  The Morava $E$-theories, $E_{n}$, are complex-oriented $2$-periodic ring
spectra, with homotopy groups $\mathbb{W}_{\mathbb{F}_{p^{n}}}[[u_{1}, u_{2},
... , u_{n-1}]][u,u^{-1}]$. Here $\mathbb{W}$ denotes the Witt vector ring.
$E_{n}$ is a Landweber exact spectrum and hence uniquely determined by this
ring as $BP_{\ast}$-algebra. Algebraic $K$-theory of $E_{n}$ is a key
ingredient towards analyzing the layers in the $p$-complete Waldhausen
$K$-theory chromatic tower. One hopes to use the machinery of trace methods to
get results towards $K$-theory once the computation for $THH(E_{n})$ is known.
  In this paper we describe $THH(E_{2})$ as part of consecutive chain of
cofiber sequences where each cofiber sits in the next cofiber sequence and the
first term of each cofiber sequence is describable completely in terms of
suspensions and localizations of $E_{2}$. For these results, we first calculate
$K(i)$-homology of $THH(E_{2})$ using a B\"okstedt spectral sequence and then
lift the generating classes of $K(1)$-homology to fundamental classes in
homotopy group of $THH(E_{2})$. These lifts allow us to construct terms of the
cofiber sequence and explicitly understand how they map to $THH(E_{2})$.
