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

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where, without loss of generality, dx/dt is discretized using Δx/Δt ≡ (x_t − x_t−1)/Δt with Δt being the time increment.¹ Function f_t, 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 f_t as a function of the state variables is exactly known, x_t for t ∈ [−∞, ∞] is in principle also known. By assuming that x_t and f_t in Eq. (1) can be considered as random variables, the spectra of x_t and f_t, which can be derived from the time series of x_t and f_t, 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, κ_xf(ω), between x_t and f_t [for the definition of κ_xf(ω), see, e.g., Jenkins and Watts 1968] is exactly one, implying a perfect spectral relation between x_t and f_t. Since this situation appears to be trivial, the spectral relation between x_t and f_t has barely been seriously studied. Almost all theoretical studies are devoted to the understanding of the deterministic processes constituting f_t.

This note questions the usefulness of knowledge about f_t in determining the spectrum of x_t at zero frequency. The consideration is motivated by the squared coherency spectrum κ_MF(ω) between the global atmospheric axial angular momentum M_t and the global axial torque F_t derived from a numerical integration of a climate model. The M_t and F_t 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 κ_MF(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 M_t and F_t 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 F_t must be, by construction, M_t − M_t−1 divided by Δt.

It is noted that M_t 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 f_t, the expected spectral relation between x_t and f_t 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 f_t is sufficient for determining everything of x (i.e., not only any value of x_t at a given time t, but also the variance and spectrum of x_t) would therefore become questionable.

2. Consequences of the constraint f_t = (x_t − x_t−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 x_t and its forcing f_t, which determines the time evolution of x_t and satisfies (1), are considered. It is assumed that f_t is exactly known. With this assumption, f_t must be exactly equal to (x_t − x_t−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

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x_t − x_t−1 is known as the first difference of x_t.

It can be shown (von Storch and Zwiers 1998), that the spectrum Γ_x(ω) of x_t and the spectrum Γ_f(ω) of f_t = (x_t − x_t−1) are related to each other via

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with

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where Γ_x(ω), Γ_f(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), Γ_f(ω) is simply a filtered version of Γ_x(ω). The filter G(ω) acts to respectively enhance and damp, rather than to alter, the spectrum of f_t in relation to the spectrum of x_t. The enhancement occurs at high frequencies with ¹/₂ ≥ ω > 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 Γ_f(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.

The efficiency of the first difference operator as a high-pass filter can cause practical difficulty in detecting the spectral relation between x_t and f_t. This is particularly true when Γ_x(ω) remains essentially constant as ω goes to zero. The values of Γ_f(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 x_t and f_t 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, Γ_x(0) represents the flat spectral level at low frequencies. The spectrum Γ_f(ω), which is then proportional to ω², has extremely small values as ω goes to zero. The smallness of Γ_f(ω) makes the detection of the spectral relation between x_t and f_t difficult. In the following, this difficulty is further illustrated by considering the squared coherency spectrum κ_xf(ω) between x_t and f_t = (x_t − x_t−1) for an arbitrary x_t, whose spectrum is essentially flat at low frequencies.

Figure 3 shows κ_xf(ω), where x_t 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 κ_xf(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, κ_xf(ω) is not distinguishable from zero at low frequencies. Taking ω₉₅ to be the frequency below which κ_xz(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 x_t and f_t at frequency ω → 0 is futile. The suggestion is consistent with the result that a finite Γ_x(0) cannot be determined from Γ_f(0).

The generality of the difficulty in detecting the spectral relation between x_t and f_t 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 κ_xf(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 x_t and the other being f_t. No manner how κ_xf(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 f_t through Eq. (1). For such a variable, the forcing f_t is, by construction, the first difference of x_t divided by the time increment Δt. This has the following consequences. Theoretically, the spectral relation between x_t and f_t, 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 x_t and f_t 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 f_t in relation to the spectrum of x_t. Since the squared coherency spectrum κ_xf(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 f_t. As frequency goes to zero, more and more variations of f_t are suppressed. At zero frequency, as long as the spectral value of x_t is finite, which is a reasonable assumption for a climate state variable, the spectral value of f_t must diminish to zero. The Γ_x(0) can therefore not be determined from Γ_f(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 Δx_t/Δt being replaced by dx(t)/dt and x_t and f_t 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, M_t, but for any arbitrary climate variable, x_t, 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 x_t, the efficiency of the difference operator as a high-pass filter will ultimately make the detection of low-frequency spectral relations between x_t 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 x_t over time, but in determining of the spectral value of x_t 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 x_t is considered.