THE UNIVERS I TY OF MI CH IGAN COLLEGE OF ENGINEERING Department of Meteorology and Oceanography Final Report A TWO-LAYER MODEL OF THE GULF STREAM Stanley Jo Jacobs Assistant Professor of Oceanography ORA Project 07544 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH CONTRACT NO. Nonr-1224(55), NR-083-204 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1966 Distribution of this document is unlimited.

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v ABSTRACT vii 1. INTRODUCTION 1 2. FORMULATION 2 3. ANALYTICAL SOLUTIONS 8 A. A Quasi-Linear Case 8 B. General Theory 22 4. NUMERICAL RESULTS 27 5. CONCLUDING REMARKS 30 ACKNOWLEDGMENTS 31 APPENDIX. AN EXISTENCE CRITERION 32 REFERENCES 35 iii

LIST OF ILLUSTRATION Table Page IE Comparison of analytical and numerical results 29 Figure 1. Flow configuration. 3 2. Meridional velocity of upper layer vs. dimensionless distance from coast at y =.75. 19 35 Meridional velocity of lower layer vso dimensionless distance from coast at y =.75. 20 4o Position of interface at y =.75o 21 5o Numerical solution for meridional velocities at y =.75, 28 v

ABSTRACT A theory is developed for a two-layer inertial model of the Gulf Stream, Both layers are in motion, but it is assumed that the ratio of the geostrophic drift in the lower layer to that of the upper layer is small, Approximate analytical solutions are obtained under this assumption. In addition, a criterion for the existence of inertial boundary currents is established, An important result is the prediction of deep and surface countercurrents to the east of the high velocity part of the Stream. These are due to the effect of bottom topography. Another important result is that the interface at the coast comes to the surface at a lower latitude if the deep water is in motion, and that the intersection of the interface and the sea surface extends out to sea in a northeasterly direction from the coast. The theory of the flow near the line of zero upper layer depth is as yet incomplete, vii

1 INTRODUCTION This paper is about the Gulf Stream, or, more precisely, that portion of the Gulf Stream which lies along the North American continent. In the theory used here (cf., Charney, 1955,and Morgan, 1956) the boundary current is considered to be driven by advection of mass at its seaward edge rather than by the local wind stress. The boundary current can thus be studied as an isolated entity, provided that the mass flux into the current is known. In Charney's and Morgan's papers a two-layer model was used, the lower layer being assumed motionless, Their results are in good agreement with observations except for an absence in the theoretical results of countercurrents east of the high velocity part of the Stream (see Stommel, 1965, p. 123). More recent papers have attempted to extend the earlier work to take account of motion of the deep water, Robinson (1965) uses what is essentially a quasi-geostrophic theory for a stream with continuous density and velocity variation with depth. His paper is concerned more with setting up a theoretical framework than with obtaining detailed results, In another recent paper (Blandford, 1965), the earlier two-layer model was modified by considering three layers, the lower one being at rest. Blandford attempted to find a deep countercurrent, presumably due to advection of warm water from low latitudes which causes a zonal temperature contrast, but in this he was unsuccessful, In the present work a two-layer hodel is employed with both layers in motion. Thus, by contrast to the other layer models, the effect of topography on the Stream can be included, An analytical solution is obtained under the assumption of small velocities in the lower layer, Some numerical results are also presented. The effect of topography proves to be very important, for under the assumption that velocities in the lower layer are small, the relative vorticity of the lower layer is negligible except very near the coast; and consequently, the requirement of conservation of potential vorticity in a region of decreasing depth of the lower layer in the shoreward direction implies a deep countercurrent. This proves to be at approximately the same location and with the approximate magnitude of that observed (Stommel, p. 188). A surface countercurrent somewhat east of the deep countercurrent is also predicted, though this is somewhat weaker than observed. In addition to the foregoing results, a criterion for the existence of inertial boundary currents is obtained, This generalizes earlier work by Greenspan (1963) but its conclusions are essentially the same, namely that a sharp northward variation of depth at the seaward edge of the Stream may be incompatible with the existence of an inertial boundary current, 1

2.a FORMULATION We consider a two-layer fluid on the P-plane, with Coriolis parameter f fo + Py, and with coordinates (xyz) which measure respectively distance to the east, the north, and in the verticals The flow is assumed to be steady, inviscid, and geostrophic in the x direction, The fluid is bounded below by bottom topography at z = b, above by a free surface at z = H, and to the east by a meridional wall at x = O0 No mass flux is allowed across the interfaces z = ho Let subscripts 1 and 2 denote quantities in the upper and lower layers, and let subscript k when it appears be either 1 or 2, Also, let D1 and D2 be the depths of the upper and lower layers at x = oo, y = O0 and let H1T and TE2 be defined by I = D1 + D2 + H, Ap/P2, (loa) h = Di + 112 - Ii pi/p2, (lob) where Ap = P2 - PIt Imposing the conditions that the pressure vanishes at the free surface and is continuous at the interface yields Pi/Pi + gz = g(Dp + D2) + gITio (2.2a) P2/P2 + gz = g(Dipi/p2 + D2) + goTr2 (2ob) where gi = gAp/p2, while the definitions of T11 and Ha lead to H - h = Di + Iz - HTa (3(a) h - b = D + X - ITipi/pa - b D2 + h2 - Hi r b < (35b) In the definition of the depth of the lower layer, (3ob), we have assumed that AP/P2 < < i~ 2

Figure 1. Flow configuration.

The equations of motion consistent with the above definitions and assumptions are -fvk + g'rkx = (4) Ukvkx + Vkvky + fuk + g' ky = 0, (5) [uj(Dj + II - Il2)] + [vi(D+ I - H2)]y = 0, (6) and [u2(D2 + 12 - 1 - b):]x + [v2(D2 + 112 - - b)]y = 0. (7) Taking ui(y) and u2 (y) to be known functions, we require that u (oo, y) = k (y), v (o,y) = 0, (8.a) k k k iLe,, we consider the boundary current to be driven by a known zonal flow at its seaward edge. In addition,.the kinematic condition at the coast implies uk (0, Y) = 0 (8,b) As a last condition, we require that there be no net transport between the coast and the point x = o, y = 0. It is convenient now to introduce non-dimensional variables. These are * = (fo/NgD)x, y* =( /fo)y, u* = (fo/g,'Dj) u, v = ( g k 0 k k k H* = (l/D), f* = 1 + y, D* = 1 - b/D2, k k D^- being the non-dimensional depth of the lower layer in the absence of flow,

The non-dimensional version of the equations of motion is, with asterisks omitted, -fv + IT =0 (9) k kx u v + v v + fu + = 0 (10) k kx k ky k ky [u (l + ni T)] + [v( - 1 I2)] = o, (11) x y and (u2[D + 7 (I2 i" n) + (v2 [D + 7(. - T )]T) = 0, (12) x y where Y = D1/D2o This scaling is convenient, but overestimates somewhat the meridional extent of the current, It also overestimates, by a large amount, velocities in the lower layer. We choose to work with first integrals of the equations of motion rather than with equations themselves. To this end, we introduce transport stream functions t1 and *2 through *lx = vi [1 + I1 - II2], y = F- u1 + Hi - IT] (15) =2X V2[D + y7(112 - r)], iy - U2 D + 7(112 - ~) ] H (14) and define Bernoulli functions cx and ac by 1 (15) k k k It is easily shown that the potential vorticity and Bernoulli function of each layer are constant on streamlines, whence = kC (t(k) 16) and f + vix = [1 + It1 - Ia2] Ci(41), (17) 5

f + v2x = [D + y(112 - 1i) ] C02(2). (18) Replacing IT1 and IT2 in these equations and in tix = vi [1 + 1 - IT21], (19) 82x = v2 [D + Y(112 - 11)] (20) through the use of (15) provides a system of four equations in the four unknowns vi, v2, 1Y, *2 in place of the original system. As boundary conditions, we have V (0,y) = 0 (21.a) k and also 1(0oo,Y) = - ti [1 + il - 12]dy, (21.b) 0 (o P' A / A A 2(OOy) = - S U2[D(y) + 7 (R2 - 1i)dy, (21. c) 0 where D(y) = D(oo,y) and where A Ad nk(y) = - | fukdy = k(coy) (22) An alternate integrated form of equations (9)-(12) which is useful for some purposes is f + (l/f)lxx = [1 + IT, - 112] F, [,i +,- rx1], (23) Pf2 f + (l/f)II2xx = [D + 7(112 - l)] F2 [n12 + -L I2x]. (24) 2f2 These follow from the fact that the potential vorticity and Bernoulli functior of each layer are constant on streamlines and hence functionally related. Use of the x momentum equations to express vi and v2 in terms of 1 and 12 leads 6

immediately to the desired result, a set of two equations in the unknowns TI and f2 in place of the original system. Since from the y momentum equations [k + f k] y = - uk [f + (l/f) Tk], (25) the boundary conditions for this last set of equations are \ + 1 H 2 = 0 at x = O, (26) k 2f2 kx and (22). 7

3. ANALYTICAL SOLUTIONS Though the process of numerically integrating the equations is difficult and time consuming, some numerical solutions have been obtained and will be presented below. Here, however, we present an analytical solution. The method of obtaining this is based on the fact that velocities in the lower layer are much smaller than in the upper layer, so that to lowest order the flow in the upper layer and the position of the interface are decoupled from the flow in the lower layer. The flow in the lower layer and higher order approximations in the upper layer are then determined using the method of matched asymptotic expansions (Van Dyke, 1964). We will obtain solutions first for a particularly simple case and then generalize. A. A QUASI-LINEAR CASE Suppose the zonal velocities at x = oo are ul (y) = - 1/f (- e +y (27.a) (1_7y)' A ~ (1-yy)3 2(y) = - e t (27.b) ( -7y) (37.b) where e is a constant, and suppose also that in the open sea the bottom is flat, so that D(y) = 1. Then A 1y 1 Ji (y) = y +j e[(it)2 (28.a) l-yy 2(y) = (l+y) y (28.b) l () = 2e [( )2 -1], (28.c) 2 l-yY and IT2 (y)=+y ) - ) -1]. (28.d) l-7y If follows that 01 ('1) = _1 (29.a) and 8

a2(22) = le [1 + 2/E)2 - 1]. (29.b) 2 For this flow, the potential vorticity of the upper layer is constant, in agreement with observations (Stommel, p. 111). We exploit this fact in using equation (23) to describe the flow in the upper layer, since (23) is linear in 1l for constant potential vorticity. The flow in the lower layer will be described by equations (18) and (20). Another fact which will be exploited is that C < < 1 if the flow is to be representative of oceanic conditions, since the velocities in the lower layer are much smaller than those in the upper layer. Consequently, the interfacial position must be only weakly dependent on the flow in the lower layer, which thus behaves much like the flow of a one-lower fluid of given depth forced by a zonal velocity at infinity of magnitude e. This motivates the introduction of a scaled stream function E and meridional velocity V through e2 = of, v2 = e V, (30.a) in which it is assumed that v and V are order unity. For notational reasons, we introduce two more new functions, and X through: = H11, X = I12. (30.b) The equations to be solved can now be written out. They are y + (1/f) = - X, (51) xx YX -D [D + 7 (X - )] V, (32) Vx = - f + [D + y (X - )] (l +) (33) 1 X = [(1 + Y)2 - 1 -V2], (34) 2 which are to be solved subject to C+1 2 = = o at x =0, (55.a) 2f2 x and 9

=y + 1 [(l+y )2 _ 1], [l+ ] y, (3.b) 2 1-yy l-yy at x = o. When these equations are solved, t1 and vl can be computed from vi = x/f, 1 = + I v. (36) Also, the depth of the upper layer can be obtained; it is 1 + - X. The fact that the small constant af~ multiplies the differentiated terms of (32) and (33) indicates that this is a singular perturbation problem. In what is obviously a boundary layer of thickness Fe near the coast the variables must depend on a stretched coordinate ~ = x//c in order that the differentiated terms enter into the balance. Away from the coast, the variables are smooth functions of x. In each region the equations may be solved by expanding in powers of c1/2; in the inner region near the coast the solutions are made to obey the boundary conditons at x = t = 0, and in the outer region the solutions are made to satisfy the boundary conditions at x = o. Any remaining ambiguity is resolved by requiring that the limit as t -> oo of inner solution matches the limit as x -> 0 of the corresponding outer solution. We turn first to the determination of:, and in this we let a subscript i denote the inner solution, a subscript o the outer solution, and a superscript the order of the term in the expansion. Thus, when x is order unity, x being the outer variable, C = C ((x ) = 1/2)() + /o() (x) + C o((x) +... (37. a) and when x is small, of order fe, = C ) ( = ) (0) + E 12 i()() + E ()() + (37.b) where x = x/fe is the inner variable. We now substitute these expansions into the equations. As X(O) = X() = 0, the equations in the outer region are y + (~)/f = 5 (~), S(i)/f = 5 (1), (8) OXX 0 OXX 0 10

(2/f = (2) (2) ( o3 /f () x etc Hence ^Oxx/ = 0 oxx o) _ y + a(O ) etf. x, f 1) = a(x) e-Nf'x, (39.a) and 00 (j2) - G(2) (x,x')X2) (x') dx', (39.b) ~0 - 0 0 where a(~) and a ) are constants, and G(2) (xx') and G(3)(x,x') are appropriate Green's functions. In order to determine these it is necessary to rewrite the outer solution in inner variables and re-arrange terms so as to match with the inner solution term by term. Thus, using the Taylor series expansion of the exponential function, we obtain the inner expansion of the outer solution, r; =y+ a( ) [1 - /+ 1 ef-2 - - (f)3/2 3];'~, ~2 +Je a(l) [1 - g + 1 ef2] + E[% (o) +f x (o) 3] + 3/2 (3) (o) + 0 (e2) (40) (o) = [y + a(o) [' fa a(0 ] -[y a ] + Fe [a() f a(o) + e[2) (o) -f a(l) + 1fa(~) 2] + E32 [43) (o) + j2) (o) + + fa fa f3/2 a ( 3] 0Qax 2 + 0(e2) The inner solution obeys = ef[ - x -y] (41) iUS i i 11

and the boundary condition 2ef2 i +i)2 = 0 at 6 = 0. (42) Introducing (57.b) and sorting out the terms, we obtain for the first four terms in the expansion (0), 0=, 0(43.a) =o, = 0 and (2) (~ ) (3) ) f) q5i=(. -Y), 1i9 (43 b) with boundary conditions i = 0, 2f2() +( 0 (44 a) and f2 (1) + )( ) =0 o (44b) r2 %~.) ~3) 2f2 2) + 2, (()) + 2) 44oc) at ~ = 0, and with the matching conditions i( o Y + a ) i ) ) f a ( o) ( 2 i 0 oX _ff al() a + I a(o) ~, (3) ()() + Ix(2) (o) i (45) 2 i o ox + f a() S2 -1 f32 a() 3 2 6 as - oo. Now (4r5,a) shows that () and (1) are linear functions of i, and by the first two of the matching conditions they are 12

t(o) = y + a(O), (1) = a(l) _f a(O) o The first of the boundary conditions (44,a) is identically satisfied, and the second implies 2f2 (y + a() )+ f(a( ) 2. Hence a(O) = f[ - 1], (46) l+y and f(O) is completely determined both in the inner and outer regions. From (43.b), 1(2) is equal to f a(~) ~2/2 plus a linear function of o The matching condition yields j(2) = 5(2) (o) - Ff a(l) + 7 f a(~) g2 i/ 0 and the boundary condition (44.b) irmplies a(l) = 0. Hence ~(1) is determined. The next term in the expansion is 3, which by virtue of the equation it satisfies and the matching condition is given by 1+ ox 6 (3) = 3)(o) + () (o) _ - 6 f3/2 a(~) 3 The boundary condition (44.c) becomes f3/2 5(2) (o) - a(0) o(2) (o) = 0, (47) and hence G(2)(x,x') is G(2)(X,Xt) fx a( 0) G(2(x,x) = - e vf x > [sinh if x + f cosh ff x< ] v1-y (48) lee ishget anxthlsyo (., hs(2) where x> is the greater and x< the lesser of (x,x'). Thus ( is determined~ Higher order approximations can be calculated if desired, but will not be obtained here. It is seen by inspection that with an error of order C the inner solu13

tion is merely the outer solution written in terms of t Consequently, the solutions (o) =y + a() e -f x (49) -(1) = 0, (50) Co00 (2) =- (2) (xx,) X(2) (x') dx (51) "0 0 are uniformly valid, To this order, the flow in the upper layer does not exhibit boundary layer phenomena. From (36) and (49)-(50) the meridional velocity and transport stream function are computed to be a(o) Zfx v = - e -fx + (ef) () + o0(3/2) (52) f and t1 = y + a(O) e x + [(a(~))2/2f] e -2 f x + E (2) - (a(O)/f3/2) x(2) e - If x) + 0 (E3/2) (53) Also, the depth at the coast of the upper layer is [1 + 0(o) + (e(2) - X(2))]x = o = 2 + (E[(2)(o) - X(2)() ]o We now turn to solution of the equations of the lower layer. With an error of order e, these equations are eT = [D -(o)], (54.a) (0) V = - f + [D - y ] (1+). (5b) In solving (54,a) and (54.b) it is expedient not to use the method of matched asymptotic expansions. Instead, let 14

X (0) (D - y( ) dx; o then ~e ~t = v, (55) Te V = 1 + - f/(D-7 ),) (56) t which yield ettt - = 1 - f/(D-y (0)), (57) which in turn has solution t= - 1 + e -t/\e + (f/2~e) {e-t/ fE et'/. dt o (D-f(~)) + et E e-t/ dt e-t/E'/ dt-. (8) t (D-y(O)) o (D-75(O) ) Repeated integration by parts yields a series in ascending powers of NE, of which we keep only the first two terms. The result is = - - 1 ] - (;) e-t-/4 + (e) (59) D-7:0O) r where r is (D-y (O)) evaluated at x = O, i.e., it is the approximate (with an error of order e) depth of the lower layer at the coast. Recalling the definitions of t,', and V, we obtain from the above the uniformly valid solutions x = - - 1] - r exp - (D- + () D- ~(O)r E (60) and Ix D (Y] (0) V2 Xe (fF()) expD- )) x] - of xx + 0(e3/2). vr 0 (D-ra(O))3 (61) Note that the stretched boundary layer variable is not x/e as assumed earlier, but rather 15

x \ (1/J) A (D-I (o)) dx. 0 Note also that away from the coast *2 = D [- - 1] + 0(~2), D-7A~) V2 = 0 + O(E)o, so the term (2) which appears in (51) is (2) f2 xo =z 7[- (o)1]. (D-7(0))2 It is in order now to discuss these results, As noted earlier, the flow in the upper layer and the position of the interface behave to lowest order as if the fluid in the lower layer were at rest. The flow in the lower layer is as if it were the flow of a one-layer fluid of given depth forced by a zonal velocity at infinity of magnitude E. The meridional velocity in the lower layer is of magnitude E except in a layer of thickness \Ec near the coast where it is of order Eo, Consequently, the relative vorticity of the lower layer is smaller than the planetary vorticity by a factor of E except very near the coast, where it is comparable to the planetary vorticity, This has very important consequences, for it means that the direction of the deep meridional velocity is greatly influenced by the topography, Away from the coast the potential vorticity is essentially equal to the planetary vorticity divided by the depth of the layer, as shown by the first term in equation (60), in which, it will be remembered, (D-zy(O)) is the approximate depth of the lower layer. Consequently, if the depth of this layer decreases in the shoreward direction, the streamlines must deviate to the south in order to conserve potential vorticity. Since in the ocean there are regions in which the depth of the lower layer does decrease in the shoreward direction and in which there is a deep countercurrent (see Stommel, p, 188-190), the present theory provides a possible explanation for the deep countercurrent. Near the coast the relative vorticity of the lower layer is not small, and in this region the deep current according to this theory is in the same direction as the surface current, Turning again to the flow in the upper layer, we note that the correction due to the motion of the lower layer must be important at least near y = 1, as seen from (48)~ Near y = 1, 16

G(2)(x,x') 2 e- 2 (x> + x<), and the non-dimensional depth of the upper layer becomes 2c J2 x 1 + - X 2 + [ 71-y - 2] ee,2 x e J1-y x - e-2 x' X() (x) dx', 0a"hsoli which vanishes on the line 00 x = log( 1-/ ty + e2 (2)(X') dx'}. \72 2 y1i-, o This intersects the coast at a value of y somewhat smaller than 1, 0 y P 1 -$2 at/a p-I- x x(2) (xI) dX', and extends in a north-easterly direction from the point of intersection with the coast. At y = 1 the method of solution is invalid, because the correction term due to motion in the lower layer is no longer small and because the nonlinear terms in the x momentum equation, which were neglected at the outset, also become large. In fact, the values of u1 and u2 become infinite like +l/l-yT as y - 1. We note also that even for y < 1 the present theory is invalid unless ve < r, for otherwise the basic approximation that ~< X does not hold true. In order to obtain numerical results we must assign numerical values to the constants, pick a representation for the bottom topography, and carry out the integration in (51). We take y = 0 to coincide with 15~ latitude, and chose D1 = 500 m, D2 = 4700 m, f = 6.14 x 10-5 sec-1, = 2.07 x 10-11 m-1 sec-1, Ap/p2 = 2 x 10-3, e = 0.04. Thus x is measured in units of 51 km, y in units of 2970 km, ul and u2 in units of 5.4 cm/sec, and vl and v2 in units of 313 cm/sec. The transports T1 and T2 for the two layers are T, = (79.8 x 106 m3/sec) *1, T2 = (750 x 106 mS/sec) W2, and with e = 0.04 this yields 17

T1 = 63.8 x 106 Mn/sec T2 = 27.0 x 106 m3/sec at y = o75, which is close to 35~ latitude. These values are at least representative of those for the Gulf Stream. For the topography, we use a smoothed representation by taking D =.5 + x/12 0 < x < 6 D = 1 x > 6. This was chosen more for convenience than for accuracy, though apart from omission of the steep part of the continental shelf it is not unreasonable. For evaluating the integral in (51), we make use of the fact that y = Di/D2 is small, of the order of 0.1. Indeed, a theory could be made (and has been, by the author, though it is not presented here) based entirely on the fact that the lower layer is much thicker than the upper. For small y, we make the approximation 1 1 1 + 2ya(~) e f'x (D-7r'O))2 (D-yy-ya()edf x)2 (D-yy)> (D-yy)3 in the integrand of (51), and then find that for a constant slope bottom f(2) can be expressed in terms of elementary functions and various types of exponential integrals,which are tabulatedo The expression for S(2) is lengthy and uninformative and is not presented hereo In Figures 2 and 3 the velocities vi and v2 are plotted. The maximum value of the deep countercurrent is 7 cm/sec., and it occurs some distance from the coast, There is also a surface countercurrent, but this is weaker than the deep countercurrent, its maximum velocity being 3 cm/sec. The location of the countercurrents appears to be in reasonable agreement with observations, but their magnitudes are too small. These could be increased by using a larger value of c or a greater amplitude for the bottom topography, but then the method of solution used here becomes invalid, The position of the interface is shown in Figure 4o As can be seen, there is no indication of a warm core, which would be characterized by a decrease of the depth of the upper layer in the seaward directiono 18

400 300 Lii 200 I, 200 2 \ u \ 100 - -100 - I,I, I I I I Ii, 0 I 2 3 4 5 6 7 x (DIMENSIONLESS) Figure 2. Meridional velocity of upper layer vs. dimensionless distance from coast at y =.75. 19

400 300. 200 I.i Iq_ U CO0 (I 100 x (DIMENSIONLESS) Figure 3. Meridional velocty of lower ler. dimensioess distance from coast at y =75.20

5.2 4 Z=h E 3 N1 Il I.... I I I I I 0 1 2 3 4 5 6 7 x (DIMENSIONLESS) Figure 4. Position of interface at y =.75. 21

B. GENERAL THEORY We turn now to the task of generalizing the previous results, and in this we assume only that the magnitude of uA is of order unity, while that of u2 is of order e. It follows that i1 (ii) will be of the form ai(i.) = A(,1) + E B(i1) + 0(2), and that 0C2(*2) = e C(42/E) + 0(e2); here A, B, and C are functions with amplitude of order unity. Defining Y, V, ~, and X as before, and again letting superscripts denote the order of a term in expansions in powers of eJ2, we have T'x = vi (1 + - x), (62) vlx = -f + (1 + - x). (A' (f) + e B' (Q1) +...), (63) /eJ.- - = V(D-7+7YX), (64) J'..". = -f + (D - 7 + 7X) (C' (Y) +.. ), (65) where = A(r1) + e B(11) + v (66) X = e[C(Y) +.. - V2], (67) to be solved subject to 1J (O,y) = r(Oy) = 0, 1 (ooy) = f(y), (68) 22

In discussing the top layer, we use the results found previously, that with an error of order e3/2 the variables in the top layer do not exhibit boundary layer character, and furthermore have no term proportional to e1/2 in their expansion. Also, we use fv2 = SX2 which comes from the x momentum equation. Now ax = Sx(l+I)/f - vx, (69) so t = ( + l2)/f + vxdx + b. (70) Here b is a function of y which is determined by the values of tf and a at x = xoo, In general, it is of the form b = bo) + b(2) +.. Inserting perturbation series and sorting out terms, we arrive at (o) = (o) (1 + I (o))/f + b(o), (71) (2) (o) 7~ 22 ( = [(1l+ + o)/f] ) + v X(2) dx (2) (72) x Since A(*_) + e B(r1) = A(r1(~)) + E[i1(2)AI (1r(~)) + B(r1( ))] + 0(2), (73) and since f + v) = (1 + (o)) A' (=1(0 ), (74) we have from (66) ( = =A(r1 ) - (O))2/2 (75) 25

and (1) = _1(2) (f + v o))/(l + ~(O)) + E(1(~O)) - vo) (2)/f. (76) Consequently, from (71) and (75), we obtain (o) + ( o))2/2f2 = A[(~)(l + (O~))/f + b()], (77) x 2 a differential equation which can be solved by quadrature, while the equation for C(2) becomes (G(2)/ v~O)) = [f/(v(o))2] (B(A0o)) + A' (t()) [ vO)x2) dx + b(2)]) ~~~X ~(78) which can also be solved by quadrature. This completes the solution for the flow in the upper layer, since from the solutions for ~(o) from (77) and for r(2) from (78) the other variables of interest can be calculated. If we let x t = | (D - y7(O)) dx To as before and neglect terms of order c, the equations describing the flow in the lower layer become ie Et = V, (79) \fE Vt = C' (E) - f/G(t), (80) where (o) G(t) = D - y() These will be solved by the method of matched asymptotic expansions, the inner variable being T = t/Ce. Thus when t is order unity, = ~) (t) + e(1) (t)+..., and when t is small, of order ~e, 24

o = )) (T) +yEl) (T) +., and V is treated in the same way. In the outer region, Vo~) -, C' ((~)) = f/G(t), (81) and v(1) = (~) ) 0. (82) VO 0ot' 0 Bearing in mind the definition of G(t), this indicates that as in the case treated in 3.A. the velocity in the lower layer is found by requiring conservation of f divided by the depth of the layer, For matching the outer solution to the inner solution, we need the results that when t is order J, To O-0)(o) + Ne T T (0) (0), -,1 _-" o G'+ (0) T/G2(), G(t) G(o) so (p) C' [. (O)] = f/G(O), (o) 2 (o) CAt [o (0)] = -fG' (O)/[G2() Ot (0~) In the inner region, giT = Vi, (83) ViT = C' (.i) - f/G (87), (84) which combine to iTT = C1 (ii) - f/G (47)o (85) Consequently, 25

(o) = c' (0) () C (=, )) - f/G(O) (86) (1) (1) (o) YiT =E() C" (y ) + fG' (O)T/G2(0). (87) (0) Multiplying (86) by YjT and integrating, we obtain 1 (o) 2 (o) (O) (~) 0Y (.)) C(i T)) - fy~) /G(O) + d. (88) Here d(~) is a constant of integration which is found by using (o) (o). i > (0) as T +- o. Using the value of d(O)obtained in this way, we can then solve (88) by quadrature, using the boundary condition at T = 0. Before solving (87), we note that as T -+ oo ) + TY ) (0), hence i1) Ct' (i~o)) + - fG' (0) T/G2 (0), as it should. Now C (yo)0) 1 [' (( ~) ) = v( )/v(0o 2. v OT i T T )~ vi hence (87) becomes () (1) (o) (0) [V1i - v-T -i T1 = fG' (O)TV~ /G2 (0), (89) and integration together with application of the matching condition yields () (o) ((o) 2 (o) [i / i ] = - [fG' (O)/G(O)Vi ) ] TV dT, (90) so the solution of this equation also is obtained by quadrature. Hence the equations of the lower layer are solved, and a uniformly valid solution can be constructed. It is important to note that as in 3,A the relative vorticity of the lower layer is negligible except very near the coast. Consequently, away from the coast, a decrease of depth of the lower layer in the shoreward direction implies southward motion of the deep watero 26

4. NUMERICAL RESULTS In order to check the analysis an attempt has been made to obtain numerical solutions for the case discussed in 3.A. The method consists of guessing values of vi and v2 at x = 0 and then integrating. A Runge-Kutta scheme with spatial steps of.005 was used. In general the initial guesses will be incorrect and the integration must be repeated with different values of vl (0) and v2 (0) until the solutions appear to obey the boundary conditions at x = o. The integration is easier if topography is ignored, for then it can be shown that A = f(*1 + 42/Y) - H1 - I2/7 - 2 (InT - 112) is independent of x. Since A is a function only of y it can be computed from the known conditions at x = o. This provides a relation between vl and v2 so that (say) only v2 (0) need be guessed. No such simplification was found for the topographic case. It is difficult to say whether the numerical integrations represent true solutions because in many cases the numerical solutions diverge with x. It is felt that this is due to incorrect values of vl (0) and v2 (0), since the solutions are extremely sensitive to the initial conditions. The results of a typical calculation are shown in Figure 5, which represents the best result that could be obtained with a reasonable amount of effort. It is apparent that for x greater than 2.5 the numerical solution is inaccurate. In Table I the values of vi (0) and v2 (0) as found analytically and numerically are compared for different values of y. The agreement is much better for the non-topographic case, in which both the analytical and numerical solutions are more accurate. In view of the great difficulty in obtaining numerical solutions the principle conclusions of this paper must rest on the analytical work of the previous section. The numerical results serve to check some of the qualitative features of the analysis, however, and for this reason have been presented. 27

400 300 I). 200 \ 100 0-100 -100 I I I I I- - - -I-I x (DIMENSIONLESS) Figure 5. Numerical solution for meridional velocities at y =.75. 28

TABLE I COMPARISION OF ANALYTICAL AND NUMERICAL RESULTS (VELOCITIES ARE GIVEN IN CM-SEC-1) Analytic Numerical Non-topographical case y vi(O) V2(0) vi(O) v2(o).25 83 16 82 16.50 174 32 171 31.75 285 46 265 45 Topographical case y vi(0) V2() v2(O) V2(o).25 105 82 79 82,50 215 114 178 111.75 364 152 284 119 29

5. CONCLUDING REMARKS An interesting result is the separation of the boundary current from the coast at a slightly smaller value of y than that predicted by the Charney-Morgan theory and the existance of large positive zonal velocities in both layers near the separation latitude. The flow in the region near this latitude is not accurately described by the present theory. In order to obtain such a description and thus to treat the portion of the Gulf Stream northward and eastward of Cape Hatteras a much more extensive theory is necessary. The present work is applicable south of the separation latitude and appears to account for a number of observed features of the Stream. However, frictional effects, which have been ignored here, are undoubtedly important at least very near the coast. An extension of the present theory by inclusion of frictional effects would serve to put the work on a firmer basiso 50

ACKNOWLEDGMENTS The numerical calculations were carried out by Mr, Yossef Sela, This research was sponsored by the Office of Naval Research under contract Nonr-1224(5), 31

APPENDIX AN EXISTENCE CRITERION A necessary condition for the existence- of solutions can be obtained by linearization about the flow at infinity. The solutions of the linearized equations are then examined to see if they decay with x as they should. We will use the dimensional form of (23) and (24) for this purpose, f + (g'/f) Ilix.H (g'/f) ixx= Fl [ii + g' Tx], (A.1) D1 + Iti - H1 2f2 f + (g'/f) nxI -- F2 [I2 + - -- x], (A.2) Df + (H2 - H12 - b 2f2 A A and linearize taking cpk = lHk - Ik to be small. We will also take b = b(y), thus assuming that there is no appreciable dependence of the topography on x far from the coast. In what follows, we let A 2 A P = f/ [g' (D1 + 1ti + 2) i P2 = f/[g (D + I2 - + 1 - b), (A.3) and remember that fuk + g'Iky = O. (A.4) Then Fk[Ik + g'/2f2 rIkx] Fk{[lk + Tl] Fk (k) + qk Fi (Ik) (A.5) = Fk (Ik) + (cpk/lky) /~y Fk(Hlk) = g' [P - (g'pk/uk)) Pky]. 52

We now let Qk = (l/uk) N/Sy log Pk (A.6) and obtain, after linearization of the left sides of (A.1) and (A.2), (l/f)cplxx + (Q1 - Pl)cp1 + P2CP2 = 0, (A.7) (l/f)cP2xx + (Q2 - P2)c2 + Pcpj = 0. (A.8) Assuming solutions of the form exp[fl/2Ax] leads to the characteristic equation,4 - [(P1 - Q1) + (P2 - Q2)] A + [P1 - Q1)(P2 - Q2) - P1P2] = 0, (A.9) which in turn leads to 2A2 = (P1-Q1) + (P2-Q2) + [(P1-QA) + (P2-Q) ]2 - 4[(Pi-Ql) x x (P2-Q2) - PP2 1/2 (A. 10o) The expression inside the curly bracket is easily shown to be positive, and hence the eigen-values A are either pure real or pure imaginary. In order that they be real, so that the solutions decay exponentially with x, (Pl-Q1) + (P2-Q2) > O, (Pl-Q) (P2-Q2) > PIP2, (A. 11) which is the desired criterion. Now let A A A A A 51 = D1 + III - E2, 2 = D2 + 12 - I2 - H11 - b; these are the depths of the upper and lower layers at x = oo. Using (A. 4) and 53

the fact that each of the terms (P1 - Q1), (P2 - Q2) must be separately positive, we arrive at UlU2 > g'UzSi/f2 (A. 13. a) a12 >g'U2(52 + fty)/f2, (A.13.b) and A A A A +fA A U1U2p( ul 2 + U2b2) + fu2by] < UU2 g1 [(P1)(f2 + fby)]/f2. (A.13.c) It is now a matter of working through the inequalities to find that: A (1) for P>2 + y > 0 A A (A.14) u < 0, u2 < 0 and (2) for 52 + fby < 0, /.A A P(ulSl + U252) + flu2by < g (PB1) (02 + fby)/f2 (U12 > 0); (A.15) and g' (PB) (s2 + fby)/f2 < ( 5 + u22) + f u2y < 0 (U1U2 < 0). (A.16) These conditions are subsumed by P(Albl + A AA P(u 2 + u22) + f u2by < 0, (A.17) which is a special case of a result proved by Pedloskey (1965) for a baroclinic A A fluid and which reduces to Greenspan's criterion if ui = u2. It should be noted that a rapid variation of topography with y can be highly important even though u2 is in general quite small. 54

REFERENCES Blandford, R., 1965, Inertial flow in the Gulf Stream. Tellus 17, ppo 69-76o Charney, J. G., 1955, The Gulf Stream as an inertial boundary layer. Proc. Nat. Acad. Scio, Wash., 41, pp. 731-740o Greenspan, H. P., 1963, A note concerning topography and inertial currents. Jour. Mar. Res., 21, pp. 147-154. Morgan, G. W., 1956, On the wind-driven ocean circulation. Tellus 8, pp. 301320. Pedloskey, J., 1965, A necessary condition for the existence of an inertial boundary current in a baroclinic ocean. Jour. Mar. Res,, 23, pp. 69-72. Robinson, A. R., 1965, A three-dimensional model of inertial currents in a variable-density ocean. J. Fluid Mech., 21, pp. 211-224. Stommel, H., 1965, The Gulf Stream. University of California Press, Van Dyke, M., 1964, Perturbation Methods in Fluid Mechanics. Academic Press. 35

Unclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report ie classified) 1. ORIGINATIN G ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION The University of Michigan, College of Engineering Unclassified Department of Meteorology and Oceanography 2b GROUP Ann Arbor, Michigan 3. REPORT TITLE A TWO-LAYER MODEL OF THE GULF STREAM 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Final Report 5. AUTHOR(S) (Last name, first name, initial) Jacobs, Stanley J. 6. REPO RT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS September 1966 35 8 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) Nonr-1224(55) 07344-1-F b. PROJECT NO. NR-083-204 _ c. 9b. OTHER REPORT NO($) (Any other numbers that may be assigned this report) d. 10. A VA IL ABILITY/LIMITATION NOTICES Distribution of this document is unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Department of the Navy Office of Naval Research Washington, D. C. 13. ABSTRACT A theory is developed for a two-layer inertial model of the Gulf Stream. Both layers are in motion, but it is assumed that the ratio of the geostrophic drift in the lower layer to that of the upper layer is small. Approximate analytical solutions are obtained under this assumption. In addition, a criterion for the existence of inertial boundary currents is established. An important result is the prediction of deep and surface countercurrents to the east of the high velocity part of the Stream. These are due to the effect of bottom topography. Another important result is that the interface at the coast comes to the surface at a lower latitude if the deep water is in motion, and that the intersection of the interface and the sea surface extends out to sea in a northeasterly direction from the coast. The theory of the flow near the line of zero upper layer depth is as yet incomplete. DDR.............ORM 17 Sci DD JAN 64 1473... Unclassified Security Classification

Unclassified Security Classification 14. KLINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Gulf Stream Inertial boundary currents INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this the report, the report. report from DDC." 2a. REPORT SECURITY CLASSIFICATION: Enter the over2a. REPORT SECURTY CLASSIFICATION: Enter the over- (2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whether ror not au ize "Restricted Data" is included. Marking is to be in accord-thozed ance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Di-users shall request through rective 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional.. markings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ~~~~~~~~~~~ized~~. ~report directly from DDC. Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified., If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qualimmediately following the title, ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of ____ report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Enter last name, first name, middle initial, tory notes. If military, show rank and branch of service. The name of the principal au.thor is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE. Enter the date of the report as day, ing for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication. 13. ABSTRACT: Enter an abstract giving a brief and factual 7a. TTAL NUMBER OF PAGES: T tosummary of the document indicative of the report, even though 7a. TOl TAL NUMBER OF PAGESi: The total page count it may also appear elsewhere in the body of the technical reshould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information. be attached. 7b. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C), or (U). the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number, subproject number, system numbers, task number, etc. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NUMBER(S): If the report has been words but will be followed by an indication of technical conassigned any other repcrt numbers (either by the originator text. The assignment of links, rules, and weights is optional. or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those Uncla ssified Security Classification

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