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## Conservation of Salt:

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**Conservation of Mass or Continuity:**Conservation of Salt: Conservation of Heat: Equation of State: Equations that allow a quantitative look at the OCEAN**Conservation of Momentum (Equations of Motion)**Newton’s Second Law: as they describe changes of momentum in time per unit mass Conservation of momentum**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**+ gravity + Coriolis + 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 in ‘x’ =**Net Force per unit mass in ‘x’ = Total pressure force/unit mass on every face of the fluid element is:**Pressure of water column at 1 (hydrostatic pressure) :**Pressure Gradient Pressure Gradient Force Hydrostatic pressure at 2 : Pressure gradient force caused by sea level tilt: BAROTROPIC PRESSURE GRADIENT Illustrate pressure gradient force in the ocean Pressure Gradient? z 1 2**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)**0**Acceleration of a particle on a rotating Earth with respect to an observer in space Coriolis Centripetal 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**Coriolis Acceleration**Ch Cv The equations of conservation of momentum, up to now look like this:**Ch**Cv**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**Eastward flow will be deflected to the south**Northward flow will be deflected to the east f increases with latitude fis 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 ~ 0.5%, so for practical**purposes, g =9.80 m/s2**Friction (wind stress)**z W Vertical Shears (vertical gradients) u**Friction (bottom stress)**Vertical Shears (vertical gradients) z u bottom**Friction (internal stress)**Vertical Shears (vertical gradients) z 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; it 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**becomes:**If viscosity is constant, And up to now, the equations of motion look like: These are the Navier-Stokes equations Presuppose laminar flow!**Reynolds Number**Inertial to viscous: Compare non-linear (advective) terms to molecular friction 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 = 0.1 m/s; L = 10 km;**kinematic viscosity = 10-6 m2/s Is friction negligible in the ocean?**- Use these properties of turbulent flows in the Navier**Stokes equation Frictional stresses from turbulence are not negligible but molecular friction is negligible at scales > a few m.**x (or E) component**0 Navier-Stokes equations • Upon applying mean and fluctuating parts to this component of motion: • -The only terms that have products of fluctuations are the advective terms • All other terms remain the same, e.g., What about the advective terms?**0**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 10-1 and 105m2/s**Azoscillates between 10-5 and 10-1m2/s Az<< Ax, Ay but frictional forces in vertical are typically stronger eddy viscosities are up to 1011 times > molecular viscosities