Abstract
A general statistical theory of the intermittency in turbulence based on short-lived coherent structures (instantons) is presented. The probability density functions (PDFs) of the flux R are shown to have an exponential scaling P (R) exp (-c Rs) in the tails, with the exponent s= (n+1) m. Here, n and m are the order of the highest nonlinear interaction term and moments for which the PDFs are computed, respectively; c is constant depending on spatial profile of the coherent structure. The results can have important implications for understanding the universality often observed in simulations and experiments.
| Original language | English |
|---|---|
| Article number | 114506 |
| Journal | Physics of Plasmas |
| Volume | 15 |
| Issue number | 11 |
| DOIs | |
| Publication status | Published - 25 Nov 2008 |
| Externally published | Yes |
Funding
Kim Eun-jin Anderson Johan Department of Applied Mathematics, University of Sheffield , Sheffield, S3 7RH, United Kingdom 11 2008 15 11 114506 03 09 2008 31 10 2008 25 11 2008 2008-11-25T15:13:38 2008 American Institute of Physics 1070-664X/2008/15(11)/114506/4/ $23.00 A general statistical theory of the intermittency in turbulence based on short-lived coherent structures (instantons) is presented. The probability density functions (PDFs) of the flux R are shown to have an exponential scaling P ( R ) ∝ exp ( − c R s ) in the tails, with the exponent s = ( n + 1 ) ∕ m . Here, n and m are the order of the highest nonlinear interaction term and moments for which the PDFs are computed, respectively; c is constant depending on spatial profile of the coherent structure. The results can have important implications for understanding the universality often observed in simulations and experiments. EPSRC-GB EP/D064317/1 A need for statistical theory of plasma turbulence has grown significantly over the past decade with accumulating evidence from simulations and experiments showing highly intermittent and bursty turbulent transport. 1–10 Probability distribution functions (PDFs) inferred from these experiments are strongly non-Gaussian, especially in the tails caused by rare events of large amplitude. For instance, exponential scalings are often observed in the tails of fluxes in a variety of tokamaks (e.g., see Refs. 7–10 ). These observations thus suggest that Gaussian statistics and average transport coefficients based on a mean field theory fail badly in capturing essential transport processes of intermittency, demanding a proper nonlinear theory for events of large amplitude. Given the potentially disastrous impact of these events on confinement, the importance of a predictive theory of the PDF tails thus cannot be overemphasized. In this paper, we present a nonperturbative theory of PDFs in turbulence, which is rather insensitive to the details of a system. Our theory is motivated by the following key experimental observations. The first is that coherent structures (which tend to form naturally in nonlinear systems) mediate fast transport, being responsible for the intermittency in the PDFs. The second is that they tend to be short-lived in time, causing bursty events (e.g., see Refs. 9 and 10 ). Examples of such short-lived structures include streamers, blobs, and vortices. This empirical fact that short-lived coherent structures are responsible for intermittency and PDF tails is precisely built into our theoretical tool to be utilized—the so-called instanton method. Note that, having originated in quantum field theory, the instanton method has been adapted to classical fluid problems including Burgers turbulence 11,12 and Kraichnan model. 13 We consider turbulence modelled by the following prototype nonlinear dynamical system driven by an external (stochastic) forcing f : ∂ t ϕ + N ( ϕ ) = f , (1) where N ( ϕ ) represents the sum of linear and nonlinear interactions with the highest nonlinearity of n . That is, N ( ϕ ) = P n ( ϕ ) is the n th -order polynomial, which can involve spatial derivatives of ϕ . Equation (1) can easily be generalized to describe the evolution of more than two dynamical variables by taking ϕ to be a vector consisting of those variables and N ( ϕ ) to be a matrix, representing the self- and cross-couplings among those variables. For simplicity, we take the statistics of the forcing in Eq. (1) to be Gaussian with delta-correlation in time as follows: ⟨ f ( x , t ) f ( x ′ , t ′ ) ⟩ = δ ( t − t ′ ) κ ( x − x ′ ) , (2) and ⟨ f ⟩ = 0 . An equivalent way of prescribing the second moment [Eq. (2) ] for the Gaussian forcing is to introduce the probability density function for f as follows: 14 d [ ρ ( f ) ] = D f e ( − 1 ∕ 2 ) ∫ d x d x ′ d t f ( x , t ) κ − 1 ( x , x ′ ) f ( x ′ , t ) . (3) The average value of a quantity Q is then computed as ⟨ Q ⟩ = ∫ d [ ρ ( f ) ] Q , (4) where the angular brackets ⟨ ⟩ represent the average over the statistics of the forcing f . By utilizing Eq. (3) , we construct the PDFs of the flux, which is the m multiple product of ϕ (i.e., m th moment), which is denoted by M ( ϕ ) = P m ( ϕ ) . For instance, for momentum flux, M ( ϕ ) = v x v y = − ∂ x ϕ ∂ y ϕ = P 2 ( ϕ ) is the second moment; for velocity gradient, M ( ϕ ) = v x = − ∂ y ϕ = P 1 ( ϕ ) is the first moment. Hereafter, M ( ϕ ) will be called the observable since we are interested in measuring its PDFs. The PDFs of M ( ϕ ) to take the value of R can then be represented in terms of a path integral as follows: P ( R ; x 0 ) = ⟨ δ ( M ∣ ( ϕ ) ∣ x 0 − R ) ⟩ = ∫ d λ e i λ R I λ , (5) where I λ = ⟨ e − i λ M [ ϕ ( x = x 0 ) ] ⟩ . Here, the PDFs of the local value of M ( ϕ ) at x = x 0 are considered for simplicity. This is sufficient for our purpose since the derivation for the nonlocal PDFs follows exactly the same procedure. By taking Q [ ϕ ] = exp { − i λ M [ ϕ ( x 0 ) ] } in Eq. (4) and by using Eq. (3) , I λ can be rewritten in terms of a path integral as I λ = ∫ D ϕ D ϕ ¯ e − S λ , (6) where S λ is the effective action given by S λ = − i ∫ d x d t ϕ ¯ [ ∂ t ϕ + N ( ϕ ) ] + 1 2 ∫ d x d x ′ d t ϕ ¯ ( x ) κ ( x − x ′ ) ϕ ¯ ( x ′ ) + i λ ∫ d x d t M ( ϕ ) δ ( t ) δ ( x − x 0 ) . (7) In Eq. (7) , ϕ ¯ is the conjugate variable to ϕ , introduced to impose the constraint given by the equation of motion of ϕ in Eq. (1) as N = ∫ D ϕ ¯ exp { i ∫ d x d t ϕ ¯ [ ∂ t ϕ + N ( ϕ ) − f ] } , with a normalization constant N . Although ϕ ¯ appears to be simply a convenient mathematical tool, it does have a useful physical meaning that should be noted; it arises from the uncertainty in the value of ϕ due to stochastic forcing. That is, the dynamical system with a stochastic forcing should be extended to a larger space involving this conjugate variable, whereby ϕ and ϕ ¯ constitute a uncertainty relation (see Fig. 1 ). The instanton solution follows from a particular path out of all possible (functional) values of ϕ and ϕ ¯ , which minimizes the action S λ . Furthermore, as shall be seen shortly, it has the interesting physical meaning of mediating the forcing κ and the flux M ( ϕ ) (observable) whose PDFs are sought for (see Fig. 2 ). The key concept underlying the instanton method is that coherent structures which are localized in time are responsible for the rare events of large amplitude, causing strong intermittency in the PDF tails with possibly significant transport. Assuming that such a coherent structure has a spatial profile ϕ 0 ( x ) with a temporal evolution governed by F ( t ) as ϕ ( x , t ) = ϕ 0 ( x ) F ( t ) , and similarly, ϕ ¯ = ϕ ¯ 0 ( x ) μ ( t ) , we rewrite the nonlinear interaction term N ( ϕ ) and the observable M ( ϕ ) as N ( ϕ ) = F n P n ( ϕ 0 ) , M ( ϕ ) = F m P m ( ϕ 0 ) , where only the highest nonlinear term that dominates the PDF tails is taken into account. Since instanton ϕ propagates forward in time and its conjugate variable ϕ ¯ backward in time while the PDF is computed at t = 0 , the boundary conditions on F and μ are (see also Fig. 1 ) F ( − ∞ ) = 0 , μ ( t > 0 ) = 0 . (8) We then rewrite the action S λ in Eq. (7) by using ϕ ( x , t ) = ϕ 0 ( x ) F ( t ) and ϕ ¯ = ϕ ¯ 0 ( x ) μ ( t ) and minimize S λ with respect to F ( t ) and μ ( t ) to obtain the equations for F ( t ) and μ ( t ) as follows: ∂ t F + c 2 F n = − i c 3 μ , (9) ∂ t μ − n c 2 F n − 1 μ = − λ c 4 m F m − 1 δ ( t ) , (10) where (11) c 1 = ∫ d x ϕ ¯ 0 ( x ) ϕ 0 ( x ) , c 1 c 2 = ∫ d x ϕ ¯ 0 ( x ) P n [ ϕ 0 ( x ) ] , c 1 c 3 = ∫ d x d y ϕ ¯ 0 ( x ) κ ( x − y ) ϕ ¯ 0 ( y ) , c 1 c 4 = P m [ ϕ 0 ( y ) ] . Note that c 1 and c 1 c 2 are the projections of the coherent structure ϕ 0 and its nonlinear interaction P n [ ϕ 0 ] onto the conjugate structure ϕ ¯ 0 , respectively; c 1 c 3 is the projection of the forcing correlation κ ( x − y ) onto the conjugate structure ϕ ¯ 0 ; the observable M ( ϕ 0 ) = P m [ ϕ 0 ] is embedded in c 4 c 1 . Some notable fact about Eqs. (9) and (10) is that the observable c 4 excites the conjugate variable ϕ ¯ ( μ ) , which in turn drives the instanton ϕ ( F ) . That is, ϕ ¯ acts as a mediator between the observables and instantons (physical variables) through stochastic forcing. This is schematically shown in Fig. 2 . For t < 0 , Eqs. (9) and (10) give the equation for F ( t ) as ∂ t t F = n c 2 2 F 2 n − 1 , (12) which can easily be solved with the boundary conditions (8) and F ( t = 0 ) = F 0 as F − n + 1 = F 0 − n + 1 + c 2 ( − n + 1 ) t . (13) To find the value of F 0 , we integrate Eq. (10) for an infinitesimal time interval t = [ − ϵ , ϵ ] by using Eq. (8) to obtain μ ( − ϵ ) = μ ( 0 ) = λ m c 4 F 0 m − 1 . (14) The use of Eq. (14) and ∂ t F = c 2 F n in Eq. (9) then gives us F 0 n − m + 1 = − i m c 3 c 4 λ 2 c 2 ≡ q λ , (15) where q = − i m c 3 c 4 ∕ 2 c 2 . To determine the PDFs of M ( ϕ ) for n − m + 1 ≠ 0 , we evaluate S λ in Eq. (5) by using the solutions (13)–(15) , S λ = Q λ a , (16) where Q = i ( c 1 c 4 ) q a − 1 ∕ a and a = ( n + 1 ) ∕ ( n − m + 1 ) . The final step requires the computation of the λ integral in the limit of large λ , which corresponds to the PDF tails. To this end, we substitute Eq. (16) into Eq. (5) and evaluate the λ integral ∫ d λ e i λ Z − S λ ≡ ∫ d λ e G ( λ ) by using the value of λ 0 which minimizes G ( λ ) = i λ Z − S λ (i.e., a saddle-point), λ 0 ( a − 1 ) = i Z a Q . (17) Thus, the desired PDF tails P ( R ) ∼ exp { G ( λ 0 ) } are found to be P ( R ) ∝ exp ( − ξ { R ∕ M [ ϕ ( x 0 ) ] } s ) , (18) s = n + 1 m , (19) ξ = 2 n + 1 ∣ ( c 2 c 1 ) c 1 ( c 3 c 1 ) ∣ , (20) where c i ( i = 1 ⋯ 4 ) are given in Eq. (11) . Note that Eq. (18) can also be shown to hold for n + 1 = m by computing Eq. (5) in a slightly different way. The results [Eqs. (18)–(20) ] clearly show that the predicted PDFs are all exponential while the exact scaling depends on the property of a nonlinear system: The exponent s depends on a nonlinear system (through the highest nonlinearity) and the observables (through the order of moments), while the overall amplitude ξ depends on the form of a coherent structure through the values of c i in Eq. (11) . It is worthwhile to examine in detail how the exponent s varies. (i) In a linear system, n = 1 and s = 2 ∕ m . Thus, the PDFs of the first moment (such as v x , T , etc.) are Gaussian with e − c R 2 ( c is constant), as expected. However, the PDFs of the second moment with m = 2 (such as momentum flux) become ∝ e − c R , which are non-Gaussian. This is an interesting result, confirming that the PDFs of the product of the two Gaussian fluctuations are not necessarily Gaussian due to the correlation between the two. In particular, this can explain the exponential scaling e − c R of particle flux found in Ref. 1 . (ii) In a nonlinear system with a quadratic nonlinearity, n = 2 and s = 3 ∕ m . Examples of such a system are numerous, including Burgers turbulence 11,12 and the Hasagawa–Mima turbulence. 15,16 For the PDFs of the positive velocity gradient in Burgers turbulence, s = 3 ( n = 2 and m = 1 ), leading to P ( R ) ∝ e − c R 3 , thereby recovering the previous result. 11,12 For the momentum flux R = v x v y in the Hasagawa–Mima turbulence, s = 3 ∕ 2 , and thus P ( R ) ∝ e − c R 3 ∕ 2 , also in agreement with our previous predictions. 17 In the ion temperature gradient turbulence studied in Refs. 18–20 , T ∝ ϕ ( T and ϕ are temperature and electric potential, respectively) was assumed. Thus, the PDFs of the heat flux as well as momentum flux also follow P ( R ) ∝ e − c R 3 ∕ 2 scaling whether the former is spatially averaged or local. In this model, the PDF of the third moment such as n v x v y would, however, become P ( R ) ∝ e − c R if n ∝ ϕ ( m = 3 ) or P ( R ) ∝ e − R 3 ∕ 4 if n ∝ ϕ 2 ( m = 4 ) ! Note that both are significantly enhanced over the Gaussian prediction. This could explain a considerable enhancement of the PDF tails of n v x v y with the exponent s = 2 ∕ 3 – 1 , observed in numerical simulations of blobs by Myra et al. 7 Note also that the cross-phase involved in the m th moment is a subtle, but important, issue. (iii) In a cubic nonlinear system (e.g., Ref. 21 ), the PDFs of the velocity gradient become P ( R ) ∝ e − c R 4 . Our theory discussed above can offer a powerful mechanism for exponential scalings which have often been observed in the tails of the flux of heat, particle, momentum, etc., in a variety of tokamaks (e.g., see Refs. 7–10 ). Furthermore, there has been accumulating supporting evidence for the importance of short-lived coherent structures in intermittency, corroborating the basic concepts underlying our method (instantons). While this exponential scaling is independent of the details of a coherent structure, the latter is crucial for the overall amplitude of the PDFs through the coefficient ξ [see Eqs. (11) and (20) ]. The key question is then, “What should we use for ϕ 0 ?” or alternatively, “What are the possible coherent structures that are likely to form in a given nonlinear system?” As noted previously, exact solutions to nonlinear equations which tend to be supported naturally are examples of such structures. In the presence of a stochastic forcing, these structures can readily be created, being localized in time. Assuming such exact solutions ϕ 0 , we can then approximate N ( ϕ 0 ) ∝ α ϕ 0 with a constant α to obtain c 2 c 1 ∼ α c 1 . (21) To estimate c 3 c 1 , we expand the forcing correlation function κ ( x − y ) in terms of the basic structure as κ ( x − y ) ∼ κ 0 ∑ m , n ϕ 0 m ( x ) ϕ 0 n ( y ) , where κ 0 is the strength of the forcing. We can then recast c 3 c 1 as c 3 c 1 = κ 0 ∫ d x d y ϕ ¯ 0 ( x ) ∑ m , n ϕ 0 m ( x ) ϕ 0 n ( y ) ϕ ¯ 0 ( y ) ∼ κ 0 β c 1 2 , (22) where β reflects the fraction of the coherent structure imprinted in the forcing. Alternatively, β is the measure of the similarity in spatial form between coherent structure and perturbation. Therefore, a kind of a “resonant excitation” of the coherent structure is expected when the forcing correlation has exactly the same spatial profile as the structure ( β = 1 ) . We now estimate ξ by substituting Eqs. (21) and (22) in Eq. (11) as ξ ∼ α κ 0 β . (23) The result [Eq. (23) ] reveals the important fact that the contribution from the coherent structures to PDF tails increases ( ξ decreases) when the forcing is strong (large κ ) and when its spatial profile is similar to ϕ 0 (large β ). The PDF, however, decreases with α as the nonlinear effect (i.e., self-regulation) becomes stronger. In summary, we presented a general statistical theory of turbulence and intermittency which is rather insensitive to the details of a dynamical system, depending only on the highest nonlinear interaction. This was motivated by various experimental results that coherent structures often associated with bursty events cause a significant transport. Our predictions are in agreement with various recent experiments (e.g., Refs. 8 and 9 ). While the leading order prediction of instantons is limited to exponential PDFs, there is much hope that the extension of this method would give more diverse scaling prediction, including the combination of exponential and power law, power law, etc., which can explain behavior not only in the tails but near the center of the PDFs. Other steps to improve the present predictability of the instanton method include (i) to keep contributions from the perturbations around instanton (i.e., higher order corrections in the action and path integral), that is, to incorporate both coherent structures and fluctuations (turbulence); (ii) to incorporate contributions from anti-instantons, multi-instantons, and multi-structures; (iii) to generalize the method to account for a finite-correlation time of the forcing and for a non-Gaussian statistics; and (iv) to derive the forcing consistently which may arise from some instabilities in a system, rather than taking it to be given. These improvements could provide a theoretical framework in which a broad range of experimental data, including finite size scaling with power-law PDFs, 22 can be understood. This research was supported by the Engineering Physical Science Research Council (EPSRC) Grant No. EP/D064317/1. FIG. 1. Uncertainty in ϕ = F ( t ) ϕ 0 and ϕ ¯ = μ ( t ) ϕ ¯ [or in F ( t ) and μ ( t ) ] due to a stochastic forcing. FIG. 2. Schematic diagram showing the relation among the observable M ( ϕ ) of a dynamical quantity ϕ , its conjugate variable ϕ ¯ , and the stochastic forcing.
ASJC Scopus subject areas
- Condensed Matter Physics