Introduction to geophysical fluid dynamics

Lecture II of VI(Claudio Piani)

Linearized shallow Water Equations, linear gravity waves, ray tracing equations, depth refraction.

LSWE: WAVES

( )0 0 0( ', ', ') ( , , ) i kx ly th u v h u v e

Linear differential equations with constant coefficients allow for solutions in the form of waves like the ones given below (you should be familiar with this kind of formalism….).

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wave solutionsIf we substitute the wave solution into the LSWE (slide 15, lecture I) we obtain:

0 0

0 0

0 0 0

u gkh

v glh

h H ku lv

They can be combined to obtain:

0 0

0 0

0 0 0

2 2 2

Hk u gkh

Hl v glh

h H ku lv

gH k l

Congratulations!, you have derived the dispersion relation for non-rotating SW-gravity waves in the absence of wind (or current).

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1. Back to the Linearized Shallow Water Gravity Waves

We have found solutions for h (henceforth we simplify our notation by writing h’ as h since we always look for solutions for the perturbed fields) in the following form:

Let us assume l=0 (wave moving in the x-direction) then k, and w must verify the following dispersion relation:

2 2gHk

( )0i kx ly th h e

dxgH

dt k k

Group velocity and phase speed can be derived (you should know how to do this) and they turn out to be equal, independent of wave number (non dispersive) but dependent on the depth of the fluid H.

This causes coastal wave refraction. Surfers are very aware of this….

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Monocromatic wave in 1DThis is an animation of the 1D SWGW. The dots represent individual fluid parcels,• what path do they describe as the wave passes

overhead? • Can you see how the flow converges (red) and diverges

(blu) ahead of the peak and trough?• When does the max velocity occur relative to the peak

and trough?

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SWGW: Depth refraction

2 2gH k

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Monochromatic wave refraction

This is a tank experiment. Notice how the wavelength is reduced as you move to shallower waters? You can calculate the rate of change of the wave number using the dispersion relation (you should know how to do this).

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dk k g H

dt x H x

Actually there is a general theory that allows you to describe how the characteristics of a linear wave packet changes in response to changes in the characteristics of the medium… say hello to:

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SWE: the joys of ray tracing….If you are the kind of student who needs to prove every theorem she/he uses then refer to “Waves in Fluids” (Lighthill 1978), pp317. Everybody else just accept the following set of ray tracing equations (RTE) that define the direction and speed of propagation of the wave and how the characteristics (wavelengths) of the wave change along the way. dx

dt kdy

dt ldk

dt xdl

dt y

Possibly some of you will notice the parallel with Hamiltonian formalism. In this case the Hamiltonian is the frequency w, which is conserved as the wave travels , while k and l are the associated momentum components. Let’s apply these equations to the dispersion relation of shallow water non-rotating gravity waves. 2 2 2gH k l

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SWE: the joys of ray tracing….The RTE applied to SWGW, assuming only H has spacial dependence:

2

2 2

2 2

2

1

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2

dx gHk

dt kdy gHl

dt ldk H

g k ldt x xdl H

g k ldt y y

0

2

0

dxgH

dt kdy

dt l

dk k g dH

dt x H dxdl

dt y

Let’s consider the case of one dimensional waves (l=0), and a sloping topography: 0( )H x H x

Phase speed and group velocity are parallel!!

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SWE: the joys of ray tracing….1 1

0

0

2 2

2ln ln

2 x

dk dk dx k dH dk dH

dx dt dt x k H dx kdx Hdx

H xd k d H c ck l

dx dx cH H x

where lx is the x wavelength and c is a constant. So this shows why linear waves slow down, steepen and break as they approach the shore.

H

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11SWE: topographic wave focusing with RTE

0; 0k k l

0; 0dk dl

dt dt

0; 0dk dl

dt dt

12SWE: linearization with a background windNow let’s look at the case where the background velocities are not zero, such that u= U+u’. The SWE become:

' ' ' ' '' '

' ' ' ' '' '

' ' ' ' ' '' '

u u u u hU u v g

t x x y x

v v v v hU u v g

t x x y y

h h h h u vU u v H h

t x x y x y

From which we can eliminate all the terms which are second order in the perturbation variables.

These are the advection terms relative to the background wind.

13SWE: linearization with a background windAgain we substitute wave type solutions and do the same old thing:

0 0 0

0 0 0

0 0 0 0

2 2 2

2 2

Hk u Uku gkh

Hl v Ukv glh

h Ukh H ku lv

Uk gH k l

Uk gH k l

Now let’s look at what happens to a wave packet as it moves into a region where there is shear in the background wind. We will use our tried and tested ray tracing equations.

Ray tracing in a windThe RTE applied to SWGW in a wind, assuming only U has spacial dependence:

2 2

2 2

dx gHk kU U gh

dt k Kgh k l

dy gHl lgh

dt l Kgh k l

dk Uk

dt x xdl U

kdt y y

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Phase speed and group velocity are NOT parallel!!

Ray tracing in a westerly wind

U

dx kU gh

dt k K

dy lgh

dt l K

dk Uk

dt x xdl U

kdt y y

15Now we assume U only depends on y (increasing with latitude) and the GW is travelling exactly north (k=0)…..

Exercise:1) A tsunami is triggered in the Mediterranean by a fault move

off north Africa. How long will it take to reach the shores of southern France? Distance 1000 km., average depth of sea = 1000 m. (Hint: Assume the tsunami can be treated as a linear SWGW ).

2) Starting from the LSWE, derive the dispersion relation for SWGW given in slide 3 (show all the steps).

3) Consider a SWGW with given k=k0,w=w0 and l=0, what is the maximum displacement of a fluid parcel that sees the wave pass overhead. (Hint: you can integrate the wave solution for u.)

4) Consider the ray tracing exercise on slide 15, what would happen if, initially, GW had k=l=l0 > 0. State clearly if and how the wavelengths would change. Also state what the ray path would look like.

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