Transcript cjones_pan-GASS_poster_cloth.pptx

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: 𝜏inv = 𝐷 −1 ~ a few days.
𝑧𝑖
• Thermodynamic: 𝜏th =
~1 day.
Fast cloud base response in MLM and LES
Fast time scale due to entrainment feedbacks
𝑤𝑒 +𝑤𝑠
• 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: 𝜏𝑒 ~ a few hours.
LES
MLM
𝝀𝟐 (day −1 )
Intermediate
𝝀𝟑 day −1
Slow
No R,P,E
-1.19
-1.19
-0.32
Incl. P; No R,E
-1.19
-1.19
-0.32
Incl. P,R; No E
-1.19
-0.68
-0.32
Incl. P,R,E (full MLM)
-5.72
-1.17
-0.11
-4.98
-1.19
-0.14
𝒘𝒆 = 𝑐1
𝒛𝒃
𝟏−
+ 𝑐2
𝒛𝒊
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 Δ𝜃ℓ = 9 K.
• D = 3.75x10-6 s-1, SST = 292.5 K.
• Constant moisture profile 𝑞𝑡+ = 1.5g kg −1 above cloud layer.
• Simplified dependence of radiative cooling on cloud structure.
Perturbation (𝚫𝐪+
𝐭 ):
• Moist layer 𝑞𝑡+ = 2.25g kg −1 above the BL that subsides into cloud
top after approximately 5 hours.
Extracting time scales from MLM linearization
Original system of equations:
𝑑𝒚
= 𝒇(𝒚; 𝜶)
𝑑𝑡
Consider a perturbation at 𝑡 = 0 to either the forcing parameters (𝜹𝜶) or the state
(𝜹𝒚𝟎 ), and linearize:
𝑑
𝜹𝒚 ≈ 𝑳 𝒚𝟎 ; 𝜶𝟎 𝜹𝒚 + 𝜹𝑭
𝑑𝑡
where
𝜕𝑓𝑖
𝜕𝒇
𝐿𝑖𝑗 =
, 𝜹𝑭 =
𝛿𝛼𝑘
𝜕𝑦𝑗
𝜕𝛼𝑘
𝒚𝟎
Models
Physical interpretation of fast entrainment response
The entrainment closure can be written as:
2.5𝐴
𝑧𝑏
𝑧
𝑧
𝑏
𝑏
′
′
′
′
𝑤𝑒 =
1−
𝑤𝑏 𝑐+
𝑤 𝑏 𝑠 = 𝑐1 𝑡 1 −
+ 𝑐2 (𝑡)
Δ𝑏
𝑧𝑖
𝑧𝑖
𝑧𝑖
The terms 𝑐1 (𝑡) and 𝑐2 (𝑡) are approximately constant over short times (see figure
below), while cloud base can respond quickly.
𝑤𝑒 ≈ 𝑐1
𝑧𝑏
1−
+ 𝑐2
𝑧𝑖
𝑘
Solution:
𝜹𝒚 𝑡 = 𝑒 𝑳𝑡 𝜹𝒚𝟎 + (𝑒 𝑳𝑡 − 𝑰)𝑳−𝟏 𝜹𝑭
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).
In terms of eigenvalues (𝜆𝑖 ) and eigenvectors (𝒗𝒋 ) of L:
𝑞𝑡+
𝑞𝑡+ = 1.5 g kg −1
𝝀𝟏 (day −1 )
Fast
Feedbacks included in
linearization
= 2.25 g kg
𝜹𝒚(𝑡) =
−1
slow
𝑎𝑗 exp 𝜆𝑗 𝑡 + 𝑏𝑗 𝒗𝒋
𝑗
Time scales:
𝜏𝑖 = −𝜆−1
𝑖
Δq+
t (LES)
CTL (LES)
MLM: LES-tuned entrainment and drizzle (More details: Caldwell and
Bretherton 2009, J. Climate; Uchida et al. 2010, ACP).
𝐵𝐿
𝑑ℎ 1
Δ𝐹
𝑅
=
𝑤𝑒 Δi ℎ + 𝑤𝑠 ℎ0∗ − ℎ −
𝑑𝑡 𝑧𝑖
𝜌0
𝑑𝑞𝑡 1
=
𝑤𝑒 Δ𝑖 𝑞𝑡 + 𝑤𝑠 𝑞0∗ − 𝑞𝑡 + 𝐹𝑃 0
𝑑𝑡
𝑧𝑖
𝑑𝑧𝑖
= 𝑤𝑒 − 𝐷𝑧𝑖
𝑑𝑡
𝑤𝑒 =
𝐴𝑤 ∗3
𝑧𝑖 Δi 𝑏
CTL
Δ𝑞𝑡+
Linearized
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.
Evolution of perturbation eigenvectors in MLM
(Entrainment Closure)
•
•
•
•
𝑤𝑒 : Entrainment rate
D: Large-scale divergence
𝑞𝑡 : Total water mixing ratio
Δ𝑖 𝜃 = 𝜃 𝑧𝑖+ − 𝜃
Thanks: Marat Khairoutdinov for SAM
Funding: NOAA MAPP CPT
𝑧𝑖
𝑑𝑧𝑏
𝑤𝑒 + 𝑤𝑠
= 𝑁 𝑧𝑖 , 𝑧𝑏 , ℎ −
+
𝑑𝑡
𝑧𝑖
𝜆fast ≈ 𝜆th
+
𝑐1 +
2 𝑧𝑏 𝑧𝑏
𝑧𝑖
𝜆𝑒 = −4.8 day
−1
Physically for this perturbation, increasing 𝑞𝑡+ implies:
 cloud base initially drops (cloud thickens rapidly, generates more turbulence)
 𝑤𝑒 increases in response (increases entrainment warming, drying)
 opposes further LWP increase
Conclusions
MLM
Variables:
• 𝑧𝑖 : Inversion height
• 𝑧𝑏 : Cloud base
• ℎ: Moist static energy
• 𝑤𝑠 : Surface exchange velocity
The thermodynamic variables can be combined in the MLM to generate:
𝑑𝑧𝑏 1
+
𝐵𝐿
=
𝑤𝑒 𝑧𝑏 − 𝑧𝑏 − 𝑤𝑠 𝑧𝑏 − Δ𝐹𝑏
𝑑𝑡
𝑧𝑖
𝑧𝑏
Using the approximation 𝑤𝑒 ≈ 𝑐1 1 −
+ 𝑐2 and rearranging yields:
Fast component dominates initial cloud thickening response through rapid cloud base adjustment
• 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.