## 1. Introduction

This paper develops the model of formation of a 3D spatial spectrum of mesoscale wind velocity and temperature fluctuations in a stably stratified atmosphere to explain the forms of the observed vertical and horizontal wavenumber spectra of these fluctuations in the upper troposphere and stratosphere (Chunchuzov 2001, 2002; Gurvich and Chunchuzov 2008). The explanation of the shaping mechanism for the 3D spectrum of mesoscale fluctuations is of independent theoretical interest, since at present, a considerable number of works is devoted to modeling and explanation of the one-dimensional (1D; horizontal and vertical) wavenumber spectra of mesoscale fluctuations (Nastrom and Gage 1985; Hostetler and Gardner 1994; Dewan 1979, 1997; Bacmeister et al. 1996; Skamarock et al. 2014; Lindborg 2006, 2007, 2015; Callies et al. 2014; Sun et al. 2016). The knowledge of the 3D spectra of the wind velocity and temperature fluctuations is of practical interest since it makes it possible to find the statistical characteristics of the fluctuations of the parameters (such as an amplitude, phase, and angle of arrival to the reception point) of infrasonic waves (Chunchuzov 2004; Ostashev et al. 2005) and light (Gurvich and Kan 2003; Kan et al. 2014) propagating in the atmosphere.

There is still a question of how internal gravity waves (IGWs) contribute to the formation of the horizontal wavenumber spectrum of the mesoscale wind velocity and temperature fluctuations in the upper troposphere and lower stratosphere (Lilly and Lester 1974; Vinnichenko et al. 1980; Bacmeister et al. 1996; Dewan 1979, 1997; Hertzog et al. 2002; Callies et al. 2014). According to the observations of Nastrom and Gage (1985), such spectrum decays as

Based on the theory of similarity and dimensionalities, Dewan (1979, 1997) suggested that the portion of the horizontal wavenumber spectrum with a −5/3 power-law decay is shaped as a result of the nonlinear wave–wave interactions that cause a cascade-like transfer of wave energy from internal waves of large scales to the wave perturbations with small scales. The analysis of numerous aircraft measurements of the horizontal wind velocity in the upper troposphere and lower stratosphere shows the existence of both the horizontal vortical motions (for which the horizontal divergence of the wind velocity field is equal to zero) and wave motions with the nonzero horizontal divergence (Callies et al. 2014; Lindborg 2015).

Lindborg (2015) used a method of Helmholtz decomposition of the wind velocity field onto a rotational component with a zero horizontal divergence and a component with the zero horizontal rotor that includes wave motions. Comparing the ratio of energies for the vortical and wave components of the velocity field in the lower stratosphere and upper troposphere, Lindborg concludes that it exceeds unity in the range of scales for which the 2/3 law is observed for structure functions (i.e., −5/3 for the spectra). Based on this, he concluded that there is an insignificant contribution of internal waves to the wind velocity fluctuations with the

From the analysis of the aircraft measurements of the wind velocity, Callies et al.(2014) makes a conclusion about the dominance of the contribution of inertia–gravity waves to the formation of a horizontal spectrum of velocity fluctuations in the range of scales from 10 to 500 km. At synoptic scales, over 500 km, the slow (in comparison with wave motions) horizontal vortex motions predominate, obeying geostrophic and static equilibrium. The importance of the wave contribution to the mesoscale fluctuations of wind velocity and temperature was also shown by Skamarock et al. (2014) using a numerical model of atmospheric circulation with high spatial resolution. It is obvious that the estimation of the ratio of energy contributions of the wave and vertical vorticity motions can be strongly influenced both by the errors in calculating these components caused by the chosen numerical model and by the errors of the very method of decomposition of the velocity field into the vortex and wave components.

It is important to pay attention to earlier studies of horizontal spectra of wind velocity and temperature fluctuations obtained from aircraft measurements under stable stratification conditions of the troposphere and the lower stratosphere (Shur 1962; Lilly and Lester 1974; Vinnichenko et al. 1980; Bacmeister et al. 1996). All of them pointed to the presence of a transition in the rate of decay of the spectra with decreasing scales from the −5/3 power law to a steeper decay in the range from about 6 km to 300 m, where the exponent ranged from −2.5 to −3. Above the mountainous terrain, such transition occurred at scales of 15–23 km (Nastrom et al. 1987). It is to this steep part of the horizontal wavenumber spectrum that an explanation will be given in this paper.

As for the observed vertical wavenumber spectra of the temperature and horizontal wind velocity fluctuations in the stratosphere and mesosphere, they follow the power law close to

The nonlinearity of the hydrodynamic equations, including advective nonlinearity, plays an important role in the formation of the space–time spectrum of mesoscale velocity and temperature fluctuations induced by internal waves in stably stratified fluid (Allen and Joseph 1989; Hines 1991a,b, 2001; Bacmeister et al. 1996; Eckermann 1999; Chunchuzov 1996, 2001, 2002; Pinkel 2008). The wave-induced advection of fluid parcels arises because of the presence of the acceleration term **k** are generated, each of these waves experiences the influence of the advection caused by the total internal wave field. Such advection along with the nonlinearities of the equation of state and of the buoyancy forces leads to the interaction between internal waves.

Considering the interaction of two internal waves with the velocity amplitudes

While the amplitudes of the interacting modes are small, the amplitude of the generated resonance mode grows linearly in time over a time interval small in comparison with the characteristic mode interaction time, reaching a maximum

If velocity shears induced by the initial modes become sufficiently large, then the generated nonresonant modes become comparable in amplitude with the resonant modes, and the wave interactions themselves cease to be weak. The frequencies of the generated nonresonant modes are not related to their wave vector by the dispersion relation; therefore, these modes are no longer linear internal waves. They can interact with any other wave mode because of the nonselective nature of the nonresonant interactions. Moreover, the characteristic time of such interactions becomes short and comparable to the periods of the original waves. Such fast nonresonant interactions of the modes give rise to the cascade character of energy transfer over the mode scale, which is analogous to the turbulent cascade of energy transfer along the spectrum of turbulent fluctuations.

In the case when the 1D spectrum of the wave modes is continuous and has a spectral energy density *E*(*k*), Phillips (1967) has written down the following condition:

Shur and Lumley followed the assumption of a cascade transfer of the kinetic energy of turbulent motions from large scales to small ones, taking into account the transformation of some of the kinetic energy into potential energy due to work performed by turbulent vortex motions against buoyancy forces in a stably stratified medium. It is this work that leads to a more rapid decay (with a slope of −3) of the spectrum of horizontal velocity fluctuations with increasing *k* in a stratified fluid compared to the −5/3 law for turbulence in the inertial range of scales. As Lumley (1964) noted, his theory is applicable in the buoyancy subrange regardless of whether random fluctuations in a stably stratified fluid are caused by wave disturbances or turbulence. Thus, the cascade energy transfer over scales caused by strong nonresonant interactions of wave disturbances (with a short interaction time) and a turbulent cascade in a stably stratified fluid lead to the same spectrum with a −3 slope in the buoyancy subrange.

The purpose of this paper is to obtain an analytic form for the 3D spectrum of the mesoscale fluctuations of the vertical displacements (or relative temperature fluctuations) and horizontal velocity in stably stratified atmosphere, which will allow us to explain their observed vertical and horizontal wavenumber spectra. To find this form, we will rely on a Lagrangian approach to the description of the shaping mechanism for the internal wave spectrum developed by Chunchuzov (1996, 2001, 2002, 2009), Eckermann (1999), Hines (2001), and on analogous approach developed by Gurbatov et al. (1991) for a high-intensity acoustic noise. Here, we generalize this approach to the case when horizontal vortical motions contribute along with internal waves to the formation of the spectrum of velocity fluctuations. In section 2, a relation between the correlation functions of the random velocity and displacement fields in Lagrangian and Eulerian coordinate systems will be obtained. This relation will be used in section 3 to find the shapes of the 3D spectra of vertical displacements and horizontal velocities in the Eulerian system. Using these forms, the vertical and horizontal wavenumber spectra will be obtained in section 4 and compared with the observed spectra.

## 2. The relation between correlation functions of the random displacements and velocities in Lagrangian and Eulerian coordinate systems

**i**

_{3}is the unit vector in the vertical direction. The linearization in this case is carried out over a small Froude number Fr =

*U*/(

*NL*) ≪ 1, where

*U*and

*L*are the characteristic velocity and length scales, respectively, on which the velocity field varies significantly.

The nonlinearity of the motion equations leads to the interactions of internal waves, internal waves with horizontal vortical motions, and horizontal vortical motions (Lelong and Riley 1991; Embid and Majda 1998). As shown by Lelong and Riley (1991), the interaction of internal waves can generate a vertical vorticity component without violating the potential vorticity conservation law along the trajectory of a fluid particle (the potential vorticity coincides with the vertical vorticity only in the linear approximation).

If at the initial moment a stationary mode of vertical vorticity (with a zero frequency) is absent, and only two internal waves are present, then their weak interaction can generate vertical vorticity modes with sum and difference frequencies and wavenumbers of the primary waves. The forcing of the vertical vorticity arises because of the so-called vortex turning mechanism, when the velocity field of each wave turns the other wave’s horizontal vorticity toward the vertical (Lelong and Riley 1991). Such vortical motions do not arise only when internal wave velocities lie in the same vertical plane or two waves have equal frequencies.

The nonresonant wave–wave interactions, the interactions between internal waves and horizontal vortical modes (that do not change the frequencies of primary waves), and horizontal vortical mode interactions (which change only modal wavenumbers under zero frequency) are nonselective, in contrast to the resonant wave–wave interactions. Such interactions can occur many times, generating new waves and horizontal vortex modes. Each of these modes experiences the influence of advection in the velocity field created by all other waves and vortical modes.

**r**in the unperturbed atmosphere (

*S*

_{i}= 0), which is in static equilibrium. The displacement

**x**and a time

*t*is caused by the instantaneous displacement

**r**to a point

**x**satisfying (2); therefore,

*N*are caused by an ensemble of upward-propagating internal gravity wave modes:where

*N*(

*z*) over vertical scales of wave modes or for two-layer atmosphere with some constant

The Jacobian *J* is the ratio of the air parcel density *J* from 1 equals the relative change in density

Another assumption simplifying the derivation of the expression for the correlation function (13) was that the components of the displacements and hydrodynamic fields

*z*= 0, which is in the midpoint between the heights of the points

## 3. The spectra of the Eulerian vertical displacements and horizontal velocities

### a. Low wavenumbers

In the linear approximation

While the displacements are caused by vertical vorticity modes and waves of small amplitudes and with small wavenumbers, the space–time spectrum

### b. 3D spectra of vertical displacements and horizontal velocities at high wavenumbers

When wavenumbers approach the ellipsoid surface

The nonlinear spectral tail at

Suppose that in the Lagrangian system, some equilibrium spectra of the displacements and velocities are formed because of the balance between the energy input to the system of waves and horizontal vortical motions from their random sources at wavenumbers

Each component of the Eulerian space–time spectrum (17) with the wavenumber

*z*]. Because of the assumed axial symmetry of the Lagrangian spectra

*M*being the ratio of the rms vertical (horizontal) displacements to the vertical scale

*M*and anisotropy

*A*

_{0}grows from zero along with the mean square of the vertical gradient

The important result that follows from the obtained asymptotic spectra (38) and (39) and (41) and (42) at high vertical wavenumbers is that they do not depend on the specific forms of the Lagrangian spectra *L*(*k*_{⊥}, *m*) = const

### c. Space–time (four-dimensional) spectra of vertical displacements and horizontal velocities at high wavenumbers

It is seen from (43) that nonlinear diffusion of the spectrum

An expression similar to (43) may be obtained for the space–time spectrum of the horizontal velocity

## 4. One-dimensional spectra of vertical displacements and horizontal velocities

### a. Vertical wavenumber spectra at high wavenumbers

*χ*= (10–100) and

The amplitude of the spectral tail (44) reaches a maximum value of 0.22 for

For the lower stratosphere, the typical rms values of horizontal wind velocity fluctuations ^{−1} (Vincent et al. 1997). Taking *N* = 0.02 rad m^{−1}, it is possible to estimate the maximal (outer) and minimal vertical sales that bound the *L** = 2*π*/*m** = 1.6–3.0 km, *l*_{c} = 2*π*/*m*_{c} = 10–16 m.

We note that under nonlinear saturation of the spectrum, *M* ~ 0.32–0.40, the anisotropy of the layered inhomogeneities with small vertical scales,

*N*= 0.02 rad m

^{−1},

*M*= 0.4, we obtain from (49)

The 1D frequency spectrum of the velocity fluctuations predicted by the model is given by

As shown by Chunchuzov (2002, 1771–1772), the contribution to the obtained *J* [given by (4)], which take into account effects of fluid compressibility, does not change the *J* = 1. However, the numeric simulation by Klaassen and Sonmor (2006) and Klaassen (2009a,b) of formation of the internal wave spectrum with the kinematic model comprising a number of specified wave modes showed that, for the typical wave amplitudes required to produce the *J* may reach zero values that mean unphysical fluid parcel volume changes.

The model by Chunchuzov (2009) with only three interacting discrete wave modes also showed the appearance of multivalues and discontinuities in the Eulerian field at some finite wave amplitudes (see the bottom of Fig. 3). Such unphysical multivalues and discontinuities in a highly nonlinear wave field arise only in the ideal (nonviscous) fluid and within the narrow layers, whose thickness is small compared to the vertical scales,

Another well-known example of the wave solutions in the ideal fluid is a plane sinusoidal acoustic wave of finite amplitude, whose wave profile distorts and undergoes nonsinusoidal steepening with increasing distance from a plane source until it becomes multivalued within a narrow wave front (Lighthill 1978; Rudenko and Soluyan 1977). By taking into account a molecular viscosity and thermal conductivity within a narrow shock front, one can stabilize the nonlinear steepening of the wave profile and prevent the arising of multivalues.

In case of internal waves propagating through realistic atmosphere, the different types of wave-induced instabilities may prevent the arising of discontinuities in the wave field profile through wave-breaking processes and transferring wave energy into the energy of turbulent eddies. Klaassen (2009b, p. 1123) suggested that “through instability atmospheric wave fields create, and possibly coexist with, a field of smaller-scale (perhaps turbulent) eddies, which act to exert a force on the mean background flow. In other words, the secondary field of eddies acts as a dissipation mechanism for the larger-scale internal waves, producing momentum deposition and perhaps saturation as the latter continue to propagate.” As shown by Chunchuzov (2009), the smaller-scale eddies are generated within local regions of space whose vertical sizes are smaller than the vertical scales of highly nonlinear wave perturbations shaping the

In the statistical model developed here for the case of a high number of randomly independent amplitudes of modes, the multivalues and unphysical volume changes are reached when the rms value Λ = [〈(*J* − 1)^{2}〉]^{1/2} becomes close to 1. As shown by Chunchuzov (2002, p. 1771), Λ = [〈(*J* − 1)^{2}〉]^{1/2} = 3.8*M*^{2} for the specific form of the Lagrangian spectrum, where *M* becomes close to 0.5. It is important to note that the nonlinear saturation of the *M* = 0.32–0.40.

### b. Horizontal wavenumber spectra in the region of large wavenumbers

Thus, the layered inhomogeneities with vertical scales

As seen from (55), the horizontal wavenumber spectrum depends on the anisotropy of the inhomogeneities *σ* = 4 m s^{−1}, *N* = 0.02 rad m^{−1}, *χ* = 10, and *M* = 0.4, then *α* = 0.12,

With the parameters chosen above, the result of calculating the contribution (53) from the inhomogeneities with small vertical scales,

The large-scale IGWs and horizontal vortical motions with

### c. The model of 3D spectrum at small wavenumbers

We assume that the sources of IGWs and vortical motions are localized at small wavenumbers,

As *p* = 5/3 according to Nastrom and Gage (1985) and *p* = 2 according to Bacmeister et al. (1996). The mechanism of formation of the

*C*is a coefficient of proportionality. Within such a wide wavenumber interval,

*x*axis decreases as

*f*is the inertial frequency. The effect of rotation becomes significant when the frequency becomes close to

*f*; that is, for the horizontal wavenumber

*m*(Gill 1982, section 7),where

The choice of the Lagrangian spectrum in the form (56) allows us to estimate the variance of the horizontal gradient

*σ*= 3 m s

^{−1},

*N*= 0.02 rad s

^{−1},

*f*= 10

^{−4}rad s

^{−1}in (46) and taking into account that, under saturation of the spectrum, the nonlinear parameter

*m*

_{0}= 0.0027 rad m

^{−1}(the corresponding vertical scale is 2.3 km), the outer vertical scale of the layered inhomogeneities

We now proceed directly to calculate the contribution

*C*(

*z*) is unknown function. In (63), we took into account that for sufficiently large horizontal wavenumbers, satisfying the condition

*C*(

*z*), we integrate (63) over

*M*= 0.32–0.40, and lies in the range 0.008–0.012. For the parameters chosen (

*σ*= 3 m s

^{−1},

*f*= 10

^{−4}rad m

^{−1}), we obtain

### d. Comparison of the theoretical and observed horizontal wavenumber spectra

The aircraft measurements of the wind velocity fluctuations show that the variances of the fluctuations over a mountainous terrain is several times higher than over a flat terrain or ocean (Lilly and Lester 1974; Nastrom et al. 1987). The horizontal wavenumber spectra of the velocity fluctuations obtained in these studies are shown in Fig. 5 along with the spectra calculated from our model. Lilly and Lester (1974) analyzed in detail the horizontal wind velocity and temperature fields over the mountain ridge of southern Colorado. These fields were measured along 150–200-km tracks in the altitude range from 13 to 20 km with a vertical separation of 600 m. The most intense wave disturbances with characteristic horizontal scales of 20–30 km, alternating with the regions of small-scale turbulence, were detected in the lee zone of the ridge. Along with wave disturbances, the alternating layers with strong and weak winds and strong and weak vertical gradients of potential temperature were found.

At the same time, the variances of the horizontal wind velocity fluctuations were high enough, *σ*^{2} = 30–68 m^{2} s^{−2}, whereas the variances of the vertical velocity fluctuations ^{2} s^{−2} (see Table 1 in Lilly and Lester 1974). The ratio *N*^{2} = (5.1–6.2) × 10^{−4} rad^{2} s^{−2}.

We note that the values of ^{−4}–10^{−3}).

The model spectra *σ* = 7 m s^{−1}, *N* = 0.025 rad s^{−1}, *m** = *N*/*σ* = 0.0036 rad s^{−1},

Using expressions (65) and (66) for the

Figure 5a shows the average horizontal spectra of meridional wind velocity fluctuations obtained by Nastrom et al. (1987) in the upper troposphere and lower stratosphere (the tropospheric spectrum is shifted down on one decade with respect to the stratospheric spectrum). The flight routes were divided into 64- and 256-km-long segments, for which there was an increase in the variance of the velocity fluctuations ^{2} s^{−2}. The spectrum calculated for *σ* = 3 m s^{−1},

## 5. Conclusions

In this paper, the analytic form for the 3D spatial spectrum of mesoscale fluctuations of the horizontal wind velocity and vertical displacements in the model of a stably stratified atmospheric layer with constant buoyancy frequency *N* and in the absence of mean wind shear was obtained. It was assumed that during some observational periods, the random displacements of fluid parcels in Lagrangian frame caused by an ensemble of weakly interacting internal gravity modes and horizontal vortical motions may be considered as a stationary random and homogeneous Gaussian process within some local volume of the atmosphere. The random wind velocity and displacement spatial gradients were supposed to be induced by the waves and horizontal vortical motions themselves.

The nonlinear transfer to the Eulerian frame leads to the deviation of random displacements from the Gaussian distribution. In this frame, the 3D spectrum of the wind velocity fluctuations was obtained, using which the observed vertical wavenumber and horizontal wavenumber spectra of the fluctuations were explained. The mechanism proposed for the formation of such a spectrum is in the nonlinear transfer of energy, coming from the sources of internal waves and horizontal vortical motions at some vertical and horizontal scales, toward velocity fluctuations with smaller vertical scales and, at the same time, larger horizontal scales. It was shown that such energy transfer caused by strong wave–wave interactions, the interactions between waves and vertical vorticity modes, and between vertical vorticity modes leads to the formation of finescale layered inhomogeneities in the velocity and displacement fields.

It was also shown that the frequency–wavenumber (4D) spectrum of the layered inhomogeneities is localized at low frequencies, including zero frequency, high vertical wavenumbers, and low horizontal wavenumbers. Because of the strong advection of wave modes and vortical modes by the wind induced by all these modes, the obtained 3D and 4D spectra of the layered inhomogeneities take universal forms regardless of the forms of the Lagrangian spectra of the random sources of these modes.

The 3D spectrum was shown to depend only on the mean-square gradients of the displacement field, which determine the rate of nonlinear diffusion with increasing wavenumbers of the spectral density in the 3D wavenumber space and the anisotropy of such diffusion. The horizontal vortices contribute to the rms gradients of the displacement and velocity fields along with internal waves, thereby increasing the rate of nonlinear spreading of the 3D spectrum in comparison with the case when the displacements are caused by internal waves only.

The obtained 1D (vertical and horizontal) wavenumber spectra of the horizontal velocity and vertical displacements fall off in accordance with the −3 power law. The ranges of the vertical and horizontal wavenumbers were found for which the anisotropy of the inhomogeneities (the ratio of their horizontal scales to the vertical ones) remains constant. In the region of large scales, in comparison with the rms displacements of the air parcels, the 3D spectrum of wave perturbations was chosen so that the corresponding 1D horizontal wavenumber spectrum was close to the observed

The horizontal wavenumber spectrum of the velocity fluctuations was shown to contain the contributions from the large-scale perturbations caused by internal waves and horizontal vortical motions and from finescale layered inhomogeneities. The horizontal scale is found at which there is a transition from the

The theoretical spectra obtained were compared with the observed vertical and horizontal spectra of the mesoscale horizontal wind velocity fluctuations in the upper troposphere and stratosphere. It was shown that the main parameters on which these spectra depend are the rms fluctuations of the horizontal velocity during the observation time, their ratio to the rms fluctuations of the vertical velocity (i.e., the anisotropy parameter of the wave sources), the Brunt–Väisälä frequency, and the Coriolis parameter.

This work was supported by Grants RFBR 16-05-00438 and 15-05-3461 (sections 2 and 3) and Grant RSF 14-47-00049 (section 4). I thank G. Golitsyn for discussion of my work.

# APPENDIX

## Contribution to the 1D Spectrum from Large Vertical Scales

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