SO335 Quantitative Methods: Assignment #1 (40 Pts)
Homework Assignment #1 Page 3
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) Name two differences between and u.
c. (4 pts) What coordinate axes will we use in oceanography, and what do they mean physically? How do the coordinate axes relate to the , , and unit vector directions? How does the z-axis relate to depth in the ocean? How does the z-axis relate to height in the atmosphere?
2. (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 , where the symbol is known as “grad” and “del”. The expression 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 , where is the density of the fluid parcel. What units must “” have to make an acceleration? (Show all work). Knowing its units will be helpful, as you will see the Greek symbol many, many times this semester.
3. Units and meridional velocity component, v.
a. (5 pts) Advection (where fluid moves a quantity of something to a different place) can take many forms. You have already heard of temperature advection in SO254 last year, and you can imagine moisture advection, salinity advection, and advection of other “scalar” quantities. The result of advection should be the change in that quantity over time. What, then, should be the units of meridional temperature advection, ? (Show your work)
b. (5 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: , where is the velocity vector with units m s-1.
i. Expand the determinant to express in components (show your work)
ii. What are the units of vorticity? (Show your work)
5. (10 pts) The following is a nonlinear form of the equation of motion of a fluid parcel:
(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 u. 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!)
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?