Forces and accelerations in a fluid: (a) acceleration, (b) advection, (c) pressure gradient force, (d) gravity, and (e) acceleration associated with viscosity υ.

(a) Centrifugal and centripetal forces and (b) Coriolis force.

Illustration of molecular processes that create viscosity. The mean flow velocity is indicated in gray (U). L' is the distance between molecules. U' is the speed of the molecules. Random molecule motions carry information about large-scale flow to other regions, thus creating (viscous) stresses. Viscous stress depends on the mean molecular speed |U' | and mean molecular free path |L' |.

(a) Observed diapycnal diffusivity (m²/s²) along 32°S in the Indian Ocean, which is representative of other ocean transects of diffusivity. (b) Average diapycnal diffusivity as a function of latitude range (color codes). Source: From Kunze et al. (2006).

(a) Observed diapycnal diffusivity (m²/s²) along 32°S in the Indian Ocean, which is representative of other ocean transects of diffusivity. (b) Average diapycnal diffusivity as a function of latitude range (color codes). Source: From Kunze et al. (2006).

Mixed layer development. (a, b) An initially stratified layer mixed by turbulence created by wind stress; (c, d, e) an initial mixed layer subjected to heat loss at the surface, which deepens the mixed layer; (f, g, h) an initial mixed layer subjected to heat gain and then to turbulent mixing presumably by the wind, resulting in a thinner mixed layer; (i, j) an initially stratified profile subjected to internal mixing, which creates a stepped profile. Notation: τ is wind stress and Q is heat (buoyancy).

(a) Bottom boundary layers and their advection away from the bottom Source: From Armi (1978). (b) Mixing of a plume of dense water as it flows out over a strait into less dense ambient waters. After Price and Baringer (1994).

Double diffusion: (a) salt fingering interface (cold, fresh water warms and rises; warm, salty water cools and sinks). (b) Diffusive interface. (c) North Atlantic Mediterranean eddy salinity profile with steps due to salt fingering (25° 23'N, 26°W). (d) Arctic temperature profile with diffusive layering. Source: From Kelley et al. (2003).

(a) Schematics of inertial currents in the Northern and Southern Hemispheres. (b) Hodograph of inertial currents at 45°N for a wind blowing in the y-direction; the numbers are in pendulum hours. Source: From Ekman (1905). (c) Observations of near-inertial currents. Surface drifter tracks during and after a storm. Source: From d'Asaro et al. (1995).

"Windrows" of foam, associated with the Langmuir circulation in Loch Ness. The surface wave field suggests the wind direction, which is parallel to the narrow bands of foam. Source: From Thorpe (2004).

Langmuir circulation, first described by Langmuir (1938). Source: From Smith (2001).

Ekman layer velocities (Northern Hemisphere). Water velocity as a function of depth (upper projection) and Ekman spiral (lower projection). The large open arrow shows the direction of the total Ekman transport, which is perpendicular to the wind.

Ekman transport convergence and divergence in the Northern Hemisphere due to variations in a zonal (eastward) wind. Ekman transport is southward, to the right of the wind. Divergent transport causes downwelling, denoted by circles with a cross. Convergent transport causes upwelling, denoted by circles with a dot.

Ekman transport divergence near the equator driven by easterly trade winds. (a) Ekman transports. (b) Meridional cross-section showing effect on the thermocline and surface temperature. (c) Coastal upwelling system due to an alongshore wind with offshore Ekman transport (Northern Hemisphere). The accompanying isopycnal deformations and equatorward eastern boundary current and poleward undercurrent are also shown (see Section 7.9).

Observations of an Ekman-like response in the California Current region. (a) Progressive vector diagrams (Section 6.5.2) at 8, 16, 24, 32, and 40 m depth. Because of the way the ADCP measures, the currents are shown relative to a deeper depth, rather than as absolute currents. The wind direction and speed for each day is shown by the small arrows on the 8 m progressive vector curve. (b) Observed mean velocities (left) and two theoretical Ekman spirals (offset) using different eddy diffusivities (274 and 1011 cm²/S). The numbers on the arrows are depths. The large arrow is the mean wind. Source: From Chereskin (1995).

Observations of an Ekman-like response in the California Current region. (a) Progressive vector diagrams (Section 6.5.2) at 8, 16, 24, 32, and 40 m depth. Because of the way the ADCP measures, the currents are shown relative to a deeper depth, rather than as absolute currents. The wind direction and speed for each day is shown by the small arrows on the 8 m progressive vector curve. (b) Observed mean velocities (left) and two theoretical Ekman spirals (offset) using different eddy diffusivities (274 and 1011 cm²/S). The numbers on the arrows are depths. The large arrow is the mean wind. Source: From Chereskin (1995).

Ekman response. Average wind vectors (red) and average ageostrophic current at 15 m depth (blue). The current is calculated from 7 years of surface drifters drogued at 15 m, with the geostrophic current based on average density data from Levitus, Boyer, and Antonov (1994a) removed. (No arrows were plotted within 5 degrees of the equator because the Coriolis force is small there.) This figure can also be found in the color insert. Source: From Ralph and Niiler (1999).

(a) Zonally integrated meridional Ekman fluxes (Sv) for the three oceans by latitude and month. (Positive is northward, negative is southward.) (b) Zonally integrated vertical Ekman volume flux (Sv) at the base of the Ekman layer per 10 degrees latitude belt by latitude and month. (Positive is up, negative is down.) Source: From Levitus (1988).

Geostrophic balance: horizontal forces and velocity. (a) Horizontal forces and velocity in geostrophic balance. PGF = pressure gradient force. CF = Coriolis force. (b) Side view showing elevated pressure (sea surface) in center, low pressure on sides, balance of PGF and CF, and direction of velocity v (into and out of page).

Example of a daily weather map for North America, showing high- and low-pressure regions. Winds are generally not from high to low, but rather clockwise around the highs and counterclockwise around the lows. Source: From NOAA National Weather Service (2005).

Geostrophic flow and thermal wind balance. (a) Schematic of change in pressure gradient force (PGF) with depth, assuming that the left column (A) is shorter and denser than the right column (B), that is, ρ_A > ρ_B and H_A < H_B. The horizontal geostrophic velocity V is into the page for this direction of PGF and is strongest at the top, weakening with depth, as indicated by the circle sizes. (If the densities of the two columns were the same, then the PGF and velocity V are the same at all depths.) (b) Same, but for density (red) increasing with depth, and isopycnals tilted, and assuming that the sea surface at B is higher than at A so that the PGF at the sea surface (h₁) is to the left. The PGF decreases with increasing depth, as indicated by the flattening of the isobars p₂ and p₃.

(a) Potential density section across the Gulf Stream (66°W in 1997). (b) Specific volume anomaly δ (X 10⁻⁸ m³/kg) at stations A and B. (c) Dynamic height (dyn m) profiles at stations A and B, assuming reference level at 3000 m. (d) Eastward geostrophic velocity (cm/sec), assuming zero velocity at 3000 m.

Mean annual dynamic topography of the Pacific Ocean sea surface relative to 1000 dbar in dyn cm (ΔD = 0/1000 dbar). Source: From Wyrtki (1975).

Dynamic topography of 100 dbar surface relative to 700 dbar surface (ΔD = 100/700 dbar) in dyn cm in the Atlantic Ocean. Source: From Stommel, Niiler, and Anati (1978).

The two-layer ocean. (a) Vertical density profile with upper and lower layers of density Ρ₁ and Ρ₂. (b) Sea surface and pycnocline for two stations, A and B, where the thickness of the layer above the "ideal level surface" is h_A and h_B and the thickness of the layer below the level surface is H_A and H_B, respectively. Both h and H are part of the "upper" layer shown in (a).

Two-layer ocean depiction of a (a) "cold," cyclonic ocean circulation showing the "ideal" sea surface and the subsurface thermocline structure and a (b) "warm" anticyclonic circulation.

Vorticity. (a) Positive and (b) negative vorticity. The right-hand rule shows the direction of the vorticity by the direction of the thumb (upward for positive, downward for negative).

Conservation of potential vorticity: changes in relative vorticity and Coriolis parameter f, if thickness is constant.

Conservation of potential vorticity: changes in thickness and relative vorticity, assuming constant latitude (constant f).

Conservation of potential vorticity: changes in thickness and latitude (Coriolis parameter f), assuming negligible relative vorticity (Northern Hemisphere).

Schematic of a long wavelength Rossby wave.

(a) Rossby deformation radius (km) for the first baroclinic mode. Source: From Chelton et al. (1998). (b) Shortest period (in days) for the first baroclinic mode, based on the deformation radius in (a). Note that the annual cycle, at 365 days, occurs around latitudes 40 to 45 degrees; poleward of this, all such waves are slower. Source: From Wunsch (2009).

(a) Rossby deformation radius (km) for the first baroclinic mode. Source: From Chelton et al. (1998). (b) Shortest period (in days) for the first baroclinic mode, based on the deformation radius in (a). Note that the annual cycle, at 365 days, occurs around latitudes 40 to 45 degrees; poleward of this, all such waves are slower. Source: From Wunsch (2009).

Dispersion relation for first mode baroclinic Rossby waves (Eq. 7.40), assuming a deformation radius R_I of 50 km, latitude 20 degrees (north or south) and y-wavenumber l=0. (a) Frequency ω versus x-wavenumber k and (b) period versus wavelength. The Rossby radius is shown with the dashed line. The highest frequency and shortest period are at the Rossby radius length scale.

Sverdrup balance circulation (Northern Hemisphere). Westerly and trade winds force Ekman transport creating Ekman pumping and suction and hence Sverdrup transport.

Stommel's wind-driven circulation solution for a subtropical gyre with trades and westerlies like the central latitudes of Figure S7.32: (a) surface height on a uniformly rotating Earth and (b) westward intensification with the β-effect. After Stommel (1965).

Munk's wind-driven circulation solution: zonal wind profiles on left and circulation streamlines in the center. After Munk (1950).

(a) Vorticity balance at a western boundary, with side wall friction (Munk's model). (b) Hypothetical eastern boundary vorticity balance, showing that only western boundaries can input the positive relative vorticity required for the flow to move northward.

Inertial circulation, in the absence of friction and wind, but in the presence of the β-effect. Source: From Fofonoff (1954).

(a) Subduction schematic (Northern Hemisphere). (b) Streamlines for idealized subduction on an isopycnal surface. The light gray regions are the western pool and eastern shadow zone, where streamlines do not connect to the sea surface. The heavy dashed contour is where the isopycnal meets the sea surface (surface outcrop); in the dark gray area there is no water of this density. After Williams (1991).

(a) Schematic of upper ocean equatorial circulation (large white arrows), surface temperature (red is warm, blue is cold), and thermocline depth and upwelling, driven by the Walker circulation ("convective loop"). Source: From NOAA PMEL (2009b). (b) "Bjerknes feedback" between the trade wind strength and zonal (east-west) difference in tropical surface temperature. (Arrows mean that increase in one parameter results in an increase in the second parameter.) In this positive feedback loop, increased trade winds cause a larger sea-surface temperature difference, which in turn increases the trade wind strength.

Processes in a deep convection region. After Marshall and Schott (1999).

The role of vertical (diapycnal) diffusion in the MOC, replacing Sandström's (1908) deep tropical warm source with diapycnal diffusion that reaches below the effect of high latitude cooling.

(a) Abyssal circulation model. After Stommel and Arons (1960a). (b) Laboratory experiment results looking down from the top on a tank rotating counterclockwise around the apex (So) with a bottom that slopes towards the apex. There is a point source of water at So. The dye release in subsequent photos shows the Deep Western Boundary Current, and flow in the interior Si beginning to fill in and move towards So. Source: From Stommel, Arons, & Faller (1958).

Global abyssal circulation model, assuming two deep water sources (filled circles near Greenland and Antarctica). Source: From Stommel (1958).

(a) Schematic of the Stommel (1961) two-box model of the meridional overturning circulation. The direction of the arrows assumes that the higher latitude box (blue) has higher density water. Each box is well mixed. (b) Schematic of the hysteresis in North Atlantic sea-surface temperature resulting from hysteresis in MOC strength. The starting point in freshwater is denoted by 1; starting at lower freshwater, hence higher salinity, in the left panel. Freshening is denoted by the blue arrow, with the same total amount in both panels. Salinification is denoted by red arrow, and should be exactly opposite to the freshwater arrow. In the left panel, starting at higher salinity, the freshening allows the system to remain on the top branch, and so subsequent evaporation returns the system to original state. In the right panel, with a fresher starting point, the same freshening causes transition to lower curve (2), and subsequent evaporation returns system to a different state, denoted by 3.