Transcript Slide 1

TURBULENCE CLOSURE PROBLEM
FOR STABLY STRATIFIED
GEOPHYSICAL FLOWS
S. Zilitinkevich1-4, N. Kleeorin5, I. Rogachevskii5
1
Finnish Meteorological Institute, Helsinki, Finland
2 Atmospheric Sciences, University of Helsinki, Finland
3 Nizhniy Novgorod State University, Russia
4 Institute of Atmospheric Physics RAS, Moscow, Russia
5 Ben-Gurion University of the Negev, Beer Sheba, Israel
February 2012
Main-stream in turbulence closure theory
Boussinesq (1877) Turbulent transfer is similar to molecular transfer
but much more efficient  down-gradient transfer  K-theory  eddy viscosity,
conductivity, diffusivity
Richardson (1920, 1922) stratification Ri (and concept of the energy cascade)
Keller - Fridman (1924) a chain of budget equations for statistical moments
Problem: to express higher-order moments through lower-order moments
Prandtl (1930s) mixing length l ~ z, velocity scale uT ~ ldU/dz, viscosity K ~ luT
Kolmogorov (1941) (quantified the cascade) closure as a problem of energetics:
• budget equation for turbulent kinetic energy (TKE)
• TKE dissipation rate expressed through the turbulent-dissipation length scale
uT ~ (КЭТ)1/2, K ~ lεuT underlies further efforts until the end of 20th century
Obukhov (1946) TKE-closure extended to stratified flows, Obukhov length scale L
Monin-Obukhov (1954) alternative  similarity theory for the surface layer
z /L
Mellor-Yamada (1974) Hierarchy of K-closures. The problem of turbulence cut-off
Turbulence cut-off problem
Buoyancy
b = (g/ρ0)ρ
Velocity shear S = dU/dz
(g – acceleration due to gravity, ρ –density)
(U – velocity, z – height)
db / dz
Ri 
(dU / dz) 2
Richardson number characterises static stability
The higher Ri (or z/L), the stronger suppression of turbulence
Key question What happens with turbulence at large Ri?
Traditional answer Turbulence degenerates, and at Ri exceeding a critical
value (Ricritical< 1) the flow becomes laminar (Richardson, 1920; Taylor,
1931; Prandtl, 1930,1942; Chandrasekhar, 1961;…)
In fact field, laboratory and numerical (LES, DNS) experiments show that
GEOPHYSICAL turbulence is maintained by shear at least up to Ri ~ 102.
Modellers are forced to preclude the turbulence cut-off ARTIFICIALLY
Milestones
Prandtl-1930’s followed Boussinesq’s idea of the down-gradient transfer (K-theory),
determined K ~ luТ , and expressed uT heuristically through the mixing length l
Kolmogorov-1942 (for neutrall stratication) followed Prandtl’s concept of eddy viscosity
KM ~ luТ ; determined uT = (ТКЕ)1/2 through TKE budget equation with dissipation
ε ~ (TKE)/tT ~ (TKE)3/2/lε; and assumed lε ~ l (grounded in neutral stratification)
Obukhov-1946 and then the entire turbulence community extended Kolmogorov’s closure
to stratified flows keeping it untouched. They only included in the TKE equation the
buoyancy term that caused cutting off TKE in “supercritical” stable stratification
In doing so, they missed turbulent potential energy (TPE interacted with TKE);
overlooked inapplicability of Prandtl’s relation K ~ luТ to the eddy conductivity KH;
and disregarded principal deference between lε and l
For practical applications Mellor and Yamada (1974) developed heuristic corrections
preventing unacceptable turbulence cut-off in “supercritical” static stability
Energy- & flux-budget (EFB) theory (2007-2012)
Budget equations for major statistical moments
EK
EP
Vertical flux of temperature
Fz= <θw> [or buoyancy (g/T)Fz ]
Vertical flux of momentum
τiz = <uiw> (i = 1,2)
New relaxation equation for the dissipation time scale tT = lε/(TKE) l  l
Turbulent kinetic energy (TKE)
Turbulent potential energy (TPE)
Accounting for TPE vertical flux of buoyancy (that killed TKE in Kolmogorov’s type
closures) drops out from the equation for total turbulent energy (TTE = TKE + TPE)
The heat-flux equation reveals a self-limitation of the vertical heat/buoyancy flux
causing essential self-preservation of turbulence up to Ri
~ 102
Physical mechanisms and concepts
• Kolmogorov’s model for the effective dissipation of the turbulent flux of momentum
• Non-gradient generation of the buoyancy flux  self-preservation of turbulence
• New, physically consistent model of the turbulent dissipation time / length scales
• Fully revised inter-component energy exchange (instead of “return to isotropy”)
Finally we got rid of misleading analogies with molecular transfer
Turbulent potential energy (analogy with
Lorenz’s available potential energy)
Fluctuation of buoyancy
g  
2



b 
 
z  N z
0
0 z
g
Fluctuation of potential energy (per unit mass)
1 z  z
( EP )   b z dz
z z
2
2
2
1 (b)
1  


2

        ( E )
2
2 N
2 N 
N
Turbulent energy budgets
1
Kinetic energy EK  ui ui
2
DE K  K

 S  Fz   K
Dt
z
1   2
Potential energy EP    
2 N 
2
DE P  P

   Fz   P
Dt
z
Total energy E  EK  EP
DE  ( K  T )

 S  ( K   P )
Dt
z
Buoyancy flux βFz drops out from the turbulent total energy budget
Budget equation for the turbulent flux
of momentum  i 3  ui w
D i 3  ( )
U i
 Φ i  2 E z
  i(3()eff )
Dt
z
z
Effective dissipation

( )
i 3(eff )
 ui w 
ui w
 i3
1
 2
  Fi 
p
 ~
xk xk
0
tT
 z xi 
LES verification of Kolmogorov closure for effective
( )

dissipation of the turbulent flux of momentum i 3(eff )
Budget equation for the vertical turbulent
flux of potential temperature Fz   w
DFz  ( F )

Fz
2
  z  C    2 Ez

Dt z
z
CF tT
The “pressure term” is shown to be proportional to the mean squared
fluctuation of potential temperature
1
0
p

~  2
z
On the r.h.s. of the equation, the 1st term (generating positive heat flux)
counteracts to the 2nd term (generating negative heat flux) and assures
self-preservation of turbulence in very stable stratification
LES verification of our parameterization of
1

the pressure term 0  p z
Turbulent dissipation time and length scales
1/ 2
By definition, time scale: tT  EK /  K , length scale: l  EK
tT
The steady-state TKE budget
Flux Ri.
number
 Fz 
Ri f 

S
SL
Shear: neutral
1/ 2
S
0
Obukhov
length
L

 Fz
Rif  R  1
 Fz  1 / 2
S

R
R L
1/ 2
 
k z
1 
 k / R  1.6
S

kz 
R L 
 1 / 2 , extreme stable (TKE)
kz
Interpolation yields empirical
law valid in any stratification
Combining the law
with TKE budget
equation yields
where l
EK
S  Fz  S 1  Rif    K  
tTE
3/ 2
l0 ( EK /  )
 EK 
lE 
 l0 

1
1  k ( R  1) z / L
  
3/ 2
 Cl z /(1  Cz / E )
1/ 2
K
3/ 2
1  Ri f / R
1  Ri f
is master length scale
Relaxation equation for dissipation time scale
Evolution of tT is controlled by the tendency towards equilibrium:
tT  tTE
l0
( EK /  )
l0  EK 
 1/ 2
 1/ 2 

1
EK 1  k ( R  1) z / L EK   
3/ 2
3/ 2
1  Ri f / R
1  Ri f
and distortion by non-stationary processes and, in heterogeneous
flows, by the mean-flow and turbulent transports.
Their counteraction is described by the RELAXATION EQUATION:
 tT

DtT 
tT
 KT
 CR   1
Dt z
z
 tTE

where CR ~ 1 is the relaxation constant.
EFB closure and M-O similarity theory
dU  1 / 2 
k z
Substituting the above empirical law
1 

S

dz
kz  R L 
  Fz
into definition of flux Richardson number Ri f 
S
kz / L
Rif 
yields CONVERTOR between Rif and z/L
1  kR1 z / L
Ri
EFB closure yields CONVERTOR between Rif and Ri: Ri f 
PrT (Ri)
where
PrT  P0  0.8
at Ri <<1,
PrT  Ri / R  4Ri
at Ri >>1
(see below empirical Ri-dependence of turbulent Prandtl number PrT)
Major results
The concept of turbulent potential energy (Z et al., 2007) analogous to Lorenz’s
available potential energy (both proportional to squared density)
New treatment of and relaxation equation for turbulent dissipation time scale
Disproved widely recognised, erroneous conclusion (from traditional turbulenceclosure theory) that shear-generated turbulence cuts off and flow becomes
laminar at Richardson numbers Ri exceeding a critical value Ric ~ 0.25-1.
Instead, in the EFB theory, a threshold value of Ri separates two regimes of the
stably stratified turbulence of principally different nature:
“Strong turbulence” KM ~ KH typical of boundary layers (at Ri< Ric)
“Weak turbulence” PrT = KM /KH ~ 4 Ri (at Ri >>Ric) – unknown until now
A hierarchy of closure models of different complexity – for use in research and
operational modelling atmospheric and oceanic flows
Principal revision of the Monin-Obukhov similarity theory (transitional asymptote)
Field, laboratory and numerical (LES, DNS) experiments confirm our theory up
to Ri ~ 102 – for conditions typical of the free atmosphere and deep ocean
Examples of empirical verification of the
steady-state version of the EFB closure
Turbulent Prandtl number PrT = KМ /KH versus Ri
Atmospheric data: (Kondo et al., 1978), (Bertin et al., 1997); laboratory experiments:
(Rehmann & Koseff, 2004), (Ohya, 2001), (Strang & Fernando, 2001); DNS: (Stretch et
al., 2001); and LES: (Esau, 2009). The curve sows our EFB theory. The “strong” turbulence
(PrT  0.8) and the “weak” turbulence (PrT ~ 4 Ri) separate at Ri ~ 0.25.…
Longitudinal Ax, transverse Ay & vertical Az TKE shares vs. z/L
Experimental data from Kalmykian expedition 2007 of the Institute of Atmospheric
Physics (Moscow). Theoretical curves are plotted after the EFB theory. The traditional
“return-to-isotropy” model overlook the stability dependence of Ay clearly seen in
the Figure, where the strongest stability, z/L =100, corresponds to Ri = 8.
The share of turbulent potential energy ЕР / (ЕР+ЕК)
Насыщение ЕР / (ЕР + ЕК) ~ 0.2-0.4
Порог Ri = 0.25
The share of the energy of vertical velocity Еz / ЕK
The dimensionless vertical flux of momentum  two plateaus
corresponding to the “strong” and “weak” turbulence regimes
The dimensionless heat flux  sharply
diminishes in the “weak” turbulence regime
The velocity gradient M   kz / u  /  U / z 
versus ζ = z/L after LES (dots) and the EFB model (curve)
The temperature gradient  H  (kT z 1/ 2 / Fz ) / z 
versus ζ = z/L after LES (dots) and the EFB model (curve)
Richardson number, Ri, versus ζ = z/L
after LES (dots) and the EFB model (curve)
Conclusions
 TKE budget equation is INSUFFICIENT
EK and EP are equally important  Е = EK + EP
 There is no Ric in the energetic sense; experimental
data confirm this theoretical conclusion up to Ri ~ 102
 There is a threshold Ri ~ 0.2-0.3 (quite close to the liner
instability limit) – separating principally different
regimes of “strong” and “weak“ turbulence
 The newly discovered “weak turbulence regime” is
typical of the free atmosphere and deep ocean,
wherein it determines turbulent transport of the energy
and momentum and diffusion of passive scalars
 A hierarchy of EFB closure models – new instruments
for research and modelling applications
References (last decade)
Zilitinkevich, S.S, Gryanik V.M., Lykossov, V.N., Mironov, D.V., 1999: A new concept of the third-order transport and
hierarchy of non-local turbulence closures for convective boundary layers. J. Atmos. Sci., 56, 3463-3477.
Mironov, D.V., Gryanik V.M., Lykossov, V.N., & Zilitinkevich, S.S., 1999: Comments on “A new second-order turbulence
closure scheme for the planetary boundary layer” by K. Abdella, N. Mc.Farlane. J. Atmos. Sci., 56, 3478-3481.
Zilitinkevich, S.S., Elperin, T., Kleeorin, N., Rogachevskii, I., 2007: Energy- and flux-budget (EFB) turbulence closure
model for the stably stratified flows. Pt.I: Steady-state, homogeneous regimes. Boundary-Layer Meteorol. 125, 167192.
Mauritsen, T., Svensson, G., Zilitinkevich, S.S., Esau, I., Enger, L., Grisogono, B., 2007: A total turbulent energy closure
model for neutrally and stably stratified atmospheric boundary layers, J. Atmos. Sci., 64, 4117–4130.
Zilitinkevich, S., Elperin, T., Kleeorin, N., Rogachevskii, I., Esau, I., Mauritsen, T., Miles, M., 2008: Turbulence energetics
in stably stratified geophysical flows: strong and weak mixing regimes. Quart. J. Roy. Met. Soc. 134, 793-799.
Sofiev M., Sofieva V., Elperin T., Kleeorin N., Rogachevskii I., Zilitinkevich S.S., 2009: Turbulent diffusion and turbulent
thermal diffusion of aerosols in stratified atmospheric flows. J. Geophys. Res. 114, DOI:10.1029/2009JD011765
Zilitinkevich, S., Elperin, T., Kleeorin, N., L'vov, V., Rogachevskii, I., 2009: Energy- and flux-budget (EFB) turbulence
closure model for stably stratified flows. Pt.II: The role of internal waves. Boundary-Layer Meteorol. 133, 139-164.
Zilitinkevich, S.S., 2010: Comments on numerical simulation of homogeneous stably stratified turbulence. Boundary-Layer
Meteorol. DOI 10.1007/s10546-010-9484-1
Zilitinkevich, S.S., Esau, I.N., Kleeorin, N., Rogachevskii, I., Kouznetsov, R.D., 2010: On the velocity gradient in the stably
stratified sheared flows. Part 1: Asymptotic analysis and applications. Boundary-Layer Meteorol. 135, 505-511.
Kouznetsov, R.D., Zilitinkevich, S.S., 2010: On the velocity gradient in stably stratified sheared flows. Part 2: Observations
and models. Boundary-Layer Meteorol. 135, 513-517.
Zilitinkevich, S.S., Kleeorin, N., Rogachevskii, I., Esau, I.N., 2011: A hierarchy of energy- and flux-budget (EFB)
turbulence closure models for stably stratified geophysical flows. Submitted to Boundary-Layer Meteorol.
Turbulence does not degenerate
up to very strong stratification
From «only TKE»
to«TKE + TPE»