# American Institute of Mathematical Sciences

August  2019, 39(8): 4533-4545. doi: 10.3934/dcds.2019186

## Shallow water models for stratified equatorial flows

 1 Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands 2 KTH Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25,100 44 Stockholm, Sweden

* Corresponding author: Ronald Quirchmayr

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin.

Received  August 2018 Revised  October 2018 Published  May 2019

Fund Project: Both authors acknowledge the support of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) during the program "Mathematical Aspects of Physical Oceanography". R. Quirchmayr acknowledges the support of the European Research Council, Consolidator Grant No. 682537.

Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a detailed analysis of the effect of a given density distribution on the depth-structure of the corresponding traveling waves.

Citation: Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186
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Fig. 1a illustrates the fluid domain in the physical $({\bar x}, {\bar z})$-plane between the flat bed at ${\bar z}= 0$ and the free surface ${\bar z} = {\bar h}_0+{\bar \eta }(\cdot, {\bar t})$ at a certain instant of time ${\bar t}$. The average water level ${\bar h}_0$ is indicated by a dashed line, ${{\bar \lambda } }$ shows the distance between two consecutive crests and ${\bar a}$ is the vertical deviation of a typical crest from ${\bar h}_0$. Fig. 1b shows a prescribed depth dependent density distribution ${\bar \rho }({\bar z})$ with a significant density increase in the region between the two dotted horizontal lines close to the surface giving rise to a pycnocline
Solitary traveling wave solutions (26) of the surface equation (20) with linear density function $\rho (z) = 1+A(1-z)$. In Fig. 2a we see that for larger values of the parameter $A>0$ the profile becomes taller and also wider. In Fig. 2b we see how the amplitude of solutions decreases with depth $z\in [0, 1]$ like $(Az)^{-1}$ according to (27)
In Fig. 3a we see a plot of the density profile $\rho (z) = a_0-a_1\arctan\left(a_2(z-a_3)\right)$ defined in (29) for suitable choices of parameter values $a_i\in\mathbb{R}$ to model a density increase of $1\%$ from surface to bed. Fig. 3b shows a schematic representation of the fact that the amplitude of solitary wave solutions $\phi(\xi-c\tau, z)$ decays with depth inversely proportional to $\rho(z)$, cf. (27)
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