Frequency power spectra of global quantities in magnetoconvection

We present the results of direct numerical simulations of power spectral densities for kinetic energy, convective entropy and heat flux for unsteady Rayleigh-B\'{e}nard magnetoconvection in the frequency space. For larger values of frequency, the power spectral densities for all the global quantities vary with frequency $f$ as $f^{-2}$. The scaling exponent is independent of Rayleigh number, Chandrasekhar's number and thermal Prandtl number.


I. INTRODUCTION
The temporal fluctuations of spatially averaged (or, global) quantities are of interest in several fields of research including turbulent flows 1-4 , nanofluids 5 , biological fluids 6,7 , geophysics 8,9 , phase transitions 10,11 . The probability density function (PDF) of the temporal fluctuations of thermal flux in turbulent Rayleigh-Bénard convection (RBC) was found to have normal distribution with slight asymmetries at the tails. The direct numerical simulations (DNS) of the Nusselt number Nu, which is a measure of thermal flux, also showed the similar behaviour in presence of the Lorentz force 12 . The power spectral density (PSD) of the thermal flux in the frequency (f ) space 3,12,13 was found to vary as f −2 . In this work, we present the results obtained by DNS of temporal signals of global quantities: spatially averaged kinetic energy per unit mass E, convective entropy per unit mass E Θ and Nusselt number Nu in unsteady Rayleigh-Bénard magnetoconvection (RBM) [14][15][16] . The kinetic energy as well as the entropy vary with frequency as f −2 at relatively higher frequencies. In this scaling regime, the scaling exponent does not depend on the Rayleigh number Ra, Prandtl number Pr and Chandrasekhar's number Q.

II. GOVERNING EQUATIONS
The physical system consists of a thin layer of a Boussinesq fluid (e.g., liquid metals, melt of some alloys (i.e., N aN O 3 melt), nanofluids, etc.) of density ρ 0 and electrical conductivity σ confined between two horizontal plates, which are made of electrically non-conducting but thermally conducting materials. The lower plate is heated uniformly and the upper plate is cooled uniformly so that an adverse temperature gradient β is maintained across the fluid layer. A uniform magnetic field B 0 is applied in the vertical direction. The positive direction of the z-axis is in the direction opposite to that of the acceleration due to gravity g. The basic state is the conduction state with no fluid motion. The stratification a) Electronic mail: kumar.phy.iitkgp@gmail.com of the steady temperature field T s (z), fluid density ρ s (z) and pressure field P s (z), in the conduction state 14 , are given as: where T b and ρ 0 are temperature and density of the fluid at the lower plate, respectively. P 0 is a constant pressure in the fluid and µ 0 is the permeability of the free space.
As soon as the temperature gradient across the fluid layer is raised above a critical value β c for fixed values of all fluid parameters (kinematic viscosity ν, thermal diffusivity κ, thermal expansion coefficient α) and the externally imposed magnetic field B 0 , the convection sets in. All the fields are perturbed due to convection and they may be expressed as: ρ s (z) →ρ(x, y, z, t) = ρ s (z) + δρ(x, y, z, t), (4) T s (z) → T (x, y, z, t) = T s (z) + θ(x, y, z, t), (5) P s (z) → P (x, y, z, t) = P s (z) + p(x, y, z, t), where v(x, y, z, t), p(x, y, x, t), θ(x, y, z, t) and b(x, y, z, t) are the fluid velocity, perturbation in the fluid pressure and the convective temperature and the induced magnetic field, respectively, due to convective flow. The perturbative fields are made dimensionless by measuring all the length scales in units of the clearance d between two horizontal plates, which is also the thickness of the fluid layer. The time is measured in units of the free fall time τ f = 1/ √ αβg. The convective temperature field θ and the induced magnetic field b are dimensionless by βd and B 0 Pm, respectively. The magnetoconvective dynamics is then described by the following dimensionless equations: where D t ≡ ∂ t + (v · ∇) is the material derivative. As the magnetic Prandtl number Pm is very small (≤ 10 −5 ) for all terrestrial fluids, we set Pm equal to zero in the above. The induced magnetic field is then slaved to the velocity field. We consider the idealized boundary (stress-free) conditions for the velocity field on the horizontal boundaries. The relevant boundary conditions 14,17 at horizontal plates, which are located at z = 0 and z = 1, are: All fields are considered periodic in the horizontal plane.
where a ± = 1 4 The kinetic energy E and convective entropy E Θ per unit is mass are defined as: E = 1 2 v 2 dV and E Θ = 1 2 θ 2 dV , respectively. The Nusselt number Nu , which is the ratio of total heat flux and the conductive heat flux across the fluid layer, is defined as: The system of equations may also be useful for investigating magnetoconvection in nanofluids with low concentration non-magnetic metallic nanoparticles 12 . A homogeneous suspension of nanoparticles in a viscous fluid works as a nanofluid. As the fluid properties depend on the base fluid and the nano-particles, their effective values may be used for the nanofluid. All fluid parameters are may be replaced by their effective values in the presence of nanoparticles in a simple model. If φ is the volume fraction of the spherically shaped nanoparticles, the effective form of the density and electrical conductivity of the nanofluid may be expressed as: where ρ f and σ f are the density and electrical conductivity of the base fluid, respectively. Here ρ p is the density and σ p is the electrical conductivity of the nanoparticles. The effective thermal conductivity K 18 is expressed as: where K f and K p are the thermal conductivity of the base fluid and that of the spherical shaped nanoparticles, respectively. Similarly, the effective specific ccapacity c V may be expressed through the following relation 19 : The effective dynamic viscosity µ of the nanofluid 20 may also be expressed as: The relevant values of effective fluid parameters may be used in the set of equations 8-11 for investigating flow properties in nanofluids.

III. DIRECT NUMERICAL SIMULATIONS
The direct numerical simulations are carried out using pseudo-spectral method. The perturbative fields are expanded as: ilk c (Q)U lmn + imk c (Q)V lmn + nπW lmn = 0.     Fig 2(a). The energy spectra are very noisy for dimensionless frequencies between 0.04 and 1.0. In this fre-quency range (0.04 < f < 1.0), the spectra is noisy and the slope of the curves E(f )−f on the log-log scale varies between −3.2 to −5.1. However, the E(f ) is found to have negligible noise 1 < f < 200. The PSD shows a very clear scaling behaviour for f > 1. The PSD (E(f )) of the energy signal scales with frequency f as almost f −α with α ≈ 2. The scaling behaviour is found to be continued for more than two decades. The scaling exponent is independent of Pr, Ra and Q in this frequency window. Table-I gives the exact values of the exponent α for different values of Ra, Pr and Q. The scaling law E(f ) ∼ f −2 was also observed in rotating Rayleigh-Bénard convection (RBC) 13 .   Fig 2(b) shows the PSDs of the convective entropy E Θ (f ) = |θ(f )| 2 of the fluid in the frequency space for different values of Ra, Pr and Q. Its power spectra is also noisy in the dimensionless frequency range 0.04 < f < 1.0. The slope on the log-log scale varies between −5.9 and −6.4. However for f > 1.0, E Θ also scales with frequency with as f −β with β ≈ 2. The numerically computed values of the exponent β are listed in Table-I Fig. 2(c). The PSDs also show the scaling behaviour. The PSDs are noisy, as in the case of energy and entropy signals, for dimensionless frequencies 0.04 < f < 1.0. The scaling exponent varies between −4.5 to −6.4 in this frequency range. However, for dimensionless frequencies range 1 < f < 200, the spectra for thermal flux Nu(f ) also shows very clear scaling: Table-I shows the values of the exponent γ computed in DNS. The measurements of the spectra of thermal flux in RBC also shows the similar scaling law 3 .
The scaling law showing the variation of the power spectra as f −2 starts at a critical frequency f c for different values of the Chandrasekhar number. Fig. 3 shows the variation of the critical frequency for E(f ), E Θ (f ) and Nu(f ) with Q two different values of Pr. The critical frequency f c (E) becomes lower as Q is increased (see Fig. 3 (a)). In addition, it is less for smaller values of Pr. Figs. 3 (b)-(c) show the variation of f c (E Θ ) and f c (Nu), respectively, with Q. The values of critical frequencies are slightly different for E(f ), E Θ (f ) and Nu(f ). However they all decrease with increase in Q. They also decrease with decrease in the value of Pr.

V. CONCLUSIONS
Results of direct numerical simulations on Rayleigh-Bénard magnetoconvection show that power spectral densities the kinetic energy E(f ), convective entropy E Θ (f ) and the Nusselt number Nu(f ) scale as f −2 for frequencies above a critical value f c . The critical values f c (E), f c (E Θ ) and f c (Nu) are different for kinetic energy, convective entropy and the Nusselt number. The critical frequency decreases with increase in the strength of the external magnetic field. However, the scaling exponent is independent of the thermal Prandtl number, Rayleigh number and Chandrasekhar number. The results may be relevant for geophysical problems, water based nanofluids and crystal growth.