Looking Again At Bessel Functions

 Among the most important special functions in mathematics are the Bessel functions.  Below I show the graphs of the two Bessel functions  J_o (aR)  and J₁ (aR)   plotted using the Mathcad software program:

Where J_o (aR)  is a Bessel function of zero order, and  J₁ (aR) is a Bessel function of first kind, order unity.  In general we have (Menzel, 1961)[1]:

J_m (x) = (1/ 2^m m!) x^m [1 -  x ²/ 2² 1! (m + 1)  +  x⁴/ 2⁴2! (m + 1) (m + 2) -  ….

..(-1)^j x^2j / 2 ^2j j! (m + 1) (m + 2)……(m + j) +  …]

which we terminate with second order terms.

The Bessel functions play a useful role in that they are solutions of Bessel’s differential equation, consistent with a cylindrical geometry.   This equation can be written in one of the more familiar forms as the Helmholtz equation:

1/r  d/dr (r d B _z  / dr  )  +  a² B _z        =   0

 If  B _z  is finite on the r = 0 axis, then the solution may be written (Lundqvist, 1951)[2]

 B _z   =   B_o J_o (aR)         and          B _f   =   B_o J₁ (aR)

 where J_o (aR)  is a Bessel function of zero order, and  J₁ (aR) is the Bessel function of first kind, order unity.   Note the DE can also be derived from the relation for curl B in cylindrical geometry, i.e.

curl B  =  [1/r e_r         e_f          e_z ]

                    [¶/¶r       ¶/¶f       ¶/¶ z]

                    [ B_r          rB_f             B_z ]

 with B = (0, B_f , B_z ) provided:    ¶/¶f =  ¶/¶ z = 0.

 And  curl B  =  [0     -¶ B_z /¶r         1/ r  ¶//¶r   ₍rB_f)]

  Which is the exact truncation value of the Lundqvist Bessel function solution.  From the preceding curl elements:

-¶ B_z /¶r   = aB _f 

1/r  ¶/ ¶r (r B _f  )  =   aB _z

whence  - for variation in one quantity:

B _f    = 1/a ( d B _z  / dr ) 

and 

1/r  d/dr (r B _f   )  -  a B _z        =   0

Now, substitute the top equation into the bottom and multiply through by (-a) to get the Helmholtz DE.    Given the cylindrical geometry it can be seen that the Bessel functions would have much relevance to solar loops with the same geometry.  Here,  the equations in B _z (r),  B _q (r),  with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube. For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r)  =  (L B _q(r))/  (r  B _z (r))

Where L denotes the length of the sunspot-flux tube dipole and r, the radius. If the twist exceeds 2p then the magnetic configuration may be approaching instability and a solar flare.   For completeness a graph showing a series of Bessel functions of 1st kind, integral order, are  shown below:

Plot of Bessel function of the first kind, J_α(x), for integer orders α = 0, 1, 2.

The Bessel functions behave like a damped sine wave for integral order n and large argument, x.  E.g.

J_o (x)  =  Ö (2 / p x )   [cos (x -  p/4   -    np/2 )] 

Problems:

1) Compute the Bessel functions for J_o(x)   and  J₁(x)  with x = 1 and then compare with the values obtained from the graph shown above.

2) A large sunspot has an equilibrium magnetic field  B_o=  0.01T.  Find its azimuthal magnetic field if  x = 2, for the associated Bessel Function, J₁(x).

3) Find the twist in a solar loop (take it to be a uniform cross section magnetic tube) if: B _q(r) = 0.1T and B _z (r) = 0.2T. Take the radius of the tube to be r = 10 ⁴  km and the length L = 10 ⁸  m.   Is the tube kink unstable or not? (Kink instability is said to obtain when: T(r) > 2p)

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[1] D. Menzel, Mathematical Physics, Dover Publications, p. 204.

[2] S. Lundqvist, Physical Review, Vol. 83, p. 307, 1951.