Ocean Eddies

What are Ocean Eddies?

Ocean eddies are deviations from mean current systems in the ocean. They are a form of turbulence, with sizes ranging from 60 to over 200 kilometers. Unlike flows directly driven by atmospheric forcing, eddies develop intrinsically from ocean flow.

Characteristics of Ocean Eddies

  • Horizontal Scales: Typically less than 100 kilometers, but can sometimes reach 200 kilometers or more.
  • Vertical Scales: Can range from shallow to the entire water depth.
  • Time Scales: From months to years.

Distribution of Ocean Eddies

Eddies are found nearly everywhere in the ocean. Regions with high eddy activity include:

  • North Atlantic
  • Southern Ocean
  • Agulhas Current
  • Gulf Stream
  • Gulf of Mexico

Some areas exhibit less eddy activity, but it's rare to find places with absolutely no eddies.

Force Balance in Ocean Eddies

The dynamics of ocean eddies can be described using equations derived from the Navier-Stokes equations. The primary balances include:

  • Geostrophic balance
  • Hydrostatic balance

The geostrophic balance implies that the pressure gradient force is balanced by the Coriolis force, which largely determines the flow within an eddy.

Geostrophic Balance

The geostrophic balance means that the pressure gradient force is balanced by the Coriolis force. This balance dictates the flow within the eddy. The relevant equations are:

\text{Acceleration} + \text{Advection of Momentum} = \text{Coriolis Force} + \text{Pressure Gradient Force} + \text{Friction/Forcing}

In the geostrophic balance:

\text{Pressure Gradient Force} = \text{Coriolis Force}

Hydrostatic Balance

Hydrostatic balance is also valid within an eddy and becomes important in relation to density stratification.

Sea Surface Tilts and Geostrophic Flow

Eddies can be observed from space due to sea surface tilts (small hills and crests). A high-pressure area corresponds to a higher sea surface. The equation relating velocity (V) to sea surface height ($\eta$) is:

V = \frac{g}{f} \frac{\partial \eta}{\partial x}

Where:

  • g is gravitational acceleration,
  • f is the Coriolis parameter, and
  • \frac{\partial \eta}{\partial x} is the gradient of the sea surface height.

Effects of Density Stratification

When density stratification is present, the water column isn't uniform. Pushing water together can cause deviations in isopycnals/isotherms. For instance, a warm core eddy results from warmer water being pushed downwards. This situation leads to the thermal wind relation.

Thermal Wind Equations

The thermal wind equations help understand flow when geostrophic balance is assumed. By differentiating the geostrophic equation by depth (dz) and using the hydrostatic equation, we relate the change in velocity with depth to the horizontal gradient of buoyancy or density.

\frac{\partial V}{\partial z} = -\frac{g}{f \rho_0} \frac{\partial \rho}{\partial x}

If \frac{\partial B}{\partial x} > 0 (buoyancy increases with x) in the Northern Hemisphere (f > 0), then \frac{\partial V}{\partial z} > 0. This means the flow is stronger in the upper ocean than in the lower ocean.

Non-Linearities

Eddies are turbulent motions, and despite the linear geostrophic balance, non-linearities are crucial for:

  • Creation of eddies
  • Propagation of eddies
  • Structure preservation

Structure Preservation

Non-linear terms help eddies maintain their shape and coherence over time. In models without non-linear terms, eddies tend to dissipate, with energy radiating away as waves.

Eddy Propagation

Eddy propagation occurs due to:

  • Background Flow: Eddies are advected by mean flows.
  • Change in Coriolis Parameter with Latitude: The Coriolis parameter (f) varies with latitude, influencing eddy movement.
Westward Propagation

Consider an eddy with pressure P and geostrophic equations for upper and lower parts:

U2 = -\frac{1}{f2 \rho} \frac{\partial P}{\partial y}

U1 = -\frac{1}{f1 \rho} \frac{\partial P}{\partial y}

Where f2 > f1. For the equation to hold, velocities must differ, causing convergence on the western side and divergence on the eastern side. This results in an increase in pressure on the western side, shifting the eddy westward.

Beta Propellers

Beta propellers describe another propagation mechanism related to the change in the Coriolis parameter with latitude. This is a non-linear effect where eddies affect surrounding parcels of water.

Consider an eddy affecting water parcels outside it. Due to the conservation of potential vorticity:

q = \frac{f + \zeta}{H}

Where:

  • f is the Coriolis parameter,
  • \zeta is the relative vorticity, and
  • H is the water depth.

If an eddy moves a parcel southward (decreasing f), \zeta must increase to conserve potential vorticity. This creates small induced eddies that propel the main eddy poleward (cyclonic eddies go north, anticyclonic eddies go south).

Eddy Formation: Baroclinic Instability

Baroclinic instability occurs when there are horizontal and vertical density gradients.

Process

  1. Small perturbations in the flow.
  2. Perturbations grow over time.
  3. Non-linear effects become significant, leading to turbulence and eddy formation.
  4. Eddies drain energy from mean available potential energy, flattening isopycnals.

Wavelength Selection

In the initial stages, the flow is unstable, and small perturbations grow. The Eady problem describes this process, where waves grow exponentially depending on their wavelength. There's a maximum growth rate for a specific wavelength, which is related to the baroclinic Rossby radius.

L_R = \frac{NH}{f}

Where:

  • N is the buoyancy frequency,
  • H is the water depth, and
  • f is the Coriolis parameter.

Eddies tend to have a horizontal scale close to the baroclinic Rossby radius because these waves grow the fastest initially.

Rossby Radius

The Rossby radius can be calculated using stratification, water depth, and the Coriolis parameter. It varies with latitude, being smaller at higher latitudes. Typical values are around 60 kilometers in the subtropics. The biggest dependence of the Rossby radius is the Coriolis parameter, which changes with latitude.

Other Generation Mechanisms

  • Barotropic Instability: Occurs with a strongly sheared horizontal flow.

Energy Transfers and Dissipation

Energy is transferred between different reservoirs due to non-linearities. The Lorenz energy cycle illustrates this.

Dissipation Mechanisms

  • Bottom Friction: Eddies interacting with the ocean sea floor lose energy due to friction, transferring it to small-scale turbulent motions.
  • Lee Wave Generation: Eddies interacting with seamounts can generate internal waves, dissipating energy to the wave field.
  • Loss of Balance: An eddy can become unstable and radiate internal waves, leading to direct dissipation.
  • Negative Wind Work: Wind interacting with eddies can either add or remove energy, depending on relative directions. This energy is transferred to the atmosphere.