THE ROLE OF MODE MIXING IN THE ABSORPTION OF p-MODES

We consider wave propagation within a two-dimensional model with cylindrical geometry, in which the ambient medium is axisymmetric and depends only on the radial distance r and depth z (Figure 1). In the absence of the magnetic field, the pressure and density in the model grow with depth z as

$$\begin{eqnarray} p_a(z) &=& p_0 \left(1 +\frac{z}{L} \right)^{q+1}, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \rho _a(z) &=& (q+1)\frac{p_0}{gL} \left(1+\frac{z}{L} \right)^{q}, \end{eqnarray} \tag{ 2 }$$

where z increases downward, L is a characteristic scale length, p₀ is a characteristic pressure, g is the constant gravitational acceleration, and the parameter q is the polytropic index, which is related to the specific heat ratio as γ = (q + 1)/q.

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In the present calculations, we adopt a model (see Figure 2(a)) with the characteristic pressure p₀ = 1.8 × 10⁴ Pa, hydrodynamic scale height L = 0.5 Mm, polytropic index q = 2.5, and gravity g = 2.7 × 10² m s⁻².

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The magnetic field is purely vertical (B = {0, 0, B_z(r)}) and in the form of a tube with field strength, varying with radius as $B_z(r)=B_0 e^{-r^2/R^2}$, where B₀ is the characteristic magnetic field strength, and R is the characteristic radius of the flux tube. In the presence of such a magnetic field, the gas pressure may be determined to be

$$\begin{equation} p(r,z) = p_0 \left(1 +\frac{z}{L} \right)^{q+1} - \frac{B_0^2}{2\mu _0} e^{-2 r^2/R^2}. \end{equation} \tag{ 3 }$$

Since the density inside a purely vertical flux tube is only a function of height (see, e.g., Gordovskyy & Jain 2007), the sound speed is given by

$$\begin{equation} c^2(r,z) = \frac{gL}{q} \left(1+ \frac{z}{L} - \frac{B_0^2}{2\mu _0 p_0} e^{-2 r^2/R^2}\left(1+ \frac{z}{L}\right)^{-q}\right). \end{equation} \tag{ 4 }$$

Let us consider linear harmonic waves in terms of complex pressure and velocity perturbations, p'(r, z) and v'(r, z), respectively, which vary with time as e^iωt and do not depend on the angle ϕ (i.e., all the considered modes have azimuthal order m = 0). Their propagation through a magnetized nonuniform plasma may be described by the following set of linearized MHD equations (see, e.g., Gordovskyy & Jain 2008):

$$\begin{eqnarray} i \omega \rho v_r^{\prime } &=& \frac{\partial p^{\prime }}{\partial r} - \frac{i}{\omega } T_r \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} i \omega \rho v_z^{\prime } &=& \frac{\partial p^{\prime }}{\partial z} + \frac{i}{\omega }\left(v_r^{\prime } \frac{\partial \rho }{\partial r}+i v_z^{\prime } \frac{\partial \rho }{\partial z}\right) g \nonumber \\ &&+\frac{i}{\omega }\rho \left(\frac{1}{r} \frac{\partial r v_r^{\prime }}{\partial r}+\frac{\partial v_z^{\prime }}{\partial z}\right) g-\frac{i}{\omega } T_z \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} i \omega p^{\prime } &=& \left(v_r^{\prime } \frac{\partial p}{\partial r}+v_z^{\prime } \frac{\partial p}{\partial z}\right)+ c^2 \rho \left(\frac{1}{r} \frac{\partial r v_r^{\prime }}{\partial r}+\frac{\partial v_z^{\prime }}{\partial z}\right). \end{eqnarray} \tag{ 7 }$$

Here, T is a vector representing the magnetic forces:

$$\begin{eqnarray} {\bf T}&=&\frac{1}{\mu }_0({\bf \nabla } \times {\bf B}) \times ({\bf \nabla } \times ({{\bf v}} \times {\bf B})) \nonumber \\ && +\frac{1}{\mu }_0({\bf \nabla } \times ({\bf \nabla } \times ({{\bf v}} \times {\bf B}))) \times {\bf B}. \end{eqnarray} \tag{ 8 }$$

Let us introduce new dimensionless variables as follows: r = L S and z = L H for radius and depth, respectively; $\omega = \sqrt{\frac{g}{L}}\, \Omega$ for frequency; $v_r^{\prime } = i V_r e^{i\omega t}\,\sqrt{gL}$, $v_z^{\prime } = i V_ze^{i\omega t}\,\sqrt{gL}$, c² = C² gL for velocity components and sound speed, respectively; p' = p₀ Q e^iωt and p = p₀ f for pressures; $\rho = \frac{p_0}{gL}\, w$ for density; $B_r = \sqrt{\mu p_0} b_r$ and $B_z = \sqrt{\mu p_0} b_z$ for magnetic field components. Substitution of these variables into Equations (5)–(8) yields three equations for dimensionless wave perturbations weighted by the nondimensional radial distance S:

$$\begin{eqnarray} \frac{\partial (SQ)}{\partial S} &=& \frac{(SQ)}{S}+ \Omega w (SV_r) + \frac{S}{\Omega }\mathcal {T}_r \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} \frac{\partial (SV_r)}{\partial S} &=& - \frac{\Omega }{C^2 w}(SQ)-\frac{1}{C^2 w}\frac{\partial f}{\partial S}(SV_r)\,{-}\, \frac{1}{C^2 w}\frac{\partial f}{\partial H}(SV_z)\quad \nonumber \\ &&- \frac{\partial (SV_z)}{\partial H} \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \hspace{-10pt}(SV_z) &=&\frac{\Omega \frac{\partial (SQ)}{\partial H}\,{-}\,\left(\frac{1}{C^2}\frac{\partial f}{\partial S}\,{-}\,\frac{\partial w}{\partial S}\right)(SV_r)\,{-}\,\frac{\Omega }{C^2} (SQ)- S\mathcal {T}_z}{\frac{1}{C^2}\frac{\partial f}{\partial H}-\frac{\partial w}{\partial H}+\Omega ^2 w},\quad \end{eqnarray} \tag{ 11 }$$

with the magnetic force term given by

$$\begin{equation} {\bf \mathcal {T}}=({\bf \nabla } \times {\bf b}) \times ({\bf \nabla } \times ({\bf V} \times {\bf b}))+ ({\bf \nabla } \times ({\bf \nabla } \times ({\bf V} \times {\bf b}))) \times {\bf b}. \end{equation} \tag{ 12 }$$

We assume that the wave perturbations SQ, SV_r, and SV_z can be decomposed into incoming and outgoing components, and that both components satisfy Equations (9)–(12). Far from the flux tube, at S = S_max, the pressure and velocity perturbations for the incoming component correspond to those within the nonmagnetic plane–parallel atmosphere. This provides a boundary condition for SQ_in, SV_r in, and SV_z in:

$$\begin{eqnarray} SQ^{(0)}_{\rm in}&=&(J_0(\xi l S) - i Y_0(\xi l S))\,S \nonumber \\ && \times (H+1)^q e^{- \xi l (H+1)} M(-n,q,2\xi l (H+1)),\quad \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} SV^{(0)}_{r\,{\rm in}}&=& \frac{1}{\Omega w} \left(\frac{\partial SQ_{{\rm{in}}}^{(0)}}{\partial S}- \frac{ SQ_{{\rm{in}}}^{(0)}}{S}\right) \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} SV^{(0)}_{z\,{\rm in}}&=&\frac{1}{\Omega w} \left(\frac{\partial SQ_{{\rm{in}}}^{(0)}}{\partial H}- \frac{ SQ_{{\rm{in}}}^{(0)}}{C^2} \right), \end{eqnarray} \tag{ 15 }$$

where M is the Kummer function. Here, l is the p-mode degree that defines the horizontal wavenumber $k_h = \sqrt{l(l+1)/R^2_{\odot }}$, and ξ is a dimensionless parameter, ξ = L/R_☉.

It can be seen that the first term on the right-hand side of Equation (9) asymptotically grows when the radial distance S tends to zero. Hence, the wave function cannot be calculated in the vicinity of the axis S = 0. Therefore, we introduce a boundary condition very close to the axis at r = r_min. This does not significantly affect the solution as far as the distance S_min = r_min/L is much smaller than the considered radius of the flux tubes and the considered p-mode wavelength. In the present calculations, the distance S_min is chosen to be 0.2 (i.e., r_min = 0.1 Mm). Hence, for the smallest considered flux tube radius of 2 Mm the ratio R/r_min = 20, while for the shortest considered wavelength of ∼3.1 Mm (n = 0, ν = 4 mHz) the ratio is λ/r_min = 33.

Thus, at the distance S = S_min, the following conditions apply:

$$\begin{eqnarray} {\rm Re} (SQ_{\rm out}) &=& {\rm Re} (SQ_{\rm in}), \end{eqnarray}\vspace*{-8pt} \tag{ 16 }$$

$$\begin{eqnarray} {\rm Im} (SQ_{\rm out}) &=& - {\rm Im} (SQ_{\rm in}), \end{eqnarray}\vspace*{-8pt} \tag{ 17 }$$

$$\begin{eqnarray} {\rm Re} (SV_{r\,\rm out}) &=& {\rm Re} (SV_{r\,{\rm in}}), \end{eqnarray}\vspace*{-8pt} \tag{ 18 }$$

$$\begin{eqnarray} {\rm Im} (SV_{r\,\rm out}) &=& - {\rm Im} (SV_{r\,{\rm in}}). \end{eqnarray} \tag{ 19 }$$

The upper and lower boundaries of the numerical domain are chosen such that the upper and the lower turning points of the considered p-modes are well within the computational domain (see Figure 2(b)); thus, we can set these boundaries as open-type boundaries.

The set of Equations (9)–(12) together with the boundary conditions (Equations (13)–(19)) is solved using a finite difference approach. A second-order Runge–Kutta scheme is used to integrate the equations in the radial direction, marching from S = S_max toward S = S_min for the incoming wave component and marching from S = S_min toward S = S_max for the outgoing wave component.

The step in the radial direction is δS = 0.002, and the step in the vertical direction is δH = 0.2. The radial distance S ranges from S_min = 0.2 to S_max ≈ 205, while the depth H ranges from H_min = −0.6 to H_max ≈ 205. Hence, the considered numerical domain has physical dimensions of ∼ 100 × 100 Mm.

The flux tube's characteristic magnetic field B₀ is set to be 1.4 kG, and the flux tube's radius is either 2 or 8 Mm.

We carry out calculations for different sets of incoming wave frequency ν = 1–4 mHz and radial order n₀ = 0–2. Throughout the paper, n₀ denotes the radial order of the incoming mode. The corresponding harmonic degrees may be calculated using the following expression (Lamb 1932):

$$\begin{equation} l \approx 4\pi ^2 \frac{\nu ^2 }{g R_{\odot }} \left(\frac{q}{q+2n} \right). \end{equation} \tag{ 20 }$$

The p-mode degrees corresponding to our chosen sets of frequency ν and order n are shown in Table 1.

The open upper and lower boundaries of the domain are implemented using asymmetrical schemes for the appropriate derivatives. Thus, the vertical derivatives $\frac{\partial f}{\partial H}$ for the internal (I =  2, ..., N − 2 ) grid points are evaluated using the fourth-order symmetrical schemes (known as CD4):

$$\begin{equation} \left(\frac{\partial f}{\partial H} \right)_I = 1/12 \left(f_{I-2}-8 f_{I-1}+8 f_{I+1}- f_{I+2}\right)/\delta H \nonumber \end{equation} \tag{ 21 }$$

while for the near-boundary grid points, the asymmetrical scheme is used

$$\begin{eqnarray} \left(\frac{\partial f}{\partial H} \right)_0 &=& 1/12 (- 25 f_0 + 48 f_1 -36 f_2 +16 f_3 - 3 f_4) /\delta H \nonumber \\ \left(\frac{\partial f}{\partial H} \right)_1 &=& 1/12 (- 3 f_0 -10 f_1 +18 f_2 -6 f_3 + f_4)/\delta H \nonumber \\ \left(\frac{\partial f}{\partial H} \right)_{N-1} &=& 1/12 (3 f_N +10 f_{N-1} -18 f_{N-2} +6 f_{N-3}\nonumber\\ &&-\, f_{N-4})/\delta H \nonumber \\ \left(\frac{\partial f}{\partial H} \right)_N &=& 1/12 (25 f_N - 48 f_{N-1} +36 f_{N-2} -16 f_{N-3}\nonumber\\ &&+\, 3 f_{N-4})/\delta H.\nonumber \end{eqnarray}$$

Here, I is the grid number (I = 0, ..., N), and δH is the grid step in the vertical direction. Therefore, no external grid points are required.

Physically, these boundary conditions mean that waves can leave the domain through upper and lower boundaries without reflection.

Figures 3 and 4 show typical wave perturbations (or wave functions) for the dimensionless pressure, calculated using the approach described in Section 2. From these figures, it can be seen, for the large degree l, that the amplitude of the outgoing component is markedly lower in comparison to the amplitude of the corresponding incoming component.

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This indicates that the mode lost a portion of its energy as it traversed the magnetic inhomogeneity. This finding is qualitatively consistent with observations and with our previous work (Gordovskyy & Jain 2008). Furthermore, most of the wave functions exhibit modes with n ≠ n₀ in the outgoing component.

Figures 3(a) and 4(a) for n₀ = 0 show that the initial oscillations are slightly modulated by a longer wavelength. However, in the case of n₀ = 0, the additional mode order has a much weaker amplitude than in n₀ = 2 case: the amplitude of the mode-mixed wave is about 10–20 times smaller than the amplitude of the incoming mode.

Figures 3(b) and 4(b) for n₀ = 2 clearly show that the solution is a combination of the incident wavelength with oscillations of a shorter wavelength (i.e., a wave with lower order n). Thus, in the case of ν = 2 mHz and n₀ = 2, the amplitude of the overtone is about 10% of the amplitude of the incoming mode. In the case of ν = 3 mHz and n₀ = 2, the amplitude of the overtone is approximately the same as that of the n = n₀ mode.

In order to study the composition of the outgoing component, we calculate power spectra using the Fourier–Hankel analysis described in Section 3.2.

We analyze the calculated wave functions using an approach analogous to the decomposition method used in helioseismic observations (e.g., Braun 1995). The dimensionless pressure wave perturbation Q is decomposed into a Hankel series for m = 0 at some fixed depth H₁:

$$\begin{equation} Q(S,H_1) = \sum \limits _{l} A_{l} \mathcal {H}^{(I)}_0(\xi l S) + \sum \limits _{l} B_{l} \mathcal {H}^{(II)}_0(\xi l S). \end{equation} \tag{ 22 }$$

Here, $\mathcal {H}^{(I)}_0$ and $\mathcal {H}^{(II)}_0$ are zeroth-order Hankel functions of the first and second kinds, respectively, and A_l and B_l are complex amplitudes of the incoming and outgoing wave components, respectively. Therefore, |A_l|² and |B_l|² are powers of the incoming and outgoing components which are proportional to the energy fluxes carried by these wave components.

The decomposition is done outside the magnetized region between S = 48 (corresponding to r ≈ 24 Mm) and S_max.

The resolution of these spectra in l depends on the width (or radius) of the domain as δl ≈ πR_☉/r_max. For the domain with radius r_max = 102 Mm, resolution in l is about 40. Hence, for the frequency of 2 mHz, peaks corresponding to a different radial order may be resolved up to n = 3 and up to n = 4 for ν = 3 mHz.

Due to the finite number of harmonics used in the discrete Fourier–Hankel transform, any δ-like peak in the power spectrum |B_l|² is distributed among neighboring mesh points (the spectral leakage effect) and, as a result, the amplitude of the peak is reduced. In order to achieve the correct power values, we use a power-correction procedure that selects peaks higher than 4% of the maximum value and sums power over the five nearest points: the central point and two points on either side.

Power spectra versus degree l for the incoming and outgoing wave components (|A_l|² and |B_l|², respectively) are shown in Figures 5 for a few sets of parameters. It is evident that some of the spectra reveal a number of easily resolved peaks. Comparison of the peak locations in degree l with those in Table 1 show that all of them correspond to modes with a different radial order for the same frequency. This suggests that scattering across the radial order n occurs when the incident wave interacts with the magnetic flux tube.

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The power-correction procedure gives good results if the distance between two peaks is more than 100–120, otherwise the powers may be substantially overestimated. Therefore, this method may not provide correct powers for peaks corresponding to high orders n.

In order to avoid this problem, we use an alternative approach by fitting the obtained wave functions with sets of Whittaker functions, which are the vertical part of the eigenfunction for the case of a nonmagnetized atmosphere:

$$\begin{eqnarray} &&p^{\prime }_{\rm out}(r,z) = \sum \limits _n b(n) \Phi (n,k(n),z) \mathcal {H}^{(II)}_0(kr_1), \end{eqnarray}\vskip-3pt \tag{ 23 }$$

where Φ(n, k(n), z) = z^qe^−kzM(−n, q, 2kz). Here, the function k(n) taken for a complete polytrope with a constant polytropic index and for a constant frequency. The fitting is done using the following two steps. At first, we use the least-squares method to fit the pressure wave perturbation of the outgoing wave p'_out(r₁, z) with the series of Φ(n, k, z) functions as $p^{\prime }_{\rm out}(r_1,z) = \sum \limits _n \tilde{b}(n,r_1) \Phi (n,k,z)$ for a wide range of the radial distance r₁ outside the magnetized region (S > 48). In the numerical domain with z_max ≈ 100 Mm the spectra can be easily resolved up to n = 10. The obtained complex coefficients are functions of radius: $\tilde{b}(n,r) = b(n) \mathcal {H}^{(II)}_0(kr)$. Second, we calculate the absolute values of these complex coefficients $|\tilde{b}|^2(n,r) = |b|^2(n) |\mathcal {H}^{(II)}_0|^2(kr)$. Since $|\mathcal {H}^{(II)}_0|^2(kr) \sim \frac{2}{\pi k r}$ for large kr, we can easily "eliminate" the Hankel function and, as a result, we deduce powers |b|²(n) for each mode in the outgoing wave function.

In order to estimate the error of this method, we use wave functions calculated for plane–parallel nonmagnetic atmosphere. In this case, obviously, the amplitudes of the outgoing modes should be |b|²(n₀) = |a|²(n₀) and |b|²(n ≠ n₀) = 0. The characteristic error in the power |b|² is about 0.5%–0.8%. It should be noted that this error contains the error of the method as well as the error of the numerical scheme.

We now calculate absorption coefficients defined as α(n) = (|a(n)|² − |b(n)|²)/|a(n₀)|², where a(n) and b(n) are the amplitudes of the incoming and outgoing wave of the mode n, respectively. It should be noted that α(n) represents absorption of an individual p-mode and includes "real" absorption as well as redistribution of energy between modes with a different radial order. Since the coefficients |a(n)|² are given, the characteristic error in α is the same as the characteristic error in |b(n)|².

Absorption coefficients α versus order n are shown in Figures 6 and 7.

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The obtained absorption coefficients α(n) reveal two interesting features. First, the absorption coefficients for modes with n = n₀ are positive, which suggests a partial absorption of the incident wave in the magnetized region. However, the absorption coefficients appear to be substantially smaller here than in Gordovskyy & Jain (2008); the maximum absorption coefficient for the f-mode is about 75%, whereas for p₂, it is only about 10%. This arises because the pressure is smaller near the top boundary of the model (≈0.3p₀), which, in turn, results in a smaller characteristic magnetic field B₀. Therefore, the equipartition layer, where the sound speed and Alfvén speed are equal, is shallow, resulting in less efficient interaction between the wave and magnetic field.

The second feature is that α is negative for n ≠ n₀, which suggests that significant scattering of acoustic energy across the radial order n occurs. Thus, in cases where the incoming wave consists solely of p₂, the outgoing wave contains not only p₂, but also the f-mode (Figures 6(c)–(d)) and p₁-mode (Figures 7(c)–(d)). The power of the f-mode excited by p₂ is about 30% of the power of an incident p₂-mode (see also Figure 5(d)).

The calculated powers do not directly represent the amount of energy carried by different modes in the outgoing wave function; there is a multiplicative function of frequency and mode order n as well. In the following section, we evaluate energy fluxes based on the calculated powers in order to understand how the scattering across n affects actual energy fluxes carried by different modes.

Let us consider the outgoing component of a p-mode with the radial order n, wavenumber k, azimuthal order m = 0, amplitude b(n, k), and phase δ.

The pressure wave perturbation is described by Equation (23), while the radial velocity wave perturbations can be found as

$$\begin{eqnarray} v^{\prime }_{r\,\rm out} &=& i\frac{b(n,k)k}{\omega \rho (z)} \Phi (n,k,z) \mathcal {H}^{(II)}_1(kr)e^{i\omega t+\delta }. \end{eqnarray} \tag{ 24 }$$

The energy flux carried by the mode outside the magnetized region through a cylindrical surface r = r₁ can be written as

$$\begin{equation} \mathcal {F} = \omega r_1 \int \limits _0^{2\pi /\omega }\int \limits _0^\infty {\rm Re}[p^{\prime }(r_1,z,t)]{\rm Re}[v^{\prime }_r(r_1,z,t)]dz dt. \end{equation} \tag{ 25 }$$

Using Equations (22)–(23), one can rewrite the integrand in Equation (24) as follows:

$$\begin{eqnarray} {\rm Re}[p^{\prime }(r_1,z,t)]&&{\rm Re}[v^{\prime }_r(r_1,z,t)] = -\frac{b^2(n,k)k}{\omega \rho (z)} \Phi ^2(n,k,z) \nonumber \\ && \times (J_0(kr_1)\cos (\omega t +\delta)+Y_0(kr_1)\sin (\omega t +\delta)) \nonumber \\ &&\times (J_1(kr_1)\sin (\omega t +\delta)-Y_1(kr_1)\cos (\omega t +\delta)),\nonumber\\ \end{eqnarray} \tag{ 26 }$$

which, in turn, after averaging over the period of oscillations 2π/ω, can be rewritten using the Wronskian of Bessel functions (Y_m(kr)J_{m + 1}(kr) − J_m(kr)Y_{m + 1}(kr) = 2/(πkr)) as follows:

$$\begin{equation} \mathcal {F} = - \frac{b^2(n,k)}{\pi } \int \limits _0^\infty \frac{\Phi ^2(n,k,z)}{\rho (z)}dz. \end{equation} \tag{ 27 }$$

The ratio $\mathcal {F} /b^2(n,k)$ versus order n is plotted in Figure 8. This function rapidly grows with the radial order n, and for the energy flux carried by p₂ is approximately 10–20 times higher than the energy flux carried by the f-mode.

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Further, we use Equation (26) to evaluate the energy flux scattered from the incoming mode with the radial order n = n₀ into other modes (n ≠ n₀). This is done by adding up the energy fluxes of modes with the radial order n from 0 to 5 (except n = n₀) in the outgoing wave. The resulting values, κ, are shown as dashed lines in Figure 9 as a function of frequency compared with the total energy flux lost by the incoming wave (α, solid lines). Hence, the difference between the two curves is the "real absorption."

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It can be seen that similar to the absorption coefficient α the energy scattered due to mode mixing, κ increases with frequency and slightly decreases with order n₀. Furthermore, the coefficient κ is slightly higher for larger flux tube radii. However, it is interesting to note that the fraction of energy flux lost due to mode mixing (i.e., the ratio κ/α) apparently sharply decreases with frequency. Thus, in the case of n₀ = 0 (Figures 9(a) and (d)) at frequency ν = 2 mHz, the amount of energy flux lost due to mode mixing is about κ/α ∼ 0.25–0.30 of the overall energy flux lost by the incoming mode. At the same time, at ν = 4 mHz, this fraction is only 0.05–0.15. In the case of higher radial orders n₀, this contrast is not so strong: the κ/α ratio for n₀ = 2 and ν = 2 mHz is about 0.3–0.5 decreasing to 0.2–0.3 at ν = 4 mHz.

However, it can be noted that the error in energy fluxes calculated for higher orders may be large. Indeed, if we assume that the characteristic error in power is the same for all orders n and is about Δb² ∼ 2% then the characteristic error in the energy flux calculations can be found as $\Delta \mathcal {F} \approx \Delta b^2 \mathcal {F} /b^2(n,k) \Phi ^2(n,k,z)$. If the error Δb² is constant and the ratio $\mathcal {F} /b^2(n,k)$ substantially increases with the order n, the error in the energy flux calculations also quickly increases with order n. Therefore, the contribution of the high-n modes into the energy flux of outgoing waves and, consequently, the part of energy lost due to mode mixing may be underestimated.