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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
Struttura e modellazione dello strato limite atmosferico – p. 2
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)
Struttura e modellazione dello strato limite atmosferico – p. 3
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.
Struttura e modellazione dello strato limite atmosferico – p. 4
flussi: qnbl
Flussi turbolenti in condizioni neutrali sul mare. Da Garratt (1992).
Struttura e modellazione dello strato limite atmosferico – p. 5
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.).
Struttura e modellazione dello strato limite atmosferico – p. 6
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.
Struttura e modellazione dello strato limite atmosferico – p. 7
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)
Struttura e modellazione dello strato limite atmosferico – p. 8
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
Struttura e modellazione dello strato limite atmosferico – p. 9
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
Struttura e modellazione dello strato limite atmosferico – p. 10
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)
Struttura e modellazione dello strato limite atmosferico – p. 11
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)
Struttura e modellazione dello strato limite atmosferico – p. 12
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)
Struttura e modellazione dello strato limite atmosferico – p. 13
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
Struttura e modellazione dello strato limite atmosferico – p. 14
sbl
As before, but for Φh.
The u∗-less hypotesis suggests Φh ∝ ζ−1/3
Struttura e modellazione dello strato limite atmosferico – p. 15
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)
Struttura e modellazione dello strato limite atmosferico – p. 16
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)
Struttura e modellazione dello strato limite atmosferico – p. 17
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
Struttura e modellazione dello strato limite atmosferico – p. 18
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
Struttura e modellazione dello strato limite atmosferico – p. 19
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)
Struttura e modellazione dello strato limite atmosferico – p. 20
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)
Struttura e modellazione dello strato limite atmosferico – p. 21
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
Struttura e modellazione dello strato limite atmosferico – p. 22
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