Governing Equations I

1. Governing Equations I by Clive Temperton (room 124) and Nils Wedi (room 128)

2. 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)

3. 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 !

4. 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

5. 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

6. 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

7. 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)

8. 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

9. 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)

10. Momentum equation (continued) For any vector , in fixed frame in rotating frame. (fixed frame) (rotating frame) Now = velocity of due to its rotation = ,etc.

11. Momentum equation (continued) Reminders: (a) is the total derivative in the rotating system. (b) Eq. (3) is true for any vector . (3)

12. Momentum equation (continued) In particular: Now substituting from Eq. (2), and finally using Newton’s Law [Eq. (1)], centrifugal Coriolis

13. 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).

14. 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:

15. Components of momentum equation with “Shallowness approximation” – take r = a = constant, where a = radius of earth.