Issue 
A&A
Volume 686, June 2024



Article Number  A293  
Number of page(s)  10  
Section  The Sun and the Heliosphere  
DOI  https://doi.org/10.1051/00046361/202449887  
Published online  20 June 2024 
A model of umbral oscillations inherited from subphotospheric fastbody modes
^{1}
Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea
email: jcchae@snu.ac.kr
^{2}
Bay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA 94035, USA
^{3}
Lockheed Martin Solar and Astrophysics Laboratory, 3251 Hanover St, Palo Alto, CA 94306, USA
^{4}
Solar and Space Weather Group, Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
Received:
7
March
2024
Accepted:
17
April
2024
Recently, complex horizontal patterns of umbral oscillations have been reported, but their physical nature and origin are still not fully understood. Here we show that the twodimensional patterns of umbral oscillations of slow waves are inherited from the subphotospheric fastbody modes. Using a simple analytic model, we successfully reproduced the temporal evolution of oscillation patterns with a finite number of fastbody modes. In this model, the radial apparent propagation of the pattern is associated with the appropriate combination of the amplitudes in radial modes. We also find that the oscillation patterns are dependent on the oscillation period. This result indicates that there is a cutoff radial mode, which is a unique characteristic of the model of fastbody modes. In principle, both internal and external sources can excite these fastbody modes and produce horizontal patterns of umbral oscillations.
Key words: magnetohydrodynamics (MHD) / waves / Sun: chromosphere / Sun: oscillations / sunspots
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Umbral oscillations are a magnetohydrodynamic (MHD) process conspicuous in every sunspot umbra. Following their initial detection (Beckers & Tallant 1969), subsequent works revealed that the umbral oscillations are the slow MHD waves propagating upwards from the photosphere to the corona with a group speed of around the speed of sound (Lites 1984; Centeno et al. 2006; Felipe et al. 2010; Krishna Prasad et al. 2017). Because the temperature minimum region between the photosphere and the chromosphere acts as a highpass filter, the wave power peaks around the cutoff frequency (ω_{c}) of around 6 mHz in the chromosphere (Roberts 1983; Centeno et al. 2006). Theoretical studies have revealed that even a portion of lowfrequency waves (ω < ω_{c}) can propagate upwards due to the effects of the temperature gradient (Schmitz & Fleck 1998; Chae & Litvinenko 2018) and the radiative relaxation of nonadiabatic heating and cooling (Roberts 1983; Centeno et al. 2006; Chae et al. 2023). The most plausible sources of the umbral oscillations have been considered to be either the magnetoconvection inside a sunspot, such as an umbral dot (Lee 1993; Jacoutot et al. 2008; Jess et al. 2012), or the absorption of the incident f and p mode waves (Braun & Duvall 1987; Spruit & Bogdan 1992; Cally & Bogdan 1997).
Interestingly, observational studies recently reported waves that appear to move across the magnetic field, forming ringlike patterns (Zhao et al. 2015; Cho & Chae 2020; Cho et al. 2021) or spiralshaped wave patterns (SWPs) (Sych & Nakariakov 2014; Su et al. 2016; Felipe et al. 2019; Kang et al. 2019). There are two models that can reproduce these patterns: a model of propagating waves excited in the subphotosphere (e.g., Zhao et al. 2015; Cho & Chae 2020), and a model of slowbody resonant modes (e.g., Edwin & Roberts 1982; Stangalini et al. 2022). The wavepropagation model assumes that there is a localized disturbance in the highβ region of the subphotosphere. This disturbance excites fast MHD waves, and these waves propagate quasiisotropically. The wavefront reaching the equipartition layer (β ∼ 1) has a time delay as a function of horizontal distance, and this time delay causes ringlike patterns to propagate across the magnetic field (Zhao et al. 2015). After reaching β ∼ 1, the fast waves are partially converted to the slow waves by the modeconversion process (Zhugzhda & Dzhalilov 1984; Cally 2001; Schunker & Cally 2006). This model successfully reproduces the ringlike wavefronts (Cho & Chae 2020; Cho et al. 2021) and the SWPs (Kang et al. 2019) observed in the chromosphere. To explain the SWPs, Kang et al. (2019) additionally assumed that the pointlike source generates nonaxisymmetric modes. However, it is not yet clear as to whether or not the exciting source assumed in this model really exists, and if it does, how such a source can generate nonaxisymmetric modes also remains unclear.
The other model to explain the complex wave patterns is the model of slowbody resonant modes. Eigenmode waves are known to appear when the flux tube resonates with external driving (Edwin & Roberts 1982, 1983; Roberts 2019). Considering the cylindrical geometry, Jess et al. (2017) successfully identified the firstorder azimuthal mode from chromospheric umbral oscillations. Recently, Albidah et al. (2022) found highermode oscillation patterns in circular and elliptical sunspots. In addition, in a largescale sunspot with complex geometry, Stangalini et al. (2022) successfully reproduced the oscillation patterns by the superposition of more than 30 slowbody resonant modes in Cartesian geometry. However, it is difficult to understand how a systematic horizontal pattern can be established from the resonance of slowbody waves that are very inefficient in transferring energy in the horizontal direction across magnetic fields.
In the present work, we propose another model: a model of umbral oscillations inherited from subphotospheric fastbody modes. In this model of fastbody modes, we suppose that the driving and resonance of the flux tube occur in the subphotosphere, resulting in the subphotospheric fastbody modes. These fast modes are partially converted to slow waves in the equipartition layer. These slow waves come to have twodimensional patterns with resonant modes because they inherit the fast modes from the subphotosphere.
In Sect. 2, we propose the simple analytic model of the subphotospheric fastbody modes. In Sect. 3, we show observations of twodimensional patterns in the pore and present a method to analyze them. In Sect. 4, we compare the observation and the model. Finally, in Sect. 5, we summarize and discuss our model.
2. The model
We regard the patterns of umbral oscillations as the inherited patterns of fastbody modes in the subphotosphere (see Fig. 1). We conjecture that the inheritance is realized by the process of fasttoslowmode conversion in the equipartition layer (e.g., Zhugzhda & Dzhalilov 1984; Cally 2001) after the fastbody modes have formed in the subsurface. These converted slow waves (blue solid line) propagate upwards to the detection layer along the magnetic field. As a consequence, the observed horizontal patterns of umbral oscillations can be identified with the patterns of fastbody modes (red solid line) in the subphotosphere.
Fig. 1. Illustration of our model of umbral oscillations inherited from subphotospheric fastbody modes. The yellow area represents the highβ region (c_{s} > v_{A}) below the sunspot surface, and the white area represents the lowβ region (c_{s} < v_{A}). The black solid line between the two regions represents the equipartition layer (c_{s} = v_{A}) where the mode conversion can occur. The gray vertical lines illustrate the magnetic field lines of the flux tube, and the vertical dashed line indicates the boundary of the flux tube. The red line demonstrates the vertical velocity v_{z} of fast body waves, and the blue line represents v_{z} of converted slow waves. The dashdotted horizontal line represents the detection layer of the chromosphere. The two panels in the bottom right corner show the vertical velocity fluctuation of two azimuthal modes of m = 0 and 2. The top right panel shows a horizontal crosssection view of the modeled oscillation patterns. 
For simplicity, we adopt a vertical magnetic flux tube with a radius R where density ρ, Alfvén speed v_{A}, sound speed c_{s}, and tube speed are uniform inside the flux tube. The magnetoacoustic body mode of vertical velocity with angular frequency ω, azimuthal mode m, radial wavenumber k_{r}, and vertical wavenumber k_{z} is given by the solution in cylindrical coordinates (Edwin & Roberts 1983; Roberts 2019),
where A_{kr, m, ω} is a complex constant and J_{m} is the first kind of the Bessel function of an azimuthal mode m. Here, the radial wavenumber k_{r} depends on ω and k_{z} as in the expression
and is to be determined from the boundary condition at r = R:
which is obtained from the continuity of total pressure and radial velocity v_{r} (Edwin & Roberts 1983). Here, K_{m} is the second kind of the modified Bessel function, the prime operator (′) denotes the derivative of the Bessel function, and the subscript e stands for quantities in the external region r > R.
For the body wave solution, k_{r} should be real and hence has to be positive (Edwin & Roberts 1983). This requirement is satisfied for fastbody waves when the vertical phase speed v_{p} ≡ ω/k_{z} lies between the internal sound speed and the external sound speed:
Similarly, we obtain the condition for the slowbody wave solution:
indicating that v_{p} should be in between the internal Alfvén speed and the tube speed.
We note that Eq. (3) has a simple solution,
in the limit of zeroexternal density (ρ_{e} = 0). This condition might be satisfied when the external medium is much hotter than the interior. As J_{m} is an oscillatory function, there are several values of k_{r}(n) with different values of radial mode n. In other words, k_{r}(n)R is the nth root of Eq. (6). Generally speaking, k_{r} is specified by m, n, and R.
The zerodensity limit is useful for understanding the characteristics of the body waves when only considering the internal atmospheric conditions such as c_{s}, and v_{A}. Eqs. (2) and (6) are solved for the expression
with the parameters
We note that the larger value, k_{z, +}, corresponds to slow waves and the smaller one, k_{z, −}, to fast waves.
The relationship in Eq. (7) can be used to calculate k_{z} as a function of ω or equivalently the oscillation period P = 2π/ω when m and n are specified in the cases of slowbody waves and fastbody waves. As an illustration, we consider a flux tube of R = 3.5 Mm, c_{s} = 6.3 km s^{−1}, and v_{A} = 0.56c_{s} at a depth of β = 3.8 below the surface (Fig. 2). Here, the choice of c_{s} has been made by extrapolating the Maltby M sunspot model (Maltby et al. 1986) to the depth of 100 km. As the condition c_{s, e} > c_{s} > v_{A} is satisfied in this flux tube, both fastbody waves and slowbody waves can occur if v_{p} is within a specified range.
Fig. 2. Period P as a function of vertical wavenumber k_{z} for (a) the slowbody waves and (b) the fastbody waves for n = 1, 2, 3, 4, 5, and 10 with m = 2 in the subphotospheric condition (c_{s, e} > c_{s} > v_{A}). The white area in panel a marks the range of v_{A} > v_{p} > c_{T} within which the slowbody wave solution can exist, and that in panel b represents the range within which the fastbody wave can exist (c_{s, e} > v_{p} > c_{s}). The inset figure in panel a shows the magnified figure in the ranges of 0.011 < k_{z} < 0.012 km s^{−1} and 155 < P < 165 s. The dashed line represents the period of 160 s. The colorshaded regions in panel b represent the regions with the same cutoff radial mode n_{M}; yellow is n_{M} = 2, green is n_{M} = 3, blue is n_{M} = 4 and purple is n_{M} ≥ 5. We have taken the internal sound speed of c_{s} = 6.3 km s^{−1}, internal Alfvén speed of v_{A} = 0.56c_{s}, and external sound speed of c_{s, e} = 1.7c_{s}. 
We first consider slowbody modes (see Fig. 2a). These waves can exist within the v_{p} range marked as a white area and satisfying v_{A} > v_{p} > c_{T}. We find that the curve representing the solution for k_{z}(P) with m = 2 of any value of n fits into this range. An important property of slow waves is that for a specified period, there exists an infinite number of solutions with n = 1, …, ∞ with any value of m.
We now consider fastbody modes. Figure 2b presents the solution for k_{z}(P) for each value of n with m = 2. Even though this solution is independent of the physical condition outside the flux tube, it can have a physical meaning as fast waves only when the condition c_{s, e} > v_{p} > c_{s} is satisfied. With the choice of c_{s, e} = 1.7c_{s}, the physically meaningful range of v_{p} is marked as a white area. A distinct property of fast waves is that for a specified period, there exists a finite number of solutions with n ≤ n_{M} for each m, where n_{M} is the highest radial mode that can be trapped in the flux tube satisfying the condition of Eq. (4). In other words, for a given period, there exists a cutoff (n_{M}) in the radial mode in fastbody waves. For example, for the waves with a period of 160 s, which corresponds to the dashed line, the cutoff radial mode is n_{M} = 4. We note that n_{M} depends on P as well as m in fast waves, whereas n_{M} = ∞ always in slow waves.
As we regard the observed pattern of velocity fluctuation as the inheritance of the fastbody waves in the subphotosphere, it can be modeled by a superposition of all modes, as in
where A_{n, m, ω} represents the real amplitude and the phase is described by θ_{m, ω} and t_{m, n, ω}. We note that k_{z} is determined from Eq. (7) for given ω, m, and n. The inheritance is simply achieved by choosing z to represent the detection layer, assuming a constant phase lag between the chromosphere and the subphotosphere in all the waves. For reasons of practicality, we set z = 0 to represent the detection layer, θ_{0, ω} = 0, and t_{0, 1, ω} = 0 for a reference without loss of generality.
3. Data and method
We now describe how to obtain v_{z}(t, r, θ; ω) in the detection layer from the observation, and how to determine the parameters n, A_{n, m, ω}, θ_{m}, and t_{m, n} from v_{z}(t, r, θ; ω) for each m with significant power. We applied our model to a small sunspot, a pore observed in NOAA 12078 at (301″, 162″) on June 3, 2014, from 16:49 UT to 17:56 UT (Fig. 3a). The data –taken with the Fast Imaging Solar Spectrograph (FISS, Chae et al. 2013) at the Goode Solar Telescope (GST)– have been used in several studies (Chae et al. 2015, 2022; Kang et al. 2019). The time cadence of the data is 20 s, the spatial sampling is 0.16″, the spectral sampling is 19 mÅ, and the spectral range is −5 Å to 5 Å of Hα. We calibrated the raw data following the reduction pipeline described in detail by Chae et al. (2013). We calculated the lineofsight (LOS) Doppler velocity v_{LOS}(t, x, y) at every pixel (x, y) at different times using the lambdameter (bisector) method (e.g., Chae et al. 2014) with a chord of 0.1 Å.
Fig. 3. Detection of a pattern of umbral oscillations. (a) Continuum intensity map of the GST/FISS constructed at the −43.5 Å of the Hα line center and chromospheric intensity map constructed at the line center at 17:44:07 UT on June 3, 2014. (b) Twodimensional wave patterns of the chromospheric Doppler velocity map temporally filtered in 5.5 − 9 mHz. The black contour displays the boundary of the pore, and the red cross symbol marks the center position of the SWP. (c) Azimuthally decomposed wave patterns for m = 0 and ±2 modes. (d) Modeled LOS velocity fluctuation map constructed by the superposition of 12 modes (m × n, where m = 0, 2, −2 and n = 1, 2, 3, 4). The dashed circle shows the boundary of the modeled flux tube. (e) All modes of trapped fastbody waves for the modeled LOS velocity map. Here m represents the azimuthal mode and n refers to the radial mode. The color limit of panels b–e is −3.5 to 3.5 km s^{−1}, where the positive sign signifies the redshift. The temporal evolutions of panels b–d are shown in Fig. 5. 
In the present study, we focus on the twoarmed SWPs that occurred around 17:44:07 UT (t = 40 s). The wavelet power spectrum of velocity fluctuations v_{LOS}(t, x_{c}, y_{c}) at the center of these SWPs indicates that most power is concentrated in the 2 − 3 min band with a peak of around P = 160 s (Fig. 4). We confined our analysis to this band, with the peak frequency ω = 2π/P = 0.04 rad s^{−1}. We extracted the velocity at this band v_{LOS}(t, x, y; ω) by applying the 5.5 − 9 mHz passband filtering to the time series of LOS velocity v_{LOS} at every pixel (x, y) on the image plane. As an illustration, Fig. 3b shows v_{LOS}(t, x, y; ω) at t = 40 s.
Fig. 4. Wavelet power spectrum of the Doppler velocity averaged over 3 × 3 pixels at the center of the oscillation patterns. The black dashed line represents the time t = 0 at 17:43:27 UT, which is the middle time of the wave packet of spiralshaped wave patterns. The red contour indicates the 95% confidence level, and the hashed region shows the cone of influence. Among the total observing duration (of −3200 s to 780 s), the diagram shows only the time range of −780 s to 780 s. 
Now we describe how we extracted the azimuthal component of mode m from v_{LOS}(t, x, y; ω). We first set the polar coordinates with the origin at the center of the oscillation patterns, and obtained v_{LOS}(t, r, θ) in these coordinates using interpolation. Second, we performed the Fourier transform of v_{LOS}(t, r, θ) over θ for every t and r, which we then multiplied by the filter function that is set to one for the specific value of m and zero for the other values. By applying the inverse Fourier transform over θ for every t and r, and mapping the data back to the image coordinates, we obtained the azimuthal component of velocity v_{LOS}(t, x, y; ω, m). We found that the velocity data are satisfactorily reproduced by the summation of m = 0 mode and m = 2 mode only, as illustrated in Fig. 3c. The m = 1 mode has not been included here because it has insignificant power (see Fig. 1d in Kang et al. 2019).
To reproduce this v_{LOS}(t, x, y; ω, m), we obtained the parameters t_{n, n, ω}, A_{n, m, ω}, and θ_{m, ω} from Eq. (11) using the following method. First, we remapped v_{z}(t, r, θ; ω) to the image coordinates, v_{z}(t, x, y; ω). Second, we obtained the values of t_{n, m, ω}, A_{n, m, ω}, and θ_{m, ω} for each m by comparison with v_{LOS}(t, x, y; ω, m), maximizing the Pearson correlation value between the model and the observation at a reference time of t = 40 s. The temporal evolution of the pattern is reproduced by changing time t in Eq. (11). In this process, we empirically find that t_{n, m, ω} mostly affects the overall shape of the pattern and A_{n, m, ω} reflects the width of the arms and their radial speed. In particular, the inner shape of the patterns is highly affected by highorder n, and the outer part is associated with loworder n. Finally, we reconstructed the patterns summing up azimuthal modes of m = 0, m = 2, and m = −2 (Fig. 3d). All of the modeling parameters are shown in Table 1 and illustrated in Fig. 3e.
4. Results
Figure 5 indicates that our model of subphotospheric fastbody modes can successfully reproduce the observed horizontal patterns of umbral oscillations. Here the observed patterns have been temporally filtered in the frequency range of 5.5 − 9 mHz. The observation displays the two rotating spiral arms that appear to propagate outwards, which is successfully reproduced by the model constructed by the superposition of a total of 12 modes; m = 0, ±2, and n = 1, 2, 3, 4. As can be seen from the figure, this model is very similar in appearance to the observation, with the Pearson correlation coefficient being as large as cc = 1.00. The standard deviation of the residual is as small as 0.13 km s^{−1} and is roughly twice the standard error of the LOS velocity of 0.07 km s^{−1}. We note that our model has the advantage that the temporal evolution is fully described with the set of model parameters computed at the reference time using Eq. (11), without having to repeat the fit at individual time steps (cf. Stangalini et al. 2022).
Fig. 5. Temporal evolution of the spiralshaped wave patterns. (a) Observed LOS velocity map temporally filtered in the frequency range of 5.5 − 9 mHz. (b) LOS velocity map additionally filtered in the azimuthal mode of m = 0, and 2. (c) Modeled velocity map constructed by the superposition of a total of 12 modes; m = 0, ±2, and n = 1, 2, 3, 4. (d) Residual between the filtered map (b) and the model (c). The model parameters are shown in Table 1. Columns show the temporal evolution of each map from t = 0 to t = 80 s. Here, the time t = 0 s is equal to 17:43:27 UT, which is the middle time of the wave packet shown in Fig. 4. The black solid contour indicates the boundary of the pore, and the black dashed circle represents the boundary of the flux tube of the model. The correlation values between (b) and (c) are shown in the top corner of each panel of (c). 
Table 1 lists the values of the model parameters. We first consider the case of m = 2. In this case, the amplitude is larger than 2 km s^{−1} for n = 1, 2, 3 but is smaller than 1 km s^{−1} for n = 4, being as small as onequarter of the value for n = 3. This means that the patterns are fairly well described by the low radial modes up to n = 3, and including n = 4 is more than sufficient. In order to clarify this point, we attempted to include n = 5. The amplitude of the n = 5 mode is found to be less than that of n = 4 by a factor of 2. Furthermore, its inclusion only slightly reduces the residual error. Therefore, we conclude that the inclusion of higher modes is physically meaningless and the observation can be successfully reproduced by a small number of modes. The rapid decrease in the amplitude over n around n = 4 indicates that the cutoff radial mode n_{M} may be equal to 4. In the case of m = 0, we may obtain a similar conclusion as in the case of m = 2 described above. In the case of m = −2, the amplitude for n = 1, 2, 3 is small, not being much larger than that for n = 4. Nevertheless, as the amplitude of n = 4 mode is, in this case, comparable to the corresponding values in the cases of m = 0 and m = 2, and the amplitude of n = 5 is found to be smaller than that of n = 4, it is likely that n_{M} = 4 for m = −2 as well.
We investigated how the cutoff radial mode depends on the frequency. To this end, we processed the observed patterns in two other bands: a lower frequency band of 2 − 4 mHz (4.2 − 8.3 min) and a higher frequency band of 12 − 20 mHz (0.8 − 1.4 min) as shown in Fig. 6. We find that in the lower frequency band (panel a), spiral arms are absent. In the higher frequency band (panel b), the spiral arms not only exist but also appear to be more tightly wrapped than the 5.5 − 9 mHz band pattern as shown in Fig. 5. The observed patterns in these two bands can also be well reproduced by the fastbody wave models as shown in the bottom panels. In each band, we tried different values of the maximum n values for each m. As a result, we found that the pattern of the 2 − 4 mHz band can be reproduced satisfactorily by setting n_{M} = 2, and the pattern in the 12 − 20 mHz band by setting n_{M} = 6. With these values, the lower frequency pattern is well modeled with 6 modes, and the higher frequency pattern with 18 modes. The Pearson correlation between each observed pattern and the corresponding model is found to be stronger than 0.97.
Fig. 6. Oscillation patterns in low and highfrequency bands. Top panels show the observed oscillation patterns at 17:44:07 UT, which are spatially filtered in m = 0 and 2 modes and temporally filtered in (a) 2.5 − 4 mHz and (b) in 12 − 20 mHz. Bottom panels show the model for each frequency band. The contour represents the boundary of the pore, and the dashed circle illustrates the boundary of the flux tube. All panels only show a 3 Mm radius from the center because the outer region is affected by the running penumbral waves and error. 
Combined with n_{M} = 4 obtained above for the 5.5 − 9 mHz band pattern, it is noteworthy that n_{M} increases with frequency. This result was empirically obtained from comparison of the models with observations and is exactly what the dispersion relation for subphotospheric fastbody modes predicts (Fig. 2b), supporting our model of umbral oscillations inherited from subphotospheric fastbody modes.
5. Discussion
In this study, we successfully reproduced the twoarmed spiralshaped wave patterns in the pore from the superposition of 12 subphotospheric fastbody modes. We find that the patterns of umbral oscillations depend on the oscillation frequency, comparing the patterns in three different bands. In addition, these oscillation patterns consist of the finite number of the radial mode: n_{M} = 4 for the 3 min period band, n_{M} = 2 for the 5 min band, and n_{M} = 6 for the 1 min band. These results are in agreement with the concept of the cutoff radial mode in the model of the subphotospheric fastbody modes.
Our model is based on several assumptions for the sake of simplicity. We believe these assumptions are not detrimental. It is assumed that the flux tube is uniform, has a circular shape, and is surrounded by an external medium of zero density. In Appendix A, we discuss these assumptions in detail and find that they are tolerable enough to allow a reasonable interpretation of the observed patterns. In addition, the magnetic field is assumed to be purely vertical. Because of this assumption, we confined the modeling region to the inside of a circular pore where magnetic field inclination is almost vertical (Chae et al. 2015).
We now consider how fast waves become trapped. In a flux tube, the fast waves can be either refracted or reflected when they reach the boundary, because the external sound speed and the internal sound speed differ from each other (c_{s, e} > c_{s}). In particular, the waves can be fully reflected if the incident angle of the waves is larger than the critical angle (θ_{c} = arcsin(c_{s}/c_{s, e})) of the total internal reflection, similar to the light in an optical fiber. Therefore, only the waves with an incident angle of larger than the critical angle can be fully reflected and finally form resonance modes.
In this regard, the cutoff in the radial mode occurs for the fastbody modes. In slowbody waves, the incident angles of all radial modes are large because these waves tend to be directed along magnetic fields (k_{z} ≫ k_{r}). As a consequence, all the slow waves are trapped inside the flux tube. In contrast, in fastbody waves, with small k_{z}, the radial directionality (k_{r}/k_{z}) becomes stronger for a higher n, because the higher n has a larger value of k_{r} by definition and has a smaller value of k_{z} for a given period (see Fig. 2b); this means that the incident angle becomes smaller. As a result, the fastbody waves have a cutoff in radial modes. We note that higher radial modes with incident angles smaller than θ_{c} can propagate across the boundary as leaky waves. In our case, for example, the n = 4 mode with a period of 160 s has an incident angle of about 43°, which is larger than θ_{c} ≈ 42°, but the incident angle of the n = 5 mode is about 26°. Thus, only the radial mode with n ≤ 4 can be trapped in the flux tube.
Our model is similar to the other two models of umbral oscillation patterns in several respects. It is like the wave propagation model in that the observed slow waves are regarded as the inheritance of the subphotospheric fast waves. It is also similar to the slow resonance model in that the oscillation patterns are described as the superposition of several modes.
Our model has a distinct advantage in describing the observed frequencydependent patterns. As shown in Fig. 6, we find that the oscillation pattern at each frequency band is a superposition of radial modes, the number of which depends on frequency. This frequency dependence is not at all expected in the wave propagation model if one single source drives the waves. In the wave propagation model, the horizontal pattern depends only on the depth of the source and the wave propagation speed (Zhao et al. 2015; Cho & Chae 2020), and so the pattern should be the same regardless of the oscillation frequency, unless the wave propagation speed depends on frequency. Of course, the waves can form frequencydependent patterns if the vertically elongated source, or more than two localized sources at different depths, generate waves of different frequencies, but this frequency dependence has not yet been reported in previous simulation works. The observed frequency dependence cannot be explained by the slow resonance model either. In this latter model, an infinite number of radial modes can exist regardless of frequency (Edwin & Roberts 1983), which is not compatible with the finding that the pattern can be described by only a few radial modes. In our model, in contrast, this finding is easily described by the existence of the cutoff radial mode increasing with frequency.
One might attribute the appearance of the cutoff radial mode to the dependence of the efficiency of the fasttoslow conversion on the radial mode. Nevertheless, this is not the case. It is likely that the efficiency decreases with the radial mode because the conversion efficiency is known to become high when the wave vector is aligned with the magnetic field in order to ensure k_{z}/k_{r} ≫ 1 (Schunker & Cally 2006). However, this expected decrease is gradual, unlike the observed sharp drop in amplitude (see Table 1), which is naturally interpreted as the occurrence of the cutoff.
Our model requires the trapping of fast waves in the subphotosphere. We note that the fastbody modes can exist at regions satisfying the condition of Eq. (4). This condition is not satisfied in the photosphere, or above it, where v_{A} > c_{s, e} > c_{s}. Nor is it satisfied in the deep region where c_{s} ∼ c_{s, e}. Therefore, it is likely that the trapping of fast waves is occurring in the shallow region in the subphotopshere. Indeed, we estimated the trapping region by calculating the sound speed and the Alfvén speed in the subphotosphere using the Maltby model (Maltby et al. 1986), and found that the fastbody modes of the observed twoarmed SWPs may be formed in a shallow region from −300 km to −50 km below the equipartition layer. In this region, plasma β is found to range from 3 to 10. We think that this region is for horizontal trapping only, and is not for vertical trapping. In the vertical direction, fast waves might come from below or from above.
Our model of fastbody modes has a couple of noticeable characteristics. First, the apparent radial propagation of the pattern is reproduced by an appropriate combination of amplitudes in the radial modes. For example, Time–distance (TD) maps in Fig. 7 clearly show that the speed of radial propagation increases with the ratios of the n = 1 mode amplitude to that of the other modes. In this regard, our model is in contrast with the wave propagation model, where the speed of the radial motion depends on the depth of the localized source (Zhao et al. 2015; Felipe & Khomenko 2017; Kitiashvili et al. 2019; Cho & Chae 2020). Second, fastbody modes, in principle, can be excited by arbitrary sources, either internally or externally. Among these two possible sources, we conjecture that the predominant source may be the external f and p modes in the subphotosphere, because f and p modes ubiquitously occur in the quiet Sun region regardless of the existence of a sunspot, and the pmode absorption is an effective mechanism to absorb incoming pmode waves (e.g., Braun & Duvall 1987; Cally & Bogdan 1997). In addition, irrespective of the excitation source, the fastbody modes, once established, display the pattern of umbral oscillations apparently propagating out of the center of the crosssection of the flux tube. In the wave propagation model, in contrast, the source is assumed to be a localized disturbance in the subphotosphere, which is related to the center of the oscillation patterns observed in the chromosphere (Cho & Chae 2020; Kang et al. 2024).
Fig. 7. Time–distance map of modeled m = 0 mode waves for three different cases. (a) TD map with the amplitudes of A_{0, 1} = 2.25, A_{0, 2} = 2.0, and A_{0, 3} = 0.5 in units of km s^{−1}. (b) TD map with the amplitudes of A_{0, 1} = 1.5, A_{0, 2} = 2.25 and A_{0, 3} = 1.5 in units of km s^{−1}. (c) TD map with the amplitudes of A_{0, 1} = 0.5, A_{0, 2} = 2.0 and A_{0, 3} = 2.25 in units of km s^{−1}. All of them have the phase differences of t_{0, 1} = 0, t_{0, 2} = −20 and t_{0, 3} = −40 in seconds. The solid lines represent the gradients of the ridges, and their values are shown in each panel. 
Our model has difficulty in explaining the observed change of the center of the pattern. From the observation of a sunspot with the light bridge, Cho et al. (2021) reported that the center of the wave pattern was located at different positions inside the same umbra at different times. It is difficult to explain this observation with our model as far as we currently understand, unless the shape of the sunspot significantly changed during the observation. This is a challenging problem for our model. To obtain a solution, further numerical and observational studies are necessary.
Acknowledgments
We appreciate the referee’s positive evaluation and the constructive comments. This research was supported by the National Research Foundation of Korea (RS202300273679). J. Chae was supported by the National Research Foundation of Korea (RS202300208117). E.K. Lim was supported by the Korea Astronomy and Space Science Institute under R&D program of the Korean government (MSIT; No. 2024185002, 2024181003). We gratefully acknowledge the use of data from the Goode Solar Telescope (GST) of the Big Bear Solar Observatory (BBSO). BBSO operation is supported by US NSF AGS2309939 and AGS1821294 grants and New Jersey Institute of Technology. GST operation is partly supported by the Korea Astronomy and Space Science Institute and the Seoul National University.
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Appendix A: Dependence of the model on the assumptions
Our model, for simplicity, assumes the zerodensity limit, circular shape, and uniformity inside the flux tube. Here we discuss how each of these assumptions affects the model.
A.1. Zerodensity limit
As a boundary condition, our model assumes that mass density is zero outside the flux tube. In this zerodensity limit, the node is always located at the boundary of the flux tube, which leads to the very simple boundary condition of Eq. (6). However, in the general case, the external density is not zero and, hence, the node does not occur exactly at the boundary. To examine the difference, we solve the full boundary condition in Eq. (3) and present the solution in Fig. A.1. As a result, we find that the node is located slightly inside the boundary. The mode has slightly larger values of k_{r}, k_{z} for a given period than in the case of zerodensity limit (see Fig. A.1). An important point here is that the differences are sufficiently small, as can be seen from the similarity of the curves shown in Fig. A.1 to those in Fig. 2b. This result suggests that the assumption of a zerodensity limit is sufficiently justified to allow interpretation of the oscillation patterns, unless precise modeling is required.
Fig. A.1. Numerical solution of Eq. (3) with the same parameters as Figure 2b but without the assumption of zerodensity limit. As the fastwave solution is confined within the range of c_{s, e} > v_{p} > c_{s}, curves are confined to the white area. 
A.2. Circular shape
The geometry of the flux tube can change the shape of the pattern (Albidah et al. 2022). If the crosssection of the flux tube is not circular, the center of the oscillation pattern is not located at the center of the flux tube. In our study, the effect of noncircular shape does not seem to be strong because the pore has a relatively circular shape. In our model, we identified the center of the twoarmed SWPs at the center of the flux tube, which is found to be 1″ from the center of the pore determined by the ellipse fitting of the morphology. We think the deviation of this amount is tolerable, and so the assumption of the circular shape is sufficiently justified for our purposes. We also note that the geometry of the flux tube does not affect the presence of the cutoff radial mode, which is the most important characteristic of our model here.
A.3. Uniformity
For simplicity, our model assumes that all the physical parameters are uniform both horizontally and vertically in the subphotosphere. The assumption of uniformity holds better in the horizontal direction than in the vertical direction. The vertical uniformity is hindered by the gravitational stratification of pressure and density. Nevertheless, it may not be detrimental in our model because the vertical extent of the subphotosphere (∼250 km) is not much larger than the local pressure scale height (∼200 km).
A.4. Trapping
The existence of the modes implicitly assumes that the waves are trapped somehow inside the flux tube. The trapping requires not only horizontal confinement but also vertical confinement. It is obvious that horizontal confinement occurs at the lateral boundaries. This leads us to question where the vertical confinement might occur. Our model of fastbody modes implicitly assumes that the vertical confinement occurs outside the subphotosphere: both above and below it. In the region above the photosphere, density rapidly decreases with height and the Alfvén speed rapidly increases with height, which causes the fast waves to be reflected downward. In the deep interior below the subphotosphere of our interest, the temperature rapidly increases with depth, and the propagation speed of the fast waves rapidly increases with depth, which causes the waves to be reflected upward.
All Tables
All Figures
Fig. 1. Illustration of our model of umbral oscillations inherited from subphotospheric fastbody modes. The yellow area represents the highβ region (c_{s} > v_{A}) below the sunspot surface, and the white area represents the lowβ region (c_{s} < v_{A}). The black solid line between the two regions represents the equipartition layer (c_{s} = v_{A}) where the mode conversion can occur. The gray vertical lines illustrate the magnetic field lines of the flux tube, and the vertical dashed line indicates the boundary of the flux tube. The red line demonstrates the vertical velocity v_{z} of fast body waves, and the blue line represents v_{z} of converted slow waves. The dashdotted horizontal line represents the detection layer of the chromosphere. The two panels in the bottom right corner show the vertical velocity fluctuation of two azimuthal modes of m = 0 and 2. The top right panel shows a horizontal crosssection view of the modeled oscillation patterns. 

In the text 
Fig. 2. Period P as a function of vertical wavenumber k_{z} for (a) the slowbody waves and (b) the fastbody waves for n = 1, 2, 3, 4, 5, and 10 with m = 2 in the subphotospheric condition (c_{s, e} > c_{s} > v_{A}). The white area in panel a marks the range of v_{A} > v_{p} > c_{T} within which the slowbody wave solution can exist, and that in panel b represents the range within which the fastbody wave can exist (c_{s, e} > v_{p} > c_{s}). The inset figure in panel a shows the magnified figure in the ranges of 0.011 < k_{z} < 0.012 km s^{−1} and 155 < P < 165 s. The dashed line represents the period of 160 s. The colorshaded regions in panel b represent the regions with the same cutoff radial mode n_{M}; yellow is n_{M} = 2, green is n_{M} = 3, blue is n_{M} = 4 and purple is n_{M} ≥ 5. We have taken the internal sound speed of c_{s} = 6.3 km s^{−1}, internal Alfvén speed of v_{A} = 0.56c_{s}, and external sound speed of c_{s, e} = 1.7c_{s}. 

In the text 
Fig. 3. Detection of a pattern of umbral oscillations. (a) Continuum intensity map of the GST/FISS constructed at the −43.5 Å of the Hα line center and chromospheric intensity map constructed at the line center at 17:44:07 UT on June 3, 2014. (b) Twodimensional wave patterns of the chromospheric Doppler velocity map temporally filtered in 5.5 − 9 mHz. The black contour displays the boundary of the pore, and the red cross symbol marks the center position of the SWP. (c) Azimuthally decomposed wave patterns for m = 0 and ±2 modes. (d) Modeled LOS velocity fluctuation map constructed by the superposition of 12 modes (m × n, where m = 0, 2, −2 and n = 1, 2, 3, 4). The dashed circle shows the boundary of the modeled flux tube. (e) All modes of trapped fastbody waves for the modeled LOS velocity map. Here m represents the azimuthal mode and n refers to the radial mode. The color limit of panels b–e is −3.5 to 3.5 km s^{−1}, where the positive sign signifies the redshift. The temporal evolutions of panels b–d are shown in Fig. 5. 

In the text 
Fig. 4. Wavelet power spectrum of the Doppler velocity averaged over 3 × 3 pixels at the center of the oscillation patterns. The black dashed line represents the time t = 0 at 17:43:27 UT, which is the middle time of the wave packet of spiralshaped wave patterns. The red contour indicates the 95% confidence level, and the hashed region shows the cone of influence. Among the total observing duration (of −3200 s to 780 s), the diagram shows only the time range of −780 s to 780 s. 

In the text 
Fig. 5. Temporal evolution of the spiralshaped wave patterns. (a) Observed LOS velocity map temporally filtered in the frequency range of 5.5 − 9 mHz. (b) LOS velocity map additionally filtered in the azimuthal mode of m = 0, and 2. (c) Modeled velocity map constructed by the superposition of a total of 12 modes; m = 0, ±2, and n = 1, 2, 3, 4. (d) Residual between the filtered map (b) and the model (c). The model parameters are shown in Table 1. Columns show the temporal evolution of each map from t = 0 to t = 80 s. Here, the time t = 0 s is equal to 17:43:27 UT, which is the middle time of the wave packet shown in Fig. 4. The black solid contour indicates the boundary of the pore, and the black dashed circle represents the boundary of the flux tube of the model. The correlation values between (b) and (c) are shown in the top corner of each panel of (c). 

In the text 
Fig. 6. Oscillation patterns in low and highfrequency bands. Top panels show the observed oscillation patterns at 17:44:07 UT, which are spatially filtered in m = 0 and 2 modes and temporally filtered in (a) 2.5 − 4 mHz and (b) in 12 − 20 mHz. Bottom panels show the model for each frequency band. The contour represents the boundary of the pore, and the dashed circle illustrates the boundary of the flux tube. All panels only show a 3 Mm radius from the center because the outer region is affected by the running penumbral waves and error. 

In the text 
Fig. 7. Time–distance map of modeled m = 0 mode waves for three different cases. (a) TD map with the amplitudes of A_{0, 1} = 2.25, A_{0, 2} = 2.0, and A_{0, 3} = 0.5 in units of km s^{−1}. (b) TD map with the amplitudes of A_{0, 1} = 1.5, A_{0, 2} = 2.25 and A_{0, 3} = 1.5 in units of km s^{−1}. (c) TD map with the amplitudes of A_{0, 1} = 0.5, A_{0, 2} = 2.0 and A_{0, 3} = 2.25 in units of km s^{−1}. All of them have the phase differences of t_{0, 1} = 0, t_{0, 2} = −20 and t_{0, 3} = −40 in seconds. The solid lines represent the gradients of the ridges, and their values are shown in each panel. 

In the text 
Fig. A.1. Numerical solution of Eq. (3) with the same parameters as Figure 2b but without the assumption of zerodensity limit. As the fastwave solution is confined within the range of c_{s, e} > v_{p} > c_{s}, curves are confined to the white area. 

In the text 
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