Symmetry Extensions In Kinetic And Hydrodynamic Plasma Models
V. B. Taranov
Institute for Nuclear Research, 47 Nauky Ave. Kyiv 03028 Ukraine,
Abstract. Symmetry transformations are found which allow us to reduce the number of equations for the collision less
plasmas containing components with equal charge to mass ratio of particles. Self similar solutions are found for Hasegawa-
Mima model and its 3D generalizations. Such solutions exist as a consequence of the symmetry extension in the vicinity of the
critical points of the drift-ion acoustic waves. Different symmetry exploration methods for the integro-differential equations
of the kinetic (Vlasov-Maxwell) theory are compared.
Keywords: plasma, kinetics, hydrodynamics, symmetry, exact solutions
PACS: 52.65.Ff, 52.65.Kj
In recent decades Lie group analysis has been applied to explore many physically interesting nonlinear problems in
gas dynamics, plasma physics etc. . Furthermore, different extensions of the classical Lie algorithm to the integrodifferential
systems of equations of the kinetic theory were proposed [2, 3, 4]. In particular, this can be done by using
an indirect algorithm [2, 3, 5] which allows us to obtain symmetries of the kinetic equations from the symmetries of
an infinite set of partial differential equations for the moments of distribution functions.
By this indirect algorithm, it was shown in  that the integro-differential equations of the kinetic theory of collision
less plasma containing particles with equal charge to mass ratio have additional symmetries. For example, alpha
particles and deuterium ions participating in a thermonuclear reaction D+ +T+ !He++ +n+17:6MeV have close
charge to mass ratios.
In the Section 2, integro differential kinetic model for the collision less plasma is presented.
In the Section 3, extended symmetry transformations are considered for the plasma containing alpha particles and
deuterium ions. It is shown that these transformations allow us to reduce the number of equations.
In the Section 4, symmetry extension is considered for the hydrodynamical Hasegawa-Mima model in the vicinity
of the critical points of the potential of drift ion acoustic waves.
Conclusions are made in the Section 5.
2. COLLISION LESS PLASMA MODEL
Let us consider N-component collision less plasma. In this case Vlasov – Maxwell integro-differential system of
= 0; (1)
= 0; Ñ¢E = 4pr; Ñ£B =
j; Ñ¢B = 0; (2)
where fa (t;r;v) is the distribution function of the a ¡th component of the plasma, a = 1; :::;N. Charge and mass
of particles of the a ¡th component are denoted by ea and ma , respectively. Charge and current densities have the
fa dv =
eaM0;a ; j =
v fa dv =
eaM1;a ; (3)
where M0;a and M1;a are the moments of the distribution functions fa . These moments are very important, since
they enter into Maxwell equations explicitly.
Higher order moments are determined in a similar way:
z fa dv;
where k = (k1;k2;k3), k1 to k3 are equal to 0, 1, 2, … .
Additional global physical conditions must be fulfilled:
a) distribution functions must be non-negative;
b) their moments, at least M0;a and M1;a , must exist.
3. EXTENDED SYMMETRIES OF THE MODEL
Let a =1 and a =2 correspond to plasma components with equal charge to mass ratio of particles, for example, alpha
particles and deuterium ion components, so we assume (and it is a good approximation:
According to (3), these components enter the Maxwell equations only as the sum
e1 f1+e2 f2; (5)
which is in fact the distribution function of the charge density of the components a = 1 and a = 2. In addition,
according to (1), (4), the functions f1, f2 and e1 f1+e2 f2 satisfy the same Vlasov equation (1). So the transformation
0 = f1¡e2F( f1; f2); f2
0 = f2+e1F( f1; f2); (6)
where F( f1; f2) is an arbitrary function of its arguments, leaves the Vlasov-Maxwell equations invariant, at least if
we perform in (3) the summation first and the integration later.
The transformation (6) does not preserve, in general, the positiveness of the distribution functions. Moreover, it can
lead to the divergence of the moments, which can be shown, for example, by choosing F = const. Another difficulty
in treating (6) as a symmetry originate from the fact that no explicit transform can be obtained, in general, from (6) for
the moments Mk;a .
It was shown in  that if there are plasma components with equal charge to mass ratio of particles,
for some m and n , kinetic model (1), (2) admits additional symmetries with infinitesimal operators
Xm;n = fm (
In the case considered below (4), the equations (1), (2) admit the following additional symmetries:
X1;2 = f1(
); X2;1 = f2(
Finite transformations generated by X1;2 , X2;1 are:
0 = f1exp(a12); f2
0 = f2+
(1¡exp(a12)) f1; (10)
0 = f2exp(a21); f1
0 = f1+
(1¡exp(a21)) f2; (11)
respectively, a12 and a21 are arbitrary constants.
It is clear that if the moments Mk;a exist for the functions f1, f2 , then the same holds for the transformed functions
0. On the other hand, for positive definite f1, f2 transformed functions f1
0 can be negative for some values
of the parameters a12 and a21.
Symmetries (10), (11) are particular examples of the general symmetry (6) determined by the special choice of the
function F, namely,
F = (1¡exp(a12))
; F = ¡(1¡exp(a21))
for (10) and (11), respectively.
It is important that in the limit a12 !¡¥ we obtain from (10) the transformation
0 = 0; f2
0 = f2+
The transformation (13) is degenerate since it does not determine f1, f2 from the transformed functions f1
Nevertheless, it allows us to reduce the number of equations (1), (2) by one.
According to (13), we can choose the invariant (5) as a new distribution function and solve the system (1), (2) for
e1 f1+e2 f2; f3; ::: ; fN (14)
and the fields E and B, so the number of equations is reduced by one.
4. HASEGAWA-MIMA MODEL
Let us consider an inhomogeneous plasma slab in the external homogeneous magnetic field. Electrons, unlike ions,
are magnetized, smoothing an electrostatic potential F along the magnetic field lines. In this case, Hasegawa–Mima
model equations hold :
; Y = F¡D?F; (15)
where Y ´ Yz is the generalized vorticity, J(F;G) ´ ¶ F
¶ y ¡ ¶G
¶ y and D? ´ ¶ 2
¶ x2 + ¶ 2
¶ y2 :
The simultaneous presence of the F and D?F in the second equation of (15) leads to the symmetry reduction and, as
a consequence, to the absence of self-similar solutions . Nevertheless, self-similar solutions can exist as asymptotic
ones. To prove this, let us first neglect and the term D?F in the system (15). In this way we obtain more symmetric
equations, which admit the following similarity transforms:
X1 = x
¶Y; X2 = t
Self-similar solutions are possible since the additional symmetry X1 allows us to try solutions in the form:
F = xF(t;y); Y = xG(t;y); (17)
Returning to the initial system of equations (15) we see that it can have solutions of the form (17) since the additional
¶ x2 = 0 (18)
is fulfilled on such solutions and D?F is reduced to ¶ 2F
¶ y2 , which leads to the symmetry extension.
For the F(t,y) and G(t,y) we obtain the equations
= 0; G = F ¡
¶ y2 ; (19)
Equations (19) have a solution:
F =a(1+b cos((w +dw)t +qy); (20)
where a is arbitrary constant amplitude, constant factor b determines the relative weight of the zonal flow and the
monochromatic wave, the frequency is w = q=(1+q2) and the frequency shift is dw = ¡aq3=(1+q2).
In the general case of the coupled drift and ion acoustic waves, self-similar solutions of the form (17) also can be
Symmetry extension for the plasma containing the components with closed values of the charge to mass ratio of
particles was considered. As an example the plasma containing alpha particles and deuterium ions was chosen.
By simple vector considerations, we see that the transformation (6) does not change the integro-differential kinetic
equations (1), (2) if we perform the summation first and the integration later. So this transform, determined by the
arbitrary function F( f1; f2), is a symmetry of the integro-differential kinetic model. On the other hand, since this
transform does not satisfy, in general, global conditions of (a) positiveness of distribution functions and (b) existence
of their moments, it is only a ’formal’ symmetry. For example, in the simplest case of this transformation, F = const,
moments M0;a diverge.
If we explore the symmetry of (1), (2) by the indirect algorithm based on the use of the infinite set of moment
equations [2, 3, 5], all transformations producing divergent moments are automatically excluded. Symmetries (10),
(11) produce simple linear transformations of all the moments, they cannot lead to the divergence of the moments.
So the condition (b) is satisfied automatically. On the other hand, these symmetries can violate the condition (a) of
perturbation functions non negativeness for some values of the constants a12 and a21. As a consequence, group orbits
can contain simultaneously physical (non-negative) and non-physical solutions. Symmetries (10), (11) are particular
examples of the general symmetry (6) determined by the special choice of the function F.
In the case of hydrodynamical Hasegawa-Mima model the symmetry extension is connected to the additional
condition (18) and leads, for example, to the existence of the self similar solutions (17) describing asymptotic behavior
of plasma waves in the vicinity of zeroes of the potential F. Namely, according to (20) we obtain:
F =ax(1+b cos((w +dw)t +qy); (21)
which is an exact solution of the full Hasegawa-Mima model. Other self similar solutions are possible for this model
and its 3D generalizations .
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