The dynamics in the ocean can be described by the equation of motion $\frac{Du}{Dt}=-{\nabla}{\Phi}- \frac{1}{p}{\nabla}p- f\cdot u$ . Consider the motion of water in a full kitchen sink, with the water slowly draining out the bottom. The height of the water surface is estimated to vary by about 1 cm over horizontal distances of order 10 cm and typical water speeds in the sink are estimated to be 0.1 m/s. (i) Using the relevant equation(s), estimate the corresponding horizontal pressure gradients.
For a question like this are we imagining two point on the surface of a draining sink, seperated by
10cm and varying in height of 1cm, The inclusion of the water speed is confusing me can it just be ignored as the two points would be moving at the same speed?
Hint: how much pressure difference is there for 1 cm in water height? If that is the variation over 10 cm, the gradient is that difference divided by 10 cm.