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2017

The shallow water equations

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.

Adjustment of meteorological fields

According to Riddaway and Hortal, the set of coupled non-linear partial differential equations solved in numerical weather prediction (NWP) describe three important dynamical processses:

  1. Advection
  2. Diffusion
  3. Adjustment

The first two are familiar to the non-meteorologist more used to working with the general equations of fluid motion. The third, adustment, is described as 'how the mass and wind fields adjust to one another'.

Fredholm integral equations and the resolvent kernel

Consider the inhomogeneous Fredholm integral equation:

\[ H \psi(\mu) = \lambda \int_a^b \sigma(\mu, \mu') \psi(\mu') d\mu' + S(\mu) \]

The unknown to be solved for is \(\psi(\mu)\) where \(a \le \mu \le b\) is the independent variable, and the known function \(\sigma(\mu, \mu')\) is known as the kernel. This is an inhomogeneous equation because the known function \(S(\mu) \ne 0\).