Summer school on Dynamics of North Indian Ocean : Equatorial beams

J) Equatorial beams (July 21)

Waves generated by periodic forcing propagate vertically as well as horizontally. Such vertically-propagating signals have been observed at several locations in the equatorial ocean. One property of vertically-propagating waves is that the propagation directions of phase and energy are opposite: phase propagates upward (downward) when energy propagates downward (upward). To obtain these “beam-like” solutions, we sum the responses to N = 25 baroclinic modes. The solutions essentially are a repeat of the EUC solution of Section I, except forced by oscillating winds.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{x} wind; P = 15 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 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, (J1a)

Y(y) = 0.5{1+cos[2π(y–30º)/30º]}, –15º ≤ y ≤ 15º, (J1b)

T(t) = sin(σt), (J1c)

where σ = 2π/P, P = 15 days, τo = 1.5 dyn/cm^{2}, and X(x) and Y(y) vanish outsider their designated ranges. The movie shows the response of the LCS model with N = 25 modes, plotting an equatorial (x,z) section of u (shading) and (u,w) vectors. It is perhaps best to view this movie at a slow speed (3%) to see the spin-up of the Kelvin-wave beam and at an intermediate speeds (12%, say) to view the equilibrium response.
During the spin-up, Kelvin waves associated with several low-order baroclinic modes radiate eastward from the forcing region, and they superpose to form a beam. The beam is fully established by the end of February, 1991, by which time all the Kelvin waves have reached the eastern boundary. In equilibrium, the Kelvin beam radiates downward from the forcing region, reflects off the ocean bottom to return to the ocean surface, then makes another top-to-bottom transit, and finally almost another bottom-to-top transit. The angle of energy propagation is given by θ = ±σ/Nb, where Nb(z) is the background Vaisala frequency. Note that |θ| increases with depth because Nb decreases. Consistent with theory, phase propagates upward (downward) for beams in which energy propagates downward (upward). In addition, there is a phase shift of 2π across each beam (i.e., one positive and one negative region).
Also present in the movie are gravity waves associated with high-order baroclinic modes. They tend to be standing oscillation (no vertical obvious propagation), with eastward (westward) phase velocity east (west) of the wind. Because their group velocities are small, they tend to remain near the forcing region.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{x} wind; P = 30 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
As in Experiment J1a, except with P = 30 days. In this case, a Kelvin beam radiates downward from the forcing region at half the angle of the one in Experiment J1a, reflects off the ocean bottom, and returns to the ocean surface. At the eastern boundary, the beam reflects as a packet of boundary waves. All of the reflected waves are boundary trapped (i.e., ℓ_{cr} defined in Eq. G1 is less than 1 for all values of n). The n = 1, ℓ = 1 Rossby wave, however, is weakly damped (see Experiment G2b), and is visible in the solution. It strengthens for 30º ≤ x ≤ 45º, a result of direct forcing by the wind.
Because it propagates at a shallower angle, the beam is narrower than in Experiment J1a. To generate a narrower beam requires the superposition of Kelvin waves for considerably more baroclinic modes. As a result, the beam spins up more slowly, and is not well developed until June, 1991.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{x} wind; P = 60 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
As in Experiment J1a, except with P = 60 days. In this case, the Kelvin beam radiates downward from the forcing region at one quarter the angle of the one in Experiment J1a, and just reaches the bottom at the eastern boundary. At this period, several Rossby waves exist (see Experiment G2c), and they are visible in the solution.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{x} wind; P = 180 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
As in Experiment J1a, except with P = 180 days and

X(x) = cos[2π(x–60º)/40º] –40º ≤ x ≤ 80º, (J2)

where X(x) = 0 outside its designated range. It is useful to view the movie slowly (3%) to see the initial response and at intermediate speeds (25%, say) to see the equilibrium state. At this period, both Kelvin and Rossby beams are apparent.
Initially, a small-vertical-scale oscillation develops underneath the forcing region, clearly visible during March, 1991. It is a resonant, equatorial inertial oscillation with near-zero group velocity; as a result, it remains underneath the forcing region, and, because it is such a high-order baroclinic mode, it decays in time. At the same time, Rossby and Kelvin waves are also visible propagating from the forcing region, and the waves associated with a number of vertical modes quickly superpose to form beams. Because P = 180 days, the angle at which the Kelvin wave descends into the ocean is small, and it intersects the eastern boundary at a depth of only about 200 m. The Rossby beam that is visible west of the forcing region is a superposition of ℓ = 1 Rossby waves for a number of vertical modes; its structure is similar to the Kelvin beam, except that it descends into the ocean westward and at a steeper angle, θ = −(σ/Nb)/3.
Subsequently, Rossby waves reflect from the eastern boundary and begin to superpose to form beams. The most obvious beam leaves the eastern boundary near 200 m, reaches the bottom near 50°, and reflects from the bottom as an upward-propagating beam that intersects the western boundary near a depth of about 1200 m. An analogous beam of ℓ = 3 Rossby waves is less clear; it extends from the eastern boundary at 200 m, descends into the deep ocean at the steeper angle, θ = −(σ/Nb)/5, and intersects the bottom near 85°.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{y} wind; P = 15 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
An equatorial section of v and (v,w) vectors, showing the response of the LCS model with N = 25 modes when P = 15 days. Compare the response to that of Experiment J1a.
As for the Kelvin beam, the angle of energy propagation for a Yanai beam is θ = ±σ/Nb, where Nb(z) is the background Vaisala frequency, and phase propagates upward (downward) for beams in which energy propagates downward (upward). As a result, the ray paths of Yanai beams are identical to those for Kelvin beams.
Although less clear than the Kelvin-wave beam in Experiment J1a, a Yanai beam radiates downward from the forcing region, reflects off the ocean bottom to return to the ocean surface, and then makes another top-to-bottom transit and partial bottom-to-top transit. (To pick out the beam, view the movie at full speed and look for regions of clear upward and downward phase propagation. They all lie in the beam path.)
The Yanai waves that contribute to the beam are those that are strongly excited by the wind. Since the wind is large-scale with respect to the equatorial Rossby radius, essentially only Yanai waves with large zonal scale are excited. For P = 15 days, the waves that have large zonal scale are the low-order baroclinic modes. Consistent with this property, the dominant mode in the beam appears to be n = 3, as indicated by the number of zero crossings in the beam from the top to the bottom of the ocean.
Gravity waves associated with high-order baroclinic modes are also visible in the movie. They appear to be standing oscillations (i.e., no vertical obvious propagation), with eastward (westward) phase velocity east (west) of the wind.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{y} wind; P = 30 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
As in Experiment J2a, except when P = 30 days. Compare the response to that of Experiment J1b.
A well-formed Yanai beam radiates downward from the forcing region, reflects off the ocean bottom, and returns to the ocean surface in the top-right corner of the movie. For P = 30 days, the waves that have a large zonal scale and hence are strongly excited by the wind, are associated with intermediate (n ≈ 12) baroclinic modes. This property is apparent in that the beam has 12 zero crossings from the top to the bottom of the ocean.
There are also weak gravity waves associated with high-order baroclinic modes. They are much weaker than in Experiment J1b, because their mode number is high so that they are strongly damped. There is no indication of waves that reflect off the eastern boundary because all of them are boundary trapped (i.e., ℓ_{cr} defined in Eq. G1 is less than 2 for all values of n).

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{y} wind; P = 60 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
As in Experiment J2a, except when P = 60 days. Compare the response to the Kelvin-beam of Experiment J1c.
The spin-up process is interesting and best viewed at an intermediate speed (25%, say). Initially, a Yanai beam quickly forms, and at later times its propagation angle, θ, continues to decrease; at the same time, the beam develops a smaller vertical scale and weakens in amplitude, eventually disappearing.
For P = 60 days, the waves that have a large zonal scale and hence are strongly excited by the wind, are associated with very high baroclinic modes. For these very high-order modes, the equatorial Rossby radius of deformation, (cn/β)½, is so small that it cannot be resolved by the numerical grid (Δy = 25 km). So, the vanishing of the Yanai beam is an artifact of the numerical model. (On the other hand, the high-order Yanai beam cannot exist in a model with realistic damping; see Experiment J2d.)
At P = 60 days, ℓ_{cr} is large enough for Rossby waves to exist for the n =1 and n = 2 baroclinic modes. Wind-generated, longer-wavelength, Rossby waves (with westward group velocity) propagate to the western boundary, where they reflect as shorter-wavelength, Rossby waves with eastward group velocity. As time progresses, the shorter-wavelength Rossby propagate into the interior ocean where they dominate the response.
Weak gravity waves associated with high-order baroclinic modes are also visible in the solution in the longitude range of the forcing.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on, oscillatory τ^{y} wind; P = 60 days Mixing:ν_{h} = 5x110^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment J2c, except with damping. In this case, the Yanai beam decays by explicit damping (A ≠ 0) as well as by numerical damping. In addition, the reflected shorter-wavelength Rossby waves are efficiently damped.

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