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Quantifying the limits of convection parameterization
Modeling Across Scales
Global circulation Cloud-scale&mesoscaleprocesses
Radiation,Microphysics,Turbulence
Parameterized
Scale Separation
“Consider a horizontal area … large enough to contain an ensemble of cumulus clouds, but small enough to cover only a fraction of a large-scale disturbance. The existence of such an area is one of the basic assumptions of this paper.”
-- AS 74
A parameterization determines the “expected” collective effects of many clouds over a large area.
One of the issues is that the sample size is not very large.
A summer afternoon in Colorado
The space scales are not sufficiently separated.
Limiting Cases
Quasi-EquilibriumConvection
Non-DeterministicConvection
Deterministic butNon-Equilibrium
Convection
Higher resolution
• Gradualist approach: dx gradually decreases, without changing parameterizations
• OK for NWP, not so good for climate
• No qualitative change until dx~5 km
• Aggressive approach: dx~5 km right now
• Currently too expensive for climate
• Super-parameterization as a compromise
Does increased resolution improve the results?
Buizza 2010:
“...although further increases in resolution are expected to improve the forecast skill in the short and medium forecast range, simple resolution increases without model improvements would bring only very limited improvements in the long forecast range.”
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“Ratio” refers to the ratio of forecast error to its saturation value. Black symbols for the T799 “perfect model,” grey symbols for real forecasts.
Error versus resolutionwithout changing the parameterizations
Error
Horizontal Grid Spacing
200 km 20 km 2 km
NWP
Climate
0
Global circulation Cloud-scale&mesoscaleprocesses
Radiation,Microphysics,Turbulence
Parameterized
Parameterize less.
Parameterizations for low-resolution models are designed to describe the collective effects of ensembles of clouds.
Parameterizations for high-resolution models are designed to describe what happens inside individual clouds.
Increasingresolution
GCM CRM
Parameterize Different.
Expected values --> Individual realizations
Todd Jones
Ensembles of CRM runsAn extension of
Xu, Kuan-Man, Akio Arakawa, Steven K. Krueger, 1992: The Macroscopic Behavior of Cumulus Ensembles Simulated by a Cumulus Ensemble Model. J. Atmos. Sci., 49, 2402-2420.
Three-dimensional model (important for sample size)
Sensitivity to forcing period
Sensitivity to domain size
Extended how?
Constant SST
Prescribed radiation
256 km square domain
~18 km depth
2-km horizontal grid spacing
Large-scale forcing by advective cooling and moistening
Some wind shear
Domain-averaged wind relaxed to “obs”
Prescribed U-Wind
-15 -10 -5 0 5[m s-1]
0
5
10
15
20
Hei
ght [
km]
Prescribed Large-Scale Forcing
-40 -20 0 20 40[k day-1]
0
5
10
15
20
Hei
ght [
km]
AdvectiveCooling
AdvectiveMoistening
Experiment Design
Series of constant forcing simulations
Series of periodically forced simulations
Periods range from 120 hours down to 2 hours
15 cycles each
Subdomains:
Fraction Whole Quarter 16th 64th 256th
Width, km 256 128 64 32 16
Experiment Design
3D Numerical Simulation
Constant Forcing
CF1Mean: 0.23 Std: 0.05
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF2Mean: 0.47 Std: 0.05
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0D
om. A
vg. P
rec.
[mm
hr-1
]
CF3Mean: 0.73 Std: 0.08
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF4Mean: 0.97 Std: 0.09
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF5Mean: 1.23 Std: 0.11
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF6Mean: 1.48 Std: 0.13
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF7Mean: 1.74 Std: 0.14
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF8Mean: 1.98 Std: 0.18
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CF9Mean: 2.23 Std: 0.19
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
CFxMean: 2.48 Std: 0.17
0 50 100 150Hours
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Dom
. Avg
. Pre
c. [m
m h
r-1]
Dependence on Period
F02 - Full Domain
0.0 0.5 1.0 1.5 2.0Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]Forcing leads Precip by:
60.0 minutes (50.00 % of the forcing period)
F08 - Full Domain
0 2 4 6 8Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 80.0 minutes (16.67 % of the forcing period)
F16 - Full Domain
0 5 10 15Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 80.0 minutes (8.33 % of the forcing period)
F30 - Full Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 80.0 minutes (4.44 % of the forcing period)
F60 - Full Domain
0 10 20 30 40 50 60Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]Forcing leads Precip by:
80.0 minutes (2.22 % of the forcing period)
120 - Full Domain
0 20 40 60 80 100 120Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 70.0 minutes (0.97 % of the forcing period)
Dependence on Domain SizeF30 - Full Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 80.0 minutes (4.44 % of the forcing period)
F30 - Quarter Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 82.5 minutes (4.58 % of the forcing period)
F30 - 16th Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 86.9 minutes (4.83 % of the forcing period)
F30 - 64th Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 113.8 minutes (6.32 % of the forcing period)
F30 - 256th Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 149.0 minutes (8.28 % of the forcing period)
0
2
4
6
0
2
4
6
256
x 256
km D
omain
0
2
4
6
-2
0
2
4
6
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
0 20 40 60 80 100 120Time [hr]
-2
0
2
4
6
8
0.0 0.5 1.0 1.5 2.0Time [hr]
-2
0
2
4
6
16 x
16 km
Dom
ain
-2
0
2
4
6
-2
0
2
4
6
128
x 128
km D
omain
2 hr Forcing Period 30 hr Forcing Period 120 hr Forcing Period
Slower ForcingSlower ForcingSm
aller
Dom
ainSm
aller
Dom
ain
Figure 2. Composite over 15 realizations for surface precipitation rate (mm h ) for the noted simulations. The black curve is the composite mean. The blue hash-filled region bounded by the dot-dashed lines denotes the ±1 standard deviation across the realizations. The red curve references the timing and relative magnitude (0%–100%) of the large-scale forcing; no specific values are implied. Note: not all precipitation axes are on the same scale.
!1
A perfect parameterization
The ensemble mean represents a
perfect deterministic non-equilibrium parmeterization.
It is, of course, a perfect parameterization of the CRM, not of the real world.
Standard Deviation / MeanWhat is the best we can do?
Subdomain Side Length (km)Subdomain Side Length (km)Subdomain Side Length (km)Subdomain Side Length (km)Subdomain Side Length (km)
Period (hr)
256 128 64 32 16
15 0.125 0.698 1.205 1.745 2.215
30 0.113 0.656 1.177 1.693 2.185
60 0.116 0.664 1.222 1.760 2.227
120 0.147 0.707 1.282 1.815 2.257
Even with a large domain and slowly varying forcing, a perfect parameterization will routinely produce ~10% errors, due to inadequate sample size.
A stochastic parameterization should be able to explain these numbers.
Stochastic Parameterization
A deterministic parameterization simulates ensemble means.
A stochastic parameterization simulates individual realizations.
Is “stochastic” equivalent to“It doesn’t do the same thing every time?”
F30 - Quarter Domain
0 5 10 15 20 25 30Time [hr]
-2
0
2
4
6
Surfa
ce P
reci
pita
tion
[mm
hr-1
]
Forcing leads Precip by: 82.5 minutes (4.58 % of the forcing period)
If so, self-sustaining, long-lived mesoscale structures may contribute to “stochastic” behavior.
Can we identify stochastic behaviorin observations of convection?
Does temporal variability reflect the spread of the ensemble?
Precipitation simulated by the six integrations SHIFT1 to SHIFT6 with (a) ensemble mean of the accumulated precipitation (mm; from 0000 UTC 25 Sep 1999 to 0600 UTC 26 Sep 1999), and precipitation rates (mm h−1) for individual ensemble members averaged over (b) the Basel and (c) the Lago Maggiore subdomain.
Cathy Hohenegger and Christoph Schär, 2007
Super-ParameterizationAn embedded CRM (super-parameterization) is a stochastic parameterization. It has a memory, and it exhibits sensitive dependence on its past history.
Because of its two-dimensionality, the MMF probably exaggerates the stochastic component of convection.
A super-parameterization can simulate the lag between the forcing and the convective response.
We don’t know whether or to what extent the successes of super-parameterization are due to these attributes.
Concluding Thoughts
• Because a super-parameterization has built-in memory and exhibits sensitive dependence on its past history, it can represent non-equilibrium, non-deterministic convection.
• “Expected values” give 10% errors even with wide (256 km) grid cells. Non-deterministic behavior becomes very strong with just slightly finer resolution.
• Do we need “stochastic backscatter” in global cloud-resolving models, or in super-parameterizations?