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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

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North Atlantic

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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

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The Denmark Strait

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

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Temperature, Salinity, and Density

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Temperature, Velocity Magnitude and Direction Time Series for CM3

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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

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Stability Analysis: Scaling

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

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Physical Constants Dimensionless Parameters Observational Parameters

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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!

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Stability Analysis: Governing Equations

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Stability Analysis: Boundary Conditions 1. @ 2. @ 3. @ 4. @

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Stability Analysis: Governing Equation

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

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

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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

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

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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

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

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

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Theory Thermal wind:

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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

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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

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Density Sections Northern lineSouthern line

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Spectral Analysis

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Coherence of Velocity

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Coherence of Cross-stream Velocity

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North-South Coherence

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Heat Flux

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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

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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

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Girton and Sanford (2001)

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The Outside Sources

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Hoyer, Quadfasel, Andersen 1999

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Girton and Sandford 2003

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Spall and Price 1998

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