Mathematical Model WARM

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Short Description

The mathematical model WARM (Wave Ray Model) is a wave model which is based on the transport equation for the two-dimensional energy density spectrum. The equation is solved in frequency directional domain. The model is able to take into account several effects related to the influence of bathymetry, current velocity and water level, namely

  • refraction by bottom friction and current variations,
  • shoaling,
  • wave current interaction (wave blocking), as well as
  • non-linear dissipation of wave energy.

In a first computational step wave rays are determined along with computation of several relevant quantities defined in frequency directional domain. Computation of wave propagation can be alternatively carried through forward or backward in time and space. One of the following combinations can be selected:

  • with variation of water level
  • with current velocity
  • steady or
  • unsteady
  • without current velocity
  • without variation of water level.

Taking the influence of current refraction into account upstream propagation of waves may be blocked. This will happen in situations when wave and current speed are equal in size but oriented in opposite directions.

From these informations the components of the wave spectrum can be calculated. The amount of wave energy present is influenced by various processes which can be described by the following sources and sinks:

  • input of energy from the atmosphere (wind shear),
  • dissipation of energy caused by turbulent diffusion as well as
  • dissipation at the bottom.

Source functions are used as presented by H. Günther and W. Rosenthal (1995). Neglecting the non-linear interactions it was possible to show that the source functions are able to describe the normal growth of wave height precisely.

For this type of model no tuning of parameters needs to be carried through. Quality of model results do only depend on input data quality and the discretization chosen to represent the two-dimensional wave spectrum.

WARM was developed on behalf of Bundesanstalt für Wasserbau, Außenstelle Küste by GKSS Research Center, Geesthacht.

Along the open boundary of the modeling domain wave spectra may be optionally used as boundary conditions. They are calculated internally by the model. This calculation is based on fetch and duration of wind speed and direction.


Programs for Simulation

Simulation of steady wave propagation:

  • WARM: simulation run.

Simulation of unsteady wave propagation taking the interaction with current velocity and water level into account:

  • TRIM-2D: two-dimensional numerical simulation of water level and current velocity;
  • TR2GEOM: generation of bathymetry and computational grid;
  • TR2DIDA: conversion of hydrodynamic simulation results;
  • ZEITR: conversion of synoptic data sets into time series representations at each grid point;
  • FD2MET: generation of meteorological data used as boundary conditions for wind induced energy input.
  • WARM: simulation run.


  • GVIEW2D: graphical representation of calculated time series of water level elevation at selected locations;
  • FDGITTER05: two-dimensional graphical representation of wave rays in combination with bathymetry;
  • HVIEW2D: two-dimensional graphical representation of wave rays and wave heights in combination with bathymetry, current velocity an water level elevation.

Example Applications

  • Marinestützpunkt Warnemünde: Untersuchung der Seegangsverhältnisse in Unterwarnow und Breitling; please refer also to Anwendung des Seegangsmodells WARM published in No. 1/1998 of Supercomputing News (text is available in German only). This text illustrates the application of the model to study wave motion in the harbour of Rostock-Warnemünde.
  • Winkel, N., 1999, Investigations of Wave Climate in the Harbour of Rostock-Warnemünde with a Spectral Wave Model, Second German-Chinese Joint Seminar on Recent Developments in Coastal Engineering - The Sustainable Development in Coastal Zone, Tainan, Taiwan.

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