THE ASTROPHYSICAL JOURNAL, 473 : L59–L62, 1996 December 10 q 1996. The American Astronomical Society. All rights reserved. Printed in U.S. A.
ON THE GENERATION OF EQUIPARTITION-STRENGTH MAGNETIC FIELDS BY TURBULENT HYDROMAGNETIC DYNAMOS P. CHARBONNEAU
AND
K. B. MACGREGOR
High Altitude Observatory, National Center for Atmospheric Research,1 P.O. Box 3000, Boulder, CO 80307-3000;
[email protected],
[email protected] Received 1996 August 12; accepted 1996 September 24
ABSTRACT The generation of a mean magnetic field by the action of small-scale turbulent fluid motions, the a-effect, is a fundamental ingredient of mean-field dynamo theory. However, recent mathematical models and numerical experiments are providing increasingly strong support to the notion that at high magnetic Reynolds numbers, the a-effect is strongly impeded long before the mean magnetic field has reached energy equipartition with the driving fluid motions. Taken at face value, this raises serious doubt as to whether the solar magnetic field is produced by a turbulent hydromagnetic dynamo after all, since it is an observed fact that the Sun does possess a structured, large-scale mean magnetic field of strength comparable to equipartition. In this Letter we demonstrate that the class of mean-field turbulent hydromagnetic models known as interface dynamos can produce equipartition-strength mean magnetic fields even in the presence of strong a-quenching. Subject headings: magnetic fields — MHD — Sun: interior — Sun: magnetic fields where h e 5 c 2 /4 ps e is the magnetic diffusivity (with s e the electrical conductivity). Solving equation (1) with ^U& and a given is a meaningful procedure only as long as the Lorentz force associated with the growing magnetic field remains small enough not to impede the driving flow, either on the large (^U&) or small (a, b) scales. When this ceases to be the case, evolution equations for ^U&, u, and b should also be solved. Since the whole point of mean-field theory is to avoid having to deal in detail with the small scale fluctuations, the quenching effect of the Lorentz force on the small-scale turbulent fluid motions is often introduced directly into equation (1) in parametric form, for example by making the a-effect itself an explicitly nonlinear function of the mean magnetic field:
1. MEAN-FIELD DYNAMOS AND THE SOLAR MAGNETIC FIELD
The magnetic fields of the Sun, planets, and Galaxy are generally believed to be contemporaneously generated, rather than being of fossil origin (e.g., Parker 1979). The underlying physical mechanism responsible for doing so is suspected to be a d ynamo, which fundamentally involves the conversion of bulk kinetic energy into magnetic energy. At its most basic level, the dynamo problem then consists in setting up a flow field U that can amplify a weak seed magnetic field B exponentially in time. In most situations of astrophysical interest U is a turbulent flow, which has motivated the use of mean-field theory (e.g., Moffatt 1978, and references therein) to construct turbulent hydromagnetic dynamo models. Briefly, the procedure consists in writing the fields in terms of a large-scale mean and small-scale fluctuating components (i.e., U 3 ^U& 1 u, and B 3 ^B& 1 b) and averaging over a suitably chosen intermediate scale. The resulting evolution equation for the mean magnetic field ^B& includes a mean electromotive force term (= 3 ^u 3 b&) which arises because even though the fluid and magnetic fluctuations individually average out to zero by assumption, their correlation ^u 3 b& in general does not. The system is closed by expressing the correlation as a tensorial expansion in the mean fields and their spatial derivatives. In the limit of isotropic turbulence, retaining the first two terms in this expansion yields ^u 3 b& 5 a^B& 1 b= 3 ^B&, with a and b being scalar functions related to the statistical properties of the small-scale fluctuations. The mean-field induction equation now includes two additional terms on its right-hand side; a source term a^B& (the “a-effect”), and an extra contribution to the dissipation term (the “turbulent magnetic diffusivity”):
a 3 a ~^B&! 5
,
(2)
where a 0 is a value characteristic of the linear regime and Beq is the equipartition field strength, based on the kinetic energy of turbulent convective fluid motions (at the base of the solar convection zone, Beq 1 10 4 G). Equation (2) “does the right thing,” in the sense that once ^B& significantly exceeds Beq , the a-effect is effectively shut off, and the growth of the mean magnetic field saturates at a level ^B& 1 Beq . Note that equation (2) implies that this happens once equipartition is reached between the small-scale turbulent fluid motions and the mean, large-scale magnetic field. It has been argued, however, that the small-scale fluctuating component of the magnetic field reaches equipartition with the turbulent fluid motions long before the large-scale mean field does, implying that equation (2) severely underestimates the efficacy of a-quenching. In particular, Vainshtein & Cattaneo (1992), Gruzinov & Diamond (1995), and Cattaneo & Hughes (1996) have presented numerical calculations and mathematical models on the basis of which they argue that a-quenching should be described instead by
^B& 5 = 3 ~^U& 3 ^B&! 1 = 3 ~ a ^B&! t 2 = 3 @~ b 1 h e!= 3 ^B&# ,
a0 1 1 ~u^B&u/Beq! 2
(1)
a 3 a ~^B&! 5
The National Center for Atmospheric Research is sponsored by the National Science Foundation. 1
L59
a0 1 1 Rm~u^B&u/Beq! 2
,
(3)
L60
CHARBONNEAU & MACGREGOR
where Rm [ ul/ h e is a magnetic Reynolds number characteristic of the microscopic properties of the flow (u being a typical turbulent velocity and l a typical length scale for the turbulence). Adopting this parameterization for a-quenching, one would now expect that in the nonlinear regime, exponential growth of the large-scale mean field ceases once ^B& 1 Beq /(Rm ) 1/2 . In terms of generating a mean magnetic field of strength significant with respect to equipartition, in the Rm .. 1 regime equation (3) is effectively a death sentence for the turbulent dynamo (see Kulsrud & Anderson 1992 for a discussion of this very same problem in the context of the galactic dynamo). The solar convective envelope (Rm 1 108 – 1010) easily meets the Rm .. 1 criterion, yet the Sun definitely manages to produce a structured large-scale magnetic field. Moreover, models of the rise of toroidal magnetic flux rope through the convective envelope suggest that the strength of that field may well need to significantly exceed equipartition for the observed properties of bipolar magnetic regions to be reproduced adequately (see Choudhuri & Gilman 1987; Moreno-Insertis et al. 1992; Fan, Fisher, & DeLuca 1993; D’Silva & Choudhuri 1993; Caligari, Moreno-Insertis, & Schu ¨ssler 1995). Broadly speaking, there are two avenues available to resolve this paradoxical situation. The first is to abandon altogether the idea of a turbulent hydromagnetic dynamo and concentrate instead on models that regenerate all components of the magnetic field directly through large-scale, essentially laminar processes; an example of these are the so-called Babcock-Leighton dynamos, which have been enjoying a revival in recent years (see Wang & Sheeley 1991; Durney 1995). The second possibility is to construct a dynamo relying on the classical turbulent a-effect, but to do so in such a way as to bypass a-quenching, even in the strong form embodied in equation (3) above. This Letter demonstrates that interface dynamos (Parker 1993; Tobias 1996; MacGregor & Charbonneau 1997) can achieve precisely this. 2. INTERFACE DYNAMOS: A MODEL
With ^B& 5 = 3 A 1 B(r, u, t)eˆf and under the assumption of axisymmetric mean components and with ^U& 5 r sin u V(r, u )eˆf , it is convenient to separate the mean magnetic field into poloidal and toroidal parts, i.e., with A 5 A(r, u , t)eˆf . Equation (1) is then be separated in poloidal and toroidal parts:
S
D
1 A 5 Ca a B 1 n ¹ 2 2 2 A, t Ã
S
(4a)
D
1 B 5 CV Ã ~= 3 A! z ~=V!1 n ¹ 2 2 2 B , t Ã
(4b)
where all lengths are expressed in units of the solar radius RJ , time in units of the diffusion time t 5 R 2J / h E , with the magnetic diffusivity normalized to its value in the envelope ( n 5 h / h E ), and with the dynamo numbers Ca 5 a 0 RJ / h E and CV 5 V 0 R 2J / h E making their usual appearance. Equations (4) define what is commonly referred to as an av dynamo. Note that the a-effect term has been dropped from equation (4b) a reasonable approximation in the case of the Sun. Specifying the angular velocity V(r, u ) and a-effect functional a (r, u ) allows us to reformulate the governing equations in terms of a linear eigenvalue problem. In a recent paper (Charbonneau & MacGregor 1996b, hereafter CM96), we have presented results of such linear calculations. Figure 1a shows one of the two solar-like internal differential rotation
Vol. 473
FIG. 1.—Solar-like angular velocity profile used for the calculations. (a) Contours of constant angular velocities in a meridional quadrants, for values 0.71 # V # 0.99 in increments DV 5 0.02. (b) Corresponding radial cuts in the region of the core-envelope interface (rE 5 0.7, indicated by a dotted line on [a]), in the equatorial plane and along the polar axis, together with the variations with depth of the radial shear in the equatorial plane. The rdependence of the a-effect source term is also shown (long-dashed line). A radial shear exists only below the base of the convective envelope and is restricted to a thin spherical layer.
profile used in CM96. It is characterized by a depth-independent rotation law in the convective envelope, matched smoothly across a thin layer onto a radiative core that is assumed to rotate rigidly at a rate comparable to the surface mid-latitudes. This is very much similar to the internal differential rotation inferred from helioseismic inversions of rotational frequency splittings (see, e.g., Tomczyk, Schou, & Thomson 1995). The magnetic diffusivity is assumed to vary discontinuously across the core-envelope interface (r 5 rE ), assuming above it a constant value ( h E ) characteristic of a turbulent regime ( h E 1 1010 –1012 cm 2 s 21 using mixing length estimates), and below it a constant value ( h C ) which should be of the order of the microscopic magnetic diffusivity for solar interior conditions, i.e., h C 1 103 –104 cm 2 s 21 . To avoid exciting dynamo modes associated exclusively with the latitudinal shear within the convective envelope, the a-effect is concentrated immediately above the core-envelope interface. A very important point to note from Figure 1 is that the radial shear and a-effect are segregated in spatially distinct portions of the spatial domain, and moreover operate in regions where the magnetic diffusivity assume markedly different values. Three distinct dynamo modes can be supported under a solar-like internal angular velocity profile. True interface modes can exist in association with either the (negative) radial shear below the core-envelope interface in the polar regions,
No. 1, 1996
GENERATION OF MAGNETIC FIELDS
or with the (positive) radial shear in the equatorial regions. Furthermore, a dynamo mode feeding on the latitudinal shear both above and below the core-envelope interface can also be excited. Because this mode requires the presence of a latitudinal shear both below the interface and above, where the a-effect operates, this is not an interface mode in the strict sense of the term. Because it does share a number of properties with true interface modes, it was labeled “hybrid mode” in CM96. In the linear regime, which of these three modes is to be preferentially excited depends on the diffusivity ratio n 5 h C / h E , dynamo numbers, assumed shear layer thickness, and adopted angular dependency for the a-effect. Representative solutions for these three dynamo modes can be viewed at the URL: http://www.hao.ucar.edu/public/research/si/ interface/interface.html. A property common to all three of these dynamo modes is their ability to produce very strong magnetic fields within the shear layer while producing much weaker fields at the base of the convective envelope, where the a-effect is presumed to operate. The ratio of peak toroidal field strengths on either side of the interface in the course of the cycle, in the linear regime, is found to scale approximately as n 21/2 for the hybrid mode, and n 21 for the interface modes in the supercritical regime, at least in the parameter range explored in CM96. The strong form of a-quenching (eq. [3]) suggests that the strongest field that can be tolerated in the a-effect region is uBu 1 Beq /(Rm ) 1/2 . It then follows that the maximum field strength in the shear layer attainable by an interface dynamo should be uBu Beq uBu Beq
1
1
ÎnR
Hybrid mode ,
(5a)
Interface mode ,
(5b)
m
1
1
n ÎRm
Assuming now that h E 1 ul in the (kinematic) turbulent regime, it follows that n 21 5 ul/ h C [ Rm if h C is of the order of the microscopic diffusivity. From these order-of-magnitude arguments one would be lead to believe that large-scale, equipartition field strengths can be attained by the hybrid mode and exceeded by true interface modes, as long as Rm . 1. 3. NONLINEAR INTERFACE DYNAMO SOLUTIONS INCLUDING a-QUENCHING
Equations (5a) and (5b) are preempted on the assumption that the scalings characterizing the linear solutions carry over to the nonlinear regime. In order to verify whether or not this is the case we have solved equations (3) and (4) as a nonlinear initial-boundary value problem, using the angular velocity and a-effect profiles shown on Figure 1. We used a few representative (supercritical) linear solutions from CM96 as initial conditions. We define the following two measures for the strength of the mean toroidal field; the unsigned maximum within a preestablished sampling volume V, and a rms average defined over the same volume: Bmax 5 max ~uBu! ,
(6a)
V
^B& 5
3E 1
V
V
B2 dV
4
1 /2
.
(6b)
L61
FIG. 2.—Evolution of the maximum (thick curves) and rms (thin curves) Toroidal field strengths above (dotted lines) and below (solid lines) the core-envelope interface, for a representative equatorial interface solution (a) and hybrid solution (b). Time is measured in units of the diffusion time t 5 R 2J / h E 1 150 yr for h E 5 1012 cm 2 s 21 . The interface solutions is defined by the parameter values CV 5 105 , Ca 5 275, d/RJ 5 0.03, n 5 1022 , and Rm 5 104 ; in the linear regime, it has a growth rate s 5 1.35 yr 21 and period P 5 7.0 yr. The hybrid solution has parameter values CV 5 105 , Ca 5 12.5, d/RJ 5 0.1, n 5 1025 and Rm 5 105 ; it has a growth rate s 5 19.3 yr 21 and period P 5 13.1 yr in the linear regime.
The sampling volume is chosen so as to cover the portion of the shear layer where the dynamo is operating. We compute similar quantities for a sampling volume located at the base of the convective envelope, where the a-effect operates. The time histories of these four quantities are shown in Figure 2 for a representative equatorial interface solution (Fig. 2a), and a representative hybrid solution (Fig. 2b). The former has Rm 5 104 and n 5 1022 ; the latter has Rm 5 105 and n 5 1025 . Early in the evolution the toroidal field grows exponentially until the mean field at the base of the envelope (thin dotted lines) becomes comparable to Beq /(Rm ) 1/2 , at which point the growth of the field is halted both above and below the core-envelope interface. This occurs at t/ t 1 0.075 and 0.7 for the interface and hybrid mode, respectively, reflecting the very different linear growth rates for these two solutions. In both cases however, after nonlinear saturation the rms and peak field strengths in the shear layer (solid curves) respectively approach well within a factor of 2 (^ B &) and exceed [max (B)] equipartition. In the saturated phase, the ratio of peak field strengths above and below the interface for the hybrid mode (Fig. 2b, thick curves) is compatible with the n 21/2 dependency characterizing the linear regime. The behavior of the equatorial interface solution (Fig. 2a) is more complex. While the solution initially saturates more or less at the level expected from equation (5b), an obvious transition occurs at t/ t 1 0.35.
L62
CHARBONNEAU & MACGREGOR
Examination of the solution reveals that the quenching of the equatorial interface mode leads to the growth of a hybrid mode, with the result that at later times the solution is effectively a combination of this hybrid mode with the original interface mode. The important point remains that the saturated field strengths in the shear layer are compatible with the estimates produced using the scalings from the linear regime (eqs. [5a] and [5b]). Interface dynamos can avoid a-quenching, even in the strong form embodied in equation (3). 4. DISCUSSION AND CONCLUSION
We have presented a few representative nonlinear interface dynamo solutions, and demonstrated that such dynamos are capable of avoiding a-quenching, even in its strong form recently put forth by various authors. Interface dynamos achieve this effect primarily by letting the shear and a-effect operate in spatially distinct portions of the spatial domain. Interface dynamos differ from conventional mean-field av dynamos (by which we are referring to models where the shear and a-effect are spatially coincident) in one important respect, namely the crucial role played by magnetic diffusion. In conventional mean-field models, magnetic diffusion is needed to destroy magnetic flux from previous cycles, even though a cursory glance at equations (4a) and (4b) may lead one to
believe that the smaller h is the better, since diffusion represents a sink in the dynamo equations. With interface dynamos, magnetic diffusion plays a second and critical role. Because the shear and a-effect regions are spatially segregated, magnetic diffusion is needed to couple the two source regions: the toroidal magnetic flux produced in the shear layer must diffuse upward into the convective envelope for the poloidal field to be regenerated, and vice versa. If this diffusive transports stops for any reason, the dynamo cannot operate. The modifications to the turbulent fluid motions caused by the Lorentz-Force associated with the small-scale, fluctuating component of the growing magnetic field is at the root of a-quenching, whether described by equation (3) or some other parameterization. These same modifications may be expected to lead to a reduction in the net magnetic diffusivity, which in mean-field theory is also of turbulent origin. Not surprisingly, nonlinear interface dynamo calculations incorporating such effects via a parameterization similar to equation (2) show that the interface modes are extremely sensitive to “h-quenching.” It is then of primary importance to conduct numerical experiments similar to those having been carried out to measure a-quenching, but aimed at measuring h-quenching in turbulent three-dimensional flows with high magnetic Reynolds number.
REFERENCES Caligari, P., Moreno-Insertis, F., & Schu ¨ssler, M. 1995, ApJ, 441, 886 Cattaneo, F., & Hughes, D. W. 1996, Phys. Rev. Lett., submitted Charbonneau, P., & MacGregor, K. B. 1996, ApJ, 454, 901 Choudhuri, A. R., & Gilman, P. A. 1987, ApJ, 316, 788 D’Silva, S., & Choudhuri, A. R. 1993, A&A, 272, 621 Durney, B. R. 1995, Sol. Phys., 160, 213 Fan, Y., Fisher, G. H., & DeLuca, E. E. 1993, ApJ, 405, 390 Gruzinov, A. V., & Diamond, P. H. 1995, Phys. Plasmas, 2, 1941 Kulsrud, R. M., & Anderson, S. W. 1992, ApJ, 396, 606 MacGregor, K. B., & Charbonneau, P. 1997, ApJ, submitted
Moffatt, H. K. 1978, Magnetic Field Generation in Electrically Conducting Fluids (Cambridge: Cambridge Univ. Press) Moreno-Insertis, F., Schu ¨ssler, M., & Ferriz-Mas, A. 1992, A&A, 264, 686 Parker, E. N. 1979, Cosmical Magnetic Fields (Oxford: Clarendon) ———. 1993, ApJ, 408, 707 Tobias, S. M. 1996, ApJ, 467, 870 Tomczyk, S., Schou, J., & Thomson, M. J. 1995, ApJ, 448, L61 Vainshtein, S. I., & Cattaneo, F. 1992, ApJ, 393, 165 Wang, Y.-M., & Sheeley, N. R., Jr. 1991, ApJ, 375, 761
