Turbulent boundary layers

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Models for the turbulent ocean surface boundary layer are partial differential equations that approximate the effects of atmospheric forcing on the turbulent vertical flux and evolution of large-scale temperature, salinity, and momentum fields.

Internal and surface fluxes of heat are due to

absorption of incoming shortwave solar radiation;
cooling by outgoing longwave radiation;
latent and sensible heat exchange with the atmosphere.

Surface fluxes of salinity occur due to evaporation and precipitation, while momentum fluxes are associated with atmospheric winds.

Vertical turbulent fluxes are typically associated with

gravitational instability and convection, and
mechanical turbulent mixing associated with currents and wind forcing.

OceanTurb.jl uses an implementation of atmospheric and radiative forcings that is shared across all models. The models therefore differ in the way they parameterize convective and mechanical mixing.

Coordinate system

We use a Cartesian coordinate system in which gravity points downwards, toward the ground or bottom of the ocean. The vertical coordinate $z$ thus increases upwards. We locate the surface at $z=0$. This means that if the boundary layer has depth $h$, the bottom of the boundary layer is located at $z=-h$.

Governing equations

The one-dimensional, horizontally-averaged boundary-layer equations for horizontal momentum $U$ and $V$, salinity $S$, and temperature $T$ are

\[\beqs U_t = f V - \d_z \overline{w u} + F_u \c \label{xmomentum} \\ V_t = - f U - \d_z \overline{w v} + F_v \c \\ T_t = - \d_z \overline{w \theta} + F_\theta \c \label{temperature} \\ S_t = - \d_z \overline{w s} + F_s \c \label{salinity} \\ \eeqs\]

where subscripts $t$ and $z$ denote derivatives with respect to time and the vertical coordinate $z$ and $f$ is the Coriolis parameter. The lowercase variables $u$, $v$, $s$, and $\theta$ refer to the three-dimensional perturbations from horizontal velocity, salinity, and temperature, respectively. In \eqref{xmomentum}–\eqref{temperature}, internal forcing of a variable $\Phi$ is denoted $F_\phi$.

Buoyancy

OceanTurb.jl uses a linear equation of state, so that buoyancy is deteremined from temperature $T$ and salinity $S$ via

\[\begin{align} B & \equiv - \frac{g \rho'}{\rho_0} \\ & = g \left [ \alpha \left ( T - T_0 \right ) - \beta \left ( S - S_0 \right ) \right ] \c \end{align}\]

where $g = 9.81 \, \mathrm{m \, s^{-2}}, \alpha = 2 \times 10^{-4} \, \mathrm{K^{-1}}$, and $\beta = 8 \times 10^{-5}$, are the default gravitational acceleration, the thermal expansion coefficient, and the haline contraction coefficient, respectively.

Surface fluxes

Turbulence in the ocean surface boundary layer is driven by fluxes from the atmosphere above. A surface flux of some variable $\phi$ is denoted $Q_\phi$. Surface fluxes include

Momentum fluxes due to wind, denoted $Q_u \b{x} + Q_v \b{y} = -\rho_0 \b{\tau}$ for wind stress $\b{\tau}$;
Temperature flux $Q_\theta = - Q_h / \rho_0 c_P$ associated with 'heating' $Q_h$;
Salinity flux $Q_s = (E-P)S$ associated evaporation $E$ and precipitation $P$.

We use the traditional convention ordinary to physics, but not always ordinary to oceanography, in which a positive flux corresponds to the movement of a quantity in the positive $z$-direction. This means, for example, that a positive vertical velocity $w$ gives rise to a positive advective flux $w \phi$. This convention also implies that a positive temperature flux at the ocean surface –- corresponding to heat fluxing upwards, out of the ocean, into the atmosphere –- implies a cooling of the ocean surface boundary layer.

Turbulent velocity scales

The surface buoyancy flux is determined from temperature and salinity fluxes:

\[\beq Q_b = g \left ( \alpha Q_\theta - \beta Q_s \right ) \p \eeq\]

The velocity scale of turbulent motions associated with buoyancy flux $Q_b$ and velocity fluxes $Q_u$ and $Q_v$ are

\[\beq w_\star \equiv | h Q_b |^{1/3} \qquad \text{and} \qquad u_\star \equiv | \b{\tau} / \rho_0 |^{1/2} \p \eeq\]

where $h$ is the depth of the 'mixing layer', or the depth to which turbulent mixing and turbulent fluxes penetrate, $\b{\tau}$ is wind stress, and $\rho_0 = 1028 \, \mathrm{kg \, m^{-3}}$ is a reference density.

Note that we also define a turbulent velocity scale for stabilizing buoyancy fluxes $Q_b < 0$, even though a stabilizing buoyancy flux suppresses, rather than generates, turbulence.