Using the Weak Temperature Gradient Approximation to Understand Self-Aggregation 1

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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

nmtlogo 1This work is supported by the National Science Foundation Sessions (NMT) Multiple Equilibria in a CRM May2011 1/21 Self-Aggregation

Bretherton et al (2005) precipitation

1 WVP = g rt dp R

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Sessions (NMT) Multiple Equilibria in a CRM May2011 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 May2011 3/21 Weak Temperature Gradients in the Tropics

Charney 1963 scaling analysis Extratropics (small Rossby number) F δθ ∼ r θ Ro Tropics (Rossby number not small)

δθ ∼ Fr θ

2 −3 Froude number: Fr = U /gH ∼ 10 , measure of stratiﬁcation −5 Rossby number: Ro = U/fL ∼ 10 /f , characterizes rotation

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Sessions (NMT) Multiple Equilibria in a CRM May2011 4/21 Weak Temperature Gradients in the Tropics

Charney 1963 scaling analysis Extratropics (small Rossby number) F δθ ∼ r θ Ro Tropics (Rossby number not small)

δθ ∼ Fr θ

2 −3 Froude number: Fr = U /gH ∼ 10 , measure of stratiﬁcation −5 Rossby number: Ro = U/fL ∼ 10 /f , characterizes rotation

In the tropics, gravity waves rapidly redistribute buoyancy anomalies nmtlogo

Sessions (NMT) Multiple Equilibria in a CRM May2011 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:

∂T u ∇ T ∂t + h T + ωS = Qtotal

∂q u ∇ ∂q q ∂t + h q + ω ∂p = Qtotal

and S = T ∂θ is the static stability θ ∂p nmtlogo

Sessions (NMT) Multiple Equilibria in a CRM May2011 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 u ∇ T ∂t + h T + ωS = Qtotal

∂q u ∇ ∂q q ∂t + h q + ω ∂p = Qtotal Steady state WTG approximation and S = T ∂θ is the static stability θ ∂p nmtlogo

Sessions (NMT) Multiple Equilibria in a CRM May2011 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

T ∂θ and S = θ ∂p is the static stability

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Sessions (NMT) Multiple Equilibria in a CRM May2011 6/21 WTG Implementation in a Cloud Resolving Model (CRM)

Raymond & Zeng 2005

∂θ ∇ v T ≡ − ∂t + (ρ θ + θ) ρ(Sθ Eθ)

∂ rt ∇ v T ≡ − ∂t + (ρ rt + r ) ρ(Sr Er )

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Sessions (NMT) Multiple Equilibria in a CRM May2011 7/21 WTG Implementation in a Cloud Resolving Model (CRM)

Raymond & Zeng 2005

∂θ ∇ v T ≡ − ∂t + (ρ θ + θ) ρ(Sθ Eθ)

∂ rt ∇ v T ≡ − ∂t + (ρ rt + r ) ρ(Sr Er )

Enforcing WTG generates wwtg

∂θ θ − θ0(z) Eθ = wwtg = sin(πz/h) ∂z tθ

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Sessions (NMT) Multiple Equilibria in a CRM May2011 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 WTG: vertical θ profile Average moisture approximately horizontally in immediate 12000 homogeneous convective vicinity

9000

height (m) 6000

3000

0 300 320 340 0 0.005 0.01 0.015 0.02 θ (K) rt (g/g) Moisture advected from environment via mass continuity nmtlogo

Sessions (NMT) Multiple Equilibria in a CRM May2011 8/21 WTG Implementation in our CRM

Raymond & Zeng 2005

∂θ ∇ v T ≡ − ∂t + (ρ θ + θ) ρ(Sθ Eθ)

∂ rt ∇ v T ≡ − ∂t + (ρ rt + r ) ρ(Sr Er )

wwtg vertically advects moisture and brings in environmental moisture via mass continuity:

(r t − rx (z)) ∂(ρwwtg ) ∂r t Er = + wwtg ρ ∂z ∂z

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Sessions (NMT) Multiple Equilibria in a CRM May2011 9/21 Environmental moisture in WTG simulations

(r t − rx ) ∂(ρwWTG ) ∂r t Er = + w ρ ∂z WTG ∂z

r0 ∂(ρwwtg )/∂z > 0 rx = height r t ∂(ρwwtg )/∂z < 0

mass flux ( ρ w wtg )

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Sessions (NMT) Multiple Equilibria in a CRM May2011 10/21 Precipitation in WTG simulations

∂θ 40 Eθ = wwtg ∂z tθ=30 sec (strict WTG) 35 t =31 min − θ [θ θ0(z)] z h tθ=1.85 hours = sin(π / ) tθ 30 tθ=∞ (RCE) 25

20 θ0(z) is RCE proﬁle for vy = 5 m/s on 2D 15

200 km domain rain rate (mm/day) 10

tθ is a measure of the 0 enforcement of WTG 5 10 15 20 horizontal windspeed (m/s)

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Sessions (NMT) Multiple Equilibria in a CRM May2011 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 May2011 12/21 Multiple Equilibria in Precipitation

Sobel, Bellon & Bacmeister (2007)

← initially moist

← initially dry

Reference proﬁle is RCE with SST of 301 K nmtlogo

Sessions (NMT) Multiple Equilibria in a CRM May2011 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 May2011 14/21 Multiple Equilibria in a CRM

Sessions, Sugaya, Raymond & Sobel (2010)

40 50km domain, tθ=1.85 hours RCE input dry initial 30 Reference proﬁle is RCE with vy = 5 m/s 20 rain rate (mm/day) 10

0 5 10 15 20 horizontal windspeed (m/s)

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Sessions (NMT) Multiple Equilibria in a CRM May2011 15/21 Normalized Gross Moist Stability (NGMS)

Deﬁne NGMS to characterize the convective environment

1 TR ∇ (sv)dp moist entropy export NGMS = g = − 1 R ∇ v moisture import L g (r )dp R

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 May2011 16/21 NGMS in the Steady State

Sessions, Sugaya, Raymond, Sobel (2010)

40 50km domain, tθ=1.85 hours RCE input 30 dry initial In the steady state, 20 10 Net precipitation =

rain rate (mm/day) net entropy forcing 0 NGMS 0.8

0.6

0.4 NGMS 0.2

0 5 10 15 20 horizontal windspeed (m/s) nmtlogo

Sessions (NMT) Multiple Equilibria in a CRM May2011 17/21 NGMS as a diagnostic

Sign of NGMS moist entropy import moist entropy export moisture import − + moisture export + −

1 TR ∇ (sv)dp moist entropy export NGMS = g = − 1 R ∇ moisture import L g (rv)dp R

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Sessions (NMT) Multiple Equilibria in a CRM May2011 18/21 Steady state precipitation and NGMS

NGMS moist entropy export 40 7 m/s = moisture import 10 m/s 15 m/s 30 20 m/s Dry equilibrium NGMS + or - net entropy forcing 20 Rain rate = NGMS

rain rate (mm/day) 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 May2011 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 May2011 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 May2011 21/21