The Godfrey Island Rule is a generalization of the Sverdrup balance that overcomes a key limitation of Sverdrup's 1947 formulation for the relation of ocean currents to wind forcing, which could not be extended west of an interrupting island. This was a particular problem with the position of Australia between the Pacific and Indian Oceans; Godfrey's result showed the dynamical connection between the wind-driven circulations of both oceans.
History The Sverdrup balance (1947) was a monumental achievement in dynamical oceanography that clarified the large-scale response of ocean currents to wind forcing. It was followed quickly by Stommel and Munk's extensions to Sverdrup's theory that showed how the great ocean gyres explained by Sverdrup could be completed by western boundary currents. These three results enabled a consistent dynamical understanding of wind-driven currents in ocean basins like the Atlantic and the North Pacific that have continuous eastern and western coastlines and could be considered largely closed at their northern ends, but the South Pacific and Indian Oceans are connected not only by the Southern Ocean but by the open channel of the Indonesian Throughflow (ITF). The Sverdrup/Stommel/Munk results provided no framework to constrain either the flows of the South Pacific, with its two exits in the west, or the Indian Ocean with its two corresponding entrances. What determined how much water could flow through the ITF? What controlled meridional transport through the South Pacific? What are the relative roles of wind vs thermohaline (density) forcing? In the 40 years after Sverdrup, Stommel and Munk, accumulating observational and water property evidence (see Thermohaline circulation) showed that the ITF carried substantial flow from the South Pacific to the Indian Ocean, but no theory was available to explain how this was controlled or how large it should be. Godfrey's (1989) methodology provided this explanation, using assumptions and dynamics of comparable simplicity to the work of Sverdrup and Stommel.
Derivation Sverdrup fundamentals
Sverdrup made simplifying assumptions by neglecting friction (except that of the wind stress on the ocean surface), time variability (thus the Sverdrup balance represents the mean state), and the advective (nonlinear) terms of the Navier-Stokes equations, and by considering only the top-to-bottom vertically integrated currents in a flat-bottomed ocean. These are consequential restrictions that remove - in addition to thermohaline influences - many factors affecting ocean circulation, including mixing, boundary currents, and the ability to describe the vertical structure, but nevertheless reproduce in broad strokes the large-scale features of the currents.
With Sverdrup's simplifying assumptions, the barotropic momentum equations become
with the continuity equation
where upper-case variables are vertically integrated, is the Coriolis parameter, and are the eastward and northward velocity components, is the (integrated) pressure, is the wind stress (direction indicated by superscripts ), subscripts indicate derivatives. For simplicity of notation, both and are understood to be divided by the background density of seawater , thus the units of and are m2/s2. Sverdrup's top-to-bottom vertical integration in steady state leads to the simplified (non-divergent) continuity equation.
The simplified momentum equations (1) and (2) can be expressed as sums of the well-known geostrophic and Ekman flow types: .
Sverdrup's analysis proceeded with the crucial first step of taking the Curl of the momentum equations, using to define the variation of the Coriolis parameter with latitude ( is approximated as a constant). Then use continuity to remove the term. (See Sverdrup balance).
In regions where the simplifying assumptions hold, the result defines the meridional transport entirely in terms of the wind:
Godfrey's approach
Momentum equations expressed as pressure (western boundary currents) Godfrey derived his extension to Sverdrup using an unconventional method, writing the momentum equations in terms of pressure. Thus (1) and (2) become:
The pressure can be evaluated from the wind using () and for the pathway ABC in Figure 1.
For pathway AB against the coast, set in because cross-shore (here, zonal) flow must be zero, thus representing an alongshore wind balanced by an alongshore ocean pressure gradient: . (Here and below, it is assumed for simplicity that the coast runs north–south, but it is straightforward to define a sloped or curved coastline).
The integration is continued from the coast at B westward to an interior-ocean point C using Sverdrup's from (4) in , thus (schematically ). This calculation, repeated for each latitude B, then along each longitude to C, gives the complete pressure field relative to point A, everywhere east of a thin western boundary layer (Figure 2 is an example).
However, without accounting for the boundary layer, where equations (1)-(4) are not valid because of the Sverdrupian neglect of frictional and non-linear terms, the integrations cannot be carried further. In the spirit of Stommel (1948), Godfrey added terms representing unspecified stresses due to lateral and bottom friction, assuming that they were negligible away from the fast western boundary currents (thus, east of the boundary layer the Sverdrup solution does not change; in effect the boundary layer is appended to the interior solution). The augmented momentum equations become
The continuity equation (3) remains as before.
Godfrey additionally argued that the boundary layer is thin enough (relative to the width of the ocean) that winds within it can be ignored. Since the western boundary current (WBC) is strictly alongshore, cross-shore flow is zero, thus there are no zonal frictional stresses . With these stipulations, (5) and (6) in the boundary layer reduce to:
The first of these says that longshore flow in the boundary layer is geostrophic; the second that the longshore momentum balance is between the longshore pressure gradient and longshore friction . As Stommel had shown, this allows western boundary currents to complete the Sverdrupian gyres (e.g., the two subtropical gyres seen in Figure 2; for example the higher pressure along the western boundary near 25°N than near 40°N supports the Kuroshio Current off Japan by balancing alongshore friction). Note that this simple formulation defines only the transport of the WBC, saying nothing about its width or structure.
Now, the pressure integration can be defined for the entire coast-to-coast path DEF:
where (which remains unknown) is the total (coast-to-coast) meridional transport, namely the sum of the Sverdrup transport across the interior path from D to E (similar to the path BC found above), plus the geostrophic transport of the western boundary current. Schematically, (7) is .
In a closed basin in steady state there can be no net transport across a section like FED, so . This is the situation for the Atlantic and the North Pacific; the transport of the WBC exactly balances the interior Sverdrup transport: . But an island allows flow to go around either side. Even though the interior-ocean circulation east of the island may be fully defined by the Sverdrup balance, in the presence of an island is unknown so the Sverdrup/Stommel formulation outlined above cannot speak to the transport contribution of the island's WBC. This is the problem that Godfrey aimed to resolve.
The Island Rule Consider an island west of a continent as sketched in Figure 3. Note that although is not known, the same transport must flow across both sections QR and TS.
Godfrey could then define the pressure difference between Q and T along the path TSRQ with the method above: use equation (7) for the two zonal sections QR and TS, and , with U=0, for the alongshore path RS. Since is multiplied by in (7), the different values of at the latitudes of and must be taken into account:
,
where and are alternately and along the path TSRQ. This gives an expression for between Q and T.
Godfrey realized that he could write a second expression for the alongshore pressure difference QT on the west side of the island, analogous to that for the path SR, again specifying in ():
Equating these two expressions for , then solving for , Godfrey obtained his "Island Rule" defining the total transport between the island and the continent to its east, entirely as a function of the wind. The resulting integral is taken counterclockwise along the closed path TSRQT (red dashes in figure 3):
where again indicates the along-path direction.
Once is defined, the western boundary transport along the island's east coast can be found as the difference between (constant over the latitude range of the island) and the Sverdrup transport at each latitude .
A more physically intuitive derivation
The connection between ocean dynamics and along-path winds around the boundary of a region as in (8) might not be immediately obvious. This is clarified by Green's Theorem, which states that the line integral of a vector quantity around a closed curve equals the area integral of its Curl over that region.
Oceanographers usually think of western boundary currents as closing the interior (Sverdrup) zonal transport (see Sverdrup balance). A more physically intuitive derivation makes this explicit.
Add and subtract the along-path wind integral along the inner path (green line east of the western boundary layer in Figure 3) to divide the region into two: First, an integral around the inner path (box from the continental coast to the offshore edge of the island's western boundary layer , thus including only the region where the circulation can be assumed Sverdrupian); second, an integral around the island including its western boundary layer:
The inner-path integral (first term in the last equality above) can be simplified using Green's Theorem, approximating , then recognizing that Sverdrup
where the overbars indicate a meridional average: is the average Sverdrup meridional transport in the inner box .
Thus the alternate expression for the Island Rule is the inner-box average Sverdrup transport plus the wind integral around the island:
In addition to being easier to calculate the area-integral of the Curl than long path-integrals of the wind (which generally involve complicated curved coastlines), the circumisland wind term is usually (not always) small and can often be ignored. from (9) for the significant islands of the South Pacific is shown in Figure 6.
Neglecting the circumisland term, (9) states that The WBC against the island is due only to the variation of with latitude ( is a single number for the island), so if is constant in y, there will be no WBC: ). According to the Mean Value Theorem, there must be a zero WBC at some point along the island: The WBC against an island must either be identically zero or include a bifurcation. Unlike the path integral of wind east of a continent, which can support a large alongshore pressure gradient, the WBC against an island will usually provide only a small part of the net transport east of the island. Since (see Sverdrup balance), (9) emphasizes the role of the Sverdrup zonal transport feeding (or removing) mass to the western boundary region, which is the intuitive picture of how a western boundary current is driven. With in (9), the Island Rule passes the east-side meridional-average wind stress curl to its west side as a single streamfunction value () that resets the subsequent westward integration; note the flat streamfunction regions west of islands in Figure 5. If is not the same as that of the streamfunction in the regions north or south of the island tips, as will commonly be the case, the discontinuity will produce jets extending west along the streamfunction gradient at the latitudes of T and Q.
Illustration from observed winds