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Some recent findings about Some recent findings about very stable boundary layersvery stable boundary layers

Branko GrisogonoBranko Grisogono, , AMGI, ZagrebAMGI, ZagrebII..KavčičKavčič, I.Stiperski,, I.Stiperski,

Mark Ž, Danijel B, Leif E, Larry M, Dale D, Thorsten M, Mark Ž, Danijel B, Leif E, Larry M, Dale D, Thorsten M, Amela J, Sergej Z, …Amela J, Sergej Z, …

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• PRANDTL MODEL PRANDTL MODEL [[ff ,, K(z)K(z)]]

• ASYMPTOTIC SOLUTION - ASYMPTOTIC SOLUTION - WKBWKB MET METHODHOD

• MIUUMIUU MODEL – CHARACTERISTIC LENGTHSCALEMODEL – CHARACTERISTIC LENGTHSCALE

• PRANDTL-NUMERICAL & MIUU SIMULATIONPRANDTL-NUMERICAL & MIUU SIMULATION

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CLASSICCLASSIC PRANDTL MODELPRANDTL MODEL

x

z• Analytic model for a simple katabatic

flow – basic, dirty & nice…

• Balance between neg. buoyancy &

turbulent diffusion

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zθK

zU αγ

tθ

zVK

zUαf

tV

zUK

z Vαfθα

θg

tU

)sin(

Pr)cos(

Pr)cos()sin(0

B. C.

U:

V:

:

00000

)V(z)U(z)θ(z)V(z)U(zC, )θ(z

NEWNEW PRANDTL PRANDTL: [: [ff , , K(z)K(z)]]

Valid even for finite-amplitude perturbations,Stiperski et al. QJRMS 2007; Kavčič & Grisogono, BLM 2007

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(z)σ(z)σCθ WKBWKB

WKB

2(z)σ

sin2(z)σ

expγsin(α)Cσ

U

2cos

2exp

WKBWKB20

WKB

Solutions: Solutions: UU && ;; K K == K(z) K(z)as for steady flow:as for steady flow:

Pr = Km/Kh= const ?; K(z)=Kh, e.g. Ko(z/h)exp(-0.5z2/h2)Grisogono & Oerlemans, JAS 2001; Parmhed et al. QJRMS 2004; etc.

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Numerical (dashed), analytic WKB (full) U i tot, (a) t = T; (b) t = 10T. (α, γ, Pr, C, f) = -4°, 4K/km, 1.1, -8°C, 1.1∙10-4s-1. T=2π/sin(α) ~ 1-5 hours.

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Solution: Solution: VV(z,t(z,t)... Boring skip it…)... Boring skip it…

2cos

2exp

Pr210

(z)σ(z)σt

I(z)-erf VV WKBWKBWKB

I(z)zK zKKc

)(2/1

TttzIerf

tKzerf zKK

c

,

Pr2)(

Pr2 )(

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Numerical (dashed) & analytic (VWKB – full), (c) t = 20T i (d) t = 50T. Other parameters as before. Vf is solution a reasonable K=const.

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MIUUMIUU (Leif’s) (Leif’s) MODELMODEL 3D, 3D, nonlinear, hydrostatic,nonlinear, hydrostatic, ff =const=const

H-O-C turb. parameterization (level 2.5 & many fine details) H-O-C turb. parameterization (level 2.5 & many fine details) 5 progn. 5 progn. Eqn’sEqn’s: : UU, , VV, , , , qq,, TKETKE (e.g. Grisogono & Enger QJRMS 2004)(e.g. Grisogono & Enger QJRMS 2004)

HereHere:: xx ≈ 1.9 km = const, slope = -2.2≈ 1.9 km = const, slope = -2.2oo, C = ≈6.5, C = ≈6.5ooCC 211 × 7 × 201 points (211 × 7 × 201 points (y=consty=const), total (nk)=402), total (nk)=402 tt = 13 s = 13 s, , zzTOPTOP = 5.6 km = 5.6 km (“s(“spongeponge”” from from zz = 4.4 km = 4.4 km)) 1 m < 1 m < zz < < 29.5 m, 29.5 m, ““staggered, terrainstaggered, terrain influencedinfluenced”” IniInitialized by:tialized by: UU i i VV ≈ ≈ 0, 0, (z) (z) 5K/km 5K/km

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LSTAB = 2A*min[ (TKE)1/2/N, (TKE)1/2/S]

Too much mixing, too elevated INV. & LLJ:=>

LSTAB = min[2A(TKE)1/2/N, A(TKE)1/2/S]

Grisogono &Enger, QJRMS 2004

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New lowered LSTAB (z,t) in MIUU following simulations…

Analytics used for model tuning – coeff. choice for:LSTAB, as from obs. data or LES => get model coeff…

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(t, z) for tot = + z MIUU model: old L (a) & new L (b) (f, , , Pr, C) = (1.03×10-4 s-1, −2.2°, 5×10-3 Km-1, 1.1, −6.5°C), T = 3.39 h, (for K: hmax = 200 m, Kmax = 2 m2s-1).

C)(θ totMIUU

t (h) t (h)

(a) (b)With old L New LNo explicit Shear in L; ~same res. if same weight in L to N & S

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(t, z) for tot = + z from simple numerical (a) & MIUU model (b) (f, , , Pr, C) = (1.03×10-4 s-1, −2.2°, 5×10-3 Km-1, 1.1, −6.5°C), T = 3.39 h, (for K: hmax = 200 m, Kmax = 2 m2s-1).

C)(θ totnum C)(θ tot

MIUU

t (h) t (h)

(a) (b)

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U(t, z) from simple numerical (a) & MIUU model (b). The rest as before. Max(Unum) ≈ 5.4 ms-1 & LLJ at ≈ 16 m. The input parameters as before.

)( 1msU num )( 1msU MIUU

t (h) t (h)

(a) (b)

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V(t, z) from simple numerical (a) & MIUU model (b).The rest as before. Min(Vnum) ≥ -2.6 ms-1. The input parameters as before.

)( 1msVnum )( 1msVMIUU

t (h) t (h)

(a) (b)

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V(t, z) from simple numerical (a) & MIUU model (b).The rest as before. Min(VWKB) ≥ -2.75 ms-1. The input parameters as before.

)( 1msVnum )( 1msVWKB

t (h) t (h)

(a) (b)

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tot WKB method & simple numerial (a) & MIUU model (b), after t = 20 h. The input parameters as before.

C)(θθ totnum

totWKB , C)(θ tot

MIUU (a) (b)

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Mauritsen et al. JAS2007, TTE = TKE + TPE…. In H-O-C Approach

IMPORTANT TOMODEL SABLMOREPROPERLYIN CLIMATEMODELS

-AFFECTS E.G. ICE &

GLACIERS MASS

BALANCE VIA MELTING, MICRO-

CLIMATE…

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CONCLUSIONCONCLUSION

Agreement among: analytic (WKB), numerical &

MIUU model solutions for ((U, VU, V,,

V(z,t) difusses upV(z,t) difusses up; ; (U, (U, θθ)) quasi-steady quasi-steady Limited data comparisons…Limited data comparisons…

ZOOM OUT :ZOOM OUT :

Very Very SSABL,ABL, Ri -> ∞, => tough to parameterize & Ri -> ∞, => tough to parameterize &

model the flows (here a partial sol. given?)model the flows (here a partial sol. given?)

typically NWP models are over-diffusivetypically NWP models are over-diffusive

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U(z) from WKB method & simple numerical (a) & MIUU model (b), after t = 20 h. The input parameters as before.

)(, 1msUU numWKB )( 1msU MIUU(b)(a)

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V(z) from WKB method & simple numerical (a) & MIUU model (b), after t = 20 h. The rest as before.

)(, 1msVV numWKB )( 1msVMIUU(b)(a)

Spare slide