Yall ep a Airy tne Eh Feil cite tte B acted verte E Hale agin To Fest Sad freee? Katha tO iomtes een Lo Malt Lis B oltee tt oe Foci PM ed nt eal hatin Se a P ial Ht Tar ago Maan ete rtne Tio SE Abed ot Shae Bank Spy Frat Sante eel eat ast atin: Fa pitts Hagth ott ha se a say Sh ae Pi LAG He Ge WM AGRE DE PDAS TCE n Lee Mw tet ee RE hag Mb = ave ae ee oe (VPx y)/(2.VT) This formula describes a quasi-horizontal geostro- phic wind tangential to the geopotential surfaces and at right angles to the temperature gradient. The wind speed V is given by the formula V = (g/22) sec@ sina where 6 is the angle between Q2 and VT and a is the angle between - Vi and VT and g is the constant for gravity at the origin (see Fig. 3). SOME ASPECTS OF WEATHER AND SOLAR ACTIVITY 19 z A VERTICAL Fig. 3. Diagram showing some important vectors and the local coordinate system. Now g/22 ~ 67 km/sec, and |cos0| ~ |sindo], where $ = latitude. So, for a wind speed of 100 km/hour and latitude $ = 30° we obtain ; zs 4 sina ~ 2 x 107 Clearly the angle a is very small even for gales and a very small change in a corresponds to a large change in wind velocity. But sina = y/|VT| where yy is the quasi-horizontal component of the temperature gradient (i.e. tangential to the geo- potential). In the upper troposphere |VT| ~ 6.5°C/km so, for latitude 30° and V = 100 km/hour Yu * 0.13°C per 100 km For undisturbed conditions in the lower stra tosphere |VT| is very small but again o must be very small at mid-latitudes so the temperature gradient must be quasi-vertical when this solu- tion of the governing equations is applicable. Also slightly different values of a above and below the tropopause could be accompanied by large differences in the speed and direction of the corresponding winds. This solution for the speed V is accurate to about 1% and can be made more accurate by not neglecting the variation of gravitation with height which affects the value of the wind speed and gives rise to a small shear in the direction of VW. These, of course, are results applicable to special conditions. But even if some restricting conditions are introduced, as, for example, wind shear, and the final value of the wind speed will be many times smaller, this result gives a strong indication that under certain circumstances this small parameter, namely a horizontal temperature gradient can exercise a controlling influence. If wind velocity changes drastically, so must the curl and consequently its vertical component, so called vorticity, that is, even small changes in temperature due to changes in solar activity could cause changes in vorticity. If these temperature changes do occur near the tropopause the drastic change in wind velo- city could change essentially laminar stratospher- ic flow just above the tropopause into a turbulent tropospheric flow, that is, the tropopause ef- fectively rises which is what happens after solar flares according to observations reported by Schuurmans and Oort (1973). The above results could, however, have an- other interesting application not related to changes in solar activity. Because of the direct mixing of the tropospheric and stratospheric air at the polar and tropical tropopause breaks (Fig. 4), the naturally occurring horizontal temperature gradient could be one of the causes and, possibly, the main cause of the formation of jet streams. So called Clear Air Turbulence (CAT) could also be caused by a similar situation at other places of the tropopause region. ‘Ihe inclination ot the tropopause to horizontal direction could also play an important role. In the previous solution it was assumed that the density of air at a point did not vary with time. On removing this restriction we obtain an approximate solution (described below) in which the velocity may have an appreciable vertical com- ponent accompanied by a pressure wave as indicated below. This solution is a supporting factor for our explanation of the experimental fact of the tropopause rising after solar flare. A Solution where 30/dt # 0 In this solution kj, kj, kz are constant direction numbers for the temperature gradient using axes Ox(East), Oy(North) and Oz(Vertical). OY is parallel to Q jaca k) Bi: etc. DY = BT T = exp[J(v)] where J'(v) =8 Vo = kyx + koy + kz2 - wt/(sind) where kycosd + kzsind = 1 w = vertical velocity (arbitrary) Geopotential ~ gz (g constant) My = (u,v,w) (a constant velocity) 20 V. KASTALSKY AND OTHERS QOPOPAUSE =18 km “ BREAK” ,-,-SUBTROPICAL oF JET STREAM LAR TROPOPAUSE _{ _\_poL AR PO AB NET STREAM 90° 60° 30° 0° Fig. 4. The tropopause complex in schematic meridional cross-section. One may add the "'third tropopause" or a "leaf" above the overlapping parts of the tropical and middle latitude tropopauses (e.g. Haltiner and Martin, 1957, Fig. 23.9, p.426). East Wind: u k2 (g/20) North Wind: v =wcot > - kj (g/2Q) de/dt o(6 + €/T)w/sind vy dv P = Po exp(-el5 =) where € = g sind/R,od = latitude. Velocity of Pressure Waves (etc.) * wif VT is quasi-vertical. This solution reduces to the Geostrophic (steady) solution on putting w = 0. V. Kastalsky and C.B. Kirkpatrick, School of Physics, A.T. Daoud, School of Mathematics, University of New South Wales, P.O. Box 1, KENSINGTON, N.S.W. 2033. AUSTRALIA. (Manuscript received in final form 2.2.85) Journal and Proceedings, Royal Society of New South Wales, Vol. 118, pp. 21-25, 1985 ISSN 0035-9173/85/010021 — 05 $4.00/ 1 Weber Transform of Certain Generalized Functions of Rapid Growth* R.S. PATHAK AND H. K. SAHOO ABSTRACT. We (x) ty) := G(y) = It is shown that the classical Weber transform b 1 | (xy) °C, (xy ,ay) g(x) dx a is a continuous Pipear Mapping from a testing function space Ga pb into Hg,p and also the inverse Weber transform W,, is a continuous linear mapping from H into Gh, b: An operational calculus is developed and is applied to the solution of certain genéralized differential equations. INTRODUCTION In this paper we consider the Weber transform G(y) of a function g(x) which is such that g(x) be- longs to L,(0,~) and g(x) = 0 for x > b and define b L W LEG) IO) := G(y) = | (xy) °C (xy, ay) 800) dx, a (1) where Ci (xy, ay) = JCy Yay) - aij play Yy, (xy) and uU is a real number. Obviously, g(x) belbngs to L, (0,%) and hence by a theorem of Titchmarsh (1924) ao ( (xy) 4c, (xy, ay) u [G(y) I(x) := g(x) = . ea G(y) dy, (2) where Qi (ay) = Jy (ay) + yi (ay) - It has been shown by Griffith (1956, 1957) that (1) can be extended to a complex variable z= y + if by G(z) = [ (x2) #C, (xz, a2) g(0) dx. (3) a The following theorem due to Griffith plays a fundamental role in the present investigation. Griffith's Theorem In order that g(x) belongs to L,(0,%) and g(x) = 0 for x > b> a, it is necessary and sufficient that (a) G(y)/Q(ay) belongs to L,(0,%); -L (b) y *G(y) is analytic in z = y + iff for S220); (c) y *6(y) is an even function of y; (d) z%6(z) = o¢e'P 9), ast 2) 32 6 200. We construct a testing function space G, sb of certain C -functions g(x) that are identically zero * Communicated by W.E. Smith for all sufficiently large x. We construct another testing function space Ha , which consists of cer- tain entire analytic functions of succhetce growth. With suitable topologies assigned to Gh sb and Hg p> it is shown by means of the Griffith théorem that the uth order Weber transformation is a continuous linear mapping from els sb into Ha, p for every real u. An application of this *transformation is given to develop an operational calculus involving the operator su P THE TESTING FUNCTION SPACES Gh AND Gh AND THEIR DUALS a,b Let b > a > O and u be a real number. Then de sb is the space of all smooth complex valued func- eens @ on a < x < © such that $(x) = 0 on b (t) = Haye u'V Hence, in view of the seminorm Nee S $,,(t) conver- ges uniformly on every {} as Vv > ©, UMoreover, we have 1 1 Ero Dares) (t) = Dt” %6 “(n/ey Pty n The left-hand side of (9) converges uniformly on every 2 as V > © since multiplication by a power of t or the application of D™* preserves this prop erty. Moreover, $,,(t) converges in the same way, ; : y in view of the seminorm Yj. We now see from (9) that D,t “ %p,,(t) also converges uniformly on every 8, which implies in turn that Deo, (t) does the same. Next, by virtue of the application of the operator S,,, it follows that D2o (t) also con- verges cnnrcoemiy on every &. i Vv (nm). (9) k We repeat this argument eee $,, replaced by S¢,, and S$, replaced by sk+lp | This shows that fbr every hon-negative integer K, DX, (t) conver- ges uniformly on every 2. Consequently, there exists a smooth function $(t) on I such that for each k and t, D4 (t) SpeOtt ass = =. Next, the assumption that {$} is a Cauchy sequence in Gh pb can be restated as follows. For each non-negative integer k and any given ec > 0, there exists a real N,, such that for every v, n > Ny, we have y (9, - >,) < €. Therefore, when n> ©, we obtain YO, - se (10) for all v > Ny. In other words Yk, - >?) > 0 as Vi> © foreach k =20, 15 2....-. = Finally, there exists a constant C, not de- pending.on v such that YK (O,)) < (,. Therefore, by (9) we have CO) = EGG) = Gee Oi ane, ee. (it) Thus $ ds in Ch ob and is the limit in Chop of {>}. This completes the proof. (anys is the dual of Gob and by Theorem 1.8-3 of Zemanian (1968) it is also complete. The members of (Go we are generalized functions. For any fixed u, Ge is defined to be the strict inductive limit (Friedman, 1963) of Gh 15 where b traverses a monotonically increasing ; sequence of positive numbers and tends to infin- ity. Thus, ¢ € Gh if and only if > © Gh p for some b. A sequence {$,} converges to in GH if and only if $ and all 9, are in Cas for some fixed b, and {o} a0 Ge fie Ga is sequentially complete. ‘ We now list some properties of Chop and (Gap 1. If c < d, then G. ology of G Cc Ga q» and the top- ny of Gh on it by Gad: is identical’to the’ topology induced 2. For each f € (Gare. there exists a non- negative integer r and a positive constant C such that || < C max ¥; (9) (12) O © Gob: Here C and r depend on f but not on 9. 3. The differential operator S,, is a continu- ous linear mapping of Gh ob into Ga he as is implied by the equation (5,9) = yu» 6 EG. Thus the adjoint Si of S,, is a generalized differ- ential operator on (eo ' into (Ge p)! defined by := <£,S o>. 13 ; ) u? (13) Since S,, = S., we also have that S!' = S"'. Because of the Symmet ry of S.,, we shall henceforth denote the generalized differential operator Sj by S,,, thereby dropping the prime. The symbol S,, denotes either a conventional or a generalized differential operator depending on the ways in which it is used. If it acts on a testing function in Gabe it isa conventional operator; and if it acts on a gener- alized function in (GE b)'» it is a generalized operator. i Let a be any real number, and let f(t) be a locally integrable function on a < t < © such that f(t) is absolutely integrable on a < t < ©. Then, f(t) generates a regular member of Gey which we shall denote by f, through the definition <£,o> := i f(t)o(t)dt, 6 Gh ,. (14) a ’ That f is truly a member of (Gone can be seen from the inequality |<#,6>| < yp () [ [f(t [at <@, gE GE. (15) a For a > 0, Ge pb can be identified with a subspace of (Gh.4)'. for, if y(t) and 6(t) are both in Gh, and differ somewhere, we can choose a non-negative testing function $(t) © D(I) whose support is con- tained in an interval on which jp - 6 is never equal to zero. In this case, # <0,6> where $(t) #0. This verifies that ~ and @, taken as members of (Go wu must differ. Thus, fora 7 0, es pb can be identified with a subspace of (G, ,,)' and we write a AA i WEBER TRANSFORM 23 THE TESTING FUNCTION SPACES H AND H a,b a Let y and w be real variables and let n=ytiw where, b > a> 0; © is a member of H, sb if and only if n~26(n) is an even analytic cane eion of n and a, (9) ee Bap ess och een NCE) n The topology of H is is one sma by a, using the seminorms oO, sb,k? kK =20,052, . An equivalent topology is eee by ie. following set of norms: max a O n To, To sles ees dS waned So= morphism from H, p onto itself. 3 Proof We have 2r ek! ®) oe see = Lemma 4 The operation f > oe so INS ae morphism from Hi p onto itself. » 1S an iso- Proof The proof follows from Theorem 1.10-2 of Zemanian (1966). Lemma 5 Let Y(n) be an even entire function such that | als c|w lyon | < gel + In where Q and c are positive constants and m is a non- negative integer, then ¢ > Y® is a continuous linear mapping of H, b into H > a,b+c’ Proof We have | | -c|w e Y(n) wa peek < sup Py ene eles a x sup] e911 14 [2 n2k49(n) n = Qo, bk i Seep een H, will denote the strict inductive limit (Friedman, 1963) of Ha, pb Where b traverses a monotonically in- creasing sequence of positive numbers tending to in- finity. The dual of Ho is denoted by Hi: Lemma 6 F + YF is a continuous linear mapping of H4 into H%. The proof follows by using Theorem 1.10-2 of Zemanian (1968). Theorem 2 The Weber transform W,, is a continuous linear mapping from ch b into Ha, p and the inverse Weber transform W,~ i$ a continuous linear mapping from H into GH a,b a,b° Proof let us assume that ¢$ © ct and 20 First, as > Then, $(x) © L., (a,b) and we set b ®(n) = | (xn) %¢,, (xn, an) (0 dx. a 24 R.S. PATHAK AND H. K. SAHOO -L By Griffith's theorem, n *6(n) is an even and ana- lytic function of n. Therefore b 1 (- 1 7" 9(n) = f (-1)™n™ xn) 4c, xn an) (x0) dx a b | S™C (xn) 2C. (xn, an) 36 (x) dx ay u b 1 zi | SL (x) J(xn) 7C. (xn, an) dx A u u (by integration by parts). Therefore m_ 2m+¥ : Sia (tans so (nes | (xn), (xn, an) [St $(x) Ix a For a-< x < b, (mn) > 0-7 > 0.. In this case xn = O(an) and = 1). |J,,Can) J, Gn)| = 0(1) Also, as xn > 0, = ae Cy (xn, an) = O(log -* Moreover, as (xn) > %, i J,,Can) J, (xn) = OL(xn) “J. Thus, (xn) C,, (xn, an) is bounded by a constant for all neEcC anda0O. Therefore a 2m+}; -(b-a) |w| oben = sup|n d(n)e n m b -4 b. Hence by the inversion formula (2), we have L C, (xy, ay) (x) = {: ty) > dy. 0 CQ (ay) J By differentiating under the integral sign, we get 1, C, (xy, ay) (-D*sKeon = [ yKocy) (xy)? 2 ay. 0 CQ, (ay) J For a 0 OL (log y) 773 as y > 0 for Wy=70). Therefore 2 k Ae 72 [QJ IS¢00 | $M sup ly 40¢y) (1ty7) | ee | Oy <2 Oy») 2k +35 2 = . (18) We can also define the inverse Weber transform f = (W')-1F, when F EH}, by u <(WA) "TF, (Wo) = (19) for all 6 € H,- The following theorem is a consequence of Theorem 2 and Theorem 1.10-2 of Zemanian (1968). Theorem 3 W), is a continuous linear mapping from el: into H4, and cw) -4 is a continuous linear mapping from Ht into GiH. AN OPERATIONAL CALCULUS The generalized Weber transform can be used in solving boundary value problems with distributional boundary conditions. The following theorem is fundamental in this connection. Theorem 4 If £ © Gi" and F = Wif © HX, then WEBER TRANSFORM 25 We (S*£) : (-1) *n"*ws, k = 0,1,2,-.. - (20) Proof From the fefinition of the operator s, and (18), we have k k = a ) i Wy) k = i? k = yeoW CS, 9) k_ 2k = iI (Dm uw = (-1) £n? Kew, >. Hence the proof. AN APPLICATION We seek the solution u © ras of the differen- tial equation Ce ge 21 wt = 8 (21) where g ©€ Gil such that Wig € H}. Applying Theorem 4, we have 2,k C-ny Un) = Gi); (22) Department of Mathematics, Banaras Hindu University, Varanasi 221005, India. where U(n) = W,,u and G(n) = Wy,g are elements of H}. Also, we may add to (22) the general solution H(n) of the homogeneous equation. REFERENCES Friedman, A., 1963. GENERALIZED FUNCTIONS AND PAR- TIAL DIFFERENTIAL EQUATIONS. Prentice-Hall, Englewood Cliffs, N.J. Griffith, J.L., 1956. On Weber transforms. Jd. and Proc. Roy. Soc. N.S.W., 89(4), 232-248. Griffith, J.L., 1957. Addendum to my paper, on Weber's transforms. J. and Proc. Roy. Soc. NiS.W.,. 91(4)., 189. Pathak, R.S. and Pandey, R.K., 1981. Distributional Weber transformation. J. and Proc. Roy. Soc. N.S.W., 114(3,4), 63-75. Titchmarch, E.C., 1939. THE THEORY OF FUNCTIONS. 2nd edn. Oxford University Press. Titchmarch, E.C., 1924. Weber's integral theorem. Proe. London Math. Soe. 22(2), 15-28. Zemanian, A.H., 1968. GENERALIZED INTEGRAL TRANS- FORMATIONS. Interscience Publishers, New York. Zemanian, A.H., 1966. A distributional Hankel trans- formation. oJ. SIAM Appl. Math., 14(3), 561-576. (Manuscript received 25.10.1984) Journal and Proceedings, Royal Society of New South Wales, Vol. 118, pp. 27-42, 1985 ISSN 0035-9173/85/010027 — 16 $4.00/ 1 The Electromagnetic Pinch: From Pollock to the Joint European Torus* R.S. PEASE INTRODUCTION This review of the electromagnetic pinch starts with an exhibit taken from Pollock's work, carefully preserved and drawn to attention of modern research by Professor C. Watson-Munro. It is a compressed and distorted length of copper tube originally part of the lightning conductor on the Hartley Vale kerosene refinery in New South Wales (Figure 1). It was known to have been struck by lightning. Pollock and Barraclough (1905) from the Department of Mechanical Engineering at Sydney University carried out an analysis to see whether or not the compression could have arisen from the flow of electric current. They concluded that the compressive forces, due to the interaction of the large current flow with its own magnetic field could have been responsible for the compression and distortion. As far as I know, this is the first identified piece of observational data on the electromagnetic pinch; and the first theoreti- cal discussion of the effect. Pollock and Barraclough did not, perhaps, have accurate data on the currents and duration of lightning strokes, and therefore believed that the compression of the tube was aided by a softening of the copper because its temperature was raised. A repetition of their analysis with modern data (Fink and Carroll, 1968) suggests that if the duration of the stroke was its normal length - not much more than 10 +s - then the heating effect would have been small. The required compressive effect in unsoftened copper of the strength they quote would have been provided by currents exceeding about 150kA. About 1% of all lightning strokes have peak currents greater than about 150kA, so that although the stroke is unusual, the basic conclusion drawn by Pollock would seem to be correct. Because the electric nature of lightning and lightning conductors were well known by 1905, it is perhaps surprising that the effect had not been observed earlier. However, the use of copper tube upon which the observation depended may have depended on the understanding of the skin effect, which was developed only in the later part of the nineteenth century. The actual arrangement sketch- ed in Figure 1 shows another interesting feature. There was a keen dispute as to whether or not the tops of lightning conductors should be made sharp or blunt (Seeger, 1973). It would appear that the top here is basically spherical but with some sharp points (some of which were dislodged by the lightning flash) added by way of compromise. * Pollock Memorial Lecture for 1984 delivered at the University of Sydney on 28th November, 1984. Section at b Figs. 1 Lightning Conductor at the Hartley Vale Kerosene Plant, N.S.W., examined by Pollock and Barraclough, 1905. Finakly, it should be remarked that only part of the copper tube collapsed, most of its length was unaffected, so that the effect must have been rather marginal in this case. DEFINITIONS The electromagnetic pinch is the constriction of a compressible conducting column when a large current is passed along it. The current generates a magnetic field B which must penetrate to the 28 R.S. PEASE region where there is a current density j. The force per unit volume there is j x B, and in general gives rise to a gradient of pressure p and an acceleration of the material of mass density pe and velocity v, which can be described by the simple magnetohydrodynamic fluid equation: j x B = grad p + >. (pv) In the simplest case of a straight cylindrical conductor of radius a with the acceleration term neglected, the pressure given by the effect can be calculated simply, provided the distribution of current as a function of radius r is known. For example, the pressure on axis p(0) is given in the Simple case of a uniform current distribution by DUO) als (nase we S| aa neaetes (2a), and for case of strong skin current plo), =.12/2mae 6 hee eal (2b). Here I is the total current in e.m.u. The differ- ence between (2a) and (2b), even though both cases have the same magnetic field on the surface of the conductor, reminds us that the concept of magnetic pressure characterized by the quantity H2/8r, which is a very useful concept, does not always yield an exact pressure in the presence of curved lines of force. A more general form of the pressure balance, which is independent of the current distribution, was first given by Bennet in 1934, for the case of a gaseous medium characterized by a gas pressure p p = nkT where n is number density and k Boltzmann constant and T the temperature. By integrating the expression over the radius and assuming negligible pressure at the outside one can obtain the general expression = a i a ee ‘ B,I* = 2k (NT, + N,T,) wees (3) where No and N. are the number of electrons and ions reSpectivély per unit length of the discharge; T. and T. are their mean temperatures; and 8, is a coefficient which we shall need later, and which is unity in the simple pinch. The inertial term in Equation (1) is important in rapidly rising current discharges through low pressure gases, where both the mass density and velocity of the ionized gas change with time as the discharge collapses. The inertial term is likely to have played a significant part in the collapse of Professor Pollock's lightning conductor. CONFIRMATORY WORK IN LIQUID METALS Shortly after Pollock's discovery, Northrupp (1907) working in America, published a paper ''Some months ago my friend Carl Hering described to me a surprising and apparently new phenomenon which he had observed, namely in passing a large alternating current through a liquid conductor contained in a trough the liquid contracted in cross sections and flowed uphill, length wise of the trough, climbing up upon the electrodes." Northrupp identified the effect as due to the self-magnetic-pinch effect, and he ascribes the use of the word "pinch" to Hering who he said "jocosely called it the pinch phenomenon", a name which has stuck ever since. Northrupp himself extensively studied the way the forces acted on this pinch, and showed by experiments with liquid potassium in kerosene that indeed this effect happened. There is nothing mysterious about it, but it was at that time a new manifestation of the electromagnetic forces between currents first analysed by Ampére. One interesting example of Northrupp's work showed that the pinch effect could be used to produce continuous motion of mercury (Figure 2). The current flows in and out of the double-walled vessel containing mercury as shown. There are insulating walls, which constrict the current channel in the centre, so that it produces a local high-pressure pinch effect. The mercury therefore is squeezed out of this pinch, axially upwards (with a manometer on the top, he could measure the mercury rising in the manometer when the currents pass). In this particular experiment he connected the mercury round through a separate channel, by which it could get back into the pinch region. This is an interesting example of the continuous motion of fluid produced just by a current flowing through it, without any use of external magnetic field (as in a conventional electromagnetic pump). The effect might seem to be the inverse of the self-excited dynamo; perhaps if one took some mercury and circulated it vigorously in the apparatus such as in Figure 2, a current could be generated. However, the analogy with geophysical processes is perhaps not sound, because of the presence of insulators. The subject of the pinch, generally speaking, did not attract much attention in the years follow- ing these reports of this effect in solids and liquids. More recently, experiments have been conducted in falling columns of sodium and of mercury, mostly to show that the pinch equilibrium is unstable, and that the magnetic forces, although they are there, very quickly can get out of equilibrium (Dattner, Lehnert and Lindquist, 1958; Bickerton and Spalding, 1962). Finally, there are now industrial applications of magnetic pressure, in which metals are formed by means of pulsed electromagnetic fields. For example, at the exhibition accompanying the Fourth U.N. Conference on the Peaceful Uses of Atomic Energy (in 1971), the Soviet Union showed magnetic crimping of aluminium cans on to uranium fuel elements. RADIATIVE COLLAPSE OF DEGENERATE PINCHES Since the magnetic forces have no mechanical upper limit, it is interesting to speculate on how large a pressure might be generated. Northrupp himself was clearly very interested in how far you might be able to carry this phenomenon. For instance, consider a one-micron-radius copper wire; such a conductor can be said to have a current- carrying capacity of about 350 megamps; that is, if all the electrons in the copper wire move with the velocity of light, then the current would be about 350 MA. Of course, it is perhaps a challenge for the engineers to get 350 MA through a one-micron radius piece of copper. But if one did there would be a very substantial pressure generated, and the question is: what would happen? THE ELECTROMAGNETIC PINCH 29 Current leads o\ \\ ~ \ 4 WANN NAVVAAAAAAAAAAA ANAND CRNA ENIAC Mercury ANAAAVAANAAAAANAAN Insulator ANVAAAAARARAALARARD LAALANRALY ORT URANRALRARRRARLRRRARRLIANANY Figure 2. Apparatus for pinch effect to produce continuous circulation of conducting fluid (Northrupp, 1907). B, D, - metal electrodes; p, - insulating plug; T, t - outer and inner insulating tubes; S, - insulating constriction; h, G, - flow channels for mercury. Arrows show direction of flow (independent of polarity). Now the copper would certainly vaporize and form a hot ionized gas or plasma essentially fully ionized. Classically the way the pressure increases as the radius decreases would mean that, provided the gas is conserved and is adiabatic, then the gas pressure rises more rapidly than the magnetic pressure, so that the compressing force of the current is resisted at some radius, although perhaps quite a small one. If there is some radiative cooling, as is believed to be responsible for the collapse of stars, then the gas would cool, and it would be unable to resist the magnetic forces. The radiated power can be calculated with a fairly straight- forward formula, at least in the non-relativistic limit, and compared with the ohmic heating, from the current flow. Because the resistivity of an ionized gas decreases as the temperature is raised, then it can be shown that, if the current is large enough, the radiative cooling will exceed the resistive heating, the gas will cool itself, and the pinch radii will collapse. If the current is fairly small, then the ohmic heating in the gas will overcome radiative loss, and the pressure prevents the collapse of the conductor. For the simplest case of radiative cooling by Bremsstrahlung, the critical current separating the two regimes is given (Pease, 1957) by mc2, -he ,45/2Z+1 1 s 45 a i at ote. (c) Here the electron charge e is in e.s.u., the ionic charge, supposedly fully stripped is Z and Log A is the coulomb logarithm which appears in ion-electron collision terms. For the case of Z=1 (hydrogen) , 8, = 1 (mo trapped magnetic field) Lt is about 1°7 MA. Numerical factors apart, this is the current carried by a conductor containing one electron per classical radius of electron moving at the speed of light, multiplied by the square root of the fine structure constant. It comes out at roughly this quantity irrespective of the number of particles per unit length, or of the radius of the discharge; and is relatively insensitive to the nuclear charge Z. So that this collapse is current dependent, and not geometry dependent. Of course, if the radiation loss is enhanced by (for example) line radiation, then the critical current will be substantially less (Ashby and Hughes, 1981). This calculation is non-relativistic. Nevertheless, at currents greatly below 1 megamp one would expect the wire to explode; whereas with much larger currents we might expect it to collapse. Some recent work is being carried out to see if a radiative collapse can be detected (Hammel, Scudder and Schlachter, 1983). Of course, as the pinch collapses the inductance of the circuit increases and so to keep the current constant additional energy has to be fed in; other- wise the current will fall off. In any case it is likely (as mentioned above) to be unstable. If, nonetheless, the filament collapses under the self-magnetic pressure, then presumably the Fermi pressure of a degenerate state would deter- mine the equilibrium radius (Fowler, 1955). If there are N electrons per unit length, moving at some fraction n of the velocity of light, then in the non-relativistic case the radius aR of the pinch is given by Here h is Planck's constant, m and e the mass and charge of the electrons; the pressure of the ions is neglected. Consequently, when N,>101° per m length, as is invariably the case in gas discharges, the radius of the pinch could be somewhat less than the Bohr radius depending on the current as represented by n. The radius given in equation (5) is the lower limit given by complete degeneracy. Some non- degeneracy must be allowed in order that the discharge can radiate, which would expand the radius. Taking the numerical example of the 1 micron radius copper wire when Ng = 7X10!8m"1, if n = 1/100, then I = 3.5 Mamps, and a = 10 2 m. Equation (3) still holds, but the electron 30 R.S. PEASE temperature is replaced by the Fermi energy. If larger currents are carried then the Fermi energy becomes relativistic. As is well known from the study of gravitational collapse, the Fermi pressure then varies more slowly with density, (Fowler, 1955), namely: p = (hc/8). ae and the pressure balance would thus require a still smaller radius 3 2 o TT -] he -3 aR 8 i (34 i If we take the same copper wire, with n = 1/10 (a current of 35 MA) equation (6b) leads to a radius of ~ 10 '3m. Such speculative radius is unlikely to be achieved because of instabilities, but this particular case has not, to my knowledge, been analysed for stability. GASEOUS PINCH DISCHARGES In the 1940s, the possibility of using the pinch to provide the temperatures and high pressures needed to produce thermonuclear reactions was realized by a number of workers, including P.C. Thonemann from Australia (1958), and G.P. Thompson (1946) in England. In the Soviet Union and in America a consider- able number of experiments were carried out in straight gaseous discharge tubes, generally made of quartz or other high quality ceramic, currents of up to about 2 MA were passed, and these produced temperatures of about 2 million degrees at relative ly modest densities, about 10!7cm™ These discharges were energized by powerful banks of capacitors and were of short duration, i.e. 1 - 10 microseconds (Figure 3). The chief results of this work, which is extensively reviewed (Artsimovich, 1964), are that the primary phenomenon observed were dynamic in nature, and rather far removed from the steady-state calculations dealt with to trigger circuit EER WUMMUMMNVIM IN; gegen \__/ pump Figure 3. Diagram of capacitor - driven electric discharge apparatus, used for studies in U.S.A.and U.S.S.R. of the pinch effect in low pressure gases. above. The collapse phase of the pinch during the rising current is well-described by the inertial terms of equation (1) developed into the so-called snow plough model, independently in the U.S. and USSR (Rosenbluth and Garwin, 1954; Leontovitch and Osovets, 1956). Thereafter, the pinch column tends to break up due to the onset of instabilities predicted by Kruskal and Schwarzchild (1954). The rapid increases of the circuit inductance produced strong electric fields which, aided by the ionization produced by the radiation from the compressed colum, induced secondary breakdown at the walls of the tube; it is these secondary breakdowns which effect- ively prevented the achievement of higher pinch currents. This simple type of linear geometry pinch discharge is still studied, but mostly in a form known as the "plasma focus'' which uses a form chosen to maximize the compression and to minimize the influence of the walls and electrodes (Filippov, Filippova and Vinogradow, 1962). A fine current carrying filament is formed, with currents in the range 1-2 MA, lasting for about 100 nanoseconds. The duration of this stage is again apparently limited by instabilities; an example of these is shown in an optical interferogram at Figure 4. In the later stages of the discharges strong beams of quite high energy ions and electrons (1 MeV - 10 MeV) are formed, a remarkable result in view of the fact that the potential applied across the electrodes does not exceed 100 kV. The detailed physics, including the question of pressure balance, remains to be resolved. In the early stages, with a filamentary discharge of 2 or 3 mm diameter, electron densities reach 10!%cm73 with temperatures up to about 10 M°K (3rd International Workshop on Plasma Focus Research, Stuttgart, 1984). When operated in deuterium or in deuterium-tritiun, these discharges are a strong source of nuclear reactions - up to about 10! reactions in D-T have been observed, and the yield increases rapidly with current. However, most of the reactions are thought to result from interaction of the acceler- ated particles, and not from thermonuclear reactions The relation of the particle motions in the plasmas to the macroscopic quantities such as current density and pressure has been elucidated by Haines (1978). The highest compressions observed occur in simple vacuum sparks, where the current carrying material is provided by the metallic electrodes. In these spark discharges, small hot spots a few microns in diameter, are formed with densities of about 107*cm 3 and temperatures of 10 - 20 MK (Vikrevet al, 1981; Negus and Peacock, 1979). These produce confinement of the particles comparable to that needed in fusion reactors, but it is probable that the power required to maintain them is relatively very large, so that the energy confine- ment time - which is the crucial factor - is much less than the particle confinement time. It cannot be excluded that these hot spots are formed from metallic particles released from the electrode, rather than from an electromagnetic compression. The phenomenon in this respect is perhaps more like that encountered when large currents are passed through thin wires. THE ELECTROMAGNETIC PINCH Figure 4. Interferogram of a pinched discharge in a plasma focus geometry. The length of the discharge shown is 3 cm. The discharge current is 0.85 MA, aes 5x10!9%cm 3, T, ~ 2keV, exposure time 1.2x1079s. The interferometer is set up to produce a series of horizontal fringes; on introducing a plasma these fringes are displaced, producing (approximately) contours of plasma density. The implosion of a current-carrying skin produces a conical plasma at one end. A-A is a region of low compression, B-B of high compression. A series of instabilities can be seen breaking out from the main core (Peacock, Hobby and Morgan, 1971). 31 32 R.S. PEASE Taken together, there is still much to be understood in these simple types of pinches. But for thermonuclear research the emphasis has for a long time now been on the quasi-steady pinch stablized by longitudinal magnetic fields. More- over, to avoid thermal conduction losses to and contamination from electrodes, these discharges are induced in toroidal chambers by pulsed transformer action, as pioneered by Cousins and Ware (1951). TOROIDAL STABILIZED DISCHARGES The instability of a simple pinch discharge, as illustrated in Figure 4, is a primary obstacle to obtaining in a pinch the high temperatures (about 100 million degrees) and good thermal insulation required to achieve a net output of energy from thermonuclear fusion reactions of hydrogen isotopes. In the 1950s, experiments on stabilizing the discharges by adding externally generated longitudinal fields and by using highly conducting metal-walled toroidal chambers were undertaken independently in several laboratories (Levine, 1955; Beztachenko et al, 1956; and Bickerton, 1958). Figure 5 illustrates the generic arrangement. The early experiments showed considerable improvements in the stability of quasi-steady state discharges. An example is the large Zeta machine built by Thonemann and Carruthers at Harwell (Butt et al, 1958), initial- ly designed for 10QkA currents, when temperatures in the range 1-2 M K were established. However, detailed observation showed that in the initial experiments, not all the fluid instabilities were suppressed, that the thermal insulation of the gas by the magnetic fields was disappointingly low, and that the fusion reactions observed were primarily due to a distortion of the Maxwellian distribution of the ion velocities, rather than to the high temperatures (Burton et al, 1962; Afrosimov et al, 1960). Later developments on Zeta, using currents of up to 1 MA, showed that the origin of the main fluid instabilities was due to shortcomings in the detailed distribution of magnetic field components (Robinson and King, 1969). The critical observations showed moreover that the required configuration could, at sufficiently high conductivity of the gas, be generated by the flow of current itself in the plasma. The fluctuation of the current shown in Figure 6 indicates the presence of instabilities; but they cease for a period of about 3 milliseconds during the discharge when the configuration relaxes to a configuration calculated to be stable on the basis of the ideal fluid equation (1). : The essential requirement in this case is that the strength of the stabilizing field as a function of minor radius should fall to zero and change sign in the outer region of the discharge. In addition, there has to be no turning point in the magnitude of the stabilizing field in the outer regions; this last corresponds to the more general requirement that the pitch length (q) of the lines of force should always be changing as a function of minor radius r, i.e. dq/dr # 0 except at r= 0. The experimental results were in accord with this theory (Rosenbluth, 1958). A more complete theory given by Taylor (1974), predicts a current density distribution determined by Poloidal Transformer iN Winding (Primary. | ron . circuit) Transformer Toroidal Toroidal (Secondary circuit) Magnetic Resultant [Field Helical Field Figure 5. The toroidal pinch, stabilized with an externally generated toroidal field. The pitch of the helical lines is short compared to the major circumference in the reverse field pinch case; and is long in the Tokamak case. ee EEEEEEEIEEESEEEEEE EEE mE j = uB where yw is a constant in space. It accounts for the self-stabilization phenomenon in the following qualitative sense. When currents flow through a flexible compressible conductor, the forces exerted on the conductor will always tend to minimise the total associated magnetic energy SH2/81 subject to the conservation of flux; in a normal wire, the resulting motion tends to increase the circuit inductance. If resistivity is small but finite, the configuration must change to one that minimizes the magnetic energy integral. The configuration found in Zeta nearly satisfies that requirement. A more modern experimental result (Ohkawa and Kerst, 1961; Bodin and Keen, 1977) shows j/B plotted against radii in a pinch experiment. In particular, theory accounts for the formation of the reverse field in the outer region. It does not explain the detailed mechanism by which the necessary current flows are driven: these are still unknown. The configuration found to arise is close to that originally calculated by Rosenbluth (1958) and which was also experimented on by trying to set up the configuration by deliberately programmed fields by a number of groups (Ohkawa and Kerst, 1961; Bodin and Keen, 1977). Both this reversed field pinch configuration and the spontaneous relaxation phenomenon are now the subject of considerable research (see references in Pease, 1985). However, from the point of view of thermo- nuclear applications, it is essential that the THE ELECTROMAGNETIC PINCH 33 gas 800 400 Longitudinal field B, (gauss) — 400 RroureeG. ZETA (Robinson and King, 1969). current Oo + va / ‘Minor Radius’ Oo Self-stabilization and field configuration in the outer region of the discharge in The inset oscillogram shows the current I rising to 420kA peak; 4.0ms 1.2ms O—5.0ms ee A 41 (cm) the fluctuation in the dI/dt trace drops to a low value for about 3ms,indicating a stable period. During stability the sign of the stabilizing field is reversed in the outer regions, and there are no turning points. configuration actually departs from the prescription given by Taylor (equation (7)) in two respects: First that there must be some current flow Jy perpendicular to the field - otherwise no pressure gradient can be sustained; and secondly that in the outer regions, where the gas is cold and/or the walls of the torus intrude, the current density is liable to fall to zero even though the magnetic field strength stays finite, i.e. yw cannot be constant everywhere as asked for by relaxation theory. A key question of modern thermonuclear research on pinches is whether or not sufficient departures can be obtained to obtain high tempera- ture plasma confinement without encountering instabilities which reduce the thermal insulation, and allow the discharge to relax back towards the force-free Taylor state. THE TOKAMAK CONFIGURATION The Tokamak configuration differs from that investigated in Zeta by having a very strong stabilizing field; it was developed primarily in the Soviet Union (Artsimovich, 1972), although there was some early work at ANU, Canberra (Liley, 1968). The field has to be sufficiently strong for the pitch length of the lines of force to exceed the major circumferences 27R of the torus; this is represented by a safety factor rB for stability. Here B, is the stabilizing field, B. the field due to the?pinch current. All the above considerations of self-stabilization apply, but the configuration apparently allows greater departures from the Taylor prescription (equation (7)), including especially the outer regions where the connection to the zero current region, necessarily present close to the chamber walls, can be sustained more easily than in the reverse field pinch. As a result, longer duration discharges are obtained (although the initial phases of the discharge normally show fluctuations indicative of instabilities). An important turning-point observation made on the Russian Tokamak T-3 installation, using the then novel technique of Thomson scattering of laser light to measure the quantity nkT_ as a function of posit- ion, enabled the first demonstration of equation (3) in quasi steady state to be achieved. In this case, the stabilizing field itself takes up some of the pinch pressure; and the coefficient 8, has the theoretical value of about %, close t® that observed (Liley et al, 1968). Furthermore, the temperatures 34 R.S. PEASE obtained exceeded 10 MK at currents of up to about 200 kA. JOINT EUROPEAN TORUS The results obtained on these toroidal discharges, and particularly on the Tokamak, restored confidence that stabilized toroidal discharges could lead to a confined thermonuclear plasma. It appeared that the stability and the high temperatures resulted from rather general principles, and therefore could be extrapolated to apparatus capable of producing conditions close to those needed in an energy-producing thermonuclear reactor. Consequently, in discussing these matters with our European colleagues in 1970 we suggested that a collaborative venture might result ina bigger advance than could be taken individually. A number of considerations were in mind at that time in discussing what eventually became the Joint European Torus (Willson, 1981). First, to create conditions in which heating of the gas by the thermonuclear reactions could take place, then it is essential that the 3.5 MeV alpha particles from D-T fusion reactions be trapped within the plasma by the magnetic fields. It turns out that to confine charged particles in a torus, the primary consideration is that the Larmor radius in the poloidal field component B should be less than the minor radius; whatever Phe strength of the stabilizing field B,, in the long mean-free-path conditions expected, ‘the radial excursion of charged particles in a toroid is about (r/R)?x Larmor radius in B,. And this requires, from straightforward Larmor radius consideration, that the current should be about 3MA.