In his papers "Isocrystals with additional structure" I and II, Kottwitz defines the notion of $\sigma$-$L$-spaces. In the first one the situation is the following

$k$ an algebraically closed field of char. $p>0$, $K$ the fraction field of the Witt ring $W(k)$, $F$ finite extension of $\mathbb{Q}_p$, $L$ the compositum of $K$ and $F$ in $\bar{K}$, $\sigma$ the Frobenius automorphism of $L$ over $F$.

He later defines a $\sigma$-$L$-space to be a finite dimensional vector space $V$ over $L$ together with a $\sigma$-semilinear bijection $\Phi: V\rightarrow V$.

In "Isocrystals with additional structure II" he considers a different situation.

Again $F$ is a finite extension of $\mathbb{Q}_p$, $F^{nr}$ the maximal unramified extension of $F$ in some algebraic closure $\bar{F}$, but now $L$ is the completion of $F^{nr}$, $\sigma$ is the continuous extension of the Frobenius automorphism of $F^{nr}$ over $F$.

He then defines $\sigma$-$L$-spaces exactly as above.

I don't see how these two definitions are the same, since the second definition makes no reference to the Witt ring. What is the relation between these two definitions?

EDIT: Kottwitz assumes in his paper that $k$ is algebraically closed. But actually I'm interested in the more general situation of $k$ just a perfect field of char. $p$. Do the definitions agree in this more general situation?

Thanks.