**4. Impulsive source model**

Radiation of internal waves by the jet turbulence occurs due to the transfer of the kinetic energy of the vertical velocity fluctuations into the initial potential energy of isopycnal displacement and its further redistribution between the kinetic and potential energy of waves and turbulence. This transfer is most pronounced at the early stage of the jet flow evolution when the vertical velocity fluctuations are significant (i.e. for *Nt* < 10, cf. Fig. 4). The results in the previous section show that the properties of IW radiated by the jet flow turbulence are to some degree analogous to the properties of the internal waves radiated by an impulsive source. Let us consider this analogy in more detail.

<sup>0</sup> 40 at the distance of 2 2 *y z* 3 from the jet streamwise axis for Richardson numbers Ri = 1 and Ri = 3. The figure shows that in both cases IW packet arrives at the observation point at time moment 10 *Nt* which is independent of the Richardson number. At the given point, the IW paket amplitude grows at times 20 *Nt* whereas the velocity *Uz* and density *ρ* oscillate with a phase shift / 2 typical of a monochromatic, linear internal wave (Phillips 1977). At times *Nt* > 30 the oscillations of *Uz* and *ρ* become incoherent and IW packet amplitude decreases. At sufficiently late times (*Nt* > 60) there remain only small-amplitude oscillations of the velocity and density with the buoyancy frequency *N*. These smallamplitude oscillations are similar to the buoyancy oscillations which supercede internal waves, which are mutually cancelled due to the destructive interference as their wavelengths become of the order of the source diameter (i.e. of order one in the considered

Figure 11 presents spatially averaged frequency spectra of the vertical velocity obtained in DNS for Richardson numbers Ri = 1 and 3. Each spectrum was obtained by Fourier transform of time series ( ) *U t <sup>z</sup>* and averaged over 100 points located uniformly at a cylindrical surface with radius 2 2 *y z* 3 in the range 0 0 0 90 . The figure shows that the velocity spectra are characterized by a well pronounced peak at the frequency / 0.7 *N* which corresponds to the IW propagation angle <sup>0</sup> arccos / 45 *N* . This value is close to the prediction of the linear theory stating that the amplitude of the waves radiated impulsively by a small-size source has a maximum at <sup>0</sup> arctan 2 55 . This is also in

Figure 12 presents the dependence of the internal waves amplitude ( *iw* ) vs. the inverse Richardson number ( <sup>1</sup> *Ri* ) obtained in DNS for two different values of the Reynolds number, Re = 400 and 700. The IW amplitude was evaluated in each DNS run as a maximum of the r.m.s. density fluctuation '( , , ) *y z t* (2.16) at the distance 2 2 *y z* 3 from the jet streamwise axis. (Note that since we consider the linear reference density profile, '( , , ) *yzt* is also equivalent to the average fluid particle vertical displacement amplitude i.e. the average IW amplitude.). The DNS results show that, for all considered Ri, the maximum of ' is observed at time moment *Nt* ≈ 15 in the range 0 0 40 60 . Figure 12 shows that IW amplitude decreases with increasing Ri. The figure also shows an asymptotic estimate

Below in Section 4 the application of the impulsive source theory to explain the observed

Radiation of internal waves by the jet turbulence occurs due to the transfer of the kinetic energy of the vertical velocity fluctuations into the initial potential energy of isopycnal displacement and its further redistribution between the kinetic and potential energy of waves and turbulence. This transfer is most pronounced at the early stage of the jet flow evolution when the vertical velocity fluctuations are significant (i.e. for *Nt* < 10, cf. Fig. 4). The results in the previous section show that the properties of IW radiated by the jet flow turbulence are to some degree analogous to the properties of the internal waves radiated by

general agreement with the data in Figs. 7 and 8 discussed above.

for the amplitude of the impulsively emitted internal waves (in dotted line).

an impulsive source. Let us consider this analogy in more detail.

IW kinematics and dynamics is discussed.

**4. Impulsive source model** 

case) (Voisin 1991).

Fig. 9. Instantaneous contours of the vertical velocity *Uz* in the horizontal (*x,y*) plane above the jet streamwise axis at *z* = 3 at time moments *Nt* = 12, 15, 18 (from top to bottom).

In the considered case, an IW radiation source function can be evaluated from the equation for the vertical velocity derived by Phillips (1977). In this equation, the non-linear terms are not discarded but can be regarded as an effective source of IW. This equation is derived in the inviscid case in the form:

$$\frac{\partial^2}{\partial t^2} \left( \nabla^2 \mathcal{U}\_z \right) + N^2 \nabla\_h^2 \mathcal{U}\_z = \mathcal{Q}(x, y, z, t) \tag{4.1}$$

where the IW source function is

$$\mathcal{Q}(\mathbf{x}, y, z, t) = \frac{\hat{\mathcal{O}}^3}{\hat{\mathcal{O}}\mathbf{x}\_{\alpha} \hat{\mathcal{O}} z \hat{\mathcal{O}} t} \left( \mathcal{U}\_{j} \frac{\partial \mathcal{U}\_{\alpha}}{\partial \mathbf{x}\_{j}} \right) - \nabla\_{h}^{2} \left\{ \mathcal{U}\_{j} \frac{\partial \mathcal{b}}{\partial \mathbf{x}\_{j}} + \frac{\hat{\mathcal{O}}}{\hat{\mathcal{O}} t} \left( \mathcal{U}\_{j} \frac{\partial \mathcal{U}\_{z}}{\partial \mathbf{x}\_{j}} \right) \right\} \tag{4.2}$$

In (4.1) and (4.2) the notations are as follows: 222 2 2 22 *xyz* , 2 2 2 *<sup>h</sup>* 2 2 *x y* are the full and horizontal Laplasian operators, respectively; summation over the repeated indexes

Internal Waves Radiation by a Turbulent Jet Flow in a Stratified Fluid 113

Ri = 1 Ri = 3

0.01 0.1 1 10

Fig. 11. Spatially averaged frequency spectra of the vertical velocity obtained in DNS for

Thus *Q* in (4.1) can be regarded as an impulsive source function which brings about IW radiation by the jet flow turbulence. The temporal growth and maximum amplitude of this source function is controlled by the non-linear flow dynamics, not by buoyancy. Let us now make some estimates provided by the impulsive source theory (Zavolskii & Zaytsev 1984, Voisin 1991, BCH 1993) concerning the kinematics and dynamics of the radiated internal

The time interval during which the IW packet, emitted impulsively by the source (S) at the angle *θ* with respect to the vertical (cf. Fig. 15), comes to the observation point (O) located at the distance 2 2 *ry z* , can be estimated with the use of the linear theory as follows. The group velocity ( *<sup>g</sup> c* ) of internal waves with streamwise wavenumber *k* and the isophase line

sin 2

Thus the time interval, *Nt*<sup>1</sup> , during which the IW wave packet comes from the source (S) to

2 sin 2

Eq. (4.4) shows that time interval *Nt*1 does not depend on *N* in agreement with the DNS results in Fig. 10. Substitution of *r* = 3, arctan 2 , and 1.3 *<sup>s</sup> k k* into (4.4) gives the

*<sup>k</sup>* . (4.3)

*kr Nt* . (4.4)

2 *<sup>g</sup> N*

1

estimate 1 *Nt* 8 , which is also in good agreement with the DNS results in Fig. 10.

*c*

/N

0

at the angle *θ* with respect to the vertical is evaluated as:

the observation point (O), is evaluated as:

0.0005

different Richardson numbers.

waves.

0.001

0.0015

0.002

0.0025

Uz

 1,2 and *j* = 1, 2, 3 is performed; 123 ( , , ) (,,) *x x x xyz* are the Cartesian coordinates; <sup>123</sup> ( , , )( , , ) *UUU UUU <sup>x</sup> <sup>y</sup> <sup>z</sup>* are the fluid velocity components; and *b Ri* is the buoyancy.

Fig. 10. Temporal development of the vertical velocity and density, *Uz* and *ρ*, obtained in DNS at *x* = 18 and *θ* = <sup>0</sup> 40 (the angle with respect to the vertical) at the distance 2 2 *y z* 3 from the jet streamwise axis for Richardson numbers Ri =1 (top frame) and Ri = 3 (bottom frame).

Figure 13 shows an instantaneous distribution of the source function *Q*(*x*, *z*, *y* = 0) evaluated in DNS in the central vertical plane at time moments *Nt* = 3 and 6. The figure shows that the distribution of *Q* is quite inhomogeneous and confined to the jet core region. Therefore, the effective source diameter is of the order of the initial jet flow diameter. The figure indicates also that the growth of the modes with wavenumbers *<sup>s</sup> k k* and 0.5 *<sup>s</sup> k k* in the velocity spectra at time moments *Nt* < 10 (Fig. 5) leads to a modulation of the source function amplitude along the streamwise coordinate . (In Fig. 13 this modulation is more pronounced at time moment *Nt* = 6.)

Figure 14 presents the temporal development of the volume-averaged dispersion of the IW source function, 2/12 *tzyxQ* ),,,( , obtained in DNS for Ri = 1 and Ri = 3. The figure shows that, for all Ri, the dispersion increases by t = 2.5 by the order of magnitude. The figure shows also that at later times 2/12 *tzyxQ* ),,,( decays, so that its amplitude is reduced by more than half by t = 5 and becomes negligible at *Nt* > 10. From the spectra in Fig. 11 the period of internal waves having the maximum amplitude can be evaluated as *NNT* 8/2 *IW* (cf. Fig. 11) and is almost twice as large as compared to the period of the IW source function half-decay.

 1,2 and *j* = 1, 2, 3 is performed; 123 ( , , ) (,,) *x x x xyz* are the Cartesian coordinates; <sup>123</sup> ( , , )( , , ) *UUU UUU <sup>x</sup> <sup>y</sup> <sup>z</sup>* are the fluid velocity components; and *b Ri* is the buoyancy.

0 40 80 120 160 200

U

z

0 40 80 120 160 200

Nt

Fig. 10. Temporal development of the vertical velocity and density, *Uz* and *ρ*, obtained in DNS at *x* = 18 and *θ* = <sup>0</sup> 40 (the angle with respect to the vertical) at the distance 2 2 *y z* 3 from the jet streamwise axis for Richardson numbers Ri =1 (top frame) and Ri = 3 (bottom

Figure 13 shows an instantaneous distribution of the source function *Q*(*x*, *z*, *y* = 0) evaluated in DNS in the central vertical plane at time moments *Nt* = 3 and 6. The figure shows that the distribution of *Q* is quite inhomogeneous and confined to the jet core region. Therefore, the effective source diameter is of the order of the initial jet flow diameter. The figure indicates also that the growth of the modes with wavenumbers *<sup>s</sup> k k* and 0.5 *<sup>s</sup> k k* in the velocity spectra at time moments *Nt* < 10 (Fig. 5) leads to a modulation of the source function amplitude along the streamwise coordinate . (In Fig. 13 this modulation is more pronounced

Figure 14 presents the temporal development of the volume-averaged dispersion of the IW source function, 2/12 *tzyxQ* ),,,( , obtained in DNS for Ri = 1 and Ri = 3. The figure shows that, for all Ri, the dispersion increases by t = 2.5 by the order of magnitude. The figure shows also that at later times 2/12 *tzyxQ* ),,,( decays, so that its amplitude is reduced by more than half by t = 5 and becomes negligible at *Nt* > 10. From the spectra in Fig. 11 the period of internal waves having the maximum amplitude can be evaluated as

8/2 *IW* (cf. Fig. 11) and is almost twice as large as compared to the period of the


0.01

0.02


frame).

*NNT*

IW source function half-decay.

at time moment *Nt* = 6.)


0


0

0.01

0.02

Fig. 11. Spatially averaged frequency spectra of the vertical velocity obtained in DNS for different Richardson numbers.

Thus *Q* in (4.1) can be regarded as an impulsive source function which brings about IW radiation by the jet flow turbulence. The temporal growth and maximum amplitude of this source function is controlled by the non-linear flow dynamics, not by buoyancy. Let us now make some estimates provided by the impulsive source theory (Zavolskii & Zaytsev 1984, Voisin 1991, BCH 1993) concerning the kinematics and dynamics of the radiated internal waves.

The time interval during which the IW packet, emitted impulsively by the source (S) at the angle *θ* with respect to the vertical (cf. Fig. 15), comes to the observation point (O) located at the distance 2 2 *ry z* , can be estimated with the use of the linear theory as follows. The group velocity ( *<sup>g</sup> c* ) of internal waves with streamwise wavenumber *k* and the isophase line at the angle *θ* with respect to the vertical is evaluated as:

$$c\_{\mathcal{g}} = \frac{N}{2k} \sin 2\theta \,\,. \tag{4.3}$$

Thus the time interval, *Nt*<sup>1</sup> , during which the IW wave packet comes from the source (S) to the observation point (O), is evaluated as:

$$Nt\_1 = \frac{2kr}{\sin 2\theta}.\tag{4.4}$$

Eq. (4.4) shows that time interval *Nt*1 does not depend on *N* in agreement with the DNS results in Fig. 10. Substitution of *r* = 3, arctan 2 , and 1.3 *<sup>s</sup> k k* into (4.4) gives the estimate 1 *Nt* 8 , which is also in good agreement with the DNS results in Fig. 10.

Internal Waves Radiation by a Turbulent Jet Flow in a Stratified Fluid 115

oscillations of the density with the frequency *N* whose group velocity is identically zero. These buoyancy oscillations are present in Fig. 10 (for *Nt* > 40) and in the distributions of the vertical velocity in the form of columnar disturbances at time moment *Nt* = 90 in Figs. 7, 8. The asymptotics for the IW amplitude in Fig. 12 is derived as follows. Using the same schematic in Fig. 15 the following estimate can be obtained for the amplitude of internal waves ( *iw* ) emitted impulsively by the source (S) and coming to the observation point (O)

QIW


*iw* <sup>~</sup> *Ri* . (Note

x

Fig. 13. Instantaneous contours of the IW source function *QIW* in the vertical central plane

*iw* <sup>~</sup> *Nt*

max

*iw* <sup>~</sup> *Nr*

Eq. (4.8) shows that IW amplitude is inversely proportional to *N*, so that max 0.5

that the jet flow Froude number equals *Fr Ri <sup>j</sup>* 1 / , so that max *iw* ~ *Frj* .) This estimate is in good agreement with the results in Fig. 12 for sufficiently small amplitudes (for *iw* < 0.03). The growth of *iw* saturates for larger amplitudes, probably due to nonlinear effects, and

In the present paper, direct numerical simulation (DNS) has been performed in order to study the process of internal waves radiation by a stratified turbulent jet flow. An initially

1/2 sin cos

. (4.7)

. (4.8)

*Nr*

1/2

1/2 sin 2

Eq. (4.7) shows that *iw* increases with time. As it was discussed above, the IW amplitude increases until time when the wavelength of incoming waves becomes of the order of the source size, i.e. for *Nt Nt* <sup>2</sup> . Thus, the maximum amplitude ( max *iw* ) can be evaluated from

at time moments *Nt* = 3 (top frame) and *Nt* = 6 (bottom frame).

0 5 10 15 20 25 30 35

at time *t* (cf. e.g. BCH):

(4.6) and (4.7) as:

z


5


increases for larger Reynolds number.

**5. Conclusions and discussion** 

0

0

5

Fig. 12. The internal waves amplitude ( *iw* ) vs. inverse Richardson number obtained in DNS for different Reynolds numbers. The dotted line shows the asymptotic solution for the amplitude derived from the linear theory for an impulsive point source.

Figure 10 shows that at sufficiently late times (*Nt* > 40) the density oscillations at the considered distance from the jet streamwise axis (*r* = 3) are damped. The time moment ( *Nt*<sup>2</sup> ) after which the IW amplitude decays at a given location can be estimated as follows. The wavelength *λ* of IW emitted impulsively by a source (S) at the angle *θ* with respect to the vertical at the distance *r* and coming to the observation point (O) at time *t* (cf. Fig. 15) can be evaluated as:

$$
\lambda \approx \frac{2\pi r}{Nt\sin\theta}.\tag{4.5}
$$

The theoretical analysis of the internal waves field generated by a sphere performed by Voisin (1994) shows that if *λ* becomes of the order of the sphere radius, the destructive interference of the waves emitted from different locations on the sphere leads to a decay of the wave amplitude. In the considered case, the cutoff wavelength is of the order of the jet initial diameter, i.e. 1 *<sup>c</sup>* . Substitution of *c* in (4.5) gives

$$Nt\_2 \approx \frac{2\pi r}{\sin \theta} \,\mathrm{.}\tag{4.6}$$

For *r* = 3 and arctan 2 we obtain from (4.6) an estimate 2 *Nt* 30 , which is also in good agreement with the DNS results (Fig. 10). The linear theory also predicts that at times *Nt Nt* <sup>2</sup> , the incoming waves are mutually cancelled, and there remain only buoyancy

Re=700 Re=400

Ri-0.5

0.1 1 10

Fig. 12. The internal waves amplitude ( *iw* ) vs. inverse Richardson number obtained in DNS for different Reynolds numbers. The dotted line shows the asymptotic solution for the

Figure 10 shows that at sufficiently late times (*Nt* > 40) the density oscillations at the considered distance from the jet streamwise axis (*r* = 3) are damped. The time moment ( *Nt*<sup>2</sup> ) after which the IW amplitude decays at a given location can be estimated as follows. The wavelength *λ* of IW emitted impulsively by a source (S) at the angle *θ* with respect to the vertical at the distance *r* and coming to the observation point (O) at time *t* (cf. Fig. 15)

> 2 sin *r*

The theoretical analysis of the internal waves field generated by a sphere performed by Voisin (1994) shows that if *λ* becomes of the order of the sphere radius, the destructive interference of the waves emitted from different locations on the sphere leads to a decay of the wave amplitude. In the considered case, the cutoff wavelength is of the order of the jet

. (4.5)

*<sup>r</sup> Nt* . (4.6)

*Nt* 

> 2 2 sin

For *r* = 3 and arctan 2 we obtain from (4.6) an estimate 2 *Nt* 30 , which is also in good agreement with the DNS results (Fig. 10). The linear theory also predicts that at times *Nt Nt* <sup>2</sup> , the incoming waves are mutually cancelled, and there remain only buoyancy

amplitude derived from the linear theory for an impulsive point source.

initial diameter, i.e. 1 *<sup>c</sup>* . Substitution of *c* in (4.5) gives

Ri -1

0.01

can be evaluated as:

0.1

iw

oscillations of the density with the frequency *N* whose group velocity is identically zero. These buoyancy oscillations are present in Fig. 10 (for *Nt* > 40) and in the distributions of the vertical velocity in the form of columnar disturbances at time moment *Nt* = 90 in Figs. 7, 8. The asymptotics for the IW amplitude in Fig. 12 is derived as follows. Using the same schematic in Fig. 15 the following estimate can be obtained for the amplitude of internal waves ( *iw* ) emitted impulsively by the source (S) and coming to the observation point (O) at time *t* (cf. e.g. BCH):

Fig. 13. Instantaneous contours of the IW source function *QIW* in the vertical central plane at time moments *Nt* = 3 (top frame) and *Nt* = 6 (bottom frame).

$$\rho\_{iw} \sim \frac{\sin \theta \boxed{Nt} \boxed{Nt} \boxed{\cos \theta \boxed{\cdot}}^{1/2}}{Nr} \,. \tag{4.7}$$

Eq. (4.7) shows that *iw* increases with time. As it was discussed above, the IW amplitude increases until time when the wavelength of incoming waves becomes of the order of the source size, i.e. for *Nt Nt* <sup>2</sup> . Thus, the maximum amplitude ( max *iw* ) can be evaluated from (4.6) and (4.7) as:

$$
\rho\_{iw}^{\text{max}} \sim \frac{\left[\pi \left| \sin 2\theta \right| \right]^{1/2}}{N r^{1/2}}.\tag{4.8}
$$

Eq. (4.8) shows that IW amplitude is inversely proportional to *N*, so that max 0.5 *iw* <sup>~</sup> *Ri* . (Note that the jet flow Froude number equals *Fr Ri <sup>j</sup>* 1 / , so that max *iw* ~ *Frj* .) This estimate is in good agreement with the results in Fig. 12 for sufficiently small amplitudes (for *iw* < 0.03). The growth of *iw* saturates for larger amplitudes, probably due to nonlinear effects, and increases for larger Reynolds number.

#### **5. Conclusions and discussion**

In the present paper, direct numerical simulation (DNS) has been performed in order to study the process of internal waves radiation by a stratified turbulent jet flow. An initially

Internal Waves Radiation by a Turbulent Jet Flow in a Stratified Fluid 117

Other waves become unstable at a lower amplitude and their breaking would explain the observed frequency selection. However, in our case the wave amplitude is much smaller than the critical value / 0.07 *AIW x* (cf. Fig. 4 where max 0.05 *IW* and max / 0.01 *IW x* for 5 *<sup>x</sup>* ) and this mechanism is not applicable. Taylor and Sarkar (2007) developed a linear viscous model to estimate the decay in wave amplitude. Waves with high and low frequencies have smaller vertical group velocities and dissipate more as compared to the waves with propagation angle in the vicinity of <sup>0</sup> 27 . Thus the viscous dissipation may cause the observed selected IW frequency range. However, our DNS results show that the selection of the IW propagation angle occurs in the close vicinity of the jet core region ( *z* 1 ) where viscous diffusion effects have not yet accumulated and are negligible as far as the frequency selection is concerned. Therefore, the impulsive source model considered in the present paper provides the most plausible explanation of the IW kinematics and

r

Fig. 15. Schematic of the radiation of the IW packet by an impulsive point source (S) with the group velocity *<sup>g</sup> c* at the angle *θ* with respect to the vertical axis at the distance *r* from the

This work is supported by RFBR (10-05-00339, 10-05-91177, 09-05-00779, 11-05-00455, 10-01-

Batchelor J.K. & Gill A.E. (1962). Analysis of the stability of axisymmetric jets. *J. Fluid Mech.*

Beckers M., Verzicco R., Clercx H.J.H., & Van Heist G.J.F. (2001) Dynamics of pancake-like

Bevilaqua P.M. and Lykoudis P.S. (1978). Turbulence memory in self-preserving wakes. *J.* 

Bonneton P., Chomaz J.M., & Hopfinger E.J. (1993). Internal waves produced by the

vortices in a stratified fluid: experiments, model and numerical simulations. *J. Fluid* 

turbulent wake of a sphere moving horizontally in a stratified fluid. *J. Fluid Mech.*

O

dynamics observed in DNS.

observation point (O).

00435).

**7. References** 

**6. Acknowledgement** 

Vol. 14, pp. 559-551.

*Mech.*Vol. 433, pp. 1-27.

Vol. 254, pp. 23-40.

*Fluid Mech.* Vol. 89, pp. 589-606.

z

s

cg

circular, turbulent jet flow with a Gaussian profile of the mean streamwise velocity component in a fluid with stable, linear density stratification is considered which models the flow created in the far wake of a sphere towed in a stratified fluid at large Froude and Reynolds numbers. The DNS results show that at early times ( *Nt* 30 , where *N* is the buoyancy frequency) there occurs a collapse of the vertical velocity fluctuations which brings about the radiation of internal waves (IW). The characteristic spatial period of these waves is found to be close to the wavelength of the spiral instability mode of a non-stratified jet flow. The IW amplitude decreases with increasing the flow global Richardson number and is well described by the asymptotics 5.0 ~ *Ri iw* . At late times (*Nt* > 60) the jet flow becomes quasi-two-dimensional and is dominated by large-scale pancake vortices. At that stage, internal waves are superseded by non-propagating, columnar, small-amplitude buoyancy oscillations confined to a central vertical layer with a thickness of the order of the jet width. A linear model is proposed where the jet turbulence collapsing under the stabilizing effect of the buoyancy forces, is regarded as an impulsive source of IW radiation. The kinematics and dynamics of the internal waves observed in DNS are found to be in good agreement with the model prediction.

Note that a relatively narrow IW frequency range ( 0 0 40 arccos / 60 *N* ), similar to the one observed in our DNS, has been also observed in mixing-box experiments (Dohan & Sutherland 2003), in a flow over a vertical obstacle (Sutherland & Linden 1998), during the collapse of a mixed patch (Sutherland et al. 2007) and in LES of a density-stratified boundary layer (Taylor & Sarkar 2007). In these works, several models were proposed to explain the observed IW frequency range, and among them, perhaps, two pertain to the considered case of IW radiation by a temporally developing turbulent jet flow. Dohan and Sutherland (2003) employed stability criteria derived by Sutherland (2001) for low- and high-frequency waves which show that the largest critical IW amplitude corresponds to the waves propagating at <sup>0</sup> 45 .

Fig. 14. Temporal development of the volume-averaged dispersion of the IW source function normalized by its initial value for different Richardson numbers. Note that the graph on the right is scaled with the buoyancy frequency.

Other waves become unstable at a lower amplitude and their breaking would explain the observed frequency selection. However, in our case the wave amplitude is much smaller than the critical value / 0.07 *AIW x* (cf. Fig. 4 where max 0.05 *IW* and max / 0.01 *IW x* for 5 *<sup>x</sup>* ) and this mechanism is not applicable. Taylor and Sarkar (2007) developed a linear viscous model to estimate the decay in wave amplitude. Waves with high and low frequencies have smaller vertical group velocities and dissipate more as compared to the waves with propagation angle in the vicinity of <sup>0</sup> 27 . Thus the viscous dissipation may cause the observed selected IW frequency range. However, our DNS results show that the selection of the IW propagation angle occurs in the close vicinity of the jet core region ( *z* 1 ) where viscous diffusion effects have not yet accumulated and are negligible as far as the frequency selection is concerned. Therefore, the impulsive source model considered in the present paper provides the most plausible explanation of the IW kinematics and dynamics observed in DNS.

Fig. 15. Schematic of the radiation of the IW packet by an impulsive point source (S) with the group velocity *<sup>g</sup> c* at the angle *θ* with respect to the vertical axis at the distance *r* from the observation point (O).
