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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
[email protected], [email protected]
COSMO General Meeting, Krakow, Poland 15-19 September 2008
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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
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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
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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
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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,
`
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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.
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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
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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.
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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.
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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.
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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
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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.
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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
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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.
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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),
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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
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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*
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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
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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.
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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
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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.
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TKE Budget in Stably Stratified PBL (weakly stable)
Black – shear, red – buoyancy,
green – third-order transport, blue – dissipation.
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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
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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)
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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.
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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.
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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.
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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.
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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
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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.
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• 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.
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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>
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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
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Thank you for your attention!
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Stuff Unused
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Appendix (Slides may be used as the case requires, e.g. to
answer questions, clarify various issues, etc.)