Abstract. 1 Introduction

Transcription

1 Diffusion of pollutants by helical vortices with subgrid turbulence R. Cieszelski Department 'Modelling of Atmospheric Processes', Institute for Tropospheric Research I.f.T., Permoserstrasse 15, Leipzig, Germany Abstract It is shown that the life cycle structure of helical vortices in grid-point modelling is improved if two-scale energy and helicity conservation is taken into account. For the large-scale regime the 'Ansatz' of Lautenschlager et.al [6] is adopted and for the subgrid turbulence regime prognostic equations of turbulent kinetic energy TKE and of turbulent helicity TH (TKEH model) are solved. Cieszelski [1] showed that grid point modelling with the closed equation set preserves the stabilizing effects of helicity of spectral models. The resulting effects on multi-scale diffusion of pollutants is discussed. 1 Introduction A problem in grid point modelling arises with the transport of energy as well as passive and reactive pollutants by helical vortices. The intensity of helical vortices is measured by kinetic energy E=0.5 UiUi and the strength of knottness by helicity #=%, if Ui is one of the three flow vector components and Ci is one of the three vorticity vector components. From observations (see e.g^ Etling [4]) it has been speculated, that the spectral helicity density H(k-, f) concentrated at large scales leads to a reduced spectral energy flux to smaller scales or at least slowes down the development of the energy cascade. This effect should be preserved in grid-point modelling and is realized by interactive spectral fluxes (a) between large scale kinetic en-

2 176 Computer Simulation ergy E=0.5 f/jc/i and turbulent kinetic energy TKE, defined by e 0.5n^ and (b) between large-scale helicity H=U i and turbulent helicity h=u' [. The spectral separation is linked to the resolution A of the grid-point model and so defines subgrid turbulence. Lautenschlager et.al [6] developed a new 'Ansatz' of subgrid scale coupling to helical vortices within complex flows as for example the PBL. In case of Reynold's averaged incompressible flow and assuming vorticity as the turbulence generating quantity, series expension of the Reynold's- tensor divergence, dimensional and symmetry arguments and derivation within the frame of turbulence theory lead to: dv _ = (i) For simplicity of presentation, the overbars have been omitted. 0 indicates the tensor product, V the flow vector, Q<> the background air density, p* the pressure anormaly, oinega^, the earth rotation vector, and <jy the arbitrary sources. The tendency of the flow on the left hand side of eqn (1) is balanced on the right hand side by large scale advection (1), pressure forces (2), Coriolis- 'force', sources/sinkes (4). Turbulence effects on large-scale flow is discribed by grad-e-term (5) and by the effect of subgrid turbulence due to rotational effects, expressed by the new rotational parametrization (6)- (8). The parametrization coefficients a, j3 and 7 in terms (6)- (8) in eqn (1) depend on length scales f%, on turbulent kinetic energy TKE=e and on turbulent helicity TH=A. In the limiting case of zero turbulent helicity TH, h > 0, turbulence terms (6)- (8) in eqn (1) reduce to the diffusion term (7), as a > 0 and 7» 0 and /3V x = fi&v. Lautenschlager et.al [6] additionally showed that non-zero turbulent helicity h / 0 can stabilize and force large-scale helical vortices in grid point modelling, if the large-scale to turbulence-scale coupling is parameterized as stated by equations (1) and (2) and e and h are assumed time invariant in afirststep. 2 Modelling of helical vortices with subgrid turbulence Cieszelski [1] extended the new 'Ansatz' to a closed set of prognostic equations together with a turbulence model for turbulent kinetic energy TKE and

3 Computer Simulation 177 in addition for turbulent helicity TH (TKEH-model). This model configuration allows energy and helicity fluxes between large-scale and subgrid-scale regimes to develop interactively, limited by conservation of total kinetic energy Etot = E + e and total helicity H^t = H + h. Todays atmospheric turbulence modelling deals with (a) SOC= Second order closure, SOC in V and SOC in <f results in interactive energy, helicity and additional rotational fluxes. Six equations for the turbulence regime are solved, which is numerically very expensive. Numerous parametrization constants must be specified for atmospheric problems. Not used in global modelling, but state-of-the art in mesoscale modelling, however rarely applied in an unsimplified form. (al) TKEH closure (simplified SOC, SOC in F, FOC in (), allows four interactive energy and helicity fluxes resulting from the grad-e, a-, /3- and 7- terms of eqn (1), which are parameterized by turbulent kinetic energy TKE=e and in addition by turbulent helicity TH=/t. This requires 2 equations for the turbulence regime, which is 30 per cent of the numerical expense of SOC. (a2) TKE closure (simplified SOC, SOC in 7, ZOC in () results in diffusiontype energy fluxes and NO helicity diffusion as in the case of FOC; however, the diffusion coefficient /3(eJ,...) is determined by turbulent kinetic energy TKE=e, so one equation is solved and minimal dynamics of the turbulence regime is taken into account. Widely used in atmospheric modelling, because of little numerical expense (15 per cent of SOC). (b) FOC= First order closure results in energy diffusion and NO helicity diffusion. The Diffusion coefficient /?($) is determined solely by large-scale variables $. NO equations for the turbulence regime are solved. The direction of the energy fluxes is similar as in the ca.se of TKE-modelling and of the diffusion-type. Widely used in atmospheric modelling. (c) ZOC= Zero order closure results in unphysically modelling of turbulence flows and will not be discussed. Distinguishing between the order of closure with respect to V and (clarifies that TKE-closure is of ZOC with respect to (. In contrast, TKEH closure is of FOC-type with respect to ( as well. The new TKEH-turbulence modelling concept was implemented in a columnar-type grid-point model based on the code described in Lautenschlager et.al [6]. It is non-hydrostatic, threedimensional, written in cylinder^coordinates (r, 0, z) and includes prognostic equations for the wind vector f, for virtual-potential temperature 0%, for arbitrary pollutants ra,, for turbulent kinetic energy TKE, e, and for turbulent helicity h. For pressure expressed by the Exner function TT, an iterative

4 178 Computer Simulation Poisson-solver based on the conjugated gradient algorithm was used (Kapitza et.al [5]). Large-scale regime dt (1) (2) (5) ATT* = 90y dt -v-(f(g)y) + (/ 8%,c, (i) -^ (2) = - V (78%) 4- (i) (2) - Ve + ac + 4-?A(,(4) (5) i = _v.(ymj +, dt (1) (2) (6) Subgrid turbulence regime (i) (2) (3 (4) (7) V-(e-C) + V ~v^- (2) (4) 2V- (8) Term (1) is the advection term thoroughly the equation set -(8). In the equation for the flow vector the remaining terms are pressure force (2),

5 Computer Simulation 179 buoyancy with g the gravity constants, k the vertical oriented unit vector, gradient-e-term (4) and the rotational parametrization (5). In eqn (5) for virtual potential temperature 8%, term (2) is turbulent diffusion and term denotes divergence of radiative flux S. In the balance equation for the chemical components m* (6), o\j denotes the source/sink due to chemical reaction with respect to the chemical component raj. In the equation for TKE e (7), the terms are shearproduction (2), diffusion (4), dissipation (5) and sources/sinks (6). In the equation for TH h (8) the terms are shearproduction due to wind shear and vorticity shear (2), divergence of vorticity-tke correlations, diffusion (4) (5), dissipation (6), and sources/sinks (7). li are additional length scales arising with SOC and C2=l//c2- The Reynolds-tensor 44 is defined by R= u(u'j and the corresponding turbulent vorticity flow tensor is defined by lt= 0.5[n^ - ^Cl]. 3 Results of numerical experiments Two cases of numerical experiments were performed with subgrid turbulence sources (term (5) in eqn (7) and term (7) in eqn (8)) assumed zero, to show the weakly non-linear effect of the new concept. 3.1 Stabilization of a barotropic model hurricane A simple columnar-type vortex with exclusively radial dependent initial structure is assumed as afirstcandidate. The radius of the column is 100 km and the model height is 10 km. The wind speed maximum is 12 m/s near the center, vertical velocity is 5 m/s at the model hurricanes center. The corresponding energy and helicity distributions were assumed to be similar in the subgrid turbulence regime, so E e and H = h initially. During thefirst24 hours, the vortex is forced by fixed inflow conditions at the bottom. At t 24 hours, the boundary conditions are changed to zero gradient, so the vortex can develop according to its internal dynamics. locally T-turbulence modelling is applied, as the mesoscale structure of the vortex (A/^ % 10 km) suggests that subgrid processes of helical convection e.g. cloud clusters and individual storms are of the local-type, resulting in no transport of the subgridfieldsof TKE e and TH h. 'l-tkeh'-modelling implies organized turbulence with turbulent helicity h non-zero. In this model assumptions, the large scale helical vortex structure keeps alive after t 24. Internal two-scale dynamics are adopted resulting in large scale vortex structures 'frozen-in'. Finally the integrals of large-scale kinetic energy and large-scale helicity remain nearly time in-

6 180 Computer Simulation variant. The solution is an approximation to the large scale mature state observed. Control run: T-TKE'-modelliiig implies random turbulence with turbulent helicity h > 0. In this case, the large scale vortex structure weakens after t=24 hours and the vortex diameter increases in time, so that the vorticity and helicity of the vortex decrease. This is the classical diffusiontype solution which is often found inadequate for modelling quasi-stationary 'mature' states of helical atmospheric vortices. 3.2 Stabilization of a model thermal with rotation A more complex ring vortex with a two-dimensional initial structure is assumed as an second example. The radius of the model column is 1 km and model the height is 4 km. The rotational wind speed is initially 0.6 m/s to keep vortex breakdown of the dust-devil type excluded from the numerical experiments. The thermal is initialized by initial heat supply up to 500 in above the ground and additionally during thefirst30 seconds of simulation at the ground. Subgrid fields of turbulent TKE and TH were set at nearly zero to initiate the turbulence model equations. Due to the smaller scale of this ring vortex, the transport of subgrid turbulence energy e and helicity h must be included. An analysis of observations by LeMone [7] and Cieszelski [2] shows that turbulence is transported by thennals. Comparison of 'TKE' and 'TKEH' modelling. In the case of TKEH modelling, the model thermal keeps its rotational flow structure much better, resulting in wind speeds around 50 per cent more intense as in the case of TKE-modelling, taken as the control run for diffusion-type modelling. The updraft speed is also enhanced by non-linear energy exchange (rotational to vertical flow), so that the rotational flow is transported somewhat higher. 3.3 Complex mesoscale PBL modelling In the case of grid-point-modelling of two atmospheric helical vortices, the life cycle behaviour could be signicantly improved if turbulence modelling accounted for organization in the subgrid scale regime. In the next step, the new turbulence modelling concept will be implemented in a complex mesoscale PBL model including more physical processes due to e.g. orography, roughness and albedo inhomogeneities at the ground, radiative forcing and chemical reactions, cloud- and aerosol physics. As to the temporalspatial relevance of helical vortices related to PBL modelling, instabilities of Ekman boundary layer flows (Etling [4]), convection effected by shear (Wu et.al [8]), thunderstorms (Droegemeier et.al [3]) and fronts are important

7 candidates. Computer Simulation Effects on diffusion of pollutants The life cycle of a chemical component m,=a is affected by different types of helical vortices. On the large scale, the advection term (1) in eqn (6) acts as a non-linear coupling term to the large-scale flow. On the subgrid turbulence scale, the diffusion coefficient 0 depends on turbulent kinetic energy, see eqn (2), which in turn depends indirectly on the large-scale flow. Figure 1: Antidiffusive effect of helical vortices on pollutants As large scale helical vortices are sensitive to turbulent helicity, and turbulent kinetic energy and helicity are affected by large scale induced shearproduction, terms (2) in eqns (7)-(8), there is a strong linkage between large-scale and turbulent scale regime determining the multi-scale diffusion and/or antidiffusion of pollutants. Figure 1 shows a sketch of diffusion scenario with two stacks s-1 and s-2. During pollutant release (a), small-scaled thermal as well as dynamic instabilities determine the helical structure of the instantaneous cloud axis. Chemical reactions due to high temperatures are most vigorous. During pollutant transport near the source (b), larger-scaled

8 182 Computer Simulation processes enter the dynamics and chemical reactions slow down. During this transport reaction with the remaining components of the pollutant spectra rrii^a of different sources has also to be taken into account. As long as the transport is traced, more sources and their long-range transport (c) becomes important, which in case of small initial errors may lead to large errors in long-term modelling. In addition, reactivity with the background chemistry has to be considered. Finally, during deposition (d), again very small scale processes must be taken into account, as the surface structure of vegetation and soil is very complex. The problem is additionally complicated if cloudaerosol- physics with radiative feed back are important. Most important influence of helical vortices in the PBL on pollutant diffusion are expected during transport near the source (b), as two different spectra of chemical components may be merged by helical flow structures, resulting in different spectra for longe-range transport of chemical components. References [1] Cieszelski,R.: Turbulenzmodellierung fuer Stroemungen mit Helizitaet, GKSS external report bf GKSS 92/E/12, Thesis University Hamburg [2] Cieszelski,R.: Zwei Fallstudien geordneter Konvektion anhand von Flugzeugmessungen waehrend des Experiments KonTur, Diploma University Hamburg, [3] Droegemeier,K.K.,Lazarus,S.M,Davies-Jones,R.: The influence of helicity on numerical simulated convective storms, Monthly Weather Review 1993, V 121, [4] Etling,D.: Some aspects of helidty in atmospheric flows, Contribution to atmospheric physics, 1985, 58,1, S [5] Kapitza,H., Eppel,D: A 3-d Poisson solver based on the Conjugated Gradient algorithm, 1985, GKSS external report 85/E/23. [6] Lautenschlager,M., Eppel,D.R, Thacker,W.C, 1988: Subgrid Parametrization in Helical Flows, Contribution to atmospheric physics 61,2, [7] LeMone, M.A.: Modulation of Turbulence Energy by Longitudinal Rolls in an Unstable Planetary Boundary Layer, Journal of Atmospheric Sciences, 1976, Vol.37, 1313ff. [8] Wu,W.S.,Lilly,D.K.,Kerr,R.M.: Helidty and Thermal convection with shear, Journal of Atmospheric Sciences, 1992, Vol.49,