Process Solutions Jay McCreary Summer School on: Dynamics of the North Indian Ocean National Institute of Oceanography Dona Paula, Goa June 17 – July 29, 2010

Introduction 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 u n, v n, and p n 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 w 1 rather than by a damping term, –A/c n 2 p n.

Solving the complete set of equations for a single one in v n, and, for convenience, dropping the subscript n, setting τ y = 0, and F = τ x / 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/fσ/f k/αk/α When σ is large, the kβ term is small, and can be neglected to first order. The resulting curve describes gravity waves. σ/fσ/f k/αk/α When σ is small, the σ 2 /c 2 term is small, yielding the Rossby- wave curve. σ/fσ/f k/αk/α 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.

σ/fσ/f k/αk/α Free waves How does the interior-ocean approximation distort the curves? σ/fσ/f k/αk/α 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 τ x 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 τ x 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 τ x 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 τ x 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 τ x (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)Y(y) Forcing by a band of alongshore wind τ y 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 τ y In a 3-d model (x, y, h) with β = 0, in addition to local upwelling by w e, 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 p y balances τ y everywhere. Response to switched-on τ y 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 τ y

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 τ y σ = 0 σ = 2π/200 days

Equations for the u n, v n, and p n 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 u n equation, neglect horizontal mixing, and ignore forcing by τ x (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 v n, and, for convenience, dropping the subscript n, setting τ x = 0, and G = τ y / 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 y cr solutions are coastally trapped (β-plane Kelvin waves) whereas equatorward of y cr they radiate offshore (Rossby waves). Free waves

σ/fσ/f k/αk/α 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/fσ/f k/αk/α

Solution forced by periodic τ y 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 k 1, rather than k 2, 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 τ y 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 G o = τ y / H. Then, the solution for total p is (4)

Solution forced by periodic τ y 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 c r. In the second limit, so that

Equatorial ocean dynamics

d (1 month) d (6 months) d (1 year) d (5 years) Equatorial jet Kelvin wave Rossby wave Reflected Rossby-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 u n, v n, and p n 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, L R = (c n /β), the equatorial Rossby radius of deformation for baroclinic mode n. (For n = 1, L R 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 v ℓ 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 P o = π¼P' o.

Free waves σ/σoσ/σo k/αok/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/σoσ/σo k/αok/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, k ℓ 1,2, associated with each m value. The wavenumbers k ℓ 1 (k ℓ 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 v ℓ, u ℓ, and p ℓ fields for equatorially trapped Rossby and gravity waves are where V ℓ is a constant amplitude, j = 1,2, and Equatorial gravity and Rossby waves

The u, v, and p fields for a Yanai wave when c n = 250 cm/s and P = 30 days. For this P, σ/σ o =.36 and λ = 7.3º. σ/σoσ/σo k/αok/αo 1 3 Equatorial gravity and Rossby waves Courtesy of Francois Ascani

The u, v, and p fields for a Yanai wave when c n = 250 cm/s and P = 360 days. For this P, σ/σ o =.03 and λ = 0.64º. σ/σoσ/σo k/αok/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 c n = 250 cm/s and P = 360 days. For this P, σ/σ o =.03 and λ = 240º. σ/σoσ/σo k/αok/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 c n = 250 cm/s and P = 360 days. For this P, σ/σ o =.03 and λ = 140º. σ/σoσ/σo k/αok/αo 1 3 Equatorial gravity and Rossby waves Courtesy of Francois Ascani

Free waves σ/σoσ/σo k/αok/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/σoσ/σo k/αok/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 (fv n = τ x / H n ) holds, whereas at the equator u n continues to accelerate (u nt = τ x / H n ). The switch from one dynamical regime to the other occurs at y ≈ α 0 –½ = (β/c n ) –½.

If only the damping term (A/c n2 )u n 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 p nx 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 k ℓ 1 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 u K field there is and it must be cancelled by the recursion relation for Moore’s famous chain rule. To eliminate u K, we use the u 1 wave and set B 1 = U o. With this choice the  0 terms are cancelled, but a  2 term is created. We use the u 3 wave to cancel this term, and so on. In general, once B ℓ 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

d (1 month) d (6 months) d (1 year) d (5 years) Equatorial jet Kelvin wave Rossby wave Reflected Rossby-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, 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 u n, v n, and p n 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. σ/σ0σ/σ0 k/α0k/α0 3 1 k/αk/α 3 1 σ/σ0σ/σ0 k/α0k/α0 Long-wavelength approximation