# Hamiltonian Fluid Mechanics VORTEX DYNAMICS OF CLASSICAL FLUIDS IN HIGHER DIMENSIONS Banavara N. Shashikanth, Mechanical and Aerospace Engineering, New Mexico State University Outline Recall some basic facts of the vorticity two-form surfaces of singular vorticity in R 2 (point vortices) and in R 3 (vortex filaments) the local induction approximation (LIA) and the self-induced velocity of filaments Surfaces of singular vorticity in R 4 ---vortex membranes Self-induced velocity field of a membrane using LIA Dynamics of ! ^! and an application to Ertels theorem Vorticity of ideal fluids in R n For n = 2; 3 vorticity is commonly identified with a

function, vector field, respectively, and defined as ! vec = r v Strictly speaking, vorticity is a two-form (Arnold (66)) ! = dv[ v[ is the velocity one-form, d is the exterior derivative By definition, vorticity is a closed two-form i.e. d! = 0 Vorticity of ideal fluids in R n For n = 2; 3 the Hodge star operator allows the identification with a function, vector field, respectively For n 4, vorticity must be considered as a two-form In Cartesian coordinates on R n , vorticity has n(n 1)=2 components. For n = 4, ! = ! 12dx1 ^ dx2 + ! 13dx1 ^ dx3 + ! 14dx1 ^ dx4 + ! 23dx2 ^ dx3 + ! 24dx2 ^ dx4 + ! 34dx3 ^ dx4 Vorticity of ideal fluids in R n

Lie-Poisson evolution of vorticity of an ideal fluid in R n (more generally, ideal fluid on an n-dimensional manifold M ) Arnold (`66), Marsden and Weinstein (`83), Morrison (`82) The vorticity two-form is an element of ---dual of the Lie algebra of divergence-free velocity fields in R n Singular distributions of vorticity Preservation of coadjoint orbits : A vorticity two-form, evolving by Lie-Poisson dynamics, remains on the same coadjoint orbit of (Marsden and Weinstein (83)) Singular vorticity distributions: in the context of classical fluids, can be viewed as idealized models of coherent vorticity. Examples of singular vorticity distributions: ----point vortices in R 2 , ----vortex filaments in R 3

Point vortex and vortex filament models are popular with engineers, mathematicians and physicists! Singular distributions of vorticity point vortices in phase space Poisson brackets Singular distributions of vorticity vortex filaments in 1 3 phase space: space of images of N smooth maps ' : S ! R (modulo re-parametrizations), infinite-dimensional space Poisson brackets f F; Gg =

XN 1 j =1 j I Cj F G ; tj Cj Cj dsj

(functional derivatives identified with normal vector fields on curves) Singular distributions of vorticity Marsden and Weinstein (83) : The Poisson brackets/symplectic structures for both models are obtained from the formula for the symplectic structure of coadjoint orbits (c.o.) of vorticity two-forms ! where are divergence-free velocity fields. M & W derived this from the general Kirillov-Kostant-Soureau formula for the symplectic structure of coadjoint orbits Singular distributions of vorticity M & W (`83) showed that this recovers the classical N point vortex symplectic structure: ! p:v: (r) =

XN j (r r j )dx ^dy; j =1 Z ) R2 ! p:v: (u; v) = giving the symplectic form point vortex phase space XN j dx ^dy (u(r j ); v(r j ))

j =1 P N j =1 j dxj ^dyj on the Singular distributions of vorticity M & W (`83) also presented a symplectic structure for N vortex filaments: ! l (r) = XN j (r r(sj ))dn1;j ^dn2;j (sj ); j =1

Z ) R3 ! l (u; v) = XN j =1 I j Cj dn1;j ^dn2;j (sj ) (u(sj ); v(sj )) dsj XN giving the symplectic form

j =1 I j Cj dn1;j ^dn2;j (sj ) ( ; ) dsj on the phase space of vortex filaments. This form acts on filament normal vector fields Singular distributions of vorticity n1;j n2;j tj Cj

Singular distributions of vorticity General geometric features of point vortices and vortex filaments: They are co-dimension 2 surfaces The vorticity two-form ! acts on planes that intersect these surfaces transversally. More precisely, the two-form ! is perpendicular to these surfaces. Equivalently, using the Hodge star operator in R n , the n-2 form ?! is tangent to these surfaces Singular distributions of vorticity Minimum dimension of the surfaces = the degree of the form ?! = n-2 Can we have singular distributions that do not satisfy the above? For example, point vortices in R 3 or `vortons (Novikov (`83), Leonard (`85)). Here, each point is also assigned a (time-varying) direction vector and

the `vorticity two-form acts on the single plane to which it is normal. This `vorticity two-form is not closed i.e. d! 6 = 0! Singular distributions of vorticity Moving on to R4 We consider two-dimensional surfaces/manifolds to which ?! ---now, a two-form---is tangent. We term these surfaces vortex membranes At each point of such a (co-dimension 2) surface there exists a plane of normals Singular distributions of vorticity The vorticity two-form ! for a membrane is ! (m) = (m p)dn1 ^ dn2(p); m 2 R 4; p 2

where dn1 ^ dn2(p) is an area form in the plane of normals at p n1 p t2 n2 t1 Dynamics of singular vorticity An important notion in the dynamics of singular vorticity is the self-induced velocity field Recall, point vortices in ---- no self-induced velocity field vortex filaments in ---- infinite self-induced velocity field!

Two ways of obtaining the expression for the self-induced velocity of a filament in ---- invert the kinematic relation r 2v = r ! vec ---- use the M & W symplectic structure and the kinetic energy Hamiltonian Dynamics of singular vorticity Both lead to the Biot-Savart integral for filaments I vS I (q(~ s)) = 4 t(s) (l(~ s) l(s)) ds C

jl(~ s) l(s)j 3 ; q(~ s) 2 C The integral is divergent due to the integrand singularity at s = s ~ The velocity has a logarithmic singularity and is infinite in the binormal direction curvature vS I (q(~ s)) = lim log (~

s) b(~ s) + O(1) terms; ! 0 4 ( =j s~ s j) Dynamics of singular vorticity To obtain a finite vS I , the integral has to be regularized One commonly used regularization method is the Local Induction Approximation (LIA) (DaRios (06), Arms and Hama (65)) The LIA is based on the observation that the leading order contribution to vS I is due to a local neighborhood of s ~. Non-local portions of the filament contribute O(1) terms only Treating as a small but fixed `cut-off parameter, the regularized velocity, according to the LIA, is vS I ;r eg(q(~ s)) = log

(~ s) b(~ s) 4 Dynamics of singular vorticity This leads to the famous filament equation (using the SerretFrenet equations for a curve in ) @C(s; t) @C(s; t) @2C(s; t) = @t @s @s2 where C : S 1 R ! R 3 or C : R R ! R 3 Zs (u) du Hasimotos transformation (s) = (s) exp i

0 gives the non-linear Schrdinger equation (NLS)! @ @2 1 i = 2 + j j2 ; @t @s 2 :R R ! C Dynamics of singular vorticity NLS is an integrable Hamiltonian system. The precise relation between the M & W Poisson structure of the filament equation and the Poisson structure of the NLS was clarified by Langer and Perline (`91). Dynamics of singular vorticity

Returning to R 4 and vortex membranes Main objective: obtain an expression for the (regularized) self-induced velocity field of a membrane Generalize the Biot-Savart expression as follows: first, generalize the kinematic relation r 2v = r ! to vec dv[ = ! ; where : k+1(R4) ! k (R4) is the co-differential operator defined as = ? d? Dynamics of singular vorticity In the standard basis f ei g, the equation is equivalent to the Poisson equation r 2vi = f i ; i = 1; ; 4

where f i := ! (ei ) Elliptic theory, Greens functions and integration by parts gives Z v(m) ~ = (?(dm G(m; m) ~ ^?! ))] m ; m; m ~ 2 R4 R4 where G = 2(j r(m) r(m) 2 1 is the ~ j ) Greens function of the Laplacian in R 4 Dynamics of singular vorticity The above is the generalization of the Biot-Savart formula to R 4 ~ 2 R4 due to any !

and gives the velocity at any field point m Substituting ! gives the expression for the self-induced velocity of a membrane Z vS I (p) = ((r G n2)n1 (r G n1)n2)ds1 ^ds2; m; p 2 where (n1; n2; t1; t2) is a moving orthonormal frame with dsi (tj ) = i j t1;t2 2 Tm ; n1; n2 2 N m and m ! p; vS I (p) As blows up due to the integrand singularity and the expression has to be regularized using LIA Dynamics of singular vorticity LIA applied to a membrane

s1; s2 : arc length parameters s2 s1 p s2 = f (s1; ); ! s1; as s1 ! 0 Choose an (s1; ) coordinate system as shown, with p at s1 = 0 Series expand position vectors r(s1; ) and moving frame basis (n1; n2; t1;t2)(s1; ) for small s1 along `diameter curves ( = constant) 1 Dynamics of singular vorticity At p(s1 = 0) there is a one-parameter family of tangent vectors (n1(0); n2(0); t1(0; ); t2(0; )) Introducing the `cut-off parameter of LIA, obtain

vS I ;r eg(p) = 2 log Z 0 2 @n2(0) t1(0;) @s1 n1(0)

@n1(0) t1(0; ) @s1 n2(0) d Dynamics of singular vorticity Another expression for vS I ;r eg(p) using the M&W coadjoint orbit symplectic formula Consider P the phase space of membranesthe space of images of maps (modulo re-parametrizations with the same image) : S 2 ! R4 or : R 2 ! R4 An element of TpP can then be identified with a field of normal vectors un on The M&W formula yields a symplectic structure on P Z

Z R4 ! (u; v) = dn1 ^dn2 (un ; vn ) Dynamics of singular vorticity Kinetic energy of the fluid flow Z 1 K :E : = v[ ^?v[ ; 2 R4 Z 1 = ! ^?A [integration by parts]

2 R4 where the vector potential two-form A satisfies dA = ! ; (A = v[ ; dA = 0) This is again Poissons equation in each of the six components of ! and A . For ! , inversion by Greens function gives Z 1 4 A (m) = ; m 2 R ;p 2 p 2

2 (j r(m) r(p) j ) Dynamics of singular vorticity The kinetic energy of the flow due to a membrane Z K :E : = A (m); m 2 2 2 Z Z 1 = p m ; m; p 2 2 2 2 (j r(m) r(p) j ) This is a functional on the membrane phase space P . However, the integral is not convergent as m ! p

Regularization by LIA gives the membrane Hamiltonian 2 K :E :r eg := log Z =: H ( ) Dynamics of singular vorticity The Hamiltonian, modulo constants, is the area functional. Recall, the Hamiltonian (regularized K.E.) for a filament is the length functional Using this Ham and the M&W symplectic structure, and taking variational derivatives, obtain the Hamiltonian vector field X H (p) vS I ;r eg as vS I ;r eg(p)

2 = log()R 0 1 B @n1 C @ n @ n @ n 1 2 2

B t1 + t2; t1 + t2C (p) =2 @ A @s @s @s @s | 1 {z 2 } | 1 {z 2 } n 1 component n 2 component Dynamics of singular vorticity

Note: in taking the variational derivatives, we use the following chartslightly different from the previous chart s2 p s1 In summary, we have two expressions for vS I ;r eg(p) , both using LIA. One directly from the (generalized) Biot-Savart integral, the other from the Ham vector field of the kinetic energy Dynamics of singular vorticity The final step is showing that each of these (modulo constants) is equal to the mean curvature vector rotated by 90 degrees in the plane of normals Recall, that for a 2D surface in R 3, the second fundamental form is defined as S (V )(p) := hh(D n(p) V ); V i i ;

V 2 Tp; p 2 where n is the unit normal, and the mean curvature as 1 (p) = (S (t1) + S (t2)) (p); t1; t2 2 Tp; 2 where (n; t1; t2) is an orthonormal frame Dynamics of singular vorticity For a 2D surface in R 4, there is a second fundamental form associated with each of the normal directions of the moving frame (n1; n2; t1; t2) S 1(V )(p) := hh(D n1(p) V ); V i i ; S 2(V )(p) := hh(D n2(p) V ); V i i ; V 2 Tp ; The mean curvature vector K(p)is then defined as X2 1 K(p) =

(S i (t1) + S i (t2)) (p)ni (p) 2 i =1 Dynamics of singular vorticity And so, for the expression obtained using M&W formula vS I ;r eg(p) 2 = log()R 0 1 B @n1 C @n1 @n2

@n2 B t1 + t2; t1 + t2C (p) =2 @ A @s @s @s @s | 1 {z 2 } | 1 {z 2 } n 1 component 2 2 =

log()R =2 K(p) n 2 component Dynamics of singular vorticity Next, showing that Z 2 1 2 0 Z 2 1

2 0 @n2(0) t1(0; ) = K(p) n2(0) @s1 @n1(0) t1(0; ) d = K(p) n1(0) @s1 the expression obtained from the (generalized) Biot-Savart law can also be written as 2 2 vS I ;r eg(p) =

log()R =2 K(p) The dynamics of the four-form ! ^! In R 2 and R 3, ! ^! is identically zero Integral laws for ! ^! derived by Arnold and Khesin (98), also discussed in papers on 4D Navier-Stokes turbulence, for ex. Gotoh, Watanabe, Shiga and Nakano (2007) The evolution equation for ! ^! is @(! ^! )

+ L v (! ^! ) = 0 @t An application to Ertels theorem in R3 Consider a divergence-free velocity field w in R 3 and a smooth function f : R 3 ! R The vector field v := (w; f )is divergence-free in R 4. In coordinates (x; y; z; o) , v[ = w1(x; y; z)dx + w2(x;y;z)dy + w3(x;y; z)dz + f (x;y; z)do Eulers equations for v (or v[ ) are Eulers equations for w and the passive advection equation for f , and ! ^! = (~ ! vec r f ) dx ^dy ^dz ^do where !~= dw[ References B. N. Shashikanth (2012), Vortex dynamics in R 4, Journal of Mathematical Physics, vol. 53, 013103, 21 pages

B. A. Khesin (2012): Symplectic Structures and Dynamics on Vortex Membranes, Moscow Mathematical Journal, Vol. 12, No. 2 Khesin generalizes these results to any R n and also presents a Hamiltonian formalism for vortex sheets References S. Haller and C. Vizman (2003): Nonlinear Grassmannians as coadjoint orbits, arXiv:math.DG/0305089, 13pp Haller and Vizmanworking in a purely geometric context show that the Hamiltonian vector field for the volume functional on the Grassmannian of codim-2 submanifolds N of a Riemannian manifold M gives an evolution equation for N which is skew trace of the second fundamental form R. L. Jerrard (2002), Vortex Filament Dynamics for GrossPitaevsky Type Equations, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, Vol. 1, No. 4, pp.733-768 In the context of superfluids and the G-P equations, Jerrard shows that in spaces of dimenison m 3, a codim-2 spherical vortex membrane evolves by skew mean curvature flow

Open questions/future directions Can Hasimotos transformation be generalized for membranes? If yes, what are the transformed PDE? Surfaces of singular ! ^! ?