Process SolutionsProcess Solutions

Jay McCrearySummer School on:

Dynamics of the North Indian Ocean

National Institute of OceanographyDona Paula, Goa

June 17 – July 29, 2010

IntroductionIntroduction

1) Interior ocean dynamics

2) Coastal ocean dynamics

3) Equatorial ocean dynamics

Interior ocean dynamics

When β = 0 (f-plane), easterly winds force northward Ekman

drift (northern hemisphere), which spins up two counter-

rotating, geostrophic gyres, a process known as Ekman

pumping.

When β ≠ 0, Rossby waves extend the response west of the

wind region, and adjust the response to a state of Sverdrup

balance.

–

Response to a switched on zonal wind

Equations for the un, vn, and pn for a single baroclinic mode are

Even this linear set of equations, though, is difficult to solve.

Interior-ocean equations

A useful (and still popular) simplification is to drop the acceleration and damping terms from the momentum equations. In addition, the horizontal viscosity terms are assumed small dropped in the interior ocean, and are only retained to represent western boundary currents.

It is often useful to view this set of equations as corresponding to a linear, 1½-layer system, in which diffusion results from an entrainment velocity w1 rather than by a damping term, –A/cn

2pn.

Solving the complete set of equations for a single one in vn, and, for convenience, dropping the subscript n, setting τy = 0, and F = τx/H, gives

whereas solving the approximate set for the interior ocean gives

(1)

(2)

Equation (2) is valid provided the first three terms of (1) are small compared to the fourth. That will be true provided that

and similarly that

Free waves

A useful way of understanding the impact of the approximations is to see how they distort the free waves of the system. Typically, one proceeds by “pretending” that f and β are both constant. Then, solutions have the form of plane waves,

providing dispersion relations for the waves.

The resulting dispersion relation for the complete system is

whereas that for the approximate one is

In the HIG Notes, σ → –σ.

To plot σ(k,l), for the moment consider a slice along l = 0.

Free waves

σ/f

k/α

When σ is large, the kβ term is small, and can be neglected to first order. The resulting curve describes gravity waves.

When σ is small, the σ2/c2 term is small, yielding the Rossby-wave curve.

When l ≠ 0, the curves extend along the l/α axis, to form circular “bowls.”

NOTE: The limits on the axes are not accurate. For example, the gravity-wave curve should bottom out near σ/f = 1.

Free waves

How does the interior-ocean approximation distort the curves?

It eliminates gravity waves and the Rossby curve becomes a straight line (non-dispersive). Moreover, σ is independent of ℓ, so that the Rossby wave curve is not a bowl in k-ℓ space, but a plane.

When are the waves accurately simulated in the interior model? When kR = k/α << 1, ℓR << 1, and σ/f << 1.

Solution forced by switched-on τx

Let the ocean be unbounded and forced by a switched-on wind patch with the separable form

A simple way to solve this equation is to use Laplace and Fourier transforms in t and x, which replaces derivatives with the numbers,

As a result, differential equation (1) reduces to an algebraic one, which can be easily solved.

(1)

Neglecting damping, the v-equation we would like to solve is

Note that y-derivatives are absent from (1), a major simplification.

Solution forced by switched-on τx

The equation for double-transformed v is then

To invert the Laplace transforms, we use the inverse Laplace transforms

where ^ and ~ over the v indicate a Laplace and Fourier transformed variable. Solving for transformed v gives

to get

(2)

Solution forced by switched-on τx

To invert the Fourier transforms, in (2), we use

to obtain the solution

The u field can be found by integrating v according to

where the integral is from +∞ because no information can appear west of the forcing since Rossby waves propagate westward.

Solution forced by switched-on τx

Term (a) – (b) advances westward as a Rossby wave, leaving a Sverdrup flow (c) behind. Note that (a) and (c) cancel when t 0, so that the initial circulation is just Ekman flow, –(b).

(a) (b) (c)

–

Although the v field of the Rossby wave separates from v of the forcing region, the u field does not because of the x-integral. It might seem from the form of (3) that u extends farther west than the front, but one can show that the integral vanishes there.

Solution forced by switched-on τx

(3)

–

f-plane

The initial response is the same as on the f-plane. Subsequently, the radiation of Rossby waves adjusts the circulation to Sverdrup balance.

Interior-ocean dynamics

β-plane

Coastal ocean dynamics

Y(y)

Forcing by a band of alongshore wind τy

All the solutions discussed in this part 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

In a 2-dimensional model (x, h), alongshore winds drive upwelling

and coastal currents only locally, in the region of the wind. The

offshore decay scale is the Rossby radius of deformation.

Response to switched-on τy

In a 3-d model (x, y, h) with β = 0, in addition to local upwelling by we, 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 to balance the wind

stress.

f-plane

1½-layer model

f-plane

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

Response to switched-on τy

β-plane

A fundamental question about eastern-boundary currents, then, is: Why do they exist at all?

There is upwelling in the band of wind forcing. There is a surface

current in the direction of the wind, and a subsurface CUC.

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

The model allows offshore propagation of Rossby waves.

A steady coastal circulation remains, however, because the offshore propagation of Rossby waves is damped by vertical

diffusion.

Response to switched-on τy

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 response to a periodic wind (right panels), the adjustment never ceases, and the coastal

currents exhibit upward phase propagation. Does all the current propagate offshore, or does some remain attached

to the coast?

Response to switched-on and periodic τy

σ = 0

σ = 2π/200 days

Equations for the un, vn, and pn for a single baroclinic mode are

which are difficult to solve.

A useful simplification for coastal oceans is to drop the acceleration and damping terms from the un equation, neglect horizontal mixing, and ignore forcing by τx (although the latter two are not necessary). In this way, the alongshore flow is in geostrophic balance.

Coastal-ocean equations

whereas solving the approximate set for coasts gives

As for the interior equations, As before, y-derivatives are absent from (2), a major simplification. Equation (2) will be valid provided that the second and third terms of (1) are small compared to the fourth. That will be true provided that

(1)

(2)

Solving the complete set of equations for single one in vn, and, for convenience, dropping the subscript n, setting τx = 0, and G = τy/H, gives

If we look for plane-wave [exp(ikx + ily – iσt)] solutions to (2), the resulting dispersion relation is

Equation (3) is quadratic in k, and has the solutions

(3)

Note that the roots are either real or complex depending on 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).

Free waves

Free waves

How does the interior-ocean approximation distort the curves?

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

It eliminates gravity waves and the Rossby curve has the correct shape for ℓ = 0. But, σ is independent of ℓ, so that the Rossby wave curve is not a bowl in k-l space, but a curved surface.

σ/f

k/α

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

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

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 ∞ and y 0. In the first limit,

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

a long-wavelength Rossby wave propagating westward at speed cr.

In the second limit,

so that

so that

Equatorial ocean dynamics

d (1 month)

d (6 months)

d (1 year)

d (5 years)

Equatorial jet

Kelvin wave

Rossby wave

ReflectedRossby-wave

packet

Sverdrup flow

Spin-up of an inviscid, baroclinic mode

If the wind oscillates at the annual (or semiannual) period, these

adjustments continue indefinitely. Coastal Kelvin waves

continuously radiate from the equator around the perimeter of

the basin, followed by the propagation of Rossby waves

into the basin interior.

In response to forcing by a patch of easterly wind, Kelvin and

Rossby waves radiate from the forcing region, reflect from basin boundaries, and eventually adjust the system to a state of Sverdrup

balance.

Steady, linear response with damping

With diffusion: When the LCS model includes diffusion, realistic steady flows can be produced near the equator.

McCreary (1981)

Oscillating wind: Suppose the wind oscillates at the semiannual or annual cycle. Then, the equatorial currents oscillate, penetrate into the deep ocean along beams, and exhibit upward phase propagation, similar to the coastal circulation.

EUC

Equations for un, vn, and pn for a single baroclinic mode are

Because f vanishes at the equator, no terms can be dropped that allow for mathematically simple solutions near the equator.

Equatorial-ocean equations

(1)

A useful assumption, though, is to set f = βy, known as the equatorial β-plane approximation. As a result, one can look for solutions as expansions in Hermite functions.

Equatorial gravity and Rossby waves

Because f varies so much near the equator, we cannot assume ℓ(y) = exp(iℓy). It is useful to introduce the non-dimensional variable, for y,

and to rewrite (2) as

The operator in parentheses is the Hermite operator. Its eigenfunctions are the solutions to

Where ℓ = 0, 1, 2, …, and they are referred to as Hermite functions.

We look for free-wave solutions to (1) of the form, ℓ(y)exp(ikx– iσt), without damping (A = 0), and, for convenience, we drop the subscript n. The resulting v equation is

(2)

(3)

NOTE: In the HIG Notes, ℓ → m.

Equatorial gravity and Rossby waves

Fedorov and Brown (2007)

The figure plots the first six Hermite functions ℓ (ℓ = 0–5). The scaling factor, LR = (cn/β), the equatorial Rossby radius of deformation for baroclinic mode n. (For n = 1, LR is roughly 250 km.) Note that the ℓ are less equatorially trapped (extend farther off the equator) as ℓ increases. Note also that they alternate between being symmetric and antisymmetric about the equator.

Ascani (2002)

The solutions to (3) can be represented as expansions in Hermite functions

where vℓ is a wave amplitude. Each term in expansion (4) is an individual equatorial wave. Inserting (4) into (3) gives

which provides the dispersion relation

for equatorial, Rossby and gravity waves.

Equatorial gravity and Rossby waves

(4)

The equatorial Kelvin wave has v = 0, and so was missed in the preceding solutions. To find it, set v = A = 0 in (1), and look for a free-wave solution of the form (y)exp(ikx– iσt).

With these restrictions, equations (1) reduce to

The first and third equations imply

Equatorial Kelvin wave

and the second then gives

(5)

The solution that grows exponentially in y, which corresponds to the root, k = –σ/c, is physically unrealistic in an unbounded basin and must be discarded. Therefore, the only possible wave is

Equatorial Kelvin wave

which describes the structure and dispersion relation for the equatorial Kelvin wave. In (6), I have used the property that

(6)

The solution to (5) is

and redefined the arbitrary constant amplitude to be Po = π¼P'o.

Free waves

σ/σo

k/αo

1

3

For each ℓ, there is a gravity wave (large σ) and a Rossby wave (small σ). The plot indicates the waves for ℓ = 1, 2, and 3.

There is a new type of wave, the mixed Rossby-gravity (Yanai) wave, which joins the Rossby (gravity) wave curves for large negative (positive) values of k.

Another additional wave is the equatorial Kelvin wave.

Free waves

σ/σo

k/αo

1

3

Also as for the coastal model, the wavenumbers are real for small σ (Rossby waves) and become complex as σ increases. Eventually, they become real again for even larger σ (gravity waves).

As for the coastal model, there are two wavenumbers, kℓ

1,2, associated with each m value. The wavenumbers kℓ

1 (kℓ

2) describe waves with eastward (westward) group velocity or decay.

The region of complex roots for ℓ = 1 waves is indicated by the shading.. Such waves exist only along boundaries, where they superpose to generate β-plane coastal Kelvin waves.

The vℓ, uℓ, and pℓ fields for equatorially trapped Rossby and gravity waves are

whereVℓ is a constant amplitude, j = 1,2, and

Equatorial gravity and Rossby waves

The u, v, and p fields for a Yanai wave when cn = 250 cm/s and P = 30 days. For this P, σ/σo = .36 and λ = 7.3º.

σ/σo

k/αo

1

3

Equatorial gravity and Rossby waves

Courtesy of Francois Ascani

The u, v, and p fields for a Yanai wave when cn = 250 cm/s and P = 360 days. For this P, σ/σo = .03 and λ = 0.64º.

σ/σo

k/αo

1

3

Equatorial gravity and Rossby waves

Courtesy of Francois Ascani

The u, v, and p fields for an ℓ = 1 Rossby wave when cn = 250 cm/s and P = 360 days. For this P, σ/σo = .03 and λ = 240º.

σ/σo

k/αo

1

3

Equatorial gravity and Rossby waves

Courtesy of Francois Ascani

The u, v, and p fields for an ℓ = 2 Rossby wave when cn = 250 cm/s and P = 360 days. For this P, σ/σo = .03 and λ = 140º.

σ/σo

k/αo

1

3

Equatorial gravity and Rossby waves

Courtesy of Francois Ascani

Free waves

σ/σo

k/αo

1

3

The first equatorially trapped waves to be discovered were gravity-wave

resonances with periods of O(10 days) (Wunsch and Gill, 1976; Deep-Sea

Res.). There are no publications that explore the possibility of Rossby-wave

resonances.

The equatorial Kelvin wave was discovered after it was predicted to be dynamically important in El Nino (Knox and Halpern, 1982, JMR).

The mixed Rossby-gravity (Yanai) wave was first observed in the atmosphere by Yanai. In the ocean, it was (probably) first

detected in the Indian Ocean by Tsai et al. (1992) using altimeter

data.Who first detected an equatorial

Rossby wave?

Free waves

σ/σo

k/αo

1

3

The first equatorially trapped waves to be discovered were gravity-wave

resonances with periods of O(10 days) (Wunsch and Gill, 1976; Deep-Sea

Res.). There are no publications that explore the possibility of Rossby-wave

resonances.

The equatorial Kelvin wave was discovered after it was predicted to be dynamically important in El Nino (Knox and Halpern, 1982, JMR).

The mixed Rossby-gravity (Yanai) wave was first observed in the atmosphere by Yanai. In the ocean, it was (probably) first

detected in the Indian Ocean by Reverdin and Luyten (1986) using

altimeter data.Who first detected an equatorial

Rossby wave?

Tom Farrar 2010

Kozo Yoshida wrote down the first solution for an x-independent equatorial current driven by zonal winds. The (more complete) theoretical solution developed somewhat later (Dennis Moore) has come to be called the “Yoshida Jet” (Jim O’Brien).

Yoshida Jet

The basic dynamics of the Yoshida Jet can be understood from the zonal-momentum equation. Neglecting the pressure-gradient and mixing terms, in the zonal momentum equation gives

Offshore, Ekman balance (fvn = τx/Hn) holds, whereas at the equator un continues to accelerate (unt = τx/Hn). The switch from one dynamical regime to the other occurs at y ≈ α0

–½ = (β/cn) –½ .

If only the damping term (A/cn2)un is included in the zonal momentume equation, the flow stops accelerating, but it is unrealistically fast (the unit is km/s) and extends to the bottom.

If the pressure-gradient term pnx is then included, the flow field has both a realistic amplitude and structure.

Zonal flows along the equator don’t continue to accelerate in reality or models. Why not?

Bounded Yoshida Jet

d (1 month)

d (6 months)

Equatorial jet

Kelvin wave

Rossby wave

Bounded Yoshida Jet

In response to forcing by a patch of easterly wind, a Yoshida Jet initially develops in the forcing region. Subsequently, Kelvin and Rossby waves radiate from the forcing region, generating an eastward jet along the equator, both east and west of the forcing region: the bounded Yoshida Jet.

Eastern-boundary reflections

Suppose the ocean is forced by a patch of oscillating zonal wind confined to the interior ocean. It generates an equatorial Kelvin wave,

that radiates to the eastern boundary of the basin.

There can be no zonal flow through the boundary. How does the system adjust to prevent this flow? Dennis Moore showed that a packet of equatorial waves with the zonal velocity field,

are generated at the eastern boundary. In (7), the wavenumbers kℓ1

correspond to waves with westward group velocity or decay.

(7)

Eastern-boundary reflections

For convenience, let the eastern boundary be located at x = 0. The uK field there is

and it must be cancelled by

the recursion relation for Moore’s famous chain rule.

To eliminate uK, we use the u1 wave and set B1 = Uo. With this choice the 0 terms are cancelled, but a 2 term is created. We use the u3 wave to cancel this term, and so on. In general, once Bℓ is known then

Eastern-boundary reflections

For low frequencies, the incoming Kelvin wave reflects as a series of Rossby waves, with the waves corresponding to larger ℓ values propagating offshore more slowly.

The zonal current of the Kelvin wave divides at the Rossby-wave front to flow along the edges of the wave packet.

1

3

5

7

d (1 month)

d (6 months)

d (1 year)

d (5 years)

Equatorial jet

Kelvin wave

Rossby wave

ReflectedRossby-wave

packet

Sverdrup flow

After multiple reflections, the solution eventually adjusts to a state of Sverdrup balance.

Eastern-boundary reflections

In response to forcing by a patch of wind in the interior ocean, Kelvin waves reflect from the eastern boundary, creating a characteristic wedge-shaped pattern. In addition, wind-generated Rossby waves reflect from the western boundary to return to the interior ocean.

Eastern-boundary reflections

Remarkably, the characteristic wedge shape and westward propagation is visible in satellite data. The figure shows global maps of filtered sea level from TOPEX/Poseidon on April 13 and July 31, 1993. It shows a Rossby-wave packet generated by the reflection of an equatorial Kelvin wave forced by intraseasonal winds in the western ocean. (After Chelton and Schlax, 1996.)

Fedorov and Brown, 2007

Equations for the un, vn, and pn for a single baroclinic mode are

In the equatorial region, there are no simplifications that allow for mathematically simple solutions (i.e., that allow y-derivatives to be dropped.

One simplification (analogous to the coastal one) is to adopt the long-wavelength approximation, which restricts the zonal flow to be in geostrophic balance and, for convenience, to drop horizontal mixing from the x-momentum equation (not necessary).

Long-wavelength approximation

Include the dispersion relation here at the top of the page.

Long-wavelength approximation