# Meridional and Zonal Wavenumber Dependence in Tracer Flux in Rossby Waves

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rossby Waves

#### 2.2. Fluid and Tracer Flux

- allows for the disturbance to have any time-variation;
- explicitly quantifies the flux as a function of time;
- does not care whether manifolds intersect zero times, a finite number of times, or an infinite number of times;
- captures both simple and complicated (chaotic) forms of transport; and
- works for compressible two-dimensional flows (geophysical fluid might need to satisfy volume preservation, and so when observing behavior on two-dimensional sheets (e.g., isopycnals), area-preservation need not be satisfied since the isopycnals can compress towards one another).

## 3. Results

#### 3.1. Formulas for Flux

#### 3.2. Optimal Wavenumbers

#### 3.3. Flux for Wave Packets

## 4. Discussion and Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Relationship of Flux Definition to Other Methods

**Relationship to relative positioning of manifolds at each time:**The flux Formula (16), in the situation in which the disturbance is small, has a connection with the width of the chaotic zone. This is because flux occurs when the distance between ${y}_{s}\left(t\right)$ and ${y}_{u}\left(t\right)$ is large, and exactly this distance can be thought of as the width of the zone. Moreover, there is a dichotomy: viewing the width as a function of time at a fixed gate location (which is the viewpoint here), is equivalent to thinking of the width as a function of gate location at fixed time [41,51]. The reason for this is that any intersection patterns that exist at a fixed time, will eventually have to move through a fixed gate when time is evolved. To state this more precisely, let ℓ be a signed arc length along Γ, chosen symmetrically such that the northernmost point of Γ, $\left(\pi /2,\widehat{y}\right)$ has $\ell =0$, and $\ell <0$ for $x<\pi /2$. Then, the displacement between the stable and the unstable manifold at a location ℓ, at time t, measured in the direction normal to Γ, can be represented by [41,87]

**Relationship to chaotic transport:**The condition ${c}_{kl}\ne c$ is equivalent to that stated for the flux to be nonzero in (25). In this case, ${M}_{kl}$ is genuinely sinusoidal. This means it has infinitely many zeros. In view of the dichotomy discussed previously, if considering the picture at fixed time, this implies that the stable and unstable manifolds intersect infinitely often. Determining a zero of a Melnikov function in this way is one of the few approaches for proving that a system has chaotic transport, e.g., [46,47]. The focus in these articles was to prove that the system was chaotic, which occurs when the Melnikov function has simple zeros in time-periodic flows. Technically, however, this furnishes a proof of chaos only in the homoclinic instance in which the two anchoring points are the same; in this case each manifold can be established to wrap around the homoclinic loop, intersecting the other manifold infinitely many times, and re-enter a region arbitrarily close to itself once again. This is a crucial ingredient in the proof of the Smale-Birkhoff theorem [79], one of few tools for proving the existence of chaotic transport. Since this situation is of a heteroclinic manifold, with $\mathbf{a}\left(t\right)$ and $\mathbf{b}\left(t\right)$ being different, the Smale-Birkhoff theorem does not directly apply. In practice, though, the fact that the form (26) implies infinitely many intersections between the stable and unstable manifolds usually does result in chaotic transport between the eddy and the jet. This is because the intersection regions between them (the lobes) must stretch without bound when approaching the ends according to (A1), and the confinement of these lobes which is ensured by the fact that the line $y=0$ is invariant for the stream function (21). Thus, stretching and folding do occur in this situation, with intersection regions eventually forced to re-enter areas from which they left.

**Relationship to lobe areas:**If ${c}_{kl}\ne c$, the amplitude ${A}_{kl}$ has a relationship to the lobes generated when viewing the perturbed system in terms of a Poincaré map in the standard approach. If ${\psi}_{\mathrm{pert}}$ is time-sinusoidal, these lobes have equal areas, and the area of each lobe can be obtained by integrating the flux function between adjacent zeros [39,52], which in this case gives

**Different gate location:**What if the gate were chosen at a different location? In this case, it can be shown [39,41] that the leading-order flux function ${M}_{kl}\left(t\right)$ simply acquires a translation. In other words, this can be accomodateded by a shift in the value of the phase ${\varphi}_{kl}$. Importantly, this has no bearing on the amplitude ${A}_{kl}$.

**A measure of fluid flux:**In view of the above discussion, the amplitude of the flux function, ${A}_{kl}$ is proposed as an appropriate quantifier of the time-varying flux. This makes sense from (26), with the added observation that it is independent of the location at which the gate was positioned. Moreover, it arises naturally from the concept of the average flux, while explicitly accounting for the time-variation in a sensible way. Thus, ${A}_{kl}$ is an excellent flux measure, consistent with either the lobe dynamics and average flux approach, or the continuous-time approach used here, if ${c}_{kl}\ne c$.

**What if**${c}_{kl}=c$

**?**Then, the disturbance has no unsteadiness, and so an initial conclusion that can be reached is that flux function is independent of time. If it is a non-zero constant, this means that the stable and unstable manifolds do not intersect. This would allow for a channel to open up, and uni-directional flux will occur between the eddy and the jet (in one direction, depending on this sign of this constant). Uni-directional flux can certainly happen in incompressible geophysically-relevant flows, e.g., Kelvin-Stuart cats-eye flows [37,41], or in kinematical eddies [38], but in these situations there is dissipation due to diffusion or viscosity, and hence the flow is unsteady. This allows for incompressibility to be preserved in the system despite the uni-directional leaking of fluid (from inside the eddy, say), by the stable/unstable manifolds which bound the eddy getting closer and thereby enclosing lesser area as time progresses. In the case examined in this article under the condition ${c}_{kl}=c$, however, the full flow is also steady, and therefore having such a channel open up would violate incompressibility. Therefore, if ${c}_{kl}=c$, the fluid flux must be zero; that is, the stable and unstable manifolds, which may move from their initial locations, do so in such a fashion as to still coincide. This is the reason why the flux is stated as being zero in (25) when the disturbance wavenumbers obey ${k}_{1}^{\prime 2}+{l}_{1}^{\prime 2}={k}_{0}^{\prime 2}+{l}_{0}^{\prime 2}$.

**Time-periodicity is not necessary:**The theoretical approach due to Rom-Kedar et al. [49] was the first to provide a method for quantifying a flux in unsteady flows, due to the intersection of stable and unstable manifolds. This, however, relies on the velocity field being time-periodic, which allows for the definition of a Poincaré map P which simply strobes the flow at the period of the velocity [49,51,79]. Fixed points of P, and their stable and unstable manifolds, are then the focus. Their intersections creates lobes, and the impact of P on these lobes (i.e., the idea of lobe dynamics) gives insight into transport occurring over each time-period. In their classical picture, Rom-Kedar et al. [49] show that the area of a lobe can be used as a transport measure, and in their case the lobes all have equal areas to make this unambiguous. Considerably more details on this are available in the book by Wiggins [51], and geophysical applications are plentiful [3,20,45,46,47,53,80].

**Lobes are not necessary:**The classical approach [49,51] requires there to be lobes present, created through intersecting stable and unstable manifolds. Moreover, to use a lobe area as a measure of transport, it is necessary that these lobes have equal areas. In contrast, the time-varying approach detailed in Section 2.2 does not require the stable and unstable manifolds to intersect at all. If they do not intersect—which is a distinct possibility when including the effects of dissipation in geophysical flows [37,38,41])—there are no lobes at all, and hence no “lobe dynamics”. On the other hand, a lack of intersection makes perfect sense from the perspective of Figure 3: this implies a uni-directional flux, either from the eddy to the jet, or from the jet to the eddy. The instantaneous flux may change with time, but its sign does not.

**Compressibility:**The flux definition (16) is still legitimate even if the fluid were compressible. The issue is that if the fluid were compressible there would no longer be a stream function ψ that describes the velocity; however, the flux definition remains legitimate with the (compressible) velocity inserted instead [41]. Thus, the only change that needs to be done is the replacement of $-\partial \psi /\partial y$ with the zonal velocity, whatever it happens to be. However, its usage in the compressible case requires a generalization of the standard Melnikov function [41,87].

**The Melnikov function and flux:**A pleasing theoretical issue of the instantaneous flux was developed by Balasuriya [39,81]: this flux, as a time-varying quantity that specifically measures the amount of fluid that is transported across per unit time, can be characterized to leading-order in ε exactly by the Melnikov function (for fluid applications, see [38,44,46,47,49], for theory, see [41,51,79,88]). The argument of the Melnikov function relates precisely to time, and it is precisely this that appears in (25) and (26). Thus, unlike in the quantification of lobe areas which requires an integral of the Melnikov function [49,50,51], the Melnikov function by itself can be used to quantify the flux as a function of time [39,81]. The quantity ${M}_{kl}\left(t\right)$ presented in this article is exactly such a Melnikov function, as are the expressions (38) and (40) for non-sinusoidal situations. This works under general time-dependence and also compressibility [41].

## Appendix B. Derivation of the Formula for A kl

## Appendix C. Numerical Scheme for Obtaining Stable and Unstable Manifolds

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**Figure 1.**One-wave Rossby wave in the moving frame, with $-1<c<0$, where Γ (magenta curve) is the barrier between the chosen eddy and the jet.

**Figure 2.**The (

**a**) Poincaré map, and (

**b**) the time-continuous, approaches for the splitting of the eddy-jet barrier into stable (green) and unstable (red) manifolds when ${\psi}_{\mathrm{pert}}\ne 0$. The dashed blue line is Γ, the unambiguous barrier when ${\psi}_{\mathrm{pert}}=0$, and the Poincaré map is valid only under certain conditions, as explained in the text. (

**a**) Discrete (strobing) time; (

**b**) continuous time.

**Figure 3.**Intersection possibilities at three instances in time. (

**a**) Positive flux; (

**b**) zero flux; (

**c**) negative flux.

**Figure 4.**The instantaneous flux Function (26) for $l=1$, at several zonal wavenumber values: $k=4$ (blue), 6 (red) and 18 (yellow).

**Figure 5.**Dependence of amplitude ${A}_{kl}$ on the wavenumbers $(k,l)$, with the dashed blue curve corresponding to ${c}_{kl}=c$.

**Figure 6.**Zoom in to smaller wavenumbers in Figure 5.

**Figure 7.**Numerically computed stable (green) and unstable (red) manifolds at $t=0$ using $\epsilon =0.02$ and $c=-0.1868$. (

**a**) $(k,l)=(0.4,0.1)$, ${A}_{kl}=0.241$; (

**b**) $(k,l)=(6,1)$, ${A}_{kl}=0.698$; (

**c**) $(k,l)=(19,6)$, ${A}_{kl}=6.025$; (

**d**) $(k,l)=(28.8,0.79)$, ${A}_{kl}=12.766$.

**Figure 8.**Forward and backward FTLE fields at $t=0$ for identical parameter values as in Figure 7c,d, with the choice of flow-time $T=12$. (

**a**) $(k,l)=(19,6)$, forward; (

**b**) $(k,l)=(19,6)$, backward; (

**c**) $(k,l)=(28.8,0.79)$, forward; (

**d**) $(k,l)=(28.8,0.79)$, backward.

**Figure 9.**Changes to the flux amplitude when one of the parameters each in (37) is changed, with the the others kept fixed. (

**a**) ${\beta}^{\prime}\to {\beta}^{\prime}/10$; (

**b**) ${A}^{\prime}\to {A}^{\prime}/2$; (

**c**) ${k}_{0}^{\prime}\to 2{k}_{0}^{\prime}$; (

**d**) ${l}_{0}^{\prime}\to {l}_{0}^{\prime}/2$.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Balasuriya, S. Meridional and Zonal Wavenumber Dependence in Tracer Flux in Rossby Waves. *Fluids* **2016**, *1*, 30.
https://doi.org/10.3390/fluids1030030

**AMA Style**

Balasuriya S. Meridional and Zonal Wavenumber Dependence in Tracer Flux in Rossby Waves. *Fluids*. 2016; 1(3):30.
https://doi.org/10.3390/fluids1030030

**Chicago/Turabian Style**

Balasuriya, Sanjeeva. 2016. "Meridional and Zonal Wavenumber Dependence in Tracer Flux in Rossby Waves" *Fluids* 1, no. 3: 30.
https://doi.org/10.3390/fluids1030030