1. April 12, 2018
Hing Ong
ATM 419/563
Cumulus parameterization: Kain-Fritsch scheme and other mass-flux schemes
This lecture is designed for students who have some background in cumulus parameterization.

2. The cumulus parameterization problem
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3. Why mass-flux schemes?
They can interact with grid-scale microphysics schemes.
For example, parameterized cumuliform clouds can produce resolved stratiform clouds through hydrometeor detrainment.
They can transport any mass-coupled quantity.
For example, momentum, aerosol contents, etc.
Betts-Miller-Janjic scheme is the only non-mass-flux scheme used in WRF, and it can do none of the above.
Cumulus parameterization
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4. Objective Students should become capable of the following:
Name the three elements comprising a mass-flux scheme.
Understand the five assumptions shared by classical mass- flux schemes.
Name the two subgrid-scale processes attributing apparent sources or sinks.
Know the iterative sequence of Kain-Fritsch scheme and how “CUDT” can affect the parameterization.
Understand the limits of classical cumulus schemes.
Cumulus parameterization
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5. The Three Elements A trigger function A one-dimensional cloud model
Determining if the subgrid-scale convection may occur.
A one-dimensional cloud model
Calculating intensive quantities (θ, qv, qc, etc.) and relative strengths between types of mass flux (𝜀, 𝛿, and 𝐽) associated with the subgrid-scale convective drafts.
A closure
Adjusting the absolute strengths of the mass flux.
Cumulus parameterization
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6. The Five Assumptions Ensemble average is equal to grid-box average.
Each convective drafts and environment are horizontally homogeneous.
Convective drafts remain in steady states within a characteristic time of convective adjustment.
Fractional area covered by convective drafts in the grid cell is negligible.
Convective mass flux is locally compensated by environmental mass flux.
Cumulus parameterization
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7. Scheme Formulation 𝜕 𝜌𝜓 𝜕𝑡 +𝛁∙ 𝜌𝜓 𝐕 = 𝜌 𝜓 + SSS 𝜓
𝜕 𝜓 𝜕𝑡 + 𝐕 𝛁∙ 𝜓 = AS 𝜓
Scheme Formulation
𝜓 unity or an intensive quantity
SSS 𝜓 subgrid-scale sources or sinks
AS 𝜓 apparent sources or sinks
𝑉 grid box volume
𝜀 𝑖 entrainment rate
𝛿 𝑖 detrainment rate
𝐽 vertical mass flow rate
𝑖 convective draft
over bar ensemble average ≅ environment (tilde)
𝑡 upper boundaries
𝑏 lower boundaries
Applying the five assumptions:
SSS 𝜓 ≅ 1 𝑉 𝑖 − 𝜀 𝑖 𝜓 + 𝛿 𝑖 𝜓 𝑖 + 𝐽 𝑖𝑡 𝜓 𝑡 − 𝐽 𝑖𝑏 𝜓 𝑏
AS 𝜓 ≅ 𝜓 + 1 𝑉 𝑖 𝛿 𝑖 𝜓 𝑖 − 𝜓 + 𝐽 𝑖 𝜓 𝑡 − 𝜓 𝑏
Cumulus parameterization
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8. Recaps
The five assumptions are applied to formulate classical mass-flux schemes in a form of either the subgrid-scale sources (SSS) or the apparent sources (AS).
The three elements are applied to determine the unknown variables in either SSS or AS formula.
Detrainment or mass compensation attributes AS while entrainment does not.
Kain-Fritsch scheme is formulated in the SSS form, but it is coupled with WRF in the AS form.
Cumulus parameterization
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9. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Every CUDT
If convection is still active then
Leave it unchanged.
Else
Start the trigger function
Determine a parcel as a mixture of the lowest 50 hPa.
Calculate 𝑇 𝐿𝐶𝐿 .
Cumulus parameterization
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10. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Then, calculate 𝛿𝑇 𝐿𝐶𝐿 .
𝛿𝑇 𝐿𝐶𝐿 = 𝑤 −𝑐 𝑧 K m-0.33 s0.33, where 𝑤 𝑡 = 𝑤 𝑡 if CUDT≤𝑑𝑡, otherwise CUDT−𝑑𝑡 CUDT 𝑤 𝑡−CUDT + 𝑑𝑡 CUDT 𝑤 𝑡 and 𝑐 𝑧 = 10 −5 𝑍 𝐿𝐶𝐿 s −1 if 𝑍 𝐿𝐶𝐿 ≤2000 m 0.02 m s −1 if 𝑍 𝐿𝐶𝐿 >2000 m
Note that the trigger function is temporally nonlocal when CUDT > dt.
Then, go on if 𝑇 𝐿𝐶𝐿 + 𝛿𝑇 𝐿𝐶𝐿 ≥ 𝑇 𝐸𝑁𝑉 . Otherwise, go back and test the next 50 hPa if it is within the lowest 300 hPa.
Cumulus parameterization
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11. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Then, compute updraft properties.
𝐽 𝑢𝑡 𝜓 𝑢𝑡 = 𝐽 𝑢𝑏 𝜓 𝑢𝑏 + 𝜀 𝑢 𝜓 − 𝛿 𝑢 𝜓 𝑖 + Ψ 𝑢 (no storage)
Lift a parcel adiabatically from the updraft source layer (USL) to LCL, and assume 𝑤 𝐿𝐶𝐿 = 𝛿𝑇 𝐿𝐶𝐿 𝑇 𝐸𝑁𝑉 𝑔 m s If it exceeds 3 m s-1, set to 3 m s-1.
𝑑 𝑑𝑧 𝑤 =𝑔 𝑇 𝑣𝑢 − 𝑇 𝑣𝑒 𝑇 𝑣𝑒 − 𝑞 ℎ − 𝛿 𝑀 𝑒 𝐽 𝑢 𝑤 2
𝑤∝𝐽
𝜓 𝑡 and 𝜓 𝑏 are upstream quantities.
Cumulus parameterization
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12. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Then, compute updraft properties.
𝐽 𝑢𝑡 𝜓 𝑢𝑡 = 𝐽 𝑢𝑏 𝜓 𝑢𝑏 + 𝜀 𝑢 𝜓 − 𝛿 𝑢 𝜓 𝑖 + Ψ 𝑢 (no storage)
𝛿 𝑀 𝑒 = 𝐽 𝑢LCL −0.03𝛿𝑝 𝑅 m Pa-1.
𝑅= m if 𝑤 −𝑐 𝑧 <0 m s − m if 𝑤 −𝑐 𝑧 >0.1 m s − 𝑤 −𝑐 𝑧 0.1 m s −1 m else
𝜀 𝑢 and 𝛿 𝑢 are diagnosed with a buoyancy sorting algorithm. Set 𝜀 𝑢 ≥0.5𝛿 𝑀 𝑒
Ψ 𝑢 is diagnosed with a single moment five species (qv, qc, qr, qi, and qs) microphysics scheme. Accordingly, 𝜓 𝑖 is updated.
𝜓 𝑡 and 𝜓 𝑏 are upstream quantities.
Cumulus parameterization
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13. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Then, compute updraft properties.
Cloud top (TOP) is where upward motion stops. Detrain all updraft mass above level of equilibrium temperature (LET).
𝐷 𝑚𝑖𝑛 = m if 𝑇 𝐿𝐶𝐿 >20 ℃ 2000 m if 𝑇 𝐿𝐶𝐿 <0 ℃ 2000 m+100 𝑇 𝐿𝐶𝐿 m ℃ −1 else
If updraft depth > 𝐷 𝑚𝑖𝑛 , deep convection is activated, and if not, go back and test the next 50 hPa if it is within the lowest 300 hPa.
If all the USLs in the lowest 300 hPa do not meet the 𝐷 𝑚𝑖𝑛 criterium, shallow convection is activated.
TOP
LET
Cumulus parameterization
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14. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
If convection is activated, …
Compute convective timescale (TIMEC)
For deep convection, TIMEC= 𝑑𝑥 𝑉 , where 𝑉 =0.5 𝑉 𝐿𝐶𝐿 + 𝑉 0.5 𝑝 𝑠 If it is beyond [30,60] mn, set to 30 or 60 mn.
For shallow convection, TIMEC=40 mn.
Setup a countdown timer so that KF will not be called again until min TIMEC, 𝑑𝑥 𝑉 .
Note that the CAPE, intensive quantities, and relative strengths between types of mass flux in the updraft has been determined.
TOP
LET
Cumulus parameterization
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15. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Then, compute downdraft properties.
Downdraft source layer (DSL) lies 150 hPa above USL.
Intensity quantities are constrained by 𝑅𝐻 𝑑 = 100 % above cloud base ( 𝑍 𝐵𝐴𝑆𝐸 ) 100 %− 20 % 1000 m 𝑍 𝐵𝐴𝑆𝐸 −𝑍
Mass flux is constrained by − 𝐽 𝑑USL 𝐽 𝑢USL =2 1− 𝑅𝐻
Precipitation is hydrometeors that are generated minus detrained and evaporated.
DSL
USL
Cumulus parameterization
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16. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
Then, compute mass compensation.
− 𝜀 𝑖 𝜓 + 𝛿 𝑖 𝜓 𝑖 + 𝐽 𝑖𝑡 𝜓 𝑡 − 𝐽 𝑖𝑏 𝜓 𝑏 =0
Retrospect
SSS 𝜓 ≅ 1 𝑉 𝑖 − 𝜀 𝑖 𝜓 + 𝛿 𝑖 𝜓 𝑖 + 𝐽 𝑖𝑡 𝜓 𝑡 − 𝐽 𝑖𝑏 𝜓 𝑏 .
All the unknowns on RHS have been determined, aren’t they?
𝜓 𝑡 and 𝜓 𝑏 are upstream quantities.
Cumulus parameterization
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17. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
The unclosed mass flux does not guarantee removal of at least 90 % of CAPE within TIMEC!
Iterate for closure.
Use the unclosed SSS 𝜓 to extrapolate 𝜓 after TIMEC, and compute adjusted CAPE.
If at least 90 % of CAPE is removed, done.
If not, multiply all the types of mass flux and precipitation by a constant and reiterate.
𝜓 𝑡 and 𝜓 𝑏 are upstream quantities.
Cumulus parameterization
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18. Iterative sequence of Kain-Fritsch scheme Trigger Updraft Downdraft Compensation Closure
WRF physics coupler does not support SSS 𝜓 feedback but does AS 𝜓 feedback.
AS 𝜓 = 𝜓 𝑓 − 𝜓 𝑖 𝑇𝐼𝑀𝐸𝐶
𝜓 𝑡 and 𝜓 𝑏 are upstream quantities.
Cumulus parameterization
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19. Recaps
The aim is to diagnose the unknowns on RHS of SSS 𝜓 ≅ 1 𝑉 𝑖 − 𝜀 𝑖 𝜓 + 𝛿 𝑖 𝜓 𝑖 + 𝐽 𝑖𝑡 𝜓 𝑡 − 𝐽 𝑖𝑏 𝜓 𝑏
CUDT controls the weight of the latest w in the trigger function, which is temporally local when CUDT ≤ dt.
𝑑 𝑑𝑧 𝑤 =𝑔 𝑇 𝑣𝑢 − 𝑇 𝑣𝑒 𝑇 𝑣𝑒 − 𝑞 ℎ − 𝛿 𝑀 𝑒 𝐽 𝑢 𝑤 2 is used to integrate one- dimensional cloud models.
CAPE closure is used.
In WRF, KF scheme adjusts θ, qv, qc, qi, qr, qs, but not u, v
Details are complicated and can vary from one version to another, this introduction is based on the code in WRFV3.7.1.
Cumulus parameterization
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20. Limits of classical cumulus schemes.
With the grid spacings shrinking from 27 km to 9 km, (1) convective draft fraction may become considerable, (2) compensating mass flux may spread to adjacent grid.
Options dealing with issue (1) are as follows:
Grell–Freitas Ensemble Scheme (03)
Multi–scale Kain–Fritsch Scheme (11)
Options dealing with issue (2) are as follows:
Grell 3D Ensemble Scheme (5) with artificial nonlocal compensation
Hybrid Mass Flux Kain-Fritsch Scheme with dynamic compensation
None of the options deal with both issues.
Cumulus parameterization
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21. Conclusion 1. Name the three elements comprising a mass-flux scheme.
A trigger function, a one-dimensional cloud model, and a closure
2. Understand the five assumptions shared by classical mass-flux schemes.
Average, top hat, steady, fraction, compensation
3. Name the two subgrid-scale processes attributing apparent sources or sinks.
Mass compensation and detrainment
4. Know the iterative sequence of Kain-Fritsch scheme and how “CUDT” can affect the parameterization.
the trigger function is temporally nonlocal when CUDT > dt
5. Understand the limits of classical cumulus schemes.
Convective draft fraction and local compensation
Cumulus parameterization
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22. Questions & Discussion
Cumulus parameterization
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23. Discussions: future development Eliminating assumptions for scheme formulation one by one, a hierarchy with six levels could be built.
To eliminate assumption (1) and build a HYMACS
These are NOT needed:
More sophisticated assumptions
Extra computer resources
These are ALL needed:
A cumulus scheme diagnosing convective mass flux
A fully compressible nonhydrostatic model
Cumulus parameterization
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