ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 12

1. Waves in an ocean with rotation, f≠0=constant;2. Waves in an ocean with a varying f: Rossby waves;

Learning objectives: should understand the effects of f on gravity waves and equilibrium state, Rossby waves, their properties and causes

1. Previous Class: Effects of rotation ( ) and Rossby radius of deformation

With a uniform rotation (f is assumed to be a constant), the equations of motion for the unforced, inviscid ocean are:

Assumptions:(i) constant;(ii)(iii)

(iv) at z=H;(v) Background state:

bottom

(Ro<<1, E<<1)

For small perturbations u,v,w,p about the resting state, we have:

The linearized first order equations for perturbation:

Apply boundary conditions

Z=0,

Z=H,and vertically integrate the perturbation equations:

H

z

Following the same procedure as in the non-rotatingcase, write a single equation in alone,

Where Relativevorticity

η

Note: is referred to as absolute vorticity;

Planetaryvorticity

Relative vorticity

a) Non-rotating case (f=0):

Assume (1-dimensional exp) Recall this is the dispersion relation for long-surfacegravity waves when f=0!

Dispersion curve

ω

κ

Today: f=0, adjustment

z

x0

x0

x0

a) Initial perturbation

b) Gravity waves

c) Equilibrium state

(state of rest!)

b) Rotating case ( )Assume wave form of solution

Substitute into the vertically-integrated perturbation equation for ,and let coefficient matrix=0,

where

Math:

€

a11 a12 a13a21 a22 a23a31 a32 a33

=

a11a22a33+ a12a23a31+ a13a21a32−a11a23a32 − a12a21a33 − a13a22a31

f=0 case

What does f do to the system?

(i) Long gravity waves become “dispersive”; (ii)Long gravity waves do not have “zero” frequency

anymore. Their lowest frequency is “f”, which hasa period of a few days in mid latitude.

κ

ω

More accurate plot

Sketch

Inertial gravity waves (IGW)

f

Adjustment with f (1-dimensional exp)

Geostrophic balance

xt=0

xt=tn

z

z

Equilibrium state

0

Solutions:

is Rossby radius of deformation.where

Previous class: f=constant, adjustment

0

0

z

0

z

z

a) Initial perturbation

b) Gravity wave radiation

c) Equilibrium state

Geostrophic balance!

2. Waves in an ocean with a varying f: Rossby waves

In real ocean, varies with latitude

zy

Mid-latitude approximation.

plane

β =∂f∂y

=2Ωcosφ

R

Planetary vorticity

Relative vorticity (vertical component)

Absolute vorticity

Potential vorticity

where H is the depth of water column.

Concepts:

Equations of motionShallow water equation (as for the f=contant case),except that here,

(i)(ii)

(v) Background state:(Ro<<1, E<<1) (iii)

Exercise: Write a single equation in v alone & assume periodic waves, we obtain:

where

Dispersion relation:

To simplify the case, let look at 1-dimensional situation. We have

and c = ± gH

(i) For high frequency waves with large , the dispersion relation can be approximated by:

or, €

ω

€

ω

This is long gravity wave under the influence of f we obtained in the previous class. It is also

called inertial gravity wave.

They are not influenced much by beta-variation of f!

f

0

(ii) For low frequency waves with small

These waves do not exist in f=constant case; their existence is due to the introduction of They are called Rossby (or planetary) waves.

or, dispersion relation:

To further understand the wave property, we simplify the above equation.

Since is small for Rossby waves because

of their low frequency,

Choose ‘-’ sign:

or,

Since is independent of frequency

and wavenumber, they are non-dispersive. They are long Rossby waves and propagate westward; speed decreases as latitude increases.

Here, c is vertical mode speed: either a baroclinic orthe barotropic mode speed.

Choose the ‘+’ sign,

They are short Rossby waves. Group velocity propagates eastward but

phase propagates westward.They are dispersive.

Dispersion curves for free waves in mid-latitudeplane:

Long RossbyShort Rossby

Short Rossby waves are hardly seen in the ocean interior because (1) they are too short to be effectively excited by large-scale winds, (2) mixing in the ocean acts strongly on short and

slow waves.

Rossby waves in mid-latitude plane

(i) Existence of Rossby waves: , the variation of Planetary vorticity ( ) with latitude ( ),

(ii) They are low frequency waves (with periods of months to decades).

Oceanic adjustment

0

0

z

0

a) Initial perturbation

b) Gravity wave radiation

c) Geostrophy

d) Rossby wavesB

e) Rossby wavesC

f) Equilibrium stateQuiescent

If there is persistent wind forcing: the equilibrium state is “Sverdrup-balance”. Will be introduced later.

Annual period Rossby Wave in eastern tropical Pacfic

Kessler 1990. JGR

Comparison of annual cycle anomalies of observed 20 °C depth (left panels) and the Rossby wave model solution (Section 4.2.1) (right panels), for four average seasons (indicated to the left of each row). The common color key is at right, with contour interval of 5 m. Positive values (red) indicate deep anomalies and negative values (blue) indicate shallow anomalies.

Observed mid-latitude Rossby waves by TOPEX satellite altimetry

Mid-latitude PacificAs you can see, they propagate westward,with a decreasing speed

with increasing latitude

https://oceanservice.noaa.gov/facts/wavesinocean.html

http://www.oceanographerschoice.com/2011/06/rossby-waves/