Scott Durski

Oregon State University

College of Oceanic and Atmospheric Sciences

Scale Evolution of Coastal Upwelling Frontal Instabilities

R. M. Samelson, J. S. Allen, G.D. Egbert

• periodic domain with alongshore uniform bathymetry representative of the Oregon shelf

• typical summer stratification

• steady upwelling favorable wind forcing

Surface density.Previous work

Durski and Allen (JPO, September 2005)

1. Disturtbances initially develop fastest at approximately an 8-10 km scale.

2. The predominant alongshore scale of the disturbances rapidly progresses to longer wavelengths.

Instabilities on a time evolving flow.

Offshore advection of the upwelling front

Alteration of the alongshore average flow by finite amplitude evolution of the instabilities

What might lead to changes in the predominant disturbance scale?

Wave-wave interactions

Model Settings: No explicit horizontal mixing.

Mellor-Yamada level 2.5 vertical mixing.

3rd order upstream advection scheme.

Domain: Periodic channel domain

Uniform alongshore resolution – 1 km.

Stretched grid in the across shore – 400m at the coast to 4000m 150 km offshore.

30 vertical sigma levels, concentrated near the surface and bottom.

Alongshore uniform domain

Experimental Setup

150 100 50 01000

800

600

400

200

0

240

kmy (km)

Experimental Concept

0

-20

-40

-60-20 -10 0

x (km)

z (m

)

tb ts

)(SNt

S

Starting point: A 5 day integration of the nonlinear model from a state of rest, with a sustained upwelling favorable wind stress and no perturbations in the alongshore direction.

The nonlinear model can be expressed as:

sS

SN

t

s

S

A)(

0

and the tangent linear model as

Consider linear disturbance growth about a set of background states

)()(),( b

t

t

Fo ttStdSNzyS

s

b

1. A Frozen field.

SStdSNtzyxS Ft

t

Do

s

)(),,,(3

3. A 3-dimensionally evolving upwelling front.

),,(),()( zyxSzySttS Fos

Fo

t

t

Do StdSNtzyS

s

)(),,(2

2. A 2-dimensionally evolving upwelling front.

Do

Do StzyS 33 ),,(

4. The alongshore average of a 3-dimensionally evolving upwelling front.

dxSL

SxL

x0

1

5. The nonlinear evolution of an additionally perturbed simulation.

SzyxSzySttS Fos ),,(),()(

Disturbance structures on the frozen field

• linear disturbances grow fastest at approximately an 11 km scale.

•Disturbance structure is shallow and largely confined to the frontal region.

Long time integration of the tangent linear model on the frozen day 0 alongshore uniform background state yields estimates of the disturbance growth rate as a function of wavenumber.

Initial disturbance structures for the experiments are composed of ‘modal structures’ at wavelengths between 4 km and 240 km. Initial disturbance energy is normalized so as to be uniform across the spectrum. Multiple perturbations are generated with random phase shifts of different spectral components.

Estimate of disturbance growth rate and disturbance amplitude structure at two wavenumbers for the TLF experiment.

NLM – relaxed winds

The nonlinear model exhibits an evolution to larger scale under relaxed winds that is similar to that under sustained winds.

0

-10

-20

y (k

m)

0

-10

-20

y (k

m)

0

-10

-20

y (k

m)

0

-10

-20y

(km

)

0

-10

-20

y (k

m)

Surface density and deviation from the alongshore average for the base case nonlinear relaxed wind experiment.

TL2D disturbance evolution

Disturbance evolution on an alongshore uniform upwelling front during relaxation.

Perturbation density evolution is comparable to that in the nonlinear model only over the first several days.

Surface perturbation density in the TL2D and NL models.

Disturbance amplitude structure at the fastest growing alongchannel wavenumber over the first 1.5 days.

Two-dimensionally evolving background state under relaxed winds.

TL<3D> disturbance evolution

Disturbance evolution on the alongshore average of a three-dimensionally evolving upwelling front during relaxation.

Comparison of So2D and So

<3D> background states under relaxed winds.

Surface perturbation density in the TL2D, TL<3D> and NL models.

TL3D disturbance evolution

Disturbance evolution on the full three-dimensionally evolving upwelling front.

Surface perturbation density in a TL3D simulation. The standard deviation among perturbations in an ensemble of 15 TL3D, and NLp simulations and the nonlinear model surface density evolution.

Quantifying the rate of scale change

dydzkzyekE pp ),,()(

Define an energy as a function of wavenumber.

),,(),(

2),,(),,(

2

1),,( 2

122 kzy

z

zygkzyvkzyukzye p

bppop

and the volume integral …

)(

)(

kE

kkEk

p

p

mean

Calculate an energy-weighted mean disturbance wavenumber.

0 2 4 6 108

t (days)

Mean alongchannel wavenumber kmean, as a function of time for the relaxed wind experiment.

Disturbance energy and it’s rate of change

Perturbation energy Ep(k), as a function of time for the relaxed wind experiment.

The time-rate of change of Ep(k) as a function of wavenumber for the relaxed wind experiment (normalized by the maximum rate of change at each time).

NLM – sustained winds

0

-25

-50

y (k

m)

0

-25

-50

y (k

m)

0

-25

-50

y (k

m)

0

-25

-50

y (k

m)

0

-25

-50

y (k

m)

Surface density and deviation from the alongshore average for the base case nonlinear sustained wind experiment.

TL-2D disturbance evolution

Disturbance amplitude structure at the fastest growing alongchannel wavenumber over the first 1.5 days.

Surface perturbation density in the TL2D and NL models.

Two-dimensionally evolving background state under relaxed winds.

TL<3D> disturbance evolution

Surface density and deviation from the alongshore average for the base case nonlinear sustained wind experiment.

Comparison of So2D and So<3D> background states under relaxed winds.

TL-3D disturbance evolution

Quantifying the rate of scale change

Mean alongchannel wavenumber kmean and Ep(k) as a function of time for the relaxed wind experiment.

Rate of change of Ep(k) as a function of wavenumber for the relaxed wind experiment (normalized by the maximum rate of change at each time).

Variable wind forcing

kmean and Ep(k) as a function of time for experiments with relaxed, partially sustained and fully sustained upwelling favorable wind stress.

Forcing for the sustained and relaxed wind experiments.

Comparison of vertical structure of disturbances

Modal structure at fastest growing alongchannel scale for the sustained and relaxed wind cases. Bottom pair of panels shows the structure in a modified sustained wind TL3D simulation with reduced vertical mixing coefficients.

RLX SWTL NL TL NL

TLSW

TL(Kv=0.01Kvo

Role of vertical mixing on disturbance structure

Conclusions

• Although the frontal disturbances in the nonlinear model exhibit a progression to larger scale over the full range of forcing scenarios considered, the mechanisms most responsible for the processes differ between wind forced and unforced cases

•Under relaxed wind conditions the tangent linear disturbance growth experiments indicate that the scale evolution over the first four days is reflected in the way linear disturbances respond to the adjustment of an alongshore uniform upwelling front to wind relaxation.

•Under relaxed winds, the continued increase in scale between days 4 and 7, relates to the linear disturbance evolution on an alongshore average flow that has been altered by wave-mean flow interaction.

•The disturbance alongshore scale change past day 7 in the relaxed wind experiment and throughout the sustained wind experiment is not captured by the scale of the disturbances in the linear growth experiment and evidently results largely from wave-wave interaction processes.