Lecture 25: Ocean circulation: inferences from geostrophic and thermal wind balance

Transcription

1 Lecture 25: Ocean circulation: inferences from geostrophic and thermal wind balance November 5, 2003 Today we are going to study vertical sections through the ocean and discuss what we can learn about the circulation by invoking geostrophic and thermal wind balance. 1 Temperature and Salinity sections Observed zonal-average mean distributions of T, S, and σ in the interior of the ocean are shown in Figs.1, 2 and 3. In the abyss, vertical gradients are weak and horizontal gradients are almost nonexistent; e.g., the deep ocean is everywhere very cold (between 0 and 1 o C) and no more that 1 o Cwarmerinthetropicsthaninhighlatitudes see Fig.1. In the upper kilometer of the ocean, however see top panel of Figs.1, 2 and Fig.3 there are strong vertical gradients (especially of temperature and density); this is the thermocline of the world s oceans having a depth of 600m in middle latitudes but shallowing to m in low latitudes. The temperature contrast between high and low latitudes is not surprising; the salinity contrast, as discussed above, reflects the pattern of E P. In summary, the zonal average picture reveals a warm, salty, light lens of fluid in the subtropics of each hemisphere, shallowing on the equatorial and polar flanks, floating on a cold, somewhat fresher abyss. These lenses exist in each ocean basin and exhibit considerable regional characteristics and variability. 1

2 Figure 1: Annual-mean cross-section of zonal-average potential temperature ( C) in the world s oceans: top shows upper km: bottom from 1km to the bottom. Light shading represents warm fluid. 2

3 Figure 2: Annual-mean cross-section of zonal-average salinity ( o / oo ) in the world s oceans: top shows upper km: bottom from 1km to the bottom. Lightly shaded fluid is salty. 3

4 Figure 3: Annual-mean cross-section of zonal average potential density anomaly for the world oceans (referenced to the surface in kgm )inthetopkm. 2 Inferences from geostrophic and hydrostatic balance Fig.4 shows the surface circulation of the ocean based on shift drift measurements. The maximum horizontal current speeds in the gyres are found at the surface in the western boundary currents, where they can reach 2 ms 1.Elsewhere, in the interior of ocean gyres, the currents are substantially weaker, typically 5 10cm s 1. Since the N-S extent of gyres is typically about 10 o latitude 1000km (the E-W scale is greater) we can estimate a Rossby number for large-scale gyres thus: R o U fl 0.1ms 1 (10 4 s 1 )( m) The Rossby number for large-scale mean motions in the ocean, then, is very small. Thus the geostrophic and thermal wind approximation is generally excellent for the interior of the ocean. This is exploited to infer ocean currents from hydrographic observations of T and S. Because ρ varies by only a few % (3%) about constant reference value, 4

5 Figure 4: Major surface currents of the world oceans. 5

6 ρ ref, we can make significant simplifications. equations are: The horizontal momentum Du Dt + 1 ρ ref p x fv = F x If R 0 << 1 then: Dv Dt + 1 ρ ref p y + fu = F y p z = g ρ ref + σ u g = 1 p fρ ref y ; v g = 1 p fρ ref x Take vertical derivative and using hydrostatic balance we obtain: u g z = g bz σ fρ ref and σ depends on both T and S. Let s look at the σ section, Fig.3. Application of thermal wind yields currents that are broadly consistent with observed current structure. Historically much of observational oceanography has been built around the thermal wind equation. Observe σ (T,S) andthenintegrateupfor(u, v) knownasthedynamic method. But there is a level of no motion problem. 2.1 Ocean surface structure and geostrophic flow Near-surface geostrophic flow Consider Fig.5. If we consider some horizontal surface of constant z, thenwe can integrate the hydrostatic relation up to the free surface (where p = p s, atmospheric pressure) to give: p(z) =p s + Z η z gρ dz = p s + g hρi (η z), (1) 6

7 Figure 5: The height of the free surface of the ocean is z = η(x, y). The depths of two reference horizontal surfaces, one near the surface and one at depth is given by z = z o and z = z 1 respectively. R η z where hρi = 1 gρ dz is the mean density in the column of depth η z. (η z) If we are interested in the near-surface region (z = z 0,say,inFig.5),fractional variations in column depth are much greater than those of density, so we can neglect the latter, setting ρ = ρ ref in Eq.(1) and leaving p(z 0 )=p s + gρ ref (η z 0 ). Horizontal variations in pressure in the near-surface region will thus depend on variations in atmospheric pressure and in free-surface height. Since here we are interested in the long time scale of the ocean circulation, we can neglect day-to-day variations of atmospheric pressure to conclude that the horizontal components of the near-surface pressure gradient are given by gradients in surface elevation bz p = gρ ref bz η. Thus, the geostrophic flow just beneath the surface is, 1 u gsurface = bz p fρ ref = g bz η. (2) f Note how Eq.(2) exactly parallels the equivalent relationship for geostrophic flow on an atmospheric pressure surface. 7

8 Because oceanic flow is weaker than atmospheric flow, we expect to see much gentler tilts of pressure surfaces in the ocean. We can estimate the size of expected η variations by making use of Eq.(2) along with observations of surface currents: if U is the eastward speed of the surface current, then η must drop by an amount η in a distance L given by: η = flu g or 1m in 1000km if U =10 1 ms 1 and f =10 4 s 1. Can we see evidence of this in the observations? Observations of surface elevation If we could observe the η field of the ocean then, just as in the use of geopotential height maps in synoptic meteorology, we could deduce the surface geostrophic flow in the ocean. Amazingly variations in ocean topography, η, even though only a few cms to a metre in magnitude, can indeed be measured from satellite altimeters and are mapped routinely over the globe every week or so. Orbiting at a height of 1000km above the Earth s surface, altimeters measure their height above the sea surface to a precision of a cm or two. And, tracked by lasers, their distance from the center of the earth can also be determined to high accuracy, permitting h to be found by subtraction. The annually-averaged surface elevation (relative to the mean geoid ) is shown in Fig Geostrophic flow at depth At depths much greater than variations of η (at z = z 1,say,inFig.5),wecan no longer neglect variations of density in Eq.(1) in comparison with those of column depth. Again neglecting atmospheric pressure variations, horizontal pressure variations at depth are therefore given by, using Eq.(1), bz p = g hρi bz η + g(η z) bz hρi, 8

9 Figure 6: The mean height of the sea surface relative to the geoid (in cms) as measuredbythetopex-poseidonsatellitealtimeter. Thepressuregradientforce associated with the tilted free surface is balanced by Coriolis forces acting on the geostrophic flow of the ocean at the surface. Note that the equatorial current systems evident in Fig.4 are only hinted at in the sea surface height. Near the equator, where f is small, geostrophic balance no longer holds. 9

10 Figure 7: and therefore the deep water geostrophic flow is given by u = = 1 bz p fρ ref g [hρi bz η +(η z) bz hρi] fρ ref ' g g(η z) bz η + bz hρi, (3) f fρ ref since we can approximate hρi 'ρ ref in the firsttermbecausewearenot taking its gradient. This has two contributions: that associated with freesurface height variations, and that associated with ocean density gradients. Note that if the ocean density is uniform, the second term vanishes and the deep-water geostrophic flowisthesameasthatatthesurface: geostrophic flow in an ocean of uniform density is independent of depth. Thisisofcourse a manifestation of the Taylor-Proudman theorem. The second term in Eq.(3) is the thermal wind term, telling us that u will vary with depth if there are horizontal gradients of density. Thus, there may be a nonzero flow at depth even if the surface is flat. Conversely, the presence of horizontal variations in surface height, manifested in surface geostrophic currents as in Eq.(2), does not guarantee a geostrophic flow at depth. Indeed, typically the flow becomes much weaker at depth the Taylor columns do not extend into the deep ocean. This must mean that the density field and height field mutually adjust in such a way that the second term in Eq.(3) cancels the first. 10