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The collision between the Gulf Stream and warm-core rings
Deep-SeaResearch. Vol. 33, No. 3. pp. 359 to 378. 1986.
0198~41149/S6$3.00 + 0.1Xl © 1986PergamonPress Ltd.
Printed in Great Britain.
T h e collision b e t w e e n the G u l f S t r e a m a n d w a r m - c o r e rings DORON NOF*
(Received 1 May 1985; in rev&ed form 4 October 1985; accepted 7 October 1985) A b s t r a e t - - A non-linear two-layer model (with one active layer and one stagnant layer) is used to explain features frequently observed during ring-Gulf Stream interaction. Attention is focused on the narrow bands ( - 10 to 15 km) of Gulf Stream water which frequently envelop the colliding ring. The Gulf Stream is represented by an upper layer with a surfacing interface (front) on the north side, and the colliding ring is represented by a solid cylinder. Conceptually, the following steps are viewed as possible analogues to the actual ring-Stream interaction. Initially, the cylinder (ring) drifts toward the Stream in the Slope Water. Subsequently, it is pushed slightly into the frontal region and is then fixed. The resulting events that occur in the vicinity of the cylinder are examined and analytical solutions are constructed using power series expansions and integrated equations. It is found that even the slightest penetration of the cylinder into the Gulf Stream frontal region causes an intrusion of Gulf Stream water along the cylinder. Namely, as the cylinder is pushed into the Stream, a band of Gulf Stream water starts flowing along the cylinder upstream side. This band continues to intrude along the cylinder perimeter (in a clockwise manner) until it ultimately reattaches itself to the Gulf Stream in the downstream side. A short time after reattachment, a steady state is attained; namely, the cylinder is completely surrounded by Gulf Stream water and time-dependent motions cease. The detailed structure of this steady state is computed using the integrated moment of momentum.
1. I N T R O D U C T I O N
(a)
Background
RECENT satellite images of the ocean surface suggest that the interaction of warm-core rings with the Gulf Stream typically begins with a collision between a warm-core ring and a growing meander (e.g. JoYcE et al., 1984; EVANS et al., 1984). As a result of this collision, a narrow band (--10 to 15 km) of Gulf Stream water starts flowing along the western side of the ring (Fig. 1) and after a number of days the band envelops most of the ring. The purpose of the present paper is to examine this process and to suggest a mechanism that might be responsible for its generation. I do not intend to simulate all the details of the real system; I merely hope to preserve enough analogy to the real system so that my results will give some insight into the phenomena in question. The analytical modeling capabilities are limited and, therefore, I explore the processes in question in simple idealized systems. The reader should be prepared to accept some drastic simplifications. Consider the following three-layer model as an idealized formulation of the problem. The first layer (whose density is p) corresponds to the Gulf Stream, which is bounded on * Department of Oceanography, The Florida State University, Tallahassce, FL 32306, U.S.A. 359
360
D. NOF
intruding Gulf Stream
water
~
warm-core
Gulf Stream Fig. 1. Schematic diagram of the features associated with a typical interaction between the Gulf Stream and a warm-core ring. The diagram was adapted (by eye) from an infrared satellite image presented by BROWN et al. (1983); the boundaries were identified by tracing the maximum temperature gradients. Note that satellite images of other rings display a very similar structure, i.e. there is usually a band of Gulf Stream water that envelops the ring's left edge (see, lot example, images for 12 to 16 April 1982 in EVANS et al., 1984).
the north by a surfacing interface. The second (lens-like) layer is the warm-core ring (whose density is p + Ap') and the third (whose density is p + Ap) is an infinitely deep fluid in which both the ring and the Gulf Stream are embedded. An important property of this model is that the warm-core ring has a greater density than the Gulf Stream even though the ring originated from the Stream (i.e. 0 < Ap' < A9). This results from the ring's relatively long stay (3 to 6 months) in the Slope Water where strong atmospheric cooling frequently occurs. The nature of the ring-Stream interaction will be examined by assuming that due to a growing meander, advective currents or wind, the cooled ring has collided with the Stream at, say, t = 0. To simplify the problem assume that, as a first approximation, the colliding ring can be represented by a solid cylinder. With the aid of this important simplification, we can view the collision as follows. Initially the solid cylinder drifts slowly in the Slope Water toward the (steady) Gulf Stream. We may suppose that during this early stage there is some (mechanical) process that is causing the movement of the cylinder. At t = 0 the cylinder collides with the ring and, because of its initial drift, it slightly pushes the Gulf Stream front as shown in Fig. 2. Because of the increasing sea-level height away from the front, the penetration of the cylinder into the frontal zone ultimately will be arrested. It is assumed that, at this point, the cylinder is (mechanically) held fixed. Subsequently, there will be a Gulf Stream response to the forced cylinder and it is this motion on which we shall focus. It will be shown that even the slightest penetration of the cylinder causes an intrusion of Gulf Stream water around the cylinder edge; this condition is referred to as an edge intrusion. After the edge intrusion completely envelops the cylinder, it reattaches itself to the Gulf Stream and a steady state, whose shape resembles the Greek letter 'omega,' is reached. The solution for this 'omega flow' is obtained using the integrated moment of
The collision between the Gulf Stream and warm-core rings
A' (/0+/%/0) y~
361
~x C'
B IO)
GulfStream (t--O)
(a) Top view
"tB
(b) Side view (t=O)
~(E
(p)
(~
to)
Fig. 2. The intuitively expected Gulf Stream response to the forced presence of the cylindcr. It is expected that the Stream will simply bend and follow the cylinder outer surface on the righthand side. Dynamical considerations reveal, however, that such a response is impossible and that, instead, a band of Gulf Stream water will flow from point A to the left-hand side (Fig. 4) until it will ultimately reach point C (Fig. 5).
momentum (Sections 2 to 4). With the establishment of the steady circulation mentioned above, the cylinder is completely surrounded by Gulf Stream water. It is not a priori obvious under what conditions the replacement of a cooled warm-ring by a solid cylinder is justified. The resulting limitations and weaknesses will be discussed in detail in Section 4. The most important similarity between a colliding warm-core ring and a colliding cylinder is that both are expected to exert a pressure on the Gulf Stream front as they collide with it. A second similarity is a geometrical one, i.e. both features are either circular or close to being circular. An important difference between the two is
362
D. NOF
that a solid cylinder does not transmit pressure nor does it adjust its shape according to the environmental forces. It is believed, however, that the similarities between a colliding ring and a colliding solid are sufficient to permit the use of the solid cylinder as an analogue. It will become clear later that, with the aid of such an analogue, the results may pinpoint some of the processes which one should look for in more 'realistic' and more cemplicated models. Note that such analogues are not entirely new in the earth sciences. For example, the differentially heated annulus has been used extensively to study atmospheric motion even though the annulus is not a direct duplication of the atmospheric environment. The pipes and hydraulic models discussed by STOMMEL(1961) and STOMMELand ROOTH(1968) are another example of such analogues. (b) Formulation o f the problem Consider again the system described above. Suppose that only the upper layer is active and that all the motions are frictionless, hydrostatic and nondiffusive. Note that the depth variations are of 0(1), so that the quasi-geostrophic theory does not apply. Let us first consider the events that will take place after the cylinder has been forced into the Stream. Intuitively, one might expect that the Stream response will simply consist of a somewhat bended flow along the cylinder right-hand side (Fig. 2). This implies that there is a streamline connecting point A ' with A and B so that we may apply the Bernoulli integral to the Stream's edge. 1 2 + g'hB = .2 I U 2o, ~UB
(1.1)
where g' is the 'reduced gravity' (gAp/p, with Ap being the density difference between the layers), U0 is the known upstream speed along the front (point A'), and UB and hB are the unknown speed and depth at B. Since hB is always positive, (1.1) implies that UB < Uo. However, if the entire Stream flows to the right of the cylinder, as has been temporarily assumed, then continuity implies some convergence across the line connecting point B and y ~ - ~ so that uB > U0. These conditions, required by the continuity equation and the Bernoulli principle, are obviously incompatible, suggesting that there cannot be a streamline connecting A', A and B. Instead, it is expected that there will be a band of Gulf Stream water flowing around the cylinder in a clockwise manner (Fig. 3). In other words, particles moving along the Gulf Stream edge (i.e. the surface front) do not have sufficient energy to rise to point B and, therefore, they must go around the cylinder where the fluid is lower. Hereafter, we shall call this narrow band flowing along the ring perimeter edge intrusion, A formal proof for the inevitable existence of the edge intrusion will be provided in Sections 2 to 4 where a computation of the omega flow (i.e. the state where the intrusion completely envelops the ring and reattaches itself to the Gulf Stream in the cylinder lee side) is presented. It should be pointed out, however, that for the special case corresponding to U0 = 0 (i.e. a stream with a zero speed along the upstream edge) no proof is really necessary because under such conditions (1.1) can never be satisfied. Before completing the present discussion, it is appropriate to mention that, by assuming a cross-Stream geostrophic balance and using a slow variation expansion, one finds a zeroth order flow that has a vanishing depth at all points around the southern side of the cylinder. This may give the mistaken impression that no flow around the north side of the cylinder must take place. This is not the case, however, because the associated
The collision between the Gulf Stream and warm-core rings
363
Gulf Stream TOP V I E W
~
½
o
SIDE VIEW
Fig. 3.
The intrusion of Gulf Stream water around a solid cylinder. The upper panel should be compared to the Gulf Stream band seen in satellite images (e.g. Fig. 1).
first-order flow must always have h > 0 on the southern side of the cylinder; otherwise there will not be any resistance to the forced cylinder which is, obviously, impossible. With h > 0 (which, as just pointed out, is inevitable) there will always be a first-order flow around the cylinder, as suggested by our earlier argument. At this stage, the reader may wonder how coastal flows get around bumps without leaking fluid to the side as proposed by the argument presented above. The answer to this question is simply that leakage will not occur unless the current has a front on the upstream side of the bump. Most coastal currents do not satisfy this requirement because they flow with the coast on their right side (looking downstream) so that their front is located on the ocean side where there are no coastal bumps. (c) Edge intrusion and the time dependent problem The structure of the transient intrusion is rather complicated because we do not even know the intrusion mass transport. However, the order of magnitude of the intrusion propagation rate can be obtained by considering the intrusion width to be small compared to the radius of the cylinder. It is not a priori obvious that this approximation is adequate because the curvature effects are not entirely negligible. For example, a cylinder with a
364
D. Nov
radius (r0) of about 45 km with an intrusion speed (u) of - 0 . 5 m s-1 and an intrusion width (w) of - 1 5 kin, will have curvature induced speeds [v - 0(u w/ro)] of about 15 cm s-1. Although such speeds are not entirely negligible compared to the intrusion speed, they constitute no more than one third of the propagation rate and, therefore, their neglect cannot alter the order of magnitude of the propagation rate. With the above simplification, the problem reduces to the intrusion of a density current along a coast. This problem received much attention in recent years mainly because of its relevance to the way that rivers empty in the ocean (e.g. STERN, 1980; STERNet al., 1982; GR1FVmt and HOPFINGER, 1983; KUBOKAWAand HANAWA, 1984; SIMPSON, 1982). A result of these investigations is that no steady migration rate can ever be achieved. There usually exists, however, a quasi-steady propagation rate whose magnitude is roughly between (g'hw)" and 2(g'h,,) ~ (where hw is the intrusion upstream depth near the wall). As far as the oceanic intrusion is concerned, it is difficult to estimate hw because this would depend on the strength of the collision. However, the satellite images presented by EVANSet al. (1984) suggest that, in the Gulf Stream, disturbances associated with the collision penetrate a few kilometers into the Stream where the Stream depth is 20 to 80 m. Such depths are consistent with the intrusion depth presented by JovcE et al. (1983). Together with a reduced gravity of 5 x 10-3 s-2 [corresponding to (Ap/p) = 5 × 10 -4 which is suggested by the horizontal temperature difference seen in the satellite images presented by EVANSet al. (1984), and the cross-sections shown by JovcE et al. (1983)], this gives a nose propagation rate of about 0.4 to 1.4 m s-1. This is of the same order as the intrusion advancement rate seen in the satellite images. It should be stressed, however, that, as already mentioned, this estimate only provides the order of magnitude of the propagation rate. 2. T H E S T E A D Y ' O M E G A '
FLOW--GOVERNING
EQUATIONS
AND CONSTRAINTS
The present section has two aims. The first is to prove that there must always be a flow around the cylinder so that the time dependent intrusion discussed earlier is inevitable. The second is to find out how the Stream responds to the presence of the cylinder. Specifically, one would like to compute the omega flow speed, width and depth as a function of the distance that the cylinder is pushed into the Stream. Because of the inherent non-linearity of the problem, it is unlikely that one will be able to find analytical solutions for the whole field. Consequently, we shall attempt to find the desired flux, speed and depth without solving for the entire field. (a) General description Consider the two-layer system shown in Figs 4 and 5. The origin of our coordinate system is located at the center of the cylinder; it will become clear later that this choice is not arbitrary. The x and y axes are oriented along and across the undisturbed Stream (respectively) and the system rotates uniformly at f/2 about the z axis. The upstream current flows on top of an infinitely deep motionless layer; it has a uniform potential vorticity f / H (where H is the upper layer depth at y --~ -oo). Consequently, its velocity (U) and depth (D) obey f U = -g'OD/Oy and -OU/Oy + f = f D / H . The current edge (D = 0) is located at y = -ro (1 - ~) and U --~ 0 as y --9 - w , so that we have U = (g'H)"-'e ly+r''(l-c)l/R,'
(2.1)
The collision between the Gulf Stream and warm-core rings
A
I// : O, h=O
~r0
U,D
region ~)
I
I region (~
I I I
I I I
=J C
B
Fig. 4. A schematic diagram of the regions under discussion and the integration area (dashed line) for the moment of momentum equation. Sections AB and BC are located several deformation radii away from the origin so that the flow there is not disturbed by the forced cylinder. Note that it is not necessary to limit our .model to cylinders that are circular along all their periphery. Cylinders with apartially 'fiat' periphery near point D can also be considered; to 0(~ ~) the two problems are identical.
Y region (~ p
F
!!!ii!!i!!ii!iii!i:
>
C
x
2tRd
region (~
region (~ Fig. 5. A sketch of the forces and torque associated with the integration area shown in Fig. 4. Overall, there are four forces acting on the boundary of the integration area (ABCDEF); three of the forces (F~; F2; F3) are associated with regions 1, 2 and 3 and the fourth is the pressure exerted on the surface of the cylinder. Only the former three have a torque (relative to the origin) because the pressure (at, say, D ' ) is perpendicular to the surface of the cylinder so that it does not have any moment relative to 0. This property enables one to connect regions 1, 2 and 3 without solving for the whole field.
365
366
D. NOF D = H(1 - ely+r"(l-~)l&'),
(2.2)
where Ra is the deformation radius, (g'H)~/f, and e is the relative penetration of the cylinder (Fig. 4). Note that the presence of e in these equations is not an indication that the upstream flow is influenced by the cylinder; it is merely a result of our choice for the origin of the coordinate system. For our model it will be assumed that the upstream structure (2.1) and (2.2) is not influenced by the presence of the cylinder because all linear waves on uniform potential vorticity flows propagate downstream. (b) Governing equations for regions 1 and 2 For both regions 1 and 2 (Fig. 4), the governing equations are the usual shallow water equations. Since the Stream is assumed to have uniform potential vorticity (f/H) we may use the equation for conservation of potential vorticity. We, therefore, have,
Ovi/Ox - OuJOy + f = &f/H; OVi
i = 1, 2
OF i
ui -~x + vi -~y + fui = -g Ohi/Oy; 0 0 - - (hiui) + (hivi) = 0; ox
i = 1, 2
i = 1, 2,
(2.3) (2.4)
(2.5a)
where u and v are the horizontal depth-independent velocity components in the x and y direction, h is the depth, and the subscripts '1' and '2' denote that the variable in question is associated with regions 1 and 2, respectively. Note that because of the symmetry of the problem (i.e. v = 0; hx = 0 at cross-sections 1 and 2) the x momentum equation and the continuity equation imply that, OUi
Ox
-- 0.
(2.5b)
The boundary conditions for region 1 are, h~ = O; y = ro(1 + Yl)
(2.6)
[Ul2]y=r,,(l+y,) = [U2]y=_r,,(l_e) = g ' m
(2.7)
i 2 +g [~ut
'h
t]y=r,, = [.,u, ~ 2_ + g ' h 2lv=-r.
(2.8)
Condition (2.6) states that the depth vanishes at some unknown location; conditions (2.7) and (2.8) reflect the conservation of energy along the streamlines that bound the flow from left and right (looking downstream), respectively. Namely, (2.7) and (2.8) are simply a result of an application of the Bernoulli integral to the streamline connecting A and F and the streamline connecting E and D. Note that (2.8) represents a connection between regions 1 and 2. The boundary conditions for region 2 are, h 2 --~ H ;
U2 --* 0;
y --* - ~ .
(2.9)
The collision between the Gulf Stream and warm-core rings
367
(c) Constraints The flows in the various regions are connected to each other via (2.7) and (2.8) but there are two additional equations that the unknown variables must satisfy. The first results simplify from continuity and can be written in the form, UDdy + -r.(1-~)
f-r,,
u2h2dy +
Ulhldy = O.
_a¢
(2.10)
r,,
The second equation will be derived from the conservation of moment of momentum. This is essentially the conservation of torque exerted by the fluid. It corresponds to the cross product of the position vector r and the momentum equations, y
u--+v---fv+ Ox Oy
-~x
-x
u--+v--+fu+ Ox Oy
-~y
=0.
(2.11)
To show that (2.11) provides an additional connection between the upstream flow and regions 1 and 2,\(2.11) is multiplied by h and the continuity equation is incorporated. This gives, 0
O (hu2y) + (huv)-fvyh Ox Y-~y 0
O
- x --Ox(huv)
g' O +-(h2y) 20x g'
(2.12)
O
-~y (hxv 2) - fuhx - --2 --Oy (h2x) = 0
(2.13)
which can be rearranged and integrated over the region shown in Fig. 4, to give,
fro( -~x
+
fs0( ~y
,
hu2y - f ~ y + -- h2y - huvx 2
,
hbll;y + f~X - - ~ hZx- hxl; 2
)
dxdy
) dxdy = O,
(2.14)
where ~ is a stream function defined by, O~ Oy
O~ -
uh;
Ox
-
(2.15)
vh.
By using Stokes' theorem, (2.14) can be written in the form, hu2y - f ~ y + ~ h2y - huvx
-
huvy - hxv 2 + fallx - -~ h2x
dy
dx = O.
(2.16)
368
D. Nov
This equation can be further simplified by, (a) defining ~g to be zero along the left edge where h -- 0, (b) noting that along any streamline udy = vdx and that v = 0 along AB, BC, CD and EF, and (c) noting that the flow along BC is identical to that which would be present without the cylinder. Namely, for AB as well as BC, the geostrophic relationship fU = -g'OD/Oy can be multiplied by D and integrated once in y to give jfg = g'D2/ 2 + A, where A is a constant to be determined from the boundary conditions. Since we have chosen ~g = 0 where D = 0, we immediately find that A = 0. Using the above conditions we can simplify (2.16) to,
hu2ydy + A
fo
(hu 2 -fag + g'h2/2)ydy +
C
+
fF
(hu 2 -f~g + g'h2/2)ydy
E
(-]Sffy+ g'h2y/2)dyD
( f ~ x - g'h2x/2)dx = 0.
(2.17)
D
The first three terms are the moments of the forces in the upstream region, region 2, and region 1. The last two terms, on the other hand, represent the torque corresponding to the pressure exerted on the cylinder by the surrounding flow. Since the origin is the center of the cylinder and the pressure is always perpendicular to the surface with which the fluid is in contact, we would expect this torque to vanish. It is easy to see that since the cylinder surface is given by x 2 + y2 _ r~, we have xdx + ydy = 0 so that the sum of the last two integrals in (2.17) equals zero as expected. Hence, the integrated moment of momentum constraint takes the simple form,
DU2ydy + A
h2u2 - f v 2 + ---~-jyuy + C
h,u~-f~, +
ydy = 0,
(2.18)
E
where we have incorporated our special notation for the various regions. Note that (2.18) does not involve any variables other than those of regions 1 and 2 and the upstream area (region 3). Its physical meaning is illustrated in Fig. 5; it corresponds to a balance between the three moments created by the three forces. At first sight, the derivation of (2.18) may give the mistaken impression that, because of the vanishing of the pressure torque related to the surrounding flow, it corresponds to a special balance of forces. This is not the case because, mathematically (and physically), the vanishing pressure torque results from the fact that the solid cylinder is round. It does not imply that there is no pressure on the cylinder--there certainly are forces acting on it. The only cases for which the pressure torque of the surrounding flow [i.e. the last two terms in (2.17)] is not expected to vanish are those associated with bodies whose crosssections are not round. Since most warm-core rings are nearly circular, the actual pressure torque in the ocean is expected to be rather small. 3.
SCALING
AND
EXPANSION
OF THE
OMEGA
FLOW
(a) The basic state Before discussing the scaling of the problem and the general structure of the expansion, it is instructive to look at the details of the basic state. The structure of the zeroth order state, corresponding to the cylinder 'kissing' the Stream, is not a priori obvious. Application of the Bernoulli principle to the cylinder surface and the upstream
369
The collision between the Gulf Stream and warm-core rings
edge [(see (2.17)] implies that even when ~--, 0 the velocity along the cylinder surface is of 0(1). This means that the basic flow around the cylinder cannot be zero; rather, it must consist of an infinitesimal ribbon flowing at a speed ( g ' H ) ' , To find the details of this ribbon flow it should be noted that even though the basic state contains only an infinitesimal strip, it must satisfy the equations of motion. To find the detailed mathematical structure, it is useful to consider the potential vorticity equation and momentum conservation in cylindrical coordinate (r, 0),
1 d (3.1)
- - - (rOo) + f = f i f t H
r dr -9
v~i --
r
dh + fro
(3.2)
= g' --,
dr
where O0 is the tangential velocity, the bar (-) indicates association with the basic state, and we have taken the flow to be purely tangential (i.e. O, = 0/00 = 0). When c goes to zero, the depth around the cylinder goes to zero as well, implying that the appropriate solution for the basic state can be found by looking at the limit of the governing equations as h ~ 0. For this situation, (3.1) and (3.2) have the most general solution, fr + _ . Oo -
2
~ = (~-
r '
r2) -
+
_
8g'
(3.3)
-- , -75- 2g '
where ct is an unknown constant and we have used the condition that h = 0 at r = r.. Since at r = r 0, the absolute value of the velocity must be (g'H)" (in order to conserve the Bernoulli function along the wall and edge), we find from (3.3) that, ( / = ~ f [ ( r o - R~I):
R~],
(3.4)
where Ra = ( g ' H ) ' / f . Namely, for any given cylinder (r0), we must take a specific value for eL. For simplicity, we shall consider only cylinders with ro = 2Rj so that cc = 0. Other cylinders can, of course, also be considered and the solution, which will shortly be derived, can be easily extended to cylinders with all diameters. However, such extended solutions do not provide any new physical insights and therefore are not presented. (b) Scaling In the subsequent analysis the following nondimensional variables will be used: upstream:
U* = U / ( g ' H ) ' ;
Region 1
.,
!
!
u~ = u l / ( g H)-';
and its
,
t
y* = y / R j . J
v~ = v J ( g H)~;
x't~ = x / R ` g
vicinity: Region 2:
D* = D / H ;
r~= rJRj
(3.5a)
y~ = y t / R , i (3.5b)
= 2
~ = ~J(g'H2/f). ,
!
!
u~' = u2/(g H)~; xg" = c'-'x/R,g
v~ = v 2 / ( c g ' H ) ' ; V2" = ~2/(g'H2/f);
y~ = y 2 / R j c ~ 1.
(3.5c)
370
D. NoF
Note that in the upstream region [i.e. variables defined by (3,5a)], the x length scale is infinite, and that in region 2 [i.e. variables defined by (3.5c)], the ratio between the x length scale and the y length scale (Ra) is given by the slope of the cylinder surface which is 0(e') (Fig. 6). In other words, due to the presence of the cylinder, a finite length scale in the x direction is generated. Since the slope of the cylinder surface in the vicinity of region 2 represents the ratio between v2 and uz, it must also represent the ratio between the y and the x scales due to continuity. For the first-order flow in region 1, it is convenient to use the governing equations in cylindrical coordinates (r*, 0). In terms of the nondimensional numbers defined by (3.5), the governing equations are, 1 d r* dr* (r*v~) + 1 = hT
(3.6a)
(v~t) 2 Oh~ - + v~l ,
(3.6b)
r*
Or*
where it has been assumed that the flow is parallel to the cylinder periphery so that Vr*~ = 0/00 = 0 and v~l is the only velocity component in the area. The nondimensional governing equations for region 2 are found from (3.5), (2.3), (2.4) and (2.5) to be,
Ov~ Ou~
e--
Or*
Ov~
eu~--
Ox *
--
Oy*
+ ev~
+ 1 =h~
Ov~ Oy *
+ u~ -
(3.7a)
Oh~
(3.7b)
Oy *
0 0 #x* (h~u~) + - - (h~v~) = 0. dy*
(3.7c)
~Y ::iii!i!i!ii!i!i!iii!i!ii!i!:i ~_ ,r£,.--O,
£ir( -
T
J~ro-(2£11/2
-.,
region
Fig. 6.
tr0
r
The geometry along the left boundary of region 2.
371
The collision between the Gulf Stream and warm-core rings
The boundary conditions (2.6) and (2.7) take the form h~'=0; [1 (U~,)2 q_
(3.8) (3.9)
y* = 2 ( 1 + 7 1 ~)
hTly*=2=
[4 (u~) 2 q- h~]y . . . . 2
and h~'---~ 1; u~--~ 0;y* --~ _0o.
(3.10)
Similarly, the constraints (2.10) and (2.18) can be expressed as, U*D*dy* + 2(I-e)
f[[
u2h2dy
(U*)2D*y*dy * + 2(1-~.)
+
f22(1+7')
ulhldy
= 0
[h2*(u~)2 - ~{ + (h~)2/2ly*dy *
(3.11) (3.12)
+ f l (l+'h) [h~'(u~')2 - I1/T+ (hT)z/z]y*dy * = O. (c) Perturbation e x p a n s i o n The expansion in e is not straightforward for two reasons. First, as already pointed out, the basic state (e = 0) contains speeds of 0(1). Secondly, the choice for the origin of the coordinates system implies that the basic flow upstream is a function of e (Fig. 4). Recall that the choice of the origin for the coordinate system was 'imposed' by the use of the moment of momentum. If the origin were in any other location, then the integrated torque associated with the pressure along DE would have remained nonzero, thus making it impossible to connect the upstream region with regions 1 and 2. As a result of this 'imposed' choice, however, the upstream variables, which do not 'know' about the presence of the cylinder downstream, are a function of a. It will become clear shortly that while these two conditions make the expansion somewhat more involved, they do not present any fundamental difficulty. It is assumed that, for region 1, the expansion has the form, v~l = -r*/2 + ev~]) + g21j~2)-'}- . . .
h~' = 1 _ (r,)2/8 + ahtl) + aeht2)+ . . . Vl = ~-'~/l 1) "[-
g2"/l 2) -Jr- ....
(3.13a) (3.13b) (3.13c)
where (3.3) and (3.4) have been used to express the terms corresponding to the basic state. For region 2, the expansion is assumed to have the form u* = U* + 8u~_x) + 82u~2~ + . . .
(3.14)
h* = D* + 8h~,t) + g2h~2) "4- . . .
(3.15)
v~ = v~j) + ev_~~) + . . . .
(3.16)
where U* and D * are the undisturbed upstream flow as given by (2.1), (2.2) and (3.5). Since U* and D* contain 8, they should be expanded in a Taylor series, U* = [1 - 28 + 282 + 0(83)] e y'~+2 + . . .
(3.17a)
372
D. NOF D* = 1 - [1 - 2g + 2c 2 + 0(~3)] e y''+2 + . . .
(3.17b)
which can be combined with (3.14) and (3.15) to, u-_s" = e y*+2 + a(-2e y°'+2 + u~,l)) + c2(2ey + 2 + u~_2)) + 0(~ 3) + . . . h* = (1 - e y*+2) + c(2e y'~+2 + h~,j~) + ~2(-2e y'+2 + h~_2)) + 0(c 3) + . . . .
(3.18) (3.19)
With the aid of the Taylor series expansion the power series are expressed in a way that clearly separates the zeroth order terms from the remaining terms; the first terms in (5.18) and (5.19) will be referred to as u~,°) and h~,J~), respectively. 4. S O L U T I O N
FOR THE OMEGA FLOW
(a) General solution for region 1 Substituting (3.13a), (3.13b) and (3.13c) into (3.6a) and (3.6b), and eliminating the terms corresponding to the basic state, we find the equation,
r* dr ~[r*vg')+ r*cvg])l -- ~:hl') +
8
+ 0(c 3)
+ ~;2h12)
(4.1)
+ 0(~3)
and 0 h i 1)
0 = ~--,
(4.2)
Or*
where the term in the square brackets in the right-hand side of (4.1) is 0(c) because r* - 2 + 0(c). It will become clear shortly that the term containing v~]) is actually 0(c) and not 0(e 2) so that it must be included in the first-order balance. T o simplify the structure of (4.1), it is recalled that the first-order flow in region 1 takes place within a distance of 0(c) from the cylinder surface so that one may introduce the transformation r* = 2(1 + ~ * ) ,
where
~* - 0(1).
In terms of this new variable, (4.l) is
---+c
2 d~*
2 d~*
v~ ~ h / ~ + ¢ *
+O(d)=O
which shows that
de*
o.
This and (4.2) give v~]) = B,;
hi I ) = A , ,
(4.3)
where AI and BI are constants to be determined from the boundary conditions.
373
The collision between the Gulf Stream and warm-core rings
Substitution of (3.13) and (4.3) into the polar version of the boundary conditions (3.8) gives, ~A~ + ½- (1 + gy/I))2/2 = 0
(4.4)
-(1 + g'//l)) + gB, = -1
(4.5)
A1 = B1 = ~/11)-
(4.6)
V~]) ---- ht 1) = yl |).
(4.7
which reduces to,
This gives
Most of the solution for region 1 has, therefore, been derived; the only part that is still missing is 7t l). (b) General solution for region 2 To find the solution for this area, (3.18) and (3.19) are substituted into (3.7a) and (3.7b). One finds that the 0(1) balance is automatically satisfied and that the first- and second-order balances are
~1~
Ov~1) Ou~l ) Ox*
Oy*
-h~');
ug_°~ v . ~
Ox*
Oh~ 1) + u~"-
(4.8)
Oy*
and
Ov~2~ Ou~2~ Ox*
--
Oy*
- h~,2);
U~0) 0V~2) 0V~ 1) -+ (-2e y*+2 + u~ 1)) - if- u p ) --
Ox*
(4.9)
0h7 )
Oy*
(4.9)
Oy*
A solution of (4.8) and (4.9) is U~ 1) =
A21ey* + B21e-Y*; h~l) _-_A21eY': + B21e-y*;
0v~ 1)
Ox*
= 0
(4.10)
Ova_2 ) U~2) = A22e-Y* + B22e-Y*;
h~_2) = -A22 ey* + B22e-Y*;
Ox*
- O.
(4.11)
Note that because of the symmetry of the problem (relative to the y axis) the continuity equation for region 2 [i.e. equation (3.7c)] is, obviously, satisfied. Substitution of (3.18) and (3.19) into (3.10) gives h~') --> O; h~2)---~ O; u~') ---~ O; u~2) --~ 0; y* ---> - ~ which, in view of (4.10), (4.11) imply that
B21 = B22 = 0.
(4.12)
374
D. NOV
There is an additional boundary condition that has not been used yet [i.e. condition (3.9)], but substitution of (4.10), (4.12), (4.7), (3.18), (3.19) and (3.13) into it shows that to 0(g) it is automatically satisfied. Overall, all the governing equations and all the boundary conditions have been used, and we are left with two first order unknowns, ~,tl) and A2j. These, as well as the second order variable A22, will be computed in the next subsection by considering the two constraints (3.11) and (3.12). (c) The mass conservation constraint Substitution of the solution, (4.10), (4.11), (4.12), (4.7), (3.13), (3.18), (3.19)and the upstream conditions (3.17) into (3.11) gives 2
(D* + gh~1) + ~32h~2))~-~(D* + 13h~1) + 132hf)) dy _oa + g2('}ttl))2 + 0(133) = 0
(4.13)
which can also be written as -1 - (D* + gh~1) + g2h~2))2
+ 2J('gt2)) 2 + 0(133) = 0.
(4.14)
By considering (4.17) again one can see that (4.14) reduces to - [ c ( 2 - A21 e-2) + 132(- 2 + A22e-2)] 2 + 2132(7tl))2 + 0033) = 0
(4.15)
2 - A21e-2 = V~ 7t 1).
(4.16)
which gives It is now clear why it is necessary to consider the 0(132) perturbations in region 2; the 0(13) perturbations in depth in region 1 cause a transport of 0(132) and, in region 2, this could be balanced by velocities or depth perturbations of 0(e2). With the derivation of (4.16), which contains two unknowns A21 and ]tll), we are left with one additional constraint that is needed to complete the solution. We shall see in the next subsection that this is provided by the integrated torque. (d) The torque constraint To obtain the various approximations for the torque, we note that according to (4.10) and (4.11) the solution for region 2 obeys, ~2 = (h~")2/2 + 0(133), so that substitution of (3.14) to (3.16), (3.13) and (4.7) into (3.12) gives (U*)2D*y*dy * + -2(l-e)
+
(U*)2D*y*dy * -2
[ -2(U, + 13A21ey* + e.2A22eY*)2(D* -13A21e y* _ g2A22eY*)y*dy* + 0(133) ~_av
(y,)2 + 22
[_2
8
q + 13~,t')[y*dy * + 0(133) = 0. 3
(4.17)
By using the transformation y* + 2 = ¢*, where ~* - 0(13), and (3.17a,b), the first
The collision between the Gulf Stream and warm-core rings
375
integral in (4.17) is found to be 4a~; similarly, the transformation y* - 2 = ~* shows that the last integral equals 282(~{tl))2. Using the above information, as well as (3.17), one finds that the 0(1) balances of (4.17) are automatically satisfied and that the 0(8) balance is a
f-~[2A2ffl-
e y*+2) - A21eY*+2]y*dy * = O.
(4.18)
This immediately gives A21 = 0.
(4.19)
~,t') = V2.
(4.20)
so that by (4.16) we find that
Similarly, one finds that the 0(82 ) terms in (4.17) yield [4 + 2(~,tl)) 2] + A22 f -2(_3e3Y* +4 + 2e2y*+2)y*dy * = 0
(4.21a)
--oo
which, with the aid of (4.20) gives, A22 = 48e 2.
(4.21b)
(e) The complete solution The complete-det,ailed solution begins with the radially symmetric solution for region 1 v~l = -r*/2 + 8X/2 + 0(82) h~' = ½ - (r*)2/8 + 8V~ + 0(82) ~,~' = 8V~ +
0(82).
(4.22)
This solution is expected to be valid, even at some distance from 1, provided that the flow is still parallel to the cylinder. In region 2 the solution is u~ = ( 1 - 28 + 5082)e y*+2 q- 0(83) v~ = 0
(4.23)
h~ = 1 - (1 - 28 + 5082)ey*+2 + 0(83). Recall that the 0(8) terms in u~' and h~ originate from the choice for the origin of the coordinate system; they are not a dynamical response to the forced cylinder. Also, as mentioned earlier, the solution in region 2 had to be carried to 0(82) because otherwise the continuity constraint and momentum constraint would not have been sufficient to close the problem. Relations (4.23) show that the flow in region 2 remains unaltered to 0(8) [since u~1) = h~ 1) = 0] and that the portion of the upstream current that is 'blocked' by the cylinder is simply diverted from its original position to the perimeter of the ring. This solution demonstrates that, no matter how small the penetration of the cylinder into the Stream, a current engulfing the cylinder must always be present. Namely, whenever 8 is finite and different from zero, h~ and "/1 are also finite (and different from zero),
376
D. NOF
implying that the response of the Stream to the presence of the cylinder cannot consist of a mere adjustment in region 2. One concludes, therefore, that an edge intrusion of the kind discussed in Section lc is inevitable. Before summarizing results, it is appropriate to comment on the neglect of motion in the lower layer and the 'replacement' of the actual ring by a solid cylinder. As far as the neglect of movement in the lower layer is concerned, it has been recently demonstrated by KILLWORTHet al. (1984) and FUERL (1984a,b) that, in certain cases, when the lower layer is of finite depth (instead of infinite depth as we have considered) then additional behavioral patterns occur. These studies do not invalidate the results of the so-called layer-and-a-half models (i.e. one moving layer and an infinitely deep lower layer) such as this one (e.g. Nov, 1985) but they do show that sometimes a complete understanding of the details of the problem cannot be achieved without considering a finite lower layer. This is particularly true when the unknown upper layer speeds are small as is the lens drift in FLIERL(1984b). Since our upper layer speeds are of 0(1), it is expected that the inclusion of a finite lower layer will only have a secondary effect on the processes under discussion. Also, an addition of a finite lower layer to an already complicated problem will, in the author's opinion, make the problem untractable and, therefore, is not considered. The use of a solid cylinder as an analogue for a cooled warm-core ring is supported by the following reasoning. As already pointed out in Section 1, there are some obvious similarities between a colliding ring and a colliding solid cylinder. In particular, both features are expected to exert a pressure on the Stream as they collide with it, and both features have similar geometry in the x, y plane. However, there are also some important differences. For instance, although both the actual ring and the solid cylinder are subject to pressure forces, the former can adjust itself to the surrounding pressure, whereas the latter remains unaltered. It is expected that the collision-induced adjustment in an actual ring will consist mainly of two processes--the first is the adjustment of the ring shape and the second is the change in the position of the ring.* The ring shape is probably not so important to the intrusion and the omega flow because the observed changes in shape are small, so that the neglected pressure torque also is expected to be small. On the other hand, the change in the position of the ring is probably more important because if the colliding ring bounces back to the Slope Water, the omega flow may never be reached. Despite the obvious differences, examination of the cylinder-Stream interaction is useful for understanding the basic processes in question. That is to say, the results of this study might pinpoint some of the effects in more complicated and more realistic models. Although the idea of replacing a warm-core ring with a solid cylinder is new and the simplifications may appear to be drastic, similar and even more severe simplifications have been used before. For example, as pointed out in Section 1, there have been a number of analytical models which considered various oceanic processes to be analogous to flows in pipe loops and hydraulic systems (e.g. STOMMEL, 1961; STOMMELand ROOTH, 1968). There is no doubt that much has been learned from these models even though their simplifications appear to be more severe than those employed in this study.
* It should be recalled that, in this model, the solid cylinder is held in a fixed position whereas the actual ring is free to move.
The collision between the Gulf Stream and warm-core rings
377
5. C O N C L U S I O N S
A new conceptual model for the Stream-ring interaction has been developed with the assumption that: (a) as a first approximation, the ring can be represented by a solid cylinder embedded in the Slope Water; (b) the Gulf Stream can be represented by a single moving layer whose lower boundary intersects the free surface; and (c) all motions are frictionless and hydrostatic. Our attention has been focused on the processes that take place after the cylinder is pushed slightly into the Stream. Such a conceptual model can help our understanding of the processes in question because, in a similar fashion to a cooled ring, the forced cylinder can support a pressure on its Gulf Stream side. Immediately after the introduction of the cylinder, an intrusion along the upstream side of the cylinder outer surface takes place. The intrusion advances clockwise along the perimeter of the cylinder at a rate of - O ( g ' h w ) ' , where g' is the reduced gravity (gAp/p) and hw corresponds to the Stream depth at the edge of the forced cylinder. Typically, g' - 5 x 10-3m s-~ for the intrusion, and hw is estimated to be roughly 20 to 80 m. This gives an intrusion advancement rate of 0(1 m s-~) which is of the same order as the intrusion propagation rates seen in satellite images [see, for example, images for 12 to 16 April 1982 (EVANSet al., 1984)]. Ultimately the intrusion completely surrounds the cylinder and reattaches itself to the Gulf Stream in the cylinder lee side. Once this has happened and a period of adjustment has elapsed [0(f)-l], a steady state is reached. In this state there is a narrow band of Gulf Stream water that flows around the cylinder and forms an omega-like pattern. The details of this steady 'omega flow' can be computed using the integrated moment of momentum and a perturbation scheme in c, the relative penetration of the cylinder into the Stream. It is found that the width is cV~ro and the near wall depth is cV~H, where ro is the radius of the cylinder and H is the undisturbed depth of the Gulf Stream. The above model suggests that the bands of Gulf Stream water which have been observed to envelop warm-core rings may not be related to the motion within the ring even though the motions of both processes have the same magnitude and direction. Since the model is inviscid, the addition of clockwise rotation to the solid cylinder will not influence the results. Most images analyzed by EVANSet al. (1984) suggest that the bands occur along the ring edge with the ring's water present only on the right-hand side of the advancing nose as suggested by the model. A few images indicate, however, that portions of the band occasionally may be entirely embedded in the ring--namely, they have ring's water on both their right and left side. With our present knowledge of the processes in question, it is impossible to tell what the causes of this behavior are. One may speculate, however, that this particular situation can be a result of the fact that the ring itself is stratified so that the intrusion penetrates along the surface corresponding to its own density. Finally, one can also speculate that the 'omega' flow is responsible for the recapture of warm-core rings by the Gulf Stream.
Acknowledgements--This study was supported by the Office of Naval Research contract No. N00014-82-C0404. It had gained significantly from a fruitful collaboration with the Warm-Core Rings Program (supported by the National Science Foundation). Discussions with B. Cushman-Roisin were also helpful. REFERENCES BROWN O. B., D. B. OLSON, J. W. BROWN and R. H. EVANS (1983) Satellite infrared observation of the
kinematics of a warm-core ring. Australian Journal of Marine and Freshwater Research. 34. 535-545.
378
D. NOF
EVANS R., K. BAKER, O. BROWN, R. SMITH, S. HOOKER, D. OLSON and the WARM CORE RINGS PROGRAM SERVICE OFFICE (1984) Satellite images of warm core ring 82-B, sea surface temperature and a chronological record of major physical events affecting ring structure. FLIERL G. R. (1984a) The structure and motion of warm core rings. Australian Journal of Marine and Freshwater Research, 35, 9-23. FLIERL G. R. (1984b) Rossby wave radiation from a strongly nonlinear warm eddy. Journal of Physical Oceanography, 14, 47-58. GRIFF1TH R. W. and E. J. HOPFINGER (1983) Gravity currents moving along a lateral boundary in a rotating fluid. Journal of Fluid Mechanics, 134, 357-399. JOYCE T. M., R. W. SCHMITTand M. C. STALCUP(1983) The influence of a warm-core ring. Australian Journal of Marine and Freshwater Research, 34, 515-525. JOYCE T. M., R. BACKLUS, K. BAKER, P. BLACKWELDER, O. BROWN, T. COWLES, R. EVANS, G. FRYXELL, D. MOUNTAIN, D. OLSON, R. SCHLITZ, R. SCHMITT, P. SMITtl, R. SMITH and P. WIEI~E (1984) Rapid evolution of a Gulf Stream warm-core ring. Nature, London, 308, 837-840. KILLWORTH P. D., N. PALDOR and M. E. STERN (1984) Wave propagation and growth on a surface front in a two-layer geostrophic current. Journal of Marine Research, 42, 761-785. KUBOKAWA A. and K. HANAWA (1984) A theory of semigeostrophic gravity waves and its application to the intrusion of a density current along a coast. Parts I and 11. Journal of the Oceanographical Society of Japan, 40, 247-270. NOF D. (1985) Joint vortices, eastward propagating eddies and migratory Taylor columns. Journal of Physical Oceanography, 15, 1114-1137. SIMPSON J. E. (1982) Gravity currents in the laboratory, atmosphere and ocean. Annual Review of Fluid Mechanics, 14, 213-234. STERN M. E. (1980) Geostrophic fronts, bores, breaking and blocking waves. Journal of Fluid Mechanics, 99, 687-703. STERN M. E., J, A. WHITEHEAD and B. L. HUA (1982) The intrusion of a density current along the coast of a rotating fluid. Journal of Fluid Mechanics, 123, 237-265. STOMMEL H. (1961) Thermohaline convection with two stable regimes of flow. Tellus, 13, 224-228. STOMMELH. and C. ROOTH (1968) On the interaction of gravitational and dynamic forcing in simple circulation models. Deep-Sea Research, 15, 165-170. VON KARMANT. (1940) The engineer grapples with nonlinear problems. Bulletin of the American Mathematical Society, 46, 615.
0198~41149/S6$3.00 + 0.1Xl © 1986PergamonPress Ltd.
Printed in Great Britain.
T h e collision b e t w e e n the G u l f S t r e a m a n d w a r m - c o r e rings DORON NOF*
(Received 1 May 1985; in rev&ed form 4 October 1985; accepted 7 October 1985) A b s t r a e t - - A non-linear two-layer model (with one active layer and one stagnant layer) is used to explain features frequently observed during ring-Gulf Stream interaction. Attention is focused on the narrow bands ( - 10 to 15 km) of Gulf Stream water which frequently envelop the colliding ring. The Gulf Stream is represented by an upper layer with a surfacing interface (front) on the north side, and the colliding ring is represented by a solid cylinder. Conceptually, the following steps are viewed as possible analogues to the actual ring-Stream interaction. Initially, the cylinder (ring) drifts toward the Stream in the Slope Water. Subsequently, it is pushed slightly into the frontal region and is then fixed. The resulting events that occur in the vicinity of the cylinder are examined and analytical solutions are constructed using power series expansions and integrated equations. It is found that even the slightest penetration of the cylinder into the Gulf Stream frontal region causes an intrusion of Gulf Stream water along the cylinder. Namely, as the cylinder is pushed into the Stream, a band of Gulf Stream water starts flowing along the cylinder upstream side. This band continues to intrude along the cylinder perimeter (in a clockwise manner) until it ultimately reattaches itself to the Gulf Stream in the downstream side. A short time after reattachment, a steady state is attained; namely, the cylinder is completely surrounded by Gulf Stream water and time-dependent motions cease. The detailed structure of this steady state is computed using the integrated moment of momentum.
1. I N T R O D U C T I O N
(a)
Background
RECENT satellite images of the ocean surface suggest that the interaction of warm-core rings with the Gulf Stream typically begins with a collision between a warm-core ring and a growing meander (e.g. JoYcE et al., 1984; EVANS et al., 1984). As a result of this collision, a narrow band (--10 to 15 km) of Gulf Stream water starts flowing along the western side of the ring (Fig. 1) and after a number of days the band envelops most of the ring. The purpose of the present paper is to examine this process and to suggest a mechanism that might be responsible for its generation. I do not intend to simulate all the details of the real system; I merely hope to preserve enough analogy to the real system so that my results will give some insight into the phenomena in question. The analytical modeling capabilities are limited and, therefore, I explore the processes in question in simple idealized systems. The reader should be prepared to accept some drastic simplifications. Consider the following three-layer model as an idealized formulation of the problem. The first layer (whose density is p) corresponds to the Gulf Stream, which is bounded on * Department of Oceanography, The Florida State University, Tallahassce, FL 32306, U.S.A. 359
360
D. NOF
intruding Gulf Stream
water
~
warm-core
Gulf Stream Fig. 1. Schematic diagram of the features associated with a typical interaction between the Gulf Stream and a warm-core ring. The diagram was adapted (by eye) from an infrared satellite image presented by BROWN et al. (1983); the boundaries were identified by tracing the maximum temperature gradients. Note that satellite images of other rings display a very similar structure, i.e. there is usually a band of Gulf Stream water that envelops the ring's left edge (see, lot example, images for 12 to 16 April 1982 in EVANS et al., 1984).
the north by a surfacing interface. The second (lens-like) layer is the warm-core ring (whose density is p + Ap') and the third (whose density is p + Ap) is an infinitely deep fluid in which both the ring and the Gulf Stream are embedded. An important property of this model is that the warm-core ring has a greater density than the Gulf Stream even though the ring originated from the Stream (i.e. 0 < Ap' < A9). This results from the ring's relatively long stay (3 to 6 months) in the Slope Water where strong atmospheric cooling frequently occurs. The nature of the ring-Stream interaction will be examined by assuming that due to a growing meander, advective currents or wind, the cooled ring has collided with the Stream at, say, t = 0. To simplify the problem assume that, as a first approximation, the colliding ring can be represented by a solid cylinder. With the aid of this important simplification, we can view the collision as follows. Initially the solid cylinder drifts slowly in the Slope Water toward the (steady) Gulf Stream. We may suppose that during this early stage there is some (mechanical) process that is causing the movement of the cylinder. At t = 0 the cylinder collides with the ring and, because of its initial drift, it slightly pushes the Gulf Stream front as shown in Fig. 2. Because of the increasing sea-level height away from the front, the penetration of the cylinder into the frontal zone ultimately will be arrested. It is assumed that, at this point, the cylinder is (mechanically) held fixed. Subsequently, there will be a Gulf Stream response to the forced cylinder and it is this motion on which we shall focus. It will be shown that even the slightest penetration of the cylinder causes an intrusion of Gulf Stream water around the cylinder edge; this condition is referred to as an edge intrusion. After the edge intrusion completely envelops the cylinder, it reattaches itself to the Gulf Stream and a steady state, whose shape resembles the Greek letter 'omega,' is reached. The solution for this 'omega flow' is obtained using the integrated moment of
The collision between the Gulf Stream and warm-core rings
A' (/0+/%/0) y~
361
~x C'
B IO)
GulfStream (t--O)
(a) Top view
"tB
(b) Side view (t=O)
~(E
(p)
(~
to)
Fig. 2. The intuitively expected Gulf Stream response to the forced presence of the cylindcr. It is expected that the Stream will simply bend and follow the cylinder outer surface on the righthand side. Dynamical considerations reveal, however, that such a response is impossible and that, instead, a band of Gulf Stream water will flow from point A to the left-hand side (Fig. 4) until it will ultimately reach point C (Fig. 5).
momentum (Sections 2 to 4). With the establishment of the steady circulation mentioned above, the cylinder is completely surrounded by Gulf Stream water. It is not a priori obvious under what conditions the replacement of a cooled warm-ring by a solid cylinder is justified. The resulting limitations and weaknesses will be discussed in detail in Section 4. The most important similarity between a colliding warm-core ring and a colliding cylinder is that both are expected to exert a pressure on the Gulf Stream front as they collide with it. A second similarity is a geometrical one, i.e. both features are either circular or close to being circular. An important difference between the two is
362
D. NOF
that a solid cylinder does not transmit pressure nor does it adjust its shape according to the environmental forces. It is believed, however, that the similarities between a colliding ring and a colliding solid are sufficient to permit the use of the solid cylinder as an analogue. It will become clear later that, with the aid of such an analogue, the results may pinpoint some of the processes which one should look for in more 'realistic' and more cemplicated models. Note that such analogues are not entirely new in the earth sciences. For example, the differentially heated annulus has been used extensively to study atmospheric motion even though the annulus is not a direct duplication of the atmospheric environment. The pipes and hydraulic models discussed by STOMMEL(1961) and STOMMELand ROOTH(1968) are another example of such analogues. (b) Formulation o f the problem Consider again the system described above. Suppose that only the upper layer is active and that all the motions are frictionless, hydrostatic and nondiffusive. Note that the depth variations are of 0(1), so that the quasi-geostrophic theory does not apply. Let us first consider the events that will take place after the cylinder has been forced into the Stream. Intuitively, one might expect that the Stream response will simply consist of a somewhat bended flow along the cylinder right-hand side (Fig. 2). This implies that there is a streamline connecting point A ' with A and B so that we may apply the Bernoulli integral to the Stream's edge. 1 2 + g'hB = .2 I U 2o, ~UB
(1.1)
where g' is the 'reduced gravity' (gAp/p, with Ap being the density difference between the layers), U0 is the known upstream speed along the front (point A'), and UB and hB are the unknown speed and depth at B. Since hB is always positive, (1.1) implies that UB < Uo. However, if the entire Stream flows to the right of the cylinder, as has been temporarily assumed, then continuity implies some convergence across the line connecting point B and y ~ - ~ so that uB > U0. These conditions, required by the continuity equation and the Bernoulli principle, are obviously incompatible, suggesting that there cannot be a streamline connecting A', A and B. Instead, it is expected that there will be a band of Gulf Stream water flowing around the cylinder in a clockwise manner (Fig. 3). In other words, particles moving along the Gulf Stream edge (i.e. the surface front) do not have sufficient energy to rise to point B and, therefore, they must go around the cylinder where the fluid is lower. Hereafter, we shall call this narrow band flowing along the ring perimeter edge intrusion, A formal proof for the inevitable existence of the edge intrusion will be provided in Sections 2 to 4 where a computation of the omega flow (i.e. the state where the intrusion completely envelops the ring and reattaches itself to the Gulf Stream in the cylinder lee side) is presented. It should be pointed out, however, that for the special case corresponding to U0 = 0 (i.e. a stream with a zero speed along the upstream edge) no proof is really necessary because under such conditions (1.1) can never be satisfied. Before completing the present discussion, it is appropriate to mention that, by assuming a cross-Stream geostrophic balance and using a slow variation expansion, one finds a zeroth order flow that has a vanishing depth at all points around the southern side of the cylinder. This may give the mistaken impression that no flow around the north side of the cylinder must take place. This is not the case, however, because the associated
The collision between the Gulf Stream and warm-core rings
363
Gulf Stream TOP V I E W
~
½
o
SIDE VIEW
Fig. 3.
The intrusion of Gulf Stream water around a solid cylinder. The upper panel should be compared to the Gulf Stream band seen in satellite images (e.g. Fig. 1).
first-order flow must always have h > 0 on the southern side of the cylinder; otherwise there will not be any resistance to the forced cylinder which is, obviously, impossible. With h > 0 (which, as just pointed out, is inevitable) there will always be a first-order flow around the cylinder, as suggested by our earlier argument. At this stage, the reader may wonder how coastal flows get around bumps without leaking fluid to the side as proposed by the argument presented above. The answer to this question is simply that leakage will not occur unless the current has a front on the upstream side of the bump. Most coastal currents do not satisfy this requirement because they flow with the coast on their right side (looking downstream) so that their front is located on the ocean side where there are no coastal bumps. (c) Edge intrusion and the time dependent problem The structure of the transient intrusion is rather complicated because we do not even know the intrusion mass transport. However, the order of magnitude of the intrusion propagation rate can be obtained by considering the intrusion width to be small compared to the radius of the cylinder. It is not a priori obvious that this approximation is adequate because the curvature effects are not entirely negligible. For example, a cylinder with a
364
D. Nov
radius (r0) of about 45 km with an intrusion speed (u) of - 0 . 5 m s-1 and an intrusion width (w) of - 1 5 kin, will have curvature induced speeds [v - 0(u w/ro)] of about 15 cm s-1. Although such speeds are not entirely negligible compared to the intrusion speed, they constitute no more than one third of the propagation rate and, therefore, their neglect cannot alter the order of magnitude of the propagation rate. With the above simplification, the problem reduces to the intrusion of a density current along a coast. This problem received much attention in recent years mainly because of its relevance to the way that rivers empty in the ocean (e.g. STERN, 1980; STERNet al., 1982; GR1FVmt and HOPFINGER, 1983; KUBOKAWAand HANAWA, 1984; SIMPSON, 1982). A result of these investigations is that no steady migration rate can ever be achieved. There usually exists, however, a quasi-steady propagation rate whose magnitude is roughly between (g'hw)" and 2(g'h,,) ~ (where hw is the intrusion upstream depth near the wall). As far as the oceanic intrusion is concerned, it is difficult to estimate hw because this would depend on the strength of the collision. However, the satellite images presented by EVANSet al. (1984) suggest that, in the Gulf Stream, disturbances associated with the collision penetrate a few kilometers into the Stream where the Stream depth is 20 to 80 m. Such depths are consistent with the intrusion depth presented by JovcE et al. (1983). Together with a reduced gravity of 5 x 10-3 s-2 [corresponding to (Ap/p) = 5 × 10 -4 which is suggested by the horizontal temperature difference seen in the satellite images presented by EVANSet al. (1984), and the cross-sections shown by JovcE et al. (1983)], this gives a nose propagation rate of about 0.4 to 1.4 m s-1. This is of the same order as the intrusion advancement rate seen in the satellite images. It should be stressed, however, that, as already mentioned, this estimate only provides the order of magnitude of the propagation rate. 2. T H E S T E A D Y ' O M E G A '
FLOW--GOVERNING
EQUATIONS
AND CONSTRAINTS
The present section has two aims. The first is to prove that there must always be a flow around the cylinder so that the time dependent intrusion discussed earlier is inevitable. The second is to find out how the Stream responds to the presence of the cylinder. Specifically, one would like to compute the omega flow speed, width and depth as a function of the distance that the cylinder is pushed into the Stream. Because of the inherent non-linearity of the problem, it is unlikely that one will be able to find analytical solutions for the whole field. Consequently, we shall attempt to find the desired flux, speed and depth without solving for the entire field. (a) General description Consider the two-layer system shown in Figs 4 and 5. The origin of our coordinate system is located at the center of the cylinder; it will become clear later that this choice is not arbitrary. The x and y axes are oriented along and across the undisturbed Stream (respectively) and the system rotates uniformly at f/2 about the z axis. The upstream current flows on top of an infinitely deep motionless layer; it has a uniform potential vorticity f / H (where H is the upper layer depth at y --~ -oo). Consequently, its velocity (U) and depth (D) obey f U = -g'OD/Oy and -OU/Oy + f = f D / H . The current edge (D = 0) is located at y = -ro (1 - ~) and U --~ 0 as y --9 - w , so that we have U = (g'H)"-'e ly+r''(l-c)l/R,'
(2.1)
The collision between the Gulf Stream and warm-core rings
A
I// : O, h=O
~r0
U,D
region ~)
I
I region (~
I I I
I I I
=J C
B
Fig. 4. A schematic diagram of the regions under discussion and the integration area (dashed line) for the moment of momentum equation. Sections AB and BC are located several deformation radii away from the origin so that the flow there is not disturbed by the forced cylinder. Note that it is not necessary to limit our .model to cylinders that are circular along all their periphery. Cylinders with apartially 'fiat' periphery near point D can also be considered; to 0(~ ~) the two problems are identical.
Y region (~ p
F
!!!ii!!i!!ii!iii!i:
>
C
x
2tRd
region (~
region (~ Fig. 5. A sketch of the forces and torque associated with the integration area shown in Fig. 4. Overall, there are four forces acting on the boundary of the integration area (ABCDEF); three of the forces (F~; F2; F3) are associated with regions 1, 2 and 3 and the fourth is the pressure exerted on the surface of the cylinder. Only the former three have a torque (relative to the origin) because the pressure (at, say, D ' ) is perpendicular to the surface of the cylinder so that it does not have any moment relative to 0. This property enables one to connect regions 1, 2 and 3 without solving for the whole field.
365
366
D. NOF D = H(1 - ely+r"(l-~)l&'),
(2.2)
where Ra is the deformation radius, (g'H)~/f, and e is the relative penetration of the cylinder (Fig. 4). Note that the presence of e in these equations is not an indication that the upstream flow is influenced by the cylinder; it is merely a result of our choice for the origin of the coordinate system. For our model it will be assumed that the upstream structure (2.1) and (2.2) is not influenced by the presence of the cylinder because all linear waves on uniform potential vorticity flows propagate downstream. (b) Governing equations for regions 1 and 2 For both regions 1 and 2 (Fig. 4), the governing equations are the usual shallow water equations. Since the Stream is assumed to have uniform potential vorticity (f/H) we may use the equation for conservation of potential vorticity. We, therefore, have,
Ovi/Ox - OuJOy + f = &f/H; OVi
i = 1, 2
OF i
ui -~x + vi -~y + fui = -g Ohi/Oy; 0 0 - - (hiui) + (hivi) = 0; ox
i = 1, 2
i = 1, 2,
(2.3) (2.4)
(2.5a)
where u and v are the horizontal depth-independent velocity components in the x and y direction, h is the depth, and the subscripts '1' and '2' denote that the variable in question is associated with regions 1 and 2, respectively. Note that because of the symmetry of the problem (i.e. v = 0; hx = 0 at cross-sections 1 and 2) the x momentum equation and the continuity equation imply that, OUi
Ox
-- 0.
(2.5b)
The boundary conditions for region 1 are, h~ = O; y = ro(1 + Yl)
(2.6)
[Ul2]y=r,,(l+y,) = [U2]y=_r,,(l_e) = g ' m
(2.7)
i 2 +g [~ut
'h
t]y=r,, = [.,u, ~ 2_ + g ' h 2lv=-r.
(2.8)
Condition (2.6) states that the depth vanishes at some unknown location; conditions (2.7) and (2.8) reflect the conservation of energy along the streamlines that bound the flow from left and right (looking downstream), respectively. Namely, (2.7) and (2.8) are simply a result of an application of the Bernoulli integral to the streamline connecting A and F and the streamline connecting E and D. Note that (2.8) represents a connection between regions 1 and 2. The boundary conditions for region 2 are, h 2 --~ H ;
U2 --* 0;
y --* - ~ .
(2.9)
The collision between the Gulf Stream and warm-core rings
367
(c) Constraints The flows in the various regions are connected to each other via (2.7) and (2.8) but there are two additional equations that the unknown variables must satisfy. The first results simplify from continuity and can be written in the form, UDdy + -r.(1-~)
f-r,,
u2h2dy +
Ulhldy = O.
_a¢
(2.10)
r,,
The second equation will be derived from the conservation of moment of momentum. This is essentially the conservation of torque exerted by the fluid. It corresponds to the cross product of the position vector r and the momentum equations, y
u--+v---fv+ Ox Oy
-~x
-x
u--+v--+fu+ Ox Oy
-~y
=0.
(2.11)
To show that (2.11) provides an additional connection between the upstream flow and regions 1 and 2,\(2.11) is multiplied by h and the continuity equation is incorporated. This gives, 0
O (hu2y) + (huv)-fvyh Ox Y-~y 0
O
- x --Ox(huv)
g' O +-(h2y) 20x g'
(2.12)
O
-~y (hxv 2) - fuhx - --2 --Oy (h2x) = 0
(2.13)
which can be rearranged and integrated over the region shown in Fig. 4, to give,
fro( -~x
+
fs0( ~y
,
hu2y - f ~ y + -- h2y - huvx 2
,
hbll;y + f~X - - ~ hZx- hxl; 2
)
dxdy
) dxdy = O,
(2.14)
where ~ is a stream function defined by, O~ Oy
O~ -
uh;
Ox
-
(2.15)
vh.
By using Stokes' theorem, (2.14) can be written in the form, hu2y - f ~ y + ~ h2y - huvx
-
huvy - hxv 2 + fallx - -~ h2x
dy
dx = O.
(2.16)
368
D. Nov
This equation can be further simplified by, (a) defining ~g to be zero along the left edge where h -- 0, (b) noting that along any streamline udy = vdx and that v = 0 along AB, BC, CD and EF, and (c) noting that the flow along BC is identical to that which would be present without the cylinder. Namely, for AB as well as BC, the geostrophic relationship fU = -g'OD/Oy can be multiplied by D and integrated once in y to give jfg = g'D2/ 2 + A, where A is a constant to be determined from the boundary conditions. Since we have chosen ~g = 0 where D = 0, we immediately find that A = 0. Using the above conditions we can simplify (2.16) to,
hu2ydy + A
fo
(hu 2 -fag + g'h2/2)ydy +
C
+
fF
(hu 2 -f~g + g'h2/2)ydy
E
(-]Sffy+ g'h2y/2)dyD
( f ~ x - g'h2x/2)dx = 0.
(2.17)
D
The first three terms are the moments of the forces in the upstream region, region 2, and region 1. The last two terms, on the other hand, represent the torque corresponding to the pressure exerted on the cylinder by the surrounding flow. Since the origin is the center of the cylinder and the pressure is always perpendicular to the surface with which the fluid is in contact, we would expect this torque to vanish. It is easy to see that since the cylinder surface is given by x 2 + y2 _ r~, we have xdx + ydy = 0 so that the sum of the last two integrals in (2.17) equals zero as expected. Hence, the integrated moment of momentum constraint takes the simple form,
DU2ydy + A
h2u2 - f v 2 + ---~-jyuy + C
h,u~-f~, +
ydy = 0,
(2.18)
E
where we have incorporated our special notation for the various regions. Note that (2.18) does not involve any variables other than those of regions 1 and 2 and the upstream area (region 3). Its physical meaning is illustrated in Fig. 5; it corresponds to a balance between the three moments created by the three forces. At first sight, the derivation of (2.18) may give the mistaken impression that, because of the vanishing of the pressure torque related to the surrounding flow, it corresponds to a special balance of forces. This is not the case because, mathematically (and physically), the vanishing pressure torque results from the fact that the solid cylinder is round. It does not imply that there is no pressure on the cylinder--there certainly are forces acting on it. The only cases for which the pressure torque of the surrounding flow [i.e. the last two terms in (2.17)] is not expected to vanish are those associated with bodies whose crosssections are not round. Since most warm-core rings are nearly circular, the actual pressure torque in the ocean is expected to be rather small. 3.
SCALING
AND
EXPANSION
OF THE
OMEGA
FLOW
(a) The basic state Before discussing the scaling of the problem and the general structure of the expansion, it is instructive to look at the details of the basic state. The structure of the zeroth order state, corresponding to the cylinder 'kissing' the Stream, is not a priori obvious. Application of the Bernoulli principle to the cylinder surface and the upstream
369
The collision between the Gulf Stream and warm-core rings
edge [(see (2.17)] implies that even when ~--, 0 the velocity along the cylinder surface is of 0(1). This means that the basic flow around the cylinder cannot be zero; rather, it must consist of an infinitesimal ribbon flowing at a speed ( g ' H ) ' , To find the details of this ribbon flow it should be noted that even though the basic state contains only an infinitesimal strip, it must satisfy the equations of motion. To find the detailed mathematical structure, it is useful to consider the potential vorticity equation and momentum conservation in cylindrical coordinate (r, 0),
1 d (3.1)
- - - (rOo) + f = f i f t H
r dr -9
v~i --
r
dh + fro
(3.2)
= g' --,
dr
where O0 is the tangential velocity, the bar (-) indicates association with the basic state, and we have taken the flow to be purely tangential (i.e. O, = 0/00 = 0). When c goes to zero, the depth around the cylinder goes to zero as well, implying that the appropriate solution for the basic state can be found by looking at the limit of the governing equations as h ~ 0. For this situation, (3.1) and (3.2) have the most general solution, fr + _ . Oo -
2
~ = (~-
r '
r2) -
+
_
8g'
(3.3)
-- , -75- 2g '
where ct is an unknown constant and we have used the condition that h = 0 at r = r.. Since at r = r 0, the absolute value of the velocity must be (g'H)" (in order to conserve the Bernoulli function along the wall and edge), we find from (3.3) that, ( / = ~ f [ ( r o - R~I):
R~],
(3.4)
where Ra = ( g ' H ) ' / f . Namely, for any given cylinder (r0), we must take a specific value for eL. For simplicity, we shall consider only cylinders with ro = 2Rj so that cc = 0. Other cylinders can, of course, also be considered and the solution, which will shortly be derived, can be easily extended to cylinders with all diameters. However, such extended solutions do not provide any new physical insights and therefore are not presented. (b) Scaling In the subsequent analysis the following nondimensional variables will be used: upstream:
U* = U / ( g ' H ) ' ;
Region 1
.,
!
!
u~ = u l / ( g H)-';
and its
,
t
y* = y / R j . J
v~ = v J ( g H)~;
x't~ = x / R ` g
vicinity: Region 2:
D* = D / H ;
r~= rJRj
(3.5a)
y~ = y t / R , i (3.5b)
= 2
~ = ~J(g'H2/f). ,
!
!
u~' = u2/(g H)~; xg" = c'-'x/R,g
v~ = v 2 / ( c g ' H ) ' ; V2" = ~2/(g'H2/f);
y~ = y 2 / R j c ~ 1.
(3.5c)
370
D. NoF
Note that in the upstream region [i.e. variables defined by (3,5a)], the x length scale is infinite, and that in region 2 [i.e. variables defined by (3.5c)], the ratio between the x length scale and the y length scale (Ra) is given by the slope of the cylinder surface which is 0(e') (Fig. 6). In other words, due to the presence of the cylinder, a finite length scale in the x direction is generated. Since the slope of the cylinder surface in the vicinity of region 2 represents the ratio between v2 and uz, it must also represent the ratio between the y and the x scales due to continuity. For the first-order flow in region 1, it is convenient to use the governing equations in cylindrical coordinates (r*, 0). In terms of the nondimensional numbers defined by (3.5), the governing equations are, 1 d r* dr* (r*v~) + 1 = hT
(3.6a)
(v~t) 2 Oh~ - + v~l ,
(3.6b)
r*
Or*
where it has been assumed that the flow is parallel to the cylinder periphery so that Vr*~ = 0/00 = 0 and v~l is the only velocity component in the area. The nondimensional governing equations for region 2 are found from (3.5), (2.3), (2.4) and (2.5) to be,
Ov~ Ou~
e--
Or*
Ov~
eu~--
Ox *
--
Oy*
+ ev~
+ 1 =h~
Ov~ Oy *
+ u~ -
(3.7a)
Oh~
(3.7b)
Oy *
0 0 #x* (h~u~) + - - (h~v~) = 0. dy*
(3.7c)
~Y ::iii!i!i!ii!i!i!iii!i!ii!i!:i ~_ ,r£,.--O,
£ir( -
T
J~ro-(2£11/2
-.,
region
Fig. 6.
tr0
r
The geometry along the left boundary of region 2.
371
The collision between the Gulf Stream and warm-core rings
The boundary conditions (2.6) and (2.7) take the form h~'=0; [1 (U~,)2 q_
(3.8) (3.9)
y* = 2 ( 1 + 7 1 ~)
hTly*=2=
[4 (u~) 2 q- h~]y . . . . 2
and h~'---~ 1; u~--~ 0;y* --~ _0o.
(3.10)
Similarly, the constraints (2.10) and (2.18) can be expressed as, U*D*dy* + 2(I-e)
f[[
u2h2dy
(U*)2D*y*dy * + 2(1-~.)
+
f22(1+7')
ulhldy
= 0
[h2*(u~)2 - ~{ + (h~)2/2ly*dy *
(3.11) (3.12)
+ f l (l+'h) [h~'(u~')2 - I1/T+ (hT)z/z]y*dy * = O. (c) Perturbation e x p a n s i o n The expansion in e is not straightforward for two reasons. First, as already pointed out, the basic state (e = 0) contains speeds of 0(1). Secondly, the choice for the origin of the coordinates system implies that the basic flow upstream is a function of e (Fig. 4). Recall that the choice of the origin for the coordinate system was 'imposed' by the use of the moment of momentum. If the origin were in any other location, then the integrated torque associated with the pressure along DE would have remained nonzero, thus making it impossible to connect the upstream region with regions 1 and 2. As a result of this 'imposed' choice, however, the upstream variables, which do not 'know' about the presence of the cylinder downstream, are a function of a. It will become clear shortly that while these two conditions make the expansion somewhat more involved, they do not present any fundamental difficulty. It is assumed that, for region 1, the expansion has the form, v~l = -r*/2 + ev~]) + g21j~2)-'}- . . .
h~' = 1 _ (r,)2/8 + ahtl) + aeht2)+ . . . Vl = ~-'~/l 1) "[-
g2"/l 2) -Jr- ....
(3.13a) (3.13b) (3.13c)
where (3.3) and (3.4) have been used to express the terms corresponding to the basic state. For region 2, the expansion is assumed to have the form u* = U* + 8u~_x) + 82u~2~ + . . .
(3.14)
h* = D* + 8h~,t) + g2h~2) "4- . . .
(3.15)
v~ = v~j) + ev_~~) + . . . .
(3.16)
where U* and D * are the undisturbed upstream flow as given by (2.1), (2.2) and (3.5). Since U* and D* contain 8, they should be expanded in a Taylor series, U* = [1 - 28 + 282 + 0(83)] e y'~+2 + . . .
(3.17a)
372
D. NOF D* = 1 - [1 - 2g + 2c 2 + 0(~3)] e y''+2 + . . .
(3.17b)
which can be combined with (3.14) and (3.15) to, u-_s" = e y*+2 + a(-2e y°'+2 + u~,l)) + c2(2ey + 2 + u~_2)) + 0(~ 3) + . . . h* = (1 - e y*+2) + c(2e y'~+2 + h~,j~) + ~2(-2e y'+2 + h~_2)) + 0(c 3) + . . . .
(3.18) (3.19)
With the aid of the Taylor series expansion the power series are expressed in a way that clearly separates the zeroth order terms from the remaining terms; the first terms in (5.18) and (5.19) will be referred to as u~,°) and h~,J~), respectively. 4. S O L U T I O N
FOR THE OMEGA FLOW
(a) General solution for region 1 Substituting (3.13a), (3.13b) and (3.13c) into (3.6a) and (3.6b), and eliminating the terms corresponding to the basic state, we find the equation,
r* dr ~[r*vg')+ r*cvg])l -- ~:hl') +
8
+ 0(c 3)
+ ~;2h12)
(4.1)
+ 0(~3)
and 0 h i 1)
0 = ~--,
(4.2)
Or*
where the term in the square brackets in the right-hand side of (4.1) is 0(c) because r* - 2 + 0(c). It will become clear shortly that the term containing v~]) is actually 0(c) and not 0(e 2) so that it must be included in the first-order balance. T o simplify the structure of (4.1), it is recalled that the first-order flow in region 1 takes place within a distance of 0(c) from the cylinder surface so that one may introduce the transformation r* = 2(1 + ~ * ) ,
where
~* - 0(1).
In terms of this new variable, (4.l) is
---+c
2 d~*
2 d~*
v~ ~ h / ~ + ¢ *
+O(d)=O
which shows that
de*
o.
This and (4.2) give v~]) = B,;
hi I ) = A , ,
(4.3)
where AI and BI are constants to be determined from the boundary conditions.
373
The collision between the Gulf Stream and warm-core rings
Substitution of (3.13) and (4.3) into the polar version of the boundary conditions (3.8) gives, ~A~ + ½- (1 + gy/I))2/2 = 0
(4.4)
-(1 + g'//l)) + gB, = -1
(4.5)
A1 = B1 = ~/11)-
(4.6)
V~]) ---- ht 1) = yl |).
(4.7
which reduces to,
This gives
Most of the solution for region 1 has, therefore, been derived; the only part that is still missing is 7t l). (b) General solution for region 2 To find the solution for this area, (3.18) and (3.19) are substituted into (3.7a) and (3.7b). One finds that the 0(1) balance is automatically satisfied and that the first- and second-order balances are
~1~
Ov~1) Ou~l ) Ox*
Oy*
-h~');
ug_°~ v . ~
Ox*
Oh~ 1) + u~"-
(4.8)
Oy*
and
Ov~2~ Ou~2~ Ox*
--
Oy*
- h~,2);
U~0) 0V~2) 0V~ 1) -+ (-2e y*+2 + u~ 1)) - if- u p ) --
Ox*
(4.9)
0h7 )
Oy*
(4.9)
Oy*
A solution of (4.8) and (4.9) is U~ 1) =
A21ey* + B21e-Y*; h~l) _-_A21eY': + B21e-y*;
0v~ 1)
Ox*
= 0
(4.10)
Ova_2 ) U~2) = A22e-Y* + B22e-Y*;
h~_2) = -A22 ey* + B22e-Y*;
Ox*
- O.
(4.11)
Note that because of the symmetry of the problem (relative to the y axis) the continuity equation for region 2 [i.e. equation (3.7c)] is, obviously, satisfied. Substitution of (3.18) and (3.19) into (3.10) gives h~') --> O; h~2)---~ O; u~') ---~ O; u~2) --~ 0; y* ---> - ~ which, in view of (4.10), (4.11) imply that
B21 = B22 = 0.
(4.12)
374
D. NOV
There is an additional boundary condition that has not been used yet [i.e. condition (3.9)], but substitution of (4.10), (4.12), (4.7), (3.18), (3.19) and (3.13) into it shows that to 0(g) it is automatically satisfied. Overall, all the governing equations and all the boundary conditions have been used, and we are left with two first order unknowns, ~,tl) and A2j. These, as well as the second order variable A22, will be computed in the next subsection by considering the two constraints (3.11) and (3.12). (c) The mass conservation constraint Substitution of the solution, (4.10), (4.11), (4.12), (4.7), (3.13), (3.18), (3.19)and the upstream conditions (3.17) into (3.11) gives 2
(D* + gh~1) + ~32h~2))~-~(D* + 13h~1) + 132hf)) dy _oa + g2('}ttl))2 + 0(133) = 0
(4.13)
which can also be written as -1 - (D* + gh~1) + g2h~2))2
+ 2J('gt2)) 2 + 0(133) = 0.
(4.14)
By considering (4.17) again one can see that (4.14) reduces to - [ c ( 2 - A21 e-2) + 132(- 2 + A22e-2)] 2 + 2132(7tl))2 + 0033) = 0
(4.15)
2 - A21e-2 = V~ 7t 1).
(4.16)
which gives It is now clear why it is necessary to consider the 0(132) perturbations in region 2; the 0(13) perturbations in depth in region 1 cause a transport of 0(132) and, in region 2, this could be balanced by velocities or depth perturbations of 0(e2). With the derivation of (4.16), which contains two unknowns A21 and ]tll), we are left with one additional constraint that is needed to complete the solution. We shall see in the next subsection that this is provided by the integrated torque. (d) The torque constraint To obtain the various approximations for the torque, we note that according to (4.10) and (4.11) the solution for region 2 obeys, ~2 = (h~")2/2 + 0(133), so that substitution of (3.14) to (3.16), (3.13) and (4.7) into (3.12) gives (U*)2D*y*dy * + -2(l-e)
+
(U*)2D*y*dy * -2
[ -2(U, + 13A21ey* + e.2A22eY*)2(D* -13A21e y* _ g2A22eY*)y*dy* + 0(133) ~_av
(y,)2 + 22
[_2
8
q + 13~,t')[y*dy * + 0(133) = 0. 3
(4.17)
By using the transformation y* + 2 = ¢*, where ~* - 0(13), and (3.17a,b), the first
The collision between the Gulf Stream and warm-core rings
375
integral in (4.17) is found to be 4a~; similarly, the transformation y* - 2 = ~* shows that the last integral equals 282(~{tl))2. Using the above information, as well as (3.17), one finds that the 0(1) balances of (4.17) are automatically satisfied and that the 0(8) balance is a
f-~[2A2ffl-
e y*+2) - A21eY*+2]y*dy * = O.
(4.18)
This immediately gives A21 = 0.
(4.19)
~,t') = V2.
(4.20)
so that by (4.16) we find that
Similarly, one finds that the 0(82 ) terms in (4.17) yield [4 + 2(~,tl)) 2] + A22 f -2(_3e3Y* +4 + 2e2y*+2)y*dy * = 0
(4.21a)
--oo
which, with the aid of (4.20) gives, A22 = 48e 2.
(4.21b)
(e) The complete solution The complete-det,ailed solution begins with the radially symmetric solution for region 1 v~l = -r*/2 + 8X/2 + 0(82) h~' = ½ - (r*)2/8 + 8V~ + 0(82) ~,~' = 8V~ +
0(82).
(4.22)
This solution is expected to be valid, even at some distance from 1, provided that the flow is still parallel to the cylinder. In region 2 the solution is u~ = ( 1 - 28 + 5082)e y*+2 q- 0(83) v~ = 0
(4.23)
h~ = 1 - (1 - 28 + 5082)ey*+2 + 0(83). Recall that the 0(8) terms in u~' and h~ originate from the choice for the origin of the coordinate system; they are not a dynamical response to the forced cylinder. Also, as mentioned earlier, the solution in region 2 had to be carried to 0(82) because otherwise the continuity constraint and momentum constraint would not have been sufficient to close the problem. Relations (4.23) show that the flow in region 2 remains unaltered to 0(8) [since u~1) = h~ 1) = 0] and that the portion of the upstream current that is 'blocked' by the cylinder is simply diverted from its original position to the perimeter of the ring. This solution demonstrates that, no matter how small the penetration of the cylinder into the Stream, a current engulfing the cylinder must always be present. Namely, whenever 8 is finite and different from zero, h~ and "/1 are also finite (and different from zero),
376
D. NOF
implying that the response of the Stream to the presence of the cylinder cannot consist of a mere adjustment in region 2. One concludes, therefore, that an edge intrusion of the kind discussed in Section lc is inevitable. Before summarizing results, it is appropriate to comment on the neglect of motion in the lower layer and the 'replacement' of the actual ring by a solid cylinder. As far as the neglect of movement in the lower layer is concerned, it has been recently demonstrated by KILLWORTHet al. (1984) and FUERL (1984a,b) that, in certain cases, when the lower layer is of finite depth (instead of infinite depth as we have considered) then additional behavioral patterns occur. These studies do not invalidate the results of the so-called layer-and-a-half models (i.e. one moving layer and an infinitely deep lower layer) such as this one (e.g. Nov, 1985) but they do show that sometimes a complete understanding of the details of the problem cannot be achieved without considering a finite lower layer. This is particularly true when the unknown upper layer speeds are small as is the lens drift in FLIERL(1984b). Since our upper layer speeds are of 0(1), it is expected that the inclusion of a finite lower layer will only have a secondary effect on the processes under discussion. Also, an addition of a finite lower layer to an already complicated problem will, in the author's opinion, make the problem untractable and, therefore, is not considered. The use of a solid cylinder as an analogue for a cooled warm-core ring is supported by the following reasoning. As already pointed out in Section 1, there are some obvious similarities between a colliding ring and a colliding solid cylinder. In particular, both features are expected to exert a pressure on the Stream as they collide with it, and both features have similar geometry in the x, y plane. However, there are also some important differences. For instance, although both the actual ring and the solid cylinder are subject to pressure forces, the former can adjust itself to the surrounding pressure, whereas the latter remains unaltered. It is expected that the collision-induced adjustment in an actual ring will consist mainly of two processes--the first is the adjustment of the ring shape and the second is the change in the position of the ring.* The ring shape is probably not so important to the intrusion and the omega flow because the observed changes in shape are small, so that the neglected pressure torque also is expected to be small. On the other hand, the change in the position of the ring is probably more important because if the colliding ring bounces back to the Slope Water, the omega flow may never be reached. Despite the obvious differences, examination of the cylinder-Stream interaction is useful for understanding the basic processes in question. That is to say, the results of this study might pinpoint some of the effects in more complicated and more realistic models. Although the idea of replacing a warm-core ring with a solid cylinder is new and the simplifications may appear to be drastic, similar and even more severe simplifications have been used before. For example, as pointed out in Section 1, there have been a number of analytical models which considered various oceanic processes to be analogous to flows in pipe loops and hydraulic systems (e.g. STOMMEL, 1961; STOMMELand ROOTH, 1968). There is no doubt that much has been learned from these models even though their simplifications appear to be more severe than those employed in this study.
* It should be recalled that, in this model, the solid cylinder is held in a fixed position whereas the actual ring is free to move.
The collision between the Gulf Stream and warm-core rings
377
5. C O N C L U S I O N S
A new conceptual model for the Stream-ring interaction has been developed with the assumption that: (a) as a first approximation, the ring can be represented by a solid cylinder embedded in the Slope Water; (b) the Gulf Stream can be represented by a single moving layer whose lower boundary intersects the free surface; and (c) all motions are frictionless and hydrostatic. Our attention has been focused on the processes that take place after the cylinder is pushed slightly into the Stream. Such a conceptual model can help our understanding of the processes in question because, in a similar fashion to a cooled ring, the forced cylinder can support a pressure on its Gulf Stream side. Immediately after the introduction of the cylinder, an intrusion along the upstream side of the cylinder outer surface takes place. The intrusion advances clockwise along the perimeter of the cylinder at a rate of - O ( g ' h w ) ' , where g' is the reduced gravity (gAp/p) and hw corresponds to the Stream depth at the edge of the forced cylinder. Typically, g' - 5 x 10-3m s-~ for the intrusion, and hw is estimated to be roughly 20 to 80 m. This gives an intrusion advancement rate of 0(1 m s-~) which is of the same order as the intrusion propagation rates seen in satellite images [see, for example, images for 12 to 16 April 1982 (EVANSet al., 1984)]. Ultimately the intrusion completely surrounds the cylinder and reattaches itself to the Gulf Stream in the cylinder lee side. Once this has happened and a period of adjustment has elapsed [0(f)-l], a steady state is reached. In this state there is a narrow band of Gulf Stream water that flows around the cylinder and forms an omega-like pattern. The details of this steady 'omega flow' can be computed using the integrated moment of momentum and a perturbation scheme in c, the relative penetration of the cylinder into the Stream. It is found that the width is cV~ro and the near wall depth is cV~H, where ro is the radius of the cylinder and H is the undisturbed depth of the Gulf Stream. The above model suggests that the bands of Gulf Stream water which have been observed to envelop warm-core rings may not be related to the motion within the ring even though the motions of both processes have the same magnitude and direction. Since the model is inviscid, the addition of clockwise rotation to the solid cylinder will not influence the results. Most images analyzed by EVANSet al. (1984) suggest that the bands occur along the ring edge with the ring's water present only on the right-hand side of the advancing nose as suggested by the model. A few images indicate, however, that portions of the band occasionally may be entirely embedded in the ring--namely, they have ring's water on both their right and left side. With our present knowledge of the processes in question, it is impossible to tell what the causes of this behavior are. One may speculate, however, that this particular situation can be a result of the fact that the ring itself is stratified so that the intrusion penetrates along the surface corresponding to its own density. Finally, one can also speculate that the 'omega' flow is responsible for the recapture of warm-core rings by the Gulf Stream.
Acknowledgements--This study was supported by the Office of Naval Research contract No. N00014-82-C0404. It had gained significantly from a fruitful collaboration with the Warm-Core Rings Program (supported by the National Science Foundation). Discussions with B. Cushman-Roisin were also helpful. REFERENCES BROWN O. B., D. B. OLSON, J. W. BROWN and R. H. EVANS (1983) Satellite infrared observation of the
kinematics of a warm-core ring. Australian Journal of Marine and Freshwater Research. 34. 535-545.
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