Summer school on Dynamics of North Indian Ocean : Coastal dynamics and Undercurrents

I) EUC and CUC (July 2)

Undercurrents are prominent aspects of both equatorial and coastal circulations. In the North Indian Ocean, for example, there are coastal undercurrents along both coasts of India, as well as along Somalia and Oman. Here, we obtain solutions that illustrate the three-dimensional spin-up of the Equatorial Undercurrent (EUC) and an eastern-boundary Coastal Undercurrent (CUC) to switched-on winds, by summing the responses of N = 25 baroclinic modes, a number that ensures the solution is well converged. To prevent the western-coastal solution from propagating to the eastern boundary, the CUC solution includes the damper described in the introduction to Section H.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on τ^{x} wind patch Mixing:ν_{h} = 5x10^{6} cm^{2}/s, A = 0.00013 cm2/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) = cos[2π(x–60º)/40º], 40º ≤ x ≤ 80º, (I1a)

Y(y) = (1 + y2/Ly2)exp(y2/Ly2), (I1b)

T(t) = θ(t), (I1c)

Ly = 10º, τo = −2 dyn/cm2, and X(x) = 0 outside the designated range. The movie shows the adjustment of the LCS model to steady state for N = 25 modes, plotting a surface map of p (expressed in centimeters as sea level) and (u,v) vectors. See Experiments F1a and F1c for movies of the surface maps for the n = 1 and n = 5 modes separately.
If the movie is slowed (to 3%, say), the initial development of the Yoshida Jet is visible. Subsequently, Kelvin and Rossby waves radiate from the wind patch, and reflect from basin boundaries. Equatorial inertial oscillations are also excited, which last throughout the movie; towards the end of the movie, they appear as a pulsing of the equatorial currents.
In steady state, the low-order baroclinic modes (n ≈ 1−3), for which damping is weak, adjust to Sverdrup balance, as in Experiment F1a. Essentially, all the sea-level response is generated by the low-order modes. Note that in steady state there is a zonal pressure gradient (sea-level tilt) along the equator that balances τ^{x}. The intermediate modes (n ≈ 4−8) adjust to a state with a strong bounded Yoshida Jet as in Experiment F1c, so they are the modes that generate the strong equatorial current. For the higher-order modes (n ≥ 8), the Yoshida Jet is weak: These modes add up to generate the upwelling circulation discussed in the next two movies.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on τ^{x} wind patch Mixing:ν_{h} = 5x10^{6} cm^{2}/s, A = 0.00013 cm2/s^{3} Movie time step: daily Description:
The same solution as in Experiment I1a, except showing an equatorial (x,z) section of u (shading) and (u,w) vectors. Slow the movie down and step through the first few frames. In the first frames, there are prominent inertial oscillations that extend throughout the water column, there is surface upwelling driven by divergent Ekman drift in the mixed layer, and the Yoshida Jet accelerates at the ocean surface. As Kelvin and Rossby waves propagate across the wind patch, a near-surface pressure gradient is established that stops the Yoshida Jet from accelerating and drives the EUC. As the waves propagate farther from the forcing region, so does the EUC.
The reflection of Rossby waves from the eastern boundary is apparent after the arrival of the Kelvin waves. As the Rossby waves propagate away from the eastern boundary, phase tends to propagate upward (an indication that energy is radiating downwards). Eventually, the reflected waves generate a westward current beneath the EUC, the model’s Equatorial Intermediate Current (EIC). Even at the end of the movie, inertial oscillations remain, visible as a pulsing of the currents.

Domain: 20ºE–100ºE, 30ºS–30ºN Resolution: 0.25º Equatorial β-plane:f = βy Number of modes:N = 25 Forcing: switched-on τ^{x} wind patch Mixing:ν_{h} = 5x10^{6} cm^{2}/s, A = 0.00013 cm2/s^{3} Movie time step: daily Description:
The same solution as in Experiment I1a, except showing a meridional (y,z) section of u (shading) and (v,w) vectors at x = 60º. The spin-up of the solution is as described in Experiments I1a and I1b. The meridional structure of the equatorial currents can be seen. At the surface, there is a meridional overturning circulation, the “Tropical Cell” (TC). It consists of equatorward, geostrophic flow into the core of the EUC, equatorial upwelling, and poleward Ekman drift at the ocean surface. The cell is closed by downwelling in the tropics (not shown). (The strong TC is a limitation of this solution since the source of the water in the EUC arises from downwelling in the subtropical ocean to for the “Subtropical Cell.”)

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30ºN Number of modes:N = 25 Forcing: switched-on τ^{y} wind band Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
The wind has the form τ^{y} = τ_{o}X(x)Y(y)T(t), where

θ is a step function, Y(y) is a “top hat” function, and τo = –2 dyn/cm2, a band of wind confined between 20ºN and 40ºN. The movie shows the adjustment of the LCS model to steady state using N = 25 modes, plotting a surface map of p (expressed in centimeters as sea level) and (u,v) vectors. See Experiment H1c for a movie of the surface map for only the n = 1 mode.
If the movie is slowed (to 3%, say), the radiation of inertial oscillations from the forcing band via β-dispersion is apparent (see Experiments H1c and B4). Subsequently, Kelvin radiate poleward along the eastern coast, and Rossby waves radiate offshore. The structure of the n = 1 Rossby-wave packet is the same as that in Experiment H1c, including the trailing oscillations of shorter-wavelength Rossby waves. The n = 2 packet clearly separates from the eastern boundary about April, 1992, and the n = 3 packet separates about July, 1993. At the end of the movie, coastal currents still remain because the offshore propagation speed of higher-order baroclinic Rossby waves is so slow. Since there is no damping, however, Rossby waves for all the modes will eventually propagate offshore (at a time much longer than the movie length), and no coastal currents will remain.

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30ºN Number of modes:N = 25 Forcing: switched-on τ^{y} wind band Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
The same solution as in Experiment I2a, except showing an alongshore (y,z) section of v (shading) and (v,w) vectors. Slow the movie down (to 3%, say) and view the solution through February, 1991. During January, inertial oscillations are prominent in the response, and their meridional scale becomes increasingly small due to β-dispersion; during February the inertial oscillations weaken, and they are essentially gone by March 1.
At the same time, an alongshore current in the direction of the wind (southward) develops. As Kelvin waves propagate poleward along the coast, an alongshore pressure gradient develops to balance τ^{y}, the surface current stops accelerating, and a CUC appears at depth. As it strengthens, the CUC rises in the water column, a consequence of coastal adjustments associated with slower-propagating, higher-order-baroclinic Kelvin waves. After an initial period of growth, the coastal circulation weakens throughout the rest of the movie, a consequence of the continual offshore propagation of Rossby waves.

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30ºN Number of modes:N = 25 Forcing: switched-on τ^{y} wind band Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0 Movie time step: daily Description:
The same solution as in Experiment I2a, except showing an across-shore (x,z) section of v (shading) and (u,w) vectors. The spin-up of the solution is as described in Experiments I1a and I1b. During the adjustment to equilibrium, upward phase propagation of the coastal currents is apparent, an indication of the passage Kelvin modes associated with higher-order baroclinic modes. In addition, the offshore radiation of low-order Rossby waves can be seen, the n = 1 Rossby (no zero crossing in the upper 1000 m), then the n = 2 wave (one zero crossing), and finally the n = 3 wave (two zero crossings).

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30ºN Number of modes:N = 25 Forcing: switched-on τ^{y} wind band Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00065 cm2/s^{3} Movie time step: daily Description:
As in Experiment I2a, except with damping that weakens higher-order-mode Rossby waves before they can very far propagate offshore. In this movie, for example, only the n = 1 Rossby-wave packet is visible, whereas in Experiment I2a packets for the first 3 modes can be seen.
In steady state, the low-order baroclinic modes (n ≈ 1−2), for which damping is weak, adjust to Sverdrup balance, as in Experiments F1a and H1c: The coastal currents associated with these modes propagate offshore, and in their wake a meridional pressure gradient (sea-level tilt) is established that balances τ^{y}. Virtually all the sea-level response is generated by the low-order modes. For the intermediate modes (n ≈ 3−8), damping is strong enough for the Rossby waves to decay before they propagate very far offshore; as a result, the coastal current associated with these modes remain coastally trapped, and they are the modes that generate the alongshore currents. For the higher-order modes (n ≥ 8), the alongshore currents are weak: They add up to generate the upwelling circulation discussed in the next two movies.

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30ºN Number of modes:N = 25 Forcing: switched-on τ^{y} wind band Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00065 cm2/s^{3} Movie time step: daily Description:
The same solution as in Experiment I3a, except showing an alongshore (y,z) section of v (shading) and (v,w) vectors. The spin-up of the coastal currents is initially the same as in Experiment I2b. In contrast to Experiment I2b, however, the coastal currents do not later weaken; instead, they adjust to a steady state because higher-order-mode Rossby waves are damped before they can propagate offshore. In steady state, the surface jet and CUC extend well poleward of the forcing band (40°N), due to the Kelvin-wave propagation.

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30ºN Number of modes:N = 25 Forcing: switched-on τ^{y} wind band Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00065 cm2/s^{3} Movie time step: daily Description:
The same solution as in Experiment I3a, except showing an across-shore (x,z) section of v (shading) and (u,w) vectors. The spin-up of the solution is similar to that described in Experiments I1c, with upward phase propagation indicating the passage Kelvin waves associated with higher-order baroclinic modes. In contrast to Experiment I2c, however, only the offshore radiation of the n = 1 Rossby wave (no zero crossing in the upper 1000 m) and the n = 2 wave (one zero crossing) are visible, owing to the damping of the higher-order baroclinic modes. Furthermore, the coastal currents do not continue to shallow but rather adjust to a steady-state profile, again due to Rossby-wave damping. In steady state, there is a subsurface, onshore, geostrophic flow generated by the Rossby waves. At the coast, part of it upwells to feed the offshore Ekman drift, another part bends northward to supply water for the CUC, and the rest supplies water for weak coastal downwelling below about 300 m.

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