2.1. The YBJ model and analogy with the Schrödinger equation

We study the propagation of NIWs in a steady geostrophic flow with a reduced-gravity shallow-water model (e.g. Cushman-Roisin Reference Cushman-Roisin1994). This slab model can be thought of as representing the dynamics of NIWs confined in a mixed layer capping an abyssal layer where the only motion is the imposed geostrophic flow, assumed to be identical in both layers. For NIWs, the horizontal velocity $(u,v)$ can be written in terms of a complex amplitude $M$ according to $u+\text{i}v=M\text{e}^{-\text{i}ft}$ . The slow (compared with $f^{-1}$ ) time evolution of $M$ is governed by the YBJ equation

(2.1) $$\begin{eqnarray}\partial _{t}M+J({\it\psi},M)-\text{i}\frac{h}{2}{\rm\Delta}M+\text{i}\frac{{\rm\Delta}{\it\psi}}{2}M=0,\end{eqnarray}$$

where ${\it\psi}$ and ${\rm\Delta}{\it\psi}$ are the streamfunction and vorticity of the steady geostrophic flow, $J$ is the horizontal Jacobian and $h=g^{\prime }H/f$ is a dispersion parameter, with $g^{\prime }$ and $H$ the reduced gravity and average depth of the mixed layer. We assume that $h>0$ , as in the Northern Hemisphere where $f>0$ . The respective terms quantify the effects due to advection, dispersion and refraction. A concise derivation of (2.1) is given in appendix A.

Two simple facts help in understanding the dynamics of (2.1). First, for any constant ${\it\alpha}>0$ the YBJ equation is invariant under the scaling transformation ${\it\psi}\mapsto {\it\alpha}{\it\psi}$ , $h\mapsto {\it\alpha}h$ , $t\mapsto t/{\it\alpha}$ . This makes it obvious that the intrinsic dynamics of (2.1) depends only on the single non-dimensional parameter $h/{\it\Psi}$ , say, where ${\it\Psi}$ is the amplitude scale of ${\it\psi}$ .

Second, without the advection term $J({\it\psi},M)$ the YBJ equation is identical to the Schrödinger equation that governs the complex wavefunction ${\it\phi}(x,y,t)$ for a single particle with unit mass and external potential $V(x,y)$ :

(2.2) $$\begin{eqnarray}\partial _{t}{\it\phi}-\text{i}\frac{\hbar}{2}{\rm\Delta}{\it\phi}+\text{i}\frac{V}{\hbar}{\it\phi}=0.\end{eqnarray}$$

Here, $\hbar$ is Planck’s constant divided by $2{\rm\pi}$ . Comparing (2.1) and (2.2) and identifying $\hbar$ with $h$ shows that the effective potential in (2.1) is

(2.3) $$\begin{eqnarray}V=h\frac{{\rm\Delta}{\it\psi}}{2}.\end{eqnarray}$$

Clearly, regions of higher $V$ repel the particle whereas regions of lower $V$ attract it. Hence, if the advection term is negligible, then the mathematical analogy between the particle probability density $|{\it\phi}|^{2}$ and the inertial wave kinetic energy density $|M|^{2}$ readily implies that cyclones repel inertial waves while anticyclones attract them (Balmforth, Llewellyn Smith & Young Reference Balmforth, Llewellyn Smith and Young1998). For the special case of an axisymmetric anticyclonic vortex, this is confirmed by the existence of axisymmetric trapped modes (Llewellyn Smith Reference Llewellyn Smith1999).

Of course, if the advection term cannot be neglected then the simple analogy with the Schrödinger equation breaks down. Hence, the main task is to understand how the advection term alters the basic Schrödinger dynamics as a function of $h/{\it\Psi}$ . For this it becomes crucial to study the full set of conservation laws associated with (2.1), as we shall do now.

2.2. Conservation laws

By multiplying (2.1) by $M^{\ast }$ and adding its complex conjugate we obtain

(2.4) $$\begin{eqnarray}\partial _{t}{\textstyle \frac{1}{2}}|M|^{2}+J\left({\it\psi},{\textstyle \frac{1}{2}}|M|^{2}\right)+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{F}=0,\end{eqnarray}$$

where $\boldsymbol{F}=\text{i}h(M\boldsymbol{{\rm\nabla}}M^{\ast }-M^{\ast }\boldsymbol{{\rm\nabla}}M)/4$ . In a finite domain with suitable boundary conditions (periodicity or $M=0$ ), integration of (2.4) gives the conservation of NIW kinetic energy, as derived by Young & Ben Jelloul (Reference Young and Ben Jelloul1997),

(2.5) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\iint \frac{1}{2}|M|^{2}\,\text{d}x\,\text{d}y=0.\end{eqnarray}$$

There is another conservation law associated with (2.1) and not mentioned in Young & Ben Jelloul (Reference Young and Ben Jelloul1997). It is derived by forming the combination $M_{t}^{\ast }(2.1)-M_{t}(2.1)^{\ast }$ and integrating the result over the domain. Using properties of the Jacobian and integrating by parts, this gives

(2.6) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}(I_{1}+I_{2}+I_{3})=0,\end{eqnarray}$$

where

(2.7a−c ) $$\begin{eqnarray}I_{1}=\iint \text{i}h{\it\psi}J(M^{\ast },M)\,\text{d}x\,\text{d}y,\quad I_{2}=\iint \frac{h^{2}}{2}|\boldsymbol{{\rm\nabla}}M|^{2}\,\text{d}x\,\text{d}y,\quad I_{3}=\iint h\frac{{\rm\Delta}{\it\psi}}{2}|M|^{2}\,\text{d}x\,\text{d}y.\end{eqnarray}$$

The terms $I_{1}$ , $I_{2}$ and $I_{3}$ stem directly from the advection, dispersion and refraction terms in the YBJ equation. The overall factors of $h$ are included for two practical reasons: to keep the values of the invariants comparable when $h$ is varied and to highlight the appearance of the effective potential $V$ from (2.3) in $I_{3}$ . The consequences of the new conservation law (2.6) are discussed in more detail in the next section. In the absence of a geostrophic flow, $I_{1}$ and $I_{3}$ are identically zero, and $I_{2}$ can be recognized as the scaled NIW potential energy averaged over the fast time scale. This is not unexpected, as explained in appendix B. In the presence of a steady flow, $I_{1}+I_{2}+I_{3}$ can be interpreted as an energy in that its conservation is associated with the time invariance of (2.1). Unlike (2.5), $I_{1}+I_{2}+I_{3}$ is not conserved for arbitrary time-dependent flows. However, when the flow and NIWs evolve in a dynamically consistent manner, an analogous conservation law holds that accounts for energy transfers between the flow and the NIWs (Xie & Vanneste Reference Xie and Vanneste2015).

We remark that, in a steady geostrophic flow, differentiation of (2.1) with respect to time shows that $M_{t}$ satisfies the same equation as $M$ , and hence the same conservation laws. In particular,

(2.8) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\iint \frac{1}{2}|M_{t}|^{2}\,\text{d}x\,\text{d}y=0\end{eqnarray}$$

means that no stationary state can be reached.

3.1. Short-time solution

With homogeneous initial conditions, the first physical effect on NIW propagation is due to refraction, and the short-time behaviour is $M(x,y,t)=\exp (-\text{i}t{\rm\Delta}{\it\psi}(x,y)/2)\equiv \bar{M}(x,y,t)$ (Danioux et al. Reference Danioux, Klein and Rivière2008). A cautious definition for short time here is $t\ll t_{s}=1/\!\max \{{\it\Psi}/(2L^{2}),h/(2L^{2})\}$ . For such times, the solution can be sought as the expansion $M=\bar{M}+M^{\prime }$ , with $M^{\prime }(x,y,t=0)=0$ and $|M^{\prime }|\ll 1$ . Introducing this into (2.1) gives

(3.1) $$\begin{eqnarray}\partial _{t}M^{\prime }+J({\it\psi},M^{\prime })-\text{i}\frac{h}{2}{\rm\Delta}M^{\prime }+\text{i}\frac{{\rm\Delta}{\it\psi}}{2}M^{\prime }=-J({\it\psi},\bar{M})+\text{i}\frac{h}{2}{\rm\Delta}\bar{M}.\end{eqnarray}$$

Because of the form of $\bar{M}$ , the short-time behaviour of the right-hand side of (3.1) behaves as $O(t)$ , hence forcing $M^{\prime }(t)=O(t^{2})$ for small $t$ . Keeping this in mind, we now compare the relative size of terms $I_{1}$ and $I_{2}$ in (2.6). First, because $J(\bar{M}^{\ast },\bar{M})=0$ , $I_{1}$ is dominated by terms of the form $\text{i}h{\it\psi}J(\bar{M}^{\ast },M^{\prime })$ , resulting in an $O(t^{3})$ dependence. Second, $I_{2}$ is dominated by $h^{2}|\boldsymbol{{\rm\nabla}}\bar{M}|^{2}/2$ , yielding an $O(t^{2})$ dependence. Hence, for short enough times, $I_{2}\gg I_{1}$ . The NIW energy concentration in regions of negative vorticity can also be deduced from (2.4). On injecting the short-time solution $\bar{M}(x,y,t)$ into $\boldsymbol{F}$ , one finds that the amplitude of $M$ obeys

(3.2) $$\begin{eqnarray}\partial _{t}\frac{1}{2}|M|^{2}\simeq \frac{ht}{4}{\it\Delta}^{2}{\it\psi}\end{eqnarray}$$

at short times, consistent with the short-time solution of Klein et al. (Reference Klein, Llewellyn Smith and Lapeyre2004). Because the vorticity field and its Laplacian are anticorrelated, (3.2) gives an increase of NIW energy in anticyclonic regions for all values of $h/{\it\Psi}$ .

3.2. Long-time scaling arguments

For long times the spatial scales of $M$ need not be equal to those of ${\it\psi}$ anymore and hence the dominant balance between the various terms in (2.1) may shift accordingly. However, the conservation law (2.8) implies

(3.3) $$\begin{eqnarray}\iint \frac{1}{2}|M_{t}|^{2}\,\text{d}x\,\text{d}y=\iint \frac{1}{2}|M_{t}(t=0)|^{2}\,\text{d}x\,\text{d}y=\iint \frac{1}{2}({\rm\Delta}{\it\psi}/2)^{2}\,\text{d}x\,\text{d}y,\end{eqnarray}$$

and together with (2.5) this means that for all times the root-mean-square magnitudes of $M$ and $M_{t}$ scale with unity and ${\it\Psi}/L^{2}$ respectively. This allows a simple scaling analysis of the four terms in (2.1), which after rearranging yields

(3.4a−d ) $$\begin{eqnarray}1,\quad \frac{L}{l},\quad \frac{h}{{\it\Psi}}\frac{L^{2}}{l^{2}},\quad 1.\end{eqnarray}$$

Here, $l$ is the long-time spatial scale of $M$ such that $\boldsymbol{{\rm\nabla}}M=O(1/l)$ . We use (3.4) to determine how $l/L$ may depend on $h/{\it\Psi}$ . First, in the ‘strong-dispersion’ regime $h/{\it\Psi}\gg 1$ the only possible balance in (3.4) is $l/L=\sqrt{h/{\it\Psi}}$ , which balances dispersion and refraction whilst advection is negligible. As expected, this reduces the YBJ dynamics to that of the Schrödinger equation, so $I_{1}$ is negligible and $I_{3}\approx -I_{2}$ . In this regime the spatial scale of $M$ is larger than that of the background flow ${\it\psi}$ by a factor of $\sqrt{h/{\it\Psi}}$ .

Second, in the opposite regime $h/{\it\Psi}\ll 1$ one scaling possibility is $l=L$ , which balances advection and refraction whilst dispersion becomes negligible. However, from (2.1) this would correspond to an advective dynamics along streamlines in which $|M|$ is conserved while its phase continues to change by the refraction as in the short-term solution derived in § 3.1. This would inevitably lead to the generation of increasingly small spatial scales in $M$ and hence defeat the assumption $l=L$ . We must therefore look at the alternative, a balance between the advective and dispersive terms based on $l/L=h/{\it\Psi}$ . The advective and dispersive terms then provide a new leading-order dominant balance of size ${\it\Psi}/h\gg 1$ to the long-term evolution of the YBJ equation. Consequently, in this scenario the refraction is weak and $I_{3}$ becomes negligible, and so $I_{1}\approx -I_{2}$ . Moreover, the spatial scale of $M$ is much smaller than that of ${\it\psi}$ , by a factor of $h/{\it\Psi}$ .

It should be noted that this small horizontal scale may invalidate the assumption of small $h/(\,fl^{2})$ that underpins the near-inertial YBJ model (see appendix A). Using $l=hL/{\it\Psi}$ , we rewrite this assumption as ${\it\Psi}/(\,fL^{2})\ll h/{\it\Psi}$ in terms of the Rossby number on the left-hand side. Hence, for small but fixed $h/{\it\Psi}$ the near-inertial approximation holds provided that the Rossby number is small enough.

Finally, in the intermediate regime $h/{\it\Psi}=1$ the scaling $l/L=1$ makes all terms equally important. It stands to reason that in this intermediate regime $I_{3}$ will take moderate negative values, consistent with a monotonic transition between its value $I_{3}\approx 0$ for $h/{\it\Psi}\ll 1$ and its negative value $I_{3}\approx -I_{2}$ for $h/{\it\Psi}\gg 1$ . However, only in this intermediate regime does the length scale of $M$ equal that of ${\it\psi}$ , which arguably is the best situation for effective concentration of NIW energy in anticyclones. This suggests that the intermediate regime might be the most effective for this purpose. Of course, this simple scaling analysis can only provide a heuristic guide to the full NIW energy dynamics, not least because the spatial scales of $|M|^{2}$ are not related in a trivial way to those of $M$ . Still, we will see that the present scaling arguments do indeed provide a useful guide for understanding the numerical simulations.

3.3. Numerical simulations

Equation (2.1) is solved numerically on a doubly periodic $256\times 256$ grid using a pseudo-spectral time-split Euler scheme. A weak biharmonic dissipation is added for numerical stability. The streamfunction ${\it\psi}$ is taken as a single realization of a homogeneous isotropic Gaussian random process, with Gaussian correlation function and a correlation length $L$ of approximately $1/5$ of the domain size. The chosen vorticity field is shown in figure 1( $b^{\prime }$ ). Because its correlation scale is much smaller than the size of the domain, the results presented here are generic. We run three simulations with the same background flow but different values of $h$ such that $h/{\it\Psi}=0.2$ , 1 and 10, representative of the strong-advection, intermediate and strong-dispersion regimes. Simulations are stopped when the various terms in (2.6) no longer evolve significantly, which occurs at $t\simeq 0.3L^{2}/h$ (figure 2). For the intermediate case, using the values given at the beginning of this section, we find a time scale of between four and eight days, which is realistic.

Figure 1.

Streamfunction ( $a^{\prime }$ ) and associated normalized vorticity field ( $b^{\prime }$ ) of the background flow, and evolution of the NIW amplitude $|M|$ for $h/{\it\Psi}=0.2$ (a,d,g), $h/{\it\Psi}=1$ (b,e,h) and $h/{\it\Psi}=10$ (c,f,i). Here, (a–c) correspond to very short times after the start of the simulations ( $1/80$ th of the length of the simulations), (d–f) to short times ( $1/16$ th of the length of the simulations) and (g–i) to the end of the simulations.

Snapshots of $|M|$ are shown in figure 1 for very short, short and long times for the three values of $h/{\it\Psi}$ . At very short times, for all values of $h$ , $|M|$ is clearly anticorrelated with the vorticity field (or correlated with its Laplacian), as predicted by (3.2). As time increases, the variance of $|M|$ increases, substantially for small $h/{\it\Psi}$ but much less for large $h/{\it\Psi}$ . This is consistent with the scaling $|M-1|=O({\it\Psi}/h)$ that holds for short time and arbitrary $h/{\it\Psi}$ (as follows from integrating (3.2) for $t\lesssim L^{2}/h$ ) and for all time and large $h/{\it\Psi}$ (as follows from a perturbative treatment of (2.1)). At the end of the simulation, $M$ has larger scales than at the very first instants for $h/{\it\Psi}=10$ , in accordance with the scaling arguments in § 3.2 and also with the ‘truncated Laplacian’ solution of Klein et al. (Reference Klein, Llewellyn Smith and Lapeyre2004). The spectrum of $M$ does not change much afterwards (not shown), in contrast to the theory of Klein et al. (Reference Klein, Llewellyn Smith and Lapeyre2004) which predicts a continued cascade towards large scales. As expected, the evolution is very different for $h/{\it\Psi}=0.2$ . The short-time solution (figure 1 d) is consistent with a passive-scalar-type scenario: after the initial generation of spatial scales in $|M|$ (figure 1 a), the $|M|$ field is stretched and folded, while its amplitude grows due to dispersion. Later on (figure 1 g), NIW scales much smaller than the flow scales develop, consistent with § 3.2.

The behaviour of $|M|$ is reflected in the evolution of $I_{1}$ , $I_{2}$ and $I_{3}$ displayed in figure 2. The generation of small scales by refraction subsequently modulated by dispersion leads to an increase of $I_{2}$ . This is balanced by the decrease of $I_{3}$ at short times for all values of $h/{\it\Psi}$ and for all times for $h/{\it\Psi}=10$ , as predicted by our scaling argument. Conversely, for $h/{\it\Psi}=0.2$ we obtain $I_{1}\simeq -I_{2}$ , indicative of the balance between advection and dispersion noted above (in this case, the amplitude of these quantities decreases slightly in time because of numerical dissipation). For $h/{\it\Psi}=1$ , all terms $I_{1}$ , $I_{2}$ and $I_{3}$ have similar final amplitudes. It should be noted that the amplitude of $I_{2}$ is roughly the same in the three simulations. From the scaling $|M-1|=O({\it\Psi}/h)$ mentioned above, we infer that $I_{2}$ is proportional to ${\it\Psi}^{2}/L^{2}$ , and similarly for $I_{3}$ . This justifies a posteriori the inclusion of the overall factor $h$ in the definition of $I_{1}$ , $I_{2}$ and $I_{3}$ .

Figure 2.

Evolution of the integrals $I_{1}$ (dotted line), $I_{2}$ (dashed line) and $I_{3}$ (thin solid line) in (2.6) for $h/{\it\Psi}=0.2$ (a), 1 (b) and 10 (c). Here, $I_{1}$ , $I_{2}$ and $I_{3}$ are scaled by the area of the domain. The inset in (a) represents a zoom on very short times.

We now turn to the concentration of NIW energy in anticyclones. This is best quantified by $I_{3}/h$ , that is, the covariance between ${\rm\Delta}{\it\psi}/2$ and $|M|^{2}$ . An alternative diagnostic is the correlation between $|M|^{2}$ and ${\rm\Delta}{\it\psi}/2$ , namely

(3.5) $$\begin{eqnarray}C=\frac{I_{3}}{h\left(\displaystyle {\it\sigma}_{M}^{2}\iint ({\rm\Delta}{\it\psi}/2)^{2}\,\text{d}x\,\text{d}y\right)^{1/2}},\quad \text{where}~{\it\sigma}_{M}^{2}=\iint (|M|^{4}-1)\,\text{d}x\,\text{d}y\end{eqnarray}$$

is the variance of the NIW energy. We emphasize that $C$ focuses on the match between the spatial patterns of NIW energy and vorticity irrespective of their amplitudes; the more intuitive colocation of high values of NIW energy with regions of anticyclonic vorticity is measured by $I_{3}/h$ . The evolution of both $I_{3}/h$ and $C$ is shown for the three values of $h/{\it\Psi}$ in figure 3. At the end of the simulations, the covariance $I_{3}/h$ is largest for intermediate values of $h/{\it\Psi}$ . Small values of $h/{\it\Psi}$ lead to a strongly heterogeneous $|M|^{2}$ (large ${\it\sigma}_{M}^{2}$ , as seen in figure 1 g) but relatively weak correlation of $|M|^{2}$ and ${\rm\Delta}{\it\psi}$ (figure 3 b, dashed line); conversely, large values of $h/{\it\Psi}$ lead to a strong correlation (figure 3 b, solid line) but weak heterogeneity of $|M|^{2}$ . The intermediate case displays both a relatively strong ${\it\sigma}_{M}^{2}$ (figure 1 h) and a strong correlation (figure 3 b, dotted line), giving the maximum covariance.

Figure 3.

(a) Spatial covariance $I_{3}/h$ and (b) correlation $C$ between $|M|^{2}$ and ${\rm\Delta}{\it\psi}/2$ for $h/{\it\Psi}=0.2$ , 1 and 10. Legend in (a) applies also to (b).