Journal of Modern Physics, 2013, 4, 505516 http://dx.doi.org/10.4236/jmp.2013.44072 Published Online April 2013 (http://www.scirp.org/journal/jmp) Postclassical Turbulence Mechanics Jaak Heinloo Marine Systems Institute, Tallinn University of Technology, Tallinn, Estonia Email: jaak.heinloo@msi.ttu.ee Received January 13, 2013; revised February 15, 2013; accepted February 26, 2013 Copyright © 2013 Jaak Heinloo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This paper surveys the formalism and applications of the postclassical turbulence mechanics (PCTM) grounded on the characterization of turbulent flow field in infinitesimal surroundings of the flow field points besides the flow velocity at these points also by the curvature of the velocity fluctuation streamlines passing these points. The PCTM applies this step to found the turbulence split into the orientated and the nonorientated constituents. The split specifies the compe tence of the classical turbulence mechanics (CTM) to the description of the nonorientated turbulence constituent and delegates the description of the orientated turbulence constituent (in the spirit of the theory of micropolar fluids) to the equation of momentofmomentum. The concurrent presence of the orientated (relatively large scale) and the nonori entated (relatively small scale) turbulence constituents enables to compile the CTM and the conception of L. F. Richardson and A. N. Kolmogorov about the cascading turbulence (RK conception) within a conjoint formalism. The compilation solves the classical conflict between the CTM and the RK conception, though evinces a conflict of another type characterized as paradigmatic. Keywords: Fluid Mechanics; Turbulence; Micropolar Fluids 1. Introduction According to the classical turbulence mechanics (CTM), the flow state in the infinitesimal surrounding of each flowfield point is uniquely characterized by the flow velocity at this point. The postclassical turbulence me chanics (PCTM) modifies this statement by constituting a flow state in the infinitesimal surrounding of each flow point characterized in addition to the flow velocity at this point also by the curvature of the velocity fluctuation streamlines passing this point. The complementation is an outcome of the analysis of the relation between the CTM, the conception of L. F. Richardson [1] and A. N. Kolmogorov [2] (RK conception) about the cascading eddy structure of turbulence and the idea about the ap plicability of the theory of micropolar fluid (MF) [37] to the description of turbulent flows [812]. The analysis follows the general principles of the statistical physics connecting the properties of the statistical ensembles to the specific conditions of their formation formulated in average terms [13,14]. The adjustment of these principles within the context of the turbulence problem has been explained in [1521] and summarized in [22] (together with the physicalhistorical background of the turbulence problem) as the physical doctrine of turbulence (PDT). The formalism of the PCTM (section 2) utilizes the suggested complemented characterization of the flow field states in the infinitesimal surroundings of the flow field points as a precondition for definition of a kinema ticaldynamical pair of the Eulerian flowfield character istics reflecting local average effect of a prevailing ori entation of the largescale turbulence constituent. The turbulence property characterized by the defined quanti ties founds the decomposition of turbulence into its ori entated (relatively large scale) and nonorientated (rela tively small scale) constituents, delegates the description of the orientated turbulence constituent (in the spirit of the theory of MF) to the equation of momentofmo mentum, specifies the competence of the CTM to the description of the nonorientated constituent of turbu lence and provides an opportunity to reflect the RK con ception (in twoscale approximation) in formulation of turbulence mechanics (TM). Besides, the turbulence properties reflected by the defined flowfield characteris tics introduce substantial particularization to the descrip tion of energetic and transport processes in turbulent me dia (A more general setup of the PCTM for the multi scale representation of turbulent flowfield can be found in [1921]). Section 3 discusses a complementation of the ex C opyright © 2013 SciRes. JMP
J. HEINLOO 506 plained in the Section 2 allpurpose formalism of the PCTM by the appropriate closure assumptions summing up in the form of the theory of rotationally anisotropic turbulence (the RAT theory) [15,16,23] (henceforth, [23]). The applied closure constitutes the generalized forces driving the motion linearly connected to the re spective generalized velocities, which ascribes the coef ficients introduced by the closure with a plain and unam biguous physical sense. The perspectives of application of the RAT theory are exemplified in the Section 4 on several examples. Section 5 (Conclusion) comments the reception of the PCTM (together with the RAT theory and its applica tions) by the scientific society engaged within the CTM. The reception is characterized as evidencing a substantial paradigmatic conflict between the expressed by the PDT physical look to the turbulence problem and the look to this problem kept by the CTM. The latter reduces the turbulence problem to either a huge number of applied tasks or to the problem of integration of equations of cla ssical fluid mechanics. The aim of the current paper is to motivate physicists to determine their own position in this conflict. 2. The Formalism of the PCTM 2.1. The Grounding Steps of Formulation of the PCTM The formalism of the PCTM begins from claiming the momentary states of the turbulent flowfield fixed in in finitesimal surroundings of each flowfield point besides the flow velocity v at this point by the curvature of the velocity fluctuation streamline passing this point. The claim is accompanied with the inclusion of the curvature characteristics of the velocity fluctuation streamlines to the arguments of the probability distribution of the me dium motion states at the flowfield points. In the fol lowing the PCTM applies this preliminary step as a nec essary precondition to determine the dynamicalkinema tical pair of the Eulerian flowfield characteristics 2 R R ee Mv (1) and vk ee , (2) complementing the average flow field characteristics introduced in the CTM. In (1) and (2) (and hereafter): angular brackets denote statistical averaging, vvu (in which uv) denotes the fluctuating constituent of the flow velocity; v ev (in which v v); k e (in which is the length of the curve of v v streamline passing a flow field point) is the curvature vector of the streamline passing the flow field point; 2 kRk (in which kk) is the curvature radius vector corresponding to k; RR; and the overdot de notes the full time derivative tv s . The defined in (1) and (2) quantities characterize the average state of motion of Lagrangian particles passing the flow field points while M has the sense of the average density (per unit mass) of the moment of fluctuating con stituent of momentum at the flow field points (hence forth—the momentofmomentum) with R standing for the arm of the moment and has the sense of the average angular velocity of rotation of the medium particles at the flow field points in respect to the random curvature cen ters of the velocity fluctuation streamlines passing these points. As the characteristics of a dynamicalkinematical state of the flowfield at the flowfield points the defined M and determine the energy 1 2 KM specified as a part of the total turbulence energy 2 1 2 v 0 rep resented as KK , (3) where 01 2 K M MvRM in which and vk . The energy split in (3) shows the me dium turbulence split into the orientated and the non orientated constituents characterized by the pair M, and by 0 , respectively. Finally, proceeding from the sense of M and , it is natural to connect them by MJ, (4) . defining the tensor of effective moment of inertia Let us emphasize the following: 1) The turbulence properties reflected by M and relate the average turbulent continua to the class of MF with their MF properties reflecting the local effect of the prevailing orientation of eddy rotation; 2) Constituting M and identically vanishing, the CTM either confines its applicability to the situation with the correlations expressed by (1) and (2) absent (which is a physical assertion) or excludes axiomatically the cur vature of the velocity fluctuation streamlines from the set of characteristics of the flowfield states in infinitesimal surroundings of the flowfield points; 3) The split of turbulence into the orientated and the nonorientated constituents reserves to the CTM the com petence of describing the nonorientated constituent of turbulence; 4) The turbulence properties reflected by (1)(3) pro vide a possibility to introduce the RK conception in an explicit form (in the twoscale approximation) to the av erage description of turbulence. 2.2. The Balance Equations The following particularizes the explained in 2.1 setup of Copyright © 2013 SciRes. JMP
J. HEINLOO 507 turbulence description in terms of differential balance equations for the average momentum, for the momentof momentum, for different energy constituents and for the concentration of scalar substance. 2.2.1. The Balance Equations for the Momentum and MomentofMomentum In the universal form the differential balance equations for the momentum and the momentofmomentum write as [24]: ,ij j σ d dt uF , (5) ,j d dkj m t Mm . (6) In (5) and (6): ddtt u ; is the medium density; ij are the components of the stress tensor; F is the density (per unit mass) of the body force (hence forth—the body force); kj are the components of the moment stress tensor describing the diffusive transport of M; m is the dual vector to the antisymmetric constitu ent of the stress tensor coupling the fields of momentum and the momentofmomentum; m is the density (per unit mass) of the body moment (henceforth—the body mo ment) acting on the medium; the index after the subscript comma denotes differentiation along the respective space coordinate while the Einstein summation is assumed and the equivalent notation arbitrary tensor or vector quan tity ≡ {components of this quantity} is applied. The characterization of the turbulent flowfield ex plained in 2.1 specifies u and M in (5) and (6) as the av erage flow velocity and the momentofmomentum de fined in (1), respectively, while Equation (5) is the bal ance equation for the average momentum and Equation (6) is the equation obtained from the averaged difference of the balance equation of instantaneous momentum and Equation (5) vectormultiplied by R from the right. The derivation procedure provides all terms in (5) and (6) with specific expressions via the momentary flowfield characteristics [23]. In particular, it specifies the stress tensor as mt ij ijij m , where ij denote the compo nents of the molecular stress tensor and tvv ij j i 12 denote the components of the turbulent stress tensor, and the body moment m as m mmm, (7) where 1t mvR 2 , muM and 1 mfR f f in which denotes the fluctuating constituent of the body force acting on the medium. Notice, that the asserted by Equations (5) and (6) asymmetry of the turbulent stress tensor has the same origin as the nontriviality of the turbulent flowfield characteristics defined in (1) and (2). 2.2.2. The Energy Balance Equations The full set of the energy balance equations of the PCTM comprise the balance equations for 2 1 2 u u (where uu ) (derived as the scalar product of Equation (5) and u), for ˆ (deduced for 1J= constJ with , where denotes the unit tensor with the components ˆ 1 ij ), for , as the scalar product of Equation (6) and 0 (derived as the difference of equations of balance of turbulence energy K and ) and for internal (thermal) energy U, written as: d d uu uuu Aq t h, (8) ΩΩ ΩΩΩ d d AB q t h, (9) 00 Ω00 d d u Bq t h, (10) d d U U t h 0 ,,, uU hhhh (11) In (8)(11): denote diffusive flux vec tors of Ku, , K0 and U, respectively; A (where 1 2 u is the vorticity) denotes the work realizing the energy exchange between Ku and ; 2 B m denotes the work realizing the energy ex change between , 0 ; (in which , ut iji j u () 1 2 ttt ijij ji and ,, ,1 2ij ji ij uuu) denotes the work realizing the energy Ku scatter into the energy K0; Ωm ,1iji j m denotes the work resulting in the scatter of energy into the energy K0; ,0 um iji j u 0 00 , and reflect the molecular dissipation of the energies Ku, and K0, respectively; 0u u q ; u q , f 0 qm and describe the effect of external fields on the energies Ku, and K0. A substantial implication of the energy balance situa tion represented by (8)(11) is the specification of the pairs of “generalized forces” and the respective “gener alized velocities” as tu ,m ,iji j, ,iji j, , and , m1 . Each of the pairs determines an inde pendent physical process realizing the scatter of the en ergies Ku and/or into the energy K0. Notice that, unlike the positive u and , the work A and the work B may be either positive or negative. In particular, positive A is related to the energy Ω feeding on the energy Ku while the negative A declares the situation of the eddytomean energy conversion accompanied with Copyright © 2013 SciRes. JMP
J. HEINLOO 508 the upgradient momentum transfer. For the stationary and homogeneous situation the latter possibility assumes Ω q positive, i.e. external fields feeding the energy . The other note relates to the energy K0 feeding on the energies Ku and c , from which the RK conception excludes the first energy source and the CTM excludes the second. The concurrent inclusion of both situations in set (8)(11) suggests the compilation of the RK concep tion (in twoscale approximation) as well as the CTM within one unique formalism. 2.2.3. The Balance Equation for the Scalar Substance The turbulence properties reflected by the defined in (1) and (2) quantities introduce a change not only into the description setup of turbulent motions explained in 2.2.1 and 2.2.2 but also into the description of turbulent trans port processes. Indeed, denoting by , Cc cc and the instantaneous, average and fluctuating concentration of an arbitrary scalar substance (concentra tion of ingredients, temperature, etc.), respectively, and using the identity C 2 RvR vR , the turbulent flux vector of the substance, Cc hv, in the balance equation for C, C CQ h Ω0 CC C hhh d dt, (12) where Q denotes the bodysource of C, becomes repre sented as , (13) In (13) 0 Cc hR and C describe the turbulent transport of C by the nonorientated and by the orientated turbulence constituents, respectively. c hR ˆ 3. The RAT Theory 3.1. The Closure Assumptions The RAT theory [23] realizes the allpurpose formalism of the PCTM explained above within a specific solution of the closure problem of the balance Equations (5), (6) and (12), formulated in three steps. The first step (already applied while deriving Equation (9)) constitutes 1J= 0 , (14) where . Notice that J determines the charac teristic length scale of eddies contributing to M and . Unlike , the length scale R is an average quantity. In the second step the “generalized forces”, revealed in the analysis of energy balance in 2.2.2, are set to linearly depend on the respective “generalized velocities”, written as t ij , 2 ij i j pu , (15) 0,1,2ijk kijij m,ji , (16) 4 , and 14 (17) m. (15)(18): p is the pressur (18) Ine; 0 is the coeffi y; ,, 0 cient of turbulence shear viscosit012 , are the diffusion coefficients of M; 0 e coefficient of turbulence rotational viscosity charhe shear stresses in the relative rotatioe. for is th acterizing t n, i. ; 0 interprets as the coefficient of decay of M due to the cascading process. Relations (15)(17) arear respective relations within the MF theory whereas the relation (18) reflects a fundamental difference in the pro perties of turbulent media and micropolar fluids—when the turbulence structure requires incessant restoration then the MF theory considers the media having a fixed structure. The third step constitutes 0 C h and similto the cR, determin ing C h in (13), expressed as 0 C0 kCh and 12 kC kC c R (with the latter derived as the depeence of ndc R on linear in res nd vanishing for 0C) re sulting in CC and C pect to the both arguments a hK. (19) In (19) is the turbulent tra as nsport tensor represented as KK, (20) specifies the sym 2 01 ˆˆ skk11K 2 as k in which metric and KE (wher or)—the antisymmetric e E is the Levi Civita tensconstituent of . Equations (19ain k0 and1 as the positive coefficients characterizing normal (downgradient) t bulent diffusion of C and k2—as the coefficient charac terizing the crossgradient turbulent diffusion of C. For C not contributing to the density field the sign of k2 speci fies as negative. Otherwise the sign of k2 becomes de pending on whether the C ) and (20) expl k ur constituent perpendicular to the gravity acceleration amplifies of depresses . 3.2. The Equations of the RAT Theory The closure relations (14)(19), turn the bal tions (5), (6) and (12) (henceforth ,,,, ance Equa , ,, 0120 k , and C k1 and k2 are constituted to be constants) into the follow ing set of equations to determine u, d2 dp t uuF , (21) 02 1 d4 d 4, f Jt J um (22) Copyright © 2013 SciRes. JMP
J. HEINLOO 509 CC Q K, (23) where d dt is specified in (20). Let us accompany Equations (21)(23 lowing comments. closresulting in (21) and (22) ) with the fol 1) Theure assumptions specify also all terms in the energy balance Equations (8)(10) providing any solution of (21) and (22) with a rigid physical sense expressed in energetic terms. In par ticular, the closure relation (17) specifies the work A in (9) and (10) as 4A which sets the prob lem of eddytomean energy conversion into the dy namical context expressed by (21) and (22) avoiding the application of negative viscosity [25] or the idealization of the 2D turbulence. 2) Delegating the description of the orientated (rela tively large scale) turbulence constituent to the equation of the momentofmomentum, the formalism of the RAT theory proves closer to the theory of MF than to the CTM, which ignores this equation in its setup. 3) The constituted in the CTM statement of symmetry of the turbulent stress tensor reads in terms of Equations (21) and (22) as the condition 40 holding either for 0 or for which enlightens the ambivalence of physical interpretation of the CTM. In the first case, if 0 fmF and if is identically zero at an initial time instant, then it appears vanishing also for all following time instants. In the second case Equation (22) should reduce to the equation for vorticity following from Equation (21) which takes place if 0 and 1 . 4) Insofar as as CCsK, in which 2 ks, Equation (23) can be rewritten also as sCQ sK explaining lent transport similar to the advection by the velocity field incomd 0 s. urbulens under the Influence of External Fields unted dCC dt , (24) the effect of the crossgradient turbu of pressible flui 3.3. Turbulent Flows in Specific Conditions 3.3.1. Description of Tt Flow The effect of external fields on turbulence is acco for in (21) and (22) through the terms F and m. In ses. the following the situation is particularized for two ca The first case is related to the flows of electrically conductive media under the influence of external mag netic field for small magnetic Reynolds number values, where [26] 2 0000 ˆ EB FEBBB u1 (25) 2ˆ 1 f JB 00 0 2 mBB1 (26) E in which B0 denotes the induction of the ext netic field, ernal mag is the coefficient of electrical ity and 01 conductiv denotes certain phenomenological characteristic of the medium electric properti n o es. Expres sion (26) evinces the effect of external magnetic field resulting for small magnetic Reynolds number values in a suppressiof . The situation changes for the me dium and/or large magnetic number values when the magnetic field may prove acting as a source of energy of the orientated constituent of turbulence or the medium turbulence may prove acting as a course of generation of magnetic field. The second case is related to the flows under the influ ence of gravity force, where within the Boussinesq ap proximation [27] we have [26] (27) an 2,k d 1ˆ fk g (28) hich 1 mg g in w is the characteristic constant density of medium and is the medium actual average density, while . Here, the integration of Equ and (22) requires a specification of the equation ations (21) of state expressing through the characteristics of medium ingredients contributing to and Equations (21) and (22) should be integrated together with the equations formulated for all medium ingredients contributing to . Notice, that expression (28) compiles in one single formula the depression of the component of perpen dicular to the gravity acceleration caused by the stable stratification as well as its generation by the unstable stratification, which substantially simplifies the descrip tion of gravitationrelated processes involving the both situations. 3.3.2. Description of Turbulent Flows in Rotating Frames For the description of motion in a frame rotating with a constant angular velocity 0 , the flow velocity u is re 0 placed by or t of ur, where r denotes a radiusvect from the arbitrary point on the rotation axes to a poin lowfield, 0 and are replaced by the f and 0 , and ddtu, ddtM are replaced by 0 ddt uu , 0 ddt MM. The changes result in the complementation of the right sides of (21) and (22), respectively, by the Coriolis force term 0 2 C Fu , 29) tiot term ( and by the addinal body momen Copyright © 2013 SciRes. JMP
J. HEINLOO 510 0000 4JJ mu (30) by and in the replacement of the expression for 00 01 ˆˆ skk 11K 20 . (31) The expressions (29)(31) evidence about a sub in tat terthe right side of (30) evinces the frame rotation pre the anticyclonic (directed opposite ection of stantial difference between the turbulence properties the ro ing and the nonrotating frames. So, the firstm on ferring 0 to the dir ) orientation of (gyration effect). The work done by the moment 0 m may serve also as an additional cause of eddytomean energy conversion etc. 4. Examples of Application of the RAT Theory 4.1. OneDimensional Flows in Plain Channels, Round Tubes, between Rotating Concentric Fonsional flows in plain channels, round tubes,  Cylinders and Boundary Layers r onedime between rotating concentric cylinders [23], and in boun dary layers [28] the Equations (21) and (22) simplify to 2p t uu F , (32) 144 Jt m (33) where and u are orientated perpendicularly pend on the coordinate perndicular to and 0 f Fm the picted by (32) and (33) ve locity profiles were compared in [23] (for steady hannels, in round tubes and between rotating con cy inr os boundary and de u. pe red For plain c cil flows in centric linders) with data in [2931] and (in case of oscillating flow in round tube) with data [32]. Fo lating layer generated by undulating free flow the predicted by (32) and (33) velocity profiles were compared in [28] with data in [33]. The predicted by (32) and (33) velocity profiles for F and m specified in (25) and (26) were compared with data in [34]. In all cases the predicted velocity profiles prove excellently matching the actual velocity data. Notice, that for 0 Equa tions (32) and (33) coincide in written form with the re spective equations of the MF theory. However, the situa tions with 0 and 0 prove reflecting physi cally different situations. In particular, for steady flows in round tubes and plain channels the solutio32) and (33) predicts the flow velocity determined in the central part of the flow region by the effective viscosity ef n of ( whints to a substantial role of the turbulence properties characterized by h chi and in this region with the property characterized by play ing (in harmony with the RK conception) a marginal role. In the central part of the flow, for 0 , the turbulence epresented by the turbulence shear viscosity only. properties appear r of concentration of the suspended 4.2. Vertical Distribution of Concentration of Suspended Sediments in a River Estuary In [35] Equations (21)(23) were applied to describe the vertical distribution matter C in a river estuary modeled as an open ch nel with the fixed bottom slope angle an and the tim ing free surface angle t . Restricting t e vary he consideration with ,1 , the quasistationary flow regime and with the concentrations small enough to not influence density field, Equations (21)(23) read as e th 2 220 ut z z g , (34) 2 12420 u z z , (35) 2 01 0 C kk Q zz , where (36) QwCz , in which w is the settl and z is the vertical coordinate directed upward. The term ing velocity g in (34) expresses the summary effect of the alongflow pressure gradient and of the gr The determined from (34)(36) vertical distributions of C ompared wit for differen avity force. were ch concentration of the resuspended sediments observed in the Jiaojiang Estuary (China) [36] t time instants of a spring tide cycle. The comparison showed that the derived analytical formula for C embraces two observed basic types of vertical dis tribution of concentration, one with a monotonic de crease of concentration gradient with distance from the bottom and the other with a gradient maximum (luto cline) located at some distance from the bottom. (The both types of vertical distribution of suspended sediments were detected also in the bottom layer of natural water body, studied in [37]). 4.3. Vertical Structure of the Upper Ocean Consider now the situation in the upper ocean in Boussi nesq approximation specified by ,,0uzuzu, xy ,,0zz 0 xy and (the righthand Cartesian coordinate system ,, yz with 0z rected downward is assumed; hereafter 0 di is the angu lar velocity of the Earth rotation). From Equations (21) (23), where is determined acco9) rding to (27) and (2 as 0 2 gu , mdetermined is according to (28) as 1z fkg m and0 we with Qe hav Copyright © 2013 SciRes. JMP
J. HEINLOO 511 20 222p tz uuu (37) 2 11 4Jkg tz 24 z , (38) pg 2 01 kk ztz . Equations (37)(39) explain the stable stratificat supressing the constituent perpendicular to the grav ity acceleration together with ancrease of vertical gra dient of density, and the unstable stratification in ampli fying the constituent perpendicular to t eration together with a decrease of vertica de (39) ion in in he gravity accel l gradient of nsity. For the constant the solution of (37) and (38) [38, 39] for the velocity sums up from two addends reflecting the Stokes drift effect [40] and the classical Ekman ver tical velocity profile [27] with the turbulence viscosity replaced by the effective viscosity ef . The solution explains the Stokes drift effect in good agreement with data in [41], in dimishing the angle between the flow velocity and the shear stress. To demonstrate the stratifi cation effect Equations (37) and (38) were solved in [39] for in constant everywhere instead of a density jump at a certain depth modeling the assumed location of ther mocline in summer and winter. The calculated vertical distribution of velocity was compared in both cases with the velocity data in [42] showing a good agreement in the dominating quality. In [43,44], Equations (37)(39) were applied to model the reaction of the upper ocean to periodical cooling and heating. Here zTzSz , where T is temperature, S is salinity, is the coefficient of ther mal expansion and characterizes the salinity contrac tion variance. It is shown that Equations (37)(39) predict the formation of a typical for the upper ocean vertical density profile with the relatively uniform density distri bution in the layer next to the ocean surface separated from lower layers by a stram of relatively abrupt den sity jump in a reasonable agreement with the observed data. 4.4. Conjoint Effect of the Baroclinic Instability and the Rotational Viscosity of Turbulence In [45,46], Equations (21), together with tu F replaced by C F in (29), and (22), with m specified in (28), were applied to agphic correction ag u to geostrophically predicted net the Antarctic  tribu calculateeostro transport of Circumpolar Current [47] from the observed spatial dis tion of . The idea lies in the determination of a correction from vertical con he f the balance condition of stituent of the Coriolis force with the vertical constituent of torce described by the term2 in (21), giving ag 02 cos cos ur , (40) where is determined from the zonal projection of (22) as 1 21 g rz 14kg k Equations (40) and (41) explain ag u for joint effect of the rotational viscosity of medium turbu . (41) med as a con lence and the baroclinic instability (characterized by 20k). The both mentioned effects are excluded within the CTM. A similar setup was applied in [48] to explain the formation of zonal winds in plan mosphere with the main interest to the formation of east er etary at lies in the equatorial zone showing a reasonable agree ment with the observed velocity data in [49]. 4.5. Gyration Effect In subsubsection 3.3.2 it was pointed out that the frame rotation prefers the anticyclonic orientation of . The represented in [50,51] zonally averaged 0,0, and 0,0, M estimated from the global surface drifter data sets [52] (the righthand coordinate system ,,z , where π 2 π 2 is latitude, is longitude ,5 tical aspects e gyration effe 4.5.1. Th and axis z of the coordinate system is directed upward is applied) confirm this conclusion, called in [501] the gyration effect. In the following some theore of thct from the position of the RAT the ory are commented. eoretical Evidenceof the Gyration Effect First consider how the gyration effect agrees with Equa tions (22) and (30). Restricting the consideration with the effects of diffusion of momentofmomentum neglected we have from (22) and (30) 0sin h 0sin z u Jz 0 44 sin J (4 where h u 2) denotes horizontal gradient operator. Insofar as from the continuity equation 0 u follows that hz uz u Equation (42) rewrites also as 0 0 sin 44sin h J u (43) For 0 u Equation (43) gives Copyright © 2013 SciRes. JMP
J. HEINLOO 512 0sin . (44) Expression (44) explains the gyration effect by the rotation of frame in balance of the shear rotation and of the decrease of a prevailing orientation of ed tation 4.5.2. Anomalous Turbulent Diffusi Paper [53] exploits (44) within a model explaining the ob ulent diffusion. This explanation follows from Equation (24) for C specified generated in relative dy ro in cascading process. ve Transport served tonguelike structure of the salinity distribution in the region of the Gibraltar Salinity Anomaly (GSA) [54]. The tonguelike structure of the anomaly is explain ed as a result of crossgradient turb as salinity S depending on (longitude) and (latitude) only, written as 2 0sinSkb S s, (45) where cosa se (in which e is the unit vector directed to the east, 0 2 ak ) and 2 0 1 bk . According to Equation (45) the gyration effect stretches the salinty distribution out in the eastwest direction 2 k i 0 and shifts the maxima e latituistribution of salinity the south with distance from the Gibraltar ive agreemese of thdinal d increasingly to Strait in good qualitatnt with the obrved situation repre sented in [54]. 4.5.3. EddytoMean Energy Transfer in Geophysical Jet Flows For 0u and for 0sin conserved in a flow 0sin0 h u from (42) we have 0sin . (46) Using (46), the work A performing the en exchange betw ergy the average flow and the orien turbutituent, exp een tatedlence consresses as 0 4sinA . (47) Paper [55] employs (46) and (47) explaining the up grsfer and e ception of 2D ce [56,57]. The situation was particularized for sections of Gulf Stream at nd at 35˚N (Onslow Bay) where, in harmony y conver additional result, the discussion evidences about m. adient momentum tranddytomean energy conversion 0A avoiding the negative viscosity problem [25] or the application of the con turbulen the cross Straits) a wi As an 26˚N (Florida th (47) and the observational data in [58], the regions with the upgradient momentum transfer and eddyto mean energsion were observed at the anticyc lonic sides 0 of the stream. the insufficiency of the velocity covariance data for un ambiguous solution of the problem of turbulent stress tensor properties. In particular, for antisymmetric stresses the velocity covariance determines the stress tensor components with the accuracy up to the sign, specified from consideration of balance of internal moments acting in the mediu 4.5.4. Topographically Generated Flows Using the continuity equation in a shallow water region, written as 1 z uzHH u [27], from Equation (42) it follows, that 00 sin sin 44J HH u. (48) 0 For sin conserved in a shallow water tant, and region we have 0sin CH , (49) where C is cons 0sin CH . essing C through (50) Expr the depth cr , where the en ergy scatterergy 0 into en obtains its minimum, gives [59] 0 cr 1sin H C and from (49) and (50) we have 0 cr in1s H H (51) and 0 cr 1sin H H (52) ith (44) for cr . H Equations (51) and (52) agree w suggesting identification of the actual depth H with cr alongshore re in the open part of the waterbody. In the gion (52) declares that 0 evidencing about a non ng velocity of flow in this region. In particular, in the alongshore region of a closed water ated flow velocity is directed anticyclon islands cyclonically. For 0 vanishi body the gener ically and around Equations (51) and (52) re potentia ped . Unl h the duce to the condition of conservation of the l vorticity predicting the motion in regions with slo bottom, though leaving open the question about the source of motion energyike the discussions in [60 62] suggesting to overcome this shortcoming within rather sophisticated theoretical constructions, the mecha nism suggested above explains the energy supply from the energy associated witgyration effect converted to the flow energy in shallow water regions. Copyright © 2013 SciRes. JMP
J. HEINLOO 513 4.6. Summary to the Commented Applications of the RAT Theory According to the described above applications, the RAT theory unites a broadened physical background of its setup with a noteworthy simplification of discussion of turbulencerelated problems. The simplification follows from the split of turbulence into the orientated (relatively large scale) and the nonorientated (relatively small scale) flowT theory, the CTM unites a flows with ex M. ts besides the flow velocity at these points also by the curvature of the streamlines passing these points. The constituents with the orientated constituent of turbulence dominating in the formation of properties of average . Contrary to the RA simplification of its setup by neglecting the orientated turbulence constituent with complications in its applica tions following from the inconsistency between the ab sence of the orientated constituent of turbulence in the CTM setup and its presence in actual flows. Despite focusing on the orientated turbulence con stituent, the commented applications demonstrate con siderable perspectives of the RAT theory. So, the exam ples show the ability of the RAT theory to describe the eddytomean energy conversion avoiding the negative viscosity or evading the actual 3D structure of turbu lence. The examples demonstrate also substantial per spectives of the RAT theory proceeding from the par ticularization of the interaction of turbulent ternal fields and from the distinguishing the turbulence properties in rotating and nonrotating frames. The latter actualizes within the available ocean surface drifter data providing a plain observational evidence to the prevailing anticyclonic orientation of eddy rotation (gyration effect) which has been excluded within the CTM. Due to the inclusion of the gyration effect in its setup, the RAT the ory sets this effect into the dynamic context required to explain the physical causes of the effect and its impact to the dynamical, energetic and to some other (like trans port) processes in the upper ocean. Despite the transparency of the applications formu lated as a direct inference from the same set of equations, the focus of the applications on the orientated turbulence constituent turns the formulated results incompatible with the respective results formulated on the bases of the CTM. The indicated discrepancy increases due to the following from the RAT theory deficiency of contempo rary methods of experimental research of turbulence ad justed to the requirements of the CT 5. Conclusions The PCTM is an implication of the physicalhistorical point of view to the turbulence problem summarized as the PDT [22]. It advances the TM realizing a small but effective in its outcome modification at the very origin of the setup of the TM. The modification stands in com plementation of characterization of the flow states in infinitesimal surroundings of flow poin velocity fluctuation introduced modification was aimed to clarify the classi cal conflict between the CTM and the RK conception actualized by the idea about applicability of the theory of MF to the description of turbulent flows. The PCTM accomplishes the task by compiling the CTM and the RK conception in a single theoretical construction. The RAT theory (complementing the universal formalism of the PCTM by the appropriate closure assumptions) justifies the applied modification from the pragmatic point of view. It compiles a substantial enlargement of the com petence of the TM with a considerable simplification of the discussion without losing the physical rigidity. Be sides grounding the RAT theory, the PCTM (especially if complemented in its setup with the method of decompo sition of turbulent flow fields discussed and applied in [1921]) makes the turbulence problem an interesting subject for theoretical discussions. As a mechanical out come of the PDT the PCTM esteems also the PDT as a whole. Unlike the PDT, the dominating uptodate look on the turbulence problem reduces it to a huge number of par ticular problems stressing rather on their particularities than on their commonness, relates the RK conception to the ideas of the past not worthy to be revived in modern time and believes the fundamental aspects of the turbu lence problem belonging rather to mathematics than to physics. The conflict emerged between the CTM and the PCTM is enforced by the criticism of the PCTM in ad dress of the CTM. The conflict has all aspects archetypal to the paradigmatic conflicts in science, always accom panied with the critics of the old paradigm from the point of view of the novel paradigm and avoiding the discus sions which may insinuate doubts about the grounding statements of the old paradigm. In the end, any paradigmatic change in the science is always preceded by the superfluous aplomb of the former paradigm in its consummation, loss of adeptness for selfcriticism and relating the unresolved problems to the solution nuances which cannot attaint the existing para digm as a whole. Though, the unresolved nuances may incidentally actualize. Not finding answers within the dominating paradigm they start looking for answers on a wider scientific background embracing also neighboring science fields. If succeeding, the new expanded point of view may develop into an independent paradigm clarify ing its relation with the former paradigm through a para digmatic conflict. This is just the situation with the for mulation of the RAT theory, the PCTM and the PDT. Started with formulation of the RAT theory initiated by the discussed in 70es idea about applicability of the the ory of MF to the description of turbulent flows, the for Copyright © 2013 SciRes. JMP
J. HEINLOO 514 mulation of this theory actualizes the RK conception as well as evinces the incompatibility of the RK conception with the CTM. The incompatibility raises several ques tions and the need to look for answers to these questions within the frames of the general principles of statistical physics collected together as the PDT. It motivates also the formulation of the PCTM within the classical formal ism with an axiomatic change in the setup. This kind of the setup turns the physical background of the PCTM absolutely transparent and mandates the opponents either to agree with the suggested change or to reject the change by applying physical arguments. The fact that neither of the possibilities has realized characterizes the emerged paradigmatic conflict so deep that usually char acterizes breaking points in the respective science fields. 6. Acknowledgements The preparation of the paper was supported by grant ETF9381 of the Estonian Science Foundation. The au thor thanks Dr. Aleksander Toompuu for the idea to de termine the gyration effect from the ocean surface drifter data providing the grounding statements of the RAT the ory with observational evidence. REFERENCES r Prediction by Numerical Pro bar, 1980. [4] J. S. Dahler and Continua,” Na , 1961, pp. 3637. [1] L. F. Richardson, “Weathe cess,” Cambridge University Press, Cambridge, 1922. [2] A. N. Kolmogorov, “The Local Structure of Turbulence in Incompressible Viscous Fluids for Very Large Rey nolds Numbers,” Doklady Akademii Nauk SSSR, Vol. 30, 1941, pp. 376387 (in Russian). [3] A. C. Eringen, “Microcontinuum Field Theories II. Fluent Media,” Krieger Pub. Co., Mala L. F. Scriven, “Angular Momentum of ture, Vol. 192, No. 4797 doi:10.1038/192036a0 [5] J. S. Dahler, “Transport Phenomena in a Fluid Composed of Diatomic Molecules,” Journal of Chemical Physics, Vol. 30, No. 6, 1959, pp. 14471475. doi:10.1063/1.1730220 [6] A. C. Eringen, “Theory of Micropolar Fluids,” Journal of Mathematics and Mechanics, Vol. 16, No. 1, 1966, pp. 1 18. [7] T. Ariman, M. A. Turk and D. O. Silvester, “Microcon tinuum Fluid Mechanics—A Review,” International Jour nal of Engineering Science, Vol. 11, No. 8, 1973, pp. 905930. doi:10.1016/00207225(73)900384 70, pp. 18. 0.1016/0022247X(72)902399 [8] A. C. Eringen and T. S. Chang, “Micropolar Description of Hydrodynamic Turbulence,” Advances in Materials Sci ence and Engineering, Vol. 5, No. 1, 19 [9] A. C. Eringen, “Micromorphic Description of Turbulent Channel Flow,” Journal of Mathematical Analysis and Applications, Vol. 39, No. 1, 1972, pp. 253266. doi:1 0729 [10] J. Peddieson, “An Application of the Micropolar Fluid Model to Calculation of Turbulent Shear Flow,” Interna tional Journal of Engineering Science, Vol. 10, No. 1, 1972, pp. 2332. doi:10.1016/00207225(72)90 Statistical Physics,” .3.05 [11] V. N. Nikolajevskii, “Asymmetric Mechanics and the Theory of Turbulence,” Archiwum Mechaniki Stosowanej, Vol. 24, 1972, pp. 4351. [12] V. N. Nikolajevskiy, “Angular Momentum in Geophysi cal Turbulence: Continuum Spatial Averaging Method,” Kluwer, Dordrecht, 2003. [13] L. D. Landau and E. M. Lifshitz, “ Pergamon Press, Oxford, 1980. [14] A. Ishiara, “Statistical Physics,” Academic Press, New YorkLondon, 1971. [15] J. Heinloo, “Phenomenological Mechanics of Turbulent Flows,” Valgus, Tallinn, 1984 (in Russian). [16] J. Heinloo, “Turbulence Mechanics,” Estonian Academy of Sciences, Tallinn, 1999 (in Russian). [17] J. Heinloo, “On Description of Stochastic Systems,” Proceedings of the Estonian Academy of Sciences, Phys ics and Mathematics, Vol. 53, No. 3, 2004, pp. 186200. [18] J. Heinloo, “A Setup of Systemic Description of Fluids Motion,” Proceedings of the Estonian Academy of Sci ences, Vol. 58, No. 3, 2009, pp. 184189. doi:10.3176/proc.2009 , Vol. 8. No. [19] J. Heinloo, “The Structure of Average Turbulent Flow Field,” Central European Journal of Physics 1, 2010, pp. 1724. doi:10.2478/s115340090015y [20] J. Heinloo, “Setup of Turbulence Mechanics Accounted for a Preferred Orientation of Eddy Rotation,” Concepts of Physics, Vol. 5, No. 2, 2008, pp. 205219. doi:10.2478/v1000500700338 [21] J. Heinloo, “A Generalized Setup of the Turbulence De scription,” Advanced Studies in Theoretical Physics, Vol. 5, No. 10, 2011, pp. 477483. [22] J. Heinloo, “Physical Doctrine of Turbulence—A Re hy view,” International Journal of Research and Reviews in Applied Sciences, Vol. 12, No. 2, 2012, pp. 214221. [23] J. Heinloo, “Formulation of Turbulence Mechanics,” P sical Review E, Vol. 69, No. 5, 2004, Article ID: 056317. doi:10.1103/PhysRevE.69.056317 [24] L. I. Sedov, “A Course in Continuum Mecha ersNoordhoff, Groningen, 1971. nics,” Wolt [25] V. P. Starr, “Physics of Negative Viscosity Phenomena,” McGrawHill, New York, 1968. [26] J. Heinloo, “The Description of Externally Influenced Tur bulence Accounting for a Preferred Orientation of Eddy Rotation,” European Physical Journal B, Vol. 62, No. 4, 2008, pp. 471476. doi:10.1140/epjb/e2008001878 [27] J. Pedlosky, “Geophysical Fluid Dynamics,” Springer, New York, 1987. [28] J. Heinloo and A. Toompuu, “A Model of Average Ve locity in Oscillating Turbulent Boundary Layers,” Jour nal of Hydraulic Research, Vol. 45, No. 5, 2009, pp. 676 680. doi:10.3826/jhr.2009.3579 [29] J. Nikuradse, “Gesetzmässigkeiten der Turbulenten Strö Copyright © 2013 SciRes. JMP
J. HEINLOO 515 mung in Glatten Rohren,” VDIForschungsheft No 356, 1932, pp. 136. [30] G. ComteBellot and A. Craya, “Écoulement Turbulent entre Deux Parois Parallèles,” Fiche Détaillée, Paris, 1965. nsfer in Round Channel with Inner Ro tal Study d N. A. Carlsen, “Experimental and The [31] V. N. Zmeikov and B. P. Ustremenko, “Study of Ener getic and Heat Tra tating Cylinder,” Problems of Thermoenergetics and Ap plied Thermophysics1, Academy of Science of Kazakh stan SSR, 1964, p. 153 (in Russian). [32] V. I. Bukreejev and V. M. Shakhin, “Experimen of Unsteady Turbulent Flow in Round Tube,” Aerome hanika, Nauka, Moscow, 1976, p. 180 (in Russian). [33] I. G. Jonsson an oretical Investigations in an Oscillatory Turbulent Boun dary Layer,” Journal of Hydraulic Research, Vol. 14, No. 1, 1976, pp. 4560. doi:10.1080/00221687609499687 [34] G. G. Branover and A. B. Tsinober, “Magnetic Hydro mechanics of Incompressible Fluids,” Nauka, Moscow, 1970 (in Russian). [35] J. Heinloo and A. Toompuu, “A Model of Vertical Dis tribution of Suspended Matter in an Open Channel Flow,” Environmental Fluid Mechanics, Vol. 11, No. 3, 2011, pp. 319328. doi:10.1007/s1065201091801 [36] J. Jiang and A. J. Mehta, “Lutocline Behavior in HighCon centration Estuary,” Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 126, No. 6, 2000, pp. 324 328. doi:10.1061/(ASCE)0733950X(2000)126:6(324) [37] J. Heinloo and A. Toompuu, “A Model of the Vertical Distribution of Suspended Sediments in the Bottom Layer of Natural Water Body,” Estonian Journal of Earth Sci ences, Vol. 59, No. 3, 2010, pp. 238245. doi:10.3176/earth.2010.3.05 [38] J. Heinloo and A. Toompuu, “A Modified Ekman Layer Model,” Estonian Journal of Earth Sciences, Vol. 60, No. 2, 2011, pp. 123129. doi:10.3176/earth.2011.2.06 [39] J. Heinloo and A. Toompuu, “A Modification of the Cla ssical Ekman Model Accounting for the Stokes Drift and Stratification Effects,” Environmental Fluid Mechanics, Vol. 12, No. 2, 2011, pp. 101113. doi:10.1007/s1065201192125 [40] O. M. Phillips, “Dynamics of Upper Ocean,” Cambridge University Press, Cambridge, 1977. [41] Y.D. Lenn and T. K. Chereskin, “Observations of Ekman Currents in the Southern Ocean,” Journal of Physical Oceanography, Vol. 39, No. 3, 2009, pp. 768779. doi:10.1175/2008JPO3943.1 [42] R. R. Schudlich and J. F. Price, “Observations of Sea sonal Variation in the Ekman Layer,” Journal of Physical Oceanography, Vol. 28, No. 6, 1998, pp. 11871204. doi:10.1175/15200485(1998)028<1187:OOSVIT>2.0.C O;2 [43] J. Heinloo and Ü. Võsumaa, “Rotationally Anisotropic Turbulence in the Sea,” Annales Geophysicae, Vol ol. 101, No. 11, 1996, pp. 2563525646. doi:10.1029/96JC01988 . 10, 1992, pp. 708715. [44] Ü. Võsumaa and J. Heinloo, “Evolution Model of the Vertical Structure of the Active Layer of the Sea,” Jour nal of Geophysical Research, V ics, 1029/96JC01988 [45] J. Heinloo and A. Toompuu, “Antarctic Circumpolar Current as a DensityDriven Flow,” Proceedings of the Estonian Academy of Sciences, Physics and Mathemat Vol. 53, No. 4, 2004, pp. 252265. doi:10. hysicae, Vol. 24, No. 12, 2006, pp. eanography, Vol. [46] J. Heinloo and A. Toompuu, “Modeling a Turbulence Effect in Formation of the Antarctic Circumpolar Cur rent,” Annales Geop 3191 3196. [47] T. Whitworth, W. D. Nowlin Jr. and S. J. Worley, “The Net Transport of the Antarctic Circumpolar Current Trough Drake Passage,” Journal of Physical Oc 12, No. 9, 1982, pp. 960971. doi:10.1175/15200485(1982)012<0960:TNTOTA>2.0.C O;2 [48] J. Heinloo and A. Toompuu, “Mod Effect in Formation of Zonal Winds,” eling of Turbulence The Open Atmos 472 pheric Science Journal, Vol. 2, 2008, pp. 249255. [49] A. H. Oort, “Global Atmospheric Circulation Statistics, 19581973,” NOAA Professional Paper 14, Rockville, 1983 [50] J. Heinloo and A. Toompuu, “Gyration Effect of the LargeScale Turbulence in the Upper Ocean,” Environ mental Fluid Mechanics, Vol. 12, No. 5, 2012, pp. 429 438. doi:10.1007/s1065201292 Global Surface Drifter Data in the Pacific Ocean,” puu, J. Heinloo and T. Soomere, “Modelling of preading at Ocean [51] A. Toompuu and J. Heinloo, “Gyration Effect Estimated from IEEE/OES Baltic International Symposium, Klaipeda, 8 10 May 2012, pp. 14. [52] A. L. Sybrandy and P. P. Niiler, “WOCE/TOGA Lagran gian Drifter Construction Manual. WOCE Rep. 63,” Scripps Institution of Oceanography, La Jolla, 1991. [53] A. Toom the Gibraltar Salinity Anomaly,” Oceanology, Vol. 29, No. 6, 1989, pp. 698702. [54] R. W. Griffiths, E. J. Hopfinger, “The Structure of Mesoscale Turbulence and Horizontal S Fronts,” Deep Sea Research Part A. Oceanographic Re search Papers, Vol. 31, No. 3, 1984, pp. 245269. doi:10.1016/01980149(84)901043 [55] J. Heinloo and A. Toompuu, “EddytoMean Energy No. 24, 1979, pp. 289325. Transfer in Geophysical Turbulent Jet Flows,” Proceed ings of the Estonian Academy of Sciences, Physics and Mathematics, Vol. 56, No. 3, 2007, pp. 283294. [56] P. B. Rhines and W. R. Holland, “A Theoretical Discus sion of EddyDriven Mean Flows,” Dynamics of Atmos pheres and Oceans, Vol. 3, doi:10.1016/03770265(79)900150 [57] J. R. Herring, “On the Statistical Theory of Two Dimen sional Topographic Turbulence,” Journal of the Atmos pheric Sciences, Vol. 34, No. 11, 1977, pp. 17311750. 1:OTSTOT>2.0.Cdoi:10.1175/15200469(1977)034<173 O;2 [58] F. Webster, “Measurements of Eddy Fluxes of Momen tum in the Surface Layer of the Gulf Stream,” Tellus, Vol. 17, No. 2, 1965, pp. 239245. [59] J. Heinloo, “EddyDriven Flows over Varying Bottom Topography in Natural Water Bodies,” Proceedings of the Copyright © 2013 SciRes. JMP
J. HEINLOO Copyright © 2013 SciRes. JMP 516 o 0.1175/15200485(1986)016<2159:MCDBTD>2.0. Estonian Academy of Sciences, Physics and Mathematics, Vol. 55, No. 4, 2006, pp. 235245. [60] D. B. Haidvogel and D. H. Brink, “Mean Currents Driven by Topographic Drag over the Continental Shelf and Slope,” Journal of Physical Oceanography, Vol. 16, N 12, 1986, pp. 21592171. . doi:1 CO;2 [61] F. P. Bretherton and D. B. Haidvogel, “TwoDimensional Turbulence above Topography,” Journal of Fluid Me chanics, Vol. 78, No. 1, 1976, pp. 129154. doi:10.1017/S002211207600236X ophic Eddies and the Mean Circulation over Large [62] S. T. Adcock and D.P. Marshall, “Interactions between Geostr Scale Bottom Topography,” Journal of Physical Ocean ography, Vol. 30, No. 12, 2000, pp. 32323238.
