Global effects of local sound-speed perturbations in the Sun: A theoretical study
aa r X i v : . [ a s t r o - ph ] N ov Solar PhysicsDOI: 10.1007/ ••••• - ••• - ••• - •••• - • Global effects of local sound-speed perturbationsin the Sun: A theoretical study
S. M. Hanasoge · T. P. Larson c (cid:13) Springer ••••
Abstract
We study the effect of localized sound-speed perturbations on global modefrequencies by applying techniques of global helioseismology on numerical simulationsof the solar acoustic wave field. Extending the method of realization noise subtrac-tion (e.g. Hanasoge, Duvall, and Couvidat, 2007) to global modes and exploitingthe luxury of full spherical coverage, we are able to achieve very highly resolvedfrequency differences that are used to study sensitivities and the signatures of thethermal asphericities. We find that (1) global modes are almost twice as sensitive tosound-speed perturbations at the bottom of the convection zone as in comparison toanomalies well in the radiative interior ( r . . R ⊙ ), (2) the m -degeneracy is liftedever so slightly, as seen in the a coefficients, and (3) modes that propagate in thevicinity of the perturbations show small amplitude shifts ( ∼ . ). Keywords:
Helioseismology, Direct Modeling; Interior, Tachocline; Interior, Con-vective Zone; Waves, Acoustic
1. Introduction
Global helioseismology has proven very successful at inferring large scale propertiesof the Sun (for a review, see Christensen-Dalsgaard, 2002; Christensen-Dalsgaard,2003). Because they are very robust, the extension of methods of global helioseis-mology to study localized variations in the structure and dynamics of the solarinterior has been of some interest (e.g. Swisdak and Zweibel, 1999). However, theprecise sensitivities of global modes to local perturbations are difficult to estimatethrough analytical means, especially in cases where the flows or thermal asphericitiesof interest possess complex spatial dependencies. To address questions relating tosensitivities and with the hope of perhaps discovering hitherto unknown phenomenaassociated with global modes, we introduce here for the first time a technique tostudy the effects of arbitrary perturbations on global mode parameters in the linearlimit of small wave amplitudes.Global modes attain resonant frequencies as a consequence of differentially sam-pling the entire region of propagation, making it somewhat more difficult (in com-parison to local helioseismology) to pinpoint local thermal asphericities at depth. W. W. Hansen Experimental Physics Laboratory, StanfordUniversity, Stanford, CA 94305 email: [email protected]@sun.stanford.edu anasoge & Larson
Exactly how difficult is one of the questions we have attempted to answer in thisarticle. Jets in the tachocline (e.g. Christensen-Dalsgaard et al., 2005) are a subject ofconsiderable interest since their existence (or lack thereof) could be very important inunderstanding the angular momentum balance of the Sun. Studying the sensitivitiesand signatures of waves to flows at depth may open up possibilities for their detection.Forward modeling as a means of studying wave interactions in a complex mediumlike the Sun has become quite favoured (e.g. Hanasoge et al., 2006; Parchevskyand Kosovichev, 2007; Hanasoge, Duvall, and Couvidat, 2007; Cameron, Gizon, andDaifallah, 2007). The discovery of interesting phenomena, especially in the realmof local helioseismology (e.g. Hanasoge et al., 2007; Birch, Braun, and Hanasoge,2007), adds motivation to the pursuit of direct calculations. With the application ofnoise subtraction (Werne, Birch, and Julien, 2004; Hanasoge, Duvall, and Couvidat,2007), we can now study the signatures of a wide range of perturbations in a realisticmultiple source picture. Here, we attempt to place bounds on the detectability ofthermal asphericities at various depths in the Sun. We introduce and discuss themethod of simulation with a description of the types of perturbations introducedin the model in Section 2. The estimation of mode parameters can prove somewhatdifficult due to restrictions on the temporal length of the simulation ( < hours;owing to the expensive nature of the computation). The data analysis techniquesused to characterize the modes are presented in Section 3. We then discuss theresults from the analyses of the simulated data in § §
2. Simulations and perturbations
The linearized 3D Euler equations in spherical geometry are solved in the mannerdescribed in Hanasoge et al. (2006). The computational domain is a spherical shellextending from . R ⊙ to . R ⊙ , with damping sponges placed adjacent to theupper and lower radial boundaries to allow the absorption of outgoing waves. Thebackground stratification is a convectively stabilized form of model S (Christensen-Dalsgaard et al., 1996; Hanasoge et al., 2006); only the highly (convectively) unstablenear-surface layers ( r > . R ⊙ ) are altered while the interior is the same as modelS. Waves are stochastically excited over a 200 km thick sub-photospheric sphericalenvelope, through the application of a dipolar source function in the vertical (radial)momentum equation (Hanasoge et al., 2006; Hanasoge and Duvall, 2007). The forcingfunction is uniformly distributed in spherical harmonic space ( l, m ) ; in frequency, asolar-like power variation is imposed. Any damping of the wave modes away fromthe boundaries is entirely of numerical origin. The radial velocities associated withthe oscillations are extracted 200 km above the photosphere and used as inputs tothe peakbagging analyses. Data over the entire 360 ◦ extent of the sphere are utilizedin the analyses, thus avoiding issues related to mode leakage. We show an examplepower spectrum in Figure 1 along with the fits.The technique of realization noise subtraction (e.g. Hanasoge, Duvall, and Cou-vidat, 2007) is extensively applied in this work. Due to the relatively short timelengths of the simulations (the shortest time series yet that we have worked withis 500 minutes long!), the power spectrum is not highly resolved and it would seemthat the resulting uncertainty in the mode parameter fits might constrain our abilityto study small perturbations. To beat this limit, we perform two simulations withidentical realizations of the forcing function: a ‘quiet’ run with no perturbations, and ms.tex; 29/10/2018; 11:49; p.2 lobal effects of local perturbations Figure 1.
Example power spectrum from a simulation with l max = 95 and the correspondingfrequency fits (symbols). This is from a 24-hour simulation. The fitting algorithm is described inSection 3. Missing modes indicate that the fit for these did not converge. We do not understandwhy this occurs in the center of the power spectrum, but these modes can be made to convergeby perturbing their initial guesses. Modes with l < mostly disappear from the computationaldomain because the lower boundary is placed at r = 0 . R ⊙ . We also do not excite the highest l modes because we wish to avoid any issues related to numerical aliasing. a ‘perturbed’ run that contains the anomaly of interest. Fits to the mode parametersin these two datasets are then subtracted, thus removing nearly all traces of therealization and retaining only effects arising due to mode-perturbation interactions(see Section 3). As an example, we show in Figure 2 how a localized sound-speedperturbation placed at the bottom of the convection zone scatters waves which thenproceed to refocus at the antipode (the principle of farside holography, Lindsey andBraun, 2000). The presence of the sound-speed perturbation is not seen in panel a,whereas it is clearly seen in the noise-subtracted images of panels b and c.In these calculations, we only consider time-stationary perturbations. The sound-speed perturbations are taken to be solely due to changes in the first adiabatic index, Γ ; we do not study sound-speed variations arising from changes in the backgroundpressure or density since altering these variables can create hydrostatic instabilities.Lastly, the amplitude of all perturbations are taken to be much smaller than thelocal sound speed ( . ).
3. Peakbagging analysis
Our first round of peakbagging is done on the m -averaged power spectrum for thequiet simulation. For each l that we attempt to fit, we search for peaks in the negative ms.tex; 29/10/2018; 11:49; p.3 anasoge & Larson Figure 2.
Noise subtraction at work. Panel a is the time averaged RMS of the radial veloci-ties of the perturbed simulation; the sound-speed perturbation (located along the equator at r = 0 . R ⊙ and 180 ◦ longitude) is invisible. In panel b, the time averaged RMS of the differ-ence between the quiet and perturbed simulations and shown in panel c is the instantaneousdifference. The scattering of waves and their refocusing at the antipode is clearly seen in b. second derivative of the power. Unlike the power itself, which has a background,the second derivative has the advantage of having an approximately zero baseline.The search is accomplished by finding the frequency at which the maximum valueof the negative second derivative occurs, estimating the mode parameters using afrequency window of width 100 µ Hz centered on this peak frequency, zeroing thenegative second derivative in this interval, and iterating. If the range of power inthe frequency window is not above a certain threshold, we check the peak frequency ms.tex; 29/10/2018; 11:49; p.4 lobal effects of local perturbations found; if it is too close to a frequency found on a previous iteration, that maximumis rejected, the same interval is again zeroed, and iteration continues. Note that sucha simple algorithm is feasible only because simulation data contains no leaks. Oncewe have found as many peaks as possible with this procedure, we assign a value of n to each one based on a model computed using ADIPACK (Christensen-Dalsgaardand Berthomieu, 1991; Hanasoge, 2007).The next step is to perform an actual fit to the power spectrum in the vicinity ofeach peak we identified. For the line profile we use a Lorentzian of the form P = Aπ w ( ν − ν ) + w + B, (1)where A is the total power, w is the half width at half maximum, ν is the peakfrequency, and B is the background power. The initial guesses for these parametersare obtained in the first step as follows: B is set to the minimum value of the power inthe frequency window around the peak, A is set to the integral under the power curveminus B times the width of the window, and w is set to / ( πP max ) where P max is the maximum value of the power in the frequency window. The fitting intervalextends halfway to the adjacent peaks, or 100 µ Hz beyond the peak frequency of themodes at the edge. The fitting itself is done using the IDL routine curvefit .Once we have fit these mode parameters for the m -averaged spectrum, we usethem as the initial guesses for fitting the individual m spectra. Then for each l and n we can fit a set of a -coefficients to the frequencies as functions of m/l . Although forthe quiet sun we would expect for all the a -coefficients to be zero, this calculation isstill necessary in order to perform the noise subtraction.We also use the mode parameters from the m -averaged spectrum of the quietsimulation as initial guesses for fitting the (unshifted) m -averaged spectrum of theperturbed simulation. Although the perturbations may lift the degeneracy in m , weexpect the splitting to be very small, so that the peaks in the m -averaged spectrumcan still be well represented by a Lorentzian. We also use those same initial guessesfor fitting the individual m spectra of the perturbed simulation, and recalculate the a -coefficients.An empirical estimate of the error in frequency differences for the sound-speedperturbation at r = 0 . R ⊙ (see Section 4.1) is computed in the following manner. Welook at the difference in mode parameters only for those modes that do not penetrateto the depth of the perturbation (all modes with ν/ ( l + 1 / < ). We then make ahistogram of these differences with a bin size of 0.001 µ Hz and fit a Gaussian to theresulting distribution. With this method we find a standard deviation of 0.000474 µ Hz or 0.47 nHz. This result is confirmed by also computing the standard deviationof 95% of the closest points to the mean.
4. Results and discussion ◦ × ◦ (in longitude and latitude)with a full width at half maximum in radius of 2 % R ⊙ (13.9 Mm) at depths of r = 0 . , . , . R ⊙ , each with an amplitude α of +5 % of the local sound speed. ms.tex; 29/10/2018; 11:49; p.5 anasoge & Larson Figure 3.
Frequency shifts ( ∆ ν ) plotted against phase speed ( ν/ [ l +1 / ) for the sound-speedperturbations of Section 4.1. Solid lines indicate the phase speed of waves that have r = 0 . R ⊙ as inner turning points. Panel a shows the shifts due to a localized hot spot (sound-speedincrease) at the bottom of the convection zone. It appears that modes whose inner turningpoints are just below the bottom of the convection zone are the ones maximally sensitiveto the perturbation. Panel b shows the frequency shifts obtained from ADIPACK for thespherically symmetric component of this perturbation. In panel c, all modes feel the presenceof the relatively large near-surface hot spot and in d, the shifts predicted by ADIPACK forthe spherically symmetric analog for this near-surface anomaly are shown. Noise subtractiondoes not remove the realization noise associated with the scattering process itself; thereforethe spread in the frequency shifts of the simulated data is greater than ADIPACK ones. Because of the fixed angular size, the perturbations grow progressively smaller inphysical size with depth; our intention was to keep the perturbation as localizedand non-spherically symmetric as possible. Despite the fact that the perturbation ishighly sub-wavelength (the wavelength at r = 0 . R ⊙ is Mm or R ⊙ ), we noticethat for these (relatively) small amplitude anomalies, the global mode frequencyshifts are predominantly a function of the spherically symmetric component of thespatial structure of the perturbation. In other words, what matters most is thecontribution from the l = 0 coefficient in the spherical harmonic expansion of thehorizontal spatial structure of the perturbation. We verify this by computing thefrequency shifts associated with a spherically symmetric area-averaged version ofthe localized perturbation (with an amplitude of . A local / (4 π ) , where A local isthe solid angle subtended by the localized perturbation, 0.05 referring to the 5%increase in sound speed). We were careful to ensure that the radial dependence ofthe magnitude of the perturbation was unchanged. The frequency shifts associatedwith the spherically symmetric perturbations were calculated independently throughsimulation and the oscillation frequency package, ADIPACK (Christensen-Dalsgaardand Berthomieu, 1991) and seen to match accurately, as shown in Figure 3. ms.tex; 29/10/2018; 11:49; p.6 lobal effects of local perturbations Figure 4.
Changes in mode amplitudes (panel a) and the a coefficients (panel b) due to thesound-speed perturbation located at r = 0 . R ⊙ shown as functions of the phase speed of thewaves. In panel a, it seen that only modes which have turning points close to and below thelocation of the perturbation show changes in the amplitude (on the order of 0.5% or so). Due tospatially localized nature of the sound-speed perturbation, the m degeneracy is lifted, creatingthe slight bump in the a coefficient. Although not shown here, we observe that several a coefficients, even and odd, show the presence of the perturbation. Because of the non-spherically symmetric nature of the perturbation, we expect tosee shifts in the a -coefficients. Similarly, it is likely that there will be slight deviationsin the amplitudes of modes that propagate in regions close to and below the locationsof the perturbation. We display these effects for the case with the perturbationlocated at r = 0 . R ⊙ in Figure 4.4.2. Scattering extentWe introduce a non-dimensional measure, κ , to characterize the degree of scatteringexhibited by the anomaly: κ = 4 παA vuut N X n,l,m (cid:18) δνν (cid:19) , (2)where α = δc/c , the amplitude of the sound-speed perturbation expressed in frac-tions of the local sound speed, A the angular area of the perturbation, and N thenumber of modes in the summation term. Essentially, this parameter tells us howstrongly perturbations couple with the wave field, with larger κ implying a greaterdegree of scatter and vice versa. Because it is independent of perturbation size ormagnitude, κ can be extended to study flow perturbations as well. This measureis meaningful only in the regime where the frequency shifts are presumably linearfunctions of the perturbation magnitude. Also, it is expected that κ will retain astrong dependence on the radial location of the perturbation since different parts ofthe spectrum see different regions of the Sun. For example, placing an anomaly atthe surface will likely affect the entire spectrum of global modes, as seen in Figure 3c.Results for κ shown in Table 1 contain no surprises; for a given size and magnitudeof the perturbation, the effect on the global frequencies increases strongly with itslocation in radius. The signature of a perturbation at the bottom of the convection ms.tex; 29/10/2018; 11:49; p.7 anasoge & Larson Table 1.
The scattering extents κ , of various perturbations. TheRoot Mean Square (RMS) vari-ation in frequencies is shown aswell.Depth RMS κ ( r/R ⊙ ) δν/ν . × − . × − . × − zone on the global modes is twice as strong as an anomaly in the radiative interior( r = 0 . R ⊙ ). The surface perturbation is a little more difficult to compare withthe others because contrary to the two deeper perturbations, it is locally far largerthan the wavelengths of the modes. The result however is in line with expectation;the near-surface scatterer is far more potent than the other two anomalies.
5. Conclusion
We have introduced a method to systematically study the effects of various localperturbations on global mode frequencies. Techniques of mode finding and parameterfitting are applied to artificial data obtained from simulations of wave propagation ina solar-like stratified spherical shell. We are able to beat the issue of poor frequencyresolution by extending the method of realization noise subtraction (Hanasoge, Du-vall, and Couvidat, 2007) to global mode analysis. These methods can prove veryuseful in the study of shifts due to perturbations of magnitudes beyond the scope offirst order perturbation theory; moreover, extending this approach to investigatesystematic frequency shifts in other stars may prove exciting. We are currentlystudying the impact of complex flows like convection and localized jets on the globalfrequencies. Preliminary results seem to indicate that flows are stronger scatterers(larger κ ) than sound-speed perturbations although more work needs to be done toconfirm and characterize these effects. Acknowledgements
S. M. Hanasoge and T. P. Larson were funded by grants HMINAS5-02139 and MDI NNG05GH14G. We would like to thank Jesper Schou, TomDuvall, Jr., and Phil Scherrer for useful discussions and suggestions. The simulationswere performed on the Columbia supercomputer at NASA Ames.
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