Wave radiation
A short course on:
Modeling IO processes and phenomena
INCOIS
Hyderabad, India
November 16−27, 2015
References
1) HIGnotes.pdf: beginnings of Sections 3−5.
McCreary, J.P., 1980: Modeling wind-driven ocean circulation.
JIMAR 80-0029, HIG 80-3, Univ. of Hawaii, Honolulu, 64 pp.
2) KelvinWaves.pdf: A write-up of the Kelvin-wave solution.
Mode equations
Let q be u, v, or p of the LCS model. To focus on free waves, neglect
forcing, friction and damping terms. Then, equations of motion for
the 2-d qn(x,y,t) fields are
Solutions
to these with
equations
describe how
Waves
associated
a superposition
of waves
verticalassociated
modes with
a single vertical mode propagate horizontally.
propagate both horizontally and vertically.
vn equation
Solving the unforced, inviscid equations for a single equation in
vn, and for convenience dropping subscripts n gives
(1)
Okay.
This equation
so important
maybe
should
Problem
#1: Solveis the
equations ofthat
motion
to we
obtain
(1).derive
it in class!
Derivation of vn equation
(−1)
Derivation of vn equation
(−1/cn2)
Derivation of vn equation
vn equation
Solving the unforced, inviscid equations for a single equation in
vn, and for convenience dropping subscripts n gives
(1)
Solutions to (1) are difficult to find analytically because f is a
function of y and the equation includes y derivatives (the term vyyt).
There are, however, useful analytic solutions to approximate
versions of (1).
Dispersion relation of free waves
The simplest approximation (mid-latitude β-plane approximation)
simply “pretends” that f and β are both constant. Then, solutions
have the form of plane waves,
Then, we can set ∂t = −iσ, ∂x = ik, and ∂y = iℓ in (1), resulting in the
dispersion relation,
The dispersion relation provides a “biography” for a model. It
describes everything about the waves it supports.
Gravity waves with f = 0
The simplest dispersion relation
has f = 0, in which case the
waves are non-dispersive
gravity waves.
For convenience, the plot shows
curves for ℓ = 0.
When ℓ ≠ 0, the curves define a
surface. At each σ, the disp. rel.
gives a circle of radius r = σ/c,
so the surface is a circular cone.
The phase speed of the waves
is σ/k = ±c. The property that
dispersion curves are linear
(straight lines) indicates that the
waves are non-dispersive.
σ/f
1-
α = f/c = R−1
−1
k/α
1
Gravity waves with constant f
When f ≠ 0 and is constant, the
possible waves are dispersive,
gravity waves. There are no
waves with frequencies < f.
For convenience, the plot shows
curves when ℓ = 0.
When ℓ ≠ 0, the curves define a
surface. At each σ, the disp. rel.
is a circle with r = (σ2−f2)½/c
and its center at k = ℓ = 0. So,
the surface is a circular bowl.
The phase speed, σ/k, is no
longer linear, indicating that the
waves are dispersive.
σ/f
f=0
1-
−1
−1
k/α
1
Gravity waves with variable f (β ≠ 0)
When f ≠ 0 and β ≠ 0, the waves
are still dispersive, gravity
waves, but the curves are
modified by the β term.
For convenience, the plot shows
σ/f
curves for ℓ = 0.
When ℓ ≠ 0, the disp. rel. still
defines a circle for each σ with
its center at k = −β/(2σ), ℓ = 0
and its radius modified from
(σ2−f2)½/c. So, the surface is
still a circular bowl.
1
-
−1
k/α
1
Rossby waves
When σ is small, the σ2/c2 term
is small relative to f2/c2, giving
the disp. rel. for RWs.
Rossby exist only for negative
k, and so propagate westward.
σ/f
When
≠ 0, the
disp. rel. still
Freq. σℓattains
a maximum
1defines
a circle
σ when
with
value when
r →for
0, each
that is,
its
at k ==−β/(2σ),
=0
σ =center
½(c/f)(β/f)
½R/Re. ℓSo,
2/(4σ2) − bowl.
and
a radiusisr an
= βinverted
f2/c2.
the surface
Typically, R/Re « 1, so that the
RW and GW bands are well
separated.
R/2Re
−1
k/α
1
Kelvin waves
To derive the dispersion relation
for GWs and RWs, we solved
for a single equation in v. So,
we missed a wave with v = 0,
the coastal Kelvin wave.
The dispersion curves shown in
σ/f
the figure and equation are for
Problem
Okay. The
#2: Solve
solution
theisequations
easy, of
Kelvin waves along zonal
1motion
insightful,
to obtain
and important,
the Kelvin-wave
so
boundaries. KWs also exist
solutions.
maybe we should derive it in class!
along meridional boundaries.
The coastal KW propagates
along coasts at speed c with the
coast to its right, and decays
offshore with the decay scale
c/f = R, the Rossby radius of
deformation.
−1
k/α
1
Derivation of KW solution
(−c2)
Derivation of KW solution
(−1)
Derivation of KW solution
Look for solutions proportional to
exp(ikx –iσt). Set ∂t = −iσ and ∂x = ik.
Phase and group speed
The figure shows the wave types
that we have discussed.
The phase speed of a wave with
wavenumber k and frequency σ
is the slope of the line that
extends from (0,0) to (σ,k).
σ/f
1
-
The group speed of a wave with
wavenumber k and frequency σ
is the slope of the line parallel
to the dispersion curve at the
point (σ,k).
R/2Re
Movies A1, A3, A2
−1
k/α
1