## 1. Introduction

It is well known that the climate is a highly nonlinear system. Nevertheless, the assumption of linearity has been successfully applied in many cases, such as in theoretical studies of instability mechanisms (e.g., the classical works of Charney 1947; Eady 1949; Kuo 1949) or by the use of linear models (e.g., Egger 1977; Roads 1980; Hoskins and Karoly 1981).

When studying the real climate system, the use of comprehensive and nonlinear general circulation models (GCM) is necessary. This is the only practicable way to study the interaction between physical processes determining the climate and to analyze large-scale fluid dynamical aspects. Atmospheric GCMs, for instance, are used to predict the response of the global circulation to sea surface temperature (SST) anomalies. When interpreting these GCM simulations, however, linearity must be assumed for the feedback. Otherwise, the results could not be generalized for, for example, a slightly different forcing. This concept has often been applied to discuss the impact of tropical (El Niño–like) SST anomalies (e.g., Rowntree 1972; Shukla and Wallace 1983; Cubasch 1985; von Storch et al. 1993). Nevertheless, the success of the idea of linearity for the Tropics does not necessarily imply its validity for other latitudes where the nonlinear transient processes dominate.

For the midlatitudes, the influence of North Atlantic SST anomalies associated with interdecadal variability of the oceanic thermohaline circulation (THC) on the atmospheric circulation has become the object of several investigations. Bjerknes (1964) and, more recently, Palmer and Sun (1985) and Kushnir (1994) have extracted patterns and corresponding amplitudes of North Atlantic interdecadal SST modes and the related atmospheric conditions from observations. Manabe and Stouffer (1988) and Delworth et al. (1993) have found similar SST anomalies in coupled GCM runs, which rise from THC variations. As a common feature, the SST anomaly patterns show a monopole structure with maximum values northeast of Bermuda, while the given magnitudes are quite different from each other and, in addition, show pronounced temporal variability. For the atmosphere, the above-mentioned investigations exhibit no consistent response. Therefore, simulations in the uncoupled mode with prescribed SST would be useful and have already performed by different authors (e.g., Houghton 1974; Chervin and Schneider 1976a,b; Chervin et al. 1976; Palmer and Sun 1985; Hense et al. 1990;Peng et al. 1995; Kushnir and Held 1996) In this context, the concept of linearity as discussed above should be checked, especially since the results of the various observational studies and model simulations show remarkable discrepancies when analyzing the atmospheric response to midlatitude SST anomalies (e.g., Palmer and Sun 1985; Pitcher et al. 1988; Kushnir and Lau 1992; Kushnir 1994). As first attempts at treating the question of linearity, Kutzbach et al. (1977), Palmer and Sun (1985), Hense et al. (1990), and Kushnir and Held (1996) have compared the results of simulations with different forcing amplitudes. However, too few case studies have been available for a systematic study of the linearity.

In the present paper, a highly simplified general circulation model (SGCM) is used for a more objective analysis of the linearity of the atmospheric response to a North Atlantic SST anomaly (hereafter referred to as NA-SSTA), corresponding to that found by Palmer and Sun (1985) or Kushnir (1994). The model has not been designed to compete with GCMs with respect to a realistic representation of the observed climate, but it has the advantage of running at low computational cost over extended periods of several centuries, which is one of the most important requirements for our investigations. The atmosphere is represented by an eddy-resolving two-level global primitive equation model to provide a satisfactory account of the budgets of mass, energy, and angular momentum. In particular, baroclinic activity is incorporated explicitly. However, the processes affecting the climate also include the interactions and feedbacks between radiation, land- and ice-covered surfaces, boundary layer effects, and the exchange of properties at the air–sea interface. Here, highly simplified parameterizations are used to represent these unresolved subgrid scale processes as simply as possible to focus on fundamental studies of fluid motion. This approach combines the advantage of a economic model with the necessary complexity of a climate model.

A large number of simulations (*N* = 17) have been performed where we have taken the NA-SSTA pattern, weighted by amplitudes varying between −4 K and +4 K, as an anomalous lower boundary condition. For our principal considerations, the analysis focuses on the linearity problem and not on the detailed structure of the responses in the individual simulations, for example, when compared to realizations with a high-resolution GCM.

The paper is organized as follows. Section 2 presents the experimental design and discusses some aspects of the simulations. In section 3 we elucidate the statistical background and explain the general procedure of how to fix some range of linearity. We use these techniques to explore the atmospheric response to the NA-SSTA pattern in section 4. A concluding section closes the paper.

## 2. Experimental design

For our investigations, we use an SGCM. The SGCM is a eddy-resolving two-layer spectral model based on the dry primitive equations formulated for a rotating sphere. The model includes simple parameterization schemes for unresolved processes such as radiative-convective heating and surface fluxes. A detailed description of the SGCM is given in the appendix. In all simulations presented here, the SGCM is integrated starting from the resting state and driven by a zonally and temporally uniform distribution of the incoming solar radiation. The forcing consists of the annual mean of the SST and of the incoming solar radiation without seasonal or diurnal cycles. For the different scenarios, the NA-SSTA pattern (Fig. 1) with varying amplitudes is superimposed onto the climatological annual mean SST. The “control run” is defined as the scenario with vanishing amplitude of the NA-SST pattern. For the other scenarios, the NA-SST pattern is weighted by amplitudes of ±0.25 K, ±0.5 K, ±0.75 K, ±1 K, ±1.5 K, ±2 K, ±3 K, and ±4 K. All simulations are integrated for 60 yr except the control run, which consists of 300 yr. To account for the spinup, the first 15 yr of all simulations are not considered in the analyses described in this paper.

It is a well-known fact that the use of annual mean conditions for SST and the incoming solar radiation results in an artificial mean flow, but this should be irrelevant for the validity of the results. Moreover, the use of an idealized forcing better matches the simplifications already present in the model. It is obvious that such a simple model cannot compete with comprehensive GCMs in representing the observed climate. The central focus of this paper, however, is to examine the linearity of the atmospheric response rather than the response itself. The results of our investigations are expected to give evidence about the behavior of a comprehensive GCM, because the SGCM meets necessary requirements: similar to a full GCM, it includes the nonlinearity of the dynamics and represents important physical processes by several parameterizations. In particular, the SST enters the model dynamics via nonlinear calculations of the boundary fluxes, the radiation, and the convection (see the model description in the appen~dix). The advantage of the SGCM compared with comprehensive GCMs is that it only needs a very small amount of computer power. This enables us to perform a large number of long simulations.

Taking into account the simplicity of the model, it simulates a reasonable climate, which seems sufficient in order to ensure the reliability of the results regarding our linearity investigations. In the control run, the SGCM reproduces the observed climatological mean land surface temperature (Oort 1983), at least in a qualitative manner, except for regions with steep mountains (Fig. 2). However, in our parameterization scheme the surface is identified with the 1000-hPa level. As usual for a two-level representation, we define the barotropic part of the streamfunction as its vertical average. Referring again to the climatological mean of the control run, the model gives a fairly reasonable reproduction of the observed stationary wave pattern and simulates the observed equatorial easterlies (Fig. 3). The barotropic streamfunction show strong gradients east of the North American continent and the Himalayas, and north of Antarctica, where the upper-level winds (not shown) exhibit jets with fairly realistic amplitudes. As found in a parallel experiment, the exclusive effect of the nonseasonality in the boundary conditions is a too strong upper-level westerly jet (≈22 m s^{−1}) west of the Tibetian Plateau.

In order to measure the transient eddy activity in the SGCM, we used the standard deviation of the baroclinic streamfunction. To obtain a climatological average as well as the variability, which is needed for our statistical evaluation methodology, we calculate the standard deviation for each 3-monthly period of the simulation separately, based on half-daily data and the individual 3-monthly average. According to the 3-monthly time interval used for the calculation, this value is referred to as “3-monthly standard deviation” in the reminder of the paper. Figure 4 shows the climatological average simulated in the control run. A quite realistic maxima of baroclinic activity is found over the North Atlantic and over the North Pacific regions as well as north of the Antarctic continent just where the barotropic streamfunction (Fig. 3) exhibits jetlike zones of strong gradients.

The sensitivity of the model to the NA-SSTA is illustrated by the climatological differences between the scenario “+2 K” and the control run. Figure 5 shows the time mean difference of the surface temperature. The model simulates a warming over the Arctic Sea extending toward Sibiria and a general cooling over the North American continent, except at the eastern coast of America where a warming is found. This response is in good agreement with results of a sensitivity simulation with a comprehensive GCM [the Hamburg Atmospheric General Circulation Model ECHAM3, (Roeckner et al. 1992)] at T42/L19 resolution. The ECHAM3 simulation was performed with a comparable amplitude of the NA-SSTA (approx +2.2 K when it was interpolated to the SGCM T21-Gaussian grid) and includes the annual cycle of the background SST and the incoming solar radiation (M. Latif, personal communication). The NA-SSTA persists throughout the entire integration (10 yr) with the same pattern and magnitude (independent from the annual cycle), as in the SGCM experiments. In particular, the Northern Hemisphere winter (December–February, DJF) response is in a qualitatively good agreement with the SGCM results. Figure 6 displays the ECHAM3 DJF surface temperature response, defined as the difference between the ECHAM3 NA-SSTA experiment (averaged over the nine simulated DJFs) and the climatological DJF average of a 100-yr ECHAM3 control simulation forced by the observed climatological annual cycle of the SST (similar to the background SST in the NA-SSTA experiment).

The response of the barotropic streamfunction in the SGCM simulation is displayed in Fig. 7. The mean features are positive anomalies across the North Atlantic, with a maximum at the eastern coast of the North American continent and downstream the NA-SSTA extending toward the pole, and a negative anomaly over Greenland. In qualitatively good agreement with Kushnir’s (1994) observations, the climatological westerly winds in the SGCM are strengthened on the northern side and weakened to southern side of the NA-SSTA. Again, the results show reasonable similarities with the full GCM (e.g., ECHAM3 DJF response of the 500-hPa streamfunction; Fig. 8).

It should be noted that some features of the response, in particular the positive anomaly of the barotropic streamfunction at the American coast, seem to be in contradiction to other simulations (e.g., Palmer and Sun 1985). However, as pointed out by Peng et al. (1995) or Kushnir and Held (1996), large discrepancies in the results among different model studies can be found. For example, based on linear theory, Egger (1977) and Roads (1980) suggested a cold high upstream and a warm low downstream a warm pool, while Palmer and Sun (1985) found a surface high downstream the warm SST anomaly. Peng et al. attribute the discrepancies to the strong dependence of the response on the basic flow. In their model, they found a response similar to Palmer and Sun (a downstream high) for November initial conditions but the opposite (a downstream low) for January conditions. The dependence of the response to the mean conditions hints to the nonlinearity of the response. In this respect, the use of annual mean conditions in our study has a further justification. A particular aspect of the nonlinearity problem, the effect of the amplitude of the forcing, is the central subject of our investigations presented in the following sections.

## 3. Statistical evaluation method

When comparing patterns of two different scenarios, we detect qualitatively very similar anomalies of the mean atmospheric flow fields. However, it is not enough merely to cast an eye over the graphical output of a certain atmospheric quantity and, thereby, gain an impression of whether the response patterns of the atmospheric feedback coincide or not. What we need is an objective intercomparison method.

To relate now two scenarios to each other, the following procedure is designed:

One simulation is selected to serve as the reference run (subscript ref) for the second, so-called experimental, realization (subscript exp).

- For the reference scenario we determine the quasi- stationary response
- It is normalized to give the response
*pattern*where the inner product of spatial distributions**a**and**b**is defined as the globally integrated product - The evolutions of each response, RSP
_{ref}and RSP_{exp}, are projected onto the spatial pattern P_{ref}. From this procedure, one gets swarms of*scalar*auto and cross amplitudes of both scenarios, that is, of the reference and experimental stateswhere*n*is the length of data records. The projection allows us to treat an originally multivariate problem in an univariate manner and rely on the ordinary significance tests. The advantage of this technique is that it alerts one to the possibility of improving the physical insight into the nature of a nonlinear atmospheric response either parallel or orthogonal to the reference pattern “**REF**.” With**r**^{⊥}denoting the orthogonal residual of the projection, the experimental response can be finally written aswhere the time average,r ^{⊥ }_{exp}^{t}, generally does not vanish. Note that we assign the sign of the response to the amplitude, which can therefore become negative. For the remainder of the paper,*amplitude*considers both, the magnitude and the sign of the response. This definition will also be used for the NA-SSTA. It has to be taken into account that signals of sensitivity experiments could be masked by noise. Therefore, we have to test the following null hypothesis for significance:

- When projecting onto the extracted response pattern and referring to time averages, auto- and cross amplitudes are taken from the same sample space. If the hypothesiscan be rejected at a certain level of significance, auto- and cross amplitudes are separated in terms of the first moment. In particular, when analyzing such a positive result for the case “reference versus control,” the response of the reference quasi equilibrium state exceeds the signal-to- noise ratio.
*H*^{A}_{0}*A*_{ref}^{t}*A*_{exp}^{t} - The response of the comparative experimental realization is orthogonal to the reference pattern;that is, the amplitude of the reference pattern is zero in the experimental simulationOtherwise it has a nonzero component in the subspace collinear to P
*H*^{B}_{0}*A*_{exp}^{t}_{ref}. - When normalized by the amplitude, Δ
*T̂,*of the NA-SSTA and temporally averaged, auto- and cross amplitudes are identical:where the control run is out of interest. In other words, a rejection would exclude linearity with respect to the reference pattern. - The comparative time mean orthogonal residual is zero. To check, its temporal evolution is rewritten aswhere the corresponding pattern,
is normalized in the same manner as is the reference pattern**p**^{⊥ }_{exp}**P**_{ref}above. The null hypothesis is, therefore,Its rejection indicates that there is a significant contribution orthogonal to the chosen reference pattern, which cannot be interpreted by the assumption of linearity.

*x*(

*t*), 1 ≤

*t*≤

*n*}, {

*y*(

*t*), 1 ≤

*t*≤

*n*}, and the Student t-distribution for the difference between the mean states. Statistically significant differences can be detected in terms of a

*t*ratio

For the nonparametric test, we do not need to adopt the assumption of normality and independence of the realizations but, rather, make use of various permutation techniques explained in detail by Preisendorfer and Barnett (1983). Only the temporal order of the compared datasets is presumed immaterial due to the absence of trends. The basic idea is that the samples are repeatedly partitioned randomly into two subsets. The corresponding differences of time means allow one to determine significance levels of separation between the time averages of the two original datasets, where we can restrict ourselves to univariate distributions.

## 4. Application to the SGCM simulations

The statistical approaches to the problem of linearity, as outlined in the previous section, are applied to two target variables, the barotropic part of the streamfunction and the 3-monthly standard deviation of the baroclinic mode, as defined in section 2 (i.e., the standard deviation of the baroclinic streamfunction calculated on a half-daily basis for 3-monthly time periods). The barotropic part of the streamfunction is chosen as an example of a variable, which one would expect to respond more linearly than other variables. The standard deviation of the baroclinic mode represents the baroclinic activity due to transient eddies. Therefore, this variable is likely to respond rather nonlinearly.

All possible pair combinations were subjected to the test procedure, in which each simulation serves as the reference scenario and the other as the experimental scenarios. The time series needed for the statistical tests are constructed using annual means. For the 3-monthly standard deviation, the annual mean is the average of the four datasets belonging to the year under consideration. The last 45 yr of each simulation are used for the statistical evaluation [i.e., *n* = 45 in (5) and (6)]. A comparison of the two different test algorithms [the nonparametric test of Preisendorfer and Barnett (1983) and the Student t test] shows that similar results arise if the evaluation is based on time averages of 1 yr or longer. Therefore, only the results of the *t* test based on the yearly means will be discussed in the following.

The results of the test of the null hypothesis *H*^{A}_{0}*A*_{ref}^{t} = *A*_{exp}^{t}) are presented in Figs. 9 and 10. In each diagram the reference simulation (the simulation from which the response pattern is calculated) is assigned to the abscissa. The values indicate the amplitude of the NASSTA pattern (Δ*T̂*_{ref}). The experimental scenarios (the simulations that are projected onto the reference response pattern) are assigned to the ordinate and are also identified by the amplitude of the forcing (Δ*T̂*_{exp}). The shading intensities of the individual boxes correspond to the significance level at which the null hypothesis is to be rejected. Three different significance levels (10%, 5%, and 1%, respectively) are displayed. For both variables, the barotropic streamfunction (Fig. 9) and the standard deviation of the baroclinic streamfunction (Fig. 10), the null hypothesis *H*^{A}_{0}*T̂*_{exp} = 0 row.

Hypothesis *H*^{B}_{0}*A*_{exp}^{t} = 0) could also be rejected in most cases (Figs. 11 and 12), except for small magnitudes of the forcing, that is, for those less than 0.5 K. Note, that a rejection of *H*^{B}_{0}*H*^{A}_{0}*H*^{B}_{0}*H*^{A}_{0}

Figures 13 and 14 show the results of the test of the“linearity” hypothesis ^{C}_{0}*A*_{ref}^{t}/Δ*T̂*_{ref} = *A*_{exp}^{t}/Δ*T̂*_{exp}. As one could already observe in the previous figures, the results are not symmetric with respect to the choice of whether a simulation serves as the reference or as the experimental scenario. This indicates that the response patterns in the different simulations are not the same. For both variables under consideration, we find certain sectors in the diagram, for which the hypothesis of a linear behavior in the reference pattern amplitude, with respect to the forcing, could not be rejected (i.e., linearity could be accepted). The extension of these sectors depends on the magnitude of the forcing in the reference simulation. If the NA-SSTA pattern has a considerable magnitude in the reference simulation (larger than 2 K, say), a large range is found where linearity could be accepted. This supports the strategy of an arbitrary enhancement of the prescribed SST anomaly, which is often applied in GCM experiments. However, linearity with respect to a change in the sign of the forcing could only be accepted in very few cases. A “warm minus cold” experiment therefore gives only limited evidence about the respective “cold minus normal” and “warm minus normal” scenarios. An unexpected result is the larger extent of the linear sectors for the standard deviation compared with those for the barotropic streamfunction, which seems to be in contrast with our first guess of more nonlinearity for the standard deviation.

The results so far only give information about the linearity of the behavior of the response pattern amplitude as identified in the reference simulation. An overall linearity with respect to the amplitude of the SST pattern can only be judged if the orthogonal patterns **r**^{⊥}_{exp}*t*) [see (6)] are also taken into account by testing hypothesis ^{D}_{0}*a*^{⊥}_{exp}^{t} = 0). For this test, quite different results emerge for the two quantities. For the barotropic streamfunction (Fig. 15), some sectors in the diagram exist where the response in the experimental simulation is (in the statistical sense) only related to the reference pattern without additional orthogonal components of statistical significance. The distribution of these sectors is, however, not as homogeneous as in the hypothesis *H*^{C}_{0}^{D}_{0}*H*^{C}_{0}^{D}_{0}

The occurrence of an orthogonal response pattern of significant amplitude is an important result, since no information about this response can be extracted from the reference simulation. That means, a local response, which arises in the reference simulation, could be masked by the additional (orthogonal) response in the experimental simulation, even if the amplitude of the (global) reference response behaves linearly.

Figure 17 illustrates the departure from linearity of the response in a linearly related and a linearly unrelated simulation. It shows the difference of the barotropic streamfunction response between the +4-K scenario and 2 times the response in the +2-K scenario (Fig. 7) as well as the difference between the −2-K response and −1 times the +2-K response. Regarding the results of the *H*^{C}_{0}^{D}_{0}^{C}_{0}^{D}_{0}^{C}_{0}*H*^{D}_{0}

## 5. Conclusions

A simplified atmospheric general circulation model has been used to investigate the linearity of the atmospheric response with respect to North Atlantic sea surface temperature anomalies. Although the model is highly simplified, it provides all the characteristics of a comprehensive GCM, like nonlinearity and a number of parameterizations. A comparison of the SGCM response with results of a comprehensive GCM shows a qualitatively good agreement. Seventeen simulations have been performed where we have prescribed different amplitudes of the same SST anomaly pattern. It represents SST anomalies related to North Atlantic interdecadal variability, which have recently received considerable attention (e.g., Hense et al. 1990; Palmer and Sun 1985;Kushnir 1994; Peng et al. 1995; Kushnir and Held 1996).

The linearity of the atmospheric response is examined using a statistical approach. Global response patterns are investigated. Two scenarios at a time are intercompared, one serves as the reference simulation, the other as the experimental run. The response pattern is defined by the time mean difference between the reference and the control (no SST anomaly) simulation. Two variables, the barotropic streamfunction and the 3-monthly standard deviation of the baroclinic part of the streamfunction, are analyzed.

In order to judge the linearity of the response, two hypotheses are checked. First, the amplitude of the reference response pattern behaves linearly with respect to the amplitude of the forcing. Second, no additional orthogonal response patterns are found in the experimental runs. Only if both hypotheses are accepted can the response be interpreted as totally “linear.” The results show that linearity with respect to the amplitude of the reference pattern are to be accepted only for a limited range of variations of the forcing amplitude. Anyway, the linearity hypothesis fails if the sign of the forcing is changed. For the barotropic streamfunction, total linearity is further limited by the occurrence of additional response patterns in the comparative experimental simulations. For the standard deviation, linearity with respect to the amplitude of the reference pattern could be accepted in a slightly more extended range compared with the barotropic streamfunction. Nevertheless, total linearity must be rejected for all cases under consideration, since the change in forcing amplitude is always connected with the occurrence of an additional orthogonal response pattern of significant amplitude.

Some general conclusions can be drawn, although the particular results (like the pattern and the amplitude) are restricted to our special experimental design (model, anomaly pattern, annual mean forcing, etc.). At any rate, the atmospheric response to midlatitude SST anomalies is nonlinear. When using linear models to describe and predict the atmospheric response, the success should be limited as found by Hannoschöck (1984) and Hense et al. (1990), even if some features of the response are found to be consistent with linear theory (e.g., Kushnir and Held 1996). The results of the present study are in agreement with the discrepancies of the results among different model studies as pointed out by Peng et al. (1995) or Kushnir and Held (1996). Some of the differences might be explained by the nonlinear character of the atmospheric response to North Atlantic SST anomalies. In this respect, however, the results of our study also depend on the particular experimental design (e.g., the anomaly pattern and the basic flow).

In order to investigate the atmospheric response to SST anomalies, GCM experiments are necessary, but the reliability of GCM scenario experiments should be improved by the performance of a *set* of several sensitivity experiments, each under slightly differing boundary conditions. To some extent, this approach has already been realized (e.g., Palmer and Sun 1985; Hense et al. 1990). However, the comparison of a warm versus normal SST and a cold versus normal SST scenario (Hense et al. 1990) or simulations with two different forcing amplitudes only (Kutzbach et al. 1977; Palmer and Sun 1985) might not be sufficient. In particular, conclusions based on an identified local response should be carefully drawn, even if they are statistically significant. One has to be aware that a local feature of a global response pattern, identified in one simulation, could be easily masked by a local feature of an additional global orthogonal response arising under slightly different forcing.

It should be pointed out that it is not the aim of this paper to contradict GCM experiments. These experiments are very much needed to improve our understanding of the real climate system. However, when discussing results of such simulations one should be careful about possible misjudgements caused by the nonlinearity of the system as discussed in this paper. In particular, it should not be forgotten that our model and the experimental design, for example, the lack of variation of the SST anomaly pattern, is more likely to underestimate the nonlinear effects compared with comprehensive GCMs or observational data.

## Acknowledgments

We wish to thank Dr. M. Latif, Dr. A. W. Robertson, and Th. Witt, who kindly made the ECHAM3 data available for us. Financial support by the German Ministery for Education and Research (BMBF 07 VKV 01/1-6/18) is gratefully acknowledged. We also wish to thank Dr. P. James for carefully reading the first version of our manuscript more than once.

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## APPENDIX

### Model Description

#### Dynamics

The general circulation model developed at the University of Munich is a conventional two-layer spectral model using the dry primitive equations formulated for a rotating sphere. With pressure as the vertical coordinate, it determines the natural logarithm of surface pressure, temperature, vorticity, and divergence for the lower (800 hPa) and upper (400 hPa) troposphere by the assumption that the vertical velocity vanishes at 200 hPa (i.e., at the tropopause). The horizontal resolution is T21 and approximately 5.6° × 5.6° in the corresponding Gaussian grid. Orography is included. The dynamical core is adopted from the Hamburg (Portable University Model of the Atmosphere, PUMA), which is based on that described by Hoskins and Simmons (1975) and modified by James and Dodd (1993). An additional scheme has been introduced to guarantee the conservation of energy and angular momentum within the finite-difference formulation using isobaric coordinates instead of *σ* coordinates as in the PUMA model.

#### Parameterizations

Hyperdiffusion

A scale-selective hyperdiffusion of order 6 is used in the prognostic equations of temperature, divergence, and relative vorticity for computational reasons. It parameterizes the cascade of energy and enstrophy onto subgrid scales and its subsequent dissipation. The

*e*-folding time for the wavenumber 21 is 6 h.Radiative–convective heating

The diabatic heating is determined by the highly parameterized radiation and convection schemes developed by Eliasen (1982a) and slightly modified by Sausen (1989). The atmosphere is driven by the incoming solar radiation*Ŝ*and the prescribed climatological SST. Following Sausen (1989), the net heating rates areandwhere {*ϵ*_{l},*l*= 0, 1, 2,*s*} and {*α*_{l},*l*= 1, 2,*s*} denote the shortwave absorptivities and the longwave emissivities. Here, { ,*α̂*_{l}*l*= 0, 1, 2} are the absorptivities for longwave radiation from the corresponding level below. The dynamically passive stratosphere (level 0) above the rigid lid at 200 hPa is assumed to be in radiative equilibrium with the upper troposphere. The numerical values of all these parameters are adopted from Sausen (1989) and do not depend on isobaric coordinates. The quantities*g,**c*_{p},*σ*_{B}, and*T*denote the earth’s gravity acceleration, the isobaric heat capacity of the air, the Stefan–Boltzmann constant, and the temperature, respectively.The heating rate connected with the convective adjustment process is represented by a linear damping term for temperature differences greater than Γ = 33 K:Eliasen (1982a) has found that decay rates

*τ*_{c}of about 2 days help to avoid any convective instability.Albedo

The shortwave absorptivity*ϵ*_{s}of the surface is calculated aswhere*α*is the planetary albedo. Its parameterization is taken from Oerlemans and Van den Dool (1978) who have treated land and sea separately. The albedo over land without snow and ice is fixed to 0.31, over open sea it depends on the sun’s annual mean elevationThe ice-covered fraction of the sea surface and land increases from 0 at the melting point to 1 within a range of −20 K, following an arctan profile. No matter whether or not the land is covered by ice, its snow cover increases quadratically from 0 in areas warmer than 288.15 K to 1 in those colder than 233.15 K. The albedo over an ice-covered area is 0.54, whereas over snow-covered wooded zones and permanent high-level ice sheets it has been set to 0.41 and 0.54, respectively.Surface fluxes

A simple Ekman drag term in the momentum equations for the lower troposphere is used to represent surface friction, where the stress is assumed to act in the direction of the low-layer wind rotated through a certain angle*γ*to the left (right) in the Northern (Southern) Hemisphere:with*τ*_{s}*ρ*_{0}*C*_{D}_{s}_{s}and*ρ*_{0}= 1.225 kg m^{−3}as the reference surface density.Following Corby et al. (1972) and Gilchrist (1979) this cross-isobar angle of the surface wind depends on the thermal stratification and the land–sea mask as well as the drag coefficient. In the simulations presented here, we have assumed in addition that the latter increases over steep mountains with topographic height*h*_{s}. For unstable stratifications, that is, for negative temperature differences between the air and the surface, we haveand otherwiseThe upward sensible heat flux is given bywith*H*_{s}*ρ*_{0}*c*_{p}*C*_{H}**v**_{s}*T*_{s}*θ*_{n}where*C*_{H}*C*_{τ}*θ*_{n}means the lower-tropospheric potential temperature.Land and sea-ice temperatures

The prognostic equation of the land and sea-ice surface temperatures iswhere*C*_{s}is the heat capacity per area. In the numerical experiments presented here, it has been treated as a constant with values of 4.32 × 10^{6}J m^{2}K^{−1}for land (Eliasen 1982b) and 2.09 × 10^{5}J m^{2}K^{−1}for a 10-cm ice layer. The heat flux into the soil is neglected, whereas the oceanic heat flux is parameterized aswhere*λ*= 2 W (m K)^{−1}is a heat transfer coefficient,*T*_{f}is the freezing temperature of sea water, and*h*_{ice}is the ice thickness fixed at 2(1) m in the Northern (Southern) Hemisphere.To solve (27), the longwave heat flux is linearized with respect to

*T*_{s}at the previous time step (see, e.g., Deutsches Klimarechenzentrum 1994).