JOURNAL DE PHYSIQUE Colloque C6, supplimenr au no 8, Tome 39, aofit 1978, page C6-1322

STUD1 ES ON SUPERFLUID TURBULENCE

K.W. Schwarz

IBM Thomas J. Watson R e s e a r c h C e n t e r , P.O. Box 218, Y o r k t a m H e i g h t s , New York 10598, U.S.A.

Rdsum6.- NOUS envisagerons les ~aractdristi~ues des turbulences superfluides et discuterons quelques rdsultats expdrimentaux concernant l'dtat compl6tement turbulent. Un modzle thCorique expliquant ces propridtds sera dCcrit.

Abstract.- The qualitative nature of superfluid turbulence and some experimentally established pro- perties of the fully turbulent state are reviewed. A theoretical model explaining these properties is described.

INTRODUCTION.- Superfluid 4 ~ e remains a liquid down

to arbitrarily low temperatures, and can therefore

always be made to move with a non-zero velocity -+ -+

field v (r,t). At temperatures above absolute zero,

this fluid is permeated by a dilute gas of elemen-

tary excitations. If conditions are such that this

excitation gas comes to local thermodynamic equili-

brium in some characteristic distance and time,then

for distance and time scales much greater than the-

se, the gas can be usefully described in terms of -+ -+

an additional velocity field vn(r,t).

For laminar flows, the hydrodynamical beha-

vior of these velocity fields is well described by

the Landau two-fluid model / I / , the success and li-

mitations of which may be illustrated in terms of

the flow experiments shown in figure 1. At low heat

inputs, V and Vs are small and the measured values

of AT and AP are in good agreement /2/ with those

computed from the two-fluid equations. As the flow

is increased to some critical value. however. a new

in terms of the macroscopic two-fluid equations, and

the study of this regime has consequently been a

matter of continuing interest. It would not be dif-

ficult to assemble a list of a hundred papers repor-

ting not only on experiments of the type shown in

figure 1 , but also on many others in which different

geometries and various ingenious methods are used to

probe the properties of the turbulent superfluid.

Despite this immense effort, our understanding of

the subject remains in a primitive and somewhat con-

troversial state, partly because it has proved ex-

tremely difficult to devise turbulent flow experi-

ments which yield any kind of s i m p l e information,

and partly because the subject has not received the

amount of theoretical attention it ~erhaps deserves.

JLATING WALL

- -

regime appears in which AT becomes much larger than

expected. The appearance of this new regime is in I

many ways reminiscent of the onset of turbulence in

a classical fluid, and it is natural to interpret it

in terms of a breakdown in the laminar character of + -+ v or vn, or both. The experimental consensus seems to

-+ be that in fact it is V which usually goes unsta-

ble, due to the appearance and growth of quantized

vortex lines which begin to fill the channel in a

more or less random way. Since elementary excita-

tions are scattered by these vortices, there is now -+ +

a new, dissipative interaction between v and v

which qualitatively accounts for the anomalously

large values of AT.

Neither the onset nor the hydrodynamical be-

BATH

I I HEATER 6

Fig. I : Schematic of a typical counterflow experi- ment. V and V are the average normal and super- fluid vglocitigs. The measured quantities are AT and AP

With these considerations in mind, and with

apologies to the many distinguished scientists whose

work will not be mentioned, I have chosen not to

give a historical survey of the field. Instead, I

will review only a few properties of the fully tur-

bulent state, and then discuss the extent to which

they are presently understood.

havior of the "turbulent regime can be understood

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786565

ASPECTS OF FULLY TURBULENT FLOW.- One reason f o r con-

c e n t r a t i n g on t h e f u l l y t u r b u l e n t s t a t e , r a t h e r than

t h e onse t r eg ion , i s t h a t one can expect i t t o be

somewhat s impler . It is assumed t h a t t h e channel be-

comes f i l l e d wi th a random t a n g l e of quant ized vor-

t e x l i n e s which move about under the combined i n f l u - *

ence of t h e i d e a l f l u i d equa t ions governing v and

t h e f r i c t i o n a l f o r c e s e x e r t e d on t h e v o r t i c e s by t h e

normal f l u i d . As t h e average d r i v i n g v e l o c i t y V -V n s

is inc reased beyond t h e c r i t i c a l va lue , t h i s t a n g l e

becomes i n c r e a s i n g l y dense , and t h e c h a r a c t e r i s t i c

i n t e r l i n e spac ing 6 becomes sma l l e r . Presumably,when

6 is much l e s s than t h e channel s i z e , t he tu rbu lence

can be desc r ibed by l o c a l equa t ions , t h a t do n o t de-

pend on t h e geometry o f t h e system, excep t through

boundary cond i t ions . I n macroscopic (d 1 o - ~ cm)

channels t h i s l i m i t i s e a s i l y a t t a i n e d : f o r V -V of n s

a few cm s - l , 6 i s a l r e a d y of o r d e r t o cm.

It is i n s t r u c t i v e t o n o t e t h a t t h i s corresponds t o a

t o t a l l i n e l e n g t h of 10 t o 1000 km per cm3 of f l u i d

131.

As f a r a s we can t e l l a t p r e s e n t , i t does

appear t h a t t h e f u l l y developed t u r b u l e n t s t a t e has

some well-defined gene ra l p r o p e r t i e s . By f a r t h e

most w e l l e s t a b l i s h e d of these was observed a s e a r l y

a s 1940 by Keesom, S a r i s , and Meyer /4 / i n an expe-

riment such a s t h a t shown i n f i g u r e l . They found

t h a t t h e excess AT due t o t h e tu rbu lence went a s the

cube of t h e h e a t c u r r e n t 6. This obse rva t ion has been

confirmed approximately by many subsequent i n v e s t i -

g a t o r s 15-111. I t can be understood phenomenological-

l y by assuming t h a t i n t h e t u r b u l e n t s t a t e a new

mutuaZ friction force d e n s i t y 151.

a c t s t o couple the normal and s u p e r f l u i d components

o f t h e motion. Here, A(T) is an expe r imen ta l ly d e t e r -

mined f u n c t i o n o f temperature , ps and pn a r e t h e su- 3 -+ -+

p e r f l u i d and normal f l u i d d e n s i t i e s V = V -V and n s n s'

v i s a smal l a d j u s t a b l e parameter. F igu re 2 shows

t h e c o e f f i c i e n t A(T) a s determined i n v a r i o u s expe-

r iments . Although t h e r e is reasonab le q u a l i t a t i v e

cons i s t ency , t he magnitudes o f A ag ree on ly ve ry ap-

proximately . ~ i f f e r e n c e s on t h e o r d e r of those seen

i n f i g u r e 2 seem t o crop up r a t h e r c h a r a c t e r i s t i c a l l y

when one t r i e s t o compare v a r i o u s f low exper iments ,

f o r reasons I s h a l l d i s c u s s s h o r t l y . At o u r p resen t

l e v e l of unders tanding, t h e s e d i f f e r e n c e s a r e of l e s s

concern than t n e common f e a t u r e s r ep resen ted by equa-

t i o n ( I ) and f i g u r e 2.

TEMPERATURE ( K)

Fig . 2 : Mutual f r i c t i o n c o e f f i c i e n t observed under va r ious c o n d i t i o n s : ( a ) d 2) 0.5 cm, counterf low 161; (b) d ?, 0.01 cm, counterf low 181 ; (c) d ?. 0.3 em, counterf low /7/ ; (d) d 1. 0.03 cm, pure normal f l u i d flow / l o / ; ( e ) d 'L 0.03 cm, pure superflow / l o / . The d o t t e d l i n e shows t h e t h e o r e t i c a l p r e d i c t i o n .

-+ A s mentioned e a r l i e r , Fsn i s j u s t a f r i c t i o -

n a l f o r c e which a r i s e s because t h e quant ized vor t ex

t a n g l e s c a t t e r s t h e thermal e x c i t a t i o n s t h a t makeup

t h e normal f l u i d . Th i s i n t e r p r e t a t i o n was f i r s t pro-

posed by Vinen 1121, i n analogy t o t h e f r i c t i o n a l

f o r c e t h a t a c t s upon second sound i n a r o t a t i n g

bucket /13/ . By measuring t h e a t t e n u a t i o n o f a small

second sound wave by t h e tu rbu lence i n a channe1,he

was a b l e t o t u r n t h e argument around and deduce t h a t

t h e t o t a l l i n e l eng th d e n s i t y obeys t h e r e l a t i o n

Equat ion (2) is c o n s i s t e n t w i th equa t ion ( I ) , but it

provides a somewhat d i f f e r e n t k ind of informat ion.

It has no t been s t u d i e d n e a r l y a s e x t e n s i v e l y a s re-

l a t i o n ( I ) , a l though i t has r ece ived some i n d i r e c t

suppor t from exper iments u s i n g i o n probes 114,151.

A n improved v e r s i o n of Vinen's experiment i s cur-

r e n t l y underway, /16/ and we expect t o hea r some of

t h e r e s u l t s a t t h i s conference.

A f i n a l f e a t u r e of v o r t e x tu rbu lence was d i s -

covered r e c e n t l y by Ashton and Northby 1171. By f o l -

lowing t h e motion of i o n s which had become.trapped

on t h e quant ized v o r t i c e s , they were a b l e t o show

t h a t t h e vo r t ex t a n g l e has a n e t average d r i f t ve-

l o c i t y w i t h r e s p e c t t o t h e s u p e r f l u i d r e s t f r ame ,o f

t h e form

Th i s impl i e s t h a t t h e v o r t e x t a n g l e i s a n i s o t r o p i c ,

a r e s u l t which, i n r e t r o s p e c t , i s n o t s u r p r i s i n g .

C6- 1324 JOURNAL DE PHYSIQUE

Only t h e one experiment has been done, and f u r t h e r

work t o e s t a b l i s h the v a l i d i t y of equa t ion (3) would

be d e s i r a b l e .

Re la t ions ( I ) t o (3) make up a r a t h e r s h o r t

l i s t , one which excludes many i n t e r e s t i n g phenomena.

There i s of cour se t h e whole ques t ion o f what hap-

pens i n t h e onse t regime ; bu t one might a l s o men-

t i o n the exper iments of Peshkov and Tkachenko /18/

on the propagat ion of tu rbu lence down ve ry long

channels , o r t he very r e c e n t obse rva t ions o f random

f l u c t u a t i o n s i n the v o r t e x l i n e d e n s i t y 119,201.

Phenomena of t h e s e va r ious types appear , however , to

be more complicated than those which I have s t ressed.

As a f i n a l comment on t h e exper imenta l as-

pec t s , I would l i k e t o r e c a l l a well-known r e s u l t

from c l a s s i c a l hydrodynamics 1211 : when a v i scous

f l u i d moyes down a channel of l a t e r a l dimension d ,

t he d i s t a n c e i t must t r a v e l downstream be fo re end

e f f e c t s become smal l i s Z * 0.1 pvd2/n, where p i s

t h e d e n s i t y , V i s t h e c h a r a c t e r i s t i c v e l o c i t y , and

n is t h e v i s c o s i t y of t h e f l u i d . Le t us apply t h i s

equa t ion t o the normal f l u i d , assuming i t s f low t o

be laminar . For t h e t y p i c a l superf low exper iments

done i n narrow (d 1. lod2 cm) channe l s , Z is ve ry

sma l l and end e f f e c t s a r e unimportant. However,

such exper iments have been l i m i t e d t o measuring AT

and A P . The second sound and t h e ion probe exper i -

ments have been done i n channels wi th d * 1 cm. Here

Z t u r n s o u t t o be on t h e o r d e r of me te r s , and t h e r e

i s every reason t o expect end e f f e c t s t o be ve ry

important . This , and t h e f a c t t h a t of cour se t h e

f low p a t t e r n s a r e always inhomogeneous a c r o s s t h e

channel, may account f o r t h e c h a r a c t e r i s t i c d i f f e -

rences between v a r i o u s exper iments shown i n f i g u r e 2 .

THEORETICAL CONSIDERATIONS.- Th i s s e c t i o n w i l l be

concerned 6 t h d e s c r i b i n g a r e c e n t model 122,231 of

t h e v o r t e x t a n g l e dynamics which, d e s p i t e i t s appro-

x imate na tu re , appears t o be r a t h e r s u c c e s s f u l i n

exp la in ing the g ross f e a t u r e s d i scussed above. Le t

i t be s a i d a t t h e o u t s e t t h a t t h e gene ra l q u a l i t a -

t i v e background and s e v e r a l o f t h e s p e c i f i c i d e a s

which e n t e r i n t o t h i s model were in t roduced by

Vinen /12/ twenty y e a r s ago. I n a d d i t i o n , Ashton

and Northby / 1 7 / have given a q u a l i t a t i v e i n t e r p r e -

t a t i o n o f t h e i r v o r t e x d r i f t experiment which has

c e r t a i n e lements i n common w i t h t h e model t o b e des-

c r ibed .

The s i t u a t i o n t o be considered is a homoge-

neous v o r t e x t a n g l e s u b j e c t t o c o n s t a n t , coun te r f lo -

wing normal and s u p e r f l u i d v e l o c i t i e s . The f a sc ina -

t i n g t h i n g about t h i s problem is t h a t t h e microsco-

p i c behavior of t h e t a n g l e i s reasonably wel l under-

s tood. To s t a r t wi th , one can always d e s c r i b e the

in s t an taneous c o n f i g u r a t i o n of the quan t i zed v o r t i -

c e s e x a c t l y by s p e c i f y i n g t h e pa ramet r i c form * * 4 s = s ( C , t ), where s deriotes a po in t on t h e l i n e a n d

5 i s t h e a r c l eng th . The t ime development o f t he

t a n g l e is then conta ined i n t h e s ta tement t h a t any

given po in t on t h e l i n e moves wi th a n in s t an taneous

v e l o c i t y

wi th r e s p e c t t o the s u p e r f l u i d r e s t frame. The f i r s t

term on t h e r i g h t , t he se l f - induced v e l o c i t y o f t he

l i n e , r e p r e s e n t s t h a t p a r t of t h e motion which a r i se s

from t h e i d e a l f l u i d equa t ions . I t i s g iven t o a

good approximation by

* * where s ' = a s l a g is t h e v e c t o r t angen t of t he l i n e

-+ a t t he po in t i n ques t ion , 2' = a2s/aC2 is t h e vec-

t o r cu rva tu re , and 0 is an approximate cons tan t of

o r d e r cm2 s-l/24/.It is easy t o show t h a t t h e

motion desc r ibed by equa t ion (5) does n o t change

t h e l e n g t h All of a l i n e element and t h u s cannot cau-

s e t h e v o r t e x l i n e t ang le t o grow o r decay. The se-

cond term on t h e r i g h t of equa t ion (4) g i v e s t h a t

p a r t of t he l i n e motion which a r i s e s from t h e f r i c -

t i o n a l f o r c e s exe r t ed on t h e l i n e by t h e no rma l f lu id .

The temperature-dependent f r i c t i o n c o e f f i c i e n t a is

l e s s than 0.1 up t o 1.8 K, b u t i n c r e a s e s r a p i d l y

near the A-point. Th i s p a r t of t h e motion does no t

conserve A l l , and can r e s u l t i n e i t h e r t h e growth o r

decay of t h e v o r t e x l i n e depending on t h e d i r e c t i o n -+ .+

and magnitude of V -U ns I '

One can t a k e t h e in s t an taneous c o n f i g u r a t i o n * s ( C , t ) , u s e equa t ions (4) and (5) t o f i n d G(C) a t

every po in t on t h e l i n e , and t e l l a computer t o ta-

ke a t ime s t e p Z ( 6 , t + d t ) = Z(C, t ) + Z ( 6 ) d t . The

t ime development o f t h e t a n g l e can t h e n be obta ined

by i t e r a t i n g t h i s procedure. A numerical s imula t ion

of t h i s kind may e v e n t u a l l y prove u s e f u l i n studying

t h e o n s e t of turbulence .Forour purposes, however, i t

makes more sense t o d e s c r i b e t h e v o r t e x t a n g l e i n

s t a t i s t i c a l terms, and t o d i s c u s s its development i n

an approximate manner. To t h i s end, we cons ide rwha t -+

can be s a i d about the d i s t r i b u t i o n A(Vl. t ) , where

A(z l , t ) d % l i s def ined a s t h e t o t a l l i n e l e n g t h

p e r u n i t volume f o r which t h e se l f - induced v e l o c i t y

l i e s between W l and v,+dvl. Because of t h e key r o l e that

3 v plays i n equation (4) , t h i s turns out t o be a ma-

1 thematically usefu l descr ip t ion . I n add i t ion , s ince

+ A(vl, t ) t e l l s how much l i n e there i s and how the

tangle i s moving, it provides j u s t the kind of in-

formation we a r e looking f o r .

How does A changes a s the tangle develops ac-

cording t o equations (4) and (5) ? To determine

t h i s i t i s necessary t o know the instantaneous r a t e

of change of the length All of every l i n e elementand 3

of t h e vl associated with it. An e n t i r e l y s t ra igh t -

forward ca lcu la t ion shows t h a t the normal f l u i d

term i n equation (4) gives r i s e t o

-t + + -t a + -t 3 v 2 - % V l vie (Vns - v l ) - vl x (vl x Vns), (6b) 1

while the i d e a l f l u i d p a r t of the motion makes a

contr ibut ion

Our only purpose i s resenting these equations ex-

p l i c i t l y i s t o i l l u s t r a t e the d i f f e r i n g natures of

t h e two cont r ibu t ions t o the evolut ion of the tan-

gle . The e f f e c t of the normal f l u i d can be determi-

ned i m e d i a t e l y , s ince equations (6a) and (6b) in- +

volve only v . I f one keeps t rack of what happensto 1

the l i n e elements i n i t i a l l y contained i n some ele- +

ment d3vl of v space, one obtains 1

A(; +; dt , t+d t )d3v ( t + d t ) = ( l + ~ i d t / A ~ ) ~ ( $ ~ , t ) d ~ v ~ ( t ) 1 1 1

This simple bookkeeping formula, when expanded t o

f i r s t order i n d t and combined with equations (6a)

and (6b), y i e l d s the normal f l u i d con t r ibu t ion t o

a A / a t . This con t r ibu t ion contains convective terms +

which move the tangle around i n v space, and sour- 1

ce terms which increase and decrease the l i n e length +

i n various p a r t s of v space, but it says nothing 1

about the random nature of the tangle. By i t s e l f , i t

does not lead t o a s teady s t a t e .

The i d e a l f l u i d p a r t of the motion c ~ ~ u a t i o n $

(7b)l is e n t i r e l y d i f f e r e n t i n nature, s ince V de- 1

pends on l o c a l p roper t i es zl", zl:"" of the l i n e ele- -f

ment about which A(vl,t) provides no information.

Thus, e x t r a physical arguments must be introduced

to deal witE the e f f e c t of t h i s term on A . Now, the

l i n e i n the vortex tangle w i l l have some characte-

r i s t i c average curvature <s"> and hence a characte-

r i s t i c self-induced ve loc i ty <v > 2 B<s">. Various 1

p a r t s of the l i n e a r e moving about randomly with

v e l o c i t i e s of t h i s order , producing a complicated

i n t e r n a l motion. This motion w i l l lead t o frequent

l i n e - l i n e crossings, which i n t u r n generate l a r g e

random d i s t o r t i o n s a t local ized regions on the l i n e .

These d i s t o r t i o n s then propagate along the l i n e , so

t h a t any given point on the l i n e w i l l receive i n t e r -

mi t ten t randomizing "signals". More detai ledconside-

r a t i o n s show t h a t i n f a c t the typ ica l randomizing

s igna l occurs i n an average time of order 6/<vl>

and t h a t i t a r i s e s from a crossing which occured a

d i s tance 6 away, where 6 again i s t h e average i n t e r -

l i n e dis tance.

The conclusion t h a t the l i n e is repeatedly

chopping i t s e l f i n t o sect ions of length 6 makes it

na tura l t o assume t h a t the t ang le remains "kinky"

on t h e s c a l e 6. Given <sf'>, t h i s implies t h e order-

of-magnitude condit ions

13f1 1 % <s1'>/6, (9)

Furthermore, 31' and 2" randomly take on new values i n

the time Ci/<v:.Equation (7b) then implies, roughly -t

speaking, t h a t the v associated with any l i n e seg- 1

ment changes by a random amount of order < V l > i n the

c h a r a c t e r i s t i c time i n t e r v a l 6/<v >. Thus, any + 1

s t r u c t u r e i n A(v , t ) w i l l spread out a d i s tance<v > + 1 1

i n v1 space i n t h e time 6/<v >. This e f f e c t may be 1

credely modelled a s a d i f f u s i v e re laxa t ion process,

a c t i n g on t h e d i s t r i b u t i o n A($ , t ) and characterized 1

by a d i f f u s i o n constant of order <V >3/6. 1

The combination of the convective and source

terms due t o the normal f l u i d , and the d i f f u s i v e

term a r i s i n g from the self-induced motion of t h e

tangle seems t o provide a l l the necessary ingredients

f o r a reasonable, a l b e i t very ~ r i m i t i v e , descrip-

t i o n of t h e vortex tang le dynamics. Indeed, it a l -

ready appears t h a t a model of t h i s type can explain

the experimentally observed f e a t u r e s discussed ear-

l i e r . A s p e c i f i c equat ion f o r a A / a t , based on these

arguments, has been in tegra ted numerically t o f i n d +

the steady s t a t e A(vl,m) f o r var ious values of V ns

and T. This equation i s extremely complicated, and

i t may be possible t o f i n d simpler versions. Never-

the less , t h e numerical work has already produced

several i n t e r e s t i n g r e s u l t s . F i r s t , k(Sl ,a) i s found

to be s t rongly peaked a t values of V. on the order 1

of B/6 and t o be s t rongly an iso t rop ic with S1 poin- -t

t ing p r e f e r e n t i a l l y i n t h e d i r e c t i o n of V This i s n' i n accord with the speculat ions of Ashton and

C6- 1326 JOURNAL DE PHYSIQUE

and <vlZ> deduced from the computed distributions

are in surprinsingly good agreement with experimen-

tal observations. Figures 2 and 3 show the calcula-

ted values of F as a function of V and T. The sn ns

predicted results are certainly within the range of

experimental variation, and similar agreement is +

found for L and <v z>. Finally, A(v~,~) is found nu- 1 merically to obey a certain scaling behavior. By

integrating over the equation for ah/at and making

use of this scaling property, it is possible to de-

rive an equation identical in form to the well-

known Vinen equation /12/.

Fig. 3 : Predicted behavior of the mutual friction F . The d-+.s are the values actually computed, the l%es are straightline fits. The experimental re- sult is F a (Vns - 2 v ), to an accuracy of 3 0

In summary, it appears that significant pro-

gress can be made in understanding the gross pro-

perties of the fully turbulent state. It must be

admitted that the theoretical model, as it now

stands, raises perhaps as many questions as it ans-

wers. However, its initial success is very encouraging

and should stimulate further theoretical a d experimen-

tal work on this interesting problem.

References

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/24/ z' , 31, make up a right-handzd, mutually orthogonaI system of vectors, with respective magnitudes 1, R-l, BR-~, where R is the local radius of curvature of the line