Geophys. J. R. astr. SOC.(1972) 30, 415-432.
Estimation of Density Currents in the Liverpool Bay
Area of the Irish Sea
N. S. Heaps
(Received 1972 July 12)
Summary
Density currents in Liverpool Bay are evaluated on the basis of a simple
hydrodynamical theory. The results are shown to account for some of the
main features of the observed residual motions in the Bay, including the
persistent landward drift at the sea bed.
1. Introduction
Recent observations by Halliwell (1973) have shown that sea-bed drifters released
at various positions in Liverpool Bay move, at all times of the year, in directions
easterly to south-easterly on to the coasts of Lancashire, Cheshire and North Wales.
Investigations by Ramster (1965) have indicated that similar bottom drifts to the
coast occur in other areas of the eastern Irish Sea. The primary purpose of the
present work is to give theoretical support to the proposition, herewith made, that a
main cause of these observed landward movements at the sea bed is an on-shore
density flow brought about by horizontal gradients of salinity and temperature
normal to the coast. That such gradients produce on-shore bottom drifts in open
water on the rotating Earth has recently been questioned, but an argument given in
the following analysis shows that the effect is consistent with dynamical principles.
The eastern Irish Sea, in particular Liverpool Bay, appears for the most part to
be very nearly vertically homogeneous throughout the whole year. As a general
feature the isohalines and isotherms run parallel to the coast, the density increasing
progressively in passing from the relatively fresh water near the shore to the more
saline water further out to sea (Bowden 1955; Jones & Folkard 1971). Calculations
carried out here, specifically for Liverpool Bay, show that the above characteristic
density distribution gives rise to horizontal currents directed across the a,-lines
towards the coast at the sea bed, turning to the left as their height above the bed
increases. As a result of this rotation through depth, the currents at the sea surface
are inclined (in the horizontal) at an obtuse angle to those at the bottom. Immediately
forthcoming is an explanation for the south-east going bottom drift and the north
going surface drift in the residual water movements of Liverpool Bay, deduced from
direct current measurements and drifter experiments (Ramster & Hill 1969; Ramster
1971). Moreover, the theoretical estimates of bottom drift are able to account substantially for the persistent transport of sea-bed drifters to the shores of the Bay.
The motion described above is steady and takes place against the alternating
background of the dominant tidal streams. These generate turbulence which evidently
causes vertical mixing of such high intensity as to produce the state of near-homogeneity
from sea-surface to bottom observed at most places. The turbulence also influences
the density distribution through horizontal eddy diffusion (Bowden 1963). Thus,
415
416
N. S. Heaps
regarding conditions as stationary, and uniform in a long-shore direction, across any
vertical section there is a balance between the salt diffused inwards towards the shore
and that convected outwards by coastal discharge, a situation which leads to the
gradation of salinity and hence density between the coast and the open sea. Functions
of the turbulence, important for the purposes of the present work, are the coefficients
of vertical eddy viscosity and bottom friction: basic parameters involved in the
calculation of the density currents.
The procedure of this paper is to set up a simple hydrodynamical theory for
density-induced motion in an idealized coastal sea, and then apply the theory to
Liverpool Bay taking numerical values relevant to that area. In the analysis a long
straight coast situation is assumed, all transverse sections normal to the coastline
being equal and conditions in each being the same. Steady motion is determined
with the geostrophic effect included. Depth, density and coefficients of friction are
prescribed general functions of the horizontal distance measured from, and perpendicular to, the coast. The formulation is a linear one leading to an analytical
solution of the dynamical equations. Similar solutions were obtained many years ago
by Nomitsu (1933), but whereas he appeared to be mainly concerned with the
development of the theory as such, the interest now is with its application to a real
sea basin.
Nomitsu pointed out that if a current driven directly by a horizontal density
gradient is checked by land, water will accumulate and a surface slope and therefore
a slope-current (or ‘ gradient current ’) will develop. In circumstances of this kind,
the presence together of horizontal flows actuated on the one hand by density gradients
and on the other hand by surface gradients produces a tendency towards differential
motion through the depth since the flows are by nature compensatory. As a consequence, vertical shears between different horizontal levels are a feature of the
current profles derived from present theory and, significantly, have been indicated by
drifter observations as existing in the residual currents of the Irish Sea (Harvey 1968).
2. Basic equations
Consider the motion of water bounded by a straight infinitely long coast. Assume
that all sections normal to the coast have the same geometry (Fig. 1). Denote by
Cartesian co-ordinates, forming a left-handed set, in which x,y are
measured in the horizontal plane of the undisturbed sea surface in the
on-shore and long-shore directions respectively, and z is depth below the
surface;
the components of current at depth z, in the directions of increasing
x,y respectively;
the components, in the directions of increasing x , y respectively, of the
frictional stress by which the water above the depth z acts on the water
below that depth;
the elevation of the water surface above its undisturbed level;
the mean depth of water;
the density of the water;
the pressure at any position in the sea;
the geostrophic coefficient, equal to 2 0 sin4, where o denotes the
angular speed of the Earth’s rotation and 4 the latitude;
the acceleration of the Earth’s gravity.
417
Estimation of density currents in the Irish Sea
(a)
!
Sea surface
,. .
Sea bed
(b)
Coast
I
I
FIG.1. Mathematical model showing (a) the vertical cross-section normal to the
coast and (b) the horizontal plane.
Then the steady-state equations of continuity and motion may be written (Proudman
1953, p. 17 and p. 96):
Here, the convective terms in the acceleration relative to the Earth are neglected and
no account is taken of internal stresses acting in a horizontal plane. The geostrophic
coefficientf is regarded as invariable. Equation (4), derived from the assumption that
pressure obeys the hydrostatic law, may be integrated to give
418
N. S. Heaps
where pa, considered to be a constant, denotes atmospheric pressure on the sea
surface.
We are concerned with the currents brought about by the existence of an xdependent density field. This field may represent a real situation in which salinity and
hence density increase progressively from coast to open sea without any significant
variation laterally or through the depth. Thus
and from
(9,
Suppose that conditions in all sections normal to the coast are the same. Then
dependency on y is eliminated and, using (6) and (7), the dynamical equations (1)
to (3) reduce to
h
”s
ax -c
vdz = 0,
Further, take
where N is a coefficient of eddy viscosity-assumed to be a known function of x.
Also, let q , a constant, denote the rate of seawards discharge of water per unit length
of coastline, representing the fresh water flows from various rivers in reality. Then
integrating (8) and combining (9) and (10) with (11) gives
[ u d z = -4,
(12)
-F
With p, N and q prescribed, these equations are to be solved for u and u subject to
appropriate boundary conditions at the sea surface and the sea bottom. Thus, let
suffix
denote evaluation at the surface z = -c and suffix Iz evaluation at the
bottom z = h. Then the requirement of zero tangential surface stress:
-c
(F)-c = ( G ) - , = 0
yields
419
Estimation of density currents in the Irish Sea
while assuming linear relations between components of bottom friction and bottom
current of the form:
F h = kpUh,
Gh = kpVh,
(17)
where the coefficient of friction k is at most some function of x, yields
Equations (12), (13), (14), (16) and (18) are basic to the present analysis.
3. Long-shore component of bottom current
Integrating through the total depth in equation (10) gives
h
/I
1
--
aG
J -&z
= f / udz
-c
-c
and therefore, using (12), we get
1
- [(G)h-(G)-(.l
P
=fq*
Incorporating (15) and (17) it follows that
which yields a simple formula for the long-shore component of bottom current:
Applying (19) to the Liverpool Bay Area of the Irish Sea (Fig. 2) we insert
f = 0-4218/hr, q = 2 m3 s-' km-',
k = 0-2cin s-l
and obtain
vh
= 0.01 cm s-'.
Here q is derived by taking the average daily discharge from the Rivers Dee, Mersey
and Ribble together to be 160 m3 s-'. The coastline corresponding to this discharge
is about 80 km in length giving a discharge per kilometre of coast of about 2 m3 s-'.
The value of k is one suggested by evidence available from Bowden (1953) and
various other sources.
By most standards the calculated value of is negligibly small, due mainly to the
smallness of q. Any significant density current at the sea bed is therefore directed
normal to the coast.
4. Mathematical solution
Introducing the complex variable
W = u+iv
equations (13) and (14) may be combined to give
a2 w - a2 w + - (z+C)-az2
N {
--
ap
ax
1
+ax
420
N. S. Heaps
where
o! =
(l+i)n/D
and
D = n(2N/f)+,
the latter being Ekman’s ‘frictional depth’ (Defant 1961, p. 401). Also, from (16)
and (18),
and
The general solution of (21) is
where A and B denote arbitrary constants. Determining these constants from (24)
and (25), and then taking real and imaginary parts in the resulting expression for
W yields u and u. Eliminating a[/ax from the u and u thus derived, using for this
purpose the relation obtained by satisfying (19) (or alternatively (12)), gives
u = ( g H / f ) ( X Q- YP)(~P/W/P
+ (fq/k)(MP-LQ)/S,
LJ
= (gH/f)(XP+
Y Q + A + W P / W / P + (fq/k)(l-LP-MQ)/S.
)
(27)
The various parameters involved here may be specified by passing sequentially through
the following definitions:
Z=z+[,H=h+[,
u = nH/D, a, = ~ (- ql) ,
q=Z/H,
= a?,
b = kH/N,
+
E = a(sinh a cos a +cosh a sin a) + b sinh a sin a,
C = a(sinh a cos a -cosh a sin a) b cosh a cos a,
L = b cosh a, cos a,, M = b sinh a, sin a,,
I
+
P = C / ( C 2+E’), Q = E/(C2 E’),
R = P c o s h a c o s a + Q s i n h a s i n a , S = 1-Rb,
A = (R-P-S)/S,
A = l+b+bA,
X = cosh a, cos a, + (b/2u)(sinh a, cos a, + cosh a, sin a,) - A cosh a, cos a,,
Y = sinh a, sin a, + (b/2a)(cosh a, sin a, - sinh a, cos a t )- 1 sinh a, sin a,.
J
The above sequence may be conveniently employed in the numerical evaluation of u
and u. The basic parameters of the list are q, a, b, these being non-dimensional and
functions of 2,H, D, N , k where 2 is depth below the sea surface and H total depth
from surface to bottom. The input values for the evaluation of u and u in (27) may
therefore be taken as q , H, N , k , f, g , (ap/ax)/p, q. In the expressions for u and u the
first term clearly represents the effect of density gradient and the second the influence
of coastal discharge. The contribution of the latter to values of current was found
to be very small (generally between 0.01 and 0.1 cm s-’) in the calculations of the
present work.
42 1
Estimation of density currents in the Irish Sea
As a further result of the mathematical analysis the slope of the sea surface is
given by
allax = AH(ap/ax)/p+fzqigks.
(29)
5. Application of the theory to Liverpool Bay
The foregoing theory is now used to estimate density currents in Liverpool Bay,
as illustrated in Fig. 2. A typical a,-distribution, quoted by Halliwell (1972), relating
to conditions of the near-bed waters for the period 1970 May 27 to 29, is shown
drawn on a chart of the Bay. The o,-lines tend to run parallel to the coast: from north
to south off the sea-boards of Lancashire and Cheshire, curving westwards following
the line of the North Wales shore. In this situation, A, A, is a cross-section normal
to the lines and the general sweep of the coastal boundary. Density currents along
the section are calculated using equation (27) with
(ap/ax)/p = -0.2 x 10-3/n.mi.,
f = 0.4218/hr,
q = 2m3 s-' km-',
k = 0-2cms-',
g = 980 cm s - ~ .
F I ~2.. Chart of Liverpool Bay showing u,-lines for a period in May 1970. Also
shown is the section A 1 A 2 used in calculating the density currents associated with
this u,-distribution. Drawn on the chart are horizontal vectors of the calculated
current field, for b = 4, at 11 equidistant depths from the surface 1) = 0 to the
bottom 7 = 1, at position P, 12 nautical miles from the coast on A 1 A 2 ;0 marks
Ramster's long-term current meter station M, and + stations at which bottom
.-.
*-. . 20 fm;
drifters were released by Halliwell. Depth contours are:
- -. 1Ofm;--------5fm.
,
3 fm.
-. . -.
4
................
-a
*-a
422
N. S. Heaps
The horizontal co-ordinate x is measured along A , A , towards the shore at A,,
consistent with the notation employed in the mathematical model (Fig. 1). All
horizontal distances are in nautical miles. The density gradient ap/dx is estimated
from the spacing of the ot-lines, while the values allocated tof, q and k are those
introduced in Section 3.
Other parameters determining the currents are H (total depth), N (eddy viscosity)
and q (the fractional depth at any position). The variation of H along A , A , is taken
to be that of chart depth (Fig. 3) incremented uniformly by 2 fathoms to yield a
representative mean-depth distribution. The choice of appropriate values for the
eddy viscosity is important, but difficult because of the lack of reliable knowledge
about this coefficient. The approach taken is to regard
b
= kH/N
(31)
as a constant. Thereupon, fixing the value of 6, and knowing k and H , the variation
of N along A , A , is completely determined.
Assuming
N = KHD
(32)
where K is a constant and U a representative mean value for the amplitude of tidal
current in the Bay, we have from (31):
b = k/Ku.
(33)
Consider the evaluation of b using (33). On the basis of information given in atlases
of tidal streams (e.g. Atlas der Gezeitenstrowe fur Die Nordsee, Den Kana1 und Die
Britischen Gewusser-published by Deutsches Hydrographisches Institut, Hamburg,
in 1963) one may reasonably take u = 70cms-'.
Estimates of K may be obtained employing values of N derived by Bowden, Fairbairn
& Hughes (1959) from observations in a tidal current off Red Wharf Bay, Anglesey,
North Wales. On the average, for times from 4 to 2 hours before high water they
found N = 273 cmz s-' with U = 44-7 cm s-' and H = 22 m, so that
K = NIHU =
2.78 x
20-
Depth
(fm)
I5 -
10 -
5-
0- ,
,
1
I
-
Estimation of density currents in the Irish Sea
423
while for times from 2 to 4 h r after high water their determinations yielded
N = 132 cm2 s-l with g = 39.4cm s-' and H = 22.5 m, whence K = 1.49 x
Inserting the first estimate of K into (33), along with k = 0-2 cm s- and U = 70 cm s- ',
gives b = 1.03 for Liverpool Bay; associated with this value of b, N increases from
31 1 cm2 s-l at a depth of 8.75 fathoms (16 m) to 71 1 cm2 s-' at 20 fm. On the other
hand, the second value of K when substituted into (33) gives b = 1-92, in which case
N increases from 167 cm2 s-' at 8.75 fathoms to 381 cm2 s-l at 20fathoms. Determinations by Bowden (1960) and Bowden & Sharaf El Din (1966a) for the Mersey
Estuary, and by Bowden & Sharaf El Din (1966b) for a nearby station in Liverpool
Bay, suggest that N-corresponding to mean conditions over a tidal period-might
have somewhat lower values than those just derived, specifically so at the smaller
depth. In fact choosing b = 4-0, yielding N = 80 cm2 s-l at 8.75 fathoms and
N = 183 cm2 s - l at 20 fm, would appear to be more in agreement with their findings.
A degree of vertical stratification in the region of observations was thought to account
for some of this reduction in N.
Jn the light of the above evidence it is judged that the range of realistic b for
Liverpool Bay is almost certainly spanned by the values b = 1,2,4. The density
currents along A , A,, in the situation described by (30), have therefore been computed
for these three cases. Consider the calculated values of the surface current and the
bottom current. Let V,, Vl denote their respective magnitudes and f?, 8, their
respective angles of inclination to A , A , at any position, as illustrated in Fig. 4.
Formally,
V, = (uO2+uo2)t-, eo = tan-'(uo/uo),
(34.1)
V, = ( U , ~ + V , ~ ) + , 8, = tan-'(ul/ul),
where
I
(34.2)
-determined from (27).
The bottom current and its direction are plotted against distance from the coast
in Fig. 5. The curves of this figure show that the current is highest in the deeper
water, its possible values (corresponding to the admitted range of b, namely
1 < b < 4) lying between 2.5 and 4-Ocms-' at a position 17 miles out to sea.
Manifestly, as the coast at A , is approached from some 12 miles off shore the current
' Y
FIG.4. Notation for the resultant surface current and the resultant bottom current
at any position, showing their respective magnitudes (Vo, V , ) and directions
(00,
01).
424
N. S. Heap
Nautical miles from the coast along sectbn
4
A,
FIG.5. Variation of the resultant bottom current along section A I A z forb = 1,2,4.
The magnitude of the current (V,)is plotted and also its inclination (0,) to the
-----,
on-shore x-direction A I A 2 .
decreases progressively in magnitude, falling below 0.2 cm s-’ within 3 miles of the
land. The variation of 0, indicates that the bottom flow may, without much error,
be regarded as directed on shore along almost the entire length of A, A,; its direction
changes appreciably on nearing the coast as the motion associated with river discharge
becomes relatively significant, but the currents in this region are so small anyhow
that such directional changes are of little importance in the present study. It is of
interest to compare the computed bottom currents at a position P on A, A,, 12 miles
out from A,, with the near-bottom residual currents derived by Ramster (1971) from
current measurements made throughout the period March 1970 to February 1971 at
a neighbouring station M (53” 31-6‘ N, 3” 3 3 4 W ) . Both P and M are marked in
Fig. 2. Ramster’s results, given in the form of frequency diagrams, show that from
Estimation of density currents in the Irish Sea
425
1;
1c
0
E
4
2
0
17C
16C
150
140
130
I20
I
I
t
8
I
I
1
I
1
16
14
12
10
8
6
4
2
0
Nautical miles from the coast along section A, A,
-
FIG.6. Variation of the resultant surface current along section A I A pforb = 1,2,4.
The magnitude of the current (Vo) is plotted and also its inclination (0,) to the
on-shore x-direction A 1 A 2 .
426
N. S. Heaps
1970 March to June reduced currents from 3 to 5 cm s-' were most common, from
October to November a broad band of velocities from 1 to 6 cm s-' prevailed, while
from 1970 December to 1971 February speeds of some 2cms-' occurred most
frequently. On the average over the entire period a bottom residual of about 3 cm s-'
seemed to be the most persistent. This compares with a range of possible values of
density current at P from 1.6 to 2.8 cm s-', deduced from Fig. 5. Residual bottom
currents derived from the measurements at M were mainly south-going, a result to
some extent at variance with the general direction of movement of sea-bed drifters
which was to the south-east (Halliwell 1972). In relation to this inconsistency, the
bottom currents from present theory are clearly in more accord with the drifter
observations.
Consider now the computed surface currents. Magnitude V, and direction B0 are
plotted along A , A , in Fig. 6. Examination of this figure, along with Fig. 5 , shows
that the velocities at the surface reach higher values than those at the bottom and are
rotated relative to the bottom vectors through angles which vary between 120 and
170 degrees. Thus, a south-east going drift at the bottom is accompanied by a somewhat larger essentially north going drift at the surface. Ramster's near-surface current
measurements at station M during 1970 and 1971 confirm the existence of a residual
flow to the north at all times of the year. Further, his analysis of observations appears
to indicate that a surface residual of some 4 cm s-' occurred most frequently, within
a range of from 0 t o 8 cm s-'. Apparently in broad agreement with this latter result,
Fig. 6 gives surface density-currents lying between 2 and 8 ems-' (for b = 1 to
b = 4) at nearby position P. At this position, located 12 miles from the coast along
A , A,, the most frequently observed speed of 4 cm s-' corresponds to b = 2.
Let us now study the motion of a particle on the sea bed under the influence of
each of the velocity fields given by Fig. 5. Suppose the particle starts from a position
17 miles out to sea on A , A , ; this position is near to the location SDI from which
Halliwell (1973) released sea-bed drifters. Its subsequent movement may be determined as a series of consecutive displacements V, Ar taking place respectively in
successive time intervals of length At; V, denotes the current vector at the origin of a
displacement. Regarding V, as an on-shore current, which is very nearly true, the
terms of
nautical
miles
from the
coast.
along
section
A, A2
0
5
10
15
20
25
x)
35
40
45
50
55
60
T i m e in days
FIG.7. Displacement (by density flow) through time, on the sea bed, along A l A 2 ,
of a particle released 17 nautical miles from the coast: for b = 1, 2,4.
65
Estimation of density currents in the Irish Sea
427
particle is displaced continually along A , A , towards the coast. This shorewards
progression has been calculated for the cases b = 1, 2 and 4, taking At = 3 day. The
results are given in Fig. 7 which shows that the particle moves towards the land
relatively quickly at first in the deeper water where the higher currents prevail, and
then more slowly as it enters shallower water where the transporting currents are
weaker. In 66 days it is brought to within between 1 and 3 miles of the coast. Halliwell
found that sea-bed drifters released from SD1 and three neighbouring positions
(Fig. 2) took some 25 days to reach the North Wales coast during the period in
May 1970 to which the present calculations particularly refer. But after 25 days
following release from the same vicinity, Fig. 7 indicates that our particle on the sea
bed is still somewhere between 3.5 and 6 miles short of this coast, assuming
1 < b < 4 as before. It seems reasonable to conclude therefore that while density
currents are of the utmost importance in determining the movement of the drifters
to the shore, they do not provide a complete explanation of the phenomenon: there
must be other processes at work, particularly in the near-shore waters, which also
contribute to the landward drift. In this respect, residual movements of tidal origin
might have a significant part to play (Johns & Dyke 1972). Currents associated with
wave action and meteorological forces must also be present to a greater or lesser
extent at any time, having an influence on bottom conditions. The horizontal excursion
of a sea-bed drifter during a period of tidal flood, followed by beaching, might account
for the final stage of its shoreward translation; an excursion on the flood of something
like 5.4 nautical miles may be expected in Liverpool Bay, assuming an amplitude of
tidal current of 70 cm s-'.
The vertical structure of the density-current field is illustrated in Fig. 8 where
current components u and u are plotted through the depth at position P, the case
b = 4 being considered. The u-component, directed along A , A , and counted positive
when acting towards the coast at A,, exhibits a differential flow-going out from the
land at the sea surface and coming into the land at the sea bottom. The opposing
forces due to surface slope and density gradient produce this situation. Note that the
incoming current reaches a maximum above the bottom. The v-component, geostrophically induced, has its greatest value at the surface and, as depth increases, reduces
I
0.2
O"
0'5P
0.6
0.7
FIG. 8. Vertical profiles of the current components ii and u at position P, 12
nautical miles from the coast on section A 1 A 2 :for b = 4, H = 17.4 fm. Resultant
current vectors through the depth at this position are shown in Fig. 2.
428
N. S. Heap
through a point of inflexion to a near-zero minimum at the bottom. The component
d
acts in the same direction, perpendicular and to the left of A, A 2 , at every level.
From (14) and (16) we obtain the vertical shear:
which, when taken with (19) and knowing the variation of u, may be used to explain
the main features of the v-profile. Combining u and u vectorially yields the resultant
horizontal current at any depth. Resultant currents thus obtained, at eleven equidistant levels q = 0.0 (0.1) 1.0 from the surface to the bottom of the vertical water
0
100
200
300
400
N ( c m 2 s-l)
FIG. 9. Variation of bottom current (V,)with eddy viscosity (N),for various
values of depth ( H ) .
429
Estimation of density currents in the Irish Sea
column at P, are drawn out in Fig. 2. The progressive contra-sole rotation of the
current vector, from being south-easterly at the bottom to northerly at the surface, is
of particular interest and illustrates an important characteristic of the density-current
field.
Figs 9, 10 and 11 show respectively the variations of V,, V, and 0, with N for
different values of H. Curves corresponding to b = 1 , 2 , 4 are marked on each figure.
The numerical data given by (30) still apply. Referring to Fig. 9, for any H it is
apparent that, as N increases, the bottom current rises rapidly to a maximum and then
diminishes slowly under the influence of mounting friction. Also for a fixed H, the
surface current V, suffers progressive reduction as N increases (Fig. 10) and this is
accompanied by a growth in 0, above 90" (Fig. 1I) implying a contra-sole rotation of
S-
=I0
FIG.10. Variation of surface current (V,) with eddy viscosity (N), for various
values of depth ( H ) .
430
N. S. Heaps
170-
160-
&(deg)
150-
140
130
I20
110,
IOC
9c
1
100
I
200
I
300
I
400
FIG.11. Variation in direction of the surface current (0,) with eddy viscosity
for various values of depth ( H ) .
the surface current vector away from the long-shore y-direction towards the open
sea. The figures show that the depth cb-ordinate H is an important parameter in
determining the density currents: the greater the value of H , the greater the current
magnitudes. Realistic values of V,, V, and 0, probably lie within the range 2 < b < 4,
for which 45 < N < 366 cmz s-' on taking 5 < H < 20 fm.
The general character of the mathematical solution much depends on the relative
importance of the geostrophic and frictional effects. When N is small the former
predominates and there tends to be a near geostrophic balance in equation (13)
except in the vicinity of the sea bed where bottom friction exerts a perturbing influence.
The current is then mainly y-directed, large at the surface, decreasing approximately
linearly with increasing depth to a small value at the bottom. On the other hand,
when N is large and the frictional effect predominates over the geostrophic, the
current from surface to bottom is small and primarily directed parallel to the x-axis,
Estimation of density currents in the Irish Sea
43 1
being an essentially differential flow as illustrated by the u-profile in Fig. 8. The
actual situation in Liverpool Bay lies somewhere between the above two extremes, both
internal friction and the Earth's rotation having a significant influence on the current
structure. The frictional effect is accentuated in shallow water ( H small) and the
geostrophic effect in deep water (H large) as indicated by the variation of 0, in Fig. 6.
The assumption that a/ay E 0 gives the hydrodynamical model a significant
characteristic, reflecting an actual measure of long-shore uniformity. Clearly, the
assumption cannot be strictly upheld-if only because in reality the depth variation
is not exactly the same at all off-shore sections and there are departures from the
idealized straight coast. One might expect that the proximity of the curving North
Wales coast will to some extent reduce the long-shore mass flux across A , A , as
determined from the theory, giving rise to a modified circulation. Such effects have
been regarded as of secondary importance here, and this appears to be justified in the
light of the ability of the present theory to account for observed residual motions.
6. Concluding remarks
1. Using a mathematical model, density currents in Liverpool Bay have been
estimated for a range of likely values of friction.
2. The theoretical results have reproduced the main persistent features of the
residual current field in the Bay as deduced from continuous current meter measurements (Ramster 1971) and sea-bed drifter experiments (Halliwell 1973) carried out
over a period of 12 months. In general, the observations indicate the presence of a
bottom drift to the coast and a north going surface drift; both are confirmed by the
theory.
3. Quantitatively, the estimated density currents are able to account in large
measure for movements of sea-bed drifters to the shores of the Bay. However, other
mechanisms must also be present contributing to these movements, particularly in
the shallow near-shore zone. Residuals of tidal origin and motion derived from the
natural sequence of meteorological disturbances might be mentioned in this connection.
4. Both internal friction and the Earth's rotation are significant factors determining
the density-current structure. In this respect, conditions in the Bay are intermediate
in type between those in an estuary where the frictional effect predominates over the
geostrophic, and those in the deep sea where the geostrophic effect predominates over
that of friction.
5. All the available evidence, observational and theoretical, tends to point to the
very considerable importance of density currents in the residual circulation of
Liverpool Bay. A principal characteristic appears to be their persistency over long
periods, which distinguishes them from other transient and perhaps temporarily
larger effects such as those, for example, which might be produced by the wind.
Accepting their importance, it follows that any change in the density field induced by
alteration in the pattern of river discharge might conceivably have a far-reaching
influence on the Bay's overall residual current system.
6. Observations by Dietrich (1953) off Texel Island in the southern North Sea
have indicated the presence there of residual currents directed similarly to those in
Liverpool Bay, with a bottom drift towards the coast and an essentially north-going
surface drift. At the sam'e time, Dietrich found strong horizontal gradients of density
normal to the Dutch coast with the a,-lines running parallel to the shore line, reflecting
further-in perhaps a more intense form-the situation in Liverpool Bay. It seems
likely therefore that the mechanism for density currents proposed here for Liverpool
Bay may also be applicable to a belt of coastal sea off Texel Island, and perhaps to
432
N. S. Heaps
other coastal waters where conditions of high freshwater discharge and tidal mixing
occur together, and a measure of long-shore uniformity exists.
Acknowledgments
I am indebted to Mr J. E. Jones for performing the numerical computations of
this investigation and for his help generally. Mrs A. Burgess takes the credit for preparing the figures for publication. Also it is a pleasure to record my thanks to Professor
A. R. Halliwell for his interest and encouragement, to Professor K. F. Bowden for
comments and a valuable discussion, and to Mr A. J. Lee of the Fisheries Laboratory,
Lowestoft, for drawing my attention to the paper by Dietrich mentioned above.
The work was carried out at the Institute of Coastal Oceanography and Tides, and
I am grateful to the Director for permission to publish the paper.
Institute of Coastal Oceanography and Tides
Birkenhead
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