Transport and mixing in Q2D-dimension atmospheric flow - - PowerPoint PPT Presentation

International Conference International Conference “ “CITES CITES-

• 2007

2007” ” Tomsk Tomsk, Russia, 20 , Russia, 20-

• 25 July, 2007

25 July, 2007

Transport and mixing in Q2D-dimension atmospheric flow

KRUPCHATNIKOFF V., I. BOROVKO Institute computational mathematics and mathematical geophysics SB RAS Novosibirsk State University, Novosibirsk,

e - mail: vkrup@ommfao1.sscc.ru The Workshop

• Profs. Kabanov and Lykosov
• Acknowledgements. The work was supported by RFFI № 05-05 - 64989

The transport and mixing in atmosphere

Transport and mixing processes in the atmosphere operate on

scales from millimeters to thousands of kilometers. In certain parts of the atmosphere the large-scale quasi-horizontal flow appears to play the dominant role in transport and in the stirring process that leads ultimately to molecular mixing at very small scales. The works in other dynamical contexts such as `chaotic advection' is also relevant. The transport and mixing properties of the atmospheric flow are of great signicance, since they play a major role in determining the distribution of atmospheric chemical species,

G Goals

• als and

and objectives

• bjectives

l The report was focused upon some studies about

turbulent diffusion of scalar (active scalar), transport and mixing that relevant to the underlying quasi - qeostrophic turbulence and 2D atmospheric flows

Contents Contents

l

Quasi -- Geostrophic Turbulence

l

Transport and mixing from 2D atmosphere dynamics point of view, we mean development of “chaotic advection” (“lagrangian turbulence”)

Quasi Quasi --

• - Geostrophic Turbulence (QGT)

Geostrophic Turbulence (QGT)

Quasi-geostrophic turbulence -- 2D or 3D turbulence

l

2D Turbulence -- law conserve of energy, enstrophy and no vortex stretching

l

3D Turbulence -- enstrophy not conserved and vortex stretching

l

QGT -- law conserve of energy, enstrophy and vortex stretching

Quasi Quasi --

• - Geostrophic Turbulence

Geostrophic Turbulence

l

The similarity of 2D and QG flows allow Charney J. (1971) to conclude that an energy cascade to small-scales is impossible in QGT.

Quasi Quasi --

• - Geostrophic Turbulence.

Geostrophic Turbulence.

Spectrum of energy in 2DT

2D turbulence theory predicts:

l

Inverse energy cascade from the point of energy input (spectral slope –5/3), (Kolmogorov, 1941)

l

Downscale cascade to smaller scales (spectral slope –3), (Kraichnan,1967)

2/3 5/3

( ) E k k ε

−



2 3 3

( ) E k k η

−



Cascades in

(a) two - dimensional vorticity dynamics, (TDV) (b) surface quasi-geostrophic dynamics, (SQG), (Held, I., et al..1995) (c) large-scale quasi-geostrophic dynamics, (LQG), (Larichev V., McWilliams J. 1991)

Surface quasi-geostrophic dynamics, (SQG), (Held, I., et al..1995) For example, we look at the potential temperature (PT) in a system of equations called the surface quasigeostrophic (SQG) equations. Not only do the SQG equations have geophysical relevance they have a strong relation to the full 3D Euler equations. In a sense the PT in the SQG system acts as a ‘‘bridge’’ from 2D to 3D turbulence.

Passive scalar transport in isotropic turbulence

The passive scalar fluctuations are introduced into the turbulence in one of two ways.

• statistically isotropic passive scalar fluctuations are introduced

directly into the fluid at the initial time. The introduced scalar fluctuations are then smoothed by turbulent mixing and molecular diffusion, and the scalar variance decays with the mean-square velocity.

• weak uniform mean scalar gradient is imposed across a turbulent
• fluid. Statistically homogeneous (but not isotropic) passive scalar fluctuations

are then created as a consequence of the turbulent motion along the mean gradient; the scalar variance is initially zero and then increases. At later times turbulent mixing and molecular diffusion act to smooth the generated scalar fluctuations.

2D Isotropic Turbulence: Top row: total (left), coherent (middle), and incoherent (right) vorticity fields. Coherent part:

0.2% N wavelet modes, 99.9% of kinetic energy, and 93.6% of enstrophy. Incoherent part: 99.8% N wavelet modes, 0.1% of kinetic energy, and 6.4% of enstrophy

Bottom row: scalar advected in the total (left), coherent (middle), and incoherent (right) flows.

The Schmidt number of the tracer is Sc = 1 and concentration is normalized between 0 and 1. (C. Beta, K. Schneider, M. Farge, 2004)

Quasi Quasi --

• - Geostrophic Turbulence.

Geostrophic Turbulence.

Spectrum of passive tracer

Quasi Quasi --

• - Geostrophic Turbulence.

Geostrophic Turbulence.

Spectrum of passive tracer

Quasi Quasi --

• - Geostrophic Turbulence.

Geostrophic Turbulence.

Spectrum of passive tracer

Quasi Quasi --

• - Geostrophic Turbulence.

Geostrophic Turbulence.

Spectrum of passive tracer

Quasi Quasi --

• - Geostrophic Turbulence.

Geostrophic Turbulence.

Spectrum of passive tracer

1/4 1/2 1/ 2 1 3

!_ Pr 1 , ( ) ( ) k k P k k

ν

ε ν ν τ χ ν ε ε

−

      ⇒ = = ⇒ = ⋅             

1/ 3 2/ 3 1/ 3 2 /

2 2 1/3 1/3 5/3 3/ 2 1/3 1/3 Pr 3/2 1/3 5/3 Pr

!_ Pr 1, ( ) _ _ 2 ( ) Pr ( ) _ , ( ) ( ) ( ) ( ) ( ) 2 , Pr ( ) ( )

k k

d k and k P k dk P k k and k k P k k k k d k e dk k k k P k k e

ν ε ν ε ν

χ ν χ χ χ τ ε χ ε τ χ ν χ ε χ χ χ χ χ ε

− −

− − −   −   −     −   − −  

  = = −     = = ⇒ = ⇒   = − ⋅ ⇒ =     = = = ⇒ = 

3

.

Spectral range of large Prandtl (Schmidt) number: Spectral range of little Prandtl (Schmidt) number:

QG Turbulence. Stratification QG Turbulence. Stratification

QG Turbulence. Stratification QG Turbulence. Stratification

QG Turbulence. Stratification QG Turbulence. Stratification

QG Turbulence. Stratification QG Turbulence. Stratification

Remarks on Charney’s Note (Geostrophic turbulence, 1971)

• n Geostropic Turbulence
• Charney’s work was motivated by the observation available at the time (Wiin-Nielsen 1967), which showed an

apparent k **(-3) power-law behavior in the energy spectrum for horizontal wavenumbers k in the synoptic scales and its similarity to the k**(-3) spectrum predicted by Kraichnan (1967) for 2D turbulence for wavenumbers higher than the excitation wavenumber.

• There is a demonstration of isomorphism between QGT and 2D turbulence, and consequently the observed

k**(-3) spectrum over the synoptic scales was explained using Kraichnan’s (1967) theory on isotropic and homogeneous 2D turbulence.

• It attempts to prove that energy cascades upscale in the net in QGT, similar to 2D turbulence.

However, both of these results contain inaccuracy

(K. K. TUNG, WENDELL T. WELCH, 2003)

Diffusivity, Kinetic Energy Dissipation: Eddy Heat Flux (G. LAPEYRE, I. M. HELD, 2003)

The theory for baroclinic eddy heat fluxes (Held I., V.Larichev, 1996) satisfies two constraints between the eddy diffusivity and the rate of baroclinic energy production. The first of these constraints arises from the assumption that the energy-containing eddies stem from an inverse energy cascade that is halted by the β effect (Rhines, 1975). From this assumption, one can relate the eddy length and velocity scales to ε, the rate per unit of mass at which kinetic energy is flowing through the inverse cascade and being dissipated at large horizontal scales by surface friction. From these length and velocity scales, one can estimate a diffusivity from their formula:

3 4 5 5

D ε β

−



Diffusivity, Kinetic Energy Dissipation: Eddy Heat Flux

The second half of the theory consists in relating ε with the production of available potential energy. In a quasigeostrophic system in which baroclinic production is the only significant eddy source, eddy energy is created through downgradient eddy heat fluxes and is typically dissipated at large scales through friction (vertical turbulent diffusion in surface boundary layers). In the Boussinesq approximation, the baroclinic production is

' ' ( ' ')

PB B

v b B v b I B y z ε     ∂       = − = − ⋅   ∂     ∂ ∂    

Diffusivity and cascading eddies Inverse cascade scalings

3 3

/

b

V k k V ε ε ⇒ = 

12 R

k V β   =    

3 1 5 5

kβ β ε

−

=

(Danilov and Gurarie, 2002, Smith et al., 2002)

1 5 2 5

_

b R

If k k k k V

β β

β ε

−

⇒ ⇒   

Diffusivity and cascading eddies

Diffusivity scaling

_

d

V k mixing scale D = −

4 5 3 5 d

k k D c

β

β ε

−

= ⇒ = [ ]

' '

b

B D v b y ∂ = − ∂

where, […] - horizontal average.

In 2 In 2-

• layer QG model

layer QG model

2

U ξ βλ =

1 ξ >

' ' 1 2 1 2

' ' (1 ) (1 ) Uv q Uv q U D U D ε β ξ β ξ = = − = + = −

• supercriticality, (Phillips, 1956)
• flow is unstable

1 2

D D ξ → ∞ ⇒ ≈

3 4 5 5

D ε β

−



Geophysical turbulence Observed Power Observed Power-

• Law Behaviour

Law Behaviour

Gage and Nastrom (1985) collected observational data indicating that the energy spectrum of atmosphere contains a `critical point’ at nearly 500 km, separating large and small scales. Their spectra cover scales ranging from 3 km to nearly 10,000 km. The observed spectrum is characterized by k**( –3) slope downscale enstrophy cascade at large scales and an -5/3 inverse energy cascade at small scales. Charney (1971) attributes the k**( –3) slope at scales above 1000 km to quasi-geostrophic turbulence. The mesoscale dynamics follow a Kolmogorov k**(–5/3) spectral slope.

Geophysical turbulence Observed Power Observed Power-

• Law Behaviour (cont.)

Law Behaviour (cont.)

Two different mechanisms have been proposed to explain the

• bserved mesoscale spectra:
• the first is strongly nonlinear and based on quasi-

2D turbulence. Lilly (1983) postulates that it is due to stratified turbulence at small scales.

• the second mechanism is based on a weakly

nonlinear wave theory involving the spectrum of internal waves.

Geophysical turbulence Observed Power Observed Power-

• Law Behaviour (cont.)

Law Behaviour (cont.)

• -- At length scales below 1000 km, Lilly (1983) suggests that small

scale sources of energy could be provided by thunderstorms.

• -- Small-scale shear instability may also contribute. The only a small

amount of this energy needs to inverse cascade in order to account for the observed mesoscale spectrum → Spectral backscatter

Observed Power-Law Behaviour The Spectral “critical point” (kink)

l

The observational evidence outlined above showed a “critical point” at 500 km

l

It is large for isotropic 3D effects?

l

Nastrom and Gage (1986) suggested the shortwave k**(–5/3) slope could be explained by another inverse energy cascade from storm scales. (after Larsen, 1982)

l

Lindborg & Cho (2001), however, could find no support for an inverse energy cascade at the mesoscales.

l

Tung and Orlando (2002) suggested that the shortwave k**(-5/3) behaviour was due to a small downscale energy cascade from the synoptic scales.

l

Tung and Orlando reproduced Nastrom and Gage spectrum using QG dynamics alone.

l

The NMM model also reproduces the spectral “critical point” at the mesoscales when physics is included. (Janjic, 2006)

Observations: Spectrum of U, V and potential temperature, (Nastrom and Gage, 1985)

Theory Observational Evidence

Atmosphere: (Nastrom and Gage, 1985) Wavenumber spectra of scalar variance and dissipation for stationary, homogeneous, isotropic turbulence forced at wavenumber k-force. Wavenumber spectra of kinetic energy and kinetic energy

dissipation for stationary, homogeneous, isotropic turbulence forced at

wavenumber k-force. Ocean: (W. D. Smyth and J. N. Moum, 2001)

The Spectral The Spectral “ “critical critical point point” ”in in Q2D turbulence Q2D turbulence

Geophysical turbulence

Observed Power Observed Power-

• Law Behaviour.

Law Behaviour. Draft Draft

l

Two power laws were evident:

l

The spectrum has slope close to –(5/3) for the range of scales up to 500 km.

l

At larger scales, the spectrum steepest considerably to a slope close to –3.

l

The spectral “critical point”

l

Spectral backscatter

Spectral backscatter

(G. Shutts, A stochastic Kinetic energy backscatter algorithm fo (G. Shutts, A stochastic Kinetic energy backscatter algorithm for use in EPS, 2004) r use in EPS, 2004)

The rate of energy backscatter to scales near the bound of truncation is controlled by a total energy dissipation function involving contributions from numerical diffusion, mountain drag and deep convection. The input of small-scale kinetic energy by the backscatter algorithm helps to correct a known problem with the energy spectrum in the ECMWF model – the absence of the

• bserved 5/3−spectral slope in the mesoscales.

Since observational estimation of atmospheric energy dissipation rates is subject to considerable uncertainty the estimating local upscale energy transfer rates will be source of error. The intensity

• f turbulent kinetic energy dissipation will vary by orders of

magnitude from place to place given its cubic dependence on velocity and so representivity will always be a problem The global-mean energy dissipation rate associated with vertical mixing can be assessed from NWM’s and is probably accurate to within a factor of ~ 2. In the ECMWF model the global-mean energy dissipation rate due to parametrized vertical mixing is ~ 2 Wm**(-2).

Physical basis for Energy

Spectral Spectral backscatter backscatter

Physical basis for Energy Spectral backscatter (cont.) Example

Suppose that the dissipation rate due to other local sources in the free atmosphere (e.g. due to gravity wave breaking) is comparable in size. If 1/10 of this is transferred upscale rather than dissipated, the associated energy backscatter rate is ~ 2. 10**(-4) m**2 s**(-3) implies an energy tendency of about ~20 m**2 s**(-2)/day equivalent to a local flow acceleration of ~ 4 - 5 m s**(-1)/day,

Physical basis for Energy Spectral backscatter (cont.) Therefore, backscatter forcing have impact on forecast in regions

• f strong dissipation.

Comparison with LES and observations of dissipation rates and spectral fluxes, can be useful in this respect, also with regard to

• btaining scaling laws for various meteorological conditions, and

for identifying the impact of energy fluxes .

Energy spectra at day 5 in forecasts run ECMWF at T799 : blue curve - with stochastic backscatter red curve

• without backscatter. log(E) is the logarithm of the energy density and n

is the spherical harmonic order.

Chaotic advection -- “lagrangian turbulence” (Aref, H., 1984)

Mixing and transport in critical layers (Stewartson, K, 1978; Warn, T., Warn, H., 1989)

%%



2 ' 3 ' ' 2 ' ' ' ' ' '

( , ) 0, , ( , , ) 2 1 ( , ) ( , ), ,

y x

q J q t where q y y x y t x y x y t t y t x x y x β λ µ λ λ β λ β µ ∂ + Ψ = ∂ = + ∆Ψ Ψ = − + Ψ → → Ψ → Ψ   ∂ ∂ ∂Ψ ∂Ψ ∂Ψ   + ∆Ψ + ⋅∆Ψ − ⋅∆Ψ + =     ∂ ∂ ∂ ∂ ∂     %

Chaotic advection -- “lagrangian turbulence” (Aref, H., 1984) (cont.)

' ' ' ' ' '

_ 1, , _ _ , sin sin

yy yyy xyy

k If where k is x wavenumber then y t x x y x y t Y Y x x x Y λ ε β µ τ ε ε ζ ζ ζ τ = −   ∂ ∂ ∂Ψ ∂Ψ ∂Ψ   + Ψ − ⋅Ψ − ⋅Ψ + =     ∂ ∂ ∂ ∂ ∂     = = ∂ ∂ ∂ = + − = ∂ ∂ ∂ 

Chaotic advection -- “lagrangian turbulence” (Aref, H., 1984) (cont.) ( )

2

1, , sin( ) 0, cos( ) 2 ( , ) : , sin( ) t x Y Y x d Y dt q Y Y x x Y x Y Y x ς ς = = = − + = = + Ψ = − + = = − & & & & &

• - is (scaled) first approximation to abs. vort. in critical layer
• - advecting stream function

Chaotic advection -- “lagrangian turbulence” (cont.)

, sin( ) sin ( ) 1) _ 1, 3; 2) _ 2, 2. 0.01,0.02,0.1 x Y Y x k k x ct x k c k c η π π η = = − − − − ≤ ≤ = = = = = & &

Chaotic advection -- “lagrangian turbulence”

Initial distribution, N=10000 (top) K=1,c=3,η=0.01; (bottom) k=1,c=3, η=0.02

Correlation dimension: Correlation dimension:

2 1

( ) ( ) log

d

F h Ch g h d h C = ⋅ + 

1

1.549..., 11.585... d C = =

1

1.526..., 10.819... d C = =

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 6 7 8 9 10 11 12 13 14 15

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 6 7 8 9 10 11 12 13 14

(top) K=1,c=3,η=0.01; (bottom) k=1,c=3, η=0.02 (top) K=1,c=3,η=0.01; (bottom) k=1,c=3, η=0.02

Chaotic advection -- “lagrangian turbulence”

Spectrum of the tracer distribution Spectrum of the tracer distribution

(Pierrehumbert R., 1994) (Pierrehumbert R., 1994)

1 1

( ) ( ) _ _( ( ) ) _ ( ) _ 2 _ _ ( )

d d

dF S k k J kh dh F h h S k k dh If d then S k k

∞ − −

⇒ →

∫

   

1 1 3

( ) P k k η χ

− −



Chaotic advection -- “lagrangian turbulence”

2D Turbulence. Spectrum of 2D Turbulence. Spectrum of passive tracer passive tracer

1 5 3 3

( ) P k k ε χ

− −



η-interval ε-interval

Lyapunov exponents Lyapunov exponents

(Dymnikov,V.P. Filatov, A.N., 1996) (Dymnikov,V.P. Filatov, A.N., 1996)

В основу модели положен полулагранжев метод, в котором характеристики индивидуальной частицы получаются вычислением точки, откуда пришла эта частица, и интерполяцией этих характеристик из ближайших узлов сетки в рассчитанную точку. В эйлеровой системе координат адвекция некоторой физической величины φ записывается в виде = ∇ ⋅ + ∂ ∂ φ φ V t r (1) 1 2

10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5

Lyapunov exponents Lyapunov exponents

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

References References

1. Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Acad. Sci. USSR 30, 299 - 303. 2. Obukhov, A.P. 1949 3. Batchelor, G. K. 1953 Theory of Homogeneous Turbulence. Cambridge University Press 4. Leith, C. E. 1967 Diusion approximations for two- dimensional turbulence. Phys. Fluids 11, 671 - 674.

5. Monin A., Yaglom, 1966. Statistical Fluid mechanics.

6. Rhines, P. B. 1975 Waves and turbulence on a -plane. J. Fluid Mech. 69, 417 - 443. 7. Kraichnan, R. & Montgomery, D. 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547 - 619.

• 8. Held, I. M. & Larichev, V. D. 1996 A scaling theory for horizontally homogeneous,

baroclinically unstable flow on a beta - plane. J. Atmos. Sci. 53, 946 - 952. 9. Dyminikov V.P., A.N. Filatov, 1998, Introduction to mathematical theory of climate

• 10. Danilov, S. D. & Gurarie, D. 2000 Quasi-two-dimensional turbulence. Usp. Fiz. Nauk.

170, 921-968.

• 11. K.S. Smith et al., J.Fluid Mech., 2002, v. 469, pp. 13-48

13.

• G. Vallis, 2003

14. Dymnikov V.P., 2004

• 15. Krupchatnikov V., I. Borovko, 2005

Many Thanks if you listened to me Today Birthday of professor G.P. Kurbatkin Gennady Pavlovich, we wish you happy birthday!