Simplified hydrodynamic analysis of superfluid turbulence in He II: Internal dynamics of inhomogeneous vortex tangle

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Simplified hydrodynamic analysis of superfluid turbulence in He II: Internal dynamics

of inhomogeneous vortex tangle

J. A. Geurst

Department of Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands H. van Beelen

Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands ~Received 30 January 1996; revised manuscript received 16 April 1996!

The hydrodynamic theory of superfluid turbulence is presented in a new simplified form. It applies to flow situations frequently encountered in practice, in which the thermohydrodynamic environment of a superfluid turbulent tangle of quantized vortices may be considered, in a first order of approximation, as given. Flow quantities like the mass density, the entropy density, and the drift velocities of mass and elementary excitations act, accordingly, as external parameters with respect to the internal dynamics of the vortex tangle. The internal dynamics is completely specified by a kinematic equation governing the time evolution of the line-length density of the quantized vortices and a dynamic equation involving the impulse density of the vortex tangle. The derivation of these equations starts from a variational principle that is reminiscent of Hamilton’s principle in classical mechanics and proceeds, in order to include dissipative effects, by using methods of the thermo-dynamics of irreversible processes. A new quantity called superfluid turbulent pressure is introduced which shows many properties that are familiar from the ordinary pressure in a classical fluid. Two important particu-lar cases are considered in more detail, viz., homogeneous superfluid turbulent flow and flow situations in which the vortex tangle is in permanent internal equilibrium. When diffusion of the vortex-tangle impulse is taken into account and dispersive effects are disregarded, the dynamic equation of the vortex tangle assumes, in the case of internal equilibrium, the form of Burgers’ equation with a nonlinear source term. This equation, which is new, may be considered as a natural generalization of Vinen’s equation to inhomogeneous superfluid turbulence. Some exact solutions which represent uniformly propagating superfluid turbulence fronts are listed in the Appendix.@S0163-1829~96!01033-8#

I. INTRODUCTION

The general hydrodynamic three-fluid theory of superfluid turbulence in He II developed for one-dimensional flow in

~Ref. 1! and for three-dimensional flow in ~Ref. 2! is fairly

involved. In fact, three scalar and three vector evolution equations are required for the determination, as a function of position and time, of three scalar quantitiesr, S, and L rep-resenting, respectively, the local densities of mass, elemen-tary excitations — or entropy — and line length of the quan-tized vortices constituting a vortex tangle, and three vector quantitiesvW,vWn, andvWl denoting, respectively, the drift

ve-locities of mass, elementary excitations, and quantized vor-tices. We are therefore interested in special cases where the three-fluid equations take a particularly simple form.

A good example of such a relatively simple form of the three-fluid theory is afforded by the homogeneous case in which the flow quantities do not vary with position. The case of homogeneous flow is considered in detail in Ref. 3. In addition, a section is devoted to it in Ref. 2. The decay of homogeneous superfluid turbulence is analyzed in Refs. 4 and 5. For the analysis of homogeneous flow it appears to be sufficient to supplement the two-fluid equations for super-fluid 4He~Landau-Khalatnikov equations!, irrespective of an additional mutual-friction force, with a slightly extended form of Vinen’s equation governing the time evolution of the line-length density L of the quantized vortices. The extension

of Vinen’s equation involves the appearance of the relative polarity of the vortex tangle, i.e., the cosine of the angle between the direction of the relative drift velocity

wW_l5vW_l2vW of the vortex tangle and the direction of the rela-tive drift velocity wWn5vWn2vW of the elementary excitations. In one-dimensional flow the relative polarity takes the form sgnw_l sgnwn. It plays an important part in the analysis of

the decay of superfluid turbulence~see, in particular, Ref. 5!. Although homogeneous superfluid turbulent flow appears to be understood reasonably well now, both experimentally6 and theoretically,7inhomogeneous flow phenomena like su-perfluid turbulence fronts still await a simple unifying treat-ment. It is the aim of this paper to provide such a treatment by applying only a few basic principles. In particular we are looking for equations governing inhomogeneous superfluid turbulent flow that generalize Vinen’s equation in a natural way.

In a recent paper8dealing with the propagation of super-fluid turbulence fronts a class of flow situations is envisaged in which the thermohydrodynamic environment~‘‘bath’’! in which the vortex tangle is embedded, is regarded as given, usually constant with respect to position and time, while the vortex tangle itself is allowed to evolve according to its own dynamics. The influence of the thermohydrodynamic envi-ronment on the dynamics of the vortex tangle is represented by the external parametersr, S,vW, andvWn. The vortex

line-length density L and the relative drift velocity wW_l of the 54

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quantized vortices constitute the internal variables character-izing the various superfluid turbulent states of the tangle.

We want to develop here, in an independent way, for the class of flow situations just mentioned, the appropriate sim-plified form of our hydrodynamic theory of superfluid turbu-lence. It should be noted that flow situations of this kind are, approximately, realized in a variety of experiments. Since it turns out to be sufficient, in many cases of practical interest, to consider one-dimensional flow only, the internal variables involved in the dynamics of a vortex tangle will be repre-sented by the scalar quantities L and w_l .

In Sec. II the nondissipative equation of motion for a superfluid turbulent tangle of quantized vortices is derived from an appropriate variational principle which is reminis-cent of Hamilton’s principle in classical mechanics. Accord-ing to that derivation the impulse density P_l of the vortex tangle is determined irrespective of an arbitrary function of the dimensionless Vinen number Vi5kL1/2/uwlu, where k5h/m. The specification of this function follows from an

analysis of the energy of the vortex tangle in Sec. III. A superfluid turbulent pressure p_l satisfying the Clebsch-Bateman principle is also introduced in that section. Dissipa-tive effects like mutual-friction forces are added in Sec. IV according to the thermodynamics of irreversible processes. The diffusion of vortex-tangle impulse receives explicit at-tention. After a review of homogeneous flow in Sec. V the particular form taken by the equations when a condition of permanent internal equilibrium is imposed on the vortex tangle, is investigated in Sec. VI. When dispersive effects are neglected that particular form reduces to Burgers’ equation with a nonlinear source term. Some exact solutions of this equation representing uniformly propagating superfluid tur-bulence fronts are listed in the Appendix. The basic equa-tions of the paper are finally reviewed in Sec. VII.

Theoretical and experimental results on homogeneous su-perfluid turbulent flow are comprehensively reviewed by Tough.6 For an introduction to the subject we refer to the book by Donnelly,9where a chapter is devoted to superfluid turbulence. An interesting review of recent results on inho-mogeneous superfluid turbulence is presented by Ne-mirovskii and Fiszdon.10

II. VARIATIONAL PRINCIPLE FOR NONDISSIPATIVE SUPERFLUID TURBULENCE

As set forth in the preceding section we like to develop, for flow situations in which the tangle of quantized vortices may approximately be taken to be immersed in a given en-vironment of He II, the appropriate one-dimensional simpli-fied version of our hydrodynamic theory of superfluid turbu-lent flow. The given environment is, in a first order of approximation, characterized by definite values of the

exter-nal parameters r, S,v, and vn which, in general, will vary

with position and time. Although the values of quantities like the temperature gradient may be affected as a result of the dynamics of the vortex tangle, they will do so only in a second order of approximation~see Appendix A of Ref. 8!. The flow conditions just mentioned are encountered in a va-riety of experimental situations. For instance, in capillary flow the external parameters may be treated as constants pro-vided the flow velocities are small compared to the

propaga-tion velocities of first and second sound and, in addipropaga-tion, the boundary conditions at the entrance and exit of the capillary are time independent~cf. Ref. 8; see also the first paragraph of Sec. III!.

The internal dynamics of the vortex tangle will be char-acterized by two time-dependent fields, viz., L(x,t) repre-senting the local length of quantized vortices per unit vol-ume, and w_l(x,t) denoting the relative drift velocity vl2v of the vortex tangle. It may be demonstrated by

start-ing from the ‘‘microscopic’’ equation of motion for a quan-tized line vortex ~see the Appendix in Ref. 3! that, when dissipative effects are disregarded, the line length of the vor-tices is conserved, i.e.,

]L ]t 1

]

]x~Lvl!50. ~1!

We shall refer to Eq. ~1! as the kinematic equation of a vortex tangle in the nondissipative case. The equation of mo-tion being valid in that case will be derived from a varia-tional principle, viz.,

d

E

t₀ t₁ dt

E

x₀ x₁ Lldx50, ~2!

where the Lagrangian density Ll is a function of the exter-nal parameters r, S, w_n and the internal variables L,

]L/]x, and w_l , i.e.,

Ll5Ll~r,S,wn;L,]L/]x,wl!. ~3!

Note that

wn5vn2v, w_l5v_l2v. ~4!

The variation of the internal variables in Eq.~2! is subject to the kinematic constraint~1!. The Lagrangian density will be written in the following more specific form which is remi-niscent of Hamilton’s principle in classical mechanics:

Ll5

1 2m˜lwl

2_2U˜

l . ~5!

The expression at the right-hand side of Eq.~5! is, however, quite general. In fact, m˜_l having the dimension of mass den-sity is an as yet unknown function of the external parameters and the internal variables L and w_l which may, formally, be expressed by

m˜_l_5m˜_l_~r,S,wn;L,wl!. ~6!

The potential density U˜_l , however, has a definite meaning; it represents the internal energy density of the vortex tangle according to U˜_l5rs k2 4pLln

S

c a0L1/2

D

11 2˜gl

S

]L ]x

D

2 , ~7!

where the mass density of the superfluid component is indi-cated by rs, the quantum of circulation k equals h/m and a₀'1.3 Å ~see Ref. 9! denotes the core radius of a quantized

line vortex. The first term on the right-hand side of Eq. ~7! represents the density of the ‘‘microscopic’’ kinetic energy associated with the circulating motion around the core of the quantized vortices@see Khalatnikov ~Ref. 11!#. This

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expres-sion is, approximately, valid provided a0L1/2!1. The

dimen-sionless constant c is of order unity. The gradient ]L/]x

appears only in the second term at the right-hand side of Eq.

~7!. This term, in fact, models, within the framework of the

macroscopic theory, the effects associated with local devia-tions from macroscopic behavior caused by very large values of the gradient ]L/]x. Large gradients may appear, e.g., in

sharp boundary layers and steep fronts. The coefficient g˜_l is assumed to depend on the external parameters and the single internal variable L. A dimensional analysis then shows that

g

˜_l5r_sk

2

L2gl , ~8!

whereg_l is, in general, a dimensionless function of the ex-ternal parameters, i.e.,

gl5gl~r,S,wn!. ~9!

In practice, however, g_l will, effectively, be a function of the absolute temperature, e.g., throughrs/r. Since the

coef-ficient m˜_l may depend on the internal variables L and w_l a dimensional analysis shows that

m ˜

l5rsmˆl , ~10!

where the dimensionless coefficient mˆ_l depends, irrespective of the external parameters, exclusively on the following di-mensionless combination Vi ~Vinen number! of the internal variables:

Vi5kL1/2/uwlu. ~11!

Possible cross effects between internal and external variables are accordingly suppressed, in view of our intention to model the internal dynamics of a vortex tangle. The functional de-pendence of mˆ

l is, in a qualitative way, expressed by

mˆ_l5mˆ_l_~r,S,w_n;Vi!. ~12!

Note that the dependence on the Vinen number Vi is as yet unspecified ~see, however, the next section!.

After putting

Ll52Ul , ~13!

where U_l may be considered as the extended internal energy density of the vortex tangle, we have in view of Eq.~5!

U_l5U˜_l21 2m˜lwl 2_, _~14! so that dU_l5m_ldL1g_ld

S

]L ]x

D

2 1 2mld~wl 2_!. _~15!

In this expressionm_l represents the line-length potential de-termined by ml5rkbv1 1 2 ]˜g_l ]L

S

]L ]x

D

2 21₂ ]m˜l ]L wl 2 , ~16! where bv5 rs r k 4pln

S

The Euler-Lagrange equations may be derived more easily from the equivalent Lagrangian density L_l* obtained from

Lˆl by means of partial integration, viz.,

L_l*5L_l2L

S

] ]t1vl

]

B5Pl

L , ~31!

we have according to Eqs. ~28! and ~31!

B5]wl

]x . ~32!

Note that B represents the local impulse of the vortex tangle per unit vortex length. Differentiating the Euler-Lagrange equation~27! partially with respect to the spatial coordinate

x yields

]B ]t 1

]

]x~vlB1mˆl!50. ~33!

We shall refer to Eq. ~33! as the dynamic equation of a vortex tangle. The nondissipative dynamics of a tangle of quantized vortices is apparently governed by Eqs. ~1! and

~33!.

It should be kept in mind that the effective mass density

m_l that enters Eq.~33! according to Eqs. ~30! and ~31! and which, in view of Eqs.~20! and ~23!, is given by

m_l5m˜_l21

2Vi ]m˜

l

]~Vi!, ~34!

involves an as yet unspecified function of the Vinen number

Vi. The further specification of this function will follow from

the analysis of the energy of the vortex tangle in the next section.

III. ENERGY AND IMPULSE OF VORTEX TANGLE It is known from the two-fluid hydrodynamics of He II that, when the flow velocitiesv and vn are small compared

to the propagation velocities of first and second sound, the external parameters r and S may be considered, in a first order of approximation, as constant @incompressibility ap-proximation; see Landau and Lifshitz ~Ref. 12!#. When, in addition, in one-dimensional flow the boundary conditions are time independent, the conservation equations for mass and entropy imply that also v and vn take values that are

independent of position and time.

When the conditions just mentioned are fulfilled, the La-grangian density L_l*is invariant with respect to translations in space and time. The application of Noether’s invariance theorem then yields equations that express the conservation of energy and impulse. They are given, respectively, by@see, e.g., Logan~Ref. 13!#

wl1Ll1gl ]L ]x 5vlm_lw_l1Lmˆ_l_2U_l_1g_l ]L ]x. ~44!

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Expression~43! for the impulse density of a vortex tangle is already familiar from the variational analysis in the preced-ing section.

When a Galilean transformation is performed according to

H_l5vP_l1H_l

8

, ~45!

Q_l5vP_l1vH_l

8

1Q_l

8

, ~46!

P_l5P_l

8

, ~47!

8

5m_lw_l21U_l , ~51! Q_l

8

5w_l~mlw_l21Lmˆ l!2gl dL dt , ~52! P_l

8

5m_lw_l , ~53! Pl

8

5mlwl 2_1L_m_ˆ l2Ul1gl ]L ]x. ~54!

It may be recognized that, unless m_l(Vi) represents a linear function, H_l

8

is a Legendre transform of U_l . In fact,

H_l

8

5wlP_l1U_l , ~55! while dU_l5mˆ_ldL2P_ldw_l1 ] ]x~gldL! ~56! and, accordingly, dH_l

8

5mˆ ldL1wld Pl1 ] ]x~gldL!. ~57!

Equations ~55! – ~57!, however, also apply in the singular case where m_l is a linear function of Vi@cf. Eq. ~73!#. Let us introduce another Legendre transform, viz., the superfluid

turbulent pressure p_l defined by

p_l5Lmˆ_l2U_l . ~58!

It follows from Eqs. ~31!, ~56!, and ~58! that

d p_l5Ldmˆ_l1P_ldw_l2 ]

]x~gldL! 5L~dmˆ_l_1Bdwl_!2 ]

]x~gldL!. ~59!

The impulse flux ~54! may, in virtue of Eq. ~58!, be ex-pressed by Pl

8

5mlwl 2_1p l1gl ]L ]x. ~60!

It is easily verified by taking Eqs.~13!, ~26!, ~27!, and ~58! into account that, in accordance with the Clebsch-Bateman principle,

p_l5L_l*. ~61!

The equations for the conservation of energy and impulse may be derived directly from the kinematic equation~1! and the dynamic equation~33! governing the evolution of a vor-tex tangle. To that end these last equations are brought in the form dL dt1 ] ]x~Lwl!50, ~62! dB dt 1 ] ]x~wlB1mˆl!50. ~63!

Since, according to Eqs.~31!, ~47!, and ~57!

dH_l

8

5~mˆ_l1w_lB!dL1LwldB1 ]

]x~gldL! ~64!

and

5m_lw_l21U_l5m_lw_l221

2m˜lwl

2_1U_˜

l . ~68!

This observation proves to be crucial for the further devel-opment of our hydrodynamic theory of superfluid turbulence. In fact, the total energy of a vortex tangle is contained in the ‘‘microscopic’’ kinetic energy associated with the circulating motion of the superfluid around the core of the quantized vortices. It is accordingly required that

H_l

8

5U˜_l ~69!

so that due to Eq. ~68!

m_l51

2m˜l . ~70!

By substituting Eq. ~34! in ~70! the following differential equation is obtained for the unknown function m˜_l of the Vinen number Vi:

Vi ]m

˜_l

]~Vi!2m˜l50. ~71!

The general solution of Eq. ~71! reads

m˜_l52r_sbVi, ~72!

wherebdenotes a dimensionless integration constant involv-ing, in general, the external parameters r, S, and wn. In

practice it will, principally, be a function of the absolute temperature, e.g., through rs/r. See in that connection Sec.

V, in particular, expression~119!. It follows from Eqs. ~11!,

~70!, and ~72! that

m_l5rsbkL1/2/uwlu. ~73!

This expression for m_l implies that

P_l5rkbˆ L1/2_sgnw

l , ~74!

where

bˆ 5rs

r b. ~75!

The modified coefficient bˆ is, likeb, a function of the ab-solute temperature. The property that expression~74! for the impulse density does not involve the absolute value of the relative drift velocity, seems to be characteristic for a vortex tangle. It is obviously related to the anisotropy of the effective-mass tensor in the three-dimensional case~see Ref. 2!.

The Lagrangian density L_l defined by Eq. ~5! may, in virtue of Eqs. ~7!, ~8!, ~70!, and ~73!, be expressed in the form Ll5rskbL1/2uwlu2rs k2 4pLln

S

c a₀L1/2

D

21 2rs k2 L2gl

S

]L ]x

D

2 . ~76!

The first term on the right-hand side of Eq.~76! represents a

velocity-dependent potential density@see Goldstein ~Ref. 14!

where a velocity-dependent potential is called Schering’s po-tential function; see also the discussion in Ref. 2#. Let us introduce the modified coefficientgˆ_l associated with disper-sive effects according to

gˆ_l₅rs

r gl . ~77!

By using Eqs. ~17!, ~18!, ~75!, and ~77! expression ~76! for

Ll may be brought in the form

Ll5rkbˆ L 1/2_uw lu2rkbvL2rs k2 8pL2 1 2r k2 L2gˆl

S

]L ]x

D

8

may take the following equivalent forms:

Q_l

8

5w_l~mlw_l21Lmˆ_l_!2gl dL dt 5wl~mlw_l21Ul1p_l!2gl dL dt 5wl~U˜ l1pl!2gl dL dt, ~81!

where, in view of Eqs.~7!, ~8!, ~17!, ~18!, and ~77!

U˜_l5rkb_vL1rs k2 8pL1 1 2r k2 L2gˆl

S

]L ]x

D

2 . ~82! Note that U˜_l5H_l

8

according to Eq. ~69!. The quantity

U_l1p_l appearing in Eq. ~81! may be considered as the extended enthalpy density of the vortex tangle.

At this point of the analysis it seems appropriate for a better understanding of our approach of superfluid turbulence to recapitulate what has been achieved and to indicate what still has to be done, in the sequel of the paper, to complete the theory.

All equations derived so far apply to a nondissipative vor-tex tangle. Such a vorvor-tex tangle, however, may be realized only at absolute zero, where the dynamics of the tangle is not affected by the presence of elementary excitations. The

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sta-tus of the nondissipative vortex-tangle equations, like Eq.

~62! expressing the conservation of line length of the

quan-tized vortices and Eq. ~50! for the conservation of vortex-tangle impulse, is comparable to that of the Euler equations in classical hydrodynamics which express the conservation of mass, momentum, and entropy in an ideal fluid without viscosity. In fact, the vortex-tangle equations just mentioned were derived from a variational principle of Hamilton’s type, while it is known that the Euler equations may be obtained by a similar procedure.

In classical fluid dynamics the Euler equations are ex-tended to the dissipative Navier-Stokes equations in order to complete the theory by taking into account real phenomena like the viscous properties of the fluid. It is known that this extension can be performed systematically by applying methods of the thermodynamics of irreversible processes. In a similar way we will introduce in the next section dissipa-tive terms in the nondissipadissipa-tive equations for a vortex tangle. These terms will prove to be essential for arriving at a sys-tem of evolution equations that is physically realistic. It will be shown in Sec. V that the well understood homogeneous case6,7is completely covered by these extended equations.

IV. DISSIPATIVE EFFECTS

The system of nondissipative evolution equations for a tangle of quantized vortices comprising the kinematic equa-tion~62! and the dynamic equation ~63! is clearly equivalent to the system composed of the kinematic equation ~62! and the impulse equation ~50!. This last system, however, is slightly more convenient as a starting point for the introduc-tion of addiintroduc-tional dissipative terms. After having been sup-plied with these terms it assumes the form

dL dt 1 ] ]x~Lwl!5rl , ~83! d P_l dt 1 ] ]x

S

wlPl1pl1gl ]L ]x

D

5Fsl1Fnl1Fl , ~84!

where r_l represents the density of the net production rate of line length of quantized vortices in the tangle, while Fsl and F_nl denote, respectively, the densities of the forces that the pure liquid and the elementary excitations exert on the quan-tized vortices. The quantity F_l is introduced in the impulse equation as the density of a force resulting from internal friction in the tangle. Exterior forces, like the ones that are associated with the pinning of quantized vortices at the wall of a capillary, are not considered. Note that a possible diffu-sive contribution q_l to the line-length flux has not been taken into account in Eq. ~83!. In fact, the physical back-ground of such a term is not completely understood. In ad-dition, its appearance in Eq. ~83! unnecessarily complicates the subsequent development of our simplified hydrodynamic theory of superfluid turbulence, in particular with a view to the special cases of homogeneous flow and internal equilib-rium to be treated in the following sections. The diffusive flux q_l , however, does appear in the analysis of Ref. 2~note in that connection Ref. 15!. It plays, in addition, an essential part in the observations on superfluid turbulence fronts by van Beelen et al.16

It is easily recognized that Eqs. ~83! and ~84! imply the following dissipative form of the dynamic equation of a vor-tex tangle: dB dt 1 ] ]x~wlB1mˆl!5 1 L~Fsl1Fnl!2 r_l L B1 1 LFl . ~85!

The three-dimensional analysis in Ref. 2 shows that the conservation of total momentum comprising the momentum of the pure liquid and the impulses of the elementary excita-tions and the vortex tangle, requires that

F_l52]Pl*

]x . ~86!

Note that F_sl and F_nl represent mutual-friction forces ex-erted, respectively, by the pure liquid and the elementary excitations on the vortex tangle. These forces are, accord-ingly, accompanied by reaction forces acting in the opposite direction on the environment of the vortex tangle ~see Ref. 2!.

In the expression for the density R_l of the entropy pro-duction rate associated with the vortex tangle two combina-tions of bilinear terms do appear, viz. ~cf. the general analy-sis in Ref. 2! 2Fsl~vl2v!2Fnl~vl2vn! ~87! and 2mˆ lrl2Pl* ]w_l ]x . ~88!

. ~90!

The matrix coefficients Ci jand Di j(i, j51,2) have to satisfy

the Onsager reciprocity relations

C125C21, D125D21. ~91!

The coefficients may be brought in the form

Ci j5rskLgi j ~i, j51,2!, D₁₁5r_skL22d₁₁,

D125rskL21d12,

D225rskd22, ~92!

where gi j and di j (i, j51,2) are dimensionless quantities

involving only the external parameters r, S, and possibly also wn. It seems natural to assume that they depend,

effec-tively, on the absolute temperature T, e.g., through rs/r.

That assumption is supported by expressions ~119! – ~123!. We shall follow the notation in Ref. 1 by usingg instead of d11, i.e.,

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g5d11. ~93!

In the sequel frequent use will be made of the modified co-efficientsgˆ ,gˆ_{i j}, and dˆ_{i j} (i, j51,2) determined by

gˆ₅rs r g, gˆi j5 rs r gi j, dˆi j5 rs r di j ~i, j51,2!. ~94!

In addition, the quantity qˆ will be employed defined by

qˆ5 g

ˆ₁₂₁_gˆ₂₂

gˆ₁₁12gˆ₁₂1gˆ₂₂. ~95!

The mutual-friction force density F_sl1Fnl appearing in Eqs.~84! and ~85! may then, in view of Eq. ~89!, be written as

F_sl1Fnl5rkL~gˆ₁₁₁₂gˆ₁₂₁gˆ₂₂_!~qˆwn2wl_{!. ~96!}

The requirement that the entropy production be non-negative implies that

gˆ₁₁_>0, gˆ₂₂_>0, gˆ₁₁gˆ₂₂₂gˆ₁₂2 _>0,

gˆ₅dˆ₁₁_>0, dˆ₂₂_>0, dˆ₁₁dˆ₂₂₂dˆ₁₂2_>0. _~97!

It follows from Eq. ~97! that

gˆ₁₁12gˆ₁₂1gˆ₂₂>~gˆ₁₁1/2₂_g_ˆ 22 1/2_!2_>0, dˆ 1112dˆ121dˆ22>~dˆ11 1/2₂_d_ˆ 22 1/2_!2_>0. _~98!

In view of Eqs. ~57!, ~83!, and ~84! we have

dH_l

8

dt 5mˆl dL dt1wl d P_l dt 1 ] ]x

S

gl dL dt

D

52_]] x

F

wl~Lmˆl1Plwl!2gl dL dt1wlPl*

G

1mˆ_lr_l1~Fsl1Fnl!wl1P_l*]wl ]x . ~99!

Accordingly, by taking account of Eq. ~52!,

dH_l

8

dt 1

]

]x~Ql

8

1wlPl*!5Fnl~vn2v!2Rl , ~100!

where the density R_l of the dissipation rate is given by@see Eqs. ~87! and ~88!#

R_l52mˆ_lr_l2P_l*]wl

]x 2Fsl~vl2v!2Fnl~vl2vn!. ~101!

Equation~100! shows that, per unit volume, some part of the power Fnl(vn2v) delivered by the mutual-force density F_nl is being used for modifying the energy of the vortex tangle, while the rest of it is being dissipated. It should be noticed that, whereas, in a first order of approximation, the external parameters r, S,v, and vn and, therefore, also the

absolute temperature T take fixed values independent of po-sition and time, the temperature gradient is, in a second order of approximation, determined by~see Ref. 1!

S]T

]x5Fln1Fn, ~102!

where F_ln52F_nl , while Fn represents the density of an

exterior force acting on the elementary excitations, e.g., the one resulting from the no-slip of the elementary excitations at the wall of a capillary which leads, at laminar conditions, to Poiseuille’s law.

V. HOMOGENEOUS SUPERFLUID TURBULENCE AND VINEN’S EQUATION

In the homogeneous case, when the spatial derivatives of the superfluid turbulent flow fields L and w_l vanish, the kinematic equation~83! reduces, in view of Eqs. ~79!, ~90!, and~92!–~94!, to the ordinary differential equation

dL

dt5~1/gˆ!@~bˆ /2!L

3/2_uw

lu2bvL2#. ~103! The evolution equation ~84! for the impulse density of the vortex tangle takes under homogeneous flow conditions, in virtue of Eqs.~74! and ~96!, the form

d

dt~L

1/2_sgnw

l!5~1/bˆ !L~gˆ1112gˆ121gˆ22!~qˆwn2wl!. ~104!

It will be clear that bˆ may be treated as a constant because the external parameters have, in a first order of approxima-tion, values that are independent of position and time. Mul-tiplying Eq.~104! by 2L1/2sgnw_l yields

dL

dt5~2/bˆ !~gˆ1112gˆ121gˆ22!L

3/2_~qˆwn_sgnw

l2uwlu!. ~105!

Since the differential equations ~103! and ~105! are valid simultaneously, their right-hand sides should be equal. This implies that ~2/bˆ !@~bˆ2_/4_g_ˆ_!1_g_ˆ 1112gˆ121gˆ22#L3/2uwlu 5~1/gˆ_!b_vL21~2/bˆ !~gˆ₁₁₁₂gˆ₁₂₁gˆ₂₂_!L3/2_qˆw nsgnwl . ~106! Accordingly w_l5~1/Gˆ!@~gˆ₁₂₁gˆ₂₂_!wn_1~bˆ /2gˆ_!b_vL1/2sgnw_l#, ~107! where Gˆ5~bˆ2_/4_g_ˆ_!1_g_ˆ 1112gˆ121gˆ22. ~108!

By taking Eqs. ~95!, ~97!, and ~98! into account it follows from Eq. ~107! that

e~gˆ₁₂1gˆ₂₂_!uwnu1~bˆ /2gˆ_!b_vL1/2>0, ~109!

where

e5sgnwlsgnwn561. ~110!

Substitution of Eq. ~107! in either Eq. ~103! or Eq. ~105! produces the differential equation

(9)

dL

dt5~bˆ /2gˆ!~1/Gˆ!~gˆ121gˆ22!euwnuL

3/2

2~1/gˆ_!~1/Gˆ!~gˆ₁₁12gˆ₁₂1gˆ₂₂_!b_vL2. ~111! Let us introduce the quantities x1 andx2 according to

x1a5~bˆ /2gˆ!~1/Gˆ!~gˆ121gˆ22!~rs/r! ~112!

and

~k/2p!x25~1/gˆ!~1/Gˆ!~gˆ1112gˆ121gˆ22!bv. ~113! The coefficient a appearing in Eq. ~112! is used by Schwarz.7It is related to the first Hall-Vinen coefficient B by means of

a5~rn/2r!B5~rs/r!av. ~114! The coefficient a_v has been introduced in Ref. 3. Corre-sponding relations with regard to the second Hall-Vinen co-efficient read

12a

8

512~rn/2r!B

8

5~rs/r!~12av

8

!. ~115! By using Eqs. ~112! and ~113! and realizing that vn2v5(rs/r)(vn2vs) the following slightly generalized

form of Vinen’s equation is obtained from Eq. ~111!:

dL

dt 5x1aeuvn2vsuL

3/2_2~_k_/2_p_!_x

2L2. ~116!

When e51, it reduces to the Vinen equation discussed by Schwarz.7It should be noticed that, in view of Eq.~113!, the quantityx2is, likebv, logarithmically dependent on L@see Eq.~17!#. This weak dependence on L is usually disregarded in applications of Vinen’s equation. It was also ignored by Vinen in his seminal papers.18 It should be emphasized that the Vinen equation~116! being valid for homogeneous flow conditions was derived by combining the two equations

~103! and ~104! governing the evolution of a homogeneous

vortex tangle. By the same procedure the algebraic expres-sion~107! was obtained for the relative drift velocity of the vortex tangle. The question has been raised how the single Vinen equation ~116! might be generalized to

inhomoge-neous flow conditions. That problem, however, is in the light

of our hydrodynamic theory of superfluid turbulence not properly posed. In fact, the Vinen equation~116! for the time evolution of L and expression ~107! for w_l should not be considered separately; they are intimately related within the context of the present theory. The generalization to the inho-mogeneous case is, accordingly, obvious: the system of equations~83! and ~84! generalizes the set of equations ~107! and~116!.

By substituting Eq. ~107! in Eq. ~89! and applying Eqs.

~92!–~94! the following expressions for the mutual-friction

forces may be derived:

Fnl /rkL5~1/Gˆ!$@~bˆ 2_/4_g_ˆ_!_g_ˆ 221gˆ11gˆ222gˆ12 2 #wn 2~bˆ /2_gˆ_!~gˆ₁₂1gˆ₂₂_!b_vL1/2sgnw_l%, ~117! Fsl /rkL5~1/Gˆ$@~bˆ 2_/4_g_ˆ_!_g_ˆ 122~gˆ11gˆ222gˆ12 2 !#wn 2~bˆ /2_gˆ_!~gˆ₁₁1gˆ₁₂_!b_vL1/2sgnw_l%. ~118!

Schwarz7has shown that an averaging procedure with re-spect to the quantized vortices in a homogeneous vortex tangle yields definite expressions for w_l , dL/dt, and Fnl .

The averaging may be based ~see Ref. 3! on a generalized ‘‘microscopic’’ equation of motion for a line element of a quantized vortex in which, in addition to the modified Hall-Vinen coefficientsa_vanda_v

8

the coefficienta_v

9

enters. This coefficient determines the tangential component of the local

velocity of the line vortex. It should be realized that the

~12Ii!#. ~123! The quantities I_i, I_l , and cLappearing in Eqs.~119!–~123!

represent coefficients that have been introduced by Schwarz.7These coefficients are defined as definite averages over the quantized vortices in a tangle ~see also Ref. 3!. Equations~119! – ~123!, obviously, express the five macro-scopic coefficients bˆ ,gˆ ,gˆ_{i j} (i, j51,2), note the reciprocity relation, in terms of the six quantities a_v, a_v

8

, a_v

9

, I_i,

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VI. INTERNAL EQUILIBRIUM AND BURGERS’ EQUATION WITH NONLINEAR SOURCE TERM In this section it is assumed that the cross effects associ-ated with the macroscopic coefficients dˆ12 anddˆ21 may be

disregarded, i.e., we take dˆ

125dˆ2150. ~126!

The following terminology will be used: ~1! the vortex tangle is in internal equilibrium if and only if

mˆ

l50; ~127!

~ii! the vortex tangle is in external equilibrium with the pure

liquid and the elementary excitations if and only if

F_sl1Fnl50. ~128!

It will be clear that in the homogeneous case external equi-librium entails internal equiequi-librium. In fact, in the case of external equilibrium the right-hand side of Eq. ~105! van-ishes. Accordingly dL/dt50 so that, by virtue of the kine-matic equation ~103! and Eq. ~79!, the vortex tangle is in internal equilibrium.21 The converse statement is also true unlessgˆ50. In fact, whengˆ vanishes the vortex tangle may

be in internal equilibrium without satisfying condition~128! for external equilibrium. The case gˆ_{50, which may be of}

substantial importance in practice, will be investigated in this section in more detail. Note that the requirement gˆ₅₀

en-tails, in view of the dissipative inequalities ~97!, condition

~126!.

2 22rk 2 L ] ]x

S

gˆl ]L ]x

D

. ~130!

In internal equilibrium relation~59! reduces to

d p_l5P_ldw_l2 ]

]x~gldL!, ~131!

so that the impulse equation~84! is given by

d P_l dt 1 ] ]x~wlPl!1Pl ]w_l ]x 5Fsl1Fnl1Fl . ~132!

By taking Eqs. ~74!, ~86!, ~90!, ~92!, ~94!, ~96!, and ~126! into account Eq. ~132! may be brought in the form

d dt~bˆ L 1/2_sgnw l!1wl ] ]x~bˆ L 1/2_sgnw l! 12bˆ L1/2_sgnw l ]w_l ]x 5L~gˆ1112gˆ121gˆ22!~qˆwn2wl! 1_]]_x

S

dˆ 22 ]w_l ]x

D

. ~133!

By substituting Eq. ~129! in Eq. ~133! and introducing the dependent variable y according to

y5L1/2sgnw_l ~134!

the following partial differential equation is obtained:

d y dt1

F

~2bv/bˆ !y1~8k/bˆ !gˆly 23

S

]y ]x

D

2 2~4k/bˆ !gˆ_ly22] 2_y ]x2

G

]y ]x12y ] ]x

F

S

]y ]x

D

2 2~4k/bˆ !gˆ_ly22] 2_y ]x2

G

5~1/bˆ !~gˆ1112gˆ121gˆ22!y 2

F

_qˆw_n2~2_b v/bˆ !y2~8k/bˆ !gˆly23

S

]y ]x

D

2 1~4k/bˆ !gˆ_ly22] 2_y ]x2

G

1~dˆ 22/bˆ ! ]2 ]x2

F

S

]y ]x

D

2 2~4k/bˆ !gˆ_ly22] 2_y ]x2

G

. ~135!

When dispersive effects are neglected, i.e.,gˆ_l_{50 and, in addition,}b_v is treated as a constant, Eq.~135! reduces to

dy dt1~6bv/bˆ !y ]y ]x5d ˆ 22~2bv/bˆ2! ]2_y ]x21~1/bˆ !~gˆ1112gˆ121gˆ22!y 2_@qˆwn2~2_b v/bˆ !y#. ~136!

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This equation represents Burgers’ equation supplemented with a nonlinear source term.22 In the Appendix some exact solutions of Eq.~136! are listed. They may be considered as direct extensions, to the case where diffusive effects are taken into account, of the marginally stable superfluid turbulence fronts investigated in Ref. 8.

In permanent internal equilibrium (gˆ50) the density r_l of the net production rate of line length is, in view of Eqs.~83! and

~129!, determined by r_l5dL dt 1 ] ]x

H

L

F

~2bv/bˆ !L 1/2_sgnw l1~2k/bˆ !L25/2gˆl

S

]L ]x

D

2 sgnw_l2~2k/bˆ !L23/2 ] ]x

S

gˆl ]L ]x

D

sgnwl

GJ

. ~137!

The superfluid turbulent pressure p_l fulfills in internal equi-librium, according to Eqs.~58! and ~127!, the relation

p_l52U_l . ~138!

Since, in view of Eq.~13!, 2U_l5L_l , this property is remi-niscent of the Clebsch-Bateman principle ~61!. Its range of validity, however, is both wider, it applies to dissipative flow processes, and smaller, it is restricted to internal equilibrium. We finally remark that internal equilibrium is associated with the dissipative limit (gˆ_{→0) of the balance equation of}

line length ~83!, while conservation of line length corre-sponds to the nondissipative limit (gˆ_{→`) of this equation.}

VII. REVIEW OF BASIC EQUATIONS

Since many equations have passed in review in the pre-ceding sections, it seems appropriate, for a right appreciation of our simplified hydrodynamic theory of superfluid turbu-lence, to list here the most relevant equations in their most accessible form for the particular but important case where dispersive effects are disregarded (gˆ_l_{50). The cross effects}

that are associated with the coefficientsdˆ₁₂anddˆ₂₁will also be neglected. In that connection it should be realized that, in general, cross effects are relatively small.

A simple analysis using Eqs. ~79!, ~90!, ~92!, and ~93! shows that the kinematic equation ~83! of a vortex tangle may be represented by d y dt1wl ]y ]x1 1 2y ]w_l ]x 5~bˆ /4gˆ!y 2_@wl_2~2_b v/bˆ !y#, ~139!

where the dependent variable y is related directly to the line-length density L according to y5L1/2_sgnw

l @see expression ~134! of the preceding section#. In a similar way, by applying

Eqs.~74!, ~80!, ~86!, ~90!, ~92!, ~94!,~96!, and ~134! the im-pulse equation~84!, or dynamic equation ~85!, takes the form

d y dt1 ] ]x

F

3 2y wl2~1/bˆ ! k 8p rs r y2

G

5~1/bˆ !~gˆ₁₁₁₂gˆ₁₂₁gˆ₂₂_!y2_~qˆwn_2wl_! 1~dˆ 22/bˆ ! ]2_w l ]x2 . ~140!

The second term between square brackets on the left-hand side of this equation has to be suppressed when the quantity bv, which depends, in view of Eq.~17!, logarithmically on

uyu, is being treated as a constant. In fact, in that

case the dimensionless constants c and c

8

introduced, re-spectively, by Eqs.~7! and ~19! should not be distinguished, so that relation ~18! has to be skipped. By eliminating the time derivative between Eqs. ~139! and ~140! a spatial dif-ferential equation is obtained that couples the ways in which

y and w_l depend on position.

In the special cases that have been considered in Secs. V and VI the system of equations ~139! and ~140! takes a par-ticularly simple form.

~i! Homogeneous case. It is easily verified ~see the

analy-sis in Sec. V! that, when homogeneous flow conditions pre-vail in a vortex tangle, the system of equations ~139! and

~140! is equivalent to an algebraic equation for wl , viz.,

w_l5~1/Gˆ!~gˆ₁₂₁gˆ₂₂_!wn1~1/Gˆ!~bˆ /2gˆ_!b_vy ~141!

and an ordinary differential equation ~generalized Vinen equation! for y given by

dy

dt5~bˆ /4gˆ!~1/Gˆ!~gˆ121gˆ22!wny

2

2~1/2gˆ_!~1/Gˆ!~gˆ₁₁₁₂gˆ₁₂₁gˆ₂₂_!b_vy3. ~142! Note thatGˆ is defined by Eq. ~108!. When written in terms of the Schwarz coefficients I_i, I_l , and cL, Eqs. ~141! and ~142! take, respectively, the form

w_l5@a_v

8

I_i1a_v

9

~12I_i!#wn1~12a_v

8

!Ilb_vy , ~143! dy dt5 1 2avIly 2_@wn_2~_b v/cL!y#. ~144!

~ii! Internal equilibrium. When the vortex tangle is in

per-manent internal equilibrium (gˆ50), the right-hand side of

Eq. ~139! vanishes so that

w_l5~2b_v/bˆ !y. ~145!

Substitution of this expression for w_l in Eq.~140! yields Eq.

~136! which constitutes a generalized form of Burgers’

equa-tion. Note that in the derivation of this equation the quantity bvhas been treated as a constant. When Eqs.~145! and ~136! are written in terms of the Schwarz coefficients, they take, respectively, the form

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d y dt13~12av

8

!Ilbvy ]y ]x5d ˆ 22 1 2@~12av

8

!Il# 2_b v ]2_y ]x2 11₂avIly2@wn2~bv/cL!y#. ~147!

It will be clear that in the case of homogeneous flow, Eq.

~147! passes into the generalized Vinen equation ~144!. The

form of the Vinen equation is, accordingly, not affected ex-plicitly by the condition of internal equilibrium~see also the discussion in Ref. 8!. Note, however, that, in view of Eq.

~120!, the coefficient of wn in Eq. ~143! vanishes when

gˆ50, so that the expression for the relative drift velocity of

a homogeneous vortex tangle reduces, in the case of perma-nent internal equilibrium, to the more simple form Eq.~146!. It may be concluded that under some slightly restricting con-ditions, viz., when internal equilibrium prevails and disper-sive effects are negligible, Eq.~147!, or similarly Eq. ~136!, may be considered as an extension of Vinen’s equation to

inhomogeneous superfluid turbulence.

We finally list some conclusions.

~1! It has been proved possible to derive from a few basic

principles a single partial differential equation for the line-length density L(x,t) of inhomogeneous superfluid turbu-lence that generalizes Vinen’s equation. In the case where Vinen’s equation takes, according to Schwarz,7 the form

~144!, the partial differential equation is given by Eq. ~147!.

Note that the dependent variable y is according to Eq.~134!, irrespective of the sign of the relative drift velocity w_l of the vortex tangle, equal to the square root of the line-length den-sity.

~2! Equation ~147! has been obtained from the impulse

equation~84! for a vortex tangle by making a few simplify-ing assumptions, viz., absence of dispersive effects and real-ization of internal equilibrium of the vortex tangle. These assumptions are expected to be fulfilled in many cases of practical interest.

~3! An interesting new contribution to the internal

dynam-ics of an inhomogeneous vortex tangle is supplied by the first term on the right-hand side of Eq. ~147! which models the diffusion of vortex-tangle impulse.

~4! In developing the theory a quantity pl called

super-fluid turbulent pressure could be introduced. Its explicit ex-pression as a function of L and w_l is given by Eq. ~80!. When the vortex tangle is in internal equilibrium, the super-fluid turbulent pressure satisfies Eq. ~138!.

APPENDIX: SOME EXACT SOLUTIONS OF EQ.„136… REPRESENTING SUPERFLUID TURBULENCE FRONTS

Equation ~136! which, as shown in Sec. VI, directly fol-lows from the impulse equation ~84! for a vortex tangle in the physically realistic case where dispersive effects may be neglected and the vortex tangle is in internal equilibrium, reads ( y5L1/2sgnw_l) d y dt1~6bv/bˆ !y ]y ]x5d ˆ 22~2bv/bˆ2! ]2_y ]x21~1/bˆ !~gˆ1112gˆ121gˆ22!y 2_@qˆwn2~2_b v/bˆ !y#.

Note that the first term on the right-hand side of Eq. ~136! models the diffusion of vortex-tangle impulse, while the sec-ond term accounts for the mutual friction like in Vinen’s equation ~144!. We are looking for solutions of Eq. ~136! that represent propagating waves of permanent form. We ac-cordingly take y5y@x2x02vf~t2t0!#. ~A1! In that case d y dt52wf ]y ]x, ~A2! where wf5vf2v. ~A3!

Let us introduce the following parameters~see Ref. 8!:

b5~bˆ /2b_v!qˆwn, ~A4! A5~6b_v/wn!~gˆ₁₂1gˆ₂₂_!21, ~A5! B5wf~3qˆwn!21, ~A6! C5dˆ22~2bv/bˆ !2~gˆ121gˆ22!21~qˆwn 2_!₂₁ . ~A7! These parameters may be expressed in terms of the Schwarz coefficients by using Eqs. ~119! – ~123!. Since Eq. ~136! is valid provided gˆ50 ~permanent internal equilibrium!, the

expressions in terms of the Schwarz coefficients should also apply to that particular case. They are given by

b5~cL/b_v!wn, ~A8!

A5~6b_v/wn!~12av

8

!av21, ~A9!

B5wf@3~12a_v

8

!Ilc_Lwn#21, ~A10!

C/A25dˆ22~1/36!~avIl /cL!. ~A11!

Whereas the parameter B is clearly dimensionless, the quan-tities b, A, and C have the dimension of, respectively, recip-rocal length, length, and length squared. It seems therefore natural to introduce dimensionless variablesh andj accord-ing to

(13)

By applying the dimensional scaling~A12!, Eq. ~136! passes, in view of Eq.~A2!, into

~h2B!]h_]j5~C/A2_!] 2_h

]j21h

2_~12_h_!. _~A13!

In the particular case where the coefficient C vanishes, Eq. ~A13! has been investigated in Ref. 8. Two types of propagating front solutions are distinguished there, viz., cold and warm fronts. It is shown in Ref. 8 that the marginally stable cold fronts are characterized by

B50, h5~1/2!$11tanh@~j2j₀!/2!#%, ~A14! while the marginally stable warm fronts are represented by

B51,

h5~j2j0!21 when j2j0>1,

h51 when j2j0<1. ~A15!

The front solutions ~A14! and ~A15! may be extended continuously to cases where C.0, by means of the follow-ing expressions:

B5~1/4!@12~118C/A2!1/2#, h5~1/2!$11tanh@~j2j0!/2l#%,

l5122B52C/A2_B _~A16!

for cold fronts and

B5~1/2!@11~112C/A2!1/2#, h5B~j2j0!21 when j2j0>B,

h51 when j2j0<B ~A17!

for warm fronts. Expressions~A16! and ~A17! represent, ob-viously, exact solutions of Eq.~A13!. An inspection of Eqs.

~A16! and ~A17! shows that the asymptotic behavior of the

cold fronts is exponential and that of the warm fronts is

algebraic. The significance of the exact solutions will be

considered in more detail in a separate paper dealing with propagating fronts of superfluid turbulence.

1_{J.A. Geurst, Physica B 154, 327}_~1989!.

2_{J.A. Geurst and H. van Beelen, Physica A 206, 58}_~1994!. 3_{J.A. Geurst, Physica A 183, 279}_~1992!.

4_{J.A. Geurst, in Proceedings of the 1st International Workshop on}

Quantum Vorticity and Turbulence in He II Flows, edited by G. Stamm and W. Fiszdon~Max-Planck-Institut fu¨r Stro¨mungsfor-schung, Go¨ttingen, 1994!.

5_{J.A. Geurst and H. van Beelen, Physica B 205, 209}_~1995!. 6_{J.T. Tough, in Progress in Low Temperature Physics, edited by}

D.F. Brewer~North-Holland, Amsterdam, 1982!, Vol. VIII.

7_{K.W. Schwarz, Phys. Rev. B 31, 5782} _{~1985!; 38, 2398}

~1988!.

8_{J.A. Geurst and H. van Beelen, Physica A 216, 407}_~1995!. 9_{R.J. Donnelly, Quantized Vortices in Helium II}_~Cambridge

Uni-versity Press, Cambridge, 1991!.

10_{S.K. Nemirovskii and W. Fiszdon, Rev. Mod. Phys. 67, 37}

~1995!.

11_{I.M. Khalatnikov, An Introduction to the Theory of Superfluidity}

~Benjamin, New York, 1965!.

12_{L.D. Landau and E.M. Lifshitz, Fluid Mechanics} _~Pergamon,

London, 1959!.

13_{J.D. Logan, Invariant Variational Principles} _{~Academic, New}

York, 1977!.

14_{H. Goldstein, Classical Mechanics, 2nd ed.} _{~Addison-Wesley,}

Reading, 1980!.

15_{In Eq.} _{~7.4! of Ref. 2 the quantity R/T should be replaced by}

R/T2¹W•(qW/T), while in Eq. ~7.5! the expression r_l2¹W•qW_l has to be substituted for r_l .

16_{H. van Beelen, W. van Joolingen, and K. Yamada, Physica B 153,}

248~1988!.

17_{S.R. de Groot and P. Mazur, Nonequilibrium Thermodynamics}

~North-Holland, Amsterdam, 1962!.

18_{W.F. Vinen, Proc. R. Soc. London, Ser. A 240, 114}_{~1957!; 240,}

128~1957!; 242, 493 ~1957!; 243, 400 ~1957!.

19_{R.J. Donnelly, Experimental Superfluidity}_{~The University of}

Chi-cago Press, ChiChi-cago, 1967!.

20_{C.J. Swanson and R.J. Donnelly, J. Low Temp. Phys. 52, 189}

~1983!.

21_{We note in passing that the well-known Gorter-Mellink mutual}

friction force only applies when the vortex tangle is in external equilibrium.

22

G.B. Whitham, Linear and Nonlinear Waves~Wiley, New York, 1974!.