S ^Rf ghXYikjlgYi QmX - Stanford Universityquake.stanford.edu/~sasha/PAPERS/review_2.pdf · 2003....

12
I review the progress that has been made in our under- standing of the physics of the upper boundary layer of the Sun, its influence on frequencies of five-minute os- cillations, and its role in excitation of the oscillations. I also discuss approaches to seismological diagnosis of the properties of the layer, important information about which will be obtained from MDI high-resolution data. Keywords: In most of the convection zone the temperature gradi- ent is nearly adiabatic because of the high efficiency of energy transfer by low-speed convective motions. How- ever, the gradient becomes superadiabatic and convective velocities substantially increase near the surface where the convective transfer is inefficient owing to low density. The superadiabatic boundary layer, which occupies only a few hundred kilometres below the surface, is the region of the greatest helioseismological uncertainty. The assump- tions about hydrostatic equilibrium of the solar structure and adiabaticity of oscillations that are commonly used in helioseismic inversions are not valid in this layer. This uncertainty may affect our conclusions about properties of the deep interior. Our potential for improving the ac- curacy of inversions is quite limited unless the physics of interaction between convection and oscillation modes and also the structure of the layer are understood better. The upper convective layer is likely to be the place where p modes are excited. Information about the excitation and damping is contained in the amplitudes and phases of the oscillations, and also in the shapes of spectral lines in the power spectra. Despite considerable efforts to understand the mechanisms of excitation and damping of the modes, the current state of the art is still far from completion and is not in full agreement with observations. Helioseismology offers the opportunity to study subsur- face flows, and therefore to test the numerical predictions and to improve the theoretical models of turbulent con- vection. Several techniques are available to measure sub- photospheric flows. They employ local seismic analysis, based on measuring some local properties of wave propa- gation, such as the dispersion relation or travel time, and are reviewed by F. Hill in these proceedings. The inho- mogeneous structure of the surface layer results in com- plicated splitting and shifting of the oscillation eigenfre- quencies, from which one could possibly infer character- istics of the structures (e.g. Lavely & Ritzwoller, 1992). However, the analysis requires accurate measurements of frequencies of individual modes, which are expected from SOHO and GONG. Presently, most of the helioseismic studies are concen- trated on the mean spherically symmetrical structure of the Sun’s interior. These studies also provide valuable information for testing theories of convection. The the- oretical background of the studies is provided by the so- called diagonal sum rule (Gilbert, 1971) which states that the average frequency of a split multiplet is the degenerate eigenfrequency of the spherically symmetric component of the solar structure, provided that the aspherical com- ponent can be considered as a small perturbation. This assumption is almost certainly not justified in the con- vective boundary layer. However, it is commonly used because a general analysis of non-linear aspherical per- turbations is very complicated (cf Kosovichev & Perdang, 1988). High-resolution observations of the Sun, such as those obtained with the Solar Optical Universal Polarimeter (SOUP) on Spacelab 2 (Title , 1989), reveal the highly inhomogeneous structure of the solar surface co- vered by convective granules. The granules display signi- ficant variations of temperature and density, accompanied by strong flows, the velocity of which may reach the lo- cal sound speed. The highest velocities are observed in convective downflows that form thin dark lanes around relatively bright areas of ascending flow. The typical size of granules is about km. Granules form two patterns of larger scale on the surface: mesogranulation ( km) ultimately related to large "exploding" granules, and

Transcript of S ^Rf ghXYikjlgYi QmX - Stanford Universityquake.stanford.edu/~sasha/PAPERS/review_2.pdf · 2003....

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I review the progress that has been made in our under-standing of the physics of the upper boundary layer ofthe Sun, its influence on frequencies of five-minute os-cillations, and its role in excitation of the oscillations. Ialso discuss approaches to seismological diagnosis of theproperties of the layer, important information about whichwill be obtained from MDI high-resolution data.

Keywords:� ´"��µ �r� � �I���L�����+�¶������� · ·¸���|�������+�¹�+��� ���+���|�������

�;´"��º�´"·¸���w�+���©»$�����w��·-���0�L�|�,���+���L�+�G�w���w�������Eº+���|�����|¼$�+���|��½

¾ n¿c�^RNdX_\�f�SaZbNdc�\�^

In most of the convection zone the temperature gradi-ent is nearly adiabatic because of the high efficiency ofenergy transfer by low-speed convective motions. How-ever, the gradient becomes superadiabatic and convectivevelocities substantially increase near the surface wherethe convective transfer is inefficient owing to low density.The superadiabatic boundary layer, which occupies only afew hundred kilometres below the surface, is the region ofthe greatest helioseismological uncertainty. The assump-tions about hydrostatic equilibrium of the solar structureand adiabaticity of oscillations that are commonly usedin helioseismic inversions are not valid in this layer. Thisuncertainty may affect our conclusions about propertiesof the deep interior. Our potential for improving the ac-curacy of inversions is quite limited unless the physics ofinteraction between convection and oscillation modes andalso the structure of the layer are understood better.

The upper convective layer is likely to be the place wherep modes are excited. Information about the excitation anddamping is contained in the amplitudes and phases of theoscillations, and also in the shapes of spectral lines in thepower spectra. Despite considerable efforts to understandthe mechanisms of excitation and damping of the modes,the current state of the art is still far from completion andis not in full agreement with observations.

Helioseismology offers the opportunity to study subsur-face flows, and therefore to test the numerical predictionsand to improve the theoretical models of turbulent con-vection. Several techniques are available to measure sub-photospheric flows. They employ local seismic analysis,

based on measuring some local properties of wave propa-gation, such as the dispersion relation or travel time, andare reviewed by F. Hill in these proceedings. The inho-mogeneous structure of the surface layer results in com-plicated splitting and shifting of the oscillation eigenfre-quencies, from which one could possibly infer character-istics of the structures (e.g. Lavely & Ritzwoller, 1992).However, the analysis requires accurate measurements offrequencies of individual modes, which are expected fromSOHO and GONG.

Presently, most of the helioseismic studies are concen-trated on the mean spherically symmetrical structure ofthe Sun’s interior. These studies also provide valuableinformation for testing theories of convection. The the-oretical background of the studies is provided by the so-called diagonal sum rule (Gilbert, 1971) which states thatthe average frequency of a split multiplet is the degenerateeigenfrequency of the spherically symmetric componentof the solar structure, provided that the aspherical com-ponent can be considered as a small perturbation. Thisassumption is almost certainly not justified in the con-vective boundary layer. However, it is commonly usedbecause a general analysis of non-linear aspherical per-turbations is very complicated (cf Kosovichev & Perdang,1988).

À nÂÁzNPX]SRZbNdS�XhQÃ\_ÄÅNdORQ«Z�\�^a`�Q0ZbNdcL`�Qe�\_SV^Rf�gVXmikjAgmi�QmX

À n ¾ nÆ\�Çtv�~�È�x�É"Ê�y|u Ëv

High-resolution observations of the Sun, such as thoseobtained with the Solar Optical Universal Polarimeter(SOUP) on Spacelab 2 (Title

������·-½, 1989), reveal the

highly inhomogeneous structure of the solar surface co-vered by convective granules. The granules display signi-ficant variations of temperature and density, accompaniedby strong flows, the velocity of which may reach the lo-cal sound speed. The highest velocities are observed inconvective downflows that form thin dark lanes aroundrelatively bright areas of ascending flow. The typical sizeof granules is about Ì�Í�Î km. Granules form two patternsof larger scale on the surface: mesogranulation ( ÏÑÐtÒ0Ì�Í5Îkm) ultimately related to large "exploding" granules, and

Ó Ì Ó

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supergranulation that has a scale of about �0ÒVÌ�Í�� km andis probably tied to deep convective flows.

À n À n ^���� ~�È�y|{EÉ��Av�y�����É�Ê�y�uwËv

The observed horizontal structure of the surface convec-tion is reproduced remarkably well in 3-D numericalsimulations (e.g. Cattaneo,

���]��·¸½, 1991; Rast,

������·-½,

1993). Quantitative agreement has been achieved be-tween the simulations and ‘surface’ observables, such asspectral line widths and intensities (Spruit,

���m��·¸½, 1990;

Atroshchenko & Gadun, 1994).

The simulations also provide an important insight intothe vertical structure of the convective boundary layer.They have demonstrated that convection consists of twoprimary components of flow: relatively slow broad con-vective updraft, relatively smooth and weak in turbulence,and fast narrow downdrafting plumes. The boundaries be-tween the downflowing plumes and the surrounding up-flows are regions of strong shear, and therefore, of strongturbulence. The "exploding" granules generate particu-larly sharp downdrafting plumes at their centres (Rast,1995). Descent velocities in such plumes can becomesupersonic, and therefore the plumes can be sites of sig-nificant acoustic excitation. Numerical simulations alsoshow organization of subsurface flows into large-scalestructures which have not yet been understood. Helio-seismology gives an opportunity to probe directly thestructure and flows beneath the surface and, thus, to testnumerical models of convection.

À n��©n� ~5É�Ë v�Ê�È�É"Ê�y��{�É"Ê�y|u Ë

While there have been attempts to deduce a model ofthe spherical structure by averaging the results of thenumerical simulations (e.g. Nordlund & Dravins, 1990;Atroshchenko and Gadun, 1994), the standard descriptionof the mean stratification of the convective layers is basedon mixing-length models. In the mixing-length approach(e.g. Bohm-Vitense, 1958; Cox & Giuli, 1968), the com-plicated situation in the actual convection zone is replacedby a group of identical convective elements, "bubbles", allof which have the same properties at a given distance, � ,from the solar centre. These convective elements are notdrastically different from the surrounding. They rise orfall under the buoyancy force through a characteristic dis-tance, � , the mixing length, and then break up and mixwith the surroundings. At any radius � , the convectiveelements are assumed all to have the same speed, ��������� ,which approximates the average speed of the actual con-vective elements.

The structure of the bulk of the convection zone is veryclose to being isentropic because the plasma density ishigh enough that convective flows of low speed are suf-ficient to transport the convective energy. However, near

the surface, even at high convective velocities, convectioncan less easily transfer the heat, owing to the low density.The stratification of the zone of ineffectual convection,which is only about 500 km thick, is substantially supera-diabatic, resulting in significant buoyancy force and rapidacceleration of the convective elements. Nevertheless,despite the high convective velocities, most of the energyin this layer is transported by radiation. The properties ofthe convective boundary layer obtained using the mixinglength theory are shown in Fig. 1.

Ä�y���6È�~ ¾:�°�,�&�����L�;�����d�/�0� ���W´��������d�&�������+���;� ���]º+��´"�¤���

���L�¢·¸���E���¶�+�� $�w´"���&�¢´5�+� �"!# ]�%$�� ��!¢·-���"!��)�8�)���+���L�'&8��(�)���d�L´������,�E�����Eº+���;���)!��&�E���������+*-,/.1032�4'53687�092:4;59<+(>=Aº�(�+��� ���+���|� ���r����·¸����� �;�>=��?(��E�����L� �|�>=R���¤����(¢�,���|���8���o� ������w���!E�A@�´'$_�|�,�����)�����L�,�+�_º��_�+��� ���+���|�����o����� ���W��������·©���B����!E�C@�´'$'=EDa�q�0�)���$�����%!��5�°�Eº+�����W�)���A�������I���|�������,�P·¸������·¸½

The most characteristic feature of the boundary layer isthe sudden change of specific entropy, F , that occurs justbeneath the photosphere (Fig.2a). The amplitude of theentropy jump is determined by the overall structure ofthe Sun and is essentially independent of the details ofthe boundary transition layer (Gough & Weiss, 1976).However, solar oscillation frequencies are sensitive to thestructure of the entropy jump, and therefore provide po-tential for testing theories of convection.

ÓHGdÓ

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Despite the significant differences between the mixing-length approach and the picture shown by the numericalsimulations, the averaged properties of the layer are qual-itatively similar (Spruit,

���l��·¸½, 1990). The mixing-length

theory also gives a qualitatively correct estimate of theaverage vertical velocity. Numerical simulations of thegranulation layer have been used to calibrate the mixinglength, � , which is usually assumed to be proportionalto the pressure scale height, � : ������� , where � isa constant. However, Deupree & Varner (1980), andAtroshchenko & Gadun (1994) found that the mixing-length theory fits the numerical simulations if � varieswith depth, being smaller at the very top of the convec-tion zone. They suggested that this decrease of � reflectsnon-local properties of convection.

One of the important properties for the helioseismic diag-nosis of the solar structure is the parameter of convectivestability, ��� � Ì

092:4;5�032�4'5 �

� 032�4'5� 032�4'5 ���

� ��where

is the gas pressure and is the adiabatic exponent.

The parameter� �

, sometimes called the Schwarzschilddiscriminant, is positive in convectivelystable regions andnegative in zones of convection. Its absolute value is verysmall in most of the solar convection zone, but it experi-ences a sharp variation with a deep minimum in the upperconvective boundary layer (Fig. 2b).

Ä�y����6È�~ À:��(��¤���¹�;���+��� �A�¢�����;�,�&�¤��� F �V���w� º?(��)���

�����,� ����I���W���0�&�������+���;� ���m�L���Eº�� ·1� �;��� � � � 0 F 7 032�4'5 � = F���q�V�)���a�|���+��� ���o�����;�,�&�¤�r� � �)���o�������Eº+���|���+��· · �¢�L�|�,���|� ���+������L�����W�)���b�&�������+���;�����������w��½

The profile of� �

in this layer is very sensitive to de-tails of the convection theory (e.g. Lydon,

���$��·¸½, 1992).

Since the definition of� �

includes only the ‘primary’seismic parameters,

, , and , the quantity

� �can be

measured seismologically without any other additionalassumptions.

À n��6n NC�©È�Ç � �|~5ˤÊ��È�~5v�v �©È�~

The effects of momentum transfer by convection (turbu-lent pressure) are often ignored in solar models. However,the turbulent pressure shown in Fig. 3, can be as muchas 10% of the total pressure (e.g. Canuto & Mazzitelli,1991), and therefore, it must be taken into account.

Ä�y���6È�~ �:�w���b�,���;���h�/�0� ���d�;´"��º�´"·-��� �¤�w�,���+�L´"�,�d�����)���

��������·t�¤�,���&�L´"�,�¹���¶�h�+´"�w���|����� ���������%!��"�]� ��� ���o´���������+��� ���+���|� ���]º&��´"�w�����L�_·¸���E���

The turbulent pressure affects the structure of the upperconvective layer in several ways. It contributes to the to-tal pressure, thus reducing the gas pressure and leading tolower density and sound speed. Turbulent pressure alsomodifies the superadiabatic gradient,

, � ,��! (Henyey,���

��·¸½, 1965), and results in changes in the properties of con-

vection. However, this effect seems to be less significantthan the direct impact on the hydrostatic balance (Canuto& Mazzitelli, 1991). Unfortunately, a consistent modelwith turbulent momentum transfer has not yet been de-veloped, mainly owing to the lack of a turbulence modelto provide the turbulence spectrum. However, severaldifferent approximations have been used. One of them,suggested by Gough (1977), is based on the Boussinesqapproximation and assumes that the turbulence is locallyaxisymmetric about the radial direction. The mean equa-tion of the hydrostatic balance is then

Ì 00 �

� #"�%$'&)(+* � " � � �-, �%$.&�(+*� � �0/21

�435�� G �

Ó � Ó

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where $.&�(+*

is the turbulent pressure given by

$.&)(.* � ���� 3�� � � � ��is the radial component of the convective velocity ��� ,

and , measures the anisotropy of the turbulence:

, .� ����� � 7���� 3 �� ��� �If the turbulence is isotropic then , � � . The second termin equation (2) represents the net radial force produced byhorizontal Reynolds stress for the case of anisotropic tur-bulence.

The turbulent pressure can be expressed in terms of themean turbulent speed, ����� ��� :

$.&)(+* � ���� 3�� ��� � 3��� ��� � � �where the constant � depends on the spectrum of theturbulence. Henyey

���G��·¸½(1965) argued that the value of� must be greater than unity because the mean value of

the square of the speed is invariably larger than the squareof the mean speed. For instance, for a normal distributionof velocity � ��� 7 G . Our preliminary computations ofthe turbulent pressure effects on p-mode frequencies givea good fit to the observed frequencies if ��� � 7 G (seeSec. 3.1).

Ä�y����6È�~ �:�¤��� �,��·¸���;� ��� ��� ¼$���,���¤�+� ��� � ��� ����´����,�+�

����´"�¤�¹�;���+�+��º+�����l�+��� � ���a�L�I���w�����,�¢����·¸���� ����E��·��?� � ��)���P�|´"��º�´"·¸�����¤�¤�,�L�+�L´"�,�]���¤���)��� ����E��·��?� �)����´"��� ��½The turbulent pressure decreases the sound speed in theboundary convective layer (Fig. 4) resulting in reductionof frequencies of p modes (Gough, 1984; Balmforth,1992; Guzik & Cox, 1992). This effect is important forhelioseismology and definitely deserves further investi-gation.

�znÂQmÄ�Ä�QWZbNbÁ \�Ä Z�\_^a`VQWZPNPc�\�^ \_^\VÁ6Zbc�jAjlg0Ndc�\_^ Ä�XVQ��_SVQY^¶Zbc�QWÁ

The upper convective layer affects frequencies of solaroscillations in several ways, which can be separated intotwo classes: ‘adiabatic’ and ‘non-adiabatic’. The formerare due to structure variations in the convective boundarylayer; and the latter arise from coupling between oscilla-tions and convection and from absorption and emission ofheat as a result of pulsations. Theoretical models of theseeffects show that the ‘adiabatic’ effects are dominant.

�zn ¾ n¿Nd}6~¶~��A~�{EÊ�v�u �v�Ê�È��6{EÊ �6È�~¶x�É"È�y�É"Ê�y�uwË

There are two types of the structure variations in theboundary layer: perturbations of the mean radial strat-ification, and horizontal inhomogeneity. The sphericallysymmetrical perturbations shift the mean frequencies ofsplit multiplets, whereas the nonradial inhomogeneitiescause variations in frequency splitting within the multi-plets (e.g. Lavely & Ritzwoller, 1992).

The shift of adiabatic eigenfrequencies caused by the ra-dial perturbations can be expressed, using a variationalprinciple, in terms of perturbations of two independentproperties of the internal structure, e.g. the parameterof convective stability,

� �, and the adiabatic exponent,

(see Kosovichev, 1995):!�"$#�%'& "$#& "$# � (*)�,+.- "$#0/132 % ��� 0 � " (*)�4+ - "$#0/5 % 0 � � ��6 �where the integral kernels + - "$#7/1 2 and + - "$#7/5 depend on theunperturbed eigenfrequencies, & "$# , and the unperturbedeigenfunctions, 8 8 8 8 8 8 8#� � 8:9 � 8<; � ; 8<9 and 8<; are the radial andhorizontal components of the Lagrangian displacement,8 8 8 8 8 8 8 , of oscillating fluid elements; and!="$# � ( )�?> 8 39 " 8 3;A@ � 3 0 � �CB �is the moment of inertia of oscillation modes.

Near the surface, the ratio of the horizontal and radialdisplacements is given by:8<;8 9 � �EDF 3"$#�G � Ì�ÍIH Î �KJ �MLON�P& "$#RQ 3 � ��S �where F "$# � G � & "'# , and D is the gravitational acceler-ation. If � � Ì�Í�Î then the oscillations of the typicalfrequency 3 mHz are almost radial near the surface. Thatmeans that near the surface the kernel functions, + 1 2 � �4�and + 5 � � � , are insensitive to the value of � . Therefore, fora given surface perturbation % � � and % , the right-handside of Eq. (6) does not depend on � and is approximatelya function of frequency & "$# alone. Thus, for a surfaceperturbation %'& "$#& "$#T�VU � & "$# �! "$# ��W �

Ó � Ó

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In other words, the perturbation of p-mode frequenciesdue to variations in the structure of the boundary convec-tive layer can be approximately represented by a functionfrequency alone, scaled with the inverse mode inertia.

Ä�y����6È�~��:��(��¤������� ¼$���,���w�+�¨º+���C�l�+���]�)�����Eº������L���+�?�+�,� ���´����¤�������+� & � *��+�d�/���"�%!�� �/�E� !E�,�+�b�1 ����E��� *��$���&� ��������

���W��·¸½¸����ª�ª�§�(¢���¤�o�)���a�+���L�,���;�����w��� ��!��+�,� ��´����¤�������+� & � $���R�r�L�����w�����,�r����·¸��� �������·$�+�� $�w´"���+� �?� � ����´"�b�|¼$�+��������d�|´"��º�´"·¸�������¤�,���&�L´"�,� *+�©�5�L�q�L�I���������B���]��· � !��E���,���¨������·-½¸���ª�ª� (�½� "$# �ÅÐ !="$# 7 !=" � � �6�����,� !="$# �q�b�)��� �� ������m�/� ������a� �w���L�|��� *)����½ �?(E½ �w���d�����������o§¹�q�V� �¤��· ´��E�+����� ������� "'# ���W�)���]���,�����P�/�W´"��� �;�E½º?( �¤���r��� ¼$���,���w�+�oº+���C�l�+��� �)���d�+�,� ��´����w�����L�+� & $'&)(+* �b�/��¶����·¸��� �������·�� ���6�"���&� �)���a�|¼$�+���d�/�_�;´"��º�´"·¸�����l�w�&�L� ��L´"�,�d���a�+�;�,���|� ���&���|�����R�q�Y���������R� �����_�E�+�+��´"� �l�E�+�+���,��� ��!���b� ��½ *�� ( *��������������&�������A� �h�w�&�I�����,���|����� (V���w���)�����+�,� ���´����¤������� & � $ ���d� ���W�L�����w�����,� ����E��·¸½�?( �¤���0�+�,� ��´����¤������� ¼$���,���w�+�oº+���C�l�+���8�)��� �������· �?� � ��)�����;´"��º�´"·¸��� �t�w�&�L�+�L´"�,�h���w�a�)���V�Eº������L���+�P�+�,� ��´����¤��������½

This property is commonly used in helioseismic structureinversions to eliminate the uncertainties of the structure ofthe convective layer from equation (6) (e.g. Basu

���$��·-½,

1995). The oscillation modes of relatively high angulardegree are particularly sensitive to the structure variationsnear the surface because their mode inertia is relative low.Therefore, they are used to probe the boundary layer.

Fig. 5a shows the scaled difference between the frequen-cies of the modes of �EÍ�Í�� ����Ì G Í�Í measured by Bach-mann

���¨��·¸½(1995) and the theoretical frequencies com-

puted for a standard solar model of Christensen-Dalsgaard���m��·-½(1992). Fig. 5b shows the difference between the

frequencies of a model in which the equilibrium structurewas corrected to include the effect of turbulent pressure,and a standard solar model that neglected turbulence ef-fects. The turbulent pressure may account for most of thedifference between the observed and model frequencies.It is important to note that the correction of the equilib-rium structure of the boundary convective layer does notaffect the frequencies of the f mode, which is essentiallya surface gravity wave. The mechanisms that can changethe frequency of that mode are discussed in Sec. 3.3. Theremaining differences between the observed and modelfrequencies are shown in Fig. 5c. The observed frequen-cies of p modes are still smaller than their theoreticalfrequencies; however, the p modes exhibit the same trendas the f mode, suggesting a common mechanism for thefrequency variations of both types of oscillations.

The turbulent pressure effect is, perhaps, the main reasonwhy Bachmann

���Y��·-½(1995) found their data in better

agreement with the solar model of Guzik & Cox (1992)than with the models provided by Guenther

������·¸½(1992)

and by Christensen-Dalsgaard���d��·-½

(1993). It is alsolikely that for the same reason Paterno

������·¸½(1993)

found a better agreement with observations when theyreplaced the standard mixing-length theory without tur-bulence effects with a new model of convection (Canuto& Mazzitelli, 1991), in which the turbulent pressure istaken into account.

The effects of horizontal inhomogeneities on frequencysplitting have not been studied, though a theoretical ap-proach to the effects was suggested by Lavely & Ritz-woller (1992). However, no measurements of the fre-quency splittings of individual multiplets have been made.Such measurements are an important goal for the MDI in-strument on board SOHO.

�zn À n¿NH�6È�Ç�����~5ˤÊ��È�~5v�v �6È�~����{�Ê �6É�Ê�y�uwËv��Ëu ËÉ��y�É"ÇtÉ"Ê�y�{a~A�l~5{�Ê�v

Variations of frequencies of the oscillation modes mayalso arise from coupling between convection and oscil-lations in the convective superadiabatic boundary layer.Such coupling results in fluctuations of the turbulent pres-

Ó Ð Ó

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sure (Reynolds stresses). It also plays an important rolein mode excitation and damping (Sec.4).

Using a time-dependent formulation of mixing-length the-ory, Gough (1984), Balmforth & Gough (1990) and Balm-forth (1992) found that the nonadiabatic effects due tomode interactions with convection and radiation are sub-stantially smaller than the alteration to the equilibriumstate caused by turbulent pressure. The effects of nonadi-abaticity and turbulent pressure fluctuations become moreimportant at higher oscillation frequencies, because inthe convectively active superadiabatic layer and in the ra-diative atmosphere high- & modes have relatively greateramplitudes than the modes of low frequency. The resultreproduced in Fig. 6 was obtained by Gough & Balm-forth (1990) for radial oscillations ( � � Í ). However, thefrequency difference is essentially the same for nonradialmodes when scaled with the mode inertia

!<"$#.

�y����6�~��:

�w���Ñ��� ¼m���,���w�+� & � ��� � � & � *I� ���©� �?(º+���C�l�+���¶� ���l�+�,� ��´����¤������� & � ��� � ���d�w���¤�E�����Eº+���;����������� · �·¸���|������� *I� � � ���# ������ �A�+� �¨�E��� �"!������ �&���w�,� $E� ����;�����'(� �w��·1´���� ��! �|´"��º�´"·¸�����_�w�,���+�+´"�&� @�´����|´����|�������+�8���¤� �)����E�����Eº+���;��� ���%!������+�,� ��´����w�������&� & � *I�%!E�w���L� ��! �;´"��º�´"·-��� ��w�&�L�+�L´"�,�3@�´����|´����|������� (��/�0� ���0���� ��]� ��´"� ·1��º��L� ´ ����E��·*����+�������¨��´'!��¹� �m��·% ������L� �5� ��ª�ª�¥ (�½

The strong fluctuations of the frequency difference at& � �EÍEÍ�Í � Hz result from turbulent pressure fluctua-tions. However, Gough & Balmforth (1990) pointed outthat since convective conditions in their theory are de-termined purely locally, the frequency sensitivity of theresponse of the convection might be artificially enhanced.

Further investigation of these effects with a more reli-able description of turbulent convection is necessary, notonly because this provides a way to study the physics ofthe convection, but also because understanding the effectsis important for helioseismic inversions. The results byGough & Balmforth (1990) indicate that it may be incor-rect to use the common assumption that the frequency er-ror caused by surface uncertainties can be approximatedas a smooth function of frequency (e.g. Basu,

���h��·¸½1995). For this reason, Dappen

������·¸½(1991) and Koso-

vichev���¨��·¸½

(1992) recommend using only modes withfrequencies below 3 mHz in the helioseismic structureinversions, because these modes do not interact stronglywith convection.

�zn �zn Q �l~5{�Ê�v�u �È�É�Ë��u�� x ~"��uz{�y)Ê�� � �{�Ê �É"Ê�y�uwË6vÉ�Ë�� Ê�}~ � ÉB� Ë~�Ê�y�{ �~"� �

The fluctuating velocity field associated with convectionis another factor in the convective boundary layer thatdecreases the frequency of oscillations. This effect is par-ticularly pronounced for the f mode which is essentiallyan uncompressible surface wave, insensitive to both struc-ture variations (see Fig. 5b) and nonadiabatic effects (e.g.Zhugzhda, 1982). However, there is systematic differencebetween the observed and theoretical frequencies of the fmode (Fig. 5a). The observed frequencies are somewhathigher than theory predicts at low & , but become lower asthe frequency (or the degree) increases. Such behaviourof the f-mode frequencies can be understood in terms ofthe combined action of a magnetic field and random con-vective flows (Murawski & Roberts, 1993a,b; Murawski& Goossens, 1993).

A magnetic field exerts an additional tension force on thissurface mode. As a consequence the phase speed of thewave is larger in the presence of the magnetic field, andthis leads to an increase of the oscillation frequency fora given wave number. On the other hand, random con-vective flows can decrease the frequency, because a wavetakes longer to travel a given distance due to the scatteringfrom random inhomogeneities (Howe, 1971).

Q

P

λ

R

Ä�y���6È�~��: �����¶���+���;�I���L� ��!V�+�,�� �,���w�E� � �5��� �� �

!����¤��� �;������½ �]���;�I�+�0·1� �¤�L�¨�&�I�¤�,�L�������©���+���|�����&�&� �l�����L��="�+´"· ··1� �¤�L�Y�,���w�,������� �l�&� ¨�����¤��� ���W���m� ���d�+�������,����� �A��·¸�>=���q��)���b�&���L�,��·¸���;�����¶���+��·¸�b�/�Y� ���3@�´����;´����;������� *����+�����$�P�$�l����Eª�� � (E½

The wave scattered from a point, P, will be scattered againat some other point, R (Fig. 7). Such a collision will pro-duce a scattered random field together with a mean fieldcomponent. The latter represents the energy scatteredback into the coherent wave field, and will be significant

Ó 6 Ó

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if the distance between P and R is less than the correlationlength,

, of the random fluctuations. As a result of the

scattering, the frequency of a wave is reduced by the orderof the mean squared amplitude of the fluctuations, and theenergy of the coherent wave is gradually absorbed.

Murawski & Goossens (1993) demonstrated that a modelwith random convective flows and a chromospheric mag-netic field can fit the observed f-mode frequencies. Itis likely that frequencies of p modes are also reduced bythe same mechanism, explaining the remaining differencebetween theory and observations (Fig. 5c). However, thecomputations for p modes have not been carried out.

Finally, we note that the fine difference between p-moderidges in Figs 5a and 5c is likely to be due to small varia-tions of the mean stratification of the convective boundarylayer. Such structure variations, which can be, in prin-ciple, inferred by frequency inversion, are of interest fortesting theories of solar convection in greater detail.

�6n \_fVQ Q��_ZPc�NAg0Ndc�\_^ gV^Rfkf�g UWc�^¶p

It is generally believed that f and p modes of solar oscilla-tion are excited randomly by the turbulence in the upperconvective boundary layer. This belief is based mainly ontheoretical models showing that the modes are most likelyto be intrinsically stable (e.g. Balmforth, 1992). How-ever, the physics of interaction between oscillations andconvection, the key element of the models, is poorly un-derstood, and the theory of mode excitation and dampingis far from being in quantitative agreement with observa-tions.

�©n ¾ n c�ˤÊ�È�y�Ëv�y|{av�Ê�É"Çy��y�Ê �

The vibrational stability of the Sun has a long history of in-vestigation (see Balmforth, 1992, and references therein).The main non-adiabatic process that contributes to desta-bilization of the p modes is related to variations of opacityin the hydrogen ionization zone and represents a gener-alized form of the conventional � -mechanism. The � -mechanism which operates in classical pulsating starsworks, in principle, on modulation of the radiative en-ergy flux. When, due to oscillation, the outer layer ofa star is compressed, the degree of ionization increases.This leads to an increase of opacity and to absorption of agreater amount of heat than would have occurred withoutoscillations. In the subsequent expansion, the extra heatstored in the ionization zone is converted to mechanicalwork, thus amplifying the oscillation. However, in theSun, the zones of ionization lie in the convection zonewhere the amount of heat transported by radiation is rel-atively small. Therefore, this mechanism can providerather weak driving.

The detailed driving or damping of the solar acousticmodes largely depends on their interaction with convec-tion. Oscillations perturb the turbulent fluxes of heat andmomentum. How these perturbations in turn influence theoscillations crucially determines their intrinsic stability.A mathematical description of these processes must in-volve a theory of turbulence in the upper convective layer.So far, a detailed investigation was carried out only for atime-dependent, non-local mixing-length theory (Gough& Balmforth, 1990; Balmforth, 1992).

The time-dependent theory of convection determines howoscillations perturb the turbulent flux and momentum.The perturbations then react back on the processes ofmode driving and damping. The coupling between theoscillations and convection takes two principal forms:thermal effects due to modulation of the convective heatflux and dynamical effects produced by perturbations ofthe turbulent pressure (Reynolds stresses). The computa-tions by Gough and Balmforth showed that the perturba-tions of the turbulent pressure are principally responsiblefor stabilizing the oscillations. The non-adiabatic pro-cesses related to opacity changes due to oscillations (the" � -mechanism") compete with the damping but are ratherweaker. The contributions of these processes to the damp-ing rate, � , are shown in Fig. 8.

Ä�y���6È�~��:�w���m���� $�w� �"!P�&�����

�*|����·1���P��´"�L���?(d���°�,��������·

���&��´5�L�|��� ������L���&� ¨�¤´"�I�+�aº�� �m��· °�����L�)� * ��ª�ª�¥�(0��������;� �� �/�E�������¤�E����� ]�%$E� �"! ��·-���"!��)� �)���+���L�8�����+��� ���+���|�����w½�¤���������;�I�+����´"�L���P�,���$�t�b� ���]�+�����;�L��º�´"�;�����¹�I�

���´��b���

� � �I���,�E���;����� º+�����l�+��� ������� · ·¸���;������� ���¤� �,���������;�����B=¢� ��������,���+��· � �w�d�/���$�t�W�)���P�+�����;�L��º�´"�;�����r�/�Y�|´"��º�´"·¸�����¤�¤�,�L� ��L´"�,�9@A´����;´����;��������½

Therefore, the theory predicts that solar p modes are in-trinsically stable. The theoretical estimate of the dampingrate is in qualitative agreement with the measurements ofthe oscillation line width (Fig. 9).

Ó B Ó

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Ä�y����6È�~��:�¤��� �Eº������L���&� ���¤�Ñ�)���+���,���;���+��·��E�� $�w� �"!

�,���I���������)���V�,��������·�������� · ·¸���;������� � � �«Í � ���V�b�+´"�w� �

�;����� ���]�+�,� ��´����w���E½ �¤��� ��+���+´"�&� ���� ���¶�/�o·1� �¤�=�?����� ���� . G �%� º��¢�t��º�º��&�&�+�5��* �EªE¦E¦ ( *����&���+���L� ( ��� � � G Í ������L�R���w��º����"�|¼$���L���L�����P��·-½ * ��ªEª ��( *;�;�L�����"!�·-�L� (��/���� Ì�ÍEÍ ������L�d���&�0���+��·¸�+�O�?� �)�a�)���l���E���I��� ! � � F � 7 ! # � F �*I� �������+���8�,��·¸���|� ��� ����E�8� �¤���L�;����(��I� �E�+�+��´"� �Y����� �)���� �/���I�����w�E���w�+���/�d�)���]·1� �¤� �?�����)����½��¤�����)���+���&���;���+��·����· �´���� *;����·1������´"�L���?( �l���,�]�Eº��I��� �w�+�hº�� �m��· °�����L�)�#* �EªEª� (´5�L� ��!����;� �� �/�E�������¤�E�����C b�%$�� ��!'��·¸����!E� �R� ���+���L�

The theory explains the characteristic depression of �(‘plateau’) observed at 2800

�Hz by the contribution of

the " � -mechanism" (see Fig. 8). However, it disagreeswith the observations at low and high frequencies. Weshould note that there is also significant uncertainty incurrent linewidth measurements which may be caused bygaps in ground-based observations and by other factors(see Harvey, these proceedings). Therefore, reliable mea-surement of the line widths of oscillations is an importantgoal of the GONG and SOI projects.

�6n À nÂÁ�Ê�uz{�}6É�v�Ê�y�{a~��z{Ey)Ê�É"Ê�y|u Ë

It is likely that acoustic radiation emitted by turbulentflows in the upper convective boundary layer is the pri-mary source of solar five-minute oscillations. Gener-ation of sound by turbulence is normally described byan inhomogeneous wave equation, in which the inho-mogeneous term represents the random body force aris-ing from convective entropy fluctuations and turbulentstresses (Lighthill, 1952; Stein, 1967; Goldreich & Keely,1977).

In such a model, mode amplitudes are described by theequation of a damped harmonic oscillator:

!�"$# J 0 3� "$#0� 3" G � "$# 0 � "$#0� " F 3"$# � "$# Q � � � � �

� Ì�Í �

where � "$# is a measure of the surface displacement ofa mode of angular degree � and radial order � ; F "'# , !�"$#and �

"$#are the frequency, inertia, and damping rate; and

� � � is a random forcing function. If �"'#�� F "$# then the

power spectrum,<��

, of a mode is given by:

< � � F ��� <�� � F � <�� � F � . Ì� F 3 � F "$# � 3 " � 3"$# <�� � F � �� ÌEÌ��where

< � � Ì 7�� � F 3 � F "$# � 3 " � 3"$#�� is a Lorentzian profilewith the half-width

� "$# � G � "$# and<�� � F � is the power

spectrum of � . Fig. 10 shows an example of the powerspectrum of an oscillation mode.

Ä�y���6È�~ ¾��: � · · ´5�+�;�,���|�����8�����)�����|¼$�+���W�����,���w���� k� $;�

��� �I���|����� ��� � �;���+���|�&��·$· � �w�r� � �)����������� · ·¸���;�����8���$�l����;���&���;�,�E½ < � �q�h�b�¤´"�,���©���,��� � ������� �,���w�,������� �|� �"!o�)���0� ������O�?� �)� ��� G Í � � �ÃÌ�Ì * & � ��§��ª�½q¬���¦ �©���L� � ���½ ��§ �©� �?(>=C< � �q�h�a�L�����+�����+�;���h����´"�,�+���;���+���;�L´ =W���¤�< � �q��� �����,���L´"·1�I�������;���+���;�L´ ���]�)��� ����E��½ < � �q��� ����¤�,����´����P�/�h�)���_�6���&��� � �������Ñ���w�¹�)���V����´"�,�+�V�;���&���;�L´ �½�¤���P���� $�w· � ��!V�,�����W�q�P¥�½ ¥�¬ �©� �

Ó S Ó

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The spectrum of the stochastic force appears in the oscil-lation power spectrum as multiplicative noise and signif-icantly complicates the solution of the basic helioseismicproblem – estimation of the spectral line parameters, suchas line widths, central frequencies and amplitudes. Keelyand Ritzwoller (1993) and Gough

�����·-½(1995) suggested

estimating smooth trends of the parameters over a largenumber of spectral lines in order to reduce the uncertaintydue to the multiplicative noise. Another approach, whichis being developed by Chang and Gough (1995), is to in-clude properties of stochastic excitation in the estimationof spectral parameters.

The total energy � "'# ���3!="$# > �� 3"'# " F 3"'# � 3"$# @ associated

with an oscillation mode satisfies0� "'#0� " � "'# � "$# � / "'# � � Ì G �

where / "'# � � �� "$# � � is the power of the driving force,averaged over a period of oscillation. If a mode is excitedstochastically then its energy fluctuates on the dampingtime scale and has the mean value

� "'# � / "$#� "$# � Ì � �Assuming that � � � is a Gaussian random process, Ku-mar

���m��·¸½(1988) calculated the probability distribution

functions for the energy, ���"'# , averaged over a time inter-val, � 6 � � 6

. If6 � "$# � Ì , then the distribution is

obviously a decaying exponential:

�� �"$# ������� J � ���"$#

� "$# Q � Ì � �

Ä�y����6È�~ ¾w¾:�d�q�L�;�L��º�´"�;����� ��� ����E�h���$�l���¹�+� ¨�����,�+��?� �)�$�w�,�+�������;�����¤½��¤���?����� ���|�°�,�I�¤�,���������©� ���$���q�L�;�L��º�´"�;�����

�+´"�w���;�������/������· ���/�����]���L�|� ��������m��� ������G���'�l���m�,��·-���;� ������ ��&��� ������¨���'�l�����Eº��I��� �w�+�$�+�&�� � ��� � � �� �� �E���I�� � ��ªE¦ � � �EªEª�£8º�����· � �l���L�)�����¨��·¸½ �¤����� �"��� �h����·1���V·1� �w��q�d�)���P� ���&���,���|���+��·��w�,�+�������;����� º����P´ ����+�°������·¸½ * �EªE¦E¦ (º+�����+�a���¢���L�����+�����+�;��� ��+�&�������q� ��� ����E����$5��� �����;�����*����+�����m�°· � �l���L�)�¶��·6��·-½¸�A�)���������w�&���+�+�+��� ��!�� (E½

Fig. 11 shows the distribution of the mode energy av-eraged over a half day (

6 � ÏkÌ�Í H 3 ) determined fromobservations by Elsworth

���d��·¸½(these proceedings) as

compared with the prediction (Eq. 14).

It is clear that the observed and theoretical distributionsare in good agreement (except,perhaps, at the high-energyend), giving a strong argument that the mechanism ofmode excitation is indeed stochastic.

The theories of stochastic excitation (Stein, 1967; Gol-dreich & Keeley, 1977; Osaki, 1990; Balmforth, 1992;Goldreich

���Y��·¸½, 1994; Musielak

���W��·¸½, 1994) evaluate

the turbulent power input, / "'# . The estimates of the in-put are uncertain because they depend upon details of themodel of turbulence used in the theories. Fig. 12 showsthe mean mode amplitude computed by Balmforth (1992)using time-dependent non-local mixing-length theory andassuming that the turbulence in the convection zone ishomogeneous with a Kolmogorov spectrum.

Ä�y���6È�~ ¾�À:¡¶�+��� ����E�0� ¨�¤·1� �|´����L�8 ��+���L´"�,�+�dº��m�t��º �

º��,�+�&�"� * ��ª�¦E¦ ( *���� �&��·¸��� ( ���w�¶�¤�,�+�������I�+� º�� �$��· ������L� �* ��ª�ª� (�´5�L� ��! � �]��· �� !����,�����;���+���|�L´ ���¹�&�������+���;� ����;´"��º�´"·-���¤�&� *;����·1������´"�L����(�*����+�������$��· ������L� �5� �EªEª� (E½

The theoretical estimates are substantially lower than theobserved mode amplitudes shown in the same figure.Musielak

���$��·¸½(1994) found that the theoretical results

do not change significantly for other plausible spectra ofthe turbulence in the convection zone. There were alsoseveral attempts to bring the theory into agreement withobservations by scaling the linewidth

� "$#in Eq.(13) (see

Gough & Toomre, 1991, for a discussion).

However, there is evidence that substantial acoustic poweris generated in isolated acoustic events, probably associ-ated with exploding granules and mesogranulation, whichare not described by the standard theories of stochasticexcitation (Brown, 1991; Goode,

���$��·¸½, 1992; Restanio,���°��·¸½

, 1993). Simultaneous high-resolution observationsof granulation and acoustic events by Rimmele

���h��·¸½

Ó W Ó

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(1995) found that the acoustic flux occurs preferentiallyin dark lanes of granulation. The lanes abruptly darkenimmediately before the acoustic power of the events peaks(Fig. 13).

Ä�y����6È�~ ¾ �:�©� ���������·1´"�;����� ��� *�� (��E�+��´5�+�;���C@�´'$����w�

*�º�(��¤��� ���·1�����+� !E�,����´"·¸���W� ���������+� �;�h�������,� !��+�V�������W����� ����,��·��"´"�w���&�&�W���+��´5�L�|���m���������|� *)»m� ���·-�L�6���6��·-½¸����ª�ª�§�(

The dark lanes observed in granulation are probably as-sociated with rapid cooling and strong downflows thatdevelop in the centres of exploding granules. Numeri-cal simulation by Stein & Nordlund (1990) and by Rust& Toomre (1993) have demonstrated the fast downward-plunging cool plumes are likely sites of significant acous-tic excitation.

��n Z]\_^¶ZbjlSRÁzc�\_^¶Á

The upper convective boundary layer, which is only a fewhundred km thick, is a subphotospheric region of supera-diabatic temperature gradient, where convection takes theform of extremely unstable turbulent eddies (granules)and is characterized by large-amplitude fluctuations of thethermodynamic state and nearly sonic velocities. It is theplace in the Sun where Reynolds stresses and nonadiabaticeffects have their greatest influence on the frequencies andstability of five-minute oscillations. A reliable theory ofthe interaction of the oscillations with the turbulent layerhas yet to be developed. Helioseismology provides avaluable tool for studying the interaction and also for un-covering errors made in the standard solar model due topoor understanding of the structure and dynamics of the

layer. Such studies are of great importance for improvingthe accuracy of diagnostics of the Sun’s deep interior.

Accurate measurements of high-degree f- and p-mode fre-quencies, such as those expected from the MDI instrumenton board SOHO, can be used to determine the averageproperties of the superadiabatic layer: the radial gradientof the specific entropy and the turbulent pressure. Theseproperties, inferred directly from oscillation frequenciesusing an inversion procedure, can be used for testing andcalibrating convection theories and 3D numerical simula-tions.

The turbulent pressure (Reynolds stresses), often ne-glected in the standard solar model, significantly affectsthe equilibrium stratification of the convective boundarylayer and is responsible for most of the difference betweenthe observed and theoretical frequencies of high-degree pmodes. Therefore, the turbulent pressure effects must betaken into account in future realizations of the standardsolar model, though a consistent approach to these effectshas yet to be developed.

The remaining frequency residuals are probably due tothe interaction between the oscillations and random fluc-tuations of velocity, density and magnetic field in thegranulation layer. These effects are most apparent in theresiduals of frequencies of the f mode, which is not sen-sitive to variations in the mean hydrostatic stratification.Therefore, accurate measurements of f-mode frequencies(and their temporal variation) over a broad range of hor-izontal wavenumbers is important for studying effects ofthe convective inhomogeneities and the magnetic field.

The upper convective boundary layer is likely to be theplace where five-minute oscillations are excited. There isobservational and theoretical evidence that the oscillationmodes are intrinsically stable and that they are excitedstochastically due to turbulent fluctuations of entropy andReynolds stresses. The most advanced theory of modeexcitation and damping, based on a time-dependent non-local mixing-length approach, predicts mode linewidthsand amplitudes that are in a qualitative, but not quantita-tive, agreement with observations. It is an important goalfor both the GONG and SOI-MDI projects to measure thelinewidths of oscillation modes reliably.

The distribution of mode energy fits well with theoreti-cal prediction based on the idea of stochastic excitation.However, the theoretical estimates of mode amplitudesare substantially lower than those measured. There isevidence that substantial oscillation power is generatedin isolated acoustic events associated with dark lanes inthe granulation pattern, which are probably associatedwith exploding granules. Such acoustic events are notdescribed by the theories of stochastic excitation, andneed to be studied both observationally and theoretically.

Ó Ì�Í Ó

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I thank K.T. Bachmann, T.L. Duvall, Y. Elsworth, P.R.Goode, D.O. Gough, J.W. Harvey, J.T. Hoeksema, K. Mu-rawski, M. P. Rast, M.P. Ryutova and J. Toomre for usefulcomments and discussions. This work was supported byNASA under contract NAS5-30386.

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