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: dTm dTm ρwcpw dt hm∆x∆y. For a unit area, it is: ρwcpw dt 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: dT ρ c m h = Q + Q + Q + Q + Q , (1) w pw dt m nr s l pr ent − − where Qent = ρwcpwwent(Tm Td) and Td is the temperature of the thermocline. dTm ∂Tm · ∇ Tm−Td Rewriting the equation by expanding dt = ∂t + V Tm + w hm we have:

iii ∂Tm Qnr + Qs + Ql + Qpr Tm − Td Tm − Td = − V · ∇Tm − w H(w) − went = Qnet. (2) ∂t ρwcpwhm hm hm

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.

− sw lw − 4 Qnr = (1 α0)Q0 + Q0 ǫ0σT0 . (3)

Figure 4: Schematic diagram showing radiative ﬂuxes.

In the above, sw Q0 - downward solar radiation flux at the surface; α0 - is the shortwave surface albedo (reflectivity); lw Q0 - is the downward infrared radiation flux at the surface. 4 -ǫ0σT0 - 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 × 108wm−2K−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. sw lw The surface downward short wave and long wave fluxes Q0 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.

iv (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 = ρacpd(w θ )0, (4a) − ′ ′ Ql = ρaLlv(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

v 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 = ρacpdCDH (Va Vo)(Ta To), (5a) − − Ql = ρaLlvCDE(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. 6 Llv = 2.44 × 10 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 = ρwcpwpr(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.

Tm−Td (e) Entrainment cooling -went hm 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”.

vi Tm−Td (f) Upwelling cooling -w hm H(w)

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

vii 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:

viii • (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 3 ρw, the volume of this column is: hm∆x∆y (m ) and mass is ρwhm∆x∆y (kg). For −2 a unit area, the mass is ρwhm (kg m ). 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 dSm amount of dt is dSm ρwhm dt . 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 P˙ either due to rainfall (P˙r) or snow (P˙s). It has a unit of m/s (speed). Salinity ﬂux due to this process can be written as:

ix ˙ ˙ -ρrPrS0 or -ρsPsS0. 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: ˙ ρwE0S0,

where E˙ 0 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: ˙ -ρrvRS0, where R˙ 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: dhi − ρi dt (S0 Si),

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: − − ρwwent(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:

dS dh ρ h m = −ρ P˙ S −ρ P˙ S +ρ E˙ S −ρ RS˙ +ρ i (S −S )−ρ w (S −S ). (6) w m dt r r m s s m w 0 m rv m i dt m i w ent m d x dSm ∂Sm · ∇ Sm−Sd Rewriting the equation by expanding dt = ∂t + V Sm + w hm we have:

dhi −ρ P˙rSm−ρ P˙sSm ρ E˙ 0Sm−ρ RS˙ m ρ Sm−Si − ∂Sm = r s + w rv + i dt ( ) − went(Sm Sd) ∂t ρwhm hm (7) − · ∇ − Sm−Sd V Sm w hm H(w).

The last two terms are salinity change due to horizontal advection (−V · ∇Sm) and − Sm−Sd upwelling ( w hm H(w)), respectively. As discussed in the Tm equation, upwelling is 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.

xi 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):

−αT FBO = g( Qnet − αsFnet), (8) cp0 ◦ −1 where αT < 0 is thermal expansion coefficient (unit: C ), αS > 0 is the salinity −1 expansion coefficient (unit: psu ), 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 (ms−2). −2 −2 −1 In the above, Qnet is the net surface heat flux (wm =J m s ), − · ∇ − Tm−Td − Tm−Td Qnet = Qnr + Qs + Ql + Qpr ρwcpwhm[V Tm w hm H(w) went hm ], −2 −1 Fnet is the net surface salinity flux (psu kg m s ), − ˙ − ˙ ˙ − ˙ dhi − − went(Sm−Sd) − Fnet = ρrPrSm ρsPsSm + ρwE0Sm ρrvRSm + ρi dt (Sm Si) ρwhm[ hm · ∇ − Sm−Sd V Sm w hm H(w)].

Thus, buoyancy ﬂux has the unit of N m−2 s−1=kg ms−2 m−2 s−1=kgm−1s−3.

As can be seen from FBO equation,

xii (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.

xiii