# Scalar Flux Kinematics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}and S

_{2}are connected by a streamline that separates the two members. If a moderately weak, time-periodic disturbance is added to the velocity field, S

_{1}and S

_{2}will survive in the form of moving hyperbolic trajectories H

_{1}and H

_{2}, for which the surrounding flow is, on average, convergent in one direction and divergent in another (Figure 1b). For example, the flow surrounding H

_{1}will tend to be convergent, on average, in the along-shore direction and divergent in the offshore direction. The positions of H

_{1}and H

_{2}will generally vary with time, but if the disturbance is not too large, both points will linger in the neighborhood of the original stagnations points. If one were to continuously introduce dye into the vicinity of H

_{1}, it would be attracted to H

_{1}in the along-shore direction, but would be repelled in the offshore direction forming a long streak. This streak would be an approximation of the unstable manifold (also called the “outgoing” manifold) consisting of material that diverges from H

_{1}(approaches H

_{1}in backward time). The unstable manifold of H

_{1}is depicted by red contour in Figure 1b. Similarly, stable and unstable manifolds can be defined for H

_{2}as material, time-dependent curves consisting of fluid elements that approach H

_{2}asymptotically in forward or backward time. All of the manifolds evolve over time and Figure 1b is just a snap shot, but if the time-dependence is periodic the shapes will recur periodically.

_{1}is also a stable manifold of S

_{2}and can be thought of as a boundary separating the cyclonic and anticyclonic members of the dipole. In the unsteady case, these manifolds are distinct, and it is common for the unstable manifold of H

_{1}to intersect the stable manifold of H

_{2}. One can identify lobes of fluid L

_{1}, L

_{2}whose boundaries consist of segments of the intersecting manifolds. As the flow evolves, the lobe L

_{1}will move from right to left, continuing to be bounded by the same sections of the material curves. If the time dependence is periodic, the shapes of the curves will recur after a single period, and the fluid in L

_{1}will have moved into the space depicted as L

_{2}in the figure. After another period the fluid will have moved to the space occupied by L

_{3}and so on, so there is clearly a transport from the cyclone to the anticyclone. In the same manner, fluid in lobe L

_{−1}evolves into L

_{−2}and so on, defining a transport process from the anticyclone to the cyclone. This mechanism leads to exchange between the two gyres, and the lobes are therefore called turnstile lobes. As implied in Figure 1, the lobes themselves tend to be continuously stretched and folded, so that property gradients within are amplified and irreversible property exchange through diffusion is enhanced. In time-periodic systems, the motion of fluid parcels is often described by a Poincare’ map, in which discrete parcel positions are plotted at the end of each period. In this description, the manifolds in Figure 1b remain frozen, fluid parcels remain on the fixed manifolds as they are mapped from one position to the next, and fluid blobs are mapped from one lobe to another. We prefer to think about the manifolds as time-continuous objects since this picture is relevant to the time aperiodic case, which can be similar in concept and geometry. An example from a numerical simulation of a strongly aperiodic, surface dipole in the Philippine Seas [9] appears in Figure 2.

_{H}is the horizontal eddy viscosity, H is the constant ocean depth, and

**u**is the horizontal velocity. The contour integrals are taken about the boundary $\partial R$ of a region R that contains the gyre. The integrals on the right-hand side of Equation (1) measure the advection of potential vorticity across $\partial R$, the tangential component of stress on $\partial R$, and the horizontal frictional stress acting along $\partial R$. The sum of these terms can cause the gyre to spin up or down, as measured by the rate of change of circulation (first term in the left-hand side), and/or cause the mean latitude y of the gyre to increase or decrease (second term in the left-hand side). By constructing a time-dependent boundary $\partial R(t)$ using pieces of stable and unstable manifolds, the authors were able to quantify the advection of q into and out of the gyre as a lobe exchange processes. The approach requires that one first computes the manifolds and lobes, then separately calculates the amount of area-integrated q for each lobe entering or leaving the gyre at irregular intervals.

## 2. Results

#### 2.1. Kinematics of a Residual Scalar Flux

**F**represents the flux of C. This form is quite general and, in fact, not unique, for one may have a choice whether to include specific terms in $\nabla \cdot F$ or assign them to S. For our purposes it will be advantageous to include in $\nabla \cdot F$ as many terms as possible. The flux vector

**F**will generally contain contributions from advection by the fluid velocity

**u**and from molecular or sub-grid scale diffusion. As we shall show, it may also contain certain types of forcing. It is not uniquely defined, as one can add to it any non-divergent vector. In many cases, the best choice of

**F**is determined by boundary conditions and/or by the considered formulation of the specific budget or science question that one wishes to address. Further discussion on this point follows below.

**F**can be nonuniquely represented as the product of an arbitrary scalar quantity γ(

**x**,t), with the same units as C, and a corresponding velocity

**u***(

**x**,t), so that

**u*** =

**F/**γ. For a given choice of γ, consider the trajectories

**x***(t;

**x***,t

_{o}_{o}) of the velocity field

**u***, as determined from

**u*** is divergent, however we will assume that

**u*** is sufficiently well behaved that the mapping of the initial condition $x*({t}_{o};{x}_{o}*,{t}_{o})={x}_{o}*$ to its destination $x*(t;{x}_{o}*,{t}_{o})$ at time t is unique and invertible.

_{γ}(t) enclosed in surface A

_{γ}(t) that is obtained by evolving a surface A

_{γ}(0) of initial conditions forward under Equation (3). Integration of Equation (2) over this volume then yields

**n**is the outward normal to A

_{γ}(t) and where Leibnitz’s rule is used in the first step. With the choice $\gamma =C$, this relation simplifies to

_{C}(t) remains conserved if its boundary A

_{C}(t) is advected with the velocity $u*=F/C$ and if S (or its integral over V

_{C}(t)) is zero. In this sense, the boundary acts as a “barrier” for the tracer in the same way that material contours act as barriers to material transport. However,

**u*** will generally be divergent and may be infinite (as in our first example below) so that finite-time invariant manifolds, hyperbolic and elliptic LCS and the like may not exist in the established sense.

**u**:

_{T}is considered constant. We have the choice of writing the right hand side as $-\nabla \cdot (F)$, where $F=uT-\kappa \nabla T$, or as $-\nabla \cdot (F)+S$, where $F=uT$ and $S={\kappa}_{T}{\nabla}^{2}T$. In first case $u*=u-\kappa \nabla T/T$ whereas in the second

**u*** =

**u**. The advantage of the fist choice is that a closed volume advected with the velocity

**u*** will conserve total heat.

_{e}is the horizontal eddy viscosity. The nondimensionalization and definition of the other scales can be found in [43]. Equation (7) attributes the rate of change of q to advection of q (second term on the left), vortex stretching due to Ekman pumping (first term on right) and bottom and lateral friction (final two terms). The equation may be rewritten in the form

**F**. The total amount of potential vorticity carried by a volume advected by the velocity field

**u*** =

**F**/q is thus conserved.

**u*** may be considerably more complex than the fluid velocity field. It then becomes desirable to consider smoothed versions of the velocity and tracer fields. We therefore consider a low-pass filter, taken over a window of length T:

**u**, usually assumed to be divergence free. Time averaging can be performed as above, resulting in the flux vector $\overline{F}=\overline{u}\overline{\mathsf{\rho}}+\overline{u\prime \mathsf{\rho}\prime}+\overline{u\prime \overline{\mathsf{\rho}}}+\overline{\overline{u}\mathsf{\rho}\prime}$ for the average mass flux (mean plus eddy), the residual velocity $u*=\overline{F}/\overline{\mathsf{\rho}}$ and statement $\frac{d}{dt}{\displaystyle {\int}_{{V}_{\overline{\mathsf{\rho}}}}\overline{\mathsf{\rho}}}dV=0$ indicating conservation of the total mean mass within a closed volume whose boundary is advected by $u*$. (The time-average version has apparently not been used within the LCS community).

#### 2.2. Diffusive Transport of Momentum, Vorticity, Energy and Enstrophy in Viscous, Laminar Flow

^{1/2}. One may now trace any contour y* = y*(x*,t) that evolves under Equation (21), an obvious first choice being the parallel contours y* = at

^{1/2}parameterized by a, with $0<\mathsf{\alpha}<\infty $. Note that the movement of the contours is independent of viscosity. All such contours initially lie at boundary (y = 0), but they explode into the fluid at t = 0

^{+}and migrate towards positive y thereafter (Figure 10). The infinite speed of the leading contour is simply a reflection of the property of solutions to the parabolic equation, in which a sudden change at a boundary has an immediate impact at all finite distances from the boundary. The distance between any two contours a

_{1}and a

_{2}grows in proportion to t

^{1/2}while the total amount of vorticity between the two remains constant. Thus, the vorticity moves away from the boundary and is diluted as it does so. Alternatively, one could have chosen a more complex contour, perhaps a circle, and seen that it becomes elongated in the y-direction as t increases.

^{2}/2 moves into the fluid. Multiplication of Equation (20) by ζ results, after some rearrangement, in

^{2}/2, resulting in the observation that energy moves twice as fast as momentum but is subject to a dissipation that momentum does not feel. In addition, v* < 0 for both scalars, so that the contours of zero flux move towards the boundary (which acts both as a sink of momentum and energy, but a source of vorticity and enstrophy).

#### 2.3. A Turbulent Model of the Antarctic Circumpolar Current

_{y}. The flow is spun up from rest through the application of a wind stress in the positive x-direction proportional to sin

^{2}(πy/L

_{y}). There is no buoyancy forcing and a linear drag is imposed at the bottom. The westerly surface winds create southward Ekman transport, leading to downwelling near the south wall (and upwelling at the north wall). The resulting isothermal tilt gives rise to a geostrophically balanced jet that becomes baroclinically unstable. In the simulation shown in Figure 11, the instability has equilibrated, and a statistically steady state is reached in which the dominant features are an eastward propagating meander and an accompanying anticyclonic eddy (lower left). Smaller coherent eddies can also be seen in both the southern and northern parts of the channel.

_{o})/ρ. Note that buoyancy flux is essentially mass flux, so we are close in some sense to the volume flux calculations that are traditionally considered in lobe analysis.

^{(8)}and κ

^{(4)}are the horizontal and vertical hyper-diffusivities, respectively.

**u**〉 = 〈u〉

**i**+ 〈v〉

**j**and the residual velocity field $\langle \overline{u}*\rangle =\langle \overline{F}\rangle /\langle \overline{b}\rangle $ computed using a Gaussian, low pass filter with a window T = 3.7 days. Snapshots of the forward FTLE fields for $\langle \overline{u}*\rangle $ and for 〈

**u**〉 are shown in Figure 12a,b. The maximizing FTLE ridges are suggestive of barriers associated with the hyperbolic-type LCSs or finite-time manifolds, whereas troughs in the FTLE fields at mid-jet (Figure 12a,c) are suggestive of barriers associated with the parabolic-type LCSs or shearless trajectories. These properties are backed up by independent calculations discussed below. Figure 12a contains information about the depth-averaged transport of mean (time-average) buoyancy whereas Figure 12b contains information about the depth-averaged transport of material. Note that although the anticyclonic eddy is apparent in the lower left corner of each panel (and more prominently in the 〈

**u**〉 fields) the meandering jet is clear only in the panels showing $\langle \overline{u}*\rangle $. Cross jet material transport is therefore clearly indicated in the 〈

**u**〉 fields, but the $\langle \overline{u}*\rangle $ fields suggest that the jet is a barrier to mean buoyancy transport. Thus, if Lagrangian drifters were released on one side of the jet, some would cross to the other side. However, the FTLE troughs of the depth–integrated time-averaged residual velocity $\langle \overline{u}*\rangle $ suggest that this parcel dispersion does not result in any residual transport of mean depth-averaged buoyancy: the associated “eddy” flux is balanced by the other terms in the residual flux vector, including the contribution from the mean cross-channel circulation and the explicit diffusion and hyper-diffusion terms. This cancellation is an expected consequence of the channel walls. It is also believed to take place in the Antarctic Circumpolar Current, where the wind and buoyancy forcing give rise to a mean meridional circulation that tends to steepen isopycnals, whereas baroclinic instability generates an eddy flux that tends to flatten isopycnals.

^{5}and y = 0.5 × 10

^{5}and the suggestion of a heteroclinic structure surrounding the anticyclonic gyre. In addition to looking at FTLE ridges, the finite-time manifolds emanating from the hyperbolic region have also been computed using a direct method, where they were “grown” from sets of initial conditions surrounding the hyperbolic core region. These are shown as color curves in Figure 12d, with blue/red representing stable/unstable directions. The red and blue curves fall right on top of FTLE ridges if the two images are superimposed. This calculation insures the attractive property of the FTLE ridges and confirms that ridges are hyperbolic in nature rather than associated with shear. The lobes formed by the intersections of the curves carry mean buoyancy into and out of the anticyclonic eddy. The exchange appears to be confined to the outer edges, and there is no evidence of penetration into the eddy center.

## 3. Materials and Methods

## 4. Discussion and Conclusions

**F**.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic of exchange in a dipole circulation. (

**a**) Here the flow is steady and there is no material exchange between the two cells; (

**b**) The flow is time-dependent and the gyres are no longer separated by a material boundary. Instead blue lobes (L

_{1}, L

_{2}, etc.) of material move left to right and are transported from the anticyclone to the cyclone. The uncolored lobes L

_{−1}, L

_{−2}, etc. are transported to the anticyclone from the cyclone. The solid arrows show the general direction of circulation. Dashed contours schematically indicate the outermost material contour marking the extent of the regular region within each gyre. (From [9]

^{©}American Meteorlogical Society. Used with permission.).

**Figure 2.**Philippine Seas dipole with the geometry resembling the idealized example shown in Figure 1. Red and blue curves are finite-time analogs of the stable and unstable manifolds computed for a realistic numerical simulation of the Philippine Seas. The red curves emanate from hyperbolic regions close to the coastlines of a peninsula and 2 islands near x = 620. The blue curves emanate from an offshore hyperbolic region not shown, and from hyperbolic regions near the coasts. (From [9]

^{©}American Meteorlogical Society. Used with permission.).

**Figure 3.**The finite volume lobes formed by intersecting manifolds in a 3D simulation of a Gulf of Mexico eddy. (Reprinted from [18], Copyright 2010, with permission from Elsevier.).

**Figure 4.**Forward and backward finite-time Lyapunov exponents (FTLEs) for the surface circulation according to a numerical simulation of the Philippine Seas. (From [9]

^{©}American Meteorlogical Society. Used with permission.).

**Figure 5.**The red filamented surface material in (

**a**) coalesces to form an eddy in (

**b**) that remains coherent as it travels westward across the middle portion of the S. Atlantic (

**c**); The distance covered between frames (

**b**) and (

**c**) is approximately 4000 km. (Courtesy of F. J. Beron-Vera, also see [4]).

**Figure 6.**The yellow surface is the boundary of a mesoscale eddy computed using the technique of [15] and based on the Southern Ocean state estimation model [32]. The boundary was found to remain coherent over at least 120 days. (From [15], reproduced with permission from Cambridge University Press.).

**Figure 7.**(

**a**) Schematic of the rotating cylinder flow, with the homogeneous fluid in the cylinder driven by a positive differential rotation of the lid. This forcing results in upwelling into a surface Ekman Layer, downwelling in sidewall layers, and collection and expulsion of trajectories in a bottom Ekman layer. Hypebolic stagnation points exist at the center of the bottom and top lids; (

**b**) A chaotic trajectory (red) and two material barriers (green and blue, both tori) that arise when the lid stress is moved slightly off center. (All from [34], reproduced with permission from Cambridge University Press.).

**Figure 8.**Turnstile lobe exchange of fluid in and out of a boundary trapped gyre in a single-layer, wind-driven gyre on a β-plane. The blue lobe contains fluid transported from the interior to the exterior of the gyre, while the red lobe contains fluid that is being transported into the gyre. Both are responsible for carrying potential vorticity into and out of the gyre, which can cause the gyre to spin up or spin down. (From [16]

^{©}American Meteorlogical Society. Used with permission.).

**Figure 9.**Snapshot of forward-time FTLEs on 6 January 1993 based on surface velocities from the time-dependent AVISO satellite altimetry superposed on the drifter-based mean flow. The ridges ostensibly approximate finite-time unstable manifolds or hyperbolic-type LCS. Backward-time FTLEs (not shown) exhibit the same level of complexity. (From [39]

^{©}American Meteorlogical Society. Used with permission.).

**Figure 10.**Contours corresponding to different values of a, and at two different times. The total vorticity between two contours of different a remains fixed. As time progresses, the contours move away from the lower boundary, the distance between contours increases, and the vorticity is diluted.

**Figure 11.**The geometry, forcing and main circulation features from the zonal channel model. The color shows the vertical component of fluid vorticity, normalized by f. (Based on the model runs presented in [46]).

**Figure 12.**(

**a**) Forward FTLE fields for $\langle \overline{u}*\rangle $; (

**b**) Forward FTLE fields for 〈

**u**〉; (

**c**) Forward and backward FTLE fields for $\langle \overline{u}*\rangle $; (

**d**) Stable (blue) and unstable (red) manifolds grown from the hyperbolic region near x = 1.5 × 10

^{5}for $\langle \overline{u}*\rangle $ .

**Figure 13.**The light green and pink contours, initially aligned along the channel, are advected by $\langle \overline{u}*\rangle $ for 71 days, resulting in the bright green and red dotted sets. No cross-jet transport of mean buoyancy occurs.

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Pratt, L.; Barkan, R.; Rypina, I.
Scalar Flux Kinematics. *Fluids* **2016**, *1*, 27.
https://doi.org/10.3390/fluids1030027

**AMA Style**

Pratt L, Barkan R, Rypina I.
Scalar Flux Kinematics. *Fluids*. 2016; 1(3):27.
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**Chicago/Turabian Style**

Pratt, Larry, Roy Barkan, and Irina Rypina.
2016. "Scalar Flux Kinematics" *Fluids* 1, no. 3: 27.
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