activity
Average over a large area
Cloud-scale
Any space/time/ensemble averages of the profiles in the right panel do NOT
give the profile in the left panel.
Slide from Akio Arakawa
In summary:
Three problems with conventional
parameterizations at high resolution:
•
The sample size is too small to
enable a statistical treatment.
•
The “resolved-scale forcing” varies
too quickly for a diagnostic
parameterization.
•
Convective transports become less
important, and microphysics
dominates.
Increasing resolution
makes these problems worse.
•
Smaller grid cells contain
fewer clouds.
•
Smaller weather systems,
resolved by finer grids, have
shorter time scales.
Issues with high resolution
•
The nature of the “sub-grid”
physical processes depends on the
grid size.
•
Parameterizations appropriate for
low-resolution models are not
appropriate for high-resolution
models, and vice-versa.
•
Can we design intelligent physical
parameterizations that can adapt
to a wide range of grid spacings?
“Scale-Aware” Parameterizations?
Equations and code unchanged
as grid spacing varies from 100
km to 1 km
Ensemble of clouds to the
interior of a single cloud
Goal:
Grow individual clouds when/
where the resolution is high.
Parameterize convection when/
where resolution is low.
Continuous scaling.
One set of equations, one code.
Physically based.
Four Ways to Use Cloud-Resolving Models
To Improve Global Models
•
Test parameterizations
•
Suggest ideas
•
Replace parameterizations
•
Become the global model
(b) without background shear. As we can see from these snapshots, these two runs represent
quite different cloud regimes. To see the grid-size dependence of the statistics, we divide the
Use a CRM to test ideas.
original CRM domain (512 km) into sub-domains of same size to repcresent the GCM grid
cells.
velocity
3 km
theheight
surface
Fig. 4 Snapshots of Vertical
the vertical
velocity
w above
at 3 km
simulated (a) withSubdomain
and (b) size,
without background shear, and examples of sub-domains used to see the used
grid-size
to analyze
dependence of the statistics.
dependence on
grid spacing
Sub-domain
Slide from Akio Arakawa
An example of resolution-dependence
atmospheric modeling is quite different from such a case because the spectrum is virtually
continuous due to the existence of mesoscale phenomena.
Fig. 2. Domain- and ensemble-averaged profiles of the "required source" for (a) moist static
energy (divided by c p ) and (b) total airborne water mixing ratio (multiplied by L / c p ) due to
cloudJ.-H.
microphysics
under strong
large-scale
forcing over
land obtained
with different
Jung,
and A. Arakawa,
2004.:
The resolution
dependency
of model
physics:
horizontal grid
sizes
and different time
intervals
of implementing
physics.Sci.,
The 61,
two 88-102.
extreme
Illustrations
from
nonhydrostatic
model
experiments.
J. Atmos.
profiles shown in red and green approximately represent the true and apparent sources,
respectively. Redrawn from Jung and Arakawa (2004).
At high resolution,
we can’t assume that sigma is small
2014
WU AND ARAKAWA
2
FIG. 1. The resolution dependence of the ensemble-averaged convective updraft fraction hsi as functions of height
with each subdomain size d for the (left) shear and (right) nonshear cases. The dashed lines show the 3-km level
analyzed in Part I.
nsemble average as defined above is a quantity that
nds on the resolution (i.e., the subdomain size).
Figure 2 shows the ensemble-averaged total and e
transports of moist static energy given by r*hwhi
Revised May 5, 2010
1:24 AM
Starting
point
Derivation of the Unified Parameterization
Notes by David Randall, based on a presentation by Akio Arakawa
For the case of a top-hat PDF, we can show that
w′ψ ′ = σ (1− σ ) ΔwΔψ ,
(1)
where
( )≡σ(
)c + (1− σ )( ! )
and Δ (
) ≡ ( )c − ( ! ) ,
(2)
σ is the fractional area covered by the updraft, an overbar denotes a domain mean, the subscript
c denotes a cloud value, and a tilde denotes an environmental value.
We expect Δw and Δψ to be independent of σ . In that case, (1) implies that w′ψ ′ is a
quadratic function of σ .
(
Define w′ψ ′
)
E
as the flux required to maintain quasi-equilibrium. The closure assumption
used to determine σ is
How much of the flux is resolved?
Transport d
Slide from Akio Arakawa
Other levels
2092
JOURNAL OF THE ATMOSPHERIC SCIENCES
VOLUME 71
FIG. 2. Vertical distributions of the (a),(b) ensemble-averaged total (r*hwhi) and (c),(d) eddy transports
(r*hw0 h0 i) of moist static energy divided by cp with each subdomain size d for the (left) shear and (right) nonshear
transport, and the difference between the total and eddy transports must be explicitly
No double counting
simulated. We note that the contribution from the eddy transport is larger for the non-shear
case although there is no significant qualitative difference between the two cases in the way
the transports depend on the resolution.
Fig. 6 The diagnosed total transport and eddy transport of moist static energy divided
Red curvesize
is d.
total flux.
by c p at z = 3 km for each sub-domain
Green curve is subgrid flux to be parameterized.
The subgrid part is dominant at low resolution, but unimportant at high resolution.
oicec denotes
to satisfy
the requirement that the eddy transports converge
a cloud value, and a tilde denotes an environmental value.
Notes by David Randall, based on a presentation by Akio Arakawa
w′ψ ′ is a
Δw and Δψ tothis
expect justifying
be independent
of σ . In that
case, (1)on
implies
We areWenow
dependence
based
thethatreasoning
g
quadratic function of σ .
the case of a top-hat
PDF,
we can show thatsatisfied.
ceForrequirement
is then
automatically
Define ( w′ψ ′ ) as the flux required to maintain quasi-equilibrium. The closure assumption
w′ψ ′ = σ (1− σ ) ΔwΔψ ,
E
used to determine σ is
e
w′ψ ′ )
(
σ=
.
ΔwΔψ + ( w′ψ ′ )
) ≡ σ ( )c + (1− σ )( ! ) and Δ ( ) ≡ ( )c − ( ! ) ,
(1
E
(
E
(3)
All of the quantities on the right-hand side of (3) are expected to be independent of σ . Eq. (3) is
(2
guaranteed to give
the fractional area covered by the updraft,
an overbar denotes a domain mean, the subs
0 ≤σ ≤1.
notes a cloud value, and a tilde denotes an environmental value.
(4)
By combining (3) and (1), we obtain
We expect Δw and Δψ to be independent of σ . In that case, (1) implies that w′ψ
2
w
ψ
=
1−
σ
′ ′ (
) ( w′ψ ′ )E .
ratic function of σ .
(5)
as the flux required to maintain quasi-equilibrium. The closure assum
(
)
This shows that the actual flux is typically less than the value required to maintain quasi-
Define w′ψ ′
E
equilibrium.σIn fact,
to determine
is the actual flux goes to zero as σ → 1 . On the other hand, for σ ≪ 1 , Eq. (5)
.
y
it
c
lo
e
v
l
a
ic
rt
e
v
f the
total
transport
is
due
to
grid-scale
vertical
velocity.
le
a
c
-s
d
ri
e to g of Eq. (1)
f the total transport is duTest
Revised May 5, 2010 1:24 AM
Derivation of the Unified Parameterization
Notes by David Randall, based on a presentation by Akio Arakawa
r the case of a top-hat PDF, we can derive
v
)
n
e
re
(g
y
d
d
e
d
n
a
)
d
e
(r
l
ta
to
n
a
e
Theeσ-dependence
of
the
ensemble-mean
total
(red)
and
eddy
(green)
ve
-m
le
b
m
,
w
ψ
≡
w
ψ
−
w
ψ
=
σ
1−
σ
ΔwΔ
ψ
′
′
se
n
(
)
e
e
th
f
o
e
c
n
e
d
Th σ-depen
n
se
re
p
re
e
n
li
e
lu
b
t
h
g
li
e
h
T
.
c
c
s ofof moist
static
energy
divided
by
.
The
light
blue
line
represent
y
b
d
e
id
iv
d
y
p
rg
e
n
p
e
c
ti
a
st
t
is
o
m
rts
“Modified”
meansfrom
that the
data
is averaged
over
updrafts
and
environment
before
t.
se
ta
a
nsport
diagnosed
the
modified
dataset.
d
d
ie
if
d
o
m
e
th
m
o
fr
d
se
o
n
g
ia
d
rt
o
anspcomputing the flux. In other words, a “top-hat” structure is imposed by averaging.
Other levels
JUNE 2014
WU AND ARAKAWA
2093
For d = 8 km
FIG. 3. (a)–(d) As in Fig. 2, but for d = 8 km with each bin of s. (e),(f) Modified eddy vertical transports of moist
the dashed curves shown by the arrows in Fig. 10 give estimates of Δw Δh / 4 . These values
Other resolutions
are also similar between different resolutions.
Fig. 10. The red and light blue lines in (a) are same as those in Fig. 7. Similar plots
for differentthe
resolutions
are shown
in (b)dependence
and (c). The blue for
dashed
lines show
best-fit grid
confirm
quadratic
sigma
three
different
σ (1 − σ ) curves. The arrows show estimated values of Δw Δh / 4 .
These results
Note that large sigma is only possible when the grid spacing is fine.
spacings.
SGS flux ratio
depends more on σ than on d
of height for
rivatives of th
the opposite
show the cor
tal and eddy t
Figs. 5a and
convergence
convergence
mutual comp
in Figs. 5c a
convergence
convergence,
which is presu
scale transpor
indicates. Thi
the horizonta
izontal direct
if either circu
that the total
the nonshear
In summary, i
the subscript c denotes a cloud value, and a tilde denotes an environmental value. We expect
Closure assumption
Δw and Δψ to be independent of σ . In that case, (1) implies that w′ψ ′ is a parabolic function
of σ .
(
Define w′ψ ′
)
E
as the flux required to maintain quasi-equilibrium. The closure assumption
used to determine σ is
w′ψ ′ )
(
σ=
ΔwΔψ + ( w′ψ ′ )
.
E
E
(3)
The quantities on the right-hand side of (3) are expected to be independent of σ . Eq. (3) is
guaranteed to give
0 ≤σ ≤1.
(4)
By combining (3) and (1), we obtain
(
w′ψ ′ = (1− σ ) w′ψ ′
2
)
E
.
(5)
This shows that the actual flux is typically less than the value required to maintain quasiequilibrium. In fact, the actual flux goes to zero as σ → 1 .
Revised May 5, 2010 1:24 AM
A model predicts grid cell means, rather than environmental values, so direct use of (3) is
not possible. Define
δ(
) ≡ ( )c − (
).
(6)
It follows that
δ(
) = (1− σ ) Δ ( ) .
(7)
If Δ (
)
is independent of σ , then δ (
)
decreases (in absolute value) as σ increases. We use
δ wδψ
ΔwΔψ =
2
(1− σ )
(8)
to express the closure assumption, (3), as
w′ψ ′ )
(
σ=
δ wδψ
+ ( w′ψ ′ )
(1− σ )
.
E
2
E
(9)
Eq. (9) can be rearranged to
(1− σ )
2
(
+ w′ψ ′
)
E
(9)
Eq. (9) can be rearranged to
λ (1− σ )
σ=
2 ,
1+ λ (1− σ )
2
(10)
where
w′ψ ′ )
(
λ≡
δ wδψ
E
(11)
can be computed using a plume model with quasi-equilibrium closure, provided that the plume
model determines the in-cloud vertical velocity, e.g., by solving the equation of vertical motion.
In fact, λ is equal to the value that σ would take with quasi-equilibrium closure, although (11)
does not guarantee that λ  1 , or even that λ < 1 .
Eq. (10) must be solved as a cubic equation for σ as a function of λ :
λ (1− σ ) + (1− σ ) − 1 = 0 .
3
(12)
How sigma
depends
on
lamda
σ =λ.
Fig.11 Plots of σ given by (21) and (19) as functions of λ.
(21)
rimarily responsibleIncluding
for the uncertainty,substructure
which could be formulated stochastically.
Fig. 14
Substructures are significant for large sigma, but on the other hand the
eddy flux becomes unimportant for large sigma.
What about microphysical processes?
e see that this conversion
medium to high resolution.
rease is primarily through
ution shown in Fig. 1 for
, as we have done for the
s dependence of this conain size. An example of the
g. 11 for d 5 8 km.
he ensemble average of this
more or less linearly at all
cause s is a measure of the
rafts if the area covered by
It is also a measure of the
tical velocity of individual
ximate linear dependence
in all of the existing models
meterization explicitly uses
determined by the method
re details of the practical
y linear dependence on s
the density-weighted vere-averaged conversion as
s show the vertical average
ociated with the ensemble
line connects zero at s 5
s 5 1. The slight deviation
ed straight line is probably
blimation) of cloud water
which vanishes at s 5 0 and
Condensation
The left and right panels of Fig. 12 show the conversions from cloud water–ice to rain and snow–graupel
identified by labels b and c in Fig. 9, respectively. As in
FIG. 11. (a) Ensemble-averaged net conversion from water vapor
to cloud water–ice for d 5 8 km as a function of z and s. (b) Densityweighted vertical mean of the values shown in (a) with the standard
deviation associated with the ensemble average. The dashed
straight line connects 0 at s 5 0 and the diagnosed value at s 5 1
(1026 kg kg21 s21).
Wu & Arakawa, 2014
UNE
2014
Evaporation and sublimation
WU AND ARAKAWA
2101
FIG. 14. As in Fig. 11, but for (a),(b) evaporation of rain and (c),(d) sublimation of snow–graupel
(1028 kg kg21 s21; note the difference in units from the previous figures).
rom rain to snow–graupel through freezing and from
Wu & Arakawa, 2014
e. Remark on parameterization of uncertainty
JUNE 2014
Horizontal eddy transports?
WU AND ARAKAWA
2095
FIG. 5. Vertical distributions of the (a),(c) ensemble-averaged vertical and (b),(d) horizontal convergence of
the (top) total and (bottom) eddy transports of moist static energy divided by cp for d 5 8 km with each bin of s
for the shear case (1023 K s21). Dashed lines represent negative values.
Wu & Arakawa, 2014
Inspection of (12) shows that σ → 1 as λ → ∞ , and σ → 0 as λ → 0 . The curve defined by the
solution of (12) can be plotted most easily by rearranging (12) to write λ as a function of σ :
Implementation
σ
λ=
3 .
(1− σ )
(13)
Inspection of (13) shows that λ ≥ σ for 0 ≤ σ ≤ 1 .
A possible “quick” implementation strategy:
1. Choose an existing conventional parameterization that includes the equation of vertical
motion for the plumes.
(
2. Using the plume model, calculate w′ψ ′
)
E
and δ w , and δψ . These will be functions of
height. To determine δ w and δψ , we have to choose a particular cloud type.
3. Evaluate λ using (11) and σ using (12). These will be functions of height.
4. Use (5) to “scale back” the convective fluxes.
5. Scale the “non-transport” parts of the tendencies (e.g., the condensation rate) with σ for
the convective part, and 1− σ for the environment.
References
Arakawa, A., J.-H. Jung, and C. M. Wu, 2011: Atmos. Chem. Phys., 11, 3731-3742, doi:
10.5194/acp-11-3731-2011.
Jung, J.-H. and A. Arakawa, 2004.: The resolution dependency of model physics:
Illustrations from nonhydrostatic model experiments. J. Atmos. Sci., 61, 88-102.
Wu, C. M., and A. Arakawa, 2014: A Unified Representation of Deep Moist Convection
in Numerical Modeling of the Atmosphere. Part II . J. Atmos. Sci., 71, 2089-2103.