2 length-scales for inclined stable boundary layers › conferences › proceedings › _cavtat ›...
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11
22 LENGTHLENGTH--SCALESSCALESFOR INCLINED STABLE BOUNDARY FOR INCLINED STABLE BOUNDARY
LAYERSLAYERS
BBrankoranko Grisogono & Danijel Grisogono & Danijel BeluBeluššiićć
Dept. of Geophysics, Faculty of Sci., Univ. of ZagrebCroatia
TNX to: Amela Jeričević, Lukša Kraljević, Iva Kavčič & Leif Enger
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22
OUTLINEOUTLINE
1) Modeled SABL over1) Modeled SABL over--diffused; 2) Turbulence for Ri < 1diffused; 2) Turbulence for Ri < 1
&& Ri > 1Ri > 1
MODIFIED PRANDTL MODELMODIFIED PRANDTL MODEL ::
MONINMONIN--OBUKHOV LENGTH vs. OBUKHOV LENGTH vs. LLJLLJ HEIGHTHEIGHT
ADJUSTING ADJUSTING ““zz--lessless”” MIXING LENGTH FOR MIXING LENGTH FOR SABLSABL
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Stable Boundary LayersStable Boundary Layers 33
Mauritsen et al. JAS 2007 |
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 44
PRANDTL MODEL
z-less Mix. length modified
L MONIN-OBUKHOV modified
(if a sfc. layer exists)
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 55
PRANDTL MODELPRANDTL MODEL
Analytical model for the
“pure” katabatic flow
Balance between the
negative buoyancy &
turbulent diffusion
α < 0 αx
z
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 66
PRANDTLPRANDTL MODEL MODEL –– contcont’’dd
Simplified N-S eqns.Heat (pot. temperature) eqn.
Assume:steady-state [later: semi-stationarity (U & θ, not V)]Boussinesqhydrostatic balancelinearity (no advection) but for finite amplitude perturb.simple K-theory [later: K = K(z)]γ = dθ/dz = constant, θ tot = θ + γz, (Θ = θ0 + θ tot)
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 77
⎟⎠⎞
⎜⎝⎛+−=
⎟⎠⎞
⎜⎝⎛+=
dzdθK
dzdUγ
dzdUK
dzd
θg
)sin(0
Pr)sin(00
α
θαU:
θ:
PRANDTLPRANDTL: CLASSICAL : CLASSICAL ANALYTICAL MODELANALYTICAL MODEL
Scale analysis: fcos(α)V small in x-eqn. for U
Stationarity for U & θ after t > 1-1.5T, T = 2π /(Nsin(α)),
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Stable Boundary LayersStable Boundary Layers 88
MIUU:MIUU: Modified Modified ‘‘zz--lessless’’ Mixing lengthMixing length
NTKEaSTABl
2/1)(=
]||
2/1)(,2/1)(min[
STKEb
NTKEaSTABl =
Replace:
By:
… and find ‘b’ (‘a’ known, in MIUU model: a ≈ 0.5)
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Stable Boundary LayersStable Boundary Layers 99
Same start| L: Correct VSABL | R: Over-Diffusive SABL
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 1010
(a) simple numerical (oscil. with T ) (b) MIUU model(f, α, γ, Pr, C) = (1·10-4 s-1, −2.2°, 5·10-3 Km-1, 1.1, −6.5 °C)
K(z): h = 200 m, Kmax = 2 m2s-1 here: T = 3.4 h
C)(θ totnum ° C)(θ tot
MIUU °
t (h) t (h)
(a) (b)
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 1111
(a) simple numerical (b) MIUU modelThe rest as before. Max(Unum) ≈ 5.5 ms-1 & LLJ height ≈ 16 mNumerical (U, θ) with elevated spurious oscillations decaying in time, T ≈ 3.4h
)( 1−msUnum )ms(U 1MIUU
−
t (h) t (h)
(a) (b)
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Inclined SABL: 'zInclined SABL: 'z--less' ENDless' END 1212
Conclusion Conclusion ½½: MIXING LENGTH: MIXING LENGTH
(U, θ) reach steady state after t > T; Vnum & VWKB diffuse upwards
Simple numerical, WKB & MIUU MODEL agree
for (θtot ,U, V) only with new Mix. Length
USE:
Very SABL, Ri → ∞, => difficult to treat in NWP & climate
models (usually over-diffusive) a solution is offered
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For an existing inclined surface layerFor an existing inclined surface layer 1313
MONINMONIN--OBUKHOV vs. OBUKHOV vs. LLJLLJ HEIGHTHEIGHT
,,
3
θθ
w
ugk
LMO∗−
= 4/122
2
j )()(sin
Pr44
zα
πN
K=
2/1
3
5
22 )()( Pr|)sin(|
)(8
RikzL
j
MO απ
=
if α small, e.g. < 0.1 rad & Pr ≈ Ri ~ 1, =>LMO < zj , =>
classical Monin-Obukhov OK; note N is external param.
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Stable Boundary LayersStable Boundary Layers 1414
For (LMO/zj ) > 1, LMO is less relevant scale for sfc. fluxes
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Stable Boundary Layers ENDStable Boundary Layers END 1515
111 −−− += jMOMOD bzaLL
CONCLUSION 2/2: A modified LMO includes the LLJ
height:
Lmod = min(LMO, c zj ) , c ~ 0.8
Or at least:
For NWP & climate models
It tells where/when classic LMO wrong in sloped very SABL
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Stable Boundary LayersStable Boundary Layers 1616
APPENDIXAPPENDIX
If discussion desires, continue If discussion desires, continue ……
Otherwise, be seated quietlyOtherwise, be seated quietly……
& Let the others continue & Let the others continue ……
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 1717
⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+−=∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+−=∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
++=∂
∂
zθK
zU αγ
tθ
zVK
zUαf
tV
zUK
z Vαfθα
θg
tU
)sin(
Pr)cos(
Pr)cos()sin(0
B. C. 0)V(z)U(z)θ(z0)0V(z)0U(zC, )0θ(z
=∞→=∞→=∞→======
MODIFIED PRANDTLMODIFIED PRANDTL:: [[ff && K(z)K(z)]]
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Stable Boundary Layers 18
( )( ) ( )( ),cosexp zσzσCθ WKBWKBWKB −=
( ) ( )( ) ( )( ),sinexpsin
20 zσzσαγ
CσU WKBWKBWKB −=
( ) ( )( ) ( )( ) ,cosexpPr2)(1
Prcot
⎥⎦
⎤⎢⎣
⎡−−⎟
⎠
⎞⎜⎝
⎛−−
≈ zσzσtzIerf
γαCfV WKBWKBWKB
( ) ),(20 zIzσWKB
σ=
( )∫ −=z
dzzKzI0
2/1 ,)(
( ) ( )2
22224
0 PrcossinPr αfαNσ +
=
PRANDTL: PRANDTL: WKB SOLUTIONS FOR WKB SOLUTIONS FOR ((θθ,, UU, V), V) ; ; K K == K(z)K(z)
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Stable Boundary LayersStable Boundary Layers 1919
• K(z) profile: ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 2
2
max 2exp
hz
hzeKK(z)
K(z) for numerical &
analytical solutions,
only 2 parameters!
e.g.
Kmax = 2 m2s-1, h = 200 m,
(JAS 2001, QJ2001,...)
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2020
a) t = T b) t = 10T, T ≈ 2.1h(α, γ, Pr, C, f ) = -4°, 4 ·10-3 Km-1, 1.1, - 8°C, 1.1·10-4s-1
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 2121
(a) t = T (b) t = 10T(Stiperski et al. QJRMS 2007; Kavčič & B.G., BLM 2007) Vf uses K = const = Kmax / 3
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Inclined Stable Boundary LayersInclined Stable Boundary Layers 2222
(a) WKB solution (b) MIUU modelThe rest as before. Min(VWKB) ≥ -2.75 ms-1
WKB sol. has no spurious (low energy) oscillations
)( 1−msVWKB )ms(V 1MIUU
−
t (h) t (h)
(a) (b)
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2323
MIUUMIUU MODEL MODEL & & SSimulationimulation PParametersarameters
nnonlinearonlinear, h, hydrostaticydrostatic, , ff--planeplane
2.52.5--turbulence closure levelturbulence closure level
5 progn. eqns: 5 progn. eqns: UU, , VV,, ΘΘ, , qq,, TKETKE (e.g. (e.g. QJRMSQJRMS 20042004))
•• ΔΔxx ≈≈ 1.9 km = const,1.9 km = const, αα = = --2.22.2oo, C , C ≈≈ --6.56.5ooCC
•• 211 211 ×× 7 7 ×× 201 201 gridpointsgridpoints ((y=consty=const), ), total(nktotal(nk)=402)=402
•• ΔΔtt = 13 s= 13 s, , zzTOPTOP = 5.6 km = 5.6 km ((““spongesponge”” for for zz ≥≥ 4.4 km4.4 km))
•• 1 m < 1 m < ΔΔz <z < 30 m30 m, , ““staggered, terrainstaggered, terrain influencedinfluenced””
•• Initialisation: Initialisation: UU && VV ≈≈ 0,0, γγ = 5= 5··1010--33 KmKm--11
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Mix. length: NEW, left; OLD, rightMix. length: NEW, left; OLD, right 2424
LSTAB = min[a(TKE)1/2/N, a/2(TKE)1/2/|S|], a ≈ 0.5