September 30, 2002
Review before first exam.
Units
x,y,z m
t s
u,v,w m/s
r kg/m3
Kh, Kv, Ah, Av (mixing coefficients) m2/s
Force 1N= 1 kg m/s2
Acceleration m/s2
Pressure kg/ms2
Pressure Gradient () kg/m2s2
Transport Q m3/s
Transport per unit width m2/s
Coriolis 1/s
Continuity Equation
3-dimensional
We’ve used it in two dimensions:
This assumes that of which a special case is v=0.
We used this to determine the vertical velocity near a front—such as done in class for a surface front and the homework problem for a bottom front.
PART C of HOWEWORK PROBLEM 2
We begin with the continuity equation
(1)
Taking the integral of 1 from the bottom to level z we obtain
(2)
We estimate as the gradient between A and B i.e.
(3)
So
And there you have it—the vertical velocity as a function of z. The result is plotted above
Other 2-D forms of the continuity are also possible. For example if we consider non-divergent depth averaged flow with a flat bottom we write.
where u and v are the depth averaged flow. One of the practice questions uses such a question.
If however, the bottom is not flat we would write:
(THINK TRANSPORT!!)
We also integrated vertically to obtain for a flat bottom.
If the bottom is not flat we would get . Note that by using the chain rule you can obtain the flat bottom equation () since for a flat bottom
If we integrated the 3-D continuity equation vertically we would obtain.
Let’s consider the homework problem 3.
Clearly in the coupling so how can the flow be non-divergent? There are two ways to think about this. One is that in the coupling region there is a flow towards the center of the pipe that converges at exactly the same rate the u is divergent. (see figure)
Upper figure shows flow through pipe drawing vectors showing how the fluid in the center of the pipe flows down the middle of the pipe, while the fluid on the walls runs parallel to the walls. If the flow is decomposed to a along pipe flow and a cross pipe flow we see that the along pipe flow is in fact divergent and the cross pipe flow is convergent.
Another way to think about this is analogous to the depth averaged version—where instead we average flows across the pipe to obtain
where A is the pipe’s cross section. This should be a fairly intuitive equation because uA=Q , where Q is the volume transport—and since water is incompressible the volume of fluid flowing past each section of the pipe must be the same. This can be stated as
The same logic could be applied to a flow in a channel – with the exception that sea level can go up and down. In fact it is this divergence in the flow that causes sea level to rise and fall each tidal cycle. The divergent of the transport (transport,Q, has the units m3/s—so the divergence has units m2/s and this equals the rate of change that sea level rises times the with of the channel (W). i.e.
An intuitive way to think about this is that rate that the hatched area shown below will rise is equal to the difference in Q, i.e.
or and letting Dx go to zero we get
Density of Sea Water (See September 16th notes)
Most of the water in the ocean varies between 1020-1030 kg/m3. Oceanographers use st to describe density where st = r(s,t,0)-1000. Note that st is calculated at zero pressure. This is done so that density observations can be viewed as if they were at the same depth. Freshwater at atmospheric pressure and 4 C weights approximately 1000 kg/m3.
Pressure:
Atmospheric pressure is approximately 14.7 pounds per square inch. This equals the total weight of the air above the surface of the earth—14.7 lbs. In a high pressure area the total weight of the air aloft weighs more than that in a low pressure region. In the ocean it is the overlying fluid that provides the mass. Thus the deeper you go the more the pressure you feel. This is why divers need to de-compress after a deep dive because they have been exposed to high pressure while under the water. How much pressure? We can calculate this! For convenience I’m going to turn the vertical coordinate system up-side down so that z= 0 at the surface and increases with depth. For a fluid of constant density the hydrostatic pressure is simply:
P(z)=rgz. (5)
Or more generally this can be written as
(6)
since r can change with depth.
At 10 meters depth pressure equals approximately 1 atmosphere (105 Pa). Using the ideal gas law a balloon at the surface would have it’s volume reduced by ½ if it were brought down to 10 meters depth. If it were brought to the bottom of the ocean (4000m), where the pressure is a crushing 400 atmospheres (nearly 3 tons per square inch) the balloon would be 0.25 % of its original size. So a balloon with a radius of 30 cm (as at the surface would be less than 4cm in radius at the bottom of the ocean.
Taking the vertical gradient of 5 we get:
(7).
This is a form of the vertical momentum equation stating that gravity is balanced by the divergence in the vertical pressure gradient. Under these conditions vertical motion in the water column is steady. A special case is that the vertical velocity is zero (w=0). Equation 7 assumes that the vertical acceleration is small compared to gravity, i.e.
and this is called the hydrostatic approximation. Under this case the pressure equals the weight of the overlying fluid. A good example of this is when you’re in an elevator. When the elevator is at rest your feet feel the pressure of the weigh of your body. When the elevator accelerates downward your feet feel reduced pressure until the elevator has reached a constant velocity at which time your feet once again feel the full weight of your body. As the elevator decelerates at the next floor your feet feel extra pressure—more than the weight of your body until the vertical deceleration stops and once again your feet feel only the weight of your body.
Pressure Gradient
In the above examples we have seen the vertical pressure gradient. We can also have horizontal pressure gradients. It is these horizontal pressure gradients that drive much of the circulation in the atmosphere and the ocean. As our review of units demonstrated if you divide the pressure gradient by density you have acceleration—and thus the pressure gradient enters the momentum equation (F=ma).
Momentum Equation:
The momentum equation for a fluid is an extension of Newton’s law F=ma. We tend to write the momentum equation as a=F/m. There are 4 forces that act on fluid in the ocean, gravity, friction, pressure gradients and Coriolis. We have talked about all of these forces in some detail expect friction. One of the complicating factors in fluid mechanics is that the fluid is a continuum—in contrast to describing the physics of a few particles (such as the planets orbiting the sun—which in it self is a complicated physical problem, but can be solved with great predictive skill). As a result the acceleration term has a local acceleration () and the field accelerations, sometimes called the advective term (such as the term ). Subsequently the momentum equation we’ve written to date looks like:
forces
1) Acceleration balanced by a pressure gradient.
2) Acceleration balanced by Coriolis
3) Pressure gradient balanced by advective term (Bernoulli flow)
4) Barotropic pressure gradient balances Coriolis (Geostrophic)
5) Baroclinic pressure gradient balances Coriolis (Geostrophic and obtain Thermal wind equation)
We introduced the Coriolis effect and provided some explanation of why it occurs and how it varies with latitude. The Coriolis frequency equals 2p/T, where T is the local inertial period. The local inertial period (T) varies with latitude and equals ½*Ts/sin(lat), where Ts is the number of seconds in a Sidereal day. It equals 11.96 hours at the pole, 18.8 hours in New Jersey and infinity at the equator. The Coriolis frequency then is maximum at the pole and zero on the equator.
Geostrophic Flow
A second example is in the atmosphere where we have high pressures and low pressure systems and therefore pressure gradients. In large scale weather systems (100’s-1000’s of km) most of the momentum balance is between the Coriolis force and the pressure gradient (figure 3). In this case winds flow along isobars rather than across them. This type of momentum balance can be written as:
and is called geostrophic flow. It is the first order momentum balance that occurs in much of the atmosphere and the ocean. It is why winds flow clockwise around a high (in the northern hemisphere) and counter-clockwise around a low.
Note that the Coriolis acceleration can balance a barotropic pressure gradient and a baroclinic pressure gradient.
In both the atmosphere and the ocean the first order balance is what we call “Geostrophic” in which the pressure gradient is balanced by the Corolis acceleration. In this balance the flow is steady—and thus if you know the pressure gradient you know the velocity. This has been useful to Oceanographers who can measure the density field in the ocean and calculate the pressure gradient and thus estimate the geostrophic velocity of the ocean. However, this requires an assumption that the pressure gradient goes to zero at some depth—otherwise all you are doing is estimating the flow relative to the speed at a lower depth. This can happen if isopycnals tilt in the opposite direction of the barotropic pressure gradient and produce a lower layer with no pressure gradient and thus no geostrophic velocities occurs this level. As such we call this the level “the level of no motion”. Since we have velocity above the level—and no velocity below—we have vertical shear. Note that this requires a tilting of the isopycnals and this corresponds to a horizontal density gradient.
Thermal Wind Equation
Writing the momentum balance once again for geostrophic flow:
(5 & 6)
Multiplying 5 by r and taking the vertical gradient we get
(7)
If we change the order if differentiation we get
(8)
Now recall the equation for hydrostatic pressure:
(9)
So vertical gradient in Pressure is simply:
(10)
By using equation 10 in 8 we get
(11)
In the ocean the change in density in the vertical is small with respect to changes in density (this is consistent with the Boussinesq approximation) and since f does not change with height 11 can be rewritten as:
(12)
Equations 11 and 12 are known as the thermal wind equations and demonstrate that in a rotating system a horizontal density gradient generates a vertical shear in the flow.
As a simple example imagine if we were able to run our gravitational adjustment problem is a large rotating tank. The surface slope—from the light fluid to the dense fluid would drive the fluid to the right—this can be determined from a geostropic flow for a tilting sea-surface
However, below this level there is a horizontal density gradient and in the case of this set up it produces a total pressure gradient at depth that is equal to the surface pressure gradient- but in the opposite direction. In the non-rotating case this produced a current in the upper layer that flowed in one direction and an equal and opposite current in the lower layer. However, if we were able to rotate the tank the upper layer would flow into the page and the lower layer flow out of the page (Figure 4) Consequently this drives a flow in the opposite direction producing what we call “Thermal Wind Shear” for it’s initial use was to determine atmospheric winds based on temperature measurements.
Good Luck on the Exam!