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arXiv:0805.0333v1 [astro-ph] 3 May 2008

Center for Turbulence Research

Proceedings of the Summer Program 2006

1

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

implemented.

1. Introduction

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

methods.

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

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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′

(2.1)

(2.2)

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:

E′= −1

cv′×˜ B,

4π∇ × B′,

(2.5)

J′=

c

(2.6)

L′=1

c

?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).

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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

buffer layer.

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.

4. Summary

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

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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:

f(r,θ,φ) =

∞

?

l=0

+l

?

m=−l

fl,m(r)Yl,m(θ,φ).(A1)

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:

f(r,θ,φ) =

∞

?

l=0

+l

?

m=−l

l

?

k=kmin

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:

r−2k+lfl,m(r) =

∞

?

p=0

αl,m,pr−2k+lrp,(A3)

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

functions, i.e.,

(A4)

F(r,θ,φ) =

∞

?

l=0

+l

?

m=−l

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)?.

(A5)

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

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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:

F(r,θ,φ) =

∞

?

l=0

+l

?

m=−l

?fX,l,m(r)Xl,m(θ,φ)+fV ,l,m(r)Vl,m(θ,φ)+fW,l,m(r)Wl,m(θ,φ)?,

(A6)

where Xl,m, Vl,mand Wl,mare defined according to Hill (1954) by

Xl,m(θ,φ) = −ˆθ

?

1

l(l + 1)

mYl,m(θ,φ)

sinθ

−ˆ φ

?

1

l(l + 1)i∂

∂θYl,m(θ,φ),(A7)

Vl,m(θ,φ) = −ˆ r

?

l + 1

2l + 1Yl,m(θ,φ) +ˆθ

?

1

(l + 1)(2l + 1)

∂

∂θYl,m(θ,φ)

+ˆφ

?

1

(l + 1)(2l + 1)

imYl,m(θ,φ)

sinθ

,(A8)

Wl,m(θ,φ) = ˆ r

?

?

l

2l + 1Yl,m(θ,φ) +ˆθ

?

1

l(2l + 1)

∂

∂θYl,m(θ,φ)

+ˆφ

1

l(2l + 1)

imYl,m(θ,φ)

sinθ

.(A9)

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

2

?

?

(l + m)(l − m + 1)Yl,m−1(θ,φ)

+ (ˆ x − iˆ y)1

2

(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

functions are:

fV ,l,m(r) = rl+1PV ,l,m(r2),

fW,l,m(r) = rl−1PW,l,m(r2).

(A12)

(A13)

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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6T. Hartlep, M. S. Miesch, & N. N. Mansour

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