Reading, references and study questions for lecture 3 - click here
What equations do physical oceanographers use to describe the ocean?
2. Motion of fluid parcels is governed by F - ma. Given all forces, can calculate acceleration a = delta u / delta t. We will ctalog a number of forces, using a = f/m to esimate the associated acceleration. Especially if a = 0, then sum (forces) = 0.
3. Thus gravity is a force/unit mass toward the earth's center. It results in an acceleration g = 980 cm/seci2 toward the center of the earth if no other forces act.
4. Pressure gradient force. What is pressure? It is the Force/area any volume of fluid exerts across its boundaries on any adjacent volume. It is not associated with motion.
5. In the ocean, pressure is hydrostatic. In ocean pressure at depth is weight of 1 m cross-section column and pressure at surface
P(z) ~ Patm + rho g (z below surface) ~ 1 atm + z(m)/10
1 atm = 14.7 lb/in2 = 10 m of water = 1000 mb = 106
dy/cm2
Remember 1 cm of water ~ 1 mb ~ 10-3 atm
6. Pressure gradient force.
Although the atmosphere exerts force (lb/in2) on every square inch of us, we don't fall over. So the net force due to pressure isn't just pressure. Net pressure force is due to the pressure difference - the Pressure Gradient Force. Why doesn't gravity move fluid downward? On a weathermap (figure).
Meteorologists measure PGF directly, with barometers. They put lots of barometers at sea level, constant isobars, and estimate horizontal pressure gradients.
In the ocean, horizontal pressure gradients are important. Typical PGF across the Gulf Stream corresponds to 1 meter height difference across 100 km. (another figure).
7. Can oceanographers measure the horizontal pressure gradients that matter for large-scale ocean flow? NO. Why not, meteorologists use barometers? Meteorologists can do it because air is so light that it doesn't matter if one barometer is on the floor and the adjacent one is slightly higher or lower. But water is so dense that putting one barometer on a 10 cm high rock by accident may reverse the estimated horizontal pressure gradient.
(hand-drawn figure showing rock)
This raises a fundamental point: how do we find a flat surface/plane in the ocean?
8. What is flat? We call a flat surface the GEOID. If it were solid, frictionless, nothing would roll downhill on it. It is not a sphere, because earth's rotation flatten's the earth. But it is not a flattened sphere either because earth's rocks are not all the same density. Simplest description that is correct: it's a spheroid whose equatorial radius is about 20 km greater than its polar radius and deviation of cm to a few hundred meters over kilometers to basin scales.
Figure: marine sea surface, compiled from 10 years of altimeter data. Figure source: AVISO (CNES, Toulouse, France).
deviation(sphere) >> deviation(spheroid) >> deviation (geoid).
The geoid would be the sea surface if the ocean didn't move. Unfortunately that difference is crucial.
9. Frictional forces. Due to velocity gradients (i.e. spatial variation, not time variation) and viscosity nu. Consider parcel in shear.
Force/area = (rho (nu) du/dz|z+dz) - (rho (nu)du/dz|z) = rho(velocity) d2u/dz2 delta zIn the Gulf Stream, of width W, d2u/dz2 ~ delta u/W2
acceleration = force/area times area/mass = rho (nu)d2u/dz2 (delta z) (delta x)(delta y)/(rho deltax deltay deltaz) = nu d2u/dz2.
Therefore, in horizontal force balance |PGF| >> |Friction|.
What balances the PGF? Something must if the flow is steady.
10. Now stop cataloging forces and consider some situations.
These notes (Hendershott's) are continued in the next lecture, which includes Coriolis and centrifugal force.
1. Radiation: We usually think of EM radiation. It carries energy from the sun and returns energy to outer space. But EM transfer inside the ocean is small because the ocean is not very transparent. Energy/momentum transport by internal/surface gravity wave radiation is important but we defer its discussion until these waves have been described. Seismologists are concerned with energy transport by acoustic/shear waves.
2. Diffusion: Occurs because molecules are always in random thermal motion even if material appears at rest. Diffusion needs a concentration gradient i.e. more stuff one place than somewhere else. Described by Fick's Law:
QA = stuff/volume at A = concentrationWe have to measure kappa. If we know it we can answer some questions.
dQ/dx = concentration gradient in direction x
Fick's Law: F = -kappa grad Q, where F is the vector flux.
Flux from A to B = -kappa (QB - QA)/ (B-A) = -kappa dQ/dx
kappa is the diffusion constant:
stuff/cm2/s = kappa stuff/cm3/cm
[ kappa ] = cm2/s (where [ ... ] means "units")
Examples:
(1) In Arctic, surface temperature varies seasonally (T = 86400 x 180 ~
107 sec). Heat diffuses at kappa = 10-3 cm2/s in dirt. How deep (L)
is the seasonal temperature change noticed?
L = sqrt(107 10-3) = 1 m.
(2) How long must year be for seasonal temperature variation to
penetrate 10 m? T = L2/kappa = (107)/10-3 = 109 sec = 30 years!
Molecular basis of momentum transport in a gas
Example: Molecules have random thermal speed u' in all directions plus mean flow v in (say) x direction, where v varies with z (height). Molecules go freely a distance L ("mean free path") between collisions. Molecules crossing zo from upper to lower layer carry extra mean momentum v ( z > zo) with it and deposit it in the lower layer after collisions. Thus the upper layer drags the lower layer along.
(figure)
That is to say, tauxz = -rho |u'|L V/??
The quantity |u'|L is the viscosity nu (Greek letter).
In a gas, increasing temperature increases |u'|, so the viscosity goes up.
In a liquid, increasing temperature increases molecular separation, decreases inter-molecular force and the viscosity goes down.
Size of molecular viscosity and diffusivity in sea water:
viscosity: nu = 0.018 cm2/s at 0C; 0.010 at 20C; that is, order (10-2)
diffusivity: kappatemperature = 0.0014 cm2/s
diffusivity: kappasalinity = 0.000013 cm2/sec
3. Advection: motion of fluid carries stuff.
Flux = velocity times concentration (stuff/cm3)
[ Flux ] = [stuff/cm2/sec]
This is so simple it seems useless to ponder.
Application in 2D: iso-tracers are streamlines of steady flow. i.e. if flow were steady in 2D, measuring one tracer and contouring its isolines would tell us the streamlines (but not the flow speed).
In the ocean, flow is often idealized as steady but is really 3D. Streamlines lie anywhere on isotracer sheets. Measure 2 tracers. Their sheets usually aren't parallel but intersect along a contour. The flow lies in both sheets i.e. along the intersection of the two sheets.
Simple velocity - another application
(figures showing how convergence - i.e. advection - can sharpen gradients)
Convergence and diffusion: What happens if both advection and diffusion happen at the same time (normal situation)? Advection can sharpen the stuff gradient, while diffusion smooths it. An equilibrium can exist: w = gamma z.
4. Turbulence, Reynolds number, eddy diffusivity
What is u (velocity) like in the atmosphere/ocean? Is it simple? Consider a flow U in a layer of thickness d, with a viscosity nu. If the ratio Ud/nu is small enough, the flow is laminar - that is, in sliding sheets. Dye in this flow would not disperse.
laminar flow <-> (Uslow)(dwide)/nubigThe quantity Ud/nu = Re is called the Reynolds number. The Reynolds number is a non-dimensional parameter. Re depends on geometry. In the ocean, typical scales are U = 102 cm/sec, d = 4x105 cm, nu = 10-2. Thus the typical Re = 4e109 - it is very large. This means that much of the ocean is turbulent. Whether it is everywhere/always is not clear. There may be nearly quiescent patches.
turbulent flow <-> same quantity is large.
The lesson is that the velocity field is probably very complicated. From satellite images, we see how complicated it can be.
Complex velocity and diffusion: consider putting milk into coffee.
(1) If
it is eased in, and left to spread molecularly, the time to stir =
L2/kappa = (102)/0.01 = 104 sec!!!
(2) If you get the coffee to rotate smoothly, the spot just rotates.
(3) If you stir vigorously, in a second or so, there is complete mixing.
Why??
Diffusion smoothes gradients, goes slowly. Stirring pulls out the spot and thins it, making stronger gradients, so diffusion goes faster.
People often say stirring plus diffusion creates big kappa diffusion. This motivates the concept of eddy diffusivity kappaeddy, which is much bigger than molecular diffusivity.
There are two attitudes in the literature about eddy diffusivity.
(A) accept it and use it
(B) look farther
(A) The accept and use procedure chooses kappaeddy to get the answer right. Munk's abyssal recipes is an example (see reading list). (DSR 1966, 707-735). Munk's argument was that water cools at high latitudes, sinking locally. It then rises over a broad area at a vertical speed w. w = (volume sinking/time)/(area rising) ~ 1 cm/day. The rising water carries heat up. To keep the deep water at a constant temperature, heat must diffuse downward. The heat flux HF is the same at every level.
HF = -(kappaeddy)dT/dz + wT
d(HF)/dz = 0 so w dT/dz = kappaeddy d2T/dz2. (kappaeddy Tzz).
Writing this in differences instead of derivatives,
kappaeddy ~ w (delta z) ~ (10-5 cm/s)(105 cm) = 1 cm2/sec.
Compare this with the molecular diffusivity kappamol = 10-2 cm2/s.
When we thus "back into" the answer, we haven't really said why
kappaeddy is what we get. Always kappaeddy >> kappamolecular.
Eddy diffusivities are different in the vertical and horizontal - they
are much larger in the horizontal (this is an empirical result).
kappaeddy_vertical ~ 0.1 to 102 cm2/sec.
kappaeddy_horizontal ~ 104 to 108 cm2/sec.
Part of this difference is due to the aspect ratio: that depth << length scale. The vertical extent and rms vertical velocity w of eddies is << horizontal extent and rms horizontal velocity u of eddies. this difference is mainly due to stratification, with light fluid over heavy.
(B) The "look further" approach is an attempt to estimate kappaeddy along the lines of the molecular argument. Image a gas (air) with a small concentration n(x,t) of some other gas. All molecules travel with thermal velocity vrms, between collisions. The distance between collisions is the mean free path mfp. Imagine that n(x,t) is spatially variable. The flux of n-stuff across a surface, where n is n- to the left of the surface and n+ to the right of the surface, is
flux (solute atom/sec/cm2) = n-vrms - n+ vrmsTo scale up to eddies by analogy, we could interpret vrms as the rms velocity of the fluid, and mfp as the eddy size and say
= -vrms(n+ - n-)
= -(mfp)(vrms) dn/dx
i.e. the diffusivity is kappa ~ (mfp)(vrms)
kappaeddy = (eddy size)(vfluid_rms)What is the eddy size? You can devise estimates of some length scale to put in, but there remain problems. yet the idea that small-scale flow fosters diffusion/dispersion is useful.
In flows such as the ocean, our ability to observe small details
synoptically is so limited that almost every observation is
really an average that smooths out small scales. Thus if we try
to measure a quantity S, we really get an estimate the average
of S or ave(S).
We might want ave(advective flux) = ave(US). But really
the field
(1) S = ave(s) + S' = mean plus rest and
(2) U = ave(U) + U'
This division between mean and remainder is called the
Reynolds decomposition. Note that the average of the
remainders is zero: ave(S') = 0 and ave(U') = 0
Thus
ave(US) = ave(ave(u)ave(S) + ave(U)S' + U ave(S) + U'S')
=ave(U)ave(S) + ave(U'S')
The advective flux has a part due to mean flow and a part due to fluctuations. The part due to fluctuations is called the Reynolds flux. Remarkably, very often, the Reynolds flux is >> mean flux.
Sometimes we can measure U' and S'. The eddy flux hypothesis says that:
ave(U'S') = -kappaeddy d ave(S)/dxHow can this be? Taylor: imagine a smooth initial salt field S(x) and let small scale motion distort it without any molecular diffusion.
Measuring U' and dx is not easy. We may try to make progress by
modeling the motion of the fluid parcels. The simplest model
is :
dx = sum(u delta t), in which ave(ui uj) = 0 if i .ne. j
and ave(ui ui) = ave(u2). Then
kappaeddy = -ave(ui dx) = - ave(u2) delta tbut what does this mean?
If all parcels move independently, like molecules, with ave(uAi uBj) = 0, then intuitively, an initial blob or parcel diffuses out with kappaeddy = -ave(u2) delta t. This is really just kappaeddy = -u(udeltat) = -u (mfp) = (mfp)2/dt.
If all parcels move exactly the same way, uA = uB, we still get kappaeddy = -ave(u2)dt but now an initial blob never changes hape, it just jiggles around. This is what we intuitively mean by diffusion even though the average blob does grow.
The ocean is in between. for parcels separated by small distance
uA = uB - it is not clear that ave(uA uB) ever is zero, although
much depends on what ave(..) hides. To be sure we are really
looking at parcels being taken apart, we need to think about the
average separation of the parcels:
ave(dxA - dxB)2).
