William B Ericson.

Hydromagnetic shocks. N.Y., 1958 online

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NEW YORK UNIVERaTY

JNSTITUTE OF MATHEMATICAL SQENCES

LIBRARY

AFCRC-TN-58-241 25 Waverly Pbce, New York 3, N. Y.

ASTIA DOCUMENT No. AD 152474

^^^EET PM^^ NEW YORK UNIVERSITY

'^ I isj M 5 Institute of Mathematical Sciences



^ ^ ' •' Division of Electromagnetic Research



RESEARCH REPORT No. MH-8



Hydromagnetic Shocks



WILLIAM B. ERICSON and JACK BAZER



CONTRACT Np. A F 1 9 (604) 2 1 3 8
JANUARY, 1958



AFCRC-TN-58-2ia

ASTIA Document No. AD l52U7h

NFW YORK UNIVF-RSin

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. MH-8



HYEROMAGNETIC SHOCKS



VJilliam B. Ericson and Jack Bazer




liam H, Ericson



JUU.



y^iicV. Bazer ^cf



Morris Kline
Project Director

January, 19 $8

The research reported in this document has been sponsored
by the Geophysics Research Directorate of the Air Force
Cambridge Research Center^ Air Research and Development
Command, under Contract No. AF19(60U)?138.

New York, 1958



- i -



Abstract
Our problem is to determine and to classify by analytic meana all
planar shock wave solutions of the hydromagnetic discontinuity relations.
The state ahead of the shock (i.e., on the low density side) is assumed to
be known; no restriction is placed on the direction of the magnetic field in
front. It is shown, apart from certain 'limit' shocks (e.g., pure gas shocks),
that the shock velocity and the quantities characterizing the state behind
hydromagnetic shocks may be expressed as simple algebraic functions of the
discontinuity in the magnetic field across the shock. A natural classifica-
tion of all hydromagnetic shocks, based on this representation of the state
behind the shock, is given. Several useful analytical properties of the
various types of hydromagnetic shocks are derived. The results are illustrated
graphically for the case of an ideal monatomic gas. The relation between
earlier schemes of classification and the pi^sent scheme is discussed.



- XI -



Table of Contents



Page



1, Introduction - Notation, definitions, fonnulstion of 1
the problem

1.1 Introduction 1

1.2 Notation, definitions, formulation of the problem 3

2, Survey of results 8

2.1 PrelJiTiinary remarks 8

2.2 Contact discontinuities 10

2.3 Non -compressive shocks (m / 0, [t^ « 0) - 11
Transverse shocks

2,[| Fast magnetic shocks (m / 0, hsinS > 0, 12
& jt 0°, 90°)

2.5 Slow magnetic shocks (m jl 0, hsinS < 0, 17

e^ / 0°, 90°)

2.6 Limit shocks 23

2.6.1 Preliminary remarks 23

2.6.2 Fast 0° -liir.it shocks 2k
Fast -limit shocks of type l(s > 1): 2k
Fast pure gas shocks

Fast 0° -limit shocks of type 2(s < 1): 25
Incomplete switch-on shock - fast gas
shock combination

2.6.3 Slow 0° -limit shocks 28
Slow 0°-limit of type l(s > 1): 28
Continuous transition

Slow 0°-limit of type 2(s < 1): Slow gas 28
shock - switch-on shock combination

2,6,b Fast 90 -limit shocks (perpendicular shocks) 30

2,6,5 Slow 90 -limit (contact discontinuity) 31

2.7 Concluding remarks - survey of the literature 31

2.7.1 Concluding remarks 31

2.7.2 Survey of the literature 32

3o The inadmipsibility of expansive shocks - the relation

between d[s]] and dm in compressive shocks 50

3.1 The inadmissibility of expansive shocks and related $0
results

3.2 The relation between d[sj and dm in a compressive 56
shock



U. Magnetic shocks - preliminary remarks 58

5o Fast magnetic shocks (h- > 0, < 6 < 90°); M- shocks 61

5.1 Fast magnetic shocks of types 1 and 2j 61
M^^^ and M^^^ shocks

5.2 Description of the curves X„/h„ versus h„ and 6?

^f/^f versus h^

5.?.1 M^-^^ curves (s > 1 - y sin^e /v-l) 67

1 o — o

5.2.2 ni^' curves (s < 1 - v sin^e /v-l) 70

I o ' o

5o3 5ome general features of M„ shocks 71

5.h Weak fast shocks 7U

5.5 Relation of the normal flew velocity (relative to 76

the shock) in front of and behind fast shocks to the

disturbance speeds in these regions

,0



6. Slew magnetic shocks (h >G, O 0.

According as m = or m /^ we shall speak of a shock or a contact discontinuity .
Since we are dealing exclusively with one-dimensional motion, it is clear that n
is directed along the positive or negative x-axis. In the following, assuming
that m / 0, we shall employ the convention that n points in the direction in which
the fluid crosses ^(t) . Under these circumstances, m is always positive, a
conclusion which we have anticipated in (1.2). The region into which n points will
be referred to as the region behind the shock; the remaining region will be
called the region ahead of or in front of the shock. Since we have labelled the
region into which n is directed with the subscript '1', it follows that the
subscript '1' always refers to the region behind the shock and the subscript 'o'
to the region in front of the shock.

In terms of the above notation the basic hs'-dromagnetic shock relations
may be expressed as follows jcf. [l]-[_U], especially [2] :

^0 W = °'

(^^3) Ag [rau + (p + nH^/2)n - nH^^ t] = 0,

A- [m] =0, or equivalently, m[t] - [u^] = 0,

A, m[u^/2 + e + nH^ t/2] + [u^(p+nH^/2)-nH^T!-^ = 0,



A^ m[s] > 0,



Kl^ is the projection of any vector Q on the shock front.



- 5-

where in d^ and ^^ the quantities p and e have the form
(1.3')



p = Ap"



e = p^/(y-1).

Here, y is the so-called adiabatic exponent and i^ is a function of the entropy
alone.

^ shock wave solution of A - A. will be said to be conpressive if
[t] < 0, non-conpressive if |r] ■ and expansive if [^] > 0, The major portion
of our results - those having to do with compressive shocks - will be expressed
in the following notation:

a) ^ = (Pi - Po)/Poi *l ' Pl/Po = ^ * 1»

c) I = [pI/Pq,

e) X » Y s^Y + h^/2,

f )** sin 9^ = "y, j^j» -^° < »j < 50°, j = 0, 1,



(l.h)



and



(1.5)



Y 1 + s. -/(l+s.) - Us.cos'^,



£2l\ = j 1 1 d 1

njoy 2 cos^.



Anticipating somewhat, these solutions are not necessarily 'admissible' since
they may not satisfy the entropy condition A^.

Since none of our formulas will depend in any essential way on the sign of
cos © ., j "= 0,1, there will be no loss of generality in limiting the 9 - range
to the interval [-50 , 90 ].



- 6 -




I K R—

1+s. + / (1+s.) - Ua .COS 9.

2 cos 9.
J



Here, H. = |^. | is the total magnetic field, and 9 is the angle between the
J J "J

positive n-direction and H. an region M., j=0,l} ^is the excess density ratio,

and T, the excess pressure ratio. The dimensionless ratio h will be emplojed

as a measure of the discontinuity in the transverse magnetic field, while the

quantity X will be used primarily as a means of simplifying the appearance of the

shock relations when the present notation is introduced. The quantity b .is

n, J

the Alfven or, as we shall sometimes say, 'intermediate' disturbance speed in
a direction making an angle 9^ = n H. with the magnetic field H.. The ratio
s. may be interpreted as the squared ratio of the sound speed



/ -»■ v 2 -1

to the Alfven disturbance speed in the direction of H., namely /]xH.p. . The

quantities Cg . and c„ . are, in Friedrichs' terminology , the slow and fast

disturbance speeds in ^.. As will be recalled, a., b . c . and c„ . satisfy the

relations cf. [2] p. 12 :

Angles measured clockwise from the positive n-direction to the positive Tf-direction
are considered positive-otherwise, negative [see Figure Ij.

The existence of three disturbance speeds was discovered by Herlofson' - ' and

\l\
independently by Van de Hulst'-^-' at about the same txme. The speed c corresponds

s

to the speed of Van de Hulst's slow mode, c^ to his fast mode and b„ to the Alfven

2 ^ "

mode. In the ilinit of small s = yp/m-H , Van de Hulst refers to c as the retarded

sound speed, c„ as the modified Alfven wave.



- 7 -



(1.7)



8,i n,j f,j



c . - a. ^ c



3,0 "f,a, J =0,1.



By the state P in front of a sho ck we shall mesn the set (p ,p ,u ,H ) or any

__— o —————————— O O o

equivalent set - e.g., Q « (p^, s^, sin ©^, u^/b^^ ^ H^). By the state f^ behind
a shock we shall mean the set |^ = (p^, p^ u^, H^j U^) or any equivalent set - e.g.,
r^ " (^ ^,h;m). The state in front of a shock will always be assumed known
at the outset.

Consider now the basic discontinuity relations (1.3) A ~Ai . Since H
is continuous across >d(t) and since f^ is known, A^-Ai constitute seven conditions
on the eight unknowns [T = ^Pi»Pi»^i>^tr i'"^* Clearly, A^-A, determine, in
general, a one-parameter family of states beliind the shock. Let e, < e < 6^
denote the generic shock strength parameter j e may be any scalar quantity - e.g.,
W,7,h etc. Then any determination of the state,

(1.8) \l - 1^(6, rj), < 6< e^,

behind the shock which a) depends continuously on the shock strength parameter e,
< 6 < 6-, and the state in front, and which b) satisfies the entropy condition
A^ will be called an admissible solution of the discontinuity relations.

Our problem is 1) to determine all admissible solutions of the discontinuity
relations, 2) to ascertain some of the general features of these solutions and 3) to
classify admissible solutions in a natural way by analytic means.



Our choice of parameters will be such that e = corresponds to a continuous
transition across ^(t). Thus, when we speak of a shock we must require that
e > Oj e, may be infinite, in which case ' and
conversely . Returning to our classification, we subdivide the class of all
compressive shocks as follows:

Fast magnetic shocks (M_ shocks): h sin 9 > 0; M

s s

shocks.

Another important means of classifying hydromagnetic shocks involves

the relation of V , the normal flow velocity (in the region behind the shock)
n,i

relative to the shock, to b , , the intermediate disturbance speed in the region
behind the shock. In this scheme, a shock is called fast if



(2-2) \,i > b^^^ , < e < e^,
intermediate if

and slow if



>



((. ■ ^ — ^ — __________

We have assumed implicitly that the transverse magnetic field ahead of and

behind a magnetic shock is parallel to the y-axisj we shall show in Section k

that this assumption entails no loss of generality.



- ID -



(2.1^) V^^^ < b^^^ , < 6 < e^i

in (2.2) and (2.1|) we require that the inequality holds in at least some proper
subinterval of < e < e, . Fast and slow magnetic shocks will turn out to be fast
and slow in this sense.

Our scheme of classification is now essentially complete. That it is a
natural one will become apparent as we proceed.

In the following, we shall exhibit all admissible solutions of the basic
discontinuity relations and carry out the remainder of the plan detailed at
the end of Subsection 1,2. Contact discontinuity solutions, non-compressive
shock solutions (which turn out to be 'transverse' shocks ), the so-called
parallel shock solutions (characterized by the condition h = O) and perpendicular
shock solutions (characterized by the condition H = O) are of course well knownj
these will therefore be accorded a summary treatment.

2,2 Contact discontinuities

Setting m = in (1.3) we find that if H ?^ then

(2.?) [u] = 0, [h] - 0, {>] - 0; [t], [s] arbitrary,

while if H = then
n

(2.6) [uj -0, [p ♦ ttl^j./2j =0i [u^y], [h^J, L^], [s] arbitrary,

Icf. [2] p. 36J, We conclude from (2.5) and (2.6) that a shear fLow discon-
tinuity is an inadmissible hydrotnagnetic motion unless the magnetic field

normal to the surface of discontinuity Z^(= rj^ + 1) may be calculated from

"f,l ^"^ "f,2^ ^n,l/^n,l' ^n,o/'n,o' Wf/^n,l ^"^ K'^fK,! "^^ ^« calculated
from the knowledge of how 7 „, ^^/h^. and >?_ depend on h„. These quantities may,

of course, be expressed directly in terms of h„j however, there is no real gain

in doing this.

The two 'types* of M- shocks - i.e., the M. and M^ shocks mentioned

above-are defined by the following relations:

(2.12) m[^^ s^ > 1 - y sin^9^/(Y-l),

,(2)



(2.13) mJ.^^ s^ < 1 - y sin^^/(Y-l).



In M^ shocks only the 'plus' branches in equations M„ ^ and M„ „
I i,J. i,


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Online LibraryWilliam B EricsonHydromagnetic shocks. N.Y., 1958 → online text (page 1 of 5)