Baroclinic Instability in the Denmark Strait Overflow and how it applies the material learned in this GFD course Emily Harrison James Mueller December 2, 2005

North Atlantic

The Overflow The East Greenland Current: -warm, light upper layer -cold, dense bottom layer The warm flow stays on the surface of the Irminger Sea The dense flow descend the East Greenland continental slope and enters the Denmark Strait

The Denmark Strait

Thin line: typical overflow density profile Thick line: mean back ground density Density Profile In the DS

Temperature, Salinity, and Density

Temperature, Velocity Magnitude and Direction Time Series for CM3

The Model Uniform cross-section Constant bottom slope, α Layer 1: ρ 1, U 1, average depth D 1 Layer 2: ρ 2, U 2, average depth D 2 Interface: φ(x,y,t) Channel Walls: ±L/2

Stability Analysis: Scaling

Stability Analysis: Parameters Rossby # Beta Effect Ekman # Internal Froude # Friction Parameter Bottom Slope Parameter Layer Depth Ratio

Physical Constants   Dimensionless Parameters Observational Parameters

Does β-effect really matter here? β is O(10 -3 ) B=.346 Topographic effect is 2 orders of magnitude greater than β-effect Conclusion: NO!

Stability Analysis: Governing Equations

Stability Analysis: Boundary Conditions

Stability Analysis: Governing Equation

Perturbation Pressure Equations with Boundary Conditions at y=±.5

Solving the Equations The eigenfunctions for the pressure perturbation equations: Where: the modes: the downstream wavenumber: k complex amplitude ratio: complex phase speed:

Solving the Equations Substituting ψ back into the equations yields an equation of the form: Which lead to a solution for c of the form: Where the coefficients are very, very messy -but are functions of k, m, F, β, γ, r, U 1, B

Solving the Equations With the complex coefficient components : The solution to the linear stability problem is complete! With:

Model: Instability Results Assumed: -inviscid -U 1 =0 Flow is Unstable if: B -1 >1 This means : -the shear is greater than geostrophic velocity -or, interface slope is greater than bottom slope

If λ =200km, B -1 =2.5, how long does it take the amplitude to increase by a factor of 10?

Mooring Array Spacing: ~ 15km With: Does the Mooring Array Resolve the Internal Rossby Radius of Deformation?

Theory Thermal wind:

Theory Stretching and squeezing of water columns Increase of relative vorticity (i.e. eddies) from potential energy Initial disturbance If unstable, eddies interact and form larger eddies Decrease of kinetic energy from friction Conservation of potential vorticity

Necessary Condition for Instability 1. Either changes sign in the domain, or 2. the sign of is opposite to that of at the top, or 3. the sign of is the same as that of at the bottom

Density Sections Northern lineSouthern line

Spectral Analysis

Coherence of Velocity

Coherence of Cross-stream Velocity

North-South Coherence

Heat Flux

Conclusions Linear, unstable baroclinic wave model predicts low frequency variability and cross-stream phase relationships Waves seem to be coherent only south of the sill Nonlinear effects are significant and thus need to be examined

Spall and Price (1998) Eddy diameter ~ 30 km separated by 70 km Period ~ 2-3 days which is close to Smith’s value of 1.8 days Mesoscale variability is considerably stronger than in other overflows Isopycnals are nearly parallel with the bottom, which implies the ratio of slopes is roughly 1 (i.e. not unstable). Therefore, baroclinic instability does not seem to be the primary process

Girton and Sanford (2001)

The Outside Sources

Hoyer, Quadfasel, Andersen 1999

Girton and Sandford 2003

Spall and Price 1998