ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 17Learning objectives: understand and appreciate“mixing processes in the ocean”:

1) Static stability: a) Stable and unstable stratification b) double diffusion & salt fingering2) Dynamical instabilities:a) Barotropic instabilityb) Baroclinic instabilityc) Kelvin-Helmholtz instability

1) Static stability

z

x

Stable stratification:

0m

-5000m

Archimedes Principle states that the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.

𝝆𝒂 < 𝝆𝒘 < 𝝆𝒃

• Stable and unstable stratification;

FN: Net force

FgFb

z

x

Unstable stratification:

𝝆𝒂 > 𝝆𝒘 > 𝝆𝒃

Stable:

We often use: buoyancy frequency N to describestratification.

N: Brunt-Vaisala frequnecy

Unstable:

Initially - stable

T diffusion=100 S diffusion

• Double diffusion and salt fingering

After certain time:

Unstable: falling/risingoccurs in thin columnsSalt fingering!

Small scale mixing: a few meters.

Obs: Mediterranean plume

Lab experiment: salt fingering

T! ≈ T" , 𝑆! > 𝑆"

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!

Upper Ocean circulation: long-term mean

https://www.youtube.com/watch?v=CCmTY0PKGDshttps://vimeo.com/85342340

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

Even though the ocean is stably stratified, motions in the ocean can induce instabilities and thus render the ocean

dynamically unstable.Þ Perturbation can grow by extracting energy from The background state (large-scale circulation pattern).

Eddy mixing

*Polling question

$%$&+ 𝑢 $%

$'+ 𝑣 $%

$(+𝑤 $%

$)= − *

+$,$(− 𝑓𝑢 + 𝐴'

$!%$!'

+ 𝐴($!%$!(

+ 𝐴)$!%$!)

Eddy heat & salt transport

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

Equatorial Current Systems: Barotropic Instabilities often Occurhttps://twitter.com/mathewabarlow/status/1073372146896912384?lang=en

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

𝑁𝑜𝑡𝑒: 𝑢# =𝜕𝑢𝜕𝑡

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

Substituting into the above equations,

𝐾! ="#𝑚 𝑢# + 𝑣# ; for unit mass (m=1kg/m3),

𝐾! ="#𝑢# + 𝑣#

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

The overbar represents “average over one wavelength in 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

(i) Necessary condition for a perturbation to grow, perturbation must tilt again the shear.

For a perturbation to grow:

y

y= -L

y=L

x

a) May not grow y

x

b) Likely grow

y

y= -L

y=L

x

Results after the instabilities:

Reduced shear!Eddy mixing effect!

• Barotropic instabilities extract energy from kinetic energy of the background flow.

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

Where

For an instability to grow,

Thus, must change sign somewhere in the flow.(ii) Necessary condition for barotropic instability:meridional gradient of absolute vorticity

changes sign somewherein the background flow.

We can derive another equation for barotropicinstability:

Tropical southeast Indian Ocean: Chen et al. 2020, JPO

https://www.dailymotion.com/video/xpqjap

Tropical instability waves (TIWs)

Tropical Pacific: TIWs

TIWs: Tropical Pacific

Decco et al. 2018: Ocean dynamics

TIWs: Tropical AtlanticSea surface Temperature (SST)

Sea surface Salinity (SSS)