## 1. Introduction

The internal variability of the zonal-mean zonal wind (

Much less attention has been paid to the transient period of negative eddy forcing that follows 5–7 days after the peak in

In this paper we propose an alternate explanation for the transient negative feedback and provide a quantitative theory for its impacts on *change* in background

## 2. Methods

### GCMs

The multilevel GCM is a standard primitive equation spectral model integrating the vorticity, divergence, temperature, and the log surface pressure. The sigma coordinate vertical differencing scheme of Simmons and Burridge (1981) is used. An eighth-order hyper-diffusion with a time scale of 0.1 day for the smallest-scale waves is applied to the model variables. The only nontypical aspect of the model is the time differencing, which is the AB3–AI2 method of Durran and Blossey (2012). The resolution of all simulations is T42 with 20 equally spaced vertical levels. The multilevel control simulation is forced with the diabatic heating and frictional damping of Held and Suarez (1994). The model is run for 6500 days and the first 500 days are discarded to allow for model spinup.

For the simulations with fixed zonal-mean, we use a two-level primitive equation model (GCM) based on Hendon and Hartmann (1985). The prognostic variables of vorticity, divergence, and potential temperature are defined on two pressure levels (250 and 750 hPa). In the equations, the vertically integrated divergence is constrained to be zero for consistency with the assumption of constant surface pressure at the lower boundary. For thermal forcing we use Held and Suarez (1994) and for mechanical damping we use Rayleigh friction at the lower-level with an *e*-folding time scale of 3 days. This is denoted the control simulation for the two-level model. In the past, this model has been run at coarse resolution (R15) with semi-implicit time differencing (Hendon and Hartmann 1985; Robinson 1991). However, at the T42 resolution used here, the speed of the winds is approximately equal to the phase speed of the fastest gravity mode (which is slower than usual due to the constant surface pressure); therefore, our version is fully explicit with third-order Adams–Bashforth time differencing (Durran 1991). The model is run for 6500 days and the first 500 days are discarded to allow for model spinup.

## 3. Results

### a. Internal variability

The time-mean zonal-mean zonal winds (*U*) with Held and Suarez (1994) forcing has a midlatitude jet centered at about 43° latitude (Fig. 1a). Next, the EOFs of the instantaneous vertical- and zonal-mean zonal wind are calculated (the vertical average is from 1000 to 100 hPa) and then the

The PC1 and PC2 autocorrelations (Fig. 2a) demonstrate that EOF1 is much more persistent than EOF2. The decay of the autocorrelation in time is not uniform: for the first 5–7 days the autocorrelation drops more rapidly than at longer time lags. The lagged correlation between the PCs and the vertically averaged (1000–100 hPa) eddy momentum flux convergence, or eddy forcing (Lorenz and Hartmann 2001), projected onto the corresponding EOF pattern is shown in Fig. 2b. At negative lags, the eddy forcing leads the *large* positive lags, the response of the eddy momentum fluxes to the *U*–eddy forcing feedback involving wave refection on the equatorward flank of the jet. Note that the lagged correlations show that this feedback is not simply proportional to

The transient negative forcing is primarily due to longer zonal wavenumbers. For example, for zonal wavenumber 4 (Fig. 2c) the negative forcing is about as large as the previous positive forcing. For zonal wavenumber 8 (Fig. 2d), on the other hand, there is no transient reduction in forcing at all and for EOF2 the transient effect is very weak. When looking at wavenumber 4, one also sees a quasi-periodic element to the transient forcing that continues to oscillate until lags 30–40 days. Apparently, the oscillations of individual wavenumbers destructively interfere past the initial negative forcing so that the total transient forcing rapidly decays to zero past 7 days.

### b. Initial value experiments

Here we repeat the initial value experiments of Lorenz (2015), which “branch off” the long control simulation of the multilevel GCM described in section 2a (Fig. 3). The eddy field of each branch simulation is exactly the same as the control simulation at that time. The zonal-mean state, on the other hand, is suddenly switched to a prescribed *u*, temperature and surface pressure state at the start of the initial value experiment. Afterward the initial value experiments evolve freely for 30 days with the same Held and Suarez (1994) forcing of the control simulation. We perform an initial value experiment for all 18 000 eddy fields that are archived for the 6000 day control run (i.e., the GCM data were saved every 8 h). The entire series of 18 000 initial value experiments is performed four times for four different zonal-mean states. The zonal-mean states consist of the time and zonal-mean *u*, temperature (*T*) and surface pressure (*p _{s}*) plus or minus the zonal-mean

*u*,

*T*, and

*p*associated with EOF1 or EOF2 of

_{s}^{1}over 18 000 initial value experiments and both hemispheres, which gives the mean eddy forcing response as a function of latitude and time (0–30 days). The response to EOF1 is defined as the difference between the mean eddy forcing response to the positive EOF1 state and the negative EOF1 state. Because the eddy field of the positive and negative EOF state is exactly the same at time = 0, the eddy forcing difference is identically zero at time = 0. Afterward, the eddy field evolves differently in response to the different zonal-mean states. The EOF2 response is defined in the same way. In Fig. 4a, the mean eddy forcing response to EOF1 is projected back onto the EOF1 pattern to get the positive feedback of the eddy forcing as a function of time (blue curve). This blue curve means that 2–3 days after EOF1 is “switched on,” the eddies strongly act to damp the

The transient eddy forcing appears to be a response to the

In Fig. 5, the eddy forcing response from Fig. 4 is divided into the contributions of the different zonal wavenumbers. The left panels show the initial value experiments and the right panels show the lagged regression response for positive lags only. Consistent with Fig. 2, the long zonal wavenumbers are responsible for the transient response. For EOF1, wavenumbers 4 and 5 are most responsible while for EOF2 wavenumber 5 is most responsible. The transient response also exists for wavenumbers longer than 4; however, in this GCM the amplitude of these waves is weak so they do not contribute much to the negative response. Overall the responses for the initial value and lagged regression experiments are similar; however, as noted above the initial negative response is stronger in the initial value experiments. In addition, it appears that the long-term positive response of the shorter waves for EOF1 takes some time (5–7 days) to develop in Fig. 4a. In the lagged regression, on the other hand, the

Watterson (2002) found similar transient negative responses in experiments with a barotropic model. Motivated by this study, we perform a large number of initial value experiments like the previous paragraph but in a nondivergent barotropic model. To convert the multi-vertical-level fields of the GCM to a single level we perform a weighted vertical average using the EOFs of the eddy streamfunction as described in Lorenz (2015). The barotropic model is inviscid except for hyper-diffusion, which is the same as the GCM. The response of the eddy forcing projected on the corresponding EOF structure for our barotropic experiments is shown in Figs. 6a and 6b in red. The analogous response in the GCM is also shown in blue. The transient response is very well captured in the barotropic model for both EOF1 and EOF2. The only missing response is the long-term positive feedback for EOF1, which is explained by the lack of baroclinic instability to maintain the amplitude of the eddy field in the barotropic model. Note this lack of long-term feedback does not imply the “baroclinic feedback” is operating because the baroclinic feedback states that anomalies in baroclinic instability are positively correlated in latitude with anomalies in *e*-folding time scale of 13 days (not shown). The long-term feedback, on the other hand, involves short waves and the EKE of zonal wavenumbers greater than five decay rapidly with an *e*-folding time scale of 3 days in these barotropic experiments (not shown). The rapid decay of the short waves is energetically consistent with the fact that the momentum flux of these waves reinforces the mean jet.

To eliminate potential mechanisms for the transient feedback, we repeat the barotropic experiments but now 1) the eddy dynamics is linearized and 2) the background

### c. Fixed zonal mean

The model experiments so far have all involved initial value problems, but the transient feedback has a profound impact on the long-term internal variability as well. In this section, we will quantify the impact of the transient feedback on

For the fixed zonal-mean simulation, the most obvious choice of zonal-mean state is that of the time mean of the control simulation; however, the global mean EKE, vertical EP flux (Edmon et al. 1980) and eddy momentum flux increase by 35%, 28%, and 22%, respectively, when this zonal-mean state is used (not shown). It appears that zonal-mean fluctuations in the control simulation limit eddy activity via a negative feedback between bursts of eddy heat flux and zonal-mean baroclinicity. This elevated eddy amplitude makes comparisons between the control simulation and the fixed zonal-mean problematic. To create fixed zonal-mean state that better captures the coupling between the zonal mean and eddies, we first run a simulation like the control but where all zonal-mean time tendencies are reduced by a factor of *S*. The specific value of *S* is does not affect our conclusions and we have tried reducing the tendencies by factors of 10–500. A value of *S* = 100, is a good compromise between damping short-term zonal-mean fluctuations while still allowing the model to equilibrate in a reasonable length of time (we spin up the model for 10 000 days before archiving data when *S* = 100). Allowing the eddy field to set its “preferred” nearly fixed zonal-mean state leads to a slight decrease in zonal-mean vertical wind shear (4%) and slight changes in low-level

A comparison between the fixed-zonal-mean and standard control simulation is shown in Fig. 8. The statistics used here are based on the eddy forcing projected on either the EOF1 or EOF2 structures from the control simulation. For EOF1 eddy forcing, the control simulation autocorrelation dips below zero at short lags and becomes slightly positive at longer lags. These features are consistent with the transient negative response and positive feedback, respectively, seen in Fig. 7c. When

The cross correlation between the EOF1 and EOF2 eddy forcing shows that negative EOF2 leads to positive EOF1 and positive EOF1 leads to positive EOF2. This relationship weakly extends to larger lags in a quasi-periodic manner. This relationship means that eddy forcing anomalies propagate equatorward in latitude at small time scales. This contrasts with longer time scales where poleward propagation dominates (e.g., James and Dodd 1996; Feldstein 1998; Lee et al. 2007; Chemke and Kaspi 2015; Sheshadri and Plumb 2017). This difference between high and low frequency eddy behavior was first noted in Sparrow et al. (2009). Unlike the autocorrelations, the cross correlation is not impacted much by the fixed zonal mean. This suggests that the equatorward propagation in the eddy field is not due to wave–mean flow interaction. A logical explanation for this eddy forcing behavior is the equatorward propagation of wave packets from the midlatitudes where baroclinic instability in greatest, to their critical levels in the subtropics. Because of the strong bias toward equatorward wave propagation on a sphere, the cross correlation is not isotropic but instead is biased in the same sense. The equatorward propagation of wave packets may also be the source of the small negative correlations in the EOF1 autocorrelation for fixed

The eddy forcing power spectra for the fixed-zonal-mean and control run are shown by the solid lines in Fig. 9. The spectra have a relative/global maximum at frequencies around 0.14 day^{−1}. From this maximum, the power decreases at lower frequencies until, for EOF1 alone, the long-term positive feedback leads to increased power at the lowest resolved frequencies. The pronounced minima around 0.05 day^{−1} for EOF1 and at the lowest resolved frequencies for EOF2 is unusual for a geophysical time series. This unusual structure in the eddy forcing power spectrum is synonymous with the transient negative feedback due to 1) the mathematical relationship between the power spectrum and auto-covariance and 2) the strong relationship between eddy forcing and ^{−1}, which is associated with the small dip below zero in the EOF1 autocorrelation (Fig. 8a). We argued above that this is due to the equatorward propagation of Rossby wave packets that are localized in latitude.

In addition to eddy forcing, quasi-periodicity is observed in the eddy heat fluxes and eddy kinetic energy associated with the baroclinic annular mode (BAM; Thompson and Woodworth 2014). Thompson and Barnes (2014) argue that the periodicity comes from negative feedbacks between baroclinicity and eddy heat fluxes. Zurita-Gotor (2017), however, showed that the phase difference between heat flux and baroclinicity are not consistent with this hypothesis. Here we show that the quasi-periodicity in eddy heat flux is remarkably similar to the eddy forcing and therefore the periodicity in BAM might also involve the adjustment of a preexisting eddy field to background flow transience. First, we show the eddy heat flux power spectrum in the two-level model (solid blue, Fig. 9c). At low frequencies (below 0.07 day^{−1}), there is a pronounced reduction in power, which is synonymous with a quasi-periodic peak at 0.07 day^{−1}. As noted by Zurita-Gotor (2017), calculating the power spectrum individually for each zonal wavenumber and then summing over all wavenumbers gives a curve with *no* peak (blue dotted line). Evidently, the reduction in eddy heat flux at the lowest frequencies is due to anticorrelation between the heat flux at a wavenumber *k*_{1} and another wavenumber *k*_{2} rather than the reduction of the heat flux by the individual waves (Zurita-Gotor 2017). The sum of the eddy forcing spectra over individual wavenumbers shows the same structure as the heat flux with no well-defined quasi-periodicity (dashed blue lines in Figs. 9a,b). The eddy forcing and heat flux are also similar in that the quasi-periodicity disappears when the zonal-mean state is fixed (solid orange in Figs. 9a–c). Moreover, the anticorrelation between different wavenumbers changes to positive correlation for all cases as demonstrated by the fact that the solid orange line is always larger than the dashed orange line at low frequencies.

From the above results, we conclude that wave–mean flow feedbacks are responsible for the quasi-periodicity because it disappears when the zonal mean is fixed. Furthermore, wave–mean flow feedbacks are also responsible for the anticorrelation between waves. We believe that stochastic eddy forcing/heat flux by a given wavenumber makes zonal-mean anomalies that suppress other wavenumbers via transient feedbacks. Unlike the theory of Thompson and Barnes (2014), which involves the zonal-mean baroclinicity, the transient feedbacks considered here involve generic wind anomalies that are not necessarily the same as the anomalies that most impact baroclinicity. The positive correlation between different wavenumbers for the fixed-zonal-mean simulations might come about because eddy fluxes are naturally localized in longitude rather than in zonal wavenumber space, which leads to positively correlated zonal wavenumber fluxes. In future work, the impacts of zonal-mean transience on eddy heat fluxes and eddy kinetic energy will be studied in more detail.

## 4. Analytic model of transient feedback

### a. General case

*ζ*′:

*ζ*′ the eddy relative vorticity which is a function of

*x*,

*y*, and

*t*,

*L*is a damped linear operator and

*F*is forcing. In our case,

*L*encapsulates the effect of the background zonal-mean flow on the eddies. If the linear operator is perturbed (

*ζ*′) and the perturbed background flow (

*ζ*′, we must also use the

*ζ*′ field from an integration of (1). In other words, both (4) and (1) must be integrated to obtain the solution to (4). Our integration of (4) is based on the final and simplest simulation in section 3b where the background

*L*is the barotropic linear operator for no background flow,

*ζ*′ is the eddy vorticity from (1) and

### b. Nondivergent barotropic model

*x*,

*y*) linearized about a background zonal wind

*ζ*′ is the eddy relative vorticity, ∂∇

^{−2}

*ζ*′/∂

*x*is the eddy meridional velocity (

*υ*′),

*D*is the Rayleigh damping coefficient,

*F*′ is the eddy forcing,

*β*is the planetary vorticity and

*y*.

*F*′ be proportional to exp(

*ikx*), where

*k*is the zonal wavenumber. Next, let the unperturbed background state be uniform in

*x*,

*y*and

*t*with a zonal-mean zonal wind,

*U*−

*c*, and background vorticity gradient,

*β*. The zonal wind is denoted

*U*−

*c*so that we account for nonzero basic-state eddy phase speeds,

*c*(i.e., the derivation is performed after a Galilean transform so that the forcing is stationary). For a uniform background state, the linear operator that corresponds to

*L*in (1) is

*ζ*is a complex constant. For the derivation below it is not necessarily to relate

*ζ*to the forcing explicitly because (4) does not explicitly contain the forcing.

*L*operator and the form of the dissipation does not change. To help understand the evolution of the (4), lets first consider the initial

*t*= 0, (4) becomes

*Z*′, is

*t*= 0 and

*ζ*′ does not depend on time [see (7)], and therefore the full vorticity also obeys (9) at

*t*= 0:

^{−2}

*Z*′ = −

*Z*′/

*k*

^{2}at

*t*= 0 and assume the perturbed background flow takes a simple sinusoidal form in

*y*:

*u*(

*t*), which depends on time, and meridional wavenumber,

*n*. The temporal dependence of

*k*is small and

*n*

^{2}/

*k*

^{2}> 1 and therefore the negative feedback from retrogression is dominant. For short waves,

*k*is large and

*n*

^{2}/

*k*

^{2}< 1 and therefore the positive feedback from advection is dominant. The dominance of the retrogression effect for small

*k*is consistent with the fact that the transient negative eddy forcing is caused by the long waves. The theory also predicts that the transient eddy forcing is positive for short waves. There is a small hint of this behavior at very short lags for zonal wavenumbers 8 and 9 in Fig. 5a. In appendix A, we solve (4) for the barotropic vorticity equation for all times (i.e., not just

*t*= 0) under the simplifying assumptions introduced in this section. In appendix A, we also find the eddy forcing (i.e., eddy momentum flux convergence) anomaly produced by this solution. By solving the system in detail, we see that the theory correctly predicts that the damping long wave response should be an order of magnitude bigger than the reinforcing shortwave response. The solution is discussed in the next section.

### c. Understanding the solution

*m*is the anomalous eddy forcing acting on the imposed sinusoidal

*u*is the time varying amplitude of the

*a*and

*b*are constants. The frequency of the oscillation, (14), is the difference between the frequency of the basic-state wave (kc) and the frequency of a free Rossby wave with scale

*k*

^{2}+

*n*

^{2}. In the GCM/observations there are a wide range of different frequencies and spatial scales, which leads to destructive interfere of the transient response at long time scales (see below). Hence, the transient response is localized at short time lags. For individual wavenumbers, however, one can see hints of the damped oscillations (Fig. 2c). Note that (13) does

*not*include the long-term positive feedback because it is derived from a state with uniform background winds and no baroclinic instability. Theories of the positive feedback involve either the configuration of reflecting/critical levels in a latitudinally varying mean state (Lorenz 2014) or changes in the source of wave activity from baroclinic instability (Robinson 2000).

*P*:

*n*, the spatial scale of the

*n*(=6.7), which is the dividing point between positive and negative transient response, agrees well with the wavenumber dependence of the transient response in Fig. 5a. EOF2 has a slightly smaller spatial scale and the negative response extends to higher zonal wavenumbers (Fig. 5c).

*P*factor, (16), to convert to the transient forcing (red line), we see that the negative forcing by the low wavenumbers strongly dominates over the high wavenumbers. In fact, the low wavenumber response shows a peak at zonal wavenumbers 1 and 2, which is not supported by the initial value experiments (Figs. 5a,c). This discrepancy might be due to the fact that the amplitude of wavenumbers 1 and 2 strongly peak in the polar regions (not shown) and therefore these waves have a more significant meridional wavenumber

*l*than the higher wavenumbers. This conflicts with the

*l*= 0 assumption of the analytic model. Nevertheless, the analytic model clearly reproduces the weak transient response for high wavenumbers. To understand, the source of the wavenumber dependence, we alternatively multiply the vorticity variance by

*n*to make it dimensionless. This factor does not explain the relative amplitude of the low and high wavenumbers (orange line in Fig. 10), therefore the amplitude dependence comes from the transformation from

*ζ*′ to

*k*factor). However, the biggest contributor involves subtle issues when calculating the in-phase component of

*ζ*′ and

*υ*′ when going from (A12) to (A13) in appendix A (contributes a ∼1/

*k*

^{2}factor).

The forcing of the eddy forcing *m* in (13) also includes the interesting expression *du*/*dt* + *Du*. If the dissipation for the waves, *D*, is the same as the dissipation for the zonal-mean, which is a reasonable approximation, then *du*/*dt* + *Du* is the same as the eddy momentum flux convergence via the vertical average momentum budget (e.g., Lorenz and Hartmann 2001). This eddy momentum flux convergence includes *m* itself, as well as the stochastic and long-term feedback component of the eddy forcing. We have explored separating the components by putting the *m* portion of *du*/*dt* + *Du* on the left-hand side with the other *m* terms. However, the biggest contribution to the forcing in (13) is the stochastic component of the eddy forcing, hence it is not unreasonable to understand the system in its current form.

The *du*/*dt* + *Du* expression in (13) also implies the transient forcing is zero when a *du*/*dt* + *Du* = 0 and therefore the transient forcing is zero. If the transient forcing depended on either *du*/*dt* or *Du* alone, the *m* response would persist onward past short positive lags. For example, if the forcing were proportional to –*du*/*dt* then the initial negative response would be followed by a positive response as the EOF2 anomaly decayed (*du*/*dt* < 0 implies positive transient response by the long waves). For EOF1, which decays slower than the Rayleigh damping rate due to the positive long-term feedback, *du*/*dt* + *Du* > 0 and therefore the transient feedback is partially offsetting the positive feedback at large positive lags.

### d. Response for a spectrum of waves

*k*and

*c*on the solution to the transient response, Eq. (13). First, we solve (13) in terms of an integral using the method of variation of parameters:

*P*is a constant defined by (16) and

*du*/

*dt*+

*Du*. In the initial value experiments, on the other hand, the transient response to a sudden impulse is of the form

*t*exp(−

*αt*), where

*α*is a constant (not shown precisely, but see Figs. 4 and 6). This discrepancy can be resolved by noting that (19) applies to a single wave of zonal wavenumber

*k*and phase speed

*c*[

*ω*is a function of

*k*and

*c*, see (14)]. In reality, there are a wide range of different waves in the GCM and therefore (19) must be integrated over

*k*and

*c*:

*c*and

*k*outside the interior integration, we get

*k*and

*c*(e.g., from a phase speed–latitude spectrum (Randel and Held 1991) and

*P*is a function of

*k*, (16). Given the approximations in the analytic model, calculating the double integral inside the brackets from the phase speed–latitude spectrum of

*t*exp(−

*αt*)]. In particular, the transient response is contaminated from an excessively large response in zonal wavenumbers 1 and 2. We believe this error due to the assumption that the background waves are independent of

*y*because the barotropic models in Fig. 6 are able to capture the observed response even under the approximations in (4).

*t*−

*s*. If we assume this function is of the form

*t*exp[−(

*α*−

*D*)(

*t*−

*s*)], then the

*m*(

*t*) response to a sudden impulse is of the desired form

*t*exp(−

*αt*). Under this assumption

*A*is a constant that represents the amplitude of the transient feedback. While the functional shape of the impulse response function in (22) is not derived from fundamentals, the

*du*/

*dt*+

*Du*forcing in (22) is analytically derived. In the next section we use (22) in a generalized feedback analysis that includes the transient feedback as well as the positive and poleward-propagating feedbacks (Lubis and Hassanzadeh 2021).

Equation (22) also explains why the transient feedback is bigger in the initial value experiment compared to the lagged regression for EOF1 (Fig. 4a). In the initial problem (blue line in Fig. 4a), *du/dt* + *Du* is a delta function at *t* = 0 so all the forcing of the transient feedback occurs at *t* = 0. For the lagged regression (red line), one should first note that, via the momentum budget, the eddy forcing equals *du*/*dt* + *Du* and therefore the plotted lagged regression is proportional to the lagged regression between *t* > 0. Unlike EOF1, most of the EOF2 lagged regression (red line in Fig. 4b), is restricted to small negative lags (note that the area of the red line from lags −5 to 0 is almost as much as the integral over all negative lags). In this case, the initial value and lagged regression transient response are similar because the *du*/*dt* + *Du* forcing occurs close to lag 0 in each.

### e. Feedback analysis

*m*is the total eddy forcing,

*t*is the time,

*u*is the

*d*=

*du*/

*dt*+

*Du*,

*w*(

*s*) is the discretized weighting based on (22) except normalized:

*t*is the temporal resolution of the data, the subscripts refer to the EOF index (i.e., EOF1 or EOF2) and the integrals are discretized. The Rayleigh damping,

*D*, which is assumed to be identical for the waves and the zonal-mean, is estimated as in Lorenz and Hartmann (2001) and is 0.1343 day

^{−1}. The constant

*a*

_{1}is the amplitude of the transient feedback for EOF1 and

*b*

_{11}and

*b*

_{12}are the feedback of EOF1 and EOF2

*u*

_{1}(

*t*−

*l*), where

*l*is a time lag, and then average over time, which converts products such as

*u*(

_{j}*t*)

*u*(

_{k}*t*−

*l*) to lagged covariances:

*M*(

_{jk}*l*) denotes the lagged covariance between

*m*and

_{j}*u*at lag

_{k}*l*(positive

*l*means

*u*leads

*m*),

*D*is the same but for

*d*and

*u*,

*U*is the same but for

*u*and

*u*, and we assume the lag,

*l*, is positive and large enough such that

*u*(so the

*u*

_{2}(

*t*−

*l*), gives a similar equation:

*w*(

*s*). In this Lubis and Hassanzadeh (2021) case, there are two equations for two unknowns and the solution is straightforward. In our case, the additional unknown requires more information, which we obtain by fitting multiple time lags together at once. Because the system is now overdetermined, we choose parameters

*a*

_{1},

*b*

_{11}, and

*b*

_{12}that minimize the sum over lags of the squared difference of the left and right-hand sides of (25) and (26). To obtain the equations for

*a*

_{2},

*b*

_{22}, and

*b*

_{21}, simply replace all 1 subscripts with 2 and all 2 subscripts with 1. The parameter

*α*, which is the temporal scale of the weights, is found by trial and error until the value that minimizes the squared error is obtained. The details are given in appendix B.

*D*so the units are day

^{−1}like the long-term feedback. The transient feedback parameters,

*a*

_{1}and

*a*

_{2}, are negative as expected and moreover the amplitudes are nearly the same (6% difference). The estimated long-term feedback parameters are

*α*in the temporal weighting function is 0.35 day

^{−1}. The signs of the long-term feedback parameters

*b*are consistent with Lubis and Hassanzadeh (2021), except for

_{jk}*b*

_{22}, which is very close to zero in both their and our analysis and therefore its sign is less robust. Given the sign convention of our EOFs (Fig. 7b), the signs of

*b*

_{12}and

*b*

_{21}imply the poleward propagation of

Next, (23) and the analogous equation for *m*_{2}(*t*) (replace all 1 subscripts with 2 and all 2 subscripts with 1) are used to create a “synthetic” time series of *u* and *m* (we use trapezoidal time differencing). For the eddy forcing without feedbacks

*u*via the term

*du*/

*dt*+

*Du*, (19), means that the transient eddy feedback impacts the mean response to an external forcing. In other words, the term “transient feedback” is a misnomer as far as the response to external forcing. For variability, on the other hand, the

*du*/

*dt*and

*Du*in the

*m*forcing cancel for positive lags (EOF2 and higher-order EOFs) or nearly cancel (EOF1), and therefore the true nature of the transient eddy forcing is hidden. The feedbacks on the time-mean externally forced response is the sum of the long-term feedbacks (28) plus

*D*times the transient feedback (27):

*b*, the total feedback in (29) is strongly negative for EOF2 and significantly weaker, although still positive for EOF1. The strong negative feedback for EOF2 is consistent with linear response function (LRF) experiments (Hassanzadeh and Kuang 2016) where we have imposed EOF2

## 5. Discussion and conclusions

In this paper we have developed a theory for the transient negative response to

We also develop an analytic model of the transient feedback, which shows that the sign of the response depends on the relative role of advection by

Experiments with a barotropic model also demonstrate that the transient response is essentially independent of the mean state. For example, an eddy field taken from the GCM gives the same eddy forcing response to a change in

Via the mathematics relating the autocovariance and the power spectrum, the negative transient response is synonymous with quasi-periodicity in the eddy forcing. This quasi-periodicity has striking similarities with the quasi-periodicity in the eddy heat flux and eddy kinetic energy associated with BAM (Thompson and Woodworth 2014). For example, Zurita-Gotor (2017) find that the quasi-periodicity in the heat flux is due to a reduction in power at the low frequencies that is the result of anticorrelation between the heat flux of different zonal wavenumbers. This same anticorrelation is the source of the eddy forcing periodicity as well (Fig. 9). Furthermore, GCM simulations with a fixed zonal-mean state show that the quasi-periodicity is mediated by the zonal-mean in both eddy forcing and eddy heat flux.

The analytic model also shows that the forcing of the transient eddy response is proportional to *du*/*dt* + *Du*, where *D* is the Rayleigh damping constant. The unique form of the forcing of the transient feedback leads to what seems like a contradiction regarding its impacts: for internal variability the transient feedback is limited to short time lags, but for the time-mean response to external forcing the “transient feedback” impacts the response about as much as the long-term feedback. The apparent contradiction is resolved by realizing that *du*/*dt* and *Du* cancel or nearly cancel for decaying variability but the *du*/*dt* is identically zero for the time mean. In addition, the fact that the transient feedback involves a prognostic rather than diagnostic equation suggests that FDT predictions based on a

Given the relationship between eddy forcing and meridional wave activity flux, the transient negative response acts as a mechanism for trapping wave activity: positive eddy forcing anomalies, which are associated with waves leaving a location, are followed by negative eddy forcing anomalies, which are associated with waves returning. This mechanism is independent of the traditional waveguide caused by Rossby wave turning latitudes (e.g., Hoskins and Ambrizzi 1993), instead the interaction between a wave and a time-varying background flow lead to trapping of wave activity. The effect of background flow transience on the trapping of Rossby waves has been studied by Keller and Veronis (1969); Pandolfo and Sutera (1991) and Monahan and Pandolfo (2001). This current work is unique because the trapping occurs via background flow changes caused by the Rossby wave itself.

The eddy forcing is defined to be the vertical- (1000–100 hPa) and zonal-average eddy momentum flux convergence.

## Acknowledgments.

The author would like to thank three anonymous reviewers for their helpful comments and suggestions on the manuscript. This research was supported by NSF Grant AGS-1557353.

## Data availability statement.

Please contact the author for the model code and data.

## APPENDIX A

### Analytic Solution of Transient Feedback

#### a. Perturbation vorticity equation

*y*with value

*U*−

*c*, the background eddy vorticity is given by (7) and the perturbed background flow takes a simple sinusoidal form, (11). The vorticity gradient associated with (11) is

*L*simply multiplies

*ny*) exp(

*ikx*) spatial dependence factor from the right-hand side [i.e., (A2)]:

^{−2}/∂

*x*= −

*ik*/(

*k*

^{2}+

*n*

^{2}) for waves with meridional wavenumber

*n*. Substituting (A2), (A3), and (A4) in (4) and cancelling the cos(

*ny*)exp(

*ikx*) factor:

#### b. Derivation of the eddy momentum flux convergence equation

*ikx*):

*y*dependence,

*ζ*are given by (A3) and (7), respectively,

*υ** denotes the complex conjugate of

*υ*and we use the fact that the zonal mean of two waves of the form exp(

*ikx*) with complex amplitudes

*x*and

*y*is

*y*dependence of the momentum flux convergence, (A7), is the same as the zonal wind perturbation, (11), so under the above approximations the eddy forcing either reinforces or damps the zonal wind anomalies but does not shift them in latitude. Therefore the solution can be completely characterized by the eddy forcing in phase with the zonal wind, which we call

*m*:

*m*.

*y*(see section 4), therefore:

*y*, (A3), therefore:

*r*to be the corresponding real part of (A14):

*ζ**:

*ζ** does not depend on time to move it inside the time derivative. Multiplying (A16) by

## APPENDIX B

### Derivation of Methodology for Estimating Feedback Parameters

*l*is

*l*

_{1}and

*l*

_{2}are the range of lags to minimize the error. Setting the derivatives of (B3) with respect to

*a*

_{1},

*b*

_{11}, and

*b*

_{12}equal to zero, we get a system of three equations and three unknowns, which can be easily solved to get

*a*

_{1},

*b*

_{11}, and

*b*

_{12}:

*a*

_{2},

*b*

_{22}, and

*b*

_{21}are the same except the 1 subscripts are replaced with 2 and the 2 subscripts are replaced with 1 in (B4).

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