# Weierstrass Representations
of Lorentzian Minimal Surfaces in $\mathbb{R}^{4}_{2}$

Ognian Kassabov, Velichka Milousheva  Department of Mathematics and
Informatics, Todor Kableshkov University of Transport, 158 Geo Milev Str.,
1574 Sofia, Bulgaria<EMAIL_ADDRESS>Institute of Mathematics and
Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. bl. 8, 1113,
Sofia, Bulgaria<EMAIL_ADDRESS>
###### Abstract.

The minimal Lorentzian surfaces in $\mathbb{R}^{4}_{2}$ whose first normal
space is two-dimensional and whose Gauss curvature $K$ and normal curvature
$\varkappa$ satisfy $K^{2}-\varkappa^{2}>0$ are called minimal Lorentzian
surfaces of general type. These surfaces admit canonical parameters and with
respect to such parameters are determined uniquely up to a motion in
$\mathbb{R}^{4}_{2}$ by the curvatures $K$ and $\varkappa$ satisfying a system
of two natural PDEs.

In the present paper we study minimal Lorentzian surfaces in
$\mathbb{R}^{4}_{2}$ and find a Weierstrass representation with respect to
isothermal parameters of any minimal surface with two-dimensional first normal
space. We also obtain a Weierstrass representation with respect to canonical
parameters of any minimal Lorentzian surface of general type and solve
explicitly the system of natural PDEs expressing any solution to this system
by means of four real functions of one variable.

###### Key words and phrases:

Lorentzian surfaces, Weierstrass formulas, canonical principal parameters,
minimal surfaces

###### 2010 Mathematics Subject Classification:

Primary 53B30, Secondary 53A10, 53A35

##  1\. Introduction

The study of minimal surfaces is one of the main topics in classical
differential geometry. In the last years, great attention is paid to
Lorentzian surfaces in pseudo-Euclidean spaces, since pseudo-Riemannian
geometry has many important applications in Physics. Minimal Lorentzian
surfaces in $\mathbb{C}^{2}_{1}$ have been classified by B.-Y. Chen [5].
Classification results for minimal Lorentzian surfaces in pseudo-Euclidean
space $\mathbb{E}^{m}_{s}$ with arbitrary dimension $m$ and arbitrary index
$s$ are obtained in [6]. Minimal Lorentzian surfaces in the pseudo-Euclidean
4-space $\mathbb{R}^{4}_{2}$ whose first normal space is two-dimensional and
whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the
inequality $K^{2}-\varkappa^{2}>0$ are studied in [1] under the name minimal
Lorentzian surfaces of general type. This class of surfaces is characterized
in terms of a pair of smooth functions satisfying a system of two natural
partial differential equations. The approach to the study of minimal
Lorentzian surfaces of general type in $\mathbb{E}^{4}_{2}$ is based on the
introducing of special geometric parameters which are called canonical
parameters.

A representation of a minimal Lorentzian surface was given by M.P. Dussan and
M. Magid in [8] where they solved the Björling problem for timelike surfaces
in $\mathbb{R}^{4}_{2}$ constructing a special normal frame and a split-
complex representation formula. The Björling problem for timelike surfaces in
the Lorentz-Minkowski spaces $\mathbb{R}^{3}_{1}$ and $\mathbb{R}^{4}_{1}$ is
solved in [4] and [7], respectively. Spinor representation of Lorentzian
surfaces in the pseudo-Euclidean 4-space with neutral metric is given in [3].
In [14], V. Patty gave a generalized Weierstrass representation of a minimal
Lorentzian surface in $\mathbb{R}^{4}_{2}$ using spinors and Lorentz numbers
(also known as para-complex, split-complex, double or hyperbolic numbers) thus
extending the Weierstrass representation of a minimal surface in
$\mathbb{R}^{3}_{1}$ given by J. Konderak [13].

A special Weierstrass representation, called canonical Weierstrass
representation, for a maximal spacelike surface in $\mathbb{R}^{4}_{2}$ is
obtained in [12]. These Weierstrass formulas give explicitly the solutions to
the system of PDEs describing maximal spacelike surfaces in
$\mathbb{R}^{4}_{2}$ in terms of two holomorphic functions. The explicit
solving of the system of PDEs and a relation between maximal spacelike
surfaces in $\mathbb{R}^{4}_{2}$ and maximal spacelike surfaces in
$\mathbb{R}^{3}_{1}$ is given in [11] \- a paper that is a follow-up to a
series of articles on explicit solving of the system of natural PDEs of
minimal surfaces in $\mathbb{R}^{4}$ and minimal spacelike surfaces in
$\mathbb{R}^{4}_{1}$ [9, 10].

In the present paper we study minimal Lorentzian surfaces in
$\mathbb{R}^{4}_{2}$. In Section 3 we find a Weierstrass representation with
respect to isothermal parameters (Theorem 3.3) of any minimal Lorentzian
surface with two-dimensional first normal space. The Weierstrass
representation formula given in Theorem 3.3 is equivalent to the
representation formula in [8]. But the formula in [8] uses Lorentz numbers
(split-complex numbers), while our formula gives a representation in terms of
real functions of one real variable which is much more convenient. Moreover,
this Weierstrass formula gives a clear analytic form of the surface.

Further, we obtain a Weierstrass representation formula with respect to
canonical parameters of any minimal Lorentzian surface of general type which
describes locally all these surfaces in terms of four real functions (Theorem
4.3). Using the canonical Weierstrass representation we solve explicitly the
system of PDEs that describes minimal Lorentzian surfaces of general type in
$\mathbb{R}^{4}_{2}$, expressing any solution to this system by means of four
real functions of one variable (Theorem 4.4).

In the last section, using the canonical Weierstrass representation formula we
give examples of minimal Lorentzian surfaces of general type parametrized by
canonical parameters.

##  2\. Preliminaries

Let $\mathbb{R}^{4}_{2}$ be the four-dimensional pseudo-Euclidean space with
the standard flat metric $\langle\,,\,\rangle$ of signature $(2,2)$, also
known as pseudo-Euclidean space with neutral metric, and consider a minimal
Lorentzian surface in it. A local parametrization of such surfaces is given by
the following fundamental result:

Theorem A [5]. Let $S$ be a minimal Lorentzian surface in
$\mathbb{R}^{4}_{2}$. Then it can be locally parametrized in the form

$\Psi(u,v)=\beta(u)+\theta(v),$

where $\beta$ and $\theta$ are null curves in $\mathbb{R}^{4}_{2}$, such that
$\langle\beta^{\prime}(u),\theta^{\prime}(v)\rangle\neq 0$ for every $u,\,v$.

Concerning generalizations of this theorem for more general situations see
[2], p. 53 and [6].

The Gauss curvature $K$ and the curvature of the normal connection (the normal
curvature) $\varkappa$ of any minimal surface in the Euclidean space
$\mathbb{R}^{4}$ and any spacelike maximal surface in $\mathbb{R}^{4}_{2}$
satisfy the inequality $K^{2}-\varkappa^{2}\geq 0$. The equality case
characterizes the so-called super-conformal surfaces. Minimal Lorentz surfaces
in the pseudo-Euclidean space $\mathbb{R}^{4}_{2}$ can be divided into three
basic classes:

  * •

surfaces characterized by $K^{2}-\varkappa^{2}>0$;

  * •

surfaces characterized by $K^{2}-\varkappa^{2}=0$;

  * •

surfaces characterized by $K^{2}-\varkappa^{2}<0$.

In the case $K^{2}-\varkappa^{2}>0$, Lorenzian minimal surfaces in
$\mathbb{R}^{4}_{2}$ have similar properties to minimal surfaces in
$\mathbb{R}^{4}$ or spacelike maximal surfaces in $\mathbb{R}^{4}_{2}$ [15].
Minimal Lorentzian surfaces from the first class, i.e. satisfying the
inequality $K^{2}-\varkappa^{2}>0$ are studied in [1]. In the special case
when the first normal space is one-dimensional at each point, the minimal
surface is either a degenerate hyperplane, a flat umbilic surface, a quasi-
umbilic surface, or lies in a non-degenerate hyperplane. In the case when the
first normal space is two-dimensional, the surfaces are called in [1] minimal
Lorentzian surfaces of general type, and we will follow this terminology. Note
that minimal surfaces satisfying the inequality $K^{2}-\varkappa^{2}<0$ do not
exist neither in the Euclidean space $\mathbb{R}^{4}$ nor in the family of
spacelike maximal surfaces in $\mathbb{R}^{4}_{2}$ and they need a different
approach to be studied with.

In what follows we consider minimal Lorentzian surface of general type, i.e.
at any point $K^{2}-\varkappa^{2}>0$ and the first normal space is two-
dimensional.

Let the minimal Lorentzian surface of general type

$S:\Psi=\Psi(u,v),\;(u,v)\in\mathcal{D},\,\mathcal{D}\subset\mathbb{R}^{2}$

be parametrized by isothermal parameters, i.e. $E=-G$, $F=0$. Then the
coefficients of the first fundamental form are expressed as
$E=f^{2}(u,v),\;F=0,\;G=-f^{2}(u,v)$ for some positive function $f(u,v)$.
Denote by $x$, $y$ the unit tangent vector fields in the directions of
$\Psi_{u}$, resp. $\Psi_{v}$, i.e. $x=\displaystyle{\frac{\Psi_{u}}{f}}$,
$y=\displaystyle{\frac{\Psi_{v}}{f}}$. Obviously, $\langle x,x\rangle=1$,
$\langle y,y\rangle=-1$, $\langle x,y\rangle=0$. We choose a local normal
frame field $\\{e_{1},e_{2}\\}$ such that $\langle e_{1},e_{1}\rangle=1$,
$\langle e_{2},e_{2}\rangle=-1$, $\langle e_{1},e_{2}\rangle=0$ and define
functions $a(u,v),b(u,v),c(u,v),d(u,v)$ by

$\begin{array}[]{l}\vspace{2mm}\sigma(x,x)=ae_{1}+be_{2};\\\
\vspace{2mm}\sigma(x,y)=ce_{1}+de_{2};\\\
\vspace{2mm}\sigma(y,y)=ae_{1}+be_{2},\end{array}$

where $\sigma$ is the second fundamental tensor. Let $A_{1}$ and $A_{2}$ be
the shape operators corresponding to $e_{1}$ and $e_{2}$, respectively. Then

$\begin{array}[]{ll}\vspace{2mm}A_{1}(x)=ax-cy;&A_{2}(x)=-bx+dy;\\\
\vspace{2mm}A_{1}(y)=cx-ay;&A_{2}(y)=-dx+by.\end{array}$

Using the above formulas and the Ricci equation we obtain that the Gauss
curvature $K$ and the curvature of the normal connection $\varkappa$ are
expressed by

(2.1) $K=b^{2}-a^{2}+c^{2}-d^{2},$ (2.2) $\varkappa=2(bc-ad).$

In [1] it is proved that in a neighbourhood of any point of a minimal
Lorentzian surface of general type we can introduce a special orthonormal
frame field $\\{x,y,n_{1},n_{2}\\}$ such that the vector fields $\sigma(x,x)$
and $\sigma(x,y)$ are collinear to $n_{1}$ and $n_{2}$, respectively, i.e.

$\begin{array}[]{l}\vspace{2mm}\sigma(x,x)=\nu n_{1};\\\
\vspace{2mm}\sigma(x,y)=\qquad\mu n_{2};\\\ \vspace{2mm}\sigma(y,y)=\nu
n_{1},\end{array}$

where $\langle x,x\rangle=1$, $\langle y,y\rangle=-1$, $\langle
n_{1},n_{1}\rangle=\varepsilon$, $\langle n_{2},n_{2}\rangle=-\varepsilon$
($\varepsilon=\pm 1$) and $\nu\mu$ never vanishes. The tangent directions
determined by the tangent vector fields $x$ and $y$ are called canonical
directions of the surface.

It follows from (2.1) and (2.2) that the following equality holds:

(2.3) $K^{2}-\varkappa^{2}=(\mu^{2}-\nu^{2})^{2}.$

Since we are studying surfaces for which $K^{2}-\varkappa^{2}>0$, we assume
that $\mu^{2}-\nu^{2}\neq 0$.

Moreover, if

(2.4) $E^{2}=G^{2}=\frac{1}{|\mu^{2}-\nu^{2}|},$

the parameters of the surface are called canonical [1]. We shall use the
following

Theorem B [1]. The Gauss curvature $K$ and the curvature of the normal
connection $\varkappa$ of any minimal Lorentzian surface of general type in
$\mathbb{R}^{4}_{2}$ in canonical parameters satisfy the system

(2.5) $\begin{array}[]{l}\vspace{2mm}\displaystyle\root 4
\of{K^{2}-\varkappa^{2}}\Delta^{h}\ln(K^{2}-\varkappa^{2})=8K\vspace{2mm}\\\
\vspace{2mm}\displaystyle\root 4
\of{K^{2}-\varkappa^{2}}\Delta^{h}\ln\frac{K+\varepsilon\varkappa}{K-\varepsilon\varkappa}=4\varepsilon\varkappa\end{array}$

where $\Delta^{h}$ is the hyperbolic Laplacian, $\varepsilon=\pm 1$.
Conversely, for any solution $(K(u,v),\varkappa(u,v))$ to this system with
non-vanishing functions $K$ and $\varkappa$, satisfying
$K^{2}-\varkappa^{2}>0$, there exists a unique minimal Lorentzian surface of
general type with Gauss curvature $K$ and curvature of the normal connection
$\varkappa$. Moreover, the parameters $(u,v)$ are canonical.

##  3\. The Weierstrass-type formula

The following statement is a trivial consequence of Theorem A:

######  Proposition 3.1.

Let $\beta$ and $\theta$ be null curves in $\mathbb{R}^{4}_{2}$, such that
$\langle\beta^{\prime},\theta^{\prime}\rangle\neq 0$. Then

$S:\Psi(u,v)=\beta(u+v)+\theta(u-v)$

is a minimal Lorentzian surface in isothermal parameters. Conversely, every
minimal Lorentzian surface in $\mathbb{R}^{4}_{2}$ can be parametrized locally
in this way.

Now let

$\beta(u)=\big{(}\beta_{1}(u),\beta_{2}(u),\beta_{3}(u),\beta_{4}(u)\big{)},\quad
u\in J\subset\mathbb{R}$

be a null curve in $\mathbb{R}^{4}_{2}$, i.e.

(3.1)
$(\beta_{1}^{\prime})^{2}+(\beta_{2}^{\prime})^{2}-(\beta_{3}^{\prime})^{2}-(\beta_{4}^{\prime})^{2}=0.$

First we suppose that
$(\beta_{1}^{\prime})^{2}=(\beta_{3}^{\prime})^{2}=(\beta_{4}^{\prime})^{2}$
in an interval $J_{0}\subset J$. Then
$(\beta_{2}^{\prime})^{2}=(\beta_{1}^{\prime})^{2}$ and hence, there exists a
function $f(u),\;u\in J_{0}$ such that

$\beta^{\prime}(u)=f(u)\big{(}1,\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\big{)},\quad\varepsilon_{i}=\pm
1,\;i=1,2,3.$

Now assume that $(\beta_{1}^{\prime})^{2}-(\beta_{3}^{\prime})^{2}\neq 0$ at a
point $u_{0}\in J$. Then the function
$(\beta_{1}^{\prime})^{2}-(\beta_{3}^{\prime})^{2}$ doesn’t vanish in an
interval $J_{0}\subset J$, $u_{0}\in J_{0}$. We introduce in $J_{0}$ the
following functions:

$f=\frac{\beta^{\prime}_{1}+\beta^{\prime}_{3}}{2},\qquad
g=\frac{\beta^{\prime}_{2}+\beta^{\prime}_{4}}{\beta^{\prime}_{1}+\beta^{\prime}_{3}},\qquad
h=\frac{-\beta^{\prime}_{2}+\beta^{\prime}_{4}}{\beta^{\prime}_{1}+\beta^{\prime}_{3}}.$

Hence,

$\beta^{\prime}_{2}=f(g-h),\qquad\beta^{\prime}_{4}=f(g+h).$

Note that the function $f$ doesn’t vanish and $g$, $h$ are well defined
because $(\beta_{1}^{\prime})^{2}-(\beta_{3}^{\prime})^{2}\neq 0$. Using (3.1)
we get also

$\beta^{\prime}_{1}=f(1+gh),\qquad\beta^{\prime}_{3}=f(1-gh).$

Finally, we obtain that if $(\beta_{1}^{\prime})^{2}-(\beta_{3}^{\prime})^{2}$
doesn’t vanish in $J_{0}$, then

(3.2)
$\beta^{\prime}(u)=f(u)\big{(}1+g(u)h(u),g(u)-h(u),1-g(u)h(u),g(u)+h(u)\big{)},\quad
u\in J_{0}.$

If $(\beta_{1}^{\prime})^{2}-(\beta_{4}^{\prime})^{2}\neq 0$ at a point
$u_{0}\in J$, then there exists $J_{0}\subset J$ such that the function
$(\beta_{1}^{\prime})^{2}-(\beta_{4}^{\prime})^{2}\neq 0$ in $J_{0}$. In such
case we can introduce in $J_{0}$ functions:

$f=\frac{\beta^{\prime}_{1}+\beta^{\prime}_{4}}{2},\qquad
g=\frac{\beta^{\prime}_{2}+\beta^{\prime}_{3}}{\beta^{\prime}_{1}+\beta^{\prime}_{4}},\qquad
h=\frac{-\beta^{\prime}_{2}+\beta^{\prime}_{3}}{\beta^{\prime}_{1}+\beta^{\prime}_{4}}.$

Using (3.1) we get

$\beta^{\prime}_{1}=f(1+gh),\qquad\beta^{\prime}_{2}=f(g-h),\qquad\beta^{\prime}_{3}=f(g+h),\qquad\beta^{\prime}_{4}=f(1-gh).$

So, in this case we obtain

(3.3)
$\beta^{\prime}(u)=f(u)\big{(}1+g(u)h(u),g(u)-h(u),g(u)+h(u),1-g(u)h(u)\big{)},\quad
u\in J_{0}.$

Now, we substitute

$\tilde{g}=\frac{1-h}{1+h},\qquad\tilde{h}=\frac{1-g}{1+g}$

and get

$g-h=\frac{2(\tilde{g}-\tilde{h})}{(1+\tilde{h})(1+\tilde{g})},\qquad
g+h=\frac{2(1-\tilde{g}\tilde{h})}{(1+\tilde{h})(1+\tilde{g})},$
$1+gh=\frac{2(1+\tilde{g}\tilde{h})}{(1+\tilde{h})(1+\tilde{g})},\qquad
1-gh=\frac{2(\tilde{g}+\tilde{h})}{(1+\tilde{h})(1+\tilde{g})}.$

Setting

$\tilde{f}=\frac{2f}{(1+\tilde{h})(1+\tilde{g})}$

we get

$\beta^{\prime}(u)=\tilde{f}(u)\big{(}1+\tilde{g}(u)\tilde{h}(u),\tilde{g}(u)-\tilde{h}(u),1-\tilde{g}(u)\tilde{h}(u),\tilde{g}(u)+\tilde{h}(u)\big{)}.$

Hence, omitting points at which $1+h=0$ or $1+g=0$ we can transform (3.3) in
(3.2).

Consequently, we have proved the following statement.

######  Proposition 3.2.

Each null curve $\beta=\big{(}\beta_{1},\beta_{2},\beta_{3},\beta_{4}\big{)}$
in $\mathbb{R}^{4}_{2}$ can be parametrized locally (in a neighbourhood of a
given point) in such a way that

(1)
$\beta^{\prime}(u)=f(u)\big{(}1,\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\big{)},\quad\varepsilon_{i}=\pm
1,\;i=1,2,3;$
or (2)
$\beta^{\prime}(u)=f(u)\big{(}1+g(u)h(u),g(u)-h(u),1-g(u)h(u),g(u)+h(u)\big{)}$
for some smooth functions $f(u)$, $g(u)$, $h(u)$, such that $f(u)$ does not
vanish.

In what follows we use Propositions 3.1 and 3.2 to find a Weierstrass
representation of a minimal Lorentzian surface in $\mathbb{R}^{4}_{2}$ with
two-dimensional first normal space.

Let $S$ be a minimal surface in $\mathbb{R}^{4}_{2}$. According to Proposition
3.1, $S$ can be parametrized in isothermal parameters by

$S:\Psi(u,v)=\beta(u+v)+\theta(u-v),$

where $\beta$ and $\theta$ are null curves in $\mathbb{R}^{4}_{2}$ such that
$\langle\beta^{\prime},\theta^{\prime}\rangle\neq 0$.

(i) Let us suppose that both $\beta$ and $\theta$ are of type (1) in
Proposition 3.2, i.e.

$\beta^{\prime}=f_{1}\big{(}1,\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\big{)},\qquad\theta^{\prime}=f_{2}\big{(}1,\varepsilon_{4},\varepsilon_{5},\varepsilon_{6}\big{)},\quad\varepsilon_{i}=\pm
1,\;i=1,\dots,6.$

We denote by $c_{1}$ and $c_{2}$ the constant vectors
$\big{(}1,\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\big{)}$ and
$\big{(}1,\varepsilon_{4},\varepsilon_{5},\varepsilon_{6}\big{)}$,
respectively. Then

$\beta^{\prime}(u+v)=f_{1}(u+v)\,c_{1},\qquad\theta^{\prime}(u-v)=f_{2}(u-v)\,c_{2}.$

Hence, the tangent space of $S$ is spanned by the vector fields

$\begin{array}[]{l}\vspace{2mm}\Psi_{u}=f_{1}(u+v)\,c_{1}+f_{2}(u-v)\,c_{2},\\\
\vspace{2mm}\Psi_{v}=f_{1}(u+v)\,c_{1}-f_{2}(u-v)\,c_{2}\end{array}$

and we get

$\begin{array}[]{l}\vspace{2mm}\Psi_{uu}=f^{\prime}_{1}(u+v)\,c_{1}+f^{\prime}_{2}(u-v)\,c_{2},\\\
\vspace{2mm}\Psi_{uv}=f^{\prime}_{1}(u+v)\,c_{1}-f^{\prime}_{2}(u-v)\,c_{2},\\\
\vspace{2mm}\Psi_{vv}=f^{\prime}_{1}(u+v)\,c_{1}+f^{\prime}_{2}(u-v)\,c_{2}.\end{array}$

The last three equalities show that $\sigma(\Psi_{u},\Psi_{u})=0$,
$\sigma(\Psi_{u},\Psi_{v})=0$, $\sigma(\Psi_{v},\Psi_{v})=0$, i.e. the surface
$S$ has vanishing second fundamental tensor $\sigma$. Hence, $S$ is part of a
plane.

(ii) Let $\beta$ be of type (1) and $\theta$ be of type (2) in Proposition
3.2, i.e.

$\begin{array}[]{l}\vspace{2mm}\beta^{\prime}=f_{1}\,c,\\\
\vspace{2mm}\theta^{\prime}=f_{2}\big{(}1+gh,g-h,1-gh,g+h\big{)},\end{array}$

where
$c=\big{(}1,\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\big{)},\varepsilon_{i}=\pm
1$, $\beta^{\prime}$ depends on $u+v$, $\theta^{\prime}$ depends on $u-v$.
Hence, the tangent space of $S$ is spanned by

$\begin{array}[]{l}\vspace{2mm}\Psi_{u}=f_{1}(u+v)\,c+\theta^{\prime}(u-v),\\\
\vspace{2mm}\Psi_{v}=f_{1}(u+v)\,c-\theta^{\prime}(u-v),\end{array}$

and

$\begin{array}[]{l}\vspace{2mm}\Psi_{uu}=f^{\prime}_{1}(u+v)\,c+\theta^{\prime\prime}(u-v),\\\
\vspace{2mm}\Psi_{uv}=f^{\prime}_{1}(u+v)\,c-\theta^{\prime\prime}(u-v),\\\
\vspace{2mm}\Psi_{vv}=f^{\prime}_{1}(u+v)\,c+\theta^{\prime\prime}(u-v).\end{array}$

The above equalities imply that
$\sigma(\Psi_{u},\Psi_{u})=-\sigma(\Psi_{u},\Psi_{v})=\sigma(\Psi_{v},\Psi_{v})$,
since $f^{\prime}_{1}(u+v)\,c$ is a tangent vector field. Hence, the first
normal space of the surface is one-dimensional.

Consequently, when the first normal space of the surface is two-dimensional,
the null curves $\beta$ and $\theta$ are both of type (2) in Proposition 3.2.
So, we have:

######  Theorem 3.3.

Up to change of the parameters any minimal Lorentzian surface in
$\mathbb{R}^{4}_{2}$ with two-dimensional first normal space has locally a
Weierstrass type representation in isothermal parameters

$\Psi(u,v)=\beta(u+v)+\theta(u-v)$

for some null curves $\beta$ and $\theta$ satisfying

$\begin{array}[]{l}\vspace{2mm}\beta^{\prime}(t)=f_{1}(t)\big{(}1+g_{1}(t)h_{1}(t),g_{1}(t)-h_{1}(t),1-g_{1}(t)h_{1}(t),g_{1}(t)+h_{1}(t)\big{)};\\\
\vspace{2mm}\theta^{\prime}(s)=f_{2}(s)\big{(}1+g_{2}(s)h_{2}(s),g_{2}(s)-h_{2}(s),1-g_{2}(s)h_{2}(s),g_{2}(s)+h_{2}(s)\big{)},\end{array}$

and $\langle\beta^{\prime}(t),\theta^{\prime}(s)\rangle\neq 0$.

##  4\. Minimal Lorentzian surfaces of general type and their canonical
Weierstrass representation

Let
$S:\Psi=\Psi(u,v),\;(u,v)\in\mathcal{D},\,\mathcal{D}\subset\mathbb{R}^{2}$ be
a minimal Lorentzian surface of general type. According to Theorem 3.3, $S$ is
parametrized by

(4.1) $\Psi(u,v)=\beta(u+v)+\theta(u-v),$

where $\beta$ and $\theta$ have the form

(4.2)
$\begin{array}[]{l}\vspace{2mm}\beta^{\prime}(t)=f_{1}(t)\big{(}1+g_{1}(t)h_{1}(t),g_{1}(t)-h_{1}(t),1-g_{1}(t)h_{1}(t),g_{1}(t)+h_{1}(t)\big{)},\\\
\vspace{2mm}\theta^{\prime}(s)=f_{2}(s)\big{(}1+g_{2}(s)h_{2}(s),g_{2}(s)-h_{2}(s),1-g_{2}(s)h_{2}(s),g_{2}(s)+h_{2}(s)\big{)}.\end{array}$

Then the tangent space is spanned by

(4.3)
$\begin{array}[]{l}\vspace{2mm}\Psi_{u}=\beta^{\prime}(u+v)+\theta^{\prime}(u-v),\\\
\vspace{2mm}\Psi_{v}=\beta^{\prime}(u+v)-\theta^{\prime}(u-v),\end{array}$

and the coefficients $E$, $F$, and $G$ of the first fundamental form of $S$
are

$\begin{array}[]{l}\vspace{2mm}E=4f_{1}(u+v)f_{2}(u-v)\big{(}g_{1}(u+v)-g_{2}(u-v)\big{)}\big{(}h_{1}(u+v)-h_{2}(u-v)\big{)},\\\
\vspace{2mm}F=0,\\\ \vspace{2mm}G=-E.\end{array}$

We consider the orthonormal tangent frame field
$x=\displaystyle{\frac{\Psi_{u}}{\sqrt{E}}}$,
$y=\displaystyle{\frac{\Psi_{v}}{\sqrt{-G}}}$. Note that (2.1) and (2.2) imply

$K^{2}-\varkappa^{2}=(a^{2}-b^{2})^{2}+(c^{2}-d^{2})^{2}+2(a^{2}-b^{2})(c^{2}-d^{2})-4(ac-
bd)^{2}$

or equivalently

(4.4)
$\begin{array}[]{rl}\vspace{2mm}K^{2}-\varkappa^{2}=&\langle\sigma(x,x),\sigma(x,x)\rangle^{2}+\langle\sigma(x,y),\sigma(x,y)\rangle^{2}\\\
\vspace{2mm}\hfil&+2\langle\sigma(x,x),\sigma(x,x)\rangle\langle\sigma(x,y),\sigma(x,y)\rangle-4\langle\sigma(x,x),\sigma(x,y)\rangle^{2}.\end{array}$

Using (4.3) we get

(4.5)
$\begin{array}[]{l}\vspace{2mm}\sigma(x,x)=\displaystyle{\frac{1}{E}\left(\beta^{\prime\prime}+\theta^{\prime\prime}-\frac{\langle\beta^{\prime\prime},\theta^{\prime}\rangle}{\langle\beta^{\prime},\theta^{\prime}\rangle}\,\beta^{\prime}-\frac{\langle\beta^{\prime},\theta^{\prime\prime}\rangle}{\langle\beta^{\prime},\theta^{\prime}\rangle}\,\theta^{\prime}\right)};\\\
\vspace{2mm}\sigma(x,y)=\displaystyle{\frac{1}{E}\left(\beta^{\prime\prime}-\theta^{\prime\prime}-\frac{\langle\beta^{\prime\prime},\theta^{\prime}\rangle}{\langle\beta^{\prime},\theta^{\prime}\rangle}\,\beta^{\prime}+\frac{\langle\beta^{\prime},\theta^{\prime\prime}\rangle}{\langle\beta^{\prime},\theta^{\prime}\rangle}\,\theta^{\prime}\right)}.\end{array}$

On the other hand, from (4.2) it follows that

(4.6)
$\begin{array}[]{l}\vspace{2mm}\langle\beta^{\prime},\beta^{\prime\prime}\rangle=0;\\\
\vspace{2mm}\langle\theta^{\prime},\theta^{\prime\prime}\rangle=0;\\\
\vspace{2mm}\langle\beta^{\prime\prime},\beta^{\prime\prime}\rangle=-4(f_{1})^{2}g_{1}^{\prime}h_{1}^{\prime};\\\
\vspace{2mm}\langle\theta^{\prime\prime},\theta^{\prime\prime}\rangle=-4(f_{2})^{2}g_{2}^{\prime}h_{2}^{\prime};\\\
\vspace{2mm}\langle\theta^{\prime\prime},\beta^{\prime}\rangle=2f_{1}f_{2}^{\prime}(g_{2}-g_{1})(h_{2}-h_{1})+2f_{1}f_{2}((g_{2}-g_{1})h_{2}^{\prime}+(h_{2}-h_{1})g_{2}^{\prime});\\\
\vspace{2mm}\langle\beta^{\prime\prime},\theta^{\prime}\rangle=2f_{2}f_{1}^{\prime}(g_{2}-g_{1})(h_{2}-h_{1})+2f_{1}f_{2}((h_{1}-h_{2})g_{1}^{\prime}+(g_{1}-g_{2})h_{1}^{\prime}).\\\
\end{array}$

Now, using (4.4), (4.5) and (4.6), by a straightforward calculation we obtain

(4.7)
$K^{2}\\!-\\!\varkappa^{2}=\frac{g_{1}^{\prime}(u\\!+\\!v)g_{2}^{\prime}(u\\!-\\!v)h_{1}^{\prime}(u\\!+\\!v)h_{2}^{\prime}(u\\!-\\!v)}{(f_{1}(u\\!+\\!v))^{2}(f_{2}(u\\!-\\!v))^{2}\big{(}g_{1}(u\\!+\\!v)-g_{2}(u\\!-\\!v)\big{)}^{4}\big{(}h_{1}(u\\!+\\!v)-h_{2}(u\\!-\\!v)\big{)}^{4}}.$

Note that the condition $K^{2}-\varkappa^{2}\neq 0$ and (4.7) imply that none
of the functions $g_{1}^{\prime}$, $g_{2}^{\prime}$, $h_{1}^{\prime}$,
$h_{2}^{\prime}$ can vanish. Moreover, since $K^{2}-\varkappa^{2}>0$ for
minimal Lorentzian surfaces of general type, then
$g_{1}^{\prime}g_{2}^{\prime}h_{1}^{\prime}h_{2}^{\prime}>0$. Hence, as a
consequence of Theorem 3.3 we obtain the Weierstrass representation of minimal
Lorentzian surfaces of general type given below.

######  Theorem 4.1.

Up to change of the parameters any minimal Lorentzian surface of general type
in $\mathbb{R}^{4}_{2}$ has locally a Weierstrass type representation in
isothermal parameters

$\Psi(u,v)=\beta(u+v)+\theta(u-v)$

for some null curves $\beta$ and $\theta$ satisfying

$\begin{array}[]{l}\vspace{2mm}\beta^{\prime}(t)=f_{1}(t)\big{(}1+g_{1}(t)h_{1}(t),g_{1}(t)-h_{1}(t),1-g_{1}(t)h_{1}(t),g_{1}(t)+h_{1}(t)\big{)};\\\
\vspace{2mm}\theta^{\prime}(s)=f_{2}(s)\big{(}1+g_{2}(s)h_{2}(s),g_{2}(s)-h_{2}(s),1-g_{2}(s)h_{2}(s),g_{2}(s)+h_{2}(s)\big{)},\end{array}$

and $f_{1}f_{2}(g_{1}-g_{2})(h_{1}-h_{2})\neq 0$,
$g_{1}^{\prime}g_{2}^{\prime}h_{1}^{\prime}h_{2}^{\prime}>0$ everywhere.

Further, we shall obtain a condition for the surface to be parametrized by
canonical parameters. A necessary condition for a canonical parametrization of
a minimal surface of general type is $\sigma(\Psi_{u},\Psi_{u})$ and
$\sigma(\Psi_{u},\Psi_{v})$ to be orthogonal [1]. It follows from (4.5) and
(4.6) that

$\begin{array}[]{l}\vspace{2mm}\langle\sigma(\Psi_{u},\Psi_{u}),\sigma(\Psi_{u},\Psi_{v})\rangle\\\
\vspace{2mm}=\langle\beta^{\prime\prime},\beta^{\prime\prime}\rangle(u+v)-\langle\theta^{\prime\prime},\theta^{\prime\prime}\rangle(u-v)\\\
\vspace{2mm}=-4(f_{1}(u+v))^{2}g_{1}^{\prime}(u+v)h_{1}^{\prime}(u+v)+4(f_{2}(u-v))^{2}g_{2}^{\prime}(u-v)h_{2}^{\prime}(u-v).\end{array}$

Hence, for a canonical parametrization we must have

$(f_{1}(u+v))^{2}g_{1}^{\prime}(u+v)h_{1}^{\prime}(u+v)=(f_{2}(u-v))^{2}g_{2}^{\prime}(u-v)h_{2}^{\prime}(u-v)=const\neq
0.$

Suppose that

$f_{1}=\frac{1}{4\sqrt{|g_{1}^{\prime}h_{1}^{\prime}|}},\qquad
f_{2}=\frac{1}{4\sqrt{|g_{2}^{\prime}h_{2}^{\prime}|}}.$

Then, using (2.3) and (4.7) we see that

$\left|\mu^{2}-\nu^{2}\right|=16\frac{g_{1}^{\prime}(u+v)h_{1}^{\prime}(u+v)g_{2}^{\prime}(u-v)h_{2}^{\prime}(u-v)}{\big{(}g_{1}(u+v)-g_{2}(u-v)\big{)}^{2}\big{(}h_{1}(u+v)-h_{2}(u-v)\big{)}^{2}}.$

So, (2.4) also holds and hence, the surface is parametrized by canonical
parameters. Consequently, we have proved:

######  Proposition 4.2.

Let $\beta$ and $\theta$ be null curves in $\mathbb{R}^{4}_{2}$ satisfying

(4.8)
$\begin{array}[]{l}\displaystyle\beta^{\prime}(t)=\frac{1}{4\sqrt{|g_{1}^{\prime}(t)h_{1}^{\prime}(t)|}}\big{(}1\\!+\\!g_{1}(t)h_{1}(t),g_{1}(t)\\!-\\!h_{1}(t),1\\!-\\!g_{1}(t)h_{1}(t),g_{1}(t)\\!+\\!h_{1}(t)\big{)}\\\
\displaystyle\theta^{\prime}(s)=\frac{1}{4\sqrt{|g_{2}^{\prime}(s)h_{2}^{\prime}(s)|}}\big{(}1\\!+\\!g_{2}(s)h_{2}(s),g_{2}(s)\\!-\\!h_{2}(s),1\\!-\\!g_{2}(s)h_{2}(s),g_{2}(s)\\!+\\!h_{2}(s)\big{)}\end{array}$

with $(g_{1}-g_{2})(h_{1}-h_{2})\neq 0$,
$g_{1}^{\prime}g_{2}^{\prime}h_{1}^{\prime}h_{2}^{\prime}>0$. Then

(4.9) $\Psi(u,v)=\beta(u+v)+\theta(u-v)$

is a minimal Lorentzian surface of general type in $\mathbb{R}^{4}_{2}$
parametrized by canonical parameters.

Now we want to see whether it is possible to change the parameters of a
minimal surface given in a Weierstrass form so that the obtained
representation be canonical. Consider a minimal surface given by (4.1), such
that the null curves $\beta$ and $\theta$ satisfy (4.2) and
$f_{1}f_{2}(g_{1}-g_{2})(h_{1}-h_{2})\neq 0$,
$g_{1}^{\prime}g_{2}^{\prime}h_{1}^{\prime}h_{2}^{\prime}>0$. We look for a
change of the parameters

$u=p(\bar{u}+\bar{v})+q(\bar{u}-\bar{v})\ ,\qquad
v=p(\bar{u}+\bar{v})-q(\bar{u}-\bar{v})$

for some functions $p$, $q$ of one variable, such that the function

$\overline{\Psi}(\bar{u},\bar{v})=\Psi\big{(}p(\bar{u}+\bar{v})+q(\bar{u}-\bar{v}),p(\bar{u}+\bar{v})-q(\bar{u}-\bar{v})\big{)}$

has the form

(4.10)
$\overline{\Psi}(\bar{u},\bar{v})=\bar{\beta}(\bar{u}+\bar{v})+\bar{\theta}(\bar{u}-\bar{v})$

for some functions $\bar{\beta}$ and $\bar{\theta}$ satisfying

(4.11)
$\begin{array}[]{l}\displaystyle\bar{\beta}^{\prime}(\bar{u})\\!=\\!\frac{1}{4\sqrt{|\bar{g}_{1}^{\prime}(\\!\bar{u}\\!)\bar{h}_{1}^{\prime}(\\!\bar{u}\\!)|}}\big{(}\\!1\\!+\\!\bar{g}_{1}(\bar{u})\bar{h}_{1}(\bar{u}),\bar{g}_{1}(\bar{u})\\!-\\!\bar{h}_{1}(\bar{u}),1\\!-\\!\bar{g}_{1}(\bar{u})\bar{h}_{1}(\bar{u}),\bar{g}_{1}(\bar{u})\\!+\\!\bar{h}_{1}(\bar{u})\\!\big{)},\\\
\displaystyle\bar{\theta}^{\prime}(\bar{v})\\!=\\!\frac{1}{4\sqrt{|\bar{g}_{2}^{\prime}(\bar{v})\bar{h}_{2}^{\prime}(\bar{v})|}}\big{(}\\!1\\!+\\!\bar{g}_{2}(\bar{v})\bar{h}_{2}(\bar{v}),\bar{g}_{2}(\bar{v})\\!-\\!\bar{h}_{2}(\bar{v}),1\\!-\\!\bar{g}_{2}(\bar{v})\bar{h}_{2}(\bar{v}),\bar{g}_{2}(\bar{v})\\!+\\!\bar{h}_{2}(\bar{v})\\!\big{)},\end{array}$

and as a consequence of Proposition 4.2 this will be a representation of the
surface in canonical parameters. It can be seen that

$\begin{array}[]{l}\vspace{2mm}\overline{\Psi}_{\bar{u}}=(p^{\prime}+q^{\prime})\Psi_{u}+(p^{\prime}-q^{\prime})\Psi_{v}\\\
\vspace{2mm}\overline{\Psi}_{\bar{v}}=(p^{\prime}-q^{\prime})\Psi_{u}+(p^{\prime}+q^{\prime})\Psi_{v}\end{array}$

imply

$\bar{\beta}^{\prime}(z)=2\beta^{\prime}(p(z))p^{\prime}(z),\qquad\bar{\theta}^{\prime}(z)=2\theta^{\prime}(q(z))q^{\prime}(z).$

Hence, we derive

(4.12)
$\begin{array}[]{l}\displaystyle(p^{\prime}(z))^{2}=\frac{1}{8|f_{1}(p(z))|\sqrt{|g_{1}^{\prime}(p(z))h_{1}^{\prime}(p(z))|}},\quad\bar{g}_{1}(z)=g_{1}(p(z)),\;\;\bar{h}_{1}(z)=h_{1}(p(z));\vspace{3mm}\\\
\displaystyle(q^{\prime}(z))^{2}=\frac{1}{8|f_{2}(q(z))|\sqrt{|g_{2}^{\prime}(q(z))h_{2}^{\prime}(q(z))|}},\quad\bar{g}_{2}(z)=g_{2}(q(z)),\;\;\bar{h}_{2}(z)=h_{2}(q(z)).\end{array}$

Consequently, we may determine not only the change of the parameters, i.e. the
functions $p(z)$, $q(z)$, but also a canonical representation of the given
surface. Finally, we proved:

######  Theorem 4.3.

Let $S$ be a minimal Lorentzian surface of general type in
$\mathbb{R}^{4}_{2}$ parametrized by (4.1), such that the null curves $\beta$
and $\theta$ satisfy (4.2), where $f_{1}f_{2}(g_{1}-g_{2})(h_{1}-h_{2})\neq
0$, $g_{1}^{\prime}g_{2}^{\prime}h_{1}^{\prime}h_{2}^{\prime}>0$. Then, $S$
has a canonical representation (4.10) where $\bar{\beta}$ and $\bar{\theta}$
are null curves satisfying (4.11). This canonical representation is determined
by equations (4.12).

We can also find the Gauss curvature $K$ and the curvature of the normal
connection $\varkappa$ of a minimal Lorentzian surface of general type in
$\mathbb{R}^{4}_{2}$ given in canonical parameters. Namely, for the surface
(4.9), where the null curves $\beta$ and $\theta$ are given by (4.8), we
obtain:

(4.13) $\begin{array}[]{l}\vspace{3mm}\displaystyle
K=-8\frac{\sqrt{g_{1}^{\prime}h_{1}^{\prime}g_{2}^{\prime}h_{2}^{\prime}}\big{(}(h_{1}-h_{2})^{2}g_{1}^{\prime}g_{2}^{\prime}+(g_{1}-g_{2})^{2}h_{1}^{\prime}h_{2}^{\prime}\big{)}}{(g_{1}-g_{2})^{3}(h_{1}-h_{2})^{3}};\\\
\vspace{2mm}\displaystyle\varkappa=-8\sqrt{g_{1}^{\prime}h_{1}^{\prime}g_{2}^{\prime}h_{2}^{\prime}}\left|\frac{(h_{1}-h_{2})^{2}g_{1}^{\prime}g_{2}^{\prime}-(g_{1}-g_{2})^{2}h_{1}^{\prime}h_{2}^{\prime}}{(g_{1}-g_{2})^{3}(h_{1}-h_{2})^{3}}\right|,\end{array}$

where $g_{1},\,h_{1}$ depend on $u+v$ and $g_{2},\,h_{2}$ depend on $u-v$.
Hence, using Theorem B we obtain

######  Theorem 4.4.

The system of partial differential equations (2.5) has solutions of type
(4.13) for arbitrary smooth functions $g_{1},g_{2},h_{1},h_{2}$, such that
$g_{1}^{\prime}g_{2}^{\prime}h_{1}^{\prime}h_{2}^{\prime}>0$ and the
differences $g_{1}-g_{2}$ and $h_{1}-h_{2}$ never vanish.

##  5\. Examples

In this section, using Theorem 3.3 we give some examples of minimal Lorentzian
surfaces in $\mathbb{R}^{4}_{2}$. Moreover, according to Theorem 4.1 they are
of general type and according to Proposition 4.2 they are in canonical
parametrization.

######  Example 5.1.

Let $g_{1}(t)=t+1$, $h_{1}(t)=t$, $g_{2}(s)=s$, $h_{2}(s)=s+1$. Using (4.8) we
have

$\begin{array}[]{l}\vspace{2mm}\displaystyle\beta^{\prime}(t)=\frac{1}{4}\big{(}1+t(t+1);1;1-t(t+1);2t+1\big{)};\\\
\vspace{2mm}\displaystyle\theta^{\prime}(s)=\frac{1}{4}\big{(}1+s(s+1);-1;1-s(s+1);2s+1\big{)}.\end{array}$

Then up to constant vectors the null curves $\beta$ and $\theta$ are given by

$\begin{array}[]{l}\vspace{2mm}\displaystyle\beta(t)=\frac{1}{24}\big{(}2t^{3}+3t^{2}+6t;6t;-2t^{3}-3t^{2}+6t;6t^{2}+6t\big{)};\\\
\vspace{2mm}\displaystyle\theta(s)=\frac{1}{24}\big{(}2s^{3}+3s^{2}+6s;-6s;-2s^{3}-3s^{2}+6s;6s^{2}+6s\big{)}.\end{array}$

Applying (4.9) we obtain the surface

$\Psi(u,v)=\left\\{\begin{array}[]{l}\vspace{2mm}\displaystyle\frac{1}{12}\big{(}2u^{3}+3u^{2}+3v^{2}+6u(v^{2}+1)\big{)}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \vspace{2mm}\displaystyle\frac{v}{2}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\vspace{2mm}\displaystyle\frac{1}{12}\big{(}-2u^{3}-3u^{2}-3v^{2}-6u(v^{2}-1)\big{)}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\vspace{2mm}\displaystyle\frac{1}{2}\big{(}u+u^{2}+v^{2}\big{)}\
.\end{array}\right.$

In this example,
$(g_{1}(u+v)-g_{2}(u-v))(h_{1}(u+v)-h_{2}(u-v))=(2v+1)(2v-1)$, so we assume
that $v\neq\pm\frac{1}{2}$.

According to (4.13) the Gauss curvature $K$ and the curvature of the normal
connection $\varkappa$ are expressed by:

$K=\frac{16(1+4v^{2})}{(1-4v^{2})^{3}}\
;\qquad\varkappa=-\left|\frac{64v}{(1-4v^{2})^{3}}\right|\ .$

######  Example 5.2.

Let $g_{1}(t)=t$, $h_{1}(t)=t$, $g_{2}(s)=e^{s}$, $h_{2}(s)=-e^{-s}$. Using
(4.8) and (4.9) we obtain the surface

$\Psi(u,v)=\left\\{\begin{array}[]{l}\vspace{2mm}\displaystyle\frac{1}{12}(u+v)^{3}+\frac{1}{4}(u+v)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\vspace{2mm}\displaystyle\frac{1}{2}\sinh(u-v)\vskip 3.0pt plus 1.0pt minus
1.0pt\\\
\vspace{2mm}\displaystyle-\frac{1}{12}(u+v)^{3}+\frac{1}{4}(3u-v)\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
\vspace{2mm}\displaystyle\frac{1}{2}\cosh(u-v)+\frac{1}{4}(u+v)^{2}.\end{array}\right.$

In this example,
$(g_{1}(u+v)-g_{2}(u-v))(h_{1}(u+v)-h_{2}(u-v))=(u+v-e^{u-v})(u+v+e^{-(u-v)})$,
so we assume that the parameters $u$ and $v$ satisfy $u+v\neq e^{u-v}$,
$u+v\neq-e^{-(u-v)}$, or equivalently $(u+v)^{2}-1\neq 2(u+v)\sinh(u-v)$.

The Gauss curvature $K$ and the curvature of the normal connection $\varkappa$
are:

$K=-16\frac{\cosh(u-v)\big{(}(u+v)^{2}+1\big{)}}{\big{(}(u+v)^{2}-1-2(u+v)\sinh(u-v)\big{)}^{3}}\
,$
$\varkappa=-16\left|\frac{\big{(}(u+v)^{2}-1\big{)}\sinh(u-v)+2(u+v)}{\big{(}(u+v)^{2}-1-2(u+v)\sinh(u-v)\big{)}^{3}}\right|\
.$

Note, that according to Theorem 4.4, the functions $K$ and $\varkappa$ from
Example 5.1 and Example 5.2 give solutions to the system of partial
differential equations (2.5).

Acknowledgments: The second author is partially supported by the National
Science Fund, Ministry of Education and Science of Bulgaria under contract DN
12/2.

## References

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1Department of Physics, Mar Thoma College, Kuttapuzha P. O. Tiruvalla, PIN
689103, Kerala, India.

# Solar activity and extreme rainfall over Kerala, India

Elizabeth Thomas1 S. Vineeth1 and Noble P. Abraham1,*

###### Abstract

In this paper, we examine the relationship between solar activity and extreme
rainfall events in Kerala, India. Kerala receives minimum and maximum rainfall
during the winter and monsoon seasons, respectively. Sunspot number, F10.7
Index, and cosmic ray intensity are the solar indices considered and their
variations with rainfall were studied over a period of 57 years (1965-2021),
i.e., starting from Solar Cycle 20. For each solar cycle, correlative studies
are performed and correlation coefficients are calculated. We find that
rainfall in Kerala is correlated with sunspot activity, but with varying
degrees of significance. During Solar Cycle 21, rainfall and solar activity
are correlated with high significance during both winter and monsoon seasons.
The variation of different solar indices with rainfall is studied. The winter
season showed a better link between the sun and rainfall than the monsoon
season. The years with rainfall surplus and deficiency are calculated and
compared with the solar indices. We find that the years with rainfall excess
and deficit mostly occur in the years around solar maximum or minimum (within
$\pm$2 years). We hypothesise a physical relationship between solar activity
and extreme rainfall events in Kerala that contributes to their
predictability.

###### keywords:

solar activity—extreme rainfall—sunspot cycle.

<EMAIL_ADDRESS>
12.3456/s78910-011-012-3 #### 000 0000 1– 1

##  1 Introduction

The change of the world climate is a serious matter as it affects the
existence of mankind. Weather and climate are significantly influenced by the
sun, in addition to anthropogenic factors. Solar activity, i.e. magnetic
activity inside the Sun, manifests itself in the form of sunspots, solar
flares, solar wind, coronal mass ejections, etc [Usoskin, 2017]. Some of the
solar indices commonly used to quantify solar activity are total solar
irradiance, sunspot number, solar radio flux, cosmic rays, etc. The sunspot
number quantifies sunspots and is widely used because of its long-term
availability. It is strongly correlated with other solar indices [Hathaway,
2015, Tiwari & Kumar, 2018]. A measure of solar radio flux at 10.7 cm is
called the F10.7 index, which originates deep in the corona and high in the
chromosphere [Tapping & Charrois, 1994, Tapping, 2013]. Cosmic rays originate
outside the solar system and are high-energy particles that reach the Earth.
It is observed that cosmic rays are negatively correlated with sunspot number
[Gupta et al., 2006].

The Sun’s influence on Earth’s precipitation has long been a topic of concern.
Solar activity affects rainfall in a wide variety of ways, which are reflected
in the varying correlations based on the time scale and region [Tsiropoula,
2003, Zhao et al., 2004, Wasko & Sharma, 2009, Mauas et al., 2011, Rampelotto
et al., 2012]. Some recent work on sun-rainfall link were carried out in China
[Zhai, 2017, Yu et al., 2019, Song et al., 2022], the United States [Nitka &
Burnecki, 2019], Europe [Laurenz et al., 2019], Argentina [Heredia et al.,
2019], Nepal [Tiwari et al., 2021] and Northeast Asia [Song et al., 2022].

Changes in rainfall patterns can severely affect a country like India, posing
challenges to its economy, farming, and ecosystem [Doranalu Chandrashekar et
al., 2017]. Over India as a whole or in different regions, numerous authors
have explored the possibility of an association between solar activity and
rainfall [Jagannathan & Bhalme, 1973, Ananthakrishnan & Parthasarathy, 1984,
Hiremath & Mandi, 2004, Bhattacharyya & Narasimha, 2005, Agnihotri et al.,
2011, Badruddin & Aslam, 2015, Warrier et al., 2017, Thomas & Abraham, 2022b].
The direct and indirect effects were studied. The results were often localised
and contradicted other authors [Jagannathan & Parthasarathy, 1973, Bhalme et
al., 1981, Hiremath, 2006, Bhattacharyya & Narasimha, 2007, Lihua et al.,
2007, Selvaraj et al., 2009a, Selvaraj & Aditya, 2011, Selvaraj et al., 2013,
Hiremath et al., 2015, Malik & Brönnimann, 2018, Thomas et al., 2023].

Kerala lies on the southwestern tip of India and is bounded on the east by the
Western Ghats and on the west by the Arabian Sea. It extends between 8∘15′ and
12∘50′ northern latitude and between 74∘50′ and 77∘30′ eastern longitude. The
climate of Kerala is subtropical, with the eastern highlands (rugged and cool
mountainous), the central midlands (rolling hills), and the western lowlands
(coastal plains). Kerala’s diverse features make it more vulnerable to climate
change. Kerala is known as the ”gateway to the summer monsoon.” Studies on
long-term rainfall variability found that southwest monsoon rainfall decreased
significantly while post-monsoon rainfall increased [Krishnakumar et al.,
2009, Kothawale & Rajeevan, 2017]. Recently, a few studies have reported the
influence of sunspot number on rainfall over Kerala [Thomas & Abraham, 2022a,
b, Thomas et al., 2023]. The location map of Kerala is shown in Figure 1.

Figure 1: Location map of Kerala Figure 2: Location of Grid points

Extreme rainfall events have recently occurred in Kerala, causing floods or
landslides that have resulted in the loss of lives and property. In India,
several studies have linked solar activity to extreme weather events (see for
e.g. Bhalme & Mooley [1981], Azad [2011]). Hence, a study of extreme rainfall
over the Kerala region with different solar parameters would be of interest.

Investigating the response of rainfall to different solar indices could reveal
subtle differences. In this paper, the influence of different solar indices,
i.e., sunspot number, F10.7 Index, and cosmic ray intensity, on the occurrence
of extreme precipitation over Kerala is evaluated using correlation studies.
Section 2 discusses the data and methodology of the analysis. Section 3
presents the results and discussion on the correlation and variation of
different solar indices with rainfall over Kerala. It also includes results on
extreme rainfall events. Section 4 contains the conclusions.

##  2 Data and Methods

###  2.1 Dataset

For this study, daily data of various solar indices (sunspot number, F10.7
Index, and cosmic ray intensity) and rainfall over Kerala were used from 1965
to 2021, i.e., from Solar Cycle 20. The sunspot number data was obtained from
the World Data Center SILSO, Royal Observatory of Belgium, Brussels. F10.7
Index is the solar flux data (in sfu, 1 sfu = $10^{-22}Wm^{-2}Hz^{-1}$) and
was downloaded from LASP Interactive Solar Irradiance Data Center. Cosmic ray
intensity (in counts/min) data was taken from Oulu Cosmic Ray station.
Rainfall (in mm) over Kerala was obtained from the India Meteorological
Department’s (IMD) daily gridded rainfall dataset of high spatial resolution
(0.25∘ × 0.25∘) [Pai et al., 2014]. 59 grid points covering the region of
Kerala were used in this study and are shown in Figure 2.

India Meteorological Department (IMD) classifies the seasons of India as
Winter (January-February), Pre-monsoon (March-May), Southwest monsoon (June-
September), and Post-monsoon (October-December). In this study, Southwest-
monsoon season, denoted as JJAS and the winter season, denoted as JF were
used, as maximum and minimum rainfall are accounted for during these seasons.
Similar grouping (JF and JJAS) was done with the solar activity features
(sunspot number, F10.7 index, and cosmic ray intensity) data. The daily data
for each parameter were averaged for each season, and the corresponding values
for sunspot number are given as SSN (/days), F10.7 Index is given as F10.7
(sfu/days), cosmic ray intensity is given as CRI (count/min/days), and
rainfall is given as RF (mm/days). Figure 3 and Figure 4 show the time series
of SSN, F10.7, CRI, and RF corresponding to the JF and JJAS seasons
respectively.

Figure 3: Time series of (a) sunspot number (SSN) (b) F10.7 index (F10.7) (c)
cosmic ray intensity (CRI) and (d) rainfall (RF), corresponding to JF season.

Figure 4: Time series of (a) sunspot number (SSN) (b) F10.7 index (F10.7) (c)
cosmic ray intensity (CRI) and (d) rainfall (RF), corresponding to JJAS
season.

###  2.2 Methodology

To observe the relationship between solar indices and rainfall for Solar
Cycles 20-24, correlative studies were carried out. Correlation coefficients
are usually computed to check whether any relationship exist between the two
data sets and how strong the relationship is. Here, Spearman Rank-Order
correlation coefficients and their significance were calculated to determine
the relationship between different solar indices (SSN, F10.7 and CRI) and
rainfall (RF) data. Since this correlation method provides information about
monotonic relationships, it is more powerful than linear correlation [Hiremath
& Mandi, 2004, Hiremath, 2006, Bankoti et al., 2011]. The p-value obtained
gives the significance of the test. The p-values $<$ 0.1 gives 10 %
significance. Corresponding to all the solar activity indices, correlation
coefficients, and their significance were determined for each solar cycle. As
the rainfall is noisy, the RF values was smoothened by moving average of two,
four and six points and the correlation studies were carried out [Bankoti et
al., 2011].

In order to study the relation of extreme rainfall events with the solar
activity, the years of excess and deficit rainfall over Kerala were
identified. For that, the mean ($\mu$) and standard deviation ($\sigma$) of
rainfall RF, during both the seasons (JF and JJAS), were determined. A year
$i$ was labelled as extreme rainfall year when $R_{i}\geq(\mu+\sigma)$ and a
year labelled as deficient rainfall year when $R_{i}\leq(\mu-\sigma)$, where
$R_{i}$ is the rainfall of that year, $i,k\in\mathbb{R}$ [Azad, 2011]. In this
study, $k$ was defined as one. The excess and deficient rainfall years were
used to study the relation of rainfall with solar activity.

##  3 Results and Discussions

The Spearman Rank-Order correlation coefficient and its significance between
different solar indices, i.e., sunspot number, F10.7 Index and cosmic ray
intensity, and rainfall is shown in Table 1, where * represents 0.1
significance level. The first column represents the solar cycle, the second
column represents different types of moving point average and original data
set for different seasonal months (JF and JJAS). The correlation coefficients
with significance(in brackets) is represented in column three, four, and five
of SSN, F10.7, and CRI with RF respectively.

Table 1: The correlation coefficients and significance of correlation (in
brackets) between different solar indices (SSN, F10.7 and CRI) and rainfall RF
for the seasonal months (JF and JJAS). Correlation coefficients for the same
solar activity indices with two, four and six point moving averages of
rainfall are also represented.

Solar Cycle | Seasonal months | Solar activity indices
---|---|---
|  | SSN | F10.7 | CRI
20 | JF(original) | 0.43 (0.16) | 0.51 (0.09)* | -0.50 (0.10)*
| 2 pt | 0.61 (0.04)* | 0.69 (0.02)* | -0.76 (0.01)*
| 4 pt | 0.83 (0.001)* | 0.83 (0.001)* | -0.83 (0.002)*
| 6 pt | 0.85 (0.001)* | 0.89 (0.0001)* | -0.83 (0.001)*
| JJAS(original) | 0.43 (0.16) | 0.34 (0.28) | -0.21 (0.51)
| 2 pt | 0.24 (0.46) | 0.17 (0.59) | -0.23 (0.47)
| 4 pt | 0.39 (0.20) | 0.41 (0.19) | -0.51 (0.09)
| 6 pt | 0.19 (0.54) | 0.34 (0.29) | -0.35 (0.27)
21 | JF(original) | -0.66 (0.04)* | -0.57 (0.09)* | 0.38 (0.28)
| 2 pt | -0.54 (0.11) | -0.51 (0.13) | 0.57 (0.09)*
| 4 pt | -0.75 (0.02)* | -0.71 (0.03)* | 0.30 (0.41)
| 6 pt | -0.87 (0.003)* | -0.76 (0.02)* | 0.15 (0.68)
| JJAS(original) | 0.56 (0.09)* | 0.68 (0.03)* | -0.39 (0.26)
| 2 pt | 0.76 (0.01)* | 0.83 (0.006)* | -0.83 (0.006)*
| 4 pt | 0.82 (0.007)* | 0.87 (0.003)* | -0.88 (0.002)*
| 6 pt | 0.95 (0)* | 0.93 (0.0001)* | -0.66 (0.04)*
22 | JF(original) | 0.06 (0.86) | 0.08 (0.3) | -0.09 (0.81)
| 2 pt | 0.2 (0.43) | 0.26 (0.47) | -0.31 (0.39)
| 4 pt | -0.07 (0.86) | -0.11 (0.76) | -0.15 (0.68)
| 6 pt | 0.16 (0.65) | 0.13 (0.73) | 0.05 (0.89)
| JJAS(original) | -0.08 (0.84) | -0.30 (0.47) | 0.06 (0.86)
| 2 pt | -0.22 (0.53) | -0.30 (0.41) | 0.09 (0.81)
| 4 pt | -0.25 (0.49) | -0.38 (0.28) | 0.23 (0.51)
| 6 pt | -0.37 (0.29) | -0.50 (0.14) | 0.27 (0.45)
23 | JF(original) | 0.44 (0.15) | 0.47 (0.13) | -0.30 (0.34)
| 2 pt | 0.65 (0.03)* | 0.65 (0.03)* | -0.56 (0.06)*
| 4 pt | 0.81 (0.002)* | 0.82 (0.002)* | -0.54 (0.07)*
| 6 pt | 0.83 (0.001)* | 0.78 (0.004)* | -0.68 (0.02)*
| JJAS(original) | -0.56 (0.06)* | -0.51 (0.09)* | 0.67 (0.02)*
| 2 pt | -0.50 (0.09)* | -0.44 (0.15) | 0.79 (0.004)*
| 4 pt | -0.48 (0.12) | -0.42 (0.17) | 0.76 (0.006)*
| 6 pt | -0.42 (0.17) | -0.36 (0.26) | 0.69 (0.001)*
24 | JF(original) | -0.02 (0.97) | 0.14 (0.70) | 0.03 (0.94)
| 2 pt | 0.16 (0.65) | 0.34 (0.33) | -0.22 (0.54)
| 4 pt | 0.27 (0.44) | 0.47 (0.18) | -0.25 (0.49)
| 6 pt | -0.33 (0.35) | -0.07 (0.83) | 0.21 (0.56)
| JJAS(original) | 0.16 (0.65) | 0.10 (0.78) | -0.001 (1)
| 2 pt | 0.15 (0.68) | 0.13 (0.73) | -0.07 (0.86)
| 4 pt | 0.44 (0.20) | 0.45 (0.19) | -0.36 (0.31)
| 6 pt | 0.42 (0.23) | 0.34 (0.33) | -0.18 (0.63)

Note. * indicates higher than 0.1 significance level.

###  3.1 Relationship between Sunspot number and Rainfall

Figure 5: Variation of sunspot number (SSN) and rainfall (RF) during (a) JF
season and (b) JJAS season.

As the dataset was collected from different locations, standardization of data
was done. Standardized anomaly was calculated by subtracting the mean and
dividing by its standard deviation, for both seasons. The variation in
standardized anomaly represents a deviation from the mean state. The anomaly
is positive when the data is increasing and negative when decreasing. For easy
comparison between the data, the variation in solar indices was represented as
a line graph and the rainfall as a bar graph, with each bar representing a
year. Figure 5 represents the time series of standardized values of sunspot
number (SSN) and rainfall (RF) during the JF and JJAS seasons. Solar cycles
20-24 were covered in this study. To study the response of rainfall over
Kerala to the sunspot number, the Spearman rank-order correlation coefficients
between SSN and RF were calculated. First, the entire 57 years were considered
and the correlation coefficients were found to be low, i.e., 0.03 and 0.002
during the JF and the JJAS seasons respectively. Solar cycle-wise correlation
coefficients were then determined and the results are given in Table 1. The
rainfall over Kerala was noted to be correlated with sunspot number, with
varying significance, irrespective of signs.

For the JF season, significant correlations, irrespective of signs, were
observed during Solar cycles 20, 21, and 23. The correlation coefficients were
noted to be negative only during Solar cycle 21, positive during Solar cycles
20 and 23, and changed signs on smoothing RF values during Solar cycle 22 and
24. It was seen that the significance of the correlation coefficients improved
on smoothing the rainfall RF values by two, four, and six-point moving
averages. Among the five solar cycles considered, Solar cycles 22 and 24
showed weak correlation. For the JJAS season, Solar Cycle 21 showed the
highest correlation between SSN and RF values with significance for original,
two, four, and six-point moving average values. The correlation coefficients
were positive during Solar cycles 20, 21, and 24 and negative during Solar
cycles 22 and 23. Comparing both seasons, the JF season revealed a better
association of sunspot number with the rainfall over Kerala.

###  3.2 Relationship between F10.7 Index and Rainfall

Figure 6: Time series of F10.7 Index (F10.7) and rainfall (RF) during (a) JF
season and (b) JJAS season.

Figure 6 shows the time series of standardized values of the F10.7 Index
(F10.7) and rainfall (RF) during the JF and JJAS seasons. Here also
correlation coefficient was computed to be 0.09 and -0.03 during the JF and
the JJAS seasons respectively, considering the entire period of study.
Correlation coefficients corresponding to Solar cycles 20-24 were worked out
and the results are given in Table 1. Similar to SSN results, the rainfall
values were observed to be correlated with F10.7 Index with varying
significance, regardless of the signs. During the JF season, Solar cycles 20,
21, and 23 showed a high correlation with significance. The correlation
appeared to improve in smoothing the rainfall values. In the case of the JJAS
season, Solar cycle 21 revealed a high significant correlation, compared to
other cycles. It can be concluded that rainfall in Kerala is related to solar
activity during the JF season, since the F10.7 Index showed a high correlation
during three solar cycles during this season, as compared to the JJAS season.

###  3.3 Relationship between Cosmic ray intensity and Rainfall

Figure 7: Time series of Cosmic ray intensity (CRI) and rainfall (RF) during
(a) JF season and (b) JJAS season.

Figure 7 shows the time series of standardized values of cosmic ray intensity
(CRI) and rainfall (RF) during the JF and the JJAS seasons. During the JF
season, it was observed that during the decreasing phase of CRI in the first
solar cycle starting from 1965, the rainfall anomaly is negative. In the next
cycle, starting from 1977, again the rainfall anomaly is observed to be
negative during the decreasing phase of CRI. Out of the five decreasing phases
observed, the first three rainfall anomalies were negative, and the remaining
two were positive. In the case of the JJAS season, the rainfall anomalies
alternate during each decreasing phase of CRI. During the first cycle,
rainfall anomaly is negative and during the next cycle, it is observed to be
positive, and so on. Chaudhuri et al. [2015] reported results showing that the
decreasing phase of GCR is an important phase to identify rainfall
variability.

Correlation coefficients computed between CRI and RF values considering the
entire period were found to be -0.08 and 0.10 during the JF and the JJAS
seasons respectively. Correlation coefficients were determined for each solar
cycle and the results are given in Table 1. Compared to sunspot number and
F10.7 Index, cosmic ray intensity showed correlation results that were often
opposite in sign. Solar cycles 20 and 23 showed a significant negative
correlation between CRI and RF during the JF season, whose significance
improved in smoothing the rainfall data. Solar cycles 21 and 23 revealed a
significant correlation during the JJAS season, for smoothed values of
rainfall.

Out of the five solar cycles considered, Solar Cycle 21 (1977-1986) was the
strongest, and Solar Cycle 24 (2009-2018) was the weakest, in terms of SSN. It
was noted that when the solar activity was maximum, a high correlation with
significance was observed in the JF season. A weak correlation between sunspot
number and rainfall was found during periods of low solar activity, i.e.,
during Solar Cycle 24. Solar cycles 20, 21, and 23 showed a good association
between SSN and RF, and after smoothing the data, the correlation improved. As
for the JJAS season, Solar Cycle 21 showed a positive correlation, but this
correlation decreased during Solar Cycle 24 when solar activity was lower.
These results are consistent with the earlier results. Thomas & Abraham
[2022b] analyzed the relationship between sunspot number and rainfall over
Kerala during varying levels of solar activity using wavelet coherence and
noted higher coherence during the high solar activity period than during low
solar activity, during winter (JF) and monsoon (JJAS) seasons. On considering
F10.7 Index and Cosmic ray intensity parameters, the results were similar and
the correlation results were weaker than those of sunspot number. When
comparing the two seasons, the JF season had a stronger solar influence on its
rainfall than the JJAS season.

In India, several correlative studies have been conducted in an attempt to
determine if solar influences affect rainfall there. Ananthakrishnan &
Parthasarathy [1984] reported both positive and negative correlation
coefficients while considering 306 stations in India. Hiremath & Mandi [2004]
investigated the correlative effects of sunspot number on the seasonal and
annual Indian monsoon rainfall and found that the pre-monsoon and monsoon
rainfall showed significant positive correlations. Hiremath [2006] analyzed
correlative effects of sunspot number over the Indian rainfall corresponding
to each solar cycle and noted correlation irrespective of the signs, with a
moderate to high significance. When solar activity was low, rainfall was
higher than when solar activity was high. Bal & Bose [2010] reported the
existence of weak positive and negative correlations during different seasons.
Bankoti et al. [2011] conducted several statistical studies between the solar
parameters (Sunspot number, solar active prominences, and H alpha solar
flares) and All India homogeneous rainfall and noted that the correlation
varied its sign with different seasons and also with different solar
parameters. Chaudhuri et al. [2015] performed seasonal correlation and
observed a possible association between cosmic rays and rainfall during the
post-monsoon season. Several correlative studies were carried out in different
states to find a possible sun-rainfall link, ie, in West Bengal [Chakraborty &
Bondyopadhyay, 1986], Rajasthan [Jain & Tripathy, 1997b], Tamil Nadu [Selvaraj
et al., 2009a, Selvaraj & Aditya, 2011, 2012a] and Kerala [Thomas & Abraham,
2022a].

###  3.4 Solar activity indices and extreme rainfall in Kerala

A study was conducted to examine the possible relationship between solar
activity and extreme rainfall events in Kerala during the JF and JJAS seasons.
For that, the years of excess rainfall and deficient rainfall were identified,
as explained in Section 2.2 [Azad, 2011]. During the JF season, six excess
rainfall events were visible during the years 1984, 1990, 1994, 2000, 2011,
and 2021. This season was not marked by deficient rainfall. In the case of the
JJAS season, both excess and deficient rainfall events were observable. Ten
years of excess rainfall were recorded during the years 1968, 1975, 1981,
1991, 1992, 1994, 1997, 2007, 2013 and 2019. Similarly, five years of
deficient rainfall were observed during the years 1976, 1987, 2002, 2012, and
2016.

A study of the relative timing of solar activity and extreme rainfall was
conducted using curves of different solar activity features, corresponding to
different seasons. Figure 8, 9 and 10 represent the extreme rainfall events
with SSN, F10.7, and CRI respectively. The black circles denote excess
rainfall years and the red circle denotes deficient rainfall years. The
present study covers five complete solar cycles (Solar cycles 20-24). Tables
2, 4 and 6 list the extreme rainfall events during the JF season, and Tables
3, 5 and 7 lists the extreme rainfall events during the JJAS season. The first
column represents the years of extreme of each solar activity feature (denoted
as $y$) and second column, the excess rainfall years, and the third column,
the deficient rainfall years. Values given in brackets indicate variation
concerning solar extremes.

####  3.4.1 Relation of extreme rainfall years with Sunspot number

Figure 8: The extreme years of JF and JJAS rainfall depicted on SSN plot

The sunspot number SSN was considered first. Figure 8 gives the extreme
rainfall years during the JF and JJAS seasons plotted on the SSN curve. In
this figure, slight variations in the maximum and minimum values of SSN during
the JF and JJAS seasons were visible. During the JF season, the SSN showed a
maximum during the years 1970, 1980, 1991, 2002, and 2014 and a minimum during
the years 1965, 1976, 1987, 1997, 2009, and 2020. In the JJAS season, maximum
SSNs occurred in 1968, 1979, 1991, 2000, and 2014, and minimum SSNs occurred
in 1965, 1976, 1986, 1996, 2008, and 2019.

  * •

Excess rainfall studies
The six excess rainfall years observed during the JF season are listed in
Table 2. Two of these occurred around $\pm$ 1 year and the other two around
$\pm$ 2 years of extreme solar activity (solar maximum/solar minimum). The
remaining two excess rainfall years were observed a bit further from the solar
minimum, i.e., three years before the solar minimum. Ten excess rainfall years
were identified during the JJAS season and given in Table 3. Three of them
occurred at solar extremes, five around $\pm$1 years of solar extremes, and
remaining two around $\pm$2 years of solar extremes.

  * •

Deficient rainfall studies
During the JF season, there were no years with deficient rainfall. Five
deficient rainfall events were noted during the JJAS season and are listed in
Table 3. One occurred at the extreme of the solar cycle, another one a year
later, and the remaining three occurred around $\pm$2 years of solar extreme.

Table 2: Extreme rainfall years along with extreme SSN years, during JF season Years of extreme SSN (y) | Excess rainfall years
---|---
1965 (min) |
1970 (max) |
1976 (min) |
1980 (max) |
1987 (min) | 1984 (y-3)
1991 (max) | 1990 (y-1)
1997 (min) | 1994 (y-3)
2002 (max) | 2000 (y-2)
2009 (min) | 2011 (y+2)
2014 (max) |
2020 (min) | 2021 (y+1)
Table 3: Extreme rainfall years along with extreme SSN years, during JJAS season Years of extreme SSN (y) | Excess rainfall years | Deficient rainfall years
---|---|---
1965 (min) |  |
1968 (max) | 1968 (y) |
1976 (min) | 1975 (y-1) | 1976 (y)
1979 (max) | 1981 (y+2) |
1986 (min) |  | 1987 (y+1)
1991 (max) | 1991 (y), 1992 (y+1) |
1996 (min) | 1994 (y-2), 1997 (y+1) |
2000 (max) |  | 2002 (y+2)
2008 (min) | 2007 (y-1) |
2014 (max) | 2013 (y-1) | 2012 (y-2), 2016 (y+2)
2019 (min) | 2019 (y) |

####  3.4.2 Relation of extreme rainfall years with F10.7 Index

Figure 9: The extreme years of (a) JF and (b) JJAS rainfall depicted on F10.7
plot

The extreme rainfall years were then analyzed in terms of the F10.7 Index.
Figure 9 shows the extreme rainfall years during the JF and JJAS seasons
plotted on the F10.7 curve. The two seasons show different maximum and minimum
values for each solar cycle. During the JF season, F10.7 showed peak values in
1968, 1979, 1991, 2002, and 2014 and trough values in 1965, 1976, 1987, 1996,
2009, and 2020. In the JJAS season, F10.7 revealed maximum values in 1970,
1981, 1989, 2000, and 2014 and minimum values in 1965, 1976, 1986, 1996, 2008,
and 2019.

  * •

Excess rainfall studies
During the JF season, six years of excess rainfall were noticed and are listed
in Table 4. It was noted that four out of these occurred around solar minimum
and the remaining two around solar maximum. The two years of excess rainfall
were around $\pm$1 years of solar extreme and the three were around $\pm$2
years of solar extreme. Details of the excess rainfall events during the JJAS
season are listed in Table 5. Out of the ten years of excess rainfall, two
occurred at the solar extreme, four of them at around $\pm$1 years of solar
extreme and the three around $\pm$2 years of solar extreme. One of them was
visible a bit further from the solar extreme, i.e.,3 years after the solar
maximum.

  * •

Deficient rainfall studies
During the JJAS season, five years of deficient rainfall was observed and
details are given in Table 5. One of them occurred at solar extreme, i.e., at
solar minimum. The remaining one is at a year after solar extreme, i.e., solar
minimum, and the other two are around $\pm$2 years of solar extreme, i.e.,
solar maximum.

Table 4: Extreme rainfall years along with extreme F10.7 years, during JF season Years of extreme F10.7 (y) | Excess rainfall years
---|---
1965 (min) |
1968 (max) |
1976 (min) |
1979 (max) |
1987 (min) | 1984 (y-3)
1991 (max) | 1990 (y-1)
1996 (min) | 1994 (y-2)
2002 (max) | 2000 (y-2)
2009 (min) | 2011 (y+2)
2014 (max) |
2020 (min) | 2021 (y+1)
Table 5: Extreme rainfall years along with extreme F10.7 years, during JJAS season Years of extreme F10.7 (y) | Excess rainfall years | Deficient rainfall years
---|---|---
1965 (min) |  |
1970 (max) | 1968 (y-2) |
1976 (min) | 1975 (y-1) | 1976 (y)
1981 (max) | 1981 (y) |
1986 (min) |  | 1987 (y+1)
1989 (max) | 1991 (y+2), 1992 (y+3) |
1996 (min) | 1994 (y-2), 1997 (y+1) |
2000 (max) |  | 2002 (y+2)
2008 (min) | 2007 (y-1) |
2014 (max) | 2013 (y-1) | 2012 (y-2), 2016 (y+2)
2019 (min) | 2019 (y) |

####  3.4.3 Relation of extreme rainfall years with Cosmic ray intensity

Figure 10: The extreme years of (a) JF and (b) JJAS rainfall depicted on CRI
plot

Lastly, the relationship of extreme rainfall years with CRI was studied.
Figure 10 depicts the extreme rainfall years during the JF and JJAS seasons
plotted on CRI curves. During the JF season, CRI displayed minimum values
during the years 1970, 1983, 1990, 2003, and 2015, and maximum values during
the years 1965, 1974, 1987, 1997, 2010, and 2020. In the JJAS season, CRI
showed minimum values in 1969, 1982, 1991, 2000, and 2015 and maximum values
in 1965, 1975, 1986, 1997, 2009, and 2020.

  * •

Excess rainfall studies
A list of the extreme rainfall years during the JF and JJAS seasons is given
in Tables 6 and 7 respectively. Out of the six years of excess rainfall during
the JF season, one was observed at solar extreme, three at around $\pm$1 years
of solar extreme, and the rest further away around $\pm$3 years of solar
extreme. During the JJAS season, three years of excess rainfall occurred at
the solar extreme, four at around $\pm$1 years of solar extreme, and the rest
at around $\pm$2 years of solar extreme.

  * •

Deficient rainfall studies
Five years of deficient rainfall was observed in the JJAS season, given in
Table 7. Out of them, three were observed at around $\pm$1 years of solar
extreme, one a year after solar extreme, i.e., solar minimum, and one three
years after solar extreme, i.e., solar minimum.

Table 6: Extreme rainfall years along with extreme CRI years, during JF season Years of extreme CRI (y) | Excess rainfall years
---|---
1965 (max) |
1970 (min) |
1974 (max) |
1983 (min) | 1984 (y+1)
1987 (max) |
1990 (min) | 1990 (y)
1997 (max) | 1994 (y-3)
2003 (min) | 2000 (y-3)
2010 (max) | 2011 (y+1)
2015 (min) |
2020 (max) | 2021 (y+1)
Table 7: Extreme rainfall years along with extreme CRI years, during JJAS season Years of extreme CRI (y) | Excess rainfall years | Deficient rainfall years
---|---|---
1965 (max) |  |
1969 (min) | 1968 (y-1) |
1975 (max) | 1975 (y) | 1976 (y+1)
1982 (min) | 1981 (y-1) |
1986 (max) |  | 1987 (y+1)
1991 (min) | 1991 (y), 1992 (y+1) |
1997 (max) | 1994 (y-3), 1997 (y) |
2000 (min) |  | 2002 (y+2)
2009 (max) | 2007 (y-2) | 2012 (y+3)
2015 (min) | 2013 (y-2) | 2016 (y+1)
2020 (max) | 2019 (y-1) |

When looking at the results related to sunspot number, F10.7 Index, and cosmic
ray intensity, we could conclude that Kerala experienced extreme rainfall,
mostly when solar activity was at its extreme, i.e. when solar maximum or
minimum occurred or within $\pm$2 years of solar extremes. It is therefore
possible to predict extreme rainfall events in Kerala by understanding the
extreme variability of solar activity. As the rainfall deficiency/excess was
observed almost equally around both extremes, it was impossible to determine
if the extreme events were more prevalent during solar maximum or minimum.

According to the SSN and F10.7 analyses, excessive rainfall was more prevalent
during solar minimum during the JF season and solar maximum during the JJAS
season. The deficient rainfall events were noted more around solar maximum
according to the SSN and F10.7 results.

The results obtained are in agreement with earlier studies carried out in
India. Bhalme & Mooley [1981] reported that Flood Area Index over India was
associated with the double sunspot cycle. During alternate solar cycles,
Ananthakrishnan & Parthasarathy [1984] observed significantly more excess
rainfall years during the ascending phase. Jain & Tripathy [1997b] considered
Udaipur subtropical region in Rajasthan to check any possible relation between
solar activity and its rainfall and noted that the periodicity of floods and
droughts are well correlated with sunspot main periods and/or quasi-periods.
Bhattacharyya & Narasimha [2005] revealed that high rainfall is linked with
high solar activity and low rainfall with low solar activity. Azad [2011]
while studying the relation of extreme Indian monsoon rainfall over sub-
divisions from west central and peninsular India with sunspots, reported that
the maxima of even sunspot cycles coincided with excess rainfall (with +1 year
error) and the minima of odd sunspot cycles coincided with deficit rainfall
(with $\pm$2 year error).

Several studies have reported how solar activity affects extreme rainfall
events around the world. In the United States, it was observed that the
drought cycle is related to the double (Hale) sunspot cycle [Mitchell et al.,
1979, Cook et al., 1997]. Vaquero [2004] evaluated the number of floods
recorded for the Tagus river basin, Central Spain, and it was noted that the
probability of floods increased during the episodes of high solar activity. A
study on the levels of Lake Victoria, East Africa revealed the influence of
solar activity on the levels through rainfall. The rainfall maxima had a
lagged relationship with the sunspot maxima by one year, leading to the lake
level maxima [Stager et al., 2007]. Sunspot number showed a direct correlation
with the flood/drought of the Second Songhua river basin, China, and flood
years appeared in Solar Maximum Year, years after Solar Maximum Year, and
Solar Minimum Year [Hong-yan et al., 2015]. Studies relating the response of
extreme precipitation to solar activity in typical regions of the Loess
Plateau, in Yan’an, China observed that the maximum precipitation occurred
mainly during solar maximum and was correlated [Li et al., 2017]. Yu et al.
[2019] also reported that the occurrences of droughts and floods in the
Southern Chinese Loess Plateau were synchronous with solar activities, at
least on decadal timescales.

There are instances of opposing results, as well, being reported. Wirth et al.
[2013] found that flood frequency in the European Alps increased during cool
periods, which coincided with low solar activity. In studies relating to River
Ammer floods in Germany, Rimbu et al. [2021] observed the frequency of flood
years is relatively high with the solar activity is low and vice versa. Li et
al. [2023] investigated the time-lagged correlations between solar activity
and summer precipitation in the mid-lower reaches of the Yangtze River, China
and it was observed that the sunspot number has a negative correlation with
precipitation, with a time lag of 11 months.

The solar influence on rainfall has been a matter of concern for long and many
possible links have been discussed in different works [Li et al., 2023]. The
presence of similar periods in the time series of rainfall and the solar
activity implied a possible relation between them [Nitka & Burnecki, 2019,
Heredia et al., 2019]. Sea surface temperature can be influenced by the total
solar irradiance (TSI), which alters atmospheric circulation and modulates
rainfall [Soon et al., 1996]. Stratospheric ozone absorbs ultraviolet
radiation from the sun, causing a temperature gradient. This, in turn, affects
Brewer-Dobson circulation and further alters the lower atmosphere through
interaction between troposphere and stratosphere [Baldwin & Dunkerton, 2005].

##  4 Conclusions

We have evaluated the possible impact of solar activity on extreme rainfall
events, over a period of 57 years, in Kerala, India. Different solar indices,
i.e., Sunspot number, F10.7 Index, and cosmic ray intensity were used, and
their variation with rainfall was studied. Correlation studies were performed
for each solar cycle starting with Solar Cycle 20. It was observed that the
rainfall in Kerala is correlated with sunspot activity, with varying
significance. Among the three solar activity indices, SSN, F10.7, and CRI, the
SSN showed the strongest correlation with rainfall. A significant correlation
exists between rainfall and solar activity during both the winter and monsoon
seasons of Solar Cycle 21. In comparison to the monsoon season, the winter
season revealed stronger solar influences on its rainfall. The years when
rainfall in Kerala was excess or deficient were identified, and its connection
with the different solar indices was studied. Most of the years with excess or
deficient rainfall have been observed to coincide with solar extremes, such as
solar maximum or solar minimum (within $\pm$2 years). It is possible to
predict extreme rainfall events in Kerala based on the coincidence and reduce
their disastrous impacts.

## Acknowledgements

First author acknowledges the financial assistance from the University Grants
Commission (UGC), India, under Savitribai Jyotirao Phule Fellowship for Single
Girl Child (SJSGC) (F. No. 82-7/2022(SA-III) dated 07/02/2023). Second author
acknowledges the financial assistance from Department of Science and
Technology (DST), Ministry of Science and Technology, India under INSPIRE
Fellowship (Award Letter No. IF180235 dated 08/02/2019).

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