Impact of ADCP motion on structure function estimates of turbulent kinetic energy dissipation rate

. Turbulent mixing is a key process in the transport of heat, salt and nutrients in the marine environment, with fluxes commonly derived directly from estimates of the turbulent kinetic energy dissipation rate, ε . Time series of ε estimates are therefore useful in helping to identify and quantify key biogeochemical processes. Estimates of ε are typically derived using shear microstructure profilers, which provide high resolution vertical profiles, but require a surface vessel, incurring costs and limiting the duration of observations and the conditions under which they can be made. The velocity structure function method 5 can be used to determine time series of ε estimates using along-beam velocity measurements from suitably configured acoustic Doppler current profilers (ADCP). Shear in the background current can bias such estimates, therefore standard practice is to deduct the mean or linear trend from the along-beam velocity over the period of an observation burst. This procedure is effective if the orientation of the ADCP to the current remains constant over the burst period. However, if the orientation of a tethered (cid:58)(cid:58) the (cid:58) ADCP varies, a proportion of the velocity difference between bins is retained in the structure function and 10 the resulting ε estimates will be biased. Long-term observations from a mooring with three inline ADCP show the heading oscillating with an angular range that depends on the flow speed; from large, slow oscillations at low flow speeds to smaller, higher frequency oscillations at higher flow speeds. The mean tilt was also determined by the flow speed, whilst the tilt oscillation range was primarily determined by surface wave height. Synthesised along-beam velocity data for an ADCP subject to sinusoidal oscillation in a sheared flow indicates that the retained proportion of the potential bias is primarily determined 15 by the angular range of the oscillation, with the impact varying between beams depending on the

2 Potential Bias
Adopting a bin-centred difference scheme, the ::: The second-order structure function, D LL , for along-beam separation distance r n = n δz / cos θ :::::::: r n = n δr, where n is the number of bins separating the observations, δz is the vertical bin size and θ is the beam angle (from the vertical), is then evaluated : is :::: then :::::::: evaluated ::::: using :: a ::::::::: bin-centred ::::::::: difference :::::: scheme : as: (2) the angle brackets again indicating the arithmetic mean across the N profiles in the burst (Wiles et al., 2006).For odd n, the mean of the two offset bin difference options is taken.This approach yields individual D LL (i, j, r n ) values, allowing a vertical profile of ε estimates to be constructed (e.g. Simpson et al., 2015).An alternative approach evaluates all possible r n for a range of bins to give a representative value for the depth range (McMillan and Hay, 2017).

Field Observations of ADCP Motion
This section examines the heading and tilt sensor data from three in-line tethered ADCP deployed on a buoyancy-tensioned mooring at a site in the Celtic Sea with a water depth of 145 m over a sixteen month period, providing data under a wide range of current and wave conditions.Details of the deployments and the data return are given in Appendix ?? and additional ::::::: together :::: with information on the heading and tilt observations is provided ::: are ::::: given in Appendix ?? : A.

Moorings
Three Teledyne RD Instruments (TRDI) 600 kHz Workhorse ADCP were deployed, with the nominal depths of the upper, middle and lower instruments being 20 m, 33 m and 50 m respectively.The upper and lower instruments were deployed upwards-looking in spherical syntactic buoys, whilst the middle instrument was deployed downward-looking in an open frame :: as :::::::: illustrated :: in :::::: Figure : 3.All had four-beam, Janus-style transducer heads, with the upper and middle instruments having a 20°b eam angle and the lower a 30°beam angle.The same configuration was used for all instruments and deployment periods, with a vertical bin size of 10 cm and the first bin centred 0.97 m vertically from the transducer head.Pulse-pulse coherent (TRDI mode 5) single-ping ensemble (no averaging) observations of along-beam velocity were made at 1 Hz for 5 min followed by 15 min sleep, yielding 3 bursts of observations per hour, each comprising 300 profiles for each beam.Velocities were typically resolved for bins 1 to 32 (1 to 29) for the 20°(30°) beam angle instruments, consistent with the expected range for the operating mode (Teledyne RD Instruments, 1999).
The along-beam velocity data for each profile was converted to earth coordinates following Teledyne RD Instruments (2010).
The burst mean horizontal velocities were depth-averaged over the ∼ 3 m range of the observations and the dominant tidal constituents identified using the U-Tide Matlab functions (Codiga, 2011).The site is characterised by clockwise rotating semidiurnal tides, with a pronounced spring-neap variation.Over the full deployment period, the horizontal current speed, U , observed by the upper instrument had a median value of 0.28 m s −1 , with U ⩽ 0.1 m s −1 for just 4.1 % of observations.The implication being that the ADCP mooring was under almost continual drag, rotating semi-diurnally about the position of the anchor weight.

Heading Variation
For each ADCP and deployment period, the instrument heading, ϕ H , typically oscillated around a burst mean that rotated with the tide.For each burst, the heading data was analysed as the burst maximum heading range, ∆ϕ H , evaluated as the absolute difference between the minimum and maximum ϕ H expressed on a continuous basis, such that if the instrument completes a full rotation during the burst ∆ϕ H ⩾ 360°; and the number of heading oscillations per burst, n ϕ H , evaluated as the number of times ϕ H increased above the burst mean heading, ⟨ϕ H ⟩, such that ϕ H − ⟨ϕ H ⟩ changed from negative to positive.
Statistics for each instrument and deployment period are included in Appendix ?? : A. The middle instrument, mounted in an open frame, exhibited the largest amplitude oscillations, with ∆ϕ H ⩾ 180 • in more than 9 % of bursts during the "autumn" deployment period 3 (22 August to 20 22 nd :::::: August :: to : 20 th November 2014) and approximately 7 % of bursts during the "winter" deployment period 4, compared with 1 % to 2 % for the upper and lower instruments.The middle instrument was also typically subject to more oscillations per burst than the other instruments.The lower instrument typically exhibited the fewest and smallest amplitude oscillations.
Figure 5 illustrates the variation of ∆ϕ H and n ϕ H with the concurrent tidal current speed, U , and spectral significant wave height, H m0 , for the "winter" deployment period 4. U is the current speed from the burst mean horizontal earth coordinate velocity components, depth averaged across the reliably resolved bin levels.H m0 is calculated from the Triaxys buoy data as per equation ( 9) and interpolated to the ADCP observation times.Bursts are aggregated based on similar U and H m0 for 0 m s −1 ⩽ U ⩽ 0.7 m s −1 and 0 m ⩽ H m0 ⩽ 12 m with aggregation bin sizes δU = 0.0175 m s −1 and δH m0 = 0.3 m.The left, centre and right columns show the data for the upper, middle and lower ADCP respectively.Panels (a) to (c) show the mean of the maximum heading range, ∆ϕ H , for the bursts in each (δU , δH m0 ) aggregation bin; panels (d) to (f ) the mean number of heading oscillations, n ϕ H ; and panels (g) to (i) the percentage of bursts in each bin.Plots for the other deployment periods (not shown) demonstrate the same basic patterns, subject to the more limited H m0 range.
For all instruments, ∆ϕ H is highest when U is low, tending to decrease with increasing U .There is also evidence of ∆ϕ H increasing with H m0 , most clearly for the middle instrument.Conversely, n ϕ H , exhibits a clear tendency to increase with U for all instruments, but is relatively insensitive to variations in H m0 .The rate at which n ϕ H increases with U varies between the instruments, but they all exhibit the same basic response.
The variation from a few large oscillations at low U to an increasing number of smaller amplitude oscillations at higher U is consistent with the oscillations being primarily a hydraulic response.The relatively higher values of ∆ϕ H and n ϕ H for the middle instrument suggests that the open frame housing is more susceptible to motion than the spherical housing used for the other instruments.

Tilt Variation
The pitch and roll data for each profile was used to compute the tilted beam angle relative to the vertical, α i (°), for each beam i, as described in Appendix ?? :: A. Panels (d) to (f ) show the mean of the beam tilt variation range, ∆α, with ∆α being the mean across the beams of the difference between the burst maximum and minimum α i values for beam i. Panels (g) to (i) show the mean beam tilt oscillations per burst, n α , where n α is the mean across the beams of n αi which is evaluated as the number of times the sign of α i − ⟨α i ⟩ changes from negative to positive during the burst.The plots for other deployment periods (not shown) are similar, subject to the more limited H m0 range.
The mean beam tilt angle, δ⟨α⟩, exhibits a clear dependence on U , increasing with increasing U for all instruments, the effect being relatively weaker for the upper instrument and strengthening with instrument depth.The mean beam tilt angle inevitably understates the tilt for individual beams e.g. for the lower instrument δ⟨α⟩ ⩾ 10 • for just 0.6% of bursts during deployment 4, although 4.6% of bursts had at least one beam with that level of tilt.In such circumstances the opposing beams will differ significantly in their orientation to the prevailing current, as well as spanning different vertical ranges.
The mean burst tilt range, ∆α, clearly increases with increasing H m0 , suggesting that the range of the rocking motion about the tilt axes is primarily driven by the surface wave forced orbital motion.This is consistent with the upper buoy on the mooring rising and falling with the wave, thereby varying the vertical angle of the mooring.Some tendency for ∆α to increase with increasing U is also apparent for the middle instrument and, to a lesser extent, the upper instrument.Large ranges are observed for both the upper and middle instrument, with the mean across the beams exceeding 20 • in 0.3% of bursts and at least one beam exceeding 20 • in 1.3% of bursts for the middle instrument during this deployment period, the equivalent figures for the upper instrument being 0.2% and 1.0% respectively.The beam tilt range is significantly reduced for the lower instrument, consistent with ∆α being influenced by surface waves.
The variation in the mean beam tilt oscillation frequency, as indicated by n α , is relatively limited.The highest values affecting the middle and lower instruments and occurring at low H m0 but with no consistent trends.

Retained Bias in Synthesised Sheared Flow
The observations demonstrate that tethered ADCP may be subject to both a mean tilt due to drag on the mooring, as well as significant oscillatory variation in both heading and tilt over the period of an observation burst.In the presence of a sheared flow, this motion will unavoidably result in a proportion of the non-turbulent velocity difference between bins being retained in b ′ , contributing to the structure function and biasing the ε estimates :::::: derived ::::: using ::: the ::::::: standard ::::::::: regression ::::::: method.
This retained bias was investigated using synthesised velocities for a range of scenarios with the ADCP subject to oscillatory variations in heading, pitch and roll.For each scenario, along-beam velocities, b, were synthesised for a burst of observations following the procedure detailed in Appendix ?? : B. The ADCP geometry was based on the TRDI Workhorse ADCP, with a default beam angle θ = 20°and a vertical bin size δz = 0.1 m, with bin 1 centred at δz 1 = 1 m and 30 bins per beam.The default observation burst comprised 300 profiles at 1 Hz.
No turbulence was introduced in either the along-beam velocities or the structure function, therefore ε s i was the retained bias due to the motion of the ADCP.
ε s i values were normalised as a proportion of the potential bias, ε b 45 , calculated from the along-beam velocity, b, for the same background flow and ADCP configuration, with the ADCP vertical, static and oriented with the heading at 45°to the background flow direction, such that each beam has the same difference angle to the flow and therefore the same potential bias.
The default background flow was specified with a speed at the ADCP transducer head U 0 = 0.25 m s −1 ; with depth constant direction (to) β = 90 °N; shear S 2 = 1 × 10 −4 s −2 ; and no surface waves.Testing confirmed that the results were insensitive to U 0 and that both ε s i and ε b 45 scaled as S 3 , such that ε s i /ε b 45 was independent of S 2 .Beam 3 (yellow line) is initially oriented directly downstream, so that b 3 (16, 0) has a maximum negative value.As the heading changes, the magnitude of b 3 (16, t) reduces as the cosine of the heading difference angle, reaching a minimum at t ϕ H /4 then increasing to regain its maximum value at t ϕ H /2; the variation repeating over the second half of the oscillation period.Compared with b 1 (16, t), b 3 (16, t) varies with double the oscillation frequency but a much smaller amplitude and has a non-zero mean.Symmetry again means that the b 4 (16, t) (purple line) has the same magnitude as b 3 (16, t) but opposite sign.

Heading Variation Example
Since the burst mean for beams 1 and 2 is approximately zero, the periodic variation in b is fully retained in b ′ , including any velocity differences between bins due to the sheared flow.Conversely, for beams 3 and 4, the variation in b is greatly reduced, so that the majority of velocity difference between bins is not retained in b ′ .This is reflected in panel (c), which shows the time series for δb ′ for bin 16 with r n = r max (19 δr) for each beam, δb ′ i (16, 19 δr) (mm s −1 ).The opposing beams in each beam pair have identical values but opposite sign, whilst the magnitude of the oscillation for beams 1 and 2 is clearly much larger than that for beams 3 and 4.
Values for the opposing beam pairs are identical, with the burst mean for beams 1 and 2 (red line overlying blue line) clearly significantly larger than that for beams 3 and 4 (purple line overlying yellow line).
Panel (e) shows the structure function D LL (i, 16) for each beam and a range of r n values, including r max indicated by the vertical green line, plotted against r 2 3 , demonstrating both the marked difference between the beam pairs and the non-linear growth of D LL with r 2 3 .Again, beams 1 (solid blue line) and 2 (red bullet markers) are identical, as are beams 3 (solid yellow line) and 4 (purple bullet markers).The dotted blue (beam 1) and yellow (beam 3) lines indicate the linear regression fit for all r n ⩽ r max with no restriction on the regression intercept.
The annotation in panel (e) shows the normalised residual bias ε s i /ε b 45 for each beam, indicating the retained fraction of the potential bias in each beam.For this scenario the residual bias arises almost exclusively in beams 1 and 2, which have a mean alignment across the current direction and are only exposed to the current by the oscillation, whilst the contribution from beams 3 and 4, which are closely aligned with the current direction, is negligible.
The results are summarised in Figure 8. Panel (a) shows the variation of the beam averaged normalised residual bias, ε s /ε b 45 , the underline indicating the mean of ε s i across the four beams, with the difference angle between the initial ADCP heading and the background flow direction, ψ = β − ϕ H (0), for selected heading oscillation ranges, ∆ϕ H , and a fixed heading oscillation period t ϕ H of 30 s.Since the heading oscillates around ϕ H (0), the burst mean heading ⟨ϕ H ⟩ ≈ ϕ H (0), with any slight difference arising from the burst period not being an exact multiple of the oscillation period.Hence ψ is also the burst mean heading offset angle relative to the background flow.

Tilt Variation Example
Figure 9 illustrates the impact of oscillation on the pitch tilt axis for a sample scenario with constant heading and no roll, with all panels as described in Figure 7.The initial pitch angle ϕ P (0) = 0°, the oscillation range ∆ϕ P = 20°and the oscillation period t ϕ P = 30 s.The heading is constant and aligned with the background flow and there is no tilt on the roll axis i.e. ϕ H (t) = β and ϕ R (t) = 0°for all t.
Panel (a) shows the sweep of the beams.At t = 0 s ::::: (circle :::::: marker), ϕ P is 0°and the instrument is vertical, such that beams 1 and 2 (blue and red) are oriented normal to the current and their along-beam velocities are zero.As ϕ P (t) becomes positive, beam 3 :::::: (yellow) : is tilted towards the vertical, so that its bins are higher in the water column than those in beam 4 ( ::::::: purple), as indicated by the position of square markers for the bin 30 positions after 2 s).This tilts beams 1 and 2 slightly upstream and b becomes positive for all bins in both beams ::: (red :::: line :::::::: overlying :::: blue :::: line), increasing to a positive maximum at t ϕ P /4 when ϕ P (t) = ∆ϕ P /2 ::: (just ::::: prior :: to ::: the ::::::: diamond ::::::: marker :: at :::::: t = 8 s), then reducing as ϕ P declines, so that both are zero again at t ϕ P /2, as shown in panel (b).As ϕ P becomes negative, beams 1 and 2 are both tilted slightly downstream and b becomes negative, reaching a maximum negative value at 3t ϕ P /4 when ϕ P (t) = −∆ϕ P /2 ::::: (close :: to :::: the :::::: triangle ::::::: marker :: at ::::::: t = 22 s), before returning to zero after a full oscillation period.Consequently, for beams 1 and 2, b is the same, oscillating in phase between positive and negative values with period t ϕ P and with the burst mean ⟨b i ⟩ ≈ 0 m s −1 .The slight differences in the depth ranges of the beams result in slight differences in δb ′ i between the beams, as can be seen in panel (c).Whilst the variation is identical for beams 1 and 2 (red line overlying blue line), the |δb ′ 3 | maximum during the positive ϕ P phase of the oscillation is larger than during the negative ϕ P phase of the oscillation, with the situation reversed for beam 4.This is clearer in panel (d), which shows [δb ′ i ] 2 .Beams 1 and 2 are identical, with the largest maxima and identical values during both the positive and negative ϕ P phases, whilst the maxima for beams 3 and 4 are lower and differ between the phases, such that the beam 3 values are larger during the positive ϕ P phases and the beam 4 values during the negative ϕ P phases.
The differences between beams 3 and 4 during the positive and negative phases of the oscillation are symmetric, therefore the burst mean values used by the D LL are identical, as shown in panel (e).Beams 3 and 4 yield identical results with ε s i /ε b values approximately 30 % lower than those for beams 1 and 2, for which the normalised residual bias as a result of the ADCP motion is ∼ 0.1.
Oscillation about the roll axis, which in this scenario is oriented along the background flow, has no impact on b i for beams 1 and 2 which remain normal to the flow throughout the burst.The roll oscillation has a minimal impact on the vertical observation range for beams 3 and 4 resulting in a normalised residual bias in these beams of O10 −6 , highlighting the significance of the instrument orientation to the background flow on the impact of oscillation around the individual tilt axes.

Discussion
The standard structure function methodology assumes that the along-beam velocities observed by an ADCP can be decomposed into a component due to the background flow and the time-varying turbulent velocities required to calculate ε.Deducting the mean or linear trend over a burst of observations for each bin therefore removes the component due to the background flow, including any non-turbulent velocity differences between bins due to shear.For this assumption to be valid, there must be no spatially-varying periodic non-turbulent velocity contribution to the observed velocity, such as that due to surface waves or, as considered here, due to the motion of the ADCP in a sheared background flow.
If the orientation of the ADCP varies, the burst mean velocity in any bin unavoidably underestimates the background flow contribution in some profiles and overestimates it in others.If the background flow is sheared, the residual velocity when the burst mean or linear trend is deducted will include a proportion of the associated non-turbulent velocity differences between bins.
The potential contribution to the second-order structure function if the velocity differences due to linear shear in the background flow were wholly retained in the residual velocity , is here shown to scale with the square of both the shear and the separation distance : , ::: see :::::::: equation ( 8).The potential bias will therefore scale as the cube of the shear and will be sensitive to both the choice of the maximum separation distance over which the structure function is evaluated and the ADCP bin size (which determines the number of resolved separation distances).
Data from long-term deployments of three ADCP mounted inline on a buoyancy-tensioned mooring demonstrates the instruments oscillating in heading, pitch and roll.The heading variation was found to vary between fewer, larger amplitude oscillations when the background flow is slowest and a higher number of smaller amplitude oscillation as the background flow speed increased.The background flow speed also directly influenced the mean tilt angle for the instruments as the drag determines the shape of the mooring.Surface waves had some influence on heading variation, however the impact was most apparent in the range of the tilt oscillation.There was also evidence that the way in which the ADCP was mounted influenced the movement, with the instruments in spherical syntactic buoys subject to less motion than that in an open frame.
Synthesised along-beam velocity data based on a standard TRDI Workhorse ADCP geometry was used to evaluate the impact of instrument motion in a linearly sheared flow.The residual bias was normalised by the potential bias for the defined geometry, background flow and with all beams having the same relative orientation to the flow.
Based on a wide range of synthesised scenarios, the normalised residual bias was found to be primarily determined by the oscillation angular range, both for heading and instrument tilt.
Testing indicated that the normalised residual bias becomes increasingly significant for heading angular oscillation ranges exceeding 50°, with the possibility of the full potential bias being retained in one or more beams if the angular range exceeded 140°.The frequency of occurrence of heading oscillations exceeding 50°in the observations examined was dependent on the instrument mounting, but affected more than 50 % of observations for the instrument mounted in an open frame during some deployments.Furthermore, since the heading oscillation was unconstrained, angular variations of over 360°occasionally occurred.
Oscillation on the tilt axes is inherently constrained by the tension in the mooring, therefore the potential angular range is limited.The synthesised scenarios suggest that the beam-averaged normalised residual bias due to tilt oscillation will reach 10 % only under exceptional circumstances.However, the maximum residual bias for an individual beam, which increases with the total of the pitch and roll angular ranges, can reach 30 % of the potential bias under exceptional circumstances.
This analysis suggests that under most circumstances the motion of a tethered ADCP is unlikely to be a significant source of errors in ε estimates derived using the ::::::: standard : structure function methodology.However, since the potential bias scales with the cube of the shear and depends on factors such as the bin size and the length scale over which the structure function is evaluated, there may be circumstances in which it is significant.Furthermore, since the level of retained bias is dependent on the motion of the ADCP, it is relevant to identify this as an issue for consideration both as part of the deployment planning and of the data quality assurance and analysis.The following suggestions may therefore be of interest to other researchers: 1. Mooring design: Mounting the ADCP in a streamlined buoy designed to maintain a fixed orientation relative to the background current is recommended for all deployments on a tethered mooring.If that isn't an option, mounting the ADCP in a spherical buoy is likely to result in less motion than using an open frame.
2. ADCP configuration: Ensure that the instrument orientation sensors (heading, pitch and roll) are working properly and that the instrument is configured to save the data at the same temporal resolution as the velocity profiles.
3. Initial QA: Check for periodic variations in heading, pitch or roll to determine whether the ADCP was subject to significant motion during the observation bursts.In particular, evaluate the heading angular variation range ∆ϕ H , with ∆ϕ H ⩾ 50°suggested as a threshold above which the possibility of bias should be considered.
4. Initial QA: Check for periodic variation in the along-beam velocity data collected.One option is to examine the burst variance of the along-beam velocity and check for any monotonic trend in the variance between bins, which may indicate a non-turbulent contribution and potential cause of bias.
5. Shear: Convert the along-beam velocity data to earth coordinates and determine the level of shear.This can be used to determine the maximum potential bias by computing the sheared structure function D b LL as per equation ( 8) based on the bin size and beam angle and then calculating the potential bias ε b for the proposed r max .
6. Structure function QA: Check for non-linearity of D LL versus r 2/3 .This is perhaps most easily achieved by examining the sensitivity of ε to increasing r max , with an increasing trend probably indicating a non-turbulent contribution to D LL and therefore a bias in the calculated ε values.
7. Structure function QA: Test whether ε values are more independent of r max when using the modified regression equation (6).If so, this suggests a non-turbulent contribution to D LL but care should be taken to determine the source of the non-turbulent contribution and verify that the associated velocity difference between bins varies linearly with separation distance before assuming the modified method is applicable.
clockwise rotation, then reduces by ∼ 60°between bursts and remains fairly constant over a 2 hour period (7 bursts), at the end of which it jumps by ∼ 90°and reverts to tracking the rotating tide.During this hiatus, both U and Φ are in excellent agreement with the other instruments, but the subsequent jump in ⟨ϕ H ⟩ introduces an offset of ∼ −30°in Φ.Approximately 4 hours later, the offset changes sign over a period of ∼ 1 hour, the transition coinciding with ⟨ϕ H ⟩ progressing through 360°/ 0°.The offset subsequently changes sign again as ⟨ϕ H ⟩ increases past 180°and again when it next progresses through 360°/ 0°.A second sudden change in ⟨ϕ H ⟩ between bursts occurs at circa 20:00 the same day, just prior to the second transition through 360°/ 0°, but affects just a single burst.
The incidence of such events was rare, with no clear periodicity apparent, albeit mostly occurring when U was low during neap tides, suggesting the possibility of a mechanical cause.However, the coincidence of the change in sign of the offset in Φ with the progression of ⟨ϕ H ⟩ through 180°and 360°/ 0°suggests a compass sensor problem is more likely.Despite this problem affecting the calculation of the earth coordinate current direction for some bursts, there is no indication of any problems with the variation of ϕ H during a burst.
Panel (b) shows that the variation in ϕ H was limited during the majority of bursts.However, in each of two successive burst at circa 20:00 on 7 February, the lower instrument completes an anticlockwise rotation over a period of ∼ 90 s, with the heading then returning to a similar value to that prior to the rotation.Over the rest of the burst, the heading varies over a range ∼ 30°as in other bursts.The events coincide with U being at a minimum and the direction of rotation is opposite to the rotation of the tide, suggesting the effect may be due to a relaxation of accumulated tension in the mooring.
Panels (c) to (e) show the time series of ϕ H , ϕ P and ϕ R for the individual burst identified by the green box in panel (b).The plots show that the instruments all oscillate throughout the period of the burst, the frequency and amplitude of the oscillation varying between instruments.The range and frequency of these oscillations are examined further in the following sections.

A1 Heading
Table A2 provides information on the heading variation for each instrument and each deployment, together with the number of observation bursts, n obs , and the mean depth, z (m).The middle instrument is subject to significantly higher levels of heading variation, both in terms of the the range of the angular variation and the number of oscillations per burst.This is interpreted as being a consequence of the different housing used for the instruments in the mooring -the upper and lower instruments being embedded within a spherical syntactic buoy, whilst the middle instrument was in an open frame.
The same housings were used for each deployment, so the differences in the ∆ϕ H and n ϕ H distributions between deployments for the individual instruments must arise either from performance differences of mooring elements e.g.swivels or wires, or from differing environmental conditions.

A2 Tilt
Pitch and roll, ϕ P and ϕ R , typically have a constant sign throughout an observation burst, with the burst mean values ⟨ϕ P ⟩ and ⟨ϕ R ⟩ tending to have a consistent sign throughout a deployment.This indicates that the initial orientation of the instruments in the mooring results in a preferred orientation relative to the plane of the mooring, which persists throughout the deployment with limited variation, despite the rotation of the mooring with the tide.
Tables A3 and A4 provide summary statistics for the pitch and roll data for each instrument during each of the deployments.
For the absolute burst mean tilts, |⟨ϕ P ⟩| and |⟨ϕ R ⟩|, the tables show the deployment mean, |⟨ϕ P ⟩| and |⟨ϕ R ⟩|, together with the percentage of bursts ⩾ 5°and ⩾ 10°.For the burst oscillation ranges, ∆ϕ P and ∆ϕ R , and the burst oscillation counts, n ϕ P and n ϕ R , the tables show the deployment mean together with the 10, 50 and 90 10 th and : 90 th percentiles.∆ϕ P and ∆ϕ R being evaluated as the absolute difference between the burst minimum and maximum ϕ P and ϕ R respectively; and n ϕ P and n ϕ R being evaluated as the number of instances during a burst when the tilt increases through the burst mean e.g. when ϕ P − ⟨ϕ P ⟩ changes from negative to positive.
The non-zero values for |⟨ϕ P ⟩| and |⟨ϕ R ⟩| suggest that the instruments were typically tilted from the vertical during a burst.
The percentage of bursts with high |⟨ϕ P ⟩| or |⟨ϕ R ⟩| tends to be highest for the lower instrument and lowest for the upper instrument, consistent with the mooring exhibiting a catenary shape due to lateral loading.
The deployment mean burst ranges, ∆ϕ P and ∆ϕ R tend to decline with instrument depth and to vary in a consistent manner between deployments, being highest during the "autumn" and "winter" deployments 3 and 4 and lowest during the "summer" deployment 2.
The oscillation frequency, as indicated by n ϕ P and n ϕ R , is consistent across all instruments and deployments, being higher than the equivalent n ϕ H , particularly for the upper and lower instruments.
In order to evaluate the combined impact of pitch and roll, the tilted beam angle relative to the vertical, α i for beam i, was calculated for each beam, following Woodgate and Holroyd (2011).The true pitch, ϕ Pt , correcting for the influence of roll, is to the background flow, u i,n (t) = [u i,n (t), v i,n (t), w i,n (t)], and surface waves, u i,n (t) = [ u i,n (t), v i,n (t), w i,n (t)].Finally, we determine b i,n (t) as the along-beam component of u i,n (t) = u i,n (t) + u i,n (t) in the rotated beam coordinates and assuming that there is no turbulence such that b ′ is zero.Note that synthesised velocities are calculated directly from the specified background flow and any surface waves, without any allowance for observational noise or the spatial and temporal averaging that will affect actual observations to differing degrees depending on the operating mode.
All scenarios were based on a Teledyne RDI Workhorse ADCP beam geometry, with a four beam Janus-style convex transducer head, such that with the instrument vertical, all beams have the same angle to the vertical, θ (°); with heading angle, ϕ H (°N), indicating the compass direction of the horizontal projection of beam 3; pitch, ϕ P (°), indicating the rotation in the plane of beams 3 and 4; and roll, ϕ R (°), indicating rotation in plane of beams 1 and 2, with both ϕ P and ϕ R being 0°indicating that the ADCP is vertical and the convention for direction of rotation being as per Teledyne RD Instruments (2010), taking account of whether the ADCP is specified as upward-or downward-facing.
A standard burst configuration of 300 profiles collected at 1 Hz was adopted, with 30 bins per beam and a default vertical bin size of δz = 0.1 m with bin 1 centred at δz 1 = 1.0 m.

B1 Bin Positions
Bin positions were calculated in Cartesian coordinates relative to the ADCP transducer head, with the x-axis oriented due East, the y-axis due North and the z-axis pointing vertically upward, such that the transducer head is at [x, y, z] = [0, 0, 0].
Unit vectors describing the orientation of each beam with ϕ H , ϕ P and ϕ R all 0°and the ADCP upward-facing are then: and the along-beam bin centre position for all bins, common to all beams, is: where N is the number of bins.The non-rotated coordinates for all of the bins in beam i are then given by X i = X i R, as illustrated in panel (a) of Figure A3.
Heading variation was prescribed as a sinusoidal oscillation, with an initial angle, ϕ H (0) (°N), an oscillation angular range, ∆ϕ H (°), and a heading oscillation period, t ϕ H (s), such that at profile time t: with t varying from 0 s to 299 s over the burst and ∆ϕ H = 0°or t ϕ H = 0 s indicating a constant heading.Similarly the pitch variation over the burst was defined as: and the roll variation as: with the only difference being the option to additionally specify δt R as a phase offset.
Bin positions at time t are then determined by rotation about the appropriate axes, with ϕ H describing rotation about the z-axis, ϕ P the x-axis and ϕ R the y-axis as: subject only to ϕ R = ϕ R + 180°if the ADCP is specified as downward-facing.The positions for all bins in beam i are then given by: B2 Velocity due to the Background Flow A steady horizontal current, u, is defined with a speed at the transducer head depth, u 0 (m s −1 ); compass direction (to), β (°N); and vertical shear-squared, S 2 (s −2 ) with S assumed to be positive such that current speed u increases towards the surface and S 2 = 0 s −2 indicating that the flow velocity is constant over the depth range.
The background flow velocity in earth coordinates at the beam i bin locations for time t is then given by: with U i (t), V i (t) and W i (t) being the velocity components along the x-, yand z-axes respectively.

B3 Orbital Velocity due to Surface Waves
For a monochromatic surface gravity wave, linear wave theory describes the orbital motion as: where u is the velocity component in the direction of wave propagation; w is the vertical velocity component; x is the distance in the direction of wave propagation; z is depth referenced to the surface and positive upwards; t is time; g is acceleration due to gravity; k is wavenumber given by k = 2π/λ where λ is the wavelength; A 0 is the surface amplitude of the wave; and ω is the radian frequency given by ω = ck where c is the wave phase speed from the wave dispersion equation: with h being the water column height, such that z = −h at the seabed (Phillips, 1977).
From equation (B10), the wave orbital motion velocity in earth coordinates at the beam i bin locations for time t is given by: where α (°N) is the wave propagation compass direction (to); Z i (t) = Z i (t) + z 0 is the beam i bin depths referenced to the sea surface given an ADCP depth z 0 ; and X i (t), is the array of rotated beam i bin positions relative to the direction of wave propagation, calculated as: being the scalar dot product of the horizontal components of the rotated beam bin positions and the horizontal unit vector for the wave propagation direction, as illustrated in panel (b) of Figure A3.

Beams 3
and 4 (yellow and purple) initially have a symmetrical orientation downstream and upstream respectively, such that for any bin, b 3 (0) = −b 4 (0).As ϕ P becomes positive, the change in the relative orientation of beam 3 to the horizontal current reduces the magnitude of the along-beam velocity component |b 3 |, as shown in panel (b), despite the change in the bin depths increasing the local current speed.In contrast, beam 4 is tilted towards the horizontal, the change in orientation resulting in |b 4 | increasing, despite the reduction in the local current speed at the new bin depths.As the pitch oscillation continues, b 3 and b 4 vary in phase with each other, with ⟨b i ⟩ ≈ b i (0).

Figure 10
Figure 10 shows the mean (black line), 25 % to 75 % range (dark grey shading) and 5 % to 95 % range (light grey shading) for the beam averaged normalised residual bias ε s /ε b 45 , together with the 95 95 th percentile (dotted line with triangle markers) and maximum (grey line with square markers) individual beam normalised residual bias ε s i /ε b 45 for scenarios aggregated by: panel (a) the heading offset angle to the background flow, ψ; panel (b) the sum of the absolute values of the initial tilt angles, |ϕ P (0)| + |ϕ R (0)|; and panel (c) the sum of the absolute values of the tilt oscillation ranges, |∆ϕ P | + |∆ϕ R |.
s i /ε b 45 (dotted line with triangle marker), which closely tracks that of the beam averaged values, and the beam maximum (grey line with square marker).They vary in anti-phase, with maximum ε s i /ε b 45 values of ∼ 0.3 occurring with ψ ∼ ±45°or ±135°.Panel (b) shows that the mean and range of ε s /ε b 45 exhibit minimal dependence on the mean tilt, as indicated by the sum of the initial tilt angles |ϕ P (0)| + |ϕ R (0)|, again recognising that the specified tilt oscillation means that ⟨ϕ P ⟩ ≈ ϕ P (0) and ⟨ϕ R ⟩ ≈ ϕ R (0).The 5 % to 95 % range actually narrowing slightly as the mean tilt increases.The 95 95 th percentile of the individual beam ε s i /ε b 45 values is also effectively constant, whilst there is a gradual increase in the maximum ε s i /ε b 45 values as |ϕ P (0)| + |ϕ R (0)| increases from 0°to 6°, above which it is relatively constant.Panel (c) indicates that for the scenarios examined, the residual bias is primarily determined by the total absolute oscillation range, |∆ϕ P |+|∆ϕ R |.The mean ε s /ε b 45 is ∼ 0 for |∆ϕ P |+|∆ϕ R | ⩽ 15°, gradually increasing to a maximum of ∼ 0.09 ::::: (black :::: line).The range of ε s /ε b 45 values is narrow for all |∆ϕ P |+|∆ϕ R | options.The 95 95 th percentile of the individual beam ε s i /ε b 45 values closely tracks that of ε s /ε b 45 for |∆ϕ P | + |∆ϕ R | ⩽ 20°, above which is increases at a slightly higher rate.This is also reflected in the beam maximum ε s i /ε b 45 values, which grows at an increasing rate, exceeding 0.3 for the extreme scenarios with |∆ϕ P | + |∆ϕ R | approaching 40°.

A 0
(m); and compass direction of propagation (to), α (°N).The depth of the ADCP, z 0 (m), was specified within the range −50 m ⩽ z 0 ⩽ −20 m and a standard water depth of h = 145 m was used for all scenarios.

Figure A1 .
Figure A1.Cumulative probability of (a) -(c) burst heading range, ∆ϕH , and (d) -(f ) mean number of oscillations per burst, n ϕ H , by instrument and deployment, as per legend in panel (a).

Figure A2 .Figure A3 .
Figure A2.Cumulative probability of (a) -(c) burst tilt range, ∆α, and (d) -(f ) mean number of tilt oscillations per burst, nα, by instrument and deployment, with line colour indicating the deployment as shown in panel (a); tables in panels (a) -(c) show the percentage of bursts when ∆α exceeded 5°and 10°; tables in panels (d) -(f ) show the percentage of bursts when nα ⩾ 50.
The burst maximum heading variation, ∆ϕ H , is evaluated as the absolute difference between the minimum and maximum ϕ H expressed on a continuous basis, such that if the instrument completes a full rotation during the burst ∆ϕ H ⩾ 360°.The table shows ∆ϕ H , being the mean ∆ϕ H for the instrument over the deployment period, together with the 10, 50 and 90 10 th , : 50 th : 90 th percentile values and the percentage of bursts for which ∆ϕ H ⩾ 360°.The number of heading oscillations per burst, n ϕ H , was evaluated as the number of times ϕ H increased above the burst mean heading, ⟨ϕ H ⟩, such that ϕ H −⟨ϕ H ⟩ changed from negative to positive.The table shows n ϕ H , being the mean across all bursts; together with the percentage of bursts for which n ϕ H ⩽ 1; and the 50 and 90 50 th and