The dynamics of three local models, for momentum transfer at the air–sea interface, is compared. The models differ by whether or not the ocean velocity is included in the shear calculation applied to the ocean and the atmosphere. All three cases are employed in climate or ocean simulations. Analytic calculations for the models with deterministic and random forcing (white and coloured) are presented. The short-term behaviour is similar in all models, with only small quantitative differences, while the long-term behaviour differs qualitatively between the models. The fluctuation–dissipation relation, which connects the fast atmospheric motion to the slow oceanic dynamics, is established for all models with random forcing. The fluctuation–dissipation theorem, which compares the response to an external forcing to internal fluctuations, is established for a white-noise forcing and a coloured forcing when the phase space is augmented by the forcing variable. Using results from numerical integrations of stochastic differential equations, we show that the fluctuation theorem, which compares the probability of positive to negative fluxes of the same magnitude, averaged over time intervals of varying lengths, holds for the energy gained by the ocean from the atmosphere.

The exchange of momentum, heat, water and chemical fluxes at the atmosphere–ocean interface is key to understanding
the dynamics of the atmosphere, the ocean and the climate, as well as their response to changes in the forcing
of the climate system (

In the present paper the exchange of momentum only is considered. It is caused by shear which depends on the atmospheric wind and many other physical quantities, such as the ocean velocity, the sea state and the density stratification in the atmosphere and the ocean. In this work three different approaches for parameterising the shear at the air–sea interface, which are all used in numerical simulations of atmosphere, ocean and climate dynamics, are compared. They differ by the extent to which the ocean velocity is considered in the calculation of the shear force at the air–sea interface. In the first, the ocean velocity is ignored and shear is calculated based on the atmospheric velocity only. Historically this was done in all atmosphere, ocean and climate simulations and is justified by the fact that atmospheric winds usually have higher speeds than ocean currents. In the second, the ocean velocity is considered when the shear force applied to the ocean is calculated, but not to the atmosphere. These two models are called “one-way” as the ocean dynamic does not act on the atmosphere; they are used, for example, whenever the atmospheric forcing is known prior to the integration of the ocean model, when an ocean-only simulation is performed. Only the third model is mechanically consistent, as the shear force, applied to the ocean and the atmosphere, is calculated based on the difference between the atmospheric and oceanic velocity vectors and respects Newton's laws. This model is called “two-way”.

The differences arising from including the ocean velocity in the shear calculation have been found to be important in observations
(

As local bulk formulas are investigated,
only the local exchange between the atmosphere and the ocean is considered, neglecting the horizontal interaction within
the atmosphere and the ocean.
Mathematically speaking the models are 0-D one-component (0D1C)
(see Sect.

The conspicuous feature of the atmosphere–ocean system is the strong difference in mass (and also heat capacity,

The major difference between Brownian motion and air–sea interaction is that the former system is conservative,
while the latter is dissipative and forced from the exterior.
Mathematically speaking, in the former, the dynamics conserves the phase space volume, while in the
latter it contracts and the dynamics takes place on a (strange) attractor of vanishing phase space volume.
A key feature of Brownian motion is the equipartition of energy between a Brownian particle and a molecule (

In the context of a purely 2-D dynamics, the energy dissipation within the atmospheric
and oceanic layers, due to horizontal friction processes, decreases with an increasing Reynolds number,
due to the inverse cascade of energy in 2-D turbulence (

Here, only linear models are considered, because the focus is on the analytic theory (where possible).
The analytic solution of a linear model gives the dependence on all parameters, while in a non-linear model
the parameter dependence has to be numerically evaluated for each parameter.
Furthermore, in the linear models, solutions with different forcing can be simply added up, but in their non-linear counterpart this is no longer true.
The prolongation to non-linear models and their numerical solutions will be discussed elsewhere.
It is furthermore important to note that the major differences between the three models already emerge in their linear versions.

The present work compares the three different models of air–sea transfer of momentum discussed above with
four different drag forcings, a linear drag law with
a constant or periodic (deterministic) forcing or a white or coloured random forcing.
This leads to

The fluctuation–dissipation theorem (FDT) (see

The fluctuation theorems (FTs) (

The use of stochastic models for air–sea interaction dates back to the pioneering work of

The turbulent friction at the atmosphere–ocean interface is commonly modelled by a quadratic friction law,
where the friction force is a drag coefficient times the product of the shear speed and the shear velocity (see e.g.

The mathematical models discussed here are non-dimensionalised.
The mass of the atmosphere per unit area is set to unity.
The mass of the ocean per unit area is

In the second model, L2, the ocean velocity is considered in the calculation of the shear force at the interface when the ocean dynamics is considered,
but not for the atmospheric velocity:

In model L3 the ocean velocity is considered in the calculation of the shear force at the interface,
when the atmosphere and ocean dynamics is considered:

For each of the linear models four different kinds of forcing are distinguished.
The first is a constant forcing starting at a time

White-in-time random forcing

In the fourth series of configurations, called

In the local linear models all solutions are analytic, for all types of forcing considered. These models are a firm testing ground for all theories on air–sea interaction.

First, the unforced evolution of an initial state in the three models is compared.
For the consistent model L3, Eq. (

Second, the solutions of the different models, subject to the same forcing, are compared.
Only the atmosphere is subject to an external forcing.
Two extreme cases can be distinguished.
The first is the short-term response and the second is the long-term evolution.
To consider the first question, only the

The long-term behaviours with constant forcing of the atmosphere are however completely different (see Appendix

When the forcing applied to the atmosphere is periodic (

Some of the models with random forcing have a dynamics which is not statistically stationary, and time averages depend on the length of the averaging interval.
Time averages are therefore replaced by ensemble averages, taken over an ensemble (

When the forcing is Gaussian in the

It is important to note that some of these models do not lead to a (statistically) stationary state, but that their ensemble averages evolve in time.
All the processes are, however, of stationary increment; that is, the time increments of random variables (

At the interface the ocean (Eqs.

As an example the FDR in the configuration L3W is considered after the initial spin-up, that is, for

Concerning the ocean (Eq.

Equations (

When considering the second-order moments, the parameters in the linear model are given by

It is straightforward to determine the FDR for LW1 and LW2 using results from Appendix

It is essential to note that in the linear models discussed in this subsection the forcing can be a linear combination
of different forcings proposed with different periods and correlation times.
The second-order moments are the sum of the individual second-order moments; that is, cross-correlations of variables with different types of forcings vanish.
When the forcing is a combination of a random forcing and a periodic forcing, it is important to note that
the periodic part does not contribute to the (linear) growth rate, and it also does not
contribute to the difference in the correlation between the ocean variance and the ocean–atmosphere correlation; both facts are related.
The periodic part is however important when it comes to evaluating the difference between the atmosphere variance and ocean variance.
This is a possible explanation why the estimation of the friction parameter was successful in

The fluxes of kinetic energy in the system are detailed in Fig.

When the forcing is periodic, averages over one period are taken, and when the forcing is stochastic,
ensemble averages are performed.
For convenience the same symbol

Schematic of energy fluxes in the atmosphere–ocean system.

In all the models the fluxes are related by

Energy fluxes for

For the L3 model, the only model that respects Newton's laws, all second-order moments have the same constant growth rate,
and so the differences of these second-order moments are constant in time. They are given in Table

In a perfect gas in equilibrium with molecules of different mass, the kinetic energy of each molecule,
measured by the temperature, is equal on average, and heat flows on average from the hotter substance to the colder substance (second law of thermodynamics).
For the forced and dissipative air–sea interaction of the L2 and L3 models, the energetic influence of the interface on
the ocean is

Differences of second-order-moments of the velocity (normalised by

The fluctuation–dissipation theorem (FDT) compares the response of a system subject to an external
perturbation to the internal fluctuations of the system.
This is related to Onsager's principle, which states that the system relaxes from a forced state to the unforced dynamics
in the same manner as if the forced state were due to an internal fluctuation of the system.
The expressions FDT, Onsager's principle and response theory are often interchanged in applications.
Precise definitions are given in
Appendixes

The processes considered here are of stationary increment and the perturbation matrices
are independent of the actual time

For the coloured-noise forcing the perturbation matrix in the augmented phase space is given
for the L1C model in Eq. (

The FDT relies strongly on Gaussian statistics (see e.g.

The average states and fluxes in the different models investigated as a function of their parameters are given in Sect.

The concepts of the FT are applied to a variety of problems and quantities and are also extended to deterministic dynamical systems.
In the present work the analysis of

The key quantity considered in the FT is the symmetry function:

The normalised time average over an interval

The FT holds when

The power the atmosphere loses at the interface

In the problem considered here the variable

First, the L3 model is discussed.
It is important to note that although

The parameters used in the numerical calculations are

For the atmosphere the probability of having a negative flux

Probability density function of

Lin–log plot of

Scaling exponents for

For calculations with the coloured noise model, the same parameters as in the white-noise calculations are used, and

Numerical integration of the L1W, L2W, L1C and L2C models show that

When ocean velocities are not considered in the models of air–sea interaction, the atmosphere loses, on average, more energy
and the ocean gains more energy, as compared to when the ocean velocities are taken into account.
Previous publications on the comparison of different models of air–sea interaction focus on quantitative differences.
This is justified when the short-term dynamics is considered, as shown above.
At longer timescales the differences are not only quantitative, but also qualitative, as for some models
stationary states in the ocean, the atmosphere or in both are reached, while in others this is not the case.
An example is the “eddy-killing” term (see

The magnitude of the constant growth rate is the typical growth rate of the ocean dynamics shortly after the turbulent forcing by the atmosphere has started and before dissipative processes develop to counterbalance it. It depends on the strength of the atmospheric forcing, its coherence in time and the thickness of the ocean (mixed) layer. Processes that lead to a saturation of the growth are of varying nature, space and time dependent, and typically non-linear and intermittent.

The discussion of the FDT establishes when the response to an external perturbation can be obtained from internal fluctuations of the system. In the simple system discussed here we can see analytically when it is verified and fails and how the failure can be removed by extending the phase space. Determining the response to a sudden change in the external forcing is key in many applications, such as the response of the atmospheric and oceanic planetary-boundary-layer dynamics to a change in the synoptic weather condition. The presented calculations can also be used to guide applications of the FDT to systems with large, but not infinite, time separation.

The FT concerns the transfer of energy between the atmosphere and the interface and the interface and the ocean
on different timescales.
The temporal down-scaling is solved when we can obtain the pdf of short-term averages from the pdf of longer-term averages.
The temporal up-scaling is solved when we can obtain the pdf of long-term averages from the pdf of short-term averages.
The FT relates temporal averages over different timescales and puts a large constraint on the pdfs of the
averaged energy transfers over different timescales.
The FT is key to understanding and modelling the climate dynamics, as in all observations and models some time-and-space averaging is present.
It is not always clear what the averaging period is associated with a variable in a model.
The FT gives us a hint of what to expect when passing, for example, from monthly averaged interaction/forcing to daily or hourly averages.
Considering a more fundamental point of view, equilibrium statistical mechanics is based on the pdfs of the canonical ensembles.
For non-equilibrium ensembles no such reference pdf is known in general (

The major difference between a 2-D model and a local model is that the former contains horizontal advection of momentum, while the latter does not. It is thus not clear which variable of the 2-D model has to be considered using the insight from the local models. Is it the local velocity or the velocity advected by the total-momentum mode or by the ocean dynamics, or do we have to consider coarse-grained variables for which the importance of horizontal advection is reduced? If this is the case, we have to define a coarse-graining scale that is sufficient or optimal in some sense. The FT can guide these choices.

The concepts presented here are not restricted to momentum transfer, but can also be employed to study heat exchange between the atmosphere and the ocean or other processes in the climate system with diverse characteristic timescales. Ongoing research is directed towards considering the concepts presented here in a hierarchy of models with increasing complexity and in observations. This research is of a different nature, numerical and observational, and will be described elsewhere.

The data used in Sect.

In this section the solutions of linear models L1, L2 and L3 are solved using linear algebra.
The linear differential equation

The system is forced and damped and the atmospheric dynamics acts on the ocean without considering the ocean velocity.
A coupling which is still used in some climate models.

The system is forced with damping and the atmospheric dynamics forces the ocean, ocean velocity is taken into account
for the ocean dynamics but not in the atmospheric dynamics (Newton's third law is not respected).

The atmosphere is forced with damping and the atmospheric dynamics forces the ocean, the ocean velocity is taken into account.

The solution is

In all experiments only the atmosphere is forced,

In this Appendix

In this Appendix

The solution is

The following identities are used:

The correlation matrix (

For the stochastic forcing straightforward calculations, based on Eqs. (

For the stochastic forcing straightforward calculations, based on Eqs. (

For the stochastic forcing straightforward calculations, based on Eqs. (

The solution is

From this follows (after dropping decaying exponentials):

In the absence of forcing an initial perturbation at

The solution is

Full interaction both ways.

The solution is

From this follows (after dropping decaying exponentials):

In the absence of forcing, an initial perturbation at

The fluctuation–dissipation theorem applies to a system if the system relaxes from a forced state to the unforced dynamics in the same manner as if the forced state were due to an internal fluctuation of the system.

The average response of a system to an external small-amplitude forcing is

The Ornstein–Uhlenbeck process (

Note that for Brownian motion (

More precisely, in 2-D space, a perturbation

The time-lagged correlation matrix is

Calculations give

As for the white-noise case, we get

When only Eq. (

The FDT applies only when the forcing correlation time vanishes, that is, when

As the system is linear, the pdfs of the variables are Gaussian.
In the unperturbed system, averages vanish and second-order moments are given in Eq. (

The author declares that he has no conflict of interest.

The comments of two anonymous referees considerably improved the quality of the paper. I am grateful to the Physique Statistique et Modélisation team at LiPhy Grenoble for their hospitality and discussion.

This research has been supported by Labex OASUG@2020 (grant no. ANR10 LABX56).

This paper was edited by Balasubramanya Nadiga and reviewed by two anonymous referees.