Hydrology I Jozsef Szilagyi, Professor of Hydrology Department of Hydraulic and Water Resources Engineering Budapest University of Technology and Economics

Lecture #2: Water balance and precipitation Derivation of the basic water-balance equation: the integral form of the continuity equation states d(∫ V ρdV)/dt + ∫ A ρvdA = 0 (1) where ρ is density (not a constant), v [=(u,v,w)] is velocity. The first term is a volume (V) integral, the second is a surface (A) one. vdA is the scalar product of v and the surface unit normal vector. (1) states that the mass within the control volume changes in time only if there is a difference in the in- and outflow fluxes across the surface of the control volume. Making use of the divergence theorem ∫ A ρvdA = ∫ V ▼(ρv)dV where ▼is the nabla operator [i.e. ▼(ρv) = ∂ x (ρu)+∂ y (ρv)+∂ z (ρw)], (1) becomes ∫ V [∂ t ρ +▼(ρv)]dV = 0 which is valid for any volume, thus ∂ t ρ +▼(ρv) = 0 (2) which is the differential form of the continuity equation. For an incompressible fluid ρ is constant, thus (2) becomes ▼v = 0.

Returning to (1), for an incompressible fluid one can write ρd(∫ V 1dV)/dt + ρ∫ A vdA = 0 i.e., dV/dt + ∫ A vdA = 0 The second term is positive for outflows, so it can be written as Q(t) – I(t), where Q is out, I is the inflow rate in time. In hydrology the stored water volume, V, is often denoted by S, called the storage, so one can write the lumped version of the continuity equation as dS/dt = I – Q. Precipitation characteristics -Incremental and cumulative rainfall hyetographs -Rainfall duration, depth, and intensity Measurement of precipitation at a point -Simple vessel: P = V/A, where A is the area of the opening (orifice) -Cylinder with a mm scale -Weighing-recording gauge -Tipping-bucket recording gauge

Measurement of snow: it is melted and weighed New: optical precipitation gauges, based on beam attenuation ● → ● emitter receiver Corrupting factors in precipitation measurements (1) Orifice size (should not be too small) (2) Orifice height (optimal is ground level) (3) Wind shielding

● (4) Distance to obstructions ● (5) Splash and evaporation -deep gauges with narrow inlet from collector to weighing vessel -nonvolatile immiscible oil in known quantity to float on water (such as paraffin) ● (6) Instrument error: 1-5% of total catch ● (7) Observer error: hard to quantify ● (8) Differences in observation times between gauges ● (9) Occult precipitation: fog and rime drip from vegetation (in Oregon ~880 mm/yr) ● (10) Low intensity rains: minimum amount to be reported is 0.25 mm/d, below it: trace

Estimation of aerial average (P*) precipitation: Thiessen polygons P* = ∑ i P i A i / ∑ i A i P1P1 A1A1 P2P2 A2A2 A3A3 A4A4 A5A5 A6A6 P3P3 P4P4 P5P5 P6P6 Homework or lab: Mark 5 points of your liking on a plotting paper, prescribe values to those points (representing P i -s), draw a shape of your liking (as the watershed boundary) and calculate P*. For that you need to construct the Thiessen polygons.

For engineering design, the Ddf (Depth-duration-frequency) and Idf (Intensity-duration-frequency) curves are very useful The frequency refers to the average return period (RP) of a precipitation event (with a predefined duration) that has a total depth or larger than a certain value. E.g., the 1-year return period rain event means that such a depth (for a given duration) or larger can be expected every year, i.e., 100 times out of a 100-year record of annual maximum precipitations. Thus, a 2-year RP event occurs every other year, or equally, 50 times out of a 100-year record, the 5-year RP event occurs every 5th year, or equally, 20 times, and so on. The 100-year RP event happens only once in every 100 years on average. Rain intensity is defined as depth divided by duration. What is the corresponding Idf curve? How do you obtain the two graphs? Use the values in Table 4-9. Homework or lab1: