Weakly-nonlinear Prandtl model for

simple slope flows

1Branko Grisogono, 1Toni Jurlina, 1Željko Večenaj & 2Ivan Gűttler

1Dept. of Geophysics, Faculty of Sci. & Math., Univ. of Zagreb, Croatia2Meteorological and Hydrological Service, Zagreb, Croatia

Kranjska Gora, ICAM2013, June

Weakly-nonlinear Prandtl model

After E. Ekhart 1948 After C.D. Whiteman 2000

…perspective…

CONTENT

� INTROMotivation & Overview

� PRANDTL MODELClassic, Now weakly-nonlinear, K(z)WKB approximation

� RESULTS & DISCUSSIONComparison with Pasterze glacier wind data,recommendations to climate modelers to parameterize …

Weakly-nonlinear Prandtl model

Winter & summer T2m “ errors” st order turb. schemeSame but with H.O.C. turbulence scheme

Typical reg. clim. simulation, RegCM, 1989-1998, dx ≈ 50 km

Winter & summer wind magn. errors, st order turb. schemeSame but with H.O.C. turbulence scheme

Clim. simulations & obs – cont’d - “errors” for wind simulations

Many avenues to cross…

� Better resolution, numerics, SEB, micro-physics, clouds, precipitation, ABL…

� Improving ABL schemes over complex terrain, especially those for the SABL

� …Thermally driven flows should be better treated in climate models…start with

Prandtl “kata-ana” model…

Weakly-nonlinear Prandtl model

Weakly-nonlinear Prandtl model

+Γ−=

+=

dz

dθK

dz

dU

dz

dUK

dz

d

θ

g

)sin(0

Pr)sin(00

α

θαU:

θθθθ:

CLASSIC PRANDTL MODEL

� Stationarity for (θθθθ , , , , U) after t > 1-1.5T, T = 2ππππ /(Nsin( αααα))

� PDE → ODE

� (α, Γ, C) => θ(z), U(z) = exponential-complex decay

Modified: weakly-nonlinear Prandtl model

u:

θ:

� �

Weakly-nonlinear Prandtl model

ε1:

-This is again damped oscillator but now forced by the ε0 state.

-Whole solution = 0TH + 1ST order � exp-decay with various

coeff’s … each (u1, θ1) has 5 terms…(see Toni’s diploma work)

�

�

Weakly-nonlinear Prandtl model

)exp(

)max(

πε

−

Γ≤

phC

)exp(

15)max(

πε

−

Γ

≤

ph

C

� Estimating the small parameter ε, … expecting ε ≤ O(0.1)

Thermodyn. Eqn → … Numerical

experiments →

−6 −5 −4 −3 −2 −1 0 1 20

5

10

15

20

25

30

35

40

POT. TEMP., 0 C

he

igh

t, m

<−−eps=0.01<−−eps=0.02

<−−eps=0.03

eps=0.04

theta0

theta0+0.01*theta

1

theta0+0.02*theta

1

theta0+0.03*theta

1

theta0+0.04*theta

1

−4 −3 −2 −1 0 1 2 3 4 50

5

10

15

20

25

30

35

40

WIND SPEED ms−1

he

igh

t, m

eps=0.04

eps=0.03

eps=0.02

eps=0.01

u0

u0+0.01*u

1

u0+0.02*u

1

u0+0.03*u

1

u0+0.04*u

1

here for Pasterze data: ε = 0.005

Weakly-nonlinear Prandtl model

WKB approximation - gradual K(z)

Solutions keep their original structure as from the const. coeff.

ODE solutions but now with local values (integrated from

below if previously multiplied by z) � 0TH order WKB

Chosen K(z) = K0 z/h exp{-z2/(2h2)}, or similar, h > LLJ

details in JAS’01, QJRMS’01, ACP’10, etc.

Weakly-nonlinear Prandtl model

−8 −6 −4 −2 0 2 4 60

5

10

15

20

25

30

35

40C=−7 °C, Pr=1.4, max[K(z)]=0.36 m2s−1, h=35 m, gamma=5 Kkm−1, eps=0.005

POT. TEMP., 0 C, WIND SPEED, ms−1

he

igh

t, m

u

0

utot

u0,WKB

utot,WKB

th0

thtot

th0,WKB

thtot,WKB

−8 −6 −4 −2 0 2 4 60

5

10

15

20

25

30

35

40C=−7.5 °C, Pr=1.6, max[K(z)]=0.36 m2s−1, h=35 m, gamma=5 Kkm−1, eps=0.005

POT. TEMP., 0 C, WIND SPEED, ms−1

he

igh

t, m

u

0

utot

u0,WKB

utot,WKB

th0

thtot

th0,WKB

thtot,WKB

PASTEX’94 data (8 level 13 m tower & balloon) vs. Prandtl model

Morning ↑ Afternoon →

tower = green - hatched

balloon = turquoise - star-dashed

Conclusions

- Climate models poorly treat diurnal mnt. cycle, coastal & mnt. flows, related precip….

- Slope flows parameterization should be included in climate models since the models’ poor Δz, Δx

-Modified Prandtl model allows for stronger & sharper inv. during weaker winds, large near-surface Rigrad # – missing in most of NWP & climate models

_____________________________________________

-Chances for diploma works, PhD ↔ parameterization ↔ postdocs, senior projects – climate modeling…

Weakly-nonlinear Prandtl model

Spare slides

• Eqns!

Weakly-nonlinear Prandtl model

Weakly-nonlinear Prandtl model

20

2

4 Pr

)(sin4

K

g

hp ΘΓ= α

)exp(~hom,1 λξθphz /2=ξ

)2

cos()2

exp( G)2

sin()2

exp( G 21hom,1

ξξξξθ −+−=

)exp( sin)exp( cos)exp( 111,1 ξξξξξθ −+−+−= CBApart

0

2

01/2

,Pr

, Pr

)sin( ,

2

ΘΓ=

ΓΘ=== g

Ng

K

Nhp µασ

σ

Czzz ztot =+= →010 |)()()( εθθθ 0|)()()( 10 =+= ∞→ztot zzz εθθθ

0|)()()( 010 =+= →ztot zuzuzu ε 0|)()()( 10 =+= ∞→ztot zuzuzu ε

0)0()0( 11 ==== zuzθ 0)()( 11 =∞→=∞→ zuzθ

Weakly-nonlinear Prandtl model

Final sol. for ε1 part (1st order), while the whole solution = 0th + 1st order:

]10

1)

2cos(

15

1)

2sin(

15

1)[

2exp(

)]cos(6

1)sin(

15

1)[exp()(1

++−+

−−−=

pppA

pppA

h

z

h

z

h

z

h

z

h

z

h

zz

θ

θθ

]10

1)

2sin(

30

1)

2cos(

30

1)[

2exp(

)]cos(15

2)sin(

3

1)[exp()(1

−+−−+

+−−=

pppA

pppA

h

z

h

z

h

zu

h

z

h

z

h

zuzu

Γ==

pA

pA h

Cu

K

Ch µαµθ

22

,)sin(