We investigate the focusing phenomenon of inertial waves generated by a vertically oscillating slender torus in a uniformly rotating fluid. Building on the previous research on linear aspects of inertial wave focusing by Liu et al. [“Focusing of inertial waves by a vertically annular forcing,” Phys. Fluids 34, 086601 (2022)], we analyze the focusing phenomenon in a fluid with a small Rossby number using the perturbation method. Our analytical solution to the weakly nonlinear problem reveals that, in addition to the primary inertial waves predicted by the linear theory, secondary inertial waves (SIW) are excited due to wave interaction if the forcing frequency is lower than the fluid rotation rate. Particularly, in our case, the SIWs induced directly by the oscillating torus form a relatively weak secondary focal area on the rotation axis below the primary focal point. The rays of the SIWs form a new double cone with a different angle than the primary cone. In addition, the SIWs are also excited at the primary focal point with a much stronger effect than that generated by the oscillating torus in an inviscid limit, but significantly suppressed even in low-viscosity fluids. A fully nonlinear analysis, especially in the primary focal zone, is necessary for further study.

Inertial waves are a type of fluid motion that occur in rotating systems. They are characterized by periodic variations in velocity and pressure due to the Coriolis force acting on fluid particles in motion, and their energy propagation depends only on the ratio of the wave frequency to the fluid rotation rate.1–3 In particular, the special dispersion relation of inertial waves allows for their energy to be focused onto a small area by using selective forcing, which can exhibit interesting nonlinear behaviors, such as the localized generation of turbulence. This focusing mechanism enhances mixing and transport processes in rotating fluids and has applications in various fields, including geophysics,4,5 astrophysics,6 and a wide range of industries.7 The focusing of inertial waves is often studied in controlled laboratory experiments8–10 as advances in particle image velocimetry have enabled the observation and quantitative study of inertial beams. Furthermore, combined with numerical simulations, unresolved scale information in experiments can be obtained. For instance, Duran-Matute et al.10 used a vertically oscillating slender torus in a cubic container to generate inertial waves that focused their energy onto a small region of the rotation axis, generating localized turbulence (see Fig. 1). Their experimental results were further confirmed by numerical simulations, which exhibited good qualitative and quantitative agreement. To fully comprehend the intricate process occurring in the focus region, additional theoretical analysis is required, which is the objective of the present investigation. We divide the entire study into several sub-tasks, starting from the linear approximation, extending to the weakly nonlinear analysis, and finally addressing the stability problem in fully nonlinear cases. Since the region of our interest in the theoretical analysis is far from the container boundaries, we can assume that the transport phenomena associated with the presence of remote walls (such as Ekman pumping) can be considered negligible.

FIG. 1.

Schematic drawing of the oscillating annular forcing and the focusing effect of the resulting weakly nonlinear inertial waves in the upper half plane, where Ω is the constant angular velocity of the rotating flow and b is the radius of the annular forcing distribution. θ0 and θ1 correspond to the propagation directions of the PIWs and the SIWs with different frequencies in the weakly nonlinear case, respectively.

FIG. 1.

Schematic drawing of the oscillating annular forcing and the focusing effect of the resulting weakly nonlinear inertial waves in the upper half plane, where Ω is the constant angular velocity of the rotating flow and b is the radius of the annular forcing distribution. θ0 and θ1 correspond to the propagation directions of the PIWs and the SIWs with different frequencies in the weakly nonlinear case, respectively.

Close modal

The classification of this problem strongly depends on the value of the Rossby number, which represents the ratio of the inertial force to the Coriolis force.11,12 In the first phase of our investigation, we considered the case where the Coriolis effect dominates, corresponding to the limit of Ro 0. In this regime, the nonlinear effect can be neglected, allowing us to analyze the basic properties of inertial wave focusing using the linearized Navier–Stokes equations.13 Under this assumption, we obtained the analytical solution of the velocity field, which corresponds to the primary inertial waves (PIW) in the context of the present investigation. It was shown that, under the axisymmetric annular forcing, the PIW rays form a double cone symmetric about the plane on which the torus is located. At the vertex of the cone, the waves are focused in a shock-like manner, causing localized energy surges. After focusing, the waves continue their propagation and form a new inverted cone with the same cone angle. These purely theoretical results are in good agreement with the experimental and numerical study by Duran-Matute et al.10 

In rotating flows with small but non-negligible nonlinear effects, weak interactions among the inertial waves occur as the Rossby number R o 1. These interactions lead to a slow but continuous transfer of energy between the waves and can also lead to the instability of monochromatic inertial waves of finite amplitude through triadic resonance.14 There are several approaches to analyzing the behavior of weakly nonlinear inertial waves. One classical method is to use a helical decomposition and kinetic equations to derive the energy and helicity for three-wave coupling.15,16 However, this approach can be complicated when analyzing particular solutions generated by an annular forcing. Another possible approach is to use a perturbation expansion, in which the solution is expanded in powers of a small parameter that characterizes the amplitude of the waves. This method allows us to understand the behavior of the waves in the weakly nonlinear regime and to explore a wide range of phenomena that can occur, such as wave interaction, resonance, and instability. In this investigation, we use the perturbation method to analyze the weakly nonlinear case based on the results obtain from the linear theory,13 and obtain the solution for the velocity field of the waves. Our analysis shows that the secondary inertial waves (SIW) can only be excited by nonlinearity when the forcing frequency is smaller than the fluid rotation rate, which is in good agreement with the experimental study by Cortet et al.17 Furthermore, the frequency of SIW is twice the frequency of PIW, which is exactly a consequence of the triadic resonance.18 Additionally, in our case, the SIWs excited directly by the oscillating torus focus onto a small area on the rotation axis and form a relatively weak secondary focal point below the primary focal point. The rays of the SIWs form a new double cone with a different angle than the primary cone. At the same time, SIWs are also excited locally at the primary focal point with a much stronger effect than that generated by the oscillating torus in an inviscid limit, but significantly suppressed even in low-viscosity fluids. As a result of wave interactions, nonlinearity becomes dominant around the primary focal point where a fully nonlinear analysis is required in the next stage to understand the mechanism of localized turbulence generation.

In this article, we first present the theoretical solutions of the weakly nonlinear inertial waves using the perturbation method and illustrate the velocity field graphically in Sec. II and then analyze the viscous effect on the focusing phenomenon in Sec. III. Finally, conclusions are drawn in Sec. IV.

We consider a uniformly rotating fluid with angular velocity Ω, density ρ, and kinematic viscosity ν. The flow is governed by the Navier–Stokes equations, which are non-dimensionalized using b as the length scale, Ω 1 as the time scale, and U as the characteristic relative fluid velocity, thus giving
(1)
u t + R o ( u · ) u + 2 n × u = p + E k 2 u + f ,
(1a)
· u = 0 ,
(1b)
where u is the dimensionless fluid velocity superimposed on the rigidly rotating flow, f is the external volume force, p is the fluid pressure, and n is a unit vector indicating the direction of the rotation axis. The Ekman number defined by
E k = ν Ω b 2
(2)
is a gross measure of the viscous force compared to the Coriolis force, while the Rossby number
R o = U Ω b
(3)
denotes the ratio of the inertial force to the Coriolis force and has a direct impact on the classification of such rotating flows. Actually, the characteristic velocity U is directly related to the forcing. A stronger forcing indicates a larger value of U. In this sense, the Rossby number represents the ratio of the forcing to the Coriolis force. A strong rotation and a weak forcing correspond to a small Rossby number. The ratio of the Rossby number to the Ekman number is the well-known Reynolds number.
For the following analysis, we introduce the cylindrical coordinate system ( r , φ , z ) with the coordinate unite vectors ( e r , e φ , e z ), with e z coinciding with the rotation axis n. It is assumed that the oscillating annular forcing f is monochromatic, axisymmetric, and has only one component in z-direction, i.e., f = ( 0 , 0 , f 0 z ), which can be described using the Dirac function as follows:
f 0 z ( r , φ , z , t ) = Λ 0 δ ( r 1 ) δ ( z ) e i σ 0 t .
(4)
Here, Λ 0 is a constant line density of the force along the torus r = 1. σ0 is the dimensionless forcing frequency defined as the ratio between the oscillation frequency of the torus and the rotation rate Ω of the fluid.

Generally, the perturbation method assumes that the solution to a problem can be expressed as a series of small changes or “perturbations” from a known or well-understood solution, represented by a power series in a small parameter. The leading term in this power series is the exact solution of a related, simpler problem, while further terms describe the deviations in the solution arising from the deviation from the original problem. For our weakly nonlinear problem, the solution obtained in our previous study from the linear theory still dominates and can be considered as the first term in this power series. The influence of the nonlinear term needs to be included as additional terms of perturbation. For this, the perturbation method is a very suitable approach for the weakly nonlinear problem.

Applying the perturbation method, we can formally express the full solution of u and p as a power series in a small parameter ϵ, as follows:

(5)
u = n = 0 ε n u n = u 0 + ε u 1 + ε 2 u 2 + ,
(5a)
p = n = 0 ε n p n = p 0 + ε p 1 + ε 2 p 2 + .
(5b)

For the present weakly nonlinear problem, it is natural to choose the Rossby number as the small parameter, i.e., R o = ε. Substituting the power series (5) into the momentum and the mass conversation equations (1) gives
[ ( t E k 2 ) u 0 + p 0 + 2 n × u 0 f 0 ] + R o [ ( t E k 2 ) u 1 + p 1 + 2 n × u 1 + ( u 0 · ) u 0 ] + O ( R o 2 ) = 0 ,
(6a)
· u 0 + R o · u 1 + O ( R o 2 ) = 0.
(6b)
The original momentum and mass conversation equations can now be split into a series of linear systems and solved consecutively. The zero-order equation system is given as
( t E k 2 ) u 0 = p 0 2 n × u 0 + f 0 ,
(7a)
· u 0 = 0 ,
(7b)
and the first-order equation system yields
(8)
( t E k 2 ) u 1 = p 1 2 n × u 1 + f 1 ,
(8a)
· u 1 = 0 ,
(8b)
where
f 1 ( u 0 · ) u 0
(9)
depends only on the solution of the PIW and acts like a forcing term that determines the particular solution of the first-order equation system, i.e., the SIW.
From the structure of the above two equation systems, we can clearly see that a great advantage of the perturbation method is that it allows us to express the weakly nonlinear problem as a series of linear systems that can be solved in a similar way. If necessary, we can repeat the process to find higher-order perturbations, which provide increasingly accurate corrections to the unperturbed solution. Usually, an approximate solution to the problem is obtained by truncating the series. In our case, we approximate the final solution as the sum of the primary solution and the first-order perturbation correction by keeping only the first two terms in the series (5) as follows:
u u 0 + R o u 1 and p p 0 + R o p 1 .
(10)
Here, u 0 and p0 represent the known solutions corresponding to the PIWs obtained by Liu et al.13 and serve as the basis for the subsequent perturbations. The corresponding solution u 0 for a inviscid fluid is also shown in Eq. (A1) in  Appendix A. In the following, we will focus on solving the first-order momentum equations (8) and analyzing the properties of the SIWs.
To analyze the basic propagation properties of the SIW, we begin by ignoring the effect of viscosity, i.e., E k = 0. Similar to the analysis of the PIW in our previous study,13 we can construct a wave equation for p1 by eliminating the velocity u 1 from the first-order equations. Specifically, we apply the operator ( 2 t 2 · ) to Eq. (8a) and then reduce the result using the continuity equation (8b). This yields a single wave-like equation for the pressure p1 as follows:
2 t 2 2 p 1 + ( 2 n · ) 2 p 1 = 4 n · ( n · ) f 1 + 2 n · t ( × f 1 ) + 2 t 2 · f 1 .
(11)
Under the axisymmetric assumption, the right-hand side of this equation can be expressed in cylindrical coordinates as
σ 1 2 r ( r f 1 r ) r i 2 σ 1 r ( r f 1 φ ) r + ( 4 σ 1 2 ) f 1 z z = Q 1 ( r , z ) · e i σ 1 t ,
(12)
where Q 1 ( r , z ) represents the spatially dependent part and σ1 is the frequency of the term f 1. According to (9), f 1 depends only on the velocity field solution of the PIWs and can be expressed as
f 1 ( r , z , t ) = ( U 0 · ) U 0 e i 2 σ 0 t = F 1 ( r , z ) e i σ 1 t .
(13)
Again, U 0 corresponds the spatially dependent part in the velocity solution of the PIWs with u 0 = U 0 ( r , z ) e i σ 0 t and F 1 represents the spatially dependent part of f 1. Then, we can get
σ 1 = 2 σ 0 .
(14)
This result corresponds exactly to Greenspan's description1 in his theory, as a nonlinear term can couple two inertial waves with different frequencies, producing additional waves at the sum of their frequencies. Specifically, a single wave can interact with itself, resulting in a steady zonal flow and an oscillation at twice the basic frequency. This phenomenon can also be explained by triadic resonance.14–16 Furthermore, according to the dispersion relation of inertial waves, we can predict that the energy of the SIWs will propagate at an angle of
θ 1 = cos 1 ( σ 1 / 2 ) = cos 1 ( σ 0 )
(15)
with respect to the horizontal, as shown in Fig. 1.
The properties of the SIWs caused by the forcing f 1 are contained in the particular solution of the wave-like equation (11), which is the key topic of the following analysis. Generally, the particular solution can be separated in time and space in the form
p 1 ( x , t ) = P 1 ( x ) e i σ 1 t .
(16)
Using Fourier transform in cylindrical coordinates, we can get the spatial part P 1 ( x ) in Fourier space
P ̂ 1 ( k r , k z ) = Q ̂ 1 ( k r , k z ) 4 k z 2 σ 1 2 k 2 ,
(17)
where kr and kz are the wave numbers in the r- and z-directions and k 2 = k r 2 + k z 2. P ̂ 1 and Q ̂ 1 are the Fourier transforms of P1 and Q1 in the cylindrical coordinate system, respectively, which can be generally defined as
χ ̂ ( k r , k z ) = 2 π 0 χ ( r , z ) · J 0 ( k r r ) rdr e i k z z d z .
(18)
Using this definition, the counterpart of the known spatial function Q 1 ( r , z ) from (12) in Fourier space Q ̂ 1 ( k r , k z ) can be obtained (details in  Appendix A). Replacing κ 2 with k r 2 + k z 2 and applying the inverse Fourier transform to P ̂, we obtain the solution of P1 in physical space
P 1 ( r , z ) = 1 4 π 2 0 Q ̂ 1 ( k r , k z ) J 0 ( k r r ) k r e i k z z ( 4 σ 1 2 ) k z 2 σ 1 2 k r 2 d k r d k z .
(19)
Depending on whether the frequency of the inertial waves exceeds twice the rotational frequency, two very different modes of responses are distinguished. Specifically, it is not possible to form and propagate inertial waves with frequencies higher than twice the rotating frequency of the fluid (for details see in Liu et al.13). To be able to excite SIWs, we have to restrict their frequency to | σ 1 | < 2. According to the relation (14), the frequency of the annular forcing has to satisfy | σ 0 | < 1. This is also a necessary condition for satisfying the dispersion relation in (15). Theoretically, if we consider higher-order perturbations in (10), e.g., n-order, nonlinear interactions between the waves can generate new waves with frequencies | σ n | = ( n + 1 ) | σ 0 |, where n = 1 , 2 , 3 . The dispersion relation admits wave radiation for n-order harmonic waves as long as | σ 0 | < 2 / ( n + 1 ). This condition can be used in experiments to control the frequency of the waves that one wants to observe. For example, by setting | σ 0 | ( 1 , 2 ), all higher-order inertial waves are evanescent and only the PIWs exist. In this case, the effect due to the focusing of the PIWs emitted by the annular forcing can be studied in isolation.
Applying the residue theorem,19 the integral in (19) with respect to kz can be evaluated, which leads to the result
p 1 ( r , z ) = i α 1 4 π σ 1 2 0 Q ̂ 1 ( k r , k z = α 1 k r ) J 0 ( k r r ) e i | z | α 1 k r d k r
(20)
with
α 1 = σ 1 2 4 σ 1 2 = σ 0 2 1 σ 0 2 .
(21)
Bringing the solution of the PIWs u 0 in (A1) obtained by Liu et al.13 step by step into Eqs. (13), (12), (18) and (19), we can finally obtain the solution of the pressure for the SIWs in physical space (details in  Appendix A)
p 1 ( r , z , t ) = e i σ 1 t Λ 0 2 α 0 2 8 σ 0 2 0 0 0 { 2 α 0 ( I ̃ 200 I ̃ 300 + I ̃ 201 I ̃ 301 ) r ̃ J 0 ( k r r ̃ ) sin ( α 1 k r z ̃ ) + i α 1 [ ( 3 α 0 2 + 1 ) ( I ̃ 201 I ̃ 300 I ̃ 200 I ̃ 301 ) r ̃ + 2 I ̃ 201 I ̃ 201 ] × J 1 ( k r r ̃ ) cos ( α 1 k r z ̃ ) } k r J 0 ( k r r ) e i | z | α 1 k r d r ̃ d z ̃ d k r
(22)
with
α 0 = σ 0 2 4 σ 0 2 .
(23)
To simplify the expression, the abbreviation I ̃ lmn has been introduced by the definition
I ̃ lmn ( r ̃ , z ̃ ) : = 0 k 0 r l J m ( k 0 r ) J n ( r ̃ k 0 r ) e i α 0 | z ̃ | k 0 r d k 0 r .
(24)
The associated velocity components in the radial direction u 1 r, in the azimuthal direction u 1 φ, and in the vertical direction u 1 z can be determined directly by substituting the solution of the pressure (22) into the momentum equation (8a) for E k = 0. This yields the following solutions for the velocity components:
u 1 r ( r , z , t ) = Λ 0 2 α 0 2 α 1 2 16 σ 0 3 e i σ 1 t 0 0 0 { i 2 α 0 ( I ̃ 200 I ̃ 300 + I ̃ 201 I ̃ 301 ) r ̃ J 0 ( k r r ̃ ) sin ( α 1 k r z ̃ ) + α 1 [ ( 3 α 0 2 + 1 ) ( I ̃ 201 I ̃ 300 I ̃ 200 I ̃ 301 ) r ̃ + 2 I ̃ 201 I ̃ 201 ] J 1 ( k r r ̃ ) cos ( α 1 k r z ̃ ) } k r 2 J 1 ( k r r ) e i α 1 k r | z | d r ̃ d z ̃ d k r + i Λ 0 2 α 0 2 α 1 2 16 σ 0 3 e i σ 1 t [ ( 3 α 0 2 + 1 ) I 200 I 301 ( 3 α 0 2 + 2 ) I 201 I 300 I 201 I 302 ] ,
(25)
u 1 φ ( r , z , t ) = Λ 0 2 α 0 2 α 1 2 16 σ 0 4 e i σ 1 t 0 0 0 { 2 α 0 ( I ̃ 200 I ̃ 300 + I ̃ 201 I ̃ 301 ) r ̃ J 0 ( k r r ̃ ) sin ( α 1 k r z ̃ ) i α 1 [ ( 3 α 0 2 + 1 ) ( I ̃ 201 I ̃ 300 I ̃ 200 I ̃ 301 ) r ̃ + 2 I ̃ 201 I ̃ 201 ] J 1 ( k r r ̃ ) cos ( α 1 k r z ̃ ) } k r 2 J 1 ( k r r ) e i α 1 k r | z | d r ̃ d z ̃ d k r + Λ 0 2 α 0 2 α 1 2 16 σ 0 4 e i σ 1 t [ 6 α 0 2 I 200 I 301 ( 6 α 0 2 + 1 ) I 201 I 300 I 201 I 302 ] ,
(26)
u 1 z ( r , z , t ) = sign ( z ) e i σ 1 t Λ 0 2 α 0 2 α 1 16 σ 0 3 0 0 0 { 2 α 0 ( I ̃ 200 I ̃ 300 + I ̃ 201 I ̃ 301 ) r ̃ J 0 ( k r r ̃ ) sin ( α 1 k r z ̃ ) + i α 1 [ ( 3 α 0 2 + 1 ) ( I ̃ 201 I ̃ 300 I ̃ 200 I ̃ 301 ) r ̃ + 2 I ̃ 201 I ̃ 201 ] J 1 ( k r r ̃ ) cos ( α 1 k r z ̃ ) } k r 2 J 0 ( r k r ) e i α 1 k r | z | d r ̃ d z ̃ d k r .
(27)
The abbreviation Ilmn in (25) and (26) follows the same definition as in (24) but with the variables (r, z) instead of ( r ̃ , z ̃ ).

So far, we have obtained the analytical solutions of the velocity field for the PIWs and SIWs. By adding them together, we can approximately describe the total velocity field of weakly nonlinear inertial waves, as expressed in (10). However, in all the above solutions, we have not been able to further calculate the multiple integrals analytically. To proceed further, we have evaluated the integrals numerically and plotted the solutions with the help of MATLAB. Next we will analyze the properties of the SIWs in combination with these plots.

To observe the focusing effect of the SIWs, we choose the forcing frequency σ 0 = 0.8 as an example to show the distribution of the velocity field. According to the analytical solution of the PIWs in (A1) and the SIWs in (25)–(27), the three velocity components are shown in Fig. 2, respectively, where a cutoff wavenumber at k ¯ r = 200 is introduced in the numerical evaluation. For symmetry reasons, we observe the wave beams only in one quadrant of the plane. The first column of Fig. 2 shows the amplitudes of the three velocity components in the vertical plane as a combination of the PIWs and the SIWs with a small Rossby number R o = 10 4. The corresponding frequency-filtered velocity fields are extracted in the second column for the PIWs and in the third column for the SIWs, respectively. Different from the completely linear case, the oscillating annular forcing can excite both the PIWs and SIWs simultaneously, and they propagate along the directions with inclination angles θ0 and θ1 according to the dispersion relation determined by their respective frequencies. This analytical result has been observed in an experiment by Cortet et al.,17 who suggested that such higher-order harmonic waves may originate either from a residual non-harmonic component of the oscillation of the wave generator or from nonlinear inertial effects in the flow in the vicinity of the wave generator. The same phenomenon also exists in internal waves.20 As a result, the SIWs excited directly by the oscillating torus form a new but much weaker focal point on the rotation axis below the primary focal point. The rays of the SIWs form a new double cone with a different angle from that of the PIWs. However, due to the focusing effect, the interaction between the PIWs in the primary focal area is globally the strongest, causing the primary focal point to act as a new point forcing that can excite SIWs with a much stronger effect than the annular forcing. Therefore, the SIWs are primarily distributed near the primary focal point, as shown in Fig. 2. This, in turn, can lead to further wave interactions between PIWs and SIWs, resulting in strong nonlinear effects, instability, and local turbulence. Such effects can only be studied by analyzing the fully nonlinear equation system, which will be the focus of future work.

FIG. 2.

Amplitude of the three velocity components in the vertical plane in (a), (d), (g) for σ0=0.8, which corresponds to the superpositions of the PIWs in (b), (e), (h) and the SIWs in (c), (f), (i), respectively.

FIG. 2.

Amplitude of the three velocity components in the vertical plane in (a), (d), (g) for σ0=0.8, which corresponds to the superpositions of the PIWs in (b), (e), (h) and the SIWs in (c), (f), (i), respectively.

Close modal

Based on the analysis of the viscous effect through the linear theory by Liu et al.,13 we know that in low-viscosity fluids, viscous attenuation has a relatively small impact on the focusing effect of the PIWs. For example, in the case of Ek = 10 5, the asymmetry of the wave packet at the focal point can be almost ignored, and the PIWs can be effectively focused onto a small area. In the present weakly nonlinear case, the nonlinear term ( u 0 · ) u 0, which arises as the forcing term in the first-order momentum equations, is determined by the solution u 0 of the linear theory. To simplify the analysis of the SIWs, we choose small Ek numbers, i.e., E k 10 5, for which the viscous effect on the PIWs is negligible. Thus, in this case, the solutions of the PIWs obtained from the linear theory at the inviscid limit can be further used to approximate the term f 1 in the first-order momentum equations without causing significant differences in the physical sense. This avoids the need for the much more complicated solution of the PIWs, including the viscous effect. Based on this, we perform a qualitative analysis of the viscous effect of the SIWs.

Using the same mathematical treatment as in the inviscid case, we can obtain the following wave-like equation for the pressure p1 in a viscous fluid:
( t E k 2 ) 2 2 p 1 + 4 2 p 1 z 2 = ( t E k 2 ) 2 ( · f 1 ) + 4 n · ( n · ) f 1 + 2 n · ( t E k 2 ) ( × f 1 ) .
(28)
Analogous to the analysis in the previous inviscid case, we can use the Fourier transform in the cylindrical coordinate system to express the particular solution for p1 in Fourier space as
P ̂ 1 ( k r , k z ) = Q ̂ 1 ( k r , k z ) ( σ 1 + iEk k 2 ) 2 k 2 4 k z 2 ,
(29)
where Q ̂ 1 ( k r , k z ) is the Fourier transform of the right-hand side in Eq. (28). Furthermore, by transforming the momentum equation (8a) in the z direction into Fourier space, we obtain the following scalar equation:
( i σ 1 + E k k 2 ) U ̂ 1 z = i k z P ̂ 1 + F ̂ 1 z .
(30)
Substituting (29) into (30), the particular solution of u z 1 in Fourier space is given as
U ̂ 1 z = k z · Q ̂ 1 ( k r , k z ) ( σ 1 + iEk k 2 ) [ ( σ 1 + iEk k 2 ) 2 k 2 4 k z 2 ] + i F ̂ 1 z σ + iEk k 2 .
(31)
Based on the assumption of a small Ekman number, we can reasonably neglect all terms with O ( E k 2 ). By using the residue theorem to evaluate the integral with respect to kz during the inverse Fourier transform, we obtain the following approximate solution for u z 1 in physical space (details in  Appendix B):
u 1 z ( r , z , t ) sign ( z ) e i σ 1 t Λ 0 2 α 0 2 α 1 16 σ 0 3 0 0 0 k r 2 J 0 ( k r r ) { [ i 2 α 1 ( α 1 3 / σ 0 3 ) E k k r 2 ] I ̃ 201 I ̃ 201 J 1 ( k r r ̃ ) cos ( α 1 k r z ̃ ) + [ i α 1 ( 3 α 0 2 + 1 ) ( α 1 3 α 0 2 / σ 0 3 ) E k k r 2 ] ( I ̃ 201 I ̃ 300 I ̃ 200 I ̃ 301 ) r ̃ J 1 ( k r r ̃ ) cos ( α 1 k r z ̃ ) [ 2 α 0 i ( α 0 α 1 4 / σ 0 3 ) E k k r 2 ] ( I ̃ 200 I ̃ 300 + I ̃ 201 I ̃ 301 ) r ̃ · J 0 ( k r r ̃ ) sin ( α 1 k r z ̃ ) } e i | z | α 1 k r e 16 ( α 1 5 / σ 1 5 ) E k k r 3 | z | d r ̃ d z ̃ d k r .
(32)
The structure of this solution is similar to that of the inviscid case presented in Eq. (27), but more complex. To illustrate, we choose the same forcing frequency as in the previous inviscid case, i.e., σ 0 = 0.8, and plot the vertical velocity distribution of the SIWs for three small Ekman numbers E k = 10 7 , 10 6, and 10 5 corresponding to very small values of viscosity, respectively, as shown in Fig. 3. It can be seen that the SIWs excited at the primary focal area respond very sensitively to viscosity. In the range of very small Ek numbers, as presented here, which have negligible effect on the focusing of the PIWs, the SIWs generated at this region are already significantly suppressed. In contrast, as the Ekman number increases, the SIWs emerging in the secondary focal area generated directly by the annular forcing decay relatively slowly, and the shape of the wave packet maintains good symmetry before and after focusing, just as the PIWs do at low Ekman numbers.13 A possible reason for this difference is that the mechanisms of SIW emergence are different at the primary and the secondary focal points. At the secondary focal point, the SIWs are mainly emitted directly by the annular forcing and are focused onto this region due to their propagation relation. During propagation, the focusing effect partially offsets the effect of viscous dissipation so that the kinetic energy of the SIWs in the focused region does not decrease greatly due to viscous dissipation. As the viscous effect increases, the wave beam gradually widens and exhibits asymmetry before and after focusing. Based on the analysis of the properties of the PIWs in viscous fluids presented in Liu et al.,13 we can also reasonably predict that there is an optimal propagation frequency for the SIW such that the corresponding velocity reaches its maximum at the secondary focal point. On the other hand, SIWs can also be generated by the nonlinear interaction of PIWs and appear mainly near the primary focal point. In this process, part of the energy of the PIWs is transferred to the SIWs. This phenomenon is especially evident in the inviscid limit case. When friction effects occur, the interaction between the PIWs is suppressed, reducing their ability to excite the SIW. Additionally, the SIWs continue to lose energy during propagation due to viscous attenuation. As a result, even in fluids with very low viscosity, the SIWs generated at the primary focal area are significantly suppressed and scarcely observed. However, the viscous effect is inevitable in practical fluids, which may be one of the reasons why SIWs were not clearly observed in the experiments of Duran-Matute et al.10 Understanding the impact of viscosity on the SIWs is crucial for accurately predicting the behavior of inertial waves in practical fluid systems. Next, we will take a closer look at the viscous effect on the SIWs from a different viewpoint.
FIG. 3.

Amplitude of the vertical velocity component of the SIWs in the vertical plane for σ 0 = 0.8 in a viscous fluid with the Ekman number (a) E k = 10 7, (b) E k = 10 6 and (c) E k = 10 5, respectively.

FIG. 3.

Amplitude of the vertical velocity component of the SIWs in the vertical plane for σ 0 = 0.8 in a viscous fluid with the Ekman number (a) E k = 10 7, (b) E k = 10 6 and (c) E k = 10 5, respectively.

Close modal
Based on the analysis in Sect. II, we know that the SIWs excited directly by the annular forcing form a new secondary focal point, and at the same time, the SIWs can also be excited by the nonlinear interaction between PIWs mainly near the primary focal point. The distinct responses of SIWs to changes in viscosity at these two focal points can be better understood by conducting a more in-depth analysis of the solution's structure at each focus. The solution for u 0 z obtained from our previous analysis for PIWs in Liu et al.13 and the solution for u 1 z for SIWs in Eq. (32) reveal that, under the assumption of a low Ekman number, the viscous damping of both PIWs and SIWs at each focal point is primarily determined by the exponential function e β k r 3, where β serves as a viscous damping factor. For different types of waves, differences in the viscous damping factors lead to different viscous effects. Specifically, the viscous damping factor of the PIWs at the primary focal point is defined by
β 0 , p = 16 E k σ 0 ( 4 σ 0 2 ) 2 .
(33)
For the SIWs, the viscous damping factor at the primary focal point is given as
β 1 , p = 16 α 1 5 E k α 0 σ 1 5 = ( 4 σ 0 2 ) 1 / 2 E k 2 σ 0 ( 1 σ 0 2 ) 5 / 2 ,
(34)
while the factor at the secondary focal point is expressed as
β 1 , s = 16 α 1 4 E k σ 1 5 = E k 2 σ 0 ( 1 σ 0 2 ) 2 .
(35)
Here, for the damping factor expressed in (33)–(35), the first subscripts “0” and “1” denote the “zero-order” solution (PIWs) and the “first-order” solution (SIWs), while the second subscripts “p” and “s” represent “primary” and “secondary” focal point, respectively. Obviously, the viscous damping factors of PIWs and SIWs are influenced not only by the value of Ek, but also by the forcing frequency σ0.
Due to the differences in the viscous damping factors in Eqs. (33) and (34), an identical value of Ek has different viscous effects on the propagation of the PIWs and SIWs at the same primary focal point. Comparing these two factors gives
β 1 , p β 0 , p = [ 1 + 3 σ 0 2 4 ( 1 σ 0 2 ) ] 5 / 2 > 1 for | σ 0 | ( 0 , 1 ) .
(36)
This result indicates that, at the primary focal point for any given forcing frequency σ0, the influence of the same viscosity on the SIWs is always stronger than that on the PIWs. For the example in Sec. II with σ 0 = 0.8, the corresponding viscous factor ratio yields β 1 , p / β 0 , p 8.3. This leads to the phenomenon that the SIWs generated locally at the primary focal point are significantly more sensitive to viscosity than PIWs. Therefore, under the assumption of a small Ekman value, it is reasonable for us to calculate the solution of SIWs by neglecting the viscous damping effect of PIWs. Furthermore, this different damping effect is also evident in the SIWs, but at different focal points. Comparing the viscous factors of SIWs at the two focal point gives
β 1 , p β 1 , s = ( 1 + 3 1 σ 0 2 ) 1 / 2 > 1 for | σ 0 | ( 0 , 1 ) .
(37)
In the previous example with σ 0 = 0.8, we obtain β 1 , p / β 1 , s 3. This result further indicates that, for any given forcing frequency σ0, the SIWs generated through the local interaction of PIWs at the primary focal point will be more strongly suppressed by the viscous effect than the SIWs excited by the annular forcing and focused to the secondary focal point, even when the viscous damping effect of PIWs has been neglected. Additionally, as the forcing frequency σ0 increases, the viscous attenuation of SIWs is significantly enhanced. At the limit σ 0 1, which corresponds to the maximum amplification of the focusing effect of PIWs, both β 1 , p and β 1 , s tend to infinity, corresponding to a complete damping of SIWs in the entire field.

We have previously conducted a theoretical analysis of the propagation properties and viscous effects of the inertial waves excited by an oscillating annular forcing based on the linearized Navier–Stokes equations.13 The aim of the present investigation is to further analyze the properties of the weakly nonlinear inertial waves based on the results of the linear theory. For this purpose, we choose the perturbation method as an analytical approach. By using this method, the original nonlinear differential equation can be decomposed into a series of linear differential equations for a small Rossby number corresponding to different orders of approximation, and the forcing-like terms in the higher order equations are determined directly from the solutions of the lower order equations. This enables us to use the same methods as in the linear theory to deal with this weakly nonlinear problem.

Our results show that due to weakly nonlinearity, the annular forcing can excite not only the PIWs with the same frequency as itself, but also the SIWs of twice the frequency, which can be understood as a result of triadic resonance. The condition for the existence of high-frequency inertial waves is that their frequency must be less than twice the rotation rate of the fluid, which is in a good agreement with the experimental result by Cortet et al.17 Similar to the PIWs, the SIWs follow their dispersion relation determined only by their own frequency and their rays form a new double cone symmetric about the plane on which the annular forcing is located. At the vertex of the cone, the SIWs are focused in a shock-like manner at the new focal point below the primary one on the rotation axis, but its focusing effect is much weaker than that of the PIWs. At the same time, due to the nonlinear interaction of the PIWs, the primary focal point acts like a new point forcing that excites much stronger SIWs in an inviscid limit. However, this phenomenon is significantly suppressed in viscous fluids even with low-viscosity, making it difficult to observe experimentally. Especially when the frequency of the annular forcing is close to the rotation rate of the fluid, the PIWs reach their maximal focusing effect, while the SIWs cannot be generated at all.

Based on the results obtained in this work, the interaction between the PIWs and SIWs at the primary focal zone in an inviscid limit can become no longer negligible. This will lead to strong nonlinear effects in this region and thus to instability and local turbulence. To understand the complex process in the fully nonlinear region around the primary focal point, it is necessary to analyze the corresponding stability problem, which is the next goal of our investigation.

This work was funded by the German Research Foundation (DFG) with Project No. 407316090. The authors acknowledge A. Delache and F. S. Godeferd for fruitful discussions.

The authors have no conflicts to disclose.

Jie Liu: Formal analysis (lead); Investigation (equal); Methodology (equal); Visualization (lead); Writing – original draft (lead). Yongqi Wang: Conceptualization (equal); Formal analysis (supporting); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (lead). Martin Oberlack: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (lead); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

First, we recall the solution of the PIWs in an inviscid fluid

(A1)
u 0 r ( r ̃ , z ̃ , t ) = sign ( z ̃ ) i Λ 0 α 0 2 2 σ 0 e i σ 0 t 0 k r 2 J 0 ( k r ) J 1 ( r ̃ k r ) e i α 0 | z ̃ | k r d k r ,
(A1a)
u 0 φ ( r ̃ , z ̃ , t ) = sign ( z ̃ ) Λ 0 α 0 2 σ 0 2 e i σ 0 t 0 k r 2 J 0 ( k r ) J 1 ( r ̃ k r ) e i α 0 | z ̃ | k r d k r ,
(A1b)
u 0 z ( r ̃ , z ̃ , t ) = Λ 0 α 0 2 σ 0 e i σ 0 t 0 k r 2 J 0 ( k r ) J 0 ( r ̃ k r ) e i α 0 | z ̃ | k r d k r ,
(A1c)
where α 0 = σ 0 2 4 σ 0 2.

Based on the solution of u 0, the forcing term can be expressed as
f 1 ( r ̃ , z ̃ , t ) = ( u 0 · ) u 0 = ( Λ 0 2 α 0 4 4 σ 0 2 ( I ̃ 201 I ̃ 300 I ̃ 200 I ̃ 301 + 1 α 0 2 r ̃ I ̃ 201 I ̃ 201 ) i Λ 0 2 α 0 4 2 σ 0 3 ( I ̃ 200 I ̃ 301 I ̃ 201 I ̃ 300 ) sign ( z ) i Λ 0 2 α 0 3 4 σ 0 2 ( I ̃ 200 I ̃ 300 + I ̃ 201 I ̃ 301 ) ) e i 2 σ 0 t = ( F 1 r ( r ̃ , z ̃ ) F 1 φ ( r ̃ , z ̃ ) F 1 z ( r ̃ , z ̃ ) ) e i σ 1 t ,
(A2)
where I ̃ is defined in (24).
Substituting (A2) into (12) and applying the Fourier transform defined in (18), we obtain the Fourier transform of Q1
Q ̂ 1 ( k r , k z ) = 4 π σ 1 2 k r 0 0 F 1 r ( r ̃ , z ̃ ) · J 1 ( k r r ̃ ) r ̃ d r ̃ cos ( k z z ̃ ) d z ̃ i 8 π σ 1 k r 0 0 F 1 φ ( r ̃ , z ̃ ) · J 1 ( k r r ̃ ) r ̃ d r ̃ cos ( k z z ̃ ) d z ̃ + 4 π ( 4 σ 1 2 ) k z 0 0 F 1 z ( r ̃ , z ̃ ) · J 0 ( k r r ̃ ) r ̃ d r ̃ sin ( k z z ̃ ) d z ̃ .
(A3)
Finally, bringing Q ̂ 1 ( k r , k z ) at k z = α 1 k r back to (20), we can get the solution of p1 in physical space as shown in (22).
In the viscous case, the Fourier transform of the right-hand side of the wave-like equation (28) is given as
Q ̂ 1 ( k r , k z ) = 2 π ( i σ 1 E k κ 2 ) 2 k r 0 F 1 r ( r ̃ , z ̃ ) · J 1 ( k r r ̃ ) r ̃ d r ̃ e i k z z ̃ d z ̃ 4 π ( i σ 1 E k κ 2 ) k r 0 F 1 φ ( r ̃ , z ̃ ) · J 1 ( k r r ̃ ) r ̃ d r ̃ e i k z z ̃ d z ̃ + i 2 π [ 4 + ( i σ 1 E k κ 2 ) 2 ] k z 0 F 1 z ( r ̃ , z ̃ ) · J 0 ( k r r ̃ ) r ̃ d r ̃ e i k z z ̃ d z ̃ ,
(B1)
where F r 1 , F 1 φ, and F 1 z are the same as given in (A2).
Substituting (B1) into (31) and neglecting all terms with O ( E k 2 ), we obtain
U ̂ 1 z ( k r , k z ) = 4 π k r G ̂ ( k r , k z ) ( 4 σ 1 2 ) k z 2 σ 1 2 k r 2 i 2 σ 1 E k κ 4
(B2)
with
G ̂ ( k r , k z ) = ( σ 1 + iEk κ 2 ) k z 0 0 F 1 r ( r ̃ , z ̃ ) · J 1 ( k r r ̃ ) r ̃ cos ( k z z ̃ ) d r ̃ d z ̃ + i 2 k z 0 0 F 1 φ ( r ̃ , z ̃ ) · J 1 ( k r r ̃ ) r ̃ cos ( k z z ̃ ) d r ̃ d z ̃ ( σ 1 + iEk κ 2 ) k r 0 0 F 1 z ( r ̃ , z ̃ ) · J 0 ( k r r ̃ ) r ̃ sin ( k z z ̃ ) d r ̃ d z ̃ .
(B3)
Applying the inverse Fourier transform on U ̂ 1 z, we can express the vertical velocity in physical space as
U 1 z ( r , z ) = 1 4 π 2 0 4 π k r G ̂ ( k r , k z ) · J 0 ( k r r ) k r e i k z z ( 4 σ 1 2 ) k z 2 σ 1 2 k r 2 i 2 σ 1 E k k 4 d k r d k z .
(B4)
By additionally considering the dispersion relation for the secondary waves, i.e., σ 1 2 k 2 4 k z 2 = 0, k2 contained in all viscous terms can be replaced with 4 k r 2 4 σ 1 2. In the complex kz-plane, the denominator of the integrand has now only two singularities, which are
k ¯ z = ± [ α 1 k r + i 16 ( α 1 / σ 1 ) 5 E k k r 3 ] .
(B5)
Applying the residue theorem (see Chap. 11.7 in Arfken19) to the integral in (B4) with respect to kz, we can further approximately express U 1 z as
U 1 z ( r , z ) sign ( z ) i σ 1 2 0 ( α 1 i β k r 2 ) k r J 0 ( k r r ) × G ̂ ( k r , k z = α 1 k r + i 16 ( α 1 / σ 1 ) 5 E k k r 3 ) × e i | z | α 1 k r e | z | 16 α 1 5 / σ 1 5 E k k r 3 d k r .
(B6)
Substituting the expression of G ̂ ( k r , k z ) at k z = α 1 k r + i 16 ( α 1 / σ 1 ) 5 E k k r 3 into the above expression, we can finally obtain the solution of u 1 z as shown in Eq. (32).
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