Chapter 2: Mathematical Methods in Fluid Dynamics Scalars and Vectors Scalar – any quantity which can be fully specified by a single number Vector – a quantity which requires both a magnitude and direction to be fully specified What are some examples of scalar and vector quantities? Coordinate systems on the Earth:

For a coordinate system with (x,y,z) we use unit vectors , , and . Vector Notation: Magnitude of a vector: Direction of a vector: For meteorological wind direction use:

, where WD0 = 180° for u > 0 and 0° for u < 0

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i

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j

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k

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u = ux i + uy

j

!u = ux2 +uy

2

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Direction = arctan uy ux( )

WD = 90! −180!

πtan−1 v

u⎛

⎝⎜⎞

⎠⎟+WD0

Algebra of Vectors Addition and subtraction of two vectors (graphic method):

Addition of two vectors and :

Subtraction of two vectors and :

Multiplication of a vector by a scalar (graphic method):

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u

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v

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u = ux i + uy

j + uz

k and v = vx

i + vy

j + vz

k

⇒ u + v = (ux + vx )

i + (uy + vy )

j + (uz + vz )

k

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u

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v

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u − v = (ux − vx ) i + (uy − vy )

j + (uz − vz )

k

Multiplication of vector by scalar c:

How does the direction and magnitude of a vector change due to multiplication by a scalar? Multiplication of two vectors Scalar product (or dot product) of and :

When will the dot product of two vectors be equal to zero? What does this tell us about the direction of the two vectors relative to each other?

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u

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c u = cux i + cuy

j + cuz

k

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u

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v

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u • v = uxvx + uyvy + uzvz = u v cosθ

Vector product (or cross product) of and :

or

What is the direction of the vector that results from the cross product?

The right hand rule What is the magnitude of this vector?

Magnitude = When will the cross product be equal to zero?

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u

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v

u × v = (uyvz −uzvy )i + (uzvx −uxvz )

j + (uxvy −uyvx )

k

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u × v =

i j k

ux uy uz

vx vy vz

= (uyvz − uzvy ) i + (uzvx − uxvz )

j + (uxvy − uyvx )

k

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u v sinθ

Scalar and Vector Fields Field – a quantity defined over a given coordinate space The field is a function of the three coordinates of position and also of time. T = f(x,y,z,t) Examples of scalar and vector fields on a weather map. Coordinate Systems on the Earth How do scalar and vector fields change when the coordinate system is changed?

How would vector change under a rotation of the coordinate system?

Meteorologists traditionally define a coordinate system relative to the Earth. What are the implications of this coordinate system accelerating through space? Non-inertial frame of reference

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u = ux i + uy

j

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" u = (ux cosα + uy sinα) " i + (uy cosα − ux sinα)

" j

Gradients of Vectors The vectors we consider in meteorology often vary in space and time (i.e. they are functions of both space and time). We can show this, for a wind velocity vector, as 𝑢"⃑ (𝑥, 𝑦, 𝑧, 𝑡). This vector, in component form, can be written as: 𝑢"⃑ (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢(𝑥, 𝑦, 𝑧, 𝑡)𝚤 + 𝑣(𝑥, 𝑦, 𝑧, 𝑡)𝚥 + 𝑤(𝑥, 𝑦, 𝑧, 𝑡)𝑘"⃑ Written in this way we see that this vector consists of zonal (u), meridional (v), and vertical (w) components of the wind that vary in all three spatial directions (x,y,z) and vary in time. The variation of the wind vector with respect to any one of the independent variables can be written as a partial derivative. 𝜕𝑢"⃑𝜕𝑡

=𝜕𝑢𝜕𝑡𝚤 +

𝜕𝑣𝜕𝑡𝚥 +

𝜕𝑤𝜕𝑡

𝑘"⃑ What does each term in this equation represent physically? What if we considered the partial derivative of 𝑢"⃑ (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢(𝑥, 𝑦, 𝑧, 𝑡)𝚤 + 0𝚥 + 0𝑘"⃑ with respect to x or y?

Eulerian and Lagrangian Frames of Reference

Eulerian frame of reference – properties of the atmosphere are defined as functions of both space (x,y,z) and time (t). In this frame of reference we can consider the properties at some fixed point, O, located at position (xO,yO,zO). The temperature, T, at this point would then be given by: Lagrangian frame of reference – define properties of the atmosphere as functions of time and of a specific parcel of air In the Lagrangian frame of reference we are now following a specific mass of air through the atmosphere rather than considering different masses of air passing a fixed point. For this case we would define the temperature, T, of an air parcel A as:

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TO =T(xO ,yO ,zO ,t)

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TA =TA t( )

Advection What processes can cause the air temperature at a fixed location to change? Advection – the change in properties at a fixed location due to the replacement of the original air parcel at that location with a new air parcel with different properties

Warm advection – warmer air is replacing cooler air at a given location Cold advection – cooler air is replacing warmer air at a given location What determines the rate at which the temperature changes due to advection?

Mathematical description of advection In a Lagrangian frame of reference the temperature of an air parcel is only a function of time and can be written as . This is known as a substantial, material or Lagrangian derivative ( ). In an Eulerian frame of reference the temperature is a function of x, y, z, and t [ ] and as such we need to consider the partial derivative with respect to time ( ) if we want to consider changes in temperature with time at a fixed location. Eulerian derivative – the rate of change of a quantity at a fixed point ( ) The relationship between the Eulerian and Lagrangian derivatives can be found from:

thus

The last three terms on the right hand side of this equation (including the minus sign) is the advection term. Is the sign of the advection term consistent with the physical interpretation of advection shown in the figure on the previous page?

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DT Dt

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D Dt

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T(x,y,z,t)

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∂ ∂t

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∂ ∂t

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DTDt

= limδt→0

δTδt

δT =∂T∂t%

& '

(

) * δt +

∂T∂x%

& '

(

) * δx +

∂T∂y%

& '

(

) * δy+

∂T∂z%

& '

(

) * δz

⇒DTDt

=∂T∂t

+∂T∂x

DxDt

+∂T∂y

DyDt

+∂T∂z

DzDt

=∂T∂t

+ u∂T∂x

+ v∂T∂y

+ w∂T∂z

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∂T∂t

=DTDt

− u∂T∂x

+ v∂T∂y

+ w∂T∂z

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% &

'

( )