GUIDE WAVE ANALYSIS AND FORECASTING - WMO

58
Time t
Initial state
E(f, θ; x, t)
Wave data
HINDCAST
numerical/manual
Wind
Atmospheric
ANALYSIS
Figure 5.1 — The elements of wave modelling
5.3 The wave energy-balance equation
The concepts described in Chapter 3 are represented in
wave models in a variety of ways. The most general
formulation for computer models based on the elements
in Figure 5.1 involves the spectral energy-balance equation
which describes the development of the surface
gravity wave field in time and space:
∂E
∂t +∇•cgE ( ) = S = Sin + Snl + Sds (5.1)
where:
E = E(f,θ,x,t) is the two-dimensional wave spectrum
(surface variance spectrum) depending
on frequency, f, and direction of propagation,
θ;
c g = c g(f,θ) is the deep-water group velocity;
S is the net source function, consisting of
three terms:
S in: energy input by the wind;
S nl: non-linear energy transfer by wavewave
interactions; and
S ds: dissipation.
GUIDE TO WAVE ANALYSIS AND FORECASTING
Wave growth
Propagation
Refraction
Shoaling
Non-linear
interactions
Dissipation
Atmospheric
forcing
∂E (f, θ; x, t)
∂t
=
-∇ • (c g E)
∂
- (∇ θ • cgE) ∂θ
+
S nl
+
S ds
+
S in
This form of the equation is valid for deep water with no
refraction and no significant currents.
5.4 Elements of wave modelling
The essence of wave modelling is to solve the energybalance
equation written down in Equation 5.1. This first
requires the definition of starting values for the wave
energy, or initial conditions which in turn requires
definition of the source terms on the right hand side of
Equation 5.1 and a method for solving changes as time
progresses.
5.4.1 Initial conditions
Time t + δt
Modified state
E(f, θ; x, t + δt)
FORECAST
Atmospheric
FORECAST
It is rare that we have a flat sea to work from, or that we
have measurements which completely characterize the
sea state at any one time.
For computer models, the usual course of action is
to start from a flat sea and “spin up” the model with the
winds from a period of several days prior to the period
of interest. We then have a hindcast derived for the
initial time. For operational models this has to be done