Cumulus Clouds

Instabilities Resulting in Vertical Overturning

1. Thermal Instability (Assuming uniform vertical pressure gradient)a) Static (Parcel buoyancy)b) Conditional (Parcel buoyancy)c) Rayleigh Benard (Parcel buoyancy, surface friction)

1. Dynamic Instabilitiesa) Shear (or inflection point) (Vorticity or shear gradient) (analogous to barotropic instability)

3. Dynamic-Thermal Instabilitiesa) Vertical Shear vs Static Stability

i. Kelvin-Helmholtz iii Gravity wave [convection (growing) , evanecent (decaying)]

b) Inertial production (Horizontal Shear) vs Static Stability i. Symmetric Instability ii. Conditional Symmetric Instability (CSI)ii Convective - Symmetric Instability(C-SI)

Γenvironment > Γdry

Γdry > Γenvironment > Γmoist

Ra > Rc ; Ra ≡

hPBL4 Γenvga e

nK

∂2V / ∂z2 =0

V + f =∂2u / ∂y2 =0

Ri ≡stability/shear = N / ∂V / ∂z( ) < 0.25

cg =cg,r + icg,i ; cg,i ≠0

V + f( )θ < 0

V + f( )θe

< 0

Γdry > Γenvironment > Γmoist ; V + f( )θoutflow

< 0 or sm all

General Classification of Moist Convection

• Shallow Cumulus (cumulus, scatted cumulus, strato-cumulus)– Depth small compared to scale height of troposphere, i.e.

– Usually confined to Planetary Boundary Layer (PBL) – Typically non-precipitating – Surface friction plays critical role to organization

• Deep Cumulus (congestus, cumulonimbi)– Depth comparable to scale height of troposphere

– Precipitating– Friction plays secondary role to organization

Δz / H = 1⇒ r' / r = 1

Δz / H : 1⇒ r' / r : 1

What goes on inside a cumulus cloud?

Evolution of a thermal from a Numerical Model

QuickTime™ and a decompressor

are needed to see this picture.

Conceptual Model

• Series of convective plumes rising to form individual turrets comprising cloud

• Each rising pulse a toroidal circulation• Successive toroids give rise to mean

upward current called updraft• Sustained downward current between

toroids, if existing, would be downdraft

Liquid Water Content

What causes liquid water content to be below adiabatic LWC?

• Lateral entrainment– Neutral mixing– Dynamic entrainment

• Cloud Top Entrainment

Bubble and JetModels of Convection

<= mixing lateral entrainment

Dynamic Lateral Entrainment

Dynamic Entrainment

Effects of Dynamic Lateral Entrainment

Effects of Dynamic Lateral Entrainment

Cloud Top entrainment

Deep Cumulus

• Must consider impact of precipitation on cumulus circulation

• Must consider pressure effects because of cloud depth– Thermodynamic pressure, ie hydrostatic pressure– Dynamic pressure due to inertia of air motions

• Friction layer small compared to cloud and we generally ignore friction

Vertical Acceleration(using Pressure)

31 1 2 2 2 1

'

3

'

( ) ( )

1i

o

u kf u f ut x

px

g

z z

r

rr

∂ ∂= − + + + −

∂ ∂

∂−

∂

−

Inertia

Pressure

Buoyancy

Vertical Acceleration(Using Total Pressure)

31 1 2 2 2 1

3

( ) ( )

1i

u kf u f ut x

px

g

z z

r

∂ ∂= − + + + −

∂ ∂∂

−∂

−

Inertia

Pressure

Buoyancy

Vertical Acceleration(using Exner function)

31 1 2 2 2 1

'

3

'

( ) ( )i

o

vv

o

u kf u f ut x

x

g

z z

πθ

θθ

∂ ∂= − + + + −

∂ ∂

∂−

∂

+

Inertia

Pressure

Buoyancy

Traditional Buoyancy

( )( )'

1 0.61 1vvv l i o

o o

gg q q qθ θ θθ θ

= + − − −⎡ ⎤⎣ ⎦| | | |

Warm/Cold air rises/sinks

Vapor less dense than dry air

Liquid water loading

Ice water loading

Anelastic Approximation• Neglect frequencies higher than those

associated with meteorological phenomena such as sound wave frequencies

• Similar to incompressible assumption, but for a compressible system

( )0 (anelastic approximation)

or

0

i

i

i i

i

i

ux

u ux tt x

ρ

ρ ρ

∂=

∂

⎛ ⎞∂ ∂⎛ ⎞∂ ∂⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠= =∂ ∂

Continuity Equation

t

j

j

j jj

j j j

uddt x

u uu

x x x

r r

rr r r

∂=−

∂

∂∂ ∂=− − =−

∂ ∂ ∂ ∂

Multiply momentum equation (momentum form) by density:

3

3

3

1( )

1

hence,

ji i ii i

j

ijk j j k ii i

ij ijk j k i

j i

i jiijk j k i

i j

uu u uu ut t t t x

k pf u gx x

u pu f u gx x

u uu p f u gt x x

rr rrr

r e z dr

r e dr

rr e r d r

∂∂ ∂ ∂∂= − = +

∂ ∂ ∂ ∂ ∂

⎡ ⎤∂ ∂= + − + − − −⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤∂ ∂= − − − −⎢ ⎥

∂ ∂⎢ ⎥⎣ ⎦

∂∂ ∂= − − − −

∂ ∂ ∂

Multiply momentum equation (vorticity form) by density:

31( )

hence,

ji i ii i

j

ijk j j k ii i

jii

j

uu u uu u

t t t t x

k pf u gx x

uuu

t x

rr rrr

r e z dr

rr

∂∂ ∂ ∂∂= − = +

∂ ∂ ∂ ∂ ∂

⎡ ⎤∂ ∂= + − + − − −⎢ ⎥∂ ∂⎣ ⎦

∂∂= −

∂ ∂ 3

3

( )

( )

ijk j j k ii i

ijk j j k ii i

k πf u gx x

k πf u gx x

e z r r d r

e z r r d r

∂ ∂− + − − −

∂ ∂

∂ ∂= − + − − −

∂ ∂

Decomposition of Pressure into Dynamic and Buoyancy Pressure

u

v

w w

u p Dynamict xv p Dynamict yw p Dynamic Buoyancyt z

r

r

r

∂ ∂= − +

∂ ∂∂ ∂

= − +∂ ∂∂ ∂

= − + +∂ ∂

Dynamics (or inertia) Terms

2 2 3 3 3 21

1 1 3 3 3 12

1 1 2 2 2 13

( ) ( )

( ) ( )

( ) ( )

u

v

w

kDynamic f u f ux

kDynamic f u f ux

kDynamic f u f ux

z r z r r

z z r r

z r z r r

∂= − + + + −

∂∂

= + + − + −∂∂

= − + + + −∂

Buoyancy Terms

( )( )1

where

w

d v l i

m l i

m d v

ll

m

ii

m

Buoyancy gg

g q q

q

q

ρρ ρ ρ ρρ

ρ ρ ρρρρρ

= −= − + + += − + +

= +

=

=

Take divergence of density multiplied by three momentum equations and then result set to zero and

solve for pressure:

2 u v u wDynamic Dynamic Dynamic Buoyancypx y z z

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞∇ = + + −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠or

2

2

''

u v ud

wb

o o d b

Dynamic Dynamic Dynamicpx y z

Buoyancyp

zp p p p p p

⎛ ⎞∂ ∂ ∂∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠

∂⎛ ⎞∇ = −⎜ ⎟∂⎝ ⎠= + = + +

Where pressure is divided into dynamic and buoyancy pressure contributions

Buoyancy vs. Dynamic Pressure• Dynamic pressure, , is zero if flow is at rest.• Buoyancy pressure, , is hydrostatic pressure for flow

at rest.• Dynamic pressure results from inertia such as:

– Rotation (cyclostrophic pressure)– Straight line accelerations – Coordinate system accelerations (coriolis)

• Buoyancy pressure results from:– Moisture anomalies– Thermal anomalies– Condensate (precipitation drag)

dpbp

Real Buoyancy Acceleration

• True buoyancy acceleration is :

• Where we see the acceleration is caused by thermal, moisture or precipitation drag anomalies

'1true buoyancy acceleration = b

o

pzr

∂−

∂

Dynamic Pressure Acceleratrion

• True dynamic pressure gradient acceleration is :

• Where we see the acceleration is caused by inertial effects of rotation, straight line movement and coordinate system movement

1dynamic pressure acceleration = d

o

pzr

∂−

∂

Conditional Instability of the First Kind

• Occurs when a parcel is statically unstable when saturated but stable when dry

• Results in the formation of moist convective thermal plumes, ie cumulus clouds

• Instability favors horizontal scales ~ vertical scale of overturning, i.e. meso-gamma scale for deep convection

Three Stages of a Deep Convective Thermal

• Simplest Case:– Conditionally unstable for deep convection– No environmental wind– Dry middle layers– Moist unstable boundary layer

Stage 1 : Cumulus Stage• Updraft only• Cloud droplets only (no precipitation)• Level of Non-divergence (LND) near top of moist Planetary Boundary

Layer (PBL) • Cloud positively buoyant throughout:

• Environment neutrally buoyant• Low pressure under updraft• High pressure throughout cloud

'' 1>0 or more precisely, 0bvv

pz

θr∂

− >∂

Stage 2 : Mature Stage

• Updraft and downdraft• Precipitation and cloud droplets throughout cloud• Level of Non-divergence (LND) at middle levels • Cloud positively buoyant at middle levels,

negatively buoyant in lower part• Cold air dome (density current) at surface• Environment neutrally buoyant but warming• Low pressure at middle levels• High pressure at surface and top of cloud

Stage 3 : Dissipating Stage

• Downdraft only• Precipitation only throughout cloud• Level of Non-divergence (LND) at upper levels • Cloud negatively buoyant throughout• Environment positively buoyant• Low pressure at middle levels and above in cloud• High pressure at surface• Low pressure at surface of environment

Reasons for Breakdown

• Water loading of updraft from precipitation drag

• Cooling due to dynamic entrainment of mid level dry air

Introduce Environmental Wind Shear to Prevent Breakdown

• Assume:– two-dimensions, i.e. infinitely long convective

line– Straight-line shear with height, I.e. wind speed

change with without direction change

Three-Dimensional Effect of Wind Shear

• As before but now assume convective plume is initially circular rather than infinitely long

• Also start by assuming a straight line shear profile again

• Assume westerly shear and veering winds in lowest 6 km

View from South

View from East

Helicity

j jH uV=

Convective Richardson Number

212 i

CAPERu

=

CAPE

( ) ( )( )

z zCAPE g dza

θ θθ−

= ∫

Wind Shear

6

06

0

km

i

i km

u dzu

dz

r

r=∫

∫