turbulence and surface-layer parameterizations for mesoscale models dmitrii v. mironov german...
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Turbulence and surface-layer parameterizations for
mesoscale models
Dmitrii V. Mironov
German Weather Service, Offenbach am Main, Germany ([email protected])
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia. 18-20 June 2012.
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
Outline
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia. 18-20 June 2012.
References
Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia. 18-20 June 2012.
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)
Recall a Trivial Fact …
Transport equation for a generic quantity f
...// ii xfudtfd
Split the sub-grid scale (SGS) flux divergence
turbiiconviiii xfuxfuxfu )/()/(/
Convection (quasi-organised)
mass-flux closure
Turbulence (quasi-random)
ensemble-mean closure
Energy Density Spectrum
ln(E)
ln(k)-1
Resolved scales
(-1 is effectively a mesh size)
Quasi-random motions (turbulence closure schemes)
Quasi-organized motions (mass-flux schemes)
Sub-grid scales
Viscous dissipation
-1
Cut-off at very high resolution (LES, DNS)
Temperature (heat) flux
iik
kkjijki
kki
k
iki
kk x
puu
xug
xuu
x
uuu
xu
t
22
iji
j
j
iikjjkijik
k
ikljlkjklilkijjik
ikj
k
jkiji
kk
x
u
x
uppupuuuu
x
uuuuugugx
uuu
x
uuuuu
xu
t
2
Reynolds stress
Second-Moment Budget Equations
22
2
1
2
1k
kkk
kk u
xxu
xu
t
Temperature variance
Turbulence kinetic energy (TKE)
Second-Moment Budget Equations (cont’d)
2
22
2
1TKE,
,2
1
2
1
iii
kikk
iik
ikii
kk
u
puuux
ugx
uuuu
xu
t
(Monin and Yaglom 1971)
Time-rate-of-change, advection by mean velocity
k
ikj
k
jkiji
kk x
uuu
x
uuuuu
xu
t
Physical Meaning of Terms
iji
j
j
iikjjkijik
k x
u
x
uppupuuuu
x
ikljlkjklilkijji uuuuugug 2
Mean-gradient production/destruction
Buoyancy production/destruction) Coriolis effects
Third-order transport (diffusion) Pressurescrambling
Viscousdissipation
Closure Assumptions: Dissipation Rates
22
,
kk
i
xx
u
Transport equation for the TKE dissipation rate
termsunderstoodpoorly many 2
k
i
kk x
u
xu
t
Simplified (heavily parameterized) ε-equation
efC
eugC
ex
uuuCDiff
xu
t iibk
ikis
kk
2
Closure Assumptions: Dissipation Rates (cont’d)
l
eC
l
eC
ee
2/1222/3
,
Algebraic diagnostic formulations (Kolmogorov 1941)
Closures are required for the dissipation time or length scales!
depth PBL theis ,,111 2
2/1h
xgN
hCeC
N
zl ii
hN
Closure Assumptions: Third-Order Terms
Numerous parameterizations, ranging from simple down-gradient formulations,
to very sophisticated high-order closures.
,,2
2
i
keki
i
kj
j
ki
k
jiuujik x
uKuu
x
uu
x
uu
x
uuKuuu
,,2
2
k
i
i
kuki
ii x
u
x
uKuu
xKu
Closure Assumptions: Third-Order Terms (cont’d)
• 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,
An “advanced” model of third-order terms (e.g. Canuto et al. 1994)
cbdadbcadcbadcba
The results is a very complex model (set of sophisticated algebraic relations) that still has many shortcomings.
Accounts for non-local transport due to coherent structures, e.g. convective plumes or rolls – mass-flux ideas! (Gryanik and Hartmann 2002)
2/32
32/12
22 ,
SuSx
Ku ii
i
Skewness-Dependent Parameterization of Third-Order Transport
Down-gradient term (diffusion)
Non-gradient term (advection)
222/1
2
2/12
i
ii w
uSuS
Skewness-Dependent Parameterization of Third-Order Transport (cont’d)
Plume/roll scale “advection” velocity
Analogies to Mass-Flux Approach
A top-hat representation of a fluctuating quantity
After M. Köhler (2005)
Updraught
Downdraught(environment)
Only coherent top-hat part of the signal is
accounted for
Transport equation for the temperature (heat) flux
iik
kkjijki
kki
k
iki
kk x
puu
xug
xuu
x
uuu
xu
t
22
iji
j
j
iikjjkijik
k
jklilkjklilkijjik
jki
k
jkiji
kk
x
u
x
uppupuuuu
x
uuuuugugx
uuu
x
uuuuu
xu
t
2
Transport equation for the Reynolds stress
Closure Assumptions: Pressure Scrambling
For later use we denote the above pressure terms by ij and i
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.
Temperature Flux Budget in Boundary-Layer Convection
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
Terms w*-2
*-1h
z/h
-15 -10 -5 0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
Terms w*-2
*-1h
z/h
Pressure term
Free convection Convection with rotation
Linear Models of ij and i
kkijji
uij uuuu
3
11
*
The simples return-to-isotropy parameterisation (Rotta 1951)
Analogously, for the temperature flux (e.g. Zeman 1981)
*
i
i
u
Linear Models of ij and i (cont’d)
.2
1
2
1
,,2
1,
3
22
,2221
i
j
j
iik
i
j
j
iik
iikkij
kk
jiij
kjijkcibjijsijsí
ti
x
u
x
uWand
x
u
x
uS
guueuu
uuawhere
uCCuWCSCu
C
,23
2
3
2321
jklilkjklilkuckkijijji
ub
kijkkjikusklklijkijkkjik
usij
us
u
ijutij
uuuuCuuuC
eWaWaCSaSaSaCSCea
C
kjijkcibjijsijs
íti uCCuWCSC
uC 22
21
Equation for the temperature flux
kjijkijijij
iik
kjjii
jj
uuWS
x
puu
xxuuu
xu
t
22
Linear Models of ij and i (cont’d)
cbsttotal ppppp
kjijki
ii
j
j
ijiji
jik
uxx
u
x
uuuuu
xxx
p
22
2
2
2
Poisson equation for the fluctuating pressure
Decomposition
Contribution to p’ due to buoyancy
Vol kiikikki
Vol ki
ki
iVol ii
ii
k
b
rr
rdv
xx
rrYYthen
rr
rdv
xx
rr
x
p
rr
rdv
x
rp
xx
p
.)()()(
4
1
,)()()(
4,
)()(
4
1,
2
,
2
2
2
Linear Models of ij and i (cont’d)
NB! The volume of integration is the entire fluid domain.
!3
1withi.e.,
3
1 21
bibbi CC
21 ikikY
The buoyancy contribution to i is modelled as
The simplest (linear) representation
… satisfying … we obtain
Cf. Table 1 of Umlauf and Burchard (2005):
Cb = (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.
.andwhere 2, iikiikikki YYYY
Linear Models of ij and i (cont’d)
!10
3with
3
2
u
bkkijijjiub
bij CuuuC
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).
.and0,where, iikkiikikjijkjikijkkij uXXXXXX
3/10?
Linear Models of ij and i (cont’d)
Non-Linear Intrinsically Realisable TCL Model
2321 mkimikikik aaaY
ij
kk
jiij
uu
uua
3
22
00 332
3 uasY kk
The buoyancy contribution to i is a non-linear function of departure-from-isotropy tensor
The representation
… together with the other constraints (symmetry, normalisation) … yields
Realisability. The two-component limit constraints (Craft et al. 1996)
2
3
1
ikki
bi a
Buoyancy contribution to i in convective boundary-layer flows (Mironov 2001).
Short-dashed – LES data,
solid – linear model with Cb=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.
Models of i against data
TCL model (sophisticated and physically plausible) still does not perform well in some important regimes.
Truncated Second-Order Closures
ijijij
kk
jiij aaA
uu
uua
2,
3
22
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.
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,
eleKKz
Kwx
u
x
uKuu u
i
j
j
iuji
2/1,,
and production-dissipation equilibrium relations for the TKE and the scalar variances.
Two-Equation TKE-Scalar Variance Model (MY74 level 3)
Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g.
22
2
1
2
1w
zzw
t
pwuuwz
wgz
vvw
z
uuw
t
eii2
1
Transport equations for the TKE and for the scalar variance(s)
,,, 221
gSz
eSwz
veSvw
z
ueSuw HHMM
egSNRiSSSS HHM /,/,of functions,, 222222221
One-Equation TKE Model (MY74 level 2.5)
Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g.
zw0
pwuuwz
wgz
vvw
z
uuw
t
eii2
1
Transport equation for the TKE
,,,z
eSwz
veSvw
z
ueSuw HMM
2222 /,of functions, SNRiSSS HM
Diagnostic formulation(s) for the scalar variance(s)
Comparison of 1-Eq and 2-Eq Models
Equation for <’2>
22
2
1
2
1w
zzw
t
Production = Dissipation (implicit in all models that carry the TKE equation only).
Equation for <w’’>
No counter-gradient term (cf. turbulence models using “counter-gradient corrections” heuristically).
2
gCz
eCw bg
1-Eq Models are Draft Horses of Geophysical Turbulence Modelling
Importance of Scalar Variance
The TKE equation
z
gNN
gw
zN
gwg
tN
g
22
22
2
22
2
2
,2
1
2
1
pwuuwz
wgz
vvw
z
uuw
t
eii2
1
The <’2> equation
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!
Given transport equation for the temperature flux,
,22
iik
kkjijki
kki
k
iki
kk x
puu
xug
xuu
x
uuu
xu
t
.i
i xKu
make simplifications and invoke closure assumptions to derive a down-gradient approximation for the temperature flux,
(Hint: the dimensions of Kθ is m2/s.)
Exercise
The Surface Layer
The now classical Monin-Obukhov surface-layer similarity theory (Monin and Obukhov 1952, Obukhov 1946).
The surface-layer flux-profile relationships
ssfc
ssfc
hh
ssm
ms
Qg
uLwQwuu
Lzz
z
u
QLz
z
zuuu
3*2
*
0*0
*
,,
,/ln,/ln
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.
,,
,,
2
1
*3*2
*
2*
3/22/3
z
u
z
u
z
u
z
uu
uCezll
eC
pwuuwz
wgz
vvw
z
uuw
t
eii
Effects of Water Vapour and Clouds
Quasi-conservative variables
Virtual potential temperature is defined with due regard for the water loading
re temperatupotential water total
humidity specific water total
ip
il
p
vt
ilvt
qc
L
Tq
c
L
T
qqqq
dvilvt RRRqqqR /,11
qt
qt qt
xΔx
xΔx
qt
x x
Neglect SGS fluctuations of temperature and humidity, all-or-nothing scheme
Account for humidity fluctuations only
Account for temperature and humidity fluctuations
no clouds, C = 0 C = 1
Cloud cover 0<C<1, although the grid box is unsaturated in the mean
tq
tq
tq
tq
sqsq
sqsq
Turbulence and Clouds
tq
after Tompkins (2002)
0
)( dssPC
0
)( dsssPqc
st qqs
cloud cover, cloud condensate = integral over supersaturated part of PDF
If PDF of s is known, then
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.
cloud cover cloud condensate
cq
Turbulence and Clouds (cont’d)
Buoyancy flux (a source of TKE),
ltlv qwDqwBwAgwg is expressed through quasi-conservative variables, where Aθ and Aq are functions of mean state and cloud cover,tqlv qwAwAw
Aq is of order 200 for cloud-free air, but ≈ 800 ÷ 1000 within clouds!
Turbulence and Clouds (cont’d)
functional form depends on assumed PDF Aθ = Aθ (C, mean state) Aq = Aq (C, mean state)
Clouds-turbulence coupling: clouds affect buoyancy production of TKE, turbulence affect fractional cloud cover (where accurate prediction of scalar variances is particularly important).
2/1222 2 ltlts qPPqa
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)
s
Surface Temperature in Homogeneous and Heterogeneous Cases
time
8h 9.75hsampling
s = (s1+ s2)
s1
s2
homogeneous case
heterogeneous case
x
s2
s1
s
y
warm stripe
cold stripe+
Mean Potential Temperature
Blue – homogeneous SBL,
red – heterogeneous SBL.
cf. Stoll and Porté-Agel (2009)
TKE and Temperature Variance
Blue – homogeneous SBL, red – heterogeneous SBL.
Large
TKE Budget
Left panel – homogeneous SBL, right panel – heterogeneous SBL.
Red – shear production, blue – dissipation, black – buoyancy destruction, green – third-order transport,
thin dotted black – tendency .
pwuwz
wgz
vvw
z
uuw
t
ei2
2
1
Decreased in magnitude
Temperature Variance Budget
Left panel – homogeneous SBL, right panel – heterogeneous SBL.
Red – mean-gradient production/destruction, blue – dissipation, green – third-order transport, black (thin dotted) – tendency .
22
2
1
2
1w
zzw
t
Net source
Key Point: Third-Order Transport of Temperature Variance
LES estimate of <w’’2> (resolved plus SGS)
222 2"
wwww
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!
a as2s1
s
s1
s2
x
z z
Third-Order Transport of Temperature Variance
0 w
0w 0
w
0
0
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
Increased
Increased Decreased (in magnitude)
2
gCz
eCw bg
downward upward
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.
Can We Improve SBL Parameterisations?
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.
Transport equation for the temperature flux
iik
kkjijki
kki
k
iki
kk x
puu
xug
xuu
x
uuu
xu
t
22
then neglecting anisotropy
iik
kniki x
Kx
ux
uu
22
3
2
3
2
(!) Using Rotta-type return-to-isotropy parameterisation of the pressure gradient-temperature covariance
,
i
i
u
x
p
yields the down-gradient formulation
,3
2
3
2
3
2 222niknikkinikki uuuuuuu
Exercise: derive down-gradient approximation for fluxes from the second-moment equations