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
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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,
(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