Modelling Turbulence Effect in Formation of Zonal Winds

Transcription

1 The Oen Atmosheric Science Journal Modelling Turbulence Effect in Formation of Zonal Winds Oen Access J. Heinloo and A. Toomuu * Marine Systems Institute Tallinn University of Technology Akadeemia tee 68 Tallinn Estonia Abstract: A turbulence-affected mechanism of formation of onal winds in the Earth s trooshere is discussed from the ersective of the theory of rotationally anisotroic turbulence (the RAT theory). The turbulence effect is exlained as an action of the turbulence rotational viscosity introduced within the RAT theory to characterie the shear in relative rotation (determined as the difference between the avere angular velocity of eddy rotation and the vorticity of the avere velocity field). The effect manifests in the form of an additive correction to the wind velocity redicted by the geostrohic aroach. It is shown that the accounted turbulence effect decreases the westerlies velocity redicted by the geostrohic aroach at lower latitudes and can be used to exlain the formation of easterlies (trade winds) in the equatorial one without any necessity of assigning the geootential a local minimum at the Equator which is required to exlain the trade winds within the urely geostrohic aroach. Keywords: Turbulence general circulation climatology.. INTRODUCTION Predominantly onal winds (ZW) in the Earth s atmoshere (westerlies at middle latitudes and trade winds in the equatorial one) are of secial interest not only because of their global significance to the Earth s climate and navigation but also because they reresent a motion attern common to all lanetary atmosheres. One of the roblems of ZW requiring additional discussion is the turbulence effect in the formation of ZW. Although the contemorary hydromechanics have develoed several methods for such discussion most of these methods are founded on caacious numerical rocedures resentings the situation with great number of details accounted [-4] which may mask the hysical essence of the effects under investigation. In order to resent more distinctly the turbulence effect in ZW formation it is aroriate to comlement numerical methods with a descrition set u immediately in terms of avere quantities. Nevertheless the conventional turbulence mechanics (CTM) suosed to be found this tye of turbulence descritions does not meet exectations. In articular CTM does not account for effects caused by the referred orientation of eddy rotation which remarkably constricts its alicability. The theory in [5-9] (concisely outlined in Section as the rotationally anisotroic turbulence (RAT) theory) removes the constriction. The RAT theory treats the turbulence according to the Richardson-Kolmogorov turbulence concetion [ ] as a stochastic form of fluids motion constituted by a hierarchy of eddies of different scales whereas the eddies of the largest scales immediately interacting with the avere flow have a referred orientation of rotation. By accounting for effects of the referred orientation of eddy rotation the RAT theory augments the turbulence avere descrition and rovides the turbulent media with *Address corresondence to this author at the Marine Systems Institute Tallinn University of Technology Akadeemia tee 68 Tallinn Estonia; Tel: ; Fax: ; heinloo@hys.sea.ee additional hysical roerties. The current aer exloits one of these roerties the turbulence rotational viscosity for outlining the turbulence effect in formation of the ZW velocity field in the Earth s trooshere. The discussion is erformed within a model similar to the model [ ] treating the turbulence effect in the formation of the Antarctic Circumolar Current velocity field. This aer discusses the turbulence effect in the ZW formation in the Earth s atmoshere which comlements former geohysical alications of the RAT theory. In addition to the mentioned Antarctic Circumolar Current roblem the roblems of the ocean uer layer dynamics [ 3] of the formation of the Gibraltar Salinity Anomaly [4] of the formation of flows over varying bottom toograhy [5] and of the momentum transfer and energy conversion rocesses in the Gulf Stream area [6] have been considered.. The RAT Theory The RAT theory [5-9] realies a generalied setu of turbulence avere descrition roceeding from distinguishing of the velocity fluctuations at each flow field oint (in addition to their mnitude and orientation) by the curvature of the velocity fluctuation streamlines assing the oint. The generaliation is formalied by inclusion of the curvature radius of the velocity fluctuation streamlines R into the set of arguments of the robability distribution secifying the alied avering. ( R e / s e / s where in which v is the velocity fluctuation e v / v v v and s is the length of the v streamline curve). The secified avering ermits the definition of quantity M v' R (angular brackets denote statistical avering in the sense secified above) having the hysical sense of internal moment of momentum (angular momentum sin) of the turbulent flow field and laying the role of a dynamical /8 8 Bentham Oen

2 5 The Oen Atmosheric Science Journal 8 Volume Heinloo and Toomuu measure of a referred orientation of eddy rotation. The definition of M is couled with definition v R R ' e v s. Quantity has the sense of the avere angular velocity of eddies rotation. It is natural to interconnect the moment of momentum M with the angular velocity of rotation by relation M J where J defines the effective moment of inertia the square root of which determines the characteristic satial scale of eddies contributing to the referred orientation of eddy rotation. The descrition of motion of turbulent media with M should be set u within the conservation laws for the avere momentum (the Reynolds equation) for the moment of momentum M and for energy K determined as the difference between the total turbulent energy K and energy K v M. The turbulent stress tensor is determined within the suggested setu of turbulence mechanics as asymmetric with the antisymmetric constituent describing the interaction of the avere flow with the turbulence constituent contributing to the condition M. The asymmetry of the turbulent stress tensor declares that the condition M turns (in general) the robability distribution non-invariant in resect to ermutation of its arguments - the velocity fluctuation comonents. The closure roblem immanent to any avere turbulence descrition is solved in the RAT theory in reement with the methodology alied in formulation of descritions of microolar fluids [7-]. The methodology connects the generalied forces of the descrition linearly with the corresonding generalied velocities. In articular the alied method connects the dual vector to the antisymmetric constituent of the turbulent stress tensor (the comonents of are defined as k eijk Rij where e ijk denote the comonents of Levi-Civita tensor R ij denote comonents of the turbulent stress tensor and the Einstein summation is assumed) to the angular velocity of relative rotation where u is the vorticity of the avere flow ( u v where v is actual velocity and the coefficient in the definition of vorticity is included for terminological convenience) by 4 ( ). () Coefficient > (characteriing the shear in relative rotation) introduced in () is the coefficient of rotational viscosity having the dimension of kinematic velocity (m s - ) but differing from the latter by its hysical sense. The quoted in the introduction geohysical models have alied the RAT theory within a simlification follow- ing from the Richardson-Kolmogorov turbulence concetion [ ]. The simlification stands in neglecting the turbulence constituent not contributing to the referred orientation of eddy rotation in the avere flow-turbulence interaction descrition. In terms of the turbulence viscosity it means considering the effects of the turbulence shear viscosity which mediates the avere flow interaction with the turbulence constituent not contributing to the referred orientation of eddy rotation negligibly small if comared with the effects of the turbulence rotational viscosity which realies the avere flow interaction with the turbulence constituent contributing to the referred orientation of eddy rotation. The resent aer alies the same simlification. 3. MODEL OF ZONAL WINDS IN THE EARTH S ATMOSPHERE 3.. Model Setu The model secifies the Earth s atmoshere as a turbulent gas layer covering the Earth (modeled as a solid shere rotating with angular velocity ) with the thickness much smaller than the Earth s radius r. The motion is considered in the right-hand sherical co-ordinate system ( ) where is longitude is latitude ( < < ) and is height (with on the solid shere). It is assumed that ( ( ) ) density and the wind velocity u determined as statistically avere quantities are secified as ( ) and u u i.e. do not deend on the onal direction. The suggested model is aimed to describe the effect of a referred orientation of turbulent eddies rotation on the onal winds formation. As including notions introduced within the RAT theory the model treats the RAT theory (Section ) as an insearable art of the discussion. 3.. Geostrohic Model of ZW Firstly consider stationary balance condition + u + g () / / neglecting the turbulence viscosity effects. In () besides the quantities exlained above ( r ( cos ) / r / / ) is the avere ressure and g ( g) is the gravitational acceleration. The geostrohic model of ZW stems from () for the vertical constituent of the Coriolis force considered as negligibly small with resect to g and. From () in this case it follows that g and u / r f where f sin is the Coriolis arameter and the index following comma denotes differentiation with resect to the corresonding coordinate. Using geootential g connected with the ressure by the exression for the wind velocity u can be written as

3 Modelling Turbulence Effect in Formation of Zonal Winds The Oen Atmosheric Science Journal 8 Volume 5 u (3) r f Differentiating (3) in resect to and accounting for and cos at the Equator we have from (3) sin ( ) u. (4) Exressions (3) and (4) show that the wind velocity u redicted by the geostrohic aroach can ree with the observed easterlies in the equatorial region only if the geootential extremum secified as the local minimum is located at the Equator. In order to estimate the effect of restrictions imosed to the geootential behavior by the geostrohic aroach let us consider the geootential distribution (Fig. ) calculated by formula Geootential ( 4 m s - ) R Td ln (5) from [3] where R 87 J kg K is the gas constant T is the onal avere air temerature (measured in K ) and is the ressure at from the data set from the eriod [4]. The estimated geootential () (Fig. ) and the calculated rofiles of its first longitudinal derivative for 5 < < 5 deicted in Fig. () show that the geootential extremum aears to be maximum and is located to the north of Equator i.e. the geostrohic aroach to the ZW calculation does not comly with the geootential distribution estimated from data hpa 45 hpa 5 hpa 55 hpa 6 hpa 65 hpa 7 hpa 75 hpa 8 hpa 85 hpa Fig. (). Latitudinal distribution of onal mean geootential () for ressure levels 4 to 85 hpa estimated from data comiled by Oort (983). Geootential derivative ( -4 m s - ) 4-85 hpa 4 hpa - Fig. (). Zonal geootential derivative in resect to latitude for ressure levels 4 to 85 hpa with ste 5 hpa. () In order to harmonie the geootential distribution with the observed meridional velocity distribution in the geostrohic aroach it has been rather common so far to accet a relatively small (in the limits of assumed estimation error) modification of geootential values in the near-equatorial region to satisfy the required criteria (see the COSPAR International Reference Atmoshere (CIRA-86) reanalysis [5] for trooshere based on data comiled by Oort [4] and the reanalysis data from rovided by the NOAA- CIRES Climate Dinostics Center Boulder Colorado on their web site at htt:// Turbulence Effect in the ZW Formation It is commonly believed [3] that accounting for the turbulence effect can overcome the shortcomings of the geostrohical model. In this case the balance condition () is relaced by + F + u + g (6) where F is the divergence of the turbulent stress tensor the comonents F i of which are defined as F i R ij j where the index following comma denote differentiation in resect to the corresonding satial coordinate and the Einstein s summation is assumed. The balance condition (6) requiring a secification of F relates the ZW descrition to the secifics of the selected turbulence model. Therefore deendent on the selected turbulence model the balance condition (6) can be used to found different articular ZW models accounting for the turbulence effect. In the model discussed in the following F is secified according to the RAT theory as F. (7)

4 5 The Oen Atmosheric Science Journal 8 Volume Heinloo and Toomuu where is determined in () ( Substituting F from (7) into (6) we have + + u + F i Rij j eijk k j ). g (8) Equation (8) is similar to the resective equation exloited [] in the context of discussion of the turbulence effect in the formation of the Antarctic Circumolar Current velocity field. Unlike the discussion in [] equation (8) is not restricted by the Boussinesq aroximation. Let u in (8) be resented as the sum u + g u u g g where u ( u ) is the geostrohic velocity constituent satisfying (8) for the turbulence effect absent and u ( u ) is the eostrohic turbulence-induced correction to u g. Accounting for () with and assuming that the medium turbulence does not violate the hydrostatic balance condition (3) from the vertical rojection of (8) we have ( ) u cos r cos. (9) Exression (9) secifies the turbulence effect on the onal wind velocity as caused by the turbulence rotational viscosity ( ). (For u from (9) the meridional rojection of (8) determines the turbulence-caused meridional ressure force constituent while the onal rojection of (8) establishes the relation between and.) Let us stress that unlike discussed in 3.. geostrohic aroach neglecting the vertical constituent of the Coriolis force the deduction of exression (9) just makes use of this Coriolis force vertical constituent. To determine equation (8) should be comlemented by the equation for the moment of momentum M. Neglecting similar to the model discussed in [] the effects of diffusion of the moment of momentum M for the body force acting on the medium secified as the gravitational force this equation in a non-rotating frame reads as [5-7] d dt M 4 + m + J( u) () d / dt / t + u is the coefficient measur- where ing the effect of M decay due to the cascade scattering and m g R. () When written in the coordinate system rotating with angular velocity the quantities d M / dt and J( u) in () are relaced by + d M / dt + M and ( u) ( J + ) + J resectively and for the steady and onally homogenous M from (9) it follows + J 4( + ) + m + J( u) ( + ) () from which for the last two terms on the left side of () considered to be small if comared to the other terms of () we shall have 4( + ) + m. (3) (The balance condition (3) differs from the corresonding balance condition alied in [] only by secification of the density which is not restricted in (3) by the Boussinesq aroximation.) The secification of R in the form [9 ] k R k (4) determines m secified by () in (3) as m k [( g ) ˆ ( ) g] + g k (5) which closes (3) to determine. In (5) ˆ is unit tensor. From (5) it follows that coefficients k and k introduced by (4) quantify the effects of stratification and baroclinic instability on the turbulent eddy rotation resectively. As far as the stable stratification suresses and the baroclinic instability feeds the turbulence the coefficients k and k are determined ositive. Using (5) and () from (3) we shall have c r c ; (6) cos c + + c + c ( sin) ; (7) (8) where c kg / 4( + ) c kg / 4( + ). Accounting for (6) exression (9) for u can be rewritten as c cos cos c u. (9) r Assuming the scaling roerty of turbulence the coefficients k and k aear roortional to the effective moment of inertia J [8] and therefore the exressions (6) (9) deend on the satial scale J of the turbulence constituent contributing to the referred orientation of eddy

5 Modelling Turbulence Effect in Formation of Zonal Winds The Oen Atmosheric Science Journal 8 Volume 53 u is ro- rotation. In articular from (9) it follows that ortional to J Model Analysis Consider a situation with horiontally homogeneous density field and with the stratification strong enough to deress and i.e. stating that. Ac- u and. The latter condition means the turbulence with the cording to (9) and (8) we have in this case avere angular velocity of eddy rotation orientated vertically. The situation changes if the density varies in meridional direction with the meridional gradient large enough to generate. Consider still remaining suressed by the stratification. It is easy to imine the generation of u and consequently of in this case as accomanied by an inclination of the referred orientation of turbulent eddies axes from their vertical direction caused by the nonhomogenous meridional density distribution. To analye the structure of u redicted by (9) consider ressure and density exonentially decreasing with height ex( ) and ( )ex( ) where and ( ) denote ressure and density on the reference level ( ) while ( ) is symmetrical in resect to the equator and increases to the oles with its gradient vanishing at the equator and at the oles. In terms of variables and we have for in this case ( ) ( ) () where ) ( ) ( ). Exression () determines the meridional density distributions at different ressure levels as self-similar. It reduces the density distributions on different ressure levels to a single density distribution on the reference level. Using () and (3) for in (9) we shall have ( ) )) g ( ( c where c g / const and exression (9) for shall become u c u f ( ) f ( ) F ( ) F ( ) () r in which F () ( cos () ) > and F () ( ()cos () cos ) are functions determined by the density distribution at the reference level and f ( ). + ( ( )) For the constituted above roerties of ( ) function F ( ) aears ositive with two maximums located at inflection oints of the density meridional distribution and with ero value at the equator. Unlike F ( ) function F ( ) is ero at the inflection oints and has three extremums at the equator and one on each side of the equator between the maximums of F ( ) and the oles. For the density increasing to the oles F ( ) is negative at the equator. Exression () secifies u as a linear combination of F ( ) and F ( ) with their relative weights determined by f ( ). The sketched structure of u dislays the turbulence effect on the ZW as increasing the wind velocity at higher latitudes and decreasing the wind velocity at lower latitudes. The decrease includes the formation of negative u in the equatorial one i.e. the turbulence can exlain the equatorial easterlies. The increase of the wind velocity at higher latitudes and its decrease at lower latitudes is accomanied with a shift of the location of the westerlies maximums to the oles if comared with the rediction of urely geostrohic aroach. Let us note that although deends on (due to the resumed deendence of on ) this deendence does not change the assertions made The Model Alied to the Actual Atmosheric Data Fig. (3) collects the data for ( ) calculated from ( + * ) ( ) () * -3 where.7 kg m hpa and ( ) is the avere density meridional distribution estimated from the temerature and humidity data comiled by Oort [4] for the ressure levels from 4 to 85 hpa with the ste of 5 hpa. The data are aroximated by 4 ( ) a a ex( b ) a ex( b ) where

6 54 The Oen Atmosheric Science Journal 8 Volume Heinloo and Toomuu Normalied density (kg m -3 ) CONCLUSION A turbulence effect in the onal winds (ZW) formation in the Earth s trooshere foreseen by the RAT theory is discussed. The analysis is erformed within a model similar to the model exlaining the turbulence effect in the formation of the Antarctic Circumolar Current velocity field []. It has been shown that the accounted turbulence effect increases the wind velocity at higher latitudes decreases the wind velocity at lower latitudes (if comared with the geostrohic ZW velocity rediction) and can exlain easterlies in the equatorial one. The increase of the wind velocity at higher latitudes and its decrease at lower latitudes is accomanied with a shift of the location of the westerlies maximums to the oles if comared with the geostrohic rediction. The suggested model does not require the geootential local minimum at the Equator to exlain the easterlies in the equatorial one Fig. (3). Normalied onal density () vs. latitude (dots) and its aroximation (curve). a b are constants (curve in Fig. (3)). Fig. (4) a b a g resents calculations for u (dashed curves) and g u u + u (curves) together with the wind data (markers) for 4 6 and 85 hpa. The calculations are erformed for the northern hemishere which is much more densely oulated by the original observation data than the southern hemishere [4 6]. The geostrohic wind is calculated from the geootential shown in Fig. () modified in the interval 5 < < 5 to make the geootential maximum location coinciding with the Equator. The geootential values alteration within the indicated latitude interval did not exceed.3%. The eostrohic wind constituent u is calculated from (9) with () by using the aroximated ( ) for the meridional deendence of on modeled 6 as cos for cg / m kg and c 6ms. Fig. (4) exemlifies the main corollaries r of the turbulence effect on the ZW discussed in subsection 3.4 an increase of the wind velocity at high latitudes and its decrease at low latitudes including the formation of the negative wind velocity in the equatorial one as well as the shift of the location of the westerlies maximums to the oles if comared with the rediction of urely geostrohic aroach. Velocity (m s - ) hpa 85 hpa 4 hpa g Fig. (4). Geostrohic u () (dashed curves) and total u (continuous curves) onal velocities with observed onal velocity data for 4 (circles) 6 (squares) and 85 (triangles) hpa from the data set comiled by Oort (983). ACKNOWLEDGEMENTS Authors are grateful to Prof. Sirje Keevallik and Prof. Ain Kallis for their helful comments. REFERENCES [] Frederiksen JS. Nonlinear studies on the effects of toograhy on baroclinic onal flows. Geohys Astrohys Fluid Dyn 99; 59 (-4): 57-8 doi:.8/ [] Huss A. Numerical integration of a quasi-geostrohic atmosheric model with an asymmetric onal current. Pure Al Geohys 966; 64 (): doi:.7/bf [3] Ellis RS Turkington B. Analysis of statistical equilibrium models of geostrohic turbulence. J Al Stochastic Anal ; 5 (4): [4] Shukla PK Stenflo L. Generation of onal flows by Rossby waves. Phys Lett A 3; 37 (-3): [5] Heinloo J. Phenomenological mechanics of turbulence. Valgus Tallinn 984;. 45 (in Russian). [6] Heinloo J. Turbulence mechanics. Introduction into general theory of turbulence. Estonian Academy of Science Tallinn 999 (in Russian). [7] Heinloo J. Formulation of turbulence mechanics. Phys Rev E 4; doi:.3/ Phys Rev E [8] Heinloo J. Setu of turbulence mechanics accounting for a referred orientation of eddy rotation. Concet Phys 8; 5 (): 5-.

7 Modelling Turbulence Effect in Formation of Zonal Winds The Oen Atmosheric Science Journal 8 Volume 55 [9] Heinloo J. The descrition of externally influenced turbulence accounting for a referred orientation of eddy rotation. Eur Phys J B 8; 6: doi:.4/ejb/e [] Heinloo J Toomuu A. Antarctic Circumolar Currant as a density-driven flow. Proc Est Acad Sci Phys Math 4; 53: [] Heinloo J Toomuu A. Modeling a turbulence effect in formation of the Antarctic Circumolar Current. Ann Geohys 6; 4: [] Heinloo J Võsumaa Ü. Rotationally anisotroic turbulence in the sea. Ann Geohys 99; : [3] Võsumaa Ü Heinloo J. Evolution model of the vertical structure of the active layer of the sea. J Geohys Res 996; : [4] Toomuu A Heinloo J Soomere T. Modelling the Gibraltar Salinity Anomaly. Oceanology 989 English ed Published by AGU in June 99; 9: [5] Heinloo J. Eddy-driven flows over varying bottom toograhy. Proc Est Acad Sci Phys Math 6; 55: [6] Heinloo J Toomuu A. Eddy-to-mean energy transfer in geohysical jet flows. Proc Est Acad Sci Phys Math 7; 56: [7] Dahler JS. Transort henomena in a fluid comosed of diatomic moleculs. J Chem Phys 959; 9: [8] Dahler JS Scriven LF. Angular momentum of continua. Nature 96; 9: [9] Eringen AC. In: Kröner E Ed. Mechanics of micromorhic continua: Mechanics of Generalied Continua Sringer-Verl: Berlin 968; [] Ariman T Turk MA Silvester DO. Microcontinuum fluid mechanics - review. Int J Eng Sci 973; : [] Richardson LF. Weather rediction by numerical rocess. Cambridge Univ Press: Cambridge 9. [] Kolmogorov AN. The local structure of turbulence in incomressible viscous fluids for very large Reynolds numbers. C R Acad Sci URSS 94; 3: [3] Holton JR. Dynamic meteorology. Academic Press 99;. 5. [4] Oort AH. Global Atmosheric Circulation Statistics Rockville MD NOAA Professional Paer 4 983;. 8. [5] Fleming EL Chandra S Barnett JJ Corney M. Zonal mean temerature ressure onal wind and geootential height as function of latitude Adv Sace Res 99; : -59. [6] Oort AH. Adequacy of the rawinsonde network for global circulation studies tested through numerical model outut. Mon Weather Rev 978; 6: Received: August 8 8 Revised: Setember 8 Acceted: October 8 Heinloo and Toomuu; Licensee Bentham Oen. This is an oen access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (htt://creativecommons.org/licenses/bync/3./) which ermits unrestricted non-commercial use distribution and reroduction in any medium rovided the work is roerly cited.