sign in sign up
<<
Home , Rotational speed

OPEN A Variational Method in

SUBJECT AREAS: Out-of-Equilibrium Physical Systems STATISTICAL PHYSICS, Mario J. Pinheiro THERMODYNAMICS AND NONLINEAR FLUID DYNAMICS Department of Physics, Instituto Superior Te´cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. PHYSICS We propose a new variational principle for out-of-equilibrium dynamic systems that are fundamentally based on the method of Lagrange multipliers applied to the total entropy of anP ensemble of particles. Received k However, we use the fundamental equation of thermodynamics dU~TdS{ kFkdx on differential 17 April 2013 forms, considering U and S as 0-forms. We obtain a set of two first order differential equations that reveal Accepted the same formal symplectic structure shared by classical , fluid mechanics and thermodynamics. 21 November 2013 From this approach, a topological torsion current emerges of the form ijkAjvk, where Aj and vk denote the components of the vector potential (gravitational and/or electromagnetic) and where v denotes the angular Published of the accelerated frame. We derive a special form of the Umov-Poynting theorem for rotating 9 December 2013 gravito-electromagnetic systems. The variational method is then applied to clarify the working mechanism of particular devices.

Correspondence and rom 1893-96, the Norwegian explorer Fridtjof Nansen, while traveling in the Arctic region, noticed ice requests for materials drifting across the polar sea, at an of 20 to 40 degrees relative to the direction of the wind. Nansen should be addressed to speculated that, in addition to the of the wind, the Coriolis Effect could be used to explain his F 1 M.J.P. (mpinheiro@ observation. In 1905, Vagn Walfrid Ekman introduced a theory of wind currents in open seas, explaining that tecnico.ulisboa.pt) sea currents change direction based on their depth as a result of the that exists according to the rotating coordinate system associated with the Earth. In addition to atracting considerable interest in geophysical flow problems2, these discoveries stimulated further investigations in fields such as magnetic geodynamics, binary stars and new-born planetary systems as well as furthering in the important problem of angular transport. The problem of enhanced angular transport in accretion disks3 and the break-down of Keplerian as well as the removal of from a vortex due to moving spiral waves, which is an important aspect of the total angular momentum balance of the core and the intensification of a tropical cyclone4, are all examples of problems that demand a clear understanding of the dynamics of gravito-electromagnetic rotating systems. Furthermore, special attention must be dedicated to the role of flux of angular momentum flux and its conservation. Building on the previous attempts to generate an accurate account of angular momentum transport5, an additional equation of conservation (besides continuity, momentum and equations) is required, which relates both the local angular momentum density and flux. To develop a consistent theory, the equation for the angular momentum balance must be included and, to the best of our knowledge, these problems were treated by Curtiss6 and Livingston and Curtiss7. These problems are also addressed in the present work using a variational method. The first attempt to obtain the general equations of of an isolated thermodynamic system K, from the equilibrium condition dS 5 0, was generated by Landau and Lifshitz8. Buchdahl and Simpson9 obtained an explicit form of the nonrelativistic motion of an isolated system in equilibrium and showed that the temperature of K is nonuniform when the system is accelerated. Diu et al.10 went a step further, but they did not attempt to build a framework to investigate the dynamics of K with an integrated procedure. In this article, we develop a standard technique for treating a physical system that is based on a previously developed information-theoretic framework11–13. The proposed technique starts with the total entropy of the system composed of N particles (or bodies). The method of the Lagrange multipliers is then applied for the k entropy differential (0-form) dS~+kSdx (k 5 x, y, z). Finally, the total entropy is inserted into the fundamental equation of entropy written using the differential form, +kU~T+kS{Fk, which is summed over the ensemble of N particles. This method leads to a set of two first order differential equations, revealing the same formal symplectic structure shared by and thermodynamics14,15. When the maximization of entropy is sought,

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 1 www.nature.com/scientificreports the well-known equations of (electro)dynamics (if electromagnetic the particles’ total angular momentum (the sum of the orbital angu- entities enter into the system) result. Our method bears a resemb- lar momentum and the intrinsic angular momentum, i.e., spin) lance to the isoentropic but non-energy-nonconserving variational XN XN hi 17 principle proposed by R. Jackiw et al. , which allows one to study L~ LðÞa ~ rðÞa |pðÞa zJðÞa : ð4Þ non-equilibrium evolution in the context of quantum field theory a~1 a~1 but with various classical analogous, such as the Schro¨dinger equa- (a) tion, giving rise to reflectionless transmission. In the above equations, r is the position vector relative to a fixed (a) This article is organized as follows. In Sec. II, we extend this reference frame R, p is the total momentum (particle 1 field) and (a) mathematical formalism to non-equilibrium information theory. L is the total angular momentum of the particle, comprising a In Sec. III, we analyze the equilibrium and stability of a rotating vector sum of the particle’s orbital angular momentum and intrinsic plasma. In Sec. IV, we apply our formalism to angular momentum angular momentum J (e.g., these momentums are contributed by the transport, obtaining, in particular, the Umov-Poynting theorem for electron spin and/or nuclear spin, because the electromagnetic rotating gravito-electromagnetic systems (e.g., rotating plasmas, momentum is already included in the preceding term through (a) magnetic geodynamics, vortex motion and accretion disks in astro- p ), see Fig. 1. The maximum entropy principle introduces physics), the applications of which may contribute to the clarification Lagrange multipliers from which ponderomotive are obtained. of still poorly understood phenomena. It is necessary to find the conditional extremum; these extremum are established not for the function S itself, but rather for the changed function S. Following the mathematical procedure proposed in Ref. Results 13 the total entropy of the system S is thus given by The information-theoretic method proposed in this paper constitutes 2 2 an alternative approach by applying the concept of maximizing XN pðÞa JðÞa ~ ðÞa ðÞa { { { entropy to the problem of out-of-equilibrium physical systems. It S S E ðÞa ðÞa a~1 2m 2I bears some resemblance to the Hamiltonian formulation of dynamics, which expresses the first order constraints of the Hamiltonian, H,ina qðÞa VðÞa zqðÞa AðÞa :vðÞa {UðÞa ð5Þ 2n-dimensional phase , revealing the same formal symplectic mec structure shared by classical mechanics and thermodynamics. hi XN Although the simplifying assumption of an isothermal system zða:pðÞa zb: rðÞa |pðÞa zJðÞa ~ ðÞa : rules out its ability to accurately explain such problems as the coher- a~1 ent transport of angular momentum in astrophysics or certain types of laboratory devices (e.g., the Ranque-Hilsch effect), the present where a and b are Lagrange multipliers (as vectors). It can be shown method attempts to further the understanding of specific trends, in that vrel 5 aT and v 5 bT (see also Ref. 8). The conditional particular, predicting the forced angular momentum transport that extremum points form the dynamical of a gen- occurs radially outward from the symmetry axis of the rotation. The eral physical system (the equality holds whenever the physical system type of Umov-Poynting theorem obtained expresses the interplay is in thermodynamic and mechanical equilibrium), which is defined between entropy and energy, where the energy and entropy trend by two first order differential equations: towards minima and maxima, respectively, while explaining the LS §0 canonical momentum; ð6aÞ formation of physical structures. In particular, it is that compressi- LpðÞa bility that is an important property in the transport of angular momentum and a possible driving mechanism for instability. This L L ðÞa development is believed to be advantageous and creates options for S ~{ 1 ðÞa { 1 ðÞa v +rðÞa U m §0 systematic improvements. LrðÞa T T Lt ð6bÞ fundamental equation of dynamics: Mathematical procedure. Let us consider a simple dynamical system consisting of a set of N discrete interacting point (a) ðÞa ðÞa ~ ~ m (a 5 1, 2, …, N) with xi and vi ðÞi 1, 2, 3; a 1, ..., N denoting the coordinates and of the in a given inertial reference frame. The Latin subscript refers to the Cartesian components and the Greek superscript distinguishes between the different masses. The gravitational potential w(a) associated with a mass a is given by X mðÞa wðÞa ~G , ð1Þ ðÞa { ðÞb b~1 jjx x b=a with G denoting the gravitational constant and x(a) and x(b) represent- ing the instantaneous positions of the mass (a), and (b). S denotes the summation of every particle in the system. The energy, momentum and angular momentum conservation laws must be verified for a totally isolated system: XN a E~ EðÞa , ð2Þ Figure 1 | An assortment of particles of mass m( ), in rotational motion a~1 around an axis OZ with v, where r(a) denotes the position vector relative to a fixed reference frame, R,p(a) denotes the total XN momentum (particle 1 field) and L(a) denotes the total angular ~ ðÞa P p , ð3Þ momentum of the particle. a~1

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 2 www.nature.com/scientificreports

Eq. 6b gives the fundamental equation of dynamics and has the ðÞa ðÞa dv ~{ a{ ðÞa ðÞa { ðÞa ðÞa form of a general local balance equation that has an entropy gradient, m T+rðÞa m +w q +V = S . 0, while Eq. 6a gives the canonical momentum (see also Eq. dt a !ð13Þ 12). At thermodynamic equilibrium, the total entropy of the body ðÞa 2 ðÞa ðÞa : ðÞa J : ðÞa ðÞa has a maximum value, constrained through the supplementary con- zq + A v {+ ðÞa {v J zF : r 2IðÞa ext ditions 2, 3, and 4, which typically occurs as a result of the minimiza- tion techniques associated with Lagrange multipliers. In the more For conciseness, the term U(a) now includes all forms of energy general case of a non-equilibrium process, according to Vander- inserted into the above Eq. 5. On the right-hand side (r.h.s.), the first linde’s proposition18, a condition required for the gravitational force term must be present whenever the mechanical and thermodynami- to exist is that the entropic gradient in Eq. 6b must be positive. cal equilibrium conditions are not fulfilled, the second term is the However, new physics may be developed by the set of two first order gravitational force, the third and fourth terms constitute the Lorentz differential equations related to the interplay between the tendencies force, the fifth term is a new term that represents the transport of of energy and entropy to attain minima and maxima, respectively. angular momentum, and the last term represents other external In non-equilibrium processes the gradient of the total entropy in forces that are not explicitly included but still act on the particle (a). momentum space multiplied by factor T is given by The present formalism was applied in a previous article13, and hi LS paÞ qðÞa therein we obtained the ponderomotive forces acting on a charged T ~{ z Azv z v|rðÞa , ð7Þ LpðÞa mðÞa mðÞa e particle. For a neutral particle or body in a gravitational field, Eq. 13 points to a type of extended fundamental equation of dynamics for a so that maximizing the entropy change in Eq. 6a leads to the well- given species (a) at equilibrium and at a given point of space- known total (canonical) momentum: (Eulerian description): hi ðÞa ðÞa ðÞa ðÞa ðÞa p ~m vezm v|r zq A: ð8Þ ðÞa ðÞa 2 ðÞa Lv ðÞa ðÞa J : ðÞa m ~{m +w {+ ðÞa {v J : ð14Þ Lt r 2IðÞa The above formulation bears some resemblance to the Hamiltonian formulation of dynamics which expresses the first order constraints Eq. 14 gains a new term because the body possesses an intrinsic of the Hamiltonian H in a 2n dimensional phase space, p_ ~{LH=Lq angular momentum. In a non-rotating , we set v ~ and q_ LH=Lp. This can be solved along trajectories, such as quasi- 5 0, wherein we use the work-energy theorem to obtain the total 2 static processes, revealing the same formal symplectic structure mechanical energy of the system: Emec~KzUzJ 2Ic. This is a 14–16 c shared by classical mechanics and thermodynamics . common approach in classical mechanics. We are interested in the ðÞa The internal mechanical energy term, Umec, appearing in Eq. 5 may effect of a given force at a given space-time coordinate, not in its effect be defined by: along the particle trajectory. It is worth noting that Eq. 14 was XN obtained through a variational procedure in contrast to the usual ðÞa ~ ðÞa ðÞa z ðÞa ðÞa,b conservation theorem used, for example, in Ref. 21, 22. Umec m w ðÞr m w : ð9Þ b~1 Included in the internal energy term is the interpressure term (see b=a Eq. 9; note that here we consider a homogeneous and isotropic fluid). The above described framework (see also Ref. 13 for additional Considering this equation, by the definition of thermodynamic tem- information) leads us to the well-known hydrodynamic equation perature, hS(a)/hU(a) ; 1/T(a), it follows that for a given species (a): ðÞa ðÞa ðÞa ðÞa p : ðÞa + ðÞa ~{ +w { + { N r U m ðÞa p ðÞa X m ðÞa dv ~{ ðÞa ðÞa { ðÞa ðÞa,b ð10Þ m +rðÞa m w m +rðÞa w ðÞa dt ~ J 2 { ðÞa ðÞa z ðÞa a: ðÞa b 1 +rðÞa q +V q + A v : b=a ð15Þ 2IðÞa ðÞa 2 J ðÞa ðÞa {+ ðÞa {v: z : Eq. 10 contains the particle’s self-energy and the particle interaction r ðÞa J T energy for the gravitational and electromagnetic fields, but it may 2I also include other terms, such as terms included in Eq. 9, represent- In the r.h.s. of Eq. 15, we explicitly introduce external force terms, ing different occurring phenomena (exemplifying energy as a book- eventually present in open systems. keeping concept). We may recall that the entropic flux in space is a Using the following correspondence from particle to fluid descrip- 19,20 type of generalized force Xa ; therefore, it can be shown that the tions following equation holds: X ð ðÞa 3 m ? d x’rvðÞx’ , ð16Þ ðÞa ~{ ðÞa ðÞa z ðÞa ðÞa : ðÞa V T+rðÞa q +rðÞa V q +rðÞa A v a ! 2 ð11Þ and an analogous relation for the electric charge J ðÞa ðÞa ðÞa : ðÞa : ðÞa ð zm v +v {+ ðÞa {v J , X r 2IðÞa qðÞa ? d3x’rðÞx’ , ð17Þ a V We can now write the fundamental equation of thermodynamics using the form of a space-time differential equation: we can rewrite Eq. 15 using the form of the Euler (governing) equa- X X tion: LvðÞa T+Sz mðÞa ~ +UðÞa : ð12Þ dv Lt r ~{r +rw{+rp{+rWJ {+rf : ð18Þ a a v dt v Taking into account the convective derivative, dv(a)/dt ; hv(a)/ht 1 Here, as usual, the total interparticle pressure term (e.g., Ref. 23) is v(a)?=v(a), we obtain: given by:

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 3 www.nature.com/scientificreports

X XN with the horizontal (see, e.g., p. 97 of Ref. 29). Eq. 15 can be used to pðÞr ~ mðÞa wðÞa,b ðÞr : ð19Þ solve this problem, where v 5 0 (the reference frame is non-rotating) a b~1 and considering that only the gravitational force acts on the rolling b=a body with an inertial moment relative to its own center of mass given by I 5 bMR2. Hence, we obtain: To simplify, we introduce a functional integral of the form of an c intrinsic angular momentum energy density (comprising the ‘‘inter- M€x~Mg sin h{Lxw: ð27Þ v W action energy term’’, ? J), J: 2 ð ðÞJc X ðÞa 2 Here, w: . Assuming that the x-axis is directed along the J a ðÞa {v: ðÞ{ D ? W ’ z ’ 3 ’, 2Ic ðÞa J ðÞF ½J ðÞx f ðÞx d x ð20Þ a 2I inclined plane and considering that the angular momentum relative to the center of mass is given by Jc 5 Icv9. We find that considering that the intrinsic angular momentum density refers to a dv’ 2 v’ given blob of fluid (with inertial momentum I, a measure of the local M€x~Max~Mg sin h{Icv’ ~Mg sin h{bMR a, ð28Þ rotation, (i.e., spin, of the fluid element), and its associated free vxdt v’R energy (per unit volume), f 5 f 2 T . Eq. 18 also suggests that the 0 s where v9 5 dh/dt, while noting that dx 5 v dt (holonomic con- function S(a) (the field integral of r(a)) is constant along the integral x straint). Because a 5 a/R, we find the well-known result curves of the space field r(a). The gradient of the free energy, f, of the out-of-equilibrium state is the source of the spontaneous change g sin h a ~ : ð29Þ from an unstable state to a more stable state while performing work. x ðÞ1zb For example, a common source of free energy in a collisionless plasma is an electric current24; in a magnetically confined plasma, Extremum conditions imposed on the entropy or internal energy not several classes of free energy sources are available to drive instabil- only constraint the evolution of the system, but determine the ities, e.g., the relaxation of a non-Maxwellian, non-isotropic velocity stability of thermodynamic systems at equilibrium. Furthermore, it 30 distribution25. At this stage, it is worth noting that our procedure has been shown that a state of mechanical equilibrium can be includes the treatment of the effect of angular momentum (through reached, if the entropy increases with distance: Eq. 5), a necessary inclusion in a consistent theory, according to TLrSw0, ð30Þ Curtiss6. Furthermore, using the mathematical identity where Lr is the Lie derivative along the vector field r acting on a scalar ðÞa : ðÞa ~ ðÞa : ðÞa z ðÞa : ðÞa z S. Therefore, we can find another extreme condition through the + A v A + v v + A general expression hihihihið21Þ ðÞa ðÞa ðÞa ðÞa ðÞa A | +|v z v | +|A , L ðÞa ðÞa ðÞa ðÞa : ðÞa T ~{q + ðÞa V zq + ðÞa v A LrðÞa r r ! we obtain, thorugh the use of algebra, the following expression: 2 ð31Þ L ðÞa JðÞa hihi { ðÞa v { { : ðÞa ðÞa m +rðÞa v J : a a LA a a a L ðÞa + AðÞ:vðÞ ~{ { v|AðÞ z vðÞ|BðÞ : ð22Þ t 2I Lt According to Noether’s theorem, the total canonical momentum is Here, B 5 [=3A]. In addition, we notice that the following equality conserved in a closed system. We can thus state the closure relation: holds: hi XN hi AðÞa :+ vðÞa ~ v|AðÞa , ð23Þ pðÞa zqðÞa AðÞa ~0: ð32Þ a~1 where It can be shown (see Ref. 12, 13 for details) that the relation that X ðÞb ðÞa ~ ðÞb v prevails in equilibrium for a rotating plasma is given by A q , ð24Þ hi rab b~1 ðÞa ðÞa z ðÞa | ðÞa ~ ðÞa ðÞa z b=a q E q v B m +rðÞa w X hi denotes the vector potential actuating on the particle (a) due to every ðÞa ðÞa,b { ðÞa ðÞa | ðÞa ð33Þ m +rðÞa w q A v : other particle, and the vorticity is defined by b~1 hi b=a ðÞa ~ | ðÞa ~ ðÞa V +rðÞa v 2v : ð25Þ We may further develop the correspondence drawn from Eqs. 16–17 Therefore, the general equation of dynamics for a physical system to obtain the general condition of equilibrium of a rotating plasma in (Lagrangian description) follows: the presence of gravitational and electromagnetic interactions (e.g., Ref. 31): dv r ~rEz½J|B {+w{+pzr½A|v : ð26Þ dt rEz½J|B ~+Wz+p{r½A|v : ð34Þ The last term in the r.h.s. of the Eq. 26 is a new term that represents a Here, to simplify the algebra, we took the averaged type of topological spin vector26, an artifact of non-equilibrium pro- v, for the entire system. We find in Eq. 34 that the vector potential A cess. We will show that the topological spin vector plays a role in is presented on the same footing as the E and B-fields. The relative plasma arcs, as well as in magnetocumulative generators27, and sug- importance of the vector potential depends on the characteristics of gests a new method for obtaining the helicity transport equation28. the B-field prevalent in the system; for instance, if the B-field is homogeneous, the vector potential field predominates in the region Discussion near the axis because rvA/JB , 1/r2. The topological spin vector We verify Eq. 14 using a standard example from classical mechanics: term is fundamental because it produces work that is responsible a rigid body of mass, M, rolling down an inclined plane of angle h for the system angular momentum modification, producing a

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 4 www.nature.com/scientificreports rocket-like rotation effect on the plasma. The theoretical frame- m Bz work delineated here may help to clarify problems related to rotat- jjr v 0 v, amperian rotation: ð38bÞ c 4p S ing-plasma systems32–34 and controlled thermonuclear plasma 35 confinement . Here, S denotes the filamentary current cross-section, with r 5 rc. The compression of an electric current by a magnetic field, the z- For negative charge carriers (rc 52jrcj52ene), we obtain an pinch effect, can be studied on the basis of Eq. 34, which gives the Amperian (clockwise) rotation for high magnetic fields and relatively condition for dynamic equilibria. Let us assume a typical geometry weak arc currents. We also find a retrograde rotation for higher for an infinitely long axisymmetric cylindrical arc (Fig. 2) with axial intensity arcs (higher S) and small transverse magnetic fields, which current density Jz 5 Jz(r). Because the current density is assumed to is in agreement with experimental evidence (e.g., Ref. 36). Here, we be constant, Maxwell’s equations in the steady state yield the azi- see the interplay between the tendencies of the energy to attain a muthal component Bh 5 m0Jzr/2 for r # R with R the outer boundary minimum value, while the entropy attempts to attain a maximum of the cylindrical arc. The vector potential is purely radial and is given value. From Eq. 37, we obtain an expression for the spot velocity in a 2 transverse magnetic field: by Az(r) 52m0R Jz/4, for r , R, where the Coriolis term plays no role. We can write Eq. 34 in the form: 4p 1 i B R v~vR~ 0 z z : ð39Þ { ~ dp m0 jjrc 2psc S JzBh : ð35Þ dr 2 Here, we have made use of the Bernoulli relation, Dp~ 0E 2, and where it follows that ð the constitutive equation, Jr 5 scE, where sc denotes the electrical conductivity of the plasma. Although Eq. 39 is not self-consistent, it ~ Rdp ~ 1 2 2{ 2 prðÞ dr m0Jz R r , ð36Þ shows that the force term [J 3 B], is not the only important term and r dr 4 that the spot velocity depends linearly on the arc current (see, e.g., which is a well-known result. Ref. 44). In out-of-equilibrium systems, a new force term is present The interaction between vacuum arcs and transverse magnetic that can suppress the Amperian force under certain conditions (see fields is used in switching devices (see e.g. Refs. 36, 37). We can also Ref. 40). instead consider a coaxial configuration with a cathode on-axis with A better understanding of this phenomenon is crucial, because arc a stabilizing magnetic induction field B directed along the axis of discharges are powerful generators of non-equilibrium atmospheric symmetry and an arc current density J flowing radially (and assum- pressure plasmas. We compare the results predicted by Eq. 39 2 ing a ‘‘filamentary’’ current with radius R9, Ar 52m0R9 Jr/4, with m0 with experimental data available in the literature in Table 1. The representing the permeability of the vacuum). In this case, we may calculations were done using the expression for the electrical con- apply Eq. 34 and obtain the pressure differential, from the axis to the ductivity in terms. of the microscopic parameters of the plasma, wall (at R): s ~en m~e v2 n , with the electronic mobility given by m 5 e/ hi0 e 0 pe c 2 m0 mn (nu denotes the electron collision frequency) because, for the DprðÞ~R ~2pR {JrBzzjjr pR Jrv Dh: ð37Þ c c c 4p majority of the data, the transverse magnetic field was below 0.1 T and is not expected to greatly influence the plasma arc electrical m B 41 jjr w 0 z v, retrograde rotation ð38aÞ conductivity . c 4p S We now address the transport of angular momentum, a phenom- enon of great importance in several fields, such as in the working mechanism of an accretion disk, the formation of a tornado, and the planetary-atmospheric circulation. Angular momentum conservation can be obtained using the fol- lowing equation: ðÞa ðÞa ðÞa dv L ðÞa ðÞa ðÞa ðÞa m ~{T {+ ðÞa m W {q + ðÞa V dt LrðÞa r r ! ðÞa 2 ð40Þ ðÞa ðÞa : ðÞa J ðÞa : ðÞa zq + ðÞa v A {+ ðÞa {v J : r r 2IðÞa

ðÞa ~ Multiplying Eq. 40 by the particle velocity vi ðÞi x,y,z , and after rearranging the terms, we obtain: X X X ðÞa { ðÞb ðÞa { ðÞb 1 d 2 1 xi xi vi vi mðÞa vðÞa ~{ G mðÞa mðÞb i ðÞa ðÞb 3 2 a dt 2 a b=a jjx {x a b a b ! X X ðÞ{ ðÞ ðÞ{ ðÞ X ðÞa 1 1 xi xi vi vi 1 LA { qðÞa qðÞb { qðÞa vðÞa i pe ðÞa ðÞb 3 i L 2 a b=a 4 0 jjx {x 2 a t X X ð41Þ z 3 ðÞa ðÞa ðÞa z 1 ðÞa ðÞa ðÞa ðÞa eijkvk q Aj vi eijk q vi vj Bk 2 a 2 a "# X a 2 X 1 L JðÞ L ðÞa z vðÞa { va:JðÞa zT vðÞa : 2 i ðÞa 2IðÞa i ðÞa a Lxi a Lxi Figure 2 | Geometry and vectorial fields in the Bennett pinch generated by an axial current Jz creating a toroidal field Bw. If, instead, we consider a Eq. 41 can be written in a more comprehensive form if we separate vacuum arc discharge with radial current Jr and magnetic field Bz, we find a the terms from their different contributions. For this purpose, we rotating arc with an angular velocity v. define the total by the expression

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 5 www.nature.com/scientificreports

Table 1 | Comparing Eq. 39 with experimental data. Electric current i 5 60 A

Low pressure DC discharge Atmospheric pressure DC discharge

a a Te(eV) 1.0 0.87 23 12 f 16 b Electron density (cm ) ne 5 2.5 3 10 ne 5 2 3 10 21 7 11 c Collision frequency (s ) nc 5 6.98 3 10 nc 5 7.2 3 10 21 10 12 d Plasma frequency (s ) vpe 5 8.94 3 10 vpe 5 8 3 10 Average (m/s) 3.0 (, 2)e 1023(5 3 1023)e

a 39 For laboratory discharges, the Coulomb logarithm is ln L , 10, see Ref. 38, for electron temperatures of the order of Te < 10000 K . bRef. 39. { c ~ | {6 3=2 The frequency of collisionffiffiffiffiffi was calculated using the standard expression nc 2:91 10 ne ln LTe , Ref. 38. d 4p We use vpe~5:65|10 ne, (see Ref. 38). eIn parenthesis are the experimental data collected for atmospheric pressure, from Ref. 40; for low pressure, see Ref. 42. f Data interpolated from Ref. 43, assuming Te 5 0.8 eV.

X forces usually dominate the magnetic forces involved, which is a 1 a a 2 : mðÞvðÞ , ð42Þ crucial aspect in the development of instabilities. 2 a From Eq. 47, we see that equilibrium ensues (neglecting thermal and we denote and configurational effects) when the rotational velocity of the fluid satisfies the local condition (see, e.g., Ref. 45): X X X ðÞa ðÞb 1 1 m m 2 : mðÞa QðÞa ~ G , ð43Þ dVðÞr g 2 2 jjxðÞa {xðÞb §0, ð49Þ a a b=a dr to be the overall gravitational energy. Similarly, the total electrostatic where we identify v with the bulk angular velocity. Eq. 49 is related to energy is given by: the conservation of energy. However, when condition 48 is not ful- X X filled, a magneto-rotational instability (MIR)45,46 occurs, which 1 1 qðÞa qðÞb : : ð44Þ appears as the result of the interplay between three different terms: v pe ðÞa { ðÞb 2 4 0 a b=a jjx x i) the angular momentum acquired by the fluid (or particles), ii) an interaction term due to the coupling between the fluid angular Following the Umov-Poynting theorem, it is observed that Eq. 40 momentum with the driven angular velocity, and iii) the fluid ther- reduces to the following: mal energy and configurational entropy. X hihi A typical experiment consists of a fluid rotating of a fluid between d ~{ d { d z 3 a ðÞa | ðÞa | ðÞ g v q v A v two concentric cylinders - related to the so called Taylor-Couette dt dt dt 2 a instability - driven by velocity gradients. In the presence of an axial "# 2 ð45Þ magnetic field, the Taylor-Couette instability develops when Eq. 49 is X JðÞa X L ðÞa z 1 ðÞa : { : ðÞa z ðÞa not satisfied. We may also expect that, owing to the fact that for two v +rðÞa v J T vi : 2 2IðÞa ðÞa a ? b a a Lxi different particle species with different inertial moment, I I , it can be expected that at some point, (given r 5 rc of the radial axis), an As before, it is convenient to introduce a new physical quantity that inversion of the sign of the inequality of Eq. 48 must take place, and represents the rotational energy: instability occurs. In particular, in the presence of two different spe- X hihi cies with different inertial moments, the fluid may be intrinsically 3 a a a ~{ q vðÞ| AðÞ|v : ð46Þ unstable at higher angular . MIR instabilities threaten the rot 2 a stability of plasma configurations. In the 1960’s, when MHD Eq. 45 can then be written using the more general form: plants were considered to be an efficient process for the conversion thermal energy into electrical energy, E. Velikhov, discovered this d 1 jjJ 2 electrothermal instability, which is the cause of strong magneto- z z z ~ v:+ {ðÞv:J {DF : ð47Þ 45,47,48 dt g v rot 2 r 2I hydrodynamic turbulence . Eq. 47 can be written in the form of an energy conservative equa- Here, v represents the velocity of an ‘‘element of fluid’’, and the last tion: term in brackets is an average local value. A simple analysis of Eq.47 L shows that the system is rotationally stable provided the following z z z ~ Lt g v rot condition is satisfied: ()"# X ðÞa 2 2 1 : ðÞa J : ðÞa ðÞa jjJ { + ðÞa v { z v J {T + {ðÞv:J {DF v0: ð48Þ 2 r 2IðÞa ð50Þ r 2I a "# 2 X JðÞa The last term is the free energy per unit volume, F 5 F0 2 TS.Itis 1 ðÞa ðÞa ðÞa z { z v: { + ðÞa : : ðÞa J T r v worth noting that Eq. 48 is consistent with the Gibbs distribution in a 2 a 2I rotating body (see, e.g., Ref. 8), which means that the radial energy flux must be positive (flowing out radially from the system’s bound- Finally, we can transform Eq. 50 into a version of Poynting’s Theorem ary). In addition, we note that the equilibrium of a gravito-electro- for rotating fluids: magnetic system depends on its mechanical rotational properties as L well as on the free energy available for intrinsically linking any mech- { ðÞU ~+:SzP’, ð51Þ Lt anical process to thermodynamic variables and revealing options for possible unconventional mechanisms for the control of instabilities. on which we define a type of Poynting vector for rotational fluids, In the domain of astrophysical plasmas, gravitational and rotational which gives the of rotational energy flow:

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 6 www.nature.com/scientificreports

"# X a 2 2 1 JðÞ ~ J { : { S~ vðÞa { z vðÞa :JðÞa {T ðÞa : ð52Þ Edriv ðÞv J DF: ð55Þ ðÞa 2Ip 2 a 2I Next, we will discuss several examples illustrating the application of The power input driving the process (source/sink term) is given by: the variational method. "# X a 2 A hurricane is a natural airborne structure that converts its kinetic 1 JðÞ ’: +: ðÞa { v: ðÞa z ðÞa , and by means of the transport of angular P v ðÞa J T ð53Þ 2 a 2I momentum from the inner core to the outer regions, conveyed either directly by moving matter, or by non-material stresses such as those The total energy is defined by summing up the different exerted by electric or magnetic fields49. We may apply Eq. 53 to this contributions: specific problem, assuming that all of the mechanical and thermal ~ z z z energy is converted into electromagnetic energy Ue, to obtain: U g v rot: ð54Þ 2 L ~ J { : { ~{ Ue ðÞa P’ v v J DF ð56Þ Here, the term T represents the thermal energy associated with 2Ip Lt the species (a) that is equal to 2DF, the free energy of the physical system. For a system in contact with a reservoir at constant temper- or ature this is the maximum amount of work extractable from the P’~vEdriv: ð57Þ system; the free energy tends to decrease for a system in thermal contact with a heat reservoir. In particular, notice that when the Let us consider the case of a hurricane in an axisymmetric configura- 2 2 2 angular velocity, v, is multiplied by Eq. 40, the driving power is tion, with J 5 vI. We can safely assume that v I Ip ? v IzDF. obtained. It is worth noting that the presence of the term v ? J, which We can now envision a simple model of a hurricane with a total mass, plays an analogous role to the slip in electrical induction motors, that M, and radius, R, approximated as a solid cylinder with I 5 MR2/2. is, the lag between the rotor speed and the magnetic field’s speed, is Hence, the total power driving the hurricane is given by provided by the stator’s rotational speed. Furthermore, we see that 1 v3 M2R4 M2R4 the power input depends on the fluid compressibility = ? v. This P’~ !v3 , ð58Þ means that compressibility is a factor that determines the amount 8 2 Ip Ip of transported angular momentum through the stress-tensor tij and or, as a function of the fluid density r: may be responsible for a new driving mechanism in addition to the p2 R6L2 well-known MRI. The driving energy of the rotating system can be P’~ v3 r2: ð59Þ expressed in the form: 4 Ip

Figure 3 | A group of particles spinning about their axes and revolving around a common axis, Oz, subject to a . Three situations typically occur. (a) An outward transport of angular momentum occurs with a larger gradient, while free energy flows to the center of the field. (b) If the inward free energy gradient is dominant relative to the angular momentum gradient, a reversal of the particle’ angular momentum may occur. (c) If the angular momentum gradient is of the same order of magnitude as in case (b) but still dominant relative to the inward free energy gradient, the particles may continue spinning in the same direction.

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 7 www.nature.com/scientificreports

Our result shows the same type of dependency that was demonstrated Once again, the problem of radiative atmospheric heating reveals an by Chow & Chey50, and, in particular, it shows that the intrinsic interesting interplay between energy and entropy, with each attempt- inertial momentum of the particles constituting the fluid plays a ing to achieve a different equilibrium condition. substantial role. It has been experimentally shown51 that periodic radiative heating 1. Ekman, V. W. On the Influence of Earth Rotation on Ocean Currents. Arkiv. of the Earth’s atmosphere transmits angular momentum to it as a Math. Astro. Fys. 2, 1–52 (1905). result of the Earth-atmosphere coupling through frictional interac- 2. Greenspan, H. P. The Theory of Rotating Fluids, (Breukelen Press, Brookline-MA, tions52. The images sent by the ESA’s Venus Express confirms this 1990). fact on Venus (Earth’s planetary twin) based on the presence of a 3. Balbus, Steven, A. Enhanced Angular Momentum Transport in Accretion Disks. ‘‘double-eye’’ atmospheric vortex at the ’s south pole and the Annu. Rev. Astron. Astrophys. 41, 555 (2003). 4. Chow, K. C. & Chan Kwing, L. Angular Momentum Transports by Moving Spiral presence of high velocity winds whirling westwards around the pla- Waves. J. Atmos. Sci. 60, 2004 (2003). net, which is characterized by a four-day period. 5. Ji, H., Kronberg, P., Prager, S. C. & Uzdensky, D. A. Princeton Plasma Physics Schubert and Withehead’s51 conducted an experiment with the Laboratory Report PPPL-4311, May 2008; accessed at web address: http://arxiv. purpose of providing an explanation for the high wind velocities org/abs/0803.1009, date of access: 7 Mars 2008. 6. Curtiss, C. F. Kinetic Theory of Nonspherical Molecules. J. Chem. Phys. 24, during apparent cloud formation in the upper atmosphere of 225–241 (1956). Venus. In this experiment, a Bunsen flame rotating below a cylin- 7. Livingstone, P. M. & Curtiss, C. F. Kinetic Theory of Nonspherical Molecules. IV. drical annulus filled with liquid mercury induced the rotation of the Angular Momentum Transport Coefficient. J. Chem. Phys. 31, 1643–1645 (1959). liquid mercury in a direction counter to that of the rotating flame. 8. Landau, L. & Lifschitz, E. M. Physique Statistique, p. 134, (Mir, Moscow, 1960). 9. Buchdahl, H. A. & Simpson, M. A. Remark on the equilibrium of moving systems. The speed of the flame was 1 mm/s and the temperature of the Am. J. Phys. 43, 1041–1045 (1975). mercury increased from room temperature at a rate of approximately 10. Diu, B., Guthmann, C., Lederer, D. & Roulet, B. Macroscopic motion of a totally 3uC per minute. After 5 minutes, a steady-state flow was established, isolated system in statistical equilibrium. Am. J. Phys. 58, 974–978 (1990). with the mercury rotating in the counter-direction of the flame, with 11. Jaynes, E. T. Information Theory and . Phys. Rev. 106,620 (1957). a speed of approximately 4 mm/s. If we consider the liquid to be a 12. Pinheiro, M. J. An information-theoretic formulation of Newton’s second law. spinning body, we can use Eq. 49 to estimate that Europhys. Lett. 57, 305 (2002). 2 2 13. Pinheiro, M. J. Information-Theoretic Determination of Ponderomotive Forces. d Ipv’ Ipv Physica Scripta 70, 86 (2004). { zT w0[IpDðÞv’zv ðÞv’{v zDðÞT w0: dr 2 2 ð60Þ 14. Peterson, M. A. Analogy between thermodynamics and mechanics. Am. J. Phys. 47, 488 (1979). Hence, the following result is obtained: 15. Pavlov, V. P. & Sergeev, V. M. Thermodynamics from the differential geometry sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi standpoint. Theor. Math. Phys. 157, 1484 (2008). 16. Abreu, R. & Guerra, V. Introducing thermodynamics through energy and DF v’ > +v { , entropy. Am. J. Phys. 80, 627–637 (2012). * 1 2 ð61Þ ´ 2Ipv 17. Eboli, O., Jackiw, R. & Pi, So-Young Quantum fields out of thermal equilibrium. Phys. Rev. D 37, 3557–3581 (1988). v < v9 18. Verlinde, E. On the Origin of Gravity and the Laws of Newton. JHEP 2011,29 and, in the limit , (2011). D 19. Glansdorff, P. & Prigogine, I. Structure, Stabilite´ et Fluctuations (Masson E´diteurs, > 1 T v’Dv * { : ð62Þ Paris, 1971). 2 Ip 20. Khinchin, A. I. Mathematical Foundations of Statistical Mechanics (Dover Publications, New York, 1949). Here, v is the angular speed imposed on the system (heat source), v9 21. Liboff, R. L. Kinetic Theory - Classical, Quantum, and Relativistic Descriptions is the mass flow induced angular speed due to a sustained source of (Prentice-Hall, New Jersey, 1990). energy; Dv ; v9 2 v. We use the following tabulated data: 22. Huang, K. Statistical Mechanics (Wiley, Hoboken, 1987). ~ {1 {1 23. Chandrasekhar, S. Selected Papers, Vol.4 - Plasma Physics, Hydrodynamic and 16:6J:K mol for mercury at temperature of the experiment Hydromagnetic stability, and Applications of the Tensor-Virial Theorem, 23 3 and r < 13.6 3 10 kg/m . We also consider that the volume of (University of Chicago Press, Chicago, 1989). mercury is contained in the channel of the experimental apparatus 24. Gary, S. P. Theory of Space Plasma Microinstabilities (Cambridge University Press, forming a rim with an average radius of R 5 30 cm. Using Eq. 62, we Cambridge, 1993). 25. Stacey, W. M. Fusion Plasma Physics (Wiley-VCH, Weinheim, 2005). estimate that after one minute, the speed must be approximately 26. Kiehn, R. M. Topological Torsion and Topological Spin. Ann. Fond. L. de Broglie 3.7 mm/s. It is clear that the sense of rotation and the speed of the bf 32, 389–408 (2007). wind depend on the latent heat stored in the planetary atmosphere 27. Onoochin, V. & Phipps, T. E. Jr. On an Additional Magnetic Force Present in a and the temperature difference between the boundaries (through System of Coaxial Solenoids. Eur. J. Elect. Phen. 3, 256 (2003). DT). Eq. 61 is consistent with the results reported by53, where a 28. Steinhauer, L. C. & Ishida, A. Relaxation of a Two-Specie Magnetofluid. Phys. Rev. Lett. 79 3423 (1997). maximum mean surface flow is observed corresponding to the angu- 29. Lamb, H. Higher Mechanics (Cambridge University Press, London, 1920). lar velocity of the heat source when the convective velocity (in our qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30. Lavende, B. H. Principles and representations of nonequilibrium ~ thermodynamics. Phys. Rev. A 9, 929 (1974). example, vstam DF 2Mm, with Mm denoting the molar mass) is 31. Chakraborty, B. Principles of Plasma Mechanics, p. 286 (Wiley, New Delhi, 1978). attained. Note that, in fact, our vstam represents the limiting speed for 32. Wilcox, J. M. Review of High-Temperature Rotating-Plasma Experiments. Rev. the transport of angular momentum. For mercury, this velocity is Mod. Phys. 31, 1045–1051 (1959). 33. Anderson, O., Baker, W. R., Bratenahl, A., Furth, H. P. & Kunkel, W. B. vstam < 24 m/s, with Mm 5 200.59 g/mol in the conditions of the Hydromagnetic Capacitors. J. Appl. Phys. 30, 188–196 (1959). moving flame experiment. The existence of a limit to the amplifica- 34. Slepian, J. Comment on ‘‘Review of High-Temperature Rotating-Plasma tion of the angular speed was also suggested in Ref. 54, which demon- Experiments’’ Rev. Mod. Phys. 32, 1032–1033 (1960). strated the effect of a heating or cooling source in the momentum 35. Hurricane, O. A. Rotating toroidal equilibria of quasi-neutral and non-neutral plasmas. Phys. Plasmas 5, 2197–2202 (1998). equilibration. Three possible cases for the clockwise or counter- 36. Antoni Klajn, A. Switching Vacuum Arc in a Pulsed Transverse Magnetic Field. clockwise motion of a fluid in a rotation frame are observed in Fig. 3. IEEE Trans. Plasma Science 27, 977 (1999). Although the initial assumptions taken in the present formulation 37. Power circuit breaker: theory and design Ed. Flurscheim, C. H. (Peter Peregrinus, require further research and considering that the interactive terms London, 1985). 38. Chen, F. F. Introduction to Plasma Physics and Controlled Fusion, Vol. 1 (Plenum with the medium, conveyed, for example, through the thermal dif- Press, New York, 1984). fusion coefficient, are not taken into account by Eq. 49, the agree- 39. Tu, X., Che´ron, B. G., Yan, J. H. & Cen, K. F. Plasma Sources Sci. Technol. 16, ment is reasonable and offers a possible explanation for this effect. 803–812 (2007).

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 8 www.nature.com/scientificreports

40. James, D. R. An Experimental Examination of Retrograde Motion of an Electric 52. Chandrasekar, A. Basics of Atmospheric Science (PHI Learning Private Limited, Arc, Ph.D. dissertation, Wright-Patterson Air Force Base, Ohio, 1962. New Deli, 2010). 41. Younian, W. & Tengcai, M. Investigation of Electric Conductivity of Plasma in 53. Douglas, H. A., Mason, P. J. & Hinch, E. J. Motion due to a moving internal heat Magnetic Rotating Arc, in Plasma Jets in the Development of New Materials source. J. Fluid. Mech. 54, 469–480 (1972). Technology, Eds. Solonenko, O. P. & Fedorchenko, A. I. pp. 27–32 (VSP BV, The 54. Stern, M. E. Generalizations of the rotating flame effect. Tellus XXIII, 122–128 Netherlands, 1990). (1971). 42. Li, L. et al. Control of cathodic arc spot motion under external magnetic field. J. Vacuum 91, 20–23 (2013). 43. Brussard, G. J. H., van de Sanden, M. C. M. & Schram, D. C. Ions densities in a high-intensity, low flow nitrogen-argon plasma. Phys. Plasmas 4, 3077–3081 Acknowledgments (1997). We are grateful to Professor Roman W. Jackiw, MIT Center for Theoretical Physics, for 44. Boxman, R. L., Martin, P. J. & Sanders, D. Handbook of vacuum arc science and keeping us informed of the fundamental paper referred to in Ref. 17. The author gratefully technology: fundamentals and applications (Noyes Publications, New Jersey, acknowledges partial financial support by the International Space Science Institute (ISSI) as 1995). 45. Balbus, S. A. & Hawley, J. F. A powerful local shear instability in weakly a visiting scientist and expresses special thanks to Professor Roger Maurice-Bonnet. magnetized disks. I. Linear Analysis. Astrophys. J. 376, 214–222 (1991). 46. Velikhov, E. P. Magnetic Geodynamics. JETP Letters 82, 690–695 (2005). Additional information 47. Velikhov, E. P. Stability of an ideally. Conducting liquid flowing between cylinders PACS: 45.10.Db, 05.45.-a, 83.10.Ff, 52.30.-q, 52.80.Mg, 47.32.C-, 47.32.Ef, 95.30.Qd, rotating in a magnetic field. Soviet Phys. - JETP 36, 1398 (1959). 52.55.Fa 48. Chandrasekhar, S. The stability of non-dissipative Couette folow in hydromagnetics. Proc. Nat. Acad. Sci. 46, 253 (1960). Competing financial interests: The authors declare no competing financial interests. 49. Boo´irevich, V., Freibergs, Ya., Shilova, E. I. & Shcherbinin, E. V. Electrically How to cite this article: Pinheiro, M.J. A Variational Method in Out-of-Equilibrium Induced Vortical Flows (Kluwer Academic Publishsers, Dordrecht, 1989). Physical Systems. Sci. Rep. 3, 3454; DOI:10.1038/srep03454 (2013). 50. Chow, K. C. & Chan, K. L. Angular Momentum Transports by Moving Spiral Waves. J. Atmos. Sci. 60, 2004–2009 (2003). This work is licensed under a Creative Commons Attribution 3.0 Unported license. 51. Schubert, G. & Whitehead, J. A. Moving flame experiment with liquid mercury: To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0 possible implications for the Venus atmosphere. Science, New Series 163, 71–72 (1969).

SCIENTIFIC REPORTS | 3 : 3454 | DOI: 10.1038/srep03454 9