The collisional drift wave instability in steep density gradient regimes

TL;DR

This paper investigates the collisional drift wave instability in steep density gradient regimes using a full-F gyro-fluid model, revealing significant corrections to previous approximations and identifying non-modal phenomena and localized turbulence.

Contribution

It introduces a full-F gyro-fluid model that relaxes the OB approximation, providing new insights into drift wave behavior in steep density gradients.

Findings

Corrections of order one to eigenvalues compared to OB approximations

Discovery of non-modal phenomena and localized growth patterns

Persistence of non-modal effects in nonlinear turbulence

Abstract

The collisional drift wave instability in a straight magnetic field configuration is studied within a full-F gyro-fluid model, which relaxes the Oberbeck-Boussinesq (OB) approximation. Accordingly, we focus our study on steep background density gradients. In this regime we report on corrections by factors of order one to the eigenvalue analysis of former OB approximated approaches as well as on spatially localised eigenfunctions, that contrast strongly with their OB approximated equivalent. Remarkably, non-modal phenomena arise for large density inhomogeneities and for all collisionalities. As a result, we find initial decay and non-modal growth of the free energy and radially localised and sheared growth patterns. The latter non-modal effect sustains even in the nonlinear regime in the form of radially localised turbulence or zonal flow amplitudes.

Equations

$$\frac{\partial }{\partial t}n+\nabla \cdot \left(n{\mathbf{u}}_E\right)$$

$$\frac{\partial }{\partial t}N+\nabla \cdot \left(N{\mathbf{U}}_E\right)$$

$$\nabla \cdot \left(\frac{N}{{\Omega }_0}\frac{{\nabla }_{\perp }\varphi }{B_0}\right)$$

$$\frac{\partial }{\partial t}n+\nabla \cdot \left(n{\mathbf{u}}_E\right)$$

$$\alpha :=\frac{T_{e0}{k}_{\parallel }^2}{{\eta }_{\parallel }{e}^2{n}_0{\Omega }_0}.$$

$$n:=n_G(1+\delta n),N:=n_G(1+\delta N).$$

$$L_n:=-{\left(\frac{\partial }{{\partial }_x}\ln (n_G(x)/n_0)\right)}^{-1},$$

$$\frac{\partial }{\partial t}\delta n+\frac{1}{B_0}\frac{1}{L_n}\frac{\partial }{\partial y}\varphi$$

$$\frac{\partial }{\partial t}\delta N+\frac{1}{B_0}\frac{1}{L_n}\frac{\partial }{\partial y}\varphi$$

$$-\frac{1}{L_n}\frac{\partial }{\partial x}\varphi +{\nabla }_{\perp }^2\varphi$$

$$\frac{\partial }{\partial t}\mathbf{v}=\mathbf{A}\mathbf{v},$$

$$\mathbf{A}$$

$$E_{k_x,k_x^{\prime }}$$

$$D_{xx,k_x,k_x^{\prime }}$$

$$D_{x,k_x,k_x^{\prime }}$$

$$\kappa :=||\mathbf{V}||||{\mathbf{V}}^{-1}||\ge 1,$$

$$\displaystyle{\mathbf{M}}:=\frac{1}{2}\left(\begin{array}[]{c c}{\mathbf{I}}&{\mathbf{0}}\\ {\mathbf{0}}&\rho_{s0}^{2}\left(-{\mathbf{D}}_{xx}+k_{y}^{2}{\mathbf{I}}\right)\\ \end{array}\right).$$

$$0=\frac{{\omega }_{OB}^2}{{\Omega }_0^2}+\frac{{\omega }_{OB}}{{\Omega }_0}\mathcal{B}+i\frac{{\omega }_{\ast }}{{\Omega }_0}\mathcal{B},$$

$${\omega }_{R,OB}(\mathbf{k},\alpha ,L_n)$$

$${\gamma }_{OB}(\mathbf{k},\alpha ,L_n)$$

$$\frac{||\mathbf{v}(t)||}{||\mathbf{v}(0)||}\le ||e^{\mathbf{A}t}||,$$

$$\beta (\mathbf{A}):=\underset{\mathbf{k}}{\max }\left\{\gamma (\mathbf{A})\right\},$$

Full text

I Introduction

In a series of seminal theoretical works in the early 1960s it has been established that low-frequency small scale instabilites are naturally immanent to magnetically confined plasmas Rudakov and Sagdeev (1961); Chen (1964a, b); Moiseev and Sagdeev (1963); Galeev, Oraevskii, and Sagdeev (1963); Krall and Rosenbluth (1965). This is due to the inherent plasma pressure gradients, which nurture these so called drift wave (DW) instabilities. These DW instabilities drive the turbulent cross-field transport of particles and heat, which exceeds predictions from classical and neo-classical theory and remains a serious barrier for sufficient plasma confinement in laboratory plasmas.

Unstable DWs are triggered by the non-adiabatic coupling of plasma density fluctuations and the electric potential, which can arise due to various physical mechanisms Kadomtsev and Pogutse (1970); Mikhailovskii (1974); Tang (1978); Horton (1999); Bellan . Two of particular importance are (i) collisional friction of electrons and ions along the magnetic field line and (ii) wave-particle resonances. The first of these mechanisms is associated to the collisional (resistive, dissipative) DW instability Chen (1964a, b); Moiseev and Sagdeev (1963), on which we focus in this contribution. The second purely kinetic mechanism results in the so called collisionless (universal) DW instability Galeev, Oraevskii, and Sagdeev (1963); Krall and Rosenbluth (1965). The latter two DW instabilities may be linearly stabilised by magnetic shear Guzdar et al. (1978); Chen et al. (1979); Pearlstein and Berk (1969); Tsang et al. (1978); Ross and Mahajan (1978); Chen et al. (1978); Antonsen (1978); Landreman, Antonsen, and Dorland (2015); Helander and Plunk (2015). However, the inherent non-modal character of density gradient driven DWs (cf. Camargo, Tippett, and Caldas (1998); Mikhailenko, Mikhailenko, and Stepanov (2000); Friedman and Carter (2015); Landreman, Antonsen, and Dorland (2015)) allows for transient amplification to sustain DW turbulence Scott (1990, 1992); Drake, Zeiler, and Biskamp (1995); Landreman, Plunk, and Dorland (2015).

After nearly 60 years density gradient driven DW instabilities remain an active theoretical research field. The latest efforts focus on a unified description of the collisional and collisionless DW instability Angus and Krasheninnikov (2012); Jorge, Ricci, and Loureiro (2018) or proof instability for collisionless DWs in sheared magnetic fields after decades of misconception Landreman, Antonsen, and Dorland (2015).

Another outstanding regime of interest is that of large inhomogeneities, in particular if the background density varies over more than one order of magnitude. This for example prevails in tokamak fusion plasmas, where steep background density gradients can emerge due to the formation of internal or edge transport barriers Shao et al. (2016); Kobayashi et al. (2016). Under these circumstances the Oberbeck-Boussinesq (OB) approximation Oberbeck (1879); Boussinesq (1903) breaks down, which is throughout applied in former studies of DW instabilites. Thus, a rigorous stability analysis for large inhomogeneities must relax the latter assumption.

In spite of that, recent OB approximated analysis of large inhomogeneity effects on trapped electron or ion temperature gradient driven DW instabilities indicate their relevance for transport bifurcation in the edge of tokamaks Xie and Xiao (2015); Xie, Xiao, and Lin (2017); Chen and Chen (2018). Initial investigations of large density inhomogeneity effects on the stability and dynamics of collisional DWs rest partly upon the OB approximation and exploit further approximations in their linear analysis Mattor and Diamond (1994); Zhang and Krasheninnikov (2017).

In this contribution we investigate the linear dynamics of the collisional DW instability for steep background density gradients. This is achieved by consistently linearising a non-Oberbeck-Boussinesq (NOB) approximated full-F gyro-fluid model Madsen (2013), which accurately accounts for collisional friction between electrons and ions along the magnetic field. We numerically solve the generalised eigenvalue problem to show that the growth rate and real frequency deviate by factors of order one from the OB approximated case in the steep background density gradient regime. In this regime, the NOB approximated model is strongly non-normal. As a consequence the eigenfunctions are spatially localised, as opposed to the OB limit. Moreover, non-modal effects arise, resulting in initial decay and transient non-modal growth of the free energy and radially localised and sheared growth of an initially unstable random perturbation. The latter NOB signature lingers into the nonlinear regime, where radially localised turbulence amplitudes appear.

II Gyro-fluid model

Our analysis is based on an energetically consistent full-F gyro-fluid model Madsen (2013), which is derived by taking the gyro-fluid moments over the gyro-kinetic Vlasov-Maxwell equations Brizard and Hahm (2007). In order to ease the following discussion we assume constant temperatures, cold ions and a constant magnetic field $B=B_{0}$ with straight and unsheared unit vector $\hat{{\mathbf{b}}}:={\mathbf{B}}/B={\mathbf{\hat{e}}}_{z}$. The resulting set of gyro-fluid equations consists of continuity equations for electron density ${n}$ and ion gyro-center density $N$ and the quasi-neutrality constraint

[TABLE]

where $\phi$ is the electric potential, $\Omega_{0}:=eB/m_{i}$ is the ion gyro-frequency and ${\mathbf{\nabla}}_{\perp}:=-\hat{{\mathbf{b}}}\times(\hat{{\mathbf{b}}}\times{\mathbf{\nabla}})$ and $\nabla_{\parallel}:=\hat{{\mathbf{b}}}\cdot{\mathbf{\nabla}}$ are the perpendicular and parallel gradient, respectively. The ${\mathbf{E}}\times{\mathbf{B}}$ drift velocity is defined by ${\mathbf{u}}_{E}:=\hat{{\mathbf{b}}}\times{\mathbf{\nabla}}\phi/B_{0}$. As opposed to this, the gyro-center ${\mathbf{E}}\times{\mathbf{B}}$ drift velocity ${\mathbf{U}}_{E}:={\mathbf{u}}_{E}+{\mathbf{U}}_{p}$ contains the ponderomotive correction ${\mathbf{U}}_{p}:=-\hat{{\mathbf{b}}}\times{\mathbf{\nabla}}{\mathbf{u}}_{E}^{2}/(2\Omega_{0})$.

Parallel collisional friction between electrons and ions is introduced on the right hand side of Eq. (1a) with parallel Spitzer resistivity $\eta_{\parallel}:=0.51m_{e}\nu_{e}/(ne^{2})$ Spitzer (1956); Lingam et al. (2017). These closures of the Hasegawa-Wakatani (HW) type are obtained from the evolution equation for the parallel electron velocity. In the electron collision frequency $\nu_{e}:=ne^{4}\ln(\Lambda)/(3\sqrt{(2\pi)^{3}\epsilon_{0}^{4}m_{e}T_{e}^{3}})$ the Coulomb logarithm $\ln(\Lambda)$ is treated as a constant Wesson (2007). This is a reasonable approximation even if the density profile varies over several orders of magnitude. Consequently, the parallel Spitzer resistivity $\eta_{\parallel}$ has no explicit dependence on the electron density $n$, since we only retain the electron density $n$ proportionality in the electron collision frequency $\nu_{e}$.

The two dimensional form of the presented full-F extension of the ordinary HW (OHW) model is obtained by rewriting $\nabla_{\parallel}^{2}\ln\left(n\right)$ to $\nabla_{\parallel}^{2}\ln\left(n/\langle n\rangle\right)$ and by replacing the parallel derivative with the characteristic parallel wave-number, so that $\nabla_{\parallel}^{2}=-k_{\parallel}^{2}$. Here, the average over the “poloidal” y coordinate $\langle h\rangle:=L_{y}^{-1}\int_{0}^{L_{y}}dy\hskip 2.84526pth$ is introduced, which is the 2D equivalent of a flux surface average. With these manipulations Eq. (1a) reduces to Held et al. (2018)

[TABLE]

where the adiabaticity parameter is

[TABLE]

The nonlinear gyro-fluid model of Eqs. (2), (1b) and (1c) allows to study NOB effects on collisional DWs, since in this regime of large collisionality $\nu_{e}\gg\omega$ and small Knudsen-number $\textit{Kn}:=k_{\parallel}\lambda_{mfp}\ll 1$ the presented fluid approach is valid.

III Linearised gyro-fluid model

The linearised gyro-fluid model is obtained by expanding the nonlinear gyro-fluid model of Eqs. (2), (1b) and (1c) around a reference background density profile $n_{G}(x)$ according to

[TABLE]

The chosen exponential reference background density profile $n_{G}$ yields a constant density gradient (e-folding) length

[TABLE]

as in $\delta{f}$ theory. Note that we found similar trends in our results for non-exponential reference background density profiles. Assuming that the relative fluctuation amplitudes $\delta{n}\propto\delta{N}\ll 1$ are small and the averaged and reference background density profiles coincide $\langle n\rangle\approx n_{G}$ yields the final form of the linearised gyro-fluid model

[TABLE]

As opposed to previously exploited linearised models Mattor and Diamond (1994); Camargo, Biskamp, and Scott (1995); Numata, Ball, and Dewar (2007); Angus and Krasheninnikov (2012); Zhang and Krasheninnikov (2017) we do not apply the OB approximation in the linearised Eqs. (6a), (6b) and (6c) or in the further course of the calculation. Thus the derived set of linearised Eqs. (6a), (6b) and (6c) differs from linearised OB approximated models in two substantial aspects. First, we retain the background density on the right hand side of Eq. (6a) instead of assuming a constant reference density $n_{G}\approx n_{0}$. Secondly, we preserve the term $-\frac{1}{L_{n}}\frac{\partial}{\partial x}\phi$ on the left hand side of Eq. (6c), which originates from the nonlinear contribution of the polarization charge density. Both of these terms, but especially the NOB approximated resistive term, produce novel linear effects for collisional DWs in steep background density gradients as is shown in section IV.

IV Linear effects

In the following we want to gain insight into the linear dynamics, in particular in stability and transient time behavior, of the linear model Eqs. (6a), (6b) and (6c). This is accomplished within a discrete approach, which utilises a Fourier transformation of the form $e^{ik_{y}y}$ in the periodic poloidal coordinate $y$ and a Galerkin approach with a sine basis in radial direction. This fulfills the chosen Dirichlet boundary conditions in radial direction. The resulting linear equation reads in matrix form

[TABLE]

with vector ${\mathbf{v}}:=({\mathbf{\delta{n}}}_{k_{y}},e{\mathbf{\phi}}_{k_{y}}/T_{e0})^{T}$ and matrix

[TABLE]

with matrix ${\mathbf{F}}:=\rho_{s0}^{2}\left({\mathbf{D}}_{xx}-k_{y}^{2}{\mathbf{I}}-\frac{1}{L_{n}}{\mathbf{D}}_{x}\right)$. The coefficients of the symmetric matrix ${\mathbf{E}}$ and ${\mathbf{D}}_{xx}$ and of the skew-symmetric matrix ${\mathbf{D}}_{x}$ are derived to

[TABLE]

with radial box size $L_{x}$. The radial wave-numbers are defined by $k_{x}:=\pi m/L_{x}$ and $k_{x}^{\prime}:=\pi m^{\prime}/L_{x}$ with mode-numbers $m\in\mathbb{N}$ and $m^{\prime}\in\mathbb{N}$, respectively.

Note, that the matrix ${\mathbf{A}}$ of Eq. (10) is in general far from normal (${\mathbf{A}}{\mathbf{A}}^{\dagger}\neq{\mathbf{A}}^{\dagger}{\mathbf{A}}$) with non-orthogonal eigenvectors but approaches a normal matrix (${\mathbf{A}}{\mathbf{A}}^{\dagger}={\mathbf{A}}^{\dagger}{\mathbf{A}}$) in the OB limit of very flat background density profiles $n_{0}/n_{G}\approx 1$, $L_{n}^{-1}\partial_{x}\phi\ll 1$ and additionally $\alpha\gg\rho_{s0}^{2}k_{y}/L_{n}$ (cf. Camargo, Tippett, and Caldas (1998)). This is best shown by characterizing the departure from normality by the the condition number

[TABLE]

of the matrix of eigenvectors ${\mathbf{V}}$ of ${\mathbf{A}}$. Here, the matrix norm is induced by the free energy vector norm $||{\mathbf{v}}||:=\sqrt{{\mathbf{v}}^{\dagger}{\mathbf{M}}{\mathbf{v}}}$ with Camargo, Tippett, and Caldas (1998)

[TABLE]

This normalises the condition number $\kappa$ to unity for a normal system. In Fig. 1 we plot the condition number $\kappa$ as a function of adiabaticity $\alpha$ and background density length $L_{n}$, respectively.

In contrast to the OB case, we universally obtain non-normality ($\kappa>1$) for the NOB case for steep background density gradients ($L_{n}=16\rho_{s0}$). In this regime the NOB condition number $\kappa$ is at least a magnitude higher than its OB equivalent and approaches extremely large values for adiabaticities below $\alpha\leq 0.01$. For a fixed adiabaticity $\alpha=0.005$ the NOB dynamics are normal in the OB limit ($L_{n}\geq 10^{4}$), but are strongly non-normal for steep background density gradients. This behavior of the condition number $\kappa$ suggests much larger non-modal effects for the NOB model than for the OB model. However, the condition number $\kappa$ does not give insight into how the departure from normality affects the linear dynamics. Thus, we analyze in the following the modal and non-modal behavior of Eqs. (7).

IV.1 Modal analysis

First, we address the eigenvalues $\omega({\mathbf{A}})$ and the radial eigenfunctions. The numerically calculated growth rate $\gamma({\mathbf{A}}):=Re{(\omega({\mathbf{A}}))}$ and real frequency $\omega_{R}({\mathbf{A}}):=-Im{(\omega({\mathbf{A}}))}$ of the NOB case are compared to the maximum of its OB counterpart. In the OB limit the dispersion relation can be simply derived analytically Camargo, Biskamp, and Scott (1995); Numata, Ball, and Dewar (2007)

[TABLE]

where we defined $\mathcal{B}:=\alpha\left(1+k_{\perp}^{2}\rho_{s0}^{2}\right)/(k_{\perp}^{2}\rho_{s0}^{2})$, the drift frequency $\omega_{*}:=\Omega_{0}k_{y}\rho_{s0}^{2}/\left[L_{n}\left(1+k_{\perp}^{2}\rho_{s0}^{2}\right)\right]$ and the perpendicular wave number $k_{\perp}:=\sqrt{k_{x}^{2}+k_{y}^{2}}$. Consequently, the real frequency $\omega_{R,OB}$ and growth rate $\gamma_{OB}$ are derived to

[TABLE]

with real part $Re(z):=-\mathcal{B}^{2}$, imaginary part $Im(z):=4\mathcal{B}\omega_{*}/\Omega_{0}$ and argument $\theta:=\arg(z)$ of the complex number $z$.

In Fig. 2 we show the normalised growth rate $\gamma/\max_{k_{x}}(\gamma_{OB})$ and real frequency $\omega_{R}/\max_{k_{x}}(\omega_{R,OB})$ as a function of the radial wave number $k_{x}$ for various background density gradient lengths $L_{n}$ and adiabaticities $\alpha$.

Here, the NOB growth rates $\gamma$ and real frequencies $\omega_{R}$ exhibit significant deviations from the OB limit in particular for steep background gradients and for a range of typical adiabaticity parameters. In particular, the magnitude of the fastest growing mode differs by up to roughly a factor five and the radial wave-number of the fastest growing mode is also different for certain parameters. However, the NOB eigenvalues resemble the OB limit for very flat background gradients.

The radial eigenfunctions for the relative density fluctuation $\delta{n}_{k_{y}}$ of the fastest growing $k_{x}$ mode are depicted in Fig. 3 for two different density gradient lengths $L_{n}$.

Remarkably, for steep background density gradients these eigenfunctions are spatially localised and do no longer coincide with the ordinary sine like eigenfunction of the OB model. Moreover, the phase shift between the real and imaginary parts of the eigenfunctions leads to shearing in the x-y plane as we illustrate in section IV.2. Again, for flat background density profiles the eigenfunctions transition into the OB approximated equivalent.

IV.2 Non-modal analysis

We now face the question how non-modal effects manifest in numerical simulations of the nonlinear full-F ordinary HW model, given by Eqs. (2), (1b) and (1c). In particular, we study if initial or transient dynamics are pronounced during the linear phase and if these non-modal effects survive into the nonlinear regime.

It is well known that for a normal matrix the time evolution of the norm of the linear Eq. (7)

[TABLE]

is bounded by the spectral abscissa

[TABLE]

for $t\geq 0$ since the matrix exponential reduces to $||e^{{\mathbf{A}}t}||=e^{\beta({\mathbf{A}})t}$. However, for a non-normal matrix the maximum growth estimate of the modal analysis of section IV.1, determined by the spectral abscissa $\beta$, only holds for $t\rightarrow\infty$ and the matrix exponential reduces to a loose upper bound $||e^{{\mathbf{A}}t}||=\kappa({\mathbf{V}})e^{\beta({\mathbf{A}})t}$ for $t\geq 0$. As a consequence, a non-normal system may exhibit pronounced initial or transient phenomena, for which also estimates and bounds exist. In particular, the numerical abscissa $\eta({\mathbf{A}}):=\max_{\mathbf{k}}\left\{\omega\left[({\mathbf{A}}^{\dagger}+{\mathbf{A}})/2\right]\right\}$111Note that the spectral and numerical abscissa coincide $\beta({\mathbf{A}})=\eta({\mathbf{A}})$ for a normal matrix (${\mathbf{A}}{\mathbf{A}}^{\dagger}={\mathbf{A}}^{\dagger}{\mathbf{A}}$). represents an upper bound for $t=0$ and the so called $\epsilon$-pseudospectral abscissa $\alpha_{\epsilon}({\mathbf{A}})$ yields estimates for transient phenomena Trefethen and Embree (2005). Although, the latter two approaches are useful to detect or quantify non-modal effects, we do not make use of them in the following discussion. Instead, we present a direct numerical approach to the initial value problem.

The numerical implementation of the latter full-F gyro-fluid model utilises the open source library Feltor Wiesenberger and Held (2018). This initial value code relies on a discontinuous Galerkin discretization, which is also used for verification of the herein presented Galerkin approach for the modal analysis of section IV.1. We limit our study to a single exemplary initial condition, but note that in general the initial condition can be optimised to produce maximum growth at small, intermediate or large time Camargo, Tippett, and Caldas (1998). The chosen initial conditions mimics a random perturbation $\delta{n}({\mathbf{x}},0)=\delta{N}({\mathbf{x}},0)=af_{bath}({\mathbf{x}})$ of amplitude $a$ with vanishing electric potential $\phi({\mathbf{x}},0)=0$.

In Fig. 4 we show the temporal behavior of the square root of the normalised free energy norm $||{\mathbf{v}}(t)||/||{\mathbf{v}}(0)||$ for various adiabaticities $\alpha$ in the steep gradient regime ($L_{n}=16\rho_{s0}$).

Here, the random bath initial condition with the small amplitude $a=10^{-5}$ limits us to linear effects only. Interestingly, two clear footprints of non-modal behavior emerge in the linear dynamics. First, initial decay of the square root of the normalised free energy norm appears for both the OB and NOB case, despite the fact that all eigenvalues are unstable (cf. Fig. 2). Secondly, the transient exponential growth at later times either surpasses (NOB) or falls below (OB and NOB) the spectral abscissa $\beta$. Both of these effects are due to the shrinking of non-orthogonal eigenvectors, which is intrinsic to non-normal systems. We refer the interested reader to Schmid (2007) for an illustrative sketch of this phenomenon. The observed initial decay of the collisional DW instability is similar to that of the collisionless DW instability Landreman, Plunk, and Dorland (2015). However, for the latter instability transient amplification can trigger subscritical turbulence in the absence of linear instability.

During this linear growth phase non-modal features may appear in the spatial structure of the relative density fluctuation $\delta{n}$. This is depicted in Fig. 5 for the turbulent bath initial condition with $a=0.01$ and for a steep background density profile ($L_{n}=16\rho_{s0}$) and typical adiabaticity ($\alpha=0.005$).

In contrast to the OB approximated model, we observe sheared and localised growth of the initial perturbation in the steep background density regime. This is in qualitativ agreement with the previously reported NOB shearing effect of DWs of Held and Kendl (2015). The radial location of the strongest growth coincides approximately with the maximum of the absolute background density gradient $|\partial_{x}n_{G}|$. These NOB effects are again reasoned in the strong non-orthogonality of the eigenvector Matrix ${\mathbf{V}}$, which is pronounced for a system with strong non-normality and by implication high condition number $\kappa$ (cf. Fig. 1)

Finally, we study how non-modality affects the turbulence intensity in collisional DW turbulence without and with zonal flows. Here, the underlying models are the NOB-extended OHW (equations (2), (1b) and (1c)) and modified HW (MHW) model Held et al. (2018), respectively. In Fig. 6 we show the spatial structure of the relative density fluctuation $\delta{n}$ for both models and the latter initial condition but during the nonlinear phase.

Without zonal flows (OHW) we observe a radial peaking of the maximum of the relative density fluctuation amplitude. Analogously, with zonal flows (MHW) a similar radial peaking appears for the zonal flow amplitude. Note that the radial localization of the turbulence intensity continues to exist at late turbulence saturation times if the radial particle transport is weak. This occurs for large adiabaticity (small collisionality). For both model cases, this constitutes a clear non-modal footprint in saturated collisional DW turbulence and shows that indeed these non-modal effects can survive into the nonlinear regime. As opposed to this, the radial peaking of the turbulence or zonal flow intensity is again absent in the OB approximated OHW or MHW model (cf. Camargo, Biskamp, and Scott (1995); Numata, Ball, and Dewar (2007); Kendl (2018)).

V Disussion and conclusion

We studied the collisional DW instability in a straight and unsheared magnetic field within a full-F gyro-fluid model, which relaxes the OB approximation. In the regime of steep background density gradients both the eigenvalues and eigenfunctions fundamentally deviated from former OB approximated investigations. In particular, our modal analysis demonstrated NOB corrections by factors of order one to the eigenvalues, highly non-orthogonal eigenvectors and spatially localised eigenfunctions for typical plasma parameters. Our non-modal analysis revealed initial damping and transient non-modal growth of the free energy of an initially unstable random perturbation. Remarkably, this linear growth is radially localised and sheared. It was numerically shown that this NOB signature subsists into the nonlinear regime, where radially localised turbulence or zonal flow amplitudes emerge.

The herein presented results emphasise the need for NOB approximated models to consistently capture the (linear) dynamics of the collisional DW instability for large density inhomogeneities. For instance, this may prove necessary for the accurate calculation of transport levels in high-confinement tokamak plasmas within quasilinear gyro-kinetic or gyro-fluid models (e.g.: Staebler, Kinsey, and Waltz (2005); Bourdelle et al. (2007)). Finally, we conclude that the study of linear effects, that is solely based on a modal approach, may give misleading predictions since this approach overlooks non-modal features like initial and transient phenomena.

VI Acknowledgements

This work was supported by the Austrian Science Fund (FWF) Y398. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC) and the EUROfusion High Performance Computer (Marconi-Fusion).