I was wondering if there was an identity concerning the material derivative of a line integral, e.g., the material derivative of the zonal barotropic wind $u_{\tau}$ (which itself is the vertical-average of the total zonal wind $u$)
$$\frac{\text{D} u_{\tau}}{\text{D}t} = \frac{\text{D}}{\text{D}t} \left[ \frac{1}{H}\,\int_{0}^{H} u\left(t,\ x,\ y,\ z\right)\,dz \right],$$
where the material derivative is
$$\frac{\text{D}}{\text{D}t} = \partial_t + u\,\partial_x + v\,\partial_y + w\,\partial_z.$$ The source here says that it is trivial and simply moves the material derivative into the integral, which I don't quite trust. What would the material derviative of $dz$ be?
The fluid I am working with is incompressible, which guarantees the existence of a stream function. In fact, the stream function for the barotropic wind $\tau$ is the vertical-average of the stream function for the total wind $\chi$
$$\tau\left(t,\ x,\ y\right) = \frac{1}{H} \int_{0}^{H} \chi\left(t,\ x,\ y,\ z\right)\,dz$$
and so the barotropic winds are entirely horizontal, and are given by
$$\overrightarrow{u}_{\tau} = \left[ -\partial_y \tau,\ \partial_x \tau,\ 0 \right]^T.$$
EDIT: I see from the preliminary responses that I can indeed just bring the material derivative into the integral. I guess I was hoping for some sort of identity relating the material derivative of $u_{\tau}$ to some function of $u_\tau$ i.e., eliminate the dependence on $z$. When you bring the material derivative into the integral, you must be certain that you don't integrate $u$ and $v$ with respect to $z$, i.e.,
$$ \begin{align} \frac{\text{D} u_{\tau}}{\text{D}t} &= \frac{\text{D}}{\text{D}t} \left[ \frac{1}{H}\,\int_{0}^{H} u\left(t,\ x,\ y,\ z\right)\,dz \right] \nonumber \\ &= \frac{1}{H}\,\int_{0}^{H} \partial_t u\left(t,\ x,\ y,\ z\right) + u\left(t,\ x,\ y,\ z'\right)\,\partial_x u\left(t,\ x,\ y,\ z\right) + v\left(t,\ x,\ y,\ z'\right)\,\partial_y u\left(t,\ x,\ y,\ z\right) + w\left(t,\ x,\ y,\ z'\right)\,\partial_z u\left(t,\ x,\ y,\ z\right)\,dz \nonumber \\ &\neq \frac{1}{H}\,\int_{0}^{H} \partial_t u\left(t,\ x,\ y,\ z\right) + u\left(t,\ x,\ y,\ z\right)\,\partial_x u\left(t,\ x,\ y,\ z\right) + v\left(t,\ x,\ y,\ z\right)\,\partial_y u\left(t,\ x,\ y,\ z\right) + w\left(t,\ x,\ y,\ z\right)\,\partial_z u\left(t,\ x,\ y,\ z\right)\,dz \end{align} $$
What I was really hoping for was an evolution equation for the barotropic voticity $\zeta_{\tau} = \partial_x v_{\tau} - \partial_y u_{\tau}$ dependent only on the barotropic wind speeds for the model given in Ogrosky, Stechmann, Hottovy 2019,
I've been told the evolution equation is
$$ \partial_t \zeta_{\tau} = -\frac{1}{\tau_u} - \beta\,v_{\tau} - u_\tau\,\partial_x \zeta_\tau - v_{\tau}\,\partial_y \zeta_\tau $$
which looks a lot like a sort of "barotropic" material derivative. If that is the correct evolution equation, how did the total wind speeds in the advection terms become barotropic wind speeds?