document: turbulence is a strong nonequilibrium phenomenon , exhibiting unpredictable behavior of the velocity field , which can result from the multiple degrees of freedom and nonlinearity of fluid systems @xcite . 
   this situation complicates our understanding of turbulence , making it one of the unresolved problems in modern physics . 
   the studies on turbulence in quantum fluids have possibility to shed new light on our understanding of turbulence . in quantum turbulence ( qt ) for quantum fluids such as superfluid helium and one - component atomic bose - einstein condensate ( bec ) @xcite , many quantized vortices with quantized circulation are nucleated , forming tangles . 
   this situation is drastically different from classical turbulence ( ct ) realized in classical fluids because the circulation of velocity field is not quantized in this system . 
   therefore , the element of qt is more obvious than that in ct , and qt is believed to be useful for the understanding of ct . 
   recently , a new trend begins to appear in turbulence study in atomic becs , which is turbulence in multi - component becs @xcite . 
   atomic becs have various characteristic features , one of which is a realization of multicomponent becs @xcite . in this system 
   , there exists not only a velocity field but also a ( quasi ) spin field . owing to the spin degrees of freedom , various topological excitations such as monopole , skyrmion , knot , domain wall , and vortex 
   therefore , in this system , one expects novel turbulence , in which both the velocity and spin fields are much disturbed and various topological excitations are generated . 
   thus , this kind of turbulence should give us new observations for turbulence not found in conventional systems . 
   previously , we studied turbulence in a spin-1 ferromagnetic spinor bec , which is a typical multicomponent bec @xcite . in this turbulence , the spin field 
   is disturbed , so that we call it spin turbulence ( st ) . in our previous study 
   , we focused on the spectrum of the spin - dependent interaction energy corresponding to the spin correlation , finding the characteristic @xmath1 power law . 
   however , this observation sees only one side of st . in st , 
   the velocity and spin fields interact with each other , so that a coupled turbulence with two fields can be realized , lending the possibility of showing a property of the velocity field not seen in conventional ct and qt . studying the both sides of turbulence completes the story .    in this paper 
   , we treat this problem , focusing on the spectrum of the superflow kinetic energy in a spin-1 ferromagnetic spinor bec . 
   the spectrum of the kinetic energy is theoretically and numerically found to show a kolmogorov spectrum through the interaction between the spin and velocity fields . 
   the kolmogorov spectrum refers to the @xmath0 power law in the kinetic energy spectrum , which is known to appear in ct and qt @xcite . 
   this spectrum is considered to be related to the vortex dynamics , so that it is significant to confirm it for understanding the turbulence addressed here . 
   furthermore , when the @xmath0 power law is sustained , the spectrum of the spin - dependent interaction energy exhibits a @xmath1 power law simultaneously . 
   therefore , we obtain the coupled turbulence with the disturbed spin and superfluid velocity fields sustaining the two power laws , anew calling it spin - superflow turbulence ( sst ) instead of st . 
   we consider a bec of spin-1 bosonic atoms with mass @xmath2 at zero temperature without trapping and magnetic fields . 
   this system is well described by the macroscopic wave functions @xmath3 ( @xmath4 ) with magnetic quantum number @xmath5 , which obey the spin-1 spinor gorss - pitaevskii ( gp ) equations @xcite given by @xmath6 in this paper , roman indices that appear twice are to be summed over @xmath7 , 0 , and 1 , and greek indices are to be summed over @xmath8 , @xmath9 , and @xmath10 . 
   the parameters @xmath11 and @xmath12 are the coefficients of the spin - independent and spin - dependent interactions , which are expressed by @xmath13 and @xmath14 , respectively 
   . here , @xmath15 and @xmath16 are the @xmath17-wave scattering lengths corresponding to the total spin-@xmath18 and spin-2 channels . 
   the total density @xmath19 and the spin density vector @xmath20 ( @xmath21 ) are given by @xmath22 and @xmath23 , where @xmath24 are the spin-1 matrices . 
   the sign of the coefficient @xmath12 drastically changes the spin dynamics . in this paper 
   , we consider the ferromagnetic interaction @xmath25 .    in this paper , we focus on the energy spectra for the superflow kinetic and the spin - dependent interaction energy . 
   the kinetic energy of superfluid velocity @xmath26 per unit mass is given by @xmath27 where @xmath28 and @xmath29 is a total particle number . 
   the superfluid velocity @xmath30 is given by @xmath31 by using the fourier series @xmath32 , we define the spectrum for the kinetic energy per unit mass : @xmath33 where @xmath34 and @xmath35 are given by @xmath36 and @xmath37 , respectively , with system size @xmath38 and spatial dimension @xmath39 . 
   similarly , the spectrum of the spin - dependent interaction energy per unit mass is defined by @xmath40 where @xmath41 is defined by @xmath42 } $ ] with the fourier transformation @xmath43 } = \int \cdot \hspace{0.5mm}e^{-i\bm{k}\cdot \bm{r } }   d\bm{r}/v$ ] and @xmath44 . 
   we discuss the possibility for a kolmogorov spectrum in sst with hydrodynamic equations obtained from eq . 
   ( [ gp ] ) . 
   these hydrodynamic equations are derived in @xcite , being composed of the equations of the total density @xmath19 , the superfluid velocity @xmath26 , the spin vector @xmath45 , and the nematic tensor @xmath46 with @xmath47/2 $ ] . 
   we apply the following three approximations to these hydrodynamic equations : ( i ) the macroscopic wave functions are expressed by the fully magnetized state , ( ii ) the total density is almost uniform ( @xmath48 ) , and ( iii ) the magnitude of velocity @xmath26 is much smaller than the density sound velocity @xmath49 . 
   these similar approximations are discussed in our previous papers @xcite . 
   the approximation ( i ) leads to the relation between the spin vector and nematic tensor @xcite : @xmath50 thus , by eliminating the nematic tensor in the hydrodynamic equations of @xcite , we obtain the following equations @xmath51 @xmath52 @xmath53 with the approximations ( ii ) and ( iii ) , eqs . 
   ( [ 1 ] ) - ( [ 3 ] ) become @xmath54 , \label{spin_motion}\end{aligned}\ ] ] @xmath55 in eq . 
   ( [ velocity_motion ] ) , the inertial term @xmath56 is smaller than the other terms @xcite , but we retain this term for the following explanation . 
   we apply a kolmogorov - type dimensional scaling analysis @xcite to eqs . 
   ( [ spin_motion ] ) and ( [ velocity_motion ] ) , obtaining the @xmath0 power law . 
   we consider the scale transformation @xmath57 and @xmath58 . 
   then , if @xmath59 and @xmath60 are transformed to @xmath61 and @xmath62 , eqs . 
   ( [ spin_motion ] ) and ( [ velocity_motion ] ) are invariant . 
   thus , the velocity field satisfies @xmath63 with a nondimensional coefficient @xmath64 
   . then the spatial and temporal dependence of @xmath65 is same as that of @xmath26 , which leads to @xmath66 with @xmath67 . 
   thus , in sst , the spectrum of the superflow kinetic energy can be determined by the kinetic energy flux @xmath68 and the coefficient @xmath69 , which , by using a kolmogorov - type dimensional analysis , leads to @xmath70 the coefficient @xmath71 is nondimensional , which corresponds to the kolmogorov constant in ct . 
   therefore , the spectrum of the kinetic energy of sst can obey the kolmogorov spectrum .    applying a similar analysis to the spin field 
   , we obtain the relation @xmath72 with a dimensional coefficient @xmath73 , which leads to the @xmath1 power law given by @xmath74 with the spin - dependent interaction energy flux @xmath75 and a dimensional coefficient @xmath76 . 
   this was discussed in the previous study @xcite . 
   we note that this @xmath0 power law in sst is much different from that in ct . in three ( two)-dimensional ct 
   , there is the direct ( inverse ) energy cascade , where the @xmath0 power law is generated by the inertial term @xmath56 in the navier - stokes equation @xcite . on the contrary , in sst , the spatial gradient of the spin vector in eq . 
   ( [ velocity_motion ] ) leads to the @xmath0 power law because , in eq . 
   ( [ velocity_motion ] ) , the order estimation @xcite finds that the inertial term is smaller than the nonlinear spin term in the range @xmath77 with the density coherence length @xmath78 . 
   this suggests that the mechanism responsible for the @xmath0 power law in sst should be different from that in ct .    finally , the assumptions used in the derivation of the @xmath0 and @xmath1 power laws are discussed . 
   we use five assumpotions , which are ( i ) the macroscopic wave functions are expressed by the fully magnetized state , ( ii ) the total density is almost uniform ( @xmath48 ) , ( iii ) the magnitude of velocity @xmath26 is much smaller than the density sound velocity @xmath79 , ( iv ) the spin and velocity fields are scale invariant , and ( v ) the energy flux is independent of the wave number . 
   it is difficult to theoretically confim the validity of these assumptions . 
   then , we consider that the validity may be indirectly confirmed if the @xmath0 and @xmath1 power laws based on these assumptions appear in the numerical result .    in the following , 
   we show our numerical method and results to confirm these theoretical considerations . 
   [ b ]     components and the rotations for the velocity and spin fields in sst at @xmath80 . ( a ) and ( b ) show the @xmath8 components for the velocity @xmath26 and spin @xmath81 fields , and ( c ) and ( d ) show the @xmath10 component of their rotation . 
   the system size is @xmath82.,width=321 ]    [ t ]          in our numerical calculation , we introduce a phenomenological small - scale energy dissipation term into eq . ( [ gp ] ) for the following reasons . in three - dimensional ct 
   , an energy cascade from low to high wave number is assumed , and the energy in the high - wave - number region is considered to dissipate @xcite . unless this kind of dissipation takes place , energy accumulates in the high - wave - number region , which can break the power law in the spectrum .    based on this description of the energy cascade in three - dimensional ct , in the previous study of qt in the one - component becs @xcite 
   , a phenomenological small - scale dissipation was added to the one - component gp equation given by @xmath83 @xmath84},\end{aligned}\ ] ] where the macroscopic wave function and the interaction coefficient for the one - component bec are denoted by @xmath85 and @xmath86 , respectively . 
   @xmath87 is the fourier component @xmath88 } $ ] of this wave function . 
   the function @xmath89 is defined by @xmath90 with the step function @xmath91 and the strength of dissipation @xmath92 , which dissipates the energy in the high - wave - number region larger than the wave number @xmath93 corresponding to the coherence length in the one - component gp equation . also , this dissipation reduces the particle number , so that , in the study of ref . 
   @xcite , the chemical potential is adjusted for it s conservation . 
   thus the chemical potential @xmath94 has a time dependence . 
   we introduce a similar phenomenological dissipation term into the spin-1 spinor gp equations ( [ gp ] ) . in a spin-1 spinor bec 
   , there are two characteristic lengths : the density coherence length @xmath95 and the spin coherence length @xmath96 , which are defined by @xmath97 and @xmath98 . in the usual experiments 
   , @xmath99 is larger than unity , which leads to the condition @xmath100 . 
   the size of the spin structure such as the spin domain wall and the spin vortex is on the order of @xmath101 . with reference to qt in the one - component bec 
   , we expect that the energy dissipation occurs for wave numbers greater than the wave number @xmath102 corresponding to the spin coherence length @xmath103 . therefore , we add a phenomenological small - scale energy dissipation to eq . 
   ( [ gp ] ) , which is given by @xmath104 @xmath105 } , \label{dgpk2}\end{aligned}\ ] ] where @xmath106 and @xmath107 is defined by @xmath108}$ ] and @xmath109 . in the previous study , 
   the chemical potential has a time dependence in order to conserve the total particle number . 
   however , in our calculation , we do not adjust the chemical potential @xmath110 , because the total particle number hardly decreases in sst . 
   we describe the parameters and the initial state in our numerical calculation of eqs . 
   ( [ dgpk1 ] ) and ( [ dgpk2 ] ) . 
   all our numerical results are obtained in a two - dimensional system , whose size is @xmath82 . to generate sst , 
   we prepare the dynamically unstable state as the initial state , which is the counterflow state @xcite . 
   this state is given by @xmath111 with a relative velocity @xmath112 . in our numerical calculation 
   , we set @xmath113 . the dependence of numerical result on @xmath112 is discussed in sec . 
   vi a. the usual experimental ratio @xmath114 for @xmath115 is about @xmath116 @xcite , but , in our numerical calculation , we set @xmath117 with a positive @xmath11 and negative @xmath12 to grow the dynamical instability quickly . 
   we use the strength of the dissipation @xmath118 and the chemical potential @xmath119 . in the initial state , we add a small white noise contribution to cause the counterflow instability . 
   figures [ fig1](a ) and 1(b ) show the distribution of the @xmath8 components of the velocity and spin fields in sst at @xmath120 with @xmath121 . through the counterflow instability 
   , these two fields are disturbed . 
   the @xmath10 components of the rotation for these two fields are shown in figs . [ fig1](c ) and 1(d ) , where the rotations are also disturbed @xcite . in a spin-1 spinor bec , the superfluid velocity is related to the spin field through the mermin - ho relation , so that the vortical field can be continuous , which is much different from the one - component bec @xcite . actually , as seen in fig . [ 
   fig1](c ) , the vortical field @xmath122_{z } $ ] has a smooth spatial dependence . 
   the time development of the spectrum of the spin - dependent interaction and the superflow kinetic energy is shown in fig . 
   [ fig2 ] . in the early stage of the instability , as shown in fig . 
   [ fig2](a ) , these spectra have a peak corresponding to the most unstable wave number @xmath123 for the counterflow instability , which was theoretically obtained by using the bogoliubov - de gennes equations for this initial state @xcite . 
   this wave number @xmath124 is the energy injection scale . 
   as time progresses , the energy is transferred from low to high wave number , as seen in fig . [ fig2](b ) . 
   after time @xmath125 , the spectra of the velocity and the spin - dependent interaction energy exhibit the @xmath0 and @xmath1 power laws in figs . 
   [ fig2](c ) and ( d ) . in fig . 
   [ fig2 ] ( d ) , the range @xmath126 of these scaling laws is not so wide , but the scaling behavior is consistent with our consideration in sec . 
   therefore , we conclude that our numerical result confirms the -5/3 and @xmath1 power laws 
   .    we comment on the scaling range . 
   this narrow scaling range may come from how to excite to the initial state . as shown in fig . 
   [ fig2](a ) , the energy is injected at the wave number @xmath127 , which roughly determines the lower limit of the scaling range . 
   therefore , if we use another method with lower @xmath124 for generating sst , the scaling range may be wider .    in our calculation 
   , the @xmath0 power law applies in the range @xmath128 . for @xmath129 , we confirm that the spectrum of the kinetic energy begins to deviate from the @xmath0 power law as shown in fig . 
   3 ; this is caused by the shortage of energy in the low - wave - number region and the energy accumulation near @xmath102 . 
   on the other hand , the @xmath1 power law in the spectrum of spin - dependent interaction energy sustains even for @xmath129 . 
   as time sufficiently passes , the spectrum in low wave number region grows , and this spectrum has a configuration similar to that in our previous study @xcite . 
   this growth is discussed in sec . 
   b.    finally , we comment on what happens if the energy dissipation is absent . 
   we perform numerical calculation without the dissipation , which shows two following behaviors : ( i ) the spectrum of superflow kinetic energy still shows the @xmath0 power law , but the time period sustaining this power law becomes shorter , and ( ii ) the fluctuation of the spectrum is larger . 
   these similar behaviors were observed in the previous study for qt of one - component bec too @xcite . 
   [ t ]     ( a ) 800 and ( b ) 1500 . the dotted lines in the upper and lower graphs are proportional to @xmath130 and @xmath131 , respectively.,width=321 ]      we now consider what structure of velocity field leads to the kolmogorov @xmath0 power law in sst . in qt , 
   the vortical velocity field seems to be important for the kolmogorov spectrum . 
   then , to investigate the vortical flow of @xmath132 in sst , we decompose the vector @xmath133 into incompressible @xmath134 and compressible @xmath135 parts @xcite . 
   the helmholtz theorem leads to @xmath136 , where the relations @xmath137 and @xmath138 are satisfied . 
   thus , the superflow kinetic energy per unit mass is expressed by @xmath139 with @xmath140 ( @xmath141 ) . using the fourier series @xmath142 ( @xmath141 ) 
   , we can define the spectra for each kinetic energy per unit mass as @xmath143 we calculate the time dependence of @xmath144 in fig . 
   [ fig4](a ) ; one can see that the incompressible superflow kinetic energy is much larger than the compressible one . 
   this can be caused by ( i ) the condition @xmath145 under which the total density is hard to disturb , and ( ii ) the dissipation , which prevents the total density modulation from accumulating . figure [ fig4](b ) shows the spectrum @xmath146 of the compressible kinetic energy at @xmath147 , which deviates from the @xmath0 power law in comparison with @xmath148 in fig . [ 
   fig2](d ) . 
   therefore , the vortical structure of @xmath133 is significant for the kolmogorov spectrum in sst , which is similar to the situation of qt . 
   [ t ]     ( @xmath149 ) and ( b ) spectrum for compressible superflow kinetic energy at @xmath150 . the dotted line in ( b ) is proportional to @xmath131.,width=321 ]      we numerically confirm that the @xmath0 power law in sst is different from that in ct . in sec . 
   iii , we point out that the @xmath0 power law in sst originates from the nonlinear spin term of eq . 
   ( [ velocity_motion ] ) . 
   the magnitude of the inertial and nonlinear spin terms is dependent on the scale , so that we perform the fourier transform for these two terms , comparing the magnitude of the fourier components in order to confirm the argument of sec . 
   specifically , we numerically calculate the following quantities : @xmath151 @xmath152 where @xmath153 and @xmath154 are defined by @xmath155 , \end{aligned}\ ] ] @xmath156.\end{aligned}\ ] ]    figure 5 shows the wave number dependence of @xmath157 and @xmath158 . 
   one sees that the nonlinear spin term @xmath159 is larger than the inertial term @xmath160 , and these ratios are about @xmath161 in the scaling range @xmath126 of fig . 2 where the @xmath0 power law appears . 
   thus we can consider that the kolmogorov spectrum in sst is generated by the nonlinear spin term . 
   [ t ]     ( @xmath162 ) at @xmath163 ( a ) 500 and ( b ) 700 . 
   , width=321 ] 
   we now discuss the dependence of the spectra of the superflow kinetic and spin - dependent interaction energy on @xmath112 . 
   when the relative velocity @xmath112 is large , the period sustaining the kolmogorov spectrum in the superflow kinetic energy is numerically confirmed to become short . 
   this can be caused by the following two reasons . 
   one is ( i ) modulation of total density , and the other is ( ii ) high energy injection wave number . as for ( i ) , when the relative velocity is large , the total density is easy to fluctuate , which leads to the growth of the compressible velocity field . 
   actually , we numerically confirm the increase of the compressible kinetic energy . in our derivation of the power law , 
   the uniformity of the total density is assumed , so that this increase shortens the time period sustaining the kolmogorov spectrum . as for ( ii ) , the most unstable wave number becomes high when the relative velocity is large @xcite , which leads to the high injection wave number @xmath124 . as discussed in sec . v. a 
   , this generates the narrow scaling range , which disturbs the appearance of the power law . because of the above reasons , in our calculation , we set @xmath113 .    on the other hand , in the spectrum of spin - dependent interaction energy , the @xmath1 power law can appear even under the condition of large relative velocity . 
   this was confirmed in our previous study @xcite . at present , we do not understood why this spectrum can exhibit the @xmath1 power law independently of the relative velocity .      in this paper 
   , we perform the two - dimensional numerical calculation for sst , so that the possibility of inverse cascade in sst is discussed .    as briefly described in sec . 
   iii , in three - dimensional ct , the kinetic energy is transported from low wave numbers to high ones , which is called direct energy cascade . however , in two - dimensional ct , the energy inverse cascade occurs , where the kinetic energy is transported from high wave numbers from low ones . 
   this inverse cascade is caused by the conservation of ensthropy as well as that of kinetic energy in two - dimensional fluid system @xcite .    at present , we consider that the inverse energy cascade for superflow kinetic energy does not occur in the two - dimensional sst because the enstrophy in ferromagnetic spin-1 spinor bec is not conserved . actually , our numerical calculation does not exhibit the sign of the energy inverse cascade in the superflow kinetic energy . 
   on the other hand , as for the spin - dependent interaction energy , the inverse energy cascade may occur because our previous study and fig . 
   [ fig3 ] seem to exhibit the growth of spectrum of spin - dependent interaction energy in low wave number region as the time sufficiently passes . 
   this growth may be caused by the ferromagnetic interaction since it tends to align the spin density vector and make the spin domain . 
   however , as shown in fig . 
   [ fig2 ] , the spectrum apparently seems to exhibit the sign of direct energy cascade . 
   thus , at the moment , we does not sufficiently understand the direction of energy cascade for the spin - dependent interaction energy . 
   we theoretically and numerically studied sst in a spin-1 ferromagnetic spinor bec at zero temperature by using the spin-1 spinor gp equations , finding that both the @xmath0 and @xmath1 power laws appear in the spectrum of the superflow kinetic and the spin - dependent interaction energy . 
   first , we discussed the possibility for the kolmogorov spectrum in sst , pointing out that this spectrum can be generated by the nonlinear spin term . 
   second , we showed the numerical results of the spin-1 spinor gp equation with the phenomenological small - scale energy dissipation , whereby sst in the two - dimensional system was obtained by the counterflow instability . 
   our numerical results indicated that both the @xmath0 and @xmath1 power laws appeared . 
   furthermore , we estimated the magnitude of the inertial term and the nonlinear spin term of eq . 
   ( [ velocity_motion ] ) in the wave number space , numerically confirming that the kolmogorov spectrum in sst can be generated by the latter term . 
   k. f. was supported by a grant - in - aid for jsps fellows grant number 262524 . 
   m. t. was supported by jsps kakenhi grant number 26400366 and mext kakenhi `` fluctuation @xmath164 structure '' grant number 26103526 . 
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summary: spin - superflow turbulence ( sst ) in spin-1 ferromagnetic spinor bose - einstein condensates is theoretically and numerically studied by using the spin-1 spinor gross - pitaevskii ( gp ) equations . 
   sst is turbulence in which the disturbed spin and superfluid velocity fields are coupled . applying the kolmogorov - type dimensional scaling analysis to the hydrodynamic equations of spin and velocity fields 
   , we theoretically find that the @xmath0 and @xmath1 power laws can appear in spectra of the superflow kinetic and the spin - dependent interaction energy , respectively . 
   our numerical calculation of the gp equations with a phenomenological small - scale energy dissipation confirms sst with the coexistence of disturbed spin and superfluid velocity field with two power laws .