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

2. ECMWF Governing Equations 1 Slide 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. ECMWF Governing Equations 1 Slide 3 Equations Newtons second law Boltzmann equations Navier-Stokes equations Euler equations individual particles statistical distribution continuum Note: Simplified view ! Mean free path number of particles kinematic viscosity ~1.x10 -6 m 2 s -1, water ~1.5x10 -5 m 2 s -1, air

4. ECMWF Governing Equations 1 Slide 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. ECMWF Governing Equations 1 Slide 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 moves with the fluid and always contains the same particles => Lagrangian viewpoint

6. ECMWF Governing Equations 1 Slide 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 is the rate of change following the motion. total derivative local rate of change advection

7. ECMWF Governing Equations 1 Slide 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. ECMWF Governing Equations 1 Slide 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: Note: Lagrangian point of view. where or equivalent, (R=gas constant) and Eq. of state:

9. ECMWF Governing Equations 1 Slide 9 Momentum equation Newtons 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 earths centre. Orthogonal unit vectors: in fixed frame, in rotating frame. (1) (2)

10. ECMWF Governing Equations 1 Slide 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. ECMWF Governing Equations 1 Slide 11 Momentum equation (continued) Reminders: (a) is the total derivative in the rotating system. (b) Eq. (3) is true for any vector. (3)

12. ECMWF Governing Equations 1 Slide 12 Momentum equation (continued) Coriolis centrifugal and finally using Newtons Law [Eq. (1)], Now substituting from Eq. (2), In particular:

13. ECMWF Governing Equations 1 Slide 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. ECMWF Governing Equations 1 Slide 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. ECMWF Governing Equations 1 Slide 15 Components of momentum equation Shallowness approximation – take r = a = constant, where a = radius of earth. with