Advanced Ocean Modelling: Using Open-Source Software

Chapter 3Basics of Nonhydrostatic ModellingAbstract This chapter introduces the reader to nonhydrostatic finite-differencesolvers of the Navier-Stokes equations. For a start, the ocean is treated as as verticalslice. Flow and gradients of variables normal to this plane are assumed to vanish, andthe Coriolis force is ignored. Exercises address deep-water (short) surface gravitywaves, bottom-attached density-driven currents, internal waves, instabilities of verticalshear flows, lee waves, double-diffusive instability, double-diffusive layeringand free convection.3.1 Level ModelsThe most common types of vertical level models used in the field of oceanographyare z-coordinate models and σ -coordinate models (Fig. 3.1). Z-coordinate modelsare based on fixed vertical levels of vertical grid spacing Δz. A shortcoming ofz-coordinate models is a step structure of bottom topography which can lead tocertain problems. In contrast to this, coordinate surfaces in σ -coordinate modelsfollow the sea floor. The σ -coordinate is defined by:σ = h o + zh o + ηwhere z is the Cartesian vertical coordinate, h o is undisturbed water depth, and ηis sea-surface elevation. Accordingly, σ varies from zero at the sea floor to unityat 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 onσ -coordinates are not considered in this book.J. Kämpf, Advanced Ocean Modelling, DOI 10.1007/978-3-642-10610-1 3,C○ Springer-Verlag Berlin Heidelberg 201021