A three-dimensional model of a thermohaline-maintained circulation is presented for a closed basin of finite depth on the beta plane. The circulation, which conserves potential vorticity, consists of a westward drift at all levels in the open ocean and a return flow that is accomplished in a swift eastward-flowing current. A comparison is made to Worthington\u27s recent water-mass analysis of deep circulation in the North Atlantic Basin

Niiler, P. P.

Robinson, A. R.

Spiegel, S. L.

English

Yale University

      P.O. BOX 208118 | NEW HAVEN CT 06520-8118 USA | PEABODY.YALE. EDU   JOURNAL OF MARINE RESEARCH The Journal of Marine Research, one of the oldest journals in American marine science, published important peer-reviewed original research on a broad array of topics in physical, biological, and chemical oceanography vital to the academic oceanographic community in the long and rich tradition of the Sears Foundation for Marine Research at Yale University.  An archive of all issues from 1937 to 2021 (Volume 1–79) are available through EliScholar,  a digital platform for scholarly publishing provided by Yale University Library at  https://elischolar.library.yale.edu/.  Requests for permission to clear rights for use of this content should be directed to the authors, their estates, or other representatives. The Journal of Marine Research has no contact information beyond the affiliations listed in the published articles. We ask that you provide attribution to the Journal of Marine Research.  Yale University provides access to these materials for educational and research purposes only. Copyright or other proprietary rights to content contained in this document may be held by individuals or entities other than, or in addition to, Yale University. You are solely responsible for determining the ownership of the copyright, and for obtaining permission for your intended use. Yale University makes no warranty that your distribution, reproduction, or other use of these materials will not infringe the rights of third parties.  This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. https://creativecommons.org/licenses/by-nc-sa/4.0/   On Thermally Maintained Circulation tn a Closed Ocean Basin 1 P. P. Niiler, A. R. Robinson, and S. L. Spiegel Pierce Hall, Harvard University ABSTRACT I A three-dimensional model of a thermohaline-maintained circulation is presented for a closed basin of finite depth on the beta plane. The circulation, which conserves potential vorticity, consists of a westward drift at all levels in the open ocean and a return fl ow that is accomplished in a swift eastward-flowing current. A comparison is made to Worthington's recent water-mass analysis of deep circulation in the North Atlantic Basin. Introduction. Worthington (1965) has recently performed a water-mass anal-ysis of a substantial portion of the deep Gulf-Stream system. The circulation consistent with his analysis is an approximately semicircular gyre with a strong north-south asymmetry. This circulation pattern was recognized by the present authors as strikingly consistent with the conservation of potential vorticity (Robinson 196 5) in the entire motion. We demonstrate this fact by presenting a class of theoretical circulations of similar form. These Bows are of interest in their own right, since, to the best of our knowledge, they represent the first analytical solutions for nonlinear, three-dimensional circulation in an enclosed basin of variable density. It is further remarkable that such diverse attacks on the deep-circulation problem yield similar results. The approach via water-mass analysis has the advantage of dealing directly with real oceanic data; the advantage in the mathematical formulation li es in our ability to provide details of the circulation patterns (e.g. vertical-velocity distributions) that are consistent with the field equations for the conservation of mass, mo-mentum, heat, and salt. We model the deep water of an ocean basin as a Boussinesq fluid on an enclosed beta plane of finite depth. W e neglect diffusive processes and hence allow no interchange of momentum or heat with the region above the thermo-cline. The quasilinear, and quasigeostrophic,2 three-dimensional problem is 1. Accepted for publication and submitted to press 9 August 1965. 2 . Here the vertical component of vorticity in the potential-vorticity equation is evaluated with velocity components that are geostrophic. 222 1965] Nii/er, Robinson, and Spiegel: Thermally Maintained Circulation 223 solved for the case of a semicircular basin of constant depth, where the po-tential vorticity is conserved in a relatively simple manner. The principal results indicate a slow westward drift at all levels, with a swift returning current over the northern arc of the basin. We are cognizant of the fact that such a simple model cannot represent the full complexity of (diffusive) processes that drive the circulation of a real ocean basin. However, it does represent the manner in which vorticity, once introduced by diffusion from the horizontal or vertical boundaries of the basin, tends to be conserved and hence is a significant isolatable process in the theory of oceanic circulation. 2. Basic Equations of the Model. We consider a semicircular ocean basin that has as the bounding diameter the latitudinal circle designated by 0o. Let the basin be of constant depth, H, and extend a maximum distance, R, north from 0o latitude [ cf. Fig. 1]. (The symbol Ro denotes the radius of the earth.) H Figure r. Model of semicircular ocean basin. We use the beta plane, Boussinesq approximation to the equations of motion, and write the Coriolis parameter as f' =Jo+ (3 y, where y is the distance meas-ured north from the latitudinal circle 0o. Let the circular-polar coordinates be r', cp, z, and, in dimensionless units, we measure distance in r = r'/Rcp, C = z/H. The corresponding dimensionless velocity components will be v = urp/Uo, w = (R/HUo)uz. In the conservation equations we also use the dimensionless pressure, p, and density anomaly, 0, defined as (3) where Lle is the total vertical density difference in the ocean basin. 224 'Journal of Marine Research We now defin e Uo == gHiJ e/eoJoR, e == Uo/JoR, /3* = (R/R o) cotan (@o), f == I + f3*rsincp, and write the conservation equations as e v·Vv+- +Ju =- -- , uvl 1 8 p r r 8cp e - - ap - ac' -:C-v0 = o. [ 23,3 (4) (s) At this point we note that Uo is a presently convenient quantity for the scaling of the velocity field, but is not, in fact, the anticipated amplitude of the motion. The baroclinic part of the latter quantity is based on the horizontal density difference. This scaling will become clear in § 3. We shall seek solution of the equations under the condition that the normal component of velocity vanishes at a level surface, C = I, the bottom, C = o, and the vertical walls enclosing the semicircular basin. The virtue of this geometry is its simplicity; its value as a model of subthermocline flow is moot. The flow is maintained by density differences across the surfaces of the basin. A detailed examination of (S) reveals a third-order system of equations that has as a first integral the potential-vorticity function. In addition to the density-boundary condition, we need to specify the potential-vorticity integral. In fact, simple forms of this integral will prove to be relevant for dealing with a closed-circulation problem. The potential-vorticity equation is most simply derived after a quasi-Lagrangian transformation of the equations that interchanges the dependent and independent roles of 0 and C. The resulting equations are [ vide Robinson 1965 for details of the transformation and derivation of (7)] UV] e v ·V,,v+-; +Ju = -8ll/r8cp, (6) 1965] Nii/er, Robinson, and Spiegel: Thermally Maintained Circulation 225 c = aII;a0, (6) ac f}() u = - 01fJ/or, where II = p + 0 C. In the above, "h is the horizontal component of vector de! operator taken at constant (), and a stream-like function, 1P, has been defined to satisfy the transformed form of the mass-continuity equation. The vertical component of the vorticity equation can now be integrated to yield the potential-vorticity equation, [ 1 a I a ] ac e -- rv - -- u +f =- P(1P,0). ror rorp f}() (7) The free functional of integration, the potential vorticity, P(1P, 0), in general depends on both of its arguments. Here we choose p = I +C1j), C=O,I,-1, (8) where all possible choices of linear functions of 1P are expressed in (8). We remark that a primary effect of the ()-dependence of P (ignored here) is to determine the vertical-density stratification that is uncoupled to the motion field . Since we are primarily interested in the horizontal streamline structure of the solution, (8) was chosen as the simplest mathematical structure. At the same time, the dependence of P on 1P couples the functional dependence to the motion field, and such a choice qualitatively illustrates the effect of this coupling. The 0-dependence of P has been discussed elsewhere (Spiegel, personal communication3). 3. Circulation in a Closed Basin. Since our problem is now homogeneous with respect to the boundary conditions, the motion is driven by the inhomo-geneous term in the potential-vorticity equation. The solution that we exhibit is based on the assumption that f3 * < I , which is consistent with the validity of the {)-plane approximation to the equations. For the case of interest in com-paring our results with Worthington's, /3* = 1/4, (cf. § 4 below). Hence a power-series expansion of the equations and boundary conditions in /3* is allowed. We define the variables 3. From Ph. D. thesis in preparation. 226 ]ournal of Marine Research [ 23,3 II = - 02/2 + (J*II, + 0((3*2), V = {J* V 1 + ...... .... . . , ' = 0+(3* ( ,+ ... ... .. . , 1P = /3* 'I/Jr + .. .. ..... .. . ' (9) u = /3* u, + . ...... . . . .. ' w = (3* 2w2 + ... . .. ... .. . Substituting the expansion (9) into (6) and (8), we obtain the quasigeostrophic potential-vorti city equation, e -- r - Il, +- - II, - CII, + -- = - r sm<p, [ 1 a a I a2 ] a2 II, . r 8r 8r r 2 O<p2 802 (1 o) where II, satisfies the boundary conditions W 2 = 2_ [all, a2 II, _ aII, a2II,J = 0 0 at = 0 ,1 . r O<p 8r80 8r 8<p80 (II) In ( 1 o) and ( 1 1) we have made an error of o ( s/J *), which is to be small. The boundary conditions for W 2 should be satisfied at two levels of(; how-ever, to the order of expansion here considered, 0 = C + o ((J*) and the fir st-order solution formally must satisfy the boundary conditions at constant 0. The substitution II, = n, (r, 0) sin <p (12) is consistent with the problem as posed. This allows the boundary conditions (11) to be integrated as (for details, see Robinson 1965: § 3.2): 8:n, n, = k, 80 at 0 = 1 , an, 0 n, = ko - at = o . 80 The constants, k,,o, will be related to details of subsequent solutions for :n, . The quasigeostrophic state of the circulation is calculated from - COS<p u, = - -- n,, r v, = sin <p8n,/8r, ( , = - sin<p&n, /80. The solution for :n, can be represented as 1965] Nii/er, Robinson, and Spiegel: Thermally Maintained Circulation 227 where :n, is any solution of ii100-Cii1 = - I' ( I 6) and satisfies boundary conditions ( 13). We have used the definition Z, (o::nr) as the first-order Bessel function of the complex argument o:: r which is ' n ' bounded at r --+ o. The complete set of functions, <pn (0), on the interval [ o, 1 J is and An are generated from , _An[-k,+ ko] tan ''n - -'--'=--------= , o::~ + (A; - C)/ c = o . I + kok,A; The coefficients .An are determined from satisfying the mass-conservation condition at r = 1 (i.e. no normal velocity), whence I I .An = - ~<pn:n,(0)d0/ ~<p;(0)d0. 0 0 At this time we point out that, for a typical ocean basin, c < < 1 . Hence, essentially two distinct types of circulation can occur in the basin that con-serves potential vorticity. If all an determined from (18) are imaginary (A~ > C), ( 1 5) implies a boundary layer of o ( c1i 2) at r --+ 1, the northern arc of the basin. A theorem has been proved (Spiegel, personal communication) that this class of solutions corresponds to a slow westward drift at all levels in a major portion of the basin, with a swift return flow over the northern arc [ cf. Fig. 2]. If, however, ko, k, are chosen such that an is real for some n = q (it can be shown that there are at most two o::q that can be real for any choice of ko, k,), then for c small, Z, ( o::q r) represents a series of small horseshoe-like gyres over the major portion of the ocean. Fig. 3 illustrates the lowest eigenfunction in this case. In general, the lowest eigenfunction will dominate all the flow in the basin, with the strongest gyre in the northern part of the basin. For certain values of ko, k,, this is not the case, and a complicated superposition of Figs. 2a and 3 will result. These complex-valued eigenvalues are particularly intriguing, because they give rise to motion of an intermediate scale in the interior basin; this motion has a wave length of the order of I oo km with a strong bias to the north-south components of circulation ( cf. Crease 1962 for measurement 228 'Journal of Mar ine R esearch (a) HORIZONTAL TRAN SPORT STREAMLINES b 80 b [ 2 3,3 ( b) SURFACE TEMPERATURE (c) SECTION 11bb11 TEMPERATURE Figure 2 . Schematic presentati on of solution. of moti ons of this scale). W e full y reali ze that the latter remarks are speculative in nature, and there is very littl e reason to believe that such small- scale circula-ti on is reall y part of a coherent structure that spans the whole basin. On the other hand, the ghost of this simplifi ed phenomenon may haunt the much more complex real ocean fl ows. 4. Deep Circulation in the North Atlantic Basin. To relate our results to W orthington's water-mass analysis, we consider the part of the basin in the North A tlantic that is directl y south of the Gulf-Stream system. T he latit udinal F igure 3. H orizontal transport of cx1 (real) eigenfunction. 1965] Nii/er, Robinson, and Spiegel: Thermally Maintained Circulation 229 circle of 30° is the diameter, and the northern arc is composed of the 4000-m sea-bottom topography on the eastern coast of North America, extending to 40°N. The eastern portion of the arc is completed by the Newfoundland Ridge at 53°W. W e consider the circulation of the water mass that lies between 2°C and I 2°C in this basin. The values of the parameters are /3*~1 /4, Under these numbers the upper bound for the magnitude of the deep flow is of the order of 20 cm/sec in the major portion of the gyre, and the returning flow is of the order of 200 cm/sec at its maximum. The width of the returning current is of the order of I oo km. The total transport is of the order of 30 x 106 m3/sec. Actual calculations for the detailed structure of the flow reveal that these velocity and transport estimates are high by at least a factor of two or three. Moreover, the numerical values of the field quantities depend on the choice of ko, k,, and C. In particular, note that the characteristic horizontal density difference (and therefore the thermal wind shear) is a function or these quantities. Fig. 4 represents Worthington's schematic circulation derived from water-mass analysis and geostrophic calculations. T o infer this pattern, Worthington 80° 70° 60° 50° 40° 400m TOPOGRAPHY // I 50° 40° 30° 20° Figure 4. Transport of water that l ies between 2° C and 12°C i~ the Gulf-Stream system. Each transport lin e represents 106 m3/sec accordmg to Worthmgton (1965). 230 J ournal of Mm-ine R esearch [ 2 3,3 has expli citl y assumed that water transported to the east by the Gulf Stream returns to the west at the same temperature range. Thus, his empirical analysis is consistent with potential-vorti city conservati on, and the resemblance to our results, though striking, is understandable. Acknowledgment. T he support of this work by the National Science Foun-dati on under a grant in Oceanography to Harvard University (GP-3533) is gratefull y recognized. R EFER ENCES CREASE, ] . 1962. Velocity measurements in the deep water of the western Nor th Atlantic. J. geophys. Res., 67: 3173- 3176. R OBINSON, A. R . 1965. Three dimensional mode of inertial currents in a variable-density ocean. J. fl uid M ech., 2r: 211-223. W ORTH INGTON, L. V. 1965. On the shape of the Gulf Stream system. Trans. Amer . geophys. Un., 46 (1): 99. TOOLS OF OCEANOGRAPHY 

On thermally maintained circulation in a closed ocean basin

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On thermally maintained circulation in a closed ocean basin

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