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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