ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 8

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

Ekman number

Learning objectives: UNDERSTAND where are the equations from

Previousclass:

Ryz

x

Substituting these to F=ma,Non-rotating coordinatesystem

We obtained the equation of motion (vector form):

Using:

We obtain the eq. of motion for x, y, z components:

ij k𝜑

𝜃

𝜃

Math: cross product of two vectors:

Dot products: 𝑨 " 𝑩 = 𝑨𝒙 𝑩𝒙+ 𝑨𝒚 𝑩𝒚+ 𝑨𝒛 𝑩𝒛

𝐴=(𝐴! 𝚤 + 𝐴" 𝚥 +𝐴# 𝑘), 𝐵=(𝐵! 𝚤 + 𝐵" 𝚥 +𝐵# 𝑘);

Writing change from Lagrangian form to Eulerianform:

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

(iPad demo)

is Coriolis parameter.

Equations of motion for x, y, z components:

Frictional effects: Reynolds work, eddies exert stressesand strains on ocean circulation: (iPad demo)

1. Continuity of volume: mass conservation

y

z

x

€

u,ρ

Rate of mass into the box (kg/s):Rate of mass out of the box (kg/s):

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.

Similarly, we can obtain the net mass changein y and z directions. Thus, total mass that flowsinto the box is the sum of the three components:

The net mass in or out must equal the mass changein the fluid parcel, which is:

Rewrite:

This is one form of continuity equation.Considering,

We obtain:

Or,

If the fluid is incompressible: then,

So,

This is continuity eqn for incompressible fluid.

2. Scale analysisMotivation: understand the underlying physics by

finding important processes that govern the motion.

Scales: based on observations.INTERIOR: away from surface & side boundaries

Watch the movie: pay particular attention to the zonal and meridional extent (scales) of currents in ocean interior away from side boundaries

https://www.youtube.com/watch?v=CCmTY0PKGDs

ScalingMotivation: 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

Laminar flow occurs at small Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; turbulent flow occurs at large Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.