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?