Oceanography: Vectors and Vector Calculus Notes

Vectors and Vector Calculus

Overview of Vectors and Scalars Related to the Ocean

  • Scalars: Quantities defined by magnitude only. Examples include:

  • Temperature

  • Salinity

  • Density

  • Pressure

  • Vectors: Quantities defined by magnitude and direction. Examples include:

  • Kinetic energy

  • Potential energy

  • Wave height

  • Velocity

  • Wind stress

  • Force (any kind)

  • Heat flux/transport

  • Salt flux/transport

  • Wave direction

Vector Addition

  • To add vectors:
  • Example: ( extbf{i} + 2 extbf{j} = extbf{i} + 2 extbf{j} )
  • Finding the Length of a Resulting Vector: If ( extbf{A} = extbf{i} + 2 extbf{j} ), the length ( || extbf{A}|| ) is calculated as:
    • ( || extbf{A}|| = \sqrt{1^2 + 2^2} )

Coordinate System in Oceanography

  • X-Direction: Eastward
  • Y-Direction: Northward
  • Z-Direction: Upward
  • Reference to poles: North Pole is at the top, South Pole at the bottom.

Wind Stress

  • Wind can be understood as shear stress, categorized as:
  • ( \tau(x) ): Wind stress in the eastward direction
  • ( \tau(y) ): Wind stress in the northward direction
  • The magnitude of wind stress depends on the square of the wind speed.

Zonal and Meridional Directions

  • Zonal Direction: East-West orientation measured in wind stress (N m²).

  • Example Data Representation (January SCOW Zonal Wind Stress): Visualizes stress levels across latitudes from 60°N to 60°S.

  • Meridional Direction: North-South orientation measured similarly.

  • Example Data Representation (January SCOW Meridional Wind Stress): Also represented across latitudes with similar visualization.

Ocean Surface Velocity

  • Velocity expressed in three components: ( u \textbf{i} + v \textbf{j} + w \textbf{k} )
  • Consideration of which component is the smallest is important in ocean studies.

Ocean Depth and Aspect Ratio

  • Earth is vast, yet oceans are relatively shallow (e.g., 4km deep compared to a 15000km radius).
  • Visual analogy: if Earth were an orange, the ocean is thinner than plastic wrap.
  • This comparative measurement is referred to as the aspect ratio.

Pressure Gradients

  • Horizontal Pressure Gradient Force: Movement from high to low pressure creates a horizontal force:
  • Calculation often involves a small control volume to relate pressure differences (
    • ( F = - \Delta p / \Delta x \times A ) where A is the area).

Derivation of Forces

  • Gradient in 3D: Grow stress from gradients defined as:
  • ( F = -V (\partial p / \partial x \textbf{i} + \partial p / \partial y \textbf{j} + \partial p / \partial z \textbf{k}) )
  • This representation highlights the relationship between pressure gradients and forces acting in a fluid.
  • The gradient operator (( ∇ )) summarizes rate changes across dimensions.

Conservation of Mass in Fluids

  • Incompressibility: Water is treated as incompressible, enforcing conservation of mass within a volume.
  • Mathematical Representation: Changes in velocity would equal net outputs:
  • ( (u{in} - u{out}) \Delta y \Delta z + (v{in} - v{out}) \Delta x \Delta z + (w{in} - w{out}) \Delta x \Delta y = 0 )
  • Divergence: Represents the net input of a fluid written as:
  • ( ∇ \cdot v = \frac{du}{dx} + \frac{dv}{dy} + \frac{dw}{dz} = 0 )

Pressure and Flow Dynamics

  • Eventually, wind effects create differences in water levels, leading to pressure gradient forces that result in water movement. Equilibrium occurs once water levels stabilize.

Presentation Guidelines for Future Sessions

  • Presentation time limits and grading criteria focused on clarity of scientific interests, defined quantities, impacts on oceanographic phenomena, and responses to questions.