Kinematic Dynamo Theory Research Articles

Analytic theory of dynamos

Recent developments in kinematic dynamo theory are described. Attention is restricted to the case of large magnetic Reynolds number dynamos. The nature of the “slow” Braginsky dynamo, which operates on the magnetic diffusive timescale, is outlined. The possible existence of “fast” dynamos, which grow on the time scale of the convective motions is discussed. Only steady flow is considered but the case of particle paths confined to stream surfaces is carefully distinguished from the case with chaotic particle paths. The underlying theme is that of flux expulsion with magnetic field confined to ropes and sheets.

Exact evaluation of the effect of an arbitrary mean flow in kinematic dynamo theory

An exact derivation is given of the dynamo equation in the presence of an arbitrary incompressible mean flow v0. A mixed representation is used specifying v0 in terms of a Lagrangian displacement, while Eulerian coordinates are employed for the turbulent velocity v superimposed on v0.When the first-order-smoothing approximation is made (FOSA; valid when the turbulent velocity v has a short correlation time τc) the usual dynamo equation is recovered, except that the turbulent velocity v in the tensors αis and βisk is replaced by . The bar represents the effect of advection and is expressed solely in terms of the Lagrangian coordinate specifying the mean flow v0. Thus the intuitive idea is confirmed that dynamo action depends only on velocity correlation functions measured at a point comoving with the mean flow. The result admits easy evaluation in actual model situations. This is illustrated with an example tailored to the solar dynamo. A shear in v0 causes a (kinematic) anisotropy in the tensors αis and βisk. This can be a large effect, which comes on top of the intrinsic (dynamical) anisotropy in the velocity correlation functions. Subsequently, the analysis is extended beyond FOSA up to arbitrary order, relevant for long correlation times τc on the basis of the work of Van Kampen (1974). It is shown that the same formalism is also applicable to the problem of turbulent transport of a scalar.Conditions for applicability of the work are (1) very large magnetic Reynolds number, (2) incompressible flows v and v0, (3) stationary mean flow v0, and (4) correlation time τc [Lt ] period of the dynamo.

Theory of the solar cycle

The properties of kinematic αω -dynamos are briefly reviewed. The mean field concept, including turbulent diffusivity, is defended against recent criticism. It is pointed out that although the Maunder minimum cannot be explained by kinematic dynamo theory alone, this does not invalidate dynamo theory in general. A special discussion is devoted to attempts to evaluate the coefficients of the mean field induction equation in the case of very large conductivity. The field then behaves intermittent, in the form of locally concentrated flux tubes, and the α-effect and the turbulent diffusivity may be determined by asymptotic techniques or with the help of an exact solution of the non-dissipative induction equation in Lagrangian co-ordinates.

The cowling anti-dynamo theorem

Abstract The Cowling anti-dynamo theorem is proved using the Bullard and Gellman spectral formulation of kinematic dynamo theory.

On conserved quantities in kinematic dynamo theory

Using a Lagrangian approach to the magnetic induction equation in an infinite medium, we demonstrate that there exist seven conserved quantities which, by analogy with classical mechanics, we label as ’’energy,’’ ’’momentum,’’ and ’’angular momentum.’’ For prescribed fluid motions we spell out the detailed conservation equations. For a fluid motion which is turbulent we also give the average conserved quantities. In a pragmatic sense it is expected that these conservation laws will be of use in attempts to obtain numerically accurate solutions to the turbulent kinematic dynamo equations. Since the magnetic induction equation is not self-adjoint, numerical attempts to date have to impose some extraneous ad hoc ’’criteria of goodness’’ at any given level of numerical truncation. The conserved quantities given here provide an internal check of the accuracy of any numerical calculation without the necessity for arbitrarily imposed external criteria of accuracy. As such they should be a powerful tool in rapidly increasing the accuracy of numerical solutions to the kinematic dynamo equations. We also point out that the conserved quantities can be used to indicate the possibility of kinematic dynamo activity ahead of any detailed calculations.

Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. III. Solution of the Kraichnan equations under Parker's ``short-sudden'' conditions

Using Parker's ``short-sudden'' conditions, we solve the Kraichnan equations exactly. We find that the normal modes of the ensemble average magnetic field have very different properties in this case than when the velocity turbulence is static, which situation was investigated in earlier papers in this series. We have done this calculation for two reasons: first because exact solutions of the Kraichnan equations are few in number, and second because the nonlinearity and singularity of the Kraichnan equations is such as to emphasize the physical difference in the properties of the normal modes of the average magnetic field under a small change in the prescription of the turbulent velocity field.

Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. II. Incompressible, helical velocity turbulence and a pair of coupled, singular, nonlinear integral equations

Using Kraichnan's direct interaction approximation, we set up the equations governing the normal modes of the ensemble average magnetic field under incompressible, nonmirror symmetric velocity turbulence. We show that (i) the Green's stress tensor enjoys equipartition of its symmetric and antisymmetric parts at the normal mode frequencies of the ensemble average field, (ii) for static velocity turbulence, including helicity, there are no growing modes, (iii) the commonly used first order smoothing theory approximation is invalid when compared to the Kraichnan equations, for the Kraichnan equations do not satisfy Hammerstein's theorem while first order smoothing theory requires the satisfaction of Hammerstein's theorem, (iv) if there is to be any growth of the ensemble average magnetic field it must come from time dependent velocity turbulence, and when the velocity turbulence is time dependent we have so far been unable to solve the Kraichnan equations. We have done these calculations for two reasons. First to illustrate, by exact solution, the manner in which the normal modes of the ensemble average magnetic field depend on the helicity and Reynolds number of the turbulent velocity field. Second to show that approximate treatments of the hydromagnetic equation (like first order smoothing theory), rather than exact solution, are liable to give rise to substantial error in view of the fact that the Kraichnan equations do not satisfy Hammerstein's theorem.

Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. I. Incompressible isotropic turbulence and a singular integral equation

Using Kraichnan's direct interaction approximation in the induction equations we set up the kinematic equations governing behavior of an ensemble average magnetic field under homogeneous, incompressible, mirror symmetric isotropic velocity turbulence. We demonstrate that the normal modes of the field depend on the solutions to a nonlinear single integral equation. For a simple form of the velocity turbulence we have investigated some of the properties of the integral equation. In particular, we have been able to construct a particular solution. We also point out what remains to be done if we are to obtain all the modes of the ensemble average magnetic field. We have done this calculation so that the behaviour and normal modes of the field can be investigated at arbitrary magnetic Reynolds' numbers. This is in contrast to customary approximations (like first-order smoothing theory) which normally are valid only for very small Reynolds numbers, if at all, and which therefore omit large regimes of considerable physical interest.

Kinematic Dynamo Theory. V. Comments on Diverse Matters Including Historical Development, Isotropic Turbulence, and Expansion Techniques

view Abstract Citations (8) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Kinematic Dynamo Theory. V. Comments on Diverse Matters Including Historical Development, Isotropic Turbulence, and Expansion Techniques Lerche, I. Abstract We show that three of the four major criticisms by Krause and Roberts of our work in turbulent kinematic dynamo theory possess fundamental errors, and are without foundation. We also correct the errors in the historical record of the development of turbulent kinematic dynamo theory presented in their paper. We agree with Krause and Roberts that violation of Bochner's theorem is not a matter to be treated lightly-particularly for velocity turbulence which includes helicity. We demonstrate violation of this theorem for the turbulence used by Steenbeck, Krause, and (1966) to calculate their "a-effect." We further point out that the ansatz &v x B) = aB, 50 often invoked by Steenbeck, Krause, and , cannot be correct except under Parker's "short-sudden" approximation. Subject headings: hydromagnetics - turbulence Publication: The Astrophysical Journal Pub Date: May 1973 DOI: 10.1086/152106 Bibcode: 1973ApJ...181..993L full text sources ADS | Related Materials (3) Part 1: 1971ApJ...166..627L Part 2: 1971ApJ...166..639L Part 3: 1971ApJ...168..115L

Nearly Symmetric Kinematic and Hydromagnetic Dynamos

Certain aspects of the hydromagnetic and kinematic dynamo theory related to the Earth's magnetic field are considered. The work of Tough and Roberts [J. G. Tough and P. H. Roberts, Physics of the Earth and Planetary Interiors (North-Holland, Amsterdam, 1968), Vol. 1, p. 288] on the hydromagnetic dynamo is extended in two respects. First, the most general form of the prescribed body force which is consistent with the model is considered. Secondly, the mean quantity 〈Sφ〉 [see Eqs. (18)] is determined up to order R−1/2, where R is the magnetic Reynolds number. It is shown that the remarkable effective variables first introduced by Braginskii {S. I. Braginskii, Zh. Eksp. Teor. Fiz. 47, 1084 (1964) [Sov. Phys. JETP 20, 726 (1965)]} are relevant in the hydromagnetic dynamo in the second approximation. It is also shown by considering a related problem concerning the heat conduction equation that effective variables are unlikely to have any relevance in the kinematic dynamo theory in the third approximation.

Kinematic Dynamo Theory: The Dyson Equation and the Large-Scale Field; the Bethe-Salpeter Equation and the Fluctuation Intensity

An exact statistical set of kinematic dynamo equations is given representing turbulent generation of both a large-scale magnetic field and a small-scale turbulent field. These equations do not rely on approximately solving the fluctuation equations and using the results in the ordered field equations as do most treatments of statistical kinematic dynamos. Instead they treat both the fluctuation equations and the ordered field equation exactly. The results obtained indicate several points. First: The fluctuation intensity equation has the character of the Bethe-Salpeter equation. Depending on whether one uses the long-slow or the short-sudden approximation there may, or may not, be an upper limit on the velocity turbulence in order that the energy density stored in the magnetic field fluctuations remain finite. No such restriction is found using approximate kinematic dynamo equations. Second: The normal modes of the large-scale field (which obeys a Dyson equation) may, or may not, be mirrored in the singular eigenmodes of the fluctuation intensity equation. Third: the structure of the exact statistical kinematic dynamo equations is very different from the structure of the approximate kinematic dynamo equations-particularly in the equation describing the fluctuation intensity. We have done this problem in order to demonstrate that the exact solution of at least one problem in statistical kinematic dynamo theory introduces new and interesting phenomena which are not brought to light in approximate treatments of the same phenomena.

Kinematic-Dynamo Theory. III. The Effect of Turbulent Diffusivity in the Dynamo Equations

view Abstract Citations (6) References (15) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Kinematic-Dynamo Theory. III. The Effect of Turbulent Diffusivity in the Dynamo Equations Lerche, I. Abstract The kinematic-dynamo equations, developed previously for a constant diffusivity, are further generalized to include the effect of turbulent diffusivity. This generalization is necessary since current ideas concerning galactic, and solar, dynamo action indicate that turbulent diffusivity is larger than "classical" diffusivity by about three orders of magnitude. Accordingly it must be included if a better understanding is to obtain of dynamo action in turbulent media. Some simple cases are worked through to illustrate the physical importance of turbulent diffusivity on dynamo action. Publication: The Astrophysical Journal Pub Date: August 1971 DOI: 10.1086/151066 Bibcode: 1971ApJ...168..115L full text sources ADS | Related Materials (3) Part 1: 1971ApJ...166..627L Part 2: 1971ApJ...166..639L Part 4: 1973ApJ...181..993L

Kinematic-Dynamo Theory. II. Dynamo Action in Infinite Media with Isotropic Turbulence

view Abstract Citations (20) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Kinematic-Dynamo Theory. II. Dynamo Action in Infinite Media with Isotropic Turbulence Lerche, I. Abstract The general kinematic-dynamo equations derived in an earlier paper are applied to infinite media. It is demonstrated that under homogeneous, isotropic, and stationary velocity turbulence, an infinite, stationary medium possesses large-scale magnetic-field components which are stationary, and of a scale much larger than the correlation length of the weak random velocity turbulence. It is further shown that in the vicinity of each stationary mode (of the mean magnetic field) there exist both growing and decaying dynamo states. And whether the growing dynamo states lie to the long-wavelength side or the short-wavelength side of the stationary states is determined by the relative sizes of the resistivity and the turbulent-velocity field. It is also shown that inclusion of a small bulk velocity converts the stationary modes, and the regenerative and degenerative modes in their vicinity, into progressive dynamo waves which propagate (the stationary modes), propagate and grow (the regenerative modes), or propagate and decay (the degenerative modes). We also give a prescription for reducing the reduced dynamo equation derived elsewhere (valid for small bulk velocities and small bulk-velocity shears) even further under the "short-sudden" approximation. This reduction is of practical significance since it enables various terms in the dynamo equations to be singled out and investigated for their direct contribution (if any) to dynamo action. Publication: The Astrophysical Journal Pub Date: June 1971 DOI: 10.1086/150989 Bibcode: 1971ApJ...166..639L full text sources ADS | Related Materials (3) Part 1: 1971ApJ...166..627L Part 3: 1971ApJ...168..115L Part 4: 1973ApJ...181..993L