Name:____________________________
SO335 Quantitative Methods: Assignment #1 (40 pts)
1. Variables used often in SO335. a. (3 pts) Identify, with words, and give the units for the following variables that we will see hundreds of times this semester. u v w b. (3 pts) What are some differences between u and u ?
c. (3 pts) What coordinate axes will we use in oceanography, and what do they mean physically?
How do the coordinate axes relate to the ˆ , j
ˆ
, and ˆ unit vector directions? How does the z -axis relate to depth in the ocean?

Homework Assignment #1 Page 2
2. Units and meridional velocity component, v . a. (4 pts) Advection (where fluid moves a quantity of something to a different place) can take many forms. You have probably already heard of temperature advection, and you can imagine moisture advection, salinity advection, and advection of other “scalar” quantities. But fluid can also advect its momentum. The result of advection should be the change in that quantity over

T time. What, then, should be the units of meridional temperature advection, v
 y
? b. (4 pts): Vorticity is a measure of spin in a fluid. As a vector field, vorticity (represented by the
Greek letter omega,

) is defined as the curl of the velocity field:
   u , where u is the velocity vector with units m s
-1
. The ˆ k
 
, is

 v x


 u y
, and this quantity is called “vertical vorticity.” In the northern hemisphere, positive vorticity is found in low pressure centers and cyclonic gyres, and negative vorticity is found in high pressure centers and anticyclonic gyres. Find the units of vertical vorticity.

Homework Assignment #1 Page 3
3. (10 pts) Fluid accelerations owing to pressure gradients are the most important in all of the atmosphere and ocean. Mathematically, pressure gradients are expressed as
 p , where the
 symbol is known as “grad” and “del”. The expression  p is then called “grad p” or “del p”. a. (3 pts) What are the units of pressure, p ? (Express in kilograms, meters, and seconds) b. (7 pts) The acceleration a fluid parcel “feels” from the pressure gradient can be written as acceleration
 p , where

is the density of the fluid parcel. What units must “
 ” have to make
1
 p an acceleration? (Show all work). Knowing its units will be helpful, as you will see the Greek symbol

many, many times this semester.
5. (10 pts) The following is a nonlinear form of the equation of motion of a fluid parcel:
 v
 t
 u
 v
 x
 v
 v
 y
 w
 v
 z

Forcing terms (1)
If you wanted to know how fast a blob of ocean water (or a parcel of air) was moving in the north-south direction, you could solve (integrate) this equation for v . We will learn much more about this equation later in the course.
For now, it is important to know that in order to add the four left-hand side terms together, their units must be the same. (That principle will be very helpful for you in this course, so remember that in order to add anything together, the units must be the same!)

Homework Assignment #1 Page 4 a. (7 pts) Prove that the four terms on the left-hand side of equation (1) have the same units. b. (3 pts) What are the units of the right-hand side Forcing Terms?