The Traditional Approximation in Geophysical Fluid Dynanicsin Geophysical Fluid Dynamics Theo...

Netherlands Institute for Sea Research, Texel, NL

The Traditional Approximation (TA)in Geophysical Fluid Dynamics

Theo Gerkema

29 November 2011

Universidad de Chile 1/28 The Traditional Approximation in GFD

Overview

1 Back to the origin: Laplace’s Mécanique Céleste

2 Deflection by the ’cosine’ terms

3 The ’Traditional Approximation’

4 Manifestations of non-traditional effects

5 I. Equatorial flows

6 II. Deep convection

7 III. Internal waves

8 Conclusion


Back to the origin: Laplace’s Mécanique Céleste

1. Back to the origin: Laplace’s Mécanique Céleste

This talk is about the effect of the Earth’s rotation on currents andwaves.

Laplace was the first to treat this problem in an exact and completeway, in his Traité de Mécanique Céleste (1798).



Laplace examined the modes of tidal propagation, in a layer of wateron a rotating planet.

To do this, he first had to derive the forces acting in such a rotatingsystem.

The rotation axis is asymmetry breaker:

neither aligned to gravity,nor horizontal.

Anisotropy in thetangential plane !



Laplace found four deflecting terms:

* two terms proportional to the sine of latitude (∼ Ω⊥)* two terms proportional to the cosine of latitude (∼ Ω‖)

NB: θ denotes co-latitude and n = Ω.



Definef = 2Ω⊥ = 2Ω sin φ

f = 2Ω‖ = 2Ω cos φ

where Ω is the Earth’s angular velocity,and φ latitude.

The four terms represent the followingeffects:

(figure by Louis Gostiaux)

Initial velocity: Induced Coriolis force (in Northern Hemisphere):eastward (u) (1) southward (−fu) & (2) vertically upward (f u)northward (v) (3) eastward (fv)upward (w) (4) westward (−f w)


Deflection by the ’cosine’ terms


The two terms proportional to the sine of latitude are familiar:

they are responsible for the deflection to the right/left in theNorthern/Southern Hemisphere.

The two cosine terms are undoubtedly less familiar. . .

Their effect can be illustrated with two simple mechanical examples:

(I) a stone dropped from a tower undergoes a small deflection to the east;

(II) the weight of an eastward moving object is slightly reduced.

Note that in both cases, the vertical direction is involved.



ad I: The debate on the eastward deflection of falling objects startedin the 17th century (Borelli, Newton, Hooke. . . ). The exact expressionfor the deflection (d) was derived by Gauss and Laplace, in 1803:

d =23

Ω cos(φ)√

2h3/g

where h is the height from which the object falls.

The effect is very small.

Several experiments were carried out todemonstrate it, for instance in 1831 byFerdinand Reich in a minepit in Freiberg(Saxony), 158.5 m deep.

Theoretically, the deflection is 2.8 cm, andthis is, on average, confirmed by Reich’sexperiments.



ad II: There is an upward force acting on eastward moving parcels.

This was noticed when gravity-measurements were made on a ship(early 20th century): on eastward moving ships smaller values were foundfor g than on westward moving ships – the Eötvös effect.

This effect is small: at the equator, on a ship moving at 13 knots to theeast, one finds a value for g that is too small by one hundredth of apercent. . . General expression: gcorr = g− 2Ω cos(φ)u.

Eötvös explained this phenomenonand invented a device to illustrate it(Ann. d. Physik, 1919):


The ’Traditional Approximation’

The ’Traditional Approximation’

To come back to Laplace: as we have seen, he derived the ’Coriolis’force in its complete form, yielding four terms in the momentumequations.

However, he went on to argue that:

1) the ocean merely forms a thin spherical shell compared to theEarth’s radius;

2) this confinement means that large-scale flows must bepredominantly horizontal;

3) hence the deflecting forces associated with vertical movements– i.e. the terms with f = 2Ω cos φ – can be neglected.

This is now known as the Traditional Approximation’ (TA).


Manifestations of non-traditional effects

Manifestations of non-traditional effects

In making this approximation, Laplace has since been followedalmost universally in studies on Geophysical Fluid Dynamics.

Yet, recent work suggests that f can be important in somephenomena !

Possible manifestations of ‘non-traditional’ effects in GFD:

I. Equatorial flows: geostrophic balance modified by f ;also: stability of zonal jets affected;

II. Deep convection: mixing occurs along slanted surfaces;III. Internal waves: behaviour of low-frequency waves in weakly

stratified layers depends strongly on f .IV. Ekman layers: characteristics of Ekman spiral depend on wind

direction, via f (not discussed in this talk);


I. Equatorial flows

I. Equatorial flows

‘Traditional’ thermal-wind balance:

f∂u∂z

=gρ∗

∂ρ

∂y; f

∂v∂z

= − gρ∗

∂ρ

∂x(TA)

Near equator: f → 0, so this notion of geostrophy breaks down. . .

Colin de Verdière & Schopp (1994) derived modified, uniformly validthermal-wind balance (φ latitude, λ longitude):

fr

∂u∂φ

+ f∂u∂r

=g

ρ∗r∂ρ

∂φ

fr

∂v∂φ

+ f∂v∂r

+f wr

= − gρ∗r cos φ

∂ρ

∂λ

At equator, horizontal density gradients now related to meridionalshear ! A first test (NIOZ-cruise 2007, Brazil basin) has however notconfirmed this balance.


II. Deep convection

II. Deep convection

In the Labrador, Greenland, and Weddell Seas, deep convectionoccurs intermittently in ‘plumes’, whose horizontal extent is ∼1 km.Vertical velocities can be as high as 10 cm/s.

Numerical simulations by Straneo et al. (2002) and Wirth & Barnier(2006, 2008) demonstrate that non-traditional terms produce a tilttowards the equator.

Left: Traditional Approximation. Right: non-traditional.


II. Deep convection

The idea is that convection takes place along lines of constant

−fy + f z = const ,

where y is the meridional coordinate (south-north), and z the vertical.

Under the TA, we would have y = const, i.e. vertical convection.

With f included, convection is tilted towards the equator (in bothhemispheres), aligned to the full rotation vector.

This tilt has been confirmed in laboratory experiments !


II. Deep convection

‘Non-traditional’ laboratory experiments on slantwise convection bySheremet (2004).

Small tank at outer end of rotating table.Free surface tilted due to centrifugal force.Hence effective gravity gc (normal to thesurface) makes an angle with the rotationaxis:

This angle introduces non-traditionaleffects.


III. Internal waves

III. Internal waves

The buoyancy frequency N is a measure of vertical stratification:

N2 = −g2(

∂ρ

∂p− 1

c2s

)where ρ is in-situ density, p pressure, and cs the speed of sound.

Strong stratification (i.e. large N Ω) suppresses non-traditionaleffects, but this condition does not hold in the abyssal ocean !


III. Internal waves

Two classes of internal waves prevail in the ocean: internal tides andnear-inertial waves.

An example of internal tides inthe Bay of Biscay.

They are generated near theshelf edge and then propagatediagonally downward.

Results from a numerical calcula-tion (Gerkema et al., 2008).

Non-traditional case agrees betterwith observed beam from Pingree& New (1991, asterisks).


III. Internal waves

Momentum equations (linear, inviscid) with full Coriolis force:

∂u∂t

= fv−f w − ∂p∂x

(horizontal, west-east)

∂v∂t

= −fu − ∂p∂y

(horizontal, south-north)

∂w∂t

= +f u − ∂p∂z

+ b (vertical, radially outward)

The coefficients f , f are here taken constant.

Assume ∼ exp(iωt), and ∂/∂x = 0 (south/north propagation only).Together with eqs for mass and energy conservation, we then find

A∂2ψ

∂y2 + 2B∂2ψ

∂y∂z+ C

∂2ψ

∂z2 = 0

withA = N2 −ω2 + f 2 , B = f f and C = f 2 −ω2 .


III. Internal waves

A∂2ψ

∂y2 + 2B∂2ψ

∂y∂z+ C

∂2ψ

∂z2 = 0

(1)

This PDE changes radically if one makes the TraditionalApproximation : under the TA, the mixed derivative disappears(B = 0), rendering the problem separable in y and z.

Two important consequences of not making the TA :

(1) Range of allowable frequencies enhanced by non-traditional terms.

(2) Singular behaviour at lower bound in non-traditional case :appearance of a low-frequency short-wave limit !


III. Internal waves

These results follow from the dispersion relation, which is is derivedas follows:

reduce Aψyy + 2Bψyz + Cψzz = 0 to a Sturm-Liouville problemvia the transformation

ψ = Ψ(z) exp ik(y− Bz/C) .(2)

Substitution in PDE gives:

Ψ′′ + k2[

B2 −ACC2

]Ψ = 0

(3)

Together with the boundary conditions Ψ = 0 at z = −H, 0(bottom, surface), we obtain the general solution

ψ = ∑n

Ψn(z) [ an sin kn(y− Bz/C) + bn cos kn(y− Bz/C) ]

whose non-separable character, due to B, is evident.Universidad de Chile 20/28 The Traditional Approximation in GFD

III. Internal waves

The dispersion relation for N = const:

Frequency range is enlarged.

At lowest frequency, wavesbecome short:

→ sub-inertial short-wave limit

– a major departure from‘traditional’ theory !


III. Internal waves

Enlargement of the frequency range : consequences in presence oflayers of different stratification with

Layer 1 : strong stratification (large N)

Layer 2 : weak stratification (small N)

Then there is a common band of frequencies around |f |– near-inertial internal waves.

For frequencies in this band, waves can exist in both layers.


III. Internal waves

(Gerkema & Exarchou, 2008)


III. Internal waves

Internal waves can become sub-inertial.

On the sphere (or β-plane)this means that waves canpropagate poleward oftheir inertial latitude,where ω = |f |(Gerkema & Shrira, 2005).

Numerical experimentson time-evolution, com-paring TA and non-traditional propagationand showing abyssaltrapping (Winters et al.,2011):


III. Internal waves

Observational evidence?

Polarization of near-inertial waves:

Current meter records from Mediterranean, Algerian Basin(Van Haren & Millot, 2004):

Convective, extremely weakly stratified layers were found beneathlayers of stronger stratification.

Band-pass filtered current-meter records yield hodographs showing thenear-inertial velocity components u (east) and v (north).

According to traditional theory, the polarization should be circular.

Is it?


III. Internal waves

N at 1800m:between 2.2-2.5f

ratio major/minor axis:between 1.25-1.5

non-traditional theory:1.24-1.34

N at 2700m:between 0-0.4f

ratio major/minor axis:between 2-3.5

non-traditional theory:3.2-4.2


Conclusion

Conclusions

Abandoning the Traditional Approximation turns out to have anoticeable effect on a number of phenomena, such deep convection,equatorial flows, Ekman layers, and internal waves.

In numerical models it is straightforward to include non-traditionaleffects. This can be consistently done in a fully nonhydrostatic orquasi-hydrostatic approach.

This holds as well for some analytical problems, for example the basictheory of internal waves.

So, there is often no reason to make the Traditional Approximation !


Conclusion

I thank the following colleagues for a non-traditional collaboration. . .

• Victor I. Shrira (Keele Univ., UK): non-traditional internal-wave theory(J. Fluid Mech. 2005, J. Geophys. Res. 2005, Geophys. Res. Lett. 2006);

• Sjef Zimmerman, Leo Maas & Hans van Haren (NIOZ): review on TA(Rev. Geophys. 2008);

• Eleftheria Exarchou: properties of near-inertial internal waves(J. Mar. Res. 2008);

• Kraig Winters (Scripps, USA) & Pascale Bouruet-Aubertot (Paris VI):numerical modelling of near-inertial internal waves(J. Fluid Mech. 2011);

• Louis Gostiaux (Coriolis, Grenoble): history of the Coriolis force(Reflets de la Physique 2009, Eur. Phys. News 2012);

• Harry Swinney (Univ. Texas, Austin): turning depths of internal tides(J. Geophys. Res. 2012).


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