ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 17

Review: Learning objectives: understand and appreciate“The mixing processes in the ocean”: Static stability & dynamical instabilities -barotropic instability

1. Mid-term2. Final project

Conserved.

Wave breaking!

€

Cp = Cg = ± gD

€

ω = ±κc

1. Mid-term:

Coastal kelvin wave.

Coast. Vertical walls.

y

v=0u

x

𝜔=- "#$%&

𝑘

𝑐) = 𝑐+ = − "#$%&

y

WestCoastPersonA

EastCoastPersonB

EquatorialPacific

EQ

Zonal wind stress

2. Final project

Mixing processes in the ocean (this and next lecture)

1) Static stability: Stable and unstable stratification, double diffusion, salt fingering (review today);

2) Instabilities:a) Barotropic instability (review);b) Baroclinic instability;c) Kelvin-Helmholtz instability;

3) Eddies in the ocean.

1) Static stability: stable and unstable stratification, double diffusion, salt fingering

z

x

z

x

a) stable b) unstable

Archimedes Principle

0

-5000m

Archimedes Thoughtful by Fetti (1620)

Stable:

Archimedes’ Principle: Buoyant upthrust=weight of surrounding fluid displaced.

We often use: buoyancy frequency N to describestratification.

N: Brunt-Vaisala frequnecy.

Unstable:

Double diffusion and salt fingeringInitially - stable

T diffusion=100 S diffusion

After certain time:

Unstable: falling/risingoccurs in thin columns

Salt fingering!

Small scale mixing: a few meters. .

Obs: Mediterranean plume

Lab experiment: salt fingering

Critical thinking: In which part of the ocean salt fingering most likely occurs?

Example: Mediterranean Overflow Water

By contrast, if a layer of colder, fresher water is above a layer of warmer, saltier water,the water just above the interface becomes lighter than its above and thus rises, while water belowthe interface gets heavier and thus sinks.

Layering: may result in homogeneous layersSeparated by thinner regions of large gradients of T and S.

The ocean is NOT static; they’re in constant motion!

2) Instabilities: Batotropic, baroclinic, and Kelvin-Helmholtz instabilities

Even though the ocean is stably stratified, motions in the ocean can induce instabilities and thus renderthe ocean dynamically unstable.Þ Perturbation can grow by extracting energy from The background state (large-scale circulation pattern).

Eddy mixinghttps://vimeo.com/85342340

Gulf-stream rings (eddies)

Climate studies: some people called it: chaotic variability;because they are unrelated to “forcing (e.g. by winds)” associated withclimate variability and change.

a) Barotropic instabilityHorizontal shears of currents are strong: say western boundary region, equatorial current systems.

Equatorial Current Systems: Barotropic Instabilities often Occur

To isolate barotropic instability, we use a homogeneous ocean with density and background current:

y

y= -L

y=L

x

The nonlinear equations for the barotropic, inviscidocean with no forcing are:

Where

Small parameterBackgroundcurrent

perturbation

Purpose: derive an energy equation to show under what conditions a “small perturbation”of current can “grow”.

Substituting into the above equations,

Obtain the kinetic energy “time variation” equation for the perturbation and integrate through the channel:

The overbar represents “average over one wavelengthin x direction”.

In order for an instability to grow, has to bePositive=> Kinetic energy of perturbation increaseswith time. Thus,

(KE )t =12(u2 + v2 )

−Ly

Ly∫

tdy = − Uy−Ly

Ly∫ uvdy

(KE )t

As

y

y= -L

y=L

x Fig. 19.4a

(a) In order for a perturbation to grow, the Perturbation must tilt again the shear.

y

y= -L

y=L

x

Results after the instabilities:

Reduced shear!Eddy mixing effect!

Total kinetic energy in the channel is conserved. Instabilities redistribute the energy via mixing effects.

Tropical Pacific:Tropical instability

waves (TIWs): observations

https://podaac.jpl.nasa.gov/OceanEvents/TropicalInstabilityWaves_Pacific_July2012

Where

For an instability to grow,

Thus,

must change sign somewhere in the flow.Necessary condition for barotropic instability:meridional gradient of absolute vorticity

changes sign somewherein the background flow.

We can derive another equation for barotropicinstability: