Multiple Equilibria in a Cloud Resolving Model:
Using the Weak Temperature Gradient Approximation to
Understand Self-Aggregation 1
Sharon Sessions, Satomi Sugaya, David Raymond, Adam Sobel
New Mexico Tech
SWAP 2011
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1
This work is supported by the National Science Foundation
Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
1 / 21
Self-Aggregation
Bretherton et al (2005)
precipitation
WVP =
1
g
R
rt dp
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Day 1
Sessions (NMT)
Day 50
Multiple Equilibria in a CRM
May 2011
2 / 21
Possible insight from limited domain WTG simulations
Bretherton et al (2005)
Day 1
Day 50
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
3 / 21
Weak Temperature Gradients in the Tropics
Charney 1963 scaling analysis
Extratropics (small Rossby number)
δθ ∼
Fr
θ
Ro
Tropics (Rossby number not small)
δθ ∼ Fr θ
Froude number: Fr = U 2 /gH ∼ 10−3 , measure of stratification
Rossby number: Ro = U/fL ∼ 10−5 /f , characterizes rotation
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
4 / 21
Weak Temperature Gradients in the Tropics
Charney 1963 scaling analysis
Extratropics (small Rossby number)
δθ ∼
Fr
θ
Ro
Tropics (Rossby number not small)
δθ ∼ Fr θ
Froude number: Fr = U 2 /gH ∼ 10−3 , measure of stratification
Rossby number: Ro = U/fL ∼ 10−5 /f , characterizes rotation
In the tropics, gravity waves rapidly redistribute buoyancy anomalies
Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
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4 / 21
Weak Temperature Gradient (WTG) Approximation
Sobel & Bretherton 2000
Single column model (SCM) representing one grid cell in a global
circulation model
Convection is not explicitly resolved
Temperature (T) and moisture (q) governed by:
and S =
T ∂θ
θ ∂p
∂T
∂t
T
+ uh · ∇T + ωS = Qtotal
∂q
∂t
q
+ uh · ∇q + ω ∂q
∂p = Qtotal
is the static stability
Sessions (NMT)
Multiple Equilibria in a CRM
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May 2011
5 / 21
Weak Temperature Gradient (WTG) Approximation
Sobel & Bretherton 2000
Single column model (SCM) representing one grid cell in a global
circulation model
Convection is not explicitly resolved
Temperature (T) and moisture (q) governed by:
∂T
∂t
T
+ uh · ∇T + ωS = Qtotal
∂q
∂t
q
+ uh · ∇q + ω ∂q
∂p = Qtotal
Steady state WTG approximation
and S =
T ∂θ
θ ∂p
is the static stability
Sessions (NMT)
Multiple Equilibria in a CRM
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May 2011
5 / 21
Weak Temperature Gradient (WTG) Approximation
Sobel & Bretherton 2000
Parameterize the large scale circulations
T
ωS = Qtotal
Conventional parameterizations: specify ω, diagnose T
WTG parameterization: specify T , diagnose ω
and S =
T ∂θ
θ ∂p
is the static stability
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
6 / 21
WTG Implementation in a Cloud Resolving Model (CRM)
Raymond & Zeng 2005
∂θ
∂t
∂rt
∂t
+ ∇ · (ρvθ + Tθ ) ≡ ρ(Sθ − Eθ )
+ ∇ · (ρvrt + Tr ) ≡ ρ(Sr − Er )
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
7 / 21
WTG Implementation in a Cloud Resolving Model (CRM)
Raymond & Zeng 2005
∂θ
∂t
∂rt
∂t
+ ∇ · (ρvθ + Tθ ) ≡ ρ(Sθ − Eθ )
+ ∇ · (ρvrt + Tr ) ≡ ρ(Sr − Er )
Enforcing WTG generates wwtg
θ − θ0 (z)
∂θ
Eθ = wwtg
= sin(πz/h)
∂z
tθ
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
7 / 21
Specifying Convective Environment
Reference profiles of potential temperature, θ0 , and moisture, r0 , are
generated by running model to radiative-convective equilibrium (RCE)
15000
height (m)
12000
WTG: vertical θ profile
approximately horizontally
homogeneous
Average moisture
in immediate
convective vicinity
9000
6000
3000
0
300
320
θ (K)
340
0
0.005
0.01
rt (g/g)
0.015
0.02
Moisture advected from environment via mass continuity
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
8 / 21
WTG Implementation in our CRM
Raymond & Zeng 2005
∂θ
∂t
∂rt
∂t
+ ∇ · (ρvθ + Tθ ) ≡ ρ(Sθ − Eθ )
+ ∇ · (ρvrt + Tr ) ≡ ρ(Sr − Er )
wwtg vertically advects moisture and brings in environmental moisture via
mass continuity:
Er =
∂r t
(r t − rx (z)) ∂(ρwwtg )
+ wwtg
ρ
∂z
∂z
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
9 / 21
Environmental moisture in WTG simulations
height
Er =
∂r t
(r t − rx ) ∂(ρwWTG )
+ wWTG
ρ
∂z
∂z
rx =
r0
rt
∂(ρwwtg )/∂z > 0
∂(ρwwtg )/∂z < 0
mass flux ( ρ w wtg )
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
10 / 21
Precipitation in WTG simulations
40
∂θ
Eθ = wwtg ∂z
[θ−θ0 (z)]
tθ
θ0 (z) is RCE profile
for vy = 5 m/s on 2D
200 km domain
rain rate (mm/day)
= sin(πz/h)
tθ=30 sec (strict WTG)
tθ=31 min
tθ=1.85 hours
tθ=∞ (RCE)
35
30
25
20
15
10
5
tθ is a measure of the
enforcement of WTG
0
5
10
15
horizontal windspeed (m/s)
20
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
11 / 21
Multiple Equilibria in Precipitation
Sobel, Bellon & Bacmeister (2007)
Single column model (SCM) with WTG approximation
Parallel experiments: identical forcing
initially moist troposphere
initially dry troposphere
Observe convective response with different SSTs
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
12 / 21
Multiple Equilibria in Precipitation
Sobel, Bellon & Bacmeister (2007)
← initially moist
← initially dry
Reference profile is RCE with SST of 301 K
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
13 / 21
Multiple Equilibria in a CRM
Sessions, Sugaya, Raymond & Sobel (2010)
2-D Cloud Resolving Model (CRM) with WTG approximation
periodic boundary conditions
horizontal domains vary from 50-200 km (0.5-1 km resolution)
vertical dimension of 20 km (250 m resolution)
Parallel experiments: Initial moisture profile of tropics
same as surrounding environment
completely dry
Observe convective response with different surface wind speeds
(surface wind speed and SST both control surface fluxes)
4 month simulations, averaged over last month of simulation
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
14 / 21
Multiple Equilibria in a CRM
Sessions, Sugaya, Raymond & Sobel (2010)
rain rate (mm/day)
40
50km domain, tθ=1.85 hours
RCE input
dry initial
30
Reference profile is
RCE with vy = 5 m/s
20
10
0
5
10
15
horizontal windspeed (m/s)
20
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
15 / 21
Normalized Gross Moist Stability (NGMS)
Define NGMS to characterize the convective environment
NGMS =
TR g1
−L g1
R
R
∇ · (sv)dp
∇ · (r v)dp
=
moist entropy export
moisture import
GMS based on moist static energy budget (Neelin & Held 1987)
Thorough discussion of NGMS (Raymond et al. 2009)
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
16 / 21
NGMS in the Steady State
rain rate (mm/day)
Sessions, Sugaya, Raymond, Sobel (2010)
40
30
50km domain, tθ=1.85 hours
RCE input
dry initial
In the steady state,
20
Net precipitation =
10
net entropy forcing
NGMS
0
0.8
NGMS
0.6
0.4
0.2
0
5
Sessions (NMT)
10
15
horizontal windspeed (m/s)
20
Multiple Equilibria in a CRM
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May 2011
17 / 21
NGMS as a diagnostic
moisture import
moisture export
NGMS =
Sign of NGMS
moist entropy import
−
+
TR g1
−L g1
R
R
∇ · (sv )dp
∇ · (rv )dp
=
moist entropy export
+
−
moist entropy export
moisture import
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
18 / 21
Steady state precipitation and NGMS
50
rain rate (mm/day)
40
30
NGMS =
7 m/s
10 m/s
15 m/s
20 m/s
moist entropy export
moisture import
Dry equilibrium
NGMS + or Rain rate =
20
net entropy forcing
NGMS
NGMS=0.3 divides equilibria
10
0
-0.5
0
0.5
1
NGMS
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
19 / 21
Multiple Equilibria & Self-Aggregation NGMS
Day 50
Bretherton et al (2005)
+ GMS in precipitating region
− GMS in dry region
Sessions et al (2010)
+ NGMS in precipitating
equilibria
small or − NGMS in dry
equilibria
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
20 / 21
Summary
WTG approximation parameterizes the large scale circulations
Multiple equilibria in CRM
determined by surface fluxes
initial moisture determines which state is realized
NGMS characterizes the steady state atmosphere
larger values for precipitating equilibrium
smaller or negative values for dry equilibrium
Investigating multiple equilibria in limited domain WTG simulations may
be a computationally economic approach to understanding self-aggregation
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Sessions (NMT)
Multiple Equilibria in a CRM
May 2011
21 / 21