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Pergamon Continental Shelf Research. Vol. 14, No. 15, pp. 1701-1721, 1994

Copyright (~) 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0278-4343/94 $7.00 + 0.00

Upwelling in broad fjords

BENOIT CUSHMAN-RoISIN,* LARS A S P L I N t and HARALD SVENDSENt

(Received 17 August 1992; accepted 26 January 1993)

Abstraet - -A broad fjord is defined here as a stratified fjord whose width exceeds the first baroclinic radius defined from the stratification. When a longitudinal wind blows along such a fjord, the response entails upwelling on one side and downwelling on the other. The physics are identical to those of the classical theory of coastal upwelling, except that here upwelling and downwelling are simultaneously present and interfere with each other.

Our solution is based on the impulse method suggested by CSANADY (1977, Journal of Geophysical research, 82,397-419) and assumes a two-layer stratification. Limiting cases of infinite lower-layer depth (toward one layer) and/or infinitely wide fjord (toward one coast) are explored to appreciate the dependency of the solution on the parameters it involves. The general conclusion is that the shallower and the narrower the fjord, the weaker the upwelling~lownwelling responsc. However, when the wind impulse is large, this is true only for upwelling.

Relevance to Porsangerfjord in northern Norway and comparison with a numerical model demonstrate the applicability of the model to observed events.

1. INTRODUCTION

DUE to their geological origin, fjords in Norway and elsewhere are generally deep and narrow waterways flanked by mountains on each side. As a result, when the wind blows, it tends to be channelled along the fjord, either toward the sea or toward the fjord's head. In a narrow fjord, i.e. a fjord sufficiently thin to preclude substantial variations from side to side, the response is t ime-dependent and entails an exchange of waters with the sea (or outer fjord, if this is the case). Such flushing may be balanced by a countercurrent at depth. By contrast, if the fjord's width largely exceeds the (baroclinic) radius of deformation corresponding to its stratification, Coriolis effects lead to upwelling and downwelling responses, which are contained within a band on the order of the deformation radius along each side, and each side behaves as a separate coastal ocean. In the intermediate case of a fjord of width on the order of the deformation radius, the situation involves both Coriolis and wall-to-wall effects. Porsangerfjord, for example, falls in this interesting category.

Porsangerfjord is a broad, nearly rectangular fjord extending in a southwest-northeast direction on the northern tip of Norway facing the Barents Sea (Fig. 1). Its latitude is 70.5°N (f = 1.37 x 10 -4 s - l ) , its mean depth H = 200 m and its mean width L = 18 km. During the summer (SVENDSEN, 1991), the stratification can be approximated to a

*Thayer School of Engineering, Dartmouth College, Hanover, NH 03755-8000, U.S.A. tGeophysical Institute, University of Bergen, Bergen, Norway.

1701

1702 B. CUSnMAN-ROISIN et al.

homogeneous upper layer of density Pl = 1022 kg m-3 and depth H l = 50 m, capping the deeper fjord waters of average density P2 = 1024 kg m -3. From these values, we estimate the lower-layer depth/42 = H - H1 = 150 m, the reduced gravity g' = g(P2 - Pl)/P2 = 0.02 m s -z, and the baroclinic radius of deformation R = ( g ' H l H J H ) V 2 / f = 6.3 km. Thus, the fjord width is approximately three times the deformation radius, placing it in the intermediate category where Coriolis and wall-to-wall effects are equally important.

The satellite image of this fjord taken on 12 August 1990 (right panel of Fig. 1) shows a peculiar surface temperature distribution: the surface water along most of the fjord is noticeably warmer on the right than on the left, especially at mid-fjord. Thermal forcing is an unlikely cause since solar radiation must be quite uniform over a body of water of this size. Also, the right side is not significantly shallower than the left side, and there is no major cold or warm river runoff on either side. A more plausible explanation is that the warmer surface water has been moved to the right, leaving colder water from below exposed on the left. A wind event could be responsible for such modification.

Analysis of weather charts over the preceding days reveals a cyclonic wind field driven by a low pressure over the Barents Sea, north of Porsangerfjord. On the morning of 10 August, a southwesterly wind was blowing over northern Norway with speeds of around 15 knots (7.7 m s- l ) . In the early afternoon, a cold front passed and the southwesterly winds intensified to a sustained speed of 30 knots (15.4 m s -1) for a few hours (Fig. 2). Over Porsangerfjord, these winds were aligned with the fjord, blowing seaward. Over the next 2 days, 11 and 12 August, winds subsided and the weather was calm. A cloud-free sky on 12 August allowed the satellite image to reveal the surface temperature distribution.

It can be reasonably inferred that the 30-knot seaward wind over the fjord on 10 August was responsible for an Ekman drift from left to right, thus displacing the warm surface waters to the right and creating upwelling on the left and downwelling on the right. Geostrophic adjustment over the course of the following day would have then led to the generation of long-fjord geostrophic currents, which would have maintained the situation in equilibrium over the third day when the satellite image was taken. This hypothesis prompts us to investigate the coupled upwelling-downwelling process in a broad fjord.

2. THE WIND IMPULSE

The theory of coastal upwelling (e.g. GILL, 1982, pp. 403--408) teaches us that as the offshore Ekman drift depletes the upper layer in the vicinity of the coast, a low pressure is formed which gradually becomes equilibrated by a longshore geostrophic current (in the same direction as the wind). This tendency toward geostrophic equilibrium continues as long as the wind is blowing; once the wind ceases, equilibrium can be reached, and the geostrophically equilibrated current can persist quite a long time (PEVVLEY and O'BRIEN, 1976). Although the complete temporal development of the phenomenon can only be modeled numerically (O'BR~EN and HURLBtJRT, 1972, for example), it is possible to derive the properties of the post-wind equilibrium by a relatively simple analytical theory. Indeed, in a study of upwelling in Lake Ontario, CSANADV (1977) introduced the concept of the wind impulse and showed how, with the aid of this concept, properties of the final upwelled state can be derived solely from those of the initial state. He also established a simple criterion telling whether the upwelling will cause the thermocline to surface or not and, if it does, how far from the coast the outcrop will be situated. Later, CUSHMAN-RoISIN (1985), as an aside to a study of interface surfacing, applied the concept again to coastal

Upwelling in broad fjords 1703

N

/ , " August 12. 1990

Fig. l. Left panel: Map of northern Norway showing the location of Porsangerfjord. Right panel: Satellite infrared image showing the surface temperature distribution on 12 August 199(I. The left- to-right temperature contrast at mid-fjord is presumed to be the result of an upwelling-

downwelling cvent.


7 1 ~

7 0 °

2 2 ° 2 4 ° 2 6 ° E 2 2 ° 2 4 ~ 1007 2 6 ~ E , I , z , - ~ / I I I I i

A 7 / ' °

P <: ?: i !

J ' L ~ 1 0 1 1 ~ m ~ , 08.10 1 2 G N 08~10 0 0 , G M T - - I 24 ~

2 2 '~ 2 4 > 2 6 2 2 2 6 "

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o .r, , ' " < % ° < Y' ~'J~' .." "7- . r

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! 08.1170_GMT z 22 ' : 2 4 ~

1012

26 ~

7 1 =

70-

N

2 2 : 2 4 :~

- ~ p., 7 1 c' , \ ?

L,~, , , : - ~ l :7 T--.T .,I..I. r

2 2 ~ 2zl-'

2 6 ~ E

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i

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I~ L ,

26 ~

7 1 °

~ 70"

N

Fig. 2. W e a t h e r m a p s s h o w i n g the e v o l u t i o n o f w e a t h e r p a t t e r n s a n d w i n d s ove r n o r t h e r n

N o r w a y d u r i n g 10 -12 A u g u s t 1990. N o t e the 3 0 - k n o t w ind b l o w i n g s e a w a r d o v e r P o r s a n g e r f j o r d on

10 A u g u s t , 12:00 G M T .

upwelling and showed how the assumption of a rigid lid considerably simplifies the analysis.

The concept of the wind impulse ( C S A N A D Y , 1977; see also C S A N A D Y , 1978, 1982) is quite straightforward and will, for the sake of clarity, now be described without regard for the remainder of the dynamics. For this, let us consider the upper layer of a multi-layer ocean of instantaneous depth h and of resting depth HI, and let us assume that there are no variations in one of the two horizontal directions (say y), such as in the alongshore direction. Then the depth h, the velocity components u~ and Vl (in the x and y directions, respectively), and the pressure in the layer depend only x and t. The momentum balance in the y direction is:

d v I + f u l _ T ( 1 )

dt plh'

where f i s the (constant) Coriolis parameter , r is the y-component of the wind stress, and the material time derivative includes only two terms:

1706 B. CUSHMAN-ROISIN et al.

d o 0 - ¢- ( 2 )

dt at Ul ~x.

The missing advective term in (2) and pressure-gradient term in (1) are identically zero because of the absence of variation in the y-direction. Since definition (2) implies dx/dt = ul, equation (1) can be written as

_ 2" -~(Vl + fx) p l h '

and a subsequent time integration following upper-layer fluid particles yields:

1 I' r[x(t)'t] V 1 + f x = V 1 + f X 1 + ~ o h[x(t),t] dt, (3)

where in the integral r and h are taken following a chosen particle with time. The constants of integration V 1 and X t are the initial y-component of the velocity and the x coordinate of the particle (at time t -- 0) now at the position x (at time t). If the wind blows only during a limited time (0 -< t -< T), the integral in equation (3) remains constant at all times following the wind event. It is defined as the wind impulse,* noted I:

1 I T r[x(t),t] I = ~ o h[x(t),t] dt, (4)

and equation (3) applied after the wind has ceased is

vl + fx = V1 + fXl + I. (5)

Obviously, the wind impulse varies from particle to particle. [I is a function of X 1.] Although it is easily defined, the wind impulse cannot be so easily determined, chiefly

because its evaluation entails a Lagrangian integration that strictly cannot be performed until the solution to the problem has been obtained. However, if the wind event is relatively brief (mathematically, if T << l/f), the upper-layer depth does not have time to vary much while the wind blows (CR~.PON, 1967), and h can be approximated by H l over the course of the integration. If, moreover, the wind stress is spatially uniform (usually a good approximation because weather patterns are far wider than the coastal processes of interest or because of orographic constraints), the integral in (4) can be easily evaluated from a time series of the wind stress:

I = 1 (rr(t)dt ' (6) plH1Jo

and the resulting wind impulse is the same for all particles. If the above conditions (of wind brevity and uniformity) are not strictly met, integral (6)

with the wind stress taken at one fixed location can nonetheless provide a very useful estimate of the wind impulse. The approximations involved may, indeed, not be worse

*Our definition of the wind impulse differs from that of CSANADY (1977) by the inclusion of the layer thickness in the denominator . We advocate our definition because it permits generalization, in principle, to wind events of finite duration.


wind,

surface

terrace at rest

~ ~ interfaCey

x--0 x = L x

Fig. 3. Sketch of the two-layer flat-bottom fjord model and the attending notation.

than other assumptions typically made in this type of problem (e.g. reduction of stratification to a layered system).

A less obvious but nonetheless important assumption made in the above derivations is that the Ekman-layer thickness not exceed the upper-layer depth at any time. If this were not the case, a portion of the wind stress would be felt in the lower layer. Taking the Ekman-layer thickness as 0.2 u , / f (CSANADY and SHAW, 1980; STIGEBRANDT, 1985), were

R l U , ) , we thus assume h > 0.2 u, is the friction velocity obtained from the wind stress (r = 2 u , / fdur ing the duration of the wind event. For a brief event, it suffices to require H 1 > 0.2 u, / f , which is usually satisfied.

It is worth making clear that while integrations (4) or (6) extend only over the wind event, equation (5) applies as well to any time after the wind event, that is during the following geostrophic adjustment and thereafter. Should an eventual steady state be reached, equation (5) relates, in a direct way, initial and final variables. We shall see how this statement, conservation of potential vorticity and geostrophic balance are sufficient to describe fully the steady state following a wind event, circumventing a numerical integration of the time-dependent response.

3. UPWELLING--DOWNWELLING IN A BROAD CHANNEL

For the purpose of modeling, a fjord can be assimilated to an infinitely long channel bounded laterally by two parallel vertical walls (separated by a distance L, the average width of the fjord) and below by a flat bottom (total depth H, the average depth of the fjord). Typically, the stratified waters of a fjord exhibit a nearly two-layer structure during the summer season (PICKARD and ROGERS, 1959; GADE, 1970; CANNON and LAIRD, 1978; SVENDSEN, 1981).

Let us denote by Pl, H1 and P2,//2 the densities and averaged thicknesses of the upper and lower layers, respectively (Fig. 3). Obviously, the s u m H 1 + H 2 is equal to the total depth H. Introducing the reduced gravity g' = g(P2 - PO/P2 and the Coriolis parameter f, taken here as constant and positive, we write the governing equations of the two-layer system as:

1708 B. CUSHMAN-RoIsIN et al.

Oul _g, Oh Op OUl jr. U 1 - - f v 1 . . . . (7) Ot ~x Ox 3x

_ _ Ovl _ r OV1 q- H1 "}-ful (8) Ot -~x Plhl

Oh+~x(hUO 0 (9) 3t

OU 2 -}- OH 2 Op Ot u2 ~x - fv2 = 3x (lO)

Ov2 3w + u, v'~ + fu2 = 0 (11)

Ot ~ Ox

oh+ o - Ot 0 G [ ( H - h) .2] = 0, (12)

where (ul, vl) and (u2, v2) are respectively the velocity components in the upper and lower layers, h is the local and instantaneous upper-layer depth and p is the pressure (divided by density) in the lower layer.

For simplicity, a rigid-lid approximation has been made. CSANADY (1977) retained the free-surface variations in his analysis and then approximated the solution using the fact that the barotropic and baroclinic radii of deformation are vastly different. The assumption of a rigid lid is the physically equivalent approximation and one that permits to take advantage of a reduced algebra right from the start of the calculations.

The above equations lead to conservation of potential vorticity for individual fluid columns in each layer, as long as there is no wind-stress curl (Or/Ox = 0). The potential vorticities take the form:

1 (f Ovl] ' (13) ql = ~ + Ox /

1 ( f + Ov2] (14) q2 - H ~ Ox J"

If the initial state is one of rest, with no currents and a levelled interface, these potential vorticities are uniform in each layer:

1 ql - (15)

H 1 '

q 2 - ~ f - f (16) H - Hi 142"

We now turn our attention to the final steady state, after the completion of the geostrophic adjustment following the wind event. With vanishing time derivatives, the continuity equations (9) and (12) lead to ul = u2 = 0, and there is no cross-fjord flow. Equations (7) and (10) yield geostrophic balance for the along-fjord flow:

- f v , = - g ' dh _ dp (17) dx dx'


--fv 2 -- @ . (18) dx

A Lagrangian time integration of equations (8) and (11) from the initial, pre-wind state of rest to the final, steady state provides

vl + fx = f X l + I, (19)

V 2 + f x = f X 2 , (20)

according to the derivations detailed in the previous section. The wind impulse I is defined exactly by (4) but can be estimated by (6); the Lagrangian coordinates Xl(X) and X2(x ) represent respectively the initial cross-fjord positions of the upper- and lower-layer water columns now at cross-fjord position x. Finally, conservation of potential vorticity provides two additional equations constraining the steady-state variables:

f + d V l - f h, (21) dx H~

f + d v 2 _ f ( H - h ) , (22) dx H 2

which result directly from equations (13) to (16). It is well known that geostrophic adjustment, of which the above problem is an example,

includes a loss of energy (BLUMEN, 1972; OU, 1986). This loss is attributed to the outward radiation of gravity waves, which contribute in a substantial way to the transient behavior during adjustment. In our fjord case, with a solid boundary on each side, internal-gravity wave energy is reflected back and forth, and one may wonder whether geostrophic adjustment can ever take place. In reality, however, the fiord's conditions and wind forcing are not strictly uniform in the down-fjord direction, and waves will zigzag their way out the fjord. Wave breaking along the sloping shores is another possible dissipative mechanism. Because waves do not transport fluid parcels, neither process compromises conservation of potential vorticity in most of the domain.

The above set of equations, (17)-(22), forms a six-by-six system for the unknowns h, p, Vl, v2, X1 and X 2. Since the latter two only enter equations (19) and (20), there is a temptation to discard these two equations and solve the remaining four by four problem separately. This, however, leads to undetermined constants of integration, and equations [(19) and (20)] that have no derivatives but are nonetheless accompanied by boundary conditions (e.g. a particle initially at the wall is still at the wall at the final state) contribute to determine the remaining constants.

The method of solution proceeds as follows. Elimination of p by subtraction of equations (17) and (18) and, then, elimination of vl - v2 from the difference of (21) and (22) yield a single equation for the upper-layer depth h:

dZh 1 h - f2H (23) dx 2 -~7 g' H-----2'

where R = (g'H1Hz/H)V2/f is the baroclinic radius of deformation. The most general solution, with two constants of integration, is:

1710 B. CUSHMAN-RoISIN et al.

Fig. 4.

02

x = O x = L x

x = a

~ P . 1 h(x) H 1

P2

x = O x = L x

Case a Case b

Possible cross-fjord profiles of the interface in the final steady state. Case a: without outcropping; Case b: with outcropping and a surface front (full upwelling).

h = H 1 + A e x/R + B e - x / R , (24)

which cor responds to the baroclinic flow field

- g ' ( A e x/R - B e - X / R ) . (25) V 1 -- V 2 - - - ~

To isolate vl and v2, we re turn to equat ions (21) and (22), now taken separately:

f R ,__ xlR B e x m ) v l = C + ~ t , a e - (26)

f R ( A e X m _ B e _ X / R ) ' (27) v 2 = C - - ~ 2

where C is a third constant of integrat ion. The lower- layer pressure p is in turn derived f rom either geost rophic balance (17) or (18):

p = C f x - g ' H 1 ( A e X / R + B e x/R), (28) H

where an additive constant of integrat ion has not been in t roduced as it plays no role. Depend ing on whe ther the interface be tween the two layers has ou tc ropped or not (Fig.

4), different boundary condit ions must be applied to de termine the three constants of integrat ion, A, B and C.

In Case a, when outcropping does not take place, the water columns along the walls in each layer have remained against their respective walls. There fore , we must enforce

at x = 0, X1 = X2 = 0 (29)

at x = L, X1 = X2 = L. (30)

By virtue of (19) and (20), this translates into the following boundary condit ions on the velocities:

a t x = O, vl = I , v2 = 0

a t x = L, v I = I, v2 = 0.

Of these four condit ions, one is r edundan t , and the three others de termine the three constants:

A = + H 1 H 2 I 1 - e - L / R H f R e L/R - e - L / R (31)


B - Hall2 I e L / R - 1 H f R e Lm - e - L m (32)

C = H1 1 (33) H "

This solution is obviously acceptable only if h is positive everywhere, i.e. the upper layer exists everywhere. For positive values of I (a choice that may always be made by directing the y-axis downwind), upwelling occurs on the left side (recall that we have taken f positive) and the least value of h is found at x = 0. Thus, the above solution is restricted to

h(x = O) >- O,

which yields H 1 + A + B -> 0 and a condition on the wind-stress impulse:

i < - f R H (34) He tanh (L /2R)"

This implies that no outcropping occurs if the wind stress is sufficiently weak or its duration is sufficiently short. Should this not be the case, the interface surfaces at some distance from the coast on the upwelling side, and the above solution must be amended.

Let us now consider such case (Case b of Fig. 4). Two modifications to the above solution must be made: first, a new solution must be derived in the region where only one layer is present; and second, boundary conditions on the left must be revisited. In the one- layer region, extending from the left wall (x = 0) to the outcrop position (x = a, a to be determined), the upper-layer thickness vanishes (h = 0), the variable vl has no meaning, and the above equations (18), (20) and (22) reduce to

d v 2 __ f H v2 + fx = fX2, f +

dp fv2- dx'

The general solution is:

dx H 2"

f n l v2 = ~-? x + D (35)

f 2 H l x e P = 2He + f D x + E. (36)

Enforcing )(2 = 0 at x = 0 (the water column at the wall has remained against the wall), we find Vz(X = 0) = 0 and therefore D = 0.

We now apply the conditions at the outcrop (x = a). The requirements are the matching of the lower-layer variables Ve and p between the one-layer and two-layer portions of the domain (continuity of v 2 is a direct consequence of continuity of the Lagrangian coordinate X2), the vanishing of the upper-layer depth (h = 0 by definition of the outcrop), and the requirement that the upper-layer fluid parcel at the outcrop be the one that was initially at the left wall (X1 = 0 at x = a). We thus write:

r e ( a - ) = v2(a+): f i l l a = C - f R (Aeam _ B e _ , m ) H2

f e l l 1 a 2 + E = Cfa - g'H1 (Aea/R + Be-a/R) p2(a--) = p2(a+): 2 H 2 H


Fig. 5.

a/R

~- L/R=4

L/R =3

UR=2

L/R=1

1 2 3 4 5 6 7 8 9 10

H21/HfR

Variation of the distance a between the coast and the upwelling front (outcrop) with the strength of wind impulse, for selected fjord widths.

h(a) = 0: H1 + Ae aIR + Be -~/R = 0

fR (Aea/R _ Be_a/R) + fa = 1. X1(a) = 0: C +

This forms a four-by-four linear system for A, B, C and E, where the still unknown quantity a plays the role of parameter. The solution is:

H,(H2 I_ a 1]e_a/t~ A = 2 \ H f R R

(37) /

B = HI ( tt2 I a ) 5 - - + f i - 1 e ° 'R (38)

H. C = - - ' I (39)

H

E : - f2Hl a 2 + f H l l a + g'H2 (40) 2H 2 H H

Finally, the constant a, which gives the distance of the outcrop to the coast, is determined by applying one boundary condition on the right: the upper-layer parcel at the right wall has remained there (X 1 = L at x = L) and thus, by virtue of equation (19), vl (L) : I. The result is a transcendental equation for a:

11-12 R) - a sinh L - a lH2 (41) f R H cosh L R R fRH"

Physically, the solution is acceptable only if a is a positive distance. This occurs when

I > f R H (42) //2 tanh (L/2R)"

Because this condition is complementary to the no-outcrop condition (34), we conclude that we have derived the solution for all wind-impulse values, low or high.

Figure 5 displays the variation of the outcrop distance a with the wind impulse and the


fjord width, in dimensionless form. Each line corresponds to a different fjord width. The intercept of the curve with the horizontal axis (a/R = 0) marks the critical value of the quantity IH2/HfR below which an outcrop does not exist [Inequality (42)]. For an infinitely wide fjord, this value is 1. Figure 5 indicates that the upwelling process is significantly inhibited in fjords of width less than 2R, because of the proximity of downwelling on the other side.

4. ASYMPTOTIC CASES

The previous section established a criterion on the wind impulse indicating whether outcropping occurs or not and provided a formula for the distance from coast to outcrop, should one be present. It is of interest to analyze the asymptotic behaviors of those expressions under the limits of a very deep fjord (H2 >> H 3 , a very narrow fjord (L << R), and a very wide fjord (L >> R).

In the case of a very deep fjord (H 2 >> H1 and consequently/42 = H), the present solution converges toward that of the one-layer, reduced-gravity equations. The salient results are as follows. Outcropping does not occur as long as the wind impulse meets

I <- ~/-g'H1 (43) tanh (L/2R1)'

where R1 = (g'H1)t/2/fis the appropriate radius of deformation. Should this condition not be satisfied, outcropping occurs and the upwelling front is found at the distance a from the coast given by

I a) - a s inhL-a I X/?H1 /~1 cosh LR1 R----~ = ~ " (44)

In the case of a fjord that is very narrow (L << R) but not necessarily very deep (H2 finite), the corresponding criterion and expression for the outcrop distance are:

I <-- 2g'H1 (45) f c

/ g'H1L (46) a = C - v f f "

As we can see, the smallness of L requires a very large wind impulse before outcropping can occur. Physically, the proximity of the two walls prevents a sizeable cross-fjord Ekman drift and thus severely restricts the occurrence of upwelling on its upstream side. Note also that, in this case, the wind criterion and outcrop distance do not depend on the depth of the lower layer. This is explained by the fact that the situation is exclusively controlled by the narrowness of the fjord and, consequently, the presence of the bottom is only peripheral.

In the opposite case of a very wide fjord (L >> R), we return to the classical case of coastal upwelling (CSANADY, 1977). Indeed, the two coasts are so far apart that the upwelling on one side is totally dissociated from the downwelling on the other. The asymptotic expressions are


/

< - / g ' (47) HH1

I 142

a - IH2 R, (48) f/-/

which, as expected, are independent of the fjord width, L, and are identical to those obtained by CSANADV (1977).

O'BRIEN and HURLBURT (1972) presented a numerical model of coastal upwelling using a two-layer model over a flat bot tom and on an f-plane. Except for the fact that they simulated a continuously developing upwelling event under a persisting wind, their model is in essence identical to ours, and a comparison between their numerical results and our theoretical predictions is of interest. In particular, the comparison may shed light on the adequacy of the impulse approximation (brief forcing) in situations with lasting forcing. O'BRIEN and HURLBURT (1972) selected the following values: H1 -- 50 m, H2 = 150 m (hence, H = 200 m), g ' = 0.02 m s -2, f = 10 -4 s--1 and r/pl = 10 -4 m 2 s -2. Their simulation was terminated after 10 days when the interface nearly reached the surface (their Fig. 5). Using T = 10 days = 8.64 × 105 s, we estimate the wind "impulse" to be

I - r T _ 1 . 7 3 m s - I . plH1

Since the model of O 'Br ien and Hurlbur t was run almost until surfacing of the interface occurred, the appropriate impulse value to be used for comparison is lcrit, for which the theory predicts surfacing right at the wall, as in the numerical model. The one-coast theory provides the critical value [see equation (47)]:

/crit =/g'HH1 = 1.15 m s - l . ~/ H 2

The two values are relatively close, especially in consideration of the fact that the wind impulse I has been only roughly approximated. According to O'Br ien and Hurlburt , the upwelling zone is about 30 km wide. Using the distance at which the exponential reaches 5% of its value at the boundary, we estimate the upwelling width at 3R or 26 km.

A more critical test of the present theory is the comparison of velocities. The numerical model (Fig. 6 of O'BRIEN and HURLBURT, 1972) predicts v 1 = v2 --~ 35 cm s - l in the wind direction far offshore, vl ~- 64 cm s - t at about 8 km from the coast (it decays to zero at the coast because of lateral friction) and v2 = 0 at the coast (no evidence of frictional boundary layer). The corresponding theoretical predictions are Vl = v2 = HII /H = 29 cm s-1 in the wind direction far offshore (i.e. no baroclinic flow as in the numerical simulations, but otherwise reduced values; use of I = 1.73 m s-1 yields v l -- v2 = 43 cm s -1 , thus erring the other way), v~ = 63 cm s -1 at 8 km from the coast and v2 = 0 at the coast. Considering the great differences in forcing (continuous vs impulsive), we find the agreement between numerical results and theoretical predictions excellent and conclude that the impulse- method can provide useful estimates even beyond its stated range of applicability.

Finally, in the case of a fjord that is both very wide (L >> R) and very deep (112 >> HI) , the common asymptotic limits of (43), (44), (47) and (48) are:

I -< ~gg'H, (49)


I a = -~ - R. (50)

1

Here , we recover the expressions obtained by CUSHMAN-RoISIN ( 1 9 8 5 ) in a study of coastal upwelling with a reduced-gravity model.

5. APPLICATION TO PORSANGERFJORD

In order to illustrate the preceding solution and to determine qualitatively the import- ance of the fjord depth and width on the character of the upwelling-downwelling pattern, we now assign to the model parameters the values of Porsangerfjord. The oceanography of this fjord is presently under investigation (SVENDSEN, 1991), and it would be premature to attempt a thorough application of the theory at this time.

Therefore , what follows should be considered solely as an illustration of how the theory may be applied and of what types of results can be derived.

Using the parameter values listed in Section 1, criterion (34) for the absence of outcropping yields: I -< 1.29 m s- 1, which in terms of a mean wind stress r and a duration T corresponds to [see equation (6)]:

rT-< 6.59 x 104 kg m -1 s -1 = 6.59 x 105 (dynes cm 2) × s.

A wind stress of 1 dyne cm-2 lasting i day (= 8.64 104 S) leads to TT = 8.64 103 kg m i s 1 somewhat below the critical value. Hence, to create an upwelling front, the wind stress must be several dynes cm -2 strong and/or last for several days. Such a situation is not unlikely but does not appear to have been the case during the period 10-12 August 1990. Indeed, using Pair = 1.20 kg m - s , wind velocities U o f 7.7 m s -1 (15 knots) and 15.4 m s i (30 knots) and drag-coefficient values given by

Co = (0.61 + 0.063 U) 10 -3

(GILL, 1982, p. 29), we evaluate wind stresses at

U = 7 . 7 7 m s -1, r = C o P ~ i r U 2 = 0 . 0 7 8 N m - 2 = 0 . 7 8 d y n e s c m 2

U = 15.4 m s -1, r = CDPai r U 2 = 0.45 Nm -2 = 4.5 dynes cm -2,

and we estimate the wind impulse by assuming a 15-knot wind blowing for 24 h followed by a 30-knot wind blowing for 12 h:

I - E r T _ 0 . 5 1 m s -1. plH1

This value falls below the theoretical threshold of 1.29 m s -1 but has nonetheless the same order of magnitude. Since the wind event was not impulsive but lasted well over an inertial period, during which the thermocline began to respond, the decrease in upper-layer depth on the upwelling side contributed to increase the wind impulse there and the local impulse might have approached the threshold value.

To investigate the interfacial profile, h, and the along-fjord barotropic ~ = [hvl + (H - h)vz]/H and baroclinic v' = vl - v2 velocities, we will select two impulse values, I = 1.0 m s -1 and I = 2.0 m s - i , corresponding respectively to situations without and with outcropping. In the figures, the most general solution (with both L and H finite--as in


0 ' - . . , , . . . . , . .

[m] ~ ............. t ~ ~ : : ' : : : : = .................

. . . . Case I I I 100" Case t V

150 i i i ~ ~ i i i j i , j i ~ i l l l l l l l ; J l l l l , l l J l , l l

5 10 [km] 15 x=L

Fig. 6. Cross-fjord profiles of the interface in the no-outcrop case (I = 1.0 m s ] ). The solid line (Case IV) corresponds to the general case and thus the most accurate theory. Case III is the limit of a very wide fjord (classical coastal upwelling), Case II is the limit of an infinitely deep bottom (reduced-gravity model), and Case I is the limit of a fjord that is both infinitely wide and infinitely

deep.

o -

[m] 5O

100

. . . . . . . . . . . . . . . Case I " * ° ' * -

. . . . . . . . . . Case I I

Case I I I

150 , , , , . , , , , i . , , , , . , , , i , , , , , , , , , = , , , , ~ ( . L 0 5 10 [km] 15

F i g . 7. S a m e as F i g . 6 , b u t i n t h e o u t c r o p case (1 = 2 . 0 m s - ] ) . N o t e h o w t h e a s y m p t o t i c l i m i t s o f

v e r y w i d e f j o r d s ( C a s e s I a n d [ I ] ) g r e a t l y o v e r e s t i m a t e t h e d i s t a n c e o f t h e f r o n t f r o m t h e c o a s t a n d

miss the influence of the downwelling from the other side.

Porsangerfjord) is denoted Case IV. The asymptotic solutions are denoted as: Case III when L >> R (very wide fjord or classical coastal upwelling); Case II when H 2 >> H I (very deep fjord or reduced-gravity model); and Case I when both L >> R and H 2 >> H 1.

We first look at the interfacial profile, h ( x ) , for the two chosen values of I. In Fig. 6, the solid line (Case IV) is the position of the interface corresponding to upwell ing- downwelling in Porsangerfjord under a wind impulse I = 1.0 m s ~. The solution behaves as expected from Fig. 4, Case a. From the curves for the asymptotic Cases I, II and I I I , it is seen that the upwelling response is restricted by the presence of both coast and bot tom.

In Fig. 7, the solid line is again the interfacial profile most relevant to Porsangerfjord. Here , I (= 2.0 m s -~) is larger than the critical value necessary to create an outcrop. The surfacing of the interface (depth h = 0 m) indicates the outcrop. Comparing the asymptotic solutions of Cases II and I I I , we notice that the presence of another coast (Case II) is more

Upwell ing in broad fjords 1717

Fig. 8.

1 0 -

0,75-

/ / ,,.,/ ..,,

'%'., ,°°°

"-.'.', ioo" -- . , /

° I ".% %. ~ ,

%'- ....... "". "~%'oc,, - ' " / i <~ '.... "-.%.. ~o~, ..o'" (t) 05. ~, ... ..... ..

E % " - " . . . . . . . . . . . . . . . . . ' . . . . .

""""--..,,, ~ ~/ ~ a l e a r o g ~ c

"''" G a~'°

025 ' _ _ a..~. . . . . . . . . . . . . . . . . . ""-,. Ca~ ~ I I b a t o t r o p ~ c

0 . 0 I I I I I I I I I I I I I i i i I I I I I I I I I i I I I I i I I I I I

5 10 15 x = t [km] Cross-f jord profiles of the barotropic and barocl inic velocities in the no-outcrop case.

2 0 - o* , /

7 1.5- .... ° oO"

,, ..°

%',..., Ca~ / ~,bat o ' ° ' "Y /

~il i l I ...,... O.a.rlla

0.5- ~ , o.o I

i i n ~ n J * i I ~ i i , , , , l , I u i u u , i , u ~ 1 1 5 1 t i i i [

0 5 1 0 [km] x=L

l ) 1 .o-

E

Fig. 9. Same as Fig. 8, but in the outcrop case,

inhibiting to the upwelling than the finite depth (Case III), although each contributes to a reduced upwelling in comparison with the reduced-gravity coastal upwelling solution (Case I). Figures 6 and 7 clearly show that the errors made by using the asymptotic solutions in applying the theory to broad fjords can be significant. In other words, the magnitude of upwelling in a broad fjord, such as Porsangerfjord, is significantly different from that in open-ocean coastal upwelling.

The barotropic and baroclinic velocities for Case IV and for Case III (L >> R) are presented in Figs 8 and 9, where I = 1.0 m s - i and I = 2.0 m s -a, respectively. The barotropic velocities are clearly proportional to h(x). The asymptotic value 0.25 I is the


0

[m]

5O

100

150-

200

I = 1 m s -1

I I I I I I I I I I

s lo

\ \

I I I I I I I

15 x=L [km]

Fig. I0. Cross-fjord density stratification according to the numerical model. This figures serves for comparison with Fig. 6 and thus validation of the theoretical model. Plotted are values of

10 30.

barotropic velocity far f rom the coast. The curve in Case IV is symmetric about this asymptotic value.

The baroclinic velocities reach their maximum values at the coasts. In Case I I I , the velocity approaches zero at large distances from the coast (where the interface displace- ment vanishes). The maximum value of the baroclinic velocity is equal to I. When the wind impulse is larger than the critical value and an outcrop exists (Fig. 9), the maximum value (equal t o / ) is obtained only at the right wall x = L. Baroclinic velocities are non-existent where there is only one layer (Fig. 9 where 0 < x < a). At the front position (x = a), the value of the baroclinic velocity is equal to/crit" This is also the maximum value in Case III . In the one-layer systems (Cases I and II), velocities are comparable to the baroclinic velocity of Cases I I I and IV, respectively.

6. COMPARISON OF THEORY WITH NUMERICAL SIMULATIONS

In order to assess the suitability of the analytical solution, a numerical simulation has been performed. The numerical model used is e c o m 3 d (BLUMBERG and MELLOR, 1987), a primitive-equation, sigma-coordinate, t ime-dependent , free-surface estuarine and coastal model with an embedded turbulence-closure submodel. This model should be most suitable for a problem like the one at hand. The model domain is a very long channel, 18 km wide and of constant depth equal to 200 m. The resolution is 1 km in the horizontal and 20 sigma-layers in the vertical (16 in the upper 75 m). The initial conditions are the same as in the analytical investigation. The model was forced by a uniform wind-field, held after a 3-h gradual increase at the constant value r = 2.85 N m -2 (corresponding to a 30 m s -1 wind). This value was chosen to be relatively high to run the model in a somewhat impulsive mode. Cross-sections were taken at different times to correspond to specific values of the wind impulse.

Figure 10 shows the situation for I near 1 m s - 1. Comparison with Fig. 6 indicates good


0

[m]

50.

100.

150-

2 0 0 i ' ~ I ' ' ' I ' '

0 5 10 [km] 15 x=L

Fig. 11. Cross-fjord density stratification according to the numerical model with a greater net value of the wind impulse than for Fig. 10. This figures serves for comparison with Fig. 7. Plotted

are values of 10 -3 (7 t.

Fig. 12.

0

[m]

5O

i . . . . I , , I I Y ~ . . . . . . m s Ioc,ity,, . . . . . . . . . . .

s lo [km] is x=L

Cross-fjord distribution of the along-fjord velocity according to the numerical model in the no-outcrop case (I = 1.0 m s 1), for comparison with Fig. 8.

agreement. Figure 11 constructed for I = 2.1 m s - t is comparable to Fig. 7. There is again good agreement in the results, but it appears that the theoretical model tends to underest imate slightly both the outcrop-distance on the left and the downwelling on the other side. It may also be that the presence of transients in the numerical model (under only modera te friction and under continuous forcing) do not permit the numerical solution to be fully adjusted, Figure 12 shows the along-fjord velocity of the numerical model for I = 1 m s -1. A comparison with Fig. 8 shows that the numerical model yields much the


same velocity pattern as the analytical, impulse approach. There appears a slight bias in speed between the analytical and numerical results (Figs 8 and 12). Adding the value 0.15 m s -~ (about 15%) everywhere to the velocity-field in the numerical results would yield a near-perfect agreement with the analytical prediction. From this, we conclude that the theory overestimates the barotropic flow.

In Porsangerfjord, both depth of the upper layer and stratification vary during the year. Accordingly, the critical value of the wind impulse necessary for outcropping is variable. Furthermore, a two-layer structure only appears in June, when the melting of snow begins, and lasts until late fall (November). During late summer and fall, the depth of the upper layer increases, and the stratification decreases. It is therefore expected that the critical value of the wind impulse is largest in the beginning of the summer. Likewise, since the stratification decreases in the out-fjord direction. It is expected that the critical value of the wind impulse is smaller in the outer part of the fjord than in its inner part.

7. CONCLUSIONS

The concept of wind impulse introduced by CSANADY (1977) is found extremely useful in the study of upwelling-downweiling patterns along a coast or in a broad fjord (two parallel coasts). The concept not only leads to a simple theory from which can be derived pressures, depths and velocities in the geostrophically adjusted state following the wind event, but it also provides a simple criterion to determine whether or not outcropping of an interface will occur. Furthermore, if outcropping occurs, the theory provides an equation for the distance of the outcrop to the coast.

Here, the wind-impulse method has been applied to a two-layer model (with one or two coasts) and for a single wind event. Generalizations to stratified systems with more than two layers, and to successive wind events is feasible, with a proportional cost in algebraic manipulations. CSANADY (1982) solved the one-coast, three-layer problem. The two main assumptions (of a relatively short wind event and of zero wind-stress curl) need, however, be kept in mind.

The application of the method to Porsangerfjord, a broad fjord of northern Norway, shows that the combination of vertical stratification and fjord width permits, under typical wind stresses, the development of upwelling on one side, downwelling on the other, and an interaction between the two. Moreover, the numbers are such that outcropping may occasionally occur on the upwelling side of the fjord. With the available data, however, we are unable to assess whether the noticeable cross-fjord temperature variation observed on 12 August 1990 resulted from outcropping (full) upwelling or not. To do so requires a better knowledge of the hydrography and wind velocities in the period preceding the event than has been available to use for this study.

Acknowledgements--This research would not have been possible without the support of the Office of Naval Research (Grants N-00014-91-J-1789 and N-00014-93-I-0391 to Dartmouth College) and the Norwegian Council, Department of Fisheries (Grant No. 1201-119.001 and scholarship to Lars Asplin), for which the authors are most grateful. This work is part of the North Norwegian Coastal Ecology Program, whose partial support is also acknowledged. The Norwegian Council for Science and Humanities provided access to the supercomputer at SINTEF in Trondheim. Finally, the authors wish to express their gratitude to Professor G. T. Csanady for comments on an earlier version of the article.


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