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Equivalence of variational principles in vortex dynamics
Volume 153, number 4,5
PHYSICS LETTERSA
4 March 1991
Equivalence of variational principles in vortex dynamics S.K. Nemirovskii and A.Ja. Baltsevich Institute of Thermophysics, Siberian Branch of the Academy ofSciences ofthe USSR, prospect Lavrent’eva 1, Novosibirsk 630090, USSR Received 26 April 1990; revised manuscript received 11 December 1990; accepted for publication 12 December 1990 Communicated by D.D. HoIm
The equivalence of Lin’s variational principle and that of Rasetti and Regge is shown. The introduction of an additional term in Lin’s principle can be considered not as a condition of the conservation ofLagrange coordinate along the trajectory of a liquid particle but as a condition of proportionality of action to the area swept by the moving vortex line.
In the paper of Rasetti and Regge (RR) [l]the variational principle for the equations of motion of a vortex filament in an incompressible inviscid liquid has been described. The corresponding action functional is of the following form: fp,c dx~cjvk ~ A = dt e~x’~ dci— ~ v2 d3x)~
J
J
~
J
r
(1) where x’(a, t) is a parametrical vortex line equation, and K the circulation. The first term in the right hand side of (1) describes the contribution of the vortex line and is proportional to the area which is swept out by the moving one. This ideology arises in Nambu relativistic string theory [21. Rewriting the second term in the form ,c2p
ç ç
~ J i
~
Ix—x’ I
,
(2)
where there are two additional terms besides the kinetic energy one. The second one expresses the condition of continuity. The third one is necessary to include the vortex motion and has no sufficiently clear physical interpretation. The functions andfare are called Clebsch with variables. The velocity andy vorticity connected them by means of v=Va+ ~ o=Vxv= -1-(Vfxvy). (4) p p The purpose of this Letter is an attempt to show the equivalence of these (at first sight completely different) formulations of the variational principles. At first, we will show that Lin’s principle for the vorticity localized on a one-dimensional singularity (vortex filament) coincides with the RR one. Taking into account expression (4) for w one can write
andvarying (l)byx(a, t) givesthewell-knownBiotSavart law, On the other hand, the so-called Lin’s variational principle takes place for an ideal liquid dynamics (see, e.g. ref. [3J). The functional in this case is of the form
~
A=$dtJd3x[9_+aV.(pv)+~(~f+V.(fv))],
A=JdtJd3x(p~~_~-).
(3) Elsevier Science Publishers B.V. (North-Holland)
0(~), y=\/~7~ 9(x~), (5) where 0(x) is the Heaviside function, and x~and x~, local coordinates on the plane transverse to the vorticity. Integrating the last two terms of (3) by parts and using expression (4), the action (3) can be rewritten in the form (6)
Further, we will prove the equality 209
Volume 153, number 4,5
J
dtjd3
xy
PHYSICS LETTERS A
of
p3
=~JdtJd3x(_x.~VY+x.~Vf).
(7)
dt
tersected so that the vorticity is directed along the normal vector of the cross section at every point. Then one can cover the obtained surface with a grid of size h. A circulation of the obtained vortex lines is of the form
(10)
Ka=JUWOJöS~.
p
Integrating (overtime) the second term in the right hand side (6) by parts and recalling that the action variations vanish at the time-interval ends, we obtain
J J
4 March 1991
where ÔS~is an oriented element of the surface ( h2). Now we use expression (1), where K iS replaced by and then sum over all surface elements, i.e. ~PJeUkxi.___da. (11) da dt
d3x x.~Vf
,~a,
=_JdtJd3xx.yV~
a
=JdtJd3xx.~fVy.
r’~
If we choose a as an arclength and change
~-.f
~
(12)
(8)
p3
sr
The last equality in (8) was obtained by integrating by parts over the whole space—time. We proceed to the curvilinear coordinates x~,xi,, X~connected with the vortex line and use (7) in (6) for this. Integrating in the new basis yields (using (5))
f d3x \~otVf— ~‘Vy)
J
p3
=
—
J
~Jd3xx.(wXv).
(13)
p3 ‘
Now we write (4) instead of w and return to the
X
functional (2). Thus we have shown the equivalence ~
) ö(x~) I— ii, .
~(X~)
ö(x~)
‘I~~ I
Finally, one can obtain expression (1) using (6)— (9). Thus, we have shown the equivalence of the presented formulations, Now we regard the case of the distributive vorticity. One can start from the fact that in an ideal incompressible fluid the flow is defined by ~kt: ~-+x [41. Here we denote by ~the Lagrangian coordinates of a fluid particle, and by ~ the coordinate mapping (x(0, ~) = ~). Then: o (x, t) = Jw 0 One can see that the vorticity is carried out along particle paths but distorted by the Jacobian matrix (J) of the flow. The next step is a discretization of the vorticity. It means that a vortex tube existing in the fluid is in-
(c).
210
Remember the normal vector of the surface is directed along the vorticity at every point. Thus, we have
of Lin’s and Rasetti and Regge’s variational principles. In other additional taking with into account vortexwords, flows the in Lin’s one isterm connected the fact that the contribution of the vortex filament to the action is proportional to the area swept by the line. This interpretation of Clebsch’s variables differs from Lin’s one. He considered the condition Of! c9t+v’Vf=O as the conservation of a Lagrangian coordinate along the trajectory of a liquid particle. References El] M. Rasetti and T.
Regge, Physica A 80 (1975) 217.
[2] Y. Nambu, Lectures at the Copenhagen Summer Symposium, 1970. [3] R.L. Seliger and GB. Whitham, Proc. R. Soc. A 305 (1968) 1. [4] J.T. Beale and A. Majda, Contemp. Math. (collection of papers) Vol. 28, p. 221.
PHYSICS LETTERSA
4 March 1991
Equivalence of variational principles in vortex dynamics S.K. Nemirovskii and A.Ja. Baltsevich Institute of Thermophysics, Siberian Branch of the Academy ofSciences ofthe USSR, prospect Lavrent’eva 1, Novosibirsk 630090, USSR Received 26 April 1990; revised manuscript received 11 December 1990; accepted for publication 12 December 1990 Communicated by D.D. HoIm
The equivalence of Lin’s variational principle and that of Rasetti and Regge is shown. The introduction of an additional term in Lin’s principle can be considered not as a condition of the conservation ofLagrange coordinate along the trajectory of a liquid particle but as a condition of proportionality of action to the area swept by the moving vortex line.
In the paper of Rasetti and Regge (RR) [l]the variational principle for the equations of motion of a vortex filament in an incompressible inviscid liquid has been described. The corresponding action functional is of the following form: fp,c dx~cjvk ~ A = dt e~x’~ dci— ~ v2 d3x)~
J
J
~
J
r
(1) where x’(a, t) is a parametrical vortex line equation, and K the circulation. The first term in the right hand side of (1) describes the contribution of the vortex line and is proportional to the area which is swept out by the moving one. This ideology arises in Nambu relativistic string theory [21. Rewriting the second term in the form ,c2p
ç ç
~ J i
~
Ix—x’ I
,
(2)
where there are two additional terms besides the kinetic energy one. The second one expresses the condition of continuity. The third one is necessary to include the vortex motion and has no sufficiently clear physical interpretation. The functions andfare are called Clebsch with variables. The velocity andy vorticity connected them by means of v=Va+ ~ o=Vxv= -1-(Vfxvy). (4) p p The purpose of this Letter is an attempt to show the equivalence of these (at first sight completely different) formulations of the variational principles. At first, we will show that Lin’s principle for the vorticity localized on a one-dimensional singularity (vortex filament) coincides with the RR one. Taking into account expression (4) for w one can write
andvarying (l)byx(a, t) givesthewell-knownBiotSavart law, On the other hand, the so-called Lin’s variational principle takes place for an ideal liquid dynamics (see, e.g. ref. [3J). The functional in this case is of the form
~
A=$dtJd3x[9_+aV.(pv)+~(~f+V.(fv))],
A=JdtJd3x(p~~_~-).
(3) Elsevier Science Publishers B.V. (North-Holland)
0(~), y=\/~7~ 9(x~), (5) where 0(x) is the Heaviside function, and x~and x~, local coordinates on the plane transverse to the vorticity. Integrating the last two terms of (3) by parts and using expression (4), the action (3) can be rewritten in the form (6)
Further, we will prove the equality 209
Volume 153, number 4,5
J
dtjd3
xy
PHYSICS LETTERS A
of
p3
=~JdtJd3x(_x.~VY+x.~Vf).
(7)
dt
tersected so that the vorticity is directed along the normal vector of the cross section at every point. Then one can cover the obtained surface with a grid of size h. A circulation of the obtained vortex lines is of the form
(10)
Ka=JUWOJöS~.
p
Integrating (overtime) the second term in the right hand side (6) by parts and recalling that the action variations vanish at the time-interval ends, we obtain
J J
4 March 1991
where ÔS~is an oriented element of the surface ( h2). Now we use expression (1), where K iS replaced by and then sum over all surface elements, i.e. ~PJeUkxi.___da. (11) da dt
d3x x.~Vf
,~a,
=_JdtJd3xx.yV~
a
=JdtJd3xx.~fVy.
r’~
If we choose a as an arclength and change
~-.f
~
(12)
(8)
p3
sr
The last equality in (8) was obtained by integrating by parts over the whole space—time. We proceed to the curvilinear coordinates x~,xi,, X~connected with the vortex line and use (7) in (6) for this. Integrating in the new basis yields (using (5))
f d3x \~otVf— ~‘Vy)
J
p3
=
—
J
~Jd3xx.(wXv).
(13)
p3 ‘
Now we write (4) instead of w and return to the
X
functional (2). Thus we have shown the equivalence ~
) ö(x~) I— ii, .
~(X~)
ö(x~)
‘I~~ I
Finally, one can obtain expression (1) using (6)— (9). Thus, we have shown the equivalence of the presented formulations, Now we regard the case of the distributive vorticity. One can start from the fact that in an ideal incompressible fluid the flow is defined by ~kt: ~-+x [41. Here we denote by ~the Lagrangian coordinates of a fluid particle, and by ~ the coordinate mapping (x(0, ~) = ~). Then: o (x, t) = Jw 0 One can see that the vorticity is carried out along particle paths but distorted by the Jacobian matrix (J) of the flow. The next step is a discretization of the vorticity. It means that a vortex tube existing in the fluid is in-
(c).
210
Remember the normal vector of the surface is directed along the vorticity at every point. Thus, we have
of Lin’s and Rasetti and Regge’s variational principles. In other additional taking with into account vortexwords, flows the in Lin’s one isterm connected the fact that the contribution of the vortex filament to the action is proportional to the area swept by the line. This interpretation of Clebsch’s variables differs from Lin’s one. He considered the condition Of! c9t+v’Vf=O as the conservation of a Lagrangian coordinate along the trajectory of a liquid particle. References El] M. Rasetti and T.
Regge, Physica A 80 (1975) 217.
[2] Y. Nambu, Lectures at the Copenhagen Summer Symposium, 1970. [3] R.L. Seliger and GB. Whitham, Proc. R. Soc. A 305 (1968) 1. [4] J.T. Beale and A. Majda, Contemp. Math. (collection of papers) Vol. 28, p. 221.