Numerical Weather Prediction in Atmospheric Sciences

Numerical Weather Prediction MUSTANSIRIYAH UNIVERSITY COLLEGE OF SCIENCES DEPARTMENT OF ATMOSPHERIC SCIENCES 2024-2025 Dr. Sama Khalid Mohammed FOURTH STAGE LECTURE3

HINT You can use this website https://glossarytest.ametsoc.net/wiki/Welcome for searching about some items and terms in Meteorology

FYI ( ) . : 1 - KINEMATICS 2 - DYNAMIC : :

Scalars and Vectors Scalars: are variables such as temperature and air pressure that have magnitude but not direction. Vectors: are variables such as velocity that have magnitude and direction.

Cartesian and Spherical Coordinates Cartesian Coordinates System: is a system of rectangular coordinates with three axis: x, y, and z. unit vectors in this system are i, j, and k, respectively. This system is used on the microscale and mesoscale. the natural axes represented by s, which is the axis parallel to the wind movement, and axis n, which is the axis perpendicular to the wind direction, through which the change in the movement of the system will be represented relative to the Cartesian axes. The Cartesian axes and their vectors, through which the mathematical representation will be used to determine the path of motion in the wind field, and the velocity components are also shown with respect to these axes.

Cartesian and Spherical Coordinates Spherical Coordinates System: The spherical coordinate system divides the Earth into longitudes (meridians), and latitudes (parallels), The Prime Meridian, which runs through Greenwich, United Kingdom, is defined to have longitude 0 . The Equator is defined to have latitude 0 . On the spherical coordinate grid, the west east distance between meridians is the greatest at the Equator and converges to zero at both poles. Conversions between increments of distance in Cartesian coordinates and increments of longitude or latitude in spherical coordinates, along the surface of Earth, are obtained from the equation for arc length around a circle. In the west east and south north directions, these conversions are: ?? = (?? ????)??? ?? = ?? ?? radius of the Earth, ??? is a west-east longitude increment (radians), ?? is a south-north latitude increment (radians). If a grid cell has dimensions ???= 5? and ?? = 5? , centered at ? = 30?? latitude, find ?? and ?? at the grid cell latitudinal center. where ??= 6371 km is the

Wind Velocity Winds are described by three parameters-velocity, the scalar components of velocity, and speed. Velocity is a vector that quantifies the rate at which the position of a body changes over time: V =iu+jv+kw (total vector) ? = ?? + ?? ( ????????? ??????) And, u=dx/dt ,v=dy/dt ,w=dz/dt are the scalar components of velocity. They have magnitude only. The magnitude of the wind is its speed. The total and horizontal wind speeds are defined as: ?2+ ?2+ ?2 ?2+ ?2 ? = , ? =

Vector Multiplication There two types of multiplication namely: The dot product: It is a product of two vectors gives a scalar. Let ? and ? are two vectors, ?.? = ????+ ????+ ????(How?) , ?.? = ? ? ???? Example 1.2: Let ? = 2? 1/2? 3?, and ? = 3? + ? 1/2? , Find ?.? and the angle between the two vectors. ?.? = 6 + ( 1/2) + 3/2= 6.5 + 1.5 = 5 Solution: ? = 2+ ?? 2+ ??2 = ?? 4 + 1/4 + 9 = 13.75 2+ ?? 2+ ??2 = 10.25 ? = ?? 9 + 1 + 1/4 = ?.? ? ? 5 5 11.52= 0.434 ???? = = 13.25 10.25=

Vector Multiplication The Cross Product: it is product of vectors gives a vector: ? ? = ? ? = ? ? = ? ? ???? The direction of ? is perpendicular on the plane of ? and ?. ? ? = ? = ???? ????? + ???? ????? + ???? ????? Del Operator: is a vector differential operator denoted by the symbol ?: ? = ?? ??+ ?? ???? ?? This operator can be used in three differential ways:

Gradient, Divergence and Curl Del Operator Gradient of Scalar (pressure) ?? = ??? ??+ ??? ????? ?? (??????) Divergence of a Vector (velocity) ?.? =?? ??+?? ??+?? ?? (??????) (????) Curl of a Vector (velocity) ? ? ?? ? ? ? ? ?? ? ? ?? ? ? ? = ?? ?? ?? ?? ?? ?? ?? ?? ?? = ? + ? + ? (??????) ?? ?? ??

Laplacian Operator If Q is any quantity then, ?.?? ?2Q where ?2 (del squared) is the scalar differential operator: ?2 ??2+ ?2 ??2+?2 ?2 ??2 ?2Q is called the Laplacian of Q and appears in several important partial equations of mathematical physics.

The general form of the equations Atmospheric models are built from: The equation of motion the mass conservation equation (or continuity equation) the energy conservation equation (or, the thermodynamic equation) the water vapor conservation equation the equation of state

The general form of the equations For unit mass, with a frame of reference attached to the Earth and having its origin located at the Earth s center, the equations take the form:

The general form of the equations The geopotential is defined as the product of height z by acceleration due to gravity g, which combines only the effects of Newtonian gravity g* and the centrifugal force (assuming that the Earth is isolated in space and so neglecting the effect of the other bodies of the solar system); it is expressed in J.kg 1 in SI units: where is the angular velocity of rotation of the Earth, r the radial distance (r = r , where r is the radius vector as measured from the Earth s center), and r cos , the distance to the Earth s rotation axis at the latitude . In equations (2.1) to (2.5), V3 represents the three-dimensional wind velocity, the angular velocity vector of the Earth, is air density, p pressure, T temperature, q specific humidity, R and Cp the ideal gas constant and specific heat at constant pressure for air. The symbol 3 denotes the gradient operator, the symbol 3. the divergence operator, and d/dt, the total derivative operator whose expression depends on the system of coordinates adopted.

The general form of the equations The term 2 V3 represents the Coriolis acceleration resulting from the choice of a frame of reference rotating with the Earth. F, Q and M represent the source and sink terms for momentum, heat, and specific humidity, respectively. Their somewhat complicated expression depends in particular on the scale of the atmospheric motion these models are intended to describe. For an adiabatic frictionless and water vapor conserving atmosphere, these quantities are equal to zero. These hypotheses are now maintained to explain the various forms of the equations and the way to solve them numerically. In this case, the water vapor equation reduces to an advection equation for a passive scalar (e.g. specific humidity) with no interaction with the other variables; it is discretized in a similar way to the advection term of temperature in the thermodynamic equation. This is why the water vapor equation will now be omitted from the system of equations, at least for the time being.