An algebraic expression for the eddy diffusivity in the residual layer

Transcription

1 An algebraic expression for the eddy diffusivity in the residual layer G. A. Degrazia^, J. P. LukaszcykP, D.Anfossi^ ^Departmento defisica, Universidade Federal de Santa Maria, Santa Maria R.S., Brazil, CEP: ^Departamento de matemdtica, Universidade Federal de Santa Maria, Santa Maria R.S., Brasil CEP: oslo. ccne. ufsm. br WC.N.R., Istituto di cosmogeofisica, C.so Fiume 4, Torino, Italy anfossi@to.infn.it Abstract In this paper the problem of the theoretical derivation of a parameterization for the eddy diflusivity in the Residual Layer (RL), the nearly adiabatic remainent of the day time boundary layer, where the dispersion of contaminants occurs in condition of decaying turbulence is addressed The model makes use of the three-dimensional dynamical equation for the classical statistical diffusion theory. The starting point is the Heisenberg's elementary decaying turbulence theory. The main assumption is related to the identification of a frequency, lying in the inertial energy transfer among eddies of different size. The novelty in this study is the derivation of an algebraic formulation for the eddy diflusivity in the RL. 1 Introduction Degrazia et al. [1], aiming at suggesting a physical parameterization of RL eddy diflusivity K^ (a = x,y,z) that could be used in dispersion models, proposed a complex formulation having the form

2 24 Air Pollution Kg m<,jl/6 = 0.15^ (1) where w* is the convective velocity scale, h is the height of the CBL, q» = 1 - expf- 4^ exp^S^ is a stability parameter and /* = "%, with /? the frequency and U the mean wind speed The AT, temporal evolution, as given by eqn (1), is in a good qualitative agreement with the corresponding evolution presented by Freedman & Bornstein[2]. In particular, the eqn (1) provides an eddy diffusivity higher than that typical of the SBL by about an order of magnitude. It is the aim of the present paper to propose an algebraic formulation for the eddy diffusivity in the RL. The great advantage of using the proposed algebraic expression unless the computational point of view is the facility of use and it will be usefull in the solution of large and complex atmospheric diffusion models. 2 Energy transfer equation for homogeneous isotropic turbulence Homogeneous isotropic turbulence, when buoyancy and shear production terms are not important, satisfies the following energy tranfer relation (Hinze[6] ; StanisicflO] ) "#,') - 2 x ^E(6,f ) (2) 0 t where k is the wavenumber, E(k,t) is the three-dimensional energy density spectrum (EDS); w(k,t) is referred to as the energy transfer spectrum function and represents the contribution due to the inertial transfer of energy among different wavenumbers or the time-rate-of-change per unit wavenumber of the energy spectrum, due to nonlinear interactions (Chandrasekhar[ll]; Batchelor[12]; Frisch[7] ), the second term on the right hand side of eqn (2) is the energy loss due to ordinary viscous dissipation. Following several physical considerations one can obtain the following equation (Degrazia et al.[l] ),f) 8^r n 2r,/,x,nx = ^r + r" *%"/) (3) where Vj represents the kinematic turbulence viscosity (KTV), v is the molecular viscosity and S(n,t) is the energy density spectrum in the frequency space.

3 Air Pollution 25 3 Decaying eddy turbulence in the PEL The starting point of our analysis consists in considering the structure of the tree dimensional turbulent spectrum S(n) in geophysical flows, such as the PEL, where Reynold's number are very large ( limit of infinite Reynold's number, Rose & Sulem[3]; Mann[4]; Wilson[5] ). In such situations, the turbulent energy spectra can be subdivided in three major spectral regions: energycontaining, inertial and dissipation subranges (Hinze[6] ). In the energycontaining subrange, where eddies make the main contribution to the turbulent kinetic energy (TKE), EDS shows its maximum, so it is possible to choose its corresponding frequency n, to characterise the size of these eddies. In the dissipation subrange, where the viscosity plays itsmajor role, it is also possible to associate a frequency»,, with the size of the eddies that provide the main contribution to the dissipation. Following a modern view point with emphasis on postulated symmetries rather than on postulated universality (i.e. independence on the particular mechanism by the turbulence is generated; Frisch [7] ) if the Reynold's number is infinitely large all the possible symmetries of the turbulent flow are restored and self-similarity occurs at small scales (where small scale is here understood as scales small compared to the energy-containing eddies scale) and away from boundaries. Therefore, for fully developed turbulence (y -> 0) the finite positive dissipation in the region of frequencies very far below the region of maximum dissipation will be negligibly small compared with the flux of energy transfered by inertial effects. In such an inertial subrange, the effect of molecular viscosity would then vanish. As a consequence we may associate a frequency rij with the size of the eddies that provide the main contribution to this inertial energy flux, that is: rie «nj «nj (4) with this assumption, the turbulence, in this subrange, is statistically independent of the range of energy-containing eddies and a relationship for KTV can be obtained. This means that, in the inertial range, the energy flux at scales % / involves predominantly scales of comparable size. The traditional argument in favour of localness can be found in the literature (Hinze[6] at page 232 and Frisch[7] at page 105). VT can be calculated directly from Taylor's statistical diffusion theory for large travel times r (Hanna[8] ; Weil [9] )as: where a * is the turbulent velocity variance calculated in the inertial subrange and /? is defined as the ratio of the Lagrangian to the Eulerian time scales. Eqn (3) can thus be solved (setting S^(n) = S(n,Q) ), obtaining:

4 26 Air Pollution Having obtained an expression for the time evolution of EDS in decaying turbulence, an equation for the eddy diffusivity in decaying turbulence may be easily derived by using the same theoretical framework used to derive Vj-(n). Let us assume that in the initial stage of the decay process, this last is predominantly determined by the decay of the energy-containing eddies (Hinze[6] ; Nieuwstadt & Brost[13] ; Sorbjan[14] ), so that these large eddies decay according to their intrinsic time scales. Therefore, as the energycontaining spectral range is characterized by n<,, an expression like eqn (5) can be written for the eddy difiusivity K, namely: K.IZL (7, where cr * = As in the case of eqn (5), for large travel time, K can be expressed as a function of local turbulence properties and in terms of the characteristic discrete spectral frequencies n^ (energy-containing eddies). Eqn (7) represents the general semiempirical method to derive eddy difiusivities in a decaying turbulence that we propose. Looking at this equation, suggests that the decrease of eddy difiusivity K with time depends on n^. As n<, refers to the large eddies, this means that a certain level of turbulence can be sustained for a long time. On the other hand, from eqn (7) it can be seen that the decrease of K with time depends also on vy. This, in turn, depends on n^, that describes the dimension of eddies providing the major contribution to the inertial energy flux. Consequently, this additional viscosity may be estimated from observed spectra. 4 Derivation of an one-dimensional KTV and K^ in the RL As the three dimensional EDS for an isotropic and homogeneous turbulent flow can be related to the one dimensional EDS, we can calculate KTV and K^ in the residual layer. Our starting point is that in the first stage the turbulence structure of the RL is the same as that of the previously existing CBL. This means that the turbulence energy spectra are assumed to have the same form as they had in the recently decayed mixed layer. The vertical turbulence energy spectrum in the inertial subrange, derived by Kaimal et al.[15] on the basis of direct observation, can be written as:

5 Air Pollution 27 sh / Where ^ = v n / wf/,-% (8) / 3 is the non dimensional molecular dissipation rate / * function and K = 0.4 is the Von Karman constant. In convective conditions % «0.65 (Caughey & Palmer[16] ;Kaimal & Finnigan[19]) By setting /?*, = 0.55%. (Degrazia & Anfossi[18], computing & from the integration from rij to infinity of eqn (8), the following expression for is obtained (from eqn (5)): KTV Uj. = wJ I 1 (9) Concerning eqn (6), let us remember that, according to the convective similarity theory (Kaimal & Finnigan[19] ; Degrazia et al.[17] ), the Eulerian vertical velocity spectra under unstable conditions can be expressed by the following relationships: 5/3 (10) (ft Since the spectral peak for the vertical component can be approximated (Kaimal et al.[15] ; Caughey & Palmer[16] by (/l^) = a*, /?#%/, the following relationships hold: / *\ z _ z _ U w. -7iT"-z^z ^ ^ -z^z and, consequently, eqn (10) becomes: ' /3w! U We remind that, according to Caughey & Palmerfl6] and Kaimal & Finnigan[19], for the vertical turbulent wind component #, = 1.8 and ^ = l-expf-4^j expf8^j, q^ is thus related to the vertical profile of more energetic eddies and accounts for the level of turbulence at each height.

6 28 Air Pollution To derive explicit relationships to be used in practical applications, it is necessary to have a good experimental estimation of rij to be inserted in eqn (9). Following Kaimal et al.[15], who found that the onset of the inertial subrange in CBL occurs at wavelenght /&*, = 0.1/z, we assui in this work "* ~ /h' ^ ^ ^ interest to point out that, using /?<, values estimated by Kaimal et al.[15] in a well mixed layer, Nieuwstadt & Brost[13] and Sorbjan[14] in large eddies simulations, the ratio *y ranges from 15 to 20. / "<? Consequently the condition n^ «rij is justified and the inertial subrange can be considered statistically independent of the subrange energy-containing eddies. With this assumption eqn (9) becomes: VT =1.98xl(TW (12) Considering typical CBL values (w* = 2.0m/s and h = 1500m), it results Vj «6.0m*/s for the w component. It is worth noting that this value verifies the above condition v^» v. By integrating eqn (6) (valid here for the one dimensional turbulent energy spectrum) from n^ to infinity yields J(l.Sg 5 An Algebraic formula for the the turbulent velocity variance and eddy diffusivity in the RL In Nieuwstadt & Brost[13] we have the following expression for the decaying rate of the normalizated turbulent velocity variance: Following this ideas we have tried an expression of the the following form - = ^A/ (15) w. V h for adequate integer m and this have resulted in very good agreement with the numerical value calculated directly by eqn (13). First, at */ = 0, we can make direct intengration of the expression on eqn (13) and obtain (14) - =0.4787?;% (16)

7 Air Pollution 29 In this way, making */ - 0 in eqn (15) we can put C, = /" ^"*/o Now we make a polinomial interpolation of seventh degree at!/ = 10 using numerical calculations obtained from eqn (13) in?/ ; 03 ; ; 0.9 and we obtain: ^s\ = ) }-) */,o h \hj \hj \h s \5 x \6 / \ l/,v lav l/,j Using this aproximation we can calculate Q *"* = 10, putting = and we obtain w* ' -" ' c = crj [ ^ coeficient in eqn (15) making After some numerical experiments, we find m - 33 to be the most appropriate. Finally, using eqn (7) we have an algebraic expression for the eddy diffusivity in the RL by putting /? U/ and /?<, = we find that is _ 035 U cr, w (17). h = 0.17 <? 3.3 (18) 6 Discussion and summary The eddy difiusivity graphs in the RL for the numerical expression (eqn (17)), with ^// given by eqn (13), is shown in Figure 1, and with algebraic K / / expression (eqn (18)), is shown in Figure 2 in a y,, x *A graph for

8 30 Air Pollution 0.8 \ z/ 0.6 / h 0.4 / Figure 1: Temporal evolution of the vertical profiles of normalised RL eddy K / / diffiisivity y, asa function of dimensionless height y^ (eqn (1)) F. ore 2: Temporal evolution of the vertical profiles of normalised RL eddy diffiisivity ^ as a function of dimensionless height ^ (eqn (18)).

9 Air Pollution 31 The comparison of these figures shows a high degree of agreement between the two expressions. The great advantage of using the proposed algebraic expression for the eddy difusivity in the RL is the fact that this formula, under the computational point of view, is faster than the numerical integration of eqn (1) and, consequently, it will be useful in the solution of large and complex atmospheric diffusion models. References 1. Degrazia, G. A., Anfossi, D., Moraes, O. L. L., Goulart, A., Trim Castelli, S., Eddy diffusivity parameterization in the decaying convective residual layer, Proc. of 23"* NATO/CCMS International technical Meeting on Air Pollution Modelling and its Applications, Varna Bulgaria, September 28-octuber 2, Freedman F. K., Bomstein, R. D., Study of the spacial and temporal structure of turbulence in the nocturnal residual layer, Proc. of 22"* NATO/CCMS International technical Meeting on Air Pollution Modelling and its Applications, Clermont-Ferrand, France, June 2-6, Rose H. A., Sulem, P. L., Fully developed turbulence and statistical mechanics, Le journal de Physique T39, pp , Mann, J., The spatial structure of neutral atmospheric surface layer turbulence, Journal Fluid Mechanics 273, pp , Keith Wilson, D., A three-dimensional correlation/spectral model for turbulent velocities in a convective boundary layer, Boundary-Layer Meteorology* 85, pp 35-52, Hinze, J. O., Turbulence, McGrall-Hill, New York, Frisch, U., Turbulence, Cambridge University Press, Hanna, S. R., Lagrangian and Eulerian time-sclale in the daytime boundary layer, Journal of Applied Meteorology, 20, pp , Weil, J. G, Dispersion in the convective boundary layer, Lectures on Air Pollution Modelling, Venkatran, A. and Wyngaard, J. C. eds, Amer. Meteor. Soc., pp , Stanisic, M., M., The mathematical theory of turbulence, Springer Verlag, Berlin and New York, Chandrasekhar, F. R. S., On Heisenberg's elementary theory of turbulence, Proc. Roy. Soc. A 200, pp Bachelor, G. K., Diffusion in a field of homogeneous turbulemce, I: Eulerian analysis, Australian Journal ofscience Research 2, pp , Nieuwstadt, F. T. M., Brost, R. A, The decay of convective turbulence, Journal of Atmospheric Sciences, Vol. 43, n- 6, pp Sorbjan, Z., Decay of convective turbulence revisieted, Boundary Layer Meteorology 82, pp , Kaimal, J. C., Wyngaard, J. C., Haugen, D. A, Cote, O. R., Izumi, Y., Caughey, S. J., Readings, C J., Turbulence structure in the convective

10 32 Air Pollution boundaruy layer, Journal of Atmospheric Sciences, Vol. 33, pp , Caughey, S. J., Palmer, S. G., Some aspects of turbulence structure throught the depth of the convective boundary layer, Quarterly Journal of Royal Meteorological Society, 105,pp , Degrazia, G. A., Moraes, O.L. L., Oliveira, A. P., An analytical method to evaluate mixing length scales for the planetary boundary layer, Journal applied Meteorology 35, pp , Degrazia, G. A, Anfossi, D., Estimation of the Kolmogorov constant Q from classical statiscal diffusion theory, Atmospheric Enviroment Vol. 32, n- 20, pp , Kaimal, J. C., Finnigan, J. J., Atmospheric Boundary Layer Flows, Oxford