Introduction to physical oceanography - NOTES

Jan 14 - Introduction

• We went over the syllabus and the course requirements.
• There was a brief presentation on how technology helped the study of the ocean. Major points:
  • Navigation - the longitude problem.
  • Subsurace measurements of ocean properties - the development of the reversing thermometer and Nansen bottles.
  • Satellite measurements - the ocean as seen from space.
  • Numerical modeling - forecasting ocean state.

Jan 16 - Motivation and history

• Why bother with physical oceanography?
  • The oceans are a source of food.
  • The oceans are used by man.
  • The oceans influence weather and climate.
• Approaches for studying physical oceanography (a blend of all three aproaches is best).
  • Mathematical (analytical) studies
  • Observational programs
  • Numerical modeling
• Eras in physical oceanography (see text).
• Sampling error
  • Sampling error includes measurement error as well as the inherent variability of the system.
  • For example, annual variability must be accounted for when trying to find a decadal mean.
  • Bias occurs when the mean of the measurements does not equal the actual mean of the feature being measured. This could occur through a bias in the instrument itself, or in the measurement approach (i.e., only measureing temerature in the summer to get a decadal mean).
• Nonlinearity
  • Nonlinearity is the largest problem for analytical modeling
  • Example, the nonlinear transport equation: u_t + u u_x = 0.
• Fram links
  • http://www.sverdrup2000.org/fram.htm
  • http://www.nrsc.no/nansen/fritjof_nansen.html
  • http://www.ocean98.org/nans.htm

Jan 21 - Physical setting

• How do we measure distances in the ocean?
  • Degrees of latitude and longitude
  • Kilometers
  • Nautical miles
• There are three recognized oceans: Atlantic, Pacific and Indian.
• Dimensions of the ocean
  • If the ocean were as wide as a sheet of paper, it would also be as thick.
• Depths of the ocean
  • Most of the ocean is between 4 and 6 km deep.
• Measuring the depths of the ocean
  • Shipboard echo sounders
  • Satelite altimetery
• Map projections
  • http://mac.usgs.gov/mac/isb/pubs/MapProjections/projections.html
  • http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html
• Maps of bathymetery
  • http://deeptow.tamu.edu
  • http://topex.ucsd.edu/marine_topo/text/topo.html
  • http://www.ngdc.noaa.gov/mgg/global/etopo5.HTML

Jan 23 - Winds over the ocean

• Large scale wind paterns
  • Hadely cells
  • Trade winds
  • Intertropical convergence zone
• Small scale wind patters
  • Sea breeze
  • Topographic effects
  • Weather and fronts
• How do we measure the wind?
  • Shipboard: Beaufort wind scale and anemometers
  • Satellite: Scatterometers
  • Numerical weather prediction: Nowcasts/forecasts and reanalysis
• Wind stress
  • tau = rho C_D U₁₀², where
    • tau is the wind stress
    • rho is the density of air (aprox 1.3 kg / m³)
    • C_D is the drag coefficient (usually about 0.001)
    • U₁₀² is the wind speed at 10 m above the water
  • There are a number of other equations that account for changes in the drag coefficient with changing wind speed (due to surface roughness, etc). See Gill's book (Atmosphere-Ocean Dynamics, 1982) for some examples.
• Wind speeds are much faster than ocean current speeds, due to the density diference between air and water. Water is higher density, and can carry the same momentum with smaller speeds.

Jan 28 - Heat flux, part 1

• Heat budget: Q_T = Q_SW + Q_LW + Q_S + Q_L + Q_V
  • Q_T: Total heat flux
  • Q_SW: Insolation (incoming shortwave radiation)
  • Q_LW: Blackbody radiation (outgoing longwave radiation)
  • Q_S: Sensible heat flux (heat conduction between things that are physically connected)
  • Q_L: Latent heat flux (freezing and evaporation)
  • Q_V: Advective heat flux (due to ocean currents)
• Differences between the incoming and outgoing radiation
  • Incoming radiation comes in at a peak of 0.5 microns wavelength (the peak is near the visible part of the light spectrum), due to the temperature of the sun (~6000 ^oK).
  • Outgoing radiation leaves at a much longer wavelength, in the infared part of the spectrum due to the temperature of the earth (~255 ^oK).
  • Water (primarily) and other gasses block segments of the spectrum differently.
  • The greenhouse effect is due to the fact that incoming radiation is not blocked, while outgoing radiation is blocked, which traps heat.

Jan 30 - Heat flux, part 2

• Global, annual mean heat budgets indicate that
  • The largest exchanges are due to short- and long-wave radiation.
  • Latent heat flux is the next largest contributor (and is actually much larger than net long-wave radiation)
  • Sensible heat flux is the smallest part of the heat budget.
• There is a surplus of heat flux in the low latitudes, and a deficit in the high latitudes
  • The excess heat in the lower latitudes is delivered to the higher latitudes through atmospheric and oceanic circulation.
  • The heat flux can be due to mean currents (like the Gulf stream) or eddy fluxes.
• Circulation movie links:
  • http://www-mount.ee.umn.edu/~okeefe/micom/ocean.html (The movies shown in class)
  • http://panoramix.rsmas.miami.edu/micom/highres/coads/High_res_contents.html (More MICOM results)
  • ftp://ftp-mount.ee.umn.edu/pub/ocean/movies/ (Not for the feint of heart..)

Feb 4 - Heat flux example, and strange the properties of water

• Heat flux example.

  • An example of advective heat flux was given in class with two insulated boxes with two connecting pipes. There is a temperature flux into each box (K (T₁ - B₁) and K (T₂ - B₂), where B₁ and B₂ are the boundary temperatures along one face of the box, T₁ and T₂ are the water temperatures in each box, so the heat flux is proportional to the temperature difference times some constant, K), and an exchange of water between the boxes. Each pipe carries water from one box to the other, where the water has the temperature of the origionating box. The temperature in each box may be solved by equating the incoming and outgoing heat fluxes.

    In the limit in which the exchange is very strong (Q >> K), the temperatures in each box are 1/2 * (B₁ + B₂). In the limit of weak exchange (Q << K) the temeratures in each box match the boundary temperatures (T₁ = B₁ and T₂ = B₂).

• Properties of water
  • Water is not a symetric molecule, and has a polarity. The side with the two hydrogen atoms has a positive charge, the oxygen side has a negative charge. This causes water to be an effective solvent, and enhances the surface tension in water.
  • The ions in salt attract the oppositely charged sides of the water molecules, such that the ions remain surrounded by water molecules. This causes salt to remain dissolved in water. It is also one of the reasons why salt increases the density of water, the attractive forces between the salt ions and water cause the molecules to be more tightly packed.
  • The densest (fresh) water is found at 4 ^oC. This is because at colder temperatures, ice crystals form in the solution, and take up more space than the free molecules would take up. Salt water does not have a similar minimum in density; colder water is always more dense.

Feb 6 - Measuring temperaure, salinity, pressure and density

• Temperature (T)
  • Modern emperature measurements are made through thermistors or by measuring resistance in a platnum wire. Both precision and accuracy are good.
• Salinity (S)
  • Modern salinity measurements are actually measurements of conductivity, which is measured by resistance of sea water, or by induction in sea water. Precision and accuracy are good, but the accuracy of instruments tend to drift with use.
• Pressure (p)
  • Pressure is made by measuring vibrations in a tungston wire, or by pressure on quartz crystals. Accuracy is good, but precision is excelent.
• Density (rho)
  • Density is found by using the equation of state: rho = rho(S,T,p). This equation is a complicated polynomial with emperically derived constants. Accuracy is even better than the measurement accuracy of the above properties.

Feb 13 - More on water properties and light

• Where do you find different kinds of water?
  • Temperature:
    • Higher temperatures are found near the poles, cooler temperatures in the high latitudes.
    • This is because of higher heating in the low latitudes.
  • Salinity:
    • Higher salinity is found in the tropics (where the higher temperatures are found), low salinities in the high latitudes.
    • This is primarily because of evaporation in the tropics.
• Light penetration in the water column.
  • Light penetration is approximated by exponential decay:
    I(z) = I_o e^{-c z} where I(z) is the light intensity at depth z, I_o is the light intensity at the sea surface, and c is the light attenuation coefficient.
  • The attenuation coefficient is dependant on the frequency (or color) of the light. Red light (and on into the infrared spectrum) is attenuated more, blue light is attenuated less. Scatter is also higher for bluer light; this combined with the low attenuation causes the ocean to look blue. Particals in the water (plankton or sediment) can alter the color of the water.

Exam 1

The equations of motion

• Mass balance example

  Consider an estuary with an inflow of river water Q, an inflow of oceanic water V_in (with salinity S₀), and an outflow of estuary water (of intermediate salinity of S_estuary) V_out. In a steady state, this system must conserve salt and mass. The bousinesq approximation allows us to convert the mass conservation to volume conservation; rho may be treated as constant. So the two equations for the system are

  • Mass: Q + V_in = V_out
  • Salt: S₀ * V_in = S_estuary * V_out
  These equations can be combined to give us information about the system. By eliminating, say, V_in, we may solve for V_out given the various salinities and the river discharge.

• Conservation of mass

  Changes in density (mass per unit volume) can be cause by advection from neighboring regions, mixing with neighboring water masses, or local sources and sinks. Mathematically, this means:

  d rho / dt + u * d rho / dx + v * d rho / dy + w * d rho / dz
  = A_H * ( d² rho / dx² + d² rho / dy² ) + A_V * d² rho / dz² ( + Local Sources/sinks )

  The mass conservation is simplifed consederably if we use the Bousinesq approximation (that density variations are ignored, except when multiplied by gravity). This reduces mass conservation to volume conservation:

  du/dx + dv/dy + dw/dz = 0

• Conservation of momentum

  Changes in momentum are reated to Newton's second law: du/dt = a = F/m. Thus, the changes in momentum in each direction (x,y and z, the three axes of the coordinate system, related to the speed of water in those directions: u, v, and w) are equal to the sum of forces acting on a particle. The three main forces are gravity, the pressure gradient, and the Coriolis force. Momentum, like mass may also be affected by advection and mixing with neighboring water masses. The three momentum equations are:

  du/dt + u*du/dx + v*du/dy + w*du/dz - f*v = -(1/rho₀) dp/dx + A_H * ( d²u/dx² + d²u/dy² ) + A_V * d²u/dz²
  dv/dt + u*dv/dx + v*dv/dy + w*dv/dz + f*u = -(1/rho₀) dp/dy + A_H * ( d²v/dx² + d²v/dy² ) + A_V * d²v/dz²
  dw/dt + u*dw/dx + v*dw/dy + w*dw/dz = -g -(1/rho₀) dp/dx + A_H * ( d²S/dx² + d²S/dy² ) + A_V * d²S/dz²

• Tracer equations

  Similar to the density equation, changes in the tracers temperature (T) and salinity (S) may be cause by advection from other regions, mixing, or local sources and sinks.

  dS/dt + u*dS/dx + v*dS/dy + w*dS/dz = A_H * ( d²S/dx² + d²S/dy² ) + A_V * d²S/dz² ( + Local Sources/sinks )
  dT/dt + u*dT/dx + v*dT/dy + w*dT/dz = A_H * ( d²T/dx² + d²T/dy² ) + A_V * d²T/dz² ( + Local Sources/sinks )

Reynolds averaging

• Rules of averaging:
  • < u > = U
  • < u' > = 0
  • < U > = U
  • < const * u > = const * < u > = const * U
  • < du/dx > = d/dx < u > = dU/dx
  • < u' * U > = < u' > * < U > = 0 * U = 0
• All linear terms convert nicely from u to U, for example: < k du/dx > = k dU/dx. Nonlinear terms to not convert nicely: < u' * du'/dx > = ?.
• Use the continuity equations to convert the nonlinear advection terms into what looks like a divergance of turbulent stresses: < u' * du'/dx > + < v' * du'/dy > + < u' * dw'/dz > = d/dx < u' * u' > + d/dy < v' * u' > + d/dz < w' * u' >
• Coherent turbulent motions cause < u' * u' > terms to be non-zero. These terms can be related to turbulent mixing, and can be given the same form as the molecular mixing (for mathematical simplicity), so that, for example, < w' * u' > = K_z dU/dz so that d/dz < w' * u' > = K_z d²U/dz² so that the turbulent mixing velocities (u', v', and w') are put in terms of the mean flow (U, V, and W).

April 1 - The Richardson number

April 3 - Inertial motions

Comparing the time derivative term (du/dt) with the coriolis term (-fv) gives a time-scale for when the earths rotation will be important: T = 1 / f. On shorter timescales, the earths rotation is not important. On longer timescale it is. Away from the equator, this timescale is on the order of a few hours.

The length scale, related to the radius of the inertial oscilations, can be interpreted as the (horizontal) size that a feature must be in order to be affected by rotation: L = V / f. L can be interpreted as the distance a particle can travel at speed V on the rotational timescale. After this time, rotation will have had a profound impact on the particle trajectory; in intertial oscillations, the velocity vector has been rotated by 90 degrees.

Click here to see an animation of an inertial oscilation from a non-rotating frame of reference.

April 8 - Ekman dynamics

• The wind driven currents are trapped to the surface by the earths rotation. The depth scale for the currents is H ~ SQRT( A_V / f )
• The Ekman transport (the vertical integral of the currents) is proportional to the magnitude of the wind stress, Tau, and the strength of the local Coriolis parameter, f. v * H ~ V = - Tau^x / rho₀ f
  u * H ~ U = Tau^y / rho₀ f The Ekman transport is always 90^o to the right of the wind stress, and does not depend on the details of the Ekman spiral (since it is an integrated property).

April 9 - Review for Exam 2

April 10 - Exam 2

April 15 - The geostrophic approximation

• Scaling the momentum equations

  Using the scales found in the geometry and circulation patterns in the ocean, we can 'scale' the momentum equations to find out which terms are dominant, and which terms are small, and can be ignored. [see the section Scaling the Equations: The Geostrophic Approximation in the text.]

• The hydrostatic balance

  Scaling the vertical momentum equation gives us the following dominant terms (the other terms are 10⁶ smaller):

  &part p / &part z = - g &rho. This is the same balance that would exist if the water were not moving, thus this is called the hydrostatic balance (hydro = water, static = not moving).

  If the water is assumed to be all the same density, &part p / &part z is constant, so that changes in pressure are due only to changes in sea surface height. Note, if we assume friction is negligable, the horizontal momentum equations (for u and v) don't have any z derivatives. If there is initially no shear (&part u / &part z = &part v / &part z = 0), there is never any way to get shear. Thus, the water column is ridged, meaning that the currents will be the same all the way through the water column.

April 17 - The geostrophic approximation, vertically averaged equations

• The shallow water equations.

  Vertically averaging the horizontal momentum equations gives

  &part u / &part t - f v = -g &part &eta / &part x
  &part v / &part t + f u = -g &part &eta / &part y Vertically averaging the continuity equation gives H ( &part u / &part x + &part v / &part y ) = &part &eta / &part t where u and v are the depth averaged currents, and &eta is the sea surface height.

April 22 - Vortiticity, Rossby/shelf waves

• The Taylor column - conservation of vorticity

  Take the 'curl' of the two horizontal momentum equations (i.e., &part / &part x [v-equation] - &part / &part y [u-equation]). Put the result into the continuity equation. This results in the conservation of vorticity

  d / dt (&zeta + f / H) = 0 where &zeta = &part v / &part x - &part u / &part y is the curl of the velocity, referred to as the relative vorticity. In this context, the Coriolis parameter is referred to as planitary vorticity, which gets larger to the north. This statement is equivilant to conservation of angular momentum, where the entire angular momentum (the local angular momentum relative to the earth plus the angular momentum of the earth itself) is considered.

  If relative vorticity is ignored (&zeta = 0), and the column of water remains at approximately the same latitude, this requires that h = const. This is sometimes refered to as a Taylor column. Taylor columns are ridged (since h is constant), so that the water column can only flow along constant depths. If a column of water is on top of a sea mount, it will resist being displaced, because moving off the seamount would require the column to increase it's depth.

• Rossby waves/topographic waves.

  The diagram drawn in class shows how conter-rotatining vorticies (created when fluid is pushed to either different depths or latitudes) can cause westward propegation of a wave form. Conceptually, the propegation of these wave forms is identical for the topographic waves (caused by changes in depth) and Rossby waves (caused by changes in the Coriolis parameter with latitude).

  
• The Stommel model of westward intensification (what causes the Gulf Stream?)

  The simplest model of a western boundary current relies on the following sequence, also shown in the figure below

  • The westerlies and trade winds are such that there is a convergence in the surface Ekman layer. This convergence causes water to be forced into the interior. The columns of water are squished. To conserve potential vorticity, they must travel south (here, the scales of flow are too large for &zeta to balance the potential vorticity). This balance (between wind stress curl and north-south transport in the ocean interior) is called the Sverdrup balance.
  • The weak southward transport in the interior of the basin must be balanced somewhere by a northward transport. This transport occurs in a narrow layer near the western boundary. The layer must be strong enough for bottom friction to matter, since when the surface friction (due to the wind) is dominant, the flow is to the south. In this boundary layer, the flow is stronger near the western boundary, so that there is a divergance in the bottom Ekman layer transport (always to the left of the flow, and proportional to the flow speed). This divergance sucks water out of the overlying water, stretching the water column. To conserve potential vorticity, the flow must move to the north.
  • So the water travels in a closed circuit: the water columns are squished and move south in the ocean interior, and the water columns are stretched and move north in the western boundary current. In both cases, the stretching and squishing are due to convergences and divergences in the frictional layers. In the ocean interior, the surface friction (wind) is the most important, in the western boundary layer, the bottom friction (due to the strong flow) is the most important.

  

Why is the western boundary current on the western side of the basin? If the boundary current were on the other side of the basin, there would be a convergence in the bottom Ekman layer. This would cause the water column to be squished. Thus, the water columns would be squished both in the interior and in the eastern boundary current. This cannot be. The boundary current must be on the western side so that the water columns are not perpetually squished. In the case of the western boundary current, the water column is alternately stretched and squished, so there is no net change in the height of the column in one loop around the basin.

April 25 - Thermal wind and the geostrophic method

• The thermal wind balance

  The thermal wind balance is derived by combining the geostrophic approximation and the hydrostatic balance. For example, take the east-west balance (with north/south currents) and the hydrostatic balance

  -f v = - (1/&rho₀) &part p / &part x
  &part p / &part z = - g &rho Taking a z-derivative of the first equation, and an x-derivative of the second equation, and eliminating the pressure terms between the two results gives &part v / &part z = - ( g / f &rho₀) &part &rho / &part x Thus, vertical shear in currents can be related to horizontal changes in density.

• Currents from hydrography

  It is much easer to measure density (through temperature and salinity) than it is to measure currents - especially large-scale, slowly varying, weak flows which may be masked by quickly varying waves in the ocean. The thermal wind balance may be used to estimate the flow in the ocean from hydrographic measurements. The problem is that the thermal wind gives vertical shear (changes in current strengh with depth) instead of absolute currents. A commonly used solution to this problem is to assume there is a deep 'level of no motion,' where the velocity is zero. At this depth the velocity is pinned down to a particular value (zero), and the thermal wind relation can be integrated vertically to get an estimate of the flow through the entire water column.

May 2, Final exam

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