Equations that allow a quantitative look at the OCEAN Equation of State: Conservation of Mass or Continuity: Conservation of Salt: Conservation of Heat:

Conservation of Momentum (Equations of Motion) Newton’s Second Law: Conservation of momentum as they describe changes of momentum in time per unit mass

Circulación típica en un fiordo x z Circulación típica en un fiordo

Aceleraciones x z

Gradiente de presión z x Debido a la pendiente del nivel del mar (barotrópico) Debido al gradiente de densidad (baroclínico)

Fricción x z Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)

Coriolis x z Debido a la rotación de la Tierra; proporcional a la velocidad

Balance de momentum x z

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

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

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:

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

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

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

Acceleration due to Earth’s Rotation Remember cross product of two vectors: and

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)

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

Coriolis Acceleration The equations of conservation of momentum, up to now look like this: Coriolis Acceleration Cv Ch

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

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

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

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

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)

Variation in g with latitude is ~ 0 Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s2

Friction (wind stress) z W u Vertical Shears (vertical gradients)

Friction (bottom stress) z u Vertical Shears (vertical gradients) bottom

Friction (internal stress) z Vertical Shears (vertical gradients) u1 u2 Flux of momentum from regions of fast flow to regions of slow flow

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

Net force per unit mass (by molecular stresses) on u

If viscosity is constant, becomes: And up to now, the equations of motion look like: These are the Navier-Stokes equations Presuppose laminar flow!

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

Low Re High Re

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?

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.,

are the Reynolds stresses arise from advective (non-linear or inertial) terms

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)

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

Equations of motion – conservation of momentum