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2 Review of Set Theory CS 336 Lecture 2 ThursdayAugust 24 2000 August 24 2000 2 Review of Set TheOIy Outline Logic Sets Relations and Functions August 24 2000 2 Review of Set TheOIy Logic August 24 2000 2 Review of Set TheOIy Some Logical Operators pqpampqand pqporq FF F 171 F FT F FT T TF F TT T TF T TT T p notp F T T F August 24 2000 2 Review of Set TheOIy Some More Logical Operators p q p gt q p q p ltgt q implieS equivalence F F T F F T F T T T F F F T F T T T T F F T T T August 24 2000 2 Review of Set TheOIy Logical Quantifiers there exists for all ex There exists an ex For all integer n integer n such that n n is an integer 4 lt n lt 10 ex For all integer n there exists uniquely n O n ex There exists a unique integer n such that 4 lt n lt 6 August 24 2000 2 Review of Set TheOIy Sets August 24 2000 2 Review of Set TheOIy Subsets X is a subset of Y ltgt for all z in X z is in Y Two sets are equal iff they have the same elements X Y ltgt X is a subset of Y and Y is a subset ofX August 24 2000 2 Review of Set TheOIy Construction of sets1 Power set PowersetX Y Y is a subset of X Notice that sizeofPowersetX 2 raised to the power sizeofX Union unionXY a a in X ora in Y Intersection intersectionXY a a in X and a in Y August 24 2000 2 Review of Set TheOIy Construction of sets2 The elements of a set have no implied order That is a b b a We need to be able to specify the order of elements so we define an ordered gair ab 8 ab Note ba b ab We define a triple by abC abC and extend the concept to ntuples x1x2x3xn x1x2x3xn1xn August 24 2000 2 Review of Set TheOIy 10 Construction of sets3 Product XxYab ainXand binY XxYxZabc ain Xb in Y cin Z Disjoint Union disjointunionXY union0xX1xY Set Difference XYxxinXandxnotinY August 24 2000 2 Review of Set TheOIy Relations and Functions August 24 2000 2 Review of Set TheOIy Binary Relations A binary relation R between sets X and Y is an element of powersetX x Y That is R is a subset of X x Y If xy in R we write ny ex is a binary relation on the natural numbers August 24 2000 2 Review of Set TheOIy 13 Partial and Total Functions A partial function ffrom X to Y is a relation for which xy in fand xy in fgt y y That is fx has a unique value A total function ffrom X to Y is a partial function which is defined for every x in X x in X gt there is a y in Y such that xy inf August 24 2000 2 Review of Set TheOIy 14 Composition of Relations and Functions Let R be a relation between sets X and Y and S be a relation between sets Y and Z Then S o R xz for some y in Y xy in R and yz in S August 24 2000 2 Review of Set TheOIy 15 Equivalence Realtions Let R be a relation on X That is R is a subset of X x X R is reflexive if xRx for all x in X R is symmetric if ny gt ny R is transitive if ny and sz gt sz R is an equivalence relation if R is reflexive symmetric and transitive August 24 2000 2 Review of Set TheOIy 16 Equivalence Classes Let R be an equivalence relation on the set X Then the Requivalence class of xisyyinXandey The equivalence classes induced on a set X are mutually inclusive and painvise joint That is x in X belongs to exactly one equivalence class August 24 2000 2 Review of Set TheOIy 17 Examples of Binary Relations On the integers is an equivalence relation lt is reflexive and transitive but not symmetric lt is transitive but not reflexive or symmetric August 24 2000 2 Review of Set TheOIy 18 PHYSICAL REVIEW E 71 026306 2005 Stability of strati ed ow with inhomogeneous shear Vladimir S Mikhailenko Kharkov National University 61108 Kharkov Ukraine and National Science Center Kharkov Institute of Physics and Technology Scienti c and Production Complex Renewable energy sources and sustainable technologies 61108 Kharkov Ukraine Earl E Scime Physics Department West Virginia University Morgantown West Virginia 26506 USA Vladimir V lLikhailenko Kharkov National University 61108 Kharkov Ukraine Received 13 July 200439 published 11 February 2005 The temporal evolution of perturbations in strati ed ow with inhomogeneous shear is examined analyti cally by an extension of the nonrnodal approach to ows with inhomogeneous shear The solutions of the equations that govern the linear evolution and the weak nonlinear evolution of perturbations of the stream function for strati ed ow with monotonic inhomogeneous shear are obtained It is shown that stabilization of perturbations arises from nonrnodal effects due to ow shear Conditions at which these nonrnodal effects may be strong enough to stabilize the RayleighTaylor instability are presented These analytical results are also compared to numerical simulations of the governing equations performed by Benilov Naulin and Rasmussen DOI 101103PhysRevE7l026306 I INTRODUCTION It is well known that inversely strati ed uids for which density increases upward are unstable due to the Rayleigh Taylor instability for all wavelengths l The Rayleigh Taylor instability in a plasma is governed by similar equa tions for perturbations of electrostatic potential and density That instability develops in plasmas embedded in an unfa vorably curved magnetic eld with a density gradient anti parallel to the magnetic eld radius of curvature In such magnetically con ned plasmas examples of RayleighTaylor modes include the ideal and resistive balooning instabilities 2 A considerable amount of research has been devoted to the study of the RayleighTaylor instability in uids and its application to ionospheric turbulence It is believed that the RayleighTaylor instability can play a major role in the onset of equatorial spread F 34 In these physical systems the essential characteristics of the RayleighTaylor instability are similar The instability arises in inhomogeneous media uid or plasma and acts to interchange high and lowdensity regions or to interchange high and lowtemperature regions In many regions of interest such as the atmosphere the ionosphere or the edge of tokamak plasmas ordinary uids or plasma may also contain inhomogeneously sheared ows ie ow elds with a spatially varying ow speed gradient It has been shown that homogeneous velocity shear can have a dramatic effect on the RayleighTaylor instability eg Mles 5 and Kuo 6 demonstrated that ow shear can sup press the Rayleigh Taylor instability in unbound systems with homogeneously sheared ows Suppression of the RayleighTaylor instability in plasmas by shear ow was also demonstrated in linear theory by Guzdar et al 7 Signi cant stabilization occurs in the linear theory for v6 2 EZ IZy where 1262 is the velocity shear and the growth Electronic address vmikhailenkokipt kharkovua 15393755200571202630682300 0263061 PACS numbers 4720k 4710g 5235Kt rate of the instability in the absence of ow shear is y Historically two different approaches have been used in the development of linear stability theories for systems with sheared ows Both techniques employ a spectral expansion in time The rst is the method of normal modes or the modal approach In a system with a zdependent ow ve locity 1202 directed along the x axis the perturbations are assumed to be harmonic in time with separable time and space dependencies described by z rt zexpiiwt ikx where 1M2 de nes the mode structure The ow is deemed unstable if at last one mode grows exponentially with time In the case of an ordinary sheared uid the mode structure 2 is governed by a secondorder differential equation that possesses a singularity at a critical level 9 where the Dopplershifted phase velocity wkvoz van ishes Because of the singularity the equations governing the modal structure are nonnormal ie the eigenfunctions as sociated with the governing differential equation are not mu tually orthogonal and experience strong interference There fore a stability analysis based on considering the only eigenvalues w obtained from the modal approach may be inappropriate for certain ranges of system parameters and a solution based on solving the initial value problem is pre ferred Several authors have pursued solutions of the initial value problem through the use of a Laplace transform in time The principal nding of the initialvalue problem ap proach is that in addition to the discrete eigenvalues linked to the normal modes there exists a continuous spectrum of eigenvalues Thus the modal approach cannot provide a complete solution for all sheared ow systems It was dem onstrated see for example Ref 8 that owing to the exis tence of the continuous spectrum of eigenvalues initial dis turbances may decay or even grow as a nonmodal pertur bation with nonseparable space and time dependencies and with tim edependent amplitudes that are powerlaw functions of time Such disturbances at certain times may overwhelm 2005 The American Physical Society MIKHAILENKO SCIME AND MIKHAILENKO the exponentially growing modes of the discrete spectrum 12 and always dominate over the exponentially decaying modes Therefore for the conditions at which the Rayleigh Taylor instability is stabilized by sheared ow nonmodal effects should be considered For those time intervals during which the nonmodal solutions dominate the nonmodal ef fects may even become strong enough to disrupt the devel opment of the typical nonlinear processes predicted by non linear theories of normal mode evolution and therefore new nonlinear processes due to the growth of the nonmodal solu tions should also be considered An alternative approach to the initialvalue problem solu tion for unbounded homogeneously sheared ow was de scribed by Hartman 13 That approach involves a transfor mation to coordinates in the local rest frame of the ow and does not invoke the normal mode ansatz or any spectral ex pansion in time This method previously used by Lord Kelvin 14 in studying the evolution of initial disturbances in parallel viscous ows with uniform shear and by Phillips 15 in investigations of internal waves in a weakly sheared thermocline yielded a successful analysis of the evolution of disturbances of the ows of uids and plasmas with homo geneous shear see for example Refs 1718 and references therein In the ow frame coordinates the evolution of an initial perturbation in homogeneously sheared ow is solv able analytically for any time of interest and is free from ambiguities arising from the mathematical singularity ap peared at a normalmode critical level Hartman s analysis obtains the same constraint 06222 12y on the velocity shear 1262 for stabilization and suppression of Rayleigh Taylor instability as was obtained with the modal approach However Hartman found that the solution to the initialvalue problem has a typically nonmodal powerlaw temporal de pendence and thus the normal mode solution is not the steadystate limit for the initialvalue problem Weak nonlin ear analysis of the RayleighTaylor instability in a plasma performed by Mkhailenko er a1 17 using a nonmodal ap proach demonstrated that homogeneous ow shear stabilizes not only linearly unstable twodimensional perturbations of electrostatic potential but also nonlinearly unstable terms including the fourth order of the perturbed potential were considered perturbations Compared to the previous linear analyses the nonlinear analysis yielded a slightly different constraint on the magnitude of the velocity shear required for stabilization of the RayleighTaylor instability Until now the nonmodal approach has only been applied to systems with homogeneous ow shear ie 1262 indepen dent of 2 In this work we develop an analytic framework for the extension of the nonmodal approach to ows with inho mogeneous shear and we also examine the stability of strati ed ows with inhomogeneous shear We will consider at the outset only twodimensional perturbations in a strati ed me dium The application of a twodimensional model is justi ed by the twodimensional nature of the RayleighTaylor instability in fusion 2 and ionospher 7 plasma as well as in internal gravity waves in ordinary strati ed uids 1011 The structure of this paper is as follows In the next section we formulate the basic nonm odal nonlinear equation that can be solved asymptotically in the case of inhomogeneous ow shear In Sec III the solutions of that equation for strati ed PHYSICAL REVIEW E 71 026306 2005 ows with inhomogeneous monotonic velocity shear are developed In Sec IV the weak nonlinear temporal evolution of the nonmodal solutions is studied and some concluding remarks are presented in Sec II THE GOVERNING EQUATIONS We choose as our model a twodimensional unbounded plane with inhomogeneously sheared ow and an ex ponential density scaling The equilibrium velocity pressure and density are given by v0v0 2ex P2 Po exp gfzpozrdzrgt and P0ZJP00 eXP ZH Where 8 is the acceleration due to gravity p00 is the mean density and H is the scale height for the density The uid is assumed to be incompressible and therefore a stream function 3133 can be de ned through 3 3 vw cw 32 3x The solution of the temporal evolution of inhomoge neously sheared ow reduces to the solution of the set of differential equations for p and density perturbation p a a H 3 3132 5voltzltiA JvoltzaxMx MZW g 5P 1 p0 3x Lumi ii i Nzg 2 31 3x 32 3x 3x 32 p0 3x where the prime denotes 3 dz The linearized version of Eqs 1 and 2 may be combined through the Boussinesq ap proximation into the equation 3 3 3 3 3 32 37U3335voltzAervgzf iNz 3 in which 49362 P0 N2 is the BruntVaisala frequency In the normal mode ap proach in which a stream function 13 is assumed to be har monic in I ie 2expilaciiwl the equation for the mode structure 132 is called the TaylorGoldstein equation and has e 011139 Q lt M 322 k2 levee 7 w levee 7 02 with singular points where the phase velocity matches the equilibrium velocity wkivo20 Such points are com monly referred to as critical levels 9 It follows from Eq 4 that estimation of the shear ow effect is related to the solution of the nonlinear eigenvalue problem with singular nonorthogonal eigenfunctions in the modal approach gt 0 4 0263062 STABILITY OF STRATIFIED FLOW WITH To avoid the mathematical dif culties involved in the nor mal mode approach here we use the nonmodal method Our approach permits the development of asymptotic methods for analysis of the linear as well as the weak nonlinear evolu tionary stages for systems in which the inhomogeneous ow shear 1262 is either a small or a large parameter The non modal approach begins with a transformation to the convec tive coordinates 636 vozt that are the coordinates in the local rest frame of the mean ow These coordinates are generalizations of the convective coordinates used previously in systems with homogeneously sheared ows 13717 In terms of the convective coordi nates the system of equations may be combined into a single nonlinear differential equation for the stream function 1p 772 11 5 1 1 i v8ltvgt J7 0677T197 0677T197 067719 3amp2 2 8T v677TIZW 7735 196377 ST Ma Ma 97 ef e v3ltm 0077Tlvo77 377 1961977 19 lt xi igtlt xi igtA 0677le 1977196 196977 1977196 3amp7 6 1 i i 0677Tl197 0677Tl197 01K A zlm rill ST dll all2 1711712 amp iztvawnzar 9 ilt v677Tlv where the terms of the order of 1 are omitted and 92 lemWi lrv b iilv877r v677lzrzilz lt10 Equation 9 includes the effects of both ow shear inhomo geneity and nonlinearity To understand the role both of these effects play in the complete solution it is instructive to con sider these effects separately It follows from Eqs 6 and 7 or Eqs 9 and 10 that ow shear is the source of the nonmodal time dependence of the stream function Nonmodal effects are negligible in the case of weak ow shear ie 0677Tlt 1 For T 71 where y Nzo is the local growth rate of the ordinary instability iltawltzlm 77 PHYSICAL REVIEW E 71 026306 2005 where J 77 N v 772 is the Richardson number Equation 6 contains two parameters One v 77T is the magnitude of the ow shear shear parameter and the second ST is an amplitude parameter which de nes a measure of the nonlin earity 7 is the dimensionless time variable de ned by I Tr where T is the time scale of interest The Laplacian operator A in the new variables is timedependent and is equal to A zz 7i i 7 3992 70307 a Tag 3772 ZIUOUTW7 U 77vo77T73 gt 92 0677le7213 3 7 In the new differential equation the shear parameter 0677 T can be considered to be a small or large asymptotic parameter to determine the qualitative behavior of the solu tions and to develop appropriate asymptotic methods for so lution in the cases of weak or strong ow shear Performing a Fourier transformations of Eq 6 over variable MTJWFJ39d expi l t n 8 the equation for M21 77 is obtained from Eq 6 and is Z rl 77 lzNZIMTJ 77 it A ml 77 7le11 77 A we 77 712 H 2 711 vg39ltnzlzzultzzbwltzzbwwgw lthzhmwivoltngtzz ltzsz lt9 of the inversely strati ed uid 919 initial perturbations de velop as are typically obtained in modal approach solutions The development of the modal instability is followed by the development of nonlinear effects which ultimately may lead to instability saturation Such modal turbulence may affect the mean ow by changing its structure through processes such as the formation Kelvin cateyes vortices in the critical layer regions 9 andor development of turbulent viscosity It is only during the long time evolution of such nonlinear processes that nonmodal effects might be important in the case of weak ow shear Is follows from Eq 10 that nonmodal effects vanish for values 770 such that 067700 Nearby these zero shear re gions nonmodal effects are weak and therefore perturbations will develop as governed by the modal equations and the effects of shear ow are minimal Note that all nonmodal 0263063 MIKHAILENKO SCIME AND MIKHAILENKO terms in Eq 9 are multiplied by the wavenumber compo nent I along the shear ow Therefore nonmodal effects also vanish for disturbances with 0 Small values of I reduce the effect of the ow shear and thus nonmodal effects are again minimized It follows from Eqs 9 and 10 that non modal effects dominate when 112677 T1 21 and the wave number I is not small Comparison of the modal and non modal terms in Eqs 9 and 10 leads to a general condition for dominance of nonmodal effects 10377l i 10607 11 where h is the scale length of the perturbation along the ow shear For T 71 Eq 11 becomes 10377l i 1mm 71 39 When Eq 12 is satis ed the initial perturbation develops according to the nonmodal constraints prior to the develop ment of the modal instability Therefore Eq 12 is akin to a bifurcation condition that separates two distinctively differ ent types of solutions to Eq 9 The temporal evolution of perturbations in the case of strong monotonic inhomogeneous ow shear for which Eq 12 is satis ed is the focus of the rest of the analytic analysis 106771Tgtmaxlt1 11 10677lgtmaxlt77 12 PHYSICAL REVIEW E 71 026306 2005 presented in this work The temporal evolution of perturba tions in the case of monotonic inhomogeneous ow shear was considered numerically by Benilov el al 19 They showed that inhomogeneous ow shear stabilizes only short scale disturbances and leaves unstable large scale distur bances ie those with small wave number I In the next sections we obtain linear and weak nonlinear solutions to Eq 9 for inhomogeneous ow shear under the constraint given by Eq 12 for the dominance of nonmodal effects We show that under these conditions the solutions for stream function are stable Therefore Eq 12 de nes a boundary for the regions of stable and unstable wave number I III LlNEAR EVOLUTION STAGE OF PERTURBATIONS 1N l39NHOMOGENEOUSLY SHEARED FLOW Here we consider the effect of inhomogeneous shear on the linear temporal evolution of perturbations in a system with strati ed ow In the case of strong ow shear it is convenient to introduce a new variable 7 77l de ned by zml1v3ltnTr12 ltrml 13 For zero order in the nonlinearity parameter 2T ie we omit the right nonlinear part of Eq 9 we obtain the fol lowing equation for g 71171771 N27 211 a3 1 Marys 1 1 4 L 4111 F 372 v 2 7l77v T 7372377 3110606D7371 0 06126739quot3 7177 067723737706T73377 v 1 ii 6 1 iltv6 1 113 1 gt132 4 ii 7Zlltv gtv T7137v6T2 74 Tl7712377 0606U2 74 T l 772377 0606792 7257 067327337 9ng vg1 19 12192 4193g7 61192 Iv 77T12 72 3 v TDZ 73 19772197 v T2 7 3772 The presence of the small parameter 12677T 1 lt1 in Eq 14 permits us to obtain a solution in terms of a power series in the parameter 12677 TTI arm 07gtlgt77 1TL77 0677T1 l W 27l77 15 By employing this new power series approach we can obtain homogeneous asymptotic solutions for times 7gtv 77T 1 in systems with inhomogeneously sheared ows for which the condition 0677T 1lt1 is ful lled for all considered values of 77 For go we have the equation lt gt lt a 1977123023792 1397 v v T2741377 vgvmzmnw 1193 6ltUg1192711a4g v3 3192 13 My 03192 72 3772372 14 32 07gtlgt 77 1 7 TJ77 07gtlgt77 0 16 obtained earlier by Hartman 13 for the case of the homo geneous ow shear The solution to Eq 16 is easily ob tained and is equal to M7177C1L77W1C2L77WZ 17 for the case J 77 7k 14 where 1 1 k12 i ZJ77 18 For J7714 the solution is 07Jgt777JZIC1L77C2L7711171 19 0263064 STABILITY OF STRATIFIED FLOW WITH PHYSICAL REVIEW E 71 026306 2005 It follows from the solution 17 that the stream function lv 77lTgtmaxlh 1 in Eq 11 violates near the boundary 1p decays with increasing time when the condition where the spatial scale h of the perturbations along the ow 712 shear tends to zero Therefore Eqs 17 and 19 are actually 007 i 2 2 7 20 asymptotic solutions which are valid for times 7 is ful lled for all 77 considered as it is in the case of the 21060371171 and for locations far from the boundaries homogeneous shear ow where the conditions given by Eq 11 or Eq 12 are valid The perturbation of the uid density in the convective For the SOhltiOh 0f Speci c initial and boundary Prehlems coordinates is given by the obtained solutions here would have to be matched with 2 the solutions for Mal 77 obtained for the times 7 p ilpofiltg7xi gag lt 12677 TTI and near the boundaries However the solution g 7 k2 k1 of the complete problem which includes the initial and boundary problem is beyond the scope of this work Here only the effect of the inhomogeneously sheared ow on the stability of the strati ed uid ie the unbounded case is Thus the perturbation of uid density grows slowly in time as 7 see also Ref 17 where 1 1 y 2 2 considered K 5 t It U77 Turning to the next term in the series according to Eq 0 77 15 the equation for 1rl 77 is given by Such a dramatic difference between the time dependencies of 32 the stream function and the density perturbations in a linear J wi g1 l 77 system is strictly a nonmodal effect arising from the velocity 37392 7392 s ear 07132 U7 1 a U 1 It is interesting to note that in convective coordinates 5 3j 7 7i 81 2 5 l 77 spatial derivatives in 77 are absent in the equation for 0 and ll0 7 372 ll0 7392 37 ll0 73 the spat1al var1able 77 only enters 1nto Eq 16 and 1nto 1ts 21 33 0 4Z 32 0 41 ago 7 21 solutions 17 and 19 as a parameter The speci c spatial 77 73721377 aw I an dependence of the solutions 17 and 19 is determined en th39ely by the 77 dependence 0f the OW velocity WW and by In that equation the derivatives of 1 over 77 are also absent the initial conditions through which the functions C172l 77 and Eq 21 is also the ordinary differential equation The are determined It is important to note that the condition solution to Eq 21 is readily obtained and is equal to 11 1rl77C1l777j 71207723kf10k18 2 a kl2k173kfi3k12h171 106146 vow mm 1 ak1kfi3k12lt 1 gt 1 wan 2139 akl 1 7 3k2710k1 8 2k173 vawwn k2 2ch 12k 3071 1 t 06771977 1kf73k1219C1 11 7 0677lk2 an to lt22 forJ77 14 and LHUWE a 112132 325 1rl77 712U 7724cl77cz7715lnr va 2 377 5lnr 23 for J 771 4 In Eq 22 the notation 1 lt gt2 indicates the is that the stabilization of the RayleighTaylor instability by additional terms similar to those presented explicitly but sheared ow persists even for inhomogeneous ow shear with relabeled suf xes in C1 and k1 according to 1H2 and The complete solution has the same structure of the stream 2H1 Equations 17 18 22 and 23 demonstrate that function as obtained in laboratory coordinates in Eq 16 solution 1r0r 1 07ln 7 ie solution 57 a1gebra of Ref 11 However our solution as given by Eqs 22 ically decreases with time more rapidly under the condition and 23 PIOVideS ah the eoef CientS explicitly and the itera given by Eq 20 than the solution 57 Therefore the so tive procedure may be easily continued to the desired order lution for the stream function 1pv677T7 2 0r 1r is 0f the Parameter 0607 T11 stable in the case of strong monotonic inhomogeneous ow shear for which Eq 12 is satis ed Perhaps most important 0263065 MIKHAILENKO SCIME AND MIKHAILENKO IV WEAK NONLINEAR EVOLUTION OF PERTURBATIONS OF THE FLOW WITH lNHOMOGENEOUS SHEAR Nonlinear studies of the RayleighTaylor instability have been carried out in recent years by Hassam 16 lIikhailenko el al 17 and others Hassam performed a onedimensional calculation and showed that the Rayleigh Taylor instability in a magnetized plasma may nonlinearly saturate because of ow shear The nonmodal approach ap plied to the study of the weak nonlinear evolution of the twodim ensional RayleighTaylor instability in plasmas with homogeneous ow shear by Mkhailenko el al 17 demon strated that homogeneous ow shear stabilizes not only lin 32 71177 1 3912 mTJ77 1zL 1117 f dll f 71607117111572 37 PHYSICAL REVIEW E 71 026306 2005 early unstable twodimensional perturbations of electrostatic potential but also the nonlinearly unstable perturbations Now we consider the effect of weak nonlinear nonmodality on the evolution of perturbation for inhomogeneous mono tonic shear ow Here we derive a solution to Eq 9 in the form of the power series in the nonlinearity parameter 2T 71L77 0011 77 6T 1TL 77H 8792111121171 77 24 In Eq 24 the solution 07l 77 is determined by Eq 17 for J77 7 14 and by Eq 19 for J77 l 4 The equation for 17 l 77 in which terms of zero order in the parameter v Tfl are included is 0 7J1 xi7 W3 quot111v311111127011m7117111211111111311111127012111111 25 77 1977 where C10Cll 77 139jl2 The solution to Eq 25 is given by 1 1g 17gtZgtU WJ39dl1J39dlz 5U l1 lzF 77 4391 19 13k gtltC11C 27 1 k2 C UC Z C 1C Zgt 7 C C 1gt kl C IC Zgt3 111 1 E 1 0 iiltc 1gtc 2gti 2 cgzgtc 1gt3 k11n 1 E 1 377 377 k1 gtltltln 7 3 lt1 lt1 1 ltClt2gt3C1 Cm i 2 lt2gt3C2 1 2 377 1 377 k1 2 1917 311272 7 81120307 2iaz l15k1 7 1 21 13 Clt1gtCltZgt1 fk1 7 J39dl J39dl 5171 71 22C2 k172gti 2 2 k17k111v301112 1 2 1 2 1 an 21 112 army szde 17171 k1 kzv 772 1 2 5 1 2 F 3cm 3C1 3cm 1 3C2 1 X 2C1174q7k1 C12 C11 2C0 2 7J 2 k1 1 H 2 377 377 377 k1 3k1 l 4il l ltln 7 gt 0077 k2 377 2 1 377 k2 2112 k1 377 k1 811112 037 210171211112 7 4112112 7 4112 74111111 216 7 k1 12607 3161 i 2 1977 3161 i 2 7sz 2 1917 311272 26 for J 77 77 l 4 In Eq 26 the notation 1 H2 indicates additional terms similar to the terms presented explicitly but with relabeled suf xes 112 in C10 for which J77 14 17l 77 is given by 41 19C 1 7 dl dl 17171 12 1 lt3 2C216 C2 17 77 Fw hmzj39 1f z l 1 2 2 2 an 1 377 2 377 3C 19C 19C ln 7lt 2 C 1C 2 6 377 377 377 19C 19C 2 C11 1 377 377 aca 3717 C 1gt3C U 111 1 0307 1 0077 1 3cm 2 cy 1112 172C 11 3 77 Cg 6 and kl according to 11 is changed to 12 and 12 is changed to 11 For the values of 77 19C 19C ltc 3c gt 19C C0 2 1977 19C 1977 C97 16 19C 19C 32 cg 1112 1 2cgl 77 1977 123C 1C 2gt 16C 1gtC 2 C CP 3C 2gt 111 1C 1C 2 C 1gtC Zgt 6C 1C 2gt 1n2 1C 1gtC 27 0263066 STABILITY OF STRATIFIED FLOW WITH Both expressions for 57 con rm that weak nonmodal nonlinearity does not eliminate the stabilization of the RayleighTaylor instability by inhomogeneous monotonic shear ow that satis es the constraint described by Eq 12 The solutions for the nonlinear nonmodal perturbation of the stream function 1rl 77v77 T7 2 17 with either J77 14 or J7714 will decrease with time provided that v10 lt28 Otherwise the perturbations will grow algebraically The constraint described by Eq 28 is more restrictive than the constraint given by Eq 20 for the linear stabilization of the stream function because in the range 1 J2ylt 0607 lt2x3 y nonlinearly excited perturbations of the stream function 101 will grow algebraically It is interesting to note that the condition 28 was ob tained earlier in Ref 17 for the case of homogeneous ow shear even though the nonlinear right side of the equation for Q1 was different in the cases of homogeneous and inho mogeneous ow shear The only differences in time depen dencies in the case of inhomogeneous shear are the appear ance of In 7 and In fr2 multipliers in Eqs 26 and 27 because of the 77 dependencies of km However these mul tiplicative factors do not affect the decaying nature of the solutions for 1v677T7 2 17 under the constraint of Eq 28 V CONCLUSIONS In this work we have developed an analytical framework that is a natural extension of the nonmodal approach for ho mogeneous sheared ows to the case of inhomogeneous shear The linear and weak nonlinear evolution of perturba tions in systems with monotonic inhomogeneous shear was determined and it was demonstrated that stabilization of the RayleighTaylor instability by shear ow can persist in the case of inhomogeneously sheared ow The temporal evolu tion of the initial perturbations depends on the relative mag PHYSICAL REVIEW E 71 026306 2005 nitudes of the shear parameter 0677 T and the nonlinearity parameter 2T In the speci c case of monotonic shear sheared ow disturbances that satisfy Eq 12 for all values of 77 are stable to the development of RayleighTaylor insta bility The stabilization arises from nonmodal effects due to the ow shear Stabilization of the RayleighTaylor instability in plasmas and in ordinary homogeneously sheared ows was reexam ined in a recent paper by Benilov el al 19 They concluded that stabilization of the RayleighTaylor instability by homo geneous shear ow is a quir of the model chosen and any deviation from a linear velocity pro le triggers instability ie inhomogeneous shear is not stabilizing Their conclusion was based on a study of the stability of small wave number I perturbations in systems with monotonic shear ow and a study of the stability of jet ow in the vicinity of a tip where 126770 0 ie the cases were considered in which shear ow effects are too small as in the case of small I or even absent as at the tip of the jet ow However it is important to note that the stabilization of the RayleighTaylor instability even by homogeneous shear ow is not a universal effect for all values of the ow shear 06 and wave number I Stabilization by homogeneous ow shear requires ful llment of the con dition 1262 2 2 12y for linearly unstable perturbations and ful llment of slightly more stringent conditions for nonlin early unstable perturbations 17 In fact homogeneous shear ow provides no stabilizing effect for perturbations with l 0 17 The stabilization condition found in this work Eq 12 is a similar constraint for inhomogeneously sheared ow Our analysis is consistent with the numerical results described in Ref 19 in that for cases where the effect of ow shear as expressed through the nonmodal terms in Eq 6 or Eq 9 is minimal or absent as it is in the two cases examined in Ref 19 the RayleighTaylor instability grows However if the perturbation characteristics and ow shear satisfy Eq 12 all perturbations of the stream function are stable ACIQVOWLEDGEMENTS VSM wishes to thank Professor AB Hassam for draw ing attention to the paper by Benilov el al 1 S F quot 1 1 39 and quot 1 39 Stabil ity Int Ser Monographs on Physics Clarendon Press Oxford 1961 2 R D Hazeltine and J D Meiss Plasma Con nement Ser Monographs Frontiers in Physics AddisonWesleyPubl Company Reading MA 1992 3 S L Ossakow J Atmos Terr Phys 43 437 1981 4 B G Fejer and M C Kelley Rev Geophys Space Phys 18 401 1980 5 J w Miles J Fluid Mech 10 496 1961 6 H L Kuo Phys Fluids 6 195 1963 7 P N Guzdar P Satyanarayana J D Hyba and S L Ossa kow Geophys Res Lett 9 547 1982 8 K M Case Phys Fluids 3 144 1960 9 R E Kelly and S A Maslowe Stud Appl Math 49 301 1970 10 J R Booker and F P Bretherton J Fluid Mech 27 513 1967 11 S N Brown and K Stewartson J Fluid Mech 100 811 1980 12 L N Trefethen A E Trefethen S C Reddy and T A Driscoll Science 261 578 1993 13 R J Hartman J Fluid Mech 7189 1975 14 Lord Kelvin Philos Mag 24 188 1887 0263067 MIKHAILENKO SCIME AND MIKHAILENKO 15 O M Phillips The Dynamics ofthe Upper Ocean Cambridge University Press Cambridge 1966 16 A B Hassam Phys Fluids B 4 485 1992 17 V S Mikhailenko V V Mikllailenko and J Weiland Phys Plasmas 9 2891 2002 PHYSICAL REVIEW E 71 026306 2005 18 V S Mikhailenko V V Mikhailenko M F Heyn and S M Mahajan Phys Rev E 66 066409 2002 19 E S Benilov V Naulin and J Juul Rasmussen Phys Fluids 14 1674 2002 0263068 1 Review of De nition of Language Syntax CS 336 West Virginia University Lecture 1 August 22 2000 Frances L Van Scoy fvansc0ywvuedu August 22 2000 1 Review of Definition of Language Semantics Alphabets and Languages An alphabet is a nite nonempty set Let S and T be alphabets STstseSt8T We ll often write ST for S T 7L empty string string of length zero SO 7L S1 S Sn Sn391 S n gt1 SS1US2US3U S S0 U S A language L over an alphabet S is a subset of S August 22 2000 1 Review of Definition of Language Semantics How Many Languages Are There How many languages over a particular alphabet are there Uncountably in nitely many Then any nite method of describing languages can not include all of them Formal language theory gives us techniques for de ning some languages over an alphabet August 22 2000 1 Review of De nition of Language Semantics Methods for De ning Languages Grammar Rules for de ning which strings over an alphabet are in a particular language Automaton plural is automata A mathematical model of a computer which can determine whether a particular string is in the language August 22 2000 1 Review of Definition of Language 4 Semantics De nition of a Grammar A grammar G is a 4 tuple G N 2 P S Where N is an alphabet of nonterminal symbols 2 is an alphabet of terminal symbols N and 2 are disjoint S is an element of N S is the start symbol or initial m of the grammar P is a set of W of the form or gt B Where ocisinNUZNNUZ B is in N U Z August 22 2000 1 Review of Definition of Language Semantics De nition of a Language Generated by a Grammar We de ne gt by 7 0c 5 gt 7B 5 if 0L gtB is in P and yand 8 are in N U 2 gt is the transitive closure of gt gt is the re exive transitive closure of gt The language L generated by grammar G N 2 P S is de ned by LLGX Sgtxandxisin2 August 22 2000 1 Review of Definition of Language Semantics Classes of Grammars The Chomsky Hierarchy Type O Phrase Structure same as basic grammar de nition Type 1 Context Sensitive 10L gt B where 0c is in N U 2 N N U 2 B is in N U Z and length0t S lengthB 2yA 8 gt yB Bwhere Ais inN B is in NU2 and yand 5 are in N U 2 Type 2 Context Free A gt B where A is in N B is in N U 2 Linear Agt X or A gt X B y where A and B are in N and X and y are in 2 Type 3 Regular Expressions 1 left linear A gt B a or A gt a where A and B are in N and a is in Z 2 right linear A gt a B or A gt a where A and B are in N and a is in 2 August 22 2000 1 Review of Definition of Language Semantics Comments on the Chomsky Hierarchy 1 De nitions 1 and 2 for context sensitive are equivalent De nitions 1 and 2 for regular expressions are equivalent If a grammar has productions of all three of the forms described in de nitions 1 and 2 for regular expressions then it is a linear grammar Each de nition of context sensitive is a restriction on the de nition of phrase structure Every context free grammar can be converted to a context sensitive grammar with satis es de nition 2 which generates the same language except the language generated by the context sensitive grammar cannot contain the empty string M The de nition of linear grammar is a restriction on the de nition of context free The de nitions of left linear and right linear are restrictions on the de nition of linear August 22 2000 1 Review of Definition of Language 8 Semantics Comments on the Chomsky Hierarchy 2 Every language generated by a left linear grammar can be generated by a right linear grammar and every language generated by a right linear grammar can be generated by a left linear grammar Every language generated by a left linear or right linear grammar can be generated by a linear grammar Every language generated by a linear grammar can be generated by a context free grammar Let L be a language generated by a context ee grammar If L does not contain 7 then L can be generated by a context sensitive grammar If L contains 7 then Lk can be generated by a context sensitive grammar Every language generated by a context sensitive grammar can be generated by a phrase structure grammar August 22 2000 1 Review of Definition of Language 9 Semantics Type 3 Regular Expressions Right Linear Grammars Left Linear Grammars Finite State Automata August 22 2000 1 Review of Definition of Language 10 Semantics A Left Linear Grammar for SgtSa SgtSb SgtSl SgtSZ Sgta Sgtb August 22 2000 Identi ers Sgta SgtSlgta1 SgtSZgtSb2 gtSlb2gta1b2 1 Review of Definition of Language 11 Semantics A Right Linear Grammar SgtaT SgtbT Sgta Sgtb TgtaT TgtbT August 22 2000 for Identi ers T gt1T Sgta Tgt2T Tgta SgtaTgta1 Tab T91 Tgt2 SgtaTgta1T gta1bTgta1b2 1 Review of Definition of Language 12 Semantics De nition of a Deterministic Finite State Automaton A deterministic nite state automaton m is a 5tuple M Q 2 5 qO F where Q is a nite nonempty set of m 2 is the nite nonempty tape alphabet 5 Q X 2 gt Q is the transition function qO is an element of Q the initial state F is a subset of Q the set of nal states August 22 2000 1 Review of Definition of Language 13 Semantics De nition of a Language Accepted by a Deterministic Finite State Automaton De ne 5 Q X 2 gt Q by 5 qa 5qaa 5 qaaX 5 5qaaX 5 q k q for qin Q ainZ Xin2 The language L acce ted b the deterministic nite state automaton M Q 2 5 qo F is de ned by L TM X X in 2 and 5 q0 X is in F We generally use 5 and 5 interchangeably August 22 2000 1 Review of Definition of Language 14 Semantics A Deterministic Finite State Automaton for Identi ers 12 Q ab12 August 22 2000 1 Review of Definition of Language 15 Semantics qO q 1 q2 q3 C14 A Deterministic Finite State Automaton for Real Numeric Literals 1 ql ql q2 error error error q2 error error error q2 q3 error error q4 error error error q4 error August 22 2000 1 Review of Definition of Language Semantics De nition of a Nondeterministic Finite State Automaton Let PS denote the powerset of S the set of all subsets of S A nondeterministic nite state automaton is de ned the same as a deterministic nite state automaton except 5 PQ X 2 gt PQ TMXXin2 lt and 5q0X intersection F 72 I August 22 2000 1 Review of Definition of Language 17 Semantics Example of a Nondeterministic Finite State Automaton a b a ajb August 22 2000 1 Review of Definition of Language 18 Semantics Example of Conversion of a NDFSA to a DFSA August 22 2000 1 Review of Definition of Language Semantics Class 2 Context Free Grammars Pushdown Automata August 22 2000 1 Review of Definition of Language 20 Semantics A Context Free Grammar SgtE EgtET EgtET EgtT TgtTF TgtTF TgtF August 22 2000 for Expressions FgtE SgtEgtET Fgta gtETT 33 Fgtc Pugtd 9aTT gtaTFT F39gte gtaFFT gtabFT gtabcT gtabcF gtabcd 1 Review of Definition of Language 21 Semantics eX grammar G1 left recursive SgtE EgtET EgtT TgtTF TgtF FgtE F gt identi er August 22 2000 S t E t E 39 T E T F 39 F identi erd T F l identi era identifierc lt quot11 identi erb 1 Review of Definition of Language 22 Semantics eX grammar G2 right recursive S S gt E t E E gt T E i T 39 E E gt T i T gt F T F T E T gt F l T 1dent1f1era T F F gt E i i i F gt identi er F identifierc F l identi erjb identi erd August 22 2000 1 Review of Definition of Language 23 Semantics eX grammar G3 ambiguous SgtE EgtEE EgtEE EgtE E gt identi er August 22 2000 1 Review of Definition of Language Semantics 24 Ambiguity Given a language L and a grammar G such that L is generated by G if X is in L and there are two distinct parse trees of X with respect to G then grammar G is an ambiguous grammar if every grammar which generates L is ambiguous then L is an ambiguous language August 22 2000 1 Review of Definition of Language 25 Semantics An Ambiguous Context Free Language albiekjkgt 1 U ajbkokjkgt 1 isan ambiguous language We can prove that regardless of grammar the string aj lj o1 j gt 1 will always have two distinct parse trees August 22 2000 1 Review of Definition of Language 26 Semantics An Ambiguous English Sentence Time ies like a green arrow subject time verb ies subject you understood verb ies subject ies verb like August 22 2000 1 Review of Definition of Language 27 Semantics August 22 2000 eX grammar G4 right linear S gt identi er T S gt identi er T gt S T gt S 1 Review of Definition of Language Semantics 28 August 22 2000 De nition of Pushdown Automaton A pushdown automaton M is a 7 tuple M Q S T qO F to d Where Q is a nite nonempty set of states S is a nitenonemptyu set of terminal symbols or tape symbols T is a nite nonempty set of stack symbols qO in Q is the initial state F subset of Q is the set of nal states to in T d QXSXTgtQXT 1 Review of Definition of Language 29 Semantics Language Recognized by FDA TM X in 8 dq0Xt0 qt0 Where q is in F August 22 2000 1 Review of Definition of Language Semantics 30 eX Recognizing Odd Palindromes With Center Marker S gt a S 21 dq0aX qOxa S gt b S b dq0bX q0Xb S gt dq0X q1X dq1aa q1lambda dq1bb q1lambda August 22 2000 1 Review of Definition of Language 31 Semantics eX Recognizing Even Palindromes Without Center Marker S gt a S a s gt b s b S gt a a s gt b b August 22 2000 1 Review of Definition of Language Semantics 32 Determinism vs Nondeterminism Nondeterministic and deterministic nite state automata have the same power That is given a nondeterministic nite state automaton I can produce a deterministic nite state automaton which recognizes exactly the same language Nondeterministic pushdown automata have greater power than deterministic pushdown automata That is there are some nondeterministic pushdown automata for which no equivalent deterministic pushdown automaton eXists August 22 2000 1 Review of De nition of Language 33 Semantics eX Some English Palindromes Madam I39m Adam Able was I ere I saw Elba Pa39s a sap Won39t lovers revolt now A man a plan a canal Panama August 22 2000 1 Review of Definition of Language 34 Semantics Class 1 Context Sensitive Grammar Linear Bounded Automata 2Stack Pushdown Automata August 22 2000 1 Review of Definition of Language 35 Semantics eX Grammar for anbncn n gt O SgtaSBC SgtaBC aBgtab bBgtbB bCgtbc cCgtcc CBgtBC August 22 2000 SgtaBCgtabCgt abc SgtaSBCgtaaBC BCgtaabCBCgt aabBCCgtaabb CCgtaabbcCgt aabbcc 1 Review of Definition of Language 36 Semantics Some Other Context Sensitive iLanguages W W Recall W WR is linear ambncmmngt0 Recall arn bn cmn m n gt O is linear ambncmdn rnngt0 Recall arn bn cn dm m n gt O is linear and arn brn cn Cln m n gt O is context free August 22 2000 1 Review of Definition of Language 37 Semantics 2Stack Pushdown Automaton input tape memory state two stacks August 22 2000 stack stack 1 1 Review of Definition of Language Semantics stack 2 38 eX Parsing wwwinZ state qo processing characters to left of if input is any symbol except or eof push it onto stack 1 and remain in state qO if input is go to state ql if input is eof error state ql processing characters to right of if input is any symbol except or eof push it onto stack 2 and remain in state ql if input is error if input is eof go to state q2 state q2 at end of input comparing contents of two stacks if top of stack 1 top of stack 2 pop both stacks and remain in q2 if both stacks are empty success AugustZZQIlJIEI39Wlse error 1 Review of Definition of Language 39 Semantics eX Parsing 0 0 n in 2 same as previous example except everything is done nondeterrninistiely August 22 2000 1 Review of Definition of Language 40 Semantics EX Parsing ambncmdn mngtO state qO if input is a push onto stack 1 and stay in qO if input is b push onto stack 2 and go to ql state ql if input is b push onto stack 2 and stay in ql if input is c don39t erase c om input string go to q2 state q2 if stack 1 is not empty pop top of stack 1 and push value onto state 2 and stay in q2 if stack 1 is empty go to q3 state q3 if input is c and top of stack is a pop top of stack 2 and stay in q3 if input is d and top of stack is b pop top of stack 2 and go to q4 state q4 if input is d and top of stack is b pop top of stack 2 and stay in q4 if input is eof and stack 2 is empty success Otherwise error August 22 2000 1 Review of Definition of Language 41 Semantics Linear Bounded Automaton if input has 11 characters then input tape has space for k 11 characters and lba can write to tape as well as read from tape memory state 1 pushdown stack August 22 2000 1 Review of Definition of Language 42 Semantics The Determinism Question Deterministic and nondeterministic nite state automata are equivalent in power There are some languages which can be recognized by a nondeterministic pushdown automaton but not by a deterministic pushdown automaton It is an open question whether deterministic and nondeterministic linear bounded automata are equivalent in power Deterministic and nondeterministic Turing machines are equivalent in power Turing machines and phrase structure grammars define the same class of languages August 22 2000 1 Review of Definition of Language 43 Semantics Class O Phrase Structure Grammar Turing Machine August 22 2000 1 Review of Definition of Language 44 Semantics Turing Machine 0 nite memory n0 stack oneway in nite tape readwrite head on tape unit August 22 2000 1 Review of Definition of Language 45 Semantics The ChurchTuring Thesis Alonso Church and Alan Turing The algorithms implementable by a Turing machine are precisely those which are computable August 22 2000 1 Review of Definition of Language 46 Semantics Summary We can de ne a language by a grammar or automaton Different classes of languages can be de ned by restrictions on the form of the productions of the grammar Sometimes the syntactic de nition of a language has semantic implications August 22 2000 1 Review of De nition of Language Semantics 47 Quiz Construct grammars for each of the following languages aibjckd1ijk121 aj bl Ii 2 1 aibkckdil jk21 aibickdkjk21 akbkckk21 August 22 2000 1 Review of Definition of Language 48 Semantics
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