Random and quasi-coherent aspects in particle motion and their effects on transport and turbulence evolution

Trajectory trapping or eddying is a consequence of the invariance of the Lagrangian potential for ${\tau }_{c}\to \infty ,$ which is approximately maintained at large ${\tau }_{c}$. The DTM and the NSA were developed having in mind the necessity of evidencing the statistical consequences of this invariance. We have also imposed the condition of performing only the approximations that do not violate this property.

The main idea is to define subensembles of the realizations of the potential that correspond to the same values of the potential, of the velocity and of its derivatives in the starting point of the trajectories ${\bf{x}}={\bf{0}},t=0.$ A system of nested subensembles is constructed. All the realization with the potential $\phi ({\bf{0}},0)={\phi }^{0}$ are grouped together in a subensemble ${S}_{0},$ then S₀ is subdivided into subensembles S₁ according to the values of the initial velocity ${\bf{v}}({\bf{0}},0)={{\bf{v}}}^{0}$, then each S₁ is again divided into smaller subensembles S₂ using the values of the derivatives of the velocity, and the procedure can continue. The trajectories in a subensemble S_n are super-determined, in the sense that they have supplementary initial condition (the potential ${\phi }^{0},$ the initial velocity ${{\bf{v}}}^{0}$ and its derivatives up to the order $n).$ This leads to a high degree of similarity of the trajectories in a subensemble S_n, that increases with the increase of the nesting order $n.$

The statistical description of the velocity in a subensemble S_n can be derived from the statistics in the whole set of realization using conditional averages. A Gaussian velocity reduced at the subensemble S_n remains Gaussian, but its average and dispersion are modified. The subensemble average ${{\bf{V}}}^{{S}_{n}}({\bf{x}},t)\equiv {\langle {\bf{v}}({\bf{x}},t)\rangle }_{{S}_{n}}$ exists even in the case of zero average fields. It is a function of the EC that also depends (linearly) on the set of parameters of the nested subensembles. The subensemble average velocity ${{\bf{V}}}^{{S}_{n}}({\bf{x}},t)$ is a space–time dependent function, which has the value ${{\bf{V}}}^{{S}_{n}}({\bf{0}},0)={{\bf{v}}}^{0},$ as imposed by the conditions that define S₁ in the nested subensembles. The subensemble dispersion ${\langle {(\delta {\bf{v}}({\bf{x}},t))}^{2}\rangle }_{{S}_{n}}$ (where $\delta {\bf{v}}({\bf{x}},t)={\bf{v}}({\bf{x}},t)-{{\bf{V}}}^{{S}_{n}}$) is much smaller than ${{\bf{V}}}^{2},$ the amplitude of the velocity fluctuations in the whole set of realizations. It is zero for ${\bf{x}}={\bf{0}}$ and t = 0 (because imposing a condition on the value of the velocity eliminates any fluctuation), and it reaches the value ${{\bf{V}}}^{2}$ at large distances and/or times ($| {x}_{i}| \gt {\lambda }_{i},t\gt {\tau }_{c}).$ The small amplitude of the velocity fluctuations in the subensemble S_n provides a second contribution to the increase of the degree of similarity of the set of trajectories that yield from the realizations that are included in ${S}_{n}.$

The existence of the subensemble average velocity ${{\bf{V}}}^{{S}_{n}}({\bf{x}},t)$ determines an average trajectory in S_n that is obtained by performing the subensemble average of the equation of motion

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{\langle {\bf{x}}(t)\rangle }_{{S}_{n}}={\langle {\bf{v}}({\bf{x}}(t),t)\rangle }_{{S}_{n}}.\end{eqnarray} \tag{ 8 }$$

The right-hand side of this equation is generally very difficult to be evaluated since it is the average of a stochastic function ${\bf{v}}({\bf{x}},t)$ of a stochastic variable, the trajectory ${\bf{x}}(t)$ that is determined by ${\bf{v}}({\bf{x}},t)$. However, the high degree of similarity of the trajectories in the subensemble S_n enables an important simplification that consists in neglecting the fluctuations of the trajectories. The average Lagrangian velocity is thus approximated in S_n by the subensemble Eulerian velocity calculated along the average trajectory

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{{\bf{X}}}^{{S}_{n}}(t)={{\bf{V}}}^{{S}_{n}}({{\bf{X}}}^{{S}_{n}}(t),t),\end{eqnarray} \tag{ 9 }$$

where ${{\bf{X}}}^{{S}_{n}}(t)$ is the approximate subensemble average trajectory in S_n, ${{\bf{X}}}^{{S}_{n}}(t)\cong {\langle {\bf{x}}(t)\rangle }_{{S}_{n}}.$ It is called the decorrelation trajectory (DT), because it shows the way toward decorrelation in each subensemble. We note that the approximation (9) of equation (8) is in agreement with the invariance of the Lagrangian potential. Equation (9) has Hamiltonian structure as the equation of particle motion (1) because ${{\bf{V}}}^{{S}_{n}}({\bf{x}},t)$ has the same structure as particle velocity. It can be derived from a function ${{\rm{\Phi }}}^{{S}_{n}}({\bf{x}},t),$ which is the subensemble average potential

$$\begin{eqnarray}&&{V}_{i}^{{S}_{n}}({\bf{x}},t)=-{\varepsilon }_{{ij}}\displaystyle \frac{\partial {{\rm{\Phi }}}^{{S}_{n}}({\bf{x}},t)}{\partial {x}_{j}}.\end{eqnarray} \tag{ 10 }$$

This leads, in the case of static velocity fields, to the invariance of the Lagrangian potential ${{\rm{\Phi }}}^{{S}_{n}}({{\bf{X}}}^{{S}_{n}}(t)),$ which is equal to ${\phi }^{0}$ for any value of the nesting order $n.$

The LVC, D_ii(t) and other Lagrangian averages are determined by summing the contribution of all subensembles. This leads to integrals over the parameters that define the subensembles.

Thus the NSM is a systematic expansion based on the nested subensembles. It is a semi-analytical method that determines the statistics of the stochastic trajectories in terms of the DTs. These trajectories and the integrals over the subensembles have to be numerically calculated. The calculations are at PC level, with run times that increase as the nesting order increases, starting from few minutes for n = 1.

The DTM [13] corresponds to the nesting order n = 1. The results obtained using the NSM of order n = 2 are presented in [14]. The DTs for the subensembles S₂ are completely different of those of the DTM. However they lead to results for the LVC that are not much different of those obtained with the DTM, which shows that the NSM converges fast. The higher order NSM provides more detailed statistical information at the expense of the increase of the number of the DTs and of the complexity of their equation. The statistics of the distance between neighbor trajectories was determined with the n = 2 NSM.

We use here the DTM that is based on the subensembles S₀ and ${S}_{1}.$ An important simplification was recently found [28]. We have shown that the number of parameters of the subensembles can be reduced to only two without significant modifications of the diffusion coefficients. The magnitude u of the normalized velocity ${v}_{i}({\bf{0}},0)/{V}_{i}$ can be eliminated from the definition of ${S}_{1}.$ The integral over u can be performed in the subensemble EC in ${S}_{1}.$ This leads to a modified average potential, which depends only on ${\phi }^{0}$ and ${\theta }^{0},$ the orientation of the normalized velocity

$$\begin{eqnarray}&&{{\rm{\Phi }}}^{{S}_{1}^{\prime }}({\bf{x}},t)={\phi }^{0}\displaystyle \frac{E({\bf{x}},t)}{E(0)}+\sqrt{\displaystyle \frac{8}{\pi }}\cos ({\theta }^{0})\displaystyle \frac{{\partial }_{2}E({\bf{x}},t)}{{V}_{1}}-\sqrt{\displaystyle \frac{8}{\pi }}\sin ({\theta }^{0})\displaystyle \frac{{\partial }_{1}E({\bf{x}},t)}{{V}_{2}}.\end{eqnarray} \tag{ 11 }$$

The parameter ${\theta }^{0}$ defines a larger subensemble ${S}_{1}^{\prime }$ that includes the subensembles S₁ with any value of $u.$

The time dependent diffusion coefficients are obtained in this case from

$$\begin{eqnarray}&&{D}_{11}(t)=\displaystyle \frac{{V}_{i}}{2\pi }\sqrt{\displaystyle \frac{\pi }{2}}{\int }_{-\infty }^{\infty }{\rm{d}}{\phi }^{0}P({\phi }^{0}){\int }_{0}^{2\pi }\ {\rm{d}}{\theta }^{0}\cos ({\theta }^{0}){X}_{1}^{{S}^{\prime }}(t),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{D}_{22}(t)=\displaystyle \frac{{V}_{i}}{2\pi }\sqrt{\displaystyle \frac{\pi }{2}}{\int }_{-\infty }^{\infty }{\rm{d}}{\phi }^{0}P({\phi }^{0}){\int }_{0}^{2\pi }\ {\rm{d}}{\theta }^{0}\sin ({\theta }^{0}){X}_{2}^{{S}^{\prime }}(t),\end{eqnarray} \tag{ 13 }$$

where ${{\bf{X}}}^{{S}^{\prime }}(t)$ is the DT in the subensemble ${S}^{\prime }$ that is the solution of

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{X}}_{i}^{{S}^{\prime }}}{{\rm{d}}{t}}={V}_{i}^{{S}^{\prime }}({{\bf{X}}}^{{S}^{\prime }},t)+{\delta }_{i2}{V}_{{d}},\end{eqnarray} \tag{ 14 }$$

where we have introduced an average velocity V_d along ${{\bf{e}}}_{2}$ axis.

Thus, the diffusion coefficients are obtained by calculating the double integral in equation (12) with the DTs determined for each value of ${\phi }^{0},{\theta }^{0}$ from equation (14), where the velocity ${V}_{i}^{{S}^{\prime }}$ is obtained from the potential (11).

Other Lagrangian quantities are obtained using expression similar to (12), which consists of using the DTs for the evaluation of the average Lagragian quantities in ${S}^{\prime }$ and of the summation of the contributions of all subensembles.

This is a very fast version of the DTM because the number of DTs is strongly reduced (from N³ to ${N}^{2},$ where N is the number of discretization points for each parameter of the subensembles).

The DTs are the main concept of this theoretical description of trajectory statistics. They are approximations of the average trajectories in the subensembles, which embed the high degree of similarity of the corresponding sets of stochastic trajectories. The DTs are in agreement with the invariance of the Lagrangian potential in the static case. Each DT is a periodic functions in this case, and it is tied to the contour line of the subensemble average potential. The Lagragian potential for the DT is ${\phi }^{0},$ the same with the Lagrangian potential for any trajectory included in ${S}_{0}.$

In the case of slow decorrelation (large ${\tau }_{c}),$ the DTs are completely different for small and large values of the initial potential. At large values of $| {\phi }^{0}|$, the DTs rotate on almost closed paths, while at small $| {\phi }^{0}|$ the DTs make long displacements out of the correlated zone. The typical structure of the real trajectories (that consists of a random sequence of eddying events and jumps) is represented by the DTs. They actually describe segments of trajectories of time intervals of the order ${\tau }_{c}$ that are either trapping events or free displacements, depending on the initial potential. More important, the statistical relevance of each type of motion and the influence of the trajectory trapping is determined by the whole set of DTs.

Typical results for D_ii(t) obtained for isotropic turbulence with the EC of the potential $E={{\rm{\Phi }}}^{2}\exp (-({x}_{1}^{2}+{x}_{2}^{2})/{\lambda }^{2}/2-t/{\tau }_{c})$ and ${V}_{d}=0$ are shown in figure 1. D_ii(t) has different shapes at small and large amplitudes of the potential fluctuations ${\rm{\Phi }}.$ The diffusion coefficient increases linearly at small time and it eventually saturates (for $t\gt {\tau }_{c})$ in the case of small Φ (dashed line), while, at large Φ (continuous line), a transitory decay appears before saturation. The decay accounts for the micro-confinement produced by trajectory trapping. This process appears when ${\tau }_{{\rm{fl}},i}\lt {\tau }_{c}$ (in general, ${\tau }_{{\rm{fl}},i}\lt {\tau }_{d}),$ where ${\tau }_{{\rm{fl}},i}={\lambda }_{i}/{V}_{i}$ is the time of flight over the correlation length. One can see in figure 1 that the variation of ${\tau }_{c}$ determines opposite effects in the two cases. The decrease of ${\tau }_{c}$ leads to the saturation of D_ii(t) at smaller time. This corresponds to smaller D_ii(t) (smaller asymptotic ${D}_{i})$ in the quasilinear case (dashed line) and to larger D_i for the nonlinear transport (continuous line). In a static potential (the dotted line for ${\tau }_{c}\to \infty$), the decay is not limited and ${D}_{{ii}}(t)\to 0$. The DTM yields subdiffusive transport for such fields where the trajectories are tied on the contour lines of the potential. The LVC (4) is negative at large time, such that its integral decays to zero. It has a negative tail that decays as a power of time. We note that similar results are obtained for more extended EC that has space decay of power low type [30].

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The existence of an average velocity V_d in the equation of motion determines the decrease of the fraction of trapped trajectories and modifications of the DTs in the nonlinear regime. A new type of open DTs that turn toward the direction of ${V}_{d}{{\bf{e}}}_{2}$ appears. Their number increases as V_d increases and, when V_d is much large than the stochastic velocity, all DTs are of this type. Thus, the average velocity eliminates progressively (as V_d increases) the trapping of the trajectories and eventually leads to quasilinear transport. The open trajectories along the average velocity lead to a transitory regime characterized by the increase of ${D}_{22}(t).$ It last for the interval $[{\tau }_{* },{\tau }_{c}],$ where ${\tau }_{* }={\lambda }_{2}/{V}_{d}$ is the characteristic time defined by the average velocity.

The time dependence of D_ii(t) for the unperturbed static potential (figure 1, the dotted curve) is the representation of the micro-confinement produced by the permanent trapping. The mixing of the close trajectories, which have periods that increase with the increase of their size, eliminates progressively the contributions of the trapped trajectories at the diffusion process. D_ii(t) decreases due to the decrease of the fraction of free particles. The LVC and D_ii(t) have long (algebraic) tails, which show that the diffusion process easily reacts at perturbations represented by the time variation of the potential or by other components of the motion. Any small perturbation (characterized by a large decorrelation time ${\tau }_{d})$ has significant effect that consists of releasing a part of trajectories. The transport reservoir is the maximum of D_ii(t) (figure 1). It provides the maximum diffusion coefficient that correspond to a decorrelation process with ${\tau }_{d}={\tau }_{{\rm{fl}}},$ which releases all the trapped trajectories.

Various aspects of the transport in fusion plasmas were studied using the DTM, that was developed and adapted to the study of models with increased complexity [32–40].

We determine here the parameters required by the test mode study and present a short review of the diffusion process.

The equation for the trajectories in the drift turbulence with a zonal flow mode potential ${\phi }_{{zf}}$ (in the frame that moves with the potential) is

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{x}}_{i}}{{\rm{d}}{t}}=-{K}_{* }^{\prime }{\varepsilon }_{{ij}}{\partial }_{j}[\phi ({\bf{x}},t)+Z\,{\phi }_{{zf}}({\bf{x}},t)]+{V}_{d},\end{eqnarray} \tag{ 15 }$$

where dimensionless quantities are used with the units: ${\rho }_{s}$ (for the distances and the correlation lengths ${\lambda }_{i},{\lambda }_{{zf}}),{\tau }_{0}={L}_{n}/{c}_{s}$ (for time) and Φ (for the potential). In this normalization, the time of flight is a decreasing function of Φ, while ${\tau }_{* }$ and ${\tau }_{d}$ are fixed (independent of the turbulence amplitude). The normalized amplitude of the zonal flows is $Z={{\rm{\Phi }}}_{{zf}}/{\rm{\Phi }}$ and the normalized diamagnetic velocity is ${V}_{d}=1.$ The dimensionless parameter ${K}_{* }^{\prime }$ is the measure of turbulence amplitude

$$\begin{eqnarray}&&{K}_{* }^{\prime }\equiv \displaystyle \frac{{\rm{\Phi }}}{B{\rho }_{s}{V}_{* }}=\displaystyle \frac{e{\rm{\Phi }}}{{T}_{e}}\displaystyle \frac{{L}_{n}}{{\rho }_{s}},\end{eqnarray} \tag{ 16 }$$

which naturally appears in connection with the diamagnetic velocity.

The spectrum of the drift turbulence has a special shape determined by the absence of the modes with ${k}_{2}=0$, which are stable. It has two symmetrical peaks for ${k}_{2}=\pm {k}_{0}.$ The ISC method shows that the evolution of the drift turbulence (see section 8) is in good agreement with the model

$$\begin{eqnarray}&&S({\bf{k}})\sim {k}_{2}^{m}\exp \left(-\displaystyle \frac{{k}_{1}^{2}}{2}{\lambda }_{1}^{2}\right)\left[\exp \left(-\displaystyle \frac{{({k}_{0}-{k}_{2})}^{2}}{2}{\lambda }_{2}^{2}\right)-\exp \left(-\displaystyle \frac{{({k}_{0}+{k}_{2})}^{2}}{2}{\lambda }_{2}^{2}\right)\right].\end{eqnarray} \tag{ 17 }$$

The Fourier transform gives the EC

$$\begin{eqnarray}&&E({\bf{x}},t)={{\rm{\Phi }}}^{2}{\partial }_{y}^{m}\left[\exp \left(-\displaystyle \frac{{x}_{1}^{2}}{2{\lambda }_{1}^{2}}-\displaystyle \frac{{x}_{2}^{\prime 2}}{2{\lambda }_{1}^{2}}\right)\displaystyle \frac{\sin {k}_{0}{x}_{2}^{\prime }}{{k}_{0}}\right],\end{eqnarray} \tag{ 18 }$$

where ${x}_{2}^{\prime }={x}_{2}-{V}_{* e}t$ accounts for the motion of the drift turbulence potential with the diamagnetic velocity. The spectrum and the EC are functions of five parameters: ${\rm{\Phi }},{\lambda }_{1},{\lambda }_{2},$ k₀ and the power m that determines the decay of the spectrum to ${k}_{2}=0.$ The EC has negative parts and possibly an oscillating decay as shown in figure 7(b). This leads to a transport process with important differences compared to the example presented in section 3.2 that are obtained for a monotonically decreasing EC. The transport coefficients in this type of EC were studied in [28, 29] for electron and ion diffusion.

Typical results obtained with the DTM using equations (10)–(14) and the EC (18) are shown in figure 2. We note that the transport is not isotropic. The special shape of the EC leads to diffusion regimes for ${D}_{11}(t)$ that have similar dependence on time in the quasilinear and nonlinear regime. This determines smaller values of the quasilinear D₁ than in a decaying EC. The shape of the EC also determines negative parts in the time dependence of ${D}_{11}(t)$ (seen as breaks in the red lines in figure 2). A large increase of the asymptotic coefficient D₂ appears at the transition from the quasilinear to the nonlinear regime, which provides a strong auto-control parameter. As seen in figure 2, the ratio of the asymptotic values in the nonlinear and quasilinear regime is much larger for D₂ than for ${D}_{1}.$

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The parameters of the trajectory structures are obtained using the DTM as weighted averages of the DTs ${{\bf{X}}}^{S}(t)$ in the static potential. The solutions of equation (14) are periodic functions in this case with the periods $T({\phi }^{0},{{\bf{v}}}^{0})$ that depend on the subensemble. The fraction of trapped trajectories at time t is

$$\begin{eqnarray}&&{n}_{{\rm{tr}}}(t)=\int {\rm{d}}{\phi }^{0}P({\phi }^{0})\int {{\rm{d}}{v}}_{1}^{0}{{\rm{d}}{v}}_{2}^{0}P({v}_{1}^{0})P({v}_{2}^{0})\ {c}_{{\rm{tr}}}(t;{\phi }^{0},{{\bf{v}}}^{0}),\end{eqnarray} \tag{ 19 }$$

where ${c}_{{\rm{tr}}}(t;{\phi }^{0},{{\bf{v}}}^{0})=1$ if $t\gt T({\phi }^{0},{{\bf{v}}}^{0})$ and ${c}_{{\rm{tr}}}(t;{\phi }^{0},{{\bf{v}}}^{0})=0$ if $t\lt T({\phi }^{0},{{\bf{v}}}^{0}).$ The sizes s_i (t) of the trajectory structures in the two directions are

$$\begin{eqnarray}&&{s}_{i}(t)=\int {\rm{d}}{\phi }^{0}P({\phi }^{0})\int {{\rm{d}}{v}}_{1}^{0}{{\rm{d}}{v}}_{2}^{0}P({v}_{1}^{0})P({v}_{2}^{0})\ {X}_{i}^{\max }(t;{\phi }^{0},{{\bf{v}}}^{0}),\end{eqnarray} \tag{ 20 }$$

where ${X}_{i}^{\max }(t;{\phi }^{0},{{\bf{v}}}^{0})$ is the dimension of ${{\bf{X}}}^{S}(t;{\phi }^{0},{{\bf{v}}}^{0})$ along the direction i if $t\gt T({\phi }^{0},{{\bf{v}}}^{0})$ and ${X}_{i}^{\max }(t;{\phi }^{0},{{\bf{v}}}^{0})=0$ if $t\lt T({\phi }^{0},{{\bf{v}}}^{0}).$

The functions n_tr(t) and s_i(t) describe the growth of the trajectory structures. These functions saturate in a time ${\tau }_{s},$ which is the characteristic time for the formation of the structure. ${\tau }_{s}$ is an increasing function of the amplitude of the turbulence ${K}_{* }^{\prime }.$ The asymptotic values of n_tr(t) and s_i(t) describe the parameters of the structures as functions of the characteristics of the turbulence. The dependence on the amplitude of the turbulence is presented in figure 3. It shows that the structures continuously increase when turbulence amplitude increases and eventually they include all the trajectories ${n}_{{\rm{tr}}}\to 1$. In the presence of a decorrelation process with characteristic time ${\tau }_{d}\lt {\tau }_{s}$, the parameters of the structures are smaller because they are destroyed during the formation at the time ${\tau }_{d}.$ They can be approximated by ${n}_{{\rm{tr}}}({\tau }_{d})$ and ${s}_{i}({\tau }_{d}),$ instead of the asymptotic values.

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The micro-probability ${P}^{{\tau }_{d}}({\bf{x}},t)$ can be evaluated using the DTM as a histogram of the DTs. It can be Gaussian, or non-Gaussian, depending on turbulence parameters and on the decorrelation time ${\tau }_{d}.$ Non-Gaussian ${P}^{{\tau }_{d}}$ are associated to the presence of trapping. The micro-confinement appears very clearly in the probability as a narrow peak in ${\bf{x}}={\bf{0}},$ which remains invariant after a formation time ${\tau }_{s},$ the same as in the time evolution of the fraction of trapped trajectories ${n}_{{\rm{tr}}}(t)$. ${P}^{{\tau }_{d}}$ has also a time dependent part around this maximum, which is determined by the free particles. For the drift turbulence case, the motion of the potential with the diamagnetic velocity leads to the separation of the two parts of the micro-probability: the narrow peak is displaced in the direction ${V}_{d}{{\bf{e}}}_{2}$ and the free particle part concentrates in the opposite direction then moves. This result is in qualitative agreement with the process of flow generation in zero divergence velocity fields (section 2), but the velocity of the free particles obtained with the DTM is smaller than the the velocity obtained from equation (2), ${V}_{f}=-{V}_{d}{n}_{{\rm{tr}}}/{n}_{f}$.

The analytical results for the test modes (section 4.2) are obtained using simplified approximations of ${P}^{{\tau }_{d}}$ in agreement with equation (2) for the ion flows (see [15] for details). The trapped particle part was represented for simplicity by a Gaussian function, but with fixed dispersion given by the size s_i of the structures ${S}_{i}={s}_{i}^{2}$. The shape of this function does not change much the estimations. The free trajectories are described by a Gaussian with dispersion that grows linearly in time: ${S}_{i}^{\prime }(\tau )={S}_{i}+2{D}_{i}(t-\tau ),i=x,y$. The initial value ${S}_{i}^{\prime }(t)={S}_{i}$ is an effect of trapping. It essentially means that the trajectories are spread over a surface of the order of the size of the trajectory structures when they are released by a decorrelation mechanism. Introducing the flows (2), the approximation for the micro-probability is

$$\begin{eqnarray}&&{P}^{{\tau }_{d}}(x,y,t)={n}_{{\rm{tr}}}G(x,y-{V}_{d}t;{\bf{S}})+{n}_{f}G(x,y-{V}_{f}t;{{\bf{S}}}^{\prime }),\end{eqnarray} \tag{ 21 }$$

where $G({\bf{x}};{\bf{S}})$ is the two-dimensional Gaussian distribution with dispersion ${\bf{S}}=({S}_{x},{S}_{y}).$ We note that this probability is non-Gaussian.

The separation of the distribution and the existence of ion flows in drift type turbulence are confirmed by numerical simulations [24].

We have studied in [15] drift type test modes on turbulent plasmas. The frequencies and the growth rates are obtained as functions of the characteristics of the turbulence. They show that ion diffusion (generated by the random component of the trajectories) has a stabilizing effect, while ion trapping (the quasi-coherent component) leads to strong nonlinear effects: increase of the correlation lengths, nonlinear damping of the drift modes and generation of zonal flow modes. The strength of each of these processes depends on the parameters of the turbulence.

We use the basic description of the (universal) drift turbulence provided by the drift kinetic equation in the collisionless and low density limit. We consider a plasma confined by an uniform magnetic field B, taken along the ${{\bf{e}}}_{3}$ axis in a rectagular system of coordinates. The density gradient with characteristic length L_n is taken along ${{\bf{e}}}_{1}$. The drift wave instability that is produced by the electron kinetic effects and the ion polarization drift velocity is studied. We have chosen this simple instability for this first study of the effects produced by trajectory trapping or eddying in order to avoid interferences with other processes. The stabilizing effect produced by a sheared magnetic field on bound eigenmodes [44], or, alternatively, the instability for wavelengths below the ion Larmor scale [45] are not considered here. The effects of the magnetic shear on transport were studied using DTM in [46], which enables the extension of the present study to more complex confining configuration.

The solution of the dispersion relation for quiescent plasma is (see [41])

$$\begin{eqnarray}&&\omega =\displaystyle \frac{{k}_{2}{V}_{* }}{1+{k}^{2}{\rho }_{s}^{2}},\ \gamma ={\gamma }_{0}\omega ({k}_{2}{V}_{* }-\omega ),\end{eqnarray} \tag{ 22 }$$

where ω is the frequency of the mode with wave number ${\bf{k}}=[{k}_{1},{k}_{2},{k}_{z}]$, $k=\sqrt{{k}_{1}^{2}+{k}_{2}^{2}}$, ${{\bf{V}}}_{* }={V}_{* }{{\bf{e}}}_{2}$, ${V}_{* }={T}_{e}/({{eBL}}_{n})={\rho }_{s}{c}_{s}/{L}_{n}$ is the diamagnetic velocity, ${\rho }_{s}={c}_{s}/{{\rm{\Omega }}}_{i},{c}_{s}=\sqrt{{T}_{e}/{mi}},$ T_e is the electron temperature, m_i is the ion mass, e is the absolute value of electron charge and ${{\rm{\Omega }}}_{i}={eB}/{m}_{i}$ is the ion cyclotron frequency. The constant ${\gamma }_{0}$ that plays the role of drive of the instability is ${\gamma }_{0}=\sqrt{\pi /2}/(| {k}_{z}| {v}_{{Te}}),$ where ${v}_{{Te}}=\sqrt{{T}_{e}/{m}_{e}}.$ The drift waves are unstable ($\gamma \gt 0$) when the phase velocity $\omega /{k}_{z}\lt {V}_{* }$.

Electron and ion responses to a small perturbation $\delta \phi$ applied on the turbulent state with potential ϕ are determined. They lead to a modified propagator

$$\begin{eqnarray}&&{\overline{{\rm{\Pi }}}}^{{i}}={\rm{i}}{\int }_{-\infty }^{t}{\rm{d}}\tau \ M(\tau )\ \exp [-{\rm{i}}\omega (\tau -t)],\end{eqnarray} \tag{ 23 }$$

which includes the effects of the turbulence in the function $M(\tau )$ defined by

$$\begin{eqnarray}&&M(\tau )\equiv \langle \exp \left[{\rm{i}}{\bf{k}}\cdot ({\bf{x}}(\tau )-{\bf{x}})-{\int }_{\tau }^{t}{\rm{d}}{\tau }^{\prime }\ {\boldsymbol{\nabla }}\cdot {{\bf{u}}}_{p}({\bf{x}}({\tau }^{\prime }))\right]\rangle ,\end{eqnarray} \tag{ 24 }$$

where the average $\langle \rangle$ is over the ion trajectories in the stochastic potential $\phi .$ The polarization drift ${{\bf{u}}}_{p}$ determines a compressibility term due to its divergence ${\boldsymbol{\nabla }}\cdot {{\bf{u}}}_{p}=-{\partial }_{t}{\rm{\Delta }}\phi /({{\rm{\Omega }}}_{i}B).$ The dispersion relation (quasi-neutrality condition) is the same as in the quiescent plasma in terms of the propagator

$$\begin{eqnarray}&&-({k}_{2}{V}_{* e}-\omega {\rho }_{s}^{2}{k}^{2}){\overline{{\rm{\Pi }}}}^{i}=1+{\rm{i}}\sqrt{\displaystyle \frac{\pi }{2}}\displaystyle \frac{\omega -{k}_{2}{V}_{* e}}{| {k}_{z}| {v}_{{Te}}}\end{eqnarray} \tag{ 25 }$$

(see [15] for details).

The micro-probability of the displacements can be Gaussian or non-Gaussian, depending on the spectrum parameters and on the decorrelation time ${\tau }_{d}.$ The simple approximation (21) includes the main features of the micro-probability. It has the advantage of enabling the derivation of simple analytical expressions for the function M (24), the renormalized propagator (23) and eventually to solve the dispersion relation (25) for the complex frequency of the drift modes (see [15]).

The drift modes have modified frequency and growth rate compared to the quiescent plasmas (22), which depend on the characteristics of the trajectory structures and on the diffusion coefficients

$$\begin{eqnarray}&&{\overline{\omega }}_{d}=\displaystyle \frac{1}{2}\left[{\overline{\omega }}_{0}+(1-n){\overline{k}}_{2}+{sg}\sqrt{{\overline{\omega }}_{n}^{2}+\displaystyle \frac{4n{\overline{k}}_{2}^{2}}{1+{ \mathcal F }{\overline{k}}^{2}}}\right],\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{\overline{\gamma }}_{d}=\displaystyle \frac{{\overline{\gamma }}_{0}({\overline{\omega }}_{d}+n{\overline{k}}_{2})[(1-n){\overline{k}}_{2}-{\overline{\omega }}_{d}]-{n}_{f}{\overline{k}}_{i}^{2}{\overline{D}}_{i}}{{[(1-n){\overline{k}}_{2}-{\overline{\omega }}_{d}]}^{2}(1+{ \mathcal F }{\overline{k}}^{2})+n{\overline{k}}_{2}^{2}}{({\overline{k}}_{2}-{\overline{\omega }}_{d})}^{2},\end{eqnarray} \tag{ 27 }$$

where ${\overline{\omega }}_{n}={\overline{\omega }}_{0}+(n-1){\overline{k}}_{2},{n={n}_{tr}/1-{n}_{tr},\overline{\omega }}_{0}={ \mathcal F }{\overline{k}}_{2}/(1+{ \mathcal F }{\overline{k}}^{2})$ is the frequency obtained for $n\to 0,{ \mathcal F }=\exp (-{s}_{i}^{2}{k}_{i}^{2}/2)$ is a factor that depends on the sizes of the trajectory structures, ${sg}={\rm{sign}}({\overline{\omega }}_{n})$ and ${\overline{\gamma }}_{0}={\gamma }_{0}{c}_{s}/{L}_{n}.$ Usual normalized quantities ${\overline{k}}_{i}={k}_{i}{\rho }_{s},\overline{\omega }=\omega {L}_{n}/{c}_{s},\overline{\gamma }=\gamma {L}_{n}/{c}_{s}$ are used.

The expressions for ${\overline{\omega }}_{d}$ and ${\overline{\gamma }}_{d}$ show that at small amplitude, when the trajectories are random and $n=0,{ \mathcal F }=1,$ the background turbulence has a stabilizing effect that consists of a negative term in the growth rate $-{\overline{k}}_{i}^{2}{\overline{D}}_{i}$ [16]. It determines the damping of the large k modes and the narrowing of the spectrum, which corresponds to the increase of the correlation lengths ${\lambda }_{i}.$ This effect persists at large amplitudes, but it is modified by the quasi-coherent component of the ion trajectories, which introduces the factors seen in equation (27) and changes the diffusion coefficients.

The quasi-coherent structures of the ion trajectories influence the modes through the factor ${ \mathcal F },$ while the flows generated in the moving potential (2) are represented by n in equations (26) and (27).

The effect of the structures can be understood by taking n = 0 in these equations, which become

$$\begin{eqnarray}&&{\overline{\omega }}_{d}={\overline{\omega }}_{0}=\displaystyle \frac{{ \mathcal F }{\overline{k}}_{2}}{1+{ \mathcal F }{\overline{k}}^{2}},\ {\overline{\gamma }}_{d}={\overline{\gamma }}_{0}{\overline{\omega }}_{d}({\overline{k}}_{2}-{\overline{\omega }}_{d})-{\overline{k}}_{i}^{2}{\overline{D}}_{i}.\end{eqnarray} \tag{ 28 }$$

The factor ${ \mathcal F }$ determines the decrease of ${\overline{\omega }}_{d}$ and the displacement of its maximum from ${\overline{k}}_{i}\cong 1$ to ${\overline{k}}_{i}\cong 1/{s}_{i}.$ The maximum of ${\overline{\gamma }}_{d}$ moves toward smaller values of ${\overline{k}}_{2}$ as s_i increase. The effect is the increase of the correlation lengths accompanied by the decrease of the diffusive damping due to the decrease of $\overline{k}.$ The physical reason for this behavior is the eddying of the ion trajectories that eliminates large $\overline{k}$ modes by averaging.

The ion flows represented by finite values of n in equations (26) and (27) formally determine a jump of ${\overline{\gamma }}_{d}$ to negative values in the range $\overline{k}\lt {k}_{d},$ where k_d is the solution of ${\overline{\omega }}_{n}=0.$ k_d increases as n increases until $n\geqslant 1$ where ${\overline{\omega }}_{n}=0$ has no solution and sg = 1 in equation (26). The opposite flows (of the trapped ions that have ${V}_{{\rm{tr}}}=1$ and of the free ions with ${V}_{f}=-n$) appear in equation (27). The instability condition that yields from this equation is ${V}_{f}\lt {\overline{\omega }}_{d}/{\overline{k}}_{2}\lt {V}_{{\rm{tr}}}+{V}_{f}$. The condition ${\overline{\omega }}_{d}/{\overline{k}}_{2}\lt 1-n$ is not fulfilled for $n\lt 1$ at small $\overline{k}$ and for $n\gt 1$ at any $\overline{k}.$ Thus, the drift modes are damped in these conditions. The physical reason is the advection of the trapped ions with the potential that moves with the diamagnetic velocity. The effective diamagnetic velocity of this part of ions is zero, which corresponds to an effective density gradient equal to zero. The drift waves and instabilities do not exist in this case and ${\overline{\gamma }}_{d}\lt 0$ in equation (27). The large $\overline{k}$ modes have the phase velocity ${\overline{\omega }}_{d}/{\overline{k}}_{2}\cong -n,$ which is in the instability domain. At $n\gt 1,$ the trapped ions dominate leading to the damping of all modes.

The compressibility term determines unstable modes completely different from the drift modes. They have ${k}_{2}=0$ and very small frequencies

$$\begin{eqnarray}&&{\overline{\omega }}_{{zf}}=-{\overline{k}}_{1}\overline{a}\displaystyle \frac{1+{ \mathcal F }{\overline{k}}_{1}^{2}{n}_{f}}{1+{ \mathcal F }{\overline{k}}_{1}^{2}},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\overline{\gamma }}_{{zf}}=\displaystyle \frac{{n}_{{\rm{tr}}}{ \mathcal F }{\overline{k}}_{1}^{2}[{\overline{\gamma }}_{0}{\overline{\omega }}_{{zf}}^{2}-{n}_{f}{ \mathcal F }{\overline{k}}_{1}^{4}{\overline{D}}_{1}]}{(1+{n}_{f}{ \mathcal F }{\overline{k}}_{1}^{2})(1+{ \mathcal F }{\overline{k}}_{1}^{2})},\end{eqnarray} \tag{ 30 }$$

where $\overline{a}=a/{V}_{* }.$ These unstable modes that are named zonal flow modes are the consequence of trapping combined with the polarization drift. The compressibility term in the function M determines correlations with the displacements

$$\begin{eqnarray}&&{L}_{i}(\tau )=-\displaystyle \frac{1}{{{\rm{\Omega }}}_{i}B}{\int }_{\tau }^{t}{\rm{d}}{\tau }^{\prime }\langle {x}_{i}(t)\,{\partial }_{{\tau }^{\prime }}{\rm{\Delta }}\phi ({\bf{x}}({\tau }^{\prime }))\rangle ,\end{eqnarray} \tag{ 31 }$$

which, as evaluated in [15], can be significant for i = 1 (in the direction of the density gradient) for the trapped ions. It can be approximated by

$$\begin{eqnarray}&&{L}_{1}(\tau ,t)\cong a(t-\tau ),\ a=2{\partial }_{y}^{2}{\rm{\Delta }}E({\bf{0}})\displaystyle \frac{{\tau }_{{fl}}{V}_{* }}{{{\rm{\Omega }}}_{i}{B}^{2}}.\end{eqnarray} \tag{ 32 }$$

This correlation determines an average velocity for the trapped ions. One can see that when trapping is negligible (${n}_{{\rm{tr}}}\cong 0),{\omega }_{{zf}}=-{k}_{1}a$ and ${\gamma }_{{zf}}=0,$ and that ${\omega }_{{zf}},{\gamma }_{{zf}}=0$ for $\overline{a}=0.$

The evolution of turbulence has an initial stage that depends on the initial condition for the spectrum. However, for all physically reasonable conditions, the shape of the spectrum changes and it always develops two symmetrical maxima in ${k}_{2}=\pm {k}_{0}$ and ${k}_{1}=0.$ They are produced by the diffusive damping of the modes, which acts very strongly on large k₂ and ${k}_{2}\cong 0$ domain and makes $\gamma \lt 0$ even at small values of the diffusion coefficient ${D}_{y}.$ The growth rate in the quasilinear limit that can be obtained from equation (27) for n = 0 and ${ \mathcal F }=1$ at ${k}_{1}=0$ is

$$\begin{eqnarray}&&{\overline{\gamma }}_{d}={\overline{k}}_{2}^{2}\left({\gamma }_{0}\displaystyle \frac{{\overline{k}}_{2}^{2}}{{(1+{\overline{k}}_{2}^{2})}^{2}}-{\overline{D}}_{2}\right).\end{eqnarray} \tag{ 33 }$$

The parentheses becomes negative in the limits of both large and small ${\overline{k}}_{2}.$ The diffusive decay at large $| {k}_{2}|$ combined with the growth rate that decay to zero at small $| {k}_{2}|$ determine the generation of the two peaks in the spectrum. This leads to the increase of the anisotropy of the turbulence. The amplitude of the normalized velocity V₁ increases while V₂ decreases in this stage (see the small time evolution in the figures 4(a) and 5(a)). The cause of this behavior is the increase of the dominant wave number ${k}_{2}^{0}$ and of the correlation lengths ${\lambda }_{i}$ (figures 4(b) and 5(b)). The amplitude of the potential fluctuations represented by ${K}_{* }^{\prime }$ increases exponentially.

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It is interesting to note that, besides the large stochastic velocity along the density gradient (${V}_{1}\gg {V}_{2}),$ the diffusion coefficient is smaller (${D}_{1}\lt {D}_{2}).$ This behavior is the effect of the special shape of the EC of the drift turbulence that yields decreasing diffusion coefficient D₁ at large ${\tau }_{d}$ even in the quasilinear regime.

The initial stage is not a regime of completely independent evolution of the modes. The influence of the background turbulence that consists of the diffusive damping is not negligible at the very small values that correspond to this stage. It determines the change of the spectrum shape.

Typical evolution at weak drive is presented in figure 4, where ${\overline{\gamma }}_{0}=1.$ After the initial stage, the diffusive damping becomes stronger and it determines important changes in the evolution. The amplitude of the potential ${K}_{* }^{\prime }$ continues to increase but with a slower rate: the exponential evolution is replaced by a roughly linear increase (figure 4(a), the back curve for ${K}_{* }^{\prime }$). Both components of the normalized velocity decrease. They have a fast decay followed by the tendency of saturation (figure 4(a)). This is the effect of diffusive damping. The diffusion coefficients increase as ${({K}_{* }^{\prime })}^{2}$ and they produce the decrease of the growth rate of the drift modes.

Besides this well known effect of damping, the diffusion determines significant changes of the shape of the spectrum. The width of the two peaks decreases in both directions, which determines the increase of the correlation lengths. It also determines the displacement of the position of the peaks toward smaller wave numbers. The evolution of the spectrum parameters is shown in figure 4(b).

The condition for the transition to the trapping regime (${K}_{* }^{\prime }\gt {\lambda }_{1})$ is not attained in this case since, as seen in figure 4(a), the increase rate is larger for ${\lambda }_{1}$ than for ${K}_{* }^{\prime }.$ The structures of trajectories are practically absent, as seen in figure 4(c) where ${n}_{{\rm{tr}}}\lt {10}^{-3}$ and the ${s}_{i}\lt {10}^{-2}.$ The normalized diffusion coefficients saturate at ${\overline{D}}_{1}\cong 0.03$, ${\overline{D}}_{2}\cong 0.2.$

Drift turbulence evolution at weak drive is essentially determined by ion diffusion, which reduces the growth rate and determines a significant change of the spectrum that prevents the development of the coherent effects produced by trapping.

The increase of the drive parameter determines a strong effect on turbulence evolution, as seen in figures 5–7, where ${\overline{\gamma }}_{0}=5.$ This large values of ${\overline{\gamma }}_{0}$ lead, since the initial stage of the evolution, to a faster increase of the amplitude of the normalized potential ${K}_{* }^{\prime }$. Ion diffusion determines the same effects as in the weak drive case, but the fast exponential increase of the potential up to large values makes ${K}_{* }^{\prime }\gt {\lambda }_{1},$ and determines ion trapping.

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Three time intervals with different characteristics of the evolution can be identified in the typical example presented in figure 5: $t\lt 2.5,2.5\lt t\lt 6$ and $t\gt 6.$ The first interval corresponds to the initial stage of quasi-independent development of the modes. The other two intervals are influence by the trapping process, but in different ways.

As seen in figure 5(a), the evolution enters in the nonlinear regime ${K}_{* }^{\prime }\gt {\lambda }_{1}$ at the end of the initial stage determining ion trajectory trapping (figure 5(c)). The potential ${K}_{* }^{\prime }$ has a fast transitory decay, followed by a regime of continuous increase of the average ${K}_{* }^{\prime }$ (figure 5(a), black curve). During this stage, ${K}_{* }^{\prime }\gt {\lambda }_{1}$ and both parameters increase with roughly the same rate. The ratio ${K}_{* }^{\prime }/{\lambda }_{1}$ is approximately constant, which corresponds to small variation of the parameters of the trajectory structures (figure 5(c), the time interval $[4,\ 6]$). The spectrum and the EC are strongly modified during this interval. The correlation lengths ${\lambda }_{i}$ increase significantly and the dominant wave number ${k}_{2}^{0}$ decreases (figure 5(b)). The evolution of the velocity is different from that of the potential as reflected in the normalized velocities ${V}_{1},\,{V}_{2},$ which are time dependent functions (see figure 5(a)). Their evolution is actually determined by the change of the characteristics of the EC, because ${V}_{1}^{2}\cong {({k}_{2}^{0})}^{2}+3/{\lambda }_{2}^{2}$ and ${V}_{2}^{2}\cong 1/{\lambda }_{1}^{2}.$

A different evolution regime appears at $t\gt 6$ in the example presented in figure 5. The normalized potential drops and it saturates (figure 5(a)). The normalized velocity V₁ saturates and V₂ increases. The characteristics of the EC, ${\lambda }_{i}$ and ${k}_{2}^{0},$ also saturate (figure 5(b)). The condition for the existence of trajectory structures is no more fulfilled (${K}_{* }^{\prime }$ is significantly smaller than ${\lambda }_{1})$. However, trapping becomes much stronger (n_tr and the size of the structures increase, as seen in figure 5(c)).

This rather non-trivial evolution of the drift turbulence can be understood by analyzing the nonlinear effects that influence the growth rate (27): the ion diffusion, the formation of ion trajectory structures and flows.

The evolution of the diffusion coefficients D_i is presented in figure 6, where the amplitudes of the drift turbulence ${{\rm{\Phi }}}^{2}$ and of the zonal flow potential ${{\rm{\Phi }}}_{{zf}}^{2}$ are also shown. One can see that the zonal flow modes become unstable when trapping appears and they grow exponentially until $t\cong 6.$ Their amplitude is negligible compared to that of the drift turbulence during this time interval. Thus, it is not expected an important role of the zonal flow modes at this stage.

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The diffusion coefficient D₂ is much larger than ${D}_{1}.$ It has a very large increase rate during the first transition into the nonlinear regime. The cause is ion trapping in the potential that drifts with the diamagnetic velocity, which determines a strong amplification of the diffusion along ${V}_{d}{{\bf{e}}}_{2},$ as discusses in section 4.1 and represented in figure 2. The strong diffusion determines the transitory decay of the potential ${K}_{* }^{\prime }$ under the nonlinearity limit. Consequently, the structures of the ion trajectories are destroyed. The fraction of trapped trajectories and the size of the structures decrease when ${K}_{* }^{\prime }$ decreases (figure 3, for $t\cong 2.7$). The diffusion coefficient D₂ decreases due to both effects (the decrease of Φ and the reduction of structures). One can see in figures 6 and 5(c) that transitory peaks and decays appear around t = 2.7 for ${D}_{2},{\rm{\Phi }},$ n_tr and s_i. After that, the potential increases again and reaches a higher level in the nonlinear regime. The diffusion coefficients D₂ recovers large values, but without driving the decay process. The strong diffusion is supported by the potential, which increases at this stage instead of having a fast decay as in the previous event. The cause of the different behavior is the evolution of the parameters of the spectrum. The dominant wave number ${k}_{2}^{0}$ (maxima of the spectrum) and the width of the spectrum $\delta {k}_{i}\cong \sqrt{2\pi }/{\lambda }_{i}$ decrease as seen in figure 5(b). As a consequence, the damping effect of the diffusion ${k}_{i}^{2}{D}_{i}$ is weaker at small wave numbers and the growth rate of the modes remains positive.

The third time interval in the evolution is characterized by the presence of the zonal flow modes at amplitudes comparable to ${\rm{\Phi }}.$ The drift turbulence decays and saturates while the zonal flow amplitude increases. The EC of the drift potential also saturates (${\lambda }_{i}$ and ${k}_{2}^{0}$ are roughly constant during this interval). The increase of V₂ shown in figure 5(a) is determined by the growing contribution of the zonal flow modes.

Zonal flow provide the explanation for the increase of trapping (figure 5(c)) that seems paradoxical in the condition of the decay of ${K}_{* }^{\prime }$ below the nonlinearity limit. The structure of the potential contour lines is modified by the zonal flow potential, which determines larger structures that capture the ions released from the micro-confinement determined by the (small scale) drift turbulence. The fraction of trapped ions and the size of the structures increase in correlation with the increase of ${{\rm{\Phi }}}_{{zf}}$ (figures 5(c) and 6). A deformation of the trajectory structures appears due to the stronger increase of s₂ that of ${s}_{1}.$ They are initially elongated in the gradient direction and eventually they acquire a shape dominated by the zonal flow modes. The fraction of trapped ions strongly increases and, at $t\cong 7,$ it overcomes the value ${n}_{{\rm{tr}}}=1/2$ ($n=1)$ that corresponds to negative growth rate of the drift modes (27) even in the absence of diffusion.

The diffusion coefficients decrease in this stage due to the decay of the potential ${\rm{\Phi }}.$ The increase of the diffusion D₂ determined by the zonal flow contribution discussed in [29] is not observed here because the negative growth rate leads to the transition in the quasilinear regime and the condition (${{\rm{\Phi }}}_{{zf}}={\rm{\Phi }}\gt {\lambda }_{1}$) for this effect is not fulfilled. The decay of the drift turbulence is caused by the zonal flow modes through the amplification of trapping rather than by an increased diffusion.

The evolution of the spectrum of the drift turbulence is presented in figure 7(a), which shows the function $S(0,{k}_{2})$ at time intervals ${\rm{\Delta }}t=1.$ The spectrum becomes narrower as time increases and its maxima move to smaller values of ${k}_{2}.$

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The EC of the potential is shown in figure 7(b). The oscillatory dependence on x₂ is determined by the narrow peaks of the spectrum. It determines the small diffusion along the density gradient ${D}_{1}\lt {D}_{2}$ despite the amplitude of the stochastic velocity that is much larger in this direction (${V}_{1}\gt {V}_{2}).$

The spectrum of the zonal flow modes ${S}_{{zf}}({k}_{1})$ is shown in figure 7(c) at the same time moments as $S(0,{k}_{2}).$ The two spectra have similar shapes, with the difference that the zonal flow modes have small tails at large k₁ while a very strong cut appears for the drift modes at large k₂. This is the consequence of the diffusive damping that is weak in the case of the zonal flow modes, as seen in equation (30). The dominant wave number of the zonal flow modes is around ${k}_{{zf}}\cong 1,$ and it has a weak decrease with time (as also seen in figure 5(b), black curve).

We note that the nonlinear processes found here are related to ion stochastic trapping in the structure of the potential. The growth and saturation of the zonal flows in drift turbulence were intensively studied in the last decades [7], but ion eddying was not identified as a source of zonal flow modes. It is completely different from the trapping of the drift modes represented as quasi-particles in the wave-kinetic approach (see the very recent papers [47, 48]).