A&A 427, 353-361 (2004)
A. Costa1 - R. González1,2 - A. C. Sicardi Schifino3
1 - Instituto de Astronomía y Física del Espacio, CONICET, 1428 Buenos Aires, Argentina
2 - Departamento de Física, FCEyN - UBA, Argentina
3 - Departamento de Física, Facultad de Ciencias, Facultad de Ingeniería, UR, Uruguay
Received 7 May 2004 / Accepted 15 July 2004
Using methods of non-equilibrium thermodynamics that extend and generalize the MHD energy principle of Bernstein et al. (1958, Proc. Roy. Soc. A, 244, 17) we develop a formalism in order to analyze the stability properties of prominence models considered as dissipative states i.e. states far form thermodynamic equilibrium. As an example, the criterion is applied to the Kippenhahn-Schlüter model (hereafter K-S) considering the addition of dissipative terms in the coupled system of equations: the balance of energy equation and the equation of motion. We show from this application, that periods corresponding to typical oscillations of the chromosphere and photosphere (3 and 5 min respectively), that were reported as observations of the prominence structure, can be explained as internal modes of the prominence itself. This is an alternative explanation to the one that supposes that the source of these perturbations are the cold foot chromospheric and photospheric basis.
Key words: Sun: prominences - instabilities - waves
Many stability criteria, based on energy principles, can be found in plasma physics literature, ranging from the classical criterion attributed to Bernstein et al. (1958) for ideal magnetohydrodynamics to more general and specific ones. Some astrophysical examples where variational principle are applied to spread a spectrum of different problems are: the modeling of pulsating stars (Costa et al. 2001; Ledoux & Walraven 1958), the calculations of stellar structures (Kennedy & Bludman 1997) or the characterization of the continuous Alfvén spectrum of line-tied coronal loops (Halberstadt & Goedbloed 1993). In relation to the subject of interest in this paper, many MHD prominence models - assuming both magnetohydrostatic and thermal equilibrium - have been proposed, e.g. Kippenhahn & Schluter (1957); Lerche & Low (1980); Oliver & Ballester (1996); Nagablushana (1998). However, stability is a crucial requirement for a model in order to give a realistic description of the problem. Thus, different stability analysis considering prominence models can be found in literature generally restricted to special type of perturbations and specific equilibrium models. A more general intent was performed by Zweibel (1982,1981) and also by Galindo Trejo (1989) who used the known Bernstein's MHD-variational principle in order to analyze the stability of different known two dimensional prominence models. They found that many of the situations described represented unstable equilibriums.
But, as was pointed out by Lerche & Low (1981) among all variational criteria there is an important and fundamental difference between them related to whether they assume adiabatical configurations or not. In the applications of Bernstein's criterion (Bernstein et al. 1958) the adiabaticity presupposes the irrelevancy of the energy balance equation and thus dissipation is impossible. A more realistic case, concerning stable configurations described by non-conservative equations with non-self-adjoint operators, was presented by Lerche & Low (1981). They proposed a Lagrangian principle in order to analyze quiescent prominences that can suffer thermal instabilities.
In this paper we provide a criterion in order to analyze the stability of prominences - considering dissipative terms - via the application of a general procedure derived recently by us (Sicardi et al. 2004; see also Sicardi et al. 1991,1989a,b, 1985). This procedure gives a general principle - based on firmly established thermodynamic laws - and can be understood as an extension of the Bernstein's MHD principle to situations far from thermodynamic equilibrium. It has then the advantage that many known results obtained by the simpler ideal MHD criterion (Bernstein et al. 1958) (e.g. Galindo Trejo 1989,1987) can be reexamined by a direct comparison with this analysis, and that, as it is obtained via a thermodynamic approach, it gives a maneuverable description of non-thermodynamic equilibrium situations.
The observational evidence of prominence seismology is reviewed in Oliver (1999): one can distinguish between the large amplitude oscillations due to a wavefront excited during a flare affecting the whole prominence, and small amplitude oscillations affecting only part of the prominence. Vial (1998) reviewed the typical periods observed in velocities which can be separated in three categories: short periods (less than 5 min), intermediate periods (between 6 and 20 min) and long periods (between 40 and 90 min) as seen recently by Régnier et al. (1999). Balthasar et al. (1999) had also established the existence of oscillations of very short period (30 s).
One of us (Blanco et al. 1999; Bocchialini et al. 2001) reported observational studies that exhibit correlation with the classification of oscillations cited above. We found that large energy content is localized in waves with periods between one and six minutes, which can be classified as intermediate and/or very short periods. These periods correspond to typical oscillations of the chromosphere and photosphere (3 and 5 min respectively) suggesting a possible source for these perturbations, which, outside from the prominence coronal observations, are known to be effectively stopped by the transition region (Gouttebroze et al. 2001).
Whether these range of oscillations represent intrinsic properties of the prominence (normal modes), or if they are a forced stable response to an external wavefront perturbation, is a fact that must be explained by theoretical prominence models that take into account thermal and mechanical stability considerations in order to guarantee that such models can exist. Thus, a crucial question of any theoretical model is one that inquires about the frequency stability range. Then, as the consideration of one or another model must be related to this analysis, we applied thermodynamic irreversible techniques in order to analyze both thermal and mechanical coupled stability. Our principle is applied to the known Kippenhahn-Schluter (hereafter K-S) model. It gives an example of the physical insight that the criterion can provide due to the advantages of the irreversible thermodynamic analysis. This is, - while more realistic cases require numerical treatment, a task that will be accomplish in next steps - this analytic example shows that the role of the different operators, that compose the equations describing the system, can be established in terms of their stability significance. Moreover, this results in the construction of a variational principle that associate stability with the sign of a quadratic form avoiding non-self-adjoint operators. In fact, obtaining a self-adjoint operator is a requirement for our principle to hold. Nevertheless, when this is accomplished the calculus result simplified due to an important mathematical property which greatly aids in the stability analysis. The self-adjoint character of an operator implies that the eigenvalues are purely real. Hence stability transitions always occur when crosses zero, rather than at some particular points of the real axis where the real part of the eigenvalue is different from zero, i.e. Re , leading to an efficient formulation for testing stability. Thus, the symmetry considerations of the self-adjoint operators, the fact that there is a diagonal form associated to this operators, and that the Rayleigh-Ritz theorem states the existence of a minimum eigenvalue, are important reasons to try to maintain self-adjointness in the consideration of stability.
The thermodynamics of irreversible processes is described in terms of phenomenological relations between conjugate pairs of thermodynamic variables: the flows and the forces that cause them. It is a subject where a large amount of work has been done on both formal theory and applications. The first works, treating small deviations from the equilibrium state and including fluctuations in the neighborhood of this state, were performed by Onsager & Machlup (1953) and by Prigogine (1967). The criterion based on these first formulations was successfully applied to several linear non-equilibrium problems (Glansdorff et al. 1974; Nicolis & Prigogine 1979).
Linear thermodynamics studies the behavior of the system around the equilibrium state or around a non-equilibrium stationary one that is linearly close to it. They are states of detail balance between flows and forces and, as Onsager showed (1931), this is due to the microscopic reversible character of equilibrium. Thus, the empirical relations between flows and forces are linear and the resistance matrix that relates them is symmetric and positive definite. It's symmetric character is guaranteed by the principle of microscopic reversibility and its positive definiteness by the proximity of the reference state to the thermodynamic equilibrium, where the entropy has a maximum.
But, there is no continuity between linear and nonlinear thermodynamical processes. When the system is beyond the immediate neighborhood of the stationary state the non-linearities become visible. Instabilities that cause dynamic transitions in open systems are responsible for the qualitative difference between linear and nonlinear thermodynamics. Therefore, dynamic cooperative phenomena can only arise in nonlinear thermodynamics. Thus, nonlinear thermodynamics is related to the stability properties of non-equilibrium stationary states, where the linear relation between flows and forces can become state dependent (i.e. is not necessarily a symmetric positive definite matrix), and the problem of having a thermodynamic theory to provide a general criterion for the stability of the system - which is not evident through the integration of the variational equations- becomes a fundamental point. The theory was then extended to situations far from thermodynamic equilibrium (Glansdorff & Prigogine 1971; Keizer 1976; Graham 1978; Lavenda 1993,1987), where the relaxation of these processes to a steady state of non-equilibrium (nonlinear state) is described.
As it is expressed by the second law of thermodynamics,
in an isolated system the entropy S growth exponentially up to
its maximum value
If the system is isolated, then and Eq. (2) reduces to Eq. (1). Thus, even when is never negative the term has not a definite sign and a less than zero entropy state can be reached as a result of the evolution (i.e. final entropy less than its value at the thermodynamic equilibrium state). Moreover, the system can remain indefinitely in one of these states if it happens that dS=0, or equivalently, that (note that must be 0). These states are known as stationary states and a coherent dynamic of the system could last if sufficient negative entropy flow is provided to it. Thus, the criterion that states the stability of this stationary state gives insight of the dynamic structures that can be found in nature.
The stability of the stationary state is determined by the thermodynamic properties of the system. In a linearized description of the system around the stationary state (a nonlinear state) the first order of Eq. (2) is reduced to the balance of the forces of the system and its second order determines the stability. The physical meaning of this statement is immediate: in a thermodynamic system the time is a parameter as in the Newtonian dynamics, then the positive sign of the time reveals the direction of the evolution. The internal production of entropy associated to the dissipation of the system is a definite positive second order quantity that manifests the irreversibility of the processes. In other words, the spontaneous evolution of the system implies and when a perturbation not satisfying this condition occurs the systems returns to the reference state. The criterion, by Onsager, that the principle of least dissipation of energy must govern the probability of a succession of non-equilibrium states towards a stationary one, results in the requirement that every perturbation that can arise must satisfy .
The mathematical expression of the
stability condition can be formulated in the following form: the
stationary solution for a physical system described by generalized
- the evolution of
which is governed by a system of known differential equations -
will be stable if a function ,
called the "Lyapunov
function'' - defined in a neighborhood of the stationary point
q1o,q2o,...,qno) in the configuration space of
the system - can be found satisfying the following conditions
Then, according to
Lyapunov's theorem a sufficient condition for the stability of a
steady state is that
Taking into account that a system is completely
described by a set of N extensive thermodynamic variables
xi=qi-qi0, and that the tendency of the system to
seek equilibrium is measured by the thermodynamic forces
(where S(x) is the entropy
of the non-equilibrium state) we find that, around the reference
state, the excess entropy (the second order entropy variation
can be written as the quadratic form
phenomenological relations between flows and forces are expressed
In the following we indicate the procedure to obtain an
energy principle from the basic equations that describe the system
of interest. A general mechanical and thermal description - the
equation of motion coupled with the equation of energy balance - of
an open system can be written as
The principle of power balance implies
that a sufficient condition for the Lyapunov function to exist is
are hermitian. Then, as we
show in what follows, (Sicardi et al. 1991,2004; Costa et
al. 2001) the general stability criterion has the form
In summary, if
a sufficient condition for stability
is that relation (14) holds. Then, Eq. (14):
In order to board the stability analysis of the
prominence problem the equations governing the dynamics of the
system must be expressed in the form shown in last section. In the
general case these equations can be written as a system of two
coupled equations: the balance energy equation and the equation of
motion (as shown in Eq. (12)). Thus, the perturbation analysis
around a stationary state will imply considering a variable state
vector of four independent components (i.e. the three component
displacement and the temperature variation). In this section we
deduce the stability principle (Eq. (14)) for the general
prominence case. In the Appendix a simpler academic example to
illustrate the procedure avoiding mathematical complications is
The fundamental magnetohydrodynamic equations
to be considered are as follows. The mass conservation equation,
The linearization procedure is performed by replacing
equations, and assuming hydrostatic conditions for the equation of
the perturbation around the equilibrium of the equation of motion
(the stationary state), also
Eqs. (16) and (18) the corresponding linearized
equations (Eqs. (23)-(27)) result:
The K-S model (Galindo Trejo 1989; Priest
1982), can be viewed as a current
sheet concentrated in the region
exposed to the magnetic field
For the ideal MHD study, and in accordance with Anzer
(1969), Galindo Trejo (1989) and Galindo Trejo &
Schindler (1984), the model results stable due to
Bernstein's MHD criterion. Also, Zweibel (1982), using
this energy principle finds a sufficient condition for the
stability of the K-S prominence model that represents an extension
to more realistic situation i.e. without making an approximation
of infinitesimal thickness of the prominence. Galindo Trejo
(1989) postulates a displacement of the form
The displacements must
satisfy boundary conditions that for the case considered here
assume the form
Then, in function
of the perturbations ((41) and (42)) the extended
Bernstein principle that predicts stability acquires the form
Here we proposed a self-adjoint stability criterion that takes into account the coupled thermal and mechanic balance conditions. This variational principle is a natural extension of the Bernstein's stability criterion for ideal MHD to more general states i.e. state far from thermodynamic equilibrium. The procedure to obtain the principle followed thermodynamic irreversible arguments and techniques.
As an example we applied
the method to the known K-S model in its extension to dissipative
situations, and in the scenario described by usual prominences
parameters. The stability criterion can be rewritten in the form
Moreover, considering both thermal and mechanical
perturbations, is easy to note from last equation, that when the
mechanical displacement vanishes (
for any thermal perturbation
T1. Note that also, for any displacement of the form
As an example, we applied the variational principle to the more general mode presented in this work (Eqs. (41) and (42)) to obtain the associated frequencies. In order to accomplish this task we followed the schematic procedure described by Galindo Trejo (1987) but we used a symbolic manipulation program to integrate the equations. and the perturbation were expanded in a two dimensional-Fourier basin that adjust to border conditions. Thus, a quadratic form for was obtained and was minimized with the Ritz variational procedure. Finally, a matrix discrete eigenvalue problem subject to a normalization constraint was obtained. The procedure is equivalent to solve Eq. (34) for Galindo Trejo's formulation and Eq. (37) for our modified principle.
Table 1: Periods and wavenumbers: comparison between the results of the K-S model considering dissipation with Galindo Trejo's calculus.
As the aim of our analysis was to find stable perturbation we were interested in the smallest eigenvalue that can be obtained by these procedure. Doing this, for the range of parameters and the displacement chosen, we obtained that stable modes with frequencies of about three and five minutes can be described as normal modes of the K-S model when dissipative terms are considered. The results are displayed in Table 1. The parameter range was chosen to be the same as in Galindo Trejo (1987). From the table it can be seen that the periods obtained range between 2 m and 6.55 m which is in agreement with reported observational data (e.g. in Blanco et al. 1999; Bocchialini et al. 2001). Moreover, comparing our results with Galindo Trejo's ones, it can be mentioned that the dissipative analysis exhibits deeper dependencies that in his results. We found a more markedly dependence of the period with the Lz dimension and also we found a more markedly dependence of the period with the wavenumber. The period augments with Lz increasing. Note that the Lz value more nearer to the 3 m period is Lz=8.88while the Lz value more nearer to the 5 m period is Lz=29.6. Also, k=0.13 represent the nearest wavenumber associated to the periods we are interested in (3 m and 5 m).
The example presented here is an illustrative one. It shows how the principle - the extension of Bernstein's principle to situations far from thermodynamic equilibrium - can be used in its application to the K-S model. It allowed us to obtain relevant information of the K-S prominence model when considered as a dissipative structure i.e., the periods associated to normal modes. Moreover, firm conclusions about type and structure of modes and the dependence between structures, wavenumber and periods must wait for a numerical analysis that explores different scales and prominence models. This point needs a complete eigenmode and eigenvalue analysis and will be accomplished in a next work.
In this Appendix we present the one-dimensional simple
text-book example of the Rayleigh-Bénard convection in order to
illustrate the procedure of the criterion. Bénard (1900) performed
experimental studies to consider the stability of a nearly
incompressible liquid and Rayleigh (1916) found the analytic
condition for stability. Convection is the response of a system
submitted to an unstable stratification of the density where the
destabilizing force is the differential buoyancy suffered by a
particle subject to a temperature fluctuation.
incompressibility means that the density does not change on the
application of pressure. However, it is necessary to take into
account that the density decreases raising the temperature
allowing the action of the differential buoyancy. With these
considerations the liquid can become unstable to convection
because the hotter lighter liquid at the bottom pushes to come on
top of the colder liquid above. A simplified approach in terms of
a temperature fluctuation
coupled to the vertical motion
vz is sufficient to account for the main features of the
instability mechanism. Moreover, since only the differential
buoyancy between fluid particles at the same altitude is involved,
a model including only the horizontal dependence is sufficient to
describe the required condition. The balance of energy equation
that describes the problem is (Manneville 1987):
Linearizing the energy equation, replacing the
expression of T0(z) and discarding second order terms we