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large scale structures like galaxies and clusters of galaxies are believed to have formed by gravitational amplification of small perturbations @xcite . observations suggest that the initial density perturbations were present at all scales that have been probed by observations . an essential part of the study of formation of galaxies and other large scale structures is thus the evolution of density perturbations for such initial conditions . once the amplitude of perturbations at any scale becomes large , i.e. , @xmath0 , the perturbation becomes non - linear and the coupling with perturbations at other scales can not be ignored . indeed , understanding the interplay of density perturbations at different scales is essential for developing a full understanding of gravitational collapse in an expanding universe . the basic equations for this have been known for a long time @xcite but apart from some special cases , few solutions are known . a statistical approach to this problem based on pair conservation equation has yielded interesting results @xcite , and these results have motivated detailed studies to obtain fitting functions to express the non - linear correlation function or power spectrum in terms of the linearly evolved correlation function @xcite . it is well known from simulation studies that at the level of second moment , i.e. , power spectrum , correlation function , etc . , large scales have an important effect on small scales but small scales do not have a significant effect on large scales @xcite . most of these studies used power spectrum as the measure of clustering . results of these simulation studies form the basis for the use of n - body simulations , e.g. , from the above results we can safely assume that small scales not resolved in simulations do not effect power spectrum at large scales and can be ignored . substructure can play an important role in the relaxation process . it can induce mixing in phase space @xcite , or change halo profiles by introducing transverse motions @xcite , and , gravitational interactions between small clumps can bring in an effective collisionality even for a collisionless fluid @xcite . thus it is important to understand the role played by substructure in gravitational collapse and relaxation in the context of an expanding background . in particular , we would like to know if this leaves an imprint on the non - linear evolution of correlation function . effect of substructure on collapse and relaxation of larger scales is another manifestation of mode coupling .    in this paper , we report results from a study of mode coupling in gravitational collapse . in particular , we study how presence of density perturbations at small scales influences collapse and relaxation of perturbations at larger scales . these effects have been studied in past @xcite but the motivation was slightly different @xcite . we believe it is important to study the issue in greater detail and make the relevance of these effects more quantitative using n - body simulations with a larger number of particles . we also study the reverse process , i.e. , how does collapse of perturbations at large scales effect density perturbations at much smaller scales . it is well known that the local geometry of collapse at the time of initial shell crossing is planar in nature @xcite , hence we model density perturbations as a single plane wave in this work . simple nature of the large scale fluctuation allows us to study interaction of well separated scales without resorting to statistical estimators like power spectrum . we are studying the same problem in a more general setting and those results will be reported in a later publication . key features of collapse of a plane wave can be understood using quasi - linear approximations , at least at a qualitative level . initial collapsing phase is well modelled by the zeldovich approximation @xcite , wherein particles fall in towards the centre of the potential well . zeldovich approximation breaks down after orbit crossing as it does not predict any change in the direction of motion for particles , thus in this approximation particles continue to move in the same direction and the size of the collapsed region grows monotonically . in a realistic situation we expect particles to fall back towards the potential well and oscillate about it with a decreasing amplitude , and the collapsed region remains fairly compact . several approximations have been suggested to improve upon the zeldovich approximation @xcite . the adhesion approximation @xcite invokes an effective viscosity : this prevents orbit crossing and conserves momentum to ensure that pancakes remain thin and matter ends up in the correct region . this changes the character of motions in dense regions ( no orbit crossing or mixing in the phase space ) but predicts locations of these regions correctly . if one assumes that the gravitational potential evolves at the linear rate @xcite , then it can be shown that the collapsed region remains confined . the effective drag due to expanding background slows down particles and they do not have enough energy to climb out of the potential well . thus the process of confining particles to a compact collapsed region results from a combination of expansion of the universe and gravitational interaction of in falling particles . none of the approximations captures all the relevant effects . therefore we must turn to n - body simulations @xcite in order to study collapse and relaxation of perturbations in a complete manner . we will consider only gravitational effects here and ignore all other processes . we assume that the system can be described in the newtonian limit . the growth of perturbations is then described by the coupled system of euler s equation and poisson equation in comoving coordinates along with mass conservation , e.g. , see @xcite . @xmath1 it is assumed that the density field is generated by a distribution of particles , each of mass @xmath2 , position @xmath3 . @xmath4 is the present value of hubble constant , @xmath5 is the present density parameter for non - relativistic matter and @xmath6 is the scale factor . in this paper we will consider the einstein - de sitter universe as the background , i.e. , @xmath7 . these can be reduced to a single non - linear differential equation for density contrast @xcite . @xmath8 where @xmath9 \delta_{\bf k ' } \delta_{{\bf k } -{\bf k ' } } \nonumber\ ] ] and @xmath10 \;\;\ ; ; \;\;\;\ ; m= \sum\limits_i m_i \nonumber\ ] ] the terms @xmath11 and @xmath12 are the non - linear coupling terms between different modes . @xmath12 couples density contrasts in an indirect manner through velocities of particles ( @xmath13 ) . the equation of motion still needs to be solved for a complete solution of this equation , or we can use some ansatz for velocities to make this an independent equation .    it can be shown that individual _ virialised _ objects , i.e. , objects that satisfy the condition @xmath14 where @xmath15 is the kinetic energy and @xmath16 is the potential energy , do not make any contribution towards growth of perturbations through mode coupling @xcite at much larger scales , i.e. , the @xmath17 term is zero . the contribution of mode coupling due to interaction of such objects is not known . approximate approaches to structure formation can be developed by ignoring interaction of well separated scales . the evolution of density perturbations can be modelled as a combination of non - linear collapse at small scales , and the collapsed objects can be displaced using quasi - linear approximations @xcite . these approaches yield an acceptable description of properties of collapsed objects and their distribution for a first estimate . pinocchio @xcite provides sufficient information about halo properties and merger trees for use with semi - analytic models of galaxy formation . the efficacy of these models puts an upper bound on the effects of mode coupling that we are studying here .    in this paper we simplify the system by starting with perturbations that have a non - zero amplitude only for two sets of scales . we simulate the collapse of a plane wave by starting with non - zero amplitude of perturbations for the fundamental mode of the simulation box along the @xmath18 axis , the wavenumber of the fundamental mode is denoted by @xmath19 . this serves as the large scale perturbation in our study . the amplitude for this mode is chosen so that shell crossing takes place when the scale factor @xmath20 . power spectrum for small scale fluctuations is chosen to be non - zero in a range of wave numbers @xmath21 with a constant amplitude across this window , i.e. , @xmath22 for @xmath23 . a gaussian random realisation of this power spectrum is used for small scale fluctuations . here @xmath24 is the power per logarithmic interval in @xmath25 contributed by small scales ( large @xmath25 ) and @xmath11 is the amplitude of the fundamental mode that gives rise to the plane wave . the ratio of @xmath26 at @xmath27 and for the plane wave is denoted by @xmath28 , thus when @xmath29 collapse of perturbations at these scales happens at the same time whereas for @xmath30 perturbations at small scales collapse before the plane wave collapses . we chose the ratio @xmath31 so that there is distinct separation in the scales involved .    , we go from a single stream region to a three stream region and so on up to seven stream region near the centre . ] collapse of a plane wave of this kind leads to formation of a multi - stream region , we will also use the term pancake to describe this region . figure  [ phase_space ] shows the phase space plot for the plane wave at late stages of collapse for the simulation pm_00l ( see table 1 for details of the simulation ) . velocity of particles is plotted as a function of position , only the @xmath18 component is plotted as there is no displacement or velocity along other directions in this simulation . regions where particles with different velocities can be found are the multi - stream regions . as we approach the centre of pancake located at @xmath32 , we go from a single stream region to a three stream region and so on up to seven stream region near the centre .    in initial stages , the mass in the pancake increases rapidly as more particles fall in . figure  [ den_plwave ] shows this in terms of over - density which increases sharply from @xmath20 to @xmath33 . a significant fraction of the total mass falls into the pancake and the infall velocities for the remaining matter are very small . in this regime the mass of the pancake is almost constant , this can be seen from the panels of figure  [ den_adh_sub ] where the mass enclosed in the pancake region is almost constant from @xmath33 to @xmath34 . .this table lists parameters of n - body simulations we have used . all the simulations used @xmath35 particles . the first column lists name of the simulation , second column lists the code that was used for running the simulation , third column gives the relative amplitude of small scale power and the plane wave , the fourth column tells us whether the large scale plane wave was present in the simulation or not , and the last column lists the distribution of particles before these are displaced using a realisation of the power spectrum . _ grid _ distribution means that particles started from grid points . _ perturbed grid _ refers to a distribution where particles are randomly displaced from the grid points , this displacement has a maximum amplitude of @xmath36 grid points . such an initial condition is needed to prevent particles from reaching the same position in plane wave collapse as such a situation is pathological for tree codes . the treepm simulations were run with a force softening length equal to the grid length . [ cols="<,<,<,<,<",options="header " , ]     in absence of any substructure the collisionless collapse retains planar symmetry and we have layers of multi - stream regions with the number of streams increasing towards the centre of the pancake . presence of small scale fluctuations can induce transverse motions and these motions are amplified in the pancake . weakly bound substructure can be torn apart due to interaction with rapidly in falling matter . on the other hand , higher average density in the multi stream region can lead to rapid growth of perturbations . it is known that pancakes are unstable to fragmentation due to growth of perturbations @xcite . the velocity field is anisotropic due to infall along one direction , hence the rate at which perturbations grows will also exhibit anisotropies . velocity dispersion along the direction of plane wave collapse is larger than the transverse direction , hence the growth of fluctuations in the transverse plane is expected to be more rapid . if the in falling material contains collapsed substructure , then gravitational interactions between these can induce large transverse velocities . this takes away kinetic energy from the direction of infall , which in turn can lead to more fragmented and thinner multi - stream region .    in the following sections we describe the numerical experiments have undertaken in detail , and test the physical ideas and expectations outlined above we used a particle - mesh code @xcite and the treepm code @xcite . some simulations were run using the parallel treepm @xcite . treepm simulations used spline softening with softening length equal to the length of a grid cell in order to ensure collisionless evolution . we used force softening assuming a spline kernel @xcite . all the simulations were carried out with @xmath35 particles . table  1 lists parameters of the simulations we have used for this paper . we have used two types of initial distribution of particles . in the _ grid _ distribution particles are located at grid points before being displaced to set up the initial perturbations . _ perturbed grid _ refers to a distribution where particles are randomly displaced from the grid points @xcite , this displacement has a maximum amplitude of @xmath36 grid length . such an initial condition is needed to prevent particles from reaching the same position in plane wave collapse as such a situation is pathological for tree codes . these small displacements do not affect the power spectrum to be realised , pm_@xmath37l and t_@xmath37l were compared to test for any systematic effects . simulations t_@xmath38p and t_@xmath39p were similar to t_@xmath38l and t_@xmath39l except that the small scale fluctuations were restricted to the direction orthogonal to the direction of plane wave . thus the small scale fluctuations had the same form for all @xmath18 . these simulations are useful for differentiating between competing explanations for results outlined below .    in addition to the n - body simulations listed in table  1 , we also carried out one dimensional simulations within the adhesion model @xcite with a finite viscosity following a method similar to the one outlined by weinberg and gunn ( 1990 ) .     and @xmath33 from the pm_00l simulation , dashed lines show the density profile from the t_00l simulation with the same profile . ] figure  [ den_plwave ] shows the density profile of the pancake for pm_00l and t_00l simulations at two epochs . these figures demonstrate that the density profiles in these simulations are almost identical , indeed the tiny differences can be attributed to the different initial distribution of particles . we have checked this assertion by running the pm_00l with the perturbed grid initial conditions . the treepm method has a slightly better resolution but it does not induce any new features . this is expected as the force softening length used in the treepm simulations is one grid length , same as the average inter - particle separation and it has been shown than such force softening does not induce two body collisions @xcite . we will mostly use treepm simulations for the remaining part of this study . an important indicator of the role played by substructure is the thickness of the pancake that forms by collapse of the plane wave . if substructure does not play an important role in evolution of large scale perturbations then the thickness of pancake should not change by a significant amount . on the other hand , if substructure does indeed speed up the process of dynamical relaxation then we should see some signature in terms of the thickness of pancake , velocity structure , or both . any such effect will be apparent only at late times as infall of matter into the pancake dominates at early times . dynamical effects of substructure will become important only at late times . figure  [ pthick ] shows a slice from some of the simulations listed in table  1 . the plane wave collapses along the vertical axis . configuration at @xmath33 is shown here , the plane wave begins to collapse at @xmath20 . different panels in this figure refer to simulation t_00l , t_10l and t_40l . the boundary of the multi stream region is visible clearly in all the slices even though this region is fragmented in the last panel ( t_40l ) . it is clear that the pancake is thinner in simulations with more substructure . a more detailed comparison of simulations with different level of substructure is shown in figure  [ den_all ] . the top panel of this figure shows the averaged over - density as a function of the @xmath18 coordinate , the plane wave collapses along this axis . over - density is averaged over all @xmath40 and @xmath41 for a given interval @xmath42 to obtain the averaged values plotted here . the peak over - density at the centre of the pancake is smaller in simulations with more substructure . the mass enclosed within a given distance of the centre of pancake ( defined here as the trough of the potential well of the plane wave ) is smaller for more substructure , even though the variation is very small at less than @xmath43 between the extreme cases ( see figure  [ den_adh_sub ] ) . potential wells corresponding to substructure prevent infall into the pancake region . as the amount of substructure is increased , there is visible reduction in the size of the region around the pancake where density is greater than average . the visual impression of figure  [ pthick ] is reinforced by the variation of over - density . the middle panel of figure  [ den_all ] shows the _ rms _ velocities of particles in direction transverse to the plane wave collapse as a function of the @xmath18 coordinate . as in the top panel , averaging is done over all @xmath40 and @xmath41 for a given interval @xmath42 . this plot shows that the transverse motions are enhanced in the dense pancake region . the amplitude of transverse motions is larger in simulations with more substructure . size of the region where these motions are significant varies with the amount of substructure , as in case of over - density ( top panel ) . the _ rms _ transverse velocities do not go to zero outside the pancake region , instead these level off to a small residual value .    , the direction of collapse for the plane wave . density profile has been averaged over the directions transverse to the collapse of plane wave . the curves are for @xmath33 , simulations used are t_00l ( solid line ) , t_05l ( dashed line ) , t_10l ( dot - dashed line ) , t_20l ( dotted line ) and t_40l ( dot - dot - dashed line ) . the middle panel shows the rms transverse velocities of particles at the same epoch for t_05l ( dashed line ) , t_10l ( dot - dashed line ) , t_20l ( dotted line ) and t_40l ( dot - dot - dashed line ) . the lower panel shows the rms transverse velocities of collapsed haloes at the same epoch for t_05l ( dashed line ) , t_10l ( dot - dashed line ) , t_20l ( dotted line ) and t_40l ( dot - dot - dashed line ) . ] transverse motions are due to motions of particles in clumps that constitute substructure , due to infall of particles in these clumps , and , transverse motions of clumps as they move towards each other . in order to delineate these effects , we have plotted the _ rms _ velocities for haloes in the last panel of figure  [ den_all ] . these haloes were selected with the friends - of - friends ( fof ) algorithm using a linking length of @xmath44 grid length . transverse component of the velocity of centre of mass for haloes with more than @xmath45 particles was used for the figure . such a high cutoff for halo members is acceptable because typical haloes have several hundred members , see the following subsection on mass functions . differences between simulations with different amount of substructure are more pronounced than in the middle panel . for simulations with a small amount of substructure , motion of clumps is subdominant and hence the transverse motions are contributed mostly by internal motions and infall . in simulations with more substructure , motions of clumps contribute significantly to the _ rms _ transverse velocity . gravitational attraction of clumps , particularly in close encounters in the pancake region induce the transverse component . collisions are enhanced in the pancake region as the number density of clumps is higher .    in order to convince ourselves that transverse motions induced by scattering / collision of clumps is the most likely reason for the reduced thickness of pancakes , we compare simulations t_@xmath38l and t_@xmath39l with t_@xmath38p and t_@xmath39p . in t_@xmath38p and t_@xmath39p simulations , the small scale fluctuations do not have any @xmath18 dependence . in these ( t_@xmath38p and t_@xmath39p ) simulations there are no clumps but streams of particles that are falling in and this reduces the number of scattering that take place  no @xmath18 dependence means that dense streams run into each other head on with grazing collisions happening only rarely . of course , in the simulation the presence of the plane wave leads to breaking of these streams into clumps as the streams are stretched inhomogeneously in the @xmath18 direction . these clumps are aligned parallel to the @xmath18 axis . in the pancake region scattering of these streams occasionally leads to complex patterns .    if presence of substructure and its growth in the pancake was the only cause for making the pancake thinner then pancake in these simulations should be thinner as well . figure  [ pic_lp ] shows slices from simulations t_@xmath39l and t_@xmath39p for @xmath33 . a slice from the simulation pm_@xmath37l is also plotted here for reference . this visual comparison shows that the pancake is thinner in t_@xmath39l as compared to t_@xmath39p . indeed , the thickness of pancake in t_@xmath39p and pm_@xmath37l is very similar . this reinforces the point that scattering of clumps in the pancake region is the key reason for thinner pancakes . the substructure is helping to confine the pancake to a smaller region . it is interesting to study the collapse of a plane wave in an n - body simulation and compare it with the collapse in the adhesion model @xcite with a finite effective viscosity . we first study the collapse of a plane wave in absence of any substructure , n - body simulations pm_00l for comparison with numerical simulations using the adhesion model with finite effective viscosity . one dimensional adhesion simulations were done using the plane wave with the same amplitude as the n - body simulations . we use the standard method for computing the trajectories of particles in the adhesion model @xcite , a summary of the basic formalism is reproduced here for reference . in adhesion approximation , the equation of motion for a particle is replaced by the burgers equation @xcite . in the one dimensional situation we are considering here , we have : @xmath46 here @xmath47 is the velocity of particles and @xmath48 is the linear growth factor . this equation can be solved by introducing the velocity potential @xmath49 , where @xmath50 coincides with the gravitational potential at the initial time . solution has the following form . @xmath51 and , @xmath52 dq   .\ ] ] here @xmath53 is the lagrangian position of the particle and @xmath40 is the eulerian position . in this method we integrate the differential equation for particle trajectories . at each time step velocity is calculated by above procedure at grid points and interpolated to particles positions . figure  [ den_adh ] shows the mass enclosed within a distance @xmath54 from the centre of the pancake . the enclosed mass is defined as : @xmath55 here @xmath56 is the density at position @xmath18 and @xmath57 is the centre of the pancake . there is no ambiguity for comparing the results with n - body simulations in case of no substructure as density depends only on @xmath18 . while comparing other simulations with the adhesion solution , we will consider density averaged over @xmath40 and @xmath41 directions  adhesion model is run only for the one dimensional problem . top panel of figure  [ den_adh ] shows the enclosed mass @xmath58 at @xmath59 , middle panel is for @xmath60 and the lower panel is for @xmath61 . the solid curve shows the enclosed mass for pm_00l . in the region with a given number of streams , the n - body curve is smooth . jumps in mass enclosed occur at transition from single stream to multi stream region , and at other transitions where the number of streams changes within the multi stream region . all other curves show @xmath58 for adhesion model : dashed curve is for @xmath62 , dotted curve is for @xmath63 and the dot - dashed curve is for @xmath64 . there is no constant effective viscosity curve that follows the n - body curve closely through the multi stream regions . in regions with a given number of streams , the n - body curve stays around a curve for constant effective viscosity in the adhesion model . a remarkable fact is that the n - body curve for the three stream region at all the epochs follows the adhesion model curve for @xmath65 . similar behaviour is seen for the five stream region which follows @xmath66 though the range of scales and epochs over which this can be resolved is somewhat limited . addition of substructure clearly changes the character of the problem and the collapse is no longer one dimensional . however , the scale of the substructure is so small compared to the wavelength of the plane wave that the large scale collapse is still very close to planar . figure  [ den_adh_sub ] shows the mass enclosed within a distance @xmath54 from the centre of the multi stream region for simulations pm_00l , t_10l and t_40l . density is averaged over all @xmath40 and @xmath41 for this plot in the same manner as for figure  [ den_all ] . also plotted in the figure are curves for the adhesion model ( @xmath63 ) , where the calculation is done without taking substructure into account . the motivation for such a comparison is to see the effect of substructure on the favoured value of effective viscosity . substructure removes the sharp change in density at the boundaries of @xmath67-stream , @xmath68-stream and @xmath69-stream regions and the curves for t_10l and t_40l are smoother in the pancake region . the finite viscosity curve matches simulations with substructure over a wider range of scales than with pm_00l . there are no other noteworthy differences . mass function of collapsed haloes in these simulations can be used to understand the influence of plane wave collapse on substructure . collapsed structures form in these simulations primarily due to initial density fluctuations at small scales , with some modulation by the collapse of the plane wave . in this section we study the effect of the collapsing plane wave on the mass function of collapsed haloes . these haloes were selected with the friends - of - friends ( fof ) algorithm using a linking length of @xmath44 grid length . the initial power spectrum has a peak at the scale corresponding to @xmath70 of the simulation box , or @xmath71 grid lengths . thus typical haloes will have a lagrangian radius of about @xmath72 grid lengths and should contain about @xmath73 particles . thus a cutoff of @xmath45 or more particles for haloes is reasonable for this study .            in absence of the plane wave , the only perturbations are at small scales . the small scale perturbations are concentrated around a given mass scale and the mass function is also peaked around this mass at early epochs . at late epochs mergers lead to formation of haloes with a larger mass and the range of masses is greater for models with a larger amplitude of fluctuations . figure  [ slices ] shows these features in the distribution of particles . these features can also be seen in figure  [ massf ] where mass fraction @xmath74 for @xmath75 , @xmath76 and @xmath77 is plotted in different panels . @xmath74 is the fraction of total mass in collapsed haloes with halo mass above @xmath78 . adding the plane wave at a much larger scale than the small scale fluctuations essentially pushes much of the mass into the pancake region , leaving a small fraction of matter in the under dense regions that occupy much of the volume . growth of small scale fluctuations in the under dense regions is inhibited whereas growth of fluctuations in the pancake region is enhanced , this is seen clearly in the slices from simulations shown in figure  [ slices ] . higher background density in the pancake region leads to rapid growth of perturbations , mergers of haloes also lead to formation of massive clumps . these effects become more pronounced at late epochs and result in a shift of mass function towards larger masses , indeed haloes at two distinct mass scales are present . low mass clumps in under dense regions have the mass expected of haloes in regions where small scale fluctuation dominate whereas haloes of a much higher mass are present in the pancake region . figure  [ massf ] shows these two mass scales very clearly . total mass in collapsed haloes does not change significantly with the addition of the plane wave . indeed for simulations t_40l and t_40 , mass function is the same at @xmath75 as small scales dominate . at late times ( @xmath33 ) , the effect of plane wave makes the mass function of t_05l , t_10l and t_20l similar . not surprisingly , presence of large scale power leads to formation of more massive haloes . however it does not seem to enhance the total mass in collapsed haloes . in this paper we studied the effect of substructure on collapse of a plane wave . the key conclusions of the present study of the role of substructure are :    * the pancake formed due to collapse of the plane wave is thinner if the in falling material is formed of collapsed substructure . * we show that collisions between clumps lead to enhancement of velocities transverse to the direction of large scale collapse . * we show that in simulations with substructure where collisions are suppressed , pancakes are not thinner . * thus collision induced enhancement of motions transverse to the collapsing plane wave takes away kinetic energy from the direction of infall and leads to thinner pancakes . * presence of large scale power shifts the mass function towards larger masses . there is , however , no change in the total mass in collapsed haloes . the points outlined above essentially relate to coupling of density fluctuations at well separated scales . each of these points refers to a measurable effect of such a coupling . the nature of large scale fluctuation , a single plane wave , does not allow us to estimate the effect in terms of statistical indicators like the power spectrum . we plan to study these aspects with larger ( @xmath79 ) simulations where the large scale collapse will also be generic . large , high resolution studies are needed as @xmath35 simulations with particle mesh code have not shown any large effect in power spectrum at late times @xcite .     from the centre of the multi stream region . the top panel shows the curves for @xmath33 . the thick solid curve is for the n - body simulation pm_00l . jumps in the mass enclosed occur at transition from multi stream region with @xmath80 streams to @xmath81 streams , with @xmath82 a non - zero positive integer . all other curves show @xmath58 for adhesion model : dashed curve is for @xmath62 , dotted curve is for @xmath63 and the dot - dashed curve is for @xmath64 . the middle panel shows the same set curves for @xmath60 and the lower panel is for @xmath34 . ]     from the centre of the multi stream region . the top panel shows the curves for @xmath33 . the solid curve is for n - body simulation pm_00l . other simulations are also plotted here t_10l ( dashed curve ) and t_40l ( dot - dashed curve ) . dotted curve shows the mass enclosed in the one dimensional adhesion model with @xmath63 . the lower panel shows the same set of curves for @xmath34 and the middle panel is for @xmath60 . ]     as a function of mass @xmath78 in n - body simulations . the top panel is for @xmath75 , the middle panel is for @xmath20 and the lower panel is for @xmath33 . curves are shown for t_05 ( solid curve ) , t_05l ( thick solid curve ) , t_10 ( dashed curve ) , t_10l ( thick dashed curve ) , t_20 ( dotted curve ) , t_20l ( thick dotted curve ) , t_40 ( dot - dashed curve ) and t_40l ( thick dot - dashed curve ) . ] another important point to consider is that we have considered two well separated scales for fluctuations and there is no infall once fluctuations at the larger scales collapse . numerical experiments that can shed light on effects of this feature are also required to improve our understanding of issues . we also compared the collapse of a plane wave in an n - body with the collapse in the adhesion model with a finite effective viscosity . we found that :    * the adhesion model predicts the variation of density very well with a constant effective viscosity in regions with a given number of streams . * regions with a given number of streams coincide with the adhesion model with the same value of effective viscosity at all epochs . jsb thanks r.nityananda , t.padmanabhan and k.subramanian for useful discussions and suggestions . jsb also thanks varun sahni and uriel frisch for a useful discussion on related issues . numerical experiments for this study were carried out at cluster computing facility in the harish - 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we study the interplay of clumping at small scales with the collapse and relaxation of perturbations at much larger scales . 
 we present results of our analysis when the large scale perturbation is modelled as a plane wave . 
 we find that in absence of substructure , collapse leads to formation of a pancake with multi - stream regions . 
 dynamical relaxation of plane wave is faster in presence of substructure . 
 scattering of substructures and the resulting enhancement of transverse motions of haloes in the multi - stream region lead to a thinner pancake . in turn , 
 collapse of the plane wave leads to formation of more massive collapsed haloes as compared to the collapse of substructure in absence of the plane wave . 
 the formation of more massive haloes happens without any increase in the total mass in collapsed haloes . 
 a comparison with the burgers equation approach in absence of any substructure suggests that the preferred value of effective viscosity depends primarily on the number of streams in a region . 
 gravitation  cosmology : theory  dark matter , large scale structure of the universe