  Recent theoretical work has explored dark matter accumulation in the Earth
and its drift towards the center of the Earth that, for the current age of the
Earth, does not necessarily result in a concentration of dark matter ($\chi$)
in the Earth's core. We consider a scenario of long-lived ($\tau_\chi\sim
10^{28}$ s), super heavy ($m_\chi=10^7-10^{10}$ GeV) dark matter that decays
via $\chi\to \nu_\tau \bar{\nu}_\tau$ or $\chi\to \nu_\mu \bar{\nu}_\mu$. We
show that an IceCube-like detector over 10 years can constrain a dark matter
density that mirrors the Earth's density or has a uniform density with density
fraction $\epsilon_\rho$ combined with the partial decay width $B_{\chi\to
\nu_\tau \bar{\nu}_\tau}\Gamma_\chi$ in the range of $(\epsilon_\rho/10^{-10})
B_{\chi\to \nu_\tau}\Gamma_\chi \lesssim 1.5\times 10^{-29}-1.5\times 10^{-28}$
s$^{-1}$. For $\chi\to \nu_\mu \bar{\nu}_\mu$, $m_\chi = 10^8-10^{10}$ GeV and
$E_\mu>10^7$ GeV, the range of constraints is $(\epsilon_\rho/10^{-10})
B_{\chi\to \nu_\mu}\Gamma_\chi \lesssim 3\times 10^{-29}-7\times 10^{-28}$
s$^{-1}$.
