Planetary-Scale Thermal Waves in Saturn's Upper Troposphere

ICARUS

119, 350–369 (1996) 0024

ARTICLE NO.

Planetary-Scale Thermal Waves in Saturn’s Upper Troposphere RICHARD K. ACHTERBERG Laboratory for Extraterrestrial Physics, NASA Goddard Space Flight Center, Code 693.2, Greenbelt, Maryland 20771, and Science Systems and Applications, Inc., 5900 Princess Garden Parkway, Suite 300, Lanham, Maryland 20706 AND

F. MICHAEL FLASAR Laboratory for Extraterrestrial Physics, NASA Goddard Space Flight Center, Code 693.2, Greenbelt, Maryland 20771 E-mail: [email protected] Received September 8, 1994; revised August 28, 1995

1. INTRODUCTION We have used temperatures retrieved from the Voyager IRIS (Infrared Interferometer Spectrometer) data at pressures of 130 and 270 mbar to search for waves in the upper troposphere of Saturn. The data are from three global mapping sequences taken by Voyagers 1 and 2, which together provide coverage between 608S and 808N latitudes. To analyze the temperatures, we grouped them within 108 wide latitude bins and applied a periodogram analysis of their zonal variance. Since the data for each map were taken over periods of 12 to 18 hours, the phase velocity of any spectral component identified is ambiguous. We estimated the phase velocity of these components by varying the assumed phase velocity to maximize the amplitude of the periodogram response at integer zonal wavenumbers. The results are dominated by zonal wavenumber 2 at 130 mbar in northern midlatitudes. In the Voyager 1 data this component is statistically significant at the 99% confidence level over latitudes 208N to 408N; the zonal phase of the wave is roughly constant over this latitude range. Estimates of the phase velocity using the combined Voyager 1 inbound and outbound maps allow a solution for the midlatitude zonal wavenumber 2 wave that is quasi-stationary in the reference frame defined by the rotation period of Saturn’s magnetic field. Raytracing models suggest that the midlatitude waves are quasi-stationary Rossby waves. The waves are confined meridionally by meridional variations of the zonal mean winds, and confined vertically by variations in the static stability of Saturn’s atmosphere. The meridional structure of these features suggests that they may be excited by overreflection of waves incident on a critical surface at midlatitudes, and that the low phase velocities are a consequence of the gradient in potential vorticity vanishing where the zonal winds are small. The waves can leak into the upper stratosphere, saturating or ‘‘breaking’’ at pressure levels p1022 –1021 mbar. The viscous damping times are p10 years, comparable to the radiative damping time in the tropopause region and in the stratosphere.  1996 Academic Press, Inc. 350 0019-1035/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

Waves seem to be nearly ubiquitous in the atmospheres of the giant planets. Visible images of Jupiter from Voyager have revealed a rich display of wave patterns over a large range of scales (Smith et al. 1979a,b; Ingersoll et al. 1979; Mitchell et al. 1979; Hunt and Miller 1979; Flasar and Gierasch 1986). Images of Saturn have shown fewer discrete features, but those observed often exhibit a high degree of regularity. An example is the so-called ribbon feature that wraps around the planet in the latitude band 458–508N (Sromovsky et al. 1983, Godfrey and Moore 1986). Another is the hexagonal feature at 778N that encircles the pole. This feature has the amazing property that it rotates with the radio rotation period, thought to characterize the interior rotation of the planet, even though it is imbedded in zonal winds that are greater than 100 m sec21 (Godfrey 1988, 1990; Allison et al. 1990). Finescale wavelike structures have also been observed in Neptune’s atmosphere, often in the vicinity of larger discrete features, such as the Great Dark Spot (Smith et al. 1989, Ingersoll et al. 1995). Visible images do not provide the only evidence of wave activity in the giant planets. Temperature profiles of Jupiter, retrieved from Voyager radio occultation soundings, show undulations with altitude that are suggestive of vertically propagating waves (Lindal et al. 1981, Flasar 1989, Allison 1990). Detailed analysis of scintillations in the Voyager radio occultation soundings of Uranus and Neptune (Hinson and Magalha˜ es 1991, 1993) have indicated the presence of vertical wave-like structure. Infrared observations have also shown waves. On Jupiter, thermal features that are nearly stationary with respect to the internal rotation of the planet have been identified from both Voyager

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data (Magalha˜es et al. 1989, 1990) and ground-based observations (Deming et al. 1989). Planetary-scale waves in Jupiter’s stratosphere are often apparent in infrared maps synthesized from ground-based observations (Orton et al. 1991). Temperature maps of Neptune, derived from Voyager IRIS data, give some indication of wave-like structure, particularly near the latitudes of the Great Dark Spot (Conrath et al. 1991, Flasar and Conrath 1991, Ingersoll et al. 1995). The identification of waves is of great interest for the study of planetary atmospheres. First, their observed characteristics depend on the properties—temperatures, winds, and rotation—of the atmosphere through which they propagate. As such they can serve as sensitive probes of atmospheric structure and dynamics, permitting the study of regions that might be otherwise inaccessible to direct observation. For example, Flasar and Gierasch (1986), from observations of numerous mesoscale wavetrains in Voyager images, inferred the existence of internal gravity waves vertically trapped in a stably stratified (i.e., subadiabatic) duct below the visible cloud tops. More recently, Ingersoll and Kanamori (1995) have interpreted the ring disturbances that have been observed around the impact sites of the comet Shoemaker–Levy 9 collision with Jupiter as further evidence of vertically trapped gravity waves within a deep tropospheric duct. The stationary thermal features on Jupiter and the polar hexagon on Saturn, discussed above, may in some way provide a probe of the deep corotating interiors of these planets (Magalha˜ es et al. 1990). Second, atmospheric waves are also of great interest because they can transport angular momentum and energy over great distances, providing a direct coupling between the troposphere and the stratosphere or higher regions. They therefore can play an important role in governing the general circulation at high altitudes. Leovy et al. (1991) have suggested that forcing by stresses associated with vertically propagating waves is responsible for the apparent 4–5-year oscillation in Jupiter’s equatorial stratospheric temperature that has been observed from groundbased observations (Orton et al. 1991). Vertically propagating waves which ‘‘break’’ in the stratosphere may also account for the decay of the zonal winds with altitude above the cloudtops that has been observed for each of the giant planets (Gierasch et al. 1986, Flasar et al. 1987, Conrath et al. 1991). Breaking waves can also locally control the vertical mixing of heat and molecular constituents (e.g., Fritts and Dunkerton 1985). In this work we search for planetary-scale waves in the temperature fields retrieved from Voyager 1 and 2 IRIS observations of Saturn. Our search is restricted to the upper troposphere and tropopause region. Heterogeneous cloud opacities render ambiguous any attempt to interpret the temperature fields at deeper levels (Conrath and Pirraglia 1983). Extension to higher levels would require the use of

radiances within the 7.7-em n4 band of CH4 , which are often used as a ‘‘thermometer’’ for the stratosphere; unfortunately, the signal-to-noise ratio in this spectral region was inadequate to permit a search for zonal structure. The plan of the paper is as follows: Section 2.1 describes the global mapping sequences used in the data analysis and the derivation of temperature fields whose zonal structure is spectrally analyzed. The spectral analysis is based on the Lomb– Scargle periodogram, described in Section 2.2, which is particularly well suited to the treatment of series that are unevenly spaced with respect to the independent variable (here longitude). Because the maps are constructed from data that extend over a period of time comparable to that of the planetary rotation, there is an ambiguity between zonal phase velocity and wavenumber in any identifiable spectral component, and we describe the procedure for dealing with this ambiguity. The Appendix discusses in some detail the statistical test used to determine the significance of peaks that appear in the periodograms of unevenly spaced data. Section 2.3 presents the results of the periodogram analysis. Significant zonal periodicities are identified at several latitudes, but the most significant is a wavenumber 2 mode in the Voyager 1 tropopause temperatures that extends from 408N toward the equator with decreasing amplitude but with a nearly constant zonal phase. The wave is almost stationary with respect to Saturn’s internal rotation. In Section 3 we interpret the wavenumber 2 mode as a planetary, or Rossby, wave and demonstrate by means of a raytracing analysis how this wave is trapped within a midlatitude waveguide that is formed by the meridional variation in Saturn’s zonal winds and vertical variation in its thermal stratification. The analysis shows that the wave can leak out into the upper atmosphere, and in Section 4 we estimate the upward wave flux of zonal momentum and the levels at which this momentum is deposited into the zonal flow. We also address the possibility that the wave may be excited by a baroclinic/barotropic instability associated with a critical surface just north of the waveguide. Finally, in Section 5, we summarize our results and conclusions. 2. DATA ANALYSIS

1. Data Set The data consist of three global mapping sequences, referred to as the ‘‘north–south maps,’’ obtained by the Voyager Infrared Interferometer Spectrometer (IRIS) experiment. During each of the mapping sequences the IRIS field of view scanned the subspacecraft meridian of Saturn, with longitude coverage provided by the rotation of the planet. Figure 1 shows the spatial coverage of each of the maps. The Voyager 1 inbound map was made between 35 and 46 hr before closest approach, the Voyager 1 outbound map was made between 30 and 41 hr after closest approach, and the Voyager 2 inbound map was made between 32

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FIG. 1. Coverage of the global mapping sequences used in the data analysis. (Top) shows the location of spectra from the Voyager 1 inbound north–south map. (Middle) shows the location of spectra from the Voyager 1 outbound map. (Bottom) shows the location of spectra from the Voyager 2 inbound map. The vertical line with horizontal arrow shows the longitude of the first data point in each map. The gray areas mark the latitude ranges where the rings obscure part of the IRIS field of view. Subspacecraft latitudes during the mapping sequences are marked to the right of the graphs.

and 43 hr before closest approach. For each of the maps, the 0.258 circular IRIS field of view projected between 88 and 128 of a great circle arc at the subspacecraft point. The actual spatial resolution is somewhat better than this, because the beam pattern is not flat but peaked at the center, although the exact pattern is not known (Gierasch et al. 1986). We inverted the measured radiances in the spectral region from 200 to 500 cm21 using a constrained linear inversion method (Conrath and Gautier 1980) to obtain temperatures in the upper troposphere at 130 and 270 mbar. At these wavenumbers, we anticipate that the effects of aerosols are small, and we assume that the opacity is caused only by collision-induced hydrogen absorption (see, e.g., Conrath and Pirraglia 1983). The effective vertical resolution of the inversion is about one-half of a pressure scale height. We calculated zonal mean temperatures using a running mean with 58 bins at 18 intervals. We then subtracted the interpolated zonal means from the retrieved

temperatures and spectrally analyzed the zonal structure of the temperature differences. The zonal mean temperatures reveal systematic offsets of a few degrees in the latitude pointing data for the maps. Comparable offsets have been observed in the north–south mapping sequences of Jupiter (Gierasch et al. 1986). These pointing offsets resulted from uniform offsets in the pointing for whole blocks of data records. We therefore adjusted the latitudes of the points in the Voyager 1 inbound and outbound maps to align the zonal mean temperatures with those from the Voyager 2 inbound map. The adjustment was performed by adding a constant to sin(f2 fsc), where f is the latitude of the data point and fsc is the subspacecraft latitude; the Voyager 1 inbound map was shifted south by 38, and the Voyager 1 outbound map was shifted north by 58. Figure 2 shows the zonal mean temperatures for each of the maps at 130 and 270 mbar. Also shown are the standard deviations of the temperature differences from the zonal mean in 108 latitude bins and typical uncertainties determined from the propagation of instrument noise through the inversion process. The standard deviations of temperature from the zonal mean are roughly a factor of two to four larger than the temperature uncertainties; this indicates that the variance is at least partially caused by real temperature variations with longitude in Saturn’s atmosphere. We find that the variance in temperature is larger at 270 mbar than at 130 mbar. Of particular interest is a local maximum in the deviation of temperatures from the zonal mean at around 408N in the 130-mbar data. 2. Method of Periodogram Analysis To search for periodic structure, we group the temperature differences into bins 108 wide in latitude with the centers of the bins every 58 of latitude, giving two sets of overlapping bins for each map. We Fourier analyze the temperature differences within each bin to estimate the power in each zonal wavenumber. Lomb (1976) and Scargle (1982) have proposed a modification of the standard periodogram to allow the direct use of unequally spaced data without the need to interpolate the data onto a uniform grid. The direct use of the unequally spaced data can reduce aliasing from high wavenumbers onto low wavenumbers (Press and Teukolsky 1988). The Lomb–Scargle periodogram for the spectral power in zonal wavenumber k (i.e., k wavelengths in 3608 of longitude), normalized by the variance of the data, is given by

S

2 1 [oj51 T9j cos k(lj 2 t)] P(k) 5 2 N 2s oj51 cos2 k(lj 2 t) N

N

1

D

[oj51 T9j sin k(lj 2 t)]2

oj51 sin2 k(lj 2 t) N

.

(1)

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FIG. 2. (a) (Upper) shows the zonal mean temperatures at 130 mbar for the Voyager 1 inbound map (dotted line), the Voyager 1 outbound map (dashed line), and the Voyager 2 inbound map (solid line). The strong temperature minimum in the Voyager 1 inbound map at 58S is an artifact caused by the rings. (Lower) shows the standard deviation in 108 latitude bins of the temperature differences from the zonal mean. The symbols indicate an estimate of the temperature uncertainties due to instrument noise. (b) As (a), but at 270 mbar.

Here T9j is the deviation of the temperature from the zonal mean, lj is the longitude of the jth data point, s2 5 N (N 2 1)21 oj51 T9j 2 is the variance of the temperature deviations, and N is the number of data points. The parameter t is defined for each wavenumber k by tan(2kt) 5

O sin 2kl @O cos 2kl N

N

j

j 51

j

(2)

j 51

and is included to make the periodogram independent of the zero point of longitude (Scargle 1982). This is a generalization of the standard periodogram to allow for data that is not evenly spaced and reduces to the standard periodogram in the case of equal spacing. It is important to note that for unequally spaced data, it is generally not possible to find a set of wavenumbers for which the powers at each wavenumber are independent. Because each of the north–south maps was taken over a period of 11 or more hours, any identifiable wave component has an ambiguity in both zonal phase velocity and

wavenumber (Conrath 1981). Consider a reference frame S that rotates from west to east about the planetary axis of rotation with period P. Let l denote west longitude in this system. The phase of a wave of frequency g and planetary wavenumber k at time t is then F 5 kl 1 gt.

(3)

The convention is such that g . 0 implies eastward propagation. For a stationary wave, g 5 0 in (3). Suppose, instead, that the wave propagates 2f radians around the latitude circle in time P0 . Then its frequency, measured in the reference frame S, is simply g 5 2f

S

D

1 1 2 k. P0 P

(4)

Consider the geometry of the north–south mapping sequence. Within each latitude band analyzed there is a sequence of times and longitudes (ti , li), which, when substi-

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tuted in (3), yields the phases Fi . Suppose, for simplicity, that the li are a linear function of the times ti . Then, suitably redefining t1 5 0 and l1 5 0, it follows that ti li 5 . 2f P

(5)

Substitution of (4) and (5) into (3) yields the relation Fi 5

P kli . P0

(6)

The Doppler-shifted wavenumber, P k, kd 5 P0

(7)

is what one would infer if the wave were assumed stationary in the coordinate system S. Equation (7) illustrates two points. First, kd increases with the ratio P /P0 . The faster a wave propagates eastward, the more crests one will count in a north–south mapping sequence as 2f radians of longitude in S sweeps by. Second, for discrete values of the ratio P /P0 , kd is integral. The set of such discrete values is infinite, although in practice other considerations—for example, the requirement that the phase velocities, relative to the background flow, be subsonic—exclude all but a finite number. In computing a periodogram, a nonintegral value of kd means that the spectral power in the mode having wavenumber k is dispersed over several integral wavenumbers. As kd approaches an integer, more and more of the spectral power is concentrated at that integral wavenumber. With this in mind, we adopt the following approach: We compute the west longitudes of the map footprints for a sequence of coordinate systems S having rotation periods P distributed between predetermined minimum and maximum values. For each of these coordinate systems, we compute periodograms; the resulting power spectrum pertains to zonal disturbances assumed stationary in that system. Finally, the power computed for each integral wavenumber is plotted as a function of P. Peaks which occur in this type of plot satisfy (7) to a good approximation, even when the data are not spaced uniformly, and determine the propagation period (around a latitude circle) of candidate modes at each wavenumber, provided their power amplitudes are sufficiently large to be statistically significant. Let us denote the wavenumber and period of a candidate mode by k 9d (here integral) and P 9, respectively. In practice, the periodogram peaks define these periods better at high wavenumbers than at low wavenumbers. This is because the peak widths are comparable to their spacing, which from (7) is proportional to 1/k9d 2 1/(k9d 1 1). In

searching for saturnian waves, we find it more convenient to use the zonal phase velocity c, instead of P, as our independent variable. We compute c relative to the coordinate system that rotates with the period of the modulation of Saturn’s kilometric radio emission (SKR), PSKR 5 10.66 hr, which presumably also characterizes the internal rotation of the planet (Desch and Kaiser 1981). The period P is related to the zonal phase velocity in a coordinate system rotating with period PSKR by the relation c 5 2f

S

D

1 1 2 a cos f, P PSKR

(8)

where a denotes the planetary radius, and c . 0 for eastward propagation. To determine if any peaks in the periodograms are statistically significant, we consider the null hypothesis that the data consist of white noise. Given this assumption, the distribution function for the maximum power in the periodogram may be determined; the details of this calculation are given in the Appendix. From this distribution function, we calculate the probability of a sampling of white noise producing a peak in the periodogram as large as or larger than the maximum power in our actual periodogram; this probability is called the false alarm probability f. A small false alarm probability indicates a statistically significant wave. 3. Results of Periodogram Analysis For each of the 108 latitude bins in each of the global maps, we evaluate the periodogram (1) for wavenumbers 1 through N/2, the nominal Nyquist wavenumber, at zonal phase velocities between 21000 and 1000 m sec21. Waves with phase velocities outside this range would be moving relative to the mean zonal flow at speeds exceeding the sound speed. Although the Lomb–Scargle periodogram can potentially measure power above the nominal Nyquist limit (Press and Teukolsky 1988), for our data, wavenumbers above this limit would have wavelengths smaller than the IRIS field of view. An example periodogram is shown in Fig. 3 for the 308N to 408N latitude bin of the Voyager 1 outbound map at 130 mbar. Figure 3a shows the temperature difference from the zonal mean for each of the data points in this bin, along with a running mean using a 408 wide triangular window, in the SKR reference frame. These data clearly show temperature variations with longitude that are larger than the scatter of the data about the running mean. Figure 3b shows the power in wavenumbers 1 through 36 (the nominal Nyquist wavenumber) at zero phase velocity. We find that wavenumber 2 has a false alarm probability f , 0.01 and is much stronger than the other wavenumbers. Figure 3c shows the power in wavenumbers 1 through 5 as a function of phase velocity; the plot also illustrates the

PLANETARY-SCALE WAVES ON SATURN

355

FIG. 3. Results of a peridogram analysis for the 308N to 408N bin of the Voyager 1 outbound map at 130 mbar. (a) The crosses show the temperature deviations and longitudes for each data point in the map. The solid line is a running mean of the data using a 408 triangular bin. (b) Power, normalized by the variance of the data, in zonal wavenumbers 1 through 36, for zero phase speed. The horizontal dotted lines are the power levels corresponding to false alarm probabilities of 0.10, 0.01, and 0.001. (c) Power in zonal wavenumbers 1 (solid line), 2 (dashed line), 3 (dot–dash line), 4 (dotted line), and 5 (dot–dot–dot–dash line) as a function of phase velocity. The horizontal dotted lines are the power levels corresponding to false alarm probabilities of 0.10, 0.01, and 0.001. (d) As (c), but with a least-squares fit to wavenumber 2 at 2210 m sec21 subtracted from the data.

poor velocity resolution at low wavenumbers. For example, the maximum in the power of the wavenumber 2 component occurs at 2210 m sec21, but the peak is so broad that the phase velocity is hardly resolved at all. If we subtract from the data a least-squares fit to wavenumber 2 at c 5 2210 m sec21 and recalculate the periodogram, we get the result shown in Fig. 3d. The strongest wavenumber in the periodogram is now wavenumber 4, with a false alarm probability f , 0.05, while the normalized power in wavenumber 1 has been cut in half. The wavenumber 5 peak is a Doppler-shifted image of the wavenumber 4 peak, illustrating the ambiguity between wavenumber and phase speed discussed earlier. The normalized power in wavenumber 4 increased because the removal of wavenumber 2 reduces the variance of the data. The decrease in the power in wavenumber 1 when wavenumber 2 is

removed indicates that at least some of the power originally in these wavenumbers is the result of aliasing from wavenumber 2. For the Voyager 1 data, we can combine the inbound and outbound maps, and this has two benefits. First, it increases the signal-to-noise ratio of the periodograms by increasing the number of data points in each latitude bin. Second, it allows an improvement in the determination of the zonal phase velocities of significant wave components. Figure 4 shows the power in wavenumbers 1 through 5 as a function of phase velocity for the 308N to 408N bin in the combined Voyager 1 inbound and outbound maps. Increasing the time coverage of the data has narrowed the peaks considerably; the widths approximately scale inversely with the total length of the time series, here about eight Saturn rotations. However, because the time period

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f0 , and coincident with S at time t1 . The longitude l9 in S 9 at time t is related to the longitude in S by l9 5 l 1

c (t 2 t1). a cos f0

(10)

A wave that is stationary in S 9 and observed at longitudes l9i corresponding to the longitudes li in S will have the phases F9i 5 kl9i 5

H

(11)

kli

for 1 # i # M,

k(li 1 c(t2 2 t1)/a cos f0) for M 1 1 # i # M 1 N.

Thus, if c5

FIG. 4. Results of a peridogram analysis for the 308N to 408N bin of the combined Voyager 1 inbound and outbound maps at 130 mbar. (Upper) shows the temperature deviations and longitudes for each data point in the inbound (1) and outbound (*) maps. The solid line is a running mean of the data using a 408 triangular bin. The dashed line is a least-squares fit to zonal wavenumbers 1, 2, and 4, assuming zero phase velocity. (Lower) shows the power in zonal wavenumbers 1 (solid line), 2 (dashed line), 3 (dot–dash line), 4 (dotted line), and 5 (dot–dot–dot– dash line) as a function of phase velocity. The horizontal dotted lines are the power levels corresponding to false alarm probabilities of 0.10, 0.01, and 0.001.

between the maps is much longer than the time required for the individual maps each wavenumber now has a series of peaks, equally spaced in phase velocity, of slowly varying amplitude over the widths of the peaks in the individual maps—the periodogram is not very sensitive to the number of times the wave propagated through 2f phase during the time between the maps. To understand why this happens, consider for simplicity a data set consisting of M observations at time t1 and N observations at time t2 . In some reference frame S, the phase of a stationary wave of wavenumber k for each observation will be Fi 5 kli .

(9)

Now consider another reference frame S 9 moving with velocity c with respect to S in a bin centered at latitude

2fja cos f0 k(t2 2 t1)

(12)

for some integer j, a stationary wave of wavenumber k in S 9 and a stationary wave of wavenumber k in S will produce identical observations. Because of the large number of latitude bins examined, there is a strong possibility of noise producing apparently significant peaks in the periodograms. For n independent bins, the probability of exactly k bins having waves with false alarm probability f # f0 is given by the binomial distribution, and the ‘‘joint false alarm probability’’ fJ —the probability of one or more waves in each map, with n independent latitude bins, having a false alarm probability f0 or less assuming that the data are white noise—is fJ( f0 ; n) 5

O SknD f n

i51

k 0

(1 2 f0)n2k 5 1 2 (1 2 f0)n.

(13)

Tables I (130 mbar) and II (270 mbar) list all of the wavenumbers in our data with fJ # 0.1. For the individual maps and the combined Voyager 1 maps the number of independent 108 bins ranged from n 5 5 to n 5 9. This means that the largest peak in each bin had to have f0 # 0.01–0.02 to be included in the tables. For latitude bins with more than one significant wave, the normalized power is calculated assuming all of the more significant waves have been subtracted from the data. The amplitudes and phase velocities have been determined by least-squares fitting of all of the significant wavenumbers to the data simultaneously. The quoted uncertainties are the 1s uncertainties of the fit, using the variance of the deviations of the data from the fit as an estimate of the uncertainty in the temperatures; changing the number of wavenumbers included in the fit can change the measured amplitudes

PLANETARY-SCALE WAVES ON SATURN

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TABLE I Observed Waves at 130 mbar

and phase velocities by a large fraction of the quoted uncertainties. Ambiguities in zonal phase velocity and wavenumber have been resolved by choosing the peak with zonal phase velocity closest to zero. For the low wavenumber results in the Voyager 1 outbound and Voyager 2 inbound maps at 130 mbar, there is only one peak in the periodogram with a subsonic phase velocity. For wavenumbers 2 and 4 in the combined Voyager 1 maps, there are peaks at subsonic phase velocities other than in those given in

Table I. For wavenumber 2 there are peaks at 2580 and 1520 m sec21 (see the bottom panel of Fig. 4), and for wavenumber 4 there are several additional peaks, beginning with those at phase velocities 2280 and 1300 m sec21. In Section 3 we will argue that the low-wavenumber features are Rossby waves and show that the properties of these waves preclude these higher velocity solutions. The wavenumber 2 component at 130 mbar is by far the most prominent feature in the data. It has a false alarm

TABLE II Observed Waves at 270 mbar

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temperature is largest (Fig. 2). It decreases rapidly north of 408N, and more slowly south of 358N. We will see in the next section that the observed behavior of wavenumber 2 is consistent with a quasi-stationary planetary-scale Rossby wave confined meridionally by the zonal mean winds. Examination of Table I reveals that the amplitudes of the midlatitude wavenumber 2 and 4 features did not change significantly during the nine months between the encounters. Those of other features did. For example, comparison of the data shows that the wavenumber 1 feature at 308N to 408N doubled in amplitude between the encounters, indicating at least some variability in wave structure on the timescale of a few months. (Although not included in Table I, a wavenumber-1 feature with amplitude 0.26 K and f 5 0.03 was detected at these latitudes in the combined Voyager 1 inbound and outbound maps.) We do not see any evidence in the IRIS data of either the ribbon-like wave observed near 468N in the imaging data (Sromovsky et al. 1983; Godfrey and Moore 1986) or the polar hexagon (Godfrey 1988, 1990). However, the absence of these features in the IRIS data is not surprising, as their widths are roughly 48 of latitude, less than one-half of the projected IRIS field of view during the mapping sequences. 3. MODELING OF PLANETARY-SCALE WAVES

FIG. 5. Observed structure of zonal wavenumber 2 in the combined Voyager 1 inbound and outbound maps. (Upper) shows a smoothed version on the velocity of the mean zonal flow measured by Ingersoll et al. (1984). (Middle and Lower) show the amplitude and phase, respectively, of zonal wavenumber 2, assuming zero phase velocity in System III.

probability f , 0.05 in the 308N to 408N bin in all three maps, and f , 0.01 in all of the latitude bins centered between 208N and 408N in the combined Voyager 1 maps. It is not statistically significant at 270 mbar ( f . 0.95). This is a consequence both of the higher zonal variance at 270 mbar (Fig. 2) and of a lower temperature amplitude. A least-squares fit to wavenumber 2 in the 308N to 408N bin at 270 mbar gives an amplitude of 0.22 6 0.15 K, slightly less than half the amplitude (or one-fourth of the power) as at 130 mbar. Figure 5 shows the observed structure of zonal wavenumber 2 at 130 mbar in the combined Voyager 1 maps, determined by least-squares fitting for the amplitude and phase in each of the 108 latitude bins, assuming the wave is stationary in System III. The phase of wavenumber 2 is nearly independent of latitude between 108N and 458N (even though wavenumber 2 is not statistically significant in the bins centered at 108N and 458N). The amplitude is largest between 358N and 408N, just south of the westward jet at 408N, and where the zonal variance of

Planetary-scale waves can be split into two general classes (e.g., Andrews et al. 1987, Chap. 4): Planetary waves (also called Rossby waves), in which the restoring force is produced by variation of the zonal mean potential vorticity with latitude, and inertio-gravity waves (including Kelvin and Yanai waves), in which the restoring force is produced by buoyancy modified by Coriolis effects. Propagating inertiogravity waves are restricted to frequencies in the range f 2 # g2 # N 2 where N 2 is the Brunt-Via¨sa¨la¨ frequency (a measure of the static stability of the atmosphere) and f 5 2V sin f is the Coriolis parameter. Inertiogravity waves of zonal wavenumber k at latitude f will therefore have a zonal phase velocity c 5 ga cos f /k (relative to the mean zonal winds) greater than u2Va sin f cos fu/k 5 (19780 m sec21)sin f cos f /k. This gives a minimum phase velocity of around 4600 m sec21 for wavenumber 2 gravity waves at 358N, well in excess of the ambient sound speed (P725 m sec21) and therefore unlikely. We will see below that Rossby waves at midlatitudes in Saturn’s upper troposphere generally have phase velocities (relative to the mean zonal flow) of order 100 m sec21 or less. These limits on the phase velocities indicate that the observed midlatitude, low zonal wavenumber waves that dominate the data are probably quasi-stationary planetary-scale Rossby waves. The near-equatorial waves with high phase velocities, and the higher wavenumber waves seen in the 270-mbar data, are possibly inertiogravity waves.

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PLANETARY-SCALE WAVES ON SATURN

S

1. Dynamics of Planetary-Scale Rossby Waves Our examination of the properties and dynamics of planetary-scale Rossby waves follows the treatment of Karoly and Hoskins (1982). Consider a wave with periodic variation of the form exp i(2kl 1 ly 1 mz 2 gt),

(14)

where l is west longitude, y 5 tanh21(sin f) is a Mercator projection of latitude, f, and z ; 2ln( p/1000 mbar) is the vertical coordinate. The dimensionless wavenumbers k, l, and m are related to the wave frequency g through the dispersion relation g 5 kf 2

kbM , k 2 1 l 2 1 a2(m2 1 n2)

(15)

where f 5 U/a cos f is the angular velocity of the background flow with respect to the SKR frame, which rotates at the rate V 5 1.638 3 1024 sec21. U(f, z) is the mean zonal velocity, which generally varies with latitude and altitude, and a 5 60367 km is Saturn’s equatorial radius. The quantity bM 5 2V cos2 f 2 cos f

S

D

F

­ 1 ­ (f cos2 f) ­f cos f ­f

­ ­f 2e z e2za 2 ­z ­z

G

(16)

is cos f times the gradient with respect to latitude of the mean zonal potential vorticity, which is the basis of the restoring force producing the Rossby-wave oscillations. The first term is the contribution of the Coriolis force. In the last term, arising from the vertical variation in the zonal winds, a2 5

sin2 f cos2 f , S

(17)

where S5

S

D

R RT ­T 1 , 2 (2Va) Cp ­z

(18)

is the dimensionless static stability, a measure of how subadiabatic the atmosphere is. Here R is the gas constant, T(z) is the horizontally averaged temperature, and Cp is the specific heat at constant pressure. The quantity n2 in (15) is given by

D

SD

1 ­S ­2S 1 3 ­S 2 2 2 1 2 n2 5 1 . 4 2S ­z ­z 4S ­z

(19)

The first term on the right arises because of the exponential decrease in atmospheric density with altitude, and the remainder from vertical variation in the static stability S. If k21, l21, and m21 are small compared to the length scales characterizing the variation of the background atmosphere (e.g., of bM , a2, and n2) with longitude, latitude, and pressure, respectively, the dispersion relation (15) holds locally, and raytracing, based on the approximations of geometric optics, can be used. In practice, raytracing can still provide insights into the dynamical behavior of waves even when the conditions prescribed above do not strictly hold. The path x 5 (x, y, z) followed by a wave, called a ray, is calculated by integrating the group velocity cg

S

D

dx ­g ­g ­g 5 cg ; , , , dt ­k ­l ­m

(20)

­ d ; 1 cg ? = dt ­t

(21)

where

is the time derivative of a point moving at the group velocity. The variation of wavenumber and frequency along the ray are given by the gradient and time derivative, respectively, of the dispersion relation (e.g., Lighthill 1978): dk dg ­g 5 2=g; 5 . dt dt ­t

(22)

If the static stability and mean zonal winds do not change with time, which we assume to be the case, the frequency of a wave will be a constant along a ray. Furthermore, if the background atmosphere is independent of a particular spatial coordinate, the corresponding wavenumber will be constant. In the present study, we assume a zonally symmetric background atmosphere, so k remains constant along each ray. In these circumstances, the zonal phase velocity c 5 g/k also remains constant along each ray. Examination of the dispersion relation (15) reveals several general properties of planetary waves. The phase velocity c is always westward relative to the zonal mean flow where bM is positive, eastward where bM is negative, and it has a maximum value relative to the zonal mean flow of bM /(a2n2). The dispersion relation can be rewritten in the form k 2 1 l 2 1 a2m2 5

bM 2 a2n2 ; n 2. f2c

(23)

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ACHTERBERG AND FLASAR

dU dT R . 52 dz 2Va sin f df

FIG. 6. (a) Nondimensional static stability S as a function of pressure, calculated from the Voyager 2 ingress radio occultation temperature profile (Lindal et al. 1985). (b) The function n2 calculated from the static stability profile in (a). The vertical dotted line is n2 5 0.25, the value of n2 when the static stability is independent of altitude.

The quantity n is called the index of refraction, by analogy with geometric optics, and depends only upon the background atmosphere and c. Waves of zonal wavenumber k can propagate where n 2 . k 2 and are evanescent (either growing or decaying exponentially) where n 2 , k 2. Karoly and Hoskins (1982) have shown that waves are refracted toward the direction of =n and thus that maxima in the index of refraction can act as waveguides. Surfaces where n 2 5 k 2 are turning points, at which waves are reflected. Surfaces where f 5 g/k, and thus un 2u R y, are called critical surfaces. When a critical surface bounds a propagating region, an approaching wave has a total wavenumber k 2 1 l 2 1 a2m2 that increases to infinity. The group velocity normal to the critical surface goes to zero sufficiently rapidly that the wave requires an infinite amount of time to reach it. If viscous effects dominate, the wave will be absorbed by the zonal mean flow (Booker and Bretherton 1967). 2. Raytracing Solutions for Saturn To solve the raytracing equations, we need to specify the mean zonal velocity and static stability profiles. The static stability S, shown in Fig. 6, is calculated from the Voyager 2 radio occultation ingress temperature profile of Lindal et al. (1985). At the level of the observed clouds, the mean zonal velocity can be measured by tracking discrete features (Sromovsky et al. 1983, Ingersoll et al. 1984). Although direct measurements of wind velocities at other altitudes is not generally possible, the vertical wind shear can be calculated from zonal mean temperature measurements through the thermal wind equation

(24)

Temperatures retrieved from high spatial resolution Voyager IRIS data by Conrath and Pirraglia (1983) indicate that the vertical wind shear is negatively correlated with the mean zonal winds at 150-mbar pressure for latitudes poleward of about 208. The positive and negative values for the wind shear are of roughly equal magnitude, but the eastward jets in the zonal mean winds are much stronger than the westward jets. Thus, if the wind shear persists into the stratosphere, the winds will decay to a roughly uniform eastward rotation. The correlation between the thermal wind shear and the zonal mean flow is also present to some degree at 290 mbar, although the amplitude variation of the temperature is much smaller. At 730 mbar the correlation is absent. At this level lateral variations in cloud opacity may be important. Figure 7 shows the cloud top zonal mean velocity Uc , along with a calculation of bM under various assumptions. The zonal velocity is taken from Table I of Sromovsky et al. (1983) and smoothed over 58 latitude. Figure 7 (top) depicts bM for the barotropic case, in which the mean zonal wind is assumed to be independent of altitude. In Figure 7 (middle) ­u/­z 5 / 0 and is computed from (24), using temperatures at 130 and 270 mbar. (We have neglected the contribution of fzz Y Tfz to the last term in (16), as temperatures were only available at these two pressure levels; its contribution to this term is relatively small, approximately 20%.) Figure 7 (bottom) depicts bM for models in which the zonal wind decays exponentially with altitude to a uniform eastward rotation f( y, z) 5 f0 1 (fc ( y) 2 f0) F (z),

(25)

where F(z) 5

H

exp 2 (z 2 z0)/H, if z $ z0 ;

1,

if z , z0 ,

(26)

and fc 5 Uc /a cos f. The zonal velocity is equal to the cloud top zonal velocity for z , z0 and decays to a constant value f0 as z R y on a scale of H pressure scale heights. It is important to point out that existing wind and temperature data do not determine bM very precisely, particularly at latitudes spanning the critical surface, 378N–418N, where bM is small. There are large uncertainties in the meridional curvature of the mean zonal winds (see, e.g., Fig. 13 of Sromovsky et al. 1983), and the meridional spatial resolution of the retrieved temperatures is several degrees of latitude, coarser than that of the measured cloud top winds themselves. For our computations we therefore use the simplified model given by (25) and (26). Once the mean

PLANETARY-SCALE WAVES ON SATURN

FIG. 7. Cloud top zonal winds U (heavy dashed lines) used in the raytracing models and zonal mean potential vorticity gradient bM (solid lines) for several cases. (Top) Barotropic case, H 5 y. (Middle) Computed from 270- and 130-mbar temperatures, neglecting fzz in (16). (Bottom) Constant decay model with H 5 5 and 10 with f0 5 1.0 3 1026 sec21. In each of the panels the lighter dashed curves depict U 2c, where the zonal phase velocity c corresponds to g/k 5 26.0 3 1027 sec21 in the upper curves—approximately 230 m sec21 at 358N—and g/k 5 16.0 3 1027 sec21 in the lower curves.

zonal velocity and static stability are chosen, the raytracing equations (20) and (22) are integrated numerically using a variable-order predictor-corrector method (routine LSODE from ODEPACK; Hindmarsh 1983). The initial condition for the ray is determined by choosing the frequency g, the zonal wavenumber k, the initial position of the ray, and the initial direction of travel for the ray l/m; with these assumptions the initial wavenumbers l and m are determined from the dispersion relation.

361

Figure 8 (upper) depicts contours of the index of refraction n for stationary waves (c 5 0) with H 5 10, f0 5 1.0 3 1026 sec21 (corresponding to a velocity of approximately 50 m sec21 at midlatitudes), and z0 corresponding to a pressure of 270 mbar. In the raytracing approximation, waves with zonal wavenumber k only propagate where n 2 . k 2 . 0. Hence, wave propagation at pressures greater than about 20 mbar is restricted to latitudes between 318N and 408N and pressures less than about 200 mbar at most latitudes. The higher the zonal wavenumber, the more restricted the latitude range is. The meridional confinement of the waves is caused by the variation of the mean zonal flow with latitude, while the restriction of waves to the upper troposphere and stratosphere is caused by the thermal structure of Saturn’s atmosphere. Stationary waves can propagate only where bM /f . a2n2 1 k 2. Thus, the increase in zonal velocity f equatorward of 31– 328N and poleward of 408N accounts for the evanescent regions at these latitudes near the tropopause (p100 mbar). (The narrow evanescent region near 378N results from a local minimum in bM , illustrated in Fig. 7 (bottom)). Wave propagation at pressures above 200 mbar is inhibited by the low static stability and by the large maxima in n2 at 500 mbar (which is caused by the large static stability gradient). Of course, wave disturbances do penetrate the evanescent regions, where n 2 , 0, but in the absence of sources within these regions they should decay with distance away from the propagation region(s). The more negative n 2 is, the faster the decay. Thus, the large, negative values of n 2 that occur at pressures greater than 150–200 mbar are consistent with the reduced amplitude and low statistical significance of the wavenumber 2 component in the 270-mbar temperatures. Furthermore, the decay of a planetary wave in the tropopause region equatorward of 338N would be expected to be slower than that northward of 408N, where the meridional gradient of n 2 is much steeper. This is consistent with the meridional structure of the wavenumber 2 feature in Fig. 5, in which the amplitude decreases rapidly north of p408N. Note also that from (23), l 2 is more negative in the evanescent regions the larger k is. This may explain why the wavenumber 2 feature is detectable to lower latitudes than higher wavenumber features, such as k 5 4, are. Figure 8 (lower) illustrates raypaths for zonal wavenumber 2 disturbances emitted from two sources in the lower stratosphere at midlatitudes. The circles mark position along the rays at one-week intervals. Many of the rays, sometimes after one or more reflections, leak out into the upper stratosphere. The paths shown for the more southern source indicate that some rays can undergo many reflections in the lower stratosphere before escaping upward, depending on the geometry of the boundaries of the propagation regions. In the present example, several of the rays emitted from the more northern source are attracted to

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ACHTERBERG AND FLASAR

FIG. 8. Results of raytracing calculations for f0 5 1.0 3 1026 sec21,

H 5 10 scale heights, z0 5 270 mbar, k 5 2, and g 5 0. (Top) Contours of the index of refraction n. The thin solid line is the contour for n 5 0, the dotted lines are contours where n 2 . 0 (n real), and the dashed lines are contours where n 2 , 0 (n imaginary). Contours are labeled with un u. Contour level are un u 5 0, 2, 4, ..., 14, 16, 32, 64, 128. The thick solid line is a critical surface, where un u R y. (Bottom) As (top), with paths of rays emitted from two sources superimposed (thick solid lines). The open circles mark positions along the rays at one-week intervals.

the critical surface. One should bear in mind the limitations of the raytracing model and note that the two propagating regions are not as isolated as Fig. 8 suggests. Waves in one propagating region very likely penetrate the thin evanescent region into the other. Figure 9 depicts the meridional and vertical wavenumbers and phase for three of the rays in the more southern propagating region in Fig. 8. Vertical wavenumbers are in the range umu & 1, corresponding to vertical scales umu21 of a scale height or greater, comparable to or larger than the vertical resolution of the Voyager IRIS data. Meridional wavenumbers are ulu & 10, which implies (dimensional) scales ulu21 * 48, comparable in magnitude to the width of the propagating region in the upper troposphere (48–58). All this seems consistent with the

constancy with latitude of the zonal phase of the wavenumber 2 feature depicted in Fig. 5. There is not a large change in zonal phase within most of propagating region itself, and bins south of 308N sample latitudes where the wave is evanescent and the phase does not change at all. The dispersion relation (15) restricts Rossby-wave phase velocities to the range 0 $ c 2 f $ 2bM /(k 2 1 a2n2). At midlatitudes in Saturn’s upper troposphere, bM P 2V P 3.28 3 104 sec21, S P 2 3 1024 (and thus a2 P 1000), and n2 P Af (see Figs. 6 and 7), for a maximum westward phase velocity of about 80 m sec21 relative to the mean zonal flow. However, midlatitude wave propagation near the tropopause is already quite restricted for more modest westward phase velocities. Figure 10 illustrates the effect of moderately small zonal phase velocities (P 630 m sec21) on the index of refraction. Westward propagating waves are generally confined to the upper stratosphere. Eastward propagating waves occur farther south than stationary waves, at latitudes where the mean zonal flow is more strongly eastward. Hence, if the observed midlatitude features are planetary waves, the observed maximum in the amplitudes at latitudes 308N–408N at the tropopause (Fig. 5) suggests that the zonal phase speeds are small, ucu ! 100 m sec21. Figure 11 illustrates the effect of changing the asymptotic zonal velocity f0 from the value in Fig. 8 to 0. With f0 5 0, wave propagation is confined to the stratosphere above the 100-mbar level, except for narrow (,0.58) propagation regions adjacent to the critical surfaces. We have also examined the effects on the raytracing solutions of changing the decay scale H of the zonal mean winds. Changing H by a factor of 2 leads to only modest changes in the behavior of

FIG. 9. Meridional and vertical wavenumbers for the rays labeled a, b, and c in Fig. 8, emanating from the more southern source: a, solid lines; b, dotted lines; and c, dashed lines. The crosses mark the initial positions of the rays.

PLANETARY-SCALE WAVES ON SATURN

FIG. 10. Index of refraction contours for the conditions described in the legend to Fig. 8, except with g/k 5 26.0 3 1027 sec21 (Top), which corresponds to a phase velocity of approximately 230 m sec21 at midlatitudes, and g/k 5 16.0 3 1027sec21 (Bottom).

363

wavelengths (Sromovsky et al. 1983, Ingersoll et al. 1984) indicate the presence of what appear to be unstable convective cloud systems. This activity is near the critical latitude, where c 5 U P 0 (Fig. 7), and it is also near the latitude where the gradient of the potential vorticity bM (Eq. (16)) vanishes. The vanishing of this gradient is well known as a necessary condition for baroclinic and barotropic instabilities (Charney and Stern 1962). Lindzen and Tung (1978) and Lindzen et al. (1980) (see also Lindzen 1988) have suggested that these instabilities are associated with the self-excitation of Rossby waves. The excitation entails overreflection of waves incident on a critical surface and confinement of waves propagating away from it. In the present case, the latter is ensured by the evanescent region bounding the propagation region equatorward of p308N, although the waves eventually can leak out into the upper atmosphere. To have overreflection, an evanescent region, where n 2 , 0 in (23), must lie between the critical surface and the region of wave propagation. This requires that the potential vorticity gradient reverse sign between the region of wave propagation and the critical layer. For the models we have examined in which there is exponential vertical decay of the winds to a uniform eastward rotation, the gradient of the potential vorticity, instead, vanishes away from the wave propagating region on the far side of the critical latitude (see Fig. 7 (bottom)). Hence the critical surface in Figs. 8 and 10 (but not 11) bounds the midlatitude propagation region. However, the 270-mbar temperatures (Fig. 7 (middle)) suggest that bM does vanish between the critical surface and the propagation region. Given the uncertainties in bM , in particular where it vanishes, it is possible that Saturn’s midlatitude atmosphere does satisfy the overreflection criterion, and the midlatitude features are forced by an instability in the vicinity of the critical latitude near 408N.

the rays depicted in Fig. 8. In any case, realistic assumptions about the variation of the winds with altitude give raytracing solutions with propagation of quasi-stationary waves in the upper troposphere confined to latitudes near 358N, and pressures less than p200 mbar. 4. DISCUSSION

The nearly stationary midlatitude k 5 2 feature (as well as the midlatitude features with k 5 1 and k 5 4 in Table I) is reminiscent of the quasi-stationary waves on Jupiter identified by Magalha˜ es et al. (1989, 1990) and Deming et al. (1989). Magalha˜ es et al. (1990) have suggested that the stationarity of the jovian waves indicates that they are not locally excited, but rather are directly forced by the deep atmosphere or interior, where the winds vanish. However, the Saturn results may admit an alternate hypothesis. The amplitude of the wavenumber 2 feature is largest near 408N (Fig. 5), where spacecraft observations at visible

FIG. 11. Index of refraction contours for the conditions described in the legend to Fig. 8, except f0 5 0.

364

ACHTERBERG AND FLASAR

This interpretation naturally accounts for why the features are quasi-stationary. The lighter dashed curves in Fig. 7 depict the quantity U 2 c near the cloudtops for phase velocities c p 630 m sec21. For westward phase velocities & 210 m sec21, U 2 c does not vanish anywhere, and there is no critical surface, hence no overreflection. This situation is also indicated in Fig. 10 (upper). For eastward phase velocities, the more southerly boundary of the critical surface moves equatorward into the propagating region at p308N–368N. Even with the uncertainties in bM , it is still evident that as c becomes more positive, it is less likely that an evanescent region will lie between the propagating region and the critical surface. Hence, the likelihood of overreflection occurring is highest for small phase velocities. In essence, the near-stationarity of the features is a consequence of bM vanishing at a latitude where the zonal winds are relatively low, in other words, a consequence of the zonal-wind and thermal structure near the tropopause. The effects of Saturn’s interior only enter indirectly, insofar as they relate to the generation and maintenance of Saturn’s zonal winds. The critical-surface instability that may be associated with the observed midlatitude Rossby waves can generate horizontal eddy momentum fluxes that act to smooth out the zonal winds with altitude (Pirraglia 1989, Orsolini and Leovy 1989). However, the waves that leak out of the tropospheric region and propagate upward can also interact with the zonal winds at higher levels. These effects can be estimated by considering the Eliassen–Palm flux of the quasi-geostrophic Rossby wave. The flux F is (e.g., Andrews et al. 1987, Section 4.5.5)

S

D

H2 F kuc9u2 5 0, l, 2 a2m , 2 r 2a a

(27)

where r is the density of the background atmosphere, c 9 is the amplitude of the streamfunction perturbations of the wave, and H is the pressure scale height (about 30 km in the upper troposphere). The amplitude of the streamfunction c 9 can be calculated from the observed amplitude of the temperature fluctuations T 9 from hydrostatic and geostrophic balance, giving

c9 5

RT 9 , 2V(im 1 (c/2)) sin f

(28)

where R is the gas constant and c 5 1 1 Sz /S. For the observed wavenumber 2 wave, T9 P 0.5 K, m P 0.5 for waves which can propagate into the stratosphere, and c P 1.25, or uc 9u 5 1.1 3 107 m2 sec21. For these waves, ulu P 5, m P 0.5 (see Fig. 9), and a2 P 1000 at the level of the observations, which gives an Eliassen–Palm flux of approximately F/ r 5 (0, 20.16, 0.008) m2 sec22 for wave-

number 2 at 130 mbar. The effect of a wave on the zonal mean flow depends upon the divergence of the Eliassen– Palm flux (e.g., Andrews et al. 1987, Section 3.5); ignoring other sources of forcing and dissipation, ­U 5 r21= ? F. ­t

(29)

We cannot determine the divergence of F directly from the data, as the wave is observed at only one pressure level. However, we can make a rough estimate by assuming that a vertically propagating wave loses its energy to the mean zonal flow over an altitude range of a pressure scale height; in this case, ­U/­t 5 r21 = ? F P 2F/H 5 22.7 3 1027 m sec22 (although the meridional component of F is larger than the vertical component, the divergence will be dominated by the vertical derivative as the vertical length scale H is three orders of magnitude smaller than the meridional length scale a). This value of the Eliassen–Palm flux divergence would cause a zonal jet with a velocity of order 100 m sec21 to decay on a time scale of 3.7 3 108 sec, or about 12 years, which is comparable to the radiative damping time in Saturn’s upper troposphere and stratosphere (Conrath and Pirraglia 1983, Conrath et al. 1990). This is consistent with the results of linear, axisymmetric dynamical models having Newtonian radiative and viscous damping, which attempt to fit observed cloud-top winds and temperatures in the upper tropospheres and lower stratospheres of the giant planets. These models obtain best fits when the ratio of the viscous damping time to the radiative damping time is approximately unity at these altitudes. (Gierasch et al. 1986, Flasar et al. 1987, Conrath et al. 1991). Where might the vertically propagating waves deposit their zonal momentum? The Eliassen–Palm flux F will generally be conserved unless a critical level is encountered or the wave amplitude grows until it ‘‘breaks’’ or saturates. For stationary waves, the former possibility is remote if the sense of the vertical shear in the zonal wind near the tropopause persists to higher levels–e.g., the winds decay to a state of uniform eastward rotation (Section 3). To estimate the effects of breaking, note that if F is conserved, (27) implies that the amplitude of a vertically propagating wave will vary approximately as r21/2, where r is the density of the ambient atmosphere at the altitude of the wave. Vertical variation in the zonal winds and thermal stratification will also affect the amplitude, although probably not as strongly as the decrease in pressure; in any case, the winds and stratification are poorly known in the upper atmosphere. The amplitude of the wave will eventually become large enough with altitude that the wave breaks, and nonlinear effects become dominant. One criterion for breaking is that the vertical tempera-

365

PLANETARY-SCALE WAVES ON SATURN

ture gradient of the background atmosphere plus wave remain exceed the adiabatic lapse rate, i.e.,

U U

­T 9 ­T RT $ , 1 ­z ­z Cp

(30)

where T is the temperature of the unperturbed atmosphere and Cp is the specific heat at constant pressure. A vertically propagating wave with the properties of the observed wavenumber 2 feature would satisfy u­T9/­zu P (m2 1 (c/2)2)1/2 uT9u P 0.4(130 mbar/p)21/2 K. If we assume that Saturn’s stratosphere is nearly isothermal (Lindal et al. 1985), then ­T/­z 1 RT//Cp P 45 K, which gives an estimated breaking level around 0.01 mbar. Alternatively, Lindzen and Schoeberl (1982) have proposed that Rossbywave amplitudes might be limited by the criterion that the meridional derivative of the potential vorticity of the wave should not exceed that of the background flow (i.e., bM). Otherwise, the meridional gradient of the total potential vorticity can change sign, satisfying the necessary criterion for barotropic and baroclinic instabilities. The amplitude of the potential vorticity variations of a Rossby wave, q9 (sec21), is related to the streamfunction variations c 9 (m2 sec22) by

uq9u 5

[k 2 1 l 2 1 a2(m2 1 n2)] uc 9u a2

(31)

(Karoly and Hoskins 1982). With l and m as before, n2 P 0.25 (see Fig. 7), and a2 P 500 in the stratosphere, the potential vorticity of a vertically propagating wavenumber 2 wave uq9u P 1 3 1026 (130 mbar/p)1/2 sec21, and the meridional gradient is ulq9u P 5uq9u. For a crude estimate of the implied breaking level, we assume that the background potential vorticity gradient is dominated by that in the planetary vorticity bM P 2V cos2 f. This assumption may actually be fairly good, if the decay of the zonal winds inferred in the upper troposphere and tropopause from the temperature field persists through the upper stratosphere. This implies a breaking level of approximately 0.07 mbar. Lindzen and Schoerbel have also proposed the more conservative criterion that the perturbation potential vorticity of the wave cannot exceed the range of potential vorticity of the background flow, which follows from the conservation of potential vorticity: in the absence of sources or sinks of potential vorticity, the potential vorticity of the zonal flow plus the wave must remain within the range of potential vorticity of the background flow. This criterion implies amplitude saturation at much higher altitudes—0.003 mbar–than even (30) implies. All this leads us to expect that if our observed waves do leak into the stratosphere, they will reach a pressure level p1022 – 1021 mbar before breaking.

5. CONCLUSIONS

We have used a set of global maps obtained by Voyager IRIS to search for planetary-scale waves in the upper troposphere of Saturn. The list of statistically significant waves (Tables I and II) is dominated by zonal wavenumber 2 at northern midlatitudes at 130 mbar. This feature is not statistically significant at 270 mbar, both because its amplitude is lower and because the zonal variance of the 270mbar temperatures is higher. The lack of observed waves in the southern hemisphere is possibly a selection effect—the only southern hemisphere data are in the Voyager 1 inbound map, which has the fewest data points in each latitude bin, and also fewer points in the southern hemisphere bins than in the northern hemisphere bins. Comparison of the Voyager 1 and Voyager 2 data shows that the amplitude of wavenumbers 2 and 4 at midlatitudes did not change significantly, but that of other features did, indicating that there is at least some variability in wave structure on the time scale of a few months. Raytracing calculations suggest that the observed structure of the midlatitude wavenumber 1, 2, and 4 features at 130 mbar is consistent with models of the dynamics of planetary-scale Rossby waves. Meridional variations of the mean zonal winds confine the propagation of quasi-stationary waves to latitudes between about 308N and 408N; variations in the static stability restrict wave propagation to pressures less than about 200 mbar. Waves will decay exponentially outside the propagating regions. Higher zonal wavenumber midlatitude Rossby waves may be present, but their regions of propagation are more restricted and their decay outside these regions more rapid, than the lower wavenumber features, that they are less likely to have been observed. The zonal wavenumber 1 waves seen near the equator have phase velocities too large for Rossby waves and are probably equatorially trapped gravity waves if real. We have only been able to determine zonal phase velocities with any resolution for waves identified in the combined Voyager 1 incoming and outgoing maps (Table I). The midlatitude wavenumber 1, 2, and 4 Rossby-like waves are nearly stationary with respect to System III and are reminiscent of the quasi-stationary waves on Jupiter that have been identified by Magalha˜ es et al. (1989, 1990) and Deming et al. (1989). Although the periodogram analysis allows other solutions with phase velocities, .100 m sec21, these are not consistent with the interpretation of the features as Rossby waves. The observed meridional structure of the waves suggests that they may be excited by an instability associated with overreflection from a critical surface that lies just north of their location. The near-stationarity of these features is then a consequence of the meridional gradient of the potential vorticity vanishing—a necessary condition for the instability—in a region where the zonal winds are small.

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ACHTERBERG AND FLASAR

The raytracing studies indicate that the stationary Rossby waves can leak upward and break at pressures p1022 –1021 mbar at midlatitudes. They may thus provide a coupling between the tropopause region and the upper atmosphere, providing a braking of the zonal winds at high altitudes. The estimated time scales for momentum damping are p10 years, comparable to the stratospheric radiative damping time. APPENDIX

Statistical Behavior of the Lomb-Scargle Periodogram 1. Distribution Function for a Single Wavenumber To estimate the significance of peaks in the periodogram, we consider the null hypothesis that the data are normally distributed noise. We assume a data set X(lj ) sampled at longitudes lj , with j 5 1, 2, ..., N, where X(lj) are independent random variables taken from a normally distributed population with zero mean and variance s 20 . We write the periodogram as P(k) 5

1 (C 2(k) 1 S 2(k)), 2s2

(A-1)

where C(k) 5

FO FO FO FO N

j51

?

S(k) 5

N

j51

N

j51

?

X(lj) cos k(lj 2 t)

G

21/2

cos2 k(lj 2 t)

X(lj) sin k(lj 2 t)

N

j51

G

sin2 k(lj 2 t)

G

,

(A-2)

G

21/2

,

(A-3)

s 2 5 (N 2 1)21 oj51 X 2j is the sample variance, and t is given by Eq. (2). C(k) and S(k) are normally distributed random variables with zero mean and variance s 20 ; furthermore, because of the t terms, the covariance of C(k) and S(k) is zero (Lomb 1976). The quantity (C 2 1 S2)/ s 20 is thus the sum of two uncorrelated, normally distributed random variables of zero mean and unit variance, and follows a x 2 distribution with two degrees of freedom. Therefore, the periodogram normalized by the population variance, Z(k) 5 (C(k)2 1 S(k)2)/2s 20 , has a cumulative distribution function N

FZ(z) 5 PrhZ(k) . zj 5

E

y

z

pZ(z9) dz9 5 exp(2z).

(A-4)

Many authors have applied the exponential distribution function to the periodogram normalized by the sample variance (Scargle 1982, Horne and Baliunas 1986, Press and Teukolsky 1988). Koen (1990) has pointed out that the ratio of sample variance to the population variance, s 2 / s 20 , is itself a random variable which follows a x 2 distribution with N 2 1 degrees of freedom. The periodogram when normalized by the sample variance is thus the ratio of two x 2 distributions. If Z(k) and s2 / s 20 were independent random variables, then P(k) would follow an F-distribution, as was suggested by Koen (1990). However, the sample variance is weakly correlated to the power normalized by the population variance. This is

FIG. A1. (a) Scatter plot of the ratio of sample variance to population variance as a function of the power in zonal wavenumber 1, for 10,000 random data sets consisting of Gaussian noise at 73 points equally spaced at 58 intervals in longitude. (b) Distribution function for the power in zonal wavenumber 2 for a data set consisting of Gaussian noise sampled at the same longitiudes as the data in the 308N to 408N bin of the Voyager 1 inbound map. The solid line is the distribution function for power normalized by the sample variance. The dashed line is the distribution function for power normalized by the population variance. The dotted line is the exponential function. The dot–dashed line is Fisher’s (1929) distribution function (A-5). shown in Fig. A1a for 10,000 data sets consisting of Gaussian noise, with zero mean and unit variance, at 73 points spaced at 58 intervals. The data sets with the highest power (when normalized by the population variance) all have sample variances larger than the population variance. Therefore, the probability of observing high powers when the periodogram is normalized by the sample variance is lower than predicted by the exponential distribution. The correct distribution function for the periodogram normalized by the sample variance was determined by Fisher (1929) for the case of N equally spaced data evaluated at the set of wavenumbers ki 5 2fi/(lN 2 l1) for i 5 1, 2, ..., (N 2 1)/2 FP(z) 5 PrhP(ki) . zj 5 (1 2 g(z))(N23)/2,

(A-5)

where g(z) 5 z

FO j51

G

21

(N21)/2

P(kj)

5

2z . N21

(A-6)

Figure A1b shows the result of Monte Carlo calculations of the distribution function for the power in a single wavenumber for unequally spaced

367

PLANETARY-SCALE WAVES ON SATURN data, which indicate that (A-5) is also a good approximation for unequally spaced data when g(z) is defined as g(z) 5 2z/(N 2 1). Because the maximum value of P(k) is (N 2 1)/2, g is restricted to the range 0 # g # 1.

2. False Alarm Probability If we have measurements of P(ki) at M different wavenumbers ki at which the power P(ki) is independent of the power at the other wavenumbers, then the probability of the largest value in a periodogram exceeding the value z, assuming a data set consisting of white noise, is given by the false alarm probability f, defined as (Scargle 1982) f ; Prhmax P(ki) . zj 5 1 2 (1 2 FP(z)) M.

(A-7)

i

If z is chosen to be the power of the largest peak in a periodogram, then 1 2 f is its statistical significance. Equation (A-7) assumes the periodogram is evaluated at M independent wavenumbers. For the case of N equally spaced data points, the set of wavenumbers kj 5

2fj (lN 2 l1)

(A-8)

are uncorrelated for 1 # j # M0 5 (N 2 1)/2. However, for equally spaced data,

O P(k ) 5 (N 2 1)/2. M0

j51

(A-9)

j

Thus, the set of M0 values of P(ki) are not completely independent but are restricted to a M0 2 1 dimensional hyperplane in the M0 dimensional space of possible sets of values. This restriction was used by Fisher (1929) to calculate the proper false alarm probability for equally spaced data (evaluated at all M0 wavenumbers): f (z) 5 Prhmax P(ki) . zj i

O L

S

D

M0! jz 12 5 (21) j21 j!(M0 2 j)! M0 j51

M021

(A-10)

FIG. A2. (a) Results of a Monte Carlo calculation of the false alarm probability f for a data set with 73 points equally spaced at 58 intervals. The solid line is the results of the Monte Carlo calculation. The dotted line (almost invisible beneath the solid line) is Fisher’s (1929) solution for f (A-10). The dashed line is Eq. (A-7), using (A-5) as the distribution function for the power at a single wavenumber. The dot–dashed lines is Eq. (A-7), using the exponential distribution for the power at a single wavenumber. (b) As (a), but for a data set with points at the same longitiudes as the data in the 308N to 408N bin of the Voyager 1 inbound map.

,

where L is the largest integer less than M0 /z. For unequally spaced data, there is in general no set of wavenumbers that are uncorrelated. However, pairs of wavenumbers that are near the multiples of the wavenumber k1 5 2f/(lN 2 l1) are often (though not always) only weakly correlated, and the restriction (A-9) does not apply, so that (A-7) provides a reasonable approximation to the true false alarm probability. Figure A2 shows Monte Carlo calculations of the false alarm probability for both equally spaced and unequally spaced data sets. For the equally spaced data (Fig. A2a), f is nearly indistinguishable from (A-10). For the unequally spaced data (Fig. A2b), f is within a few percent of (A-7) if (A-5), and not an exponential, is used as the distribution function for a single wavenumber. When calculating the periodogram for unevenly spaced data, it is common to oversample the spectrum on a finer wavenumber grid than that given by (A-8). For our Voyager IRIS data we have evaluated the periodogram at integer wavenumbers for a range of assumed phase velocities, which is approximately (but not exactly) equivalent to oversampling the spectrum at fractional wavenumbers. The effects of oversampling on the false alarm probability are shown in Fig. A3a for the 308N to 408N bin of the Voyager 1 inbound map with the periodogram oversampled by factors of 1, 2, 4, 8, 16, and 32, and with the assumed phase velocity varied between 21000 and 1000 m sec21. Oversampling of the periodo-

gram increases the false alarm probability, by up to a factor of about 4 to 5. When the periodogram is oversampled, the power in two different wavenumbers can be strongly correlated, so that (A-7) is not generally a good approximation to the true false alarm probability when M is set equal to the number of wavenumbers at which the periodogram has been calculated. Horne and Baliunas (1985) and Press and Teukolsky (1988) have suggested that an appropriate value for the ‘‘number of independent wavenumbers’’ M can be determined by Monte Carlo calculations with a representative data spacing. We can write (A-7) in the form M5

ln(1 2 f ) . ln(1 2 FP)

(A-11)

Figures A3b and A3c show M/N for the 308N to 408N bin of the Voyager 1 inbound map for oversampling factors of 1, 2, 4, and 8, as well as for variable assumed phase velocity, using both Fisher’s distribution (A-5) and the exponential distribution (A-4) for FP(z). Assuming an exponential distribution, M becomes small (,0.1N) for large powers. Thus (A-7), using an exponential distribution and M P N (as suggested by Horne and Baliunas 1985 and Press and Teukolsky 1988) can overestimate the

368

ACHTERBERG AND FLASAR 1.5N, depending upon how heavily the periodogram is oversampled. It should also be noted that if the periodogram is extended to wavenumbers above the nominal Nyquist wavenumber Nf/(lN 2 l1), then M will need to be increased accordingly. If a more accurate estimate of the false alarm probability is desired, it needs to be calculated by Monte Carlo simulation. The false alarm probabilities used in this paper were calculated by Monte Carlo simulation.

ACKNOWLEDGMENTS We thank B. Conrath and C. Barnet for much useful advice in retrieving temperatures from the Voyager IRIS spectra. This work was supported by the NASA Planetary Atmospheres Program. R.K.A. was supported by a National Research Council Resident Research Associateship at NASA Goddard Space Flight Center during part of this research.

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FIG. A3. (a) False alarm probabilities for the 308N to 408N bin of the Voyager 1 inbound map, with the periodogram oversampled by factors of 1, 2, 4, 8, 16, and 32 (dashed lines), and with the periodogram evalued at phase velocities between 21000 and 1000 m sec21 (solid line). The lines for 8, 16, and 32 times oversampling are nearly indistinguishable. The dotted lines are false alarm probabilities predicted by Eq. (A-7), with M 5 N/2, M 5 N, and M 5 2N, and FP(z) given by Eq. (A-5). (b) ‘‘Number of independent frequencies’’ M/N as a function of power for the false alarm probabilities shown in (a), with FP(z) as given by Eq. (A5). (c) As (b), but with FP(z) 5 e2z, with the power normalized by the sample variance.

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