Eddy kinetic-energy redistribution within real and idealized extratropical storms

1 LMD-ENS, IPSL/CNRS, Paris

2 CNRM-GAME, Météo-France & CNRS, Toulouse

Gwendal Rivière12

Collaboration: Philippe Arbogast, Alain Joly

A 3D easterly view of the storm Christian (Oct 2013)

Forecast from Arpege, 00 UTC 28 October, step 9h

w 0w 0w

Max u (900 mb): poisonous tail of the bent-back warm front

L Warm

front

Cold frontSouth

North

Another example: the storm Klaus (24/01/09)

12 UTC 23 Jan

Lθe (900hPa)

Max wind speed (650-850 mb)

Warm front

Cold front

Warm front

Col

d fr

ont

L00 UTC 24 Jan

EKE=u’2/2 (650-850 mb)

Approach: separation into a background flow and a perturbation by time filtering

L LPerturbation wind u'

Mean westerlies u

Early stage: strongest wind speeds are reached along the WCB

Later stage: strongest wind speeds are reached in the frontal fracture zone

(e.g. Gronas, 1995)

Data: ERAinterim (0.75° grid spacing)

Idealized study: use of the two-layer quasi-geostrophic model

lull

luuu

fq

fq

2

2

0 kkt qq .uk

Rossby radius of deformation

0' u

0' l

ζ’ (300mb) >0

ζ’ (900mb) >0

uu

luUpper layer

Lower layer

Isolated cyclonic anomaliesBasic state

Klaus case

u 300mb

The eddy kinetic energy (EKE) equation

')(2

2

au''uuu'u'

t

'')'('

''

pp aa u'u'

Horizontal ageostrophic geopotential fluxes

Vertical ageostrophic geopotential fluxes

Baroclinic conversion

Pressure work

u geostrophic windΦ geopotential

ua ageostrophic wind

af kua

0

1

p

p

vf

pfs

f

yvuJa

2

02

220

22

0

2

2

),(2

Q Omega equation

Q-vector

Diagnostic equation forageostrophic scalar

a

EKE redistribution at the lower layer in presence of the vertical shear

t=12H

2

2u'

dt

d

2

2u'

t=0H

f-plane, linear

'au'

X (1000km)

t=0H

β-plane, linear

t=12H EKE accumulation east of the low

X (1000km)

y (1

000k

m)

y (1

000k

m)

EKE accumulation west of the low

South

North

)'( au'

EastWest

Role of β and the vertical shear in EKE redistribution

'au'

k''

No βWith β

)''()'')(()2(

'

'')()''()()2(

'

200221

22

2222

20022

0

2

lululu

ullulual

vvK

vvuuKs

K

KK

uuKf

mu

Wavelike disturbance assumption (wavenumbers m, K)

β increases the upward transfer and decreases the downward transfer, so a sink of energy for the lower layer !!!

Role of nonlinearities in EKE redistribution

t=12H

2

2u'

dt

d

2

2u'

t=0H

Linear

'au'

f-plane

X (1000km)

t=0H

Nonlinear

t=12H EKE accumulation southwest of the low

X (1000km)

y (1

000k

m)

y (1

000k

m)

)'( au'

Role of background lateral shear in EKE redistribution

Linear

Cyclonic shear

Anticyclonic shear

2

2u'

dt

d

f-planet=12H

X (1000km)

y (1

000k

m)

y (1

000k

m)

Nonlinear

Cyclonic EKE redistribution

EKE accumulation

2

2u'

X (1000km)

Combined effects of anticyclonic shear / PV gradient in EKE redistribution

t=12h

X (1000km)

y (1

000k

m)

2

2u'

X (1000km)

f-plane, nonlinear

Anticyclonic shear

EKE accumulation to the east

β-plane, nonlinear EKE accumulation to the west

2

2u'

dt

d

The stronger the vertical average of the PV gradient, the more important the EKE accumulation to the east

Isolated cyclonic anomalies initiated south of a westerly jet

uu

lu

Upper layer

Lower layer

Isolated cyclonic anomalies

Basic state

Isolated cyclonic anomalies initialized south of a meridionally confined zonal jet

ζ’ (300mb) >0

ζ’ (900mb) >0

0' u

0' l

u 300mb

Evolution of the idealized jet-crossing cyclone

The key finding: once the cyclone has crossed the large-scale jet, EKE is cyclonically redistributed by ageostrophic geopotential fluxes in regions where the eddy and mean winds align with each other !

Jet axis u'u

u

t=12H

Jet axis

t=24H

Jet axis

t=36H

Jet axis

t=48H

)''()(

)'')(()2(

'

'2

'')()(

)''()(

)2('

2

2

221

22

222

22

2222

2

2

220

2

luluy

lulu

llyuy

ulluy

lulu

al

vvK

uuvvuu

Ks

K

uuK

K

KK

uu

uu

Kf

mu

'au'

)'( au'

Data: ERAinterim (0.75° grid spacing)

00 UTC 23 Jan 2009

Interpreting EKE distribution during for the storm Klaus

06 UTC 23 Jan 200912 UTC 23 Jan 200918 UTC 23 Jan 200900 UTC 24 Jan 2009

environment 2/EKE 2u' u'

'au'

)'( au'

L

u

A 3D view of Klaus at the later stage

Mid-tropospheric baroclinic conversion

Downward EKE transfer –ω’Φ’k

Lower-tropospheric cyclonic EKE redistribution u’aΦ’

Spain

Atlantic Ocean

EKE

p '

'

Confirmation of dry simulations of NWP model (Papritz and Schemm, 2013)

Conclusions

• Before jet crossing: both the background anticyclonic shear and PV gradient explain EKE accumulation to the southeast

• After jet crossing: rapid cyclonic EKE redistribution to the southwest by ageostrophic geopotential fluxes (also nonlinear advection) due to the action of the cyclonic shear

EKE

Schematic of EKE redistribution processes for jet-crossing cyclones following the Shapiro-Keyser conceptual model

Max |u|

Rivière et al (2014a,b; QJRMS)

Outlook

• What is the role played by the synoptic-scale EKE redistribution in the formation of mesoscale jets (sting jets, cold-conveyor-belt jets)

(Browning, 2004; Gray et al. 2011; Martinez-Alvarado, 2014) ?

• Relation with jet-crossing cyclones.

Friedhelm at 12 UTC 8 Dec 2011

From Baker et al (2013)

L

Other slides

EKE budget for the idealized cyclone at the lower layer EKE ct

'- au' EKE

'-

c

uuu'

EKE budget in the cyclone frame of refrence (c phase speed of the cyclone)

'' p

)''( p )'( au'

uuu' '-

EKE'- u EKE)-(c u

EKE budget for the storm Klaus (650-850 hPa) EKE ct

'- u' EKE

'- 33

c

uuu'

EKE budget in primitive equations in the cyclone frame of refrence (c phase speed of the cyclone)

'' p

)''( p )'( au'

33 '- uuu'

EKE'- 33u EKE)-(c 33 u

EKE)'('- cuuu'u' 33

Data: ERAinterim (0.75° grid spacing, 3 hour time steps)

EKE budget for the storm Friedhelm (650-850 hPa) EKE ct

'- u' EKE

'- 33

c

uuu'

EKE budget in primitive equations in the cyclone frame of refrence (c phase speed of the cyclone)

'' p

)''( p )'( au'

33 '- uuu'

EKE'- 33u EKE)-(c 33 u

EKE)'('- cuuu'u' 33

Data: ERAinterim (0.75° grid spacing, 3 hour time steps)

3D view of Klaus

(u’a,ω’ k)Φ’

u’aΦ’

EKE

p '

'