Analogies between the Mass-Flux and the Ensemble-Mean Second-Moment Modelling Frameworks

1
Turbulence and surface-layer parameterizations
for mesoscale models
Dmitrii V. Mironov German Weather Service,
Offenbach am Main, Germany (dmitrii.mironov_at_dwd.d
e)
Croatian - USA Workshop on Mesometeorology,
Ekopark Kraš Resort near Zagreb, Croatia. 18-20
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-heterogeneo
  us surface LES and prospects for improving
  parameterizations
• Conclusions and outlook

Croatian - USA Workshop on Mesometeorology,
Ekopark Kraš Resort near Zagreb, Croatia. 18-20
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, 161-221. doi
10.1007/978-3-540-78961-1 5)
Croatian - USA Workshop on Mesometeorology,
Ekopark Kraš Resort near Zagreb, Croatia. 18-20
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)
Quasi-random motions (turbulence closure
schemes)
ln(E)
Resolved scales (?-1 is effectively a mesh size)
Viscous dissipation
Sub-grid scales
?-1
ln(k)
?-1
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 (contd)
Turbulence kinetic energy (TKE)
(Monin and Yaglom 1971)
8
Physical Meaning of Terms
Mean-gradient production/destruction

• Time-rate-of-change,
• advection by mean velocity

Coriolis effects
Buoyancy production/destruction)
Viscous dissipation
Pressure scrambling
Third-order transport (diffusion)
9
Closure Assumptions Dissipation Rates
Transport equation for the TKE dissipation rate
Simplified (heavily parameterized) e-equation
10
Closure Assumptions Dissipation Rates (contd)
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 (contd)
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
Non-gradient term (advection)
Down-gradient term (diffusion)
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 (contd)
Plume/roll scale advection velocity
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
Convection with rotation
Free convection

• Budget of ltu3?gt 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
  ?ltu3?gt?lt?gt/?x3, green third-order transport
  ?ltu3u3?gt/?x3, black buoyancy g3?lt?2gt, blue
  pressure gradient-temperature covariance ?lt??
  p/?x3gt. 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 (contd)
20
Linear Models of ?ij and ??i (contd)

• Equation for the temperature flux

21
Linear Models of ?ij and ??i (contd)
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 (contd)
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 (contd)
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?b0.5,
• 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 (contd)
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 lt?2gt

Production Dissipation (implicit in all models
that carry the TKE equation only).
Equation for ltw?gt
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 ltui2gt (kinetic energy
of SGS motions) and for lt?2gt (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 lt?2gt 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 K? 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 (contd)

• 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
Neglect SGS fluctuations of temperature and
humidity, all-or-nothing scheme
qt
qt
no clouds, C 0
C 1
x
x
?x
?x
Account for humidity fluctuations only
Account for temperature and humidity fluctuations
qt
qt
x
x
Cloud cover 0ltClt1, although the grid box is
unsaturated in the mean
37
Turbulence and Clouds (contd)
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 (contd)
Buoyancy flux (a source of TKE),
is expressed through quasi-conservative
variables,
where A? and Aq are
functions
of mean state and cloud
cover
A? A? (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
Surface Temperature in Homogeneous and
Heterogeneous Cases
?s
?s1
??s? ?(?s1 ?s2)
homogeneous case
heterogeneous case
?s2
time
8h
9.75h
sampling
y
?s1
cold stripe
? ??
x
??s?
?? ??
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 ltw?2gt (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 lt?2gt due to
divergence of third-order transport term!
46
Third-Order Transport of Temperature Variance
?s1
??s?
x
?s2
z
z
?a
?a
?s2
?s1
47
Enhanced Mixing in Horizontally-Heterogeneous SBL
An Explanation
increased lt?2gt near the surface ? reduced
magnitude of downward heat flux ? less work
against the gravity ? increased TKE ? stronger
mixing
Decreased (in magnitude)
Increased
Increased
downward
upward
48
Can We Improve SBL Parameterisations?

• In order to describe enhanced mixing in
  heterogeneous SBL,
• an increased lt?2gt 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. 18-20
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. 18-20
June 2012.
52
Croatian - USA Workshop on Mesometeorology,
Ekopark Kraš Resort near Zagreb, Croatia. 18-20
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