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A stratified model of flow in a coastal trench
A stratified model of flow in a coastal trench J. A. Johnson School of Mathematics (Received
December
and Physics,
University
of East Anglia, Norwich
NR4
7TJ, UK
1984)
A stratified model of the circulation in a deep, narrow trench adjacent to a coast is described. The flow in the trench is driven by surface wind stress, coastal runoff and inflow at one end. The model is being developed to investigate the Norwegian coastal current flowing through the Norwegian Trench.
Key words: mathematical
The Norwegian Trench is an interesting topographic feature that follows the coastline of Norway from the outflow of the Kattegat through the Skagerrak, and then along the eastern boundary of the North Sea as far as the continental shelf. Its maximum depth in the Skagerrak is 500 m and its average depth is between 250 m and 300 m, compared with a depth of less than 100 m over the rest of the North Sea. The currents in the Norwegian trench are along the coast, with the coast on the right-hand side, flowing out from the Kattegat into the Norwegian Sea. Observations’ suggest that large eddies are common and these drift northward with the current. The water close to the coast is fresher due to the large amounts of runoff from fjords and rivers. A review of theories and observations of the Norwegian coastal current, but not of topography, has been presented by Mark,’ including the effects of variability. He noted that the observed cross circulation appears to be dominated by wind effects. Furnes,3 investigating wind effects in the North Sea, showed that the observed wind-induced fluxes in the Norwegian Trench are strongly coupled to the fluxes in the shallower North Sea and that this coupling is controlled by the topography. A numerical model by Davies and Heaps4 looked at the influence of the Norwegian Trench on the wind-driven circulation in the North Sea. They showed that the presence of the trench greatly affects the circulation in the entire northern part of the North Sea. A theoretical paper by Djurfeldt’ considered divergent topographic waves over a variety of topographic features and applied the theory to two sections of the Norwegian Trench. This paper describes a three-dimensional mathematical model that takes account of trench topography and includes the effect of stratification. The model is driven This paper was originally presented at the JONSMOD held 2-4 July 1984, Bergen, Norway
0307-904X/85/060403-6/$03.00 0 1985 Butterworth & Co. (Publishers)
Ltd
Conference,
model, Norwegian Trench, flow
by a combination of surface wind stress, coastal runoff and throughflow representing the inflow from the Kattegat. Although the model is variable in time, it does not allow the introduction of the time and length scales of eddies, but aims rather to provide a description of the quasi-steady circulation that might serve as a basis for a perturbation eddy analysis.
The model The model consists of a long, deep trench bounded, on one side by a coastline where runoff occurs, and on the other side by a broad shallow shelf. The horizontal axes are chosen so that x’ and y ’ are measured across and along the trench respectively, with primes indicating dimensional variables. The z’-axis is vertically upward from the surface at z’ = 0 which is an acceptable simplification as tidal effects do not significantly affect the circulation in the Norwegian trench. A typical cross-section is illustrated in Figure 1 with the bottom of the sea at z’ = --dH(x’/Z) and the coastline along the trench at x’ = I. Variations with the coordinate y’ along the coast of the trench depth and profile, and in the shape of the coastline, are now under investigation. In the discussion later in the paper it is assumed that y increases in a northerly direction. Three length scales occur in the model: the ‘width’ of the trench I, the ‘depth’ of the trench d, and the ‘length’ of the trench L. The circulation in the model is driven by a combination of imposed surface wind stress rX, rY, by runoff R,&‘, c ‘) along the coast representing outflow from fjords and rivers, and by prescribed throughflow entering the model at a fixed y’ = ub, say, representing the outflow from the Kattegat. The distribution of the circulation is determined by a combination of analytical and numerical methods after the cross-section has been split into an interior region sandwiched between surface and bottom Ekman layers.
Appl.
Math.
Modelling,
1985,
Vol. 9, December
403
Flow in a coastal trench: J. A. Johnson geostrophic terms. This balance exists if T = T1 = L/(v). Baroclinic changes in the density or potential temperature fields occur, however, if T = T2 = L/V so that the et term in equation (5) balances the advection terms. The ratio of T1 to T2 gives T1 = RoT2 and, therefore, the barotropic changes happen on a relatively short time scale, whereas the baroclinic changes occur only slowly on a longer time scale. Typical lengths for these time scales are derived below.
Typical numerical values To relate the model sea to the Norwegian trench requires the following typical values: L = 800km
I= 1OOkm V= 0.3ms-’
g=
v = 10-2mZs-’
10mse2
d = 500m f=
10-4s-’
(YAB= 5 x 1o-4
Using these values, the Ekman and Rossby numbers and the stratification parameter have the following magnitudes: Figure
1
Geometry
Introduce
of model trench
dimensionless
x’ = lx = Lx(I/L)
coordinates y’ = Ly
by setting:
z’ = dz(l/L)
t’ = Tr
vt + u = -Py
pz = se
(2) (3)
u,+vy+w,=o
(4)
(L/(TV))
(5)
Bt + ~0, + ~0, + w0, = 0
where u, v and w are the velocity components corresponding to x, y and z respectively, p is the reduced pressure, and 0 is the potential temperature. I/ is a characteristic horizontal velocity scale, and the stratification parameter S = ga(AB) d/(fVJ) where g is the acceleration due to gravity, (Yis the coefficient of thermal expansion, A0 is a potential temperature scale, and f is the local Coriolis parameter. The simplifications involved in equations (l)-(5) have been described in detail elsewhere6 but may be summarized as follows. All frictional and diffusive effects are confined to the bottom and surface Ekman layers and only affect the interior region through Ekman suction. The thicknesses of the Ekman layers are O(Er”d) where E = v/(fd2) is the vertical Ekman number and v is the vertical eddy viscosity. The hydrostatic balance equation (3) is a very good approximation in the vertical distribution of p. The assumption that the Rossby number (RO= V/(p)) of the flow is small allows the nonlinear terms in the momentum equations to be neglected compared with the geostrophic terms. The flow along the trench, which is the largest velocity component, is in geostrophic balance by equation (1). Two natural time scales appear in equations (2) and (5). Barotropic changes of the velocity field are associated with the balance of the vt term in equation (2) with the
404
Appl. Math. Modelling, 1985, Vol. 9, December
S=O.7
l/L = E114
(1)
v=px
1O-2
so that the Ekman and Rossby numbers are suitably small. Note that E”4 = 0.14 and Z/L = 0.125 have similar sizes. Johnson6 has shown that if the model sea is sufficiently narrow so that l/L G 0(E”4), then the whole sea is affected by the frictional bottom boundary layer. In this paper only the case:
where d, I, and L are the three scales, and T is a characteristic time scale. It has been shown by Johnson6 that, in the interior region, the dominant terms in the dimensionless equations of motion are :
(L/(W))
R,,=
E=4~10-~
(6)
is discussed. The time-scales introduced above have the following typical values. The barotropic scale Tl = L/(Zf) = 0.9 day, indicating that significant barotropic changes of the velocity field may be expected to happen over periods of a few days, with fairly rapid spin-up due to onset of wind. However, the baroclinic scale T2 = L/V = 62 days, which suggests that significant changes to the density or temperature fields are likely to occur much more slowly over periods of months.
The interior region The large difference between the time scales T1 and Tz permits a simplification in the method of solution of equations (l)-(5). In particular if the characteristic time scale T is chosen as T1 so that the rapid spin-up of the velocity field may be dealt with, then equations (l)-(5) may be written: v=p,
vt + u = -py
(7)
pZ = se
(8)
u, + vy + w, = 0
(9)
et +R~(uB,
f vey + ~0,) = 0
(10)
with the last equation demonstrating the slow change in potential temperature 0 by advection with the rapidly changing velocity field. To reflect the difference in time scales for barotropic and baroclinic changes: P = P*(x,.Y, t) +%,~,z,Rot) where p* and I; are the barotropic and baroclinic ponents of pressure respectively. Similarly: u=u*+ri
v=v*+3
com-
Flow in a coastal trench: J. A. Johnson
To allow for the variation of w with depth: w = w*(x,y,
2, t) + G(x,y,
As the temperature clinic time scale:
the bottom Ekman layer that is proportional balance requires:
z,Rot)
P,*+Px=~=(rY+RE)d2
field only changes on the longer baro-
at
to u. This
x=
1
(18)
At the far edge of the shallow shelf a similar balance of transport must be maintained for the offshore flow. This reauires:
e = @x,y, z,R*t)
1
The method of solution consists of separate numerical integrations for the barotronic and baroclinic comuonents taking account of the coupling between the components.
The barotropic component To simplify the bottom boundary
condition:
5 = (l/L) z/H(x) so that the surface is at { = 0 and the bottom is at { = -1. Then the barotropic components of equations (7)-(9) reduce to: v*=p;
v~+u*=-p;
p;=o (11)
H(uZ + vy*)+ w; = 0 from which it may be deduced that: HP&,, = Hz& = w$
and
(12)
Hence w* may be determined from the Ekman suction conditions that match the surface and bottom boundary layers onto the interior region. It has been shown6 that for the case Z/L = E 1’4, the Ekman suction conditions are: w”=rXy
at
{=O at
{=-1, A*= 2(1+xH:)
(14)
+ P& d2 + H, P; = 7:
(15)
This equation has to be solved subject to various boundary and initial conditions. The numerical calculations are commenced at time t = 0 when it is assumed that: p*=O
at
t=Oforallx,y
(16)
so that the initial barotropic fields are zero and the early stages of the calculation represent the spin-up of the barotropic circulation. To deal with the alongshore y derivative in equation (15) the inflow from the Kattegat is prescribed by setting: P* = Pinflow(X, t)
at
ps
components ti = -by
satisfy, from es = SHAH
tiX- (s‘H,/H) is + ;ly + H-‘Gr = 0
(20) (21) (22)
where t^=Ro t. Although the barotropic field is spun-up from p* = 0 at t = 0, a different approach is adopted for the baroclinic field. In order to include stratification in the model sea, it is assumed that the potential temperature 0 = B(x,y,z,Rot)is prescribed at t = 0 and that subsequent changes in ~9due to advection are calculated using equation (22). An integral of equation (20) is:
(23) Hence P is also specified at t = 0 in terms of I? and its subsequent changes are determined from equations (22) and (23). Then equations (20) and (21) give the slow changes in velocity fields li, V, 6.
-z&/2)
Hence the equation for p* may be obtained from equations (11) and (12) as: HP&X,+ H,P:,
The slowly varying baroclinic equations (7) to (9):
-((v*+~)~y-(w*+6)H-‘&=0
where X = EH/E and EH is the horizontal Ekman number. For values typical of the Norwegian Trench h = 0( 10e3) and therefore, in this paper, it is assumed that AZ = 2. The first suction condition, equation (13), is derived from the curl of the applied surface wind stress. The second condition, equation (14), states that the normal velocity into the bottom Ekman layer is proportional to the divergence of the Ekman transport (Au*). Using equations (12)-(14) it may be shown that: w* = (I+ 0 r,’ + c(H,u*
The baroclinic component
e^; = -(u* + ti) {& - (s‘H,/H) if}
(13)
w*+H,u*=(Av*),
x=--1(19)
where the third term represents Hu, the flow in the interior. If the shelf is closed bv a coast at x = -1 at which H = 0 then, clearly, equation (19) will be simplified. Boundary condition equations (18) and (19) both contain feedback from the baroclinic pressure P‘to the barotropic pressure P*. The barotropic circulation is obtained by integrating equation (15) subject to the conditions of equations (16)-(19).
i, = r;, - ({HJH)
w& = 0
at
p~+~x+H(py*+Ijy+p~,)=~y
Y =Yo
(17)
The boundary condition at the Norwegian coast is found by matching the runoff RE(y), and the onshore surface Ekman transport ry, with the downwelling transport in
The numerical solution The barotropic component p* is determined from equation (15) using a semi-implicit method. In equation (15) xderivatives are approximated by central differences whereas y and t derivatives are replaced by forward differences. If p* and P are known at time t, then equation (17) gives p* at time t + At at y =yo. The solution of equation (15) is then calculated at time level t + At at ye + Ay, ye + 2Ay and so on by using the Crank-Nicolson method to find all the x-values across the trench for each y as the solution is stepped along the trench from y = y. to the northern end. When this is completed the calculation resumes at y = y. for time level t + 2At. The boundary conditions given by equations (18) and (19) with the most recent values of P are incorporated in the Crank-Nicolson tridiagonal matrix. At each time step in the calculation the slow change in 0 (and hence in P) is determined from equation (22) using the latest values of u*, v*, w*. The updated values ofP are used in the next time step of the p* calculation.
Appl. Math. Modelling, 1985, Vol. 9, December
405
Flow in a coastal
trench:
J. A. Johnson
Special care has to be taken at the northern outlet end (Y = YN) of the trench and shelf if there happens to be inflow at that end. When the flow is outward at y =YN there is no numerical problem. If, however, the velocity at y = YN is inward with UN < 0 then some decision has to be made about the heat advected southward into the trench and onto the shelf. It is assumed that the trench connects with a large deep sea or ocean at y = YN and that, in this deep sea, changes occur more slowly than in the trench. This assumption is built into the numerical scheme by assuming that, at one grid point beyond y = YN, the temperature is held fixed at the initial distribution (chosen at time t = 0) unless it is changed by (i) outward advection from the trench or (ii) slow sideways mixing. In mathematical terms these assumptions require the temperature ON+r at the grid pointy = YN+i to satisfy: aeN+
-=
ae
at
~
ay
a2
aeN+
where effect lateral p2EH
p2EH
at
a2eN+1
~
a2
b 0.3
O0
0
P
0
C
Figure 2
(a), Initial potential temperature distribution with depth at y= 1 .4 given by equation (24); (b), inflow at y =yO. That is %nput /ax from equation (25); (c), depth of sea bed given by equation (26)
ifUN >0
The topography
of the trench and shelf are defined by
H = (0.5 + 0.6 exp(-20(x
Numerical results
i-f
$+i*anh2(r+0.3) >
>(
(24)
which represents a slightly warmer layer above z = -0.3 with a slightly cooler layer below, and a general cooling with increasing y along the trench. The southern entry to the trench is at ye = 0.8, and the northern exit at YN = 1.5. The inflow at y = y. is: Pinflow=-Ij(X1,Yo,-1,t)+2+ +
f3-ix
tanh(lO(x-0.8))}(1-exp(-t))
(25)
where the exponential t dependence gives a smoother start to the calculation. The p^contribution allows for flow associated with the prescribed density field. The tanh term provides an input jet over the trench topography at Y ‘Yo.
406
Appl.
Math.
-0.8)*)}
x{l-exp(lO(x-1)))
The numerical solution described above has been carried out for a variety of examples with different strengths to the driving mechanisms and different depths of trench. To illustrate the circulations obtained, three particular cases are discussed. Further improvements to the model and other interesting variations of the prescribed boundary conditions are introduced later. In all three examples the step lengths in the numerical calculations were h, = 0.025, h, = 0.05 and h, = 0.01 with 41 grid points across the trench and shelf and 16 grid points along the trench. This distribution allowed for the sharper gradients in the x-direction across the trench compared with the y-direction along the trench. In the vertical the 5 coordinate has step length hf = 0.1 and 11 grid points. More grid points in { would be desirable to permit the inclusion of stronger stratification but this is impractical at present. In each case considered here the density stratification is fairly weak and is prescribed at t = 0 as:
(
Modelling,
1985,
1
z=-/+(x) 4
a.
p1 and p2 are constants. The ccl terms represent the of outward advection and the p2 term allows some mixing. In the numerical calculations pi = 4.0 and = 0.05.
i=
X
I
I
-0.5
ifUN< -=
0
a2eN+1
-=+/.l&
-plR,,vN
8
Vol. 9, December
(26)
which is illustrated in Figure 2 with the distributions shown in equations (24) and (25). The flow is partly driven by the surface wind stress: rx = 0
rY = T tanh(2.5t)
{l-exp(-2(y-yo))} (27)
where r is a constant. They dependence simplifies the calculations at y = y. where the numerical scheme commences for each value of 1. Equation (27) is a simple spin-up wind which spins up the barotropic field by about t = 2. The results presented below are after 600 time steps when t = 6 at a time when the barotropic field is well developed but the baroclinic field is still changing. The other driving mechanism is the run-off at the coast. This is prescribed as: RE(~)
= R tanh2t
at
x = 1
(28)
where R is a constant. This simple distribution represents uniform runoff along the coast after a short spin-up time. More interesting distributions of RE are discussed in the final section of this paper. Case A (7 = -0.35, R = 0.2) The first example presented in Figure 3 has a wind along the trench from a northerly direction which produces an Ekman transport in the surface boundary layer away from the Norwegian coast. The runoff from the coast provides the majority of this Ekman transport with very little needed from upwelling along the coast. Most of the flow in the bottom Ekman layer is away from the coast, in association with the longshore velocity u*, towards the north. All this requires a large return flow in the interior towards the coast and, hence, a large positive u*. Near the coast the alongshore flow is just southward to fit equation (18). This requirement, however, depends on
Flow in a coastal
trench: J. A. Johnson
coastal region and offshore flow in the interior over the trench. The convergence of the interior flow near the edge of the trench is associated with the increased northward flow with increasing y as the extra runoff is absorbed into the longshore current.
Further developments
a
0
x
1
’ d
\
Figure 3 Case A (7 = -0.35, R = 0.2). (a) shows contours for px and B at time t = 6; (bj section of potential temperature s^; (cl, diagrammatic representation of cross-trench flow; (d) distribution of u*, v* across trench, all at y = 1.4
the relative strengths of the wind induced Ekman transport and the runoff, both chosen arbitrarily here. The predominantly onshore flow in the interior causes the contours of p* (which are essentially streamlines for the horizontal interior motion) to lean towards the coast after the initial spreading out near y = yo. This spreading of the p* contours represents the adjustment of the imposed inflow to the geometry and dynamics of the model. There is a tendency for warmer water to be carried along the trench resulting in the isotherm distribution illustrated in Figure 3. Increased throughflow and more runoff would both enhance this effect and tend to cause the isotherms to line up more along the coast. This is discussed further in a later section. Case B (T = 0.7, R = 0.2) This example, illustrated in Figure 4, has the wind direction reversed and its magnitude strengthened compared with case A. Hence the direction of the surface Ekman transport is reversed and the coastal runoff has to enter the bottom Ekman layer and begin to descend. A more refined model (see later) would have the runoff RE at a lower density than the ambient density of the coastal water and then the runoff would collect in the corner of the cross-section rather than descending so far. The onshore U* is much reduced compared to case A as the cross-trench flow can be largely completed by the bottom boundary layer transport. u* is hardly changed compared with case A as it is mainly determined by the inflow at y = y. and the coastal runoff, except near the coast where the boundary condition (equation (18)) includes rY. The streamlines are more nearly parallel to the coast due to the reduction in u*. CaseC(T= 1.05,R=0.5) In this last example shown in Figure 5, both the wind stress and the runoff are increased compared with case B, but the inflow at y = y. remains unchanged. The enhanced surface Ekman transport and runoff combine to force increased downwelling at the coast, in excess of that needed in the bottom boundary layer to support the longshore flow u*. Hence there is input to the interior from the
As has been mentioned in preceding sections there are a number of ways in which this model should be improved to provide a better description of the conditions found in the Norwegian Trench. Clearly, changes to the coastline by including a curved coastal boundary, but retaining the trench hugging the Norwegian coast by having a broadening of the shelf with latitude, are desirable. Topographic maps of the trench suggest that inclusion of changes in depth along the trench may be interesting. These geometrical alterations are relatively easy to make to the model and are in progress at present.
RE ---------
I I
L 1
a
0
x
Figure 4
Case 6 (7 =0.7,
R =0.2).
(a), (b), (cl, (d) as in Figure3
,,;-----
b
6
a Figure 5
Math.
0.1
i
x
Case C (7 = 1.05,
Appl.
v
R =0.5).
Modelling,
(a), (b), (cl, (d) as in Figure 3
1985.
Vol. 9, December
407
Flow in a coastal trench: J. A. Johnson
Another modification, that is relatively straightforward, is the use of more realistic time-varying and spatially varying winds. Such winds are expected to generate both shelf waves along the coast and trench waves along the offshore edge of the trench. A more significant improvement and one that requires a lot more thought is concerned with the modelling of the density stratification and the runoff of fresh water along the coast. The use of a more intense density stratification (including perhaps a pycnocline) would be an advantage but needs many more grid points in the vertical to resolve it satisfactorily. The runoff Rn needs to be improved in two ways. Firstly by making the input more discrete at particular points along the coast representing outflow from specific fjords and rivers rather than a continuous distribution along the coast. More difficult, and probably more important, is to allow the runoff to bring in water of its own density rather than assuming it has the ambient local density. Inclusion of this effect will tend to produce a fresh surface layer near the coast, as indicated in Mark,’ and once again the grid point limit in the vertical is a problem. In this paper an attempt has been made to show how a particular mathematical model can be applied to a small part of the North Sea, and to indicate where further modelling effort is required to provide a more realistic
408
Appl.
Math.
Modelling,
1985,
Vol. 9, December
model of the stratified circulation Trench.
of the Norwegian
Acknowledgements The final version of this paper was prepared after collaboration with Professor Gjevik of Oslo University sponsored by NATO Research Grant No. 453/84.
References McClimans, T. A. and Nilsen, .I. H. ‘Whirls in the Norwegian coastal current’, Proc. Norw. Coastal Current Symp., Geilo, Norway, Bergen Univ. Press, 1981 Mork, M. ‘Circulation phenomena and frontal dynamics of the Norwegian coastal current’, Phil. Trans. R. Sot. Land. 1981, A302,635 Furnes, G. K. ‘Wind effects in the North Sea’, J. Whys. Oceanogr. 1980,10, 978 Davies, A. M. and Heaps, N. S. ‘Influence of the Norwegian Trench on the wind-driven circulation of the North Sea’, Tellus 1980,32,164 Djurfeldt, L. ‘A unified derivation of divergent second-class topographic waves’, Tellus 1984,36A, 306 Johnson, J. A. ‘A stratified model of a partially enclosed narrow sea’, Geophys. Astrophys. Fluid Dynamics (to appear), 1985
December
and Physics,
University
of East Anglia, Norwich
NR4
7TJ, UK
1984)
A stratified model of the circulation in a deep, narrow trench adjacent to a coast is described. The flow in the trench is driven by surface wind stress, coastal runoff and inflow at one end. The model is being developed to investigate the Norwegian coastal current flowing through the Norwegian Trench.
Key words: mathematical
The Norwegian Trench is an interesting topographic feature that follows the coastline of Norway from the outflow of the Kattegat through the Skagerrak, and then along the eastern boundary of the North Sea as far as the continental shelf. Its maximum depth in the Skagerrak is 500 m and its average depth is between 250 m and 300 m, compared with a depth of less than 100 m over the rest of the North Sea. The currents in the Norwegian trench are along the coast, with the coast on the right-hand side, flowing out from the Kattegat into the Norwegian Sea. Observations’ suggest that large eddies are common and these drift northward with the current. The water close to the coast is fresher due to the large amounts of runoff from fjords and rivers. A review of theories and observations of the Norwegian coastal current, but not of topography, has been presented by Mark,’ including the effects of variability. He noted that the observed cross circulation appears to be dominated by wind effects. Furnes,3 investigating wind effects in the North Sea, showed that the observed wind-induced fluxes in the Norwegian Trench are strongly coupled to the fluxes in the shallower North Sea and that this coupling is controlled by the topography. A numerical model by Davies and Heaps4 looked at the influence of the Norwegian Trench on the wind-driven circulation in the North Sea. They showed that the presence of the trench greatly affects the circulation in the entire northern part of the North Sea. A theoretical paper by Djurfeldt’ considered divergent topographic waves over a variety of topographic features and applied the theory to two sections of the Norwegian Trench. This paper describes a three-dimensional mathematical model that takes account of trench topography and includes the effect of stratification. The model is driven This paper was originally presented at the JONSMOD held 2-4 July 1984, Bergen, Norway
0307-904X/85/060403-6/$03.00 0 1985 Butterworth & Co. (Publishers)
Ltd
Conference,
model, Norwegian Trench, flow
by a combination of surface wind stress, coastal runoff and throughflow representing the inflow from the Kattegat. Although the model is variable in time, it does not allow the introduction of the time and length scales of eddies, but aims rather to provide a description of the quasi-steady circulation that might serve as a basis for a perturbation eddy analysis.
The model The model consists of a long, deep trench bounded, on one side by a coastline where runoff occurs, and on the other side by a broad shallow shelf. The horizontal axes are chosen so that x’ and y ’ are measured across and along the trench respectively, with primes indicating dimensional variables. The z’-axis is vertically upward from the surface at z’ = 0 which is an acceptable simplification as tidal effects do not significantly affect the circulation in the Norwegian trench. A typical cross-section is illustrated in Figure 1 with the bottom of the sea at z’ = --dH(x’/Z) and the coastline along the trench at x’ = I. Variations with the coordinate y’ along the coast of the trench depth and profile, and in the shape of the coastline, are now under investigation. In the discussion later in the paper it is assumed that y increases in a northerly direction. Three length scales occur in the model: the ‘width’ of the trench I, the ‘depth’ of the trench d, and the ‘length’ of the trench L. The circulation in the model is driven by a combination of imposed surface wind stress rX, rY, by runoff R,&‘, c ‘) along the coast representing outflow from fjords and rivers, and by prescribed throughflow entering the model at a fixed y’ = ub, say, representing the outflow from the Kattegat. The distribution of the circulation is determined by a combination of analytical and numerical methods after the cross-section has been split into an interior region sandwiched between surface and bottom Ekman layers.
Appl.
Math.
Modelling,
1985,
Vol. 9, December
403
Flow in a coastal trench: J. A. Johnson geostrophic terms. This balance exists if T = T1 = L/(v). Baroclinic changes in the density or potential temperature fields occur, however, if T = T2 = L/V so that the et term in equation (5) balances the advection terms. The ratio of T1 to T2 gives T1 = RoT2 and, therefore, the barotropic changes happen on a relatively short time scale, whereas the baroclinic changes occur only slowly on a longer time scale. Typical lengths for these time scales are derived below.
Typical numerical values To relate the model sea to the Norwegian trench requires the following typical values: L = 800km
I= 1OOkm V= 0.3ms-’
g=
v = 10-2mZs-’
10mse2
d = 500m f=
10-4s-’
(YAB= 5 x 1o-4
Using these values, the Ekman and Rossby numbers and the stratification parameter have the following magnitudes: Figure
1
Geometry
Introduce
of model trench
dimensionless
x’ = lx = Lx(I/L)
coordinates y’ = Ly
by setting:
z’ = dz(l/L)
t’ = Tr
vt + u = -Py
pz = se
(2) (3)
u,+vy+w,=o
(4)
(L/(TV))
(5)
Bt + ~0, + ~0, + w0, = 0
where u, v and w are the velocity components corresponding to x, y and z respectively, p is the reduced pressure, and 0 is the potential temperature. I/ is a characteristic horizontal velocity scale, and the stratification parameter S = ga(AB) d/(fVJ) where g is the acceleration due to gravity, (Yis the coefficient of thermal expansion, A0 is a potential temperature scale, and f is the local Coriolis parameter. The simplifications involved in equations (l)-(5) have been described in detail elsewhere6 but may be summarized as follows. All frictional and diffusive effects are confined to the bottom and surface Ekman layers and only affect the interior region through Ekman suction. The thicknesses of the Ekman layers are O(Er”d) where E = v/(fd2) is the vertical Ekman number and v is the vertical eddy viscosity. The hydrostatic balance equation (3) is a very good approximation in the vertical distribution of p. The assumption that the Rossby number (RO= V/(p)) of the flow is small allows the nonlinear terms in the momentum equations to be neglected compared with the geostrophic terms. The flow along the trench, which is the largest velocity component, is in geostrophic balance by equation (1). Two natural time scales appear in equations (2) and (5). Barotropic changes of the velocity field are associated with the balance of the vt term in equation (2) with the
404
Appl. Math. Modelling, 1985, Vol. 9, December
S=O.7
l/L = E114
(1)
v=px
1O-2
so that the Ekman and Rossby numbers are suitably small. Note that E”4 = 0.14 and Z/L = 0.125 have similar sizes. Johnson6 has shown that if the model sea is sufficiently narrow so that l/L G 0(E”4), then the whole sea is affected by the frictional bottom boundary layer. In this paper only the case:
where d, I, and L are the three scales, and T is a characteristic time scale. It has been shown by Johnson6 that, in the interior region, the dominant terms in the dimensionless equations of motion are :
(L/(W))
R,,=
E=4~10-~
(6)
is discussed. The time-scales introduced above have the following typical values. The barotropic scale Tl = L/(Zf) = 0.9 day, indicating that significant barotropic changes of the velocity field may be expected to happen over periods of a few days, with fairly rapid spin-up due to onset of wind. However, the baroclinic scale T2 = L/V = 62 days, which suggests that significant changes to the density or temperature fields are likely to occur much more slowly over periods of months.
The interior region The large difference between the time scales T1 and Tz permits a simplification in the method of solution of equations (l)-(5). In particular if the characteristic time scale T is chosen as T1 so that the rapid spin-up of the velocity field may be dealt with, then equations (l)-(5) may be written: v=p,
vt + u = -py
(7)
pZ = se
(8)
u, + vy + w, = 0
(9)
et +R~(uB,
f vey + ~0,) = 0
(10)
with the last equation demonstrating the slow change in potential temperature 0 by advection with the rapidly changing velocity field. To reflect the difference in time scales for barotropic and baroclinic changes: P = P*(x,.Y, t) +%,~,z,Rot) where p* and I; are the barotropic and baroclinic ponents of pressure respectively. Similarly: u=u*+ri
v=v*+3
com-
Flow in a coastal trench: J. A. Johnson
To allow for the variation of w with depth: w = w*(x,y,
2, t) + G(x,y,
As the temperature clinic time scale:
the bottom Ekman layer that is proportional balance requires:
z,Rot)
P,*+Px=~=(rY+RE)d2
field only changes on the longer baro-
at
to u. This
x=
1
(18)
At the far edge of the shallow shelf a similar balance of transport must be maintained for the offshore flow. This reauires:
e = @x,y, z,R*t)
1
The method of solution consists of separate numerical integrations for the barotronic and baroclinic comuonents taking account of the coupling between the components.
The barotropic component To simplify the bottom boundary
condition:
5 = (l/L) z/H(x) so that the surface is at { = 0 and the bottom is at { = -1. Then the barotropic components of equations (7)-(9) reduce to: v*=p;
v~+u*=-p;
p;=o (11)
H(uZ + vy*)+ w; = 0 from which it may be deduced that: HP&,, = Hz& = w$
and
(12)
Hence w* may be determined from the Ekman suction conditions that match the surface and bottom boundary layers onto the interior region. It has been shown6 that for the case Z/L = E 1’4, the Ekman suction conditions are: w”=rXy
at
{=O at
{=-1, A*= 2(1+xH:)
(14)
+ P& d2 + H, P; = 7:
(15)
This equation has to be solved subject to various boundary and initial conditions. The numerical calculations are commenced at time t = 0 when it is assumed that: p*=O
at
t=Oforallx,y
(16)
so that the initial barotropic fields are zero and the early stages of the calculation represent the spin-up of the barotropic circulation. To deal with the alongshore y derivative in equation (15) the inflow from the Kattegat is prescribed by setting: P* = Pinflow(X, t)
at
ps
components ti = -by
satisfy, from es = SHAH
tiX- (s‘H,/H) is + ;ly + H-‘Gr = 0
(20) (21) (22)
where t^=Ro t. Although the barotropic field is spun-up from p* = 0 at t = 0, a different approach is adopted for the baroclinic field. In order to include stratification in the model sea, it is assumed that the potential temperature 0 = B(x,y,z,Rot)is prescribed at t = 0 and that subsequent changes in ~9due to advection are calculated using equation (22). An integral of equation (20) is:
(23) Hence P is also specified at t = 0 in terms of I? and its subsequent changes are determined from equations (22) and (23). Then equations (20) and (21) give the slow changes in velocity fields li, V, 6.
-z&/2)
Hence the equation for p* may be obtained from equations (11) and (12) as: HP&X,+ H,P:,
The slowly varying baroclinic equations (7) to (9):
-((v*+~)~y-(w*+6)H-‘&=0
where X = EH/E and EH is the horizontal Ekman number. For values typical of the Norwegian Trench h = 0( 10e3) and therefore, in this paper, it is assumed that AZ = 2. The first suction condition, equation (13), is derived from the curl of the applied surface wind stress. The second condition, equation (14), states that the normal velocity into the bottom Ekman layer is proportional to the divergence of the Ekman transport (Au*). Using equations (12)-(14) it may be shown that: w* = (I+ 0 r,’ + c(H,u*
The baroclinic component
e^; = -(u* + ti) {& - (s‘H,/H) if}
(13)
w*+H,u*=(Av*),
x=--1(19)
where the third term represents Hu, the flow in the interior. If the shelf is closed bv a coast at x = -1 at which H = 0 then, clearly, equation (19) will be simplified. Boundary condition equations (18) and (19) both contain feedback from the baroclinic pressure P‘to the barotropic pressure P*. The barotropic circulation is obtained by integrating equation (15) subject to the conditions of equations (16)-(19).
i, = r;, - ({HJH)
w& = 0
at
p~+~x+H(py*+Ijy+p~,)=~y
Y =Yo
(17)
The boundary condition at the Norwegian coast is found by matching the runoff RE(y), and the onshore surface Ekman transport ry, with the downwelling transport in
The numerical solution The barotropic component p* is determined from equation (15) using a semi-implicit method. In equation (15) xderivatives are approximated by central differences whereas y and t derivatives are replaced by forward differences. If p* and P are known at time t, then equation (17) gives p* at time t + At at y =yo. The solution of equation (15) is then calculated at time level t + At at ye + Ay, ye + 2Ay and so on by using the Crank-Nicolson method to find all the x-values across the trench for each y as the solution is stepped along the trench from y = y. to the northern end. When this is completed the calculation resumes at y = y. for time level t + 2At. The boundary conditions given by equations (18) and (19) with the most recent values of P are incorporated in the Crank-Nicolson tridiagonal matrix. At each time step in the calculation the slow change in 0 (and hence in P) is determined from equation (22) using the latest values of u*, v*, w*. The updated values ofP are used in the next time step of the p* calculation.
Appl. Math. Modelling, 1985, Vol. 9, December
405
Flow in a coastal
trench:
J. A. Johnson
Special care has to be taken at the northern outlet end (Y = YN) of the trench and shelf if there happens to be inflow at that end. When the flow is outward at y =YN there is no numerical problem. If, however, the velocity at y = YN is inward with UN < 0 then some decision has to be made about the heat advected southward into the trench and onto the shelf. It is assumed that the trench connects with a large deep sea or ocean at y = YN and that, in this deep sea, changes occur more slowly than in the trench. This assumption is built into the numerical scheme by assuming that, at one grid point beyond y = YN, the temperature is held fixed at the initial distribution (chosen at time t = 0) unless it is changed by (i) outward advection from the trench or (ii) slow sideways mixing. In mathematical terms these assumptions require the temperature ON+r at the grid pointy = YN+i to satisfy: aeN+
-=
ae
at
~
ay
a2
aeN+
where effect lateral p2EH
p2EH
at
a2eN+1
~
a2
b 0.3
O0
0
P
0
C
Figure 2
(a), Initial potential temperature distribution with depth at y= 1 .4 given by equation (24); (b), inflow at y =yO. That is %nput /ax from equation (25); (c), depth of sea bed given by equation (26)
ifUN >0
The topography
of the trench and shelf are defined by
H = (0.5 + 0.6 exp(-20(x
Numerical results
i-f
$+i*anh2(r+0.3) >
>(
(24)
which represents a slightly warmer layer above z = -0.3 with a slightly cooler layer below, and a general cooling with increasing y along the trench. The southern entry to the trench is at ye = 0.8, and the northern exit at YN = 1.5. The inflow at y = y. is: Pinflow=-Ij(X1,Yo,-1,t)+2+ +
f3-ix
tanh(lO(x-0.8))}(1-exp(-t))
(25)
where the exponential t dependence gives a smoother start to the calculation. The p^contribution allows for flow associated with the prescribed density field. The tanh term provides an input jet over the trench topography at Y ‘Yo.
406
Appl.
Math.
-0.8)*)}
x{l-exp(lO(x-1)))
The numerical solution described above has been carried out for a variety of examples with different strengths to the driving mechanisms and different depths of trench. To illustrate the circulations obtained, three particular cases are discussed. Further improvements to the model and other interesting variations of the prescribed boundary conditions are introduced later. In all three examples the step lengths in the numerical calculations were h, = 0.025, h, = 0.05 and h, = 0.01 with 41 grid points across the trench and shelf and 16 grid points along the trench. This distribution allowed for the sharper gradients in the x-direction across the trench compared with the y-direction along the trench. In the vertical the 5 coordinate has step length hf = 0.1 and 11 grid points. More grid points in { would be desirable to permit the inclusion of stronger stratification but this is impractical at present. In each case considered here the density stratification is fairly weak and is prescribed at t = 0 as:
(
Modelling,
1985,
1
z=-/+(x) 4
a.
p1 and p2 are constants. The ccl terms represent the of outward advection and the p2 term allows some mixing. In the numerical calculations pi = 4.0 and = 0.05.
i=
X
I
I
-0.5
ifUN< -=
0
a2eN+1
-=+/.l&
-plR,,vN
8
Vol. 9, December
(26)
which is illustrated in Figure 2 with the distributions shown in equations (24) and (25). The flow is partly driven by the surface wind stress: rx = 0
rY = T tanh(2.5t)
{l-exp(-2(y-yo))} (27)
where r is a constant. They dependence simplifies the calculations at y = y. where the numerical scheme commences for each value of 1. Equation (27) is a simple spin-up wind which spins up the barotropic field by about t = 2. The results presented below are after 600 time steps when t = 6 at a time when the barotropic field is well developed but the baroclinic field is still changing. The other driving mechanism is the run-off at the coast. This is prescribed as: RE(~)
= R tanh2t
at
x = 1
(28)
where R is a constant. This simple distribution represents uniform runoff along the coast after a short spin-up time. More interesting distributions of RE are discussed in the final section of this paper. Case A (7 = -0.35, R = 0.2) The first example presented in Figure 3 has a wind along the trench from a northerly direction which produces an Ekman transport in the surface boundary layer away from the Norwegian coast. The runoff from the coast provides the majority of this Ekman transport with very little needed from upwelling along the coast. Most of the flow in the bottom Ekman layer is away from the coast, in association with the longshore velocity u*, towards the north. All this requires a large return flow in the interior towards the coast and, hence, a large positive u*. Near the coast the alongshore flow is just southward to fit equation (18). This requirement, however, depends on
Flow in a coastal
trench: J. A. Johnson
coastal region and offshore flow in the interior over the trench. The convergence of the interior flow near the edge of the trench is associated with the increased northward flow with increasing y as the extra runoff is absorbed into the longshore current.
Further developments
a
0
x
1
’ d
\
Figure 3 Case A (7 = -0.35, R = 0.2). (a) shows contours for px and B at time t = 6; (bj section of potential temperature s^; (cl, diagrammatic representation of cross-trench flow; (d) distribution of u*, v* across trench, all at y = 1.4
the relative strengths of the wind induced Ekman transport and the runoff, both chosen arbitrarily here. The predominantly onshore flow in the interior causes the contours of p* (which are essentially streamlines for the horizontal interior motion) to lean towards the coast after the initial spreading out near y = yo. This spreading of the p* contours represents the adjustment of the imposed inflow to the geometry and dynamics of the model. There is a tendency for warmer water to be carried along the trench resulting in the isotherm distribution illustrated in Figure 3. Increased throughflow and more runoff would both enhance this effect and tend to cause the isotherms to line up more along the coast. This is discussed further in a later section. Case B (T = 0.7, R = 0.2) This example, illustrated in Figure 4, has the wind direction reversed and its magnitude strengthened compared with case A. Hence the direction of the surface Ekman transport is reversed and the coastal runoff has to enter the bottom Ekman layer and begin to descend. A more refined model (see later) would have the runoff RE at a lower density than the ambient density of the coastal water and then the runoff would collect in the corner of the cross-section rather than descending so far. The onshore U* is much reduced compared to case A as the cross-trench flow can be largely completed by the bottom boundary layer transport. u* is hardly changed compared with case A as it is mainly determined by the inflow at y = y. and the coastal runoff, except near the coast where the boundary condition (equation (18)) includes rY. The streamlines are more nearly parallel to the coast due to the reduction in u*. CaseC(T= 1.05,R=0.5) In this last example shown in Figure 5, both the wind stress and the runoff are increased compared with case B, but the inflow at y = y. remains unchanged. The enhanced surface Ekman transport and runoff combine to force increased downwelling at the coast, in excess of that needed in the bottom boundary layer to support the longshore flow u*. Hence there is input to the interior from the
As has been mentioned in preceding sections there are a number of ways in which this model should be improved to provide a better description of the conditions found in the Norwegian Trench. Clearly, changes to the coastline by including a curved coastal boundary, but retaining the trench hugging the Norwegian coast by having a broadening of the shelf with latitude, are desirable. Topographic maps of the trench suggest that inclusion of changes in depth along the trench may be interesting. These geometrical alterations are relatively easy to make to the model and are in progress at present.
RE ---------
I I
L 1
a
0
x
Figure 4
Case 6 (7 =0.7,
R =0.2).
(a), (b), (cl, (d) as in Figure3
,,;-----
b
6
a Figure 5
Math.
0.1
i
x
Case C (7 = 1.05,
Appl.
v
R =0.5).
Modelling,
(a), (b), (cl, (d) as in Figure 3
1985.
Vol. 9, December
407
Flow in a coastal trench: J. A. Johnson
Another modification, that is relatively straightforward, is the use of more realistic time-varying and spatially varying winds. Such winds are expected to generate both shelf waves along the coast and trench waves along the offshore edge of the trench. A more significant improvement and one that requires a lot more thought is concerned with the modelling of the density stratification and the runoff of fresh water along the coast. The use of a more intense density stratification (including perhaps a pycnocline) would be an advantage but needs many more grid points in the vertical to resolve it satisfactorily. The runoff Rn needs to be improved in two ways. Firstly by making the input more discrete at particular points along the coast representing outflow from specific fjords and rivers rather than a continuous distribution along the coast. More difficult, and probably more important, is to allow the runoff to bring in water of its own density rather than assuming it has the ambient local density. Inclusion of this effect will tend to produce a fresh surface layer near the coast, as indicated in Mark,’ and once again the grid point limit in the vertical is a problem. In this paper an attempt has been made to show how a particular mathematical model can be applied to a small part of the North Sea, and to indicate where further modelling effort is required to provide a more realistic
408
Appl.
Math.
Modelling,
1985,
Vol. 9, December
model of the stratified circulation Trench.
of the Norwegian
Acknowledgements The final version of this paper was prepared after collaboration with Professor Gjevik of Oslo University sponsored by NATO Research Grant No. 453/84.
References McClimans, T. A. and Nilsen, .I. H. ‘Whirls in the Norwegian coastal current’, Proc. Norw. Coastal Current Symp., Geilo, Norway, Bergen Univ. Press, 1981 Mork, M. ‘Circulation phenomena and frontal dynamics of the Norwegian coastal current’, Phil. Trans. R. Sot. Land. 1981, A302,635 Furnes, G. K. ‘Wind effects in the North Sea’, J. Whys. Oceanogr. 1980,10, 978 Davies, A. M. and Heaps, N. S. ‘Influence of the Norwegian Trench on the wind-driven circulation of the North Sea’, Tellus 1980,32,164 Djurfeldt, L. ‘A unified derivation of divergent second-class topographic waves’, Tellus 1984,36A, 306 Johnson, J. A. ‘A stratified model of a partially enclosed narrow sea’, Geophys. Astrophys. Fluid Dynamics (to appear), 1985