New scaling for the alpha effect in slowly rotating turbulence
Using simulations of slowly rotating stratified turbulence, we show that the effect responsible for the generation of astrophysical magnetic fields is proportional to the logarithmic gradient of kinetic energy density rather than that of momentum, as was previously thought. This result is in agreement with a new analytic theory developed in this paper for large Reynolds numbers. Thus, the contribution of density stratification is less important than that of turbulent velocity. The effect and other turbulent transport coefficients are determined by means of the test-field method. In addition to forced turbulence, we also investigate supernova-driven turbulence and stellar convection. In some cases (intermediate rotation rate for forced turbulence, convection with intermediate temperature stratification, and supernova-driven turbulence) we find that the contribution of density stratification might be even less important than suggested by the analytic theory.
Nordita, KTH Royal Institute of Technology and Stockholm
University, Roslagstullsbacken 23,
SE-10691 Stockholm, Sweden
Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden
Department of Physics, Gustaf Hällströmin katu 2a (PO Box 64), FI-00064 University of Helsinki, Finland
Department of Mechanical Engineering, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel
Turbulent dynamos occur in many astrophysical situations. They tend to develop large-scale magnetic structures in space and time that are generally understood in terms of mean-field dynamo theory (e.g. Moffatt 1978; Parker 1979; Krause & Rädler 1980; Zeldovich et al. 1983; Ruzmaikin et al. 1988; Rüdiger & Hollerbach 2004; Brandenburg & Subramanian 2005). Central to this theory is the effect, which denotes a contribution to the mean electromotive force that is given by a pseudo-scalar multiplying the mean magnetic field. Such a pseudo-scalar can be the result of rotation, , combined with stratification of density and/or turbulence intensity, and/or , respectively. Here, is the mean gas density and is the rms value of the turbulent velocity, .
There have been a number of analytic studies quantifying the effects of rotating stratified turbulence on the mean electromotive force (Krause & Rädler 1980; Kichatinov 1991; Rüdiger & Kichatinov 1993; Kitchatinov et al. 1994; Rädler et al. 2003; Kleeorin & Rogachevskii 2003). In particular, it was found, using the quasi-linear approach (or second-order correlation approximation), that the diagonal components of the tensor for slow rotation rate (or small Coriolis numbers) are given by (Steenbeck et al. 1966; Krause & Rädler 1980)
where is a relevant length scale and is the characteristic turbulent time related to the turnover time.
In the solar convective zone the mean fluid density changes by seven orders of magnitudes, while the turbulent kinetic energy () changes by only three orders of magnitudes and, according to stellar mixing length theory (Vitense 1953), would be approximately constant in the solar convective zone. Here the fluid density is the sum of the mean and fluctuating density. This issue has become timely, because there is a new numerical technique that allows the different proposals to be examined with sufficient accuracy. The so-called test-field method (Schrinner et al. 2005, 2007) allows one to determine all the relevant turbulent transport coefficients in the expression for the mean electromotive force without the restrictions of some of the analytic approaches such as the quasi-linear approach, the path-integral approach or the approach. With the test-field method one solves sets of equations for the small-scale fields resulting from different prescribed mean fields — the test fields. These equations resemble the usual induction equation, except that they contain an additional inhomogeneous term.
This method is quite powerful because it has been shown to be rather accurate and it gives not only the tensor coefficients of effect and turbulent diffusivity, but it also allows the scale-dependence to be determined, which means that these coefficients are actually integral kernels that allow the effects of neighboring points in space and time to be taken into account. For details regarding scale separation, see Brandenburg et al. (2008) and Hubbard & Brandenburg (2009). We apply this method to numerical simulations of forced turbulence in a stably stratified layer in the presence of rotation and a prescribed vertical dependence of the turbulence intensity. We also use test-field method in simulations of turbulent convection and supernova-driven turbulence of the interstellar medium (ISM).
The goal of the present paper is to determine the correct scaling of the effect with mean density and rms velocity for slow rotation, i.e., when the Coriolis number is much less than unity, and large Reynolds numbers. In addition to the parameter , we determine the exponent in the diagonal components of the tensor,
Such an ansatz was also made by Rüdiger & Kichatinov (1993), who found that in the high conductivity limit, for slow rotation and for rapid rotation. However, as we will show in this paper, both numerically (for forced turbulence and for turbulent convection with stronger temperature stratification and overshoot layer) as well as analytically, our results for slow rotation and large fluid and magnetic Reynolds numbers are consistent with . In some simulations we also found .
2 Theoretical predictions
We consider the kinematic problem, i.e., we neglect the feedback of the magnetic field on the turbulent fluid flow. We use a mean field approach whereby velocity, pressure and magnetic field are separated into mean and fluctuating parts. Unlike in earlier derivations, and to maintain maximum generality, we allow the characteristic scales of the mean fluid density, , the turbulent kinetic energy, , and the variations of to be different. We also assume vanishing mean motion. The strategy of our analytic derivation is to determine the dependencies of the second moments for the velocity and for the cross-helicity tensor , where are fluctuations of magnetic field produced by tangling of the large-scale field. To this end we use the equations for fluctuations of velocity and magnetic field in rotating turbulence, which are obtained by subtracting equations for the mean fields from the corresponding equations for the actual (mean plus fluctuating) fields.
2.1 Governing equations
The equations for the fluctuations of velocity and magnetic fields are given by
where Equation (3) is written in a reference frame rotating with constant angular velocity , and is the fluctuating density. We consider an isothermal equation of state, , so that the density scale height is then constant. Here, is the sound speed, is the mean magnetic field, and are the full (mean plus fluctuating) fluid pressure and density, respectively. The terms and , which include nonlinear and molecular viscous and dissipative terms, are given by
where is the molecular viscous force and is the magnetic diffusion due to the electrical conductivity of the fluid.
2.2 The derivation procedure
To study rotating turbulence we perform derivations which include the following steps:
(i) use new variables and perform derivations which include the fluctuations of rescaled velocity and the mean magnetic field ;
(ii) derive equations for the second moments of the velocity fluctuations and the cross-helicity tensor in space;
(iii) apply the spectral closure, e.g., the spectral approximation (Pouquet et al. 1976; Kleeorin et al. 1990) for large fluid and magnetic Reynolds numbers and solve the derived second-moment equations in space;
(iv) return to physical space to obtain formulae for the Reynolds stress and the cross-helicity tensor as functions of . The resulting equations allow us then to obtain the dependence of the effect. In these derivations we only take into account effects that are linear in both and , where .
where . Furthermore, is the nonlinear term. We consider two cases: (i) low Mach numbers, where the fluid velocity fluctuations satisfy the equation in the anelastic approximation, and (ii) fluid flow with arbitrary Mach numbers, where velocity fluctuations satisfy continuity equation Equation (5). To derive Equation (8) we use the identities given in Appendix A.
2.3 Two-scale approach
We apply the standard two-scale approach with slow and fast variables, e.g., a correlation function,
(see, e.g., Roberts & Soward 1975). Hereafter we omit the argument in the correlation functions, , where
and we have introduced the new variables , , , . The variables and correspond to large scales, while and correspond to small scales. This implies that we assume that there exists a separation of scales, i.e., the maximum scale of turbulent motions is much smaller than the characteristic scale of inhomogeneity of the mean magnetic field.
2.4 Equations for the second moments
where , , and similarly for , , and similarly for . The source terms and in Equations (10) and (11) contain large-scale spatial derivatives of , which describe the contributions to turbulent diffusion ( and are given by Equations (A4) and (A6) in Rogachevskii & Kleeorin (2004)). The terms and are related to the third-order moments that are due to the nonlinear terms and are given by
where and .
The equations for the second-order moments contain higher order moments and a closure problem arises (Orszag 1970; Monin & Yaglom 1975; McComb 1990). We apply the spectral approximation or the third-order closure procedure (Pouquet et al. 1976; Kleeorin et al. 1990, 1996; Rogachevskii & Kleeorin 2004). The spectral approximation postulates that the deviations of the third-order-moment terms, , from the contributions to these terms afforded by the background turbulence, , are expressed through similar deviations of the second moments, , i.e.,
and similarly for the tensor . Here the superscript corresponds to the background turbulence (i.e., non-rotating turbulence with a zero mean magnetic field), is the characteristic relaxation time of the statistical moments, which can be identified with the correlation time of the turbulent velocity field for large Reynolds numbers. We also take into account that . We apply the -approximation (12) only to study the deviations from the background turbulence. The statistical properties of the background turbulence are assumed to be known (see below). A justification for the approximation in different situations has been obtained through numerical simulations and analytical studies (see, e.g., Brandenburg & Subramanian 2005; Rogachevskii et al. 2011).
2.6 Model for the background compressible turbulence
We use the following model for the background turbulence:
is the degree of compressibility of the turbulent velocity field, the terms take into account finite Mach numbers compressibility effects, , , is the exponent of the spectrum function for Kolmogorov spectrum), , is the maximum scale of turbulent motions, and is the characteristic turbulent velocity at scale . The motion in the background turbulence is assumed to be non-helical. For low Mach numbers (, Equation (13) satisfies to the condition: .
2.7 Contributions to the effect caused by rotation
Since our goal is to determine the effect, we solve Equations (10) and (11) neglecting the sources and with large-scale spatial derivatives of . We subtract from Equations (10) and (11) the corresponding equations written for the background turbulence, and use the spectral approximation. We only take into account the effects which are linear in and in . We also assume that the characteristic time of variation of the second moments is substantially larger than the correlation time for all turbulence scales. This allows us to get a stationary solution for Equations (10) and (11) for the second-order moments. Using this solution we determine the contributions to the mean electromotive force caused by rotating turbulence:
After performing the integration in space, we get:
For low Mach numbers and for , the tensor depends only on , i.e.,
where we used , and is the symmetric part of . Furthermore, the pumping velocity, is independent of rotation for small Coriolis numbers, because the rotational contribution to the pumping velocity vanishes. In this case the pumping velocity is determined only by the inhomogeneity of turbulent magnetic diffusivity.
For arbitrary Mach numbers and when
the diagonal part of the effect depends only on , i.e.,
The latter equation can be rewritten in the following form:
where is determined by Equation (18). In the next section we will determine the exponent from numerical simulations.
3 Numerical simulations
We now aim to explore the universality of the derived scaling by comparison with results from very different astrophysical environments. To do this, we performed three quite different types of simulations of:
(i) artificially forced turbulence of a rotating stratified gas, where the density scale height is constant and the turbulence is driven by plane wave forcing with a given wavenumber;
(ii) supernova-driven interstellar turbulence in a vertically-stratified local Cartesian model employing the shearing sheet approximation;
(iii) turbulent convection with and without overshoot layers.
The advantage of the first approach is that it allows to impose a well-defined vertical gradient of turbulent intensity, i.e., of the rms velocity of the turbulence such that is approximately constant over a certain interval, excluding the region near the vertical boundaries. Naturally, the physically more complex scenarios (ii) and (iii) are less-well controlled but allow to demonstrate the existence of the effect scaling in applications of direct interest to the astrophysical community. We use different variants of the test-field method to measure the effect (and other turbulent transport coefficients).
3.1 Basic equations
In the following we consider simulations where the turbulence is either driven by a forcing with a function , in which case we assume an isothermal gas with constant sound speed , or it is driven through heating and cooling either by supernovae in the interstellar medium or through convection with heating from below. In all cases, we solve the equations for the velocity, , and the density in the reference frame rotating with constant angular velocity and linear shear rate :
where is a combined Coriolis and tidal acceleration for pointing in the direction and a linear shear flow , is the kinematic viscosity, is a forcing function, is gravity, and is the traceless rate-of-strain tensor, not to be confused with the shear rate , and is the advective derivative with respect to the total (including shearing) velocity. In the isothermal case, the pressure is given by , while in all other cases we also solve an energy equation, for example in terms of the specific entropy , where and are respectively the specific heats at constant pressure and constant volume, their ratio is chosen to be 5/3, and obeys
where the temperature obeys , is the radiative flux, is a heating function, and is a cooling function. In the isothermal case, the entropy equation is not used, and the forcing function consists of random, white-in-time, plane, non-polarized waves with a certain average wavenumber, .
The simulations are performed with the Pencil Code (http://pencil-code.googlecode.com) which uses sixth-order explicit finite differences in space and a third-order accurate time stepping method (Brandenburg & Dobler 2002). The simulations of supernova-driven turbulence in the ISM have been performed using the Nirvana-III code (Ziegler 2004) with explicit viscosity and resistivity.
3.2 The test-field method
where is the magnetic gradient tensor. The test-field method works with a set of test fields , where the superscript stands for the different test fields. The corresponding mean electromotive forces are calculated from , where with
Here, and taken from the solutions of the momentum equation. In the case with shear, we replace by . On the top and bottom boundaries we assume perfect conductors, and for the and directions periodic boundary conditions. These small-scale fields are then used to determine the electromotive force corresponding to the test field . The number and form of the test fields used depends on the problem at hand.
We either use planar () averages, which depend only on and (hereafter referred to as test-field method I), or, alternatively, we assume that the mean field varies also in the and directions, but that the turbulence is homogeneous in those two directions, and that the direction constitutes a preferred direction of the turbulence (test-field method II). In the former case, I, only the and components of are important for dynamo action, and the magnetic gradient tensor has only two non-vanishing components which can be expressed in terms of the components of the mean current density alone. We put such that , with being the magnetic permeability, is the mean current density. Thus, we have
with and being either 1 or 2.
Alternatively, in case II, the mean electromotive force is assumed to be characterized by only one preferred direction which we describe by the unit vector . Then, can be represented in the form
with nine coefficients , , , . Like , also is determined by the gradient tensor . While is given by its antisymmetric part, is a vector defined by with being the symmetric part of . For details of this method, referred to below as test-field method II, see Brandenburg et al. (2012). Equation (2) is expected to apply to , while can, in certain cases, have opposite sign (Brandenburg et al. 1990; Ferrière 1992; Rüdiger & Kichatinov 1993). This is why we associate in the following in this equation with the defined in Equation (3.2).
Errors are estimated by dividing the time series into three equally long parts and computing time averages for each of them. The largest departure from the time average computed over the entire time series represents an estimate of the error.
3.3 Simulations of forced turbulence
We begin by studying forced turbulence. We consider a domain of size in Cartesian coordinates , with periodic boundary conditions in the and directions and stress-free, perfectly conducting boundaries at top and bottom, . The gravitational acceleration, , is chosen such that the density scale height is small compared with the vertical extent of the domain, i.e., . The smallest wavenumber that fits into the cubic domain of size is , so the density contrast between bottom and top is and the mean density varies like , where is a constant. In all cases, we use a scale separation ratio , a fluid Reynolds number between 60 and 100, a magnetic Prandtl number of unity. We use a numerical resolution of mesh points for all forced turbulence runs.
We perform simulations for different values of the rms velocity gradient, , but fixed logarithmic density gradient, . Thus, we have
i.e., can be obtained conveniently as the value of for which vanishes. By arranging the turbulence such that and are approximately independent of , the value of is also approximately constant. In that case, however, all the other turbulent transport coefficients are -dependent. However, by normalizing by and the other coefficients by , we obtain non-dimensional quantities that are approximately independent of . We denote the corresponding non-dimensional quantities by a tilde and quote in the following their average values over an interval , in which these ratios are approximately constant.
In Figure 1, we plot the normalized profiles of and as functions of . Note that within the range with and , both functions are approximately constant. For , they are of opposite sign; see the lower panel of Figure 1 and Table LABEL:Tresults. As discussed above, this behavior has been seen and interpreted in earlier calculations (Brandenburg et al. 1990; Ferrière 1992; Rüdiger & Kichatinov 1993).
In Figure 2 we plot the dependence of the normalized mean values of on the value of for two values of Co. The value of can then be read off as the zero of that graph. We find for low values of Co, and a somewhat smaller value () for larger values of Co.
In Figure 3 we show the -dependence of the remaining 7 normalized coefficients. They are all approximately independent of within the same range as before. In Figure 4 we show the scaling of these turbulent transport coefficients (including now also ) with . The results for and suggest a dependence proportional to the gradient of with between 0 and 0.3 for and between 0.7 and 1 for . On the other hand, all the other coefficients seem to be independent of and we find , , , , and . These results are quantitatively and qualitatively in agreement with those of Brandenburg et al. (2012). The fact that turned out to be essentially zero was addressed earlier (Brandenburg et al. 2012), where it was found that significant values are only found for scale separation ratios around unity, i.e., when the scale of the mean field is comparable to that of the turbulent eddies.
3.4 The case of supernova-driven ISM turbulence
We now turn to simulations of supernova-driven ISM turbulence (see Gressel et al. 2008a, b, for a detailed description of the model), similar to those of Korpi et al. (1999) and Gent et al. (2012), but extending to larger box sizes, thus allowing for better scale separation. In these simulations, expansion waves are driven via localized injection of thermal energy, . Additionally, optically thin radiative cooling with a realistic cooling function and heating lead to a segregation of the system into multiple ISM phases.
Here we solve the visco-resistive compressible MHD equations (supplemented by a total energy equation), using the Nirvana-III code (Ziegler 2004); for the full set of equations, we refer the reader to Equations (2.1) in Gressel (2010). For the present run, we chose a resolution of mesh points, and apply a value of . The fluid Reynolds number, defined as , varies within the domain and takes values –, which is somewhat larger than for the forced turbulence case, where . The Coriolis number is here .
Despite the aforementioned differences, the basic properties of the turbulence producing an effect are in fact quite similar: rotation of the system together with stratification in the mean density and turbulence amplitude. Notably, the ISM simulations are strongly compressible with peak Mach numbers of up to ten, corresponding to a typical value of , and with peak values up to five. We note that the ISM simulations also include shear with , which may affect the production of vorticity.
To obtain an estimate for from the time-series of a single simulation run, we apply a method distinct from the one described above: We treat the profile (inferred with test-field method I) as the data to be modeled and obtain error estimates by means of the standard deviation within four equal sub-intervals in time (see gray line and shaded areas in Figure 5). We then compute time averages of the profiles for and (here without error estimates), from which we compute a model prediction for based on the expression