# On the noncollapsedness of positively curved Type I ancient Ricci flows

Liang Cheng and Yongjia Zhang

###### Abstract

In this article, we study complete Type I ancient Ricci flows with positive
sectional curvature. Our main results are as follows: in the complete and
noncompact case, all such ancient solutions must be noncollapsed on all
scales; in the closed case, if the dimension is even, then all such ancient
solutions must be noncollapsed on all scales. This furthermore gives a
complete classification for three-dimensional noncompact Type I ancient
solutions without assuming the noncollapsing condition.

##  1 Introduction to the Main Results

The study of ancient solutions to the Ricci flow, ever since Hamilton had
published his program [12], has been an important topic in the field of Ricci
flow. Ancient solutions are Ricci flows whose existing intervals extend to
negative infinity. They are of great importance because they usually arise as
blow-up limits at finite-time singularities of the Ricci flow, and to this
kind of blow-up limits, a term not improper, “singularity models”, is
assigned.

Perelman [21] proved that a Ricci flow on a closed manifold cannot become
locally collapsed within finite time. Subsequently one may conclude that every
singularity model must be $\kappa$-noncollapsed on all scales. This precisely
means the following.

######  Definition 1.1 ($\kappa$-noncollapsing).

A Ricci flow $(M^{n},g(t))$ is called $\kappa$-noncollapsed on all scales,
where $\kappa$ is a positive number, if for any point $(x,t)$ in space-time
and any positive scale $r$, it holds that
$\displaystyle\operatorname{Vol}_{g(t)}\big{(}B_{g(t)}(x,r)\big{)}\geq\kappa
r^{n}$ whenever $R\leq r^{-2}$ on $B_{g(t)}(x,r)$. Here $R$ stands for the
scalar curvature

The noncollapsing notion defined above is sometimes called the _strong_
noncollapsing in the literature. The _weak_ noncollapsing notion is defined
similarly with only the “whenever $R\leq r^{-2}$ on $B_{g(t)}(x,r)$” statement
replaced by “whenever $|Rm|\leq r^{-2}$ on $B_{g(t)}(x,r)\times[t-r^{2},t]$”.
It is known that for an ancient solution with bounded and nonnegative
curvature operator, the weak noncollapsing is equivalent to the strong
noncollapsing condition, with possibly a different $\kappa$. The noncollapsing
condition which we use throughout this paper is the strong one. We remark that
these two notions are not equivalent in general. For instance, a closed
nonflat and Ricci-flat (static) Ricci flow is weakly noncollapsed but not
strongly noncollapsed.

Since, according to Hamilton [12], ancient solutions are critical to the
understanding of the singularity formation in the Ricci flow (see, for
instance, Perelman’s proof of the canonical neighborhood theorem [21]), it
makes sense to assume the noncollapsing condition when studying ancient
solutions. With this assumption, many groundbreaking works are done, and the
most outstanding one is of Perelman [21]. See also [3], [4], [5], and [16],
etc., to list but a few.

It is well-known that not all ancient solutions are noncollpased. But what if
some further conditions are added? Concerning this a question is proposed in
[8]:

> Problem 9.41. _Are nonflat Type I ancient solutions with nonnegative
> curvature operator $\kappa$-solutions?_

Recall that an ancient solution $(M,g(t))_{t\in(-\infty,w)}$ is called _Type
I_ if

$\displaystyle\limsup_{t\rightarrow-\infty}|t|\big{|}Rm_{g(t)}\big{|}<\infty.$

Without any further qualification, the answer to the above question is
obviously “no”, either in the closed case or in the complete and noncompact
case. One may immediately think of $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$ or
$\mathbb{S}^{n-2}\times\mathbb{R}\times\mathbb{S}^{1}$ as counterexamples.
There are also much more sophisticated counterexamples constructed. For
instance, Fateev [10] constructed an ancient solution on the Hopf fiber
bundle, and Bakas-Kong-Ni [1] generalized this construction to all odd-
dimensional spheres; all these ancient solutions are Type I, collapsed, and
with positive curvature operator.

In this article, we give a relatively satisfactory answer to Problem 9.41 in
[8] as quoted above. First of all, to rule out the possibility of a compact
flat factor, with which the ancient solution is always collapsed, we would
like to assume that the sectional curvature is strictly positive. This
condition also largely simplifies the geometry in the complete and noncompact
case, since the underlying manifold must be diffeomorphic to the Euclidean
space by the Gromoll-Meyer theorem.

######  Theorem 1.2.

Let $(M^{n},g(t))_{t\in(-\infty,w)}$, where $0<w\leq\infty$, be a complete and
noncompact Type I ancient solution with positive sectional curvature. Then
$(M^{n},g(t))_{t\in(-\infty,0]}$ is $\kappa$-noncollapsed on all scales for
some $\kappa>0$.

The Type I, closed, and collapsed examples constructed in [1] are only in odd
dimensions, while the cases of even dimensions are yet open. The following
theorem shows that there are no such collapsed examples in even dimensions.

######  Theorem 1.3.

Let $(M^{n},g(t))_{t\in(-\infty,w)}$, where $n$ is an even number and
$0<w\leq\infty$, be a closed Type I ancient solution with positive sectional
curvature. Then $(M^{n},g(t))_{t\in(-\infty,0]}$ is $\kappa$-noncollapsed on
all scales for some $\kappa>0$.

The critical observations applied to the proofs of Theorem 1.2 and Theorem 1.3
are some injectivity radius estimates resulted from the Gromoll-Meyer theorem
and the Klingenberg theorem. These classical theorems imply that an ancient
solution as described in Theorem 1.2 or Theorem 1.3 has a Type I injectivity
radius lower bound. This is sufficient to conclude the existence of an
asymptotic shrinker, which in turn implies noncollapsedness; the authors used
a similar idea in [7] to prove the noncollapsedness of a more general type of
ancient Ricci flows—the locally uniformly Type I ancient solutions. Here we
emphasize that the Type I injectivity radius lower bound itself does not
directly imply the noncollapsedness; see section 2 for more details concerning
this point.

An immediate application of Theorem 1.2 is the following classification of
three-dimensional Type I ancient solutions without assuming the noncollapsing
condition. This is a generalization of [23] (or [15]). In consequence, all
noncompact three-dimensional Type I ancient solutions must be noncollapsed.

######  Corollary 1.4.

A three-dimensional noncompact Type I ancient solution must be the standard
cylinder $\mathbb{S}^{2}\times\mathbb{R}$,
$(\mathbb{S}^{2}\times\mathbb{R})/\mathbb{Z}_{2}$, or
$\operatorname{\mathbb{R}P}^{2}\times\mathbb{R}$. Hence it must also be
$\kappa$-noncollapsed on all scales for some $\kappa>0$.

In [8], it is asked in Problem 9.40 whether a Type I ancient solution with
positive curvature operator is closed. While we are not yet able to give an
answer to this question, the following Corollary rules out one possibility of
its asymptotic shrinker—the standard cylinder. This also means that if such an
ancient solution did exist, then its geometry could not be very simple (though
topologically it is the Euclidean space).

######  Corollary 1.5.

A Type I complete and noncompact ancient Ricci flow with nonnegative curvature
operator and positive sectional curvature cannot have the standard cylinder
$\mathbb{S}^{n-1}\times\mathbb{R}$ as its asymptotic shrinker.

Ni [19] classified all closed Type I ancient and noncollapsed Ricci flows with
nonnegative curvature operator. It is noted in [1] that, because of the
examples therein, the noncollapsing condition in the classification of [19]
cannot be dropped. Nonetheless, because the examples in [1] are only in odd
dimensions, this conclusion is not decisive in even dimensions. Indeed, the
noncollapsing condition can be dropped in even dimension, at least for the
strictly positive curvature operator case.

######  Corollary 1.6.

An even-dimensional closed Type I ancient Ricci flow satisfying the strict
$\operatorname{PIC}-2$ curvature condition (and in particular, with positive
curvature operator) must be a standard shrinking round space form.

Here we remark that, unlike [19], we are not able to deal with the case when
the curvature operator admits a zero eigenvalue, since in this case we can no
longer prove the noncollapsedness. For instance, one may think of
$\mathbb{S}^{1}\times\mathbb{S}^{2m+1}$, where the $\mathbb{S}^{1}$ factor is
static and the $\mathbb{S}^{2m+1}$ factor is the standard shrinking sphere.

This paper is organized as follows. In section 2 we use the Gromoll-Meyer
theorem and the Klingenberg theorem to derive the Type I injectivity radius
lower bound for the ancient solutions in question. In section 3 we review the
fact that Perelman’s entropy and the Nash entropy converge to the entropy of
the asymptotic shrinker. In section 4 we show that the existence of asymptotic
shrinker implies noncollapsedness. In section 5 we prove all the corollaries.

_Acknowledgment._ The second author would like to thank Professor Jiaping
Wang, Professor Lei Ni, and Professor Bennett Chow for many helpful
discussions.

##  2 The injectivity radius

The injectivity radius estimates are provided by the following classical
theorems of Gromoll-Meyer and Klingenberg.

######  Proposition 2.1 (Gromoll-Meyer; c.f. Theorem 1.168 in [8]).

Let $(M^{n},g)$ be a complete and noncompact Riemannian manifold satisfying
$0<\operatorname{sec}\leq K$, where $K$ is a positive number. Then the
injectivity radius of $(M^{n},g)$ satisfies

$\displaystyle\operatorname{inj}(g)\geq\frac{\pi}{\sqrt{K}}.$

Moreover, $(M^{n},g)$ is diffeomorphic to the standard Euclidean space.

######  Proposition 2.2 (Klingenberg; c.f. Theorem 1.115 in [8]).

Let $(M^{n},g)$ be an even-dimensional closed orientable manifold satisfying
$0<\operatorname{sec}\leq K$, where $K$ is a positive number. Then the
injectivity radius of $(M^{n},g)$ satisfies

$\displaystyle\operatorname{inj}(g)\geq\frac{\pi}{\sqrt{K}}.$

The results above imply that the ancient solutions in question have Type I
injectivity radii lower bounds.

######  Lemma 2.3.

Let $(M^{n},g(t))_{t\in(-\infty,w)}$ be an ancient solution as described in
either Theorem 1.2 or Theorem 1.3. Then there exists a constant $c>0$, such
that the injectivity radius of $g(t)$ satisfies

$\displaystyle\operatorname{inj}(g(t))\geq c\sqrt{|t|},$

for all $t\in(-\infty,0)$.

###### Proof.

By the Type I and the positive sectional curvature conditions, we have that
there exists $C>0$, such that

$\displaystyle 0<\operatorname{sec}(g(t))\leq\frac{C}{|t|},$

for all $t\in(-\infty,0)$. The lemma then follows from Proposition 2.1 and
Proposition 2.2. Note that though Proposition 2.2 requires orientability,
which is not assumed in the statement of Theorem 1.3, yet one may always
consider the double cover if necessary. ∎

Here we remark again that this Type I injectivity radius lower bound does not
imply $\kappa$-noncollapsedness on all scales, since at some point on the
manifold, the curvature could decay faster than Type I. Nevertheless, this
estimate is sufficient for the existence of an asymptotic shrinker, which in
turn implies $\kappa$-noncollapsedness on all scales.

##  3 The Asmptotic Shrinker

Perelman [21] and Naber [18] proved the existence of the asymptotic shrinker
for ancient solutions under the assumptions of nonnegative curvature operator
and of Type I curvature bound, respectively. They both also assumed the
noncollapsing condition. However, it turns out that the only place where they
applied this condition was to obtain an injectivity radius lower bound for a
blow-down sequence, and they need this injectivity radius lower bound only at
base points to conclude the convergence. This, of course, can be covered by
Lemma 2.3. A more important fact is that Perelman’s entropy and the Nash
entropy on the ancient solution converge to the entropy of the asymptotic
shrinker; see Proposition 3.1 below. Let us first review these notions of
entropy.

Let $(M,g(t))_{t\in(-\infty,w)}$ be an ancient solution, where
$0<w\leq\infty$. Let $(x_{0},t_{0})\in M\times(-\infty,w)$ be a fixed point in
space-time and $u:M\times(-\infty,t_{0})\rightarrow\mathbb{R}_{+}$ the
fundamental solution to the conjugate heat equation
$\displaystyle-\partial_{t}u-\Delta u+Ru=0$ based at $(x_{0},t_{0})$. We write
$u$ as

$\displaystyle u:=(4\pi\tau)^{-\frac{n}{2}}e^{-f},$

where $\tau=t_{0}-t\in(0,\infty)$. Then Perelman’s entropy and the Nash
entropy based at $(x_{0},t_{0})$ are respectively defined as

$\displaystyle\mathcal{W}_{x_{0},t_{0}}(\tau)$ $\displaystyle:=$
$\displaystyle\int_{M}\Big{(}\tau\big{(}|\nabla
f|^{2}+R\big{)}+f-n\Big{)}udg_{t},$
$\displaystyle\mathcal{N}_{x_{0},t_{0}}(\tau)$ $\displaystyle:=$
$\displaystyle\int_{M}fudg_{t}-\frac{n}{2}.$

It is a well known fact that both $\mathcal{W}_{x_{0},t_{0}}(\tau)$ and
$\mathcal{N}_{x_{0},t_{0}}(\tau)$ are increasing in $t$ (or decreasing in
$\tau$). Furthermore, we have

$\displaystyle\lim_{\tau\rightarrow
0+}\mathcal{W}_{x_{0},t_{0}}(\tau)=\lim_{\tau\rightarrow
0+}\mathcal{N}_{x_{0},t_{0}}(\tau)=0.$

The following result is already well-established in literature.

######  Proposition 3.1.

Let $(M,g(t))_{t\in(-\infty,w)}$, where $0<w\leq\infty$, be an ancient
solution as described in Theorem 1.2 or Theorem 1.3. Let $(x_{0},t_{0})\in
M\times(-\infty,w)$ be an arbitrarily fixed point and $\displaystyle
u:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}$ the conjugate heat kernel based at
$(x_{0},t_{0})$. Let $\tau_{i}\nearrow\infty$ be an increasing sequence of
positive numbers. Then, the following sequence of tuples

$\displaystyle\Big{\\{}\big{(}M,g_{i}(t),(x_{0},-1),f_{i}\big{)}_{t\in(-\infty,0)}\Big{\\}}_{i=1}^{\infty}$

converges, possibly after passing to a subsequence, to the canonical form of a
shrinking gradient Ricci soliton, called the _asymptotic shrinker_

$\displaystyle\big{(}M_{\infty},g_{\infty}(t),(x_{\infty},-1),f_{\infty}\big{)}_{t\in(-\infty,0)},$

where $f_{\infty}$ is the potential function, satisfying

$\displaystyle Ric_{g_{\infty}}+\nabla^{2}f_{\infty}=\frac{1}{-2t}g_{\infty}.$

The convergence $g_{i}\rightarrow g_{\infty}$ is in the pointed Cheeger-
Gromov-Hamilton [13] sense, and the convergence $f_{i}\rightarrow f_{\infty}$
is in the locally smooth sense. Here $g_{i}$ and $f_{i}$ are obtained by time-
shifting and parabolic scaling as follows

$\displaystyle g_{i}(t)$ $\displaystyle:=$
$\displaystyle\tau_{i}^{-1}g(\tau_{i}t+t_{0}),$ $\displaystyle f_{i}(\cdot,t)$
$\displaystyle:=$ $\displaystyle f(\cdot,\tau_{i}t+t_{0}).$

Furthermore, Perelman’s entropy and the Nash entropy converge to the entropy
of the asymptotic shrinker. By this we mean

$\displaystyle\int_{M}(4\pi|t|)^{-\frac{n}{2}}e^{-f_{\infty}}dg_{\infty}=1,$
(3.1)
$\displaystyle\lim_{\tau\rightarrow\infty}\mathcal{W}_{x_{0},t_{0}}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{N}_{x_{0},t_{0}}(\tau)=\mu_{\infty},$

where

$\displaystyle\mu_{\infty}$ $\displaystyle=$
$\displaystyle\int_{M}\Big{(}|t|(|\nabla
f_{\infty}|^{2}+R_{g_{\infty}})+f_{\infty}-n\Big{)}(4\pi|t|)^{-\frac{n}{2}}e^{-f_{\infty}}dg_{\infty}$
$\displaystyle=$
$\displaystyle\int_{M}f_{\infty}(4\pi|t|)^{-\frac{n}{2}}e^{-f_{\infty}}dg_{\infty}-\frac{n}{2}$

is a negative constant independent of time $t$, which we call _the entropy of
the asymptotic shrinker_.

###### Proof.

The proof of this proposition can be modified from, for instance, [6] or [22].
First of all, the Type I condition implies that there exists a positive number
$C$, such that

$\displaystyle|Rm_{g(t)}|\leq\frac{C}{|t|}$ (3.3)

for all $t\in(-\infty,0)$. We then observe that for all $(x,t)\in
M\times(-\infty,0)$, it holds that

$\displaystyle\operatorname{Vol}_{g(t)}\big{(}B_{g(t)}(x,\sqrt{|t|})\big{)}\geq
c|t|^{\frac{n}{2}},$ (3.4)

where $c>0$ is a constant. Suppose this is not true, then one may find a
sequence of counterexamples $\\{(x_{i},t_{i})\\}_{i=1}^{\infty}\subset
M\times(-\infty,0)$, such that the scaled Ricci flows
$(M,g_{i}(t))_{t\in(-\infty,0)}$, where
$g_{i}(t):=|t_{i}|^{-1}g_{i}(t|t_{i}|)$, all satisfy (3.3), but

$\displaystyle\operatorname{Vol}_{g_{i}(-1)}\big{(}B_{g_{i}(-1)}(x_{i},1)\big{)}\rightarrow
0.$ (3.5)

By Lemma 2.3, we have

$\displaystyle\operatorname{inj}(g_{i}(-1),x_{i})\geq c>0.$

Hence, by [13], the sequence of Ricci flows
$\displaystyle\big{\\{}(M,g_{i}(t),(x_{i},-1))_{t\in(-\infty,0)}\big{\\}}_{i=1}^{\infty}$
converges, possibly after passing to a subsequence, to a smooth ancient
solution $(M_{\infty},g_{\infty}(t),(x_{\infty},-1))_{t\in(-\infty,0)}$. In
particular, we have

$\displaystyle\operatorname{Vol}_{g_{\infty}(-1)}\big{(}B_{g_{\infty}(-1)}(x_{\infty},1)\big{)}>0,$

and this contradicts (3.5).

One may then follow the proofs in [6] or [22] to conclude this Proposition.
Obviously, the noncollapsing condition in these proofs can be replaced by
(3.4).

∎

##  4 The Nash Entropy and Noncollapsing

It turns out that from Proposition 3.1 it is sufficient to conclude that the
ancient solution is noncollapsed on all scales everywhere. This follows from
an observation made in [20]. Let $(M,g(t))_{t\in(-\infty,w)}$, where
$0<w\leq\infty$, be an ancient solution as describe in Theorem 1.2 or Theorem
1.3. One may generally regard $(-\infty,w)$ as the maximum existing interval
of $g(t)$, in which case $t=w$ is the singular time (whether it is infinity or
not). The following lemma says $g(t)$ has bounded geometry as long as it is
regular.

######  Lemma 4.1.

For all $t\in(-\infty,w)$, it holds that

$\sup_{M}\big{|}Rm_{g(t)}\big{|}<\infty\text{ and }\displaystyle\inf_{x\in
M}\operatorname{Vol}_{g(t)}\big{(}B_{g(t)}(x,1)\big{)}>0.$

###### Proof.

If $w<\infty$, then by the definition of finite singular time, this is the
first instance at which $g(t)$ has unbounded curvature. If $w=\infty$, then
this means that $g(t)$ has bounded curvature for all $t$. The volume lower
bound for unit balls follows from a straightforward volume distortion
estimate. ∎

From this time-wise geometry bound, we may conclude the following proposition;
this is a combination of Proposition 3.1 above and Proposition 4.6 in [20],
where the second author together with Zilu Ma proved (as a consequence of
Corollary 5.11 in [2]) that on an ancient solution with bounded geometry on
each time-slice, Perelman’s entropies and the Nash entropies based at all
points converge to the same number as the time approaches negative infinity.

######  Proposition 4.2.

Let $(M,g(t))_{t\in(-\infty,w)}$, where $0<w\leq\infty$, be an ancient
solution as describe in Theorem 1.2 or Theorem 1.3. Then for all $(x,t)\in
M\times(-\infty,w)$, the following holds

$\displaystyle\lim_{\tau\rightarrow\infty}\mathcal{W}_{x,t}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{N}_{x,t}(\tau)=\mu_{\infty},$

where $\mu_{\infty}$ is the entropy of any one of asymptotic shrinkers based
at any point (for their entropies are all equal). In particular, we have

$\displaystyle\mathcal{N}_{x,t}(\tau)\geq\mu_{\infty}$ (4.1)

for all $(x,t)\in M\times(-\infty,w)$ and for all $\tau>0$.

Though in the original proof of Perelman [21], he uses the bound of the $\mu$
functional to show the no local noncollapsing theorem, yet the second author
showed that the boundedness of the Nash entropy could also be used to prove
the noncollapsedness at its base point; this is the following Proposition.

######  Proposition 4.3 (Theorem 6.1 in [2]).

Let $(M,g(t))$ be a Ricci flow and $(x,t)$ a point in the space time. Let $r$
be a positive scale such that $[t-r^{2},t]$ is in the existing interval and
$R\leq r^{-2}$ on $\displaystyle B_{g(t)}(x,r)$. Then, it holds that

$\displaystyle\operatorname{Vol}_{g(t)}\big{(}B_{g(t)}(x,r)\big{)}\geq
c\exp\big{(}\mathcal{N}_{x,t}(r^{2})\big{)}r^{n}.$

Here $c$ is a positive constant depending only on the dimension.

_Remark:_ Though Bamler [2] proved the above result for Ricci flows on closed
manifolds, yet one may check the proof of Theorem 6.1 in [2] and easily verify
its validity for Ricci flows with bounded geometry on each time-slice; one may
need to apply Theorem 4.4 of [20] in this verification. Fortunately, all the
Ricci flows we work with in this paper satisfy this condition. On the other
hand, the second author proved that bounded Nash entropy implies weak
noncollapsing, and this proof does not need bounded geometry on each time-
slice; see Proposition 3.3 in [23].

###### Proof of Theorem 1.2 and Theorem 1.3 .

Let $(M,g(t))_{t\in(-\infty,w)}$ be an ancient solution as described in either
Theorem 1.2 or Theorem 1.3. Let $(x,t)\in M\times(-\infty,w)$ be an arbitrary
space-time point, and $r$ any scale that satisfies

$\displaystyle R\leq r^{-2}\text{ on }B_{g(t)}(x,r).$

Since, by Proposition 4.2, we have

$\displaystyle\mathcal{N}_{x,t}(r^{2})\geq\mu_{\infty}\in(-\infty,0),$

where $\mu_{\infty}$ is the entropy of one of the asymptotic shrinkers, it
then follows from Proposition 4.3 that

$\displaystyle\operatorname{Vol}_{g(t)}\big{(}B_{g(t)}(x,r)\big{)}\geq
ce^{\mu_{\infty}}r^{n};$

this finishes the proof.

∎

##  5 Applications

In this section, we prove all the corollaries proposed in the introduction
section.

###### Proof of Corollary 1.4.

Let $(M^{3},g(t))$ be a three-dimensional noncompact Type I ancient solution.
By Chen [9], $g(t)$ has nonnegative sectional curvature everywhere. If its
sectional curvature is strictly positive, then, by Theorem 1.2, it is also
noncollapsed. It follows from [24] (or [15]) that such ancient solution does
not exist.

If $g(t)$ ever attains zero sectional curvature somewhere, then by the strong
maximum principle of Hamilton [11], $(M^{3},g(t))$ splits locally and hence
its universal cover must be the standard shrinking cylinder
$\mathbb{S}^{2}\times\mathbb{R}$. Furthermore, the only noncompact quotients
of $\mathbb{S}^{2}\times\mathbb{R}$ are the $\mathbb{Z}_{2}$ quotients. The
reason is that the projection of a group action on the $\mathbb{R}$ factor can
only be either the reflection or the identity—if it is ever a translation,
then this action will generate an infinity group action on
$\mathbb{S}^{2}\times\mathbb{R}$, and the quotient space must be compact. This
finishes the proof of the corollary. ∎

###### Proof of Corollary 1.5.

We argue by contradiction. Assume one of the asymptotic shrinkers is the
standard cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$, then all the results
obtained in section 3 of [17] can be applied to this ancient solution. In
particular, it satisfies a canonical neighborhood theorem and hence always has
a non-neck-like region at each time (c.f. Theorem 1.3 in [17]), it always
splits as $\mathbb{S}^{n-1}\times\mathbb{R}$ at space infinity (c.f.
Proposition 3.9 in [17]), it satisfies the neck stability theorem of Kleiner-
Lott [14] (c.f. Theorem 3.11 in [17]), and all such ancient solutions form a
compact space (c.f. Theorem 1.2 in [17]). Knowing all these facts, one may
follow the arguments in [24] line by line to conclude that such an ancient
solution does not exist. ∎

###### Proof of Corollary 1.6.

The $\operatorname{PIC}-2$ condition implies the positive sectional curvature
condition. Hence, according to Theorem 1.3, such ancient solution must be
$\kappa$-noncollapsed on all scales for some $\kappa>0$. The conclusion then
follows from Corollary 0.4 in [19]. ∎

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School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical
Sciences, Central China Normal University, Wuhan, 430079, P.R.China
E-mail address<EMAIL_ADDRESS>

School of Mathematics, University of Minnesota, Twin Cities, MN, 55414, USA
E-mail address<EMAIL_ADDRESS>


#  In-situ observation of Alfvén waves in ICME shock-sheath indicates
existence of Alfvénic turbulence

Anil Raghav1∗, Zubair Shaikh2, Omkar Dhamane1, Kalpesh Ghag1, Prathmesh Tari1,
Utsav Panchal1

1Department of Physics, University of Mumbai, Mumbai, India
2Indian Institute of geomagnetism, Panvel, Navi Mumbai, India
<EMAIL_ADDRESS>

###### Abstract

The dynamic evolution of coronal mass ejection (CME) in interplanetary space
generates highly turbulent, compressed, and heated shock-sheath. This region
furnishes a unique environment to study the turbulent fluctuations at the
small scales and serve an opportunity for unfolding the physical mechanisms by
which the turbulence is dissipated and plasma is heated. How does the
turbulence in the magnetized plasma control the energy transport process in
space and astrophysical plasmas is an attractive and challenging open problem
of the 21st century. For this, the literature discusses three types of
magnetohydrodynamics (MHD) waves/ fluctuations in magnetized plasma as the
magnetosonic (fast), Alfvénic (intermediate), and sonic (slow). The
magnetosonic type is most common in the interplanetary medium. However,
Alfvénic waves/fluctuations have not been identified to date in the ICME
sheath. The steepening of the Alfvén wave can form a rotational discontinuity
that leads to an Alfvénic shock. But, the questions were raised on their
existence based on the theoretical ground. Here, we demonstrate the observable
in-situ evidence of Alfvén waves inside turbulent shock-sheath at 1 AU using
three different methods desciribed in the literature. We also estimate
Elsässer variables, normalized cross helicity, normalized residual energy and
which indicate outward flow of Alfvén waves. Power spectrum analysis of IMF
indicates the existence of Alfvénic turbulence in ICME shock-sheath. The study
has strong implications in the domain of interplanetary space plasma, its
interaction with planetary plasma, and astrophysical plasma.

##  1 INTRODUCTION

Coronal mass ejection (CME) is a huge cloud of solar plasma ( mass $\sim
3.2\times 10^{14}~{}g$, kinetic energy $\sim 2.0\times 10^{29}~{}erg$)
submersed in magnetic field lines that are blown away from the Sun which
propagates and expands into the interplanetary medium (Vourlidas et al., 2010;
Howard, 2011). Their studies are of paramount importance given their natural
hazardous effects on humans and the technology in space and ground (Schrijver
and Siscoe, 2010; Moldwin, 2008; Schwenn, 2006). The propagation speed of CMEs
is often higher than the ambient solar wind which causes the formation of
fast, collision-less shocks ahead of CMEs (Kennel et al., 1985). These shocks
cause heating and compression of the upstream (anti-sunward side) slow solar
wind plasma, forming turbulent sheaths between the shocks and the leading edge
of the CMEs (Sonett and Abrams, 1963; Kennel et al., 1985; Zurbuchen and
Richardson, 2006; Jian et al., 2006; Echer et al., 2011; Richardson and Cane,
2011; Kilpua et al., 2017).

The shock and sheath are responsible mostly for (i) acceleration of solar
energetic particles (Tsurutani and Lin, 1985; Manchester IV et al., 2005;
Gosling, 1983; Giacalone et al., 1994; Zank et al., 2000; Verkhoglyadova et
al., 2015; Zank et al., 2007; Li et al., 2003), (ii) significant geomagnetic
activity (Tsurutani et al., 1988; Shen et al., 2018; Oliveira and Raeder,
2014, 2015; Lugaz et al., 2016), (iii) Forbush decrease phenomena (Raghav et
al., 2014, 2017, 2020; Bhaskar et al., 2016; Shaikh et al., 2017, 2018,
2020a), (iv) accelerate pickup ions (Giacalone and Jokipii, 1995; Gloeckler et
al., 1994; Zank et al., 1996), and (v) auroral lightning (Baker and
Lanzerotti, 2016) etc. Besides this, the shock initiates a magnetosonic wave
in the magnetosphere and associated electric field accelerates electrons to
MeV energies (Foster et al., 2015; Kanekal et al., 2016).

Recent theoretical (Zank et al., 2014, 2015; Li et al., 2003; Le Roux et al.,
2015, 2016; le Roux et al., 2018) and observational (Khabarova et al., 2015,
2016, 2017; Zhao et al., 2018, 2019; Shaikh et al., 2018, 2019, 2020a, 2020b;
Raghav and Shaikh, 2020) studies clearly indicate the local generation of
quasi-2D structure (Zank et al., 2017; Adhikari et al., 2017), flux ropes
(Shaikh et al., 2017) or magnetic islands in sheath region, and they may
responsible for the acceleration of charged particles. Recently, the loss of
electron flux from the radiation belt has been observed during the shock-
sheath encounter with Earth’s magnetosphere (Hietala et al., 2014; Kilpua et
al., 2015a, 2017). This may be caused due to an increase in ultra-low
frequency (ULF) wave power and dynamic pressure which is further responsible
for pitch angle scattering and radial diffusion of the electron flux. The
precipitated high energy electron flux from the radiation belt is used as a
key parameter in climate models and the understanding of atmospheric chemistry
and associated climatological effects (Verronen et al., 2011; Andersson et
al., 2014; Mironova et al., 2015). In addition to this, the other planets and
their atmospheres are highly affected by the shock-sheath of CME, for example,
in the case of Mars, loss of the ions flux ($>~{}9~{}amu$) is observed which
might be caused by its high dynamic pressure (Jakosky et al., 2015).

CME induced shock-sheath provides a unique opportunity to investigate the
nature of plasma turbulence, plasma energy/fluctuation-dissipation, and plasma
heating process. The plasma turbulence demonstrates the features such as
Alfvén waves, Whistler waves, ion cyclotron waves, or ion Bernstein waves, etc
(Krishan and Mahajan, 2004; Gary and Smith, 2009; Schekochihin et al., 2009;
Shaikh, 2010; He et al., 2011; Sahraoui et al., 2012; Salem et al., 2012). In
fact, sometimes plasma fluctuations do not exhibit any wave-like configuration
at all but resemble nonlinear structures such as current sheets (Sundkvist et
al., 2007; Osman et al., 2010). Various studies related to the nature of
turbulence and generation of waves in the shock-sheath region have reported in
the recent past. Liu et al. (2006) observed the mirror mode wave within the
shock-sheath region. Kilpua et al. (2013) observed that the power of ultra-low
frequency fluctuations (in the dynamic pressure and magnetic field) peaks
close to the shock-front and sheath-magnetic cloud boundary. Furthermore, a
large-amplitude magnetic field fluctuation, as well as intense irregular ULF
fluctuations and regular high-frequency wave activity is also observed in the
downstream of CME shocks (Kataoka et al., 2005; Kajdič et al., 2012; Goncharov
et al., 2014). Moreover, Whistler waves associated with weak interplanetary
shocks are also observed (Ramírez Vélez et al., 2012).

The solar wind is predominantly associated with turbulent plasma (Bruno and
Carbone, 2005, 2013), which contributes in acceleration of the solar wind
(Verdini et al., 2009; Lionello et al., 2014), solar wind heating (Freeman,
1988; Usmanov et al., 2011; Adhikari et al., 2015), and the scattering of the
solar energetic particles (Li et al., 2003; Zank et al., 2007). The turbulence
of the solar wind plasma increases due to interplanetary shock (Burlaga, 1971;
Richter et al., 1985; Jian et al., 2011). The particle acceleration rate is
controlled by the shock strength, the turbulence level, the magnetic field
strength, and the shock obliquity (Zank et al., 2000, 2006). It has been noted
that the turbulence behind quasi-perpendicular shocks is more sporadic than
that behind quasi-parallel shocks (Macek et al., 2015). Note that, for quasi-
perpendicular shocks, the cross-field currents are strong, produces
significant levels of downstream plasma wave turbulence. Also, the shock
steepening and the structure of shocks highly depends on the properties of the
associated turbulence (Adhikari et al., 2016). Moreover, Zank et al. (2015)
demonstrated that the shock downstream turbulent, including vortical
turbulence and Alfvénic like fluctuations is generated by the impact of
upstream Alfvénic fluctuation disturbances. Recently, Zank et al. (2018) and
Adhikari et al. (2016) studied the interaction between turbulence and
termination shock and showed that quasi-two-dimensional turbulence dominates
and slab-like turbulence plays a secondary role in the downstream of the shock
wave.

Besides this, various studies investigated Alfvénic fluctuations and Alfvén
waves inside an ICME from $1~{}AU$ to $5.4~{}AU$ (Li et al., 2017; Marsch et
al., 2009; Yao et al., 2010; Haoming et al., 2012; LI et al., 2013; Li et al.,
2016a; Gosling et al., 2010; Raghav and Kule, 2018a) and calculated their
contribution to local plasma heating(Li et al., 2017). ICME driven shocks are
faster, stronger and show a larger distribution of shock parameters as compare
to stream interaction shock (Kilpua et al., 2015b). Therefore, sheath plasma
is expected to be highly compressed, hence pure Alfvén wave is not at all
expected. Moreover, the turbulent nature of the sheath suggests the existence
of Alfvénic fluctuations (Kataoka et al., 2005) but explicitly not found to
date. Besides, the literature indicates a theoretical debate on the existence
of Alfvénic shock existence (Wu, 1987; Kunkel, 1966; Taniuti and Jeffrey,
1964) which may be expected during the steepening of Alfvén waves. Therefore,
it is necessary to study the characteristics of Alfvén wave/fluctuations
inside the highly turbulent sheath. Here we demonstrate the unambiguous in-
situ evidence of Alfvén wave suggests nature of Alfvénic turbulence within the
shock-sheath region of CME.

##  2 Event details

The shock-sheath under investigation is engendered by a CME which crossed the
WIND and ACE spacecraft on $06^{th}$ November $2000$. Figure 1 demonstrates
the temporal variations of various in-situ plasma parameters and the
interplanetary magnetic field (IMF) measured by the Wind spacecraft (The ACE
spacecraft measurements are also studied, however not presented here). The
commencement of the shock at spacecraft is identified as a sudden enhancement
in the total IMF ($B_{mag}$), plasma beta ($\beta$), plasma density ($N_{p}$),
plasma temperature ($T_{p}$), and plasma speed ($V_{p}$), it is indicated by
the first vertical black dashed line. In general, Rankine-Hugoniot condition
is used to confirm the shock. The same condition is employed to the shock
events observed by Wind spacecraft and its characteristics are given online at
https://www.cfa.harvard.edu/shocks/ac_master_data/00076/ac_master_00076.html.
The shock is followed by large fluctuations in IMF (See $\delta B$ variations
in the fifth panel of Figure 1) with enhanced magnetic field strength; high
$N_{p}$, & $T_{p}$ which is manifested as a shock-sheath region. The second
shaded region shows the least fluctuations in $B_{mag}$ and its components,
the slow variation in $\theta$ and $\phi$, the slow steady trend in $V_{p}$,
and low $\beta$. This indicates the presence of a ICME magnetic cloud region
(Zurbuchen and Richardson, 2006). The studied event is also listed in ICME
catalogs available online e.g.,
https://wind.nasa.gov/list_plot_Wind/20001106_311_wind.png,
http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm, and
http://space.ustc.edu.cn/dreams/wind_icmes/web/png/WIND_20001106_223050_20001107_174216.png.

Figure 1: The CME observed by the Wind spacecraft on
$06^{th}~{}November,~{}2000$ with time cadence of 92 sec. The top to bottom
panels represents different interplanetary parameters such as: total
interplanetary field strength IMF $B_{mag}$,elevation
($\theta^{\circ}=\arccos(\frac{-B_{z}}{B})-90^{\circ}$) $\&$ azimuth
($\phi^{\circ}=\arctan(\frac{-B_{y}}{-B_{x}})+180^{\circ}$) angle, IMF vectors
i.e. $B_{vec}$, Plasma velocity ($V_{p}$), absolute value of IMF fluctuation
i.e. $\delta B_{i}=\frac{B_{i+1}-B_{i-1}}{2}$, Proton density ($N_{p}$), and
Temperature ($T_{p}$) $\&$ plasma beta ($\beta$) respectively. The shaded
regions represents the shock-sheath of CME (cyan) and its magnetic cloud
(purple). All observations are in GSE coordinate system.

##  3 Alfvén wave identification

In literature, an Alfvén velocity is defined as:

$V_{A}=\frac{B}{\sqrt{\mu_{0}\rho}}$ (1)

where B is a magnetic field and $\rho$ is proton density. A typical Walén test
is employed to confirm the presence of the Alfvén wave in the solar wind. The
Walén relation is described as (Walén, 1944; Hudson, 1971; Yang and Chao,
2013; Yang et al., 2016; Raghav and Kule, 2018a, b):

$\Delta V=R_{w}\Delta V_{A}$ (2)

where, $R_{w}$ is the Walén slope, $\Delta V$ and $\Delta V_{A}$ are the
fluctuations in solar wind speed and fluctuations in Alfvén speed (magnetic
field) respectively. The presence of Alfvén wave/ variations in the solar wind
is suggested by a high correlation between the corresponding components of
$\Delta V$ and $\Delta V_{A}$ as well as $R_{w}=\pm 1$. The estimation of the
correct background magnetic field and solar wind speed is essential to deduce
fluctuations in their respective components. In this study, we confirmed the
presence of Alfvén waves /fluctuations in the sheath region using three
different methods as follow;

###  3.1 Method 1

In the first method, for the shock sheath region, the average values are
estimated for each component of the magnetic field and solar wind vector
respectively. We obtain $\Delta B$ by subtracting a mean value of the
corresponding B component from each measurement. As a result, the Alfvén
velocity fluctuation is given as:

$\Delta V_{A}=\frac{\Delta B}{\sqrt{\mu_{0}\rho}}$ (3)

Similarly, we calculate $\Delta V$ by subtracting averaged proton flow
velocity from measured values of each component respectively. In Figure 5 top
three panels represents comparisons of $x$, $y$, and $z$ of components of
$\Delta V_{A}$ and $\Delta V$ respectively. It clearly shows correlated
variations between their respective components within the shock-sheath region
and indicates the possibility of an Alfvén wave. Their correlation and
regression analysis is depicted in Figure 2. The Pearson correlation
coefficients (R) of each x, y, and z components are –0.83,–0.44, and –0.75 and
the corresponding regression slopes are –0.90, –1.1, and –0.78. (For ACE
spacecraft data with 64 sec time resolution, the correlation coefficients are
–0.80, –0.92 and –0.91 and the corresponding regression slopes are –0.50,
–0.72, and –0.68 respectively. It should be noted that the corresponding
figures are not displayed here. The Anti-Sunward Alfvén wave is confirmed by
the significant negative correlation and regression coefficient $\approx~{}-1$
confirms the presence of Anti-Sunward Alfvén wave in the shock-sheath region
of the examined CME (Gosling et al., 2010; Raghav and Kule, 2018a).

###  3.2 Method 2

In general, the mean values for selected regions or the average value of de
Hoffmann-Teller (HT) frame are utilized as background quantities (Raghav and
Kule, 2018a; Raghav et al., 2018; Raghav and Kule, 2018b; Yang and Chao, 2013;
Gosling et al., 2010). However, Gosling et al. (2009) and Li et al. (2016b)
suggested that the HT frame can change in high-speed solar wind streams and
the solar wind fluctuations are pertinent to a slow varying base value of the
magnetic field. In order to reduce the uncertainty in Alfvén wave
identification, the fourth-order Butterworth bandpass-filters are applied to
each component of plasma velocity and magnetic field data. The equally spaced
10 logarithmic frequency bands are selected. The applied filters are 10s-15s,
15s-25s, 25s-40s, 40s-60s, 60s-100s, 100s-160s, 160s-250s, 250s-400s,
400s-630s, and 630s-1000s. The Walén relation for each band-passed signal is
analyzed as follows:

$V_{i}=\pm R_{w}V_{Ai}$ (4)

The band-passed $V$ and $V_{A}$ components with the $i^{th}$ filter are
represented here by $V_{i}$ and $V_{Ai}$. The value of the correlation
coefficient between the respective components of $V_{i}$ and $V_{Ai}$ for each
frequency band-passed signal confirms the presence of Alfvén waves or Alfvénic
oscillations in the region under study. A similar approach is used by Li et
al. (2016b). The complete region under study is separated with 15 minutes of
time window into the sub-regions. The frequency-time distribution contour map
with a 15-minute bin size is displayed in Figure 3. The value of the
correlation coefficient for each sub-region is shown as a colour map on the
contour plot. Dark blue shed shows negative correlation, while dark red shed
shows a significant positive correlation. As a result, the presence of
dominating Alfvénic flow along the x, y, and z components are shown in Figure
3 contour map by the dark color-strips, particularly inside the sheath region.

Figure 2: The correlation and regression analysis between the respective
$\Delta V$ and $\Delta V_{A}$ components. The scattered black circle with
filled red color represent observations from Wind spacecraft with time cadence
of $92~{}s$. The $R$ is the coefficient of correlation. The equation in each
panel indicate the linear fit relation between respective components of
$\Delta V$ and $\Delta V_{A}$. Figure 3: Time-frequency distribution of
correlation coefficient between $V_{Ai}$ and $V_{i}$ for complete event. Solar
wind speed and magnetic field are shown in last panel. Wind satellite $3s$
observations are utilized for the analysis.

Figure 4: Wavelet coherence analysis between solar wind velocity and magnetic
field components during ICME sheath and MC transit. The color bar represents
magnitude-squared coherence and angle of black arrows from the x + direction
gives phase $\psi$ angle between the two signals. The cone of influence is
represented by the white dashed line marks. The onset of shock is treated as
$0~{}hours$, the front edge of MC is $\approx~{}5.5~{}hours$

Very recently, (Chen and Boldyrev, 2017) utilized a wavelet coherence test to
study the nature of plasma turbulence at kinetic Alfvén scales in the Earth’s
magnetosheath. The wavelet coherence test is generally employed to identify
regions in time-frequency space where the two-time series co-vary (but does
not necessarily have high power). We have also used this test to double-check
the existence of Alfvénic fluctuations/waves in the ICME sheath region. Figure
4 demonstrate the magnitude squared wavelet coherence ($\gamma$), between
$V_{Ax}$ & $V_{x}$ (top panel), $V_{Ay}$ & $V_{y}$ (middle panel), and
$V_{Az}$ & $V_{z}$ (bottom panel) and the phase lag $\psi$ (black arrow) for
$\gamma>0.75$ (i.e. for highly significant correlated values), measured by
Wind spacecraft. We observed a strong anti-correlation in all the components
of $V_{A}$ & $V$ within the spacecraft frame frequencies range
$0.25~{}mHz<f_{sc}<64~{}mHz$. This significant anti-correlation (see the
direction of the black arrow in Figure 4) clearly indicates that the shock-
sheath region is dominated by the Anti-sunward deviation from anti-correlation
is also visible for some frequency regions, especially towards the higher
frequency side. Kindly note that for a certain interval of time (see middle
and trailing edge of the sheath region), we observed the absence of Alfveńic
fluctuations for certain frequency bands (for consistency, see Figure 3 and
4).

##  4 Properties of Alfvén wave

Here, we opine that the Walén test and definition of Alfvénicity are based to
a large extent, on the approximate incompressibility of the background. The
top three panels of figure 5 make it very evident how closely correlated the
components of $\Delta V$ and $\Delta V_{A}$ are to one another.

Presence of Alfvénic fluctuation within the solar wind (especially in
corotating high velocity streams) demands to use Els$\ddot{a}$sser variables
to separate out the contribution of “outward” and “inward” to the turbulence.
Els$\ddot{a}$sser variables are used in the theoretical studies (Elsasser,
1950; Dobrowolny et al., 1980a, b; Veltri et al., 1982; Marsch and Mangeney,
1987; Zhou and Matthaeus, 1989) as well as for the first time in
interplanetary space (data analysis) by Grappin et al. (1990); Tu et al.
(1989); Tu and Marsch (1990). The Els$\ddot{a}$sser variables are defined as:

$\vec{Z}^{\pm}=\vec{V}\pm\frac{\vec{B}}{\sqrt{4\pi\rho}},$ (5)

here, $\vec{V}$ and $\vec{B}$ are proton velocity and magnetic field
fluctuation, measured in the GSE co-ordinate system. The $\pm$ sign in front
of $\vec{B}$ depends on the sign of [$-k~{}\cdotp B_{0}$]. The
Els$\ddot{a}$sser variables are defined in such a way that $\vec{Z}^{+}$ and
$\vec{Z}^{-}$ always refers to the waves propagating outward and inward
direction (Roberts et al., 1987a, b). $\theta_{VB}$ is estimated as

$\theta_{VB}=cos^{-1}(\frac{-B_{x}}{B_{mag}})$ (6)

The equation 5 get modified for $\theta_{VB}\leq 90^{\circ}$ as
$\vec{z^{+}}=\vec{V}-\vec{V}_{A}$ and $\vec{z^{-}}=\vec{V}+\vec{V}_{A}$ , for
$\theta_{VB}>90^{\circ}$ the equation will remains the same. The energy
associated with $z^{+}$ and $z^{-}$ is defined as:

$e^{\pm}=\frac{1}{2}\langle(z^{\pm})^{2}\rangle,$ (7)

Moreover, the normalized cross helicity is estimated as

$\sigma_{c}=\frac{e^{+}-e^{-}}{e^{+}+e^{-}},$ (8)

also, the normalized residual energy defined as

$\sigma_{R}=\frac{e^{v}-e^{b}}{e^{v}+e^{b}}$ (9)

where $e^{v}$ $\&$ $e^{b}$ is kinetic and magnetic energy respectively.
$\sigma_{R}$ is measure of the excess magnetic field energy with respect to
kinetic energy or vice versa Bruno and Carbone (2013). Where,
$e^{v}=0.5<v^{2}>$ and $e^{b}=0.5<b^{2}>$ are kinetic and magnetic energy
associated with Elsässer variables ($z^{\pm}$). Here, $v=0.5(z^{+}+z^{-})$ and
$b=0.5(z^{+}-z^{-})$ respectively.

Here, the $\sigma_{c}>0$ $\&$ $\sigma_{R}<0$ indicates dominant flow of
outward propagating waves (Matthaeus and Goldstein, 1982; Tu et al., 1989). In
our study, we got highly positive value of normalized cross helicity in the
shock-sheath region indicates dominance of outward Alfvénic turbulence.

Figure 5: Top three panels demonstrate the temporal variation of Alfvén
velocity fluctuation vector $\Delta V_{A}$ (red) and that of the proton flow
velocity fluctuation vector $\Delta V$ (blue). It demonstrates Alfvénic and
shock-sheath characteristic of the studied region of an ICME. The fourth panel
represents the ratio of inward to outward Elässer variable. The fifth panel
gives appearance of the angle between solar wind velocity and magnetic field.
The Sixth panel and seventh panels represents the temporal variation of the
normalized cross helicity ($\sigma_{c}$) and normalized residual energy
($\sigma_{R}$) respectively. The analysis is performed using wind spacecraft
data with time cadence of $3~{}sec$. Figure 6: The scatter plot of normalized
residual energy ($\sigma_{r}$) and normalized cross helicity ($\sigma_{c}$) ,
estimated by Wind spacecraft data of 2000 November 06.

The Figure 6 demonstrate scattered plot of the normalized residual energy
($\sigma_{R}$) Vs normalized cross helicity ($\sigma_{c}$). These defined
parameters are valid only if $\sigma_{R}^{2}+\sigma_{c}^{2}\leq 1$ i.e., the
data points should lie within the circle (Bavassano et al., 1998). Note that
most of the observed data points has positive $\sigma_{c}$. This implies that
identified Alfvén wave within the ICME sheath region dominantly propagating
towards the earth.

##  5 PSD Analysis

We performed a power spectral density (PSD) analysis to study the
characterization of the multi-scale nature of shock-sheath
turbulences/fluctuations. Figure 7 represents the spectral output for the
$B_{x}$, $B_{y}$ and $B_{z}$ components of the IMF. The common feature of the
incompressible MHD plasma turbulence i.e. Kolmogorov-like turbulence is
observed in the intermediate frequency range which is consistent with $\sim
f^{-1.66}$ (Bruno and Carbone, 2005). We believed that the entire turbulent
interactions within these regimes are governed by the Alfvénic cascade.

Figure 7: Power spectra of magnetic fluctuations along the $B_{x}$ direction
in GSE coordinate (black color) as a function of frequency in the spacecraft
frame as measured by Wind on 2000 November 06, from 09:15 to 14:16 UTC,
computed with FFT (black) algorithms. We have used WINDS high resolution (11
Hz) IMF data.

We estimate cyclotron frequencies for proton and alpha particle for the
observed magnetic field of $5-12~{}nT$ range, which turn out to be
$0.47~{}Hz-1.10~{}Hz$ for proton and $0.24~{}Hz-0.55~{}Hz$ for alpha. The
various studies reveal that at length-scales beyond the MHD regime, the power
spectrum shows spectral break which halts the Alfvénic cascade (Goldstein et
al., 1994; Leamon et al., 1999; Bale et al., 2005; Alexandrova et al., 2008;
Sahraoui et al., 2009; Shaikh and Shukla, 2009). At $\sim 0.5-0.6$ Hz, the PSD
spectrum “breaks" from a $\sim f^{-1.6}$ power-law inertial range to a $\sim
f^{-3.1}$ dissipation range (see Figure 7). However, Perri et al. (2010)
suggest that the spectral break in the solar wind is independent of the
distance from the Sun and that of both the ion-cyclotron frequency and the
proton gyro-radius. Therefore, it is also possible that the observed high-
frequency break in our study is caused by a combination of different physical
processes as a result of high compression within the shock-sheath region. The
other possible mechanism for the spectral break may result from energy
transfer processes related to; 1) kinetic Alfvén wave (KAW) (Hasegawa and
Chen, 1976), 2) electromagnetic ion-cyclotron-Alfvén (EMICA) waves (Wu and
Yoon, 2007; Gary et al., 2008), or the fluctuation associated with the Hall
magnetohydrodynamics (HMHD) plasma model (Alexandrova et al., 2007, 2008).

At higher frequencies, the spectrum of the magnetic field fluctuations has
power-law dependence as $\sim f^{-\alpha}$, where, the value of $\alpha$ may
range from 2 to 4. The average value of the $\alpha$ is close to 7/3 (Leamon
et al., 1998; Smith et al., 2006; Alexandrova et al., 2008). In our study, it
is about $\sim{-3.1}$. These higher frequency part of the spectrum may be
associated either with a dissipative range (Leamon et al., 1998; Smith et al.,
2006) or with a different turbulent energy cascade caused by dispersive
effects (Stasiewicz et al., 2000; Sahraoui et al., 2006; Galtier and Buchlin,
2007; Alexandrova et al., 2008; Sahraoui et al., 2009; Li et al., 2016b).
Stawicki et al. (2001) proposed that suppression of the Alfvénic fluctuations
are due to the ion cyclotron damping at intermediate wave frequency
(wavenumber), hence the observed power spectra are weakly damped dispersive
magnetosonic and/or whistler waves (unlike Alfvén waves). The presence of the
whistler wave mode in the high-frequency regime was proposed by the (Beinroth
and Neubauer, 1981). Goldstein et al. (1994) found out the existence of multi-
scale waves (Alfvénic, whistlers, and cyclotron waves) with a single
polarization in the dissipation regime of the spectrum. Observation of the
obliquely propagating KAWs (in the $\omega<\omega_{ci}$ regime or Alfvénic
regime) puts a question about the spectral breakpoint due to damping of ion
cyclotron waves (Howes et al., 2008). The Kinetic (Howes et al., 2008) and
Fluid (Shaikh and Shukla, 2009) simulations show that the ion inertial length-
scale is comparable to that of the spectral breakpoint near the characteristic
turbulent length-scales. For the length-scales larger than the ion inertial
length-scales, the simulations demonstrate Kolmogorov-like spectra. Moreover,
for smaller ion inertial length-scales, they observed the steeper spectrum
that is close to $f^{-7/3}$.

##  6 Discussion

Using three distinct techniques, we found a substantial correlation between
the variations in the magnetic field and velocity vectors. This suggests that
the magnetic field and fluid velocity are oscillating simultaneously and
propagating in the same direction as the magnetic tension force. The various
plasma features (see Figure 2, 3,4,5) confirms the presence of Alfvén wave in
shock-sheath region. Similar analysis has been performed for various ICME-
sheath regions observed by Wind spacecraft and listed in
https://wind.nasa.gov/ICMEindex.php. It is important to note that Alfvén
waves/fluctuations have not been found in all the studied ICME-sheath events.
Moreover, only following events shows inward or outward flow of Alfvén
waves/fluctuations out of studied events; e.g Very soon we will analysed all
the events listed in aforementioned catalogue to study their possible origin,
propagation and dissipation in sheath plasma.

The overall distinguishable features of Alfvén waves during the shock sheath
strongly support dominant Alfvénic turbulence. Moreover, the Alfvén waves are
pervasive in the solar wind, and it is important to note that the method 2
(section 3.2, Figure 3) shows the presence of Alfvén wave in up-stream of
shock. However, their transmission in shock-sheath is questionable. The solar
wind at 1 AU is overwhelmingly Alfvénic, therefore it is possible that the
same Alfvénic background is present in the ICME shock-sheath region too and a
significant compressible component is just superposed on that (Chen et al.,
2013).

The Alfvén waves lead to non-linear interactions (Dobrowolny et al., 1980a)
which are crucial for the dynamical evolution of a Kolmogorov-like MHD
spectrum (Bruno and Carbone, 2013). We have also performed a power spectral
density (PSD) analysis for all IMF vector components. It depicts Kolmogorov-
like turbulent nature (The PSD analysis follows $f^{-1.66}$ spectrum) for the
frequency range between $0.4\times 10^{-3}$ Hz to 0.5 Hz in the studied the
shock-sheath region. Thus the existence of Alfvén waves with the Kolmogorov-
like turbulence depicts Alfvénic turbulent nature of the shock-sheath. Thus,
we observed the continued cascade of energy from large scales to smaller
scales of wavelengths and eventually to such small scales that the plasma no
longer behaves like a fluid due to a change in velocity and magnetic field
fluctuations. At this scale, the particle distribution is affected by the
magnetic field which may lead to plasma heating through resonant interactions
(Tsurutani and Lakhina, 1997). We opine that the plasma heating in shock-
sheath could be associated with an above-discussed process.

Alfvénic-like fluctuations may also occur in the impact sheath, but the
dominant one will be the quasi-2D structure (Zank et al., 2018). For example,
Zheng and Hu (2018) used the Grad-Shafranov reconstruction method to show the
flux rope in the impact sheath area. However, our observations clearly
demonstrate the existence of Alfvén waves in the sheath region. It does not
mean that Alfvénic fluctuations dominantly seen in the sheath, rather a
statistical study will put some light on the aforementioned issue.

##  7 Implications

Several open questions need to be addressed in view of turbulent nature in
highly compressed and heated shock-sheath such as, (i) What is the origin of a
turbulent cascade in shock-sheath? Is it the coronal plasma or local driving?;
(ii) How does the cascade modify the shock-sheath plasma?; (iii) How do the
turbulent fluctuations get dissipated into heat?; (iv) What is more important
for energy dissipation, non-linear turbulent heating, or resonant wave-
particle interactions?; (v) Can shock-sheath turbulence be parameterized and
included in heliospheric models for space weather prediction?

Recently, the presence of the Alfvén wave has been seen in a magnetic cloud of
CME (Raghav and Kule, 2018a). It is manifested that the Alfvénic oscillations
in a magnetic cloud of CME may cause the internal magnetic reconnection and/or
thermal anisotropy in plasma distribution which leads to the disruption of the
stable magnetic structure of the CME (Raghav and Kule, 2018b). Their presence
in the magnetic cloud of CME also controls the recovery phase of the
geomagnetic storms (Raghav et al., 2018, 2019). In the introduction section of
the article, we emphasize that the shock-sheath of CME not only affects the
interplanetary plasma characteristics but also affects the dynamics of the
magnetosphere, ionosphere, radiation-belt, and upper atmosphere of the Earth.
It affects the other planetary exospheres as well. Therefore how the typical
configuration such as Alfvén fluctuations embedded shock-sheath influence the
overall solar-terrestrial plasma will be intriguing and may activate the
possible direction of future studies. One can also expect similar features of
shock-sheath in interstellar medium as well e.g. supernovae shocks and
associated sheaths.

…

##  8 acknowledgement

We acknowledge that the NASA/GSFC Space Physics Data Facility makes the
OMNIWeb (or CDAWeb, or ftp) service and OMNI data available. We thank SERB,
India for supporting AR and OD through project reference file number
CRG/2020/002314. We also acknowledge the Department of Physics (Autonomous),
University of Mumbai, for supplying the resources necessary to complete this
research.

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