Download 1 Module 8 Ocean Circulation and Surface Processes 8.1

Module 8
Ocean Circulation and Surface Processes
8.1
Introduction
Winds are produced in response to radiative heating of the atmosphere. These winds
constitute an important forcing for ocean currents, which are generated due to momentum
transfer into the ocean by winds. The pressure gradients generated by radiative heating
could produce wind speeds of about 10 ms −1 in the atmosphere just above the ocean. Yet,
there will be no momentum transfer by winds to ocean layers if there were no friction at
the surface. Because of the frictional contact, no slip condition will be satisfied by the
airflow at solid surface boundary; that is, the air in immediate contact with the boundary
attains zero velocity. This will set up a velocity gradient (or shear) near the solid
boundary. The shear flow set up in this manner is not stable at higher windspeeds because
small disturbances can grow at the expense of mean motion to turn the flow turbulent.
The turbulent eddies are responsible for the gusty nature of the flow, modify shear for a
well-defined mean velocity structure to develop after sufficiently long time. The flow
velocity is a function of z , i.e. the distance from the surface in vertical direction. The
shear depends on mean stress τ, density ρ and distance z from the ground, and one
obtains a logarithmic mean velocity profile.
Near the ground, wind shear varies as (∂u / ∂z)  1 / z ; that is, the inverse law holds
only sufficiently close to the ground. The turbulent eddies in the shear flow close to the
surface affect the transfer of momentum, heat and moisture at the interface of ocean and
atmosphere. What are the other processes that affect the evolution of the surface layer
besides the dynamical forcing of winds? How are their effects included in the analysis?
There exists an extensive literature for an elaborate response to these questions. The other
thermodynamically mediated processes are primarily the solar heating deposited in a few
tens of meters in the upper ocean; evaporation at the sea surface; cooling by evaporation
and sensible heat transfer at the surface of the ocean. Atmospheric circulations of
different temporal and spatial scales produce horizontal gradients in the seawater
properties in the open ocean affecting the current strengths below the interface. Such
horizontal gradients, in conformity with nonlinearity of the equation of state of seawater,
could impact the ocean stratification in different latitudes.
The ocean waters are inherently stratified with higher density waters arranged at the
ocean bottom where temperatures are cold and salinity is low. The uniform distribution of
potential temperatures in deep ocean signifies constant mixing forced by friction in the
bottom Ekman layer. At the ocean surface, atmospheric processes constantly change the
distribution of sea surface temperature and salinity through horizontal transport and
stirring by winds, rain and evaporation, while moving air parcels pick up moisture from
the sea surface and distribute it around the globe.
For the same solar heating rate, land temperatures rise faster in summer than the sea
temperatures and also land cools rapidly during winter with the diminishing radiative
heating. As a result of this contrast, low-pressure systems during summers and high
pressures during winters develop over land areas. Such a flip-flop in atmospheric
pressures over land will result in the alteration of pressure systems over the oceans as
well. For example, there is a high-pressure system (the Azores high) in the north Atlantic
in summer, which is replaced by the Icelandic low during winter. Since the winds will be
almost parallel to isobars over the oceanic regions, the wind induced surface ocean
1
currents (Fig. 8.1) will have anticyclonic orientation below the atmospheric high pressure
cell and cyclonic below the low pressure cell. Due to Ekman transport, light waters are
on the right relative to the direction of the wind in the northern hemisphere, a lens of
warm water (Fig. 8.1) will form the core of anticyclonic surface current cell, and a cold
core for the cyclonic cell. In this manner, interaction forced by atmospheric circulation at
the ocean surface contributes to variability of the climate system, which will finally affect
the transfer processes at the ocean surface wherever air interacts with the sea.
High
Pressure Cell
sea level
Low
Pressure Cell
Atmosphere
sea level
Warm lens
convergence
divergence
Warm
Ther
mocl
ine
Ocean
Cold
Atmosphere
Ocean
Thermocline
Cold
Fig. 8.1 Development of a warm lens on the sea surface under the action of anticyclonic wind stress and
cooling of surface waters under cyclonic wind stress. Note the vertical movement of thermocline under the
influence of a circulation associated with a low and a high in the atmosphere.
These processes also play a dominant role in changing the density of seawater, and
any loss of buoyancy would engender convective overturning in the ocean depth. From
the equation of the state of seawater, it may be inferred that temperature and salinity
generally play an opposing role on density of ocean waters. In the context of dynamics of
oceans and atmosphere, it may be said that atmospheric winds drive ocean circulation;
and evaporation from sea surface contributes water vapour to the atmosphere, which
drives atmospheric circulation in combination with the radiative heating and convective
overturning in the troposphere. Therefore, interaction of the oceans and atmosphere is
key to the understanding of climate and its variability at shorter time scales and to climate
change at longer time scales.
8.2
Drag Coefficient
The atmosphere and ocean exchange momentum, heat and mass including
aerosols/tracers, which have a bearing on biological productivity of the oceans. The
dynamical interaction between the ocean surface and wind is incorporated through
roughness length, z0 , which depends on the wind at standard height (10 m) above the
surface, wave formation and breaking and momentum dissipation at the surface. In the
absence of waves all the momentum will be transferred to the sea surface. However, for
momentum transfer, viscosity of a fluid defines its diffusive capacity. Therefore at the
ocean-atmosphere interface, the ratio of the diffusive capacities of air and water directly
gives the surface velocity in terms of the geostrophic wind U g . For given density and
2
kinematic viscosity ρa , ν a (= µa / ρa ) of air and ρw , ν w (= µw / ρw ) of seawater, the
surface drift velocity U 0 on the ocean side of the interface is given by
0.5
ρ ⎛ν ⎞
1
(8.1)
U0 = a ⎜ a ⎟ × Ug =
U g = 0.005U g
ρw ⎝ ν w ⎠
200
U 0 has the same direction as that of geostrophic velocity U g . However, the Ekman
theory predicts the interface velocity in the direction of the resultant of geostrophic wind
and the current vectors. This amounts to a contradiction between the two views and
further studies are necessary. Below the interface into the sea, the mean velocity is much
smaller than the interface drift velocity. However, for air-sea interaction, it is the 10-m
wind ( U10 ), which is used in the calculations. Moreover, because surface roughness of
the ocean waters changes with windspeed, a drag coefficient C D may therefore be
defined to deal with the changing winds and wave interaction. For windspeeds exceeding
8 ms −1 we have
u ∗2
(8.2)
CD =
; U = u 2x + u 2y
2
(U10 − U 0 )
where u* is the friction velocity, U10 and U 0 are respectively the windspeeds at 10m
height and at the earth surface. The drag coefficient is an appropriate parameter for the
momentum transfer under different atmospheric conditions and the state of the sea
surface − stable, unsteady or dominated by waves. Once C D is known, then from the
boundary layer theory, the wind stress ( τ ) at the surface is defined as,
τ = ρau∗2 = ρaC D (U10 − U 0 )2 = ρaC DU102 as U 0 << U10 .
(8.3)
That is, the wind stress τ is directly related to the 10m windspeed from (8.2). The
formula (8.2) is one of the bulk aerodynamic formulae for computing the fluxes of
momentum, heat and moisture at the atmosphere-ocean and atmosphere-land interface.
On the ground U 0 = 0 , but over the ocean U 0 is the velocity of the wind at the sea
surface. Though ocean is not a solid surface, yet the surface velocity U 0 is sufficiently
small (typically U 0 ≈ 0.03U10 ). This is basically due to the density differences between
the air and water and the same momentum of air masses at air-sea interface can be carried
with much smaller velocities of water masses. Hence the turbulence production over
ocean is similar to land because the shear over the oceans is as large as that over the land.
The drag coefficient CD increases with wind speed (U ) for the ocean surface as well, thus
⎧⎪1.1
10 C D = ⎨
⎪⎩0.61+ 0.063 U10 ,
3
U10 ≤ 6.0 ms −1
6.0 < U10 < 22.0 ms −1
(Smith 1980)
Charnock (1981) gave the following expression for the drag coefficient:
⎧ ⎡ ρ gz ⎤ ⎫
C D = κ ⎨ln ⎢
⎥⎬
⎩ ⎣ aτ ⎦ ⎭
−1
; κ = 0.4, a = 0.0185
The parameter κ is the von Karman constant. U10 = u102 + v102 is the windspeed at 10 m
height and U 0 is the component of ocean surface velocity along the wind direction. Uo is
3
generally neglected in comparison to U10 ; but it should not be neglected in the regions of
ocean where surface currents are strong and winds are weak.
8.3
Bulk parameterization of heat and moisture
The expression (8.3) relates the wind stress τ with windspeed at 10 m height, which
can be further simplified by neglecting U0 , which gives the following expression for the
friction velocity
(8.4)
u ∗2 = C DU102
The bulk aerodynamic formulae for the sensible heat flux QH and the moisture flux E are
given by
QH = − ρa C p C HU10 (T − T0 )
(8.5)
E = − ρa CE U10 [ q(T ) − q(T0 )]
(8.6)
The latent heat flux is then calculated as
(8.7)
QE = Lv E
These parameterizations are applicable both for land and sea with the coefficients C H for
sensible heat and CE for water vapour. These parameters must be known for stable,
neutral and unstable atmospheric conditions. Both C H (Stanton number) and CE (Dalton
number) are the dimensionless aerodynamic exchange coefficients for temperature and
humidity transfer respectively. When one uses these formulae for air-sea exchange to
compute the fluxes of sensible heat and evaporation, then T and q in (8.5) and (8.6) are
respectively replaced by the sea surface temperature Ts and saturated specific humidity
qs . Since the primary goal is to estimate the fluxes of momentum, heat and moisture
using mean observations at one height, the formulae (8.4) – (8.6) fulfil this requirement.
The value of both C H and CE in neutral conditions is 1.2 × 10 −3 which remains
independent of windspeed within 5 − 20 ms −1 , for windspeeds exceeding 20 ms −1
possibly CE may increase due to enhanced evaporation. Under unstable conditions, the
transfer coefficients have large values for light wind conditions, which decrease with
increasing windspeeds. For C H and CE , Smith (1980) found a good fit of data with the
following values
⎧⎪ 0.83 × 10 −3
CH = ⎨
−3
⎩⎪ 1.10 × 10
for stable conditions
for unstable conditions
,
CE = 1.50 × 10 −3
The bulk aerodynamic formulae are based on the premise that the near-surface
turbulence arises from mean wind shear over the surface and that turbulent fluxes of heat
and moisture are proportional to their gradients just above the sea surface. Like the
shear, the temperature and humidity gradients also increase, as the ocean surface is
approached, in inverse proportion (~ 1 / z) with the distance (z) from the surface. Both
heat and moisture are vertically transferred at the air-sea interface by bodily movement of
air parcels. Hot and moist air parcels move upward and relatively cold and dry air would
be transferred downward.
4
8.4
Ekman layers
One may notice from Fig. 8.1, that low-level winds have a direct influence on the
surface currents; besides this, the net effect of wind action on the surface is seen in the
vertical movement of thermocline. The momentum transfer to the ocean surface can be
calculated if the components of the wind stress are known. For this purpose, the wind
stress components from (8.3) can be written as
τ = (τ x , τ y ) = ρaC DU10 (u10 , v10 )
(8.8)
Here (u10 , v10 ) are the components of the wind at 10 m height. Another simplification
arises if the Rossby number is small: one can neglect the acceleration terms in the
equation of motion. In this situation, the Coriolis force, the horizontal pressure gradient
and force due to wind stress would balance in the top layer of the ocean, which reads as
1 ∂ p 1 ∂τ x
(8.9)
− fv = −
+
ρ0 ∂x ρ0 ∂z
fu = −
1 ∂ p 1 ∂τ y
+
ρ0 ∂y ρ0 ∂z
(8.10)
In the above equations ρ0 (= 1027 kg m −3 ) is the reference density of seawater, (u, v) are
the components of horizontal velocity of ocean currents. The effect of surface wind stress
will propagate downwards by the action of turbulence and wind induced stirring at the
sea surface. The balance of wind stress and the frictional force arising from vertical
shear-induced turbulence will happen in a thin boundary layer which is known as the
Ekman layer in the ocean. This suggests that wind stress can be modelled using fluxgradient relation by employing an eddy viscosity coefficient K as
τy
τx
∂u
∂v
.
(8.11)
=K
and
=K
ρ0
∂z
ρ0
∂z
In view of (8.11), the governing equations (8.9) and (8.10) of the Ekman layer take the
following form
∂ ⎛ ∂u ⎞
− f (v − vg ) = ⎜ K ⎟
(8.12)
∂z ⎝ ∂z ⎠
∂ ⎛ ∂v ⎞
f (u − ug ) = ⎜ K ⎟
(8.13)
∂z ⎝ ∂z ⎠
The above equations underscore the role of ageostrophic components of the wind in the
Ekman layer dynamics. The boundary conditions for the above set of equations are
(u, v) → 0 as z → − ∞
(at the bottom)
∂v ⎞ ⎛ τ x (0) τ y (0) ⎞
⎛ ∂u
,
at z = 0 (at the sea surface)
⎜⎝ K , K ⎟⎠ = ⎜
∂z
∂z
ρ ⎟⎠
⎝ ρ
0
(8.14)
(8.15)
0
Note that in (8.12) and (8.13), the pressure gradient terms have been replaced by the
velocity components of the geostrophic current. Equations (8.12) and (8.13) can be
further simplified if the velocity components (u, v) are decomposed in to a geostrophic
part and an ageostrophic part as follows:
5
(u, v) = (ug + uag , vg + vag )
(8.16)
Using (8.16) in (8.12) and (8.13), we obtain the following equations for the ageostrophic
velocity components of the ocean currents
∂ ⎛ ∂u ⎞
(8.17)
− fvag = ⎜ K ag ⎟
∂z ⎝ ∂z ⎠
∂ ⎛ ∂vag ⎞
K
∂z ⎜⎝ ∂z ⎟⎠
The boundary conditions for the above system now read as
(uag , vag ) → 0 as z → − ∞ or z → − δ (Ekman layer depth)
fuag =
(8.18)
(8.19)
∂vag ⎞ ⎛ τ x τ y ⎞
∂ug
∂vg
⎛ ∂uag
(8.20)
K
,
K
=
,
at
z
=
0
(assuming
=
0;
= 0)
⎜⎝ ∂z
∂z ⎟⎠ ⎜⎝ ρ0 ρ0 ⎟⎠
∂z
∂z
The Ekman layer thickness in the ocean is about 30m , i.e. δ = 30m . The condition
(8.19) requires ageostrophic velocity to vanish at the lower limit of the Ekman layer and
(8.20) does not allow shear in the geostrophic currents.
An important derivation: Before attempting the solution of the Ekman boundary value
problem (8.17 – 8.20) we derive an important relation. Vertically integrating the
governing equations for the ageostrophic components (8.17) and (8.18), we get
z=0
0
∂u
τ
⎡ ∂uag ⎤
− f ∫ vag dz = ⎢ K
= K ag (0) = x
⎥
∂z
ρ0
⎣ ∂z ⎦ z=−δ
−δ
τy
∂v
⎡ ∂v ⎤
f ∫ uag dz = ⎢ K ag ⎥
= K ag (0) =
∂z
ρ0
⎣ ∂z ⎦ z=−δ
−δ
Since the wind stress vanishes at the lower boundary ( z = − δ ) of the Ekman layer, the
above equations can be written in terms of integrated mass transport through the Ekman
layer as
0
− f My = τ x
and
z=0
f M x = τ y ; Mx =
0
∫
ρ0 uag dz and My =
−δ
0
∫ρ v
0 ag
dz
(8.21a)
−δ
Hence, we write the mass transport M as follows
M =−
k̂ × τ
f
; M = (M x , M y )
(8.21b)
In (8.21b), k̂ is the unit vector along the vertical. The important inference from (8.21b)
is that the lateral mass transport is directed at 90 0 on the right to the direction of wind
stress in the northern hemisphere. Note that the direction of the wind stress is same as
that of the wind.
Determination of Ekman solution and Ekman spiral: We now present the solution of the
Ekman boundary value problem (BVP) described by (8.17) – (8.20). The equations (8.17)
and (8.18) can be combined into one equation by multiplying by i (= −1) the eq. (8.18)
and then adding the result to (8.17) and assuming constant eddy viscosity coefficient K ,
we get the following steady state equation,
6
K
d 2 (uag + i vag )
= i f (uag + i vag )
dz 2
with boundary conditions
uag + i vag → 0 as z → −δ
(8.22)
(8.23)
d(uag + ivag ) τ x (0) + i τ y (0)
(8.24)
=
at z = 0
dz
ρ0
The solution of Ekman BVP (8.22) – (8.24) is straightforward but involves some
algebraic and trigonometrical manipulations. The steady solution of the above BVP is
K
eλ z
uag =
ρ0 f K
π⎞
π ⎞⎞
⎛
⎛
⎛
⎜⎝ τ y (0 )cos ⎜⎝ λ z + 4 ⎟⎠ + τ x (0 )cos ⎜⎝ λ z − 4 ⎟⎠ ⎟⎠
(8.25)
eλ z
ρ0 f K
π⎞
π ⎞⎞
⎛
⎛
⎛
⎜⎝ τ y (0 )cos ⎜⎝ λ z + 4 ⎟⎠ − τ x (0 )cos ⎜⎝ λ z − 4 ⎟⎠ ⎟⎠
(8.26)
vag =
f
; also λ −1 has dimension of length, therefore, for every
2K
increase in depth equal to λ −1 , the magnitude of the velocity decreases by a factor e−1 .
The hodograph of the steady solution (8.25) and (8.26) depicts the Ekman spiral, with
maximum current speed at the surface inclined at angle of 45 0 on the right to the
direction of wind stress vector in the northern hemisphere. Indeed, E.W. Ekman (1904)
produced this solution overnight when Fridtjof Nensen suggested this problem to him
while Ekman was a student of Vilhelm Bjerknes, the famous Swedish meteorologist who
first recognized weather forecasting as an initial value problem in mathematical physics
in 1904 and also proclaimed that the system of equations to be solved were already
known, at least in general form. (The Bjerknes family has made important contributions
to meteorology in its recognition as a science. C. Bjerknes, father of V. Bjerknes,
formulated the equations of motion with rotation; and, this knowledge helped V. Bjerknes
to make several landmarks in meteorology along with his son J. Bjerknes, who is famous
for providing a deeper understanding of El Niño Southern Oscillation).
In the above solution, λ =
Ekman pumping and suction: Away from the geographical boundaries and equator, the
wind stress forcing can produce vertical motions in the open ocean, when horizontal
gradients in wind stress are present. A simple assumption that seawater may be
considered incompressible in the upper layer, then the continuity equation reads,
∂u ∂v ∂w
+
+
=0 .
∂x ∂y ∂z
Integrating the above equation over the depth of the Ekman layer, one obtains
0
⎡∂ 0
⎤
∂v ⎞
⎛ ∂u
∂ 0
w(0) − w(−δ ) = − ∫ ⎜ ag + ag ⎟ dz = − ⎢ ∫ uag dz +
vag dz ⎥
∫
⎝ ∂x
∂y ⎠
∂x −δ
−δ
⎣ ∂x −δ
⎦
7
Assuming that w(0) = 0 and writing w(−δ ) = we as the vertical velocity at the bottom of
the Ekman layer, the above equation can be written as
we =
τy ⎞
⎛ τx
∂ ⎛ Mx ⎞ ∂ ⎛ My ⎞ ∂ ⎛ τy ⎞ ∂ ⎛ τx ⎞
+
=
+
−
=
k̂.∇
×
,
⎜⎝ ρ f
∂x ⎜⎝ ρ0 ⎟⎠ ∂y ⎜⎝ ρ0 ⎟⎠ ∂x ⎜⎝ ρ0 f ⎟⎠ ∂y ⎜⎝ ρ0 f ⎟⎠
ρ0 f ⎟⎠
0
Hence we have the following expression for Ekman pumping and suction
⎛ τ ⎞
(8.27)
we = k̂.∇ × ⎜
⎝ ρ0 f ⎟⎠
Thus, in the open ocean the vertical velocity at the bottom of the Ekman layer is
proportional to wind stress at the surface divided by the Coriolis parameter.
As an example of air-sea interaction where surface wind induced Ekman pumping
and suction play a key role, is the triggering of El Niño–Southern Oscillation (ENSO)
event in the Pacific Ocean. It is one of the most outstanding phenomena in the tropics that
produce global impacts such as large-scale droughts and flooding in many parts of the
world. The El Niño (warm SST) anomalies have been explained as the response of the
ocean to surface winds. The westerly winds induce downwelling over the equator and
upwelling on the northern and southern latitudes (Fig. 8.2a) in accordance to the Ekman
dynamics. The Ekman layer is thought to be infinitely thin on the ocean surface where
waters are collected horizontally by Ekman drift; and these surface waters sink down by
Ekman pumping. As a result, thermocline will deepen allowing charging of warm waters
on the equator in the west Pacific. Further, the westerly wind anomalies depress the
thermocline in the east and reduce equatorial upwelling in the Pacific. The warm waters
from the western Pacific subsequently set out eastwards under the forcing of the
westerlies over the equatorial region. Warmer than normal SSTs, developed in the
eastern Pacific, induce upward atmospheric motions accompanied by moisture
convergence, which increases the conditional instability of the column above. The
instability is accompanied by the release of large amount of latent heat in the deep cloud
columns, which will increase atmospheric heating. This is the Bjerknes positive feedback
mechanism of the anomalies in the ocean and the atmosphere (J. Bjerknes, Atmospheric
teleconnections from the equatorial Pacific. Monthly Weather Rev., vol.97, 1969). Thus,
ocean dynamics changes sea-surface temperatures in the equatorial region, which change
atmospheric heating and induce changes in the circulation; and the altered atmospheric
circulation, in turn, changes the ocean dynamics.
(a) Equatorial downwelling
10N
Upwelling
Westerlies
10N
Downwelling
Ekman drift
Ekman drift
Equator
Sinking
Westerlies
(b) Equatorial upwelling
Ekman drift
Upwelling
Ekman drift
Upwelling
Easterlies
Equator
Easterlies
Downwelling
10S
10S
Fig. 8.2 Illustration of downwelling and upwelling in the equatorial latitudes
This mechanism has been successfully verified in the Cane-Zebiak model, a simple
couple ocean-atmosphere model (SE Zebiak and MA Cane, A model of El Niño–Southern
8
Oscillation, Monthly Weather Rev., vol.115, 1987). The model could also demonstrate that
prior to the onset of ENSO, the equatorial heat content increases which, after the onset of
ENSO, actually decreases. The Cane-Zebiak model could also predict, in the Pacific, return
to normal conditions (non-El Niño case) in much agreement with the observations. Once the
normal conditions are established in the Pacific, the easterlies reappear and, as shown in
Fig. 8.2b, the associated upwelling of cold waters over the equator produce a tongue of cold
water in east Pacific extending westward from the Peru coast. The winds force a coastal
current that also brings cold Southern Ocean waters to the eastern Pacific equatorial region.
8.5
Western boundary currents
The western boundary currents have been mentioned earlier in Module 2 while
discussing the global current systems in the ocean. These currents also arise in response
to the forcing due to wind stress and flow northward in the northern hemisphere. The
current speeds exceed 100 cm s −1 and both types of currents, i.e. warm and cold, are
driven by the winds. The Gulf Stream in the North Atlantic and Kuroshio Current in
Western Pacific are the warm currents and transport large amounts of heat poleward.
However, the Somali Current produces strong upwelling on the east coast of Somalia and
therefore it is a cold current that gives rise to significant Arabian Sea cooling during
monsoon months.
To discuss the large-scale response of the ocean to winds, it is pertinent to begin with
the vorticity equation as discussed in Module 2. It was pointed out that divergent motions
produce significant amount of relative vorticity which will give rise to meridional
movement of rotating column as explained in Fig. 2.13 of Module 2. The equation (2.13)
can be reduced to a simpler form where stretching of the vortex tube (planetary vorticity)
by divergent motion is balanced by the meridional advection of planetary vorticity; that
is,
df
v = − f ∇.V
dy
df
∂w
and ∇.V = −
, V is the mean horizontal velocity; the above equation
Since β =
dy
∂z
can be written in the following form
∂w
(8.28)
βv = f
∂z
The eq. (8.28) is the most simplest equation, known as the Sverdrup relation, which has a
deeper physical interpretation: production of vorticity by stretching of the planetary
∂w
vorticity ( f
) equals to loss due to advection of planetary vorticity (β v) by the
∂z
meridional velocity v . This means that positive vorticity production will force northward
movement of water masses to reach the latitude (φ ) with correct value of Coriolis
parameter ( f = 2Ωsin φ ).
The western boundary currents can be explained using the vorticity model of winddriven circulation. In this model it is recognised that the torque of the atmospheric wind
field would generate motion of similar kind in the ocean. Thus, between trades
(easterlies) and westerlies, anticyclonic vorticity is generated in the oceanic cells;
analogously, cyclonic vorticity in cells between westerlies and polar easterlies. Between
9
the western and eastern boundaries of the ocean, the motion will be symmetric about the
centre of the cell. Now, consider an anticyclonic oceanic cell on the west side of the
basin, which is displaced by the curl of the wind stress poleward. Due to this action the
anticyclonic relative vorticity of fluid column will further intensify to conserve its
absolute vorticity. While on the eastern side the anticyclonic vorticity will be balanced by
the positive relative vorticity as the fluid columns move equatorward. Essentially, on the
western side of the ocean, there is an excess of anticyclonic vorticity, which continues to
get augmented as the fluid columns move further poleward. This cannot continue and a
breaking mechanism is necessary. The friction at the west boundary could provide brakes
to the current. Therefore, when the strong poleward moving current interacts with
western boundary, a strong shearing flow will be produced with equal and opposite
component of frictionally generated cyclonic vorticity; consequently, the current will
become narrow and strong along the west coast. In this manner the lateral boundaries
provide a balancing mechanism, but strong asymmetry in the motion of water masses will
be noticeable: strong northward moving current on the west side of the basin and much
weaker equatorward current on the eastern side, though the torque of the wind driving the
currents is symmetrical.
The above mechanism explains all those very strong and narrow currents in the
major ocean basins, viz., the Gulf Stream in the North Atlantic, Kuroshio Current in the
western Pacific near Japan, Somali Current in the Indian Ocean, Brazil Current in the
South Atlantic, Agulhas Current in the Indian Ocean and the Australian Current. Henry
Stommel (1951) called these currents as the western boundary currents.
Based on eq. (8.28), Stommel also explained the abyssal circulation in the global
ocean. There exist western boundary currents in the deep ocean also. The sources of
deep water formation are the North Atlantic Ocean and the Weddell Sea in the South
Atlantic Ocean. The theory explaining abyssal circulation is not included here but one
can follow it from the standard textbooks on oceanography.
8.6
Mixed Layers in the ocean
In the shallow top layer, directly in contact with the atmosphere, both salinity and
temperature are well mixed due to stirring of the ocean surface by the action of wind
gusts and turbulence. The seawater properties have practically uniform profiles there.
This region is known as the mixed layer with near-neutral stratification (N = 0) in the
ocean. An oceanic mixed layer is bounded below by seasonally changing thermocline.
One of the key features of this layer is that it responds strongly to seasonal changes.
Incident solar radiation of all wavelengths is absorbed in the top 100m with nearinfrared and infrared being absorbed in just a centimetre thick layer on the sea surface.
The action of wind at the surface enhances evaporation, which cools the surface; and cold
parcels will then sink and would be replaced by lighter fluid from below. Hence in the
mixed layer, there is net upward flux of energy due to convective overturning which
mixes the fluid parcels from different depths. Turbulent mixing is another mechanism,
which produces a uniform profile in the mixed layer. During summers, intense solar
heating of seawater at the interface produces a warm layer. This will reduce the buoyancy
of parcels, which will result in a shallow mixed layer. On the contrary, if surface layer is
cooled at a faster rate, the buoyancy induced convective overturning will be strong
because the cold parcels could sink as deep as one kilometre. This will result in a deeper
10
mixed layer. However, in the tropics mixed layer depth (MLD) is generally taken as the
depth of the 20 0 C isotherm in the ocean. The mixed layer depth in tropics varies in the
range 50 − 100 m as seasons change.
cooling
cooling
SW
z=0
−30
heating
LW
Evap/Precip
Turbulence
Wind
mixing
Mixed layer
BLD
−100
/ ML
D
Depth (m)
T
Thermocline
Barrier layer
Entrainment
Fig. 8.3 Temperature profile, different layers of the
ocean, surface and vertical processes, and the
coordinate system are depicted here. In the upper
part of the mixed layer both temperature and
density are well mixed. But in the barrier layer part
of the mixed layer, density gradients are dominant
(curved line illustrates density gradients in BL).
Abyssal Waters
The situation in the Bay of Bengal is rather unique as it receives large volumes of
freshwater discharge from the Brahmaputra River. Besides the river runoff, the region
also receives excess precipitation during the monsoons. The freshwater flux makes the
near surface waters less saline and creates strong salinity stratification in the isothermal
mixed layer. As a consequence, an internal layer forms within the mixed layer above the
thermocline. This intermediate layer is referred to as the barrier layer (BL), which
inhibits turbulent entrainment of cool thermocline water at the bottom of the mixed layer
(Girishkumar, Ravichandran, McPhadan and Rao, JGR, vol.116, C03009, 2011). The
formation of barrier layer would thus impact the vertical heat flux and thereby the sea
surface temperature (SST). Ekman pumping and suction will modulate the barrier layer
depth (BLD) and it would vary between 40 − 100 m . Thicker BL will make the wellmixed layer shallow (MLD ~ 10 − 20 m) at the top. Thus, barrier layer is bounded above
by a region of mixed layer with no vertical gradients of temperature and density, and
below by thermocline or the layer dominated by strong temperature gradients. Note that
salinity stratification could as well give rise to temperature inversions in the mixed layer.
In tropics, depth of the isothermal layer could be less than the depth of the 200C isotherm.
This observation could easily be confirmed from the Argo Data of the Bay of Bengal.
There are various models of oceanic mixed layer, which are based on physical
processes of entrainment of fluid from surrounding stable layers. Along with the
temperature profile, we have schematically shown in Fig. 8.3 the surface vertical
processes and locations of mixed layer and the barrier layer in the interior of the oceans.
The thickness of the barrier layer and therefore of mixed layer depends much on the
large-scale variability of the ocean. The curved line represents BLD if the barrier layer
present, else the mixed layer depth (MLD). Besides entrainment from bottom of mixed
layer, other important processes that determine the deepening and shoaling of mixed
layer are the shortwave heating, longwave cooling, turbulent mixing induced by wind
stirring, momentum transfer by winds, evaporation, freshwater flux and the large scale
11
dynamics that produces Ekman pumping and suction. In our discussion, both the top
layer and the barrier layer (if present) are the two regions of the composite mixed layer
with different salinity structures that could even give rise to temperature inversions in the
mixed layer. To summarize, in mixed layer both temperature and salinity are well mixed
(or uniform), while salinity is stratified and temperature is uniform in the barrier layer.
This is a distinct feature in the salinity and temperature profiles of the mixed layer that
allows identification of the barrier layer.
In the coordinate system chosen here, z = 0 is at the surface and decreases
downward as shown in Fig. 8.3. Let h represent the mixed layer depth, then the local
changes in h are determined from the following equation
∂h
(8.29)
= −h∇.V + we
∂t
where V is mean horizontal velocity and we is the entrainment velocity at the base of the
isothermal mixed layer. The first term in (8.29) on the right hand side, represents the
changes in the mixed layer depth arising from processes such as the Ekman pumping and
internal gravity waves. The second term in the above equation represents the changes in
h arising from turbulent entrainment of the fluid in the mixed layer from below. That is,
the changes in the mixed layer depth (h) are caused by entrainment (turbulent) of fluid
from its bottom and convective overturning caused by sinking plume from surface
cooling. One of the characteristics of the ML is that both temperature (T ) and salinity
(S) are well mixed. Therefore, due to thorough mixing in the ML, the vertical density
gradients will also vanish. The density gradients will produce buoyancy of fluid parcels,
which is needed to compute local changes in h . Using the Boussinesq approximation (i.e.
density variations are important only when occur in combination with acceleration due to
gravity), the buoyancy b per unit volume can be calculated as
ρ − ρ0
b = −g
= g[α T (T − T0 ) − β (S − S0 )]
(8.30a)
ρ
(8.30b)
α T ≈ 77.5 + 8.70 T
and
β ≈ 779.1− 1.66 T ( T > 5 0C )
In writing the right hand side of (8.30a), we have used the equation of state (6.11) for
seawater. The quantities with subscript 0 are the reference values; α T (T , S) is the
thermal expansion and β (T , S) is the haline contraction. Linear approximations (8.30b)
for α T and β have been found to be useful for mixing studies at atmospheric pressure.
The evolution of b is described by a conservation equation of the form
db ∂wb'
gα ∂Fpen ∂Sb
(8.31)
+
=− T
=
dt
∂z
ρCp ∂z
∂z
where Fpen (z) is the penetrating solar radiation flux, Cp is the specific heat of seawater.
Hence the term ∂Sb / ∂z represents the source of buoyancy arising from solar radiation
absorbed in the interior of the mixed layer. Since the vertical gradients of b will vanish
(∂b / ∂z = 0) if density is well mixed; and if the horizontal variations of b are assumed to
be negligibly small, then the Lagrangian time derivative in (8.31) can be replaced by the
local derivation; and (8.31) can be written as
12
∂b ∂wb' ∂Sb
(8.32)
+
=
∂t
∂z
∂z
On assuming z increasing downward and integrating (8.32) vertically from 0 to h , we
get the following result
∂b
(8.33)
h + wb'(h) − wb'(0) = Sb (h) − Sb (0)
∂t
Set wb'(0) = B , the buoyancy flux at the interface in the ocean; neglect the small amount
of penetrating flux Sb (h) ; and represent mathematically wb'(h) = we Δb , then (8.33)
takes the following form
∂b
(8.34)
h = B − we Δb − Sb (0)
∂t
In equation (8.33), the surface buoyancy flux B can be parameterized as
⎤
g ⎡α
(8.35)
B = ⎢ T Qw + β S(E − P) ⎥
ρ ⎢⎣ Cp
⎥⎦
where Cp is the specific heat of seawater; and the surface heat flux Qw is given in terms
of net radiative flux ( Fnet ), sensible heat flux ( Qs ) and latent heat flux ( Lv E ) as
(8.36)
Qw = Fnet + Qs + Lv E
It may be noted that B has different numerical values in the ocean and the atmosphere
but it has the same sign. The buoyancy flux at the bottom ( we Δb ) is indeed the power
required for seawater at the bottom to drive it into the mixed layer, that is
we Δb = we (b − bh )
(8.37)
Thus Δb is the buoyancy discontinuity at z = h . The evolution equation of bh reads
∂bh
∂b(z)
(8.38)
= we
∂t
∂z
From turbulent kinetic energy (TKE) considerations, Krauss and Businger (AtmosphereOcean Interaction, Oxford University Press, 1994) derived the following expression for
the entrainment velocity we , which reads
we = −2
m1u *3 + m2 w*3
c12 + q 2 − m3 (ΔU )2
(8.39)
where q 2 = u j u j is the mean squared turbulent velocity, c1 is the velocity of the internal
gravity wave along the density discontinuity at z = h and it is possible to estimate it as
ρ − ρh
(8.40)
hΔb = hg
≈ c12 .
ρ0
In (8.39) ΔU is the horizontal velocity change across h which arises due to shear. If we
is positive, then the mixed layer will grow as time derivative of h will be positive. This
is deepening of the mixed layer. However, if the denominator and the numerator in
(8.39) are positive, then entrainment will occur from the bottom of the mixed layer as
we < 0 (because downward direction is positive). This means the thermocline waters will
be forced into the mixed layer and shoaling of the mixed layer will happen. The values of
constants m1 , m2 and m3 (Kraus and Businger 1994) are as follows:
13
0.55 < m1 < 0.7 ; m 2 = 0.2 ;
In (8.39), w* and u* are estimated as
1
hB
w*3 = hB ; u*3 =
2
2m1
m3 = 0.65
(8.41)
Another expression for calculating we : The turbulent entrainment also velocity we reads
⎛ B⎞
C1u*3 − C2 ⎜ ⎟ h
⎝ ρ⎠
(8.42)
we =
; C1 = 2.0 and C2 ≈ 0.2
h g(α T ΔT − β ΔS)
In (8.42), the entrainment velocity depends on surface buoyancy flux B , friction
velocity u* and the jumps in temperature ΔT and salinity ΔS at the base of the mixed
layer (i.e. at the depth h ). If mixed layer surface cools, then convective instability will
provide mixing energy; on the other hand, heating would cause density to decrease.
Hence, some of the mechanical energy (C1u*3 ) will be used to overcome the stabilizing
effect of surface heating. Therefore, h will increase with increasing surface windspeeds
and decreasing surface buoyancy flux B .
There are two regions in the World Ocean where thermodynamic extremes are
present: (i) the ocean below the sea ice cover in the Arctic Ocean; and (ii) the Warm Pool
in the Western Pacific and. The mixed layer characteristics in these regions are now
described in what follows.
(i) Mixed layer in the Arctic Sea: The presence of sea ice cover in the Arctic Sea totally
prevents the transfer of momentum by winds; therefore, only buoyancy fluxes maintain
the mixed layer in this basin. The buoyancy fluxes arise from the processes of ice growth
and melting. The salinity of the mixed layer below the Arctic ice cover is between
30 − 31 psu due to large amount of river runoff in the Arctic basin. Maximum salinity
reaches in early spring and a minimum in late summer. Thus the annual cycle of salinity
in this region is dominated by the freeze-melt cycle of the Arctic sea ice. The mixed
layer temperatures do not depart significantly from freezing temperature, but relative
velocities of ice (zero velocity) and ocean may cause mechanical mixing, which together
with buoyancy effects could modulate the mixed layer depth. During freezing, brine
rejection is also significant, which increases salinity during autumn. Brine drainage
continues with ice cover growth in the Arctic Sea. The rise in salinity will increase
density; consequently, static instability shall be generated which will enhance mixing
(turbulent) in the layer. This causes the mixed layer depth to grow (i.e. mixed layer
deepens) to a depth of 60 m in mid-May. On the contrary, melting snow and ice during
the melt season will add fresh water and density would reduce in the top layer. This will
cause stabilization (i.e. reduces turbulent mixing). Therefore the mixed layer in the
Arctic Sea will collapse continuously to reach a minimum depth of 15 meters due to
combined effect of stabilization arising from increased freshening of saline waters and
layer warming due to increased flux of solar radiation. The strong increase in salinity
with depth in the Arctic Ocean produces a very stable density structure up to a depth of
200 meters. Consequently, heat and salt from layers below 200 meters do not upwell to
14
the surface. Because of this reason, the surface waters, augmented by river runoff in
Arctic Sea, remain relatively fresh and cold.
The development of a simplified, convective, oceanic mixed layer in winter in the
Arctic could serve as an example of a simple model of mixed layer (Marshal and Plumb
2006). The key paradigm of such a model is that the water column responds to heat loss
at the sea surface by adjusting its heat content. That is, by matching the changing heat
content of water column and heat lost from surface, deepening of the mixed layer can be
∂T
calculated from prescribed surface temperature Ts and lapse rate of temperature
.
∂z
Both Ts and the lapse rate change due to cooling caused by the loss of heat. Though ocean
surface cooling (heating) happens due to (i) emission (absorption) of radiation and (ii)
exchange (loss/gain) of sensible heat between ocean surface and atmosphere, yet both can
be combined into a prescribed rate of cooling so that the mixed layer model could be be
easily understood. Convection sets in due to sinking of colder water as the ocean surface
cools mixes the waters that results in the deepening of the mixed layer.
To develop a mathematical model of the mixed layer based on above paradigm,
assume a temperature profile in the ocean at the start of winter as
T ( z ) = Ts + Λ z,
(8.43)
where z is depth (which is zero at the surface and increase upwards) and the gradient
Λ > 0 . Also, suppose that heat is lost from the surface at a prescribed rate Q W/m2 during
the winter. As surface cools, convection sets in and mixes the developing cold mixed
layer of depth hm(t) to reach a uniform temperature Tm(t). Further assume that
temperature is continuous across the base of the developing mixed layer. Since the depth
hm(t) is constantly increasing due to surface cooling, ocean waters arranged in the column
according to (8.43) would mix up to a depth hm(t). Hence the mean temperature Tm(t) of
the layer is
hm (t )
hm (t )
1 hm (t )
Tm (t) = ∫
T (z)dz / ∫
dz =
T (z)dz .
(8.44)
0
0
h(t) ∫0
The density of the well-mixed waters ignoring the salinity changes in the layer can be
obtained from density-temperature relationship of the following form
ρ m = ρ0 [1− α T (Tm − Ts )] .
(8.45)
Substituting T(z) from (8.43) into (8.44) and integrating the resulting expression, we get
Λ
Λ
(8.46)
Tm (t) = Tm (0) + hm (t) = Ts + hm (t)
2
2
The heat content of the layer of depth hm(t) at any time level t is
H (t) = ρ m hm (t)CwTm (t) .
(8.47)
The heat content of the layer at the next time level t + Δt will become H (t + Δt) , hence
by equating the change in heat content ΔH = H (t + Δt) − H (t) to cooling of the sea
surface, the requite equation can be formulated as
ΔT
(8.48)
ΔH = QΔt or ρ m hm (t)Cw m = Q
Δt
In the above equation, it is assumed that hm remains constant within the interval Δt .
Taking the limit Δt → 0 , the discrete form (4.48) is written in the analytic form as
15
ρ mCw hm (t)
dTm
=Q
dt
(8.49)
On integrating (8.49), on obtains
Qt
(8.50)
ρ mCw hm (t)
However, the main aim of this mathematical model is to obtain the mixed layer depth,
which be obtained by substituting the expression (8.46) of Tm in (8.50). After some
rearrangement, the expression the hm is obtained as
2Q t
(8.51)
h 2m =
(assuming ρ m = ρ0 = const)
ρ mCwm Λ
If density of the mixed water is calculated from the Tm , then combining (8.45) and
(4.46), we get the expression of seawater density as
α Λ
(8.52)
ρ m = ρ0 [1− T hm ]
2
Substituting ρ m from (8.52) in (8.51), we obtain the following cubic equation in hm
αT Λ 3 2
2Q t
(8.53)
h −h +
=0
2
ρ 0 Cw Λ
However, the magnitude of the first term is much less that the second one in (8.53), hence
the computation of hm(t) is sufficiently accurate , and it is thus estimated from
Tm (t) = Ts +
2Q t
(8.54)
ρ mCwm Λ
Note that in the estimate of mixed layer depth, surface temperature does not appear
in the above expression but the lapse rate of temperature Λ does. This means the growth
rate of the mixed layer depends inversely upon the square root of Λ and directly to the
square root of the cooling rate Q. We have already described a very elaborate model
earlier, but a simple model like this helps comprehension of an evolving complex
phenomenon. Though the model applies to regions like Artic, yet the formula (8.54)
offers some deeper insights on the depth of mixed layer in global oceans.
If Q and Λ are specified, then one can calculate the time that is required for the
hm =
mixed layer to grow to a given depth. For example, with ρ m = 1025kgm −3 and specific
heat capacity Cw = 3980J kg −1K −1 , if Q = 25Wm −2 and Λ = 10 0C km −1 , then mixed layer
would become 100m deep in 91 days. That is, the mixed layer develops 1 m deep
everyday with a given rate of cooling 25 Wm-2 at the surface of the ocean. Further, this
simple model explains that when lapse rate of temperature increases in the ocean, the
mixed layer depth reduces. A consequence of this fact is that in regions of appreciable
lapse rates (i.e. warm surface waters) mixed layers are shallow.
(ii) Mixed layer in the Pacific warm pool: In the tropical warm pool, skin temperatures
(i.e. temperatures at the interface between the atmosphere and ocean) could reach as high
as 34 0 C , when light winds (5 ms −1 ) prevail. Due to deep convection, precipitation is
(
)
also high in the warm pool region P − E ~ 2 m yr −1 . Due to net surface heating, the
16
mixed layer would be relatively shallow ( ~ 30 m deep ) . However, large amounts of
precipitation could produce a steady, constant density mixed layer ( ~ 40 m deep ) and
below the salinity profile shows sudden increase with depth (i.e. presence of barrier layer
below 40 m ). However, the temperature profile remains almost uniform further down in
the deeper layer. It is also observed that the nearly isothermal layer extends down in the
warm pool up to a depth of 100 m around 155°E in the equatorial Western Pacific
region. This discussion clarifies that the layer above the barrier layer (i.e. top layer)
satisfies condition of well-mixedness. The presence of barrier layer below the well-mixed
layer affects the heat flux and mixing of water further down.
The warm pool mixed layer responds strongly to daily weather conditions. Under
low surface wind conditions, both surface temperatures and heat content of the mixed
layer rise significantly in response to surface heating because of the presence a relatively
fresh isohaline layer ( ~ 40 m deep ) produced by heavy precipitation. A diurnal cycle
with amplitude of 1− 2 0 C has been observed in the surface temperatures. During TOGACOARE, it has also been observed that if windspeeds decrease (~ 4 ms −1 ) , mixed layer
becomes very shallow (~ 20 m deep); on the contrary, if the windspeeds increase, mixed
layer depth also increases. This implies that the action of the winds at the interface is
mainly responsible for the deepening and shoaling of mixed layer when an isohaline layer
is present just below the surface.
17