Continental Shelf Research, Vol. 8, No. 2, pp. 153--158, 1988. 0278-4343188 $3.00 + 0.00 Printed in Great Britain © 1988 Pergamon Journals Ltd.

Mooring design to minimize Savonius rotor overspeeding due to wave action

DAVID A . GRIFFIN*

(Received 19 September 1986; in revised form 24 March 1987; accepted 4 June 1987)

Abstract A theoretical formula providing an estimate of the surface wave induced overspeeding of non-vector-averaging recording current meters is derived for shallow water (continental shelf) single meter moorings having subsurface buoyancy directly above the meter. It is shown that the common practice of placing meters close to the bottom in an attempt to escape wave action can result in greater overspeeding, even though the wave induced currents are weaker at depth. In cases where a variety of meters are available to be used at various depths, it is therefore recommended that the newer vector-averaging meters be used near the bottom as well as near the surface, with non-vector-averaging meters being used only at mid-depth on moorings whose uppermost buoyancy is also at mid-depth.

I N T R O D U C T I O N

IT HAS long been recognized that non-vector-averaging Savonius rotor type current meters, such as the reliable and popular Aanderaa RCM4, are ill suited to the task of measuring tidal or longer period currents in the presence of surface wave induced currents of period less than the meter's sampling interval. The problem is that these devices integrate speed but only record instantaneous measurements of direction. If the current's direction oscillates during the sampling interval then an erroneously high estimate of the water velocity is recorded because the sign changes go undetected. To reduce this "overspeeding" (or "rotor pumping") due to surface wave action, this paper recommends that a single point mooring with subsurface floats immediately above the meter be used because the meter is then drogued somewhat to the water, reducing the amplitude of high frequency oscillatory currents seen by the meter and hence the contamination of the data. This type of mooring has become more common following the increasingly widespread use of acoustically activated release devices which obviate the need for surface floats. It is appropriate, therefore, that the argument above be quantified to some extent.

There has been much discussion in the literature on the effect of the float's depth on meter overspeeding when the floats are much nearer to the surface than is the meter. However, this is due to a different process to that considered here and discussion has been largely restricted to intercomparisons of data and to data self-consistency tests. For example, HALPERN and PILLSBURY (1976) compared currents at 43 m depth in 50 m of water recorded by Aanderaa RCM4s on two nearby moorings whose uppermost floats

* Faculty of Science, University of New South Wales, Kensington, N.S.W., 2033, Australia. Present address: Depar tment of Oceanography, University of British Columbia, 6270 University Boulevard, Van- couver, B.C. Canada V6T lW5.

153

154 D . A . GRIFFIN

were at 3 m and 18 m depth, respectively. They showed that serious overspeeding had occurred on the mooring with the shallow floats and deduced that this was due to the vigorous vertical motion of the floats being transmitted down the mooring line to the meter. PEARSON et al. (1981) estimated the overspeeding of Aanderaa RCM4s deployed directly below floats on the Alaskan shelf at depths greater than 20 m in water at least 60 m deep. The winter mean wave height was 3.3 m and period 7.9 s. Pearson et al. deduced that overspeeding was not significant because the observed M2 amplitude was not correlated with wave height and changed by less than 10% from summer to winter. They henceforth concluded, after comparing their results with those of Halpern and Pillsbury, that the depth of the floats must be a critical factor. Given, then, that one should avoid the use of near-surface floats to avoid overspeeding due to transmission down the mooring line of the vigorous surface motions, the purpose of the present work is to estimate the overspeeding due to the horizontal component of the waves' orbital currents by considering the basic physics of a simple model. Several approximations will be made to facilitate the analysis but these are not believed to compromise to any serious degree the validity of the conclusions.

T H E O R Y

The simplest case to consider is for when the "quasi steady" (period greater than the meter's sampling interval) current Uqs is zero while the "oscillatory" (shorter period) current Uc (due to surface waves) is not. In this case, an ideal vector-averaging meter would record zero current but an RCM4 would record a finite speed S (with a random direction) because it integrates speed rather than velocity.

A first estimate of S is 21 ucl/ , the time-average of the rectified oscillatory current. But this estimate assumes that the meter remains fixed in space, hardly a plausible assumption. A better estimate of S results from also calculating Urn, the meter's own horizontal oscillatory velocity. Consider a mooring as an inverted simple pendulum driven by the viscous drag of the surrounding moving medium. Let it comprise a thin, massless mooring line of length L (account for its mass and width will be made later) anchoring a single meter (with flotation attached) of mass M, cross-sectional area A and net buoyant force B. The equation of motion of the meter is accordingly written

CDpA(U~ - Urn) 2 BX~ M dUn 2 (L 2 - X2) 1/2 dt

- - - 0 , ( 1 )

where Xm is the horizontal displacement of the meter from the vertical, Co is the drag coefficient and p is the density of seawater. The first term is the drag force on the meter due to the relative current, the second is the horizontal component of the mooring line tension and the third represents the acceleration of the meter.

Several assumptions and simplifications must now be made to render (1) more tractable. Primarily, a simpler parameterization of the drag is sought. Unfortunately, Co is known to vary markedly over the range of Reynolds numbers (Tin,ON, 1977) characterizing current meter deployments. The drag will therefore be arbitrarily written linear in Uc - U m with coefficient r because of the considerable analytical simplification this affords while still preserving the essential nature of the problem. The second term of

Savonius rotor overspeeding 155

(1) is simplified by assuming Xm a L (which will be true if the wave height is appreciably less than L). Equation (1) may now be written

which with

allows solutions of form

Substitution yields

BX., M dU~ r(Uc- Um) - - -- 0 (2)

L dt

giving

where

U~ = Re[Ue i°'t] (3)

Um= Re[aUe i('°t+~)]

X, .= Re[aUei(°~t+~)]

i B

(4)

(5)

(6)

where

o32 = gk tanh kH, (11)

g is the acceleration due to gravity, H the water depth and k and a are the wavenumber and amplitude of the wave at the surface. The recorded speed S is then given by

2U sin 6 S - - - (12)

RESULTS

It is apparent from (12) (and intuitively obvious) that it is desirable to minimize 151, the phase difference between the meter's and the water's velocity in order to minimize

5 = arctan((Bo3 Mo3)/r). (8)

The relative (oscillatory) current seen by the meter is hence given by

Uc- Um = Re[Ue i('°t+r'-'v2) sin 5]. (9)

If Uc is due to surface gravity waves then its amplitude is given (LEBLONO and MVSAK, 1978) by

cosh kL U = ao3 sinh kH" (10)

a = cos 8, (7)

156 D.A. GRIFFIN

the relative oscillatory current seen by the meter. From (8) it is seen that this is achieved when m 2 = B/(LM), i.e. when the mooring is forced at its natural frequency. At frequencies off resonance, it is evident that ~ is approximately inversely linear in r. Clearly then, r is a critical parameter but unfortunately it must be determined empirically for the tackle used if an accurate estimate of the overspeeding is sought.

To provide some example calculations of the overspeeding of some typical continental shelf deployments of current meters, an estimate of r based on tabulated drag data provided in the Aanderaa RCM4 manual will be used. This table lists the drag on an RCM4 and on a VINY 2090 float for several current speeds. Choosing to use four such floats (each provides 20 kg of buoyancy) and then attempting to include the drag on the mooring line by increasing the linear fit drag coefficient by 50%, an estimate of r of 180 N m -1 s results. The mass of this equipment is ~35 kg (being 19, 12 and 4 kg, respectively, for an RCM4 meter, floats and wire) and it has a net buoyant force of ~560 N.

The resulting estimate [using (12)] of the overspeeding per meter of swell amplitude as a function of instrument depth for various water depths and swell periods is shown in Fig. 1. Perhaps the most significant feature of Fig. 1 is the dramatic increase in the overspeeding as the mooring line is shortened to less than =20 m. This is because the shorter the mooring line, the greater is the buoyant restoring force per meter of horizontal displacement. With less horizontal freedom, the current meter is drogued less completely to the oscillating current so the overspeeding is greater. At intermediate

v

" r

r=/_ i , , a

OVERSPEEDING AT ZERO MEAN CURRENT (CM/S)

O"

I0-

20-

30-

40.

50-

60-

70.

aoJ

90-

100- II0-

120-

-5 5 -5 5 -5 _ _ . - ~ I i I

X \ "~i I I ""~\I '~ \ \ "i I

,-~, 77 ~ \ \

B=560 N /71,

R=I80 NI(MIS) M= 35 KG

SWELL PERIOD (S) 6 9 12 15

: ' I I I

I ' I I I

: I

5 -5 5

I

)\

77 k

\ \ \

/-~ 77

Fig. 1. The swell induced overspeeding per meter of swell amplitude as a function of current meter depth for various water depths and swell periods. The buoyancy, linear drag coefficient and

mass of the instrument package is shown.

Savonius rotor overspeeding 157

depths, for several of the cases shown, there is no overspeeding at all. This only occurs when the period of the swell equals the natural period of the mooring "pendulum", as discussed above. Closer to the surface, the overspeeding is apparently negative. This does not mean that "underspeeding" occurs but just that the mooring's own velocity lags the water's rather than leading it (as it does when the buoyant restoring force exceeds the inertia) so the sign of the relative current is reversed with respect to the phase of the wave passing overhead. As discussed in the Introduction, however, this paper has not set out to include the effects of vertical currents, which are strongest near the surface, so the near-surface estimates of Fig. 1 are to be treated with caution.

Finally, the reader is reminded of the further calculations necessary for an accurate estimate of the overspeeding for a real deployment: firstly, one must know the amplitude of the surface waves, as a function of time and frequency, in order to calculate a time- series of the overspeeding due to the integrated effect of the entire wave spectrum. Secondly, it should be pointed out that the presence of a non-zero "quasi steady" current Uqs will reduce the absolute overspeeding (and therefore greatly reduce the relative error), as discused by SHERWIN (1988). This effect will not be quantified here but briefly: if Uqs is parallel with Uc and of greater magnitude, then there will be no overspeeding at all because the current never reverses (but if Uqs < Uc then the overspeeding will also clearly be reduced). If Uq, is perpendicular to Uc then the overspeeding will be reduced because I Uqs --t- Uc{ < ] Uqs I + I Uc]. Obviously a blend of these two effects will generally apply.

DISCUSSION

The present paper has discussed how the overspeeding of a non-vector-averaging current meter deployed in continental shelf waters may be estimated and, indeed, how the mooring can be designed to minimize the overspeeding due to the horizontal component of surface wave induced currents. Primarily, the danger has been highlighted of placing the meter too close to the bottom in the mistaken belief that surface wave effects will be reduced at maximum depth. Secondly it has been shown that a reduced drag coefficient will also result in greater overspeeding (the price of reduced mooring lay- over for a given steady current). The optimum choice of buoyancy and mass depends on the mooring line length and expected wave frequency. This paper has not attempted to analyse the case where several meters are deployed on the one mooring line but it is clear that the importance of sympathetic mooring motion, as a means of reducing overspeed- ing, will carry over. It is often desirable for the purposes of the experiment to deploy meters at several depths throughout the water column. With vector-averaging current meters becoming more numerous, the trend has been to use these near the surface and the older meters at greater depth. The recommendation of this paper, however, is that vector-averaging meters should also be used near the bottom in shallow water deploy- ments.

Acknowledgements--This work was commenced with the support of Australian Marine Sciences and Technolo- gies Grants Scheme grant 85/994 and the Commonwealth Postgraduate Research Awards Scheme and was completed with the support of Natural Sciences and Engineering Research Council grant A7490. I would like to thank my Doctoral supervisor Dr J. H. Middleton for his enthusiastic guidance during my candidature.

158 D . A . GRIFFIN

R E F E R E N C E S

HALPERN D. and R. D. PILLSBURY (1976) Influence of surface waters on subsurface current measurements in shallow water. Limnology and Oceanography, 21,611-616.

LEBLOND P. H. and L. A. MYSAK (1978) Waves in the ocean, Elsevier, Amsterdam, 602 pp. PEARSON C. A., J. D. SCHUMACHER and R. D. MUENCH (1981) Effects of wave-induced mooring noise on tidal

and low-frequency current observations. Deep-Sea Research, 28, 1223-1229. SHERWIN T. J. (1988) Measurements of current speed using an Aanderaa RCM4 current meter in the presence

of surface waves. Continental Shelf Research, in press. TR1TTON D. J. (1977) Physicalfluid dynamics, Van Nostrand Reinhold (U.K.), Berkshire, 362 pp.