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20 3 Basics of Geophysical Fluid Dynamicsu = dx∗dtv = dy∗dtw = dz∗dt(3.1)where (x ∗ , y ∗ , z ∗ ) is the location of a parcel. Speed is the magnitude of velocity andis given by:Speed = √ u 2 + v 2 + w 2For example (u,v,w) = (0.0, −10.0m/s, 0.0) refers to a parcel moving at aspeed of 10 m/s into the negative y-direction; whereas (u,v,w) = (2 m/s, 2 m/s, 0.0)describes movement at a speed of √ 8 ≈ 2.83 m/s diagonally across the x-y plane.3.4 Types of Motion3.4.1 Steady-State MotionsA steady state is a situation in which currents do not show any time variations.This implies that there is a balance between all forces involved. Considerations ofsteady-state force balances are useful tools in geophysical flui dynamics, leadingto important relations such as the geostrophic balance, the thermal-wind relations orthe Sverdrup balance, to be discussed below.3.4.2 WavesWaves are another type of motion being oscillations in time and space. Individualwaves can be classifie in terms of a period T (don not confuse this with temperature)and a wavelength λ (using the Greek symbol “lambda”). The wave periodis the time lapse between successive peaks of a wave, whereas the wavelength isthe distance between these peaks. One could measure the wave period with a stopwatch, whereas the wavelength can be derived from instant photographs of the waveshape.3.4.3 The Sinusoidal WaveformIt is convenient to use the sinusoidal function to describe waves in a mathematicalmanner. This function is based on radians and a complete cycle relates to a change of