![](media/image1.png){width="2.979861111111111in"
height="1.2770833333333333in"}

1\. Introduction

Over the last ten years remarkable data from space-based observatories
(*Yohkoh, SoHO, TRACE, RHESSI*), combined with advances in ground-based
observing techniques, have revolutionized our understanding of the Sun.
As such, the investigation of the Outer Solar Atmosphere (OSA; a
collective term for the photosphere, chromosphere and corona) remains
central to astronomy and physics in general, as it is the only
spatially-resolved star and provides a natural laboratory for plasma
conditions impossible to recreate on Earth. Furthermore, the effect of
the Sun on our solar system makes space weather an increasingly
important subject area for satellite operations and space exploration.

The OSA is a complex natural system whose dynamics encompasses numerous
non-linear processes across an enormous range of spatial, temporal, and
energy scales ranging from a few km to several solar radii, from seconds
to years, and from 10^26^ to \> 10^33^ ergs. This complexity is best
addressed by modelling the system across these scales and comparing
models to observed parameters. In the chromosphere and corona, the
plasma β (the ratio of the magnetic pressure to the plasma pressure) is
much smaller than unity; hence the plasma and magnetic field are closely
coupled and the complex atomic calculations can be reduced to a few
magnetohydrodynamic (MHD) parameters. It is now generally accepted that
the magnetic field rooted in the turbulent photosphere is vital to the
understanding of the OSA, creating the majestic features evident in the
TRACE image above. The most important parameter in this system is the
balance between the time for information to propagate along a magnetic
field line (the Alfven time, t~A~) and the driving timescale, t~D~. For
t~A~\> t~D~ the magnetic structure is slowly stressed, leading to local
energy build up and explosive events (*DC mechanism*). For t~A~\< t~D~
the energy exists in waves which must be generated, propagate and
release energy via resistivity and viscosity (*AC mechanism*). Both
these mechanisms are probably present, with each one dominating over
different spatial, temporal and energy scales.

The highly inhomogeneous, complex, and dynamic nature of the OSA demands
new data analysis techniques. Spatial and temporal *multiscale* and
*multifractal* analysis techniques developed as part of the applicant's
National Research Council Research Associateship will be used to achieve
this. It is envisaged that over the next few years these techniques will
become more commonplace, in the same way that wavelet analysis has
replaced Fourier analysis as the tool of choice for detecting
non-stationary periodic components in light curves. Furthermore,
statistical techniques, generally unknown and unused by solar
physicists, will also be developed to complement well-known data
analysis methods. Specific examples of where these will be used are
presented below. In general, these will be of great interest to a
scientist of any field who has encountered the problems associated with
structured, inhomogeneous physical systems.

2\. AC Mechanisms; waves in the OSA

Despite the wealth of existing data and advancement of complex models,
solar physicists have yet to solve the long-standing chromospheric and
coronal heating problem. However the study of the counter intuitive
existence of a hot (compared to the surface) solar atmosphere has been
redefined over the last 10 years; observations of a highly-structured
atmosphere show that the problem is now both spatially and temporally
localized.

***2.1 The Heating Problem***

In the 'field-free' chromosphere, acoustic waves generated in the
photosphere may steepen and shock. However, for magnetized regions,
further heating is necessary. It is thought that kink-mode MHD waves
generated in the photosphere can travel up through the chromosphere and
couple to slow-mode waves, which can then shock. The transient nature of
these waves means a wavelet analysis is necessary to provide time
localization, and oscillations can then be followed as a function of
height through the OSA by simultaneous imaging in different wavelengths
(McAteer et al. 2002, 2003; Bloomfield et al. 2004). In the corona,
Alfven waves have long been suggested as a possible heating mechanism
(Alfven 1947), but sufficient power has not been observed to explain the
energy losses. This may be due to the lack of full magnetic field
information and synchronous imaging and spectroscopy. Short period
fast-mode waves (\<10 sec, Williams et al. 2002) detected in solar
eclipse data were also shown not to contain sufficient energy to heat
their surroundings.

Over the next few years, the high spatial resolution of the vector
magnetogram and PPARC-funded Extreme Ultra-Violet Imaging Spectrometer
(EIS) onboard Solar-B (due to be launched in 2006), combined with
simultaneous high-cadence multiple passband imaging by Queen's
University instruments (Rapid Dual Imager -- RDI; Rapid Oscillations in
the Solar Atmosphere - ROSA) will allow observations of the generation
and propagation of these waves. As it is not possible to directly
measure the coronal magnetic field we cannot directly observe the
oscillating field; instead we infer its presence.

-   One approach to this is study the effect of line-of-sight on line
    widths, as Alfven waves (with wave vector perpendicular to the field
    lines) will produce a larger line width when the line-of-sight is
    normal to the field lines and vice versa. Acoustic (longitudinal)
    waves will have the opposite effect.

-   On the other hand, the photospheric magnetic field can be measured,
    and compared to velocity measurements. High temporal and spatial
    observations are necessary, as any oscillations could be due to flux
    moving in/out of a pixel element, or missed entirely if the
    observation timescale is too slow.

If these waves do exist, a further problem is found in the dissipation
mechanisms. By considering the induction equation, it is clear that, as
in most astrophysical cases, the magnetic Reynolds number (the ratio of
the convective to the diffusive term) is much greater than unity for the
OSA. Hence dissipation only becomes important for small lengths scales
(\~100km in the photosphere) and energy must be rapidly transferred from
large to small lengths scales. Presently, the two main candidate
mechanisms for this transfer are phase mixing and resonant absorption,
both of which can be tested by observations. For phase mixing, an
inhomogeneous magnetic field will support vertically propagating waves
which will quickly become out of phase, causing gradients along the wave
front and energy dissipation. This can be important in coronal holes and
solar wind acceleration. For resonant absorption, MHD surface waves in a
loop disturb the field lines inside the loop, some of which react
resonantly. In both cases the spatial scales involved are beyond the
resolution of current instruments but observed light curve parameters
(e.g. damping scales, amplitudes) will be compared to models. This is
commonly achieved via the χ^2^ minimisation method. However the observer
has to be careful not to fit too many parameters; new expectation
maximization methods can be used to complement this.

***2.2 Loop Oscillations***

With current observations, it is possible to detect waves as predicted
over 20 years ago by MHD models. For example, post-flare oscillations
have been detected on the Sun using spatially localized methods
(M^c^Ateer et al. 2005a) and also in spatially unresolved distant stars
(Mathioudakis et al. 2003). This has developed to such an extent that
wave theory can now be used to diagnose unobservable quantities in the
OSA (i.e. coronal seismology). Furthermore, explosive events cause
global kink oscillations (oscillation normal to longest dimension of
loop), implying a Reynolds number which is 8 - 9 orders of magnitude
smaller than expected, and removing the need to transfer energy from
large to small scales. Estimates of the energy carried by these waves
are dependent on amplitudes which in turn depend on the line-of sight.
This will be resolved by the Solar Terrestrial Relations Observatory
(STEREO) which will provide 3-dimensional views of loops to better
determine these quantities. Furthermore;

-   Wavelet Analysis studies of oscillatory phenomena may be
    semi-automated; as demonstrated for chromospheric (M^c^Ateer et
    al. 2004) and coronal longitudinal waves (de Moortel & M^c^Ateer
    2005).

-   Observations at high cadence may test predictions of Porter et al.
    (1994a, b), whereby slow-mode waves with periods of \<100s can decay
    sufficiently quickly to heat the corona. To date, these have not
    been observed, and if they exist they must have very small amplitude
    (\<2%). Tracking disturbances also suffer from non-simultaneous
    imaging, which affects the phase analysis. The multi-CCD synchronous
    imaging capabilities of RDI and ROSA will solve this problem for the
    upper photosphere and chromosphere.

3: DC Mechanisms; self-organised criticality

***3.1: Nanoflares and Microflares.***

At small spatial scales, magnetic field lines can diffuse through the
plasma and reconnect. For coronal heating, Parker (1988) suggested a
large number of small reconnections (nanoflares, 10^24^ ergs) could be
vital (likewise for the chromosphere a large number of microflares may
contribute to heating). A frequency distribution power index stepper
than -2 is needed for nanoflare heating to dominate (Hudson 1991). These
small events are spatially, temporally and temperature localized; hence
the choice of thresholds (in each dimension) used to identify them have
a large effect on the index. Depending on these selection thresholds,
large or small events may be preferentially observed and/or identified.
Furthermore, the standard χ^2^ fitting is highly dependent on binsize
and fit range; as before expectation maximization can be used to
optimize this process.

-   Further work on this can be carried out by studying the
    chromospheric component of small scale brightenings, requiring high
    spatial resolution data at high cadence using RDI and ROSA.

-   Another vital component here is the spatial comparison of events to
    the magnetic field, as reconnection requires both polarities at
    small scales.

***3.2: Large Eruptive Events.***

Solar eruptions range over several orders of magnitude in size, duration
and energy, and can have significant effects on Earth. The power law
frequency distribution of flares is a clear signature of the
self-organised criticality (SOC) of an externally driven, dissipative,
dynamical system (Bak et al. 1987). This system arises from competition
between the photospheric driver (input) and redistribution at some
critical value (output). Although localized, this redistribution affects
neighbouring regions and an avalanche occurs. The energy build-up is
powered by the photosphere, but the reconnection occurs in the corona,
resulting in electron beams which propagate down into the dense
chromosphere and undergo Bremstrahlung interactions. These chromospheric
footpoint (FP) brightenings can be directly observed, and show a good
correlation with hard X-ray (HXR) sources. The spatial and temporal
behaviour of the footpoints provides information about the conditions
during a flare. Possible studies using current and future instruments
are:

-   The standard model of solar flares predicts that FPs will move
    apart, away from the magnetic neutral line, as the flare progresses.
    However, for many flares this does not occur. As the position and
    angle of the neutral line is dependent on the spatial resolution of
    the data, a study of the FP motions compared to multiscale neutral
    lines in the photosphere is needed.

-   SOC models predict multifractal light curves; essentially a
    quantification of the 'bursty' nature of the light curves. A study
    of the multifractal nature of light curves at different energies
    (e.g. HXR using RHESSI; optical light curves using RDI, ROSA) will
    test these models. This has further applications in astrophysics
    where other 'bursty' light curves are studied (e.g. gamma-ray
    bursts).

-   A fractal study of the photospheric magnetic field is a direct
    indication of the complexity of flare-productivity (M^c^Ateer et al.
    2005b). A large scale multifractal study is necessary to develop
    this.

-   As flares are powered from the stressed field rooted in the
    photosphere, a study of the photospheric magnetic energy spectra
    will be used to predict flares, and understand the physics of the
    magnetic field. Flare productive regions of the Sun tend to have a
    much steeper spectrum than the classical Kolmogorov value of -5/3,
    and turbulent wavelet analysis will improve our understanding of not
    just the Sun, but any turbulent system.

4: The Institute; Queen's University Belfast

QUB has an excellent record of solar physics publications, as cited
throughout this proposal. Recently, the award of a Royal Society grant
resulted in the development of RDI, successfully tested at the Dunn
Solar Telescope, where is it now a common user instrument. Further
funding of £200K has been awarded by QUB for the development of ROSA (a
6 camera version of RDI), in collaboration with the National Solar
Observatory. This places QUB in an excellent position for joint
observing proposals with SoHO and Solar-B over the next 5 years. In
order to present this work at national and international conferences, a
nominal travel sum of ₤2000 per year is requested. A high end computer
(₤5000) and IDL license fees are also requested (₤1000 per year). QUB
has committed a PhD studentship for this project.

REFERENCES