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Motivation

In the fields of surface water waves and plasma, it is well-known that the oblique interaction of two solitary waves could enhance the amplitude by a factor of up to 4 (i.e., twice the linear superposition) through the development of a (Mach) stem^{1),8),12),14),15),17)}(Fig. 1). However, this process has seldom been investigated for internal solitary-like waves (ISWs, or internal solitons) in the ocean, despite observations of its surface signature by remote sensing^{4),11),16),18),19)}. We investigate oblique ISW interaction in the Andaman Sea with particular focus on the effects of topography and Earth's rotation, because they make the oblique ISW interaction in the ocean different from that in other fields of physics. This study began when my collaborator, Prof. Keisuke Nakayama, identified the occurrence of oblique ISW interaction in early results of my MITgcm ISW modelling, and introduced me to the topic.

Methods

To analyse oblique ISW interaction under realistic oceanic conditions, we use the Miles theory¹³⁾ but with the following extensions:

• Small-but-finite incident angles, based on studies on surface water waves^7),9),20);
• Unequal incident wave amplitudes, based on mathematical studies⁷⁾; and
• Transitional response, based on studies in plasma physics⁶⁾.

The theory for the asymptotic state (after sufficiently long interaction) depends on the following parameters: \begin{align} a_s&=\frac{A_s}{\bar{A}}, \\ \kappa&=\frac{\text{tan} \Psi}{\sqrt{2\alpha \bar{A}/c} \text{cos} \Psi}, \\ \sigma&=\frac{1}{2}\sqrt{\frac{\Delta A}{\bar{A}}}, \\ \sqrt{\alpha \bar{A}}&=\left( \sqrt{\alpha A_2}+\sqrt{\alpha A_1} \right)/2, \\ \sqrt{\alpha \Delta A}&= \sqrt{\alpha A_2}-\sqrt{\alpha A_1}, \end{align}

where $A_1$, $A_2$, $A_s$ are respectively the amplitude of the first incident, second incident, and stem-like ISWs, $\alpha$ is the quadratic nonlinear coefficient in the KdV theory, and $\Psi$ is the incident angle. The theory shows that the maximum amplification $a_s=4$ occurs when $\kappa=1$ (Fig. 2).

For our 3-D ISW modelling, we use MITgcm¹⁰⁾, realistic topography and stratification, and (spatially-varying) monochromatic M₂ tidal forcing. To investigate the effects of Earth’s rotation, we consider the following 3 cases:

• Full Coriolis (including both vertical and horizontal components of Earth’s angular velocity);
• Traditional Coriolis (including only the vertical component)²⁾; and
• No Coriolis.

Results

The model output for Full Coriolis case (Fig. 3) shows that the interaction of ISWs with ~10-m and ~20-m amplitudes ('Incident 1' and 'Incident 2') produced an ISW with >50-m amplitude (labelled as 'Stem-like'), confirming the nonlinear enhancement of ISW amplitudes under realistic topographic and Coriolis effects. After adjusting the amplitudes for significant lateral topographic amplitude variation (by a factor of up to 2) and considering the transitional response, the simulated interaction is found to be approximately consistent with the extended Miles theory (i.e., $a_s \approx 4 $ when $\kappa \approx 1$ after sufficiently long interaction in Fig. 4). Although Earth's rotation has little effect on ISW propagation due to the low latitude (~9°N), Earth's rotation has considerable effects on the oblique interaction.

Implications

• Amplitude variation due to lateral topographic effects does not appear to have been studied, and we had to resort to empirical amplitude adjustment. It needs to be investigated in the future.
• Since the horizontal component of Earth's rotation is unimportant, the effects of Earth's rotation can be studied using rotation-modified Kadomtsev-Petviashvili (KP) equation^3),5).
• Oblique ISW interaction is potentially important for mixing, sediment resuspension, and offshore structures and operations, because it could decrease the gradient Richardson number, and increase bottom shear stress and forces on offshore structures, by an order of magnitude.
• Although it appears difficult to estimate the spatial and temporal distribution of stem-like ISW development with the current observational and modelling capabilities, potential observation by the next-generation swath-wide satellite altimeter might begin to shed light on the distribution (Fig. 5).

Related Publications

Details are available in

Shimizu, K., and K. Nakayama. 2017. Effects of topography and Earth's rotation on the oblique interaction of internal solitary-like waves in the Andaman Sea. Journal of Geophysical Research - Oceans, 122: 7449–7465, doi:10.1002/2017JC012888.

Data are available in

Shimizu, K., and K. Nakayama. 2017. Data from: Effects of topography and Earth's rotation on the oblique interaction of internal solitary-like waves in the Andaman Sea. Zenodo, doi:10.5281/zenodo.817616.

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