\section{Introduction}
Coronal lines are collisionally excited forbidden transitions within
low-lying levels of highly ionized species (IP $>$ 100 eV). As such,
these lines  form in extreme energetic environments and thus
are unique tracers of AGN activity; they are not seen in starburst
galaxies.  Coronal lines appear from X-rays to IR and are common in
Seyfert galaxies regardless of their type (Penston et al.  1984;
Marconi et al. 1994; Prieto \& Viegas 2000).
The strongest ones  are seen  in the IR; in the
near-IR they can even dominate the line spectrum (Reunanen et al. 2003).

Owing to the high ionization potential, these lines
are expected to be
limited to few tens to hundred of parsec around the active nucleus.
On the basis of spectroscopic observations,
Rodriguez-Ardila et al. (2004, 2005)
unambiguously established the size of the coronal line region (CLR)
in NGC~1068 and the Circinus Galaxy, using  the coronal
lines [SiVII] 2.48~$\rm \mu m$\,, [SiVI] 1.98~$\rm \mu m$\~,, [Fe\,{\sc vii}] 6087\AA,
[Fe\,{\sc x}] 6374 \AA\ and [Fe\,{\sc xi}] 7892 \AA\. They find these
lines  extending up
to 20 to 80 pc from the nucleus, depending on ionization potential.
Given those sizes, we started
an adaptive-optics-assisted imaging program with the
ESO/VLT aimed at
revealing the detailed morphology of the CLR in some of nearest
Seyfert galaxies. We use as a tracer the isolated IR line [Si VII]
2.48~$\rm \mu m$\  (IP=205.08 eV).
This letter presents the resulting narrow-band images of the [Si VII]
emission line, which
reveal for the first time the detailed morphology of
the CLR, and with suitable resolution for comparison with radio and
optical-
lower-ionization-gas images. The morphology of the CLR is sampled
with a spatial resolutions almost a factor 5 better than any
previously obtained, corresponding to scales $\sim <$10 pc.  The
galaxies presented are all Seyfert type 2: Circinus, NGC~1068,
ESO~428-G1 and NGC~3081. Ideally, we had liked to image type 1
objects, but, in the Southern Hemisphere, there are as yet no known,
suitable type~1 sources at sufficiently low redshift to guarantee the
inclusion of [Si VII] 2.48~$\rm \mu m$\ entirely in the filter pass-band.

\section{Observations, image registration and astrometry}

Observations  were done  with the adaptive-optics assisted IR
camera NACO at the ESO/VLT.  Two narrow band filters, one centered on
the coronal [SiVII] 2.48~$\mu m$ line and  an adjacent band centered on
2.42~$\mu m$ line-free continuum, were used.  The image scale was
0.027 arcsec pixel$^{-1}$ in all cases, 0.013  arcsec  pixel$^{-1}$
in NGC\,1068.
Integration times
were chosen to keep the counts within the linearity range: $\sim 20$
minutes per filter and source.
For each filter, the photometry was
calibrated against standard stars observed after each science target.
These stars were further used as PSF when needed and for deriving  a
correction   factor that
normalizes    both narrow-band filters to provide
  equal number of counts
per a given flux. In deriving this factor it is assumed that
the continuum level
in the stars  is the same in both filters and not emission lines are
not present.
The wavefront sensor of the adaptive optics system followed the optical
nucleus of the galaxies to determine seeing corrections.

The achieved spatial resolution was estimated from stars available in
the field of the galaxies when possible; this was not possible in
NGC 3081 and NGC 1068 (cf. Table 1).
The resolutions were  comparable in both filters
within the reported errors in Table 1.
Continuum-free [SiVII]2.48~$\rm \mu m$\ line images are shown
in Figs. 1 and 2 for each
galaxy. These were produced after applying the normalization factor
derived from the standers stars.
The total integrated coronal line emission derived from these
images is listed in Table 2.  For comparison, [SiVII] 2.48~$\rm \mu m$\ fluxes
derived from long-slit spectroscopy are also provided.

Also in these figures, images
 with the  2.48~$\rm \mu m$\~ filter of the   standard stars -also used as PSF's control- are
 shown. The images provide a rough assessment of
the image quality/resolution achieved in the
 science frames. For the case of Circinus and  ESO 428-G014, a more
 accurate evaluation  is possible   from  the images of
a field star. One of these field star is shown  in both  filters
in Figs. 1e  and 2b respectively..  To get an assesment of the
 image quality at the lowest signal levels, the images of the field stars in particular are
 normalized to  the galaxy peak at the corresponding filter. These
are much fainter than the galaxy nucleus, thus, the
 star peak  is a mere $\sim$5\% of the galaxy peak.

\begin{table*}
\centering
\caption{Galaxies scale and achieved NACO angular resolution. $*$: in NGC 1068,  is given the
size of the nucleus as K-band interferometry  sets an upper limit for the
core of 5 mas (Weigelt et al. 2004);  in NGC 3081, the size of a PSF  star
taken after the science frames is given}
\begin{tabular}{cccccccc}
 \hline
AGN & Seyfert & 1 arcsec & Stars & FWHM & FWHM & Size of nucleus \\
 &type &in pc & in field & arcsec &   pc &   FWHM arcsec \\
 \hline
Circinus	& 2 & 19   & 2	& 0.19$\pm$0.02	  & 3.6 &  0.27 \\
NGC 1068        & 2 & 70   & 0  & 0.097$^*$        & 6.8 & $<$0.097 \\
ESO\,428-G014   & 2 & 158  & 3  & 0.084$\pm$0.006 & 13  & 0.15$\pm$0.01 \\
NGC\,3081       & 2 & 157  & 0  & 0.095$^*$       & 14 & $<$0.32 \\
 \hline
 \end{tabular}

\end{table*}

\begin{table*}
\centering
\caption{Size and photometry of the 2.48 $\mu m$ coronal line region.  $*$: from a 1'' x 1.4'' aperture  \label{flxTb}}
 \begin{tabular}{ccccc}
\hline
{AGN} &  {Radius from nucleus} &  {Flux NACO} &   {Flux long-slit} &   {Reference long-slit} \\
&pc &\multicolumn{2}{c}{in units of $10^{-14}$~erg~s$^{-1}$ cm$^{-2}$} \\

\hline
Circinus  & 30   & 20 & 16 &  Oliva et al. 1994 \\
NGC 1068 & 70  &  72 & 47$^*$ & Reunanen et al. 2003 \\
ESO\,428-G014 & 120 - 160  & 2.0	& 0.7$^*$ & Reunanen et al. 2003 \\
NGC\,3081 & 120	& 0.8	& 0.8$^*$ & Reunanen et al. 2003 \\
\hline
  \end{tabular}
\end{table*}

Radio and HST images were used, where available, to establish an
astrometric reference frame for the CLR in each of the galaxies.
For NGC~1068 (Fig 1a, b, \& c), the registration of radio,
optical HST and adaptive-optics IR images by Marco et
al. (1997;  accuracy $\sim$ 0.05'') was adopted.  The comparison of the [SiVII]
2.48~$\rm \mu m$\ line image with the HST [OIII] 5007 \AA\ followed by
 assuming the peak emission in Marco's et
al. K-band image to coincide with that in the NACO 2.42~$\rm \mu m$\ continuum
image.  The comparison with the MERLIN 5~GHz image of Gallimore et
al. (2004) was done assuming that the nuclear radio source 'S1' and
the peak emission in the NACO 2.42~$\rm \mu m$\ image are coinciding.

In Circinus (Fig. 1d, e \& f), the registration
of NACO and HST/$H\alpha$ images was done on the basis of  3--4
stars or unresolved star clusters available in all  fields.
 That provides an accurate registration better
than 1 pixel (see Prieto et al. 2004).  No radio image of comparable
resolution is available for this galaxy.

For ESO\,428-G014 (Fig. 2a, b, \& c), NACO images were registered on
the basis of 3 available stars in the field. Further registration with
a VLA 2~cm image (beam 0.2''; Falcke, Wilson \& Simpson 1998) was made
on the assumption that the continuum peak at 2.42~$\mu m$ coincides with
that of the  VLA core.  We adopted the astrometry provided by
Falcke et al. (uncertainty $\sim$ 0.3")
who performed the registration of the 2~cm and the
HST/H$\alpha$ images, and plotted the HST/H$\alpha$ atop the NACO
coronal line image following that astrometry.

NGC~3081 (Fig. 2d, e, \& f) has no stars in the field.  In this case
NACO 2.42~$\rm \mu m$\ and 2.44~$\rm \mu m$\ images, and an additional NACO deep
Ks-band image, were registered using the fact that the NACO adaptive
optics system always centers the images at the same position of the
detector within 1 pixel (0.027'').  The registration with a HST/WFPC2
image at 7910\AA\ (F791W), employed as a reference the outer isophote
of the Ks-band image which show very similar morphology to that seen
in the HST 7910\AA\ image.  Further comparison with an HST PC2
$H\alpha$ image relied on the astrometry by Ferruit et al. (2000).
The registration with an HST/FOC UV image at 2100\AA\ (F210M) was
based on the assumption that the UV nucleus and the continuum peak emission at 2.42 2.42~$\rm \mu m$\  coincides.
The radio images available for this galaxy have a beam
resolution $>0.5''$ (Nagar et al. 1999), which includes all the
detected coronal extended emission, and are therefore not used in this
work.

\section{The size and morphology of the coronal line region}

In the four galaxies, the CLR resolves into a bright
nucleus and extended emission along a preferred position angle, which
usually coincides with that of the extended lower-ionization gas. The
size of the CLR is  a factor 3 to 10 smaller than the extended
narrow line region (NLR). The maximum radius (Table 2) varies from 30
pc in Circinus to 70 pc in NGC 1068, to $\sim >$ 120 pc in NGC 3081
and ESO~428-G014. The emission in all cases is diffuse or filamentary,
and it is difficult to determine whether it further breaks
down into compact knots or blobs such as those found in H$\alpha$, [OIII]
5007\AA\ or radio images even though the resolutions are comparable.

In Circinus, [SiVII]2.48~$\rm \mu m$\ emission extends across the nucleus and
aligns with the orientation of its one-sided ionization cone, seen in
H$\alpha$, or in [OIII] 5007 \AA. In these lines, the counter-cone is not seen (Wilson et
al. 2002), but in [SiVII], presumably owing to the reduced extinction,
extended diffuse emission is detected at the counter-cone position
(Fig. 1f; Prieto et al. 2004).
This has been  further confirmed with
VLT/ISAAC  spectroscopy
which shows both [SiVII]2.48~$\rm \mu m$\ and [SiVI] 1.96~$\rm \mu m$\ extending up to
30 pc radius from the nucleus (Rodriguez-Ardila et al. 2004).
In the coronal line image, the  North-West  emission
is  defining an   opening cone angle larger than that in  $H\alpha$.
The  morphology of [SiVII] in this region is suggestive of
 the coronal emission tracing the
walls of the ionization cone (see fig. 1f).

In ESO~428-G014, the coronal emission is remarkably aligned with
the radio-jet (Fig. 2c). The 2~cm emission is stronger in the
northwest direction, and [SiVII] is stronger in that direction too.
H$\alpha$ emission is also collimated along the radio structure, but
the emission spreads farther from the projected collimation axis and
extends out to a much larger radius from the nucleus than the coronal
or radio emission (Fig. 2b).  Both H$\alpha$ and the 2 cm emission
resolve into several blobs but the coronal emission is more diffuse.

In NGC 3081, the coronal emission resolves into a compact nuclear
region and a detached faint blob at $\sim$120 pc north of it.  The HST
[OIII] 5007 and H$\alpha$ images  show rather collimated structure
extending across the nucleus along the north-south direction over
$\sim$ 300 pc radius (Ferruit et al. 2000).
Besides the nucleus, the second brightest region in those lines
coincides with the detached
[Si VII] emission blob (Fig. 2d).  At this same position, we also find
UV emission in a  HST/FOC image at 2100 \AA.

NGC~1068 shows the strongest [Si VII] 2.48~$\rm \mu m$\ emission among the four
galaxies, a factor three larger than in Circinus, and the only case
where the nuclear  emission shows detailed structure. At
$\sim 7 ~pc$ radius from the radio core S1, [Si VII] emission
divides in three bright blobs.
The position of  S1 falls in between the blobs. The southern blob looks
 like a concentric shell. The northern blob coincides with the
central [OIII] peak emission at the vortex of the ionization cone; the
other  two  blobs are not associated with a particular
enhancement in [OIII] or radio emission (Fig. 1b \& c).  [Si
VII] depression at the position of S1 may indicate a very high
ionization level at already 7 pc radius (our resolution)  from the center; the
interior region might instead be filled with much higher ionization-level
gas, e.g.  [FeX], [Si IX] and higher.  This central structure, $\sim$14~ pc
radius in total, is  surrounded in all directions by much lower
surface brightness  gas, extending up to at least 70 pc
radius.  The presence of this diffuse region is
confirmed by VLT/ISAAC spectra along the north-south direction, which
reveal [SiVI]1.96~$\rm \mu m$\ and [Si VII]2.48~$\rm \mu m$\ extending at both sides of
the nucleus up to comparable radii (Rodriguez-Ardila et al. 2004, 2005).  This diffuse
emission shows slight enhancement at both sides of the 5 GHz jet, but
there otherwise appears no direct correspondence between the CLR and
radio morphology.

\section{Discussion}

ESO~438-G014 and NGC~3081 show the largest and best collimated [SiVII]
emission, up to 150 pc radius from the nucleus. To reach those
distances by nuclear photoionization alone would require rather low
electron densities or a very strong (collimated) radiation field.
Density measurements in the CLR are scarce: Moorwood et al. (1996)
estimate a density $n_e \sim 5000~cm^{-3}$ in Circinus on the basis of
[NeV]~14.3~$\rm \mu m$\ /24.3~$\rm \mu m$; Erkens et al. (1997) derive $n_e < 10{^7}~
cm^{-3}$ in several Seyfert 1 galaxies, on the basis of several
optical [FeVII] ratios. This result may be uncertain because the
optical [Fe VII] are weak and heavily blended.  Taking  $n_e \sim 10^4
cm^{-3}$ as a reference value, it  results in an ionization parameter U
$<\sim 10^{-3}$ at 150~pc from the nucleus, which is far too low to produce strong [SiVII]
emission (see e.g Ferguson et al. 1997; Rodriguez-Ardila et
al. 2005).

We argue that, in addition to photoionization, shocks must
contribute to the coronal emission.  This proposal is primarily
motivated by a parallel spectroscopic study of the kinematics
of the CLR gas of several Seyfert galaxies
(Rodriguez-Ardila et al. 2005), which reveals
coronal line profiles with  velocities  500 $\rm km\, s^{-1}$ $< v <$ 2000 $\rm km\, s^{-1}$ .
Here we assess the proposal in a qualitative
manner, by looking for evidence for shocks from the morphology of the
gas emission.

In ESO~428-G014, the remarkable alignment between [Si VII] and the radio emission
is a strong indication of the interaction
of the radio jet with the ISM. There is  spectroscopic
evidence of a highly turbulent ISM in this object: asymmetric
emission line profiles at each side of the nucleus indicate gas velocities
of up to 1400 $\rm km\, s^{-1}$ (Wilson \& Baldwin 1989).  Shocks with those
velocities heat the gas to temperatures of $>\sim 10^7 K$, which will
locally produce bremsstrahlung continuum in the UV -- soft X-rays
(Contini et al. 2004) necessary to produce coronal lines.  [Si VII]
2.48~$\rm \mu m$\ with IP = 205.08 eV eV will certainly be enhanced in this process.

The concentric shell-like structure seen in NGC 3081 in [OIII] 5007
\AA\ and H$\alpha$ (Ferruit et al. 2000)
is even more suggestive of propagating shock
fronts.  From the [OIII]/H$\alpha$ map by Ferruit et al., the
excitation level at the position of the [Si VII] northern blob is similar to
that of the nucleus, which points to similar ionization parameter
despite the increasing distance from the nucleus. The cloud density might then
decrease with distance to balance the
ionization parameter, but this would demand a strong radiation field
to keep line emission efficient. Alternatively, a local source of
excitation is needed. The presence of cospatial UV
continuum, possibly locally generated bremsstrahlung, and [Si VII]
line emission circumstantially supports the shock-excitation proposal.

In the case of Circinus and NGC~1068, the direct evidence for shocks
from the [Si VII] images is less obvious. In NGC~1068, the orientation
of the three blob nuclear structure does not show an obvious
correspondence with the radio-jet; it may still be  possible  we missed
the high velocity coronal gas component measured in NGC 1068
in our narrow band filter.
 In Circinus, there are not radio
maps of sufficient resolution for a meaningful comparison.  However,
both galaxies present high velocity nuclear outflows, which are
inferred from the asymmetric and blueshifthed profiles measured in the
[OIII] 5007 gas in the case of Circinus (Veilleux \& Bland-Hawthorn
1997), and in the Fe and Si coronal lines in both. In the latter,
velocities of $\sim$500 $\rm km\, s^{-1}$ in Circinus and $\sim$ 2000 $\rm km\, s^{-1}$ in NGC
1068  are inferred from the coronal profiles (Rodriguez-Ardila et
al. 2004, 2005).

An immediate prediction for the presence of shocks is the production
of free-free emission, with a maximum in the UV- -- X-ray,
from the shock-heated
gas. We make here a first order assessment of this contribution using
results from photoionization - shocks composite models by Contini et
al. (2004), and compare it with the observed soft X-rays.
  For each galaxy, we derive the 1 keV emission due to
free-free from models computed for a nuclear ionizing flux, $F_h = 10^{13}
photons~cm^{-2} s^{-1} eV^{-1}$, pre-shock density $n_o=300 cm^{-3}$ and shock velocity closer
 to the gas velocities measured in these galaxies (we use
figure A3 in Contini et al.). The selection of this high-
ionizing-flux  value
has a relative low impact does  on   the 1 keV emission estimate
 as the    bremsstrahlung emission from this flux  drops sharply shortwards the Lyman
limit;  the results are  more depending on the strength of the
post-shock bremsstrahlung  component, this being  mainly dominated
by the shock  velocity and peaks in the soft X-rays (see fig. A3 in Contini et al.
for illustrative examples). Regarding selection of densities,
 pre-shock densities of a few hundred  $ cm^{-3}$
actually imply densities downstream (from where
 shock-excited lines are  emitted) a factor of 10 - 100 higher,
the higher for higher velocities,
and thus  whitin the range of those estimated from  coronal line measureemnts (see above).

Having selected the model parameters, we further assume that the estimated   1 keV emission
comes from a region with size that of the observed [Si VII] emission.
Under those premises, the results are as follows.  For NGC 1068,
assuming the free-free emission extending uniformly over a $\pi
\times (70 pc)^2 cm^{-2}$ region (cf. Table 1), and models for shock
velocities of 900 $\rm km\, s^{-1}$, the inferred X-ray flux is larger by a factor
of 20 compared with the nuclear 1 keV Chandra flux derived by Young et
al. (2001).  One could in principle account for this difference by
assuming a volume filling factor of 5-10\%, which in turn would
account for the fact that free-free emission should mostly be produced
locally at the fronts shock.

In the case of Circinus, following the same procedure, we assume a
free-free emission size of $\pi \times (30 pc)^2 cm^{-2}$ (cf. Table
1), and models of shock velocities of 500 $\rm km\, s^{-1}$ (see above). In this
case, the inferred X-ray flux is lower than the 1 keV BeppoSAX flux,
as estimated in Prieto et al. (2004),  by an order of magnitude.  For
the remaining two galaxies, we assume respective free-free emission
areas (cf. Table 1) of 300 pc x 50 pc for ESO 428-G014 --  the width of [Si VII] is  $\sim 50 pc$ in the direction
perpendicular to the jet -- and $2 \times (\pi \times (14~pc)^2
cm^{-2})$ for NGC 3081 -- in this case, free-free emission is assumed to
come from
the nucleus and the detached [Si VII] region North of it only. Taking
the models for shocks velocities of 900 $\rm km\, s^{-1}$, the inferred X-ray
fluxes, when compared with 1 KeV fluxes estimated from BeppoSAX data
analysis by Maiolino et al. (1998), are of the same order for ESO
428-G014 and about an order of magnitude less in NGC 3081.

The  above results are clearly  dominated by the assumed size of the
free-free emission region, which is unknown.  The only purpose of this
exercise is to show that under reasonable assumptions of shock
velocities, as derived from the line profiles,
the   free-free emission  generated by these shocks  in the
 X-ray  could be accommodated within the observed soft
X-ray fluxes.

We thank Heino Falcke who provided us with the 2 cm radio image
of ESO 428-G014, and Marcella Contini for a thorough review of the manuscript.


\section{Small $x$ Resummation}
\subsection{Motivation}
Current and forthcoming particle collider experiments involve very high energies, such that the momentum fractions $x$ of initial state partons are extremely small. The splitting functions that govern the evolution of parton densities $f_i(x,Q^2)$ with momentum scale $Q^2$, together with the coefficients that relate these partons to proton structure functions, are unstable at low Bjorken $x$ values due to terms behaving like $x^{-1}\alpha_S^n\log^m(1/x)$ where $n\geq m+1$. Although the standard DGLAP theory (where the splitting and coefficient functions are considered at a fixed order in $\alpha_S$) works well in parton fits, there is some evidence that a resummation of small $x$ logarithms is necessary. Previous work has shown that a LL analysis fails to describe data well. One resums small $x$ logarithms in the gluon density by solving the \emph{BFKL equation}~\cite{BFKL}, an integral equation for the unintegrated gluon 4-point function. One then relates this gluon to structure functions using the $k_T$ factorisation formalism~\cite{CollinskT,CatanikT} to obtain the resummed splitting and coefficient functions.
\subsection{Solution of the BFKL equation}
Introducing the double Mellin transformed unintegrated gluon density:
\begin{equation}
f(\gamma,N)=\int_0^\infty (k^2)^{-\gamma-1}\int_0^1 dx x^N f(x,k^2),
\label{Mellin}
\end{equation}
the NLL BFKL equation in $(N,\gamma)$ space is a double differential equation in $\gamma$:
\begin{align*}
\frac{d^2f(\gamma,N)}{d\gamma^2}&=\frac{d^2f_I(\gamma,Q_0^2)}{d\gamma^2}-\frac{1}{\bar{\beta}_0 N}\frac{d(\chi_0(\gamma)f(\gamma,N))}{d\gamma}\notag\\
&+\frac{\pi}{3\bar{\beta}_0^2 N}\chi_1(\gamma)f(\gamma,N),
\end{align*}
with $\bar{\beta}_0=3/(\pi\beta_0)$. The derivatives in $\gamma$ arise from the use of the LO running coupling $\alpha_S(k^2)=1/(\beta_0\log{k^2/\Lambda^2})$ in momentum space, and $\chi_n(\gamma)$ is the Mellin transform of the $n^{\text{th}}$-order BFKL kernel. One may solve this to give:
\begin{equation}
f(N,\gamma)=\exp\left(-\frac{X_1(\gamma)}{\bar{\beta}_0 N}\right)\int_\gamma^\infty A(\tilde{\gamma})\exp\left(\frac{X_1(\tilde{\gamma})}{\bar{\beta}_0 N}\right)d\tilde{\gamma}
\label{sol}
\end{equation}
for some $A(\tilde{\gamma})$ and $X_1(\tilde{\gamma})$. One would ideally like to factorise the perturbative from the non-perturbative physics to make contact with the collinear factorisation framework. This can be achieved (up to power-suppressed corrections) by shifting the lower limit of the integral in equation (\ref{sol}) from $\gamma\rightarrow0$. Then one finds for the integrated gluon:
\begin{equation}
{\cal G}(N,t)={\cal G}_E(N,t){\cal G}_I(Q_0^2,N),
\end{equation}
where the perturbative piece is:
\begin{equation}
{\cal G}_E^1(N,t)=\frac{1}{2\pi\imath}\int_{1/2-\imath\infty}^{1/2+\imath\infty}\frac{f^{\beta_0}}{\gamma}\exp\left[\gamma t-X_1(\gamma,N)/(\bar{\beta}_0 N)\right]d\gamma,
\end{equation}
where $X_1$ can be derived from $\chi_0(\gamma)$ and $\chi_1(\gamma)$, and $f^{\beta_0}$ is a known function of $\gamma$. Structure functions have a similar form:
\begin{equation}
{\cal F}_E^1(N,t)=\frac{1}{2\pi\imath}\int_{1/2-\imath\infty}^{1/2+\imath\infty}\frac{h(\gamma,N)f^{\beta_0}}{\gamma}\exp\left[\gamma t-X_1(\gamma,N)/(\bar{\beta}_0 N)\right]d\gamma,
\end{equation}
where $h(\gamma,N)$ is a NLL order impact factor coupling the virtual photon with the BFKL gluon. If all impact factors are known, one can derive all necessary splitting and coefficient functions in double Mellin space (within a particular factorisation scheme) by taking ratios of the above quantities. The non-perturbative dependence then cancels, and one obtains results in momentum and $x$ space by performing the inverse Mellin integrals either numerically or analytically. The exact NLL impact factors are not in fact known, but the LL results supplemented with the correct kinematic behaviour of the gluon have been calculated~\cite{Peschanski,WPT}. We have shown that one expects them to approximate well the missing NLL information in the true impact factors~\cite{WT1}. \\
Consistent implementation of small $x$ resummations in the massive sector requires the definition of a variable flavour number scheme that allows the massive impact factors to be disentangled in terms of heavy coefficient functions and matrix elements. We have devised such a scheme, the DIS($\chi$) scheme~\cite{WT2}. With resummations in both the massive and massless sectors, one has everything necessary to carry out a global fit to DIS and related data. First, the resummed splitting and coefficient functions are combined with the NLO DGLAP results using the prescription:
\begin{equation*}
P^{tot.}=P^{NLL}+P^{NLO}-\left[P^{NLL(0)}+P^{NLL(1)}\right],
\end{equation*}
where the subtractions remove the double counted terms, namely the LO and NLO (in $\alpha_S$) parts of the resummed results. Then the resulting improved splitting and coefficient functions interpolate between the resummed results at low $x$, and the conventional DGLAP results at high $x$.
\section{Results}
The resummed splitting functions $P_{+}$ ($\simeq P_{gg}+4/9 P_{qg}$ at small $x$) and $P_{qg}$ are shown in figure \ref{pgg}. One sees that the LL BFKL results are much more divergent than the standard NLO results, which are known to describe data well. The addition of the running coupling to the LL BFKL equation suppresses this divergence, but it is still unacceptable. Inclusion of the NLL BFKL kernel, however, leads to a significant dip of the splitting functions below the NLO results. This dip is also observed in other resummation approaches~\cite{ABF,CCSS} and has an important consequence in the global fit in that it resolves the tension between the Tevatron jet data (which favour a larger high $x$ gluon) and the H1 and ZEUS data (which prefer a higher low $x$ gluon). By momentum conservation, one cannot increase the gluon at both low and high $x$ in a standard NLO DGLAP fit. This is possible in the resummed approach, due to the dip in the splitting functions.  \\
\begin{wrapfigure}{}{0.6\columnwidth}
\centerline{\includegraphics[width=0.6\columnwidth]{white_chris.fig1.ps}}
\caption{Splitting functions in the DIS scheme for $n_f=4$, $t=\log(Q^2/\Lambda^2)=6$: NLL+NLO (solid); LL with running coupling + LO (dashed); LL + LO (dot-dashed); NLO (dotted).}\label{pgg}
\end{wrapfigure}
Indeed, the gluon distribution at the parton input scale of $Q_0^2=1\text{GeV}^2$ is positive definite over the entire $x$ range. This is in contrast to a NLO fit, where the gluon distribution is negative at small $x$ for low $Q^2$ values. Whilst a negative gluon is not disallowed, it can lead to negative structure functions which are unphysical. The resummed gluon, however, leads to a prediction for the longitudinal structure function that is positive and growing at small $x$ and $Q^2$, in contrast to fixed order results which show a significant perturbative instability.\\
A consequence of a more sensible description for $F_L$ is that a turnover is observed in the reduced cross-section $\tilde{\sigma}=F_2-y^2/[1+(1-y)^2]\,F_L$ at high $y$. As seen in figure \ref{redxsec}, this is required by the HERA data. Furthermore, this feature is missing in NLO fits (but present at NNLO). Thus the resummations lead to qualitatively different behaviour, consistent with known consequences of higher orders in the fixed order expansion. Overall, we find very compelling evidence of the need for BFKL effects in describing proton structure \cite{WT3}.
\begin{figure}[t!]
\begin{center}
\scalebox{0.5}{\includegraphics{white_chris.fig2.ps}}
\caption{Reduced cross-section data, compared with both resummed and fixed order predictions.}\label{redxsec}
\end{center}
\end{figure}

\begin{footnotesize}


\section{Introduction}

One of the more surprising results of the Compton Gamma-Ray Observatory (CGRO) was the EGRET discovery of an 18 GeV photon associated with GRB 940217 (Hurley 1994). About a half-dozen bursts were seen over the course of the CGRO mission with photons above 100 MeV (Catelli 1998, Dingus 2003). Since the GRB spectral energy distribution at lower energies has been well characterized by a modified power law with peak fluxes at energies of the order of 200 KeV, the existence of photons at energies $10^4$ times higher puts a significant constraint on any viable model of the GRB phenomenon. This has been
a subject of great interest for missions that followed EGRET. Prior to launch of the Fermi Gamma-ray Space Telescope, it was possible to speculate that the LAT instrument would detect more than 200 GRB events per year (Dingus, 2003). In the two year period since the launch of the Fermi Gamma-ray Space Telescope, the Gamma-ray Burst Monitor (GBM) has reported approximately 475 GRBs, ie. a rate of about 250 per year. Over essentially the same period, only 17 bursts have been identified by the Fermi/LAT. We now see that the range of GRB photon energies extends over a scale of $10^6$ but
the physical dynamics of these phenomena are still not understood. This is coupled to the question of whether high energy photons are associated with all GRBs or only with a small sub-class. Since the Fermi mission is unlikely to be duplicated any time soon, there is some urgency to assuring that the maximum information is being extracted from
this valuable facility. Thus, our group has set about
developing techniques for enlarging the number of gamma-ray bursts identified with high energy photon emission, ie. above 100 MeV.

The first result of this effort has established the correlation of two $Swift$/XRT-localized bursts, GRB 080905A and
GRB 091208B, with high energy photons in the $Fermi$/LAT detector (Akerlof el al. (2010), hereafter A10). The statistical technique
employed is the $matched~filter$ method, most familiar to those detecting signals in the time domain. The underlying assumption is that the characteristics of both the signal and background are $a~priori$ known functions of one or more variables. Since the matched filter maximizes the signal-to-noise ratio, moderate departures from optimality degrade the filter performance relatively slowly, making this a valuable tool for investigating the possible existence of faint signals. The details of the filter algorithm are explicitly described in A10. In this paper, we take the next harder step of dropping our reliance on precision burst coordinates provided by
Swift or other similar high resolution instruments. Instead, we use the approximate localization of the $Fermi$/GBM
to map a region of interest on the $Fermi$/LAT field of view. By identifying high energy photon clusters, provisional
burst coordinates can be determined with significantly smaller errors than available from the GBM. From there, the
burst identification follows along lines set out in A10.

\section{Sample Selection}

As a first step in this program, a list of all GBM triggers was obtained from the
$fermigbrst$ catalog maintained by the $Fermi$ Science Support
Center\footnote{http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html}. The
catalog contains 497 GRB triggers from launch to July 9, 2010.  This list was
cross-matched with Table 1 in Guetta \& Pian (2009) and Table 2 in Guetta et al.
(2010) to identify the burst GCN designations and the low energy fluences.
For triggers occuring after February 18, 2010, fluences were obtained from individual GCN circulars.
GBM triggers were also checked against XRT locations from
$Swift$\footnote{http://heasarc.gsfc.nasa.gov/docs/swift/archive/grb$\_$table/} to remove
events already considered in A10.

Using data from the Fermi spacecraft attitude file, we further selected those triggers with a boresight angle
less than 52$^\circ$ and an estimated GBM error circle less than 10$^\circ$. Events without
GBM fluence information or previously claimed LAT detections\footnote{
http://fermi.gsfc.nasa.gov/ssc/observations/types/grbs/grb$\_$table/} were also discarded.
Applying a final cut on GBM fluence (8 - 1000 KeV) requiring greater than 5.0 $\mu$erg/cm$^2$ reduced
the number to 22 events (see Table 1). These are termed the ``GBM'' data. 464 additional fields were
taken at random on the sky
with similar criteria to study the background behavior and are identified as the``random'' data in the following text.

\begin{deluxetable}{llrrr}
 \tabcolsep 0.4mm
 \tablewidth{0pt}
 \tablecaption{List of 22 GBM trigger GRBs}
  \tablehead{\colhead{GRB} & \colhead{Trigger} & \colhead{RA} & \colhead{Dec} & \colhead{$S_{GBM}$ } \\
  \colhead{} & \colhead{} & \colhead{($^{\circ}$)} & \colhead{($^{\circ}$)} & \colhead{$\mu$erg/cm$^2$} }
\startdata
080830  & 080830368  & 160.10  &  30.80  &   9.2 \\
080904  & 080904886  & 214.20  & -30.30  &   5.0 \\
080906B & 080906212  & 182.80  &  -6.40  &  10.9 \\
080925  & 080925775  &  96.10  &  18.20  &  19.4 \\
081122A & 081122520  & 339.10  &  40.00  &   9.6 \\
081231  & 081231140  & 208.60  & -35.80  &  12.0 \\
090112A & 090112332  & 110.90  & -30.40  &   5.2 \\
090131  & 090131090  & 352.30  &  21.20  &  22.3 \\
090227A & 090227310  &   3.30  & -43.00  &   9.0 \\
090228A & 090228204  & 106.80  & -24.30  &   6.1 \\
090319  & 090319622  & 283.30  &  -8.90  &   7.5 \\
090330  & 090330279  & 160.20  &  -8.20  &  11.4 \\
090514  & 090514006  &  12.30  & -10.90  &   8.1 \\
090516B & 090516137  & 122.20  & -71.62  &  30.0 \\
090829A & 090829672  & 329.23  & -34.19  & 102.0 \\
090829B & 090829702  & 354.99  &  -9.36  &   6.4 \\
090922A & 090922539  &  17.16  &  74.30  &  11.4 \\
091120  & 091120191  & 226.81  & -21.79  &  30.2 \\
100122A & 100122616  &  79.20  &  -2.71  &  10.0 \\
100131A & 100131730  & 120.40  &  16.45  &   7.7 \\
100423B & 100423244  & 119.67  &   5.78  &  12.3 \\
100511A & 100511035  & 109.29  &  -4.65  &   7.1 \\
\enddata
\end{deluxetable}

\section{Signal Detection Technique}

The core task of this search procedure is the identification of triplet clusters of photons in the $Fermi$/LAT instrument
whose spatial accuracy is considerably better than the GBM. The set of photon data for each candidate
burst is confined to lie within a $16^\circ$ cone angle of the GBM direction and a time window extending from zero to
47.5 s after the GBM burst trigger. The procedure first computes a signal weight for each photon pair based on
photon energy, detection time, photon event class and angular separation relative to the expected LAT PSF errors. The
photon pair weights are subject to a weak threshold cut designed to avoid combinatorial overload should large
photon numbers be encountered. In practice, this was not a severe problem and can be ignored. The formula for the pair
weights is given by:
\begin{equation}
Q_{ij} = w_i{\ }{\cdot}{\ }w_j{\ }{\cdot}{\ }\Delta_{ij},
\end{equation}
where
\begin{equation}
w_i = w_E(i){\ }{\cdot}{\ }w_t(i){\ }{\cdot}{\ }w_c(i){\ }{\cdot}{\ }4\pi {\sigma}^2_{PSF}(E_i),
\end{equation}
\begin{equation}
\Delta_{ij}=\frac{e^{-\delta_{ij}}}{4\pi ({\sigma}^2_{PSF}(E_i) + {\sigma}^2_{PSF}(E_j))},
\end{equation}
\begin{equation}
\delta_{ij} = \frac{1}{2}\frac{{\theta}^2_{ij}}{{\sigma}^2_{PSF}(E_i) + {\sigma}^2_{PSF}(E_j)},
\end{equation}
and ${\theta}_{ij}$ is the angle between the $i$'th and $j$'th photon. The definitions of $w_E$, $w_t$
and $w_c$ can be found in equation 1,3 and 4 of A10.

The next step is to link photon pairs so that the three pairs, \{i, j\}, \{j, k\} and \{i, k\}, become identified
as the triplet, \{i, j, k\}. The triplet weight value is computed by the formula:
\begin{equation}
R_{ijk} = (w_i{\ }{\cdot}{\ }w_j{\ }{\cdot}{\ }w_k{\ }{\cdot}{\ }\Delta_{ij}{\ }{\cdot}{\ }\Delta_{jk}{\ }{\cdot}{\ }\Delta_{ik})^{\frac{1}{3}}
\end{equation}

The triplet weights are ranked by value and the set is pruned by the condition that a triplet element, $R_{ilm}$, is removed if $R_{ilm} < R_{ijk}$. This leaves a set of triplet clusters, each with a discrete complement of three photons. For
each of these clusters, a PSF-weighted estimate of the burst direction is performed and the matched filter angle weight,
$w_\theta$ is computed with respect to this vector as described in equation 2 of A10. At this point, an event weight
for each cluster is computed by the formulas given in A10 with one small modification. In the scheme described here,
the GRB direction is not initially defined with any precision. Thus, it is inappropriate to include a $1/\sigma^2_{PSF}$
factor for all values of $w_E$. For the highest energy photon in each triplet, the $4\pi\sigma_{PSF}^2$ factor is
removed to reflect that this leading photon plays the principal role in fixing the apparent GRB direction. Although the
calculations carry each cluster through the same computational path, the expectation is that the cluster with the highest
matched filter weight is the most probable identification.

The 22 GBM fields described earlier were the target of this investigation. We recognized
that the most convincing argument for a true LAT identification should rely on the statistical distributions
for the matched filter weights in LAT fields with similar characteristics. To increase that number
as much as possible, the LAT fields of view were segmented into 12 circular tiles embedded on a spherical surface. Each tile subtends a cone with a half-angle of 16.0$^\circ$. This tiling scheme was applied
to both the GBM and random field data sets to realize 182 and 3440 independent directions in space satisfying all the
criteria described previously. Taking advantage of the fact that each field observation was blocked into a 250-s segment,
the number of independent observations was multiplied by five by regarding each 50-s time slice as a separate sample.
Thus, there are 910 background measurements taken from LAT observations obtained simultaneously with the
candidate GBM fields and an additional 17200 samples taken under similar but not identical conditions. One particular
concern for an analysis of this type is that false positives will selectively occur as the sample photon rate
rises substantially above the mean. Evidence that this is not the case here is shown in Figure 1 which plots the
cumulative distribution of the total number of photons within the LAT field of view over a 250-s interval. These rates
explicitly exclude contamination from photons beyond the 105$^\circ$ zenith angle cut. As shown in the plot,
the prominent GBM trigger event reported here is not associated specifically with fields with high ambient
background rates. The
similarity of the distributions for GBM and random fields also shows that the GBM data are not pathological as far as
rates are concerned.

\begin{figure}
  \includegraphics[scale=0.475]{fgst_GBM_graph_b.eps}
  \caption{Cumulative distributions of LAT photon rates over 250-s intervals for the GBM (blue) and random
background (red) fields. These rates reflect the entire LAT FoV except for photons that lie outside the
105$^\circ$ zenith angle cut. The rate corresponding to the most prominent GBM event is indicated by the arrow.}
\end{figure}

\section{Results}

Our statistical localization and weighting scheme identified one outstanding candidate for high
energy photon emission, GRB 090228A. The best estimate for the
probability of such an occurence by chance alone was obtained by performing identical searches on
random LAT fields with the same criteria. Thus, 11 out of 17200 random fields generated matched filter weights
exceeding the value for our candidate event. Multiplying by a trials factor of 22 for the number of GBM localized fields considered yields a false positive probability of 1.4\%. To check that these correlations were simply not due
to preferentially higher background rates for the GBM exposures, we also performed similar calculations for
the LAT data confined to an average of 8 uncorrelated directions per exposure and five independent time intervals from the same GBM data sets.
In this case, 2 fields out of 910 exceed the candidate signal for a false positive rate of 4.8\%. The cumulative distributions are plotted in Figure 2. The statistical similarity of the
GBM off-axis and random field data demonstrates that the GBM data set is not correlated with anomalous environmental
conditions such as higher cosmic ray background rates. A list of photons associated with this burst is provided in Table 2.

The GBM data for GRB 090228A is described in GCN 8918 (von Kienlin, et al., 2009). According to this note, the
burst was localized to a $1-\sigma$ accuracy of better than $1^\circ$ with an additional systematic uncertainty
of the order of $2.5^\circ$. The coordinate values obtained by the GBM group and this analysis are listed in Table 2.
The GBM value generated some concern since our cluster finder position disagreed by 8.7$^\circ$. Fortuitously
as this manuscript was being drafted, a paper was posted to astro-ph (Guiriec et al., 2010) providing a burst direction
with an estimated accuracy of $0.2^\circ$ and lying $0.5^\circ$ from our own estimate. We believe that this establishes
the validity of our identification to near certainty. In addition, the positive GRB correlation of Event Class 2 and 3
photon rates discussed in A10 was observed for the ensemble of GBM fields as well. The photon clustering is easily observed in the sky map shown in Figure 3.

\begin{figure}
  \includegraphics[scale=0.475]{fgst_GBM_graph_a.eps}
  \caption{Complements of the cumulative distributions for ${\zeta}{\sum}w_i$
           for 22 GBM fields (blue), 910 random fields obtained nearly simultaneously with
           the GBM data (red) and 17200 random fields obtained at random times (green).}
\end{figure}

\begin{deluxetable}{crrrrr}
 \tabcolsep 0.4mm
 \tablewidth{0pt}
 \tablecaption{~ GRB 090228A high energy photon list and celestial coordinate estimates}
  \tablehead{\colhead{$i$} & \colhead{$t$} & \colhead{$\theta$} & \colhead{$E$} & \colhead{$c$} & \colhead{$w_i$} \\
  \colhead{} & \colhead{(s)} & \colhead{($^{\circ}$)} & \colhead{(MeV)} & \colhead{} & \colhead{} }
\startdata
  1  &     2.007   &    1.692  &   125.241 & 2  &    52.814   \\
  2  &     3.752   &    2.611  &   206.983 & 3  &    46.545   \\
  3  &    25.141   &    0.592  &   308.638 & 3  &    44.278   \\
  4  &     3.243   &    1.978  &   638.692 & 3  &     1.356   \\
  5  &     4.966   &    0.063  &  2787.028 & 3  & $^*$0.376   \\
  6  &    33.621   &    5.063  &   340.623 & 1  &     0.002   \\
\\
\multicolumn{6}{c}{$\zeta$ = 0.99722~~~~~$\zeta\sum w_i$ = 144.969} \\
\\
\hline
\hline
\\

& source & $\alpha$   & $\delta$   & $\sigma_\theta$ & $\theta_{i-1, i}^d$  \\
&        & ($^\circ$) & ($^\circ$) & ($^\circ$)      &   ($^\circ$)         \\
\\
\hline
\\
& GBM$^a$  &     106.80   &    -24.30  &   $\lesssim3.0$   &   \\
& LAT$^b$  &      98.56   &    -28.86  &   0.18         & 8.66 \\
& IPN$^c$  &      98.30   &    -28.40  &   0.02         & 0.51 \\
\enddata
\tablenotetext{*}{indicates diminished $w_E$ for highest energy photon}
\tablenotetext{a}{von Kienlin et al. (2009)}
\tablenotetext{b}{this paper}
\tablenotetext{c}{Guiriec et al. (2010)}
\tablenotetext{d}{$\theta_{i-1, i}$ is the angle between the spatial directions for the GBM and LAT or
the LAT and IPN}
\end{deluxetable}

\begin{figure}
  \includegraphics[scale=0.475]{fgst_GBM_map_h.eps}
  \caption{Sky map of $>100$ MeV photons for GRB 090228A. The diameter of each dot is proportional to its
statistical weight. Thus, the largest diameters represent Event Class 3, etc. The dotted circles
around each point indicate the $1-\sigma$ errors. The figure is centered on the nominal
coordinates furnished by the GBM; the blue dot on the lower left shows the GRB coordinates computed by the
cluster algorithm described in the text. The large green circle depicts the boundaries of the 16.0$^\circ$ cone
that defines the fiducial boundaries for the cluster search. North is up and East is to the right.}
\end{figure}

\section{Discussion}

The one event identified in this paper establishes the validity of our statistical techniques to a level of near certainty. By using GBM triggers to guide the discovery of photon clusters in the LAT, the phase space
for finding counterparts can be reduced from hundreds of square degrees to one square degree or less. This
makes a very significant difference for those seeking to identify GRB optical counterparts. If the algorithms
used here could be adapted to the real time environment, the number of bursts with high energy associations
could be increased appreciably. The additional computational load is negligible - about 30 ms per day.
For such real time applications, the high selectivity employed here is overkill - any
identification that can be corroborated optically will suffice. Thus, effective signal-to-noise rates of the order of unity are extremely valuable. As shown here, these techniques greatly enhance the dynamic range over which high
energy radiation can be explored. It is not too outrageous to claim that this is the equivalent of making the LAT
three to ten times larger in size.

As was noted in A10, the most surprising aspect of our recent work is the very small number of GRBs that can be positively identified with high energy emission despite the substantially lower fluence thresholds. This is a mystery which deserves serious consideration. We hope that raising new questions is sometimes more useful than answering old
problems.

\vspace{0.1cm}

\acknowledgments

We thank Chris Shrader, director of the $Fermi$ Science Support Center for his considerable help in obtaining and interpreting the $Fermi$ mission data products. Fang Yuan provided valuable assistance in the initial process of learning
how to access and manipulate the $Fermi$ data. This research is supported by NASA grant NNX08AV63G and NSF
grant PHY-0801007.


\section{\label{sec:INTRO}Introduction}

One of the simplest and most commonly studied systems for investigating convection dynamics is the so-called Rayleigh-B\'{e}nard configuration, consisting of a Boussinesq fluid layer of depth $H$ confined between plane-parallel boundaries, and heated from below.  The constant gravity vector $\mathbf{g} = - g {\bf\widehat z}$ points vertically downwards.  Two limiting cases for thermal boundary conditions are often considered when posing the problem mathematically: (1) `perfectly conducting', or fixed temperature (FT), boundary conditions in which the temperature is held fixed along the bounding surfaces; and (2) `perfectly insulating', or fixed flux (FF), boundary conditions in which the normal derivative of the temperature is fixed at the boundaries \citep[e.g.][]{cC80}.  Thermal boundary conditions of geophysical and astrophysical relevance are often considered to reside somewhere between these fixed flux and fixed temperature limits.

For a Newtonian fluid of constant thermal expansivity $\alpha$, kinematic viscosity $\nu$, thermal diffusivity $\kappa$, the non-dimensional Rayleigh number quantifies the strength of the buoyancy force.  For the FT and the FF cases we have, respectively,
\begin{equation}
Ra_{FT} = \frac{\alpha g \Delta T H^3}{\nu \kappa}, \quad Ra_{FF} = \frac{\alpha g \beta H^4}{\nu \kappa},
\end{equation}
where $\Delta T$ is the fixed temperature difference between the top and bottom boundaries and $\beta$ is the fixed temperature gradient maintained at the boundaries.
The Prandtl number quantifies the relative importance of viscous and thermal diffusion as $Pr = \nu/\kappa$.
Upon defining the non-dimensional measure of heat transfer via the Nusselt number,
\begin{equation}
Nu = \frac{\textnormal{total heat transfer}}{\textnormal{conductive heat transfer}} = \frac{\beta H}{ \Delta T} ,
\end{equation}
it is straightforward to show that the two Rayleigh numbers defined above are related simply by $Ra_{FF} = Nu Ra_{FT}$.  We thus see that for linear convection in which $Nu \equiv 1$ the two Rayleigh numbers are equivalent.  For nonlinear convection in which the critical Rayleigh number has been surpassed, $Nu > 1$ is achieved by adjustment of the temperature gradient $\beta$ at fixed $\Delta T$ for FT boundaries, and vice versa for FF boundaries.

Linear stability shows that for the case of non-rotating convection the most unstable wavenumber is finite for FT boundary conditions \citep[e.g.][]{sC61}, but is zero for FF boundary conditions \citep{dH67}. Although previous work suggests that these differences for linear convection also hold for nonlinear convection \citep{cC80}, numerical simulations of convection show that the statistics for the two cases converge as the Rayleigh number is increased and the flow becomes turbulent \citep{hJ09}.

When the system is rotating with rotation vector $\mathbf{\Omega} =  \Omega {\bf\widehat z}$, the Ekman number, $E_H = \nu/2 \Omega H^2 $, is an additional non-dimensional number required to specify the strength of viscous forces relative to the Coriolis force.  The rapidly rotating, quasi-geostrophic convection limit is characterized by $E_H \rightarrow 0$.  As of this writing, only two investigations of FF boundary conditions for the rotating plane layer geometry have been published in the literature, with \cite{tD88} and \cite{sT02} examining the weakly rotating and rapidly rotating linear cases, respectively.  \cite{sT02} utilized a modal truncation approach to show that the critical parameters for the two cases should converge as $E_H\rightarrow 0$; the present work confirms this suggestion.

In the present work we distinguish between `interior' and 'boundary layer' dynamics, and show that the interior governing equations are identical for the two different thermal boundary conditions upon a simple rescaling of the Rayleigh number and temperature.  Because the $E_H\rightarrow 0$ limit is a singular perturbation of the governing equations, the interior equations cannot satisfy the FF boundary conditions at leading order; a double boundary layer structure is necessary to adjust both the horizontal viscous stresses and the normal derivative of the temperature fluctuation to zero \cite[c.f.][]{wH71}.  It is shown that the boundary layer corrections are asymptotically weak, however, showing that to leading order the interior quasi-geostrophic convection dynamics are equivalent for both thermal boundary conditions.

In section \ref{S:Lin} we present the linear stability of the full Boussinesq Navier-Stokes equations.  In section \ref{S:Asymp} we present the asymptotic reduction of the Navier-Stokes equations in the rapidly rotating limit and concluding remarks are given in section \ref{S:discuss}.

\section{Linear stability of the Navier-Stokes equations}
\label{S:Lin}

In the present section we briefly present the linear stability of the Boussinesq Navier-Stokes equations for both FT and FF thermal boundary conditions.  Upon scaling lengths with the depth of the fluid layer $H$ and time with the viscous diffusion time $H^2/\nu$, the linear system becomes
\begin{equation}
  {\partial_t} \mathbf{u} + \frac{1}{E_H} \, {\bf\widehat z} \times \mathbf{u} = - \frac{1}{E_H} \nabla p +  \frac{Ra}{Pr} \, \vartheta^{\prime} \, {\bf\widehat z} + \nabla^2 \mathbf{u}, \label{E:momlin}
\end{equation}
\begin{equation}
{\partial_t} \vartheta^{\prime} - w =  \frac{1}{Pr} \nabla^2 \vartheta^{\prime}, \label{E:heatlin}
\end{equation}
\begin{equation}
{\nabla \cdot} \mathbf{u} =  0 , \label{E:contlin}
\end{equation}
where the velocity vector is denoted by $\mathbf{u}=(u,v,w)$, and the temperature is decomposed into mean and fluctuating variables according to $\vartheta = \overline{\vartheta} + \vartheta^{\prime}$.  For both sets of thermal boundary conditions $\overline{\vartheta} = 1-z$ and the fluctuating thermal boundary conditions therefore become
\begin{equation}
\vartheta^{\prime} = 0, \quad \textnormal{at} \quad z = 0, 1, \quad (FT) \label{E:FT2}
\end{equation}
\begin{equation}
{\partial_z} \vartheta^{\prime} = 0, \quad \textnormal{at} \quad z = 0, 1 . \quad (FF) \label{E:FF2}
\end{equation}
Stress-free, impenetrable mechanical boundary conditions on the top and bottom boundaries are assumed throughout and given by
\begin{equation}
w = {\partial_z} u = {\partial_z} v = 0, \quad \textnormal{at} \quad z = 0, 1.
\end{equation}

The system \eqref{E:momlin}-\eqref{E:contlin} is discretized in the vertical and horizontal dimensions with Chebyshev polynomials and Fourier modes respectively, and formulated as a generalized eigenvalue problem.  We solve the system in primitive variable form and enforce boundary conditions via the tau method.  The eigenvalue problem is solved with Matlab's `sptarn' function.  For further details of the numerical methods the reader is referred to \cite{mC13}.

Figure \ref{F:linNS} shows results from the linear stability calculations. Results are given for both steady ($Pr=1$) and oscillatory ($Pr=0.1$) convection; we note oscillatory convection does not exist for $Pr\ge1$ and becomes the primary instability for $Pr\lesssim 0.68$ \citep{sC61}.  For $E_H \lesssim 10^{-5}$, both the asymptotically scaled critical Rayleigh number $Ra_c E^{4/3}$ (Figure \ref{F:linNS}a)  and wavenumber $k_c E^{1/3}$ (Figure \ref{F:linNS}b) obtained from FT and FF boundary conditions are observed to converge to nearly equivalent values.  The open circle shows the Ekman number $E_H=0.0745$ calculated by \cite{tD88} at which the instability becomes characterized by $k_c \ne 0$ with $Ra_c = 341.05$.

Figure \ref{F:linNS}c shows the horizontal ($x$) velocity eigenfunction for $E_H=10^{-4}$ where Ekman layers can be seen at the top and bottom boundaries for the FF (dashed curve) case; a magnified view of the bottom Ekman layer is shown in the inset figure.  The temperature perturbation eigenfunctions shown in Figure \ref{F:linNS}d show that both the FT and FF cases have identical structure in the fluid interior, whereas a thermal boundary layer is observed in the FF case.  In the following section we present the asymptotic reduction of the Navier-Stokes equations to better understand and quantify this behavior.

\begin{figure}
  \begin{center}

   \subfloat[]{
      \includegraphics[height=5cm]{Rc_vs_E}}
      \qquad
    \subfloat[]{
      \includegraphics[height=5cm]{kc_vs_E}} \\
   \subfloat[]{
      \includegraphics[height=5cm]{u_eigTa1e8P1}}
      \qquad
    \subfloat[]{
      \includegraphics[height=5cm]{T_eigTa1e8P1}}
  \end{center}
\caption{Linear stability of the Navier-Stokes equations for fixed temperature (FT) and fixed flux (FF) thermal boundary conditions.  (a) Asymptotically scaled critical Rayleigh number and (b) critical wavenumber as a function of the inverse Ekman number for both steady ($Pr=1$) and oscillatory ($Pr=0.1$) convection.  (c) Horizontal velocity and (d) temperature eigenfunctions for $Pr=1$ and $E_H=10^{-4}$; the inset figure in (c) shows a magnified view of the Ekman layer along the bottom boundary.  In (a) and (b) the Ekman number of $E_H=0.0745$ at which the critical wavenumber for FF boundary conditions becomes non-zero is shown by the open circle, as first determined by \cite{tD88}.}
\label{F:linNS}
\end{figure}

\section{Asymptotics}
\label{S:Asymp}

To proceed with the asymptotic development, we follow the work of \cite{mS06} and write the governing equations using a generic non-dimensionalization such that
\begin{gather}
D_t \mathbf{u}  + \frac{1}{Ro} {\bf\widehat z} \times \mathbf{u}   =  - Eu \nabla p + \Gamma \theta {\bf\widehat z}  + \frac{1}{ Re} \nabla^2 \mathbf{u},  \label{E:mom1}  \\
\nabla \cdot \mathbf{u}  =  0, \label{E:mass1}  \\
D_t \vartheta =  \frac{1}{Pr Re} \nabla^2 \vartheta ,  \label{E:energy1}
\end{gather}
where $D_t (\cdot) = {\partial_t}(\cdot) + \mathbf{u} \cdot \nabla(\cdot)$ and the velocity, pressure and temperature are denoted by $\mathbf{u}$, $p$, and $\vartheta$, respectively.  The above system has been non-dimensionalized utilizing the velocity scale $U$, length $L$, time $L/U$, pressure $P$, and temperature $\widetilde{T}$. For the FT and FF cases the temperature scale becomes $\Delta T$ and
$Nu  \Delta T$,
respectively.  The Rossby, Euler, buoyancy and Reynolds numbers are defined by
\begin{equation}
Ro = \frac{U}{2 \Omega L}, \quad
Eu = \frac{P}{\rho_0 U^2}, \quad \Gamma = \frac{g \alpha \widetilde{T}}{U^2}, \quad Re = \frac{U L}{\nu} .
\end{equation}

In the present work we are interested in the $\epsilon \equiv Ro \rightarrow 0$ limit. In the fluid interior we employ multiple scales in the axial space direction and time such that
\begin{equation}
{\partial_z} \rightarrow {\partial_z} + \epsilon {\partial_Z} , \quad
{\partial_t} \rightarrow {\partial_t} + \epsilon^2 {\partial_{\tau}} ,
\end{equation}
where $Z = \epsilon z$ is the large-scale vertical coordinate and $\tau = \epsilon^2 t$ is the `slow' timescale.  It has been shown that the following distinguished limits can be taken to reduce the governing equations to accurately model quasi-geostrophic convection \citep[e.g.][]{mS06}
\begin{equation}
Eu = \frac{1}{\epsilon^2}, \quad \Gamma = \frac{\widetilde{\Gamma}}{\epsilon}, \quad Re  = O(1) , \quad Pr = O(1),
\end{equation}
where $\widetilde{\Gamma}=O(1)$.  Scaling the velocity viscously such that $U = \nu/L$ we have
\begin{equation}
\epsilon = E^{1/3}, \quad \widetilde{\Gamma} = \frac{E^{4/3} Ra}{Pr}, \quad Re = 1 ,
\end{equation}
where the $L$-scale Ekman number is related to the $H$-scale Ekman number via $E=E_H\epsilon^{-2}$, i.e.~$L=E^{1/3} H$.  We keep the notation for the Rayleigh number generic in the sense that $Ra$ denotes either $Ra_{FT}$ or $Ra_{FF}$ depending upon the particular boundary conditions employed.  Hereafter, we define the asymptotically reduced Rayleigh number as $\widetilde{Ra} \equiv E^{4/3} Ra$.

We utilize a composite asymptotic expansion approach \citep[e.g.][]{aN08} and, following \cite{wH71}, decompose each variable into interior $(i)$, middle $(m)$, and Ekman layer $(e)$ components.  For instance, the dependent variable $f$ can be written as
\begin{equation}
f= f^{(i)}({\bf x},Z,t,\tau) + f^{(m)}(x,y,\xi,t) + f^{(e)}(x,y,\eta,t) , \label{E:decomp}
\end{equation}
where $\xi=z$ and $\eta = \epsilon^{-1/2}z$ are boundary layer variables.  The above representation ensures that each dependent variable is uniformly valid throughout the domain.  The boundary layer variables consist of a sum of contributions from the top and bottom boundary layers; for brevity, we focus on the bottom boundary layers. In the present work we make use of the following limits and notation
\begin{equation}
\lim_{\xi \rightarrow \infty} \left (  f^{(m)} \right )  = \left (  f^{(m)} \right ) ^{(i)} = \lim_{\eta \rightarrow \infty}  \left (  f^{(e)} \right )  = \left (  f^{(e)} \right ) ^{(i)} = 0 , \label{E:limits1}
\end{equation}
\begin{equation}
\lim_{\xi \rightarrow 0} \left (  f^{(i)} \right )  = \lim_{\eta \rightarrow 0} \left (  f^{(i)} \right )  = f^{(i)}\left (  Z = 0\right )  = f^{(i)}(0) . \label{E:limits2}
\end{equation}
We then expand each variable in a power series according to
\begin{equation}
f^{(i)}({\bf x},Z,t)  = f_0^{(i)}({\bf x},Z,t,\tau) + \epsilon^{1/2} f_{1/2}^{(i)}({\bf x},Z,t,\tau) + \epsilon f_{1}^{(i)}({\bf x},Z,t,\tau) + O(\epsilon^{3/2}) . \label{E:expand}
\end{equation}
Each dependent variable is further decomposed into mean and fluctuating components such that
\begin{equation}
f^{(i)}({\bf x},Z,t,\tau) = \overline{f}^{(i)}(Z,\tau) + f^{\prime (i)}({\bf x},Z,t,\tau) ,
\end{equation}
where the averaging operator is defined by
\begin{equation}
\overline{f}(Z,\tau) = \lim_{[\tau], [A] \rightarrow \infty} \, \frac{1}{[\tau] [A]} \int_{[\tau],[A]} f \, dx dy , \qquad\mbox{and}\qquad \overline{f'} \equiv 0.
\end{equation}

\subsection{The interior equations}

By substituting decompositions for each variable of the form \eqref{E:decomp} into the governing equations and utilizing the limits \eqref{E:limits1}-\eqref{E:limits2}, equations for each region can be derived; expansions of the form \eqref{E:expand} are then utilized to determine the asymptotic behavior of each fluid region.  Because the derivation of the interior equations has been given many times previously, we present only the salient features and direct the reader to previous work \citep[e.g.][]{mS06} for details on their derivation.  The main point is that the interior convection is geostrophically balanced and horizontally divergence-free to leading order
\begin{equation}
{\bf\widehat z} \times \mathbf{u}^{(i)}_{0} = - \nabla_\perp p^{(i)}_{1}, \quad {\nabla_\perp \cdot} \mathbf{u}^{(i)}_{0,\perp} = 0,
\end{equation}
where $\nabla_{\perp} = ({\partial_x},{\partial_y},0)$.  The above relations allow us to represent the geostrophic velocity via the geostrophic streamfunction $\psi^{(i)}_0 \equiv p^{\prime(i)}_1$ such that  $\mathbf{u}^{(i)}_{0,\perp} = -\nabla \times \psi^{(i)}_0{\bf\widehat z}$.  The vertical vorticity is then $\zeta^{(i)}_{0} = \nabla_\perp^2 \psi^{(i)}_0$.  The interior vertical vorticity, vertical momentum, fluctuating heat, and mean heat equations then become
\begin{gather}
  D^{\perp}_{t} \zeta^{(i)}_{0}  - {\partial_Z} w^{\prime(i)}_0 = {\nabla_\perp^2} \zeta^{(i)}_{0}, \label{E:vortint} \\
  D^{\perp}_{t} w^{\prime(i)}_0 + {\partial_Z} \psi^{(i)}_0   = \frac{\widetilde{Ra}}{Pr} \vartheta_1^{\prime(i)}  + {\nabla_\perp^2} w^{\prime(i)}_0, \label{E:momint} \\
  D^{\perp}_{t}\vartheta_1^{\prime(i)}  + w^{\prime(i)}_0 {\partial_Z} \overline{\vartheta}^{(i)}_0  =  \frac{1}{Pr} {\nabla_\perp^2}   \vartheta_1^{\prime(i)}, \label{E:heatint} \\

  {\partial_{\tau}} \overline{\vartheta}_0^{(i)} + {\partial_Z} \overline{\left (  w^{\prime(i)}_0 \vartheta_1^{\prime(i)} \right ) } = \frac{1}{Pr} \partial^2_{Z} \overline{\vartheta}_0^{(i)}, \label{E:mheatint}
\end{gather}
where $D_t^{\perp} (\cdot) = {\partial_t}(\cdot) + \mathbf{u} \cdot \nabla_{\perp}(\cdot)$ .  The mean interior velocity field $\overline{\bf u}_0^{(i)}$ is zero and the mean momentum equation reduces to hydrostatic balance in the vertical, ${\partial_Z} \overline{p}^{(i)}_0 = (\widetilde{Ra}/Pr) \overline{\vartheta}_0^{(i)}$.

The interior system is fourth order with respect to the large-scale vertical coordinate $Z$.  Two boundary conditions are supplied by impenetrability such that $w^{\prime(i)}_0(0) =  w^{\prime(i)}_0(1) = 0$.  Although no $Z$-derivatives with respect to $\vartheta^{\prime(i)}_1$ are present in equation \eqref{E:heatint}, evaluating this equation at the boundaries shows the FT conditions $\vartheta^{\prime(i)}_1(0)= \vartheta^{\prime(i)}_1(1)=0$ are satisfied implicitly for the fluctuating temperature. Evaluating equation \eqref{E:momint} at either the top or bottom boundary with the use of impenetrability shows that stress-free boundary conditions are implicitly satisfied as well since
${\partial_Z} \psi^{(i)}_0(0) = {\partial_Z} \psi^{(i)}_0(1) = 0$.

For the case of FT thermal boundary conditions, we have
\begin{equation}
\overline{\vartheta}_0^{(i)}(0) = 1, \quad \textnormal{and}  \quad \overline{\vartheta}_0^{(i)}(1) = 0.  \quad (FT)
\end{equation}
Thus, for the FT case the boundary layer corrections are identically zero and the above system is complete.  Numerous investigations have used the above system of equations to investigate rapidly rotating convection in the presence of stress-free mechanical boundary conditions and have shown excellent agreement with direct numerical simulations (DNS) of the Navier-Stokes equations and laboratory experiments \citep{sS14,jmA15}.

For the FF case the mean temperature boundary conditions become
\begin{equation}
{\partial_Z} \overline{\vartheta}_0^{(i)}(0) = -1, \quad \textnormal{and}  \quad {\partial_Z} \overline{\vartheta}_0^{(i)}(1) = -1. \quad (FF)
\end{equation}
We further require ${\partial_Z} \vartheta_1^{\prime(i)}(0) = {\partial_Z} \vartheta_1^{\prime(i)}(1)= 0$; boundary layer corrections are therefore required since these conditions are not satisfied by \eqref{E:heatint}.  In the following two subsections we determine the magnitude of these boundary layer corrections.

\subsection{The middle layer equations}

The first non-trivial fluctuating middle layer momentum equation occurs at $O(\epsilon)$ to yield the thermal wind balance
\begin{equation}
{\bf\widehat z} \times \mathbf{u}^{\prime (m)}_2 = - \nabla p^{\prime (m)}_{3} + \frac{\widetilde{Ra}}{Pr} \vartheta^{\prime(m)}_2 {\bf\widehat z} , \label{E:mommid2}
\end{equation}
such that ${\nabla_\perp \cdot} \mathbf{u}^{\prime (m)}_{2} = 0$ and $w^{\prime (m)}_{2} \equiv 0$.
The mean velocity field $\overline{\bf u}^{(m)} \equiv 0$.

The leading order temperature equation for the middle layer is
 \begin{equation}
{\partial_t} \vartheta^{\prime(m)}_2 + \mathbf{u}^{\prime (i)}_0 (0) \cdot {\nabla_{\perp}} \vartheta^{\prime(m)}_2  = \frac{1}{Pr} \nabla^2 \vartheta^{\prime(m)}_2 ,
\end{equation}
with corresponding boundary conditions
\begin{equation}
{\partial_Z} \vartheta^{\prime(i)}_{1}(0) + {\partial_\xi} \vartheta^{\prime(m)}_{2}(0) = 0, \quad \vartheta^{\prime(m)}_{2}(\xi \rightarrow \infty) \rightarrow 0 . \label{E:midthermBC}
\end{equation}
We find the first non-trivial mean temperature to be of magnitude $O(\epsilon^5)$ and therefore omit any further consideration of this correction.

The first three orders of the stress-free mechanical boundary conditions along the bottom boundary become
\begin{equation}
{\partial_Z} \mathbf{u}^{\prime (i)}_{0,\perp}(0) = 0, \quad
{\partial_Z} \mathbf{u}^{\prime (i)}_{1/2,\perp}(0) = 0, \quad
{\partial_Z} \mathbf{u}^{\prime (i)}_{1,\perp}(0) + {\partial_\xi} \mathbf{u}^{\prime (m)}_{2,\perp}(0) + {\partial_\eta} \widetilde{\mathbf{u}}^{\prime (e)}_0(0) = 0 . \label{E:stressBC}
\end{equation}
Thus, the first two orders of the interior velocity satisfy stress free conditions on their own and therefore need no boundary layer correction.  Here we have rescaled the Ekman layer velocity according to $\mathbf{u}^{\prime (e)}_{5/2} = \epsilon^{5/2} \widetilde{\mathbf{u}}^{\prime (e)}_0$; this rescaling is simply highlighting the fact that the Ekman layer velocities are significantly weaker than those in the interior.

\subsection{The Ekman layer equations}

The Ekman layer equations have been studied in great detail in previous work \citep[e.g.][]{hG68}, so we simply state the leading order continuity and momentum equations as
\begin{equation}
{\nabla_\perp \cdot} \widetilde{\mathbf{u}}^{\prime (e)}_0 + \partial_{\eta} \widetilde{w}^{\prime (e)}_{\frac{1}{2}} = 0, \quad {\bf\widehat z} \times \widetilde{\mathbf{u}}^{\prime (e)}_0 = \partial^2_{\eta} \widetilde{\mathbf{u}}^{\prime (e)}_0
\end{equation}
where $w_3^{\prime(e)}=\epsilon^3  \widetilde{w}^{\prime (e)}_{\frac{1}{2}} $. All of the mean Ekman layer variables can be shown to be zero.  A key component in the present analysis that differs from previous work is the middle, thermal wind layer that enters the Ekman layer solution via the stress-free boundary conditions \eqref{E:stressBC}.  Utilizing the thermal wind relations for the middle layer that follow from equation \eqref{E:mommid2},
\begin{equation}
{\partial_\xi} u^{\prime (m)}_2 = -\frac{\widetilde{Ra}}{Pr} {\partial_y} \vartheta^{\prime(m)}_2, \quad {\partial_\xi} v^{\prime (m)}_2 = \frac{\widetilde{Ra}}{Pr} {\partial_x} \vartheta^{\prime(m)}_2 ,
\end{equation}
the stress-free boundary conditions along the bottom boundary can be written as
\begin{equation}
{\partial_Z} \mathbf{u}^{\prime (i)}_{1,\perp}(0) + \frac{\widetilde{Ra}}{Pr} \nabla^{\perp} \vartheta^{\prime(m)}_2(0) + {\partial_\eta} \widetilde{\mathbf{u}}^{\prime (e)}_0(0) = 0 ,
\end{equation}
where $\nabla^{\perp} = (-{\partial_y},{\partial_x},0)$.  Solving the Ekman layer momentum equations for the horizontal components of the velocity field with the additional requirement that $(\widetilde{u}^{\prime (e)}_0,\widetilde{v}^{\prime (e)}_0) \rightarrow 0$ as $\xi \rightarrow \infty$, the continuity equation is then used to find the Ekman pumping velocity
\begin{equation}
 \widetilde{w}^{\prime (e)}_{\frac{1}{2}} = \left [  {\partial_Z} \zeta^{(i)}_1(0) + \frac{\widetilde{Ra}}{Pr} {\nabla_\perp^2} \vartheta^{\prime(m)}_2(0) \right ]  e^{-\frac{\eta}{\sqrt{2}}} \cos \left (  \frac{\eta}{\sqrt{2}} \right )  . \label{E:ekpump}
\end{equation}
Thus, vertical velocities of magnitude $O(\epsilon^{3})$ are induced by FF thermal boundary conditions and result from both finite viscous stresses within the fluid interior and horizontal variations of the temperature within the middle layer.  This finding is closely analogous to the Ekman pumping effect first reported by \cite{rH64} for shallow layer quasi-geostrophic flow in the presence of lateral temperature variations along a free surface.  Evaluating equation \eqref{E:ekpump} at $\eta=0$ provides a parameterized boundary condition for the effects of Ekman pumping.

The small magnitude of the Ekman pumping velocity \eqref{E:ekpump} results in very weak $O(\epsilon^5)$ temperature fluctuations within the Ekman layer.  Because of this, the dominant correction of the FF thermal boundary conditions occurs within the middle layer and we do not consider the Ekman layer temperature any further.

\subsection{Synthesis}

The thermal boundary layer correction given by equation \eqref{E:midthermBC} is passive in the sense that $\vartheta^{\prime(m)}_2$ can be calculated a posteriori with knowledge of $\vartheta^{\prime(i)}_1$.  Thus, the leading order interior dynamics are insensitive to the thermal boundary conditions.  The Ekman layer analysis shows that the first six orders of the interior vertical velocity satisfy the impenetrable mechanical boundary conditions $w^{\prime(i)}_i(0) = 0$, for $i=0,\ldots, 5/2$.
At $O(\epsilon^3)$ we have the Ekman pumping boundary conditions
\begin{equation}
w^{\prime(i)}_3(0) = - w^{\prime(m)}_3(0) - {\partial_Z} \zeta^{(i)}_1(0)  - \frac{\widetilde{Ra}}{Pr} {\nabla_\perp^2} \vartheta^{\prime(m)}_2(0) ,
\end{equation}
where we have used the Ekman pumping relation \eqref{E:ekpump} evaluated at $\eta=0$.  From the standpoint of linear theory, the first correction to the critical Rayleigh number will therefore occur at $O(\epsilon^3)$; this explains the linear behavior previously discussed in section \ref{S:Lin}.

\section{Discussion}
\label{S:discuss}

In light of the boundary layer analysis, we conclude that the leading order quasi-geostrophic dynamics are described by equations \eqref{E:vortint}-\eqref{E:mheatint} for \textit{both} FT and FF thermal boundary conditions.  Indeed, inspection of the system shows that it is invariant under the following rescaling of the Rayleigh numbers and temperature variables,
\begin{equation}
\widetilde{Ra}_{FT} = \frac{\widetilde{Ra}_{FF}}{Nu}, \quad \vartheta_{1,FT}^{\prime(i)} = Nu \, \vartheta_{1,FF}^{\prime(i)}, \quad \overline{\vartheta}_{0,FT}^{(i)} = Nu \, \overline{\vartheta}_{0,FF}^{(i)} . \label{E:scaling}
\end{equation}
Integrating the time-averaged mean heat equation with respect to $Z$ yields
\begin{equation}
Pr \overline{\left (  w^{\prime(i)}_0 \vartheta_{1,FT}^{\prime(i)} \right ) } = {\partial_Z} \overline{\vartheta}_{0,FT}^{(i)} + Nu, \quad (FT) \label{E:mFT}
\end{equation}
\begin{equation}
Pr \overline{\left (  w^{\prime(i)}_0 \vartheta_{1,FF}^{\prime(i)} \right ) } = {\partial_Z} \overline{\vartheta}_{0,FF}^{(i)} + 1, \quad (FF) \label{E:mFF}
\end{equation}
for the FT and FF cases, respectively.  The appropriate thermal boundary conditions have been applied at $Z=0$ in the above relations.  Taking either equation \eqref{E:mFT} or \eqref{E:mFF} and utilizing \eqref{E:scaling} shows that the mean interior temperature gradient is described by identical equations for the two cases.  This leading order correspondence is the result of the anisotropic spatial structure of rapidly rotating convection.

The above results indicate that the findings of previous work on low Rossby number convection employing FT thermal boundary conditions can be accurately applied to the case of FF thermal boundary conditions by use of the rescalings given by equations \eqref{E:scaling}.  \cite{kJ12} identified four flow regimes that occur in rapidly rotating convection as a function of the Prandtl and (FT) Rayleigh numbers.  The so-called `convective Taylor column' (CTC) regime is distinguished by coherent, vertically aligned convective structures that span the depth of the fluid.  Figure \ref{F:nonlin}(a) shows a volumetric rendering of the temperature perturbation for $Pr=7$ and $\widetilde{Ra}_{FT}=46.74$, or $\widetilde{Ra}_{FF}=1000$ and $Nu=21.39$. The CTC regime occurs over the FT Rayleigh number range of  $20\lesssim \widetilde{Ra}_{FT} \lesssim 55$, corresponding to a FF Rayleigh number range of $82\lesssim \widetilde{Ra}_{FF} \lesssim 1656$ \citep[e.g.~see][]{dN14}.   Figure \ref{F:nonlin}(b)  shows mean temperature profiles obtained utilizing the FT and FF thermal boundary conditions, along with the remapped FF mean temperature profile.  The Nusselt number $Nu=21.39$ corresponds to a mean temperature difference of $0.0468$ between the top and bottom boundaries for the FF case.

Of particular interest in convection studies is the dependence of the heat transfer scaling with the strength of the thermal forcing input via Nusselt-Rayleigh number scalings of the form $Nu \sim \widetilde{Ra}_{FT}^\alpha$.  With the rescaling given in \eqref{E:scaling} the FF equivalent of this relation becomes $Nu \sim \widetilde{Ra}_{FF}^\beta$ where $\beta  = \alpha/(\alpha+1)$.  For the CTC regime the exponent is $\alpha \approx 2.1$ \citep{kJ12}, yielding $\beta \approx 0.68$.  Additionally, the final regime of geostrophic turbulence achieves a dissipation-free scaling law with $\alpha = 3/2$ such that $\beta = 3/5$ \citep{kJ12b}. Similarly, the dependence of all other variables of interest on the Rayleigh number (e.g. mean temperature gradient, vorticity, etc.) can also be remapped to the case of FF thermal boundary conditions.

\begin{figure}
  \begin{center}
    \subfloat[]{
      \includegraphics[height=5cm]{R1000P7_T}}
      \hspace{1.5cm}
    \subfloat[]{
      \includegraphics[height=4.5cm]{theta_bar_profiles}}
  \end{center}
 \caption{(a) An example volumetric rendering of the temperature perturbation from a simulation of the quasi-geostrophic convection equations showing the `convective Taylor column' (CTC) regime.  (b) Mean temperature profiles obtained with both FT (solid blue) and FF (dashed black) boundary conditions, and the rescaled FF temperature profile (red open circles). The parameters are $\Pr=7$, $\Ra_{FT} = 46.74$, $\Ra_{FF} = 1000$, and $Nu=21.39$.

 }
 \label{F:nonlin}
\end{figure}

\section{Conclusion}
\label{S:conclude}

In the present work we have shown that the leading order dynamics of rapidly rotating convection in a plane layer geometry are equivalent for both FT and FF thermal boundary conditions.  FF thermal boundary conditions give rise to a double boundary layer structure in the limit of rapid rotation that is asymptotically weak.  Our findings suggest that all previous work employing FT thermal boundary conditions also accurately describes FF thermal boundary conditions within the regime of asymptotic validity, i.e. $Ro\sim E^{1/3}\ll1$ and $\widetilde{Ra} \lesssim \mathcal{O} (E^{-1/3})$ \citep{kJ12b}.  This result adds to the robustness of the reduced, quasi-geostrophic model in geophysical  and astrophysical applications where stress-free boundary conditions are typically assumed; recent work has also extended the model to the case of no-slip boundary conditions \citep{kJ15} that are of relevance for laboratory experiments and planetary interiors \citep[e.g.][]{jmA15,jC15}.

An interesting consequence of these findings is that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading order influence on the interior convection.  This finding helps to explain the results of previous spherical convection studies investigating the role of temperature variations along the outer boundary \citep[e.g.][]{cD09}.

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