Etudes et évaluation de processus océaniques par des hiérarchies ...
Etudes et évaluation de processus océaniques par des hiérarchies ... Etudes et évaluation de processus océaniques par des hiérarchies ...
218 44 CHAPTER 7. MULTI-LAYER OCEAN DYNAMICS 7.2 Conservation of Potential Vorticity Exercise 45: Show that the linearised version of eqs. (7.1) – (7.6) conserve the linear potential vorticity at every horizontal location and for every layer. Exercise 46: Show that eqs. (7.1) – (7.6) conserve the potential vorticity at every horizontal location and for every layer (when friction is neglected). Wow! This means that if we describe our ocean by more and more layers, then potential vorticity is conserved for every fluid particle! 7.3 Geostrophy in a Multi-Layer Model tel-00545911, version 1 - 13 Dec 2010 As we have seen in section 5.8 the geostrophic equilibrium is a balance between the pressure and the Corilolis force, neglecting time-dependence, non-linearity, friction and using the Boussinesq approximation, eqs. (7.1) – (7.6) then read: fv 1 = g∂ x (η 1 + η 2 ) (7.7) −fu 1 = g∂ y (η 1 + η 2 ) (7.8) fv 2 = g∂ x (η 1 + η 2 ) + g ′ ∂ x η 2 (7.9) −fu 2 = g∂ y (η 1 + η 2 ) + g ′ ∂ y η 2 (7.10) (7.11) It is now interesting to consider the differences between eqs. (7.7) – (7.9) and (7.8) – (7.10) which are: v 1 − v 2 = − g′ f ∂ xη 2 (7.12) u 1 − u 2 = g′ f ∂ yη 2 , (7.13) (7.14) which are called the thermal wind relation, as they were first discovered in, and applied to, atmospheric dynamics. They show that in the geostrophic limit the horizontal gradient of the height of the interface is related to the velocity difference across the interface perpendicular to the gradient of the height of the interface. This finding can of course be generalised to models with several layers and also to the limit of an infinity of layers, that is, to a continuous variation of density and velocity. Which than means in the geostrophic limit: if we know the density structure of the ocean, we know the vertical gradient of the horizontal velocity every where. If we knew the velocity at a certain depth we could use the thermal wind relation to calculate the velocity every where. As the geostrophic velocities in the deep ocean are usually smaller than near the surface, oceanographers conjecture a level of no motion which is set rather arbitrarily to, for example, 4000m depths, to calculate the geostrophic velocities every where. The thermal wind relation was of paramount importance in the past, when it was difficult to measure velocities from a ship at open sea. The density structure on the contrary was much easier to determine precisely. Today with the help of satellites the measurements of velocities have become much more precise, and comparisons with the density structure show the good agreement with the thermal wind relations. Exercise 47: Where would the velocities be directed in fig. 7.1 on the southern hemisphere?
219 7.3. GEOSTROPHY IN A MULTI-LAYER MODEL 45 A B C tel-00545911, version 1 - 13 Dec 2010 × surface layer bottom layer × flow into the page flow out of the page × Figure 7.1: Geostrophy in a two layer model: in region A the pressure gradient of the interfacial slope compensates the pressure gradient of the surface slope and the lower layer is inert; in region B the surface is level and there is no geostrophic motion in the surface layer, the inter-facial slope corresponds to a velocity in the bottom layer; in region C the slope of the surface and the interface lead to a higher velocity in the bottom layer. The slope of the surface is exaggerated with respect to the interface slope, the vertical variations of the interface are of the order of 1m, while the interface varies hundreds of meters. The situation presented corresponds to the northern hemisphere.
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- Page 183 and 184: 177 Contents 1 Preface 5 tel-005459
- Page 185 and 186: 179 Chapter 1 Preface tel-00545911,
- Page 187 and 188: 181 Chapter 2 Observing the Ocean t
- Page 189 and 190: 183 Chapter 3 Physical properties o
- Page 191 and 192: 185 3.3. θ-S DIAGRAMS 11 3.3 θ-S
- Page 193 and 194: 187 3.6. HEAT CAPACITY 13 tel-00545
- Page 195 and 196: 189 3.7. CONSERVATIVE PROPERTIES 15
- Page 197 and 198: 191 Chapter 4 Surface fluxes, the f
- Page 199 and 200: 193 4.2. FRESH WATER FLUX 19 water.
- Page 201 and 202: 195 Chapter 5 Dynamics of the Ocean
- Page 203 and 204: 197 5.2. THE LINEARIZED ONE DIMENSI
- Page 205 and 206: 199 5.4. TWO DIMENSIONAL STATIONARY
- Page 207 and 208: 201 5.6. THE CORIOLIS FORCE 27 Whic
- Page 209 and 210: 203 5.8. GEOSTROPHIC EQUILIBRIUM 29
- Page 211 and 212: 205 5.10. LINEAR POTENTIAL VORTICIT
- Page 213 and 214: 207 5.13. A FEW WORDS ABOUT WAVES 3
- Page 215 and 216: 209 Chapter 6 Gyre Circulation tel-
- Page 217 and 218: 211 6.1. SVERDRUP DYNAMICS IN THE S
- Page 219 and 220: 213 6.2. THE EKMAN LAYER 39 In the
- Page 221 and 222: 215 6.3. SVERDRUP DYNAMICS IN THE S
- Page 223: 217 Chapter 7 Multi-Layer Ocean dyn
- Page 227 and 228: 221 7.5. EDDIES, BAROCLINIC INSTABI
- Page 229 and 230: 223 Chapter 8 Equatorial Dynamics t
- Page 231 and 232: 225 Chapter 9 Abyssal and Overturni
- Page 233 and 234: 227 9.2. MULTIPLE EQUILIBRIA OF THE
- Page 235 and 236: 229 9.3. WHAT DRIVES THE THERMOHALI
- Page 237 and 238: 231 Chapter 10 Penetration of Surfa
- Page 239 and 240: 233 10.2. TURBULENT TRANSPORT 59 If
- Page 241 and 242: 235 10.5. ENTRAINMENT 61 instabilit
- Page 243 and 244: 237 Chapter 11 Solution of Exercise
- Page 245 and 246: 239 65 Exercise 32: The moment of i
- Page 247 and 248: 241 INDEX 67 Transport stream-funct
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219<br />
7.3. GEOSTROPHY IN A MULTI-LAYER MODEL 45<br />
A<br />
B<br />
C<br />
tel-00545911, version 1 - 13 Dec 2010<br />
×<br />
surface layer<br />
bottom layer<br />
×<br />
flow into the page<br />
flow out of the page ×<br />
Figure 7.1: Geostrophy in a two layer mo<strong>de</strong>l: in region A the pressure gradient of the interfacial<br />
slope compensates the pressure gradient of the surface slope and the lower layer is inert;<br />
in region B the surface is level and there is no geostrophic motion in the surface layer, the<br />
inter-facial slope corresponds to a velocity in the bottom layer; in region C the slope of the<br />
surface and the interface lead to a higher velocity in the bottom layer. The slope of the surface<br />
is exaggerated with respect to the interface slope, the vertical variations of the interface are<br />
of the or<strong>de</strong>r of 1m, while the interface varies hundreds of m<strong>et</strong>ers. The situation presented<br />
corresponds to the northern hemisphere.
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