Discreteness and resolution effects in rapidly rotating turbulence

Transcription

1 PHYSICAL REVIEW E 78, Dicretene and reolution effect in rapidly rotating turbulence Lydia Bourouiba* McGill Univerity, Montréal, Québec H3A 2K6, Canada Received 2 June 8; publihed 2 November 8 tating turbulence i characterized by the nondimenional by number, which i a meaure of the trength of the Corioli term relative to that of the nonlinear term. For rapid rotation, nonlinear interaction between inertial wave are weak, and the theoretical approache ued for other weak wave turbulence problem can be applied. The important interaction in rotating turbulence at mall become thoe between mode atifying the reonant and near-reonant condition. Often, dicuion comparing theoretical reult and numerical imulation are quetioned becaue of a peculated problem regarding the dicretene of the mode in finite numerical domain veru continuou mode in unbounded continuou theoretical domain. Thi argument find it origin in a previou tudy of capillary wave, for which reonant interaction have a very particular property that i not hared by inertial wave. Thi poible retriction on numerical imulation of rotating turbulence to moderate ha never been quantified. In thi paper, we inquire whether the dicretene effect oberved in capillary wave turbulence are alo preent in inertial wave turbulence at mall. We invetigate how the dicretene effect can affect the etup and interpretation of tudie of rapidly rotating turbulence in finite domain. In addition, we invetigate how the reolution of finite numerical domain can affect the different type of nonlinear interaction relevant for rotating inertial wave turbulence theorie. We focu on by number ranging from to 1 and on periodic domain due to their relevance to direct numerical imulation of turbulence. We find that dicretene effect are preent for the ytem of inertial wave for by number comfortably maller than thoe ued in the mot recent numerical imulation of rotating turbulence. We ue a kinematic model of the cacade of energy via elected type of reonant and near-reonant interaction to determine the threhold of below which dicretene effect become important enough to render an energy cacade impoible. DOI:.13/PhyRevE PACS number : i, i, y I. INTRODUCTION tation affect the nonlinear dynamic of turbulent flow. The by number i =U/2 L, with U a typical flow velocity, the rotation rate, and L the characteritic length cale of the flow. It i the dimenionle ratio of the magnitude of the nonlinear term in the Navier-Stoke equation u u, where u i the velocity vector to the Corioli term 2 u, where i the rotation vector. When =, the Navier-Stoke equation in a rotating frame are linear and admit inertial wave olution with aniotropic diperion relation k = k 2 k/ k, where k =, and k i the wave number 1. The mode with zero frequency k are mode that correpond to vertically averaged real-pace velocity field for example, columnar vertically averaged mode aligned with the rotation axi. When threedimenional iotropic rapidly rotating turbulent flow have been oberved to generate two-dimenional columnar tructure experimentally 2 4, and numerically in decay turbulence 7 and in forced turbulence When, matched aymptotic expanion method can be ued for thi weakly nonlinear problem. At firt order in the expanion, energy tranfer are retricted to interaction between triad of inertial wave that atify a condition of reonance 12. Similar to other three-wave reonance ytem, rotating turbulence with inertial wave i a ytem in which near-reonant interaction play an important role. *lydia.bourouiba@mail.mcgill.ca They were found to be reponible for the generation of an aniotropy oberved 13. When comparing imulation and experiment to theory, one hould keep in mind that imulation and experiment are necearily carried out in finite domain, wherea wave turbulence theorie often aume an infinite domain. More pecifically, numerical imulation uually aume periodic boundarie, and experiment are obviouly carried out in bounded domain. In unbounded domain, the component of the wave vector atifying reonant and near-reonant condition are real number, wherea they are retricted to the et of integer in bounded and periodic domain. Thi eemingly benign difference turn out to have major implication. In fact, Ref. 14 invetigated the peculiar characteritic of everal reonance planetary wave, gravity wave, capillary wave, and drift wave in plama atified only by integer wave number olution in periodic domain. Reference conidered the exitence of olution for variou wave ytem including gravity, by, and capillary wave and howed that capillary wave in a finite domain integervalued wave vector cannot be reonant. Their reonance condition only ha real-valued wave vector olution. Thi kinematic reult turn out to have major implication on the applicability of wave-turbulence reult which are derived in infinite domain. In fact, everal numerical tudie howed that a ufficient level of nonlinearity i needed for the amplitude of the capillary wave decribed by integer-valued wave vector to overcome the dicretene and atify an approximate reonance, i.e., a near-reonance 16,17. In other word, in a dicrete domain, a ufficient threhold nonlinear broadening of the capillary wave reonance condition 39-37/8/78 / The American Phyical Society

2 LYDIA BOUROUIBA i needed for the near-reonant condition to generate nonlinear energy tranfer. Below thi threhold, the nearreonant interaction are not eaily formed. Thu, given that no reonant interaction are poible, the energy cacade i topped altogether. Thi regime i referred to a frozen turbulence 16. Similar dicretene effect were alo invetigated for water wave 18. Nazarenko 21,22 explicitly addreed the iue of applicability of wave turbulence correponding to the continuou wave number limit to dicrete wave number domain for the cae of water-wave and MHD turbulence, repectively. The main ditinction between inertial wave and capillary wave i that for the former, integer-valued wave number reonance olution do exit. The reonant inertial wave interaction are thu alway preent in bounded and periodic domain. In contrat to capillary wave, there i no a priori need for a ufficient level of nonlinearity in order to trigger nonlinear energy tranfer. However, the diperion relation of inertial wave can lead to complicated reonant urface. Therefore, a imilar effect to that of frozen capillary wave turbulence effect could exit for inertial wave. If that wa the cae, the dicretene effect would be expected to reduce the poibility of energy tranfer both via the reonant interaction alone or combined with near-reonant interaction. In fact, an effect of thi nature ha been peculated although not verified. For example, Ref. 24 peculated that the ue of numerical imulation for the tudy of mall flow could mi the low mode dynamic. Uing decaying turbulence imulation, Bourouiba and Bartello 7 howed the exitence of three rotating turbulent regime. 1 A weakly rotating by regime, for which the turbulent flow i not affected by rotation. 2 An intermediate by regime characterized by a trong tranfer of energy from the wave to the zero-frequency mode, with a peak at A mall, or aymptotic by regime, for which the zero-frequency mode receive le and le energy from the wave mode tending to confirm decoupling theorie. When forcing their imulation at intermediate cale, Ref. 13 peculated that their own imulation uing =.1 guaranteed a ufficient nonlinear amplitude to allow for the capture of the near-reonant interaction while other imulation of Ref. 11 invetigating a lower, were not able to accurately capture the neceary near-reonant interaction. Thu, it wa implied that the dynamic oberved by Ref. 11 for their maller wa a numerical artifact. Although Ref. 13 tudied the effect of near-reonant interaction for O.1, they did not quantify the effect of dicretene. Neverthele, they did call for the need of doing thi work: although the latter tatement capturing the nearreonance in finite domain beg to be quantified, uch a tudy i beyond the cope of the preent paper. The finite domain effect on rapidly rotating flow i inherent to both numerical and experimental tudie, yet tudying uch an effect uing experiment i delicate. One could enviion comparing tatitic obtained in a erie of different ized rotating tank. However, at which the aymptotic theorie apply found to be, for example, =.1 in Ref. 2 are coniderably maller than thoe typical of rotating tank experiment for example, =2 wa the lowet by TABLE I. Propertie of the interaction of interet contributing to the energy tranfer. Type of triad PHYSICAL REVIEW E 78, Reonant R and/or Near-reonant N : 1 = 22 2 R R 33 2 N with C 33 2 vanihe with C 33 2 = 23 3 R and N R catalytic: no energy tranfer to the 2D mode 33 3 R and N R: energy tranfer to the 3D mode with mallet k number obtained in experiment of Ref. 26. The difficulty i that boundary layer effect quickly become important when rotation rate are high Ref. 27, for example. Moreover, in a more recent tudy, the confinement effect wa argued to inevitably generate inhomogeneou turbulence ee Ref. 28. A poible alternative to rotating tank i the etup ued by Ref. 29, in which a flow of air wa paing through a large rotating cylinder and a dene honeycomb. Thi produce elongated tructure which could be le affected by boundary layer and inhomogeneity effect. Iolating the dicretene effect could be poible in uch a etting in which dominant boundary layer effect could be reduced. We now focu on the dicretene effect in numerical domain. In particular, our goal i to clarify thi iue and to invetigate how the reduction of affect the ditribution and number of reonant and near-reonant interaction a in periodic domain. When carrying out and interpreting the reult of imulation of rotating turbulence, there are two apect to the dicretene effect. The firt i the difference between bounded and unbounded domain in general. Unbounded domain lead to real-valued wave number in pectral pace. Bounded and periodic domain lead to integer-valued wave number. The econd i related to the patial reolution of a given bounded domain of interet. When a bounded domain of interet i elected, the patial reolution determine the truncation wave number. We inquire whether a dicretene effect leading to a phenomenon imilar to the frozen turbulence oberved with capillary wave i alo preent here with inertial wave. In addition, we invetigate how the reolution affect the different type of interaction relevant for rotating inertial wave turbulence theorie ee TableI. We find that dicretene effect are preent for the ytem of inertial wave. We quantify the minimum nonlinear broadening by number min below which the domain doe not contain interaction coupling the zero-frequency twodimenional 2D mode to the 3D inertial wave mode. Thu, a regime of decoupling would be oberved for imulation with min. We alo find the critical nonlinear broadening c below which the theorie predicting the freezing of vertical tranfer in inertial wave turbulence would be oberved in bounded domain. Finally, given a domain of interet for numerical imulation, we invetigate whether a freezing of the energy cacade i preent for iner

3 DISCRETENESS AND RESOLUTION EFFECTS IN tial wave and find thi to be the cae. We quantify the threhold nonlinear broadening f below which uch a freezing occur. Thi i done for variou reolution. Thi freezing of the energy cacade i found to be an inherent property of bounded ytem. Our reult can allow a more accurate interpretation and comparion of the reult obtained in bounded domain with the reult derived in unbounded domain. In particular, our reult alo aim at improving the etup of future numerical imulation of rapidly rotating turbulence. Firt, the notation ued to claify the variou type of interaction are preented in Sec. II. The propertie of the interaction and their interaction coefficient are recalled. In Sec. III, the number of near-reonant interaction and reonant interaction are quantified a a function of the nonlinear broadening and the truncation wave number. In Sec. IV, we invetigate the effect of the nonlinear broadening on the capacity of the reonant and near-reonant interaction to generate a propagation of energy tranfer to all the mode of the model. Threhold value of nonlinear broadening are obtained for variou reolution. In Sec. V we dicu the reult obtained in the light of recent dynamical numerical imulation of rotating turbulent flow. II. MODAL DECOMPOSITION AND TYPES OF INTERACTIONS We focu on a triply periodic domain a it i the domain mot relevant for direct numerical imulation of homogeneou turbulence. In a triply periodic domain of ize L L L the velocity field of an incompreible fluid can be repreented by a Fourier erie u r,t = u k,t e ik r, k where r= x,y,z i the poition vector in real-pace with three component i= 1, k= k k x,k y,k z, with k =2 /L, and with k x,y,z integer wave number taking the value, 1, 2,.... The continuity equation in Fourier pace i u k k=. Solution of the linear rotating flow equation with = can be expreed a a uperpoition of inertial wave with nondimenional inertial frequencie k = k ẑ k/ k, where k =, and ẑ i the unit vector for the direction correponding to the rotation axi. See Ref. 1 for detail. The inertial wave turn out to have the ame tructure a the helical mode eigenmode of the curl operator, which are defined a 1,3 N k = ẑ k ẑ k k k + i ẑ k k 3 ẑ k. When i mall but nonzero, a two-time-cale aymptotic expanion can be performed with a fat time cale =t aociated with the rotation time cale, and a low time cale 1 = t aociated with the nonlinearity. The velocity field become 1 2 u k,, 1 = A k k, 1 N k k e i k. k = When carrying the expanion to the econd order 1,31 the equation governing the evolution of the amplitude A k1 i obtained. The nonlinear interaction are retricted to interaction between mode k 1,k 2,k 3 atifying the reonant condition 4 k1 + k2 + k3 =, k 1 + k 2 + k 3 =. The nonlinear interaction that are reonant are thu the interaction contributing on the long-time cale 1 to the longterm evolution of the wave amplitude; omitting vicoity: 1 A k1 k 1, 1 = 1 4 k1 + k2 + k3 = k 1 +k 2 +k 3 = k1, k2 C k2 k3 k1 k2 k 3 k 1 A * k2 k 2, 1 A * k3 k 3, 1, 6 where C k2 k3 k1 k2 i the interaction coefficient of the triad k 3 k 1 k 1,k 2,k 3 that contribute to the equation governing the amplitude of the mode k 1. Reference 3 howed that C k2 k3 k1 k2 k 3 k = k2 k 2 k3 k 3 N* k 2 N* k 3 N* k Each interacting triad atifying k 1 +k 2 +k 3 = conerve energy and helicity implying that and C k3 k2 k1 k3 k 2 k + k1 k3 k2 Ck1 1 k 3 k + k1 k2 k3 Ck1 2 k 2 k = 8a 3 k1 k 1 C k3 k2 k1 k3 k 2 k + k2 k 2 C k1 k3 k2 1 k1 k 3 k + k3 k 3 C k1 k2 k3 2 k1 k 2 k =. 3 8b We decompoed the mode into two categorie: the mode correponding to the nonzero inertial wave with k z, denoted 3 or 3D, If k W k = k k and k z then u k = u 3D k and the mode with a null inertial wave frequency with k z =, denoted 2 or 2D, which correpond to a real-pace velocity field that i averaged vertically If k V k = k k and k z = then u k = u 2D k h, 9 where k h = kx 2 +k y 2 i the horizontal wave number. The total energy E= 1 2 k u k 2 become E=E 2D +E 3D, with E 2D = 1 2 k V k u 2D k 2, PHYSICAL REVIEW E 78, E 3D = 1 2 k W k u 3D k Uing Eq. 9 and, the nonlinear interaction between triad in Eq. 6 can be claified a 33 2, 23 3, 33 3, and 22 2, detailed in Table I. The notation jk i tand for interaction between mode j, k, and i contrib

4 LYDIA BOUROUIBA uting to the equation of evolution for the mode i, with an interaction coefficient C j, k, i jk i. It i ymmetric in j and k. In the limit, where the two-timecale expanion i valid, only the reonant interaction atifying Eq. make a ignificant contribution to the nonlinear energy tranfer. The interaction 22 2 are all trivially reonant. The reonant interaction 33 2 were hown by Waleffe 31 to have no contribution to the energetic of the 2D mode. That i, if k 1,k 2,k 3 i a reonant triad, with k 2,k 3 W k and k 1 V k, then the interaction coefficient C k3 k2 k1 k3 k 2 k =, and the 1 k1 k2 k3 = two other interacting coefficient atify C k1 k 2 k 3 C k3 k1 k2 k3 from Eq. 8a. The reonant interaction 32 3 k 1 k 2 are referred to a catalytic. If thi property i combined with our definition of interaction ij k a an interaction with ijk C ij k, then the reonant interaction 33 2 do not exit are inactive. The energy pectra equation accounting only for reonant interaction can then be written a E 3D k W k,t = T 33 3re + T 32 3re k W k,t, t E 2D k V k,t = T 22 2re k V k,t, t where T repreent the Fourier-pace energy tranfer term and the term T 33 2re = doe not appear. Note that in contrat to the reonant 33 2, the near-reonant interaction 33 2 are active. They exchange energy with the 2D mode, i.e., C The near-reonant interaction denoted 33 2 and 32 3 are eentially denoting the ame active triad. The reonant 33 2 property i at the origin of the decoupling theorie, which predict a decoupling between the inertial wave and the two-dimenional coherent tructure in a rapidly rotating flow. Thi i coherent with the work of Ref. 32,33, who averaged the equation and eparated the fat wave and the low mode. They obtained an equation governing the vertically averaged 2D tructure of the flow decoupled from the wave dynamic in the limit of mall. However, Ref. 34 later argued that for an unbounded domain, coupling term between the 2D and the wave mode remain active even at =. That i, no decoupling i achievable in unbounded domain. Concerning the 3D dynamic governed by Eq. 12, the reonant 33 3 interaction tranfer energy to the W k mode of the triad with the lowet frequency k 31. Cambon et al. 24 oberved an inhibition of the overall energy tranfer including both 2D and 3D mode, but no invere cacade. They argued that an invere cacade could only be achieved by reonant interaction. Uing a cloure model, they later confirmed the tendency of the reonant interaction to tranfer energy toward the low frequency mode 23. Another dynamic wa propoed for the reonant 3D mode. Reference 33 uggeted that more 32 3 reonant interaction are poible compared to the number of A a reult, PHYSICAL REVIEW E 78, energy tranfer in Eq. 12 would be dominated by 32 3 interaction and thi would lead to a reduced vertical 3D energy tranfer a. Uing a kinematic approach, Ref. 3 howed that nearreonant interaction in dicrete domain are much more numerou than exact reonance in three-wave interaction ytem inertial wave are an example of uch a three-wave reonant interaction ytem. Thi i not necearily the cae for four- and more -wave interaction ytem, where the number of exact-reonance in a dicrete domain can be very large even in mall pectral domain. The reult of the forced numerical imulation of rotating turbulence in periodic domain by Ref. 13 howed that the near-reonant interaction atifying an approximate reonance modeled by k1 + k2 + k3, k 1 + k 2 + k 3 = 14 played an important role in the energy tranfer at moderately mall value of. Note that here, the frequencie k are nondimenional a defined in Eq. 2. The mall parameter in Eq. 14 i the nonlinear broadening of the reonant interaction. It depend on the pectrum of the flow and the level of nonlinearity of the turbulence for the cale aociated with the interacting triad. Recall that due to the diperion relation of capillary wave, Ref. howed that only near-reonant interaction are poible in a dicrete domain. In an unbounded domain, both reonance and nearreonance would contribute to nonlinear tranfer. In contrat to capillary wave, inertial wave mode can be both exact reonance and near reonance 14 in both a dicrete and continuou pectral domain. III. NUMBER OF EXACT AND NEAR-RESONANCES A a firt tep in evaluating the influence of the 1 dicretene and the 2 reolution effect, we conider a range of pectral domain correponding to a fixed ize L L L but with variou patial reolution N , 3,133 3, 3, Thee correpond to pectral truncation wave number of k t 21,33,, k t N/3 for dealiaing 36. Due to the aniotropy of the inertial diperion relation 2, we conider a pectral domain with a cylindrical truncation uch that max k z =k max, and max k h =k max. We ignore the range of mode correponding to the diipation range k d k k t. Auming that k d.9k t, we only conider the mode of the pectral domain with k z k max and k h k max, with k max =,3,4,4,. To give ome perpective we recall that the numerical tudie of rotating turbulence performed by Ref. 8 were obtained in a 32 3 numerical domain which would correpond to k t =. Simulation of Ref. 13 ued a reolution of 64 3, which correpond to k t =21. Simulation of Ref.,11 ued reolution of 128 3, which correpond to k t =42. Reference 7 ued reolution of 3 and 3, which correpond to k t =33 and 66. The recent large-cale forcing imulation of Ref. 37 ued a reolution of 12 3, which correpond to k t =17. We counted the number of interaction atifying Eq. 14 for a fixed domain ize with varying truncation number cor

5 DISCRETENESS AND RESOLUTION EFFECTS IN Number of 33 3 interaction Number of 33 2 interaction kt kt3 kt4 kt4 kt (a) (b) Number of 32 3 interaction kt kt 3 kt 4 kt 4 kt kt kt3 kt4 kt4 kt PHYSICAL REVIEW E 78, (c) FIG. 1. Color online Total number of active reonant and nearreonant 33 3, 33 2 and 23 3 ee Table I and Eq. 14 a a function of nonlinear broadening and the patial reolution ued. reponding k max =,3,4,4,. For each triad, we conider the eight poible type of interaction defined by k1, k2, k3. We ummarize in Table I the type of interaction on which we focu. When calculating the number of 3 a 3 b 3 c and 2 a 2 b 2 c interaction, we only conider the interaction with all three non-null interacting coefficient C 3a 3b 3c 3b 3c 3a 3a 3c 3b,C3b,C3a 3a 3 b 3 c 3 c 3 a 3 c 3 and b C 2a 2b 2c 2b 2c 2a 2a 2c 2b,C2b,C2a 2a 2 b 2 c 2 c 2 a 2 c 2, repectively. The 32 3 interaction are thoe with at mot one zero interaction coeffi- b cient. The 3 a 3 b 2 are retricted to the interaction with C 3a 3b 2 3a. In other word, the number of 32 3 contain 3 b 2 both reonance and near-reonance, wherea the 33 2 are only near-reonance. Figure 1 how the reulting number of 33 3, 33 2, and 32 3 atifying Eq. 14 for a range of nonlinear broadening between and 1. The number of 33 2 and the 33 3 interaction top and lower panel both how a decreae with until they reach a plateau for mall, wherea, 33 2 middle panel near-reonance decreae with and vanih for ufficiently mall. A the nonlinear broadening i decreaed, the number of 33 3 interaction reache a nonzero plateau that varie with the truncation wave number conidered. Thi plateau correpond to the number of exactly reonant interaction atifying Eq.. The number of 33 3 reonance varie from R 33 3 =232 for k max = to R 33 3 =748 for k max =. Thi remain quite mall compared to the number of nearreonance N 33 3 obtained for higher value of nonlinear broadening. Thi latter increae rapidly with and thi i true for all truncation. For example, for a truncation of k max =, the number of near-reonance N 33 3 increae dramatically from 74 for =1 7 to for =.8. The plateau correponding to the number of exactreonance followed by a rapid increae of the number of near-reonance a increae i reminicent of the reult obtained for near-reonant urface gravity wave in Ref. 18. They found that the number of near-reonance reache a nonzero plateau a their nonlinear broadening decreaed. However, the number of reonant interaction on the plateau wa found to be ufficiently large to preclude a regime imilar to capillary wave frozen turbulence. The decoupling between the 3D and 2D mode wa predicted by decoupling theorie derived in continuou domain for infiniteimally mall. Thee theorie rely on the vanihing coupling coefficient between the 2D and the 3D mode in the 33 2 interaction, leading to decoupled Eq. 12 and 13. In other word, we know that in unbounded domain C 33 2 a implie a decoupled dynamic. However, in bounded domain the for which thee theorie would apply i finite and undetermined. We counted the number of 33 2 near-reonance in a given bounded domain. Figure 1 how clearly that there i a minimum finite nonlinear broadening, below which the 33 2 interaction vanih. We will refer to thi minimum a min. It varie with the reolution ued. min change from min =6.6 at pectral truncation k max = to min =3.3 6 for a truncation of k max =. For min the only interaction between the V k and the W k mode are thoe labeled 32 3, which become catalytic, reulting in a decoupling between the 2D and the 3D dynamic. Recall that for min, the catalytic interaction do not exchange energy between cale. Although, they do reditribute the 3D energy horizontally between variou k x,k y mode of fixed k h. Fi- 639-

6 LYDIA BOUROUIBA Number of 33 3 and 32 3 interaction kt 32 3kt 33 3kt4 32 3kt4 33 3kt 32 3kt FIG. 2. Color online Comparion of the number of reonant and near-reonant interaction 33 3 and 32 3 a a function of the nonlinear broadening and the truncation of the pectral domain ued. nally, the value of min from Fig. 1 are far lower than by number found in the literature on numerical imulation of rotating turbulence. For example, with a reolution of with a maximum wave number cloe to 4 numerical imulation performed with a nonlinear broadening of min =7.8 6 would automatically produce a decoupling between 3D and 2D mode. For min, near-reonance tranferring energy between 2D and 3D mode are till preent, hence, if a decoupling i oberved numerically at thee by number, it i an intrinic dynamical property of the flow a oppoed to being due to a lack of capture of key interaction by the numerical domain. In the context of reonant interaction theorie, Ref. 33 uggeted that reduced vertical energy tranfer would be oberved. The reaon for thi lie with the aumption that the number of exact reonance R 32 3 would be greater than R A imilar talling of vertical tranfer wa predicted and oberved for weak Alfvén turbulence 38. Thee prediction contrat with the reult uggeting the creation of aniotropy by the predominant reonance 33 3 tranferring energy toward mode with mall frequencie k. Figure 2 how that both dynamic exit. In one cae, the 33 3 are dominant and in the other, the 32 3 are dominant. A critical nonlinear broadening, c, delimit the two cae. For c, the number of 33 3 near-reonance N 33 3 i maller than the number of 32 3 near-reonance N For c, N 33 3 N c i a function of the truncation wave number. The higher the k t, the maller the c. For c, the reduction of vertical energy tranfer dicued in Ref. 33 could be obervable in bounded domain, wherea for c, the dominant number of 33 3 lead to energy tranfer toward the mode with maller frequencie k. We find that c min for all the k max ued here. Thi implie that for a nonlinear broadening of c min, the intercale energy tranfer in a given imulation would be due to the relatively mall number of 33 3 interaction. For example, in the cae of k t =, for = 6 c min, PHYSICAL REVIEW E 78, we have N 33 2 =O 3. A a reult, if one i intereted in the mall limit, the quetion become whether reonance, even in thee mall number, are ufficient to guarantee that energy tranfer occur in uch bounded domain. In a continuou pectral pace truncated at k t, the number of 33 3 interaction are expected to increae with k t 6 with the truncation wave number. Similarly, the number of 32 3 and 33 2 i expected to increae a k t. Uing thee caling, and for different truncation wave number, we normalized the reult of Fig. 1. Figure 3 how the number of 33 3 interaction normalized with k t 6 and the number of 32 3 and 33 2 normalized with k t. For ufficiently large, the normalized curve are inditinguihable. In other word, for large, there are no difference expected to arie between reult in bounded domain and thoe in unbounded domain. Below a certain value of the nonlinear broadening, thi caling i weaker: the caled curve differ from one another for mall. The dicrepancy in the collape of the caled curve i particularly triking for the nearreonance N For example, k t = curve tart diverging from the larger k t curve at O 3. Thi i larger than min which wa identified a threhold for 2D-3D decoupling. Hence, in the domain conidered here, dicretene effect are increaingly important for a nonlinear broadening larger than min. To ummarize, in the preent ection we dicued the reult of wave turbulence derived for unbounded domain. We examined whether thee theoretical reult e.g., 2D-3D decoupling and freezing of the vertical energy tranfer would be obervable in bounded domain. For domain typically ued in numerical imulation we found and quantified a threhold min below which the aumption behind the 2D-3D decoupling theorie i valid in finite domain. We found and quantified a nonlinear broadening threhold c below which the vertical freezing prediction of Ref. 33 would be obervable in bounded domain. A nonlinear broadening above c correpond to a dynamic dominated by triple-wave interaction. We invetigated the dicrepancie between the number of interaction reolved in unbounded and bounded domain for a given truncation wave number. We found that for all dynamical regime dicued decoupling, vertical freezing, etc., dicretene effect lead to difference between the number of near-reonance obtained in bounded and unbounded domain for relatively mall. For the domain conidered here, we found that dicretene effect become important for nonlinear broadening larger than min. However, baed on thee reult alone, we cannot determine thi tranition with more preciion. We denote thi tranition nonlinear broadening a f, and now turn to a kinematic model of cacade in order to obtain a more precie aement of f. Below f the dynamic could become ignificantly affected by dicretene effect. IV. KINEMATIC MODEL In hi tudy of near-reonant capillary wave, Puhkarev 16 built two-dimenional map k x,k y of mode atifying the capillary three-wave near reonance. Uing thi, he 639-6

7 DISCRETENESS AND RESOLUTION EFFECTS IN Normalized number of 33 3 (a) Normalized number of 33 2 (b) Normalized number of 32 3 (c) kt kt3 kt4 kt4 kt howed that the denity of active mode changed ignificantly with the nonlinear broadening 17. A kinematic model wa propoed by Ref. 17 to tudy the dicretene effect on the tranfer of energy via near-reonant capillary kt kt3 kt4 kt4 kt kt kt3 kt4 kt4 kt FIG. 3. Color online Effect of the dicretene on the variation of the number of reonant and near-reonant 33 3, 33 2, and 23 3 a a function of nonlinear broadening. The number of 33 3 interaction are normalized with k t 6 and the number of 32 3 and 33 2 normalized with k t. wave. They tarted from an initial map of excited or activated mode, then given a level of nonlinear broadening, obtained ubequent map of activated mode interacting with the initial et. The erie of ubequent et of active mode atified the near-reonant condition. Thi approach allowed them to determine the nonlinear broadening threhold below which no energy cacade could occur in a dicrete pectral pace. Following a imilar approach, we contruct an iterative kinematic model of energy cacade through near-reonant inertial wave interaction 14. The diperion relation of inertial wave i aniotropic and we therefore ditinguih between the vertical and the horizontal wave number. The map of mode i contructed for three-dimenional wave number k x,k y,k z. We conider a dicrete pectral domain of truncation k t and tart by auming that only the mode in the initial et S are active. For a given value of nonlinear broadening, we contruct the et of mode k of generation 1, S 1 uch that k S 1 if k S or k + k 1 + k 2 = PHYSICAL REVIEW E 78, and k + k1 + k2 with k 1,k 2 S S. That i, S 1 include all mode k interacting with two mode of S through near-reonance and with at mot one zero interacting coefficient and S S 1. The procedure can then be repeated uing S 1 a the initial et of active mode. We obtain ucceive generation of active mode S,...,S n,...,s N. The lat et S N include all the mode that were activated by the cacade. All the et with n N are identical to S N, i.e., they do not contain any new activated mode. When S N i reached, there are two poibilitie. The firt i that S N contain all available mode of the domain. In other word, the propagation of the initial excitation reache all mode. The econd poibility i that not all the mode of the domain are contained in S N, thu a ubet of mode doe not participate in the dynamic. Hence, dicretene effect clearly are preent and can compromie numerical repreentation of the dynamic in the domain conidered. In thi lat cae, we ay that the cacade i halted or that a freezing of the kinematic model cacade i taking place. In addition to the general propagation of active mode throughout the pectral domain, we are alo intereted in invetigating the role played by the variou type of nearreonant interaction decribed in Table I. When going through the interactive tep decribed in Eq. we alo tet for the type of mode involved V k or W k. For the 22 2 and 33 3 interaction we further require all three interaction coefficient to be nonzero. The 32 3 interaction have at leat one nonzero interaction coefficient. Finally, the 2D interaction coefficient of the 33 2 interaction i nonzero. We invetigated two et of initial mode: S a and S b. S a wa choen uch that k and contain a mall number 424 of mode Fig. 4 top panel. A uch, thi erie S na, n=,1,2,..., mimic a forward cacade in a pectral domain. S b contain 286 mode and i defined by S b = k:11 k 12 in order to mimic a cacade initiated with a forcing at intermediate to mall pectral cale

8 LYDIA BOUROUIBA (a) k x (c) k x k y k y (b) k h - (d) PHYSICAL REVIEW E 78, k h k z k z FIG. 4. Initial map of active mode S a top panel and S b bottom panel. Here the domain i truncated at k t =. The map of the horizontal component of the active mode are hown on a k x -k y plane left panel and the vertical component of the active mode on a k h -k z plane right panel. Figure how the maximum number of generation in the kinematic model when initialized with S a and for k t =. For mall value of the nonlinear broadening, the equence i frozen after three generation. At = f.9 4, there i an abrupt tranition. For larger than thi freezing threhold, S Na include all mode. For only marginally larger than f, a larger number of generation Generation of mode of the cacade Frozen cacade k t = Full domain cacade FIG.. Maximum number of iteration N of generation in the equence S an initiated with S a and for k t =. For a nonlinear broadening le than about 6 4 the erie wa frozen after a few interaction. For larger, S Na included all mode. are needed before thi occur. Neverthele, for all f all mode are eventually activated and we conclude that dicretene effect are not precluding an energy cacade. It i alo noteworthy that f appear to be only weakly dependent on reolution. For example, a doubling of k t reulted in only a mall decreae in f, i.e., from f =.9 4 for k t = to f =.87 4 for k t =4. It thu appear that, at leat in thi intance, dicretene effect become problematic at coniderably larger than min. However, the by number typically ued in imulation e.g., 2 are comfortably larger than f. A wa the model of Ref. 17, our model i kinematic only. It doe not give information about the direction of exchange of energy between mode involved. However, from previou numerical tudie of rapidly rotating turbulence, we know that mode with low linear frequencie receive energy preferentially. Thee correpond to mode with k h k z. Alo, in the decaying numerical imulation of Ref. 7, horizontal 3D energy tranfer were more efficient than 3D energy vertical tranfer. It i intereting that the kinematic model how a imilar aniotropy. That i, fewer generation are needed before all horizontal wave number are excited, wherea more are needed to excite all available vertical wave number. Thi i hown in Fig. 6, which plot the maximum horizontal and vertical wave number reached at each generation for variou value of and for both reolution 64 3 and 3. The point i further illutrated in Fig. 7, which how the horizontal and vertical mode activated for the firt few gen

9 DISCRETENESS AND RESOLUTION EFFECTS IN PHYSICAL REVIEW E 78, = - = -4 =. -4 f =.9-4 = 1-3 = 1-2 = 1-1 = = - = -4 =. -4 f =.9-4 = 1-3 = 1-2 = 1-1 = 1 k hmax > f f k h =k t k zmax > f k z =k t f (a) < f Cacade generation (b) < f Cacade generation =. -4 =.8-4 =.9-4 = 1-3 = =. -4 =.8-4 =.9-4 = 1-3 = k h =k t 3 k z =k t k hmax 2 f k zmax 2 f (c) Cacade generation (d) Cacade generation FIG. 6. Maximum horizontal left panel and vertical right panel wave number at each tep of the cacade generation initiated with the et of excited mode S a, and for a domain of k t = top panel and k t =3 lower panel. The critical freezing nonlinear broadening i the mallet value to allow the cacade to extend to the mallet horizontal and vertical cale of the domain. eration of the kinematic model initialized with S a and for k t =. It i clear that large horizontal wave number are activated prior to the activation of large vertical wave number. Thi aniotropy i viible in both the 33 3 and the 32 3 interaction. However, the 33 3 interaction lead to a fater excitation i.e., an excitation in fewer iteration of the kinematic model than the 32 3 interaction. A an illutration, we diplay the paucity of S Na for a cae where f for which the propagation wa halted at N=3 in Fig. 8. Finally, 22 2 interaction do not how any freezing of the propagation of the excitation among 2D mode not hown. In fact, the propagation of the active mode among the et of 2D mode alway take only three generation. V. DISCUSSION AND CONCLUSION The dicretene of wave number in a periodic domain ha proven to be at the origin of the freezing of the nonlinear tranfer between mode dominated by reonant capillary wave interaction. A imilar effect wa alo upected in rotating turbulence. In contrat to capillary wave, however, inertial wave can atify the reonant condition for integervalue wave number, uggeting that a freezing might not occur in that ytem. We tudied the dicretene effect in order to clarify thi iue. The dicretene effect are twofold. One apect i the integer-valued wave number, i.e., the finitene intrinic property of bounded domain veru unbounded domain. The other apect i the reolution ued in a particular bounded domain of interet. We examined how the reduction of the nonlinear broadening affect the ditribution and number of reonant and near-reonant interaction on a range of in periodic domain relevant for numerical tudie. We contructed a kinematic model of reonant and near-reonant interaction and ued it to how that dicretene effect are detected for mall nonlinear broadening. We alo howed that, a with capillary wave, a freezing of the energy tranfer i alo poible in dicrete inertial wave, provided wa ufficiently mall. Thi latter effect wa not enitive to a change of reolution. Concerning the decoupling between the dynamic of the 2D and 3D mode predicted for the limit of, we howed the exitence of a minimum nonlinear broadening min below which 3D-2D interaction do not exchange energy. In a continuou domain, only the = limit trictly prevent the reonant 3D-2D interaction from exchanging energy. However, in a dicrete domain, thi value i nonzero. We quantified min for different reolution. For example, 639-9

10 PHYSICAL REVIEW E 78, 639!8" LYDIA BOUROUIBA (a) (c) kh kh kz kz (b) kh - kz kz (d) kh - kz (e) (g) kh - (f) (h) FIG. 7. kh-kz map of mode activated at generation S2!top panel" to S!bottom panel" by the 32 3 interaction!left panel" and the 33 3 interaction!right panel" with =.1, kt = and initial active mode Sa!ee Fig. 4". The final tate of the cacade correpond to the activation of all the mode of the domain. 639-

11 DISCRETENESS AND RESOLUTION EFFECTS IN - (a) k h (b) k h FIG. 8. Map of the mode of generation S Na, with N=3 at which the cacade i halted when =. 4 and k t =. k z k z PHYSICAL REVIEW E 78, min varie from min =6.6 for a reolution 64 3 to min =3.3 6 for a reolution Thu, a reolution of with a minimum nonlinear broadening of min =7.8 6 would automatically how a decoupling between the 3D and the 2D mode. In addition, another threhold of nonlinear broadening i found and denoted c. For c, the number of 32 3 interaction i larger than the triple wave 33 3 interaction. In thi regime, the freezing of the vertical energy tranfer predicted by Ref. 33 would be poible. Above c no uch regime would be poible. The kinematic model alo howed that both 32 3 and triple wave interaction 33 3 favor the propagation of the cacade to mall horizontal cale and large vertical cale. We find that it take a maller number of generation for thee interaction to excite the maximum horizontal wave number of the domain compared to the number of generation needed to reach the maximum vertical wave number. Thi reinforce waveturbulence prediction of aniotropy and concentration of energy into mall inertial frequency mode. We found a regime of freezing of energy tranfer to be preent in the ytem of inertial wave. We quantified the threhold nonlinear broadening f below which no energy cacade develop throughout the entire domain. f depend on the initial et of excited mode cale of the forcing. Surpriingly, it only varie lightly with change of domain reolution. The kinematic model howed that a imulation in a domain with a reolution of about 64 3 would not be ubject to an energy tranfer freezing except for nonlinear broadening value f a mall a.9 4. Thi threhold decreae only lightly to f =.87 4 for a reolution of Thee value of f are much maller than by number ued in mot numerical imulation of rotating turbulence. The exitence of the f threhold and exitence of exact reonance in dicrete domain i reminicent of water-wave ytem, where an intereting dicretene phenomenon i een in a weakly forced etting. In that ytem energy wa oberved to accumulate around the forcing cale until the nonlinear broadening of reonant interaction attained a critical value ufficient to overcome the dicretene of the mode. Thi wa followed by a udden dicharge or avalanche of energy in the form of an energy cacade after which the ytem ocillated between tage where the energy build up and tage where it i dicharged 21. Weak forcing of inertial wave turbulence could lead to a imilar ocillation. We would oberve alternating phae: a freezing in which energy build up until reache f, above which a phae of energy tranfer could then occur. Thi would need further invetigation of forced rotating flow with f. In the domain examined here, we found that f min c. From a practical tandpoint, thi implie that dynamical theorie related to the propertie of the reonance alone that i the decoupling theorie and the vertical energy freezing would not be applicable in imulation with f, for which the number of interaction cale a in an infinite domain. In other word, if a decoupling or a reduced vertical energy tranfer i oberved in thee domain, they are intrinic dynamical propertie of the flow in the bounded domain conidered. We howed that thee feature cannot be directly explained by the reult on continuou reonance interaction and epecially cannot be explained by the notion that dicretene effect do not allow for ome interaction to be reolved. When carrying out imulation and conidering f, it i important to ue the relevant definition of. Several definition of by number are ued in the literature. Some calculate baed on the forcing cale, which guarantee a contant value. Some ue a by number baed on the tatitic of the evolving flow. Thee include, for example, the mean velocity or vorticity 29. A an illutration, conider the example of Ref. 13, who etimated their baed on the forcing of the flow. They obtained a = However, they found that a reduced model of nearreonance calculated uing in 14 better reproduced the increae of 2D energy of the full flow. Uing their full-imulation pectra Fig. 3, we etimate that their mot energy containing cale larger cale ha a macro- by of =U/2 L.2. Thi value matche the nonlinear broadening.28 of the near-reonance found to reproduce better the dynamic of the full equation. Thi ugget that when comparing of imulation with value uch a f or min identified in our tudy, one hould conider that the relevant i that obtained from averaged velocity or vorticity quantitie of the evolving flow e.g., macro- or micro-. Note that Ref. 7 found that the macro and micro were equivalent in the intermediate regime imulation, where energy i concentrated in the larger cale of the flow, thi i not the cae for other rotating regime, uch a the mall or large regime

12 LYDIA BOUROUIBA In um, given the relatively mall value of f and it weak dependence on reolution, we ubmit that mot numerical imulation on rotating turbulence carried out o far would have been free of dicretene effect. However, it i important for furture tudie of rotating turbulence to ae in which kinematic regime they fall. Thi can be done by comparing their with the threhold identified in thi paper taking into account the domain of tudy conidered. When comparing the by number of a imulation to the freezing etimated in thi tudy, the relevant definition of by number i that related to the averaged evolving quantitie of the flow. PHYSICAL REVIEW E 78, ACKNOWLEDGMENTS The author i grateful for the financial upport from the Natural Science and Engineering Reearch Council of Canada and acknowledge the Conortium Laval-UQUAM- McGill et l Et du Québec upercomputer center. The author i grateful to Dr. D. Straub for fruitful dicuion and comment and alo thank Dr. M. Mackey, Dr. S. Nazarenko and other reviewer for helpful comment. 1 H. P. Greenpan, The Theory of tating Fluid Cambridge Univerity Pre, Cambridge, G. I. Taylor, Proc. R. Soc. London, Ser. A 4, A. D. McEwan, Nature London 26, C. N. Baroud, B. B. Plapp, Z. S. She, and H. L. Swinney, Phy. Rev. Lett. 88, J. Bardina, J. H. Ferziger, and R. S. gallo, J. Fluid Mech. 4, P. Bartello, O. Métai, and M. Leieur, J. Fluid Mech. 273, L. Bourouiba and P. Bartello, J. Fluid Mech. 87, M. Hoain, Phy. Fluid 6, P. K. Yeung and Y. Zhou, Phy. Fluid, L. M. Smith and F. Waleffe, Phy. Fluid 11, Q. Chen, S. Chen, G. L. Eyink, and D. D. Holm, J. Fluid Mech. 42, A. C. Newell, J. Fluid Mech. 3, L. M. Smith and Y. Lee, J. Fluid Mech. 3, E. A. Kartahova, Phy. Rev. Lett. 72, E. Kartahova, in Nonlinear Wave and Weak Turbulence, edited by V. E. Zakharov American Mathematical Society, Providence, R.I., 1998, pp A. Puhkarev, Eur. J. Mech. B/Fluid 18, C. Connaughton, S. Nazarenko, and A. Puhkarev, Phy. Rev. E 63, M. Tanaka and N. Yokoyama, Fluid Dyn. Re. 34, V. E. Zakharov, A. O. Korotkevich, A. N. Puhkarev, and A. I. Dyachenko, JETP Lett. 82, 44. Y. V. Lvov, S. Nazarenko, and B. Pokorni, Phyica D 218, S. Nazarenko, J. Stat. Mech.: Theory Exp. 6 L2. 22 S. Nazarenko, New J. Phy. 9, F. Bellet, F. Godeferd, J. Scott, and C. Cambon, J. Fluid Mech. 23, C. Cambon, N. N. Manour, and F. S. Godeferd, J. Fluid Mech. 337, L. Bourouiba, Phy. Fluid, C. Morize and F. Moiy, Phy. Fluid 18, A. Ibbeton and D. J. Tritton, J. Fluid Mech. 68, G. P. Bewley, D. P. Lathrop, L. R. M. Maa, and K. R. Screenivaan, Phy. Fluid 19, L. Jacquin, O. Leuchter, C. Cambon, and J. Mathieu, J. Fluid Mech. 2, F. Waleffe, Phy. Fluid A 4, F. Waleffe, Phy. Fluid A, A. Babin, A. Mahalov, and B. Nicolaenko, Eur. J. Mech. B/Fluid, A. Babin, A. Mahalov, and B. Nicolaenko, Theor. Comput. Fluid Dyn. 11, C. Cambon, R. Rubintein, and F. S. Godeferd, New J. Phy. 6, E. Kartahova, Phy. Rev. Lett. 98, J. P. Boyd, Chebyhev & Fourier Spectral Method Springer- Verlag, Berlin, W.-C. Müller and M. Thiele, Europhy. Lett. 77, S. Galtier, S. Nazarenko, A. C. Newell, and A. Pouquet, J. Plama Phy. 63, The intermediate regime in decaying turbulence hare mot of the propertie oberved in forced turbulence imulation retricted to near-reonant interaction uch a ued in Ref. 13. The exitence of thi regime i certainly due to the particular property of near-reonant interaction for thi range of