Transports of water masses through the Faroese Channels determined by an inverse method

Deep-Sea R~earch, Vol. 35, No. 4, pp. 595-617, 1988. lhrlnted in Great Britain.

0198-0149/88 $3.00 + 0.00 ~) 1988 Pergamon Press plc.

Transports of water masses through the Faroese Channels determined by an inverse method HENDmK M. VAN A K ~ * (Received 29 September 1986;/n revisedform 19 September 1987; accepted 16 November 1987) Abstract--The transport of water between the Atlantic Ocean and the Norwegian Sea through the Faroese Channels is studied by means of geostrophic calculations and use of an inverse method. The inverse method is used to evade the arbitrary character of a pr/or/choice for a level of no motion. The resulting transport of Atlantic water into the NorwegianSea (~8 × lfft ms s-1) agrees with existing estimates based on water and heat budgets of the Norwegian Sea, but the horizontal structure of the flow differs from literature. The flowof intermediate and deep water from the Norwegian Sea towards the Atlantic Ocean (~10~ m3 s-1) agrees with reported values.

INTRODUCTION

THE exchange of water and heat between Arctic seas and the North Atlantic Ocean occurs in the Denmark Strait, across the Faroe--Iceland Ridge and through the Faroese Channels between the Faroe Islands and the Hebridian Shelf. These transports are important to t h e c l i m a t e system since they bring large amounts of warm and salty Atlantic water to the high latitudes (WoRtHInGTON, 1970). This paper deals mainly with the transports through the Faroese Channels (see Fig. la). The outflow of the different water masses through the Faroese Channels has been estimated by a number of authors using different methods. CREASE (1965) estimated the outflow of Norwegian Sea Deep Water through the Faroe Bank Channel to be 106 m 3 s-1 using dynamic computations and observations of neutrally buoyant floats. I4_~RUAr~ (1967) obtained the same estimate, based on observations with m o o r e d current meters. DooL~Y and MEINCr~ (1981) used dynamic computations and observations with m o o r e d current meters to estimate the outflow through the Faroe Bank Channel of deep and intermediate water to be 1.4 x 106 m 3 s-1. The inflow of Atlantic water through the Faroe--Shetland Channel was estimated by TArr (1957) using dynamic computations relative to the S = 35 isohaline; he found a mean inflow of 2.5 x 106 m 3 s-1 Atlantic water into the Norwegian Sea. DOOLEY and MEISCI~'S (1981) dynamic computations with a level of no motion at 550 m gave an inflow of 2 x 106 m 3 s--1 Atlantic water. T h e y also concluded that 1.2 x 106 m 3 s--1 of fresher Modified Atlantic Water recirculates clockwise around the Faroe Islands towards the Iceland Basin. WORrmNGTON (1970) studied the heat and water budgets of the Norwegian Sea and concluded that a total inflow of 8 x 106 m 3 s--x Atlantic water was needed between Scotland and Iceland. H e supposed that nearly all this transport is * Institute for Meteorology and Oceanography, University of Utrecht, The Netherlands. Present address: Netherlands Institute for Sea Research, P.O. Box 59, 1790AB Den Burg/Texel, The Netherlands. 595

596

H . M . VAN AKEU

Faroe

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Fig. 1. (a) Depth contours (in m) around the Faroe Islands, and (b) the CTD stations used in this. Cross-channel sections are indicated with a letter, the sea areas used for the inverse calculations are numbered.

Transport of water through Faroese Channels

597

concentrated in the Faroe-Shetland Channel. HA~SEN(1985) reviewed existing estimates of the transport through the Faroe Channels and concluded that there is "at present no reliable number for the total transport of Atlantic water into the Norwegian Sea". He blamed this on the fact that the arbitrary choice of a level of no motion strongly influences the result of the calculations. The mean transport could easily be doubled by modifying the assumptions. In this paper I try to improve the estimated transports through the Faroese Channels by an objective choice of an optimum reference level and by allowing a reference velocity to be present at the reference level. This reference velocity will be calculated with an inverse method based on mass conservation. THE OBSERVATIONS

In the summer of 1983, a quasi-synoptic hydrographic survey of 173 CTD casts in the seas around the Faroe Islands was carded out on board R.V. Tyro (Fig. lb). Sections O, C, D, E, H, F, J and K were cross-channel sections perpendicular to the Faroe-Shetland Channel and Faroe Bank Channel. The station distance in the deeper parts of the channels was about 10 km with the exception of section O, where the mean station distance was 17.5 kin. Sections G and I were in the North Rockall Trough just south of the Wyville Thomson Ridge. Sections L and M were in the northeastern part of the Iceland Basin and section Y was on top of the Iceland-Faroe Ridge. Sections C to M were carried out from 14 to 25 June, section O from 9 to 11 July and section Y on 14 and 15 July. The CTD data were de-spiked, corrected for temperature and salinity calibrations and low-pass filtered with a Bartlett filter with a width of 0.25 MPa (25 dbar). The data set thus appeared to have an overall absolute accuracy after the calibration of <0.02°C for temperature and at <0.01 (pss-78) for salinity. The precision of the data after filtering, two times r.m.s., is estimated to be 0.002°C for temperature and 0.004 (pss-78) for salinity. These last values are of importance for the dynamic computations discussed in this paper since a fixed systematic error (accuracy), which is equal for successive CTD casts, hardly influences the horizontal density gradient used in the geostrophic calculations.

The hydrography of the Faroese Channels An extensive analysis of the hydrography of the whole Faroe area, based on all CTD casts obtained in 1983, has been published by VAN Arran and EISMA (1987), but a summary of the results for the Faroese Channels is given in this section. A clear feature is the lowering of the main thermocline in the Faroe Bank Channel relative to the Faroe-Shetland Channel (Fig. 2a); see also Hansen (1985). The structure of the water masses (Fig. 2b), however, is coherent throughout the Faroese Channels in agreement with MOILER e t al. (1979). The upper layer, which is formed by the mixed layer and the seasonal thermocline, contains North Atlantic Water (NA) with S > 35.3, Modified North Atlantic Water (MNA) with 35.2 < S < 35.3, as well as fresher water types originating from either the Hebridian and Faroe Shelves or from the East Iceland Current. The lower boundary of this upper layer coincides with the potential density excess To = 27.47 kg m -3 (To = p(0,S,o) - 1000 kg m-a). The horizontal structure of the upper layer is quite complicated as shown in Fig. 3a.

598

H.M. VANAKEN Potential Temperature

I

Salinity

°C

~

pss-78

'to

HPa 10-

1 15-

3~.9

Sa(inity

I

pss-78

3~1

I

• ~i 3

I J

°C

m5

E

zo.

Fig. 2. Cross-channel mean diagrams for section C (full line) and section J (dashed line). The mean values were obtained by averaging along isopycnals; (a) profiles of potential temperature and salinity; (b) cross-channel mean 0-S diagram.

The layer below the upper layer, with 27.47 < Y0 < 27.80 kg m -a, coincides with the main thermodine. This layer contains a mixture of upper layer water (mainly N A and MNA) and the underlying water types and is designated Lower Atlantic Water (LAW). At the lower boundary of the main thermocline a distinct layer of intermediate water is found in the Faroe-Shetland Channel. This intermediate water, generally termed North Icelandic/Arctic Intermediate Water (NI/AI) (Sa'EFANSON, 1962), shows a salinity minimum in the Faroe-Shetland Channel. In the Faroe Bank Channel this intermediate

rn

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Fig. 3. Horizontaldistributionof salinityin representative layers; (a) the verticalmean salinity of the surfacelayer with )'e < 27.47 kg m-3. This layer is situated above the main haloclineand is representative for the Atlantic surface waters; (b) salinity at the isopyenal surface with 70 = 27.9 kg m-a which is representative for the intermediatewater; (c) salinityat the isopyenal surface with ~e = 28.03 kg m-a which is representative for the upper part of the Norwegian Sea Deep Water. water appears to mix with the overlying water from the main thermocline (Fig. 2b). The intermediate water is generally found at potential density levels with 27.8 < Y0 < 28.0 kg kg m -3. The salinity distribution in the Y0 = 27.9 plane (Fig. 3b), shows that the fresh intermediate water type originates from the Norwegian Sea. Much higher salinities along the slope of the Hebridian Shelf and the Wyville Thomson Ridge probably represent the overflow water found at the sill in the Faroe Bank Channel (VAN AICEN and EISMA, 1987). The water encountered in the North Rockall Trough appears to have the same origin. The Norwegian Sea Deep Water (NSDW) is found below Y0 = 28.0 kg m -3. The upper part of the deep water shows only a limited variation in salinity (Fig. 3c), but the highest salinities are found along the Hebridian Shelf and the Wyville Thomson Ridge as well as in the overflow into the Iceland Basin. The salinity of deeper parts of the deep water varies only within the precision obtained from our CTD observations. THE DYNAMIC COMPUTATIONS

The geostrophic method For the dynamic computations the dynamics of the motion can be approximated by the hydrostatic and geostrophic balances. This gives, within the pressure p as independent

Transport of water through Faroese Channels

601

vertical co-ordinate and x as horizontal co-ordinate along a CTD section, the following expression for the geostrophic shear:

Ov(p) _ op

10a

(i)

fox

In this expression v is the velocity component in the y-direction perpendicular to the CTD section, f is the Coriolis parameter and ct is the specific volume to be determined from temperature and salinity observations. By replacing the differentials in (1) by finite differences the mean geostrophic shear at pressure p between a pair of CTD stations can be determined. Integration of (1) gives the velocity v(p) according to v(p)=-

fp , flOa ~ x x d P ' + vr,

(2)

where Pr is a reference level and vr is a reference velocity at that level. The choice of both these reference parameters introduces a subjective element in geostrophic calculations. The classical solution is the introduction of a level of no motion, that is v~ = 0 at pr = Po. The choice of the level of no motion, Po, is generally quite arbitrary, in most cases based on an interpretation of the available hydrographic information. The geostrophic transport between two stations, situated at xi and Xi+l, in a water layer, j, between pressure levels P1 and P/+I, Trii, can be calculated by

rr,j=

1 x,., x,

.p,

v(p) dp d x ,

(3)

where g is the gravitational acceleration. If the correct SI units are used, Tr0 is in kg s-1. A transport Tr0 = 109 kg s-1 is about equal to the traditional volume transport unit 1 Sv or 106 m 3 s-1. For the computation of (3) the vertical integration is approximated by the trapezoidal sum over vertical intervals of 125 kPa (12.5 dbar).

Extrapolation The vertical geostrophic shear between a pair of CTD stations, A and B, can only be calculated down to the deepest observational level of the shallower station of each pair, PB. Our observations have been carried out until the bottom was reached. Therefore a procedure is needed to extrapolate the geostrophic shear in the interval between the different bottom depths of each CTD pair, PA-PB [see FouI~ (1964), section 22 and FIADEIRO and VEROmS (1983)]. Four alternative approaches have been studied: (1) suppose that the vertical geostrophic shear is zero from PB towards PA (used by FIADEIRO and VERONIS, 1983); (2) decrease the geostrophic shear linear to zero from PB to PA (used by FIADEIROand VEROmS, 1982); (3) maintain the geostrophic shear constant from PB to PA, and (4) calculate the geostrophic shear between A and B from the density structure, obtained by maintaining the slope of the isopycnals from PB downwards to the bottom while below the bottom an imaginary no-shear water mass with horizontal isopycnals is assumed (Fig. 4). In all four methods the bottom above the reference level is thought to be a reference level or level of no motion. Method 1 only uses information on the velocity at level PB, while methods 2 and 3 add information on the shear at level PB. Method 4 moreover uses extra information on the varying vertical density structure at Sta. A between PB and PA.

602

H . M . vm~ AKEN

mX

Px A

¢

at2

0c3 o4

~5

/ 7 ot6

~2 Qt3 o~ ~5

PA

P

Fig. 4. Schematic presentation of the method used to extrapolate the vertical geostrophic shear as used in our computations (method 4) for two different situations. The isopycnals cti have the slope of the isopycnal ul which intersects the bottom in Sta. B. Below the bottom an imaginary no shear water mass is supposed. In this method the shear is reduced to zero at the level PA, with a rate determined by the density structure in Sta. A.

If the reference level, Pr, is chosen to lie above PB the effect of the extrapolation procedure is confined to the region between PB and PA, but if p, is situated below PB the extrapolation procedure will affect the velocity calculations in the whole water column. A large value of PA--PB will yield relatively large differences between the different procedures. In this study I use as reference level pr = 4 MPa (400 dbar), which is shallow compared with the depth of the Faroese Channels. Therefore, the effects of a reference level below PB are confined to the calculated flow over the Hebridian Shelf, the Faroe Shelf and the Faroe Bank. Because the distance between neighbouring stations is relatively small, the typical value ofpA--pB is 1.2 +_ 1.2 MPa. The value ofpA--PB hardly ever exceeds 5 MPa. This limits the effect of the extrapolation procedures on the transport calculations. Extrapolation methods 2, 3 and 4 appear to give total transports between station pairs with a mutual difference of only 0.02 109 kg S- 1 (r.m.s.), both for station pairs with P~ > PB and with p~ < PB. Method 1, which did not use any information on the shear, differed from methods 2 to 4 with a typical value of transport difference between station pairs of 0.1 109 kg s-I. Method 4 is used for the transport calculations since it used a maximum of available hydrographic information. Method 4 is a variation of the method already used in the 19th century by Mohn and in the 1930s by Helland Hansen (FOMIN, 1964, p. 151). Noise due to internal waves

Internal waves and internal tides make the isopycnals move up and down. Since the CTD stations are not synoptic this causes an uncertainty in the geostrophic shear. This effect appears to be larger than the errors due to limited precision of our data set. A number of CTD stations have been occupied repeatedly. The difference in density structure between successive casts at these stations has been used to simulate the effect of temporal changes of the density structure on the calculated geostrophic transport in the four layers defined in the following sections of this chapter. The estimated error in the

Transport of water through Faroese Channels

603

geostrophic transport between two stations due to the temporal change of the density structure is 0.4 109 kg s-1 (r.m.s.), but the effect of errors in one CTD station has a tendency to be cancelled in the total transport through two neighbouring station pairs which surround that station. The total error in the geostrophic transport across a CTD section in the Faroese Channels due to internal waves and extrapolation is estimated to be of the order of 0.1 109 kg s-1.

The inverse method An inverse method can replace the concept of the level of no motion by adding extra physical constraints to the hydrostatic and geostrophic balances (Wtmscri, 1978). Mass conservation is supposed for a number, m, of well-defined layers or boxes, separated by, for example, isopycnal surfaces in a sea area surrounded by either hydrographic sections or the coast line. If the hydrographic section consists of n station pairs, i, the mass conservation for box j is written as n

E

Trij = 0

j=l,...,m.

(4)

i=1

Suppose Vo(p) to be the geostrophic velocity relative to a level of no motion at Pr. Then the transport matrix Trq, according to (2) and (3), can be written as

l f~x'+' Vo(p) dpdx + Aqv~ . , -[P'+' p,

Trij = g

(5)

In this expression Aq is the vertical surface of layer j between stations of station pair i, expressed in horizontal distance time pressure units divided by g. The reference velocity for station pair i is v,~. Substitution of (5) into (4) gives in matrix notation:

Avr = c

with

c~ = - Tr/,

(6)

where Tr/is the net mass flux through the boundaries into box j due to the geostrophic flow relative to a level of no motion in p,. Equation (6) gives a generally incomplete set of m linear equations with the reference velocities v,~ for the station pairs i = 1 to n as unknowns. By increasing the number of layers, m, to n one can construct a complete set of n equations with n unknowns. But, generally such an increased set will contain nearly dependent equations. In that case one has an ill-posed problem, and the solution will be dominated by the noise due to internal waves and tides and instrumental errors. If one adds the conservation equations for boxes in bordering sea areas, the number of equations will increase but so will the number of unknowns. Therefore (6) will remain underdetermined and the reference velocities at levelp~ cannot be determined uniquely. However, addition of the extra constraint that ~.n=l v,~ 2 has to be minimized makes it possible to find a unique solution for all reference velocities v,~ by means of a MoorePenrose solution method (Wtmscn, 1978). In fact this constraint replaces the concept of a "level of no motion" by a "level of slow motion". One can also interpret the set of reference velocities v,~ obtained in this way as a vector which all possible solutions of (6) have in common. The unresolved part of the motion is found in a n-m dimensional solution space orthogonal to the solution vector v,. Wtmscn (1978) and STOMMELand VERO~S (1981) showed that for simple geometries the solution v,/ obtained by an inverse analysis is determined by the thickness of the layers

604

H . M . VAN AKEN

involved as well as by the variable station distance. To evade a solution, solely determined by bottom topography and station distance, WUNSCH (1978) proposed to filter (6) by normalizing the columns of A by the square root of the length of the columns of A. In that case (6) is replaced by AW-1 Wv, = c ,

(7)

where W is the weighting matrix describing the effect of the normalization. The MoorePenrose solution for (7) will give an estimate of Vr for which the length of Wvr is minimized. Since the Moore-Penrose solution of (6) minimizes the length of v,, solution of (7) will supply a larger length of v,. APPLICATION

OF T H E I N V E R S E M E T H O D F O R T H E F A R O E S E C H A N N E L S

Defining the boxes In each of the 10 sea areas shown in Fig. lb, specific layers or boxes have to be defined for which conservation can be assumed. For each of the sea areas 1-4, the four layers described in the section on the hydrography were chosen as conservative boxes, that is the upper layer with 7o < 27.47 kg m -3, the main thermocline with 27.47 < 'ta < 27.8 kg m -a, the intermediate water with 27.8 < 7o < 28.0 kg m -a, and the deep water with "to > 28.0 kg m -3. From section F into the Faroe Bank Channel the intermediate water dearly appears to mix with the water from the main thermocline. Therefore, these two layers for sea areas 5-8 are combined into one box with 27.47 < 'to < 28.0 kg m -3. That gives a total of 32 boxes, 25 in the Faroese Channels, three in the Iceland Basin and four in the North Rockall Trough, where the NSDW layer is absent. For all these boxes mass conservation is assumed. Realistic estimates have revealed that the extra mass deficit or excess in the surface boxes due to a rotating wind stress was of the order of 0.02 10 9 kg s-1, which is less than the estimated error in the calculation of the baroclinic mass transport through a section. Therefore, effects due to the divergence of a wind-driven Ekman transport are ignored.

The reference level For the sections in the Faroese Channels a constant pressure level is chosen as reference level. Another possible choice is an isopycnal surface, but since the layers for which we assume mass conservation are also defined by means of potential density surfaces, layer thickness and reference level may be correlated. This can easily lead to biased results, although any choice is arbitrary. However, for the sections outside the Faroese Channels (sections G, I, L, M and Y) the outflowing water is dearly concentrated in a sub-horizontal layer, parallel to the sloping bottom (VAN ArraN and EISMA, 1987). For these sections an isopycnal surface has been chosen for reference level, since the reference level at any constant pressure gives unrealistic results. In order to determine an optimum reference pressure for the dynamic computation in the Faroese Channels a procedure is followed similar to the one suggested by FIADEIRO and VEROmS (1982), but with the addition of a normalization procedure. Let Trj(po) be the mass influx excess into box j for a level of no motion at a pressure Po. Mass conservation should imply Trj = 0. The sum ~=1 T~(po) (m=25) is a measure of the extent to which the net mass influx into the boxes deviates from mass conservation given

Transport of water through Faroese Channels

605

a uniform level of no motion, Po, in the Faroese Channels. To optimize the choice ofpo, by minimizing Y. T~, the method of FIADERO and VERO~IS (1982) has been applied for single boxes as well as for sea areas 1-7. This does not reveal any tendency for an optimum Po to vary systematically from section to section. Therefore, this method has been applied to all boxes in the Faroese Channels together (m = 25). However, only a few of the 25 boxes appear to dominate the result due to much larger T~ values compared with other boxes. This effect is mainly caused by the relatively large thickness of the deepest layer (70 > 28.0 kg m-3), especially in sea areas 1 and 2, due to large depth in these areas. Therefore T~(po) has been normalized by dividing it by the maximum value it obtained from all possible po'S. That is

(8)

=

With this Tn2 it is possible to define a normalized measure for the deviation from mass conservation the parameter, F(po):

F(po) = 1 ~

Tn2(po) .

(9)

m j=l This parameter F will vary between 0 (ideal level of no motion) and 1 (maximum deviation from mass conservation for all boxes). F(po) has been computed with 1 MPa intervals from Po = 0 to 11 MPa (Fig. 5). The optimum mass conservation appears to be found for a level of no motion at 4 MPa, and therefore the reference pressure for the inverse calculations in the Faroese Channels is chosen at that level. In the Iceland Basin as well as in the North Rockall Trough the choice of p, for further calculations is more arbitrary. For sections L, M and Y, Pr is chosen to be at the 7o = 27.8 kg m -3 level, or the bottom if that density is not present in a station pair. With this choice the reference level is slightly above the cold, thin, westward flowing layer of water near the bottom that clearly originates from overflow of intermediate and deep water. For sections G and I, Pr is chosen at 70 = 27.47 kg m -3 or, if absent, at the bottom. This reference level is always slightly above the thin layer of cold and fresh overflow water, mainly found at section I. These choices appear to be of minor importance for the resulting reference velocities in the Faroese Channels. F(P')0.5 ~ I

•

1,0 I

Q

5--

P= HPa

J

lO--

Fig. 5. The parameter F (Po) as a function of the pressure of the level of no motion, Po. F (Po) is a normalized measure of the deviation from mass conservation for a choice ofpo as defined in (9).

606

H. M. VAN AKEN

Experiments with the inverse method With the above-defined reference pressures, inverse computations based on the Moore-Penrose solution method (WtmscH, 1978; FIAD~RO and VERoms, 1982) have been carried out in a number of experiments. In experiment I reference velocities are obtained for sections C and D, based on inverse computations with mass conservation for the four layers in sea area 2 only. For the computations the unfiltered equation (6) is used. In experiment II (6) is also used, but here mass conservation is assumed for all 32 boxes. In experiment III mass conservation for all 32 boxes is supposed, but in order to prevent dominance of the solution by only a few equations or by only a few station pairs both the rows and the columns of the equation have been normalized (WtJNSCH, 1978, 1985). In experiment IV these normalized equations are also used, but mass conservation is only applied for the 21 boxes in sea areas 2, 3, 4, 5a, 6a and 7 in the Faroese Channels. In all four experiments the equations appear to be linearly independent. No cut-off has been applied and the full solutions are used. In experiment I, the r.m.s, value of Vr along sections C and D amounts to 2.0 cm s-1. Comparison of experiment I with experiment II shows the influence of the extra constraint of mass conservation in neighbouring areas (Fig. 6a). The distribution of the reference velocity has the same shape for both experiments, but the magnitude of vr in experiment II is larger due to the addition of a number of equations for the other sea areas. The r.m.s, value in experiment II of Vr along sections C and D is 2.6 cm s-x. The influence of normalization can be seen in Fig. 6b. The effect of the normalization can be observed over the Faroe and Hebridian Shelves, where the water depth is shallow. There the reference velocities dearly increase, but the effect of the normalization in the deep parts of the Faroe-Shetland Channel is quite limited. In these deeper parts all four layers for which mass conservation has been applied are present, whereas over the shelves only the surface and thermocline layers are found. The limited horizontal extent of the intermediate and deep layers due to topographic constraints appears to dominate the distribution of the reference velocities, that is not influenced by the normalization procedure. The r.m.s, value along sections C and D obtained in experiment III is 2.6 cm s-x, the same value as is found in experiment II. The influence of areas 1, 5b, 6b and 8, where the main inputs and outputs of the current system in the Faroese Channels are located, can be seen in Fig. 6c. The only station pairs where clear differences between experiments III and IV are observed, are located over the Faroe Shelf. The variation of the total transports through section C in the different layers obtained from the different experiments is shown in Table 1. A striking feature is the strong reduction of the transport in the deep layer due to the application of the inverse method. The positive transport in the intermediate layer in experiment I contradicts the hydrographic evidence for transport of intermediate water from the Norwegian Sea towards the Iceland Basin. Addition of the extra constraints from other sea areas in experiments II, III and IV, however, forces the total transport of intermediate water to be directed towards the Iceland Basin. The differences between the total transports in experiments II, III and IV appear to be of the order of the estimated error in the geostrophic calculations. The small differences between experiments III and IV (Fig. 6c and Table 1) indicate that the fact that sections O and Y are less quasi-synoptic than the other sections did not influence our results strongly. In the following section results will be discussed for the whole research area as

607

Transport of water through Faroese Channels ref. Velocity

ref. Velocity

(era/s)

(¢ml=)

.rl

j--]

. . . . . .

.....

Lr

O" r._.j------t_..1 ..... J /'7

-5"

LJ 5'o

16o

I~o

-5

L.I' 26o

b

o

2~

distonce (kin]

~

60

1~o

26o

z~o

dtstonce (kin)

ref. Velocity (cmls)

i

O'

-5' C

o

5'0

I~o

1~o

2bo

2~o

distance (km)

Fig. 6. Comparison of the reference velocities obtained for section C in different experiments. Positive velocities are directed into area 2: (a) full line, experiment I; dashed line, experiment II; (b) full line, experiment II; dashed line, experiment III; (c) full line, experiment III; dashed line, experiment IV.

Table 1. Mass transport through section C in four layers for different experiments described in the text. The mass transport is given in 109 kg s- 1 and/s positive towards the Norwegian Sea, and negative towards the Iceland Basin

Level of no motion at 4 MPa Experiment I Experiment II Experiment HI Experiment IV

Surface layer

Thermocline layer

Intermediate layer

Deep layer

0.64 0.84 0.76 0.89 0.84

0.00 0.34 0.38 0.18 0.23

-0.33 0.04 -0.16 -0.18 -0.12

-2.58 -0.37 -0.26 -0.31 -0.31

608

H . M . VAN AKEN

obtained in experiment III. The 32 normalized equations appear to be finearly independent, and the 32 eigenvalues vary from 0.40 to 4.38. The eight smallest eigenvalues together contribute only 7.3% of the total summed squares of v,~, indicating a low noise content of the full solution. Therefore, no cut-off is applied and the full filtered solution for the 178 reference velocities v,~ is used for the transport calculations. The r.m.s, value of v, on all sections in the Faroese Channels amounts to 4.7 cm s-1. On the sections in the North Rockall Trough and the Iceland Basin, that is sections G, I, L, M and Y, the r.m.s. value of v, is larger, 13.7 cm s-1. These magnitudes of the reference velocity appear to be at least an order of magnitude larger than the reference velocities found by FIADEIROand VERONIS(1982) for a much simpler area east of Australia, but they are of the same order of magnitude or less than the reference velocities obtained by FIADEIRO and VERONIS (1983) in the western North Atlantic. The relatively large values of v, in the North Rockall Trough and the Iceland Basin are probably caused by the more complicated hydrography with thin layers of overflow water and a clearly different vertical density structure compared with the Faroese Channels. Changes in the reference level and in the layers for which the mass conservation was applied did not result in essential changes in these areas. R E S U L T S AND D I S C U S S I O N

The horizontal structure of the transports

Since the mass transport within each of the above-defined layers is supposed to be free of divergence due to mass conservation, it is possible to define a transport stream function ~ , for each layer, so that Mx--

O5/ Oy '

My-

0~ Ox "

(10)

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The net flow from area 8 into the Faroe Bank Channel is negligible and is hardly influenced by the reference velocities. It consists of an inflow and an outflow, both of the order of 1 x 10 9 kg s-1. The main inflow of Atlantic water with NA and LAW characteristics is found over the Wyville Thomson Ridge and has a net value of 1.7 × 10 9 kg s-1. In the Faroe-Shetland Channel this transport of Atlantic water is split into several branches due to the presence of a number of baroclinic eddies. Over the Hebridian Shelf near the Shetland Islands a return flow with a transport of the order 0.3 × 10 9 kg s-1 is found, mainly in the fresher coastal water southeast of the high salinity core. In the northwestern half of the Faroe-Shetland Channel recirculation of water with MNA characteristics is observed, but that water does not enter the Faroe Bank Channel from the east, as supposed by DOOLEYand MEINCrm (1981). These results agree with HANSEN'S(1985) proposal of a complete return flow of MNA in the Faroe-Shetland Channel. The total transport into the Norwegian Sea of Atlantic water computed here, appears to be of the same order as the value suggested by WORaa-IINGTON (1970), but 75% of this transport took place across the Iceland-Faroe Ridge and only 25% through the Faroese Channels. There is hydrographic evidence that some more MNA and LAW flows along the polar front across the Faroe-Iceland Ridge at more western positions, not covered by the CTD survey used here, but since the polar

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front shifted back to a position south of the Iceland-Faroe Ridge just west of section M, that net transport is thought to be small (VAN ArEs and EISMA, 1987). The computed flow of MNA and LAW across the Faroe-Iceland Ridge is of the same order of magnitude as the 14 days mean flow in the upper 500 m within 150 km north of the Faroe Islands reported by HASSEN et al. (1986). They found a transport of about 4.7 x 109 kg s-1. The transport of NA water through the Faroe-Shetland Channel is of the same order as the transport given by DOOLEY and M~.n~cr~ (1981). This result suggests an explanation for the contradicting views on the transport of Atlantic water in the Faroe area. While the heat and salt budgets of the Arctic seas require an influx of about 8 x 10 9 kg S- 1 Atlantic water (WORrrm~OTON, 1970) current observations and geostrophic calculations (TAn', 1957; DOOLEYand MEINCKE, 1981) indicate that only 1530% of that transport could be sustained by the flow through the Faroe--Shetland Channel. The results presented here and those of HANSENet al. (1986) indicate that the remaining 70-85% of the required transport probably flows across the Faroe-Iceland Ridge into the Norwegian Sea, a possibility that was overlooked by Worthington. The flow of intermediate water is dominated by a number of eddies in the Faroe-Shetland Channel (Fig. 7d). Due to these eddies the intermediate water, originating from the Norwegian Sea, recirculates in the Faroe-Shetland Channel. A net amount of only 0.2 x 10 9 kg s-1 was found to flow towards the Faroe Bank Channel. The high salinity water found along the Hebridian shelf slope appears to flow towards the Faroe Bank Channel while the fresher water recirculates. This agrees with the hydrographic evidence presented by VAN Armr~ and EISMA (1987). About 0.4 X 109 kg S- 1 of the thermocline and intermediate water spills over the Wyville Thomson Ridge into the Rockall Trough, following the contours of the Faroe Bank. An equal amount of water flows from the Rockall Trough into the Faroe-Shetland Channel (Figs 7b and 8c). Also, minor amounts of intermediate water (about 0.2 x 109 kg s-1) spill across the IcelandFaroe Ridge under the polar front (Fig. 8d) causing a second core of overflow water along the Iceland-Faroe Ridge (VAN Ar,.EN and EISMA, 1987). The flow of deep water (Fig. 7e) shows recirculation in eddy-like and topographically determined cells in the Faroe-Shetland Channel and the deeper parts of the Faroe Bank Channel. The net transport of NSDW towards the Iceland Basin amounts to 0.3 x 109 kg s-1. The proportionality of the outflow of intermediate and deep water is reflected in the overflow water. The overflow water west of section M originating from the Faroe Bank Channel contains about 24% intermediate and 31% deep water (VAN AKEN and EISMA, 1987) which gives, within the accuracy of the estimates, the same ratio as the transports. The total flow of water with potential density excess over 27.8 kg m -3, that is the intermediate and deep water density range, more than doubles between sections H and M, probably due to entrainment of thermocline water into the overflowing intermediate water. This mass exchange between the thermocline layer and intermediate water is possible because in the Faroe Bank Channel we have combined these two layers into one box. The direction of the mass exchange from thermocline layer into intermediate water agrees with the hydrographic evidence presented by VAN Armr~ and EISMA (1987). The influx of the total overflow water from the Faroe Bank Channel through section M into the Iceland Basin amounts to 1.1 x 10 9 kg s-1, a value which agrees with the literature values cited in the Introduction.

Transport of water through Faroese Channels

615

The "reality" of inverse solutions One can question here the physical reality of the transports derived by means of inverse dynamic computations. Recently, VeRoms (1986) and WuNscrI (1985, 1986) argued about the interpretation of results of inverse methods in tracer studies. Their discussion is not yet decided, but clearly acceptance of the results of inverse computations depends strongly on one's opinion of "physical reality". It is worthwhile to keep in mind in these discussions that the use of a survey with only a finite number of density profiles at discrete positions always filters the velocity field derived from these density profiles, suppressing the high wavenumber part of the flow field. However, large wavenumber flow variations leak into the larger scale motion due to aliasing of baroclinlc structures with scales smaller than the station distance. Since our survey is not fully synoptic but only quasi-synoptic, the derived flow field also may contain spatial variations due to aliasing of temporal changes into the spatial field. Therefore, even without the problems of the inverse calculations, the physical reality to be obtained from discrete, quasi-synoptic surveys is already somewhat ill-defined. But what is the physical content of the reference velocity derived by an inverse method? Why do we not "trust the intuition of a traditional hydrographer", as one of the referees of this paper proposed, and simply use a level of no motion for our calculations instead of using inverse procedures. I-IANsEN(1985) clearly showed that the traditional level of no motion, based only on hydrographic experience, until now only has produced uncertain and contradicting transport estimates. Therefore, we have added a number of physical constraints to our dynamic computations, instead of adding another subjective choice to the literature based on hydrographic intuition and prejudice. The inverse method is a data assimilation method in which different types of physics can be combined, not unlike the objective analysis methods used in meteorology. The fact that (6) has an infinite number of solutions reflects that there is no unique relation between an observed density distribution and a single geostrophic flow field. The results obtained here have to be valued as the simplest possible solution for the transport distribution in terms of the length of vr or Wvr at an optimum reference level and as such, as the smallest possible deviation from the concept of a level of no motion. There is, however, no physical reason why the length of v~ or Wv, should be minimized. But the "physical reality" is that the solution agrees with the observed density structure and the imposed restrictions of mass conservation. Since the observed hydrographic fields agree qualitatively with the distribution of the computed transport stream functions and since the total transports in the different layers do not deviate unrealistically from other existing estimates our results seem to be at least not far from a more comprehensively defined "physical reality". Probably our transport values are reliable for the period of observation, but that should not imply a priori that they are representative for the annual mean flow. Data may exist from previous experiments in the Faroese Channels, which are fit for dynamic calculations using an inverse method like the one used in this paper. Study of the transports obtained from those historical data by means of an inverse method and comparison with existing current meter records can give insight whether the inverse method gives reliable and representative results. Then it can be determined whether the inverse method is one of the modern techniques HANSEN(1985) asks for.

616

H . M . VAN AKEU CONCLUSIONS

The determination of the geostrophic transport through the Faroese Channels is known to be strongly influenced by the arbitrary assumptions concerning the existence and the depth of the level of no motion (I-IANs~, 1985). By determining an optimum reference level and application of an inverse method, as proposed by FIADEIROand VEROmS (1982), we have tried to improve the arbitrary character of the geostrophic calculations. The transport of Atlantic water determined by this method agrees with the amount of water needed by WOR~nNOrON (1970) to close the heat and salt budgets of the Norwegian Sea. But contrary to his suggestion only 25% of this transport went through the Faroese Channels, the rest of the flow was diverted west of the Faroe Islands over the Iceland-Faroe Ridge. The overflow from the Faroe Bank Channel into the Iceland Basin agrees with the amounts cited from literature, while also overflow across the Iceland-Faroe Ridge is observed. The flow patterns, derived from the inverse calculations, agree with the hydrographic structure of the area, described in detail by VAN Ar~.~ and EISMA (1987). It seems worthwhile to repeat this type of inverse calculation of geostrophic transport for existing data from the Faroese Channels. Acknowledgements,--The cruise of R.V. Tyro was financially supported by the Netherlands Council for Oceanic Research. The data have been made available by permission of Dr D. Eisma. I thank the referees whose critical remarks on the first version of this paper forced me to use a more systematic approach to the problems discussed here. This has deepened my understanding of the questions related to the use of inverse models in general and to the use of inverse modelling in dynamic computations in particular. I thank Marjofijn Verhoeven for typing the successive versions of this manuscript. REFERENCES A z ~ H. M. VAN and D. EISMA(1987) The circulation between Iceland and Scotland derived from water mass analysis. Netherlands Journal of Sea Research, 21, 1-15. CREASEJ. (1965) The flow of Norwegian Sea water through the Faroe Bank Channel. Deep-Sea Research, 12, 143-150. DooI~y H. D. and J. MEn~CKE (1981) Circulationand water masses in the Farocse Channels during Overflow '73. Deutsche HydrographiacheZei~chrifl,34, 41-54. FIADEmO M. E. and G. V E R O m S (1982) O n the determination of absolute velocitiesin the ocean. Journalof Marine Research,Suppl. 40, 159-182. FIADEntO M. E. and G. VEROW_S (1983) Circulationand heat fluxin the Bermuda Triangle.Journalof Physical Oceanography, 13, 1158-1169. FoMnq L. M. (1964) The dynamic method in oceanography. Elsevier,Amsterdam, 212 pp. H A N S ~ B. (1985) The circulationof the northern part of the North-east Atlantic.RitFiskideildar,9, 110-126. HANSEN B., S. A. ~ E R O , O. H. SAturN and S. ~s'r~RtlUS (1986) Measurement of flow north of the Faroe Islands, June 1986. ICES Hydrographic Committee C.M. 1986/C:12, 14 pp. (mimeo). H E R M A ~ F. (1967) The T-S diagram of the water masses over the Iceland-Faroc Ridge and the Faroc Bank Channels. Rapports et Proces-Verbaux des Reunions, Conseil International pour l'Exploration de la bier, 157, 139-149. M~ZJZR T. J., J. ME~CI~ and G. A. B E e r , s (1979) Overflow '73: The distribution of water masses on the Greenland--seotland Ridge in August/September 1973. Berichte aus den Institut ffir Meereskunde, Nr. 62, Kiel, 172 pp. S3"EFANsoNU. (1962) North Icelandic Water. Rit Fiskideilder, 5, 269 pp. S'rOM~i H. and G. VEROmS (1981) Variational inverse method for study of oceanic circulation. Deep-Sea Research, 28, 1147-1160. TAn" J. B. (1957) Hydrography of the Faroe-Shetland Channel 1927-1952. Mar. Res. Scot., H.M. Stationary Office, Edinburg, 309 pp.

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617

VERONIS G. (1986) Comments on "Can a tracer field be inverted for velocity?". Journal of Physical Oceanography, 16, 1727-1730. WORTHINOTONL. V. (1970) The Norwegian Sea as a mediterranean basin. Deep-Sea Research, 17, 77--84. WUNSCH C. (1978) The North Atlantic general circulation west of 5ff'W determined by inverse methods. Reviews of Geophysics and Space Physics, 8, 583--620. WUNSCH C. (1985) Can a tracer field be inverted for velocity.'? Journal of Physical Oceanography, 15, 1521-1531. WLrNSCI-IC. (1986) Reply. Journal of Physical Oceanography, 16, 1731-1732.