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Article Type:
Research Article

A New Rotating Axes Method for Processing High-Resolution Horizontal Velocity Measurements on EM-APEX floats

Je-Yuan Hsu aInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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https://orcid.org/0000-0002-4229-2633
Online Publication:
26 Mar 2024
Print Publication:
01 Mar 2024
DOI:
https://doi.org/10.1175/JTECH-D-23-0014.1
Page(s):
319–339
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Abstract

A new rotating axes method (RAM) is developed to improve the vertical resolution of the horizontal current velocity measurements u at EM-APEX floats. Unlike the traditional harmonic fitting method (HFM), which yields u averaged in 50-s intervals, RAM decodes and interprets 1-Hz measurements of horizontal seawater velocity u˜, and averages u˜ in 12-s windows for removing wind waves with a typical peak frequency ∼ 0.12 Hz. Estimates of u from RAM agree with those from HFM but with a higher vertical resolution of ∼1.5 m, 4 times better than HFM. Note that extracting float signals due to seawater motion needs to assume slow-varying voltage offset ΔΦoffset. The typical variations of estimated ΔΦoffset do not affect the results of u significantly. Estimates of u are excluded when ΔΦoffset fluctuates strongly in time and scatter significantly. RAM is applied to float measurements taken near Mien-Hua Canyon, Taiwan. Composite vertical shear spectra Ψ computed using u from RAM exhibit a spectral slope of −1, as expected for the saturated internal waves in the vertical fine-scale range. The RAM provides EM-APEX float’s horizontal velocity measurements into fine vertical scales and will help improve our understanding of energy cascade from internal wave breaking and shear instability into turbulence mixing.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Je-Yuan Hsu, jyahsu@ntu.edu.tw

Abstract

A new rotating axes method (RAM) is developed to improve the vertical resolution of the horizontal current velocity measurements u at EM-APEX floats. Unlike the traditional harmonic fitting method (HFM), which yields u averaged in 50-s intervals, RAM decodes and interprets 1-Hz measurements of horizontal seawater velocity u˜, and averages u˜ in 12-s windows for removing wind waves with a typical peak frequency ∼ 0.12 Hz. Estimates of u from RAM agree with those from HFM but with a higher vertical resolution of ∼1.5 m, 4 times better than HFM. Note that extracting float signals due to seawater motion needs to assume slow-varying voltage offset ΔΦoffset. The typical variations of estimated ΔΦoffset do not affect the results of u significantly. Estimates of u are excluded when ΔΦoffset fluctuates strongly in time and scatter significantly. RAM is applied to float measurements taken near Mien-Hua Canyon, Taiwan. Composite vertical shear spectra Ψ computed using u from RAM exhibit a spectral slope of −1, as expected for the saturated internal waves in the vertical fine-scale range. The RAM provides EM-APEX float’s horizontal velocity measurements into fine vertical scales and will help improve our understanding of energy cascade from internal wave breaking and shear instability into turbulence mixing.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Je-Yuan Hsu, jyahsu@ntu.edu.tw
Keywords: Mixing; Algorithms; In situ oceanic observations; Measurements; Oceanic profilers

1. Introduction

In the turbulent kinetic energy (TKE) budget, the vertical shear of horizontal velocity plays an important role in converting the kinetic energy of mean flow to turbulence via shear production (Strang and Fernando 2001). The wind-driven shear turbulence may act to cool the temperature in the surface mixed layer (e.g., Price et al. 1994; Hughes et al. 2020). In most of the interior ocean, shear instability is the major mechanism for energy transfer from internal waves to microscale turbulence (Gregg et al. 1993; Thorpe 1999; van Haren and Gostiaux 2010; Kunze 2019; Munk and Wunsch 1998). The vertical resolution of velocity measurements strongly affects the estimation of vertical shear in the studies of turbulence mixing because oceanic vertical shear is often stronger at vertical fine scales, <10 m. Therefore, observing the fine-scale horizontal current velocity is crucial for understanding the processes of internal wave breaking, shear instability, and turbulence mixing.

In the modern technology for exploring the ocean current, vertical profiles of horizontal current velocity are typically measured by acoustic Doppler current profilers (ADCPs) either mounted on research vessels (RVs), moorings, or other oceanic platforms, such as Seagliders and Remus (Todd et al. 2017). Different frequencies of ADCPs will affect the vertical resolution of velocity measurements and the corresponding measurement range, e.g., 8-m vertical resolution and ∼500-m depth range for 75-kHz ADCP, and 2-m vertical resolution and ∼100-m depth range for 300-kHz ADCP. Choosing which types of ADCPs on the platforms depends on the needs in the vertical resolution and depth range of velocity measurements, higher vertical resolution with a shorter depth range, or lower vertical resolution with a longer depth range. Lowered ADCP (LADCP) on the RVs has been used to collect the high vertical resolution of measurements to the deeper ocean but might suffer from operational challenges and instrumental noises. The horizontal velocity measurements taken by the ADCP also suffer from uncertainties from various factors (Mueller et al. 2007). The velocity error of single-ping ADCP measurements is about 0.3 m s−1 (Rennie 2008). Because the uncertainty of velocity measurements can be reduced by averaging multiple pings, the typical velocity uncertainty of ADCPs for 10-pings average is ∼0.01–0.02 m s−1 under low-vertical-shear environments (Chereskin and Harding 1993; Klema et al. 2020). The magnitude of velocity errors also increases when the vertical shear becomes stronger.

Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats as autonomous vehicles measure electric current due to the conductive seawater motion in the geomagnetic field based on the motional induction theory (Longuet-Higgins et al. 1954; Weaver 1965; Sanford 1971). The floats can operate stably even under strong turbulence environments (Sanford et al. 2011). Two pairs of electrodes are mounted on the float. An array of vertically tilted vans is attached to the body of the float, so the relative vertical motion to the surrounding seawater causes the floats to rotate during profiling. The voltage induced by the seawater motion, measured by each pair of electrodes in a 1-Hz sampling rate, exhibits sinusoidal variations due to the float’s rotation.

Sanford et al. (1978) develop the harmonic fitting method (HFM; appendix A) to convert the measured voltage signals on EM-APEX floats to the oceanic horizontal current velocity. Amplitudes of sinusoidal variations are fitted by the least squares method in a 50-s window to derive the averaged horizontal current velocity u. The root-mean-squared error of u measured by EM-APEX floats is ∼1–2 cm s−1 (Sanford et al. 1978). The vertical resolution of independent velocity measurements is approximately 5–7 m at the typical profiling speed ∼0.1–0.14 m s−1. EM-APEX float velocity measurements have been used to study wind-driven current (e.g., Sanford et al. 2011), internal tidal energy flux (Lien et al. 2013), small-scale potential vorticity (Lien and Sanford 2019), wind-forced near-inertial waves modulated by eddies (Essink et al. 2022). Other sensors mounted on EM-APEX floats include SBE41 CTD, pressure sensor, and microstructure temperature sensor FP07 (Sanford et al. 2005; Lien et al. 2016).

Here, a new rotating axes method (RAM) is presented for deriving horizontal current velocities using EM-APEX float measurements at a vertical resolution of up to 1.3 m, a factor of 4–5 times better than that of HFM. This new method permits the vertical resolution of the horizontal velocity to the oceanic fine scales and should become helpful for understanding internal wave breaking, vertical shear instability, small-scale vortical motions, and anisotropic turbulence. It can also help improve the studies of the fine-scale turbulent mixing parameterization schemes, such as reduced shear parameterization (Kunze et al. 1990), fine-scale turbulence scaling for internal waves (Polzin 1996), and gradient Richardson number scaling (Large et al. 1994; Forryan et al. 2013). Section 2 will describe the analytical forms of voltage and magnetic field measurements on the EM-APEX floats. The new method RAM will be presented in section 3. Uncertainties of estimates of u will be discussed in section 4. In section 5, vertical shear spectra are examined using RAM-derived measurements of u.

2. EM-APEX float measurements

Two pairs of orthogonal electrodes (E1 and E2) and a magnetometer (hereafter EM sensors; Fig. 1; Hsu 2021) are equipped on each EM-APEX float, and take measurements of voltage and magnetic field at 1 Hz (Sanford et al. 2005). Floats move vertically by adjusting their buoyancy relative to the surrounding seawater. As the float profiles, an array of slanted blades forces the float and EM sensors to rotate. The rotation frequency Ω is proportional to the profiling speed of the floats W (Hsu et al. 2018).

Fig. 1.
Fig. 1.

(left) Photo of the EM-APEX float and sensors and (right) the schematic illustration of two pairs of electrodes E1 and E2 viewed from the top of the float. The motion-induced electric current (Jx is the zonal component i^ and Jy is the meridional component j^) is captured by the electrode pairs as voltage. The dashed arrows labeled hx and hy are the orthogonal axes for the magnetometer measurements. The angle θΩ is the angle from each electrode pair to magnetic east (i = 1 for the E1 pair and i = 2 for the E2 pair), and θΩ0 = Ωt + ϕ0 is the angle between hx and the magnetic east, assuming hx is aligned with E2 pair. The Ω is the rotation rate of the electrodes, L is the separation between two electrodes, and ϕ0 is the angle at a reference time t = 0. Fy is the magnetic north of Earth.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

The voltage ΔΦi (i = 1 for the E1 pair and i = 2 for the E2 pair; unit: V) captured by the electrode pairs is associated with the horizontal electric current J (=Jxi^+Jyj^, where i^ and j^ are the unit vectors in the x and y directions, respectively) induced by the seawater velocity (appendix B; Sanford et al. 1978). After including the effect of floats’ vertical motion on the measured voltage measurements, the analytical form of ΔΦi can be expressed (Hsu et al. 2018) as
ΔΦi(t)L=(FyC2W−C1Jx*)cos⁡θΩ(t)−C1Jy*sin⁡θΩ(t)+ΔΦoffsetL+δnL,
where L is the separation between electrodes, Fy the magnetic north component on Earth, θΩ=θΩ0+(2−i)(π/2), θΩ0(t) = Ωt + ϕ0 the angle between the electrode pair of E2 and the geomagnetic east (i.e., θΩ0 = ϕ0 at t = 0), i = 1 for the E1 pair and i = 2 for the E2 pair, Jx* and Jy* are the electric current Jx and Jy divided by the conductivity of seawater (same as the unit of an electric field), respectively, t is the time, C1 = 1.5 the head factor due to the insulated surface (Sanford et al. 1978), C2 = 0.8 the head factor due to W (Sanford et al. 1978), ΔΦoffset the unknown voltage offset (appendix C, section a), and δn the instrumental noise of voltage measurements.

A vertical profile of measurements taken by the EM sensors on the float f9207 (appendix D) is used to illustrate the typical voltage measurements (Fig. 2). As expressed in Eq. (1), the voltage measurements are constituted by the sinusoidal-oscillating voltage C1Jx*cosθΩ and C1Jy*sinθΩ due to the seawater motions (termed motion-induced voltage), the voltage offset ΔΦoffset, and the measurement error δn. The rotation of EM sensors (i.e., Ω ≠ 0) allows us to separate the motion-induced signals from the slow-varying voltage offset. It should be noted that the voltage offset ΔΦoffset may vary with temperature and salinity in depth (appendix C, section a). Different electrode pairs have different ΔΦoffset. The instrumental noise of voltage measurements δn may yield the uncertainty of ∼0.007 m s−1 for the estimated current velocity (appendix C, section b).

Fig. 2.
Fig. 2.

(a) Voltage measurements taken by two pairs of electrodes E1 and E2, (b) magnetometer measurements on hx and hy, after removing the mean of hx and hy and then being divided by the standard deviation of hx and hy, respectively, and (c) temperature [blue line in (c)] and salinity [red line in (c)] from one profile of the float f9207. The mean and standard deviation of hx and hy in (b) are computed using the whole 550-s data. Because the offset of the magnetometer measurements is nearly constant in time, using the whole 550-s magnetometer measurements for computing the mean and standard deviation will derive the same result from the moving 50-s windows.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

The magnetic field captured by the magnetometer is projected onto the orthogonal axes (i.e., hx and hy in Fig. 1). Unlike the voltage offset ΔΦoffset, the offset of the magnetic field measurements remains nearly constant with time and depth. The angle between the magnetometer of hx or E2 and the magnetic east [i.e., θΩ0 in Eq. (1)] can be estimated as
θΩ0˜(t)=tan⁡−1(hx−⟨hx⟩hy−⟨hy⟩),
where the tilde represents the estimated variables in our processing step (i.e., θΩ0˜ is the “estimated” angle between hx and the magnetic east), and 〈⋅〉 is the mean within a selected time window. The θΩ0˜ is crucial for estimating the horizontal velocity (to be discussed in section 3).

3. Rotating axes method

The horizontal seawater velocity u˜ in the upper ocean is mainly constituted by low-frequency currents u and high-frequency surface waves near the surface (Hsu et al. 2018). The motion-induced voltage associated with the u [the first two terms on the right of Eq. (1)] is often estimated by fitting the measured voltage in 50-s windows using the HFM (Fig. 2) (appendix A; Sanford et al. 1978, 2005). The 50-s windows including more than two rotation cycles of voltage measurements will cause the vertical resolution of current velocity results ∼5–7 m. Improving the vertical resolution by using a shorter window may need to be compromised by a larger velocity error in the fit (section 3c). Here, this study presents a new RAM for processing float’s voltage measurements that can provide estimates of u at a vertical resolution finer than HFM without compromising velocity errors significantly. Abbreviations and definitions of symbols are shown in Table 1.

Table 1.

List of abbreviations and symbols used in this paper. Units are in parentheses where applicable.
Table 1.

a. Estimation of horizontal seawater velocity

Because the voltage offset ΔΦoffset is inherent to electrodes and does not exhibit sinusoidal variations when float rotates [Eq. (1)], the key of RAM is to estimate and remove the slow-varying ΔΦoffset from the measured voltage measurement. The 1-s signals caused by the horizontal seawater motion can then be computed as residuals, appearing in a sinusoidal form due to the float’s rotation, the first two terms on the right of Eq. (1) (Fig. 2).

The voltage measurements on each pair of electrodes are first least squares fitted [Eq. (3); Fig. 3a] to a superposition of sinusoidal variations associated with the rotation of EM sensors θΩ, and the temporal-varying voltage offset, assuming ΔΦoffset˜/L=∑n=0Nbn˜tn. By using the estimated angle θΩ0˜ between E2 and magnetic east, the analytical form of the least squares fit on either E1 or E2 pairs of electrodes can be expressed as
ΔΦiL(t)=a1˜cos⁡θΩ0˜(t)+a2˜sin⁡θΩ0˜(t)+∑n=0Nbn˜tn+ϵi;for i=1,2,
where t is the time relative to the first point of the windows, a1˜ and a2˜ the fitted coefficients of motion-induced voltage, bn˜ the fitted coefficients of voltage offset using the N-order polynomial function tn in time, L the separation between electrodes, and ϵi the residuals (V m−1). The N = 4 should be the largest order for the polynomial function orthogonal to the two sinusoidal terms during the fit to two rotation cycles of measurements. This should prevent the polynomial fit from grabbing too much sinusoidal signal. I estimate the time series of ΔΦoffset˜  in a 50-s moving window, with a 45-s overlap with adjacent windows. Throughout the least squares fitting of the entire vertical profile, 10 estimates of ΔΦoffset˜ are obtained at each second because of the overlapping windows. All estimates of ΔΦoffset˜ at each 1-s point are averaged to obtain a mean ΔΦoffset˜ except for those having less than five estimates at the edge of the profiles. Similarly, the magnetometer orientation θΩ0˜  is computed following Eq. (2) in the same moving windows as for the estimation of voltage offset. All estimates of θΩ0˜  at each 1-s point are averaged to obtain a mean θΩ0˜ . Therefore, a profile of averaged estimates of ΔΦoffset˜ and θΩ0˜  and their standard deviation (σoffset is the standard deviation of ΔΦoffset˜) can be obtained at 1-s time intervals.
Fig. 3.
Fig. 3.

(a),(b) Illustration on the points within the profile that are used for estimating voltage offset ΔΦoffset˜ (black dots). Because the 50-s windows [blue dots in (a)] are moving in the increments of 5 s, each 1-s point has 10 estimates of ΔΦoffset˜ except near the edge of the profile. The estimates [along the red lines in (a)] are used for computing the mean and standard deviation σoffset [error bars in (b)]. The mean ΔΦoffset˜ at the points having less than 5 estimates are excluded (the sequence of data at the time prior to the black dashed line). The plot in (b) also demonstrates an alternative way for evaluating the uncertainty of the estimated ΔΦoffset˜, assuming two different profiles of ΔΦoffset˜. Compared to the slowly fluctuating ΔΦoffset˜ (green solid line), the temporal change of ΔΦoffset˜ (Φt; unit V s−1) at the highly fluctuating ΔΦoffset˜ (magenta solid line) will yield higher standard deviation σΦt of Φt in each 12-s window [dashed lines with values labeled on the right of (b)].

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

The estimated motion-induced voltage Δ Φ˜i (i = 1 and 2 for E1 and E2 pairs, respectively), i.e., the first two terms on the right of Eq. (1), can then be computed by removing the mean ΔΦoffset˜ from the raw 1-s voltage measurements and expressed as
ΔΦ2˜L≈(FyC2W−C1Jx*)cos⁡θΩ0˜−C1Jy*sin⁡θΩ0˜+δ2L ΔΦ1˜L≈−(FyC2W−C1Jx*)sin⁡θΩ0˜−C1Jy*cos⁡θΩ0˜+δ1L},
where the errors δ1 and δ2 may consist of the measurement noise δn and the unremoved voltage offset, and W is the vertical profiling speed of the floats. Next, the 1-s Δ Φ˜i taken from the rotating EM sensors E1 and E2 is converted to the fixed Cartesian coordinates (=ΔΦx˜i^+ΔΦy˜j^), using a rotation matrix
M=[cos⁡θΩ0˜−sin⁡θΩ0˜sin⁡θΩ0˜cos⁡θΩ0˜],
i.e.,
[ΔΦx˜LΔΦy˜L]=M[ΔΦ2˜LΔΦ1˜L]=[(FyC2W−C1Jx*)−C1Jy*]+M[δ2Lδ1L].
The motion-induced electric current J and the horizontal seawater velocity u˜ (=u˜i^+υ˜j^) are linearly related via Eq. (B4).
By substituting Eq. (B4) on Eq. (5), the horizontal velocity components can be estimated using ΔΦx˜ and ΔΦy˜ as follows:
u˜=ΔΦy˜FzC1L+δu υ˜=FyFzC2C1W−ΔΦx˜FzC1L+δυ }.
The velocity errors δu and δυ include the contribution from instrument noise δn, unremoved voltage offset (oscillated due to the rotation matrix M), and a depth-independent current u*¯ due to the unknown background electric field (Sanford et al. 1978), i.e.,
(δu,δυ)=1FzC1L[δ0cos⁡(θΩ0˜+ϕ′),δ0sin⁡(θΩ0˜+ϕ′)]+u*¯,
where δ0=δ12+δ22 and ϕ′=tan⁡−1(δ2/δ1); δ0 may be randomly distributed (due to δn) with a nonzero mean [due to the imperfect estimation of voltage offset; Eq. (7)]. The value of u*¯ is depth independent and may vary slowly in time (Sanford et al. 1978). It can be estimated by matching the float’s measured velocity with the GPS-fixes-derived velocity (Lien and Sanford 2019).

The same sequence of voltage measurements in Fig. 2 is used for illustrating the steps in RAM (Fig. 4). The ΔΦoffset˜ estimated via the least squares fit changes slowly in depth [Eq. (3); Figs. 4b,c]. The σoffset (the scattering of 1-s estimates of ΔΦoffset˜) is significant near the sea surface because surface waves can affect the fitted coefficients of ΔΦoffset˜ (Hsu et al. 2018). The profiles of mean ΔΦoffset˜ are derived by averaging the estimates of ΔΦoffset˜ at every 1 s. For comparison, I also estimate the mean ΔΦoffset˜ via a low-pass filter (assuming cutoff time = 50 s). Using either overlapped windows or a low-pass filter can derive similar results of ΔΦoffset˜ except near the sea surface, presumably due to the transient effect in the low-pass filter.

Fig. 4.
Fig. 4.

Raw voltage measurements (lines with sinusoidal variations) and the mean (solid lines between the sinusoidal oscillation) and standard deviation (nearly invisible vertical lines are error bars) of (a) estimated ΔΦoffset˜ on E1 and E2, (b),(c) estimates of voltage offset ΔΦoffset˜ on E1 [blue lines in (b)] and E2 [red lines in (c)] in each 50-s window, (d) voltage measurements after removing the mean of estimates of ΔΦoffset˜ , and (e) estimates of horizontal seawater velocity u˜ in zonal (blue) and meridional directions (red). The estimated orientation of electrode pairs ΔθΩ0˜ is shown as black dots in (d), labeled on the right axis of (d). The lines with dots in (e) are u˜ averaged in 12-s intervals as horizontal current velocity u. The upper axis of (a) displays the depth of measurements. The results from the low-pass filter are compared to the fitted results in each window in (b) and (c).

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

After removing the mean ΔΦoffset˜ from voltage measurements, the remaining voltage fluctuations Δ Φ˜i (i = 1 and 2 for E1 and E2 pairs, respectively) exhibit a sinusoidal variation due to the measured seawater motion on the rotating electrode pairs [Fig. 4d; Eq. (4)]. The magnitude of the sinusoidal variation is proportional to the magnitude of the horizontal current velocity u. The orientation of EM sensors determined by 1-s magnetometer measurements is then used to convert the seawater-motion induced voltage Δ Φ˜i to the zonal and meridional components ΔΦx˜ and ΔΦy˜ in the Cartesian coordinates [Eq. (5)], from which the zonal and meridional horizontal velocity of seawater u˜ is computed [Fig. 4e; Eq. (6)].

b. Results of horizontal current velocity

The results of horizontal seawater velocity u˜ [Eq. (6)] via RAM consist of a low-frequency horizontal ocean current u, high-frequency surface waves uw, and 1-s errors (δu and δυ). To extract the low-frequency components u, RAM averages the results of u˜ in independent windows whose intervals equal ΔT. This step as a low-pass filter can also reduce the sinusoidal-fluctuating velocity errors δu and δυ [Eq. (7)] due to the unremoved offset. The mean of the δu and δυ should be nearly zero when ΔT is longer than one rotation period of EM sensors 2π/Ω s. The ΔT = 12 s is set in this study, yielding the vertical resolution of u as small as 1.3 m, one-fourth the default for HFM. In contrast to HFM requiring 50-s windows for fitting horizontal velocity, RAM is more flexible in choosing the length of windows for deriving independent results of u, favorable to an improvement in the vertical resolution.

The 1-s results of horizontal seawater velocity u˜ at each profile of the float f9208 are used for computing the frequency spectra at different layers (Fig. 5). In the upper 50-m ocean, except for the spectral energy due to the low-frequency current at <0.04 Hz, the frequency of spectral peak is about 0.11 to 0.13 Hz, similar to the typical wind waves. Most spectral peak in the deep layer (from 80 to 100 m) <1.0 × 10−2 m2 s−1, is less than that from 20 to 50 m (>1.0 × 10−2 m2 s−1), consistent with the depth-decaying feature of the surface waves. The uw as the high-frequency component of u˜ has been used to compute the directional spectra of surface waves at EM-APEX floats (Hsu 2021). It should be noted that the uw captured by a vertical-moving float is presented as depth-varying signals in u˜. To study the roles of fine-scale vertical shear in turbulent mixing, the horizontal current velocity u needs to be reliably separated from the uw of wind waves, or it will cause an overestimated vertical shear of u. Theoretically, setting ΔT = 12 s can exclude the wind waves whose frequency is >0.09 Hz. If the sea state is dominated by strong propagating swell, such as near tropical cyclones, setting ΔT = 14 s will be more appropriate for excluding the uw near the sea surface (peak frequency of waves > 0.08 Hz; Hu and Chen 2011). Because the magnitude of uw decays exponentially in depth, part of u near the sea surface may include the Stokes drift velocity of surface waves.

Fig. 5.
Fig. 5.

Frequency spectra computed using estimates of horizontal seawater velocity in different layers—(a) 20–50-m depth; (b) 50–80-m depth; (c) 80–100-m depth—for float f9208. The color of the lines represents the time of the individual profiles in the yearday of 2021. The black dashed line indicates frequency of (1/12) Hz, i.e., period of 12 s.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

c. Comparison to the HFM

RAM projects the 1-s signals of motion-induced voltage to the fixed Cartesian coordinates using 1-s magnetometer measurements and then temporal averages the horizontal seawater velocity results as current velocity. In comparison, the traditional HFM uses 50-s magnetometer measurements to determine the frequency and phase for the sinusoidal oscillations and then fits magnitudes of the sine and cosine functions and one electrode ΔΦoffset with a linear trend in every 50-s interval (appendix A). The oceanic current velocity is computed using the fitted magnitudes of sine and cosine functions in the 50-s windows. In other words, RAM processes the 1-s velocity signals as horizontal current velocity, and HFM extracts the mean of 50-s velocity signals as horizontal current velocity.

The results of horizontal current velocity near Mien-Hua Canyon are computed using two data processing methods (RAM: subscript r; HFM: subscript h). The uh from the traditional HFM (appendix A) is estimated using successive 50-s windows with 25-s overlapped measurements. After deriving the 1-s horizontal seawater velocity u˜ from RAM, the urs (superscript s represents a shorter window) is computed as the mean of u˜ in independent windows of ΔT = 12 s, and the url (superscript l represents a longer window) is the mean result using ΔT = 50 s with 25-s overlapped measurements (the same windows as uh). Because of the overlapped measurements, the vertical resolution of uh and url equals the products of 25 s and the vertical speed of the floats W. The results of urs are interpolated to the grids of uh.

Comparing the interpolated urs with uh (Fig. 6), the RMS difference is 0.02 m s−1, regardless of the descending (W < 0) or ascending (W > 0) profiles. The RMS between the url and uh is about 0.017 m s−1, slightly smaller than that between the urs with uh. Because the uh used in this study is the weighted-averaging horizontal current velocity from two independent electrode pairs (appendix A), the results from two different methods should not be identical even averaging over the same window length. Because of the negligible RMS difference between RAM and HFM-derived velocity, it is sufficient to conclude that RAM can derive the general structure in the profiles of horizontal current velocity as the traditional HFM. On the other hand, one may suspect the velocity resolution from RAM should equal the product of floats’ vertical speed W and the window length, instead of the temporal-averaging intervals ΔT and W. From the perspective of deriving 1-s seawater velocity results, removing the 50-s results of the voltage offset from 1-s voltage measurements can be regarded as excluding linear-interpolated 50-s velocity errors. The vertical resolution of ur should still be determined by the temporal-averaging intervals ΔT and W.

Fig. 6.
Fig. 6.

(a),(b) Mean vertical resolution Δz of ur (red lines with dots; rotating axes method) and uh (blue lines with dots; harmonic fitting method) from each profile at the two floats. (c),(d) The results are then compared (x axis: uh; y axis: ur) and marked in the color based on the vertical speed of the floats W.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

A simulated profile of voltage measurements (appendix C, section c) is used to demonstrate why RAM can derive velocity results in a finer vertical resolution than HFM (Fig. 7). A rotation period of simulated EM measurements of 22 s is used for simulating voltage measurements on two electrode pairs, corresponding to the vertical speed of the float ∼0.1 m s−1 (not shown in this study). The uh (HFM) and u˜ are derived by least squares fitting the simulated measurements in independent windows whose length equals 18, 36, and 50 s, respectively (the increment between windows in RAM is still 5 s). The ur (RAM) is the bin-averaged result of u˜ in 12-s windows. When the length of fitting windows includes at least two rotation cycles of measurements (Sanford et al. 2005), both RAM and HFM can reproduce the simulated profile of current velocity reliably. With less than two rotation cycles of voltage measurements used in each fit, the discrepancy of uh to the reference profiles of current velocity increases more significantly than ur. When the uh is derived by using less than one rotation cycle of voltage measurements, the errors of uh increase to more than 0.03 m s−1, much higher than the ur.

Fig. 7.
Fig. 7.

(a)–(c) Voltage offset ΔΦoffset˜ and (d)–(f) current velocity u estimated by using least squares fit windows in RAM whose length equals 18 s (blue dots and lines), 36 s (yellow dots and lines), and 50 s (green dots and lines), respectively. The black lines are the reference profiles used for simulating the profile of voltage measurements at an EM-APEX float, including the (a),(b),(c),(h) voltage offset and (d)–(f) horizontal current velocity. The red dots in (d)–(f) are the current velocity results from HFM. Dashed lines between two dots in (a)–(c) are different sequences of estimated voltage offset. Shorter window length of the least squares fit will increase the scattering of voltage offset between overlapped windows. (g) The results in the overlapped windows are averaged and used for computing their discrepancy to the reference profile of voltage offset.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

In the least squares fit, shortening the window length will increase the uncertainties of the fitted coefficients, regardless of the motion-induced voltage for HFM or voltage offset for RAM. HFM requires two rotation cycles of voltage measurements for reliably estimating the coefficients of motion-induced voltage thereby uh (appendix A). For RAM, although imperfectly estimating the fitted coefficients of voltage offset will also cause an unremoved voltage offset as “errors” in the seawater velocity results u˜ [δu and δυ in Eq. (7)], temporal averaging u˜ as ur will reduce the sinusoidal-fluctuating errors. It allows RAM to derive the ur reliably even using less than two rotation cycles of voltage measurements in each fit. The next section will also use in situ float data to demonstrate how the change in the window length affects the RAM’s velocity results.

4. Error analysis on the performance of RAM

Estimating voltage offset ΔΦoffset˜ reliably is important for RAM to derive horizontal current velocity u, achieved by using the successive least squares fit windows. The following analysis will discuss how the parameters in the least squares fit can affect the coefficients of ΔΦoffset˜ and u using the float data from an experiment near Mien-Hua Canyon. A quality control procedure will be proposed by examining the assumption of a slow-varying voltage offset in the observations. The uncertainty of u due to the scattering of estimated voltage offset in overlapped windows will then be studied using the simulated profiles of voltage measurements.

a. Parameters for estimating voltage offset

Three parameters are used in the fit of low-frequency ΔΦoffset˜, including the length of the fitting windows (default = 50 s), the overlapped period (default = 45 s), and the order of the polynomial function (default = 4). RAM will compute the horizontal current velocity uref using the default parameters, and uexp after each parameter is adjusted, respectively. The sensitivity tests will be performed by computing the root-mean-squared difference (RMS) between the uref and uexp.

The uexp is first estimated by varying the length of the fitting windows from 30 to 80 s (the window increment of 5 s), which is greater than the typical rotation cycle of EM sensors 20 s (Figs. 8a–c). A window length of less than two rotation periods of EM sensors will increase the RMS slightly, as the analysis using the simulated voltage measurements in section 3. However, the RMS does not change significantly when the length of the fitting windows is longer than 50 s. Using 50-s windows should be sufficient for capturing the slow-varying voltage offset in the observations. Different overlapped periods ranging from 30 to 10 s are then tested (Figs. 8d–f). Decreasing the overlapped period has negligible influences on the horizontal current velocity results with the consistent RMS between uref and uexp, ∼ 0.017 m s−1. Finally, the effect of the polynomial functions in the least squares fit is studied (Figs. 8g–i). Even the least squares fit assumes the linear change of voltage measurements in 50-s windows, the RMS between uref and uexp is still less than 0.005 m s−1, which is even smaller than the instrumental noise (∼0.007 m s−1). High-order polynomial functions do not affect the estimates of uexp significantly.

Fig. 8.
Fig. 8.

Comparisons between uref and uexp by using the results from the two floats, where uref is the horizontal current velocity derived from the default setting in the rotating axes method, and uexp is the processed horizontal current velocity by changing (a)–(c) the length of the fitting windows, (d)–(f) the number of overlapped measurements in each window, and (g)–(i) the order of polynomial functions in the estimation of voltage offset ΔΦoffset˜. The mean and RMS of the difference between uref and uexp are labeled on the upper-left corner of the panels.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

b. Quality control based on the estimated voltage offset

The primary principle of RAM is to assume the temporal variation of offset ΔΦoffset much slower than the motion-induced voltage on the rotating electrode pairs. If the results of ΔΦoffset˜ fluctuate rapidly in time, RAM may have failed to estimate and separate ΔΦoffset˜ from the motion-induced voltage. The standard deviation σΦt between the 1-s change rate of offset [=d(ΔΦoffset˜)/dt] is computed in the same temporal-averaging window of u to quantify the magnitude of the fluctuation (Fig. 3b). Due to the unknown dependence of ΔΦoffset˜ on the surrounding seawater properties, it is challenging to propose a constant value of σΦt as criteria for all electrode pairs. The absolute difference in σΦt between E1 and E2 is used as the first quality control criterion, assuming a similar depth variation of ΔΦoffset˜ at two electrode pairs of the same floats.

Besides, in the overlapped windows, a significant σoffset may occur when high-frequency signals have influences on the fitted coefficients of ΔΦoffset˜ (Fig. 3b). The variance of motion-induced voltage at the rotating electrode pairs should be the same [Eq. (1)], which will yield a similar σoffset at E1 and E2 (Hsu et al. 2018). Meanwhile, an unknown noise may increase the scattering of estimated ΔΦoffset˜ at only one of the electrode pairs, thereby a difference in σoffset between E1 and E2. After computing the mean of σu = σoffset/FzC1L (unit: m s−1) in the same intervals ΔT with velocity results, the absolute difference in σu (i.e., Δσu) between E1 and E2 can be derived and used as the second quality control criterion.

The probability distributions of ΔσΦt and Δσu for the two floats near Mien-Hua Canyon are used for demonstrating the quality control procedure (bars in Fig. 9; f9207 and f9208). When the high ΔσΦt and Δσu occur simultaneously, the least squares fit may have difficulties in capturing the highly fluctuating offset at one of the electrode pairs. To reduce the velocity errors due to the uncertainty of voltage offset results, the results of u at each float are excluded from this analysis (<0.5% of the total velocity results at f9207 and f9208) if their ΔσΦt and Δσu are in the upper 5% tier at the same time.

Fig. 9.
Fig. 9.

Probability distribution (bars) and cumulative distribution (lines) of (a) ΔσΦt and (b) Δσu using the results at two floats f9207 (blue) and f9208 (red). The dots indicate the intersection between the cumulative distribution and 95% tier line (black dashed lines).

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

c. Effect of 1-s voltage noise

At two electrode pairs of each float, the voltage signals due to seawater motion should affect the coefficients of voltage offset in the fit similarly, yielding a similar scattering between the estimated voltage offset (i.e., Δσu ∼ 0). According to the cumulative distributions of Δσu between E1 and E2, more than 95% Δσu is within 0.025 m s−1. The Δσu should be mainly caused by an unknown noise plus different instrumental noises between two electrode pairs. It is crucial to estimate the uncertainty of 12-s current velocity results due to the 1-s voltage noise. Below, the profile of voltage measurements is first simulated by using a reference profile of current velocity (black line in Fig. 10) captured by the rotating EM sensors (appendix C, section c). The Δσu = 0.025 m s−1 is assumed and used for simulating 1-s voltage offset errors in a normal distribution with zero mean. The realizations of 1-s voltage offset errors are added to the results of voltage offset ΔΦoffset˜ as the apparent voltage offset profiles. The 100 realizations of current velocity results can then be derived after excluding the realizations of apparent voltage offset profiles from the simulated voltage measurements, respectively.

Fig. 10.
Fig. 10.

Mean (sold lines, bottom axis) and standard deviation (dashed lines, top axis) of horizontal current velocity u, computed using the voltage offset results plus 100 realizations of voltage offset errors. The voltage offset results are derived by using the simulated voltage measurements captured by EM sensors whose rotation period (colored lines) varies from 10 to 22 s. The black solid line is the reference profile of horizontal current velocity for simulating motion-induced voltage.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

The mean and standard deviation of velocity results at each depth are computed using the 100 realizations of current velocity profiles (color lines in Fig. 10). Despite the additional errors added to the results of voltage offset at one of the electrode pairs, the mean of u is still nearly the same as the reference velocity profile (black line in Fig. 10). A constant Δσu of 0.025 m s−1 may cause a standard deviation of u ∼ 0.005 m s−1, regardless of the rotation rate of EM sensors. Because the standard deviation of simulated u is less than the uncertainty that can be caused by the instrumental noise of voltage measurements (appendix C, section b), it should be sufficient for demonstrating the negligible influence of 1-s noise on the 12-s velocity results. On the other hand, if future studies can propose a more reliable method for estimating the 1-s voltage offset, the vertical resolution of u can become finer by shortening ΔT. One possible approach is to study the change in ΔΦoffset˜ due to the vertical structure of temperature and salinity. Collecting high-resolution CTD measurements at the floats can help predict the depth variations of voltage offset and then favor the estimation of current velocity and shear at the fine scale.

5. Horizontal current velocity and vertical shear from RAM

a. Structure of horizontal current velocity and vertical shear near Mien-Hua Canyon

Strong currents associated with Kuroshio and tides at Mien-Hua Canyon have been reported by previous studies (Lien et al. 2013; Chang et al. 2019). Two EM-APEX floats (f9207 and f9208) were deployed near Mien-Hua Canyon for 2 days at the end of February 2021 (appendix D). Floats took measurements of electrode voltage and magnetic field in the upper 100 m and CTD measurements in the upper 150 m. At f9207, the flow in the upper ocean moved northward in the upper ocean before yearday 55.5 at a speed > 1 m s−1 and the flow moved westward >1 m s−1 after the floats entered the Mien-Hua Canyon (Fig. 11). The change from northward to westward current is consistent with the westward trajectories of the float.

Fig. 11.
Fig. 11.

(a),(f) Zonal (u) and (b),(g) meridional (υ) current velocities, (c),(h) squared vertical shear [(∂u/∂z)2 + (∂υ/∂z)2], (d),(i) buoyancy frequency N2, and (e),(j) inverse gradient Richardson number (Ri−1) at the two EM-APEX floats near Mien-Huan Canyon: (left) f9207 and (right) f9208. The vertical trajectories of the floats (black dots) from the CTD measurements are shown in (a). Magenta dots marks the results where N2 < 0.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

The 2-m CTD measurements at the floats were used to compute the buoyancy frequency N2 [=−(g/ρ)(∂ρ/∂z)], where g is the gravity and ρ the seawater density (Fig. 11). The gradient Richardson number Ri = N2/(∂u/z)2 could then be derived after interpolating the results of u to the grids of CTD measurements. In other words, RAM can be used for estimating Ri in a vertical resolution of 2 m at APF-11 EM-APEX floats. The N2 was ∼5.0 × 10−5 s−2 from 20- to 80-m depth. The negative N2 as the density inversion near the sea surface might result from atmospheric forcing. In the subsurface layer, a strong shear squared of 1.0 × 10−4 s−2 was found at f9207 between 40 and 80 m around yearday 55.5, yielding a Ri < 0.25. Otherwise, the Ri was mostly greater than 0.25 at the two floats. The magnitude of vertical shear might not be strong enough for destratifying the upper ocean structure (Smyth and Moum 2013), so the N2 in the thermocline remained nearly the same during this period.

b. Vertical shear spectra

I compute the vertical wavenumber spectra of vertical shear Ψ (subscript r from RAM and subscript h from HFM) using the profiles of vertical shear ∂u/∂z (Fig. 12). The profiles are gridded in the intervals of floor(5Δzmin)/5 m, where Δzmin is the minimum interval between the results of u in each profile unless Δzmin < 0.5 m (set Δzmin = 0.5 m if Δzmin < 0.5 m). The spectral level of Ψ above the Nyquist wavenumber [=1/(2Δz) cpm] is excluded, where Δz is the mean vertical resolution of u in each profile. The spectral slope m of composite Ψ is fitted in the logarithmic scale (i.e., Ψ ∝ k−m). The Ψh from HFM is available at the range of wavenumber bands narrower than Ψr from RAM due to the difference in the mean vertical resolution of u in each profile Δz.

Fig. 12.
Fig. 12.

(a),(b) Individual spectra of vertical shear using each profile of ur (colored lines) and (c),(d) composite spectra of vertical shear (blue lines: Ψh from HFM; red lines: Ψr from RAM) at the two float: (left) f9207 and (right) f9208. The lines of each profile in (a) and (b) are colored based on its time in the upper 100 m. The red shading area in (c) and (d) covers the mean ± one standard deviation of RAM’s Ψr results. The black double-headed arrows display the range of vertical wavenumber band k for fitting the composite spectra on the logarithmic scale, and the slopes of log(Ψh) and log(Ψr) are labeled. The black thick lines are the empirical GM model. The black dashed lines extended from the black thick lines with the −1 slope are the saturated spectra. The vertical black dashed lines marks the typical rolloff wavenumber of 0.1 cpm in the GM spectrum.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

The composite spectra of vertical shear are computed by averaging all individual spectra unless the number of spectra at high wavenumber is less than one-third of the total spectra (Figs. 12c,d). At f9207, the spectral slope m of log(Ψr) from RAM is about −0.8 at k = 0.1–0.2 cpm, similar to the slope of the saturated spectra (Gargett et al. 1981; Kunze 2019). The spectral slope m of −1.5 at k > 0.1 cpm at f9208 is within the range of the slope reported by previous studies (e.g., Gregg et al. 1993). Compared to the empirical GM spectrum rolling off at 0.1 cpm (Garrett and Munk 1975, 1979), it is difficult to identify the rolloff wavenumber k0 in the observed Ψr. However, the Ψr at the low wavenumbers (<0.1 cpm) still follows the extrapolation of the −l slope of the saturated spectra (black dashed lines; Gregg et al. 1993; D’Asaro and Lien 2000). Strong internal waves near Mien-Hua Canyon (Lien et al. 2013) may increase the spectral energy at the low wavenumbers and thereby lower the k0 (Gregg et al. 1993, 1996). Interestingly, although the composite Ψh is similar to the Ψr at the low wavenumbers k < 0.07 cpm, the spectral slope of Ψh from HFM drops rapidly at k > 0.07 cpm, associated with the resolution of HFM at Nyquist wavenumber. High Δz of u from RAM should be more useful than HFM for studying the fine-scale structure of vertical shear. It should be noted that the spectral level of Ψr at f9207 may increase more rapidly than f9208 at high wavenumbers k > 0.2 cpm. Due to the limited results in the narrow bands of wavenumber, it is hard to conclude the flat spectral level due to the more energetic fine-scale turbulence or noise in velocity results. Fortunately, compared to the previous EM-APEX floats with the APF-9 firmware, the latest APF-11 EM-APEX floats can transmit the raw 1-Hz voltage measurements via Iridium satellites in the near–real time. Future studies may deploy APF-11 floats in regions ubiquitous with strong internal waves, such as the South China Sea, to study the turbulence effect on the spectral slope at high wavenumbers > 0.2 cpm.

c. Effect of temporal-averaging intervals on the vertical shear spectra

RAM derives horizontal current velocity u by averaging the horizontal seawater velocity results u˜ in the time interval ΔT. Because it is challenging to accurately estimate and remove the voltage offset from the voltage measurements, the mean of the errors (due to unremoved offset) may not be negligible when ΔT is much shorter than one rotation period of EM sensors 2π/Ω. The variance of velocity errors at the fine-scale grids will then increase the spectral level of Ψ at high wavenumbers. Temporal averaging the horizontal velocity uw of a surface wave in intervals shorter than one wave period may also cause errors in the estimation of vertical shear, especially near the sea surface. Exploring the effect of ΔT (or grid size of u) on the spectral slope of log(Ψ) is crucial.

I compute u thereby composite Ψ using the intervals ΔT ranging from 6 to 20 s (Fig. 13; section 5b). All estimates of Ψ are similar at low wavenumbers k < 0.1 cpm. When the ΔT is adjusted from 10 to 8 s, the spectral level of Ψ becomes blue at high wavenumbers k > 0.3 cpm. The high spectral level at k = 0.4 cpm (=1/Δz) implies the shear significantly increases at the resolution Δz of 1.25 m, corresponding to the product of the vertical speed in the ascending profiles (∼0.12 m s−1) and ΔT of 8 s. Because the peak frequency of surface waves during this field experiment is around 0.12 Hz, setting ΔT = 8 s cannot completely separate the low-frequency u from uw. Decreasing ΔT to 6 s may further include surface waves and bias the spectral level of vertical shear at k = 0.5 cpm. Even within the layer from 80 to 100 m, surface waves may still affect the frequency spectra at around 0.12 Hz (Fig. 5c), because the significant wave height at yearday ∼ 55.2 can be more than 3 m. The biased u then increases the spectral level at the high wavenumbers. The observed blue spectral level at k > 0.3 cpm at intervals ΔT < 10 s should be caused by the contamination of surface waves rather than the imperfect estimation of the offset. The ΔT = 12 s should be a reliable temporal-averaging interval for deriving the fine-scale structure of vertical shear.

Fig. 13.
Fig. 13.

(a),(b) Vertical shear spectra Ψ at the two EM-APEX floats, computed by using the results of horizontal current velocity from different intervals ΔT (colored lines) in the rotating axes method. The black double-headed arrows display the range of vertical wavenumber band k for fitting the composite spectra on the logarithmic scale, and the slopes of log(Ψ) at different ΔT are labeled. The black solid lines are the empirical GM model, and the black dashed lines are the saturated spectra.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

6. Summary and conclusions

This study introduces a new rotating axes method (RAM) for processing the EM-APEX float measurements of voltage as horizontal current velocity u. The rotation of electrode pairs on the floats causes all motion-induced voltage signals to oscillate sinusoidally (i.e., on a rotating frame), differing from the slow-varying offset ΔΦoffset˜ inherent in the measurements. The main focus is to decode and interpret these 1-s voltage signals as the horizontal seawater velocity u˜, achieved by estimating and excluding the ΔΦoffset˜ from the measurement via the 50-s least squares fit windows. The results of ΔΦoffset˜ derived from the least squares fit are nearly the same as those via a low-pass filter. Because the results of u˜ mainly consist of a low-frequency ocean current u and high-frequency surface waves uw, temporal averaging the profiles of u˜ in the intervals of ΔT = 12 s (as a low-pass filter) can separate the u from uw, whose peak frequency ∼ 0.12 Hz near the sea surface. Compared to the traditional harmonic fitting method (HFM) which fits the magnitudes of motion-induced voltage in 50-s windows, i.e., within a narrow high-frequency band of raw voltage measurements, RAM high-pass filters the raw voltage measurements by demodulating slow-varying voltage offset. When RAM and HFM use the same windows for deriving the horizontal current velocity, the results are similar, confirming the quality of RAM.

A simulated profile of voltage measurements is used to demonstrate why RAM can derive velocity results in a finer vertical resolution than HFM. When the least squares fit window contains less than two rotation cycles of voltage measurements, the uncertainty in the estimation of motion-induced voltage will cause velocity errors in HFM. Although shortening the window length also causes errors in the voltage offset results of RAM, these errors are presented as sinusoidal-oscillating signals in the results of u˜. Extracting the low-frequency components (i.e., u) from u˜ via bin averaging will effectively reduce errors due to imperfect estimation of voltage offset at the same time. It allows RAM to derive horizontal current velocity results in a vertical resolution finer than HFM.

The uncertainties in the RAM’s velocity results due to the estimation of voltage offset are then studied using in situ data. Three parameters of the fit are adjusted in the sensitivity tests, including the length of the fitting windows, the number of overlapping measurements with adjacent windows, and the order of the polynomial functions in the fit. None of these parameters can affect u significantly. To control the quality of u, if the mean ΔΦoffset˜ is highly fluctuating in time, and the estimates of ΔΦoffset˜ in the overlapped windows are scattered significantly, RAM may have failed to estimate the voltage offset in this sequence of float data. The results of u are excluded if their temporal fluctuations of ΔΦoffset˜ and scattering of estimated ΔΦoffset˜ are in the upper 5% tier simultaneously. The scattering of estimated ΔΦoffset˜ from the float data is used for simulating errors in the results of voltage offset. The fluctuating 1-s voltage offset errors below the 95th percentile should have negligible effects on the 12-s velocity results.

RAM is used for deriving the u at two floats (f9207 and f9208) near Mien-Hua Canyon. Combined with the 2-m CTD measurements at the floats, the vertical shear from RAM can be used for estimating 2-m results of gradient Richardson number Ri. Most Ri is greater than 0.25, except some low Ri at f9207 between 40 and 80 m around yearday 55.5 in 2021. The vertical shear spectra Ψ are then computed using the profiles of ∂u/∂z. Despite limited results at k > 0.1 cpm, the composite Ψ at both floats is nearly proportional to k−1 at vertical wavenumber k = 0.1–0.2 cpm, consistent with the spectra of vertical shear (Dewan and Good 1986). Interestingly, Ψ remains tied with the saturated spectra at k < 0.1 cpm. Because of the qualitative agreement between the estimated Ψ and saturated spectra, the effect of ΔT on the spectral slope of Ψ is further discussed. According to the spectrum results estimated using ΔT ranging from 6 to 20 s, shortening ΔT from 10 to 8 s significantly increases the spectral level of Ψ at high wavenumbers k > 0.3 cpm. If a short ΔT fails to separate the u from surface wave signals reliably, the errors in the estimated vertical shear will increase the spectral level at high wavenumbers.

In short, RAM uses the 1-s voltage and magnetometer measurements to convert EM-measured voltage into the Cartesian coordinate as 1-s results of u˜. The high-frequency signals of u˜ have been used for deriving surface wave spectra (Hsu 2021), so the horizontal current velocity u as the low-frequency signals of u˜ can be derived by temporal averaging the u˜ in the intervals ΔT ≥ 12 s. The qualitative agreement between the estimated Ψ and saturated spectra may also support the robustness of RAM in deriving u at the fine scale. Of course, the present analysis focuses on the method for increasing the resolution of horizontal velocity on the EM-APEX floats, so the intercomparison to the velocity measurements from other platforms, such as LADCP (Kunze et al. 2006), is still required in the following studies. Because microstructure sensors can be mounted on EM-APEX floats to measure the TKE dissipation rate ϵ (Lien et al. 2016; Kunze et al. 2021), it will be interesting for the floats to simultaneously measure ϵ and the high-resolution Ri. These results will be useful for guiding the parameterizations of turbulent mixing in the surface boundary layer (Polzin et al. 1995), which is critical to the simulated air–sea interactions in the coupled models.

Acknowledgments.

The author appreciates the National Science and Technology Council in Taiwan for funding this work (112-2611-M-002-029). This manuscript is a tribute to the contribution of Dr. Sanford, who invented the EM-APEX floats. The author also appreciates the comments from Dr. Ren-Chieh Lien and several reviewers for improving the structure of the manuscript.

Data availability statement.

The processed float measurements are available upon request to the author. The codes of the rotating axes method are currently stored at https://github.com/namon123x/emhighvel.git.

APPENDIX A

Harmonic Fitting Method

Assuming the linear change of offset ΔΦoffset in time, the float-measured ΔΦi(t) in Eq. (1) is often least squares fitted in 50-s windows with 25-s overlapped measurements (Sanford et al. 1978, 2011), i.e.,
ΔΦiL(t)=a1˜cos⁡θΩ˜(t)+a2˜sin⁡θΩ˜(t)+a3˜t+a4˜+ϵi,for i=1,2,
where a1˜ to a4˜ are the fitted coefficients, L the separation between electrodes, and ϵi the residuals (V m−1). θΩ˜=θΩ0˜+(2−i)(π/2)  is computed by using the measurements of the magnetic field [Eq. (2)]. The horizontal velocity of a low-frequency current (<0.02 Hz) can then be computed by using the coefficients of a1˜ and a2˜ (Sanford et al. 1978) at each pair of electrodes separately, i.e.,
{ui+u*¯=a2˜FzC1 υi+υ*¯=FyFzC2C1W−a1˜FzC1,for i=1,2,
where u*¯ (=u*¯i^+υ*¯j^; Sanford et al. 1978) is a depth-independent current due to the unknown background electric field, often negligible in the whole water column. The i = 1 and i = 2 represent the results of fitted coefficients from the E1 or E2 pair of electrodes, respectively. Although the residuals ϵi in the least squares fit should not directly result from the unreliable estimate of sinusoidal oscillation of motion-induced voltage, the HFM often computes the variable Verr using the standard deviation of ϵi (σϵ) in the 50-s windows, i.e.,
Verr=σϵFzC1L.
Some studies use the mean horizontal current velocity between E1 and E2 for the analysis (e.g., Shay et al. 2019). Because the velocity estimates on the individual pair of electrodes may be biased when the Verr is large [Eq. (A3)], the estimates of horizontal current velocity u (=ui^+υj^) from the harmonic fitting method are computed by weighted averaging ui and υi as uh=[ϵ22/(ϵ12+ϵ22)]u1+[ϵ12/(ϵ12+ϵ22)]u2 and υh=[ϵ22/(ϵ12+ϵ22)]υ1+[ϵ12/(ϵ12+ϵ22)]υ2 (Hsu et al. 2017). The HFM method is applied to a sequence of datasets on f9207. The fitted coefficients of the sinusoidal functions [i.e., a1˜ and a2˜ in Eq. (A1)] result from the zonal and meridional current velocities (Fig. A1). Because the change in voltage offset within this sequence of datasets is close to a linear trend, the fitted results agree well with the raw voltage measurements at both electrode pairs. The residuals on E2 are slightly larger than those on E1 because the voltage offset on E2 changes more nonlinearly than E1.
Fig. A1.
Fig. A1.

Demonstration of how the harmonic fitting method processes the voltage measurements [blue lines in (a) and (e)] on (left) E1 and (right) E2 at float f9207 in the 50-s window. The fitted results of the voltage [red lines in (a) and (e)] are constituted by (b),(f) two sinusoidal functions and (c),(g) voltage offset. The residuals as the difference between the observations and fitted results are shown in (d) and (h).

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

APPENDIX B

Electric Current Induced by Seawater Motion

In motional induction theory, electrons brought by the motion of seawater in Earth’s magnetic field F (=Fyj^+Fzk^) can induce an electric field and electric current in the ocean (Sanford 1971). We can define the induced horizontal electric current J near the sea surface constituted by that due to a low-frequency ocean current (J¯) and surface waves (J′), respectively (Hsu et al. 2018), i.e.,
J=J¯+J′.
Because the horizontal scale of most ocean currents, such as inertial waves, is much larger than their vertical scale, the horizontal gradient and vertical velocity have negligible effects on J¯. The electric field J¯/σ due to the current velocity v (=u+wk^=ui^+υj^+wk^) is therefore simplified (Sanford et al. 1978) as
J¯σ=Jx¯σi^+Jy¯σj^=v×F−∇Φ≈Fz(υ+υ*¯)i^−Fz(u+u*¯)j^,
where σ is the conductivity of seawater, −∇Φ is the background electric field, and i^, j^, and k^ are the unit vectors in the x, y, and z directions, respectively. The depth-independent current u*¯=u*¯i^+υ*¯j^ is caused by an unknown background electric field ∇Φ (Sanford et al. 1978), often negligible at the deep ocean. On the other hand, because of the orbital velocity of surface waves, the wave-induced J′/σ in deep water can be expressed (Hsu et al. 2018) as
J′σ=Jx′σi^+Jy′σj^=Fz(υwi^−uwj^),
where uw=uwi^+υwj^ is the apparent horizontal velocity of surface waves whose magnitude has been modulated by the geomagnetic field’s inclination and propagation direction of surface waves (Hsu et al. 2018). The uw also decays exponentially in depth.
Based on Eqs. (B1)–(B3), the relationship between the motion-induced electric current J and u˜ can be written as
Jσ=Jx*i^+Jy*j^=v˜×F−∇Φ≈Fz(υ˜+υ*¯)i^−Fz(u˜+u*¯)j^,
where v˜ (=u˜+w˜k^=u˜i^+υ˜j^+w˜k^) is the seawater velocity, and Jx* and Jy* are the electric current J that have been divided by the conductivity of seawater σ as an electric field (section 3). The horizontal seawater velocity u˜ near the sea surface is constituted by a low-frequency current and high-frequency surface waves, i.e., u˜ =u+uw.

APPENDIX C

Voltage Measurements at EM-APEX Floats

a. Voltage offset at each electrode pair

RAM fits the voltage offset ΔΦoffset˜ using the 50-s windows at two floats near Mien-Hua Canyon, respectively (section 4). At each float (Fig. C1), the mean of ΔΦoffset˜ at each electrode pair can change by more than 300 μV within 1 day. That is, the ΔΦoffset˜ at each float can vary with time significantly. Because the density stratification in the upper ocean does not change significantly during the experiment, there may be some unknown factors causing the temporal change of ΔΦoffset˜. Instead of assuming a constant dependence of voltage offset on the ocean structure for all floats, least squares fitting the voltage offset in each profile of the floats should be more reasonable.

Fig. C1.
Fig. C1.

Results of voltage offset ΔΦoffset˜ on two pairs of electrodes (a),(b) E1 and (c),(d) E2 at two floats: (left) f9207 and (right) f9208.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

b. Instrumental noise estimates

The time series of voltage measurements are used for computing the frequency spectra of voltage after excluding the results of ΔΦoffset˜ from the raw measurements (Fig. C2). The estimated spectra are averaged in the frequency intervals of 0.01 Hz for smoothing. The composite frequency spectra of voltage at two pairs of electrodes Ψi (i = 1 for the E1 pair and i = 2 for the E2 pair) are then computed by using all profiles at the floats (section 4). Two peaks are observed below 0.15 Hz, which is similar to the rotation rate of the EM sensors (rotation period ∼20 and 10 s during ascending and descending, respectively). The spectra Ψ1 and Ψ2 are nearly flat above 0.3 Hz. The flat spectral level (white noise) of 9 × 10−5 m2 s−1 at the high-frequency bands may bias the voltage measurements. By assuming the upper limit of the spectral level of the white noise is ∼9 × 10−5 m2 s−1, the standard deviation of instrument noise of voltage measurements will yield the uncertainties of velocity results ∼0.007 m s−1.

Fig. C2.
Fig. C2.

Composite frequency spectra using the voltage measurements from all profiles at the two floats, captured by two pairs of electrodes (E1 and E2). The unit of the spectral level is converted to the speed of seawater motion based on the motional induction theory.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

c. Simulation of voltage measurements

In this study, several profiles of voltage measurements at EM-APEX floats are simulated and used for validating the robustness of RAM. A reference profile of horizontal current velocity is used for simulating the electric current in the ocean (appendix A). Because the vertical float motion will affect the rotation rate of EM sensors, the measurements at two floats near Mien-Hua Canyon are used for deriving the dependence of rotation rate on the vertical speed of the floats (not shown in this study). The simulated electric current is then captured by two rotating electrode pairs of a vertical-moving float as voltage [Eq. (1)], assuming no signals due to surface waves, instrumental noise, and an unknown depth-independent current u*¯. Because factors to the change of voltage offset are still unclear, a voltage offset with depth variations is assumed and added to the simulated voltage measurements at floats. That is, the simulated profiles of 1-s voltage measurements are the sum of the motion-induced voltage and a slow-varying voltage offset.

APPENDIX D

Experiment near Mien-Hua Canyon

Around the end of February 2021, a 6-day field experiment was conducted near Mien-Hua Canyon to test two APF-11 EM-APEX floats (f9207 and f9208; Fig. D1). All raw voltage and magnetic field measurements are transmitted via Iridium satellite communication. The wind speed and significant wave height can be more than 10 m s−1 and 3 m, respectively. Float f9208 was deployed at approximately 25.1°N, 122.4°E at 1030 UTC 23 February 2021. Because of the westward tidal current, the float initially drifted to the continental shelf within the first half day. After recovery (i.e., the end of the first leg), the float was redeployed (i.e., the start of the second leg) at approximately 1400 UTC 24 February 2021. On the other hand, float f9207 drifted northward due to the Kuroshio and then turned west due to the tidal current. The trajectory between the second leg of f9208 and f9207 was similar. Two floats were recovered by the ship before 0600 UTC 25 February 2021.

Fig. D1.
Fig. D1.

Trajectories of the two EM-APEX floats f9207 (blue line with dots) and f9208 (black lines with dots) around the end of February 2021 near Mien-Hua Canyon. The trajectory of f9208 from 1000 UTC 23 Feb (yearday 53.44) to 0200 UTC 24 Feb (yearday 54.11) is defined as the first leg and that from 1400 UTC 24 Feb (yearday 54.61) to 0200 UTC 25 Feb (yearday 55.11) is defined as the second leg. The time interval between each dot is 2 h.

Citation: Journal of Atmospheric and Oceanic Technology 41, 3; 10.1175/JTECH-D-23-0014.1

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    • Search Google Scholar
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Journal of Atmospheric and Oceanic Technology

  • Chang, H., and Coauthors, 2019: Generation and propagation of M2 internal tides modulated by the Kuroshio northeast of Taiwan. J. Geophys. Res. Oceans, 124, 2728–2749, https://doi.org/10.1029/2018JC014228.

    • Search Google Scholar
    • Export Citation

  • Chereskin, T. K., and A. J. Harding, 1993: Modeling the performance of an acoustic Doppler current profiler. J. Atmos. Oceanic Technol., 10, 41–63, https://doi.org/10.1175/1520-0426(1993)010<0041:MTPOAA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., and R.-C. Lien, 2000: The wave–turbulence transition for stratified flows. J. Phys. Oceanogr., 30, 1669–1678, https://doi.org/10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Dewan, E. M., and R. E. Good, 1986: Saturation and the “universal” spectrum for vertical profiles of horizontal scalar winds in the atmosphere. J. Geophys. Res., 91, 2742–2748, https://doi.org/10.1029/JD091iD02p02742.

    • Search Google Scholar
    • Export Citation

  • Essink, S., E. Kunze, R.-C. Lien, R. Inoue, and S.-i. Ito, 2022: Near-inertial wave interactions and turbulence production in a Kuroshio anticyclonic eddy. J. Phys. Oceanogr., 52, 2687–2704, https://doi.org/10.1175/JPO-D-21-0278.1.

    • Search Google Scholar
    • Export Citation

  • Forryan, A., A. P. Martin, M. A. Srokosz, E. E. Popova, S. C. Painter, and A. H. H. Renner, 2013: A new observationally motivated Richardson number based mixing parametrization for oceanic mesoscale flow. J. Geophys. Res. Oceans, 118, 1405–1419, https://doi.org/10.1002/jgrc.20108.

    • Search Google Scholar
    • Export Citation

  • Gargett, A. E., P. J. Hendricks, T. B. Sanford, T. R. Osborn, and A. J. Williams, 1981: A composite spectrum of vertical shear in the upper ocean. J. Phys. Oceanogr., 11, 1258–1271, https://doi.org/10.1175/1520-0485(1981)011<1258:ACSOVS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Garrett, C., and W. Munk, 1975: Space‐time scales of internal waves: A progress report. J. Geophys. Res., 80, 291–297, https://doi.org/10.1029/JC080i003p00291.

    • Search Google Scholar
    • Export Citation

  • Garrett, C., and W. Munk, 1979: Internal waves in the ocean. Annu. Rev. Fluid Mech., 11, 339–369, https://doi.org/10.1146/annurev.fl.11.010179.002011.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., D. P. Winkel, and T. B. Sanford, 1993: Varieties of fully resolved spectra of vertical shear. J. Phys. Oceanogr., 23, 124–141, https://doi.org/10.1175/1520-0485(1993)023<0124:VOFRSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gregg, M. C., D. P. Winkel, T. B. Sanford, and H. Peters, 1996: Turbulence produced by internal waves in the oceanic thermocline at mid and low latitudes. Dyn. Atmos. Oceans, 24, 1–14, https://doi.org/10.1016/0377-0265(95)00406-8.

    • Search Google Scholar
    • Export Citation

  • Hsu, J.-Y., 2021: Observing surface wave directional spectra under Typhoon Megi 2010 using subsurface EM-APEX floats. J. Atmos. Oceanic Technol., 38, 1949–1966, https://doi.org/10.1175/JTECH-D-20-0210.1.

    • Search Google Scholar
    • Export Citation

  • Hsu, J.-Y., R.-C. Lien, E. A. D’Asaro, and T. B. Sanford, 2017: Estimates of surface wind stress and drag coefficients in Typhoon Megi. J. Phys. Oceanogr., 47, 545–565, https://doi.org/10.1175/JPO-D-16-0069.1.

    • Search Google Scholar
    • Export Citation

  • Hsu, J.-Y., R.-C. Lien, E. A. D’Asaro, and T. B. Sanford, 2018: Estimates of surface waves using subsurface EM-APEX floats under Typhoon Fanapi 2010. J. Atmos. Oceanic Technol., 35, 1053–1075, https://doi.org/10.1175/JTECH-D-17-0121.1.

    • Search Google Scholar
    • Export Citation

  • Hu, K., and Q. Chen, 2011: Directional spectra of hurricane-generated waves in the Gulf of Mexico. Geophys. Res. Lett., 38, L19608, https://doi.org/10.1029/2011GL049145.

    • Search Google Scholar
    • Export Citation

  • Hughes, K. G., J. N. Moum, and E. L. Shroyer, 2020: Evolution of the velocity structure in the diurnal warm layer. J. Phys. Oceanogr., 50, 615–631, https://doi.org/10.1175/JPO-D-19-0207.1.

    • Search Google Scholar
    • Export Citation

  • Klema, M. R., A. G. Pirzado, S. K. Venayagamoorthy, and T. K. Gates, 2020: Analysis of acoustic Doppler current profiler mean velocity measurements in shallow flows. Flow Meas. Instrum., 74, 101755, https://doi.org/10.1016/j.flowmeasinst.2020.101755.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., 2019: A unified model spectrum for anisotropic stratified and isotropic turbulence in the ocean and atmosphere. J. Phys. Oceanogr., 49, 385–407, https://doi.org/10.1175/JPO-D-18-0092.1.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., A. J. Williams III, and M. G. Briscoe, 1990: Observations of shear and vertical stability from a neutrally buoyant float. J. Geophys. Res., 95, 18 127–18 142, https://doi.org/10.1029/JC095iC10p18127.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., E. Firing, J. M. Hummon, T. K. Chereskin, and A. M. Thurnherr, 2006: Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles. J. Phys. Oceanogr., 36, 1553–1576, https://doi.org/10.1175/JPO2926.1.

    • Search Google Scholar
    • Export Citation

  • Kunze, E., J. B. Mickett, and J. B. Girton, 2021: Destratification and restratification of the spring surface boundary layer in a subtropical front. J. Phys. Oceanogr., 51, 2861–2882, https://doi.org/10.1175/JPO-D-21-0003.1.

    • Search Google Scholar
    • Export Citation

  • Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, 363–403, https://doi.org/10.1029/94RG01872.

    • Search Google Scholar
    • Export Citation

  • Lien, R.-C., and T. B. Sanford, 2019: Small-scale potential vorticity in the upper-ocean thermocline. J. Phys. Oceanogr., 49, 1845–1872, https://doi.org/10.1175/JPO-D-18-0052.1.

    • Search Google Scholar
    • Export Citation

  • Lien, R.-C., T. B. Sanford, S. Jan, M.-H. Chang, and B. B. Ma, 2013: Internal tides on the East China Sea continental slope. J. Mar. Res., 71, 151–185, https://doi.org/10.1357/002224013807343461.

    • Search Google Scholar
    • Export Citation

  • Lien, R.-C., T. B. Sanford, J. A. Carlson, and J. H. Dunlap, 2016: Autonomous microstructure EM-APEX floats. Methods Oceanogr., 17, 282–295, https://doi.org/10.1016/j.mio.2016.09.003.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., M. E. Stern, and H. Stommel, 1954: The electrical field induced by ocean currents and waves, with applications to the method of towed electrode. Pap. Phys. Oceanogr. Meteor., 13, 1–37, https://doi.org/10.1575/1912/1064.

    • Search Google Scholar
    • Export Citation

  • Mueller, D. S., J. D. Abad, C. M. García, J. W. Gartner, M. H. García, and K. A. Oberg, 2007: Errors in acoustic Doppler profiler velocity measurements caused by flow disturbance. J. Hydraul. Eng., 133, 1411–1420, https://doi.org/10.1061/(ASCE)0733-9429(2007)133:12(1411).

    • Search Google Scholar
    • Export Citation

  • Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.

    • Search Google Scholar
    • Export Citation

  • Polzin, K., 1996: Statistics of the Richardson number: Mixing models and finestructure. J. Phys. Oceanogr., 26, 1409–1425, https://doi.org/10.1175/1520-0485(1996)026<1409:SOTRNM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Polzin, K., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterizations of turbulent dissipation. J. Phys. Oceanogr., 25, 306–328, https://doi.org/10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Price, J. F., T. B. Sanford, and G. Z. Forristall, 1994: Forced stage response to a moving hurricane. J. Phys. Oceanogr., 24, 233–260, https://doi.org/10.1175/1520-0485(1994)024<0233:FSRTAM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Rennie, C. D., 2008: Uncertainty of ADCP spatial velocity distributions. Proc. Sixth Int. Symp. on Ultrasonic Doppler Method for Fluid Mechanics and Fluid Engineering, Prague, Czech Republic, ISUD, 147–150.

  • Sanford, T. B., 1971: Motionally induced electric and magnetic fields in the sea. J. Geophys. Res., 76, 3476–3492, https://doi.org/10.1029/JC076i015p03476.

    • Search Google Scholar
    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (left) Photo of the EM-APEX float and sensors and (right) the schematic illustration of two pairs of electrodes E1 and E2 viewed from the top of the float. The motion-induced electric current (Jx is the zonal component i^ and Jy is the meridional component j^) is captured by the electrode pairs as voltage. The dashed arrows labeled hx and hy are the orthogonal axes for the magnetometer measurements. The angle θΩ is the angle from each electrode pair to magnetic east (i = 1 for the E1 pair and i = 2 for the E2 pair), and θΩ0 = Ωt + ϕ0 is the angle between hx and the magnetic east, assuming hx is aligned with E2 pair. The Ω is the rotation rate of the electrodes, L is the separation between two electrodes, and ϕ0 is the angle at a reference time t = 0. Fy is the magnetic north of Earth.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) Voltage measurements taken by two pairs of electrodes E1 and E2, (b) magnetometer measurements on hx and hy, after removing the mean of hx and hy and then being divided by the standard deviation of hx and hy, respectively, and (c) temperature [blue line in (c)] and salinity [red line in (c)] from one profile of the float f9207. The mean and standard deviation of hx and hy in (b) are computed using the whole 550-s data. Because the offset of the magnetometer measurements is nearly constant in time, using the whole 550-s magnetometer measurements for computing the mean and standard deviation will derive the same result from the moving 50-s windows.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a),(b) Illustration on the points within the profile that are used for estimating voltage offset ΔΦoffset˜ (black dots). Because the 50-s windows [blue dots in (a)] are moving in the increments of 5 s, each 1-s point has 10 estimates of ΔΦoffset˜ except near the edge of the profile. The estimates [along the red lines in (a)] are used for computing the mean and standard deviation σoffset [error bars in (b)]. The mean ΔΦoffset˜ at the points having less than 5 estimates are excluded (the sequence of data at the time prior to the black dashed line). The plot in (b) also demonstrates an alternative way for evaluating the uncertainty of the estimated ΔΦoffset˜, assuming two different profiles of ΔΦoffset˜. Compared to the slowly fluctuating ΔΦoffset˜ (green solid line), the temporal change of ΔΦoffset˜ (Φt; unit V s−1) at the highly fluctuating ΔΦoffset˜ (magenta solid line) will yield higher standard deviation σΦt of Φt in each 12-s window [dashed lines with values labeled on the right of (b)].

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Raw voltage measurements (lines with sinusoidal variations) and the mean (solid lines between the sinusoidal oscillation) and standard deviation (nearly invisible vertical lines are error bars) of (a) estimated ΔΦoffset˜ on E1 and E2, (b),(c) estimates of voltage offset ΔΦoffset˜ on E1 [blue lines in (b)] and E2 [red lines in (c)] in each 50-s window, (d) voltage measurements after removing the mean of estimates of ΔΦoffset˜ , and (e) estimates of horizontal seawater velocity u˜ in zonal (blue) and meridional directions (red). The estimated orientation of electrode pairs ΔθΩ0˜ is shown as black dots in (d), labeled on the right axis of (d). The lines with dots in (e) are u˜ averaged in 12-s intervals as horizontal current velocity u. The upper axis of (a) displays the depth of measurements. The results from the low-pass filter are compared to the fitted results in each window in (b) and (c).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Frequency spectra computed using estimates of horizontal seawater velocity in different layers—(a) 20–50-m depth; (b) 50–80-m depth; (c) 80–100-m depth—for float f9208. The color of the lines represents the time of the individual profiles in the yearday of 2021. The black dashed line indicates frequency of (1/12) Hz, i.e., period of 12 s.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a),(b) Mean vertical resolution Δz of ur (red lines with dots; rotating axes method) and uh (blue lines with dots; harmonic fitting method) from each profile at the two floats. (c),(d) The results are then compared (x axis: uh; y axis: ur) and marked in the color based on the vertical speed of the floats W.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    (a)–(c) Voltage offset ΔΦoffset˜ and (d)–(f) current velocity u estimated by using least squares fit windows in RAM whose length equals 18 s (blue dots and lines), 36 s (yellow dots and lines), and 50 s (green dots and lines), respectively. The black lines are the reference profiles used for simulating the profile of voltage measurements at an EM-APEX float, including the (a),(b),(c),(h) voltage offset and (d)–(f) horizontal current velocity. The red dots in (d)–(f) are the current velocity results from HFM. Dashed lines between two dots in (a)–(c) are different sequences of estimated voltage offset. Shorter window length of the least squares fit will increase the scattering of voltage offset between overlapped windows. (g) The results in the overlapped windows are averaged and used for computing their discrepancy to the reference profile of voltage offset.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Comparisons between uref and uexp by using the results from the two floats, where uref is the horizontal current velocity derived from the default setting in the rotating axes method, and uexp is the processed horizontal current velocity by changing (a)–(c) the length of the fitting windows, (d)–(f) the number of overlapped measurements in each window, and (g)–(i) the order of polynomial functions in the estimation of voltage offset ΔΦoffset˜. The mean and RMS of the difference between uref and uexp are labeled on the upper-left corner of the panels.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Probability distribution (bars) and cumulative distribution (lines) of (a) ΔσΦt and (b) Δσu using the results at two floats f9207 (blue) and f9208 (red). The dots indicate the intersection between the cumulative distribution and 95% tier line (black dashed lines).

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Mean (sold lines, bottom axis) and standard deviation (dashed lines, top axis) of horizontal current velocity u, computed using the voltage offset results plus 100 realizations of voltage offset errors. The voltage offset results are derived by using the simulated voltage measurements captured by EM sensors whose rotation period (colored lines) varies from 10 to 22 s. The black solid line is the reference profile of horizontal current velocity for simulating motion-induced voltage.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    (a),(f) Zonal (u) and (b),(g) meridional (υ) current velocities, (c),(h) squared vertical shear [(∂u/∂z)2 + (∂υ/∂z)2], (d),(i) buoyancy frequency N2, and (e),(j) inverse gradient Richardson number (Ri−1) at the two EM-APEX floats near Mien-Huan Canyon: (left) f9207 and (right) f9208. The vertical trajectories of the floats (black dots) from the CTD measurements are shown in (a). Magenta dots marks the results where N2 < 0.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    (a),(b) Individual spectra of vertical shear using each profile of ur (colored lines) and (c),(d) composite spectra of vertical shear (blue lines: Ψh from HFM; red lines: Ψr from RAM) at the two float: (left) f9207 and (right) f9208. The lines of each profile in (a) and (b) are colored based on its time in the upper 100 m. The red shading area in (c) and (d) covers the mean ± one standard deviation of RAM’s Ψr results. The black double-headed arrows display the range of vertical wavenumber band k for fitting the composite spectra on the logarithmic scale, and the slopes of log(Ψh) and log(Ψr) are labeled. The black thick lines are the empirical GM model. The black dashed lines extended from the black thick lines with the −1 slope are the saturated spectra. The vertical black dashed lines marks the typical rolloff wavenumber of 0.1 cpm in the GM spectrum.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    (a),(b) Vertical shear spectra Ψ at the two EM-APEX floats, computed by using the results of horizontal current velocity from different intervals ΔT (colored lines) in the rotating axes method. The black double-headed arrows display the range of vertical wavenumber band k for fitting the composite spectra on the logarithmic scale, and the slopes of log(Ψ) at different ΔT are labeled. The black solid lines are the empirical GM model, and the black dashed lines are the saturated spectra.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Demonstration of how the harmonic fitting method processes the voltage measurements [blue lines in (a) and (e)] on (left) E1 and (right) E2 at float f9207 in the 50-s window. The fitted results of the voltage [red lines in (a) and (e)] are constituted by (b),(f) two sinusoidal functions and (c),(g) voltage offset. The residuals as the difference between the observations and fitted results are shown in (d) and (h).

  • <sc>Fig</sc>. C1.
    View in gallery
    Fig. C1.

    Results of voltage offset ΔΦoffset˜ on two pairs of electrodes (a),(b) E1 and (c),(d) E2 at two floats: (left) f9207 and (right) f9208.

  • <sc>Fig</sc>. C2.
    View in gallery
    Fig. C2.

    Composite frequency spectra using the voltage measurements from all profiles at the two floats, captured by two pairs of electrodes (E1 and E2). The unit of the spectral level is converted to the speed of seawater motion based on the motional induction theory.

  • <sc>Fig</sc>. D1.
    View in gallery
    Fig. D1.

    Trajectories of the two EM-APEX floats f9207 (blue line with dots) and f9208 (black lines with dots) around the end of February 2021 near Mien-Hua Canyon. The trajectory of f9208 from 1000 UTC 23 Feb (yearday 53.44) to 0200 UTC 24 Feb (yearday 54.11) is defined as the first leg and that from 1400 UTC 24 Feb (yearday 54.61) to 0200 UTC 25 Feb (yearday 55.11) is defined as the second leg. The time interval between each dot is 2 h.

Fig. 1.

(left) Photo of the EM-APEX float and sensors and (right) the schematic illustration of two pairs of electrodes E1 and E2 viewed from the top of the float. The motion-induced electric current (Jx is the zonal component i^ and Jy is the meridional component j^) is captured by the electrode pairs as voltage. The dashed arrows labeled hx and hy are the orthogonal axes for the magnetometer measurements. The angle θΩ is the angle from each electrode pair to magnetic east (i = 1 for the E1 pair and i = 2 for the E2 pair), and θΩ0 = Ωt + ϕ0 is the angle between hx and the magnetic east, assuming hx is aligned with E2 pair. The Ω is the rotation rate of the electrodes, L is the separation between two electrodes, and ϕ0 is the angle at a reference time t = 0. Fy is the magnetic north of Earth.
Fig. 2.

(a) Voltage measurements taken by two pairs of electrodes E1 and E2, (b) magnetometer measurements on hx and hy, after removing the mean of hx and hy and then being divided by the standard deviation of hx and hy, respectively, and (c) temperature [blue line in (c)] and salinity [red line in (c)] from one profile of the float f9207. The mean and standard deviation of hx and hy in (b) are computed using the whole 550-s data. Because the offset of the magnetometer measurements is nearly constant in time, using the whole 550-s magnetometer measurements for computing the mean and standard deviation will derive the same result from the moving 50-s windows.
Fig. 3.

(a),(b) Illustration on the points within the profile that are used for estimating voltage offset ΔΦoffset˜ (black dots). Because the 50-s windows [blue dots in (a)] are moving in the increments of 5 s, each 1-s point has 10 estimates of ΔΦoffset˜ except near the edge of the profile. The estimates [along the red lines in (a)] are used for computing the mean and standard deviation σoffset [error bars in (b)]. The mean ΔΦoffset˜ at the points having less than 5 estimates are excluded (the sequence of data at the time prior to the black dashed line). The plot in (b) also demonstrates an alternative way for evaluating the uncertainty of the estimated ΔΦoffset˜, assuming two different profiles of ΔΦoffset˜. Compared to the slowly fluctuating ΔΦoffset˜ (green solid line), the temporal change of ΔΦoffset˜ (Φt; unit V s−1) at the highly fluctuating ΔΦoffset˜ (magenta solid line) will yield higher standard deviation σΦt of Φt in each 12-s window [dashed lines with values labeled on the right of (b)].
Fig. 4.

Raw voltage measurements (lines with sinusoidal variations) and the mean (solid lines between the sinusoidal oscillation) and standard deviation (nearly invisible vertical lines are error bars) of (a) estimated ΔΦoffset˜ on E1 and E2, (b),(c) estimates of voltage offset ΔΦoffset˜ on E1 [blue lines in (b)] and E2 [red lines in (c)] in each 50-s window, (d) voltage measurements after removing the mean of estimates of ΔΦoffset˜ , and (e) estimates of horizontal seawater velocity u˜ in zonal (blue) and meridional directions (red). The estimated orientation of electrode pairs ΔθΩ0˜ is shown as black dots in (d), labeled on the right axis of (d). The lines with dots in (e) are u˜ averaged in 12-s intervals as horizontal current velocity u. The upper axis of (a) displays the depth of measurements. The results from the low-pass filter are compared to the fitted results in each window in (b) and (c).
Fig. 5.

Frequency spectra computed using estimates of horizontal seawater velocity in different layers—(a) 20–50-m depth; (b) 50–80-m depth; (c) 80–100-m depth—for float f9208. The color of the lines represents the time of the individual profiles in the yearday of 2021. The black dashed line indicates frequency of (1/12) Hz, i.e., period of 12 s.
Fig. 6.

(a),(b) Mean vertical resolution Δz of ur (red lines with dots; rotating axes method) and uh (blue lines with dots; harmonic fitting method) from each profile at the two floats. (c),(d) The results are then compared (x axis: uh; y axis: ur) and marked in the color based on the vertical speed of the floats W.
Fig. 7.

(a)–(c) Voltage offset ΔΦoffset˜ and (d)–(f) current velocity u estimated by using least squares fit windows in RAM whose length equals 18 s (blue dots and lines), 36 s (yellow dots and lines), and 50 s (green dots and lines), respectively. The black lines are the reference profiles used for simulating the profile of voltage measurements at an EM-APEX float, including the (a),(b),(c),(h) voltage offset and (d)–(f) horizontal current velocity. The red dots in (d)–(f) are the current velocity results from HFM. Dashed lines between two dots in (a)–(c) are different sequences of estimated voltage offset. Shorter window length of the least squares fit will increase the scattering of voltage offset between overlapped windows. (g) The results in the overlapped windows are averaged and used for computing their discrepancy to the reference profile of voltage offset.
Fig. 8.

Comparisons between uref and uexp by using the results from the two floats, where uref is the horizontal current velocity derived from the default setting in the rotating axes method, and uexp is the processed horizontal current velocity by changing (a)–(c) the length of the fitting windows, (d)–(f) the number of overlapped measurements in each window, and (g)–(i) the order of polynomial functions in the estimation of voltage offset ΔΦoffset˜. The mean and RMS of the difference between uref and uexp are labeled on the upper-left corner of the panels.
Fig. 9.

Probability distribution (bars) and cumulative distribution (lines) of (a) ΔσΦt and (b) Δσu using the results at two floats f9207 (blue) and f9208 (red). The dots indicate the intersection between the cumulative distribution and 95% tier line (black dashed lines).
Fig. 10.

Mean (sold lines, bottom axis) and standard deviation (dashed lines, top axis) of horizontal current velocity u, computed using the voltage offset results plus 100 realizations of voltage offset errors. The voltage offset results are derived by using the simulated voltage measurements captured by EM sensors whose rotation period (colored lines) varies from 10 to 22 s. The black solid line is the reference profile of horizontal current velocity for simulating motion-induced voltage.
Fig. 11.

(a),(f) Zonal (u) and (b),(g) meridional (υ) current velocities, (c),(h) squared vertical shear [(∂u/∂z)2 + (∂υ/∂z)2], (d),(i) buoyancy frequency N2, and (e),(j) inverse gradient Richardson number (Ri−1) at the two EM-APEX floats near Mien-Huan Canyon: (left) f9207 and (right) f9208. The vertical trajectories of the floats (black dots) from the CTD measurements are shown in (a). Magenta dots marks the results where N2 < 0.
Fig. 12.

(a),(b) Individual spectra of vertical shear using each profile of ur (colored lines) and (c),(d) composite spectra of vertical shear (blue lines: Ψh from HFM; red lines: Ψr from RAM) at the two float: (left) f9207 and (right) f9208. The lines of each profile in (a) and (b) are colored based on its time in the upper 100 m. The red shading area in (c) and (d) covers the mean ± one standard deviation of RAM’s Ψr results. The black double-headed arrows display the range of vertical wavenumber band k for fitting the composite spectra on the logarithmic scale, and the slopes of log(Ψh) and log(Ψr) are labeled. The black thick lines are the empirical GM model. The black dashed lines extended from the black thick lines with the −1 slope are the saturated spectra. The vertical black dashed lines marks the typical rolloff wavenumber of 0.1 cpm in the GM spectrum.
Fig. 13.

(a),(b) Vertical shear spectra Ψ at the two EM-APEX floats, computed by using the results of horizontal current velocity from different intervals ΔT (colored lines) in the rotating axes method. The black double-headed arrows display the range of vertical wavenumber band k for fitting the composite spectra on the logarithmic scale, and the slopes of log(Ψ) at different ΔT are labeled. The black solid lines are the empirical GM model, and the black dashed lines are the saturated spectra.
Fig. A1.

Demonstration of how the harmonic fitting method processes the voltage measurements [blue lines in (a) and (e)] on (left) E1 and (right) E2 at float f9207 in the 50-s window. The fitted results of the voltage [red lines in (a) and (e)] are constituted by (b),(f) two sinusoidal functions and (c),(g) voltage offset. The residuals as the difference between the observations and fitted results are shown in (d) and (h).
Fig. C1.

Results of voltage offset ΔΦoffset˜ on two pairs of electrodes (a),(b) E1 and (c),(d) E2 at two floats: (left) f9207 and (right) f9208.
Fig. C2.

Composite frequency spectra using the voltage measurements from all profiles at the two floats, captured by two pairs of electrodes (E1 and E2). The unit of the spectral level is converted to the speed of seawater motion based on the motional induction theory.
Fig. D1.

Trajectories of the two EM-APEX floats f9207 (blue line with dots) and f9208 (black lines with dots) around the end of February 2021 near Mien-Hua Canyon. The trajectory of f9208 from 1000 UTC 23 Feb (yearday 53.44) to 0200 UTC 24 Feb (yearday 54.11) is defined as the first leg and that from 1400 UTC 24 Feb (yearday 54.61) to 0200 UTC 25 Feb (yearday 55.11) is defined as the second leg. The time interval between each dot is 2 h.
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