An introductory overview of observations in geophysical flows

Transcription

1 Diffusion, mixing, dispersion An introductory overview of observations in geophysical flows Alberto Maurizi Institute of Atmospheric Sciences and Climate - CNR, Bologna, Italy stratified - rotating turbulence p.

2 Turbulent dispersion at different scales Volcanic eruption: overturning time typical of mesoscale structures: few days stratified - rotating turbulence p.

3 Turbulent dispersion at different scales Volcanic eruption: overturning time typical of mesoscale structures: few days Sigarette smoke: overturning time of the order of seconds stratified - rotating turbulence p.

4 Turbulent dispersion at different scales Volcanic eruption: overturning time typical of mesoscale structures: few days Sigarette smoke: overturning time of the order of seconds The interpretation depends on the scale of interest! stratified - rotating turbulence p.

5 A case study: the Etna 2002 eruption Animation of the thermal view of the Etna plume from to (Courtesy of Fred Prata, CSIRO, Division of Atmospheric Research, Australia) See also the 500hPa animation (by BOLAM) and the particle trajectories (Villani et al., 2006) stratified - rotating turbulence p.

6 Etna 2002 Cross section of the thermal image of the plume on H09:45 (Tiesi et al., 2006) stratified - rotating turbulence p.

7 Etna 2002 Increase of the horizontal spread (with respect to the center of mass) stratified - rotating turbulence p.

8 Etna 2002 Lidar backscatter map measured at Potenza (Italy) on and Vertical dispersion occurs at scales smaller than horizontal. stratified - rotating turbulence p.

9 Turbulent dispersion and atmospheric energy spectrum Observed dispersion properties strongly depend on the ratio between the observation time T and the integral correlation time T I = 0 R(t) dt. Depending on this ratio, the energy spectrum is divided into two parts: scales for which correlation time is larger that t, that do not contribute to dispersion but can be regarded as responsible for advection of the center of mass scales for which the correlatin time is smaller than t for which many realizetion accur during the observed phenomenon so that their effect is a dispersion about the center of mass stratified - rotating turbulence p.

10 outline observations on spectra of velocity in the atmospheric flows dispersion measurements: absolute, relative, meandering diffusion and diffusion equation (classical results and limitations) lagrangian stochastic modelling: theory and practice fractional diffusion equation: a new approach stratified - rotating turbulence p.

11 Atmospheric Spectra: the Atmospheric Bondary Layer Horizontal velocity variance spectrum in the PBL, from Van der Hoven (1957). Observe the gap between mesoscale and fully 3D boundary-layer turbulence stratified - rotating turbulence p.

12 Small scale spectra: Neutral BL Most of the PBL studies concentrate on the high frequency part of the spectrum: Spectra (single cases reported with different symbols) measured at NIS at 2 m and at 10 m for the (a) longitudinal, (b) lateral and (c) vertical components. The lines show the reference spectra (Kaimal et al., 1972). stratified - rotating turbulence p. 1

13 Small scale spectra: Convective Boundary Layer Vertical velocity in the CBL, from Alberghi et al. (2002) show a similar scaling k 1 as the horizontal components in the neutral BL. stratified - rotating turbulence p. 1

14 Small scale spectra: Stable Boundary Layer Stably stratified BL: a classical paper by Finnigan et al. (1984) shows that the horizontal components contain more energy than the vertical one in the lower layers, whereas at higher layers turbulence tends to become isotropic, and also shows the slope typical for gravity waves. stratified - rotating turbulence p. 1

15 Atmospheric spectra: From FT to BL (and vice versa) The relative energy spectra of horizontal wind components (u and v) for Dickson. Constants are added to the energy spectra amplitude at successive heights to separate them on the graph. The corresponding height of a spectrum can be read from the left axis. (c) and (d) Horizontal wind component spectra at the lowest (120 m) and 2.05-km range gates for Dickson. (gupta et al., 1997) stratified - rotating turbulence p. 1

16 Atmospheric spectra: the Troposphere Composite of wavenumber spectra of zonal and meridional velocity (shifted one decade on the right, after Nastrom et al. (1984)). stratified - rotating turbulence p. 1

17 Atmospheric spectra: near the Tropopause Ensemble-averaged vertical wind speed spectra meassured in a frame of reference relative to air (left) and ratio between vertical component and total wind spectra, showing the departure from isotropy at different wavelenghts, after Duck and Whiteway (2005). stratified - rotating turbulence p. 1

18 An atmospheric summary Different mechanisms: the balance between momentum and heat flux, and the wall effect, in the surface layer shear, convection and gravity waves in the well developed PBL deep convection, waves and 2D large scale motions in the troposphere and all lead to spectra characterized by an inertial-like range. stratified - rotating turbulence p. 1

19 Horizontal dispersion observations Horizontal dispersion data (for tropospheric cases only, on the right) from Gifford (1982). Note that the standard deviation increases at an intermediate rate between t 1 and t 3/2, a signature of enhanced (Richardson) dispersion. stratified - rotating turbulence p. 1

20 Absolute dispersion and meandering a set of N particles, positions x (n) i (t), i = 1, 2, 3, n = 1,..., N. (1) x i = 1 N N n=1 x (n) i Relative positions y (n) i (2) = x (n) i x i. y i = 0. y i y j = x i x j x i x j M sets of N particles each (3) x i x j = y i y j + 1 M M x i (m) x j (m) = y i y j + x i x j m=1 stratified - rotating turbulence p. 1

21 Relative dispersion and meandering Let us consider the separation between two particles, for each m set: x i = x (j) i x (l) i. (4) x 2 i 1 N 2 N j=1 N l=1 [x (j) i x (l) i ] 2 = 2 x 2 i 2 x i 2 (5) x 2 i 2 y 2 i namely, the variance of the distance between pair of particles is two times the dispersion about the mean position. stratified - rotating turbulence p. 1

22 Surface layer: meandering from field data A field experiment carried out in Denmark: plume lateral meandering (Mikkelsen et al., 1987): the figure shows the instantaneous observed standard deviation about the center of mass, averaged over the ensemble of realizations (open circles) with error bars; the dispersion associated with the movement of the centreline (triangles); the total lateral dispersion (full circles). stratified - rotating turbulence p. 2

23 CBL meandering: laboratory data Single realizations and ensembe average for a puff released in the model CBL, after Snyder et al. (2002). stratified - rotating turbulence p. 2

24 CBL meandering: laboratory data Centerplane concentration averaged over the 33 realizations, at different times, after Snyder et al. (2002). stratified - rotating turbulence p. 2

25 CBL meandering: laboratory data RMS meandering, relative and total dispersion in lateral direction for plumes with different initial buoyancy in the model CBL, after Weil et al. (2002) stratified - rotating turbulence p. 2

26 CBL meandering: laboratory data RMS meandering, relative and total dispersion in vertical direction for plumes with different initial buoyancy in the model CBL, after Weil et al. (2002) stratified - rotating turbulence p. 2

27 Turbulent disperson basics results Taylor (1921) Richardson (1926) asymptotic limit and diffusion concept diffusion equation diffusion coefficient in atmospheric flows stratified - rotating turbulence p. 2

28 Dispersion by continuous movements After Taylor (1921) (6) x i (t) = t 0 v i (t 1 ) dt 1 + x i (0) (7) x i x j = t 0 dt 1 t 0 dt 2 v i (t 1 )v j (t 2 ) = 2 t 0 R ij (s)(t s) ds (8) T ij = 1 v i v j 0 R ij (s) ds stratified - rotating turbulence p. 2

29 Dispersion by continuous movements For t << T ij the ballistic regime occurs: (9) x i x j = v i (0)v j (0)t 2 For t >> T ij diffusion occurs: (10) x i x j 2T ij u i u j t and the eddy diffusion coefficient (computed as a limit) is (11) D ij = lim t x i x j 2t = u i u j T ij stratified - rotating turbulence p. 2

30 Relative dispersion In the inertial subrange where energy scales as k 5/3, the mean square separation scales as x 2 i εt 3 This is equivalent to have a diffusion coefficient that scales as D ul l 1/3 l = l 4/3 stratified - rotating turbulence p. 2

31 Some evaluations of the eddy dispersion coefficient Assuming the existence of an inertial subrange characterised by the spectrum (12) E(k) = C K ε 2/3 k 5/3 ε may be derived ( C 0 = 2, C K =.5) fitting the data, leading to an estimate of the eddy dispersion coefficient as a function of the scale (wavenumber) (13) D(k) = 2σ4 (k) C 0 ε stratified - rotating turbulence p. 2

32 Some evaluations of the eddy dispersion coefficient note that σ 2 (k) k 2/3 and D(k) k 4/3 Eddy dispersion coefficient as a function of wavenumber, from different spectra. stratified - rotating turbulence p. 3

33 A short cruise in the ocean Horizontal dispersion coefficient as a function of scale, from diffusion experiments in the ocean, after Monin and Ozmidov (1985). stratified - rotating turbulence p. 3

34 Dissipation estmates Estimates of the dissipation ε are: high frequency, near surface data: ε = m 3 s 2 low frequency, near surface data: ε = 10 4 m 3 s 2 troposphere: ε = 10 4 m 3 s 2 including statosphere: ε = 10 5 m 3 s 2 stratified - rotating turbulence p. 3

35 Other estimates of D surface layer data: Maryon (1998): m 2 s 1 large scale horizontal dispersion: Gifford (1982): m 2 s 1 for tropospheric data only, 10 5 m 2 s 1 including stratospheric data long range tracer dispersion in the PBL (Sorensen et al., 1998): m 2 s 1 stratospheric balloons (Lacorata et al., 2004): 10 7 m 2 s 1 horizontal growth of Etna plume (Tiesi et al., 2006): 10 3 m 2 s 1 PUFF model (Searcy et al., 1998): horizontal m 2 s 1, vertical 10 2 m 2 s 1 ; but using high resolution simulations D h = 10 2 (Tanaka and Yamamoto, 2002). A fundamental warning: is it correct to use this coefficient? Is it justified in terms of scale separation? stratified - rotating turbulence p. 3

36 Diffusion equation and diffusion coefficient (14) t c + u c = κ 2 c taking the Reynolds (ensemble) average (C = c : c = C + c, u i = U i + u i ): (15) t C + U C = u c and, assuming the validity of Fick s law, (16) u ic = D ij xj C D ij = u i u j T L is the diffusion coefficient. stratified - rotating turbulence p. 3

37 A sign of warning Quoting Stanley Corrsin (1974): [... ] a gradient transport model requires (among other things) that the characteristic scale of the transporting mechanism [... ] must be small compared with the distance over which the mean gradient of the transported property changes appreciably [... ] nearly all traditional turbulent transport problems violate this requirement [... ] There has been an astonishing lack of amazement at the partial success of these models. stratified - rotating turbulence p. 3

38 Fick s law limitations CBL: strong asymmetry, scales on transp. mech. domain size concentration gradient scale near source dispersion: t < T L (no asymptotic behaviour) finite size domain (Boffetta et al, 2000: Finite Size Lyapunov Exponents: statistics are better computed at fixed size rather than at fixed time) stratified - rotating turbulence p. 3

39 Update An updated and refined version can be found at ginevra.bo.isac.cnr.it/turbdiff/?qq=lecture_notes stratified - rotating turbulence p. 3

40 References Alberghi, S., A. Maurizi, and F. Tampieri, 2002: Relationship between the vertical velocity skewness and kurtosis observed during seabreeze convection. J. Appl. Meteorol., 41, Corrsin, S., 1974: Limitations of gradient transport models in random walks and in turbulence. Advan. Geophys., 18A, Duck, T. and J. Whiteway, 2005: The spectrum of waves and turbulence at the tropopause. Geophys. Res. Letters, 32, L L Finnigan, J., F. Einaudi, and D. Fua, 1984: The interaction between an internal gravity wave and turbulence in the stably stratified nocturnal boundary layer. J. Atmos. Sci., 41, Gifford, F., 1982: Horizontal diffusion in the atmosphere: a Lagrangiandynamical theory. Atmos. Environ., 15, gupta, R. McNider, M. Trainer, R. J. Zamora, K. Knupp, and M. Singh, 1997: Nocturnal wind structure and plume growth rates due to inertial oscillations. journal of applied meteorology, 36, Kaimal, J., J. Wyngaard, Y. Izumi, and O. Coté, 1972: Spectral characteristics of surface-layer turbulence. Quart. J. Roy. Meteor. Soc., 98,

41 Lacorata, G., E. Aurell, B. Legras, and A. Vulpiani, 2004: Evidence for a k 5/3 spectrum from the EOLE Lagrangian balloons in the low stratosphere. J. Atmos. Sci., 61, Maryon, R., 1998: Determining cross-wind variance for low frequency wind meander. Atmos. Environ., 32, Mikkelsen, T., S. Larsen, and H. Pecseli, 1987: Diffusion of Gaussian puffs. Quart. J. Roy. Meteor. Soc., 113, Monin, A. and R. Ozmidov, 1985: Turbulence in the ocean, D. Reidel Publ. Co., Dordrecht. Nastrom, G., K. Gage, and W. Jasperson, 1984: Atmospheric kinetic energy spectrum, km. Nature, 310, Searcy, C., K. Dean, and W. Stringer, 1998: Puff: a high resolution volcanic ash tracking model. J. of Volcanology and Geothermal Research, 80, Snyder, W., R. Lawson, jr., M. Shipman, and J. Lu, 2002: Fluid modelling of atmospheric dispersion in the convective boundary layer. Boundary-Layer Meteorol., 102, Sorensen, J., A. Rasmussen, T. Ellermann, and E. Lyck, 1998: Mesoscale influence on long range transport. Evidence from ETEX modelling and observations. Atmos. Environ., 32,

42 Tanaka, H. and K. Yamamoto, 2002: Numerical simulation of volcanic plume dispersal from usu volcano in japan on 31 march 2000 using puff model. Earth Planets and Space, 54, Taylor, G., 1921: Diffusion by continuos movements. Proc. London Math. Soc., 20, Tiesi, A., M. Villani, M. D Isidoro, A. Prata, A. Maurizi, and F. Tampieri, 2006: Estimation of dispersion coefficient in the troposphere from satellite images of volcanic plumes: application to mt. etna, italy. Atmos. Environ., 40, , doi: /j.atmosenv Van der Hoven, J., 1957: Power spectrum of horizontal wind speed in the frequency range from to 900 cycles per hour. Journal of Meteorology, 14, Villani, M., L. Mona, A. Maurizi, G. Pappalardo, A. Tiesi, M. Pandolfi, M. D Isidoro, V. Cuomo, and F. Tampieri, 2006: Transport of volcanic aerosol in the troposphere: the case study of the 2002 etna plume. J. Geophys. Res., 111, doi: /2006jd Weil, J., W. Snyder, R. Lawson, jr., and M. Shipman, 2002: Experiments on buoyant plume dispersion in a laboratory convection tank. Boundary-Layer Meteorol., 102,