the dynamics of convection

1. Cumulus cloud dynamics

• The basic forces affecting a cumulus cloud– buoyancy (B)– buoyancy-induced pressure perturbation gradient acceleration (BPPGA)– dynamic sources of pressure perturbations– entrainment: Simple models of entraining cumulus convection (E)

Suggested readings: M&R Section 2.5: pressure perturbationsBluestein (Synoptic-Dynamic Met Pt II, 1993) Part III

Houze (Cloud Dynamics, 1993), Chapter 7 Holton (Dynamic Met, 2004) sections 9.5 and 9.6 Cotton and Anthes (Storm and Cloud Dynamics, 1982) Emanuel (Atmospheric Convection, 1994)

Essential Role of Convection

• You have learned about the role of baroclinic systems in the atmosphere: they transport sensible heat and water vapor poleward, offsetting a meridional imbalance in net radiation.

• Similarly, thermal convection plays an essential role in the vertical transport of heat in the troposphere:– the vertical temperature gradient that results from radiative

equilibrium exceeds that for static instability, at least in some regions on earth.

Cumulus Clouds• Range in size from

– cumuli (less than 1000 m in 3D diameter)– congesti (~1 km wide, topping near 4-5 km) – cumulonimbus clouds (~10 km) to– thunderstorm clusters (~100 km) to– mesoscale convective complexes (~500

km).

• All are ab initio driven by buoyancy B• vertical equation of motion:

• PPGF: pressure perturbation gradient force

topping near the tropopause

Bz

p

Dt

Dw

'1

PPGF

buoyancyforce

buoyancy: we derive buoyancy from vertical momentum conservation

equation

dvpvv

cR

ovd

hp

vvh

v

v

RccqTTp

pTTRp

relationsbasic

qp

p

c

cqq

p

p

T

T

g

B

gz

p

z

p

z

p

z

p

z

p

z

pg

z

p

Dt

Dw

then

gdz

pdppassume

ztzyx

zptzyxptzyxp

gz

p

Dt

Dw

p

d

)61.01(

:

'61.0

''

'''

'''

''

)2(;'')1(:

)(),,,('1

)(),,,('),,,(

1

'''

vertical equation of motion, non-hydrostatic

basic state ishydrostatically balanced

terms cancel small term

define buoyancy B:

• Complete expression for buoyancy:

where qh is the mixing ratio of condensed-phase water (‘hydrometeor loading’).

In terms of its effect on buoyancy (B/g), 1 K of excess heat ‘ is equivalent to ...

... 5-6 g/kg of water vapor (positive buoyancy)

... 3-4 mb of pressure deficit (positive buoyancy) ... 3.3 g/kg of water loading (negative buoyancy)

This shows that ’ dominates. All other effects can be significant under some conditions in cumulus clouds.

scaling the buoyancy force

h

p

vv q

p

p

c

cqgB

'61.0 '

'

see (Houze p. 36)

h

p

vv q

p

p

c

cq

g

B

'61.0 ''

buoyancy:

outside of cloud, buoyancy is proportional to the virtual pot temp perturbation

'''

''

61.0

:)0(

'61.0

'

'1

1

v

v

v

h

hp

vv

qg

B

qcloudofoutside

qp

p

c

cq

g

B

Bz

p

Dt

Dw

gz

p

Dt

Dw

• To a first order, the maximum

updraft speed can be estimated

from sounding-inferred CAPE:

This updraft speed is a vast over-estimate, mainly b/o two opposing forces.

• pressure perturbations: Buoyancy-induced (or ‘convective’) ascent of an air parcel disrupts the ambient air. On top of a rising parcel, you ‘d expect a high (i.e. a positive pressure perturbation), simply because that rising parcel pushes into its surroundings. The resulting ‘perturbation’ pressure gradient enables compensating lateral and downward displacement as the parcel rises thru the fluid. Solutions show the compensating motions decaying away from the cloud, concentrated within about one cloud diameter.

• entrainment

The buoyancy force

this ignores effect of water vapor (+B) and the weight of hydrometeors (-B).

CAPEdzg

Bdzw

thus

gBz

w

z

ww

Dt

Dw

LNB

LFC

LNB

LFC

2'2

2

2

1 '2

w= sqrt(2CAPE) is the thermodynamic updraft strength limit

this pressure field contains both a hydrostatic and a non-hydrostatic

component

B > 0 B < 0

Fig. 2.2 in M&R

discuss pressure changes in a hydrostatic atmosphere (M&R 2.6.1)

• mass conservation:– pressure tendency = vertically integrated mass

divergence

• hypsometric eqn: – pressure tendency = vertically integrated temperature

change

2nd force: pressure perturbation gradient acceleration (PPGA)

i

ji

jD

B

DB

DB

x

vv

xvvp

z

Bp

where

ppppartition

FFp

forceCoriolistheignoringthus

fvvz

Bp

v

now

fvvz

Bp

t

v

takebymultiply

vkfvvkBpt

v

'2

'2

''

2

2

2

':

'

,,

'

0

'

_,__

'1

FB: buoyancy source FD: dynamic source

anelastic continuity eqn

(M&R section 2.6.3)

** These equations are fundamental to understand the dynamics of convection, ranging from shallow cumuli to isolated thunderstorms to supercells.

For now we focus on FB.

Later, we ‘ll show that FD is essential to understand storm splitting and storm motion aberrations.

tensornotation

Bz

p

z

p

Dt

Dw DB

'' 11

)0(0'

)0(0'

''

'2

2

ncirculatioicanticycloninp

ncirculatiocyclonicinp

ppnote

fieldpressurescalesynoptictheisfp

syn

syn

synsyn

syn

222'

222

'2

2

)(,,

)(,,

)(,,

22

Dp

vectorvorticityy

u

x

v

x

w

z

u

z

v

y

wwith

vectorndeformatioy

u

x

v

x

w

z

u

z

v

y

wDwith

vectordivergencez

w

y

v

x

uwith

D

vvp

D

D

It can be shown that

Also, p’D> 0 (a high H) on the upshear side of a convective updraft, and p’D< 0 (a low L) on the downshear side

Analyze: This is like the Poisson eqn in electrostatics, with FB the charge density, p’B the electric potential, and p’B show the electric field lines.

The + and – signs indicate highs and lows: where L is the width of the buoyant

parcel

the buoyancy-induced pressure perturbation gradient acceleration(BPPGA):

Shaded area isbuoyant B>0

z

BFp BB

'2

2' Lz

BpB

'Bp

x

z

Where pB>0 (high), 2pB <0, thus the divergence of [- pB] is positive, i.e. the BPPGA diverges the flow, like the electric field.

The lines are streamlines of BPPGA, the arrows indicate the direction of acceleration.

Within the buoyant parcel, the BPPGA always opposes the buoyancy, thus the parcel’s upward acceleration is reduced.

A given amount of B produces a larger net upward acceleration in a smaller parcel

for a very wide parcel, BPPGA=B (i.e. the parcel, though buoyant, is hydrostatically balanced)(in this case the buoyancy source equals d2p’/dz2)

'BpBPPGA

0

__

1

01

..

__

2

'2

2

'2

'2

2

'2

'

'

y

p

x

pthatimpliesthis

z

Bp

becausez

p

z

Bor

Bz

pthen

Bz

p

Dt

Dwei

balancechydrostatiassume

BB

B

B

B

B

(10 km) (3 km)t=13 min t=8 min

(no entrainment)

Fig. 3.1 in M&R

H

L

H

L L

pressure field in adensity current(M&R, Fig. 2.6)

pressure units: (Pa)

Note that p’ = p’h+p’nh = p’B+p’D

z

BFp BB

'2p’B is obtained by solving with at top and bottom.p’D = p’-p’B

0

z

Bp’h is obtained from and p’nh = p’-p’h B

z

p h '1

H

L

H

L

H

LH

L

H

H

interpretation: use Bernoulli eqn along a streamline

.'

2

2

constBzpv

pressure field in a cumulus cloud(M&R, Fig. 2.7)

pressure units: (Pa)

2K bubble, radius = 5 km, depth 1.5 km, released near ground in environment with CAPE=2200 J/kg. Fields shown at t=10 min

H

L

HL

H

L H

L

H

L

Note that p’ = p’h+p’nh = p’B+p’D

z

BpB

'2p’B is obtained by solving with at top and bottom.

p’D = p’-p’B

0

z

B

p’h is obtained from and p’nh = p’-p’h

Bz

p h '1

Third force (also holding back buoyancy): entrainment• entrainment does two things:

(a) both the upward momentum and the buoyancy of a parcel are dissipated by mixing

(b) cloudy air is mixed with ambient dry air, causing evaporation

• No elegant mathematical formulation exists for entrainment E. The reason is that we are entering the realm of turbulence. We are reduced to some simple conceptual models of cumulus convection.

– Thermals or Bubbles– Plumes or Jets

• A general expression for the vertical momentum eqn for continuous, homogenous entrainment (Houze p. 227-230) (1D, steady state) is:

2'

2

2

''

12

1

:

:

11

wdz

dpB

dz

wd

dz

dww

dz

dm

m

wEplume

kwEthermal

Edz

dp

dz

dpB

Dt

Dw

e

DB

simplify to 1D, steady state & solve

(assumed)

mixing

H

penvparcel

p Dt

Dq

c

Lw

Dt

Dq

c

L

Dt

D

)(

M&R Figure 3.2. A possible trajectory (dashed) that might be followed by an updraft parcel on a skew T-log p diagram as a result of the entrainment of environmental air.

effect of entrainment on a skew T-log p diagram

Thermals or Bubbles

• Laboratory studies• negatively buoyant, dyed parcels are released, with

small density difference relative to the environmental fluid

• basic circulations look like this:

At first vorticity is distributed throughout the thermal.Later it becomes concentrated in a vortex ring.

The thermal grows as air is entrained into the thermal, via:• turbulent mixing at the leading edge;• laminar flow into the tail of the thermal

Results: shape oblate, nearly spherical; volume=3R3

• note shear instability along leading boundary

• entrainment rate seems small at first

• undilute core persists for some time, developing into a vortex ring

Sanchez et al 1989

Example of a growing cu on Aug. 26th, 2003 over Laramie. Two-dimensional velocity field overlaid on filled contours of reflectivity (Z [dBZ]); solid lines are selected streamlines. (source: Rick Damiani)

Cumulus bubble observation

dBZ

• Two counter-rotating vortices are visible in the ascending cloud-top.

• They are a cross-section thru a vortex ring, aka a toroidal circulation (‘smoke ring’)

20030826, 18:23UTC

8m/s

(Damiani et al., 2006, JAS)

Cumulus bubble observation

2.6 sounding analysis

• CAPE• CIN

• DCAPE (D for downdraft)

implications: * use Tv (not T) to compute CAPE/CIN * plot Tv (not T) in soundings

2.7 hodographs

• total wind vh

• shear vector S• storm motion c

• storm-relative wind vr = vh-c

height AGL (km)

z

v

z

u

z

vS h ,

storm-relative flow vr

vh

S

c

2.7.5 true & storm-relative wind near a supercell storm

real example(M&R, Fig. 2.14)

M&R, Fig. 2.13: hypothetical profiles: different wind vh, but identical vr

c

2.7.6 horizontal vorticity

storm-relative flow

horizontal vorticity

z

vO

z

uO

x

wO

x

wOassume

Skz

u

x

w

y

w

z

vh

,,

ˆ,,

streamwise vorticity• streamwise

• cross-wise

stre

amw

ise

cross-wise

sin

cos

hcr

hr

r

rc

hsr

hrs

v

v

v

v

v

v

errornote error in book

definition of helicity (Lilly 1979)

• the top is usually 2 or 3 km (low level !)• H is maximized by high wind shear NORMAL to the storm-

relative flow strong directional shear

• H is large in winter storms too, but static instability is missing

dzvSkdzSkvdzvdzvHtop

r

top

r

top

hr

top

sr 0000

ˆ)ˆ(

storm-relative flowhorizontal vorticity