Numerical investigation of co- and counter-propagating internal wave-vortex interactions.
Internal waves play an integral role in oceanic and atmospheric dynamics by affecting global circulation patterns, maintaining environmental energy budgets, and acting as a source of turbulence and mixing. Accurate estimates of the generation, propagation, and evolution of internal waves through the ocean and atmosphere are necessary to characterize their affects on larger scale dynamics. Common situations include internal wave propagation through a mean current, spatially varying current or shear [16, 25, 26, 1], vortex or vortices [8, 12, 17], and other waves [20, 19, 18, 3, 4, 5, 6, 23, 24, 7]. As internal waves propagate through these fields they can refract and redistribute energy to different scales or break causing irreversible mixing.
Mesoscale horizontally rotating vortices are common in rotating fluids, such as the ocean and atmosphere. Often these vortices are vertically constrained due to the stable stratification of the medium. Propagation of internal waves through these vortices can result in spectral spreading of wave energy, enhancing the vortical modes, or wave breaking [14, 11]. Moulin and Flor  used three-dimensional ray-tracing of interactions between large-scale internal waves and a Rankine-type vortex having a Gaussian vertical distribution of vertical vorticity and found relatively weak vortices caused wave refraction while relatively strong vortices trapped some waves in the rotating motion of the vortex. The aspect ratio (a three-dimenionsional feature of the vortex) and the strength of the Doppler shift measured by the horizontal wavenumber multiplied by the Froude number, kFr, are used to determine where wave trapping occurs and for which waves in the wave field. Regardless of wave refraction or trapping, wave propagation after the vortex was not limited to a single horizontal plane.
[FIGURE 1 OMITTED]
Godoy-Diana et al. explored internal wave beams interacting with a horizontal Lamb-Chaplygin pancake vortex dipole bounded vertically by a Gaussian, as can be seen in figure 1. In one experimental case, beams of internal waves propagating in the same horizontal direction as the translation of the vortex dipole (co-propagating) were observed bending to the horizontal and the wave energy was presumably absorbed by the dipole at a critical level. However when internal waves propagated horizontally opposite to the direction of dipole translation (counter-propagating) they were reflected vertically at a turning point. These phenomena both occurred in the vertical symmetry plane separating the dipole's counter-rotating vortices. Outside of this plane, the horizontal structure of the dipole seemed to generally cause spanwise divergence, or defocusing, of the internal wave beam in co-propagating interactions and spanwise convergence, or focusing, in counter-propagating interactions. Though the experiment was fully three dimensional, the observations were two dimensional because the viewing techniques were limited to two orthogonal planes: the vertical symmetry plane and the horizontal midplane of the dipole.
This paper considers the three-dimensional nature of the co- and counter-propagating interactions of internal waves with a Lamb-Chaplygin pancake vortex dipole of constant rotation and translation in a linearly-stratified fluid, as was experimentally investigated by Godoy-Diana et al. . We utilize ray theory to explore further the dynamics of the interaction of an internal wave with a vortex dipole, including the fate of off-center rays and current estimates of critical levels and turning points. These are three-dimensional thus increasing comprehension of results found in plane through the experiments of Godoy-Diana et al.. Section 2 reviews the experimental setup of Godoy-Diana et al.  and provides the numerical setup of the current study, including ray theory and details of the Lamb-Chaplygin pancake vortex dipole. Sections 3 and 4 present and compare the results of three-dimensionally ray tracing co- and counter-propagating internal wave-vortex dipole interactions, respectively. Section 5 closes the paper with a discussion of the results and main conclusions.
2.1 Review of Experimental Setup
The experiments by Godoy-Diana et al.  completed the aforementioned experiments in a tank of linearly stratified saline solution. An overview of their setup is described here as the numerical ray tracing will be similar. A columnar Lamb-Chaplygin vortex dipole was generated at one end of the tank by closing two flaps that spanned the tank's depth. The view was set such that the dipole translated right to left, as shown in Fig. 1 (as can be seen also in Fig. 1 and 2 of Godoy-Diana et al. ). Downstream of the newly generated dipole a thin screen prevented all but a horizontal slice of the columnar dipole to pass into the interaction region of the tank and overlaid the pancake dipole with an approximately Gaussian vertical distribution of velocity. The top half of this velocity profile is sketched in Fig. 1.
Above the interaction region of the experiments and spanning the width of the tank, a horizontal cylinder was oscillated to generate beams of internal waves. The translating dipole then intersected a beam, initiating an interaction. The co-propagating wave beam (moving left) may be absorbed into the dipole jet at a critical level at depth [z.sub.c] (where the frequency of the internal wave approches zero) and the counter-propagating wave beam (moving right) may be reflected vertically away at a turning point at depth [z.sub.T] (where the frequency of the internal wave approaches the buoyancy frequency, N).
2.2 Review of Ray Theory
Ray theory is a linear theory that traces in time and space the propagation of internal wave energy by assuming an internal wave is a point convected along a path, or ray . Though the solution is not representative of all wave-vortex interactions, under the Wentzel-Kramer-Brillouin-Jeffreys and Boussinesq approximations it is realistic when slowly-varying internal waves interact with larger-scale background flows that are assumed unaffected by the interaction. The method used to check validity of the slowly varying approximation within the WKBJ approximation is incomplete for multiple dimensions and does not validate every ray in the internal wave-vortex dipole interactions of this study (See Appendix 6 for more details). Nevertheless, even those interactions which do not obey this assumption generally follow the trajectory patterns observed experimentally by Godoy-Diana et al.  and the horizontal and temporal scale separation is generally adequate.
The following is a review of ray theory. For a more extensive description, see Lighthill . The dispersion relation
[[omega].sup.2.sub.r] = [N.sup.2] ([k.sup.2] + [l.sup.2]) + [f.sup.2][m.sup.2] / [k.sup.2] + [l.sup.2] + [m.sup.2] (1)
defines the relative frequency, [[omega].sub.r], of the internal waves in a frame of reference moving with the wave energy. Where k, l, and m are the two horizontal and single vertical components of the wavenumber vector k , N [greater than or equal to] [[omega].sub.r] is the buoyancy frequency of the fluid, and f [less than or equal to] [[omega].sub.r] is the system's frequency of rotation (e.g., the Coriolis frequency in geophysical flows). In the case of a non-rotating system, as in the study for this paper, f = [0s.sup.-1].
As it is often inconvenient to track the interaction of internal waves in the frame of reference of the internal waves, the Doppler relation shifts the frame of reference to a stationary one as follows:
[OMEGA] = k x V + [[omega].sub.r] (2)
where [OMEGA] is the total frequency of the internal waves and V = (U,V,W) is the background velocity. In this frame of reference [OMEGA] is approximately constant.
To determine the total velocity of the energy propagation of the internal wave along a ray, the group velocity, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is defined by
dx / dt = [partial derivative][omega] / [partial derivative]k = V + [[nabla].sub.k][[omega].sup.r] (3)
for which x = (x, y, z) defines the domain space and [[nabla].sub.k] = ([partial derivative]/ [partial derivative]k, [partial derivative]/[partial derivative]l, [partial derivative]/[partial derivative]m) is the spectral gradient operator. Equation 3 and the law governing wave refraction,
dk / dt = - [partial derivative][OMEGA]/[partial derivative][OMEGA] = - k x [nabla]V - [nabla] [[omega].sub.r] (4)
are used to calculate the ray path and wavenumber along the ray. Changes of the relative frequency with respect to time may be calculated by combining Equations 3 and 4 or by using Equation 1 after calcuating all other parameters such that the change in frequency over time is
[d[omega].sub.r] / dt = dk / dt x [[nabla].sub.k][[omega].sub.r] + dx / dt x [nabla][[omega].sub.r] (5)
Due to conservation of wave action (A= E/[[omega].sub.r]) along a ray, changes in the total energy of a wave packet, assuming the energy is not a function of the volume of the wave, can also be tracked through the ratio
E / [E.sub.0] = A / [A.sub.0] [[omega].sub.r] / [[omega].sub.r,0] = [[omega].sub.r] / [[omega].sub.r,0] (6)
2.3 Numerical Setup of Lamb-Chaplygin Vortex Dipole
Three nondimensional control parameters are used to define the vortex dipole: the horizontal Froude number [Fr.sub.h] = [U.sub.0]/NR, where [U.sub.0] is the translational speed of the dipole and R is the dipole radius; the Reynolds number Re = [R.sup.2][NFr.sub.h] /v , where v is the kinematic viscosity of the fluid; and the aspect ratio [alpha] = [L.sub.v]/R, where [L.sub.v] is the vertical length scale of the dipole defined by a Gaussian envelope. The values for all of these constants for each of the interaction cases are given in Table 2.3 (the value for the kinematic viscosity is assumed to be that of salt water and an experimentally- conducive value is assumed for buoyancy frequency).
Dipole Parameters and Properties
Re [Fr.sub.h] [alpha] R(cm) co-propagating 182 0.18 1.27 5.03 counter-propagating 131 0.06 0.40 7.40 [L.sub.v](cm) [U.sub.0](cm/s) N ([s.sup.-1]) co-propagating 6.39 0.4 0.447 counter-propagating 2.96 0.2 0.447 v([m.sup.2]/s) co-propagating 1.12[e.sup.-6] counter-propagating 1.12[e.sup.-6]
The mechanics of the Lamb-Chaplygin vortex dipole were thoroughly described by Chaplygin , in which the equation of motion is the piecewise stream function in polar-coordinate notation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where [J.sub.0] and [J.sub.1] are, respectively, the zero- and first-order Bessel functions of the first kind; and [[mu].sub.1] =3.8317 is the first zero of [J.sub.1]. The sign in each piece of the function enables right-to-left translation of the numerically-simulated dipole in a rightward-positive coordinate system to facilitate comparisons with Godoy-Diana et al. . The vertical velocity profile is numerically simulated as the product of the dipole's local velocity U(r,6) and an exact Gaussian curve [e.sup.[-(z/[L.sub.v]).sup.2]], where z is the domain's range of depth.
[FIGURE 2 OMITTED]
Fig. 2 displays the local horizontal divergence and shear of the co-propagating dipole at the horizontal midplane as light (positive) and dark (negative) contours. The contours in Fig. 1 have a range of about [+ or -] 0.6m/[s.sup.2] and the others have an approximate range of [+ or -] 0.25m/[s.sup.2] (the difference was for aesthetic purposes and so that the contours are visible in each frame). The overlain velocity vectors correspond to the dipole velocity at the horizontal midplane (where the vorticity ranges from [+ or -] 0.8[s.sup.-1]). Note the strongest velocities, shown by the longest vectors, are found where the vortices merge to form the jet of the dipole in the vertical symmetry plane (i.e., y = 0m). Though velocity and divergence or shear magnitudes are different for the dipole in the counter-propagating interactions, the flow is visually identical and Fig. 2 can be referenced in discussions specific to both co- and counter-propagating interactions.
3 Co-Propagating Results
This section is divided into two subsections. First, ray tracing results are presented for co-propagating internal wave interactions with the dipole jet in the vertical symmetry plane. These results are compared and contrasted with the experimental results and with an internal wave interacting with a two-dimensional steady shear flow. Second, ray tracing results are presented for the co-propagating interactions away from the vertical symmetry plane. The increased qualitative and quantitative awareness offered by the three-dimensional ray tracing assists in explaining certain phenomena not clearly seen or understood by the experiments alone, including focusing and defocusing.
3.1 Internal Wave-Dipole Jet Interactions
The series of two-dimensional images in Fig. 9 of Godoy-Diana et al.  shows the evolution of co-propagating internal wave-vortex dipole interactions observed in the vertical symmetry plane during the experiment. Above and to the right of the images, a horizontal cylinder was oscillated to generate the wave beams seen in the figure as diagonal lines. The direction of the wave energy propagation is along the beams, away from the cylinder. The black and white shading within each beam shows the phases of the internal waves. After a copropagating interaction with the dipole jet, some of the wave beams bend to the horizontal and at a later time have disappeared entirely from view, their energy absorbed by the jet.
The initial conditions of the experiment were used in the ray tracing. The initial relative frequency was [[omega].sub.r],0 = 0.2[s.sup.-1]. Godoy-Diana et al.  provided a spectrum for the initial streamwise horizontal wavenumber used. A value near the end of this spectrum, [k.sub.0] = 60[m.sup.-1], provided numerical results most similar to those of the experiment. Because the wave beams did not initially propagate in the spanwise direction, the initial horizontal spanwise wavenumber is [l.sub.0] = 0[m.sup.-1]. The initial vertical wavenumber is calculated from Equation 1 to be approximately [m.sub.0] =120[m.sup.-1]. The components of the wavenumber vector are reevaluated at each time step and Equation 5 is integrated to solve for [[omega].sub.r].
Out of a range of results qualitatively similar to those shown in Fig. 9 of Godoy-Diana et al. , two cases were selected because of the insight they offer into the mechanics of internal wave-dipole jet interactions. These ray paths are both shown in Fig. 3. The velocity profile used the translational speed of the dipole to define the steady shear and is displayed at [10.sup.3] times its magnitude to show its relative shape and position. The ray of the first case, represented by the heavy dashed line, is inititated at x = 0.7m (note the dipole was initiated at x = 1m). This interaction occurs at the front of the dipole and the dipole will later overtake the wave packet in the x-direction. For that reason, this case is hereafter referenced as the "Dipole Front" interaction. Like the wave beams in Fig. 9 of Godoy-Diana et al. , the direction of wave propagation bends to the horizontal near the end of the fourteenth buoyancy period. The wave energy remains at the front of the dipole jet at an approximate depth of z = 0.041m for the remainder of the simulation. The group speed is then the same as the dipole's translational speed.
[FIGURE 3 OMITTED]
The second interaction, represented by the dotted line in Fig. 3, is initiated at x = 0.85m. This packet enters the dipole from behind and interacts at the rear; thus, the interaction is hereafter referenced as the "Dipole Rear" interaction. A brief refraction decreases the vertical propagation of the packet, accelerating it to the front of the dipole. Then, before the vertical propagation reaches zero, a second refraction causes the packet to descend again, though not as steeply as during the initial approach to the interaction. The packet descends deeper into the dipole until refracting a third time, bending to the horizontal like the "Dipole Front" ray and in the experiment. Reaching a depth of about z = 0.028m, the wave packet remains at the front of the dipole until the end of the simulation, its group speed equal to the translational speed of the dipole.
In bending to the horizontal, these ray tracings resemble those of internal waves approaching critical levels. Such an interaction has been traced through a two-dimensional steady shear (in the form of the same Gaussian as for the dipole) and is labeled "2D Steady Shear" in Fig. 3. This ray approaches a critical level located at a depth of about z = 0.036m above the horizontal midplane of the shear profile. Although both the Dipole Front and Rear rays reach nearly the same critical level location, they are dynamically different situations.
Because V = 0m/s in the dipole jet and W = 0m/s everywhere in the domain, Equation 4 takes on a two-dimensional simplification, which allows direct comparison with the results of the "2D Steady Shear" ray:
dk / dt = - k [partial derivative]U / [partial derivative]x
dl / dt = 0
dm / dt = - k [partial derivative]U / [partial derivative]z (8)
where [partial derivative]U/[partial derivative]x is the streamwise divergence in the streamwise direction (see Fig. 1) and [partial derivative]U/[partial derivative]z is the vertical shear induced by the Gaussian envelope. Equation 1 also simplifies for two-dimensional flow:
[[omega].sup.2.sub.r] = [N.sup.2][k.sup.2] / [k.sup.2] + [m.sup.2] (9)
The time evolution of the relative frequency normalized by its initial value is given in Fig. 4. The corresponding non-dimensional wavenumbers are shown in Fig. 5a and 5b with only the first 20 buoyancy periods shown, after which the wavenumbers approach infinity. The local divergence and shear experienced by the wave packets during their respective interactions can be seen in Fig. 2. These aid in understanding the reasons for the changes to the wave properties according to the refraction equation.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The relative frequency of the "2D Steady Shear" ray shown in Fig. 4 steadily and asymptotically approaches zero, as is expected for a critical level. Consideration of Equation 8 shows that k remains constant because [partial derivative]U/[partial derivative]x = 0 in a non-accelerating flow. So m approaches infinity due entirely to an increasing [partial derivative]U/[partial derivative]z and causes the decrease of [[omega].sub.r] according to Equation 9.
Per Equation 8, [partial derivative]U/[partial derivative]x is the only divergence term affecting any wave refraction for these interactions with the dipole jet. As seen in Fig. 1, [partial derivative]U/[partial derivative]x is strongly negative in the region where the "Dipole Front" ray enters the dipole because energy of the dipole jet is transitioning to spanwise motions (i.e., |V| increases as U decreases), causing k to increase in magnitude after time t/[T.sub.N] =6.26 and resulting in [[omega].sub.r] increasing. Because k grows negative and [partial derivative]U/[partial derivative]z increases positive, m increases after time t/[T.sub.N] = 8.39. These property changes lead to the strong refraction seen in Fig. 3 at t/[T.sub.N] =13.80 .
As each wavenumber increases exponentially, the numerator and denominator of Equation 9 balance in such a way that the relative frequency plateaus at [[omega].sub.r] / [[omega].sub.r,0] = 2.12 , short of reaching the value of the fluid's buoyancy frequency. This unbounded growth of k and subsequent finite plateauing of [[omega].sub.r] lead to wave capture [8, 9]. During wave capture, the group velocity of the wave packet approaches the translational speed of the dipole and is literally caught into and pushed along by the flow.
According to Fig. 1, the "Dipole Rear" ray first experiences a positive [partial derivative]U/[partial derivative]x, which causes the magnitude of k to decrease, causing the initial descent of [[omega].sub.r]. At this point, the ray appears to approach a critical level much like the "2D Steady Shear" ray.
However, the positive [partial derivative]U/[partial derivative]x experienced by the ray while it is at the rear of the dipole accelerates it to the front where it experiences a negative [partial derivative]U/[partial derivative]x as did the "Dipole Front" ray. During this transition, and after about t/[T.sub.N] =7.26, the vertical wavenumber increases slowly because [partial derivative]U/[partial derivative]z is nearly zero, though positive. The vertical wavenumber only reaches 9% of its initial value when |k| and [[omega].sub.r] each reach a minimum at t/[T.sub.N] = 14.73 .
At this same time, the second of three refractions causes |k| to grow quickly to infinity, driving [[omega].sub.r] to more than double its initial value. However, as the horizontal wavenumber exceeds its initial value, the vertical wavenumber approaches infinity and balances the effects of the rapidly increasing k such that the relative frequency plateaus at [[omega].sub.r]/[[omega].sub.r] = 2.16, just greater than the final relative frequency of the "Dipole Front" ray but still less than the fluid's buoyancy frequency. This last refraction causes the same effect of wave capture as the "Dipole Front" ray. Because the magnitudes of k and m approach infinity during these wave interactions with the dipole jet, the horizontal and vertical wavelengths decrease to zero and the waves have become a part of the background.
3.2 Internal Wave-Vortex Dipole Interactions
[FIGURE 6 OMITTED]
Rays interacting with the dipole outside of the vertical symmetry plane experience spanwise shear and divergence due to the dipole and so refract in all three dimensions. Fig. 5 and 5 show the "Dipole Front" and "Dipole Rear" interactions, respectively, for all time of the simulation and on the positive side of the vertical symmetry plane. Each plot shows the outermost rays nearly outside the influence of the vortices. Rays were initiated at the same position in x and z , given the same initial wave properties, and were simulated for the same duration of time as were the rays of the respective dipole jet interactions presented in Section 3.1. The spanwise velocity profiles of U and V as they exist in the horizontal midplane and across the cores of the vortices are displayed at [10.sup.3] times their magnitudes to give their positions and shapes relative to the rays. Spanwise defocusing of the wave energy was seen in the co-propagating experiments of Godoy-Diana et al. . Although the vertical velocity is generally consistent before and after the interaction, except for those rays closest to the center of the dipole which increase in vertical velocity, the co-propagating ray tracings here show generally the same results, but with some variation.
During the "Dipole Rear" interactions, the off-center rays principally diverge as expected based on the co-propagating experiment. But the ray tracing clearly shows that the spanwise refractions during the "Dipole Front" interactions vary depending on the initial position relative to the dipole. Rays initiated at [y.sub.0] > 0.11m, which seems to be just outside the reach of the strong rotational motion of the dipole, actually focus while those initiated at [y.sub.0] [less than or equal to] 0.11m experience multiple spanwise refractions before ultimately defocusing. The available experimental data is given within a domain width spanning y [approximately equal to] [+ or -]0.1275m, so the outermost wave-vortex interactions and their eventual focusing in a different plane may not have been easily observed in the experiment. Also, the data collected from a single horizontal plane made it impossible to see the detail of multiple refractions as they do not occur in a single plane.
[FIGURE 7 OMITTED]
In a three-dimensional, non-rotational frame of reference equation 1 simplifies to
[[omega].sup.2.sub.r] = [N.sup.2] ([k.sup.2] + [l.sup.2]) / [k.sup.2] + [l.sup.2] + [m.sup.2] (10)
Fig. 7 shows the evolution of [[omega].sub.r] / [[omega].sub.r,0] for each of the off-center rays (as well as that of the center ray for comparison) during the "Dipole Front" (6) and "Dipole Rear" (6) interactions. N/[[omega].sub.r,0] is again plotted for reference. Due to the spanwise symmetry, each of the relative frequency curves associated with an off-center ray corresponds to the curve with the same symbol as in either Fig. 5 or Fig. 5. Each of the "Dipole Front" rays experience similar changes to relative frequency, most of them decreasing early on while being affected by the rotational motion of the dipole, then increasing as the acceleration of the dipole dominates and finishing the simulation at a higher final relative frequency than initial regardless of whether the ray exited the interaction converging or diverging. The relative frequency of the "Dipole Rear" rays likewise increases, but never decreases as the rotational motion counterbalances the expected decrease in frequency associated with the dipole jet region. Equation 6 shows that this increase in frequency directly relates to an increase in energy and waves interacting with a co-propagating vortex dipole will increase in energy. However all waves outside of the dipole jet reach lower final frequencies, resulting in less energy gain. This may be expected due to the slower velocities outside this region.
Changes to wavenumbers of off-center rays are explained by the following simplification to Equation 4:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [partial derivative]V / [partial derivative]x is the spanwise shear in the streamwise direction (see Fig. 1), [partial derivative]U / [partial derivative]y is the streamwise shear in the spanwise direction (see Fig. 1), [partial derivative]V / [partial derivative]y is the spanwise divergence in the spanwise direction (see Fig. 1), and V/ z is the spanwise shear with respect to depth. To give an idea of what a single ray in each of these off-center interactions experiences in three dimensions, dipole divergence and shear components of the rays initiated at [y.sub.0] = 0.05m (open circle symbols) are shown in Fig. 8. Resultant changes to wavenumber are not shown, but as can be seen by the magnitude of the divergence and shear, the rays which encounter the highest vorticity regions will have the largest changes in wavenumber. This is due to the rays experiencing multiple spanwise refractions and closely following streamlines of a vortex in the frame of reference of the translating dipole. That is, they circle the core of the vortex until they escape the interaction, not unlike the wave trapping researched by Moulin and Flor  mentioned in Section 1. However here a calculation of the local k[theta], as was discussed by Moulin and Flor , does not increase by an order of magnitude as would be expected for the stricter definition of wave trapping.
[FIGURE 8 OMITTED]
4 Counter-Propagating Results
As was done in Section 3, this section is divided into two subsections. The first details ray tracings in the vertical symmetry plane of counter-propagating internal wavevortex dipole interactions. A range of interactions are discussed, including wave capture and turning points. The second subsection focuses on rays traced outside the vertical symmetry plane and the corresponding focusing and defocusing.
4.1 Internal Wave-Dipole Jet Interactions
A turning point was found by Godoy-Diana et al.  when an internal wave beam of initial relative frequency [[omega].sub.r,0] = 0.2[s.sup.-1] propagated counter to the directon of vortex dipole translation. Fig. 7a of Godoy-Diana et al.  provides a series of images from the experiment of this interaction in the vertical symmetry plane of the dipole. The internal wave is propagating downward from the top left and turning up-right, but with some energy emerging at approximately the same frequency below the dipole.
[FIGURE 9 OMITTED]
The results from the experiment are numerically replicated using a range of inital positions. Initiated at [z.sub.0] = 0.3m and every five centimeters in the range [x.sub.0] = [-0.1,0.1]m, the five rays in Fig. 8 demonstrate virtually the complete spectrum of possible results within the vertical symmetry plane. In addition a sixth ray, labeled "2D Steady Shear" demonstrates a turning point in a steady shear and contrasts the turning point of the ray initiated at [x.sub.0] = 0.05m. Though this spectrum of results can also occur by varying the initial relative frequency or wavenumber vector, the initial wave properties used here are the same as for the co-propagating interactions discussed above. The principal reason for doing this is that ray theory only realizes a turning point in the dipole jet for [k.sub.0] [greater than or equal to] 60[m.sup.-1] when [[omega].sub.r,0] = 0.2[s.sup.-1] .
[FIGURE 10 OMITTED]
The ray initiated at [x.sub.0] = 0.05m exhibits a turning point as its path loops in space over time. This loop is consistent with turning points that occur in a shear flow, as described by Sutherland . Changes to relative frequency for the rays interacting with the dipole jet are displayed in Fig. 8. Here, it is confirmed that the turning point occurs at approximately t/[T.sub.N] =15 when the relative frequency approaches the buoyancy frequency. By this time, the dipole jet has already horizontally turned the wave. However, the horizontal wavenumber does not change sign (see Fig. 9), so the leftward movement is entirely due to the dipole translating the wave. Also, the horizontal and vertical wavenumbers (shown in Fig. 10) now quickly increase in magnitude much like during the wave capture the co-propagating rays experience in the vertical symmetry plane. The relative frequency sharply decreases as the wave energy works against the translation of the dipole and the negative [partial derivative]U / [partial derivative]x in the jet. At about t/[T.sub.N] = 22 , the ray passes the dipole's strongest velocities and enters the rear region of the dipole where it experiences positive [partial derivative]U / [partial derivative]x. Escaping wave capture, the ray experiences a decreasing [partial derivative]U/ [partial derivative]z because it is propagating upward and counter to the dipole translation. As [partial derivative]U/ [partial derivative]z approaches zero, the ray exits the dipole and slowly propagates rightward, as dictated by the positive horizontal wavenumber. The ray's continued propagation is in a quiescent medium. Thus the completion of the turning point does not happen while the wave packet is still interacting with the dipole jet, but rather after the interaction. (This occurs after about t/[T.sub.N] = 35; the remaining portion of the ray required to close the loop required an additional 142 buoyancy periods.) This particular turning point lacks resemblance to the shape and size of the turning point reported by Godoy-Diana et al.  (Fig. 7a of Godoy-Diana et al. ). This may likely be due to the limitations of ray theory as the wavenumbers are not slowly varying in this region.
The rays initiated at [x.sub.0] = 0m and [x.sub.0] = -0.05m reflect horizontally and then experience wave mechanics like those experienced by the co-propagating "Dipole Front" and "Dipole Rear" rays in the vertical symmetry plane. The wavenumbers are unbounded, the relative frequencies plateau well above their initial values, the vertical group speeds asymptotically approach zero, and the horizontal group speeds approach the translational speed of the dipole. Indeed, these rays experience wave capture. The ray initiated at [x.sub.0] = 0m also experiences a vertical turning point which causes the vertical wavenumber to change sign and approach negative infinity before the ray plateaus a little above the horizontal midplane. The corresponding change in sign of [partial derivative]U / [partial derivative]z causes the positive exponential growth of the vertical wavenumber. The ray initiated at [x.sub.0] = -0.05m does not reflect vertically and plateaus below the horizontal midplane of the dipole.
Each of the other rays enter the front of the dipole, and therefore initially interact with positive [partial derivative]U / [partial derivative]x and experience an increasing horizontal wavenumber. One ray, intitiated at [x.sub.0] = 0.1m, ultimately experiences a decrease in k because it passes briefly through the front of the dipole and then slows due to the positive [partial derivative]U / [partial derivative]x in the rear. This short interaction is sufficient for the dipole to absorb nearly one-half the wave energy, which is enough to cause the steepness to become negligible for the majority of the simulation.
The ray initiated at [x.sub.0] = -0.1m has many things in common with the ray just discussed. One difference of interest, though, is that the relative frequency approaches the initial value. Though it interacts with the dipole to a greater extent than the other ray, having shifted nearly 10cm along the length of the domain, it ultimately experiences little to no energy exchange with the background and thus exits the dipole at virtually the same angle of propagation at which it entered.
Counter-propagating rays in the vertical symmetry plane do not experience a uniform increase or decrease of wave energy. Rather, their final values of relative frequency nearly span the possible spectrum. In addition a variety of internal wave phenomena are possible, including turning points and wave capture.
4.2 Internal Wave-Vortex Dipole Interactions
[FIGURE 11 OMITTED]
Fig. 10 displays the spanwise refractions in time of rays initiated at [x.sub.0] = -0.1m (for brevity, results of only this initial position will be presented). It shows that both focusing and defocusing are possible for a given set of initial wave properties and position. This is the general result found despite the initial horizontal ray location. The relative frequency of the five rays highlighted with symbols are shown in Fig. 10.
The focusing predicted by ray theory generally agrees with the experiment, although initially these rays defocus as they are caught in the dipole. The focussing is less severe for each ray traced farther outward until rays actually defocus. Defocusing was not reported for the experimental counter-propagating interactions, but the available experimental data does not extend to the outer spanwise boundaries. For the rays closest to the vertical symmetry plane, ray theory agrees with Godoy-Diana et al. , in which focusing of wave energy was observed and ray theory confirms these rays converge below the diple in the same region as found in the experiments. These results support the focusing results, however the increased field of view shows rays initiated at [y.sub.0] [greater than or equal to] 0.08m, where the effects of the dipole are severely diminished, ultimately defocus, contrary to the experimental results.
Ray theory also calculates multiple spanwise refractions not captured during the experiments, which were limited to a single horizontal plane in the middle of the dipole. Rays initiated at [y.sub.0] [less than or equal to] 0.07m, due to the rotation of the vortex, defocus before sharply focusing toward the vertical symmetry plane, where they cross their symmetric counterparts (not traced in the figure). These rays cross where Godoy-Diana et al.  find energy below the dipole. These focusing rays, i.e., those nearest the dipole jet, experience a notable increase in relative frequency, and thus expected energy, and seem to be captured. Defocusing rays complete the interaction with virtually the same frequency (and thus energy).
Experimental analysis has found that when an internal wave beam propagates along the vertical symmetry plane of a Lamb-Chaplygin pancake vortex dipole, critical levels occur when co-propagating with the dipole and turning points when counter-propagating. Outside of the vertical symmetry plane, spanwise spreading of wave energy was seen in the horizontal midplane of the dipole, defocusing during co-propagation and focusing during counter-propagation. The ray tracing of this work generally agrees with the available experimental data: both critical levels and turning points are possible, respectively, in co- and counter-propagating interactions; and, where ranges of initial spanwise ray positions match the views provided by the experiment, defocusing and focusing respectively correspond to co- and counter-propagating interactions. The location where waves enter the dipole jet may play a critical part in the final outcome of the waves since changes to the local divergence and shear experienced by the various rays cause wave properties to change in diverse ways.
Ray tracing expands understanding of these interactions where the experiment may be limited. In the vertical symmetry plane and relative to the dipole, a range of initial positions (approximately [x.sub.0] = 0.7m to [x.sub.0] = 0.85m at [z.sub.0] = 0.3m in this work) for rays co-propagating with the dipole yield results similar to critical levels, refracting to the horizontal as the vertical group speed approaches zero. However, not every case meets the requirement of a critical level, namely that the relative frequency approaches zero. Some rays, especially those interacting with the front of the dipole, experience wave capture during which the relative frequency increases to values near the buoyancy frequency as the vertical and streamwise horizontal wavenumbers are unbounded and the horizontal group speed approaches the background translational speed. Rays that enter the dipole from behind may approach critical levels, but from which they may escape, accelerating to the front of the dipole where they may experience wave capture.
Co-propagating internal wave interactions with a vortex dipole jet resemble two-dimensional wave interactions with an accelerating shear flow . This is especially so when rays approach the horizontal, as if at a critical level, despite increasing intrinsic frequency. Indeed, the components of group velocity behave like those of waves in an accelerating shear, with the vertical asymptotically approaching zero and the horizontal approaching the speed of the background. As this work shows, this indicates wave capture, which can be distinguished more fully by considering the exponential growth of the horizontal and vertical wavenumbers as waves interact with a vortex dipole . There, it is also suggested that [square root of ([k.sup.2] + [l.sup.2] /m | = [L.sub.v]/H)], though confirmation in this study has thus far been unsuccessful.
When counter-propagating in the vertical symmetry plane, rays experience a variety of possible outcomes depending on the initial streamwise position relative to the dipole. Turning points are possible, but so is wave capture after the background horizontally reflects the propagation of a ray.
Ray tracing also shows that, away from the vertical symmetry plane, internal wave energy is not limited to defocusing when internal waves co-propagate with a vortex dipole or to focusing when internal waves are counter-propagating. The location of entrance into the vortex-dipole by off-center internal waves affects the subsequent spanwise refractions. For example, the co-propagating "Dipole Front" rays initiated at [y.sub.0] [less than or equal to] 0.11m refract multiple times in the spanwise direction, not unlike wave trapping (again, however, not strictly trapped), and ultimately defocus, whereas rays initiated farther from the vertical symmetry plane primarily focus but would do so further downstream and not have been seen in the previous experiments.
Rays outside the vertical symmetry plane for the counter-propagating cases considered ([x.sub.0] = -0.1m to [x.sub.0] = 0.1m) also experienced focusing and/or defocusing depending on the initial spanwise position relative to the dipole. Outer rays initiated at about [y.sub.0] [greater than or equal to] 0.1m , depending on the initial streamwise position, defocused while those closer to the vertical symmetry plane focused. These correspond to counter-propagating interactions of the experiment where wave energy was observed below the turning point (see Fig. 7a of Godoy-Diana et al. ) and below the vortex dipole. On a second horizontal plane located at this depth, PIV measurements recorded wave energy generated off-center converging on the vertical symmetry plane . All cases of counter-propagating ray tracing considered in this work agree with this finding. For example, Fig. 10 shows that rays initiated at about [y.sub.0] [less than or equal to] 0.07m sharply focus. They cross the vertical symmetry plane below the ray interacting in the vertical symmetry plane where the wave energy would be seen relative to the dipole during the experiment. The experiment also suggests that only a portion of the wave beam energy was reflected at the turning point while some was transmitted through the dipole jet, appearing at the same location as the focused wave energy just mentioned . This transmission, or tunnelling , of energy is not confirmed using ray theory due to the relative scales of the internal waves and the vortex dipole.
Moreover, in crossing the vertical symmetry plane, focusing internal wave energy would not only interact with the vortex dipole jet at that location, but it would interact with its symmetric counterpart. Such an intersection of wave energy would occur, of course, during the experiment. However, the effects of this are not calculable using ray theory because each ray is independently traced. Therefore, this work has not considered the dynamics of the expected interaction.
6 Validity of Assumptions
Two assumptions of ray theory deserve mention in relation to the internal wavevortex dipole interactions presented and discussed. They are the scale separation hypothesis and the slowly varying approximation.
The scale separation hypothesis requires that the temporal and spatial scales of internal waves be sufficiently smaller than those of the background in which they interact such that the background remains unaffected by the internal wave propagation.
Initially, the spatial scaling here violates the scale separation hypothesis. However, as rays experience turning points or wave capture in the vertical symmetry plane the scale factors sharply approach infinity. For a co-propagating ray approaching a critical level in the vertical symmetry plane, the vertical scale factor also quickly increases to infinity, but the streamwise horizontal scale factor decreases unless the ray escapes the critical level and approaches wave capture, at which time the streamwise scale factor approaches infinity. Wave trapping of co- and counter-propagating rays outside the vertical symmetry plane does not significantly affect the scale factors any more than other horizontal refractions. In any case, as long as internal waves do not experience nonlinearities, such as energy absorption by the dipole or wave breaking, the ray trajectories remain correct .
The slowly varying approximation requires that fractional changes in a given wave property are small relative to the reciprocal of that wave property ; that is, [[omega].sub.r.sup-2] [partial derivative][[omega].sub.r] / [partial derivative]t| <<1, |[k.sup.-2] [partial derivative]k / [partial derivative]| <<1, | [l.sup.-2] [partial derivative]l / [partial derivative]y| <<1 and |[m.sup.- 2] [partial derivative]m / [partial derivative]z| <<1 (the horizontal factors were interpreted based on the others). And thus, for example, the horizontal wavenumber changes slowly over the horizontal scale. However, it is uncertain what the proper forms of the fractional changes are in multiple dimensions .
In general, rays in this work are within these bounds, though exceptions arise when some rays, co- and counter-propagating, initiated within 10 or 15 centimeters of the vertical symmetry plane, including [y.sub.0] = 0cm , experience a sudden increase in one or more of these factors. These occur during moments of stronger refraction, such as when approaching a critical level, wave capture, wave trapping, or a turning point. Though the slowly varying approximation is necessary for accurate calculations in ray theory, the results of this study generally agree with the experiment  and the assumptions regarding these fractional changes may not always be appropriate for internal wave-vortex dipole interactions.
This research was supported in part by the National Science Foundation (Grant No. CBET-0854131).
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Tyler D. Blackhurst and Julie C. Vanderhoff (1)
Brigham Young University, Provo, Utah, USA
Mon Sep 24 17:11:15 2012
(1) Corresponding author address: Julie C. Vanderhoff, Mechanical Engineering Department, Brigham Young University, 435 CTB, Provo, UT, 84602-4201. (email: email@example.com).
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|Author:||Blackhurst, Tyler D.; Vanderhoff, Julie C.|
|Publication:||International Journal of Dynamics of Fluids|
|Date:||Jun 1, 2012|
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