At infinity ($z \to \infty$), the flow tends to a uniform geostrophic velocity $ \boxed{ u = U, v = 0, w = 0} $
At the rigid wall ($z=0$), by the no-slip condition that both the normal and tangential velocities vanish on $z = 0$ $ \boxed{ u = v = w = 0} $

In the absence of friction, the balance would be $$ u = U = -\frac{1}{\rho f} \frac{\partial p}{\partial y}, \quad v = w = 0 $$ which is just the geostrophic balance. But since friction and the wall condition require $u,v=0$ at $z=0$, the velocity must significantly depart from geostrophy balance near the wall

Assume exact solution form $ u = u(z), \quad v = v(z), \quad w = w(z) $
Substitute into continuity \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0\) $$ \frac{\partial w}{\partial z} = 0 $$ With boundary condition $w(0) = 0$ $$ w(z) = 0 \quad \forall z $$ The vertical velocity vanishes everywhere

With $w=0$ and no $x,y$ dependence, the momentum equations reduce to
$x$-momentum \( \cancel{\frac{\partial u}{\partial t}} + \cancel{u \frac{\partial u}{\partial x}} + \cancel{v \frac{\partial u}{\partial y}} + w \cancel{\frac{\partial u}{\partial z}} - fv = -\frac{1}{\rho}\frac{\partial p}{\partial x} + A_v \frac{\partial^2 u}{\partial z^2} + \cancel{A_H \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)} \Rightarrow -fv = -\frac{1}{\rho}\frac{\partial p}{\partial x} + A_v \frac{d^2 u}{dz^2} \)
$y$-momentum \( \cancel{\frac{\partial v}{\partial t}} + \cancel{u \frac{\partial v}{\partial x}} + \cancel{v \frac{\partial v}{\partial y}} + w \cancel{\frac{\partial v}{\partial z}} + fu = -\frac{1}{\rho}\frac{\partial p}{\partial y} + A_v \frac{\partial^2 v}{\partial z^2} + \cancel{A_H \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right)} \Rightarrow fu = -\frac{1}{\rho}\frac{\partial p}{\partial y} + A_v \frac{d^2 v}{dz^2} \)
Hydrostatic balance \( \cancel{\frac{\partial w}{\partial t}} + \cancel{u \frac{\partial w}{\partial x}} + \cancel{v \frac{\partial w}{\partial y}} + w \cancel{\frac{\partial w}{\partial z}} = -\frac{1}{\rho} \frac{\partial p}{\partial z} - g + \cancel{A_v \frac{\partial^2 w}{\partial z^2}} + \cancel{A_H \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right)} \Rightarrow g = -\frac{1}{\rho}\frac{\partial p}{\partial z} \)