Transcript of Myrl Hendershott's notes from 2003 on dynamics, diffusion

1. This lecture is a primer in fluid mechanics. All in words. Can't fully solve problems this way but can understand solved ones. This mode of thinking usually has to go before detailed analysis.

2. Motion of fluid parcels is governed by F = ma. Given all forces, can calculate acceleration a = delta u / delta t. We will catalog 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/sec^{2} 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) ~ P_{atm} + rho g (z below surface) ~ 1 atm + z(m)/10

1 atm = 14.7 lb/in^{2} = 10 m of water = 1000 mb = 10^{6} dy/cm^{2}

Remember 1 cm of water ~ 1 mb ~ 10^{-3} atm

6. Pressure gradient force.

Although the atmosphere exerts force (lb/in^{2}) 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) d^{2}u/dz^{2} delta z

acceleration = force/area times area/mass = rho (nu)d^{2}u/dz^{2} (delta z) (delta x)(delta y)/(rho deltax deltay deltaz) = nu d^{2}u/dz^{2}. In the Gulf Stream, of width W, d^{2}u/dz^{2} ~ delta u/W^{2}

acceleration = nu delta u/W^{2} = (0.02)(100)/(100e5)^{2} = 1e-14 cm/s^{2} << PGF

Even if we use the eddy viscosity kappa_{eddy} = 10^{8}, this quantity is 1e-4 cm/s^{2} << PGF

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.

(a) Fluid at rest in gravity field. PGFs push up, gravity pulls down. No vertical acceleration. Balance is hydrostatic. If pressure is constant at some level (such as free surface) then there is no horizontal acceleration either.

What happens if the parcel is a bubble?

What happens if the parcel is a crown of gold?

(b) Explosion. Just after combustion. Rapid acceleration = PGF/mass. Parcel is accelerated outward. Unsteady flow.

(c) Bubble. Pinside > Poutside. Surface acts like a string, "tension" makes Pinside > Poutside.

(d) Pipe. Pump forces fluid through. PGF is balanced by friction.

(e) Tilted pipe. Pump and PGF forces fluid through. PGF is balanced by friction.

(f) Collapse of "heaped up" free surface. Water drains out from under initial mound because the PGF pushes it horizontally outward. In a stratified fluid, the same thing happens if isopycnals are disturbed. YET IN THE OCEAN across the Gulf Stream, the surface doesn't collapse! Why?

These notes (Hendershott's) are continued in the next lecture, which includes Coriolis and centrifugal force.

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4. Ocean transport processes, viscosity, eddy diffusivity:
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Scanned pdf of Myrl Hendershott's transport processes lecture, from fall, 2001

Transcript of Hendershott lecture.

"Stuff" gets from one place to another by radiation, diffusion and advection.

Transport is quantified by the idea of a flux: magnitude and direction.

^{2} in direction perpendicular to A in one second.

Note flux is in a direction, normal to A

Examples:

(1) geothermal heat flux through the ocean floor = 10^{-6} cal/cm^{2}/s

(2) Top of atmosphere solar flux = 1300 Joule/m^{2}/s = 1300 W/m^{2} away from sun

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 = concentration

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/cm^{2}/s = kappa stuff/cm^{3}/cm

# [ kappa ] = cm^{2}/s (where [ ... ] means "units") We have to measure kappa. If we know it we can answer some questions. How far (L) does stuff diffuse in time T? L^{2}/T = kappa so L = sqrt(kappa T)

# If stuff has diffused to L, how long (T) has it been diffusing? T = L^{2}/kappa.

Examples:

(1) In Arctic, surface temperature varies seasonally (T = 86400 x 180 ~ 10^{7} sec). Heat diffuses at kappa = 10^{-3} cm^{2}/s in dirt. How deep (L) is the seasonal temperature change noticed? L = sqrt(10^{7} 10^{-7}) = 1 m.

(2) How long must year be for seasonal temperature variation to penetrate 10 m? T = L^{2}/kappa = (10^{7})/10^{-3} = 10^{9} 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 z_{o} from upper to lower layer carry extra mean momentum v ( z > z_{o}) 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 cm^{2}/s at 0C; 0.010 at 20C; that is, order (10^{-2})

diffusivity: kappatemperature = 0.0014 cm^{2}/s

diffusivity: kappasalinity = 0.000013 cm^{2}/sec

3. Advection: motion of fluid carries stuff.

Flux = velocity times concentration (stuff/cm3)

[ Flux ] = [stuff/cm^{2}/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)/nubig

turbulent flow <-> same quantity is large. The 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 = 10^{2} cm/sec, d = 4x10^{5} cm, nu = 10^{-2}. Thus the typical Re = 4 x 10^{9} - 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.

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 = L^{2}/kappa = (10^{2})/0.01 = 10^{4} 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 kappa_{eddy}, 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 kappa_{eddy} 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 = -(kappa_{eddy})dT/dz + wT

d(HF)/dz = 0 so w dT/dz = kappa_{eddy} d^{2}T/dz^{2}. (kappa_{eddy} T_{zz}).

Writing this in differences instead of derivatives,

kappa_{eddy} ~ w (delta z) ~ (10^{-5} cm/s)(10^{5} cm) = 1 cm^{2}/sec.

Compare this with the molecular diffusivity kappa_{mol} = 10^{-2} cm^{2}/s.

When we thus "back into" the answer, we haven't really said why kappa_{eddy} is what we get. Always kappa_{eddy} >> kappa_{molecular}. Eddy diffusivities are different in the vertical and horizontal - they are much larger in the horizontal (this is an empirical result).

kappa_{eddy_vertical} ~ 0.1 to 10^{2} cm^{2}/sec.

kappa_{eddy_horizontal} ~ 10^{4} to 10^{8} cm^{2}/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 kappa_{eddy} 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 v_{rms}, 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/cm^{2}) = n_{-}v_{rms} - n_{+} v_{rms}

= -v_{rms}(n_{+} - n_{-})

= -(mfp)(v_{rms}) dn/dx

i.e. the diffusivity is kappa ~ (mfp)(v_{rms}) To scale up to eddies by analogy, we could interpret v_{rms} as the rms velocity of the fluid, and mfp as the eddy size and say

kappa_{eddy} = (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') = -kappa_{eddy} d ave(S)/dx How can this be? Taylor: imagine a smooth initial salt field S(x) and let small scale motion distort it without any molecular diffusion.

(figure)

A parcel starts at x_{o}, and goes over to x_{o} + dx. Then S'over = Sover - Sstart = S(x_{o}) - S(x_{o} + dx) = -dx dS(x)/dx.

Average over small scale displacements: ave(U'S') = - ave(U'dx)dS/dx. That is, to get kappa_{eddy}, measure U' = local velocity and dx, the "recent" displacement, and then kappa_{eddy} = ave(U'dx). You could measure these quantities with fluid-following drifters. Note that this is very conceptual - i.e. what does "recent" mean? Nevertheless this approach is useful.

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(u_{i} u_{j}) = 0 if i .ne. j and ave(u_{i} u_{i}) = ave(u^{2}). Then

kappa_{eddy} = -ave(u_{i} dx) = - ave(u^{2}) delta t but what does this mean?

If all parcels move independently, like molecules, with ave(u_{Ai} u_{Bj}) = 0, then intuitively, an initial blob or parcel diffuses out with kappa_{eddy} = -ave(u^{2}) delta t. This is really just kappa_{eddy} = -u(udeltat) = -u (mfp) = (mfp)^{2}/dt.

If all parcels move exactly the same way, u_{A} = u_{B}, we still get kappa_{eddy} = -ave(u^{2})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 u_{A} = u_{B} - it is not clear that ave(u_{A} u_{B}) 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:

eve(dx_{A} - dx_{B})^{2}).

SIO 210 HOME October 17, 2006