First Steps toward Seismic Holography of the Tachocline
Manuel Díaz Alfaro , Fernando Pérez Hernández , Irene González Hernández , Charles Lindsey
1
1,2
3
4
(1) Instituto de Astrofísica de Canarias, Tenerife, Spain; (2) Departamento de Astrofísica, Universidad de La Laguna, Tenerife, Spain;
(3) National Solar Observatory, Tucson, USA; (4) Northwest Research Associates, Boulder, USA
The solar dynamo is thought to be generated at the tachocline and the deepest layers of the convection zone. Yet much about these layers or how this mechanism works remains unknown. In this work we present the first of a series of steps in order to apply helioseismic holography to the study of the tachocline.
We also present the theoretical background to use a spherical polar expansion to calculate the Green's functions instead of the usual acoustic ray path approximation. These new Green's functions will allow to reach the deepest layers of the convection zone with more accuracy.
Greens's functions calculated through spherical polar­expansion Seismic holography compares Green's functions, a model of wave propagation, with the observed wave field in an area of the solar surface (the pupil).
In the quiet Sun the wave field can be well represented by a superposition of these Green's functions. Yet, when the focus is located within an active region, the ingression H­ and egression H+ functions, representing waves in and out of the Imaging the frontside
Traditionally a plane­parallel approximation has been used for earthside seismic holography, but this approximation does not work for deep layers. In this work we have used a spherically symmetric approach to map an active region in the frontside of the Sun to test the potential for reaching the tachocline.
focus, show a phase shift, which can be studied through the phase of the correlation 2
in the Fourier domain :
∗
C r 0 ≡∫ H  r 0 ,  H − r 0 , d 
1
The method has traditionally used Green's functions calculated using the ray­path approximation, e.g., Lindsey & Braun (2000). However, this approximation is formally only valid for high frequencies and high angular degrees and therefore, not apropriate for studying deeper layers in the Sun. Pérez Hernández & González Hernández (2010) introduced Green's functions calculated through their spherical polar expansion, with the asymptotic approximation for their radial factors, but no asymptotic approximation for their angular factors:
1 ∞ r 
G  r∣
r 0 , =
G l Gl
∑
4
l=0
More accuracy is expected for low and intermediate degrees, which will map deeper layers of the Sun.
Acoustic ray­path vs spherical polar­expansion
Testing the Green's functions at deeper layers
These Greem's functions may also be used for the farside of the Sun. A better signal­to­noise ratio is obtained and better results are achieved for weaker active regions.
25 Nov 2006
27 Nov 2006
Left, image of the frontside of the Sun including NOAA's active region 10808 (13 Sep 2005) calculated through seismic holography using the ray approx. for the Green's functions. Right, a MDI (SoHO) magnetogram shows the same active region for comparison. We expect better results with the spherical polar­expansion Green's functions.
29 Nov 2006
In order to test the possibility of the application of these new Green's functions to the deeper layers of the convection zone, we computed the correlation between the ingression and egression functions with different theoretical signals Ψ and perturbations L'. The egression and ingression functions are given by:
H ±=∫ G l±  r∣r0 ,  L '  r , dV
V
Test of the technique for anomalies at 0.71R⊙ (tachocline) left and 0.90R⊙ right for a Dirac Delta (above) and a wide gaussian­shaped perturbation (below). The position of the perturbation is recovered by moving the position of the focus of the Green's functions. The red solid line shows the position of the introduced perturbation.
Farside images for three different days showing a weak active region. Above, using the new spherical polar­expansion for the computation of the Green's functions compared to the ray approx. in the middle. Bottom, the plot shows cuts at the latitudinal center of the active region: solid lines represent the new spherical polar expansion and dashed lines, the (acoustic) ray­path approximation. The signal­to­noise ratio at the active region is higher using the Green's functions calculated through a spherical polar­expansion.
References:
[1] Lindsey, C. & Braun, D. C. 2000, Sol. Phys., 192, 261
[2] Pérez Hernández, F. & González Hernández, I. 2010, ApJ, 711, 853
This work utilizes data obtained by the Global Oscillation Network Group (GONG) Program. GONG is managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísica de Canarias and Cerro Tololo Interamerican Observatory.
This work has been supported by the Spanish National Research Plan under project AYA2010­17803 and the NASA Living with a Star – Targeted Research and Technology program.