Bjerknes’ hypothesis on the coldness during AD 1790–1820 revisited

Abstract

The aim of this paper is to re-examine and quantify a hypothesis first put forward by J. Bjerknes concerning the anomalous coldness during the AD 1790–1820 period in western Europe. Central to Bjerknes’ hypothesis is an anomalous interaction between ocean and atmosphere studied here using an ocean-atmosphere coupled climate model of intermediate complexity. A reconstruction of the sea-level pressure pattern over the North Atlantic sector averaged over the period 1790–1820 is assimilated in this model, using a recently developed technique which has not been applied to paleoclimatic modelling before. This technique ensures that averaged over the simulation the reconstructed pattern is retrieved whilst leaving atmospheric and climatic variability to develop freely. In accordance with Bjerknes’ hypothesis, the model results show anomalous southward advection of polar waters into the northeastern North Atlantic in the winter season, lowering the sea-surface temperatures (SSTs) there with 0.3–1.0°C. This SST anomaly is persistent into the summer season. A decrease in western European winter surface air temperatures is found which can be related almost completely to advection of cold polar air. The decrease in summer surface air temperatures is related to a combination of low SSTs and anomalous atmospheric circulation. The modelled winter and summer temperatures in Europe compare favourably with reconstructed temperatures. Enhanced baroclinicity at the Atlantic seaboard and over Baffin Island is observed along with more variability in the position of the North Atlantic storm tracks. The zone of peak winter storm frequency is drawn to the European mid-latitudes.

Appendix

1.1 Forcing singular vectors

An introduction to forcing singular vectors applied to the context of this study follows here. A more extensive account is given by Barkmeijer et al. (2003), where the application used in this study is termed ‘forced sensitivity calculations’.

Write the climate model in the following abstract form,

$$ \frac{{dx}} {{dt}} = G(x), $$

(3)

with x the vector containing all prognostic variables of the model and G(x) the model formulation. In numerical weather prediction models, the forecast quality will depend on e.g. the parametrization of physical processes in the model formulation, which motivates studying the model response to small perturbations of the model tendencies. Here, we assume that the (external and/or internal) forcings which gave rise to a historic atmospheric state, anomalous to a modern climatology, can be regarded as a perturbation f of the model tendencies (whose spatial pattern and amplitude is kept constant in time). The linear evolution equation of this anomaly in the atmospheric state ɛ satisfies:

$$ \frac{{d\varepsilon }} {{dt}} = {\mathbf{L}}\varepsilon + {\mathbf{f}}, $$

(4)

where L denotes the time dependent Jacobian of G, evaluated along a solution of equation (Eq. 3). Solutions of equation (Eq. 4) take the form:

$$ \varepsilon (T) = {\mathbf{M}}(0,T)\varepsilon (0) + \int\limits_0^T {M(s,T){\mathbf{f}}ds,} $$

(5)

where M(s, T) is the propagator from time s to time T of equation (Eq. 4) without forcing: f = 0. For an arbitrary forcing f, the vector ${\mathbf{y}} = \mathcal{M}{\mathbf{f}},$ where

$$ \mathcal{M} = \int\limits_0^T {{\mathbf{M}}(s,T)ds,} $$

(6)

is simply determined by integrating (Eq. 4) to time t = T with initial condition ɛ (0)=0.

The forcing perturbation is now determined by minimization of the cost function

$$ J(f) = \left| {P(\mathcal{M}{\mathbf{F}} - \psi } \right| $$

(7)

where ψ is the target pattern and P is a projection operator onto the extra-tropical North Atlantic sector (taken to be bounded by 90°W-40°E, 15°N-90°N). The gradient of J(f), required in the quasi-Newton conjugate gradient method (NAG’s E04DGF), is:

$$ \nabla J = 2\mathcal{M}*P*P(\mathcal{M}f - \psi ) $$

(8)

To derive the adjoint of $\mathcal{M}$ it is instructive to write (Eq. 4) as:

$$ \frac{d} {{dt}}\left( {\begin{array}{*{20}c} \varepsilon \\ {\mathbf{f}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\mathbf{L}} & {\mathbf{I}} \\ 0 & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} \varepsilon \\ {\mathbf{f}} \\ \end{array} } \right), $$

(9)

where I and 0 are the identity and zero operator respectively. The adjoint of (Eq. 9) is:

$$ - \frac{d} {{dt}}\left( {\begin{array}{*{20}c} {\hat \varepsilon } \\ {{\mathbf{\hat f}}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\mathbf{L}}*} & {\mathbf{I}} \\ 0 & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\hat \varepsilon } \\ {{\mathbf{\hat f}}} \\ \end{array} } \right). $$

(10)

By writing (Eq. 10) as a coupled system:

$$ - \frac{d} {{dt}}\hat \varepsilon = {\mathbf{L}}*\hat \varepsilon $$

(11)

$$ - \frac{d} {{dt}}{\mathbf{\hat f}} = \hat \varepsilon $$

(12)

it follows how to determine $\mathcal{M}*{\mathbf{y}}$ for a given inputvector y. First: integrate the regular adjoint model as given by equation (Eq. 11) backward in time t = T to time t=0, with $\hat \varepsilon (T) = {\mathbf{y}}.$ Second: integrate equation (Eq. 12) backward in time from time t = T using the intermediate fields of the adjoint integration (Eq. 11) as tendencies for the corresponding time step and ${\mathbf{\hat f}}(T) = 0.$ Integrating to time t=0 yields $\mathcal{M}*{\mathbf{y}} = {\mathbf{\hat f}}(0)$