\documentclass[twoside]{article}
\usepackage[silent,pdf,hyper]{elxshort}
\usepackage{color}
\usepackage{hyperref}
\journalid{ES}{19}{3}{19}
\citnumber{}{4}
\paperid{2018ES000653}
\papertype{RAR}
\cpright{}{2019}
\lefthead{bekryaev}
\righthead{north atlantic climate variability characteristics}
\received{23}{October}{2018}
\accepted{30}{November}{2018}
\published{3}{June}{2019}
\definecolor{brown}{RGB}{157,23,4}

\authaddr{Roman V. Bekryaev,
Earth Science Institute, St. Petersburg State University, St. Petersburg, Russia.
(bekryaev@mail.ru)}


\begin{document}

\title{Interrelationships of the North Atlantic multidecadal climate variability characteristics}

\author{1}{Bekryaev}{Roman V.}{}{1}{2}{}

\affil{1}{Earth Science Institute, St. Petersburg State University}{St. Petersburg}{}{Russia}
\affil{2}{Voeikov Main Geophysical Observatory}{St. Petersburg}{}{Russia}

\keywd{North Atlantic}
\keywd{stochastic forcing}
\keywd{variability}
\keywd{Atlantic Multidecadal Oscillation}
\keywd{cross-correlation}

% Interrelationships of the North Atlantic multidecadal climate variability characteristics
% Roman V. Bekryaev
% 1 Earth Science Institute, St.-Petersburg State University, St.-Petersburg, Russia
%2 Voeikov Main Geophysical Observatory, St.-Petersburg, Russia bekryaev@mail.ru



\abstract{The North Atlantic is one of the key regions, where low-frequency climate
variability is formed. However, despite numerous studies related to this topic, some
issues still remains unsolved. One of them is the ambiguous cross-correlation of the
North Atlantic sea surface temperature (SST) and the intensity of Atlantic Meridional
Overturning Circulation (AMOC). A widely accepted concept suggests that the long-term climate variability is a result of the atmospheric stochastic forcing transformed by
the inertial ocean. Existence of negative and positive feedback mechanisms suggests
that the long-term North Atlantic dynamics may be considered as a damped
stochastically forced oscillator in which both SST and AMOC are the elements of the
same process. In this study, we analyze the cross-correlation functions of the main
North Atlantic climatic indexes derived from a simple box-like stochastic model. The
random forcing simulates the air-sea interface heat fluxes and excites both the SST and
the AMOC. Stochastic excitation of the meridional circulation implies the leading
AMOC and stochastic forcing of SST implies the leading Atlantic Multidecadal
Oscillation (AMO). Connection of the AMOC and AMO indexes depends on the
principal oceanic feedbacks and the dissipation intensity.}


\section{1. Introduction}

   One of the key regions where the climate multidecadal variability (MDV) originates
is the North Atlantic. The most impressive example of long-term changes may be
related to the Atlantic Multidecadal Oscillation. The AMO index is defined by spatially
averaged detrended anomalies of the North Atlantic SST
[\itc{Enfield et al.,} \reflink{Enfi01}{2001};
\itc{Knight et al.,} \reflink{Knig05}{2005}, \reflink{Knig06}{2006}].
However, there are some studies which use slightly different AMO
definition, e.g.
[\itc{d'Orgeville and Peltier,} \reflink{Orge07}{2007};
\itc{Trenberth and Shea,} \reflink{Tren06}{2006}].\vspace*{.6ex}

   The North Atlantic SST variability is closely connected to dynamics of the large-scale oceanic components, such as AMOC and Subpolar Gyre (SPG). The AMOC plays
an important role in the meridional oceanic heat transfer. The largest Atlantic
meridional energy transport reaches $\sim 1.2$~PW across 26\deg N
[\itc{Johns et al.,} \reflink{John11}{2011};
\itc{Ushakov and Ibrayev,} \reflink{Usha18}{2018}].\vspace*{.6ex}

   The leading role of low-frequency variability with the scale of 50--80 years in the
North Atlantic climate system was found in
[\itc{Enfield et al.,} \reflink{Enfi01}{2001};
\itc{Polyakov et al.,} \reflink{Poly10}{2010};
\itc{Schlesinger and Ramankutty,} \reflink{Schl94}{1994}].
 Modeling experiments by
\itc{Delworth and Mann} [\reflink{Delw00}{2000}]
 showed the pronounced 7--8 decades spectral maxima. Spectral analysis
led
\itc{Wei and Lohmann} [\reflink{Wei12}{2012}]
to similar conclusions.
\itc{Wouters et al.} [\reflink{Wout12}{2012}]
found the
dominance AMOC variability with a scale of 50--60 years.
\itc{Frankcombe and Dijkstra} [\reflink{Fran11}{2011}]
and
\itc{Frankcombe et al.} [\reflink{Fran10}{2010}]
analyzing the 500-years GFDL CM 2.1 control
experiments have shown two broad extremes of variability on the time scales of 2--3 and
5--7 decades.
\itc{Chylek et al.} [\reflink{Chyl12}{2012}]
came to similar results analyzing the isotope $\delta^{18}$O
samples of the Greenland ice sheet drilling. MDV of the AMOC index covers the time
range from 50 to 200 years
[\itc{Danabasoglu et al.,} \reflink{Dana12}{2012}].
Using the Kiel Climatic Model
experiments
\itc{Ba et al.} [\reflink{Ba13}{2013}]
analyzed the AMOC and AMO indexes and showed a
roughly 60-year oscillation mode. Empirical Mode Decomposition of the AMO time
series also demonstrated the prevalence of MDV
 [\itc{Chen et al.,} \reflink{Chen16}{2016}].
  The same
conclusions follow from the analysis of the North Atlantic SST and turbulent surface
heat fluxes
[\itc{Gulev et al.,} \reflink{Gule13}{2013}].

   In spite of the well-established leading role of the MDV, frequency structure of the
North Atlantic climatic indexes still remains ambiguous. Moreover, the existence of
spectrum extremes seems doubtful
[\itc{Cane et al.,} \reflink{Cane17}{2017}].
For example,
 \itc{Medhaug and Furevik} [\reflink{Medh11}{2011}]
 found no statistically significant spectral maxima in the AMO and
AMOC time series derived from CMIP3 model runs. The same indexes calculated using
CMIP5 experiments also demonstrated an absence of the spectral density maxima in any
frequency ranges
[\itc{Zhang and Wang,} \reflink{ZhWa13}{2013}].
 Analysis of the AMO variability by
  \itc{Ba et al.} [\reflink{Ba14}{2014}]
  and
 \itc{Clement et al.} [\reflink{Clem15}{2015}]
 confirmed these conclusions. Multidecadal and
multicentury AMO variability studied by
 \itc{Park and Latif} [\reflink{Park08}{2008}]
 and
 \itc{Dommenget and Latif} [\reflink{Domm08}{2008}]
 showed the spectral density continuum without pronounced extremes. Lack
of the spectral maxima can be explained as a feature of the nonlinear chaotic dynamics
or as a linear transform of a stochastic external forcing.

	The possible mechanisms of the MDV were in a focus of numerical studies.
Several hypotheses explaining mechanisms of the MDV have been proposed. One of
them associates the formation of MDV with long-term external forcing, first of all to the
aerosol of volcanic origin
[\itc{Ottera et al.,} \reflink{Otte10}{2010};
\itc{Booth et al.,} \reflink{Boot12}{2012}].
However, this idea
was criticized later by
\itc{Zhang et al.} [\reflink{Zhan13}{2013}]
who showed that historical forcing was too
weak to serve as the main driver of MDV. Alternative point of view was outlined by
\itc{Bellomo et al.} [\reflink{Bell18}{2018}]
who stressed the prevailing role of external factors at the
timescale from two decades and longer.

   The other concept suggests the leading role of nonlinear processes in the ocean-atmosphere system
   [\itc{Dijkstra and Ghil,} \reflink{Dijk05}{2005};
   \itc{Dijkstra et al.,} \reflink{Dijk08}{2008}].
   The proposed
mechanism assumes the existence of two quasi-stable stationary states of the Atlantic
thermohaline circulation
[\itc{Cessi,} \reflink{Cess94}{1994};
\itc{Rahmstorf,} \reflink{Rahm95}{1995}]
or SPG
 [\itc{Born and Stocker,} \reflink{Born14}{2014};
 \itc{Born et al.,} \reflink{Born15}{2015}].
 Dynamical coupling between the atmospheric and oceanic
circulation can serve as an alternative mechanism responsible for the low-frequency
climate variability
[\itc{Wills et al.,} \reflink{Will19}{2019}].
 Finally, the most accepted hypothesis explains
the formation of MDV via response of the highly inertial ocean system to the random
atmospheric forcing
[\itc{Clement et al.}, \reflink{Clem15}{2015};
\itc{O'Reilly et al.,} \reflink{Reil16}{2016}].

\begin{sloppypar}
   The circuit of several cause-and-effect feedbacks argues in favor of describing the
process as a damped oscillator
[\itc{Park and Latif}, \reflink{Park10}{2010};
\itc{Sevellec and Huck,} \reflink{Seve15}{2015}].
Intensification of the AMOC means the advection of warm waters into the northern part
of the North Atlantic, which leads to the positive AMO phase. Warm SST anomalies in
the subpolar North Atlantic region prevent deep winter convection. Weakening of the
deep water formation leads to the AMOC decrement. Corresponding deficit of the
meridional heat transport implies in turn the development of negative SST anomalies
and transition to the negative AMO phase. Cold surface waters in the region of
Greenland and Labrador seas initiate formation of the deep waters and increase of the
AMOC.
\end{sloppypar}

   The majority of the Coupled General Circulation Models (GCM) experiments
demonstrated that the AMOC index leads AMO by several years. As a rule, the indexes
are positively correlated at zero time lag.
\itc{Gastineau et al.} [\reflink{Gast16}{2016}]
analyzed the control
IPSL-CM5A-LR 500-year climate run and found that the cross-correlation is positive
with AMOC leading AMO by 5--12 year. The correlation $r \approx 0.42$   is maxima with the
time lag of 8 years.
 \itc{Van Oldenborgh et al.} [\reflink{Olde09}{2009}]
 and
 \itc{Sun et al.} [\reflink{Sun15}{2015}]
 obtained similar
estimate of the AMOC and AMO pair correlation, $r\approx 0.55 \div 0.60$, analyzing the GCM
simulations. On the other hand, the COSMOS model generated the AMOC and AMO
indexes changing almost in phase
[\itc{Wei and Lohmann,} \reflink{Wei12}{2012}].
 For five CMIP3 model
runs the AMOC-AMO correlation coefficient ranges between 0.1 and 0.5, with the
strongest correlation found when AMOC leads AMO by 0 to 3 years
[\itc{Day et al.,} \reflink{Day12}{2012}].
\itc{Roberts et al.} [\reflink{Robe13}{2013}]
investigated 10 CMIP5 model runs and demonstrated similar
results. The highest correlation was found when AMOC leads by 1--5 years.

   However, there is much complexity in the analysis of the AMOC-AMO connections.
The cross-spectral analysis carried out by
\itc{Marini and Frankignoul} [\reflink{Mari14}{2014}]
 displayed that
the low-frequency AMOC and AMO indexes variability resides in phase. Contrary, the
phase shift between the two indexes in the model IPSLMC5 was close to 180\deg
 [\itc{Marini and Frankignoul,} \reflink{Mari14}{2014}].
  The estimates derived from 26 CMIP5 models
  [\itc{Muir and Fedorov,} \reflink{Muir15}{2015}]
  revealed large scatter of the AMOC and AMO characteristics.
   \itc{Zhang and Wang} [\reflink{ZhWa13}{2013}],
   \itc{Wang and Zhang} [\reflink{Wang13}{2013}]
   demonstrated the ambiguous AMOC and
AMO relationship, when the correlation in some cases was negative.

\begin{sloppypar}
   What causes the uncertain AMOC-AMO interconnections?
   \itc{Tandon and Kushner} [\reflink{Tand15}{2015}],
   analysing the CMIP3 and CMIP5 control and historical runs, suggested that the
external atmospheric forcing can destroy the dynamical AMOC-AMO relationships. As
a rule, the leading role belongs to the AMOC. An increase of the AMOC index initiates
the positive SST anomalies in the subpolar North Atlantic region. However, when an
external forcing switches the scenario, the AMO takes the leading role. Now the SST
increase induces attenuation of the meridional thermohaline circulation and the
correlation of the two indexes becomes negative.
\end{sloppypar}


   In this study we further develop the idea of
   \itc{Tandon and Kushner} [\reflink{Tand15}{2015}].
   We
hypothesize that the atmospheric random forcing is responsible for the MDV formation.
In that, the leading role of AMOC is determined by the oceanic system feedbacks. The
heat fluxes at the air-sea interface can impact both SST and meridional overturning
circulation via intensification of the deep oceanic convection. In case of random
excitation of the meridional circulation, the AMOC and AMO indexes are positively
correlated. However, the stochastic atmospheric forcing affecting the SST means that
correlation between the indexes changes to negative. Theoretical justification of this
hypothesis is in the focus of this study.

  \begin{sloppypar}
 In the second section, we describe the box stochastic model of the North Atlantic.
The third section is devoted to cross-correlation analyses of the model. Finally, we
discuss the results and draw the conclusions.
\end{sloppypar}



\section{2. A Simple North Atlantic Model}

   \itc{Legatt et al.} [\reflink{Lega12}{2012}]
   formulated a box North Atlantic model. Built on the
   [\itc{Marshall et al.,} \reflink{Mars01}{2001}]
   concept, the model associates the excitation of low-frequency oceanic
variability with stochastic atmospheric forcing which is spatially coherent. The main
role in the forcing belongs to the North Atlantic Oscillation (NAO) strongly impacting
the deep winter convection
[\itc{McCarthy et al.,} \reflink{McCa15}{2015}].
 On the other hand, NAO also
controls the energy exchange between the ocean and the atmosphere forming the large-scale SST anomalies.

\enlargethispage{-0.85\baselineskip}

   \itc{Legatt et al.} [\reflink{Lega12}{2012}]
   suggested that the principal factors forming the North Atlantic
MDV include the meridional $\psi_m$  and horizontal  $\psi_g$ components of stream function, and
the spatially averaged subpolar North Atlantic SST $T$. The intensities of  $\psi_m$ and $T$  can
serve as the analogues of AMOC and AMO indexes, respectively. We have modified
\itc{Legatt et al.} [\reflink{Lega12}{2012}]
model constraining it for simplicity to two equations. In that, the
horizontal component of the stream function was neglected. Besides, the dissipative
term was added to the evolutionary equation of the meridional steam function. The
dimensionless model equations can be written as

\begin{equation}\label{1}
\frac{d T}{dt} = m\psi_m - \lambda T + F_T (t)
\end{equation}   	 			%	(1)

\begin{equation}\label{2}
\frac{d \psi_m}{dt} = - s T - \alpha \psi_m  + F_m (t)
\end{equation}   	 			%		(2)
where  $F_T(t)$ is the atmospheric forcing determined by the net heat surface fluxes, $F_m(t)$
is the forcing responsible for the deep water formation and AMOC intensification. We
assume the external forcing as a linear function of a stationary Gaussian random process $X(t) =\partial W/\partial t$
  (the derivative of the standard Wiener process, $W$);
  $F_T(t) = \sigma_{F_T} X (t)$ and
  $F_m(t) = -\sigma_{F_m} X (t)$,
  where $\sigma^2_{F_T}$  and  $\sigma^2_{F_m}$ are variances of the corresponding fluxes. The
negative sign in the last relationship is due to the fact that surface heating of the North
Atlantic waters damps the deep oceanic convection. The coefficients $\lambda$   and  $\alpha$  describe
the heat and impulse dissipation, respectively; the parameters $m$  and  $s$ are positive and
simulate the AMOC-AMO linear feedbacks. Unit of the dimensionless time corresponds
to $\sim 4$ years of physical time, the product $ms \approx 3$,
[\itc{Legatt et al.,} \reflink{Lega12}{2012}].
Note, that the
heat fluxes are considered positive when energy is gained by the ocean (in contrast to
[\itc{Gulev et al.,} \reflink{Gule13}{2013}]).

   In general, the model formulation assumes that the main peculiarities of the North
Atlantic climate variability can be described in the framework of randomly exited
damped oscillator.

\section{3. Correlation and Cross-Correlation Analyses of AMO and AMOC}

\subsection{3.1. Stochastic Forcing of the Meridional Thermohaline Circulation}

  In the event of the non-zero meridional thermohaline circulation forcing, i.e.
$F_m (t) \neq 0$, $F_T (t) = 0$ the equation for SST takes the form of the linear damping
oscillator

\begin{equation}\label{3}
\frac{d^2 T}{dt^2}  + 2a \frac{d T}{dt} + b T= mF_m (t)
\end{equation}   	 			%	(3)

\begin{equation}\label{4}
\frac{d \psi_m}{dt} = - s T - \alpha \psi_m  + F_m (t)
\end{equation}   	 			%		(4)
The coefficients are defined as

\begin{eqnarray*} % \begin{equation}\label{5}
2a = \alpha + \lambda, \qquad b= ms + \alpha \lambda
\end{eqnarray*} % \end{equation}   	 			%		(5)


   This way, the evolution of the SST is described by the linear differential equation of
the second order with a stochastic forcing. The statistically stable solutions of
 \eqref{3}        % Eqns. (3)
and
\eqref{4}        % (4)
under the assumptions of $a>0$  and  $b-a^2 >0$ yields (see e.g.
[\itc{Yaglom,} \reflink{Yagl87}{1987}])

\begin{equation}\label{5}
T(t) = \int\limits_0^{\infty} h_T (u) F_m (t-u) du
\end{equation}   	 			%		(6) N5
where $h_T (u)$  is weight function

\begin{equation}\label{6}
h_T (u) = \frac{m}{\beta} \exp (-au) \sin (\beta u)
\end{equation}   	 			%		(7) N6

\begin{eqnarray*} % \begin{equation}\label{8}
\beta= \sqrt{b-a^2}
\end{eqnarray*}  %\end{equation}   	 			%		(8)

\begin{eqnarray*}
\psi_m(t) =
\end{eqnarray*}
\begin{equation}\label{7}
\int\limits_0^{\infty}\exp(-\alpha u) (- sT(t-u) + F_m (t-u)) du
\end{equation}   	 			%		(9) N7


%%%%%%%%%%%%%%%%%%   After simplifications (see Appendix A for details),
After simplifications (see \hyperref{}{section}{apnd1}{Appendix A} for details),
   \eqref{7}        % Eqn. (9) N7
   can be presented in the
form of the Duhamel integral

\begin{eqnarray*}
                  % \begin{equation}\label{10}
\psi_m(t) = \int\limits_0^{\infty}h_m( u)  F_m (t-u) du
\end{eqnarray*}       % \end{equation}   		%		(10)
where  $h_m (u)$ is a weight function

\begin{eqnarray*}
h_m (u) = \exp(-\alpha u) +
\end{eqnarray*}
\begin{eqnarray*}
% \frac
\Bigl\{ ms(- \exp(-\alpha u) \beta +  \exp(-a u) \times
\end{eqnarray*}
\begin{eqnarray*}
[\beta \cos (\beta u) + (a-\alpha) \sin (\beta u)] )\Bigr\} \Bigl/
\end{eqnarray*}
\begin{equation}\label{8}
\Bigl\{ \beta ((a-\alpha)^2 + \beta^2)\Bigr\}
\end{equation}   	 			%		(11)		N8


   The correlation and cross-correlation functions can be written
   (see \hyperref{}{section}{apnd2}{Appendix B},
equation \hyperref{}{equation}{eqB2}{B2})  as

\begin{equation}\label{9}
B_T(\tau) = \sigma^2_{F_m} \int\limits_0^{\infty}h_T( \xi) h_T (\xi+ \tau)
d \xi
\end{equation}   	 			%		(12) N9

\begin{equation}\label{10}
B_{\psi_m}(\tau) = \sigma^2_{F_m} \int\limits_0^{\infty}h_m( \xi) h_m (\xi+ \tau) d \xi
\end{equation}   	 			%		(13) N10

\begin{equation}\label{11}
B_{T, \psi_m}(\tau) = \sigma^2_{F_m} \int\limits_0^{\infty}h_T( \xi) h_m (\xi+ \tau) d \xi
\end{equation}   	 			%		(14) N11
The weight functions
\eqref{6}        % (7)
and
\eqref{8}        % (11)
allow analytical integration of the
\eqref{9}--\eqref{11}        % Eqns. (12--14).
The
correlation functions can be presented in the following normalized form

\begin{eqnarray*}
r_T(\tau) = \frac{B_T (\tau)}{\mathrm{Var} [T]} =
\end{eqnarray*}
\begin{eqnarray*}         % \begin{equation}\label{15}
\exp(-a|\tau| ) \bigl( \cos (\beta \tau) + \frac{a}{\beta} \sin (\beta|\tau|) \bigr)
\end{eqnarray*}         % \end{equation}   	 			%		(15)

\begin{eqnarray*}
r_{\psi_m}(\tau) = \frac{B_{\psi_m} (\tau)}{\mathrm{Var} [\psi_m]} =
\end{eqnarray*}
\begin{eqnarray*}
\Bigl\{\exp(-a|\tau| ) \bigl( \beta( b+ \lambda^2) \cos (\beta \tau) -
\end{eqnarray*}
\begin{eqnarray*}  % \begin{equation}\label{16} %
a(b-\lambda^2) \sin (\beta|\tau|) \bigr) \Bigr\}  \Bigr /
\Bigl\{ \beta (b + \lambda^2) \Bigr\}
\end{eqnarray*}         % \end{equation}   	 			%		(16)

   Note, that only random process  $T(t)$ is differentiable.

   The variances of SST and the meridional component of stream function are defined
as
\begin{eqnarray*}       % \begin{equation}\label{17}
\mathrm{Var} [T] = \frac{\sigma^2_m m^2}{4a (\beta^2 + a^2)}
= \frac{\sigma^2_m m^2}{4a b}
\end{eqnarray*}  %\end{equation}   	 			%		(17)

\begin{eqnarray*}   % \begin{equation}\label{18}
D[\psi_m] = \frac{\sigma^2_m (b+ \lambda^2)}{4a b}
\end{eqnarray*}         %\end{equation}   	 			%		(18)

   The normalized cross-correlation function of $T(t)$  and $\psi_m (t)$  can be written

\begin{eqnarray*}
r_{T, \psi_m}(\tau) =
%\frac{
\Bigl\{\exp(-a|\tau| ) ( 2 \beta \lambda \cos (\beta \tau) -
\end{eqnarray*}
\begin{eqnarray*}
2 b \sin (\beta \tau ) + \lambda (\alpha +\lambda) \sin (\beta |\tau|))\Bigr\} \Bigl/
\end{eqnarray*}
\begin{equation}\label{12}
\Bigl\{ 2\beta \sqrt{b+ \lambda^2}\Bigr\}
\end{equation}   	 			%		(19) N12

   The normalized cross-correlation function at the zero time lag, $\tau =0$  is described by
the relationship

\begin{eqnarray*} % \begin{equation}\label{20}
r_{T, \psi_m} (\tau=0) = \frac{\lambda}{\sqrt{b+\lambda^2}}
\end{eqnarray*} % \end{equation}   	 			%		(20)

   Positive correlation of $T(t)$ and $\psi_m (t)$ is found in many estimates derived from the
GCM numerical simulations. However, the magnitude of the correlation coefficient $r_{T, \psi_m} (0)$
  can be relatively modest suggesting that the GCM-based estimates of
  $r_{T, \psi_m} (0)$
cannot serve as a strong argument in the AMOC-AMO interrelationship discussion. The
derivative of the cross-correlation function by the zero lag, is always negative

 \begin{equation}\label{13}
\frac{ d r_{T, \psi_m}}{d \tau} \Bigl|_{\tau=0} = \frac{- b}{\sqrt{b+\lambda^2}}
\end{equation}   	 			%		(21)

   Analysis of the
formula
\eqref{12}                % (19)
provides simple equation defining the time delay
corresponding to the largest AMO-AMOC correlation
	
\begin{eqnarray*}     %  \begin{equation}\label{22}
\tau_{ex} = \frac{-\arccos \Bigl(
\displaystyle{\frac{\alpha + 3 \lambda}{2 \sqrt{ ms + 2\lambda (\alpha+\lambda)}} \Bigr) }}{\beta}
\end{eqnarray*}    %\end{equation}   	 			%		(22)

Applying the parameters values  $ms=3$, $\alpha=\lambda=0.5$ one can obtain an estimate of
dimensionless time $\tau_{ex} \approx - 0.6$. It corresponds approximately to 2.5 years and, therefore, is
in agreement with the GCM simulations (e.g.
[\itc{Day et al.,} \reflink{Day12}{2012};
\itc{Roberts et al.,} \reflink{Robe13}{2013}]).

\subsection{3.2. Stochastic Forcing of the SST}

\begin{sloppypar}
   The second choice for stochastic atmospheric forcing is the direct excitation of the
SST. For the sake of simplicity, we will consider here the case when,
$F_T (t) \neq 0$, $F_m (t) = 0$. The normalized correlation functions and variances of the SST
and the meridional component of stream function (we apply here upper asterisk to
distinguish from the random excitation of the stream function $\psi_m$) can be represented
now as
\end{sloppypar}


\begin{eqnarray*}
r_{T}^*(\tau) =
% \frac{
\Bigl\{ \exp(-a|\tau| ) [ (b+ \alpha^2)  \beta \cos (\beta \tau) -
\end{eqnarray*}
\begin{eqnarray*}
 a(b-\alpha^2) \sin (\beta|\tau|)]\Bigr\} \Bigl/
 \end{eqnarray*}
\begin{eqnarray*}     % \begin{equation}\label{23}
\Bigl\{ (b+ \alpha^2) \beta\Bigr\}
\end{eqnarray*}           %\end{equation}   	 			%		(23)

\begin{eqnarray*}
r_{\psi_m}^*(\tau) =
\end{eqnarray*}
\begin{eqnarray*}     % \begin{equation}\label{24}
\exp(-a|\tau| ) \bigl( \cos (\beta \tau) + \frac{a}{\beta} \sin (\beta|\tau|) \bigr)
\end{eqnarray*}      % \end{equation}   	 			%		(24)

\begin{eqnarray*}  % begin{equation}\label{25}
\mathrm{Var}^* [T] = \frac{\sigma^2_T(b+\alpha^2)}{4a b}
\end{eqnarray*}     %\end{equation}   	 			%		(25)


\begin{eqnarray*}  %\begin{equation}\label{26}
\mathrm{Var}^* [\psi_m(t)] = \frac{\sigma^2_Ts^2}{4a b}
\end{eqnarray*}     % \end{equation}   	 			%		(26)

   The normalized cross-correlation function takes the form

\begin{eqnarray*}
r^*_{T, \psi_m}(\tau) =
\Bigl\{ - \exp(-a|\tau| ) ( 2 \alpha\beta  \cos (\beta \tau) +
\end{eqnarray*}
\begin{eqnarray*}
2 b \sin (\beta \tau) + \alpha (\alpha +\lambda) \sin (\beta |\tau|))\Bigr\} \Bigl/
\end{eqnarray*}
\begin{eqnarray*} % \begin{equation}\label{27}
\Bigl\{2 \sqrt{b+ \alpha^2}\beta \Bigr\}
\end{eqnarray*} %\end{equation}   	 			%		(27)

   Analysis of the last equation shows that at zero time lag the pair correlation of $T$
and  $\psi_m$ is always negative

\begin{eqnarray*} % \begin{equation}\label{28}
r^*_{T, \psi_m}(0) =
\frac{- \alpha}{\sqrt{b+ \alpha^2}}
\end{eqnarray*} % \end{equation}   	 			%		(28)

   Stochastic forcing of the meridional overturning circulation leads to a positive
synchronous AMOC-AMO correlation. On the contrary, stochastic forcing of the SST
leads to a negative regression. The magnitude of the coefficient can be small and its
analysis based on relatively short time series may be problematic. On the other hand, for
any stochastic forcing (see also
equation
\eqref{13})          % 21) N13
the derivative of the cross-correlation
function is negative

 \begin{eqnarray*} % \begin{equation}\label{29}
\frac{d r^*_{T, \psi_m}}{d\tau} \Bigl|_{\tau=0} =
\frac{- b}{\sqrt{b+ \alpha^2}}
\end{eqnarray*}   % \end{equation}   	 			%		(29)

\section{4. Discussion}

   Some important properties of the North Atlantic MDV can be described in the
stochastic model framework
(equations
\eqref{1}--\eqref{2}).          %(Eqs. (1-2)).
Particularly, the sign of AMOC-AMO
correlation depends on the nature of external forcing. The stochastic excitation of the
meridional overturning circulation leads to a positive correlation; the SST random
forcing leads to a negative correlation. However, the AMOC-AMO correlation is
modest at zero time lag.

   The excitation of the meridional overturning circulation in the case of moderate
negative time delays  $\tau$ leads to a positive correlation of $T(t)$  and  $\psi_m(t)$
 (\figref{1}).           % (Figure.1).
That provides evidence for AMOC leading role, similar to the results of
 \itc{Sun et al.} [\reflink{Sun19}{2019}].
 The examples of  $r_{T, \psi_m}(\tau)$ for several different sets of parameters
 (\figref{1}a--\figref{1}d).           %  .           % (Figure 1. a-d)
demonstrate that behaviour of the cross-correlation function is determined by the values
of dissipation parameters $\alpha$  and $\lambda$.
\figref{1}           % Figure 1
a corresponds to e-folding time of 4
dimensionless time units or, respectively, to 16 years and is characterised by slowly
decaying oscillating behaviour. Increase of the dissipation parameter $\alpha$  and  $\lambda$ leads to
faster decay of oscillations
(\figref{1}b,c,d).           % (Figure 1 b c d).

   The SST stochastic excitation is characterised by important distinctions of cross-correlation function. A comparison between
   $r^*_{T, \psi_m}(\tau)$  and  $r_{T, \psi_m}(\tau)$ reveals some
interesting features. For example, if the dissipation parameters
$\alpha$  and $\lambda$
are small then
the behaviour of
 $r^*_{T, \psi_m}(\tau)$  and  $r_{T, \psi_m}(\tau)$
  is similar
  (\figref{1}a,b).           % (Figure 1 a b).
  Increase of
  $\alpha$  and $\lambda$
results in significantly different dependences of
 $r^*_{T, \psi_m}(\tau)$  and  $r_{T, \psi_m}(\tau)$
   on time delay $\tau$
(\figref{1}c,d).           % (Figure 1, c and d).
The strongest negative cross-correlation
 $r^*_{T, \psi_m}(\tau)$
   is observed at the
positive time lags. Now the cross-correlation functions
  $r^*_{T, \psi_m}(\tau)$
   can be interpreted as
evidence for the leading role of the AMO. The positive SST anomalies prevent deep
water formation and damp the meridional oceanic circulation.

   We note that caution should be exercised when interpreting the results of the cross-correlation analysis of the observational and/or GCM modeling data. Often the
interpretation of this analysis and especially establishment of the cause-effect
relationships seems doubtful and ambiguous. Besides, limited and, as a rule, strongly
auto-correlated time series of GCM simulations often cannot guarantee statistically
significant estimates. As a consequence, establishment of the physically-based
conclusions requires formulation of an adequate background model.

\def\dbltopfraction{.95}

\begin{figure*}[t]                     % Fig 2
\figurewidth{35pc}
\setimage{}{}{41pc}{}{2018es000653-f01}
\caption{Normalized cross-correlation functions of  $T(t)$ and $\psi_m(t)$.
$ r_{T, \psi_m} (\tau)$ (blue line),  $ r^*_{T, \psi_m} (\tau)$ (red line).
   The positive time lags correspond to the leading SST (AMO index). $ms=3$,
   a) $\alpha=\lambda = 0.25$,
   b)  $\alpha=\lambda = 0.5$,
 c)  $\alpha=\lambda = 1.0$,
 d)  $\alpha=\lambda = 2.0$.
   The parameter  values  $\alpha=\lambda = 0.5$ corresponds to e-folding time equal to 2 units of
dimensionless time and, respectively, to 8 years.}
\end{figure*}


\section{5. Conclusions}

\begin{enumerate}
 \item
\begin{sloppypar}
Some important properties of the North Atlantic MDV can be reproduced within
the stochastically forced oceanic oscillator model. Stochastic excitation of the
meridional overturning circulation corresponds to a positive synchronous AMOC-AMO
correlation, random forcing of the SST leads to a negative correlation. These differences
can explain some discrepancies of the correlation analyses made in the past using GCM
outputs.
\end{sloppypar}

   \item
\begin{sloppypar}
Stochastic excitation of the meridional overturning circulation implies the leading
AMOC and stochastic forcing of SST implies the leading AMO. However, in both cases
the time derivative of AMOC and AMO cross-correlation function is always negative at
zero time lag.
\item
Significant effect of the e-folding time on the AMOC-AMO interrelationship has
been found. Cross-correlation function demonstrates highly oscillating behaviour in the
case of relatively small dissipation coefficients. Decrease of the e-folding time is
characterized by almost monotonous decay of the cross-correlation function.
\end{sloppypar}
\end{enumerate}

  \acknowledgments{The author thank anonymous reviewer for helpful remarks. The
author is also grateful to Igor Polyakov for help and constructive criticism.}


\hyperdef{section}{apnd1}{}

\appendix{}    %%%%%%%%%%%%%%%%%%%%AAAAAAAAAAAAAAAAAAAAAAAAAAA

   Using the expression
\eqref{5}       % (6)
   and change of variables, $\xi + u = t'$, we can transform
equation
\eqref{7}      % (9)

\begin{eqnarray*}
\psi_m(t) = - sm
 \int\limits_0^{\infty}\exp(-\alpha \xi) \times
\end{eqnarray*}
\begin{eqnarray*}
 \int\limits_\xi^{\infty} h_T (t' -\xi) F_m (t-t') dt' d\xi +
\end{eqnarray*}
\begin{eqnarray*}
 \int\limits_0^{\infty}\exp(-\alpha \xi)  F_m (t-\xi) d\xi =
\end{eqnarray*}
 \begin{eqnarray*}
 - \frac{sm}{\beta} \int\limits_0^{\infty}\exp(- \alpha \xi)
 \int\limits_\xi^{\infty}\exp(-a (t' - \xi))
 \times
 \end{eqnarray*}
  \begin{eqnarray*}
 \sin (\beta(t' -\xi))  F_m (t-t') dt' d\xi +
\end{eqnarray*}
\begin{equation}\label{A1}%%%\hypertarget{A1}{}    %%%%A1A1A1A1A1
 \int\limits_0^{\infty}\exp(-\alpha \xi)  F_m (t-\xi) d\xi
\end{equation}   	 			%		(A1)
	
	
The trigonometric identities allow rewriting \eqref{A1}  as

\begin{eqnarray*}
\psi_m(t) =  - \frac{sm}{\beta} \int\limits_0^{\infty}\exp(( a
-\alpha) \xi) \times
\end{eqnarray*}
\begin{eqnarray*}
\cos (\beta \xi)
 \int\limits_{\xi}^{\infty}  ( -at' )\sin (\beta t')  F_m (t-t') dt' d\xi +
\end{eqnarray*}
 \begin{eqnarray*}
  \frac{sm}{\beta} \int\limits_0^{\infty}\exp((- \alpha + a)\xi) \sin (\beta \xi) \times
 \end{eqnarray*}
\begin{eqnarray*}
 \int\limits_{\xi}^{\infty}\exp(-a t')\cos(\beta t')  F_m (t-t') dt' d\xi +
\end{eqnarray*}   	
\begin{equation}\label{A2}\hypertarget{A2}{}      %%%%% A2A2A2A2
 \int\limits_0^{\infty}\exp(-\alpha \xi)  F_m (t-\xi) d\xi
\end{equation}   	 			%		(A2)


   Now we can integrate \eqref{A2} by part. After some simplifications
\eqref{A2} can be presented in the form of Duhamel integral

\begin{eqnarray*}
\psi_m(t) =   \int\limits_0^{\infty} h_m (u) F_m (t-u) du
\end{eqnarray*}
where  $h_m(u)$ is a weight function

\begin{eqnarray*}
h_m(u)		= \exp (-\alpha u) +
\end{eqnarray*}
\begin{eqnarray*}
\Bigl\{ ms (- \exp (- \alpha u) \beta +  \exp (- a u) \times
\end{eqnarray*}
\begin{eqnarray*}
[\beta \cos (\beta u) + (a-\alpha) \sin (\beta u)])\Bigr\} \Bigl/
\end{eqnarray*}
\begin{eqnarray*}
\Bigl\{\beta ((a-\alpha)^2 + \beta^2)\Bigr\}
\end{eqnarray*}  			%		(A4)

\hyperdef{section}{apnd2}{}

\appendix{}%%%%%%%%%%%%%%%%%%%%%%%%%%  BBBBBBBBBBBBBBBBBBBBBBBBB

   Let us represent the solutions for the two components $Y(t)$  and $Z(t)$  of the
stochastically forced differential equation system in the form of Duhamel integrals

\begin{eqnarray*}
Y(t)= \int\limits_0^{\infty} h_Y (u) X (t-u) du
\end{eqnarray*}
\begin{eqnarray*}
Z(t)= \int\limits_0^{\infty} h_Z (u) X (t-u) du
\end{eqnarray*}  	 			%	(B.1)
   where  $h_Y (t)$ and $h_Z (t)$  are the weight functions of the $Y (t)$  and  $Z (t)$, respectively.
The input signal $X (t)$  represents stationary stochastic process and is characterized by
the correlation function $B_X (\tau)$. In the assumptions
$M[Y(t)] = M[Z(t)] =0$   cross-correlation function of the $Y(t)$   and   $Z(t)$ can be written as

\begin{eqnarray*}
B_{Y,Z}(\tau)= M[Y(t) Z (t+\tau)] =
\end{eqnarray*}
\begin{eqnarray*}
M
[\int\limits_0^{\infty} \int\limits_0^{\infty} h_Y (\xi) X (t-\xi)
h_Z (\eta) \times
\end{eqnarray*}
 \begin{eqnarray*}
 X(t+\tau -\eta) d\eta d\xi] =
\end{eqnarray*}
\begin{eqnarray*}
\int\limits_0^{\infty} h_Y (\xi) \int\limits_0^{\infty} h_Z (\eta)
B_X (\tau + \xi - \eta) d\eta d\xi 	
\end{eqnarray*} 	 			%	(B.2)

   Using Parseval's theorem we can write
      [\itc{Bekryaev,} \reflink{Bekr16}{2016}]
    cross-correlation function
in the form of convolution integral


\begin{eqnarray*}
B_{Y,Z}(\tau)= \int\limits_{-{\infty}}^{\infty} B_X(s) \times
\end{eqnarray*}

\hyperdef{equation}{eqB1}{ }

\begin{equation}\label{B1}    %%\hypertarget{B1}{}        %%%%%%%%%%%B1B1B1B1B1B1B1B1
 \int\limits_0^{\infty} h_Y (\xi)
  h_Z ( \xi +s +\tau)
d\xi ds   	
\end{equation}   	 			%	(B.3)

   If the input process $X(t)$  is a derivative of the standard Wiener process, $W(t)$, i.e.
\begin{eqnarray*}
X(t) = \sigma^2_X \frac{dW}{dt}
\end{eqnarray*}
equation \hyperref{}{equation}{eqB1}{B1}   %%\eqref{B1}
  can be represented in a form

\hyperdef{equation}{eqB2}{ }

%%\begin{eqnarray*}
\begin{equation}\label{B2}\hypertarget{B2}{}       %%%%%%%%B2B2B2B2B2B2B2B2B2B2
B_{Y,Z}(\tau)= \sigma^2_X
 \int\limits_0^{\infty} h_Y (\xi)
  h_Z ( \xi +\tau)
d\xi
\end{equation}
%%\end{eqnarray*}   	 			

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\vspace*{.3pc}

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\end{document}

