the differentially heated rotating fluid annulus is a widely studied experimental apparatus for modeling large - scale features of the mid - latitude atmosphere ( planetary waves , cyclogenesis , etc . ) . in the classic set - up ,first studied by the group of fultz ( 1959 ) and , in parallel by fowlis and hide ( 1965 ) , a rotating cylindrical tank is divided into three coaxial sections : the innermost domain is kept at a lower temperature , whereas the outer rim of the tank is heated .the working fluid in the annular gap in between thus experiences lateral temperature difference in the radial direction , which initiates a ` sideways - convective ' circulation .these boundary conditions imitate the meridional temperature gradient of the terrestrial atmosphere between the poles and the equator .the coriolis effect arising due to the rotation of the tank modifies this axisymmetric basic flow markedly , and leads first to the formation of an azimuthal flow and then to baroclinic instability with cyclonic and anticyclonic eddies in the full water depth . in the present work ,we study a modified version of this experiment , in which besides the aforementioned radial temperature difference vertical salinity stratification is also present .  whereas the baroclinic instability experiments are classical models of mid - latitude atmospheric dynamics , our aim is to create a juxtaposition of convective and motionless stratified layers in the laboratory .we expect therefore that the lessons learned from our original experimental set - up may well be useful for future laboratory arrangements ( in somewhat different geometries and material properties ) that could mimic the convective and radiative zones of stars , the tropospheres and stratospheres of planetary atmospheres or the surface turbulent sea layers above deeper stratified waters .let us stress in particular that the exchange of momentum and energy between these layers ( in particular by the propagation of internal gravity waves ) is a major issue in astrophysics , in atmospheric sciences as well as in oceanography .therefore , we claim that our present set - up may lead to new insights in geo- and astrophysical applications ( plougonven and snyder , 2007 ; nettelmann , 2015 ) .small variations in the density of a fluid parcel ( ) are connected to those of salinity ( ) and temperature ( ) in the form of , where and denote the coefficients of thermal expansion and saline contraction , respectively .the global thermohaline ocean circulation is driven by the endless competition between the thermal and saline contributions to the buoyancy of each and every parcel of seawater ( vallis , 2006 ) .better understanding the nature of this complex interplay has been the key motivation of the present study .large enough ( stable ) vertical salinity gradients may inhibit the formation of full - depth ( unicellular ) overturning flow , even in such laterally heated systems as the rotating annulus experiments . the vertical extent of a convective cellis naturally limited by the condition that the initial ( saline ) density difference between the top and bottom of the cell can not exceed the ( thermal ) horizontal density difference between the lateral sidewalls , i.e. : where is the average density within the cell .this scale can be expressed with buoyancy frequency , in the form this characteristic vertical scale of ( non - rotating ) _ double - diffusive convection _ is referred to as the chen scale , since the pioneering experimental analysis of chen ( 1971 ) .within each cell the convective flow yields mixing and the initial stratification vanishes .thus , a so - called ` double - diffusive staircase ' develops with jump - wise density changes between these locally homogenized cells .double - diffusive staircases are formed in natural water bodies , ranging from saline lakes to oceans ( boehrer , 2012 ; schmitt , 1994 ) . however , due to the inevitable mixing between the cells , such configurations are mostly transient . hopefully ,as the spatial scales involved in the baroclinic as well as in the convective instabilities are quite large , the transient characteristic times calculated on these lengths are long enough to permit a full investigation .some of the experimental runs that we will describe later lasted several days .the paper is organized as follows : after having recalled the basic principles of double - diffusive convection ( and its response to rotation ) and then of the baroclinic instability , in sections [ ddconv ] and [ baro ] respectively , in section [ setup ] we describe our experimental arrangement and measurement methods .section [ results ] is devoted to the presentation of the results and their analysis that will concern two sets of experimental runs . in the first set ( subsection [ simple ] ) , the salinity profiles were mainly linear ( with a single buoyancy frequency in the core of the fluid ) with two non - stratified regions at the top and bottom of the full water layer .the corresponding results permit to test the possibility to confine the baroclinic instability in two shallow layers separated by a motionless zone . in a second part of this result section ( subsection [ complex ] ) , more complex salinity profilehave been realized with vertical variations of the buoyancy frequency . in this case ,multilayered convection sets in .the analysis is thus much more complex and necessitates a particular method to discriminate and quantify the dynamics inside these independent layers .section [ conclusion ] summarizes our main results and opens discussions on the future possibilities of our original experimental set - up .for the sake of qualitative demonstration , in fig .[ sim_profiles]a - c we present typical vertical profiles of total density , temperature , and horizontal velocity , from two - dimensional solutions of the non - rotating boussinesq equations , acquired via _ numerical simulation _ using the open - source software package advanced ocean modeling of kmpf ( 2010 ) .no - flux conditions were prescribed at the top and bottom boundaries of the rectangular solution domain , and differential heating was applied at its lateral walls : the temperature at the vertical sidewall at the left ( right ) hand side of the domain was prescribed to be k colder ( warmer ) that the initially uniform temperature of the fluid .the fluid had a stable and continuous vertical density gradient and was at rest at time zero .each profile of fig .[ sim_profiles ] was extracted from the vertical column at the middle of the domain .since these graphs serve demonstration purposes only , no units are given .in panel ( a ) it is clearly visible that the initial density stratification ( dashed line ) has developed into the aforementioned double - diffusive staircase ( solid line , red online ) ; see also the corresponding velocity profiles in panel c ) . in qualitative agreement with the formula ( [ lambda ] ) , the largest cells both in terms of vertical size and of water flux appear at mid - depths , where the initial salinity profile exhibited the smallest gradient .note , however , that in the regions characterized by the largest density gradients , the flow is basically blocked ( the horizontal velocities are practically zero ) : in the spirit of ( [ lambda ] ) , the large local would yield very shallow convective layers , but their formation is inhibited by viscous effects .this `` viscous cut - off '' bounds the cell thickness from below : it can not be smaller then a certain , or , in terms of the buoyancy frequency , there is a maximum , above which convection is locally blocked .  in their experiments , chen ( 1971 )investigated the development of convective rolls in a rectangular vertical tank filled up with saline water of initially uniform vertical stratification with differential heating at two opposing lateral walls .his results are expressed in terms of a certain version of rayleigh number , where the vertical scale is given by  as derived from ( [ lambda ] ) and the characteristic temperature difference is measured between the lateral walls ; thus incorporates the effects of both the horizontal and the vertical density gradients : where is the thermal diffusivity and is the kinematic viscosity of the working fluid .chen ( 1971 ) found that the critical value of or , equivalently , is encountered around , below which convection is inhibited .if exceeds this threshold , simultaneously or successively formed multicellular double - diffusive convection rolls can develop ( kranenborg and dijkstra , 1995 ) .  introducing _ rotation _ around the vertical axis modifies the dynamics of double - diffusive convection markedly , as demonstrated in the theoretical work of kerr ( 1995 ) .his conceptual set - up involves vertical salinity gradient and differentially heated lateral sidewalls with gapwidth , which rotate at angular velocity ( i.e. coriolis parameter ) .unlike chen , kerr defines separate rayleigh numbers for the vertical and horizontal directions ( and , respectively ) , both taken with the same length scale , which is , in fact , the only geometrical scale here , since the vertical extent of the domain is assumed to be infinite in the calculations .the horizontal rayleigh number reads as , whereas its vertical counterpart is given by .note , that based on these formulae , the chen scale ( defined in ( [ lambda ] ) ) , can be expressed as .kerr studied the -dependence of the minimum , below which the flow is blocked , at different fixed values of and found two types of solutions in all cases .a schematic sketch of a typical stability diagram is shown in fig .[ bifurc ] .one branch ( green curve ) is only slowly changing with : this result is consistent with the findings of chen ( 1971 ) in the limit case .   as a function of coriolis parameter , for a given .for slow rotations , the slowly changing branch here , referred to as ` chen branch ' is stable , but above a certain threshold this branch vanishes , and only the sharply decreasing kerr s `` rotating branch '' survives.,width=377 ]  this branch vanishes at a certain threshold , whose value depends on : is an increasing function of . in the regimethe other branch of solutions describes the appearance of a new regime ( orange curve in fig .[ bifurc ] ) .this branch decays in an inversely proportional manner with  thus it is non - existent in the limit and its parameter depends on as well : .    for a qualitative understanding of the effect of rotation on the double - diffusive background flowlet us consider an experiment with counter - clockwise , i.e. ` northern hemisphere'-like , rotation .as observed from the co - rotating frame of reference , the coriolis force tends to push the horizontal flow branches to their right : at large enough values of rotation rate and/or horizontal velocity this action can yield strong zonal flows in the system .their directions are alternating with depth , corresponding to the convective branches of double - diffusive convective cells ; the coriolis effect yields _ zonal _ propagation in the presence of _ radial _ buoyancy gradient .this is demonstrated in the composite photo of fig .[ dye_demo ] taken during one of our experimental runs corresponding to the multi - layered cases that will be detailed in section [ complex ] .the figure presents the side view ( through the transparent glass walls of the rotating tank ) of dye patterns propagating away from a painted wooden stick ( black ) in ` eastward ' ( rightward ) and ` westward ' ( leftward ) zonal directions at rotation rate rpm , and average buoyancy frequency rad / s .it is important to emphasize , however , that the presence of baroclinic instability resulting in zonal wave propagation in the annular tank may largely add to the complexity of the emerging flow fields .note also that even if kerr ( 1995 ) never mentioned it , the solutions for seem very close in shape and wavelength to the classical baroclinic eddies .the most important non - dimensional parameter , traditionally used to describe the flow regimes in a rotating system with a finite kinematic viscosity is the taylor number , which reads as follows : where is the vertical scale and is the ` gapwidth ' of the apparatus ( or more generally , the horizontal scale ) , as in the earlier formulae .another relevant parameter , the thermal rossby number is defined as the ratio of the characteristic velocity of the flow driven by the relative density change in the lateral direction ,  to the rotation rate , in the form of where is the acceleration due to gravity .comparing the above formulae it is obvious that for a fixed ( as in the experiments to be discussed here ) , and are inversely proportional to each other and therefore the combination can sufficiently parametrize the flow in the given experimental setup .  in a certain region of the parameter space, the slope of the isopycnal surfaces in the water ( which are not horizontal in this rotating system , due to thermal wind balance ) surpasses a critical threshold and _ baroclinic instability _ emerges ( vallis , 2006 ) . the extra potential energy stored in these tilted layersis then released via the excitation of complex wavy patterns , consisting of meandering zonal jets , accompanied by cyclonic and anticyclonic eddies . for k ( that is the typical case for this study , as will be discussed later ) the taylor number corresponding to the onset of baroclinic instability is ( or equivalently , ) ; above ( in terms of , below ) this threshold wave propagation is expected in the zonal direction ( von larcher and egbers , 2005 ) .  in the classic parametrization of homogeneous ( i.e. thermally driven ) baroclinic annulus experiments( vincze , 2014 ; vincze 2015 ) the vertical scale that appears in the definitions of non - dimensional numbers ( [ ta ] ) and ( [ ro ] ) corresponds to the total water level in the tank , since the sideways convective unicellular flow penetrates through the whole depth of the domain ( ) .the presence of vertical salinity stratification however , yields a system of shallow - layer convective cells ( cf .[ dye_demo ] ) , whose thicknesses may be determined by chen s formula ( [ lambda ] ) , and by the cut - off value .therefore it is useful to introduce _ local _ versions of the taylor and thermal rossby numbers by taking as vertical scale , corresponding to the local .hence , the local thermal rossby number takes the form and the local is written as it is reasonable to assume that such parameter combinations exist , where the global taylor number calculated with the full water depth does not exceed , but the local taylor number , taken with the thickness of a given convective cell ( ) is already above the threshold. the multicellular state can therefore enhance baroclinic instability within the cells , and thus open the way for wave propagation in such configurations , which ( given their values of and ) would be stable in the homogeneous case . also , due to differences in cell thicknesses , the ` baroclinicity ' ( i.e. the deviation of the flow field from the wave - free , axially symmetric state ) in the set - up may vary from cell to cell , thus different flow states may be observed at different depths .the experiments of the present study were conducted in the laboratory set - up shown in fig . [ setup ] . the working fluid consisted of sodium - chloride / de - ionized water solution , with lewis number , where and are the diffusion coefficients for heat and salt , respectively .the radii of the inner ( cooling ) and outer ( heating ) cylinders were cm and cm , yielding a cm wide annular cavity that was filled up to heights ranging from cm to cm in the different runs .the lateral temperature difference was set between 6 and 10 k in all cases .a continuously stratified salinity profile was prepared in the cavity before each measurement with the standard two - bucket technique ( oster and yamamoto , 1963 ) two vessels containing saline and fresh water are connected with a pipe so that the freshwater inflow to the saline bucket yields a mixture of ever decreasing salinity in time . by raising or lowering one of the buckets during this filling - up procedure , we were able to adjust the water fluxes into the mixing compartment and , thus , create arbitrarily non - uniform stable salinity profiles .cm , cm , cm , rpm , k. the direction of the tank s rotation is also indicated.,width=377 ]  our experiments investigated variations of the flow field with height for different types of initial salinity stratification .thus , the fluid was seeded with m - diameter tracers for particle image velocimetry ( piv ) . after filling up the tank ,the initial salinity profiles were measured using a hand - held conductivity sensor . then the differential heating was switched on .  even before starting rotation ,the onset of double diffusive convection could be observed visually .generally , most of the -values corresponding to the prepared vertical profiles yielded smaller s than the aforementioned critical threshold of convection in a non - rotating system , .therefore , the convective flow in the bulk was mostly inhibited .however , the no - flux conditions for salinity at the bottom and the top of the water body translate to conditions at these regions in terms of buoyancy frequency .thus in thin layers at the vicinity of these boundaries where holds , localized convective cells could still partially invade the water column , as seen in fig .[ no_rot_cells ] , visualized with fluorescein dye .is larger.,width=302 ]  rotation rate was increased gradually before reaching its target value ( in increments of rpm in every 5 minutes ) . at this pointthe system was left undisturbed for 30 - 40 minutes , thus the early transients of the rotating double - diffusive cell formation were not observed .then , piv measurements were conducted using horizontally illuminated laser sheets at different water depths .two slightly different arrangements were used for the two sets of experiments .for the first set , a camera together with a laser were fixed on an arm from the sidewall of the container ( ` narrow field ' observations ) .close - ups of the velocity fields could be acquired in this manner . in the second set of experiments , complete ` wide field ' views of the velocity fields in the full annuluscould be observed from a co - rotating camera platform situated above the tank ( as depicted in fig .[ setup ] ) . alongside these piv cameras , for both sets of experiments an infrared thermographic camera was also mounted on the platform to enable the simultaneous observation of the temperature field at the free water surface . at the end of the piv measurements , rotation rate lowered gradually ( in the same manner as in the spin - up phase ) , and after the flow reached a full stop , a second salinity profile was also recorded .the observation of surface temperature patterns in the tank using infrared camera revealed that contrarily to the initial nave expectations of the authors , but in full agreement with our hypothesis of baroclinic destabilization via the scale selection of , discussed subsection [ ddconv ] in experiments with salinity stratification the layer at the water surface becomes baroclinic unstable at much lower rotation rates than in the homogeneous system .this phenomenon is demonstrated in fig .[ infrared ] . in panela ) the surface temperature field of a homogeneous control experiment is shown , where the vertical scale is set by the total water depth cm .the standard temperature difference k and the applied rotation rate rpm yields , that is below the aforementioned , or equivalently , the corresponding is above , as determined by von larcher and egbers ( 2005 ) .consistently , the temperature distribution is axisymmetric indeed , implying that the fluid stays in the stable regime .rad / s ) with the same ` global ' parameters : k , cm , and rpm.,width=377 ]  panel b ) shows the temperature pattern emerging in the stratified case for the same setting of parameters , and , resulting in a fourfold symmetric baroclinic wave , generally referred to as in the literature .von larcher and egbers ( 2005 ) found using the same experimental tank and as here , but with homogeneous working fluid that the lower limit of the regime in terms of taylor number is around ( or somewhat below , at if ` spin - down ' initial conditions are applied ) . from the formula ( [ ta ] )one can therefore calculate that the maximum characteristic vertical scale of the convection has to be approximately cm to explain the observed wave number .taking this value as the chen scale of ( [ lambda ] ) yields a local buoyancy frequency rad / s .this is consistent with the measured salinity profile , although due to the finite size of used the hand - held conductivity sensor there are no measured data points from the uppermost layer of the initial profile . yet , based on the fact that 2 cm below the surface rad / s was measured , and on the aforementioned no - flux condition for salinity prescribing at the top , it is reasonable to interpolate rad / s for the considered region .moreover , cm also matches the observed thickness of the spontaneously forming convection cells at the boundaries , cf .[ no_rot_cells ] .in order to obtain a more quantitative relationship between and the ` baroclinicity ' of the flow , one needs to investigate the bulk of the working fluid , where the buoyancy frequency profiles can be properly determined .this domain , however , is not accessible for infrared thermography ( since the penetration depth of the applied infrared wavelength range in water is of the order of a millimeter ) .thus , the buoyancy frequencies directly could only be compared with piv data .   in this sectionwe will present experiments which we started from particularly simple density profiles as illustrated in figure [ nprofileexp4]a .the buoyancy frequency is plotted versus the vertical coordinate in figure [ nprofileexp4]b . we performed different experiments at the following values of global parameters : k and rpm ( denoted as exp .4 ) , k and rpm ( exp .5 ) , k and rpm ( exp .6 ) , k and rpm ( exp . 7 ) . in the most detail the first of these runs ( i.e. ` exp . 4 ' ) will be discussed .one can see in fig .[ nprofileexp4 ] that the vertical density profile is almost linear . at the upper and lower boundaries the no - flux boundary condition imposes constant concentration and therefore constant density .thus , close to these boundaries approaches zero .is fairly constant around a value of rad / s , width=302 ]  contrarily to the non - stratified case , as explained in section [ ddconv ] , the imposed horizontal temperature gradient has to reach a certain threshold before sideways convection sets in .figure [ visuconv ] shows the top and bottom convective layers , where the fluid has been colored by the gentle injection of dye at the water surface . as visible and as expected , convection is confined in layers of water where stratification is smallthe thickness of this convective cell progresses slowly in time from around cm at threshold .    to obtain velocity vectors of the flow ,the piv method is suitable .as mentioned in section [ setup ] , in this series of experiments the narrow field camera ( gopro , 1080p , 30fps ) and the co - rotating laser sheet were utilized to acquire video information . for calculating the velocities we used successive frames taken at equally distributed time intervals with increment of s and we processed them with the public license matpiv toolbox .the advantages of using a co - rotating measurement system are that the camera can be positioned near the free surface and thus the resolution will increase , and we do nt need to correct the calculated velocities by subtracting the tank s rotation speed .as soon as rotation is started , a zonal flow is created by the action of the coriolis force on the convective currents .piv maps have been acquired for height levels .depending on the rotation rate , and also on the intensity of the convective velocities , these zonal flows can be either stable or unstable in terms of baroclinic instability .[ pivmap ] illustrates two horizontal close - ups of the velocity fields .the first one ( panel a ) is taken near the surface , inside the layer where convection is allowed to set in . herea baroclinic wave can be observed . at the convective region close to the bottom of the tank a convective cell appears too , but in contrast with the one near the surface it is baroclinic stable , possibly because the zonal flow is weakened by the bottom boundary layer .the next question is then to explore the possible generation of inertial - gravity waves ( igws ) by the instability as it can be the case for instance in unbalanced dynamics of atmospheres ; see for instance osullivan and dunkerton ( 1995 ) or plougonven and snyder ( 2007 ) .for the first time jacoby ( 2010 ) looked for igws in a classical continuously stratified , differentially heated rotating annulus .they found signatures for high - frequency waves but surprisingly the waves were related to boundary layer instabilities and not to the fronts of baroclinic waves .recently , borchert ( 2014 ) and randriamampianina and del arco ( 2015 ) numerically investigated the occurrence of igws by the baroclinic instability .the former claimed to have found igws generated spontaneously in accordance with the waves detected in a two - layer baroclinic annulus by lovegrove ( 2000 ) and williams ( 2003 ) .the latter showed that in the continuously stratified case with small prandtl number , igws are generated by kelvin - helmholtz instability and not via boundary layer instability .although we can not prove yet that the present instability can generically generate igws , the velocity field of fig .[ pivmap]a exhibits a wave train whose characteristics are compatible with igws .the second close up ( panel b ) corresponds to the calm region in the center of the tank , where convection is blocked due to the effect of the strong initial stratification .driven by the strong flow in the upper and lower convective layers there is still some azimuthal symmetric flow , but the velocities are an order of magnitude lower than the ones from the upper and lower convective layers .cm and in the middle region at cm ) showing the azimuthal flow with or without the baroclinic instability .the small arrows represents the velocity vectors in the horizontal plane , the colored maps are the horizontal divergence fields .the stream lines are plotted in red .close to the surface the velocities are an order of magnitude higher ( mm / s ) than in the center ( mm / s).,width=302 ]  this flow structure demonstrates the features that was illustrated by our numerical observation presented in section [ ddconv ] .[ pivmean ] presents the vertical azimuthal velocity profile as measured in the experimental runs denoted exp.4 to exp.7 , as mentioned above .for each point we calculated the mean velocity over the whole investigated plane , which corresponds to approximatively one sixth of the cylindrical gap in the azimuthal direction . since the presence of baroclinic instability breaks the axial symmetry of the flow , temporal averagingwas performed upon a sufficient timespan to permit that the flow from different segments can contribute to the final azimuthal mean ( between one and minutes ) .although the vertical resolution is rather poor ( data points in the full water layer ) one can easily recognize the opposite signs of the zonal flows at the borders of the convective cells . in the small region between themonly weak flow was measured .it is to be emphasized , that despite the differences in rotation rate and temperature difference , all four experiments yielded very similar results .it is also worth mentioning here that this motionless layer can possibly sustain inertia - gravity waves ( igws ) that would be launched from the top or the bottom unstable zones.   of about k and different values of showing three distinct zones : two unstable convective cells at bottom ant top of the full layer and a motionless layer that separate them in the middle plane.,width=377 ]   in this section we present our findings from the second set of experiments , where more complex initial salinity profiles were studied , two of which are presented in panels a ) and d ) of fig .[ profiles ] . in panela ) both the initial ( dashed line ) and final ( solid line ) profiles are shown . similarly to the simulated final profile in fig .[ sim_profiles ] , the presence of double - diffusive staircase is apparent .panels b ) and e ) show the values of buoyancy frequency , as calculated from polynomial fits of the density profiles , and evaluated at the height levels where the ` wide field ' piv measurements were conducted .as an example of the piv fields evaluated in this series of experiments fig .[ piv_field ] shows the acquired velocity vectors of a fully developed fourfold symmetric wave pattern obtained from ca. 1 cm below the water surface , superimposed onto the simultaneously obtained surface temperature field .k , rpm).,width=264 ]  at each of these levels , the horizontal velocity field was calculated on a lattice of grid points , yielding a spatial resolution of around cm . in order to obtain a proper measure of ` baroclinicity ' in such a slice of the flow , firstly , the angle between each velocity vector and the respective position vector  directed from the center of the tank , i.e. the axis of rotation was computed ( see the explanatory sketch in the inset at the bottom right corner of fig . [ cdf_fig ] ) .our approach is based on the distribution of these -values . in the absence of baroclinic instability ,the temperature pattern is axially symmetric and the flow is zonal ( see fig . [infrared]a or state a in fig .[ cdf_fig ] ) , thus the velocity field is highly _ anisotropic _ in the sense that the vast majority of the velocity vectors are perpendicular to their position vectors ( ) . at larger values of meandering jet appears ( state b in fig .[ cdf_fig ] ) : the values of scatter in a broader range around . in a fully developed baroclinic unstable case ( states c and d )the flow field is dominated by vortices , making the density distribution close to uniform within the interval , i.e. , in a certain sense , _isotropic_.    the right panel shows the cumulative density functions ( cdf ) of the values of four measured vector fields , whose streamlines ( states b through d ) were sketched , as discussed above .each count of the cdfs was calculated with a certain weight factor , assigned to each velocity vector proportionally to their relative magnitudes compared to the mean ( after the careful removal of outliers ) , in order to compensate for the inevitable uncertainties in the values when measuring small vectors . in a perfectly isotropic case, the cdf curve would follow the linear function .the more anisotropic the velocity field , the larger the deviation of the graph from this straight line .we quantify the anisotropy of the flow by defining a parameter as where is the total number of the obtained ` useful ' velocity vectors of the piv data , which lie within the measurement domain ( the annular cavity covers only a fraction of the rectangular image ) and remain after the removal of clear outliers .this process typically left us with values of for a given horizontal piv slice . serves as the desired measure ( order parameter ) of baroclinic instability : generally , the smaller is , the more baroclinic unstable the system appears to be in the considered horizontal section .a perfectly axisymmetric flow , with all the velocity vectors aligned perpendicularly to their position vectors would yield ( independently of whether the flow is clockwise or counterclockwise ) , whereas in a fully developed geostrophic turbulent state , where all directions are equally probable would hold .the vertical profiles of anisotropy parameter obtained for the two measurements of fig . [ profiles ] are shown there in panels c ) and f ) .we emphasize that the determination of these -values is completely independent from that of the density profiles , the former being obtained directly from the piv data , and the latter via a hand - held conductivity probe . yet, there is an apparent anticorrelation between and in both shown cases , as seen when comparing panels b ) to c ) , and e ) to f ) in fig .[ profiles ] .the acquired data points from four experiments with different vertical density profiles and slightly different rotation rates ( ranging from rpm to rpm ) are combined in fig .[ alldata]a , with colors accounting for the corresponding values of local thermal rossby number , as in ( [ ro_loc ] ) : here each point is calculated from a complete horizontal piv velocity slice .the overall trends are apparent : firstly , it is reassuring that anisotropy parameter seems to behave consistently with .generally , the smaller is , the closer the system gets to geostrophic turbulence , yielding smaller s , as expected .secondly , markedly anticorrelates with in itself too , despite of the differences in the rotation rates , underlining the robustness of the -dependence : the steeper the stratification , the smaller the vertical convection scale becomes , supporting the development of shallow layer geostrophic turbulence , i.e. smaller .[ alldata]b shows the same data set , now colored with respect to the corresponding relative average absolute horizontal velocities , based on the piv fields .these values have been normalized so that corresponds to the largest observed average velocity .we found that two of the three weakest observed flow layers ( the two blue and green data points on the right of the plot ) appear at the largest buoyancy frequencies ( rad / s ) , which is in sharp contrast with the large velocities measured at only slightly smaller s .this may well imply that here the blocking threshold is approached , and above rad / s ( marked by a solid vertical line ) the convection is inhibited ; thus the corresponding parameters do not have actual physical meaning here , being results of the angle statistics of noise - dominated piv fields .the consistency of this finding with the theoretical background is addressed in the next section .chen ( 1971 ) and kerr ( 1995 ) have analyzed the conditions under which double diffusive convection can set on in laterally heated , initially stratified set - ups . for a non - rotating casethe criterion ( as reported in the experimental study of chen ( 1971 ) ) was found to be that the ` combined ' rayleigh number , which incorporates the effects of both the salinity and the temperature gradients as defined in ( [ ra_chen ] ) , must exceed a certain threshold . from here, one can calculate the critical buoyancy frequency above which the viscous effects inhibit the formation of convective cells in our experimental tank . this estimation yields rad / s , which appears to be consistent with the observation that in the _ non - rotating _control experiments , cell formation was limited to the vicinity of the top and bottom boundaries , where is small ( see fig . [ no_rot_cells ] ) , and is also in fairly good agreement with the results of the first set of our _ rotating _ experiments , descussed in section [ simple ] .  based on this , we would barely see any flow in the bulk of the fluid at all , because most of our actual -values scatter in the interval rad / s , as shown in figs .[ profiles ] and [ alldata ] . yet , in our second set of experiments , where more complex initial salinity profiles were investigated ( section [ complex ] ) , once rotation was turned on , an alternating zonal flow emerged at mid - depths as well ( see fig . [ dye_demo ] ) , implying convection .therefore it is reasonable to assume that in the rotating case the system can also exhibit another type of dynamics , which we referred to as the kerr branch .the states of this branch only exist in the presence of rotation , as seen in fig .[ bifurc ] .kerr s analysis is based on a horizontal rayleigh number and a vertical one , as discussed in subsection i.b .the average of our multilayered experiments yields , and their typical rotation rate rpm translates to in the dimensionless units applied in the analysis ( using as time units , cf .kerr ( 1995 ) ) . from here , using the data presented by kerr ( 1995 ) , we can also estimate that is the corresponding critical horizontal rayleigh number .knowing now the ratio , we can directly obtain the actual critical chen scale as , and finally , applying ( [ lambda ] ) , we arrive at the theoretical prediction of in the form of : that is in perfect agreement with the observed rad / s , implying that our multilayered experiments followed the ` kerr ( rotating ) branch ' of solutions , depicted in fig .[ bifurc ] .note , however , that as mentioned above our first set of experiments , where only two separate convective layers were observed at the small- regions of the top and bottom of the tank ,convection was blocked already at rad / s , and remained blocked , even when higher values of and were applied .this observation implies that the dynamics in these experiments stayed on the ` chen branch ' of fig .[ bifurc ] .the present study lacks an empirical determination of the breaking point , where according to kerr s linear stability analysis the chen branch is expected to vanish and the kerr to remain the only possible solution .this transition between the two types of solutions needs to be clarified by future experiments .to the best of our knowledge , we have reported here the very first experimental results on the combined effect of double convective and baroclinic instabilities in a rotating stratified layer and we decided to coin this phenomenon the ` barostrat instability ' . following the previous analyses of chen ( 1971 ) and kerr ( 1995 ) we have observed the formation of patterns with different azimuthal wave numbers . as expected ,these structures are confined in layers where the initial vertical density stratification was smaller .the thresholds for convective cell formation in terms of buoyancy frequency and the observed typical vertical length scales were found to be in agreement with the theoretical predictions .we hope that this study will open new routes in the study of atmospheric baroclinic instabilities : in particular let us remark that the stable stratified zone that lies under the unstable shallow layer in the experiment mimics the stratosphere above the baroclinic unstable layers of the troposphere .the next question is then to explore the possible generation of inertial - gravity waves by the barostrat instability as it can be the case for instance in the unbalanced dynamics of atmospheric fronts .  as closing remarks, we emphasize that it was somewhat counter - intuitive for the authors to find that the presence of moderately strong vertical salinity stratification can increase isotropy , i.e. enhance the development of baroclinic instability in certain shallow layers via the formation of double - diffusive cells .this is seemingly in contrast with reasoning that stable stratification ( large ) generally blocks baroclinic instability .our findings imply that although , indeed , the unicellular full - depth overturning may be inhibited the steeper the stratification is , the shallower the double - diffusive cells may be , which then decreases the local value of thermal rossby number , and by doing so , pushes the system ( i.e. the given sideways - convective layer ) towards quasi - geostrophic turbulence , as long as the stratification does not exceed a critical value above which the flow is indeed completely blocked .  with presenting the new findings of the present pilot study ( which clearly needs to be followed by further experimental and numerical investigations ) the authors hope to increase awareness within the community of the fact that the velocity field , and thus , horizontal _ mixing _ can vary in such a nontrivial way with .this may be of interest in ocean dynamics , where horizontal ( e.g. meridional ) temperature difference and vertical salinity gradients co - exist .this work was supported by the european high - performance infrastructures in turbulence ( euhit ) program and by otka grant no .the authors are grateful for christian maa , yongtai wang , and torsten seelig for the technical support during the measurement campaigns .the fruitful discussions with tamas tel and yossi ashkenazy are also highly acknowledged .boehrer b 2012 double - diffusive convection in lakes _ encyclopedia of lakes and reservoirs _ 223 - 224     fultz d , long r , owens g , bohan w , kaylor r and weil j 1959 studies of thermal convection in a rotating cylinder with implications for atmospheric motions _ meteor .soc . _ * 4 * 1 - 104  jacoby tn l , read p l , williams p d , and young r m b 2010 , generation of inertia - gravity waves in the rotating thermal annulus by a localized boundary layer instability _ geophys .fluid dyn . _ * 105 * 161 - 181       nettelmann n , fortney j , moore k and mankovich c 2015 an exploration of double diffusive convection in jupiter as a result of hydrogen - helium phase separation _ monthly notices of the royal astronomical society _ * 447 * 34223441        vincze m , harlander u , von larcher t and egbers c 2014 an experimental study of regime transitions in a differentially heated baroclinic annulus with flat and sloping bottom topographies _ nonlin .processes geophys . _* 21 * 237 - 250  vincze m , borchert s , achatz u , von larcher t , baumann m , liersch c , remmler s , beck t , alexandrov k , egbers c , froehlich j , heuveline v , hickel s and harlander u 2015 benchmarking in a rotating annulus _ meteorol .z. _ * 23 * 611 - 635

Abstract: a water - filled differentially heated rotating annulus with initially prepared stable vertical salinity profiles is studied in the laboratory . based on two - dimensional horizontal particle image velocimetry ( piv ) data , and infrared camera visualizations , we describe the appearance and the characteristics of the baroclinic instability in this original configuration . first , we show that when the salinity profile is linear and confined between two non stratified layers at top and bottom , only two separate shallow fluid layers can be destabilized . these unstable layers appear nearby the top and the bottom of the tank with a stratified motionless zone between them . this laboratory arrangement is thus particularly interesting to model geophysical or astrophysical situations where stratified regions are often juxtaposed to convective ones . then , for more general but stable initial density profiles , statistical measures are introduced to quantify the extent of the baroclinic instability at given depths and to analyze the connections between this depth - dependence and the vertical salinity profiles . we find that , although the presence of stable stratification generally hinders full - depth overturning , double - diffusive convection can yield development of multicellular sideways convection in shallow layers and subsequently to a multilayered baroclinic instability . therefore we conclude that by decreasing the characteristic vertical scale of the flow , stratification may even enhance the formation of cyclonic and anticyclonic eddies ( and thus , mixing ) in a local sense .


due to the advance of the web 2.0 techniques , various kinds of websites have become parts of our life .most people nowadays are used to sharing , seeking and obtaining information , and interacting with others through various social networking websites such as digg , slashdot , facebook , and twitter , to name a few . asmore and more people join these social websites , the phenomenon of immense quantity of information flow and influence spread becomes prominent .this then motivates the utilization of this phenomenon , known as the `` word - of - mouth '' effect , to bring significant potential benefits in many types of business such as viral marketing and innovation promotion .for example , a company can effectively promote its brand and new products by involving a group of influential people spreading the good words over social websites .hence , in very recent years , there is active research exploring the problem of influence maximization .existing works studied the influence maximization problem generally based on two kinds of diffusion models , the linear threshold ( lt ) and the independent cascade ( ic ) models . in the lt model ,the influence strength between a pair of individuals is modeled as a fixed weight , and the influence on an individual is described as cumulative .an individual is activated when the sum of the weights propagated from his / her activated neighbors exceeds his / her own threshold .the ic model assumes that each influence of one to another is an independent chance to activate the latter with a probability . on both models, the problem of influence maximization aims at identifying a proper group ( of a specified size ) of individuals as seeds to have the number of activated people maximized. however , note that such influence maximization problems have not taken into account the monetary aspect , which usually plays a crucial factor in people s purchase decisions .that is , maximizing the influence spread does not necessarily bring maximum revenue , as an individual may be very interested in a new smartphone , but does not purchase the smartphone due to its high price .  according to the studies in the fields of social psychology and economics , one s valuation towards a commoditywill be positively influenced by those that have purchased the commodity , which is known as the herd mentality or positive network externalities .in other words , people tend to imitate others in their consumer behaviors , which will result in irrational valuations of the commodity , i.e. , higher valuations beyond their inherent valuations . based on these phenomena , mirrokni et al . addressed the * revenue maximization * problem by incorporating the monetary concept into the maximization problem of influence spread based on the lt model . in the monetary lt model , the influence strength ( or said weight )is transformed into the valuation concept , and the individual threshold is regarded as the inherent valuation .one s valuation of a commodity is thus the sum of his / her inherent valuation and the valuation increment from social influences .an individual then pays for a commodity and propagates his / her influence whenever her / his valuation is larger than or equal to the pricing of the commodity .consider the network shown in figure [ fig : ex ] as an example , where the number contained in each node is the inherent valuation and the number on an edge from one node to another is the weight , i.e. , the influence strength . for the ease of illustration , without loss of generality , the function is used to transform the effects of the social influences cumulated on an individual into the valuation concept in this example . if purchases the commodity , s valuation is increased to . if , , and all purchase the commodity , the valuation increment for is and s valuation thus becomes .the objective of the revenue maximization problem thus aims at determining the pricing of the commodity and identifying a seed group of individuals for promotion so that the total revenue is maximized .however , to the best of our knowledge , none of the existing works has taken a quantity constraint on commodities into account .note that due to the consideration of marketing strategies or practical conditions , quantities of commodities are often constrained in the real world .for example , the scarcity fallacy usually makes commodities more desirable , as a company may want to release limited edition commodities to boost customer desire .another example is the case of promoting a concert , where the quantity of the concert tickets is constrained by the number of available seats . in this case ,due to the schedules of the musicians or singers , it is hard to add more shows and the total number of concert tickets .hence , selecting seeds to receive free commodities will reduce the quantities of commodities that can be sold .this makes the prior works unapplicable to determine the proper pricing and obtain maximum revenue .for example , suppose that the network shown in figure [ fig : ex ] is considered , where the diffusion model with the monetary concept in the work is adopted and the function is used to transform the effects of the social influence into the valuation concept . without consideration of the commodity quantity ,the maximum revenue is obtained at the pricing of with the seed group . with the influence from , and the first to purchase the commodity , since both their valuations are equal to the pricing of .then , and will subsequently purchase the commodity due to their valuations and , respectively .however , if the quantity of commodities is limited to 4 , the revenue will decrease to since two free commodities given to the seeds and should be taken off first . in this case , letting the commodity be priced at and choosing only as the seed will be the best . under the influence of , first purchases the commodity since its valuation is larger than the pricing of .then , both and purchase the commodities due to their valuations and , respectively .finally , since , , and all pay for the commodity sold at the price of , the maximum revenue is .therefore , when the quantity of commodities is limited , we argue the need of new approaches for the revenue maximization problem .\(a ) ; ( c ) [ right=2 cm of a ] ; ( b ) [ above of = c ] ; ( d ) [ below of = c ] ; ( e ) [ right=2 cm of b ] ; ( f ) [ below of = e ] ;  \(a ) edge [ - > , bend left=20 ] node 2 ( b ) edge [ - > , bend right=20 ] node [ swap ] 3 ( c ) ( b ) edge [ - > ] node 1 ( a ) edge [ - > , bend left=20 ] node [ swap ] 2 ( f ) ( c ) edge [ - > ] node 3 ( d ) ( d ) edge [ - > , bend left=40 ] node 4 ( b ) edge [ - > , bend right=20 ] node [ swap ] 2 ( f ) edge [ - > , bend left=20 ] node 5 ( a ) ( e ) edge [ - > ] node [ swap ] 4 ( b ) edge [ - > , bend left=20 ] node 2 ( c ) ( f ) edge [ - > , bend left=20 ] node 1 ( c ) edge [ - > ] node [ swap ] 5 ( e ) ;  specifically , we address the revenue maximization problem with a quantity constraint on commodities . given a limited quantity of commodities and a social network with monetary concept regarding to the commodity , the problem is to determine the pricing of the commodity and identify a small group of people , i.e. , a seed group , to be the initial customers receiving freebies to help promote the commodity at the beginning , so that the total revenue is maximized . in this paper, we investigate this problem on the monetary - concept incorporated lt - model , and propose algorithm prub ( pricing searching strategy using revenue upper bound ) to derive the optimal solutions .then , we further revise prub to propose a heuristic algorithm prub+if ( pricing searching strategy using revenue upper bound with importance feedback ) for better efficiency . experiments on real social networks show the good effectiveness of prub and prub+if , demonstrating the revenue maximization with and without a quantity constraint differs from each other .  in summary ,the contributions of this paper are :  1 .  we are the first to address the general case of the revenue maximization problem .note that the previous work of the revenue maximization without a quantity constraint is a special case of our problem with the quantity set as the total number of individuals on the social network .2 .  for the addressed problem, we proposed the optimal prub as well as a heuristic prub+if .3 .  experiments on real social networks demonstrate the effectiveness of the proposed approaches for both general and special cases of the revenue maximization problem .note even in the case without a commodity constraint , the proposed approach outperforms the state of the art approach that incorporates the monetary concept .to formulate the revenue maximization problem , in this section , we first describe the social network with the monetary concept , then explain how the social influences propagate over the network using the monetary - concept incorporated lt - model , and finally define the revenue and the revenue maximization problem formally .  given a commodity , a social network with the monetary concept , referred to as a * monetizing social network * ,is a weighted digraph , where is the set of individuals and for each individual , his / her inherent valuation is carried in .usually , the valuation information can be estimated and learned from questionnaires or historical sales data , with regard to the commodity . for any individuals , if s purchase will directly encourage s desire for the commodity , an edge from to , denoted as ( standing for the edge set ) , represents the existance of the influence , and represents the influence strength as the weight on . adopting the concave graph model in the work , we consider a non - negative , non - decreasing , and concave function with to transform the weights into the valuation concept .that is , given a set of individuals directly exerting influences on an individual , s valuation towards a commodity will become .based on the monetary - concept incorporated lt - model , the propagation of social influences over the monetizing social network happens whenever individuals _ adopt _ the commodity .an individual _ adopts _ the commodity if and only if 1 ) the pricing of the commodity is less than or equal to his / her valuation , or 2 ) this individual is an initial customer receiving a freebie .after an individual adopts the commodity , he / she will then exert influences on his / her out - neighbors to encourage them and raise their valuations for adoption .the following shows an example of social influences propagating over the monetizing social network .[ ex : ex1 ] given the monetizing social network in figure [ fig : ex ] and the concave influence function , consider the promotion of a concert .suppose that the pricing of the concert tickets is and the seed group is . influenced by , the valuations of , , and are increased by , , and , respectively .the valuations of , and thus become , , and , respectively .then , note that as the valuation of is equal to the pricing of the concert tickets , also adopts the concert ticket and exerts influences on its out - neighbors and .the valuations of and thus increase to and , respectively .after that , as no other individuals adopt the concert ticket , the influence propagation stops .consequently , under s influence , will also adopt the concert ticket , and the valuations of , and become , , and , respectively .consider another example of the seed group . under the influence of and ,the valuations of , , and are raised to , , , and , respectively .( here s influence on is neglected since as an initial customer has already adopted the concert ticket . )hence , and adopt the concert tickets and exert influences .consequently , the valuations of and become : in the end , , , , and all adopt the concert tickets since their final valuations are larger than or equal to the pricing of the concert tickets .9999  here we define that the revenue comes from the number of people paying for the commodity and the pricing of the commodity .recall example [ ex : ex1 ] where the pricing is and the seed group is , and suppose that the quantity of the concert tickets is 4 .the revenue is , since two concert tickets are used as free tickets given to and , and only two concert tickets are left for sale ( even though , , , and all want to purchase these concert tickets ) .formally , given the quantity of commodities , the pricing of the commodity , and the seed group , the revenue is defined as follows .* revenue function .* given the quantity , the revenue at the pricing of with the seed group is where is the set of individuals adopting the commodity under the influences of .to increase the revenue , finding the proper pricing and identifying a proper seed group is important .consider the same setting mentioned above , but with an empty seed group and the pricing set as .the revenue is , which is less than .since any company will expect to earn as higher revenue as possible given a fixed amount of commodities , in this paper , we are interested in looking for such pricing and a seed group as initial customers that can bring maximum revenue .the proposed problem is formally defined as follows .* the rm problem : revenue maximization with a quantity constraint . * given a monetizing social network , a set of input prices , and a quantity of commodities , the problem is to determine the pricing of the commodity and find a seed group as initial customers , where , such that the revenue is maximized , i.e. , .note that the revenue function is not single - peaked with respect to the pricing , which means the approaches such as the binary search and the gradient decent search are not applicable to finding out the optimal pricing .for example , following the setting of example [ ex : ex1 ] and letting the quantity of the concert tickets be 4 , we obtain the highest revenue at pricing of , , and as , , and , respectively ( with the seed groups , , and , accordingly ) .obviously , with respect to the pricing , is not single - peaked .furthermore , the rm problem can be proved to be np - hard . to show this, we first introduce a special case of rm , and then prove that this special case is np - hard .as the special case is np - hard , rm is thus np - hard .* specialrm .* this problem is a special case of rm , and asks for only one input price and a sufficient quantity of commodities for the population .that is , given , , and , specialrm is to determine a seed group such that is maximized .[ theo : np ] the problem of specialrm is np - hard .we reduce the minimum vertex cover ( mvc ) problem to the specialrm problem . given every instance of mvc involving an undirected graph , we can construct a corresponding instance of specialrm as follows .( 1 ) set ; ( 2 ) set ; ( 3 ) direct all edges in both directions , i.e. , for each , and ; ( 4 ) for each , define , where is the in - degree of in ; and ( 5 ) set . for the instance of specialrm ,if there exists a minimum set that maximizes the revenue , we can obtain another set . then , it can be shown that is the minimum vertex cover of since theorem [ theo : np ] is thus proved .in this section , for the revenue maximization problem , we first propose algorithm prub ( pricing searching strategy using revenue upper bound ) that is able to derive the optimal solutions .note that finding the optimal pair of and in the revenue maximization problem is exhausted .then , for better efficiency , a heuristic algorithm prub+if ( prub with importance feedback ) is introduced following the framework of prub .   to tackle the rm problem ,algorithm prub is designed to obtain the optimal solutions . as mentioned previously ,finding a proper pair of pricing and seed group is critical in this problem , since the revenue comes from the pricing of the commodity and the number of people paying for the commodity .a nave approach for finding the optimal pair of pricing and seed group is to search at all , enumerate all seed groups , and calculate the corresponding revenue .the rationale of algorithm prub is based on this nave approach and prunes the search space by 1 ) progressively filtering out non - candidate pricing and 2 ) deriving an upper bound of the size of seed groups for each price .we illustrate the two ideas of pruning in the following .* non - candidate pricing filtering . *the first idea is to prune the searching on non - candidate pricing .motivated by the concern of only maximum revenue , this pruning derives an upper bound of maximum revenue at each price , and utilizes an achievable global revenue that records the maximum revenue discovered so far to progressively filter out the pricing that will not result in higher revenue. that is , if an upper bound of maximum revenue at each price can be derived , prub then does not need to seek the seed group for maximum revenue at a specific price when the upper bound at this price is less than or equal to . when is updated and increased progressively after the successful searching of the seed group at each price , the non - candidate pricing is also progressively detected and filtered out .  in order to infer the upper bound of maximum revenue at a specific price, the maximum number of individuals who have potential for adopting the commodity at this price is needed . to find out whether or not an individual has this potential , it needs to estimate the valuation of the individual .therefore , in the following , we first introduce the definitions of _ maximum valuations _ and _ potential buyers _ for defining the upper bound of maximum revenue .* maximum valuation . *the maximum valuation of an individual is the valuation under the influences from all s in - neighbors , which is denoted as   * potential buyer .* an individual is regarded as a potential buyer at a specific price if has potential for adopting the commodity at , i.e. , .  *upper bound of maximum revenue . * given a quantity constraint , the upper bound of maximum revenue at a price is where is the number of potential buyers at , i.e. , given the monetizing social network in figure [ fig : ex ] , the concave influence function , and a set of input prices , consider the promotion of a concert .table [ t : maximalbudgets ] shows the maximum valuation of each individual .suppose the quantity of the concert tickets is set as 4 .the corresponding upper bounds of maximum revenue for all prices are listed in table [ t : maximalrevenues ] .9999  in addition , to have better effectiveness of pruning by utilizing the upper bound of maximum revenue , prub searches the prices in a descending order of the upper bounds .then , once prub discovers a price , at which the upper bound is less than or equal to the achievable global revenue , the searching for the following prices ( including the current price ) can be ignored .therefore , in the above example , prub will start the examination of the prices from , , , , and so on .* bound of seed group s size . * the second idea for pruning search space is to avoid enumerating useless seed groups under a specific price .the inspiration is , under the consideration of a specific price , it involves more than quantities of commodities to be sold for finding the revenue higher than the achievable global revenue obtained so far . when considering the price , algorithm prub then bounds the size of the seed groups as since the quantity of commodities is limited .follow example [ ex : framework ] , and suppose the achievable global revenue . in order to find the revenue higher than ,when searching at the price , prub expects more than concert tickets left for sale and enumerates only the seed groups of sizes less than or equal to 2 ( since the quantity of the concert tickets is set as 4 ) .the searching on the seed groups of sizes 3 and 4 is thus pruned .note that incorporating the two pruning methods is satisfactory to the accuracy .therefore , algorithm prub is still able to derive the optimal solutions , even though the search space is reduced .the details of prub are presented in algorithm [ algo : framework ] . in the following , we show an example to illustrate how prub figures out the optimal pair of and .[ algo : framework ]   , , derive for all sort all descendingly by [ code : fram : search1 ]  [ ex : exprub ] follow example [ ex : framework ] , and initialize , , and as , an empty set , and , respectively . according to the upper bounds of maximum revenue in table [ t : maximalrevenues ] , prub will consider the prices in the following order : , , , , and so on .beginning from the price , prub checks whether or not . since , prub looks for the maximum revenue at the price by enumerating all the seed groups whose size is bounded by equation ( [ e : abound ] ) ( including the size 0 ) .first , for the empty seed group , no individual adopts the concert ticket at .so prub goes to the size 1 .then , prub lists all the seed groups of size 1 since . among all the seed groups of size 1, prub finds the highest revenue ( referred to example [ ex : ex1 ] ) .as is higher than , prub updates , , and . before seeking the seed groups of size 2 for maximum revenue, prub ensures .after all seed groups of size 2 are enumerated , the highest revenue is found ( referred to example [ ex : ex1 ] ) . because , prub updates , , and .later , note that .the searching for the maximum revenue at the price stops .the next price considered is .a similar process is performed until prub finds such a price that . in the end , the maximum revenue is , where the pair of and is the optimal solution .9999  [ theo : complexity ] the time complexity of prub is .we first show that prub costs time at a specific price for searching the maximum revenue .given the quantity of commodities , the largest size of the seed group is . in total , there are thus seed groups . for a seed group of size , there are seeds exerting influences on the other o( ) individuals that are not seeds . due to the influence cascade ,it costs o( ) time to figure out the influence spread of a seed group of size . therefore , the total time complexity for searching the maximum revenue by enumerating seed groups is as searching at a price costs , the total time complexity of prub is thus .theorem [ theo : complexity ] is proved .note that the high complexity of algorithm prub indicates the poor scalability of the optimal algorithm for real social networks which are usually large .therefore , in the following subsection , we propose a heuristic based on the framework of prub to obtain the feasible solutions on larger instances .   in this section, the heuristic algorithm prub+if ( prub with importance feedback ) is proposed as a feasible solution .the heuristic prub+if differs from prub at the way of finding the most proper seed group at each price . following the framework of prub, prub+if introduces the concept of _ pricing - sensitive importance _ , accompanying the contribution feedback from the out - neighbors , for selecting seeds in a heuristic manner , instead of listing all seed groups .the main idea of prub+if is that , an individual with greater potential for making others adopt the commodity should be regarded as more important . selecting the most important individuals as seeds greedily is able to lead to a feasible solution .then , the question is how to evaluate one s potential for making others adopt the commodity .the intuition is about how many others will be encouraged and how much their valuations can be increased for approaching the pricing , under one s influence .we then accumulate these effects to calculate one s importance . hereonly the effects on potential buyers should be included in the accumulation since only the potential buyers have possibilities of paying for the commodity . in addition , note that the individuals who newly adopt the commodity under one s ( direct and indirect ) influence will further spread their influences to encourage more others in adoption . in order to estimate one s importance more carefully ,the effects through influence cascades should also be included in the accumulation .therefore , prub+if introduces the pricing - sensitive importance that sums up the consideration of the normalized weights , the influence propagation , and the potential buyers in the measurement of one s advantage in the commodity promotion at the given pricing .then , the strategy is to select seeds in accordance with the pricing - sensitive importance .  in the following ,we introduce the three key points , normalized weight , feedback of influence propagation , and potential - buyer filtering , in measuring the pricing - sensitive importance .briefly , for each individual , 1 ) the normalized weight is used to evaluate s importance according to s direct effect on another individual s adoption ; 2 ) the feedback of influence propagation is incorporated to evaluate s importance towards by considering also the indirect effects through influence cascades ; 3 ) the potential - buyer filtering is considered to derive s pricing - sensitive importance from accumulating all s direct and indirect effects on all the other potential buyers .  1 .  * normalized weight . * + by intuition , for commodity promotion , an individual s importance towards another individual can be evaluated from how much s valuation approaches the pricing due to s effect .inspired by this intuition , prub+if calculates the normalized weight , indicating s importance towards s adoption , as follows .+ & _ uv = & + + where is s valuation of the commodity under the current seed group s effects . in the case of , it means has adopted the commodity .so the normalized weight , indicating that s importance towards is none due to no impact from on s adoption of the commodity . for the other case , the normalized weight from to bounded by 1 , and is the portion of s impact on s adoption of the commodity .+ [ ex : wnorm ] given the monetizing social network in figure [ fig : ex ] and the concave influence function , consider the promotion of a concert .given and , no one has adopted the concert tickets . under this condition , is less than the pricing of , and the normalized weight from to is .this means that selecting into the seed group can make adopt the concert ticket immediately .similarly , 2 .* feedback of influence propagation . * + note that the effect of influence propagation is important in the commodity promotion . to carefully estimate one s advantage in the promotion, prub+if further takes the importance feedback from those individuals into account on s importance , if adopts the commodity due to s direct or indirect effect and thus devotes to the propagation of s effect .therefore , s importance towards will be derived recursively as follows .+ & if^(k+1)(u , v)= & + where and contains the individuals who newly adopt the commodity by s effects at propagation , i.e. , . as the influencepropagates , there are multiple updates for s importance towards .when the propagation stops , i.e. , , the final importance of towards is denoted as . in other words ,equation ( [ e : if ] ) evaluates the advantage of selecting into the current seed group from the aspect of s adoption , by considering the direct effect derived by equation ( [ e : nw ] ) and the indirect effects propagated through individuals .in addition , equation ( [ e : if ] ) ensures that the importance of towards is bounded by 1 , even though there may be many direct and indirect impacts from to .+ [ ex : propagate ] follow example [ ex : wnorm ] and consider the individual .initially , , because . for all individuals ( excluding itself ) are thus calculated as follows . since no other individuals newly adopt the concert ticket , i.e. , , the propagation stops .finally , are as above .* potential - buyer filtering .* + once the advantage of selecting an individual as a seed can be estimated from the aspect of another s adoption , intuitively , the importance of an individual can be derived by summing up his / her importance towards all the others . however , counting one s importance towards all the others may be misleading , since the advantage estimated from those who will never adopt the commodity at this price is also counted . to avoid overestimating the importance of an individual in the promotion, prub+if thus incorporates the concept of potential buyers into the pricing - sensitive importance which is formally defined below . where is the set of potential buyers at this price , i.e. , the individuals whose maximum valuation is larger than or equal to the price ( the same as introduced in section [ sec : framework ] ) . as a result ,only if an individual is a potential buyer who has potential for adopting the commodity at the price , the increase of his / her valuation will be regarded as important .+ [ ex : psi ] follow example [ ex : propagate ] . since the potential buyers at the pricing of are , , , and ( referred to table [ t : maximalbudgets ] ) , the pricing - sensitive importance of is .9999   , , [ code : fram : init1 ] derive for all sort all descendingly by [ code : fram : init2 ]  the algorithm of prub+if is presented in algorithm [ algo : contribution ] .prub+if uses the framework of prub as a base , and differs from prub at the strategy for seed selection ( algorithm [ algo : contribution ] lines [ code : fram : differ3]-[code : fram : differ4 ] ) . in the following , we go through the details of prub+if . similarly to prub , prub+if initializes , , and , compute the upper bounds of maximum revenue for all prices , and then considers the prices one - by - one in a descending order of the upper bounds of maximum revenue ( lines [ code : fram : init1]-[code : fram : init2 ] ) . at a specific price , prub enumerates all seed groups while prub+if iteratively picks up a seed until the size of the seed group reaches the bound in equation ( [ e : abound ] ) ( line [ code : fram : prune6 ] ) .specifically , when considering a price , prub+if first checks whether or not is non - candidate pricing that has no chance to yield higher revenue than the achievable global revenue ( line [ code : fram : prune4 ] ) . if passes the pruning , prub+if first calculates the revenue regarding to the empty seed group ( lines [ code : fram : empty1]-[code : fram : empty3 ] ) . after that , prub+if expands the seed group by one seed at a time according to the pricing - sensitive importance in equation ( [ e : psi ] ) ( line [ code : fram : psi ] ) .each time the individual who has not yet adopted the commodity and has the greatest pricing - sensitive importance will be selected into the current seed group ( lines [ code : fram : differ3]-[code : fram : differ4 ] ) . in the case of the newly obtained revenue higher than ,prub+if updates , , and similarly to prub ( lines [ code : fram : update3]-[code : fram : update4 ] ) .once the size of the seed group reaches the bound in equation ( [ e : abound ] ) ( line [ code : fram : prune6 ] ) , prub+if considers the next price .after all prices are examined , and are returned as the solution .the following example demonstrates the searching process of prub+if .follow example [ ex : framework ] , and initialize , , and as , an empty set , and , respectively . according to the upper bounds of maximum revenue in table [ t : maximalrevenues ] ,the first price considered is . since , prub+if tries to seek for higher revenue at the price .first , for , there is no revenue since the inherent valuations of all individuals are less than .after that , for selecting the first seed , prub+if computes the pricing - sensitive importance of all individuals in a similar way as shown in example [ ex : psi ] : hence , the first seed selected is and the corresponding revenue will be ( referred to example [ ex : ex1 ] ) . as , prub+if performs the updates of , , and . until now , the process for the first seed finishes , and prub+if checks whether or not to continue selecting the second seed . since , the pricing - sensitive importance of all individuals , except the individuals who have adopted the concert tickets , i.e. , and , is calculated with respect to the current seed group : accordingly , will be selected as the second seed and the revenue is ( referred to example [ ex : ex1 ] ) , which leads to the updates of , , and . until now ,the process for the second seed finishes , and prub+if checks whether or not to continue selecting the third seed .however , reaches the bound so that the searching at the price stops .the searching at the remaining prices , , , and so on , is performed in the same manner as that at the price . in the end, prub+if will report the pricing of and the seed group as the answer .9999  [ theo : complexity2 ] the time complexity of prub+if is .we first show that prub+if costs time at a specific price for searching the maximum revenue .given the quantity of commodities , the largest size of the seed group is .when seeds have been selected , it takes time to compute the pricing - sensitive importance for each of the other nodes .therefore , the total time complexity for searching the maximum revenue is as searching at a price costs , the total time complexity of prub+if is thus .theorem [ theo : complexity2 ] is proved .[ t : summary ]  [ cols="^,>,>,>,>,>,^,^,^,^,^,^ " , ]   in this section , we first provide the evidence that the revenue maximization problem with a quantity constraint differs from that without a quantity constraint .besides , we also show the approximation performance of prub+if to prub . in order to demonstrate the effectiveness of prub+if , we further conduct experiments on larger real social networks and compare prub+if with other heuristics .all the programs are implemented in java with version 1.6.0_27 .the experiments are performed on a quad - core intel xeon x5450 3.00ghz pc with 8 gb ram using ubuntu 8.04.2 .* datasets .* we use three real social networks as follows .1 ) the dataset _ highschool _ ( sampled from the dataset due to poor scalability of prub and the compared approach ) is a friendship network for the boys in a small highschool in illinois .the nodes are boys in the highschool , and each directed edge represents a boy choosing another as a friend in questionnaires .2 ) the dataset _ digg _ is a reply network of digg .each node is a user in digg and each directed edge stands for a user replying to another .3 ) the dataset _ facebook _ is the communication network of facebook .the nodes are the users in facebook while each directed edge represents that a user posts an article on another user s wall . for these datasets , the self - loops and isolated nodesare removed , and the edge weights are derived from the number of choices , replies , and articles of one to another , respectively .both the degree distributions and weight distributions of all the datasets follow the well - known power law .more details are shown in the left five columns of table [ t : summary ] .  * valuation distributions .* note that there is lack of valuation information in these publicly available datasets . a nave way is to learn the valuation information through questionnaires or historical sales data toward a specific commodity .however , note that the valuation distributions vary with commodities . in this paper, we then explore the performance under different valuation distributions as follows .* normal distribution ( n ) : most people have similar valuations and a few people have extremely higher or lower valuations toward a commodity .* m - shape distribution ( m ) : people do not have consensus on the value of a commodity . one group of people have higher valuations and the other group of people have lower valuations toward the commodity .  in our experiments ,we implement the m - shape distribution using two normal distributions , where the valuation of each individual follows one of the normal distributions with equal probabilities .the parameters for generating the valuations are listed in the right three columns of table [ t : summary ]. * concave influence function .* as introduced in section [ sec : pf ] , any non - negative , non - decreasing , and concave function can be applied to the concave influence function . in our experiments , we consider the function to transform the weights into the valuation concept for simplicity .  * input prices . * by considering different currencies , costs , and cultures in different countries , the pricing of a commodity may be different . for simplicity , we set the prices as input parameters in the experiments . for _ highschool_ , the input prices are all integers ( as the unit of the dollars ) in  ] .   in this subsection , we first show that the consideration of a quantity constraint in the revenue maximization problem makes it differ from the original problem by comparing prub with the previous work , referred to as randomizedusm in this paper . due to poor scalability of prub and randomizedusm , these experiments are conducted on the small dataset _ highschool _ with normal and m - shape valuation distributions .in addition , note that since randomizedusm does not consider a quantity constraint , we derive the revenue with two kinds of post - processes as follows .* randomizedusm : given the solution pair of and randomizedusm reports , this post - process considers the quantity constraint by calculating the quantities left for sale and computing the revenue as , where is the set of individuals adopting the commodity under s effects . *randomizedusm(p ) : this post - process considers the quantity constraint at each price and then returns the highest revenue among all prices . for each price and the corresponding seed group obtained by randomizedusm, the revenue is , where is the set of individuals adopting the commodity under s effects .figure [ fig : small ] shows the maximum revenue reported by randomizedusm , randomizedusm(p ) , prub+if and prub with respect to different ratios of the quantity of commodities to the number of population , i.e. , .accordingly , the revenue reported by randomizedusm and randomizedusm(p ) is , on average , less than 70% of that obtained by prub .moreover , note that randomizedusm has the poorest performance and even brings zero revenue at the cases of the supply ratio below 0.2 .these results demonstrate that the capability of the previous approach ( even with the post - processes ) is limited for the rm problem . on the other hand , even for the special case that the quantity is not constrained , i.e. , the ratio of the quantity of commodities to the number of population , prub+if still outperforms randomizedusm and randomizedusm(p ) .this is because prub+if selects seeds in order of the pricing - sensitive importance while randomizedusm and randomizedusm(p ) consider the seeds in arbitrary order .  on the same datasets, we then also explore the approximation of prub+if to prub ( which derives the optimal solutions ) . according to the results in figure [ fig : small ], the performance of prub+if is very close to that of prub , i.e. , about 96% on average , and outperforms both randomizedusm and randomizedusm(p ) .this implies that the proposed pricing - sensitive improtance of prub+if is highly effective to identity the crucial individuals as seeds .   in this subsection ,we conduct experiments on larger datasets _ digg _ and _ facebook _ to further show the performance of algorithm prub+if in terms of effectiveness , pricing , and efficiency . hereprub+if is compared with two other heuristics prub+r and prub+sw .these two heuristics also take the prub approach as the framework while applying different strategies for seed selection .specifically , prub+r ( r standing for random ) selects seeds randomly ; prub+sw ( sw meaning sum of the weights ) always picks up the individual with the maximum sum of out - weights from those who have not yet adopted the commodity . besides , in order to demonstrate the advantage of each part of prub+if , the heuristics prub+n , prub+f , and prub+p represent the approaches that incorporate normalized weight , feedback of influence propagation , and potential - buyer filtering , respectively , for seed selection .  [ fig : effectiveness ]  [ fig : advantage ]  [ fig : pricing ]  * effectiveness .* here we use the maximum revenue obtained without consideration of social influences ( nosocial ) , i.e. , the adoption of an individual only depends on his / her inherent valuation , as the baseline , and show the effectiveness of the heuristics in ratios to the baseline .figure [ fig : effectiveness ] reports the effectiveness with respect to different ratios of the quantity of commodities to the number of population , i.e. , . for these results , we have two observations as follows . first , on _ digg _ and _ facebook _ with both normal and m - shape valuation distributions , prub+if is the best , and prub+sw outperforms prub+r only in the cases of small .the reason is that , to estimate the advantage of selecting an individual in the promotion , prub+if introduces the pricing - sensitive importance , which considers the normalized effects of the influence propagated through the social network , while prub+sw only takes the sum of weights on one s direct out - neighbors .rrub+r picks the seeds blindly .second , as decreases , the performance of prub+if becomes even more noticeable ( compared to prub+sw and prub+r ) .this is because the number of potential buyers reduces when the pricing of the commodity increases due to scarcity .the less the number of potential buyers is , the more challenging the seed selection is .hence , the results demonstrate that the strategy of prub+if for the seed selection outperforms those of prub+sw and prub+r .  in order to demonstrate the advantage of each part of prub+if , figure [ fig : advantage ]shows the effectiveness of prub+if , prub+n , prub+f , and prub+p with respect to different ratios of the quantity of commodities to the number of population , i.e. , .accordingly , the three parts of prub+if are more or less all equally important .* pricing . * in addition to the demonstration of effectiveness by maximum revenue , whether or not the suggested pricing complies with the common sense is also an interesting question . for answering this question, we plotted the trend of the pricing suggested by prub+if , prub+sw , prub+r , and nosocial with respect to different ratios of the quantity of commodities to the number of population , i.e. , , in figure [ fig : pricing ] . accordingly, prub+if suggests higher pricing in most cases than the other approaches to gain higher maximum revenue ( as shown in figure [ fig : effectiveness ] ) in all cases .this implies that prub+if performs the most elegant strategy for seed selection .the smart utilization of word - of - mouth effects can help the promotion of commodities in higher pricing , which complies with the common sense .furthermore , for all methods , there is a upward trend in the pricing as decreases .the less the quantity is , the more precious the commodity could be , which also complies with the common sense .there are some exceptions to this pricing trend : prub+if at 0.25 on _digg_(m ) , and both prub+sw and prub+r at 0.25 or 0.3 on _facebook_(n ) .the reason is that a quantity of supply is used as freebies .this may allow higher pricing in some cases by leaving less quantities for sale .the tiny example in figure [ fig : exp ] explains such a case in more details .( is used as the concave influence function . ) for the case of the quantity of the concert tickets as 2 , the maximum revenue is ( i.e. , and adopt the concert tickets ) . now consider that the quantity of the concert tickets is increased to 3 .the maximum revenue is ( i.e. , adopts the concert ticket under the impact of and ) .the pricing is higher even though the quantity increases .this is because 2 out of 3 concert tickets are used as free tickets and only 1 concert ticket is left for sale .hence , according to these results , prub+if is capable of suggesting elegant marketing strategies .  * efficiency .* now we compare the efficiency of the three heuristics .figure [ fig : runtime ] shows the runtime with respect to different ratios of the quantity of commodities to the number of population , i.e. , , on _digg_(n ) , _ digg_(m ) , _facebook_(n ) , and _facebook_(m ) .prub+if spends more execution time than prub+sw and prub+r in all cases , since prub+if considers the influence propagation in the derivation of pricing - sensitive importance for seed selection .prub+r generally spends a little bit more time than prub+sw due to the blind seed selection ( that makes prub+r need to search more prices for finding maximum revenue ) .nevertheless , the runtime of all the three heuristics is not sensitive to . for prub+if , other factors such as the local structure of potential buyers and the influence cascades will also affect the runtime .\(a ) ; ( b ) [ right=1 cm of a ] ; ( c ) [ right=1 cm of b ] ;  \(a ) [ - > ] edge node 2 ( b ) ( c ) [ - > ] edge node 2 ( b ) ;  [ fig : runtime ]* influence maximization . *the influence maximization problem aims at identifying a group of seeds so that the number of the active people is the largest . to the best of our knowledge, the first study could be traced to the work , where domingos et al . studied the influence maximization problem as an algorithmic problem. then , kempe et al . formally modeled the problem as a discrete optimization , which is to identify seeds for maximizing the influence spread over the social network .meanwhile , kempe et al . also showed that the influence maximization problem is np - hard under both the linear threshold ( lt ) and the independent cascade ( ic ) models , and proposed a greedy strategy with approximation guarantees .specifically , in the lt model , the influences on an individual are cumulative , and each individual has his / her own activation threshold .an individual becomes active if the sum of weights from his / her active neighbors is larger than or equal to his / her activation threshold .then , this active individual will also propagate influences to his / her inactive neighbors . in the ic model ,each influence an individual receives is independent . for each influence, there is one and only one chance to activate the individual with a probability .if the individual becomes active , he / she futher spreads his / her influence in the same mannar . in the work , chen et al . showed that , given seeds , computing the influence spread under the lt model is # p - hard , and provided a scalable heuristic based on the fast computation for directed acyclic graphs ( dags ) . singer the inherent costs of making individuals as seeds and followed the incentive compatible mechanism to extract the true information about individuals costs for identifying and rewarding the seeds . as mentioned in section [ sec : intro ] , all these influence maximization problems do not take the monetary effects on people s purchase decisions into account , and thus differ from our problem.  * revenue maximization . *the problem of revenue maximization is to derive the marketing strategy that brings the optimal revenue under the social influence , and was first addressed by hartline et al . . as a solution , hartline et al . proposed a marketing strategy , named influence - and - exploit ( ie ) , to tackle the problem in two steps . in the influence step , ie identifies a group of customers as seeds . after that , in the exploit step , ie visits the remaining customers in a random order and offers each of them the optimal ( myopic ) pricing of the product in order to maximize the revenue .the decision whether or not an individual purchases the product depends on the pricing offered and the individual s valuation , which takes the social influences into consideration .hartline et al. also discussed the inferences of the valuations from social influences by uniform additive model , symmetric model , and concave graph model .later , fotakis et al . proposed a polynomial - time approach to approximate the maximum revenue in the uniform additive model .another work distinguishes between activation and adoption . when an individual is active by the social influence , he / shewill then determine whether or not to adopt the product with respect to the pricing and his / her valuation .the marketing strategy of all these works is to determine customized pricing for different customers .mirrokni et al . argued that offering the product at fixed pricing is more reasonable since offering customized pricing is hard to implement in the real world .hence , their objective is to discover a seed group with the fixed pricing to maximize the revenue .mirrokni et al . adopted the concave graph model proposed in the work to incorporate the social influence into the valuation , and presented a ( 1/2)-approximation based on the randomized linear time approximation algorithm for the unconstrained submodular maximization problem .however , all the works above assumed that there is an unlimited supply of commodities , whereas we consider the scenario with limited supply such as limited edition commodities and concert tickets . as shown in sections [ sec : intro ] and[ sec : exp1 ] , it is unapplicable for these works to determine the proper marketing strategies .in this work , we addressed the revenue maximization problem with a quantity constraint on a monetizing social network . to maximize the revenue ,the marketing strategy is to determine the pricing of the commodity and find a seed group of individuals as the initial customers .hence , we provided algorithm prub to derive the pricing and the seed group for the optimal revenue . for better efficiency , we further proposed algorithm prub+if as a heuristic algorithm based on the framework of prub for a feasible solution .the experiments on real datasets with different valuation distributions showed the effectiveness of prub and prub+if .10 d. arthur , r. motwani , a. sharma , and y. xu .pricing strategies for viral marketing on social networks . in _ wine _ , 2009 .s. e. asch .effects of group pressure on the modification and distortion of judgments . in h.guetzkow ( ed . ) , _ groups , leadership and men_(pp .177 - 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Abstract: recently the influence maximization problem has received much attention for its applications on viral marketing and product promotions . however , such influence maximization problems have not taken into account the monetary effect on the purchasing decision of individuals . to fulfill this gap , in this paper , we aim for maximizing the revenue by considering the quantity constraint on the promoted commodity . for this problem , we not only identify a proper small group of individuals as seeds for promotion but also determine the pricing of the commodity . to tackle the revenue maximization problem , we first introduce a strategic searching algorithm , referred to as algorithm prub , which is able to derive the optimal solutions . after that , we further modify prub to propose a heuristic , algorithm prub+if , for obtaining feasible solutions more efficiently on larger instances . experiments on real social networks with different valuation distributions demonstrate the effectiveness of prub and prub+if .