Homework #8

Name______Due on Halloween!

Woo HA HA HA!

1) Figure 1 shows a 10 second time series of along-channel current speed in an estuary at two locations (call the locations red and blue). Assume that the along channel salinity gradient is the same at both locations.

A) What is the approximate mean current speed at each location?

Red 10 cm/s

Blue 0 cm/s

B) What is the approximate velocity scale of the turbulent fluctuations at each location.

Red < 1 cm/s

Blue ~ 10 cm/s

C) Which location would have the larger along channel advection of salt? Why?

Red—because the mean current speed is larger so is larger

D) Which location is likely to have the larger turbulent transport of salt? Why?

Blue-- because the turbulent fluctuations are larger.

2) A steady wind blows at 10 m/s to the north. The wind stress can be parameterized as Cd * U2 where Cd = .0015

A) Write down the momentum equation assuming that the momentum balance is between the wind friction and the Coriolis force.

The momentum equation can be written vertically averaged over the Ekman layer

Where De is the Ekman Layer thickness

Or we can write the depth dependent equation:

B) If this wind forcing occurred over a 100 km square box where f= 10-4 s-1 what would be the transport of water through each side of the box?

Flow is to the east. Flow vertically averaged over the Ekman layer on the north and south side of the box is zero. Flow through the east and west side of the box is:

=1.5*105 m3/s

C) Now consider the fact that the Coriolis parameter varies with latitude. How will the transport and Ekman depth vary with latitude? Is the flow divergent or non-divergent?

Both the Ekman transport and Ekman Depth will increase with latitude. However, since the volume transport that enters the box on the west side is exactly equal to the volume transport that leaves on the east side—the flow is non-divergent.

A second way to think about this would be mathematically (of course!) the divergence of the horizontal flow is:

where u and v are the vertically averaged flow in the Ekman layer.

Since v is zero the second term is zero.

The first term is also zero because

and while f and De vary with y—they do not vary in x- so and thus the flow is non-divergence

D) If the wind were from the west and we considered the fact that f changes with latitude would the Ekman flows be divergent or non-divergent? Would the flows lead to a change in the thickness of the mixed layer? Would it get thicker, thinner or stay the same?

Here the flow will be convergent, because the Ekman tranport, which is now to the south, increases with decreasing latitude. Therefore the transport leaving the south side of the box is greater than the transport entering the north side of the box and the flow is divergent. The divergent surface flows produce an upwelling and this tends to decrease the thickness of the mixed layer.

3)Figure 2 shows a sea-level that tilts downward to the left 1 cm over a 10 km distance. The water column is 100 meter deep it is well mixed but has a mean horizontal density gradient.

A) If the baroclinic pressure gradient is in the opposite direction of the barotropic pressure gradient should or 

Because for the water to slump to the right, which is in the opposite direction of the barotropic pressure gradient, heavier water needs to be on the left

B) What is the barotropic pressure gradient?

(if this is written without the it is the acceleration—which we’ll also accept as a correct answer)

C) Write an expression for the baroclinic pressure gradient in terms of the horizontal density gradient, (hint—since the baroclinic pressure gradient is depth dependent the expression needs to include z in it)

Recall that the pressure can be written as

P=gz

Therefore

DIf the density increases by 0.2 kg/m3 over 10 kilometer, what is the total horizontal pressure gradient as a function of z in the x direction due to both barotropic and baroclinc forcing

So the pressure gradient force (recall that the pressure gradient force is the negative of the pressure gradient) is 10-3 times gravity and directed to the left at the surface, and the same magnitude – but in the opposite direction at the bottom.

Since the pressure gradient is linear with z—the vertical structure of the pressure gradient will look like the figure to the left

E) Estimate the speed and direction of the flow as a function of depth if the flow were geostrophic. Draw it graphically.

For geostropic flow the current speed is simply the pressure gradient divided by density times the Coriolis parameter or.

So the vertical structure of the flow is identical to the pressure gradient and its magnitude is (1/f)=10 times larger. Thus currents speeds are g/100 m/s to the north at the surface (9.8 cm/s) and g/100 m/s to the south at depth.

F) How would the inclusion of bottom friction impact the direction of the near-bottom flows?

The inclusion of friction will cause flow in the bottom layer to flow towards the lower pressure. This occurs because friction reduces the velocity and thus the Coriolis term (which is equal to f time the velocity) is reduced. With the Coriolis term reduced it no longer balances the pressure gradient and the flow begins to accelerate down the pressure gradient until it is balanced by friction. In the bottom layer, because the pressure gradient there is to the right, friction will cause the flow to go the right.