Chapter 3
Basics of Nonhydrostatic Modelling
Abstract This chapter introduces the reader to nonhydrostatic finite-difference
solvers of the Navier-Stokes equations. For a start, the ocean is treated as as vertical
slice. Flow and gradients of variables normal to this plane are assumed to vanish, and
the Coriolis force is ignored. Exercises address deep-water (short) surface gravity
waves, bottom-attached density-driven currents, internal waves, instabilities of vertical shear flows, lee waves, double-diffusive instability, double-diffusive layering
and free convection.
3.1 Level Models
The most common types of vertical level models used in the field of oceanography
are z-coordinate models and σ -coordinate models (Fig. 3.1). Z -coordinate models
are based on fixed vertical levels of vertical grid spacing Δz. A shortcoming of
z-coordinate models is a step structure of bottom topography which can lead to
certain problems. In contrast to this, coordinate surfaces in σ -coordinate models
follow the sea floor. The σ -coordinate is defined by:
σ =
h o + z
h o + η
where z is the Cartesian vertical coordinate, h o is undisturbed water depth, and η
is sea-surface elevation. Accordingly, σ varies from zero at the sea floor to unity
at the sea surface. Note that σ -coordinates respond to fluctuations of the sea level.
Hydrodynamic models based on σ -coordinates are suitable for coastal applications,
but steep bottom slopes can cause substantial truncation errors. Models based on
σ -coordinates are not considered in this book.
J. K¨ ampf, Advanced Ocean Modelling, DOI 10.1007/978-3-642-10610-1 3,
C
Springer-Verlag Berlin Heidelberg 2010
21
Basics of Nonhydrostatic Modelling
Abstract This chapter introduces the reader to nonhydrostatic finite-difference
solvers of the Navier-Stokes equations. For a start, the ocean is treated as as vertical
slice. Flow and gradients of variables normal to this plane are assumed to vanish, and
the Coriolis force is ignored. Exercises address deep-water (short) surface gravity
waves, bottom-attached density-driven currents, internal waves, instabilities of vertical shear flows, lee waves, double-diffusive instability, double-diffusive layering
and free convection.
3.1 Level Models
The most common types of vertical level models used in the field of oceanography
are z-coordinate models and σ -coordinate models (Fig. 3.1). Z -coordinate models
are based on fixed vertical levels of vertical grid spacing Δz. A shortcoming of
z-coordinate models is a step structure of bottom topography which can lead to
certain problems. In contrast to this, coordinate surfaces in σ -coordinate models
follow the sea floor. The σ -coordinate is defined by:
σ =
h o + z
h o + η
where z is the Cartesian vertical coordinate, h o is undisturbed water depth, and η
is sea-surface elevation. Accordingly, σ varies from zero at the sea floor to unity
at the sea surface. Note that σ -coordinates respond to fluctuations of the sea level.
Hydrodynamic models based on σ -coordinates are suitable for coastal applications,
but steep bottom slopes can cause substantial truncation errors. Models based on
σ -coordinates are not considered in this book.
J. K¨ ampf, Advanced Ocean Modelling, DOI 10.1007/978-3-642-10610-1 3,
C
Springer-Verlag Berlin Heidelberg 2010
21