Comprehensive analysis of physical processes in shallow coastal waters (100m depth) including upwelling, stratification, mixing, waves, and circulation dynamics by Claudio Iturra
🌬️ Wind Stress & Ekman Transport
Wind Stress:
τ = ρₐ × Cᴅ × |U₁₀| × U₁₀
Ekman Transport:
Mₑ = τ / (ρ × f)
Ekman Depth:
Dₑ = π × √(2Aᵥ / f)
Where:
• τ = Wind stress (N/m²)
• ρₐ = Air density (1.225 kg/m³)
• Cᴅ = Drag coefficient
• U₁₀ = Wind speed at 10m
• f = Coriolis parameter
• Aᵥ = Vertical eddy viscosity
Physical Explanation: Wind stress drives surface currents and Ekman transport. In coastal regions, this creates upwelling (offshore transport) or downwelling (onshore transport) depending on wind direction relative to the coast.
Where:
• w = Vertical velocity
• τₓ, τᵧ = Wind stress components
• UI = Upwelling index (m³/s/100m)
• H = Water depth
• Kᵥ = Vertical diffusivity
Physical Explanation: Coastal upwelling occurs when alongshore winds drive offshore Ekman transport, causing deep, cold, nutrient-rich water to rise to the surface. Downwelling is the opposite process. The relaxation time indicates how quickly the system responds to wind changes.
🌀 Rossby Deformation Radius
Internal Rossby Radius:
Rᵢ = NH / f
Barotropic Rossby Radius:
R₀ = √(gH) / f
Burger Number:
Bu = (Rᵢ / L)²
Brunt-Väisälä Frequency:
N² = -(g/ρ₀) × (∂ρ/∂z)
Where:
• N = Brunt-Väisälä frequency
• H = Water depth
• f = Coriolis parameter
• g = Gravitational acceleration
• L = Horizontal length scale
• ρ = Density
Physical Explanation: The Rossby deformation radius determines the horizontal scale over which geostrophic adjustment occurs. It's crucial for understanding coastal jet formation, eddy scales, and the transition between geostrophic and ageostrophic dynamics in shallow water.
Where:
• uₘ, vₘ = Geostrophic velocities
• η = Sea surface height
• U = Characteristic velocity
• R = Radius of curvature
• p = Pressure
Physical Explanation: Geostrophic balance dominates large-scale coastal flows, but ageostrophic effects become important in shallow water, near boundaries, and during rapid acceleration. The Rossby number indicates the relative importance of inertial vs. Coriolis forces.
🌡️ Density Stratification
Density Equation of State:
ρ = ρ₀[1 - α(T-T₀) + β(S-S₀)]
Brunt-Väisälä Frequency:
N² = -(g/ρ₀) × (∂ρ/∂z)
Richardson Number:
Ri = N² / (∂u/∂z)²
Potential Density:
σₜ = ρ(S,T,0) - 1000
Where:
• ρ₀ = Reference density
• α = Thermal expansion coefficient
• β = Haline contraction coefficient
• T, S = Temperature, Salinity
• Ri = Richardson number
Physical Explanation: Density stratification controls vertical mixing and stability. Strong stratification (high N²) suppresses vertical mixing, while weak stratification allows turbulent mixing. The Richardson number determines mixing stability - Ri < 0.25 indicates unstable, turbulent conditions.
Physical Explanation: Turbulent mixing is driven by wind stress, bottom friction, and shear instabilities. The mixing length determines the size of turbulent eddies, while the Ozmidov scale represents the largest overturning eddies that can exist in stratified flow.
Physical Explanation: The bottom boundary layer is where bottom friction significantly affects the flow. The logarithmic velocity profile develops near the bottom, and the boundary layer height scales with the friction velocity and Coriolis parameter. Bottom roughness elements enhance mixing and energy dissipation.
🌊 Surface Gravity Waves
Dispersion Relation:
ω² = gk × tanh(kh)
Phase Speed:
c = ω/k = √(g/k × tanh(kh))
Group Velocity:
cₘ = ½c × [1 + 2kh/sinh(2kh)]
Wave Energy:
E = ½ρg × H²
Energy Flux:
F = E × cₘ
Where:
• ω = Angular frequency
• k = Wavenumber
• h = Water depth
• H = Wave height
• c = Phase speed
• cₘ = Group velocity
Physical Explanation: Surface gravity waves are the most visible ocean waves. In shallow water (kh < π/10), waves become non-dispersive with c = √(gh). Wave energy propagates at the group velocity, which differs from phase speed in dispersive waves. Wave breaking occurs when H/h > 0.78.
🌀 Internal Waves
Internal Wave Dispersion:
ω² = N² × k²/(k² + m²)
Vertical Wavenumber:
m² = (N² - ω²)k²/(ω² - f²)
Internal Wave Speed:
c = ω/k
Tidal Internal Wave:
c = NH/π
Critical Slope:
γc = (ω² - f²)/(N² - ω²)
Where:
• N = Brunt-Väisälä frequency
• m = Vertical wavenumber
• γc = Critical slope
• ω = Wave frequency
• f = Coriolis frequency
Physical Explanation: Internal waves propagate along density interfaces and are crucial for vertical mixing in stratified waters. They can be generated by tidal flow over topography, wind stress, or flow instabilities. The critical slope determines where internal waves can propagate vs. reflect.
Where:
• u, v = Velocity components
• η = Surface elevation
• r = Bottom friction coefficient
• A = Tidal excursion
• U₀ = Tidal velocity amplitude
Physical Explanation: Tidal dynamics in shallow water are governed by the shallow water equations, which include Coriolis effects, pressure gradients, and bottom friction. The tidal excursion length determines how far water parcels move during a tidal cycle, important for transport processes.
Where:
• Ω = Earth's rotation rate
• φ = Latitude
• U = Initial velocity
• R = Inertial radius
• τ = Damping time
Physical Explanation: Inertial oscillations occur when water parcels are displaced from geostrophic equilibrium, causing them to move in circular or elliptical paths at the local inertial frequency. These are common after storm events and contribute to upper ocean mixing through shear-induced turbulence.
Physical Explanation: Coastal circulation is driven by wind stress, density gradients, and topographic effects. Alongshore winds create coastal jets, while cross-shore density gradients drive thermal wind shear. The interplay between these forces determines the complex three-dimensional circulation patterns.
🌊 Cross-shelf Exchange
Ekman Pumping:
wₑ = (1/ρf) × curl(τ)
Bottom Ekman Transport:
Mᵦ = τᵦ/(ρf)
Ramp Relaxation:
u(t) = u∞[1 - exp(-t/T)]
Exchange Time Scale:
T = L²/(2Kₕ)
Flushing Time:
Tf = V/Q
Where:
• wₑ = Ekman pumping velocity
• Mᵦ = Bottom Ekman transport
• T = Adjustment time scale
• L = Cross-shelf scale
• Kₕ = Horizontal diffusivity
• V = Volume, Q = Exchange rate
Physical Explanation: Cross-shelf exchange transports water, nutrients, and organisms between coastal and offshore waters. It's driven by Ekman transport, density-driven flows, and tidal mixing. The exchange time scale determines how quickly coastal waters are renewed.
📊 Integrated Coastal Analysis
🎯 Coastal Ocean Dynamics Summary
Coastal physical oceanography involves the complex interaction of multiple processes operating on different spatial and temporal scales. The 100m depth represents a transition zone where both shallow water and deep water dynamics can be important.
Key Dimensionless Parameters:
Rossby Number: Ro = U/(fL) - Inertial vs. Coriolis forces
Richardson Number: Ri = N²/(∂u/∂z)² - Stratification vs. shear
Burger Number: Bu = (NH/fL)² - Stratification vs. rotation
Froude Number: Fr = U/√(gh) - Inertial vs. gravitational forces
Reynolds Number: Re = UL/ν - Inertial vs. viscous forces