Download presentation

Presentation is loading. Please wait.

Published bySamantha Crowley Modified over 5 years ago

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

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google