in the last fifteen years network science has exploded , revealing a world composed by interconnected systems ubiquitously found both in natural sciences and in socioeconomic disciplines . since the very beginning of network science ,many different network representations have been adopted in order to study the particular system at hand .however , the class of networks represented by bipartite networks has been recognized to provide a particularly insightful representation of many different systems : ecological networks , trade networks , citations and collaboration networks represent only few examples.  one could thus expect a relevant amount of work aimed at identifying the statistically - relevant patterns observed in real bipartite networks , at least comparable to the mass of results obtained so far for monopartite networks : however , quite surprisingly , little work has been done so far to implement null models on real bipartite networks . generally speaking ,null models are statistical models used to make inference on a real system on the basis of partial information .the latter usually corresponds to some observable property of interest as the number of trade partners of a country , its exports and imports , the total exposure of a bank , etc .in particular , null models for bipartite networks being _ real - data rooted _ and showing the desirable features of _ general applicability _ and _ analytical character _ are currently missing .more in detail , the algorithms proposed so far show several limitations , ranging from being purely numerical ( thus lacking the analytical character ) , to assuming an _ a priori _ functional form either for the distribution of the quantities of interest or for the model parameters ( thus not being real data - rooted ) or , lastly , using approximate analytical models .moreover , almost all the aforementioned approaches are tailored on ecological networks , thus lacking the character of general applicability .the lack of such models is , maybe , also due to the misconception that bipartite networks could be analysed by , firstly , projecting them on one of the layers and , secondly , analysing the projection with one of the models currently available for monopartite networks .as we will show in what follows , the monopartite and the bipartite representations enclose different kinds of information , irreducible to each other ( in the most general case ) .the aim of the present paper is to fill this gap , proposing a theoretical framework guaranteeing the three aforementioned properties . in order to do this ,we extend a recently - proposed method to randomize monopartite networks to bipartite networks .the method rests upon the sequential maximizations of shannon entropy and the network likelihood function , a combination which has been proven to be rather effective both for detecting patterns and to reconstruct the structure of several real - world networks . to the best of our knowledge , the only other paper proposing a method satisfying the three requirements above is : we will comment on the differences with the one proposed here in the discussion section .  while the proposed formalism is perfectly general ,in this paper we apply our method to the binary , undirected , bipartite representation of the world trade web ( hereafter wtw ) .we focused on this particular system precisely because of its popularity among network scientists , who have applied null models to all its possible representations , with the exception of the bipartite one . as we will show in what follows , representing the wtw as a bipartite network allows to gain a substantially new insight into an already deeply explored system .the rest of the paper is organized as follows : data section is devoted to the description of the dataset used for the present analysis , methods section reports the detailed description of our method and results section illustrates the results which are discussed in discussion section , where conclusions are also drawn .the wtw can be represented in many different ways , depending on the level of information that we want to process .the most popular ones represent it via an adjacency matrix with nodes playing the role of world - countries and links indicating the presence of ( any kind ) of trade exchange between them .this framework has been recently extended to analyse the wtw as a multiplex , where trade exchanges corresponding to different commodities are distinguished .  herewe represent the wtw as a bipartite network , i.e. by considering the set of world - countries and the set of products as different entities and linking a given country to a given product if ( and only if ) the former exports the latter _ above _ a certain threshold ( the so - called rca ) . applying the latter rises the probability that the exported commodity is actually produced by the exporting country . in this representation , any two countries ( as well as any two products )can not be directly linked ( i.e. links connecting nodes of the same set are not allowed ) : thus , any two nodes of the same set can be still thought as `` interacting '' but only indirectly , via a connection with the same nodes of the other family .this way of representing the wtw allows us to analyze the global economy from a different perspective , by making the _ productivity relations _ between countries explicit ( i.e. _ which country produces which product _ ) .the dataset we have considered for the present analysis is the nber database , collecting data for the 38 years 1963 - 2000 and categorizing products according to the sitc revision 2 at four - digits level .data have been further processed , building upon the data - mining procedure adopted in , to produce a dataset with 538 products across all years and a number of countries varying from 130 to 151 .the distinction between countries and products leads naturally to the definition of an biadjacency matrix , which will be indicated with . in the present paper we focus on the binary , undirected representation of the wtw : thus , the matrix entries will be either , indicating that country exports an amount of product above the rca threshold , or , indicating that the production of by country is below the rca threshold and , thus , has been ignored . as a consequence ,each row represents the export basket of a given country , while each column represents the subset of producers of a given product . a pictorial representation of the wtw biadjacency matrix in the year 2000 is shown in fig .[ one ] , with the blue dots representing the ones and the white dots the zeros .if we indicate with the total number of countries and with the total number of products , the total number of elements of the biadjacency matrix ( i.e. its volume ) is , also representing the maximum observable number of connections .in fact , unlike the usual square representation , the problems arising from the presence of self - connections are not encountered here . moreover, the presence of two different subsets ( also known as _ layers _ ) induces a measure of `` rectangularity '' of our matrix  , i.e. , ranging in , with values closer to 1 indicating a large asymmetry between the number of countries and the number of products and values closer to 0 indicating equivalence between the two layers cardinality ( notice that the information on the sign would be based on the arbitrary choice of the layers ordering ) .the definitions of other topological quantities of interest easily follow from the usual ones , as the _ number of links _ ( i.e. the total number of connections )    and the _ connectance _ , measuring the percentage of observed connections .fundamental properties are represented by the number of node - specific connections , i.e. the _ degree of countries _ , also named _ diversification _ , measuring the number of products exported by each country    and the _ degree of products _ , also named _ , measuring the number of countries exporting each product    definitions [ cdeg ] and [ pdeg ] induce the notions of _ countries mean degree _ and _ products mean degree _  the last passage follows from noticing that .  in order to make the connections between nodes of the same family explicit, a bipartite network can be projected on its layers , thus recovering two traditional , monopartite representations .this operation can be straightforwardly implemented by considering the matrix products    where is the transpose of the biadjacency matrix . while the dimensions of are , the dimensions of its transpose are .this implies that results in a matrix whose generic element , with , counts the number of patterns of length two between countries and .the generic , diagonal element is precisely the degree of country .similarly , results in a matrix whose generic element , with , counts the number of patterns of length two between products and .as before , the generic , diagonal element is the degree of product .remarkably , the entries of matrices and have a clear macroeconomic interpreation : while counts the number of products shared by countries and , counts the number of countries exporting both products and .since nodes of the same layer can not be directly linked , it is enough that a path of length two ( i.e. the minimum allowed length ) connects any two nodes of the same family to directly link them in the corresponding monopartite projection .thus , by first applying the heaviside step - function  ] , where  ] , with indicating the tendency of links to connect nodes with similar degrees and indicating the tendency of links to connect nodes with different degrees .[ [ nestedness . ] ] nestedness .+ + + + + + + + + + +  on the basis of the two aforementioned measures and , one can reorder the matrix rows and columns ( i.e. countries and products ) by , respectively , decreasing the fitness along rows ( from top to bottom ) and increasing the complexity along columns ( from left to right ) , thus obtaining the triangular structure shown in fig .[ rnest ] . in order to quantify the shape of such a matrix ,several measures have been recently proposed , under the common name of _nestedness_. here we adopt the one proposed in ( called nodf - see also the supplementary information ) . notice that the measure of nestedness adopted here does nt depend on the rows and columns ordering criterion ( in what follows we will adopt the one based on and measures ) .   in order to implement suitable null models to detect the statistically - relevant patterns of real bipartite networks ,the lines of the method proposed in can be followed . in particular ,an ensemble of binary , undirected , bipartite networks must be considered , in order to maximize shannon entropy    under a given set of constraints  .notice that the probability coefficient is assigned to every adjacency matrices in the esemble and the constraints are defined in terms of the entries of .the result is the well - known exponential distribution :    with the hamiltonian compactly expressing the imposed set of constraints , being the vector of lagrange multipliers associated to the vector of constraints and being the normalization .  in the monopartite case ,one of the most insightful null models has been proven to be the so - called configuration model ( cm ) .let us now implement the bipartite extension of the cm ( bicm , in what follows ) , by constraining the degree sequence of the binary , undirected , bipartite wtw and analyzing the system beyond the information contained into it .since now we have two different layers of nodes , the hamiltonian reads    now we can calculate the probability coefficient ( [ probcoef ] ) , associating a probability to each network in the ensemble on the basis of the specific degree sequences and :    the notation being equivalent to ( see the supplementary information for the detailed calculations ) .the coefficient , with and , is the ensemble probability of having a link between country and product , as .our null model provides the analytical expression of a network probability as a product over all the accessible pairs of nodes .in other words , the bicm interprets the links as independent random variables , thus defining a grandcanonical probability measure where links correlations are discarded .notice also that no probability coefficients controlling for the presence of links between nodes in the same layer appear in the expression ( [ probcoef2 ] ) .this is a consequence of having considered an _ ensemble of bipartite networks _ as the support of our probability distribution : in so doing , the forbidden intra - layer links are automatically excluded by the choice of the allowable configurations volume .the probability distribution in ( [ probcoef2 ] ) depends on unknown parameters ( i.e. the lagrange multipliers ) , also called_ hidden variables _ .the recipe provided by statistical mechanics to estimate the hidden variables is summed up by the equations    however , no indication about the numerical value to be assigned to the ensemble average of constraints is provided .thus , in order to estimate the hidden variables from data , let us first note that can be rewritten solely in terms of the observed constraints value , i.e.  .then , let us consider the log - likelihood function :    the recipe provided by statistics to estimate the unknown parameters of a given probability distribution prescribes to maximize  .this means solving the system of equations in unknowns :    in what follows the vector of solutions satisfying the system ( [ sys ] ) , for given and as degree mean values , will be indicated as .notice that the coefficients appearing at the second member of the system equations have the same functional form both for countries and products .this is a consequence of assigning only one lagrange multiplier to each node but in such a way to distinguish the nodes in the first layer from the nodes in the second layer .   in the previous subsectionsseveral quantities of interest to be measured on binary , undirected , bipartite networks have been listed . in this subsectionwe will show how our method can be implemented to calculate their expected value ( to be compared with the observed one ) and the relative errors ( to quantify the discrepancies ) in order to assess up to what level our null model is able to explain the higher - order structure of the network .our method allows us to proceed in a two - fold way .the first one is analytical . using the link - specific probability coefficients and the passages sketched in , we are able to analytically calculate both the expected value and the standard deviation of the ( analytically - definable ) quantities of the previous subsections . however , because of the impossibility to perform analytical evaluation of the average for some key quantities , we have adopted a different strategy : we have sampled the grancanonical ensemble of binary , undirected , bipartite networks induced by the bicm according to the probability coefficients , measured the aforementioned properties on our sample and calculated the statistical moments , as average and standard deviation , of the generic quantity as    i.e. as sampling moments according to the sampling frequencies ( being the number of networks in the ensemble having biadjacency matrix equal to ) .since our method is unbiased , numerically sampling provides a faithful representation of the whole ensemble .we have also calculated the probability distribution ( induced by ) of some of the properties of interest , in order to quantify the statistical significance of their observed value ( via the -score , for example ) .nevertheless , the analytical expressions of the expected value and standard deviation of the quantities explicitly defined in the previous subsections has been derived in the supplementary information .let us first show our results on the temporal snapshot of the wtw corresponding to the year 2000 .the number of nodes is and , causing the index to be ( see section methods ) .the high asymmetry of our network is also pointed out by the different mean degrees , and , indicating that countries are , on average , almost three times more connected than products .however , the connectance is : thus , our bipartite wtw is much sparser than its monopartite counterpart .notice that our null model , constraining ( on average ) the degree sequence , exactly reproduces any network s connectance by definition , spanning the domain of applicability of both the sparse and the dense network reconstruction algorithms .[ knn2000 ] shows the comparison between observed and expected values of our coefficients of assortativity .having plotted vs and vs , we firstly observe that the bipartite wtw shows a _ disassortative _ behavior , signalled by a globally decreasing trend of our measures .more detailedly , two distinct behaviors seem to characterize as a function of : while countries with _ low diversification _ are preferentially linked to products with _ high ubiquity _( left side of panels [ knn2000]a and [ knn2000]b ) , countries with _ high diversification _ are linked to _ almost all products _( right side of panels [ knn2000]a and [ knn2000]b ) .this is also reflected in the triangular structure of the matrix ( see fig . [ one ] ) . for products ,this distinction is less sharp ( panels [ knn2000]c and [ knn2000]d ) : in fact , while _ high - ubiquity _ products are linked to _almost all countries _ , _ low - ubiquity _ products can be found connected to _ both high- _ and _ low - diversification countries_.    as can be seen from fig .[ knn2000 ] , the bicm captures the disassortative behavior of both and ; however , only part of the observed points lies within the standard deviations region .this means that the mechanism shaping the disassortative behavior of the wtw is not completely explained by our null model , signalling a non - trivial origin of the wtw degree correlations .what is strikingly surprising is the prediction based on the random graph model ( birg ) : the corresponding trend is closer to the bicm prediction than in the monopartite representation of the wtw .moreover , since disassortativity is more pronounced in real data , our results indicate that the bicm performs better than birg for small values of and , while the birg correctly capture their flat behavior at large and ( i.e. for competitive countries and ubiquitous products , for which , ) .this seems to indicate that the explanatory power of the degree sequence is far more limited in the bipartite representation than in the monopartite one and that additional information is required to improve the agreement between observations and predictions ( even at the simplest level of binary , undirected networks ) .[ knnevot ] extends our assortativity analysis to the entire dataset . in order to condensate the information of 38 scatter plots , we have computed the barycenter and sparseness of both the observed and expected clouds of points . in particular , we have calculated the arithmetic mean of both the observed values     and , the expected values and and the corresponding confidence intervals ( ci ) at 95% level . as for the motifs ,also and can be interpreted in macroeconomic terms .in fact , measures the country - specific number of competitions , thus quantifying the ( average ) presence of a country on the global market .further averaging over all countries provides a measure of the integration of world - countries production .what emerges is that the evolution of expected points closely follows the evolution of the observed ones , pointing out that the bicm correctly describes the temporal trend of the assortativity measures .notice that , even if observed points are systematically more concentrated on higher levels ( as shown in panels [ knnevot]a and [ knnevot]c ) , the confidence intervals are still close enough to let us interpret the bicm predictions as correct .moreover , the constancy of the amplitude of the confidence intervals for both observed and expected anpu values indicates that the corresponding clouds of points maintain the same sparseness across our 38 years dataset ; on the other hand , the amplitude of the observed ancd confidence intervals slightly reduces , indicating a shrinkage of the corresponding cloud of points ( compare panels [ knn2000]b and [ knn2000]d ) .  the temporal trends of and interesting differences .in fact , while keeps increasing across the whole dataset , does not ( and from 1975 starts decreasing ) . since the countries mean degree keeps rising as well ( and ) , the increasing trend is probably due to the birth of new links , indicating that while existing countries have enlarged their production , new - born countries have started theirs .the results seem also to be compatible with the picture of several `` appealing '' products behaving as hubs and attracting links , including the ones of the new - born countries which in turn , having a low degree , reduce the value of the .since ranges in the interval  ] ) .remarkably , our index can be now interpreted a `` genuine '' measure of globalization , not affected by any spurious effect .very interestingly , the temporal trend of after 1970 becomes now almost flat .this means that the wtw evolution does not actually affect the value of countries integration which organize in such a way to maintain the same value of , irrespectively of the rising number of countries , their higher diversification , etc .this seems to confirm the stationary evolution of such network , recently pointed out .a similar reasoning leads us to interpret as a measure of products homogeneity .we have also calculated the pearson correlation coefficient between the vectors and ( panel [ knnevot]b ) and between the vectors and ( panel [ knnevot]d ) , in order to quantify the agreement on the `` shape '' of the clouds of points .the correlation of the latter is lower than the correlation of the former : this is due to the shape of the empirical cloud of ancd which is less linear than the empirical anpu , thus worsening the agreement with the corresponding expectations ( which show an almost perfectly linear trend ) .complexity and fitness can be obtained only numerically , as the result of the convergence of the algorithm proposed in .panels [ qf2000]a and [ qf2000]b show the comparison between observed and expected complexity ( plotted vs ubiquity ) for the years 1963 and 2000 ; panels [ qf2000]c and [ qf2000]d show the comparison between observed and expected fitness ( plotted vs diversification ) for the same years . our null model capture both trends with a larger accuracy than in the measure of assortativity : notice how the expected trend under the bicm reproduces the `` beak '' of the observed complexity in real data and the vast majority of the observed cloud lies within the standard deviations region .similarly , the expected trend of reconstructed fitness captures the different growth regimes of the observed fitness in the wtw data , showing few sparse points outside the same error region ( clearly visible in the log - log plots of fig .[ qf2000 ] ) .the regime with lower slope ( left side of panels [ qf2000]e and [ qf2000]f ) represents the so - called `` poverty trap '' , i.e. the area populated by the group of countries with lowest fitness : notice how all such countries lie within the standard deviation region ( or immediately outside ) .similar considerations hold for all the remaining years , indicating a constant performance of our method across our 38-years dataset .  the average trends in fig .[ qf2000 ] are computed differently from those in fig .[ knn2000 ] : while the latter represent the node - specific , ensemble averages and , the former represent averages taken over ranked nodes , ordered according to their complexity - panels a and b - and fitness - panels c and d. generally speaking , ordering nodes on the basis of such procedure will produce a different ranking for different bipartite networks of the ensemble. moreover , the ranking operation guarantees neither that the identity of ranked nodes remains the same ( e.g. two different countries can be ranked first for two different networks ) , nor that the corresponding complexity and fitness maintain their value across our sample ( i.e. the nodes ranked first will , in general , have different values of and ) : this in turn implies that each ranked node degree may change as well ( i.e. the nodes ranked first for different networks will , in general , have different degrees ) . from these considerations , the need of quantifying 1 ) the variation of any country diversification as a function of its fitness and 2 ) the variation of any product ubiquity as a function of its complexity follows .this is in line with the spirit of the research in : trying to establish a biunivocal relation both between ubiquity and complexity and between fitness and diversification , in order to unambiguously rank countries and products .this kind of analysis represents a highly non - trivial test bench of our model which appear to perform very well .the motifs analysis has been carried on by calculating two different quantities .the first one has been defined as    and named _ similarity _ : it quantifies the goodness of our prediction , measuring the difference between the observed and expected abundances . beside similarity, we have also considered the traditional -scores , defined as the ratio of the difference between the observed and expected abundances and the corresponding standard deviation    with and indicating the particular motif considered .even if -scores have been recognized to be dependent on the network size ( at least for monopartite networks ) , our dataset collects matrices with very similar volume (  ] .this allows us to calculate the difference between and simply as the total number of links variance    where . in order to calculate the standard deviation , we use eq .( [ motsigma ] ) to find    and finally obtain    the same procedure can be applied to all the motifs belonging to the v and families .more explicitly , in the cases and the following results hold :    and  }{\sqrt{\sum_{p=1}^p(4u_p^3 - 18u_p^2 + 22u_p-6)^2\sigma^2_{u_p}}}.\ ] ]  we have also tested the agreement between the analytical expressions of the aforementioned -scores and the values obtained by explicitly sampling the grandcanonical ensemble induced by the cm . as fig .[ zed ] shows , despite the presence of two approximations , our analytical estimates work quite satisfactorily .  figs .[ mot4 ] and [ mot5 ] show the analysis of the x , m and w - motifs .as for the other motifs previously considered , the three distributions follow a gaussian very closely , whose mean and variance have been calculated on the networks sample ( 5000 matrices ) .again , this can be ascribed to the ( generalized ) central limit theorem . as a general remark , the three ,most complex motifs show higher fluctuations and are less accurately reproduced than the simpler ones ( i.e. v and ) .  , and abundance in the year 2000 ( fits are obtained by superimposing normal distributions with the sample average and variance ; red points represent the observed motifs abundance).,scaledwidth=49.0% ]  -scores and similarity evolution across our database years of ( ) , ( ) , ( ).,scaledwidth=49.0% ]  beside having provided a simple expression for the -scores of the v and families of motifs , we have also shown that the latter may have a definite sign ( negative , in our case ) : while quite surprising , in cases like this -scores might still be used to test the agreement between observations and predictions but should be considered _one - sided _ statistical tests of significance .this implies that the values enclosing the probabilities of , and no more coincide with , and , because no more computable on both tails of the reference gaussian distribution .the right values for one - sided tests are , enclosing a probability of , and , enclosing a probability of  .the three more complex motifs ( x - motifs , m - motifs and w - motifs ) do not have a definite sign .the reason for the sign definiteness lies in the explicit dependence of the quantities of interest from the chosen constraints , as shown below .let us consider the taylor expansion of a quantity of interest around the expected value :    after calculating , the expression can be rewritten as    whenever represents a given , chosen constraint whose expected value on the ensemble is , by definition , equal to the observed one , we obtain    now , if higher - order moments can be ignored or the function is quadratic in , the right hand side of the above equation is proportional to the numerator of the function -score , whose sign is negative .a simple example is provided by the function , with : we obtain , as it should . in the v - 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Abstract: within the last fifteen years , network theory has been successfully applied both to natural sciences and to socioeconomic disciplines . in particular , bipartite networks have been recognized to provide a particularly insightful representation of many systems , ranging from mutualistic networks in ecology to trade networks in economy , whence the need of a pattern detection - oriented analysis in order to identify statistically - significant structural properties . such an analysis rests upon the definition of suitable null models , i.e. upon the choice of the portion of network structure to be preserved while randomizing everything else . however , quite surprisingly , little work has been done so far to define null models for real bipartite networks . the aim of the present work is to fill this gap , extending a recently - proposed method to randomize monopartite networks to bipartite networks . while the proposed formalism is perfectly general , we apply our method to the binary , undirected , bipartite representation of the world trade web , comparing the observed values of a number of structural quantities of interest with the expected ones , calculated via our randomization procedure . interestingly , the behavior of the world trade web in this new representation is strongly different from the monopartite analogue , showing highly non - trivial patterns of self - organization .


the presence of turbulence impacts the evolution of a wide variety of plasma environments , from galaxy clusters to accretion disks around compact objects , to the solar corona and solar wind , and to the laboratory plasmas of the magnetic confinement fusion program .establishing a thorough understanding of plasma turbulence is a grand challenge that has the potential to impact this wide range of research frontiers in plasma physics , space physics , and astrophysics .ultimately , such efforts are aimed at developing the capability to predict the evolution of any turbulent plasma system .the _ quasilinear premise _ is a hypothesis for the modeling of plasma turbulence with the potential to lead to a quantitative , predictive theory of plasma turbulence .the quasilinear premise states simply that _ some _ characteristics of the turbulent fluctuations in a magnetized plasma may be usefully modeled by a superposition of randomly - phased , linear wave modes .the nonlinear interactions inherent to the turbulent dynamics may be considered to transfer energy among these linear wave modes therefore , the model is quasilinear .this premise is hotly debated at present , with significant questions raised by the heliospheric physics community about the validity of using the theory of linear plasma waves to analyze and interpret the turbulent fluctuations measured in the solar wind plasma .on one hand , a large body of work on plasma turbulence either explicitly or implicitly assumes the relevance of some linear plasma wave properties . on the other hand ,the nonlinearity inherent in turbulent interactions raises obvious questions about the relevance of linear theory .  in this paper , we define precisely the concepts encapsulated by the quasilinear premise and identify the limitations of such an approach .we outline the theoretical arguments that justify the application of linear plasma wave theory to the study of plasma turbulence , and review supporting and conflicting evidence from theory , simulation , and observation .the quasilinear premise proposes a fundamental picture of plasma turbulence in which the turbulent fluctuations are represented by a superposition of randomly - phased linear wave modes , and energy is transferred among these wave modes via nonlinear interactions .although this simple model can not capture all of the known characteristics of turbulence , it does provide a foundation upon which a quantitatively predictive model of turbulent nonlinear energy transfer and plasma heating may be constructed .below we describe the features of a model for turbulence in a magnetized plasma based on the quasilinear premise , focusing in particular on the aspects of turbulence that can and can not be described by such an approach .several important properties of plasma turbulence are described by a model adopting the quasilinear premise .first , and most important , is that the characteristic eigenfunctions of the turbulent fluctuations ( the amplitude and phase relationships among the electric , magnetic , velocity , and density fluctuations of a single spatial fourier mode ) are given by the linear wave eigenfunctions .second , in a weakly collisional plasma such as the solar wind , the fluctuations associated with each wave mode are damped at the appropriate instantaneous collisionless damping rate given by linear kinetic theory .third , the nonlinear energy transfer is described by a phenomenological prescription derived from modern theories for anisotropic plasma turbulence .finally , the distribution of the power of the turbulent fluctuations in three - dimensional wavevector space is guided by theory and numerical simulations of plasma turbulence , and is chosen to satisfy key constraints , such as the observed wavenumber spectrum of magnetic energy .thus , the quasilinear premise is a marriage of the quantitative linear physics of waves in a kinetic plasma with a phenomenological prescription for the nonlinear turbulent dynamics given by modern turbulence theories .it is important to note that the quasilinear premise is not the same as _ quasilinear theory _ in plasma physics , the rigorous application of perturbation theory to explore the long - time evolution of weakly nonlinear systems .rather , the quasilinear premise employs a phenomenological prescription for the nonlinear energy transfer in strong turbulence , rather than calculating it rigorously from first principles . as described below in [ sec : theory ] , the mathematical properties of the equations that describe turbulence in a magnetized plasma , in conjunction with a phenomenological understanding of the properties of the turbulence , provide the motivation for this quasilinear approach .  a model based on the quasilinear premisecan be used to explore the second - order statistical properties of plasma turbulence , such as the energy spectrum of the turbulence or the correlations between two different fields , for example the cross correlation between the density and parallel magnetic field fluctuations .the _ synthetic spacecraft data method _ is an application of the quasilinear premise for the interpretation of spacecraft measurements of plasma turbulence . in this method ,a synthetic plasma volume is filled with turbulent fluctuations in the form of a distribution of randomly - phased , linear wave modes .this synthetic plasma volume is then sampled along a single trajectory , generating single - point time series of turbulent plasma and electromagnetic fluctuations that may be analyzed with the same procedures as spacecraft measurements .a large ensemble of such synthetic data sets has proven useful in gaining new understanding of the compressible fluctuations in the inertial range of solar wind turbulence and of the magnetic helicity of solar wind turbulent fluctuations as a function of the angle between the solar wind flow and the local mean magnetic field .in addition , one may also use the quasilinear premise to construct a turbulent cascade model to predict the nonlinear transfer of energy from large to small scales , enabling predictions of the resulting energy spectra of the turbulent fields and of the plasma heating resulting from the dissipation of the turbulent energy via kinetic mechanisms at small scales .since the nonlinear interactions under the quasilinear premise are not computed from the nonlinear terms in the governing equations , but rather are given by a phenomenological prescription , the third - order and higher order correlations observed in solar wind turbulence can not be investigated using a turbulence model based on the quasilinear premise .such higher order statistics depend critically on the phase relationships between different linear wave modes , and these phase relationships are determined by the nonlinear interactions responsible for the turbulent cascade of energy from large to small scales . for a collection of randomly phased linear waves , such higher order statistics will average to zero , yielding no useful information .in other words , the random phases between the constituent wave modes can not capture the intermittency and coherent structures , such as current sheets , that are observed in plasma turbulence simulations and inferred from observations of solar wind turbulence .in addition , inherently nonlinear fluctuations fluctuations that can not be expressed as a superposition of linear eigenfunctions cannot be described by a model based on the quasilinear premise . such an inherently nonlinear fluctuation has recently been derived in the asymptotic analytical solution of the nonlinear interaction between counterpropagating alfvn waves ; this mode has been shown to play an important role in the nonlinear transfer of alfvn wave energy to smaller scales . however , if a phenomenological picture can be devised to describe the distribution of such nonlinear modes , such as that suggested by , it may be possible to produce refined turbulence model by extending the quasilinear premise to incorporate such modes .in addition to specifying which properties of the turbulence this approach can and can not describe , it is also important to state which conditions are necessary , and which are not , for the quasilinear premise to be applicable .first , the quasilinear premise is expected to become valid at length scales significantly smaller than the scale of turbulent energy injection , where the amplitude of the magnetic field fluctuations becomes small relative to the local mean magnetic field , .it remains valid at all smaller scales , including the scales at which dissipation mechanisms act to terminate the cascade .second , the application of the quasilinear premise requires both guidance from turbulence theories and constraints from simulations and observations .the most important element required from turbulence theory is the prescription for the nonlinear transfer of energy among wave modes that underlies the turbulent cascade from large to small scales .third , the division of turbulent power among the possible linear wave modes , and the possibly anisotropic distribution in wavevector space of the turbulent power for each wave mode , also must be guided by theoretical considerations as well as numerical and observational constraints .the amplitude of the spectrum of linear modes as a function of wavenumber can be specified such that the scaling of the turbulent power in the model is consistent with the observed turbulent power spectrum .finally , we emphasize that the fruitful investigation of the nature of plasma turbulence using the quasilinear premise does _ not _ require the turbulent fluctuations derived from the linear wave mode properties to persist for many wave periods , nor does it require evidence of a `` linear dispersion relation '' to be apparent in the measured turbulent power spectrum , as will discussed in more detail in [ sec : theory ] below .the quasilinear premise , as defined above , is motivated by the mathematical properties of the equations that describe turbulence in a magnetized plasma .the key concepts are most easily explained for the  simplified case of turbulence in an incompressible mhd plasma , the minimal model containing the essential physics of plasma turbulence , specifically the development an anisotropic alfvnic cascade and the generation of current sheets at the smallest resolved scales .it is important to note , however , that the qualitative properties illustrated by the case of incompressible mhd turbulence are expected to persist under more general plasma conditions , specifically carrying over to the kinetic plasma physics describing the dissipation range of turbulence in the weakly collisional solar wind .the equations of ideal incompressible mhd can be expressed in the symmetric form , where the magnetic field is decomposed into equilibrium and fluctuating parts , is the alfvn velocity due to the equilibrium field , is total pressure ( thermal plus magnetic ) , is mass density , and are the elssser fields given by the sum and difference of the velocity fluctuation and the magnetic field fluctuation expressed in velocity units .the elssser field ( ) represents either the alfvn or pseudo - alfvn wave traveling down ( up ) the equilibrium magnetic field .the second term on the left - hand side of equation ( [ eq : elsasserpm ] ) is the linear term representing the propagation of the elssser fields along the mean magnetic field at the alfvn speed , the first term on the right - hand side is the nonlinear term representing the interaction between counterpropagating waves , and the second term on the right - hand side is a nonlinear term that ensures incompressibility .two properties of incompressible mhd turbulence that support the relevance of linear wave properties to plasma turbulence are evident upon inspection of the form of equation ( [ eq : elsasserpm ] ) .first , as initially recognized in the 1960s , if two waves are propagating in the same direction along the mean field ( yielding either or ) , the nonlinear interaction is zero . in this case , an arbitrary waveform of finite amplitude provides an exact nonlinear solution to equation ( [ eq : elsasserpm ] ) .therefore , turbulent cascade of energy from large to small scales in an incompressible mhd plasma is caused by the nonlinear interaction between counterpropagating alfvn waves .the physics of this fundamental building block of astrophysical plasma turbulence has recently been solved analytically , validated numerically using gyrokinetic simulations , and verified experimentally in the laboratory .second , the strength of the nonlinearity may be quantified by the magnitude of the nonlinear term relative to the linear term , denoted by the _ nonlinearity parameter _ therefore , in the limit of weak nonlinearity , , it is possible to identify a regime of weak mhd turbulence , motivating a quasilinear approach using perturbation theory .note that , in the case of incompressible hydrodynamic turbulence ( euler or navier - stokes ) , the absence of a linear term disallows the possibility of a perturbative approach , a fundamental distinction between turbulent hydrodynamic and magnetohydrodynamic systems .  in the absence of the nonlinear terms( setting the right - hand side of equation ( [ eq : elsasserpm ] ) to zero ) , the behavior of the plasma is entirely determined by the linear term .if the right - hand side of the equation is considered to be an arbitrary perturbing source term , the linear term determines the instantaneous response of the plasma to the imposed perturbation . in the case of weak turbulence ,  ,the nonlinear terms on the right - hand of equation ( [ eq : elsasserpm ] ) are indeed a small perturbation to the linear system , representing the nonlinear transfer of energy among the linear wave modes .perturbation theory may be applied to the study of the turbulent dynamics in this limit , so the quasilinear premise is clearly valid for the case of weak turbulence .but the turbulent magnetic fluctuations at large scales in the solar wind typically satisfy , so , at the scale of energy injection , solar wind turbulence is believed to be in a state of strong turbulence , .the theory of strong incompressible mhd turbulence suggests that the turbulent fluctuations at small scales become anisotropic , where the nonlinear cascade of energy generates turbulent fluctuations with smaller scales in the perpendicular direction than in the parallel direction , .this inherent anisotropy of magnetized plasma turbulence has long been recognized in laboratory plasmas and in the solar wind , as well as in early numerical simulations .it has been conjectured that strong turbulence in incompressible mhd plasmas maintains a state of _ critical balance _ between the linear timescale for alfvn waves and the nonlinear timescale of turbulent energy transfer .therefore , the nonlinearity parameter is expected to maintain a value of , so the linear term and nonlinear term in equation ( [ eq : elsasserpm ] ) are of the same order .this property of strong plasma turbulence motivates the quasilinear premise .  for strong turbulence, the condition of critical balance implies that the energy in a particular linear wave mode may be transferred nonlinearly to other modes on the timescale of its linear wave period .but since the linear and nonlinear terms are of the same order in critical balance , the linear term still contributes significantly to the instantaneous response of the plasma , even in the presence of strong nonlinearity .therefore , the fluctuations in a strongly turbulent magnetized plasma are expected to retain at least some of the properties of the linear wave modes . in particular , for a turbulent fluctuation with a given wavevector , the amplitude and phase relationships between different components of that fluctuation are likely to be related to linear eigenfunctions of the characteristic plasma wave modes . and, the frequency response of the plasma to a turbulent fluctuation is likely to be similar to the linear wave frequency , at least to lowest order , the key concept used in two recent papers used to evaluate the validity of applying the taylor hypothesis to linear kinetic wave modes in the solar wind plasma and to determine qualitatively and quantitatively the effect of the violation of the taylor hypothesis on the magnetic energy frequency spectrum measured _ in situ _ by spacecraft .it is important to address here a common misconception that plasma turbulence with small amplitude fluctuations , , is necessarily weak turbulence . at small scales in the inertial range , and throughout the dissipation range , turbulent fluctuations of the magnetic field satisfy the condition .the small amplitude of the fluctuations has lead many researchers to conclude that the turbulent interaction must necessarily be weak .however , as can be seen from the inspection of equation ( [ eq : nlparam ] ) , the ratio of the nonlinear to the linear term in the equation is given by .therefore , if the turbulent fluctuations are sufficiently anisotropic , , then the nonlinearity parameter may easily yield the condition of strong turbulence , , even when fluctuations have a small amplitude , .therefore , without knowledge of the anisotropy of the turbulent fluctuations , it is incorrect to conclude that turbulence is weak simply because the fluctuation amplitudes are small .  given that the idea of critical balance in strong plasma turbulence is essentially a quasilinear concept that the frequency of the nonlinear energy transfer remains of the same order as the _ linear _ wave frequency evidence in support of critical balance indirectly supports the quasilinear premise .although the concept of critical balance was initially applied only to the case of mhd turbulence , the theory has been extended to the dispersive wave regime of the dissipation range .evidence in support of or in conflict with the conjecture of critical balance from numerical simulations and solar wind observations is reviewed in [ sec : num ] and [ sec : obs ] below .the simple physical example of a damped simple harmonic oscillator illustrates the concept , central to the quasilinear premise , that the linear terms of an equation contribute significantly to the evolution of the system , even in the absence of oscillatory ( or wave - like ) behavior .the equation of evolution for a damped harmonic oscillator is given by where is the undamped frequency of the harmonic oscillator and is a frictional damping coefficient .the dimensionless parameter that characterizes the strength of the damping relative to the linear restoring term responsible for oscillatory behavior is .this term is analogous to the nonlinearity parameter in plasma turbulence which characterizes the strength of the nonlinear to the linear term , given by equation ( [ eq : nlparam ] ) .the system is underdamped for , critically damped for , and overdamped for .  specifying the initial conditions at to be and , we consider the evolution of the system in time , as shown in panel ( a ) of figure [ fig : dsho ] . in the underdamped case ( dashed ), the oscillatory behavior of the system is evident .for either critically damped case ( solid ) or overdamped case ( dotted ) , the system does not demonstrate oscillatory behavior . in either of these non - oscillatory cases, however , the linear restoring term still plays an important role in governing the evolution of the system .  to  if the restoring term is eliminated by setting in equation ( [ eq : dsho ] ) , the initial velocity ( not shown ) monotonically decreases to zero and the position asymptotes to a value , as shown in panel ( b ) of figure [ fig : dsho ] .but when the linear restoring term is present , the evolution is qualitatively different , as seen by comparing the left and right panels of figure [ fig : dsho ] .for the critically damped case ( right panel , solid ) , the position reaches maximum value before returning monotonically to zero , while the velocity ( not shown ) decreases from the initial velocity to a minimum , before returning monotonically to zero . therefore , even though the damped simple harmonic oscillator for exhibits no oscillatory behavior , the presence of the linear restoring term still significantly influences the evolution of the system .  in a turbulent magnetized plasma, the balance is between the nonlinear and linear response terms , rather than linear damping and linear restoring terms , but the lesson is analogous : even in a strongly turbulent plasma in which the nonlinear term is of the same order as the linear term , the linear terms continue to play a significant role in governing the turbulent dynamics .the phenomenon of critical damping in this harmonic oscillator system is analogous to the concept of critical balance in plasma turbulence .it is important to point out that this analogy does _ not _ carry over to the case of hydrodynamic turbulence the absence of a linear wave term in the governing equations precludes the identification of an equivalent critical parameter .this highlights a fundamental difference in the nature of hydrodynamic and plasma turbulence .the concept central to the quasilinear premise that the turbulence consists of a broadband spectrum of alfvn waves is frequently criticized with the argument that , if waves are indeed present in the turbulence , one should be able to see clear evidence of the linear dispersion relation ( meaning a plot of vs.  ) in the analysis of turbulence simulations .we argue here that there are two good reasons why one should not necessarily be able to see clearly such a dispersion relation .the first reason is related to the semantic difference between an alfvn wave and an alfvnic fluctuation .the second reason is that changes in the amplitude or phase of a wave with a constant frequency will significantly broaden the frequency content determined by a fourier transform of a time series .  in the weak turbulence limit , , it requires nonlinear interactions with many counterpropagating alfvn waves before the bulk of the energy of a given alfvn wave is cascaded to a smaller scale . in this limit , therefore , it is likely that the signature of wave - like motions may be relatively easy to observe .but for a case of strong turbulence , in which the nonlinearity parameter , the energy of an alfvn wave is completely cascaded to smaller scale through a collision with a single counterpropagating alfvn wave , so that the timescale for nonlinear energy transfer is the same order as the linear wave period . in this case , if the interacting turbulent fluctuations persist for only a single wave period , does it make sense to refer to the interacting fluctuations as waves ?in our view , this point is largely semantic , for the following reasons .  in the simplified context of incompressible mhd , any finite amplitude fluctuation with and ( or with and ) , is an exact nonlinear solution of the equations of evolution , corresponding to an arbitrary waveform propagating unchanged in one direction along the mean magnetic field at the alfvn velocity .this solution is clearly a finite amplitude alfvn wave , but it need not oscillate sinusoidally as one typically envisions when discussing a wave - like motion .a fourier transform of the time series measured at a fixed eulerian point as the wave passes by may yield a broad frequency response ; for example , an isolated `` wavepacket '' consisting of a single parallel wavelength of a perfectly sinusoidal signal will indeed return a frequency response that is significantly broadened about the `` linear '' frequency .some may prefer the more general term `` alfvnic fluctuation '' in place of `` alfvn wave '' in this case , be the difference is merely semantic .that an alfvn wave may not be purely sinusoidal , and thus not have a well defined frequency , does not change the nature of the fluctuation , which is determined by the linear response of the plasma to an applied perturbation , as dictated by the linear term in the equation of motion equation ( [ eq : elsasserpm ] ) .it is in this sense that we speak of turbulence supported by alfvn waves .therefore , we hereby clarify that our use of the term alfvn wave pertains to any fluctuation whose linear response is the same as that of an alfvn wave thus , we refer to any alfvn fluctuation as an alfvn wave .the second reason that evidence of a linear dispersion may not be apparent in measurements or simulations of strong plasma turbulence , even for turbulence consisting precisely of a sum of linear wave modes , is that changes in the amplitude or phase of a wave will broaden the frequency response substantially , as shown in figure [ fig : phase_amp ] .here we plot the frequency retrieved from a fourier transform of the time series for a pure sinusoidal signal ( black ) , a sinusoidal signal with linearly increasing amplitude ( red ) , and a sinusoidal signal with shifting phase ( blue ) .it is clear that the frequency response when the amplitude or phase of a mode changes , one obtains a very broadened response in frequency , even when the basic signal has a well - defined frequency .therefore , it is likely to be very difficult to recover a clean vs. dispersion relation , even for turbulence simulation data .  to make matters more complicated , for spacecraft measurements, the presence of a broad spectrum of alfvn wave power in three - dimensional wavevector space further complicates the interpretation of a time series of measurements at a fixed point in space in terms of the frequency of the underlying fluctuations .it is worthwhile to raise one final point regarding the frequency spectrum of the fluctuations in plasma turbulence .one may view the dynamical evolution of plasma according to equation ( [ eq : elsasserpm ] ) as the linear response of the plasma ( given by the left - hand side of the equation ) to perturbations generated by the nonlinear terms ( given by the right - hand side of the equation ) . for a particular wavevector , the characteristic frequency of the linear plasma responseis given by , where the component of the wavevector parallel to the mean magnetic field is .as in the case of a driven , damped single harmonic oscillator , one may obtain a response at a frequency only by sustained driving at another frequency .but since the nonlinear terms perturbing a given wavevector mode are dominated by the interactions between nearby wavevectors , the frequency driving a particular spatial fourier mode in plasma turbulence is likely to be very near the linear frequency of the mode being driven .the power generated by the nonlinear interaction between two counterpropagating alfvn waves can not have a frequency higher than the sum of the frequency of those two modes , so the turbulent power will not contain frequencies significantly higher than the characteristic linear alfvn wave frequency for a particular wavevector .in addition , there would be a significant impedance mismatch if the plasma is driven at a frequency well above the linear alfvn wave frequency , as seen in simulations .an important component of the quasilinear premise that enables the construction of predictive models of the turbulent dissipation and resultingplasma heating is that the turbulent fluctuations associated with each wave mode are damped at the appropriate instantaneous collisionless damping rate given by linear kinetic theory .the applicability of these linear kinetic damping rates , of course , depends on the central concept of the quasilinear premise that the turbulent fluctuations have the properties of the linear waves supported by the weakly collisional plasma .it is important to note that the terms in the kinetic equation for weakly collisional plasmas that are responsible for collisionless wave - particle interactions with the equilibrium distribution function are linear , and therefore the same arguments for the relevance linear wave modes in a turbulent plasma also apply to the linear collisionless damping . herewe discuss specifically two important issues regarding the collisionless dissipation : ( i ) possible nonlinear saturation or inhibition of the collisionless damping and ( ii ) the spatial distribution of damping via wave - particle interactions .there are two possible ways that the linear collisionless damping rate may be altered : ( i ) quasilinear or nonlinear evolution of the equilibrium distribution function may flatten the slope at the resonant velocity for a particular wave , thus reducing or entirely inhibiting collisionless damping of that wave ; and ( ii ) forcing by strong turbulence in a stochastic manner may break the coherence between a particular wave mode and a particle necessary for collisionless damping to extract significant energy from the wave .it has been suggested that electron landau damping is the dominant collisionless mechanism for the damping of fluctuations in the dissipation range of solar wind turbulence . in this regime , it has been proposed that the turbulent fluctuations are kinetic alfvn waves , a hypothesis now strongly supported by a number of numerical and observational studies of solar wind turbulence .see for a recent review of evidence for kinetic alfvn waves in the solar wind .flattening of the distribution function , and the associated reduction in collisionless damping rates , due to quasilinear evolution of the equilibrium is a well - understood process in plasma physics .the key question in the context of the turbulent solar wind plasma is whether this mechanism operates effectively to quench collisionless damping , as recently suggested .an argument against this quasilinear quenching of the landau damping of kinetic alfvn waves arises from the variation in the resonant velocity in the electron distribution function .since the kinetic alfvn wave is a dispersive wave mode , its phase velocity ( which is nearly parallel to the magnetic field direction ) increases linearly with in the dissipation range at .therefore , the parallel velocity of electrons that are resonant with the turbulent kinetic alfvn waves . ] increases from a velocity at to at .therefore , without flattening the electron velocity distribution over the entire range , it seems unlikely that a quasilinear flattening of the distribution function could completely suppress landau damping for .a second possible way to suppress collisionless wave - particle interacts is to interfere with the coherence time between electromagnetic waves and particles so that the particles are unable to experience a net gain of energy .for example , in strongly collisional plasmas , such as mhd plasmas , electromagnetic waves are undamped by collisionless wave - particle interactions because frequent collisions prevent a single particle from maintaining a sufficiently long coherence time to interact resonantly with the waves .a recent study has shown that , when the linearized vlasov equation is perturbed by a stationary random force , the effective landau damping rate can be significantly reduced under appropriate conditions .work continues to understand the effect of landau damping in a turbulent setting .nonlinear kinetic simulations of plasma turbulence will hopefully be able to quantify any suppression of the linear kinetic damping rate .another issue associated with the collisionless damping of plasma turbulence is the spatial distribution of the plasma heating resulting from the dissipation of the turbulent energy .based on the decomposition of turbulent fluctuations into component plane waves , it has frequently been asserted that landau damping must lead to spatially uniform heating .recently , there has been significant work investigating non - uniform distribution of heating in the solar wind plasma , especially focusing on enhanced heating in the neighborhood of current sheets , where it has been suggested that heating associated with current sheets can not be a consquence of landau damping . however , the assertion that landau damping must lead to spatially uniform heating is simply incorrect .  consider the case of a current sheet with a strong guide magnetic field , meaning that the current sheet is associated with an abrupt rotation of the angle of the magnetic field .the spatially localized structure of the current sheet necessarily implies significant energy in small wavenumber components .if one views the situation in wavevector space , each of the fourier components landau damps at its linear kinetic damping rate .but , the spatial distribution of this heating will not be uniform , but instead will be localized in the vicinity of the current sheet .this argument has recently been made to resolve the apparent contradiction in recent findings from gyrokinetic simulations of turbulence at the electron scales where a clear correlation between electron heating rate and the presence of current sheets is observed , but the heating as a function of wavenumber is completely explained , with no fitting parameters , by landau damping of each spatial fourier mode at it linear kinetic damping rate .besides the feasibility arguments for the validity of the quasilinear premise outlined above , we give no _ a priori _ proof for its validity in strongly nonlinear plasma turbulence .nonlinear simulations of plasma turbulence provide a powerful and direct avenue for testing the validity of the premise . in this section ,we present evidence from numerical simulations that support the relevance of linear plasma wave properties in a strongly turbulent plasma , including evidence of the relevance of linear wave eigenfunctions , linear collisionless damping rates , and linear frequency response as well as support for critical balance .first , a gyrokinetic simulation of the transition from the inertial to the dissipation range found that the perpendicular magnetic , parallel magnetic , and perpendicular electric field energy spectra are well fit by a turbulent cascade model that assumes the relationships between these energy spectra are determined solely by the linear kinetic eigenfunctions of the alfvn and kinetic alfvn waves .second , a gyrokinetic simulation spanning the entire dissipation range from the ion to the electron larmor radius was able to predict the parallel magnetic and perpendicular electric field energy spectra observed in the simulation by using the perpendicular magnetic energy spectrum and assuming that the relationships between the turbulent field components are described by the linear eigenfunction of the kinetic alfvn wave .these energy spectra were also well fit by a refined cascade model that again assumed the applicability of linear eigenfunctions and kinetic damping rates .finally , an analysis of a simulation of kinetic alfvn wave turbulence finds that the measured electron heating as a function of perpendicular wavenumber is well estimated , with no fitting parameters , by assuming all dissipation is provided by collisionless wave particle interactions at the linear landau damping rate .another proposed technique for evaluating the importance of linear wave modes is to compute the frequency spectrum for particular fourier modes or the eulerian frequency spectrum at a particular point in space from numerical simulations . as discussed above in [ sec : disp ] , except for the case of weak turbulence , however , it has not been established that one should indeed expect to see evidence of a `` linear dispersion relation '' in the frequency spectra of strong , driven plasma turbulence since an individual wave mode is not expected to persist for more than a single wave period .a study of driven turbulence in 3d incompressible mhd simulations found little evidence that linear waves play a significant role in mhd turbulence , but it has been pointed out that the method of driving used in the simulations may have had a dominant impact on the measured frequency spectrum . a particle - in - cell ( pic ) simulation of decaying 2d magnetosonic turbulence over the - plane found a cascade consistent with the properties of fast magnetosonic waves , and that little energy appeared to be nonlinearly transferred to the slow magnetosonic or ion bernstein waves .two studies of driven , 2d hybrid kinetic ion and fluid electron simulations of turbulence over the perpendicular plane found a low level of wave activity , with the dynamics dominated by nonlinear activity .the 2d geometry of these hybrid simulations , however , admits only linear wave modes with , and therefore these simulations can not support alfvn waves , kinetic alfvn waves , or whistler waves , calling into question their relevance to the study of solar wind turbulence .finally , 3d decaying simulations of turbulence at the high - frequency end of the inertial range using both the hall mhd and landau fluid theory find that the peak of the turbulent magnetic and kinetic energy frequency spectra for particular fourier modes show excellent agreement with the linear wave mode frequencies for the fast , alfvn , and slow modes .this collection of apparently contradictory findings suggests that this line of investigation of the relevance of linear wave properties in numerical simulations of turbulence will remain an active area of research .given that the idea of critical balance in strong plasma turbulence is essentially a quasilinear concept that the frequency of the nonlinear energy transfer remains of the same order as the _ linear _ wave frequency numerical evidence in support of critical balance indirectly supports the quasilinear premise .although the concept of critical balance was initially applied only to the case of mhd turbulence , the theory has been extended to the dispersive wave regime of the dissipation range . in numerical simulations ,critical balance is generally tested by a a measure of the turbulent power on the - plane in wavevector space in numerical simulations .the results of these numerous studies are contradictory , with some claiming to support critical balance , and others , to refute it .the lines of conflict , however , coincide with the method used to determine the direction of the magnetic field : studies using a _ local _mean magnetic field are consistent with the predictions of critical balance , while studies employing a global magnetic field are inconsistent with critical balance .it appears that , as long as the direction of the magnetic field is determined locally , there exists significant evidence in support of critical balance .a recently proposed alternative method for testing critical balance eliminates the need to define the direction of the magnetic field by noting that the linear wave frequency for alfvnic plasma waves is proportional to the parallel wavenumber ,  . in this case, one may compute the distribution of turbulent power on the - plane and look for the predicted scaling , where  or  for the mhd alfvn wave cascade and for the kinetic alfvn wave cascade .gyrokinetic simulations of driven , 3d kinetic alfvn wave turbulence support the predicted critical balance scaling of in this regime .  finally , a recent study of ion temperature gradient driven turbulence in gyrokinetic simulations of magnetic confinement fusion plasmas has found support for the turbulent fluctuations satisfying critical balance .spacecraft measurements in the turbulent wind plasma have been used to evaluate the characteristic nature of the turbulent fluctuations , seeking agreement with the typical frequencies of alfvn and kinetic alfvn wave modes using a k - filtering analysis of multi - spacecraft data and searching for evidence of critical balance through the spectral indices measured at different angles with respect to the direction of magnetic field . in addition, laboratory experiments of plasma turbulence have begun to uncover evidence suggesting the turbulence indeed satisfies critical balance .multi - spacecraft observations of solar wind turbulence may also be used to determine the fluctuation frequency in the solar wind plasma frame for each spatial fourier mode .the first -filtering analysis of _ cluster _ multi - spacecraft data in the unperturbed solar wind showed that the turbulent fluctuations in the inertial and transition range , , are consistent with the dispersion relation of the alfvn / kinetic alfvn wave branch , and are inconsistent with the fast / whistler branch . a subsequent study that performed a similar -filtering analysis of _ cluster _ multi - spacecraft data obtained contradictory results , finding little agreement with any particular linear wave mode . a number of issues , however , cast serious doubts on the validity of the latter study : the error in the plasma - frame frequency is not estimated , the results of this study at large scales are inconsistent with the observations that demonstrate the largely incompressible , alfvnic nature of large scale fluctuations in the solar wind , inspection of the cis and peace data show that all of the intervals studied suffer either electron or ion foreshock contamination , and the spacecraft configuration for each of these intervals shows significant levels of planarity and elongation , indicating a poor tetrahedron .a subsequent k - filtering study found further evidence in support of low frequency kinetic alfvn waves in _ cluster _multi - spacecraft data .it is clear that new multi - spacecraft studies will provide valuable guidance in assessing the relevance of linear wave modes in the solar wind plasma .  in measurements of solar wind turbulence, one can test the idea of critical balance by measuring the 1d magnetic energy spectrum as a function of the angle with respect to the magnetic field . as in the case for numerical simulations , there exist contradictory findings , with support for critical balance found when the magnetic field direction is determined locally and conflict with critical balance when the study employs a global magnetic field to determine the parallel direction .finally , recent experimental measurements of strongly magnetized plasma turbulence driven by the ion temperature gradient in the mega amp spherical tokamak ( mast ) have been found to be consistent with the predictions of critical balance .the question of the validity of the quasilinear premise that linear wave properties are relevant to the dissipation range of solar wind turbulence clearly remains to be settled . on balance, however , the bulk of the evidence appears to argue in favor of its validity .the utility of the quasilinear premise for the study of plasma turbulence , however , may also be judged _ a posteriori _ by the insights gained from such an approach .below we briefly discuss a number of applications of the quasilinear premise to the study of plasma turbulence .  the _ synthetic spacecraft data method _ is a technique for producing artificial plasma turbulence measurements that can be directly compared to _ in situ _ measurements of turbulence in the solar wind .the turbulence in the synthetic plasma volume is constructed assuming the quasilinear premise .such an approach has proven to be very successful , discovering that the compressible fluctuations in the inertial range of solar wind turbulence are anisotropically distributed slow waves , and illuminating the nature of the fluctuations responsible for the observed magnetic helicity signature of turbulent fluctuations as a function of the angle between the magnetic field and the solar wind flow .additionally , based on the fundamental concepts embodied by the quasilinear premise , a number of simple models for the turbulent cascade of energy in weakly collisional plasma turbulence have been devised , with the ability to model the dissipation of the turbulent cascade based on kinetic damping mechanisms . in addition to successfully modeling the energy spectra in nonlinear kinetic simulations of turbulence , such cascade models can be applied to develop a predictive capability for the plasma heating resulting from the dissipation of turbulence in weakly collisional space and astrophysical plasmas .finally , the characteristic eigenfunctions of the linear kinetic plasma waves can be exploited in an attempt to identify the characteristic nature of the turbulent solar wind fluctuations .there are also a couple of other lines of argument that support the relevance of linear wave properties to the strongly turbulent solar wind plasma .first , an important feature of solar wind turbulence is the observation that linear kinetic temperature anisotropy instabilities appear to constrain the limits of the temperature anisotropy of ions in the solar wind .second , from fundamental theorems about dimensional analysis , _i.e. _ the pi theorem , the dimensionless parameters upon which the linear theory depends will necessarily also be parameters upon which the nonlinear theory depends ( since linearization involves dropping terms , never adding ) , although the nonlinear theory undoubtedly depends on additional dimensionless parameters .the quasilinear premise is a hypothesis for the modeling of plasma turbulence in which the turbulent fluctuations are represented by a superposition of randomly - phased linear wave modes , and energy is transferred among these wave modes via nonlinear interactions .although a large body of work on plasma turbulence either explicitly or implicitly assumes the relevance of some linear plasma wave properties , the nonlinearity inherent in turbulent interactions raises obvious questions about the relevance of linear theory .this papers attempts to present a broad range of theoretical , numerical , and observational evidence in the attempt to evaluate the validity of the quasilinear premise .  after defining the quasilinear premise precisely ,we highlight the the aspects of turbulence that can and can not be described by such an approach : turbulent fluctuation properties such as the eigenfunction , frequency , and collisionless damping rate as well as second - order statistics such as the energy spectra or magnetic helicity of the turbulence can be described by a model of turbulence based on the quasilinear premise ; third - order and higher order statistics , such as intermittency and coherent structures , such as current sheets , can not be investigated using such an approach .we present a wide range of theoretical arguments in support of the relevance of linear wave properties even in a strongly turbulent plasma , motivated by the mathematical properties of the nonlinear equation of evolution for an incompressible mhd plasma .we present an analogy with the case of a critically damped simple harmonic oscillator , and suggest that it is neither necessary nor expected that one should see evidence of a linear dispersion relation ( vs.  ) in measurements of turbulence .in addition , we present argument that linear collisionless damping may persist even in the strongly turbulent regime and the resulting plasma heating certainly can be spatially non - uniform , despite frequent claims to the contrary .we review evidence in support of the quasilinear premise from numerical simulations , including results supporting the applicability of linear eigenfunctions and linear collisionless damping rates in the turbulent plasma .frequency diagnostics of plasma turbulence simulations have yielded contradictory findings , but simulations employing all three dimensions in space appear to largely find the frequency of the turbulent dynamics consistent with the linear wave frequencies . given that the idea of critical balance in strong plasma turbulence is essentially a quasilinear concept , evidence in support of critical balance indirectly supports the quasilinear premise .simulations again yield contradictory results , but the line dividing these conflicting results appear to coincide with the method used to determine the direction of the magnetic field : studies using a _ local _mean magnetic field are consistent with the predictions of critical balance , while studies employing a global magnetic field are not .finally , we discuss observational evidence from turbulence in the solar wind that supports the quasilinear premise , including multi - spacecraft k - filtering analyses that find plasma - frame frequencies of the turbulent fluctuations consistent with alfvn and kinetic alfvn waves .measurements of the the magnetic energy spectrum as a function of the angle of the solar wind flow with respect to the magnetic field also find support for critical balance found when the magnetic field direction is determined locally .the question of the validity of the quasilinear premise that linear wave properties are relevant to strong plasma turbulence clearly remains to be settled . on balance , however , we argue here that the bulk of the evidence appears to support it as a valuable means of modeling turbulence .the utility of the quasilinear premise for the study of plasma turbulence , however , may also be judged _ a posteriori _ by the insights gained from such an approach , and we review a number of studies , including those using the novel _ synthetic spacecraft data method _ , that have succeeded in more strongly constraining the fundamental nature of plasma turbulence .the ultimate goal of turbulence models based on the quasilinear premise is to develop the capability to predict the evolution of any turbulent plasma system , including the spectrum of turbulent fluctuations , their dissipation , and the resulting plasma heating .                 , c. h. k. , howes , g. g. , bonnell , j. w. , mozer , f. s. , klein , k. g. , & bale , s. d. 2013 , in american institute of physics conference series , vol . 1539 , american institute of physics conference series , ed . g. p. zank , j. borovsky , r. bruno , j. cirtain , s. cranmer , h. elliott , j. giacalone , w. gonzalez , g. li , e. marsch , e. moebius , n. pogorelov , j. spann , & o. verkhoglyadova , 143146

Abstract: the quasilinear premise is a hypothesis for the modeling of plasma turbulence in which the turbulent fluctuations are represented by a superposition of randomly - phased linear wave modes , and energy is transferred among these wave modes via nonlinear interactions . we define specifically what constitutes the quasilinear premise , and present a range of theoretical arguments in support of the relevance of linear wave properties even in a strongly turbulent plasma . we review evidence both in support of and in conflict with the quasilinear premise from numerical simulations and measurements of plasma turbulence in the solar wind . although the question of the validity of the quasilinear premise remains to be settled , we suggest that the evidence largely supports the value of the quasilinear premise in modeling plasma turbulence and that its usefulness may also be judged by the insights gained from such an approach , with the ultimate goal to develop the capability to predict the evolution of any turbulent plasma system , including the spectrum of turbulent fluctuations , their dissipation , and the resulting plasma heating .