Systematische Untersuchung höherer statistischer Momente und ihrer Bilanzen

Budgets of second order moments for cloudy boundary layers 1Systematische Untersuchung höherer statistischer Momente und ihrer Bilanzen 1LES der atmosphärischen Grenzschichtturbulenz –systematische Untersuchung höherer statistischer Momente und ihrer Bilanzen

5 September 2011

Rieke Heinze, Siegfried RaaschInstitute for Meteorology und Climatology

Leibniz University Hannover, Germany

Dmitrii MironovGerman Weather Service

Budgets of second order moments for cloudy boundary

layers derived from LES

COSMO General Meeting, Rome, September 2011

Budgets of second order moments for cloudy boundary layers 2

Background (I)

• Atmospheric boundary layer is characterized by turbulent motions

• In weather prediction and climate models turbulence and shallow convection have to be parameterized

• A lot of closure models are based on the truncated ensemble-mean budget equations for the statistical moments of second order:

...

...2

li

l

ut

t

...

...2

1 2

ji

i

uut

ut

...

...2

qut

qt

i

...

lqt

Motivation LES Results Summary Outlook


Background (II)

• Budgets are needed for– Estimation of relative importance of the budget terms– Testing of parameterizations and further development– Determination of disposable/tuning parameters

Budgets have a fundamental meaning So far: lack of systematic and comprehensive analysis for cloudy

boundary layers Usage of large eddy simulation (LES) to determine all the budgets

of second order moments and to examine pressure correlation terms like

ix

pq

il x

p

or



x

z

k

E | Production | Inertial subrange | Dissipation |

Introduction• Scales of turbulence

– Large eddies: 103 m, 1 h– Small eddies: 10-3 m, 0.1 s

• Spectrum of TKE– Large scales: production– Small scales: dissipation

• Filtering– Large, energy containing

eddies are resolved – Small eddies are modeled

Calculation of a turbulent boundary layer flow

LES: volume average

/1

Resolved scale

Subgridscale



Budgets from LES

• Closure models are often based on ensemble-mean budgets

• LES: approximation by averaging over horizontal planes and afterwards over several time steps

• LES is based on filtering: Application to budget equations as well:

Explicit consideration of SGS budgets


ttt


mixed layer

cloud layer

inversion

BOMEX − first order moments

• BOMEX LES study (Siebesma et al., 2003, 10 LES models, GCSS)– Stationary case of shallow cumulus convection over ocean (no precip.)– Domain size: 6.4 x 6.4 x 3.0km3, grid spacing: Δ=10m, time: 6h



BOMEX − TKE and its budget



BOMEX − temperature variance and its budget



BOMEX − variance of total water content and its budget



BOMEX − temperature flux and its budget



DYCOMS − first results

• DYCOMS LES study (Stevens et al., 2005, 10 LES models, 16 sim.)– Homogeneous, nocturnal stratocumulus over ocean (no drizzle)– Domain size: 3.4 x 3.4 x 1.5km3, grid spacing: Δ=35m, Δz=5m, time: 4h



Summary

• Budgets for all second order moments (16 equations) for cloud topped boundary layers are determined with very high resolution LES (2560 x 2560 x 1280 grid points)

• PALM is able to simulate GCSS cases

• Shallow cumulus (BOMEX)– Budgets of

• TKE and scalar variances: production and dissipation largest but transport is also very important

• Scalar fluxes: importance of buoyancy and pressure scalar cov.

• Stratocumulus (DYCOMS)– First results look promising



Future plans

• Simulation of a SC-CU transition case

• Detailed examination of pressure-scalar and -velocity covariance

• Testing of existing parameterizations against LES data

• LES data set will be made available – Testing and evaluation of currently used closure schemes– Further development of new schemes (→ e.g. TKE-scalar variance

scheme)

• Data is in netCDF format and preparation of a detailed documentation is ongoing


Benefit for COSMO community


Supplementary material

Budgets of second order moments for cloudy boundary layers 15Systematische Untersuchung höherer statistischer Momente und ihrer Bilanzen 1515

LES basic equations (I) • Filtering: with

• Basic system of equations consists of the filtered Navier-Stokes equations in Boussinesq form

Volume balance approach (Schumann, 1975)

VdtVzyx

tVV

,1

,

j

rij

vigkikjijkij

iji

x

gufuf

xx

uu

t

uk

*

0333

*

0

1

Equations of motion

ijjirij euu

3

2

2

2

1iue

ep 0**

3

2 Modified pressure

SGS stress

SGS TKE 0

j

j

x

uContinuity equation


Conservation equation

PRECj

qj

j

j

t

q

xx

qu

t

q

1616

LES basic equations (II)• Filtering: with

• Basic system of equations consists of the filtered Navier-Stokes equations in Boussinesq form

Volume balance approach (Schumann, 1975)

VdtVzyx

tVV

,1

,

First principle

PREC

l

RAD

l

j

j

j

ljl

ttxx

u

t

Potential liquid water temperature:

ljj u

lp

vl q

Tc

L

SGS temperature flux:

Total water contentlv qqq

qu jqj SGS humidity flux


LES SGS turbulence parameterization (I)• Turbulence closure based on Deardorff,1980

– Flux-gradient relationship for SGS fluxes– Prognostic equation for SGS TKE

• Diffusion coefficients

• The mixing length depends on– Grid spacing– Distance to nearest solid wall– Stratification

e

i

j

j

im

rij x

u

x

uK

j

lhj x

K

jhqj x

qK

elcK mm mh Kl

K )2

1(

3/1)( zyx mitund

l

02,,7.0min

02,7.0min

Nsld

Nd

ld

N


03

0

peu

xu

g

x

uuu

x

eu

t

ej

jv

j

iji

j

j

• The prognostic equation for ise

1818

LES SGS turbulence parameterization (II)• Turbulence closure based on Deardorff,1980

– Flux-gradient relationship for SGS fluxes– Prognostic equation for SGS TKE

• Diffusion coefficients

e

i

j

j

im

rij x

u

x

uK

j

lhj x

K

jhqj x

qK

elcK mm mh Kl

K )2

1(

3/1)( zyx mitund


BOMEX − second order moments

Buoyancy flux Total water flux


BOMEX − skewness

Cumulus convection shows strong updrafts of moist and cold air within cloud layer

2/32

3

S

(important for e.g. statistical cloud schemes)


BOMEX − covariance and its budget


BOMEX − flux of total water content and its budget

Systematische Untersuchung höherer statistischer Momente und ihrer Bilanzen

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