Ocean models simulate the state of the ocean, which is determined by the variables described in Sect. 21.1, analogous to atmospheric models, which simulate the state of the atmosphere. Compared to the atmosphere, ocean modeling has three major differences:
The thermodynamic equation of state is considerably more complicated. In numerical terms, however, this is not a major problem.
In the ocean there are no phase transitions. This is a major simplification, since it removes the need to compute phase transition rates and latent heat, and fewer tracers have to be advected.
While the upper boundary of the atmosphere is set more or less arbitrarily, the water column has a clear vertical extent $[-H,\eta]$, where $H$ is the depth and $\eta$ is the displacement of the water surface due to currents (tides, wind-driven circulation, and waves). The difficulties here are that $H$ can become zero (land), and that $\eta$ is time dependent.
The prognostic variables of ocean models are essentially those described in Sect. 21.1, albeit with a few peculiarities. The following is a list of prognostic variables in typical use; other variables may of course be employed instead, as long as those listed in Sect. 21.1 can be derived from them:
The horizontal velocities and the surface elevation are used directly as such.
Since ocean models almost always make use of the hydrostatic approximation, the vertical velocity is expressed in the p-system or with respect to another generalized vertical coordinate.
Instead of the temperature, the potential temperature is used, since it is a passive tracer and satisfies a simple continuity equation.
The density is used directly as such, since, in contrast to the potential density, it satisfies a simple continuity equation.
The salinity is used either as a specific quantity $q_s$ (mass of salt per mass of seawater) or as a volume-based quantity $\rho q_s$ (mass of salt per volume of seawater).
All other thermodynamic quantities can be derived from the potential temperature, the density, and the salinity.
Due to the challenge described in the previous section, the vertical coordinate plays a particularly important role in ocean modeling. In general, a vertical coordinate $\kappa$ is a bijective mapping
\[ \begin{align} z = z\left(\kappa\right). \end{align} \]
With z-coordinates (also called geopotential coordinates), the coordinate surfaces are exactly horizontal. The main advantage is the absence of metric terms in horizontal derivatives. At the coasts, the coordinate surfaces intersect the ocean floor, so that the number of layers depends on the horizontal coordinates. Owing to the surface displacement $\eta$, the thickness of the first layer is moreover time dependent. The bathymetry is thereby represented in a step-like fashion.
With $\sigma_z$-coordinates, the layers are fitted to the bathymetry so that they do not intersect the ocean floor. In shallow waters the layers thereby become very thin, and on slopes the coordinate surfaces run very steeply. The mathematical formulation is analogous to the terrain-following coordinates in the atmosphere described in Sect. 12.3.
Within the deeper ocean, the motions proceed predominantly along isopycnals. Aligning the coordinate surfaces with these is numerically advantageous, since vertical motions then almost vanish. This is, however, only possible if the ocean is stably stratified (otherwise the potential density within a water column cannot be mapped uniquely onto the geometric height), which is not the case everywhere, which is why these coordinates are not globally applicable. This is analogous to the isentropic coordinates in the atmosphere considered in Sect. 12.2.
$z^\star$-coordinates („z-star coordinates“) are a generalization of z-coordinates that resolves the problem that the thickness of the topmost layer depends on the surface displacement. The transformation to geometric coordinates reads
\[ \begin{align} z = \eta + z^\star\frac{\eta + H}{H}. \end{align} \]
The coordinate surfaces are therefore no longer exactly horizontal, but they still intersect the ocean floor.
All of the vertical coordinates listed so far have substantial advantages and disadvantages. Hybrid coordinates are an attempt to combine the advantages of the various coordinates with one another. Within the upper few tens of meters, the so-called thermocline, the ocean is well mixed and weakly stratified. Here, z- or $z^\star$-coordinates are appropriate. Further down, the ocean is dominated by quasi-geostrophic motions along the isopycnals (surfaces of equal potential density). In coastal regions, by contrast, terrain-following coordinates are appropriate in order to be able to represent the bathymetry. Hybrid coordinates are accordingly a mixture of $\rho-$, $\sigma_z-$, and $z^\star-$coordinates. They give the ocean model HyCOM (Hybrid Coordinate Ocean Model) its name.
Sea state is another word for water surface waves. The prognostic variable considered here is the displacement of the water surface
\[ \begin{align} h = h\left(x, y, t\right) \end{align} \]
from the mean position of the water surface.$h$ cannot simply be defined as a deviation from the geoid, since the dynamic topography would still be superimposed on it. This does not contribute to the displacement associated with the waves. The averaging interval must therefore be chosen accordingly. In many situations, $h$ is not a function of the horizontal coordinates, for example in surf or in a rough sea, because in such cases the position of the water surface is no longer uniquely defined. Such effects are accounted for later in the form of energy-dissipating source terms.
One could now use the shallow-water equations derived in Sect. 13.8.1 as a set of prognostic equations, possibly supplemented by some semi-empirical additional terms. However, this has several disadvantages:
In order to capture a significant portion of the energy of the wave field, a very small time step and a very fine spatial resolution would be required.
In most cases, the phases of the waves are not known and are not particularly relevant. Instead, quantities such as wave direction and wave height are decisive.
Therefore, water surface waves are usually described by means of a radiative transfer equation (RTE). The prognostic variable is then the spectral radiance
\[ \begin{align} N = N\left(\mathbf{k},\mathbf{r},t\right) \end{align} \]
Here $\mathbf{r}$ is a two-dimensional position vector and $\mathbf{k}$ is a two-dimensional wave vector. Usually, $\mathbf{k}$ is given not in Cartesian coordinates but in polar coordinates $\left(k,\theta\right)$. As the dispersion relation $\omega = \omega\left(k, \theta\right)$, one uses that of water surface waves
\[ \begin{align} \omega^2 &= gk\tanh\left(kD\right) \Rightarrow \omega = \sqrt{gk\tanh\left(kD\right)}, \end{align} \]
where $D$ is the mean water depth, i.e. the water depth without waves. From this it follows that
\[ \begin{align} c_\text{ph} &= \frac{\omega}{k} = \sqrt{\frac{g\tanh\left(kD\right)}{k}},\\ c_\text{gr} &= \frac{1}{2\omega}\frac{\partial\omega^2}{\partial k} = \frac{1}{2\omega}\left[g\tanh\left(kD\right) + \frac{gkD}{\cosh^2\left(kD\right)}\right] = \frac{g\tanh\left(kD\right)}{2\omega}\left[1 + \frac{kD}{\sinh\left(kD\right)\cosh\left(kD\right)}\right]\nonumber\\ &= \frac{c_\text{ph}}{2}\left[1 + \frac{2kD}{2\sinh\left(kD\right)\cosh\left(kD\right)}\right] = \frac{c_\text{ph}}{2}\left[1 + \frac{2kD}{\sinh\left(2kD\right)}\right]. \end{align} \]
Thus the group velocity is isotropic but not homogeneous,
\[ \begin{align} c_\text{gr} = c_\text{gr}\left(k,\mathbf{r},t\right). \end{align} \]
The spatial and temporal dependence arises through the spatial and temporal dependence of $D$.
From this, one can derive a spectral radiative flux density
\[ \begin{align} N\left(k, \theta,\mathbf{r},t\right)\left(c_\text{gr}\left(k,\mathbf{r},t\right) + \mathbf{v}\left(\mathbf{r}, t\right)\right) \end{align} \]
where $\mathbf{v} = \mathbf{v}\left(\mathbf{r}, t\right)$ is the current velocity. The radiative transfer equation is a kind of continuity equation for $N$, which is conceptually related to the conservation of energy:
\[ \begin{align} \frac{\partial N\left(k, \theta\right)}{\partial t} + \nabla\cdot\left(N\left(k, \theta\right)c_\text{gr}\left(k\right)\right) &= S_\text{nl}\left(k, \theta\right) + S_\text{ws}\left(k, \theta\right) + S_\text{wc}\left(k, \theta\right)\nonumber\\ & + S_\text{diss}\left(k, \theta\right) + S_\text{bd}\left(k, \theta\right)\tag{22.10}\label{eq:rte_water_surface} \end{align} \]
The spatial and temporal dependence is no longer written explicitly here. In addition, five source terms have been included:
$S_\text{nl}\left(k, \theta\right)$ is the energy source resulting from nonlinear effects. This term is particularly relevant for interactions across scales. One example is the emergence of long waves, the so-called swell, from the shorter-wave wind sea produced directly by the wind, via upscaling.
$S_\text{ws}\left(k, \theta\right)$ is the energy source resulting from interaction with the wind field. This is the main energy source of the wave field.
$S_\text{wc}\left(k, \theta\right)$ is the energy source resulting from spray, the so-called whitecapping. This is a dissipative and therefore usually negative term.
$S_\text{diss}\left(k, \theta\right)$ is the energy source associated with internal friction, i.e. viscosity. Since viscosity acts quadratically more strongly on the smaller scales, this effect causes wind sea to be dissipated faster than swell.
$S_\text{bd}\left(k, \theta\right)$ is the energy source associated with interaction with the bathymetry, bottom drag. This, too, is a dissipative and therefore usually negative term.
Depending on the particular situation, further source terms can be included. Eq. (22.10) is also called the wave action equation.
The total energy of the wave spectrum $E$ at a given location and time is the integral over the entire spectrum:
\[ \begin{align} E = \int_0^{2\pi}\int_0^\infty N\left(k,\theta\right)dkd\theta.\tag{22.11}\label{eq:wave_spectrum_total_energy} \end{align} \]
The energy-weighted spectral mean of a quantity $\psi$ is therefore given by
\[ \begin{align} \newoverline{\psi} \coloneqq \frac{1}{E}\int_0^{2\pi}\int_0^\infty\psi\left(k,\theta\right)N\left(k,\theta\right)dkd\theta.\tag{22.12}\label{eq:wave_spectral_average} \end{align} \]
The significant wave height is a kind of representative wave height for which
\[ \begin{align} H_s = 4\sqrt{E}. \end{align} \]
The mean wave direction $\theta_m$ is calculated from Eq. (22.12) as
\[ \begin{align} \theta_m = \arctan2\left(b,a\right) \end{align} \]
with
\[ \begin{align} a &\coloneqq \frac{1}{E}\int_0^{2\pi}\int_0^\infty\cos\left(\theta\right)N\left(k,\theta\right)dkd\theta,\nonumber\\ b &\coloneqq \frac{1}{E}\int_0^{2\pi}\int_0^\infty\sin\left(\theta\right)N\left(k,\theta\right)dkd\theta. \end{align} \]
For the mean wavelength $L_m$, one has
\[ \begin{align} L_m = \newoverline{\left(\frac{2\pi}{k}\right)} = 2\pi\newoverline{k^{-1}}. \end{align} \]
There are different ways of calculating the mean wave period:
\[ \begin{align} T_{m,1} &= \frac{2\pi}{\newoverline{\sigma}},\\ T_{m,2} &= \frac{2\pi}{\sqrt{\newoverline{\sigma^2}}},\\ T_{m,-1} &= \newoverline{\left(\frac{2\pi}{\sigma}\right)} = 2\pi\newoverline{\sigma^{-1}}. \end{align} \]
Here $\sigma$ is the angular frequency measured relative to the seabed.
The most common sea-state forecast model is Wavewatch III. Models based on Eq. (22.10) are so-called third-generation sea-state forecast models. They solve this equation on a grid and are therefore grid-point models, even though they are sometimes referred to as spectral models because their prognostic variable has spectral meaning. They solve Eq. (22.10) by introducing, at each grid point, a spectral direction-dependent grid in $\left(k,\theta\right)$ space. In most cases, a fairly large variety of semi-empirical source terms $Q_i$ is also included.