1 Introduction

Air temperature and water vapor gradients in transition zones between the land and the ocean shelf as well as between different water masses substantially affect the propagation of electromagnetic waves (EM ducting) in the marine atmospheric boundary layer (Thompson and Haack 2011; Geernaert 2007). As for the latter, an example is the sharp oceanic fronts, such as the Gulf Stream (GS) cold wall, where the rate of evaporation may greatly differ on the warm and cold sides of the front, being dependent on the wind speed, the air humidity, and the sea surface temperature (SST). The humidity and SST gradients in the atmospheric marine boundary layer affect the development of evaporative ducts, which is critical in radar operations, both communications and object detection. The SST depends not only on the atmospheric fluxes, state of the sea surface, and horizontal and vertical advections of sea water but also on turbulence and associated mixing in the water interior. The ocean surface layer can be well-mixed or stratified, and its depth is mainly determined by wind and convective mixing, lateral advection, stratification, and turbulent mixing in the underlying pycnocline. Therefore, microstructure measurements in the upper 100–200 m of the ocean or through the entire water column at shallow depths can provide valuable information on the turbulent kinetic energy (TKE) dissipation rate ε and vertical eddy diffusivity K z (Osborn 1980) in various mesoscale (30–100 km) and sub-mesoscale (1–10 km or less) dynamical structures (e.g., Thomas et al. 2008; Gula et al. 2016), depending on prevailing energetic processes. Comprehensive measurements shedding light on small-scale turbulence and associated frontal dynamics (down to sub-mesoscale features) have been reported by D’Asaro et al. (2011) for Kuroshio, Thomas et al. (2016) for Gulf Stream, and Jinadasa et al. (2016) for Bay of Bengal.

The Coupled Air Sea Processes and Electromagnetic Ducting Research (CASPER) program has been developed to study EM ducting in coastal areas. The CASPER-East field campaign (http://met.nps.edu/∼qwang/casper/home/home.php) was conducted on the North Carolina (NC) shelf and in contiguous waters of Gulf Stream during October–November 2015, and this paper discusses several aspects of microstructure measurements during its oceanographic component. The CASPER microstructure measurements were carried out by R/V Atlantic Explorer (Bermuda Institute of Ocean Sciences; http://www.bios.edu/ship-ops) on the shelf to the east of the coastal village Duck, NC and also in deep waters of the Gulf Stream. A bathymetric map of the region showing locations of the measurements is in Fig. 1. The analysis of microstructure data presented in this paper is focused on observations at two shelf stations R02 (75.61° W, 36.18° N) and R30 (75.14° W, 36.18° N) conducted on October 13–21, 2015 and located 2 and 30 miles away from the coast, respectively. Three blue water stations (GS_S, GS_N, and R56; October 31 and November 1, 2015) were taken at the southern and northern ends of the hydrographic transect across the Gulf Stream and at the slope, where oceanic microstructure could be influenced by various mesoscale and sub-mesoscale processes associated with the Gulf Stream. Note that sub-mesoscale structures are controlled by the Rossby number Ro = U/Lf that is close to unity (e.g., McWilliams 1985, 2008). Here, U and L are characteristic velocity and length scales and f is the Coriolis parameter. For characteristic f ≈ 8.6 × 10−5/s in the region and surface currents on the shelf U shl  ∼ 0.3 − 0.4 m/s, the horizontal scales of the structures pertained to Ro ~ Ο(1) are L shl  ~ 3 − 4 km. For much higher velocities in the Gulf Stream, U gs  ∼ 1 − 2 m/s, the scales of sub-mesoscale structures may grow to 10–20 km, which could be resolved along the transect.

Fig. 1
figure 1

The Etopo-2 bathymetry of the Casper-East measurement site with the locations of VMP stations on the North Carolina shelf (R02, φ = 36.18° N, λ = 75.61° W, ∼17 m depth; R30, φ = 36.18° N, λ = 75.14° W, ∼30 m depth, stars), near the shelf break (R56, red snowflake, φ = 36.25° N, λ = 74.76° W), and in the deep ocean at the southern (GS_S, φ = 35.83° N, λ = 74.1° W, magenta triangle) and northern (GS_N, magenta diamond for 10 a.m. and square for 8–9 p.m.; Nov 1, φ = 36.15° N, λ = 74.53° W) ends of the hydrographic transect (GSc–GS7, yellow circles) crossing the Gulf Stream front

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Note that the estimates of TKE dissipation rate for NC shelf have been reported first by St. Laurent (2009). The data were obtained in the Onslow Bay, and are compared with our shelf measurements in Sect. 3. The history of the dissipation measurements in various sections of the Gulf Stream is richer, but mostly has been carried out decades ago (Oakey and Elliott 1977; Gregg and Sanford 1980; Osborn 1980; Gargett and Osborn 1981; Lueck and Osborn 1984; Winkel et al. 2002). Newer comprehensive field campaigns that allow linking small-scale turbulence with some features of mesoscale and sub-mesoscale GS dynamics were taken in 2007 (Inoue et al. 2010) and 2012 (Thomas et al. 2016) in the regions hundreds of miles to the northeast from our GS measurements, which are described in Sect. 4. Section 5 summarizes major findings of the current study.

2 Instrumentation, data collection, and processing

The turbulence and stratification data were collected using a vertical microstructure profiler (VMP)-500 (http://rocklandscientific.com/products/profilers/vmp-500/) from the sea surfaces down to ∼17 m (at R02; 38 casts total) and 30–33 m (R30; 37 casts) on the shelf and to ∼130–150 m in deep waters (14 casts in 3 locations shown in Fig. 1). The VMP carries two airfoil probes to estimate small-scale shear and the dissipation rate ε, a three-component accelerometer, a pressure sensor (depth), and a Seabird temperature–conductivity unit to obtain precise temperature, salinity, and potential density profiles. Turbulence in the upper 3–5 m near the sea surface could have been contaminated by the ship movement; thus, the dissipation rate measurements were trusted in the depth range below 5 m. The data processing of each individual cast followed the methodology of Roget et al. (2006); however, the small-scale shear spectra were fitted to Nasmyth (1970) rather than to the Panchev and Kesich (1969) canonical spectrum to calculate ε averaged over 0.5 m (on the shelf) or 1 m (deep water) segments.

During October 13–21, 2015, VMP profiling was conducted once per day in the morning around 9–10 a.m. (R30) and in the afternoon around 2–3 p.m. (R02) local time (LT). Most of the time, the ship sailed along φ = 36.18° N back and forth between these two focal stations, transmitting EM signals to another ship that sailed specific routes based on the experimental design. The ship continuously collected standard meteorological data, underway water temperature T w3m and near-surface salinity S 3m at z = −3 m by the shipboard water intake system. Occasional VMP measurements were also taken between R02 and R30. The measurement cycle at R02 and R30 contained a number of consecutive VMP casts (mostly five casts in each series) to obtain the burst-averaged profiles of temperature T (z), salinity S (z), specific potential density σ θ (z), and the TKE dissipation rate ε(z) during ∼15–30 min interval (the exact number of VMP casts are shown in Fig. 4b). The averaging reduced the short-term fluctuations of stratification and turbulence at various depths (the vertical coordinate z is positive upwards). The estimates of vertical eddy diffusivity used the expression K z  ≡ K N  ∼ γε/N 2, where N 2(z) =  − (g/ρ 0) × (dρ θ /dz) is the squared buoyancy frequency, γ a mixing efficiency, and g ,  ρ θ , and ρ 0 are the gravity, potential, and reference densities, respectively. A canonical value of γ = 0.2 was used for fully developed turbulence, although it is possible that it depends on a variety of background parameters (e.g., Lozovatsky and Fernando 2013).

The R/V Atlantic Explorer was equipped with a 75-kHz Acoustic Doppler Current Profiler (ADCP). Owing to its low vertical resolution (16 m), it was not possible to infer useful estimates of the mean vertical shear \( Sh={\left[{\left(\Delta \overline{u}/\Delta z\right)}^2+{\left(\Delta \overline{v}/\Delta z\right)}^2\right]}^{1/2} \) and the gradient Richardson numbers Ri = N 2/Sh 2 at shallow stations R02 and R30. Here, \( \overline{u} \) and \( \overline{v} \) are the zonal and meridional components of mean currents. In blue water, however, the ADCP currents and vertical shear were calculated using CODAS (http://currents.soest.hawaii.edu/docs/adcp_doc/codas_doc/) software and then interpolated to a 10 m grid; the Sh(z) and Ri(z) profiles started at z = −36 m, which is the mid-point of the first unbiased bin below the sea surface.

Shipboard meteorological measurements at 12 m above the sea surface were used to calculate the wind stress and friction velocity u ∗(t) adjusted to a standard 10 m height. The buoyancy flux J b (t) was estimated following Shay and Gregg (1986), using the bulk-formula scripts of Matlab air–sea toolbox (http://woodshole.er.usgs.gov/operations/sea-mat/air_sea-html/index.html). It appears (Fig. 2) that during the first 3.5 days of observations, warm easterly winds (mean wind speed W 10 = W a  = 5.9 m/s) prevailed over the shelf. The averaged air temperature \( {\overline{T}}_a={20.6}^{{}^{\circ}}\mathrm{C} \) was less than only one degree below the water temperature \( {\overline{T}}_{w3 m}={21.45}^{{}^{\circ}}\mathrm{C} \). At mid-day of October 16, the wind direction sharply changed toward the south, and this northerly wind dominated over the shelf until October 20. On October 18–19, the wind speed W a increased to 12–14 m/s and the air temperature dropped to 11–12 °C. Only at the end of October 20, the cold northerly wind ceased to 2–3 m/s. Warmer easterly and southerly winds returned after this mild storm, and on October 21, the air temperature T a rose to 17–18 °C. The water temperature \( {\overline{T}}_{w3 m} \) gradually decreased during the storm, almost catching up with T a on October 21 (Fig. 2). The surface salinity S 3m , in contrast to T w3m , depicts substantial spatial variations when the ship sailed along 36.18° N between R02 and R30 (see Fig. 2). The difference of salinity between relatively fresh coastal (R02) and more saline mid-shelf (R30) waters was about 4 psu on the average. On the mid-shelf, the salinity did not change much during the first 3.5 days of observations (S 3m ∼33–33.1 psu), increasing to ~33.6–34 psu after the storm. Near the coast, however, S 3m was steadily increasing day by day from 28.8 to ~30 psu, but on October 18, when the northerly winds ceased, S 3m rapidly decreased again to 28 psu on October 20.

Fig. 2
figure 2

The ship track between R02 (open triangles) and R30 (filled triangles) (the longitude λ along 36.18°N); the sea surface temperature Tw3m and salinity Sw3m (at z = −3 m); the air temperature T a , wind direction ϕ 10 and wind speed W a  = W 10 , at 10 m above the sea surface; buoyancy J b flux and friction velocity u * at the sea surface (every second sample is plotted). All meteorological data are 30 min averages

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The observed spatial-temporal variations of salinity on the inner shelf were mostly caused by alternative advection of relatively fresh estuarine and highly saline ocean water on shelf, depending on the direction of dominant winds. Indeed, the HYbrid Coordinate Ocean Model (HYCOM; https://hycom.org/hycom) simulated surface currents (Fig. 3; the maps were adopted from http://assets.maracoos.org/) clearly show that on October 16, the freshwater outflow from Albemarle Sound was partially blocked near R02 by along shore (north-west directed) shelf currents, causing the observed increase of S 3m on inner shelf. On October 19, however, the opposite inflow of saline water into Albemarle Sound was driven by northerly stormy winds. On October 20–21, fresh estuarine water restored its flow out of Albemarle Sound onto the shelf (Oct 21; Fig. 3), being supported by the restored easterly winds. As a result, the near coast surface salinity rapidly reduced. On the other hand, relatively gradual increase of S 3m on mid-shelf near R30 was probably associated with currents entering on shelf from the shelf break to the east of R30 (Oct 19; Fig. 3).

Fig. 3
figure 3

The HYCOM simulated surface currents for October 16, 19, and 21, 2015 at 9 a.m. local time. The distance between R02 and R30 is about 50 km. The data are from http://assets.maracoos.org

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Fig. 4
figure 4figure 4

a Temperature T, salinity S, and specific potential density σ θ contour plots overlaid by the averaged TKE dissipation rate ε profiles at R02 (the upper 5 m are blanked as ε in this layer could be affected by ship movement). The number of VMP casts in each series is given in the upper panel below black triangles. The time averaged (every 3 h) wind vectors W a (t) are on the very top; below are the wind vectors for the duration of dissipation measurements. b Temperature T, salinity S, and specific potential density σ θ contour plots overlaid by the averaged TKE dissipation rate ε profiles at R30 (the upper 5 m are blanked as ε in this layer could be affected by ship movement). The number of VMP casts in each series is given in the upper panel below black triangles. The time averaged (every 3 h) wind vectors W a (t) are on the very top; below are the wind vectors for the duration of dissipation measurements

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This brief analysis of hydrometeorological conditions for the duration of our VMP measurements points to wind-driven aspects of mesoscale dynamics on the NC shelf. In addition to wind forcing and freshwater discharge in the surface layer, barotropic tidal currents and episodic intrusions of Gulf Stream water mainly in the bottom boundary layer (BBL) shaped the mesoscale background conditions for turbulence and mixing in the region. Note that the mean currents and stratification on the NC shelf have been thoroughly explored during Coastal Ocean Processes Inner Shelf (CoOP94) study, leading to an extensively list of publications (e.g., Lentz et al. 1999; Austin and Lentz 1999; Rennie et al. 1999; Lentz 2001; Lentz et al. 2001, to name a few).

3 Turbulence and mixing on the NC shelf

As mentioned, to our knowledge, the only prior turbulence measurements on the NC shelf have been taken on November 5–11, 2009 in Onslow Bay between ~34°–34° 25′ N and 76° 55′–76° 45′ W (St. Laurent 2009), about 100 miles to the south of the CASPER site at shallow depths of 30–35 m. The TKE dissipation rate ε therein ranged from ~10−9 to 10−6 W/kg, being influenced by the variations of stratification, from well-mixed conditions to multi-layered structures, mainly due to the variable amount of freshwater discharge to the bay. The temperature and salinity profiles reported by St. Laurent (2009) show structures similar to those obtained in our measurements, which are shown in Fig. 4a, b for R02 and R30, respectively. The most prominent thermohaline feature in the figures is the along-bottom intrusion of a warmer, saline GS water, which occupied the lower ~10 m at R30 and ∼7 m at R02 between October 13 and October 18. The highest temperature 24.48 °C and salinity 34.79 psu of this intrusion occurred on October 16 at R30, and the corresponding maxima at R02 were 21.32 °C, with 32.15 psu on the same day. The intrusion was stably stratified N 2 ∼ (4 − 5) × 10−4 /s2 at R30 and N 2 ∼ (8 − 9) × 10−5/s2 at R02, indicating that bottom friction did not produce substantial turbulent mixing in the BBL. The dissipation profiles in Fig. 4a, b are consistent with this notion, showing a low level of the dissipation rate (ε <  ∼ 10−8 W/kg) on the mid-shelf as well as at the shallower depths near the coast.

During the storm (see wind vectors for October 18–19 in Fig. 4), the intrusion was annihilated. The water column became completely mixed closer to the coast (R02), but at R30, fragments of a weak near-bottom thermohalocline remained above the BBL because of two opposite influences of the storm. First, wind-driven horizontal advection of colder more salinity stratified water from the north (see the wind and current vectors in Fig. 4b and Fig. 3 for October 18–19) tends to strengthen stratification at R30 at all depths. Thus, the wind generated turbulence at the sea surface was forced to work against this stronger stratification, producing less effective vertical mixing. At the beginning of the storm, downward penetration of TKE was impeded by strong near-surface pycnocline (the mean buoyancy frequency N p  ∼ 4.5 × 10−2/s) developed a day earlier due to the appearance of a lower-saline lens in the upper 10 m of the mixed layer, h 0 = 10 m. This effect in combination with lateral advection at all depths could explain the observed incomplete homogeneity of the water column on the mid-shelf. The mixed layer model of Pollard et al. (1972) can be used to estimate (Lozovatsky et al. 2005) the approximate final depth z mx of a 10-m-deep mixed layer with underlying stable stratification subject to a wind stress, viz.,

$$ {z}_{mx}={2}^{3/4}{\mu}_{{}_{\ast }}/\sqrt{N_p f,} $$
(1)

where the Coriolis parameter f = 0.86 × 10−4/s and the mean friction velocity during October 18u * = 1.3 × 10−2 m/s (see Fig. 2). The calculated value is z mx  = 11.3 m, leading to a mixed layer depth (MLD) of h mx  = h 0 + z mx  ≈ 21 − 22 m, which agrees reasonably well with what observed on October 19 (Fig. 4b).

The averaged dissipation rate profiles \( \tilde{\varepsilon}(z) \) shown in Fig. 4b for the mid-shelf suggest that turbulence below the surface layer was generally weak under light and mild winds. On October 15, for example, the characteristic values of \( \tilde{\varepsilon}(z) \) in the upper quasi-homogeneous layer (between 5 and 22 m) were in the range (2 − 3) × 10−9 − (1 − 2) × 10−8 W/kg. However, after a mild storm on October 19, the dissipation rate in the same almost unstratified depth range increased up to \( \tilde{\varepsilon}\sim \left(0.6-3\right)\times {10}^{-7}\ \mathrm{W}/\mathrm{kg} \), approximately in accordance with the boundary layer (law of the wall) scaling \( {\varepsilon}_{lw}(z)={u}_{\ast}^3/\kappa z \) for unstratified flow; here, κ = 0.4 is the Karman constant. The observed \( \tilde{\varepsilon}(z) \) profiles on October 15 and 19, however, were underestimated by ε lw (z), about 1.2 and more than two times under moderate and stormy winds, respectively. This could be explained following the work of Anis and Moum (1995); during the growing phase of wind waves (during the storm, in our case), certain amount of wind energy and momentum is consumed by the wave generation. Thus, the mixed layer turbulence ε lw (z) below the wind waves should be scaled by the residual fraction of the original friction velocity u * estimated at 10 m above the sea surface.

The wind-induced turbulence, which was dominant on the NC shelf, produced vertical mixing that can be quantified by the diffusivity profiles K N (z) shown in Fig. 5a, b for R02 and R30, respectively. During non-stormy periods, the diffusivities near the coast were relatively small in the lower part of the water column (z > −9 m), varying from almost molecular values up to ∼10−5 m2/s. Under stormy winds, however, K N (z) jumped to ∼10−1 m2/s on October 18, decreasing during the next days to ∼10−3 m2/s (below z = −10 m). Only on October 20, the diffusivity at R02 relaxed to pre-stormy values. Such high values of diffusivity (10−1 − 10−3m2/s), nevertheless, should be viewed with caution because variable, but stable density gradient is almost completely destroyed by the storm, leading to very small and possibly uncertain estimates of the buoyancy frequency.

Fig. 5
figure 5

The eddy diffusivity profiles K N (z) at R02 (a) and R30 (b) (the upper 5 m are blanked as this layer could be affected by ship movement). The cumulative distribution function CDF of K N for R30 (c) approximated by lognormal distribution with the parameters μ and σ given in the legend. The arrow points to the median value \( {\widehat{K}}_N\sim 5\times {10}^{-4}{\mathrm{m}}^2/ s \)

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On the mid-shelf, the diffusivities in the water interior spanned over five decades during the observational period, from ∼10−6 to ∼10−1 m2/s. To analyze this wide variability of K N , the cumulative distribution function CDF (K N ) was calculated and roughly approximated by a lognormal distribution with parameters μ and σ given in the legend of Fig. 5c. The application of lognormal model to the probability distribution of eddy diffusivity has been shown and discussed in Roget et al. (2006). The diffusivities on mid-shelf are characterized by the median value \( {\widehat{K}}_N\sim 5\times {10}^{-4}{\ \mathrm{m}}^2/\mathrm{s} \), which only slightly exceeds the median estimate of lognormal approximation e μ = 4.2 × 10−4 m2/s, and it is comparable with the values in other regions (e.g., Bastida et al. 2012). The most significant departure of the empirical CDF(K N ) from the lognormal fit is observed around K N  ∼ 10−5 m2/s, which is probably due to relatively short sampling record (464 K N samples below 5 m during 8 days) used for the CDF calculation.

4 Mixing in the Gulf Stream and on the slope near the shelf break

The majority of direct measurements of the TKE dissipation rate and temperature microstructure in waters affected by the Gulf Stream have been conducted closer to the northern edge of the current, in the region approximately bounded by 38°–40° N and 62°–69° W (e.g., Oakey and Elliott 1977; Gregg and Sanford 1980; Osborn 1980; Gargett and Osborn 1981; Lueck and Osborn 1984; Thomas et al. 2016). Inoue et al. (2010) analyzed the dissipation data obtained along a hydrographic section centered at ∼66° W, which crossed the GS between 37° and 39° N. Turbulence measurements in the Florida Current, which is the most southern segment of the GS, have been discussed in Winkel et al. (2002), but no estimates of mixing intensity in the central part of GS have been reported so far.

Note that the vertical diffusivity in the GS pycnocline has been found to be relatively small, which, according to Oakey and Elliott (1977) and Winkel et al. (2002), is ∼(2 − 4) × 10−5m2/s. Slightly higher diffusivities, K N  ∼ 5 × 10−5 m2/s, have been reported for the GS pycnocline by Osborn (1980), Gregg and Sanford (1980), and Gargett and Osborn (1981) under moderate winds. However, strong winter winds and unstable buoyancy flux at the sea surface may induce strong mixing below the surface mixed layer with diffusivities exceeding 10−4 m2/s (Inoue et al. 2010).

4.1 Stratification and currents along a Gulf Stream transect

In Figs. 6 and 7, several contour plots are shown that utilize our hydrographic data across a central section of the GS. The transect (see Fig. 1) started approximately near the GS core (station GS7, SST = 27.9 °C), crossed the GS north wall (GSNW) between GS3 (SST = 26.5 °C) and GS1 (SST = 21.6 °C), and ended close to the continental slope (the lowest SST = 19.6 °C, however, was observed slightly to the south of station GSa). Several intrusions of colder less saline shelf water were observed to the north of the GSNW. Note that the underway SST and salinity measurements across the GSNW revealed an extremely sharp and narrow local front, which was visible at the sea surface and detected by the ship’s radar. The temperature and salinity differences over a distance of 100 m across the front exceeded 0.5 °C and 0.6 psu,respectively (Lozovatsky et al. 2016). A sub-mesoscale low-salinity lens/meander occupied the upper ∼50 m layer of the transect, extending to the north of the GSNW for about 15 km. Salinity in the lens core dropped below 33.5 psu, indicating shelf origin of this water.

Fig. 6
figure 6

The temperature (a), salinity (b), and geostrophic velocity (c) contour plots along the hydrographic section (Fig. 1), extending from the GS warm core (GS7) to the continental slope while crossing the GS cold wall (GS2–GS1). The distances in nautical miles start from the most northern station GSc

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On the contrary, the main salinity maximum (S max > 36.7 psu) was observed in the thermocline, at z = −120–130 m in the GS core (Fig. 6b). The depth of this maximum outcropped toward the sea surface along the sloping isothermal/isopycnal surfaces (σ θ contours are shown in Fig. 7), losing salinity on the way. Strong horizontal density (pressure) gradients along the GS transect must induce significant geostrophic currents directed northeastward. Figure 7c shows that near the southern end of transect a high-amplitude geostrophic component, U gs  = 0.7 − 1.1 m/s, occupied the upper 90–100 m layer, indicating the proximity to the GS core. A geostrophic flow in this region with U gs  > 0.1 m/s occupied the upper ∼250 m layer. The thickness of the northeastern geostrophic current decreased to about 70–100 m near the GSCW, and it almost vanished between GSa and GSb. An indication of a weak southwestern counter current can be found near the very northern station GSc. Note the possibility of a strong peripheral branch of Gulf Stream in the vicinity of the GSCW, which is seen in the upper ∼70 m layer, with high-speed geostrophic flow having U gs  > 1 m/s near the sea surface.

Fig. 7
figure 7

The ADCP currents along the GS section over the Richardson number log10 Ri color palette and σ θ contours (a); the vector’s magnitude is proportional to the flow speed shown near the arrows by numbers. Regions with Ri <1 are encircled by thin magenta lines. The vertical shear log10 Sh plot (b) is overlaid by the same σ θ contours (labeled only here) as those shown in a

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The geostrophic component is expected to dominate the oceanic currents in the GS region, but other dynamical phenomena such as drift currents, inertial oscillations, tides, and internal wave-induced currents can also play role in determining the variables that sway the spatial structure of turbulence and vertical mixing. To this end, the total measured ADCP current vectors at every station of the CTD transect are shown in Fig. 7. The northeastward flow to the south of GSNW (GS1) corresponds well with the direction of the GS; the amplitudes of the vectors in Fig. 7a are up to 2 m/s close to the GS core (GS4–GS7, z = −30–40 m). Direct current measurements show much deeper penetration of GS flow compared to U gs , which was calculated using the end point of each CTD cast (z end =  − 300 m) as the zero level of geostrophic velocity (which is the usual approach, but it appears to be inadequate for current work). On the other hand, spatial structure of U gs in Fig. 6c and current vectors in Fig. 7a are in excellent qualitative agreement (but the vector amplitudes are about twice of U gs ), even depicting a very weak southward current near the northern end of the transect. The only clear difference between the currents shown in Figs. 6c and Fig.7a is the absence of a peripheral GS branch near the GSCW in the ADCP data, which can be attributed to unresolved upper 30 m layer by a 75-kHz shipboard ADCP.

The mean vertical shear in the GS pycnocline appears to be not high (see Fig. 7b), which corroborates several previous GS studies (e.g., Gregg and Sanford 1980; Gargett and Osborn 1981). As a result, the gradient Richardson numbers calculated with 10 m vertical resolution exceed not only the linear limit of 0.25 but also the non-linear critical value of 1 almost everywhere along our transect, let alone near the upper boundary of the pycnocline (the depth range ∼30–80 m). Only a few patches with Ri < 1 were observed deeper in the pycnocline, centering at z = −190 m (GS3) and z = −240–260 m (GS5 and GS6). Thus, the Ri-number color section shown in Fig. 7a may indicate a low probability of shear and convective instabilities (Ri < 1) in the GS pycnocline. However, several white-colored Ri segments inside the inclined portion of the pycnocline (Fig. 7a), with 1 < Ri < 1.8, suggest the instability of Holmboe waves that usually occurs in stratified shear layers at Ri > 1.3 (Strang and Fernando 2001). These waves can be a possible source of the turbulence generation therein.

Direct measurements of turbulence and the estimates of eddy diffusivity are examined next.

4.2 Mixing parameterizations

The diffusivities \( {\overline{K}}_N=0.2\overline{\varepsilon}/{\overline{N}}^2 \) were calculated using the original 1-m sampled estimates of ε(z) obtained during a series of VMP casts at a particular station, and then averaged over consecutive 10 m depth intervals. The calculated \( {\overline{N}}^2(z) \) were also averaged over the same 10 m segments that corresponded to \( \overline{Sh}(z) \) profiles and the gradient Richardson number \( \overline{Ri} \) = \( {\overline{N}}^2(z) \) / \( \overline{Sh}(z) \) 2. This approach allowed to obtain pairs of \( \overline{\varepsilon}-\overline{Ri} \) and \( {\overline{K}}_N-\overline{Ri} \) for further scaling analysis.

4.2.1 Near the Gulf Stream core

The relationship between \( {\overline{K}}_N \) and \( \overline{Ri} \) is shown in Fig. 8b for station GS_S located near the southern end of the GS transect in the vicinity of GS core, and the corresponding T(z), S(z), and σ θ (z) profiles are presented in Fig. 8a. The \( {\overline{K}}_N-\overline{Ri} \) samples shown in the regression plot are naturally partitioned into two distinct groups. The first one consists of five large blue snowflakes that are located along the best least square fit \( {\overline{K}}_N={K}_{sl}{\overline{Ri}}^{-1.2} \), with the coefficient of determination r 2 = 0.97 and the highest diffusivity recorded in the surface layer is K sl  = 5.8 × 10−3 m2/s at z = −36 m. Indeed, all these samples belong to the upper turbulent, weakly stratified part of the water column. A transition layer below MLD was occupied by an intrusion of slightly less saline, colder water atop of the pycnocline. The absolute values of \( {\overline{K}}_N \) near the core of GS flow are high, ranging between ∼10−2 and 10−5 m2/s. The rest of the diffusivity samples (smaller black snowflakes) are from the GS pycnocline, where a weak mixing at all depths is quantified by \( {\overline{K}}_N\sim 2\times {10}^{-6}\ {\mathrm{m}}^2/\mathrm{s} \) with only one exception at z = −126 m, where \( {\overline{K}}_N \) rises to 8.3 × 10−6 m2/s. This indicates that a power function dependence \( {\overline{K}}_N\left(\overline{Ri}\right) \) similar to that reported in several previous studies is applicable to the surface layer of the GS core flow with shear generated turbulence (see, for example, an overview and discussion in Lozovatsky et al. 2006). On the other hand, although the mean shear is still relatively high in the GS pycnocline (Fig. 7), strong stratification therein prevents shear induced turbulence. Mixing is very weak, and the diffusivities do not depend on the gradient Richardson number.

Fig. 8
figure 8

Left panel: temperature (red), salinity (green), and specific potential density (blue) profiles in the GS core near GS7. Right panel: the diffusivity K N (Ri) as a function of the gradient Richardson number in the pycnocline (z > −80 m, smaller black symbols) and above it (−30 < z < −80 m, large blue snowflakes; the corresponding mean depths of the samples are labeled in red)

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4.2.2 Waters to the north of the GSNW

The VMP stations marked as GS_N in Fig. 1 were taken close to the northern end of the GS transect in the morning (10 a.m., diamond) and in the evening (8–9 p.m., square) of November 1. The mesoscale features in this region is much more active and variable than that near the GS core. It is affected by the GS waters and waters originated on the shelf and continental slope. A large number of narrow local frontal zones, plenty of multiple intrusions (large and small) induced by GS filaments, and meanders interacting in vertical and horizontal planes are exemplified in the GS_N VMP profiles (Fig. 10a, b). Despite the rich variety of mesoscale features to the north of GSNW, the functional dependence between \( {\overline{K}}_N \) and \( \overline{Ri} \) generally holds everywhere in the shallow pycnocline, which starts almost at the sea surface (Fig. 9a) or covered by a thin (less than 25 m) mixed layer (Fig. 9b). The main trend \( {\overline{K}}_N\sim {\overline{Ri}}^{-1} \) (Fig. 9c) suggests that turbulent mixing in the layered pycnocline was mainly caused by shear generated turbulence. The two outliers in Fig. 9c at z = −106 and −116 m belonging to the 10 a.m. measurement Fig. 9a) may indicate a non-shear induced mixing in the lower part of a large, about 50 m thick, cold fresh intrusion, where shear near its inflection point (around z = −106 m, see Fig. 9a) should be negligibly small. This powerful intrusion may have had sufficient capacity to induce double-diffusive instability, generating diffusive mixing at its lower boundary under high gradient Richardson numbers (low shear). Indeed, the development of the diffusive regime of double diffusion (Turner 1973) is most probable in the depth range between ∼95 and ∼120 m, a claim supported by the values of Turner angle (Ruddick 1983), which are close to −90° (stars in Fig. 9a). The Turner angle is related to the stability ratio R ρ  =  − tan(Tu + 45o), where R ρ  = α∂ z T/β∂ z S, α and β are temperature expansion and salinity contraction coefficients, and ∂ z T and ∂ z S are vertical gradient of temperature and salinity, respectively. Note that double diffusive convection can also be developed in the upper part of this large intrusion where 45o < Tu < 90o (salt fingers regime, circles in Fig. 9a), but with a lower probability than the occurrence of diffusive regime below the intrusion. This is because Tu angles close to 90° occupy only a narrow depth range around z = −85 m, and importantly, there is a relatively high vertical shear at this depth that would overshadow molecular phenomena of double-diffusion.

Fig. 9
figure 9

Temperature (red), salinity (green), specific potential density (blue), and squared shear (black triangles) profiles at GS_N, to the north of the GSCW taken on November 1 at 10 a.m. (a) and 8–9 p.m. (b). The Turner angle (Tu) is in a: −90∘ < Tu < 45∘ (stars), −45∘ < Tu < 90∘ (open circles), see text for details. The eddy diffusivity—\( {\overline{K}}_N \)—as a function of the gradient Richardson number \( \overline{Ri} \) is in c. The depths of two outliers (−z = 116 and 126 m) are close to the velocity inflection point of a large cold intrusion (a), in which the vertical shear is very low but the possibility of diffusive convection is high (Tu ≈  − 90∘)

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The general outcome of above analysis is that mixing in the GS waters is mainly caused by regular shear instability, which generates turbulence almost everywhere in the pycnocline at the north frontal edge of the GS, but in the core of the GS current, mixing is only effective in a weakly stratified surface layer.

4.2.3 Continental slope near the shelf break

A special VMP station R56 (Fig. 1) was setup in deep waters on the continental slope near the NC shelf break. The measurements were taken on November 1 between the morning and the evening VMP casts at GS_N. The stratification at R56 was quite striking, exhibiting a strong narrow (∼10–15 m deep) near-surface pycnocline (Fig. 10a) originated due to a low saline, cold shelf water surface lens. A stably stratified layer (\( {N}_{up}^2 \) ∼1.2 × 10−4/s2) with decreasing downward temperature and increasing salinity occupied the depth range between ∼15–20 and ∼55–60 m. A narrow density interface separated this upper stratified layer form a deeper, somewhat less stratified pycnocline with \( {N}_{lp}^2\approx \left(4-5\right)\times {10}^{-5}{/\mathrm{s}}^2 \). A regression plot \( {\overline{K}}_N\left(\overline{Ri}\right) \) shown in Fig.10b depicts a power fit (with r 2 > 0.99) to three samples from the upper layer, \( {\overline{K}}_N=1.82\times {10}^{-6}{\overline{Ri}}^{-0.89} \) (Ri < 1), suggesting continuous decrease of the diffusivity with depth. However, further down, below z = −60 m, the diffusivities increased several times compared to \( {\overline{K}}_N \) (z = −56 m), forming a cloud of samples around \( {\overline{K}}_N\approx {10}^{-5}\ {\mathrm{m}}^2/\mathrm{s} \) without any particular dependence on \( \overline{Ri} \). Thus, regular shear instability of the mean currents appears to play a minor role for mixing in the lower pycnocline.

Fig. 10
figure 10

Temperature (red), salinity (green), and specific potential density (blue) profiles on the continental slope near the NC shelf break at R56 (a). The diffusivity \( {\overline{K}}_N(\overline{Ri}) \) as a function of the gradient Richardson number (b) and the TKE dissipation rate \( \overline{\varepsilon} \) as a function of the IW-turbulence parameter G iw (c); the large blue snowflakes are juxtaposed by the mean depths of samples (numbers in red)

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To this end, we explored the influence of internal waves (IWs) on turbulence generation at the continental slope. This mechanism is especially plausible at R56, which is near the shelf break where barotropic tide often generates tidal internal waves and packets of higher frequency non-linear waves, which finally degenerate into small-scale turbulence (e.g., St. Laurent et al. 2011). We explored the IW-based parameterization of turbulence (MacKinnon and Gregg (2003) with respect to our data. It links the TKE dissipation rate \( \overline{\varepsilon} \), not the oft-used diffusivity \( {\overline{K}}_N \), but with the so-called IW-dissipation parameter \( {G}_{iw}={\overline{\varepsilon}}_0\overline{N}\overline{Sh}/{N}_0{Sh}_0 \), where the reference values are N 0 = Sh 0 = 3 cph. The background dissipation \( {\overline{\varepsilon}}_0=6.9\times {10}^{-10}\ \mathrm{W}/\mathrm{kg} \) was derived by Henyey et al. (1986) from an analytical model. In later studies, alternative values of N 0, Sh 0, and \( {\overline{\varepsilon}}_0 \) have been used as adjustable parameters.

The dependence between \( \overline{\varepsilon} \) and G iw is shown in Fig. 10c. It appears that G iw in the upper pycnocline (−36 < z < −56 m) does not change much, and therefore, the dissipation rate \( \overline{\varepsilon} \) in this depth range (large snowflakes) is almost independent of G iw . All other \( \overline{\varepsilon} \) samples depict growing trend of \( \overline{\varepsilon} \) with increasing G iw . The least square fit to these data, \( \overline{\varepsilon}={\overline{\varepsilon}}_0{G}_{iw}^{1/2} \) with \( {\overline{\varepsilon}}_0=4.57\times {10}^{-10}\ \mathrm{W}/\mathrm{kg} \), is markedly weaker than the linear dependence suggested by MacKinnon and Gregg (2003), but it still indicates a substantial impact of internal waves on stratified turbulence in blue water on the continental slope to the north of the Gulf Stream front. The diffusivity \( {\overline{K}}_N \), however, is only indirectly (via \( \overline{\varepsilon} \)) affected by G iw . No specific functional dependence could be found between these two variables.

5 Conclusions

During October–November 2015, microstructure measurements were collected by R/V Atlantic Explorer on a shallow North Carolina shelf as well as in deep waters on the continental slope near the Gulf Stream northern wall and in the Gulf Stream core.

On the shelf, stratification was influenced by highly variable surface salinity, 28–31 psu near the coast and 31–34 psu on mid-shelf. The observed spatial–temporal variations of salinity on the inner shelf were caused by the advection of relatively fresh estuarine and highly saline ocean water, which was governed by the direction of dominant winds. The wind-driven dynamics on the NC shelf was the major source of small-scale turbulence and mixing in the water interior. In addition to wind forcing and freshwater discharge to the surface layer, barotropic tidal currents and episodic intrusions of Gulf Stream water, mainly in the bottom boundary layer, shaped the mesoscale background conditions for turbulence and mixing in the region.

The daytime wind-induced turbulence on the shelf can be scaled using the law of the wall parameterization, which is satisfied in the mixed layer below the wind waves. During the growing stage of surface waves, however, the modeled dissipation rate was found to be overestimated by the bulk-formula-based friction velocity u *. At depths below ∼5 m, the dissipation rate ε was generally low, not exceeding ∼2 × 10−8 W/kg. Only after a day-long stormy winds (with hourly averaged wind ∼14 m/s), the water column on mid-shelf (∼33 m depth) became turbulent with a characteristic dissipation rate ε ∼10−7 W/kg.

The vertical diffusivities K z near the coast were relatively small in the lower half of the water column, ranging from almost molecular values up to ∼10−5 m2/s during non-stormy periods. Under stormy winds, K z jumped to ∼10−3 − 10−1 m2/s at all depths. On mid-shelf, the diffusivities in the water interior vary over five decades with a median value of ∼5 × 10−4 m2/s.

The mean vertical shear in the Gulf Stream pycnocline was found to be fairly weak, consistent with data reported in previous studies. The gradient Richardson numbers calculated with 10 m vertical resolution exceeded the canonical critical values almost everywhere in the Gulf Stream pycnocline. Only a few patches with Ri < 1 were observed near the upper boundary of the pycnocline; however, in the upper mixed layer in the vicinity of the Gulf Stream core, the gradient Richardson number was sub-critical, favoring shear-induced turbulence. For shear induced turbulence, the diffusivity in GS waters shows a power dependence on Ri, K z  ∼ Ri −p, with p varying around unity.

The stratified turbulence on the slope, presumably generated by instabilities of high-frequency internal waves, showed an increase of ε in the pycnocline with increasing internal wave dissipation parameter \( {G}_{iw}={\overline{\varepsilon}}_0\overline{N}\overline{Sh}/{N}_0{Sh}_0 \). The empirical relationship so obtained, \( \overline{\varepsilon}={\overline{\varepsilon}}_0{G}_{iw}^{1/2} \) with \( {\overline{\varepsilon}}_0=4.57\times {10}^{-10}\ \mathrm{W}/\mathrm{kg} \), however, had a much weaker dependence between \( \overline{\varepsilon} \) and G iw than that originally suggested by MacKinnon and Gregg (2003).

More extensive data sets and theoretical studies are needed to analyze the interplay between mesoscale, sub-mesoscale, and small-scale (turbulence) dynamics more thoroughly. As reiterated by the referees, sub-meso and small-scale structures are variable in time and intermittent in space, which require substantial efforts to observe wide scales of processes simultaneously. Recent large oceanographic projects such as LATMEX (Shcherbina et al. 2015, Gulf Stream) and ASIRI (Wijesekera et al. 2016, Bay of Bengal) as well as preceding measurements in one of the Kuroshio local fronts (D’Asaro et al. 2011) are some antecedent comprehensive studies on the interdependence of sub-mesoscale dynamics and turbulence in the regions with high mesoscale activity.