Development of the two-equation second-order turbulence-convection model (dry version):

analytical formulation, single-column numerical results, and problems encountered

Dmitrii Mironov1 and Ekaterina Machulskaya2

1 German Weather Service, Offenbach am Main, Germany2 Hydrometeorological Centre of Russian Federation, Moscow, Russia

dmitrii.mironov@dwd.de, km@ufn.ru

COSMO General Meeting, Krakow, Poland 15-19 September 2008

Recall … (UTCS PP Plan for 2007-2008) Task 1a: Goals, Key Issues, Expected Outcome

Goals

• Development and testing of a two-equation model of a temperature-stratified PBL

• Comparison of two-equations (TKE+TPE) and one-equation (TKE only) models

Key issues

• Parameterisation of the pressure terms in the Reynolds-stress and the scalar-flux equations

• Parameterisation of the third-order turbulent transport in the equations for the kinetic and potential energies of fluctuating motions

• Realisability, stable performance of the two-equation model

Expected outcome

• Counter gradient heat flux in the mid-PBL • Improved representation of entrainment at the PBL top

Outline

• Governing equations, truncation, closure assumptions

• One-equation model vs. two-equation model – key differences

• Formulations for turbulence length (time) scale

• Numerical experiments: convective PBL

• Numerical experiments: stably stratified PBL, including the effect of horizontal inhomogeneity of the surface with respect to the temperature

• Problems encountered

• Conclusions and outlook

Governing Equation, Truncation, Closure Assumptions

• Prognostic equations are carried for the TKE (trace of the Reynolds stress tensor) and for the potential-temperature variance

• Equations for other second-order moments (the Reynolds stress and the temperature flux) are reduced (truncated) to the diagnostic algebraic relations (by neglecting the time-rate-of-change and the third-order moments)

• Slow pressure terms in the equations for the Reynolds-stress and for the temperature-flux are parameterised through the Rotta return-to-isotropy formulations; linear parameterisations for the rapid pressure terms are used

• The TKE dissipation rate is parameterised through the Kolmogorov formulation • The temperature-variance dissipation rate is parameterised assuming a constant ratio of

the temperature-variance dissipation time scale to the TKE dissipation times scale (alternatively, the time scale ratio can be computed as function of the temperature-flux correlation coefficient)

• The third-order transport terms in the TKE and the temperature-variance equations are parameterised through the simplest isotropic gradient-diffusion hypothesis (alternatively, a “generalised” non-isotropic gradient-diffusion hypothesis can be used)

• The system is closed through an algebraic formulation for the turbulence length (time) scale that includes the buoyancy correction term in stable stratification

Prognostic and Diagnostic Variables

Prognostic variables TKE and potential-temperature variance,

.,,,,, wvwuwwwvvuu

.,2

1 2 iiuue

Diagnostics variables components of the Reynolds stress and the potential-temperature flux,

`

Prognostic Equations

The TKE equation,

.2

1

2

1 22

wzz

wt

,2

1

pwuuwz

wgz

vvw

z

uuw

t

eii

The potential-temperature variance equation,

where is the thermal expansion coefficient (=1/ref), and g is (the vertical component of) the acceleration due to gravity.

The TKE dissipation rate,

Dissipation Rates

,3

10,,, 2/3

2/1

2/1

ee CCC

eC

lee

l

eC

e

where is the TKE dissipation time scale, and Ce is a constant that relates the TKE with the square of the surface friction velocity in the logarithmic boundary layer.

,2

1,

2,

22 2/1

22

2/12

R

eC

lReR

l

e

R

C

where is the temperature-variance dissipation time scale, and R is the dissipation time-scale ratio.

The temperature-variance dissipation rate,

Alternatively, R can be computed as function of the temperature-flux correlation coefficient,

.,13

222

2

e

uuA

AR ii

Third-Order Transport Terms

.09.0,2

1 2/1

dede

deii Cz

ele

C

C

z

eeCpwuuw

The third-order transport (diffusion) term in the TKE equation,

.09.0,2

2/12

2

dd

d Cz

leC

C

zeCw

The third-order transport term in the temperature-variance equation,

A higher value of Cd can also be tested, e.g. Cd=0.15.

Third-Order Transport Terms (cont’d)

,135.0,2

1

dedeii Cz

ewwCpwuuw

A generalised non-isotropic gradient diffusion hypothesis,

.135.0,2

2

dd Cz

wwCw

A higher value of Cd can also be tested, e.g. Cd=0.20.

Diagnostic Equations for the Reynolds Stress and for the Potential-Temperature Flux

.09.0,, 2/12/1

uuuu

uuuu

uu Cz

vle

C

C

z

veCvw

z

ule

C

C

z

ueCuw

The off-diagonal components of the Reynolds stress,

The potential-temperature flux,

,

1

3

21

3

2 22/1

2/12

ge

l

C

CC

zle

C

CgCC

zeCw

pbuup

buu

.3

1,2.0 p

bu CC

Notice that Cuu=Ce-2 is suggested by the log-layer relations.

Diagnostic Equations for the Reynolds Stress (cont’d)

,

1

3

4,

3

2** u

t

ub

C

CCwCeww

The diagonal components of the Reynolds stress,

.2

1

3

2,

2

1

3

2** wCevvwCeuu

Boundary Conditions for the Second-Order Moments

,,02

2/12/1

F

zle

C

C

z

ele

C

C dde

.0,0 2 e

At the top of the domain (well above the boundary layer),

At the underlying surface,

where F is the flux of the potential-temperature variance through the underlying surface. Setting F>0 should account for the horizontal inhomogeneity of the underlying surface and should make it possible to maintain turbulence in a strongly stable PBL.

One-Equation Model vs. Two-Equation Model – Key Differences

Equation for <’2>,

.2

1

2

1 22

wzz

wt

Production = Dissipation (implicit in all models that carry the TKE equations only).

Equation for <w’’>,

No counter-gradient term.

.13

2 2

gCCz

eCw pbuu

Turbulence Length Scale

.m 200,76.0,40.0,111

2/1

lCeC

N

lzl NN

An algebraic expression for l,

Estimates of l range from 100 m to 500 m.

Other estimates of CN should be tested, ranging from 0.76 to 3.0.

Formulations for Turbulence Length (Time) Scale

. s,600Min,01.0exp101.0exp 02/1

0

N

Czezzl N

.0,01.0exp101.0exp 12/10 zezzl

Teixeira and Cheinet (2004), Teixeira et al. (2004),

Does not satisfy the logarithmic boundary layer constraint, l=z as z0. This defect is easy to fix, e.g.

.

3

10,,Minwhere

,/10exp1/10exp2/1

0

10

12/10

h

eN

e

edzhwN

C

w

h

hzehzzl

A more flexible formulation (cf. Teixeira and Cheinet 2004),

Formulations for Turbulence Length (Time) Scale (cont’d)

. ,Min, 02/120

2

2/10

N

C

w

h

ez

ezl N

e

A simple interpolation formula (cf. Teixeira and Cheinet 2004, Teixeira et al. 2004),

Asymptotic behaviour

tion.stratifica stable (strongly)in surface thefromaway

tion,stratifica unstablein surface thefromaway

surface, near the

2/1

2/1

N

eCl

ew

hl

zl

N

e

Outline of Test Cases (CBL)

Convective PBL • Shear-free (zero geostrophic wind) and sheared (10 m/s geostrophic wind)• Domain size: 4000 m, vertical grid size: 1 m, time step: 1 s, simulation

length: 4 h • Lower b.c. for : constant surface temperature (heat) flux of 0.24 K·m/s • Upper b.c. for : constant temperature gradient of 3·10-3 K/m • Lower b.c. for U: no-slip, logarithmic resistance law to compute surface

friction velocity • Upper b.c. for U: wind velocity is equal to geostrophic velocity • Initial temperature profile: height-constant temperature within a 780 m

deep PBL, linear temperature profile aloft with the lapse rate of 3·10-3 K/m • Initial TKE profile: similarity relations in terms of z/h • Initial <’2> profile: zero throughout the domain • Turbulence moments are made dimensionless with the Deardorff (1970)

convective velocity scales h, w*=(g<w’’>sfc)1/3 and * =<w’’>sfc/ w*

Mean Temperature in Shear-Free Convective

PBL

One-Equation and Two-Equation Models

Red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar (1962) formulation for the turbulence length scale. Black curve shows the initial temperature profile.

Mean Temperature in Shear-Free Convective PBL

(cont’d)

One-Equation and Two-Equation Models

vs. LES Data

Potential temperature minus its minimum value within the PBL. Black dashed curve shows LES data (Mironov et al. 2000), red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar (1962) formulation for the turbulence length scale.

Potential-Temperature (Heat) Flux in Shear-Free Convective PBL

One-Equation and Two-

Equation Models vs. LES Data

<w’’> made dimensionless with w**. Black dashed curve shows LES data, red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar formulation for the turbulence length scale.

TKE in Shear-Free Convective PBL

One-Equation and Two-

Equation Models

vs. LES Data

TKE made dimensionless with w*

2. Black dashed curve shows LES data, red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar formulation for the turbulence length scale.

Potential-Temperature Variance in Shear-Free

Convective PBL

One-Equation and Two-Equation Models

vs. LES Data

<’2> made dimensionless with *

2. Black dashed curve shows LES data, red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar formulation for the turbulence length scale.

Budget of TKE in Shear-Free Convective PBL One-Equation and Two-Equation Models vs. LES Data

Dashed curves – LES data, solid curves – model results. Left panel – one-equation model, right panel – two-equation model. Red – mean-gradient production/destruction, green – third-order transport, blue – dissipation. The budget terms are made dimensionless with w*

3/h.

Budget of Potential-Temperature Variance in Shear-Free Convective PBL

One-Equation and Two-Equation Models vs. LES Data

Dashed curves – LES data, solid curves – model results. Left panel – one-equation model, right panel – two-equation model. Red – mean-gradient production/destruction, green – third-order transport, blue – dissipation. The budget terms are made dimensionless with *

2w*/h.

Counter-gradient heat flux

Mean Temperature in Sheared Convective PBL

One-Equation and Two-

Equation Models

Red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar (1962) formulation for the turbulence length scale. Black curve shows the initial temperature profile.

TKE and Potential-Temperature Variance in Sheared Convective PBL

TKE (left panel) and <’2> (right panel) made dimensionless with w*2 and *

2, respectively. Red – one-equation model, green – two-equation model, blue – one-equation model with the Blackadar formulation for the turbulence length scale.

Budget of TKE and of Potential-Temperature Variance in Sheared Convective PBL

Left panel – TKE budget, terms are made dimensionless with w*3/h. Black – shear, red – buoyancy, green

– third-order transport, blue – dissipation.

Right panel – <’2> budget, terms are made dimensionless with *2w*/h. Red – mean-gradient

production/destruction, green – third-order transport, blue – dissipation.

Outline of Test Cases (SBL 1)

Weakly stable PBL

• Wind forcing: 5 m/s geostrophic wind

• Domain height: 2000 m, vertical grid size: 1 m, time step: 1 s, simulation length: 24 h

• Lower b.c. for <’2>: zero flux, <w’’2>sfc=0

• Lower b.c. for : radiation-turbulent heat transport equilibrium, Tr

4+Ts4+<w’’>sfc=0, logarithmic heat transfer law to compute the surface heat

flux as function of the temperature difference between the surface and the first model level above the surface

• Upper b.c. for : constant temperature gradient of 3·10-3 K/m

• Lower b.c. for U: no-slip, logarithmic resistance law to compute surface friction velocity

• Upper b.c. for U: wind velocity is equal to geostrophic velocity

• Initial temperature profile: log-linear with 5 K temperature difference across a 200 m deep PBL, linear temperature profile aloft with the lapse rate of 3·10-3 K/m

• Initial profiles of TKE and <’2>: similarity relations in terms of z/h

Potential-Temperature Boundary Condition at the Underlying Surface

,044 sfcsr wTT

Radiation-turbulent heat transport equilibrium (cf. Brost and Wyngaard),

where Tr is the “radiation-equilibrium” temperature that the surface temperature Ts achieves if <w’’2>sfc=0.

Mean Potential Temperature and Mean Wind in Stably Stratified PBL (weakly stable)

Left panel – mean potential temperature, right panel – components of mean wind.

Red – one-equation model, green – two-equation model.

=26

TKE and Potential-Temperature Variance in Stably Stratified PBL (weakly stable)

Left panel – TKE, right panel – <’2>.

Red – one-equation model, green – two-equation model.

TKE Budget in Stably Stratified PBL (weakly stable)

Black – shear, red – buoyancy,

green – third-order transport, blue – dissipation.

Outline of Test Cases (SBL 2)

Strongly stable PBL • Wind forcing: 2 m/s geostrophic wind • Domain height: 2000 m, vertical grid size: 1 m, time step: 1 s, simulation

length: 24 h • Lower b.c. for <’2>: (a) zero flux, <w’’2>sfc=0 K2·m/s, (b) non-zero flux,

<w’’2>sfc=0.5 K2·m/s • Lower b.c. for : radiation-turbulent heat transport equilibrium,

Tr4+Ts

4+<w’’>sfc=0, logarithmic heat transfer law to compute the surface heat flux as function of the temperature difference between the surface and the first model level above the surface

• Upper b.c. for : constant temperature gradient of 3·10-3 K/m • Lower b.c. for U: no-slip, logarithmic resistance law to compute surface friction

velocity • Upper b.c. for U: wind velocity is equal to geostrophic velocity• Initial temperature profile: log-linear with 15 K temperature difference across a

200 m deep PBL, linear temperature profile aloft with the lapse rate of 3·10-3 K/m

• Initial profiles of TKE and <’2>: similarity relations in terms of z/h

Effect of Horizontal Inhomogeneity of the Underlying Surfacewith Respect to the Temperature

Equation for <’2>,

.2

1

2

1 22

wzz

wt

Within the framework of one-equation model, <w’’2> is entirely neglected

Within the framework of two-equation model, <w’’2> is non-zero (transport of <’2> within the PBL) and may be non-zero at the surface (effect of horizontal inhmomogeneity)

Mean Potential Temperature and Mean Wind in Stably Stratified PBL (strongly stable)

Left panel – mean potential temperature. Red – one-equation model, solid green – two-equation model, dashed green – two-equation model with non-zero <’2> flux.

=42

=35

Right panel – components of mean wind. Green – two-equation model with zero <’2> flux, red – two-equation model with non-zero <’2> flux.

TKE and Potential-Temperature Variance in Stably Stratified PBL (strongly stable)

Left panel – TKE, right panel – <’2>. Red – one-equation model, solid green – two-equation model, dashed green – two-equation model with non-zero <’2> flux.

TKE Budget in Stably Stratified PBL (strongly stable)

Solid curves – two-equation model, dashed curves – two-equation model with non-zero <’2> flux. Black – shear, red – buoyancy, green – third-order transport, blue – dissipation.

Potential-Temperature Variance Budget in Stably Stratified PBL (strongly stable)

Solid curves – two-equation model, dashed curves – two-equation model with non-zero <’2> flux. Red – mean-gradient production/destruction, green – third-order transport, blue – dissipation.

where ε is the TKE dissipation time scale.

• Formulation of turbulence length (time) scale

• (The so-called) stability functions

Problems Encountered

,,

1

3

4,

1

1 222 z

gN

C

CC

NC refu

ub

Stability functions in the shear-free convective PBL,

,13

2 2

gCCz

eCw pbuu

Potential-Temperature

Flux in Shear-Free Convective PBL

Stability Functions

<w’’> made dimensionless with w**. Black dashed curve shows LES data (Mironov et al. 2000), green – two-equation model with “new” formulation for turbulence length scale and no stability functions, red – two-equation model with the Blackadar (1962) formulation for the turbulence length scale and with stability functions.

• A generalised gradient-diffusion hypothesis for the third-order moments

Problems Encountered (cont’d)

… does not improve the model performance so far due, among other things, to problems with the realisability of <w’2> near the entrainment zone.

Diagonal Components of the Reynolds Stress Tensor in Shear-Free Convective PBL. Realisability Problem

<u’2> and <w’2> made dimensionless with w*2. Black dashed curves show

LES data (Mironov et al. 2000), green solid curves – two-equation model with “new” formulation for turbulence length scale and no stability function.

negative <w’2>

Conclusions and Outlook

• A dry version of a two-equation turbulence-convection model is developed and favourably tested through single-column numerical experiments

• A number of problems with the new two-equation model have been encountered that require further consideration (sensitivity to the formulation of turbulence length/time scale, consistent formulation of “stability functions”, realisability)

Ongoing and Future Work• Consolidation of a dry version of the two-equation model (c/o

Ekaterina and Dmitrii), including further testing against LES data from stably stratified PBL (c/o Dmitrii in co-operation with NCAR)

• Formulation and testing of a moist version of the new model

Thank you for your attention!

Stuff Unused

Appendix (Slides may be used as the case requires, e.g. to

answer questions, clarify various issues, etc.)