GFD – 2 Spring 2007 P.B. Rhines (Lecture 3 03) 12 April ...But this large transport is a rather...

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1 GFD – 2 Spring 2007 P.B. Rhines (Lecture 3 03) 12 April 2007 3.1 θ-S diagrams, water masses and baroclinic circulation, barotropic circulation. So far we have looked at some thermodynamics, a bit of the equation of state for seawater, and a stratified geostrophic adjustment problem which gives us a flow in geostrophic balance. The thermal-wind equation relates the lateral variations in density to vertical shear of the horizontal velocity: hence water masses to general circulation. We spent the effort to follow this path because it expresses the enigma of ocean- and atmosphere circulations: they are part of a great heat engine whose energy almost entirely originates in radiant sunshine. And yet, the mechanical (kinetic-) energy that might be created with some considerable efficiency cannot be: Coriolis forces put a limit on it, and the thin geometry of the ocean and atmosphere inhibits the creation of general circulation by buoyancy forcing. It is as if ocean water, at the time- and length-scales of the general circulation, were not water at all, but some more elastic substance that resists many kinds of easy circulation patterns. We will try to explore this idea dynamically while at the same time looking at the observations to see where water masses are produced, where they go, and what happens to them on the way (this was the answer given by a Ph.D. student some years ago in an oral exam, when asked the question, ‘what is physical oceanography about’). We have argued that buoyancy is related to both temperature and salinity, and hence to heat- and fresh-water dynamics. So we see that both atmosphere and ocean have intrinsically interactive heat- and fresh-water fields. In a tour of the world-ocean, the θ-S diagram migrates rather smoothly: it is remarkably simple really. Surface waters tend to be rather fresh at high latitude, where the water is cool enough that evaporation is weak, sunshine is weak, skies are cloudy, and there are ample sources of fresh-water (precipitation, run-off from continents and melting sea-ice). The strongest source of high salinity is seen in the subtropics where, as described above, summer sun is strong and skies are clear. Most often, temperature wins out in determining the vertical structure (temperature decreases downward) yet there are many small counterexamples, regions where salinity wins. High salinity waters are thus produced by solar heating-produced evaporation, yet shouldn’t the sunshine at the same time warm the water, preventing it from becoming dense? The answer is in the feedback structure of air/sea interaction. The surface temperature of the ocean is strongly damped by the surface air temperature, yet the surface salinity has only a more subtle feedback; it is much less constrained by the atmosphere. While observations suggest that the heating and cooling of the top of the ocean by the atmosphere and sunshine are a stronger flux of

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GFD – 2 Spring 2007 P.B. Rhines (Lecture 3 03) 12 April 2007 3.1 θ-S diagrams, water masses and baroclinic circulation, barotropic circulation. So far we have looked at some thermodynamics, a bit of the equation of state for seawater, and a stratified geostrophic adjustment problem which gives us a flow in geostrophic balance. The thermal-wind equation relates the lateral variations in density to vertical shear of the horizontal velocity: hence water masses to general circulation. We spent the effort to follow this path because it expresses the enigma of ocean- and atmosphere circulations: they are part of a great heat engine whose energy almost entirely originates in radiant sunshine. And yet, the mechanical (kinetic-) energy that might be created with some considerable efficiency cannot be: Coriolis forces put a limit on it, and the thin geometry of the ocean and atmosphere inhibits the creation of general circulation by buoyancy forcing. It is as if ocean water, at the time- and length-scales of the general circulation, were not water at all, but some more elastic substance that resists many kinds of easy circulation patterns. We will try to explore this idea dynamically while at the same time looking at the observations to see where water masses are produced, where they go, and what happens to them on the way (this was the answer given by a Ph.D. student some years ago in an oral exam, when asked the question, ‘what is physical oceanography about’). We have argued that buoyancy is related to both temperature and salinity, and hence to heat- and fresh-water dynamics. So we see that both atmosphere and ocean have intrinsically interactive heat- and fresh-water fields. In a tour of the world-ocean, the θ-S diagram migrates rather smoothly: it is remarkably simple really. Surface waters tend to be rather fresh at high latitude, where the water is cool enough that evaporation is weak, sunshine is weak, skies are cloudy, and there are ample sources of fresh-water (precipitation, run-off from continents and melting sea-ice). The strongest source of high salinity is seen in the subtropics where, as described above, summer sun is strong and skies are clear. Most often, temperature wins out in determining the vertical structure (temperature decreases downward) yet there are many small counterexamples, regions where salinity wins. High salinity waters are thus produced by solar heating-produced evaporation, yet shouldn’t the sunshine at the same time warm the water, preventing it from becoming dense? The answer is in the feedback structure of air/sea interaction. The surface temperature of the ocean is strongly damped by the surface air temperature, yet the surface salinity has only a more subtle feedback; it is much less constrained by the atmosphere. While observations suggest that the heating and cooling of the top of the ocean by the atmosphere and sunshine are a stronger flux of

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buoyancy than the evaporation and precipitation which alter the saline buoyancy, nevertheless we see θ and S having comparably large effects on the density ρ. This apparent paradox shows how subtle the ocean circulation is, and the answer There are some important, special geographic regions: the Mediterranean Sea has great evaporation and produces some of the densest water in the world-ocean. This salty water mass flows out into the Atlantic through the Straits of Gibralter with salinity approaching 39 psu. It plummets down the slope so rapidly that it mixes with surrounding waters and settles out at about 1200 m depth. Another salt-conduit is the Gulf Stream and, to a lesser extent the other western boundary currents of the oceans. The Gulf Stream moves saline, warm waters northward transporting greater than 100 Sverdrups just northeast of Cape Hatteras. But this large transport is a rather local event, and the remnant Gulf Stream that reaches the subpolar North Atlantic (north of 500N) has transport that is smaller, less than 30 Sverdrups. The bottom-topographic ridge running from Greenland to Scotland between 600N and 700N marks (Fig. 3.1) the transition from the Atlantic to the Nordic Seas (Greenland, Norwegian, Icelandic Seas) and the warm, saline water transport there is roughly 6 Sverdrups.

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Figure 3.1. The North Atlantic ocean basin, showing contours of constant f/h (explained later in this lecture). The Erika Dan section line is shown near the tip of Greenland. The depth generally shoals (decreases) northward, with dominant ridges: Mid-Atlantic and Iceland-Scotland. This long transition from tropics to Arctic passes through many dynamical regimes, but it is still useful to think of it as one great meridional overturning circulation (MOC) whose mission is complementary to that of the atmospheric generalized Hadley cell: the transport of heat, fresh-water and bio-geochemical tracers to satisfy the climate system and ‘feed’ the ecosystems of Earth. While we cannot present a reliable MOC plot purely from observations, Fig.3.2 shows a believable plot from Dr. Sirpa Häkkinen’s numerical model of the Atlantic and Arctic.

Fig. 3.2 Meridional overturning circulation streamfunction (MOC), integrated east and west along z=constant lines, from Häkkinen numerical ocean/ice model. The model includes Arctic and Atlantic to 30S. The volume transport reaches 24 Sverdrups but the transport of the NADW is more like 15 Sverdrups. Sinking in this model is dominated by the Labrador Sea, with the deep overflows from farther north failing to sink completely to the bottom. On the world-tour of θ-S you will see regions where properties are relatively constant for a thick mass of water…a water-mass. This suggests a common origin, and our lab experiment is the cue. Strong forcing of buoyancy change at the sea-surface, perhaps due to cold wintertime winds at high latitude, or strong sunshine and clear skies at low latitude, can produce intense down-flows which color the θ-S diagram. This may involve sinking of surface waters immediately, or it may first appear as deep convection

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where the surface waters become denser than deeper waters, producing a deep overturning in a small geographical region like the Labrador Sea. We can speak of sinking circulation in physical space (downward along the z-axis) as in Fig. 3.2 or as a complement to that, water-mass conversion, or ‘sinking’ in density space (downward along the σθ -axis) as in Fig. 3.3. Both are important, and both relate to different aspects of the conversion of potential energy of the mass field to kinetic energy of the circulation. We make plots of the oceanic overturning circulation by averaging the north-south (v-) and up-down (w-) velocities east-west between lateral boundaries. This ‘overturning streamfunction’ Ψ(y, z) is the most common plot in publications based on ocean-climate models, where (v,w) = (-Ψz, Ψy) . We also can produce as in Fig. 3.3, an overturning diagram Ψ(y, σθ) using the idea of movement of water across density surfaces due to convection and small-scale mixing. Thus we have in the ocean circulation both sinking to greater z-depth and transformation from less dense to more dense, which can be called sinking in density space. This latter transformation of density is the true maker of water masses.

Fig. 3.3. MOC in density/latitude space; the same numerical simulation as in Fig. 1. This is from averaging the north-south and vertical velocity east-west along isopycnal surfaces rather and level z=const. surfaces. Note that there is considerable flow along constant density surfaces, and that the sinking regions are at different latitude than in Fig. 1. With isopycnals sloping from east to west, the two circulation patterns may have different amplitude. In this diagram the upper 100m of the ocean is quite exaggerated, with shallow overturning cells near the Equator. These ideas come to life when looking at a section down the middle of the Atlantic, like KNSV25W4.DAT. Plotting salinity as a color field, and following the θ-S diagram

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as we progress south from Iceland to Antarctica we see interleaved water-masses: layered as in a simple lab experiment with waters of slight different density competing for space. Water masses typically have fairly uniform properties characteristic of their formation regions. Ray Montgomery, working with Carl Gustav-Rossby, and later Joe Reid at Scripps, developed the idea that the tilted stable layers, that is the surfaces of constant potential density, σθ, are gravitationally free pathways on which water masses can move without paying a buoyancy penalty. Water properties (θ, S, trace chemical concentration) mapped on surfaces of constant σθ become very revealing, for water wants to move along these surfaces. Thus our picture of the stably layered ocean now becomes much more interesting: the layers are not horizontal, but tilt up to touch the sea surface at high latitude. And as the thermal wind equation shows, these tilts imply horizontal circulation. It is a fascinating puzzle: the water masses are thermodynamically created at the sea-surface, and somehow slide downward along sloping isopycnal surfaces traveling great horizontal distances while doing so… in fact traveling throughout the world ocean before surfacing once again. And yet it is the slope of these isopycnal surfaces that is necessary to provide geostrophic circulation. The zonally averaged sections of either circulation (MOC) or water-mass properties like salinity lull us into thinking that the ocean is simply rolling over like the laboratory experiment of interleaving water layers of different density in a thin (east and west), long (north and south) plexiglass tank. In fact the circulation is fully 3-dimensional. It snakes in and out of these meridional (north-south) sections in the form of wind-driven horizontal gyres, boundary currents and other ‘jets’. Does this detail matter? We think that it does, yet a broad-brush understanding of the oceanic meridional circulation may not require it. The 3D structure is visible in individual slices of ocean running east and west (Fig. 3.4). Thermal wind balance connects what we see in these hydrographic sections with the currents themselves. Consider one important artery of the deep branch of the MOC, located on fig.3.4 near the western boundary of the subtropical North Atlantic.

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Fig. 3.4 Erika Dan section showing temperature (0C) along 590 N latitude. Labrador is at the left, Cape Farewell, the southern tip of Greenland at center, the Reykjanes (mid-ocean) Ridge east of that, and then Rockall Plateau and the Scottish continental rise. This is one of the most active regions of the global ocean-climate system, where the dominant deep waters of the world ocean are produced. Warm water from the subtropics is seen pressing against Europe, at the east, as it flows northward as part of the cyclonic subpolar gyre. Some 6 Sverdrups of this water makes its way across the ridge between Iceland and Scotland, into the Nordic Seas and Arctic…to return as the deep blue bottom waters seen leaning westward on the lower slopes. The deep, cold, rapidly flow waters feed the Deep Western Boundary Current (DWBC) that extends the entire length of the Atlantic to the Southern Ocean. The Labrador Sea at the left, is an in-and-out passage for these deep waters, with a top-to-bottom transport of about 40 Sverdrups (including the upper-ocean waters) circulating as boundary currents. Some of this transport recirculates round the subpolar gyre, while some eventually heads south along the N.American continental slope. North Atlantic Deep Water, the dominant deep water mass of the world ocean, lies between 1000m depth and the bottom. It appears in zonally averaged MOC diagrams with a transport of about 20 Sverdrups. Important, though small in area, are the waters on the shallow continental shelves: these are very low in salinity and rather cold. Their transport is important to the MOC, for one thing in sealing off the top of the open ocean with buoyant surface waters, inhibiting deep convection. Let us look at the actual velocity field in one branch of this circulation. Measuring velocity with current meters is expensive and difficult, but we set out an array of moorings in the deep western boundary current near 300N and looked at the flow for 12 months (Fig. 3.5)

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Fig. 3.5 Potential temperature section east-west across the Blake-Bahama Outer Ridge (see location in Fig. 3.1). As in the Erika Dan section the thermal wind flow is southward (out of the page to the east of the ridge and northward its west. It increases downward toward the sea-floor.

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Fig. 3.6 12-month mean currents (8/1977-7/1978) from four current meter moorings in the deep western boundary current. Depth of the instruments labeled in km. The deep jet follows the bottom topographic contours. The weaker flow at the upper levels veers in the sense of a stratified Taylor column. The 0.6 km-deep flow is consistent with an inflow joining the Gulf Stream to its west.

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Fig. 3.7 Time series over 12 months of horizontal velocity at 2000, 3000 and 3800m depth. at the mooring on the eastern flank of the Ridge (Fig. 3.6). The Deep Western Boundary Current is faster near the bottom, in accord with thermal wind balance. Note also the change in direction with depth.

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Fig. 3.8. Tritium tracer concentration in the section shown in Fig. 3.6. The maximum hugging the ridge shows the origins of this water at the ocean surface far to the north. This is Denmark Strait Overflow Water encountered at 59N in Fig. 3.5 (Jenkins and Rhines, Nature 1980). Tritium was injected into the atmosphere in the nuclear bomb tests of the 1950s/60s, and it entered the surface ocean both as rain and through water vapor, and run-off from ice-locked land surfaces. This demonstrated directly the ‘breathing’ of the ocean and the appearance of this signal some 3000 km to the south. There is much more analysis ahead regarding these circulations, but now we want to step back and look at the other side of the coin: barotropic circulation.

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3.2 Dynamics of barotropic, geostrophic circulation Consider an unstratified (constant-density) fluid, rotating about the z-axis. The thermal-wind equations

( / ) , ( / )z x z yfv g fu gρ ρ ρ ρ= − = become simply (1) 0, 0.z zu v= = This is known as the Taylor-Proudman approximation or ‘theorem’. Combine this with the mass-conservation/incompressibiliy equation (2) 0x y zu v w+ + = and you find (3) 0zzw = The horizontal velocity components are independent of z, and the vertical velocity varies at most linearly in z. The physical origin of this strangely constrained flow is in the 'stiffness' endowed to the fluid by rapid rotation of the Earth, which has a peculiarly strong sense along the axis of rotation. GI Taylor’s laboratory experiments nearly a century ago showed how homogeneous fluid tends to move in vertical columns, as suggested by (1). Dye forms ‘curtains’, and viewing the dye from above shows fine twists and whirls that are vertically coherent. Our words 'rapid rotation' means more rapid than the fluid circulation. The time scale of the flow is L/U, and that of the planet is 1/f. The ratio of these two times is the Rossby number, Ro = U/fL, which must be small if the flow is to be geostrophic. In some cases we have a wave-related time-scale, T, rather than an advection-related time scale. T might be the period of a wave (divided by 2π). We thus require also that 1/fT be small for the flow to be geostrophic. For horizontal flow speeds of 10 cm sec-1, we thus require that the horizontal scale, L be greater than 10 km and the time scale T be greater than about 4 hours for the flow to be geostrophic. Can Earth's rotation affect the circulation of a small pond in your back-yard? Yes, certainly: they may not be geostrophic currents but if Ro and 1/fT are not very large (say 5 or so) rotation will still be felt. This would be the case for example if U ~ 5 cm sec-1, L ~ 100m, T ~ 1 hour. For this choice of L we probably mean that the pond is about 300m wide (L being the length scale of the flow, or the wavelength over 2π). A time-lapse video of such a pond might be revealing. Flow along geostrophic contours. Consider now the kinds of flow that can occur in a basin of rotating fluid, with vertical rotation axis, under constraints (1) and (2). If the depth of the fluid, h(x,y) varies, with mountains and valleys, the 'stiffness' property will keep fluid moving in columns along ‘altitude contours’, h=const. We have to add more

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information to see whether these columns can stretch vertically (as they might under constraint (2)). If their stiffness prevents them from stretching, the flow must lie parallel with contours of constant depth, which are known as geostrophic contours.

A streamfunction for the horizontal velocity can be written down. Supposing the upper surface to be rigid and horizontal, conservation of mass (2) can be integrated vertically to give (using (1))

0[ ]z

x y z huh vh w ==−+ =

and the boundary condition at the bottom is 0| | 0z h zw u h w=− == − •∇ = . Together these give (4) 0hu∇• = based on the horizontal velocity vector. This non-divergence property allows us to define a streamfunction hu z ψ= ×∇ which automatically satisfies the conservation of mass (4). Now the horizontal momentum equations show that with geostrophic flow the pressure acts as a streamfunction for the horizontal velocity, fu z pρ = ×∇ As we suggest above, circulations that are nearly in geostrophic balance will thus have both ψ=ψ(h), p=p(h); that is flow along contours of constant depth. Tilted rotation vector. Geostrophic contours are the same as depth contours when the rotation axis is vertical. However the spherical shape of the Earth comes into play: the rotation vector is no longer aligned with the local vertical direction. The above remarks described the horizontal velocity, but in this case of a constant-density fluid, horizontal really means the velocity normal to the rotation vectory, Ω , and the depth of the ocean refers to the depth measured not vertically but parallel with Ω . Call this new depth d, while the vertical depth is still h. A sketch shows that for a thin spherical shell d is the larger, and they are related by d ≅ h/sin ϕ or, noting that the Coriolis frequency, f has the same latitude variation, f = 2Ω sin ϕ, we have d = 2Ω h/f where ϕ is the latitude. Now we have the possibility of currents freely flowing along curves d = constant, or f/h = constant in which case we have streamfunction and pressure that are functions of f/h (ψ=ψ(f/h); p=p(f/h)).

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When we first mapped these geostrophic contours (Fig. 3.1), they turned out to be an interesting combination of latitude circles and bottom topographic contours. Over small horizontal distances and at high latitude topography, h, tends to dominate but over longer distances or in the tropics, the latitude-variation of f dominates. The problem is that the geostrophic contours bump into continents, so that ocean currents running along them have a serious difficulty there. Actually all such f/h contours head toward the Equator as they run up into shallow water (Fig. 3.1; as h => 0 f => 0 also, hence ϕ => 0). We will soon see how the this rudimentary theory of ocean circulation builds into a relatively complete picture with the addition of more ideas about vorticity dynamics. Wind-driven circulation, I. Notice what happens if we add some forcing to the problem. Horizontal variations in this flux cause vertical Ekman pumping which presses down (or Ekman suction which pulls up) the water below the Ekman layer. The vertical velocity at the base of the Ekman layer (typically about 10m below the surface), we is 1

0 ( / )ew z fρ τ−= •∇× τ being the known field of wind-stress, and z a vertical unit vector. Where the pumping is downward (negative curl of the wind-stress), fluid columns are pushed downhill, and conversely uphill for Ekman ‘suction’ (positive wind-stress curl). Down- or uphill here refers to planetary hills and valleys of f/h. Suppose for simplicity that h=constant, so the only ‘topography’ is that of the spherical shape of the ocean. This produces a simple version of one of the dominant equations of ocean circulation: the Sverdrup relation; some nasty trigonometry yields ( )ew u depth measured parallel to= •∇ Ω or

(5) // ( / )

evh f wvh f z f

ββ τ ρ

== •∇×

With topographic variations this generalizes to

(6) 2

ˆ( ) ( / )h fu z ff h

τ ρ•∇ = •∇× .

Once again, the fluid velocity can be found by integrating the wind forcing term along f/h curves. ¤ The sense of the Sverdrup relation is that in the subtropical Atlantic (roughly 15N to 50N), with westerly winds to the north and easterly ‘trade-‘ winds to the south, the wind-curl is negative. This leads to Ekman pumping downward, and southward geostrophic currents. The equation predicts only the meridional velocity, but the zonal velocity component can be found from this by mass conservation (plus some dynamics we will

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encounter later, regarding boundary conditions on the continental coasts). Once we devise a method of closing the circulation (bringing the southward moving fluid back north again), we will find a great anticyclonic wind-driven gyre, with high central pressure and sea-surface height, in the subtropics. In the subpolar gyre (50N-65N), the wind-curl tends to be positive, producing Ekman suction (upward we). This drives geostrophic interior flow northward. Again if we find that this flow returns southward at the western boundary rather than the eastern boundary, the subpolar gyre will be cyclonic, with low pressure and depressed sea-surface. ¤ It is important that the Ekman flux ˆ ( / )z fτ ρ− × varies with both wind-stress and Coriolis frequency. Even a uniform westerly windstress blowing on the Atlantic would cause Ekman suction. Indeed, the Ekman transport observed at 10N is very large, of order 12 Sverdrups (after integrating east-west across the ocean), whereas the stronger winds far to the north achieve a much smaller Ekman transport, of order 4 Sverdrups. We could rewrite Eqn. 5 to include both the Ekman transport and interior geostrophic transport together, which reads

0

1z

z h

v dz zβ ρ τ=

=−

= •∇×∫

This is the classical form of the Sverdrup relation.