Jax Approximate Ocean Models

Shallow Water Equations

J. Emmanuel Johnson
CNRS
MEOM

Examples

We have a few examples of how one can use the Shallow water equations to generate some simulations under different parameter regimes.

Linear Shallow Water Model

Jupyter Notebook

In this example, we look at the linearized shallow water model given by:

∂th+H(∂xu+∂yv)=0∂tu−fv=−g∂xh−κu∂tv+fu=−g∂yh−κv\begin{aligned} \partial_t h &+ H \left(\partial_x u + \partial_y v \right) = 0 \\ \partial_t u &- fv = - g \partial_x h - \kappa u \\ \partial_t v &+ fu = - g \partial_y h - \kappa v \end{aligned}
(1)

We demonstrate how we can use this for generating a simulation with idealistic Kelvin waves that move around the boundary and an idealistic jet across the East-West extent.

NonLinear Shallow Water Model

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In this example, we look at the nonlinear shallow water model given by:

Height:∂th+∂x(hu)+∂y(hv)=0Zonal Velocity:∂tu+u∂xu+v∂yu−fv=−g∂x(h+ηB)+Fx+Bx+Mx+ξxMeridonal Velocity:∂tv+u∂xv+v∂yv+fu=−g∂y(h+ηB)+Fy+By+My+ξy\begin{aligned} \text{Height}: && && \partial_t h + \partial_x (hu) + \partial_y (hv) &= 0\\ \text{Zonal Velocity}: && && \partial_t u + u \partial_x u + v\partial_y u - fv &= - g \partial_x (h + \eta_B) + F_x + B_x + M_x + \xi_x \\ \text{Meridonal Velocity}: && && \partial_t v + u \partial_x v + v\partial_y v + fu &= - g \partial_y (h + \eta_B) + F_y + B_y + M_y + \xi_y \end{aligned}
(2)

We demonstrate how we can use this for generating a simulation with an idealistic jet across the East-West extent.

Vorticity Formulation

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We can also write the shallow water equation as

Height:∂th=−∂x(hu)−∂y(hv)=0Zonal Velocity:∂tu=qhv−∂xp+Fx+Mx+Bx+ξxMeridonal Velocity:∂tv=−qhu−∂yp+Fy+My+By+ξy\begin{aligned} \text{Height}: && && \partial_t h &= -\partial_x (hu) - \partial_y (hv) = 0\\ \text{Zonal Velocity}: && && \partial_t u &= qhv - \partial_x p + F_x + M_x + B_x + \xi_x \\ \text{Meridonal Velocity}: && && \partial_t v &= - qhu - \partial_y p + F_y + M_y + B_y + \xi_y \end{aligned}
(3)

which is the vector invariant formulation that uses vorticity. We also showcase how we can use this formulation for the same parameter regime but with slightly different end results due to numerical computation differences.

Quasi-Geostrophic Model
QG Inversion
Shallow Water Model
Formulation