September 16, 2002
Reading Ocean Circulation 230 (sections 3.3.2 & 3.3.2 & 6.4)
Pond & Pickard (Chapter 2 and sections 8.6 & 8.7)
Begin with a more correct derivation of the continuity equation to calculate sea-level.
(1)
Consider the example in figure 1 of a flat bottom ocean whose undisturbed depth is H. Changes in depth while h is variations in the sea level around H. So the actual depth of the water is H+h.
Integrating 1 with z we obtain.
w(h)-w(0)=- (2)
In equation 2 we assume that w(0) (the vertical velocity at the bottom) is equal to zero and by definition the vertical velocity of the sea-surface is . Note that equation 2 is non-linear in that sea-level is a product of both current velocity gradients and sea level. Consider the case of tidal motion where tidal currents (and their horizontal gradients) and sea level both vary sinusoidally in time (i.e. sin( wt)) has a component that varies with (sin( wt)2) which equal 1+sin(2wt) (figure 2). This is how a non-linear system can generate other frequencies while linear systems cannot. If we are in an environment where H >> h we can linearize equation 2 to read
(3)
However, if h becomes of order H, such as may happen in a shallow estuary where the tidal range is not small compared to the mean water depth this non-linearity cannon be ignored. In the Navesink River in Northern New Jersey (Figur3) the effect of this non-linearity is clearly evident in the distortion of sea-level that rises more that it falls. Estuaries with quick rising tides are known as flood dominated systems and this asymmetry in the tide traps sediment and larvae in the estuary. A similar steepening of wave motion though this non-linearity also occurs as a wave approaches the beach and the leading edge of the wave steepens (Figure 4).
Back to the Momentum Equation
Now that we’ve completed derivations of the advection/diffusion equations and the continuity equation we are going to return to the momentum equation. Recall that at this point we’ve written:
(4)
(note that I’ve written this as the total derivative of u, which contains both the local time rate of change of and the advective terms such as ) where r is the density of sea water. In the ocean there are 4 only four kinds of forces that acting on its fluid and they are gravity, pressure gradients, friction and Coriolis. Thus we can write equation four quite simply in words to describe the equation of motion for fluid in the ocean.
Density of water x particle acceleration = gravity + pressure gradient + Coriolis + friction.
Pretty simply eh? Can we all go home now? Of course not! As we’ve already seen even the particle acceleration is complicated because of the advective terms, and we still need to describe in detail the nature of these four forces and express them mathematically so that we can develop simple (and complicate) mathematical models of ocean circulation to provide an intuitive understanding of the ocean’s circulation that is grounded in physics.
Since we all have an intuitive understanding of gravity I’ll skip over that for now. Obviously gravity effects the momentum equation for vertical motion (i.e. ) —this is why things like apples fall to the ground. But as we will see gravity also works it way into the pressure gradient term for horizontal motion as well. However, before we discuss pressure gradients we need to discuss the density of sea-water for it is the overlying weight of the fluid that produces pressure in the ocean.
The Density of Sea water.
The density of sea water is determined by its pressure, salinity and temperature so we write r(p,s,t). The density of water in the ocean actually varies only very little. Most of the fluid in the ocean weighs between 1020 and 1030 kg/m3.And 50% of the ocean’s water is between 1027.7 and 1027.9 kg/m3. Yet these small differences in density significantly impact the circulation and mixing of the world’s oceans. Figure 5 depicts the range that temperature and salinity vary in the worlds oceans on a T-S diagram.
In oceanography density is often written out in sigma (s) units whereby:
s=r-1000
Thus for r=1025.5 s=25.5
Furthermore, the density is defined three ways.
s=r(s,t,p)-1000
st=r(s,t,0)-1000
sq=r(s,q,0)-1000
s is simply the in situ density. Sigma-t is the density that the same particle of water would have if it were brought to the surface, while Sigma-theta is the density that the fluid would have if it were brought to the surface adiabatically (meaning that we include the change in temperature that would occur due to changes in pressure felt by the fluid s it moves to the surface). In the oceanographic literature density is usually defined by st or sq for they characterize what the density of two water masses would be if they were at the same level. In coastal and estuarine systems sigma-t is used. However in the deep ocean sigma-theta is often.
In the ocean water masses obtain their t,s properties at the surface and tend to retain these properties for centuries as they are subducted into the ocean’s interior and away from any sources or sinks of heat. Later we will discuss the formations and transport these various water masses that are apparent on a T-S diagram (Figure 6)-- but for now we want to move onto the momentum equation and a discussion of pressure.
Pressure
We all know pressure from filling up tires. In the US tire pressure is lbs/in2 or force/area. In the ocean Pressure is in pascals (Pa) after the Parisian Blasé Pascal (1623-1662) who among other things experimented with pressure. One pascal = 1 N/m2. One atmosphere is approximately 105 Pa. You can feel pressure by putting a weight on your hand. Two weights of equal mass, but of different surface area’s apply different pressure. While your hand can handle the weigh of a 10 kg brick because it’s load would be distributed over your entire hand— but that same weight would cut through your hand if it were applied over a thin area, such as by a knife.
In the ocean it is the overlying fluid that provides the mass. Thus the deeper you go the more the pressure. 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 pressure is simply:
P(z)=rgz. (5)
Or more generally this can be written as
(6)
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 at the surface would be less than 2mm in radius at the bottom of the ocean.
Taking the vertical gradient of 5 we get:
(7)
However, recall that earlier we wrote a different equation whereby
(8)
How can two different things = g? Are they the same? No! Equation 7 is for a body in free fall, such as someone jumping out of an airplane before reaching terminal velocity. They are accelerating and fell no pressure! Equation 6 is what we call the hydrostatic pressure and assumes that vertical accelerations are small relative to gravity. A more complete equation could be written as
In the case of the skydiver eventually friction = gravity. In the Ocean under most cases it is the vertical pressure gradient that balances gravity—and thus we make the hydrostatic assumption.
Similarly we can write an equation for the horizontal momentum balance
Note that gravity is not used here, because gravity only acts in the vertical—but as we’ll see that gravity works its way into the pressure gradient for the u and v momentum equation.
We talk of two types of pressure gradients in the ocean—a barotropic pressure gradient and a baroclinic pressure gradient (section 3.3.2 in Ocean Circulation).
Barotropic Pressure Gradient
The barotropic pressure gradient is generated by a sloping sea-surface and the pressure gradient is depth independent. For a fluid that is homogenous (i.e. the fluid’s density is constant everywhere) pressure gradients will only be barotropic. Pressure gradients can also be purely barotropic if the lines of constant pressure (isobars) are parallel to lines of constant density (see figure 3.11 a in Ocean Circulation). An example of a sloping sea level driving a barotropic pressure gradient is depicted in figure 7. The pressure gradient between P(1,1) and P(2,1) occurs because there is more water above P(1,1) than P(2,1). At P(1,2) the pressure is equal to the pressure at P(1,1) plus the weigh of the fluid between P(1,1) and P(1,2). Likewise at P(2,2) the pressure is equal to the pressure at P(2,1) plus the weight of the fluid between P(2,1) and P(2,2). Since the weight of the fluid between P(1,1) and P(1,2) is the same as the weight of the fluid between P(2,1) and P(2,2) the difference between pressure at this location is only due to the pressure difference at P(1,1) and P(2,1)—thus the pressure gradient is the same. As we move further down the the water column the same applies—pressure increments moving downward are the same at P(1,z) and P(2,z) and therefore the pressure gradient remains constant.
Consider the case where the sea surface at P(1,1) were 1 cm higher than at P(2,1) apart the pressure at P(1,1) would be equal to P(2,1) + rgDh=1000*9.8*.01~100 Pa. If these two stations were separated by 10 km the pressure gradient would be 100/1000=0.1 N/m3. The same pressure gradient would be felt between P(1,2) and P(2,2) between P(1,3)and P(2,3) and between P(1,4) and P(2,4). Neglecting other forces this pressure gradient would accelerate the fluid to the right at rate
=10-4 m/s2.
which by integrating with time we can find an exact solution
u(t)=u(t0)+10-4 t
Of course in the ocean other forces would come to play the fluid velocity would not follow this simple balance. One important point, however, is that the fluid accelerates at all depths at the same rate. Subsequently a barotropic pressure gradient will not generate vertical shear in the flow—but rather a depth averaged flow. In contrast, as we will see, a a baroclinic pressure gradient drives vertical shear and by decomposing the pressure gradient into a barotropic component and a baroclinic component we can come up with a baroclinic pressure gradient that drives vertical shears—but no depth averaged flow.
Baroclinic Pressure Gradient
Consider a channel filled with fluid on the left side with fluid of density r1 and fluid of density r2 on the right side and that r2 = r1+dr> r1 (Figure 8). On the left size the pressure at depth z is P1(z)= gr1z, while on the right hand side the pressure at depth z is P2(z)= gr2z=g(r1+dr)z. The pressure difference between the two fluid is P2(z)- P2(z)=g drz. Therefore the pressure gradient increases with z. The pressure gradient is zero at the surface and equals gdrH at the bottom. Considering a momentum equation between the local acceleration and the pressure gradient we write
Thus the fluid accelerates more rapidly on the bottom than on the at the surface and as a consequence a vertical shear develops in the water column. Note that the pressure gradient is directed to the left throughout the water column. This would produce a depth averaged acceleration of fluid to the left, which in turn would cause sea level on the left to stand higher than on the right. If the channel is closed on both ends sea level rises high enough to cancel out the depth averaged pressure gradient so that the depth averaged flow is zero. This occurs because when depth averaged flow is not zero sea level continues to rise on the left and the resulting sea level slope eventually produces a pressure gradient that exactly equals the depth averaged baroclinic pressure gradient. We’ll calculate this in the next class. But what happens is that a 2 –layer flow develops that drives fluid to the right in the surface layer and to the left in the lower layer. This 2-layer exchange flow is what drives estuarine circulation (Figure 9).