Today we’re going to complete the momentum equation by adding the frictional term. The frictional term is probably the most complex term in the momentum equation and certainly the one that we understand the least.
Start with writing the momentum equation
But don’t write friction immediately. Talk about each term talk about Friction as things rubbing against each other.
Ultimately friction is the loss of energy to heat due to the rubbing of molecules—(not enough to heat the ocean) –
But we’ll see that this “rubbing” is complex in fluid flow
In your high school physics book friction was often introduced in the “inclined plane”
Here R has units.
Show movies of boundary layers—and discuss boundary stresses versus interfacial stress
Osborne Reynolds found in studies of pipe flow that at sufficient low flow speed (U) or pipe diameter (d) dyed fluid parcels followed straight lines—termed laminar flow. But as U of d increased there came a point of abrupt transition to a turbulent regime in which fluid particles no longer stayed on wall-parallel streamlines. Thus U and d are important quantities and since they operate in the same sense (i.e. as the flow speed or pipe diameter increases turbulent is increase) they should be multiplied. Thus to get another non-dimensional number U*d needs to be divided by a quantity with units m^2/s. One thing that has this is viscosity, so the ratio U*d/ is called the Reynolds number and it represents the ratio of inertia to friction. For Reynolds numbers greater than 2000 the fluid becomes turbulent. Much of the ocean, particularly the surface and the coastal regions, are highly turbulent and this has important implications for both the mixing of tracers (salt, heat, nutrients) and frictional forces.
We are all familiar with friction. It occurs when one thing is rubbed against another--line your hand against the table. So in the ocean we can get friction when the water is rubbed by the atmosphere (wind friction) where the water rubs the bottom (bottom friction) and when water rubs against itself (internal friction).
Friction is characterized as a tangential stress.
The wind inputs energy into the ocean via a surface wind stress. This energy is exchanged between the layers via friction (although internal wave motion also does this) and is drained from the ocean at the bottom via mixing or bottom friction/
In fluid mechanics we often use the Greek Letter for friction
Reynold’s stress parameterization
More Rigor (even more on page 103)
We call these stresses Reynolds Stress after the famous fluids guy Osborne Reynolds (1842-1912). A more formal derivation of the Reynolds stress comes from a decomposition of the flow from a mean and turbulent part.
Following Knauss page 102-104
Consider the averaging of separate components
If we average this over a time period we get
When is the second term zero? When is it not zero?
On page 103 – if you can follow the derivation – you can see that this term arises from the advective term in the momentum equation.
Now if we move the Reynolds Stress terms to the other side of the equation – they can be
thought of as Friction
“Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.”
― Lewis Fry Richardson
<u’w’> (where brackets indicate the average over time)
Note that if there is as mean vertical shear that when w is upwards this will advect slower velocity upward and thus reduce u.
Thus when w’ is positive, u’ will tend to be negative.
Likewise when the w’ is downward this will tend to bring faster fluid down and thus increase u’
Thus when w’ is negative u’ tends to be positive.
Thus u’ and w’ are correlated and thus the average of their product over time is is not zero:
Draw time series that are correlated:
Multiply them together and show average is not zero:
Similar analogy can be made with a tracer (such as salinity or temperature)
DO THIS AND WRITE (w’s’) represents a flux of salt associate with turbulent fluctuations.
Recall that the diffusion of material is k * dC/dx: does anybody remember this?
A similar analogy is used for the transform of momentum (stress) using an eddy viscosity
If the transfer of momentum is actually done by the friction of water molecules rubbing against each other then
Internal Friction—is the friction between the fluid elements and in principle it can either laminar or turbulent though for all practical purposes friction in the ocean is due the interaction of turbulent eddies. In laminar flow (movie in class) friction arises due to the actual rubbing of molecules— and the frictional force is equal to the density times the kinematic viscosity times the vertical
where is the molecular kinematic viscosity of sea water where has a value of 1*10-6 m2/s (note those units—we’ll talk about them later). We’ll see later that turbulent friction, while considerably more complicated can be expressed in an identical form expect is replaced with the turbulent eddy viscosity (Av) (though this is a leap of faith) also with unites of m2/s--what makes turbulent friction more complicated is that Av varies by orders of magnitude in the ocean depending on the state of the flow. For numerical models of ocean circulation this requires a “turbulent closure” since we know have another variable in the equation.
Mention the 7 equations—but now I’ve introduced an 8th variable.
Discussion of turbulent closures is beyond the scope of this class.
So we write:
The transition between Laminar flow and Turbulent flow is determined by a Reynolds number—which is essentially the ratio of the advective term to the viscous term.
Tomorrow I’ll show some movies of turbulent flows and laminar flows and the transition.
Now that you know the units of viscosity you can calculate the units of friction—and they are kg/(m*s2) which is a force per unit area
F=ma/AREA= (kg m/s^2)/ m^3=kg (m s^2)
Thus friction has the same units pressure (force per unit area like your tires – pounds per square inch) and can be expressed in Pascals, but we call frictional forces stress.
1 Pascal = 1 Nm-2 = 10 dyne cm-2
Question: How does Pressure enter the momentum equation? (i.e. how is pressure related to acceleration?). Since Stress has the same units as pressure how then must the stress (frictional) term enter the momentum equation. Answer. Gradients in stress dt/dz dt/dy dt/dx
In general the greatest stress gradients are in the vertical. Q Why? Recall the aspect ratio of the Ocean. Lots of structure in the vertical—scales are much smaller and there fore gradients tend to be much larger!
Internal friction transfers energy between layers of fluid. Consider the case of wind blowing on the surface of the ocean (later we’ll quantify how much friction the wind imparts on the ocean’s surface’)
By doing dimensional analysis we can estimate how long it would take for wind friction to penetrate down 100 meters via molecular viscosity alone.
Note that the depth squared divided by the viscosity has units of time.
What time scale would this be? It would be the time scale for the wind forced motion to be transferred vertically by friction . In the molecular viscosity case it would be the top layer of molecules accerating the layer beneath it and then that layer accelerating the layer beneath it—and so on. Given the low value of molecular viscosity
100/1.e-6=1.e8 seconds ~ 3 years
It take three years for the wind to effect the water 10 meters down. Does anyone belive this result?
From observations we know that this is not the case. In the 1800’s a researcher named Zoppritz argued that the winds could not be a driving mechanism for ocean circulation because laboratory measurements of water’s viscosity were so low that it would take 1000’s of years for the wind to drive ocean circulation. Obviously Zoppritz spent too much time in the lab. In contrast Nansen, who spent years stuck on a boat in trapped in the ice in the Arctic, who was contemporary Zoppritz ,noted that Icebergs—which can extend 100 meters below the ocean’s surface—are driven by the wind and suggested that the ocean is in fact driven by the wind. It was Nansen’s observations that gave birth to the modern field of physical oceanography by the development of a theory for wind-driven flow’s by Ekman who was inspired by Nansen’s observations.
Rather than using the molecular viscosity Ekman proposed that the vertical transfer of momentum occurs through turbulent eddies and proposed a vertical eddy viscosity that was many times the magnitude of molecular viscosity. Here the value of the viscosity can be thought of as the size of the turbulent eddy times the current velocity in the turbulent eddy i.e.
where L and q are characteristic length and velocity scales of turbulent eddies. (Explain what characteristics scales are), and Av is the vertical eddy viscosity. In the ocean this can vary from near molecular levels to over 1 m2/s in unstratified turbulent flows.
Incorporating friction into the momentum equation (Jim did this last time)
How do we incorporate frictional stresses into the momentum equation? The stress represents the force applied to an area—but to have it enter the momentum equation we need to consider the force on a mass of fluid. To do this consider the box
Consider a case where the flow is vertically sheared and that the flow is stronger near the surface. Intuition would tell you that the frictional forces on the top of the middle box would drag the box to the right thus increasing its speed , while the slower moving fluid beneath the middle box would tend to drag the fluid to the left and decelerate the flow. The total force on the box then would be the stress on top of the box trying to drag the box to the right minus the stress on the bottom of the box which is trying to drag the box to the left
where is the surface area of the box. Recall the Taylor expansion whereby
Putting 2 into 1 yields
where is the volume of the box. The force per unit volume becomes
Which is the vertical stress divergence.
An identical procedure can be used to characterize the stresses on the vertical sides of the box—and they would take the form
Note that Av and Ah are fundamentally different—because in the ocean the vertical turbulent eddies are suppressed by stratification, while horizontal eddies are not. This is why isopyncal mixing is stronger than diapycnal mixing.
EMPHASIZE THIS POINT—DIFFERENCE BETWEEN ISOPYCNAL AND DIAPYCNAL
Now we’re ready – for the first time—to write the full blown momentum equation that includes the effects of the local and field accelerations, pressure gradients, the earth’s rotation and the vertical and horizontal stress divergence.
similar equation for v
One way to think of surface and bottom friction would be to depth average the momentum equation. This week’s homework problem (due next week) involves the depth averaged momentum equation (sometimes called the BAROTROPIC MODE) Most of the terms have a pretty simple form in the depth averaged sense (in detail both the pressure gradient term and the field acceleration term require more care than I’m going to give them here) and read.
the stress term becomes.
Where sand b are the surface wind stress and bottom stress respectively.
Observations suggest are surface currents are 3% of the wind—a 20 knot wind would drive a .6 knot current and that the stress applied to the ocean’s surface increases with the speed of the wind squared i.e. =aC w2.
is the surface wind stress where a is the density of air (1.2 kg/m3) , C is a “constant” and w is wind speed. The reason that constant is in quotes is that it’s not really a constant but varies with wave height and shape, but typically is somewhere between 1-2 10-3.
Assume a 10 m column of fluid starts at rest and a 10 m/s wind begins blowing and Cd is .001. If we neglect all other terms the water will begin accelerating at a rate of
=1.2*10-5m/s2 So after 10 hours (36000 seconds) the current speed would be .43 m/s
But after 100 hours the current speed would be 4.3 m/s
But observations tell us that after 4 days of a 20 knot wind the current does not flow at 4.3 m/s (8 knots!!).
Because the assumption of other terms not being important is not valid. In this case both the earth’s rotation, bottom friction and in the case of enclosed seas—the pressure gradient will become important.
The term Av du/dz has units m2/s2 and its square route is called a friction velocity u*.
Thus stress is:
u*2 (we can also use a u* at the surface for the surface friction velocity)
The breakthrough by Von Karman was that the eddy viscosity would vary proportionally to u*z
Because larger the friction velocity—the larger the turbulent velocity flucutations—and the larger the distance from the bottom (z) the larger the size of the turbulent eddies.
Thus the stress can written as
which can be integrated to yield the famous “Law of the Wall”
where zo the bottom roughness is a constant of integration obtained by assuming that the velocity goes to zero at height zo above the bottom.
If you square the above equation and rearrange the terms you find
So like the wind the bottom friction is proportional to the current speed squared/.
Frictional forces on the bottom will tend to slow down the fluid motion.
note that we use u times the absolute value of u so that the bottom frictional force always opposes the flow. While numerous things can impact the value or k it is also tends to fall in the range of .002- .003.