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Governing Equations I. by Clive Temperton (room 124) and Nils Wedi (room 128). Overview. Introduction Fundamental physical principles Eulerian vs. Lagrangian derivatives Continuity equation Thermodynamic equation Momentum equation (in rotating reference frame) Spherical coordinates.

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## Governing Equations I

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**Governing Equations I**by Clive Temperton (room 124) and Nils Wedi (room 128)**Overview**• Introduction • Fundamental physical principles • Eulerian vs. Lagrangian derivatives • Continuity equation • Thermodynamic equation • Momentum equation (in rotating reference frame) • Spherical coordinates Recommended: An Introduction to Dynamic Meteorology, Holton (1992) An introduction to fluid dynamics, Batchelor (1967)**Equations**Mean free path Navier-Stokes equations Newton’s second law Boltzmann equations kinematic viscosity ~1.x10-6 m2s-1, water ~1.5x10-5 m2s-1, air number of particles Euler equations individual particles statistical distribution continuum Note: Simplified view !**Continuum assumption**All macroscopic length (and time) scales are to be taken large compared to the molecular scales of motion. Mean free path length l of molecules in atmosphere: Surface ~ 10-7 m 16 km ~ 10-6 m 100 km ~ 0.1 m 135 km ~ 15 m In ocean: ~ 10-9 m**Fundamental physical principles**• Conservation of mass • Conservation of energy • Conservation of momentum • Consider budgets of these quantities for a control volume: (a) Control volume fixed relative to coordinate axes => Eulerian viewpoint (b) Control volume moveswith the fluid and always contains the same particles => Lagrangian viewpoint**Eulerian vs. Lagrangian derivatives**Particle at temperature T at position at time moves to in time . Temperature change given by Taylor series: i.e., then Let local rate of change is the rate of change following the motion. advection total derivative**Mass conservation**Inflow at left face is . Outflow at right face is Difference between inflow and outflow is per unit volume. Similarly for y- and z-directions. Thus net rate of inflow per unit volume is = rate of increase in mass per unit volume = rate of change of density => Continuity equation(N.B. Eulerian point of view)**Thermodynamic equation**First Law of Thermodynamics: where I = internal energy, Q = rate of addition of heat (energy), W = work done by gas on its surroundings by expansion. For a perfect gas, ( = specific heat at constant volume), Alternative forms: (R=gas constant) and Eq. of state: Note: Lagrangian point of view. or equivalent , where**Momentum equation**(1) Newton’s Second Law in fixed frame of reference: N.B. use D/Dt to distinguish the total derivative in the fixed frame of reference. We want to express this in a reference frame which rotates with the earth: = angular velocity of earth, = velocity relative to earth, =position vector relative to earth’s centre. Orthogonal unit vectors: in fixed frame, in rotating frame. (2)**Momentum equation (continued)**For any vector , in fixed frame in rotating frame. (fixed frame) (rotating frame) Now = velocity of due to its rotation = ,etc.**Momentum equation (continued)**Reminders: (a) is the total derivative in the rotating system. (b) Eq. (3) is true for any vector . (3)**Momentum equation (continued)**In particular: Now substituting from Eq. (2), and finally using Newton’s Law [Eq. (1)], centrifugal Coriolis**Momentum equation (continued)**Forces - pressure gradient, gravitation, and friction Where = specific volume (= ), = pressure, = sum of gravitational and centrifugal force, = friction. Magnitude of varies by ~0.5% from pole to equator and by ~3% with altitude (up to 100km).**Spherical polar coordinates**: = longitude, = latitude, = radial distance Orthogonal unit vectors: eastwards, northwards, upwards. As we move around on the earth, the orientation of the coordinate system changes:**Components of momentum equation**with “Shallowness approximation” – take r = a = constant, where a = radius of earth.

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