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Reflection of a point disturbance from a conducting wall in two-dimensional magnetohydrodynamics online

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MF 81



Courant Institute of
Mathematical Sciences

Magneto-Fluid Dynamics Division



Reflection of a Point Disturbance
from a Conducting Wall in
Two-Dimensional Magnetohydrodynamics



Harold Weitzner



Air Force Office of Scientific Research Report
June 1975



New York University



NEW YORK UNIVERSITY
COURAMT INSTITUTE-UBnARY
251 Meraer St. M«w York, H.Y. 1«012



New York University
Coursmt Institute of Mathematical Sciences
Magneto-Fluid Dynamics Division



MF-81

REFLECTION OF A POINT DISTURBANCE FROM A CONDUCTING
WALL IN TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS

Harold Weitzner
June 1975



Research sponsored by the Air Force Office
of Scientific Research, Office of Aerospace
Research, United States Air Force under
Grant No. AFOSR-71-2053 •



Abstract
In order to stuay the effects of tied magnetic lines in
ideal magnetohyarodynamics , the simple problem is treated of
the reflection of an initial point disturbance, i.e. delta
function, froni a conducting wall in two-dimensional magneto-
hydrodynamics linearized about a constant state. The lowest
oraer magnetic field is taken normal to the conducting wall.
The conducting wall is the line x = and the initial dis—
turoance isatx=x >0, y=0. Tne solution separates
into three parts, the ordinary wave generated by a point

disturoance atx=x,y=0,a reflected wave centered at

o

tne image point x = -x , y = 0, and a third uncentered wave.
The first reflectea wave contains a disturbance similar to
the ordinary wave and a centered simple wave. The uncentered
wave ana its non-self-similar wave fronts are described. The
presence of lacunae, line singularities, and the nature of
the singularities at the wave fronts are discussed.



Ill



I. Introduction

It is well— known that the boundary conditions associated with
problems in ideal magnetohydrodynamics are very different if the
boundary is a flux surface, n'B = 0, or if the magnetic field
lines enter the boundary, n'B 5^ 0, where n is the normal to the
bounding surface and B is the magnetic field vector^ .
In the former case, the boundary condition is n*u = 0, where u is
the flow velocity vector, while in the latter, usually referred to
as tied magnetic lines, the condition is u = 0. The case of tied
magnetic field lines is of particular interest in stability analy-
ses of various configurations, as it is generally agreed that
tied lines should be much more stable than untied lines. Unfor-
tunately, there are almost no concrete comparisons of tied and
untied lines available. The potential interest in tied magnetic
lines leads us to consider a much simpler problem in two dimen-
sional magnetohydrodynamic wave propagation of interest in its
own right for which explicit solutions and comparisons are possi-
ble. We shall see that the solutions for tied and untied lines
are totally different in structure.

We consider a plasma ±1 the half— plane, x >_ 0, -« £ y £ «
and we assume the line x = to be a perfect conductor. We lin-
earize the equations of ideal magnetohydrodynamics about a state
of constant magnetic field, plasma mass density and zero flow
velocity. If we take the unperturbed magnetic field in x direc-
tion, then the field lines are tied and the perturbed flow veloc-
ity vector vanishes at x = 0. If the unperturbed magnetic field



is in the y direction, then the field lines are not tied and only
the X component of the perturbed velocity vector vanishes at
X = 0. We solve for the Green's Functions for these problems,
corresponding to giving delta function initial values for the
perturbations and then following the solution in time. The solu-
tion for the general initial value problem Is a convolution
Integral of the Green's Functions with the given initial data.

Some years ago the Green's Function was given for the case
of a whole plane of plasma -oo , (l6a)

while for x < x we have

-(-'^'p) = J^ A3i:.(k,p),k,pj [aiKP)[p'-^y^i^,p) - ^'^-K^)

+ ika^sj(k,p)e(k,p) [ . (I6b)

We could verify explicitly from (16) that u(x,k,p) tends to zero
as IpI -»- 0, as a Laplace transform must.

We may combine (l6a) and (l6b) into one formula by the use of
(13) and the relations s-|^(k,p)+ST (k,p) = = S2 (k,p )+S2^ (k,p)



14



^ Sj(k,p) |x-x^| ^
J - J s J I,



+ ika^s.(k,p)g sgn(x-x^)
s . ( k , p ) X



^3 A3[s.(k,p),k,pJ { «(K>P)(p'-A^j'(k,p).k^(a^A^))



+ ikaS (k,p)B(k,p)



(17)



-s (k,p)
Since a(k,p) and 6(k,p) are linear combinations of e
-S2(k,p)x^ s-,(k,p)x^ S2^(k,p)x^

and e or equivalently e and e we see

that u(x,k,p) is a linear combination of a function of x-x , a

function of x+x , and a more general function of x and x . In

(17) the function of x-x is given by the first summation and the

other two functions are given by the second summation. We set

u(x,y,t) = u-|^(x-x^,y ,t) + U2(x+x^,y,t) + U2(x,x^,y,t) ,

where the transform of u-, is given by the first summation in (17)
and the sum of the transforms of u^ and Uo is given by the second
summation in (17)- We treat the terms separately and start with
the simpler u, (x-x ,y,t).

The representation of u, (x-x ,y,t) is



u-|^(x-x^,y,t) = Re



277 1



«> + 1°°

dk
a-l^o



^ Sj(k,p)|x-x^



, iky pt V e

'P ' ' ^3 A3is.(k,p),k,pJ



(18)



< f(p^-A^sJ(k,p)+k^(a^+A^)) + ika^Sj(k,p)g sgn(x-x^)
where a > 0. We now deform the contour of integration in the



15



K7

p plane to CI or C2 [see Figure 2] as t - / ^ g" I ^~^o ' ^^ greater

o o
than or less than zero. We readily verify that on these contours

Re (pt+s . (k,p) |x-x I) 10, j = 3,^- If we denote the integrand in

(18) as I, we find

u-L(x-x^,y,t) = Re —



2TT^i



dk



dp I



CI or C2



27r



dk e



Iky



/a^A^ / /a^A^



a A'
o o



a A'
o o



In the first integral we Introduce a new variable p = p/k and we
find that we may perform the k integration by means of the fol-
lowing distribution identity

ika



dk e-



TT(5(a) + i/a



where i/a is understood to be in a principal value sense:



u-,^(x-x^,y,t)



= Re



2.h



CI or C2



4 r f [p^-A^s^(l,p)+(a^+A^)+ia^s.(l,p)g sgnCx-x^))



s.(p)
• (TT6(y+ I t + -^^— |x-x^|)) +



s.(p)

V + ?- + -^. Ix-x

" 1 1 ' o



o o \ o o



16



The delta function under the Integral sign does not contribute to

the integral, for either it has non— zero argument or where its

argument is zero, the contribution is pure imaginary. We deform

the p integration to a contour just to the right of the imaginary

s . (p) _-,
axis. We recall that (y + ?■ + -^ — ' ^"■'^n ' -^ "^^ defined in a

principal value sense, so we must add in one— half the residue at
the points at which it has a pole. As with the delta function,
the residues yield a purely imaginary term, which does not con-
tribute to u, . Thus, we obtain



u-,^(x-x^,y,t)



o+i- 4 f(p2-A2s2(i,p)+(a^+A2)) + ia^gs . (1 ,p)sgn(x-x^ )



Re



2tt



0-1°



dp I
. = 3



o J



o" J
s,(l,p)



A^(s.(l,p),l,p)(y + £ + -^ l^-^oD



o o



t - x-x



a +A
o o

a A
o o



where a > and there are no singularities in < Re p and j = 1,2. Hence we nay add into the summation
the terms with j = 1,2, after observing that the branch line in
Re p > disappears when we combine the two terms with j = 1 and
j = 2. We find



u-|^(x-x^,y,t) = Re — ^

2Ti



°^^°° , f(P^-AoC^(p)+a^A^) + ia^gap)sgn(x-x^)
'^^ A[?(p),l,pJ|x-x^|



0_loo



/ 2 2
o o



a'^+A'^
o o



..y_j_.. J p 2° I ^"^n ' ' ^^ "°^ present. We recall

o o
from (2) that for any magnetohydrodynamics problem f(x,y) = ^^'^

Thus, if we take the convolution of the delta function part of

the solution with f(x,y) we find



dx dy f(x-x ,y-y )H|t - j 2 2° '^'"^oM ^^^ ^

^0^0



\ O O / \ 0/

Thus, for a magnetohydrodynamics problem only the end points of
the above line segment are in the support of the distribution.

We analyze the behavior of the fundamental solution near the

r"2 2
a +A



points y = 0, t - / ?, ° | x-x | = in greater detail in the next

^a-A'^ °
o o

section and we show that if one solves a magnetohydrodynamics
problem for which the initial data f(x,y) = '^^^'^ , then the


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Online LibraryH WeitznerReflection of a point disturbance from a conducting wall in two-dimensional magnetohydrodynamics → online text (page 1 of 3)