## 1. Introduction

*τ*= 0 with decay of the wave amplitudes upstream and downstream. This pattern is normally moving eastward, intensifies with increasing negative lag, and becomes weaker for positive increasing lag. An example is presented in Fig. 1 where potential vorticity (PV)is selected as a variable with density

*ρ*, vorticity

*ζ*, potential temperature

*θ*, and Coriolis parameter

*f*. The data evaluation is based on the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) data (see section 2). Although

*q*is not strictly conserved in adiabatic and frictionless flow, the deviations of

*q*from the correct formulation in isentropic coordinates are too small to be of interest here. Let us denote by

*C*(

*b*,

*c*|

*τ*) the covariance of variable

*b*and variable

*c*where

*b*leads with lag

*τ*. Figure 1 shows the regression

*z*= 8 km where

**v**= (

*u*,

*υ*,

*w*), where the overbar denotes the time mean state and the prime deviations from the mean. It is assumed in (1.2) that the flow is incompressible and that dissipative effects can be represented by a simple damping term. Forcing by heating is excluded. The deviations contain all available time scales.

To solve (1.3) one has to “invert”

On the other hand, Frederiksen and Kepert (2006) and Zidikheri and Frederiksen (2010a,b) used data from numerical simulations to represent the contributions from subgrid motions to the nonlinear eddy terms in a quasigeostrophic model. A specific linear model that includes white noise forcing is used for these contributions. Thus, a fairly detailed treatment of the forcing was found to be necessary for successful simulations. Jin et al. (2006a,b) demonstrated that the vorticity transports by synoptic eddies can be modeled on the basis of the so-called synoptic eddy and low-frequency flow (SELF) closure.

Further progress in the field may profit from information based on data. It is the purpose of this paper to explore the statistical characteristics of the nonlinear term in (1.3) by considering the situation for the correlation point in Fig. 1.

## 2. Observed forcing characteristics

*λ*is longitude,

*φ*is latitude,

*a*is the earth’s radius, and the mean density is

*Dz*= 1000 m and for a horizontal mesh of 2.5° × 2.5°. The values of

*F*are calculated at all grid points as daily means. The correlation point

*P*is located in the North Atlantic storm-track region (47.5°N, 45.0°W) at a height

*z*= 8 km. Note that the forcing contains all time scales. It is left for further work to introduce various filters. On the other hand, (1.2) does not contain any preference for a time scale. Of course, the forcing has to be adapted to the model where it is incorporated. A model for slow dynamics needs forcing by fast motions only.

Figure 2 shows the normalized covariance *τ* = −1, (b) *τ* = 0, and (c) *τ* = 1 day. The basic pattern at *τ* = 0 consists of a domain of positive values centered at *P* with negative areas in the east and west. This forcing is quite strong with a maximum of ~4 PVU day^{−1} (1 PVU = 10^{−6} m^{2} K s^{−1} kg^{−1}). The zonal extent of this positive domain is ~500 km; the meridional one is somewhat less. There are weak indications of a wave pattern in meridional direction. A simple dynamical explanation of this structure is not available. For example, unstable Eady waves are generally accepted as simple models of active baroclinic systems but *F* = 0 for these modes because their perturbation PV vanishes. The pattern moves eastward with a speed of ~5 m s^{−1}. There is rapid increase from *τ* = −1 to *τ* = 0 days and equally rapid decay for positive lag. These rapid changes, however, do not continue with increasing |*τ*|. Appreciable covariances are found for lags of a few days. For example, ^{−1} at *τ* = 0, but the value of −0.86 (−0.9) for *τ* = −2 (2) days is almost as large as for *τ* = −1 (1) day. It is likely that this tail of the decay is due to the interaction between synoptic eddy and low-frequency variability (e.g., Jin et al. 2006a,b).

The covariance *z* = 10 km (Fig. 3) is quite similar to that at *z* = 8 km in Fig. 2. The same is true at *z* = 6 km except that the amplitudes are less than one-tenth of those in Fig. 2 (not shown). There is a clear signal even at *z* = 4 km (Fig. 4) although the amplitudes are quite small. No appreciable covariances are found at *z* = 2 km (not shown). Altogether it is seen that the covariance

The “response” of PV to the forcing can be seen from Fig. 5, where *z* = 8 km is displayed. The basic pattern is quite simple. It is kind of a replica of the forcing pattern but shifted by ~250 km westward. At *τ* = 0, there is a prominent maximum slightly west of the correlation point. The minimum in the east is somewhat stronger than that in the west. Motion is eastward with increasing lag. The related amplitude changes are less dramatic than in Fig. 2. The maximum amplitude at *τ* = 0 is ~0.3 PVU and 0.15 PVU at *τ* = 1 day. The meridional structure of the pattern is richer as in Fig. 2. The response at *z* = 10 km (Fig. 6) is quite strong with separate extrema farther north. The intensity of the maximum varies little with lag. The response pattern at *z* = 6 km (not shown) is similar to Fig. 5, but that at *z* = 4 km is too noisy and weak to be shown.

The similarity of Figs. 1 and 2b is obvious and the responses in Fig. 5 are rather close to the forcing at least as far as the patterns are concerned. On the other hand, the forcing in Fig. 2b is quite strong with a maximum amplitude of 4 PVU day^{−1}, but the observed response value of 0.3 PVU in Fig. 5b appears to be too small to be explained as a simple response to the forcing. We can test this impression by invoking an advection model where zonal advection by the mean flow is the only mechanism that affects PVU in addition to the forcing.

## 3. Advection model

*F*is now a stochastic forcing that represents the nonlinear eddy term in (1.2). Of course, (3.1) is just one-dimensional in space. Thus,is the equation to be solved where the right-hand side must be specified according to the observations. Givenwhere

*D*vanishes for

*τ*→ ∞, we can write down the solution

*D*= 0 for |

*x*| > 1, |

*τ*| > 1. A scaling adapted to Fig. 2 is 1500 km for length and 2 days for time. There is a prominent maximum at

*x*= 0 in (3.5) and two adjacent minima upstream and downstream. The covariances drop to 0 within the time 2/

*π*. The

*x*integral over

*D*vanishes, so that there is no mean forcing of PV. The eastward motion of the forcing is represented by a speed

*c*~ 0.5. That corresponds with a speed of 5 m s

^{−1}, in rough agreement with the observations.

The solution is shown in Fig. 7 for a (*x*, *τ*) plane, the forcing (3.5) and a mean flow *d* = 1/2.5 [nondimensional units; *d* = 1/(5 days) in dimensional units]. There is no response for *τ* < −1, of course, nor for *x* < −1, but there is a rapid buildup of the standard structure with a maximum near *x* = 0 and minima upstream and downstream. The situation at *τ* = 0 corresponds reasonably well with Fig. 5b. The pattern moves then downstream with the mean flow. There is little activity at *x* > 0 for, say, *τ* > 2. The advection solution, however, is not satisfactory farther downstream because the decay is dictated by the damping. Atmospheric damping times are certainly not shorter than 5 days (e.g., Whitaker and Sardeshmukh 1998). The response in Fig. 7 is small for, say, *τ* ≥ 3 (i.e., after two weeks). That is too long compared to the observations.

The ratio of maximum response to maximum input is ~0.8 day taking into account (3.5), Fig. 7, and the scaling. The observed ratio of ~0.1 day is much smaller. One can bring down the result of the advection model somewhat by choosing specific parameter values but we have not been able to match the observations for a reasonable choice. In other words, the response in the advection model is too strong and too long-lived. Atmospheric dynamics has more possibilities to react than an advection model. This finding is in line with the result of Whitaker and Sardeshmukh (1998) that the white noise forcing provides only a minor fraction of the flow’s energy. On the other hand, we can test the assumption of white noise in time by replacing cos(*πτ*/2) by a delta function. It can be seen from (3.4) that *τ* < 0. There is then a jump to the full profile of *D* with respect to *x* at *τ* = 0. This profile moves then with speed *c* + *U* downstream and is damped.

## 4. Discussion and conclusions

The three-dimensional divergence of the PV flux has been evaluated from data because this is the dynamic forcing needed in linear models of point-correlation maps. It is found that the autocovariance *τ* = 0 but the values for lags of a few days are negative and not negligible as requested by a white noise model.

The forcing is almost white in meridional direction but not zonally where pronounced up- and downstream minima are found. The extent of this wave pattern is about 3000 km. Thus the forcing has a zonally propagating structure.

The response pattern *τ* = 0 shows reasonably good agreement with that resulting from the advection equation for a forcing that mimics the observed forcing. The advective solution, however, is not acceptable for large *τ* and *x* because the observed decay cannot be captured with realistic damping rates. Moreover, the amplitudes of the response are too large. This means that the forcing is too effective in “generating” PV compared to the atmosphere. This deficit of the advection model may be explained by the fact that the advection model assumes a one-layer atmosphere without meridional PV gradients. The PV in this model responds differently to a forcing than does the atmospheric PV. On the other hand, we have to stress that although *F* is treated like a forcing in the paper, the divergence of the eddy PV flux is just a term in the PV equation. Nevertheless, if *τ <* 0 in strong contradiction to the observations. Nonlinear eddy terms should not be assumed to be white.

Our work can only be seen as a first step toward a global forcing climatology. The correlation point selected here is presumably representative of the upper tropospheric part of a storm-track region but many more points would have to be selected before a reasonably complete stochastic climatology of the forcing emerges. Moreover, no effort has been made to distinguish between resolved and subgrid motions. That is appropriate in view of the linear theories where such a separation does not make sense. It would, however, make sense if a global model is run with a certain resolution and if data with higher resolution were available. Another possibility would be the restriction of *F* to certain time scales. This way one could investigate, for example, the interaction of synoptic eddies and low-frequency motion (e.g., Ren et al. 2009). The ultimate step would be the incorporation of the observed forcing statistics into a linear model like that of Whitaker and Sardeshmukh (1998). In principle, the solution of the advection equation shows how this problem can be solved but one would hesitate to solve the linear equations of a global model just to obtain point-correlation maps for one grid point. A more attractive method is to analyze the forcing for many grid points and to construct a stochastic model for the forcing similar to Frederiksen and Kepert (2006) and Zidikheri and Frederiksen (2010a,b). One would have to integrate in time a global linear model with this forcing to extract the required statistics.

The calculations of the forcing covariances depend, of course, on the resolution of the data. With a grid distance of 2.25° we can be certain that the zonal structures are captured quite well. It may be, however, that the meridional width of the forcing distribution is somewhat overestimated.

## Acknowledgments

The authors thank F.-F. Jin for his constructive and encouraging comments. Remarks by two anonymous referees helped to improve the paper.

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