When the wind blows over an ocean, a stress is exerted on its surface. The applied wind stress produces a horizontal mass transport in a thin surface layer, which is mostly between 10 and 100 meters deep. If the applied stress were spatially uniform, the ocean below this layer would be little affected by the wind. However, spatial variations of the wind also cause spatial variations in the horizontal mass transport near the surface. These horizontal divergences or convergences of matter result in vertical motions, with water flowing upward or downward to replace the displaced surface water. Deep in the ocean, however, this vertical mass flux must move so as to preserve the specific angular momentum of each fluid particle. As we shall see in Section 2.6.2, this may be accomplished by large-scale horizontal motions in the deep interior, where the basic momentum balance for this flow is given by the geostrophic approximation. Such a simple solution is clearly not complete, however, since those terms in the vorticity equation that are negligible in the open ocean become important near the ocean's lateral boundaries. In Section 2.6.3 we shall thus present a geostrophic flow that is frictionally eroded by horizontal eddy viscosity acting near these boundaries but is maintained by wind stresses acting over the ocean's surface.
2.6.1 Ekman layer at the ocean-atmosphere interface
We suppose that the ocean is bounded by the horizontal surface z = 0 at which the wind stress
is applied. Since we are making allowance for a large-scale geostrophic motion in the inviscid interior, the basic flow in the surface layer is described by Eqs. (2.87) and (2.88).
However, the presence of an applied wind stress requires that du d v
— = Sx and p KV — dz dz p Kv— = Sx and p Kv — = Sy, at z = 0, (2.103)
where we took into account that the vertical scale of the boundary layer is much smaller than that of the horizontal scale on which the wind stress varies (see Eqs. [2.62] and [2.63]). The flow in the surface layer must also merge with the geostrophic flow at depth. Hence, we let u ^ ug and v ^ vg, as z ^ -<x, (2.104)
where ug and vg are defined in Eq. (2.79). As was originally shown by Ekman (1905), the boundary-layer solution satisfying these conditions is u = ug + e+z/A[(Sy + Sx)cos(z/A) - (Sy - Sx)sin(z/A)] (2.105) 2 p KV
and v = Vg + 1 e+z/A[(Sy - Sx) cos(z/A) + (Sy + Sx)sin(z/A)], (2.106) 2 p K V
where A is defined in Eq. (2.93).* Let us define uB = (u - ug)i + (v - Vg) j, (2.107)
which is the friction velocity in the surface boundary layer. Figure 2.4 illustrates the spiral distribution of this vector. At the surface, the velocity uE is 45° to the right of the applied wind stress. As the depth below the free surface increases, the direction of this vector rotates uniformly in a clockwise sense and its magnitude falls off exponentially. The horizontal mass flux associated with the velocity uE is f0 1
J-™ Pf which does not depend on the eddy viscosity. Note also that the vector ME is orthogonal to the applied stress. This is a consequence of the fact that a net mass flux in that direction would give rise to a net Coriolis force that would remain unbalanced.
For further reference, we can also integrate the continuity equation for the velocity uE over the entire depth of the Ekman layer to obtain the vertical velocity wE flowing into that layer. Taking into account that the surface wind usually varies much more rapidly
* This solution was motivated by observations made by FridtjofNansen (1861-1930) during the Norwegian North Polar Expedition of 1893-1896. Looking at observations of the wind and of ice drift taken from his ship Fram (i.e., "Forward") while she drifted in the arctic ice, he saw that the direction of the ice drift showed a systematic deviation to the right relative to the wind direction. Nansen correctly guessed that this deviation was in some way related to the Earth's rotation. The problem was given to the young Swedish scientist Vagn Walfrid Ekman (1874-1954), who came out in 1905 with a full scale theory of the so-called Ekman spiral. For a detailed account of these and related matters, see Arnt Eliassen, "Vilhelm Bjerknes and his Students," Annual Review of Fluid Mechanics, 14, 1, 1982.
than f, one finds that
which is the relation between the vertical velocity at the lower edge of the surface boundary layer and the z component of the curl of the surface wind stress.
To illustrate the basic features of wind-driven currents in the oceans, we shall assume that our model, of constant depth H, is of uniform density. The rotating fluid thus consists of three regions: a thin surface Ekman layer, the inviscid interior, and a thin bottom frictional layer. For low Rossby numbers, the basic momentum balance is given by the geostrophic approximation. By making use of Eqs. (2.78) and (2.97), we can also write
Dt d z where Z is the z component of the relative vorticity of the interior flow and fiv is the change with latitude of the planetary vorticity f. For steady motions and low Rossby numbers, this equation becomes dw fv = f—. (2.111)
Since the interior flow is homogeneous and geostrophic, the functions u, v, and Z must be independent of z (see Eq. [2.83]). Hence, Eq. (2.111) may be integrated over the whole depth of the ocean to give pv = f [wE(x, y, 0) - wE(x, y, —H)], (2.112)
where wE(x, y, 0) and wE (x, y, — H) are the vertical velocities entering the interior flow on its upper and lower edge, respectively (see Eqs. [2.109] and [2.96]). Neglecting the bottom contribution, we obtain pv = — k ■ curl S. (2.113)
This is Sverdrup's (1947) solution for large-scale oceanic currents.
The interpretation of these relations is as follows. Wherever the horizontal mass flow in the surface layer is divergent, with upwelling taking place to replenish the surface water transported away, the right-hand sides of Eqs. (2.109) and (2.113) are positive so that the change of planetary vorticity along the motion, pv, is also positive. Since the planetary vorticity f increases poleward, it follows at once that the interior flow must move poleward to maintain the balance defined by Eq. (2.113). Conversely, wherever the horizontal surface transport is convergent, with downwelling as a result of the accumulation of mass, the interior flow must move in the direction along which the planetary vorticity f decreases, that is to say, equatorward. Such a mass transport is observed over much of the circulation pattern in the North Atlantic Ocean, with equatorward flowing currents on the eastern side of the ocean basin, north of approximately 20°N (e.g., Pedlosky , Fig. 5.1.1, or Apel , Fig. 2.21). The southward Sverdrup flow cannot apply to the whole ocean basin, however, since it must necessarily return to the north so as to ensure continuity and conservation of specific angular momentum of each fluid particle. As we shall see below, this circumstance results in the formation of the intense and narrow Gulf Stream current, which flows along the western edge of the North Atlantic Ocean, from Florida to Cape Hatteras, where it rejoins the generally clockwise oceanic circulation. Although the Gulf Stream is the best-known example of a western boundary current, such a vorticity-balancing and mass-balancing flow is present off the east coasts of continents everywhere in the world.
2.6.3 Western boundary currents: TheMunk layer
Although the southward Sverdrup flow occurs over a very large portion of an ocean basin, Eq. (2.113) alone cannot satisfy all lateral boundary conditions, nor can it satisfy mass conservation for the basin as a whole. The inadequacy of this solution strongly suggests the existence of turbulent boundary layers adjacent to the perimeter of the basin. Making allowance for horizontal friction in Eqs. (2.74) and (2.75), we can rewrite the vorticity equation in the form
where KH = AH/p (see Eq. [2.72]). Again for steady motions and low Rossby numbers, this equation may be integrated over the thickness of the ocean to yield pv—+0)=pHk'c"rls- (2115)
(Compare with Eq. [2.113].) Since the horizontal motion is geostrophic, the functions u and v may be written in terms of a stream function ^. We can thus write u =--and v = +--, (2.116)
9y 9 x so that
To be specific, we shall consider the case where the meridional boundaries x — Xw and x = XE are independent of latitude. In the interior of the ocean, away from these boundaries, we have p— = — k ■ curl S, (2.118)
where is the interior stream function, which depends on the applied wind stress. In the boundary layers, however, Eq. (2.115) can be written as
since only the highest derivative with respect to x will be retained in the boundary-layer analysis.
Near the eastern boundary, it is convenient to make use of the stretched variable
In order to solve Eq. (2.119), we shall also let
V = Vi(x, y) + Ve(£, y), (2.122) where VE must go to zero as £ ^ œ. Inserting this relation into Eq. (2.119), one obtains d 4Ve 9Ve
As usual, the complete solution must satisfy the conditions of no normal flow and no slip at the boundary x = XE (see Eqs. [2.17] and [2.18]). Thus, we also have
The solution that satisfies these conditions is
pfi H Jxe
It is immediately apparent from Eq. (2.125) that this solution acts only to satisfy the no-slip condition on the eastern boundary, having little effect on the large-scale mass transport in the ocean.
The situation is quite different on the western side of the ocean. Here we shall define the stretched variable x — Xw n = ^XK , (2.127)
where S is still defined in Eq. (2.121). We shall also let
where Vw must go to zero as n ^ to. By virtue of Eq. (2.119), the function Vw satisfies d4Vw dVw dn4 dn
The boundary conditions at the western boundary x — Xw are
The appropriate solution is
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