For example, for values of A x = 0 m /s, f 0 s, and L = 0 km, then E h = 0. and the motion may be influenced by horizontal friction if Corioli

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1 Lecture. Equations of Motion Scaling, Non-dimensional Numbers, Stability and Mixing We have learned how to express the forces per unit mass that cause acceleration in the ocean, except for the tidal forces which will be studied later. These expressions form a set of 7 equations with seven unknowns that has to be solved in order to study the dynamics of a given area. Before embarking on an attempt to solve the full set of equations, one must determine the relative importance of the forces that act on the particular area of interest. A useful approach is to scale each term of the equations of motion, according to typical length, time, and flow velocity scales. The scaled terms are then compared to each other to obtain non-dimensional numbers, in the same way as we did to obtain the Reynolds number. This scaling approach gives an idea of what forces may be most important in driving the motion of a particular area. For example, the u momentum equation can be scaled as follows: (.) With this scaling, we can compare, for example, the ratio of inertial to rotation forces/mass: gives the non-dimensional Rossby number (Ro). Note than when Ro is very small (Ro M ), then the Coriolis acceleration is more important than the inertial (non-linear) acceleration and the -4 - motion is linear. For instance for typical values of U = 0. m/s, f 0 s, and L = 0 km, then Ro = 0. and the motion may be influenced by non-linear (or inertial) accelerations. Another comparison of terms in (.) is the ratio of friction to rotation, which gives the horizontal Ekman number (E ) and the vertical Ekman number (E ): h v Analogous to Ro, when the Ekman number is very small (M ), then the frictional forces (per unit mass) acting on the flow are negligible compared to those produced by Coriolis acceleration.

2 For example, for values of A x = 0 m /s, f 0 s, and L = 0 km, then E h = 0. and the motion may be influenced by horizontal friction if Coriolis acceleration is important. Likewise, -2 for A z = 0, and H = 0 m, E v = and vertical friction is at least as important as rotation influences. For the w momentum component, one has to consider the consequences of density variations with depth, i.e., whether the water column is stable or not. The stabilizing forces in the water column can be compared to the destabilizing forces brought about by vertical shears in the horizontal flow. This comparison yields another non-dimensional number, the Richardson number. But before learning about the Richardson number, let us first consider the consequences of density variations with depth. Stability and Mixing Consider a negative vertical density gradient in a given area of the ocean (,#/,z < 0). Then the density increases with depth, and the water column is said to be stable because relatively heavier water lies underneath relatively lighter water. On the other hand, if,#/,z > 0, it means that lighter water lies underneath heavier water, and the water column is unstable. This is usually a transient situation because the water column will tend to stabilize. With these ideas in mind, a formal concept of static stability (E) is presented as: (.2) - Note that E has units of m and uses the vertical gradient in % t. To determine stability from insitu density, the expression has to allow a correction for compressibility. This is because at great pressures, compressibility effects make in-situ density look stratified where it actually may be vertically homogeneous. Then, for in-situ density: where c is the speed of sound, and. The water column is stable when E > 0 and unstable when E < 0. It is neutrally stable when E = 0. An example of a stability profile obtained from a density profile is presented in the next figure. 2

3 E % t The greatest water column stability appears in the upper 30 m, where the density gradient is the strongest. In the ocean, there are flow perturbations (remember that oceanic flows are in general turbulent, i.e., display fluctuations or perturbations from a mean state) that cause oscillations to stable conditions in the water column. In other words, flow variations acting on a vertical density gradient will cause oscillations or waves (internal waves). The maximum frequency of such oscillations is given by: (.3) 2 2 which is called the Brunt-Väisälä frequency and has units of radians /s. It is more common to express this frequency in cycles per second or hertz or, The profile of N/2! for the density profile shown above is illustrated in the following figure 3

4 % t Note that the highest values of N are found at the pycnocline. The pycnocline typically coincides with the thermocline in the open ocean, and with the halocline in coastal and estuarine waters. Also note that the vertical density gradient has to remain negative for oscillations to exist. If the vertical density gradient becomes positive, then the oscillations break, and the water column becomes unstable (square root of a negative number). The fact that the water column is stable does not necessarily mean that the exchange of heat and salt between layers is null. In fact, in some areas of the ocean, there are double diffusion processes that allow salt and heat to be transferred from layer to layer. We will now talk about double diffusion. Double Diffusion This phenomenon also helps understanding the concept of molecular diffusion. There are two types of double diffusive processes that take place in the ocean, namely salt fingering and layering. They both occur under stable or neutrally stable conditions. Let us examine both. Salt Fingering Assume two stable layers in the interior of the water column. One layer with temperature T and salinity S is above the second layer with T and S in such a way that T > T and S > S,

5 according to the following diagram: a b 2 2 The layer distribution is such that #(S, T, p) v #(S, T, p). Water parcel "a" is in contact with 2 2 the colder layer underneath and loses heat faster than salt because the rate of heat diffusion is approximately 00 times greater than that of salt diffusion. By losing heat, water parcel "a" becomes colder and heavier than its surroundings, it acquires unstable buoyancy and sinks. Similarly, water parcel "b" gains heat faster than it gains salt. Then "b" becomes warmer and lighter than its surroundings, becomes unstable and rises. In general, sinking and rising of water parcels occurs in thin columns. This double diffusive instability is called salt fingering. Notice that for salt fingering to develop, both the temperature vertical gradient (,T/,z), and the salinity vertical gradient (,S/,z) are positive, i.e., both T and S decrease with depth. An example of this phenomenon occurs with the Mediterranean outflow, which is saltier and warmer than the Atlantic water underneath. Salt fingering is not frequently observed in coastal regions because externally generated mixing may hinder its formation. Layering Now assume two stable layers in the interior of the water column. This time, the layer with temperature T and salinity S is above the second layer with T and S in such a way that T < 2 2 T 2 and S < S 2, according to the following diagram: 5

6 a b 2 2 Once more, #(S, T, p) v #(S, T, p). Water parcel "a" is in contact with the warmer layer 2 2 underneath and gains heat faster than salt. By gaining heat, water parcel "a" becomes warmer and lighter than its surroundings and rises. Similarly, water parcel "b" loses heat faster than it loses salt. Then "b" becomes colder and heavier than its surroundings and sinks to its neutral buoyancy. This double diffusive effect is called layering. For layering to develop, both the temperature vertical gradient (,T/,z), and the salinity vertical gradient (,S/,z) are negative, i.e., both T and S increase with depth. An example of this phenomenon occurs underneath Arctic ice. Sea water is colder and less saline near the ice than deeper into the water column. Layering could be observed in early fall in temperate estuaries and coastal regions, where surface heat loss causes T and S vertical gradients to be of the same sign. Strong shears in the flows normally hinder development of layering in coastal regions and generate a monotonic increase of properties with depth. Salt fingering and layering lead to vertical transport of heat and salt that is higher than molecular diffusion. Hence these double diffusive phenomena produce enhanced mixing that results in distinct layers in the ocean. These phenomena are studied from the point of view of stratification patterns only, regardless of the interactions that those patterns may have with the ocean flows. Flow shears tend to transfer momentum, salt, and heat between layers at higher rates than those involved in double diffusive phenomena. We will now look at the competition between buoyancy forces that enhance stratification and flow shear forces, which tend to destratify or mix the water column. Richardson Number and Mixing Even if the water is stably stratified and double diffusion is not permitted by the T and S vertical gradients (are of opposite sign) vertical mixing may occur. Mixing will develop if motion is initiated causing the perturbation at the interface between the stratified layers to become unstable and break into turbulence. A sheared flow on an interface will produce waves just like the wind causes waves on the sea surface. These waves may grow and become unstable and break. One mechanism of wave breaking and mixing in the interior of the ocean is through the so-called Kelvin-Helmholtz instabilities. These instabilities consist of filaments of heavy fluid that curl over regions of light fluid and eventually break and cause mixing. 6

7 What will determine whether the waves become unstable or not? It will be determined by the competition between the stabilizing forces, expressed by the Brunt-Väisälä frequency, and the flow vertical shears. This ratio is called the Richardson number Ri: or expressed quantitatively, (.4) This is called the gradient Richardson number. In practice, we use the overall Richardson number Ri, which involves measured parameters at discrete points and can be expressed as: o It has been found from laboratory experiments that the necessary but not sufficient condition for instability to occur in stratified flows is when Ri is lower than Instabilities in ocean flows will be caused mainly by the forces that produce friction. We have seen that there are mainly three types of such forces: wind stress, bottom stress, and internal friction. These mechanisms are also responsible for generating instabilities and mixing in the ocean. Wind stress produces flow vertical shears and mixing near the surface. Bottom stress favors mixing near the bottom, and internal shears stimulate mixing in the interior of the water column. Now that we understand some aspects of the competing mechanisms among the forces that cause acceleration in oceanic flows, let us examine the different types of motion that may arise from the dominance of two or more forcing mechanisms. 7