ICARUS
135, 501–517 (1998)
IS986000
ARTICLE NO.
Thermal Structure and Para Hydrogen Fraction on the Outer Planets
from Voyager IRIS Measurements
Barney J. Conrath, Peter J. Gierasch, and Eugene A. Ustinov
Astronomy Department, Space Sciences Building, Cornell University, Ithaca, New York 14853
E-mail: [email protected]
Received January 23, 1998; revised June 22, 1998
initial para hydrogen fraction rather than immediately assuming the thermodynamic equilibrium value associated
with the lower temperature. The degree of disequilibrium
that occurs will depend on the para hydrogen equilibration
time relative to the dynamic transport time. The equilibration time is not well constrained. For pure hydrogen under
conditions of pressure and temperature encountered in
Jupiter’s upper troposphere, the equilibration time is approximately 3 3 108 s. However, this time can be significantly shortened in the presence of various catalytic processes, as discussed by Massie and Hunten (1982) and
Conrath and Gierasch (1984). A mechanism of particular
interest in the context of the atmospheres of the jovian
planets is ortho–para conversion occurring at paramagnetic sites produced by solar photons on the surfaces of
cloud particles.
Lagged conversion of para hydrogen can play a significant role in the thermodynamics of convective processes,
and this possibility was examined by Conrath and Gierasch
(1984) and Gierasch and Conrath (1987). This subject has
recently been more extensively investigated by Smith and
Gierasch (1995) within the framework of convective adjustment. Para hydrogen can also serve as a tracer of atmospheric motion. If the equilibration time can be estimated,
then measurements of the spatial distribution of disequilibrium para hydrogen can be used to diagnostically study
upper tropospheric circulation. These possibilities have
motivated the present study.
A number of attempts have been made to obtain information on the ortho–para ratio of hydrogen in the observable layers of the atmospheres of the giant planets. These
include analyses of both reflected solar radiation in nearinfrared hydrogen lines (see for example Baines and Bergstralh 1986, Smith and Baines 1990, Baines et al. 1995) as
well as emission in the thermal infrared. In the present
work, we are concerned with the analysis of thermal infrared spectra obtained from the Voyager IRIS experiment.
The first attempts at extracting information on the para
hydrogen fraction from these data (Conrath and Gierasch
Voyager infrared spectra from Jupiter, Saturn, Uranus, and
Neptune are used to infer latitude-height cross sections of temperature and para hydrogen fraction. A new inversion algorithm is developed that simultaneously retrieves both quantities.
It uses all portions of the spectra containing information about
temperature and para fraction and not influenced by other
properties, such as cloud opacity. The sensitivity of spectra to
temperature and para hydrogen values at different heights is
calculated and presented. Retrievals based on artificial data
are carried out. The sensitivity studies and the artificial retrievals are used to determine the information content of the spectra.
Temperature and para hydrogen retrievals are presented for a
layer about a scale height deep, centered near the 300-mb level,
but varying in position from planet to planet. Temperature
cross sections show influence of seasonal solar forcing on Saturn
and Uranus. On all the planets except Uranus there are also
thermal anomalies correlated with zonal flows in the lower
tropospheres. Para hydrogen cross sections show complicated
patterns. On Uranus there is north–south hemispheric asymmetry. On Neptune there is correlation with latitudinal gradients
of the zonal flow. Fractional para hydrogen anomalies are larger
than those of temperature, and the ratio suggests a para hydrogen relaxation time of about a century near the 200-mb pressure
level.  1998 Academic Press
Key Words: outer planets; atmospheres; atmospheric structure.
I. INTRODUCTION
The ratio of molecular hydrogen in the ortho state (odd
rotational quantum number) to that in the para state (even
rotational quantum number) is a parameter of considerable
interest in the study of the structure and dynamics of the
upper tropospheres of the giant planets. At deeper atmospheric levels where temperatures exceed approximately
300 K, the ortho–para ratio is expected to lie near the high
temperature limit of 3 : 1 (the so-called ‘‘normal’’ value).
Since transitions between ortho and para states are strongly
forbidden, a parcel initially at deeper, warmer levels that
is moved upward to cooler levels will tend to retain its
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Copyright  1998 by Academic Press
All rights of reproduction in any form reserved.
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CONRATH, GIERASCH, AND USTINOV
1983, 1984) indicated the existence of disequilibrium para
hydrogen in Jupiter’s upper troposphere at low latitudes.
In the work of Conrath and Gierasch, a highly simplified
retrieval scheme was used to estimate both temperature
T and para hydrogen fraction fp within an atmospheric
layer about one pressure scale height thick, nominally centered near 300 mbar. With this algorithm, no attempt was
made to obtain information on the vertical profile of para
hydrogen. The mean value within the layer was estimated.
This approach was necessitated by the need to analyze
several thousand IRIS spectra within the constraints of the
computational capabilities available at that time. Large
numbers of retrievals were required in order to apply the
results to studies of dynamics on a global scale. Using these
retrievals, Gierasch et al. (1986) investigated the zonal
mean properties of Jupiter’s atmosphere.
Recently, Carlson et al. (1992) have carried out a more
complex analysis of the distribution of the para hydrogen
fraction in selected regions of Jupiter’s atmosphere. Forward integration of the radiative transfer equation was
used to calculate synthetic spectra from parametric models
of the relevant atmospheric properties, including temperature, para hydrogen fraction, gaseous ammonia, and ammonia ice clouds. A parameterized representation of the
para hydrogen profile was assumed, and the parameters
were adjusted to bring the synthetic and measured spectra
into agreement to within specified limits. In this manner,
they attempted to obtain information on the para hydrogen
profile between 100 and 900 mbar. Carlson et al. applied
this method to averages of several spatially inhomogeneous
ensembles of spectra selected using constraints on the behavior of the spectra themselves. They conclude that fp
and its vertical gradient vary spatially, at least at low latitudes, and that the variations are related to catalytic equilibration of para hydrogen on the surfaces of cloud particles.
An interested reader not familiar with the Voyager IRIS
spectra might wish to consult Carlson et al. (1992) for
plotted displays of spectra, noise estimates, and synthetic spectra.
The objective of the present study is to extend previous
work on the jovian para hydrogen fraction by examining
the zonal mean meridional temperature and para hydrogen
structure on Jupiter, Saturn, Uranus, and Neptune and to
discuss in a preliminary manner the meridional circulations
that are indicated by the results. The derived structures
will permit quantitative diagnostic study of the large scale
mean meridional circulation in the upper troposphere of
each planet, but this will not be the subject of the present
paper. To infer para hydrogen fractions and temperatures,
an inversion approach is applied to Voyager IRIS data to
simultaneously retrieve information on the temperature
and para hydrogen profile. Our goal is to extract the information contained in the spectral measurements with a
minimum of ad hoc model assumptions. It is essential to
understand the intrinsic information content of the measurements and the limitations on the uniqueness of solutions. In particular, it is necessary to establish the atmospheric height range over which both fp and T can be
unambiguously determined.
We first develop an algorithm for the simultaneous retrieval of two atmospheric profiles. This is described in
Section 2 below where we also present calculations of the
functional derivatives of the spectral radiance with respect
to the temperature and para hydrogen profiles and present
numerical results for all four planets. The retrieval algorithm is used to make a systematic investigation of the
information content of the spectral measurements in Section 3. In Section 4, the retrieval algorithm is applied to
zonal mean spectra from each planet, and the results are
discussed. In Section 5 qualitative conclusions are presented concerning the zonal mean circulation and/or seasonal time dependence in the upper troposphere of each
planet. Finally, our conclusions are summarized in Section 6.
2. RETRIEVAL ALGORITHM FORMULATION
We now consider the problem of formulating an algorithm suitable for the simultaneous retrieval of profiles
of temperature and para hydrogen from Voyager IRIS
measurements. As in previous studies, we attempt to exploit the fact that the relative sensitivity of the spectral
radiance to ortho and para hydrogen varies with wavenumber. For example, the S(0) collision-induced H2 absorption
line originates from transitions between para states while
the S(1) line results from ortho transitions.
Assume that we have spectral measurements at m frequencies within the S(0) and S(1) hydrogen absorption
lines, and we wish to retrieve the profiles of para hydrogen
fraction fp(z) and temperature T(z) where z 5 2ln p. To
calculate the radiance I(n), numerical quadrature must be
used, and T(z) and fp(z) are defined at n atmospheric levels.
We first linearize the radiative transfer equation about the
reference profiles f 0p(z) and T 0(z),
DIi 5
O ddTI DT 1 O ddfI Df
n
n
i
i
j
j51
j
j51
pj
(1)
pj
where dIi /dTj and dIi / d fpj are values of the functional derivatives of the radiance at ni with respect to T and fp at
level zj . The perturbations with respect to the reference
profiles are
DTj 5 T(zj) 2 T 0(zj),
(2)
Dfpj 5 fp(zj) 2 f 0p(zj),
(3)
DIi 5 I(ni) 2 I (ni),
(4)
0
where I 0(ni) is the radiance calculated using T 0 and f 0p .
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PARA HYDROGEN ON OUTER PLANETS
The direct solution of (1) for T(z) and fp(z), given measurements of I(ni), is an ‘‘ill-posed’’ problem. Arbitrarily
small changes in Ii can map into finite changes in the retrieved profiles, so in the presence of even quite small
measurement errors, the results are usually catastrophic.
In addition, there are usually more parameters to be determined than independent measurements, even when the
profile of only one parameter is to be retrieved. To obtain
physically meaningful solutions, it is necessary to introduce
constraints. Usually, these take the form of strong lowpass filtering of the solutions. This general approach to
inverse problems is reviewed in detail by Craig and
Brown (1986).
Define the matrices
Kij 5
dIi
,
dTj
dI
Mij 5 i .
d fpj
hM̂ TE 21[1 2 K̂(K̂ TE 21K̂ 1 c1)21K̂ TE 21]M̂ 1 h1jb
(13)
5 M̂ TE 21[1 2 K̂(K̂ TE 21K 1 c1)21K̂ TE 21] DI.
This expression can be simplified using the matrix identity
(ATB21A 1 C 21)21ATB21 5 CAT(ACAT 1 B)21, (14)
(5)
where A is an m 3 n matrix, B is an m 3 m matrix, and
C is an n 3 n matrix (see for example Westwater and
Strand 1968). The inverse of B and C are assumed to exist.
By applying (14) to both sides of (13), we obtain
(6)
[cM̂ T(K̂K̂ T 1 cE )21M̂ 1 h1]b 5 cM̂ T(K̂K̂ T 1 cE )21 DI.
(15)
Applying (14) once again to (15) gives
Equation (1) can then be written in the form
DI 5 M Dfp 1 KDT.
In these and subsequent expressions, 1 denotes a unit matrix, either of dimensions n 3 n or m 3 m, depending on
the context. Solving (11) for a, substituting into (12), and
rearranging yields an equation for b of the form
b 5 cM̂ T(cM̂M̂ T 1 hK̂K̂ T 1 chE )21 DI.
(7)
(16)
In general, Dfp and DT can be expressed in terms of sets
of basis vectors, i.e.,
If we define the two-point correlation matrices of the basis
vectors as
DT 5 Fa,
(8)
U 5 GG T,
(17)
(9)
S 5 FF ,
(18)
Dfp 5 W DI,
(19)
W 5 bUM T(aKSK T 1 bMUM T 1 E )21,
(20)
Dfp 5 Gb,
where F and G are matrices whose columns are the basis
vectors, and a and b are column vectors of expansion coefficients.
Now consider minimization with respect to Tj and fpj of
the quadratic form
Q 5 (DI 2 K̂a 2 M̂b)TE 21(DI 2 K̂a 2 M̂b)
1 caTa 1 h bT b,
T
then
where
(10)
where E is the measurement error covariance matrix (usually diagonal), and the superscript T denotes matrix transposition. Here we have introduced the definitions K̂ 5 KF
and M̂ 5 MG. In the case when c 5 h 5 0, (10) reduces
to the usual quadratic merit function that is minimized in
least squares fitting. The last two terms in (10) impose the
constraint that T and fp lie close to the reference profiles;
c and h are the weights with which the constraints are
imposed relative to the least squares fitting of the measurements. Carrying out the minimization yields the following
set of equations:
(K̂ TE 21K̂ 1 c1)a 1 K̂ TE 21M̂b 5 K̂ TE 21 DI,
(11)
M̂ TE 21K̂a 1 (M̂ TE 21M̂ 1 h1)b 5 M̂ TE 21 DI.
(12)
and the definitions a 5 1/c and b 5 1/ h have been introduced. Similarly, an expression for DT can be obtained in
the form
DT 5 V DI,
(21)
V 5 aSK T(aKSK T 1 bMUM T 1 E )21.
(22)
where
The nonlinearity of the problem can be taken into account
through iterative application of (19) and (21). Note that
only the correlation matrices of the basis vectors appear
in the expressions and not the basis vectors themselves.
In the present application, it can be assumed that the
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CONRATH, GIERASCH, AND USTINOV
random measurement errors at any two points in the spectrum are uncorrelated so E can be taken as a diagonal
matrix with the diagonal elements equal to the square of
the noise equivalent spectral radiance (NESR) associated
with the IRIS instrument. The rms error of the retrieved
value of fp at the ith atmospheric level can be estimated
using
s fpi 5
SO
D
1/2
W 2ij Ejj
j
.
(23)
Similarly, an estimate of the rms error in T at the ith level
is given by
s Ti 5
SO D
1/2
V 2ij Ejj
.
Kij 5
­B[ni , T(zj)] ­t̃ (ni , zj)
,
­Tj
­zj
(22)
where B(n, T ) is the Planck radiance, and t̃ (n, z) is the
transmittance at wavenumber n from level z to the top
of the atmosphere. In this formulation, the temperature
dependence of the atmospheric opacity is neglected since
it is generally much weaker than the temperature dependence of the Planck function. An analytic expression for
the functional derivative of the spectral radiance with respect to fp is derived in Appendix A.
(24)
j
3. INFORMATION CONTENT OF MEASUREMENTS
When analyzing ensemble averages of spectra, the effective
NESR for the average spectrum is assumed to be given by
the NESR of an individual spectrum divided by the square
root of the number of spectra in the ensemble. It should
be emphasized that the error estimates (23) and (24) pertain only to the precision or reproducibility of the retrievals
in the presence of random measurement error.
The magnitudes of the factors a and b determine the
degree of damping imposed on the solutions, and their
ratio determines the relative emphasis placed on the temperature versus the para hydrogen fraction. Their values
can most easily be determined through numerical experiment, as discussed further in the following section. The
values of S and U must also be specified. If the reference
profiles are chosen as the means of statistical ensembles
and the basis vectors are chosen as the departures from
the means of the individual profiles, then S and U represent
the statistical correlations of the profiles between pairs
of levels, and (19) and (20) become statistical estimation
algorithms. This approach is generally not relevant for
planetary applications. If it is assumed that no correlation
exists between levels, then S and U become proportional
to unit matrices, and Eqs. (19) and (20) become identical
to linearized maximum entropy algorithms. In the present
application, we find it useful to specify S and U as Gaussians
of the form
Sij 5 Uij 5 exp[2(zi 2 zj)2 /2c 2],
tion of the functional derivatives K and M. The functional
derivative with respect to temperature can be directly evaluated from the expression
(21)
where c is the correlation length in scale heights. This
simply provides a convenient means for filtering the solutions, and based on estimates of the resolution from K and
M as discussed in the following section, a value of c 5 0.5
is found to be appropriate.
The remaining task in the development of the inversion
algorithm is to provide formulations for the rapid calcula-
To gain an understanding of the information content
of the spectral measurements used in this study, we first
examine the behavior of the functional derivatives (5) and
(6). In calculating these quantities, we include collisioninduced absorption by molecular hydrogen due to H2 –H2
and H2 –He interactions. The necessary absorption coefficients were obtained using the algorithms of Borysow et al.
(1985, 1988). Compositions for Jupiter, Saturn, Uranus,
and Neptune were assumed to be 89, 93, 85, and 85%
hydrogen. Calculations for Jupiter were carried out prior
to publication of the Galileo results on the jovian helium
abundance (Von Zahn and Hunten 1996, Niemann et al.
1996). Retrievals made for selected test cases assuming a
hydrogen mole fraction of 85% indicate changes in fp of
p15%, independent of latitude or temperature. Examples
of the functional derivatives for Jupiter, Saturn, and Neptune are shown in Figs. 1–3, represented as contour plots
in a space defined by the spectral wavenumber and atmospheric pressure level. The functional derivative is a measure of the sensitivity of the spectral radiance at a given
wavenumber to a perturbation in T or fp at a given atmospheric level. Note that a vertical slice at a given wavenumber through the contour representation of a functional
derivative with respect to temperature is equivalent to the
usual temperature ‘‘contribution function’’ at that wavenumber.
In order to simultaneously determine both T(z) and
fp(z), it is necessary for two spectral regions to exist that
are sensitive to essentially the same atmospheric layer,
while possessing different relative sensitivities to the temperature and para hydrogen fraction in that layer. If this
condition is met, then in principle only one combination
of T and fp can simultaneously satisfy the measurements
in both spectral regions. In practice, limited vertical resolution and measurement errors will permit acceptable solutions over some range of combinations of T and fp .
PARA HYDROGEN ON OUTER PLANETS
505
over the accessible portions of the spectrum, information
on both temperature and fp can be obtained between 200
and 500 mbar.
A similar analysis of the functional derivatives for Saturn
(Fig. 2) indicates that T and fp can be simultaneously
obtained between about 100 and 350 mbar, again with
limited vertical resolution. The upper and lower limits
of the sampled spectral range are set equal to those for
Jupiter, but with no excluded region between the S(0)
and S(1) lines, since there is no evidence there for cloud
opacity effects.
Because of the low atmospheric temperatures of Uranus
and Neptune, the signal to noise ratio is adequate for
this analysis only between about 200 and 350 cm21. The
functional derivatives for both planets are similar. Those
for Neptune are presented in Fig. 3. They demonstrate
that simultaneous information on both T and fp can be
obtained between about 200 and 800 mbar. Note that above
about 245 cm21, dI/ dfp is negative, again due to the presence
FIG. 1. Sensitivities for Jupiter. (a) The functional derivative dI/ dT.
(b) The functional derivative dI/ dfp . The functional derivatives have been
normalized by their maximum absolute values. Note the change in sign
with wavenumber of fp sensitivity near 520 cm21. See text for discussion.
First considering Jupiter (Fig. 1), the spectral region
shown is sensitive to both T and fp in the atmospheric layer
between approximately 200 and 500 mbar. An increment
in temperature produces an increase in radiance throughout this part of the spectrum; however; in the case of para
hydrogen, an increment in fp results in a decrease in radiance at wavenumbers below 500 cm21 but an increase in
radiance above 500 cm21. This behavior is a direct consequence of the fact that the S(0) line, centered at 354 cm21
results from transitions between para states while the S(1)
line, centered near 600 cm21 is due to ortho transitions. In
practice, it is necessary to exclude the portion of the Jupiter
spectrum below 320 cm21 and in the region 430–520 cm21
between the S(0) and S(1) lines because of the possible
presence of cloud opacity (Carlson et al. 1992). As a consequence of the variation of relative sensitivity to T and fp
FIG. 2. As Fig. 1, but sensitivities for Saturn.
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CONRATH, GIERASCH, AND USTINOV
FIG. 3. As Fig. 1, but sensitivities for Neptune.
of the S(0) line. However, at the low temperatures of these
planets, the collision-induced translational absorption
dominates below 245 cm21 and the functional derivative
becomes positive. In all cases, since the vertical half-widths
of the functional derivatives are comparable to the total
thickness of the region from which unambiguous information can be obtained, only minimal vertical resolution can
be anticipated.
In order to achieve an understanding of the capabilities
and limitations of the inversion approach, and to ‘‘calibrate’’ the algorithm, we have applied (19) and (21) to
simultaneously retrieve estimates of fp(z) and T(z) from
spectra calculated from model atmospheres. A series of
such numerical experiments were used to establish the
optimum values for each planet of the parameters a and
b controlling the strength of the low-pass filtering imposed
on the solution profiles. Values were picked that provided
sufficiently strong filtering to prevent the occurrence of
nonphysical, high-frequency structure in the solutions
while still achieving residuals comparable to or less than
the anticipated random measurement error. In addition,
calculations of the random error in the solutions resulting
from the propagation of the measurement errors were
made to ensure that the precision of the retrievals fell
within acceptable limits. Examples for Neptune are shown
in Figs. 4 and 5. The retrievals of T(z) and fp(z) are shown
in the first and third panels from the left in each case,
along with the ‘‘true’’ profiles assumed for the models and
the reference or ‘‘first-guess’’ profiles. In the case of para
hydrogen, the first-guess profile corresponds to the equilibrium value of fp at each level. The second and fourth panels
show the random errors of the retrievals due to propagation of a noise level representative of an average of 50
IRIS Neptune spectra. The model para hydrogen profile
for the case shown in Fig. 4 is quite smooth, and the solution
is good down to about 800 mbar; at deeper levels the
spectra no longer contain significant information on fp . At
deeper levels, the solution tends back toward the first
guess, which is a characteristic of the type of inversion
algorithm used in this study, and is a direct consequence
of the presence of the last term in the merit function (10).
Figure 5 shows the results of the retrieval for a case in
which the true fp profile changes rapidly with height over
a relatively small vertical scale. The retrieval is a highly
smoothed approximation to the true profile, but both profiles yield the same spectrum to within the assumed measurement error. This illustrates the very limited vertical
resolution obtainable from the spectra. Our algorithm is
designed to give the smoothest solutions that can adequately reproduce the spectrum. This is consistent with
our goal to extract only that information that is actually
contained in the spectral data with a minimum of assumptions. To infer profiles with finer scale structure, it is necessary to introduce additional constraints, either implicitly
or explicitly. While the use of such constraints may sometimes be justifiable on physical grounds, it is important to
clearly distinguish between the information actually contained in the measurements and that introduced through
other considerations. The introduction of constraints beyond the low pass filtering contained in the inversion algorithm lies outside the scope of this investigation.
The emission angle of the measured spectrum is fully
taken into account in the radiative transfer code and the
inversion algorithm. The functional derivatives shown in
Figs. 1–3 were calculated assuming normal viewing. The
principal effect of increasing the emission angle is to move
the maxima at each wavenumber upward in the atmosphere, making the measurements sensitive to a somewhat
different altitude range. This remapping of the functional
derivatives goes approximately as e21/2, where e is the
cosine of the emission angle, due to the p2 dependence of
the optical depth. To investigate this effect on the retriev-
PARA HYDROGEN ON OUTER PLANETS
507
FIG. 4. Neptune artificial retrievals for a case with smooth fp(z).
als, we have inverted synthetic spectra over a range of
emission angles. Retrievals were carried out for values of
e ranging from 0.5 to 1.0. The maximum dispersion in the
solutions for fp in the region of significant information
content is about 0.01. This is negligible compared to other
limitations in the retrieval.
As discussed above, we have selected spectral regions
that are believed to be largely free of opacity due to clouds.
In doing so, we have omitted portions of Jupiter’s spectrum
that are sensitive to deeper levels of the atmosphere. For
example, use might be made of the entire spectral region
between 200 and 900 cm21; however, extension to this
broad region introduces sensitivities to additional heightdependent parameters, including ammonia abundance,
cloud particle number density, single scattering albedos,
etc. In general, the number of unknowns will exceed the
number of independent measurements added, and a
unique separation of temperature and para hydrogen cannot be achieved at levels deeper than about 500 mbar
without the imposition of additional constraints. At levels
shallower than 500 mbar, H2 is the only gas influencing
the spectral regions we employ.
FIG. 5. Neptune artificial retrievals for a case with a large jump in fp(z).
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CONRATH, GIERASCH, AND USTINOV
4. RESULTS
We have applied the algorithms developed in the previous section to selected sets of data to obtain simultaneous
retrievals of temperature and para hydrogen fraction on
all four planets. Our goal in this study is to attempt to
estimate the zonal mean behavior of these parameters.
Because of the limited height range over which information
can be obtained, our analyses are confined to the tropopause and upper troposphere regions. The available data
coverage differs from planet to planet as does the signalto-noise ratio of the spectra. This has necessitated the use
of a somewhat different strategy in estimating zonal means
on each planet. Although the retrievals from mean spectra
are not strictly equivalent to the mean of retrievals from the
individual spectra in the ensemble because of the nonlinear
relationships involved, they can be regarded as a firstorder approximation to the mean profiles. The selection
of spectra for each planet is discussed below.
Jupiter. IRIS spectra from the incoming north–south
mapping sequence acquired from Voyager 1 were used. In
this sequence, the planet was mapped over 3608 of longitude in the planetographic latitude range 608N to 608S.
The spatial resolution on the planet corresponded to about
108 of latitude near the equator. The data were averaged
in 108-wide latitude bins on 58 centers. Spectra from beam
locations near the edge of the disk with the cosine of the
emission angle less than 0.4 were omitted to avoid the
chance of contamination by empty space and to ensure
that the emission angle is uniform within the beam. There
are typically about 50 spectra in each bin.
Saturn. The IRIS north–south mapping sequences on
Saturn did not provide sufficient latitude coverage for purposes of this study, so individual latitude scans were used.
These corresponded to spatial resolution in the range of
4–88 in latitude near the equator. Beams that might have
been contaminated by the rings were avoided. As in the
case of Jupiter, latitude bins 108 wide on 58 centers were
selected. Each bin contains 10–20 spectra.
Uranus. Because of the constraints of the Voyager flyby geometry of Uranus, it was not possible to make a
systematic north–south mapping of the planet. However,
multiple latitude scans were made in each hemisphere, and
these provide some indication of the meridional structure
although the range in longitude sampled is small. Complete
latitude coverage was obtained between 908N and 908S.
The spatial resolution of the individual fields of view near
mid-latitudes is about 58 of great circle arc. Because of the
low signal associated with the cold atmosphere and the
relatively sparse spatial coverage, latitude bins of 208 width
on 108 centers were employed. The bins contained 10–20
spectra each.
Neptune. A north–south mapping sequence similar to
that on Jupiter was acquired for Neptune. Complete longitude coverage was obtained between approximately 308N
and 808S. In order to ensure adequate signal to noise ratio,
averages over 208-wide latitude bins on 108 centers were
constructed. These bins contained approximately 60–90
spectra.
The mean latitude and emission angle cosine were calculated for each bin for use in the retrievals and the subsequent construction of meridional cross sections. The variation of the gravitational acceleration with latitude was
taken into account. Quality control was exercised by monitoring the standard deviation of spectra associated with
each mean and by visually examining plots of individual
spectra.
The retrievals from each planet have been used to construct the meridional cross sections of T and fp shown in
Figs. 6 and 7. In all cases, the contours have been extended
somewhat beyond the vertical regions of maximum validity
of the retrievals. The information content decreases gradually rather than abruptly in moving away from these regions
toward both higher and lower pressures. Because of the
constraints imposed by the inversion algorithms, the retrievals tend toward the reference profiles or first guesses
as the information content decreases, while in the regions
of maximum information content, the solutions are essentially independent of the first guesses. The extended region
of contouring has been retained to better facilitate the
visual identification of patterns.
The temperature cross sections (Fig. 6) are essentially
consistent with previous analyses of IRIS data for all four
planets. For the case of Jupiter, the region of maximum
information content extends from approximately 80 to 500
mbar. The high-pressure limit is imposed primarily by the
existence of particulate opacity while at lower pressures
contributions to the radiances decrease exponentially with
increasing height. For Saturn, the region of validity extends
from about 80 mbar down to 700 mbar with the increased
depth of penetration resulting from presumably decreased
cloud opacity at these levels relative to Jupiter. For both
Uranus and Neptune, the solutions are valid from about
70 mbar down to 800 mbar under the assumption that
particulate opacities in the upper tropospheres of these
planets are negligible. The regions of validity for the para
hydrogen fraction (Fig. 7) are more limited on all four
planets. For Jupiter the fp retrievals are most reliable between about 100 and 500 mbar, for Saturn between about
100 and 350 mbar, while for Uranus and Neptune reliable
information on fp is obtained between approximately 200
and 800 mbar.
Because of the limited spatial resolution of the data used
here, the jet structure of Jupiter and Saturn is not well
resolved, but large-scale latitude structure is captured.
Only very large scale structure is seen in the Uranus cross
PARA HYDROGEN ON OUTER PLANETS
509
FIG. 6. Temperature cross sections. Height range of maximum information is indicated by ‘‘1’’ symbols near left side of plot. See text for
discussion.
sections because of the necessity of using large averaging
bins to achieve adequate signal to noise with the sparse
numbers of spectra available and cold atmospheric temperatures. In the case of Neptune, the coverage is limited
to the southern hemisphere and low northern latitudes;
however, the known jet structure is fully resolved.
5. DISCUSSION
In the upper tropospheres and lower stratospheres of
the outer planets, thermal anomalies can be caused by
circulations induced by the jets in the lower atmosphere
or by radiative forcing. Temperature observations by the
Voyager IRIS instrument have been discussed previously
(Gierasch et al. 1986, Conrath and Pirraglia 1983, Flasar
et al. 1987, Conrath et al. 1991). It will be useful to review
the major conclusions before discussing the new para hydrogen cross sections.
Figure 6 shows temperature minima near the 100-mbar
level on Jupiter and Neptune that appear to be associated
with jets. On Jupiter, the poor spatial resolution hampers
interpretation, but we know from higher resolution analysis (Gierasch et al. 1986) that the minima near 6208 latitude
are associated with the decay with height of the easterly
jets at about 188 north and south latitude. On Neptune,
the minimum near 458S latitude is associated with decay
with height of the flank of the easterly equatorial current.
Latitudinal temperature gradients, assuming geostrophic
and hydrostatic balance, imply vertical wind shears. Observed jet velocities refer to the drift of clouds that are
deeper in the atmosphere than these temperature measurements. The sign of the shear shows that the zonal jets are
decaying with height. The magnitude of the shear shows
that the scale height for decay of the wind is a few pressure
scale heights. Simple linear axisymmetric modeling can
reproduce this behavior if a Rayleigh friction drag is invoked, with time constant approximately equal to the radiative time constant (Conrath et al. 1990).
In contrast, Fig. 6 shows that Uranus and Saturn exhibit
hemispheric asymmetry in thermal structure, suggesting
seasonal radiative control. Table I presents estimates of
the radiative time constant tR near the 300-mbar level
510
CONRATH, GIERASCH, AND USTINOV
FIG. 7. Para hydrogen fraction cross sections. Height range of maximum information is indicated by ‘‘1’’ symbols near left side of plot. See
text for discussion.
(Conrath et al. 1990), compared with the orbital period tO
divided by 2f. For all four planets, the ratio r 5 2ftR /tO
is greater than unity, implying a large phase lag of approximately a quarter of a period and an amplitude reduced
by Ï(1 1 r 2) relative to seasonal changes in instantaneous
radiative equilibrium (Conrath and Pirraglia 1983). Conrath and Pirraglia (1983) show that the Saturn temperatures are consistent with these ideas. Conrath et al. (1990)
calculate Uranus radiative forcing at solstice conditions
and find approximately 10 K difference between winter
TABLE I
Radiative and Orbital Time Constants
Parameter
Radiative time constant
tR (years)
Orbital period tO (years)
2ftR /tO
Jupiter
5
11.86
2.6
Saturn
30
29.46
6.2
Uranus
130
84.01
10
Neptune
100
164.79
3.7
and summer hemispheres. Reducing this by a factor of
Ï(1 1 r 2) P 10 gives an expected response amplitude of
about 1 K, in rough agreement with Fig. 6. The phasing is
not as expected, however. The Voyager flyby occurred near
solstice, and if the phase lag is indeed a quarter of a season
then the hemispheric asymmetry should be near zero. It
is possible that the influence of longer time constants at
deep levels alters the phasing. This discrepancy is a major
question for future work.
Since temperatures vary from place to place, the thermodynamic equilibrium value of the para hydrogen fraction
also varies. The first question one might ask about fp is
whether its pattern simply reflects thermodynamic equilibrium in the presence of the observed temperatures. Figure
8 displays the equilibrium value of the para fraction, fpe ,
and Fig. 9 shows the difference fpe 2 fp . The difference is
larger than lateral variations of fpe , indicating that vertical
advection is the most likely cause of fp variations.
To prepare for interpretation of the para hydrogen data,
we would like to make estimates of the dynamical overturning rates on all four planets. Our approach will be a scaling
511
PARA HYDROGEN ON OUTER PLANETS
FIG. 8. Equilibrium para fraction.
analysis of the axisymmetric governing equations used by
Conrath et al. (1990), but extended to include the conservation equation for para hydrogen. The heat equation is
Dh fp
1 DhT uo 2 up
2
XH2
T Dt
cp T
Dt
1w
S
D
dfp
1 dT R uo 2 up
q̇
1 2
XH2
, (23)
5
T dz cp
cpT
dz
cpT
where Dh /Dt denotes the horizontal Lagrangian advective
time derivative, uo and up are the internal energies of ortho
and para hydrogen, respectively, cp is the heat capacity of
the atmosphere at constant pressure and composition, XH2
is the number fraction of hydrogen, w 5 Dz/Dt 5 2D(ln
p)/Dt, and q̇ is the heating rate by radiation or other
sources (but not hydrogen conversion, which is explicitly
written on the left-hand side). The radiative heating rate
can be estimated by writing
q̇
1 TE 2 T
,
5
cp T T
tR
(24)
where TE is the radiative equilibrium profile that would be
produced by the diurnal average insolation at a particular
seasonal time and at a particular latitude. Since w is in
units of pressure scale height per second, a dynamical time
can be estimated by focusing on the vertical advection term
on the right-hand side of (23) and writing tD 5 1/w. Defining
=5
dfp
1 dT R uo 2 up
1 2
XH2
,
T dz cp
cp T
dz
(25)
we can write
=
1 TE 2 T
p
,
tD T tR
(26)
= can be calculated from the observed T and fp fields.
To estimate TE 2 T it is necessary to determine whether
seasonal or jet forcing dominates. On Saturn and Uranus
Fig. 6 shows that there are irregularities in the isotherms
that show an influence of zonal jets superposed onto the
512
CONRATH, GIERASCH, AND USTINOV
FIG. 9. Difference fpe 2 fp . These plots show clearly that the para fraction is far from equilibrium.
large-scale seasonal gradients. The data suggest that on
Jupiter and Neptune, jets are strong enough and the seasonal forcing is weak enough so that thermal anomalies
forced by the lower tropospheric jets dominate the upper
tropospheric and stratospheric temperature field. On Saturn and Uranus the seasonal and dynamical forcings are
both important.
We estimate that at the times of the Voyager flybys, the
dominant heating for Jupiter, Saturn, and Neptune is due
to radiative perturbations produced by the thermal anomalies due to atmospheric jets. The best estimate of TE 2 T
is given by the lateral variations in the observed T. For
Uranus, which was near solstice conditions, the heating is
dominated by the seasonal forcing, and we have estimated
TE 2 T by using the lateral variations in the TE field
calculated and displayed by Conrath et al. (1990). The
adopted values are given in Table II. Figure 10 displays
calculated fields of =, and Table II gives values adopted
as typical of the upper troposphere for scaling purposes.
Results for the dynamical time constant tD are calculated
from (26) and displayed in Table II.
The dynamical time constants tD are extremely long by
terrestrial standards. This inference is a direct consequence
of the long radiative time constants on the outer planets
and the observed stratification. If vertical motions were
more rapid, corresponding to smaller tD , the atmospheres
would be more closely adiabatic than the observations
show. The dynamical time constants can also be used to
place an upper bound on the effective diffusivity for vertical heat transfer by writing KH p H 2 /tD , where H is the
TABLE II
Scaling Parameters and Time Constants
Parameter
Jupiter
Saturn
Uranus
Neptune
(TE 2 T )/T
The coefficient =
tD (years)
fpe 2 fp
dfp /dz
tp (years)
0.02
0.3
70
0.03
0.02
110
0.03
0.2
200
0.04
0.05
160
0.04
0.2
700
0.02
0.11
120
0.10
0.2
200
0.06
0.11
105
PARA HYDROGEN ON OUTER PLANETS
513
FIG. 10. The coefficient of w in (23), called = in the text and defined in (25).
pressure scale height. Using scale heights of 20, 36, and
25, and 25 km for Jupiter, Saturn, Uranus, and Neptune,
respectively gives KH p 0.2, 0.2, 0.03, and 0.1 m2 s21, respectively, as upper limits.
The para hydrogen fraction is governed by an equation
analogous to (23) and (24),
dfp fpe 2 fp
Dh fp
,
1w
5
Dt
dz
tp
(27)
where tp is the relaxation time constant for fp toward thermodynamic equilibrium. As discussed in the introduction,
tp is not well known. With the values of tD estimated just
above, it is possible to use (27) to reach an independent
new estimate of tp . As in the case of the heat equation,
assume that the vertical advection term on the left-hand
side of (27) can be used for scaling purposes to estimate
the advective time derivative. Using w 5 1/tD , (27) gives
tp p tD
fpe 2 fp
.
dfp /dz
(28)
Values of fpe 2 fp and of dfp /dz for the upper tropospheres
can be read from Figs. 9 and 7. Values adopted for our
estimates are given in Table II, along with the derived tp .
The derived para hydrogen relaxation times on Jupiter
and Saturn are longer than the radiative time constant,
and on Uranus and Neptune they are of the same order
as the radiative time constant. This reflects the observation
that fractional para hydrogen anomalies are larger than
those of temperature on Jupiter and Saturn but not on
Uranus, where fp is closest to equilibrium. The para
hydrdogen relaxation times are, within the uncertainties,
all of the same order of magnitude, near a century. If they
are applicable at the 200-mbar level, which is approximately where the temperature and fp contrast estimates of
Table II originate, the time constants are about a factor
of 5 larger than the estimate by Conrath and Gierasch
(1984). They estimated tp p 10 years at the 300-mbar level,
with the rate being approximately proportional to the pressure. Their estimate was based on extrapolating laboratory
measurements of liquid hydrogen conversion rates, assuming that the density ratio gives the rate ratio.
514
CONRATH, GIERASCH, AND USTINOV
The sign and patterns of fp can be qualitatively interpreted. On Jupiter, Fig. 7 shows that the dominant feature is
a minimum in fp at low latitudes. Figure 9 shows that fp is
less than fpe throughout this minimum. Of the four planets,
Jupiter is the only one for which the estimated dynamical
time constant tD is smaller than the hydrogen relaxation
time. The interpretation is that overturning advects low fp
gas from deep regions, and the overturning is rapid enough
so that equilibrium is never achieved. At high latitudes on
Jupiter, especially in the north, there is a suggestion of the
opposite behavior, with downward advection leading to fp
larger than equilibrium in the upper troposphere. Notice
that Fig. 9 suggests that fp is larger than equilibrium at 608
latitude at the tropopause temperature minimum (0.1 bar)
but this should be viewed with caution because the vertical
resolution of the retrieval is limited. It is impossible by
advective processes to produce fp greater than the local
equilibrium value at a location where the temperature is
minimum. On Saturn, the patterns of both fp and of T
(Figs. 6 and 7) are complicated and not easy to interpret.
The difference fpe 2 fp changes sign from place to place,
consistent with tD . tp . Near 200 mbar, there is an fp
minimum near 2608 and a maximum near 2158. This is
consistent with upward motion producing a relatively low
temperature at 200 mbar near 2608, and downward motion
producing a warm thermal anomaly near 2208. Seasonal
effects produce cooler temperatures in the northern hemisphere and higher fp in the northern hemisphere, consistent
with downward displacement in the hemisphere which has
just passed through winter. Patterns on Uranus are dominated by the seasonal effects. It appears from Figs. 6 and
7 that the northern hemisphere, which at the time of the
Voyager flyby was near winter solstice, has lower temperatures and higher fp , consistent with radiative cooling and
downward motion. On Neptune the patterns are dominated by the influence of the strong tropospheric flow.
Low temperature at latitudes between 2208 and 2608 are
consistent with upward motion and adiabatic cooling, and
relatively low fp between 2108 and 2508 accompanies the
upward displacements.
The thermodynamic importance of para hydrogen conversion can be evaluated by examining the size of the dfp /dz
term on the right-hand side of (25). Figure 11 displays
=N 5
5
1 dT R
1
T dz cp
(29)
HN 2
.
g
Here N is the Brunt frequency relevant to high frequency
displacements, cp is the heat capacity at fixed p and fp , and
g is the acceleration of gravity. Figure 11 shows that the
mapped regions on all four planets are dynamically stable,
with =N . 0, although the stability becomes small at deep
levels. The difference between Fig. 11, =N , and Fig. 10,
=, gives a measure of the importance of hydrogen conversion in the heat balance. In the upper tropospheres, at
pressures less than about 200 mbar, there is little difference
on any of the planets. On Jupiter there is little difference
at any height. In the lower tropospheres of Saturn, Uranus,
and Neptune, at pressures greater than 300 mbar, hydrogen
conversion introduces large effects. This is also indicated
by the behavior of =, which becomes very small at deep
levels on these three planets. Small values indicate that
hydrogen conversion is balancing temperature advection,
with radiative heating of negligible importance. This behavior is consistent with shorter hydrogen conversion time
constants and larger radiative time constants at higher pressures.
6. SUMMARY
The Voyager IRIS spectra in the region of the S(0) and
S(1) hydrogen absorption lines contain information on vertical structure of both temperature and hydrogen ortho/
para ratio. Detailed examination of the sensitivity of spectra to T(z) and fp(z) demonstrates that both profiles can
be retrieved over limited pressure ranges within the upper
tropospheres and lower stratospheres. For Jupiter it is necessary to exclude portions of the spectra that might be
affected by aerosol opacity. In the cases of Saturn, Uranus,
and Neptune it is also necessary to average large numbers
of spectra. For Jupiter it may be possible to use the new
retrieval algorithm on individual spectra, and this is an
important goal for future work because it will permit high
spatial resolution study of local regions.
Latitude–height cross sections of T and fp show evidence
of seasonal effects on both Saturn and Uranus. On Jupiter
and Neptune the cross sections show T and fp anomalies
that appear to be associated with the zonal flows at deeper
levels. These cross sections will be valuable constraints on
detailed modeling of the structures of the outer planetary atmospheres.
Scaling analysis based on the axisymmetric flow model
of Conrath et al. (1990) but extended to include para hydrogen conversion leads to an estimate of the hydrogen equilibration time constant of about a century at the 200-mbar
level on all four planets. The estimate is highly uncertain,
because the assumptions of the scaling cannot be tested
and are not unique. Better determination of the conversion
rate is an important goal for future detailed modeling.
APPENDIX: DERIVATION OF dI/ dfp
The intensity of outgoing monochromatic thermal radiation from the
top of an optically thick purely absorbing planetary atmosphere in the
direction with a local zenith angle arccos e may be written in the form
PARA HYDROGEN ON OUTER PLANETS
515
FIG. 11. Stratification with respect to high frequency displacements. These are contours of =N , defined in (29). The Brunt frequency is given
by N 2 5 (g/H )=N .
I(e) 5
E
y
2y
S D
2
­t̃
B(T(z)) dz,
­z
(A1)
where t̃ (z, e) is transmissivity and B(T(z)) is the Planck function. Both
t̃ and B are functions of frequency but this argument is suppressed for
brevity. The transmission function depends on the vertical profile of the
volume absorption coefficient of the atmosphere, k(z). This dependence
can be expressed by
S
D
1
t̃ (z, e) 5 exp 2 t (z) ,
e
t (z) 5
E
y
z
k(z9) dz9.
p2(z)
.
T(z)
b(T ) 5 fp bp(T ) 1 (1 2 fp)bo(T )
(A5)
c(T ) 5 fp cp(T ) 1 (1 2 fp)co(T ),
(A3)
RT 20
XH2 hXH2 a(T(z)) 1 XHe b(T(z))
gp 20
1 XCH4 c(T(z))j
a(T ) 5 f 2p ap(T ) 1 2fp(1 2 fp)aop(T ) 1 (1 2 fp)2ao(T )
(A2)
In this appendix we shall treat the more general case of a three component mixture, H2 –H2 , H2 –He, and H2 –CH4 . The expression for the volume absorption coefficient k(z) may be written in the form
k(z) 5
Here R is the gas constant for the atmosphere, g is gravity, T0 and p0 are
standard temperature and pressure, respectively, and Xi is mole fraction of
the ith gas. The coefficients a, b, and c are dependent on the para hydrogen
fraction fp ,
(A4)
where subscripts p, o, and op refer to para–para, ortho–ortho, and ortho–
para interactions, respectively.
The relations (A1) through (A5) may be considered as defining a
composite operator relation between the vertical profile fp(z) and the
spectral/angular dependence of intensity of the outgoing thermal radiation I(e) (Ustinov 1990). It should be emphasized that it is an operator
relation between functions fp(z) and I(e) because it involves integration
over altitude. Thus the sensitivity of I(e) to variations of fp(z) is a functional derivative dI/ dfp .
The functional derivative involves the application of the chain rule
with integration over the intermediate argument. We start by rewriting
(A1) in terms of the transmittance t̃ (z, e) instead of its derivative:
516
CONRATH, GIERASCH, AND USTINOV
y
I(e) 5 B(T(z))t̃ (z, e)u2y
2
E
5 B(T(y)) 2
y
2y
E
y
2y
t̃ (z, e) dB(T(z))
ACKNOWLEDGMENTS
(A6)
t̃ (z, e) dB(T(z)).
Taking the functional derivative of this expression with respect to the
absorption coefficient k(z) we have
dI(e)
52
dk(z)
E
y
2y
dt̃ (z9, e)
dB(T(z9)).
dk(z)
This work has been supported by the NASA Planetary Atmospheres
Program. E.A.U. was a National Research Council Research Associate
at NASA Goddard Space Flight Center during a portion of this work.
The authors are grateful for comments and suggestions from A. Weir
and two anonymous reviewers.
(A7)
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From (A2) we have
dt̃ (z9, e)
dt (z9)
1
5 2 t̃ (z9, e)
.
dk(z)
e
dk(z)
(A8)
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dI(e)
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52
dk(z)
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E
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(A11)
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U
dI(e) dI(e) dk
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,
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(A12)
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