A&A 399, 723-730 (2003)
F. Schmitz1 - B. Fleck2
1 - Institut für Theoretische Physik und Astrophysik der Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
2 - ESA Research and Scientific Support Department, NASA/GSFC, Mailcode 682.3, Greenbelt, MD 20771, USA
Received 4 November 2002 / Accepted 6 December 2002
New standard forms of the time-independent linear adiabatic wave equation of plane atmospheres are presented. The main objective is to obtain equations with invariants as simple as possible so that oscillation theorems can be applied effectively. By transformations of both the independent and the dependent variables, equations with simple invariants are formulated. We present a standard form of the wave equation the invariant of which depends only on the first derivative of the equilibrium density, as opposed to the common standard form the invariant of which depends also on second derivatives. Further, we discuss a procedure which replaces the wave equation by a system of two simple second order differential equations. In this case we try to draw conclusions on the general behavior of solutions by use of oscillation theorems. In addition, a re-formulation of the wave equation is presented, which eliminates terms with first derivatives of atmospheric quantities. The independent variable of the resulting equation depends not only on the geometrical height but also on the ratio . In this case, it is necessary to use a diagnostic diagram the axes of which are given by and instead of the common diagram. Therefore we discuss the meaning of the parameter for the representation of dispersion curves. Finally, for the VAL-atmosphere (Vernazza et al. 1981), regions of certainly nonoscillatory waves are considered.
Key words: Sun: atmosphere - Sun: oscillations - stars: atmospheres
The present paper is a continuation of a paper by Schmitz & Fleck (1998), called Paper I, which considered vertically propagating adiabatic waves in a plane atmosphere. There, common standard forms of the 1-dimensional wave equation are discussed critically and a wave equation with a very simple invariant is presented. We now have performed extensions of the strategy of Paper I to the three-dimensional propagation of linear adiabatic waves in plane atmospheres. Here, the problem of the reformulation of the time-independent wave equation is significantly more complex than in the one-dimensional case.
To solve the wave equation numerically, one takes a corresponding system of two first order equations. In general, one uses the two linearized hydrodynamical equations from which the wave equation is formed. If we take appropriate dependent variables (vertical displacement and Lagrangian pressure perturbation), then this system is uncomplicated, and can easily be integrated numerically.
To obtain solutions in closed form for simple analytical model atmospheres, the representation of the wave equation and the choice of the variables are crucial. In that case one tries to transform the wave equation to known second order equations, in particular equations of special functions (e.g. Schmitz & Steffens 1999).
To understand the behavior of waves without solving the wave equation explicitely or to interprete results of numerical calculations, one writes the wave equation in standard form and studies the behavior of the invariant. Besides, the standard form is also used for a numerical integration, as there are particular numerical integration methods for so-called Schoedinger-type equations. It is clear that a simple invariant is convenient. Further, a simple invariant is useful in connection with approximation methods of solutions as the WKB-approximation (e.g. Mathews & Walker 1970).
Recent investigations of resonances of standing waves in the VAL-atmosphere and modifications (cool chromosphere or shifted transition layer) of this atmosphere by Steffens et al. (1997) have yielded a wealth of features in the diagnostic diagram. Further, investigations of the influence of atmospheric layers (VAL-atmosphere and modifications) on the p-modes of the sun by Steffens (1998), and Steffens & Schmitz (2000) have led to complicated dispersion curves. The corresponding ridges in the diagnostic diagram show strong bending or indicate avoided crossing. For the VAL-atmosphere, the frequency range where such effects occur is mHz. The results are not yet interpreted and understood.
A frequently discussed, but not commonly accepted phenomenon is that of a chromospheric mode, presumably due to an acoustic cavity between the temperature minimum and the transition region (cf. e.g. Leibacher & Stein 1981). For a detailed discussion refer to Steffens et al. (1985), who report on signatures of a chromospheric mode at 6 mHz in their observations. However, Steffens & Schmitz (2000) demonstrated that the corresponding features can appear in a diagram even without a chromosphere, i.e. without a temperature minimum and a chromospheric temperature increase. It therefore does not necessarily require invoking a chromospheric mode to explain these features. Further, their results show that, in order to understand the diagnostic diagram of the Sun, it is useful to consider modifications and variations of the outer layers. It remains open, however, how to best explain certain features in diagnostic diagrams - by physical systems (e.g. cavities) or by a discussion of the equations. A nice example can be found in Aizenman et al. (1977). They have investigated the behavior of l=2-modes of stars "evolving away from the main sequence''. There, in a frequency-age diagram mode bending (bumping) similar to that in the solar diagram occurs. The authors have shown that the bending of the ridges can be explained by avoided crossing of the ridges of two single systems. These systems however, are fictitious systems defined by equations which are obtained by simplifying the full equations.
However that may be, the common standard form of the adiabatic wave equation is not suitable to explain and understand the features. Further, it is not clear which combinations of the wave number k and the frequency should be taken to understand the behavior of structures and ridges in the diagnostic diagram: the pairs ( ) or ( ) or ( ) etc.?
Paper I was closed by the presentation of a simple standard form of the equation of vertically propagating waves. In the case of the VAL-atmosphere, the numerical behavior of the invariant of this equation is very simple. The three-dimensional case is much more complicated and the optimal standard form of the equation is not unique. Thus, in the present paper we concentrate on the mathematical tools. A subsequent paper by co-workers will deal with features and ridges in diagnostic diagrams.
The present paper is organized as follows: in Sect. 2 we present the linearized hydrodynamic equations and the resulting wave equation. Section 3 deals with the common standard form of the wave equation. This form is critized in Sect. 4 where also oscillation theorems are given. In Sect. 5, we present and discuss some new standard forms of the wave equations. The equation given in Sect. 5.1 is obtained by introducing the logarithmic mass as the independent variable. In Sect. 5.2 we separate the wave equation into two second order differential equations and apply oscillation theorems to obtain general statements on the behavior of solutions. A standard form of the wave equation the invariant of which is free of first derivatives of atmospheric quantities is presented in Sect. 6. In Sect. 7 we discuss criteria for nonoscillatory waves with applications to the VAL-atmosphere.
Let z be the vertical, outwards directed geometrical coordinate and g the constant gravity. Let p(z) and be the pressure and the density of the equilibrium atmosphere, c(z) the adiabatic sound speed, the vertical component of the Lagrangian displacement and the Lagrangian pressure perturbation. The horizontal position and the time t are separated by , where is the real frequency, and the horizontal wave vector.
From the linearized hydrodynamic equations we obtain two first
order differential equations (cf. e.g. Schmitz & Fleck 1994)
From Eqs. (1) and (2) we obtain the
equation of the Lagrangian pressure perturbation :
For this form (see e.g. Deubner & Gough 1984),
the independent variable is the geometrical height z.
With the transformation
The standard form given here is the most common. Occasionally, other forms of the wave equation are used. Mihalas & Toomre (1981) used a standard form which was obtained by the wave equation of the vertical displacement .
The common standard form of the wave equation is not suitable, as it requires the calculation of the second derivative of the density, which is nearly impossible to do in the case of an empirical atmosphere.
The invariant used by Mihalas & Toomre (1981) contains the second derivative of the adiabatic sound speed (comparable to the second derivative of the density in Eq. (7)). The authors write: "however, it was found to be essential to smooth the model in order to obtain continuos derivatives of the sound speed and density scale height. Otherwise small scale fluctuations in the vertical wave number produce numerous partial reflections which are artifacts of the model and have no physical basis''. Numerical differentiation is usually a difficult operation, particularly in the case of empirical data. In Paper I we calculated the second derivative of the adiabatic sound speed of the VAL-atmosphere. There, small deviations from the VAL-data strongly influenced the second derivative.
Besides, the numerical behavior of the invariant is far too complicated to draw conclusions to the form of the solutions. In Paper I we have extensively discussed this problem for the vertical case k=0. The so-called invariant of a second order differential equation is not a physical invariant, i.e. a scalar field, but it changes when the variables are transformed. In Paper I we have compared some different representations of the wave equation. We have seen how the invariants undergo strong changes when the variables are transformed. It was shown that the behavior of waves can be described by a wave equation, the invariant of which contains only one coefficient, which is the quantity . In the VAL atmosphere, and there in particular in the chromosphere, the behavior of this quantity differs markedly from the behavior of the temperature or the adiabatic sound speed.
An uncomplicated and computationally simple invariant plays a role also in the following case: for a numerical integration, a differential equation of higher order is replaced by a system of first order equations. There is an exception to this rule. For the numerical integration of second order differential equations whithout the first derivative, there are particular methods. Mihalas & Toomre (1981) have used such a method to solve the wave equation.
Oscillation and nonoscillation theorems for second order equations are given in Gradshteyn & Ryzhik (1980). Only a few oscillation theorems for the solutions of second order differential equations are really useful. Most of such theorems concern infinite intervals. (Mathematically, solutions are said to be oscillatory if they possess an infinite number of zeros in the interval , nonoscillatory, if they possess only a finite number of zeros.)
For this paper, we need only the following two nonoscillating theorems for a differential equation y'' + f(x) y = 0:
1. If on an interval (a,b) or on , then y(x) has not more than one zero in this interval. In this paper, we call this case nonoscillatory.
2. There are strictly increasing and decreasing linearly independent positive solutions if f < 0 is continuous in and if x f(x) is not integrable on . This property corresponds to the existence of the solutions e-x and e+x of the equation y'' - y = 0 where only these solutions, apart of constant factors, are strictly monotonous and positive as opposed to the solutions or . As we shall consider atmospheric layers with finite extension the integrability of the function x f(x) is unimportant. For f < 0 we can always find a monotonic and positive solution.
When the invariant is negative, the solutions are nonoscillatory. This statement is sufficient, but not necessary. So, f(x)=0does not separate exponential and oscillatory behavior as it is often claimed. Only criteria for certainly nonoscillatory behavior can be given.
Using the column mass m given by
We define a new independent variable x by
We consider only two cases. First, we put
More important than gravity waves are acoustic waves. In this case, we put
Therefore, for a given atmospheric stratification,
only waves with frequencies
In the following we combine Eqs. (1) und (2) to a
wave equation the invariant of which does not contain the
derivative of the density. As the coefficients of these equations
are free of derivatives, such a procedure should be possible.
We found that only one way is practicable.
We may assume that
By the transformations of the displacement
and the pressure perturbation
First, we introduce a new dependent variable w and an new coordinate x by
In the case of the isothermal atmosphere we have:
Finally, for k=0 the wave equation reduces to Eq. (26), the
equation recommended in Paper I. The pressure perturbation is given by
|Figure 1: The adiabatic sound speed and the derivative of the density as functions of the height z.|
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Let us now consider the VAL-atmosphere (Vernazza et al. 1981). We take the temperature stratification T(z) and the pressure p(z)of this atmosphere. We calculate the density and the adiabatic sound speed c(z) including dissociation and ionization of hydrogen and ionization of helium: H , H, H+, He, He+, He++, e-. We have used the LTE-code described by Wolf (1983). The calculated densities coincide with the tabulated densities of the VAL-atmosphere. Only in the upper chromosphere (z > 1500 km) there are deviations, because of the NLTE-effects of the VAL-atmosphere. We therefore may assume that also the calculated adiabatic sound speed is correct. In the paper of Steffens & Schmitz (2000), the VAL- atmosphere was matched to the convection zone of Spruit (1977). There an equidistant data set was generated from the tabulated points of the convection zone and the VAL-atmosphere. This set, which was used to interpolate the coefficients of Eqs. (1) and (2) for the numerical integration, is not suitable to calculate the derivative of the density. We have therefore calculated the derivative of the density at the tabulated points of the VAL-atmosphere using a simple second order formula for non-equidistant points. Figure 1 shows the adiabatic sound speed and the derivative of the density as functions of the geometrical height. At the four lowest points of the VAL-atmosphere, the stratification is superadiabatic so that Eq. (70) yields complex frequencies. Also the HSRA-atmosphere (Gingerich et al. 1971) has this property, and even a density inversion. In the following we omit the first four points of the VAL-atmosphere.
For different wave numbers k, we now consider
frequencies with certainly nonoscillatory solutions.
Figure 2 shows boundary curves
defined by Eq. (70)
k= 0, 1, 2, 2.485, and 3.
In the range
2.485 < k < 7.6085, where the regions are not connected,
instead of the inequality (68) the inequalities (67)
can be used. Therefore, the figures show the atmospheric f-mode with
and the so called "Lamb-mode''
denoted by f and L.
In the limit
|Figure 2: The VAL-atmosphere: regions with certainly nonoscillatory waves. The solid curves limit the regions where Eq. (68) holds. The dotted curves are the f-mode and the "Lamb-mode'' and limit the regions given by Eqs. (67).|
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Studies of the propagation behavior of linear adiabatic waves in empirical or theoretical model atmospheres usually comprise two steps: the numerical integration of the wave equation and the interpretation of the results. The first order Eqs. (1) and (2) are well suited for numerical integration. They are simple and the coefficients can be calculated without complications. However, to understand the results and get a deeper insight into the underlying physical effects, it is necessary to use a standard form of the wave equation. For standard forms, oscillation theorems and comparison theorems are powerful tools.
The invariant of the common standard form of the 3-dimensional wave equation of plane atmospheres, given in Sect. 3 depends on second derivatives. This makes the wave equation too complicated for heuristic or interpretative studies. Even first derivatives of atmospheric quantities in the invariant can cause complications. It is for these reasons that we have sought suitable transformations that yield new standard forms of the wave equation with simple invariants.
The standard form (25) is still conventional. It offers the advantage that the independent variable, the logarithmic mass, is a very simple quantity. The invariant of this equation contains only the first derivative of the density, which makes this equation superior to previous standard forms of the wave equation.
In Sect. 5.2 we have adopted an unconventional approach and have replaced the wave equation by a non-linearly coupled system of two linear second order differential equations. The behavior of the invariants of these equations indicates an uncomplicated behavior of high frequency acoustic waves and of gravity waves. It is worth pointing out, that, for the latter, this includes the whole range of gravity waves.
In Sect. 6, by some unconventional transformations, we were able to find a wave equation the invariant of which does not even contain the first derivative of the density. The disadvantage of this wave equation, however, is that the independent variable is no longer a function only of the geometrical height, but also a function of the parameter . One thus has to discuss the invariant for a given . We have therefore given some arguments for such a procedure which are related to Duvall's law. Like the wave equation given in Sect. 5.1, this wave equation, too, offers the advantage that, in the isothermal case, it reduces directly to the familiar wave equation of the isothermal atmosphere.
The invariant of the wave equation presented in Sect. 5.1 defines regions with certainly nonoscillatory solutions. For selected values of the horizontal wave number k we have presented such regions in the plane.
A future paper will address specific features in the diagnostic diagram such as the p-mode ridges. We also hope that future studies in this area will clarify which of the two wave equations is the most effective.