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in recent years several lagrangian type closures have been developed to model fluctuations of turbulent incompressible fields @xcite .
the starting point for all these models goes back to the so - called _ restricted euler _
closure @xcite which models the time evolution of the gradient tensor @xmath0 @xmath1 this model is derived using the incompressibility constraint and neglects the viscous term and the anisotropic part of the pressure hessian .
the _ restricted euler _ model is already able to capture geometric features of vortex stretching and alignment but fails as a robust model to obtain stationary statistics due to the appearance of finite time singularities .
several strategies have been applied to regularize the restricted euler model where the most prominent ones are the _ tetrad model _
@xcite which models the anisotropic part of the pressure hessian and the _ recent fluid deformation _ approach which models both the viscous term and the anisotropic part of the pressure hessian @xcite .    in this paper
we extend the _ recent fluid deformation _ approach to the case of the time evolution of the gradient of passive scalar and to kinematic magnetohydrodynamic turbulence .
comparing these two examples is especially interesting since both examples differ mainly by a sign in the stretching term .
thus structures which are expanded in one equation ( e.g. gradient of passive scalar ) are compressed in the other ( e.g. mhd ) .
comparison with direct numerical simulations are performed to test the range of applicability of the extended _ recent fluid deformation _ models .
before we apply the _ recent fluid deformation _ approach to the gradient of a passive scalar and to kinematic mhd turbulence , we first recall the main steps in the derivation of this model .
detailed information can be found in @xcite .
starting with the navier - stokes equation @xmath2 one takes the gradient @xmath3 and denotes the velocity gradient @xmath4 as @xmath5 . using the lagrangian material derivative @xmath6 eqn .
( [ eq : gradns ] ) reduces to @xmath7 assuming the pressure hessian to be isotropic @xmath8 and neglecting the viscous term would result in the restricted euler model given by eqn .
( [ eq : re ] ) . in order to improve the model for the pressure hessian , chevillard and meneveau @xcite consider the pressure hessian @xmath9 expressed in the lagrangian frame and postulate that the lagrangian pressure is isotropic at some given time @xmath10 . applying
the change between eulerian and lagrangian coordinates @xmath11 twice - while neglecting higher orders - one obtains @xmath12 where @xmath13 is the _ short - time cauchy - green tensor _ resulting from the integration of the flow map .
the time @xmath14 is chosen of the order of the kolmogorov time so that it is possible to linearize the integration from @xmath15 to @xmath16 .    applying a similar reasoning to the viscous term results in the approximation @xmath17 , @xmath18 being a characteristic friction time such that the reynolds number @xmath19 is proportional to @xmath20 @xcite . in all simulations described below
, the parameters @xmath21 and @xmath22 were chosen which result in a taylor - reynolds number of @xmath23 .
this value is in the range of the performed dns simulations ( see table [ table : paramscalar ] ) .    finally , one obtains a stochastic ordinary differential equation ( sde ) @xmath24 where the stochastic forcing term @xmath25 is added to represent large scale forcing .
the sde ( [ eq : final ] ) can be integrated with standard methods for stochastic equations @xcite .
in this section we apply the _ recent fluid deformation _ theory to the case of an admixture @xmath26 which is passively advected by the fluid according to @xmath27 in order to study the dynamics of the gradient of the passive field @xmath26 we take the derivative of eqn .
( [ eq : ps ] ) and obtain @xmath28 using again the lagrangian material derivative @xmath6 eqn .
( [ eq : gradps ] ) reduces to @xmath29 in this equation , the gradient tensor @xmath5 is obtained from integrating eqn .
( [ eq : final ] ) . modeling the viscous term as for eqn .
( [ eq : sde_ns ] ) and adding a stochastic forcing @xmath30 , we obtain the stochastic differential equation ( sde ) @xmath31\mathrm{d}t +      \mathrm{d}v_i\ ] ] this equation is integrated together with the equation ( [ eq : final ] ) for the velocity gradient tensor .
the forcing is chosen to be vectorial gaussian noise , delta - correlated in time , as for the velocity gradient .
the forcing amplitude is @xmath32 as for a standard random walk .
the parameters for the result shown here are @xmath33 and @xmath22 .    in order to test the applicability of the _ recent fluid deformation _ approximation
, we performed direct numerical simulations for passive scalar turbulence .
for this , we utilized the pseudo - spectral simulation framework latu @xcite .
table [ table : paramscalar ] summarizes the relevant parameters of the simulation .
this model extension has been analyzed in @xcite in great detail , considering important phenomena like amplification of the gradient s norm and alignment to strain principal axes .
but in contrast to this publication , we focus on the comparison of different turbulent systems .    [
cols="^,^,^,^,^,^,^,^,^ " , ]     the top of figure [ fig : gradtheta ] shows the probability distribution function ( pdf ) of fluctuations of @xmath34 obtained from integrating eqns .
( [ eq : final ] ) and ( [ eq : sde_ps ] ) . in the bottom of figure  [ fig : gradtheta ]
the pdf obtained from the direct numerical spectral simulation is shown .
the qualitative agreement is excellent and the tails of the pdfs of @xmath34 approximately follow an exponential decay behavior as predicted by shraiman and siggia @xcite for a gaussian velocity field .
we start with the incompressible mhd equations in the kinematic regime neglecting the back - reaction of magnetic fluctuations on the velocity field .
the equation for the time evolution of the magnetic field @xmath35 reads @xmath36 where the velocity field obeys the navier - stokes equations  ( [ eq : ns ] ) . in the lagrangian frame
this equation takes the form @xmath37 again , using the same _ recent fluid deformation _
approximation for the resistive term and adding a stochastic forcing , which is as above gaussian and white - in - time , we obtain @xmath38    note that this equation differs from by the transpose in @xmath39 and - more important - the sign of the first term on the right hand side . thus directions , where the magnetic field is stretched , are directions , where the gradient of the passive scalar is compressed and vice versa .
this very different influence of the stretching term results in a qualitatively different pdf for the magnetic field fluctuations compared to the @xmath34 fluctuations . in order to test
whether the _ recent fluid deformation _ approach is able to reproduce the different role of the stretching term , we performed direct numerical simulations of kinematic mhd turbulence . here
, we again performed simulations using the framework latu and the relevant parameters are given in table [ table : paramscalar ] .
the simulation was started with random initial condition for the magnetic field @xmath35 and an already fully turbulent velocity field .
external forcing was only applied to the velocity field by keeping the large scale fourier - modes @xmath40 constant .
the top of figure [ fig : b ] shows the pdf of magnetic field fluctuations calculated using equations ( [ eq : final ] ) and ( [ eq : sde_mhd ] ) .
this pdf has to be compared to the pdf obtained from direct numerical simulation shown in the bottom of figure [ fig : b ] .
it is again remarkable that the simple stochastic ode model ( [ eq : sde_mhd ] ) is able to capture correctly the shape of the pdf of magnetic field fluctuations .
since we considered the kinematic mhd equations in the dynamo regime , the pdf of the stochastic model could not be obtained from a time series as in the case of the passive scalar since the magnetic field is growing exponentially .    to obtain meaningful statistics ensemble averaging
was used instead of time sampling .
numerically , a random initial state was generated and integrated a sufficiently large number of time steps , to retain no statistical influence of the initial state .
the final state enters the statistics and a new realization is generated via another random initial condition .
the pdf was obtained by sampling over @xmath41 initial conditions .    for illustration of the kinematic dynamo effect and a consistency check with direct numerical simulations of the kinematic mhd equations we tracked the local magnetic energy @xmath42 in figure [ fig : dynamo ] for @xmath43 , a value well in the dynamo regime of the modell .
for this simulation only the velocity gradient itself was driven by stochastic forcing , as outlined above , while the magnetic field was undriven .
the forcing amplitude was scaled with @xmath32 , analog to the scaling for a standard random walk .
+ the parameter @xmath14 of the model was chosen to be @xmath44 .
+ the magnetic diffusion time @xmath45 scale is varied to explore the reaction of the dynamo effect . below the critical parameter of approximately @xmath46 we observed no growing magnetic energy .
the corresponding growth rate , estimated from an exponential growth @xmath47 , is about @xmath48 .
the growth rate compares well with the growth rate of our dns simulation , where we obtain @xmath49 .
although the identification of the correct kolmogorov timescale @xmath14 is not without uncertainties , it is remarkable that this value of the growth rate is in the range of data obtained by kinematic dynamo simulations ( see also table i in @xcite and references therein ) .
in this paper we have shown that the natural extension of the _ recent fluid deformation _ model for navier - stokes turbulence @xcite to the case of fluctuations of the gradient of a passive scalar and to magnetic field fluctuations is able to produce probability distribution functions that agree well with pdfs obtained from direct numerical simulations .
the pdfs from the stochastic odes ( [ eq : final ] , [ eq : sde_ps ] , [ eq : sde_mhd ] ) are obtained with a fraction of the computing resources necessary for direct numerical simulations .
the next step is to include the back - reaction of the magnetic field on the fluid flow .
if this can be managed , then generating magnetic field fluctuations using the _ recent fluid deformation _ model is a tempting alternative for e.g. the problem of cosmic ray propagation where magnetic field fluctuations are generated by other means which are not able to capture intermittency effects ( see @xcite ) .
+ r.g . acknowledges stimulating discussions with l. chevillard during the workshop `` euler equations : 250 years on '' .
access to the jugene bluegene / p computer at the fz jlich was made available through project hbo22 .
this work benefited from support through dfg - for1048 .
in this paper we present an extension of the _ recent fluid deformation ( rfd ) _ closure introduced by chevillard and meneveau @xcite which was developed for modeling the time evolution of lagrangian fluctuations in incompressible navier - stokes turbulence . 
 we apply the rfd closure to study the evolution of magnetic and passive scalar fluctuations . 
 this comparison is especially interesting since the stretching term for the magnetic field and for the gradient of the passive scalar are similar but differ by a sign such that the effect of stretching and compression by the turbulent velocity field is reversed . 
 probability density functions ( pdfs ) of magnetic fluctuations and fluctuations of the gradient of the passive scalar obtained from the rfd closure are compared against pdfs obtained from direct numerical simulations .