Effect of Symmetry/Asymmetry of Shear Rotation of a Plasma Column in a Radial Electric Field on the Level of Turbulent Density Fluctuations

Abstract

The influence of the properties of the profile of a radial static electric field E(r) on the evolution of an unstable ion temperature–gradient (ITG) drift wave in a nonuniformly rotating plasma column in a magnetic field is studied. The effect of symmetry on the decrease in the level of turbulent fluctuations, which are associated with the limiting state of the ITG wave during its destruction, is discussed. The level of turbulence is estimated in the framework of approximation of finite amplitudes depending on the electric field structure. It is shown that the maximum decrease in the level of fluctuations occurs with a symmetrical configuration of the radial electric field.

1. Introduction

The drift turbulence of magnetically confined plasma, on the one hand, differs from the hydrodynamic turbulence of a neutral fluid. In the first case, the fluctuations of all quantities are small, i.e., drift turbulence is small-scale. On the other hand, the origin of pulsations in both cases is due to similar reasons, namely, the growth of unstable waves. However, the nature of these instabilities is different. Drift instabilities are of an electromagnetic nature (at low plasma pressure in a strong magnetic field, they are described in the electrostatic approximation). What is common in both cases is that turbulence is associated with the characteristics of the shear flow. In the case of plasma, such a shear flow (in the case under consideration, shear rotation of the plasma column) is formed by a radial static electric field, which causes the well-known effect of the E × B drift (E is the radial electric field, B is the magnetic induction). Therefore, the analysis presented in this paper is of a general nature, i.e., some conclusions can be extended not only to the considered plasma configurations, but also to some hydrodynamic flows [1,2].

Drift turbulence in magnetized plasma determines the transport properties of such a plasma [3,4]. Therefore, the search for methods to reduce the level of turbulent fluctuations is important for improving the quality of plasma confinement in magnetic traps, including the well-known tokamaks and stellarators [5,6]. Fundamental discoveries in this direction are associated with the so-called H-mode, the improved confinement mode [7,8]. One of the features of the improved regime is the partial suppression of small-scale plasma turbulence and a significant decrease in the level of plasma density fluctuations. Studies have shown that the improved H-mode is accompanied not only by the formation of an inhomogeneous radial electric field, but also by the inhomogeneous rotation of the plasma column associated with it. This rotation is considered to be the cause of partial suppression of turbulence and a decrease in the level of plasma density fluctuations in the wavelength range corresponding to the ion temperature-gradient (ITG) drift mode [9,10].

One way to achieve improved performance is to apply neutral beam injection (NBI) heating [11,12]. Having established a direct connection between the radial static electric field formed in the plasma and the H-mode, another method was used in the experiments to obtain it by directly creating an inhomogeneous radial electric field by introducing an electrode with an adjustable potential into the edge of the plasma. This method was used on CCT [13] and TEXTOR [14] tokamaks. On a GAMMA-10 mirror trap [15], an inhomogeneous field was created by adjusting the potential on segmented plates installed at both ends of the setup.

Different ways to achieve the H-mode (NBI, electrodes, segmented plates) form different shapes of the radial electric field profiles and, accordingly, different configurations.

The purpose of this work is to study the effect of various configurations of the radial electric field on the possible level of turbulent fluctuations in plasma density associated with the development of ITG instability. We consider the stage of initial perturbation growth from the small amplitude harmonic drift wave to its conventional decay. Such decay does not mean the disappearance of the wave, but such strong distortion of the wave compared to the original harmonic form, so it causes the non-uniformity of the environment and makes the further existence of the wave impossible. The corresponding amplitude value is typical in terms of limitation of the linear growth of the perturbation and the transition to the nonlinear stage.

2. The Model and Results

Previously, we showed that the inhomogeneous rotation of the plasma, due to shear, affects the unstable drift ITG wave already at the initial stage of its formation [16,17]. In the present study, we use this result and extend the model to the case of an arbitrary electric field profile. Below, the resulting model and explanations for it are presented.

The wave is considered in the finite amplitude approximation under the following assumptions. It is assumed that the wave propagates in the plasma layer along the Oy-axis in local Cartesian coordinates. Figure 1 presents the geometry of the problem of the drift wave propagation in the plasma and the motion of a medium in the xOy-plane. The magnetic field induction B, directed along the Oz-axis, and the external (with respect to the wave) radial density gradient (dn₀/dx) and temperature gradient (dT₀/dx) are set constant within the layer and unchanged in time. Next, in the xOy-plane, a thin curved plasma layer was considered, characterized by a constant density n₀ and plasma temperature T₀, which reflects the motion of wave elements and is associated with its profile.

Each wave element (Figure 1) has displacement $\tilde{x}$ along Ox-axis and displacement $\tilde{y}$ along Oy-axis. Radial density gradient dn₀/dx (absolute value) is assumed to be constant, so density deviation δn is related to the displacement $\tilde{x}$ as

$$\delta n=-\tilde{x}(d{n}_0/dx).$$

(1)

The initial stage of the wave was set in the following form:

$$\tilde{x}(y_0)=x_0\sin (k_y{y}_0),$$

(2)

$$\tilde{y}(y_0)=y_0,$$

(3)

where k_y = (2π)/λ is the wavenumber, λ is the wavelength, x₀ is the initial displacement amplitude (x₀ << λ), y₀ is auxiliary parameter that characterizes the initial position of the wave in Lagrangian variables.

It is assumed that the arising wave is associated with ITG instability. Classic linear theory of ITG instability [18] implies infinitesimal amplitude, the increase of which is characterized by growth rate $\gamma =\frac{1}{\delta n}\frac{\partial \delta n}{\partial t}$. A typical value of the growth rate of the ITG instability is proportional to diamagnetic drift frequency, which leads to the following characteristic dependence [19]: $\gamma \approx C{v}_{Ti}/L_n$, where v_Ti in the thermal velocity of the ions, $L_n=n_0/(d{n}_0/dx)$ is so-called gradient length, C is a numerical factor of order 0.1..0.2 for the conditions considered here. The numerical value of γ under the conditions under consideration is of the order of 10⁶ s⁻¹. In our model, the growth rate γ is assumed to be constant so it is affects the velocity v_x of wave element as $v_x=\gamma \tilde{x}(y_0,t)$.

In the theory of ITG instability, it is assumed that the motion of electrons ensures the fulfillment of the Boltzmann relation for their distribution in the electric field of the wave:

$$n=n_0+\delta n=n_0\exp \left(\frac{e\varphi }{k_B{T}_0}\right),$$

(4)

where n is the electron density, n₀ is unperturbed density, e is the electron charge, φ is the deviation of the electric potential, k_B is the Boltzmann constant.

Linearization of this relation leads to the dependence

$$\frac{\delta n}{n_0}=\frac{e\varphi }{k_B{T}_0}.$$

(5)

With the development of ITG instability, the quasi-neutrality of the oscillating plasma is preserved, since these are low-frequency and long-wave waves. Because the reason of the latter, the use of the drift approximation is valid for waves of this class. In a magnetic field, charged plasma particles move along the so-called Larmor circles, the centers of which (the so-called guiding centers) drift at constant speeds. In crossed electric and magnetic fields, a drift of the guiding center of the Larmor circle is performed by the E × B drift velocity

V_E = E × B/B²,

(6)

where E = −grad φ is the static electric field, B is the magnetic field.

In the coordinates we have chosen, the E × B drift affects the velocity component v_x of the wave elements as follows

$$v_x(y_0,t)=\frac{E_y}{B}=-\frac{1}{B}\frac{\partial \varphi (y_0,t)}{\partial y}.$$

(7)

Applying Equation (4) it can be established that the fluctuations form the velocity v_x changing according to

$$v_x(y_0,t)=\frac{k_B{T}_0}{eB{n}_0}\frac{\partial \delta n(y_0,t)}{\partial y}.$$

(8)

The plasma rotation velocity profile determines the velocity v_y, which is considered as some given function f(x). Relationships are obtained for determining the waveform, expressed as a displacement $\tilde{x}(y_0,t)$ along the Ox-axis and a displacement $\tilde{y}(y_0,t)$ along the Oy-axis, given through the parameter y₀ and time t,

$$v_x(y_0,t)=\frac{\partial \tilde{x}(y_0,t)}{\partial t}=\gamma \tilde{x}(y_0,t)-\frac{k_B{T}_0}{qB{n}_0}\frac{d\delta n(y_0,t)}{dy},$$

(9)

$$v_y(y_0,t)=f(x).$$

(10)

It was shown previously [17] that due to the growth of the wave amplitude in plasma, the growth of local gradients in plasma leads to violation of the initial conditions for the appearance and existence of ITG instability. The linear theory of ITG instability works while the amplitudes in the plasma are small or in other words it implies amplitudes tending to zero. Therefore, at small but finite amplitudes, a situation may arise in which the conditions for the onset of ITG instability predicted by the linear theory are violated. Of course, our statement is an assumption, which partly justified by the results of our work. The transition to essentially finite amplitudes means a nonlinear stage in the development of a drift wave, which is difficult from the point of view rigorous physical consideration yet. ITG instability arises under the condition that the plasma has a radial density gradient (dn₀/dx) and a temperature gradient (dT₀/dx). Under the conditions of the problem, with an increase in the wave amplitude, an increase in the local value of the density gradient d(δn)/dy is inevitable, which at some point in time reaches a value comparable to the initial value (dn₀/dx). As a result, a condition was introduced to limit the growth of the wave, which means the equality of the average effective value <d(δn)/dy> of the derivative of the wave form to the given value dn₀/dx of gradient of unperturbed density, i.e.,

<d(δn)/dy> = dn₀/dx.

(11)

The average effective value <d(δn)/dy> of the derivative is calculated over the entire wave structure. Since d(δn)/dy can take both positive and negative values, we propose to calculate its effective value over the entire wavelength as

$$<d(\delta n)/dy>=\sqrt{{\frac{1}{\lambda }}_{\underset{0}{\int }}^{\lambda }{(d(\delta n)/dy)}^{2}dy}.$$

(12)

The shear effects caused by the inhomogeneous radial electric field and the velocity v_y lead to stronger local gradient d(δn)/dy and the violation of the conditions occurs at lower values of the wave amplitude compared to the case when there is no non-uniformity (i.e., dv_y/dx ≠ 0). As a consequence, smaller amplitudes of plasma density fluctuations are observed.

Usually, theoretical studies assume a linear symmetric electric field profile and the associated velocity profile [11]. In experiments, profiles are obtained that can be considered linear. In particular, such profiles were observed on TEXTOR [14], DIII-D [20], and T-10 [21] tokamaks. More complex profiles were obtained on the ASDEX-upgrade [11], JET [12], CCT [13] tokamaks and a number of experiments on the T-10 tokamak [21] and the W7-AS stellarator [11]. In experiments, plasma rotation is usually associated with the E × B drift velocity (Equation (6)). In the above-mentioned experiments on different devices, different profiles of the radial electric field and the associated profiles of the plasma rotation velocity V_E were obtained. Notably, the velocity profile V_E, experimentally obtained on the TEXTOR tokamak [22], which has certain pronounced features of interest. Figure 2 shows a schematic draw of the profile, which qualitatively reflects what is observed in mentioned experiment. We proceed from this experiment, since there are characteristic profile dependences v_y = V_E = f(x) in different parts of the velocity profile V_E, which we consider in more details.

Several theoretical works were analyzed in [22] for the case of a linear symmetric electric field profile (Figure 2, case 1). A generalized approximation of the relative level of density fluctuations was presented in the following form:

$$\Theta =\frac{{\left(\delta n/n_0\right)}_{{\gamma }_S}^2}{{\left(\delta n/n_0\right)}_0^2}\approx \frac{1}{1+K{\left({\gamma }_S/\gamma \right)}^s}.$$

(13)

where <δn> is the rms value of density fluctuations, n₀ is the unperturbed plasma density, ${\left(\delta n/n_0\right)}_{{\gamma }_S}^2$ is the level of fluctuations at γ_S ≠ 0, ${\left(\delta n/n_0\right)}_0^2$ is the level of fluctuations at γ_S = 0, K = 1..2 (the estimates [17] led to a same-type dependence with K = 1.67), s = 2, γ_S = dV_E/dr is the inhomogeneity of the plasma shear flow velocity.

At the initial state amplitude is assumed to be infinitesimal, but during the growth, it is considered as finite. For quantitative estimates of growth, the linear growth rate of ITG mode is used. The influence of flow shear on the fluctuation level is determined by varying the ratio between the growth rate γ and shear parameter γ_S. Wave tilting is not considered; as for the drift waves, it would mean a discontinuity of the medium. It is assumed that the state called the decay corresponds to the close conditions. Turbulent fluctuation level is associated with amplitude value before the decay stage.

Of particular interest is the question of the influence of the sign of the velocity shear γ_S on the level of fluctuations (Figure 2, case 1). Similar experiments were carried out on the CCT [13] and TEXTOR [14,22] tokamaks and on the GAMMA-10 mirror trap [15]. In all these experimental works, it was found that the sign of the shear parameter γ_S does not matter in principle for the magnitude of the observed level of plasma density fluctuations and the conditions for achieving improved confinement. This was confirmed in theoretical works [17,22].

In our work, the calculations were performed as follows. The initial stage of the wave is set according to Equations (2) and (3) (given $\tilde{x}$ and $\tilde{y}$). The shift is related to the density deviation δn according to Equation (1). Further, the position of the waveform is determined by numerical methods at the following time points, using Equations (9) and (10) for the velocities of the wave elements v_x and v_y. The function v_y = V_E = f(x) is chosen based on the considered case of the profile velocity V_E, as shown in Figure 2. When calculating the level of density fluctuations Θ, we used the effective value of the density deviation δn/n₀ related to Equation (1) and defined as

$$\delta n/n_0=\frac{1}{n_0}\sqrt{{\frac{1}{\lambda }}_{\underset{0}{\int }}^{\lambda }\delta n^{2}dy}=\frac{d{n}_0/dx}{n_0}\sqrt{{\frac{1}{\lambda }}_{\underset{0}{\int }}^{\lambda }{\tilde{x}}^{2}dy}.$$

(14)

Figure 3 shows the wave profile shapes $\tilde{x}(y)$ obtained from in calculations at the moment of time corresponding to the achievement of condition presented by Equation (11). In the case 1 (Figure 2), i.e., for the velocity v_y = const·x (Figure 3a) and v_y = −const·x (Figure 3b), the waveform and velocity profile V_E are symmetrical with respect to the direction of wave propagation, so the sign of γ_S does not matter in the above dependence. Note that despite the fact that the sign of the velocity shear does not affect the decrease in the level of fluctuations, the radial plasma density profile can differ depending on the method of formation of the electric field profile.

For the plasma region corresponding to the limit of γ_S sign reversal (Figure 2, case 2), estimates of the level of fluctuations lead to the same result as in the case when the sign does not change, provided that the velocity gradient V_E changes its sign abruptly. Figure 3c shows the resulting wave profile. If we assume that the velocity V_E changes smoothly in the region where the sign of γ_S changes (Figure 3d), then higher values of the wave amplitude are obtained in the calculations, and the level of fluctuations Θ turns out to be higher. Moreover, if the region of smooth change in velocity V_E is wider, then the level Θ is even higher.

Next, we consider the velocity profile, for which the velocity gradient V_E is present only in one half-plane xOy (on the boundary of V_E change, Figure 2, case 3). The considered profile corresponds to the asymmetric configuration of the radial electric field. In this case, the wave propagates simultaneously in two parts of the flow of different types (Figure 3e). Calculations have shown that shear effects affect the waveform only in the region where the velocity gradient is present, despite the fact that the media elements oscillating across the wave repeatedly cross the boundary of the flow sub-regions. The approximation dependence of the relative level of density fluctuations in this case is similar to expression (10) with the parameters K = 1.33 and s = 2.5.

Two cases were considered: (i) a sharp transition and (ii) a smooth transition. A sharp transition means a break in the velocity profile at x = 0. With a smooth transition, the transition region is localized in a narrow section Δx. With such a velocity profile, even higher values of the wave amplitude are obtained in the calculations, and the level of fluctuations Θ turns out to be even higher. Moreover, as in the previous case, if the region of smooth change in velocity V_E is wider, then the level Θ is even higher.

Note that Equations (9) and (10) describe the velocities of the oscillating elements of the plasma. The streamlines are the trajectories of such elements. The corresponding velocity field can be represented by the stream function $\psi (x,y)$ which is related to the velocity components as follows: $v_x=-\partial \psi /\partial y$, $v_y=\partial \psi /\partial x$. It is convenient to consider streamlines in the reference frame of coordinates, moving together with a perturbation (drift wave), the phase velocity of which includes the E × B drift velocity V_E of the plasma in the x direction. In such a reference frame, the drift velocity V_E = 0 at some coordinate (here taken as x = 0). As an example, such streamlines are shown in Figure 4 for the case of Figure 3b, i.e., for linear symmetrical distribution of V_E, which in the chosen coordinate system has the following form: $V_E(x)={\gamma }_{S}x$. These streamlines are shown for the final stage of disturbance growth, i.e., for the point in time at which the condition expressed by Equation (11) is satisfied.

In Figure 5, for all the above cases with different velocity profile v_y = V_E shows the dependence of the relative level Θ of density fluctuations on the velocity shear related to the instability increment γ_S/γ. These mechanisms are in accordance with the well-known criterion γ_S > γ and results of theories [23].

Note that the presented analysis is based on the evolution of a single-mode perturbation, while really, many modes coexist in turbulent medium. The estimates of the perturbation amplitudes involve the use of some typical parameters of instability with a certain wave number k_y. Such an estimate gives a typical average value, not a spectrum.

3. Conclusions

In all existing theoretical works, including [11,22], only the linear shear flow profile formed by the linear profile of the radial electric field is considered, whereas, at present, a wide variety of electric field profiles has been obtained in experiments, which are interesting to study from the point of view of suppressing turbulence, or more specifically, reducing the level of density fluctuations.

The present study showed that in the case of a symmetrical configuration of the radial electric field, the calculations yield smaller values of the relative level of density fluctuations, and this case can be considered more attractive from the point of view of the possibility of suppressing small-scale turbulence associated with the propagation of ITG waves in the plasma.

The values of plasma density fluctuations directly depend on the value of the shear flow inhomogeneity parameter γ_S. The greater inhomogeneity leads the lower fluctuation amplitude level. For the considered symmetric configurations, it was found that the greatest effect of fluctuation limitation is achieved for linear profiles (γ_S = const). In the case of asymmetric configurations, the level of density fluctuations is higher compared to symmetrical configurations with the same level of maximum inhomogeneity.