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Published byTodd McCormick Modified over 9 years ago
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Equations that allow a quantitative look at the OCEAN
Equation of State: Conservation of Mass or Continuity: Conservation of Salt: Conservation of Heat:
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Conservation of Momentum (Equations of Motion)
Newton’s Second Law: Conservation of momentum as they describe changes of momentum in time per unit mass
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Circulación típica en un fiordo
x z Circulación típica en un fiordo
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Aceleraciones x z
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Gradiente de presión z x
Debido a la pendiente del nivel del mar (barotrópico) Debido al gradiente de densidad (baroclínico)
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Fricción x z Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)
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Coriolis x z Debido a la rotación de la Tierra; proporcional a la velocidad
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Balance de momentum x z
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Pressure gradient + Coriolis + gravity + friction + tides
Forces per unit mass that produce accelerations in the ocean: Pressure gradient + Coriolis + gravity + friction + tides Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS
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Pressure gradient + Coriolis + gravity + friction + tides
Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS
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Net Force per unit mass in ‘x’ =
Net Force in ‘x’ = Net Force per unit mass in ‘x’ = Total pressure force/unit mass on every face of the fluid element is:
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Illustrate pressure gradient force in the ocean
z Pressure Gradient Force 1 2 Pressure of water column at 1 (hydrostatic pressure) : Hydrostatic pressure at 2 : Pressure gradient force caused by sea level tilt: BAROTROPIC PRESSURE GRADIENT
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Precipitación pluvial y Ríos
Aporte aproximado por lluvia: 2000 mm por año area superficial: 350 km por 10 km = 3.5x109 m2 200 m3/s Dirección Meteorológica de Chile Aporte aproximado por ríos: 1000 m3/s Milliman et al. (1995) Descarga de Agua Dulce Precipitación pluvial y Ríos
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Pressure gradient + Coriolis + gravity + friction + tides
Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS
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Acceleration due to Earth’s Rotation
Remember cross product of two vectors: and
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Now, let us consider the velocity of a fixed particle on a rotating body at the position
The body, for example the earth, rotates at a rate , To an observer from space (us): This gives an operator that relates a fixed frame in space (inertial) to a moving object on a rotating frame on Earth (non-inertial)
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This operator is used to obtain the acceleration of a particle in a reference frame on the rotating earth with respect to a fixed frame in space Acceleration of a particle on a rotating Earth with respect to an observer in space Coriolis Centripetal
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Coriolis Acceleration
The equations of conservation of momentum, up to now look like this: Coriolis Acceleration Cv Ch
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f is the Coriolis parameter
Making: f is the Coriolis parameter This can be simplified with two assumptions: Weak vertical velocities in the ocean (w << v, u) Vertical component is ~5 orders of magnitude < acceleration due to gravity
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f increases with latitude
Eastward flow will be deflected to the south Northward flow will be deflected to the east f increases with latitude f is negative in the southern hemisphere
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Pressure gradient + Coriolis + gravity + friction + tides
Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition
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Pressure gradient + Coriolis + gravity + friction + tides
Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS
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Centripetal acceleration and gravity
g has a weak variation with latitude because of the magnitude of the centrifugal acceleration g is maximum at the poles and minimum at the equator (because of both r and lamda)
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Variation in g with latitude is ~ 0
Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s2
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Friction (wind stress)
z W u Vertical Shears (vertical gradients)
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Friction (bottom stress)
z u Vertical Shears (vertical gradients) bottom
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Friction (internal stress)
z Vertical Shears (vertical gradients) u1 u2 Flux of momentum from regions of fast flow to regions of slow flow
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Shear stress is proportional to the rate of shear normal to which the stress is exerted
at molecular scales µ is the molecular dynamic viscosity = 10-3 kg m-1 s-1 for water is a property of the fluid Shear stress has units of kg m-1 s-1 m s-1 m-1 = kg m-1 s-2 or force per unit area or pressure: kg m s-2 m-2 = kg m-1 s-2
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Net force per unit mass (by molecular stresses) on u
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If viscosity is constant,
becomes: And up to now, the equations of motion look like: These are the Navier-Stokes equations Presuppose laminar flow!
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Compare non-linear (advective) terms to molecular friction
Inertial to viscous: Reynolds Number Flow is laminar when Re < 1000 Flow is transition to turbulence when 100 < Re < 105 to 106 Flow is turbulent when Re > 106, unless the fluid is stratified
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Low Re High Re
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Consider an oceanic flow where U = 0
Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10-6 m2/s Is friction negligible in the ocean?
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Frictional stresses from turbulence are not negligible but molecular friction is negligible
at scales > a few m. - Use these properties of turbulent flows in the Navier Stokes equations The only terms that have products of fluctuations are the advection terms All other terms remain the same, e.g.,
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are the Reynolds stresses arise from advective (non-linear or inertial) terms
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This relation (fluctuating part of turbulent flow to the mean turbulent flow) is called a
turbulence closure The proportionality constants (Ax, Ay, Az) are the eddy (or turbulent) viscosities and are a property of the flow (vary in space and time)
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Ax, Ay oscillate between 101 and 105 m2/s
Az oscillates between 10-5 and 10-1 m2/s Az << Ax, Ay but frictional forces in vertical are typically stronger eddy viscosities are up to 1011 times > molecular viscosities
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Equations of motion – conservation of momentum
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