Global Kelvin waves in the upper atmosphere excited by tropospheric forcing at midlatitudes

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jd007235, 2007 Global Kelvin waves in the upper atmosphere excited by tropospheric forcing at midlatitudes Murry L. Salby, 1 Ludmila Matrosova, 2 and Patrick F. Callaghan 2 Received 27 February 2006; revised 6 September 2006; accepted 5 October 2006; published 22 March [1] Nonlinear integrations with a 3-D primitive equation model of the middle and upper atmosphere reveal global-scale Kelvin waves. Owing to their meridional extent, these eastward propagating disturbances can be excited by tropospheric fluctuations over much of the globe. Stochastic forcing in the midlatitude troposphere produces an eastward response that involves the Kelvin normal mode, with barotropic vertical structure, as well as a continuum of vertically propagating Kelvin waves. Having periods of order a day and shorter, those disturbances are all global. Transient fluctuations representative of midlatitude weather systems reproduce observed Kelvin structure and amplitude near the tropopause. Vertical amplification then leads to wind fluctuations at mesospheric and thermospheric altitudes of 5 15 m/s. Approaching tidal amplitudes, global Kelvin waves should therefore represent a measurable if not prominent feature at those altitudes. Citation: Salby, M. L., L. Matrosova, and P. F. Callaghan (2007), Global Kelvin waves in the upper atmosphere excited by tropospheric forcing at midlatitudes, J. Geophys. Res., 112,, doi: /2006jd Introduction [2] Kelvin waves figure importantly in the momentum budget of the tropical atmosphere. Trapped within a neighborhood of the equator, these eastward disturbances propagate upward from deep convection, wherein they are excited through the release of latent heat [Holton, 1972; Salby and Garcia, 1987; Bergman and Salby, 1994]. Their absorption in the middle atmosphere contributes to the easterly phase of the QBO and likewise to the SAO [Holton and Lindzen, 1972; Hitchman and Leovy, 1988; Dunkerton, 1979; Sassi and Garcia, 1997]. [3] Satellite observations, buttressed by radiosonde and rocket measurements, reveal that equatorial Kelvin waves actually appear in a spectrum, in which their structure and frequency change with altitude: From narrow meridional and vertical structure (concentrated within 15 of the equator with vertical wavelengths of 10 km) and periods of days that prevail in the lower stratosphere, they shift to broader meridional and vertical structure (extending into midlatitudes with vertical wavelengths of km) and periods as short as 2 days [Wallace and Kousky, 1968; Hirota, 1978; Salby et al., 1984; Hayashi et al., 1984; Canziani et al., 1994; Canziani, 1999]. The shift of Kelvin wave activity to higher frequency and broader scale is consistent with slower components of the Kelvin wave spectrum being selectively absorbed, leaving faster components to propagate to higher altitude [Garcia and Salby, 1987]. The fastest components observed by satellite (the socalled ultrafast Kelvin wave) continue to amplify upward, propagating through the roof of satellite measurements (60 km). Accordingly, they should become increasingly prevalent at yet higher altitude. [4] A relative of equatorial Kelvin waves is the Kelvin normal mode, which is globally extensive with barotropic vertical structure (infinite vertical wavelength). Having theoretical period of only 32 hours, the Kelvin normal mode is too fast to be observed in satellite measurements from an individual platform. However, its period has surfaced in meteor winds in the mesosphere, in meteorological analyses, even in records of sea level pressure [Salby and Roper, 1980; Matsuno, 1980; Hamilton, 1984; Hamilton and Garcia, 1986; Mathews and Madden, 2000]. Unlike equatorial Kelvin waves, the Kelvin normal mode has global 1 Department of Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado, USA. 2 Atmospheric Systems and Analysis, Broomfield, Colorado, USA. Copyright 2007 by the American Geophysical Union /07/2006JD Figure 1. Raw power spectrum of wavenumber 1 component of stochastic forcing applied near the tropopause (normalized), after 80 days. 1of11

2 structure. It can therefore be excited by transient fluctuations outside the tropics, as characterize the noisy weather regime of the midlatitude troposphere. The same feature must apply to high-frequency Kelvin waves in general, which have global structure, but not necessarily the barotropic vertical structure associated with the Kelvin normal mode. [5] An earlier study investigated the 2-day wave in a 3-D nonlinear model that solves the primitive equations in the middle and upper atmosphere [Salby and Callaghan, 2003]. Integrations were forced stochastically by fluctuations inside the troposphere. Accompanying the 2-day wave response is a robust signature at eastward periods of a day and shorter. Reported here, those eastward propagating disturbances have the structure and dispersion characteristics of Kelvin waves, not the equatorial Kelvin waves familiar in the stratosphere, but global Kelvin waves that can be excited at midlatitudes as efficiently as elsewhere. Section 2 provides an overview of the 3-D model and how it is forced. The eastward response to stochastic forcing in the troposphere is then presented in section 3, with implications to observing systems drawn in section 4. Figure 2. Power spectrum of wavenumber 1 zonal wind, averaged vertically over the equator (unweighted) between 15 and 88 km. 2. Numerical Framework [6] The nonlinear model is described by Salby and Callaghan [2003]. It is derived from the lower atmosphere model of Callaghan et al. [1999], wherein the numerical formulation is developed in detail Domain [7] Formulated in isentropic coordinates, the model is three-dimensional and global. The vertical domain extends upward from near the tropopause to 185 km. Well into the thermosphere, the upper boundary is buried inside a deep thermal sponge layer that extends upward from 142 km. The useful domain thus spans altitudes from the tropopause to 142 km. [8] Eddy diffusion prevails in the middle atmosphere, especially in the mesosphere, where gravity wave breaking drives vertical mixing. Above the turbopause (105 km), eddy diffusion decays, replaced by molecular diffusion that increases sharply with altitude. Behavior within a few tens of kms above the turbopause then falls under diffusive control. The short timescale of molecular diffusion at those altitudes must be treated semi-implicitly, to preserve numerical stability while averting the computational burden becoming intractable. [9] Accompanying molecular diffusion at thermospheric altitudes is ion drag. Like diffusion, it acts to dissipate large- Figure 3. Amplitude (solid) and phase (dashed) of the eastward response in wavenumber 1 Montgomery stream function divided by g (units of meters). Composited against eastward propagation at an individual reference site, which reflects the average structure by eliminating day-to-day changes introduced through sporadic interference with stochastic forcing. Phase contoured at increments of 45. 2of11

3 Figure 4. Composite structure of Y/g (contours) and horizontal motion (vectors) at (a) 14.5 km, (b) 50.7 km, and (c) km. 3of11

4 Figure 5. time. Maximum equatorial zonal wind, as function of the real atmosphere, which has a rigid lower boundary at the Earth s surface. [12] The response in zonal wavenumber m is considered from stochastic fluctuations at the lower boundary of wavenumber m and of meridional structure that is indiscriminate: simply constant over latitude. Much the same behavior is recovered if stochastic forcing is concentrated (sinusoidally) in the winter hemisphere, where it reflects baroclinic weather systems that involve a broad spectrum of transient fluctuations. It is from such forcing that the results below are drawn. [13] Temporal variability of the stochastic forcing is dictated by its frequency spectrum. Defined to be Gaussian, the frequency spectrum is uniquely specified by the forcing variance and the autocorrelation time [see Salby and Callaghan, 2003]. The stochastic forcing then represents broadband variance that is red, characteristic of tropospheric fluctuations. scale organized motion. Beneath 142 km, however, its influence is comparatively minor. [10] In the integrations presented below, the model is executed under conditions of northern winter, with horizontal resolution equivalent to rhomboidal 10 (comparable to triangular 20 with grid spacing of 5 ) and vertical resolution of 3.5 km Stochastic Forcing [11] The model is forced at its lower boundary by stochastic fluctuations of prescribed statistics. Representative of tropospheric fluctuations, they are applied to the geopotential (e.g., height) of the lower-bounding isentropic surface, which lies near the tropopause. Specifying isentropic geopotential at the lower boundary admits the same normal modes, with vertical structure of a Lamb wave, as in 3. Eastward Propagating Response [14] Figure 1 plots the power spectrum of wavenumber 1 forcing, positive frequencies n representing eastward propagation. After 80 days of integration, the raw spectrum (although noisy) has converged to Gaussian form with an autocorrelation time (reflecting an e-folding time) of approximately 2 days. The associated variance has been scaled to recover wave amplitudes representative of those observed at lower levels [Mathews and Madden, 2000] Wavenumber 1 [15] Plotted in Figure 2 is the spectrum of wavenumber 1 zonal wind, averaged vertically over the equator between 15 and 88 km. The response in u EQ 1 is dominated by power at eastward frequencies of cpd. Included is a sharp peak at n ffi 0.75 cpd = (32 hours) 1. It corresponds to the dispersion characteristics of the Kelvin mode for an equivalent depth of 10 km, wherein the Kelvin mode assumes the barotropic vertical structure of a normal mode [Salby, 1984; Figure 6. As in Figure 3 but filtered to frequencies of cpd. 4of11

5 Figure 7. As in Figure 3 but filtered to frequencies of cpd. Mathews and Madden, 2000]. (Also apparent in Figure 2 is a peak at 0.20 cpd, the westward frequency of the 5-day wave [Madden and Julian, 1972]. Likewise of wavenumber 1, it too corresponds to a normal mode that induces zonal wind perturbations along the equator.) Although prominent, the discrete spectral feature at n = 0.75 cpd accounts for only about half of the eastward response (variance represented by area under the curve). The remainder lies at lower frequencies, which correspond to smaller equivalent depths and, although still globally extensive, baroclinic (vertically propagating) structure. [16] Figure 3 plots, as a function of latitude and altitude, the composite structure of wavenumber 1 Montgomery stream function Y 1 /g. Normalized by the complex amplitude of Y 1 at an individual reference site, the composite behavior excludes day-to-day fluctuations of structure that arise through sporadic interference with the random forcing. Figure 3 thus reflects the average structure, seen from a reference frame moving eastward with Y 1 at the reference site; see Salby and Callaghan [2003] for details. [17] The eastward response has amplitude (solid) with meridional structure that is symmetric about the equator, decays monotonically poleward, and is globally extensive. Maximizing over the equator, jy 1 /gj amplifies upward. Near the tropopause, Y 1 /g is of order 1 m, broadly consistent with the observed amplitude of the Kelvin normal mode [Mathews and Madden, 2000]. Its fast frequency limits Doppler shifting by mean wind shear, enabling the eastward response to amplify upward exponentially. By the lower thermosphere, its amplitude approaches 100 m. (Real amplitude equals twice the magnitude of the complex amplitude plotted; compare Figure 4.) [18] Phase (dashed) has meridional structure that is likewise symmetric about the equator. Vertical structure in the middle atmosphere is nearly barotropic (no phase tilt). The virtual absence of phase tilt below 110 km reflects the substantial contribution to the eastward response from the Kelvin normal mode (Figure 2). A steeper phase tilt seen above 110 km reflects accelerated absorption of wave activity [Salby, 1980], which follows from strong molecular diffusion that prevails at those altitudes. [19] Plotted in Figure 4 is the horizontal structure of eddy stream function and motion at levels between the tropopause and the thermosphere. At 15 km (Figure 4a), just above the tropopause, the wavenumber 1 anomaly in Y (contours) extends from pole to pole, even though the forcing is confined to the winter hemisphere. The emergence of global structure not far above tropospheric forcing reflects the prevailing contribution from the Kelvin normal mode, which assumes the form of a standing oscillation between the poles. The computed response in wavenumber 1 is in reasonable agreement with that composited from observations [Mathews and Madden, 2000]. Eddy motion in Figure 4 (vectors) is, throughout, nearly zonal and in phase with Y. It too is consistent with the structure of the Kelvin mode. Similar but amplified structure prevails at 50 km (Figure 4b), near the stratopause, and, again, at 110 km (Figure 4c), in the thermosphere. The slight eastward shift at 110 km reflects eastward phase tilt and upward transmission of wave activity. [20] Figure 5 plots, as a function of time, the amplitude of wavenumber 1 zonal motion over the equator at that level where u 1 EQ maximizes (Figure 3). Following a spin up of a couple of days, ju 1 EQ j varies sporadically (through constructive and destructive interference with the stochastic forcing). It remains bounded, achieving amplitudes of 5 10 m/s. [21] Additional insight into the Kelvin structure follows by discriminating between the contribution from the Kelvin normal mode and other contributions. Filtering the stochastic forcing to n = cpd leads to a response spectrum for u 1 EQ similar to that in Figure 2, but exclusive of the discrete peak at n = 0.75 cpd (not shown). The corresponding structure of Y 1 is plotted in Figure 6. Amplitude (solid) has much the same form as that due to 5of11

6 Figure 8. As in Figure 2 except for wavenumber 2. all frequencies (Figure 3), but reduced in magnitude in proportion to variance in the band admitted. Phase (dashed), however, exhibits a systematic and steeper tilt with altitude. Corresponding to a vertical wavelength of 200 km, the phase tilt is characteristic of meridionally broad but vertically propagating Kelvin waves. Analogous behavior characterizes the ultrafast Kelvin wave, which has been reported above tropical convection at periods of 2 4 days in satellite observations and GCM integrations [Salby et al., 1984; Hayashi et al., 1984; Canziani et al., 1994]. The vertically propagating Kelvin waves in Figure 6, however, are excited, not by tropical convection, but by midlatitude variability, which has intentionally been excluded from the tropics. [22] Plotted in Figure 7 is analogous structure, but with the stochastic forcing now filtered to n = cpd, which discriminates to the discrete peak associated with the Kelvin normal mode. Amplitude has the same essential form as the vertically propagating response, but stronger in proportion to the variance carried by the Kelvin normal mode (Figure 2). Phase structure, however, is virtually barotropic: No phase tilt until levels of strong molecular diffusion are encountered. Comparison with Figures 3 and 6 implies that what phase tilt is evident in the overall eastward response follows chiefly from the broadband response at frequencies below 0.75 cpd (periods longer than 32 hours), composed of vertically propagating Kelvin waves Wavenumber 2 [23] As only wavenumber 1 is well documented in observations at lower levels, wavenumber 2 is forced by the same stochastic spectrum. Figure 8 plots the power spectrum of wavenumber 2 zonal motion, averaged vertically over the equator between 15 and 88 km. It has the same form as the response of wavenumber 1 (Figure 2), but with a greater contribution from broadband variance. A sharp peak appears at n ffi 1.5 cpd = (16 hours) 1. Having twice the frequency of counterpart structure in wavenumber 1, this discrete spectral feature is consistent with the dispersion characteristics of the Kelvin mode for an equivalent depth of 10 km: Frequency increases linearly with wavenumber. The sharp peak represents the response of the wavenumber 2 Kelvin normal mode. At frequencies less than 1.5 cpd, the spectrum reflects broadband variance associated with smaller equivalent depths and vertically propagating structure. In contrast to wavenumber 1, the vertically propagating response represents most of the variance in u EQ 2. [24] Plotted in Figure 9 is the composite structure of Y 2 for the eastward response. Maximizing over the equator and amplifying upward, it is similar to the structure of wavenumber 1. However, the amplitude of wavenumber 2 is weaker, reflecting the red dependence on wavenumber of the response [see, e.g., Salby and Garcia, 1987]. The amplitude structure is also narrower than that of wavenumber 1. Similarly, phase exhibits a steeper vertical tilt than wavenumber 1, corresponding to a shorter vertical wavelength. The shorter meridional and vertical scales apparent in Figure 9 reflect smaller equivalent depths, which comprise the broadband response of wavenumber 2 at frequencies less than 1.5 cpd (Figure 8). Figure 9. As in Figure 3 but for wavenumber 2. 6of11

7 Figure 10. As in Figure 4 but for wavenumber 2. 7of11

8 Figure 11. As in Figure 9 but filtered to frequencies of cpd. Figure 12. As in Figure 9 but filtered to frequencies of cpd. 8of11

9 Figure 13. As in Figure 2 except for wavenumber 3. [25] Figure 10 plots the horizontal structure of eddy stream function and motion at levels between the tropopause and the thermosphere. The structure is similar to that of wavenumber 1 (Figure 4). However, it is distinguished by behavior just above the troposphere (Figure 10a), where Kelvin structure does not dominate. Concentrated in the winter hemisphere, the response at 15 km reflects the stochastic forcing, which has been confined to northern latitudes. By 50 km (Figure 10b), however, the symmetric Kelvin structure prevails; compare Figure 9. Notice: The meridional structure of wavenumber 2 is distinctly narrower than that of wavenumber 1 (Figure 4b). Nevertheless, like wavenumber 1, it has eddy motion that is nearly zonal and in phase with Y. Much the same structure appears at 110 km (Figure 10c), in the thermosphere. At the level where eddy zonal motion maximizes, ju 2 EQ j max attains values of 1 3 m/s (not shown). [26] Figure 11 plots the structure of Y 2 when the forcing has been discriminated to frequencies of n = cpd. Excluding the discrete response at 1.50 cpd, this band captures most of the broadband response evident in Figure 8. Amplitude structure is similar to that in Figure 9. Phase, however, exhibits a systematic and steeper tilt with altitude, corresponding to a vertical wavelength of 100 km. [27] Analogous information is plotted in Figure 12 for frequencies of n = cpd, which discriminate to the discrete response at 1.50 cpd. Amplitude structure is similar, albeit broader than that in Figure 11. Phase structure, however, is virtually barotropic below 105 km, consistent with normal mode structure. Horizontal structure (not shown) is likewise broader meridionally than the vertically propagating response, which corresponds to shorter equivalent depths Wavenumber 3 [28] Figure 13 plots the power spectrum of wavenumber 3 zonal wind, averaged vertically over the equator between 15 and 88 km. The response at eastward frequencies is now dominated by broadband variance between 0.5 and 2.0 cpd. A discrete response associated with the Kelvin normal mode, which has a theoretical frequency of n ffi 2.25 cpd, is overshadowed by the vertically propagating response that carries most of the eastward variance. [29] The composite structure of Y 3 (Figure 14) has the same essential structure seen earlier for wavenumbers 1 and 2. However, in the meridional direction, the amplitude is narrower yet. Phase likewise exhibits a systematic and steeper tilt with altitude, corresponding now to a vertical wavelength of only 65 km. The narrower meridional and vertical structure reflects smaller equivalent depths, which comprise the vertically propagating response of wavenumber 3. [30] The narrower meridional scale of wavenumber 3 is plainly evident in horizontal structure, plotted in Figure 15 at 110 km. There, horizontal structure has the symmetric Figure 14. As in Figure 3 but for wavenumber 3. 9of11

10 Figure 15. As in Figure 4c but for wavenumber 3. form of the Kelvin mode, but clearly narrower than counterpart structure in wavenumbers 1 and 2 (compare Figures 4c and 10c). Nonetheless, wavenumber 3 also has eddy motion that is nearly zonal and in phase with Y. At these altitudes, ju 3 EQ j max attains values of order 1 m/s. [31] While sharing these features with wavenumbers 1 and 2, the symmetric Kelvin structure emerges only above 50 km, as can be inferred from Figure 14. The higher altitude at which the Kelvin structure prevails reflects the dominant contribution to the eastward response coming from vertically propagating Kelvin waves: Although of small amplitude at lower altitude, they amplify with altitude faster than the barotropic (normal mode) response. Consequently, they eventually surface above other variance that is excited by the broadband tropospheric forcing. 4. Conclusions [32] Transient forcing representative of fluctuations in the midlatitude troposphere excites planetary-scale Kelvin waves that attain significant amplitudes in the mesosphere and thermosphere. The eastward response to such forcing involves the Kelvin normal mode, as well as a continuum of vertically propagating Kelvin waves. Having periods of order a day and shorter, those disturbances are all global in meridional extent. This feature enables them to be excited by fluctuations, not only in the tropics, but over much of the Earth. Kelvin wave activity can therefore be excited by broadband variability that is characteristic of midlatitude weather systems. [33] The computed response reproduces observed Kelvin structure and amplitude near the tropopause. Vertical amplification then leads to wind fluctuations at mesospheric and thermospheric altitudes of 5 15 m/s. Approaching tidal amplitudes, Kelvin wave activity should therefore represent a measurable if not prominent feature at those altitudes. In the integrations presented, forcing was confined to one hemisphere. However, transient fluctuations like those considered here operate in both hemispheres, albeit incoherently. The variance is then additive, so the actual response variance should be even greater, by as much as a factor of 2. [34] The discrete spectral response associated with the Kelvin normal mode is sharp in wavenumbers 1 and 2. Spectrally distinct, it should be identifiable in spectra from an individual site (e.g., in records from individual radar or lidar), provided that those records are sufficiently long. On the other hand, the broadband response associated with vertically propagating Kelvin waves of individual wavenumbers overlap in frequency. Accordingly, they will blend together in observations from an individual site. Isolating those components, enabling the structure and dispersion characteristics of individual wavenumbers to be identified, will require observations of horizontal structure. Such information can be collected through coordinated groundbased measurements or through satellite measurements. The latter afford global coverage, enabling individual wavenumbers to be discriminated to. Reaping the benefit of such coverage, however, will have to await multiple satellite platforms, which can sample the Earth frequently enough to resolve the short periods of global Kelvin waves. [35] Acknowledgments. The authors are grateful for constructive comments provided during review. Figures were prepared by Jackie Gratrix. This work was supported by NSF Grant ATM References Bergman, J., and M. Salby (1994), Equatorial waves derived from fluctuations in observed convection, J. Atmos. Sci., 51, Callaghan, P., A. Fusco, G. Francis, and M. L. Salby (1999), A Hough spectral model for 3-dimensional studies of the middle atmosphere, J. Atmos. Sci., 56, Canziani, P. (1999), Slow and ultraslow equatorial Kelvin waves: The UARS-CLAES view, Q. J. R. Meteorol. Soc., 125, Canziani, P. O., J. R. Holton, E. Fishbein, L. Froidevaux, and J. Waters (1994), Equatorial Kelvin waves: A UARS MLS view, J. Atmos. Sci., 51, Dunkerton, T. (1979), On the role of the Kelvin wave in the westerly phase of the semiannual zonal wind oscillation, J. Atmos. Sci., 36, Garcia, R., and M. Salby (1987), Transient response to localized episodic heating in the tropics, part II: Far-field behavior, J. Atmos. Sci., 44, Hamilton, K. (1984), Evidence of a normal mode Kelvin wave in the atmosphere, J. Meteorol. Soc. Jpn., 62, Hamilton, K., and R. Garcia (1986), Theory and observation of short-period normal mode oscillations in the atmosphere, J. Geophys. Res., 91, 11,867 11, of 11

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