Download Chapter 7: Thermodynamics

Chapter 7: Thermodynamics
7.1 Sea surface heat budget
In Chapter 5, we have introduced the oceanic planetary boundary layer-the Ekman layer.
The observed T and S in this layer are almost uniform vertically, thus it is also referred to
as the surface mixed layer. This layer is in direct contact with the atmosphere and thus is
subject to forcings due to windstress (which enters the ocean as momentum flux), heat flux,
and salinity flux. Heat and salinity fluxes combine form buoyancy flux.
Below, we will discuss the heat fluxes that force the ocean, and examine the processes
that can cause mixed layer temperature changes by introducing the mixed layer temperature
equation.
Why is the surface heat budget important? Heating and cooling at the ocean surface
determine the sea surface temperature (SST), which is a major determinant of the static
stability of both the lower atmosphere and the upper ocean. For example, the wintertime
cold SST in the North Atlantic and in the GIN Seas (Greenland, Ice land, and Norwegian
Seas) increase density, destabilizing the stratification of the ocean, resulting in deep water
formation and therefore affecting the global thermohaline circulation. On the other hand, in
the equatorial Western Pacific and eastern Indian Ocean warm pool region, SST exceeds 29◦ C
and thus destabilize the atmosphere (because the atmosphere is heated from below), causing
convection. Convection in the warm pool region is an important branch for the Hadley and
Walker circulation and therefore is important for the global climate. The surface heat fluxes
at the air/sea interface are central to the interaction and coupling between the atmosphere
and ocean.
Before we discuss the processes that determine the SST variation, let’s first look at the
annual mean SST distribution in the world oceans (Figure 1).
Why SST is generally warm in the tropics and cold poleward? Solar shortwave flux is
high in the tropics and low near the poles. There is net heat flux surplus at lower latitudes
and deficit at high latitudes (Figure 2). Why the SST is cold in the eastern Pacific (cold
tongue)? Upwelling - Ocean processes. Therefore, SST distribution is determined from both
surface heat flux forcing and from the oceanic processes.
For simplicity, we will examine the temperature equation for the surface mixed layer, and
assume solar shortwave radiation is completely absorbed by the surface mixed layer. In fact,
this is a mixed layer model for temperature. [RECALL that some light can penetrate down
to the deeper layers, depending on the turbidity of the water.]
The processes that determine the temperature change of a (Lagrangian) water parcel in
the surface mixed layer are:
• net surface radiation flux Qnr ;
• the surface turbulent sensible heat flux Qs ;
• the surface turbulent latent heat flux Ql ;
i
Figure 1: Annual mean SST in the Pacific, Atlantic, and Indian Oceans.
Figure 2: Latitudinal distribution of net surface radiative fluxes.
ii
• heat transfer by precipitation (usually small) Qpr ;
• entrainment of the colder, subsurface water into the surface layer Qent .
The first law of thermodynamics tells us that heat absorbed by a system is used to increase
the internal energy of the system and used to do work to its environment. An example is
a metal box that is full of air with a sliding door on one side. Initially air pressure on
both sides of the door are the same, which equals the atmospheric pressure. When the box
is heated up from below, air temperature inside the box will increase because its internal
energy increases and molecules motion increases. This will increase the air pressure on the
inner side of the door and thus pushes it to move outside. If the sliding door is fixed, all the
heat will be used to increase the internal energy of the air inside the box.
For the oceanic mixed layer, energy absorbed by the mixed layer per unit area is used to
increase the internal energy (temperature) of the water column. Now, let’s apply the first
law of thermodynamics to the oceanic mixed layer with depth hm for a unit area (Figure 3).
Figure 3: Schematic diagram showing the oceanic mixed layer and heat fluxes that act on
the ocean.
For a water column of the mixed layer with an area of ∆x × ∆y, internal energy increase
is:
ρw cpw dTdtm hm ∆x∆y. For a unit area, it is: ρw cpw dTdtm hm ,
where ρw is water density, cpw is specific heat of water (J/kg/◦C).
This energy increase will be caused by the net heat flux due to both heating from the
surface and cooling from the bottom of the mixed layer. That is:
dTm
hm = Qnr + Qs + Ql + Qpr + Qent ,
dt
where Qent = −ρw cpw went (Tm − Td ) and Td is the temperature of the thermocline.
d
Rewriting the equation by expanding dTdtm = ∂T∂tm + V · ∇Tm + w Tmh−T
we have:
m
ρw cpw
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(1)
∂Tm
Qnr + Qs + Ql + Qpr
Tm − Td
Tm − Td
=
− V · ∇Tm − w
H(w) − went
= Qnet .
∂t
ρw cpw hm
hm
hm
(2)
Next, we’ll discuss each term in detail and Qnet is the net surface heat flux.
(a) Qnr
The net surface radiation flux, Qnr , is the sum of the net solar and long wave fluxes at
the surface.
lw
4
Qnr = (1 − α0 )Qsw
0 + Q0 − ǫ0 σT0 .
(3)
Figure 4: Schematic diagram showing radiative fluxes.
In the above,
Qsw
0 - downward solar radiation flux at the surface;
α0 - is the shortwave surface albedo (reflectivity);
Qlw
0 - is the downward infrared radiation flux at the surface.
-ǫ0 σT04 - outgoing longwave radiation of the ocean. This is from the Stefan-Boltzman’s
law of radiation. To a fairly high accuracy, a black body (100% emmisivity) with temperature
T emits radiative flux as E = σT 4 where σ = 5.67 × 108 wm−2 K −4 .
T0 is the skin temperature at the very surface; but if we consider the mixed layer is well
mixed, T0 represents the mixed layer temperature Tm . ǫ0 - surface longwave emissivity (0.97
for the ocean). The ocean is close to a black body.
lw
The surface downward short wave and long wave fluxes Qsw
0 and Q0 depend on the
amount of radiation incident at the top of the atmosphere and on the atmospheric conditions:
Temperature profile, gaseous constituents, aerosols, clouds. Radiative transfer processes and
models are covered by the radiation class. So we will not get deep into this part here.
.
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(b) Qs and Ql
The surface turbulent sensible and latent heat fluxes. Turbulent is a small-scale irregular
flow that often occurs in atmospheric and oceanic planetary boundary layers (PBL). It is
characterized by eddy motion. It has a wide range of spectra in spatial and temporal scales.
Unlike the large scale deterministic flow whose horizontal scale is much larger than its vertical
scale, turbulent flow has comparable horizontal and vertical scales and thus is bounded by the
planetary boundary depth 1km. Its smallest scale is 10−3m. These eddies produce efficient
mixing in the PBL, bring heat from the oceanic surface to the top of the PBL and bring the
cooler air from the PBL top to the surface. Since it is not possible to predict the behavior
of the wide range of eddies using analytical or numerical methods, we usually determine the
turbulent motion using statistical approximations. To do so we separate the total flow into
mean (deterministic) and the turbulent component, and obtain empirical formulae. That is,
uT = u + u′ where uT , u, and u′ represent total, mean, and turbulent flow.
Qs = −ρa cpd (w ′θ′ )0 ,
(4a)
Ql = −ρa Llv (w ′ qv′ )0 ,
(4b)
where w ′ - turbulent vertical velocity; θ′ is the turbulent potential temperature, overline
“-” is time mean, qv is air specific humidity, and Llv is the latent heat of evaporation.
Potential temperature of a water or air parcel is defined to be the temperature of the parcel
when it is adiabatically bring to the sea level pressure. It is used here rather than in situ
temperature for convenience (so that we don’t have to worry about the temperature change
due to pressure).
Figure 5: Schematic diagram showing eddy sensible and latent heat transport.
In Figure 5, SST is higher than air temperature and thus it warms up the air right above
the sea surface. Eddies bring the warm air from the surface upward and bring the colder
air down to the sea surface, producing the mixing. As a result, the ocean loose heat to the
atmosphere. The latent heat flux in fact is turbulent moisture transport. Why moisture
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transport is related to heat flux? Because evaporation, which produces the moist air, needs
to cost the internal energy of the ocean to overcome the molecular attractions of sea water to
become water vapor. As a result, SST decreases and the ocean looses heat to the atmosphere.
The covariances (w ′ θ′ )0 and (w ′ qv′ )0 can be determined from high-frequency measurements of w, θ, and specific humidity qv . However, such measurements are rarely available.
Therefore, we usually use bulk aerodynamic formulae to estimate them. The bulk formulae are based on the premise that the near-surface turbulence arises from the
mean wind shear near the surface, and that the turbulent fluxes of heat and
moisture are proportional to their gradients just above the ocean surface.
According to these assumptions, we obtain the bulk formulae:
Qs = ρa cpd CDH (Va − Vo )(Ta − To ),
(5a)
Ql = ρa Llv CDE (Va − Vo )(qva − qvo ),
(5b)
where cpd = 1004J/kg/◦C-specific heat of air, CDE is close to CDH under ordinary conditions. Va - 10m windspeed, Vo oceanic surface current in the wind direction, Ta surface
air temperature, and To is the SST and is Tm is we consider the surface layer is well mixed.
In fact, potential temperatures should be used but at the oceanic surface (sea level), potential temperature is equivalent to in situ temperature so people often use T instead of θ.
Llv = 2.44 × 106 J/kg - latent heat of evaporation. qva is surface air humidity, and qvo is
saturation specific humidity when Ta = To .
(c) Qpr
Heat transfer by precipitation occurs if the precipitation has a different temperature than
SST. It is small for a long term mean (say monthly mean),maybe is important during a short
rainfall period.
Qpr = ρw cpw pr (Twa − To )
where pr is precipitation rate, Twa is atmospheric wet bulb temperature (rain drop temperature).
(d) Horizontal advection -V · ∇T
This processes is significant only when SST gradient is strong and current speed is large.
d
(e) Entrainment cooling -went Tmh−T
m
where Td is the temperature of the thermocline water. Entrainment of colder, subsurface
water (thermocline water) into the surface mixed layer. Entrainment rate we is a function
of windspeed and buoyancy. When windspeed is strong, we is large and the ocean tends
to entrain the colder subsurface water into the mixed layer. When the ocean is weakly
stratified, we also tends to increase. When winds is strong and the ocean is weakly stratified,
instability (K-H) is favored and thus mixing is strong. Even the ocean is stable, strong wind
input mechanical energy into the ocean and thus produce entrainment. This cooling is due
to the “mixed layer process”.
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d
H(w)
(f) Upwelling cooling -w Tmh−T
m
Figure 6: Schematic diagram showing eastern Pacific upwelling.
In a continuously stratified model, strong upwelling in the eastern Pacific Ocean makes
the thermocline outcrop and thus directly cools the oceanic surface (Figures 6 and 7). This
cooling process is due to dynamic reason: Surface Ekman divergence shoals the surface
mixed layer and the thermocline, resulting in the colder, thermocline water entering the
surface mixed layer. Meanwhile, strong winds produce entrainment and thus still maintain
a well-mixed surface layer. That is, the mixed layer depth is not zero but the mixed layer
water is replaced by the colder thermocline water. In the mixed layer model we’re discussing,
this process can be represented by assuming a minimum mixed layer thickness hmin . When
hm goes to or is smaller than hmin (say hmin = 5m) due to strong divergence (and thus
w > 0), we let the thermocline water upwell to the mixed layer to maintain hm = hmin . The
heaviside function H(w) = 1 when w > 0 and otherwise H(w) = 0. This syas that upwelling
cools the SST; downwelling should not affect the SST directly. Note, however, that in a
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Figure 7: Observed SST in the eastern tropical Pacific.
mean upwelling zone, anomalous downwelling associated with surface Ekman convergence
will increase SST by “reducing” the mean upwelling cooling.
Note that the eastern equatorial Pacific cold tongue SST has a significant annual cycle.
The processes that determine the annual SST variability have been well studied (e.g., Wang
B. and X. Fu, Journal of Climate, 2001; Swenson and Hansen, 1999, Journal of Physical
Oceanography).
7.2 Sea surface salinity budget
Except for the SST, the sea surface salinity (SSS) budget also plays an important role
in determining the stability of the upper ocean because its variation will cause density
change. The saline surface water in the high-latitude North Atlantic Ocean (say MOW) is
a key factor that allows surface water to sink deep into the ocean. In all the concentration
basins (Mediterranean Sea, Red Sea, Persian Gulf), evaporation is greater than precipitation,
increasing SSS and thus increasing density, resulting in deep water formation and therefore
affect global thermohaline circulation. On the other hand, fresh surface water acts to stabilize
the mixed layer in the Arctic Ocean (dilution basin) and in the tropical western Pacific and
east Indian Ocean warm pool region.
Heat and salinity fluxes combine form buoyancy flux.
For simplicity, we will examine the salinity equation for the surface mixed layer.
The processes that contribute to the salinity change in the surface mixed layer are:
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• (i) precipitation;
• (ii) evaporation;
• (iii) river runoff;
• (iv) formation and melting of sea ice;
• (v) oceanic transport below the surface mixed layer due to entrainment.
Figure 8: Schematic diagram showing salinity budget in the surface mixed layer.
Next, we will quantify the effects for each of the above processes. The combination
of these processes determines the mixed layer salinity change (also called salinity storage).
First, we need to quantify the salinity change in the surface mixed layer.
• Change of salinity. For a mixed layer water column with area ∆x × ∆y and density
ρw , the volume of this column is: hm ∆x∆y (m3 ) and mass is ρw hm ∆x∆y (kg). For
a unit area, the mass is ρw hm (kg m−2 ). Recall that the definition of salinity is the
number of grams of dissolved matter per kilogram of seawater. Therefore the salinity
flux (kg m−2 s−1 ) that is required to increase the salinity of a water column by the
amount of dSdtm is
ρw hm dSdtm .
If we take salinity as no unit as we discussed in the ocean observation section, salinity
flux has a unit of (kg m−2 s−1 ) and is produced by the combination of all the five
processes listed above. If we use psu as salinity unit, salinity flux has a unit of psu kg
m−2 s−1 .
• (i) Precipitation induced salinity flux. Assume the precipitation rate is Ṗ either due
to rainfall (Ṗr ) or snow (Ṗs ). It has a unit of m/s (speed). Salinity flux due to this
process can be written as:
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-ρr Ṗr S0 or -ρs Ṗs S0 .
As we shall see later, the negative sign indicates that precipitation will reduce the SSS,
S0 . If we consider the mixed layer is well mixed, S0 is the mixed layer salinity Sm .
• (ii) Evaporation induced salinity flux. Similar to the precipitation, salinity flux due to
evaporation can be written as:
ρw Ė0 S0 ,
where Ė0 is the evaporation rate (m/s) at the oceanic surface. Note that evaporation
tends to increase salinity, as the situation in the concentration basins of the world
ocean.
• (iii) River runoff induced salinity flux. Similar to the precipitation, river runoff induced
salinity flux is:
-ρrv ṘS0 ,
where Ṙ is the river runoff rate (m/s).
• (iv) Salinity flux due to sea ice melting and freezing. Sea ice melting and freezing affect
the salinity in the ocean. When sea ice melts, it increases fresh water in the ocean
and thus decreases salinity. When sea ice freezes, fresher water freezes first because
of its low freezing point (0◦ C for fresh water and -2◦ C for salty water), and therefore
increases SSS. Consequently, sea ice melting and freezing can affect salinity and thus
density in the ocean, influencing global thermohaline circulation. Salinity flux due to
this process can be parameterized as:
i
(S0 − Si ),
ρi dh
dt
where ρi, hi , and Si represents the sea ice density, thickness, and salinity.
• (v) Salinity flux due to entrainment. Similar to the mixed layer temperature equation,
entrainment due to surface wind-stirring and cooling will entrain the subsurface water
into the surface mixed layer, affecting the SSS. This process is due to “mixed layer
physics”, which is different from the upwelling process caused by ocean dynamics.
Salinity flux due to entrainment can be written as:
−ρw went (S0 − Sd ),
where went is the entrainment rate as discussed in the previous class. It is determined
by wind-stirring and surface cooling. Entrainment is strong in regions with strong
winds and weakly stratified ocean. Sd is the salinity below the surface mixed layer,
which represents the salinity in the thermocline layer.
If we consider a uniform salinity in the surface mixed layer, we can use Sm to replace S0
and thus give rise to the following salinity equation in the surface mixed layer:
ρw hm
dhi
dSm
= −ρr Ṗr Sm −ρs Ṗs Sm +ρw Ė0 Sm −ρrv ṘSm +ρi
(Sm −Si )−ρw went (Sm −Sd ). (6)
dt
dt
x
Rewriting the equation by expanding
dSm
dt
=
∂Sm
∂t
d
+ V · ∇Sm + w Smh−S
we have:
m
dh
∂Sm
∂t
=
−ρr Ṗr Sm −ρs Ṗs Sm +ρw Ė0 Sm −ρrv ṘSm +ρi dti (Sm −Si )
ρw hm
d
−V · ∇Sm − w Smh−S
H(w).
m
−
went (Sm −Sd )
hm
(7)
The last two terms are salinity change due to horizontal advection (−V · ∇Sm ) and
d
upwelling (−w Smh−S
H(w)), respectively. As discussed in the Tm equation, upwelling is
m
caused by surface Ekman divergence, which is a dynamical process, whereas entrainment is
due to mixed layer process.
From the above equation, we can see that salinity change in the surface mixed layer
is determined by the following processes: (i) Precipitation (rain or snow) reduces SSS; (ii)
Evaporation increases SSS; (iii) Fresh water from river runoff reduces salinity; (iv) Sea ice
melting (freezing) reduces (increases) SSS; (v) entrainment can either increase or decreases
SSS depends on the value of Sd ; (vi) Horizontal advection can affect SST in regions where
salinity gradients are strong; (vii) Oceanic upwelling can bring the subsurface water into the
surface layer and thus change SSS.
Processes (i) and (ii) (Precipitation and evaporation: P-E) play a deterministic role in
the open ocean. The latitudinal distribution of P-E agrees well with the salinity distribution
in the subtropical-mid latitude oceans and to a lesser degree, in the tropics (Figures 9 and
10). Process (iii) river runoff can be important in coastal regions, such as the Bay of Bengal
in the Indian Ocean where Ganges-Bramaputra rivers discharge a large amount of fresh
water into the Bay (Figure 10). In the Arctic Ocean, river runoff is also very important.
Process (iv) is important at high latitudes and in the Arctic Ocean. Oceanic processes due
to entrainment, advection, and upwelling can have large influence in certain regions of the
ocean, depending on the ocean dynamics and mixed layer process.
Figure 9: Latitudinal distribution of P-E and sea surface salinity.
7.3 The ocean surface buoyancy flux.
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Figure 10: Observed mean sea surface salinity distribution in the World’s oceans.
The net surface heat flux combine with the net surface salinity flux produces the ocean
surface buoyancy flux, FBo , which can be written as (Curry and Webster book, Chapter 9):
FBO = g(
−αT
Qnet − αs Fnet ),
cp0
(8)
where αT < 0 is thermal expansion coefficient (unit: ◦ C −1 ), αS > 0 is the salinity
expansion coefficient (unit: psu−1 ), and αS is greater than the absolute value of αT , cp0 is
the specific heat of surface water (J kg −1 ◦ C −1 ), and g is the acceleration of gravity (m s−2 ).
In the above, Qnet is the net surface heat flux (wm−2 =J m−2 s−1 ),
d
d
Qnet = Qnr + Qs + Ql + Qpr − ρw cpw hm [V · ∇Tm − w Tmh−T
H(w) − went Tmh−T
],
m
m
−2 −1
Fnet is the net surface salinity flux (psu kg m s ),
i
(Sm − Si ) − ρw hm [ went (Shmm −Sd ) −
Fnet = −ρr Ṗr Sm − ρs Ṗs Sm + ρw Ė0 Sm − ρrv ṘSm + ρi dh
dt
d
V · ∇Sm − w Smh−S
H(w)].
m
Thus, buoyancy flux has the unit of N m−2 s−1 =kg m s−2 m−2 s−1 =kgm−1 s−3 .
As can be seen from FBO equation,
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(i) when there is surface heating Qnet > 0 and fresh water input (Fnet < 0), FBO > 0 and
the ocean is stabilized.
(ii) When there is surface cooling (Qnet < 0) and salty water input (Fnet > 0), FBO is
negative and the ocean is destabilized.
These are quantitative expression for what we have discussed in earlier classes. Buoyancy
is an upward force exerted by a fluid, that opposes the weight of an immersed object. The
stronger the stratification, the larger the buoyancy forcing.
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