Baroclinic Instability in the Denmark Strait Overflowand 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

Magnitudeand Direction

Time Series for CM3

The Model

• Uniform cross-section• Constant bottom

slope, α• Layer 1:

ρ1, U1, average depth D1

• Layer 2: ρ2, U2, average depth D2

• Interface: φ(x,y,t)

• Channel Walls: ±L/2

Stability Analysis: Scaling

),,(*),,( '''2 iiiiii wvuUwvu

),,(*),,( ''' zyxLzyx

'

2

tU

Lt

' d

L

D2

rg

fLUd 2

ggr2

'222222112 )()( fLpUzDgzDDgP r

'1212111 )( fLpUzDDgP

Stability Analysis: Parameters

12 fL

U

)cot

(2

2

RfU

Dgr

1

2

D

D

22fD

E

)1(2

2

1

OE

r 2

22

Df

LfF

r

)( 1

2

FOfU

sgB r

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

izzixiiiit uE

pvyFuu2

)1()( u

izziyiiiit vE

puyFvv2

)1()( u

izp0

Stability Analysis: Boundary Conditions

1. @

2. @

3. @

4. @

011 pw

1z

)( yiziti vuFw

12 ppFz 1

)( 2222 yx uvrFBvw FByz

0iv

02

1 2

1

2

1lim

dxux

y

y

x

x

itx

2

1y

Stability Analysis: Governing Equation

i12

1x1y1 wy]Fp)[

yp

xp

t(

βγ

22

i12

2x2y1 prwB)y]F(βp)[

yp

xp

t(γ

)p)(py

px

pt

F(w 122x2yi

),,(),,( tyxyUtyxp iii

0)]1([)]()[( 111212

1

xUFFx

Ut

22

211212 )]1([)]()[(

rUBFFxt x

Stability Analysis: Governing Equation

02

1lim

2

1

2

1

dxx

y

y

x

x

iytx

0

xiPerturbation

Pressure Equations with Boundary

Conditions at y=±.5

12 U

Solving the Equations

The eigenfunctions for the pressure perturbation equations:

)]cos(Re[

)]cos(Re[)(

2

)(1

ymAe

ymAectxik

ctxik

Where:

the modes:

the downstream wavenumber: k

complex amplitude ratio:

complex phase speed:

,...3,2,1m

ir i

ir iccc

Solving the Equations

Substituting ψ back into the equations yields an equation of the form:

02 dba cc

Which lead to a solution for c of the form:

2

12 ]4[

2

1

2adb

aa

biccc ir

Where the coefficients are very, very messy-but are functions of k, m, F, β, γ, r, U1, B

Solving the EquationsWith the complex coefficient components:

The solution to the linear stability problem is complete!

2

1

1

)1)](1([1

c

ck

rcUB ir

r

2

1

1

)]1([)1(

c

cUBck

rir

r

With:222

)1(1 ir ccc F

mk 222

Model: Instability Results

• Assumed: -inviscid

-U1=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?

s

m

km

s

k

fc

ii 5.5

5.7*200

210*3.1 13

hrskc

t

e

i

ktci

76.310*35.1)10ln(

10

4

• Mooring Array Spacing: ~ 15km

With:

Does the Mooring Array Resolve the Internal Rossby Radius of Deformation?

25.4

s

mgr mH 150

2

2

2 3.0

sH

g

H

gN r

155.0 sN

)105(35.610*3.1

150*55.014

1

kmOkms

ms

f

NHL

oD

Theory

Thermal wind:

xf

g

z

v

0

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

0ic

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

y

q

y

q

y

q

z

u

y

b

f

N

z

u

2

Density Sections

Northern line Southern 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