arXiv:0805.0333v1 [astro-ph] 3 May 2008
Center for Turbulence Research
Proceedings of the Summer Program 2006
Wave propagation in the magnetic sun
By T. Hartlep, M. S. Miesch†, AND N. N. Mansour
This paper reports on efforts to simulate wave propagation in the solar interior. Pre-
sented is work on extending a numerical code for constant entropy acoustic waves in
the absence of magnetic fields to the case where magnetic fields are present. A set of
linearized magnetohydrodynamic (MHD) perturbation equations has been derived and
The evolution of the solar interior is actively studied theoretically, numerically, and
observationally. Many recent advancements in our understanding of the structure and
dynamics of the sun have been made by studying solar oscillation. This is the science of
helioseismology. Through ground- and satellite-based telescopes, oscillations on the solar
surface are being observed and, through appropriate techniques, are used to infer infor-
mation about the sun’s internal structure and composition. One interesting technique
provides so-called far-side images of the sun (Lindsey & Braun 2000), where maps of ac-
tive regions on the far side (the side of the sun facing away from earth) are inferred from
the oscillations observed on the front side (the one facing earth). For a review of other
helioseismic methods, as well as general properties of the observed solar oscillations, see
Christensen-Dalsgaard (2002). In general, these inferences are based on simplified mod-
els of wave propagation, and researchers in the field (e.g., Werne, Birch & Julien 2004)
have expressed their need for wave propagation simulations to test and calibrate these
Of course, wave propagation can be studied by simulating the full compressible equa-
tions; for instance, simulations of the shallow upper layer of the solar convection zone
by Stein & Nordlund (2000) demonstrate excellent agreement with existing analytical
theories and observations. But due to the enormous computational requirements, these
types of simulations are able to simulate only a very small part of the sun. For global sim-
ulations such an approach is not expected to be feasible in the immediate future. Except
for the very near surface region, though, flow velocities in the sun are much smaller than
the speed of sound and a perturbation approach is applicable. Thus, acoustic waves can
be treated here as small perturbations traveling through a base flow. This is the foun-
dational idea of the this project. The base flow itself can be either artificially prescribed
or it can be provided by other simulations such as three-dimensional global simulations
of solar convection in the anelastic approximation (Miesch 1998; Brun & Toomre 2002;
Brun, Miesch & Toomre 2004). Our work extends that of Hartlep & Mansour (2005) by
including the effects of a prescribed magnetic field on the propagation of waves, which
is especially of interest for testing the far-side imaging technique previously mentioned.
We present here the perturbation equations we derived and our numerical method, and
comment on the current status of this project.
† High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO
2T. Hartlep, M. S. Miesch, & N. N. Mansour
2. Model equations
The idea underlying the derivation is to split the dependent variables (density, pressure,
magnetic field strength, etc.) into base variables, denoted by ˜ ·, and wave perturbations,
·′, and then to derive appropriate equations for the perturbations. We will take the base
state to be non-rotating, without flows (˜ v = 0), and with prescribed three-dimensional
sound speed cs, density ˜ ρ, magnetic field˜ B, and electric current distribution˜ J. The real
sun, of course, is rotating and has non-vanishing flow velocities. We plan to add these
influences at a later stage, but the current focus is on the effects that magnetic fields and
spatial variations in the speed of sound have on the propagation of acoustic waves. The
linearized perturbation equations arising from the continuity and Euler’s equation are:
∂tρ′= −∇ · m′
∂tm′= −∇p′+ ρ′˜ g + L′
with m′= ˜ ρv′, ˜ g = −ˆ r˜ g, and L′being the mass flux, the acceleration due to gravity,
and the perturbation of the Lorentz force, respectively. The background state is assumed
to be in magnetohydrostatic balance: ∇˜ p = −˜ ρ˜ g +˜L. In principle, another term in the
momentum equation (2.2) appears to account for the perturbation of the gravitational
acceleration caused by the waves, −˜ ρg′. This term can be expected to be very small,
though, and is neglected here. Needed next is an equation that relates the pressure to
the density perturbation. Assuming adiabatic processes, we have p′= c2
with Cp and γ being the specific heat at constant pressure and the adiabatic index,
respectively. Only isentropic waves are considered at this point, and therefore s′, the
entropy perturbation, is set to zero. The magnetic field, of course, is divergence free,
sρ′+ γ˜ p/Cps′
∇ · B′= 0, (2.3)
and is given by Faraday’s law of induction:
∂tB′= −c∇ × E′.(2.4)
Also included are Ohm’s law (in the infinite conductivity limit), Ampere’s law, and the
Lorentz force given by:
4π∇ × B′,
?J′×˜ B +˜ J × B′?. (2.7)
The displacement current ∂tE′in the equation for the electric current has been ne-
glected because the velocities involved are small compared to the speed of light c. These
linearized equations do not account for the initial excitation of the waves, which is a
non-linear process. Acoustic waves are expected to be generated by the vigorous turbu-
lent convection close below the solar surface. We simulate these stochastic excitations by
artificially adding a random forcing term to Eq. (2.1).
Wave propagation in the magnetic sun3
3. Numerical method
The preceding equations are solved in spherical coordinates with a pseudo-spectral
method. Scalar quantities such as pressure and density are expanded in terms of spher-
ical harmonic basis functions for their angular structure, and B-splines (de Boor 1987;
Loulou, Moser, Mansour & Cantwell 1997; Hartlep & Mansour 2004, 2005) for their ra-
dial dependence. Vector fields such as the magnetic field, the Lorentz force, and the mass
flux are expanded in vector spherical harmonics and B-splines. Vector spherical harmon-
ics were selected because of the coordinate singularities in spherical coordinates, which
are most easily treated in those basis functions (see Appendix A). Typically, B-splines of
polynomial order 4 are used, and the spacing of the generating knot points is chosen to
be proportional to the local speed of sound. This results in higher radial resolution near
the solar surface, where the sound speed is low (less than 7 km/s) compared to the deep
interior, where the sound speed surpasses 500 km/s. The numerical domain is chosen to
be larger than the solar radius of approximately 696 Mm by an additional 2–4 Mm to
account for an absorption layer as a means of realizing non-reflecting boundary condi-
tions. This layer is implemented by adding damping terms −σρ′and −σm′to Eq. (2.1)
and (2.2), where σ is a positive damping coefficient that is independent of time, which is
set to be zero in the interior, and from the solar surface up increases smoothly into the
The equations are then recast using an integrating factor exp(σt), and advanced using
a staggered leapfrog scheme in which ρ′and B′are advanced at the same time, and m′
is offset by half of a time step. The basic properties of the background state, ˜ ρ, ˜ g, and cs,
are obtained from solar models, which provide shell-averaged properties of the quiet sun.
Solar model S by Christensen-Dalsgaard et al. (1996) is used for radii below 696.5 Mm,
the highest radius considered in that model. Properties above this radius are taken from
chromosphere model C by Vernazza et al. (1981). Models for 3-D spatial variations of
the temperature, and therefore the sound speed, and the magnetic field, in structures
such as sunspots are then added to this basic, spherically symmetric background.
We have derived a set of linearized MHD perturbation equations for the propagation
of waves in a simplified sun that is non-rotating and has no background flows, and we
have implemented these equations as an extension to our previous non-magnetic wave
propagation code (Hartlep & Mansour 2005). This represents a considerable extension
and required developing routines for vector spherical harmonics, including fast trans-
formations between spectral and physical space (which were not present in the original
code). The new implementation is still in the testing phase, but we hope to be able to
perform numerical experiments (e.g., a test of the far-side imaging technique) soon.
Appendix A. Treatment of coordinate singularities in spherical coordinates
The sun is very well approximated by a sphere, and spherical coordinates are therefore
the obvious choice in the numerical method. This requires, however, that special care is
taken to ensure regularity of physical quantities on the polar axis and at the coordinate
center. The use of a spherical harmonics expansion guarantees smoothness at the polar
axis, but additional constraints on the expansion arise from requiring regularity at the
origin. Fortunately, in many applications the origin can be excluded from the numerical
4T. Hartlep, M. S. Miesch, & N. N. Mansour
domain and therefore those constraints are irrelevant. In the present case though, acoustic
waves travel through the whole sphere, and the center cannot be excluded, requiring that
we enforce these constraints. The constraints are different for scalars such as density and
pressure, and for vector quantities such as velocity and magnetic field. The following
briefly outlines their derivation.
We start with a scalar variable, which is expanded in terms of spherical harmonics
Yl,mfor its angular dependence and, for each spherical degree l and azimuthal index m,
some radial functions fl,m:
Of course, in the actual implementation the infinite sum is truncated at a chosen maxi-
mum value of l. Using the definition of the spherical harmonics and the transformation
to Cartesian coordinates (x = rsinθcosφ, y = rsinθsinφ, z = rcosθ) we can rewrite
Eq. (A1) into:
ck,l,m(x ± iy)|m|z2k−(l+|m|)r−2k+lfl,m(r), (A2)
with some constants ck,l,m, and where kmin= (l +|m|)/2 for even values of l +|m|, and
kmin= (l + |m| + 1)/2 for odd values of l + |m|. The plus and minus signs in Eq. (A2)
are used for positive and negative values of m, respectively. The terms involving x, y,
and z are always regular at the center since |m| and 2k−(l+|m|) are non-negative. The
only constraint is that r−2k+lfl,m(r) must also be regular. This function, expanded in a
Taylor series around r = 0, reads as:
where individual terms are only smooth at the origin if the combined exponent −2k+l+p
is positive and even. For all terms for which this is not the case, the expansion coefficient
αl,m,pmust vanish. In other words, the constraint is that the radial function fl,mmust
be of the form:
fl,m(r) = rlP(r2),
with P(r2) being a smooth function in r2. An alternative derivation of this constraint
can be found in Stanaway (1988). For similar constraints that arise in Fourier expansions
in cylindrical coordinates see Lewis & Bellan (1990).
Additional complications arise for vector quantities from the choice of unit vectors. A
rather simplistic way would be to write a vector field in terms of r-, θ-, and φ-components,
and then to expand these components like scalars individually in spherical harmonic
Yl,m(θ,φ)?fr,l,m(r)ˆ r(θ,φ) + fθ,l,m(r)ˆθ(θ,φ) + fφ,l,m(r)ˆφ(θ,φ)?.
Such an expansion is fine if the computational domain does not include the origin r = 0,
as is the case in the simulations of the solar convection layer by Brun et al. (2004). In our
case, however, it is not an appropriate choice, since, for the vector field to be regular at
Wave propagation in the magnetic sun5
the origin, fr,l,m, fθ,l,m, and fφ,l,mare not independent of each other and the regularity
condition becomes hard to enforce. The alternative here is to use so-called vector spherical
harmonics as angular basis function:
where Xl,m, Vl,mand Wl,mare defined according to Hill (1954) by
Xl,m(θ,φ) = −ˆθ
l(l + 1)
l(l + 1)i∂
Vl,m(θ,φ) = −ˆ r
l + 1
2l + 1Yl,m(θ,φ) +ˆθ
(l + 1)(2l + 1)
(l + 1)(2l + 1)
Wl,m(θ,φ) = ˆ r
2l + 1Yl,m(θ,φ) +ˆθ
l(2l + 1)
l(2l + 1)
Here, the regularity conditions for the radial functions decouple and are actually very
similar to what is found for scalar fields. For instance, Xl,mcan be rewritten as
Xl,m(θ,φ) = (ˆ x + iˆ y)1
(l + m)(l − m + 1)Yl,m−1(θ,φ)
+ (ˆ x − iˆ y)1
(l − m)(l + m + 1)Yl,m+1(θ,φ)
+ ˆ zmYl,m(θ,φ),(A10)
and therefore, following the arguments above, the radial function fX,l,m must behave
just like a scalar, i.e.,
fX,l,m(r) = rlPX,l,m(r2),(A11)
with again PX,l,m(r2) being a smooth function in r2. The results for the other two
fV ,l,m(r) = rl+1PV ,l,m(r2),
fW,l,m(r) = rl−1PW,l,m(r2).
In our numerical method, all radial functions are expanded in B-splines, which are piece-
wise polynomials. Conditions (A4) and (A11)–(A13) result in a linear coupling of the
expansion coefficients for B-splines near the center. These linear systems are very small
and easy to implement. Only the first k +1 coefficients are coupled, where k is the poly-
nomial order of the B-splines used (here usually k = 4), since these are the only B-splines
that are non-zero close to boundary r = 0.
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