## 1. Introduction

In this note, the thermodynamics and energetics of the ocean in Boussinesq approximation are revisited. The aim is to formulate a consistent energy cycle for the ocean in Boussinesq approximation with a nonlinear equation of state and to formulate explicit conservation equations of the relevant thermodynamic quantities that have been previously discussed involving the correct exchange terms and molecular fluxes of heat and salt. Such a formulation is a prerequisite for energetically consistent ocean models, both numerical and analytical ones, but it is also necessary for conceptual reasons since almost any theory of the ocean is based on the Boussinesq approximation.

In general, the total energy of the ocean is given by the sum of kinetic energy *u* (e.g., Olbers et al. 2012). However, it is discussed below that a peculiar feature of the Boussinesq approximation is that the gravitational potential energy *u* adds to enthalpy

Instead of *u*, another physically meaningful split of enthalpy *h* in the Boussinesq approximation is into potential enthalpy *u*, for which dynamic internal energy

## 2. Energetics of the Boussinesq approximation

*ρ*, with

*p*:The key to interpret the energetics of the Boussinesq approximation is to note that the term

### a. Thermodynamics in the Boussinesq approximation

Recall that the thermodynamics of the compressible equations are based on the first and second law of thermodynamics for salty water parcels. The combination of both laws yields the Gibbs relation, which can be interpreted as the total differential change of internal energy by changes in its canonical state variables entropy *η*, salinity *S*, and specific volume *η*, *S*, *T* follow from appropriate Legendre transformations (Olbers et al. 2012). The analogous concept starting from the first law is used here to derive the thermodynamics of the Boussinesq approximation. The pressure work is modified in the first law for a fluid parcel in Boussinesq approximation. The other thermodynamic potentials are then derived by Legendre transformations from this approximate first law, rather than by applying the Boussinesq approximation directly to the exact expressions for the potentials.

*u*denotes the internal energy for the Boussinesq approximation, and

*h*is enthalpy, also for the Boussinesq approximation. While

*S*due to a mass exchange with the surrounding environment. This mass exchange is also assumed to take place at constant pressure

*h*with respect to

*S*is taken at constant pressure

*T*(Olbers et al. 2012). The replacement of the pressure work is the only necessary modification of the energetics in Boussinesq approximation. The thermodynamic potentials and all thermodynamic relations including the equation of state follow now in the same way as for the compressible equations.

*η*of the system by

*η*changes because of the heat exchange

*η*with respect to

*S*is again taken at constant pressure

*T*. Combining both relations yields the Gibbs relation in the Boussinesq approximation:with the relative chemical potential of seawater

^{1}The Gibbs relation [Eq. (5)] states that the canonical state variables for the thermodynamic potential

*u*in the Boussinesq approximation are

*υ*is used as a state variable. The other thermodynamic potentials are now given by an appropriate Legendre transformation using Eq. (5). Thus, instead of setting

*h*and

*η*,

*μ*, and

*ρ*in Eq. (8) are evaluated at the Boussinesq reference pressure

^{2}becomeswhere the derivative with respect to

*z*is taken at constant salinity

*S*and temperature

*T*. The other thermodynamic relations follow in an analogous way.

### b. Internal energy of the Boussinesq approximation

*η*. These processes will be specified below in Eq. (16). The conservation equation for

*η*will be replaced later by an equation for internal energy or potential enthalpy.

*u*, which follows from the Gibbs relation [Eq. (5)] for a water parcel replacing the total change in salinity

^{3}

*u*in the Boussinesq approximation allows for a consistent description of the energetics of ocean models based on this approximation. Total energy

### c. Dynamic and potential enthalpy

*u*as basic thermodynamic quantity, it is often more instructive to consider enthalpy, which becomes

*h*becomesNote that because

*h*again into different parts, namely, dynamical and potential enthalpy.

In the compressible equations, the canonical state variables of the thermodynamic potential *S*, entropy *η*, and pressure

### d. Dynamic and potential internal energy

To obtain a relation between dynamic enthalpy and gravitational potential energy, it is useful to consider an analogous split into a dynamical and a potential part for internal energy. In the compressible equations, the canonical thermodynamic state variables of internal energy are salinity *S*, entropy *η*, and specific volume

*u*as potential are

*S*,

*η*, and

*ρ*(or

*υ*), which follows from the Gibbs relation in the form Eq. (5). The resulting thermodynamic relation

*μ*and

*T*taken at the reference density

*μ*and

*T*taken at the reference depth

^{4}However, it was noted by McDougall (2003) that the magnitude of

*υ*to the Boussinesq reference pressure

## 3. Conclusions and discussion

It is the aim of this note to provide closed and explicit conservation equations for all forms of energy of a salty water parcel in Boussinesq approximation previously discussed by Young (2010) and Nycander (2011) based on molecular fluxes of enthalpy and salt and kinetic energy dissipation. The momentum, continuity, and salt equation in the Boussinesq approximation, Eqs. (10), (11), and (12), are complemented by a consistent thermodynamic equation obtained by a modification of the pressure work in the first law as suggested by Tailleux (2012). This modification leads to a thermodynamic conservation equation formulated either for internal energy *u* in Eq. (17) or for enthalpy *h* in Eq. (18) and also to a consistent equation of state for the Boussinesq approximation [Eq. (9)]. Adding either kinetic energy *h*, or gravitational potential energy *u*, yields both a conserved total energy variable and a consistent description of the energetics of the ocean in Boussinesq approximation is obtained.

*h*or

*u*. This yields a simple and consistent description of the energetics of the ocean in the Boussinesq approximation given byFor simplicity, it is assumed here in Eq. (33) that

It was proposed by McDougall (2003) to neglect

This means that while the dissipation *u* is in general much larger than mechanical energy ^{5} and is also related to the separation between reversible and irreversible energy conversions by the definition of dynamic and potential internal energy or enthalpy that is only approximate. Nevertheless, a closed and consistent energy cycle can be defined for the Boussinesq approximation as for fully compressible equations based on the full internal energy *u* or enthalpy *h* as outlined above.

Instead of splitting the energy variables *h* and *u* into dynamic and potential reservoirs, it can also be useful to consider the total energy

## Acknowledgments

The author is grateful for discussions with Dirk Olbers, Jürgen Willebrand, Remi Tailleux, and Fabien Roquet. Jonas Nycander and an anonymous reviewer helped to improve the manuscript.

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^{1}

In the definition for *μ*, enthalpy *h* is differentiated with respect to *S* at constant pressure *T*. In the following, partial differentials are always taken at constant (remaining) canonical state variables of the respective quantity. The value *μ* are the only exceptions since the canonical state variables of *h* are

^{2}

^{3}

To obtain an expression for

^{4}

The term

^{5}

The only differences besides the role and definition of the pressure work