the conditions under which planetary formation is initiated in the protoplanetary disk remain poorly understood despite several decades of astrophysical research .furthermore , the continued discovery of extrasolar planets ( over two hundred currently ) further emphasizes the need to understand the formation of not only our own solar system , but of these numerous strange and diverse systems as well . at the forefront still remains the basic question of how growth occurs from dust to planet .the key properties of protoplanetary nebulae remain controversial , yet grain growth from dust to larger agglomerates ( cm - m ) , and from planetesimals ( km ) to planets must have occurred in some manner . while numerous studies of -body dynamics have addressed the growth of terrestrial planets from asteroid - size planetesimals which are largely decoupled from the gas ( e.g. , * ? ? ?* ; * ? ? ?* ) , it is the transition from agglomerates to planetesimals which continues to provide the major stumbling block in planetary origins .grain growth begins with sticking of sub - micron sized grains , which are dynamically coupled to the nebula gas and collide at low relative velocities which are size - dependent .these relative velocities can be caused by a variety of mechanisms , such as brownian motion for the smallest grains , differences in coupling to eddies in a turbulent nebula , and/or vertical , radial , and azimuthal drift driven by global gas pressure gradients .for the smallest sized grains and aggregates , sticking is caused by weak van der waals interaction forces ( although stronger forces may also act ; see , e.g. , * ? ? ?this sticking forms larger and larger aggregates until particles grow large enough to decouple from the gas and settle to the midplane where they may , barring strong turbulence ( although see * ? ? ?* for an exception ) , grow into larger objects , even planetesimals .there is considerable observational evidence supporting dust growth in protoplanetary disks , at least from sub - micron to millimeter scales , based on observations of the dust continuum emission originating from protoplanetary disks .the wavelength dependence of dust emission from ir to radio wavelengths suggests , in many cases , that typical grain sizes are in the millimeter range , which is orders of magnitude larger than interstellar grains .measurements of the contrast of the orion trapezium silicate emission feature at 10 m relative to the local continuum implies an increase in size from interstellar dust ( sub - micron ) to micron sizes in the disk photospheres .theoretical approaches to the study of dust coagulation normally involve solving the cumbersome collisional coagulation equation ( see eq .1 ) at each spatial location ( ) in the disk .the collisional kernel for particles of mass and usually involves some sort of sticking efficiency , and particle cross - sectional area , in addition to the particle - particle relative velocities .although the _ systematic _ particle - particle velocities that arise from gas pressure gradients are somewhat easy to incorporate into the collisional kernel , relative velocities in turbulence are more complicated . if one assumes that nebula turbulence follows a kolmogorov inertial range in which energy cascades from the largest , slowly rotating eddies , down to some minimum lengthscale where the intrinsic molecular viscosity can dissipate the macroscopic gas motions ( and turbulence ceases ) , one can obtain closed - form expressions for the particle - particle , and particle - gas turbulent relative velocities , even for particles of different sizes . numerical models of particle growth in the inner regions of protoplanetary disks indicate that inclusion of these turbulence - induced relative velocities is essential , as it provides a source of energetic collisions that , in the presence of fragmentation and destruction , leads to the ongoing existence of small grains even after several million years .  a full scale solution to the problem of dust coagulation for a size spectrum at every vertical and radial location in the protoplanetary disk , as a function of time ,is thus prohibitive even with today s computational capabilities .so it is advantageous to find alternative methods of extracting information from the coagulation process , such as time scales of growth , the largest particle size , the size that dominates the area ( radiative transfer ) , and the size that dominates the mass ( migration and redistribution ) without having to track the behavior of the entire mass distribution .  in this paper , we describe a method that utilizes the moments of the coagulation equation to accomplish this task , in particular when the functional form of the mass distribution can be assumed . in 2 ,we derive the moment equations from the coagulation equation , and compare direct integration of the coagulation equation with solutions using the moments approach in the case of a simple turbulent coagulation kernel .we also present two different approaches to obtaining solutions using a realistic kernel , when the form of the size distribution is assumed to be a powerlaw . in 3, we evaluate the different approaches developed in 2 for realistic kernels by numerical integration of the moments equations and compare these moments to those obtained by direct integration of the coagulation equation .we also compare the moments solution to an alternate analytical approach by . in 4 , as a demonstration of the practical application of the moments method , we calculate the wavelength - independent opacity , and discuss generalization to wavelength - dependent planck or rosseland mean opacities . in 5, we indicate how porosity may be included . in 6 , we present our conclusions .the moments method approach to modeling particle growth allows an attractive alternative to solving the full coagulation equation when it is only necessary to know a particular property , or properties of the evolving particle size distribution .the coagulation equation in its integro - differential form is given by    where is the particle number density per unit mass at mass , and is the collision kernel connecting the properties of interacting masses and , which typically takes into account mutual particle cross section , relative velocity , and a particle sticking efficiency .the last two terms on the rhs of eq .( 1 ) represent sources and sinks such as particle erosion , fragmentation and subsequent redistribution of the fragmented population , and gravitational growth .although we will not specifically address these mechanisms , we will indicate how they may be treated , and save implementation for a forthcoming paper .the motivation behind using the moments method is to avoid the computational cost inherent in eq .( 1 ) which entails solving the convolution integral ( the first term on the rhs of eq .1 ) at every location , and at every time step . depending on the functional form of the kernel , the second integral on the rhs may also need to be integrated repeatedly .given that the typical mass spectrum may involve bins to acquire the desired accuracy , the computational burden required becomes a detriment to any study involving a wide range of parameter space .the so - called brute force solution of the coagulation equation thus becomes the primary bottleneck in running 2d evolutionary models over extended periods of time .we define the -th moment of the distribution , where need not be an integer , as    where the units of are such that represents the total number density of particles , and is the total volume mass density of solids .the essence of the moments method is as follows .we multiply both sides of the coagulation equation ( eq .1 ) by , where is an integer , and then integrate both sides over mass :    we introduce a step function , such that for and otherwise , to extend the limits of the integral over from ( ) to ( ) . the convolution integral ( the first integral on the rhs of eq .3 ) then becomes    where on the rhs of eq .( 4 ) we have switched the order of integration and made the substitution .the rhs side of eq .( 4 ) may then be combined with the last double integral in eq .( 3 ) to yield the set of ordinary differential equations ( odes ) for the integer moments :   k(m , m^\prime ) f(m , t ) f(m^\prime , t ) dm dm^\prime,\ ] ]  where we have substituted for with no loss of generality . depending on the mass dependence of the kernel ( as illustrated below ) the right hand side can readily be expressed as products of moments of order or less , leading to a closed system of equations which may be solved using standard techniques . in this definition of the coagulation equation in which it is assumedthere are no sources and sinks , the first moment is constant in time , that is will not be constant due to , e.g. , size - dependent advection terms in the equations . such effects can be treated separately from the `` coagulation '' step . see 2.2.2 . ] .exact solutions to the coagulation equation have been obtained for some specific choices of the kernel ( e.g. , * ? ? ?* ; * ? ? ?* ) , the most simple being that of constant kernel . although the exact solution for can not be obtained from the moments equations , the time rate of change of the zeroth moment ( ) , can easily be obtained from eq .( 5 ) which reduces to and has the trivial solution ( see , e.g. , * ? ? ?* )    independent of the initial choice of distribution .likewise , the ode for , which can be associated with the density - weighted mean particle size ( ) , yields the simple solution . despite not knowing the exact solution, we are able to understand the behavior of general properties of the mass distribution with time through the moments equation without tracking the behavior of the full mass spectrum . thus , if it is only desired to know , for example , the time evolution of the particle representing most of the _ area _ ( first moment ) or most of the _ mass _ ( second moment ) of the distribution , then the advantage of the moments method becomes clear .a small number of moments is all that is necessary to determine the behavior of the system . in particular, we will be interested in the size of the largest particle in the entire mass distribution as a function of time ( which we show below can be computed from the integer moments ) , because these are usually the most rapidly drifting and most violently colliding particles .we can illustrate the moments method approach using a very simple turbulent coagulation kernel where  , and no sources or sinks .physically , this represents the product of for area , and for relative velocity of two particles in a laminar shear flowfield .if a sticking coefficient were desired then we would specifically have where . here, is a constant that depends on the reynolds number of the gas flow . inserting this kernel into eq .( 5 ) , we find the set of equations    we note that the physical derivation of kernels in terms of particle radius leads to _ fractional _ moments in terms of .these fractional moments look complicated , but can be solved for by simple interpolation using lagrange polynomials in terms of the more familiar integer moments .thus , any fractional moment may be expressed compactly in the normalized form :  ^{l^n_j(p)},\ ] ]  where is the number of integer moments , , and the exponent is defined as    with , and the are the binomial coefficients .the important thing to note here is that the order of the moments must remain less than or equal to the largest moment in order for the system to remain closed .in general , this will be true for realistic collision kernels ( see 2.2 ) .  to test the accuracy of the moments method, we integrated equations ( 7 ) using a fourth order runge - kutta scheme , and compared the results to a brute force integration of the coagulation equation ( eq .1 ) . for the latter , we integrated the distribution function at each timestep to determine the moments of the distribution , , and .we chose as our initial distribution for simplicity . since the form of the mass distributionis only assumed at , we consider this to be an example of an implicit assumption ( see 2.2.2 ) .the results of this calculation are presented in figure 1 .we have used two different resolutions for the brute force calculation in order to demonstrate how the higher resolution ( and much more computationally expensive ) case ( solid symbols ) approaches the moments approach solution .notice that the first moment remains constant as expected . giventhat general numerical errors can arise from the finite mass grid and coarse timesteps ( years ) used for the calculation , and that systematic errors may be introduced by the lagrangian interpolation , the fit is quite good .it is important to point out here that while the coagulation calculation required as many as hours of cpu time to complete on a 2 ghz machine , the moments calculation of eq .( 7 ) is essentially instantaneous .the saffman - turner turbulent coagulation kernel we used as an example in 2.1 is simple to utilize and is expressible explicitly in powers of the mass . in practice , however , the realistic coagulation kernels that we will be using will be more complicated than this simple example. a realistic coagulation kernel will be , at the very least , a product of a mutual cross section and a relative velocity    where , , and are the systematic ( pressure gradient driven ) particle velocities , and is the turbulent velocity contribution which in general is not separable into distinct functions of and  .the systematic velocities , which are derived for a discretized particle size distribution in the appendix , are complicated functions of the particle size through the stopping time ,    where is the relative velocity between the particle and the gas , and is the drag force on the particle of size which depends on the size of the particle relative to the mean free path of the gas molecule ( e.g. , see * ? ? ?depending on whether or determines whether the stopping time itself depends on the instantaneous relative velocity between the particle and the gas ( stokes regime ) or does not ( epstein regime ) . in the latter case ,calculation of the systematic velocities ( ) and gas velocities ( ) for a particle size distribution is straightforward . in the former case ,iterations are required to correctly calculate ( see appendix ) .the turbulent velocities are less straightforward to implement than their systematic counterparts because of the different coupling that exists between particles and eddies of different sizes ( and thus stopping and decay times , respectively ) .it has not been until very recently that closed form expressions for the turbulence - induced velocities ( for a particle size distribution ) were derived .these expressions can be written separately for the so - called `` class i '' and `` class ii '' eddies is much shorter than the eddy decay time and the time it takes to cross the eddy ; thus , they will align themselves with the gas motions of the eddy prior to its decay .class ii eddies then are defined as ones in which the eddy decay time is fast compared to the particles , and thus only provide a small perturbation to the particle motion . ] as :      where , is the reynolds number , is the turbulent gas velocity with the sound speed ( e.g. , * ? ? ? * ; * ? ? ? * ) , and the particle stokes numbers , where is the turnover time of the largest eddy , typically taken to be .the boundary between class i and class ii is defined by the `` combined '' stokes number and the values of the are obtained from the equation :    where . in our calculations, we will make use of these expressions when comparing cases with turbulence - induced velocities .given that the moments method requires that we express the kernel in terms of fractional or integer moments , the problem becomes expressing the relative velocity ( eq .10 ) in a form that satisfies this requirement . as an example , the systematic velocities , , and can each be fit easily by a finite series in fractional powers of as where the coefficients can be found by fitting points to the function ( an integer ) .this leads to the system of equations   coefficients then follow from matrix inversion .similarly , one can find expressions , and so that we may construct the laminar expression for in terms of the variables and by multiplying out the individual terms , e.g. , and so on .we find then that we can express the square of eq .( 10 ) as  ,\ ] ]  where , and for closure .this determines the matrix of coefficients which operate on the finite power series in and .although we can not use the same approach for solving for in eq .( 10 ) because of the radical , we were motivated to express in a similar form , i.e. , as $ ] , and .the problem reduces to solving for the coefficients directly , by similar matrix inversion techniques .unfortunately , the number of points needed to obtain an accurate representation of the two - variable function this way exceeds the limitations of the inversion . in practice we found that we could solve a system for with at most points ( matrix elements ) , whereas eq .( 16 ) carries much less restriction in the number of points needed because we could construct by the product of accurate representations of single variable functions .furthermore , due to the coupling evident in the turbulence induced velocities ( equations 12 and 13 ) , the turbulent velocities are not separable functions of the masses and , further complicating the analysis ._ powerlaw assumption _ : fortunately , it turns out that , although the direct calculation approach to described above would be mathematically appealing in the sense that the moment equations would remain implicit ( i.e. , the form of the mass distribution would only be assumed at ) , it is not necessary . in the next sections we describe two alternative approaches to dealing with realistic coagulation kernels that employ the moments method under the assumption that the form of the mass distribution is a powerlaw . such an assumption is not entirely unfounded .a number of detailed models by have shown that powerlaw size distributions result , which have nearly constant mass per decade radius to an upper limit which grows with time until the frustration limit of around a meter is reached , and ( under turbulent conditions , at least ) growth stalls .similar trends are found by in which different assumptions about collisional ejecta are made .these authors do find minor fluctuations in the distribution , but if one is primarily interested in general properties of the distribution , such as how the largest particle size changes with time , and not the fine structure of the mass distribution itself , the approach is quite advantageous .there are reasons to believe that a powerlaw is a natural end - state , especially those with equal mass per decade , because they have self - preserving properties for the collision kernels of interest .the significance of the upper mass cutoff depends on the assumed slope of the powerlaw . in all real distributions, there will be a rapidly decreasing abundance of particles for masses exceeding some threshold , even though the abundance may not drop immediately to zero as in our assumed model .the rare , extremely large particles might be of interest for some applications , but our focus will be the particle size carrying most of the area , or most of the mass ( the first or second moments ) . for powerlaw distributions with , is itself the mass of the particle carrying most of the mass and is independent of the selection of a lower particle size cutoff . for ,the distribution contains equal mass per decade , and for , most of the mass is in the small particles and the value of depends on the lower particle size chosen .most realistic distributions , and those most commonly treated in the literature , have ; here we assume , a widely used fragmentation powerlaw . in this case , a strict cutoff at represents a well - defined distribution with easily - understood moments where most of the mass is at and the area is nearly equally distributed per decade with a mass dependence .although we do not include either imperfect sticking or fragmentation at this stage in the model , we believe that the moment equations as expressed in eq . ( 5 ) will remain valid up to a `` fragmentation barrier '' , which may be defined as that size for which the typical disruption energy of a particle is on the same order as the energy of identical colliding particles .the fragmentation barrier will also depend on one s choice of nebula parameters .this treatment ( up to the fragmentation size ) is consistent with recent work by ( their fig .13 ) which shows a constant powerlaw mass distribution up to a cutoff size which then falls off abruptly .this `` knee '' in the distribution represents that efficient fragmentation size .once reaches the fragmentation barrier , growth beyond this stage would need to be treated in a different manner , for example , by simple sweepup of small , less disruptive particles by large ones .creation and disruption of these particles can be handled as part of the source and sink terms in which for example , disrupted particles are assumed to be fragmented back into a powerlaw distribution ( which is suggested by experimental evidence and widely assumed by other modelers ) as opposed to monomers .the model as presented within this paper , however , should be useful for the early stages of protoplanetary nebula particle growth relevant to spectral energy distributions and mri suppression .we leave the incorporation of growth stages beyond the efficient fragmentation stage for a later paper .motivated by our discussion of the previous section , if we assume that the form of the particle mass distribution function remains a powerlaw at all times , we may express , such that is the growing upper limit of the distribution , is the slope which is assumed to be constant ( although see below ) , and is a normalization coefficient . taking the lower limit of the mass distribution to be , then moments as expressed in eq .( 2 ) are explicitly given for by    we then take the time derivative of eq . (17 ) for the zeroth and second moments , and substitute these expressions on the lhs of the corresponding moment equations in eq .( 5 ) to get      where eq . ( 18 ) is valid for , and and are the integrals on the rhs of eq .( 5 ) for and , respectively .that is      where the kernel is given by , with , , and is the particle material density , assumed constant .note that for the special case of , eq . ( 18 ) has a slightly different form which depends on .after some simple algebra , eq.s ( 18 ) and ( 19 ) can be written as  ,\ ] ]  ,\ ] ]  which we integrate using a fourth order runge - kutta scheme .equation ( 17 ) then gives the moments and as a function of time , which can be directly compared with direct integration of the same conditions using the coagulation equation ( eq .the advantage of this approach ( in which a powerlaw is assumed at all times ) is its transparency ; that is , the variables being sought ( and ) are solved for directly .furthermore , the change in the coagulation kernel as the particle size distribution changes is included because the kernel is updated and explicitly integrated into and with every time step .this will prove advantageous when additional effects such as sticking are included in the kernel .in addition , source and sink terms need not parameterized in terms of integer moments and can be implemented directly .an unfortunate disadvantage of this approach is that because the kernel must be integrated ( in fact several times ) over both and every time step to get and , the cpu time involved is significantly longer than a fully implicit case ( e.g. , 2.1 ; also see 2.2.2 ) ; however , it remains a much faster approach ( orders of magnitude ) than solving eq .( 1 ) directly since the cumbersome convolution has been eliminated .   here , we present an alternative moment - based approach to solving the realistic coagulation kernel in which the integer moments appear directly . unlike the previous case of 2.2.1 , where the double integral of eq .( 5 ) is used to get the functions of , and , we only integrate over one mass variable ( i.e. , only integrate one of the integrals ) , defining the functions      where the form of the kernel is the same that in 2.2.1 .we then fit and with a finite series in fractional powers of in the same manner as given in eq .( 15 ) . substituting these functions in place of one of the integrals ( say over ) , we may integrate over to get  ^{\frac{1}{2}(p_i-1 ) ( p_i-2)}\left[m^\prime_1\right]^{-p_i(p_i-2)}\left[m^\prime_2\right]^ { \frac{1}{2}p_i(p_i-1)},\end{aligned}\ ] ]  where we have made use of equations ( 8) and ( 9 ) to express the solution in terms of the integer moments . here , , , , and .the are fairly smooth functions over a large range of particle radii ; , however , the accuracy of fitting a single series in fractional moments over a very broad range of particle sizes ( i.e. , over many orders of magnitude ) may drop off significantly as the broadness of the range increases .this issue may be circumvented by employing a piecewise fit to the integrated kernels .it is interesting to note that by this definition of , we have effectively accomplished what we set out to do in our discussion at the beginning of 2.2 , that is , defining the coagulation kernel in terms of finite series in powers of the mass .the difference here is that we have done so through the first integral of the kernel , and not the kernel itself .this means that the mass distribution is expressed in the calulation of the , but remains implicit in the definition of the odes ( eq .thus , the method is semi - implicit , because the rhs of the equations above can be expressed in terms of the moments ( as defined in eq .2 ) . equations ( 26 ) are then integrated using the fourth order runge - kutta method , and may be compared with the results of 2.2.1 and the direct integration of eq . ( 1 ) .this semi - implicit approach tracks the evolving kernel through the integration of equations ( 24 ) and ( 25 ) after every timestep , thus the computational time involved is similar to the explicit case . in order to update the kernel, one may solve for the new after each using the equation ( )    the new normalization coefficient can then be found from the definition of . we then reintegrate equations ( 24 ) and ( 25 ) under the powerlaw assumption , and then proceed to fit the with a finite series in fractional powers of .although , in principle , any other two moments could be used to obtain , , which lies between and , and because it roughly characterizes the evolution of the largest particle ( see discussion at the end of 2.2 ) , seems the most consistent choice .the -th moment is calculated using the lagrange polynomial interpolation scheme ( eqs . 8 and 9 ) .the advantage of the moments method lies in the ability to express the differential equations in terms of the moments of the distribution ( i.e. , their integrated properties ) .if a more explicitly mass - dependent approach is adopted ( as is the case in 2.2.1 , and the semi - implicit approach described here ) , then the computational time significantly increases .one can improve the speed of computation by calculating the periodically , or in the extreme case , only at which would make the approach truly _ implicit _( e.g. , 2.1 ) .the advantage of an implicit approach is that it becomes fully general ( the form of is only assumed at the onset ) , and also in the time it takes to solve ( minute ) .the bulk of the time is spent in the integration of equations ( 24 ) and ( 25 ) which would occur only once .the disadvantage , of course , is that the particle velocity distribution is not updated as it changes with time ( due to , e.g. , changes in the bounds of the size distribution ) .we present examples of both extremes in 3 .if one wanted to implement a mass- or velocity - dependent sticking coefficient , it can readily be included in the integration to obtain the .the additional inclusion of source and sink terms due to erosion , fragmention , or gravitational growth in this semi - implicit formalism would require that we fit these terms in a similar manner to the so that their subsequent integration over all will yield sums over integer moments weighted by different sets of coefficients .caution must be exercised in fitting , e.g. , the gravitational growth term to ensure that the system of equations remains closed . under such circumstances , a fully explicit approach such as that of 2.2.1 may be preferred .alternatively , these effects may be included in particle - histogram space in between coagulation interations .finally , we should point out that allowing for other parameters ( such as the index of the powerlaw ) to vary with time , does not pose a problem in either of the approaches we have presented . in both cases , one would simply need an additional moment , e.g. , to determine . similarly , a bifurcated distribution in which the powerlaw exponent changes at a particular particle size ( see , e.g. , * ? ? ?* ) may also be studied .  in 3, we will compare the two approaches to the direct integration of the coagulation equation for cases in which there are only systematic velocities ( ) , as well as cases in which the velocity differences are induced by turbulent motions .we carried out several calculations in order to demonstrate the accuracy of each alternative method compared to the brute force integration of the collisional coagulation equation .for the purposes of comparison , unless otherwise noted , we chose the initial conditions to be a minimum mass solar nebula at 1 au at a height of km above the midplane , and a particle size distribution with a minimum initial radius of 1 cm and a maximum initial radius of 10 cm .the powerlaw exponent , which is assumed to be constant in these calculations , is representative of a fragmentation population .standard integrations were carried out with a timestep of years .   in figure 2, we compare the case in which there are only systematic ( pressure - gradient driven ) velocities between particles ( ) .the solid curves represent the explicit assumption ( invariant powerlaw slope assumed in integration of equations 22 and 23 ) , while the dashed curves represent the implicit assumption . that is , in the integration of eq .26 , the form of the mass distribution is assumed only at .as before , the symbols represent the brute force calculation for two different grid resolutions ( solid = 100 bins , open = 1000 bins ) .since with the explicit assumption we do not solve for the moments specifically , the values for the solid curves were obtained by substituting the time integrated values of and back into eq .( 17 ) for only .this is because although we have used the moment equations to obtain these results , the explicit assumption of an invariant powerlaw size distribution means that only the equations for the moments are needed ( although see 3.2 ) .this is not true , however , for the implicit ( and semi - implicit ) assumption , where all the moments appear in the differential equations .we see that there is excellent agreement between both approaches and the numerical values obtained with the highest resolution case .in fact , the agreement between the explicit and implicit assumptions is also quite good .however , at this stage there has not been a great deal of growth ( the largest particle size in the distribution has only grown to cm in size by the end of the simulation for the explicit case ) .note that even though in the implicit case the are calculated only once , the estimate for the largest mass for the evolving distribution is still calculated using eq .( 27 ) . at least in the case of minimal or slow growth, it appears that an implicit approach , or even periodic calculation of the , may be sufficient .numerical glitches in the low resolution brute force case arise from the interpolation scheme for sampling the kernel .in particular , these glicthes are likely enhanced because the systematic relative velocities quickly approach zero for identical particle sizes ( with the effect much more prominent for larger particles , hence not appearing so much in ) .the higher resolution case contains enough points to smooth out this effect .note that ( in all simulations ) the direct integration of the coagulation equation gives a constant as is to be expected given that we found in the derivation of the moment equations in the absence of sources and sinks ; this further validates the numerics of the brute force solution even for the complicated collisional kernel being utilized , and also provides a posteriori validation of our result that from , e.g. , eq .( 5 ) , which further validates derivation of equations ( 24 ) and ( 25 ) which was based on symmetry of the kernel .next , we explored cases in which the systematic velocities were set to zero so that velocity differences between particles are due only to those induced by turbulence . depending on the magnitude of the turbulence parameter , these induced velocities can be either large or small relative to the systematic velocities . to demonstrate, we ran cases for three different values , and . in the absence of any mechanism to counter growth ( e.g. , fragmentation ) , larger translates to faster rates of growth ( due to larger relative velocities ) .  in figure 3we plot the results for , for the explicit ( solid curves ) and implicit ( short dashed curves ) approaches .the growth rate is more rapid than in the laminar case , with the second scaled moment % larger ( compared to fig .2 ) at the end of the run , meaning that the largest particle size achieved is cm .we note that , as was the case for the curves in fig .2 , the explicit and implicit cases are very similar ; however , this is not the case for the curves .the explicit approach overestimates the the value of compared to the highest resolution brute force calculation , whereas the implicit approach significantly underestimates the zeroth moment .both approaches understimate the value of . when we compare integrations of for smaller values of as we do in figure 4 , we find that both approaches fit the coagulation calculation quite well. the growth rate for these two values of are more in line with growth rate found when there are only systematic velocities , so the agreement should not be surprising .the long - dashed curves in figures 3 and 4 are the implementation of the semi - implicit approach ( 2.2.2 ) in which the are updated at every time step . in this case, the semi - implicit approach provides a much a much better fit to the brute force calculation , indicating that the kernel is evolving fast enough that an implicit approach can not capture this effect .we consider the slight discrepencies between the semi - implicit approach and the brute force calculation to be as much a result of grid resolution as inaccuracy in using the lagrange polynomial fits to the fractional moments .the explicit and fully implicit approach values of apparently understimate the largest particle size relative to the semi - implicit approach ( which gives a value of cm ) .finally , we explored a variation of the explicit approach in which the condition was strictly enforced ( recall that does not appear in equations 18 - 19 ) .this amounts to replacing eq .( 18 ) ( derived for ) with the corresponding equation for . as a result, we can not simultaneously fit both and .alternatively , we can use only eq .( 19 ) for the second moment , by substituting ( from the definition of , eq .17 ) in eq .( 19 ) so that eq .( 19 ) alone determines the growth of the largest particle .the differential equation for then becomes  .\ ] ]  the results of the integration of eq .( 28 ) are shown as the dotted curves in figure 3 and for the case in figure 4 .it is clear that this modification provides a much better fit to , perhaps even better than the semi - implicit case above .however , using calculated from eq .( 28 ) , and solving for does a poor job fitting ( see figure 3 ) .thus we conclude that although the modification of the explicit approach matches the brute force calculation of quite well , only the semi - implicit case is able to provide a simultaneous fit to both the zeroth and second moments ( under the condition that ) . has developed a simplified analytical approach for dealing with the growth of particles in a turbulent regime , in which the particle size distribution is parameterized by a powerlaw in particle mass with the same exponent we have used in this paper ( ) .similar to what we have presented in previous sections , the underlying assumption of is that collisions between particles occur frequently enough that a steady - state balance is reached in the form of the particle size distribution , but with an upper size cutoff that varies with time .thus , the model of is _ explicit _ and follows only two parameters : the growth of the largest particle , and the normalization factor .  given the similarity of the underlying assumptions for the garaud model and the examples we have presented , it is a useful exercise to compare the results of the two approaches directly . the growth of the largest particle in the garaud model ( her eq .36 ) , expressed in terms of the notation used in this paper , is given by    in eq .( 29 ) is the stokes number for the largest particle , is the scale height of the gas , is the adiabatic index of the gas , and in this expression it is assumed that the sticking coefficient , and that .the above equation is similar to the formula for grain growth proposed by to factors of order unity . finally , we point out that the garaud model for particle growth is restricted to the value in order to preserve its completely analytical nature . as a means of a fair comparison , we chose to compare the garaud model to our explicit case ( eq .28 ) for reasons explained below . in the limit of and ,( 28 ) becomes    we present the results of our comparison in figure 5 for the same initial conditions as described at the beginning of 3.2 ( upper curves ) .we find quite generally that our explicit calculation ( eqs .28 and 30 ) leads to a faster growth rate than what is predicted from the garaud model , initially .we note that the garaud expression steepens quickly for cm , suggesting that the growth rates of the two approaches are more comparable at later times . however , the minor `` kink '' in the long - dashed curve is due to the shift from epstein to stokes flow .a much more subdued kink is visible in the explicit approach which uses the full expressions for the turbulent velocity , whereas in the derivation of eq .( 29 ) , it is assumed that the stopping time in the stokes regime is defined by some mean characteristic velocity which leads to a much more noticeable discontinuity .regardless , the overall more subdued growth rate elicited by eq .( 29 ) ( despite the steepening at later times due to a shift in flow regimes ) is apparent from the lower set of curves in fig .5 where the initial conditions were chosen with cm . for this case ,growth occurs only in the epstein regime , but still , the curves for the explicit approach and that calculated from eq .( 29 ) begin to diverge .thus , the garaud expression apparently underestimates the growth rate relative to our approach .we emphasize that the treatment of particle growth by requires several approximations in order to derive a purely analytical expressions for .besides the aforementioned restriction of , garaud approximates the full expressions for the turbulent velocities that we use here by partitioning into seperate cases dependent on the particles stopping time relative to the turnover times of the smallest and largest scale eddies .furthermore , some question may be raised as to the comparability of the moment equations used here to derive eq .( 28 ) , versus the particle growth equation used by ( * ? ? ?* her eq . 29 ) . the equation used by is more akin to a `` sweep - up '' equation , with no sources or sinks , than to a formal coagulation equation .because it bears some resemblance to the equation for ( with some algebra , to factors of order unity ) , we concluded that our eq .( 28 ) is the appropriate analog . despite the differences in growth rate, we find the agreement in the general trend of growth of the two approaches reassuring .evolutionary models of protoplanetary nebulae , giant planet atmospheres , etc . must somehow treat the escape of thermal radiation . in most cases ,particles provide the primary opacity for these models .observations of these and similar objects often rely on spectral energy distributions ( seds ) which can be compared to a model once the model s internal temperature distribution is known ; clear evidence is seen for grain growth in many cases ( see review by * ? ? ?because of the nearly insurmountable computational burden involved with performing a fully self - consistent calculation of particle growth by coagulation along with an already difficult fluid dynamical calculation , most modelers simply assume some invariant particle size distribution , such as the mrn interstellar grain distribution , or make arbitrary assumptions about particle growth .  in the simplest regime ( monodisperse particle radius larger than a wavelength ) , the particle opacity can be written as the area per unit mass :    thus growth in radius from 0.1 to 1 mm leads to a factor of change in opacity . to the degree that this wavelength - independent regime holds ,including particle size evolution by the moments method in one s evolutionary models would allow a very simple way to track particle growth and decreasing opacity .for instance , equation ( 31 ) above is easily generalized to the area per unit mass integrated over the size distribution :    as an application of the moments method in figure 6 , we have calculated the decrease in opacity ( given by eq .32 ) with time using the semi - implicit approach .an initial particle size distribution with a lower bound of cm and was used .both the pressure gradient driven systematic velocities and the turbulence - induced velocities were used . in the absence of any mechanism to hinder particle growth , larger values of to more steeply decreasing opacities with time .  in a regime where the particle extinction efficiency is wavelength - dependent ( say, if the particles are comparable to or smaller than the wavelength ) , one simply integrates over the powerlaw mass distributions resulting from the moments model .for example ,   these opacities can be used to calculate planck or rosseland ( wavelength - averaged ) means for use in radiative transfer models .recall that the powerlaw slope can be freely adjusted within a small but plausible range to explore different growth regimes .fractal growth of particles by low - velocity sticking of small solid monomers with radius and mass ( and/or aggregates of such monomers ) causes them to have a density much less than the material density of the monomers .these porous particles can be described as fractals with dimension , such that the particle mass increases proportionally to where is some effective radius and is the fractal dimension .thus the particle s internal density is a function of particle size :    for a typical situation where  , and thus the product is a constant across a wide range of particle sizes ( until compaction sets in ) .these more complex but quite plausible particle density - size relationships complicate the expressions for particle stopping time and stokes number ( 2.2 , eq .11 ; also , see appendix ) .because the particle stopping times enter in through the collisional kernel , the fractal nature of particles can be accounted for using the method of moments in a straightforward manner while maintaining the criterion for closure of the system .we can verify this by noting that the mutual particle cross section    is proportional to , whereas the stopping time ( eq .11 ) in the epstein regime ( the regime that would apply to fluffy fractal aggregates ) where the drag force ( e.g. , * ? ? ?* ) is given by    in the limit of small stokes number ( ) , both the systematic and turbulence - induced velocities are proportional to , so that , and the entire kernel is proportional to . for larger , the kernel has a shallower powerlaw dependence .thus , in general , the dependence of on the mass is , preserving closure of the system , even for fractal particles .even though the variation in the properties of the evolving distribution due to porous particles are incorporated directly into the kernel , and are folded into the integration of the explicit approach , the effects of fractal aggregates can , nonetheless , affect which moments characterize what properties in the semi - implicit ( or implicit ) approach .for example , the wavelength - independent opacity expressed in eq .( 32 ) takes the form .it should be noted that the value represents a special case in that the particle - to - particle relative velocities in the epstein regime do not depend on the mass of the fractal particle .indeed , this would seem to indicate that fractal growth can proceed unabated with the corresponding stopping time of the fluffy aggregate remaining the same as that of a single monomer , which would have a significant effect on other particle properties .in particular , the wavelength - independent opacity for is constant .however , impacts will eventually lead to compaction or even fragmentation depending on the relative velocities ( e.g. , * ? ? ?both fractal grains and non - fractal particles in the same mass distribution can be treated in the explicit approach without any modifications , while a piecewise fit to the integrated kernel ( 2.2.2 ) can be used in order to account for the change in regimes in the semi - implicit approach .we have demonstrated an approach to solving the collisional coagulation equation with an arbitrary collisional kernel which should be useful in cases when it is only necessary to keep track of general properties of the distribution .this approach involves solving a finite set of coupled differential equations in terms of the integer moments of the particle size distribution .the number of equations ( and thus moments ) needed depends on the number of properties being tracked .the advantage of the moments method approach is that it allows for considerable savings in computational time compared to direct integration of the coagulation equation , which requires keeping track of every particle size at every spatial location and timestep .  in this paperwe have specifically studied the growth of the largest particle under the assumption that the particle size distribution is a powerlaw ; however , the technique can be extended to track other properties of the distribution that may change with time .there are many reasons to believe that a powerlaw size distribution is a natural end - state of particle growth , especially those with equal mass per decade , because they have self - preserving properties . with the assumption of a powerlaw distribution , we have provided two different approaches to solving the moment equations , one explicit in which the powerlaw assumption is enforced rigorously at all times , and a semi - implicit approach in which the kernel is integrated over one of the mass variables as much as once every time step .the latter approach can be made fully implicit by only assuming the form of the mass distribution at .these approaches are significantly faster than solutions of the coagulation equation because , in particular , the convolution integral ( first term on rhs of eq .1 ) has been eliminated . in realistic evolutionary models ,intermediate steps performed in particle `` histogram '' or `` size - distribution '' space may be interleaved with moments - based coagulation steps in order to account for , e.g. , advective / transport terms .we have compared these alternate approaches to the brute force integration of the full coagulation equation for cases in which there are only systematic velocities , and cases in which the differences in velocity between particles is induced by turbulence .if the growth rate is gradual , the explicit and implicit approaches match the brute force calculation well ( fig .2 ) , whereas faster growth rates are more difficult to model ( fig .we find that we are able to use the semi - implicit approach in which the are updated at every timestep , and a modification to the explicit approach , in which we solve the equation for only with the assumption of strictly enforced , in order to compensate for the faster growth rates .the modification to the explicit approach is useful if one is not particularly interested in following the evolution of the number density of particles , or other properties which may be decribed by ( fractional ) moments ( e.g. , see 4 ) .our results also suggest that a fully implicit approach is probably most useful under circumstances in which the kernel depends on the mass in a straightforward manner ( e.g , the saffman - turner kernel , 2.1 ) .we have compared the approaches developed in this paper to an alternative model for particle growth in a turbulent nebula .we have found that there is fairly good agreement in general , but that the curves diverge as time proceeds .this does not appear to be particle - size dependent , or due to a shift in the flow regime .the garaud expression ( eq .29 ) underestimates the growth rate of the largest particle size relative to our approach by only % in our comparison .we note that the advantage of the method of is that it is purely analytical ; however , preservation of her analytical approach requires , amongst other things , the powerlaw exponent be restricted to .our approach has no such restriction on the choice of exponent , nor for to even be a constant .  as a sample application, we show how the moments method can be used in one s evolutionary model to track particle growth and opacity . as a specific case , we calculated the change in wavelength - independent opacity with time for an initial particle size distribution with upper and lower bounds of cm .both systematic and turbulent velocities were included . in the absence of any mechanism to counter growth ,the opacity decreases sharply for higher choices of the turbulent parameter .extension to cases in which the extinction efficiency is wavelength dependent is straightforward .such opacities can be used to calculate planck or rosseland wavelength - averaged means for use in radiative transfer codes .finally , we indicate how porous particles with fractal dimension can be accounted for in the moments method .the particle - size density relationships that arise affect the particle stopping times and stokes number , both of which appear only in the collisional kernel .thus implementation is straightforward .we can treat mass distributions composed of only fractal grains , or both fractal and non - fractal particles . in either case, relatively little modification is needed in the explicit approach , whereas with the semi - implicit ( and implicit ) approach , a piecewise fit to the integrated kernel using the method as described in 2.2.2 can be used to account for the change in particle growth regime when both types of particles are included.  the computational burden of directly solving the coagulation equation makes it quite prohibitive to explore large regions of parameter space , and thus it serves as the primary bottleneck in evolutionary growth models .the approach demonstrated herein is intended to obtain robust , quantitative results for disk properties such as particle growth timescales and `` typical '' particle sizes that may be used in modeling efforts that are focused more on the larger problem of planetesimal formation .although we have not included sticking or fragmentation in this paper , we believe that the moment equations ( eq .5 ) will remain valid at least up to the fragmentation barrier .this size will depend on the assumed particle strengths and choice of nebula parameters .the treatment of growth up to the fragmentation size is consistent with recent work by for the case in which additional effects such as radial drift are not included .the model as presented in this paper , however , should be useful for the early stages of protoplanetary nebula particle growth relevant to spectral energy distributions and mri suppression .  in a forthcoming paperwe will explore the effects of a variable sticking coefficient .furthermore , we will explore the addition of source and sink terms such as gravitational growth , erosion , sublimation , and condensation in addition to fragmentation. the ultimate goal will be to apply this methodology to a global model that studies the evolution of both the gas and solids in nebular and subnebular ( giant planetary ) environments .in this appendix , we generalize the basic equations of ( * ? ? ? * also , see tanaka et al .2005 ) for a two - component fluid by extending the particle component to incorporate a size distribution .we adopt cylindrical coordinates , designating the corresponding gas velocity components as and the particle velocity components as .the equations of motion of particles and gas for the radial and tangential velocities , generalized for a particle size distribution , are given by   where the system is assumed to be axysimmetric . here, with the stopping time of a particle of radius , and are the gas mass density and pressure , and is the material density of particles of radius . we have not included the vertical component equations ; here , generalization to a particle size distribution is straightforward since it remains true in general that ; that is , the vertical gas component of the velocity is negligibly small compared with ( see appendix , * ? ? ? * ) .  if the dust stopping time is short compared to the orbital period , we can seek steady - state solutions for the velocity components . setting ,this system can be solved exactly in the epstein regime . by solving eq .( a1 ) for and inserting into eq .( a2 ) , we obtain    where was obtained by insertion of back into eq .( a1 ) . in eqs .( a5)-(a6 ) we have expressed the last equality in terms of the stokes number , with being the local kepler frequency .these expressions for the individual particle velocity components can be inserted into eqs .( a3 ) and ( a4 ) to yield expressions for radial and tangential gas velocity components . with little difficulty , one finds        from these expressions ,one can easily obtain expressions for the relative velocities with respect to the gas and , or the relative velocities between particles of different sizes and .these expressions are also applicable for the stokes regime in which the stopping times are a function of the relative particle - 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Abstract: we demonstrate an approach to solving the coagulation equation that involves using a finite number of moments of the particle size distribution . this approach is particularly useful when only general properties of the distribution , and their time evolution , are needed . the numerical solution to the integro - differential smoluchowski coagulation equation at every time step , for every particle size , and at every spatial location is computationally expensive , and serves as the primary bottleneck in running evolutionary models over long periods of time . the advantage of using the moments method comes in the computational time savings gained from only tracking the time rate of change of the moments , as opposed to tracking the entire mass histogram which can contain hundreds or thousands of bins depending on the desired accuracy . the collision kernels of the coagulation equation contain all the necessary information about particle relative velocities , cross - sections , and sticking coefficients . we show how arbitrary collision kernels may be treated . we discuss particle relative velocities in both turbulent and non - turbulent regimes . we present examples of this approach that utilize different collision kernels and find good agreement between the moment solutions and the moments as calculated from direct integration of the coagulation equation . as practical applications , we demonstrate how the moments method can be used to track the evolving opacity , and also indicate how one may incorporate porous particles .


analysis of geophysical flows often employs techniques that decompose the velocity field in a manner that will yield a desired insight .a commonly used technique is the mean - eddy decomposition  [ eq : dyn ]   where the field is described by the unsteady eddy activity around a mean state ; henceforth , and denote the time average ( mean ) and the residual ( unsteadiness or eddy ) , respectively . here is the velocity field and represents any property of the flow such as temperature , chemical or biological properties . in the absence of unsteadiness , the kinematic transport occurs only along the mean streamlines on which is everywhere tangent .an effect of the unsteadiness is to stir the flow instantaneously and induce mixing across the mean streamlines over time .it is also well accepted that instantaneous pictures of the unsteady flow themselves do not indicate transport explicitly .a variety of eulerian and lagrangian methods have been developed to study transport observationally , analytically , and numerically .the companion paper presented a new transport method that is a hybrid combination of lagrangian and eulerian methods .this paper develops a framework for the analysis and diagnosis of transport processes based on this new method .our main focus is on two - dimensional geophysical flows that may be compressible . to introduce our method , we begin with a brief discussion of lagrangian and eulerian methods specific to our needs .the basis of any lagrangian method involves the tracking of individual fluid particles by solving the initial value problem of .starting from at time , a particle trajectory at time is given by the temporal integral of the local velocity field along itself :    a very rudimentary description of lagrangian transport may be obtained from a so - called `` spaghetti diagram '' which is constructed by simply plotting the trajectories .typically this results in a complex tangle of curves from which detailed a detailed assessment of lagrangian transport may prove difficult . in recent years the mathematical theory of dynamical systems has provided a new point of view and tools for classifying , organizing , and analyzing detailed and complex trajectory information by providing a theoretical and computational framework for an understanding of the geometric properties of `` flow structures '' .recent reviews of the dynamical systems approach to transport are given in .nevertheless , there still remain many challenging problems to be tackled by lagrangian methods .quantifying lagrangian transport is extremely elaborate in general .while techniques based on dynamical systems theory are conceptually ideal for tracking transport of fluid particles , they have not proven as useful for studying transport of , unless is uniform and passive . moreover , lagrangian methods are not suitable for separating and/or isolating the roles played by the mean state and the unsteadiness in the trajectory .in contrast to lagrangian methods , transport quantities computed with eulerian methods utilize information taken at pre - selected stationary points . at a station , the most basic eulerian transport may be given by the temporal integral of the local velocity during a time interval  ] over a spatial segment  ] starting from when a minimum of occurs first time after the spin - up ; the number in the subscript of represents time in day from here on .we also define } ] as the first and the second half of } ] , increases from a minimum to a maximum while reaches a minimum after about .conversely during } ] .the eddy streamfunction field associated with provides an instantaneous pattern of the eddy activity . in the double - gyre application, shows two types of eddy activity ( figure [ fg : dg_me]b ) .one is the westward propagation of rossby waves , whose latitudinally elongated structures are especially visible in the eastern basin .typically there are three waves in the entire basin , although the one in the western basin is distorted around the dipole region .the travel time of the wave from the eastern to the western boundary is 453(=151)days .each wave has the longitudinal width 333(=1000/3)km and travels with the propagation phase speed 2.2(=1000/453)km / day .as mentioned briefly in the introduction , for our study of inter - gyre transport in the wind - driven ocean , the axis of the eastward jet is chosen to be .it is the separatrix that connects the western boundary to the eastern boundary .we will discuss this in more detail in section [ sec : bsc_c ] .although the westward propagation of rossby waves is seen in almost the entire ocean basin , the dipole region has the eddy activity that is more energetic .the left panels of figure [ fg : dg_ph4_psie_phi ] show the phases of these eddy vortices during } ] or } ] or } ] ,  ] between the eddy vortex centers with the middle one stronger than the other two . in between consecutive intensifications ,the flux zones weaken and quickly propagate along as reaches minimum ( ) or maximum ( ) . the propagation speed and direction of the three flux zonesare the same as those of the circulating eddy vortices .  as the eddy vortex leaves the mean jet axis near at , a weak fourth flux zoneis generated over  ], the flux zones associated with them also merge there .the time method uses the streamlines associated with as the eulerian boundaries across which the transport is estimated . along ,the flight - time is a natural choice of the coordinate variable because is obtained by solving for with a choice of initial position .particle advection along in the mean flow is referred to as the _ reference trajectory_. starting from at , the reference trajectory is uniquely parameterized by and can be written as using the flight - time coordinate .some key locations on are shown in figure [ fg : dg_me ] . in , is the location that is far enough from the hyperbolic stagnation point of on the western boundary point in so that becomes non - negligible to induce the instantaneous flux ; and are the locations where the ( meandering ) jet axis makes sharp turns . in , and the locations where the centers of the eddy vortices pause to intensify ; is where the eddy centers leave ; around , two types of variability , the circulating eddy vortices in the upstream and the rossby waves in the downstream , meet on .accordingly , , , , and are the boundary points of the flux zones along in as the eddy vortices pause to intensify and meet the rossby waves in .the flight - time coordinates of , , , , , and are , , , , , and in the unit of days , respectively , starting with located very close to the western boundary .   the hovmller diagram of the instantaneous flux in the space : is fundamental to the time method because it contains the stirring information locally and instantaneously extracted from along .we refer to it as the _ flux diagram_. at a given instance , a continuous segment of with corresponds to a positive flux zone where the instantaneous flux goes from the right to the left across with respect to the direction of increasing .the direction of the flux is reversed for .the reference trajectory is a diagonal line going through with the unit slope ( see the main panel of figure [ fg : dg_must_u ] ) .the nature of the signals in is dependent on both the system and the choice of .these signals can be complex , as we shall observe in the double - gyre application ( figure [ fg : dg_must_u ] ) .nonetheless , having systematically examined the mean ( in terms of ; section [ sec : bsc_mean ] ) , dynamic variability ( in terms of ; section [ sec : bsc_dyn ] ) , flux variability ( section [ sec : bsc_flx ] ) , and the geographic location of in the flow field ( section [ sec : bsc_c ] ) , the physical interpretation of the signals in is straightforward .any signals can be traced back to certain flux zones in , and hence the mean - eddy interaction process between and .  by the construction of the flux diagram along , the propagation speed and direction of the signals in defined with respect to the particle advection along the reference trajectory of the mean flow .signals with positive slopes correspond to the downstream propagation of the coherent structures in .in contrast , signals with negative slopes are related to the upstream propagation of the coherent structures . if the slope is steeper than 1 , then the propagation speed is slower than the particle advection along in .  in the double - gyre application , is almost periodic with period ( figure [ fg : dg_must_u ] ) , i.e. , , due to the periodic dynamics in . within one period ,the positive and negative phases are almost anti - symmetric , i.e. , . for , is very small because of near - zero and changes the sign in synchrony with } ] for .the strongest signals in are concentrated over the segment  ] and } ] connects to during } ] in a sequence ; conversely a negative flux zone over during } ] , and then to during } ] is mainly caused by the meander of ( figure [ fg : dg_me ] ) . over ,the signals of the rossby waves and those of the circulating eddy vortices are mixed because the two types of variability merge between and in ( sections [ sec : bsc_dyn ] and [ sec : bsc_flx ] ) .accordingly , the space can be divided into sub - domains based on the types of the mean - eddy interaction .table [ tbl : d ] summarizes the main domains for the double - gyre application .two types of the variability are associated with the two domains ; where the signals are associated with the circulating eddy vortices in the dipole region and where the signals are associated with the rossby wave in the downstream region of the eastward jet .the total domain is .temporally periodic variability leads to the temporal decomposition of based on } ] .another useful definition is transient that covers the space up to the present time .the companion paper introduced the time functions .the main focus in was the mathematical formulation , the validity of perturbation approximations , as well as the verification of the method by comparing with the lagrangian method based on the lobe dynamics for the case of chosen as a heteroclinic connection of the reference state .this section refines the time functions for a much more detailed analysis of the transport processes due to the variability in the flow .we note that although the time functions as developed in are based on a perturbation approach in the sense that the fluctuating part of the velocity field is small compared to the reference state , none of the development up to this point in the paper requires this smallness requirement all that has been required is the decomposition of the velocity field into a ( steady ) reference state and a fluctuation about the reference state .the accumulation function along is defined as the left - hand side of ( [ eq : mc_org ] ) denotes the amount of fluid transport that occurs during the accumulation time interval  ] .it can be extended to study `` when '' , `` where '' , and how variability of the flow contributes to the transport .this is done by restricting the integration in ( [ eq : mc_org ] ) to a specific space - time domain that contains the particular signals of interest ( section [ sec : bsc_mu ] ) .formally and practically this leads to the slight modification to the accumulation function :  [ eq : mac ] where   acts as a switch to turn on and off the instantaneous flux depending on whether or not is in at time .for example using as , is the transient transport by accounting for the accumulation up to the present time ( see table [ tbl : d ] for the definition of ) .for the analysis of transport processes , there are properties of that are useful ( see for technical details ) .one is the _ invariance _ property along the individual reference trajectory by advection of the accumulation : for any , but with fixed .the invariance property implies a conservation law of the accumulation by the advection . at any along a reference trajectory, is independent of as long as is independent of .the other useful property is the _ ( piece - wise ) independence _ property of . by breaking up the domain into non - overlapping sub - domains , the piece - wise independence property implies that the accumulation function can be written as    for example , in the double - gyre application , the effect of the circulating eddy vortices and that of the rossby wave propagation can be examined separately , while the overall transport is the sum of the two , i.e. , .transient transport can be also decomposed into the sub - domains as long as is the same for all , e.g. , }(t)) ] , we analyze the inter - gyre transport processes of the lagrangian lobes and examine the impact of variability during } ] as progresses .this pseudo - lobe first gains positive accumulation over where a positive flux zone pulsates during } ] at the end of } ] . hence , a small , narrow ( in terms of ) positive pseudo - lobe develops mostly over during } ] when the positive flux zone over moves quickly to , the positive pseudo - lobe follows it . in figure[ fg : dg_mdt_pl ] , at is shown by the dashed line and this positive pseudo - lobe is indicated by .  during } ] when the positive flux zone quickly moves to , the positive pseudo - lobe once again follows it . in figure [ fg : dg_mdt_pl ] , is plotted at by the solid line .the spread of the pseudo - lobe is observed over .  during } ] gradually pushes this positive pseudo - lobe from the upstream region . at the end of } ] , the positive pseudo - lobe first gains accumulation from a positive flux zone induced by the rossby waves over , but loses it quickly because the negative flux zone induced by the rossby waves moves into . in the meantime, the positive pseudo - lobe continues to spread into the downstream region .once the accumulation passes , it simply advects along the reference trajectory . near the end of } ] .accumulation processes are synchronized with the evolution of the circulating eddy vortices , and are fed mostly by the three flux zones over in while they are mostly the same signed over the spatial segments through ; in the subsequent in , the rossby waves help form the final shape of the pseudo - lobe with the width in .after in , the pseudo - lobe moves by the advection along the reference trajectory because there are no further accumulation in the downstream direction of .thus , it takes almost in to develop a fully - grown pseudo - lobe of the width in .while developing , the propagation speed of the pseudo - lobe is much slower than the particle advection along the reference trajectory : over in , the pseudo - lobe moves about in starting from .the analysis also reveals that the rossby waves have very little impact on the inter - gyre transport for two reasons .one reason is that the signals of in are weak ( see section [ sec : bsc_mu ] ) .the other reason comes from the fact that the upstream propagation of the signal in in is nearly periodic in time ( see section [ sec : fnc_m ] ) .figure [ fg : dg_mdt_pl_allcvrw ] shows the decomposition of the transient inter - gyre transport into the part induced by the circulating eddy vortices and that induced by the rossby waves at ( end of } ] where the signals of the circulating eddy vortices and that of the rossby waves are mixed in ( section [ sec : bsc_mu ] ) .lagrangian lobes have proven to be extremely useful and insightful in a variety of transport studies and they provide precise amounts of lagrangian transport of fluid . in the companion paper , it was shown that provides a good approximation to the amount of transport carried by individual lobes in the inter - gyre transport ( see also ) .figure [ fg : dg_md_pl ] shows by the dash line at the end of } ] yet .a slight difference occurs over  ] of the unsteady eddy activity in , figure [ fg : dg_md_pl ] shows }) ] of the length in , }) ] .accordingly , during } ] with width in , but no negative pseudo - lobe can grow to the final form .if the period is shifted by , then only one negative pseudo - lobe grows into its final form over the same  ] differs from because of the accumulation in that has occurred prior to } ]are indicated by the two horizontal lines ( solid ) at and , while the boundary between and is shown by the vertical line ( dashed ) at .the two diagonal lines ( solid ) are examples of the reference trajectory , i.e. , with and .the left panel shows ( solid line ) and ( dashed line ) vs.  .the top panel shows vs.  with the same abscissa as . ]   ( left ) and ( right ) during } ] with .,height=396 ]   at the end of } ] ( ; solid line ) for ; the figure is made using .the positive pseudo - lobe that starts developing from the begining of } ] ) ; the figure is made at using .dark diamonds and dark triangles are the same as in figure [ fg : dg_mdt_pl ] . ]   for the total transient transport ( solid line ; same as in figures [ fg : dg_mdt_pl ] ) , }) ] ( dash - dot line ) , for total transport ( dash line ) at ( i.e. , the end of }$ ] ) ; the figure is made at using .diamonds and triangles are the same as in figure [ fg : dg_mdt_pl ] . ]

Abstract: we present a framework for the analysis of transport processes resulting from the mean - eddy interaction in a flow . the framework is based on the * t*ransport * i*nduced by the * m*ean-*e*ddy * i*nteraction ( time ) method presented in a companion paper . the time method estimates the ( lagrangian ) transport across stationary ( eulerian ) boundaries defined by chosen streamlines of the mean flow . our framework proceeds after first carrying out a sequence of preparatory steps that link the flow dynamics to the transport processes . this includes the construction of the so - called `` instantaneous flux '' as the hovmller diagram . transport processes are studied by linking the signals of the instantaneous flux field to the dynamical variability of the flow . this linkage also reveals how the variability of the flow contributes to the transport . the spatio - temporal analysis of the flux diagram can be used to assess the efficiency of the variability in transport processes . we apply the method to the double - gyre ocean circulation model in the situation where the rossby - wave mode dominates the dynamic variability . the spatio - temporal analysis shows that the inter - gyre transport is controlled by the circulating eddy vortices in the fast eastward jet region , whereas the basin - scale rossby waves have very little impact .  and  eulerian transport , lagrangian transport , mean - eddy interaction , dynamical systems approach , wind - driven ocean circulation 47.10.fg , 47.11.st , 47.27.ed , 47.51.+a , 92.05.-x , 92.10.a- , 92.10.ab , 92.10.ah 92.10.ak , 92.10.lq , 92.10.ty , 92.60.bh