Struttura e modellazione dello strato limite atmosferico · altezza dello strato limite stabile 1...

Struttura e modellazione dello stratolimite atmosferico

Francesco Tampieri

CNR ISAC, Bologna

Struttura e modellazione dello strato limite atmosferico – p. 1

definizioni

’friction velocity’ (scala delle velocita’):

u∗ =“

u′

1u′

3

2+ u′

2u′

3

2”1/4

(1)

scala di temperatura:

ϑ∗ = u′

3ϑ′|0/u∗(2)

lunghezza di Obukhov:

LMO = −ϑ00u3

∗

κgu′

3ϑ′|0(3)

altezza dello strato limite h


similarita’

nondimensional variables ζ = x3/L and ξ = x3/h

nondimensional vertical gradients of mean velocity and temperature

du1

dx3=

u∗

κx3Φm(ζ, ξ) ;

dϑ

dx3= −

ϑ∗

κx3Φh(ζ, ξ)(4)

profiles: e.g.

u(z) − u(z0) =u∗

κ

»

ln(z/z0) +

Z ζ

ζ0

Φm(ζ′) − 1

ζ′dζ′–

=u∗

κΥ(ζ, ζ0)(5)

nondimensional nth order moments

u′ni

un∗

= Φ(n)i (ζ, ξ) ;

ϑ′2

ϑ2∗

= Φ(2)ϑ (ζ, ξ)(6)


il prototipo

strato limite su terreno piatto ed omogeneo

∂ui

∂t= εij3f(uj − ugj) +

∂

∂x3

„

ν∂ui

∂x3− u′

iu′

3

«

(7)

∂ϑ

∂t=

∂

∂x3

χ∂ϑ

∂x3− u′

3ϑ′

!

(8)

Se i flussi non sono costanti con la quota lo strato limite non puo’ essere contemporaneamente

stazionario e unidimensionale.


flussi: qnbl

Flussi turbolenti in condizioni neutrali sul mare. Da Garratt (1992).


flussi: cbl

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

z/h

(uw2+vw2)1/2/u*2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

z/h

wt/wt|0

Momentum and heat vertical fluxes in the CBL, normalised to the surface values: u2∗

obtained as a

best fit from the (u′w′2

+ v′w′2)1/2 data, and ϑ′w′|0, from the observations by Hartmann

(pers.comm.).


flussi: sbl

Flussi di quantita’ di moto e di calore in condizioni stabili (da Nieuwstadt, 1985). Qui

τ =q

(−u′w′)2 + (−v′w′)2.


altezza dello strato limite stabile

1

10

100

1000

0.01 0.1 1 10 100

h

z/LMO

z=3 m; z0=z0t=0.01 m

Zilitinkevic and Esau,2007

(after Zilitinkevich and Esau, 2007)


unstable surface layer

0.1

1

10

100

0.001 0.01 0.1 1 10 100

u3’2

/u*2

-z/LMO

ARTOVEq. 43

SGS2000

0.01

0.1

1

10

100

1000

10000

0.001 0.01 0.1 1 10 100

t’2 /T*2

-z/L

’../artov-tot-3-26062008.dat’ u ($37<0?(-$21):1/0):($18>0.01? ($12/$20)**2:1/0)’’ u ($37<0?(-$21):1/0):($18<0.01? ($12/$20)**2:1/0)

f(x,a,b)h(x,d)

u′23 /u2

∗and ϑ′2/ϑ∗ for unstable conditions


free convection scaling

w∗(x3) =

„

g

ϑ00u′

3ϑ′|0x3

«1/3

; ϑ∗∗ = u′

3ϑ′|0/w∗(9)

0.1

1

10

100

0.001 0.01 0.1 1 10 100

w’2

/w*2 (z

)

-z/LMO

ARTOVARTOV mean free convection value

SGS2000

0.1

1

10

100

1000

10000

0.1 1 10 100t’2 /T

**2

-z/L

’../artov-tot-3-26062008.dat’ u ($21<-0.1?(-$21):1/0):(($12/$24)**2)’’ u($21<-0.1?(-$21):1/0):($37<0.?(($12/$24)**2):1/0)

u′23 /w2

∗(z) and ϑ′2/ϑ∗∗ as function of stability ζ


vento medio

1

10

100

0.001 0.01 0.1 1 10 100

u/u *

-z/L

ARTOV z=3m

’../artov-tot-3-26062008.dat’ u ($37<0.? -$21:1/0):($18>0.01?($3/$19):1/0)1./0.4*log(3./0.022)

’../../../parametrizzazioni/codici/u_t_bel_hol-artov.dat’ u ($1<0? -$1:1/0):3’../../../parametrizzazioni/codici/u_t_kad-artov.dat’ u ($1<0? -$1:1/0):3

Da Kader and Yaglom (1990):

u(z) =u∗

κlog

„

z

z0

«

, − ζ0 < −ζ < ζA(10)

u(z) =u∗

κ

»

log

„

ζA

−ζ0

«

+ 3A“

ζ−1/3A − (−ζ)−1/3

”

–

, ζA < −ζ < ζB(11)

u(z) =u∗

κ

»

log

„

ζA

−ζ0

«

+ 3A“

ζ−1/3A − ζ

−1/3B

”

+ 3B“

(−ζ)1/3 − ζ1/3B

”

–

, − ζ > ζB(12)


vento medio (2)

1

10

100

0.001 0.01 0.1 1 10 100

u/u *

-z/L

ARTOV z=3m

’../artov-tot-3-26062008.dat’ u ($37<0.? -$21:1/0):($18>0.01?($3/$19):1/0)1./0.4*log(3./0.022)

’../../../parametrizzazioni/codici/u_t_bel_hol-artov.dat’ u ($1<0? -$1:1/0):3’../../../parametrizzazioni/codici/u_t_kad-artov.dat’ u ($1<0? -$1:1/0):3

Da Beljaars and Holtslag (1991):

u(z) =u∗

κ

»

log

„

z

z0

«

+ log

»

(1 + ξ)2(1 + ξ2)

(1 + ξ0)2(1 + ξ20)

–

+ 2 [arctan(ξ) − arctan(ξ0)]

–

(13)

ξ = (1 − 16ζ)1/4(14)


scaling with z/h (ζ ≫ 1)

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1

u 3’2 /w

*2 (h)

z/h

ARTOVKader, 1994, Eq. 40

HartmannLenschow et al, 1980, Eq. 42

0.1

1

10

100

1000

0.001 0.01 0.1 1

t’2 */t *

*2 (h)

z/h

’artov-tot.dat’ u($21<-0.5? (3./$32):1/0):($20>0.2?($12/$40)**2 :1/0)h(x,a,b)

from Hartmannsigma1(x)sigma2(x)

u′23 /w2

∗(h) and ϑ′2/ϑ∗∗(h) for free convection conditions:

from Lenschow et al. (1980):

u′23

w2∗(h)

= 1.8ξ2/3(1 − 0.8ξ)2(15)

from Strunin et al. (2004):

ϑ′2

ϑ2∗∗

(h)= 1.8ξ−2/3 (1 − ξ)4/3 + 1.4ξ4/3 (1 − ξ)−2/3(16)


sbl

from Yague et al. (2006): Φm vs. ζ from the field experiment SABLES98: a: z = 5.8 m, b:z = 13.5m, c: z = 32 m.z-less parameterisation (Wyngaard and Coté, 1972; Nieuwstadt, 1985): local values to determineLMO , du/ dz constant with z

u∗-less hypotesis (Grachev et al., 2007): Φm ∝ ζ1/3


sbl

As before, but for Φh.

The u∗-less hypotesis suggests Φh ∝ ζ−1/3


sbl: u/u∗ from ARTOV data (1)

1

10

100

1000

0.001 0.01 0.1 1 10 100

u/u *

z/L

ARTOV z=3m

’../artov-tot-3-26062008.dat’ u ($37>0.? $21:1/0):($3/$19)a*x**(1./3.)

1./0.4*log(3./0.022)’../../../parametrizzazioni/codici/u_t_bel_hol-artov.dat’ u ($1>0? $1:1/0):3

’../../../parametrizzazioni/codici/u_t_cb-artov.dat’ u ($1>0? $1:1/0):3

Beljaars and Holtslag (1991):

u(z) =u∗

κ

»

log

„

z

z0

«

+ a(ζ − ζ0) + b [(ζ − c/d) exp(−dζ) − (ζ0 − c/d) exp(−dζ0)]

–

(17)

Cheng and Brutsaert (2005):

u(z) =u∗

κ

"

log

„

z

z0

«

+ a logζ + (1 + ζb)1/b

ζ0 + (1 + ζb0)1/b

#

(18)


sbl: u/u∗ from ARTOV data (2)

1

10

100

1000

0.001 0.01 0.1 1 10 100

u/u *

z/L

ARTOV z=3m

’../artov-tot-3-26062008.dat’ u ($37>0.? $21:1/0):($3/$19)a*x**(1./3.)

1./0.4*log(3./0.022)’../../../parametrizzazioni/codici/u_t_hog_mod-artov.dat’ u ($1>0? $1:1/0):3’../../../parametrizzazioni/codici/u_t_yag_mod-artov.dat’ u ($1>0? $1:1/0):3

Högström (1996):

u(z) =u∗

κ

»

log

„

z

z0

«

+ αm3 (ζ − ζ0)

–

(19)

u∗-less modification (ζ > ζm):

u(z) =u∗

κ

»

log

„

ζm

ζ0

«

+ αm3 (ζm − ζ0) + 3“

ζ−1/3m + αm3ζ

2/3m

”“

ζ1/3 − ζ1/3m

”

–

(20)


sbl: u′23/u2

∗

0.1

1

10

100

0.001 0.01 0.1 1 10 100

u3’2

/u*2

z/LMO

ARTOVEq. 43

SGS2000

left: Yague, pers. comm.; right: ARTOV data


sbl

0

0.5

1

1.5

2

2.5

3

3.5

0.1 1 10

pdf(

w2 /u

*2 )

w2/u*2

dati ARTOV

|z/L|<0.1lg3(x)

0

0.5

1

1.5

2

2.5

0.1 1 10 100

pdf(

(w2 /u

*2 )(z/

L)-2

/3)

(w2/u*2)(z/L)-2/3

dati ARTOV

1<z/L<10lg3(x)

’../codici/istog-log-wust-05-50.dat’ u 5:3

w′2/u∗2 = 1.25 per |z/L| < 0.1

w′2/u∗2 = 2.06(z/L)2/3 per 1 < z/L < 10


Richardson numbers (1)

Flux Richardson number Rf :

Rf =

gϑ00

w′ϑ′

u′w′ dudz

(21)

Gradient Richardson number Rg :

Rg =

gϑ00

dϑdz

“

dudz

”2(22)

Bulk Richardson number Rg :

Rb =g

ϑ00

∆ϑ

z2 − z1

(z3 − z0)2

u2(23)

L−1MO = Rb

Υ2m (ζ3, ζ0)

Υh (ζ2, ζ1)

z2 − z1

(z3 − z0)2(24)


Richardson numbers (2)

0.01

0.1

1

10

100

0.1 1 10 100

Rf

z/LMO

z=3 m; z0=z0t=0.01 m

Hogstrom, 1996 (modificato)Beljaars and Holstag, 1991Cheng and Brutsaert, 2005

from Yague et al, 2006 (modificato)

0.01

0.1

1

10

0.1 1 10 100

Rb

z/LMO

z=3 m; z0=z0t=0.01 m

Hogstrom, 1996 (modificato)Beljaars and Holstag, 1991Cheng and Brutsaert, 2005

from Yague et al., 2006 (modificato)


sommario

leggi di similarita’: consistenza delle leggi di potenza per i momenti primo e secondo(almeno!)

estensione a condizioni di forte instabilita’ (free convection) e forte stabilita’: problema deidati

altezza dello strato limite stabile e flussi non costanti

effetti della scelta delle parametrizzazioni sulla stima della stabilita’ al suolo nei modelli


References

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22-3

Struttura e modellazione dello strato limite atmosferico · altezza dello strato limite stabile 1...

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