The shallow water equations

By 0x7df, Fri 17 February 2017, modified Fri 17 February 2017, in category Atmospheric science

The governing equations in geophysical fluid dynamics can be simplified somewhat if the horizontal length and velocity scales are much larger than the vertical length and velocity scales, respectively.

They are commonly used to describe both the ocean and atmosphere, as well as estuaries, rivers and channels.

A set of assumptions are made:

  1. The fluid is incompressible.
  2. Vertical acceleration is small compared with horizontal accelerations, and hydrostatic balance can be assumed.

Making these assumptions results in the 3D shallow water equations (de Boer (2003). However it is much more common to make the additional assumption that:

  1. The horizontal velocity is constant with height/depth (there is no vertical shear).

Given these assumptions, the 2D shallow water equations are then derived by integrating over height to remove the vertical velocity, and therefore the vertical dimension altogether. (Note that this is not the same as assuming the vertical velocity is zero, which is not necessarily the case. If the bathymetry accounted for, the vertical velocity must be non-zero where the floor changes depth. The vertical velocity can be recovered from solutions of the shallow water equations via the continuity equation.)

In the following equations we will ignore viscosity (i.e we will work from the Euler fluid equations not the Navier-Stokes equations, although in some disciplines the fluid viscosity is important and can be included in the shallow water equtions). This renders the non-linear PDEs hyperbolic rather than parabolic. We shall also ignore diffusion.

Three-dimensional shallow water equtions

In general in 3D the conservation of momentum is expressed by:

$$ \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} + f\mathbf{k}\times\mathbf{v} = -\frac{1}{\rho}\nabla p - g\mathbf{\hat k} $$

where \(f\) is the Coriolis parameter, \(\mathbf{\hat k}\) is a unit vector pointing away from the centre of the planet. Also, \(\mathbf{u}\) is the three-dimensional velocity vector and \(\mathbf{v}\) is the two-dimensional horizontal velocity vector, and we have assumed the Coriolis force acts solely in the horizontal, since its vertical component is small compared with the gravitational and pressure gradient forces.

Alternatively:

$$ \frac{\partial u}{\partial t} + \frac{\partial uu}{\partial x} + \frac{\partial uv}{\partial y} + \frac{\partial uw}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + fv $$
$$ \frac{\partial v}{\partial t} + \frac{\partial vu}{\partial x} + \frac{\partial vv}{\partial y} + \frac{\partial vw}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial y} - fu $$
$$ \frac{\partial w}{\partial t} + \frac{\partial wu}{\partial x} + \frac{\partial wv}{\partial y} + \frac{\partial ww}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial z} - g $$

The conservation of mass under the assumption of incompressibility is:

$$ \nabla\cdot\mathbf{u} = 0 $$

or:

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$

The condition of hydrostatic balance, which is the first key assumption of the shallow water equations, is given by:

\begin{equation} \label{HydrostaticBalance} \frac{1}{\rho}\frac{\partial p}{\partial z} = -g \end{equation}

This is derived from the momentum equation describing \(w\) by assuming that the left- hand side is zero:

$$ \frac{Dw}{Dt} = \frac{\partial w}{\partial t} + \frac{\partial wu}{\partial x} + \frac{\partial wv}{\partial y} + \frac{\partial ww}{\partial z} = 0 $$

The hydrostatic balance equation can be immediately integrated from some arbitrary depth \(z\) up to the free surface, to give:

$$ p = p_0 + \rho g\left(h - z\right) $$

where \(p_0\) is the fixed pressure at the free surface (e.g. zero for the atmosphere, or atmospheric pressure for the ocean), and \(h\) is the elevation of the free surface above \(z = 0\).

Then:

$$ \frac{\partial p}{\partial x} = \rho g\frac{\partial h}{\partial x} $$

and similarly for \(\partial p/\partial y\). I.e.:

$$ -\frac{1}{\rho}\nabla_h p = -g \nabla_h h $$

The implication is that the horizontal pressure gradient force is a result only of horizontal variatons in the free surface height.

The remaining two components of the momentum equation are consequently:

\begin{equation} \label{3Du} \frac{\partial u}{\partial t} + \frac{\partial uu}{\partial x} + \frac{\partial uv}{\partial y} + \frac{\partial uw}{\partial z} = -g\frac{\partial h}{\partial x} + fv \end{equation}
\begin{equation} \label{3Dv} \frac{\partial v}{\partial t} + \frac{\partial vu}{\partial x} + \frac{\partial vv}{\partial y} + \frac{\partial vw}{\partial z} = -g\frac{\partial h}{\partial y} - fu \end{equation}

Equations \(\ref{3Du}\), \(\ref{3Dv}\) and \(\ref{3Dh}\) are the 3D shallow water equations.

Two-dimensional shallow water equations

To obtain the 2D shallow water equations we integrate the 3D equations over \(z\), to obtain depth- (or height-) averaged forms of the equations. The Liebniz integral rule is required since \(h\), which is one of the limits of the integration over \(z\), depends on \(x\), \(y\) and \(t\).

In 2D, the shallow water equations (neglecting diffusion, viscosity and friction) are:

$$ \frac{\partial h}{\partial t} + \frac{\partial hu}{\partial x} + \frac{\partial hv} {\partial y} = 0 $$
$$ \frac{\partial hu}{\partial t} + \frac{\partial hu^2}{\partial x} + \frac{\partial huv}{\partial y} = -g\frac{\partial h}{\partial x} + fhv $$
$$ \frac{\partial hv}{\partial t} + \frac{\partial huv}{\partial x} + \frac{\partial hv^2}{\partial y} = -g\frac{\partial h}{\partial y} -fhu $$

The shallow water equations can be written more compactly thus:

$$ \frac{\partial phi}{\partial t} + \mathbf{v}\cdot\nabla\phi + \phi\nabla\cdot\mathbf{v} = 0$$
$$ \frac{\partial\mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v} + \nabla\mathbf{\phi} + f\mathbf{\hat k}\times\mathbf{v} = 0 $$

where \(\phi = gh\).

The first equation is obtained by integrating the continuity equation:

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$

with respect to \(z\) to give:

$$ \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)h + w_h - w_0 = 0 $$

The vertical velocity disappears at the lower boundary (\(w_0 = 0\)) and the vertical velocity at the free surface (at height \(h\)) is given by:

$$ w_h = \frac{Dh}{Dt} $$

Thus:

$$ \frac{Dh}{Dt} = \frac{\partial h}{\partial t} + \frac{\partial hu}{\partial x} + \frac{\partial hv}{\partial y} = w_h = \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)h $$

One-dimensional shallow water equations

In 1D, which is often used for river management and hydraulics, the equations are:

$$ \frac{\partial u}{\partial t} = -u \frac{\partial u}{\partial x} -g \frac{\partial h}{\partial x} $$
$$ \frac{\partial h}{\partial t} = -u \frac{\partial h}{\partial x} -h \frac{\partial u}{\partial x} $$

where \(h\) is the height of the free surface. Accelerations in the \(y\) direction have also been assumed negligible.

In both equations the first term on the right represents advection and the second term on the right represents adjustment. (More complex variants are used in river modelling and hydraulics that account for, for example, the cross-sectional area of the channel. These are often referred to as the Saint Venant equations, having originally been derived by Adhemar Jean Claude Barre de Saint-Venant, in 1871.)

In hydraulics studies Coriolis force is ignored but in geophysics applications it is included, making the shallow water equations a useful simplistic framework in which to study rotational flow under gravity.

Comments

Add comment