Nicholas Hall University of Oklahoma nicholashall@ou.edu
REU Program - Summer 2006 |

It is known that the inclusion of an external magnetic field in the turbulence of an incompressible fluid brings about many substantial complications and can drastically modify the turbulent motions. The inclusion of compressibility also causes significant changes in the properties of turbulence.

got02 thoroughly and successfully investigated the velocity field statistics of high-resolution direct numerical simulations for IHD turbulence. The purpose of this paper is to address the question, are the velocity field statistics completely different from IHD when a magnetic field and compressibility are included? We extend the analysis of **got02** to incompressible magnetohydrodynamic (hereafter IMHD) and compressible magnetohydrodynamic (hereafter CMHD) turbulence in an attempt to answer this question. This is considered mostly through the use of high order scaling exponents of the velocity structure functions.

Emphasis is also placed on the differences between the scaling exponents of longitudinal and transverse structure functions. For high order scaling exponents the findings of **got02** are confirmed for structure functions measured in the global reference frame for IHD turbulence. Scaling exponents are then explored for IMHD and CMHD turbulence in the global frame and the local frame, defined by the local mean magnetic field. The scaling exponents are then decomposed into Alfvenic, fast, and slow modes and the contributions of individual modes are explored. The paper is organized as follows. Sec. II presents the numerical aspects of the three numerical simulation models considered (IHD, IMHD, and CMHD). Sec. III contains analysis of the kinetic energy spectra of the three models. In Sec. IV probability distribution functions are considered. Structure functions are explored in Sec. V. Sec. VI presents scaling exponents of the structure functions. In Sec. VII the scaling exponents are decomposed into the three MHD modes. Finally, Sec. VIII includes a discussion and summary.

The IHD data used in this study came from simulations with a grid size of $256^3$. For both MHD models, a high resolution grid size of $512^3$ was used and the external magnetic field is in the positive x-direction.

Figure 1 - Kinetic energy spectra for the three models as well as Kolmogorov's power-law dependence in the inertial range for comparison.

The three-dimensional kinetic energy spectrum for each model is shown in figure 1. The inertial range of the spectrum, the flat part of the curve, is of most interest for this study. The shape of the curves at wave numbers less than those of the inertial range depends on how the turbulent motions are excited. The shape of the curves at wave numbers greater than those of the inertial range is affected significantly by viscosity, the bottleneck effect (fal94), and the resolution of the model. Despite the fact that the Kolmogorov phenomenology, E(k) proportional to k^{-5/3}, is derived for IHD, in the inertial range all three curves demonstrate very similar slopes, suggesting that the energy transfer process in the inertial range is consistent for all three models. This may be surprising considering the many changes resulting from the inclusion of a magnetic field, such as Alfvenic waves propagating parallel to the magnetic field and elongation of eddies in the velocity field. The inclusion of compressibility introduces significant changes as well, such as shocks. gol95 provide an explanation of this behavior. See cl05 also for a review. Deviations from the 5/3 power law have become an issue of debate (see

Figure 2 - PDF's of the velocity field parallel (x) and perpendicular (y \& z) to $B_{ext}$ for the three models.

The probability distribution function (hereafter PDF) is used to obtain information about one-point velocity statistics. The single-point PDF's of different components of the velocity field for each model shown in figure 2 are histograms, normalized to the grid size, of the values of the velocity at each point. For the IHD model the x, y, and z components were averaged. For the two MHD models, the histograms were separated into components parallel (x direction) and perpendicular (y \& z directions) to the external magnetic field, ($B_{ext}$); therefore, the PDF's for the x components are presented separately and the y and z components are averaged.

The PDF for the IHD model is somewhat asymmetric. Any asymmetry or skewness in the PDF's should disappear when averaged over multiple time steps (see

The PDF's of the IMHD and CMHD models perpendicular to $B_{ext}$ approximately follow each other and are even more similar to each other than the parallel components. They are very symmetric and gaussian-like, have the most shallow slopes of all, and therefore have the most high velocity points.

When an external magnetic field is introduced into the turbulent motions, an Alfvenic mode is produced with perturbations perpendicular to the direction of the field. This additional contribution to the velocity is responsible for the higher velocities found in the perpendicular PDF's. The additional kinetic energy given to the perpendicular direction explains the separation of these two components seen in the top of figure 2. This appears to be equally valid for both positive and negative values of the velocity.

Figure 3 - PDF's, normalized to match their maxima, of the velocity field parallel and perpendicular to $B_{ext}$ for the one time step of the CMHD simulation used throughout the paper and the average over 10 time steps of a 256 resolution simulation with error bars showing expected random fluctuations.

An average over 10 time steps of the velocity PDF's for a CMHD simulation with 256 resolution was performed. The 10 parallel time steps were averaged as well as the 20 perpendicular time steps, 10 for y and 10 for z. These PDF's are plotted in figure 3 along with the CMHD PDF's mentioned above. The PDF's are normalized to match their maxima in order to better compare their tails. The parallel and perpendicular PDF's for the single time step lie sufficiently close to the expected random errors obtained from the PDF's averaged over multiple time steps to justify using a single time step and averaging the perpendicular components.

Figure 4 - PDF's of the transverse increments of the x component of the velocity field taken along y and z averaged together for the three models. |
Figure 5 - PDF's of the transverse increments of the y component of the velocity field taken along x and z presented separately for the three models. |

Figures 4 and 5 show PDF's of the transverse increments of the velocity field which are histograms, normalized to the grid size, of the values of the difference between the velocity field at each point and the field at an adjacent point along the direction specified. Positive values suggests that $v$ is increasing in the positive direction while negative values suggest that it is decreasing. In figure 4 the increments of the x-component of the velocity field taken along the y and z directions, parallel to $B_{ext}$ for the MHD models, are averaged. Figure 5 shows the increments of the y-component along the x and z directions. The results are not averaged given that one is parallel to $B_{ext}$ while the other is perpendicular.

In both figures, all three models demonstrate near perfect symmetry. This is true despite the orientation with respect to $B_{ext}$. In figure 5 as expected for the IHD model, x and z are nearly identical because there is no magnetic field to bring about differences. On the other hand for the MHD models, x extends to higher values of the velocity increment than z.

Because of the fact that the difference in range between the x components for the IMHD and CMHD models is much greater than the difference in range between the z components for the IMHD and CMHD models, it is possible that most of the increase in range in going from IMHD models to CMHD models by introducing compressibility is coming from the components parallel to $B_{ext}$. This is supported by the fact that for the PDF's in figure 2 the perpendicular components follow each other closely while the parallel components are quite different from each other.

For each separation length $l$ from 1 to 128, $\delta v$ is calculated at 100,000 points randomly distributed throughout the entire periodic velocity field. In the global reference frame the direction of $\hat{l}$, or the direction of the longitudinal structure function, is also random. The structure functions for the IHD model are calculated in the global reference frame. For the MHD models $\hat{l}$ can be chosen randomly (global frame), parallel to the local mean magnetic field, or perpendicular to the local mean field.

Figure 6 - For (a) points are chosen along the local mean field, hereafter referred to as local parallel. For (b) points are chosen in the plane perpendicular to the local mean field so that both the longitudinal and the transverse structure functions lie in this plane, hereafter referred to as local perpendicular.

According to

Figure 7 - Second order structure functions of the velocity field for the three models. The top row are longitudinal, the middle row are transverse, and the bottom row are total structure functions. The first column is in the global reference frame, the middle row is the local parallel frame, and the bottom row is the local perpendicular frame. In the local frames the HD structure functions from the global frame are shown for comparison.

Figure 7 shows the second order longitudinal, transverse, and total structure functions of the velocity field. The column on the left is the global frame, the middle column is the local parallel frame, and the column on the right is the local perpendicular frame. In the local frames the structure functions of the IHD model calculated in the global frame are shown for comparison.

The inertial range of the turbulent motions manifests itself in the structure functions as the range of $l$ over which the slope is constant in a log-log plot. The first maximum of a structure function is related to the size of the largest structures in the corresponding field, the radius of the largest eddies for example. It will be found at the separation length at which the largest $\delta v$ is observed.

We confirm the result of

Figure 8 - The sixth order longitudinal structure function as a function of the third order longitudinal structure function both calculated in the local parallel frame for the three models, along with their values of scaling exponents obtained from a minimum chi-square linear fit. Compare to figure 10 (a).

Despite the fact that the origin is not yet fully understood, the scaling behavior of structure functions is very useful and interesting (see

Figure 9 - Scaling exponents, normalized to the third order, as a function of the order of the structure functions in the global reference frame for the three models as well as K41, SL1, and SL2 ((a) - longitudinal, (b) - transverse, and (c) - total). |
Figure 10 - Scaling exponents, normalized to the third order, as a function of the order of the structure functions in the local parallel reference frame for the three models as well as K41, SL1, and SL2 ((a) - longitudinal, (b) - transverse, and (c) - total). For the IHD model the global scaling exponents are shown for comparison. |
Figure 11 - Scaling exponents, normalized to the third order, as a function of the order of the structure functions in the local perpendicular reference frame for the three models as well as K41, SL1, and SL2 ((a) - longitudinal, (b) - transverse, and (c) - total). For the IHD model the global scaling exponents are shown for comparison. |

Figures 9 through 11 contain the scaling exponents as a function of the order of the longitudinal, transverse, and total structure functions normalized to the third order. For the MHD models the figures are organized as follows. In figure 9 the global reference frame is used, in figure 10 the local parallel reference frame is used, and in figure 11 the local perpendicular reference frame is used. For the IHD model the global scaling exponents are also presented in figures 10 and 11 for comparison, given that the local reference frame has no meaning without a magnetic field. Despite the fact that the error in the scaling exponents increases with order from the increasing fluctuation in the structure functions, emphasis is placed on high order structure functions ($p>4$) given that at low orders the theoretical predictions considered here differentiate themselves very little.

Beginning with figure 9 in the global reference frame the longitudinal scaling exponents of all three models follow SL1 quite well. In the transverse case although the slope of the IHD model initially follows SL1 it begins to flatten at around $p=7$. Its shape is in agreement with the results of

In figure 10 in the local parallel reference frame for the longitudinal component, parallel to the local mean magnetic field in this case, the IMHD model is between SL1 and SL2 while the CMHD model mostly follows SL2. For the transverse component, perpendicular to the local mean magnetic field in this case, the IMHD model is between SL1 and SL2 while the CMHD model follows SL2.

In figure 11 in the local perpendicular reference frame for the longitudinal component, perpendicular to the local mean magnetic field in this case, the best agreement to SL1 is found for all three models at all values of $p$. For the transverse component, also pependicular to the local mean magnetic field in this case, we do not find a similar agreement; the IMHD model is more SL2-like while the CMHD model remains in between.

We confirm the results of

The most conspicuous SL1 cases above are both longitudinal, namely $\zeta^{glo}_{L}(p)$ and $\zeta^{loc, \bot}_{L}(p)$ in which all models follow SL1 suggesting physically that the dissipation structures are one dimensional hydrodynamic-like vortices. The most conspicuous SL2 cases are both the transverse cases corresponding to the longitudinal SL1 cases above, namely $\zeta^{glo}_{T}(p)$ and $\zeta^{loc, \bot}_{T}(p)$ in which the IMHD model mostly follows SL2 suggesting physically that the dissipation structures are two dimensional sheets dominated by MHD processes. It is possible and perhaps more likely that the dissipation structures only appear one or two dimensional in slices of the three dimensional MHD turbulence (see

HD and MHD turbulence are similar over high order longitudinal scaling exponents measured in the global reference frame as well as for motions perpendicular to the local mean field. In the local magnetic system the motions parallel to the local mean field for both MHD models are different from hydrodynamic motions.

Figure 12 - Separation method. We separate Alfven, slow, and fast modes in Fourier space by projecting the velocity Fourier component ${\bf v_k}$ onto bases ${\bf \xi}_A$, ${\bf \xi}_s$, and ${\bf \xi}_f$, respectively. Note that ${\bf \xi}_A = -\hat{\bf \varphi}$. Slow basis ${\bf \xi}_s$ and fast basis ${\bf \xi}_f$ lie in the plane defined by ${\bf B}_0$ and ${\bf k}$. Slow basis ${\bf \xi}_s$ lies between $-\hat{\bf \theta}$ and $\hat{\bf k}_{\|}$. Fast basis ${\bf \xi}_f$ lies between $\hat{\bf k}$ and $\hat{\bf k}_{\perp}$ (from Cho \& Lazarian 2003.

Figure 13 - Decomposed scaling exponents, normalized to the third order, as a function of the order of the structure functions of the velocity field in the global reference frame for the IMHD and CMHD models as well as K41, SL1, and SL2 ((a) - Alfvenic, (b) - fast, and (c) - slow). |
Figure 14 - Decomposed scaling exponents, normalized to the third order, as a function of the order of the structure functions in the local parallel reference frame for the IMHD and CMHD models as well as K41, SL1, and SL2 ((a) - Alfvenic, (b) - fast, and (c) - slow). |
Figure 15 - Decomposed scaling exponents, normalized to the third order, as a function of the order of the structure functions in the local perpendicular reference frame for the IMHD and CMHD models as well as K41, SL1, and SL2 ((a) - Alfvenic, (b) - fast, and (c) - slow). |

The compressible MHD equations support three types of linear waves or modes known as Alfvenic, fast, and slow. In the incompressible limit the fast mode does not exist. The velocity fields can be decomposed into these three components and scaling exponents for each component can be obtained, as explained in

Beginning with figure 13 in the global reference frame, for the Alfv\'en mode the longitudinal components of both models follow SL1 while the transverse follow closer to SL2. For the the fast mode the CMHD longitudinal and transverse components are almost indistinguishable and follow SL2. For the slow mode the transverse components of both models are more SL2-like.

In figure 14 in the local parallel reference frame, for the Alfv\'en mode the CMHD transverse component follows SL2 quite well while the longitudinal component falls between SL1 and SL2. For the fast mode the CMHD transverse component falls well below SL2 while the longitudinal component falls between SL1 and SL2. For the slow mode the IMHD longitudinal component falls between SL1 and SL2 while the CMHD longitudinal component follows SL2 quite well. The transverse components of both models fall well below SL2 and for the IMHD model the curve actually becomes flat.

In figure 15 in the local perpendicular reference frame, for the Alfv\'en mode all cases follow SL1 except the IMHD transverse component which falls between SL1 and SL2. For the CMHD model the longitudinal and transverse components are indistinguishable. For the fast mode both components of the CMHD model follow closer to SL2, the transverse component more so. For the slow mode the IMHD longitudinal component follows closer to SL1 while the transverse component follows SL2 fairly well. For the CMHD model this is the only case where the transverse component is significantly greater than the longitudinal component. The transverse component follows SL2 while the longitudinal component falls well below SL2.

There are three cases in which scaling exponents fall well below SL2, namely the CMHD longitudinal component of $\zeta^{loc, \|}_{fast}(p)$, the transverse component for both models of $\zeta^{loc, \|}_{slow}(p)$, and the CMHD longitudinal component of $\zeta^{loc, \bot}_{slow}(p)$, suggesting that intermittency is greater in these cases.

Also, $\zeta_L(p)\geq\zeta_T(p)$ for all cases except for the CMHD model of $\zeta^{loc, \bot}_{slow}(p)$. This is not reflected in figure 11 where for the CMHD model $\zeta_L(p)>\zeta_T(p)$. This suggests that the amplitude of the slow mode for this model is small compared to the amplitude of the Alfvenic and fast modes. The two cases where $\zeta_L(p)=\zeta_T(p)$ are for the CMHD models of $\zeta^{glo}_{fast}(p)$ and $\zeta^{loc, \bot}_{alfv}(p)$.

In agreement with the above-mentioned property, $\zeta_L(p)\geq\zeta_T(p)$, it seems that the longitudinal components are more likely to be SL1-like while the transverse components are more likely to be SL2-like. This is the case for $\zeta^{glo}_{alfv}(p)$ and $\zeta^{loc, \bot}_{slow}(p)$, which is in direct agreement with the conspicuous cases mentioned in the discussion of figures 9 through 11. Combining the results of figures 9 and 13 it seems that in the global reference frame the Alfvenic mode is mostly responsible for the fact that the longitudinal components of both MHD models follow SL1. For the IMHD model both the Alfvenic and slow modes seem to contribute significantly to the fact that the transverse components appear to follow SL2. Combining the results of figures 11 and 15 it seems that in the local perpendicular reference frame once again the Alfvenic mode is mostly responsible for the fact that the longitudinal components of both MHD models follow SL1. For the IMHD model in this case the slow mode seems to contribute most to the fact that the transverse components appear to follow SL2.

It is also true that for the decomposed scaling exponents of the two MHD models the longitudinal components are more alike than the transverse components, in every case except for $\zeta^{loc, \bot}_{slow}(p)$. This makes sense in the local parallel reference frame where there is a bias in that the longitudinal components are chosen in the direction of the local mean field while the transverse components are chosen randomly to be perpendicular to the longitudinal components, but this observation also holds in the global reference frame where there is no bias because the longitudinal components are chosen randomly. If the dissipation structures are similar for compressible and incompressible MHD then it is possible that the longitudinal scaling exponents would do a better job of probing this.

- Substantial similarity was shown between the spectra for all three models.
- The PDF's of the velocity components perpendicular to the external magnetic field are IHD-like while those parallel to the field have a significantly smaller range of velocities. The PDF's of the transverse velocity increments for the MHD models decay slower than the IHD model.
- The structure functions of the MHD models in the global frame and in the plane perpendicular to the local mean field are IHD-like in that the values of the transverse structure functions are greater than those of the longitudinal. In these two frames the shapes of the structure functions of the IMHD model are more IHD-like than the CMHD model.
- The similarity of HD, IMHD, and CMHD turbulence persists over high order longitudinal structure function scaling exponents measured in the global reference frame as well as for motions perpendicular to the local mean field. In these two frames the longitudinal scaling exponents of both MHD models seem to follow the predictions for one dimensional IHD-like dissipations structures while the transverse scaling exponents of the IMHD model seems to follow the predictions for two dimensional IMHD-like dissipation structures. In the local magnetic system the motions parallel to the local mean field for both MHD models are different from hydrodynamic motions.
- In the global reference frame and for motions perpendicular to the local mean field the Alfvenic mode is mostly responsible for the fact that the longitudinal components of both MHD models follow the IHD model and the IHD theoretical dissipation structure predictions. In the global reference frame for the IMHD model both the Alfvenic and slow modes seem to contribute significantly to the fact that the transverse components follow the IMHD theoretical predictions. For motions perpendicular to the magnetic field for the IMHD model the slow mode seems to contribute most to the fact that the transverse component seems to follow the IMHD theoretical predictions. It is also true that for the decomposed scaling exponents of the two MHD models the longitudinal components are more alike than the transverse components, in every case except for the slow mode in the plane perpendicular to the local mean field.