Wind-forced solutions:
Coastal ocean
A short course on:
Modeling IO processes and phenomena
INCOIS
Hyderabad, India
November 16−27, 2015
References
1) HIGNotes.pdf: Section 5, pages 53−60.
McCreary, J.P., 1980: Modeling wind-driven ocean circulation.
JIMAR 80-0029, HIG 80-3, Univ. of Hawaii, Honolulu, 64 pp.
2) CoastNotes.pdf
3) Shankar_notes/coastal_ocean.pdf
Coastal phenomena: steady currents
Mittelstadt et al. (1975), Huyer (1976)
The coastal circulation along Northwest Africa at 21° 40′N during
February−April, 1974. The prevailing winds are southward. The
coastal response includes a surface flow in the direction of the wind,
a poleward undercurrent, and coastal upwelling.
Coastal phenomena: steady currents
Nearly all eastern-boundary
coastal currents have a similar
structure, with upward sloping,
near-surface isopycnals,
equatorward surface flow in
the direction of prevailing
equatorward winds, and a
poleward undercurrent.
The Leeuwin Current (and
coastal current off Portugal) is
remarkable because it goes the
“wrong way,” flowing against
the prevailing equatorward
winds.
Godfrey & Ridgway (1985)
Coastal phenomena: steady currents
32ºS
Feng et al. (2003, 2008)
ORCA025
Leeuwin Current solution w/wo shelf/slope
Much of the source water for the Leeuwin Current is eastward flow
across the south Indian Ocean. It bends to the south along the
shelf/slope to form the southward Leeuwin Current.
It is driven by a southward increase in density. The density gradient
lowers sea level to the south, thereby generating a shallow, eastward,
geostrophic current. This current is strong enough to overwhelm the
westward, Ekman drift.
Coastal phenomena: wave radiation
In the NIO, steady coastal currents like the above are not
apparent, because they are forced by the highly variable, monsoon
winds. Instead, the radiation of waves along coasts (Kelvin waves
and shelf waves) and into or offshore from coasts (Rossby waves)
is prominent.
Sea-level movies
Questions
What forcing mechanisms drive coastal currents?
alongshore wind stress τy; surface heat flux Q
What waves are generated at coasts?
Kelvin and Rossby waves; shelf waves
What are the key differences between 2-d and 3-d theories
of coastal circulation?
wave radiation; establishment of py to balance τy
Why do eastern-boundary currents exist at all?
vertical mixing; shelf trapping
How does the shelf/slope impact the dynamics of coastal
currents?
topographic β effect; shelf trapping
Introduction
1) Coastal-ocean equations
2) Solutions for switched-on winds
3) Solutions for periodic winds
4) Solutions with a shelf/slope
Coastal-ocean equations
Coastal-ocean equations
A useful set of equations for the coastal ocean is
A key
simplification
is to drop
the acceleration
and damping
terms
As for
the interior-ocean
equations,
this approximation
is useful
from
theitzonal
In addition,
since it is the
wellslowly
because
filtersmomentum
out gravityequation.
waves. Thus,
it only describes
known
theofcoastal
ocean responds
much
moreforced
strongly
to Rossbyvaryingthat
parts
the response,
that
is,
the
directly
and
x forcing. Finally, for simplicity neglect the
alongshore
winds
drop
τ
wave (if β ≠ 0) parts of the response.
horizontal mixing terms. In this way, the alongshore flow is in
geostrophic balance, a property consistent with observations.
Coastal-ocean equations
A useful set of equations for the coastal ocean is
A key
simplification
is to drop
the acceleration
and damping
terms
As for
the interior-ocean
equations,
this approximation
is useful
from
theitzonal
In addition,
since it is the
wellslowly
because
filtersmomentum
out gravityequation.
waves. Thus,
it only describes
known
theofcoastal
ocean responds
much
moreforced
strongly
to Rossbyvaryingthat
parts
the response,
that
is,
the
directly
and
x forcing. Finally, for simplicity neglect the
alongshore
winds
drop
τ
wave (if β ≠ 0) parts of the response.
horizontal mixing terms. In this way, the alongshore flow is in
geostrophic balance, a property consistent with observations.
Conditions of validity
Under what conditions is the approximate equation set valid? For
convenience, drop subscripts n, mixing and damping terms, and τx
forcing. Then, the v equation for the complete set of equations is
In contrast, the v equation for the approximate set lacks vyyt, vttt, and (τy)tt
terms.
So, the approximation is valid provided the second and third terms
are small compared to the fourth. That will be true provided that
and similarly that
Mathematical usefulness
Under what conditions is the approximate equation set valid? For
convenience, drop subscripts n, mixing and damping terms, and τx
forcing. Then, the v equation for the complete set of equations is
Mathematically, a great advantage of this approximate equation is that
it lacks the vyyt term. As a result, it is possible to find simple solutions
even when f varies with y.
Distortion of free waves
How does the approximate equation set distort the dispersion
relation? To focus on free waves, neglect forcing in the approximate v
equation to get
Assuming a sinusoidal wave form
gives the dispersion relation
Since the disp. rel. is nonlinear, the Rossby waves are dispersive.
Distortion of free waves
How does the coastal-ocean
approximation distort the disp.
curves?
It eliminates gravity waves and
the Rossby curve has the
correct shape for ℓ = 0 except σ
is independent of ℓ, so that the
RW curve is not a bowl in k-l
space, but a curved surface.
σ/f
1
-
When are
are also
the RWs
There
KWsaccurately
along basin
simulated in the
coastal
boundaries.
When
β ≠ model?
0, their
When ℓRand
<< 1,
and σ/f << 1.
easternwestern-boundary
KWs have a more complicated R/2R
e
structure (see below).
−1
k/α
1
Solutions for
switched-on winds
Forcing by a band of alongshore wind τy
All the solutions discussed in the rest of my talk are forced by a
band of alongshore winds of the form,
Since this wind field is x-independent, it has no curl. Therefore,
the response is entirely driven at the coast by onshore/offshore
Ekman drift. The time dependence is
either switched-on
or periodic
Y(y)
the latter case discussed
in the next section.
2-d response to switched-on τy
R
hm
we
Consider the 2-dimensional (x, h) coastal response of a 1½-layer
model when the wind is independent of y.
If the alongshore winds are directed southward, they force offshore
Ekman drift. Since there can be no flow through the coast, the
thermocline must rise to conserve mass. It rises until it intersects the
surface mixed layer, and then subsurface water entrains (upwells)
into surface layer.
The offshore decay scale of the circulation is the Rossby radius of
deformation, R. There is a geostrophic coastal current v in the
direction of the wind.
2-d response to switched-on τy
It is easy to solve the coastal equations for the initial rise of the
thermocline. At that time, the response is inviscid, and the coastal
equations written in terms of a 2-d, 1½-layer model are
Solving for a single equation in h gives
(1)
where R2 = g'H/f2 is the square of Rossby radius of deformation. The
forcing term vanishes because τy is independent of x.
2-d response to switched-on τy
The general solution to (1) is
The coast is at x = 0 and the ocean lies in the region x < 0, so we have
to drop the B term to ensure the solution is bounded as x → −∞.
To evaluate A, we impose the boundary cond. that u = 0 at x = 0.
Using the v-momentum equation to write u in terms of h gives
and then
2-d response to switched-on τy
The solution is then
For southward winds (τy < 0), h thins at the coast, and the coastal
response weakens exponentially offshore with width scale R.
There is a meridional geostrophic current associated with h,
a coastally trapped jet flowing in the direction of the wind.
How long does it take for h to thin to the surface at the coast?
For the parameter choices
the time is 29 days.
3-d response to switched-on τy (β = 0)
Two-dimensional coastal upwelling is altered dramatically when 3-d
processes are included. Specifically, the propagation of Kelvin waves
along the coast stops the rise of h.
In a 3-d model (x, y, h) with
β = 0, in addition to local
upwelling, coastal Kelvin
waves extend the response
north of the forcing region.
The pycnocline tilts in the
latitude band of the wind,
creating a pressure force that
balances τy and stops the
coastal jet from accelerating.
f-plane
f-plane
3-d response to switched-on τy
To see these properties, we solve the coastal equations keeping the vy
and hy terms. Then, the inviscid coastal equations written in terms of
a 1½-layer model are
Solving for a single equation in h gives
(1)
where R2 = g'H/f2 is the square of Rossby radius of deformation. The
forcing term vanishes because τy is independent of x.
3-d response to switched-on τy
The general solution to (1) is
(1)
The coast is at x = 0 and the ocean lies in the region x < 0, so we have
to drop the B term to ensure the solution is bounded as x → −∞.
To evaluate A, we impose the boundary cond. that u = 0 at x = 0.
Using the v-momentum equation to write u in terms of h gives
which, using (1), provides an equation for A,
3-d response to switched-on τy
We obtain the solution for A by splitting it into particular (steadystate) and homogeneous (Kelvin-wave) responses,
where Λ(x,y) is an as yet unspecified function.
To satisfy the initial condition that h = H at t = 0, we must choose
Λ(y) = −χ(y), so that
Initial adjustment
To determine the response a short time after the wind switches on,
we expand χ(y − ct) in a Taylor series about t = 0 to get
t+
Thus, at small times, the response is just the 2-d response!
The response does not change from the 2-d response until the Kelvin
waves have propagated across the wind band.
Final adjustment
At longer times the solution for all the fields is
Movies H1a and H1b.
A packet of Kelvin waves propagates poleward. Note that, consistent
with Kelvin waves, there is no u field associated with the packet.
After its passage, the solution adjusts to a steady-state balance.
Key properties of the steady solution are: 1) a pressure gradient that
balances the wind along the coast (x = 0), that is, py = g'hy = τy/H; 2) a
coastal jet with a transport HRv that supplies the Ekman transport
from the coast; and 3) Ekman drift that weakens to zero at the coast.
3-d response to switched-on τy (β ≠ 0)
When β ≠ 0, Rossby waves
carry the coastal response
offshore, leaving behind a state
of rest in which py balances τy
everywhere.
β-plane
The RW speed is
Movie H1c
So, RWs propagate faster
closer to the equator (cr ~ f−2).
A fundamental question, is:
Given offshore Rossby-wave
propagation, why do easternboundary currents exist at all?
Multi-baroclinic mode adjustment with damping
The
Howplot
does
shows
the LCS
the response
model adjust
of
the
when
n =many
1 mode
baroclinic
without modes
damping.
are
included?
It also illustrates the n > 1
responses, except that the
currents are narrower because the
Rossby-wave speed is smaller
since (cn < c1).
With damping, the responses of
the n > 1 modes are increasingly
damped since ν = A/cn2. In that
case, the Kelvin and Rossby waves
that radiate from the forcing
region are weakened for larger n.
For sufficiently large n, then, the
response is confined to the forcing
region.
β-plane
Multi-baroclinic mode adjustment with damping
McCreary (1981) obtained a
steady-state, coastal solution to
the LCS model with damping.
The model allows Rossby waves
to propagate offshore. A steady
coastal circulation remains,
however, they are damped by
vertical diffusion.
There is upwelling in the band of
wind forcing. There is a surface
current in the direction of the
wind, and a subsurface CUC.
3-d response to switched-on τy (OGCM)
In an OGCM solution forced
by switched-on, steady winds
(left panels), coastal Kelvin
waves radiate poleward and
Rossby waves radiate offshore,
leaving behind a steady-state
coastal circulation.
Movies I2 and I3
Philander and Yoon (1982)
Solutions for
periodic winds
Evanescent waves
The gravity and Rossby waves
we have discussed so far have
the sinusoidal form
that is, are “trigonometric.”
Because there is a coast more
waves are possible. Suppose the
coast is oriented north-south.
Then, “evanescent” waves that
decay offshore are also possible.
σ/f
1
-
R/2R
R/Ree
−1
k/α
1
Rossby and β-plane Kelvin waves
To see this property, solve the coastal dispersion relation
for the zonal wavenumber k.
Note that the roots are either real or complex depending on the size of
the last term under the radical, which defines a critical latitude,
Poleward of ycr solutions are coastally trapped (β-plane Kelvin waves)
whereas equatorward of ycr they radiate offshore (Rossby waves).
Alternately, suppose you are observing coastal signals at latitude y.
Then, you will note that there is a critical frequency
Signals with frequencies σ > σcr are coastally trapped (β-plane Kelvin
waves) whereas those with σ < σcr radiate offshore (Rossby waves).
Rossby and β-plane Kelvin waves
Taking the limit of the expression for large σ (σ » σcr) gives
demonstrating that the high-frequency waves are Kelvin-like. Note that
these β-plane KWs oscillate, as well as, decay offshore.
Alternately, the limit for small σ (σ « σcr) is
so that low-frequency waves behave like long-wavelength RWs.
Rossby and β-plane Kelvin waves
The red curves are the disp.
curves for the new evanescent
waves (real part solid,
imaginary part dashed).
At an eastern-ocean boundary,
we must choose the root with the
positive imaginary part, so that
the wave decays westward.
σ/f
1
-
Movies H2
R/Ree
R/2R
−1
k/α
1
Vertical propagation (KW beams)
Recall that the vertical structure of waves in the LCS model satisfy
Rather than to look for solutions as expansions in vertical modes, ψn(z),
another way of studying solutions to the LCS model is to look for
approximate solutions of the form,
under the restriction that the background stratification, Nb(z) varies
slowly with respect to the vertical wavelength of the wave, m(z) (the
WKB approximation). In that case,
and cn can be replaced by
Vertical propagation (KW beams)
The dispersion relation for Kelvin waves along a southern (east-west
oriented) boundary then becomes
Group theory states that a packet of Kelvin waves (that is, a superposition
of several waves associated with different k and m values) propagates at
the “group” velocity
Thus, the energy of the packet propagates to the east with the slope
So, if phase propagates upwards (m > 0), energy propagates downwards,
and vice versa.
Vertical propagation (KW beams)
In a solution to an OGCM forced
by switched-on, steady winds (left
panels), coastal Kelvin waves radiate
poleward and Rossby waves radiate
offshore, leaving behind a steady-state
coastal circulation.
In contrast, in an OGCM solution
driven by periodic forcing, Kelvin
and Rossby waves are continually
generated.
Furthermore, the coastal currents
exhibit upward phase propagation,
indicating that energy propagates
downward from the surface.
σ = 2π/200 days
σ=0
Movies K
Solutions with a
shelf/slope
Shelf waves (Australia)
Buchwald and Adams (1968) first drew attention to the importance
of shelf waves along coasts. They noted their existence along the
Australian continental shelf/slope, and developed a lovely (simple and
Need
insightful) theory for them in the framework of a 1-layer
(barotropic) model.
The real ocean is stratified, which allows the existence of baroclinic
Kelvin waves. Later studies extended shelf-wave theory to allow for
stratification.
Topographic β
Shelf waves exist because of the continental slope. To illustrate the
basic physics of the slope, consider wave solutions to a 1-layer model
when the bottom slopes meridionally (Hy ≠ 0). Then, the linearized
equations are
where
Variable d is the change in sea level due to the wave. Hence, it is small
and the terms (du)x and (dv)y are negligible.
Topographic β
An equation for v can be obtained just as we did for baroclinic waves.
where c2 = gH. For simplicity, we set β = 0.
We are interested in low-frequency waves (σ « f). In the open ocean
where bottom slopes are typically gentle, it then follows that vyt « fvx
so that the v equation simplifies to
This equation has the same form as the v equation for baroclinic
waves, except with β replaced by
hence its label, “topographic β.”
Shelf waves
BA68 considered an idealized shelf
independent of y and extending into
the region x > 0.
Need
They separated the domain into two
regions: shelf and open-ocean
regions. Then, they found solutions
for each region and matched them
at x = λ).
λ
0
The shelf topography had the
simple form
z
which has the very useful property
x
where b is a constant.
Shelf waves
Because d « H, BA68 dropped the dt term to get the equation set
This simplification allows the streamfunction ψ
to be defined.
Plugging these expressions into the momentum equations quickly
leads to an equation for ψ
Shelf waves
Compare the ψ and v equations. The former differs in that it lacks ψttt
and (f2/c2)ψt.
Because the shelf λ is narrow, both terms are small with respect to the
ψxxt term and can be neglected. Specifically,
The first inequality holds because we consider shelf-wave frequencies that
are much less than f (σ2 « f2). Let R02 = c2/f2 = gH/f2 be the square of the
Rossby radius of deformation for the 1-layer system: With g = 1000 cm2/s,
H = 200 m, and f = 10−4 s−1, R0 = 447 km. Then, the second inequality
holds because λ2 « R02.
Shelf waves
Compare the ψ and v equations. The ψ-equation has two bottomslope terms, whereas the v-equation only has one.
For topographic waves, we argued that vyt « fvx because the bottom
slope was small. For a narrow shelf, ψxt ≈ fψy because the bottom slope
is steep, and BA68 retained both terms.
Shelf waves
BA68 looked for a shelf solution of the form
Then, the ψ equation gives the dispersion relation
which can be rewritten
Since b > 0, for f > 0 cp is negative and the wave propagates
southward. More generally, shelf waves propagate like KWs with the
coast to their right (left) in the northern (southern) hemisphere.
Shelf waves
For our purposes, we don’t need to go any further. To complete their
solution, however, BA68 solved for the response in the offshore region.
There, b = Hx/H = 0, and the ψ-equation simplifies to Laplace’s
equation.
Laplace’s equation has simple solutions of the form
BA68 required that ψ and ψx for the shelf and offshore solutions match
at x = λ, thereby determining k for the offshore solution.
Eastern-coastal currents w/wo shelf/slope
The fundamental way in which a shelf/slope impacts coastal currents
can be understood using a 1½-layer model.
ρ1
ρ2
Without a shelf/slope, the model is a 1½-layer system everywhere in
the domain. The coastal currents all radiate offshore due to the
westward radiation of Rossby waves.
Eastern-coastal currents w/wo shelf/slope
The fundamental way in which a shelf/slope impacts coastal dynamics
can be understood using a 1½-layer model.
offshore
regime
ρ1
ρ2
coastal
regime
ρ1
ρ1
ρ2
With athe
shelf/slope,
theoffshore
model ispropagation
a 1½-layer system
offshoreβfrom
Over
slope, then,
due planetary
is the
grounding line
regime),
but is a due
1-layer
model inshore
overwhelmed
by(offshore
alongshore
propagation
to topographic
β, of the
grounding
line (coastal
regime).
allowing
coastally
trapped
currents.
Leeuwin Current solution w/wo shelf/slope
Benthuysen et al. (2013)
Consistent with the above theory, without a shelf/slope the coastal
currents are very weak. They exist at all due to vertical diffusion.
With a shelf/slope the Leeuwin Current is an order of magnitude
larger. The current is generated by the slope, and the strongest current
speed lies at the shelf break. The dynamics of the current over the
flat shelf are those of a Munk layer.
Solution forced by periodic τy
Neglecting damping terms, an equation in p alone is
It is useful to split the total solution (q) into interior (q') and coastal
(q") pieces. The interior piece (forced response) is x-independent,
and so is simply
The coastal piece (free-wave solution) is
where we choose k1, rather than k2, because it either describes waves
with westward group velocity (long-wavelength Rossby waves) or
that decay to the west (eastern-boundary Kelvin waves).
Solution forced by periodic τy
To connect the interior and coastal solutions, we choose P so that that
there is no flow at the coast,
To solve for P, it is useful to define the quantity (integrating factor)
in which case,
Define Go = τy/H. Then, the solution for total p is
(4)
Solution forced by periodic τy
The solution has interesting limits when y >> ycr and y << ycr. In
the first limit,
so that
a β-plane Kelvin wave with an amplitude in curly brackets.
In the second limit,
so that
a long-wavelength Rossby wave propagating westward at speed cr.
This solution is derived in the HIGNotes.pdf and CoastalNotes.pdf.