ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 8

1.  The continuity of volume; 2.  Scale analysis (or scaling) 3.  Physical meaning and usefulness of the Rossby number and Ekman number

Learning objectives: UNDERSTAND where are they from

Previous class:

R

y z

x Substituting these to F=ma,

We obtain the equation of motion (vector form):

Using:

We obtain the eq. of motion for x, y, z component: Board demo cross product and dot product

i j k

Writing change from Lagrangian form to Eulerian form:

property change local zonal meriodinal vertical of water parcel change advection advection advection Advection: nonlinear terms, inertial terms

is Coriolis parameter.

Equations of motion for x, y, z components:

Frictional effects: Reynolds work, eddies exert stresses and strains on ocean circulation: (board demo)

1. Continuity of volume: mass conservation

y

z

x

€

u,ρ

In x-direction, rate of mass that flows into the box:

(kg/s)

The mass flows out of the box:

To the lowest order, net mass that flows into the box: Mass in - mass out.

Note: the second order term is neglected. (board demo)

Similarly, we can obtain the net mass change in y and z directions. Thus, total mass that flows into the box is the sum of the three:

The net mass in or out must equal the mass change in the fluid element, which is:

Rewrite:

This is one form of continuity equation. Considering,

We obtain:

Or,

If the fluid is uncompressible: then, So,

This is continuity eqn for uncompressible fluid.

2. Scaling Motivation: understand the underlying physics by finding important processes.

Scales: based on observations.

for ocean interior

INTERIOR: away from surface & side boundaries

=>

(10~100 days)

(upper bound)

/s

Scaling vertical equations of motion.

Pressure gradient force balances gravitational force

This is: hydrostatic equation. To the lowest order, large-scale interior Ocean obey hydrostatic balance.

Scaling zonal momentum equaiton

If ignore mixing we call it “inviscid”.

Rewrite:

Similarly, from the equation for v:

These are: geostrophic balance

To the lowest order, large-scale, interior ocean, horizontal equations of motion obey: Geostrophy! This builds the scientific justification for using “geostrophic method” to infer current

3. Rossby number, Ekman number, Reynolds Number

Large-scale, interior ocean:

AdvectionCoriolis

=inertialCoriolis

Large-scale, interior ocean:

Rex =inertialviscous

=ULAx