aa r X i v : . [ phy s i c s . g e o - ph ] J a n Global fluctuations in magnetohydrodynamicdynamos
V. Tanriverdi, A. Tilgner
Institute of Geophysics, University of G¨ottingen, Friedrich-Hund-Platz 1, 37077G¨ottingenE-mail: [email protected]
E-mail: [email protected]
Abstract.
The spectrum of temporal fluctuations of total magnetic energy for severaldynamo models is different from white noise at frequencies smaller than the inverseof the turnover time of the underlying turbulent velocity field. Examples for thisphenomenon are known from previous work and we add in this paper simulations ofthe G.O. Roberts dynamo and of convectively driven dynamos in rotating sphericalshells. The appearance of colored noise in the magnetic energy is explained by simplephenomenological models. The Kolmogorov theory of turbulence is used to predict thespectrum of kinetic and magnetic energy fluctuations in the inertial range.PACS numbers: 91.25.Cw, 47.65.-d, 47.27.-i, 47.27.Gs lobal fluctuations in magnetohydrodynamic dynamos
1. Introduction
Several experiments have been carried out in the last decade in liquid sodium athigh magnetic Reynolds numbers and in highly turbulent flows. Measurements ofmagnetic field fluctuations, either due to an externally imposed magnetic field or due tomagnetic field generated through a dynamo effect by the sodium flow itself, generallyreveal spectra with a 1 /f -noise at low frequency [1, 2, 3]. Numerical simulations ofmagnetohydrodynamic (MHD) flows showed the same phenomenon [4, 5] althoughsometimes the precise form of the low frequency noise varies [6]. The precise value of theexponent in the algebraic decay in a spectrum is not so important, but a behavior otherthan white noise, i.e. colored noise, always invites closer inspection [7]. For example, a1 /f -noise, frequently called flicker noise, cannot extend down to zero frequency if thepower integral is to stay finite. The low frequency cutoff to the 1 /f -spectrum shouldcarry some informations about the dynamics of an MHD dynamo.Measurements of velocity in a fixed point of a turbulent flow frequently find whitenoise at low frequency (see for example the data compilation in fig. 6.14 of [8]), but thereare also examples of colored noise [9]. The simultaneous presence of colored noise inboth magnetic and velocity spectra looks like a plausible combination: If flow velocitiesincrease in a dynamo, the magnetic field is amplified more rapidly and the amplitude ofthe magnetic field increases. If the velocity has no white noise, so should the magneticfield have a nontrivial spectrum. In the Karlsruhe experiment on the other hand [1]the 1 /f -fluctuations in magnetic signals had no correspondence in pump pressures orvolumetric flow rates. This prompts us to investigate in more detail the mechanismsleading to colored noise in magnetic spectra.This paper will deal with the fluctuations of total magnetic energy, which is aglobal measure of the field amplitude. A theory for the fluctuations of a global quantityis of interest even though laboratory experiments usually report local measurements.Numerical simulations always compute total magnetic and kinetic energies. Theirstatistics are less noisy than those of field amplitudes at a given point because ofthe spatial averaging inherent to the computation of total energies. Spectra of thegeomagnetic field are frequently plotted as variations of another global quantity, thedipole moment [10], because the geomagnetic field is dominated by the dipole componentand low frequency variations reflect variations of the dipole moment. Secular variationsof the Earth’s magnetic field again exhibit colored noise [11, 12]. In astrophysics, we dohave local measurements of solar magnetic fields, but we only know a global amplitudefor most stellar magnetic fields.The goal of this paper is to explore conditions leading to colored noise in the totalmagnetic energy and to show that this type of noise can appear even if the kineticenergy has white noise. Three simplified models of MHD flows are presented in section2. The first two model the slow fluctuations per se, the last one by contrast computes thespectrum of inertial range fluctuations in homogeneous and isotropic turbulence. Section3 presents numerical simulations of the G.O. Roberts dynamo [13] and compares the lobal fluctuations in magnetohydrodynamic dynamos
2. Phenomenological models
The basic problem in this paper is to find the temporal spectrum of a quantity derivedfrom a magnetic field B ( r , t ), such as the magnetic energy. The induction equationgoverns the evolution of B in a liquid conductor with magnetic diffusivity λ whosemovement is given by the velocity field v ( r , t ): ∂ t B + ∇ × ( B × v ) = λ ∇ B (1)Exact solutions of this equation are difficult to obtain, so that we resort tophenomenological models. Mean field magnetohydrodynamics have proven a most fruitful simplification of theinduction equation [14]. In this approach, the effect of small scale fluctuations on thelarge scales are not computed exactly but are modeled, in the simplest case as an α − effect. The number of magnetic degrees of freedom which need to be retained isthus reduced and in an extreme simplification, only one mode remains. If we call B the amplitude of that mode, ˜ α ( t ) and β the coefficients describing the α − effect andits quenching, respectively, and µ a coefficient related to the magnetic dissipation, thesimplest model reproducing the main features of the induction equation is: ∂ t B = ˜ α ( t ) B − βB − µB. (2)˜ α is allowed to be time dependent in order to reflect a time dependent velocity field.This time dependence will be essentially random for a turbulent velocity field. Thereduction of the α − effect by the term βB models the retroaction of the magnetic fieldon the velocity field via the Lorentz force (which is quadratic in the magnetic field) inthe Navier-Stokes equation. We now consider α ( t ) = ˜ α ( t ) − µ to be a random processwith mean square h α i and remove the dimensions from eq. (2) by expressing timein multiples of h α i − and the magnetic field amplitude in multiples of ( h α i /β ) / .The adimensional quantities t ′ , α ′ and B ′ are given by t ′ = t h α i , α ′ = α/ h α i and B ′ = B q β/ h α i . In the remainder of this section, all quantities are understood tobe nondimensional and the primes are omitted for convenience. The nondimensionalvariables then obey the equation: ∂ t B = α ( t ) B − B (3)in which α ( t ) is a random variable with h α i = 1. As long as B is small, the solution tothis equation is B ( t ) = B (0) exp (cid:18)Z t α ( τ ) dτ (cid:19) . (4) lobal fluctuations in magnetohydrodynamic dynamos B ( t ) − B (0) B (0) ≈ Z t α ( τ ) dτ. (5)Taking the Fourier transform of this equation, it follows that the spectrum of B is,apart from frequency independent prefactors, the same as the spectrum of α divided bythe square of the angular frequency, ω . For example, if the spectrum of α is a whitenoise, the spectrum of B behaves as ω − . For large times t , eqs. (4) and (3) becomea poor approximation, which means that the ω − will not be observable below somecutoff-frequency. If the mean of α , h α i , is different from zero, B will be large enough forthe nonlinear term in eq. (3) to become dominant after a time on the order of h α i − .In that regime, and concentrating on slow fluctuations, eq. (3) reduces to B = α .Considering again the example of α ( t ) with a white noise, one finds a spectrum of B which is a white noise, too. The transition in the spectrum of B from ω to ω − occursat a frequency which increases with increasing h α i , because eq. (5) fails at earlier times t . In order to test these ideas, we solved eq. (3) numerically. The random α ( t )was generated by sending the output of a gaussian deviate random number generatorthrough a Butterworth filter [15, 16]. The filter was adjusted such that its output hada spectrum as a function of frequency f = ω/ (2 π ) in 1 / (1 + ( f /f ) ) with f = 50. Thespectrum of B is shown in fig. 1 for different h α i . As expected, this spectrum decays as ω − for ω > ω = 2 πf , and as ω − for ω c < ω < ω . Below the cut-off ω c , the spectrumis independent of ω , and ω c ∝ h α i .Even though the spectra in fig. 1 differ only in the value of ω c , the qualitativeappearence of the underlying time series is quite variable: The time series consists ofintermittent bursts for small h α i and of random fluctuations around a well defined meanfor large h α i . A study of the probability distribution function of the solutions of eq. (2)together with a few graphs of representative time series is presented in [17]. We will now investigate under which conditions the single mode model is applicable tomore general systems and extend the discussion of the previous section to include severalmodes. It will be shown that the predictions of the single mode model are recovered inthe limit of small fluctuations.The precise form of the dynamical system used as a model does not matter muchfor the following analysis, but a specific system has to be chosen for the numericalexamples. In order to stay as close as possible to the previous section, let us assume ∇ · v = 0 and rewrite the left hand side of eq. (1) as ∂ t B i + P j v j ∂ j B i − P j B j ∂ j v i . Wethen proceed through the same steps as before and replace the combination of velocityand derivation by a random variable in which we absorb the dissipative term and theright hand side, remove dimensions, and model saturation through a cubic term. This lobal fluctuations in magnetohydrodynamic dynamos Figure 1.
Spectral power density of B , the solution of eq. (3), as a function offrequency f = ω/ (2 π ) for < α > = 1 (red), 5 (green) and 15 (blue). The straight linesindicate the power laws ω − and ω − . leads to the following system: ∂ t B i +( α ( t )+ α ( t )+ α ( t )) B i − α i ( t )( B + B + B ) = − B i , i = 1 , , , (6)in which the α i ( t ) are random variables. This system bears only a metaphorical relationwith the original induction equation and will be used to exemplify two limiting cases:If the fluctuations of the α i ( t ) are small compared with the mean of the α i ( t ), thesolution of (6) will be close to the solution of the time independent system in which each α i ( t ) in (6) is replaced by its mean h α i i . Let us assume h α i = h α i = h α i <
0. Aneigenvalue analysis of the left hand side of (6) then reveals one neutral mode and twomodes with equal and positive growth rate. In the presence of small fluctuations, theneutral mode will not contribute significantly to the dynamics. If the initial conditionsand the nonlinear term select an arbitrary direction in the space spanned by the twodegenerate growing modes, we expect (6) to behave the same as the single mode model.This is a fortiori true for a dynamical system with a single non-degenerate growingmode.If on the other hand the fluctuations of the α i ( t ) are large compared with theirmeans, the dynamics is not dominated by a single mode anymore and the analysis ofthe previous section breaks down. The spectrum of the fluctuations of qP B i can nowbe different and must be found from numerical computation.Fig. 2 shows some examples of solutions of eq. (6) in which the spectrum of thefluctuations of the α i ( t ) is in 1 /f [18] and h α i i = 1. Let us first consider the case inwhich the fluctuations of the α i are small compared with their mean. The spectrum of qP B i shown in fig. 2 should then behave as predicted by the single mode model: Atthe smallest frequencies, the spectrum of qP B i must decay the same as the spectrumof the α i , i.e. as 1 /f in the present example. Above a frequency on the order of h α i i , lobal fluctuations in magnetohydrodynamic dynamos f between the power laws followed by the spectra of qP B i andthe α i , which implies a spectrum in 1 /f for qP B i in the example considered here. Allthese predictions fit well the spectrum shown in fig. 2 for h α i i = 1 and h α i i = −
5. Inorder to further support the applicability of the single mode model, we also computedthe angle between the instantaneous vector B ( t ) = ( B ( t ) , B ( t ) , B ( t )) and its mean h B i . The cosine of that angle, h B i · B ( t ) / q |h B i| | B ( t ) | in the statistically stationarystate stays larger than 0.99 for h α i i = − h α i i = − .
01. In the latter case the fluctuations of the α i are large compared with theirmean and different exponents unrelated to the single mode model become possible. Adecay in 1 /f appears in fig. 2 which will become important again in connection withthe convectively driven dynamos in spherical shells discussed below. Figure 2.
Spectral power density of pP B i obtained from the solution of eq. (6), asa function of frequency f = ω/ (2 π ), for h α i i = 1 and h α i i = − .
01 (red), -0.3 (green)and -5 (blue). The straight lines indicate the power laws ω − , ω − and ω − . The main focus of this paper is on slow fluctuations, but it is also worthwhile to havea look at the fast fluctuations in order to see where the differences are. We also wouldlike to check whether it was reasonable in section 2.1 to assume a spectrum for α whichis flat at low frequencies and steeper at large frequencies. We will specifically look atfluctuations of total kinetic and magnetic energy. The spectra of these fluctuationsmust be different from flow to flow, but we can expect a unique behavior in flows towhich the Kolmogorov phenomenology (K41 in short after [19]) of homogeneous andisotropic turbulence applies. We will employ this phenomenology in a cartesian domainwith periodic boundaries, so that it is useful to expand all fields in Fourier series. Mucheffort has been spent in the past to theoretically deduce the wavenumber dependenceof the Fourier coefficients, the best known theory being of course K41. But the Fourier lobal fluctuations in magnetohydrodynamic dynamos h E kin i of a turbulent flow is decomposed into contributionsmade by wavevectors of modulus k in the form h E kin i = R ∞ h E k i dk . According to K41, h E k i ∝ k − / . The number of modes in a thin shell of radius k in spectral space isproportional to k , so that the contribution of a single mode with wavevector k tothe total mean kinetic energy is proportional to k − / . We now consider the rootmean square of the fluctuations of the total kinetic energy: σ tot = h ( E − h E kin i ) i / .The angular brackets denote average over time. The total root mean square is againdecomposed into contributions of different wavevector shells: σ tot = R ∞ σ k dk . In orderto compute σ k , we assume that the fluctuations of the k modes in the wavevector shellof radius k are all uncorrelated so that the mean squares of the fluctuations of eachsingle mode simply add to give σ k .We next invoke the concept of self-similarity, which is a central tenant of theK41 theory: All structures (or eddies or modes) in the inertial range are statisticallyindistinguishable from each other after a rescaling of length and time. This implies thatthe histogram of the fluctuations in every mode has the same shape if the fluctuations aregiven as multiples of the mean amplitude. We then have to conclude that the root meansquare of the fluctuations scales the same as the mean amplitude, i.e. is proportional to k − / . It follows that σ k ∝ √ k k − / = k − / .Note that it is not possible to deduce the scaling of σ k from the usual dimensionalarguments of K41. The ratio σ k / h E k i tends to zero if the number of modes in awavevector shell tends to infinity because the fluctuations of the modes in that shellaverage out to zero. The rms of the total fluctuation depends therefore on the numberof degrees of freedom, which in turn depends on the ratio of integral to dissipative lengthscale. These two length scales are assumed to be irrelevant in the K41 theory, however.The energy of the modes in the wavevector shell of radius k is now written in theform E k ( t ) = h E k i + σ k h k ( t ) (7)in which h k ( t ) has to obey h h k i = 0 and h h k i = 1. The Fourier transform of thefluctuations of the total energy is Z ( E k ( t ) − h E k i ) e − iωt dt ∝ Z σ k ˆ h k ( ω ) dk (8)with ˆ h k ( ω ) ∝ R h k ( t ) e − iωt dt . We now use again the hypothesis of self-similarity and notethat the typical time scale for a mode with wavenumber k is its turn-over time which is lobal fluctuations in magnetohydrodynamic dynamos k − / [20, 8]. Restricting ourselves to frequencies ω and wavenumbers k in the inertial range, we expectˆ h k ( ω ) ∝ k − / g ( ωk − / ) (9)with an unknown but universal function g . The form of the argument reflects thatall ˆ h k ( ω ) should have the same form once ω is expressed in multiples of the turn-overfrequency, and the amplitude factor k − / follows from the requirement that h h k i = 1,which implies that R | ˆ h k ( ω ) | dω is a constant independent of k . Substituting z = ωk − / into R | ˆ h k ( ω ) | dω = R k − / | g ( ωk − / | ) dω shows that this is indeed fulfilled.We can now proceed to finally compute the spectrum of the energy fluctuations,using the same substitution z = ωk − / , to find: Z ( E k ( t ) −h E k i ) e − iωt dt ∝ Z k − / k − / g ( ωk − / ) dk ∝ ω − Z z g ( z ) dz. (10)The spectrum of the energy fluctuations, i.e. the square of the Fourier spectrum above,must thus decay as ω − for ω in the inertial range. This prediction will be verified below.Spectra in MHD turbulence can have a variety of shapes, depending on the presenceof Alfv´en waves, an applied magnetic field, the magnetic Prandtl number etc. [21, 22]The above calculation can be directly reproduced for the magnetic energy as long asthe magnetic energy density follows a Kolmogorov spectrum in k − / . This happens inhomogenous and isotropic flows at high magnetic Reynolds numbers without appliedexternal field [23]. In that case, magnetic field fluctuations should decay as ω − , too.
3. The G.O. Roberts dynamo
Dynamos based on periodic flows first investigated by G.O. Roberts [13] have beenuseful for a number of basic studies of the dynamo effect [24, 25, 26, 27, 28] and haveinspired the Karlsruhe dynamo experiment [29]. In order to verify the ideas developedin the previous section, we wish to numerically simulate the Navier-Stokes and inductionequations in the following non-dimensional form: ∂ t B + ∇ × ( B × u ) = PmRe ∇ B (11) ∇ · B = 0 (12) ∂ t u + u · ∇ u = −∇ p + 1Re ∇ u + F + ( ∇ × B ) × B (13) ∇ · u = 0 (14)Re and Pm are two control parameters standing for Reynolds and magnetic Prandtlnumbers, respectively. The forcing F will take the form F = 8 π Re u (15)One solution to the above equations is then B = 0, u = u with u = √ πx ) cos(2 πy ) −√ πx ) sin(2 πy )2 sin(2 πx ) sin(2 πy ) (16) lobal fluctuations in magnetohydrodynamic dynamos B = 0. The velocityfield (16) is a cartesian arrangement of helical eddies with their axes along z . Periodicboundary conditions will be applied in all cartesian directions x , y , z with periodicitylengths l x = 1, l y = 1, and three different choices of l z , namely l z = 1, 1 . L x the periodicity length in x − direction, the laminarvelocity v = V u , and the kinematic viscosity and magnetic diffusivity ν and λ ,respectively, the nondimensional parameters are given by Pm = ν/λ and Re = V L x /ν .The above problem is conveniently discretized with a spectral method. However,we need very long time series if we are interested in small frequencies and the availabilityof computer hardware for long runs becomes an important consideration in the choiceof the numerical method. The most readily available platform happened to be a set ofmachines equipped with CUDA (compute unified device architecture) capable graphicprocessing units. These architectures are not well adapted to spectral methods, nor toPoisson solvers. That is why a fully explicit finite difference scheme was implementedto solve the following set of equation: ∂ t B + ∇ × ( B × u ) = PmRe ∇ B (17) ∇ · B = 0 (18) ∂ t u + u · ∇ u = − c ∇ ρ + 1Re ∇ u + F + ( ∇ × B ) × B (19) ∂ t ρ + ∇ · u = 0 (20)These equations describe a compressible fluid with the equation of state p = c ρ and alinearized continuity equation. Solutions of the incompressible Navier-Stokes equationwill be recovered in the limit of the sound speed c tending to infinity. The sound speedwas chosen large enough so that the peak Mach number never exceeded 0 .
12 in thecomputations presented below. The typical Mach number was more like 0 .
05. Thefinite difference code was validated against a spectral code [30] by comparing the resultsof short runs, but all the heavy computation was done with the finite difference code.We will first look at non-magnetic flows. Figure 3 shows spectra of kinetic energydensity E kin for l z = 2, 1 . E kin is defined as E kin = 12 V Z u dV, (21)where V is the computational volume. In all cases, the spectra of E kin follow an ω − over at least a decade in ω . This power law identifies the inertial range of aKolmogorov cascade according to section 2.3 (It was not checked directly whether thespatial spectrum of the velocity fields follows the K41 law because the simulations werenot done with a spectral code and the evaluation of the spatial spectra would have beencumbersome). An even steeper decay follows at higher frequencies before the spectrumdives under the noise level introduced by roundoff error.On the low frequency side, the inertial range ends in a conspicuous hump. Thefrequency of the local maximum in the spectrum, f max , scales with what one may identify lobal fluctuations in magnetohydrodynamic dynamos Figure 3.
Spectral power density of the kinetic energy as a function of frequency f = ω/ (2 π ) for flows without magnetic field. The panels show spectra for l z = 2, 1 . (red), 3 × (green), 10 (blue)and 3 × (pink). The straight lines indicate the power laws ω − and ω − . lobal fluctuations in magnetohydrodynamic dynamos l z Re Re Re f max
341 272 3 ×
630 292 10 ×
353 281.5 3 ×
640 331.5 10 ×
382 311 3 ×
670 351 10 × Table 1. l z , Re, Re and Re f max for the simulations in fig. 3. as the energy injection scale: Re is a control parameter in eq. (14), but the Reynoldsnumber determined a posteriori is Re = √ E kin . The values of Re given in table 1 varyfrom 340 to 2100 indicating that all flows are turbulent. The product Re f max variesfrom 27 to 41. Despite this variation, we identify f max with the injection scale becausethe definition of Re does not take into account that the flow is anisotropic so that wedo not expect Re f max to be strictly constant.The spectra at frequencies smaller than the injection frequency are different for l z = 1 and l z = 2 or 1 .
5: For l z = 1, the spectrum is purely white noise, whereas asection of 1 /f -noise is visible for l z = 2 and 1 .
5. The section of 1 /f -noise shrinks withincreasing Re.We next turn to dynamos. For Pm slightly above the critical value given in table1, the spectra of the magnetic energy density E B , defined as E B = V R B dV , show asegment of spectrum in ω − for l z = 2 and 1 . ω − for l z = 1. A white spectrumis always found at the smallest frequencies (figure 4). This matches the prediction ofsection 2.1 in as far as the velocity spectrum without magnetic field contains an intervalwith a spectrum close to ω − for l z = 2 and 1 . l z = 1. According to section 2.1, there should be a factor ω between magnetic and velocity spectra at low frequencies, which is compatible with theresults in figures 3 and 4.The transition from white noise to either ω − or ω − occurs at a higher frequencyfor larger Pm in figure 4. This is again in agreement with section 2.1 because a largerPm at constant Re corresponds to a flow farther above onset, which corresponds to alarger h α i in the model of section 2.1.Experiments use local probes to characterize the magnetic field. In the presentsimulations, the magnitude | B | of the field at position x = y = 0, z = l z / lobal fluctuations in magnetohydrodynamic dynamos Figure 4.
Spectral power density of the magnetic energy E B as a function of frequency f = ω/ (2 π ). The panels show spectra for l z = 1 . .The different lines in the top are for Pm = 0 .
03 (red), 0.05 (green) and 0.1 (blue), andin the bottom panel for Pm = 0 .
27 (red), 0.5 (green) and 1 (blue). The straight linesindicate the power laws ω − and ω − (top) and ω − and ω − (bottom). field has been measured in the Karlsruhe experiment [1]. The spectra of the magneticfield amplitude in one point show a power law at low frequencies over a larger intervalthan the spectra of E B , but with an exponent which bears no obvious relation with theexponent of the energy spectra (figure 5). There does not seem to be any theoreticaltool for predicting local fluctuations, so that we simply note the spectra in figure 5 asan empirical fact. However, they also approach a white noise spectrum as Pm increases,just as the magnetic energy spectra do in fig. 4.
4. Dynamos in spherical shells
Simulations of convectively driven dynamos in spherical shells reported in [6] yieldedspectra with power laws which are not compatible with the single mode model. lobal fluctuations in magnetohydrodynamic dynamos Figure 5.
Spectral power density of the local magnetic field amplitude | B | at theposition x = y = 0, z = l z / f = ω/ (2 π ). The panels showspectra for l z = 1 . and the same Pm as in fig.4. The straight lines indicate the power laws ω − . (top) and ω − / (bottom). According to the phenomenology developed in section 2.2, this simply means that thefluctuations in these dynamos are large. It is expected that at sufficiently low Rayleighnumbers and in sufficiently quiet convection, one finds again spectra in agreement withthe single mode model. In order to test this idea, we also performed simulations ofdynamos in spherical shells. Because these computations are more expensive, the spectraare more noisy and not as extended in frequency as for the G.O. Roberts dynamos, butthey demonstrate the main effect.We consider the same physical model as in [6] of a rotating spherical shell with itsgap filled with fluid driven by convection. The variable quantifying buoyancy will becalled temperature here, but the employed boundary conditions better fit compositionalconvection: Fixed temperature at the inner boundary and zero heat flux through theouter boundary. Both boundaries are assumed no slip and electrically insulating. The lobal fluctuations in magnetohydrodynamic dynamos ∂ t u + ( u · ∇ ) u + 2ˆ z × u = ∇ φ − E RaPr T r r + ( ∇ × B ) × B + E ∇ u (22) ∂ t T + ( u · ∇ ) T = EPr ∇ T − E ǫ (23) ∂ t B + ∇ × ( B × u ) = EPm ∇ B (24) ∇ · u = 0 , ∇ · B = 0 (25)are solved in a spherical shell for the velocity, magnetic and temperature fields u , B and T . The control parameters are the Ekman number E, the Rayleigh number Ra, thePrandtl number Pr, and the magnetic Prandtl number Pm. The ratio of inner and outerradii is fixed at 0 .
35 with the dimensionless outer radius being r = 1 / .
65, and ǫ , thevariable modeling the buoyancy source [6] is fixed at ǫ = 1. The equations have beensolved using the spectral method described in [31] with a resolution of 33 Chebychevpolynomials in radius and spherical harmonics of degree up to 64.Fig. 6 shows two calculations in which all parameters are held constant except forthe Rayleigh number. In going from the low to the high Rayleigh number, the spectrumof kinetic energy changes from white noise to approximately f − . The spectrum ofmagnetic energy shows a decay in f − in both cases. This means that at low Ra, theexponents in the power laws for the kinetic and magnetic energies differ by 2, just asthey should according to the single mode model. At higher Ra, i.e. in the flow withlarger fluctuations, the exponents are different as expected from section 2.2 and differby 1. Incidentally, the combination of f − and f − for kinetic and magnetic energies isidentical with a combination of exponents found in the low dimensional dynamic systemof section 2.2.
5. Conclusion
Colored noise has been noted in fluctuations of magnetic field strength in bothexperiments and simulations. According to common experience with turbulent velocityfields, the fluctuations of mechanical quantities have on the contrary a white noise at lowfrequencies, a low frequency being a frequency smaller than the inverse of the integraltime scale. It has been shown in the present paper that even the simplest phenomenologybased on a single mode model predicts a spectrum in 1 /f for the total magnetic energyif the velocity field is characterized by white noise. The single mode model is justifiedas long as the fluctuations in the velocity field are small enough in amplitude. Thesesimple models are corroborated by numerical solutions of the dynamo equations for 2Dperiodic dynamos and for convectively driven dynamos in spherical shells.Colored noise must have a low frequency cut-off for the power integral to converge.This low frequency is independent of diffusive time scales according to the models studiedhere. Instead, it depends on how strongly a dynamo is driven. The more supercriticala dynamo is, the larger is the cut-off frequency. lobal fluctuations in magnetohydrodynamic dynamos Figure 6.
Spectral power density of the kinetic energy E kin (top) and the magneticenergy E B (bottom) as a function of frequency f = ω/ (2 π ) for Pm = 13, Ek =6 . × − , Pr = 1 and Ra = 10 (red) or Ra = 10 (green). The straight lines indicatethe power laws f and f − in the top panel and f − in the bottom panel. It is also possible to deduce the spectrum of turbulent magnetic fluctuationsfrom the Kolmogorov phenomenology. Within the inertial range, fluctuations of thetotal kinetic and magnetic energy should decay as f − , which is verified by numericalsimulation.It is left to future work to deal with fluctuations of the magnetic field at a givenpoint in space. If the magnetic field is dominated by a large scale component, localfluctuations are of course similar to global fluctuations. At large magnetic Reynoldsnumbers, small scale fluctuations exceed in magnitude the fluctuations of the large scalefield and the spectrum of the fluctuations of local magnetic fields can be different fromthe spectrum of total magnetic energy. The spectra computed for the 2D periodicdynamos at high magnetic Reynolds numbers reveal power laws at low frequencies inthe range f − . to f − . . lobal fluctuations in magnetohydrodynamic dynamos References [1] U. M¨uller, R. Stieglitz, and S. Horanyi. A two-scale hydromagnetic dynamo experiment.
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