Numerical Representation of MHD Turbulence Prior to the Ultimate Phase of Decay
Ripan Roy^{*}, M. Abu Bkar Pk
Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh
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To cite this article:
Ripan Roy, M. Abu Bkar Pk. Numerical Representation of MHD Turbulence Prior to the Ultimate Phase of Decay. Pure and Applied Mathematics Journal. Vol. 5, No. 2, 2016, pp. 32-38. doi: 10.11648/j.pamj.20160502.11
Received: February 26, 2016; Accepted: March 8, 2016; Published: March 21, 2016
Abstract: Following Deissler’s approach the magnetic field fluctuation in MHD turbulence prior to the ultimatephase of decay is studied. Two and three point correlation equations have been obtained and the set of equations is made determinate by neglecting the quadruple correlations in comparison with second and third order correlations. The correlation equations are changed to spectral form by taking their Fourier transforms. The decay law for magnetic field fluctuations is obtained and discussed the problem numerically and represented the results graphically.
Keywords: Correlation Function, Deissler’s Method, Fourier-Transformation, Matlab, Navier-Stokes Equation
1. Introduction
Magneto hydrodynamic (MHD) turbulence is characterized by nonlinear interactions among fluctuations of the magnetic field and flow velocity over a range of spatial and temporal scales. It plays an important role in the transport of energetic particles. Magneto hydrodynamics turbulence has been employed as a physical model for a wide range of applications in astrophysical and space plasma physics. The fundamental aspects of MHD turbulence include spectral energy transfer, non-locality, and anisotropy, each of which is related to the multiplicity of dynamical time scales that may be present. These basic issues can be discussed based on the concepts of magnetic Prandtle number of the small scales in the magnetic field. The magnetic Prandtle number defined as the ratio between the kinematic viscosity and the magnetic diffusivity. Boyd (2001) discussed Chebyshev and Forier spectral methods. Shebalin (2002) explained the statistical mechanics of ideal homogeneous turbulence. Biskamp (2003) obtained magneto hydrodynamic turbulence. Islam and Sarker (2001) developed the first order reactant in MHD turbulence before the final period of decay for the case of multi-point and multi-time. Shebalin (2006) also oriented ideal homogeneous magnetohydrodynamic turbulence in the presence of rotation and a mean magnetic field. Deissler (1958, 1960) developed a theory ‘on the decay of homogeneous turbulence for times before the final period.’ By considering Deissler’s theory, Loeffler and Deissler (1961) studied the decay of temperature fluctuation in homogeneous turbulence before the final period. Bkar Pk et al. (2013) illustrated the decay of MHD turbulence prior to the ultimate phase in presence of dust particle for four-point correlation. Chandrasekhar (1951) obtained the invariant theory of isotropic turbulence in magneto-hydrodynamics. Rahaman (2010) obtained the decay of first order reactant in incompressible MHD turbulence before the final period for the case of multi-point and multi-time in a rotating system. Corrsin (1951) considered the spectrum of isotropic temperature fluctuations in isotropic turbulence. In their approach they considered the two- and three-point correlation equations and solved these equations after neglecting the fourth and higher order correlation terms analytically. Here, two- and three-point correlation equations have been considered, and the same approach of Deissler (1960) is applied to a theory of decaying homogeneous turblence. Sarker and Kishore (1991) derived the problem decay of the MHD turbulence before the final period analytically. In this chapter, we have discussed the problem numerically and represented the results graphically. We have shown that if the magnetic diffusivity is constant and kinematic viscosity is transferable then the fluctuation of the decay curves is linear and is parallel to the direction of -axis and the decay is very small at constant time.
2. Mathematical Formulation
For two points, we need two equations. Let the induction equation of magnetic field at the point is
(1)
and at the point will be
(2)
where
is the total MHD pressure,
is the hydrodynamic pressure,
is the fluid density,
is defined as the magnetic prandtle number,
is the kinematic viscosity,
is the magnetic diffusivity,
is the magnetic field fluctuation,
is the turbulent velocity,
is the time, is the space coordinate, and the repeated subscripts are summed from 1 to 3.
Multiplying the equation (1) by and (2) bywe get respectively
(3)
(4)
Adding equation (3) and (4) and taking ensemble average with transformations
and Chandrasekhar’s relation (1951)
we obtain
(5)
We have three dimensional Fourier transforms
(6)
(7)
Integrating the subscripts and then integrating the points , we have
(8)
Putting these three equations in (5) becomes
The above tensor equation becomes a scalar equation by contraction of the indices
(9)
The term on the right hand side of the equation (9) is called energy transfer term while the second term on the left hand side is the dissipation term.
We consider the momentum equation of MHD turbulence at the point , and the induction equations of magnetic field fluctuation at and as follows
(10)
(11)
(12)
Multiplying equation (10) by , (11) by and (12) by , we obtain
(13)
(14)
(15)
Combining equations (13), (14), (15), and taking time average and using the transformation, we obtain
,
and
We get
(16)
Using Fourier transforms
(17)
and
(18)
with transformation
and
The equation (17) reduces into the form
(19)
Taking contraction of the indicesin equation (19), we get the spectrum equation at three point correlation as:
(20)
Taking derivative of equation (10) at with respect to we get
(21)
For independent variables multiplying equation (21) by , taking time averages we get
(22)
Applying Fourier transforms of the equation (22), we get
(23)
where is the pressure correlation.
3. Results and Discussion
The solution is obtained by considering the two-point correlation after neglecting the third order correlations the three point correlation equations is considered and the quadruple correlations are neglected. The terms associated with the pressure correlations must be neglected. Thus, neglecting all the terms on the right hand side of the equation (2), we have
(24)
Integrating the equation (24) between with inner multiplication by and gives
(25)
Where is the angle between
Now, by letting in the equation (18) and comparing with the equation (7) and (8), we obtain
(26)
(27)
Substituting the equations (25), (26) and (27) in equation (9), we get
(28)
Now, can be expressed in terms of .
Following Sarker and kishore (1991) and using Loeffler and Deissler (1961) and assumption and then integrating w.r.to, we get,
(29)
where is the magnetic energy spectrum function and is the energy transfer term given by
(30)
Integrating equation (30) with respect to , we have
(31)
Solving the linear equation (29)and using Corrsin(1951) relation is a constant,we get
(32)
where
,
By setting
,
in the equation (20), we get the expression for magnetic energy decay as
(33)
Substituting equation (32) into (33) and after integration w. r.to time, we can write
(34)
where
=
=
which is the decay law for magnetic energy fluctuation before the final period.
Now we are going to discuss the problem in numerical analysis:
If (magnetic diffusivity) is fixed and (kinematic viscosity) varies from 0.05 to 0.15 in Table 1, Magnetic Prandtle number is increased for increasing of kinematic viscosity () because they are proportional to each other.
For fixed time (0.4, 0.7, 0.9) and for different values of and , the total energy is increasing from Figure 1 to Figure 3. Here time has been taken in the direction of -axis and total energy in the direction of -axis. In the direction of -axis for time the limit has been taken from 1 to 2 and in the direction of -axis for total energy the limit has been taken from 0 to 7.
When λ is fixed and ϑ varies from 0.20 to 0.30, different values of A & B in Table 2, the change of total energy is very small from Figure 4 to Figure 6.
Similarly, if is fixed and varies from 0.35 to 0.45 in Table 3, the straight line is smaller than that of Figure 5 which has been shown in the Figure 9.
But for fixed time and different values of which has been indicated by Table 3, the total energy is decreasing slowly from Figure 7 to Figure 9.
4. Conclusion
From the above tables, figures and discussion we conclude that the following results:
• When magnetic diffusivity is constant and kinematic viscosity is changeable then the Magnetic Prandtle number is proportional to the kinematic viscosity.
• In the absence of non-dimensional quantity the total energy of magnetic field fluctuation is decaying very rapidly.
• The magnetic field fluctuation of total energy is gradually increased at fixed time = 0.4, 0.7, and 0.9.
• The energy decay is very small at constant time which has been shown in the Figure 6 to Figure 8.
Acknowledgements
Authors (Ripan Roy & M. Abu Bkar Pk) are grateful to the Department of Applied Mathematics, University of Rajshahi, Bangladesh for giving all facilities and support to carry out this work.
References