The Workshop International Conference “ “CITES CITES- -2007 2007” ” International Conference Profs. Kabanov and Lykosov Tomsk, Russia, 20 Tomsk , 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 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,
Goals oals and and objectives objectives G 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 Quasi -- Geostrophic Turbulence l Transport and mixing from 2D atmosphere dynamics point of l view, we mean development of “chaotic advection” (“lagrangian turbulence”)
Quasi -- -- Geostrophic Turbulence (QGT) Geostrophic Turbulence (QGT) Quasi Quasi-geostrophic turbulence -- 2D or 3D turbulence 2D Turbulence -- law conserve of energy, enstrophy and no vortex l stretching 3D Turbulence -- enstrophy not conserved and vortex stretching l QGT -- law conserve of energy, enstrophy and vortex stretching l
Quasi -- -- Geostrophic Turbulence Geostrophic Turbulence Quasi The similarity of 2D and QG flows allow Charney J. (1971) to l conclude that an energy cascade to small-scales is impossible in QGT .
Quasi -- -- Geostrophic Turbulence. Geostrophic Turbulence. Quasi Spectrum of energy in 2DT 2D turbulence theory predicts: Inverse energy cascade from the point of energy input (spectral slope –5/3), l (Kolmogorov, 1941) ε − 2/3 5/3 E k ( ) k Downscale cascade to smaller scales (spectral slope –3), l (Kraichnan,1967) 2 − η 3 3 E ( k ) k
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 -- -- Geostrophic Turbulence. Geostrophic Turbulence. Quasi Spectrum of passive tracer
Quasi -- -- Geostrophic Turbulence. Geostrophic Turbulence. Quasi Spectrum of passive tracer
Quasi -- Quasi -- Geostrophic Turbulence. Geostrophic Turbulence. Spectrum of passive tracer
Quasi -- -- Geostrophic Turbulence. Geostrophic Turbulence. Quasi Spectrum of passive tracer Spectral range of large Prandtl (Schmidt) number: ε 1/4 ν 1/2 ν 1/ 2 − ⇒ = τ = ⇒ = χ ⋅ 1 !_ Pr 1 k , ( ) k P k ( ) k ν ν ε ε 3 Spectral range of little Prandtl (Schmidt) number: χ ν d χ = χ = − 2 !_ Pr 1, ( ) _ k and _ 2 k P k ( ) Pr dk P k k ( ) − − − χ = τ = ε ⇒ = χ ε ⇒ 2 1/3 1/3 5/3 and _ , ( ) k ( k ) P k ( ) ( ) k k τ ( ) k ν χ ν − − ε 1/ 3 2/ 3 d 3/ 2 k − = − χ ε ⋅ ⇒ χ = χ 1/3 1/3 Pr 2 k e , 0 dk Pr ν − − ε 1/ 3 2 / 3 3/2 k − − χ = χ = = ⇒ = χ ε 1/3 5/3 Pr ( k k k ) P k ( ) k e . ν 0 0 0
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) on 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 i naccuracy (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 ε 3 β 5 − 4 5 D 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 ε β 5 5 D
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 ∂ ' ' v b B ε = − = − ⋅ ( ' ') v b I ∂ ∂ PB B B y ∂ z
Diffusivity and cascading eddies Inverse cascade scalings 12 β = ε ⇒ = ε 3 3 / k V k k V 0 b R V (Danilov and Gurarie, 2002, Smith et al., 2002) − 3 1 = β ε k β 5 5 ⇒ ⇒ β − ε 1 5 2 5 If _ k k k k V β β b R
Diffusivity and cascading eddies Diffusivity scaling = − − V = ⇒ = β ε 4 5 3 5 k mixing _ scale k k D c β d D d ∂ [ ] B = − D v b ' ' where, […] - horizontal average. ∂ b y
In 2- In 2 -layer QG model layer QG model U ξ > ξ = 1 - supercriticality, (Phillips, 1956) - flow is unstable βλ 2 ξ → ∞ ⇒ ≈ ε = = − = β + ξ = β − ξ D D ' ' Uv q ' Uv q ' U D (1 ) U D (1 ) 1 2 1 2 1 2 − 3 4 ε β 5 5 D
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