M.Sc. in Meteorology Synoptic Meteorology [MAPH P312] Prof Peter - PowerPoint PPT Presentation

M.Sc. in Meteorology Synoptic Meteorology [MAPH P312] Prof Peter Lynch Second Semester, 2004–2005 Seminar Room Dept. of Maths. Physics, UCD, Belfield.

Part 8 The Quasigeostrophic System These lectures follow closely the text of Holton (Chapter 6). 2

The Quasi-Geostrophic Equations 3

The Quasi-Geostrophic Equations We will derive a system of equations suitable for qualitative analysis of mid-latitude weather systems. 3

The Quasi-Geostrophic Equations We will derive a system of equations suitable for qualitative analysis of mid-latitude weather systems. We will assume that the motion is hydrostatically balanced and approximately in geostrophic balance. 3

The Quasi-Geostrophic Equations We will derive a system of equations suitable for qualitative analysis of mid-latitude weather systems. We will assume that the motion is hydrostatically balanced and approximately in geostrophic balance. In that case, the three-dimensional flow is de- termined by the pressure field. 3

The Quasi-Geostrophic Equations We will derive a system of equations suitable for qualitative analysis of mid-latitude weather systems. We will assume that the motion is hydrostatically balanced and approximately in geostrophic balance. In that case, the three-dimensional flow is de- termined by the pressure field. Since meteorological measurements are generally referred to constant pressure surfaces and since the equations are simpler in pressure coordinates than in height coordinates, we will use pressure as the vertical variable. 3

The Quasi-Geostrophic Equations We will derive a system of equations suitable for qualitative analysis of mid-latitude weather systems. We will assume that the motion is hydrostatically balanced and approximately in geostrophic balance. In that case, the three-dimensional flow is de- termined by the pressure field. Since meteorological measurements are generally referred to constant pressure surfaces and since the equations are simpler in pressure coordinates than in height coordinates, we will use pressure as the vertical variable. In that case, the three-dimensional flow is de- termined by the geopotential field. 3

The Primitive Equations 4

The Primitive Equations The dynamical equations in pressure coordinates are d V dt + f k × V + ∇ Φ = 0 ∂ Φ ∂p = − RT p ∇· V + ∂ω ∂p = 0 � ∂ ˙ � Q ∂t + V · ∇ T − Sω = c p 4

The Primitive Equations The dynamical equations in pressure coordinates are d V dt + f k × V + ∇ Φ = 0 ∂ Φ ∂p = − RT p ∇· V + ∂ω ∂p = 0 � ∂ ˙ � Q ∂t + V · ∇ T − Sω = c p Here the total time derivative is dt = ∂ d ∂t + ( V · ∇ ) p + ω ∂ ∂p 4

Notation: 5

Notation: The vertical velocity is ω = dp/dt , 5

Notation: The vertical velocity is ω = dp/dt , S = − T∂ ln θ/∂p is the static stability parameter The typical scale of S is about 5 × 10 − 4 K Pa − 1 in mid-troposphere). 5

Notation: The vertical velocity is ω = dp/dt , S = − T∂ ln θ/∂p is the static stability parameter The typical scale of S is about 5 × 10 − 4 K Pa − 1 in mid-troposphere). The primitive equations will now be simplified based on the assumption that the flow is close to geostrophic balance and the vertical velocity is much smaller than the horizontal. 5

The Momentum Equation 6

The Momentum Equation We first partition the horizontal component of the wind into geostrophic and ageostrophic parts V = V g + V a with the geostrophic wind defined by V g = 1 k × ∇ Φ f 0 6

The Momentum Equation We first partition the horizontal component of the wind into geostrophic and ageostrophic parts V = V g + V a with the geostrophic wind defined by V g = 1 k × ∇ Φ f 0 In component form this is u g = − 1 ∂ Φ v g = + 1 ∂ Φ ∂y , ∂x . f 0 f 0 6

The Momentum Equation We first partition the horizontal component of the wind into geostrophic and ageostrophic parts V = V g + V a with the geostrophic wind defined by V g = 1 k × ∇ Φ f 0 In component form this is u g = − 1 ∂ Φ v g = + 1 ∂ Φ ∂y , ∂x . f 0 f 0 We take a constant “central” value f 0 of the Coriolis param- eter here. This is consistent with the assumption that the horizontal scale L of the motion is small compared to the Earth’s radius, L ≪ a . 6

We note also that the geostrophic divergence vasnishes: � 1 � � � δ g = ∇· V g = ∂ − 1 ∂ Φ + ∂ ∂ Φ = 0 ∂x f 0 ∂y ∂y f 0 ∂x 7

We note also that the geostrophic divergence vasnishes: � 1 � � � δ g = ∇· V g = ∂ − 1 ∂ Φ + ∂ ∂ Φ = 0 ∂x f 0 ∂y ∂y f 0 ∂x The continuity equation may now be written ∇· V a + ∂ω ∂p = 0 This implies that ω is determined by the ageostrophic com- ponent of the wind. 7

We note also that the geostrophic divergence vasnishes: � 1 � � � δ g = ∇· V g = ∂ − 1 ∂ Φ + ∂ ∂ Φ = 0 ∂x f 0 ∂y ∂y f 0 ∂x The continuity equation may now be written ∇· V a + ∂ω ∂p = 0 This implies that ω is determined by the ageostrophic com- ponent of the wind. The geostrophic vorticity is given by � 1 ζ g = k · ∇× V g = ∂ ∂ Φ � − ∂ � − 1 ∂ Φ � = 1 ∇ 2 Φ ∂x f 0 ∂x ∂y f 0 ∂y f 0 so that ζ g is determined once Φ is given. 7

We note also that the geostrophic divergence vasnishes: � 1 � � � δ g = ∇· V g = ∂ − 1 ∂ Φ + ∂ ∂ Φ = 0 ∂x f 0 ∂y ∂y f 0 ∂x The continuity equation may now be written ∇· V a + ∂ω ∂p = 0 This implies that ω is determined by the ageostrophic com- ponent of the wind. The geostrophic vorticity is given by � 1 � � � ζ g = k · ∇× V g = ∂ ∂ Φ − ∂ − 1 ∂ Φ = 1 ∇ 2 Φ ∂x f 0 ∂x ∂y f 0 ∂y f 0 so that ζ g is determined once Φ is given. Moreover, if ζ g is given, the Poisson equation ∇ 2 Φ = f 0 ζ g may be solved for the geopotential. Then V g follows imme- diately. 7

Scale Analysis 8

Scale Analysis We can introduce characteristic scales for the motion. Thus, L is the typical horizontal scale, H the vertical scale and T = L/V the advective time scale. 8

Scale Analysis We can introduce characteristic scales for the motion. Thus, L is the typical horizontal scale, H the vertical scale and T = L/V the advective time scale. Then the size of the advection relative to the Coriolis term is V · ∇ V f k × V ∼ V f 0 L ≡ Ro where Ro is the Rossby Number . 8

Scale Analysis We can introduce characteristic scales for the motion. Thus, L is the typical horizontal scale, H the vertical scale and T = L/V the advective time scale. Then the size of the advection relative to the Coriolis term is V · ∇ V f k × V ∼ V f 0 L ≡ Ro where Ro is the Rossby Number . For the systems of interest | V a | ≪ | V g | or, more specifically, | V a | | V g | ∼ Ro 8

Scale Analysis We can introduce characteristic scales for the motion. Thus, L is the typical horizontal scale, H the vertical scale and T = L/V the advective time scale. Then the size of the advection relative to the Coriolis term is V · ∇ V f k × V ∼ V f 0 L ≡ Ro where Ro is the Rossby Number . For the systems of interest | V a | ≪ | V g | or, more specifically, | V a | | V g | ∼ Ro We can then replace the velocity by its geostrophic com- ponent, and ignore the vertical advection in the total time derivative: � d � ∂ � � d V dt ≈ V g = ∂t + V g ·∇ V g dt g 8

Exercise: Using the vector relationship k · ∇ × V = ∇ · V × k and the def- inition V g = (1 /f 0 ) k × ∇ Φ , derive the above expression for the geostrophic vorticity. ⋆ ⋆ ⋆ 9

Exercise: Using the vector relationship k · ∇ × V = ∇ · V × k and the def- inition V g = (1 /f 0 ) k × ∇ Φ , derive the above expression for the geostrophic vorticity. ⋆ ⋆ ⋆ We wish to retain the variation of the Coriolis parameter with latitude y = a ( φ − φ 0 ) as it has important dynamical consequences. 9

Exercise: Using the vector relationship k · ∇ × V = ∇ · V × k and the def- inition V g = (1 /f 0 ) k × ∇ Φ , derive the above expression for the geostrophic vorticity. ⋆ ⋆ ⋆ We wish to retain the variation of the Coriolis parameter with latitude y = a ( φ − φ 0 ) as it has important dynamical consequences. Expanding in a Taylor series, we write the first two terms f = f 0 + βy where β = ( d f/dy ) 0 = 2Ω cos φ 0 /a with y = 0 at φ = φ 0 . 9

Exercise: Using the vector relationship k · ∇ × V = ∇ · V × k and the def- inition V g = (1 /f 0 ) k × ∇ Φ , derive the above expression for the geostrophic vorticity. ⋆ ⋆ ⋆ We wish to retain the variation of the Coriolis parameter with latitude y = a ( φ − φ 0 ) as it has important dynamical consequences. Expanding in a Taylor series, we write the first two terms f = f 0 + βy where β = ( d f/dy ) 0 = 2Ω cos φ 0 /a with y = 0 at φ = φ 0 . This is the mid-latitude β -plane approximation. 9