The Structure of Nocturnal Urban Boundary Layer: Study with Nonlocal - - PowerPoint PPT Presentation

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The Structure of Nocturnal Urban Boundary Layer: Study with Nonlocal Mesoscale Model

Albert F. Kurbatskiy

Institute of Theoretical and Applied Mechanics of Russian Academy of Sciences, Siberian Branch and Novosibirsk State University, Russia

Ludmila I.Kurbatskaya

Institute of Computational Mathematics and Mathematical Geophysics of Russian Academy of Sciences, Siberian Branch , Russia

ENVIROMIS_ 2008, Tomsk

O U T L I N E:

Introduction Improved turbulence model for stratified turbulent flows Modeling and Simulation of Stratified Boundary Layer

• ver urban-like roughness:

ŒStructure features of turbulent transport in an thermal unstable stratified boundary layer ŒStructure features of turbulent transport in a stable thermal stratified boundary layer

Conclusion

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RANS (Reynolds Average Navier-Stokes) approach for Turbulence Modeling

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i ij i ijk j k j i

DU 1 P g 2 U ; Dt x x ∂ ∂ = − τ − − − ε Ω ∂ ρ ∂

j j

D h , Dt x Θ ∂ = − ∂

ij i j

τ u u are the Reynolds stresses ≡ 〈 〉

i i

h u θ a re the hea t f lu xes ≡ 〈 〉

.

GOVERNING EQUATIONS FOR TURBULENT STRATIFIED FLOWS

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ŒE-L model: К-theory for all (!) turbulent fluxes: ŒE- model:

• Richardson flux number

‘Traditional’ turbulent models for stratified

boundary layers

( )

z z i i

wa K A / z ; K E; (E u u / 2 TK E) −〈 〉 = ∂ ∂ = = 〈 〉 ⇒ l

2 z

K c E /

µ

= ε

f

0,09 x direction c the function of Ri number z direction

µ

→  =  → 

f

Ri

ε

ENVIROMIS_ 2008, Tomsk

‘Traditional’ turbulent models for stratified

boundary layers

Usually used the semi-empirical modification of K- theory, with an additional countergradient term that incorporates the contribution of large scale eddies to total flux: This non-local flux correction is applied only to temperature for convective mixing case, but not for the stable case! n Deardorff (1972): n The MRF scheme:

( )

′ ′ = − ∂ ∂ −

C

/ ,

C

w K z where is the countergradient term

φ

φ γ γ φ

1 5 2 ' 2 ' C

cm K 10 5 . w g

− −

× ≈ θ Θ = γ

h w w b

S C ' '

) ( φ γ =

1 * −

Φ = u wS

8 . 7 = b

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Turbulence equations

ij ij ij i j j i ij ij

D D P h h П Dt τ + = +β +β − − ε

        ∂ ∂ + ∂ ∂ − =

k i jk k j ik ij

x U x U P τ τ

〉 〈 ∂ ∂ − 〉 ∂ ∂ 〈 + 〉 ∂ ∂ 〈 =

k k ij i j j i ij

pu x 3 2 x p u x p u П δ

ε δ ν ε

ij k j k i ij

3 2 x u x u 2 = 〉 ∂ ∂ ∂ ∂ 〈 =

i i

g β β ≡

      〉 〈 + 〉 〈 ∂ ∂ ≡

k ij k j i k ij

pu 3 2 u u u x D δ

h 2 i i i j ij i i j j

Dh U D h П , Dt x x

θ

∂ ∂ + = − − τ +β 〈θ 〉 − ∂ ∂ Θ

, x p П

i i

〉 ∂ ∂ 〈 ≡ θ

θ

〉 〈 ∂ ∂ = θ

j i j h i

u u x D i j

Reynolds stresses, u u

• τ ≡〈

〉

ij

i i

Heat fluxes, h u θ

• ≡〈

〉

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Full Explicit Algebraic Models for Reynolds Stresses and Scalar Fluxes

( )

ij 3 ij ij 2 ij 1 ij

B S E b α Ζ Σ α τ α + + − − =

• (

)

ij ij ij ij ij ij ij

Ï B S E 3 4 D b Dt D − + Ζ + Σ − − = = + ,

θ

− 〉 θ 〈 β + ∂ Θ ∂ τ − ∂ ∂ − = = +

i 2 i j ij j i j h i i

Ï x x U h D Dt Dh

〉 〈 + ∂ ∂       + − =

• 2

3 i 4 j ij ij j ij

g x E 3 2 b h A θ δ β τ α Θ δ τ

Coupled algebraic system equations for and

〉 〈

j iu

u

: u h

i i

〉 θ 〈 ≡

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Improved Full Explicit Algebraic Models for Reynolds Stresses and Scalar Fluxes : 2D case

( )

, , ∂ ∂   < > < > = −   ∂ ∂  

M

U V uw vw K z z

∂Θ < θ >= − + γ ∂

H c

w K z

M M

K E S τ =

H H

K E S τ =

( )

2 H

G N τ ≡

( )

2 M

G S τ ≡

2

N g z β ∂Θ = ∂

2 2 2

U V S z z ∂ ∂     ≡ +     ∂ ∂    

H H M H H H M H M

G G G d G d G d G G d G d G d D ) ( 1

6 2 5 2 4 3 2 1

− + + + + + =

ε τ E =

( ) [ ] ( )

( )

      〉 〈 + × × + − + = E g G s s s G s s G s s D S

H H H M

/ 1 1 1

2 6 5 4 3 2 1

θ β τ

( )

6 1

1 2 1 1 3

H H

S s G D c θ   = +    

2 2 2 6 5

1 2 1 ( ) 3   = + α + α τβ 〈θ 〉    γ 

M H c

G s G g D

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Three-parametric turbulence model

i ii ij i i j

DE 1 U D τ β h ε, Dt 2 x ∂ + = − + − ∂

Turbulent kinetic energy

i i

E (1/2) u u = 〈 〉

2 ε

Dε ε D Ψ, Dt E + = −

2

2 i θ θ i

D θ Θ D 2h 2ε , Dt x 〈 〉 ∂ + = − − ∂

, ε TKE dissipation

2

〈θ 〉 Temperature variance,

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Thermal Stratified Boundary Layer over Flat Terrain The potential temperature θ and velocity U are shown for

the convective and stable boundary layers.

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Boundary Layer over Urban-Like Roughness

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Parameterization of Urban-Like Roughness

˘ Numerical models of urban boundary layer must be able to resolve two main scales: ‘urban’ and ’meso’. ˘ Horizontal dimensions of domain are on order of meso’scale (~100 km) and for keeping number of grid points compatible with CPU time cost, the horizontal grid resolution

• f such (meso’scale) models

ranges between several hundreds of meters and a few kilometers. ˘ Because is not possible to resolve city structure in detail (buildings or blocks), the effects of urban surfaces must be parameterized.

• v

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Governing Equations for SBL 2D case:

0,

x z

U W + = , +

t x z x z

1 U +UU +WU = - P wu +fV ρ

• u

D

, − +

t x z z

V +UV + WV = - wv fU

v

D

2

, 1 + + = − βΘ ρ − +

t x z z z

W UW WW P w g

. Θ + Θ + Θ −〈 θ θ = 〉 − +

t x z x z

U W u w

θ

D

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Parameterization of Aerodynamical Roughness Effects

The extra terms DA in the Governing Equations are:

DU = turbulent momentum flux (horizontal surfaces) + drag (vertical surfaces) Dθ = turbulent heat fluxes from horizontal surfaces, and the temperature fluxes from the vertical surfaces DE = increasing of mean kinetic energy conversion into the TKE [for example, Raupach and Shaw (1982), Raupach et al.(1991), Raupach (1992)]

“ a ‘second’ dissipation linked with the turbulence lengthscale L, and induced by the presence of the roughness elements

pε

ˆ ( 0.7) c

ε

=

1/2 pε

E D =c L ε

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Thermal Effects of Urbanized Area : ‘Heat Island‘

ŒIn this modeling, the UHI effect was specified by an

urban-environs temperature difference.

ŒThe ground temperature was specified as

This is the only nonstationary boundary condition of the problem, which models the 24-hour cycle of solar heating of the Earth's surface.

ŒAt the ground turbulent fluxes are computing using

the MOST according to the non-iterative formulation.

(x,0,t) 6 sin( t/43200) Θ = ⋅ π

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Convective Mixing Case

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Thermal circulation above the roughness area

• 1.0
• 1

.

• 0.5

0.0 . 0.0 1.0 1 . 1.0 1.0 1 . 1 . 1 . 1.0 1.0 1 . 1.0 1 . 1 . 5 1 . 5 3.0

• 1.5

2.5 2.0 1 . 5

1.0

1.0 1 .

• 0.5

1.5 2.0 2.0

• .

5

• 1.0
• 1.5
• 2

.

• 2.5

.

X, km Z, km

40 60 0.5 1 1.5 2 2.5

UG=1 ms

• 1

12:00

roughness area

• .

4

• 0.2
• .

2

• .

2

• 0.2

0.4 . 4 0.4 . 4 0.6 0.6 0.8 1.0 1.0 1.0 1.2 1.2 1 . 2 1.4 1.4 1.6 1 . 6 1.8 . 8

X, km Z / Zi

45 50 55 0.5 1 1.5

U=3 mc

• 1

12:00

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Vertical profiles of local friction velocity

u* / u*max Z / ZH

• 1

1 2 3 1 2 3 4 5 6 7

1 2 Rotach M. Oikawa and Meng Feigenwinter C. UG=3 ms

• 1

UG=5 ms

• 1

Real scale data Computation

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Vertical profiles of ratio u*/U

u* / U Z / ZH

0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 6 7

1 2 Real scale data from Roth (2000) Computation: UG=3ms-1 UG=5ms

• 1

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Vertical velocity variance

(<w

2>) 1/2/u*max

Z / ZH

1 1.5 2 2.5 1 2 3 4 5 6 7

1 2

1- U=3ms

• 1

2- U=5ms

• 1

Experimental data from Roth's review Simulation:

• Q. J. R. Meteor. Soc.
• 2000. Vol.126, 941-990.

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Airflow Velocity Sensitivity to Large Scale Roughness and Thermal Inhomogeneity

2.25 2.25 2.25 2.75 2.75 2.75 2.75 3 . 2 5 3 . 2 5 3 . 2 5 3.75 3.75 4 . 2 5 4.75 2.25 4 . 2 5 2.75 2 . 2 5 4 . 7 5

X, km Z, km

20 40 60 80 100 120 0.5 1 1.5 2

( a )

2.25 2.25 2.25 2.75 2.75 2.75 2.75 2.75 3.25 3.25 3.25 3 . 7 5 3 . 7 5 3.75 2.75 2.25 2.75 2.25 4 . 2 5 2.25

X, km Z, km

20 40 60 80 100 120 0.5 1 1.5 2

( b )

¯

(a)

¯

(b)

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Some Results about Stable Boundary Layer Structure

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“TRADITIONAL” Boundary Layer

Velocity, mc

• 1

Z, km

1 2 3 4 0.1 0.2 0.3 0.4

TKE, m

2c

• 2

Z, km

1 2 3 4 0.5 1 1.5

TKE

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“UPSIDE DOWN” Boundary Layer

Velocity, mc

• 1

Z, km

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5

22:00 24:00 03:00

center of large-scale roughness area (x=50)

TKE Height, km

0.5 0.5

TKE

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“UPSIDE DOWN” Boundary Layer

Velocity, mc

• 1

Z, km

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5

22:00 24:00 03:00

center of large-scale roughness area (x=50)

Temperature deviation Z, km

1 2 3 4 5 6 0.1 0.2 0.3 0.4

22:00 24:00 03:00

center of large-scale roughness area (x=50)

ENVIROMIS_ 2008, Tomsk

Thermal Stratified Boundary Layer over Flat Terrain The potential temperature θ and velocity U are shown for

the convective and stable boundary layers.

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Idealized nocturnal velocity and TKE profiles

• n the left side:

enhanced of the TKE by the presence of large-scale structures;

• n the right side:

shear and wave-induced TKE generation occurring in the upper ABL in the presence of the low-level jet propagated downward to augment near-surface TKE.

Vertical Velocity Profiles Generated by a Nocturnal Low-Level Jet: Modeling and Simulation

1.0 order model: Steeneveld,

van de Weil and Holtslag (2006)

WRF 2.5 order model:

Lundquist and Mirocha (2008)

‘nose’ of jet

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Stable boundary layer: low-level jet velocity profiles

Measured averaged velocity profile (CASES-99 data: Banta et al., 2006) Velocity profile above rough area computed with use the nonlocal turbulence model

U/Ux Z/Zx

0.25 0.5 0.75 1 1.25 0.5 1 1.5 2 2.5 3

normalized velocity profile at the center rough area (00:03)

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Upside-down boundary layer: vertical variance profiles behind rough area

Observations behind Rough Area

(from Lundquist and Mirocha, 2008)

Computing by the nonlocal turbulence model

<w

2> 1/2 [m/s]

Z, km

0.02 0.04 0.06 0.08 0.1 0.1 0.2 0.3 0.4 0.5

0024 0003 x=58

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Structural Features of Turbulent Transport in Strong Stable Boundary Layer

Experimental data

Rig =N

2/(gradzU) 2

PrT= KM/ KH

10

• 1

10 10

1

10

2

0.5 1 1.5 22:00 24:00 03:00

Simulation:

Medium Range Forecast MRF Scheme

1.3

employed in MM5

Simulation: nonlocal turbulence model

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Ohya’s experiment, BLM, 2001, vol. 98, 57-82.

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Countergradient term of vertical heat flux

• Countergradient term,

c

γ

h c

wθ γ Θ K z ∂ 〈 − ∂ + 〉 =

( )

c c 2

S, N, θ E γ γ ,τ /ε 〈 〉 = =

2 2 2

U V S z z ∂ ∂     = +     ∂ ∂    

( )

2

N βg Θ/ z = ∂ ∂ c

γ

2 i

1 E u TKE 2 = 〈 〉→

ε TKEdissipation →

Countergradient term, ms

• 1 0K

Z, km

10

• 7 10
• 6 10
• 5 10
• 4 10
• 3 10
• 2 10
• 1

10 0.05 0.1 0.15 0.2 0.25 0.3

20:00 22:00 24:00

center of large scale rougness area unstable case

SBL

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Vertical heat flux in strong stable boundary layer

<w >/U0(

• 0.2
• 0.1

0.1 0.2 0.2 0.4 0.6 0.8 1 θ

Θ Θ )

0-

s

x 10

• 3

δ

z/

Ohya's laboratory experiment (BLM. 2001. V. 98)

S8 S7

< θ >,

0.2 0.4 0.6 0.8 1 m

2 c

• 2

Z, km

Simulation

0.5

• 0.5

0.1

• 0.1

x 10

• 3

at the center of roughness area at 03:00

w

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Countergradient vertical heat flux

• The aircraft-based measurements disclosed slightly positive

potential temperature gradient with upward heat flux (from levels of 150 m until 1250m):

• and an eddy coefficient for heat defined by

it is NEGATIVE in a counter-gradient region.

• Therefore, the usual down-gradient eddy coefficient expression for

the heat flux has no usefulness in a counter-gradient region where it is negative!

( )

1 H

K w / z 0(?)

−

= −〈 θ〉 ∂Θ ∂ <

0 , / w z θ > ∂Θ ∂ > ( )

Z Z

wa K A/ z ,K const E 〈 〉 = − ∂ ∂ = ⋅ > l

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Countergradient heat transfer : study by means of nonlocal turbulence model

• Understanding the countergradient heat flow

˘

¤ when turbulent diffusion is negligible, the heat flux must

be d i r e c t e d d o w n the g r a d i e n t

¤ when turbulent diffusion term is positive and exceed the

smoothing term, the countergradient heat flux to be

• expected. Only then is

2

2 D P D Dt

θ θ θ

θ = + −ε

/

θ ≡ −

θ ⋅ ∂Θ ∂ > P w z " "

θ

ε − smoothing term

2

2

θ

∂ θ ≡ − ∂ w D turbulent diffusion z

! / w z 〈 θ〉 ⋅∂Θ ∂ >

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i

Z / Z

2 2

w / w 〈 θ 〉 θ

Æ Æ

• 0.5

0.5 1 1.5 2 0.5 1 1.5

1 2 3 the gradient model LES LES

• data

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The balance of temperature variance

i

Z / Z

• 0.0002
• 0.0001

0.0001 0.0002 0.8 0.8 1 1 1.2 1.2 1.4 1.4

1 2 3

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Countergradient heat transfer in Convective Mixing Case

• 0.4
• 0.2
• .

2

• .

2

• 0.2

0.4 . 4 0.4 0.4 0.6 0.6 . 8 1.0 1.0 1.0 1.2 1 . 2 1.2 1.4 1.4 1 . 6 1.6 1.8 0.8

X, km Z / Zi

45 50 55 0.5 1 1.5

U =3 mc

• 1

12:00 80.0

• 80.0
• 8

.

• 80.0
• 80.0
• 8

.

• 80.0
• 80.0

80.0

X, km Z, km

45 50 55 0.5 1 1.5

large scale aerodynamical roughness

'Negative' eddy thermal diffusivity Unstable Boundary Layer 12:00

Kh=-<w >/gradz θ Θ

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Countergradient heat transfer in

Strong Stable Boundary Layer

0.1 0.1 0.1 . 1 5 . 1 5 0.15 0.15 . 1 5 . 1 5 0.2 . 2 . 2 0.2 0.2 0.2 0.2 0.25 0.2 . 2 5 0.25 0.25 0.25 0.3 0.3

X, km Z, km 45 50 55 0.2 0.4 0.6 0.8 1 1.2 1.4

TKE

(03:00)

• 45
• 3

7

• 29
• 2

9

• 2

5

• 2

5

• 21
• 21
• 2

1

• 17
• 17
• 9
• 9
• 5
• 5
• 45
• 45
• 1
• 1

X, km Z, km 40 45 50 55 60 0.2 0.4 0.6 0.8 1 1.2 1.4

Countergradient vertical heat flux:

large scale roughness area

Kh=-<w >gradz θ 03:00 Θ < 0

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CONCLUSIONS

■Using the updated expressions for the pressure- velocity and pressure-temperature correlations, we have derived a improved nonlocal turbulence model for describing the PBL over the aerodynamically rough surface. ■ In simple 2D case are investigated the modifications in global structure of the PBL caused by the urban- like roughness and the thermal inhomogeneous. ■ Results of simulation show that the improved nonlocal turbulence model allows to obtain the basic structural features of the SSBL in qualitative agreement with the observations data.

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THANK YOU!