A mathematical model is developed to study the free (frictionless) steady horizontal flow which can occur in a homogeneous ocean of constant depth. The flow satisfies the dynamic constraint that the vertical component of absolute vorticity is constant along a streamline...

Fofonoff, N. P.

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/   STEADY FLOW IN A FRICTIONLESS HOMOGENEOUS OCEAN 1 BY N. P. FOFONOFF2 Brown University, Providence 12, R. I. ABSTRACT A mathematical model is developed to study the free (frictionless) steady hori-zontal flow which can occur in a homogeneous ocean of constant depth. The flow satisfies the dynamic constraint that the vertical component of absolute vorticity is constant along a streamline. The conclusion is reached that in an enclosed ocean a free steady circulation cannot have any slow broad eastward currents. The east-ward currents must occur as narrow streams of high velocity and high relative vorticity. Intensified currents are present along the eastern and western coasts. The theory which is developed for the homogeneous ocean of constant depth can be applied to the two-layer ocean if the horizontal divergence of flow is negligible. If the horizontal divergence is not negligible, then the intensification of poleward currents is more pronounced and that of equatorward currents less pronounced as compared with the homogeneous ocean. INTRODUCTION The theoretical models of the wind-induced oceanic circulation de-veloped by Stommel (1948), Munk (1950), and others have contributed a great deal to the understanding of some large-scale features of the circulation in the major oceans, but these models are limit ed by the fact that they are based on linearized equations and on the artificial concept of eddy viscosity. It is possible that significant features of oceanic circulation cannot be simulated by a model in which the non-linear terms, i. e., the relative-acceleration terms, are neglected. At-tempts to improve and extend these models are hindered by the analytical complexity of a model which includes both relative acceler-ations and dissipative effects. In the present paper, simple types of ideal fluid motion are studied in order to describe characteristic features of steady free circulation in an enclosed ocean. By considering an ideal fluid , the analytical diffi-culties presented by the nonlinear terms in the hydrodynamic equations 1 Technical Report 8, Office of Naval Research contract N7onr-358(11), NR 083-050 with Brown University. Paper presented 5 May 1954 at American Geo-physical Union annual meeting, Washington, D. C. s Now with Pacific Oceanographic Group, Nanaimo, B. C. 254 1954] Fof onojf: Flow in Homogeneous Ocean 255 are avoided. It is possible to show that some features of oceanic circulation, e.g., the westward intensification of the circulation, exist in the absence of friction. The simplest model that includes accelerations in the flow is a frictionless homogeneous constant-depth ocean in which the water moves horizontally in the absence of driving forces. This model does not contain any mechanism by which free flow can be established from a state of rest, but it does contain certain dynamic restrictions which must be satisfied by an existing free flow. For steady flow, the most important constraint is that the vertical component of absolute vorticity is constant along a streamline. This constraint on the abso-lute vorticity is stringent, because the change of the Coriolis parameter over a few degrees of latitude is, over most of the real ocean, about two orders of magnitude greater than the relative vorticity of the flow. It is apparent, therefore, that the possible types of free flow are severely restricted in the ocean. Rossby (1936) has considered the dynamic constraint on the vertical component of absolute vorticity in his study of the Gulf Stream, but his analysis did not extend to the entire ocean. In the study of free atmospheric flow, there has been more progress and some attempts have been made to adapt the results of these studies to circulation in the oceans. H(lliland (1950) found particular examples of free flow which he applied to the ocean. Although these examples agree qualitatively with the free flows discussed in the present paper, H0iland fails to point out the pronounced characteristics that the free flow would exhibit. ASSUMPTIONS AND NOTATION The ocean model chosen in this analysis consists of a homogeneous body of water of constant depth bounded by smooth vertical lateral boundaries. The motion is assumed to be horizontal and uniform with depth. The effects of thermohaline structure, compressibility, and frictional forces are neglected everywhere. Vertical motions and 5-ccelerations are also neglected so that the pressure is simply deter-mined by the hydrostatic equation. The unit vector k is directed vertically upward, and z is the vertical distance measured upward from a level undisturbed ocean surface. The ocean surface in the model is at z = 1/· The horizontal velocity is designated by v and the stream function of the horizontal circulation by y;. The vertical component of absolute vorticity fa equals f + f, where f is the Coriolis parameter and r is the vertical component of relative vorticity. The quantity Q is introduced for pf p + ½(v •v) + gz, where p is pressure, p density, and g acceleration of gravity. 256 Journal of Marine Research (13, 3 EQUATIONS OF MOTION The equations of motion, consistent with the assumptions made above, may be written (v. V) V + j(k X v) = - V p/ p ' V•V = 0, p = pg(TJ - z) ' (1) (2) (3) where the differential vector operator V refers to the horizontal coordinates only. Equation (1) may be written .i"a(k X v) = - VQ or where .i"aV = - V X (Qk) , Q = gT] + ½ (v ·V) . The velocity stream function if; is defined by the equation v = V X (if;k). Elimination of v from equation (4) yields r.vf = - vQ, so that Q = Q(if;) ' .i"a = - dQ/dif;. (4) (5) (6) (7) (8) (9) Equation (9) is a statement of the well known result that the vertical component of absolute vorticity is constant along a streamline. This equation expresses the basic dynamic constraint to be satisfied by the steady free motion in a homogeneous ocean. DIMENSION AL AN AL YSIS The following dimensional constants are chosen as being representa-tive of a typical ocean: Lo = 2000 km, a typical horizontal length; Jo = 10-4, a typical value of the Coriolis parameter, TJo = 100 cm, a typical displacement of the free surface from the undisturbed level z = 0; g = 103 cm sec-2, gravity. The equations are converted into nondimensional form by intro-ducing the nondimensional variables v' = v/U, where U is a charac-1954] Fofonoff: Flow in Homogeneous Ocean 257 teristic velocity, if/ = t/t/ ULo, Q' = Q/go, ta' = ta/fo, f' = f/f'o, !;' = !;/ ( U /Lo), and V' = Lo V, the nondimensional differential operator. By choosing U = g710/f0Lo = 5 cm sec- 1 and defining o = U /f0L0 = 2.5 X 10-4, the nondimensional form of (6), (8), and (9) may be written v' = V' X (it,'k) , Q' = Q'(t/t') = 711 + ½ o(v' •v'), - dQ'/#' = f' + o!;' = f' - oV' 2it,'. THE VORTICITY EQUATION (10) (11) (12) The vorticity equation (12) contains two unknown functions, it,' and !; a', whereas (11) contains three unknown functions, it,', Q', and 71'. Therefore the motion is restricted only by (12), because 71' can always be chosen to satisfy (11) once it,' and ta' are determined. Equation (12) does not determine the motion uniquely, because the vertical component of absolute vorticity can be any function of the stream function. It is evident from (12) that, unless relative vorticities comparable with the Coriolis parameter (i.e., t' ~ l/o) are present in the flow , the stream function will be a function off' only and the flow will be along latitude circles only. It follows, therefore, that in the vicinity of meridional boundaries where the flow must be across latitude circles, the relative vorticity and hence the velocity must be much larger than the average for the ocean. It is not possible to have in an enclosed ocean a steady free circulation which does not contain relatively large velocities unless the circulation is confined to a narrow range of lati-tude. For certain choices of ta'(it,'), the circulation cannot be slow, e.g., t 0 ' = constant. In order to obtain a steady free circulation that is slow over most of the ocean, restrictions must be imposed upon the choice of ta'(t/t'). The nature of these restrictions is illustrated by considering a simple example. A SIMPLE EXAMPLE OF FREE FLOW A rectangular nondimensional coordinate system x', y' is introduced with the positive x'-axis eastward and the positive y'-axis northward. A rectangular region with sides a', b' is chosen to represent he ocean. The function ta'(t/t') is assumed to be a linear function of it,', so that (12) becomes f' - ov' 2ip 1 = Co + C11f1 , (13) where c0 and c1 are constants. The boundary condition on it,' is that the ocean boundary 1s a 258 Journal of Marine Research [13, 3 streamline. This condition is satisfied by requiring the stream function to be zero along the ocean boundary. The Coriolis parameter is assumed to be a linear function of y' of the form f' = Jo' + {3y'. + 1000 km --------------4 &-;. ____________________________ - ---------------------~ ~------------------------------------------------------- J _____________________________ ..,  _________________________  4 1000 km Figure 1. Streamlines of free flow In a homogeneous ocean, computed from equation (15) for 110' = b'. b' = I in the upper diagram and for II• ' = 0, b' = I in the lower diagram. The number b' is the ratio of the meridional extent of the ocean to Lo. 1954] Fof onojf: Flow in Homogeneous Ocean By defining Uo1 and Yo' so that uo' = (3/c1 and y0' equation (13) may be written (uo'o/(3)V'2,f/ + y/ = Uo'(y' - Yo'). 259 (co - Jo')/c1 , (14) In equation (14), uo' is a nondimensional velocity, o a ratio of relative vorticity to the Coriolis parameter, and (3 the nondimensional north-ward variation of the Coriolis parameter. Wherever the relative vorticity is negligible, the velocity is given by uo'; eastward if uo' is positive and westward if negative. A study of (14) reveals (a) that the relative vorticity in an enclosed ocean can be negligible only if uo' is negative and (b) that all eastward currents Or---,----,-----,-----::=='I""""-, 100 0 20 -utu.• 40 •o Figure 2. Velocity distiibution across the eastward jet. computed for Yo' = 0 , b' = 1, and Lo = 2000 km. The number b' is the ratio of the meiidional extent of the ocean to Lo. must occur as narrow streams of high velocity and high relative vorticity. Although the exact solution of (14) can be obtained, the simpler approximate solution obtained by boundary-layer methods is more convenient. The boundary-layer solution may be written y/ = uo'{(y' - Ya')+ Yo'exp(- y'/c.) - (b' - Yo')exp[- (b' - y')/c.]) X / I - exp(- x'/c) - exp[- (a' - x')/c]) , (15) where E = (lu0'o/f31)½. This solution is valid so long as b'/c » 1 and a'/E » 1. In Fig. 1, streamlines of the free flow are shown for Yo' = b', b' = 1 in the upper diagram and for y 0' = 0, b' = I in the lower diagram. These diagrams may be joined to give a symmetrical jet along the middle of the ocean, flanked by broad westward currents to the north 260 Journal of Marine Research [13, 3 and south. The velocity distribution across the eastward jet for y0' = 0, b' = I is computed from the equation u'/uo' = I - (1/E) exp [ - (I - y')/E] and is shown in Fig. 2. All the figures are based on computations for which £2 = 2.5 X 10-4_ DISCUSSION It is apparent from the preceding analysis that the fr ee steady circulation in an enclosed ocean has distinctive features due to the variation of Coriolis parameter with latitude. The circulation, char-acterized by a slow westward drift over the major portion of the ocean, is accelerated and concentrated into a swift current along the western boundary of the ocean. The eastward current occurs as a narrow stream of high velocity and high relative vorticity. This eastward jet which crosses the ocean at constant speed is decelerated and spread out along the eastern boundary of the ocean. A slow, broad, eastward current cannot exist in the steady state. The east-west symmetry of the stream function for the homogeneous ocean cannot exist if any dissipation is present. The swift current along the eastern boundary of the ocean, which must be supplied by a jet, is very sensitive to losses of relative vorticity in the jet. The concentration of the westward drift into an eastward jet would occur in the real ocean if frictional stresses were insufficient to de-celerate the current along the western boundary of the ocean. The presence of the Gulf Stream in the open ocean northeast of Cape Hatteras suggests that a considerable portion of the momentum and relative vorticity of the Stream actually reaches the open ocean and is dissipated there. EXTENSION TO THE TWO-LAYER OCEAN Most of the real ocean has a more or less clearly defined upper layer which contains the relatively strong motion and which represents the portion of the ocean influenced by wind stress. Below this layer there is a much larger body of water of greater density which shows little motion and little response to the prevailing atmospheric state. This picture of the ocean has often been represented in theoretical considerations, in idealized form, by the two-layer ocean consisting of two homogeneous layers of water of different densities with the motion confined entirely to the upper less dense layer. The two-layer ocean in its undisturbed state may be completely specified by giving the lateral boundaries, the densities p0 and p1 of the upper and lower layers respectively, and the volume V0 of the 1954] Fofonojf: Flow in Homogeneous Ocean 261 upper layer. In order to adapt the analysis of the homogeneous ocean to the two-layer ocean, the additional quantities 77 1, h, y.,, u, -y are introduced and defined as follows: z = 771, the equation for the interface between the two layers; h = 77 + 771, the thickness of the upper layer; if; , the stream function of volume transport; u = V •V , the divergence of the horizont al velocity; -y = p0/( p1 - p0) . The pressure in the lower layer is given by (16) The condition for no motion in the lower layer requires the horizontal pressure gradient to be zero everywhere in the lower layer, so that V771 = -yV77 , 771 = 1'77 + ho , h = (1 + -y)'l7 + ho. (17) The arbitrary constant ho is determined from the specified volume Vo of the upper layer, or from an equivalent condition. The equations for the two-layer ocean, analogous to equations (5), (6), and (9) for the homogeneous ocean, are Q vh = V X (ij;k ) , Q(if; ) = g(h - h0)/(l + -y) + ½(v •v) , fa'/h = - dQ/#. (18) (19) (20) The corresponding nondimensional equations for the two-layer ocean are formed by introducing the addit ional nondimensional variables h', if; ', u', defined ash' = h/ho, if;' = if; /UhoLo, u' = u/(U/Lo), and the nondimesnional parameter a, defined as a = (1 + -y)'l7o/ho . The nondimensional equations which form the dynamic constraints for free motion in the two-layer ocean are ta'/h' = - dQ'/dif;', Q'(if;' ) = (1 /a ) (h' - 1) + ½o(v' •V' ) , (21) (22) where ta' = f' + o( and h' = l + a771 • Equations (21) and (22) contain three unknown functions each and must be solved simultaceously. Thus, in the two-layer ocean, (22) is restri ctive in contrast with (11) whi ch is not restrictive. As in the homogeneous ocean, the cir culation is not uniquely determined, be-cause fa' /h' can be any function of if; '. No solutions of the equations for the t wo-layer ocean have been obtained. The circulation in t he upper layer will resemble the circu-lation in the homogeneous ocean if t he divergence u' can be nrglected. 262 Journal of Marine Research [13, 3 The divergence is small if the change in thickness of the upper layer along a streamline is small compared with the total thickness of the upper layer, i.e., if a is small compared with unity. In high-velocity regions of the two-layer ocean the thickness of the upper layer is reduced, so that the absolute vorticity is smaller in magnitude than the absolute vorticity in a corresponding region of the homogeneous ocean. For a two-layer ocean located in the north-ern hemisphere, the relative vorticity must be less than the relative vorticity in the homogeneous ocean for corresponding flow s. There-fore, the intensification of a current flowing toward the pole is more pronounced and that of a current flowing toward the equator is less pronounced than the intensification of a corresponding current in the homogeneous ocean. REFERENCES H¢ILAND, E. 1950. On horizontal motion in a rotating fluid . Geofys. Pub!., 17(10): 5-26. MUNK, w. H. 1950. On the wind-driven ocean circulation. J. Meteorol., 7(2): 79-93. RossBY, C.-G. 1936. Dynamics of steady ocean currents in the light of experimental fluid mechanics. Pap. phys. Oceanogr. Meteorol., 5(1): 1-43. STOMMEL, HENRY 1948. The westward intensification of wind-driven ocean currents. Trans. Amer. geophys. Un., 29(2): 202-206. 

Steady flow in a frictionless homogeneous ocean

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Steady flow in a frictionless homogeneous ocean

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