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 R2 (point vortices) and in R3
(vortex filaments)
 the local induction approximation (LIA) and the
self-induced velocity of filaments
• Surfaces of singular vorticity in R4 ---vortex membranes
• Self-induced velocity field of a membrane using LIA
• Dynamics of ! ^ ! and an application to Ertel’s theorem
Vorticity of ideal fluids in Rn
• 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 Rn
• 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 Rn , vorticity has n( n ¡ 1) =2
components. For n = 4,
! = ! 12 dx 1 ^ dx 2 + ! 13 dx 1 ^ dx 3 + ! 14 dx 1 ^ dx 4
+ ! 23 dx 2 ^ dx 3 + ! 24 dx 2 ^ dx 4 + ! 34 dx 3 ^ dx 4
Vorticity of ideal fluids in Rn
• Lie-Poisson evolution of vorticity of an ideal fluid in Rn
(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 Rn
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 R2 ,
----vortex filaments in R3
• 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
phase space: space of images of N smooth maps ' : S !
(modulo re-parametrizations),
infinite-dimensional space
Poisson brackets
f F; Gg
XN
=
j=1
1
¡j
¿
I
Cj
±F
±G
£
;tj
±Cj
±Cj
À
dsj
(functional derivatives identified
with normal vector fields on curves)
R3
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:
XN
!
p:v : (r )
=
¡ j ±(r ¡ r j )dx ^ dy;
j=1
Z
)
XN
!
p:v :
(u; v) ¹ =
R2
giving the symplectic form
point vortex phase space
¡ j dx ^ dy (u(r j ); v(r j ))
j=1
P
N
j=1
¡ j dx j ^ dyj on the
Singular distributions of vorticity
• M & W (`83) also presented a symplectic structure for N
vortex filaments:
XN
!
¯l (r )
=
¡ j ±(r ¡ r (sj ))dn 1;j ^ dn 2;j (sj );
j=1
Z
)
!
R3
I
XN
¯l
(u; v) ¹ =
¡j
j=1
dn 1;j ^ dn 2;j (sj ) (u(sj ); v(sj )) dsj
Cj
I
XN
giving the symplectic form
j=1
¡j
dn 1;j ^ dn 2;j (sj ) ( ; ) dsj
Cj
on the phase space of vortex filaments. This form acts on
filament normal vector fields
Singular distributions of vorticity
n 1;j
n 2;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 Rn , 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 R3 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) dn 1 ^ dn 2 ( p) ;
m 2 R4 ; p 2 §
where dn 1 ^ dn 2 ( 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 2 v = ¡ 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
¡
vSI (q(~
s)) =
4¼
I
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
¡ ¡
vSI (q(~
s)) = lim log ²
· (~
s) b( s~) + O(1) t erms;
²! 0
4¼
(² = j s~ ¡ s j)
Dynamics of singular vorticity
• To obtain a finite vSI , 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 vSI is due to a local neighborhood of s~.
Non-local portions of the filament contribute O(1) terms only
• Treating ² as a small but fixed `cut-off’ parameter, the
regularized velocity, according to the LIA, is
¡ ¡
vSI ;r eg (q(~
s)) = log ²
· (~
s) b(~
s)
4¼
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) @2 C(s; t)
=
£
@t
@s
@s2
where C : S1 £ R !
R3 or C : R £ R ! R3
µ Zs
¶
¿(u) du
• Hasimoto’s transformation Ã(s) = · (s) exp i
0
gives the non-linear Schrödinger equation (NLS)!
@Ã
@2 Ã 1
2
¡ i
=
+
j
Ã
j
Ã; Ã : R £ R ! C
2
@t
@s
2
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 R4 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 2 v = ¡ r £ !
±dv[ = ±! ;
where ± : - k + 1 (R4 ) ! - k (R4 ) is the
co-differential operator defined as ± = ¡ ? d?
vec
to
Dynamics of singular vorticity
 In the standard basis f ei g, the equation is equivalent to
the Poisson equation
r 2 vi = f i ;
i = 1; ¢¢¢; 4
where f i := ±! (ei )
 Elliptic theory, Green’s functions and integration by parts
gives
Z
v( m)
~ = ¡
(?(d m G(m; m)
~ ^ ?! )) ] ¹ m ; m; m
~ 2 R4
R4
¡
2
2
¢¡
where G = ¡ ¼ (j r (m) ¡ r ( m)
~ j )
Green’s function of the Laplacian in R4
1
is the
Dynamics of singular vorticity
 The above is the generalization of the Biot-Savart formula to R4
and gives the velocity at any field point m
~ 2 R4 due to any !
• Substituting ! § gives the expression for the self-induced
velocity of a membrane
Z
vSI (p) = ¡
((r G ¢n 2 )n1 ¡ (r G ¢n1 )n2 )ds1 ^ ds2 ; m; p 2 §
§
where (n1 ; n2 ; t 1 ; t 2 ) is a moving orthonormal frame with
t 1; t 2 2 Tm § ; n1; n2 2 Nm and dsi (t j ) = ±i j
• As m ! p; vSI (p) 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 ¡ lengt h paramet ers
s2
s1
p
s2 = f (s1 ; µ);
! s1 µ; as s1 ! 0
• Choose an (s1 ; µ) coordinate system as shown, with p at s1 = 0
• Series expand position vectors r (s1 ; µ) and moving frame
basis (n1 ; n2 ; t 1 ; t 2 )(s1 ; µ) for small s1 along `diameter’
curves (µ1 = constant)
Dynamics of singular vorticity
• At p(s1 = 0) there is a one-parameter family of tangent
vectors (n1(0); n2(0); t 1 (0; µ); t 2 (0; µ))
• Introducing the `cut-off’ parameter ² of LIA, obtain
vSI ;r eg (p)
¡ ¡
= 2 log ²
¼
Z
0
2¼ · µ
@n 2 (0)
t 1 (0; µ) ¢
@s1
¶
µ
n 1 (0) ¡
@n 1 (0)
t 1 (0; µ) ¢
@s1
¶
¸
n 2 (0) dµ
Dynamics of singular vorticity
• Another expression for vSI ;r eg (p) using the M&W coadjoint
orbit symplectic formula
• Consider P the phase space of membranes—the space of
images of maps (modulo re-parametrizations with the same
image) § : S2 ! R4 or § : R2 ! R4
• An element of Tp P can then be identified with a field of
normal vectors un on §
• The M&W
a symplectic structure on P
Z formula yields Z
!
R4
§
(u; v) ¹ = ¡
dn1 ^ dn 2 (un ; vn ) º
§
Dynamics of singular vorticity
• Kinetic energy of the fluid flow
Z
1
K :E : =
v[ ^ ?v[ ;
2 R4
Z
1
=
! ^ ?A [int egrat ion by part s]
2 R4
where the vector potential two-form A satisfies
d±A = ! ;
(±A = v[ ; dA = 0)
• This is again Poisson’s equation in each of the six components
of ! and A . For ! § , inversion by Green’s function gives
Z
1
4
A § (m) = ¡
º
;
m
2
R
;p2 §
p
2
2
§ ¼ (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)º ; m 2 §
2 §
2 Z Z
¡
1
=
º p º m ; m; p 2 §
2
2
2 § § ¼ (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
Z
2
¡ ¡
K :E :r eg :=
log ²
º = : H (§ )
¼
§
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) ´ vSI ;r eg as
vSI ;r eg (p)
¡2
=
log(²)R ¡
¼
0
1
B @n 1
C
@
n
@
n
@
n
1
2
2
B
¢t 1 +
¢t 2 ;
¢t 1 +
¢t 2 C
(p)
¼=2 ¢@
A
@s
@s
@s
@s
| 1
{z 2 } | 1
{z 2 }
n 1 com p onent
n 2 com p onent
Dynamics of singular vorticity
• Note: in taking the variational derivatives, we use the
following chart—slightly different from the previous chart
s2
p
s1
• In summary, we have two expressions for vSI ;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 ¾in R3, the second fundamental
form is defined as
S (V )(p) := ¡ hh(D n(p) ¢V ); V i i ;
V 2 Tp ¾; p 2 ¾
where n is the unit normal, and the mean curvature · as
1
· (p) = (S (t 1 ) + S (t 2 )) (p);
2
t 1 ; t 2 2 Tp ¾;
where (n; t 1 ; t 2 ) is an orthonormal frame
Dynamics of singular vorticity
• For a 2D surface § in R4, there is a second fundamental
form associated with each of the normal directions of the
moving frame (n1 ; n2 ; t 1 ; t 2 )
S 1 (V )(p) := ¡ hh(D n 1 (p) ¢V ); V i i ;
S 2 (V )(p) := ¡ hh(D n 2 (p) ¢V ); V i i ;
V 2 Tp § ;
• The mean curvature vector K(p)is then defined as
X2 1
K(p) =
(S i (t 1 ) + S i (t 2 )) (p)n i (p)
2
i= 1
Dynamics of singular vorticity
• And so, for the expression obtained using M&W formula
vSI ;r eg (p)
¡2
=
log(²)R ¡
¼
0
1
B @n 1
C
@n 1
@n 2
@n 2
B
¢t 1 +
¢t 2 ;
¢t 1 +
¢t 2 C
(p)
¼=2 ¢@
A
@s
@s
@s
@s
| 1
{z 2 } | 1
{z 2 }
n 1 com p onent
2¡ 2
= ¡
log(²)R ¡
¼
¼=2
¢K(p)
n 2 com p onent
Dynamics of singular vorticity
• Next, showing that
Z
2¼ · µ
1
2¼ 0
Z 2¼ · µ
1
2¼
0
¶¸
@n 2 (0)
= ¡ K(p) ¢n 2 (0)
@s1
¶¸
@n 1 (0)
t 1 (0; µ) ¢
dµ = ¡ K(p) ¢n 1 (0)
@s1
t 1 (0; µ) ¢
the expression obtained from the (generalized) Biot-Savart
law can also be written as
2¡ 2
vSI ;r eg (p) = ¡
log(²)R ¡
¼
¼=2
¢K(p)
The dynamics of the four-form ! ^ !
• In R2 and R3,
! ^ ! 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 Ertel’s theorem in R3
• Consider a divergence-free velocity field w in R3 and a
smooth function f : R3 ! R
• The vector field v := (w; f )is divergence-free in R4 . 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
• Euler’s equations for v (or v[ ) are Euler’s 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 R4, 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 Rn 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 Vizman—working 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 Hasimoto’s transformation be generalized for
membranes?
• If yes, what are the transformed PDE?
• Surfaces of singular ! ^ ! ?