1. Introduction
The earth’s climate system is characterized by variability coupled across a broad range of space and time scales (e.g., Palmer et al. 2008; Hurrell et al. 2009). The variability of the faster parts of the system is both modulated by and feeds back on the slower components of the system. It is conventional to refer to the slower parts of the system as “climate” and the faster parts as “weather”, keeping in mind that (i) this distinction is operational in that the climate on one time scale may be weather on another, and (ii) the time-scale separation between weather and climate may not be large. Nevertheless, this fast–slow separation is useful conceptually and generally underlies the development of subgrid-scale parameterizations (either implicitly or explicitly). As ocean–atmosphere processes on larger spatial scales tend to evolve more slowly, we can think of fast and slow variability as respectively being spatially small and large scale.


This perspective from statistical physics suggests that if the time-scale separation between weather and climate is very large, then the net effect of the fast weather on the slow climate averages out on climate time scales and can be represented as a deterministic function of the climate state. However, for a finite time-scale separation, a given climate state is not associated with a unique upscale influence and stochastic corrections must be accounted for. This line of reasoning underpins the motivation for the representation of unresolved scales by stochastic processes (e.g., Palmer and Williams 2010). Since the early work by Mitchell (1966) and Hasselmann (1976), stochastic differential equations (SDEs) have been used to model effective climate dynamics influenced by fast weather variability (general introductions to SDEs are presented in Gardiner 1997 and Horsthemke and Lefever 2006; an introduction in the context of climate modeling is presented in Penland 2003). These stochastic climate models have generally assumed the form of the reduced model, estimating parameters from observations (e.g., Penland 1996; Kravtsov et al. 2005); or replaced unresolved variables with stochastic processes in mechanistic models (e.g., Farrell and Ioannou 1993; DelSole 2001; Monahan 2002c, 2006; Sura and Newman 2008). Such studies have displayed considerable success in accounting for a broad range of observational results, in settings as diverse as El Niño and the Southern Oscillation, extratropical atmospheric dynamics, and atmosphere–ocean boundary layer variability. These successes provide ample evidence for the utility of the stochastic ansatz. However, relatively little attention has been paid to the explicit development of effective models of the slow climate dynamics from coupled fast–slow systems. Such a program has been carried out by Majda et al. (2003, 2005, 2010) (the so-called Majda–Timofeyev–Vanden-Eijnden (MTV) theory] for the case of fast–slow systems in which the strength of the forcing of slow by fast variables increases with the scale separation so that SDEs are obtained in the strict limit of an infinite scale separation. This earlier analysis, which yields explicit formulae for the effective dynamics of the slow system, has been applied to idealized atmospheric models in Franzke et al. (2005) and Franzke and Majda (2006) and has demonstrated that even in the absence of a large scale separation between fast and slow variables, an effective stochastic model of the climate can be produced that captures essential aspects of the full coupled system (although Culina 2009 and Strounine et al. 2010 demonstrate examples in which features of interest are not captured by the MTV reduction).
The MTV reduction method is a particular case of stochastic averaging, a more general framework for the reduction of coupled fast–slow systems to an effective dynamics of the slow variable (e.g., Freidlin and Wentzell 1998; Rödenbeck et al. 2001; Arnold et al. 2003; Vanden-Eijnden 2003; Fatkullin and Vanden-Eijnden 2004). Stochastic averaging generalizes classical averaging procedures (e.g., Sanders and Verhulst 1985), by which the slow dynamics is averaged over the fast variables to produce an effective deterministic dynamics in the slow variables alone. Stochastic corrections to the averaged forcing are given by explicit formulae in terms of the statistics of the fast variables and the dynamics of the slow variables. Stochastic averaging provides a general systematic framework for the derivation of effective stochastic climate models from coupled weather–climate systems.
An introduction to stochastic averaging is presented in section 2. The discussion in this section does not introduce any new mathematical results; rather, it presents established results in a context familiar to researchers in atmosphere, ocean, and climate dynamics. The focus is intuition building rather than mathematical detail. Subsequently, this study considers the application of stochastic averaging to two idealized climate models. First, section 3 considers the reduction of a simple Stommel (1961)-type model of the meridional overturning circulation (MOC), for which an effective stochastic dynamics of the (slow) meridional salinity gradient is obtained by averaging over the variability of the (fast) meridional temperature gradient. A coupled model of sea surface winds, surface air temperature (SAT), and sea surface temperature (SST) is considered in section 4, in which effective stochastic dynamics of the coupled SAT–SST system are derived. A discussion and conclusions are presented in section 5. A number of illustrative “tutorial” examples of the reduction procedure are presented in the appendix. These examples are strictly didactic and are included to illustrate the application of the reduction method. The models considered in this study and in a previous study by Arnold et al. (2003) are highly idealized; the application of stochastic averaging to a much more complex system (a two-layer quasigeostrophic channel model) is presented in Culina (2009) and Culina et al. (2011).
2. Stochastic averaging

a. Averaging approximation (A)








b. Linear diffusion approximation (L)

In this approximation, the stochastic process ζ(t) describes a correction superimposed on the (A) trajectory


In the (L) approximation, the stochastic correction to x(t) is described by a linear SDE driven by additive noise (i.e., it is a multivariate Ornstein–Uhlenbeck process). The linear operator corresponds to the linearization of the averaged dynamics around the (A) trajectory. The “bare truncation” linearized dynamics has been augmented by that part of the averaged influence of the fast variable, which can be described as a linear operator. The (L) model is a more general statement of the ansatz that the influence of fast weather variability on the slow climate variable can be represented as linear dynamics augmented with white noise variability (e.g., Penland and Sardeshmukh 1995). Models based on this ansatz have seen broad application in stochastic modeling of climate processes such as the El Niño–Southern Oscillation (e.g., Penland 1996; Kleeman 2008), midlatitude atmospheric variability (e.g., Farrell and Ioannou 1993; Newman et al. 1997; Whitaker and Sardeshmukh 1998; DelSole 2001), and dynamics of the meridional overturning circulation (e.g., Tziperman et al. 2008; Alexander 2008). A major motivation for the consideration of these models is their practical utility: they are amenable to both analytic solution and estimation from observations. However, such models cannot capture features in ζ such as non-Gaussian pdfs or long time-scale behavior like transitions between metastable circulation regimes that are observed in some climate processes. Such effects require that the stochastic corrections feed back on x through dynamical nonlinearities or state-dependent (multiplicative) noise, a point discussed in detail in Sardeshmukh and Sura (2009). It is to such approximations that we now turn.
c. Nonlinear diffusion approximations (N+), (N)

As with the (L) approximation, the stochastic terms in the (N+) approximation vanish as τ → 0 unless in the full system the upscale influence of the fast variables on the slow is rescaled appropriately as the time-scale separation increases. Such rescaling (as in MTV theory) allows the stochastic terms to survive in the strict τ = 0 limit. Without this rescaling, the (N+) approximation with σ ≠ 0 represents a “small τ” theory, rather than a τ = 0 theory.
Note that the hierarchy of approximations (A), (L), and (N+) are not to be interpreted as corresponding to successive terms in a formal series expansion in τ; rather, they represent approximations appropriate on different time scales. The (A) and (L) approximations are valid over shorter time scales than (N+); what defines “short” in this context is problem dependent. For many practical problems, (A) and (L) may be sufficient on time scales of interest. The (N+) approximation is more general, but it is also more difficult to implement.
3. Idealized “Stommel type” model of the meridional overturning circulation
The Atlantic MOC (AMOC) is an important component of the global climate system, transporting heat from the South Atlantic to the North Atlantic and playing a central role in the deep circulation of the World Ocean. Variability in the strength of the AMOC has been implicated in climate variations on time scales from decades to millennia: in particular, evidence (from models and paleoclimate proxies) that the AMOC can undergo transitions between large-scale “circulation regimes” suggests that it plays a fundamental role in variability such as Dansgaard–Oeschger oscillations (e.g., Rahmstorf 2002). Many fundamental questions remain regarding the mechanisms driving the AMOC and the character of its variability; idealized stochastic models have played a useful role in addressing some of these issues (e.g., Monahan et al. 2008).
In particular, Timmermann and Lohmann (2000, hereafter TL) considered a version of the Stommel (1961) two-box model in which the full two-dimensional dynamics (of meridional gradients of temperature and salinity) were reduced to a one-dimensional SDE (with multiplicative noise) in salinity alone through the assumption that the time scale of temperature variability is very much shorter than that of salinity. As was discussed in Monahan et al. (2002), the interpretation of the multiplicative noise terms in TL as arising from temperature fluctuations is problematic as it involves the simple transfer of the fluctuating temperature variable from inside an absolute value function to outside. In Monahan et al. (2002) and Monahan (2002c), the multiplicative noise term was instead interpreted as arising from variable large-scale diffusion (or horizontal gyre transport, cf. Monahan 2002b). The question remains regarding the correct limiting salinity dynamics when the time scale of temperature variability becomes very short; stochastic averaging provides a natural tool for addressing this question.




Contour plots of the averaged deterministic salinity tendency
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1



Contour plot of the noise strength rescaled by the square root of the time-scale separation
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1


Contour plots of the (N+) approximation stationary pdf Eq. (38) of the Stommel model (in the TL approximation) as a function of μ and σM.
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
A comparison of ps(x) as predicted from stochastic averaging [Eq. (38)] with that estimated from direct numerical simulations of the SDE [Eqs. (29) and (31)] over a range of τ is presented in Fig. 4 for σA = 0.05 and σM = 0.25. For small τ, the agreement between the averaged and numerically simulated pdfs is excellent. For larger τ, the averaged solution diverges from that of the full system, but even for τ = 1 the stationary pdf ps(x) of the averaged and full systems is qualitatively similar.
Comparison of the reduced (N+) stationary pdf Eq. (38) (black contours) and the stationary pdf of the full system Eqs. (29) and (31) (red contours), for σA = 0.05 and σM = 0.25, and τ = 0.1, 0.5, 1.
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1



Contour plots of
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
4. Coupled atmosphere–ocean boundary layers
The atmosphere and ocean interact through their respective boundary layers, exchanging momentum, energy, and material substances; these exchanges mediate the dynamics of the coupled climate system on seasonal to decadal time scales (e.g., Taylor 2000). The character of turbulence in the atmospheric boundary layer exercises a primary control on the strength of these exchanges; this turbulence both influences and is influenced by surface winds and thermal stratification. Because the atmospheric and oceanic boundary layers evolve over very different time scales, they are a canonical example of a coupled fast–slow geophysical system (e.g., Sura and Newman 2008). Frankingoul and Hasselmann (1977), in one of the first applications of a stochastic climate model, used a model of upper-ocean heat content driven by rapidly fluctuating surface winds as a basic “null hypothesis” explanation of the generically red spectrum of SST. Subsequent studies have addressed physical controls on the probability distributions of SAT and SST (Sura et al. 2006; Sura and Newman 2008) and of sea surface winds (Monahan 2004, 2006, 2010), informally deriving SDEs from the boundary layer heat and momentum budgets (respectively).


The dynamics of Ta and To, adapted by Sura and Newman (2008) from the earlier model of (Frankingoul and Hasselmann (1977), describes the anomaly heat budget of the coupled atmospheric column (through the depth of the troposphere) and oceanic upper mixed layer, linearized around the climate-mean state. Heat (both sensible and latent) is exchanged between the atmosphere and ocean with a net exchange coefficient β = ρacscp(1+B) (where cp is the specific heat capacity of air at constant pressure, ρa is the air density, cs is the bulk sensible heat transfer coefficient, and B is the Bowen ratio; the standard model value of β corresponds roughly to B = 10). A nonzero mean air–sea temperature gradient θ results in a nonzero average heat flux. Processes internal to the atmosphere and ocean (e.g., radiative fluxes, turbulent exchanges with the free atmosphere, or submixed layer ocean) relax Ta and To back toward their equilibrium values with rate coefficients λa and λo, respectively. The parameters γa and γo denote the effective heat capacities of the respective atmospheric and oceanic layers. The heat budgets of both the atmospheric and oceanic layers are perturbed by fast fluctuations
Model parameters were set to reproduce the observed variability at Ocean Station Papa (OSP; 50°N, 145°W). Daily-mean extended wintertime (November–March) data from 1949 to 1981 were considered. For Ta and To daily anomalies were calculated relative to a smooth annual cycle fit by three sinusoids (periods of 12, 6, and 4 months) to a raw mean seasonal cycle. These model parameter values (Table 1) are based on values determined in Sura and Newman (2008), slightly tuned to improve the agreement between the model and observations. Note that because Sura and Newman (2008) fit parameters of a reduced stochastic model of Ta and To alone, their best-fit parameters are not necessarily those of the model Eqs. (40)–(43). The stratification/boundary layer height regression parameter α was estimated from the data presented in Fig. 6. The mean boundary layer depth
Scatterplot of monthly-mean extended winter (November–February) boundary layer height h against (anomaly) surface stratification Ta − To (from 1989 to 2007) for a 1.5° × 1.5° grid box centered at (49.5°N, 145.5°W). The solid line is a best-fit linear regression curve. Data are taken from the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim reanalysis, available for download at http://data.ecmwf.int/data/d/interim_daily/. While these data are reanalysis products and should be treated with caution, they at least represent a characterization of boundary layer variability in a comprehensive, state-of-the-art general circulation model constrained to some extent by observations.
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
Observed and simulated [Eqs. (40)–(43), with parameter values in Table 1] statistics of ocean–atmosphere variables at Ocean Station Papa. (top) Autocorrelation functions and joint pdf of surface air temperature Ta and sea surface temperature To; and (bottom) autocorrelation function and pdf of wind speed w.
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
Parameter values for the coupled atmosphere–ocean boundary layer model [Eqs. (40)–(43)] set to reproduce observations of winds, sea surface temperature, and surface air temperature at Ocean Station Papa in the northeastern subarctic Pacific Ocean.




Components of the averaged deterministic temperature tendency
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1

Plot of the function ψ(Ta − To) characterizing the state dependence of the reduced model noise strength [Eq. (49)].
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
Autocorrelation functions and joint pdfs of Ta and To for the full and reduced models. (top) Reduced model with σ(Ta − To) from Eq. (48); and (bottom) as in top panels with σ reduced by a factor of 2.
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
This analysis has demonstrated how in an idealized, but physically relevant system, stochastic averaging can both produce accurate reduced models of the slow dynamics and provide insight as to how these are influenced by the feedback from the modulation of the statistics of the fast variables. Because of the poor time-scale separation between the slow and fast variables, some ad hoc retuning of the model was necessary. Application of these ideas to more complex systems (such as the two-layer quasigeostrophic channel model considered in Culina et al. 2011) requires the use of more sophisticated algorithms, but the underlying approach is the same as in this relatively simple case.
5. Discussion and conclusions
The mathematical theory of stochastic averaging discussed in this study provides a precise quantitative framework for investigating the effective dynamics of the slow (resolved) climate variables in the presence of fast (unresolved) weather variability. This framework characterizes a hierarchy of approximations to the full coupled fast–slow system. In the limit of infinitely large scale separation, the slow dynamics are deterministic, with tendencies given by those of the full system averaged over the statistics of the fast variable (conditioned on a fixed value of the slow variable). In general, these deterministic dynamics will differ from those associated with ignoring fluctuations in the fast variables and replacing them with their mean values (the so-called bare truncation). For large but finite scale separations, stochastic corrections emerge. A better approximation to the deterministic averaged system includes Gaussian corrections described by a linear stochastic differential equation (such as those considered in, e.g., Farrell and Ioannou 1993; Penland 1996; DelSole 2001; Kleeman 2008). At this level of approximation, the climate variable is described by the deterministic averaged dynamics with superimposed fluctuations that do not feed back on the averaged trajectory. A more accurate approximation describes the climate variable with a full stochastic differential equation, which may be linear or nonlinear and may involve additive or multiplicative noises. The drift and diffusion parameters of this SDE are predicted explicitly in terms of the original dynamics and the statistics of the fast variable.
Two examples of stochastic averaging were considered in this study: first, a simplified Stommel-type model of the meridional overturning circulation; and second, an idealized model of the coupled atmosphere–ocean boundary layers. The first example demonstrated how the averaged dynamics can differ substantially from the bare-truncation dynamics; in particular, the number and location of deterministic steady states can be changed by the averaging. Furthermore, as the time-scale separation is made arbitrarily large it was demonstrated that the full and reduced models come into good agreement. The second example, in which model parameters were set to reflect observed wind and temperature variability at Ocean Station Papa in the northeastern subarctic Pacific Ocean, we did not have the luxury of being able to adjust the time-scale separation between the fast and slow variables (respectively, wind and temperatures). In fact, in this example the time-scale separation was weak. Nevertheless, the statistics of the reduced model predicted by stochastic averaging were in reasonable agreement with those of the full model; this agreement was substantially improved by some slight ad hoc retuning of the stochastic corrections. Furthermore, this analysis demonstrated how the dynamics of sea surface temperature and surface air temperature are influenced by the dependence of wind speed statistics on the surface stratification—that is, how the modulation of the fast variables by the slow feeds back on the effective slow dynamics.
Other studies have considered the application of reduction techniques for deriving effective stochastic dynamics from a coupled fast–slow system. The so-called MTV theory is an example of precisely such a framework (e.g., Majda et al. 2001, 2003, 2005, 2010; Franzke et al. 2005; Franzke and Majda 2006; Strounine et al. 2010). In fact, MTV theory can be understood as a special case of stochastic averaging in which the influence of the fast variables on the slow is rescaled as the time-scale separation increases in such a way that an effective stochastic dynamics is obtained even in the strict limit of infinite time-scale separation (a simple exactly solvable case is presented in example 4 of the appendix). This assumed form of the dynamics has the advantage of dramatically simplifying computation of the parameters of the SDE; furthermore, the assumed quadratically nonlinear form of the dynamics assumed in MTV theory provides closed-form analytic expressions for the reduced dynamics. In practice, these simplifications are tremendously useful. However, they may not be consistent with the dynamics of the fast–slow system, and these inconsistencies may account for some of the biases that have been found in MTV reduced models (e.g., Culina 2009; Strounine et al. 2010). We further note that a reduction strategy similar to stochastic averaging known as “optimal prediction” was introduced by Chorin et al. (1998, 1999, 2000); the essential similarity between optimal prediction and stochastic averaging was demonstrated in Park et al. (2007).
In this study, stochastic averaging has been applied to highly idealized models. While the application of this reduction framework to more complicated models is the same in principle, in practice more sophisticated algorithms must be used. An application of stochastic averaging to a high-dimensional problem (a two-layer quasigeostrophic channel model) is presented in Culina et al. (2011), along with a detailed discussion of the algorithms required for this more complex system. As in the present study, it was found that the absence of a large time-scale separation requires some ad hoc a posteriori retuning of the stochastic corrections to obtain good agreement between the full and reduced models (as has been found to be the case with MTV theory; e.g., Culina 2009; Franzke et al. 2005; Franzke and Majda 2006). An important direction of future research is the investigation of the physical origin of such reduced model biases. Furthermore, to make practical the application of stochastic averaging to coupled weather–climate systems with modest time-scale separations, it is important to develop general strategies for the determination of such tuning parameters. The recent study of Strounine et al. (2010) presents such a strategy in the context of MTV theory; it would be interesting to see such an approach applied in the more general stochastic-averaging framework.
A potential benefit of stochastic reduction in systems with large time-scale separations between fast and slow variables is a significant increase in computational efficiency (e.g., Fatkullin and Vanden-Eijnden 2004). As discussed above, such a large separation often does not exist in atmosphere or ocean dynamics. The present study has focused on the existence and structure of the reduced dynamics; these are of interest on their own, in terms of the light they shed on the effective climate dynamics without explicitly modeled weather. A more detailed discussion of computational efficiency is given in Culina et al. (2011).
We will never in the foreseeable future be able to represent all dynamically relevant scales in models of the climate system, particularly as we ask questions about climate processes over ever longer time scales. Strategies for the construction of reduced models of the resolved variables alone will continue to be important. Mathematically rigorous strategies for the reduction of coupled fast–slow systems provide a potentially important direction of study, but these are only strictly valid in conditions that rarely obtain for real climate models. The great challenge of this program is to square the circle of balancing rigor with practical utility in the development of tools for systematically exploring the coupled dynamics of weather and climate.
Acknowledgments
The Ocean Station Papa data were kindly provided by Philip Sura. We thank Christian Franzke for bringing “optimal prediction” to our attention. We would also like to thank Philip Poon, Cécile Penland, and two anonymous reviewers for their very helpful comments on the manuscript.
APPENDIX
Toy Examples of Stochastic Averaging
The following are presented as toy “tutorial” examples of the application of averaging to simplified systems. Note that while the equations for the fast variables are linear in all of these cases for ease of analytic solution, this is not required by the averaging approach (cf. section 4).
a. Example 1


b. Example 2





Stationary pdfs of x from the full system Eqs. (A10)–(A11) for τ = 0.01 (solid black pdf), τ = 0.1 (dot-dashed pdf), and τ = 1 (dashed pdf), for Στ2 = 0.25 fixed. The gray pdf is the stationary pdf of the (N+) approximation [Eq. (79)] for Στ2 = 0.25.
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
c. Example 3
Stationary pdfs of numerically simulated x(t) from the full model [Eqs. (A30)–(A31)] (black pdf) and from the reduced model [Eq. (85)] (gray pdf), as functions of the time-scale separation τ (with τΣ2 = 1 in all cases).
Citation: Journal of Climate 24, 12; 10.1175/2011JCLI3641.1
For the system in example two the multiplicative noise of the (N+) approximation allowed the reduced model to capture the skewness of x(t). In the present example, the (N+) approximation does not differ from (L), so the skewness of x in the full model is a feature that the reduced model cannot capture (except in the limit τ → 0, in which the skewness vanishes).
d. Example 4
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