A wirewalker exploits the difference in vertical motion between a wire attached to a surface buoy and the water at the depth of a profiling body to provide the power to execute deep profiles: when the wire’s relative motion is upward, the profiler lets go; when it is downward, the profiler clamps on, and the weight attached at depth pulls the wire down, dragging the profiler downward against its buoyancy. The difference between the upward wire and profiler motion has to exceed the buoyancy-driven upward acceleration of the profiler body for this to work. Because the relative motion of the wire and water decreases as the surface is approached, the profiler might get stuck near the surface, especially when it is calm. However, two things mitigate this: 1) the system has a damped resonant response (~1.3 Hz), which induces relative motion between the buoy and water even at the surface; and 2) for waves too gentle to directly exceed the required acceleration, drag on the profiler can pull the clamped-together system down sufficiently that the buoy and wire without the profiler attached can suddenly release and bob upward faster than the profiler. For system parameters as estimated here, the latter requires submersion of less than 0.005 m below its equilibrium depth. Several such “bounces” can occur over a portion of the wave phase. These two effects explain why, in practice, the profiler does not stay long near the surface (although it does proceed downward a bit more slowly there).
Wirewalkers exploit wave motion to profile continually to arbitrary depths [see Pinkel et al. (2011) for a more detailed description of a prototype currently in use; or Rainville and Pinkel (2001) for an earlier description of the idea]. This means that all onboard power can be devoted to operating oceanographic instruments rather than having to waste a significant fraction on the profiling itself. Thus, the wirewalker is a simple, inexpensive, energy-efficient profiling system to which a suite of instruments of choice may be mounted.
The basic system consists of a buoy, wire, and weight (BWW) suspended above the bottom, plus a profiling body (PB) surrounding the wire with an internal cam that can grip it securely when the relative motion is in one sense, but release it in the other. Normally, this one-way cam is engaged to draw the profiler downward with each wave by clamping on when the orbital motion is downward and letting go when the cable returns upward. Upon hitting a bottom stop the cam is disengaged, and the profiler, being slightly positively buoyant, floats up smoothly to the surface, where the upper stop reengages the cam. While at the top, the data can be transferred to the surface buoy quickly and relayed ashore over the course of the next full profile cycle. Clearly, a significant requirement for high performance is a cam that is reliable, does not slip when it is supposed to be clamped on yet is practically frictionless in “release mode,” and does not damage the wire; however, this is not the subject of study here, and we simply assume an ideal cam. In practice, current designs have been successful in gathering tens of thousands of profiles over roughly 2-week-long deployments (limited most often by the battery requirements of the carried instruments operating continuously). Typical profiling speeds achieved to date are of order 0.5 m s−1, on both the downward (erratic) and upward (smooth) profiling directions.
Here we explore the dynamics of a conceptual wirewalker system in order to determine which aspects of the design affects the net profiling speed, with the hope of improving performance in future designs. We can think of the wirewalker system as alternating between two states: one where the BWW component and PB are disengaged (“free”) and the other when they are attached together (“clamped”). In the free state, the buoy–wire–weight component acts like a damped oscillator forced by the wavy surface, while the profiler body simply accelerates upward until it reaches terminal velocity relative to the local water, as determined by the balance between its positive buoyancy and drag. In the clamped state, the system is again a damped forced oscillator, but with different values of mass and drag, the latter of which also depends on the depth at which the profiler is clamped. The unforced oscillatory modes of these two states accommodate the transitions between the free and clamped states.
The conceptual model is based on an existing prototype, as described in Pinkel et al. (2011), with a few idealizations, for example, a perfect cam, no wire friction in the free state, no mean flow parallel to the wire, and sufficient buoyancy at the surface and weight at the deep end to ensure the wire always remains taut. At the end we shall also briefly consider another smaller prototype, the “mini-wirewalker,” which is intended to be much easier to deploy (but only holds a few small instruments). This also illustrates how the results should scale with size.
2. Motion of the BWW component
A good starting point is to model the motion of the BWW component. To keep it simple, we assume that the wire remains taut between the buoy and weight, and consider the dynamic response of the BWW to forcing by a deep-water monochromatic surface wave. The wave is described by two input parameters: the surface amplitude A and wave frequency ω (rad s−1). The wave’s vertical displacement and velocity at the surface is (to first order in wave steepness)
and as a function of depth the vertical displacements and velocities are
where is the corresponding wavenumber via (deep water) linear dispersion, with g being gravity (e.g., Lighthill 1978). Henceforth, we omit the real() notation, taking it as implied.
Next consider the characteristics of the BWW. Assume that the buoy is cylindrical at the waterline, so the restoring force versus vertical displacement is a linear relation, with a “spring constant” set by the diameter of the buoy and the water density times gravity ρg. Let the total mass of the BWW be m, and the upward force per meter of submersion be F. At rest, the buoy is pulled to a depth d where the upward force exactly counters gravity,
Now consider the behavior with respect to the perturbation depth zB of the BWW from this value (in the absence of waves), including the inertia of the BWW, but neglecting drag as yet,
which is a simple harmonic oscillator with resonant frequency .
To incorporate drag, first assume the drag occurs mainly around the weight at sufficient depth that wave motion is negligible, so it acts against the buoy motion in proportion to its actual vertical velocity (as we shall see, it is easy to change this later). For simplicity, we assume a linear drag law, with a linear drag constant 2c.
Forcing is provided by waves moving the surface zS(t), which acts through the “spring constant” F to drive zB closer to zS. The equation describing the cable motion then becomes a linear inhomogeneous second-order differential equation; in terms of acceleration of the BWW,
or, in standard form,
which corresponds to a forced damped oscillator. Solutions of the homogeneous equation (zS = 0) are found by assuming an answer of the form , with the rN being the two roots of the resulting quadratic equation,
Here we assume , that is, “subcritical damping,” so the BWW bounces a few times before the motion damps out as observed with the existing prototype. We define r1 as the (+) root and r2 as the (−) one.
Now consider forcing by the single harmonic wave [as in (2.1)]. This yields a particular solution for the BWW displacement of the form
which together with the two homogeneous solutions describes the BWW motion.
To estimate the BWW component parameters, we use the prototype described in Pinkel et al. (2011) as the exemplar. First we add up the masses: 32 kg for the surface buoy, 18 kg for the weight, and the weight of the cable. With 200 m of metal cable of diameter (say), this brings the total to about 71 kg. With a cylindrical buoy having a waterline radius of 0.356 m (28″ diameter), the force per meter of displacement is
or , or a resonant frequency of about 1.2 Hz. Finally, we estimate a drag coefficient based on the anecdotal observation that the buoy “bounces a couple times” upon provocation (when first dropped into the water, e.g.). That implies a damping rate c that is smaller than (rad), but not negligible; here we set
The effect of this value on the results will be investigated.
3. Motion of the PB
Next we consider the motion of the PB. To help keep the two wirewalker system components distinct, we associate lowercase variables with the BWW (as above) and uppercase variables with the PB, or (as appropriate) with the combined or “clamped” system.
The upward profile is simple to describe: the PB accelerates to terminal velocity over just a few seconds, and then proceeds smoothly upward until it hits the top stop, which activates the cam. On the downward profile, the cam clamps on whenever the downward velocity of the wire would otherwise exceed that of the PB, and then releases it again when the upward orbital acceleration of the wire exceeds the free-floating buoyant acceleration of the PB, so the PB “ratchets” down the wire.
For present purposes the wirewalker PB is characterized by two independent parameters, which can be determined from any two of three quantities easily estimated from the recorded PB motion: the terminal upward velocity WT (relative to the local fluid), the upward buoyancy acceleration B, and/or the drag acceleration −QWP (say). The WT is well determined from upcasts (e.g., 0.5 m s−1, based on the prototype), while the latter two are found from the downcasts by examining a “phase plot” of the estimated velocity WP relative to the local fluid (from dP/dt where P is pressure, see below) versus acceleration (from dWP/dt). Because velocity and acceleration are in quadrature for a linear surface wave, the phase plot for a wave forms a circle, with larger waves forming larger circles. Figure 1 shows example phase plots: the right panel shows an idealized phase plot for a monochromatic linear surface wave, while the left shows an example from a prototype deployment. From this, we estimate either the value of acceleration B, where WP = 0, or the value of Q from the slope of the straight line fit.
As with the BWW, we assume linear drag in the form D = −QWP, where Q is a drag constant (with units m s−1). We can evaluate Q either from the slope of the straight line fit to the “up-acceleration trajectory” in Fig. 1, or from B and WT; at the terminal velocity WP=WT, we know B + D = 0,
A quadratic drag law may seem a more appealing approximation, but as seen in Fig. 1 the linear approximation appears adequate over the range of velocities encountered here. Even for steady flows, the drag coefficient is a function of the Reynolds number (Tennekes and Lumley 1972), which varies over a moderate range in this scenario (of the order of 1000–50 000). In the strongly nonstationary conditions considered here (especially considering the radical variations as the PB clamps and unclamps from the wire), our knowledge about how a drag law should work is not well founded. In any case, a linear analysis permits some straightforward insights, and provides verification of numerical integration schemes against analytic solutions (when there are no transitions between the “free state” described above and the “clamped state” described below).
A useful depth estimate is one derived from the easily measured instantaneous pressure, providing an equivalent pressure depth ZP. The static pressure is , so this equivalent pressure depth is defined here as
For surface wave motion the pressure is constant on a material surface moving up and down with the fluid, so this corresponds to an equivalent depth in the absence of the waves, or to semi-Lagrangian coordinates that move with the waves in the vertical. Because other properties of the water (salinity, temperature, etc.) also move up and down with the wave motion, this can effectively remove or significantly reduce wave effects, which can be beneficial.
The pressure at the actual depth Z of the PB is found by integrating from the surface (where P = 0, say) to Z, including both gravity and local vertical acceleration of the water by the surface waves
Here the local displacement field is as in Eq. (2.2). Neglecting density variations (as in a Boussinesq approximation), and using the deep-water linear dispersion relation, we have
The equivalent pressure depth becomes
where, consistent with linear wave dynamics, higher-order terms in kA are neglected. The equivalent pressure vertical velocity of the PB can be written as
where W(Z) is the true vertical velocity of the PB at true depth Z, and again some higher-order terms are discarded. We see that WP is just the time derivative of the equivalent pressure depth, that is, effectively the pressure signal, which to a good approximation eliminates the local wave motion. Because the same pressure gradient works on the PB as on the surrounding water, a neutrally buoyant body would move exactly with the water in the absence of other forces (and ignoring the finite length of the PB), so it is only this relative motion WP that responds to, for example, the slight buoyancy of the body (in keeping with our Boussinesq approximation) or applies to the drag law. Technically there should be a correction for the overacceleration of the body, resulting from it being slightly lighter than the displaced water, but this is small enough to neglect.
To anticipate a bit: in a numerical integration, the depth of the PB is found by integrating the velocity in time. In this case, any wave-correlated part of the computed velocity can result in secular terms that cause the results to drift even when the PB is supposed to be clamped to the wire. The last exponential term in (3.6) can introduce such drift, because the PB depth Z includes wave-correlated motion. This is most important when the system is clamped near the upper stop, located at a fixed distance down the wire, say zB–z0. Because the BWW depth zB has finite phase lag with respect to the wave motion (because of damping), the out-of-phase part leads to secular drift. To eliminate this drift, it is necessary to use a value for Z in the exponential term of (3.6) that has no such (or a much reduced) out-of-phase wave motion. This is consistent with our approximations, because this drift is a rectified higher-order effect and may even be false. At this upper stop, a good value for the exponential argument in (3.5) would be the constant −kz0. As the wirewalker moves away from −z0, this correction rapidly becomes unimportant; so this adjustment need only be accurate at −z0. Thus, let a wave-free PB depth be defined as
This corresponds ideally to the position of the PB relative to the moving wire. The corresponding velocity of the PB relative to the BWW is then estimated as
which can be integrated without secular drift. If more accuracy is needed, then z0 can be iteratively replaced by Zf in the exponential term of (3.7); in practice just using the value from the last time step in (3.7) is adequate. Also, because these quantities are in principle defined relative to the wire, they only change when the two wirewalker components are decoupled (free state); when clamped, Zf is fixed and Wf = 0. Further, in the free state, the wave-induced motion of the PB does not have as much out-of-phase motion as the BWW (relative to the surface wave), so the secular drift problem is greatly reduced. Thus, the motion relative to the wire can be enforced to behave reliably.
4. Motion of the combined system
Next, consider the motion of the combined system in the clamped state. The system parameters change depending on whether the profiler (PB) is either clamped to the wire or free. The solution for the wire motion (section 2) is appropriate to the free state. When the PB cam clamps on, the total effective mass goes up, and both the slight PB buoyancy and its drag through the water get applied as additional forces on the system. Here we take a straightforward path to adapt the section 2 results to the clamped state by deriving the equivalent solution for the always-clamped case.
The equations of section 2 are based on acceleration; that is, the forcing is “normalized” by the mass of the BWW m. The combined system has total mass m + M, so the acceleration equations have to be renormalized. Here, we choose to renormalize the PB inertia by the BWW mass m, consistent with the previously estimated terms, yielding new PB terms multiplied by a mass ratio Rm. Based on the estimated volume of the PB structure of about 0.087 m3 and a standard density of 1025 kg m−3, the mass M of the PB as configured is about 89 kg, so
This renormalization applies to the PB buoyant and drag-induced accelerations as well as the net acceleration of the combined system, so the resulting equation analogous to (2.5) can be written
where Zf, B, and Q are as described in the last section. The left side (but with more mass) and first two terms on the right come directly from (2.5); the third term is the PB drag, and the last term is the slight positive buoyancy of the PB. This last constant term results in a slight change to the resting depth d of the system, because the slight buoyancy pushes the equilibrium upward by an amount (about 0.0055 m for weights as described and B ≈ 0.25 m s−2). We eliminate this term by defining a new vertical coordinate offset by this small amount zC = zB + Δd, and we renormalize everything by the total mass (1 + Rm). The new resonant frequency of the combined unforced system is
Defining a renormalized linear drag parameter C for the PB-induced drag as
and renormalizing the BWW drag parameter c by the combined weight also,
Splitting up this drag term and rearranging into standard form, we now have
For a single-frequency wave as defined in (2.1), the surface velocity is , and the solution is found just as before, but with (C + c′) in place of c, in place of , and the remaining part of the new drag term appearing in the forcing amplitude,
Note that, should we wish to include drag at the surface buoy rather than at the weight, this can be easily done with another pair of terms just like the new ones for the PB drag, but evaluated at the surface, where ekz = 1. In the simulations discussed below, we actually do place the buoy drag at the surface in this way, although the difference this makes is not large.
5. Transition dynamics
The transitions between clamped and free states always occur with the PB and BWW velocities being equal, so the velocity is continuous. Also, while the resting depth d changes, the actual location zB (say) of the BCW is clearly continuous. However, the acceleration can change instantaneously in this idealized model (and nearly so in reality), in particular at the clamping-on transitions.
Even though the velocity and location are continuous at the transition, they do not often match the values for the newly appropriate single-state solution for the motion of the BWW. The inevitable incompatibilities between the two single-state solutions at each transition (2.8 versus 4.8) are absorbed by the appropriate pair of homogeneous (damped oscillator) solutions. Here we use the free-state functions to develop the concept; translation of the results to the clamped state is straightforward, so we shall mostly skip it for brevity.
First, write the two homogeneous solutions in the form
where the roots r1 and r2 are as given in (2.7) and just after [for the clamped state, the frequency and damping c as used in (2.7) are changed to and (c′ + C)]. Then, at the transition time the discrepancies in location (including Δd) and velocity between the two particular solutions [(2.8) versus (4.8)] are matched by the homogeneous solutions,
Because of the denominator , this looks unstable as (critical damping); however, the solution remains well behaved. Defining a difference term in the form
the homogeneous part z′ of the solution for the wire elevation can be rewritten as
and the homogeneous part of the matching solution for velocity w′ becomes
The first (cosine) terms smoothly connect across the changes in elevation and velocity, and then decay in time. As ɛ becomes small, the last (sine) terms introduce another delayed transient bump, but this is bounded, and the exponential decay rate soon damps this out as well.
For a spectrum of waves, the linear problems described above can be applied to each spectral component using the values of f and k(f) at each frequency. The wire displacement response [either (2.8) or (4.8)] for each single-state solution can be computed by Fourier transforms of the appropriately weighted spectra, as can the equivalent solutions for the velocity. To keep it simple enough that these linear solutions were easily found, we have assumed a linear drag on the BWW and PB. These solutions are also convenient to use as initial conditions in a numerical integration.
The analytic solutions of sections 2 and 4, and the transient response as described in this section to an initial condition that has only a 0.005-m elevation discrepancy at the start, are illustrated in Fig. 2 for a simple case with no state transitions.
6. Numerical approach to the wirewalker problem
Because the actual system changes abruptly from one state to the other, forward numerical integration seems natural and promises more flexibility in implementing quadratic drag, etc.
The basic wirewalker rule is that the vertical velocity WN of the PB (relative to the local fluid, like WP of section 3) at time tN is the minimum (most negative) between the two options, either clamped to the wire or free, as
where is the difference between the wire velocity and the water at the depth ZfN of the PB [the relative velocity, analogous to (3.6) but for the wire at the PB depth], and is the relative velocity the wirewalker would have at time step N if it accelerated upward freely from the last time step. It is easiest to estimate the latter using the local relative velocity, because then we can ignore the surface wave pressure gradient forces on the PB.
Consider the simpler BWW system first (free state), with the drag deep below the wave motion. We start with the expression for acceleration at time step N [paraphrasing (2.5)],
where aN is the vertical acceleration, wN is the velocity, and zN is the depth of the BWW at step N. To estimate the new values wN and zN, we use an average acceleration and velocity over the previous time step,
Substituting these into (6.2) and gathering all of the aN terms on the left side, we obtain
Now we evaluate aN−1 from (6.2) to put this in a slightly simpler form. Note that it is important to recalculate a new from (6.2) (or the analog for the clamped system) with the new spring constant and drag values before using it after a transition, because acceleration can change discontinuously when the system changes state. Thus, we write
The terms are integration constants that can be precalculated. For sufficiently small time steps, we should be able to discard the second-order term in ; however, because the spring constant is very large, this may not be wise: in fact, for a drag , these two terms are equal for , and the last term is still 10% as large as the middle one for . In contrast, retaining this term provides a better than 1% accuracy even at for a single-state integration, compared to the analytic solution. In any case, these similar-looking factors acting on aN and aN−1 mainly implement damping, while the last term on the right represents the incremental forcing resulting from the relative motion of the surface zS(t) and BWW. We then evaluate wN and zN from (6.3) and (6.4).
For completeness, and because this also shows how to place the buoy drag at an arbitrary depth, the equivalent equation for the clamped state [from (4.6)] is
Again, the damping acts to reduce acceleration by essentially a constant factor per time step, with incremental forcing by the relative motion at the surface and now also by drag at a depth Zf where there is a finite wave orbital velocity.
In similar fashion, we evaluate the implicit free-rising PB velocity time step
where, as in section 3, the PB drag acceleration constant is . Note that when the wirewalker is traveling down (W negative), the drag becomes positive and reinforces B in trying to accelerate the profiler upward; however, the cam will only allow this if the free wire would accelerate upward even faster without the PB attached.
On occasion, the unburdened wire can accelerate upward faster than either the clamped state or ; in this case the system can switch to free mode somewhat unexpectedly. This can happen with a fairly low, long-period swell, for example, because over part of the upward acceleration phase of the wave the drag on the PB in the clamped system pulls the buoy farther underwater than where the free state would be (see Fig. 3). This stands in contrast to the anticipated behavior, which is illustrated by the action at 100-m depth (see Fig. 4). Thus, both candidate velocities in (6.1) are calculated for the decoupled state from the last time step; if the clamped state is indicated, then the integration step is recalculated for the combined system and applied to both components.
Finally, as noted above, it is attractive to integrate the depth of the PB relative to the cable, Zf [as defined in (3.7)]. If the criterion (6.1) indicates a clamped state, then Wf = 0 and ΔZf = 0 so Zf remains unchanged. If the free state is indicated, then (which will be negative), and .
This implicit approach is simple with a linear system. Implementing quadratic drag or other nonlinear dynamics would be easier in a forward finite difference scheme. One way to do this is to iterate (6.2)–(6.3) starting with the last (recalculated) acceleration as an initial guess, . In practice, just one or two iterations is enough to bring the answers very close to the implicit result, with the difference quickly approaching zero as the size of the time step is decreased. Relative to the implicit result with as a reference, the overall errors in the PB motion are 6.0% at , 5.34% at , 1.17% at , and 0.68% at . The big drop as is reduced below 0.05 s occurs because the system is sensitive to getting the transition times right to better than that margin (compared to the ~1.3-s resonant period of the BWW). The results discussed next are derived from runs at the reference resolution .
The upward profile time is set entirely by the terminal velocity WT, which is attained within a few seconds after the release at the bottom stop. With linear damping, the upward velocity after release from the bottom stop (with zero velocity at t = 0) goes as
so the time it takes to rise a vertical distance Z from the bottom stop to the top stop (say) is
(neglecting an infinitesimal term proportional to ). The last term is also relatively small, so this upward transit time is reduced for smaller drag Q or more buoyancy acceleration B.
In contrast, the downward average profiling rate is a function of several different wirewalker characteristics. The initial descent rate from 1-m depth is where the profiler progresses most slowly (as in Fig. 3), while the asymptotic “deep descent rate” (deep enough that the local wave motion in the fluid is negligible, although the wire still moves with the surface) is generally much larger (as in Fig. 4). These descent rates are illustrated in Fig. 5 for a range of driving wave periods from 1 to 20 s, with a fixed wave steepness ak = 0.1. For linear waves, the maximum accelerations at the surface are , so fixing ak is a good way to compare results across different frequencies. The resonant period of the BWW system shows up clearly in these rates, peaking at a period of 1.29 s with descent rates of the order of 40 cm s−1 both initially and at depth. As the wave period increases beyond 7 s or so, the rates quickly asymptote to a fixed initial rate of about 0.9 cm s−1, and deep rates increase linearly with the wave period roughly as (4.26 cm s−1)T, where T is the wave period in seconds.
a. Monochromatic forcing
With a working model of the BWW and PB behavior, we can explore what affects the descent rates, especially the slow initial rates. We consider first a monochromatic forcing wave; the following changes and results are for an 8-s wave with ak = 0.1 (see Tables 1 and 2 for changes at both 8 and 16 s):
Somewhat surprisingly, doubling the BWW weight decreases the initial rate by 40%–60%, while doubling the PB weight increases it by 60%–80%.
The restoring force varies with the water level diameter of the float: cutting the cross-sectional area by half (and hence F) increases the initial rates by 40%. Because this also moves the resonant frequency to a lower value, where a typical wave spectrum has more energy, this should be a doubly favorable change.
Doubling the terminal velocity but not changing the buoyancy (i.e., by reducing drag only) has surprisingly little effect on either the initial or deep descent rates.
In contrast, changing the buoyancy acceleration but not WT (corresponding to again decreasing drag Q by half, but reducing the buoyancy to compensate) has a relatively large effect: cutting B to B/2 adds 70%–90% to both the initial rates, and 30%–55% to the deep descent rates.
Cutting the buoyancy in half but not changing Q (so WT is also cut in half) adds about the same to the initial rates (consistent with WT not having much effect on the descent rates; however, note that the upward ascent rate depends directly and only on WT).
Another one of the larger effects on the initial descent rate comes from reducing the BWW drag (which is the larger of the drag terms): cutting c to c/2 increases the initial rates by 60%–80%. This again argues for a slimmer PB.
In contrast, moving the BWW drag from the surface float to the deep weight has only a small effect.
Finally, the effects on the deep descent rates resulting from all these changes to the system parameters are smaller, with the largest effect resulting from reducing both B and Q so that WT stays constant (i.e., cutting drag too), being the largest at 33%–55%. Incidentally, the performance estimate given in Pinkel et al. (2011) corresponds to this asymptotic deep monochromatic case and is very close to these results.
The results indicate that reducing the buoy diameter should be helpful, both by lowering the resonant frequency and (most likely) by reducing the BWW drag, both of which should speed up the initial descent rates. They also indicate adjusting B may have the next largest effect.
b. Broad-spectrum forcing
While interesting from a scientific point of view, particularly as it was unanticipated, the spontaneous “bouncing” action at mild wave slopes seen in Fig. 3 is not a game changer, since the resulting 1 cm s−1 descent rates are still too slow to make much difference. The important game changer is the resonant period action, which results in 40 cm s−1 rates even at the surface. We expect this to be especially true in considering a more realistic spectrum of waves, which will certainly include action at the 0.78-Hz resonant frequency of the BWW system. Further, because the clamp–unclamp transitions are strongly nonlinear, depending on extrema in the vertical accelerations in particular, we anticipate that the results may be quite different when we include a full spectrum of waves. Thus, to determine an optimal strategy for short cycle times, we need to simulate these cycles with a full wave spectrum, in order to see both the resonant (1.3 s) wave effects as well as the larger speeds as the PB gets more than a few meters from the buoy.
Both the implicit and forward time stepping models can be applied to an arbitrary surface motion time series. For initialization, the linear analytic solutions [e.g., (4.8)] can be applied to a spectrum of waves: the complex factor in front of in (2.8) and (4.8), which is a function of frequency and (fixed initial) depth, is applied to the spectral coefficients of the Fourier transform of z(t) as a spectral transfer function, and the result is inverse transformed back to time. Because the action is sensitive to mismatched conditions, initializing with the clamped analytic solution can eliminate a significant artificial startup bounce that could affect the initial descent rates. In the following, we examine total down-and-up cycle times for a model spectrum with an 8-s peak period and an f −5 frequency dependence. The high-frequency cutoff is set by the buoy diameter: we taper with a factor that drops smoothly (as sin2) from unity, for waves longer than twice the buoy diameter (2 × 71 cm, yielding f = 1.05 Hz), to zero for waves shorter than the diameter (71-cm wavelength, or 1.48 Hz). Because these are beyond the resonant frequency (near 0.78 Hz), the results are not sensitive to this choice. For a reduced diameter buoy, the resonant frequency shifts to a lower frequency, while these cutoffs shift upward, so they would become even farther apart.
A model spectrum with peak period of 8 s is shown in terms of elevation, vertical velocity, and acceleration in Fig. 6 (solid lines), along with the spectrum of the resulting BWW motion for the clamped state (dashed). The resonant response is sufficiently energetic that it creates a global peak in the acceleration spectrum, in particular. As anticipated, including the resonant response is the real game changer. Table 3 shows the round-trip average speeds for the same set of parameter variations as those in Tables 1 and 2 for spectra having an 8- or 16-s peak with (ak)peak = 0.06. The results are very different from the monochromatic case: the biggest effect here comes from reducing the PB drag Q, with some dependence also on the buoyancy B. In practice, Q is not very adjustable: we can make the body longer and thinner, but the thinness is limited by the size of the instrumentation to be carried. Other than that, we make the body as streamlined as practical and that is all. However, B is very much a matter of choice, and can even be adjusted in the field, so it is worthwhile to investigate how the cycle times vary as B is adjusted with all of the other parameters fixed. Figure 7 shows the round-trip cycle time versus B for two model spectra, one with an 8-s peak (and rms amplitude of 0.95 m, so akpeak = 0.06) and one with 16 s (and the same steepness, or rms amplitude 3.8 m).
The results show broad minima in total cycle times versus B. We also find that the rise times are consistent with 7.2, and the descent times are consistent with values found by subtracting the rise times from the totals. An empirical linear fit to the descent times versus B yields Td08 ≈ 126 + 324B for the 8-s peak spectrum, and Td16 ≈ 100 + 268B for the 16-s peak. A quadratic yields a slightly better fit for the 8-s results, but not for the 16-s peak. We have not yet deduced an objective, nonempirical way to predict these descent times for broadband forcing, although the simple estimate given in Pinkel et al. (2011) does match up well with the “deep descent rates” of the monochromatic case.
c. Extension to a mini-wirewalker
In addition to the wirewalker analyzed here, we are also developing a mini-wirewalker that is about 2.2 times smaller in each linear dimension (roughly of a 10-cm diameter by 75 cm long). The appeal of a smaller design is mainly the ease of deployment: with this smaller size, several complete systems can be deployed from a small boat or skiff in a single trip. Assuming roughly the same shape, the only important change is the drag-to-mass ratio Q. Because drag should be proportional to area, while mass is proportional to volume V (via the density of water), the drag ratio goes like V−1/3, so Q becomes about 2.2 times larger. Simulations analogous to those shown in Fig. 7 indicate again a broad minimum in cycle times near values of B yielding about 0.5 m s−1 upward terminal velocity. While the resulting upward transit time is thus the same, the increased drag has the effect of roughly doubling the downward profile time, resulting from the profiler slowing down more rapidly after releasing the clamp each time.
Because this analysis preceded a working prototype, this constituted a testable prediction. Preliminary tests (performed since the first draft) indicate close agreement in terms of round-trip times, although they are somewhat better in practice because we took advantage of the suggestion to make it longer and thinner.
d. Mean flow and a tilted wire
When a wirewalker is deployed in shallow water, the weight at the bottom is often tethered to an anchor to maintain its position; in this case, a mean flow will cause the working section to tilt away from the flow. The simplest case is if the wire remains essentially straight, but is tilted at some angle α from vertical (which can happen if the buoy and weight are sufficiently large). Then, the projection of the mean flow U parallel to the wire is a constant U sin α, and the effect is as if the buoyancy acceleration B were modified by QU sin α (still ignoring cable friction). At the same time, the effect of the buoyancy would be reduced to B cos α, so these two effects can partially cancel. The action of a sheared mean flow, and/or curved wire, is more complicated: it would be as if B became a function of depth. We have not extended the analysis to include these complications.
The dynamic model of wirewalker motion permits us to quickly and easily evaluate different strategies to best serve the interests of science. On one hand, rapid complete cycle times (with combined down and up profiles) yield better temporal resolution; on the other hand, slower up profiles permit higher vertical resolution. The current value of B = 0.25 yields roughly equal down and up speeds, near 0.5 m s−1 as observed, which yields decent vertical resolution (5 cm at 10 samples per second). This is toward the low B end of the broad minimum, so there is plenty of room to experiment with larger buoyancy values; this suggests an alternative strategy to conserve power: if we increase B to where the rise takes only half as long as the descent (say, 0.4–0.5), the power to the instruments could be shut off ⅔ of the time (since the descent is rough and turbulent, and thus less useful for data). Because, to date, the instruments have been recording all the time, this change would mean deployments could be extended to up to 3 times as long.
Our model shows that the wirewalker behavior is robust, with decent cycle times even for waves that are gentle, as observed. Exploration of the BWW parameter space indicates that there may be some advantage to making the surface buoy slimmer and taller, and that our choice of buoyancy for the body lies within, but toward the low-value end, of a broad minimum.
The authors thank Luc Rainville, Jon Pompa, Achintya Madduri, Jody Klymak, Jennifer MacKinnon, Andrew Lucas, and Melissa Omand for assistance with the development and operation of the Wirewalker. Support for Wirewalker development and for this dynamic analysis was provided by Grants NSF OCE05-01783, ONR N00014-08-1-1022, and ONR N00014-10-1-0692.
Current affiliation: Woods Hole Oceanographic Institution, Woods Hole, Massachusetts.