Turbulence Energetics in Stably Turbulence Energetics in Stably Stratified Atmospheric FlowsStratified Atmospheric Flows

Sergej Zilitinkevich Division of Atmospheric Sciences, Department of Physical Sciences

University of Helsinki and Finnish Meteorological Institute Helsinki, Finland

Tov Elperin, Nathan Kleeorin, Igor Rogachevskii

Department of Mechanical EngineeringThe Ben-Gurion University of the Negev

Beer-Sheva, Israel

Victor L’vov Department of Chemical Physics

ITI Conference on Turbulence III, 12-16 October 2008, Bertinoro, ItalyITI Conference on Turbulence III, 12-16 October 2008, Bertinoro, Italy

Stably Stratified Atmospheric FlowsStably Stratified Atmospheric Flows

Production of TurbulenceProduction of Turbulence

)(zU

z

UTF

¦ tot = ¦ ¡ ¯ jFzj

¦ = ¡ huiuj i r j ¹Ui = K M S2

HeatingHeating

CoolingCooling

by Shearby Shear

Destruction of TurbulenceDestruction of Turbulence

by Heat Fluxby Heat Flux

Laminar and Turbulent FlowsLaminar and Turbulent Flows

Laminar Boundary Layer

Turbulent Boundary Layer

Boussinesq ApproximationBoussinesq Approximation

0

,p

t

v v β v

vt

Budget Equation for TKE Budget Equation for TKE

DTΠt

Etot

K

Balance in R-spaceBalance in R-space

totΠ D

ΠD

KE

k

)(zU

z

UTF

TtC

EFSK

t

E

TK

KzM

K

||2

¦ tot = ¦ ¡ ¯ jFzj

¦ = ¡ huiuj i r j ¹Ui = K M S2

D = ºh(cirlu)2i ¼EK =(CK tT )

T = div©u

TKE Balance for SBLTKE Balance for SBL

TtC

EFSK

t

E

TK

KzM

K

||2 |)|( 2

zMTK FSKtE

B ´ ¯ Fz = ¡ (2Ez tT )@¹£@z

Observations, Experiments and LESObservations, Experiments and LES

Blue points and curve – meteorological field campaign SHEBA (Uttal et al., 2002); green – lab experiments (Ohya, 2001); red/pink – new large-eddy simulations (LES) using NERSC code (Esau 2004).

Budget Equations for SBLBudget Equations for SBL

Turbulent kinetic energy:Turbulent kinetic energy:

Potential temperature fluctuations:Potential temperature fluctuations:

Flux of potential temperature :Flux of potential temperature :

div ( )Ku z K

DEF D

Dt Φ

DF

N

Dt

DEz

2

)(div Φ

2

div ( ) ( ) 2F Fij ij i ij j i i

DF NΦ U e A e E D

Dt

F

Budget Equations for SBL Budget Equations for SBL

Total Turbulent EnergyTotal Turbulent Energy

Tu tC

EΠ

Dt

DE Φ∇

The source:

EN

EP

2

The turbulent potential energy:

Budget Equations for SBLBudget Equations for SBL

2Fz Fz z z

DF ΦD u u A E

Dt z z

Critical Richardson NumberCritical Richardson Number

B ´ ¯ Fz = ¡ (2Ez tT )@¹£@z

No Critical Richardson NumberNo Critical Richardson Number

Deardorff (1970)

B ´ ¯ Fz = ¡ (2Ez tT )@¹£@z

Budget Equations for SBLBudget Equations for SBL

Turbulent Prandtl Number vs.Turbulent Prandtl Number vs.

P rT ´K MK H

Budget Equations for SBL with Budget Equations for SBL with Large-Scale Internal Gravity Waves Large-Scale Internal Gravity Waves

WP

W

¦ W = ¦ W ( EWS2H 2;N (z)N (z0)

)

+¦ WF

! = khk N

Turbulent Prandtl Number vs. Ri Turbulent Prandtl Number vs. Ri (IG-Waves) (IG-Waves)

Meteorological observations: slanting black triangles (Kondo et al., 1978), snowflakes (Bertin et al., 1997); laboratory experiments: black circles (Strang and Fernando, 2001), slanting crosses (Rehmann and Koseff, 2004), diamonds (Ohya, 2001); LES: triangles (Zilitinkevich et al., 2008); DNS: five-pointed stars (Stretch et al., 2001). Our model with IG-waves at Q=10 and different values of parameter G: G=0.01 (thick dashed), G= 0.1 (thin dashed-dotted), G=0.15 (thin dashed), G=0.2 (thick dashed-dotted ), at Q=1 for G=1 (thin solid) and without IG-waves at G=0 (thick solid).

G / EWS2H 2

Q = N 2(z)N 2(z0)

P rT ´K MK H

vs. (IG-Waves)vs. (IG-Waves)

Meteorological observations: slanting black triangles (Kondo et al., 1978), snowflakes (Bertin et al., 1997); laboratory experiments: black circles (Strang and Fernando, 2001), slanting crosses (Rehmann and Koseff, 2004), diamonds (Ohya, 2001); LES: triangles (Zilitinkevich et al., 2008); DNS: five-pointed stars (Stretch et al., 2001). Our model with IG-waves at Q=10 and different values of parameter G: G=0.01 (thick dashed), G= 0.1 (thin dashed-dotted), G=0.15 (thin dashed), G=0.2 (thick dashed-dotted ), at Q=1 for G=1 (thin solid); and without IG-waves at G=0 (thick solid).

G / EWS2H 2

Q = N 2(z)N 2(z0)

Rif ´ ¡¯Fz¦

vs. Ri (IG-Waves) vs. Ri (IG-Waves)

Meteorological observations: squares [CME, Mahrt and Vickers (2005)], circles [SHEBA, Uttal et al. (2002)], overturned triangles [CASES-99, Poulos et al. (2002), Banta et al. (2002)], slanting black triangles (Kondo et al., 1978), snowflakes (Bertin et al., 1997); laboratory experiments: black circles (Strang and Fernando, 2001), slanting crosses (Rehmann and Koseff, 2004), diamonds (Ohya, 2001); LES: triangles (Zilitinkevich et al., 2008); DNS: five-pointed stars (Stretch et al., 2001). Our model with IG-waves at Q=10 and different values of parameter G: G=0.01 (thick dashed), G=0.04 (thin dashed), G= 0.1 (thin dashed-dotted), G=0.3 (thick dashed-dotted), at Q=1 for G=0.5 (thin solid); and without IG-waves at G=0 (thick solid).

Q = N 2(z)N 2(z0)

G / EWS2H 2

¿ ´ K M S

vs. Ri (IG-Waves) vs. Ri (IG-Waves)

Meteorological observations: squares [CME, Mahrt and Vickers (2005)], circles [SHEBA, Uttal et al. (2002)], overturned triangles [CASES-99, Poulos et al. (2002), Banta et al. (2002)], slanting black triangles (Kondo et al., 1978), snowflakes (Bertin et al., 1997); laboratory experiments: black circles (Strang and Fernando, 2001), slanting crosses (Rehmann and Koseff, 2004), diamonds (Ohya, 2001); LES: triangles (Zilitinkevich et al., 2008); DNS: five-pointed stars (Stretch et al., 2001). Our model with IG-waves at Q=10 and different values of parameter G: G=0.001 (thin dashed), G=0.005 (thick dashed), G= 0.01 (thin dashed-dotted), G=0.05 (thick dashed-dotted), at Q=1 for G=0.1 (thin solid); and without IG-waves at G=0 (thick solid).

Q = N 2(z)N 2(z0)

G / EWS2H 2

Fz = ¡ K H@¹£@z

vs. (IG-Waves) vs. (IG-Waves)

Meteorological observations: overturned triangles [CASES-99, Poulos et al. (2002), Banta et al. (2002)]; laboratory experiments: diamonds (Ohya, 2001); LES: triangles (Zilitinkevich et al., 2008). Our model with IG-waves at Q=10 and different values of parameter G: G=0.2 (thick dashed-dotted), at Q=1 for G=1 (thin solid); and without IG-waves at G=0 (thick solid for ) and (thick dashed for ).Ri1f = 0:2Ri1f = 0:4

G / EWS2H 2

Q = N 2(z)N 2(z0)

E = EK + Ep

Anisotropy vs. (IG-Waves) Anisotropy vs. (IG-Waves)

Meteorological observations: squares [CME, Mahrt and Vickers (2005)], circles [SHEBA, Uttal et al. (2002)], overturned triangles [CASES-99, Poulos et al. (2002), Banta et al. (2002)], slanting black triangles (Kondo et al., 1978), snowflakes (Bertin et al., 1997); laboratory experiments: black circles (Strang and Fernando, 2001), slanting crosses (Rehmann and Koseff, 2004), diamonds (Ohya, 2001); LES: triangles (Zilitinkevich et al., 2008); DNS: five-pointed stars (Stretch et al., 2001). Our model with IG-waves at Q=10 and different values of parameter G: G= 0.01 (thick dashed), G=0.1 (thin dashed-dotted), G=0.2 (thin dashed), G=0.3 (thick dashed-dotted), at Q=1 for G=1 (thin solid); and without IG-waves at G=0 (thick solid).

Q = N 2(z)N 2(z0)

Az = E zEK

G / EWS2H 2

ReferencesReferences Elperin, T., Kleeorin, N., Rogachevskii, I., and Zilitinkevich, S. Elperin, T., Kleeorin, N., Rogachevskii, I., and Zilitinkevich, S. 2002 2002

Formation of large-scale semi-organized structures in turbulent Formation of large-scale semi-organized structures in turbulent convection. convection. Phys. Rev. EPhys. Rev. E, , 6666, 066305 (1--15), 066305 (1--15)

Elperin, T., Kleeorin, N., Rogachevskii, I., and Zilitinkevich, S. Elperin, T., Kleeorin, N., Rogachevskii, I., and Zilitinkevich, S. 2006 2006 Tangling turbulence and semi-organized structures in convective Tangling turbulence and semi-organized structures in convective boundary layers. boundary layers. Boundary Layer MeteorologyBoundary Layer Meteorology, , 119119, 449-472. , 449-472.

Zilitinkevich, S., Elperin, T., Kleeorin, N., and Rogachevskii, I, Zilitinkevich, S., Elperin, T., Kleeorin, N., and Rogachevskii, I, 2007 2007 Energy- and flux-budget (EFB) turbulence closure model for stably Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Boundary Layer Meteorology, Part 1: steady-state stratified flows. Boundary Layer Meteorology, Part 1: steady-state homogeneous regimes.homogeneous regimes. Boundary Layer Meteorology, Boundary Layer Meteorology, 125125, , 167-191167-191..

Zilitinkevich S., Elperin T., Kleeorin N., Rogachevskii I., Esau I., Zilitinkevich S., Elperin T., Kleeorin N., Rogachevskii I., Esau I., Mauritsen T. and Miles M., Mauritsen T. and Miles M., 2008, Turbulence energetics in stably 2008, Turbulence energetics in stably stratified geophysical flows: strong and weak mixing regimes. stratified geophysical flows: strong and weak mixing regimes. Quarterly Quarterly Journal of Royal Meteorological Society, Journal of Royal Meteorological Society, 134134, 793-799. , 793-799.

ConclusionsConclusions Budget equation for the Budget equation for the total turbulent energytotal turbulent energy (potential (potential

and kinetic) plays a crucial role for analysis of SBL flows. and kinetic) plays a crucial role for analysis of SBL flows.

Explanation for Explanation for nono critical critical Richardson numberRichardson number. .

Reasonable Reasonable Ri-dependenciesRi-dependencies of the turbulent Prandtl of the turbulent Prandtl number, the anisotropy of SBL turbulence, the normalized number, the anisotropy of SBL turbulence, the normalized heat flux and TKE which follow from the developed theory. heat flux and TKE which follow from the developed theory.

The scatter of observational, experimental, LES and DNS The scatter of observational, experimental, LES and DNS data in SBL data in SBL are explained by are explained by effects of large-scale internal effects of large-scale internal gravity wavesgravity waves on SBL-turbulenceon SBL-turbulence. .

THE ENDTHE END