# Investigation of a Multiple-Timescale Turbulence-Transport Coupling Method in the Presence of Random Fluctuations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background: Multiple-Scale Turbulence and Transport Framework

## 3. A Numerical Method for Multiple-Timescale Turbulence and Transport

`Tango`[17].

## 4. Review of Relevant Mathematical Considerations in a Statistical Framework

## 5. Simple Iteration Problem with Relaxation and Noise

## 6. Generating Fluctuations with Certain Properties for Testing Purposes

#### 6.1. Generating Spatially Correlated Gaussian Noise

#### 6.2. Generating Temporally Correlated Non-Gaussian Noise

## 7. Behavior of the Multiple-Timescale Coupling Method in the Presence of Fluctuations

#### 7.1. Statement of the Problem

#### 7.2. Gaussian Noise: Spatially Correlated, Temporally White

#### 7.3. Non-Gaussian Noise: Temporally Correlated, Spatially Uniform

## 8. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Estimating the Autocorrelation Time

## References

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**Figure 1.**Schematic diagram of the coupling between the transport solver and turbulence simulation for a self-consistent solution.

**Figure 2.**(

**a**) Plots of the error $|{\delta}_{n}|$ and residual $|{\eta}_{n}|$ as a function of the iteration number for one realization of the system in Equations (14)–(16). For this example, we take $f\left(x\right)=2-{x}^{2}$, ${x}_{0}=0.7$, $\alpha =0.001$ and $\sigma =0.1$. The horizontal dashed lines are the theoretical standard deviations of $\delta $ and $\eta $ as $n\to \infty $. Observe that as ${x}_{n}\to {x}_{*}=1$, the residual reaches its noise floor at an earlier iteration number ($n\approx 700$) than the actual error reaches its noise floor ($n\approx 1700$). (

**b**) The scaling of $\mathrm{Var}\left(\delta \right)$ and $\mathrm{Var}\left(\eta \right)$ as the relaxation parameter $\alpha $ varies. In the small-$\alpha $ regime, $\mathrm{Var}\left(\delta \right)\sim \alpha $ and $\mathrm{Var}\left(\eta \right)\sim {\alpha}^{2}$.

**Figure 4.**(

**a**) Time trace of the ion heat flux (arbitrary units) from a typical gyrokinetic turbulence simulation with the GENE code. Time is measured in ${R}_{0}/{v}_{Ti}$. The time trace was obtained in a statistically steady state. The ion heat flux is averaged over a magnetic surface and averaged over a small radial window with a width of several gyroradii. In this time trace, the mean value is about 12.3, although bursts up to several times larger occur. (

**b**) Histogram of the heat flux values; the distribution is non-symmetric with a long tail. (

**c**) Same as (b), except the histogram is of the logarithm of heat flux values. The distribution of the logarithm of the heat flux looks roughly symmetric and much closer to normally distributed than the distribution of the heat flux itself.

**Figure 5.**Generated signals using ARMA-estimated models of the heat flux in Figure 4a. (

**a**) Realization of an ARMA model applied to the heat flux directly. The output is Gaussian distributed and has negative values. (

**b**) Realization of an ARMA model applied to the logarithm of the heat flux, followed by reverse transforming as in Equation (28). The output is log-normal distributed, with no negative values and with regular large bursts.

**Figure 6.**Residual and error of realizations of the system with added Gaussian noise, spatially correlated and temporally white. (

**a**) Two values of variance ${\sigma}^{2}$ of added noise, with fixed relaxation parameter $\alpha $. (

**b**) Two values of the relaxation parameter, with a fixed noise amplitude. Note that the x-axis in (b) uses a logarithmic scale because the system converges in fewer iterations at $\alpha =0.1$ than at $\alpha =0.01$.

**Figure 7.**Residual and error of realizations of the system with added non-Gaussian noise, spatially uniform and temporally correlated. ${\sigma}^{2}$ is the variance of the ARMA-generated Gaussian signal $\tilde{n}$ in Equation (38). (

**a**) Two values of normalized averaging time $T/\tau $, where τ is the autocorrelation time, with fixed relaxation parameter and noise amplitude. (

**b**) Two values of the relaxation parameter, with fixed averaging time and noise amplitude. Note that the x-axis in (b) uses a logarithmic scale because the system converges in fewer iterations at $\alpha =0.1$ than $\alpha =0.01$.

**Figure 8.**Residual and error at the noise floor. (

**a**) Scaling with normalized averaging time $T/\tau $, where τ is the autocorrelation time. Both the residual and the error scale as ∼${T}^{-1/2}$. This scaling is consistent with the simple model in Section 5 because the standard deviation of the averaged noise is proportional to ${T}^{-1/2}$. (

**b**) Scaling with the relaxation parameter $\alpha $. The error scales as ∼${\alpha}^{1/2}$, and the residual scales as ∼$\alpha $. The two different scalings are consistent with the simple analytic model in Section 5.

**Figure 9.**At $T=5\tau $ and ${\sigma}^{2}=0.21$. (

**a**) Relaxation parameter $\alpha =0.1$, showing the profile in the final 60 iterates (red lines), the average profile over those 60 iterates (black line) and the exact solution (blue lines). The exact solution and averaged profile are nearly visually overlapping. (

**b**) Same as (a), but with $\alpha =0.01$ and showing 600 iterates. Here, the averaged profile and the exact solution are visually indistinguishable. In each iterate, the error is zero at $x=1$, where a Dirichlet boundary condition is applied, and largest at $x=0$, where a Neumann boundary condition is applied.

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**MDPI and ACS Style**

Parker, J.B.; LoDestro, L.L.; Campos, A.
Investigation of a Multiple-Timescale Turbulence-Transport Coupling Method in the Presence of Random Fluctuations. *Plasma* **2018**, *1*, 126-143.
https://doi.org/10.3390/plasma1010012

**AMA Style**

Parker JB, LoDestro LL, Campos A.
Investigation of a Multiple-Timescale Turbulence-Transport Coupling Method in the Presence of Random Fluctuations. *Plasma*. 2018; 1(1):126-143.
https://doi.org/10.3390/plasma1010012

**Chicago/Turabian Style**

Parker, Jeffrey B., Lynda L. LoDestro, and Alejandro Campos.
2018. "Investigation of a Multiple-Timescale Turbulence-Transport Coupling Method in the Presence of Random Fluctuations" *Plasma* 1, no. 1: 126-143.
https://doi.org/10.3390/plasma1010012