Wind-forced solutions:Coastal ocean

A short course on:

Modeling IO processes and phenomena

INCOISHyderabad, India

November 16−27, 2015

References1) 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

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.

Mittelstadt et al. (1975), Huyer (1976)

Coastal phenomena: steady currents

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)

Feng et al. (2003, 2008)32ºS

ORCA025

Coastal phenomena: steady currents

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

QuestionsWhat 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

A useful set of equations for the coastal ocean is

Coastal-ocean equations

A key simplification is to drop the acceleration and damping terms from the zonal momentum equation. In addition, since it is well known that the coastal ocean responds much more strongly to alongshore winds drop τx forcing. Finally, for simplicity neglect the horizontal mixing terms. In this way, the alongshore flow is in geostrophic balance, a property consistent with observations.

As for the interior-ocean equations, this approximation is useful because it filters out gravity waves. Thus, it only describes the slowly varying parts of the response, that is, the directly forced and Rossby-wave (if β ≠ 0) parts of the response.

A useful set of equations for the coastal ocean is

Coastal-ocean equations

A key simplification is to drop the acceleration and damping terms from the zonal momentum equation. In addition, since it is well known that the coastal ocean responds much more strongly to alongshore winds drop τx forcing. Finally, for simplicity neglect the horizontal mixing terms. In this way, the alongshore flow is in geostrophic balance, a property consistent with observations.

As for the interior-ocean equations, this approximation is useful because it filters out gravity waves. Thus, it only describes the slowly varying parts of the response, that is, the directly forced and Rossby-wave (if β ≠ 0) parts of the response.

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

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

Mathematical usefulness

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

σ/f

k/α

-

-1

-

−1 1

R/2Re

How does the coastal-ocean approximation distort the disp. curves?

σ/f

k/α

-

-1

-

−1 1

R/2Re

When are the RWs accurately simulated in the coastal model? When ℓR << 1, and σ/f << 1.

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.

There are also KWs along basin boundaries. When β ≠ 0, their eastern- and western-boundary KWs have a more complicated structure (see below).

Solutions for switched-on winds

Y(y)

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

the latter case discussed in the next section.

Forcing by a band of alongshore wind τy

Consider the 2-dimensional (x, h) coastal response of a 1½-layer model when the wind is independent of y.

2-d response to switched-on τy

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.

we

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.

hm

R

Solving for a single equation in h gives

where R2 = g'H/f2 is the square of Rossby radius of deformation. The forcing term vanishes because τy is independent of x.

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

2-d response to switched-on τy

(1)

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 → −∞.

2-d response to switched-on τy

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

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.

2-d response to switched-on τy

How long does it take for h to thin to the surface at the coast? For the parameter choices

the time is 29 days.

There is a meridional geostrophic current associated with h,

a coastally trapped jet flowing in the direction of the wind.

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 (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.

Solving for a single equation in h gives

where R2 = g'H/f2 is the square of Rossby radius of deformation. The forcing term vanishes because τy is independent of x.

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

3-d response to switched-on τy

(1)

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 → −∞.

3-d response to switched-on τy

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,

(1)

To satisfy the initial condition that h = H at t = 0, we must choose Λ(y) = −χ(y), so that

We obtain the solution for A by splitting it into particular (steady-state) and homogeneous (Kelvin-wave) responses,

where Λ(x,y) is an as yet unspecified function.

3-d response to switched-on τy

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

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.

Initial adjustment

t +

At longer times the solution for all the fields is

Final adjustment

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.

Movies H1a and H1b.

When β ≠ 0, Rossby waves carry the coastal response offshore, leaving behind a state of rest in which py balances τy everywhere.

β-plane

A fundamental question, is: Given offshore Rossby-wave propagation, why do eastern-boundary currents exist at all?

Movie H1c

The RW speed is

So, RWs propagate faster closer to the equator (cr ~ f−2).

3-d response to switched-on τy (β ≠ 0)

How does the LCS model adjust when many baroclinic modes are included?

Multi-baroclinic mode adjustment with damping

With damping, the responses of the n > 1 modes are increasingly damped since ν = A/cn

2. 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.

The plot shows the response of the n = 1 mode without damping.

It also illustrates the n > 1 responses, except that the currents are narrower because the Rossby-wave speed is smaller since (cn < c1).

β-plane

The model allows Rossby waves to propagate offshore. A steady coastal circulation remains, however, they are damped by vertical diffusion.

McCreary (1981) obtained a steady-state, coastal solution to the LCS model with damping.

There is upwelling in the band of wind forcing. There is a surface current in the direction of the wind, and a subsurface CUC.

Multi-baroclinic mode adjustment with damping

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.

Philander and Yoon (1982)

Movies I2 and I3

3-d response to switched-on τy (OGCM)

Solutions for periodic winds

Evanescent waves

σ/f

k/α

-

-1

-

−1 1

R/Re

σ/f

k/α

-

-1

-

−1 1

R/2Re

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.

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).

Rossby and β-plane Kelvin wavesTo see this property, solve the coastal dispersion relation

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).

Alternately, the limit for small σ (σ « σcr) is

so that low-frequency waves behave like long-wavelength RWs.

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.

Rossby and β-plane Kelvin waves

σ/f

k/α

-

-1

-

−1 1

R/Re

The red curves are the disp. curves for the new evanescent waves (real part solid, imaginary part dashed).

σ/f

k/α

-

-1

-

−1 1

R/2Re

At an eastern-ocean boundary, we must choose the root with the positive imaginary part, so that the wave decays westward.

Rossby and β-plane Kelvin waves

Movies H2

and cn can be replaced by

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,

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,

Recall that the vertical structure of waves in the LCS model satisfy

Vertical propagation (KW beams)

The dispersion relation for Kelvin waves along a southern (east-west oriented) boundary then becomes

Vertical propagation (KW beams)

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.

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.

σ = 0σ = 2π/200 days

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.

Vertical propagation (KW beams)

Movies K

Solutions with a shelf/slope

Need

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 insightful) theory for them in the framework of a 1-layer (barotropic) model.

Shelf waves (Australia)

The real ocean is stratified, which allows the existence of baroclinic Kelvin waves. Later studies extended shelf-wave theory to allow for stratification.

where

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

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.

This equation has the same form as the v equation for baroclinic waves, except with β replaced by

hence its label, “topographic β.”

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

where c2 = gH. For simplicity, we set β = 0.

Shelf waves

Need

z

x

BA68 considered an idealized shelf independent of y and extending into the region x > 0.

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).

The shelf topography had the simple form

which has the very useful property

where b is a constant.

0 λ

Shelf waves

Because d « H, BA68 dropped the dt term to get the equation set

Plugging these expressions into the momentum equations quickly leads to an equation for ψ

This simplification allows the streamfunction ψ

to be defined.

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 R0

2 = 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 « R0

2.

Shelf waves

Compare the ψ and v equations. The ψ-equation has two bottom-slope 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

which can be rewritten

Then, the ψ equation gives the dispersion relation

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.

ρ2

ρ1

The fundamental way in which a shelf/slope impacts coastal currents can be understood using a 1½-layer model.

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

ρ2

ρ1ρ1

ρ2

ρ1

offshoreregime

coastalregime

The fundamental way in which a shelf/slope impacts coastal dynamics can be understood using a 1½-layer model.

With a shelf/slope, the model is a 1½-layer system offshore from the grounding line (offshore regime), but is a 1-layer model inshore of the grounding line (coastal regime).

Eastern-coastal currents w/wo shelf/slope

Over the slope, then, offshore propagation due planetary β is overwhelmed by alongshore propagation due to topographic β, 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.

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

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).

The coastal piece (free-wave solution) is

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

a β-plane Kelvin wave with an amplitude in curly brackets.

a long-wavelength Rossby wave propagating westward at speed cr.This solution is derived in the HIGNotes.pdf and CoastalNotes.pdf.

In the second limit,

so that

so that

The solution has interesting limits when y >> ycr and y << ycr. In the first limit,

Solution forced by periodic τy