anomalous scaling of a passive scalar advected by the synthetic compressible flow abstract the field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a gaussian nonsolenoidal velocity field with finite correlation time in the presence of large scale anisotropy the energy spectrum of the velocity in the inertial range has the form e k k1 e and the correlation time at the wave number k scales as k 2 it is shown that depending on the values of the exponents e and the model exhibits various types of inertial range scaling regimes with nontrivial anomalous exponents explicit asymptotic expressions for the structure functions and other correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal independent of the anisotropy anomalous exponents calculated to the first order in e and in any space dimension these anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators the anomalous exponents depend explicitly on the degree of compressibility

a compact design scheme of adaptive output feedback control for uncertain nonlinear systems abstractmultiple uncertainties noises frequently exist in a plant usually require multiple compensation techniques which renders feedback controllers highly dynamic and nonlinear this motivates us to search for a compact design scheme of compensation to reduce the complexity of controllers in this paper global output feedback stabilisation is investigated for a class of uncertain nonlinear systems with unknown unmeasured states dependent growth and input matching uncertainty to solve the problem a compact scheme is proposed to design a global adaptive output feedback controller which combines the technique of dynamic gain and extended state observer together particularly only one dynamic gain rather than two dynamic gains is introduced to deal with the unknown polynomial of output growth rate which makes our controller to have lower dynamics than those in the related works moreover the input matching uncertainty is asymptotically estimated by the extended state observer and thus its effec

passive advection of a vector field anisotropy finite correlation time exact solution and logarithmic corrections to ordinary scaling in this work we study the generalization of the problem considered in phys rev e 91 013002 2015 to the case of finite correlation time of the environment velocity field the model describes a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow inertial range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian with finite correlation time and preassigned pair correlation function due to the presence of distinguished direction n all the multiloop diagrams in this model vanish so that the results obtained are exact the inertial range behavior of the model is described by two regimes the limits of vanishing or infinite correlation time that correspond to the two nontrivial fixed points of the rg equations their stability depends on the relation between the exponents in the energy spectrum e k 1 and the dispersion law k 2 in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale l

asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials a general doubly adaptive biased coin design is proposed for the allocation of subjects to k treatments in a clinical trial this design follows the same spirit as efron s biased coin design and applies to the cases where the desired allocation proportions are unknown but estimated sequentially strong consistency a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions for two treatments a new family of designs is proposed and shown to be less variable than both the randomized play the winner rule and the adaptive randomized design also the proposed design tends toward a randomization scheme with a fixed target proportion as the size of the experiment increases

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

anomalous scaling in the n point functions of a passive scalar a recent analysis of the four point correlation function of the passive scalar advected by a time decorrelated random flow is extended to the n point case it is shown that all stationary state inertial range correlations are dominated by homogeneous zero modes of singular operators describing their evolution we compute analytically the zero modes governing the n point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the two point function of the advecting velocity field the implications of these calculations for the dissipation correlations are discussed textcopyright 1996 the american physical society

improved epsilon expansion for three dimensional turbulence two loop renormalization near two dimensions an improved epsilon expansion in the d dimensional d 2 stochastic theory of turbulence is constructed at two loop order which incorporates the effect of pole singularities at d 2 in coefficients of the epsilon expansion of universal quantities for a proper account of the effect of these singularities two different approaches to the renormalization of the powerlike correlation function of the random force are analyzed near two dimensions by direct calculation it is shown that the approach based on the mere renormalization of the nonlocal correlation function leads to contradictions at two loop order on the other hand a two loop calculation in the renormalization scheme with the addition to the force correlation function of a local term to be renormalized instead of the nonlocal one yields consistent results in accordance with the ultraviolet uv renormalization theory the latter renormalization prescription is used for the two loop renormalization group analysis amended with partial resummation of the pole singularities near two dimensions leading to a significant improvement of the agreement with experimental results for the kolmogorov constant

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

calculation of the anomalous exponents in the rapid change model of passive scalar advection to order epsilon 3 the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the gaussian velocity field with zero mean and correlation function approximately equal to delta t t k d epsilon inertial range anomalous exponents identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients are calculated within the epsilon expansion to order epsilon 3 three loop approximation including the exponents in anisotropic sectors the main goal of the paper is to give the complete derivation of this third order result and to present and explain in detail the corresponding calculational techniques the character and convergence properties of the epsilon expansion are discussed the improved inverse epsilon expansion is proposed and the comparison with the existing nonperturbative results is given

anomalous scaling of passive scalar fields advected by the navier stokes velocity ensemble effects of strong compressibility and large scale anisotropy the field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities the density and the tracer fields advected by a random turbulent velocity field the latter is governed by the navier stokes equation for compressible fluid subject to external random force with the covariance t t k 4 d y where d is the dimension of space and y is an arbitrary exponent the original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models the corresponding renormalization group equations possess infrared attractive fixed points it is shown that various correlation functions of the scalar field its powers and gradients demonstrate anomalous scaling behavior in the inertial convective range already for small values of y the corresponding anomalous exponents identified with scaling critical dimensions of certain composite fields operators in the quantum field terminology can be systematically calculated as series in y the practical calculation is performed in the leading one loop approximation including exponents in anisotropic contributions it should be emphasized that in contrast to gaussian ensembles with finite correlation time the model and the perturbation theory presented here are manifestly galilean covariant the validity of the one loop approximation and comparison with gaussian models are briefly discussed

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

simple models of non gaussian statistics for a turbulently advected passive scalar we study the probability distribution of a passive scalar undergoing turbulent mixing in the presence of a mean scalar gradient kerstein s model which describes the turbulent mixing process as a collection of instantaneous local rearrangements of the scalar is argued to be a plausible approximation when the size of the system is much larger than the velocity correlation length and after the description has been coarse grained over a correlation volume in the physical range of parameters we find numerically that the fluctuations of the scalar ensuremath theta are close to exponentially distributed when a linear mean scalar gradient is imposed the phenomenological mean field like theory of pumir shraiman and siggia phys rev lett 66 2984 1991 is derived heuristically beginning with the kerstein model this theory predicts strictly exponential tails for the probability distribution of the scalar fluctuations p ensuremath theta i e p ensuremath theta ensuremath sim exp ensuremath vert ensuremath theta ensuremath vert we also consider a simplified version of the kerstein model which is analytically fully tractable and gives qualitatively similar results numerically under conditions of spatial homogeneity and imposed linear mean scalar gradient this simplified model and the full kerstein model are described by the same mean field theory however for the simplified model the large ensuremath vert ensuremath theta ensuremath vert asymptotic behavior of p ensuremath theta is poisson like i e p ensuremath theta ensuremath sim exp ensuremath vert ensuremath theta ensuremath vert ln ensuremath vert ensuremath theta const ensuremath vert

anomalous scaling in two models of passive scalar advection effects of anisotropy and compressibility the problem of the effects of compressibility and large scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields the advecting velocity field is gaussian delta correlated in time and scales with a positive exponent epsilon explicit inertial range expressions for the scalar correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal anomalous exponents dependent only on epsilon and alpha the compressibility parameter the complete set of anomalous exponents for the pair correlation functions is found nonperturbatively in any space dimension d using the zero mode technique for higher order correlation functions the anomalous exponents are calculated to o epsilon 2 using the renormalization group as in the incompressible case the exponents exhibit a hierarchy related to the degree of anisotropy the leading contributions to the even correlation functions are given by the exponents from the isotropic shell in agreement with the idea of restored small scale isotropy as the degree of compressibility increases the corrections become closer to the leading terms the small scale anisotropy reveals itself in the odd ratios of correlation functions the skewness factor slowly decreases going down to small scales for the incompressible case but starts to increase if alpha is large enough the higher odd dimensionless ratios hyperskewness etc increase thus signaling persistent small scale anisotropy this effect becomes more pronounced for larger values of alpha

anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time two loop approximation the renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the gaussian self similar velocity field with finite and not small correlation time the inertial range energy spectrum of the velocity is chosen in the form e k ensuremath propto k 1 ensuremath 2 ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta inertial range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions identified with anomalous exponents for ensuremath eta g ensuremath varepsilon these exponents are the same as in the rapid change limit of the model for ensuremath eta l ensuremath varepsilon they are the same as in the limit of a time independent quenched velocity field for ensuremath varepsilon ensuremath eta local turnover exponent the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter the ratio of the velocity correlation time and the scalar turnover time the nonuniversality reveals itself however only in the second order of the ensuremath varepsilon expansion and the exponents are derived to order ensuremath varepsilon 2 including anisotropic contributions it is shown that for moderate order of the structure function n and the space dimensionality d finite correlation time enhances the intermittency in comparison with both the limits the rapid change and quenched ones the situation changes when n and or d become large enough the correction to the rapid change limit due to the finite correlation time is positive that is the anomalous scaling is suppressed it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero

inverse cascade and intermittency of passive scalar in one dimensional smooth flow random advection of a lagrangian tracer scalar field ensuremath theta t x by a one dimensional spatially smooth and short correlated in time velocity field is considered scalar fluctuations are maintained by a source concentrated at the integral scale l the statistical properties of both scalar differences and the dissipation field are analytically determined exploiting the dynamical formulation of the model the gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales x ensuremath gg l these scales are shown to be excited by an inverse cascade of ensuremath theta 2 and the probability distribution function pdf of the corresponding scalar differences to approach the gaussian form as larger and larger scales are considered small scale x ensuremath ll l statistics is shown to be strongly non gaussian a collapse of scaling exponents for scalar structure functions takes place moments of order pg 1 all scale linearly independently of the order p smooth scaling x p is found for ensuremath 1lpl1 tails of the scalar difference pdf are exponential while at the center a cusped shape tends to develop when smaller and smaller ratios x l are considered the same tendency is present for the scalar gradient pdf with respect to the inverse of the p eclet number the pumping to diffusion scale ratio the tails of the latter pdf are however much more extended decaying as a stretched exponential of exponent 2 3 smaller than unity this slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale

anomalous scaling regimes of a passive scalar advected by the synthetic velocity field the field theoretic renormalization group rg is applied to the problem of a passive scalar advected by the gaussian self similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient the energy spectrum in the inertial range has the form e k ensuremath propto k 1 ensuremath ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta it is shown that depending on the values of the exponents ensuremath varepsilon and ensuremath eta the model in the inertial convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the rg equations diffusive type regimes for which the advection can be treated within ordinary perturbation theory and three nontrivial convection type regimes for which the correlation functions exhibit anomalous scaling behavior the explicit asymptotic expressions for the structure functions and other correlation functions are obtained the anomalous exponents determined by the scaling dimensions of the scalar gradients are calculated to the first order in ensuremath varepsilon and ensuremath eta in any space dimension for the first nontrivial regime the anomalous exponents are the same as in the rapid change version of the model for the second they are the same as in the model with time independent frozen velocity field in these regimes the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator for the last regime the exponents are nonuniversal they can depend also on the amplitudes however the nonuniversality can reveal itself only in the second order of the rg expansion a brief discussion of the passive advection in the non gaussian velocity field governed by the nonlinear stochastic navier stokes equation is also given

anisotropic nonperturbative zero modes for passively advected magnetic fields an analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field the velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent xi inertial range anisotropic contributions to the scaling exponents zeta j of second order magnetic correlations are associated with zero modes and have been calculated nonperturbatively for d 3 the limit xi 0 yields zeta j j 2 xi 2j 3 j 2 5j 4 2 4j 2 1 where j j or 2 is the order in the legendre polynomial decomposition applied to correlation functions conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

renormalization group operator product expansion and anomalous scaling in a model of advected passive scalar field theoretical renormalization group methods are applied to the obukhov kraichnan model of a passive scalar advected by the gaussian velocity field with the covariance mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or dangerous composite operators powers of the local dissipation rate whose negative critical dimensions determine anomalous exponents the main technical result is the calculation of the anomalous exponents in the order ensuremath varepsilon 2 of the ensuremath varepsilon expansion generalization of the results obtained to the case of a slow velocity field is also presented

turbulence with pressure anomalous scaling of a passive vector field the field theoretic renormalization group rg and the operator product expansion are applied to the model of a transverse divergence free vector quantity passively advected by the synthetic turbulent flow with a finite and not small correlation time the vector field is described by the stochastic advection diffusion equation with the most general form of the inertial nonlinearity it contains as special cases the kinematic dynamo model linearized navier stokes ns equation the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic ns equation the statistics of the advecting velocity field is gaussian with the energy spectrum e k ensuremath propto k 1 ensuremath ensuremath varepsilon and the dispersion law ensuremath omega ensuremath propto k ensuremath 2 ensuremath eta k being the momentum wave number the inertial range behavior of the model is described by seven regimes or universality classes that correspond to nontrivial fixed points of the rg equations and exhibit anomalous scaling the corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field which allows one to construct a regular perturbation expansion in ensuremath varepsilon and ensuremath eta the actual calculation is performed to the first order one loop approximation including the anisotropic sectors universality of the exponents their in dependence on the forcing effects of the large scale anisotropy compressibility and pressure are discussed in particular for all the scaling regimes the exponents obey a hierarchy related to the degree of anisotropy the more anisotropic is the contribution of a composite operator to a correlation function the faster it decays in the inertial range the relevance of these results for the real developed turbulence described by the stochastic ns equation is discussed

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field

renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the gaussian non solenoidal compressible velocity field with the covariance propto delta t t x x epsilon convective range anomalous scaling for the structure functions and various pair correlators is established and the corresponding anomalous exponents are calculated to the order epsilon 2 of the epsilon expansion these exponents are non universal as a result of the degeneracy of the rg fixed point in contrast to the case of a purely solenoidal velocity field obukhov kraichnan model the correlation functions in the case at hand exhibit nontrivial dependence on both the ir and uv characteristic scales and the anomalous scaling appears already at the level of the pair correlator the powers of the scalar field without derivatives whose critical dimensions determine the anomalous exponents exhibit multifractal behaviour the exact solution for the pair correlator is obtained it is in agreement with the result obtained within the epsilon expansion the anomalous exponents for passively advected magnetic fields are also presented in the first order of the epsilon expansion