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 ...
208 34 CHAPTER 5. DYNAMICS OF THE OCEAN Balanced equations explicitly resolve the Rossby waves which play a key role in the response of balanced dynamics to forcing and the adjustment to a geostrophic state. Kelvin waves , which are also explicitely resolved by the balanced equations, are important near boundaries and in the vicinity of the equator. The very fast dynamics is the dynamics that happens at a time scale smaller than f −1 and it is usually three dimensional turbulent dynamics. To model it the full three dimensional Navier-Stokes equations have to be considered. tel-00545911, version 1 - 13 Dec 2010
209 Chapter 6 Gyre Circulation tel-00545911, version 1 - 13 Dec 2010 The ocean is forced at its surface by a wind-stress ˜τ which is measured in Newton/m 2 . A typical value for the ocean is in the order of 0.1N/m 2 . In the present manuscript we work with τ = ˜τ/ρ which has units of m 2 /s 2 . 6.1 Sverdrup Dynamics in the SW Model (the math) In all the ocean basins an almost stationary large scale gyre circulation is observed. We suppose that this circulation is a consequence of the wind shear at the ocean surface. We thus add some (wind) forcing to the linearized stationary shallow water equations on the β-plane. Adding −∂ y (6.1) and ∂ x (6.2) leads to: −fv + g∂ x η = τ x /H (6.1) +fu + g∂ y η = τ y /H (6.2) H(∂ x u + ∂ y v) = 0 (6.3) +boundary conditions. Hβv = (∂ x τ y − ∂ y τ x ) (6.4) So at every point the meridional component of the fluid transport (vH) is completely determined by the vorticity of the surface stress! Equation (6.4) is called the Sverdrup relation. It says that the if vorticity is injected into the fluid parcel it can not increase its vorticity as this would contradict stationarity, so it moves northward where planetary potential vorticity (f/H) is larger. So the Sverdrup relation is a statement of conservation of potential vorticity in a forced and stationary situation. When knowing the wind field, the Sverdrup relation gives v, using the zero divergence of geostrophic flow we can calculate ∂ x u. If we know u at one point in a ocean basin at every latitude we can determine u in the whole basin by integrating in the zonal direction, u(x1,y1) = − ∫ x1 x0 ∂ y v(x,y1)dx + u(x0,y1). (6.5) But u is prescribed at the two boundaries of the ocean basin (as the velocity vector at the boundary is directed parallel to the boundary), which makes u over-determined. What does this mean in “physical terms?” Take a look at fig. 6.1, where a caricature of the North Atlantic with a simplified wind-stress (independent of longitude) is given. The corresponding v component of the velocity is also given. If we start by imposing a vanishing zonal velocity at 35
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- Page 183 and 184: 177 Contents 1 Preface 5 tel-005459
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- 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
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- 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
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- Page 213: 207 5.13. A FEW WORDS ABOUT WAVES 3
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- 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
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208<br />
34 CHAPTER 5. DYNAMICS OF THE OCEAN<br />
Balanced equations explicitly resolve the Rossby waves which play a key role in the response<br />
of balanced dynamics to forcing and the adjustment to a geostrophic state. Kelvin waves , which<br />
are also explicitely resolved by the balanced equations, are important near boundaries and in<br />
the vicinity of the equator.<br />
The very fast dynamics is the dynamics that happens at a time scale smaller than f −1<br />
and it is usually three dimensional turbulent dynamics. To mo<strong>de</strong>l it the full three dimensional<br />
Navier-Stokes equations have to be consi<strong>de</strong>red.<br />
tel-00545911, version 1 - 13 Dec 2010