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# Transition from geostrophic turbulence to inertia–gravity waves in the atmospheric energy spectrum

Edited* by Kerry A. Emanuel, Massachusetts Institute of Technology, Cambridge, MA, and approved October 17, 2014 (received for review June 14, 2014)

## Significance

High- and low-pressure systems, commonly referred to as synoptic systems, are the most energetic fluctuations of wind and temperature in the midlatitude troposphere. Synoptic systems are a few thousand kilometers in scale and are governed by a balance between the pressure gradient force and the Coriolis force. Observations collected near the tropopause by commercial aircraft indicate a change in dynamics at horizontal scales smaller than about 500 km. Smaller-scale fluctuations are shown to be dominated by inertia–gravity waves, waves that propagate on vertical density gradients but are influenced by Earth’s rotation.

## Abstract

Midlatitude fluctuations of the atmospheric winds on scales of thousands of kilometers, the most energetic of such fluctuations, are strongly constrained by the Earth’s rotation and the atmosphere’s stratification. As a result of these constraints, the flow is quasi-2D and energy is trapped at large scales—nonlinear turbulent interactions transfer energy to larger scales, but not to smaller scales. Aircraft observations of wind and temperature near the tropopause indicate that fluctuations at horizontal scales smaller than about 500 km are more energetic than expected from these quasi-2D dynamics. We present an analysis of the observations that indicates that these smaller-scale motions are due to approximately linear inertia–gravity waves, contrary to recent claims that these scales are strongly turbulent. Specifically, the aircraft velocity and temperature measurements are separated into two components: one due to the quasi-2D dynamics and one due to linear inertia–gravity waves. Quasi-2D dynamics dominate at scales larger than 500 km; inertia–gravity waves dominate at scales smaller than 500 km.

The midlatitude high- and low-pressure systems visible in weather maps are associated with winds and temperature fluctuations that we experience as weather. These fluctuations arise from a baroclinic instability of the mean zonal winds at horizontal scales of a few thousand kilometers, commonly referred to as the synoptic scales (1⇓–3). The combined effects of rotation and stratification constrain the synoptic-scale winds to be nearly horizontal and to satisfy geostrophic balance, a balance between the force exerted by the changes in pressure and the Coriolis force resulting from Earth’s rotation. It is an open question whether the same constraints dominate in the mesoscale range (i.e., at scales of 10–500 km), or whether qualitatively different dynamics govern flows at these scales.

The synoptic-scale flows are turbulent in the sense that nonlinear scale interactions, which lie at the core of the difficulty to predict the weather, exchange energy between different scales of motion (4⇓⇓–7). Under the constraints of rotation and stratification, the synoptic-scale winds are approximately 2D and nondivergent (8, 9). In 2D flows, nonlinear scale interactions tend to transfer energy to larger scales, that is, the synoptic-scale pressure anomalies often merge and form larger ones, contrary to nonlinear scale interactions in 3D flows, which tend to transfer energy to smaller scales (10). Little energy is thus transferred to scales smaller than those at which the synoptic-scale fluctuations are generated through instabilities. Theory and numerical simulations predict that the energy per unit horizontal wavenumber *k* decays as rapidly as

Long-range passenger aircraft have been instrumented to collect velocity and temperature measurements as part of the Global Atmospheric Sampling Program in the 1970s and the Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) project in the 1990s and 2000s. The resulting dataset, described in *Materials and Methods*, consists of tens of thousands of flights. Because aircraft travel at altitudes between 9 and 14 km, the data largely reflect the upper troposphere and lower stratosphere, near the tropopause. These measurements confirm that the kinetic energy spectrum drops as *A*). In the mesoscale range, at scales smaller than 500 km, the kinetic energy spectrum decays more slowly, roughly like

The measured kinetic energy spectrum is intriguing, because it agrees so well with Charney’s theory of geostrophic turbulence at the synoptic scales (9) but deviates from that prediction at the mesoscale. The transition to the flatter

A rotating and stratified atmosphere, however, supports an additional, much faster set of motions: inertia–gravity waves. These are internal gravity waves, modified by the effect of rotation, that have periods of several minutes to a few hours. In contrast to the strongly nonlinear, turbulent synoptic-scale flow, these motions are wave-like and at small amplitude they are approximately governed by linear dynamics (20). It has been proposed that the mesoscale energy is dominated by inertia–gravity waves (21, 22), which are easily excited by any fast fluctuation of the atmospheric flows (23). In this explanation of the mesoscale part of the spectrum, linear inertia–gravity waves and nonlinear synoptic-scale turbulence coexist with little interaction.

In this paper, we present an analysis of the MOZAIC data that uses a decomposition method recently developed by Bühler et al. (24). This analysis provides compelling evidence that linear inertia–gravity waves indeed dominate the observations in the mesoscale.

## Theories for the Synoptic-to-Mesoscale Transition

Dewan (21) first suggested that the mesoscale energy is dominated by a continuum of weakly nonlinear inertia–gravity waves. VanZandt (22) showed that the mesoscale spectra of horizontal wind fluctuations as a function of horizontal wavenumber, vertical wavenumber, and frequency are related through the dispersion and polarization relations of inertia–gravity waves. The dominance of inertia–gravity waves at the mesoscale, however, seemed inconsistent with vertical velocity frequency spectra measured from radar in light-wind conditions (25). More recently, Vincent and Eckerman (26) showed that the signature of inertia–gravity waves is recovered after correcting for Doppler-shift effects in radar observations.

In the last two decades, the interpretation of the synoptic-to-mesoscale transition in terms of inertia–gravity wave dynamics has received little attention. The

The earliest turbulent theory argues that the mesoscale spectrum is due to an inverse cascade of energy injected at even smaller scales, for example by convective activity (14, 16, 17). The hypothesis is that nonlinear mesoscale interactions, much like at synoptic scales, transfer energy to larger scales because of the constraints of rotation and stratification. The kinetic energy spectrum in such a quasi-2D inverse cascade scales like

The second turbulent theory proposes that the flat mesoscale spectrum is the signature of sharp temperature fronts, which develop when synoptic-scale flows intersect a rigid boundary, like the Earth’s surface, or a strongly stratified layer, like the tropopause (18). Importantly, the winds associated with the temperature fronts are still largely in geostrophic balance. In this view, the

The third turbulent theory proposes that the

## Inertia–Gravity Waves and Geostrophic Flow

The understanding that atmospheric winds are composed of slow flows in approximate geostrophic balance and fast inertia–gravity waves has been the foundation for much progress in atmospheric science. The first numerical weather predictions were based on quasi-geostrophic dynamics, an approximation to the more complete primitive equations that filters out inertia–gravity waves (27⇓–29). In the troposphere and lower stratosphere, inertia–gravity waves typically have small amplitudes and therefore interact only weakly with geostrophic flows. Although there is a growing appreciation of rare instances of inertia–gravity waves directly influencing sensitive weather patterns (30), strong interactions between inertia–gravity waves and the geostrophic flow are typically confined to the middle and upper atmosphere, where the wave amplitude becomes large enough to allow for breaking of inertia–gravity waves and the concomitant drag force on the geostrophic flow that is well known to be crucial for the global angular momentum budget of the atmosphere (23).

In the deep ocean, breaking inertia–gravity waves mix heat and carbon. This leading-order effect has led to intensive study of the oceanic inertia–gravity wave field. It is composed of a continuous spectrum of linear waves together with isolated peaks at the inertial and tidal frequencies (31). Similar to the lower atmosphere, these linear waves interact only weakly with the geostrophic flow. Only at small vertical scales of a few tens of meters do the waves break.

Callies and coworkers (24, 32) have recently shown that the energy spectra of oceanic flows are dominated by geostrophic flows at large scales and by inertia–gravity waves at small scales. The transition between the two classes of motion occurs at scales of 10–100 km, depending on the relative strength of the geostrophic eddies and the waves. In what follows, we show that the synoptic-to-mesoscale transition in the atmospheric energy spectrum is likely an equivalent transition from geostrophic to inertia–gravity wave dynamics.

## Decomposition

From the MOZAIC aircraft observations of wind near the midlatitude troposphere, we compute power spectra of the longitudinal (along-track) velocity *u* and transverse (across-track) velocity *v*, *g* = 9.81 ms^{−2} is the gravitational acceleration, *N*^{2} is the average vertical gradient of *b*. Potential temperature *θ* is the temperature of an air parcel corrected for dynamically irrelevant compression effects. We use a typical stratification of the lower stratosphere, *N* = 0.02 s^{−1}, estimated from the ERA-Interim reanalysis (33). Fig. 1*A* shows that these MOZAIC spectra display the transition from a steep synoptic range to a flat mesoscale range at about 500 km.

If simultaneous wind and temperature observations were available in space and time, one could directly test whether the dispersion and polarization relations of inertia–gravity waves are satisfied by mesoscale motions. One could further separate out inertia–gravity waves and geostrophic flows, because inertia–gravity waves are restricted to frequencies between the Coriolis frequency *f* (equal to twice the rotation rate of the Earth multiplied by the sine of the latitude) and the buoyancy frequency *N* (the frequency at which a vertically displaced parcel of air will oscillate within the stably stratified atmosphere), whereas geostrophic flows evolve on much longer time scales. However, it is extremely difficult to collect simultaneous measurements of mesoscale fluctuations of winds and temperature in space and time.

Bühler et al. (24) have recently shown that the decomposition can be achieved from space-only measurements, provided that concurrent observations of horizontal velocities and temperature are available. Applying this new decomposition to the MOZAIC data produces two powerful arguments in support of the hypothesis that the mesoscale spectrum is dominated by inertia–gravity waves.

First, assuming that the flow is an uncorrelated superposition of a geostrophic flow and inertia–gravity waves, we diagnose the inertia–gravity wave component of the total energy, the sum of the kinetic and potential energies, solely based on the observed horizontal velocities. We then show that the thus predicted inertia–gravity wave energy spectrum closely matches the observed total energy spectrum in the mesoscale range. This indicates that the mesoscale potential energy spectrum, predicted by this procedure, is consistent with linear wave theory.

Second, assuming that the geostrophic component of the flow obeys Charney’s isotropy relation for geostrophic turbulence (9), we decompose the three observed individual spectra of longitudinal kinetic energy, transverse kinetic energy, and potential energy into a geostrophic component and an inertia–gravity wave component. The diagnosed inertia–gravity wave spectra closely match the observed spectra in the mesoscale range. This is another powerful test of the mesoscale flow’s consistency with the dispersion and polarization relations of inertia–gravity waves.

## Helmholtz Decomposition

Fig. 1*A* shows that in the synoptic range the spectra approximately satisfy Charney’s prediction for geostrophic turbulence:

Any horizontal flow field can be decomposed into its rotational and divergent components: *ψ* is the streamfunction, *ϕ* is the velocity potential, *x* is the along-track coordinate, and *y* is the across-track coordinate. If the flow is statistically isotropic horizontally and *ψ* and *ϕ* are uncorrelated (as is the case for a superposition of geostrophic flow and linear inertia–gravity waves), the spectra *ψ* and *ϕ* (*Materials and Methods*):**1** and **2** (*Materials and Methods*). Using

The Helmholtz decomposition of the MOZAIC kinetic energy spectrum in Fig. 1*B* shows that the rotational component dominates in the synoptic range, whereas the divergent component becomes of the same order at the transition to the mesoscale range. In the mesoscale range, the divergent component slightly dominates over the rotational component.

The dominance of the rotational component in the synoptic scales is consistent with Charney’s geostrophic turbulence, because quasi-geostrophic flow is to leading order horizontally nondivergent. The significant divergent component in the mesoscale, however, is inconsistent with the mesoscale theories that rely on a leading-order geostrophic balance, namely the inverse-cascade theory and the frontogenesis theory. Instead, it points to the dominance of ageostrophic dynamics.

Lindborg (35) also found that the rotational and divergent components of the flow are of the same order in the mesoscale range. In his analysis, based on curve fitting and selective Fourier transforming, the rotational component slightly dominated the divergent component in the mesoscale range. He argued that this was inconsistent with inertia–gravity waves, for which he expected the divergent component to be much larger than the rotational component. If the inertia–gravity wave field is dominated by near-inertial waves, however, as suggested by balloon measurements in the lower stratosphere (36), the rotational component is expected to be of the same order as the divergent component. In the following section, we show that the mesoscale signal is indeed consistent with linear inertia–gravity wave dynamics (i.e., with the dispersion and polarization relations of hydrostatic inertia–gravity waves).

### Decomposition of the Total Energy Spectrum into Geostrophic and Inertia–Gravity Wave Components.

Geostrophic flows are horizontally nondivergent and therefore only have a rotational component, whereas inertia–gravity waves have both a rotational and a divergent component. To perform the decomposition into these two classes of motion, we note that the component of the total energy spectrum *Materials and Methods*):

The decomposition into geostrophic flow and inertia–gravity waves of the MOZAIC data are shown in Fig. 1*C*. The mesoscale range is dominated by inertia–gravity waves, which do not contribute much energy at the synoptic scales. The residual spectrum (i.e., the total spectrum minus the inertia–gravity wave component) dominates at the synoptic scales. This component can be confidently attributed to geostrophic flows, because we have shown that the spectrum in the synoptic range is purely rotational and hence horizontally nondivergent. At the transition scale, the geostrophic component of the total energy keeps falling off steeply—the transition seems to be due to inertia–gravity waves becoming dominant in the mesoscale range. This is the main result of this paper.

Notice that the geostrophic spectrum keeps falling off steeply past the transition at 500 km but eventually flattens out at smaller scales. This flattening is likely an artifact, because at these scales the geostrophic component makes up a small fraction of the observed signal. It is quite possible that the flattening is due to noise or biases introduced by the interpolation procedure or by truncation errors in the reported wind and temperature data.

### Decomposition of Kinetic and Potential Energy Spectra.

We have shown that the total energy spectrum can be decomposed into its geostrophic and inertia–gravity wave components. To confirm that the observed spectra are consistent with geostrophic dynamics at synoptic scales and with inertia–gravity wave dynamics at mesoscales, we now decompose into its two components each of the atmospheric spectra, the longitudinal and transverse kinetic energy spectra, *A*).

This decomposition—obtained by applying Eqs. **15**–**17** and **20**–**22**, given in *Materials and Methods*—confirms the main conclusion that the observed synoptic-scale flow is consistent with geostrophic dynamics and that the observed mesoscale flow is consistent with inertia–gravity wave dynamics. In the synoptic range, the observed spectra match the diagnosed geostrophic components (Fig. 2*A*). At the transition scale, the diagnosed geostrophic spectra start deviating from the observed spectra and keep falling off steeply. At this scale, the diagnosed inertia–gravity wave spectra become comparable to the observed spectra and start matching them in the mesoscale range (Fig. 2*B*).

## Discussion

Our analysis shows that the aircraft observations are consistent with a geostrophic flow that dominates the synoptic range and with inertia–gravity waves that dominate the mesoscale range. This conclusion is predicated on the assumption that the total observed flow is a superposition of geostrophic flow and inertia–gravity waves and that these two components are uncorrelated and horizontally isotropic.

In accord with Lindborg’s result (34), our analysis conclusively shows that mesoscale flows are not in geostrophic balance, thus falsifying previous suggestions that mesoscale spectra represent geostrophic eddies generated by atmospheric convective events or geostrophically balanced fronts at the tropopause. Our analysis then further shows that the observed mesoscale spectra

A similar transition between a dominant geostrophic flow at large scales and dominant inertia–gravity waves at small scales is well established in the oceanic spectra (32), providing further support for our interpretation of the synoptic-to-mesoscale transition. The ocean, like the atmosphere, is a strongly rotating and stratified fluid. The most energetic large-scale fluctuations are generated by baroclinic instabilities in both fluids, whereas small-scale inertia–gravity waves are triggered by any fast perturbation. A posteriori, it should not be surprising that both fluids exhibit a transition from geostrophic dynamics at large scales to inertia–gravity wave dynamics at small scales.

That inertia–gravity waves dominate the mesoscale spectrum does not mean that the atmosphere is filled with a nearly uniform and stationary wave field, as seems to be the case in the ocean (31). The atmospheric wave field is likely highly intermittent in both space and time (23). Our result merely suggests that, on average, inertia–gravity waves dominate the mesoscale range. A good understanding of what sets the shape of the average atmospheric inertia–gravity wave spectrum is lacking. The oceanic inertia–gravity wave spectrum, however, has been shown to be an equilibrium solution of weakly interacting inertia–gravity waves with slopes close to

The spatial patterns of mesoscale energy are also consistent with the dominance of inertia–gravity waves at the mesoscales. The aircraft spectra in the mesoscale range are up to six times larger in mountainous regions than over flat terrain (39). The inertia–gravity wave activity is expected to be enhanced over mountains, where lee waves are excited by large- and synoptic-scale flows impinging on topography. Stratified turbulence, however, is fed directly by synoptic-scale energy and therefore is not expected to be enhanced over mountainous regions.

High-resolution numerical models reproduce the synoptic-to-mesoscale transition (40). Model results are consistent with the observation that the synoptic range is dominated by rotational flow, whereas the rotational and divergent components are of the same order in the mesoscale range (41, 42). Models also show a pattern of enhanced mesoscale energies in regions of high topography (42). The simulations further show that the mesoscale spectrum is not the result of stratified turbulence (43), in agreement with our conclusion that they are the signature of inertia–gravity waves.

The emergent picture is relatively simple. Geostrophic synoptic-scale baroclinic disturbances force a forward enstrophy cascade that continues through the synoptic-to-mesoscale transition. The

The result of this paper may also have some implications for the theoretical predictability of atmospheric flows. Lorenz (6) argued that a turbulent flow with a

If the flat mesoscale spectrum were due to a turbulent cascade, that would pose a limit on predictability of synoptic systems (7, 44). Observing and modeling the atmosphere beyond the transition at 500 km would yield rapidly diminishing returns in predictability. If this part of the spectrum is due to inertia–gravity waves, however, improving observational systems and forecast models may not prove as futile. Inertia–gravity waves do not propagate errors in the same way as the turbulent flows discussed by Lorenz. If the geostrophic component of the mesoscale flow were to be observed, despite the dominance of inertia–gravity waves, the forecast times of synoptic systems could potentially be extended considerably.

It should be noted, however, that other processes not considered in this theoretical argument may affect predictability. Moist convective processes, for example, lead to rapid growth of errors that can leak into the geostrophic flow (45). Currently, the practical predictability of the weather is likely limited by inadequate representation of such processes.

## Materials and Methods

### Aircraft Data.

The spectra shown in Fig. 1*A* are calculated from wind and temperature measurements obtained by the MOZAIC program, which equipped commercial aircraft with instrumentation to measure trace gases, but also records wind speed and direction from the onboard computer. The data used here were obtained in 2002–2010 and are restricted to the northern hemisphere midlatitudes (30–60° latitude). Great circles are fit to the flight paths and segments are discarded if they are shorter than 6,000 km, the average sample spacing is coarser than 1.2 km, or the deviation from the great circle is greater than 2°. The data are then linearly interpolated onto a regular grid with 1-km spacing. Data at pressures larger than 350 hPa are discarded. Subsequently, for each flight, data deviating more than 1 km in altitude from the mean altitude are also discarded. Temperature data are adjusted to account for remaining variations in flight altitude, assuming a constant stratification *N* = 0.02 s^{−1}, but this correction is of no consequence for the results discussed in the main paper. Nastrom and Gage (13) and Cho and Lindborg (15) showed that the spectral shapes are qualitatively the same in the upper troposphere and lower stratosphere, and thus we do not separate the data into vertical bins. Spectra are computed by applying a Hann window, compensating for the variance loss, performing a discrete Fourier transform, and averaging over all 458 segments. The windowing is necessary to prevent spectral leakage of synoptic-scale energy into the mesoscale. Spectra at wavelengths smaller than 20 km are discarded because they are potentially affected by the interpolation procedure or by truncation errors in the reported data. Locations were reported in longitude/latitude with an accuracy of 0.01°, zonal and meridional winds were reported with an accuracy of 0.01 m s^{−1}, and temperatures were reported with an accuracy of 0.01 K. These noise levels do not affect the spectra on scales larger than 20 km. We also discard the largest resolved wavelength, because the power at this wavelength is reduced artificially by the window.

### Details of Decomposition.

We here give more detail on the decomposition techniques used to analyze the aircraft spectra. Bühler et al. (24) give a more comprehensive description of these techniques and illustrate their skill to analyze oceanic spectra.

Let *u* and *v* be horizontal velocity components defined in the *x–y* plane plane with *x* aligned with the aircraft track, so *u* is the longitudinal (along-track) component and *v* is the transverse (across-track) component. The time *t* and altitude *z* are considered fixed during the measurement, so they will be ignored. A general 2D flow has a Helmholtz decomposition into rotational and divergent components of the form *ψ* and *ϕ* are uniquely determined in terms of the velocity field with doubly periodic boundary conditions.

Progress with the statistical theory is possible if *u* and *v* as

where *l* is the across-track wavenumber. Integration over *l* and some manipulation gives Eqs. **1** and **2** with

where *ψ* and *ϕ*, related to the 2D spectra by

Eqs. **1** and **2** can be solved explicitly, given the boundary conditions

where for convenience the coordinate was transformed to

Relation **5** follows from the dispersion and polarization relations of hydrostatic inertia–gravity waves. Combining the vorticity and continuity equations of the linearized primitive equations yields

and thus, with the use of Eqs. **6** and **7**,

The linear buoyancy equation is

if uncorrelated plane waves or equivalently vertical homogeneity is assumed. A slowly varying background is allowed. Adding Eqs. **13** and **14** then eliminates the dependence on *ω*, so that Eq. **5** follows by integrating over *l* and *ω*.

The decomposition of the three individual spectra can be achieved by decomposing

where Charney’s assumption

Because **5**, this can be solved for

where the boundary condition

where **5** was used to deduce the last equation.

## Acknowledgments

We thank the principal investigators of the MOZAIC project, the European Commission for supporting the MOZAIC project, ETHER [Centre National d’Etudes Spatiales (CNES)–CNRS/Institut National des Sciences de l’Univers (INSU)] for hosting the database, and the participating airliners for transporting the instrumentation free of charge. J.C. and R.F. thank Glenn Flierl for offering useful feedback during the preparation of the manuscript and acknowledge financial support under Grants ONR-N-00014-09-1-0458 and NSF-CMG-1024198. O.B. gratefully acknowledges financial support under Grants NSF-CMG-1024180, NSF-DMS-1312159, and NSF-DMS-1009213.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: joernc{at}mit.edu.

Author contributions: J.C. designed research; J.C., R.F., and O.B. performed research; J.C. analyzed data; and J.C., R.F., and O.B. wrote the paper.

The authors declare no conflict of interest.

↵*This Direct Submission article had a prearranged editor.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1410772111/-/DCSupplemental.

Freely available online through the PNAS open access option.

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