Near-inertial waves (NIWs) with frequency in the vicinity of the inertial frequency [Ferrari and Wunsch (2009); Alford et al. (2016)] are the energy-dominant high-frequency fluctuations in ocean waves with spatial scales up to 1000km. They lead to strong mixing in the upper ocean by inducing large vertical shear, and therefore contribute to the large-scale exchange of materials and energy [Alford (2001); Rimac et al. (2013)] and influence biological activities and climate processes in relevant regions [Granata et al. (1995); Jochum et al. (2013)]. NIWs are also believed to play an essential role in the energy transfer of mesoscale eddies and resolve the energy puzzle [Xie and Vanneste (2015); Rocha et al. (2018); Xie (2020)].

During propagation, NIWs change scales due to the influence of the large-scale planetary vorticities (the β-effect) and the mesoscale vorticities [van Meurs (1998)]. Evidence from ocean storm experiments suggests that the β-effect provides a global impact on the evolution of NIWs [D’Asaro et al. (1995)], resulting in a significant propagation perpendicular to the meridian. The local behavior of NIWs is more determined by the impact of relative vorticities [Weller (1982)], undergoing a scale decrease when encountering the background flows. An interesting phenomenon is that NIWs concentrate in anticyclones, which is justified by both numerical simulations [Lee and Niiler (1998); Zhai et al. (2005); Danioux et al. (2008)] and observations [Kunze and Sanford (1984); D’Asaro et al. (1995); Elipot et al. (2010); Joyce et al. (2013)].

Early studies on this phenomenon identified two regimes: the “trapping” regime and the “strong dispersion” regime [Kunze (1985); Wang (1991); Klein and Tréguier (1995); van Meurs (1998)], determined by the relative order of magnitude of the refraction and dispersion effects. In the “trapping” regime where the refraction dominants, using the Wentzel-Kramers-Brillouin (WKB) method, Kunze (1985) derived that the NIWs tend to move away from positive vorticities and towards negative ones. Here, the NIW frequency is modified by the background vorticity with a ζ/2 shift where ζ is the relative vorticity, which is the so-called Kunze’s effect. On the other hand, in the “strong dispersion” regime, NIWs are rapidly dispersed and less affected by the vorticity [Klein and Tréguier (1995)].

Subsequently, many new insights were proposed benefiting from the NIW model proposed by Young and Jelloul (1997), hereafter YBJ, which captures the effects of wave dispersion, vortical flow’s advection and refraction. A crucial advantage of the YBJ model is that it does not rely on the assumption of horizontal scale separation between waves and background flows which is required by the WKB method. However, this scale separation is usually not valid for NIWs. Balmforth et al. (1998) explored the time scale and spatial modulation of decaying inertial oscillations influenced by the geostrophic flow. The demarcation line between the “trapping” regime and the “strong dispersion” regime in the YBJ model is determined by Ψ / f 0 R n 2 , where Ψ is the magnitude of the background streamfunction, f ₀ is the inertial frequency, R_n is the deformation radius of the nth vertical mode [Young and Jelloul (1997); Balmforth et al. (1998)]. As to a reduced-gravity shallow-water system, this parameter is reduced to Ψ/h where h = g’H/f₀ with g’ and H the reduced gravity and horizontally averaged depth of the top layer. When Ψ/h >>1, the “trapping” dominants; in the opposite case, dispersion dominates. With a large-scale initial condition where the advection can be ignored compared with the refraction term, Klein and Smith (2001) investigated the spatial structure of inertial energy and suggested that the large-scale components contribute to the trapping regime in anticyclones. Introducing an extra short-time assumption, the temporal evolution of NIW energy is found to be proportional to the Laplacian of the vorticity field, i.e. Δζ [Klein et al. (2004)]. So the inertial energy is concentrated in the structure where Δζ is positive. Danioux et al. (2015) argued that the conservations in the YBJ equation lead to the concentration of NIWs in the anticyclone. With homogeneous initial conditions, they observed the long-time saturation scale of waves: in the “trapping” regime, the wave scale is much smaller than the vorticity scale, while in the “strong dispersion” regime, the wave scale is much larger than the vorticity scale. Nevertheless, this does not mean smaller-scale NIWs are easier concentrate in anticyclones for a given background flow. In this paper, we will show that for a fixed vorticity field, the larger the scale of the waves, the more favorable the concentration. However, then the concentration is suppressed by the increasing number of newly generated small-scale waves, eventually reaching saturation with an average scale shown by Danioux et al. (2015).

In this paper, we systematically study the scale dependence of NIW’s concentration in anticyclones and interpret the reason behind the concentration from a perspective of uncertainty principle borrowed from quantum mechanics. The paper is structured as follows. In section 3, we discuss the dynamics and scaling characteristics of the YBJ equation. In section 4, we provide exact and approximate solutions for a sinusoidal background shear flow and indicate the scale effect of NIWs concentration in anticyclones. In sections 5-6, numerical simulations are performed to confirm the scale effect with vortex patches and random vortexes. Section 7 shows that the combined effect of uncertainty relation and energy conservation leads to the NIW’s concentration in anticyclones. It is a new understanding of the concentration mechanism drawing on the basic concepts of quantum mechanics. Finally, we summarize and discuss our results in section 7.

We study the evolution of NIWs in a background vorticity field by the shallow-water YBJ model (Young and Jelloul, 1997; Danioux et al., 2015):

where M(x, y, t) is a complex amplitude of the horizontal velocity (u, v), u+iv=Me ^{−if ₀ t} , describing the slow spatial and long-time modulation of the NIW field. f₀ is the local Coriolis frequency and h=g ^′ H/f ₀ is a dispersion parameter with g ^′ and H the reduced gravity and horizontally averaged depth. ψ and Δψ are the barotropic geostrophic flow’s streamfunction and vorticity field. For simplicity, we only focus on the barotropic case. The operator J is the horizontal Jacobian. In this paper, we are concerned with the long-time [more than 30 days, e.g. Klein and Smith (2001); Danioux et al. (2015)] evolutionary nature of NIWs and assume that the background flow is steady.

The YBJ equation captures the advection, dispersion and refraction effects. The refraction term controls the capture of NIWs by the vorticity field, while the dispersion term promotes the escape of waves [Kunze (1985); Rocha et al. (2018)]. The relative strength of the dispersion and refraction can be measured by the dimensionless parameter h/Ψ (Young and Jelloul, 1997; Balmforth et al., 1998). Typical observation data from the North Atlantic imply that h/Ψ may range in (0.2, 8) (Danioux et al., 2015).

When the advection term is omitted, the YBJ equation is similar to the Schrödinger equation describing the motion of a single particle. In this analogy, M(x,y) corresponds to the particle’s complex wavefunction, h corresponds to the reduced Planck constant ℏ , and hΔψ/2 corresponds to the potential field subjected by the particle. The particle prefers the lower potential region; accordingly, NIWs concentrate in negative relative vorticities. This concentration in anticyclone should still be valid when the advection term is non-zero but much smaller than the refraction. The ratio between the advection and refraction is only related to the spatial scales. So we can define L _ψ /L _M , where L _ψ and L _M are the spatial scale of the background streamfunction and wave amplitude, respectively, to capture this relative importance. However, the interpretation of the energy concentration via analogy to the Schrödinger equation fails when L _ψ /L _M ≫1 . In this paper, we will show that the advection prevents the NIW’s concentration, which is weak for large-scale waves (small L _ψ /L _M ).

By introducing the amplitude and phase of M , M=M ₀ e ^iΘ where M ₀ and Θ are both real numbers, we define the local wavenumber

or equivalently

which we practically use in analyzing our numerical results.

We further define an averaged local wave-vector k _ave

with corresponding NIW’s mean spatial scale L _M =2π/k _ave .

To reveal the scale dependence of NIW’s concentration, we first study a simple case with a sinusoidal background shear flow, which can be solved analytically. Setting the core of negative vorticity as y=0 , the stream function of the background shear flow reads ψ = ( ζ 0 / k 0 2 ) cos   k 0 y , where ζ ₀>0 is the intensity of local relative vorticity, and the amplitude Ψ of ψ is defined as its root-mean-square that Ψ = ζ 0 / 2 k 0 2 . Because of the translational symmetry in the x -direction, we seek solutions with an ansatz that

Substituting it into the YBJ (1) [Young and Jelloul (1997)] we obtain

Defining A = ( ζ 0 k x / k 0 ) 2 + ( ζ 0 / 2 ) 2 and φ=arctan 2k _x /k ₀ , we obtain

which is the typical Mathieu equation, and the solutions are Mathieu functions of the first kind ([3]):

where ω ′ = 8 ( ω − h k x 2 / 2 ) / h k 0 2 , ξ = − 4 A / h k 0 2 , y ^′=(k ₀ y+φ)/2 . M _C and M _S are even and odd functions of y ^′ , respectively. C ₁ and C ₂ are arbitrary constants. The period of background shear flow is 2π/k ₀ , then the period of y ^′ in Mathieu functions is π , which determines the value of the eigenvalues ω ^′ . For the even functions M _C (ω ^′,ξ,y ^′) , the eigenvalues ω ^′ satisfy the relation in continued fractions that

For the odd functions M _S (ω ^′,ξ,y ^′) , the eigenvalues ω ^′ satisfy

Interestingly, the centers of the waves, for both the eigenfunctions M _C (ω ^′,ξ,y ^′) and M _S (ω ^′,ξ,y ^′) , are

which is only scale-dependent and independent of the kinetic parameter h/Ψ . For the even functions M _C , y _c locates at the wave peaks, while for the odd functions M _S , y _c is the position of the nodes. Distributions of the first few modes of M _C and M _S are plotted in Figures 1 – 4 . NIWs concentrate in the negative vorticity when L _ψ /L _M is small where L _ψ =2π/k ₀ and L _M =2π/k _x . With the increase of L _ψ /L _M , the center of the waves gradually deviates from the core of the negative vorticity. For a large enough L _ψ /L _M , the waves tend to be localized at the boundary (y/L _ψ =−sgn(k _x )/4 ) between positive and negative vorticities.

If the original YBJ equation has no advection term, the sine term in Eq.(6) (or φ in Eq.(7)) would be zero, which would result in y _c =0 . Therefore, the scale effect on the deviation from negative vorticity results from advection. The kinetic parameter h/Ψ controls the variance of the wave packet, with smaller h/Ψ≪1 corresponding to more compact wave packets; as h/Ψ increases, the wave packets widen. The oscillatory behavior of the waves grows when the order number of the eigenmodes increases, which would be seen visually by the approximate analytical solutions in the following subsection.

With the help of the revelation from the above analytical solutions, we study the scale effect of NIWs further in a doubly sinusoidal vortex quadrupole whose streamfunction reads

where ζ ₀>0 . Thus, the spatial scale is L _ψ =2π/k ₀ .

For a given background field, we can obtain the eigenmodes of the system by a finite difference method, as shown in Figure 7 . As can be seen from the figure, low-order modes concentrate near the core of anticyclones ( Figures 7A, B ), while high-order ones can concentrate near the boundaries or the saddle points ( Figures 7C, D ). For each eigenmode, we define the mean radius

r _ave of NIWs concentrated in anticyclones as

in which the core of the anticyclone (x ₀,y ₀) locates at (L _ψ /4,−L _ψ /4). We use the dimensionless ratio r _ave /L _ψ to measure the concentration of NIWs, and it depends on the parameter h/Ψ and the order number of the eigenmodes. In Figure 8A we plot r _ave /L _ψ of the eigenmodes with low frequency for different h/Ψ, and the corresponding spatial scale L _ψ /L _M is calculated according to Eq.(4). There is a clear positive correlation between r _ave /L _ψ and L _ψ /L _M , consistent with the trend of y _c and y _ave given by the analytical solutions shown in Figures 6 (A, B) . Hence, a larger L _M is associated with a more concentrated NIW in anticyclone for a fixed background vortex quadrupole. It should be noted in Figure 8A that for each h/Ψ, a minimum value of L _ψ /L _M appears, which is inversely correlated with h/Ψ and determined by the most concentrated mode of the system.

Using Eq.(24), we can define the degree of energy concentration σ . Figure 8B also shows that a large wave scale facilitates the concentration. Moreover, a color map of σ as a function of h/Ψ and L _ψ /L _M is shown in Figure 9 . With small h/Ψ and L _ψ /L _M , there is a “trapping” regime with a deep negative σ . When h/Ψ or L _ψ /L _M is large enough, the concentration of waves in anticyclones becomes insignificant.

With the initial state of the velocity field set as

where n is an adjustable parameter, we investigate the long-time (more than 30 days) behavior of NIWs. We plot the time average across the second half of the simulation (about 15-30 days) of r _ave /L _ψ and σ as a function of L _ψ /L _M in Figure 10 . A similar scale effect emerges that the concentration of NIW favors larger wavelength L _M defined from Eq.(4). Typically, a larger initial wavelength gives a larger L _M , resulting in a greater concentration. When L _ψ /L _M is large enough, the NIWs are no longer concentrated in anticyclones but tend to the boundary, which leads to a saturation of r _ave /L _ψ and σ, which resembles the results shown by the analytical solutions presented in Figure 6 . When h/Ψ increases, the dispersion is enhanced, weakening the concentration of the waves in anticyclones. Note that r _ave /L _ψ =0.25 is the vorticity boundary. Considering the dispersed distribution of M, r _ave /L _ψ can hardly reach the minimum value of 0 or the maximum value of 0.25.

To be more realistic, we explore the scale effect of NIWs in the random vortexes, with Gaussian covariance (cf. Danioux et al., 2015). The amplitude scale Ψ of the stream function ψ is defined as its root-mean-square. The numerical simulation of the YBJ equation is carried out on a doubly periodic 256×256 grid of a domain size 4π×4π using the pseudo-spectrum method. In order to ensure numerical stability, a weak biharmonic dissipation ν=10⁻¹⁰ is introduced. Figures 11 (A, B) shows the streamfunction and the associated vorticity field.

Because it is not easy to directly get the spatial scale L _Ψ of the random streamfunction, we define the local wavenumber of ψ as

So an averaged local wavenumber k ψ ¯ is then given as

which corresponds to the spatial scale L Ψ = 2 π / k ψ ¯ .

With an initial wave field described by Eq.(28) and illustrated in Figure 11C , the long-time (more than 30 days) behaviors of NIWs are plotted in Figures 11 (D–F) for different h/Ψ . For a larger h/Ψ , the long-time evolution yields a larger saturation scale L _M on average, which is consistent with the result in Danioux et al. (2015) for a homogeneous initial condition M(x,y,t=0)=C where C is a non-zero constant. When h/Ψ is fixed, i.e. for a steady background field, one can find the same positive correlation between the energy concentration σ and the scale factor L _ψ /L _M defined from Eq.(4), as shown in Figure 12 . Danioux et al. (2015) argues that NIWs are most concentrated in anticyclones when h/Ψ∼1. However, we point out that this depends on the wave scale in the initial condition and, therefore, on the saturation scale of the wave under long-time evolution. When h/Ψ=5 which belongs to the “strong dispersion” regime, the geostrophic flow has little effect on NIWs, and the concentration of NIWs in anticyclones is reduced. Correspondingly, the scale effect of concentration becomes less obvious. When h/Ψ=0.2 , the system enters the “strong advection” regime with a large L _ψ /L _M ≈10 and a weak concentration σ∈(−0.1,0) , which is not presented in the figure.

By analogy with the Schrödinger equation, the YBJ equation can be rewritten as

where the Hamiltonian-like operator H ^ reads

with

When H 3 ^ = 0 , one obtains the Schrödinger equation that governs the complex wave function M(x,y,t) for a single particle with unit mass and external potential hΔψ/2 . Similar to the conservation of energy for particles, i.e. ∂ t ∫ ∫ M * H ^ M d x d y = 0 , one can prove that the equation has the following conservation law using properties of the Jacobian and integrating by parts [Danioux et al. (2015)]:

Where

Here, I ₁ is non-negative, I ₂ is the covariance between Δψ and |M|² reflecting the concentration of NIW energy, and I ₃ is an effect of the advection.

In analogy to quantum mechanics, we define the position and momentum operators as

Analogously to the uncertainty relation of matter waves, we obtain

where A ^ denotes the weighted average following ∫ ∫ M * A ^ M d x d y , and Δ r ^ and Δ p ^ are defined as

The weighted averages 〈 ( Δ r ^ ) 2 〉 and 〈 ( Δ p ^ ) 2 〉 are used to measure the uncertainty in position and momentum, acting like the variance of a data set. If 〈 ( Δ r ^ ) 2 〉 ( 〈 ( Δ p ^ ) 2 〉 ) gets smaller, we learn that the waves become more concentrated in position (momentum) space. The uncertainty relation (37) tells us that the waves cannot be overly concentrated simultaneously in both position and momentum spaces, which is a natural consequence of the Fourier transform.

When the background field traps the waves, i.e. , small 〈 p ^ 〉 guaranteeing that the waves do not tend to escape. Then the uncertainty in momentum becomes

Supposing the NIWs are initially uniformly distributed, i.e. I ₂=0 . Gradually, the waves become concentrated, corresponding to a decrease in the uncertainty in position 〈 ( Δ r ^ ) 2 〉 , and then the uncertainty in momentum 〈 ( Δ p ^ ) 2 〉 increases according to Eq.(37). Thus, I ₁ increases according to Eq.(39). When L _ψ /L _M ≫1 , one gets I ₃≫I ₂ according to a scaling analysis of Eq.(35). This corresponds to the “strong advection” regime where the NIW’s concentration is insignificant. On the contrary, when L _ψ /L _M ≪1 , we have I ₂≫I ₃ , and the conservation law implies that an increase in I ₁ leads to negative I ₂ . Thus, when large-scale NIWs are concentrated, the uncertainty relation guides them to concentration in anticyclones. This scale-dependent of NIW’s concentration is inconsistent with our analytical and numerical results presented in Section 4–6.

Based on the YBJ equation, we analyze the scale effect of NIW’s concentration by both analytical derivations and numerical simulations. We start from the exact and approximate solutions for a sinusoidal background shear flow and indicate that a larger wave scale facilitates the concentration. The particular forms of approximate solutions, consisting of envelopes and order-dependent oscillations, give us intuitions about the wave shapes and approximate frequency expressions. Numerical simulations with background vortex quadrupoles and random vortexes confirm the large scale’s preference in enhancing the NIW’s concentration.

Based on the two dimensionless parameters, h/Ψ and L _ψ /L _M , in the YBJ equation, we classify three dynamic regimes: a strong “dispersion” regime with h/Ψ≫1 , a “trapping” regime with small h/Ψ and L _ψ /L _M , and an “advection” regime with a small h/Ψ and a large L _ψ /L _M . Figure 13 illustrates this classification. It is worth noting that for each h/Ψ , there exists a minimum value for L _ψ /L _M , which is determined by the scale of the most concentrated eigenmode of the system. Moreover, the smaller h/Ψ is, the larger the minimum will be. Unlike in Danioux et al. (2015), where with a homogeneous initial state, they attribute the energy concentration to the effect of only one parameter h/Ψ , we consider variable initial conditions and obtain a phase diagram about h/Ψ and L _ψ /L _M , which leads to a classification of “advection” regime.

The scale effect works mainly in the “trapping” regime. When NIWs concentrate in negative vorticities, their centers do not coincide precisely with the core of the vorticities, leaving a displacement originating from the advection. For strong “trapping”, this displacement is proportional to the local wavenumber; However, when the advection effect becomes stronger, waves approach the boundaries between positive and negative vorticities, and the displacement is inversely proportional to the local wavenumber. Thus, the advection prevents the concentration of NIWs, and NIWs with large local wavenumbers (small scales) are more likely to appear at the boundaries. As small-scale structures continue to increase, the system enters a “strong advection” regime. In contrast to the two regimes mentioned above, in the “strong dispersion” regime, NIWs quickly disperse and are slightly influenced by the background vorticity. Therefore, the concentration of the NIWs is very weak in the “strong advection” and “strong dispersion” regimes (Llewellyn Smith, 1999).

Based on the similarity between the YBJ equation to the Schrödinger equation (Balmforth et al., 1998; Danioux et al., 2015), we present a new perspective for the NIW’s concentration in the anticyclone using the uncertainty principle in quantum mechanics. Ignoring the advection term, these two equations are identical, so considering the higher probability of the particle being in the lower potential region, the NIWs prefer to concentrate in negative relative vorticities. Considering the advection’s effect, which hinders NIW’s concentration, this concentration trend could still be true if the change in the advection-related conservation, I ₃ in Eq.35, is small enough. Based on the uncertainty relation, wave concentration means a decrease in the uncertainty of the wave’s position, which leads to an increment in the uncertainty of its momentum. This will enhance the “particle” kinetic-like energy term, defined as I ₁ in Eq.35. Then, due to the conservation of energy, it could reduce the vorticity-related energy term if |ΔI ₃|<ΔI ₁ , leading the concentration towards negative vorticities. Thus, a link between the down-scale waves in space and the distribution of energy in anticyclones is naturally established.

We only consider some modes with low frequencies when studying eigenmodes in the sinusoidal shear flow and vortex quadrupole. This is reasonable because they contribute the most to the mode projection of a realistic initial condition (Balmforth et al. (1998)). For the low-frequency solutions, corresponding to small wavenumbers, the center of symmetry of the solutions can be regarded as the center of the energy distribution. While, for the modes with high frequencies, the Riemann-Lebesgue Lemma implies that the strong spatial oscillation induces only weak concentration if there is any. In addition, too high a frequency is already far from the near-inertial regime, which may make the YBJ equation invalid.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

FZ conducted the analytical derivations and numerical simulations, and wrote the first draft of the manuscript. J-HX conceived the idea and revised the manuscript. All authors contributed to the article and approved the submitted version.