Eddy Dissipation Rate Improvement of EDR forecast by ... and Kolmogorov using a turbulent length...

Improvement of EDR forecast by consideration of horizontal shear and SSO wake production using

the concept of scale separation:

Matthias Raschendorfer, FE 14

COSMO user seminar Offenbach: 09-11.03.2009Matthias Raschendorfer

DWD-Project:

Improvement of turbulence forecast for civil aviation

Eddy Dissipation Rate

Motivation:

Turbulence has major impacts on aviation:

Impacts of turbulenceTurbulence has major impacts on aviation. According to a review of National Traffic Safety Board data from 1992 to 2001 by the National Aviation Safety Data Analysis Center, turbulence was a factor in at least 509 accidents in the United States, including 251 deaths (mostly in the general aviation sector). Additionally, the FAA Joint Safety Analysis Team estimated that there are more than 1,000 minor turbulence-related injuries on commercial aircraft annually. Airlines lose millions of dollars every year due to turbulence because of injury claims, delays, extra fuel costs, and aircraft damage.

NCAR News Release

NCAR Teams with United Airlines to Pinpoint Turbulence in Clouds;Research Can Help Reduce Delays, Injuries, CostsSeptember 6, 2007


( )k32EDR 3

2

lnln ⋅−⎟⎟⎠

⎞⎜⎜⎝

⎛⋅α

turbulent peak wavelength

ln [wave number k]

( )[ ]( )aircraftoffunctionnattenuatio

TKEofdensityspectralklnln

+⋅

( )Vfln

frequency of aircraft oscillations

aircraft velocity with respect to mean wind

TKECKE

model resolution

turbulence

attenuation functionand velocity

of the aircraft

spectrum of vertical oscillations

inertial sub range spectrum of atmosphere

EDR by regression of the Kolmogorov spectrum

Aircraft measurements of EDR:( )[ ]energywindofdensityspectralk ⋅ln


Turbulence index = 1 (light) Turbulence index = 4 (moderate)

Turbulence index = 5 (severe)Colours for measurement height in [m]


EDR as part of the budget for sub grid scale kinetic energy SKE:

∑=

′=31i

2i

2 vq,

: = summation of the velocity variances for all directions

( )′∇⋅′−θ′′θ

+′∇υ−∇⋅′′−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′′+⋅−∇≈⎟

⎠⎞

⎜⎝⎛∂ ∑∑∑

===

pvwgvvvvvvvq21q

21

hhvv31i

2ii

31ii

31i

2i

22t

,,,

SKE SKE flux density Shear-Production (PS)

Buoyancy-Production (PA)

Eddy DissipationRate (EDR)

Sub gr. Scale Orography Production (PSSO)

0≥ 0< 0≥unstable:neutral:stable: 0<

0>0=

Time tendency of SKE +Convergence

of SKE= + ++

= 2 SKE

v


circvturbv ww θ′′+θ′′

Circulation production (PC) 0≥

Turbulence closure on level 2.5 (according to MY):

• We solve the system of reduced second order equations (after substituting the turbulent stress tensor by its traceless form)

using special closure assumptions in order to express the unknown statistical moments in those equations

• They are in accordance only with a special class of sub grid scale structures that we call turbulence (stochastic isotropic)

- In particular we neglect all transport terms and time tendency terms

- but we use a

- closure of pressure correlation and dissipation terms according to Rotta and Kolmogorov using a turbulent length scale according to Blackadar

yielding in particular:lEDR

3qEDRα

−≈

All reduced second order budgets degenerate to a system of linear equations

prognostic equation for turbulent SKE (TKE)

l


• Neglect of all horizontal gradients (BLA) in the second order budgets:The only relevant vertical flux densities have got flux gradient form

Cross section of model output over the chosen USA domain

from North to South cutting the jet stream

Turbulenzvorhersage für die Luftfahrt Matthias Raschendorfer 16.04.2008

Horizontal wind speed [m/s] Eddy dissipation parameter [m2/3/s]

06.02.2008 00UTC + 12h -90 E

moderate light

S N S N

Appalachian mountains

Jetstream

scale interaction by (positive definite circulation term)

Physical based solutionalso above the BL

EDR from turbulence model

- improved by consideration of horizontal shearand related length scales as an additional TKE production

- Improved by considering the interactions between turbulence and mountain blocking

- Improvement by considering the interactions between turbulence and convection

Other proper turbulence indicators, like- Ri-number, vertical gradients of temperature und wind- Horizontal shear- large scale surface roughness- …

regression model EDR from aircraft

measurements

( ) mesaa EDRScherungZahlRiEDRfn1

≈KK ,,_,mod,,

Aim of the DWD-Project:


( )∑∑==

∂⋅∂+∂⋅≈∇⋅′′−=3

1jijijiij

M3

1iiiS vvvKvvvP

,:

( )jiijM

ji vvKvv ∂+∂⋅−≈′′

3D-shear production for turbulent-isotropic structures:

isotropic-turbulent contribution to the stress tensor

l⋅⋅= MM SqK isotropic-turbulent diffusion coefficient

Isotropic-turbulent 3D shear term:

( ) ( )22z2

1zHBA vv ∂+∂⎯⎯→⎯

( ) ( ) ( )( ) ( ) ( )[ ]2

332

222

11

22332

21331

21221

vvv2

vvvvvv

∂+∂+∂+

∂+∂+∂+∂+∂+∂


m9000≈COSMO user seminar Offenbach: 09-11.03.2009Matthias Raschendorfer

horizontal grid scale

z

pL

Prandtl-layer

roughness layer

Eckmann-layer

height of the troposphere

boundary layer height

Blackadar

HD0

Separated horizontal shear mode:

Offenbach: 09-11.03.2009Matthias RaschendorferCOSMO user seminar

( ) ( ) ( )[ ]222

211

21221HHHSH vv2vvDqP ∂+∂+∂+∂⋅β⋅=:

Separated horizontal shear term:

effective mixing length of diffusion by separated horizontal shear eddies:

velocity scale of separated horizontal shear eddies

1H <β related scaling parameter

Equilibrium with scale transfer towards the turbulence mode:

HH

3HM

HHHH DqFDq

α=⋅β⋅

MHF:=

1H <α scaling parameter similar to

23

MH

2H

23

H21

HMHHHHSH FDFDqP ⋅βα=⋅β=

2Sα=:

EDRα


effective scaling parameter

Parameterization of the separated horizontal shear mode:

out_usa_shs_rlme_a_shsr_0.2


m9000≈COSMO user seminar Offenbach: 09-11.03.2009Matthias Raschendorfer


20

40

60

Pot. Temperature [K]

S N


The wake production term:

( ) ( )[ ] ( )∫∈

′−∂−ρ−∇υ−′′+ρ⋅−∇=ρ∂B

2iiiiiit sdnp

G1pgvvvvvv

ss

Filtered momentum budget:

B

Q∂

n

ivSSOQ blocking

In the TKE-budget:

( ) ( ) ∑∫=∈

⋅−=′⋅≈′∇⋅′−≈2

1i

vSSOi

B

2h

hhhSSO

iQvsdnpGv

pvPs

s

hv

21x ,

3x




m350≈

out_usa_shs_sso_turb_rlme_sso

out_usa_shs_sso_turb_rlme_sso


10X10 GP above Appalachian mountains

out_usa_shs_rlme_ssoout_usa_shs_rlme_a_shsr_0.2


• 3D-shear terms have got a significant effect only, when formulated as a scale interaction term producing TKE by shear of a separated horizontal shear mode with its own length scale

• Wake production of TKE by blocking can be formulated as a scale interaction termas well and can be described by scalar multiplication of the horizontal wind vectorwith its SS0-tendencies yielding some effect above mountainous terrain.

Conclusion:

• Next we plan to introduce a vertically resolved roughness layer including a more sophisticated interaction between turbulence and topographic blocking.

Prospect:

• We intent to derive a similar scale interaction term from the convection scheme as well.



Thanks for your attention!

Second order closure for the unknown statistical moments:

2-nd order budgets in TA:

( ) ( ) ( ) ( ) ( )( )( ) φψψφ

ψφ

ψφφφ

+φ′′+ψ′′+

φ∇⋅+ψ∇⋅+

φ∇⋅′′ψ′′ρ+ψ∇⋅′′φ′′ρ−φ∇⋅+ψ∇⋅−=φ′′+ψ′′+′′ψ′′φ′′ρ+ψ′′φ′′ρ⋅∇+ψ′′φ′′ρ∂=ψ′′φ′′ρ

prs

tt

QQQ

eeD

ee

vveevv ˆ~ˆ~ˆ~ˆ~~~~ˆ:

neglected outside the laminar layer

shear production

molecular dissipation

sub grid scale macroscopic transport

( )( ) φ∂′∂′+φ ′′∂′+

∂′∂φ ′′+∂φ ′′−

′φ ′′∂−

σ

σ

ˆzii

zii

i

zpp

pzp

p

⎭⎬⎫

⎩⎨⎧

=ψ∂ψ ′′−∈ψ

+⎭⎬⎫

⎩⎨⎧

=φ∂φ ′′−∈φ

jjii vp0

vp0

,H,

,H,

φ∇−= φφ a:emolecular flux density phase change production

pressure transport

pressure destruction

buoyancy source≈θ′′φ ′′ρθ v

v

gˆ

Projekt: „Turbulenzvorhersage für die Luftfahrt“ Offenbach: 17.04.2008Matthias Raschendorfer

Solution in horizontal boundary layer approximation:

• Neglect of all horizontal gradients in the second order budgets:

The only relevant vertical flux densities have got flux gradient form:

φ∂⋅−=′φ′ φzKw

l⋅⎭⎬⎫

⎩⎨⎧⋅=φ

M

H

SSqK turbulent diffusion coefficient for

vertical shear production( )2zu uKSP ∂⋅=

vzv

gKAP θ∂θ⋅−= θ

scalars

momentum

stability function dependent on and

buoyant production

( )2zu∂ vzv

gθ∂

θ

TKE2 ⋅ turbulent length scale

( )2zM

vzH

v uSSg

SPAPRi

∂θ∂

θ=−=:

Richardson-number


Neglect of ( )ψ ′′φ ′′ρtD , in particular all transport terms (equilibrium) except in the SKE equation

Neglect of sub grid scale phase change production ( )ψφ φ ′′+ψ ′′ QQ

Neglect of pressure transport pi ′φ ′′∂−

Bernoulli approximation constgz2

pptot ≈ρ+⋅

ρ+=vv:

Confluence/diffluence model of local isotropy for a single mode of wave length kL : ( )k

jjj

vv

l

φ ′′⋅′′≈″φ∂⋅′′

φ′′

jx

jv ′′

kl0 kL

1. Primary turbulent closure assumptions:

1x

2x3x

What is the idea of turbulence closure?


2. Secondary turbulent closure assumptions:

- spectral density of contributing modes follows a power law in terms of wave length in each direction:

- whole spectrum in a given direction is determined by a single peak wave length

- the peak wave length is the same for samples in all directions: isotropic length scale

- pressure correlation and dissipation can be closed using a single turbulent master length scalefor each location

inertial sub range spectrum

Turbulence is that class of sub grid scale structures being in agreement with turbulence closure assumptions!


What is the remaining difficulty with circulations?

- they are related with at least one additional spectral peak

- or they cause different peak wavelengths in vertical direction compared to the horizontal directions:

anisotropic peak wave length- larger peak wave length in vertical direction in case of labile stratification

kln

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ ψ ′′φ ′′ρ⋅

dkdkkln

32

− - slope in case of TKE

gD1ln

TurbulenceCirculations

Microphysics

ResolvedStructures

pL : largest turbulent wave length

convective peak in vertical spectrum

catabatic peak

neutralstabile

labile

pL1ln

at least a two scale problem


Eddy Dissipation Rate Improvement of EDR forecast by ... and Kolmogorov using a turbulent length...

Documents

Transcript of Eddy Dissipation Rate Improvement of EDR forecast by ... and Kolmogorov using a turbulent length...