Summer school on Dynamics of North Indian Ocean : Boundary reflections

G) Boundary reflections (June 30)

Equatorial waves reflect from the eastern and western boundaries of the basin. This set of experiments focuses on reflections from the eastern boundary, which can impact the ocean well off the equator. The solutions consider the reflections of a Kelvin-wave pulse (Experiment G1) and of Kelvin waves oscillating at a frequency σ (Experiments G2 and G3). At an eastern boundary, the boundary reflections consist of a set of equatorially-trapped waves (designated by the index ℓ), all linked by Moore’s chain rule.
For the oscillating solutions, there is a critical value of ℓ, given by

ℓ_{cr} = (a – a^{-1}-1)^{2}/4 (G1)

where a = 2½σ′ and σ′ = (βcn)½ is the equatorial inertial period of the nth baroclinic mode. Boundary waves with ℓ values smaller (larger) than ℓcr have real (complex) zonal wavenumbers. As a result, only the lower-order modes of the set (ℓ < ℓcr) propagate offshore, and they sum to form a packet of reflected Rossby waves; the higher-order waves (ℓ > ℓcr) all decay offshore, and they sum to generate a β-plane, coastal Kelvin wave that propagates poleward along the boundary.
It is often useful to restate the concept of ℓcr in terms of a critical latitude

y_{cr} = c_{n}/(2σ). (G2)

The critical latitude defines the location where reflected waves change from being Rossby-like (y < y_{cr}) to Kelvin-like (y > y_{cr}).

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: switched-on τx wind patch Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
The wind has the form τ^{x} = τ_{o}X(x)Y(y)T(t), where

X(x) = 0.5{1+cos[2π(x–37.5º)/15º]}, 30º ≤ x ≤ 45º, (G3a)

Y(y) = 0.5{1+cos[2π(y–30º)/30º]}, –15º ≤ y ≤ 15º, (G3b)

τ_{o} = 1.5 dyn/cm2, and X(x) and Y(y) are zero outside their designated ranges. The wind is located in the western ocean to allow Kelvin waves and Rossby waves reflected at the eastern boundary to be more visible. To weaken inertial oscillations, the time dependence is taken to be the smooth function,

T(t) = 0.5[1 – cos(2πt/δt)]θ(t)θ(δt − t). (G3c)

According to (G3c), the wind smoothly rises to a maximum at 15 days, weakens to zero at 30 days, and is switched off thereafter.
To see the initial response, slow down the viewer considerably (to 6%, say). A Kelvin-wave pulse (red) radiates from the forcing region, and begins to reflect from the eastern boundary by February. In addition, wind-forced Rossby waves have reached the western boundary of the basin where they reflect as a Kelvin wave (darker green and blue), and during February it crosses the basin. At this time, there are also small-scale oscillations behind (west of) the Kelvin-wave pulse, which are weak inertial oscillations that are still excited despite the smoothness of (G3c).
The first Rossby wave to emerge from the eastern boundary is an ℓ = 1 Rossby wave. By March, it is clearly visible as two westward-propagating (red) patches. At that time, the western-boundary Kelvin wave has crossed the basin, and also begins to reflect from the eastern boundary as an ℓ = 1 Kelvin wave. By mid-May, both of these Rossby waves have propagated to the middle of the basin. Closer to the eastern boundary, the structure of response is more complex, having more reversals in latitude and spreading farther from the equator. These properties indicate the presence of higher-order Rossby waves (ℓ > 1), but individual waves are difficult to identify because they are separate from each other.
Two other interesting features are apparent when the movie is played at full speed (100%). First, there are alternating bands of Rossby waves radiating away from the eastern boundary. They are clearly generated from the equator by multiple reflections of equatorial Kelvin waves and ℓ = 1 Rossby waves: The period of the bands is P = 4L/c1, where L is the basin width, that is, the time it takes a Kelvin wave to cross the basin and an ℓ = 1 Rossby wave to return. Second, short-wavelength Rossby waves with westward group velocity reflect from the western boundary. They have eastward group velocity, and as time passes they extend farther into the interior ocean. They exist in the solution because horizontal mixing is weak, so that they are not strongly damped.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: oscillatory τx wind; P = 15 days Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment G1, except that

T(t) = sin(σt)θ(t), (G2)

where σ = 2π/P. Accordingly, the wind switches on at t = 0 and thereafter oscillates with a period of P = 15 days.
At this period, ℓcr = .016, so that no boundary-reflected waves with real wavenumbers (i.e., Rossby waves) are possible at this frequency. As a result, the eastern-boundary reflected waves all superpose to form a β-plane coastal Kelvin wave. Note, however, that there are weak transient Rossby waves, generated because the wind is not a pure sinusoidal oscillation in time, but is switched on. They are visible during the first half of the movie, radiating from the eastern boundary as a sequence of positive and negative pulses (see the discussion of Experiment F1a).

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: oscillatory τx wind; P = 30 days Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment G2a, except with P = 30 days. At this frequency, ℓcr = 0.83, so again no boundary-reflected waves with real wavenumbers (i.e., Rossby waves) are possible, and hence the eastern-boundary reflected waves all decay offshore (i.e., are evanescent), and superpose to form a β-plane coastal Kelvin wave. At this longer period, however, the decay of the lowest-order (ℓ = 1) evanescent wave is weak. Consequently, it is visible in the movie as a westward-decaying packet at two locations: west of the wind-forced region, and near the eastern boundary. There are also transient Rossby waves like those in Experiment G2a, radiating off the eastern boundary; they are almost decayed away by the time the movie is half over.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: oscillatory τx wind; P = 60 days Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment G2a, except with P = 60 days. In this case, ℓcr = 4.6, so the ℓ = 1 and ℓ = 3 boundary-reflected waves have real wavenumbers (i.e., they are Rossby wave). (Boundary waves with even values of ℓ are not allowed because they are not symmetric about the equator.) All the other eastern-boundary reflected waves have complex wavenumbers (ℓ = 5, 7, 9, ∙∙∙), and they superpose to form a β-plane coastal Kelvin wave. The reflected Rossby waves interfere so strongly with the incoming Kelvin wave that it is difficult (impossible) to see the Kelvin wave in the equatorial response.
Since there are several reflected Rossby waves at this period, the concept of a critical latitude now makes sense. Its value is ycr = 1090 km = 9.8°, in good agreement with the latitudinal extent of the response in the movie. (The response appears to extend a bit beyond ycr in the longitude band from 30° to 40°, but that is due to the directly-forced response, not to the wave packet radiating from the eastern boundary.)

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: oscillatory τx wind; P = 120 days Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment G2a, except with P = 120 days. In this case, ℓcr = 20, so that eastern-boundary reflected waves with ℓ ≤ 19 have real wavenumbers (i.e., are Rossby waves). As a result, there is a critical latitude at ycr = 2180 km = 20º. Equatorward of ycr, the eastern-boundary reflections radiate back into the interior ocean as Rossby waves, whereas poleward of ycr they decay to the west and, hence, superpose to form a β-plane coastal Kelvin wave.
Surprisingly, the response does not show a clear critical latitude at 20°, below which the ocean is filled with Rossby waves. Instead, the Rossby-wave packet appears to bend equatorward as it propagates away from the eastern boundary. Indeed, group theory suggests the packet does bend equatorward, to an approximate focal point on the equator a distance,

xf = (π/4)(c1/σ) = c1P/8, (G4)

from the eastern boundary. For P = 120 days, xf = 3420 km = 31°, and this property is consistent with the response in the movie. (During the summer school, the concept of a focal point was mentioned but we did not cover this topic in any detail. See Problem 18 of the Equatorial Notes for a detailed discussion. Also see Experiment H2d.)

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: oscillatory τy wind; P = 15 days Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment G2a, except for a meridional wind of the form τy = τoX(x)Y(y)T(t), where the structure functions are given in (G1a), (G1b) and (G2). With P = 15 days, ℓcr = 0.016 and no boundary reflected waves with real wavenumbers (i.e., Rossby waves) are possible. As a result, the eastern-boundary reflected waves superpose to form a β-plane coastal Kelvin wave.
Because σ′ = 0.62 < 1, the Yanai waves generated by the wind have westward group velocity. Initially, however, a transient packet of Yanai waves radiates eastward, the leading edge of which has eastward phase velocity (see the discussion of Experiment F2).

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Characteristic speed:c_{1} = 264 cm/s Forcing: oscillatory τy wind; P = 30 days Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment G3a, except that P = 30 days. At this period, ℓcr = 0.83 so that no boundary reflected waves with real wavenumbers (i.e., Rossby waves) are possible. As a result, the eastern-boundary reflected waves superpose to form a β-plane coastal Kelvin wave.
There is a prominent packet of transient Yanai waves, much more visible than in Experiment G3a. Its leading edge has eastward phase velocity, its trailing edge has westward group velocity, and its center is a region with a large wavelength where k ≈ 0 (see the discussion of Experiment F2).
Because σ′ = 0.31 < 1, the Yanai waves in the equilibrium response have eastward group velocity. Their amplitude is much weaker than in Experiment G3a because their wavelength, λ = 2π/k = 2π/(σ/c1 – β/σ) = 6.7°, is much less than the zonal width of the forcing region (15°).
There is a curious pulsing of the Yanai waves in the equilibrium response. Such a pulsing can only occur due to interference between the Yanai waves and waves with a different wavelength. The most likely possible waves are the ℓ = 2 evanescent waves, which have the weakest damping rates.

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