Besides the tide itself, which two physical effects mainly raise coastal sea level in a cyclone storm surge?

Low atmospheric pressure (inverse barometer) and onshore wind stress

Increased salinity and strong heating of the whole surface layer by the storm

Coriolis force and a deeper thermocline beneath the cyclone core

Gravitational pull of the cyclone itself and the arrival of tsunami waves

For the same wind stress and shelf width, a storm-surge wind setup Δη = τL/(ρgh) is largest on a shelf that is

shallow, since Δη varies as 1/h

deep and narrow, since deep water holds more energy

shallow, since Δη varies as 1/h

of any depth, since depth cancels

deep, since Δη increases with h

A place has K1 = 10 cm, O1 = 6 cm, M2 = 40 cm and S2 = 20 cm amplitudes. Using F = (K1 + O1)/(M2 + S2), the tide type is

mixed, mainly semidiurnal: 0.25 < F < 1.5

diurnal, since F is greater than 3

mixed, mainly diurnal, since F is between 1.5 and 3

purely semidiurnal, since F is less than 0.25

mixed, mainly semidiurnal: 0.25 < F < 1.5

Why are spring tides largest around new Moon and full Moon?

Solar and lunar forces act along one line, so M2 and S2 are in phase

Earth is then closest to the Sun, so that the solar tide exceeds the lunar tide

Solar and lunar forces act along one line, so M2 and S2 are in phase

The Moon is then at its highest declination, which strengthens only K1 and O1

The Moon is then at its closest point to the Earth in its orbit

A tsunami with a wavelength of 200 km crosses an ocean of uniform depth 4000 m. Which statement describes its behaviour?

It is a shallow-water wave with c = c_g = √(gh), so it is non-dispersive

It is a deep-water wave whose speed is c = gT/2π, set by its period

It is a shallow-water wave with c = c_g = √(gh), so it is non-dispersive

Its group velocity is only half of its phase speed, as for deep-water waves

Its speed depends on the wavelength, with longer waves slower

In the northern hemisphere, a warm-core ring shed by the Gulf Stream has

anticyclonic (clockwise) rotation and a raised sea surface

cyclonic (anticlockwise) rotation and a raised sea surface

anticyclonic rotation and a depressed sea surface

cyclonic rotation and a depressed sea surface

Which statement about deep-water masses in the Atlantic is correct?

AABW lies below NADW because it is denser although less saline

NADW lies below AABW because it is the saltier of the two

Both water masses are formed mainly in the Weddell Sea

NADW forms by brine rejection beneath Antarctic sea ice

Which of the following statements about tides are correct?

The tide-generating force varies as 1/d³ of the distance to the body

The M2 constituent has a period of about 12.42 h

Spring tides have range of order M2 + S2 and neap tides of order M2 − S2

The tide-generating force varies as 1/d³ of the distance to the body

The M2 constituent has a period of about 12.42 h

Spring tides have range of order M2 + S2 and neap tides of order M2 − S2

K1 and O1 are the principal semidiurnal constituents

Which of the following hold for a deep-water surface gravity wave of period T?

Longer-period waves travel faster

The wave energy per unit area is ½ρga²

Longer-period waves travel faster

The group velocity equals the phase speed

The wave energy per unit area is ½ρga²

Which of the following statements about gyres and the thermohaline circulation are correct?

Subtropical gyres are anticyclonic in both hemispheres and have western boundary currents

Deep water forms in a few small regions and returns to the surface by slow upwelling over wide areas

Freshwater input to the North Atlantic can weaken the overturning

Subtropical gyres are anticyclonic in both hemispheres and have western boundary currents

Deep water forms in a few small regions and returns to the surface by slow upwelling over wide areas

Freshwater input to the North Atlantic can weaken the overturning

The conveyor turns over the deep ocean in a few years

Storm Surges, Tides, Tsunamis, Waves, Eddies and Gyres, and Thermohaline Circulation

This chapter finishes the ocean physics of the Engineering Sciences (XE) paper, Section XE7.2 of XE7 Atmospheric and Oceanic Sciences. It answers the last part of the heading: “Storm Surges, Tides, Tsunamis and Wind Waves; Eddies and Gyres; Thermohaline Circulation”. It teaches the physics of surge, the equilibrium tide with spring and neap cycles and the constituents M2, S2, K1 and O1, tsunami propagation with c = √(gh), the deep-water dispersion relation with phase and group velocity and significant wave height, mesoscale eddies, gyres, and the density-driven overturning with NADW and AABW. The numericals are inverse-barometer rise, tidal beat, tsunami travel time, wave group velocity and a flushing time; every constant is given in the stem.

1. Storm surges

A storm surge is the abnormal rise of sea level above the predicted tide, produced by a cyclone or intense storm. Two mechanisms act.
(1) The inverse barometer effect: sea level rises where atmospheric pressure is low, by Δη = −Δp/(ρ g), about 1 cm for each 1 hPa fall.
(2) Wind setup: onshore wind stress pushes water against the coast; in steady, shallow water with bottom friction and Coriolis force ignored the sea-surface slope balances the stress, g h dη/dx = τ/ρ, so Δη = τ L/(ρ g h) over a shelf of width L and depth h. A third contribution is wave setup by breaking waves; the surge adds to the astronomical tide, and the danger is greatest if the peak surge coincides with high (especially spring) tide.

Worked example. τ = 2.0 N m⁻² onshore, L = 100 km, h = 20 m, ρ = 1025 kg m⁻³, g = 9.81 m s⁻²: Δη = 2.0 × 10⁵/(1025 × 9.81 × 20) = 0.99 m. The setup grows with fetch and falls with depth, which is why wide, shallow shelves and funnel-shaped bays, as at the head of the Bay of Bengal, suffer the largest surges. A pressure fall of 60 hPa adds 6000/(1025 × 9.81) = 0.60 m.

⚠️ Trap: surge versus tsunami

A storm surge is meteorological and lasts hours with the storm; a tsunami is a long wave from a seafloor disturbance. Neither is a tide, although a surge is measured against the tide.

2. Tides: equilibrium tide, spring and neap, constituents

Tides are the response of the ocean to the tide-generating force of the Moon and Sun, the difference between their gravitational pull at a point and at the Earth’s centre. It scales as M/d³, not M/d², so the nearby Moon dominates: the solar tide is about 46 % of the lunar one. The equilibrium tide is the hypothetical shape an ocean-covered Earth would take if it responded instantly: two bulges, one facing the Moon and one opposite, so a place on the equator sees two highs a day. Because the Moon moves round the Earth, the lunar day is 24 h 50 min and the main lunar semidiurnal period is 12 h 25 min. Real tides are forced oscillations of ocean basins, with amplitudes and phases set by basin geometry, so the equilibrium tide gives the frequencies but not the heights.

Principal tidal constituents
Name	Origin	Period (h)	Type
M2	principal lunar	12.42	semidiurnal
S2	principal solar	12.00	semidiurnal
K1	luni-solar declination	23.93	diurnal
O1	principal lunar declination	25.82	diurnal

Spring and neap. M2 and S2 drift in and out of phase because their periods differ. The beat period is 1/(1/T_S2 − 1/T_M2) = 1/(1/12.00 − 1/12.42) ≈ 355 h ≈ 14.8 days, half a synodic month. At new and full Moon the Sun and Moon pull along the same line and the range is large (spring, amplitude ≈ M2 + S2); at the quarters they are at right angles and the range is small (neap, ≈ M2 − S2). The diurnal pair K1 and O1 beat with a period of 13.7 days, related to the Moon’s declination. The form factor F = (K1 + O1)/(M2 + S2) classifies the tide: F < 0.25 semidiurnal, 0.25 to 1.5 mixed mainly semidiurnal, 1.5 to 3.0 mixed mainly diurnal, F > 3.0 diurnal. Most of the Indian coast is semidiurnal or mixed semidiurnal.

⚠️ Trap: spring is not a season

Spring tides occur twice a month at new and full Moon, whatever the season, and neap tides at the quarters. The equilibrium tide has a 12.42 h cycle of high water, not 12 h: high water is about 50 min later each day.

3. Tsunamis

A tsunami is a series of long gravity waves generated by sudden displacement of the sea floor (thrust earthquakes mostly, also submarine landslides and volcanic collapse). Its wavelength (tens to hundreds of km) is far greater than the ocean depth, so it is a shallow-water wave: c = √(g h), independent of wavelength (non-dispersive). In the deep ocean, h = 4000 m gives c = √(9.81 × 4000) = 198 m s⁻¹ ≈ 713 km h⁻¹, with an amplitude under a metre that ships do not notice. As the wave moves into shallower water it slows, its wavelength shortens and its amplitude grows, following Green’s law, a ∝ h^(−1/4) (for a change of depth with no energy loss and no change of width). The wave period is unchanged.

Worked example. Travel time over a path of 3000 km with h = 4000 m throughout: t = 3.0 × 10⁶/198 = 1.5 × 10⁴ s = 4.2 h. Shoaling from 4000 m to 10 m multiplies the amplitude by (4000/10)^(1/4) = 4.5 and the speed falls to √(9.81 × 10) = 9.9 m s⁻¹. The 26 December 2004 Sumatra tsunami crossed the Indian Ocean in hours; the Indian Tsunami Early Warning Centre at INCOIS, Hyderabad, now issues warnings from seismic and sea-level data. The first sign at a coast may be a drawdown of the sea if the trough arrives first.

4. Wind waves

Small-amplitude surface gravity waves obey the dispersion relation ω² = g k tanh(k h), with ω = 2π/T and k = 2π/λ. In deep water (h > λ/2, kh large) tanh → 1 and ω² = g k: phase speed c = ω/k = g/ω = gT/2π ≈ 1.56 T m s⁻¹, wavelength λ = gT²/2π ≈ 1.56 T² m, and group velocity c_g = dω/dk = c/2 = gT/4π. Energy travels at c_g, so a wave group advances at half the speed of its crests, and long waves outrun short ones (dispersion): swell from a distant storm arrives with the longest period first. In shallow water (h < λ/20) tanh(kh) → kh, c = c_g = √(g h) and waves are non-dispersive like a tsunami.

Worked example. T = 10 s in deep water: λ = 9.81 × 100/2π = 156 m, c = 15.6 m s⁻¹, c_g = 7.8 m s⁻¹. Wave height. The energy per unit area of a wave of amplitude a is E = ½ ρ g a². A real sea is a spectrum of many waves; the significant wave height H_s is the mean height of the highest third of the waves, and H_s ≈ 4 √m₀, where m₀ is the variance of the sea-surface elevation (the area under the wave spectrum). It grows with wind speed, fetch and duration until the sea is fully developed. Locally generated, steep and irregular waves are sea; long-period regular waves that have left the generating area are swell.

⚠️ Trap: which speed?

In deep water c_g = c/2, in shallow water c_g = c. A tsunami (shallow-water wave) uses c = √(g h); a 10 s swell in 4000 m of water is deep-water, with c = gT/2π. Check h against λ/2 before choosing.

5. Eddies and gyres

The ocean is full of mesoscale eddies, 50 to 200 km across, living for months and carrying far more kinetic energy than the mean flow. Their size is set by the Rossby radius of deformation, about N H/f (some 10 to 50 km at mid-latitudes), the scale at which rotation and stratification balance. Eddies are in near-geostrophic balance: warm-core (anticyclonic, high sea level) eddies rotate clockwise in the northern hemisphere and cold-core (cyclonic, low sea level) eddies anticlockwise. They form from unstable meanders of strong currents (Gulf Stream rings, Agulhas rings) and move water, heat, salt and nutrients, stirring the ocean much as weather systems do in the atmosphere.

Gyres are the basin-scale circulations of Chapter 2. The five subtropical gyres (North and South Atlantic, North and South Pacific, Indian) lie under negative curl in the northern and positive in the southern hemisphere (both anticyclonic: clockwise in the north, anticlockwise in the south), with a strong western boundary current and slow, broad eastern return flow. Poleward lie cyclonic subpolar gyres, with positive curl in the northern hemisphere. The North Indian Ocean has no closed subtropical gyre; its currents reverse with the monsoon, as in the Somali Current. Surface waters in the middle of an anticyclonic gyre are depressed and nutrient-poor; waters in the cyclonic gyres, where Ekman pumping brings thermocline water up, are richer.

6. Thermohaline circulation

Wind drives the upper ocean, but the abyss is ventilated by density-driven (thermohaline) circulation: surface water cooled or made saltier becomes dense and sinks, and is replaced elsewhere by slow upwelling. The pieces are deep-water formation in a few small regions where surface water is cooled to freezing or brine is rejected by sea-ice formation: North Atlantic Deep Water (NADW), formed in the Labrador and the Greenland-Iceland-Norwegian seas (T about 2 to 4 °C, S about 35), and Antarctic Bottom Water (AABW), formed on the Antarctic shelves, especially the Weddell and Ross seas (T below about 1 °C, S about 34.7), the densest water in the open ocean. AABW spreads along the bottom beneath NADW, so the densest water is not the saltiest.

The global conveyor is the picture of one connected overturning: warm, salty surface water flows north in the Atlantic (the Gulf Stream and its extension), cools, sinks, spreads south as NADW, joins the Southern Ocean, and is returned to the surface by upwelling in the Indian, Pacific and Southern oceans. Overturning is of order 15 to 20 Sv, and the deep-water ages of hundreds to about a thousand years or more are why the deep ocean responds slowly. Worked example. If 20 Sv sinks and the ocean volume is 1.3 × 10¹⁸ m³, the flushing time is 1.3 × 10¹⁸/(2.0 × 10⁷) = 6.5 × 10¹⁰ s ≈ 2060 yr (1 yr = 3.156 × 10⁷ s), an upper bound because water recirculates. Freshwater input in the North Atlantic reduces surface density and can weaken the overturning, which is why it matters for climate. Winds also contribute, notably to upwelling in the Southern Ocean, so “thermohaline” is not the whole of the overturning.

Key takeaways

Surge = inverse barometer (about 1 cm per hPa) + wind setup τL/(ρgh), added to the tide.
Tides: forcing ∝ 1/d³, M2 12.42 h, S2 12.00 h, K1 23.93 h, O1 25.82 h; spring-neap beat of 14.8 days; form factor classifies the type.
Tsunami: shallow-water wave c = √(gh), about 200 m/s at 4000 m, and Green’s law gives growth as a ∝ h^(−1/4) on shoaling.
Deep-water waves: ω² = gk, c = gT/2π, c_g = c/2, period fixed; Hs ≈ 4√m₀.
Warm-core eddies are anticyclonic with high sea level; subtropical gyres are anticyclonic with western intensification.
Thermohaline circulation: NADW and AABW form in few places, and the overturning is 15 to 20 Sv with ventilation times of centuries to a millennium.

Practice questions (17)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Besides the tide itself, which two physical effects mainly raise coastal sea level in a cyclone storm surge?
1. Low atmospheric pressure (inverse barometer) and onshore wind stress
2. Increased salinity and strong heating of the whole surface layer by the storm
3. Coriolis force and a deeper thermocline beneath the cyclone core
4. Gravitational pull of the cyclone itself and the arrival of tsunami waves
Show answer
Answer: A — Low atmospheric pressure (inverse barometer) and onshore wind stress
The inverse barometer effect (about 1 cm per hPa) and the wind setup are the standard contributions. Salinity and heating change density, not sea level at this scale, and a cyclone’s own gravity is negligible.
For the same wind stress and shelf width, a storm-surge wind setup Δη = τL/(ρgh) is largest on a shelf that is
1. deep and narrow, since deep water holds more energy
2. shallow, since Δη varies as 1/h
3. of any depth, since depth cancels
4. deep, since Δη increases with h
Show answer
Answer: B — shallow, since Δη varies as 1/h
Δη ∝ 1/h, so halving the depth doubles the setup. Depth does not cancel in this balance, and deeper water gives less setup.
A place has K1 = 10 cm, O1 = 6 cm, M2 = 40 cm and S2 = 20 cm amplitudes. Using F = (K1 + O1)/(M2 + S2), the tide type is
1. diurnal, since F is greater than 3
2. mixed, mainly diurnal, since F is between 1.5 and 3
3. purely semidiurnal, since F is less than 0.25
4. mixed, mainly semidiurnal: 0.25 < F < 1.5
Show answer
Answer: D — mixed, mainly semidiurnal: 0.25 < F < 1.5
F = 16/60 = 0.27, which lies between 0.25 and 1.5, so the tide is mixed, mainly semidiurnal. Using only M2 against K1 would wrongly suggest a pure semidiurnal tide.
Why are spring tides largest around new Moon and full Moon?
1. Earth is then closest to the Sun, so that the solar tide exceeds the lunar tide
2. Solar and lunar forces act along one line, so M2 and S2 are in phase
3. The Moon is then at its highest declination, which strengthens only K1 and O1
4. The Moon is then at its closest point to the Earth in its orbit
Show answer
Answer: B — Solar and lunar forces act along one line, so M2 and S2 are in phase
At syzygy the Sun and Moon pull on the same axis, so M2 and S2 add; at quarter phases they partly cancel, giving neaps. Perigee gives larger tides only when it coincides, and does not set the fortnightly cycle.
In deep water, the group velocity of surface gravity waves is related to the phase speed by
1. c_g = 2c
2. c_g = c
3. c_g = c/2
4. c_g = c/4
Show answer
Answer: C — c_g = c/2
From ω² = gk, c = √(g/k) and c_g = dω/dk = ½√(g/k) = c/2. c_g = c is the shallow-water, non-dispersive limit.
A 12 s swell is travelling from deep water into progressively shallower water. Which quantity stays constant during the approach?
1. the wavelength
2. the phase speed
3. the wave period
4. the wave height
Show answer
Answer: C — the wave period
The frequency is fixed by the source and does not change as the depth changes. Wavelength and speed fall as the wave shoals and the height changes by shoaling.
A tsunami with a wavelength of 200 km crosses an ocean of uniform depth 4000 m. Which statement describes its behaviour?
1. It is a deep-water wave whose speed is c = gT/2π, set by its period
2. It is a shallow-water wave with c = c_g = √(gh), so it is non-dispersive
3. Its group velocity is only half of its phase speed, as for deep-water waves
4. Its speed depends on the wavelength, with longer waves slower
Show answer
Answer: B — It is a shallow-water wave with c = c_g = √(gh), so it is non-dispersive
λ/h = 50, so kh is small and the wave is in the shallow-water limit with c = c_g = √(gh) ≈ 198 m s⁻¹. The depth, not the wavelength, sets the speed here.
In the northern hemisphere, a warm-core ring shed by the Gulf Stream has
1. anticyclonic (clockwise) rotation and a raised sea surface
2. cyclonic (anticlockwise) rotation and a raised sea surface
3. anticyclonic rotation and a depressed sea surface
4. cyclonic rotation and a depressed sea surface
Show answer
Answer: A — anticyclonic (clockwise) rotation and a raised sea surface
A warm core has a high sea surface, and geostrophy keeps high elevation on the right, so the flow is clockwise (anticyclonic). Cold-core rings are the opposite.
Which statement about deep-water masses in the Atlantic is correct?
1. AABW lies below NADW because it is denser although less saline
2. NADW lies below AABW because it is the saltier of the two
3. Both water masses are formed mainly in the Weddell Sea
4. NADW forms by brine rejection beneath Antarctic sea ice
Show answer
Answer: A — AABW lies below NADW because it is denser although less saline
AABW is colder and so denser, and it sinks to the bottom even though NADW is saltier. NADW forms in the North Atlantic, and brine rejection around Antarctica produces AABW.
Which of the following statements about tides are correct?
1. The tide-generating force varies as 1/d³ of the distance to the body
2. The M2 constituent has a period of about 12.42 h
3. Spring tides have range of order M2 + S2 and neap tides of order M2 − S2
4. K1 and O1 are the principal semidiurnal constituents
Show answer
Answer: A — The tide-generating force varies as 1/d³ of the distance to the body; B — The M2 constituent has a period of about 12.42 h; C — Spring tides have range of order M2 + S2 and neap tides of order M2 − S2
The first three are standard. K1 and O1 are the principal diurnal constituents.
Which of the following hold for a deep-water surface gravity wave of period T?
1. λ = gT²/2π
2. Longer-period waves travel faster
3. The group velocity equals the phase speed
4. The wave energy per unit area is ½ρga²
Show answer
Answer: A — λ = gT²/2π; B — Longer-period waves travel faster; D — The wave energy per unit area is ½ρga²
From ω² = gk, λ = gT²/2π and c = gT/2π, so longer periods travel faster, and E = ½ρga². The group velocity is half of the phase speed.
Which of the following statements about gyres and the thermohaline circulation are correct?
1. Subtropical gyres are anticyclonic in both hemispheres and have western boundary currents
2. Deep water forms in a few small regions and returns to the surface by slow upwelling over wide areas
3. Freshwater input to the North Atlantic can weaken the overturning
4. The conveyor turns over the deep ocean in a few years
Show answer
Answer: A — Subtropical gyres are anticyclonic in both hemispheres and have western boundary currents; B — Deep water forms in a few small regions and returns to the surface by slow upwelling over wide areas; C — Freshwater input to the North Atlantic can weaken the overturning
The first three are correct. Deep-water ventilation times are hundreds to a thousand years or more.
An atmospheric pressure fall of 70 hPa over the sea produces a sea-level rise by the inverse barometer effect, Δη = Δp/(ρg). With ρ = 1025 kg m⁻³ and g = 9.81 m s⁻² (1 hPa = 100 Pa), the rise is ____ m (two decimal places).
Numerical answer — type the value.
Show answer
Answer: 0.70
Δp = 7000 Pa. Δη = 7000/(1025 × 9.81) = 7000/10055 = 0.696, i.e. 0.70 m. Using ρ in g cm⁻³ or omitting g gives nonsense; the rule of thumb is 1 cm per hPa.
Take the semidiurnal periods T_M2 = 12.42 h and T_S2 = 12.00 h. The interval between successive spring tides (the beat period of M2 and S2) is ____ days (two decimal places).
Numerical answer — type the value.
Show answer
Answer: 14.79
1/T_beat = 1/12.00 − 1/12.42 = 0.083333 − 0.080515 = 0.002818 h⁻¹, so T_beat = 354.9 h = 14.79 days. Averaging the periods or subtracting them directly gives wrong answers.
A tsunami travels through an ocean of uniform depth 4000 m, with g = 9.81 m s⁻². The time to cover 2400 km is ____ hours (two decimal places).
Numerical answer — type the value.
Show answer
Answer: 3.37
c = √(9.81 × 4000) = √39240 = 198.09 m s⁻¹. t = 2.4 × 10⁶/198.09 = 1.2116 × 10⁴ s = 3.37 h.
A deep-water swell has a period of 12 s. Take g = 9.81 m s⁻². Its group velocity is ____ m s⁻¹ (two decimal places).
Numerical answer — type the value.
Show answer
Answer: 9.37
c_g = gT/4π = 9.81 × 12/12.566 = 9.37 m s⁻¹. The phase speed is 18.7 m s⁻¹; quoting that would be the error.
Deep water sinks at 20 Sv (1 Sv = 10⁶ m³ s⁻¹) and the ocean volume is 1.3 × 10¹⁸ m³. Take 1 yr = 3.156 × 10⁷ s. The time to replace the ocean volume at this rate is ____ thousand years (two decimal places).
Numerical answer — type the value.
Show answer
Answer: 2.06
t = V/Q = 1.3 × 10¹⁸/2.0 × 10⁷ = 6.5 × 10¹⁰ s = 6.5 × 10¹⁰/3.156 × 10⁷ = 2059.6 yr = 2.06 thousand years.