Internal Gravity Waves in the Magnetized Solar Atmosphere. II. Energy Transport

IGWs are identified and distinguished from acoustic waves by their properties in the ${k}_{h}-\omega$ dispersion relation. In a compressible stratified nonmagnetic medium, the two types of waves are separated into two distinct branches with a band of evanescent disturbance separating them in the diagnostic diagram. A characteristic feature that differentiates the two waves is the opposite nature of their phases for waves propagating in the same direction. Phase difference spectra (or phase spectra) computed from the cross-spectrum of the velocity measurements at two heights in the atmosphere clearly reveal this difference in the k_h − ω diagnostic diagram. In what follows, we shall compare the four models with respect to the phase spectra they show. We select two representative pairs of heights, with the first pair in the lower atmosphere close to where these waves are thought to be generated and a second pair higher up in the atmosphere in order to study their propagation. The velocity–velocity (v − v) phase spectra are determined from the vertical component of the velocity for these two pairs of heights.

First, we look at the pair of heights, z = 100 km and z = 140 km, for all four models in Figure 1(a). The boundary that separates the two different regimes of wave propagation is shown as black curves. The dashed black curves represent the propagation boundaries computed from the dispersion relation for the lower height, and the solid curves are those for the upper height. Without the effect of magnetic fields, the IGW branch of the acoustic-gravity wave spectrum occupies the lower part of the diagram, below the lower propagation boundary marked by black curves running from the origin to around 4 mHz. In the rest of the paper, we will focus on this region of the diagnostic diagram. The gray curve is the dispersion relation, $\omega =\sqrt{{{gk}}_{h}}$, of the surface gravity waves. It is clear from Figure 1(a) that all the models, irrespective of magnetic fields, have the same distribution in the IGW regime, shown as the greenish region below the IGW propagation boundary, showing their characteristic downward phase. The only difference one can note is in the surface gravity waves along their dispersion relation (marked in light gray), which we will not discuss in this paper.

When we look at a slightly higher pair of heights, z = 560 km and z = 600 km as shown in Figure 1(b), we notice that there are differences in the phase spectra between the models, unlike what we see in the case of the lower pair of heights (compare with Figure 1(a)). The nonmagnetic model and the weak field model still show upward-propagating (negative phase difference) IGWs, whereas the models with stronger magnetic fields show downward-propagating IGWs as is evident from the positive phase difference (reddish region) in the IGW region of the phase spectra. In summary, despite showing similar phase spectra in the lower atmosphere, irrespective of the average flux density, we see that the phase spectra are modified by the fields in the higher layers and show downward-propagating waves in the stronger-field case.

To understand the energy transport by waves, we will now turn our attention to the energy flux spectra represented on the k_h − ω dispersion relation diagram. The time-averaged active component of the mechanical flux is given by the perturbed pressure–velocity (Δp–v) co-spectrum (see Equation (4) of Paper I). In Figure 2, we show the vertical flux of the active mechanical energy density for the four models at two representative heights. We have chosen z = 360 km and z = 700 km, in order to reveal the effects of the strongly varying average plasma-β (the ratio of gas pressure to magnetic pressure) with the magnetic flux density at the upper of the two heights. The leftmost panel in each figure shows the energy flux spectrum for the nonmagnetic model (Sun-v0), and the remaining panels are for the magnetic models Sun-v10, Sun-v50, and Sun-v100, from left to right. Examining the region below the lower propagation boundary of the k_h − ω diagram in the lower atmosphere (z = 360 km; Figure 2(a)), we see that all the fluxes are positive (yellow−red color scale). This clearly shows that the energy transport is predominantly in the upward direction on the order of 1–10 W m⁻² in the given frequency wavelength range. Comparing this to Figure 1(a), we clearly see here the signature of IGWs with their vertical component of the phases being opposite to the propagation direction. This is in contrast to the acoustic waves, occupying the region above the upper propagation boundary, which show the same direction for the phase and energy transport.

However, as we go higher up in the atmosphere, at z = 700 km (shown in Figure 2(b)), we notice that for stronger fields the energy flux is downwardly directed (green region) in regions where the excited IGWs (see Figure 1(a)) are located. The height of 700 km corresponds to a low plasma-β region for the case of strong magnetic fields (for B_z typically greater than 50 G) and a high-β region for the case of weak magnetic fields, which is an important difference that will be examined later in Section 5.1. This shows us that as the fields get stronger, there are more downward-propagating IGWs in the upper atmosphere transporting their energy downward. We would like to caution the reader that for this height and the stronger-field cases the propagation boundary in Figure 2(b) is just shown to guide the eye but otherwise has lost its physical meaning. Instead, one would have to consider the complete MAG dispersion relation, which is different from the acoustic-gravity dispersion relation. The lower part of the ${k}_{h}-\omega$ dispersion relation diagram corresponds to the slow-MAG branch, where the flux remains upward directed and gets stronger with increasing field strength (the red regions below ν = 1 mHz).

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We will now look at the active component of the Poynting flux, which can reveal more information about propagating Alfvén waves in the domain. The time-averaged flux of electromagnetic energy is given by the real part of the complex Poynting vector (see, e.g., Landau et al. 1984). In our case, this can be readily obtained by taking the co-spectrum of the vector product of the perturbed field, ${{\boldsymbol{B}}}^{{\prime} }$, and ${\boldsymbol{v}}\times {{\boldsymbol{B}}}_{0}$. In Figure 3, we show the spectra of the vertical component of the Poynting flux from the magnetic models at two different heights, z = 360 km and z = 700 km. Although the fluxes are a few orders of magnitude smaller than the mechanical flux presented in Figure 2, it is interesting to look at how they behave as the magnetic fields become stronger, as well as consider what this may tell us about propagating Alfvén waves.

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Looking at Figure 3(a), we see that for the lower layer, which is located in the high-β regime for all models, the Poynting flux in the IGW/gravito-Alfvén regime (below the lower propagation boundary) shows a positive sign, meaning upwardly directed. Compare this to the phase difference and the mechanical flux transport, both of which show that the energy is predominantly transported in the upward direction.

But as we go higher up in the atmosphere (see Figure 3(b)), we notice that these fluxes are negative in the regions where the IGWs are excited by the convection, which means that the Poynting flux, similar to the mechanical flux, is downwardly directed. This effect gets stronger the stronger the field (compare Sun-v50 and Sun-v100). This clearly shows us that there are barely any upward-directed Alfvén waves. Here again, we remind that the complete MAG dispersion relation is not considered.

We will first look at the physical structure and dynamics of the magnetoconvection occurring in the different models that we have computed. As noted in Section 2, the model atmospheres differ in the initial magnetic flux density that was added to an otherwise nonmagnetic medium. We build all the models from the same thermally relaxed initial state, as a result of which the different models show very similar thermal properties. Significant differences are evident only when comparing the properties of the atmosphere where magnetic effects dominate. The effect of magnetic field is seen in the propagation properties of the waves only in the higher layers, while the generation of the waves in the near-surface region remains the same across the different models as evident from the phase and energy flux spectra presented in the previous section.

Let us first look at the thermal profile of the models that we study here. Figure 4 shows the mean temperature (left panels) and specific entropy (right panels) as a function of height (black curve) for a single snapshot taken 4 hr after the start of the simulation for the nonmagnetic model (top panels) and a magnetic model with the largest average magnetic flux density (bottom panels). In the background we show the excursion of the corresponding temperature and specific entropy observed at the same time instance over the whole domain. The curves are color-coded to show the vertical velocity (normalized by the local sound speed) in the voxels along the z-direction, where blue means an upward velocity and red means downward velocity. For clarity, the position of z = 0 km height level is shown with a gray vertical line to mark the separation of the convective and radiative regions.

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Just below the optical surface, i.e., below z = 0 km, the ascension of hot buoyant convective parcels is clearly visible as blue shading above the mean temperature profile. Regions with lower temperatures than the mean, appearing red, are the cool denser parcels flowing down in the intergranular lanes. Above the surface, from z = 0 to about 100 km, we see that the hot voxels have overshot slightly into the stable layers, beyond which they radiatively cool down and are dominated by downward parcels as evident from the red shade. This overshooting hot material, which we see in all the models, is thought to result in the excitation of the IGWs. Further up, we see a mixture of upward- and downward-flowing plasma.

The sudden drop in temperature near the surface due to the switch from a convective interior to a radiative atmosphere is clearly evident in all the models. In the right panels of Figure 4, we can see the entropy-rich (above the mean value) upwellings of the convective parcels in blue shading below the z = 0 km level. Once these parcels enter the stably stratified radiation zone, they radiate away their heat and thereby lose entropy and fall back. This can be seen in the entropy-deficient parcels moving downward (red shade) in the convection zone. Just above the surface, the downfalling matter is entropy rich as a result of heating from below. In the magnetic models, however, there is additional, presumably magnetic heating that results in slightly higher entropy of the downward-falling material relative to the nonmagnetic model. The steady increase in entropy as a function of height in the atmosphere is visible in all models as a consequence of radiative absorption. It is clear from Figure 4 that the launching regions of IGWs are the near-surface overshooting, which generates predominantly upward-propagating waves above this layer. However, these regions are the places where we also see kilogauss fields accumulated in intergranular lanes. The question is how they influence the wave generation.

Figure 5 shows the emergent bolometric intensity from the four models taken 4 hr after the start of the simulation. Also shown is the absolute magnetic flux density in color scale for the magnetic models. The accompanying animation shows the time evolution of the granular pattern and the magnetic map over the entire duration of the simulation. One notices that the weakest magnetic model, Sun-v10, has very few concentrations of kilogauss fluxes, mostly in the vertices where several granules meet, where most of the fluxes accumulate due to strong convergent flows. Models with stronger initial field strengths have fluxes in the entire periphery of individual granules as well. Comparing this with the phase spectra of the IGWs, which do not differ much among models, clearly shows that the kilogauss fields that are present in the intergranular lanes do not have any influence on the generation of IGWs.

However, this is not the case for the upper layers, where we do see an influence of the magnetic fields on the phase spectra and the energy flux spectra. The most important property of a magnetic atmosphere relevant to the study of waves is the plasma-β. For magneto-atmospheric waves, it is well known that the β = 1 surface is a crucial layer across which mode coupling can occur. At locations of strong magnetic field the β surface dips, so that waves propagate differently within small-scale flux concentrations compared to the propagation in nearly field-free quiet regions. For IGWs, changes on these small length scales do not have a major effect; rather, it is the average properties that matter. Since we consider different magnetic field strengths, a clear distinction occurs in the average plasma-β of these models. Figure 6 shows the height of the plasma-β = 1 layer from the same snapshot for each models. This shows us the height above which the IGWs are likely to be strongly affected and how the height varies with varying average magnetic flux density.

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Figure 7 shows the plasma-β profiles as a function of height in the three models taken from the same snapshot as used in Figure 4. The average value as a function of height is marked by the black curve, and the β = 1 value is marked by the dotted line to aid the reader. The background curves show the profiles in all the domains and are color-coded to represent the inclination of the field with respect to the vertical direction. For the weak magnetic field case, Sun-v10, the gas pressure dominates magnetic pressure nearly everywhere in the atmosphere except in the very top part, resulting in a significant portion of the atmosphere being a high-β atmosphere. But as the magnetic fields get stronger, an extended part of the atmosphere in the top layers becomes low-β. The average plasma-β profile moves down in the atmosphere. This is evident from Figure 7, where we see that the average level of β = 1 is at z = 800 km for the Sun-v50 case and at z = 600 km for the Sun-v100 case, compared to z = 1100 km for the Sun-v10 case. These heights are indicated as red vertical lines to aide the reader. A consequence of this is that the IGWs in the low field strength case can propagate unhindered as far out as 1000 km, whereas for the stronger field strength case the height limit is around 700 km. It is clear from Figure 7 that above the β = 1 layer the fields are mainly vertical (green shade), so that with increasing field strength, models show an extended region of predominantly vertical magnetic fields, which is a consequence of the top boundary condition.

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The propagation properties of IGWs can be clearly understood in terms of the phase spectra presented in Section 4.1. Despite the presence of kilogauss fields in the surface layers, the similar nature of the phases in the IGW regime between nonmagnetic and all magnetic field models confirms that the granular updrafts are the main source region of these waves, as the differences between the models concern the intergranular lanes only. However, the higher layers show a completely different behavior in the phase spectra, which is a result of the differences in the average plasma-β with height of the three magnetic models.

Let us now turn to the main objective of this paper. We are interested in estimating the energy transported by wave-like motions, particularly internal wave fluctuations, and in gaining insight on how magnetic fields may influence this transport. The primary reason why waves are considered important is because they are a carrier of mechanical energy and momentum between the deep photosphere and the upper solar atmosphere without the need of any material transport. Knowing the upward flux of mechanical energy associated with the waves as a function of height can tell us at which layer the energy gets utilized. As we are looking at magneto-atmospheric waves, we also need to consider the electromagnetic work done by the waves that is given by the associated Poynting flux. The Poynting flux is then likely converted into thermal and bulk kinetic energy by magnetic dissipation and by the action of the Lorentz force (Spruit & Drenkhahn 2004). In the following, we will discuss how wave energy fluxes are estimated from real observations and compare them to the fluxes calculated using a different approach from our simulations.

The flux estimates for various observed wave phenomena rely on the fact that the energy flux can be determined directly from the mean energy density and the group velocity of the wave under consideration. Although this is generally true for any type of waves, it is often the quantities that go into this simple expression that are prone to uncertainties. First, a reliable estimate of the mean energy density depends on how closely the model atmosphere, from where the mass densities are determined, matches the observed region. Second, the group velocity of a simple observed wave is often difficult to estimate, let alone for a dispersive wave for which the concept of group velocity breaks down. In most cases, it is not explicitly stated as to where the group velocity estimates come from (Worrall 2012).

For the acoustic waves, the group velocity can be obtained using the dispersion relation. The vertical component of the group velocity is given as (Fossum & Carlsson 2006; Bello González et al. 2009)

$$\begin{eqnarray}&&{v}_{g,z}=\displaystyle \frac{\partial \omega }{\partial {k}_{z}}={c}_{s}\sqrt{1-{({\omega }_{\mathrm{ac}}/\omega )}^{2}},\end{eqnarray} \tag{ 1 }$$

which asymptotes to the local sound speed (c_s) for large frequencies, while becoming zero for lower frequencies as one approaches the acoustic cutoff (ω_ac). Assuming a speed of sound and the acoustic cutoff frequency from a model atmosphere, one can estimate the energy flux in the desired frequency bin. Bello González et al. (2009) showed that at a height of 250 km the low-frequency range (5–10 mHz) contributes more to the cumulative acoustic flux of  3000 W m⁻² than the flux in the high-frequency (10–20 mHz) range. However, Fossum & Carlsson (2005, 2006) report that the fluxes in the acoustic range are insufficient by a factor of 10 to balance the radiative losses.

For the IGWs, it is quite tricky, as these waves do not propagate purely vertically. The vertical component of the group velocity (${v}_{g,z}$) is given in the incompressible limit as (Press 1981; Pinçon et al. 2016)

$$\begin{eqnarray}&&{v}_{g,z}=\displaystyle \frac{\partial \omega }{\partial {k}_{z}}={\left(\displaystyle \frac{\omega }{N}\right)}^{2}\displaystyle \frac{\sqrt{{N}^{2}-{\omega }^{2}}}{{k}_{h}},\end{eqnarray} \tag{ 2 }$$

which can be written in terms of the vertical phase velocity (${v}_{\mathrm{ph},z}$) as (Kneer & Bello González 2011)

$$\begin{eqnarray}&&{v}_{g,z}=-{v}_{\mathrm{ph},z}{\sin }^{2}(\omega /N).\end{eqnarray} \tag{ 3 }$$

Thus, to estimate the energy flux, the main ingredient is the vertical component of the phase velocity. Straus et al. (2008) determine the energy flux of the IGWs using the v − v phase spectra from Interferometric BIdimensional Spectrometer (IBIS; Cavallini 2006) observations. The v − v phase difference spectra are first converted to phase travel time spectra and then to phase velocity spectra under the assumption that the difference between the two line formation heights is known. The group velocity spectra are then determined from the phase velocity according to Equation (3). The energy flux spectra are finally obtained by using the plasma density from the VAL model C (Vernazza et al. 1976) and the mean squared velocity from Doppler shifts. Straus et al. (2008) found the energy flux in the IGW region to be 20800 W m⁻² at a height of 250 km, dropping to 5000 W m⁻² at a height of 500 km in the atmosphere. Kneer & Bello González (2011) find a lower flux of 4100–8200 W m⁻² at 380 km and 700–1400 W m⁻² at 570 km. The main reason for this discrepancy is thought to stem from the fact that Straus et al. (2008) overestimate the vertical group velocity because they do not consider the atmospheric transmission (Kneer & Bello González 2011).

With the help of numerical simulations, we can get an estimate of the energy fluxes without needing the group velocity, since we have access to all the physical quantities everywhere in the model atmosphere. We can assume that the fluctuations caused by waves are weak and that there are no strong background flows present. In such a scenario, the nonlinear MHD equations can be linearized by expanding quantities in terms of a static background and perturbations and neglecting terms greater than first-order powers of the perturbed quantities. By algebraic manipulation and casting the equations for perturbations in the form of a conservation law, one can derive an energy density and a flux of energy density associated with linear wave motion. Even for a weak fluctuation, this assumption is not valid for a long time span, as nonlinearities eventually arise due to the sole property of the governing equations. Also, the wave energy density is conserved only for the case of a uniform stratification (Bretherton & Garrett 1968), i.e., for an atmosphere with constant Brunt-Väisälä frequency (N). In the models that we study here the N varies with height. Therefore, we would like to caution the reader that the definition of linear wave energy density and flux does not represent the total energy density and flux associated with a wave. To get a better estimate of the energy density and flux, we would have to include the second-order terms (Leroy 1985; Tarr et al. 2017). Having a time-dependent background and spatial inhomogeneity, it is evident that the total wave energy is not conserved. The wave can continuously exchange energy with the background, raising concerns on casting the wave energy equation in the form of a conservation law. Despite all these concerns, one can say that the linear energy flux considered here gives the leading-order contribution to the conserved pseudoenergy flux (Andrews & McIntyre 1978; Grimshaw 1984).

The "linear" wave energy flux in an MHD system is made up of mechanical energy flux and the Poynting flux,

$$\begin{eqnarray}&&{F}_{\mathrm{wave}}={p}^{{\prime} }{\boldsymbol{v}}+\displaystyle \frac{1}{4\pi }\left[{{\boldsymbol{B}}}^{{\prime} }\times ({\boldsymbol{v}}\times {{\boldsymbol{B}}}_{0})\right],\end{eqnarray} \tag{ 4 }$$

where p' is the pressure perturbation, ${\boldsymbol{v}}$ is the velocity, ${{\boldsymbol{B}}}_{0}$ is the equilibrium magnetic field, and ${{\boldsymbol{B}}}^{{\prime} }$ is the field perturbation. The first term on the right-hand side of Equation (4) represents the rate of mechanical work done per unit area by a volume on an adjacent volume due to the excess pressure as a wave propagates across the volume interface. Often quoted as the acoustic flux, in this context we would rather call it the mechanical flux, since it represents the total mechanical work done by all waves, as most waves in the medium cause pressure fluctuations as they propagate. This is different from the way in which the mechanical fluxes are estimated from observations. The second term in Equation (4) represents the Poynting flux, the flux of electromagnetic energy, which is also carried by all the waves in the magneto-acoustic-gravity coupled system that we consider. There is mechanical and Poynting flux associated with all the waves in the medium, and therefore, in the time domain, it is not possible to distinguish between these fluxes and map them to a specific wave motion. However, it is worthwhile to estimate the contribution to the total mechanical flux and the total Poynting flux that arise exclusively from wave-like motions in the domain.

We first look at the vertical mechanical energy flux density (F_M) as expressed by the first term on the right-hand side of Equation (4). Figure 8 shows the horizontally averaged mean F_M (black curve) as a function of height calculated from the four models. In the background we show the excursion of mean F_M for individual snapshots over the duration of the simulation. It is clear from the height dependence of mechanical energy flux that we do not see much difference between the nonmagnetic and the magnetic cases. However, we notice that there is a small scatter of locations with the mean flux directed downward (negative energy flux) in the middle part of the atmosphere around z = 0.6 Mm. It is not obvious whether this scatter can be related to waves that are present in the box. The energy fluxes obtained from observations (Straus et al. 2008; Bello González et al. 2010; Kneer & Bello González 2011) are marked as asterisks on all the plots. This includes the energy flux for the acoustic (marked in black) as well as the IGWs (red). As can be seen, close to the surface the fluxes obtained from observations (∼10⁴–10⁵ W m⁻²) are close to an order of magnitude less than what we see in the simulations (10⁶ W m⁻²). However, this difference is less pronounced as we go higher up in the atmosphere. Here we also show the observed fluxes with the negative sign as a reference for comparing the negative (downward) fluxes that we obtain in simulations.

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The horizontally averaged, vertically directed Poynting flux (F_P; black curve), as defined by the second term on the right-hand side of Equation (4), as a function of height, is shown in Figure 9. In the background we show the excursion of mean F_P for individual snapshots over the duration of the simulation for all three magnetic models. In all the magnetic models, the Poynting flux close to the surface at z = 0 km and below is directed downward as a result of the strong downdrafts in the intergranular lanes that carry with them magnetic flux, whereas close to the surface and above, the overshooting material carries magnetic flux upward, resulting in a net upward Poynting flux (Steiner et al. 2008). This can be clearly seen for all three models in Figure 9. However, we notice that there is a large scatter of fluxes with a downward contribution. This scatter seems to depend on the magnetic field strength and consequently on the location of the mean plasma-β = 1 layer. Models with a strong field show a downward-directed Poynting flux close to the layer of plasma-β = 1, extending over a broad height range below this layer. Models with weak fields show considerably less downward-directed Poynting flux. It is clear that a major contributer to the downward-directed Poynting flux comes from the region of IGWs, as evident from the energy flux spectra presented in Section 4.3.

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In this section, we will briefly mention some of the limitations of the current study and future direction. One of the major drawbacks of the present work is that the results obtained here are based on spectral analysis done on geometrical heights. This makes it hard to compare with observational data, as they are mainly obtained from spectral maps that probe constant optical depths. A further exploration of this will require performing the cross-spectral analysis on velocity data obtained from constant optical depths, or doing a spectral synthesis of different lines and performing the cross-spectral analysis on the estimated Dopplergrams, which will be explored further in a forthcoming paper.

The current simulation setup with a rather coarse grid resolution of 80 km is sufficient to capture the emergent IGW spectra. Increasing the resolution may have an effect on the overall spectra of IGWs, as the granulation pattern is seen to be more structured in higher-resolution simulations. However, due to the computationally expensive nature of such simulation, a study on the effect of grid resolution on the emergent IGW spectra was not pursued in this paper. In terms of boundary conditions, we have not studied the effect of how changing the lateral and top boundary conditions will affect the propagation of waves. For the lateral boundary, we are restricted to use a periodic boundary condition, as the FFT algorithm assumes that the boundaries are periodic. However, imposing a periodic side boundary condition restricts the maximum wavelength of the waves to the length of the box. The top boundary condition forces the field to be vertical, thereby restricting us to study a magnetic environment that has only predominantly vertical magnetic fields. This, however, is the case for all the models and thus makes it easy to do a comparative study. Further analysis with predominantly horizontal fields may reveal whether a coupling to low-β gravito-Alfvén waves is possible. In this case the angle of attack of these obliquely propagating waves will be mainly parallel to the fields and therefore have a chance to propagate into the upper layers.

We would like to mention here the effect of the Alfvén speed limiter that has been used in this study. First, this speed limiter may affect waves with horizontal phase velocity greater than 40 km s⁻¹, which fall mainly in the high-frequency part of the k_h − ω diagram, which does not include the IGW region. Second, waves with vertical phase velocity larger than 40 km s⁻¹ fall close to the propagation boundary according to Equation (1) in Paper I and are insignificant. Therefore, we expect that the major part of the IGW spectrum is unaffected by the use of the Alfvén speed limiter.

Finally, we would like to caution the reader that the diagnostic diagram for the complete spectra of MAG waves in a uniform magnetic field medium is drastically different from that of the acoustic-gravity waves (considered here and in Paper I). The fact that we have three distinct branches of coupled waves, viz., the fast MAG, the gravito-Alfvénic, and the slow MAG, and that their behavior in the diagnostic diagram varies depending on the magnetic field inclination relative to the wavevector and depending on the plasma-β, brings more variety and complexity to the diagnostic diagram (for a detailed reference see Goedbloed & Poedts 2004, Chapter 7, Section 7.3.3). In the low photosphere we still have a predominantly high-β atmosphere, which makes it easier to use the diagnostic diagram for acoustic-gravity waves. As we move higher in the atmosphere, we approach the surface β = 1 and move into the low-β regime, where the slow- and fast-MAG branches become more relevant and the three different branches look very different for different orientations of the magnetic field relative to the propagation vector. These effects are not taken care of in the present study since we focus mainly on regions with high β where the dispersion relation approximates that of the acoustic-gravity waves.

More recent work by Beck et al. (2017) has quoted a value of 35 G as the average magnetic flux density of the quiet-Sun region. Comparing it with the cases that we study in this paper, this would coincide with a model intermediate between Sun-v10 and Sun-v50, for which the average height of the plasma-β = 1 layer would correspond to around z = 900 km. Therefore, we argue that IGWs in the quiet Sun should be easily detectable up to a height of 900 km. Quasi-simultaneous measurements of 2D velocity fields at multiple heights, obtained from photospheric lines, with a reasonably large field of view (40 × 40 Mm²) and an observational duration of 3 hr will provide a good spectral resolution in the ${k}_{h}-\omega$ to identify the waves. Continuous observation with a cadence of less than 60 s (ν_max = 8.33 mHz) and with a moderate spatial resolution of up to 300 km (${k}_{h,\max }=10$ rad Mm⁻¹) can sufficiently capture the spectral region of the emergent IGWs. We predict that the observed spectra of the IGWs obtained from the same spectral lines will likely show differences among internetwork and the more magnetically dominated plage regions. Future observations using DKIST could help elucidate and validate our models.