Summer school on "Dynamics of the North Indian Ocean" Interior-ocean waves

A) Interior-ocean waves (June 23)

This set of solutions illustrates properties of mid-latitude (off-equatorial) gravity, Rossby, and Kelvin waves. In each solution, an initial condition on p, and on u or v in two cases, is specified. The model is then integrated to determine how waves radiate from the initial state.

Domain: 80ºE–100ºE, Eq.–20ºN Resolution: 0.1º f-plane:f = 2Ωsinθ, θ = 10ºN Characteristic speed:c_{1} = 264 cm/s Forcing: initial, δ-function-like, p field located in the center of the domain Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: 15 minutes (each model time step) Description:
The initial state of the ocean is a pressure anomaly,

where R = 4Δx, Δx = 0.1º and p_{0}/g = 10,000/980 cm = 10.2 cm.In response, gravity waves radiate away from the center of the basin. Eventually, they reflect from basin boundaries, and propagate back into the interior of the basin.
According to the dispersion relation, gravity waves with the shorter (longer) wavelengths have faster (slower) group and phase speeds, with the fastest (slowest) speeds approaching c1 (0) as the wavelength goes to zero (infinity). As a result, at any point in the domain away from the forcing region, the wavelengths of the gravity waves increase in time, as faster-propagating, shorter-wavelength waves leave behind slower-propagating, longer-wavelength ones.
The radiation pattern in Exp A1 is consistent with these group-velocity properties. The leading wave front is composed of wavelengths of the order of the scale of the initial p field, R, and it advances at a speed very near c1. Subsequently, weaker oscillations with wavelengths that increase in time radiate from the basin center, as indicated by the expanding circles with different green shadings. These oscillations are weak because disturbances with long wavelengths are not a significant part of the initial, δ-function-like p field.

Domain: 20ºE–100ºE, 10ºN–50ºN Resolution: 0.25º f-plane:f = 2Ωsinθ, θ = 10ºN Characteristic speed:c_{1} = 264 cm/s Forcing: initial, large-scale p field in the center of the domain Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
In this case, the initial p field has the form p(x,y,0) = p0X(x)Y(y), where

and both X(x) and Y(y) are zero outside the designated ranges. This solution illustrates the adjustment of an initial, large-scale p field to geostrophic balance. When the movie is played very slowly, the radiation of weak gravity waves away from the initial disturbance is evident. Short-wavelength gravity waves are clearly followed by longer ones, consistent with group theory. After the radiation of the waves, there is an anticyclonic, geostrophic flow around the patch of high p. It decays very slowly due to horizontal viscosity, but that weak decay is not visible in the movie.

Domain: 20ºE–100ºE, 10ºN–50ºN Resolution: 0.25º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30º Characteristic speed:c1 = 264 cm/s Forcing: initial, large-scale p field in the center of the domain Mixing:ν_{h} = 10^{7} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As for Experiment A2a, except on the β-plane. The response is initially very similar to that of Experiment A2a in that gravity waves radiate away from the initial region, leaving behind a geostrophically-balanced circulation. Subsequently, however, the geostrophic circulation propagates westward as a Rossby wave. The Rossby-wave propagation speed is faster closer to the equator, so the Rossby wave tilts as it propagates westward.
The solution also has more subtle and interesting properties. 1) Slow the movie to about 3% of its initial speed. Then, a striking packet of gravity waves is visible shortly after t = 0. It first propagates southward to the southern boundary, and then reflects there to propagate northward; subsequently, individual gravity waves reflect at their critical latitudes to propagate southward again, and so on. This process only occurs on the β-plane, and is considered further in Experiments B. 2) After some time, shorter-wavelength Rossby waves begin to appear on the eastern edge of the main packet, because their group speed is slower than that of the longer waves that make up the main Rossby-wave packet.

Domain: 80ºE–100ºE, 0ºN–20ºN Resolution: 0.1º f-plane:f = 2Ωsinθ, θ = 10ºN Characteristic speed:c_{1} = 264 cm/s Forcing: initial, p field for a Kelvin wave at the eastern boundary Mixing:νv = 10^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
The initial p field has the form p(x,y,0) = p_{0}X(x)Y(y), where

X(x) = exp[(x–100º)(f/c1)], (A3a)

Y(y) = 0.5{1+cos[2π(y–10º)/10º]}, 5º ≤ y ≤ 15º, (A3b)

and Y(y) = 0 outside the designated range. The offshore decay scale of p(x,y,0) is R = c1/f, the width of a coastal Kelvin wave.
Although p(x,y,0) has the structure of a coastal Kelvin wave, the condition that v(x,y,0) = 0 is not consistent with a Kelvin wave (see Experiment A3b). Therefore, the adjustment necessarily involves the radiation of gravity waves, as well as a coastal Kelvin wave. Gravity waves clearly radiate away from the forcing region, and, at a fixed distance from the forcing region, their wavelength increases in time. After the Kelvin and gravity waves have radiated from the initial region, a geostrophic circulation remains, circulating around a patch of high p.

Domain: 80ºE–100ºE, 0ºN–20ºN Resolution: 0.1º f-plane:f = 2Ωsinθ, θ = 10ºN Characteristic speed:c_{1} = 264 cm/s Forcing: initial, p and v fields for a Kelvin wave at the eastern boundary Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment A3a, except that the initial state also includes v = p/c_{1}, the zonal velocity field that accompanies the Kelvin wave. With this addition, a pure Kelvin wave is produced, and no gravity waves and geostrophic circulation are generated.

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