The purpose of this chapter is to discuss the tracer budget in the continuum as well as finite volumes. Discrete budgets are described in Section 30.6.
8.1 The continuum tracer concentration budget
Before starting, it is useful to summarize the terms appearing in a local budget for the tracer concentration, as determined by the tracer equation
Tt = −∇ ·(uT+F). (8.1)
In this equation,T represents the amount of a substance (generically called atracer) per unit volume; i.e., it is a concentration. Equation (8.1) is the conservation equation for this tracer concentration (e.g., Gill 1982, chapter 4 or Apel 1987, chapters 3,4). Within the Boussinesq approximation employed by MOM, T = ρos, representing the mass of salt per volume of seawater, satisfies this equation. Note that salinity s, which is a prognostic variable carried in MOM, is dimensionless since it represents the grams of salt per kilogram of seawater (ppt).
Additionally, with the same Boussinesq approximation, the amount of potential heat per vol-umeT=ρocpθ, satisfies this equation (in this context,heatis considered a “substance”). Note thatcp, the specific heat at constant pressure, is held fixed within the Boussinesq approxima-tion (e.g., Chandrasekhar 1961). Passive tracers, whereTrepresents the amount of tracer per volume, also satisfy this equation.
The terms on the right hand side were discussed in Section 4.2.4. In particular,F= Fh,Fz are the horizontal and vertical tracer flux components distinct from the advective fluxuTdue to the resolved velocity field. Typically in the mixed layer, which includes the region between the free surface z = η and the bottom of the upper ocean model boxz = z1, the fluxes take the formFh(T) = −Ah∇T, and Fz(T) = −κhTz. More general closures can be considered. In the following, these fluxes are not specified explicitly and they are generally dependent on the space-time resolution used in the model. For purposes of terminology, they will be referred to asdiffusivefluxes.
8.2 Finite volume budget for the total tracer
The time tendency for the total amount of tracer in a cell is given by
∂t[[T]k] = [[Tt]k]+[T(η)ηt]δk1, (8.2) 85
where the square brackets denote volume integration. For example, within the Boussinesq approximation, the volume integral of T = ρos represents the total mass of salt within this volume. Likewise, the volume integral of T = ρocpθ represents the total heat (in units of energy) within this volume. For T representing the mass/volume of a passive tracer, the volume integral ofTis the total mass of this tracer in the volume.
The first term on the right hand side of equation (8.2) arises from the time tendency of the tracer concentration inside the box. The second term occurs only for a surface cell and comes from the time tendency of the surface height multiplied by the surface tracer concentration.
This term alters the tracer budget through changing the volume of the box, and it is absent in the rigid lid approximation for whichη=0. Note that the value of the tracer in this expression is that at the ocean side of the free surfaceT(η).
Use of the surface kinematic boundary condition (7.16) brings the tracer budget equation (8.2) to the form
∂t[[T]k] = [[−∇ ·(uT+F)]k]+
T w+qw−uh· ∇hη
δk1, (8.3)
where all the terms multiplyingδk1are evaluated atz=η. Integration by parts leads to [[∇h·(uhT+Fh)]k] = [∇h·[uhT+Fh]k]−[∇hη·(uhT+Fh)]δk1, (8.4) where again the terms multiplyingδk1are evaluated atz=η. Using this result in equation (8.3) brings about a cancellation of the∇hη·uhTterm appearing in theδk1part of the budget. The vertical divergence integrates to
[[∂z(w T+Fz)]k] = [(w T+Fz)z=zk−1 −(w T+Fz)z=zk], (8.5) which is the divergence of the area integrated flux across the horizontal cell faces. These results render
∂t[[T]k] = −[∇h·[uhT+Fh]k]−[(w T+Fz)z=zk−1 −(w T+Fz)z=zk]
+ [T(w+qw)+∇hη·Fh]δk1 (8.6)
For a surface cell withk=1, there is a cancellation of theT(η)w(η) term to yield the budget
∂t[[T]k=1] = −[∇h·[uhT+Fh]k]+[(w T+Fz)z=z1 +(qwT−Fz+∇hη·Fh)z=η] (8.7)
8.3 Surface tracer flux
From the budget equation (8.7), it is possible to identify the total concentration flux (dimensions tracer/volume×velocity) across the air-sea interface as
QwT =−Fz(T)+Fh(T)· ∇hη+T qw z=η. (8.8) The area integral [QwT] represents the area integrated tracer concentration flux entering or leaving the ocean across the free surface interface. Equivalently, it is the total amount of tracer substance crossing the ocean surface per unit time (dimensions tracer/time). The subscript wsignifies that the flux is measured at the ocean or “water” side of the interface. The sign convention is thatQwTis counted as positive if it is directedintothe ocean.
8.4. COMMENTS ON THE SURFACE TRACER FLUXES 87 It is useful to note that the total surface tracer flux (8.8) has a direct analogue in the equation (7.50) for the surface momentum stress, which is rewritten here for convenience
τsur f
ρo
= κmuz−A(∇hη)·(∇hu)+qwuh z=η. (8.9) For momentum, the friction termsκmuz−A(∇hη)·(∇hu) are formally associated with param-eterized turbulent momentum fluxes across the air-sea interface. Likewise, diffusive terms
−Fz +Fh· ∇hηare formally associated with parameterized turbulent tracer fluxes across the air-sea interface. These fluxes can be derived from a boundary layer model and are discussed in Section 8.4.
8.4 Comments on the surface tracer fluxes
The diffusive or turbulent part of the surface tracer flux
Qdi f fwT =Fh(T)· ∇hη−Fz(T) (8.10)
must be specified by a boundary condition. This flux enters the ocean after passing a sequence of boundary layers between atmosphere and ocean. The tracer transport through these layers is governed by a complex superposition of several processes, as molecular and turbulent diffusion, wave breaking, Langmuir circulation, radiation processes, chemical reactions and biological processes. Strong local gradients as in a thermal skin layer may be built up. Thus, the calculation ofQdi f fwT is a problem in nonequilibrium thermodynamics, turbulence theory, physical chemistry, and/or biophysics and is a rather complex problem by itself. For an introduction see the book by de Groot and Mazur (1962), or the articles by Forlandet al. (1988) or Doney (1995).
The set of basic equations of MOM does not provide information on the turbulent tracer transport, and the complicated vertical structure of the air-sea boundary layer is not resolved.
However, especially for long time integrations, the surface boundary conditions are cruical for the accuracy of MOM integrations. Therefore it is useful to make a few general statements here in hopes of exposing the basic issues.
First, recall that the dimensions of the fluxQwTare tracer concentration times velocity (see the definition ofQwTin equation (8.8)). As such,Qdi f fwT represents the total mass (or energy for the case of heat) of a tracer passing through the sea surface per unit area and per unit time. The amount of the tracer substance crossing the air-sea interface fulfills certain conservation laws.
For example, if the tracer is a substancei, the mass of the tracer passing the air-sea interface must be conserved, i.e., the flux at the ocean and the air side of the interface is the same,
Qdi f fwi =Qdi f fai (8.11)
If chemical reactions at the sea surface are possible, more general expressions can be found from the conservation of mass for the involved chemical elements. If the tracer is an energy, the reaction heatsQRfrom phase transitions or chemical reactions must be included into the total energy balance at the air-sea interface,
Qdi f fwi =Qdi f fai +QR. (8.12)
At the ocean side of the boundary layer, the flux (equation (8.8))
QwT=T qw+Fh(T)· ∇hη−Fz(T) z=η (8.13) appears at the top of the uppermost ocean box. As mentioned earlier, the first contribution, T(η)qw, brings about a change in tracer concentration due to changes in ocean volume upon introducing fresh water. The air-sea interface acts on a tracer like a filter with a transparency depending on the difference of the chemical potentials of the tracers in the air and in water. For example, ionic tracers such as dissolved salt have a total air-sea fluxQwTwhich is zero due to the large hydration energy. On the other hand, many weakly dissolved trace gases leave the ocean together with evaporating water. For this reason, the remaining termsQdi f fwT =Fh(T)·∇hη−Fz(T) in equation (8.13) arenotindependent of the fresh water flux termT(η)qw. Rather, they describe the turbulent diffusive tracer flux at the top of the surface box, and this flux is established by both the tracer concentration gradients across the air-sea interface, and tracer gradients within the ocean boundary layer coming from the chemical tracer kinetics in connection with the fresh water flux.
The next issue concerns how approximations for the tracer fluxes can be found. The complexity of the boundary layer prevents a direct coupling of an ocean circulation model with a boundary layer model which resolves the genuine dynamics of the boundary layer.
Alternately, the tracer fluxes are often calculated from empirical approximations. The difference of the bulk tracers in the atmosphere and the ocean, Ta−Ts, is taken as the thermodynamic force for the diffusive tracer flux. Then the tracer flux has the general form
Qdi f fwT ∼CTuwind(Ta−Ts) (8.14)
The empirical tubulent kinetic coefficient CT summarizes the complex dynamics within the boundary layer. It depends mainly on the tracer properties, on the wind velocity and on the stability of the boundary layer.
In the following, the boundary conditions for the fresh water flux, heat flux, and salt flux are specified in more detail.
8.4.1 Fresh water flux into the free surface model
The calculation of the fresh water flux requires a boundary layer model to be coupled with MOM. Here, a very simple version is described which permits a first order guess for the fresh water flux. Fresh water flux may have two different components, rainQRwand vapour from condensation or evaporation at the sea surface,QVw,
Qw = QRw+QVw, (8.15)
The fresh water velocity,qw, which is needed in the boundary condition is qw = Qw
ρ (8.16)
The amount of rain can be provided by an atmosphere model or field data might be used.
If the boundary layer is in a turbulent steady state the water vapour flux through the boundary layer can be parameterized as
QVw = ρaCwuwind(ha−hs). (8.17)
8.4. COMMENTS ON THE SURFACE TRACER FLUXES 89 Here the thermodynamic forcing is the difference of the specific humidityhain some reference height (usually 10 m) and at the sea surface,hs. The specific humidity is defined as the mass ratio
h = mw
m = ρw
ρa (8.18)
wheremis the total mass of the air in a volume element andmwthe mass of water vapour in the same volume. Alternatively the difference of the water vapour pressure or the partial density of water vapour can be used. The kinetic coefficientCwuwinddescribes the vertical turbulent diffusion in the boundary layer and is a function of the wind speeduwind in the reference height and of the stability of the atmospheric boundary layer. There is a considerable literature on empirical parameterizations ofCw from experimental data sets. Details can be found for example in Large and Pond (1982), Smith and Dobson (1984), Rosati and Miyakoda (1988) and in the literature cited there.
As a result for the calculation of the fresh water flux the specific humidityha and the wind velocity uwind must be known in some reference height. This information may come from an atmosphere model. For the calculation of the drag coefficientCw additional information on the stability of the boundary layer, i.e, on the atmosphere temperature is necessary. The specific humidity at the sea surface, hs, can be calculated from the assumption of saturated water vapour immediately over the sea surface.
8.4.2 Heat flux into the free surface model
For temperature, the heat balance in the upper box has to be considered. The heat flux enters the ocean through a boundary layer which has an atmospheric and an oceanic component.
There are four major contributions to the heat flux,
• the insolation,
• the infrared radiation balance between ocean and atmosphere,
• the sensible heat flux, which is basically a turbulent diffusion of heat,
• the heat transfer in connection with a fresh water flux. This effect involves a direct energy flux in connection with the flux of matter and a latent heat due to the liquid-vapour phase transition at the sea surface.
The insolation and the infrared radiation are not discussed here. For simple parameterization see e.g. Smith and Dobson (1984) or Rosati and Miyakoda (1988).
The enthalpy fluxQaethrough the top of the atmosphere-ocean boundary layer is,
Qae = Qaef resh+Qradae +Qsensae . (8.19) The radiative componentQradae includes insolation and the infrared radiation from the ocean and the atmosphere. The thermal radiation is emitted or absorbed in a thin skin layer at the sea surface and the approximation of a surface flux is justified. For the structure of the thermocline it may be important to resolve the vertial absorption profile of short wave radiation. To do this, the short wave radiation must be removed from the surface flux and the vertical divergence of the short wave radiation must be included in the source term. Qsensae describes the turbulent diffusion of heat,Qaef reshis the heat flux in connection with the heat capacity of the fresh water
advected relative to the sea surface. Under the assumption that the heat flux in the boundary layer has no vertical divergence, the enthalpy flux from the bottom of the boundary layer into the ocean,Qwe, is
Qwe = Qae+Qlate , (8.20)
whereQlate is the latent heat from that amount of fresh water which undergoes a phase transition at the air-sea interface and can be calculated from the water vapour fluxqVw,
Qlate = LQVw. (8.21)
Lis the evaporation heat of fresh water. Qlate is positive if the ocean gains heat by condensation and negative if heat is used for evaporation. It is a common approximation that the latent heat flux goes directly into the ocean and leaves the atmosphere temperature unaffected.
For a simple parameterization of the sensible heat flux the difference of the bulk virtual po-tential temperature of the atmosphere,θvaand the ocean,θvs, is assumed as the thermodynamic forcing function,
Qsensae = ρacapCTuwind(θva−θvs). (8.22) As for the fresh water flux the kinetic coefficient CTuwind describes the turbulent vertical diffusion of heat and can be parameterized in terms on the wind speed and the stability of the atmosphere. cap is the specific heat of air at constant pressure,ρathe density of air. The sign convention is to count a heat flux directed into the ocean as positive.
The heat flux between atmosphere and air-sea boundary layer due to the heat capacity of the fresh water is
Qaef resh = ρwcpTRQRw+ρacapθaQVw. (8.23) TRis the temperature of the liquid fresh water flux, i.e. of rain,ρwthe fresh water density,Ta
the temperature of vapour, which should be the atmosphere temperature. Usually,TRis not known and simpler approximations are necessary.
Finally, the boundary condition for the potential temperatureθis Qdi f fwθ = θ(η)Qw+∇hη·Fh(θ)−Fz(θ),
= (cpρ)−1Qwe
= (cpρ)−1
Qradae +Qsensae +LQVw+cpTrQRw+capθaQVw
. (8.24)