Shaykin 1
Effect of a vortex on a wave-packet
We present a study, given several assumptions, of the behavior of a wave-packet in a two-dimensional fluid medium in the vicinity of a vortex in the medium. The vortex is assumed to be strong enough to change gradually the direction and speed of the wave-packet, but to be too weak to break up the wave-packet. It is further assumed that the wave-packet travels with intrinsic group velocity of constant magnitude. Given a vortex, unchanging with time, with tangential velocity inversely proportional to the square of the distance from the vortex center, we investigate under what conditions a wave-packet gets drawn into the vortex’s center. Additionally, we address the suitability of a forward-Euler numerical scheme for modeling the behavior of this system. We borrow heavily from the work of Bühler and McIntyre referenced at the end of this paper. Equations below that are not derived are explained in far more detail in that paper.
1. Governing Equations and Ray-Tracing Equations
For simplicity, we assume a vortex centered at the origin. The vortex has local velocity given bywhere u and v are the x-component and y-component, respectively, of the vortex’s velocity, and 0 is the circulation strength of the vortex.
The wave-packet is modeled by a particle, with x and y location given by x = (x , y). The wave-packet faces a direction with x- and y- components given by k = (k , l).
We would ideally like to find an explicit equation for x as a function of time to investigate the behavior of the wave-packet. While it is unclear how to find such an equation, we do know how to find an equation for the velocity of the packet. We observe that , where is the group velocity of the wave-packet and is the intrinsic group velocity of the wave-packet.
The dispersion relation governing the wave-packet is:
where is the absolute wave frequency, t is time, , and the wavenumber . Hamilton’s equations give ray-tracing equations for x and k:
and .
Taking these derivatives, we get , noting that and . Differentiating k with respect to time we get
and .
Assuming constant , the final evolution equations for x, y, k, and l are:
,
,
, and
. D
2. Modeling with a Forward-Euler Scheme
Unclear about how to solve this system of four differential equations in time for x, y, k, and lto get an explicit equation for x(t), we use a forward-Euler numerical scheme and MATLAB to ray-trace the approximate path of the wave-packet. The forward-Euler scheme makes use of the following equations, derived from the evolution equations above, by discretizing time, multiplying both sides by the timestep t, and subtracting the previous value of x,y,k,and l , respectively, from both sides, to get the present value at t = n:
,
,
, and
,
where
,
,
,
,
,
, and
.
We expect the ray-tracing results to become increasingly accurate with decreasing timestep. A timestep, a total time duration T for the path, a value for , a value for c0, and initial conditions for x, y, k, and l must be set. Figure 1 shows typical wave-packet paths generated by the forward-Euler method explained above. In this particular example, the initial values for k and l are selected so that the paths of the wave-packets begin unchanging in y from the packets’ release at x = -20.
In Figure 1, we see a scattering effect: the wave-packets closer to the center of the vortex feel the effect of the vortex more strongly, as we would expect from the equations for u. Note that the wave-packet released from x = -3 is sufficiently close to the center of the vortex that it gets sucked around the origin several times before getting sent back out to infinity. The question now becomes whether these results are accurate.
Accuracy can be determined by invariants and convergence tests. There are two invariants in our system. The first is M = ky-lx, and the second is , the absolute wave frequency as defined above. If M and are calculated at each point x for which the wave-packet is plotted, we can get a sense of the accuracy of the plot by examining how little the invariants change due to numerical error. Calculating gives us a sense of the percentage numerical error on the first invariant, and calculating gives us a sense of the percentage numerical error on the second invariant. By averaging these two percentage errors, which are quite close anyway, we can determine that the three wave-packet paths in Figure 1 farthest from the origin are quite accurately depicted. The error on the packet released from y = -18 is approximately 0.0135%, the error on the packet released from y = -13 is approximately 0.0300%, and the error on the packet released from y = -8 is approximately 0.1250%. Furthermore, Figures 2 and 3 show the linear convergence to zero of the two error checks with decreasing timestep, for the packet released from y = -8, the least accurate of the three paths examined so far. Therefore, it seems that our encouragingly low numerical errors are not coincidental.
The accuracy of the path as depicted in Figure 1 of the wave-packet released from y = -3 by the ray-tracing equations is seriously called into question, however, due to the numerical error on that path being approximately 1000%. The reason for the sudden, enormous jump in error for the path nearest the origin must be determined.
3. Redefining Invariants
We would like to determine what happens as by looking at the invariants M and . Since we will be looking at r frequently, switching to cylindrical coordinates will be helpful at this point.
We then redefine the invariants in cylindrical coordinates:
.
For , noting that
,
we get
.
By plugging the equation for M into the equation for ,
. (1)
We now have equations for M and in convenient coordinates. We will look at the behavior of these two invariants as r0 to learn about the path the wave-packet takes near the center of the vortex. We will see that the behavior of the wave-packet depends heavily upon the sign of M.
4. The Case of M = 0
If the wave-packet is launched facing directly into (or away from) the vortex center, then initial = 0, and initial M = 0. As M is invariant, M will always equal 0, and therefore so must , indicating that the wave-packet will always face directly into (or away from) the vortex center. From the equation for above, we get , indicating that remains constant as well. Now in general, the wave-packet’s radial velocity is composed of the radial component of its intrinsic group velocity and the (non-existent) radial component of vortex velocity. This gives us
(for all M).
In the case of M = 0, this simplifies to
(for M = 0).
If the wave-packet is launched directly away from the vortex center, then it will move away from the origin with constant radial velocity equal to . If the wave-packet is released facing directly into the vortex center, then it will travel with constant radial velocity equal to -, and reach the center of the vortex at time .
5. The Case of M > 0
If M > 0,we notice that M can remain constant as if . Additionally, solving equation (1)for , we get
(for all M). (2)
This equation tells us that as , as well. Because and blow up, k and l blow up as well, as . For this reason, no finitely small timestep will yield accurate results in a forward-Euler scheme of the type described above as . This was precisely the problem with the path nearest the origin in Figure 1. For that particular path, M = 3.6046 > 0. The numerical error from the forward-Euler scheme obscured the fact that the wave-packet path actually should end in the center of the vortex, as the following consideration will demonstrate.
The equation above for indicates that as , the term will dominate and will go to infinity like . Recalling that
,
we now see that and in fact . The case of M > 0 simplifies to the case of M = 0 as . In both cases, if the initial is negative, the wave-packet will get drawn into the center of the vortex. In neither case can a forward-Euler numerical scheme demonstrate this, due to k going to infinity.
6. The Case of M < 0
If M0, then as r0, equation (1) tells us that necessarily goes to infinity. Since is an invariant of the system, it is therefore not possible for r0 if M0. So if the wave-packet is released into the vortex with < 0, must eventually become positive so the packet can move away from the center of the vortex. The moment at which = 0, therefore, is the moment at which the minimum distance from the origin is achieved. Setting = 0 in equation (2),
(for M < 0).
Since r does not go to 0, k need not blow up due to the invariants. So it should be possible to determine this minimum radius with the forward-Euler scheme.
Figure 4 shows the path of a wave-packet for which M < 0. In this example, M = -1, Ω = 2.1811, and Γ = 350. Plugging these values into the equation for , we get . We can neglect the negative possibilities for , since a distance from the origin must be positive. To determine which value for is correct for the path shown in Figure 4, consider the graph of versus radius in Figure 5, obtained from the equation for above and the specific initial conditions used to generate Figure 4. At r = 4.8297 and r = 5.2881, = 0. Between these points, is imaginary; and outside these points is real. If r = 4.8297 were the true minimum radius, then there would be radii larger than it for which is imaginary, which is impossible. Therefore, the true minimum radius is the larger = 5.2881.
By recording for each x it plots, MATLAB can determine numerically. Figure 4 was generated with and has an approximately 1% error as determined by the error checks for the invariants. We hope that with decreasing timestep the forward-Euler scheme will improve numerical accuracy forwhen M < 0, rather than give us completely inaccurate results as it did for the earlier case of M > 0.
Figure 6 demonstrates that the numerical approximation of converges linearly with timestep and that it does indeed converge to our desired = 5.2881. The points plotted for the two smallest timesteps are (.003, 5.3480) and (.0015, 5.3182). The line connecting these points crosses t = 0 at = 5.2885. This value for is very good; it represents only a 0.0061% numerical error from the true = 5.2881. Figure 7 indicates that the time needed to reach converges linearly with timestep as well, which makes sense.
7. Conclusion
The model of the wave-packet path starting from y = -3 in Figure 1 is completely inaccurate, while the models of the other three paths in Figure 1 are very accurate. The path beginning at y = -3 should end at the origin, not at infinity. On that path, M > 0. The path shown, getting sucked into the vortex until some is reached and then getting thrown out to infinity, does accurately describe the behavior of wave-packets that feel the vortex strongly in a case where M < 0. An example of a path with M < 0 can be seen in Figure 4. The enormous numerical error for M≥ 0 is due to the extremely large values for and k near the origin. The forward-Euler method only accurately models the path of wave-packets that feel the vortex strongly under the assumptions provided, when M < 0.
8. Note about the Physics
It should be noted that in truth, no value for M will allow the wave-packet to reach the center of the vortex. The dispersion relation and the invariance of M are derived from physical laws that only describe the wave-packet under the original assumptions, namely that the vortex is too weak to break up the wave-packet. As r0, gets arbitrarily large and will surely break up the wave-packet before it reaches the center of the vortex. When we discussedr0, we meant only that the wave-packet gets as close to the center of the vortex as possible before the strength of the vortex current breaks it up.
- Acknowledgements
The author offers many thanks to Professors Oliver Buhler and Alex Barnett for their patience and kind help, and for teaching him a lot of math.
REFERENCE
BUHLER, O. & MCINTYRE, M. E. 2003 Remote recoil: a new wave-mean interaction effect. J. Fluid Mech. 492, 207-230