Turbulence and surface-layer
parameterizations for
mesoscale models
Dmitrii V. Mironov
German Weather Service, Offenbach am Main, Germany
(dmitrii.mironov@dwd.de)
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia. 18-20 June 2012.
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. 18-20 June 2012.
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.
Recall a Trivial Fact …
Transport equation for a generic quantity f
d f / dt  ui f  / xi  ...
Split the sub-grid scale (SGS) flux divergence
ui f  / xi  (ui f  / xi )conv  (ui f  / xi )turb
Convection (quasi-organised)
Turbulence (quasi-random)
mass-flux closure
ensemble-mean closure
Energy Density Spectrum
Quasi-organized motions
(mass-flux schemes)
ln(E)
Quasi-random motions
(turbulence closure schemes)
Resolved scales
Viscous
dissipation
(-1 is effectively
a mesh size)
Sub-grid scales
-1
Cut-off at very high resolution
(LES, DNS)
-1
ln(k)
Second-Moment Budget Equations
Reynolds stress
u j


u
 
  uk
uiu j   uiuk
 u j uk i
xk 
xk
xk
 t


 

  g i uj   g j ui   2  ilk  l uk uj   jlk  l uk ui

 ui uj


uk uiu j   ki uj p   kj ui p  p


xk
 x j xi



   ij


Temperature (heat) flux

u
 


p
2
  uk
ui   uk  i  uiuk







 gi  2 ijk j uk 
uk ui  
xk 
xk
xk
xk
xi
 t
Temperature variance
1 
  2
 1 
  uk
   uk 

uk 2   
2  t
xk 
xk 2 xk

Second-Moment Budget Equations (cont’d)
Turbulence kinetic energy (TKE)
u
1 
  2

  uk
ui  uiuk i  g i ui  
2  t
xk 
xk
xk
1 2
   ii , T KE  ui
2
1

2




 uk ui  uk p    ,
2

(Monin and Yaglom 1971)
Physical Meaning of Terms
u j


ui 
 
  uk
uiuj   uiuk

 uj uk
xk 
xk
xk 
 t

Time-rate-of-change,
advection by mean velocity

Mean-gradient production/destruction
 
 gi uj   g j ui   2  ilkl uk uj   jlkl uk ui

Coriolis effects
Buoyancy production/destruction)
 ui uj 

   ij

uk uiuj   ki uj p   kj ui p  p

 x

xk

x
i 
 j

Third-order transport (diffusion)

Pressure
scrambling
Viscous
dissipation
Closure Assumptions: Dissipation Rates
2
 ui 
   
 ,     

   
 xk 
 xk 
2
Transport equation for the TKE dissipation rate
2

   ui 
  uk
 
  manypoorlyunderstoodterms
xk   xk 
 t
Simplified (heavily parameterized) ε-equation

ui 
 

2
  uk
  Diff  Cs uiuk
 Cb g i ui   C f 
xk 
xk e
e
e
 t
Closure Assumptions: Dissipation Rates (cont’d)
Algebraic diagnostic formulations (Kolmogorov 1941)

e

 Ce
e

 e
,  
 C
l

l
3/ 2
2
2 1/ 2
Closures are required for the dissipation time or length scales!
1 1
N
1

2
 

, N   gi
, h is thePBL depth
1/ 2
l z CN e
Ch h
xi
Closure Assumptions: Third-Order Terms
Numerous parameterizations, ranging from simple downgradient formulations,
ui 2   K
 uk  ui  
 2
,
, uiuk    K u 


xi

x

x
i
k 

 uiuj uiuk uj uk
uk uiuj   K uu 


 xk
x j
xi

2


, uiuk 2   K e uk ,

xi

to very sophisticated high-order closures.
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,
        
abcd   ab cd   ac bd   ad  bc
The results is a very complex model (set of sophisticated
algebraic relations) that still has many shortcomings.
Skewness-Dependent Parameterization of
Third-Order Transport
 
2



ui   K
 S 
xi
2
2
Down-gradient term
(diffusion)
1/ 2
ui , S 


2
3
3/ 2
Non-gradient term
(advection)
Accounts for non-local transport due to coherent structures,
e.g. convective plumes or rolls – mass-flux ideas!
(Gryanik and Hartmann 2002)
Skewness-Dependent Parameterization of
Third-Order Transport (cont’d)
S  
2
1/ 2
 S ui   2
2


ui  
   wi  
1
/
2
 2 
 

Plume/roll scale
“advection” velocity
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)
Closure Assumptions: Pressure Scrambling
Transport equation for the Reynolds stress
u j
u j 


 
  uk
uiu j   uiuk
  g i uj   g j ui   2  ilk  l uk uj   ilk  l uk u j
 uiuk
xk 
xk
xk 
 t


 
 ui u j


uk uiuj   ki u j p   kj ui p  p


xk
 x j xi



   ij


Transport equation for the temperature (heat) flux

u
 


p
2
  uk
ui   uk  i  uiuk







 gi  2 ijk j uk 
uk ui  
xk 
xk
xk
xk
xi
 t
For later use we denote the above pressure terms by ij and i

Temperature Flux Budget in Boundary-Layer Convection
1
1.2
1
0.8
0.8
z/h
z/h
0.6
0.6
0.4
0.4
Pressure term
0.2
0
0.2
-3
-2
-1
0
1
Terms  w-2
-1h
* *
Free convection
2
3
0
-15
-10
-5
0
5
10
15
Terms  w-2
-1h
* *
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.
Linear Models of ij and i
The simples return-to-isotropy parameterisation (Rotta 1951)
1 
1

 ij  
 uiuj   ij uk uk 
 u* 
3

Analogously, for the temperature flux (e.g. Zeman 1981)
i  
ui 
*
Linear Models of ij and i (cont’d)
 u

2

u 
u


 ij  C
e  Cs1Sij  Cs 2  aik S kj  a jk S ki   ij akl S kl   Cs 3 aikWkj  a jkWki  e
u
3




2


 Cbu   i uj    j ui    ij  k uk    2Ccu  ilk l uk uj   ilk l uk uj ,
3


u
t
aij


i  Ct
uí 

where aij  2



 Cs1Sij  Cs2Wij uj   Cb  i  2  2Cc  ijk  j uk ,
uiuj
2
1
  ij , e  uk uk ,  i  g i ,
2
uk uk 3
1  ui uj
Sik 


2  x j xi

 ui uj
1
 and Wik  



2  x j xi


.


Linear Models of ij and i (cont’d)
Equation for the temperature flux




 uj
ui   uiu j    u k ui     p
 t


x
x j xk
xi
j


 S ij  Wij u j    i  2  2 ijk  j u k 

i  Ct
uí 



 Cs1Sij  Cs2Wij uj   Cb i  2  2Cc  ijk j uk 
Linear Models of ij and i (cont’d)
Poisson equation for the fluctuating pressure
ui uj
 2 p
2
 

uiuj  uiuj  2
 i
 2 ijk j uk
2
xk
xi x j
x j xi
xi

Decomposition

  pt  ps  pb  pc
ptotal
Contribution to p’ due to buoyancy
 2 pb
 



,
i
xk2
xi
then i   k Yik ,
1
p 
4
k
 (r ) dv(r )
p


,






i
Vol i xi r  r
xi
4
1
Yik  
4
 2 (r ) (r ) dv(r )
Vol xixk r  r .
 2  (r ) (r ) dv(r )
Vol xixk r  r ,
NB! The volume of
integration is the entire
fluid domain.
Linear Models of ij and i (cont’d)
The buoyancy contribution to i is modelled as
i   k Yik ,
where
Yik  Yki
and
Yii   2 .
The simplest (linear) representation
2

Yik   1 ik 
… satisfying … we obtain
1
 1  , i.e. bi  Cb  i  2
3
with
1
Cb  !
3

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.
Linear Models of ij and i (cont’d)
Similarly for the buoyancy contribution to ij (Reynolds stress equation)
ij   k X ijk  X jik , where X ijk  X ikj , X iik  0 and X ikk  ui .
… satisfying … we obtain
2
3


u






  C  i u j   j ui   ij  k uk  with Cb  !
3
10


b
ij
u
b
Table 1 of Umlauf and Burchard (2005):
Cub = (0.5, 0.0, 0.0, 0.5, 0.4, 0.495, 0.5).
3/10?
Non-Linear Intrinsically Realisable TCL Model
The buoyancy contribution to i is a non-linear function of
departure-from-isotropy tensor
uiuj 2
aij  2
  ij
uk uk 3
The representation
Yik   1 ik   2 aik   3aimamk  2
Realisability. The two-component limit constraints (Craft et al. 1996)
3 2   kY3k  0 as u3  0
… together with the other constraints (symmetry, normalisation) …
yields
1
 2
    i   k aik  
3

b
i
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.
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 secondmoment budget equations and to develop a hierarchy of
turbulence closure models for PBLs.
uiuj
2
aij  2
  ij ,
uk uk 3
A2  aij aij
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 downgradient formulations.
The most simple algebraic model
consists of isotropic down-gradient formulations for fluxes,
 ui uj
uiuj   K u 

 x j xi


, w    K  , K u  K  le1/ 2  e

z

and production-dissipation equilibrium relations for the TKE and
the scalar variances.
Two-Equation TKE-Scalar Variance Model
(MY74 level 3)
Transport equations for the TKE and for the scalar variance(s)
e
u
v 
 1


  wu
 wv   g w    wuiui  wp   
t
z
z 
z  2


1  2
 1 
 w 

w 2  
2 t
z 2 z
Algebraic formulations for the Reynolds stress components and
for the scalar fluxes, e.g.
wu   S Me
u
v





, w v   S Me , w    S H 1e
 S H 2g 2 ,
z
z
z
SM , SH1, SH 2 functionsof  2 S 2 , Ri  N 2 / S 2 , 2 g   2 / e
2
One-Equation TKE Model (MY74 level 2.5)
Transport equation for the TKE
e
u
v 
 1


  wu
 wv   g w    wuiui  wp   
t
z
z 
z  2


Diagnostic formulation(s) for the scalar variance(s)

0   w 
 
z
Algebraic formulations for the Reynolds stress components and
for the scalar fluxes, e.g.
wu  S Me
SM , S H
u
v





, w v   S Me , w    S He
,
z
z
z
functionsof  2 S 2 , Ri  N 2 / S 2
Comparison of 1-Eq and 2-Eq Models
Equation for <’2>
2

1 
 1 
2




 w 

w   
2 t
z 2 z
Production = Dissipation (implicit in all models that carry the TKE
equation only).
Equation for <w’’>

w   Cg  e
 Cb  g 2
z
No counter-gradient term (cf. turbulence models using “countergradient corrections” heuristically).
1-Eq Models are Draft Horses of Geophysical Turbulence Modelling
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
e
u
v 
 1













  w u
wv
  g w    w ui ui  w p   
t
z
z 
z  2


The <’2> equation
2
2
2


1 g   2
1 g  
g


2
2




  g w  
w 
 , N  g
2
2
2
2 N
t
2 N z
N
z
Exercise
Given transport equation for the temperature flux,

u
 


p
  uk
ui   uk  i  uiuk
 gi 2  2 ijk j uk  
uk ui     ,
xk 
xk
xk
xk
xi
 t
make simplifications and invoke closure assumptions to derive
a down-gradient approximation for the temperature flux,
ui    K
(Hint: the dimensions of Kθ is m2/s.)

.
xi
The Surface Layer
The now classical Monin-Obukhov surface-layer similarity
theory (Monin and Obukhov 1952, Obukhov 1946).
The surface-layer flux-profile relationships

 Qs
u*  z
u  u s  ln
 m z / L ,    s 
  z0 m
u*

 z

 h z / L  ,
ln
 z0 h

u*3
2
u*  uw , Qs  w  , L 
sfc
sfc
 gQs
MOST breaks down in conditions of vanishing mean
velocity (free convection, strong static stability).
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
(i) turbulence is continuous, stationary and horizontally-homogeneous,
(ii) third-order turbulent transport is negligible, and
(iii) changes of fluxes over the surface layer are small as compared to their
changes over the entire PBL.
e
u
v 
 1













  w u
wv
  g w    w ui ui  w p   
t
z
z 
z  2


e3 / 2
  C
, l  z , e  C 2 / 3u*2
l
3

u
u
u u*
2
*
u*
 ,
 ,
z z
z z
Effects of Water Vapour and Clouds
Quasi-conservative variables
qt  qv  ql  qi
 Lv
 Li
t   
ql 
qi
T cp
T cp
totalwater specifichumidity
totalwater potentialtemperatu
re
Virtual potential temperature is defined with due regard for
the water loading
t   1  R 1qv  ql  qi , R  Rv / Rd
Turbulence and Clouds
Neglect SGS fluctuations of temperature and humidity, all-or-nothing scheme
qt
qt
qs
qt
qs
qt
no clouds, C = 0
C=1
x
qt
Δx
Account for humidity
fluctuations only
x
Δx
Account for temperature
and humidity fluctuations
qt
qt
qs
qs
qt
x
Cloud cover 0<C<1, although the grid box is unsaturated in the mean
qt
x
Turbulence and Clouds (cont’d)
s  qt  qs
If PDF of s is known, then
cloud cover, cloud condensate =
integral over supersaturated part of PDF

cloud cover C   P ( s )ds
0
cloud condensate

qc   sP ( s ) ds
0
However, PDF is generally not known!
SGS statistical cloud schemes
assume a functional form of PDF
with a small number of parameters.
after Tompkins (2002)
Input parameters (moments predicted by turbulence scheme) →
Assumed PDF → Diagnostic estimates of C, qc , etc.
Turbulence and Clouds (cont’d)
Buoyancy flux (a source of TKE),

g wv  g A  wl  B  wqt  D  wql

is expressed through quasi-conservative variables,
where Aθ and Aq are functions
wv  A  wl  Aq  wqt,
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).

 s  a qt  P l  2 P qt l
2
2
2

1/ 2
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)
Surface Temperature
in Homogeneous and Heterogeneous Cases
s
s1
s =
(s1+ s2)
homogeneous case
heterogeneous case
s2
time
8h sampling 9.75h
s1
s
y
+ 
warm stripe
s2
cold stripe
 
x
Mean Potential Temperature
cf. Stoll and Porté-Agel
(2009)
Blue – homogeneous SBL,
red – heterogeneous SBL.
TKE and Temperature Variance
Large
Blue – homogeneous SBL, red – heterogeneous SBL.
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 .
e
u
v 
 1


2
  wu
 wv   g w    wui  wp   
t
z
z 
z  2


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 .
1  2
 1 
 w 

w 2  
2 t
z 2 z
Key Point: Third-Order Transport
of Temperature Variance
LES estimate of <w’’2> (resolved plus SGS)
w"   w  
2
2


2






2  w  w
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!
In heterogeneous SBL,
the third-order transport
of temperature variance is
non-zero at the surface
Third-Order Transport of Temperature Variance
s1
   0
s
x
   0
s2
z
z
s1
a
s2



w 0



w 0

 w   0
a
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

w   Cg  e
 Cb  g 2
z
downward
upward
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 MoninObukhov 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.
Tiled TKE-Temperature Variance Model: Results
Blue – homogeneous SBL,
red – heterogeneous SBL.
(Mironov and Machulskaya 2012, unpublished)
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.
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.
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia. 18-20 June 2012.
Exercise: derive down-gradient approximation for fluxes
from the second-moment equations
Transport equation for the temperature flux

ui
 


p
2
  uk
ui   uk 
 uiuk
 g i   2 ijk j uk  
uk ui    
xk 
xk
xk
xk
xi
 t
(!) Using Rotta-type return-to-isotropy
parameterisation of the pressure gradienttemperature covariance
p ui 


,
xi  
then neglecting anisotropy
2
2
2

2
2
2
uiuk   ik un   uiuk   ik un    ik un ,
3
3

 3
yields the down-gradient
formulation
2
2

2 
2 




ui      ik un
    uk
  K
3
xk
3
xi
xi