1. Turbulence and surface-layer parameterizations for mesoscale models
Dmitrii V. Mironov
German Weather Service, Offenbach am Main, Germany
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

2. Outline Budget equations for the second-order turbulence moments
Parameterizations (closure assumptions) of the dissipation, third-order transport, and pressure scrambling
A hierarchy of truncated second-order closures – simplicity vs. physical realism
The surface layer
Effects of water vapour and clouds
Stably stratified PBL over temperature-heterogeneous surface – LES and prospects for improving parameterizations
Conclusions and outlook
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

3. References
Mironov, D. V., 2009: Turbulence in the lower troposphere: second-order closure and mass-flux modelling frameworks. Interdisciplinary Aspects of Turbulence, Lect. Notes Phys., 756, W. Hillebrandt and F. Kupka, Eds., Springer-Verlag, Berlin, Heidelberg, doi: / )
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

4. Recall a Trivial Fact … Transport equation for a generic quantity f
Split the sub-grid scale (SGS) flux divergence
Convection (quasi-organised)
mass-flux closure
Turbulence (quasi-random)
ensemble-mean closure

5. Energy Density Spectrum
Quasi-organized motions
(mass-flux schemes)
ln(E)
Quasi-random motions
(turbulence closure schemes)
Resolved scales
(-1 is effectively
a mesh size)
Viscous
dissipation
Sub-grid scales
-1
-1
ln(k)
Cut-off at very high resolution
(LES, DNS)

6. Second-Moment Budget Equations
Reynolds stress
Temperature (heat) flux
Temperature variance

7. Second-Moment Budget Equations (cont’d)
Turbulence kinetic energy (TKE)
(Monin and Yaglom 1971)

8. Physical Meaning of Terms
Time-rate-of-change,
advection by mean velocity
Mean-gradient production/destruction
Buoyancy production/destruction)
Coriolis effects
Third-order transport (diffusion)
Pressure
scrambling
Viscous
dissipation

9. Closure Assumptions: Dissipation Rates
Transport equation for the TKE dissipation rate
Simplified (heavily parameterized) ε-equation

10. Closure Assumptions: Dissipation Rates (cont’d)
Algebraic diagnostic formulations (Kolmogorov 1941)
Closures are required for the dissipation time or length scales!

11. Closure Assumptions: Third-Order Terms
Numerous parameterizations, ranging from simple down-gradient formulations,
to very sophisticated high-order closures.

12. Closure Assumptions: Third-Order Terms (cont’d)
An “advanced” model of third-order terms (e.g. Canuto et al. 1994)
take transport equations for all (!) third-order moments involved,
neglect /t and advection terms,
use linear parameterizations for the dissipation and the pressure scrambling terms,
use Millionshchikov (1941) quasi-Gaussian approximation for the forth-order moments,
The results is a very complex model (set of sophisticated algebraic relations) that still has many shortcomings.

13. Skewness-Dependent Parameterization of Third-Order Transport
Down-gradient term
(diffusion)
Non-gradient term
(advection)
But there exists another possibility, namely, the possibility to account for non-local advective nature of the third-order transport due to the presence of coherent structures in convective flows.
This can be done through introduction of an additional skewness-dependent non-gradient (ballistic) term into the formulation of the third-order transport term.
Physically, skewness…
In order to determine skewness… -> next slide
Accounts for non-local transport due to coherent structures, e.g. convective plumes or rolls – mass-flux ideas!
(Gryanik and Hartmann 2002)

14. Skewness-Dependent Parameterization of Third-Order Transport (cont’d)
Plume/roll scale
“advection” velocity
But there exists another possibility, namely, the possibility to account for non-local advective nature of the third-order transport due to the presence of coherent structures in convective flows.
This can be done through introduction of an additional skewness-dependent non-gradient (ballistic) term into the formulation of the third-order transport term.
Physically, skewness…
In order to determine skewness… -> next slide

15. Analogies to Mass-Flux Approach
A top-hat representation of a fluctuating quantity
Updraught
Only coherent top-hat part of the signal is accounted for
Downdraught
(environment)
After M. Köhler (2005)

16. Closure Assumptions: Pressure Scrambling
Transport equation for the Reynolds stress
Transport equation for the temperature (heat) flux
For later use we denote the above pressure terms by ij and i

17. Temperature Flux Budget in Boundary-Layer Convection
Pressure term
Free convection
Convection with rotation
Budget of <u’3’> in the surface buoyancy flux driven convective boundary layer that grows into a stably stratified fluid. The budget terms are estimated on the basis of LES data (Mironov 2001). Red – mean-gradient production/destruction <u’3’><>/x3, green – third-order transport –<u’3u’3’>/x3, black – buoyancy g3<’2>, blue – pressure gradient-temperature covariance <’ p’/x3>. The budget terms are made dimensionless with the Deardorff (1970) convective scales of depth, velocity and temperature.

18. Linear Models of ij and i
The simples return-to-isotropy parameterisation (Rotta 1951)
Analogously, for the temperature flux (e.g. Zeman 1981)

19. Linear Models of ij and i (cont’d)

20. Linear Models of ij and i (cont’d)
Equation for the temperature flux

21. Linear Models of ij and i (cont’d)
Poisson equation for the fluctuating pressure
Decomposition
Contribution to p’ due to buoyancy
NB! The volume of integration is the entire fluid domain.

22. Linear Models of ij and i (cont’d)
The buoyancy contribution to i is modelled as
The simplest (linear) representation
… satisfying … we obtain
Cf. Table 1 of Umlauf and Burchard (2005):
Cb = (1/3, 0.0, 0.2, 1/3, 1/3, 1/3, 1.3).
NB! The best-fit estimate for convective boundary layer is 0.5.

23. Linear Models of ij and i (cont’d)
Similarly for the buoyancy contribution to ij (Reynolds stress equation)
… satisfying … we obtain
Table 1 of Umlauf and Burchard (2005):
Cub = (0.5, 0.0, 0.0, 0.5, 0.4, 0.495, 0.5).
3/10?

24. Non-Linear Intrinsically Realisable TCL Model
The buoyancy contribution to i is a non-linear function of departure-from-isotropy tensor
The representation
Realisability. The two-component limit constraints (Craft et al. 1996)
… together with the other constraints (symmetry, normalisation) … yields

25. Models of i against data
Buoyancy contribution to i in convective boundary-layer flows (Mironov 2001).
Short-dashed – LES data,
solid – linear model with Cb=0.5,
long-dashed – non-linear TCL model (Craft et al. 1996).
3 is scaled with the Deardorff (1970) convective scales of depth, velocity and temperature.
TCL model (sophisticated and physically plausible) still does not perform well in some important regimes.

26. Truncated Second-Order Closures
Mellor and Yamada (1974) used “the degree of anisotropy” (the second invariant of departure-from-isotropy tensor) to scale and discard/retain the various terms in the second-moment budget equations and to develop a hierarchy of turbulence closure models for PBLs.

27. Truncated Second-Order Closures (cont’d)
The most complex model (level 4 of MY74)
prognostic transport equations (including third-order transport terms) for all second-order moments are carried.
Simple models (levels 1 and 2 of MY74)
all second-moment equations are reduced to the diagnostic down-gradient formulations.
The most simple algebraic model
consists of isotropic down-gradient formulations for fluxes,
and production-dissipation equilibrium relations for the TKE and the scalar variances.

28. Two-Equation TKE-Scalar Variance Model (MY74 level 3)
Transport equations for the TKE and for the scalar variance(s)
Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g.

29. One-Equation TKE Model (MY74 level 2.5)
Transport equation for the TKE
Diagnostic formulation(s) for the scalar variance(s)
Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g.

30. Comparison of 1-Eq and 2-Eq Models
Equation for <’2>
Production = Dissipation (implicit in all models that carry the TKE equation only).
Equation for <w’’>
Summarizing, the main differences between the one-equation and two-equation model formulations are the following…
Then, if we want to use the two-equation model, two topics appear – how to model this term and this one.
No counter-gradient term (cf. turbulence models using “counter-gradient corrections” heuristically).
1-Eq Models are Draft Horses of Geophysical Turbulence Modelling

31. Importance of Scalar Variance
Prognostic equations for <ui’2> (kinetic energy of SGS motions) and for <’2> (potential energy of SGS motions).
Convection/stable stratification =
Potential Energy  Kinetic Energy.
No reason to prefer one form of energy over the other!
The TKE equation
The <’2> equation

32. Exercise Given transport equation for the temperature flux,
make simplifications and invoke closure assumptions to derive a down-gradient approximation for the temperature flux,
(Hint: the dimensions of Kθ is m2/s.)

33. The Surface Layer
The now classical Monin-Obukhov surface-layer similarity theory (Monin and Obukhov 1952, Obukhov 1946).
The surface-layer flux-profile relationships
MOST breaks down in conditions of vanishing mean velocity (free convection, strong static stability).

34. The Surface Layer (cont’d)
The MO flux-profile relationships are consistent with the second-moment budget equations. In essence, they represent the second-moment budgets truncated under the surface-layer similarity-theory assumptions
turbulence is continuous, stationary and horizontally-homogeneous,
third-order turbulent transport is negligible, and
changes of fluxes over the surface layer are small as compared to their changes over the entire PBL.

35. Effects of Water Vapour and Clouds
Quasi-conservative variables
Virtual potential temperature is defined with due regard for the water loading

36. Turbulence and Clouds qt qt x x Δx Δx Account for humidity
Neglect SGS fluctuations of temperature and humidity, all-or-nothing scheme qt qt
no clouds, C = 0
C = 1 x x Δx Δx
Account for humidity
fluctuations only
Account for temperature
and humidity fluctuations qt qt x x
Cloud cover 0<C<1, although the grid box is unsaturated in the mean

37. Turbulence and Clouds (cont’d)
If PDF of s is known, then
cloud cover, cloud condensate =
integral over supersaturated part of PDF
cloud cover
cloud condensate
However, PDF is generally not known!
SGS statistical cloud schemes
assume a functional form of PDF
with a small number of parameters.
Input parameters (moments predicted by turbulence scheme) →
Assumed PDF → Diagnostic estimates of C, , etc.
after Tompkins (2002)

38. Turbulence and Clouds (cont’d)
Buoyancy flux (a source of TKE),
is expressed through quasi-conservative variables,
where Aθ and Aq are functions
of mean state and cloud cover
Aθ = Aθ (C, mean state)
Aq = Aq (C, mean state)
functional form depends on assumed PDF
Aq is of order 200 for cloud-free air, but ≈ 800 ÷ 1000 within clouds!
Clouds-turbulence coupling: clouds affect buoyancy production of TKE, turbulence affect fractional cloud cover (where accurate prediction of scalar variances is particularly important).

39. LES of Stably Stratified PBL (SBL)
Traditional PBL (surface layer) models do not account for many SBL features (static stability increases  turbulence is quenched  sensible and latent heat fluxes are zero  radiation equilibrium at the surface  too low surface temperature)
No comprehensive account of second-moment budgets in SBL
Poor understanding of the role of horizontal heterogeneity in maintenance of turbulent fluxes (hence no physically sound parameterization)
LES of SBL over horizontally-homogeneous vs. horizontally-heterogeneous surface [the surface cooling rate varies sinusoidally in the streamwise direction such that the horizontal-mean surface temperature is the same as in the homogeneous cases, cf. GABLS, Stoll and Porté-Agel (2009)]
Mean fields, second-order and third-order moments
Budgets of velocity and temperature variance and of temperature flux with due regard for SGS contributions (important in SBL even at high resolution)
(Mironov and Sullivan 2010, 2012)

40. in Homogeneous and Heterogeneous Cases
Surface Temperature
in Homogeneous and Heterogeneous Cases s
s1
s =
(s1+ s2)
homogeneous case
heterogeneous case
s2
time 8h
sampling
9.75h y
s1
+ 
cold stripe
s x
warm stripe
 
s2

41. Mean Potential Temperature
cf. Stoll and Porté-Agel (2009)
Blue – homogeneous SBL,
red – heterogeneous SBL.

42. TKE and Temperature Variance
Large
Blue – homogeneous SBL, red – heterogeneous SBL.

43. TKE Budget
Decreased in
magnitude
Left panel – homogeneous SBL, right panel – heterogeneous SBL.
Red – shear production, blue – dissipation, black – buoyancy destruction, green – third-order transport,
thin dotted black – tendency .

44. Temperature Variance Budget
Net source
Left panel – homogeneous SBL, right panel – heterogeneous SBL.
Red – mean-gradient production/destruction, blue – dissipation, green – third-order transport, black (thin dotted) – tendency .

45. Key Point: Third-Order Transport of Temperature Variance
LES estimate of <w’’2> (resolved plus SGS)
In heterogeneous SBL,
the third-order transport
of temperature variance is
non-zero at the surface
Surface temperature variations modulate local static stability and hence the surface heat flux  net production/destruction of <’2> due to divergence of third-order transport term!

46. Third-Order Transport of Temperature Variancex
s2 z z
s1 a
s2 a

47. Enhanced Mixing in Horizontally-Heterogeneous SBL An Explanation
increased <’2> near the surface 
reduced magnitude of downward heat flux 
less work against the gravity  increased TKE  stronger mixing
Decreased
(in magnitude)
Increased
Increased
Как смоделировать этот эффект? --- Переход к следующему слайду.
downward
upward

48. Can We Improve SBL Parameterisations?
In order to describe enhanced mixing in heterogeneous SBL,
an increased <’2> at the surface should be accounted for.
Elegant way: modify the surface-layer flux-profile relationships. Difficult – not for nothing are the Monin-Obukhov surface-layer similarity relations used for more than 1/2 a century without any noticeable modification!
Less elegant way: use a tile approach, where several parts with different surface temperatures are considered within an atmospheric model grid box.

49. Tiled TKE-Temperature Variance Model: Results
Blue – homogeneous SBL,
red – heterogeneous SBL.
(Mironov and Machulskaya 2012, unpublished)

50. Conclusions and Outlook
Only a small fraction of what is currently known about geophysical turbulence is actually used in applications … but we can do better
Beware of limits of applicability!
TKE-Scalar Variance turbulence scheme offers considerable prospects (IMHO)
Improved models of pressure terms
Interaction of clouds with skewed and anisotropic turbulence
PBLs over heterogeneous surfaces
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

51. Thanks for your attention!
Acknowledgements: Peter Bechtold, Vittorio Canuto, Sergey Danilov, Stephan de Roode, Evgeni Fedorovich, Jean-François Geleyn, Andrey Grachev, Vladimir Gryanik, Erdmann Heise, Friedrich Kupka, Cara-Lyn Lappen, Donald Lenschow, Vasily Lykossov, Ekaterina Machulskaya, Pedro Miranda, Chin-Hoh Moeng, Ned Patton, Jean-Marcel Piriou, David Randall, Matthias Raschendorfer, Bodo Ritter, Axel Seifert, Pier Siebesma, Pedro Soares, Peter Sullivan, Joao Teixeira, Jeffrey Weil, Jun-Ichi Yano, Sergej Zilitinkevich.
The work was partially supported by the NCAR Geophysical Turbulence Program and by the European Commission through the COST Action ES0905.
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

52. Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

53. Exercise: derive down-gradient approximation for fluxes from the second-moment equations
Transport equation for the temperature flux
(!) Using Rotta-type return-to-isotropy parameterisation of the pressure gradient-temperature covariance
then neglecting anisotropy
yields the down-gradient formulation