Internal Gravity Waves and Internal Gravity Waves and Turbulence Closure Model for SBLTurbulence Closure Model for SBL

Sergej Zilitinkevich Division of Atmospheric Sciences, Department of Physical Sciences

University of Helsinki and Finnish Meteorological Institute Helsinki, Finland

Tov Elperin, Nathan Kleeorin and Igor RogachevskiiDepartment of Mechanical EngineeringThe Ben-Gurion University of the Negev

Beer-Sheba, Israel

Victor L’vovDepartment of Chemical Physics, Weizmann Institute of Science, Israel

L. N. Gutman Conference on Mesoscale Meteorology and Air Pollution,

Odessa, Ukraine, September 15-17, 2008

Boussinesq ApproximationBoussinesq Approximation

0

,p

t

v v β v

TTt

1v

0

v vdiv 0 Re P e

L L

v

Laminar and Turbulent FlowsLaminar and Turbulent Flows

Laminar Boundary Layer

Turbulent Boundary Layer

Why Turbulence?Why Turbulence?

Number degrees of freedom

Why Not DNS?Why Not DNS?

Turbulent EddiesTurbulent Eddies

0l

l

l

Laboratory Turbulent ConvectionLaboratory Turbulent Convection

Before averagingAfter averaging

Velocity FieldsVelocity Fields

SBL EquationsSBL Equations

( )gft z

U τ

U U e

z

F

tz

2K KM z

K T

E EK S F

t C t

?? MH KK

)(zU

z

U

gU

TF

Total EnergyTotal Energy

Total Budget Equations: BL-case Total Budget Equations: BL-case

Total Budget Equations for SBL Total Budget Equations for SBL

2 2Fz Fz z

DF ΦD u C E

Dt z z

Total Budget Equations: BL-case Total Budget Equations: BL-case

Total EnergyTotal Energy

E

DEΠ D

Dt Φ

The source:

The turbulent potential energy:

EN

EP

2

Steady-state of Budget Equations Steady-state of Budget Equations for SBLfor SBL

2 2 0;Fz zu C E D

z

Total EnergyTotal Energy

Deardorff (1970)

Steady-State Form of the Budget EquationsSteady-State Form of the Budget Equations

( ) (1 Ri )K K T z K T FE C t F C t

P K T zE C t F

2NKF Hz

Turbulent temperature diffusivity

Our model

Old classical theory

vs. vs.

0PrTF

C

C

2 ,M z zK C l E

Turbulent Prandtl NumberTurbulent Prandtl Number

Τ

Ri3(1 )1Pr (Ri)

32 1 Ri

r

rF K rf

r

C CCCC C C C

C

Total Budget Equations: BL-caseTotal Budget Equations: BL-casein Presents of Gravity Waves in Presents of Gravity Waves

WP

2 2F Wz Fz z F

DF ΦD u C E

Dt z z

W

Wz

vs. (Waves)vs. (Waves)

2 ,M z zK C l E

0PrTF

C

C

Turbulent Prandtl NumberTurbulent Prandtl Number

Τ

1

Ri3(1 )1Pr (Ri)

32 1 Ri ( , )

r

rF K rf z

r

C CCCC C C C G Q A

C

0PrTF

C

C

Anisotropy vs.Anisotropy vs.

vs. vs.

vs. (Waves)vs. (Waves)

ConclusionsConclusions

- Total turbulent energy (potential and kinetic) is

conserved

- No critical Richardson number

- Reasonable turbulent Prandtl number from theory

- Reasonable explanation of scattering of the

observational data by the influence of the large-

scale internal gravity waves.

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. Boundary Layer Meteorology, homogeneous regimes. Boundary Layer Meteorology, 125125, , 167-191167-191..

Zilitinkevich S., Elperin T., Kleeorin N., Rogachevskii I., Esau I., Mauritsen Zilitinkevich S., Elperin T., Kleeorin N., Rogachevskii I., Esau I., Mauritsen T. and Miles M.,T. and Miles M., 2008, 2008, "Turbulence Energetics inStably Stratified "Turbulence Energetics inStably Stratified Geophysical Flows: Strong and Weak Mixing Regimes". Quarterly Journal Geophysical Flows: Strong and Weak Mixing Regimes". Quarterly Journal of Royal Meteorological Societyof Royal Meteorological Societyv. 134, 793-799. v. 134, 793-799.

Many Thanks toMany Thanks to

THE ENDTHE END

TTturbulence turbulence and Anisotropyand Anisotropy

iii uUU

IsotropyIsotropy AnisotropyAnisotropy

U

iu

U

iu

Total EnergyTotal Energy

Anisotropy in ObservationsAnisotropy in Observations

Isotropy

ww

vvuu

Equations for Atmospheric FlowsEquations for Atmospheric Flows

1 div divDT

TDt v F

fv

tD

D

vdiv

t

Budget Equation for TKEBudget Equation for TKE

DTΠDt

DEtot

K

Balance in R-spaceBalance in R-space

totΠ DBalance in K-spaceBalance in K-space

0)( kT

ΠD

KE

k( Heisenberg, 1948 )( Heisenberg, 1948 )

IsotropyIsotropy

Mean ProfilesMean Profiles

Turbulent Prandtl NumberTurbulent Prandtl Number

Τ

Ri3 (1 )Pr (Ri)

31 Ri

r

rF rF

r

C ACCC C

C

Total Budget EquationsTotal Budget Equations

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

Boundary Layer HeightBoundary Layer Height

Momentum flux derived

Heat flux derived

CalculationCalculation

vs. vs.

Total Budget EquationsTotal Budget Equations Turbulent kinetic energy:Turbulent kinetic energy:

Potential temperature fluctuations:Potential temperature fluctuations:

Flux of potential temperature :Flux of potential temperature :

DFUDt

DEzji

Riju

K )(div Φ

DF

N

Dt

DEz

2

)(div Φ

2

div (Φ ) ( ) 2F R Fij ij i ij j i i

DF NU e A e E D

Dt

F

vs. vs.

1.7

2.7

(1 36Ri)Ri (Ri) 1.25Ri

(1 19Ri)f

43

2

Ri(Ri) (Ri=0) 1

Ri

M z z

fz z

f

K C l E

l l

TemperatureTemperature Forecasting CurveForecasting Curve

Anisotropy vs.Anisotropy vs.