Wednesday, 4 February 2015

Acoustic Gravity Waves and Global Resonance in the Solar Atmps

There were a couple of talks last year concerning the theory of magneto-acoustic wave propagation in gravitationally stratified and cases were considered in which wave propagation was considered in the presence of a vertical magnetic field. Alex's talk focused on the global resonance in a two layer solar atmosphere model, for example waves may be trapped by the atmospheric temperature variation, these trapped waves may be subject to a global resonance. The theoretical model presented by Alex demonstrated how the resonance arises and presented solutions for high beta cases in the upper atmosphere. The model presented by Freddie was of magneto-acoustic wave in a vertical field, Freddie presented earlier solution techniques along with his solutions using hypergeometric functions for an isothermal atmospheric model. His solutions provide an insight into how energy is propagated into the upper solar atmosphere. The motivation for such studies is to understand:
  • The basic models for wave propagation
  • The influence of a strong gravitational field on stratification (for example of density) and the subsequent influence on wave propagation.
  • How energy is transferred through and into the upper solar atmosphere
  • Since the sun is a source of widely varying and changing magnetic field it is necessary to understand the influence of the magnetic field, for example the vertical magnetic field is a frequently occuring scenario.
To   model these phenomena we use the linearised ideal MHD equations, with an ideal gas and using the adiabatic and Cowling approximation. We consider a static background plasma for which the velocities are zero and by definition derivatives with respect to time is zero. For the vertical B field case a uniform magnetic field is aplied in the z-direction. Stratification results from a uniform magnetic field in the z direction homogeneous in the x and y directions. v_1 and B_1 are the perturbed velocity and magnetic field's, zero subscripted quantities are the unperturbed, static background quantities, the equations are:



The solutions may be expressed in a number of ways. 
Using the Lagrangian displacement. defined by
Note here we are assuming a fourier decomposition of the solutions with solutions proportional to exp(-i omega t).

1. Lagrangian displacement form


2. An equivalent but more explicit form (which Freddie used) is as follows

The form use by Alex for studying global resonance was that used by Campbell and Roberts (1989ApJ...338..538C) in their study of the influence of the chromospheric magnetic field on p and f modes.

3. Velicity divergence formulation

Above, delta is the divergence of the velocity, c_s is the speed of sound, gamma is the ratio of specific heats, c_A is the Alfven speed. k_x, k_y and k_z are the wave numbers in the x, y and z direction respectively.
Assuming no y dependence we have also used normal modes of the form
There are many possible solutions for example the two layer model shown below is a model of the solar interior using a polytropic model with an isothermal atmosphere.
With 
We briefly summarise the solutions discussed by Hague (for full details see reference 1).
  1. High beta polytropic atmosphere
  2. Constant density high beta interior
  3. Isothermal atmosphere
  4. Two layer model with isothermal layer (z>0) connected to a polytropic layer (z<0), see diagram above.
  5. Additiona of an isothermal layer to model 4 give a 3 layer model

1.High beta polytropic atmosphere

For this model
and

1a.Slow Temperature Variation

In this case we have a standing wave cavity of thickness L_1, with solution
A_i(Q) and B_i(Q) are Airy functions (and wikipedia) solving the equation

where
N-0 is the Brunt-Vaisala term, expressing the convective stability or bouyancy
This is valid for

1b.Rapid Temperature Variation

For this case we could consider the region from the solar core to the surface.

In this case the solution is given by

C_1 and C_2 are constants and the 0_F_1 functions are the confluent hypergeometric limit functions (see wikipedia). In this expression \kappa is
Care should be taken with this solution near z=0, such a case would require the addition of a transition layer.

2.Constant Density High Beta Interior

For this case 
and we solve
Where
We get wave propagation for

This is solved using the Whittaker functions (see wikipedia)

3. The Isothermal Atmosphere

H is the constant atmospheric scale height
The solutions for this model can be written as
f(z) is a function which can be determined from the confluent Heun equation  (see also digital library of mathematical functions). Defining R^(+/-) as
For the above equation, we use the following definitions
For the low \beta case the equation we need to solve becomes
With the solution
Where R_+/- are as defined above. To obtain this we take the limit c_A->infinity. The modelling becomes interesting when 2 and 3 layer models are considered

4. Connected Polytropic Layer (z<0) to Isothermal Layer (z>0) L_1<<z_0

4a. Continuous Temperature

The boundary conditions are
The next boundary condition is the continuity of the Eulerian pressure perturbation at the discontinuity. Solutions for this problem are in the form of Airy functions. This model does not take account of coronal heating there is no sharp temperature rise representing the transition zone.

4b. Discontinuous Temperature

The boundary conditions are the same as the previous case and we also have
In this case solutions are in the form of Airy functions. The temperature discontinuity acts as a barrier with separate eigenvalues on either side of the discontinuity, for example
The second branch of eigen values are challenging due to the transcendental nature and are not easily solve for \omega. The resulting algebraic equation has to be solved numerically, Hague obtained good agreement and demonsrated the Sturmian behaviour, i.e. \omega increase with n.

5. Connecting the polytropic layer (z<-\delta z) to the isothermal layer (z>-\delta z) for the case L_1>>z_0

The solution for the polytropic layer

 is connected to the solution for the isothermal layer

5a Continuous temperature profile

Here, at 
 
The boundary conditions are slightly different
With R_+, R_- and \alpha as denoted in section 3 (above), the dispersion relations can be written using the confluent hypergeometric limit functions.
with

5b. Discontinuous Temperature Profile

The temperature profile is


T_0i <>T_0i, the boundary conditions are as given by the case 4 (above) and we replace z=0 with z=-\delta z . Again two branches of eigen frequencies are found from the dispersion relations (due to the temperature discontinuity), the atmospheric branch as fond in section 4b is,

The branch of frequencies for the interior  is found to be

6. Adding an isothermal layer to the interior

The temperature profile for this three layer model is 
with
The differential equation becomes
Using the definition
the differential equation becomes
Where the solutions are

After applying the boundary conditions to these equations the complex dispersion relations can be obtained, these were written down by Hasan, S. S.; Christensen-Dalsgaard, J. 1992ApJ...396..311H.

7. Acoustic Gravity Waves

Returning to the case of acoustic gravity waves modelled using the velocity divergence formulation (3 above), using
And with the Brunt-Vaiasalla term N,
The dispersion relation is
Above, the + sound corresponds to the propagation of acoustic (sound) waves (p-modes) modified by gravity and the - sign corresponds to internal gravity waves(g-modes) modified by compression.

A discussion of p-modes is presented in our blogpost entitled "Our Wobbling Star". The dispersion relation for these is highlighted in the blogpost entitled "The Solar Global Eigenmodes of Oscillation".

Appourchaux et al (2010A&ARv..18..197A) sumarise a search for the solar g-modes. Owing to the existence of the convection zone, the g modes have very low amplitudes at photospheric levels, which makes the modes extremely hard to detect. It was concluded that there was no undisputed detections. However Stodilka (2008MNRAS.390L..83S) using temperature perturbations of the solar photosphere, reports properties consistent with the presnece of internal gravity waves. Using a combination of high-quality observations and 3D numerical simulations Straus et al (2008ApJ...681L.125S)  reported the first unambiguous detection of propagating gravity waves in the solar (and hence a stellar) atmosphere.

Campbell and Roberts (1989ApJ...338..538C) in their study of the influence of the chromospheric magnetic field on p and f modes. They modelled the shift of p-mode frequency resulting from a horizontal b_field. Measurements ver 30yrs and by the BISON experiment confirm frequency shifts of the p-modes increasing with the strength of the chromospheric B-field.

Using the model for the isothermal atmosphere (presented in section 3) 

The above equation describes wave motion  governed by the Klein Gordon equation, this contains a cutoff frequency introduced by stratification. Using this equation, Taroyan and Erdelyi 2008SolarPhysics..251..523T studied the upward propagation of linear acoustic waves in a gravitationally stratified solar atmosphere.  The acoustic cutoff may act as a potential barrier when the temperature decreases with height. They showed that waves trapped below the barrier could be subject to a resonance extending into the entire unbounded atmosphere of the Sun. The characterised the parameter space determining the resonance phenomena.

Driving the lower boundary, the two layer model is characterised as follows


Where the scale height, H, is and the polytropic index, m, is
The Shwarzchild solution for convective stability suggests that \gamma=5/3 when m>3/2. For the lower layer the solution is
Where
For the upper layer the solution is
and

Connecting the layers by continuity of displacement and the Lagrangian pressure at z=0, results in expressions for A_2, C_1 and C_2. Driving the solution with an amplitude I(\omega) at frequency \omega, gives
A resonance will arise when waves are driven at the frequency
Assuming the lower layer is thin L/z_0<<1. In the upper layer there are frequencies blow the cut-off which are singular and correspond to a global resonance.

8. Adding a Horizontal Magnetic Field

Consider the work of  Campbell and Roberts (1989ApJ...338..538C) with the addition of the horizontal B-Field in the chromosphere. Using the Lagrangian displacement form of the expressions the equations may be written

The equations above describe the motion of slow magneto-acoustic waves, fast magneto-acoustic waves and the Alfven waves. In the limit k_x, K-y->0 these equations may be reduced to
Consider the case of constant background temperature i.e. the speed of sound is constant consider also the case of constant Alfven speed. The equation becomes
The magnetically modified scale height H_B is written
The solutions are
\kappa is
The firswt term with the constant D_1 corresponds to the outgoing wave. For L/z_0<<1 and computing D_1 using the boundary conditions, the global resonance frequency is
With continuous sound speed at z=0, we have c_{si}=c_{se}=c_{s0} and the global resonance frequency is

The resonant frequencies are lower than the frequencies in the hydrodynamic case. The resonant frequency decreases for given (L-0 and c_{s0}) as the Alfven speed increases. 

In the case of variable alfven speed with
The governing equation is
Using the definition
The solutions are expressed in terms of Legendre polynomials
For low plasma \beta and satisfyng the BC's requires D_2=0 and
With,
\Omega is the cutoff frequency for the upper layer. The resonant frequency does not tend to the hydrodynamic frequency when the Alfven speed becomes zero. This is because the solution is valid only around the singular point \nu=1. The resonant frequency is real for all L. Unlike the hydrodynamic and constant Alfven speed model there is no requirement that the resonant frequencies are located below the acoustic cut-off frequency. As L->infinity \omega->0 as in the hydroynamic case.

Weak field solutions have also been considered i.e. for 
Here the generating equations become
With
The Frobenius expansion (see wikipedia) can be used with the following solution
We leave discussion of this method to reference 2.

9. Magneto-acoustic Gravity Waves in a vertical Field

In his study of magneto-acoustc gravity waves in a vertical B-field, Mather considered a two layer model with two isothermal layers. The figure above illustrates the two layer model used

Consider first the limiting cases in equation set 2 and make k_x=0
For an isothermal atmosphere the solutions are
With,
Standing modes are obtained if the plasma is line tied so that vx = vz = 0 at z = 0; L. The coupled differential equations in equation set 2 can be manipulated so that they form a fourth order differential equation in terms of v_x. Using the following variable variable defnitions
Using this representation the fourth order DE becomes
The solution can be written in a series using the Frobenius method
\mu can be determined by substituting this into the DE the following relations are obtained
and
Solutions can be written as hypergeometric functions (the mu indices below should be labelled with a j index)
Solution for v_x using Htpergeometric functions with the Frobenius method
The pochammer symbol (see wikipedia) is, for any a,

The solutions for these expressions were first obtained by Ferraro & Plumpton 1958ApJ...127..459F, theuy also determined the solutions for the coupled system of equations for set 2. They were attempting to model wave propagation contributions to coronal heating and sunspot magnetic fields. As well as modelling the characteristics of the Alfven waves they studied the slow waves resulting from gas compressibility and magnetic field variations. They identified two types of slow wave one behaving like a sound wave and another behaving like an alfven wave. In so doing they corroborated Cowlings idea that vertical motions are supressed by a vertical gravitational field. As well as attempting to explain solar spicules using the s modes they studied the implication of the results for hydromagnetic wave propagation in the solar chromosphere

The work of  Leroy, B.; Schwartz, S. J.  1982A&A...112...84L found Frobenius series solutions for both vx and vz components (see also Leroy, B.; Schwartz, S. J   1982A&A...112...93S). Their study concerns the oscillations and associated energy flux in a horizontally stratified medium permeated by a uniform vertical magnetic field parallel to a uniform gravitational field. Particular attention is given to the obvious applications to sunspots and the corona. In addition to the Alfven mode, the inhomogeneous analogs of the fast and slow MHD waves which propagate in a homogeneous medium are found. The boundary conditions and energy flux calculations that are appropriate for these modes are discussed, as are some general considerations of the series solutions themselves. The theory is then applied to the solar atmosphere. Numerical parameters typical of both sunspots and coronal hole regions are used in the investigations over a wide frequency range of the energy flux carried by such oscillations from the photosphere to the corona. It is concluded that the combined effects of stratification, partially transverse propagation, and the transition region make it unlikely that such modes dominate coronal energetics. Series solutions were obtained using the Frobenius method.

Cally, P. S. 2001ApJ...548..473C again showed it was possible to write vx in terms of hypergeometric functions and was able to complete work on the computation of the reflection and transmission coefficients of waves propagating to a single layer. Solutions for magnetoatmospheric waves in an isothermal plane stratified atmosphere with uniform vertical magnetic field had been expressed in terms of Meijer G-functions. Cally pointed out that they may alternatively be expressed using the more familiar hypergeometric 2F3 functions. This gives significant advantages for ease of use and physical interpretation. The exact solutions are useful in interpreting observational results and numerical simulations of more complex magnetoatmospheric waves. Cally identified further simplification to solutions using hypergeometric functions.

Hindman et al 1996ApJ...459..760H examined the effects of a vertical magnetic field on p-mode frequencies, line widths, and eigenfunctions. They employed a simple solar model consisting of a neutrally stable polytropic interior matched to an isothermal chromosphere. The p-modes were produced using a spatially distributed driver. The atmosphere is threaded by a constant vertical magnetic field. The frequency shifts due to the vertical magnetic field are much smaller than the shifts caused by horizontal fields of similar strength. A large vertical field of 2000 G produces shifts on the order of 1 muHz while a weak field of 50 G produces very small shifts of several nanohertz. Hindman found that the frequency shifts decrease with increasing frequency and increase with field strength. The shifts are positive, except at high frequency and low field strength, where small negative shifts are possible. Coupling of the acoustic fast mode to escaping slow modes is extremely inefficient. Constant vertical magnetic field models are therefore incapable of explaining the high level of absorption observed in sunspots and plage. The damping due to this mode conversion process produces very narrow line widths. For a 2000 G field the line widths are several microhertz and for a 50 G field the line widths are several nanohertz.

Freddie's objectives were to simplify the work of the earlier authors and to derive the line of sight variable v_z which is important for observational work. He used a new variable p_1/rho_0 (calling it \theta) and write the governing equations in terms of \theta. Solutions for v_x were obtained and solutions for v_z in terms of hypergeometric functions were obtained using the Frobenius method. A two layer bounded problem with a density discontinuity between the layers was set up and solved.

Using the definition
We can write the governing equation as
A solution can be derived in the form of Hypergeometric functions can be derived using the Frobenius method. The relation between v_x and \Theta is
Eventually using the Frobenius method with the Hypergeometric functions provides a solution in eactly the same form as we obtained earlier e.g. see section 9 and the equation labelled "solution for v_x using Hypergeometric functions with the Frobenius method". The relationship between v_z and \Theta is
In terms of the hypergeometric functions the solution for v_z is
For the two layer model which was solved using this method

v_x and v_z and their derivatives are continuous across their boundary at z=0.
  • There is a density decrease of a factor of 50 at z = 0.
  • The plasma Beta at the discontinuity is 0.1 is continuous across the boundary
  • The density in the bottom layer decreases by a factor of exp(7), D1/H_L=7
  • The density in the top layer decreases by a factor of exp(2), D_2/H_U=2
  • The upper layer is far larger than the lower layer such that 
Eight equations are obtained for the constants of integration and these can be written in the form of a matrix equation. A zero determinant is required to obtain the dispersion relations. For further details about the matrix equations please refer to reference 3 and 4. A study of the wave energy distribution requires an integration of the total wave energy in the upper and lower layers i.e. in the ranges [D_2,-D_1] and  [0,D_2] for the lower layer. The resulting energy distribution is shown below. 

The above figure shows the normalised wave energy distribution for the two-layer model with kx=40.204 and omega=9.845. ME is the magnetic enrgy, KE is the kinetic energy, IE is the internal energy density, GE is the gravitational potential energy and E_T is the total energy.  The expressions are shown below
From this figure it is concluded that a standing wave for the speci c frequency and wave-number
combination is set up, with mostly magnetic character in the bottom layer, that is evanescent with
increasing height in the bottom layer. In the top-layer the wave then has the character of a vertically
propagating sound wave, with only energy contributions from the kinetic energy and internal energy
perturbations of the plasma.

Conclusions

  • From Freddie and Alex's work we have presented a wide range of scenarios for modelling wave propagation in strongly gravitationally stratified atmospheres and with magnetic fields in both the horizontal and vertical directions. More representative models and greater insight has been achieved using 2-layer and 3 layer models, these models lead to more complex solutions. These studies have been compared with brief reviews of earlier research into this diverse area.
  • It was seen that a global acoustic resonance resulted when waves are trapped in an atmosphere with variable temperature profiles. This resonance persists when a B field is added to the upper layer.
  •  For models with constant Alfven speed the resonant frequency is lower than that for the purely hydrodynamic case. However, variations in the background magnetic field may result in preventing the global resonance.
  • The resonance frequency has been computed for the case of a fixed but strong magnetic field. In the case where the plasma-\beta is initially large ut decreases with height it s possible to obtain asymptotic solutions.
  • For the two layer isothermal model with temperature discontinuity  and vertical feld it was clearly shown that standing waves and wave propagation can be established for the two layer model even with a large density continuity
The presentations by Freddie and Alex were a really helpful insight to help understand wave propagation in the solar atmosphere. The work connects well with our numerical studies using SAC/SMAUG of the influence of solar global eigenmodes on energy propagation in the solar atmosphere. In particular we are using the two layer model approach described by Mather and Hague to look at the connection between the analytical models and our numerical studies of hydrodynamics in the strongly gravitationally stratified solar atmosphere.

References

  1. Magneto-acoustic gravity waves in an inhomogeneous medium with a vertical manetic field - hague 2013
  2. Global Resonance in a two layer solar atmosphere
  3. Magnetoacoustic Gravity Waves in a Vertical Field -Freddie Mather 2014
  4. First Year Report - Freddie Mather 2014

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