1. Cloud-topped boundary layer response time scales in MLM and LES
Christopher R. Jones, Christopher S. Bretherton, Peter N. Blossey
University of Washington, Seattle USA
Classic MLM arguments (e.g., Schubert et al, 1979) suggest two characteristic time scales for a well-mixed CTBL evolution:
Inversion-deepening: 𝜏 inv = 𝐷 −1 ~ a few days.
Thermodynamic: 𝜏 th = 𝑧 𝑖 𝑤 𝑒 + 𝑤 𝑠 ~1 day.
We use a MLM and LES to show that, in addition to these time scales, a separate fast time scale exists, which is associated with entrainment-cloud thickness feedbacks:
Entrainment-LWP: 𝜏 𝑒 ~ a few hours.
Fast cloud base response in MLM and LES
Fast time scale due to entrainment feedbacks
Feedbacks included in linearization
𝝀 𝟏 (da y −1 )
Fast
𝝀 𝟐 (da y −1 )
Intermediate
𝝀 𝟑 da y −1
Slow
No R,P,E
-1.19
-0.32
Incl. P; No R,E
Incl. P,R; No E
-0.68
Incl. P,R,E (full MLM)
-5.72
-1.17
-0.11
𝒘 𝒆 = 𝑐 1 𝟏− 𝒛 𝒃 𝒛 𝒊 + 𝑐 2
-4.98
-0.14
LES
MLM
Eigenvalues of the linearization L with or without including the direct impact of radiation (R), precipitation (P) or entrainment feedbacks (E).
Idealized Test Case for MLM and LES
Control (CTL): GCSS DYCOMS-II RF01 case (Stevens et al. 2005, MWR).
Non-precipitating, well-mixed nocturnal Sc.
Entrainment rate, liquid water path, turbulence well-observed.
Initialized with cloud-topped mixed layer, zi = 840 m, N = 150 cm-3.
Linear 𝜃, moisture profiles above cloud layer, initial Δ𝜃 ℓ = 9 K.
D = 3.75x10-6 s-1, SST = K.
Constant moisture profile 𝑞 𝑡 + =1.5g k g −1 above cloud layer.
Simplified dependence of radiative cooling on cloud structure.
Perturbation (𝚫 𝐪 𝐭 + ):
Moist layer 𝑞 𝑡 + =2.25g k g −1 above the BL that subsides into cloud top after approximately 5 hours.
Models
LES: SAM6.7, Δ𝑥=Δ𝑦=25 m, Δ𝑧=5 m up to 1500 m, Lx= Ly = 6.4 km, periodic BCs (More details: Uchida et al. 2010, ACP).
MLM: LES-tuned entrainment and drizzle (More details: Caldwell and Bretherton 2009, J. Climate; Uchida et al. 2010, ACP).
Extracting time scales from MLM linearization
Original system of equations:
𝑑𝒚 𝑑𝑡 =𝒇(𝒚;𝜶)
Consider a perturbation at 𝑡=0 to either the forcing parameters (𝜹𝜶) or the state (𝜹 𝒚 𝟎 ), and linearize:
𝑑 𝑑𝑡 𝜹𝒚≈𝑳 𝒚 𝟎 ; 𝜶 𝟎 𝜹𝒚+𝜹𝑭
where
𝐿 𝑖𝑗 = 𝜕 𝑓 𝑖 𝜕 𝑦 𝑗 𝒚 𝟎 , 𝜹𝑭= 𝑘 𝜕𝒇 𝜕 𝛼 𝑘 𝛿 𝛼 𝑘
Solution:
𝜹𝒚 𝑡 = 𝑒 𝑳𝑡 𝜹 𝒚 𝟎 +( 𝑒 𝑳𝑡 −𝑰) 𝑳 −𝟏 𝜹𝑭
In terms of eigenvalues ( 𝜆 𝑖 ) and eigenvectors ( 𝒗 𝒋 ) of L:
𝜹𝒚(𝑡)= 𝑗 𝑎 𝑗 exp 𝜆 𝑗 𝑡 + 𝑏 𝑗 𝒗 𝒋
Time scales:
𝜏 𝑖 =− 𝜆 𝑖 −1
Important points:
Time scales determined by eigenvalues of linearization.
Eigenvectors indicate which perturbation vectors are associated with a given time scale.
Response to a given perturbation depends on how that perturbation projects onto each eigenvector.
Linearization works remarkably well for predicting initial evolution of perturbation.
Physical interpretation of fast entrainment response
The entrainment closure can be written as:
𝑤 𝑒 = 2.5𝐴 Δ𝑏 1− 𝑧 𝑏 𝑧 𝑖 𝑤 ′ 𝑏 ′ 𝑐 + 𝑧 𝑏 𝑧 𝑖 𝑤 ′ 𝑏 ′ 𝑠 = 𝑐 1 𝑡 1− 𝑧 𝑏 𝑧 𝑖 + 𝑐 2 (𝑡)
The terms 𝑐 1 (𝑡) and 𝑐 2 (𝑡) are approximately constant over short times (see figure below), while cloud base can respond quickly.
𝑤 𝑒 ≈ 𝑐 1 1− 𝑧 𝑏 𝑧 𝑖 + 𝑐 2
fast
slow
CTL (LES)
Δq t + (LES)
𝑞 𝑡 + =1.5 g k g −1
𝑞 𝑡 + =2.25 g k g −1
The thermodynamic variables can be combined in the MLM to generate:
𝑑 𝑧 𝑏 𝑑𝑡 = 1 𝑧 𝑖 𝑤 𝑒 𝑧 𝑏 + − 𝑧 𝑏 − 𝑤 𝑠 𝑧 𝑏 −Δ 𝐹 𝑏 𝐵𝐿
Using the approximation 𝑤 𝑒 ≈ 𝑐 1 1− 𝑧 𝑏 𝑧 𝑖 + 𝑐 2 and rearranging yields:
𝑑 𝑧 𝑏 𝑑𝑡 =𝑁 𝑧 𝑖 , 𝑧 𝑏 ,ℎ − 𝑤 𝑒 + 𝑤 𝑠 𝑧 𝑖 𝑐 1 𝑧 𝑖 2 𝑧 𝑏 + 𝑧 𝑏
𝜆 fast ≈ 𝜆 th 𝜆 𝑒 =−4.8 da y −1
𝑑ℎ 𝑑𝑡 = 1 𝑧 𝑖 𝑤 𝑒 Δ i ℎ+ 𝑤 𝑠 ℎ 0 ∗ −ℎ − Δ 𝐹 𝑅 𝐵𝐿 𝜌 0
𝑑 𝑞 𝑡 𝑑𝑡 = 1 𝑧 𝑖 𝑤 𝑒 Δ 𝑖 𝑞 𝑡 + 𝑤 𝑠 𝑞 0 ∗ − 𝑞 𝑡 + 𝐹 𝑃 0
𝑑 𝑧 𝑖 𝑑𝑡 = 𝑤 𝑒 −𝐷 𝑧 𝑖
𝑤 𝑒 = 𝐴 𝑤 ∗3 𝑧 𝑖 Δ i 𝑏 (Entrainment Closure)
CTL
Δ 𝑞 𝑡 +
Linearized
Physically for this perturbation, increasing 𝑞 𝑡 + implies:
cloud base initially drops (cloud thickens rapidly, generates more turbulence)
𝑤 𝑒 increases in response (increases entrainment warming, drying)
opposes further LWP increase
Evolution of perturbation eigenvectors in MLM
Conclusions
MLM shows three well-separated time scales:
Slow BL-deepening scale.
Intermediate thermodynamic adjustment scale.
Fast entrainment-LWP adjustment scale.
Fast scale exhibited in both LES and MLM.
Magnitude of fast scale in MLM dependent on entrainment closure.
LES time scale supports use of cloud-thickness sensitive entrainment closure in MLM.
MLM
Variables:
𝑧 𝑖 : Inversion height
𝑤 𝑒 : Entrainment rate
𝑧 𝑏 : Cloud base
D: Large-scale divergence
ℎ: Moist static energy
𝑞 𝑡 : Total water mixing ratio
𝑤 𝑠 : Surface exchange velocity
Δ 𝑖 𝜃=𝜃 𝑧 𝑖 + −𝜃
Thanks: Marat Khairoutdinov for SAM
Funding: NOAA MAPP CPT
Fast component dominates initial cloud thickening response through rapid cloud base adjustment