Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics

After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.


Introduction to isentropic compressible magnetohydrodynamics (ICMHD)
Consistent with the topic of this focus issue, this paper will deal with the equations of ICMHD [1]- [3]. Not only are these sufficiently general to encompass most of the interest in this area but they also offer a significant challenge even without dissipation. The ICMHD equations determine the dynamics of a conducting fluid flow with magnetic field B satisfying div B = 0 and current J = curl B. The equations for the fluid velocity field u, mass density ρ, specific entropy σ, pressure p(ρ, σ) and divergenceless magnetic field B are, respectively, the motion equation, Faraday's law of frozen-in magnetic flux, isentropy along flow lines and the continuity equation (1.2) with the Lagrangian time derivative defined as 3) The equation of state for specific internal energy e(ρ, σ) and the thermodynamic first law yield (1.4) so the specific internal energy changes only because of mechanical work. Combining Faraday's law with the continuity equation yields DB ρ Dt − B ρ · ∇u = 0 with B ρ := ρ −1 B. (1.5) This is the condition for the vector field B ρ · ∇ to be frozen into the flow, i.e. D Dt B ρ · ∂ ∂x = 0, along Dx Dt := u. (1.6) We denote Of course, the invariant vector field d/ds = B ρ · ∇ may be applied to any fluid quantity. For example, applying d/ds to the equations in (1.9) and using the equality of cross derivatives in d/ds and D/Dt yields the following exact nonlinear wave equations for ICMHD When linearized, these equations yield Alfvén-sound waves. The present work emphasizes the alignment dynamics that is inherent in these equations, rather than their wave properties.

Slow plus fast decomposition and nonlinear waves
It seems natural to define slow and fast aspects of the ICMHD solutions in the sense of the Lagrangian time derivative. In particular, there are two relations for slow variables where β is defined as The latter follows from equality of cross derivatives, by computing Thus, the ICMHD equations preserve the projection of B ρ on ∇σ along Lagrangian flow lines, that is, along the path x(t) of a Lagrangian fluid particle determined from dx/dt = u(x, t).

DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Equivalently, one finds the Eulerian conservation law, Of course, the evolution of the component of B ρ perpendicular to ∇σ is not so simple! One may decompose B ρ into components parallel and perpendicular to ∇σ as One then computes the auxiliary equations, and D Dt |∇σ| 2 = −∇σ · S · ∇σ. (1.21) The fluid strain-rate tensor is defined as S = 1 2 (∇u + (∇u) T ). Consequently, the evolution equation for the component of B ρ perpendicular to ∇σ is determined from which is clearly neither fast nor slow.

Lagrangian dynamics of specific volume
Two other likely candidates for fast variables are the specific volume ρ −1 and the velocity divergence div u, which satisfy The second of these equations is found using the identity where |∇u| 2 ≡ u i ,j u j ,i . As a consequence of (1.23) and (1.24), the specific volume ρ −1 satisfies the 'Lagrangian oscillation equation,' upon substituting the definition of force F := dB ds − ρ −1 ∇ . Thus, the divergence of the force combines with compressibility of the flow to drive either Lagrangian oscillations or exponential variations of the specific volume, depending on the sign of the term on the right-hand side of equation (1.26). Of course, the sign of this term could be varying rapidly in some physical applications, and observations of such behaviour would have corresponding implications for the dynamics of specific volume. In any case, the form of equation (1.26) is universal and one may conclude that the divergence of the total force drives the second Lagrangian time derivative of the specific volume.
In what follows, we shall seek equations for alignment dynamics in MHD which have the same universal property as the Lagrangian oscillation equation (1.26). In particular, we shall find that gradients of the total force (rather than its divergence) drive the Lagrangian dynamics of alignment in MHD of the frozen-in magnetic vector field B ρ relative to its projection B ρ · ∇u on to the shear tensor ∇u of the fluid velocity.

Stretching and alignment
The magnetic flux equation implies that the contravariant vector B ρ undergoes stretching to the extent it aligns with the shear ∇u. Likewise, this alignment evolves according to which recovers the nonlinear wave equation (1.12) above. We shall examine this equation from an alignment viewpoint, rather than as a wave propagation phenomenon. We begin by decomposing the vector B ρ · ∇u into its components parallel and perpendicular toB ρ : where α and χ are defined by This decomposition is explained more fully in section 2.2. In fact, α is the Lagrangian amplification rate of the magnitude |B ρ | and χ is the Lagrangian frequency of rotation of the unit vectorB ρ under the forcing by shear in the flux conservation equation (1.27), Note that no confusion should arise between this α and the one appearing in the α-dynamo equations! 6 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

A mathematical framework for magnetic fluid alignment dynamics
The quantity α defined in equation (1.30) is the alignment of B ρ with ∇u, which determines the rates of change of magnitude |B ρ | whereas χ is the misalignment, which determines the frequency of rotation of the directionB ρ . The question asked in this situation is, 'How long will B ρ remain aligned with ∇u, so it can continue to be stretched under the fluid shear?'Answering this question requires an analysis of Dα/Dt and Dχ/Dt using the alignment dynamics (1.28).
In principle, this analysis could be performed by direct computation and algebraic manipulation. However, the Lagrangian evolution equations for α and χ happen to fit perfectly into a mathematical framework that was especially designed for analysing orientation dynamics and for systematically interpreting the results. The key for recognizing this framework is to notice that the decomposition of a vector into its components parallel and perpendicular to another vector defines a type of product, or multiplication, that was first discovered by Hamilton [4]. This product reveals itself when we write the parallel-perpendicular vector decomposition equation (1.29) as though it were the pure vector components of a equation involving the fourcomponent scalar-vector object (tetrad) [α, χ] in the form, This organization of the decomposition equation (1.29) summons the product defined by which is the multiplication rule that Hamilton invented for the field of quaternions [4]. The origin of this rule and its connection to the definition of a quaternion is given in section 2.1. Of course, quaternions have everything to do with orientation [5,6]. The remainder of this paper sets-up the quaternionic framework for alignment dynamics of a general fluid flow and applies it to the ICMHD equations. The plan of this paper is as follows: section 2 summarizes the results of [7,8] and shows how the evolution of the ortho-normal frame can be calculated in general terms. Section 3 then discusses the ICMHD equations (1.9) and (1.10) in these terms, while section 4 examines alignment and growth properties to begin answering the question raised above for magnetic dynamos using these results. Finally, section 5 discusses the potential for other applications of this general method in compressible MHD turbulence.

Background for quaternions
The resurgence of practical interest in quaternions during the last two decades has been stimulated by progress in the computer animation and inertial navigation industries because of the ease with which quaternions handle moving objects undergoing three-axis rotations [5,6]. The tracking of aircraft and satellites and the animation of tumbling objects in computer graphics are obvious examples. Quaternion methods have also been recently been applied to the three-dimensional Euler equations for incompressible fluid motion [9,10] and to passive tracer particles transported by an underlying Lagrangian flow field; see [11]- [21] and references in [7,8]. The final result of these endeavours is that equations of motion can be derived for an ortho-normal frame. This 7 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT 'quaternion frame'follows the evolution of particles in a Lagrangian flow whose evolution derives from the Eulerian equations of motion.
Three-axis rotations lie at the heart of the definition of a quaternion [22]. In terms of any scalar p and any three vector q, the quaternion q = [p, q] is defined (using gothic fonts to denote quaternions) as where {σ 1 , σ 2 , σ 3 } are the Pauli spin-matrices and I is the unit matrix. The relations between the Pauli matrices σ i σ j = −δ ij I − ijk σ k then give a non-commutative multiplication rule It is easily demonstrated that quaternions are associative. In fact the individual elements of a unit quaternion provide the Cayley-Klein parameters of a rotation. This representation is a standard alternative to Euler angles in describing the orientation of rotating objects, as the books by Whittaker and Klein show [23,24].

Quaternions and Lagrangian alignment dynamics in fluids
A general quaternionic picture of the process of Lagrangian flow and acceleration in fluid dynamics is explained in this section by considering the abstract Lagrangian flow equation whose Lagrangian acceleration equation is given in general by These are the rates of change of these vectors following the characteristics of the velocity generating the path x(t) of a Lagrangian fluid particle determined from dx/dt = u(x, t). Given the Lagrangian equation (2.3) one defines the scalar α a and the three vector χ a as × a), (2.5) in which w = wŵ with w = |w|. As observed in (1.29), the three vector a is decomposed into parts that are parallel and perpendicular to w as and thus the quaternionic product (2.6) is summoned in a natural manner. By definition, the growth rate α a of the scalar magnitude w = |w| obeys Moreover, if a is Lagrangian-differentiable as in (2.4) then it is clear that a similar decomposition for b as that for a in (2.6) gives Using the associativity property, compatibility of (2.11) and (2.10) implies that which establishes a Riccati relation between the quaternions q a and q b Dq a Dt + q a q a = q b . where the Darboux vector D a defined as is the angular frequency of rotation of the ortho-normal frame F .

Remark. 1. The frame orientation is controlled by the Darboux vector
which lies in the (ŵ,χ a ) plane and is so named for its similarity to the Darboux vector in the Frenet-Serret (FS) equations for a space curve. Note that the vector D a depends on χ b but is independent of α b . 2. The frame dynamics equations (2.14)-(2.17) may also be re-written in matrix form by defining the 3 × 3 skew-symmetric matrix C a ∈ so(3) with entries [C a ] ij = − ijk D k a as, Proof. Finding an expression for the Lagrangian time derivatives of the components of the frame (ŵ,ŵ ×χ a ,χ a ) T requires the derivative ofχ a . For this, one first recalls that the three vector b may be expressed in this ortho-normal frame as the linear combination where c b is defined in (2.17) and d b = − (χ a · χ b ). The three vector product To find the Lagrangian time derivative ofχ a , we use the three vector part of the equation for the quaternion q a = [α a , χ a ] in theorem 1  (ŵ ×χ a ), which gives equations (2.14)-(2.17).

Application of the quaternionic alignment theorem to ICMHD
The way is now clear to apply theorem 1 of section 2 to the ICMHD equations (1.9) and (1.10). First, we identify from (1.27) and (1.28) The latter result is achieved through the use of Ertl's theorem (see [10,28,29]). Although messy, this provides an explicit expression for b that depends upon the pressure p, B ρ , J and their derivatives, as determined from the fluid evolution equations for u, ρ and σ, The alignment parameters {α, χ, α b , χ b } are now identified as These parameters appear in the vector decompositions, where the Darboux angular velocity vector D is defined as which depends explicitly on the ICMHD force gradients through χ b but is independent of α b . The frame dynamics equations (3.7)-(3.10) for ICMHD may also be re-written in matrix form using the 3 × 3 skew-symmetric matrix C ∈ so(3) with entries C ij = − ijk D k as,

Alignment and growth properties in ICMHD
The growth rate α(x, t), defined in (3.3), satisfies (see equations (2.7) and (2.8)) Here α can take either sign and this is the key to how fast the magnitude |B ρ | increases (or decreases) at each point in the flow. In contrast, the three vector χ(x, t) is the key to the alignment 12 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT properties of the system, because it satisfies As such, it can be interpreted as the swing rate of the unit vectorB ρ about a = B ρ · ∇u. Clearly, if B ρ is aligned with a then χ = 0 and the quaternion q involves only the scalar α. Violent corkscrew-like motions of the magnetic field lines would be consistent with significant values of χ and such motions can therefore be regarded as a diagnostic for the misalignment ofB ρ with a.
One of the messages of this paper is that putting α and χ together as the quaternion q = [α, χ] is a natural way to approach this problem because the full quaternion q with χ = 0 is summoned whenever vortex or magnetic field lines bend or tangle. The Darboux vector D is the angular frequency of rotation of the orthonormal frame F mag = (B ρ ,B ρ ×χ ,χ) T , but is itself controlled by the gradients of forces in the expression (3.2) for b. When written out in terms of α and χ, the Riccati relation (2.13) becomes Some years ago, these four equations were first expressed in this form for the incompressible Euler equations without any recourse to quaternions [25]. The quaternionic form first appeared in [8] (see [7]- [9], [28,29] for a history). Equations (4.3) appear to behave as Lagrangian ODEs driven by If the latter terms remain roughly constant equations (4.3) can easily be shown to have two fixed points [25]; one has a negative value of α and the other has a positive value. The negative one is associated with an unstable spiral and the positive one with a stable spiral in the phase plane for this simplified ODE system. Equations (4.3) with constant q = [α b , χ b ] have also recently been studied as a remarkably interesting kinematic model for the creation of non-Gaussian statistics in hydrodynamic turbulence [26]. This simple picture would seem not apply if the driving terms α b , χ b were to vary on the same timescales as α, χ, or faster. This breakdown in applicability of equations (4.3) with constant q = [α b , χ b ] would be indicated if the force gradients in (3.2) were observed to undergo rapid changes.

Conclusion
The quaternionic approach to Lagrangian frame dynamics developed here for ICMHD applies generally in fluid dynamics. Because the form of these equations is universal; that is, independent of the specific choice of forces, one may expect them to have many other applications. For example, in MHD one may use these equations to consider the effects on frame dynamics of rotation, or the effects of various kinds of subgrid-scale models, simply by identifying the corresponding expressions for the vectors a and b in equations (2.3) and (2.4). Subgridscale models of MHD turbulence may be a particularly fruitful arena for these applications, especially for those models derived from Lagrangian averaging, because Faraday's law for such models is preserved for the averaged field [27]. Depending as it does on Faraday's law, the quaternionic method for MHD is fundamental, but it is also mainly kinematic. Thus its best role may be as a means of developing diagnostics for determining the effects of total force gradients. Therefore, additional applications of the quaternionic approach may also be foreseen, as improved Lagrangian diagnostic methods are developed in the future. For example, future diagnostics may be able to distinguish between effects described by equations (4.3) with fixed values of the quaternion q b = [α b , χ b ], versus its self-consistent exact dynamics when the vector b is determined from the varying force gradients in equation (3.2).