The Extratropical Signal Generated by a Midlatitude SST Anomaly. Part I: Sensitivity at Equilibrium

Nicholas M. J. Hall Department of Atmospheric and Oceanic Sciences, and Centre for Climate and Global Change Research, McGill University, Montreal, Quebec, Canada

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Jacques Derome Department of Atmospheric and Oceanic Sciences, and Centre for Climate and Global Change Research, McGill University, Montreal, Quebec, Canada

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Hai Lin Department of Atmospheric and Oceanic Sciences, and Centre for Climate and Global Change Research, McGill University, Montreal, Quebec, Canada

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Abstract

A simple GCM (SGCM) is constructed by adding empirically derived time-independent forcing terms to a dry primitive equation model. This yields a model with realistic time-mean jets and storm tracks. The SGCM is then used to study the equilibrium response to an imposed heating anomaly in the midlatitude Pacific, meant to represent an anomaly in the sea surface temperature. Using the SGCM’s own climatology as a basic state, the same model is then used to find the time-independent linear response to the same heating anomaly. The difference between the two responses is clearly attributed to the forcing due to anomalous transient eddies.

The sensitivity of the response to the strength and vertical profile of the heating, and to the presence of the wind speed in the surface flux parameterization, is explored. It is found that for a reasonable range of heating amplitude the transient eddy forcing is proportional to the heating and the responses to heating and cooling are almost antisymmetric. The antisymmetry breaks down at large amplitude. The vertical profile of heating has a small but systematic effect on the response: deeper heating leads to stronger equivalent barotropic features. The inclusion of wind speed in the surface flux parameterization alters the response mainly by virtue of altering the basic model climatology, rather than by any local effect on the heating.

The position of the heating anomaly is varied in both latitude and longitude to gain insight into the possible effects of systematic errors in GCMs. The time-independent linear response tends to move with the heating, but the eddy-driven nonlinear part remains relatively fixed and varies only in amplitude. The heating perturbation slightly modifies the first empirical orthogonal function of the model’s internal low frequency variability. The response projects strongly onto this pattern and the probability distribution function of the projection is significantly skewed.

Corresponding author address: Nicholas M. J. Hall, Laboratoire des Ecoulements Géophysiques et Industriels, BP53, 38041 Grenoble Cedex 9, France.

Email: nick.hall@hmg.inpg.fr

Abstract

A simple GCM (SGCM) is constructed by adding empirically derived time-independent forcing terms to a dry primitive equation model. This yields a model with realistic time-mean jets and storm tracks. The SGCM is then used to study the equilibrium response to an imposed heating anomaly in the midlatitude Pacific, meant to represent an anomaly in the sea surface temperature. Using the SGCM’s own climatology as a basic state, the same model is then used to find the time-independent linear response to the same heating anomaly. The difference between the two responses is clearly attributed to the forcing due to anomalous transient eddies.

The sensitivity of the response to the strength and vertical profile of the heating, and to the presence of the wind speed in the surface flux parameterization, is explored. It is found that for a reasonable range of heating amplitude the transient eddy forcing is proportional to the heating and the responses to heating and cooling are almost antisymmetric. The antisymmetry breaks down at large amplitude. The vertical profile of heating has a small but systematic effect on the response: deeper heating leads to stronger equivalent barotropic features. The inclusion of wind speed in the surface flux parameterization alters the response mainly by virtue of altering the basic model climatology, rather than by any local effect on the heating.

The position of the heating anomaly is varied in both latitude and longitude to gain insight into the possible effects of systematic errors in GCMs. The time-independent linear response tends to move with the heating, but the eddy-driven nonlinear part remains relatively fixed and varies only in amplitude. The heating perturbation slightly modifies the first empirical orthogonal function of the model’s internal low frequency variability. The response projects strongly onto this pattern and the probability distribution function of the projection is significantly skewed.

Corresponding author address: Nicholas M. J. Hall, Laboratoire des Ecoulements Géophysiques et Industriels, BP53, 38041 Grenoble Cedex 9, France.

Email: nick.hall@hmg.inpg.fr

1. Introduction

The idea that knowledge of the sea surface temperature (SST) in middle and high latitudes might be useful in predicting the state of the atmosphere some months in advance dates back to the work of Namias (1959, 1964). In these papers he draws attention to a relationship between Pacific SSTs and anomalous flow over North America, and between Atlantic SSTs and anomalous flow over Europe. The nature and strength of these relationships, and their utility for forecasting on monthly to seasonal timescales has been the object of a great deal of observational and modelling work in the ensuing years. Observational analyses, starting with the work of Namias and also Ratcliffe and Murray (1970), have evolved using increasingly sophisticated techniques (Wallace et al. 1992; Peng and Fyfe 1996; Czaja and Frankignoul 1999) to extract a significant signal. However, a difficulty remains in obtaining reliable information about the causal links between the SST and the atmospheric flow. A statistical forecasting approach (Barnett et al. 1984; Barnett and Preisendorfer 1987) yields low levels of skill attributable to the SST alone. A clear picture of the nature of the problem has yet to emerge.

To provide some perspective and fix ideas, it is useful to compare this problem with the more familiar problem where the midlatitude flow is forced by tropical heating. The El Niño signal accounts for 43% of the variance of monthly mean Pacific SST anomalies (Desser and Blackmon 1995). The atmospheric response manifests itself in midlatitudes as a positive “Pacific–North American” (PNA) pattern (Horel and Wallace 1981). The cause and the effect are well separated and the tropical SST can thus be thought of as an “external” forcing. The signal is relatively strong and can potentially benefit seasonal forecasts in a statistical sense (Kumar and Hoerling 1995; Derome et al. 2001).

Interannual variability of midlatitude SSTs also constitutes a fairly strong signal. Desser and Blackmon (1995) find that the second most important pattern of SST variability is centerd in the northwest Pacific, and explains 11% of the variance. Can this also be viewed as “external forcing?” In this case the relevance for forcing the atmosphere is not as clear. The variance is found to increase when the atmosphere leads the ocean by two to three weeks (Desser and Timlin 1997). Modelling studies in which the extratropical SST is specified and the atmospheric response is sought are therefore unable to provide a complete picture, although they might provide useful information to a forecast system for a limited time, since the characteristic decay timescale for the SST anomaly is on the order of months (Frankignoul 1985), which is longer than the predictability limit for atmospheric flows. Barsugli and Battisti (1998) have pointed out that this type of study tends to give an underestimation of the strength of atmospheric variability, and an error in the sign of the vertical heat flux over the SST anomaly. Bretherton and Battisti (2000) have further cautioned that although ensemble integrations with atmospheric GCMs forced by observed time-varying SSTs are likely to perform well in terms of correlations with observed atmospheric signals, this does not necessarily imply useful forecast skill beyond seasonal timescales.

These problems notwithstanding, there is still a large and growing body of experimental work in which some form of extratropical SST anomaly has been specified and a GCM has been integrated to equilibrium under its influence. The level of agreement between these experiments is poor. Most give a fairly weak response for a realistic SST anomaly. That is to say the signal to noise ratio is low, and long integrations or large ensembles are needed for significant results. The results themselves differ widely in their spatial patterns. Some authors find that the response to a cold SST anomaly is the opposite of the response to a warm SST anomaly (Ferranti et al. 1994) while others find that this is not the case (Pitcher et al. 1988; Kushnir and Lau 1992). Some find a robust equivalent barotropic response (Palmer and Sun 1985) while others find the response to be baroclinic (Kushnir and Held 1996). Some find that results are not reproducible with different versions of the same GCM (Lau and Nath 1990, 1994) and some find that even with a fixed GCM, the response depends critically on the time of year (Peng et al. 1995; Peng et al. 1997).

The simplest expectation for the midlatitude response to a low-level midlatitude heating comes from the quasigeostrophic scaling arguments of Hoskins and Karoly (1981), which imply a low-level trough downstream, with a baroclinic structure giving an upper-level ridge. Many GCM experiments do not conform to this linear picture and have large amplitude equivalent barotropic features appearing both in the vicinity of the heating and in the far field. The existence and maintenance of these features is often attributed to fluxes of momentum by anomalous transient eddies.

If the results from equilibrium experiments with specified SSTs display such a large degree of sensitivity, it is pertinent to ask where this sensitivity comes from. Some of the following factors may be important:

  • the strength and sign of the heating or the degree of nonlinearity in the response;

  • the extent to which the response modifies the forcing;

  • the nature of the surface heat flux parameterization and the vertical profile of heating;

  • the basic state or model climatology (the jets);

  • the model representation of transient eddies (the storm tracks);

  • the position of the heating relative to the model’s climatological jets and storm tracks;

  • the model’s internal modes of low frequency variability.

There is clearly a large degree of overlap among these factors, especially when considering the position of the forcing and the related issues of transient eddies and low-frequency variability. To explore all these sensitivities comprehensively would require many standardized experiments with many GCMs. Furthermore, some of these points are difficult to analyze in GCM experiments. It may be difficult, for example, to find out precisely how the results differ from the linear limit. It may be difficult to isolate the influence of transient eddy forcing. Such difficulties have led some authors to supplement their GCM work with diagnostic modelling using reduced dynamical models. Ting (1991) and Ting and Peng (1995) have used stationary wave models to calculate the time independent linear responses to certain forcing profiles, effectively removing the effect of transient eddy fluxes from the solution. Peng and Whitaker (1999) use a combination of linear models to diagnose the transient eddy forcing and its effect independently of the GCM, successfully accounting for a seasonal dependence in the GCM’s response.

In this paper we use a system of reduced dynamical models instead of a GCM to explore further some of the sensitivities listed above. The advantage of this approach is that dynamical mechanisms are more easily isolated, and since the model used is computationally cheap, many experiments can be performed. The limitation of this approach is that the physical parameterizations present in GCMs are replaced by empirically derived terms, so some potentially important physical feedback mechanisms may be absent from our analysis.

Experiments are performed with an idealized heat source placed in the midlatitude Pacific. The equilibrium response to this heat source is found using a “simple GCM” (SGCM) based on the dry primitive equations with linear damping and time independent forcing [details of the SGCM can be found in Hall (2000)]. The time-independent linear response to the same heating is then found using the same model in a “linear perturbation” mode. This is achieved by calculating a new basic forcing with reference to the SGCM’s own time mean flow. It leads to a particularly clean comparison between the SGCM equilibrium solution and the time-independent linear solution. [A similar hierarchical modelling approach has been adopted recently by Hall and Derome (2000) using the SGCM to study the remote response to tropical heating.]

Section 2 of this paper describes the model and gives details of how the forcing terms are calculated. The SGCM’s climatology is shown and compared with observational analyses. In section 3 we begin to explore the sensitivities listed above, with results pertaining to the strength of the heating, its vertical structure, the effect of cooling and the potential effect of the low-level wind on the vertical heat flux. Further results are given in section 4 concerning the position of the heating relative to features of the model climatology, and in section 5 the response is characterized in terms of its projections onto the model’s internal modes of low frequency variability. The results are summarized and discussed in section 6.

In the companion paper, Hall et al. (2001) consider the nonequilibrium problem by using the same model to perform a large number of ensemble integrations from a wide variety of initial conditions. Conclusions are drawn in that paper about the practical issue of the influence of midlatitude SST anomalies on forecasts of up to 3 months and about the relevance of equilibrium results in that context.

2. Model perturbation experiments

a. A simple GCM

All the numerical experiments presented in this paper, and in the companion paper (Hall et al. 2001), are based on the dry spectral primitive equation model developed by Hoskins and Simmons (1975). The version used here differs from the original in some details. It has a split time scheme, so that physical processes such as diffusion and diabatic forcing are not passed through the semi-implicit timestep. It also has an angular momentum conserving vertical scheme due to Simmons and Burridge (1981). The resolution used here is triangular 31, with 10 equally spaced sigma levels. Dissipation is in the form of a scale selective ∇6 hyper diffusion with a timescale of 12 h at the smallest scale, applied to vorticity, divergence, and temperature. A level-dependent linear damping is also imposed on temperature and momentum. The damping is strongest below 800 mb, where subgrid scale turbulent transfer of heat and momentum in the atmosphere is strongest. Table 1 gives damping timescales for temperature and momentum at each model level. These values compare well with deductions from budget studies such as Klinker and Sardeshmukh (1992) and bring the time mean atmospheric state close to neutral stability (Hall and Sardeshmukh 1998). The sensitivity to all these damping parameters is explored fully by Hall (2000).

The only other diabatic term in the model equations is an empirically derived time-independent forcing, applied to all model variables at all levels. For the SGCM the forcing is based on correcting the systematic errors arising from a sequence of one time step integrations of the primitive equations when initialized with observational analyses, following a method first proposed by Roads (1987) and also used in a quasigeostrophic setting by Marshall and Molteni (1993) and Lin and Derome (1996).

Consider an observed atmospheric state, represented by a vector of coefficients Φ in some basis. The time evolution of Φ is given by
i1520-0442-14-9-2035-e1
where N is a nonlinear operator, in our case the primitive equations with some linear damping, and f represents all the processes not captured by N, including the effect of external (boundary) forcing and internal processes such as condensation heating. We approximate Eq. (1) for a model state vector Ψ, by assuming that f(t) can be replaced by the constant vector g:
i1520-0442-14-9-2035-e2
Suppose we set g = −N(Φ), where overbar denotes the mean over many observations Φ, which may come from a long time series. Provided there is no significant trend in the time series, this is equivalent to setting g = f since the left-hand side of (1) has a time-mean of zero. The constant forcing g can be evaluated as a function of space by integrating the primitive equation model (including the damping terms) for one time step from a sequence of observed states. Further details are given in Hall (2000). The data are taken from 9 yr (1980/81–1988/89) of half-daily observational analyses for the winter season, December, January, February (DJF) from the European Centre for Medium-Range Weather Forecasts (ECMWF). The seasonal trend is negligible compared to the other terms in the time-mean of (1).

The long runs of the SGCM presented below are based on 1700-day integrations, starting from the 9-yr mean observed DJF climatology (the results do not differ significantly from those obtained by starting from a resting isothermal state). Results are shown using the mean of the last 1400 days, discarding the first 300. This is ample time for the model climate to reach a statistical equilibrium. Figure 1 shows how well the SGCM performs compared to the ECMWF analyses. The jets and storm tracks, shown here by the 250-mb zonal wind and the standard deviation of the high-pass filtered 550-mb geopotential height, are well positioned and have close to the right amplitude. The Atlantic jet is slightly too strong and the transient eddies are slightly too weak, but the performance of the model compares well with that of many GCMs and is clearly adequate for our purposes. The main systematic error is in the strength of the southern hemisphere storm track, which is unlikely to affect our results.

If a heating perturbation h is added to the right-hand side of (2) a long-term mean at statistical equilibrium will give
i1520-0442-14-9-2035-e3
where Ψc and Ψp describe the atmospheric state for the control and perturbed experiments. Expanding the model operator N about the mean state Ψ we can define
NΨNΨLΨΨEΨΨ
where Ψ′ = ΨΨ, LΨ is a linear operator (the linearization of N about the state Ψ) and EΨ is nonlinear to the same degree as N but contains no linear terms. Equation (3) can now be written as
i1520-0442-14-9-2035-e5
The second term on the right-hand side can be viewed as an additional perturbation forcing arising from anomalous transient eddies. In the SGCM this can be diagnosed purely from knowledge of the two time-mean climates, by evaluating the left-hand side of (5). This is because all the “physics” in the model is linear.

b. Time-independent linear solutions

The equilibrium climate perturbation ΨpΨc satisfies a complicated balance [Eq. (5)] in which nonlinear time-independent terms, and nonlinear “forcing” by time-dependent transient eddies may be important. To understand this balance further it is useful to reduce it to the linear form, based on the SGCM’s climatology Ψc. This is achieved by setting the basic forcing g = −N(Ψc). An integration of (2) with initial condition Ψ = Ψc and a perturbation forcing h will then conform to
i1520-0442-14-9-2035-e6
Provided Ψ′ = ΨΨc remains small, the second term on the right-hand side can be neglected. Dropping the subscript Ψc we obtain the time dependent linear model,
i1520-0442-14-9-2035-e7
With this choice for the forcing g, and by ensuring that h remain small, the full nonlinear model can be used to integrate Eq. (7).
Time-independent solutions to (7), on the other hand, are harder to find. The linear inversion,
ΨL−1h
requires knowledge of L. Such linear “stationary wave” models have been developed by Valdes and Hoskins (1989) and Ting and Held (1990) but are only feasible with truncated resolution. It is also possible to arrive at the solution of (8) by time stepping (7), but only when the basic state Ψc is stable, that is, when L has no eigenvalues with positive real parts. However, the damping parameters listed in Table 1 result in a basic state that is slightly unstable. A modification to the time stepping method was therefore implemented in which the operator was effectively stabilized, without affecting its eigenmodes, by subtracting multiples of the identity matrix. This procedure is described in the appendix. The solutions obtained will be referred to as the time-independent linear solutions (TILS).

3. Sensitivity to heating specification

a. Strength and sign

The heating anomaly imposed is meant to simulate the effect of a strong midlatitude SST anomaly, with magnitude of the order of 3°C. How this translates into a vertical profile of heating depends on the parameterization of the vertical heat flux and also on the response of the atmosphere, in terms of temperature and winds. Peng and Whitaker (1999) point out that the initial heating rate for an SST anomaly (SSTA) acting on an unperturbed climatological flow is greater than the equilibrium heating rate established once the atmosphere has adjusted to the SST anomaly. They accordingly use an accentuated heating rate for their linear modelling compared to the rate diagnosed from their GCM. We follow suit, and impose a heating with a maximum vertical average value of 2.5°C day−1. This would correspond to a precipitation anomaly of 10 mm per day if all the heating came from condensation. The heating is centerd in the north west Pacific at 40°N, 160°E and has an elliptical squared cosine distribution in latitude and longitude, with a north–south extent of 40° and an east–west extent of 80°. The vertical profile is proportional to σ4.

Figures 2a,b shows contours of the heating that defines the fixed forcing perturbation h. In response to this heating, the SGCM adjusts to a new equilibrium climate. This is best illustrated by writing simplified temperature equations for the control climate, with temperature Tc and the perturbed climate with temperature Tp:
i1520-0442-14-9-2035-e9a
where A represents local temperature tendencies due to adiabatic processes such as advection and expansion and also due to diffusion, D is the linear damping of temperature (Table 1), and g and h denote the local heating to maintain the basic state and the perturbation, contained in g and h, respectively. Time averaging and subtracting we obtain
i1520-0442-14-9-2035-e10
The right-hand side of (10) is the effective heating perturbation felt by the SGCM once the damping terms have equilibrated. It is appropriate to include this damping term in our definition of the effective heating. Most surface flux parameterizations actually include it implicitly. To illustrate this point it is sufficient to note that the linear terms in on the right-hand side of (9a) could have been written as D(T∗ − Tc) where DT∗ = g. The perturbation heating could have been added in (9b) in terms of a temperature anomaly D(T∗ + SSTA − Tp) where DSSTA = h. The result is the same. The heating anomaly is thus parameterized as D[SSTA − (TpTc)], which has a time mean equal to the right-hand side of (10). This is shown in Figs. 2c,d for our SGCM equilibrium response. The magnitude is approximately half that of h and the vertical profile is slightly deeper. Note also the appearance of weak downstream cooling. This could be viewed as the interaction of warm air with a relatively cool ocean to the east of the SST anomaly.

The SGCM response itself is shown in Fig. 3 for the geopotential height anomaly at two levels, 550 mb and 950 mb. The low-level downstream trough is a shallow feature, extending up to about 800 mb. At upper levels there is a ridge, somewhat farther downstream than the trough, with a peak amplitude at about 300 mb and an almost equivalent barotropic structure at its center. Farther afield there are more equivalent barotropic structures of the same amplitude as the local response: a low over the pole and a ridge over Europe.

The time-independent linear solution (TILS) is also shown in Fig. 3 for comparison. Many of the features of the SGCM response appear in the linear solution, including the low-level trough and the upper-level ridge. The main difference between the two is that the ridge is stronger in the SGCM and farther east, and that the nonlocal response seen in the SGCM is not reproduced in the TILS.

To discover how these differences in the response are maintained we must diagnose the transient eddy forcing. As shown in Eq. (5), this can be done relatively easily with the SGCM by looking at the action of the nonlinear operator N on the perturbed and unperturbed climate states Ψp and Ψc. This yields tendency terms for all atmospheric variables contained in the state vector. For example, the momentum flux convergence calculated this way is illustrated in Fig. 4a in terms of a transient eddy “source” of geopotential height. Is this source consistent with the generation of the extra features of the SGCM response noted above? This is clearly not a question that can be answered by casual inspection, even though some features of the source look as though they might give rise to anomalies of the right sign. To get a clearer idea, we must use the entire right-hand side of (5), including sources of heat and divergent flow, as a perturbation forcing for the linear model. The corresponding TILS for the 550-mb height is shown in Fig. 4b. Comparing this with Fig. 3a, we see that the agreement is very good, implying that the convergence of anomalous transient eddy fluxes is the dominant player in the full nonlinear equilibrium solution. Any differences between the two solutions must arise either from the time-independent nonlinearity in the left-hand side of (10) or from inaccuracies in the extrapolation used to find the TILS (see appendix). In this case the response is slightly too strong, perhaps implying that the time-independent nonlinearity acts to damp the Pacific ridge.

The manner in which the SGCM response departs from linearity can be further examined by changing the amplitude of the forcing. In Fig. 5 experiments are shown with the perturbation heating set at h, 2h,h, and −2h. The contour interval has been doubled in the cases with double-amplitude forcing so the four panels are directly comparable with one another and also with the TILS shown in Fig. 3c. Reversing the sign of the heating anomaly gives a response of opposite sign that has some features in common with the standard heating experiment. The main centre of the response is shifted to the east and intensified in both cases compared to the TILS. Eddy-forced structures of opposite sign appear with a similar spatial distribution, although they are not as strong in the cooling experiment. In this context, it would be misleading to label the difference between the heating and cooling experiments as the “linear part” of the signal, and their sum as the “nonlinear” part, as has been done in the case of El Niño versus La Niña (Hoerling et al. 1997). The nonlinearities in this case are for the most part antisymmetric, and therefore all the more difficult to expose through traditional diagnosis of GCM output. When the amplitude of the heating is increased, the amplitude of the response increases in direct proportion, and the pattern remains almost exactly the same. It appears that the nonlinear effects are truly saturated at these amplitudes and the transient eddy forcing is a linear function of the heating anomaly. This is not so for the cooling experiments. When the cooling is doubled the amplitude of the downstream trough is more than doubled, and the remote teleconnections effectively disappear. Further linear solutions show that the transient eddy forcing, as diagnosed from each of these runs, is not responsible for the intensification of the trough. Selected contours from eddy-forced TILSs have been added to Fig. 5, and even for double-amplitude perturbations it can be seen that the symmetry between heating and cooling is preserved in the TILS for the local response, if not for the teleconnections. The intensification of the trough must therefore be attributed to time-independent nonlinear interactions.

b. Vertical structure

The linear response to a shallow midlatitude heat source in a westerly flow is a warm surface trough downstream of the heating as described by Hoskins and Karoly (1981). This is reproduced in our TILS and to some extent in our SGCM experiments, and in both cases it gives way to a ridge at upper levels. In this section we briefly examine the sensitivity of this result to the depth of the heating profile. Previous workers have used several shallow heating profiles for stationary wave studies, from the very shallow profile proportional to σ8 used by Peng and Whitaker (1999) to the deeper profile used by Ting (1991) and Ting and Peng (1995) proportional to σ4 simπσ, which has a maximum at σ = 0.82 to represent condensation in shallow convection, and the deep convective profile used in studies of tropical forcing proportional to (1 − σ) sinπσ (Ting and Held 1990; Hall and Derome 2000), which has a maximum at σ = 0.35.

Results using all these profiles and our standard σ4 profile are shown in Fig. 6 as a vertical section of the geopotential height anomaly at 40°N. In each case the central vertical average heating rate is fixed at 2.5°C per day. Figures 6a–c represent a reasonable range of shallow profiles. The solutions have the same basic structure. As the heating gets deeper the surface trough gets slightly stronger and moves slightly eastward. At the same time the upper-level ridge also strengthens and moves west, aligning itself with the surface trough. In the deep convective heating case, Fig. 6d, the trough extends into the midtroposphere where northerly flow is still needed to balance the heating. A ridge appears in the upper troposphere, above the heating maximum. The pattern of global response is not very sensitive to the changes in shallow heating profile, but as the heating becomes deeper the response is stronger both locally and in the equivalent barotropic teleconnections in the polar and Atlantic regions. For the deep convective limit the teleconnections are stronger than the local response.

c. Flux parameterization

In section 2a it was pointed out that specifying a fixed forcing in a model with linear damping is equivalent to specifying a forcing based on linear restoration to some fixed profile. It was further argued that adding a fixed heating anomaly is equivalent to adding a heat flux proportional to the difference between an SST anomaly and the resulting atmospheric temperature anomaly. However, most surface flux parameterizations depend on bulk aerodynamic formulas in which the surface wind speed also plays a part. The flux is usually proportional to the wind speed. Adding this factor into the model introduces a nonlinearity into the forcing, making the equations more complicated, but it is worthwhile examining the sensitivity of the solution to this effect. Let the surface heat flux be proportional to the wind speed s in the model’s lowest layer. So the term D(T∗ − T) now becomes (α + βs)D(T∗ − T) where α and β are constants. Equation (9a) now becomes
i1520-0442-14-9-2035-e11
The extra term q is a constant source term for temperature. A similar source must be calculated for all model variables in the same manner that g was calculated before, to ensure that the model still conforms to the forcing strategy embodied in Eqs. (1) and (2) now that the model operator N has effectively been changed. The basic model climatology Ψc will also change, and is denoted here as Ψd. Having found Ψd, we can use it as a reference for further anomaly experiments. However, the two interpretations given above for Eq. (9b) are no longer equivalent. A “fixed forcing perturbation” experiment now corresponds to
i1520-0442-14-9-2035-e12
whereas a “fixed SST anomaly” experiment corresponds to
i1520-0442-14-9-2035-e13
The difference between the climatologies Ψf and Ψs tells us how important the addition of the wind speed effect is. This difference can be expressed as a heating perturbation as in Eq. (10), by time averaging and subtracting (12) from (13), giving
i1520-0442-14-9-2035-e14
The right-hand side of (14) is the modification to the right-hand side of (10) made by introducing the wind speed effect. The first term shows that the heating perturbation is increased in porportion to the wind speed anomaly. The second term shows that the change of temperature associated with the inclusion of the wind speed effect gives rise to an additional damping. The third term involves changes in wind speed and their products and covariances with changes in temperature.
The balance of terms in (14), and the resulting solution will depend on the model and on the choice of parameters α and β. Results for a practical test with the SGCM are shown in Fig. 7. Here, α and β have been chosen such that
i1520-0442-14-9-2035-eq1
where s0 is the observed local climatology and s2, the wind speed anomaly at which the flux is doubled, is set to 12 m s−1. This gives maximum sensitivity under the constraint that s2 > max(s0). Figure 7a shows the difference the wind speed effect makes to the control climatology ΨdΨc [Eqs. (11) and (9a)]. Figure 7b then shows the effect of our standard “forcing perturbation” ΨfΨd [Eqs. (12) and (11)] and Fig. 7c shows the additional change ΨsΨf when an “SST perturbation” is considered instead [Eqs. (13) and (12)]. This last effect is small compared to the signal, and is only statistically significant at around the 80% level. Furthermore, comparing Fig. 7b with Fig. 3a we can see that the biggest change comes not from the additional forcing terms associated with the wind speed effect, but from the change in basic climatology shown in Fig. 7a. The consequences of including the wind effect are therefore likely to be very indirect, and difficult to distinguish from other factors affecting model climatology.

4. Sensitivity to heating location

A major factor in the variations seen in GCM responses to SST anomalies is undoubtedly the variations in their climatologies. Different GCMs have different systematic errors, and offer different jets and storm tracks for the perturbation to interact with. It is possible in principle to mimic the behavior of a range of GCMs with the SGCM, but such an endeavor is beyond the scope of this study. Instead, some pertinent information can be found by moving the heating perturbation, so that it interacts with different parts of the SGCM’s climatology. The heating can be moved north or south, for example, to see how it interacts with the two flanks of the jet. Stationary wave modelling studies by Ting (1999, personal communication) show that there is some degree of sensitivity to such a rearrangement, so it is natural to try and extend these results to a model that includes transient eddies, but still in an equilibrium setting. Nonequilibrium experiments with a fixed heat source but with multiple initial conditions address a similar question, and are the subject of the companion paper, Hall et al. (2001).

Eight more long integrations of the SGCM have been carried out in addition to the standard experiment shown in Fig. 3. The results for the 550-mb geopotential height anomaly are shown in Fig. 8 and the corresponding TILSs are shown in Fig. 9. The experiments are identical except that the center of the heating anomaly has been moved in each case, to cover a grid with latitudes 30°, 35°, and 40°N (the southern flank, middle, and northern flank of the climatological jet) and longitudes 160°E, 180°, and 200°E (the beginning, middle, and end of the storm track).

A great variety of responses can be seen depending on the position of the heating. As the heating is moved south across the Pacific jet core, the downstream ridge-like response at 550 mb moves south and west so that it sits directly over the heating at 30°N. At this latitude the response has become more baroclinic. This is also seen in the TILS. The equivalent barotropic low at the pole also moves south, but it appears in the linear part of the solution at this latitude rather than as a purely eddy-driven feature.

Moving the heating eastward at each latitude gives rise to relatively little change in the position of the upper-level ridge in the SGCM. It appears to lock on to a preferred location in most cases. In the TILS, on the other hand, the upper-level ridge response shifts eastward by about the same amount as the heating anomaly. In both cases, the ridge response is increasingly baroclinic as the heating is moved south and east, as a strong surface trough emerges, linked to a deep equivalent barotropic low in the Gulf of Alaska. This feature also appears in the TILS, so it does not depend on anomalous transient eddies for its maintenance.

The dependence of eddy-driven structures (those that do not appear in the TILS) on the position of the heating is better characterized by their amplitude than their location, which is relatively invariant. They tend to be excited when the heating is either in the center or on the northern flank of the jet. When the heating is on the southern flank of the jet the results resemble the TILS more closely, as wave activity is directed equatorward, and presumably absorbed.

5. Projection onto internal modes of variability

The response patterns shown in Fig. 8 are derived from the time-means of two long integrations, each in statistical equilibrium. They show how the mean climatology has changed in response to anomalous heating, but they do not show how the time variability has been affected. Internally generated low-frequency variability in the model is driven by the higher frequency transient eddies. Some of the features of the response patterns shown are also driven by transient eddies, with the result that the response patterns do not shift their position systematically as the heating anomaly is moved. These two facts imply that there may be a connection between the model’s preferred patterns of low-frequency variability and the response to midlatitude heating perturbations, as discussed by Ferranti et al. (1994) and Peng and Robinson (2001). In this section we explore this possibility.

First, it is necessary to find the dominant patterns of low-frequency variability in the model. To this end, an extended integration was carried out to give 1000 monthly mean anomalies from the time-mean climatology. The first three EOFs (the three eigenvectors of the spatial correlation matrix with the three largest eigenvalues) of these anomalies are shown in Fig. 10, normalized for unit spatial variance. The first EOF explains 21% of the temporal variance and has a strong zonally symmetric component, with a dipole over northern Europe and centers of action over Canada and eastern Siberia. It has features in common with many of the the response patterns shown in Fig. 8. The second EOF explains 19% of the variance and consists of a wavetrain over the Pacific–North American region. The response patterns to heating in the east Pacific also display a wavetrain with similar phase and therefore have some projection onto this EOF. The third EOF explains 18% of the variance and contains an intense centre of action in the Gulf of Alaska. This is the same as the position of the equivalent barotropic low generated when the heating is south of the Pacific jet. This latter feature is also present in the TILS, suggesting that in this case the associated mode of variability may be excited directly by the heating perturbation rather than by transient eddies.

The similarities described above between low-frequency variations and the response to heating perturbations are quantified in Table 2, which gives pattern correlations coefficients for the three EOFs with the response to heating perturbations at each of the nine locations. The correlations with EOF1 are large. The response to heating projects strongly onto this pattern for all heating locations. The projection is strongest when the heating is at the latitude of the jet. Correlations are weak for EOF2, reaching maximum amplitude when the heating is downstream of the jet. Correlations for EOF3 are negative when the heating is north of the jet and positive when the heating is south of the jet.

To investigate further the relationship between the mean response to heating and the low-frequency variability, another 1000-month integration was carried out with a heating perturbation in the standard position: 160°E, 40°N. Using both unperturbed (control) and perturbed integrations, monthly mean anomalies were calculated relative to the control climatology and projected onto the first EOF pattern shown in Fig. 10. The frequency of occurrence (number of months) is plotted in Fig. 11 as a bar graph for different amplitudes of the projection. For the control experiment the distribution is approximately symmetrical about zero. The effect of the heating perturbation is to shift the distribution toward positive values as expected from the fact that the mean response to heating projects positively onto EOF1. In addition to the shift, the distribution is skewed so that there is an effective cut-off for large positive values (the skewness is twice the threshold value for significance at 95% confidence for a sample of 1000). This effect is not seen for EOFs 2 and 3 (not shown).

The question remains as to whether the temporal behavior of the low-frequency variability has actually changed, or whether it is just the patterns of variability themselves that have changed. To check this, EOFs were calculated from the 1000-month perturbed run and are shown in Fig. 12. Some differences can be seen, particularly in EOF1, in which the center of action over Alaska has intensified. Using this new EOF as a basis for projection, the probability distribution function of monthly mean anomalies from the perturbed integration, referenced this time to its own climatology is shown in Fig. 11c. Naturally the distribution now has a mean of zero, but it remains significantly skewed in the same sense as before.

It appears, therefore that the introduction of a heating perturbation has altered patterns of low-frequency variability, and has also pushed the climate into a state where it populates positive and negative projections onto its leading EOF unequally. The almost linear behavior of the transient eddy feedbacks shown earlier leads us to speculate that it is longer timescale or “stationary” nonlinearities that govern this process. A similar effect has been observed by Branstator (1992) in which stationary nonlinearity limits the maximum amplitude of projection onto one phase of an EOF but not the other.

6. Summary and discussion

The fact that there is a disparity between different GCMs’ responses to midlatitude SST anomalies calls for some lower-level modelling work to be done, to quantify and explain the various sensitivities involved. This paper is an attempt to do this with a self-consistent modelling approach. The same model with the same parameters is used for linear and nonlinear experiments, so the comparison between the two is reliable. The possible factors affecting the response listed in the introduction are dealt with one by one so we can see their relative importance. Of course, other GCMs might have different sensitivities, but by using the simplest model that includes all the relevant dynamical processes, it is hoped that this paper provides a yardstick by which to measure them.

The basic response to a heating anomaly in the northwest Pacific takes the form of a low-level downstream trough, with a ridge in the middle and upper troposphere. The difference between the linear response and the nonlinear (finite amplitude) response has been shown to be mainly attributable to the anomalous transient eddies. More specifically, we find that the direct response to the heating results in a weakening of the Pacific jet, particularly in the eastern Pacific, which, in turn, leads to weakened momentum flux convergence at the jet exit, reinforcing the anomaly. This is the way in which nonlinear dynamics changes the pattern of the response. However, once the heating amplitude is large enough for the transient eddies to play their part, we find that if the heating is increased further, the action of the anomalous transient eddies increases in direct proportion. So in the case of heating, the fully developed nonlinear response behaves almost linearly in terms of the forcing. In the case of cooling, to a first approximation it can be said that the transient eddy fluxes have an equal and opposite effect, and thus reinforce the response to cooling in a similar way. So the finite amplitude response to cooling looks much like the opposite of the response to heating. However, as the cooling is increased further, and the momentum fluxes get stronger, the antisymmetry breaks down, and the circulation shifts into a self-maintaining state with very strong westerlies in the eastern Pacific. In summary, the sensitivity to the magnitude of the heating perturbation follows three stages: linear (no transient eddies); nonlinear antisymmetric (the transient eddy response behaves linearly) and nonlinear asymmetric (stationary nonlinearity becomes important). The last phase is probably unrealistic for observed SST anomalies, but the behaviour seen in our SGCM may provide clues towards understanding the behavior of GCMs.

We have gone to some lengths to stress the fact that our approach is equivalent to a model forced by linear“restoration” to an equilibrium state, and that putting a constant anomalous heat source in a model with linear damping is equivalent to parameterizing a flux proportional to the difference between an SST anomaly and the atmospheric temperature anomaly. Having specified the anomalous heat source, we note that the damping term is an important part of the heat budget, and reduces the effective heating by about half. The main difference between our experiments and GCM experiments is that we are able to impose the vertical profile of heating, whereas in a GCM the vertical profile will depend on other parameterizations, such as deep and shallow convection and vertical turbulent heat transfer. Experiments with vertical profiles of different depths show that there is some sensitivity to this factor. Deeper profiles tend to give a bigger response, particularly in the remote, equivalent barotropic part, suggesting that three-dimensional fields of the diabatic heating are an essential part of any attempt to compare GCM results. A more complicated representation of the heating shows that including the wind speed in the parameterization can make an appreciable difference to the response. However, a careful analysis reveals that the difference is unlikely to be easily explained in terms of the heat budget. The effect on the basic model climatology of adding new flow-dependent terms, and the subsequent impact on the response turns out to be more important than any local modification of the anomalous heat flux. If this result carries over to more realistic models, then it may be difficult to draw clear conclusions from experiments with those models in which the surface parameterizations are changed. When considering altered responses to SST anomalies, the effect of a relatively small change in model climatology may overwhelm the effect of a local change in surface flux.

The position of the heating relative to the fixed features of the model climatology (the jets and the storm tracks) is of great relevance to a consideration of the effects of systematic errors in GCMs. The experiments shown in this paper concern rather large displacements of the heating anomaly, and the changes in the response are correspondingly large. The results give a flavor of the type of variations that might be seen with differing GCM climatologies. They can be understood to some extent as two components overlaid: a linear part that moves with the heating and a nonlinear part that is fixed in space, but whose amplitude varies with the position of the heating. In fact the two components interact, but this basic description is consistent with the notion that transient eddy momentum flux convergence anomalies occur in preferred locations. Indeed, this is what leads to preferred patterns of low-frequency variability in the SGCM. The link between low-frequency variability and the response to anomalous heating is illustrated in section 5 in terms of the projection of the latter onto EOFs of the former. The fact that the projections are strong suggests that the link is important. We find that the distribution function for projections onto the leading EOF is skewed in the presence of anomalous heating, as well as shifted. We postulate that it is stationary nonlinearity that controls the process rather than asymmetry in the transient eddy feedback. The appearance of asymmetries in population statistics such as these suggests a link between the variability and the equilibrium response. Whether the one can affect the other remains to be established in cases where the probability distribution function is unimodal and the mechanism put forward by Palmer (1999) does not apply. When interpreting the results from GCMs it should be remembered that differences in their variability may be as important as differences in their mean climates.

In this paper we have concentrated exclusively on the equilibrium problem. It is duly noted here that long integrations are required to achieve statistically significant results when the heating anomaly approaches a realistic magnitude. It is clear from this that any practical benefit that may be derived from knowledge of the midlatitude SST will only be derived in a statistical sense, for people whose interests are long term (i.e., many seasons). We view this work as being more focused on the problem of understanding GCMs and the implications of their systematic errors. In the next paper, Hall et al. (2001) consider some experiments that are directed toward the practical seasonal forecasting problem.

Acknowledgments

We thank Grant Branstator and Prashant Sardeshmukh for stimulating conversations, and the Reading University UGAMP group for originally supplying the model. We also thank the two reviewers for helpful comments. This work was funded by the Meteorological Service of Canada through the Canadian Institute for Climate Studies, and by the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche through the Centre for Climate and Global Change Research.

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APPENDIX

Time-Independent Linear Solutions

The solution to Eq. (7) can be written down by expressing Ψ′ in terms of its projection coefficients Ψj onto the eigenvectors of L, and then expanding in terms of the eigenmodes of L as discussed, for example in the introduction of Ting and Sardeshmukh (1993). If h is constant and there is no initial perturbation Ψ′, then for each eigenmode of L we have
i1520-0442-14-9-2035-ea1
where fj is the projection of h onto the corresponding eigenvector of L and λj is the jth eigenvalue of L, and can be written as λj = σj + j to describe the growth of a normal mode in terms of a growth rate σ and a frequency ω. If σj is negative for all j then all the normal modes of L, associated with the basic state Ψc, are stable and will decay in time. In this case the solution Ψ′ can be found by integrating (7) for a suitably long time. If σj has any positive values then an integration of (7) will eventually be dominated by the mode with the largest value of σ, regardless of the details of h.
The latter is the case for our SGCM climatology Ψc, for which the most unstable normal mode takes the form of a midlatitude baroclinic wave train with a growth rate of σ = 0.085 days−1. Now if an extra damping of strength R is included in the model, and applied equally to all degrees of freedom, this is equivalent to subtracting RI from L, and provided R > 0.085 it will stabilize the system without affecting the structure or order of any of the eigenmodes of L. The jth solution for t → ∞ will be
i1520-0442-14-9-2035-ea2
Now that the system is stable it is possible to arrive at this modified solution by time stepping the model. If ΨR is found for a number of different values of R, we can extrapolate back to R = 0 to find an approximation to the true solution Ψ′.

There is a danger in doing this extrapolation in that the right-hand side of (A2) may not be well behaved in the vicinity of R = ℜ(λj) = σj making the extrapolation potentially unreliable, particularly if ωj is close to zero for σj > 0. Such “resonant” modes also pose problems for stationary wave models which work by direct inversion of L, making the results quite sensitive to the precise form of the damping. In any case, if the forcing h does project strongly onto any fast growing modes, then the relevance of the time independent linear solution becomes questionable.

In this paper we proceed using three values: R = 0.2, 0.3, and 0.4 days−1, to enable a quadratic extrapolation back to R = 0. In all cases the structure of the solution is quite insensitive to R and only the magnitude is affected, giving us some confidence that, to the extent that it is relevant, our extrapolated solution is correct.

Fig. 1.
Fig. 1.

Features of the observed (a)–(c) and modeled (d)–(f) climate. (a) and (d): 250-mb zonal wind, contours 8 m s−1, zero dotted, negative dashed. (b) and (e): 550-mb geopotential height, contours 100 m, tick marks point downgradient at extrema. (c) and (f): high-pass filtered (contribution of 3-day means removed giving approximate 6-day cutoff) root-mean-square variations in 550-mb geopotential height, contours 10 m.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 2.
Fig. 2.

Horizontal structure of heating perturbation at 950 mb and vertical structure at 40°N. Contours degrees per day, zero dotted, negative dashed. (c) and (d) show effective heating, which includes the effect of damping in the equilibrium response (see text).

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 3.
Fig. 3.

Equilibrium SGCM geopotential height response to the heating perturbation shown in Fig. 2 at (a) 550 mb and (b) 950 mb. Contours 20 m, zero dotted, negative dashed. In shaded areas the response is statistically significant at the 99% confidence level. (c) and (d) Show time-independent linear solutions for the same heating perturbation about the SGCM climatology.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 4.
Fig. 4.

(a) Geostrophic source of 550-mb geopotential height due to anomalous transient eddies in response to heating. Calculated as (f/g)∇−2 (vorticity flux convergence). Contours 2 m day−1, zero dotted, negative dashed. (b) Time-independent linear solution as in Fig. 3c but with anomalous transient eddy forcing added.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 5.
Fig. 5.

Equilibrium SGCM 550-mb geopotential height response to heating perturbation (a) as in Fig. 3, (b) with amplitude doubled, (c) with negative amplitude, (d) with doubled negative amplitude. Contours 20 m (a) and (c) or 40 m (b) and (d), zero omitted, negative dashed. Shading denotes significance at 99%. Also shown in (a) nd (c) are the 20 (dotted) and −20 (dot-dashed) contours from time-independent linear solutions with appropriate anomalous transient eddy forcing added. Similar 40 and −40 contours are shown in (b) and (d).

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 6.
Fig. 6.

Vertical structure of geopotential height response at 40°N for four different heating profiles, proportional to (a) σ8, (b) σ4, (c) σ4 sinπσ, and (d) (1 − σ) sinπσ. Contours 20 m, zero dotted, negative dashed. In shaded areas the response is statistically significant at the 99% confidence level.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 7.
Fig. 7.

(a) Change in climatology of SGCM 550-mb height when the wind effect is included in the basic forcing [Eq. (11)] before any heating anomaly is added. (b) Effect of heating perturbation alone [Eq. (12)]. (c) Difference made when the wind effect is allowed to interact with the perturbation heating [Eq. (13)]. Contours 20 m (a) and (b) and 5m (c), zero dotted, negative dashed, shading denotes significance at 99%.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 8.
Fig. 8.

Equilibrium SGCM 550-mb geopotential height response for heating at nine different locations. Contours 20 m, zero dotted, negative dashed. In shaded areas the response is statistically significant at the 99% confidence level. Position of heating denoted by a black circle, with a bar showing the range of positions in latitude and a latitude circle drawn at the latitude of the heating.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 9.
Fig. 9.

Time-independent linear solutions about the SGCM mean climatology for 550-mb response to heating in the same nine locations as in Fig. 8. Contours 20 m, zero dotted, negative dashed. Position of heating marked as in Fig. 8.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 10.
Fig. 10.

First three EOFs of monthly mean anomalies for a 1000-month integration of the SGCM with no heating perturbation. Zero contour dotted, negative dashed.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 11.
Fig. 11.

Frequency distributions for projection of monthly mean anomalies onto the first EOF from 1000-month integrations. (a) Anomalies from unperturbed (control) run relative to mean of control run, projected onto EOF1 from control run. (b) Anomalies from perturbed run relative to mean of control run, projected onto EOF1 from control run. (c) Anomalies from perturbed run relative to mean of perturbed run, projected onto EOF1 from perturbed run.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Fig. 12.
Fig. 12.

First three EOFs of monthly mean anomalies for a 1000-month integration of the SGCM with the standard heating perturbation. Zero contour dotted, negative dashed.

Citation: Journal of Climate 14, 9; 10.1175/1520-0442(2001)014<2035:TESGBA>2.0.CO;2

Table 1.

Vertical profile of linear damping timescales for momentum and temperature.

Table 1.
Table 2.

Pattern correlation coefficients between EOFs of low-frequency variability and time-mean response patterns to perturbation heating at different locations.

Table 2.
Save
  • Barnett, T. P., and R. Preisendorfer, 1987: Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis. Mon. Wea. Rev.,115, 1825–1850.

  • ——, H.-D. Heinz, and K. Hasselmann, 1984: Statistical prediction of seasonal air temperatures over Eurasia. Tellus,36A, 132–146.

  • Barsugli, J. J., and D. S. Battisti, 1998: The basic effects of atmosphere-ocean thermal coupling on midlatitude variability. J. Atmos. Sci.,55, 477–493.

  • Branstator, G., 1992: The maintenance of low frequency atmospheric anomalies. J. Atmos. Sci.,49, 1924–1945.

  • Bretherton, C. S., and D. S. Battisti, 2000: An interpretation of the results from atmospheric general circulation models forced by the time history of the observed sea surface temperature distribution. Geophys. Res. Lett.,27, 767–770.

  • Czaja, A., and C. Frankignoul, 1999: Influence of the North Atlantic SST on the atmospheric circulation. Geophys. Res. Lett.,26, 2969–2972.

  • Derome, J., and Coauthors, 2001: Seasonal predictions based on two dynamical models. Atmos.–Ocean, in press.

  • Desser, C., and M. L. Blackmon, 1995: On the relationship between tropical and north Pacific sea surface temperature variations. J. Climate,8, 1677–1680.

  • ——, and M. S. Timlin, 1997: Atmosphere–ocean interaction on weekly time scales in the North Atlantic and Pacific. J. Climate,10, 393–408.

  • Ferranti, L., F. Molteni, and T. N. Palmer, 1994: Impact of localised tropical and extratropical SST anomalies in ensembles of seasonal GCM integrations. Quart. J. Roy. Meteor. Soc.,120, 1613–1645.

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  • Fig. 1.

    Features of the observed (a)–(c) and modeled (d)–(f) climate. (a) and (d): 250-mb zonal wind, contours 8 m s−1, zero dotted, negative dashed. (b) and (e): 550-mb geopotential height, contours 100 m, tick marks point downgradient at extrema. (c) and (f): high-pass filtered (contribution of 3-day means removed giving approximate 6-day cutoff) root-mean-square variations in 550-mb geopotential height, contours 10 m.

  • Fig. 2.

    Horizontal structure of heating perturbation at 950 mb and vertical structure at 40°N. Contours degrees per day, zero dotted, negative dashed. (c) and (d) show effective heating, which includes the effect of damping in the equilibrium response (see text).

  • Fig. 3.

    Equilibrium SGCM geopotential height response to the heating perturbation shown in Fig. 2 at (a) 550 mb and (b) 950 mb. Contours 20 m, zero dotted, negative dashed. In shaded areas the response is statistically significant at the 99% confidence level. (c) and (d) Show time-independent linear solutions for the same heating perturbation about the SGCM climatology.

  • Fig. 4.

    (a) Geostrophic source of 550-mb geopotential height due to anomalous transient eddies in response to heating. Calculated as (f/g)∇−2 (vorticity flux convergence). Contours 2 m day−1, zero dotted, negative dashed. (b) Time-independent linear solution as in Fig. 3c but with anomalous transient eddy forcing added.

  • Fig. 5.

    Equilibrium SGCM 550-mb geopotential height response to heating perturbation (a) as in Fig. 3, (b) with amplitude doubled, (c) with negative amplitude, (d) with doubled negative amplitude. Contours 20 m (a) and (c) or 40 m (b) and (d), zero omitted, negative dashed. Shading denotes significance at 99%. Also shown in (a) nd (c) are the 20 (dotted) and −20 (dot-dashed) contours from time-independent linear solutions with appropriate anomalous transient eddy forcing added. Similar 40 and −40 contours are shown in (b) and (d).

  • Fig. 6.

    Vertical structure of geopotential height response at 40°N for four different heating profiles, proportional to (a) σ8, (b) σ4, (c) σ4 sinπσ, and (d) (1 − σ) sinπσ. Contours 20 m, zero dotted, negative dashed. In shaded areas the response is statistically significant at the 99% confidence level.

  • Fig. 7.

    (a) Change in climatology of SGCM 550-mb height when the wind effect is included in the basic forcing [Eq. (11)] before any heating anomaly is added. (b) Effect of heating perturbation alone [Eq. (12)]. (c) Difference made when the wind effect is allowed to interact with the perturbation heating [Eq. (13)]. Contours 20 m (a) and (b) and 5m (c), zero dotted, negative dashed, shading denotes significance at 99%.

  • Fig. 8.

    Equilibrium SGCM 550-mb geopotential height response for heating at nine different locations. Contours 20 m, zero dotted, negative dashed. In shaded areas the response is statistically significant at the 99% confidence level. Position of heating denoted by a black circle, with a bar showing the range of positions in latitude and a latitude circle drawn at the latitude of the heating.

  • Fig. 9.

    Time-independent linear solutions about the SGCM mean climatology for 550-mb response to heating in the same nine locations as in Fig. 8. Contours 20 m, zero dotted, negative dashed. Position of heating marked as in Fig. 8.

  • Fig. 10.

    First three EOFs of monthly mean anomalies for a 1000-month integration of the SGCM with no heating perturbation. Zero contour dotted, negative dashed.

  • Fig. 11.

    Frequency distributions for projection of monthly mean anomalies onto the first EOF from 1000-month integrations. (a) Anomalies from unperturbed (control) run relative to mean of control run, projected onto EOF1 from control run. (b) Anomalies from perturbed run relative to mean of control run, projected onto EOF1 from control run. (c) Anomalies from perturbed run relative to mean of perturbed run, projected onto EOF1 from perturbed run.

  • Fig. 12.

    First three EOFs of monthly mean anomalies for a 1000-month integration of the SGCM with the standard heating perturbation. Zero contour dotted, negative dashed.

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