Tests of a numerical algorithm for the linear instability study of flows on a sphere

 

Ismael Pérez-García and Yuri N. Skiba

Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito Exterior, CU, CP. 04510, México, D.F., México

 

(Manuscript received July 4, 2000; accepted in final form Oct. 9, 2000)

 

RESUMEN

Ha sido desarrollado un algoritmo numérico para estudiar la inestabilidad lineal de un flujo estacionario no divergente en una esfera en rotación. La precisión del algoritmo se prueba con soluciones zonales de la ecuación no lineal de vorticidad barotrópica (polinomios de Legendre, ondas zonales Rossby-Harwitz y modones monopolares).

 

ABSTRACT

A numerical algorithm for the normal mode instability study of a steady nondivergent flow on a rotating sphere is developed. The algorithm accuracy is tested with zonal solutions of the nonlinear barotropic vorticity equation (Legendre polynomials, zonal Rossby-Harwitz waves and monopole modons).

 

1. Introduction

The instability caused by the existence of a sufficiently large horizontal shear of the basic flow is known as barotropic instability (Pedlosky, 1979). The barotropic instability of atmospheric flows has been studied for half a century (Kuo, 1949; Lorenz, 1972; Simmons et al, 1983; Haarsma and Opsteegh, 1988; etc.). Lorenz (1972) noted that this type of instability accounted for the existence of a limit in the atmospheric predictability. On the other hand, Simmons et al. (1983) showed that barotropic instability can be responsible for a low frequency variability of the barotropic atmosphere. Usually a flow is said to be stable if any its perturbation remains bounded with time in some norm. The two most commonly used types of the flow stability are: the linear (exponential or algebraic) stability when the flow is subjected to infinitesimal perturbations (Demidovich, 1967) and the nonlinear stability when the flow is subjected to small but finite perturbations (Liapunov, 1950, 1966; Arnold, 1965).

Many papers have been devoted to the linear instability of the zonal flows since the famous work by Rayleigh (1880) (see, for example, Pedlosky, 1979; Drazin and Reid, 1981; Green, 1995). However, there are a lot of important questions that remain to be answered. The application of numerical methods permits further insight into this problem. However, the basic challenge is the accuracy of the numerical stability results (Skiba, 1998). Therefore, testing of numerical algorithms used in the stability study is of paramount importance. In the present work we describe a numerical algorithm developed for the normal mode (exponential) instability of arbitrary steady flow in an ideal nondivergent fluid on a rotating sphere. In the first stage of the test we analyze only zonal solutions (Legendre polynomial flows, zonal Rossby-Haurwitz waves and monopole modons) of the equation

It is the nonlinear barotropic vorticity equation (BVE) written here in a non-dimensional form in the spherical coordinates (λ, μ) where λ is the longitude, μ = sin Φ, Φ is the latitude, Ψ is the streamfunction, Δ is the Laplacian, and J is the Jacobian (Silberman, 1954; Adem, 1956; Platzman, 1962). The stability of a steady solution to problem (1) is of considerable hydrodynamic and geophysical interest. It should be noted that only the exponential (normal mode) stability can be analyzed numerically, whilst computer calculations are unusable in the algebraic stability study. Indeed, every nondefective matrix can be diagonalized by a similarity transformation. Therefore, the algebraic growth of perturbations occurs just in the case when the stability matrix is defective, that is, when its Jordan canonical form has at least one Jordan block of dimension k x k where k > 1 (Stewart and Sun, 1990; Demidovich, 1967). Since the Jordan canonical form is too sensitive, and can be destroyed even by an infinitesimal perturbation of the elements of the stability matrix (Wilkinson, 1965), the algebraic instability cannot be analyzed numerically. The numerical stability study algorithm has been tested here using a few theoretical normal mode stability results obtained for exact solutions to the BVE (1). In particular, we use the Rayleigh-Kuo necessary condition for the instability (Rayleigh, 1880; Kuo, 1949) which has been almost the only useful instability condition known for zonal solutions to the BVE for half a century. Notice that the utility of the Rayleigh-Kuo condition is limited, since each Legendre polynomial (LP) flow (being a basic function for the zonal flows) satisfies this condition if n ≥ 3 and amplitude a is sufficiently large (Baines, 1976; Skiba, 2000). We do not use the Tung (1981) instability condition, since its application in practice is not easier than that of the famous Liapunov (1966) theorems; instead, we apply the necessary normal mode instability conditions recently obtained for the LP flows, steady Rossby-Haurwitz waves and modons in Skiba (2000) and Skiba and Strelkov (2000a,b). These conditions are related to the spectral structure of unstable normal modes and state that Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a special number. For a LP flow , the new instability condition complements the Rayleigh-Kuo condition in the sense that while the latter deals with the basic flow structure; the former characterizes the structure of a growing perturbation. Also, we have checked the stability of the LP flows with n = 1 and n = 2 (Baines, 1976) and taken into account that zonal wavenumber m of unstable modes of each LP flow of degree n must satisfy the condition 0 < |m| < n (Skiba, 1989; Skiba and Adem, 1998). All these assertions are given in the next section (see Cases 1-5).

 

2. Normal mode stability method

We now describe the normal mode method for the study of the exponential instability of a two-dimensional steady flow on a rotating sphere (for more details about the normal mode method on a sphere, see Hoskins (1973), Baines (1976), Simmons et al. (1983), Dymnikov and Skiba (1987), Skiba (1989), Verkley (1990), Skiba and Adem (1998)). Let be the streamfunction of a solution to equation (1) on the unit sphere S. The evolution of an infinitesimal perturbation to is described by

where is the perturbation vorticity,

is the linear operator defined on sufficiently smooth complex-valued functions, and is the absolute vorticity of the basic flow. The inner product and the norm of functions on S are defined as

and

respectively, where g^* is the complex conjugate of g. We will use the spectral method (Silberman, 1954; Machenauer, 1979) when both the solution and the perturbation Ψ are expanded on S in an infinite series of the orthonormal spherical harmonics where is the associated Legendre function of degree n and zonal wavenumber m. We denote by a subspace of spherical polynomials of the degree N generated by N(N + 2) harmonics with –n ≤ m ≤ n and n = 1,2, ...,N. In order to construct the matrix representing the operator of problem (2)-(3) in the subspace , we assume that belongs to , and and belong to , that is,

For the sake of simplicity, hereinafter we use the following generally accepted notations: , and

(Platzman, 1962; Merilees, 1968; Hoskins, 1973; Machenauer, 1979; Skiba, 1989; Pérez-García and Skiba, 1999). Substituting (4) in (2), (3) and taking the inner product of the equation obtained with a spherical harmonic Y_α leads to

where

is the (α, γ)-element of the matrix L representing operator in the subspace . Substituting (3) in (6) gives

Thus, in subspace , problem (2) is reduced to

The matrix element (7) can be written as

(Skiba, 1989) where is the pure imaginary diagonal element of L, since is the product of two Kronecker deltas, and

is the triad nonlinear interaction coefficient, calculated with Gaussian quadrature formula (Machenauer, 1979). Looking for infinitesimal perturbation (4) in the form of a normal mode

leads to the spectral problem

for the operator whereω = ω_r + 𝑖ω_𝑖 is the eigenvalue, and is the corresponding eigenfunction. The streamfunction of a normal mode (10) can be written as

where

is the amplitude, and is the initial phase of the mode. Thus, a mode (10) is unstable if ω_r ＞ 0, decaying if ω_r ＜ 0, neutral if ω_r = 0, and stationary if ω_𝑖 = 0. The e-folding time and period T of the mode are determined (in days) as ,respectively.

The differential spectral problem is reduced to

in the subspace where is the eigenvector whose components are Fourier coefficients V_γ of the eigenfunction V with 𝑛_γ ≤ N.

 

3. Instability conditions

We will test the numerical normal mode stability algorithm with three types of zonal flows being exact BVE solutions: LP flows, zonal Rossby-Haurwitz waves and monopole modons.

 

a) Legendre-polynomial (LP) flow:

This zonal flow is described by the single Legendre polynomial of degree 𝑛 ≥ 1 and amplitude α.

 

b) Zonal Rossby-Haurwitz (RH) wave:

Here

𝑛 > 1, and α ≠ 0 is arbitrary real amplitude. It is a particular (zonal) case of the stationary RH wave

being an exact solution to the BVE (1) for arbitrary coefficients .

 

c) Monopole modon with uniform absolute vorticity in the inner region :

(Verkley, 1990). This zonal flow is defined in different ways in two regions of the sphere S: the inner modon region and outer modon region . Here ; B_o, B_𝑖, D₀ and D_𝑖 are constants; μ = μ_α is the boundary between the regions S_𝑖 and S₀; is the colatitude; .

Note that for a zonal flow, the structure of mode (11) is

In trials of the numerical normal mode stability study algorithm we have used the following theoretical results:

Case 1. (Kuo, 1949). Let be a zonal flow on S. Then a normal mode (18) may be unstable only if the derivative of the absolute vorticity of flow changes its sign at least in one point of the interval (-1,1).

For LP flow (13), . Thus, there is a critical amplitude α for developing the instability, due to the sphere rotation.

Case 2. (Baines, 1976). Any mode (18) of LP flow (13) of the degree 𝑛 = 1 or 𝑛 = 2 is linearly stable.

Case 3. (Skiba and Adem, 1998). Let 𝑛 ≥ 3. Any mode (18) of LP flow (13) may be unstable only if zonal wavenumber m of the mode satisfies the condition 0 < |m| < 𝑛.

Case 4. (Skiba, 2000). Let be a LP flow (13) or RH wave (16). Then the amplitude Ψ of each unstable or decaying mode (10) must satisfy the condition

where is the square of Fjörtoft's (1953) average spectral number of the mode amplitude Ψ and and are the total kinetic energy and enstrophy of Ψ.

Case 5. (Skiba and Strelkov, 2000a,b). Let be a monopole or dipole modon by Verkley (1990) with uniform absolute vorticity in the inner region S_i. Then the amplitude Ψ of each unstable or decaying mode (10) must satisfy the condition

where σ is the degree of the modon in the outer region. Moreover, the vorticity of each unstable mode must be zero in S_𝑖.

Also we have used the equation

that describes the evolution of the total energy of an infinitesimal perturbation to a zonal flow on the sphere S where

are the velocity components of the basic flow and the perturbation, respectively. This equation is useful in the analysis of the spatial structure of local unstable perturbations on the sphere. Usually, important localized perturbations are in the regions related with distinguishing features of the basic flow. Unlike the perturbation energy equation on the β-plane (Pedlosky, 1979), its spherical analogue (21) contains one more integral of the product of the basic velocity with uυ. Whereas the first integral dominates principally at the sides of the basic jets located in the tropics and mid-latitudes (where both and are large), the second integral can be significant in the central parts of strong jets (where velocity is large), and especially when such jets are located in the polar regions (where is also large).

By (21), the sign of depends on the signs of the products and in various regions of the sphere. In the case when the first integral is dominant, one can say that the growth (or decrease) of the perturbation energy takes place in the regions where the inclination of the isolines of the perturbation streamfunction is opposite to (coincides with) the inclination of the profile of the basic velocity , that is, in the regions where product is positive (negative) (Pedlosky, 1979).

 

4. Numerical experiments

We now describe the numerical normal mode stability study results obtained with our algorithm for the LP flows, zonal RH waves and monopole modons. In all the experiments, the spectral problem (12) is solved with the QR-factorization (Wilkinson, 1965). The skill of the normal mode stability study algorithm has been checked with the following facts:

a) the eigenvalues must usually appear in groups of 4 eigenvalues located symmetrically about the origin on the complex plane (Arnold, 1965);

b) any LP flow of degree one or two must be stable and have only neutral modes (Case 2);

c) for a basic LP flow of degree 𝑛, the zonal wavenumber m of each unstable mode must belong to interval 0 < |m| < 𝑛 (Case 3);

d) any LP flow of degree 𝑛 as well as any zonal RH wave may have unstable mode only if Rayleigh-Kuo condition is satisfied (Case 1);

e) if degree 𝑛 of the LP flow is odd then all the modes are divided into two groups that contain just symmetric or antisymmetric (with respect to the equator) modes (Skiba, 1989; Skiba and Adem, 1998);

f) spectral number of the amplitude of each unstable mode must be equal to a special number (Cases 4 and 5);

g) the amplitude of each unstable mode must be orthogonal to the basic flow in the energy inner product (Skiba, 2000);

h) the growth rate of the most unstable mode must be limited from above by the theoretical estimations (Skiba, 2000);

i) for a modon by Verkley (1990), the vorticity of each unstable mode must be zero in the inner modon region S_𝑖;

j) the generation and dissipation of the perturbation energy of a zonal flow should be explained with formula (21).

The requirements a) and h) have been satisfied in all the experiments. By the requirement g), the amplitude of each unstable mode must be orthogonal to the basic flow in the energy inner product: . This property is always obeyed for the zonal flows, since all the calculated modes have the form (18), that is, have a harmonic structure in the λ-direction:

 

a) Legendre-polynomial flow (13).

For such flows, the structure of the stability matrix L (see (9)) was analytically analyzed by Skiba and Adem (1998).

1. Let 𝑛 = 1. The stability matrix L is then diagonal with pure imaginary diagonal elements L_αα. In this case, the eigenvalues coincide with the diagonal elements, and hence, ω_r = 0 for any eigenvalue. Therefore all the modes are neutral regardless of the amplitude a of the LP flow. The amplitude of each mode coincides with the corresponding spherical harmonics Y_α (λ, μ), and hence is symmetric or antisymmetric about equator depending on whether 𝑛_α — m_α is even or odd.

2. Let 𝑛 = 2. Then stability matrix L is tridiagonal, and the calculations give ω_r = 0 for all eigenvalues, that is, all the modes are neutral. Thus, any LP flow of the degree one or two is exponentially stable in full agreement with Case 2.

3. Let 𝑛 = 3 and α = 0.08. Here 𝑛 is odd, and in accordance with requirement e), the amplitude of all the modes can be just of the two types: symmetric or antisymmetric around the equator μ = 0. The most unstable mode (18) is symmetric around the equator, has ω_r = 0.2073 and zonal wavenumber m = 2, and spectral number of its amplitude is equal to 11.999 (the corresponding number provided by Case 4 is 𝒳₃=12). The second unstable mode has ω_r = 0.1576 and zonal wavenumber m = 1, is antisymmetric with respect to equator, and spectral number of its amplitude is 11.999 as well. The profile of the basic flow velocity is shown in Figure 1a, while the real parts of the amplitudes of the first and second most unstable modes are presented in Figure 1b and Figure 1c, respectively. For convenience in using formula (21), the figures are marked with the symbols "+ + +" and "⋅⋅⋅" at the positions of the basic flow jets with positive and negative velocities , respectively. Indeed, the profile of has the same inclination inside each region (channel) between two neighboring lines marked with different symbols, and this inclination is changed in passing from one region to another. It should be noted that for all the modes, nonzero amplitude values are captured in some latitudinal bands (Tung, 1981). An analysis of the fields shows that both integrals of the formula (21) contribute to the instability for the most unstable mode, while the first integral is dominant in generating the instability for the second mode. It should also be noted that values of uv are much smaller in the second case. In the experiments carried out, the magnitude |ω_r| increases with the basic flow amplitude α. This fact is also in full accordance with the theoretical estimates (requirement h). We use this requirements to control the growth rate of unstable modes. The main parameters of the two unstable modes (zonal wavenumber m, the real part ω_r and imaginary part ω_𝑖of the eigenvalue, spectral number , e-folding time , and period T) are given in Table 1.

4. Let 𝑛 = 4 and α = 0.06. The profile of the basic flow velocity is shown in Figure 2a, while isolines of the real parts of the amplitudes of the four most unstable modes are presented in Figures 2b-e (c, d). Since n is even, the amplitude of all the modes are asymmetric around the equator. Moreover, the mode perturbations are basically located in one of the two hemispheres. Spectral numbers of their amplitudes almost coincide with the number 𝒳₄=20 provided by Case 4. An analysis of the fields shows that both integrals of formula (21) contribute to the instability for the first three most unstable modes (Figures 2b-d) (c, d), while the contribution of the first integral in generating the instability for the fourth mode is weaker (Figure 2e). The main parameters of these modes are given in Table 2. It is seen that the requirements c), f) - h) are fulfilled.

5. Let 𝑛 = 5 and α = -0.06. The profile of the basic flow velocity is shown in Figure 3a, while isolines of the real part of the most unstable mode amplitude are presented in Figure 3b. The zonal wavenumber of the mode is m = 2, and mode amplitude is symmetric around the equator (𝑛 is odd). The perturbations are located in a neighborhood of the two subtropical jets, and both the integrals of formula (21) contribute to the mode instability with the dominant role of the first one. The main parameters of the seven most unstable modes are given in Table 3. It is seen that the requirements c), f) - h) are fulfilled, and the dominant zonal wavenumbers of the modes are m = 2 and m = 3.

6. Let 𝑛 = 6 and α = 0.03. The main parameters of the seven most unstable modes are given in Table 4. Unstable perturbations are located in the Northern Hemisphere (modes 2, 3 and 4), in the Southern Hemisphere (modes 1, 6 and 7), and in the tropics (mode 5). It is seen from Table 4 that the requirements c), f) - h) are fulfilled.

7. Let 𝑛 = 7 and α = 0.01. The main parameters of the twelve most unstable modes are given in Table 5. The majority of the modes (modes 1, 2, 6, 7 and 8-12) are symmetric around equator (requirement e) and located in the middle latitudes.The requirements c), f) - h) are fulfilled, too.

 

b) Zonal Rossby-Haurwitz waves (I4)

1. Let 𝑛 = 5 and α = 0.0052. The profile of the basic flow velocity is symmetric around equator (Figure 4a) and has a jet in the middle latitudes. There are just four different unstable modes, and the first of them is quasi-stationary (Table 6). Spectral numbers of the amplitudes of these modes coincide with the theoretical number 𝒳₅ =30 provided by Case 4. The real part of mode (18) can also be written as

or

where is the phase velocity, and the initial phase depends on μ. For positive (or negative) the mode moves to the west (east). Isolines of the real part (23) of the most unstable mode are given in Figures 4b-d (c) at the moments t = 0, t = T/8 and t = T/4, respectively. The mode moves very slowly to the west. Spatial variations of the field (23) with time are due to the dependence of the initial phase on μ. The amplitude of the most unstable mode is antisymmetric about the equator (n is odd). An analysis shows that both integrals of formula (21) contribute to the mode instability. The requirements f) - h) are fulfilled.

2. Let 𝑛 = 6 and α = 0.002. There are just three different unstable modes, besides, the first two of them are quasi-stationary again (see Table 7). Spectral numbers of the amplitudes of these modes coincide with the number 𝒳₆ = 42 provided by Case 4.

3. Let 𝑛 = 7 and α = —0.004. The profile of the basic flow velocity is symmetric about equator (Figure 5a) and has four jets. The main parameters of the eight most unstable modes are given in Table 8. Almost all of them are quasi-stationary. Spectral numbers of the amplitudes of these modes are practically coincide with the number 𝒳₇ = 56 provided by Case 4. Isolines of the amplitude of the most unstable mode corresponding to ω_r = 0.0515 are presented in Figure 5b.

c) Zonal flow in the form of a monopole modon (17)

As one more test we now consider the zonal flow that has the form of the monopole modon (17) with uniform absolute vorticity in the inner region S_𝑖 (Verkley, 1990). The main parameters of the modon are σ = 8.06 and . The profile of the modon velocity (Fig.6a) has three jets in the outer modon region S₀. The most unstable mode has been calculated with three different truncation numbers: T₂₁, T₃₁ and T₄₂. Isolines of the real part of the amplitude of this mode are presented in Figures 6b, 6c and 6d, respectively. The basic parameters of the mode are given in Table 9. In Fig. 6b-d (c, d), nonzero perturbations are in a latitudinal belt located in the outer modon region S₀, as it must be according to Case 5. In this belt, the inclination of the principle axes of the vortices is opposite to that of the modon velocity profile, the result being in full accordance with the first integral of the formula (21). The second integral in (21) contributes in the energy growth much less than the first one, and basically in the vicinity of the modon jet center that contains important perturbations. The polar regions contain no perturbations and make no impact on the energy behavior.

It should be noted however that to achive a good agreement of the numerical and theoretical results the spectral resolution up to T₄₂ is still insufficient. Indeed, although Table 9 shows a convergence of the numerical results as truncation numbers M and N of the Fourier series (4) increase, this convergence is rather slow. For the most unstable mode, for example, one can still see a difference between the theoretical value 𝒳_σ = σ (σ +1) = 73.02 of the spectral number (according to the Case 5) and its numerical value which is equal to 80.72 at resolution T₂₁, 70.56 at resolution T₃₁, and 74.60 at resolution T₄₂. For the second unstable mode, the corresponding numerical values are 78.81 at T₂₁, 73.95 at T₃₁, and 73.11 at T₄₂, that is, they are closer to the theoretical value. Besides, as it is seen from Figure 6b-d (c), the most unstable mode is of the same structure at resolutions T₂₁ and T₄₂, but its structure is changed at resolution T₃₁. This situation occurs when the modon has a few (at least two) most unstable modes with approximately equal real parts ω_r. Dikii (1976) showed that unstable modes of a zonal flow always correspond to the isolated eigenvalues. However, if the distance between two isolated eigenvalues with maximum real parts ω_r is very small then a high resolution is required to approximate well a concrete eigenvalue and the corresponding mode. Thus, the change of the structure of the most unstable mode is due to insufficient resolution and quite large truncation errors (Simmons et al. 1983, Skiba, 1998). And although spectral numbers for the first two most unstable modes calculated tend to the theoretical value 𝒳_σ = 73.02 as truncation numbers M and N of the Fourier series (4) increase, the convergence is slow, and a higher resolution is required to get more precise results. This phenomenon is explained by the fact that the μ-derivative of the modon vorticity is discontinuous at the boundary μ=μ_α separating the regions S_𝑖 and S₀ on the sphere (Verkley, 1990; Neven, 1992), that is, the modon is not so smooth as LP flows or RH waves. As a result, the series of the spherical harmonics for the modon and its perturbation converge (in Sobolev's norms) slower than the corresponding fields for the LP flows and RH waves (Topuria, 1987; Skiba, 1989, 1994, 1997). Besides, due to the Gibbs phenomenon (Davis, 1963), the convergence of Fourier series is more rapid inside the regions S_𝑖 and S₀ than in a vicinity of the boundary μ=μ_α, so that, no matter how large M and N may be, at some points of S the finite sums (4) are very nearly the same as the functions , and ξ, but at other points these sums differ substantially from , and ξ. Thus, truncation numbers M and N considerably greater than 42 must be used in (4) to reduce numerical errors. The importance of using a high resolution for the solution of the spectral problem (12) was recently shown theoretically by Skiba (1998) and numerically by Neven (2000a,b) who has used the resolution T₃₄₁ and the power method in the normal mode stability study of modons on a sphere.

 

5. Conclusions

In the present work, we have tested a numerical algorithm developed for the normal mode (exponential) instability of arbitrary steady flow of an ideal nondivergent fluid on a rotating sphere. Only zonal solutions to the BVE (Legendre polynomial flows, steady zonal Rossby-Haurwitz waves and monopole modons) have been examined here. On trials, the skill of the normal mode stability study algorithm has been checked by comparing the numerical stability results with the theoretical requirements a)-j) formulated in the beginning of Section 4. For the LP flows and zonal RH waves the numerical stability results have a very good accuracy. All the theoretical requirements a)-j) have been satisfied. In particular, in all the figures representing the unstable mode streamfunction , the inclination of -isolmes is opposite to that of the basic velocity profile. According to the first integral of the perturbation energy equation (21), this leads to the perturbation energy growth (requirement j)). As to the requirements c), we note that in almost 50% of the experiments carried out, the most unstable mode has zonal wavenumber m = 2. The next two popular zonal wavenumbers are m = 3 and m = 1. This is also in good agreement with the results previously obtained by Baines (1976).

The numerical results on the monopole modon instability are also satisfactory. The most part of the requirements a)-j) has been satisfied. However, it should be noted that the modon is not so smooth as LP flows and RH waves (the derivative of the modon vorticity is discontinuous on a sphere), and as a result, the series of the spherical harmonics for the modon and its perturbation converge slowly as the truncation numbers M and N increase (see Table 9). We have found two manifestations of the slow convergence: 1) the structure of the most unstable mode is changed at resolution T₃₁ being the same at resolutions T₂₁ and T₄₂; and 2) in comparison with the LP flows and RH waves, spectral number the amplitude of an unstable mode converges rather slowly to its theoretical value. Thus, to get the modon stability results with a high degree of accuracy, a resolution considerably higher than T₄₂ is required.

 

Acknowledgements

This work was supported by the PAPIIT grant IN122098 (UNAM, México), CONACyT project 32247-T (México), Silicon Graphics-UNAM research projects SC-006196 y SC-011800, and SNI grant (Mexican National System of Investigators). We are grateful to Mr. Jorge Zintzun and Mr. Alejandro Aguilar for their help in using graphical programs, Mrs. Ma. Elena Castillo for typing the draft of the manuscript, Mrs. Thelma del Cid for editing the text, and Mrs. M. E. Grijalva for the final preparation of the paper.

 

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