2 LENGTH-SCALES FOR INCLINED STABLE BOUNDARY LAYERS › Conferences › Proceedings › _Cavtat ›...

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

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

Stable Boundary LayersStable Boundary Layers 33

Mauritsen et al. JAS 2007 |

Inclined Stable Boundary LayersInclined Stable Boundary Layers 44

PRANDTL MODEL

z-less Mix. length modified

L MONIN-OBUKHOV modified

(if a sfc. layer exists)


PRANDTL MODELPRANDTL MODEL

Analytical model for the

“pure” katabatic flow

Balance between the

negative buoyancy &

turbulent diffusion

α < 0 αx

z


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)


⎟⎠⎞

⎜⎝⎛+−=

⎟⎠⎞

⎜⎝⎛+=

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)),


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)


Same start| L: Correct VSABL | R: Over-Diffusive SABL


(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)


(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)

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

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.


For (LMO/zj ) > 1, LMO is less relevant scale for sfc. fluxes

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


APPENDIXAPPENDIX

If discussion desires, continue If discussion desires, continue ……

Otherwise, be seated quietlyOtherwise, be seated quietly……

& Let the others continue & Let the others continue ……


⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+−=∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+−=∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

++=∂

∂

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)]]

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)


• 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,...)

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


(a) t = T (b) t = 10T(Stiperski et al. QJRMS 2007; Kavčič & B.G., BLM 2007) Vf uses K = const = Kmax / 3


(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)

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

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

2 LENGTH-SCALES FOR INCLINED STABLE BOUNDARY LAYERS › Conferences › Proceedings › _Cavtat ›...

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