TV Norm - What Would Be the Formula?
Your definition looks fine to me, if your goal is to treat the signal at every fixed $(x,y)$ location as a function of time and then compute 1D total variation along time. Is that what you intend to do?

Here are a couple of other ways you can define it (again, I have no clue what your application is, and why you want to compute the TV norm in the first place, so I don't know which of these make sense in your application).

To fix notation, let's denote your signal of interest as $z(n_x,n_y,n_t)$ where $z$ is the quantity measured at discrete locations indexed by $(n_x,n_y)$ in space and at a time sample number $n_t$.

TV norm for each fixed spatial location:
Fix a location $(n_x^*,n_y^*)$.
$$
TV = \sum_{n_t} |z(n_x^*,n_y^*,n_t+1) - z(n_x^*,n_y^*,n_t)|
$$

TV norm for each 2D time snapshot:
Fix a time instant $n^*_t$.
$$
TV = \sum_{n_x,n_y} |z(n_x+1,n_y,n^*_t) - z(n_x,n_y,n^*_t)| + | z(n_x,n_y+1,n^*_t) - z(n_x,n_y,n^*_t)|
$$

TV norm in 3D space-time:

$$
TV = \sum_{n_x,n_y,n_t} |z(n_x+1,n_y,n_t) - z(n_x,n_y,n_t)| + | z(n_x,n_y+1,n_t) - z(n_x,n_y,n_t)| + |z(n_x,n_y,n_t+1) - z(n_x,n_y,n_t)|
$$

See also: https://en.wikipedia.org/wiki/Total_variation_denoising#2D_signal_images