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Tidal Variations - The Influence of Position and Distance

Ingersoll, A. Keldysh, M. Icarus 30 , — Munk, W. Download references. Reprints and Permissions. Nature Communications Nature The Moon and the Planets By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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Rent or Buy article Get time limited or full article access on ReadCube. References 1 Shapiro, I. Google Scholar Download references. Rights and permissions Reprints and Permissions. Comments By submitting a comment you agree to abide by our Terms and Community Guidelines. Article Tools. Article metrics Citations The temperature data were analyzed as follows to extract the tidal components, which represent 3-yr averages over the 1 November —27 October period. A standard deviation was computed for each hourly data point, primarily providing a measure of geophysical variability.

At each altitude, latitude, and longitude, Fourier least squares fits were performed with respect to UT to determine amplitudes and phases of diurnal, semidiurnal and terdiurnal tidal components. One-sigma uncertainty estimates were computed for each frequency-component amplitude and phase, based upon the hourly standard deviations. Uncertainty estimates were computed for each zonal wavenumber component, taking into account the frequency-component uncertainties from the previous stage of analysis. Average temperatures in the longitude and UT bins were also subjected to a two-dimensional FFT, determining the frequency and zonal wavenumber decompositions simultaneously, with little change in the results.

The above sampling and binning procedure must be set into the context of the constraints of instrument sampling imposed by the yaw cycle maneuvers. Figure 1 illustrates the spatial-temporal coverage for the MLS temperature data covering the complete 1 November —27 October data interval; as such, it is slightly different than the yaw cycle coverage for any given year.

On the other hand, there are some months i. There are also some months i. It is these latter months in one hemisphere or the other that may be most subject to aliasing of the type discussed in the appendix. However, the reader is reminded that the results described here are climatological in the sense that they represent multiyear averages. The multiyear averaging utilized here should ameliorate the shortcomings associated with this chosen methodology.

As is evident from Eq. As satellite sampling precesses through local time, these aliasing effects are expected to decrease and ultimately disappear until all waves are fully sampled; that is all local times are sampled at all longitudes. In the present method, where we utilize 3-yr climatological averages filling all longitude and UT bins several times over at a given height and latitude, we expect our space—time decomposition to be alias-free from this point of view.

However, see below for aliasing effects due to nonstationarity of the dynamical fields. Multiyear and monthly averaging also has the tendency to underestimate amplitudes when the dynamical fields exhibit year-to-year variability or nonstationarity during the fitting interval. It is difficult to assess the impact of these effects, but any future attempts at comparing model outputs with our results should address this problem through appropriate averaging of the model fields.

In at least two cases, nonstationarity of the dynamical fields can also lead to aliasing effects. For instance, as illustrated in the appendix , an SPW1 that evolves during the fitting interval can alias into all of the nonmigrating tidal components mentioned in the previous paragraph. A method for estimating potential aliasing contributions from this source is also provided in the appendix , and the resulting aliasing estimates are introduced throughout the following text and in the figure captions.

In addition, as shown by Forbes et al. It is possible to greatly reduce this effect by taking for UARS a day running mean through the data, subtracting this running mean from the original data, and then analyzing the residuals to extract the tidal components Forbes et al. However, it is only possible to apply this method when the data are continuous in time i. We suspect that the binning method without removal of the running mean may be more problematic when applied to a satellite that takes much longer than 36 days to precess in local time; for instance the day precession period of the Thermosphere, Ionosphere, Mesosphere, Energetics and Dynamics satellite TIMED.

Given that the present study focuses on solar thermal tides, it is natural to first examine the temperature structures revealed by the MLS experiment ordered in local time. These data are typical of other months as well. This figure also offers an opportunity to provide a measure of the geophysical variability in the data, as manifested in the displayed standard deviations.

The thick solid lines in Fig. It is readily apparent that the local time structures as well as mean values vary significantly with longitude; the former implies the existence of nonmigrating tides, the latter with stationary planetary waves. The changes in local time structure with longitude imply the presence of diurnal, semidiurnal, and perhaps terdiurnal nonmigrating tides, as discussed in connection with Eq.

The following sections primarily focus on the depiction of these tidal components and their interpretation. As noted previously, at a given longitude the local time structure is reasonably approximated by a superposition of diurnal, semidiurnal, and terdiurnal harmonics [i. The variation with longitude of the local time structure is embodied in a spectrum of zonal wavenumbers [i.

In theory an infinite sum is required to capture the longitude variation of each harmonic, but in practice relatively few harmonics are found to dominate. This point is illustrated in Fig. Migrating tides are omitted from this figure in order to highlight the smaller-amplitude nonmigrating components. We note that results in Fig. Figure 3 provides insight into potential sources for the observed nonmigrating tides. For instance, it is now generally accepted i. Existence of DW2 and D0 in Fig. These semidiurnal nonmigrating tides were found by Manson et al. These tidal components have significantly longer vertical wavelengths than their westward-propagating counterparts, are thus less susceptible to dissipation, and hence more likely to penetrate to the MLT Ekanayake et al.

The maximum in DW1 between 40 and 60 km is probably an in situ response to UV absorption by ozone, whereas the elevated amplitudes above 80 km may be due to the absorption of shorter-wavelength UV radiation by O 2 , perhaps augmented by an effect of electrodynamic origin penetrating downward from the auroral ionosphere. The origin of the peak amplitudes near 70 km is unknown, but may be a signature of chemical heating Mlynczak and Solomon ; Smith et al.

While SPW1 exhibits maximum values in Fig. The lack of similarity between the SPW1, D0, and DW2 amplitudes over the whole domain is consistent with the absence of the type of aliasing addressed in the appendix. Note also that the diurnal nonmigrating tides tend to show their largest amplitudes where signatures of both DW1 and SPW1 are relatively large, possibly reflecting the nonlinear generation of nonmigrating tides in these regions. The counterpart of Fig. Note that SW2 amplitudes are of order 2.

Based on these results, one would expect the nonlinear generation of the sum and difference waves SW1 and SW3 to be confined to the height versus month regime of significant SPW1 and SW2 amplitudes. Distributions of the SW1 and SW3 amplitudes are broadly consistent with this. However, the alternate possibility of aliasing due to time evolution of the SPW1 amplitudes during the fitting intervals must be considered.

Considering this possibility, the following depictions of nonmigrating tides are confined to 86 km, where the amplitudes are generally largest and the aliasing effects due to SPW1 are minimized. It is sometimes illustrative to examine tidal structures in terms of Hough functions, the eigenfunctions of Laplace's tidal equation Chapman and Lindzen In particular, for the diurnal tide, which consists of both propagating and trapped components, some insight into the possible existence of in situ excitation may be revealed.

In addition, the relative importance of various Hough functions for propagating components can also provide some insight into vertical structures by virtue of the connection between eigenfunctions, eigenvalues, and vertical wavelengths Chapman and Lindzen For each wavenumber, Hough functions for the first symmetric and antisymmetric propagating index m positive in Fig. In this depiction aliasing contributions to D0 and DW2 amplitudes due to SPW1 variability see the appendix are estimated to be no more than 0. DW1 amplitudes maximize at about 9 K near the equator, reflecting dominance of the first symmetric propagating component of the diurnal oscillation.

However, the broadness of this structure and to some degree the nonsymmetric phase structure suggests the presence of higher-order modes. Of course, much better agreement could be obtained by adding more Hough modes, as these functions form a complete orthogonal set. The DW2 amplitudes are reasonably represented by the first four Hough modes. Examination of the Hough mode amplitudes reveals that the most important contributions to DW1 and DW2 are the first symmetric propagating modes, but that the trapped modes also make important contributions, as might be expected from the measured amplitudes at middle to high latitudes.

Lunar and Solar tides vector illustration diagram poster. Spring and Neap tide.

This implies that part of the excitation lies at lower altitudes see the introduction , but that there is an in situ excitation mechanism for generating evanescent or trapped tidal components as well. One possibility might be broadening i. Another possibility is an in situ heat source, possibly chemical heating Mlynczak and Solomon ; Smith et al. In principle, of course, D0 and DW2 could result from a longitude-dependent heat source as well, but there is no known evidence for expecting significant zonal asymmetries in the background atmospheric state at these altitudes.

Also included in Fig. For D0 and DW2, the GSWM only includes excitation due to latent heating in the tropical troposphere, whereas for DW1 forcing due to insolation absorption by H 2 O, O 3 , and O 2 in the troposphere, stratosphere, and mesosphere, respectively, are also accounted for Hagan et al.

However, at higher latitudes the GSWM significantly underestimates the observed temperature amplitudes of all three tidal components in Fig. This may imply omission of an in situ heat source of unknown origin in the GSWM. The GSWM significantly underestimates amplitudes for D0 and DW2 at low latitudes as well, suggesting that latent heat release is unimportant for exciting these oscillations, and that nonlinear interaction between DW1 and SPW1 is the dominant forcing mechanism Hagan and Roble ; Lieberman et al.

Aliasing contributions due to SPW1 variability see the appendix are estimated to be no more than 0. For the semidiurnal tide, and the terdiurnal tidal component that will be discussed momentarily, we have only fit the first symmetric and first antisymmetric Hough modes to the data, as reasonable global fits can be accomplished this way, and differences between the observations and fits provide a measure of the importance of higher-order modes. This is not surprising, as the 2, 4 , 2, 5 , and even 2, 6 Hough modes have often been cited as contributing to global semidiurnal tidal wind structures Lindzen ; Forbes et al.

On the other hand, the SW1 and SW3 amplitude and phase structures are represented well by the first symmetric and antisymmetric Hough modes. The amplitudes of these nonmigrating tidal components maximize near 2. These differences are likely associated with inadequate treatment of mode coupling in the model i. For SW1 and SW3, the model values significantly underestimate the observations, suggesting that the primary forcing mechanism for these oscillations is nonlinear interaction between SW2 and SPW1 Teitelbaum and Vial ; Forbes et al.

The corresponding depiction for the terdiurnal tides is provided in Fig. Here the amplitudes are smaller, of order 0.

Tidal Variations - The Influence of Position and Distance

Some degree of coherence in phases between latitudes and altitudes, not shown lends some credibility to the existence of these structures as independent propagating oscillations. These oscillations are not important to the dynamics of the upper mesosphere. However, due to their long vertical wavelengths, they can be expected to achieve significant amplitudes in the —km region and above, and possibly contribute to the dynamo generation of electric fields and other aspects of the variability of the region.

Since experimental data for the atmosphere above km is particularly sparse, efforts like the present one, supplemented by tidal models or GCMs with lower boundaries in the mesosphere, can provide some insight into dynamical consequences at upper levels. In addition, the existence of nonmigrating tidal oscillations can provide clues to nonlinear interactions that may be occurring at lower levels of the atmosphere.

A perspective on seasonal—latitudinal variability of the diurnal tidal oscillations is provided in Fig. Latitude versus month contours of diurnal temperature amplitudes at km altitude for D0, DW1, and DW2 are depicted. Upper bounds on aliasing contributions to D0 and DW2 amplitudes due to SPW1 variability see the appendix are estimated to be no more than 0.

These estimates are based upon an average SPW1 amplitude of 4. Similar features are found in the amplitudes of D0 and DW2, but curiously, the high-latitude maxima are confined to the Southern Hemisphere and show relatively little dependence on time of year. The diurnal tidal wind amplitudes for D0 at 95 km, as displayed in Forbes et al. The origin of this latitudinal asymmetry remains unknown, but appears to be a real and persistent feature of the 85—km height region. A similar depiction for the semidiurnal tidal component is provided in Fig.

Amplitudes for the semidiurnal tide are generally of order 2—6 K for SW2 and 1—2 K for SW1 and SW3; that is, roughly half of those of the diurnal tides depicted in Fig. This is consistent with the observation made in connection with Fig. The source for these higher-order modes is mode coupling due to interactions between the fundamental modes and the zonal mean wind structure Lindzen and Hong ; Forbes At least in the Southern Hemisphere, this amplitude distribution is different than what one would expect on the basis of wind observations near 94 km over South Pole Forbes et al.

Modeling work is apparently needed to explain these differences. DE3 is a prominent oscillation in the sample spectra of Fig. Modeling studies Forbes et al. DE3 was found to be the largest of all the nonmigrating diurnal tidal components in the tidal analysis of UARS winds at 95 km by Talaat and Lieberman , Forbes et al. As noted in the introduction, this type of exercise develops confidence in utilizing different data types together in assimilation schemes aimed at specifying the dynamical state of the MLT. At these altitudes the maximum amplitude is about 2.

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It is clear that during some months of the year the DE3 amplitude structure is very asymmetric with respect to the equator, whereas during Northern Hemisphere summer DE3 is predominantly symmetric with a Kelvin wave latitudinal structure. The degree of asymmetry during the various months is in fact consistent with the eastward and northward wind structures at 95 km displayed in Forbes et al.

We will now examine the consistency between the temperature perturbations at 86 km and the wind perturbations at 95 km in a more quantitative way, which involves a set of basis functions called Hough mode extensions. A brief description of HMEs will now be provided, and their use in providing new information on measurements in the MLT region will be explored.

Due to lack of knowledge of in situ excitation sources associated with trapped components in the DW1 and DW2 fields, and the large uncertainties in our D0 results, we will confine our attention concerning application of HMEs to the DE3. DE3 has no known sources of excitation in the mesosphere or lower thermosphere, making it a viable candidate for application of the HME technique. As will become evident below, this is an important consideration for the application at hand. The concept of HMEs was developed by Lindzen et al. A Hough mode extension represents the global solution pole-to-pole, 0-tokm altitude to the linearized dynamical equations of the atmosphere for an oscillation of given frequency and zonal wavenumber, taking into account dissipative effects i.

The HMEs are forced with a conveniently normalized heat source confined to the troposphere, and with latitude shape given by the corresponding classical Hough mode. The methodology for fitting HMEs to observational data is fully described in Forbes et al. In more recent work, Svoboda et al. Although zonal mean winds are neglected in the computation of HMEs, this does not imply that the effects of mean winds are neglected in fitting or reconstructing tidal structures.

To first order, the distortion of tidal structures due to mean winds can be viewed as mode coupling Lindzen and Hong ; that is, the excitation of higher-order modes that combine together in a linear sense to approximate the distortion. In the same sense, a few HMEs can be fit to a tidal field in a way that the weighted superposition of these HMEs reproduces the observed distortion. The methodology assumes that all of the important mean-wind effects occur below the region in which the HME fitting occurs; that is, in association with the mesospheric jets; in other words, that the 80—km winds are too weak to produce any significant distortion of the tidal structures.

The reader is referred to Forbes et al. Figure 12 compares the latitude versus time evolution of DE3 temperature amplitude at 86 km from the work of Svoboda et al. The similarity between DE3 amplitudes in Fig. In addition, this result lends confidence to the combined utilization of space-based temperature and wind measurements in assimilation schemes to specify the dynamical state of the upper atmosphere, at least in terms of variations about the zonal mean state are concerned. In the interpretation of Fig. In extrapolating tidal wind fields at 95 km to tidal temperature fields at 86 km, the vertical structure of the HMEs, and hence the dissipation assumed in the model, assumes importance.

Molecular diffusion becomes important above km. Dissipation relates to the interpretation of the top two panels in Fig. The DE3 response can be decomposed into two primary Hough mode components, the first symmetric component, which is a Kelvin wave with inviscid vertical wavelength of 56 km, and the first antisymmetric component which is an inertio-gravity wave with 30 km inviscid vertical wavelength. Between the In addition, growth of the first symmetric HME is the same factor of 3. This difference reflects the shorter vertical wavelength of the latter, and the fact that the time constant for dissipation is proportional to the square of the vertical wavelength.

If we focus on the symmetric part of the response i. However, any change in this direction would be a weak function of the eddy diffusion coefficient due to the long vertical wavelength of the Kelvin wave, and moreover, a significantly larger value of eddy diffusion coefficient is unlikely. A smaller eddy diffusion coefficient would not change the comparison between the symmetric parts of the response in Fig.

Thus, we view this as a robust result. The comparison in Fig. These results pertain to both instantaneous overflight and climatological comparisons between the two datasets. Of particular relevance in the present context are the results of Forbes et al. Possible reasons for all of the above differences are discussed in the aforementioned papers, including differences in radar types, but the issue remains unresolved. Recall that the temperatures depicted in the lower panel of Fig. The fact that the HME temperatures agree well with and in fact slightly underestimate the MLS temperature amplitudes upper panel of Fig.

Of course, this assumes that the MLS temperatures variations reflect those of the atmosphere, and the HMEs to embody the correct temperature—wind relationship between 86 and 95 km, which depends to some degree on the assumed background temperature, density and dissipation assumed in the HME calculations. We have demonstrated that our result is robust with respect to dissipation, which is by far the issue of greatest importance. Thus, we consider our result to represent a credible and important contribution to the debate surrounding the discrepancy between GB and SB wind measurements, in addition to contributing to measurement validation in support of data-model assimilation efforts that have yet to be applied to the MLT region.

Analyses of temperatures measured between 25 and 86 km by the Microwave Limb Sounder MLS experiment on UARS reveal the presence of migrating sun-synchronous and nonmigrating solar tides. Emphasis is placed on the MLS upper altitude limit of 86 km where amplitudes are largest and aliasing effects are minimized.

The former feature is indicative of propagating modes, while the latter is associated with trapped components for DW2 and mainly trapped components for D0. This wave is probably generated by latent heat release due to deep tropical convection. Within the confines of dissipative tidal theory, and subject to some caveats, internal consistency is established between the MLS DE3 temperatures at 86 km and previously derived DE3 winds at 95 km.

This result lends confidence to the combined use of space-based temperature and wind measurements in assimilative modeling of MLT dynamics, at least in terms of variations about the zonal mean are concerned. In addition, the integrity of space-based wind measurements demonstrated within this context may have some bearing on the debate surrounding inconsistencies sometimes noted between winds measured from the ground and space near 95 km. The authors thank Ms. Xiaoli Zhang for her assistance in data processing and figure preparation.

MLS data coverage for the period 1 Nov —27 Oct Power spectra for temperature K 2 at 86 km for left stationary, middle diurnal, and right semidiurnal components as a function of latitude and zonal wavenumber positive westward for top July and bottom January. The migrating tidal components are excluded in order to more clearly illustrate the nonmigrating components.

Aliasing contributions due to SPW1 variability see the appendix may be contained in the illustrated D0 and DW2 amplitudes. However, it is possible that these amplitudes may represent, in whole or in part, signatures of nonlinear interaction between SW2 and SPW1, which would reduce this aliasing estimate accordingly. The index m is related to the number of nodes in latitude, and is positive for propagating modes and negative for trapped modes.

Latitude structures of January diurnal left temperature amplitude and right phase at 86 km altitude for top D0, middle DW1, and bottom DW2. The solid lines represent Hough mode fits to these data taking into account the first two symmetric and antisymmetric propagating and trapped modes. Aliasing contributions to D0 and DW2 amplitudes due to SPW1 variability see the appendix are estimated to be no more than 0. Same as Fig.

Venus' rotation and atmospheric tides | Nature

Terdiurnal results from the GSWM do not currently exist. Upper bounds on aliasing contributions to SW1 and SW3 amplitudes due to SPW1 variability see the appendix are estimated to be no more than 1. Latitude vs month contours of DE3 diurnal temperature amplitudes at 86 km. The tidal components primarily reflect aliasing due to SPW1 variability over the fitting intervals. Contour spacing is 0. Contours are at 0. A concern that naturally arises in space-based sampling of atmospheric structures, particularly those that are nonstationary, is that of aliasing; that is, when the energy of one sampled component leaks into another.

The reverse process also holds. That is, time variations in the above nonmigrating tides can alias into SPW1. However, the nonmigrating tidal amplitudes are considerably smaller in magnitude, hence the scenario A1 — A3 is of the greatest concern. To gain insight into the aliasing effect, the following experiment was performed. Note that the MSISE90 model used in this fashion contains only time-varying stationary planetary waves and zonal means, and no tides. The resulting data were analyzed for nonmigrating tides in a fashion identical to that described previously for the MLS temperature data.

The SPW1 amplitudes are of order 2—10 K with maximum values during local winter and spring. The nonmigrating tidal amplitudes are of order 0. Maxima tend to occur during periods of greatest variation in SPW1 amplitudes over a yaw cycle; that is, November, January, and April—June. A1 and Fig. Therefore, to obtain better quantitative estimates of aliasing contributions to the nonmigrating tides, we utilize the MLS data itself to make this estimate. The method is described below. First, it must be recognized that what is important in determining aliasing amplitudes is how well the tidal Fourier harmonics project onto the SPW1 variability over the whole span of MLS sampling.

Because we are compositing multiple years of data into a single effective yaw cycle prior to analysis, year-to-year variability diminishes the coherence of the seasonal evolution of SPW1 over the fitting interval. This irregular sampling helps to reduce the aliasing. The contributions of both of these effects reduce the potential for aliasing that would exist during a single season.

Paul Williams Solar Tides

One cannot determine the SPW1 variability in the MLS data over time scales less than a day yaw cycle, because at these time scales the SPW1 variations cannot be separated from the nonmigrating tides. However, following on the example provided above and illustrated in Figure A1 , we now propose a method for arriving at quantitative aliasing estimates.

The method is based on the following assumptions: i tidal amplitudes at 20—40 km in the stratosphere are negligibly small, and ii variability of the SPW1 in the rest of the domain follows that in the stratosphere. Note that i implies that any tidal amplitudes recovered in the MLS analysis between 20—40 km are due to aliasing by SPW1 variability.

Comparison of the nonmigrating tidal amplitudes with those of the SPW1 in the stratosphere provides a measure of how well the SPW1 variability projects onto the nonmigrating tidal components over the 3-yr climatology of the MLS temperature measurements. Assumption ii implies that this amplitude ratio extends throughout the mesosphere, and can be used to estimate the aliasing contributions to nonmigrating tides provided we know the magnitude of SPW1 at those levels.

This is equivalent to saying that the SPW1 variability projects onto the nomigrating tides with the same efficiency everywhere. This is of course not strictly true, but an assumption of this nature is required to arrive at a quantitative estimate.


  1. High and Low Nearly Twice a Day!
  2. Solar Tides.
  3. Lunar and Solar tides vector illustration diagram - VectorMine?
  4. Lunar and Solar tides vector illustration diagram!

Furthermore, since some fraction of the observed nonmigrating tidal signals in the stratosphere may be due to SPW1 interactions with migrating tidal components whose amplitudes are nonnegligible in the regions of interest , we consider our aliasing estimates to represent an upper bound. An example of how these aliasing estimates were obtained is now provided. Refer to Fig. Significant amplitudes are even found in the Southern Hemisphere at 86 km. Recall from section 2 that January is one of those months where the potential for aliasing is suspected to be particularly high, since the 15th of the month falls in the gap between yaw cycles, and about half the local times binned together originate in the previous and following yaw cycles.

We now assume that the same proportionality applies throughout the domain, and obtain aliasing estimates at for example 86 km of order 0. This is the method used to provide upper bounds on aliasing contributions to nonmigrating tidal components throughout the text and in the figure captions. In some cases these average values are used to provide rough upper limits on the aliasing contributions due to SPW1 variability cf. It is interesting to note the significantly larger aliasing estimates for the semidiurnal nonmigrating tides, similar to the results illustrated in Fig.

Next Article. Previous Article. Jeffrey M. Forbes x. Search for articles by this author. Dong Wu x. The experimental data and method of analysis. Local time structures. Spectral overview. Hough components. Seasonal—latitudinal structures. Acknowledgments The authors thank Ms. View larger version 61K Fig. View larger version 47K Fig. View larger version 49K Fig.

View larger version 73K Fig. View larger version 83K Fig. View larger version 55K Fig. View larger version 50K Fig. View larger version 45K Fig. View larger version 68K Fig. View larger version 53K Fig. View larger version K Fig.

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