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by

Tore Furevik

Geophysical Institute University of Bergen, Norway

Bergen, December 2005

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Contents

1 The equation of state 1

2 Hydrostatic balance and static stability 3

3 Local and total time derivatives 6

4 Diffusion 7

5 Coriolis force 9

6 Inertial Oscillations 11

7 Short and long waves 13

8 Dispersion 15

9 Shallow water waves 17

10 Tides in channels and bays 19

11 Internal waves 22

12 A floating bridge 25

13 Internal waves in a continuously stratified fluid 27

14 Lee waves 29

15 Normal modes 31

16 Geostrophic adjustment in a two-layer fluid 35

17 Long waves in a fluid that is rotating 38

18 Tsunami 40

19 Thermal wind I: The Denmark Strait overflow 42

20 Thermal wind II: The Norwegian Coastal Current 44

21 Available potential energy 47

22 Vorticity dynamics 49

23 Ekman spiral 51

24 Ekman transports and Ekman pumping 53

25 Storm surge 55

26 Coastal upwelling with weak winds 56

27 Coastal upwelling with strong winds 57

28 Topographic waves 59

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1 The equation of state

The equation of state is fundamental in both atmospheric and ocean dynamics. Usually it relates the density to the pressure, temperature and the chemical composition in the fluid. We will here look at some of the properties for the atmosphere and ocean. The motivation for this exercise is to get used to functions of more than one variable, and to assess some of the fundamental properties of air and sea water density.

A. Air density

Use Figure 1(a) to answer the following questions:

1a) How is air density changing with increasing temperature (T) or humidity (q)? Mathe- matically, that is ∂T∂ρ and ∂ρ∂q.

1b) Can you by looking at the figure explain why the air is rising in the low latitudes (near Equator), and sinking at the poles?

1c) What is (approximately) the virtual temperature of saturated air at 25C?

1d) What do you think happens if saturated air at 5C comes in contact with (mixes with) saturated air at 15C?

B. Ocean density

Use Figure 1(b) to answer the following questions:

1e) How is ocean density changing with increasing salinity, temperature or pressure (∂S∂ρ, ∂T∂ρ, or ∂ρ∂p)?

1f) How is ∂T∂ρ changing with salinity, temperature or pressure (∂T ∂S2ρ , ∂T2ρ2, or ∂T ∂p2ρ )?

1g) For what temperatures are the density change with pressure, the compressibility, largest (this effect is know as the thermobaric effect)?

1h) In the figure we have marked two water masses with crosses. They represent typical TS values for water masses in the Greenland Sea (cold, fresh) and in the Mediterranean (warm, salty). Which of the two is heaviest at surface pressure? Which is heaviest at 3000m depth? Do you think this has any implications for the temperatures in the deep ocean?

1i) What do you think may happen if two water masses having the same density, but different temperatures and salinities, mix (this effect is know as caballing)?

1j) In the Antarctica melting and freezing processes exist down to almost 2000m depths beneath the floating ice shelves. What can you say about the temperatures of the water masses formed under these extreme conditions?

1k) The deepest water masses in the World Oceans are formed in the Antarctica (Weddell Sea). Can you suggest why this is so?

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−400 −30 −20 −10 0 10 20 30 40 0.005

0.01 0.015 0.02 0.025 0.03 0.035

Temperature (°C)

Specific humidity (kg/kg)

1.1

1.15

1.25 1.2

1.3 1.35

1.4

1.45

(b)

33 34 35 36 37 38 39

−4

−2 0 2 4 6 8 10 12 14 16 18

Salinity (psu)

Temperature (°C)

Ocean density (kg m−3) at surface and 3000 db (~3000m) 1025

1026

1027 1028 1029

1030

1031 1037

1038

1039 1040

1041

1041

1042

1043

1044

1045

Figure 1: (a) Air density (thin curves) as functions of temperature and specific humidity at 1000 mb (near sea level). Thick line shows the humidity at saturation. (b) Ocean density (thin curves) as functions of salinity and temperature at surface pressure (solid lines) and at approximately 3000m depths (dashed lines). The two solid lines show the freezing point temperatures at the two depths. The two stars represent water masses in the Greenland Sea (cold, fresh) and in the Mediterranean (warm, salty).

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2 Hydrostatic balance and static stability

By the standard definition of a (x, y, z) coordinate system, the acceleration of gravity (g) is always pointing in the negative z direction. For an atmosphere or an ocean at rest, or where vertical accelerations are small, there will be a balance between the gravity and the pressure forces on the fluid. In this exercise we will learn more about the hydrostatic balance and the concept of static stability.

A. Hydrostatic balance

We start by making a sketch of the pressure forces acting on a small volume element with length, width and height given by δx, δy, and δz (see Fig. 3.1 in Gill).

2a) Show that the net pressure force in thez direction will be given by−δxδyδz∂p∂z, and that the net pressure force per unit mass is −1ρ∇p, where∇ ≡(∂x ,∂y ,∂z ).

2b) Explain that if the fluid parcel (volume element) is going to be at rest, there has to be a balance between pressure forces and gravity given by ∂p∂z =−ρg, or written as gradients of pressure and geopotential as ∇p+ρ∇Φ = 0.

2c) What is the air pressure 1000m above ground, if mean air density is 1.2 kgm3 and the air pressure at the ground is 1000 mb (= 105Nm2)?

2d) What is the water pressure at 1000 m depth if mean ocean density is 1030 kgm3 and the water pressure at the surface is 1000 mb?

B. Static Stability

Often it is of great interest to get a measure for the stability of the air column or of the ocean column. If for instance the air column is heated from below (during a sunny day), the air near the ground will loose density, and may start to rise and mix with the air above. Similarly, if the water column is cooled from the top (winter cooling), the water near the surface will gain density, and may start to sink. The process of mixing air or water by vertical motion, is known as convection.

2e) Can you think of other meteorological or oceanographic processes that may contribute to change the stability of the air or water column?

2f) Consider the forces acting on a small fluid parcel that is vertically displaced from its equilibrium level, and show that a fluid column is statically stable if

α(dT

dz + Γ)−βdS dz >0,

whereα≡ −1ρ∂T∂ρ is the thermal expansion coefficient,β ≡ 1ρ∂S∂ρ the haline expansion coef- ficient, and Γ≡ −dTdz is the adiabatic lapse rate (a measure for how fast the temperature will decrease due to decreased pressure with height).

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N2 =gα(dT

dz + Γ)−gβdS dz. What do we call this frequency?

2h) In Figure 2 we have shown measurements of (potential) temperature, salinity, and (po- tential) density from a station between Greenland and Spitsbergen. Use the approximate expression for the frequency above,

N2 =−g ρ

dρ dz,

whereρ is the potential density, to calculate the frequency (N) over the intervals 0-50 m, 50-80m, and 2000-2500m (suggestion: instead of using dρ/dz in the equation above, use finite differences, for instance δρ≈0.1 kgm3 andδz = 50 m for the first case).

2i) What are the three corresponding period of oscillations?

2j) What can you say about the stability in the three cases?

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(a)

4 5 6

0 10 20 30 40 50 60 70 80 90 100

Depth (m)

Pot. temperature (°C)

34.8 35 35.2

0 10 20 30 40 50 60 70 80 90

100

Salinity (psu) CTD profile in the Fram Strait

27.5 27.6 27.7 27.8 0

10 20 30 40 50 60 70 80 90

100

σt (kg m−3)

(b)

−2 0 2 4 6

0

500

1000

1500

2000

2500

3000

Depth (m)

Pot. temperature (°C)

34.8 35 35.2

0

500

1000

1500

2000

2500

3000

Salinity (psu) CTD profile in the Fram Strait

27.6 27.8 28 28.2 0

500

1000

1500

2000

2500

3000

σt (kg m−3)

Figure 2: Measured profile of potential temperature, salinity and potential density (in terms of σt units, i.e. ρ−1000 kgm3) at a station in the Fram Strait. (a) shows the upper 100 meters, and (b) shows the entire water column.

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be a function of both spatial position (x≡xi+yj+zk) and time (t). Here we will illuminate the differences and properties of the local and total time derivatives1.

3a) Show that the total derivative of a propertyγ is given by Dγ

Dt ≡ ∂γ

∂t +u· ∇γ.

3b) At a station the temperature is measured to be 15C, and it is falling at a constant rate of 2C per hour. The wind is 15ms1 straight from the north. At the same time a station 100 km to the north measures 5C. The temperature is falling with the same rate everywhere.

Estimate the rate of temperature change (C per hour) of the air particles as they move towards south.

3c) Another day the station above measure the wind speed to be 10ms1 straight from the north, and it is increasing at a rate of 5ms1 per hour. At the same time the station 100km to the north measures a wind speed of only 6ms1, again straight from the north.

The increase in wind speed is the same everywhere. What is the mean accelleration of the particles as they move towards the south?

3d) During a 1-hour period, two boats pass close to a fishing boat who is laying still. The speed of the boats and the recorded pressure changes at the three boats during the passage (1 hour interval) are:

Boat Speed Pressure change

1 5m/s straight north no change 2 10 m/s straight east -2 mb

3 no speed +1 mb

What is the magnitude and the direction of the maximum change in pressure (suggestion:

calculate the pressure gradient ∇p and use Pythagoras)?

1This exercise is partly from Wallace and Hobbs.

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4 Diffusion

Despite the fact that this course is mainly about dynamics, atmospheric and ocean dynamics are closely connected to the distribution of the density field. Horizontal density contrasts will set up horizontal pressure forces which again will lead to accelerations and motion. At the same time diffusion and mixing between the different water masses will always act to reduce density gradients and thus the pressure forces. Here we will have a brief look at the diffusion equations2.

4a) Consider the balance of salt (or specific humidity) for a small volume element fixed in space. If the density is ρ, the salinity (or specific humidity) is S, and the diffusivity of salt in sea water (or water vapour in the atmosphere) is κD, show that the mass balance for the salinity in the fixed volume becomes

∂ρS

∂t +∇ ·(ρSu−ρκD∇S) = 0.

4b) Show that when variations in density and in the diffusivity can be neglected, the equation can be written

DS

Dt =κD2S.

Typical values for the diffusivity is 1.5×109m2s1 for salt in water and 2.4×105m2s1 for water vapour in the atmosphere.

4c) Similar to the equation for the diffusion of salinity, a simplified equation for heat (or temperature) diffusion may be written

DT

Dt =κH2T,

where κH is the diffusivity of heat. Typical values are 2×105m2s1 for air and 1.4×107m2s1 for sea water. Show that

T =T0+T1eαzcos(αz+γt+φ),

where α, γ, andφ are constants, is a solution of the heat diffusion equation (suggestion:

substitute the solution into the equation and differentiate).

4d) Atz= 0, the solution above will represent a periodic boundary condition, T =T0+T1cos(γt+φ).

For the temperatures in the ocean, the period will typically be a year, that is γ= 2π/yr = 1.99×107s1.

From the plots given in Figure 3a and b, estimate T0, T1, andφ for the two positions.

4e) Use the time of maximum temperature at different depths to estimate the constant α.

What is the diffusivity of heat κH for the two cases? Why do you think the values are much larger than the molecular diffusivities cited above?

2This exercise is partly from previous exercises in GFO110.

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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan 50

100

150

200

250

300

Month of year

Depth (m)

10 10

10 10

11

11

11 11

11 12

12

12 13

13

14 14

15

(b)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

0

50

100

150

200

250

300

Month of year

Depth (m)

Annual cycle in temperature at 70°N, 0°W

3 4

4

4

4

4

4 4

5

5

5 5

6

6 7

7

8 9

Figure 3: Annual cycle in temperatures at a position in (a) The North Atlantic, and (b) The Norwegian Sea.

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5 Coriolis force

In this exercise we investigate the relations between acceleration in a rotating coordinate system and acceleration in a fixed system. In particular, we will look at the two terms known as the centrifugal and Coriolis accelerations.

We letΩbe the vector defining the rotation of the Earth. Thus the magnitude of the vector denotes the angular velocity and the direction is defined such that the rotation is clockwise when looking in the direction of the vector. A point P is defined by the position vector xf relative to a fixed coordinate system (non-rotating), and by the position vector xr relative to a rotating system (see Figure 4).

5a) Explain why we have the relationship between the velocities Dxf

Dt = Dxr

Dt +Ω×xr.

The mean radius of the Earth is 6371 km, and the rotation rate isΩ= 7.292×105s1. What is the speed of a hard working, still-sitting student in Bergen (60.40N), relative to a fixed coordinate system?

5b) Explain why we have the relationship between the accelerations D2xf

Dt2 = D2xr

Dt2 + 2Ω×Dxr

Dt +Ω×(Ω×xr).

What is the physical interpretation of each term? What will be the acceleration for the same student as above? How large is this compared to the gravitational acceleration?

5c) Show that for a local coordinate system, having the x, y, z axes pointing respectively east, north and upward, the Coriolis term becomes

2Ω×Dxr

Dt = (2Ω cosφw−2Ω sinφv)i+ 2Ω sinφuj−2Ω cosφuk,

where (i,j,k) are the unit vectors and (u, v, w) the velocities along the three axes of direction, and φis the latitude.

5d) Assume that a cannonball is shot straight west from Bergen at an average horizontal speed of 200 ms1. After 30s the cannonball hits the water. Neglecting friction, how far from the latitude of Bergen does the cannonball hit the water? (suggestion: Assume constant u, and calculate the acceleration and the distance travelled in the north-south direction during the flight).

5e) The Sotra Bridge outside Bergen (same latitude as Bergen) has a height of 30m. If you drop a coin from the bridge, how far from the vertical line will the coin hit the water?

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P

xr r

Ω × xr

Figure 4: A pointP with fixed positionxrin a frame of reference rotating with angular velocity Ωabout an axis throughO, will move in the circular path shown. Velocity will be 2Ω×xrand acceleration −∇(122r2) (redrawn from Gill, fig. 4.5).

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6 Inertial Oscillations

In some of the previous exercises we looked on the effects of a rotating Earth on the motions of a particle relative to the Earth. When looking at relatively small time scales with original velocity being along axis 1 (x direction), we only have to consider the effect the velocity in this direction has on the acceleration and velocity in the directions normal to it, for instance along axis 2 (y direction). If, however, time scales are large, the velocity along axis 2 will give a feedback to the velocity along axis 1. This give rise to circular motion, known as inertial oscillations.

6a) The equation of motion is Du

Dt + 2Ω×u=−1

ρ∇p−g+ν∇2u.

Explain what the different terms in the equation represent.

6b) Show that if no external forces are acting on a particle, and we assume no vertical motion, the equation of motion simplifies to

∂u

∂t −f v= 0, and ∂v

∂t +f u= 0, where f = 2Ω sinφis the Coriolis parameter (φis latitude).

6c) Show that the equation of motion for each of the components may be written

2u

∂t2 +f2u= 0, and ∂2v

∂t2 +f2v= 0.

6d) Show by inserting into the equations above that the solutions become u=Usin(f t+δ), and v=Ucos(f t+δ), where U and δ are constants.

6e) Integrate the two solutions foruandvto get the variations inxandypositions as function of time (this we call trajectories). Show that the trajectories become circles, given by the equation

(x−x0)2+ (y−y0)2 = (U f)2. Such motion is known as inertial oscillation.

6f) At the latitude of Bergen (60.40N), what is the period of such rotations? And what is the radius of the circles if the flow speed is 0.1 ms1? What is the radius at the North Pole or at the Equator?

6g) In Figure 5 we show some famous current meter measurements taken in the Baltic Sea (at 57.8N) in 1936. What is the theoretical period of the inertial oscillations at this location?

What is the typical speed of the current? Why do you think the radius of the inertial oscillations decreases with time?

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Figure 5: Current measurements in the Baltic in 1936. The plot show trajectories (progressive vector diagram) based on current meter measurements in a fixed point. The time interval between each tick is 12 hours (Gill, fig. 8.3).

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7 Short and long waves

We will in this exercise look at the properties of non-rotating surface gravity waves. As all surface perturbations (deviations from equilibrium) can be viewed as a superposition of an infinite number of trigonometric functions (Fourier components), many general properties of waves can be derived from looking at an individual component.

For simplicity, we put the x axis along the direction of the wave propagation, and assume constant air pressure (p0), sea water density (ρ0), and depth (H). A sketch of the system is given in Figure 6.

7a) The equations that determine this problem, are the equation of motion for the x and z directions, and the continuity equation,

∂u

∂t = −1 ρ0

∂p

∂x,

∂w

∂t = −1 ρ0

∂p

∂z −g,

∂u

∂x+ ∂w

∂z = 0.

Explain what assumptions that have been made to simplify the general equations of motion to the three equations above.

7b) Write the pressure in the fluid p as a sum of the equilibrium pressure p0(z) and the perturbation pressure p0(x, z, t), and derive the Laplace equation for the perturbation pressure

2p0

∂x2 +∂2p0

∂z2 = 0.

7c) Give the physical explanation for the kinematic and dynamic boundary conditions for this system, that is w(0) =∂η/∂t, w(−H) = 0, andp0(0) =ρ0gη.

7d) We write the surface elevation on the form η = η0cos(kx−ωt), where η0 is amplitude, k = 2π/λ the wavenumber (λ is the wave length), and ω is the frequency. We further assume that the pressure perturbation is proportional to η, that is

p0 =F(z)η0cos(kx−ωt).

Show that Laplace equation gives

F(z) =Ccosh[k(z+δ)],

where C and δ are integration constants to be determined by the boundary condition.

7e) Use the dynamic boundary condition to determine the constant C.

7f) Find the solution for the vertical velocity w, and use the kinematic boundary condition at the bottom to show that the total solution for the perturbation pressure becomes

p0 = ρ00cosh[k(z+H)] cos(kx−ωt)

cosh(kH) .

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z=0

z=−H

ρ0 p0

x

η=η(x,t)

Figure 6: Sketch of a surface gravity wave.

7g) Use the last boundary condition to derive the dispersion relation ω2 =gktanh(kH).

What is the velocity of the wave?

7h) We shall now look at two extreme cases, one where wave lengths are long compared to bottom depth,kH1, and one where wave lengths are short compared to bottom depth, kH 1. What are the frequencies and phase speeds of the two types of waves?

7i) What are the two velocity components u, and wfor the short waves (kH 1)?

7j) Integrate the two velocity components with respect to time, and show that the paths of the fluid particles become circles with radius decreasing exponentially with depth. At what depth is the radius reduced to exp(−1) of the surface value (reduced to 37%)? How deep will a 10 m long wave reach? What about a 100 m long wave?

7k) Sitting at the coast west of Bergen, you notice that waves coming in from a storm center in the North Atlantic are typically 100 m long. 12 hours later, wave lengths have decreased to 50 m. Assuming that the waves are short compared to the depth, what is the distance to the storm center?

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8 Dispersion

In the previous exercise we looked at a single wave. We will now extend the analysis to two waves, define the group velocity, and investigate more properties of short and long waves.

8a) We will look at two waves with almost equal wave numbers, k1 and k2 (see Figure 7).

What are the corresponding frequencies ω1 andω2 for a depth of 10 m (short waves) and for a depth of 1 cm (long waves)?

8b) What are the phase velocities cof the two waves for the deep- and shallow water cases?

8c) Let k1 = k +δk, k2 = k −δk, ω1 = ω +δω, and ω2 = ω −δω, and show that the superposition of the two waves

η10cos(k1x−ω1t), and η20cos(k2x−ω2t) can be written

η = 2η0cos(δkx−δωt) cos(kx−ωt).

What are the group velocities (cg) of the superposition of the two waves for the deep- and shallow water cases? How is the group velocity cg compared to the phase velocity c. Do your results agree with Figure 7?

8d) Use the definition of the group velocity,

cg ≡δω/δk and the dispersion relation

ω2 =gktanh(kH)

to derive the group velocity for short and long waves. What can you say about the dispersion properties of the two types of waves?

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0 10 20 30 40 50 60 70 80 90 100

Two waves, k1=2.0m−1 and k2=2.1m−1

Superposition of the two waves

10 s later for H=10m

10 s later for H=0.01m

Distance (m)

Figure 7: Two waves with almost equal wave numbers (upper plot), the superposition of the two waves (second plot), the group after 10 seconds for short waves (third plot) and for long waves (last plot).

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9 Shallow water waves

We will now study the adjustment processes that take place when we have an initial disturbancy of the surface elevation. We will only consider waves that are long compared to depth. For simplicity, we look at the two-dimensional case, and neglect friction, non-linear terms, and rotation.

9a) Start with the general equations of momentum and mass conservation and explain the assumptions we use to get the following set of equations

ρ∂u

∂t =−∂p

∂x, ρ∂v

∂t =−∂p

∂y, ∂p

∂z +ρg= 0, and ∂u

∂x+ ∂v

∂y+∂w

∂z = 0.

9b) We will now study waves with motion in thexz plane (v = 0 and ∂y = 0). Show that the equations above can be written as the shallow water equations

∂u

∂t =−g∂η

∂x and ∂η

∂t +H∂u

∂x = 0, where η is surface elevation,g gravity andH the constant depth.

9c) Derive a single equation for η and seek a solution on the formη=η0cosk(x−ct). What is the constant c? Are such waves dispersive?

9d) Show that the equation for the energy associated with this wave, is given by

∂t(1

2ρHu2+1

2ρgη2) + ∂

∂x(ρgHuη) = 0.

Explain (using few words!) what the terms represent.

9e) Average over a wave length to show that the sum of the kinetic and the potential energy equals

E=Ek+Ep = 1 2ρgη20.

How is the energy divided between the kinetic (Ek) and the potential (Ep) forms?

9f) Consider the situation illustrated in Figure 8. Initially the fluid is at rest (u = 0) and the surface elevation is given by η = G(x) where G(x) = η0 for −L ≤ x ≤ L and zero elsewhere. Show that every functions on the formη=G1(x−ct)+G2(x+ct) are solutions to the shallow water equations and calculate the velocity component u.

9g) Use the initial conditions to show that

G1(x) =G2(x) = 1 2G(x), and that the evolution with time is given by

η =

0 : |x−ct|> L and |x+ct|> L η0 : |x−ct|< L and |x+ct|< L η0/2 : otherwise

9h) Find the corresponding expressions for u and make a sketch of the surface elevation at times ct=L/2 and ct= 3L/2.

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z

x

z=−H

x=−L x=L

η0

ρ=const u=0

Figure 8: The initial conditions for the shallow water waves exercise. The fluid is at rest (u= 0) and the surface elevation is given by η = G(x) where G(x) = η0 for −L ≤ x ≤ L and zero elsewhere.

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10 Tides in channels and bays

For narrow gulfs and channels, the propagation of tidal waves is to a good approximation described by the shallow water equations. We will here look at a special case known as resonance.

Resonance occurs when the forcing has a frequency that matches the frequency of one of the natural modes of oscillations for the system. In certain areas, like the Bay of Fundy (see map in Figure 9a), resonance effects make the tides become almost 10 meters in amplitude.

10a) We will model the Bay of Fundy system as a two-dimensional problem, with the coordinate system located at the head (inner end) of the bay, where we have a vertical wall, the mouth of the bay is located at x =L, and we assume constant air pressure, ocean density and ocean depth. The governing equations for the problem may be written

∂u

∂t =−g∂η

∂x, and ∂η

∂t +H∂u

∂x = 0,

where η is surface elevation and u the velocity component along the horizontal axis.

Explain what simplifications we have made.

10b) A general wave solution of a wave propagating from the mouth of the bay towards it inner wall is η=η1cos(kx+ωt). What is the relation between wave numberk and frequency ω?

10c) A wave that has been reflected at the inner wall, will propagate out (towards right) given by η = η2cos(kx−ωt). Show that when η1 = η2 = η0/2, the superposition of the two waves is given by

η=η0coskxcosωt, and u= c

0sinkxsinωt, where cis the phase speedc=ω/k.

10d) Now we will look at the boundary conditions at the open boundary. To avoid that the pressure gradient or the divergence are going towards infinity, both the perturbation pressure and mass (volume) transports must be continuous across the mouth of the bay.

Show that this leads to a criteria for the impedance Z, Z = ρgη

ρAu = c

AcotkLcotωt,

where cot = cos/sin is the cotangent function andAthe area (width times height) at the mouth of the bay.

10e) If we assume that the ocean just outside the bay is infinitely wide and deep compared to the bay, Z = 0 outside the bay. Show that this implies that waves oscillating freely will have distinct wave lengths and periods given by

λ= 4L

2n+ 1, and T = 4L

√gH(2n+ 1),

wheren= 0,1,2,3, ... is a counter. Make a sketch of the solution for the first three modes (n= 0,1,2).

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10g) Now we go back to the solution for the channel (in 10c) and assume that the tides produce an oscillating surface elevation at the mouth of the bay given by ηTFcosωFt, where ηF and ωF is amplitude and frequency of the forcing. Show that with this boundary condition at the mouth of the bay, the equation for the surface elevation at the head of the bay becomes

η0 = ηFcosωFt coskLcosωt.

What happens if the frequency of the forcing approaches one of the natural frequencies of the system, that is ωF ≈ω?

10h) The period of the semi-diurnal (=twice a day) tides is 12.42 hours. Which of the first three modes found above does this period correspond to?

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(a)

68oW 67oW 66oW 65oW 64oW 63oW 62oW 43oN

30’

44oN 30’

45oN 30’

46oN

USA

Canada

Nova Scotia Bay of Fundy

(b)

z=0

z=−H

ρ0 p0

x=L x=0

η=η(x,t)

Figure 9: (a) Map of the Bay of Fundy area west of Nova Scotia. Depth contours are drawn at 20 m intervals, with every second dashed. (b) Model of the Bay of Fundy area. The length of the bay is approximately L= 150 km and the depth is typically H = 20 m.

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of different densities. Ordinary surface waves van be viewed as internal waves, but as since the density difference between the ocean and atmosphere is very large (ρ0 ≈ 1000ρa), we tend to neglect the atmospheric density and the atmospheric motion set up by the waves, in that particular kind of problems. Now we will investigate waves on an interface between two fluids, and where the density difference between the fluids are small. Examples may be waves propagating on a boundary between fresh and more salty waters, or between warm air overlying cold air (inversion)3.

11a) Give the linearised equations for long (hydrostatic) waves.

11b) Show that the horizontal velocities are independent of depth, and integrate the continuity equation over depth.

11c) We will now look at a model that consists of two, homogeneous, incompressible and fric- tionless layers with densities ρ1 and ρ2. The thickness of the two layers are H1 and H2 respectively, and the total depth is H =H1+H2. The interface is perturbed vertically, given byh(x, t). We further use the rigid lid approximation. That means that we include the pressure disturbances the wave at the interface will generate at the surface (the baro- clinic part), but ignore the velocities set up by any travelling waves at the surface (the barotropic part). Thus the pressure at z= 0 will now depend on distance and time, that is p0 =p0(x, t) (see Figure 10a). Find an expression for the pressures in the two layers, p1 and p2.

11d) Show that the equations for the horizontal velocity (momentum equation) in the two layers become

∂u1

∂t =−1 ρ1

∂p0

∂x, and ∂u2

∂t =−1 ρ2

∂p0

∂x −g0∂h

∂x, where g0 =g(ρ2−ρ1)/ρ2 is the reduced gravity.

11e) Integrate over each layer and linearise to show that the continuity equation gives H1∂u1

∂x −∂h

∂t = 0, and H2∂u2

∂x +∂h

∂t = 0.

11f) Combine the continuity equations with the momentum equations to find two expressions for∂2h/∂2t, and by adding the two equations show that the equation for the disturbance (wave on the interface) becomes

2h

∂t2 −c2i2h

∂x2 = 0, where c2i = ρ2H1H2 ρ1H22H1g0.

3This exercise is partly based on a master thesis by Birgitte Rugaard Furevik, ”Interne bølger i norske farvande observert med ERS-1 SAR” published in 1995.

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11g) Assume that the wave on the interface can be written h = h0cos(kx−ωt), derive the expressions for the horizontal velocity in the two layers, and show that

H1u1+H2u2= 0.

11h) Use the continuity equation and the linearised kinematic boundary condition at the in- terface (w1(−H1) =w2(−H1)≈∂h/∂t) to derive the expressions for the vertical velocity component in each the two layers,

w1=−cih0k

H1 zsin(kx−ωt), and w2 = cih0k

H2 (z+H) sin(kx−ωt).

11i) Make a sketch of the wave and the associated motion, and show where there is convergence or divergence at the surface.

11j) In this exercise we used the long-wave approximations. Without this approximation, the corresponding equation for the waves would have become slightly more complicated,

c2i = g

k(ρ2−ρ1) tanh(kH1) tanh(kH2) ρ1tanh(kH2) +ρ2tanh(kH1).

Show that the phase speed for long waves become the same as given above, and that the expression for short waves may be written

c2i = ρ2g0 k(ρ21). Are these waves dispersive?

11k) If you look at the internal waves depicted in Figure 10b, can you use the long wave approximation to explain what you see?

11l) What information would have been needed in order to find the area where these waves are generated?

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h=h(x,t) z=0

z=−H1

z=−H

ρ1

ρ2 P0(x,t)

(b)

Figure 10: (a) A simple model of a two-layer system with an fixed ice cover on top. (b) A synthetic aperture radar (SAR) image taken from the European ERS1 satellite. The image is from the Mediterranean side of the Strait of Gibraltar (Spain and Gibraltar is the upper left area, and Morocco the lower left area), and covers roughly 60 km × 40 km. The SAR measures the roughness of the surface, which is essentially the small surface ripples of a few centimetres. Where the surface water is converging, the ripples become steeper which is seen as bright areas in the image. Opposite, where the surface water is diverging, the ripples become smoother, which is detected as darker areas. Note the internal waves propagating eastward from the strait. The dark winding line is due to oil release from a boat, which destroys the surface tension and the ripples can not develop.

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12 A floating bridge

Again we look at a two-layer system. However, this time we will look at the combined effect of a barotropic (surface) wave and a baroclinic (internal) wave4.

12a) In a fjord the tidal wave will have the form of a progressive wave, where the surface elevation is given byη =η0sin(k0x−ωt), and the total depth isH. Use the shallow water equations to find the corresponding barotropic velocity u0, and express the wave number k0 in terms of the frequencyω, which for the tidal wave is a known constant.

12b) The fjord has a upper layer of thicknessH1 and densityρ1, and a lower layer of thickness H2 and densityρ2. Across the fjord there is going to be built a floating bridge, where the depth of each of the floating pontoons is reaching below the mixed layer, that is below the interface between the light surface water and the denser deep water (see Figure 11). We assume that the barotropic wave can continue without being affected by the pontoon (this is a good approximation if the upper layer is shallow compared to the total depth), but that the bridge sets up internal progressive waves on each side of the bridge. We assume that the horizontal velocities of the internal waves on each side is given by

u1=Asin(k1x+ωt) for x <0, and u1 =Bsin(k1x−ωt) for x >0.

Integrate the equation of continuity over the upper layer, and find an expression for the displacement h of the interface.

12c) Integrate the equation of continuity over the lower layer, and show that the horizontal velocity in the lower layer becomes

u2 =−H1 H2u1.

12d) Use rigid lid approximation to express the wave number for the waves at the interface k1 in terms of ω Suggestion: Use the results you found in part f) in the previous exercise.

12e) A boundary condition is that the total velocity, which is the sum of the barotropic velocity (u0) and the upper layer velocity associated with the internal wave (u1) is zero atx= 0.

Physically u =u0+u1 = 0 at x= 0, means that there is no flow through the pontoon.

Use this boundary condition to determine the strength of the internal wave, show that A= ωη0

k0H, and B =−ωη0 k0H,

and find the corresponding expressions for the amplitude of the interface displacement.

12f) Use the values η0 = 1 m, H1 = 5 m,H2 = 100 m, ρ1 = 1020 kgm3, ρ2 = 1030 kgm3, and ω= 1.3×104 s1, and estimate the wave numbersk0 andk1 (or wave lengths), and the amplitude of the interface displacement h. Make a sketch of the wave.

4This exercise is modified from exercises in GFO210 by Martin Mork and Frank Nilsen.

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z=0 z=−H1

z=−H

ρ1

ρ2

x=0

Figure 11: A two layer model with a floating bridge on top. The bridge has a floating device which goes below the upper mixed layer in the position x= 0.

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13 Internal waves in a continuously stratified fluid

Internal waves can always be present in a stratified fluid, even without a discontinuity in density.

In this exercise we will therefore study waves in a continuous, stratified, incompressible fluid.

13a) We will look at a situation where the density and pressure is close to their mean depth- depending values, that isρ=ρ0(z) +ρ0(x, y, z, t) andp=p0(z) +p0(x, y, z, t), whereρ0 ρ0 and p0 p0. Show that the Boussinesq approximation together with the assumption of incompressibility give the following five equations to describe the motion in the fluid

ρ0∂u

∂t =−∂p0

∂x, (1)

ρ0∂v

∂t =−∂p0

∂y, (2)

ρ0∂w

∂t =−∂p0

∂z −ρ0g, (3)

∂ρ0

∂t +wdρ0

dz = 0, (4)

∂u

∂x +∂v

∂y+ ∂w

∂z = 0. (5)

13b) Combine equations (1) and (2) with (5), and (3) with (4), to find the following two equations for the relationship between the vertical velocity and the pressure perturbations

ρ02w

∂z∂t = ∂2p0

∂x2 +∂2p0

∂y2, (6)

ρ02w

∂t2 =−∂2p0

∂t∂z−ρ0N2w, (7)

where N2 = −ρg0

0

dz. What is the name of the parameter N and what is its physical meaning?

13c) Differentiate equation (6) with respect to z and t, (7) twice with respect to x, and (7) twice with respect to y, assume that the vertical variation in ρ0 is much less than the vertical variations inw, and show that a single equation for the vertical velocity is

2

∂t22w+N22Hw= 0, (8) where ∇2 = ∂x22 +∂y22 +∂z22 and ∇2H = ∂x22 +∂y22.

13d) Assume a wave component on the form w = w0cos(kx+ly+mz−ωt) to derive the dispersion relation for internal waves in a continuous stratified fluid.

13e) Show that the frequency only depends on the stratification and the angle between the wave number vector and the horizontal plane, ω=Ncosφ (see Figure 12). Explain what you are seeing in the figure. Give a physical explanation for why the frequency should depend on the direction of the wave propagation, and what are the reasons for the frequency to be between 0 and N?

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Figure 12: Internal waves in a continuously stratified fluid. The waves are generated with an oscillating cylinder with periods (a) 4 seconds and (b) 8 seconds. In both cases the stratification corresponds to 2π/N ≈ 5s. After P. B. Rhines and E. G. Lindahl, School of Oceanography, University of Washington.

13f) What is the angle between the direction of the phase propagation and the direction of the energy propagation (suggestion: use the definition of the group velocity and show it graphically).

13g) Assume that we have a vibrating cylinder (or oscillating membrane) that oscillates at a period of 1 hour in waters with the stratification corresponding to N = 0.005s1, N = 0.01s1, orN = 0.0005s1. What are the corresponding angles between the energy propagation in the waters, and the horizontal plane.

13h) What do you think may happen if an internal wave with frequencyωgenerated in shallow waters, propagates downward towards waters with N < ω? What happens to the group velocity whenω approaches N?

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14 Lee waves

We will now look at waves in a continuously stratified, incompressible fluid that are generated at a boundary. Such waves are called forced waves, in opposition to the free waves we looked at in the previous exercise. An impressive example of such waves are the mountain waves that regularly occurs downstream from the island Jan Mayen at the boundary between the Norwegian and the Greenland Seas (see Figure 13b).

14a) We will first try to model the flow over varying topography as depicted in Figure 13a. Here the flow is uniform in the x direction with the speedu0. The topography may be viewed as a series of trigonometric functions (Fourier components), where one of the components are given as h(x) =h0sin(kx). The wave numberk is given by 2π/λ whereλis the wave length of the topography. With a coordinate system that follows the flow, the topography will appear to move towards left with a speed−u0, and the function for the bottom depth relative to the moving coordinate system will be

h=h0sink(x+u0t) =h0sin(kx−ωt),

where ω = −ku0 will be the frequency imposed by the bottom topography. Show that the linearised boundary condition at the bottom will give w(0) =u0kh0cos(kx−ωt).

14b) Using complex notation, a wave component on the form w = w0cos(kx +mz−ωt) is writtenw=w(0) exp[i(kx+mz−ωt)], where the real part of the expression is the physical part. Insert this solution into the dispersion relation for internal waves in a continuously stratified incompressible fluid (found in the previous exercise), and show that the wave number m is given by

m2= (N

u0)2−k2, where N2=−ρg0

0

dz and ω=−u0k.

14c) Give a physical interpretation of the solution when m2 >0. What does this criteria tell us about the relationships between stratification, velocity and the wave lengths of the topographic feature? Make a sketch of the solution.

14d) Give a physical interpretation of the solution when m2 < 0. What is the frequency imposed by the bottom topography compared to the N? Make a sketch of the solution.

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(a)

z x

u0

h=h(x)

(b) (c)

Figure 13: (a) Model of a uniform flow of speedu0over a mountain where amplitude of a Fourier component of the topography is h. (b) Satellite picture of a mountain wave cloud patterns in the wake of Jan Mayen on 25th of January 2000. The wind is blowing from southwest, which is from the upper left corner of the satellite picture in (b). Note how the waves penetrate hundreds of kilometres downstream of Mount Beerenberg at Jan Mayen, where it is generated.

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15 Normal modes

The ocean and the atmosphere are thin sheets of fluid in the sense that their horizontal scales are much larger than the depth or height scales. Also most of the energy associated with motion lies in components with horizontal scales much larger than the vertical. For such motions we can use the hydrostatic approximation. In this exercise we will also introduce a common technique known as separation of variables.

We will here study an example with a tidal flow entering a basin at an open boundary, setting up motion in a continuously stratified fluid (see Figure 14)5.

15a) We assume that we have a continously stratified, incompressible fluid in a non-rotating frame, and that we have only motion in the (x, z)-plane. Use Boussinesq and hydrostatic approximations and set up the four linearized equations that determine the flow.

15b) Show by elimination that two equations relating perturbation pressure p0 and vertical velocity ware

ρ02w

∂z∂t = ∂2p0

∂x2 and ρ0N2w=−∂2p0

∂t∂z, where N2=−ρg0

0

dz. What is the physical interpretation of N?

15c) Combine the two equations above to derive one equation for the vertical velocity,

2

∂t2

∂z(ρ0∂w

∂z)

0N22w

∂x2 = 0.

15d) Assume a solution on the form

w= ˜w(x, t)φ(z), and show that a necessary consequence is

2w˜

∂t2

2w˜

∂x2

= −ρ0N2φ

∂z0∂φ∂z) =c2,

where cis a constant. Explain why the general solution for ˜wbecomes

˜

w(x, t) =F(x−ct) +G(x+ct).

What is the physical interpretation of the constant c?

15e) Show that the boundary condition at the bottom gives φ(−H) = 0,

and that the boundary conditions at the free surface together with the first equation under b) gives

ρ03w

∂t2∂z −gρ02w

∂x2 = 0 at z= 0, and thus

dφ dz − g

c2φ= 0 at z= 0.

5This exercise is modified from exercises in GFO210 by Martin Mork and Frank Nilsen.

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∂z 0∂z 0 c

and explain why this to a very good approximation can be written

2φ

∂z2 + (N

c )2φ= 0.

15g) Together with the boundary conditions in e), the equation for φabove forms a so-called Sturm-Liouville problem. As we will see, these kind of problems will give an infinite number of solutions for c, so-called eigenvalues, and for each eigenvalue, there will be a solution for φ, which is known as theeigenfunction. A general solution for this problem is

φ=Csinκ(z+γ),

where C is the constant amplitude, and κ and γ are constants to be determined. Show that κ = N/c and γ = H, and use the surface boundary condition to show that an equation for c is given by

tanN H

c = N2H/g N H/c .

15h) Show that for small values of the argument for the tan function above, the phase speed is given by c2 =c20 ≈gH, and that for larger values of the argument,c=cnN H , where n = 1,2,3, .... Suggestion: Make a sketch of the tanx and 1/x functions an show where the curves cross.

15i) Show that the phase speed of the barotropic mode (c0) in h) corresponds to the homoge- neous case withN = 0, and that the phase speeds for the baroclinic modes can be found by using the rigid lid approximation w(0) = 0. Make a sketch showing how the vertical velocity of the barotropic and the first 3 baroclinic modes depend on depth.

15j) Measurements show that the density profile is given by ρ0= ¯ρ

1−z

H

,

where ¯ρ is the mean density, = 0.001, and H = 1000m. Calculate the phase speeds for the barotropic and the first 3 baroclinic modes.

15k) Explain why the general solution can be written w=

n=

X

n=0

Cnsinκn(z+H)[F(x−ct) +G(x+ct)].

In order to determine the amplitudes of each of the modes, the constants Cn, we must use the boundary conditions. Use the equation of continuity to show that the horizontal velocity must be on the form

u=

n=

X

n=0

˜

un(x, t) d

dz[φn(z)],

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where

∂xu˜n=−w˜n.

15l) Use the simplified equation for φ(under f) to show that each of the eigenfunctions for u are orthogonal to each other, that is

Z z=0 z=H

n dz

m

dz dz = 0, for n6=m.

Suggestion: Write the equation in f as c2nd2φn/d2z+N2φn = 0, do the same for a com- ponent φn, multiply the two equations with dφm/dz and dφn/dz respectively, subtract and intergrate.

15m) At a location x = 0 a tidal current on the form u0 = U(z) cosωt is measured. Assume that U(z) can be written on the form

U(z) =

n=

X

n=0

Ann dz , and show that the constants An are given by

An = Rz=0

z=HU(z)dzn dz Rz=0

z=H(dzn)2dz .

15n) Show that with U(z) =U0 = constant andH1=H/2, the values for theAn’s become An= −2U0Hsin[H(H−H1)]

(nπ)2 ,

and U(z) becomes

U(z) = −2U0

n=

X

n=0

sin2

nπ cosnπ

2 (z+H)

= −2U0H(1 π

1 dz + 1

9π dφ3

dz + 1 25π

5

dz +...+ 1 n2π

n dz ).

15o) The assumed solution fits with a tidal current over a sill. For x >0 the solution becomes u=

n=

X

n=0

An

dz cosω(t− x cn).

Find the corresponding expression for the vertical velocity w and the maximum vertical displacement associated with each of the modes. With a tidal current of U(z) = 0.1ms1 at the inflowing boundaryx = 0, what are the maximum displacements associated with each of the first three modes?

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z x

u0(z,t)

η=η(x,t)

N2=const

depth=H1 depth=H

Figure 14: Model of a basin with a flow entering at an open boundary. The open boundary is located at x=, and the flow here is given by u0(z, t). The depth of the inflow isH1 and total depth H. The density is increasing linearly with depth, given a constant buoyancy frequency N.

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16 Geostrophic adjustment in a two-layer fluid

We are going to look at a simple model for geostrophic adjustment. We assume an infinitely deep ocean, where the density in the surface in one part of the ocean is reduced due to a warming or a freshening. The density anomaly reaches down to a depthH1. Initially, we have a two-layer system where everything is at rest (u=v= 0), and where the thickness of the upper layer is given by h =H1 forx <0, and h = 0 for x >0. The density in the two layers are ρ1 and ρ2. There is no variations in they direction, so we have ∂y = 0 everywhere in the fluid6.

16a) The equations for the motion in the upper layer, is almost the same as those given in the previous exercise, with the exception that ∂y = 0, ρ is replaced by ρ1, and we keep the non-linear terms since the perturbation of the interphase is large (it goes to the surface).

∂u

∂t +u∂u

∂x−f v = − 1 ρ1

∂p

∂x (1)

∂v

∂t +u∂v

∂x+f u = 0 (2)

∂p

∂z = −ρg (3)

∂u

∂x+∂w

∂z = 0, (4)

Now using the assumption of a motionless abyss (∇p = 0 in the lower layer), show that the pressure term can be written−ρ11

∂p

∂x =−g0∂h∂x, whereg0 = ρ2ρ2ρ1 is the reduced gravity.

Integrate the continuity equation over the upper layer, and show that the three equations that describe the system are:

∂u

∂t +u∂u

∂x−f v = −g0∂h

∂x (5)

∂v

∂t +u∂v

∂x +f u = 0 (6)

∂h

∂t + ∂

∂x(hu) = 0 (7)

16b) Due to the non linear terms (u∂u∂x, u∂v∂x and hu), the equations above are impossible to solve analytical. However, much can be said about the state of the fluid when the flow has become stationary. After the adjustment process, the equation of continuity (7) yields

∂x(hu) = 0, thushumust be a constant. Sinceh= 0 at some points, the constant must be zero. But, ash6= 0 at other points, it follows thatu= 0 everywhere after the adjustment.

Use this argument to show that the velocity after the adjustment is given by v= g0

f

∂h

∂x. (8)

16c) All flows that are in geostrophic balance are solutions to the equation above. In order to find an exact solution, we must use our most powerful tool, that is the conservation

6This exercise is mainly from Benoit Cushman-Roisin: Introduction to geophysical fluid dynamics.

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