B. M. Shergelashvili - S. Poedts
Centre for Plasma Astrophysics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Received 28 May 2002 / Accepted 5 April 2005
Abstract
The observed power spectrum of high-degree solar
p-modes ()
shows discrepancies with the power spectrum
predicted by the stochastic excitement and damping theory. In an
attempt to explain these discrepancies, the present paper is
concerned with the influence of the observed subsurface flows on
the trapped acoustic modes (p-modes). The effect of these
inhomogeneous background flows is investigated by means of a
non-modal analysis and a multi-layer model. It is shown that the
rotational and meridional components of the velocity field change
the wavelengths of the oscillation modes which, in turn, results
in modifications of the corresponding modal frequencies. The
magnitudes of the frequency residuals depend on the spatial scales
of the modes and on the gradients of the different components of
the flow velocity. Together with other mechanisms (e.g. the
scattering of modes by the large-scale convection), the non-modal
effect of the variation of the frequencies in time may contribute:
1) to the observed widening of the corresponding peaks in the
observed power spectrum with increasing angular degree; 2) to the
partial dissipation of spectral power, and, as a result; 3) to the
discrepancies between the predicted and the observed power
spectrum of solar p-modes.
Key words: Sun: oscillations - Sun: rotation - hydrodynamics - waves
One of the main goals of helioseismology is to understand the
excitation and damping mechanisms that yield the observed power
spectrum of the solar oscillation modes. The theory of the random
generation of modes by convective turbulence has been developed by
Goldreich et al. (1994) and Goldreich & Kumar
(1990) in an attempt to quantitatively understand the
structure of mode sources and the generation rate of the
oscillation modes. Observations show (see e.g. Libbrecht,
1988; Woodard et al. 2001) that the energy of the
p-modes with frequencies below the photospheric cut-off grows
with the angular degree
for intermediate values of
,
i.e.
.
This observational fact is in good agreement
with the predictions from the theory of turbulent excitement and
damping of p-modes mentioned above. However, for
there are substantial discrepancies between the theory and the
observations. Recent observations by Woodard et al. (2001)
show that for
the wave energy distribution function
decreases rapidly with increasing angular degree
.
The
latter authors remark that the discovered decline in mode energy
at high
values may indicate the existence of some
"unmodeled mechanism of damping''. In the present paper, the
viability of such a mechanism is studied, viz. mode damping due
to a sheared background flow. There exist numerous
helioseismological measurements of solar subsurface flows (see González Hernández et al. 1999; González Hernández & Patrón 2000). These observations show
that the velocity field of these flows has both rotational and
meridional components and that the gradients of these flow
velocity components depend on the spatial coordinates. We will
present a multi-layer model for these flows that allows us to
quantify their effect on the solar p-modes.
Several other candidate mechanisms for non-turbulent wave damping have been suggested, e.g. the nonlinear coupling between trapped and propagating modes (Kumar & Goldreich 1989) or mode scattering by convective motions (Goldreich & Murray 1994). Moreover, the direct influence of the inhomogeneous temperature, the magnetic field, and/or the velocity fields on the solar p-modes and f-modes have been studied intensively Murawski & Roberts (1993a,b); Murawski & Goossens (1993); Vanlommel & Cadez (1998), Vanlommel & Goossens (1999), while "resonant absorption'' (or "continuum damping'') has been introduced as a candidate damping mechanism for p-modes by Tirry et al. (1998). Clearly, the physical characteristics of the solar medium affect and systematically distort the main frequencies and the travel times of the modes. This insight has led to the development of the time-distance technique in helioseismology (Duval et al. 1993) which became a powerful tool for the observation of 3D structures with inhomogeneous magnetic and temperature profiles.
As pointed out by Goldreich et al. (1994), when the widths of the frequency peaks in the power spectrum are greater than the inverse of the observation time, these peaks contain information about the damping rates of the modes. These authors also indicated that the excess width of the peaks (with respect to the inverse observation time) can result from a variation of either the frequency or the amplitude of the modes and that it is reasonable to assume that both these types of velocity modulation yield comparable contributions in the line width. Observations show that the line widths of the oscillation modes increase with increasing frequency and angular degree. Goldreich & Murray (1994) suggested the scattering of modes by turbulent velocity fluctuations as a possible mechanism supporting this observational fact.
The main properties of the solar oscillation modes are well described by the standard normal mode analysis. For a static equilibrium and for appropriate boundary conditions, the linear MHD (magnetohydrodynamics) operator is self-adjoint. However, in neutral fluids and in plasmas with nonuniform background flows there are often discrepancies between the analytical results of the standard modal analysis and experimental data. These discrepancies are usually ascribed to the mathematical incompleteness of the normal mode spectrum due to the non-self-adjoint nature of the governing equations (Trefethen et al. 1993; Criminale & Drazin 2000, 1990). One of the alternative and complementary methods to solve the initial value problem in sheared plasma flows is the non-modal analysis. In this formalism a transformation of the variables is performed and, as a result, the spatially inhomogeneous terms in the governing equations are replaced by time-dependent terms. Therefore, in general, this method enables one to study the non-exponential temporal evolution of linear perturbations of sheared flows. The non-modal analysis was introduced originally by Lord Kelvin (1887) and the theoretical basis for studying the wave dynamics in hydrodynamical and plasma flows has been developed by Goldreich & Linden-bell (1965), Craik & Criminale (1986); Chagelishvili et al. (1997); Rogava et al. (2000,2001). Many authors showed that the shear in the flow velocity field can have substantial effects on the wave dynamics, e.g. causing time-dependent frequencies and wave numbers (Butler & Farrell 1992; Chagelishvili et al. 1994) and coupling of different wave modes (Chagelishvili et al. 1996; Rogava & Mahajan 1997). The non-modal approach has also been applied to the study of several aspects of the excitement of gravitational and acoustic waves on the Sun Chagelishvili et al. (2000); Pataraya & Pataraya (1997).
The significant discrepancies between the observed and predicted
power spectra of trapped modes with high angular degree may be due to sheared flow effects. These modes mostly propagate
in the thin cavity below the photosphere (
)
in
which plasma flows with strongly pronounced velocity gradients
have been observed. Hence, the aim of our study was to investigate
the possible role of the non-exponential temporal evolution of the
modes (due to the nonuniform flows) in the formation of the
observed power spectra. We applied a non-modal analysis to study
the effect of the observed sub-photospheric shear flows on the
frequencies and the propagation characteristics of the solar
p-modes. In order to distinguish the effect of velocity shear
from all other effects, we adopt a very simple model and consider
the modes locally, in plane slab geometry. Our model allows us to
separate the effects related to the spatial inhomogeneity of the
background flow from the inhomogeneity coming from the
gravitational stratification and the temperature gradient and
enables us to focus on the evolution of the wave characteristics
associated with the flow inhomogeneity itself.
In the next section, we present the model we applied to investigate the interaction between the inhomogeneous flows beneath the photosphere and trapped acoustic modes. In the third section, we describe and analyze our theoretical results. In the fourth section, we discuss the obtained numerical results and we present and discuss some phenomenological points of view in the framework of the obtained results. Then, we discuss the possible observational consequences of the shear flow effects on the p-modes. In the final section, we formulate our conclusions.
The aim is to study the temporal evolution of small perturbations in a stratified medium with a non-uniform two-dimensional "background'' flow. The problem is considered in plane-parallel geometry and is described in Cartesian coordinates. The x-axis denotes the "toroidal'' direction and its positive direction is chosen as the direction in which the Sun rotates. The y-axis coincides with the "meridional'' direction and is directed towards the north pole. The z-axis covers the "radial'' direction and is directed outwardly, to the solar atmosphere. We study the properties of modes with small amplitudes which are considered as linear perturbations of the equilibrium state.
In this subsection, the main parameters characterizing the
equilibrium are introduced, in particular the parameters
characterizing the equilibrium flow velocity. These parameters are
determined in such a way that they yield a good approximation of
the above-mentioned measurements of the solar subsurface flow
profiles. In general, the subsurface flows can be very
complicated. In the non-modal analysis, the nine components
describing the shear rates of the background flow velocity
components in the three spatial directions are collected in the so-called "Shear Matrix'' S (for details e.g. see Mahajan & Rogava (1999)). In its most general form this matrix reads:
In Sect. 4 we will model the subsurface flows with a linear
approximation of the velocity profiles measured by González Hernández & Patrón (2000). Hence, in the local frame of reference the components of the flow velocity read as:
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Figure 1: Schematic view of the modelled subsurface spherical shell (in plain slab geometry). The dashed horizontal lines schematically represent the boundaries of the thin layers considered in our multi-layer model. Within each of these thin layers the sound speed is assumed to be approximately constant corresponding to the piecewise constant temperature profiles shown in Figs. 3 or 7. |
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In order to consider the propagation of small perturbations in the
gravitationally stratified medium we adopt a local analysis,
similar to the formalism used by Whittaker (1963) and
Ulrich (1970). It should be noted, however, that we here
focus on the effects of the inhomogeneous background flow and thus
we excluded the effects related to the magnetic field, radiative
flux and convective motions. The latter issue has already been
addressed by several authors (e.g. see Swisdak & Zweibel 1999). In addition, the influence of the random velocity fields on f-modes has been studied by Murawski & Roberts
(1993a,b) and by Murawski & Goossens (1993)
also including the effect of the chromospheric magnetic field.
Within each of the thin shells in our model (schematically shown
by the horizontal dashed lines in Fig. 1 the temperature (i.e. sound speed) is assumed to be constant. Hence, the temperature profile of the standard solar model is
approximated by a step-function. In other words, locally, in each
thin layer we have an equilibrium density and pressure profile of
the form
,
where H is the local scale height, so that in each layer the
temperature is constant.
Let us now study the propagation of small perturbations in the
considered plasma shell and focuss on the effects related to the
inhomogeneity of the background flow(s). We consider the waves as
linear perturbations of the stationary equilibrium described
in the previous section. The following set of linearized equations
then governs the dynamics of the small disturbances in a
gravitationally stratified isentropic gas:
![]() |
(4b) |
Clearly, in the case of a static medium, i.e. without the
background flows, we get a simplified set of equations without the
inhomogeneous terms coming from the convective derivatives. In
this case, the normal-mode solution of this set of equations takes the form:
However, due to the inhomogeneous background flow the governing equations have a non-self-adjoint character. Hence, we apply the non-modal technique, which enables us to find an additional class of "non-normal'' solutions of the above set of Eqs. (4). These solutions describe the evolution of the perturbations in time which, in general, can be non-exponential.
Following the non-modal technique we represent the perturbation
quantities
,
,
and
in the
following form:
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(10b) |
![]() |
(14b) |
![]() |
(14c) |
Our first aim is to qualitatively understand the temporal behavior of the p-modes driven by the joint effect of the inhomogeneous rotation and the meridional flow appearing in the thin shell just below the photosphere. Therefore, in the next section we will introduce some simplifying assumptions. This will enable us to derive an approximate analytical "dispersion relation'' which governs the temporal evolution of the "effective frequencies'' of the modes. We will also demonstrate the shear flow effect in a simple model with a linear velocity profile, viz. for a mode with l = 951 which is confined within a very thin cavity above the fractional radius 0.997. In Sect. 4 we then try to quantify this effect in first order with a multi-layer model. The measured velocity profiles are approximated linearly in each layer and the cumulative effect of the layers is calculated taking into account the time spent by the modes in each layer.
In general, the application of the non-modal approach means that,
in the governing equations, the spatial inhomogeneity (related to
the terms containing the nonuniform background flows) is replaced
by a temporal one. After this transformation of the perturbation
variables the temporal behavior of the spatial Fourier harmonics
(containing the time-dependent wavenumbers) can be studied. In
most cases, the solution of the resulting ODEs (in time) requires
the application of numerical techniques. In this particular case
of trapped acoustic-gravity modes, the initial value problem has a
specific feature. On one hand, the time dependence of the
wavevector results in a variation of the phase speed and the
direction of the wave propagation, in time. On the other hand, the
temperature gradient refracts the propagating mode. Formally
speaking, the local approach outlined in the previous section is
the "non-modal equivalent'' of the so-called ray approximation,
commonly used for the local description of p-modes (Whitaker
1963; Ulrich 1970). But, in this case we do not
perform a Fourier transform with respect to time. Instead, one has
to study the propagation of the modes by solving the governing
Eqs. (14) in each layer separately and then
matching the solutions at the interfaces between the different
layers. However, at this stage we aim to qualitatively estimate
the characteristic changes in the modal properties in order to
understand the principle of the mechanism of these changes induced
by the inhomogeneity of the background flows. Therefore, we adopt
an approximative analysis to study the propagation properties of
the modes and to construct the shape of their ray paths. The part
of the ray path confined within a given layer with constant
temperature is locally linear (a schematic view of this kind of
piecewise straight ray path is shown in Fig. 2) and
within this layer the equations governing the motion of a point of
the wave front can be written as follows:
![]() |
Figure 2: Schematic view of the piecewise straight ray path. |
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Generally speaking, the effect of an inhomogeneous flow on waves manifests itself in a temporal variation of the modal properties. Using different analytical methods (among which the non-modal approach) it has been shown that strong gradients in the background flow velocity can cause very rapid changes of the modal frequency and the amplitude, even in time intervals as short as an oscillation period. As a result, the power spectrum of the waves is affected in such a way that it may become impossible to distinguish the individual peaks in the spectrum.
In the Sun, however, such strong gradients of the flow velocity are not observed. Therefore, in this first attempt to model this "small'' effect qualitatively, we assume that the sheared flows below the surface of the Sun just slightly transform the oscillation modes and that this transformation is represented by a small deviation of the mode properties found by the standard modal analysis. As a result of the flow inhomogeneity there appear terms of the form Sijui (where a summation over repeated indices is meant) in the Eqs. (14b)-(14c). Here the indices i,j=1,2,3 represent the x-, y-, and z-direction, respectively, and S is the shear matrix. From Table 1 it is clear that the gradients of the flow velocity components are two to four orders smaller than the characteristic angular frequencies of the oscillation modes. Hence, we expect that the contribution from the Sijui terms in the solution of the governing equations is very small. Therefore, at this stage we neglect these terms, at least in Eqs. (14b)-(14c).
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Figure 3: Piecewise constant radial profile of the temperature (panel A) and scale height (panel B) corresponding to the case of the simple velocity profile addressed in Sect. 3.2. |
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![]() |
Figure 4:
The variation of the modal frequency in time. The curves
correspond to the mode ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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The above approximation results in a mathematical simplification
of the problem as it makes the frequency purely real, i.e. oscillatory. In other words, it is equivalent to the assumption that the amplitude of the mode varies much slower in time than the
rapid oscillation itself so that the amplitude remains
approximately constant. This approximation thus results in
perturbations of the form
.
The set of Eqs. (14) then
leads the dispersion equation for
,
which has a similar form as the one given by Whitaker (1963) for waves propagating in a static medium:
In order to numerically calculate the changes of the modal
frequencies purely due to the non-modal effects we developed a
numerical code making use of a spatial discretisation on the basis
of the Standard solar model (www.ap.stmarys.ca/~guenther/solar/ssm455.sink) and the
approximate "dispersion relation'' (Eq. (17)). In our
calculations we used also the observed p-mode frequency data
(published online by Libbrecht et al. and Bachmann et al., see
the web page http://www.gong.noao.edu/teams/data/jwl_freqs.html). As an example we examined the mode with
.
In Fig. 3 we show the piecewise constant
radial profiles of the temperature (panel A) and scale height
(panel B). The shown range of the fractional radius includes the
very thin cavity in which the
mode is confined.
Notice that these illustrative calculations involve a sample mode
with ,
n=2 and
mHz. The curves in
Fig. 4 correspond to a simple velocity profile. In
particular, it was assumed that the profiles of both the
rotational and the meridional flows are linear in the entire
(narrow) cavity in which the mode is confined. As one can see from
Eq. (17) the non-modal change in wavelength directly
manifests itself in a variation of the effective frequency in
time. However, the problem we are considering here has one specific feature: the modes are trapped in the acoustic cavity between the upper and inner turning points. As a result, if the
modal frequency increases (decreases), when it propagates from the
upper turning point to the inner one, then the frequency decreases
(increases) when mode propagates in the backward direction.
Therefore, any observable changes in modal frequencies (on the
surface) can only arise when the net influence of the flow, during
the propagation of the mode in the inward and outward directions,
is nonzero. Hence, the nonzero residuals in Fig. 4
result from the asymmetry of the background flow. The example
shown in Fig. 4 corresponds to a case with a simple
flow velocity profile: for this calculation we assumed linear
rotational and meridional flow profiles in the entire cavity. This
simple profile is appropriate for the modes with very high angular
degree. The total residual during the observed modal lifetimes is
thus the sum of the net frequency changes appearing during each
passage from the upper turning point to the inner one and backward
as shown in Fig. 4 (panel A). The curve a (solid
line) corresponds to the mode with kx>0 and ky0>0 and
the curve b to the kx>0 and ky0<0 (dashed line). The
points 1a and 2a correspond to the moments when the mode areaches its upper and inner turning points, respectively.
Similarly we have points 1b and 2b for the curve b. The
panel B in Fig. 4 illustrates the changes in modal
frequency for different ratios
kx/ky(0), i.e. for different
directions of the horizontal wavevector in the initial momentum.
The curves in this panel in fact are the lines connecting the
residuals in frequencies corresponding to the moments when the
considered mode is close to its upper turning points. We denote
the curves for modes with ky0>0 by the characters with the
index 1 and for the modes with ky0<0 by characters with
index 2. If we represent the ky0 components of the
wavevector as
,
where
is the
angle between the
and x-axis, then the curves on
panel B correspond to the values of
in the following
order: curves a1 and a2 (crossed lines) correspond to
,
b1 and b2 (dashed lines) correspond
to
,
c1 and c2 (solid lines)
correspond to
,
d1 and d2 (doted
lines) correspond to
and e1 and e2(dash-dotted lines) correspond to
.
Clearly,
the residuals in modal frequency due to the non-modal effects are
greater for modes propagating in an oblique direction with respect
to the both rotational and meridional flows and the effect is
maximal for the angle
(i.e.
kx=ky0)
(the angle corresponding to maximum rates of the frequency
residuals can differ from this, it can be
or
,
depending on the values of shear rates and wavevector components).
![]() |
Figure 5:
Results of the calculations for the same mode as
in Fig. 4 for a smaller value of the shear parameter
a=-0.1 ![]() ![]() ![]() ![]() ![]() ![]() |
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In Fig. 5 we show results for a rather small value of the parameter a (a=-0.1
10-6 s-1) obtained for the same mode as in
Fig. 4. The curves in panel A correspond to the
angle
.
The solid curve corresponds to the shear
rate c=2.0
10-5 s-1 and the dashed line
corresponds to c=3.0
10-5 s-1. Similarly, we show
the results for the angle
in panel B, where the
dashed-dotted curve corresponds to the smaller value of parameter c and the dotted line to the larger one. From both panels A and B one can easily see that the amplitude of the frequency
variation along the ray path increases as c increases. Finally,
we combine the results for both angle values
(solid and dashed lines) and
(dashed dotted and
dotted lines) calculated for a 10 h period. In this panel the
line styles correspond to the same values of the shear parameter c as in the previous panels. From this panel we conclude that the rate of the overall frequency residuals drops down by a factor
of 10 compared with the case shown in Fig. 4 since
now the value of parameter a is 10 times smaller. This seems to
indicate that the frequency residual rates depend nearly linearly
on the value of this parameter. We will come back on this point in the next section.
We adopted a simplified velocity profile with fixed constant
values of the shear parameters within the entire cavity (above the
fractional radius 0.997), in which the examined oscillation mode
with
is confined. This simple model yields the
following main conclusions: (1) The joint effect of the
non-uniform rotational and meridional flows can yield finite
differences between the normal mode frequencies and the "effective
frequencies''; (2) These "residuals'' can be of the order of a few
tens of
Hz. It is thus worthwhile to improve the model in an
attempt to better "quantify'' the effect of the shear flows. This
is done in the next section, where we consider a more advanced
model including observed (not simplified) velocity profiles, and
perform calculations for different sample modes including the one
considered in the current subsection.
In this section we consider cases of modes with different angular
degrees. For these purposes, in general more complicated, the
observed velocity profiles should be taken into account in the
model. Modes with
significantly lower than that considered
in the previous subsection penetrate deeper in the solar interior
and correspondingly we expand the width of cavity as is shown in
Figs. 6 and 7, where we consider the
propagation of modes. We explain the model in detail.
As a basis for the present study we use the results of
helioseismological measurements of the subsurface velocity
profiles (see e.g. González Hernández et al. 1999; González Hernández & Patrón
2000). These observations show that the gradients of the flow velocity components are functions of position. This is why we present our results for a number of magnitudes of the shear
parameters confined within some ranges of values. The model "shear
parameters'' included in the shear matrix are estimated as follows.
The photospheric latitudinal variation of the rotation rate is
usually modelled as:
To estimate the values of the parameter
b=V0x,z, which
locally represents the radial variation of the rotational
component of the flow velocity, and the radial shear c of the
meridional (y-) component of the velocity field, we used results
of helioseismic inversions given in Gonsález Hernández et al. (1999), González Hernández & Patrón (2000).
In Fig. 6 we show the rotational profile for latitude
taken from González Hernández & Patrón (2000) (dotted line). We provide a linear approximation of
the observed profile. The approximate profile is also plotted in
Fig. 6 (solid line). This approximate profile corresponds
to specific values of the shear parameter b. The results of our
estimates, for different fractional distances from the solar
center, are shown in the Table 1. At depths below the thin
sub-photospheric layer, i.e. for
,
we assume the
radial gradient of the rotational velocity to be negligibly small.
We only consider oscillation modes with angular degree
,
which have their lower turning points far above the solar
tachocline. These parameters a, b, and c determine the
inhomogeneity of the background flow. Together with the spatial
scales of the waves themselves and with the timescales of the
interaction between the modes and the flows, they determine the
strength of the influence of the inhomogeneous flow on the waves.
![]() |
Figure 6:
Profile of the rotational velocity in the solar
subsurface layers. The solid line shows the sample profile for
latitude angle
![]() |
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Figure 7: As in Fig. 3 corresponding to the range of the fractional radius values r=0.95-1.00. The squares in the right bottom corners show the areas plotted in respective panels of Fig. 3. |
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Table 1: Values of the shear parameters b and cat different fractional radii.
We now address the timescales of the interaction between the flow and the oscillation modes. One of the basic temporal parameters characterizing the solar p-modes is the lifetime. Several observational methods have been used by numerous authors to estimate the lifetimes TL of the solar p-modes, see e.g. the work of of Chou et al. (2001) and Chen et al. (1996). There exists a discrepancy between the observational measurements of the modal lifetimes. Nevertheless, from the above-mentioned measurements one can estimate the range of the lifetimes as 2-10 h.
Another characteristic temporal parameter of an oscillation mode
is its travel time
along the ray path, from one reflection
in the upper turning point to another one (see Fig. 1). These
travel times are observable by methods of local helioseismology.
The radial gradient of the temperature and, therefore, of the
sound speed inside the Sun leads to the refraction of waves.
Hence, to properly calculate the changes in the modal frequencies
as the modes propagate along their ray path, we radially
discretize the plain-parallel slab. We thus consider a set of very
thin shells so that within each of these thin plasma layers the
local sound speed, the cut-off frequency and the
Brünt-Väisäla frequency can be assumed to be constant.
Therefore, the spatial orientation of the wavevector also
approximately remains constant when the waves propagate within the
given layer. In our calculations we use the equilibrium parameters
taken from the standard solar model (see e.g. www.ap.stmarys.ca/~guenther/solar/ssm455.sink). We denote
the part of the total ray path confined within the given layer by
where,
is the modal phase
velocity and
is the time interval during which the wave front
stays within the given layer. This distance is related to the
distance between the top and bottom boundaries of the layer
as follows:
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Figure 8:
The variation of the modal frequencies in time calculated in the
case of the velocity shear profile described in Sect. 4.1 (see
Table 1). The direction of the horizontal wavevector is taken as
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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In this subsection, results of calculations are presented based
on the model discussed in the previous subsections, i.e. the more
general case with realistic approximations of the velocity
profiles. The shear parameters used for these calculations depend
on depth and are given in Table 1. In Fig. 8 the
results of these calculations are shown for different values of
the shear parameter a. The panels A, B, C, D and E
correspond to
and to the values a=-1
10-8 s-1, a=-1
10-7 s-1, a=-3
10-7 s-1, a=-6
10-7 s-1 and a=-1
10-6 s-1, respectively. Again the curves labelled with characters with index 1 correspond to modes with ky0>0,
while the labels with index 2 correspond to modes with ky0<0. The curves a1 and a2 (dashed lines) correspond to the mode with angular degree
,
n=4 and a basic modal
frequency
mHz. Clearly, the influence of shear
flow is more pronounced as time proceeds. It yields changes in the
frequency of the order of a few
Hz to, at most, 5 to
Hz after 10 h (the maximal life time of the mode). The values of the shear parameter a used in the calculations
correspond approximately to the latitudes
,
,
,
and
,
respectively. The overall frequency residual (after 10 h) for
the considered mode is plotted in panel F of Fig. 8 (dashed line) as a function of the parameter a.
From Eqs. (10) and (17)-(19) it is clear
that the rate of change of the modal wavelength in time strongly
depends on the modulus (and consequently on the angular degree of
modes as
)
and the direction of the
wavevector, i.e. on its horizontal (
)
and radial (kz)
components. In particular, the larger the component
of the
wavevector is, the more substantial the changes of the modal
properties are. Therefore, we can expect that the deviation of the
modal properties (from those of normal modes) increases with
angular degree. Our calculations show larger residuals from the
basic modal frequency for the sample modes (see Fig. 8):
,
n=6,
mHz (curves b1 and b2 (dash-dotted lines)),
,
n=2,
mHz (curves c1 and c2 (dotted lines)), and
the mode considered in Sect. 3.2 (curves d1 and d2 (solid
lines)). These plots demonstrate that the residuals of the modal
frequency due to the non-modal effects increase with growth of the
angular degree. The overall residuals after 10 h versus
parameter a are also plotted for these sample modes in panel F
in the respective order of line styles. This confirms that the
frequency residuals depend on the values of the parameter a, and
this dependence has a nearly linear character.
In addition, the modes experience a more substantial influence of
the flow when they are closer to their inner turning points. As a
result, modes with inner turning points within the area with
strongly pronounced gradients of the flow velocity (i.e. mostly
the modes with
)
change more effectively in
time. This also contributes to the appearance of more significant
non-modal effects for modes with higher angular degree. Comparing
the results shown in Fig. 8 for the sample mode with
(solid lines in panels B and E) with those obtained
with the simplified velocity shear profiles (see panel B in
Fig. 4 and panel C in Fig. 5) it is
easily seen that our preliminary estimates given in Sect. 3.2
that frequency residuals up to values of a few tens of
Hz
should be expected were true. For a=-1
10-7 s-1 the
solid line d1 in panel B in Fig. 8 should be compared
with the solid and dashed lines in panel C of
Fig. 5, which correspond to the angle
.
On the other hand, from the panel E of
Fig. 8 one can see that (comparing the solid curves d1
and d2 with the similar curves in panel B of Fig. 4)
the characteristic frequency residuals with the observed velocity
profiles are somewhat smaller than those obtained with the
constant shear rates in Sect. 3.2. This is due to the
complicated radial profile of the observed rotational and
meridional flows leading to alternating signs of the shear rates
along the radial direction.
Non-uniform flows can influence the linear wave modes in fluids
and plasmas in two different ways. On one hand the flow advects
the wave front producing a Doppler shift of the modal frequencies
(the scalar product on the RHS of the Eq. (17)). On the
other hand, the shear background flow causes (in general) changes
in the modal wavelength, which in turn results in a temporal
variation of the characteristic frequencies and/or amplitudes of
the linear disturbances (for the case of modes considered here see
the last term in RHS of Eq. (17)). Under the standard
normal mode formalism it is a priori assumed that the
perturbations evolve as purely exponential (
)
functions of time. The Doppler shift of the modal frequencies
can be described under the normal mode analysis very well. On the
other hand, as we have shown, the temporal changes in wave
characteristics related to the time dependence of the wavenumbers
can be studied as well. The normal mode decomposition of the
perturbations neglects this latter characteristic of the temporal
evolution of the wave modes. For the particular case of p-modes
trapped in the solar interior this assumption is true in most
cases. Indeed, there are no sharp flow velocity gradients observed
in the solar interior. Even in narrow shells below the convection
zone and just beneath the photosphere, where significant gradients
of the flow velocity are observed, the shear rates still are by
several orders of magnitude lower than the characteristic
frequencies of the oscillation modes. Therefore, we expected that
the inhomogeneous flows in the subsurface shell, which is under
consideration here, could only slightly deviate (non-modally) the
temporal behavior of the modes from that described under the
standard normal mode approach. The approximative analysis
performed in this work leads to the conclusion that these
expectations were true. Nevertheless, the considered non-modal
effects still turn out to be "strong'' enough to support
different observational evidence. In particular, from the global
point of view the non-modal effects can cause the "blurring''
of the global p-mode power spectra and the mixing of the modal
power with the noise. On the other hand, these effects contribute
to a slight "deformation'' of the modal ray paths. We address
these observational aspects of the non-modal effects in details in
the following subsections.
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Figure 9: Schematic view of the peak(s) in the power spectrum corresponding to the oscillation mode with a given angular degree. a) In static medium; b) in the case of uniform flow; c) inhomogeneous flow without the non-modal effects; d) inhomogeneous flow including the non-modal effects. |
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How do the effects related to the flow velocity gradients manifest
in the observed power spectrum of the solar p-modes? It is
believed that the solar oscillations are excited by the stochastic
motions in the upper part of the convection zone. From the global
point of view, the solar p-modes are observed as global
oscillation modes of the Sun by measuring the characteristic
frequencies of the intensity or the velocity field oscillations.
The observational data are then concentrated in the power spectrum
which enables the identification of different oscillation modes by
fitting the peaks arising in the spectrum with the peaks in the
analytically obtained Fourier power spectrum. The formation of
each peak involves the contribution of many modes with similar
properties. The oscillation modes are characterized by their
frequency, their radial order n, their angular degree
and
their azimuthal order m. As is well known from the theory of
stellar non-radial oscillations (Unno et al. 1989), in the
case of a spherically symmetric star the eigenfrequencies are
degenerate with respect to the azimuthal order. This means that it
is not possible to distinguish modes (in the observed power
spectrum) with the same radial order and angular degree but
propagating in different horizontal directions. In
Fig. 9, panel a schematically shows one peak
corresponding to a given radial order and angular degree and
containing the power of all modes with azimuthal order
.
Any phenomenon that breaks the spherical symmetry (for example the axi-symmetric rotation of the star, magnetic field etc.) results in lifting of the degeneracy with respect to m. In the non-spherically symmetric case, instead of one peak several peaks appear in the power spectrum corresponding to the different orientations of the horizontal wave vector. In panel b of Fig. 9 this is illustrated schematically by two peaks in the power spectrum corresponding to two modes with the same radial order and angular degree, but with horizontal wave vectors directed in opposite directions. This effect is the well-known phenomenon of "frequency splitting''. Clearly, frequency splitting observations allow us to perform the inversion of the observational data into the rotational profile inside the Sun. However, the eigenfrequency splitting can also be caused by any other factor breaking the spherical symmetry such as, for example, the meridional flows.
The effect of inhomogeneous flows on the power spectrum can be
complicated. The flow velocity gradients contribute to the
frequency splitting just as a uniform flow would do. On the other
hand, we have shown above that the influence of the flow
inhomogeneity can be important, able to serve as a mechanism
causing changes in the modal properties which can contribute to
the observed deficit of modal power for very high degree
p-modes. In the non-modal approach we have described these
changes by a time dependent wavevector and a time dependent modal
frequency. Hence, to describe the effect of inhomogeneous flows on
the power spectrum we need to use additional characteristic
temporal parameters: viz. the widths of the peaks in the power
spectra
and the observation time, which relates to
the resolution of the observations. If the changes in the modal
frequencies
occurring due to the non-modal effects
during the lifetime of the mode are much smaller than the
half-width of the corresponding peak in the power spectrum, then
one can assume that the changes in frequency are negligible. In
this case the background flow only contributes to the observed
frequency splitting. This is the case considered under the
formalism of the standard modal analysis by Ulrich et al. (1979). A schematic view of this situation is given in panel c in Fig. 9.
When
the changes in frequency
contribute to the line width (along with different damping
mechanisms causing a variation of the mode amplitudes). When
the frequencies of the individual modes
"drift'' along the frequency axis. In this case a part of their
power is mixed with the acoustic noise (leading to a decrease of
the signal to noise ratio) and a part of their power contributes
to the neighboring peaks. Under these circumstances, even in the
conservative case where we assume that the average power exchanged
by two neighboring peaks are approximately equal to each other (in
the real situation this cannot be true, see Goldreich et al. 1994), the non-modal effects could be the cause of the partial dissipation of the power because of mixing with noise.
The last two situations are schematically displayed in panel d
in Fig. 9. The modes that contribute to a given peak
in the power spectrum are randomly excited by different convective
sources at different time intervals. But, as the flow velocity
gradients are small on the Sun, the changes in frequencies of the
particular mode by the non-modal effects need a significant time
to occur. That is why the peaks in the power spectrum remain
concentrated around the basic central frequency and why the
frequency splitting is still observable.
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Figure 10: Schematic view of the modal ray paths: A) without non-modal effects; B) same as panel A (solid line) and the non-modal deformation of the ray path due to the temporal evolution of the radial component kz of the wavevector, which first increases and then decreases in length (dashed line) and vice versa (dash-dotted lines); C) the non-modal deformation of the ray path because of the temporal evolution of the meridional component ky of the wavevector (dashed and dash-dotted lines). The modes do not stay in one plane; the trajectories become 3D curves. |
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How do the non-modal changes in modal properties affect the observational data obtained by using local helioseismology (such as time-distance techniques, ring diagrams, etc.)? The expressions (15) determine the shape of the trajectory of a point with a constant phase, within the framework of our model with a piecewise constant temperature. In the shearless (zero shear matrix) limit the horizontal components (kx and ky) remain constant. Therefore, in this case the modes propagate in the horizontal direction along straight lines (x and y coordinates of the point are linear functions of time) and the ray path is confined in one plane (the problem is two dimensional). The variation of the radial component of the wavenumber kz arises from the refraction of waves by the temperature gradient along the radius. The schematic view of a given modal ray path between two neighboring reflections at the upper turning points is shown in Fig. 10 (panel A). The effects that can systematically distort the ray paths (when we exclude non-modal changes) relate only to the inhomogeneous temperature and/or magnetic field profiles and Doppler shift of frequencies due to the advection of modes by the flows (this effect is represented by the scalar product on the RHS of Eq. (17)). These systematic distortions are immediately detectable by using local methods of helioseismology. On the one hand, observations detect changes in modal travel times and distances. On the other hand, changes of modal frequencies result in a transformation of the so-called acoustic ring shape from a ring to an elliptical one.
For the case of nonzero shear rates, it can be seen from Eq. (15) that the components of the wavevector are time dependent, in general. The character of this dependence is determined by the profiles of the background non-uniform velocity field. In the particular case which we consider in this paper (rotational velocity depending on y and z coordinates and the meridional flow varying only radially) we get the "dispersion relation'' (17) by introducing an "effective frequency'' because of the rather small shear rates and, consequently, a very slow and small variation of the wavenumbers and frequencies in time. The non-modal changes of the "effective frequency'' and wavevector result in a slight deformation of the ray path. In particular, the radial component of the wavevector kz now varies along the ray path for two reasons. The first one is the usual variation due to the temperature gradient. The second reason is related to the non-modal variation (see Eq. (12)). If kz increases (decreases) within a given layer, when the mode propagates from the upper turning point to the lower one, then the non-modal variation of the radial wavenumber turns and extends (shortens) the wavevector. The wavevector behaves oppositely when the mode propagates from the lower turning point to the upper one: within the same layer it turns in the opposite direction and its length decreases (increases) resulting in a compensation of the residual of the "effective frequency'' produced within the same layer before. Therefore, the combined action of the refraction and non-modal effects produces the shape of the ray path in the kh0-kz0 plane schematically shown in Fig. 10 (panel B).
The uncompensated finite residuals in modal wavelength (frequency) can arise only in the case of nonzero a and c coefficients. This is the reason why the slight non-modal evolution of the p-modes occurs only under the joint effect of the gradients of the rotational (parameter a) and meridional (parameter c) flows. The nonzero value a leads to variation of ky(Eq. (11)) in time. Because of this, the ray paths do not stay in the kh0-kz0 plane, but instead they become 3D curves. The schematic view of these curves are shown in Fig. 10 (panel C).
The residuals occurring during two neighboring reflections are very small and they can not be detected in the observational data of the modal travel times and distances. The non-modal variation of the modal properties is a cumulative effect in the case of trapped modes, occurring only after a significant number of reflections, while the effect related to the Doppler shift of the frequencies is immediately detectable as it systematically shifts all frequencies at once. Formally, the non-modal variation of the frequency is a "second order'' effect and it only becomes significant some time after the modal excitation. The effect considered in this paper cannot be detected by the ring diagram method either. The acoustic rings represent slices of the 3D local power spectrum at a given fixed frequency and deformation of the ring demonstrates the shift of frequencies due to advection (for example see Hill 1988). Because of the non-modal variation the "effective frequency'' slowly migrates along and/or across the corresponding ridge. This is why we do not expect that the non-modal variation will cause any additional deformation of the slices of the spectrum by fixed frequency planes. However, some signature of non-modal variation can arise because of the azimuthal variation of the height (or width of the ridges). This issue is become the subject of a separate study to verify whether the order of magnitude of the mentioned azimuthal variation is compatible with the resolution of the employed method (in this case a FFT). To detect the non-modal effects efficiently, it is convenient to develop a different kind of multidimensional spectral method involving different methods of data analysis with variable temporal and spatial scales (such as a window Fourier transform or a wavelet analysis). This latter issue is out of the scope of the current work.
The influence of the observed subsurface flow inhomogeneity on
solar p-modes has been studied in the framework of a non-modal analysis. Particular attention was given to the
possible role of the non-modal time-dependent effects in the
formation of the observed p-mode power spectra. We suggest that
several properties of the high-degree p-modes could be
attributed to the slight ("non-modal'') deviation of the temporal
behavior of this kind of mode from a purely ("modal'') exponential
(
) evolution. This effect is related to the
non self-adjointness of the governing equations which leads to the
existence of alternative solutions for these equations describing
the temporal evolution (in general non-exponential) of
perturbations. This statement is based on the fact that high
degree p-modes are mostly confined in the upper thin subsurface
shell with strongly pronounced inhomogeneous flows. It was found that:
Acknowledgements
The authors thank Dr. A. D. Rogava for informative discussions and suggestions. This work has been developed in the doctoral program of B. M. Shergelashvili at the Centre for Plasma Astrophysics, K. U. Leuven (scholarship OE/02/20). These results were obtained in the projects OT/02/57 (K. U. Leuven) and 14815/00/NL/SFe(IC) (ESA Prodex 6). We express our gratitude to Prof. K. G. Libbrecht for allowing us to use the p-mode frequency data. We are also grateful to the anonymous referee whose comments and suggestions led to significant improvement of the manuscript.