Chapter 5: spectral models

Question 1

Spectral models are also based on

Answer 1

the Primitive Equations.

Question 2

Spectral models are also based on the Primitive Equations.

 But their

Answer 2

mathematical formulation and numerical solutions are quite different from Grid Point Models for some of the forecast variables

Question 3

Spectral Models were developed as a means of

Answer 3

increasing the speed while enhancing the resolution for global forecasting.

Question 4

spectral Models were developed as a means of increasing the speed while enhancing the resolution for global forecasting.

 However, as the resolution of the grid point formulations is also increasing day- by-day with the advent of

Answer 4

powerful computers

Question 5

However, as the resolution of the grid point formulations is also increasing day- by-day with the advent of powerful computers, the use of spectral methods may lessen due to

Answer 5

their complex mathematics

Question 6

However, as the resolution of the grid point formulations is also increasing day- by-day with the advent of powerful computers, the use of spectral methods may lessen due to their complex mathematics and more global models may begin using

Answer 6

grid point formulations.

Question 7

Spectral Models use

Answer 7

a combination of continuous waves of differing wavelengths and amplitudes

Question 8

Spectral Models use a combination of continuous waves of differing wavelengths and amplitudes to

Answer 8

specify the forecast variables and their derivates at all locations.

Question 9

Instead of using …………………., Spectral Models use a combination of continuous waves of differing wavelengths and amplitudes to specify the forecast variables and their derivates at all locations.

Answer 9

grid points

Question 10

Conceptually, Spectral Models follow the process of

Answer 10

drawing contours through a data field to represent the forecast variables.

Question 11

Since spectral models represent

Answer 11

some of the FV’s with continuous waves rather than at separate points along a wave

Question 12

since spectral models represent some of the FV’s with continuous waves rather than at separate points along a wave, they can use

Answer 12

more accurate numerical techniques to solve some of the equations on much longer forecast time steps than the finite difference methods used by grid point models.

Question 13

Since spectral models represent some of the FV’s with continuous waves rather than at separate points along a wave, they can use more accurate numerical techniques to solve some of the equations on much longer forecast time steps than the finite difference methods used by grid point models.

 However, some grid point calculations are

Answer 13

required (for model physics) in spectral models

Question 14

However, some grid point calculations are required (for model physics) in spectral models and this introduces

Answer 14

some computational errors associated with grid point models.

Question 15

However, some grid point calculations are required (for model physics) in spectral models and this introduces some computational errors associated with grid point models.

 Thus the more ………………………….. the …………………………………..

Answer 15

the more physics that is involved in the evolution of the forecast, the less the advantage in spectral model forecasts compared to comparable resolution grid point forecasts.

Question 16

Spectral methods are most commonly used for

Answer 16

global numerical weather prediction

Question 17

Spectral methods are most commonly used for global numerical weather prediction, with the primitive equations solved in

Answer 17

spherical rather than Cartesian coordinates, at operational forecast centers.

Question 18

examples of spectral models

Answer 18

the NCEP Global Forecast System and the

ECMWF Integrated Forecast System models

Question 19

global models need not be

Answer 19

spectral models

Question 20

However, global models need not be spectral models, nor is the use of spectral models limited to

Answer 20

global applications.

Question 21

basis functions

Answer 21

Spectral modeling for NWP is based on the Galerkin Method, in which the dependent variables are approximated by a finite sum of linearly independent functions

Question 22

The general form of approximating some function f(x) by a finite sum of basis functions is given by:

Answer 22

Question 23

The general form of approximating some function f(x) by a finite sum of basis functions is given by

For example, if the function f is a function of

Answer 23

both time t and space x, the above equation takes the form:

Question 24

in this equation

Answer 24

the spatial variability is captured by the basis functions, while temporal variability is captured by the coefficients of the base functions.

Question 25

In spectral models, the basis functions are referred to as

Answer 25

spherical harmonics

Question 26

in spectral models, the basis functions are referred to as spherical harmonics.

• In these models, the basis functions take the form of

Answer 26

Fourier series

legendre function

Question 27

In these models, the basis functions take the form of Fourier series, where

Answer 27

zonal variability in the dependent variables is represented by series of sine and cosine waves of varying wavelength,

Question 28

Legendre functions

Answer 28

depict meridional variability in the dependent variables.

Question 29

in spectral models, the basis functions are referred to as spherical harmonics.

• In these models, the basis functions take the form of Fourier series, where zonal variability in the dependent variables is represented by series of sine and cosine waves of varying wavelength, and
• Legendre functions, which depict meridional variability in the dependent variables. These basis functions are

Answer 29

continuous across the sphere

Question 30

Fourier series, with sines and cosines serving as the basis functions, are used to represent

Answer 30

zonal variability in the dependent variables

Question 31

For a generic function A(x) its Fourier series expansion is given by:

Answer 31

Question 32

Legendre functions serve as the basis functions to represent

Answer 32

meridional variability in the dependent variables.

Question 33

for a generic function A(lamda, o|) where lamda is …….. and o| is …………….. its spherical expansion is given by:

Answer 33

lamda is longitude

o| is latitude

Question 34

what does the following represent

m

M

n

N

Answer 34

m is the zonal wavenumber (no. of waves on a circle of latitude)

M is the highest-permitted zonal wavenumber

n (a positive integer) is the order of the associated Legendre function,

N is the highest-permitted order of the associated Legendre function,

Question 35

what does the following represent

a^m_n

Y^m_n​

Answer 35

spectral coefficients

spherical harmonics

Question 36

For example, the zonal variation of a meteorological variable, say T(x) can be represented in terms of a zonal mean plus a finite Fourier series as :

where T_i is

Answer 36

temperature at ith point on the latitude circle

If we assume that 3 points are sufficient to capture the information contained in each of a series of continuous waves, i varies as 1,2,3,……,m (=3n)

Question 37

For example, the zonal variation of a meteorological variable, say T(x) can be represented in terms of a zonal mean plus a finite Fourier series as :

where n is

Answer 37

Total number of waves used to describe the temperature field

In spectral models, the horizontal resolution is designated by a T number (e.g., T80), which indicates the number of waves used to represent the data

Question 38

For example, the zonal variation of a meteorological variable, say T(x) can be represented in terms of a zonal mean plus a finite Fourier series as :

where k is

Answer 38

Zonal wave number – the number of waves along a latitude circle For example, k=10 implies a wavelength of L=360^o/10=36^o.

Question 39

For example, the zonal variation of a meteorological variable, say T(x) can be represented in terms of a zonal mean plus a finite Fourier series as :

where A₀​ is

Answer 39

Zonal mean, which is considered to be wave number zero.

Question 40

For example, the zonal variation of a meteorological variable, say T(x) can be represented in terms of a zonal mean plus a finite Fourier series as :

where A_k and B_k​

Answer 40

Coefficients which are obtained using the discrete Fourier transforms

Question 41

For example, the zonal variation of a meteorological variable, say T(x) can be represented in terms of a zonal mean plus a finite Fourier series as :

…………………. can be used to obtain A_k and B_k

Answer 41

The observed temperature data

Question 42

In spectral models, the horizontal resolution is designated by

Answer 42

a “T” number (for example, T80)

Question 43

In spectral models, the horizontal resolution is designated by a “T” number (for example, T80), which indicates the

Answer 43

Number of waves used to represent the data.

Question 44

In spectral models, the horizontal resolution is designated by a “T” number (for example, T80), which indicates the number of waves used to represent the data. The “T” stands for

Answer 44

triangular truncation

Question 45

The “T” stands for triangular truncation, which indicates

Answer 45

the particular set of waves used by a spectral model.

Question 46

Spectral models represent data precisely

Answer 46

out to a maximum number of waves, but omit all, more detailed information contained in smaller waves.

Question 47

Spectral models represent data precisely out to a maximum number of waves, but omit all, more detailed information contained in smaller waves.

 This accurate representation up to

Answer 47

a cut-off and then complete omission is in contrast to grid point models,

Question 48

This accurate representation up to a cut-off and then complete omission is in contrast to grid point models, which try to represent

Answer 48

all scales but poorly handle waves only a few grid points across.

Question 49

The wavelength of the smallest wave in a spectral model is represented as:

Answer 49

minimum wavelength = 360^o​/N where N is the total no. of waves (T number)

Question 50

We can approximate the grid spacing to obtain

Answer 50

equivalent accuracy to a spectral model with a fixed number of waves using a very simple approach.

Question 51

We can approximate the grid spacing to obtain equivalent accuracy to a spectral model with a fixed number of waves using a very simple approach

Answer 51

Question 52

In fact, the dynamics of spectral models retain far better

Answer 52

wave representation than grid point models with this grid spacing.

Question 53

In fact, the dynamics of spectral models retain far better wave representation than grid point models with this grid spacing.

 However, the spectral model physics is calculated on

Answer 53

a grid, with about three times as many grid lengths as number of waves used to represent the data

Question 54

However, the spectral model physics is calculated on a grid, with about three times as many grid lengths as number of waves used to represent the data.

since it takes …………………. to represent

Answer 54

five to seven grid points to represent ‘wavy’ data well and even more for features that include discontinuities

Question 55

Since it takes five to seven grid points to represent ‘wavy’ data well and even more for features that include discontinuities, the resolution of the physics is

Answer 55

poorer than the above formulation indicates and degrades the quality of the spectral model forecast.

Question 56

In triangular truncation,

Answer 56

the highest degree of the various spherical harmonics Y^m_n (m denotes the zonal wave number and n the order) is fixed and is set equal to the highest order of these a waves.

Question 57

in triangular trunctation the highest degree of the various spherical harmonics Y^m_n (m denotes the zonal wave number and n the order) is fixed and is set equal to the highest order of the sea waves.

under this truncation, ….

Answer 57

different wavenumbers m have different degrees of freedom along a latitude.

Question 58

Having the advantages of better resolution, attempts have been made to apply

Answer 58

the spectral method to a limited area also.

Question 59

Sine functions which form a basis of

Answer 59

orthogonal functions on [0,2pi] can however be used for expanding only periodic functions over the plane domain under consideration.

Question 60

Sine functions which form a basis of orthogonal functions on 0,2􏰚 can however be used for expanding only periodic functions over the plane domain under consideration.

 One of the techniques proposed for using a basis of sine functions effectively is to

Answer 60

use doubly period fields in spectral LAMs.

Question 61

One of the techniques proposed for using a basis of sine functions effectively is to use doubly period fields in spectral LAMs.

In this technique, the working domain is

Answer 61

enlarged

Question 62

In this technique, the working domain is enlarged (Figure) so as to

Answer 62

extend the fields so that they, and their space derivatives, are periodic in both directions over the new domain

Question 63

so as to extend the fields so that they, and their space derivatives, are periodic in both directions over the new domain; the spectral method can then be applied to

Answer 63

these doubly periodic fields.

Question 64

The procedure consisting in extending the

Answer 64

domain and constructing doubly periodic fields on the domain R provides a simple way of using the spectral method on a limited domain.

Question 65

The procedure consisting in extending the domain and constructing doubly periodic fields on the domain R provides a simple way of using the spectral method on a limited domain.

 This allows

Answer 65

the space derivative to be calculated very accurately

Question 66

The development of operational weather forecasting models requires

Answer 66

a very big investment.

Question 67

The development of operational weather forecasting models requires a very big investment. We can consider using

Answer 67

a single basis code to obtain both a spherical model and a limited area model.