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Oceanic Boundary Layer

Introduction and Sverdrup Transport

Typically, the ocean can be separated into 3 distinct layers: 1) Mixed (surface) layer, 2) Thermocline, and 3) Deep layer.

The first layer (surface) can be thought of as the oceanic boundary layer, which is analogous to the atmospheric planetary boundary layer (PBL); this layer tends to have its temperature constant with depth (since it is a well-mixed layer) and can vary in depth from 25 to 500 meters in the vertical (with larger values occurring during the winter season as a result of more intense sea activity during this time of year). The second layer (thermocline) starts near the bottom of the mixed layer, as temperature begins to decrease with increasing depth, making the thermocline a very stable oceanic layer with slow vertical mixing processes. The third layer (deep layer) can be characterized using the concept of Sverdrup transport, where there is a balance between the advection of planetary vorticity (the beta effect) and divergence:

$\beta v=f{\frac {\partial w}{\partial z}}$,

where "β" is the variation of the Coriolis parameter ("f") with latitude, "v" is meridional wind, and "w" is vertical velocity. This interior transport is driven by the surface wind stress (as is the transport within the Ekman layer); any clockwise (anticyclonic) wind stress will result in northerly flow in the deep ocean layer. Therefore, the anticyclonic subtropical gyres give rise to southward flow, which can be explained using the concept of the conservation of Ertel's potential vorticity, which can be written as

$P=-g\left(\zeta _{\theta }+f\right){\frac {\partial \theta }{\partial p}}$,

where "P" is conserved during adiabatic motions (note that "ζ" is relative vorticity, "f" is planetary vorticity or Coriolis parameter, and "θ" is potential temperature). Thus, as fluid descends to the bottom of the ocean, the spacing between the potential temperature surfaces begins to decrease such that ${\frac {\partial \theta }{\partial p}}$ increases in magnitude. Therefore, either relative or planetary vorticity must decrease to compensate for the increase in stability. If the relative vorticity changes, then the depth of the fluid must change as well; thus, the planetary vorticity would be required to decrease, causing a net southward flow.

Primitive Equations

There are several equations that tend to govern ocean dynamics; these equations are almost identical to those that govern atmospheric motions. They are as follows (neglecting frictional or viscous effects, which are only important near thin layers at boundaries):

• the U-momentum equation:

${\frac {Du}{Dt}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}+fv+{\frac {1}{\rho }}{\frac {\partial \tau _{x}}{\partial z}}$

• the V-momentum equation:

${\frac {Dv}{Dt}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}-fu+{\frac {1}{\rho }}{\frac {\partial \tau _{y}}{\partial z}}$

• the W-momentum equation (accounting for the fact that large-scale flow within the ocean is approximately in hydrostatic balance):

${\frac {\partial p}{\partial z}}=-\rho g$

• the Continuity equation (accounting for the incompressibility of the ocean):

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$

• the Thermodynamic Energy equation:

${\frac {\partial T}{\partial t}}+u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}+w{\frac {\partial T}{\partial z}}={\frac {J}{c_{p}}}$.

Note that "u" is zonal velocity, "v" is meridional velocity, "w" is vertical velocity, "p" is pressure, "ρ" is density, "T" is temperature, "g" is acceleration due to gravity, "τ" is wind stress, and "f" is the Coriolis parameter.

Oceanic Boundary Layer

The mixed layer dynamics can be derived using Ekman's theory, which utilizes the U-momentum and V-momentum equations. These 2 equations can be simplified with the following assumptions:

• No pressure gradient force
• Steady-state approximation (time-dependent derivatives are negligible)
• No viscous forces.

The above assumptions yield the following momentum equations:

$fv+{\frac {1}{\rho }}{\frac {\partial \tau _{x}}{\partial z}}=0$

$fu-{\frac {1}{\rho }}{\frac {\partial \tau _{y}}{\partial z}}=0$.

If the Reynolds stress terms (those involving "τ") are parameterized with Eddy viscosity (K) such that

$\tau _{x}=\rho K{\frac {\partial u}{\partial z}}$

$\tau _{y}=\rho K{\frac {\partial v}{\partial z}}$,

then the momentum equations become the following 2nd-order ordinary differential equations:

${\frac {d^{2}u}{dz^{2}}}+{\frac {f}{K}}v=0$

${\frac {d^{2}v}{dz^{2}}}-{\frac {f}{K}}u=0$,

where the partial derivatives become regular derivatives due to "z" being the only independent variable. After defining a "complex vertical velocity" (w = u + iv, where "i" is imaginary), assuming that the vertical velocity vanishes as depth increases, and including a surface current such that w = W_0 (and assume the surface current is strictly zonal), the following horizontal wind fields are determined:

$u=U_{0}e^{\gamma z}\cos {\gamma z}$

$v=U_{0}e^{\gamma z}\sin {\gamma z}$,

where "γ" is equal to the square root of f/(2K).

These wind fields result in what is known as the "Ekman Spiral," in which a hodograph (diagram showing how velocity changes with depth) depicts an oceanic current in a decaying clockwise spiral as depth increases. The depth of so-called Ekman layer is that at which the Ekman spiral has completed one full rotation, also known as the e-folding scale of decay:

$D_{E}={\frac {1}{\gamma }}={\sqrt {\frac {2K}{f}}}$.

The net transport of water throughout the Ekman layer can be determined via integration of the Ekman velocities (u and v) through the entire depth of the layer (from the bottom to the surface):

$M_{x}=\int _{-\infty }^{0}\rho u\,dz=\rho U_{0}\int _{-\infty }^{0}e^{\gamma z}cos{\gamma z}\,dz={\frac {\rho U_{0}}{2\gamma }}$

$M_{y}=\int _{-\infty }^{0}\rho v\,dz=\rho U_{0}\int _{-\infty }^{0}e^{\gamma z}sin{\gamma z}\,dz=-{\frac {\rho U_{0}}{2\gamma }}$.

This net transport is directed at a 45 degree angle to the right of the surface, but the surface current is at a 45 degree angle to the right of the surface wind. Therefore, the net transport is directed at a 90 degree angle to the right of the surface wind; and the total Ekman transport becomes the following:

$M={\sqrt {M_{x}^{2}+M_{y}^{2}}}={\frac {\rho U_{0}}{\gamma {\sqrt {2}}}}={\frac {\left|\tau \right|}{f}}$.

Ekman Transport and the Ocean Circulation

Ekman transport has the ability to affect ocean circulation in a number of ways. First, the anticyclonic wind-driven gyres throughout the ocean basins (subtropical gyres) will have a net transport toward the gyres' centers. Therefore, there will be convergence at the surface in the center of each gyre, ultimately resulting in the elevation of sea level heights in the center, not to mention the pushing of cold, deep water to greater depths (toward the bottom of the basin). This surface convergence results in downwelling and the associated downward vertical velocities (with divergence at the bottom of the gyre), which is known as Ekman pumping. The downward vertical velocity can be represented via the integration of the continuity equation:

$W_{E}=\int _{-D_{E}}^{0}{\frac {\partial w}{\partial z}}\,dz$,

which, when integrated, becomes proportional to the curl of the surface wind stress. Reults indicate that a cyclonic wind stress will produce upwelling, while an anticyclonic stress will produce downwelling.

Furthermore, Ekman transport can have significant impacts on the climate along the west coasts of North and South America as well. First, along the western coasts of North America and South America (during the spring and summer months, March through August), the semi-permanent Pacific surface high-pressure system moves offshore with prevailing flow from the northwest to north-northwest. Thus, the Ekman transport is directed towards the Pacific Ocean, resulting in coastal surface water being pulled out to sea. There is horizontal divergence at the surface (near the coast), which results in upwelling, ultimately bringing colder and deeper water to the surface along the coast. Therefore, in spring and summer, the water along the western coasts of North America (California, Oregon, and Washington) and South America is colder than normal. During the fall and winter months, the Pacific high pressure moves farther northwest, resulting in little or no upwelling along the western coasts of both continents.