Tidally induced residual current over the Malin Sea continental slope

Author’s Accepted Manuscript Tidally induced residual current over the Malin Sea continental slope Nataliya Stashchuk, Vasiliy Vlasenko, Hosegood, W. Alex M. Nimmo-Smith

Phil

www.elsevier.com/locate/csr

PII: DOI: Reference:

S0278-4343(16)30467-8 http://dx.doi.org/10.1016/j.csr.2017.03.010 CSR3572

To appear in: Continental Shelf Research Received date: 6 September 2016 Revised date: 3 March 2017 Accepted date: 24 March 2017 Cite this article as: Nataliya Stashchuk, Vasiliy Vlasenko, Phil Hosegood and W. Alex M. Nimmo-Smith, Tidally induced residual current over the Malin Sea continental slope, Continental Shelf Research, http://dx.doi.org/10.1016/j.csr.2017.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Tidally induced residual current over the Malin Sea continental slope Nataliya Stashchuka , Vasiliy Vlasenkoa , Phil Hosegooda , W. Alex M. Nimmo-Smitha , a

School of Biological and Marine Sciences, University of Plymouth, Drake Circus, Plymouth, PL4 8AA, UK

Abstract Tidally induced residual currents generated over shelf-slope topography are investigated analytically and numerically using the Massachusetts Institute of Technology general circulation model. Observational support for the presence of such a slope current was recorded over the Malin Sea continental slope during 88-th cruise of the RRS “James Cook” in July 2013. A simple analytical formula developed here in the framework of time-averaged shallow water equations has been validated against a fully nonlinear nonhydrostatic numerical solution. A good agreement between analytical and numerical solutions is found for a wide range of input parameters of the tidal flow and bottom topography. In application to the Malin Shelf area both the numerical model and analytical solution predicted a northward moving current confined to the slope with its core located above the 400 m isobath and with vertically averaged maximum velocities up to 8 cm s−1 , which is consistent with the in-situ data recorded at three moorings and along cross-slope transects. Keywords: Tides, slope currents, Malin shelf

Preprint submitted to Continental Shelf Research

March 25, 2017

1

1. Introduction

2

Warm and saline North Atlantic Waters moving northward along the Eu-

3

ropean coast are an important element of the global meridional overturning

4

circulation (White and Bowyer, 1997; Pingree and Le Cann, 1989; Huth-

5

nance and Gould, 1989). According to Huthnance (1995), Pingree and Le

6

Cann (1989), this slope current is nearly barotropic. It is confined to the con-

7

tinental slope with its core typically located above the 500 m isobath and with

8

maximum velocities ranging from 3 to 30 cm s−1 . It was suggested by Huth-

9

nance (1984, 1986) and Hill et al. (1998) that the relatively steady character

10

of this current reflects a density driven origin. The generation mechanism is

11

associated with less-dense lower-latitude waters “standing high” compared to

12

the northern basin. Note, however, that some other driving forces, e.g. wind,

13

tides, horizontal pressure gradients can also contribute to the formation of

14

the slope current (Huthnance et al., 2009).

15

The first systematic numerical studies of the slope current in the Malin

16

Sea shelf/slope area were undertaken by Pingree and Le Cann (1989) who

17

used a model domain with 10 km horizontal resolution for the region 40-64◦ N

18

x 13-23◦W. Vertically integrated model equations were used in a spherical

19

polar coordinate system with a meridional variable Coriolis parameter. The

20

model was forced by the M2 tidal harmonic and a 10 m s−1 steady south-west

21

wind blowing over 16 days. It was found that with such model settings the

22

combination of tides and wind produces quite a weak residual slope current.

23

Only the activation of horizontal buoyancy forcing which took into account

24

the meridional density gradient allowed the generation of a slope current with

25

velocity varying between 0.05 m s−1 on the Celtic Sea slope to 0.15 m s−1 off 2

26

the Hebrides Island slope.

27

The most recent field experiments with Lagrangian drifters in the area

28

(Charria et al., 2013; Porter et al., 2016) have shown that the structure

29

of the slope current has seasonal periodicity. In the Bay of Biscay over the

30

observational period from 2004 to 2009, the slope current was persistently di-

31

rected poleward during the autumn-winter season (from October to March),

32

but transported water in the opposite direction during the rest of the year

33

Charria et al. (2013). In contrast, the drifter experiments conducted by Bur-

34

rows and Thorpe (2002) above 55◦ N latitude showed the slope current to be

35

strongly directed poleward during the whole year.

36

It is commonly believed that tides do not contribute greatly to the net wa-

37

ter transport (Huthnance et al., 2009). Exception can be made for nonlinear

38

effects when strong tidal motions generate internal solitary waves, as in the

39

Celtic Sea, where these waves can transport water at the level of O(1 m2 s−1 ).

40

This result is based on the theory by Huthnance (1986) who concluded that

41

tidally driven slope currents account for only a small fraction of the slope

42

currents usually observed. With respect to the Malin Sea shelf-slope area the

43

principal baroclinic tidal effects recorded here were tidally generated internal

44

solitary waves and bottom trapped internal Kelvin waves (see, for instance,

45

Stashchuk and Vlasenko (2005, 2016) and references therein). However, as

46

shown by Xing and Davies (2001) in a series of numerical experiments, non-

47

linear tidal effects at the Hebrides shelf edge can be responsible for the gen-

48

eration of an along slope current with velocities of up to 5 cm s−1 .

49

In this paper we show that the role of tides in the formation of the slope

50

currents in the Malin Sea shelf/slope area has been underestimated. Results

3

51

of the numerical experiments conducted using the Massachusetts Institute

52

of Technology general circulation model, MITgcm, (Marshall et al., 1997)

53

reported here suggest tides to have a much greater role in the production of

54

the along-slope transport than previously thought.

55

The paper is organised as follows. All historical observational data avail-

56

able for the area are discussed in Section 2.

Section 3 describes high-

57

resolution numerical experiments conducted for the Malin Shelf/Slope area.

58

Section 4 presents an analytical solution for tidally rectified flows and numer-

59

ical analysis that can prove this solution. Generalisation of the developed

60

theory to the Malin Sea slope and its comparison with the MITgcm output

61

are given in section 6. The paper finishes with a Discussion and Conclusions

62

section.

63

2. Observational data

64

The observational data on the characteristics of the slope current over

65

the Hebridean Slope are available from several field experiments. Measure-

66

ments of White and Bowyer (1997) were conducted at two locations to the

67

north-west of Ireland between 54.5◦ N and 55◦ N in April-December 1994. A

68

persistent poleward along-slope current was recorded at both cross-sections

69

with peak values up to 0.5 m s−1 at the position of steepest slope.

70

The first large-scale multidisciplinary oceanographic study of the Malin

71

shelf/slope area, the Shelf Edge Study (SES), was conducted between March

72

1995 and September 1996 (Souza et al., 2001). The observational area covered

73

a rectangle with meridional range from 55◦ N to 58◦ N, and zonal range from

74

8◦ W to 10◦ W. The intensity, spatial structure, and temporal variability of the 4

75

slope current were recorded at a number of moorings using Acoustic Doppler

76

Current Profilers (ADCP). It was found that the slope current in the area

77

was predominately barotropic with velocity in the core around 0.2 m s−1 . The

78

flow was stronger in winter than in other seasons. The cross-slope velocity

79

was typically at the level of 0.02 m s−1 in summer and 0.04 m s−1 in winter.

80

Quite useful information on the spatial characteristics of the slope current

81

around the Hebrides was obtained during the ARGOS tracked drifter experi-

82

ment (Burrows and Thorpe, 2002). A number of drifters with drogues at 50 m

83

depth were released over the slope at 56◦ 15N on a line between the 200 m

84

and 1000 m isobaths. Based on the drifters’ trajectories, the slope current

85

was initially directed poleward along the continental slope in a laterally con-

86

strained jet-like flow. Depending on the position with respect to bathymetry,

87

the recorded slope current velocity ranged from 0.05 to 0.7 m s−1 .

88

The most recent measurements of the slope current were obtained in July

89

2013 during the JC88 cruise aboard the RRS “James Cook” as part of the

90

NERC-funded project, Fluxes Across Sloping Topography in the North-east

91

Atlantic (FASTNEt). A number of moorings equipped with ADCPs were

92

deployed along the slope as shown in Figure 1 with the aim of resolving

93

the coherence of the slope current as it encountered a canyon orientated

94

perpendicular to the slope. Vertically averaged ADCP time series recorded at

95

Lb, Sb, and Sd moorings are presented in Figure 2. They reveal a dominant

96

tidal periodicity in all recorded time series with a superposition of semi-

97

diurnal and diurnal tidal harmonics (panel SD in Figure 2, for instance)

98

which we refer to as diurnal intermittency. The TPXO8.1 model output

99

calculated for the positions of mooring deployment and presented to the

5

100

right in Figure 2 shows quite consistent intensity of the observational and

101

model predicted tidal signals with obvious diurnal intermittency. The only

102

significant difference between observational and predicted time series is the

103

presence of quasi-stationary currents recorded at the moorings. Low pass

104

25 h running average filtering of the depth-averaged time series revealed a

105

residual current with a velocity of between 0.05 and 0.2 m s−1 (the red lines

106

in Figure 2).

107

In addition to moorings, a number of drifters were released in the area

108

during the JC88 experiment. Analysis of the drifters’ tracks has also con-

109

firmed the presence of the along-slope current with maximum velocities up

110

to 0.2 m s−1 (find drifter tracks shown in Figure 1 by red, green, and blue

111

lines).

112

During a dye release experiment that sought to identify the circulation

113

associated with the slope current, a cross-slope transect was completed whilst

114

tow-yoing a microstructure profiler behind the ship at approximately 1 knot.

115

The position of the transect is indicated by the red straight line in Figure 1.

116

The vessel-mounted RDI 75 kHz Ocean Surveyor broadband acoustic Doppler

117

current profiler (VMADCP) data collected during this transect provides in-

118

formation on the cross-slope structure of the depth-dependent currents. Hor-

119

izontal currents were measured with a 5 second ping interval in 16 m bins.

120

We present here the 2 minute short-term averaged data which have subse-

121

quently been cleaned with a 7 point median filter and smoothed with a 5

122

point running average over both time and depth (Figure 3 b). Shallow-water

123

conditions at the transect (400-700 metre depth) allowed bottom-tracking of

124

the ship’s position. This fact was used for calculations of an absolute velocity

6

125

relative to the Earth using the method described in Joyce (1989) in which

126

the raw data were corrected by misalignment angle (0.1874◦) and amplitude

127

scaling factor (1.000816).

128

The vessel completed the transect downstream of the wall of the canyon

129

during 7.43 hours on 13th July 2013. The dye release targeted the slope

130

current core that was found in a water depth of 600 m. The dye moved into

131

water of 800 m depth such that the transect extended from the upper slope

132

(≤200 m depth) to depths ≥1000 m after crossing the canyon.

133

Within the transect, the poleward current is intensified near the bed in

134

the slope region. A core of the slope current is located above the bed in

135

the area of 400 m isobath, Figure 3 b. The figure represents both spatial and

136

temporal variability of the velocity field and is thus difficult to interpret when

137

the tidal signal and slope currents are comparable. Note, however, the time

138

series of tidal currents predicted by the TPXO8.1 tidal model (Egbert and

139

Erofeeva, 2002) for the time span 15:00-22:00 on 13 July 2013 at the similar

140

positions of the vessel across the slope, Figure 3 a, clearly shows barotropic

141

tidal velocity below 0.05 ms−1 , which implies that Figure 3 b can be treated as

142

observational evidence of slope current. Thus, no filtering of the tidal signal

143

is required in order to identify the slope current with velocity 0.2 m s−1 which

144

is clearly seen in Figure 3 b.

145

Note that the velocity field obtained using the VMADCP data, Figure 3 b,

146

can be aliased by tidally generated internal waves. This effect is a matter of

147

great concern in areas of strong bottom currents, specifically, at positions of

148

tidal beam generation. The model predicted amplitude of horizontal velocity

149

of the along-slope current presented in Figure 3 c shows that the area with

7

150

strong tidal activity does not coincide with the position of intensification of

151

the bottom current found from the VMADCP data.

152

3. Numerical solution for the Malin Sea slope current

153

The MITgcm was applied to investigate the Malin Sea slope current. The

154

model domain is shown in Figure 1. The grid resolution was 150 m in the

155

horizontal and 10 m in the vertical directions. Eight principal tidal harmonics

156

were activated in the model in the right hand side of the momentum balance

157

equations. The parameters of tidal forcing were taken from the TPXO8.1

158

inverse tidal model (Egbert and Erofeeva, 2002) and using the ADCP time

159

series recorded during the JC88 experiment, Figure 2. The parameters of

160

the tidal current discharge and tidal phases of each harmonic are presented

161

in the Table.

162

The vertical turbulent closure for the coefficients of vertical viscosity ν

163

and diffusivity κ was provided by the Richardson number dependent parametri-

164

sation, PP81, (Pacanowski and Philander, 1981): ν =

ν0 + νb , (1 + αRi)n (1)

κ =

ν + κb . (1 + αRi)

165

Here Ri is the Richardson number, Ri = N 2 (z)/(u2z + vz2 ), and N 2 (z) =

166

−g/ρ(∂ρ/∂z) is the buoyancy frequency (g is the acceleration due to grav-

167

ity, and ρ is the density), u and v are the components of horizontal ve-

168

locity; νb =10−5 m2 s−1 and κb =10−5 m2 s−1 are the background parameters,

169

ν0 =1.5·10−2 m2 s−1 , α=5 and n=1 are the adjustable parameters. Such a 8

170

parametrisation increases coefficients ν and κ in the areas where the Richard-

171

son number is small which should take into account the mixing processes in-

172

duced by the shear instabilities and breaking internal waves. We set no-slip

173

boundary condition for the velocities at the bottom without activation of any

174

bottom drag parametrisation.

175

The model was run for five days in order to spin-up all tidally induced

176

processes. Barotropic and baroclinic responses were investigated separately.

177

The fluid stratification for the baroclinic mode was taken from CTD profiles

178

acquired during the JC88 observations. The buoyancy frequency profile for

179

these experiments was averaged over all CTD stations conducted in the area.

180

It is shown in the inset in Figure 1.

181

The primary target of these numerical experiments was the identifica-

182

tion of the residual currents generated by tides. The residual currents were

183

calculated by averaging the model output over four days; for transects 1-3

184

depicted in Figure 1 the residual currents are shown in Figure 4. The baro-

185

clinic response is shown in the left column, and the barotropic response to

186

the right.

187

Both barotropic and baroclinic outputs demonstrate evidence of a pole-

188

ward water flux with maximum velocity of 0.2 ms−1 in the area of the shelf

189

break. A comparison of the barotropic and baroclinic cases reveals the pre-

190

dominance of the barotropic component, as was also found by Souza et al.

191

(2001). An intermediate conclusion from these experiments is that setting

192

the tidal potential as the only forcing in the numerical model leads to the for-

193

mation of a northward directed residual current in the Malin Sea slope/shelf

194

area. In order to make our analysis more general in terms of hydrodynamic

9

195

conditions, we aim to find an analytical solution for a tidally induced slope

196

current that can be applicable for a wide range of input parameters.

197

4. Analytical solution for a tidally generated slope current

198

Consider a two-dimensional slope-shelf bottom topography H(x) with

199

isobaths parallel to the y−axis directed to the north and eastward x−axis

200

directed across the isobaths. The upward looking vertical z−axis starts at

201

the free surface. With such an arrangement the depth-averaged momentum

202

balance equations in hydrostatic approximation with linear dissipation read: ∂u ∂u ∂ζ ku +u − f v = −g − , ∂t ∂x ∂x H

(2)

∂v ∂v kv +u + fu = − , ∂t ∂x H 203

It is assumed here that there is no pressure gradient applied along the slope,

204

and that all variables are a function of the cross-slope coordinate x and time

205

t only. In these equations u(x, t) and v(x, t) are depth-averaged horizontal

206

velocities, ζ(x, t) is the surface elevation, H(x) is the water depth, f is the

207

Coriolis parameter, g is the acceleration due to gravity, and k(x) is a friction

208

coefficient.

209 210

We assume that developing dynamical processes are a superposition of tidal motions (superscript t) and a stationary slope current (superscript c): u = ut + uc , v = vt + vc, ζ = ζ t + ζ c.

10

(3)

211

Tidal currents ut and v t are periodic functions ut = a cos(ωt − φa ), v t = b cos(ωt − φb ),

(4)

212

where a, φa and b, φb are amplitudes and phases of ut and v t velocities,

213

respectively, and ω is the tidal frequency.

214 215

After substitution of (3) into (2) and averaging over one tidal cycle the governing system is reduced to the following: ut

∂ut ∂ζ c  − f v c  + g  = 0, ∂x ∂x t ∂v kv c   = − ut ∂x H

(5)

216

Here  means temporal averaging. In the derivation of (5) it was assumed

217

that the cross topography water transport is negligibly small, i.e. uc  ≈ 0.

218

The velocity of the along-slope rectified flow v c  can be found from the

219

second equation of system (5). After the definition of v c  the free surface

220

elevation ζ c  can be derived from the first equation. Taking into account

221

that the amplitudes a and b of tidal velocities (3) are depth dependent, it is

222

sensible in further analysis to operate with the tidal discharges U t = ut H,

223

V t = v t H instead of velocities ut and v t : U t = aH cos(ωt − φa ) = A cos(ωt − φa ), V t = bH cos(ωt − φb ) = B cos(ωt − φb ).

(6)

224

Note also that in long tidal waves the amplitudes of water discharge A and B

225

are less sensitive to the water depth, so one can assume here their invariance

226

for the whole slope-shelf area (non-divergent tidal wave, ∇· (H u t) = 0, where

227

u t = (ut , v t )). 11

228

After averaging the left hand side of the second equation, (5) reads ut

∂v t AB cos(φa − φb ) 1 ∂H =− . ∂x 2 H 3 ∂x

230

According to Loder (1980), the linear friction coefficient k does not depend √ on time but it is spatially variable, i.e. k(x) = CD A2 + B 2 /H(x), where

231

Cd is a drag coefficient. With this assumption the second equation of system

232

(5) is reduced to

229

v c  =

AB cos(φ) 1 ∂H √ . 2CD A2 + B 2 H ∂x

233

Here φ = φa − φb is the phase lag between ut and v t tidal velocities.

234

5. Numerical investigation of tidally induced slope currents

(7)

235

In this section we check the applicability of the analytical solution (7)

236

to real oceanographic conditions. As a test bed for the analysis we took a

237

topography profile averaged over the whole model domain shown in Figure

238

1 and approximated it by a sine function as follows H(x) = 1500 − 1300 · sin2 (π(x − L)/2L).

(8)

239

Here 0 ≤ x ≤ L, L is the measure of the topography width. The obtained

240

two-dimensional bottom profile was extended in the y−direction for 75 km,

241

so the new model domain covered the area 75×75 km2 .

242

A series of numerical experiments was conducted for a wide range of in-

243

put parameters of tidal discharge, topography scale, tidal ellipses orientation,

244

stratification, and tidal frequency. At the first stage we compared the ana-

245

lytically predicted slope current with the model output for L =24 km (the

246

basic case run, hereafter BCR). The tidal forcing was limited to only the 12

247

M2 tidal harmonic. The amplitudes of tidal discharge were A=100 m2 s−1

248

and B=40 m2 s−1 , the phase lag φ =114◦ that corresponds to cos(φ) =-0.41

249

(stronger and weaker tidal forcing as well as different phase lags are discussed

250

in sensitivity runs). The water stratification was taken as that shown in the

251

inset to Figure 1. The model was run for ten tidal periods, and the residual

252

currents were found from the last four tidal periods by time averaging.

253

5.1. Basic Case Run (BCR)

254

The results of the BCR are presented in Figure 6. Panel a) shows the 3D

255

structure of the residual current flowing northward. The track of an ad hoc

256

model drifter drogued at 70 m depth (shown in blue) depicts the trajectory

257

of water particles. The tidal current that generates the residual flow is shown

258

in Figure 6 b as a tidal ellipse at the 485 m isobath. Velocity amplitudes for

259

this ellipse were recalculated using the discharge values.

260

Figures 6 c and 6 d show the cross-slope time averaged distributions of ver-

261

tical viscosity and diffusivity coefficients predicted by the PP81 parametrisa-

262

tion (1). It was found that numerical coefficients ν and κ are consistent with

263

that measured over the Malin shelf break area by Inall et al. (2000). They

264

found that the largest tidally averaged values of vertical eddy diffusivity were

265

at the level of 12 ×10−4 m2 s−1 in a vertically integrated 100 m thick bottom

266

boundary layer (BBL). A similar value calculated by averaging the MITgcm

267

output is shown in Figures 6 c and 6 d and is equal to ν¯=10 ×10−4 m2 s−1 .

268

This consistency suggests that the set of adjustable parameters taken in (1)

269

is good enough to reproduce the background mixing correctly.

270

Having the BC run as a reference point for further calculations, we can

271

now investigate the slope current generated for a wide range of input param13

272

eters in order to compare the model predicted velocities with the analytical

273

solution (7).

274

5.2. Effect of topography

275

We start the comparative analysis by considering the sensitivity of the

276

generated slope current to the width of the topography. The velocity fields

277

calculated for L=24, 42, and 60 km of profile (8) are presented in Figure

278

7. Comparing panels a-c it is clear that the slope current weakens with an

279

increase of the topography width L. This result is in agreement with formula

280

(7) which predicts the slope current dependence on the bottom topography

281

as

1 ∂H(x) . H(x) ∂x

282

Another conclusion from Figure 7 a-c is that the model predicted slope

283

current is a superposition of barotropic and baroclinic modes. It resembles a

284

barotropic flow being mostly located over the shelf break, however, vertical

285

variability of the current is also apparent.

286

It is worth mentioning here that analytical solution (7) was developed

287

using vertically averaged equations (2), but the MITgcm numerical solution

288

is based on the full set of primitive equations. To make the comparative anal-

289

ysis more accurate, it is sensible to compare the depth integrated numerical

290

velocities as well, i.e.

291

1 v¯(x) = H(x)



0 −H(x)

vdx.

(9)

The blue lines in Figures 7 d-f show normalised profiles of the bottom term

292

1 ∂H(x) H(x) ∂x

293

normalised velocity v¯(x) is shown in red. It is seen that three pairs of curves

294

coincide nearly perfectly on the shelf including the positions of their maxima.

that appears in (7), and the vertically averaged, model predicted,

14

295

In the deep part of the basin the discrepancy between two solutions is obvious,

296

and this difference increases with the increase of L.

297

In the theory presented in Section 4 there is an uncertainty in setting the

298

value of the drag coefficient Cd . In different hydrodynamical applications it

299

varies over quite a wide range. The bottom friction for a propagating sinu-

300

soidal wave was first investigated by Putnam and Johnson (1949) who found

301

an equation that defines the change of the wave energy depending on the

302

parameters of wave and topography. Bretschneider and Reid (1954) applied

303

this equation to the Gulf of Mexico and found Cd =0.01. Hasselmann and

304

Collins (1968) calculated Cd =0.015 for the area offshore Florida assuming a

305

Gaussian type of surface wave spectrum. In the most recent study Warner et

306

al. (2013) used pressure sensors for measuring currents over rough topogra-

307

phy at the Puget Sound, Washington. They found that the drag coefficient

308

should be at the level of 9×10−2 .

309

In fact, one should distinguish the drag coefficient calculated based on

310

the observational data discussed above from that used in modelling. In nu-

311

merical circulation models, the drag coefficient Cd is normally taken in the

312

range Cd =0.0025-0.005. According to Blumberg and Mellor (1987) its value

313

depends on the grid size, von Karman constant and local bottom roughness.

314

The usage of the bottom friction parameterization in circulation models with

315

coarse vertical resolution is necessary to introduce a sink of energy in the BBL

316

which is not reproduced in the models directly. However, in cases with fine

317

resolution vertical grid (Δz = 5-10 m) the BBL and its damping effect can

318

be resolved.

319

Note, that the purpose of the present study is to investigate the slope

15

320

currents and to validate formula (7). In doing so, it is suggested here to

321

calibrate the analytical solution by finding the drag coefficient Cd that reveals

322

the best fit between formula (7) and the model output. If the values of the

323

drag coefficient Cd are consistent over a wide range of input parameters

324

(which is the case, see below), one can then recommend the use of this Cd

325

value for all further oceanographic applications on the investigation of tidally

326

induced slope currents.

327

The procedure of defining the Cd coefficient can be as follows. The MIT-

328

gcm predicted vertically averaged velocity is substituted into the left hand

329

side of formula (7). The numerical velocity is taken at the point where

330

|

331

maximums of the model velocity and the bottom term

332

Figure 7 d-f. After that Cd is calculated using formula (7).

1 ∂H(x) H(x) ∂x

| has its maximum. As it was shown above, the positions of the 1 ∂H(x) H(x) ∂x

coincide,

333

As it is seen from the bottom panels of Figure 7, the suggested method

334

works reasonably well in the considered case. Specifically, the values of the

335

drag coefficient in all three cases are quite consistent, i.e. Cd =0.014 for the

336

BCR, and Cd =0.012 for two other considered cases.

337

5.3. Influence of stratification and tidal frequency

338

There are a few more points we have to pay attention to in our analysis.

339

First of all, formula (7) was developed for a homogeneous fluid, but slope cur-

340

rents can be sensitive to fluid stratification. Another point is that the tidal

341

frequency ω is not included in the analytical solution (7), which confirms the

342

independence of the slope current to this parameter. The aim of the next se-

343

ries of experiments was to clarify the role of stratification and tidal frequency

344

in setting the tidally driven slope currents. The bottom topography in these 16

345

experiments was the same as in the BCR.

346

Figure 8 shows cross-slope transects of residual currents for the M2 tide

347

(a,b) and K1 tide (c, d) in stratified (a, c) and homogeneous (b, d) fluids. It is

348

seen that the core of the slope current in homogeneous fluid is mostly uniform

349

in the vertical direction for both tidal frequencies. However, the K1 current

350

occupies a slightly larger area than M2 current and its maximum velocity

351

v¯max =0.1 m s−1 slightly exceeds the similar velocity of v¯max =0.085 m s−1

352

generated by the semi-diurnal tide.

353

The comparison of stratified and homogeneous fluid cases shows that the

354

slope currents are mostly barotropic (Figure 8), although fluid stratification

355

introduces some distortions into the residual flows; the core of of the stratified

356

slope currents are not vertical but slightly tilted. The direction of inclination

357

depends on tidal frequency: in the case of the M2 tide it is inclined towards

358

the shelf, but for the K1 tide it is inclined towards deep water. It is interesting

359

that for the M2 tide the maximum of the current velocity v¯max in stratified

360

fluid is smaller than in homogeneous fluid, but this is quite opposite for the

361

K1 tide. Note, however, that all these variations do not exceed 20%. The

362

value of the drag coefficient Cd in this series of experiments was found to be

363

from 0.01 to 0.014.

364

5.4. Sensitivity to the tidal discharge

365

We continue the validation of formula (7) with the analysis of the sen-

367

sitivity of the analytical and numerical solutions to the tidal discharge. In √ formula (7) the latter appears as a nonlinear term AB/ A2 + B 2 . If solution

368

(7) is correct, then the increase or decrease of both amplitudes, A and B, in

369

the forcing term for the numerical model should lead to a similar tendency

366

17

370

in intensity of the rectified flows.

371

This statement was tested in a series of numerical experiments with dif-

372

ferent values of discharge A and B. The cross-slope transects of the residual

373

along-topography current calculated for three different values of the discharge

374

A and B are shown in Figure 9. It is seen that the spatial structure of the

375

slope current in all cases is nearly identical. However, in terms of intensity, a

376

twofold increase or decrease of A and B values compared with the BCR re-

377

sults in a nearly similar response of the slope current maximum. The values

378

v¯max for the three panels shown in Figure 9 (from left to right) are 0.2 m s−1 ,

379

0.079 m s−1 (BC run), and 0.036 m s−1 . In other words, a twofold increase or

380

decrease of the tidal forcing leads to an increase of 2.5 times ,or a decrease

381

of 2.2 times, the values of the maximum velocity, respectively.

382

Finally, we found that the best fit between analytical and numerical so-

383

lutions takes place when Cd is equal to 0.011, 0.014, and 0.016 in three

384

considered cases, which is consistent with the values obtained in previous

385

experiments, see Figures 7 and 8.

386

5.5. Sensitivity to the phase lag and direction of the velocity vector rotation

387

Another parameter that should be investigated in the validation of for-

388

mula (7) is the phase lag φ = φa − φb between the tidal velocities ut and v t ,

389

see (4). In the BCR the function cos(φ) returns the value -0.41. In conjunc-

390

tion with the negative term of the bottom function

391

solution (7) predicts a positive slope current directed northward, which is

392

confirmed by the MITgcm control runs.

1 ∂H(x) H(x) ∂x

the analytical

393

It is worth mentioning here that two principal parameters of the tidal

394

stream (4), i.e. the tidal phases φa and φb , do not show explicitly the in18

395

clination of the tidal ellipse γ, see Figure 10. It thus appears sensible for

396

further analysis to find an explicit relationship between the phase lag and

397

tidal ellipse inclinations.

398 399

To present the tidal stream in terms of tidal ellipses let us operate with the tidal current as a complex function: w = ut + iv t , √

(10)

−1. After substitution of (4) into (10) and conducting a series of

400

where i =

401

routine mathematical procedures one can present w as a sum of two vectors

402

rotating in opposite directions, see Figure 10: w=

403

Wa Wb exp[i(ωt + θa )] + exp[−i(ωt − θb )]. 2 2

Here Wa = Wb =

 

a2 + b2 + 2ab sin(φb − φa ), a2 + b2 − 2ab sin(φb − φa ),

 −a sin(φa ) + b cos(φb ) , = arctan a cos(φa ) + b sin(φb ) 

θa

(11)

(12)

404

 a sin(φa ) + b cos(φb ) . θb = arctan a cos(φa ) − b sin(φb ) When two circular radial vectors are aligned in the same direction, the tidal

405

current reaches its maximum. It is clear from (11) that this situation happens

406

when ωt + θa = −ωt + θb + 2kπ, where k = 0, ±1, ±2, . . .. This relation gives

407

the moment of time when the maximum is achieved:



tmax =

θa − θb kπ + . 2ω ω 19

(13)

408

The velocity vector (11) at this moment of time is expressed as follows   Wa + Wb θa + θb wmax = exp i + kπ , (14) 2 2

409

The vector length |wmax | in terms of the tidal stream parameters a, b, and

410

φ reads |wmax | =

411

(15)

Finally, the angle γ with respect to 0x axis (see Figure 5) is: a>b γ

a
 1  2 a + b2 + 2ab sin φ + a2 + b2 − 2ab sin φ . 2

⎧ ⎨ n = 0 if cos φ > 0 2ab cos φ 1 arctan + nπ, = ⎩ n = 1 if cos φ < 0 2 a2 − b2 



(16)

  2ab cos φ π 1 arctan + . = 2 2 2 a −b 2

Here the phase lag φ = φa − φb .

413

Formula (16) presents the relationship between the phase lag φ and the

414

tidal ellipse inclination γ. An obvious conclusion from this analysis is that if

415

the phase lag φ exceeds π/2, i.e. cos φ is negative, then the inclination angle

416

γ exceeds π/2. In this case formula (7) predicts the slope current as being

417

directed northward. Alternatively, the slope current should flow southward

418

when γ < π/2 (when cos φ > 0).

419

The principal outcome from this analysis is the relationship (16) between

420

the phase lag φ and the ellipse inclination γ. If the value of cos φ does control

421

the direction of the slope current as the analytical solution predicts, a similar

422

tendency should also be found in the numerical model output. 20

423

A series of numerical experiments was conducted to study the sensitivity

424

of the slope current to the phase lag φ (or tidal ellipse inclination γ) and the

425

direction of the tidal current vector rotation. A comparison of Figures 11 a,

426

11 a1 calculated for cos φ=-0.41 (φ = −114◦ , the BCR), with Figures 11 b,

427

11) b1 , for cos φ = 0.41 (φ = −63◦ ) confirms the hypothesis that the model

428

is sensitive to the parameter cos φ. In case cos φ = −0.41 and cos φ=0.41

429

it reproduces nearly the same residual current but flowing in the opposite

430

directions.

431

Note, however, that two compared vertical cross-sections are not fully

432

asymmetric. The core in the BCR current is inclined on-shelf, whereas the

433

south directed current is inclined off-shore. In addition, its maximal velocity

434

v¯max =0.087 m s−1 slightly exceeds v¯max =0.079 m s−1 found for the BCR,

435

although the calculated drag coefficient Cd values are nearly the same.

436

Input parameters in the two considered cases were the same except for

437

the phase lag φ. The latter, in fact, controls the inclination of the tidal

438

ellipse (see, for instance, Figures 11 a, and 11 b). It also sets the direction

439

of the velocity vector rotation. Negative values in the range −π < φ < 0

440

result in clockwise rotation, but a positive phase lag 0 < φ < π produces

441

counter-clockwise rotation. Taking into account that the cosine is a symmet-

442

ric function, i.e. cos(α) = cos(−α) for any angle α, formula (7) predicts that

443

the residual current should be independent of the velocity vector rotation.

444

The confirmation of this conclusion is shown in Figure 11 c which rep-

445

resents the model predicted slope current calculated for φ = 114◦ . As the

446

analytical formulae (7) predicts, the MITgcm produces nearly identical re-

447

sults to the BCR for both clockwise and counter-clockwise rotated velocity

21

448

vectors (compare Figure 11 a and 11 c). The direction of the two currents

449

and their spatial structure are similar although the maximum velocities are

450

slightly different, i.e. v¯max =0.074 m s−1 for the last case, and 0.079 m s−1

451

for the BCR. The drag coefficients in both cases are nearly the same.

452

The next test of the analytical solution also concerns the analysis of its

453

sensitivity to the combination of the tidal discharges A and B in Ox and Oy

455

directions, respectively. Formula (7) predicts that the slope current should √ be identical in cases when the term, AB/ A2 + B 2 , has the same value, re-

456

gardless A > B or vice versa. The next numerical experiment was conducted

457

with A =40 m2 s−1 , B =100 m2 s−1 , Figure 11 d, (instead of A =100 m2 s−1 ,

458

B =40 m2 s−1 , as in the BCR) and cos φ=-0.41. Figure 11 d1 shows the

459

northward generated slope current with maximum velocity v¯max =0.1 m s−1 ,

460

which slightly exceeds v¯max =0.079 m s−1 calculated for the BCR, and returns

461

the drag coefficient Cd =0.011. Note that the orientation of tidal ellipses with

462

respect to the bottom in two considered cases is completely different (com-

463

pare Figures 11 a and 11 d).

454

464

All numerical experiments discussed so far were conducted with cos φ =

465

±0.41. Formula (7) predicts strong sensitivity of the slope current character-

466

istics to the phase lag φ. Two extra runs were performed with cos φ=-0.866

467

and cos φ=0. They are presented in Figures 11 e, e1 and 11 f, f1 , respectively.

468

All other input parameters were kept the same as in the BCR. It is seen that

469

when the cosine of the phase lag changes from -0.412 to -0.866, the mean

470

velocity v¯max is nearly doubled from 0.079 m s−1 , as in the BCR (see Figure

471

11 a1 ), to 0.157 m s−1 . However, the drag coefficient Cd remains nearly the

472

same in the whole range coinciding with the BCR value.

22

473

The numerical experiment conducted with cos φ=0 (inclination γ=0) also

474

confirms formula (7): no slope current is generated by the MITgcm when

475

cos φ=0, see Figure 11 f, f1 .

476

5.6. Coordinate transformation

477

Formula (7) was derived for the bottom topography oriented in the merid-

478

ional direction with u-velocity directed across and v-velocity along the iso-

479

baths. However, in reality the continental slope can be oriented randomly.

480

To generalise the analytical solution for an arbitrarily oriented bottom con-

481

sider a simple topography scheme as that shown in Figure 12. To make

482

analytical solution (7) applicable to this situation, the coordinate system

483

should be rotated clockwise by the angle β, Figure 12. The analysis below

484

shows the relationship between the tidal ellipse parameters in the new and

485

old coordinate systems.

486

The task is to transform the topography to a new coordinate system

487

(O, xr , yr , zr ) by a simple rotation of the coordinate system (O, x, y, z) by an

488

angle β. The relationship between two coordinate systems reads: ⎡ ⎤ ⎡ ⎤⎡ ⎤ x cos β sin β x ⎣ r ⎦=⎣ ⎦⎣ ⎦. yr − sin β cos β y

(17)

489

Applying (17) to vector (4), the components utr and vrt of the tidal flow in

490

the new co-ordinate system are: utr = ar cos(ωt − φar ), vrt = br cos(ωt − φbr ),

23

(18)

491

where



a2 cos2 β + b2 sin2 β + 2ab sin(2β) sin(φb − φa ),

ar

=

br

=

φar

 a cos β sin(φa ) + b sin β sin(φb ) , = arctan a cos β cos(φa ) + b sin β cos(φb )

φbr

 −a sin β sin(φa ) + b cos β sin(φb ) . = arctan −a sin β cos(φa ) + b cos β cos(φb )



a2 sin2 β + b2 cos2 β − 2ab sin(2β) sin(φb − φa ), 

(19)



492

Using (15) and (19) one can find for the major X and minor Y axis of the

493

tidal ellipse, as well as for the ellipse inclination angle γ (see Figure 10) the

494

following Xr =

X,

Yr

=

Y,

γr

= γ − β.

(20)

495

Thus, the major and minor axis of the tidal ellipses in the new and old

496

coordinate system coincide, although the new inclination angle should be

497

corrected by the additional angle β. The angle of rotation β can be positive

498

or negative depending on the direction of rotation: β is positive in the case

499

when the coordinate system is rotated counter-clockwise and negative when

500

it rotates clockwise.

501

6. Analysis of the Malin Shelf slope current

502

The upgraded formula (7) that takes into account the coordinate trans-

503

formation was applied to the Malin shelf/slope area presented in Figure 1. 24

504

As it was shown above, the position of the core of the slope current in formula

505

(7) is controlled by the term

506

term value shown in Figure 13 a, points out that its largest absolute value

507

is located in the area above the 400 m isobath. The white line in Figure

508

13 a shows a smoothed isobath profile that was used to calculate the angle

509

of rotation β at every single point.

1 ∂H . H ∂x

The spatial distribution of the bottom

510

Substituting the discharge values for the M2 tide from the Table and

511

after the coordinate transformation, formula (7) predicts the residual tidal

512

velocity shown in Figure 13 b. The slope current is located in quite a narrow

513

band; it is stronger in the northern part of the model domain. Similar results

514

but for the depth integrated values of the along-slope current predicted by

515

the MITgcm are shown in Figure 13 c. The comparison of both figures shows

516

their consistency, both for the current position, and for its strength, although

517

the model predicted residual flow occupies a slightly larger area.

518

The stability of barotropic slope currents v¯ similar to that presented in

519

Figure 13 was discussed by Li and McClimans (2000). They found that

520

stability of v¯ is controlled by the following cross-slope function:   ∂¯ v ∂ 2 v¯ 1 ∂H f+ − 2. χ= H ∂x ∂x ∂x

521

The current is stable when χ is positive or negative, but it loses its stability

522

when χ changes sign across the jet. In our case χ is negative everywhere in

523

the Malin Sea slope area and thus the current v¯ is proven to be stable.

524

7. Discussion and conclusions

525

The theoretical analysis of tidally rectified slope currents (Section 4) pre-

526

sented here is a part of the theory developed by Huthnance (1973), Loder 25

527

(1980), and Zimmerman (1980). In particular, Huthnance (1973) considered

528

tidally rectified flows as a result of a balance between drag terms and the net

529

momentum transport into a region. To find a solution, a perturbation the-

530

ory that included multiple tidal harmonics was applied to vertically averaged

531

shallow water equations. It was found that the speed of the residual current

532

is proportional to the slope of the bottom. Unfortunately, the final formula

533

of Huthnance (1973) solution did not include the tidal ellipse parameters

534

explicitly, which is why its practical application is not straightforward. In

535

addition, the solution contains a term with the ratio of frictional to inertial

536

forces, whose estimation is not always obvious.

537

Loder (1980) modified the theory by including a feedback of the gener-

538

ated residual currents to the tidal velocity fields. He obtained an analytical

539

solution for the stepwise bottom profile that was by Huthnance (1973). The

540

complete version of the solution included not only the principal tidal con-

541

stituent with frequency ω, but also took into account its nonlinear interaction

542

with multiple 2ω and 3ω tidal harmonics. Zimmerman (1980) generalised the

543

theory to the case of an arbitrary bottom profile. His solution is based on a

544

number of special mathematical functions which makes its practical applica-

545

tion a little cumbersome with not always transparent final results.

546

Surprisingly, the simplified theory presented in Section 4 captures very

547

well all of the main features of the tidally rectified flows over the slope though

548

it does not account for higher harmonics. Formula (7) is relatively simple;

549

however, as it was shown above, it predicts properties and quantitative char-

550

acteristics of the MITgcm replicated slope currents quite accurately. Its

551

most striking feature is that the direction and strength of the rectified flow

26

552

is mostly controlled by the phase shift φ between the ut and v t tidal compo-

553

nents, which along with the amplitudes of the velocity components a and b,

554

formula (4), set the tidal ellipse inclination γ (see formulae (16)).

555

A convincing example of the fundamental sensitivity of the rectified flows

556

to the phase shift φ is shown in Figures 11 a and 11 b. An absolute value of

557

cos(φ) in both cases is the same, but signs are opposite. In such a situation,

558

formula (7) predicts that the residual current in both cases must have the

559

same in structure but should flow in opposite directions. Importantly, the

560

same behaviour is also demonstrated by the MITgcm which produces slope

561

currents flowing in opposite directions when cos(φ) changes sign. Note, that

562

the MITgcm is a fully nonlinear nonhydrostatic model which is free from

563

most assumptions used in modelling. In light of the results found above,

564

the consistency between the model output and the analytical solution looks

565

promising.

566

A reasonable explanation of the sensitivity of the slope current direction

567

to the sign of cos(φ) can be found in terms of equations (5). With the

568

assumptions of stationarity of the residual flow and zero meridional pressure

569

gradient, the only term remaining in the left-hand-side of the second equation

570

(5) is the residual advection of the v-momentum in x-direction, ut ∂v . Given ∂x

571

that ut and v t are defined by formula (6) we have

t

u

t ∂v

t

∂x

∼ cos(φa − φb ) + cos(2ωt − φa − φb )

572

In this equation the time averaging of the second term gives a zero result,

573

which means that the residual momentum advection depends on the phase

574

shift φ = φa − φb (on cos φ to be more specific) that controls the ellipse

575

inclination γ, equation (16). In the time-averaged equation (6), it can be 27

576

compensated by the dissipative term −

577

included linearly.

kv c  in which the slope current v c is H

For negative values of the bottom derivative ∂H/∂x, v c is positive when

578 579

1 2

580

clearly demonstrated in the MITgcm output (Figures 11 a and 11 b). It is

581

important that no restrictions, except of a two dimensionality of the bot-

582

tom topography, were introduced into the numerical model, but consistency

583

between theoretical and numerical results is obvious.

cos φ is negative and the ellipse inclination γ exceeds 90◦ , which is also

584

The theory presented in Section 4 is formally valid for a standard tidal

585

(west-north) co-ordinate system with meridionally oriented topography and

586

tidal amplitudes a, b and tidal phases φa and φb taken from any prediction

587

tidal model, TPXO8.1, for instance. That makes such applications quite

588

straightforward. Note, however, that in reality isobaths are oriented ran-

589

domly. In such situations, the orientation of isobaths with respect to tidal

590

ellipses would be a more representative parameter for making an operational

591

oceanographic prognosis. Subsection 5.6 provides all necessary details for the

592

co-ordinate transformation that allows application of formula (7) (with obvi-

593

ous corrections related to tidal ellipse-topography orientation) to all possible

594

oceanographic situations.

595

One of the principal outcomes from this analysis is the high level of consis-

596

tency of the drag coefficient Cd found in a wide range of input parameters. In

597

all experiments considered above the drag coefficient Cd varied in the range

598

between 0.01 and 0.016. Thus, with a 95% confidential interval one can rec-

599

ommend the usage of Cd =0.0128±0.0012. Applied to the Malin shelf/slope

600

area, formula (7) predicted a slope current with maximum velocity up to 28

601

0.08 ms−1 . The MITgcm generated a similar rectified flow, which justifies

602

application of formula (7) to many other regions worldwide.

603

Acknowledgements

604

This work was supported by the Natural Environment Research Council

605

grants FASTNEt (award NE/I030259/1) and Deep Links (award NE/K011855/1).

606

We thank two reviewers, especially Dr. Andrew Dale, for their attention to

607

this paper and a number of valuable comments.

29

608

Blumberg, A. F., and G.L. Mellor (1987) A description of a three-dimensional

609

coastal ocean circulation model. Three dimensional ocean models (N.S

610

Heaps Editor). American Geophysical Union, Washington, DC, 1-16.

611

Bretschneider, C. L., and R.O. Reid (1954) Changes in wave height due to

612

bottom friction, percolation, and refraction. Tech. memo., Beach Erosion

613

Board, U.S. Army Corps of Engineers, 45, 36pp.

614

Burrows, M., and S.A. Thorpe (1999) Drifter observations of the Hebrides

615

slope current and nearby circulation patterns. Ann. Geophysicae, 17, 280-

616

302.

617

Garria, G., P. Lazure, B. Le Cann, A. Serpette, G. Reverdin, S. Louazel,

618

F. Batioulier, F. Dumas, A. Pichon, and Y. Morel (2013) Surface layer

619

circulation derived from Lagrangian drifters in the Bay of Biscay. J. Mar.

620

Systems, 109-110, 60-76.

621 622

623 624

Egbert, G.D., and S.Y. Erofeeva (2002) Efficient inverse modeling of barotropic ocean tides, J. Atmos. Oceanic Technol., 19(2), 183–204. Hasselmann, K. and J.I. Collins (1968) Spectral dissipation of finite-depth gravity waves due to turbulent bottom friction. J. Marine Res., 26, 1-12.

625

Hill, A.E., B.M. Hickey, F.A. Shillington, P.T. Strub, K.H. Brink, E.D. Bar-

626

ton, and A.C. Thomas (1998) Eastern ocean boundaries: Coastal Segment

627

(E). The Sea, 11, edited by A.R. Robinson and K.H. Brink, 29-67.

628 629

Huthnance, J.M.(1973) Tidal current asymmetries over the Norfolk sandbanks. Estuar. Coastal Mar. Sci., 1,89-99. 30

630 631

632 633

Huthnance, J.M.(1984) Slope current and “JEBAR”. J. Phys. Oceanogr., 14, 795-810. Huthnance, J.M. (1986) The Rockall slope current and shelf-edge processes. Proc. R. Soc. Edin., 88B, 83-101.

634

Huthnance, J.M., and Gould (1989) On the northeast Atlantic slope current

635

in:Poleward flows along eastern ocean boundaries, Ed. S.J. Neshyba et al.,

636

Coastal and Estuarine Studies, 34, 76-81.

637

Huthnance, J.M. (1995) Circulation, exchange and water masses at the

638

ocean margin: the role of the physical processes at the shelf edge. Progr.

639

Oceanogr., 35, 353-431.

640 641

642 643

Huthnance, J.M., J.T. Holt, and S.L. Wakelin (2009) Deep ocean exchange with west-European shelf seas. Ocean Sci., 5, 621-634. Joyce, T.M. (1989) On the in situ “calibration” of shipboard ADCP. J. Atm. Oceanic Technol., 6, 169-172.

644

Inall, M.E., T. P. Rippeth, and T. J. Sherwin (2000) Impact of non-linear

645

waves on the dissipation of internal tidal energy at a shelf break. J. Geo-

646

phys. Res., 102, 5733-5752.

647

Li, S., and T. A. McClimans (2000) On the stability of barotropic prograde

648

and retrograde jets along a bottom slope. J. Geophys. Res., 105, 8847-8855.

649

Loder, J.W. (1980) Topographic rectification of tidal currents on the sides of

650

Georges Bank. J. Phys. Oceanogr., 10, 1399-1416.

31

651

Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey (1997) A finite-

652

volume, incompressible Navier-Stokes model for studies of the ocean on

653

the parallel computers, J. Geophys. Res., 105(C4), 8687-8705.

654

Pacanowski, R.C., and S.G.H. Philander (1981) Parameterisation of vertical

655

mixing in numerical models of Tropical Oceans. J. Phys. Oceanogr., 11,

656

1443-1451.

657 658

Pingree, R.D., and B. Le Cann (1989) Geltic and Armonian Slope and shelf residual currents. Prog. Oceanogr.,23, 303-338.

659

Porter, M., M.E. Inall, J.A.M. Green, J.H. Simpson, A.C. Dale, and P.I.Miller

660

(2016) Drifter observations in the summer time Bay of Biscay slope current.

661

J. Mar. Sys., 157, 65-74.

662 663

Putnam, J.A., and J. W. Johnson (1949) The dissipation of wave energy by bottom friction. Trans. Amer. Geophys. Un., 30, 67-74.

664

Souza, A.J., J.H. Simpson, M. Harikrishnan and J. Malarkey (2001), Flow

665

structure and seasonality in the Hebridean slope current. Oceanologica

666

Acta, 24, 63-76.

667

Stashchuk, N., and V. Vlasenko (2005) Topographic generation of internal

668

waves by nonlinear superposition of tidal harmonics Deep Sea Res. I, 52,

669

605-620.

670

Stashchuk, N., and V. Vlasenko (2016) Bottom trapped internal waves over

671

the Malin Sea continental slope Deep Sea Res. I, In press. Available at:

672

http://dx.doi.org/10.1016/j.dsr.2016.11.007 32

673

Wakelin, S.L., J.T. Holt and R. Proctor (2009) The influence of initial con-

674

ditions and open boundary conditions on shelf circulation in an3D ocean-

675

shelf model of the North East Atlantic. Ocean Dyn., 59, 67-81.

676

Warner, S.J., P. MacCready, J.N. Moum, and J.D. Nash (2013) Measurement

677

of tidal form drag using seafloor pressure sensors. J. Phys. Oceanogr., 43,

678

1150-1172.

679 680

White, M., and P. Bower (1997) The shelf -edge current north-west of Ireland. Ann. Geophysicae, 15, 1076-1083.

681

Xing, J., and A.M. Davies (2001) Non-linear effect of internal tides on the

682

generation of the tidal mean flow at the Hebrides shelf edge. Geophys. Res.

683

Lett., 28, 3939-3942.

684 685

Zimmerman, J.T.F. (1980) Vorticity transfer by tidal currents over irregular topography. J. Mar. Res., 38, 601-630.

33

Table: Zonal (A) and meridional (B) discharge of eight principal tidal harmonics used in the model setup. φa and φb are the tidal phases.

686

M2

S2

N2

K2

K1

A(m2 s−1 )

30

14

8.2

3.5

1.78 0.98 0.67 0.56

φa (degr)

143 127

75

102

170

B(m2 s−1 )

25

3

7.18 2.79

3.11 1.32 1.09

φb (degr)

28

0

125

115

146

charac

34

O1

0

63

P1

39

Q1

18 0.9

78.5 47.5

56.5

0

500

56

14

Sd

1000

section 3

N

45 37 55.5

6 La

1500

section 2

Sb Lb

2000

section 1 2500

55 -10.5

-10

-9.5

-9

-8.5

W

Figure 1: The bathymetry of the Malin Sea with the plan view of the JC88 field experiment. Cyan closed circle show positions of the CTD stations, red hexagrams depict the position of the moorings. Blue rectangle shows the model domain. An average buoyancy frequency profile is shown in the inset.

35

u (m s-1)

u (m s-1)

JC88 0.2

LB

0.1 0 180

185

190

195

TPXO8 0.1

LB

0 -0.1 180

200

185

v (m s-1)

v (m s-1)

0.2

LB

0.1 0 180

185

190

195

0.1

u (m s-1)

u (m s-1)

0 190

185

195

0.1

v (m s-1)

v (m s-1)

0 190

185

195

0.1

u (m s-1)

u (m s-1)

0 190

185

195

0.1

v (m s-1)

v (m s-1)

0 190

195

200

195

200

-0.1 185

190

Day 2013

SD

185

200

SD

180

200

0.1 180

190

0

Day 2013 0.2

195

-0.1

Day 2013

SD

185

200

SB

180

200

0.1 180

190

0

Day 2013 0.2

195

-0.1

Day 2013

SB

185

200

SB

180

200

0.1 180

190

0

Day 2013 0.2

195

-0.1

Day 2013

SB

185

200

LB

180

200

0.1 180

195

0

Day 2013 0.2

190

Day 2013

Day 2013

195

200

Day 2013

0.1

SD

0 -0.1 180

185

190

Day 2013

Figure 2: Zonal (u) and meridional (v) vertically averaged velocities recorded at moorings Lb, Sb, and Sd. Red lines show stationary currents. Left column: ADCP time series recorded during the JC88 cruise. Right column: TPXO8 predicted tidal currents.

36

Longitude -9.6

a)

-9.58

-9.56

-9.54

-9.52

-9.5

-9.48

0.06 u v

0.04

m s -1

-9.46

0.06

0.04 0.02

0.02

0

0 -0.02 23:00

22:00

21:00

20:00

22:00

21:00

20:00

19:00

18:00

17:00

16:00

-0.02 15:00

17:00

16:00

15:00

Time, 13 July 2013

b)

23:00 0

19:00

18:00

0.2

100

Depth (m)

0.1

200 300

0

400 -0.1

500 600

-0.2

-9.6

c)

-9.58

-9.56

-9.54

-9.52

-9.5

-9.48

-9.46

Longitude

0

0.3

Depth (m)

100

0.25

200

0.2

300 0.15

400 0.1

500 0.05

600 -9.6

-9.58

-9.56

-9.54

-9.52

-9.5

-9.48

-9.46

Longitude

Figure 3: a) Zonal (u) and meridional (v) tidal currents predicted using TPXO8.1 tidal model for the time span (15.00-22.00) 13 July 2013 (bottom axis) at the positions across the slope shown by the upper axis. b) Along slope current recorded by a vessel mounted ADCP at the transect shown in Figure 1 by the red line. The time span of the transect is depicted at the top axis. c) Model predicted amplitudes of the along-slope horizontal velocities found for one tidal cycle. Magenta line depicts position of baroclinic tidal beam.

37

Sd

200

0.15

400

0.1

600

0.05

Section 3

800 1000 -9.4

-9.3

0.15

400

0.1

600

0.05

Section 3

1000 -9.4

Sb

0.1

600

0.05

800

Section 2

1000

Depth (m)

Depth (m)

0.15

400

0.15

400

0.1

600

0.05

Section 2

1000 -9.7

0.1

600

0.05

Section 1

0 -0.05

-9.8

-9.7

Depth (m)

Depth (m)

400

-9.9

0

0.2 0.15

-10

-9.6

-9.5

Longitude

200

1000 -10.1

0 -0.05

-9.8

-9.5

Lb

800

0.2

200

800

0 -0.05

-9.6

Sb

Longitude 0

-9.2

0

0.2

200

La

-9.3

Longitude

Longitude

-9.7

0 -0.05

-9.5

-9.2

0

-9.8

0.2

200

800

0 -0.05

-9.5

Sd

0

0.2

Depth (m)

Depth (m)

0

La

Lb

0.2

200

0.15

400

0.1

600

0.05

800 1000 -10.1

Section 1

0 -0.05

-10

-9.9

-9.8

-9.7

Longitude

Longitude

Figure 4: Model predicted along-slope residual current calculated by averaging of the model output over four days. Appropriate cross-sections with moorings Sd, Sb, La, Lb are shown in Figure 1. Left column stands for stratified fluid, the right column shows pure barotropic response.

38

Figure 5: Schematic presentation of a two-dimensional slope-shelf topography oriented in south-north direction.

39

Figure 6: a) Spatial structure of the along-slope tidally induced rectified flow with the blue line showing a three-days trajectory of a passive tracer. b) Spatial structure and the scales of the tidal ellipse in the reference run. Panels c) and d) show spatial distributions of the viscosity ν and diffusivity κ coefficients set in the MITgcm by the KPP parameterization.

40

a)

0

0.05

0.1

0.15

b)

0 200

-0.05

BC Depth (m)

Depth (m)

400 600 800 1000

c)

800 1000

0

BC

0.2

20

0

60

20

Max Values

BC

0

0

Distance (km)

f)

1

1 0.8

0.6 0.4

0.6 0.4 0.2

0 40

Distance (km)

20

1400 40

0.2

0

40

800

Normalized units

0.6

0.15

1000

0.8

Normalized units

Normalized units

0.8

0.1

1200

60

e)

0.05

600

Distance (km)

1

60

0

0

400

600

Distance (km)

0.4

-0.05

400

1400

d)

0.15

200

1200

20

0.1

0

1400 40

0.05

200

1200

60

0

Depth (m)

-0.05

0 60

40

20

Distance (km)

Max Values

0

60

40

20

0

Distance (km)

Max Values

Figure 7: Panels a), b), c) illustrate spatial structure of the slope currents calculated for L =48, 84, and 120 km, respectively. Middle panels depict normalized depth integrated velocity (red) and the topographic term

1 dH H dx

(blue) for the same bottom profiles as above.

Bottom panels show the maximum values of the parameters shown in panels d), e), and f).

41

-0.05

0.05

0.1

b)

200 400 600 800

v¯max =

1000

Depth (m)

Depth (m)

a)

0

0

0.079m s-1

1200

0.15

0 200 400 600 800

v¯max =

1000

0.085m s-1

1200 1400

1400 0

10

20

30

40

50

60

0

70

10

20

0

d)

200 400 600 800

v¯max =

1000 1200

0.116m s

1400

-1

Depth (m)

Depth (m)

c)

30

40

50

60

70

Distance (km)

Distance (km) 0 200 400 600 800

v¯max =

1000 1200

0.1m s

-1

1400 0

10

20

30

40

50

60

70

0

Distance (km)

10

20

30

40

50

60

70

Distance (km)

Figure 8: Spatial structure of residual currents induced by M2 (panels a and b) and K1 (panels c and d) tidal harmonics for the stratified (left column) and homogeneous (right column) fluids.

42

150

150

150

100

100

100

50

50

50

-50

BC V(m 2 s-1)

0

0 -50

0 -50

-100

-100

-100

-150

-150

-150

-100

0

100

-100

0

U(m 2 s-1)

-100

100

0

0

0

200

200

200

600

0.15

800

0.1

1000

0.05

0.1

1200

1200 0

1400 0

20

40

Distance (km)

60

Depth (m)

0.2

1000

Depth (m)

800

100

400

400 0.3

600

0

U(m 2 s-1)

U(m 2 s-1)

400

Depth (m)

Discharge

Discharge

V(m 2 s-1)

V(m 2 s-1)

Discharge

600

0.08

800

0.06

1000

0.04

1200

0.02

0

0

1400

1400 0

20

40

60

Distance (km)

0

20

40

60

Distance (km)

Figure 9: Residual currents induced by M2 over 48 km wide continental slope in cases of strong, moderate, and weak tidal forcing (left, middle, and right columns, respectively). Appropriate tidal ellipses are shown in three top panels.

43

y 0.2

0.15

0.1

u-velocity (m/s)

0.05

X

γ

x

0

-0.05

Y

-0.1

-0.15

-0.2 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

v-velocity (m/s)

Figure 10: Tidal ellipse (red) presented as a superposition a clockwise (blue) and counterclockwise (green) tidal velocity components. Angle γ shows the ellipse’s inclination.

44

cos(φ) =0.41194

cos(φ) =-0.41194 V(m 2 s-1)

-50

0

-50

-50

50

500

v¯max =

1000

0.079m s

1250 1500 40

50

60

-1

500

V(m 2 s-1)

V(m 2 s-1) 0

50

50

500

C =0.011

750

d

v¯max =

1000

0.1m s

1250 1500 40

50

60

50

-1

50

f)

0 -50

50

-50

500

C =0.015 d

750

v¯max =

1000

0.157m s

1500 40

Distance (km) -0.1

50

60

-1

0

500 750

v¯max =

1000 1500 40

70

0.05

50

f1)

0.0m s -1

1250

Distance (km)

-0.05

0

U(m 2 s-1) 250

1250

70

-0.15

e1 )

70

cos(φ) =0

0

250

60

-1

Distance (km)

U(m 2 s-1) 0

d1 )

0.074m s

1500 40

70

-50

0

v¯max =

1000 1250

0

-50

Cd=0.013

750

e)

50

Depth (m)

Depth (m)

0

-1

Distance (km)

U(m 2 s-1) 250

60

50

c1 )

500

cos(φ) =-0.86603

d)

-50

0.087m s

1500 40

Distance (km)

-50

v¯max =

1250

70

0

0 250

1000

0

U(m 2 s-1)

Cd=0.015

750

cos(φ) =-0.41194 50

-50

-50

b1 )

250

Cd=0.014

750

0

50

V(m 2 s-1)

Depth (m)

0

a1 )

Depth (m)

0 250

0

c)

50

U(m 2 s-1)

U(m 2 s-1)

Depth (m)

-50

0

Depth (m)

V(m 2 s-1)

0

b)

50

V(m 2 s-1)

a)

50

cos(φ) =-0.41194

0.1

50

60

70

Distance (km)

0.15

Figure 11: Influence of the tidal ellipses orientation (panels with blue ellipses) on the structure and strength of the along-slope rectified flow (shown in the second and fourth rows). Appropriate cos(φ) value and maximum of the residual currents are depicted in the graphs.

45

56.8

56.6

y

56.4

y



Latitude

56.2

γ

56

x

β 55.8

x



55.6

55.4

55.2 -9.8

-9.6

-9.4

-9.2

-9

-8.8

-8.6

-8.4

-8.2

Longitude

Figure 12: Schematic diagram showing the changes of the ellipse inclination angle γ after rotation of the co-ordinate system (x0y) on the angle β that transforms it into system (xr 0yr ) with the axis (0x) perpendicular to isobaths.

46

-4

-4

0

2

×10 -4 4 x10

-0.15

a)

55.9

-0.1

-0.05

0

0.05

0.1

-0.15

b)

55.9

55.8

55.8

55.7

55.7

55.7

55.6

Latitude

55.8

Latitude

Latitude

55.9

-2

55.6

55.5

55.5

55.4

55.4

55.4

55.3

55.3

55.3

-10

-9.8

-9.6

-9.4

-9.2

-10.4 -10.2

-10

Figure 13: a) Bottom term

-9.8

-9.6

Longitude

Longitude

1 ∂H H ∂x .

-9.4

-9.2

-0.05

0

0.05

0.1

c)

55.6

55.5

-10.4 -10.2

-0.1

-10.4 -10.2

-10

-9.8

-9.6

-9.4

-9.2

Longitude

b) Rectified along-slope current v c predicted by formula

(7). c) Residual vertically average meridional current reproduced by the MITgcm for the Malin shelf/slope area.

47