Projections of Southern Hemisphere atmospheric circulation interannual variability

Abstract

An analysis is made of the coherent patterns, or modes, of interannual variability of Southern Hemisphere 500 hPa geopotential height field under current and projected climate change scenarios. Using three separate multi-model ensembles (MMEs) of coupled model intercomparison project phase 5 (CMIP5) models, the interannual variability of the seasonal mean is separated into components related to (1) intraseasonal processes; (2) slowly-varying internal dynamics; and (3) the slowly-varying response to external changes in radiative forcing. In the CMIP5 RCP8.5 and RCP4.5 experiments, there is very little change in the twenty-first century in the intraseasonal component modes, related to the Southern annular mode (SAM) and mid-latitude wave processes. The leading three slowly-varying internal component modes are related to SAM, the El Niño–Southern oscillation (ENSO), and the South Pacific wave (SPW). Structural changes in the slow-internal SAM and ENSO modes do not exceed a qualitative estimate of the spatial sampling error, but there is a consistent increase in the ENSO-related variance. Changes in the SPW mode exceed the sampling error threshold, but cannot be further attributed. Changes in the dominant slowly-varying external mode are related to projected changes in radiative forcing. They reflect thermal expansion of the tropical troposphere and associated changes in the Hadley Cell circulation. Changes in the externally-forced associated variance in the RCP8.5 experiment are an order of magnitude greater than for the internal components, indicating that the SH seasonal mean circulation will be even more dominated by a SAM-like annular structure. Across the three MMEs, there is convergence in the projected response in the slow-external component.

Appendix

The statistical significance of the covariances between the associated time series of the SH 500 hPa geopotential height interannual modes of variability and the corresponding components of a climate field (Sect. 3.2) is able to be estimated through testing of likelihood ratios. Following Grainger et al. (2011a), the monthly anomalies of the associated time series, p _sym , and climate field anomalies, x _sym ′, are respectively defined as

$$p_{sym} \left( i \right) = \beta_{y} \left( i \right) + \delta_{sy} \left( i \right) + \varepsilon_{sym} \left( i \right),$$

(12)

and

$$x_{sym}^{{\prime }} \left( j \right) = \beta_{y}^{{\prime }} \left( j \right) + \delta_{sy}^{{\prime }} \left( j \right) + \varepsilon_{sym}^{{\prime }} \left( j \right),$$

(13)

where i is the index of (1,…, I) modes of variability for a component and j the index of (1,…, J) geographical locations for the climate field. Note that the climate field need not be on the same spatial grid as the modes of variability. The alternative hypotheses are that the covariance between the S-mode, SI-mode and SE-mode associated time series and the corresponding components of the climate field are represented by Eqs. (5), (8) and (9) respectively. For convenience, these are written here as

$$\hat{V}\left\{ {\beta_{y} \left( i \right) + \delta_{sy} \left( i \right),\beta_{y}^{{\prime }} \left( j \right) + \delta_{sy}^{{\prime }} \left( j \right)} \right\} = \hat{V}\left\{ {p_{syo} \left( i \right),x_{syo}^{{\prime }} \left( j \right)} \right\} - \hat{V}\left\{ {\varepsilon_{syo} \left( i \right),\varepsilon_{syo}^{{\prime }} \left( j \right)} \right\},$$

(14)

$$\hat{V}\left\{ {\delta_{sy} \left( i \right),\delta_{sy}^{\prime } \left( j \right)} \right\} = \hat{V}\left\{ {\delta_{sy} \left( i \right) + \varepsilon_{syo} \left( i \right),\delta_{sy}^{{\prime }} \left( j \right) + \varepsilon_{syo}^{{\prime }} \left( j \right)} \right\} - \hat{V}\left\{ {\varepsilon_{syo} \left( i \right),\varepsilon_{syo}^{{\prime }} \left( j \right)} \right\},$$

(15)

and

$$\hat{V}\left\{ {\beta_{y} \left( i \right),\beta_{y}^{{\prime }} \left( j \right)} \right\} = \hat{V}\left\{ {p_{oyo} \left( i \right),x_{oyo}^{{\prime }} \left( j \right)} \right\} - \frac{1}{S}\hat{V}\left\{ {\delta_{sy} \left( i \right) + \varepsilon_{syo} \left( i \right),\delta_{sy}^{{\prime }} \left( j \right) + \varepsilon_{syo}^{{\prime }} \left( j \right)} \right\}.$$

(16)

The null hypotheses are that the estimated covariances are solely due to the intraseasonal (for the slow and slow-internal components) or internal (for the slow-external component) terms in Eqs. (14)–(16). Assuming statistical independence of the components this implies, respectively, that

$$\hat{V}\left\{ {\beta_{y} \left( i \right) + \delta_{sy} \left( i \right) - \beta_{y} \left( i \right) - \delta_{so} \left( i \right),\beta_{y}^{{\prime }} \left( j \right) + \delta_{sy}^{{\prime }} \left( j \right) - \beta_{y}^{{\prime }} \left( j \right) - \delta_{so}^{{\prime }} \left( j \right)} \right\} = 0,$$

(17)

$$\hat{V}\left\{ {\delta_{sy} \left( i \right) - \delta_{oy} \left( i \right),\delta_{sy}^{{\prime }} \left( j \right) - \delta_{oy}^{{\prime }} \left( j \right)} \right\} = 0,$$

(18)

and

$$\hat{V}\left\{ {\beta_{y} \left( i \right) - \beta_{o} \left( i \right),\beta_{y}^{{\prime }} \left( j \right) - \beta_{o}^{{\prime }} \left( j \right)} \right\} = 0.$$

(19)

For each (i, j) pair for each component, the expectation values for each hypothesis in Eqs. (14)–(19) can be expressed as a 2 × 2 cross-covariance matrix. Thus for the slow component alternative hypothesis, from Eq. (14) we obtain

$${\mathbf{V}}_{\mu } = \begin{bmatrix}{\hat{V}\left\{ {\mu_{sy} + \varepsilon_{syo} - \mu_{so} - \varepsilon_{soo} } \right\}} & {\hat{V}\left\{ {\mu_{sy} + \varepsilon_{syo} - \mu_{so} - \varepsilon_{soo} ,\mu_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \mu_{so}^{{\prime }} - \varepsilon_{soo}^{{\prime }} } \right\}} \\{\hat{V}\left\{ {\mu_{sy} + \varepsilon_{syo} - \mu_{so} - \varepsilon_{soo} ,\mu_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \mu_{so}^{{\prime }} - \varepsilon_{soo}^{{\prime }} } \right\}} & {\hat{V}\left\{ {\mu_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \mu_{so}^{{\prime }} - \varepsilon_{soo}^{{\prime }} } \right\}} \end{bmatrix},$$

(20)

where μ _sy  ≡ β _y  + δ _sy and μ _sy ′ ≡ β _y ′ + δ _sy ′, and the (i) and (j) notation has been dropped. Similarly from Eq. (17)

$${\mathbf{V}}_{\mu 0} = \begin{bmatrix}{\hat{V}\left\{ {\mu_{sy} + \varepsilon_{syo} - \mu_{so} - \varepsilon_{soo} } \right\}} & {\hat{V}\left\{ {\varepsilon_{syo} - \varepsilon_{soo} ,\varepsilon_{syo}^{{\prime }} - \varepsilon_{soo}^{{\prime }} } \right\}} \\ {\hat{V}\left\{ {\varepsilon_{syo} - \varepsilon_{soo} ,\varepsilon_{syo}^{{\prime }} - \varepsilon_{soo}^{{\prime }} } \right\}} & {\hat{V}\left\{ {\mu_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \mu_{so}^{{\prime }} - \varepsilon_{soo}^{{\prime }} } \right\}} \end{bmatrix}.$$

(21)

For the slow-internal component, Eqs. (15) and (18) are used to obtain

$${\mathbf{V}}_{\delta } =\begin{bmatrix}{\hat{V}\left\{ {\delta_{sy} + \varepsilon_{syo} - \delta_{oy} - \varepsilon_{oyo} } \right\}} & {\hat{V}\left\{ {\delta_{sy} + \varepsilon_{syo} - \delta_{oy} - \varepsilon_{oyo} ,\delta_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \delta_{oy}^{{\prime }} - \varepsilon_{oyo}^{{\prime }} } \right\}} \\{\hat{V}\left\{ {\delta_{sy} + \varepsilon_{syo} - \delta_{oy} - \varepsilon_{oyo} ,\delta_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \delta_{oy}^{{\prime }} - \varepsilon_{oyo}^{{\prime }} } \right\}} & {\hat{V}\left\{ {\delta_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \delta_{oy}^{{\prime }} - \varepsilon_{oyo}^{{\prime }} } \right\}} \end{bmatrix},$$

(22)

and

$${\mathbf{V}}_{\delta 0} = \begin{bmatrix}{\hat{V}\left\{ {\delta_{sy} + \varepsilon_{syo} - \delta_{oy} - \varepsilon_{oyo} } \right\}} & {\hat{V}\left\{ {\varepsilon_{syo} - \varepsilon_{oyo} ,\varepsilon_{syo}^{{\prime }} - \varepsilon_{oyo}^{{\prime }} } \right\}} \\{\hat{V}\left\{ {\varepsilon_{syo} - \varepsilon_{oyo} ,\varepsilon_{syo}^{{\prime }} - \varepsilon_{oyo}^{{\prime }} } \right\}} & {\hat{V}\left\{ {\delta_{sy}^{{\prime }} + \varepsilon_{syo}^{{\prime }} - \delta_{oy}^{{\prime }} - \varepsilon_{oyo}^{{\prime }} } \right\}} \end{bmatrix}.$$

(23)

For the slow-external component, Eqs. (16) and (19) are used to obtain

$${\mathbf{V}}_{\beta } =\begin{bmatrix}{\hat{V}\left\{ {p_{oyo} - p_{ooo} } \right\}} & {\hat{V}\left\{ {p_{oyo} - p_{ooo} ,x_{oyo}^{{\prime }} - x_{ooo}^{{\prime }} } \right\}} \\ {\hat{V}\left\{ {p_{oyo} - p_{ooo} ,x_{oyo}^{{\prime }} - x_{ooo}^{{\prime }} } \right\}} & {\hat{V}\left\{ {x_{oyo}^{{\prime }} - x_{ooo}^{{\prime }} } \right\}} \end{bmatrix},$$

(24)

and

$${\mathbf{V}}_{\beta 0} =\begin{bmatrix}{\hat{V}\left\{ {p_{oyo} - p_{ooo} } \right\}} & {\hat{V}\left\{ {\delta_{oy} + \varepsilon_{oyo} - \delta_{oo} - \varepsilon_{ooo} ,\delta_{oy}^{{\prime }} + \varepsilon_{oyo}^{{\prime }} - \delta_{oo}^{{\prime }} - \varepsilon_{ooo}^{{\prime }} } \right\}} \\ \\{\hat{V}\left\{ {\delta_{oy} + \varepsilon_{oyo} - \delta_{oo} - \varepsilon_{ooo} ,\delta_{oy}^{{\prime }} + \varepsilon_{oyo}^{{\prime }} - \delta_{oo}^{{\prime }} - \varepsilon_{ooo}^{{\prime }} } \right\}} & {\hat{V}\left\{ {x_{oyo}^{{\prime }} - x_{ooo}^{{\prime }} } \right\}} \end{bmatrix}.$$

(25)

All terms in Eqs. (20)–(25) are able to be calculated from monthly or seasonal mean moments of p(i) and x′(j); see Zheng et al. (2004) for examples of the relevant equations. The local Chi squared statistic, with one degree of freedom, for each (i, j) pair for each component is

$$\chi^{2} \left( {i,j} \right) = 2\left( {{\text{LH}}_{0} \left( {i,j} \right) - {\text{LH}}_{\text{A}} \left( {i,j} \right)} \right),$$

(26)

where LH₀ and LH_A are the log-likelihoods of the null and alternative hypotheses respectively, and are calculated using the multivariate normal distribution (e.g. Wilks 2006) with the appropriate cross-covariance matrices V ₀ and V.