## SIO 210 Talley Topic 3: Dynamical quantities, transports and forcing

Lynne Talley, 2000
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### 1. Dynamical quantities for the general circulation

1.1 Velocity. Horizontal velocity is measured directly using various current-measuring devices, or calculated from the horizontal pressure gradient. In the latter case, for the large-scale circulation, geostrophy is assumed. Vertical velocity for the large-scale circulation cannot be measured as it is too small, only inferred from the horizontal velocities.

Direct current measurements - require time and space averaging and/or filtering to remove unwanted signals such as those from tides and internal waves. The remaining signal still contains all of the time scales of the large-scale circulation. It is more or less the case that long time scales go with large horizontal spatial scales. There is no hard and fast rule about how long the time average must be that corresponds with a velocity estimate associated with the pressure gradient. (current meter results, float trajectories and averages - see aquarium lecture .)

Pressure gradients can be used to calculate geostrophic flows. Pressure can be measured in several ways. Altimetry provides the sea surface height to within several centimeters, which is not adequate for the "mean" circulation, but is adequate for determining the variability of the surface currents about an unknown mean. Pressure gauges on the ocean bottom can be calibrated with direct current measurements (say from current meters), and then used in pairs to calculate geostrophic flow. As with direct current measurements, time series signals must be averaged or filtered.

Finally, pressure gradients can be determined from dynamic height (density field), again to within an unknown pressure at each position (next subsection). The unknown can be thought of as due to the unknown exact sea surface height relative to the geoid (and which altimetry is mainly not quite adequate to measure).

Velocities associated with the general circulation are then mostly calculated from the pressure gradients, with a number of techniques used to estimate the absolute pressure at each point and hence the absolute velocity.

1.2 Dynamic height. To calculate pressure at a point relative to a deeper or shallower point at the same horizontal location, from the measured density field.

Use hydrostatic balance, which is the force balance between the vertical pressure gradient and gravity*density.

The practical details of the calculation are beyond the scope of these notes and class, but an example is worked through in class. (Figure in class: dynamic height at the sea surface relative to a deeper pressure for the Pacific, from Wyrtki , and adjusted steric height for the Pacific, from Reid .)

1.3 Transports. We often wish to express how much total water or heat or salt or carbon or "stuff" is moving around. Transports are the total flux of whatever property through a given area. The general expression for transports is the (integral of the velocity times the property computed over the area). The area we use for most transports associated with general circulation is a vertical plane (e.g. for horizontal velocities). We can also calculate a vertical transport through a horizontal or isopycnal surface. However, since vertical velocities associated with the general circulation are tiny, vertical transports based on observations (as opposed to model results) are calculated as residuals.

Volume transport: integral of velocity over the area. Units are m^3/sec. A typical transport for a major current is on the order of 10 to 100 x 10^6 m^3/sec. We use the unit "Sverdrup" where 1 Sv = 1 x 10^6 m^3/sec.

Mass transport: integral of density*velocity over the area. Units are kg/sec. A typical transport for a major current is order 10 to 100 x 10^9 kg/sec (since density is roughly 1000 kg/m^3).

Heat transport: integral of (heat*velocity) over the area. Units are Watts (e.g. Joules/sec). Heat must be calculated using Kelvin degrees, and potential temperature rather than temperature. Heat transport through a closed area, through which there is no net mass transport, is basically proportional to the temperature difference between water entering and water leaving across the closed boundaries, so degrees Celsius are fine for this practical use. (This can easily be proven by writing the temperature T_k in degrees K in terms of temperature T_c in degrees C and the constant offset between them of 273.15. Then when the heat transport integral is calculated, the portion with the constant 273.15 drops out if the total mass transport through the area is 0.) Net heat transports across various latitude lines which close at boundaries range from -1.0 to 1.0 x 10^15 Watts. We often use the unit 1 Petawatt = 1 x 10^15 Watts.

Freshwater transport: Salt in the ocean is not gained or lost over the time scales of interest to us. However freshwater is exchanged with the atmosphere and land. A kilogram of seawater consists of a portion of water and a portion of salt. The relative proportions are expressed in the salinity, which is taken for these transport calculations to be (gm salt/kg seawater) or (gm salt/(kg salt + kg freshwater)). Using the salinity measured along a section, we can calculate the freshwater transport across the section. Typical values are order 1 Sv.

Other substance transports (carbon, oxygen, etc etc): If the property is given as a concentration ( gm property/gm seawater), then the transport is integral(concentration*density*velocity) over the area, which yields gm/sec of the substance.

(Figure in class: mass transports in the North Atlantic from Schmitz and McCartney; heat transports for the globe from various sources.)

### 2. Forcing of general circulation

How do forces work on the ocean's general circulation? Most of them create pressure gradients which cause water to try to flow from high pressure to low pressure. The earth's rotation introduces a Coriolis acceleration which induces flow in the frame of reference of the earth which is to the right of the motion in the northern hemisphere and to the left in the southern hemisphere, if the flow persists for about a day or longer such that it is affected by the rotation. Details for each were given in Hendershott's lectures.

• What equations do physical oceanographers use to describe the ocean?
ma=F (vector form; F = pressure gradient, body forces and dissipation and a includes time change of velocity, advection and Coriolis acceleration)
mass conservation (whatever amount goes into a box must come out)
equation of state (dependence of density on temperature, salinity and pressure)
density changes as a function of heating/cooling and evaporation/precipitation, and pressure

2.1. Wind forcing. As described in full in Hendershott's lectures, the wind acts directly and frictionally on the surface 50 to 100 meters of the ocean, in the "Ekman layer". The net transport in the Ekman layer is to the right of the wind in the northern hemisphere and to the left in the southern hemisphere. This effect has been demonstrated by Ralph and Niiler (1999) using surface drifter data from the the Pacific (drogues at 15 m).

Figure. Annual average surface wind stress (blue arrows in stress/unit area) and the average flow at 15 m depth (red arrows in cm/sec) which arises directly in response to the winds. The latter are the average velocity based on thousands of satellite-tracked surface drifters after the average flow resulting from the ocean's pressure field (geostrophic flow) is subtracted out. Hence the red arrows are the ageostrophic component of the velocity, and correspond to the Ekman response at 15 m. (Figure from Ralph and Niiler, 1999.)

Westerlies and trades (see example or Hellerman and Rosenstein  figures in lecture handout or other wind products)

Example of global wind field FNOC monthly winds, December, 1990. Taken from the plot package "ferret"'s examples.

For a more complete and general description of how the wind drives the general circulation, see section 5 in http://sam.ucsd.edu/talley_papers/talley_tropical_pacific.html

Talley, L. D., G. Fryer, and R. Lumpkin, 1997. Physical oceanography of the tropical Pacific. In Geography of the Pacific Islands, ed. M. Rapaport. Bess Press, Honolulu, HI. In press.

The Ekman layer transports drive circulation in the ocean interior through the Sverdrup transport mechanism, also described in Hendershott's lectures. That is, where there is convergence in the Ekman transport, there is downward Ekman pumping which deforms the water columns by "squashing" them. Since the water columns are on a rotating earth, they must preserve their angular momentum (referred to as "potential vorticity" for fluids, beyond the scope of the class). This can be accomplished by either spinning more slowly in the location of the water column, or moving to another latitude where the local rotation rate (parallel to the local vertical) is slower - this would be equatorward. If on the other hand there is Ekman divergence, then the water columns are "stretched", and the column must spin faster, either in place or by moving to higher latitude. Since the general circulation is very large scale and not filled with ever-increasing vortices, the response is to move in latitude rather than to spin up locally.

This response to Ekman pumping or suction is called the "Sverdrup response". and the flow associated with the response is called "Sverdrup transport". Sverdrup transport is equatorward in subtropical regions of Ekman pumping and poleward in subpolar regions of Ekman suction. The net flow added up across the ocean (along a parallel of latitude) must be returned in a narrow flow somewhere. For the reasons discussed in Hendershott's lectures, the return flow must ALWAYS be on the western side, regardless of northern or southern hemisphere. This results in a narrow strong western boundary current returning all of the interior transport back to the latitude where it started. The Gulf Stream, Kuroshio, Oyashio, Labrador Current, etc etc that you were asked to memorize for Lecture 1 are examples of such wind-driven circulation western boundary currents. Because of various other factors, including just inertia, the western boundary currents are generally stronger than is required to simply return the interior flow. 2.2. Thermohaline forcing. The gross aspects of the salinity and temperature distributions, and the driving force for the slow overturning circulation, are due to surface heating/cooling and surface evaporation/precipitation/ continental runoff. Maps of heat flux and evaporation - precipitation are presented. Components of the heat flux are described (incoming shortwave solar radiation, outgoing longwave radiation, latent heat loss due to evaporation, sensible heat change due to conduction between air and water). Input of fresh water takes place through precipitation and runoff from land. I don't have a net buoyancy flux map for the lecture notes.

Thermohaline forcing takes several forms.
(1) cooling/evaporation in a region (cooling is often due to large evaporation), which creates dense water at the sea surface which then overturns in convective cells. These cells are very narrow (order 1 km), and are referred to sometimes as "chimneys", or a whole patch of convective cells might be referred to as a chimney. Flow in the convective cells is "ageostrophic" - that is, not geostrophic, since vertical velocities and vertical length scales are of the same order as in the horizontal.
(2) Brine rejection resulting from sea ice formation. This releases salt into the water column, which increases its density and can create overturn.
(3) Any downward flow due to increase of density must be compensated somewhere by upwelling and inflow into the overturning regions. The upwelling must be accompanied by vertical mixing as the water parcels migrate upward. This upwelling is an essential part of the thermohaline circulation and is what forces most of the flow which we call "thermohaline".
(4) Salt and temperature do not actually mix at exactly the same rates, especially at the molecular scale - temperature mixes much faster. This phenomenon is referred to as double diffusion. This difference in mixing rates can create layering or overturn - in the ocean, on vertical scales of several to 10's of meters. This is a form of thermohaline forcing.

Net surface heat flux (W/m^2) from Hsiung, 1985. Positive is heat into the ocean. Estimates of global oceanic meridional heat transport. J. Phys. Oceanogr., 15, 1405-1413.
I have superimposed on the map the directions and magnitudes of meridional heat transport at selected subtropical latitudes, based on direct oceanic measurements.

ECMWF map of surface heat flux (W/m^2) (positive means heat into ocean) from Bernard Barnier , using the European Center for Medium Range Weather Forecasting analyses for the years 1986-1988 to compute surface fluxes. Rollowing publication. Fields were made available by Barnier for this map which reproduces a similar map in the following reference: Barnier, B., L. Siefridt and P. Marchesiello, 1994. Thermal forcing for a global ocean circulation model using a three- year climatology of ECMWF analyses. J. Marine Systems, 6, 363-380.

NCEP map of surface heat flux (W/m^2).

Annual mean E-P in cm/year constructed from climatology data in DaSilva's online Atlas of Surface Marine Data, from: DaSilva, A. M., C. C. Young and S. Levitus, 1994. Atlas of surface marine data 1994.

NCEP map of annual mean evaporation minus precipitation (cm/yr).

2.3. Heat transport estimates. Heat transport was defined above in section 1.3. We calculate it for the ocean in three ways. Typically we calculate the transports across a given latitude for each ocean - it is easiest to understand the following calculations if you think about them this way first. Of course the direct and indirect #1 estimates described below can also be done for any closed box (surface area) of the ocean.
Direct.Calculate the integral given in section 1.3 for a vertical section (or closed box surrounded by vertical planes) using measurements of temperature (and salinity in order to calculate density, specific heat and potential temperature) and estimate of velocity, usually based on geostrophy and assumptions about the unknown reference velocities. The quantity is only useful if mass is balanced across the vertical section, which must therefore be a closed box. The "box" can be closed by being a section which goes from one coast to another (so could be a straight line across an ocean), or it could be a closed box within the ocean. (Why is heat transport only useful if mass is balanced?). In practice this can only be done well where geostrophic velocities are well constrained.

Indirect #1. Use estimates of the heat flux at the sea surface from the atmosphere to the ocean. These fluxes are calculated as a sum of the incoming solar radiation (based on earth orbit and cloudiness), loss of heat due to evaporation since evaporation takes work and hence removes energy from the ocean ("latent heat flux"), exchange of heat due to temperature difference between air and water right at the air-sea interface ("sensible heat flux"), and loss of heat due to back radiation from the ocean to the atmosphere (similar to black body radiation).

Using the net air-sea flux in Watts/(meter^2) over an area, calculate the net heat exchange in Watts by summing the flux over the area. Then the net heat transport by the ocean across the closed box surrounding that area is equal to the net heat exchange out of the surface of the box. (Assuming that geothermal flux from the ocean bottom is negligible, which it is.)

Common method for doing this: integrate air-sea heat flux across latitude bands to get the net ocean heat transport across those latitudes.

Indirect #2. Use calculations of the net heat exchange between space and the very top of the atmosphere, and direct estimates of net heat transport across latitude bands in the atmosphere (based on atmosphere winds and temperatures). Subtract atmosphere's heat transport from the net earth-space estimate to get the ocean heat transport across the latitude (all the way around the earth - not possible to separate oceans).

2.4. Thermohaline circulation.

The factors that affect buoyancy (density) at the sea surface (and to a much lesser extent the direct factors in the ocean interior such as geothermal heating) drive a circulation, called the thermohaline circulation. Our simple picture of this is that there are quite isolated sources of deeper waters - that is, that convection due to loss of buoyancy (gain of density) at the sea surface, are very localized. Examples are small regions in the Greenland Sea, Mediterranean Sea, Labrador Sea, Weddell Sea, Japan Sea, etc. and areas of dense shelf water formation due to sea ice formation. These newly-formed deeper waters sink and fill the oceans at depth. The result must be upwelling of these waters. We are still working as a community on exactly where the upwelling occurs. We assume it is very widespread, although there may be some local areas of more intense upwelling such as the equator. With widespread upwelling, the deep water columns are stretched. The deep water then responds in a fashion much like the Sverdrup response to Ekman pumping - the water columns "stretch" and must therefore move poleward. The net result is that the deep circulation should be more or less cyclonic (counterclockwise northern hemisphere, clockwise southern hemisphere, with low pressure in the center).

On a more global scale, there is a net transport of upper layer water into the isolated water mass formation sites and net transport of lower layer water away from the isolated sites. This is accomplished through a combination of the gentle deep gyres of the previous paragraphs, and western boundary currents that complete the mass balance.

The overall thermohaline circulation is about 1/10 the magnitude of the wind-driven circulation. That is, the net transports associated with it are on the order of 10-20 Sverdrups. However, the THC extends to the ocean bottom, whereas the wind-driven circulation by and large does not (except in the strong western boundary currents and Antarctic Circumpolar Current, where it does go to the bottom). Therefore almost all of the interior ocean circulation seen below about 2000 m is due to thermohaline forcing.