Unformatted text preview: M.Sc. in Computational Science Fundamentals of Atmospheric Modelling
´ Peter Lynch, Met Eireann
Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belﬁeld. January–April, 2004. Lecture 4 The Shallow Water Equations 2 Physical Laws of the Atmosphere
NEWTON’S LAWS OF MOTION Describe how the change of velocity is determined by the pressure gradient, Coriolis force and friction GAS LAW, or EQUATION OF STATE Relates the pressure, temperature and density CONSERVATION OF MASS Continuity Equation: air neither created nor distroyed CONSERVATION OF WATER Continuity Equation for water (liquid, solid and gas) CONSERVATION OF ENERGY Thermodynamic Equation determines changes of temperature due to heating, compression, etc.
Seven equations; seven variables (u, v, w, ρ, p, T, q ). The Primitive Equations
1 ∂p u tan φ du + Fx = 0 v+ − f+ ρ ∂x a dt 1 ∂p u tan φ dv + Fy = 0 u+ + f+ ρ ∂y a dt ∂p + gρ = 0 ∂z p = RρT ∂ρ + · ρV = 0 ∂t ∂ρw + · ρw V = [Sources − Sinks] ∂t ˙ dT Q + (γ − 1)T · V = dt cp These equations are suitable for a forecast or climate model. For understanding the dynamics, we need to simplify them.
Check: Look at the above equations. How many have we got so far?
3 4 Equations in Component Form
The equations of motion in the rotating frame are dV 1 + 2Ω × V + p − g = 0. dt ρ Next we split the equation of motion into components. We introduce local cartesian coordinates (x, y, z ). V dV/dt g p Ω 2Ω × V = = = = = = (u, v, w) (du/dt, dv/dt, dw/dt) (0, 0, −g ) (∂p/∂x, ∂p/∂y, ∂p/∂z ) (0, Ω cos φ, Ω sin φ) (2wΩ cos φ − 2v Ω sin φ, 2uΩ sin φ, −2uΩ cos φ) The variables x and y are distances eastward and northward on the globe. We will ignore the eﬀects of sphericity except in the Coriolis term (see below). Then (x, y, z ) are equivalent to Cartesian coordinates. Note: Certain trigonometric terms have been omitted from the acceleration. We assume w is much smaller than u and v , and we can neglect the term 2wΩ cos φ in the Coriolis force.
5 Coordinate System Components of Ω
6 The horizontal components of the equation of motion may now be written: du 1 ∂p − fv + =0 dt ρ ∂x 1 ∂p dv + fu + =0 dt ρ ∂y where f = 2Ω sin φ is called the Coriolis parameter. The vertical component of the equation of motion is dw 1 ∂p − 2Ωu cos φ + + g = 0. dt ρ ∂z In the absence of motion, this reduces to the hydrostatic equation, ∂p + ρg = 0 , ∂z expressing a balance between the vertical pressure gradient and gravity. Note that horizontal component of Ω no longer enters the equations, and we write 2Ω = f k.
7 For large-scale motions, the hydrostatic equation is an excellent approximation to the full vertical equation, and we adopt it from now on. As already remarked, the majority of numerical models assume hydrostatic balance. However, as the grid-scales are reﬁned below about 5 km, this assumption becomes les justiﬁed. Thus, non-hydrostatic models have been gaining popularity in recent years. The equations will now be further simpliﬁed, and we will derive the system known as the Shallow Water Equations. For a review, read Pedlosky, §§3.1, 3.2 and 3.3. As a consequence of spherical geometry, there are additional small terms involving trigonometric functions. These will be omitted, as the resulting errors are small.
8 The Beta-plane approximation
We restate the momentum and continuity equations: du 1 ∂p − fv + =0 dt ρ ∂x dv 1 ∂p + fu + =0 dt ρ ∂y ∂p + ρg = 0 ∂z ∂u ∂v ∂w + + =0 ∂x ∂y ∂z These are four equations for four dependent variables. The β -plane approximation: We neglect sphericity except in the Coriolis parameter f = 2Ω sin φ. Thus, the geometric terms arising from the sphericity of the earth are omitted. Only the dynamical eﬀect, the variation of the vertical component of Ω, is included.
9 Eliminating the Vertical Velocity
We will now eliminate the vertical velocity w, thereby reducing the system to three equations for three variables. Let h(x, y ) be the height of the free surface at point (x, y ). We integrate the hydrostatic equation between z and h: h ∂p h dz + ρg dz = 0 or p(z ) − p(h) = ρg (h − z ) z ∂z z (recall that density ρ is assumed to be constant). Thus, the pressure is given by the weight of ﬂuid above a point. We assume that the pressure p0 = p(h) at the top of the ﬂuid layer is a constant. Then p0 does not enter the dynamics: p(z ) = p0 + ρg (h − z ) , for each point (x, y ) . The gradient of pressure may now be related to the slope of the free surface: 1 ∂p ∂h 1 ∂p ∂h =g ; =g . ρ ∂x ∂x ρ ∂y ∂y
10 In vector notation, this is 1 p=g h ρ We can now write the (horizontal) equations of motion as dV + 2Ω × V + Φ = 0 . dt where Φ = gh is the geopotential.
N.B. From now on, V denotes the horizontal velocity (u, v, 0). Integrated Continuity Equation
Next, we integrate the continuity equation through the full depth of the ﬂuid. Since u and v are constant with z ,
h 0 ∂ u ∂v ∂ u ∂v + dz = h + ∂x ∂y ∂x ∂y . The third term of the continuity equation integrates to ∂w dh dz = w(h) − w(0) = . ∂z dt 0 Here we have assumed that the bottom boundary is ﬂat, so that the vertical velocity there vanishes: w(0) = 0. Combining terms, the integrated continuity equation is:
h • We next assume that, at some initial time, the velocity (u, v ) is independent of depth, z . • Examining the equations for u and v , we note that the accelerations do not vary with depth z . • Therefore, the velocity will remain independent of depth for all time.
11 dh ∂ u ∂v +h + dt ∂x ∂y =0
12 We are now in a position to write down the full set of Shallow Water Equations: Exercise: Vertical Velocity
Show that the vertical velocity is a linear function of depth. du ∂Φ − fv + =0 dt ∂x dv ∂Φ + fu + =0 dt ∂y dΦ ∂ u ∂v +Φ + =0 dt ∂x ∂y (1) (2) (3)
We deﬁne the horizontal divergence:
H ·V = ∂ u ∂v + ∂x ∂y . we note that H · V is independent of z . This is a set of three equations for (u, v, Φ). The independent variables are (x, y, t). The vertical velocity does not appear. The total time derivative is now given by: d ∂ ∂ ∂ = +u +v . dt ∂t ∂x ∂y This is a nonlinear operator. The Shallow Water Equations are, in general, impossible to solve analytically.
13 The continuity equation may be written ∂w =0 H·V+ ∂z We integrate this between 0 and z , noting that the ﬁrst term is independent of z : ( H · V)z + w(z ) − w(0) = 0 . But we assume a ﬂat bottom, so w(0) = 0. Therefore, w (z ) = ( which increases linearly with z .
14 H · V )z Scale Analysis.
We now introduce characteristic scales for the independent and dependent variables, and non-dimensionalize the equations. This enables us to examine the relative sizes of the terms. Let L and V be typical length and velocity scales. For example, we replace u by Vu∗, so that u∗ is of order unity. Thus, ∂u V ∂ u∗ ∂u∗ = , with = O(1) , ∂x L ∂x∗ ∂x∗ and similarly for the other terms. We assume an advective time scale T: ∂ 1V L ∼V· ; =; T= . ∂t TL V Also, f = 2Ω sin φ ∼ 2Ω, provided we are not too close to the equator.
15 Typical values for the atmopshere are L = 1000 km for the horizontal scale of synoptic weather systems, and V = 10 m s−1 for the wind speed. The mean depth, H, over an area A is: 1 H= h(x, y ) dxdy. AA This is assumed to be equal to the scale-height of the atmosphere. Thus, we choose H = 10 km, about the depth of the troposphere. Just as it is not the absolute pressure which determines the dynamics, but pressure gradients, similarly the dynamically important quantity is the deviation of depth from the mean: h = h − H. We denote this vertical scale by D. Thus h = H + h = H + Dh
∗ with h = O(1) . ∗ But what value should we choose for D?
16 What value should we choose for D? The typical value of surface pressure is p0 = 105 Pa. However, it is the deviation from p0 that is important. The characteristic variation of surface pressure is about 10 hPa. We set the scale of pressure variation as P = 103 Pa. This gives a scale for D: 1 P gD p=g h =⇒ = ρ ρL L Thus, the scale for depth variation is P 103 = = 102 m . ρg 1 · 10 When we examine the sizes of the terms in the momentum equation, this will be seen to be appropriate. D= We now deﬁne the scale values: L = 106 m ; H = 104 m ; V = 10 m s−1 ; D = 102 m ; T = (L/V) = 105 s ≈ 1 day g = 10 m s−2 . f ≈ 10−4 s−1 ; The momentum equations may be written in vector form: dV + fk × V + Φ = 0 . dt The magnitudes of the three terms are as follows: dV V2 ∼ ; dt L 10−4 f k × V ∼ 2ΩV ; 10−3 Φ∼ gD . L 10−3 . The size of each term (in units m s−2) is indicated. We note that the acceleration is an order of magnitude smaller than the remaining terms. The Coriolis term and the pressure gradient term are of the same order of magnitude. This is called Geostrophic Balance.
18 17 The Rossby Number.
The ratio of the acceleration to the Coriolis term is Acceleration dV/dt V2/L V = ∼ = . Coriolis term fk × V 2ΩV 2ΩL This ratio is called the Rossby Number, denoted Ro: Aside: The Froude Number
There is another non-dimensional number which depends on the depth scale D but not on the Coriolis parameter. The Froude Number is the ratio of the ﬂuid ﬂow to the speed of gravity waves: Froude Flow Velocity = Number Gravity Wave Speed We will show later that the characteristic speed of gravity √ waves is g H, so the Froude number is V Fr = √ gH With the characteristic scale values already chosen, we have 10 m s−1 1 Fr = √ ≈ 10 m s−2 · 104 m 30 Thus, for large-scale geophysical ﬂows, both the Rossby number and the Froude number are small: Ro 1 Fr 1.
20 Ro ≡ V . 2ΩL It is a fundamental number in geophysical ﬂuid dynamics. Substituting the chosen values for V, f and L, we get 10 Ro = −4 = 10−1 1. 10 · 106 The smallness of this non-dimensional parameter allows us to make various approximations and perturbation analyses. 19 The Geostrophic Wind
The momentum equation dV + fk × V + Φ = 0 dt is of the form A + B + C = 0. If assume that one term is smaller than the other two, we get various special cases. The most important of these is geostrophic balance. We saw that the acceleration term is relatively small. Omitting it, we get a diagnostic relationship called geostrophic balance: 1 fk × V + Φ = 0 ; V = k × Φ. f In terms of the pressure ﬁeld, the geostrophic wind is 1 1 fk × V + p = 0; V = k × p. ρ ρf So, the wind ﬁeld is determined by the pressure ﬁeld.
21 In terms of coordinates, the geostrophic wind is 1 ∂p 1 ∂p u=− , v=+ . f ρ ∂y f ρ ∂x For geostrophic balance, the ﬂow is perpendicular to the gradient of presure. The existence of a ﬂuid ﬂow along the isobars, rather than towards areas of low pressure, is characteristic of geophysical ﬂows, and in dramatic contrast to the situation for ﬂuid ﬂow in a non-rotating framework. 22 ECMWF Forecast Chart
72 hour forecast of sea level pressure and 850 hPa wind Web Exercise
Download and study a selection of weather charts. Find charts with both pressure and winds. Study the relationship between the wind and pressure ﬁelds. Use stuﬀ from Met Eireann web-site http://www.met.ie Use stuﬀ from ECMWF web-site http://www.ecmwf.int Search for other sites (There are hundreds) 23 24 ...
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