What Determines the Spectrum of a Climate Variable at Zero Frequency?

Jin-Song von Storch Institute of Meteorology, University of Hamburg, Hamburg, Germany

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Abstract

In order to understand the spectrum Γx(ω) of a climate variable xt, the relation between Γx(ω) and its forcing has to be considered. If the evolution of xt over (discretized) time is determined by ft, that is, Δxt ≡ (xtxt−1)/Δt = ft, the only existing relation is the one between Γx(ω) and the spectrum Γf(ω) of ft. The gain function G(ω) of the difference operator Δ/Δt, which acts as a high-pass filter, controls the relation between Γx(ω) and Γf(ω). For Γx(ω), which is bounded at zero frequency, G(ω) completely suppresses the variations of ft at zero frequency, so that Γx(0) cannot be related to Γf(0). In practice, the efficiency of the difference operator as a high-pass filter can make the detection of the low-frequency spectral relation between xt and ft difficult.

Corresponding author address: Dr. Jin-Song von Storch, Institute of Hydrophysics, GKSS Research Center, 21502 Geestacht, Germany.

Email: jin@gkss.de

Abstract

In order to understand the spectrum Γx(ω) of a climate variable xt, the relation between Γx(ω) and its forcing has to be considered. If the evolution of xt over (discretized) time is determined by ft, that is, Δxt ≡ (xtxt−1)/Δt = ft, the only existing relation is the one between Γx(ω) and the spectrum Γf(ω) of ft. The gain function G(ω) of the difference operator Δ/Δt, which acts as a high-pass filter, controls the relation between Γx(ω) and Γf(ω). For Γx(ω), which is bounded at zero frequency, G(ω) completely suppresses the variations of ft at zero frequency, so that Γx(0) cannot be related to Γf(0). In practice, the efficiency of the difference operator as a high-pass filter can make the detection of the low-frequency spectral relation between xt and ft difficult.

Corresponding author address: Dr. Jin-Song von Storch, Institute of Hydrophysics, GKSS Research Center, 21502 Geestacht, Germany.

Email: jin@gkss.de

1. Introduction

Climate systems can be described by trajectories of climate state variables over time. The equation describing how a state variable x evolves over time is generally a first-order differential equation in time. If time is discretized, the corresponding difference equation may be written as
i1520-0442-12-7-2124-e1
where, without loss of generality, dx/dt is discretized using Δxt ≡ (xtxt−1)/Δt with Δt being the time increment.1 Function ft, which is a function of x and the other state variables of the system, represents the deterministic processes controlling the time evolution of x. This note concentrates on the discrete representation (1). The continuous case is briefly discussed at the end of this note.

It is generally believed that if ft as a function of the state variables is exactly known, xt for t ∈ [−∞, ∞] is in principle also known. By assuming that xt and ft in Eq. (1) can be considered as random variables, the spectra of xt and ft, which can be derived from the time series of xt and ft, are then also known. Because of relation (1), the two spectra are related to each other in such a way that the squared coherency spectrum, κxf(ω), between xt and ft [for the definition of κxf(ω), see, e.g., Jenkins and Watts 1968] is exactly one, implying a perfect spectral relation between xt and ft. Since this situation appears to be trivial, the spectral relation between xt and ft has barely been seriously studied. Almost all theoretical studies are devoted to the understanding of the deterministic processes constituting ft.

This note questions the usefulness of knowledge about ft in determining the spectrum of xt at zero frequency. The consideration is motivated by the squared coherency spectrum κMF(ω) between the global atmospheric axial angular momentum Mt and the global axial torque Ft derived from a numerical integration of a climate model. The Mt and Ft are related to each other through an equation of type (1), so that the squared coherency between the two is expected to be one at any frequency. Figure 1 shows κMF(ω), as derived from 200 years of 12-hourly data generated by the coupled ECHAM1/LSG atmosphere–ocean general circulation model (GCM; von Storch et al. 1997). The “chunk” method is used to estimate the squared coherency spectrum. At extremely high frequencies (i.e., ω > 10−1), the low coherence is likely caused by sampling errors. Significant coherence is only obtained for high frequencies with 10−1 > ω > 10−3. At low frequencies with ω < 10−3, no significant relationship between Mt and Ft is found. Note that for 12-h data considered, the frequency of one cycle per year corresponds to ω = 1/720 = 1.39 × 10−3, and one cycle per 10 yr corresponds to ω = 1.39 × 10−4.

At the first glance, Fig. 1 suggests that something is wrong with the angular momentum budget in the model, that is, with the representation of the processes constituting F. The purpose of this note is to show that the low coherence at low frequencies as shown in Fig. 1 has little to do with the representation of processes constituting F. Instead, it is caused by the constraint that Ft must be, by construction, MtMt−1 divided by Δt.

It is noted that Mt is a state variable whose forcing can be, at least in a GCM, exactly specified. For many other variables of the climate system, such as velocity, temperature, or pressure at a given grid point, or time series of a given spatial pattern or mode, the forcing f has generally a complicated functional form and can therefore not be easily estimated. Without a precise estimation of ft, the expected spectral relation between xt and ft cannot be tested. Thus, Fig. 1 is, to some extent, the only spectral estimation ever obtained at such low frequencies for a climate variable and its forcing. If indeed Fig. 1 reflects a general property of Eq. (1), the lack of coherence at low frequencies would be expected to also be found for other pairs of climate variables and their forcings. The general belief that the knowledge of ft is sufficient for determining everything of x (i.e., not only any value of xt at a given time t, but also the variance and spectrum of xt) would therefore become questionable.

2. Consequences of the constraint ft = (xtxt−1)/Δt

In order to demonstrate that the lack of coherence shown in Fig. 1 reflects a general property of Eq. (1), an arbitrary climate variable xt and its forcing ft, which determines the time evolution of xt and satisfies (1), are considered. It is assumed that ft is exactly known. With this assumption, ft must be exactly equal to (xtxt−1)/Δt. The consequence of this constraint is discussed below. For the sake of simplicity, the constant time increment Δt is set to 1. Equation (1) then reduces to
i1520-0442-12-7-2124-e2
xtxt−1 is known as the first difference of xt.
It can be shown (von Storch and Zwiers 1998), that the spectrum Γx(ω) of xt and the spectrum Γf(ω) of ft = (xtxt−1) are related to each other via
i1520-0442-12-7-2124-e3
with
i1520-0442-12-7-2124-e4
where Γx(ω), Γf(ω), and G(ω) are defined for ω ∈ [0, ½].

The G(ω) results from the first difference operator and is known as the gain function of the first difference operator (Jenkins and Watts 1968). Because of the form of G(ω) (Fig. 2), the first difference operator is often used as a high-pass filter (Jenkins and Watts 1968; von Storch and Zwiers 1998).

Following Eq. (3), Γf(ω) is simply a filtered version of Γx(ω). The filter G(ω) acts to respectively enhance and damp, rather than to alter, the spectrum of ft in relation to the spectrum of xt. The enhancement occurs at high frequencies with 1/2ω > 1/6, while the damping occurs at low frequencies with 0 ⩽ ω < 1/6. At any nonzero low frequency, the damping can never completely diminish the amplitude of Γf(ω) [as long as Γ(ω) ≠ 0]. Since the squared coherency is a normalized quantity, it will remain one, no matter how efficient the filter is.

On the other hand, Fig. 2 also clearly demonstrates the efficiency of the first difference operator in suppressing the variations of ft at low frequencies. The efficiency appears more dramatic in Fig. 2, where both G(ω) and ω are plotted in a linear fashion, rather than in a logarithm representation. For low frequencies, ω < 10−3, at which the squared coherency in Fig. 1 is low, the values of G(ω) appear to be located at the origin where ω = 0; that is, G(ω) is approximately zero for small ω. This is because that, for πω ≪ 1, G(ω) can be approximated by
Gωπω,
which is, in a linear representation, approximately zero. Together with Eq. (3), Eq. (5) implies that, as frequency goes to zero, more and more variations of ft are filtered out by G(ω). The amplitude of Γf(ω) relative to that of Γx(ω) drastically decreases. At zero frequency, if the spectral value of xt is finite, which is a reasonable assumption for a climate variable, G(ω) would completely suppress the variations of ft. The Γf(0) would be zero regardless of Γx(0). Consequently, Γx(0) cannot be determined from Γf(0). In this sense, Eq. (3) is ill at zero frequency.2

The efficiency of the first difference operator as a high-pass filter can cause practical difficulty in detecting the spectral relation between xt and ft. This is particularly true when Γx(ω) remains essentially constant as ω goes to zero. The values of Γf(ω) with ω → 0 are then extremely small and can therefore not be easily estimated. This fact makes the detection of the spectral relation between xt and ft difficult.

It is noted that observations and data obtained from numerical integrations of GCMs indicate that many climate variables, in particular the atmospheric variables, have spectra that are finite at zero frequency. Moreover, the spectra tend to be flat for sufficiently small frequencies. For these variables, Γx(0) represents the flat spectral level at low frequencies. The spectrum Γf(ω), which is then proportional to ω2, has extremely small values as ω goes to zero. The smallness of Γf(ω) makes the detection of the spectral relation between xt and ft difficult. In the following, this difficulty is further illustrated by considering the squared coherency spectrum κxf(ω) between xt and ft = (xtxt−1) for an arbitrary xt, whose spectrum is essentially flat at low frequencies.

Figure 3 shows κxf(ω), where xt is generated by a first-order autoregressive process, with the process coefficient being 0.5. As in Fig. 1, the chunk method is used to estimate κxf(ω). The length T of the time series varies from l03 to 106. With increasing T, lower and lower frequencies can be resolved.

For all considered Ts, κxf(ω) is not distinguishable from zero at low frequencies. Taking ω95 to be the frequency below which κxz(ω) is essentially lower than the 95% confidence level, Fig. 3 shows that ω95 decreases with increasing T. This suggests that if the power of the computer is increased (so that squared coherency at much lower frequencies can be resolved), the frequency range over which low coherence is found would decrease. However, Fig. 3 also shows that the rate of decrease of ω95 is smaller for larger T. Further, ω95 decreases by one order of magnitude as T increases from 103 to 104, but only by about factor of 2 or so as T increases from 105 to 106. As the lowest resolved frequency approaches zero, it becomes increasingly difficult to move ω95 toward ω = 0 by increasing computer power. This example suggests that the attempt to detect significant coherence between xt and ft at frequency ω → 0 is futile. The suggestion is consistent with the result that a finite Γx(0) cannot be determined from Γf(0).

The generality of the difficulty in detecting the spectral relation between xt and ft at frequency ω → 0 is further confirmed by the fact that the results shown in Fig. 3 are insensitive to the number of chunks (when the chunk spectral estimator is considered) and to the method used to estimate the squared coherency spectrum κxf(ω). The latter issue is verified by repeating the calculations using two additional methods, the Daniell spectral estimator and a method based on an autoregressive model that is fitted to the bivariate time series with one component being xt and the other being ft. No manner how κxf(ω) is estimated, it is indistinguishable from zero for sufficiently small frequencies.

For the reader who likes to redo the calculation, it is noted that the squared coherency spectrum cannot be estimated from the raw bivariate periodogram, since this equals unity for any two time series (see, e.g., von Storch and Zwiers 1998). Using the chunk or Daniell method, the problem is solved by choosing the number of the chunks, or the length of Daniell smoother, being larger than one.

3. Conclusions and discussion

This note considers the spectral property at and near zero frequency of a climate variable, x, whose time evolution is determined by ft through Eq. (1). For such a variable, the forcing ft is, by construction, the first difference of xt divided by the time increment Δt. This has the following consequences. Theoretically, the spectral relation between xt and ft, as measured by the squared coherency spectrum, is perfect at any nonzero frequency but degenerated at zero frequency. Practically, it can be difficult to detect the spectral relation between xt and ft at sufficiently low frequencies. The perfectness is caused by the fact that the gain function of the difference operator damps, but does not alter, the low-frequency spectrum of ft in relation to the spectrum of xt. Since the squared coherency spectrum κxf(ω) is a normalized quantity, it remains one, no matter how strong the damping is. The illness at zero frequency, on the other hand, is caused by the efficiency of the difference operator in suppressing low-frequency variations of ft. As frequency goes to zero, more and more variations of ft are suppressed. At zero frequency, as long as the spectral value of xt is finite, which is a reasonable assumption for a climate state variable, the spectral value of ft must diminish to zero. The Γx(0) can therefore not be determined from Γf(0). The difficulty in detecting perfect coherence at low frequencies is more severe for those variables whose spectra remain essentially flat as frequency goes to zero.

The result of this note can be easily generalized for the continuous case. Equation (1) then becomes a differential equation with Δxtt being replaced by dx(t)/dt and xt and ft by x(t) and f(t), which are now continuous functions in time. The gain function of the differential operator d/dt is denoted by G*(ω). The difference between G*(ω) and G(ω) is only noticeable for large ω, in particular for ω near the frequency 1/Δ. At low frequencies, G(ω) approaches 2πω, which is identical to G*(ω). Thus, G*(ω) acts as a high-pass filter for low-frequency variations in the same way as G(ω). The result of this note applies therefore also to a continuous state variable described by a differential equation that is the continuous representation of Eq. (1).

Altogether, one can draw the conclusion that Fig. 1 describes a general phenomenon not only for the global axial atmospheric angular momentum, Mt, but for any arbitrary climate variable, xt, which is described by an equation of type (1) and has finite and essentially constant spectral values at sufficiently low frequencies. For such a variable xt, the efficiency of the difference operator as a high-pass filter will ultimately make the detection of low-frequency spectral relations between xt and its forcing difficult, even when all the processes involved in the forcing are known and have been taken into account.

Finally, it is worthwhile to emphasize that this paper does not question the ability of Eq. (1) in determining the complete evolution of xt over time, but in determining of the spectral value of xt at zero frequency. The constraint in form of G(ω), which makes the spectral representation of (1) degenerated at zero frequency, arises only when the forcing of the spectrum of xt is considered.

Acknowledgments

I thank Hans von Storch for his critical comments, in particular those that led to the consideration of the spectral behaviors of the first difference. I am also grateful to Reiner Schnur for his useful comments and his help concerning calculations using additional spectral estimators. Thanks also to Eduardo Zorita for carefully reading the manuscript.

REFERENCES

  • Jenkins, G. M., and D. G. Watts, 1968: Spectral Analysis and its Application. Holden-Day, 525 pp.

  • von Storch, H., and F. Zwiers, 1998: Statistical Analysis in Climate Research. Cambridge University Press, 528 pp.

  • von Storch, J.-S., V. Kharin, U. Cubasch, G. C. Hegerl, D. Schriever, H. von Storch, and E. Zorita, 1997: A description of a 1260-year control integration with the coupled ECHAM1/LSG general circulation model. J. Climate,10, 1526–1544.

Fig. 1.
Fig. 1.

Squared coherency spectrum between the global atmospheric angular momentum Mt and the global torque Ft. The spectrum is estimated using the chunk method. The number of the chunks is 5. Here, 200 yr of 12-h data obtained from an integration with the coupled ECHAM1/LSG model are used for the calculation. Frequency (period) is in units of cycles per half day (half day). The three dashed lines indicate the 90%, 95%, and 99% confidence levels for the null hypothesis that the coherence is zero.

Citation: Journal of Climate 12, 7; 10.1175/1520-0442(1999)012<2124:WDTSOA>2.0.CO;2

Fig. 2.
Fig. 2.

Gain function G(ω) = 2 sin(πω) of the first difference operator with 0 ⩽ ω ⩽ 0.5. This is a reproduction of Fig. 1.4 in Jenkins and Watts (1968)

Citation: Journal of Climate 12, 7; 10.1175/1520-0442(1999)012<2124:WDTSOA>2.0.CO;2

Fig. 3.
Fig. 3.

Squared coherency spectrum κxf(ω) between xt and ft = xtxt−1 as a function of the length of time series T. Here, xt is generated by a first-order autoregressive process with process coefficient being 0.5. Five chunks are used to estimated κxf(ω). The dashed lines in each panel are, respectively, the 90%, 95%, and 99% confidence levels for the null hypothesis that the coherence is zero.

Citation: Journal of Climate 12, 7; 10.1175/1520-0442(1999)012<2124:WDTSOA>2.0.CO;2

1

The result of this note can be easily generalized for Δxt ≡ (xt+nxtm)/[(n + mt], where n and m can be any finite integers, except n = m = 0. For n = 1 and m = 1, the above expression represents the centered difference.

2

This result holds also for the difference operator as in defined in footnote 1. In this case, the gain function becomes G(ω) = {2 − 2 cos[2π(n + m)ω]}1/2, which is, as the gain function shown in Eq. (4), a high-pass filter that completely suppresses the variations of ft at zero frequency when Γx(0) is finite.

Save
  • Jenkins, G. M., and D. G. Watts, 1968: Spectral Analysis and its Application. Holden-Day, 525 pp.

  • von Storch, H., and F. Zwiers, 1998: Statistical Analysis in Climate Research. Cambridge University Press, 528 pp.

  • von Storch, J.-S., V. Kharin, U. Cubasch, G. C. Hegerl, D. Schriever, H. von Storch, and E. Zorita, 1997: A description of a 1260-year control integration with the coupled ECHAM1/LSG general circulation model. J. Climate,10, 1526–1544.

  • Fig. 1.

    Squared coherency spectrum between the global atmospheric angular momentum Mt and the global torque Ft. The spectrum is estimated using the chunk method. The number of the chunks is 5. Here, 200 yr of 12-h data obtained from an integration with the coupled ECHAM1/LSG model are used for the calculation. Frequency (period) is in units of cycles per half day (half day). The three dashed lines indicate the 90%, 95%, and 99% confidence levels for the null hypothesis that the coherence is zero.

  • Fig. 2.

    Gain function G(ω) = 2 sin(πω) of the first difference operator with 0 ⩽ ω ⩽ 0.5. This is a reproduction of Fig. 1.4 in Jenkins and Watts (1968)

  • Fig. 3.

    Squared coherency spectrum κxf(ω) between xt and ft = xtxt−1 as a function of the length of time series T. Here, xt is generated by a first-order autoregressive process with process coefficient being 0.5. Five chunks are used to estimated κxf(ω). The dashed lines in each panel are, respectively, the 90%, 95%, and 99% confidence levels for the null hypothesis that the coherence is zero.

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