Geophysical Fluid DynamicsHorizontal stratification and rotation

Paul LindenDepartment of Mechanical and Aerospace Engineering

University of California, San Diego

pflinden@ucsd.edu

Tieh-Yong KohSchool of Physical and Mathematical Sciences

Nanyang Technological University

Singapore

KohTY@ntu.edu.sg

Singapore April 24, 2009

Gravity currents and rotatingflows

Avalanche in Tuca, Spain

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Objectives

The objectives of this lecture are

1. Introduce flows driven by horizontal density gradients

2. Discuss properties of gravity currents and intrusions

3. Discuss motion in a rotating frame

4. Introduce the Rossby deformation radius

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Horizontal stratification

Fluid of density ρ(x) at rest under gravity g

∂p

∂z= −gρ

Unless ρ = ρ(z) only, the pressure p will, in general,be different at different horizontal locations resulting inmotion.

Thus a fluid can only be at rest if it is vertically stratified;horizontal stratification always drives a flow.

This is result of baroclinic generation of vorticity which occurs when ∇ρ ×∇p 6= 0

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Gravity currents

A laboratory gravity current - the fluid density increases from blue to yellowto red. As the current progresses the densest fluid eventually leads (Linden &

Simpson 1986)

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Dimensional analysis

Schematic of a finite volume release

Speed U depends on g′, H, L0, t

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Constant volume phase

Initially U constant and independent of the horizontalscale L0

U = F√

g′H

F is the Froude number

Similarity phase

Later release size L0 becomes important: total buoyancyB = g′HL0 is constant. Dimension [B] = L3T−2 ⇒

U = FB1/3t−1/3

Speed decreases with time until viscous forces becomeimportant.

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Laboratory verification

The current initially travels at constant speed until the bore from the rear wallcatches up with the front – this brings the information that the release has a

finite volume (Rottman & Simpson 1983)

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The front Froude number

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Other theories give values usually less than√

2, but in allcases

F = O(1)

The Froude number can be interpreted in various ways

1. Energy KE of current is ∼ ρU 2 and PE is g∆ρH. F ∼ 1implies rough equipartition

2. Hydraulics Information at the front of the current mustbe able to propagate to the rear. The flow is subcritical,but critical F ∼ 1 at the front

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ROTATION

Rotation of the Earth - on this scale the oceans and atmosphere are too thin tobe seen!

The Earth’s rotation Ω plays an important role in manyenvironmental flows. Generally any flow structure thatpersists for a day is affected by the rotation of the earth.

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Since the oceans and atmosphere are thin layers of fluid(compared to their horizontal scale and the radius of theearth) only the vertical component of Ω is dynamicallyimportant.

Vertical component

At latitude θΩ sin θ

Rotation unimportant near the equator (θ ≃ 0)

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Coriolis force

Consider a particle at r moving with velocity u relative aframe rotating with angular velocity Ω. Then its absolutevelocity is

U = u + Ω × r

Hence

d

dt≡ d

dtR+ Ω × r

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d2r

dt2≡ d2r

dt2R+ 2Ω × dr

dtR+ Ω × (Ω × r)

Coriolis force 2Ω × drdtR

At latitude θ this is written as

f × u

where the Coriolis parameter is

f = 2Ω sin θ

The Coriolis force acts to the right in the Northernhemisphere and to the left in the Southern hemisphere

Centrifugal force

Ω× (Ω× r) - this term can be written in terms of pressureand can be ignored

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Restoring force of rotation

Effect of expansion and contraction of a horizontal ring of fluid - the framerotates anticlockwise

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Ring expands: conservation of angular momentum ⇒ring rotates anticyclonically (clockwise in NH)

Coriolis force acts to the right (in NH) - provides arestoring force.

Restoring force

1. Supports waves: inertial waves - equivalent in manyways to internal waves (e.g. frequencies < f comparedwith < N for IGW)

2. Inhibits horizontal motion - in contrast to stratificationthat inhibits vertical motion

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Inertial oscillations in the ocean

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Effects on gravity currents

The adjustment of a region of dense fluid (dyed) in a rotating fluid

No rotation The dense fluid will propagate as a gravitycurrent spreading indefinitely (until viscosity!) as a circle

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With rotation As the dense fluid spreads, conservationof angular momentum means it rotates anticyclonically.This anticyclonic motion produces an inward Coriolisforce that stops the spreading

The only place further radial spreading occurs is alongthe radial barrier. Here the anticyclonic flow is blockedby the barrier, so there is no Coriolis force parallel to thebarrier and the dense fluid flows along it as a gravitycurrent.

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The East Greenland Current - responsible for much of the ice transport fromthe Arctic

This is how coastal currents propagate - in the NHkeeping the coast on the right

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Rossby radius of deformation

The distance the circular patch spreads before it isstopped is called the Rossby radius of deformation LR

For a gravity current buoyancy g′ and depth H in a systemrotating at 1

2f

LR =

√g′H

f

It is the distance the current travels in a rotation period– consistent with the idea that rotation is only importanton time scales comparable with a day

LR is also the width of the coastal current - it is the scale atwhich the buoyancy force and the Coriolis forces balance

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Eddies in the atmosphere andoceans

In a continuously stratified fluid

LR =NH

f

This scale sets the horizontal dimensions of eddies in theatmosphere and oceans. In both cases N ∼ 10−2 s−1 andf ∼ 10−4 s−1, so

LR ∼ 100H

Typical scales are

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• Ocean H ∼ 1 km ⇒ LR ∼ 100 km

• Atmosphere H ∼ 10 km ⇒ LR ∼ 1000 km

Ocean eddies in the Atlantic

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−120 −110 −100 −90 −80

−36

−30

−30

−30

−24−24

−24

−18

−12

−6

500−Millibar Height Contours at 7:00 A.M. E.S.T.

510522

522534

534

546

558

570

582

LOW LOW

LOW

LOW

LOW

LOW

−120 −110 −100 −90 −80

Isobars - note the much larger scale than the ocean eddies

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Hurricanes and tornadoes

Both rotating convective flows – the upward convective motion stretches fluidcolumns vertically and, by conservation of angular momentum, makes themrotate faster. This leads to intense winds.

Hurricane Katrina

Hurricane The time and length scales of a hurricane (days, 100s kms) meansthat the rotation of the earth is crucial to their dynamics. The radius iscomparable to the Rossby deformation radius

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Tornado

The tornado is much smaller and lasts only a small part of a day - so the earth’srotation is not important.

In this case the rotation results from stretching rotating vortex columns causedby the wind, which have much higher rotation rates. Comparable to a bathtubvortex. Unless your bath takes a day to drain the influence of the rotation ofthe earth is negligible and you can’t use it to tell you which hemisphere you arein!

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Hurricane Katrina

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Tornado

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