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Doron Nof

Abstract

The behavior of outflows resulting from channels cutting through broad continents and emptying into wedgelike oceans, or channels cutting in wedgelike continents and emptying into broad oceans, is examined analytically. The model is nonlinear and inviscid, and the vertical structure is approximated by two layers; the upper layer is active and the lower is passive.

Examination of the governing equations shows that, since outflows are externally driven (by gravity and mass flux), there exists an “outflow length scale” in the open ocean. This length scale (l) is given by [gHb/fU 0]½, where b is half the emptying channel width, g′ the “reduced gravity,” H the channel depth, f the Coriolis parameter, and U 0 the flow speed within the channel. Solutions are constructed using this new length scale and a power series expansion.

It is found that, due to the earth's rotation, an outflow can be deflected toward one of the coasts or bifurcate into two branches, depending on the basin geometry. When the outflow results from a channel cutting through a broad continent and emptying into a wedgelike ocean, there are two possibilities. If the wedge opening is less than 90°, the outflow deflects to the right (looking downstream); if the wedge opening is larger than 90°, the outflow deflects to the left. In contrast, when the channel is cutting through a deltalike continent and emptying into a broad ocean, the outflow bifurcates. If the angle between the two walls bounding the ocean is less than 270°, the outflow splits into a narrow band that flows to the right and a broad current that veers to the left and penetrates into the ocean interior as an isolated ocean. A mirrored picture is established when the angle between the walls is large than 270°.

Possible application of this theory to the two outflow modes observed near the Tsugaru Strait is mentioned.

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Doron Nof

Abstract

The author considers two oceanic basins separated by a meridional wall. The wall contains a gap that is initially blocked by a gate; westward winds are allowed to blow over the two-layered oceans creating western boundary currents and a sea level difference between the basins. The conceptual gate is then removed and the resulting nonlinear flow is computed.

The analytical calculations are based on a simple wind-driven general circulation model and a nonlinear integrated momentum constraint. Two classes of nonlinear solutions are constructed. One corresponds to a situation where the flow through the gap originates from the right-hand side (looking upstream) of the inner Pacific basin and the other to a situation where the flow originates from the left-hand side. It is suggested that the actual Indonesian Throughflow is composed of both of these classes of flows; that is, the throughflow corresponds to an exchange via two adjacent gaps.

Computations suggest that approximately 6 Sv (Sv ≡ 106 m3 s−1) enter the passages from the North Pacific and 1 Sv from the South Pacific giving a total of 7 Sv. This may resolve the apparent difficulty associated with existing linear theories (and nonlinear theories that neglect western boundary currents), which predict that without strong turbulent diffusion only South Pacific water can enter the passages.

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Doron Nof

Abstract

In this paper a mechanism is proposed which could be responsible for the formation of sharp horizontal density gradients such as those observed in shallow seas, from fluid which initially has weak horizontal density gradients. The sharp density gradients result from the mutual intrusion of several stratified bodies of water which were exposed to various degrees of vertical mixing for a limited amount of time. The dynamics of the intrusion are examined by a simplified nonrotating, frictionless multilayer model. The results are compared quantitatively to laboratory experiments and qualitatively to field observations.

The theoretical model contains an upper and lower portion, each of which consists of several bodies of fluids with different densities corresponding to various degrees of mixing. It predicts that in both the upper and the lower portions, fluids which were exposed to intermediate mixing sink rapidly from the surface, rise from the bottom, and after a finite amount of time concentrate in mid-depth. This results in a formation of density discontinuities (fronts) near the surface, bottom, and in the boundary between the upper and the lower portions.

Rotation is excluded from the simplified model, but it is expected that mutual intrusion will take place even if rotation is included, provided that the flow is not in an exact geostrophic balance. The theoretical predictions were tested in the laboratory in a tank which contained several bodies of water with different densities separated initially by a number of gates. The experimental results compare favorably with the theoretical predictions. Observations which suggest the existence of mutual intrusion in frontal zones are discussed.

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Doron Nof

Abstract

Organized depth discontinuities involving a balance between steepening and dissipation are usually referred to as shock waves. An analytical “educed gravity” model is used to examine a special kind of shock wave. The wave under study is a depth discontinuity associated with a transition between a supercritical and subcritical flow in a channel. Even though the wave itself is highly nonlinear, the adjacent upstream and downstream fields are exactly geostrophic in the cross-stream direction. For this reason we term the wave a geostrophic shock wave. We focus on a stationary shock wave whose horizontal projection is a straight line perpendicular to the side walls. Solutions for the entire field are constructed analytically using power series expansions and shock conditions equivalent to the so-called Rankine-Hugoniot constraints.

It is found that, for particular upstream conditions, a geostrophic shock wave can be formed if the particle speed exceeds the surface gravity wave speed (i.e., the flow is “supercritical”). Specifically, in addition to supercriticality, a stationary geostrophic wave requires the upstream velocity to have a particular structure which depends on the strength of the shock and the channel width. When the latter condition is not met, a shock wave is still possible, but its adjacent fields will not be geostrophic and its shape will correspond to an “S” rather than a straight line.

Being the only known analytical solution for the entire field of shock waves on a rotating earth, the geostrophic shock provides useful information on the wave structure. For instance, it is shown that even though momentum is conserved across the shocks, relatively large changes in potential vorticity take place. For depth discontinuity of O(I) (i.e. high “amplitudes”), there is a generation of potential vorticity that is also of O(I). Such a phenomenon does not occur on a nonrotating plane where the (zero) potential vorticity may be altered through the action of shock waves in channels and passages. Possible application of this theory to various oceanic situations is mentioned.

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Doron Nof

Abstract

The interaction of two isolated lens-like eddies is examined with the aid of an inviscid nonlinear model. The barotropic layer in which the lenses are embedded is infinitely deep so that there is no interaction between the eddies unless their edges touch each other. It is assumed that the latter is brought about by a mean flow which relaxes after pushing the eddies against each other and forming a “figure 8” structure.

Using qualitative arguments (based on continuity and conservation of energy along the eddies’ edge) it is shown that, once a “figure 8” shape is established, intrusions along the eddies’ peripheries are generated. These intrusions resemble “arms” or “tentacles” and their structure gives the impression that one vortex is “hugging” the other. As time goes on the tentacles become longer and longer and, ultimately, the eddies are entirely converted into very long spiral-like tentacles. These spiraled tentacles are adjacent to each other so that the final result is a single vortex containing the fluid of the two parent eddies. It is speculated that the above process leads to the actual merging of lens-like eddies in the ocean.

Because of the inherent nonlinearity and the fact that the problem is three-dimensional (x, y, t), the complete details of the above process cannot be described analytically. Therefore, one cannot prove in a rigorous manner that the above process is the only possible merging mechanism. It is, however, possible to rigorously show analytically and experimentally that the intrusions and tentacles are inevitable. For this purpose, one of the interacting eddies is conceptually replaced by a solid cylinder. Initially, the cylinder drifts toward the eddy; subsequently, it is pushed slightly into the eddy and is then held fixed. The subsequent events are examined in a rigorous mathematical and experimental manner.

It is found that as the cylinder is forced into the eddy, a band of eddy water starts enveloping the cylinder in the clockwise direction. This tentacle continues to intrude along the cylinder parameter until it ultimately reattaches itself to the eddy, forming a “padlock” flow. Simple laboratory experiments on a rotating table clearly demonstrate that a “padlock” flow is indeed established when a lens is interacting with a solid cylinder. Using the details of this process it is argued that, in the actual eddy–eddy interaction case, intrusions must be established and that, consequently, merging of the two eddies is inevitable.

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Doron Nof

Abstract

In this brief note it is demonstrated that the question of what is the mechanism(s) responsible for the southern migration of the Gulf Stream front during wanter–is still open.

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Doron Nof

Abstract

A nonlinear one-layer model is considered in order to describe the way that water with a relative vorticity intrudes into an otherwise stagnant channel. The channel has a uniform depth (D) and width (L) and the fluid is taken to be inviscid. The intruding fluid is separated from the (initially stagnant) water in the channel by a free dividing streamline that corresponds to a “vorticity front.” This front intersects the channel wall (at the head of the intrusion) and extends backwards upstream. As the fluid with relative vorticity is intruding into the channel, the fluid with no relative vorticity (i.e., the fluid present in the channel prior to the intrusion) escapes in the opposite direction. This flow compensates for the fluid displaced by the advancing intrusion. Solutions for steadily propagation intrusions are obtained analytically by equating the flow-force ahead of and behind the for steadily propagating intrusions are obtained analytically by equating the flow-force ahead of and behind the intrusion. Namely, steady state solutions correspond to a balance between the forward momentum flux and the form drag exerted on the intrusion by the escaping fluid. The nature of the intersection of the front with the wall is analyzed by methods similar to those employed by Stokes for analyzing the maximum steepness of surface gravity waves.

It is found that the vorticity in the intruding fluid “controls” the amount of fluid that flows through the channel. When the vorticity (ζ) of the intruding fluid is uniform, the width of the intrusion is always 2/3 of the channel width and the net volume flux of the intruding fluid is (2/27)ζDL 2. In the presence of weak dissipation, the channel can can transfer an amount less than (2/27)ζDL 2, but, under no circumstances can the channel the so-called hydraulic control {∼O[(gD)½ DL]}, which corresponds to the flux of an intrusion without any relative vorticity. When ζ∼O(f), the ratio between the maximum flux allowed by the vorticity control to the flux allowed by the hydraulic control is equivalent to about 1/10 of the ratio between the channel width and the barotropic deformation radius. Hence, for midlatitude channels, the vorticity control may limit the flux to a few percent of that associated with the hydraulic control.

Possible application of this theory to various oceanic situations is mentioned.

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Doron Nof

Using hydrographic data and box models, it is shown that the presently discussed diversion of rivers such as the Yellow or the Yangtze for agricultural use is likely to cause the renewal of Bottom Water formation in the Japan/East Sea. Such formation was common (near the Siberian coast) in the 1930s, 1940s, and 1950s, but subsided since that time due to a warming trend (accompanied by a decreased salinity due to the melting of ice). Since a diversion of freshwater is analogous to evaporation, a (diversion induced) increase of salinity is expected and the increase is large enough to allow Bottom Water formation even at the present-day cooling rates. Even a modest diversion of “merely” 3000 m3 s−1 (which is 10% of the total freshwater flux) will probably cause Bottom Water formation at a rate of roughly 750 000 m3 s−1. This is the first study that predicts anthropogenic reversal of an existing vertical structure in a semienclosed sea.

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Doron Nof

Abstract

The diagnostic quasi-island model of Nof addressing the exchange between the South Atlantic and the Southern Ocean is extended to the exchange between the Pacific–Indian Ocean system and the Southern Ocean. The new calculations suggest that, in a similar fashion to the Atlantic Ocean, the Indian and Pacific Oceans have a meridional overturning cell with a transport of 18 ± 5 Sv (1 Sv ≡ 106 m3 s−1). In contrast to the Atlantic in which there is deep water formation resulting in a cell that extends all the way from the surface to the bottom, however, the Indian and Pacific circulation cell is shallow in the sense that it does not occupy the entire water column. As in the Atlantic case, the cell is driven by both winds and thermohaline processes, but the calculation does not require solving the complete wind–thermohaline problem. The computational method takes Africa, Asia, and Europe to compose a “pseudo island”; that is, the combined continent is entirely surrounded by water but has no net circulation around it. The continuation of sea level around the continent allows one to compute analytically the zonal upper-layer transport that is first forced meridionally from the Southern Ocean to the Pacific and Indian Oceans and then forced down to lower levels. Although there are no direct observations to support or refute the idea of a shallow cell in these oceans, the concept is consistent with earlier inverse calculations, with the observed distribution of silicate, and with earlier general circulation experiments. The main weaknesses of the calculation are the level-of-no-motion assumption (which is particularly questionable in high latitudes) and the neglect of form drag on the Bering Strait sill.

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Doron Nof

Abstract

A frictionless nonlinear model with allowance for motions which are far from a state of geostrophic balance is considered in order to describe the dynamics of outflows consisting of two layers of fluids. The governing equations are solved by means of perturbation expansions, conformal mapping and Fourier series. The theory is compared with laboratory experiments.

The model predicts that an outflow from a channel with uniform velocity distribution deflects to the right in the Northern Hemisphere. The parameters of the problem are combined in such a way as to show that rotational effects are important whenever the ratio between the internal Froude number to the Rossby number is not negligible; the inverse of this ratio has a “critical” value, below which the flow separates from the left basin bank. The mathematical analysis shows that an outflow from a channel with initial negative relative vorticity approximately equal to the Coriolis parameter deflects to the left. As in the uniform flow case the flow separates from one of the banks under certain “critical” conditions.

Two experimental systems which included an abrupt cross-sectional variation in a rotating channel consisting of two layers were used. The experimental results compare favorably with the direction of deflection predicted by the mathematical model. Possible application of this study to the Straits of Gibraltar and other outflows are discussed.

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