2.1 Non-dimensionalization and scaling assumptions

We set the asymptotic reduction in motion by non-dimensionalizing the Boussinesq equations (2.2)–(2.6). We choose a spatial scaling that isolates NIWs at leading order and a velocity scaling that ensures that the back-rotated velocity and the QG potential vorticity share the same evolutionary time scale. Specifically, this requires that NIW dispersion acts on the same time scale as advection and refraction by the balanced flow. We use a single horizontal length scale, $L$ , and denote the scale of the near-inertial horizontal velocity with $\tilde{U}$ . The NIW ‘amplitude parameter’

is crucial: $\unicode[STIX]{x1D716}\ll 1$ implies that nonlinear terms are small so that the NIW field is governed by linear dynamics to leading order.

The amplitude and importance of nonlinearity to balanced-flow evolution are measured by the Rossby number. We assume that the balanced flow is weak relative to the NIWs and that $\bar{U}=\unicode[STIX]{x1D716}\tilde{U}$ , where $\bar{U}$ is the characteristic velocity of the balanced flow. Under this scaling assumption the Rossby number is

The NIW amplitude parameter and Rossby number have superficial similarity but different physical interpretations. The NIW amplitude can be interpreted as the ratio between the length scale $L$ and the radius of particle orbits in an inertial circle, $\tilde{U} /f_{0}$ . The Rossby number, on the other hand, is the ratio of the rotation time scale $1/f_{0}$ and advective time scale $L/\bar{U}=(\unicode[STIX]{x1D716}^{2}f_{0})^{-1}$ . The NIW envelope and the balanced flow co-evolve on the slow time scale $(\unicode[STIX]{x1D716}^{2}f_{0})^{-1}$ .

We denote the vertical scale of the NIWs by $\tilde{H}$ , and we use $\tilde{U}$ , $L$ and $\tilde{H}$ to non-dimensionalize the horizontal and vertical velocities,

where non-dimensional variables are distinguished by $\check{\,}$   . Introducing $f_{0}$ , the local Coriolis or inertial frequency, and $N_{0}$ , the characteristic magnitude of $N(z)$ , we non-dimensionalize the buoyancy field with

We adopt a geostrophic scaling for the pressure such that

The inertial frequency is scaled so that

where $(x,y)=L(\check{x},\check{y})$ and

Finally, we define an aspect ratio

By assuming $\unicode[STIX]{x1D6FC}=O(1)$ , we imply that $f_{0}/N_{0}=O(\unicode[STIX]{x1D716})$ and justify the hydrostatic approximation in the vertical momentum equation at all relevant orders in the perturbation theory.

2.2 Multiple scales: time and space

To describe both internal waves and slowly evolving balanced flow, we use the two-time method with a ‘fast’ time $\,\tilde{t}\,=f_{0}t$ and a ‘slow’ time $\bar{t}=\unicode[STIX]{x1D716}^{2}f_{0}t$ . Time derivatives are mapped according to

We use an Eulerian time average denoted with an overbar and defined as

to separate fast and slow flow components. Thus, any field $\unicode[STIX]{x1D719}$ can be represented as

where $\bar{\unicode[STIX]{x1D719}}$ is the slowly evolving time-mean part and $\tilde{\unicode[STIX]{x1D719}}$ is the wavy part with $\overline{\tilde{\unicode[STIX]{x1D719}}}=0$ .

A multiple-vertical-scale expansion in the vertical is motivated by the disparity in aspect ratio between NIWs, and both observed $2f_{0}$ scales as well as standard QG flow. Denoting the vertical scale of the NIWs by $\tilde{H}$ , the internal wave dispersion relation implies that internal waves are near-inertial when the Burger number of the wave is small, or when

On the other hand, the standard QG equations assume that the Burger number of the balanced flow is of order unity. We use this requirement to define the vertical scale of the balanced flow, $\bar{H}$ , as

We make the scaling assumption that

This prescription for $\tilde{H}$ unifies the slow NIW dispersion time scale with the balanced-flow advection time scale. To capture both vertical scales in the expansion we split the vertical coordinate into a fast component, $\tilde{z}$ , and a slow component, $\bar{z}$ . Under this two-scale splitting, vertical derivatives become

The vertical-scale splitting requires the introduction of a vertical average, which we define as

The increase in complexity incurred by the multiple space-scale expansion is justified by a systematic explanation of the prominence and impact of the $2f_{0}$ harmonic on NIW evolution.

2.3 Complexifed non-dimensionalized equations

The derivation is greatly simplified by defining the complex horizontal coordinate and velocity field,

Spatial derivatives are expressed in terms of $s$ and $s^{\ast }$ via

These imply that $\unicode[STIX]{x0394}=4\unicode[STIX]{x2202}_{s}\unicode[STIX]{x2202}_{s^{\ast }}$ , and that

Using the scaling assumptions outlined above, and dropping decorations on non-dimensional variables, the complexified non-dimensional Boussinesq equations become

The bracketed terms in (2.27) are included for completeness, but never appear in the theory that follows. In terms of complex velocity the advection operators in (2.26) and (2.28) are

The system in (2.26)–(2.29) is the basis for our asymptotic derivation.

3.1 Leading order: NIWs

The leading-order terms in (2.26)–(2.29) are

We write the solution to the horizontal momentum equation (3.2) in terms of an NIW envelope $M$ or $A$ ,

where $\tilde{\text{L}}$ is a second-order differential operator,

Both $A(x,y,\tilde{z},\bar{z},\bar{t})$ and $M=(f_{0}^{2}/N^{2})A_{\tilde{z}}$ prove useful for confronting the algebra that ensues. The representation in (3.6) ensures that the leading-order horizontal velocity is inertial over short times; small deviations in wave field frequency from $f_{0}$ are captured by the dependence of $M$ or $A$ on the slow time $\bar{t}$ . The construction in (3.6) also implies that the vertical average of the NIW horizontal velocity is zero at this order.

With the representation in (3.6), we can integrate the continuity equation (3.5) over the fast vertical coordinate $\tilde{z}$ to yield

where the $\tilde{z}$ -independent function of integration ${\hat{w}}_{0}(x,y,\bar{z},\,\tilde{t}\,,\bar{t})$ is necessary to ensure solvability of the perturbation expansion at next order. If ${\hat{w}}_{0}$ is not included in (3.8), then the $O(\unicode[STIX]{x1D716})$ velocity field cannot satisfy continuity and the boundary conditions. At $O(\unicode[STIX]{x1D716})$ in (2.26)–(2.29), we find that ${\hat{w}}_{0}$ oscillates on the fast time scale with frequency $2f_{0}$ and is forced nonlinearly by NIW horizontal self-advection.

The leading-order buoyancy $b_{0}$ follows from integration of the buoyancy equation (3.4) using $w_{0}$ in (3.8),

where as in (3.8) we include the function of integration $\hat{b}_{0}(x,y,\bar{z},\,\tilde{t}\,,\bar{t})$ . The vertical momentum equation (3.3) implies that the leading-order pressure $p_{0}$ does not depend on the fast vertical scale $\tilde{z}$ , or that

The leading-order pressure $p_{0}$ is eventually determined by (3.28) and (3.29) below and oscillates on a fast time scale with frequency $2f_{0}$ . An important feature eventually revealed by this expansion is that the large-vertical-scale $2f_{0}$ fields ${\hat{w}}_{0}$ , $\hat{b}_{0}$ and $\hat{p}_{0}$ appear at leading order in (3.8), (3.9) and (3.28). The magnitudes of these $2f_{0}$ fields follow from the two assumptions that the $2f_{0}$ horizontal velocity and vertical scales are respectively $O(\unicode[STIX]{x1D716})$ and $O(\unicode[STIX]{x1D716}^{-1})$ compared with their near-inertial counterparts. This scaling produces a description of NIW– $2f_{0}$ interaction that neglects the complicating effect of the mean flow $\unicode[STIX]{x1D713}$ on $2f_{0}$ evolution.

3.2 First order: wave-averaged geostrophic balance and $2f_{0}$ harmonic

The $O(\unicode[STIX]{x1D716})$ terms in (2.26)–(2.29) are

These equations describe wave-averaged geostrophic balance and the nonlinearly forced $2f_{0}$ harmonic.

3.3 Second order: an NIW amplitude evolution equation

The $O(\unicode[STIX]{x1D716}^{2})$ terms in the horizontal momentum equation (2.26) are

Here, we finally apply the solvability condition arising from the introduction of multiple time scales. The solvability condition prevents the disordering of terms that would result from secular growth in ${\mathcal{U}}_{2}$ : we isolate resonant forcing terms on the right of (3.33) and set them collectively to zero. The amplitude equation yielded by this procedure governs the dependence of the NIW envelope $A$ on the slow time $\bar{t}$ . We note that the vertical average of (3.33) has no resonant terms.

We construct the amplitude equation piece by piece, starting at the far-right end of (3.33) and proceeding to the left. The final term $w_{0}\,{\mathcal{U}}_{0\bar{z}}$ in (3.33) has no parts proportional to $\text{e}^{-\text{i}f_{0}\,\tilde{t}\,}$ and so does not contribute to the amplitude equation. The next three terms from the left side of (3.33) are

where $\text{NRT}$ stands for ‘non-resonant terms’. Next in line is

It should be noted that to find (3.35) we need only consider the time-mean velocity $\bar{\boldsymbol{u}}_{1}$ , since ${\mathcal{U}}_{0}$ is proportional to $\text{e}^{-\text{i}f_{0}\,\tilde{t}\,}$ . The first term on the right of (3.33), involving the zero-order advection of the first-order velocity, is the most complicated. Carefully compiling the terms, we find

Adding (3.35) to (3.37) yields

The absence of terms cubic in $M$ is a remarkable aspect of (3.38): all 16 cubic $M$ terms in (3.35) and (3.37) conspire in collective cancellation. This simplification was previously noted by Falkovich, Kuznetsov & Medvedev (Reference Falkovich, Kuznetsov and Medvedev1994) and Zeitlin, Reznik & Ben Jelloul (Reference Zeitlin, Reznik and Ben Jelloul2003), and is the reason why no cubic terms appear in XV.

It is thus notable that our expansion identifies a surviving ‘honorary’ cubic term, proportional to $M_{\tilde{z}}^{\ast }B_{s^{\ast }s^{\ast }}$ , in (3.38). This new term results from the interaction of NIWs with both forced and freely propagating $2f_{0}$ waves. The requirement for $2f_{0}$ fields arises when the first-order continuity equation (3.27) is averaged over the small NIW vertical scale: if the vertical average of ${\mathcal{U}}_{1s}+{\mathcal{U}}_{1s^{\ast }}^{\ast }$ is non-zero, for example, then continuity can only be satisfied if ${\mathcal{U}}_{1}$ is permitted its own independent evolution. This solvability issue is addressed by introducing the $2f_{0}$ fields ${\hat{w}}_{0}$ , $\hat{b}_{0}$ and $\hat{p}_{0}$ at leading order in (3.8) and (3.9). This is the non-obvious step that ultimately produces the new term in (3.38).

The amplitude equation is then obtained from the sum of (3.34) and (3.38). In Cartesian coordinates and in terms of $A$ , the amplitude equation is

The amplitude equation (3.39) is the YBJ equation except for the final $2f_{0}$ interaction term on the left-hand side. The $2f_{0}$ interaction term is identical to the one found by Young et al. (Reference Young, Tsang and Balmforth2008) in their analysis of energy transfer from prescribed $2f_{0}$ motions to NIWs by PSI.

6.1 Wave action

The first conservation law follows from the wave equations (1.8) and (1.9). We multiply (1.8) with $(1/2)\text{L}\,A^{\ast }$ , add the complex conjugate and integrate over the domain. Using the $2f_{0}$ equation (5.3), and a liberal application of integration by parts, we find

The appearance of the $B$ terms in this first conservation law is a consequence of the time derivative in the $2f_{0}$ equation (5.3) and corresponds to the total energy in the freely propagating part of the near- $2f_{0}$ wave field. The conservation law (6.1) thus implies that resonant generation of propagating $2f_{0}$ waves extracts near-inertial kinetic energy.

6.2 Coupled energy

The second conserved quantity is a wave–mean coupled energy. We derive the associated conservation law by multiplying the potential vorticity equation (4.1) with $\unicode[STIX]{x1D713}$ and integrating over the domain. The Jacobian term $\unicode[STIX]{x1D713}\,\text{J}(\unicode[STIX]{x1D713},q)$ can be written as an exact derivative and integrates to zero. Applying integration by parts, we are left with

where $q^{w}$ is the wave potential vorticity defined in (4.2) and

is the total balanced QG energy. Next, we multiply (1.8) by $\text{i}\text{L}\,A_{t}^{\ast }/2f_{0}$ , add the complex conjugate and integrate over the domain to obtain

where

is the sum of the NIW potential energy and an action-like term associated with the $\unicode[STIX]{x1D6FD}$ -effect. The first term on the right of (6.4) corresponds to the term on the right of (6.2) and will cancel when these equations are added. Substitution of the $2f_{0}$ equation (1.9) and its complex conjugate into the second integral on the right of (6.4) followed by persistent integration by parts produces

where

An additional substitution of (1.9) and its complex conjugate transforms equation (6.7) into (6.8). The conservation of coupled energy emerges from the combination of (6.2), (6.4) and (6.6),

The conservation law in (6.9) is identical to XV’s conservation law except for the addition of the $\mathscr{E}_{2f}$ term. Equation (6.9) is the analogue of a conservation law found by Danioux, Vanneste & Bühler (Reference Danioux, Vanneste and Bühler2015) that relates the evolution of NIW potential energy to advection and refraction by steady geostrophic flows.

A thought experiment due to XV illuminates an important implication of (6.9). Envision the rapid and stormy deposition of a horizontally extensive surface-concentrated current in a region of geostrophic turbulence. As the storm passes, the unbalanced current develops into a surface-concentrated NIW that is almost horizontally uniform, and therefore has little potential energy, and thus $\mathscr{E}_{f}\approx 0$ . Over subsequent inertial periods, NIW refraction and advection by the geostrophic flow drives the creation of near-inertial horizontal scales and potential energy, which leads to vertical propagation of a refracted NIW signal and the generation of $2f_{0}$ internal waves. Because the wave action in (6.1) and coupled energy in (6.9) are distinct and independent conservation laws, the total NIW and $2f_{0}$ wave energy increases in this process at the sole expense of energy in the geostrophic flow. The role of $\mathscr{E}_{2f}$ in (6.9) is unfortunately obscure in this thought experiment, although the diagnosis of (6.9) presented in figure 7 below shows that the effect is minor in some cases.

7.1 A surface-concentrated NIW in random barotropic flow

The initial value problem involves the interaction of a surface-concentrated NIW with random barotropic balanced flow. The two-dimensional physical domain is bounded by rigid lids in $z$ with height $H=4$ km and is periodic in $x$ with width $L=800$  km. The stratification is uniform, with buoyancy frequency $N=2\times 10^{-3}~\text{s}^{-1}$ , and the inertial frequency is $f_{0}=10^{-4}\;\text{s}^{-1}=N/20$ , with $\unicode[STIX]{x1D6FD}=0$ . As for the problem considered in § 1, the NIW initial condition is

with $h=100$  m. We consider initial NIW surface velocities of $U_{0}=0.4,0.2$ and $0.1~\text{m}~\text{s}^{-1}$ .

The initial balanced $v$ velocity is

where $k_{n}\stackrel{\text{def}}{=}2\unicode[STIX]{x03C0}n/L$ and the $\unicode[STIX]{x1D719}_{n}$ are random phases between 0 and $2\unicode[STIX]{x03C0}$ for each component of the geostrophic flow. We choose $V_{0}=0.1~\text{m}~\text{s}^{-1}$ for all simulations. This produces a maximum velocity of $\max (v)\,\approx \,0.2~\text{m}~\text{s}^{-1}$ , a maximum Rossby number of $\max (v_{x})/f_{0}\,\approx \,0.1$ and a root-mean-square Rossby number of $\text{r.m.s.}(v_{x})/f_{0}\approx 0.05$ . The balanced flow and its associated vorticity field are plotted in figures 2(a,b)–4(a,b).

Figure 2.

Comparison of wave speed in numerical solutions to the three-component and Boussinesq models. (a,b) The $x$ dependence of the initial balanced velocity $v$ (a) and the balanced vorticity normalized by $f_{0}$ , $v_{x}/f_{0}$ (b). (c–h) The wave speed defined in (7.8) at $t=10$ , 40 and 80 inertial periods in the Boussinesq model (c,e,g) and the three-component model (d,f,h). The initial NIW surface velocity is $U_{0}=0.4~\text{m}~\text{s}^{-1}$ .

The numerical solutions we report are listed in table 1. We choose simulation parameters both for ease of numerical integration and for consistency with oceanic scenarios. In particular, $V_{0}=0.1~\text{m}~\text{s}^{-1}$ leads to ‘reasonable’ NIW vertical propagation in 10 or 20 inertial periods; smaller values of $V_{0}$ lead to slower vertical propagation. It should be noted, however, that $V_{0}=0.1~\text{m}~\text{s}^{-1}$ violates assumptions made in § 2 to justify the asymptotic derivation of the three-component model: the expansion assumes that $\bar{U}/\tilde{U} =\tilde{H}/\bar{H}=\unicode[STIX]{x1D716}\ll 1$ . In the following simulations, $\tilde{H}/\bar{H}=\unicode[STIX]{x03C0}h/H=0.08\ll 1$ while $\bar{U}/\tilde{U} =V_{0}/U_{0}$ ranges between $0.25$ and $1$ , so that the balanced flow seems to be too strong given that the non-dimensionalization in § 2.1 assumes $\tilde{H}/\bar{H}\sim \bar{U}/\tilde{U} =\unicode[STIX]{x1D716}\ll 1$ . Nevertheless, we find generally good agreement between the predictions of the three-component model and full Boussinesq solutions. Other calculations with $V_{0}$ reduced to $0.025~\text{m}~\text{s}^{-1}$ not shown here also validate the three-component model.

We note finally that that our two-dimensional computations do not test all aspects of the three-component model: with $\unicode[STIX]{x2202}_{y}=0$ the advective terms $\text{J}(\unicode[STIX]{x1D713},\text{L}\,A)$ and $\text{J}(\unicode[STIX]{x1D713},q)$ vanish and the available potential vorticity (Wagner & Young Reference Wagner and Young2015) $q$ does not evolve from its initial distribution. The two-dimensional simulations reported here are interesting primarily as a test of the nonlinear NIW– $2f$ interaction and the associated production of small vertical scales. Probing of the advective terms requires either three-dimensional solutions like those in Danioux et al. (Reference Danioux, Klein and Rivière2008) or solutions of the two-dimensional model introduced in § 6.2 of XV.

Table 1.

Parameters and models for the numerical simulations reported in §§ 7 and 8. A resolution of ‘ $1\times$ ’ is $nx\times nz=1024\times 2048$ , and ‘ $2\times$ ’ is twice that. In all runs $\unicode[STIX]{x1D713}_{x}^{q}=v(t=0)$ is given by (7.2) with $V_{0}=0.1~\text{m}~\text{s}^{-1}$ .

7.3 Points of comparison

We use the horizontal velocity, vertical velocity and domain-integrated vertical kinetic energy to compare the Boussinesq and three-component models. Because $v$ is initially balanced, the unbalanced part of $v$ is approximately isolated with

On the other hand, $u$ is unbalanced because $p_{y}=0$ . We thus define the unbalanced horizontal ‘wave speed’ as

The wave speed in (7.8) includes NIW and $2f_{0}$ components as well as a much smaller wave-induced mean component. In figure 2, we compare the wave speed from the Boussinesq solution with $|\tilde{{\mathcal{U}}}|\approx |\,{\mathcal{U}}_{0}+\tilde{{\mathcal{U}}}_{1}|$ diagnosed from the three-component solution, where ${\mathcal{U}}_{0}=\text{e}^{-\text{i}f_{0}t}\text{L}\,A$ and $\tilde{{\mathcal{U}}}_{1}$ is the wavy part of (3.31). The comparison is made at $t=10,40$ and 80 inertial periods. The initial NIW magnitude in figure 2 is $U_{0}=0.4~\text{m}~\text{s}^{-1}$ , and the initial balanced barotropic $v$ and local Rossby number $v_{x}/f_{0}$ are plotted in (a) and (b).

The wave speed shown in figure 2 indicates good agreement between the three-component model and the Boussinesq equations. A close inspection of the fields is required to discern differences that arise between the two models at late times. It is our consistent experience that the wave speed field is well estimated by the three-component model for the two-dimensional initial value problems examined here; we therefore focus the following discussion on the more interesting and worst-case comparison of vertical velocity.

The vertical velocities in the Boussinesq and three-component solutions are compared in figures 3 and 4 for initial NIW magnitudes of $U_{0}=0.4$ and $0.2~\text{m}~\text{s}^{-1}$ . Vertical velocity is plotted from (c,d) to (g,h) at $t=10$ , 40 and 80 inertial periods. For both cases, the agreement is good at $t=10$ inertial periods but degrades progressively thereafter. Conspicuous aspects of the Boussinesq solution absent from the three-component solution are features with small horizontal scales and steep characteristic angles. These features are especially prominent in figure 3 for the most nonlinear case with $U_{0}=0.4~\text{m}~\text{s}^{-1}$ at $t=40$ and 80 inertial periods.

Figure 3.

Comparison of the vertical velocity in numerical solutions with the Boussinesq and three-component models. (a,b) The $x$ dependence of the initial $v$ (a) and $v_{x}/f_{0}$ (b). (c–h) The vertical velocity $w$ at $t=10$ , 40 and 80 inertial periods for the Boussinesq model (c,e,g) and the three-component model (d,f,h). The initial NIW surface velocity is $U_{0}=0.4~\text{m}~\text{s}^{-1}$ .

Figure 4.

Like figure 3 but with initial NIW surface velocity $U_{0}=0.2~\text{m}~\text{s}^{-1}$ . The agreement between the Boussinesq and three-component models is better than for $U_{0}=0.4~\text{m}~\text{s}^{-1}$ .

We dissect this failure of the three-component model in figure 5, which compares vertical kinetic energy (VKE) spectra between the three-component and Boussinesq models for $U_{0}=0.4~\text{m}~\text{s}^{-1}$ at $t=10$ and 40 inertial periods. The five lines indicate internal wave frequencies based on the linear dispersion relation; proceeding clockwise from the vertical axes these frequencies are $1.01f_{0}$ , $1.08f_{0}$ , $2f_{0}$ , $3f_{0}$ and $4f_{0}$ , with the dashed line corresponding to $2f_{0}$ . The dynamics are clear: in the Boussinesq simulations, substantial VKE leaks into higher harmonic frequencies $3f_{0}$ and $4f_{0}$ . By $t=40$ inertial periods, the fraction of VKE contained in frequencies greater than $2.8f_{0}$ is 49 %. This transfer of VKE to higher harmonics decreases with $U_{0}$ : for $U_{0}=0.2$ and $0.1~\text{m}~\text{s}^{-1}$ , the fraction is 10 % and just over 1 % respectively at $t=40$ inertial periods.

Figure 5.

Snapshots of VKE spectral components, $|\breve{w}|^{2}$ , where $\breve{w}$ denotes the Fourier and vertical mode transform of $w$ , for $U_{0}=0.4~\text{m}~\text{s}^{-1}$ at $t=10$ and 40 inertial periods. The spectral components are normalized by the total Boussinesq VKE, and horizontal modes include energy from both positive and negative horizontal wavenumbers. The five lines show the linear dispersion relation for five internal wave frequencies; proceeding clockwise from the vertical axes these frequencies are $1.01f_{0}$ , $1.08f_{0}$ , $2f_{0}$ , $3f_{0}$ and $4f_{0}$ , with the dashed line corresponding to $2f_{0}$ . By $t=40$ inertial periods, 49 % of the Boussinesq VKE is in frequencies higher than $2.8f_{0}$ .

The effect of the energy transfer to NIW harmonics on total VKE is demonstrated in figure 6, which shows the evolution of total VKE, $\int w^{2}/2\,\text{d}x\,\text{d}z$ , for (a) $U_{0}=0.4~\text{m}~\text{s}^{-1}$ and (b) $U_{0}=0.2~\text{m}~\text{s}^{-1}$ . Four models are considered: Boussinesq (solid lines), three-component model (dashed lines), a two-component model that neglects $2f_{0}$ (dash-dotted lines), and a modification of the three-component model with PSI suppressed by removing $B_{xx}\text{L}\,A^{\ast }$ from the NIW equation (7.4) (dotted line, figure 6(b) only). The three-component model underestimates the amplitude of the VKE, having 54 % of the Boussinesq solution at $t=40$ inertial periods and 43 % of the total at $t=80$ inertial periods. The ‘extra’ Boussinesq VKE is thus similar to that contained in frequencies greater than $2.8f_{0}$ , implying that it originates in a transfer of horizontal NIW kinetic energy to high NIW harmonics not accounted for in the three-component model. It is not surprising that this transfer to $3f_{0}$ and $4f_{0}$ is strongest in the most nonlinear case with $U_{0}=0.4~\text{m}~\text{s}^{-1}$ .

Figure 6.

Evolution of total VKE, $\int w^{2}/2\,\text{d}x\,\text{d}z$ , for (a) $U_{0}=0.4~\text{m}~\text{s}^{-1}$ and (b) $U_{0}=0.2~\text{m}~\text{s}^{-1}$ . Results are diagnosed from the Boussinesq model (solid lines), the three-component model (dashed lines), a ‘no $2f_{0}$ ’ two-component model with $B\mapsto 0$ (dash-dotted lines) and a ‘no PSI’ three-component model with the term $B_{xx}\text{L}\,A^{\ast }$ removed from the NIW equation (7.4) (dotted line in (b) only). Black colours are used for (a)  $U_{0}=0.4~\text{m}~\text{s}^{-1}$ and blue colours for (b) $U_{0}=0.2~\text{m}~\text{s}^{-1}$ here and in figures 7 and 9.

Figure 7.

The evolution of (a) wave action and (b) coupled energy in the three-component system with initial NIW velocity in (7.1) and $U_{0}=0.4,0.2$ and $0.1~\text{m}~\text{s}^{-1}$ , and initial balanced velocity in (7.2) with $V_{0}=0.1~\text{m}~\text{s}^{-1}$ , as shown in figures 2–4.

Figure 8.

Comparison of velocity magnitude and shear magnitude between the Boussinesq equations, the three-component model and a two-component model with $B\mapsto 0$ and thus no $2f_{0}$ waves. The snapshots are taken at $t=80$ inertial periods and comprise a portion of the full domain shown in figure 2.

For the case $U_{0}=0.2~\text{m}~\text{s}^{-1}$ the three-component model correctly estimates the amplitude, but not the phase of the VKE. Unsurprisingly, given the impact of NIW–harmonic interactions on the VKE, the two-component solutions with $B\mapsto 0$ and thus no $2f_{0}$ cannot capture the evolution of the VKE for either $U_{0}=0.4$ or $0.2~\text{m}~\text{s}^{-1}$ . In figure 6(b), the suppression of PSI leads to an unrealistic accumulation of VKE in $2f_{0}$ motions starting at approximately $t=20$ inertial periods. This indicates that the transfer of energy from $2f_{0}$ back to NIWs must be accounted for to accurately capture VKE evolution.

7.4 Summary

The comparison presented in this section shows that the three-component model well describes NIW evolution and nonlinear NIW– $2f_{0}$ interaction. That the three-component model describes NIW evolution in the cases shown here is not too surprising, since it is probably driven by a linearized YBJ-type flow-induced refraction. The success of the three-component model in describing NIW– $2f_{0}$ interaction is more surprising and vindicates the heuristic derivation of $2f_{0}$ dynamics. On the other hand, the model grossly underestimates the vertical velocity magnitude when the NIWs are strong, which follows from the neglect of NIW harmonics higher than $2f_{0}$ . We stress that this two-dimensional comparison cannot test whether the three-component model correctly captures the impact of NIWs on balanced flow evolution.

8.1 Energy transfer between flow components

The two conserved quantities in the three-component model are wave action and coupled energy defined in (6.1) and (6.9) and plotted in figure 7(a,b). Figure 7(a) illustrates the transfer between NIW kinetic energy and the total energy of the $2f_{0}$ field, defined respectively as

Figure 7(a) shows the components of wave action change $\unicode[STIX]{x1D6FF}\mathscr{A}_{f}(t)\stackrel{\text{def}}{=}\mathscr{A}_{f}(t)-\mathscr{A}(0)$ and $\mathscr{A}_{2f}$ . Figure 7(a) also shows the very small change in total wave action $\unicode[STIX]{x1D6FF}\mathscr{A}=\unicode[STIX]{x1D6FF}\mathscr{A}_{f}+\mathscr{A}_{2f}$ due to hyper-dissipation with dotted lines. All curves are normalized by the initial wave action $\mathscr{A}(0)$ , which is equal to the kinetic energy in the near-inertial initial condition. Three cases corresponding to different initial amplitudes of the NIW are shown: $U_{0}=0.1,0.2$ and $0.4~\text{m}~\text{s}^{-1}$ in red, blue and black. The action transferred from $\mathscr{A}_{f}$ to $\mathscr{A}_{2f}$ increases initially to a maximum value and thereafter decays to a constant asymptotic value as $t\rightarrow \infty$ . Although the short-term maximum transfer increases with the initial NIW amplitude $U_{0}$ , the fraction as $t\rightarrow \infty$ is independent of $U_{0}$ and indicates that less than 1 % of the near-inertial action is ultimately transferred to the $2f_{0}$ field.

Figure 7(b) shows the evolution of $\unicode[STIX]{x1D6FF}\mathscr{E}_{\unicode[STIX]{x1D713}}(t)\stackrel{\text{def}}{=}\mathscr{E}_{\unicode[STIX]{x1D713}}(t)-\mathscr{E}(0)$ , $\mathscr{E}_{f}$ and $\mathscr{E}_{2f}$ following the definitions in (6.3), (6.5) and (6.7) respectively. All energies are normalized by the initial near-inertial kinetic energy $\mathscr{A}(0)$ , thus revealing an uncanny correspondence between cases: the energy transferred from balanced flow to NIWs is a constant fraction of the initial NIW kinetic energy, $\mathscr{A}_{f}(0)$ .

8.2 The $2f_{0}$ motions are a stepping stone to small vertical scales

The evolution of $\mathscr{A}_{2f}$ in figure 7(a) is unspectacular and suggests that NIW– $2f_{0}$ interaction is not important because at most a mere 3 % of the initial NIW kinetic energy is transferred to $2f_{0}$ when $U_{0}=0.4~\text{m}~\text{s}^{-1}$ . Yet the possibility for a PSI-type energy transfer from $2f_{0}$ to NIW hints that the inclusion of $2f_{0}$ and nonlinear NIW– $2f_{0}$ interaction may be necessary to capture the production of small NIW vertical scales.

We isolate the effect of this process by computing a ‘no $2f_{0}$ ’ solution of (7.3) and (7.4). In this solution we set $B\mapsto 0$ , thus removing $2f_{0}$ waves and the $2f_{0}$ -mediated transfer of energy. Figure 8 gives a qualitative impression of the results, where the wave speed (a–c) and wave shear magnitude (d–f) are plotted for three model solutions with $U_{0}=0.4~\text{m}~\text{s}^{-1}$ : Boussinesq (a,d), three-component model (b,e) and the two-component ‘no $2f_{0}$ ’ solution of (7.3) and (7.4) with $B\mapsto 0$ (c,f). Both the Boussinesq and the three-component results have small scales in the vertical velocity which are lacking when $2f_{0}$ is removed, and thus must be created by nonlinear NIW– $2f_{0}$ interaction. Without $2f_{0}$ the magnitude of the vertical shear is also underestimated near $(x,z)=(-0.1,40)~\text{km}$ . At the same time, the overall flow structure agrees between the three models.

A more quantitative estimate of small vertical scales is provided by the metric $\mathit{Ri}^{\dagger }(t)$ , which measures the smallest Richardson numbers and thus the potential for wave breaking and mixing were such processes resolved. Here, $\mathit{Ri}^{\dagger }$ is defined as the average of the smallest 0.1 % of Richardson numbers:

The evolution of $\mathit{Ri}^{\dagger }$ normalized by its initial value is shown in figure 9 for the cases $U_{0}=0.4$ and $0.2~\text{m}~\text{s}^{-1}$ . Results are compared between the Boussinesq, three-component and two-component models. The comparison reveals that small values of $\mathit{Ri}^{\dagger }$ and thus small vertical scales are produced by at least two distinct physical mechanisms: $\mathit{Ri}^{\dagger }$ first decreases to a minimum value between $t=15$ and 20 inertial periods and rises gradually thereafter. The early-time agreement between all five cases means that $\mathit{Ri}^{\dagger }$ is controlled by refraction of the NIW field by balanced flow during this first stage.

Figure 9.

The evolution of $\mathit{Ri}^{\dagger }$ defined in (8.2) and normalized by its initial value for the cases $U_{0}=0.4~\text{m}~\text{s}^{-1}$ (black) and $U_{0}=0.2~\text{m}~\text{s}^{-1}$ (blue or grey) and in three models: Boussinesq (solid lines), three-component model (dashed lines) and the two-component model with $B\mapsto 0$ and thus no $2f_{0}$ (dashed-dotted lines). The inset shows the numerical values of $\mathit{Ri}^{\dagger }$ approaching the critical value $\mathit{Ri}=1/4$ for the case $U_{0}=0.4~\text{m}~\text{s}^{-1}$ in the Boussinesq and three-component models. The metric $\mathit{Ri}^{\dagger }$ is a measure of the smallest vertical scales in the flow, whose evolution cannot be captured without $2f_{0}$ .

However, at approximately $t=40$ inertial periods, the results diverge and $\mathit{Ri}^{\dagger }$ is smaller for $U_{0}=0.4~\text{m}~\text{s}^{-1}$ in both the Boussinesq and the three-component model. It is conspicuous that in the two-component model with $U_{0}=0.4~\text{m}~\text{s}^{-1}$ , $\mathit{Ri}^{\dagger }$ is overestimated and stays close to the more linear $U_{0}=0.2~\text{m}~\text{s}^{-1}$ results. At this stage, the smallness of $\mathit{Ri}^{\dagger }$ and thus small NIW vertical scales in the Boussinesq and three-component models must be controlled by nonlinear NIW– $2f_{0}$ interaction. Strikingly, and despite the fact that they contain little instantaneous energy, $2f_{0}$ motions provide a crucial stepping stone through which NIW energy is transferred to small vertical scales. The surprisingly accurate description of this process by the three-component model suggests that it is controlled by the interaction of relatively large-vertical-scale $2f_{0}$ motions with small-scale NIWs, which figures 2 and 3 show to be well captured by the three-component model.