© ESO 2020

1. Introduction

Acoustic waves in the solar atmosphere and interior are well established, and studying these waves is a fundamental subject in solar physics. The cutoff frequency of these waves plays an important role in their propagation in the solar atmosphere because the Sun is a stratified medium. Acoustic waves whose frequency is higher than the cutoff frequency can pass through the photosphere and propagate into higher atmospheric layers. In contrast, waves whose frequency is lower than the cutoff value are called evanescent waves. They are trapped inside the solar interior. The acoustic cutoff has significant implications for several topics in solar physics. Acoustic waves have been proposed as one candidate mechanism to explain the heating of the outer atmospheric layers (Ulmschneider 1990; Narain & Ulmschneider 1996; Ulmschneider & Musielak 2003), and the cutoff frequency determines which waves contribute to the energy flux. The cutoff leads to global p-mode oscillations that are produced by waves trapped below the solar surface. It thus enables the field of helioseismology (Christensen-Dalsgaard 2002). It is also a key parameter for understanding the shift in the dominant period in some solar magnetic structures from 5 min at the photosphere to 3 min at the chromosphere (Fleck & Schmitz 1991; Centeno et al. 2006; Felipe et al. 2010a) and the more puzzling propagation of long-period waves at chromospheric layers (e.g., Orrall 1966; Giovanelli et al. 1978). Observations of these long-period waves, which are expected to be evanescent in the solar atmosphere, has kindled the interest in them (De Moortel & Rosner 2007; Jefferies et al. 2006; Centeno et al. 2009)

The acoustic cutoff frequency is dependent on the local environment. In the literature, many equations are used to compute the cutoff frequency. Lamb (1909) carried out the initial study to derive the cutoff frequency for an isothermal atmosphere. Many studies have been carried out for nonisothermal media (e.g., Deubner & Gough 1984; Musielak et al. 2006) because the Sun has a stratified atmosphere. However, it has been shown that the choice of different independent and dependent parameters can generate different analytical equations for the cutoff frequency (Schmitz & Fleck 1998, 2003). The above studies (Lamb 1909; Deubner & Gough 1984; Musielak et al. 2006; Schmitz & Fleck 1998) do not incorporate the effects of the magnetic field. There have been some attempts to evaluate the effect of the magnetic field in the cutoff frequency of the so-called magnetoacoustic waves (e.g., Thomas 1982, 1983; Roberts 1983, 2006; Stark & Musielak 1993; Perera et al. 2015). Theoretical considerations indicate that in the low-β regime, where $\beta =c_{\text{s}}^2/v_{\text{A}}^2$, a magnetic field that is inclined from the solar vertical an angle θ reduces the cutoff frequency by a factor cosθ (Bel & Leroy 1977; Jefferies et al. 2006). This effect, commonly referred to as “ramp effect” or “magnetoacoustic portals”, is due to the field-guided propagation of the slow-mode waves, which experience a reduced gravity along field lines. It has been observationally confirmed that the cutoff frequency of magnetoacoustic waves is reduced in atmospheres with inclined magnetic fields (Jefferies et al. 2006; McIntosh & Jefferies 2006; Rajaguru et al. 2019).

Comparative studies of the height dependence of the cutoff in solar observations are rare, although several works have confirmed the stratification of the cutoff in the solar atmosphere (Wiśniewska et al. 2016; Felipe et al. 2018). These works also pointed out significant disagreements between the theoretically computed and observed cutoff values. In this paper we follow a numerical approach to determine the cutoff frequency in several solar models. The results are compared with the cutoff values derived from analytical calculations. We also compare our simulation results with recent observational evidence of a low cutoff frequency at the photosphere (Rajaguru et al. 2019). The paper is structured as follows. The numerical methods, solar models, and analytical equations of the cutoff frequency are discussed in Sect. 2. The cutoff frequency derived for various quiet-Sun and sunspot models is discussed in Sects. 3 and 4, respectively. The results are discussed and compared with recent observations in Sect. 5. Finally, the conclusions are stated in Sect. 6.

2. Methods

We have performed numerical simulations of wave propagation in a set of well-known and broadly used standard one-dimensional (1D) solar models. The simulation result was employed to determine the vertical stratification of the cutoff frequency for each of the models, including a parametric study of the dependence of the cutoff on some model parameters, such as the magnetic field strength and radiative losses. The numerically determined cutoff stratifications were compared with the cutoff values derived from various formulae described in the literature. In the following sections, we describe the procedures we carried out for each of these steps.

2.1. Numerical simulations

Numerical simulations in a 2D domain were developed using the code MANCHA (Khomenko & Collados 2006; Felipe et al. 2010b). In this work, we restricted the use of the code to the magnetohydrodynamic approximation. The code computes the evolution of the perturbed variables, which are retrieved after explicitly subtracting the background state from the equations. We imposed a different solar atmosphere as background for each simulation. In the vertical direction we set the stratification of the modified solar models (see next section), whereas the backgrounds are constant in the horizontal direction. Periodic boundary conditions were imposed in the horizontal direction, and at the top and bottom boundaries, we set a perfectly matched layer (PML, Berenger 1994) to damp the waves and avoid undesired reflections.

We employed a 2.5D approximation, which means that we used a 2D domain (in the plane X − Z, derivatives are not computed in the y direction), but the vectors maintained the three spatial coordinates. In the vertical direction, the computational domain spans from z = −1140 km to z = 2420 km, with z = 0 defined at the height where the optical depth at 5000 Å is unity, and we used a constant spatial resolution of 10 km. The PML medium is established in the top ten grid points of the domain. The horizontal domain covers 4800 km with a spatial step of 50 km. The size of the numerical grid is 96 × 356.

Waves were excited by a vertical force added to the equations. The temporal evolution and spatial dependence of the driving force were taken from temporal series of photospheric Doppler velocity in the Si I 10 827 Å line reported in Felipe et al. (2018). The details of how the driver is introduced can be found in Felipe et al. (2011), although in this work we performed some variations. Here, we are not interested in a one-to-one reproduction of the observed velocities. Instead, we only need to introduce waves that propagate from below the photosphere to the higher chromospheric layers whose properties qualitatively match those found in actual sunspot observations. The driver was imposed below the photosphere (z = −180 km), around 500 km below the formation of the Si I 10 827 Å line (Bard & Carlsson 2008; Felipe et al. 2018), where the oscillations were measured. In addition, the response of the atmosphere to an oscillatory force depends on the frequency of the oscillations (Felipe et al. 2011). To generate waves with a realistic photospheric power distribution, the amplitude of the power spectra of the driving force was modified concerning the reference observational velocity. The amplitude of the driver in all the simulations presented in this work is low enough to retain the computations in the linear regime. The duration of each simulation is 65 min of solar time.

2.2. Solar models

A total of seven different solar atmospheres were analyzed, including three quiet-Sun models and four sunspot umbral models (Fig. 1). The quiet-Sun atmospheres correspond to those published in Vernazza et al. (1981), Fontenla et al. (1993), and Avrett et al. (2015). In the following, we refer to them as VALC, FALC, and Avrett2015QS, respectively. For umbral atmospheres, we explored the three models presented by Maltby et al. (1986), which correspond to the darkest part of large sunspots at the early (eMaltby), middle (mMaltby), and late (lMaltby) phases of the solar cycle, and the umbral model introduced by Avrett et al. (2015) (Avrett2015spot).

Fig. 1. Temperature stratification of the solar models. Left panel: quiet-Sun temperature distribution as given by the VALC (black line), FALC (red line), and Avrett2015QS (blue line) models. Right panel: sunspot temperature stratification from eMaltby (green line), mMaltby (red line), lMaltby (black line), and Avrett2015spot (blue line) atmospheres. In both panels, the solid lines represent the original models and dashed lines correspond to the temperature profiles we employed to compute the numerical simulations.

All of these solar models extend from the photosphere through the chromosphere into the transition region. In this work, we are interested in determining the vertical stratification of the cutoff frequency. At each atmospheric height, we aim to measure the lower frequency where waves begin to propagate upward. The strong temperature gradients of the transition region reflect the upward-propagating waves, and theory predicts that a resonant cavity can be formed between that height and the temperature minimum (Zhugzhda & Locans 1981; Zhugzhda 2008; Botha et al. 2011; Felipe 2019; Jess et al. 2019). For our numerical simulations, we partially removed the top transition region. The temperature was set constant above z = 1800 km for all the solar models. Then, the pressure and density distributions above that height were computed for the modified temperature stratification following Santamaria et al. (2015). With this approach, we avoided most of the downward-propagating waves coming from the reflections at the transition region, which may affect the estimation of the propagating or nonpropagating nature of waves at certain frequencies.

The solar models were also extended to deeper layers because the bottom boundary of the computational domain employed by our simulations is below the lower limit of these atmospheric models. Below z = 0, the temperature of the models was smoothly joined with the temperature stratification of the solar interior from Model S (Christensen-Dalsgaard et al. 1996). We took a constant adiabatic index γ = 5/3.

2.3. Estimating the cutoff frequency stratification

2.3.1. Determination of the cutoff frequency in numerical simulations

The cutoff frequency was measured based on the phase difference (Δϕ) spectra between the vertical velocity at two heights. Low-frequency waves, whose frequency is below the cutoff value, form evanescent waves. In this situation, the signals from the two atmospheric heights oscillate in phase and the phase spectrum exhibits values near zero. From a certain frequency value onward, the phase difference (obtained from the subtraction of the phase of the velocity signal at the higher height from the phase at the lower height) shows a progressively increasing tendency, indicating that these frequency modes propagate from deeper to higher layers. The starting point of the increasing trend corresponds to the cutoff frequency.

Figure 2 illustrates the phase difference between two atmospheric heights in one of the numerical simulations performed using the VALC model as a background. In this example, the lower height is z = 880 km, and the upper layer corresponds to z = 900 km. That is, the height difference between the two velocity signals is Δh = 20 km. The solid line in Fig. 2 shows the horizontal average of the phase-difference spectra, and the vertical bars are the standard deviation of these points. The temporal series of 65 min of a simulation were padded with zeros up to 341 min. Based on these data, we estimate that the cutoff frequency of this solar model at z = 890 km is 4.30 mHz. At this frequency value, the increasing trend of the phase difference starts and we are sure that there is upward wave propagation because the phase difference (taking the uncertainty estimated from the standard deviation into account) is undoubtedly positive.

Fig. 2. Average phase-difference spectra between the vertical velocity signal at z = 880 km and z = 900 km measured from the simulation of the VALC model with a vertical magnetic field of 130 G. A positive phase difference indicates upward wave propagation. Error bars show the standard deviation of the averaged data. The vertical dotted line marks the value of the cutoff frequency as determined from examining the phase spectra.

We computed the cutoff frequency for all the models presented in the previous sections, including various values of the magnetic field, at all heights between z = 0 and z = 1800 km. In all cases, we employed a Δh = 20 km, and the obtained cutoff value was assigned to the mid-point between the two heights we used for the analysis. A script was written to automatically determine the cutoff values. To adapt it to provide a good performance despite the peculiarities of each case, the procedures performed by this script slightly depart from the description of the cutoff estimation discussed in the previous paragraph. First, the mean phase-difference spectrum was smoothed by averaging all the phase differences in a box with a width of 0.5 mHz. Then, we determined the region where the smoothed phase difference, after adding to it the standard deviation, is higher than zero. At these frequencies, we consider that waves are propagating. Finally, the lowest frequency of that region was selected as the cutoff value.

2.3.2. Analytical formulae for the cutoff frequency

The cutoff frequency is a local quantity that depends on the properties of the surrounding medium. The solar atmosphere exhibits significant inhomogenities, and changes in the cutoff value both in the horizontal direction as in the vertical direction would be expected. We here focus on the stratification of the solar atmospheric models because the background model from our simulations is constant in the horizontal axis. Theory predicts a wide variety of cutoff formulae depending on the wave equation derived from different selections of independent and dependent variables (Schmitz & Fleck 2003).

The main goal of this work is to bypass this limitation in deriving the cutoff frequency by providing alternative values for the cutoff in the solar atmosphere from the examination of numerical simulations. We also computed the theoretical cutoff stratification in the analyzed models by applying several expressions commonly used in the literature. This allows us to compare the analytical cutoff frequency with the numerical estimations and distinguish between different theories.

We chose to examine the same four cutoff formulae whose performance was compared with sunspot observational measurements in Felipe et al. (2018). They are described in the following lines. In the case of an isothermal atmosphere, the correct formula for the cutoff is obtained from the original work by Lamb (1909), and it is given by

$$\begin{array}{c}\hfill {\omega }_{\mathrm{C}1}=\frac{c_{\mathrm{s}}}{2H_{\mathrm{p}}},\end{array}$$(1)

where H_p is the pressure scale height and c_s is the sound velocity. This expression is only valid for isothermal atmospheres, where it does not change with height. However, in this analysis we assumed that it is a local quantity, and we computed its vertical variation according to the temperature stratification of the solar models.

The second formula that we considered is

$$\begin{array}{c}\hfill {\omega }_{\mathrm{C}2}=\frac{c_{\mathrm{s}}}{2H_{\rho }}{(1-2\frac{\mathrm{d}{H}_{\rho }}{\mathrm{d}z})}^{1/2},\end{array}$$(2)

which is the most commonly used in helioseismology. Here, H_ρ is the density scale height. We also evaluated one of the cutoff formulae from Schmitz & Fleck (1998), which reads¹

$$\begin{array}{c}\hfill {\omega }_{\mathrm{C}3}={\omega }_{\mathrm{C}1}{(1+2\frac{\frac{\mathrm{d}{c}_{\mathrm{s}}}{\mathrm{d}z}}{{\omega }_{\mathrm{C}1}})}^{1/2}.\end{array}$$(3)

These three cutoff frequencies were derived for purely acoustic waves, neglecting the effect of the magnetic field. They are usually employed in magnetized media because the behavior of slow-mode waves in regions dominated by magnetic pressure (high atmospheric layers or even the photosphere in active regions) is similar to that of acoustic waves. The last formula that we evaluated is extracted from Roberts (2006), and it is specifically derived for slow magnetoacoustic waves in isothermal atmospheres permeated by a uniform vertical magnetic field. It is computed as

$$\begin{array}{c}\hfill {\omega }_{\mathrm{C}4}=c_{\mathrm{t}}{[\frac{1}{4H_{\mathrm{p}}^2}{\left(\frac{c_{\mathrm{t}}}{c_{\mathrm{s}}}\right)}^4-\frac{1}{2}\gamma g\frac{\partial }{\partial z}\left(\frac{c_{\mathrm{t}}^2}{c_{\mathrm{s}}^4}\right)+\frac{1}{v_{\mathrm{A}}^2}(N^2+\frac{g}{H_{\mathrm{p}}}\frac{c_{\mathrm{t}}^2}{c_{\mathrm{s}}^2})]}^{1/2},\end{array}$$(4)

where $c_{\mathrm{t}}=c_{\mathrm{s}}{\mathit{v}}_{\mathrm{A}}/\sqrt{c_{\mathrm{s}}^2+{\mathit{v}}_{\mathrm{A}}^2}$ is the cusp speed, N² is the squared Brunt-Väisälä frequency, and g is the gravity.

3. Cutoff frequency in quiet-Sun models

Following the method described in the previous section, we derived the stratification of the cutoff frequency in the quiet Sun from the examination of numerical simulations and the application of several formulae described in the literature. The two approaches and the different theoretical cutoff values are compared in Sect. 3.1, and a parametric study of the dependence of the numerical cutoff on the magnetic field is presented in Sect. 3.2. In Sect. 3.3 we discuss the effect of the radiative losses.

We explored three different quiet-Sun models: VALC, FALC, and Avrett2015QS. In all the numerical simulations the atmosphere is permeated by a constant vertical magnetic field. Some of the quiet-Sun atmospheres we imposed as background for the simulations are convectively unstable. That is, at some heights, the square of the Brunt-Väisälä frequency is below zero. However, magnetic fields inhibit convection. We have found that a magnetic field strength as low as 5 G is enough to stabilize the three quiet-Sun models we studied here.

The simulations were analyzed with a focus on oscillations in the vertical velocity. Waves are driven below the photosphere, at a depth where the sound speed is much higher than the Alfvén speed (β ≫ 1). In these layers, the oscillatory signal is produced by fast magnetoacoustic waves, whose properties are similar to that of sound waves. At the height where both characteristic velocities are comparable, mode conversion takes place, and part of the energy of the fast magnetoacoustic wave is converted into fast and slow magnetoacoustic waves in the low-β region (Cally 2006, 2007; Schunker & Cally 2006; Khomenko & Collados 2006). There, the target of our analysis is the slow magnetoacoustic mode. It is a field-guided wave that behaves like an acoustic wave. In the case of inclined magnetic fields, mode conversion certainly complicates our approach of determining the cutoff frequency. When phase difference between two layers is measured in the low-β region, the horizontal displacement of the wavefront needs to be taken into account. However, it is not trivial to define the lowest height where waves are guided by the field because wave modes in the β = 1 region exhibit some mixed properties. For simplicity, we chose to restrict the analysis of quiet-Sun atmospheres to vertical magnetic fields. In this situation, we can confidently quantify the cutoff frequency from phase-difference spectra computed between two heights at the same horizontal position. For an evaluation of the effects of magnetic field inclination on the cutoff, see Sect. 4.3.

3.1. Comparison with analytical models

Figure 3 illustrates the stratification of the cutoff frequency measured in numerical simulations for the VALC, FALC, and Avrett2015QS models, and their comparison with the values obtained from theoretical expressions. The atmospheres are permeated by a vertical magnetic field of B = 5 G or B = 300 G.

Fig. 3. Variation of the cutoff frequency with height in the quiet-Sun models VALC (panel a), FALC (panel b), and Avrett2015QS (panel c). The lines with asterisks show the cutoff values determined from the examination of the phase-difference spectra in a numerical simulation with a vertical magnetic field of 5 G (black) and 300 G (red). Color lines indicate the analytical cutoff frequency computed using Eq. (1) (blue line), Eq. (2) (green line), Eq. (3) (violet line), Eq. (4) with 5 G magnetic field strength (black line), and Eq. (4) with 300 G magnetic field strength (red line). The vertical dashed lines mark the height where the plasma-β is unity for the models with a field strength of 5 G (black) and 300 G (red).

In the case with a weak magnetic field strength, the cutoff frequency (ν_c) of VALC (Fig. 3a) shows a peak at around z = 800 km, where it increases up to ∼6 mHz. Above that height, the cutoff smoothly decreases, from ν_c = 5.4 mHz slightly above z = 800 km to ν_c ∼ 4.0 at z = 1800 km. At the deep photospheric layers, the cutoff exhibits an approximately constant value of ∼5.4 mHz between z = 0 and z = 400 km. It is reduced to ν_c = 5.0 mHz at z = 500 km, just before starting to increase to the maximum peak mentioned previously.

A comparison between the numerically estimated cutoff and the analytical formulae for VALC reveals some qualitative similarities, but significant differences. Strong differences are also found in the four theoretical cutoff estimations. All the analytical stratifications show a maximum peak between z = 500 km and z = 600 km. This peak is shifted around 200 km toward deeper layers for the location of the maximum found in the weakly magnetized simulation. The values of the cutoff at that peak in the analytical expressions are close to the numerically determined value, except for the value obtained from Eq. (1) for an isothermal atmosphere. At the deep photosphere, the cutoff from Eq. (2) shows a reasonable agreement with the numerical cutoff, although in the latter the sharp spike around z = 0 is missing. At chromospheric heights, the cutoff from Eq. (4) provides the best match with the numerical value. However, this expression was derived for slow magnetoacoustic waves, and it is therefore not applicable below the β ∼ 1 layer.

The numerical cutoffs obtained for FALC (Fig. 3b) and Avrett2015QS (Fig. 3c) models share several features with the cutoff of VALC: when the models are permeated by a 5 G magnetic field strength, they exhibit a peak around z = 800 km with a maximum value ν_C ∼ 6 mHz, at chromospheric layers, the cutoff value decreases with height, and at the photosphere, the cutoff is around 5.3 mHz. The main difference presented by FALC is that the cutoff values of the weakly magnetized case at the deep photosphere are high; they lie within the bottom 200 km of the plot. This increase agrees with the increase produced by the cutoff from Eq. (2). The Avrett2015QS model shows a more pronounced reduction of the cutoff with height at chromospheric layers than the other quiet-Sun atmospheres. Interestingly, in the two models plotted in Figs. 3b and c, the numerically determined cutoff stratifications for the cases with B = 300 G agree well with those computed with Eq. (1). The match is almost perfect above z = 800 km, and the cutoff value at the maximum (∼5.3 mHz) also agrees, although it is shifted 200 km to deeper layers.

3.2. Dependence on the magnetic field

The solar atmosphere is permeated by magnetic fields even in the regions known as quiet Sun (Trujillo Bueno et al. 2004). In this section we study the effects of these weak magnetic fields on the quiet-Sun wave propagation by analyzing numerical simulations of the VALC, FALC, and Avrett2015QS models with various values of the vertical magnetic field strength.

Figure 4 shows the stratification of the cutoff frequency derived from the numerical simulations for the three quiet-Sun models and magnetic field strengths between 5 G and 300 G. The effects of an increasing magnetic field are the same in the three atmospheric models. At most heights, the atmospheres permeated by stronger magnetic fields exhibit lower cutoff frequencies. Some regions depart from this tendency. Deep atmospheric layers are dominated by magnetic pressure, and a weaker effect of the magnetic field on the waves is expected at those heights. Surprisingly, in regions with β >  1, the effect of the magnetic field on the derived cutoff frequency is clearly noticeable, and significant differences are found between strongly and weakly magnetized models.

Fig. 4. Variation of the numerically determined cutoff frequency with height in the quiet-Sun models VALC (panel a), FALC (panel b), and Avrett2015QS (panel c). Each color corresponds to atmospheres permeated by a different strength of the vertical magnetic field: 5 G (black), 10 G (violet), 50 G (green), 130 G (blue), and 300 G (red). The vertical dashed line marks the height where the plasma-β is unity following the same color code as the cutoff values.

As the magnetic field increases, the cutoff peak exhibited by all quiet-Sun models progressively decreases and is shifted to deeper layers, from z ∼ 800 km for a strength of 5 G to z ∼ 250 km for a strength of 300 G. In addition, quiet-Sun regions with stronger magnetic fields (above ∼50 G) also show a more remarkable minimum in the photospheric cutoff. The location of this minimum is also shifted to deeper layers as the magnetic field strength increases. In the case of VALC with B = 300 G, this minimum is beyond the range of atmospheric heights we probed.

3.3. Effect of radiative losses on the cutoff

Radiative losses were implemented according to Newton’s cooling law,

$$\begin{array}{c}\hfill Q_{\mathrm{rad}}=-c_{\mathrm{v}}\frac{T_1}{{\tau }_{\mathrm{R}}},\end{array}$$(5)

where T₁ is the perturbation in the temperature, c_v is the specific heat at constant volume, and the radiative cooling time is given by the Spiegel (1957) formula as

$$\begin{array}{c}\hfill {\tau }_{\mathrm{R}}=\frac{\rho c_{\mathrm{v}}}{16\chi {\sigma }_{\mathrm{R}}{T}^3}·\end{array}$$(6)

In the latter expression, ρ is the density, T is the temperature, σ_R is the Stefan-Boltzmann constant, and χ is the gray absorption coefficient. Following Felipe (2019), the radiative cooling time given by Eq. (6) was only applied between z = 200 km and z = 1100 km. The Spiegel (1957) formula was derived for optically thin plasma and assuming local thermodynamic equilibrium. Its range of applicability is restricted to photospheric heights. Beyond this region, we imposed adiabatic propagation. Our approach at layers deeper than z = 200 km is similar to that employed by Ulmschneider (1971), who assumed a completely optically thick medium (adiabatic propagation) below z ∼ 140 km. At the chromosphere, the effect of the radiative dissipation on wave propagation is negligible (Schmieder 1977). Based on the chromospheric radiative cooling time determined by Giovanelli (1978), we estimated that the value computed from Eq. (6) is reliable up to z = 1100 km, and set adiabatic propagation above that height. A minimum value of τ_R ∼ 10 s is found at z = 200 km, and it increases to a maximum of τ_R ∼ 500 s at z = 700 km.

Figure 5 compares the stratification of the cutoff frequency in the quiet-Sun model VALC in the adiabatic case (asterisks) with that measured for simulations where radiative losses are taken into account (diamonds). The results are illustrated for three different values of the magnetic field, including a very quiet atmosphere (B = 5 G), a model with average quiet-Sun field strength (B = 130 G), and a case with a stronger magnetic field (B = 300 G). Radiative transfer is known to reduce the cutoff frequency (Roberts 1983; Centeno et al. 2006; Khomenko et al. 2008). Our measurements indicate that this reduction can be striking. When the radiative losses are turned on, the maximum value of the cutoff frequency is ∼5 mHz, whereas in the adiabatic cases, it shows peaks as high as 6 mHz (B = 5 G) or 5.7 mHz (B = 300 G). Radiative losses also lead to low cutoff values around 4 mHz at the photosphere (between z = 150 km and z = 350 km) and the low chromosphere (z ∼ 1000 km). Interestingly, with the radiative losses on, all the simulations show a similar trend in the cutoff stratification, regardless of their magnetic field strength. However, they conserve some dependence on the magnetic field. The relation between the field strength and the cutoff is similar to that found for the adiabatic case, with a general reduction of the cutoff with magnetic field strength for simulations in the range B = [5, 130] G and an increase for B = 300 G.

Fig. 5. Variation in numerically determined cutoff frequency with height in the quiet-Sun models VALC with the radiative looses turned on (dashed lines) and off (solid lines with asterisks). The color indicates the magnetic field strength, following the same color code as in Fig. 4: 5 G (black), 130 G (blue), and 300 G (red).

A magnetic field permeating the quiet-Sun atmosphere can produce significant variations in the cutoff, including the presence and location of peaks or minimums (see previous section). These changes are overcome by the effect of radiative transfer, however.

4. Cutoff frequency in umbral models

We evaluated the variation of the cutoff frequency with height in four different umbral models, the three models from Maltby et al. (1986) (eMaltby, mMaltby, and lMaltby) and the sunspot model presented by Avrett et al. (2015) (Avrett2015spot). The atmospheres were permeated by a set of values of a constant vertical magnetic field, from weakly magnetized umbrae (500 G) to strongly magnetized umbrae (3000 G). We followed the same approach as in the previous section and first compared the four atmospheres permeated by a chosen value of the magnetic field with the cutoffs derived from the analytical expressions, and then evaluated the variation in cutoff frequency produced by changes in the magnetic field and radiative losses.

4.1. Comparison with analytical models

Figure 6 shows the cutoff frequency of the umbral models with a vertical magnetic field of 3000 G. Figure 1 shows that the three atmospheres from Maltby et al. (1986) present a similar temperature stratification, but with some small differences. The minimum value of the temperature is related to the intensity of the umbra. Dark umbrae (eMaltby) have lower temperatures, but the minimum value increases for average (mMaltby) and bright (lMaltby) umbrae. In addition, the height where the temperature starts to rise above the temperature minimum is also shifted to higher layers as brighter umbrae are considered. These differences in the atmospheric stratification are captured by our cutoff measurements. Models with a lower temperature exhibit a higher maximum in the cutoff frequency. In the case of eMaltby, a maximum cutoff of 5.9 mHz is found around z = 400 km, whereas the maximum cutoff frequencies for models mMaltby and lMaltby are 5.6 mHz and 5.4 mHz, respectively. This dependence of the cutoff on temperature is predicted by theory, and all the analytical expressions explored in this work show a reduction in the photospheric cutoff from eMaltby to lMaltby, with the value of mMaltby located in between. However, all these theories overestimate the cutoff at the umbral photosphere.

Fig. 6. Variation in cutoff frequency with height in the umbral models eMaltby (panel a), mMaltby (panel b), lMaltby (panel c), and Avrett2015spot (panel d). The red lines with asterisks show the cutoff values determined from examining phase-difference spectra in numerical simulations with a vertical magnetic field of 3000 G. Color lines indicate the analytical cutoff frequency computed using Eq. (1) (blue line), Eq. (2) (green line), Eq. (3) (violet line), and Eq. (4) (red line). All the plotted heights are in the β <  1 region.

The temperature gradients above the temperature minimum (around z ∼ 800 km) also leave an imprint on the cutoff frequencies. For dark umbrae (eMaltby, Fig. 6a) the cutoff value at z ∼ 800 km is sharply reduced. In the case of mMaltby and lMaltby, the cutoff is reduced at slightly higher layers, in agreement with the differences presented in the temperature stratification of the models. This sudden cutoff variation is also captured by the theoretical formulae, but there are again significant differences between the analytically estimated values and those measured in the simulations.

In the umbral model Avrett2015spot, the general properties of the cutoff stratification are similar to those described for the umbral models from Maltby et al. (1986). As expected from its temperature profile, the cutoff is reduced at a lower height than in the other models (z ∼ 500 km). For this atmosphere, the cutoffs from Eqs. (2)–(4) predict a very high peak at photospheric layers. This peak has not been measured in the numerically determined cutoff. Equation (2) gives imaginary cutoff values at z ∼ 450 km. At the chromosphere, different theories estimate a wide range of cutoff values, and they converge toward the higher layers. The measured cutoff is somewhat in between the predictions from the analytical models.

4.2. Dependence on the magnetic field

The effect of the vertical magnetic field on the umbral cutoff frequency was evaluated by performing a set of 24 numerical simulations. They correspond to six simulations with different values of the field strength for each of the four umbral models under study. The chosen sample of magnetic field strengths spans from 500 to 3000 G with a step of 500 G. Figure 7 illustrates the measured cutoff stratifications and their comparison with the theoretical estimates of the slow-mode cutoff given by Eq. (4).

Fig. 7. Variation in numerically determined cutoff frequency with height in the umbral models eMaltby (panel a), mMaltby (panel b), lMaltby (panel c), and Avrett2015spot (panel d). Each color corresponds to atmospheres permeated by a different strength of the vertical magnetic field: 500 G (black), 1000 G (violet), 1500 G (light blue), 2000 G (green), 2500 G (orange), and 3000 G (red). The vertical dashed lines mark the height where the plasma-β is unity, following the same color code as the cutoff values. For some of the atmospheres, the line indicating the β = 1 height is not visible because it is below z = 0. Solid lines show the cutoff frequency of slow magnetoacoustic waves in atmospheres, as given by Eq. (4). Their color indicates the strength of the magnetic field.

At the deep photosphere, the effect of the magnetic field on the cutoff of the sunspot models is negligible. At these layers, the plasma-β is around unity. Its value mainly depends on the magnetic field strength. In the case of the Maltby et al. (1986) models, the atmospheres with a magnetic field below 2000 G exhibit a β >  1 in at least some of the analyzed photospheric heights, whereas atmospheres with stronger fields are in the β <  1 regime at all heights above z = 0. Our results indicate that at layers in which magnetic pressure and gas pressure are comparable, the cutoff frequency does not depend significantly on the magnetic field. In the same way, mode conversion is not relevant for the cutoff because at a fixed photospheric height, strongly magnetized atmospheres (with β <  1, i.e., the vertical velocity signal is a slow-mode wave) and weakly magnetized atmospheres (β >  1, the vertical velocity corresponds to oscillations of the fast mode) exhibit similar cutoff values.

The differences in the cutoff associated with the magnetic field are found at higher layers. As previously discussed, the cutoff in sunspot models decreases from a relatively high photospheric value to lower values at the chromosphere. The location of this variation depends on the atmospheric stratification. In addition, the magnetic field changes the height where this decrease in the cutoff starts. The lower the magnetic field, the deeper the beginning of the cutoff reduction. In the Maltby et al. (1986) models, between z = 500 km and z = 1000 km the atmospheres with a weaker magnetic field show a lower cutoff frequency. For Avrett2015spot, the region where magnetic field affects the cutoff is shifted to heights between z = 200 km and z = 500 km. This is in contrast with the analytical results from Eq. (4), where the main differences produced by the magnetic field are found at lower layers, around z = 300 km for the Maltby et al. (1986) atmospheres and at z = 100 km for Avrett2015spot. However, the effects of the magnetic field on the cutoff go in the same direction in both numerical and theoretical estimations, with a reduced cutoff associated with lower magnetic field strengths.

At the high chromosphere (above z ∼ 1400 km), the cutoff frequencies measured for the Maltby et al. (1986) models show striking differences between the weakly magnetized umbrae (B = 500 G) and the remaining cases (with magnetic fields stronger than or equal to 1000 G). The cases with B = 500 G exhibit an increase in the cutoff. This is produced by the steep temperature gradient at the beginning of the transition region. In these simulations, only a few hundred kilometers of the transition region are included in the Maltby et al. (1986) background models because above z = 1800 km a constant temperature is set (see the dashed lines in Fig. 1b). The simulation using the umbral model eMaltby contains the larger temperature contrast (around 1400 K temperature difference between z = 1600 km and z = 1800 km), and it exhibits the larger increase in the chromospheric cutoff (black line in Fig. 7a). In contrast, for lMatlby, our simulation only includes a temperature contrast of 600 K, and the chromospheric cutoff increase is more modest (black line in Fig. 7c). The presence of magnetic fields stronger than 1000 G inhibits the cutoff increase associated with the steep gradients at the base of the transition region. The simulation of the Avrett2015spot umbra does not include any steep gradient at the high chromosphere (dashed blue line in Fig. 1b). In this case, at high layers where the atmosphere is strongly dominated by the magnetic field, the cutoff is similar for all the simulations, despite their different field strengths. In this sense, this agrees with the predictions of the cutoff from Eq. (4), which do not show a dependence on the magnetic field at the chromosphere.

4.3. Dependence on the field inclination

We chose to focus on the umbral model mMaltby because it represents an average umbra and is probably the most widely used umbral atmospheric model. For these simulations, we set a magnetic field strength of 3000 G in order to ensure a β <  1 in the whole umbral atmosphere. We compared two simulations, one of them with a vertical magnetic field (θ = 0°), and the other with θ = 10°. Because we study umbral atmospheres, where the magnetic field is mostly vertical, we restricted the analysis to moderate field inclinations.

The numerical cutoff frequency was determined following the same procedures as described in Sect. 2.3.1. In the case with an inclined magnetic field, we took the horizontal displacement of the waves as they propagate upward along magnetic field lines into account. In order to be consistent with the cases with a vertical magnetic field, the distance between the velocity signals used for the computation of the phase spectra was maintained at Δh = 20 km. However, it was measured along the path of the waves, that is, departing at an angle θ from the vertical. Thus, the horizontal displacement is Δhsinθ, and the height difference is Δhcosθ. The velocity field obtained as the output of the simulation was interpolated to these locations, in order to extract the velocity signal required for computing the phase-difference spectra.

Figure 8 illustrates the comparison of the cutoff values determined for the umbral atmosphere with vertical (red) and inclined (blue) magnetic field. The dashed blue line shows the cutoff of the vertical case multiplied by cosθ. This is the expected cutoff frequency of the inclined case when the reduced gravity of the ramp effect alone modifies the wave propagation. The dashed blue line agrees with the measured cutoff around z = 500 km and above z ∼ 1000 km. These are the regions where the stratification of the umbral model presents small temperature gradients (red line in Fig. 1b). The region around z = 500 km corresponds to the plateau of constant temperature around the temperature minimum, whereas between z ∼ 1000 km and the transition region, the temperature shows a progressive increase. Conversely, at heights where sudden changes in the temperature are found, the measured cutoff is lower than expected from the ramp effect. This is clearly seen at z ∼ 120 km and especially between z = 700 km and z = 1000 km. Slow-mode waves propagate along field lines. Along this path, the temperature contrast that they experience is more gentle than that of vertically propagating waves. This way, the effects of temperature gradients in the cutoff are reduced for waves propagating at a certain angle from the vertical. This reduction in the cutoff is added to that predicted by the reduced gravity along inclined field lines.

Fig. 8. Variation in numerically determined cutoff frequency with height in the umbral model mMaltby permeated by a 3000 G magnetic field. Red asterisks show the measurements for the vertical magnetic field and blue asterisks those for a field inclination from the vertical of θ = 10°. Dashed blue line corresponds to the cutoff of the case with a vertical magnetic field multiplied by the cosine of the inclination θ = 10°.

4.4. Effect of radiative losses on the cutoff

Radiative losses were implemented in an umbral simulation following the same approach as described in Sect. 3.3. According to Eq. (6), the radiative cooling time is significantly higher in the umbra than in the quiet Sun, which means that umbral wave propagation is closer to adiabatic propagation. A minimum τ_R ∼ 110 s is found at z = 200 km, and a maximum value of τ_R ∼ 1800 s is obtained at z = 900 km.

Figure 9 shows the measurements of the cutoff for simulations in the adiabatic regime and with radiative losses, all of them using the umbral model mMaltby. As an example, the figure illustrates an umbra with a weak magnetic field (B = 500 G) and an umbra with a strong magnetic field (B = 2500 G). In both cases the field is vertical.

Fig. 9. Variation in numerically determined cutoff frequency with height in umbral model mMaltby with the radiative looses turned on (dashed lines) and off (solid lines with asterisks). Black lines correspond to atmospheres permeated by a 500 G vertical magnetic field, and orange lines to atmospheres with a 2500 G field strength.

The introduction of radiative losses reduces the cutoff in the deep photosphere (around z = 200 km) and at the base of the chromosphere (between z ∼ 950 km and z ∼ 1100 km, above this height, adiabatic propagation is implemented). This agrees with the effects found in the quiet-Sun simulations. However, around z = 600 km, the radiative transfer produces a small increase in the cutoff, independently of the magnetic field strength of the umbra. In addition, the increase in cutoff found in the weakly magnetized umbra associated with the temperature gradients at the base of the transition region is more pronounced in the simulation where radiative losses are turned on, although in these regions the wave propagation is adiabatic. The causes of this result are not clear.

5. Discussion

It is fundamental for understanding wave propagation in atmospheres stratified by gravity to determine the cutoff frequency. Theoretically, different representations of the wave equation have been derived, and they lead to various forms of the cutoff frequency, which yield significant differences in the cutoff stratification of solar models (Schmitz & Fleck 1998). In this work, we have carried out an alternative approach to determine the cutoff based on the use of numerical simulations. The cutoffs were determined from the analysis of phase-difference spectra between the velocity signals measured at two different atmospheric heights. This method has previously been employed in the context of multi-height ground-based observations (e.g., Centeno et al. 2006; Wiśniewska et al. 2016; Felipe et al. 2018) and differs from other works that determined the cutoff in numerical simulations from the evaluation of the height-dependent dominant frequency in power spectra (Murawski et al. 2016; Murawski & Musielak 2016). A small height difference of Δh = 20 km was chosen between the two heights employed for the computation of the phase spectra, which allowed a detailed sampling of the vertical variation of the cutoff. Our quantification of the cutoff is not affected by details in the derivation of the wave equation. In addition, it allows an easy addition of several layers of physics, such as the magnetic field or the radiative losses, which can be difficult to model analytically. We found significant differences between the numerically determined cutoff and the cutoff derived theoretically.

Our results do not clearly favor any of the analytical formulae. Neglecting the effects of magnetic fields and radiative losses, the cutoff measured for the deep photospheric layers of quiet-Sun models is better captured by Eq. (2), which is commonly employed in helioseismic works. However, the sharp spike exhibited by this expression around z = 0 is absent in the numerical cutoff. Our measurements validate the use of alternative cutoff expressions where the spike is absent. This is the approach followed by several works using the Wentzel, Kramers, and Brilloiuin (WKB) approximation (Cally 2007; Moradi & Cally 2008), which is only applicable for wavelengths shorter than the characteristic scales of the medium. It is therefore inconsistent with the sudden variations in the cutoff given by Eq. (2). We found that in quiet-Sun atmospheres permeated by moderate magnetic field strengths (between 50 and 300 G), the application of the simple formula for isothermal atmospheres (Eq. (1)), assuming that it is stratified according to the local temperature, provides the best estimate of the actual cutoff values for adiabatic waves.

Some recent works have measured the stratification of the cutoff in quiet-Sun (Wiśniewska et al. 2016) and umbral (Felipe et al. 2018) atmospheres based on the analysis of phase-difference spectra from multi-height observations employing several spectral lines. Wiśniewska et al. (2016) estimated that waves with frequency as low as ∼4.2 mHz can propagate in the quiet-Sun photosphere (around z = 270 km) and that the cutoff increases up to ∼5 mHz in the next 100 km. According to our findings, this qualitatively agrees with the propagation of adiabatic waves in an atmosphere permeated by a magnetic field with a specific strength around 130 G (blue lines in Fig. 3) or with nonadiabatic wave propagation, regardless of the magnetic field. The second alternative seems more plausible. For umbral atmospheres, Felipe et al. (2018) found a maximum in the cutoff frequency of 6 mHz around z ∼ 500 km. At deeper photospheric heights, this decreases to 5 mHz at z ∼ 250 km. This variation in the umbral photospheric cutoff is not captured by any model with adiabatic propagation (Fig. 7). In contrast, if radiative losses are turned on, the cutoff stratification of the model agrees significantly well with the observational results (Fig. 9). Our cutoff estimate also reproduces the chromospheric value slightly above 3 mHz measured by Felipe et al. (2018). Both in quiet-Sun and umbral atmospheres, the role of radiative losses is fundamental for understanding the cutoff frequency of the observed waves. We neglected the effect of wave reflections at the transition region. This strategy allows a direct evaluation of the analytical solutions from theoretical models, but challenges the comparison with observational results. A comparison like this must therefore be interpreted with care.

Our results are apparently in contrast with the findings from Heggland et al. (2011). These authors performed numerical simulations with a sophisticated treatment of radiative losses and found that the losses have little effect on the propagation of low-frequency waves. In this work, we approximated the radiative losses with Newton’s cooling law. This method offers a simple parameterization of the radiative transfer because it is characterized by the radiative cooling time, and suits our purpose of comparing the numerical cutoffs with those derived from the theory. Our findings agree with the simulations from Khomenko et al. (2008), who also employed Newton’s cooling law and found that the energy exchange by radiation can reduce the cutoff frequency. Here, we employed a variable cooling time as given by Spiegel (1957). At the photosphere, where this expression is reliable, we obtain a significant reduction in the cutoff. At the high photosphere and low chromosphere, the effect of radiative transfer on the cutoff is lower (at these heights, the radiative cooling time is higher, therefore propagation is closer to adiabatic), and waves still need to bypass a cutoff of 5 mHz to reach the chromosphere. This agrees with the results from Heggland et al. (2011). However, many observations have reported long-period waves in regions with vertical magnetic field (e.g., Centeno et al. 2009; Stangalini et al. 2011; Rajaguru et al. 2019). The reduction of the cutoff produced by radiative transfer is the best candidate to explain those observations.

Heating by waves is one of the mechanisms proposed to balance the radiative losses of the outer solar atmospheric layers (Biermann 1946; Schwarzschild 1948). It remains the solid candidate to explain the chromospheric heating (Jefferies et al. 2006; Kalkofen 2007; Bello González et al. 2010; Kanoh et al. 2016; Grant et al. 2018; Abbasvand et al. 2020), although some works have claimed that the acoustic wave flux is insufficient to heat quiet-Sun (Fossum & Carlsson 2005, 2006; Carlsson et al. 2007) and sunspot (Felipe et al. 2011) chromospheres. The determination of the propagating nature of compressive waves has fundamental implications to estimate their contribution to the heating. Recently, Rajaguru et al. (2019) measured an acoustic energy flux in the 2−5 mHz frequency range between the upper photosphere and lower chromosphere of quiet-Sun regions that was higher than previous estimates. These results agree with those reported by Jefferies et al. (2006), who showed that an inclined magnetic field can facilitate the upward propagation of low-frequency waves (below 5 mHz) through the ramp effect, providing a significant source of energy to the solar chromosphere. However, Rajaguru et al. (2019) determined that the frequency of the waves that carry this additional energy flux is below the theoretical cutoff value even when the effect of magnetic field inclination is taken into account. Our analyses can qualitatively explain the observational measurements from Rajaguru et al. (2019) as a reduction of the cutoff produced by radiative losses. We found that at heights between z = 200 km and z = 400 km, radiative losses can reduce the cutoff to values below 4 mHz, although Rajaguru et al. (2019) detected propagation at even lower frequencies, in the range 2−4 mHz.

Our results indicate that quiet-Sun regions with a stronger magnetic field can show a lower cutoff frequency (Fig. 4), in agreement with Jefferies et al. (2019), although in our measurements, the detailed variation in cutoff with field strength depends on the atmospheric height and stratification. In addition, our simulations prove that the lower cutoff in regions with an inclined magnetic field (and β <  1) is due to the reduced gravity and the lower temperature gradients along the field lines (Fig. 8). The effect of the gravity was predicted by Bel & Leroy (1977). It has been confirmed through observations (McIntosh & Jefferies 2006; Jefferies et al. 2006; Rajaguru et al. 2019) and numerical simulations (De Pontieu et al. 2004; Hansteen et al. 2006; Heggland et al. 2011). To the best of our knowledge, the additional cutoff reduction produced by inclined magnetic fields in regions with significant gradients in the temperature is reported here for the first time. Although the reduced cutoff associated with magnetic fields in relatively quiet regions can be seen as a mechanism to supply additional acoustic flux to the chromosphere, several works have found that in the surrounding regions the upward energy flux is reduced (Vecchio et al. 2007; Rajaguru et al. 2019). Jefferies et al. (2019) identified a larger cutoff around magnetic regions. This is consistent with the increase in cutoff frequency as solar activity increases, as measured from low-degree modes (Jiménez et al. 2011). The cutoff frequency is a fundamental parameter for helioseismology analyses because it determines the upper boundary of the p-mode resonant cavities. It has been found to be a major contributor to the travel-time shifts measured in sunspots using local helioseismic methods (Lindsey et al. 2010; Schunker et al. 2013; Felipe et al. 2017).

6. Conclusions

We have examined the wave propagation between the solar photosphere and chromosphere using numerical simulations. We focused on the evaluation of the cutoff frequency stratification, that is, we determined the minimum frequency of propagating waves as a function of height in various solar models. The cutoff frequency was derived from examining phase-difference spectra, by detecting the lowest frequency where a positive phase difference is measured. All these analyses were performed for a set of standard solar atmospheric models representing quiet-Sun and umbral regions. We evaluated several theoretical expressions that are commonly employed to derive the cutoff. Our results show that although the analytical cutoff frequencies exhibit a qualitative agreement with the actual values measured in the numerical simulations, none of them provides a notable match. The use of more refined cutoff formulae (e.g., Eqs. (2)–(4)) does not lead to significant improvements over the original cutoff expression for isothermal atmospheres (Eq. (1)). The latter expression, when applied accounting for its local variations in atmospheres with nonconstant temperature, gives a better result in quiet-Sun regions with moderate magnetic field strength. The validity of the analytical expressions is also challenged by the assumptions employed in their derivation. Some of the most commonly used acoustic cutoff formulae neglect the effects of magnetic fields or radiative losses, which are fundamental for understanding wave propagation in the solar atmosphere. We found that radiative losses greatly reduce the photospheric cutoff frequency, and can partially explain some recent observations of propagation of low-frequency waves at photospheric heights in the quiet Sun (Wiśniewska et al. 2016; Rajaguru et al. 2019) and sunspots (Felipe et al. 2018).

Acknowledgments

Financial support from the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation and Universities (MCIU) and the European Regional Development Fund (FEDER) under grant with reference PGC2018-097611-A-I00 is gratefully acknowledged. The authors wish to acknowledge the contribution of Teide High-Performance Computing facilities to the results of this research. TeideHPC facilities are provided by the Instituto Tecnológico y de Energías Renovables (ITER, SA). URL: http://teidehpc.iter.es.

References

All Figures

Fig. 1. Temperature stratification of the solar models. Left panel: quiet-Sun temperature distribution as given by the VALC (black line), FALC (red line), and Avrett2015QS (blue line) models. Right panel: sunspot temperature stratification from eMaltby (green line), mMaltby (red line), lMaltby (black line), and Avrett2015spot (blue line) atmospheres. In both panels, the solid lines represent the original models and dashed lines correspond to the temperature profiles we employed to compute the numerical simulations.

In the text

Fig. 2. Average phase-difference spectra between the vertical velocity signal at z = 880 km and z = 900 km measured from the simulation of the VALC model with a vertical magnetic field of 130 G. A positive phase difference indicates upward wave propagation. Error bars show the standard deviation of the averaged data. The vertical dotted line marks the value of the cutoff frequency as determined from examining the phase spectra.

In the text

Fig. 3. Variation of the cutoff frequency with height in the quiet-Sun models VALC (panel a), FALC (panel b), and Avrett2015QS (panel c). The lines with asterisks show the cutoff values determined from the examination of the phase-difference spectra in a numerical simulation with a vertical magnetic field of 5 G (black) and 300 G (red). Color lines indicate the analytical cutoff frequency computed using Eq. (1) (blue line), Eq. (2) (green line), Eq. (3) (violet line), Eq. (4) with 5 G magnetic field strength (black line), and Eq. (4) with 300 G magnetic field strength (red line). The vertical dashed lines mark the height where the plasma-β is unity for the models with a field strength of 5 G (black) and 300 G (red).

In the text

Fig. 4. Variation of the numerically determined cutoff frequency with height in the quiet-Sun models VALC (panel a), FALC (panel b), and Avrett2015QS (panel c). Each color corresponds to atmospheres permeated by a different strength of the vertical magnetic field: 5 G (black), 10 G (violet), 50 G (green), 130 G (blue), and 300 G (red). The vertical dashed line marks the height where the plasma-β is unity following the same color code as the cutoff values.

In the text

	Fig. 5. Variation in numerically determined cutoff frequency with height in the quiet-Sun models VALC with the radiative looses turned on (dashed lines) and off (solid lines with asterisks). The color indicates the magnetic field strength, following the same color code as in Fig. 4: 5 G (black), 130 G (blue), and 300 G (red).
In the text

Fig. 6. Variation in cutoff frequency with height in the umbral models eMaltby (panel a), mMaltby (panel b), lMaltby (panel c), and Avrett2015spot (panel d). The red lines with asterisks show the cutoff values determined from examining phase-difference spectra in numerical simulations with a vertical magnetic field of 3000 G. Color lines indicate the analytical cutoff frequency computed using Eq. (1) (blue line), Eq. (2) (green line), Eq. (3) (violet line), and Eq. (4) (red line). All the plotted heights are in the β <  1 region.

In the text

Fig. 7. Variation in numerically determined cutoff frequency with height in the umbral models eMaltby (panel a), mMaltby (panel b), lMaltby (panel c), and Avrett2015spot (panel d). Each color corresponds to atmospheres permeated by a different strength of the vertical magnetic field: 500 G (black), 1000 G (violet), 1500 G (light blue), 2000 G (green), 2500 G (orange), and 3000 G (red). The vertical dashed lines mark the height where the plasma-β is unity, following the same color code as the cutoff values. For some of the atmospheres, the line indicating the β = 1 height is not visible because it is below z = 0. Solid lines show the cutoff frequency of slow magnetoacoustic waves in atmospheres, as given by Eq. (4). Their color indicates the strength of the magnetic field.

In the text

Fig. 8. Variation in numerically determined cutoff frequency with height in the umbral model mMaltby permeated by a 3000 G magnetic field. Red asterisks show the measurements for the vertical magnetic field and blue asterisks those for a field inclination from the vertical of θ = 10°. Dashed blue line corresponds to the cutoff of the case with a vertical magnetic field multiplied by the cosine of the inclination θ = 10°.

In the text

	Fig. 9. Variation in numerically determined cutoff frequency with height in umbral model mMaltby with the radiative looses turned on (dashed lines) and off (solid lines with asterisks). Black lines correspond to atmospheres permeated by a 500 G vertical magnetic field, and orange lines to atmospheres with a 2500 G field strength.
In the text