On the bicoherence analysis of plasma turbulence

The bicoherence of ﬂuctuations in a system of drift waves and zonal ﬂows is discussed. In strong drift-wave turbulence, where broadband ﬂuctuations are excited, the bicoherence is examined. A Langevin equation formalism of turbulent interactions allows us to relate the bicoherence coefﬁcient to the projection of nonlinear force onto the test mode. The dependence of the summed bicoherence on the amplitude of zonal ﬂows is clariﬁed. The importance of observing biphase is also stressed. The results provide a basis for measurement of nonlinear interaction in a system of drift waves and zonal ﬂow. © 2005 American Institute of Physics . # DOI: 10.1063/1.2062627 $


I. INTRODUCTION
Plasma turbulence has been subject to intensive study in the last decades. [1][2][3] This is because the turbulent transport is a key in realizing the controlled thermonuclear fusion, and is because the plasma turbulence plays a key role in structure formation. Such efforts in understanding the structure formation in laboratory as well as natural plasmas are explained in, e.g., Refs. 4-6. In addition to progress in the theoretical understanding of plasma turbulence, efforts have also been focused to the direct measurement of the elementary nonlinear interactions. The identification of mesoscale structures ͓such as zonal flow 4 and geodesic acoustic modes 7 ͑GAMs͔͒ and their interaction with ambient turbulence is a highlight of the experimental study of plasma turbulence. The identification of a mesoscale zonal flow has been in progress, 8 and the efforts in the measurement of the nonlinear interactions are also ongoing. One routine method in measuring the nonlinear interactions among the fluctuating quantities is the bicoherence method. 9,10 This allows us to measure the strength and spectra of triplet correlations. The application of this method to plasma turbulence has been widely discussed. [11][12][13][14][15][16][17][18][19] Very recently, the bicoherence method is applied to the experimental study of GAMs and background turbulence. 19 Although the bicoherence method is routinely applied to the plasma physics experiments, the interpretation of the bicoherence data has not been thoroughly considered. The progress of modeling plasma turbulence, and, in particular, the importance of the nonlinear interaction between the mesoscale structure and broadband turbulence have stimulated the efforts to understand the measurement of the bicoherence of signals. 15 More detailed study in understanding the bicoherence data is required.
In this article, we discuss the bicoherence of plasma turbulence in the presence of broadband drift-wave turbulence.
The assumption of a large degree of freedom has given a formulation of Langevin equation of a dressed-test mode. 20 Based on this picture, the bicoherence of fluctuating fields is formulated as a projection of the nonlinear force onto the dressed-test mode. Bicoherence coefficients are evaluated in terms of the spectrum of the fluctuating field, the coefficient of nonlinear interaction, and the autocorrelation time of the fluctuations. Two cases are investigated. The first is the case where a large number of unstable modes are excited and are in a stationary state due to the mutual nonlinear interactions. The second example is when the zonal flow and GAMs exist in broadband fluctuations. The properties of bicoherence data are explained. A clear contrast of biphase between these two cases is demonstrated. A brief discussion on the statistical convergence is also presented. This study provides an interpretation of the bicoherence signal in understanding the nonlinear interaction process.

II. RESPONSE OF TEST WAVE WHICH IS TARGET OF THE EXPERIMENTAL STUDY
An example of the dynamical equations of fluctuation fields, in the range of drift-wave frequency of strongly magnetized plasmas, is expressed in Ref. 2. Among many issues in the nonlinear processes of drift-wave turbulence, the importance of the E ϫ B nonlinearity and the phase relations between different fluctuating quantities ͑such as the density and electric field͒ have been recognized. The former is es-sential in nonlinear stabilizing process of drift waves and in driving zonal flows from drift-wave fluctuations. The latter is the key for driving turbulence and turbulent transport. The details of the theories covering both mechanisms are explained in Ref. 6. Despite the importance of cross-correlation function between different fluctuating fields, focus is made on the E ϫ B nonlinearity in this article, and the fluctuating fields are represented by a scalar variable g ͑such as electrostatic potential͒. This simplification is accepted as the first step, because this nonlinearity has an essential role in the interaction of the drift wave and zonal flow. One can use a one-field model such as Hasegawa-Mima equation. 21 The nonlinear dynamical equation may be written in a form ‫ץ‬ ‫ץ‬t where ␥ is a linear growth rate, L 0 represents the linear frequency, and N denotes the coefficient of nonlinear interaction. N may include operators, as is explicitly shown in Sec.

3.3.
In this chapter, we discuss a response of a test mode against a nonlinear interaction between a particular pair of modes in turbulent fluctuations which are composed of a large number of excited modes. This response is a basis for clarifying the relation between the bicoherence and nonlinearity in dynamical equations.
The nonlinear terms for drift-wave turbulence are modeled as

͑2͒
where T is the nonlinear damping rate of the target mode and S is a random fluctuating force ͑noise͒. 2,20 It has been shown that the nonlinear term can be separated into the memory term ͑coherent term͒ which is coherent to g and into the fluctuating force ͑incoherent term͒, the projection of which onto g vanishes in a long-time average. 22 The spectral functions satisfy the relation 2,3,20 The eddy-damping rate T is a function of the turbulence spectrum, as is explained in Refs. 2 and 3 The response of a test mode against a nonlinear effect from a particular pair of modes is deduced by the use of Eq. ͑2͒. The fluctuation spectrum is expressed, in general, by the space-time Fourier decomposition, e.g., a power spectrum I͑k , ͒. However, such a complete data set is not easily accessible, because experimental data are usually taken by onepoint ͑or few-points͒ measurements. The bicoherence analysis of experimental data has often been performed on the temporal Fourier spectrum. Such studies have relevance for cases where the condition of the wave-number matching is approximately satisfied if the frequency matching condition is fulfilled. In studies where only frequency spectrum is used, the effective nonlinear interaction is analyzed, in which matching conditions of wave numbers are treated as an average. Although limited in accuracy, this simplified data analysis has a relevance in investigating the interactions between drift waves and zonal flows as a first step. Considering these experimental situations, we introduce Fourier components as We choose one particular frequency for a test wave g . An imposition of the test mode g affects the p-Fourier component g p via the nonlinear term N p, g p− g . ͓Note again that the matching conditions of wave numbers are included ͑as an average͒ in calculating the nonlinear coupling coefficient N p, .͔ A response of g p to the imposition of the nonlinear term g is evaluated as follows. 1 We separate one term N p, g p− g from the total nonlinear terms ͚Ngg, and express the rest in terms of the nonlinear damping term and fluctuating force as according to the same theoretical argument that is used in deriving Eq. ͑2͒. The response of g p against the imposition of g is written as ‫ץ‬ ‫ץ‬t

͑6͒
This process has been employed in statistical theories ͑see, e.g., Refs. 1 and 2͒. The meaning of this equation is that, in calculating the dynamics of g p , the nonlinear effects except N p, g p− g are combined into T Јg and SЈ. That is, the lefthand side ͑LHS͒ of Eq. ͑6͒ denotes the response of the "dressed mode." Because of a large degree of freedom of fluctuations excited in the plasma of interest, we employ the test wave approximation The meaning of Eq. ͑7͒ is that ͚Ngg and ͚Ngg − N p, g p− g e −ipt are approximately equal to each other because large numbers of modes are excited in broadband turbulence. The concept of the dressed mode and the validity of the approximation are discussed in Ref. 1. Equation ͑6͒ is solved as where p = T,p − ␥ p + i͑L 0 − p͒ with the help of Eq. ͑7a͒. The first term in the right-hand side ͑RHS͒ of Eq. ͑8a͒ represents the response against the imposition of the test mode, and g p represents the response against the noise excitation through nonlinear interaction with other modes. ͓Equation ͑8b͒ is a Brownian motion if S is Gaussian white noise.͔ Because Eq.
͑7b͒ holds for the broadband turbulence, an approximate relation holds. The integrand g p− g in Eq. ͑8a͒ loses the coherence as t − tЈ becomes longer than the autocorrelation time, a,p = min͑ c, , c,p− ͒, where c,p is the autocorrelation time of the fluctuation c,p −1 = T,p − ␥ p . By noting this fact, the integral in Eq. ͑8a͒ is evaluated as That is, A similar argument applies to g p− , and we have where the relation g − * = g is used. Equations ͑12͒ and ͑13͒ show the responses of the Fourier components g p and g p− against the imposition of the test mode g . The amplitude g p is separated into p N p, g p− g and g p ; the former is the result of the nonlinear interaction g p− g , and the latter, g p , is statistically independent from the former.

III. BICOHERENCE ANALYSIS
The bispectrum estimator B ͑ , p͒, the squared bicoherence b 2 ͑ , p͒, and the summed bicoherence ͚b 2 are defined as We see that this bispectrum estimator is in proportion to the projection of the response g p to the nonlinear force N p, g g p− . Relations between the bicoherence and nonlinear interactions are discussed in this chapter.

A. Case of broadband turbulence
We first study the case where fluctuations are composed of a broadband spectrum, as is shown in Fig. 1͑a͒. In this case all of the three components g p , g p− , and g follow similar relations such as Eqs. ͑12͒ and ͑13͒. We have From Eqs. ͑12͒, ͑13͒, and ͑17͒, the bicoherence is expressed in terms of the nonlinear terms. The derivation is given in Appendix A and the results are summarized here.

Summary of results
a. Bispectrum indicator. The bicoherence indicator, which is the third-order correlation function, is expressed in terms of the second-order correlation functions and the nonlinear coupling coefficient N as In order to have more explicit interpretations, we employ an estimate for the RHS of Eq. ͑18͒. Three terms of spectral functions in the RHS of Eq. ͑18͒, ͉͗g p− ͉ 2 ͉g ͉ 2 ͘, ͉͗g p ͉ 2 ͉g ͉ 2 ͘, and ͉͗g p− ͉ 2 ͉g p ͉ 2 ͘, depend on p and but have similar magnitude for broadband fluctuations. One can have a simplified evaluation as b. Squared bicoherence. Substitution of Eq. ͑19͒ into Eq. ͑15͒ gives the squared bicoherence. In order to have comparisons with experimental observations, a crude estimate, is employed here. This approximation is employed when g , g p , and g p− belong to the broadband spectrum. As is shown FIG. 1. Schematic drawing of the spectrum. A broadband spectrum ͑a͒ and that with a sharp peak ͑b͒.

102301-3
On the bicoherence analysis of plasma turbulence Phys. Plasmas 12, 102301 ͑2005͒ in Appendix A, the squared bicoherence is given, with the help of Eq. ͑20͒, as If phases among p N p, * , p− N p−,p , and N ,p are randomly distributed, one has The randomness of phases among p N p, c. Summed bicoherence. Equations ͑21͒ and ͑22͒ provide the expression for the summed bicoherence

Interpretation
Equation ͑18͒ shows that the magnitude of B ͑ , p͒ is an indicator of the nonlinear force. The bispectrum estimator is composed of the terms which are proportional to the projection of the nonlinear term g p− g onto the response of g p to the nonlinear force N p, g p− g . Thus, the bispectrum indicator provides the evaluation of nonlinear interaction in the observed data. The squared bicoherence shows the magnitude of the three-mode interaction.
In addition, Eq. ͑19͒ shows that the phase of B ͑ , p͒, the biphase, is directly related to the phase of the nonlinear coefficient N p, * . The biphase indicates the phase of p N p, * . That is, the biphase shows the relation between the nonlinear force and the test mode. Thus, the magnitude as well as the biphase give information about the aspects of the nonlinear interactions. For instance, the measurement of the phase of B ͑ , p͒ gives the phase of N once the real frequency and the decorrelation rate are measured.
The interpretation of Eq. ͑23͒ is as follows. The term ͉N p, ͉͉g͉ represents a nonlinear force ͑in a normalized unit in a dimension of the "frequency"͒, and p ͉N p, ͉͉g͉ indicates the competition between this nonlinear force and the effective correlation time p . Equation ͑23͒ is rewritten as

͑25͒
The RHS is composed of three terms, ͉g͉, p , and ͚b 2 ͑͒. The fluctuation level ͉g͉ is measurable, and the correlation time p is evaluated by the autocorrelation time c,p , which is measured from the fluctuation data. Thus, once the summed bicoherence ͚b 2 ͑͒ is measured, the magnitude of the nonlinear coupling coefficient ͉N p, ͉ is evaluated.

B. Case of a sharp peak within broadband fluctuations
When the drift-wave fluctuations coexist with the mesoscale fluctuation, such as zonal flow or geodesic acoustic modes ͑GAMs͒, the interaction between the modes in the sharp peak and broadband fluctuations attracts attentions. Here, the suffix indicates the mode which belongs to the sharp peak of the spectrum, and ͑p , p − ͒ denotes the broadband background turbulence ͓see Fig. 1͑b͔͒. The test mode in a sharp peak is denoted by here.

Response of a test mode
The amplitude of the modes in a sharp peak is considered to be strongly influenced by a self-nonlinear interaction, not solely determined by the fluctuating force from broadband turbulence. In the case of zonal flow dynamics, the negative eddy-viscosity-like effect by the drift-wave turbulence destabilizes the zonal flows, contrary to the case of drift waves for which Eq. ͑2͒ is used. Self-interaction is effective for the saturation of the zonal flow. 6 We introduce the amplitude of the sharp spectral mode, g ,0 , which is assumed to be determined by the self-nonlinear effects and by the excitation by turbulence force ͚ q p Ng q g q− * ͑i.e., the g p g p− * term is subtracted͒. Imposing the nonlinear interaction term g p g p− * on the test mode, one has the response of g after the similar procedure that gives Eq. ͑12͒. Thus, where the first term in the RHS represents the response against the beat interaction N ,p g p g p− * , and is calculated after Eq. ͑11͒. The autocorrelation time of the test mode c, is much longer than those of background turbulence, c,p− , so that in Eq. ͑26͒ is replaced by the autocorrelation time of background fluctuations c,p− . That is, one has an expression g Ӎ c,p− N ,p g p g p− * + g ,0 . ͑27͒ In other words, g is composed of a component g ,0 ͑which is independent of g p g p− * ͒ and a fluctuating component owing to the kick g p g p− * .

Bicoherence
The bicoherence is given from Eqs. ͑12͒, ͑13͒, and ͑27͒, as is explained in Appendix B. The result is summarized here.
a. Bicoherence indicator. The bicoherence indicator is evaluated as The first term ͑parentheses͒ in the RHS of Eq. ͑28͒ is due to the modulation of the background fluctuation by the imposition of the test mode ͑e.g., zonal flow͒. The last term in the RHS comes from the influence on the test mode by background fluctuations. As is explained in Sec. III B 2 b the phases of p N p, * and p− N p−,p are close to each other for the interaction between the zonal flow and drift-wave fluctuations. 6 Based on the estimate p N p, * ͉g p− ͉ 2 Ӎ p− N p−,p ͉g p ͉ 2 for components g p− and g p , which belong to the broadband spectrum, a simplified form of B may be used for convenience as

͑29͒
The term which is proportional to ͉g ,0 ͉ 2 in the bicoherence indicator has been pointed out in Ref. 15. The second term is the contribution of the broadband turbulence, and B ͑ , p͒ at does not vanish even in the limit of ͉g ,0 ͉ 2 → 0. b. The squared bicoherence. A simplified expression for the squared bicoherence is given in Appendix B as In obtaining Eq. ͑30͒, the approximate relations ͉͗g p g p− ͉ 2 ͘ Ӎ͉͗g p− ͉ 2 ͉͗͘g p ͉ 2 ͘ and ͉͗g p− ͉ 2 ͘Ӎ͉͗g p ͉ 2 ͘ are used because g p− and g p belong to the broadband spectrum. In addition, ͉g ,0 ͉ 2 Ӎ͉͗g ͉ 2 ͘ is employed as in Eq. ͑9͒. c. Summed bicoherence. The summed bicoherence coefficient is then expressed as where M is the number of Fourier component, M = ͚ p 1, the overbar¯is an average over the Fourier component, M p− 2 ͉N p−,p ͉ 2 = ͚ p− 2 ͉N p−,p ͉ 2 , ͉g 2 ͉ is the level of background turbulence as is given in Eq. ͑24͒, and ⌬g 2 is the variance of the amplitude of g , owing to the kicks from background fluctuations, ͑Note that the variation ⌬g 2 is defined in a time scale which is longer than c,p but is shorter than the time scale that g ,0 varies.͒

Interpretation
It should be emphasized that the phase of the bicoherence indicator B ͑ , p͒ in Eq. ͑29͒ can be different from that for the case of broadband turbulence. For instance, when is chosen to be the frequency of zonal flows ͑zonal flow, GAMs͒, the phase of the B ͑ , p͒ weakly depends on p. This is particularly noticeable when one study the coupling between the drift wave and zonal flows.
The result, Eq. ͑31͒, is interpreted as follows. ͑i͒ First, the summed bicoherence ͚b 2 ͑͒ has sharp and broad components: The first term in the RHS indicates a peak in the summed bicoherence, and the second and third terms a broad distribution in a wide frequency region. That is, the peak in the summed bicoherence appears at the peak of the power spectrum. ͑ii͒ Second, the magnitude of the peak in the summed bicoherence is in proportion to the magnitude of the mode, the nonlinear interaction coefficient, ͉N p−,p ͉ 2 , the autocorrelation time of the background fluctuations, and by the number of Fourier components, M, which are used in the data analysis. The first term in the RHS of Eq. ͑31͒, which comes from the modulation of background drift-wave fluctuations by imposed zonal flows, is proportional to M. This is because the majority of the drift waves responds to the imposed quasicoherent oscillation in a similar way. As a result of this, the summed bicoherence becomes larger as the number of Fourier components increases. ͑iii͒ Third, the detection of the first term of Eq. ͑31͒ is possible, in the data analysis, as follows: When the peak in the summed bicoherence ͚b 2 ͑͒ is obtained, ͑a͒ the dependence of ͚b 2 ͑͒ on the amplitude of the sharp mode ͉g ͉ 2 must be studied, and ͑b͒ the peak height of ͚b 2 ͑͒ must be investigated by observing the effects of the choice of M. ͑iv͒ Fourth, the second term is the contribution of the broadband fluctuations, and is given by the same response as Eq. ͑23͒. The difference in the numerical coefficients 3 and 4 is due to ͑i͒ the difference in the number of combinations and to ͑ii͒ the difference in the phase difference among nonlinear coefficients. ͑v͒ Fifth, the last term is a small correction, when the self-nonlinear effects for g are strong.
Some further comment may be made on the peak of the summed bicoherence. When the peak is apparent in ͚b 2 ͑͒, it is approximated as in the vicinity of the peak of ͚b 2 ͑͒. The Fourier decomposition is usually made as discretizing the frequency rage as p = n⌬, where ⌬ is the width of the frequency, and n =0±1, ±2, ... ± M. When the half-width of the test mode at frequency is narrower than ⌬, then the peak in the Fourier series ͉g ͉ 2 does not depend on the choice of ⌬. In this case, if one performs a convergence study such as increasing M and decreasing ⌬, the peak value of ͚b 2 ͑͒ is in proportion to M. If ⌬ is smaller than the half-width of the test mode, then M͉g ͉ 2 converges to a finite number. Then ͚b 2 ͑͒ also converges. Equation ͑33͒ shows that the magnitude of the nonlinear coefficient ͉N͉ is measured by observing the total bicoherence ͚b 2 ͑͒ together with the spectral variables ͉g ͉ 2 and p . The information of the phase of N is also obtained from the biphase.

IV. EXPLICIT FORMS
An example is discussed for drift-wave fluctuations. A normalized electrostatic potential is introduced, where ñ is the density perturbation, n 0 is the average density, is the electrostatic potential fluctuation, s is the ion gyroradius at electron temperature, and L n is the density gradient scale length. ͑The normalized variables n and are of the order unity in a stationary drift-wave turbulence. 1 ͒ The Hasegawa-Mima model gives the response of drift wave in the presence of zonal flow ͑pure zonal flow or GAMs͒ as 6 where the suffixes d and Z indicate drift waves and zonal flow, respectively, q x is the radial wave number of zonal flow, and k denotes the wave vector of drift waves. The second and third terms in the LHS of Eq. ͑35͒ stand for the linear response and nonlinear self-interaction of drift waves, respectively. The RHS represents the coupling between the zonal flow and drift waves. The interaction of drift waves has the coupling coefficient This coefficient has a magnitude like It should be noticed that the phase of Eq. ͑36͒ can take a value in a wide range. This is because the sign of the wave number in the poloidal direction k y is determined by the diamagnetic drift direction, but the wave number in the radial direction k x can have a wide variety ͑including complex val-ues͒ for drift waves. The interaction between the zonal flow and drift waves has the coefficient In this form, one sees that k x is not included and is replaced by q x . The sign of k y is dominated by the propagation of drift waves relative to the diamagnetic drift velocity. Therefore, the coefficient N keeps a same phase for components of driftwave fluctuations. The decorrelation time of drift waves through selfnonlinear interaction has been evaluated as in the strong turbulence limit, where h͑k Ќ s ͒ stands for a numerical coefficient of the order of unity. For the case of Eq. ͑23͒, one has

͑40͒
That is, the summed bicoherence has a weak dependence on the drift-wave amplitude so long as the wave numbers are unaltered. Equation ͑36͒ shows that the biphase of B spreads over the range of 0 and 2.
In the case of the GAMs and drift waves, Eqs. ͑31͒ and ͑38͒ gives the expression for Eq. ͑31͒ ͑where the first and second terms are kept͒ as so long as the frequency width for decomposing the Fourier series is wider than the half-width of the GAMs peak. The wave number q x for zonal flows is smaller than k x for drift waves. However, the dependence on M possibly gives a larger value of the summed bicoherence. From Eqs. ͑40͒ and ͑41͒, the total bicoherence at the frequency of zonal flows and that at the drift wave range of frequencies is compared as It is also noted that the total bicoherence of Eq. ͑41͒ is dependent on the local gradient of the zonal flow, The gradient has been reported in Ref. 23. It is also useful to compare Eq. ͑42͒ with the estimate of the theory. In the predator-prey model, one has the ratio of the zonal flow amplitude and the fluctuation amplitude of drift waves as where ␥ L and ␥ nd are the linear growth rate and nonlinear damping rate ͑via drift-wave-drift-wave interactions͒ of drift waves, respectively, and ␥ damp is the ͑collisional͒ damping rate of zonal flow ͑see Sec. II of Ref. 6 for more details͒. Substituting Eq. ͑43͒ into Eq. ͑42͒, one has The zonal flow is excited when ␥ L Ͼ ␥ damp holds, so that the first term on the RHS is usually much greater than unity when the zonal flows are excited.

V. SUMMARY
In this article, we discussed the bicoherence spectrum for drift-wave turbulence in strongly magnetized plasma. The case without zonal flows and that with zonal flows were analyzed. In the presence of a broadband turbulence, the nonlinear interactions are theoretically formulated in a form of the Langevin equation, and the bicoherence spectrum was shown to indicate the projection of the nonlinear force onto the test mode. Based on this formalism, the magnitude of the bispectrum was investigated for the drift-wave-zonal flow systems. It was shown that the total bicoherence for the zonal flows ͑zonal flow and GAMs͒ increases as the amplitude of the zonal flows increases. Comparison between the bispectral data for zonal flows and for drift waves was also given. These findings generalized the result in Ref. 15.
Explicit formulas for the bispectrum are summarized in the text, such as Eqs. ͑19͒, ͑21͒, and ͑23͒ for the interaction of broadband fluctuations, and Eqs. ͑29͒ and ͑31͒ for the interaction of a sharp peak with broadband fluctuations. In these expressions, the bicoherence is expressed in terms of the coefficient of the nonlinear interaction and quantities which are given by quadratic spectral functions. Therefore, by measuring the fluctuation spectrum, autocorrelation time, and bispectral functions, the nonlinear interaction of each three-wave coupling is quantitatively estimated from experimental data. Thus, the study of bicoherence will provide a fruitful understanding of nonlinear interactions in turbulent plasmas.
It might be useful to add a few comment on the statistical variance, which is caused by a finite number of realizations. The statistical error for the bicoherence indicator is estimated as where g p is a typical value of Fourier amplitude in the broadband spectrum and N R is the number of realizations employed in the analysis. The variance for the total bicoherence is given as where M is the number of Fourier components. In order to have statistically admissible estimates, the bicoherence indicator and total bicoherence must be larger than Eqs. ͑45͒ and ͑46͒, respectively. Equations ͑40͒ and ͑41͒ combined with Eq. ͑46͒, provide the necessary number of realizations N R . By observing the dependence of the total bicoherence on the amplitude of zonal flows, one can directly measure the nonlinear interaction of zonal flows and background drift waves directly. The dependence on the number of Fourier component was also clarified. The other issue is the phase of the bispectrum estimator. The importance of observing the biphase was also demonstrated. When is chosen at the frequency of zonal flows, the phase of the bispectrum estimator B ͑ , p͒ has a weak dependence on p. These properties will be used in the experimental study of turbulence. It should be noticed that in the regime of the Dimits upshift, where the majority of fluctuation energy is converted into the zonal flows, the ratio Eq. ͑44͒ becomes very large.
It should be noted that the analysis in this article is valid for cases where the condition of the wave-number matching is approximately satisfied if the frequency matching condition is fulfilled. This means that the coefficient of nonlinear interaction N p, is an effective value, in which averaging over the wave-number space is included. Experimental estimates of N p, , e.g., Eqs. ͑25͒ or ͑33͒, provide effective values. This shortcoming is due to limitations that only fewpoints measurements are usually available. It is necessary to measure the space-time Fourier decomposition, e.g., a power spectrum I͑k , ͒ and more complete bicoherence studies are necessary in order to establish a better understanding of the system of drift waves and zonal flows.
The result in this article is limited to a single-field model, and the crossphases between multiple fluctuating fields ͑e.g., ñ, , T , etc.͒ are not considered. This simplification is relevant as the first step, because the v · ٌv nonlinearity has the essential role in the interaction of the drift wave and zonal flow. The result here is applied to the study of coupling between the zonal flow and drift waves. Nevertheless, the other nonlinear interactions ͑e.g., v · ٌp and other nonlinear terms͒ can also be influential in the quantitative determination of the turbulence level. Experimental studies on cross-bicoherence analysis may be possible in the near future, and theoretical interpretation for them is required as well. Such analysis on multiple fields is left for future studies. It is noted that one-point measurement has limitation in measuring the absolute value of nonlinear interactions. When the coherence lengths of triplet modes g p , g p− and g are different, additional care is necessary.

ACKNOWLEDGMENTS
In order to have more explicit interpretations, we employ an estimate for the RHS of Eq. ͑A4͒. Three terms of spectral functions in the RHS of Eq. ͑A4͒, ͉͗g p− ͉ 2 ͉g ͉ 2 ͘, ͉͗g p ͉ 2 ͉g ͉ 2 ͘, and ͉͗g p− ͉ 2 ͉g p ͉ 2 ͘, depend on p and but have similar magnitude for broadband fluctuations. One can have a simplified evaluation as

͑A5͒
The squared bicoherence is defined as Eq. ͑15͒. Substitution of Eq. ͑A5͒ into Eq. ͑15͒ gives A crude estimate, is employed here, because all of g p− , g p , and g belong to the broadband spectrum. This allows direct comparisons with experimental observations. This approximation gives If phases among p N p, * , p− N p−,p , and N ,p are randomly distributed, one has

APPENDIX B: BICOHERENCE IN A CASE OF A SHARP PEAK WITHIN A BROADBAND TURBULENCE
The triplet product is given from Eqs. ͑12͒, ͑13͒, and ͑27͒ as g p * g p− g = g p * g p− g ,0 + p N p, * ͉g p− ͉ 2 ͉g ,0 ͉ 2 + p− N p−,p ͉g p ͉ 2 ͉g ,0 ͉ 2 + c,p− N ,p ͉g p− ͉ 2 ͉g p ͉ 2 . ͑B1͒ where the relation g q g q * Ӎ͉g q ͉ 2 is also used. The average of the first term ͗g p * g p− g ,0 ͘ is considered to vanish because g p * and g p− are responses to independent noises. Thus one has the evaluation of the bicoherence indicator by the use of the lowest-order correlation as B ͑,p͒ = ͗g p * g p− g ͘ = ͑ p N p, * ͉g p− ͉ 2 + p− N p−,p ͉g p ͉ 2 ͒ ϫ͉g ,0 ͉ 2 + c,p− N ,p ͉g p− ͉ 2 ͉g p ͉ 2 . ͑B2͒ The first term ͑with parentheses͒ in the RHS of Eq. ͑B2͒ is due to the modulation of the background fluctuation by the imposition of the test mode ͑e.g., zonal flow͒, and the last term in the RHS comes from the influence on the test mode by background fluctuations. As is explained in Sec. III, the phases of p N p, * and p− N p−,p * are common for the interaction between the zonal flow and drift-wave fluctuations. Based on the estimate p N p, * ͉g p− ͉ 2 Ӎ p− N p−,p ͉g p ͉ 2 , a simplified form of B may be used for convenience as B ͑,p͒ Ӎ 2 p− N p−,p * ͉g p ͉ 2 ͉g ,0 ͉ 2 + c,p− N ,p ͉g p− ͉ 2 ͉g p ͉ 2 .

͑B3͒
The term which is proportional to ͉g ,0 ͉ 2 in the bicoherence indicator has been pointed out in Ref. 15. The second term is the contribution of the broadband turbulence, and B ͑ , p͒ at does not vanish even in the limit of ͉g ,0 ͉ 2 → 0. The squared bicoherence is calculated from Eq. ͑B3͒ as For further transparency of argument, the approximate relations ͉͗g p g p−w ͉ 2 ͘Ӎ͉͗g p− ͉ 2 ͉͗͘g p ͉ 2 ͘ and ͉͗g p− ͉ 2 ͘Ӎ͉͗g p ͉ 2 ͘ are used, because g p− and g p belong to the broadband spectrum. In addition, an approximation ͉g ,0 ͉ 2 Ӎ͉͗g ͉ 2 ͘ is employed as is in Eq. ͑9͒. By the help of these approximations, the first term in Eq. ͑B4͒ is estimated as and is dependent only weakly on the choice of p. The second term in Eq. ͑B4͒ may be approximately as 4 p− c,p− Re͑N p−,p * N ,p ͉͒g p− ͉ 2 ͉g p ͉ 4 ͉g ,0 ͉ 2 ͉͗g p g p− ͉ 2 ͉͗͘g ͉ 2 ͘ Ӎ 4 p− c,p− Re͑N p−,p N ,p ͉͒g p ͉ 2 . ͑B6͒ The last term in Eq. ͑B4͒ is rewritten as ͉ c,p− N ,p * ͉ 2 ͉g p− ͉ 2 ͉g p ͉ 2 ͉͗g ͉ 2 ͘ . ͑B7͒ Thus, simplified expressions for the squared bicoherence and total bicoherence are given as Eqs. ͑30͒ and ͑31͒.