21 Ocean circulation

21.1 Wind-driven circulation

The derivation of the Ekman spiral carried out in Section 17.3 can be transferred to a flat ocean floor without modifications. On the ocean surface, however, the adhesion condition $\lim_{\mathbf{r} \to \partial V}\mathbf{v} = \mathbf{0}$ is missing; rather, it would make sense to assume that on the ocean surface wind speed and current speed converge towards a common limit value. The Ekman spiral would then have to be solved simultaneously in the ocean and atmosphere. However, there is an easier way to fundamentally study the influence of wind on ocean circulation. To do this, first set the balance of forces in the equations (17.35) - (17.36):

\[ \begin{align} f\mathbf{k}\times\mathbf{v}_{h} = -\nabla\phi + \frac{1}{\rho_0}\frac{\partial\mathbf{\tau}}{\partial z} \end{align} \]

It is now assumed that the vertical density gradients are so small that the term $\frac{1}{\rho_0}$ can be included in the partial derivative, i.e

\[ \begin{align} f\mathbf{k}\times\mathbf{v}_{h} = -\nabla\phi + \frac{\partial}{\partial z}\left(\frac{\mathbf{\tau}}{\rho_0}\right). \end{align} \]

From here you now apply the rotation, so

\[ \begin{align} \nabla\times f\mathbf{k}\times\mathbf{v}_{h} = \nabla\times\frac{\partial}{\partial z}\left(\frac{\mathbf{\tau}}{\rho_0}\right).\tag{21.3}\label{eq:wind-driven_circ_deriv_1} \end{align} \]

With Eq. (B.53) follows

\[ \begin{align} \nabla\times f\mathbf{k}\times\mathbf{v}_{h} &= \left(\mathbf{v}_{h}\cdot\nabla\right)f\mathbf{k} - \mathbf{v}_{h}\left(\nabla\cdot f\mathbf{k}\right) + f\mathbf{k}\nabla\cdot\mathbf{v}_{h} - \left(f\mathbf{k}\cdot\nabla\right)\mathbf{v}_{h} \end{align} \]

Projection onto the vertical direction results

\[ \begin{align} \mathbf{k}\cdot\nabla\times f\mathbf{k}\times\mathbf{v}_{h} &= \mathbf{k}\cdot\left(\mathbf{v}_{h}\cdot\nabla\right)f\mathbf{k} + f\nabla\cdot\mathbf{v}_{h}\nonumber\\ \Rightarrow \mathbf{k}\cdot\nabla\times f\mathbf{k}\times\mathbf{v}_{h} &= v\beta - f\frac{\partial w}{\partial z}. \end{align} \]

Integration over the height interval $\left[z_B, z_T\right]$ thus yields

\[ \begin{align} \beta\int_{z_B}^{z_T}vdz - f\left(w_T - w_B\right) = \mathbf{k}\cdot\frac{1}{\rho_0}\nabla\times\left(\mathbf{\tau}_T - \mathbf{\tau}_B\right). \end{align} \]

With the definitions

\[ \begin{align} D \coloneqq z_T - z_B, & {} & \newoverline{v} \coloneqq \frac{1}{D}\int_{z_B}^{z_T}vdz \end{align} \]

you can do this shorter than

\[ \begin{align} \beta D\newoverline{v} - f\left(w_T - w_B\right) = \mathbf{k}\cdot\frac{1}{\rho_0}\nabla\times\left(\mathbf{\tau}_T - \mathbf{\tau}_B\right) \end{align} \]

note down. Since this concerns the entire water column, one can use the simple boundary condition

\[ \begin{align} w_T = w_B = 0 \end{align} \]

apply. This gives you

\[ \begin{align} \beta D\newoverline{v} = \mathbf{k}\cdot\frac{1}{\rho_0}\nabla\times\left(\mathbf{\tau}_T - \mathbf{\tau}_B\right) \end{align} \]

To parameterize the friction term at the bottom, one uses the Stommel model, which reads

\[ \begin{align} \mathbf{\tau}_B = r\newoverline{\mathbf{v}_{h}} \end{align} \]

with a constant $r > 0$. With $w = 0$ we also have $\nabla\cdot \mathbf{v}_{h} = 0$, which is why the average wind can be expressed by a current function $\psi$:

\[ \begin{align} \newoverline{u} = -\frac{\partial\psi}{\partial y}, & {} & \newoverline{v} = \frac{\partial\psi}{\partial x} \end{align} \]

So you can take notes

\[ \begin{align} \beta D\frac{\partial\psi}{\partial x} + \mathbf{k}\cdot\frac{1}{\rho_0}\nabla\times\mathbf{\tau}_B &= \mathbf{k}\cdot\frac{1}{\rho_0}\nabla\times\mathbf{\tau}_T\nonumber\\ \Leftrightarrow \beta D\frac{\partial\psi}{\partial x} + \frac{r}{\rho_0}\Delta\psi &= \mathbf{k}\cdot\frac{1}{\rho_0}\nabla\times\mathbf{\tau}_T\nonumber\\ \Leftrightarrow \frac{\partial\psi}{\partial x} + \frac{r}{\rho_0\beta D}\Delta\psi &= \frac{1}{\rho_0\beta D}\mathbf{k}\cdot\nabla\times\mathbf{\tau}_T\nonumber \end{align} \]

with

\[ \begin{align} \epsilon \coloneqq \frac{r}{\rho_0\beta D}. \end{align} \]

Eq. (21.13) is the equation of motion of the Stommel model. It applies

\[ \begin{align} \frac{\frac{\partial\psi}{\partial x}}{\epsilon\Delta\psi} \sim \frac{L\rho_0\beta D}{r} \end{align} \]

with $L$ as the horizontal length scale. For $r$ you scale

\[ \begin{align} fV \sim \frac{Vr}{D\rho_0} \Rightarrow r \sim D\rho_0 f. \end{align} \]

Thus follows

\[ \begin{align} \frac{\frac{\partial\psi}{\partial x}}{\epsilon\Delta\psi} \sim \frac{L \beta}{f} \sim \frac{L}{a} \end{align} \]

As a rough approximation, one can conceptually say for very large scales

what is called Sverdrup balance. From this, the ocean current can be derived diagnostically for a given wind effect.

Now Eq. (21.13) solved on the set $\left[0, L\right] \times \left[0, L\right]$ with $L > 0$, i.e. for a square basin. This is only possible analytically approximately. For this one starts from a wind stress of the form

\[ \begin{align} \tau_T^{(x)} &= -\rho_0U^2\cos\left(\pi\frac{y}{L}\right),\\ \tau_T^{(y)} &= 0 \end{align} \]

out of. From this it follows

\[ \begin{align} \frac{\epsilon}{r}\mathbf{k}\cdot\nabla\times\mathbf{\tau}_T = -\frac{\epsilon\pi}{rL}\rho_0U^2\sin\left(\pi\frac{y}{L}\right) \end{align} \]

You make the approach

\[ \begin{align} \psi = \psi_I + \phi, \end{align} \]

where $\psi_I$ should approximate the solution inside the basin and $\phi$ should add the edge effects. Inside the basin, the larger scales are relevant, so the Sverdrup balance is used for $\psi_I$:

\[ \begin{align} \frac{\partial\psi_I}{\partial x} &= \frac{\epsilon}{r}\mathbf{k}\cdot\nabla\times\mathbf{\tau}_T = -\frac{\epsilon\pi}{rL}\rho_0U^2\sin\left(\pi\frac{y}{L}\right) \end{align} \]

to. This is solved by

\[ \begin{align} \psi_I = \left(C - x\right)\frac{\epsilon\pi}{rL}\rho_0U^2\sin\left(\pi\frac{y}{L}\right). \end{align} \]

with a constant $C$. The remaining part of the differential equation should be solved by $\phi$:

\[ \begin{align} \epsilon\Delta\phi = 0 \end{align} \]

This is solved by

\[ \begin{align} \phi = -B\frac{\epsilon\pi}{rL}\rho_0U^2\sin\left(\pi\frac{y}{L}\right)\exp\left(\pm\pi\frac{x}{L}\right) \end{align} \]

with a constant $B.$ The overall solution is therefore

\[ \begin{align} \psi = \psi_I + \phi = \left(C - x - B\exp\left(\pm\pi\frac{x}{L}\right)\right)\frac{\epsilon\pi}{rL}\rho_0U^2\sin\left(\pi\frac{y}{L}\right). \end{align} \]

This follows for the vertically averaged velocities

\[ \begin{align} \newoverline{u} &= -\frac{\partial\psi}{\partial y} = -\frac{\pi}{L}\left(C - x - B\exp\left(\pm\pi\frac{x}{L}\right)\right)\frac{\epsilon\pi}{rL}\rho_0U^2\cos\left(\pi\frac{y}{L}\right),\tag{21.28}\label{eq:stommel_result_u}\\ \newoverline{v} &= \frac{\partial\psi}{\partial x} = \left(-1 \mp B\frac{\pi}{L}\exp\left(\pm\pi\frac{x}{L}\right)\right)\frac{\epsilon\pi}{rL}\rho_0U^2\sin\left(\pi\frac{y}{L}\right).\tag{21.29}\label{eq:stommel_result_v} \end{align} \]

$B$ und $C$ werden aus den Randbediungungen abgeleitet. Diese lauten

\[ \begin{align} \newoverline{u}\left(x = 0, L\right) = 0, & {} & \newoverline{v}\left(y = 0, L\right) = 0. \end{align} \]

The boundary conditions for $\newoverline{v}$ are automatically fulfilled. From which it follows for $\newoverline{u}$

\[ \begin{align} C = B, & {} & B - L - B\exp\left(-\pi\frac{L}{L}\right) = 0 \Rightarrow B = \frac{L}{1 - \exp\left(-\pi\right)}, \end{align} \]

whereby the negative sign in the exponential function was arbitrarily determined. This follows

21.2 Thermohaline circulation

21.3 Ocean models

21.4 Sea state forecast

21.4.1 Prognostic variables

Sea state is another word for water surface waves. The prognostic variable targeted is the deflection 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 here. This does not count towards the deflection associated with the waves. The length of the averaging interval must be chosen accordingly. In many situations $h$ is not a function of the horizontal coordinates, such as in the case of surf or turbulent seas, since in these cases the position of the water surface is no longer clearly defined. Such effects are taken into account later in the form of energy dissipating source terms.

The shallow water equations derived in Section 13.8.1 could now be used as a set of prognostic equations, possibly with 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 very relevant. Rather, variables such as the direction and height of the waves are crucial.

Therefore, to describe water surface waves, a radiation transfer equation (radiative transfer equation (RTE)) is usually used. The spectral radiance is therefore used as a prognostic variable.

\[ \begin{align} N = N\left(\mathbf{k},\mathbf{r},t\right) \end{align} \]

used. Here $\mathbf{r}$ is a two-dimensional position vector and $\mathbf{k}$ is a two-dimensional wave vector. Usually $\mathbf{k}$ is not given in Cartesian coordinates, but in polar coordinates $\left(k,\theta\right)$. The dispersion relation $\omega = \omega\left(k, \theta\right)$ is that of the water surface waves

\[ \begin{align} \omega^2 &= gk\tanh\left(kD\right) \Rightarrow \omega = \sqrt{gk\tanh\left(kD\right)}, \end{align} \]

used, where $D$ is the average water depth (water depth without waves). Follow from this

\[ \begin{align} c_\text{ph} &= \frac{\omega}{k} = \sqrt{\frac{g\tanh\left(kD\right)}{k}},\\ c_\text{gr} &= \frac{1}{2\omega}\frac{\partial\omega^2}{\partial k} = \frac{1}{2\omega}\left[g\tanh\left(kD\right) + \frac{gkD}{\cosh^2\left(kD\right)}\right] = \frac{g\tanh\left(kD\right)}{2\omega}\left[1 + \frac{kD}{\sinh\left(kD\right)\cosh\left(kD\right)}\right]\nonumber\\ &= \frac{c_\text{ph}}{2}\left[1 + \frac{2kD}{2\sinh\left(kD\right)\cosh\left(kD\right)}\right] = \frac{c_\text{ph}}{2}\left[1 + \frac{2kD}{\sinh\left(2kD\right)}\right]. \end{align} \]

So 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 location and time dependence arises via the location and time dependence of $D$.

21.4.2 Wave action equation

From this one can derive a spectral radiation flux density

\[ \begin{align} N\left(k, \theta,\mathbf{r},t\right)\left(c_\text{gr}\left(k,\mathbf{r},t\right) + \mathbf{v}\left(\mathbf{r}, t\right)\right) \end{align} \]

derive, here $\mathbf{v} = \mathbf{v}\left(\mathbf{r}, t\right)$ is the current velocity. The radiation transfer equation is a kind of continuity equation for $N$, which is conceptually related to the conservation of energy:

The location and time dependency was no longer noted. Furthermore, five additional source terms were included:

• $S_\text{nl}\left(k, \theta\right)$ is the energy source resulting from nonlinear effects. This term is particularly relevant for the interaction of scales. An example is the creation of long waves (so-called swell) from the wind sea (waves arising directly from the action of wind), which consists of shorter waves, via upscaling.
• $S_\text{ws}\left(k, \theta\right)$ is the energy source resulting from the interaction with the wind field. This is the main source of energy in the wave field.
• $S_\text{wc}\left(k, \theta\right)$ is the energy source that results from the 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 related to the internal friction (viscosity). Since the viscosity has a stronger square effect on the smaller scales, this effect leads to wind sea being dissipated faster than swell.
• $S_\text{bd}\left(k, \theta\right)$ is the energy source that is related to the interaction with the bathymetry (bottom drag), this is also a dissipative and therefore usually negative term.

Depending on the specific situation, additional source terms can be included. Eq. (21.40) is also called wave action equation.

The total energy of the wave spectrum $E$ at a certain location at a certain time is the integral over the entire spectrum:

\[ \begin{align} E = \int_0^{2\pi}\int_0^\infty N\left(k,\theta\right)dkd\theta.\tag{21.41}\label{eq:wave_spectrum_total_energy} \end{align} \]

The energy-weighted spectral mean of a quantity $\psi$ is therefore calculated via

\[ \begin{align} \newoverline{\psi} \coloneqq \frac{1}{E}\int_0^{2\pi}\int_0^\infty\psi\left(k,\theta\right)N\left(k,\theta\right)dkd\theta.\tag{21.42}\label{eq:wave_spectral_average} \end{align} \]

21.4.3 Diagnostic variables

21.4.3.1 Significant wave height

The significant wave height is a kind of representative wave height for which the following applies

\[ \begin{align} H_s = 4\sqrt{E}. \end{align} \]

21.4.3.2 Medium wave direction

The mean wave direction $\theta_m$ is calculated according to Eq. (21.42) as

\[ \begin{align} \theta_m = \arctan2\left(b,a\right) \end{align} \]

with

\[ \begin{align} a &\coloneqq \frac{1}{E}\int_0^{2\pi}\int_0^\infty\cos\left(\theta\right)N\left(k,\theta\right)dkd\theta,\nonumber\\ b &\coloneqq \frac{1}{E}\int_0^{2\pi}\int_0^\infty\sin\left(\theta\right)N\left(k,\theta\right)dkd\theta. \end{align} \]

21.4.3.3 Medium wavelength

For the mean wavelength $L_m$ applies

\[ \begin{align} L_m = \newoverline{\left(\frac{2\pi}{k}\right)} = 2\pi\newoverline{k^{-1}}. \end{align} \]

21.4.3.4 Mean wave period

There are different ways to calculate 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.

21.4.4 Sea state forecast models

The most common sea state forecast model is Wavewatch III. Models that follow Eq. (21.40) are so-called third generation sea state prediction models. They solve this equation on a grid and are therefore grid point models, although they are sometimes referred to as spectral models because their prognostic variable has spectral significance. You solve Eq. (21.40), where at each grid point a spectral direction-dependent grid is spanned in $\left(k,\theta\right)$ space. In most cases, a fairly large variety of semi-empirical source terms $Q_i$ are also included.