JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 1–14, doi:10.1029/2012JA017962, 2013
Lunar semidiurnal tide in the thermosphere under solar
minimum conditions
Jeffrey M. Forbes,1 Xiaoli Zhang,1 Sean Bruinsma,2 and Jens Oberheide3
Received 17 May 2012; revised 27 October 2012; accepted 29 November 2012.
[1] Renewed interest in lunar tidal influences on the ionosphere-thermosphere (IT) system
has emerged in connection with recent studies of possible connections between
stratospheric warmings and enhanced lunar tidal perturbations of the equatorial ionosphere.
By virtue of its gravitational force, the Moon produces perturbations in the temperature,
density, pressure, and winds throughout Earth’s atmosphere. Lunar tidal winds in the
dynamo region (~100–150 km) can furthermore generate electric fields that map into the
F-region and redistribute ionospheric plasma. Direct penetration (propagation) of lunar
tides to F-region heights can also transport ionospheric plasma. Decades-long satellite data
sets now exist that can provide a global perspective on lunar tidal oscillations, but this
resource has not yet been exploited for this purpose. In this paper, we examine the global
structure of the main M2 (period = 12.42 h) lunar tide through examination of temperatures
measured by the Thermosphere Ionosphere Mesosphere Energetics and Dynamics SABER
instrument at 110 km and densities at 360 and 480 km inferred from accelerometers on the
CHAMP and Gravity Recovery and Climate Experiment satellites, respectively. Ten year
mean SABER M2 temperature amplitudes are of order 5–10 K while the corresponding
density perturbations during the 2007–2010 solar minimum period approach amplitudes of
order 5% at 360 km and 10% at 480 km. The observed amplitudes are large enough to
impose non-negligible day-to-day variability on the IT system. Global-Scale Wave Model
simulations provide a theoretical and modeling context for interpreting these data, and
moreover enable estimates of E- and F-region winds.
Citation: Forbes, J. M., X. Zhang, S. Bruinsma, and J. Oberheide (2013), Lunar semidiurnal tide in the thermosphere
under solar minimum conditions, J. Geophys. Res. Space Physics, 118, doi:10.1029/2012JA017962.
accumulated multiple decades of observations. Modeling
work by Lindzen and Hong [1974] clarified the role of zonal
mean winds and temperatures in controlling the atmosphere’s
response to tidal forcing, while Forbes [1982a, 1982b]
emphasized the propagation of tidal perturbations into the
thermosphere. Stening et al. [1997] later explored the
consequences of anomalous background atmospheric conditions (e.g., stratospheric warmings) on lunar tide propagation.
[3] Study of the lunar atmospheric tide is fundamentally
interesting, because the lunar forcing is known reasonably
well, and thus comparisons between numerical simulations
and data provide important insights into the veracity of
atmospheric models. Up until now, lunar tides were mainly
derived from ground-based observations because long data
records were required. Restriction to ground-based observations thus precluded truly global perspectives of lunar tidal
effects in the atmosphere. However, there are now decadelong observations from satellites that in principle can
provide such a global perspective. In many cases, these
observations are also available at high altitudes where the
lunar tidal variability in the more tenuous upper atmosphere
might be more easily retrieved. This paper is motivated by the
availability of such data, as well as the theoretical knowledge
and numerical modeling capabilities that have accumulated
over the past several decades.
1. Introduction
[2] Studies of the effects of the gravitational forcing of the
Moon on the solid Earth, oceans, and atmosphere have a
long history. In this paper we are interested in the effects
of gravitational forcing throughout the atmosphere. Classic
reviews on observational studies of atmospheric tides
include those by Chapman and Lindzen [1970] for lunar tide
signals in surface pressure, and Matsushita’s [1967a, 1967b]
reviews of lunar geomagnetic tides and tidal variations in the
F-region ionosphere. Additional works [e.g., Stening et al.,
1994, and references therein] examined wind measurements
at altitudes (~90–100 km) where meteor and MF radars have
1
Department of Aerospace Engineering Sciences, University of
Colorado, Campus Box 429, Boulder, Colorado, 80309-0429.
2
Department of Terrestrial and Planetary Geodesy, Centre National
d’Etudes Spatiales, 18, Avenue E. Belin31401, Toulouse, France.
3
Department of Physics and Astronomy, Clemson University, 118
Kinard Laboratory, Clemson, South Carolina, USA.
Corresponding author: J. M. Forbes, Department of Aerospace
Engineering Sciences, UCB 429, University of Colorado, Boulder, CO
80309-0429 USA. ([email protected])
©2012. American Geophysical Union. All Rights Reserved.
2169-9380/13/2012JA017962
1
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
TIMED. In this study CHAMP densities are normalized to
360 km and GRACE densities are normalized to 480 km
using the NRLMSISE00 model [Picone et al., 2002]. Because
the model is independent of lunar time and residual densities
are used, lunar tidal signals cannot be artificially introduced
by this normalization process. Due to the obvious solar cycle
influence at these altitudes, we use CHAMP and GRACE
density data from 2007 to 2010 which is considered a solar
minimum period.
[4] In this paper we seek to characterize and understand
the seasonal-latitudinal structure of the lunar tide at several
altitudes in the thermosphere. In the following section we
describe data that will employ in this study from the TIMED
(Thermosphere Ionosphere Mesosphere Energetics and
Dynamics), CHAMP (CHAllenging Minisatellite Payload),
and GRACE (Gravity Recovery and Climate Experiment)
missions, and outline the methodologies that we use to isolate
the lunar tidal signal. In section 3 we present the monthlymean lunar tide in the neutral temperature field between
50 latitude from TIMED-SABER (Sounding of the Atmosphere using Broadband Emission Radiometry) observations
in the 80–110 km height region and use a linear tidal model
(The Global-Scale Wave Model or GSWM) with realistic tidal
forcing and background atmospheric conditions to interpret
the salient features of the observed temperature response.
We furthermore employ GSWM predictions of lunar tidal
perturbation densities to interpret those derived near 360 and
480 km from the CHAMP and GRACE satellites, respectively, and to assess our ability to model the lunar tidal
response in the upper thermosphere.
2.2. Method of Analysis
[7] The semidiurnal lunar tidal forcing (M2) is the most
significant of the tides that arise due to the gravitational
potential between the Earth and the Moon [e.g., Chapman
and Lindzen, 1970; Pugh, 1987]. Under the conditions that
the Moon’s orbit around the Earth is a perfect circle and in
the same plane as Earth’s equator, then M2 would solely
comprise the total lunar potential. But since the Moon’s orbit
has a small eccentricity and an angle with respect to Earth’s
equatorial plane, M2 is not the only lunar-related periodicity.
The gravitational potential between the Earth and the Sun
represents a similar situation, and so the two pairs of gravitational potentials produce many sinusoidal forcing periodicities.
The most significant ones are (in cm2s 2):
2. Data, Model, and Method of Analysis
2.1. SABER, CHAMP, and GRACE Data Sets
[5] Satellites provide exceptional spatial coverage for the
study of global waves such as solar and gravitational tides.
However, the slow precession in local time, which usually
translates to months for a high-inclination satellite to cover
a 24 h local time cycle, does introduce problems in terms
of aliasing with trends in background mean conditions
[Forbes et al., 1997], and temporal averaging of evolving
tidal amplitudes [Forbes et al., 2008]. On the other hand,
local time precession of a satellite-based measurement has
an advantage in terms of distinguishing tides with very close
oscillation periods, e.g., the 12.0 h solar semidiurnal tide and
the 12.42 h lunar semidiurnal tide, whereas can be problematic in the analyses of measurements made from the ground.
This aspect of lunar tidal analysis of data from space-based
platforms is discussed further in section 2.2.
[6] The basic data to be employed in this study are from
three satellites/instruments: TIMED/SABER, CHAMP/STAR
and GRACE/SuperSTAR. The SABER Version 07 temperature measurements analyzed here cover March 2002 through
March 2011. For this version of SABER temperatures, each
vertical profile begins with a climatological value from
NCAR’s Thermosphere Ionosphere Mesosphere-General
Circulation Model and a retrieval is performed from 140 km
downward. The retrieved temperatures begin to become independent of the climatology at about 110 km [Russell, 2012],
and this serves as a nominal upper limit for validity of the
SABER temperatures (see also Mertens et al. [2001];
Remsberg et al. [2008]). Because we consider SABER data
equatorward of 50 latitude where sampling is unaffected
by yaw maneuvers, and relatively few data gaps exist, we have
almost continuous (but asynoptic) coverage in UT and
longitude. The CHAMP and GRACE total density data are
introduced in Forbes et al. [2009, 2011] and the reader is
referred there for relevant information on these data sets
including additional references. With their higher orbit inclinations, CHAMP and GRACE measurements extend nearly pole
to pole and have even longer local time precession rates than
(quasi-)diurnal lunar (25.819 h)
O1 ¼ 6585:P21 ðθÞsin sLr sLR t þ f
(quasi-)diurnal solar (24.066 h)
P1 ¼ 3067:P21 ðθÞsin sSr sSR t þ f
diurnal luni-solar (23.934 h)
K1 ¼ þ9268:P21 ðθÞ sin½s0 t þ f
large lunar elliptic semidiurnal (12.658 h)
N2 ¼ 1518:P22 ðθÞ cos 2sLr m t þ 2f
semidiurnal lunar (12.421 h)
M2 ¼ 7933:P22 ðθÞ cos 2 sLr t þ f
semidiurnal solar (12.00 h)
S2 ¼ 3700:P22 ðθÞ cos 2 sSr t þ f
semidiurnal luni-solar (11.967 h)
K2 ¼ 1005:P22 ðθÞ cos½2ðs0 t þ fÞ
where s0 = 2p/(sidereal day), sLR ¼ 2p=ðsidereal lunar monthÞ,
sSR ¼ 2p=ðsidereal yearÞ . The angular speed of the Earth’s
self-rotation with respect to the Moon is sLr ¼ s0 sLR , and
with respect to the Sun is sSr ¼ s0 sSR . In the Sun-EarthMoon system, the Moon’s elliptic angular speed is m = 2p/
(anomalistic month). The associated Legendre Polynomials
are P21 ðθÞ ¼ 1:5 sin2θ, which peaks at 45 latitudes, and
P22 ðθÞ ¼ 3 sin2 θ, which peaks at the equator with θ here being
colatitude. Here t is UT and f is longitude. These were
calculated by Siebert [1961] based on the work of Doodson
2
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
perspective is from satellites. Let the subscript e denote
from the Earth and subscript s denote from the satellite
perspective. Subtracting these two cases applied to sL = sS C, one gets
sLs ¼ sSs þ sLe sSe
(1)
where sLe is one of the gravitational modes’ diurnal frequencies listed in the prior subsection such as sO1 ¼ sLr sLR ,
sP1 ¼ sSr sSR , sK1 ¼ sK2 =2 ¼ s0 , sN2 =2 ¼ sLr m=2 or
sM2 =2 ¼ sLr ; sSs is the satellite precession rate and sSe is simply 1 cycle/day. As long as sLe 6¼ sSe = 1 cycle/day, we can
get the sLs for a gravitational tide of interest without aliasing
with sSs . That is, the difference between the periods of two
gravitational modes from the ground-based point of view
can be magnified if the two modes are viewed from a satellite. For example, a M2 lunar day ( 1=sLs ) from either the
ascending or descending leg of TIMED, CHAMP or GRACE
is 23.7, 26.6, and 27.1 days, respectively, which are well apart
from the corresponding solar day lengths (1=sSs ) of 120, 261,
and 324 days, respectively, although sLe and sSe are very close
(1.035 day vs 1.000 day period as aforementioned) from the
perspective of a ground-based station on the Earth. The calculation is based on equation (1) above where in most cases sSs
has a negative value, which reflects a satellite precessing in
the opposite direction that the Earth rotates with respect to
the Sun. For instance for the TIMED satellite, 1=sLs ¼
1=ð1=120: þ 1=1:035 1Þ ¼ 23:7 days; for CHAMP,
1=sLs ¼ 1=ð1=261: þ 1=1:035 1Þ ¼ 26:6 days; and
for GRACE, 1=sLs ¼ 1=ð1=324: þ 1=1:035 1Þ ¼ 27:1
days. The negative sign in any of these results indicates that
the lunar hour decreases as UT goes forward. The M2 semidiurnal period is thus 11.86, 13.28, or 13.56 from either the
ascending or descending leg of the TIMED, CHAMP or
GRACE satellite orbit, respectively, since the semidiurnal
period is half the diurnal period of the same mode. As another
example, an N2 lunar day is 1.055 (based on the N2 frequency
sLr m=2) solar days from a ground-based point of view, so its
semidiurnal period is 1/(1/120. + 1/1.055 1)/2 = 8.23 days,
1/(1/261. + 1/1.055 1)/2 = 8.93 days, or 1/(1/324. + 1/
1.055 1)/2 = 9.05 days, respectively, from the perspective
of the TIMED, CHAMP, or GRACE satellite. Note, although a
M2 lunar day or a N2 lunar day is mentioned in the above
calculation, the diurnal mode of neither exists. As a brief
summary, by taking advantage of slow satellite precession
rates (including zero), the common aliasing problem between
gravitational tides and solar thermal tides can be avoided to a
significant degree.
[10] We restrict our attention here to the largest gravitational
tide excited in the atmosphere, the migrating (longitudeindependent) M2 component. For the climatological study at
hand, we use a binning and averaging method that yields
composite lunar half-day variations of temperature or density
from which the semidiurnal component can be derived. It is
analogous to the method employed by Forbes et al. [2008]
to determine solar tides from TIMED-SABER temperature
measurements. In that study 60 days of data were required to
form hourly-mean values during a composite 24 h solar day
(including both ascending and descending parts of the orbit),
thus enabling extraction of both diurnal and semidiurnal tides.
The tidal analysis was performed on residuals from 60 day
running mean temperatures in order to minimize aliasing due
Figure 1. This figure illustrates the relation between solar
local time (t), lunar local time t, and the lunar phase angle n
for a point P on the Earth. The circle represents the plane of
the Earth’s equator with the center being the Earth’s center.
[1922] and Bartels [1957], and revisited by Chapman and
Lindzen [1970].
[8] For investigations of gravitational tides, methods and techniques are particularly important, since extracting the M2 lunar
tide from surface observations historically suffered from aliasing
with the solar semidiurnal tide. That is, if significant local time
(or other) variations occur within the time interval that is being
fitted for the lunar tide, then these variations can alias into the
lunar tide determination, possibly rendering it invalid. This is
compounded by the fact that the lunar tide is typically so much
smaller than solar tides. From the ground, one cannot separate
a 12.0 h oscillation from a 12.4 h oscillation without sufficiently
abundant and frequent sampling, and data accuracy. However, it
is well-known that from the perspective of a satellite platform the
wave periods associated with the lunar and solar semidiurnal
tides can be widely different (e.g., Ray and Luthcke [2006]; their
equation (1) and Table 1 and example for GRACE). In the
following, we describe how we use this advantage along with
abundant sampling and detrending to characterize a relatively
small semidiurnal tide in the upper atmosphere.
[9] Figure 1 illustrates the relation between solar local time
t, lunar local time t for a point P on the Earth (observed by a
ground-based station or by a satellite) and the lunar phase angle n, after Sugiura and Fanselau [1966] and Chapman and
Lindzen [1970]; that is, t = t n. Because n, the lunar phase,
is almost a linear function of universal time, the lunar phase
speed is a constant for each sinusoidal mode; let it be denoted
by C. Then sL = sS C , where sL and sS denote the Earth’s
self-rotation frequency (corresponding to the ‘sr’s in the last
paragraph when gravitational forcing is discussed from
the ground-based perspective) with respect to the Moon
(L) or to the Sun (S). From the perspective of a groundbased station, sS = 1 cycle/day. For M2, C = 1 cycle/29.53
days, so 1/sL = 1.035 days or 24.84 h, which is confirmed as
a lunar day or two times the M2 period. Another practical
3
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
response: surface friction, mean winds and meridional gradients in scalar atmospheric parameters, radiative cooling, eddy
and molecular diffusion, Rayleigh friction and ion drag. All
parameterizations and properties of the background atmosphere are described for GSWM-02 in Hagan and Forbes
[2002]. In GSWM-09 [Zhang et al., 2010], solar thermal forcings are updated with ISCCP (International Satellite Cloud
Climatology Project) radiative heating and TRMM (Tropical
Rainfall Measuring Mission) latent heating. New specifications of the mean zonal wind field derived from SABER geopotential data and the corresponding mean temperatures are
also features of GSWM-09. In this study, we utilize GSWM09 and obtain the GSWM lunar tidal results by introducing
the M2 migrating lunar forcing in a manner identical to that described in Forbes [1982a, 1982b] and turning off the solar thermal forcing. All simulations shown here correspond to solar
minimum conditions (F10.7 = 70).
[13] A much less effective source of lunar tidal forcing on
the atmosphere occurs as the result of vertical movements of
the solid Earth and oceans, serving as a dynamic boundary
condition for the tidal equations [Vial and Forbes, 1994].
While the lunar gravitational potential is known with reasonable precision, there are assumptions and uncertainties
associated with specification of Earth and ocean tide effects
on the atmosphere. The main motivation for including these
effects in the GSWM would be to examine the longitudedependent lunar tides in the atmosphere, but since the focus
of the present work is on the longitude-independent M2 tide,
Earth and ocean tidal forcing is not included in the GSWM
simulations presented here. There is an M2 component that
arises in connection with Earth and ocean tidal forcing, but
its influence on the atmospheric response is not completely
straightforward to estimate. Based on discussion of this
issue in Vial and Forbes [1994], GSWM lunar tidal
responses in following sections could underestimate the
total M2 response by up to 30%, and may reflect some small
phase effects as well. This level of uncertainty does not
appreciably impact our use of the GSWM to diagnose the
seasonal-latitudinal aspects of tidal responses presented in
the following, as these arise solely as a result of changes
in the background atmosphere.
to variations in the background (zonal-mean) temperature
(see also Forbes et al. [1997]). Similarly, the lunar tide is
superimposed on a background that evolves due to changes
in the zonal mean and local solar time. In the present application each lunar half-day encompasses 11.85, 13.26, and
13.55 days’ worth of measurements from the TIMED,
CHAMP and GRACE satellites, respectively, during which
the local time changes by about 2.37, 1.22, and 1.00 h. Our
procedure for the lunar tide analysis of SABER data similarly
begins with forming residuals, except in this case it is from the
12 day running-mean background temperature centered on
that day, at any given height and latitude. Detrending the data
this way helps to remove variations that could otherwise
project onto the semidiurnal tide within the fitting interval.
At each height and latitude we then bin 15 orbits per day of
bi-monthly SABER temperature residuals over 9 years into
twelve 1 h lunar local time bins centered at the 15th of each
month. Least-squares sinusoidal fits are then performed to
extract the semidiurnal lunar tidal signal for that
month (see sections 3.1 and 3.2 for examples). Note that over
9 years we effectively average about 15 9 5 = 675 data
points (minus some missing points due to data gaps) in each
1 h bin, which greatly reduces random errors and averages
out other variations not ordered in lunar time. Our attempts
to extract the N2 lunar component yielded amplitudes much
smaller than M2, so aliasing contributions from N2 are
considered to be small. The CHAMP and GRACE data
were analyzed using the same method, except that density
residuals from 14 day means were binned over the 4 year
2007–2010 period.
[11] Apart from aliasing with the solar tide which is
addressed above, it is not unreasonable to assume that the
lunar atmospheric tide might have a local time dependence,
due for instance to day-night differences in propagation conditions. With the SABER binning described above, the five
12 day segments comprising each bimonthly period average
out measurements from five evenly spaced different solar
local times for each of the 12 lunar local hours. With
CHAMP or GRACE, four 14 day segments for a given
bimonthly period in each year from 2007–2010 average
out four evenly spaced different solar local times. In both
cases, the four to five evenly spaced local times cover
roughly a solar local time cycle. Therefore, the climatological results to be presented below represent local-time
averaged lunar tides.
3. Results
3.1. SABER and GSWM Lunar Tide Temperatures in
the Lower Thermosphere
[14] Figure 2 illustrates hourly and longitude means of the
temperature residuals defined in section 2.1, for the
bimonthly period centered on 15 January, and plotted vs. lunar
local time between 0 and 12 h. Also shown are 12 h sinusoidal
least squares fits to these data. We use 12 h fitting intervals
(asopposed to a full lunar day) in order to double the data
available for averaging. Each fitting interval is 12 days in
length, so every bimonthly average includes five such intervals per year of observations, for a total of 45 intervals over
the TIMED-SABER data set. In addition, the atmosphere is
sampled on both ascending and descending portions of
the orbit at about 15 longitudes per day. This accounts for the
acceptably small (~1 K) standard deviations illustrated in
Figure 2, in comparison with the derived wave amplitudes.
A prominent feature of the results in Figure 2 is the asymmetric
phase relationship between the lunar tide north and south of the
2.3. Global Scale Wave Model
[12] The Global-Scale Wave Model [Hagan et al., 1995,
1999; Hagan, 1996; Hagan and Forbes, 2002, 2003] is the
model adopted for the present study. The GSWM solves
the linearized tidal equations; given the frequency, zonal
wave number and excitation of a particular oscillation, and
given a specification of the zonally-averaged atmospheric
state, the height versus latitude distribution of the atmospheric response is calculated. The linear approximation is
not considered to be a shortcoming of any significance in calculating the wave response to any given forcing. However,
the linear approximation precludes excitation of some tidal
oscillations by wave-wave interactions. The model includes
in some form or another all other processes of known importance to the calculation of the global atmospheric tidal
4
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 2. Zonal-mean lunar semidiurnal M2 fits to multi-year binned SABER residual temperatures for
January at altitudes of 90 km (left) and 110 km (right) and selected latitudes. The vertical bars are the standard deviations of the data within each hourly bin.
nonequinox periods there is generally one maximum at
low to middle latitudes on each side of the equator. During
Northern Hemisphere summer SABER maxima of about
5 K (2 K) occur near + 20 ( 25 ) latitude as compared with
the GSWM maxima, which are 12 K (7 K) and occur near
+ 25 ( 10 ). During Northern Hemisphere winter, SABER
maxima of 4–6 K occur at 20 latitude whereas the
GSWM maxima are of order 7–8 K near 30 latitude and
4 K near + 40 latitude.
[16] One notes that the scatter in the data as well as the
standard deviations of individual data points in Figure 2 vary
with latitude and height. There is also some variability from
equator. This may seem surprising given that the M2 gravitational forcing is symmetric about the equator, but the reason
is well known [Lindzen and Hong, 1974; Forbes, 1982] and
is explained below.
[15] The seasonal-latitudinal structures of lunar tidal temperature amplitudes from SABER observations and GSWM
simulations that complement those in Figure 2 at 110 km are
illustrated in Figures 3a and 3b. There are some common
features between the observations and the model, namely
the occurrence of minima around April and October, and
in the vicinity of the equator during the November–March
and May–September periods. Furthermore, during these
5
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 3. M2 temperature amplitudes and phases from multi-year-mean SABER data and GSWM simulations. (a) SABER amplitudes versus latitude and month at 110 km. (b) GSWM amplitudes versus latitude and month at 110 km. Panels (c)-(f) are height vs. latitude depictions of amplitude for SABER and
GSWM for January and July, respectively, and panels (g)-(j) are the corresponding phases. Note the different amplitude scales for panels (a) and (b) and for panels (e) and (f).
are smaller. Amplitude (phase) uncertainties at 100, 90,
and 80 km, respectively, are of order 0.25 K (0.3 h), 0.15–
0.20 K (0.7–1.2 h), and 0.10–0.14 (0.7–1.8 h), but these
must be weighed against the progressively smaller amplitudes at lower altitudes. For a normal distribution, 1s
bounds 68% of the points about the mean value. Therefore, the data-model comparisons provided above must
be tempered by these uncertainties which naturally arise
month to month (not shown). If we take the displayed
standard deviations (s) as a measure of “uncertainty”, then
the corresponding 1s uncertainties in the amplitude and
phase of the sinusoidal fit can be calculated using standard
formulas. Doing this, we find typical 1s values at 110 km
of 0.3–0.4 K in amplitude and 0.3 h in phase where
amplitudes exceed 0.8 K in Figure 3a, and 1s phase uncertainties up to 1.5 h at the higher latitudes where amplitudes
6
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 4. Top panel (a) shows the first four M2 Hough functions normalized to a maximum value of
unity: (2,2) (solid), (2,3) (dotted), (2,4) (dashed), and (2,5) (dashed-dot). Panels (b)-(d) show the vertical
profiles of amplitudes from Hough function decomposition of the GSWM simulations for January, July
and October, and panels (e)-(g) are the corresponding phases.
phases are provided in Figures 3g–3j. The January SABER
and GSWM amplitudes (Figures 3c and 3d) depict the exponential growth with height that is expected for verticallypropagating waves in the atmosphere. This growth and the
overall amplitudes for GSWM are quite similar to those of
SABER in the Southern Hemisphere. The Phase structures
(Figures 3g and 3h) are also similar in the Southern
Hemisphere, except that the contours are more compressed
for SABER. Note that phases progress to later
times as one moves downward in altitude (“downward phase
progression”), consistent with upward group velocity and a
in the extraction of a relatively small signal from an
extended data set with other geophysical variability. There
are also uncertainties in the model specification, which are
discussed later on in this paper.
[17] Comparing SABER and GSWM seasonal-latitudinal
amplitude structures at a single altitude provides only a
narrow perspective on the lunar tidal responses reflected
in both the observations and model. Consequently, we
compare the height vs. latitude structures of SABER and
GSWM temperature amplitudes for January in Figures 3c
and 3d and for July in Figures 3e and 3f. The corresponding
7
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 5. Zonal-mean background zonal winds (top three panels) and temperature (bottom three panels)
used by GSWM-09 for January (left), July (middle) and October (right).
but apart from that the phase structures share a lot of
similarity.
[18] The salient seasonal-latitudinal and vertical structure
features noted above, and the relationships between them,
can be explained in the context of linear tidal theory (see
Chapman and Lindzen [1970] for details). First, consider
the orthogonal Hough modes (Yn,s(θ)) of classical tidal
theory, solutions of Laplace’s Tidal Equation, where θ is
latitude or colatitude, n = 2 denotes a tidal frequency that is
twice Earth’s rotation frequency, and s is a latitudinal index.
A few Hough modes [usually denoted (n,s)] are depicted in
Figure 4 for the lunar semidiurnal tide [Flattery, 1967].
These include the first and second symmetric modes (2,2)
and (2,4), and the first two antisymmetric modes (2,3) and
(2,5). Note that the (2,2) mode is very similar in latitudinal
shape to that of the lunar M2 geopotential. In practice the
semidiurnal temperature field dT2(θ) at any given altitude
can usually be closely approximated by the sum of these
four functions:
wave source at lower altitudes according to atmospheric
wave theory. Thus, due to the different phase gradients with
height, while phase agreement is good near 80 km, by
110 km there is a 3 h discrepancy in phase between GSWM
and SABER in the Southern Hemisphere. In the Northern
Hemisphere, exponential growth is clearly slower during
January for GSWM as compared with SABER, and in fact
the SABER tidal structures are quite similar between hemispheres except between 80–90 km where the amplitudes
are small in any case. Focusing on the Northern Hemisphere
phases during January, we see that at +20 latitude that
GSWM undergoes a phase shift of about 5 h, from 6.0 h to
about –1.0 h from 110 to 80 km. The SABER phases at
+20 increase from about 1.0 h to + 6.0(or 6.0)h from
110 to 95 km, and continue to increase to 3.0 h at 80 km,
for a total phase shift of 8 h for SABER. Again, the SABER
data suggest presence of a shorter vertical wavelength wave
than the GSWM. Note that there is a phase shift of about
6.0 h between hemispheres at middle latitudes for SABER
(cf. Figure 2), whereas this phase difference is of order 3.0 h
for GSWM. In July, the SABER amplitudes (Figure 3e)
achieve much smaller maximum amplitudes (3–5 K) as
compared to GSWM (7–12 K, Figure 3f) in the 80–
110 km regime, and peak at a lower altitude. However,
the relative amplitudes between Northern and Southern
Hemispheres and the placement of the maxima in each
hemisphere are about the same for SABER and GSWM,
although SABER amplitudes peak in the 105–110 km altitude
regime compared to about 120 km for GSWM. If one
views the SABER (Figure 3i) and GSWM (Figure 3j)
phases diagonally from 110 km altitude and 50 latitude
to 80 km and +50 latitude, one can see a slight compression of the phase contours in SABER relative to GSWM,
dT2 ðθÞ ¼ AðθÞ cos½2Ωt fðθÞ
4
X
¼
Y2;s ðθÞ a2;s cos2Ωt þ b2;s sin2Ωt
s¼1
(2)
4
X
¼
A2;s Y2;s ðθÞ cos 2Ωt f2;s
s¼1
where the a2,s, b2,s, A2,s, f2,s are single constants for
each latitude-dependent Yn,s(θ). In fact, the structures in
Figures 3a and 3b can actually be well approximated by only
the sum of (2,2) and (2,3) modes (not shown). This makes
it simple to understand the origins of the latitudinal structures
in Figure 3, and the similarities and differences between them.
8
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 6. GSWM M2 tidal amplitudes of zonal wind (U, top left), meridional wind (V, top right), relative density (Δ r/r, bottom left) in percent, and temperature (T, bottom right) in January.
For instance, note that the (2,3) mode maximizes at about
25 latitude with one hemisphere being out of phase with
the other, and at these same latitudes the (2,2) mode is about
0.70 of its equatorial maximum value and is in phase
between hemispheres. Furthermore, each mode is multiplied
by a complex constant that determines its amplitude and
phase. One can easily imagine that, depending on the relative
amplitudes and phases of these complex constants, that a
variety of latitudinal structures are possible, each one with
maxima in the vicinity of 25 latitude, but covering a range
of relative amplitudes and phases between hemispheres. For
instance, the near-equal maxima in SABER amplitudes at
110 km in December–January and the more asymmetric amplitude structures in June–August (see Figure 3a), and the
asymmetric structures for GSWM during both solstices in
Figure 3b, can all be realized by combining the (2,2) and
(2,3) modes in this way. (Note however that the existence
of a Southern Hemisphere maximum closer to 10 latitude
for GSWM during July is due to presence of the (2,4) mode
with non-negligible amplitudes.)
[19] Understanding the vertical structures and SABERGSWM differences in Figure 3 requires some appreciation
for the effects of the background atmosphere through which
the lunar tide propagates between the lower atmosphere and
the thermosphere, which in turn translates to the relative
magnitudes and phases of the (2,2) and (2,3) modes. The
GSWM represents a good vehicle for demonstrating these
concepts. Consider the zonal-mean zonal winds and temperatures utilized by the GSWM in this study for January,
July, and October in Figure 5, and the corresponding Hough
mode decompositions of the lunar tidal temperature field
provided in Figure 4. For January, the middle atmosphere
winds between 20 and 90 km are characterized by strong
asymmetry with jet maxima of order +45 ms–1 and –60 ms 1
in the Northern and Southern Hemispheres, respectively. In
Figure 4, the Hough mode decomposition contains a strong
(2,3) antisymmetric component in addition to the symmetric
(2,2) and (2,4) modes. It is this (2,3) component that accounts
for the significant asymmetry during November-March in the
GSWM response at 110 km in Figure 3. The (2,3) mode
arises as a result of the latitudinal distortion that the asymmetric mean winds produce in the tidal response; in the context
of linear tidal theory this distortion is accommodated by
exciting the (2,3) mode as a result of “mode coupling”
[Lindzen and Hong, 1974]. Once this tidal mode is generated,
it propagates freely into the thermosphere as an independent
oscillation, carrying the “signature” of the middle atmosphere zonal jets to much higher altitudes. During July the
(2,3) mode is even larger and the (2,2) mode is reduced, consistent with the even greater asymmetry in zonal mean winds
depicted in Figure 5. This is reflected in the more distinct
two-peaked structure during May–July at 110 km, with low
amplitudes at the equator. During October the zonal mean
winds are much less asymmetric, with eastward winds jets
in both hemispheres. In this case the (2,2) mode is dominant
and little asymmetry is seen in the total tidal temperatures
around the equinoxes (Figures 3a and 3b). Also, the overall
amplitudes are smaller during equinox periods, consistent
9
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 7. Zonal-mean M2 fits to multi-year binned CHAMP residual data for July at 360 km and selcted
latitudes. The vertical bars are the standard deviations of the data within each hourly bin.
zonal mean winds and temperatures are not seasonally symmetric, many of the salient features are, although they yield
much different combinations of tidal modes (cf. Figures 3b
an 3c). One can therefore appreciate the difficulty of obtaining more exact agreement between theory and observation
in Figure 3, despite that fact that the lunar geopotential forcing is so well known.
[21] There are a few reasons why better agreement may
not be obtainable between the GSWM and SABER depictions of the lunar atmospheric tide. First, the SABER results
correspond to a 10 year mean, and phase variability between
half lunar day intervals within each bi-monthly period or
from year to year can produce interference effects that tend
to diminish the amplitude of the observed tide, and which
are not necessarily uniform with respect to latitude or month.
These phase differences can arise due to the lunar tide’s
sensitivity to the middle atmosphere zonal mean winds, a
feature which we have already noted exists in the GSWM
simulations, and which we presume carries over to the actual
atmosphere. Second, the background temperature field
between 20 and 100 km in the GSWM is constructed from
with the seasonal-latitudinal patterns depicted in Figures 3a
and 3b.
[20] The latitude structures and vertical structures displayed
in Figure 3 are in fact coupled to each other. This is
because the (2,2), (2,3), and (2,4) modes each possess their
own characteristic vertical wavelengths, which are roughly of
order 100–200 km, 45–70 km, and 30–45 km, respectively.
Because these are dependent on both temperature
and vertical temperature gradient, their local values can vary
considerably in the mesosphere-lower thermosphere height
region between 60–140 km, as depicted in Figures 3e–3g.
In particular, the theoretical vertical wavenumber for the
(2,2) mode becomes imaginary between about 60–90 km
where the mean temperature decreases with height. This
retards growth of the (2,2) mode with height, thus facilitating dominance of the (2,3) mode. It is clear that since the
relative amplitudes and phases of the tidal modes vary with
height, the aggregate vertical structures of amplitude and
phase will vary with latitude, or equivalently, the horizontal
structures will vary with height. As noted previously in
connection with Figure 5, although the January and July
10
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 8. Similar to Figure 7 except for GRACE data at 480 km.
drive observable changes in the F-region ionosphere.
The density perturbations are only 1–2.5% at 200 km,
however, which suggests that extraction of the lunar tidal signal
from, e.g., CHAMP and GRACE accelerometer-based
measurements, might be difficult. This concern turns out to be
unwarranted, however, as we show in the following section.
a 10 year mean climatology of SABER temperatures, and
the winds are calculated from the same temperatures using
the thermal gradient wind relationship. Inherent in this type
of averaging is a smoothing of vertical and horizontal and
wind gradients that are known to affect tidal propagation
[cf. Mclandress, 2002]. Shortcomings in the temperature
and wind specification are in turn manifested in the mixture
of tidal modes, which as we have seen controls the aggregate
horizontal and vertical structures of amplitudes and phases.
[22] The horizontal and vertical wind components and
perturbation density are additional outputs from the GSWM,
which extends from the surface to 400 km altitude. Figure 6
illustrates the lunar tidal zonal and meridional wind amplitudes,
percent relative density perturbation and temperature amplitudes, as a function of height and latitude for January. For all
parameters, the maxima occur between about 110 and 120 km
and on both sides of the equator, and within 30 for density
and temperature and near 50–60 latitude for the horizontal
winds. The zonal wind amplitudes are of order 10–15 ms 1
within the 100–150 km dynamo region, which compares with
20–30 ms 1 for the upward-propagating solar semidiurnal tide
computed by the GSWM under the same conditions. The lunar
tide is thus capable of producing dynamo electric fields that
3.2. CHAMP, GRACE, and GSWM Lunar Tide
Densities in the Upper Thermosphere
[23] As noted previously, the CHAMP and GRACE data
at 360 and 480 km, respectively, are analyzed using the same
method as for the SABER data, except residuals from 14 day
running means are used. Since vertically-propagating tides
penetrate into the upper thermosphere with larger amplitudes
during solar minimum vs. solar maximum (see Oberheide
et al. [2011], for some recent examples), we consider only
the 2007–2010 solar minimum period for our analysis here.
[24] Examples of fits to the CHAMP and GRACE density
residuals are depicted in Figures 7 and 8, respectively for the
July results. Note that the amplitudes are typically 2–5% for
CHAMP and 5–10% for GRACE, and are sufficiently large
compared to scatter and standard deviations that some
confidence is warranted, especially in the case of GRACE.
11
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Figure 9. GRACE (top panels) and CHAMP (middle panels) M2 amplitudes (left panels) and phases
(right panels) of relative density compared with GSWM (bottom panels). The GRACE data correspond
to an altitude of 480 km, and the CHAMP and GSWM results correspond to 360 km.
and around 0 to 50 and + 40 to + 60 , respectively, for
CHAMP. The GSWM maxima are generally in the 1.5–
2.5% range, whereas the CHAMP maxima are about 3.0–
3.5%. However, while the phase of the CHAMP density perturbation during July decreases from 1.2–2.4 h in the Southern
Hemisphere to 0–1.2 h in the Northern Hemisphere, the
corresponding GSWM phase changes are more dramatic,
from about 3.6–4.8 h to 1.2–2.4 h from Southern Hemisphere to Northern Hemisphere. If one considers the single
CHAMP and GSWM maxima during the November to
March period, there is a 1–2 h net difference in phase. These
results are consistent with the Hough decomposition in
Figure 4, which demonstrates a significant predominance
of the (2,3) mode over the (2,2) mode during July, whereas
these waves are more nearly equal in magnitude during
January. These results tell us that the (2,3) mode generation
and (2,2) mode suppression during Northern Hemisphere
summer months may be too great in the GSWM, which
in turn informs us that there may be shortcomings in the
specification of middle atmosphere winds during this period.
Note also that the phases differ by about 3 h (or 9 h) between
the two altitudes. It seems likely that GRACE leading
CHAMP by 3 h is the correct solution, as this would be consistent with downward phase progression for an upwardpropagating wave. Although the phases of GSWM lunar tidal
temperatures and horizontal winds at midlatitudes are roughly
constant with height (not shown) above 300 km due to the
effects of molecular diffusion, phases of perturbation densities
are decreasing at the rate of about 1.2 h/100 km at these
altitudes, about half the rate indicated by the CHAMP and
GRACE observations.
[25] Figure 9 provides latitude versus month depictions of
the GRACE (480 km), CHAMP (360 km) and GSWM
(360 km) M2 perturbation density amplitudes and phases.
We focus on the CHAMP-GSWM comparisons first. The
CHAMP and GSWM results are similar in terms of the
single maximum that occurs just south of the equator
between November and March, and the double maxima that
occur between May and September. The double maxima
occur around 10 to 40 and + 30 to + 50 for GSWM
12
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
gravitational (M2) forcing using satellite-based observations
and a steady state global wave model (the GSWM). The
model forcing does not vary with time and is symmetric
about the equator. Any deviations of the response from these
characteristics are thus due to specification of background
zonal-mean zonal winds in the model, and to a much lesser
degree the background temperatures, both of which have
some basis in observational data.
[28] The GSWM is able to explain the salient characteristics of the seasonal-latitudinal distribution of lunar tidal
temperature amplitudes revealed in the 9 year mean tidal
climatology obtained from TIMED-SABER observations.
These characteristics are in large part explicable in terms
of the (2,2) and (2,3) Hough modes of linear tidal theory;
the former is latitudinally symmetric and resembles the latitude shape of the M2 forcing, while the latter arises as a
result of “mode coupling” introduced by the zonal-mean
zonal jets in the middle atmosphere. There are additional
contributions due to the higher-order (2,4) and (2,5)
modes, but these serve more to account for details than
as drivers of the salient characteristics. Although the zonalmean winds are observationally-based, there are differences
between the SABER and GSWM results that suggest that
there is room for improvement in the background atmosphere
specification.
[29] Because we are primarily interested in the migrating
(longitude-independent)M2 lunar tide in this paper, the
effects of solid Earth and ocean tides on the atmospheric
lunar response [Forbes and Vial, 1994] were not considered.
While these contributions, which are open to some
uncertainty, are of primary interest with respect to the longitudinal variability that they might produce, it is possible
that we could have underestimated the M2 response in this
paper by up to 30% by omitting them. However, this does
not appreciably affect the conclusions that we draw with
respect to the influences of middle atmosphere mean
winds on the seasonal-latitudinal and vertical variations
of the lunar tide in the thermosphere.
[30] The lunar tide propagates into the thermosphere and
achieves maximum amplitudes in the ionospheric E-region,
and under solar minimum conditions propagates with sufficient strength to produce density perturbations at 360 and
480 km that are easily recovered from analyses of densities
based on accelerometer measurements on the CHAMP and
GRACE satellites. The GSWM captures many features of
the seasonal-latitudinal characteristics of lunar tidal perturbations revealed by these data. However, there are features
of the observed upper thermosphere density variations,
such as near-constancy of phase with latitude during
May–September, which suggest that the (2,3) mode might
be overestimated in the GSWM in comparison to (2,2)
during these months. This may translate to a shortcoming
in the specification of the zonal mean wind field in the
middle atmosphere. At any given altitude it is important
to recognize that any model discrepancies represent the
integrated effect of such shortcomings introduced at much
lower altitudes, compounded with the differential dissipation that the (2,3) and (2,3) modes experience as a result
of their differing vertical wavelengths. Matching observed
and modeled tidal amplitude and phase structures in the
thermosphere thus remains a significant challenge. In addition, there are uncertainties inherent in the lunar tide
It is also possible that the (2,3) mode may not be sufficiently
dissipated within the GSWM.
[26] The GRACE density perturbation maxima in the top
panels of Figure 9 are greater than CHAMP, of order 10%,
which is perhaps not surprising given the more tenuous
atmosphere at 480 km. The South Hemisphere maximum
between January and March in GRACE is shifted about
1 month earlier and 20–30 southward compared with
CHAMP. During June–July a Northern Hemisphere maximum
is the major feature in the GRACE results, but at CHAMP altitudes the major maximum is between 0-50 S latitude with a
secondary maximum between 40–60 N latitude. Similar to
CHAMP, the GRACE phases are nearly constant with latitude,
and as noted previously, lead CHAMP by about 3 h during July,
consistent with an upward-propagating wave. The CHAMPGRACE phase difference is less (1.5–2.5 h) during January–
February, but becomes large (~7 h) in May. Similar to the interpretation of model-data results in Figure 3, uncertainties in the
lunar tide fits (cf. Figures 7 and 8) need to be accounted for
when drawing conclusions from, e.g., Figure 9. For CHAMP,
we find that 1s values for amplitude are largest (1.0–1.4%)
during August–December and generally less than 1.0% between January–July; 1s values for phases during these same
periods are of order 1.5–3.0 h and less than 0.6 h, respectively.
For GRACE, 1s values for amplitude tend to maximize around
February–April and September–December, but the total range
(1.2–2.5%) over all months is not large. A similar pattern
occurs for the GRACE phases, with maximum 1s values of
0.7 h between 0–40 latitude during the above months, but
values generally of order 0.3–0.5 h otherwise. Given these
uncertainties, it appears that phase gradients are more reliable
between January and July than during the rest of the year,
and thus the large CHAMP-GRACE phase difference during
May remains inexplicable. Visual inspection of the individual
fits did not reveal any anomalies. The overall trend, though,
is for the lunar tidal variations derived from GRACE measurements to lead those derived from CHAMP, consistent with
downward phase progression for an upward-propagating wave,
although the phase gradient predicted by the GSWM between
300 and 400 km altitude (the upper boundary of the GSWM)
is only about half the 3 h phase differences between 360 and
480 km indicated in Figure 9. We do suspect, though, that there
might be some unanticipated effects reflected in these results
that are attributable to the low solar minimum conditions within
which these data were collected. For instance, Bruinsma and
Forbes [2010] noted significant differences between CHAMP
and GRACE density behaviors that they attributed to greater
than usual winter Helium abundances at GRACE altitudes.
Note that the minimum M2 amplitudes depicted in Figure 9
for GRACE occur in the winter hemisphere. It is possible that
He-O mutual diffusion may tend to damp tides at these altitudes similar to the way that O-N2 mutual diffusion tends to
damp vertically-propagating tides in the lower thermosphere
[Forbes and Hagan, 1980]. The fact that the GRACE and
CHAMP satellites might have been above and below the exobase, respectively, under these solar minimum conditions
could also be a factor.
4. Summary and Conclusions
[27] In this paper, we investigate the month-to-month
climatology of the response of Earth’s atmosphere to lunar
13
FORBES ET AL.: LUNAR SEMIDIURNAL TIDE
Forbes, J. M., S. L. Bruinsma, X. Zhang, and J. Oberheide (2009), Surfaceexosphere coupling due to thermal tides, Geophys. Res. Lett., 36, L15812,
doi:10.1029/2009GL038748.
Forbes, J. M., X. Zhang, S. Bruinsma, and J. Oberheide (2011), Sunsynchronous thermal tides in exosphere temperature from CHAMP and
GRACE accelerometer measurements, J. Geophys. Res., 116, A11309,
doi:10.1029/2011JA016855.
Hagan, M. E., Forbes, J. M., and F. Vial (1995), On modeling migrating solar tides, Geophys. Res. Lett., 22(8), 893–896.
Hagan, M. E. (1996), Comparative effects of migrating solar sources on
tidal signatures in the middle and upper atmosphere, J. Geophys. Res.,
101, 21213–21222.
Hagan, M. E., M. D. Burrage, J. M. Forbes, J. Hackney, W. J. Randel, and
X. Zhang (1999), GSWM-98: Results for migrating solar tides, J.
Geophys. Res., 104(A4), 6813–6828.
Hagan, M. E., and J. M. Forbes (2002), Migrating and nonmigrating
diurnal tides in the middle and upper atmosphere excited by
tropospheric latent heat release, J. Geophys. Res., 107(D24), 4754,
doi:10.1029/2001JD001236.
Hagan, M. E., and J. M. Forbes (2003), Migrating and nonmigrating semidiurnal tides in the middle and upper atmosphere excited by tropospheric latent
heat release, J. Geophys. Res., 108(A2), 1062, doi:10.1029/2002JA009466.
Lindzen, R. S., and S. S. Hong (1974), Effects of mean winds and horizontal
temperature gradients on solar and lunar semidiurnal tides in the
atmosphere, J. Atmos. Sci., 31, 1421–1466.
Matsushita, S. (1967a), Solar quiet and lunar daily variation fields, in
Physics of Geomagnetic Phenomena, edited by S. Matsushita, and
W. H. Campbell, p. 301, Academic, San Diego, Calif..
Matsushita S. (1967b), Lunar tides in the ionosphere, Handb. Phys., 49/2, 547.
McLandress, C. (2002), The seasonal variation of the propagating diurnal
tide in the mesosphere and lower thermosphere. Part II: The role of tidal
heating and zonal mean zonal winds. J. Atmos. Sci., 59, 907–922.
Mertens, C. J., et al. (2001), Retrieval of mesospheric and lower
thermospheric kinetic temperature from measurements of CO2 15 mm
earth limb emission under non-LTE conditions, Geophys. Res. Lett., 28,
1391–1394.
Oberheide, J., J. M. Forbes, X. Zhang, and S. L. Bruinsma (2011), Climatology of upward propagating diurnal and semidiurnal tides in the thermosphere, J. Geophys. Res., 116, A11306, doi:10.1029/2011JA016784.
Picone, J. M., A. E. Hedin, D. P. Drob, and A. C. Aikin (2002), NRLMSISE00 empirical model of the atmosphere: Statistical comparisons and scientific
issues, J. Geophys. Res., 107(A12), 1468, doi:10.1029/2002JA009430.
Pugh, D. T. (1987), Tides, Surges, and Mean Sea Level, 486 pp., John
Wiley, Chichester, U. K..
Ray, R. D., and S. B. Luthcke (2006), Tide model errors and GRACE gravimetry; towards a more realistic assessment, Geophys. J. Int., 167, 1055–1059.
Remsberg, E. E., et al. (2008), Assessment of the quality of the Version 1.07
temperature versus-pressure profiles of the middle atmosphere TIMED/SABER,
J. Geophys. Res., 113, D17101, doi:10.1029/2008JD010013.
Russell, J. (2012), SABER Principal Investigator, Private Communication.
Siebert, M. (1961), Atmospheric tides. Advances in Geophysics, Vol. 7.
New York: Academic. Press, 105–183.
Stening, R. J., A. H. Manson, C. E. Meek, and R. A. Vincent (1994),
Lunar tidal winds at Adelaide and Saskatoon at 80 to 100 km heights,
1985-1990, J. Geophys. Res., 99, 13,273–13,280.
Stening, R. J., J. M. Forbes, M. E. Hagan, and A. D. Richmond (1997),
Experiments with a lunar atmospheric tidal model, J. Geophys. Res.,
102, 13,465–13,471.
Sugiura, M., and Fanselau, G. (1966), Lunar phase numbers n and n0 for
years 1850 to 2050. Goddard Space Flight Center Report X-612-66-401.
Vial, F., and J. M. Forbes (1994), Monthly simulations of the lunar semidiurnal tide, J. Atmos. Terr. Phys., 56, 1591–1607.
Zhang, X., J. M. Forbes, and M. E. Hagan (2010), Longitudinal variation
of tides in the MLT region: 1. Tides driven by tropospheric net
radiative heating, J. Geophys. Res., 115, A06316, doi:10.1029/
2009JA014897.
amplitudes and phases, which therefore limit our comparative conclusions to the most salient features reflected in
both the data and model.
[31] On the other hand, there is an alternate explanation
for existence of model-data discrepancies that remains an
important consideration for any interpretation of climatological lunar tidal analyses in the context of a numerical model.
The observational results displayed in Figures 3 and 9 represent averages of the lunar tidal signal corresponding to many
realizations of background wind conditions. Variability in
background wind conditions would very likely change the
phase of the (2,3) mode, which in fact owes its existence
to the very presence of the background winds. The ensemble
averages in Figures 3 and 9 could very well suppress the
(2,3) mode to some degree due to phase cancellations
effects, while the GSWM simulation adopted for comparison holds for one single climatological realization of the
background wind field. The same applies, in fact, for any
comparison between models and climatological mean data
sets, or mean tidal fields that are constructed using data from
slowly-precessing satellites [e.g., Forbes et al., 2008]. As
our models grow in sophistication, and as long data sets
become more plentiful, these considerations need to be taken
into account.
[32] Acknowledgments. The involvement of J. Forbes and X. Zhang
in this work was supported by Grants NNX10AE62G from NASA and
ATM-0719480 from the NSF to the University of Colorado. J. Oberheide
was supported by Grants NNXAJ13G from NASA and AGS-1139048 from
the NSF. Computational support was provided by the National Center for
Atmospheric Research.
References
Bartels, J. (1957) Gezeintenkräfte, Handbuch der Physics Vol. XLVIII,
Springer, Berlin, pp. 734–774.
Bruinsma, S. L., and J. M. Forbes (2010), Anomalous behavior of the thermosphere during solar minimum observed by CHAMP and GRACE, J.
Geophys. Res., 115, A11323, doi:10.1029/2010JA015605.
Chapman, S., and R. S. Lindzen (1970) Atmospheric Tides: Thermal and
Gravitational, 200 pp., Gordon and Breach, New York.
Doodson, A. T. (1922), The harmonic development of the tide generating
potential, Proc. R. Soc. Lond. A 100, pp. 305–329.
Flattery, T. W. (1967), Hough functions, Tech. Rept. 21, Dept. of Geophys.
Sci., Univ. of Chicago, Chicago, Ill.
Forbes, J. M. (1982a), Atmospheric Tides. I. Model description and results
for the solar diurnal component, J. Geophys. Res. 87, 5222–5240.
Forbes, J. M. (1982b), Atmospheric Tides. II. The solar and lunar semidiurnal components, J. Geophys. Res. 87, 5241–5252.
Forbes, J. M., and M. E. Hagan (1980), Tidal dynamics and composition
variations in the thermosphere, J. Geophys. Res., 85, 3401–3406,
doi:10.1029/JA085iA07p03401.
Forbes, J. M., M. Kilpatrick, D. Fritts, A. H. Manson, and R. A Vincent
(1997), Zonal mean and tidal dynamics from space: an empirical examination of aliasing and sampling issues, Ann. Geophys., 15, 1158–1164.
Forbes, J. M., X. Zhang, S. Palo, J. Russell, C. J. Mertens, and M. Mlynczak
(2008), Tidal variability in the ionospheric dynamo region, J. Geophys.
Res., 113, A02310, doi:10.1029/2007JA012737.
14