Gap Winds

Introduction

• gap winds are a low level flow of dense air constrained vertically by an inversion, and horizontally by channel walls, forced by synoptic- to meso- scale pressure gradients.

• can potentially occur wherever deep topographic valleys exist

• events along the B.C. coast occur mainly during the winter when cold arctic air moves into southern B.C. from Alaska.

• the Coast Mountains can block the cold air from reaching the coast, except through ``gaps'' cutting through the mountains

• results in strong pressure gradients across the Coast Mountains

• air is strongly forced to flow down the pressure gradient through the gaps / fjords to the coast

• position and internal structure are determined by topography of gap and hydraulic effects (short temporal and spatial scales).

• resemble flow of water in an open channel (hydraulic flow)

BC Location map.

Howe Sound Location map.

Schematic visualization of gap flow in Howe Sound.

Wind roses at stations in Howe Sound.

Typical Synoptic Conditions

[OH: synoptic evol]

Synoptic conditions during a Gap wind episode.

Synoptic scale versus Boundary layer dynamics

• synoptic conditions determine the large-scale pressure gradients which force the flow
• if the air is stratified (as is the case with cold arctic air), then variations in the boundary-layer depth will also result in pressure gradients which are superimposed upon the synoptic gradients
• the wind responds to the total pressure gradient which is a combination of large-scale (synoptic) and small-scale (due to variations in boundary-layer depth).

Simple models of gap winds

Rotational effects

• shallow, strongly stratified flow forced through a channel by a PGF oriented along the channel, after a long time in a long channel will undergo geostrophic adjustment:
  • the flow will be forced up the right side of the channel due to Coriolis turning
  • the BL depth will increase on the right side so that the along-channel PGF is changed into a cross-channel PGF and the down channel wind is in geostrophic balance with the cross-channel PGF
• the time-scale for geostrophic adjustment is $\propto$ ${\frac{1}{f}}$ $\approx$ 3 hours
• in a 100 km straight channel with winds less than 30 kmh, adjustment could occur
• for stronger gap wind cases, the flow will not become geostrophic

``Bernoulli'' and ``Friction'' winds

• Wind in channels has long been modelled using simple relationships with along-channel pressure gradients.
• This seems a reasonable thing to do, based upon observations from a number of places:[OH: Eisacktal]

Example of relationship between $\Delta$ P and wind.

• The equation of motion is:

  $$\underbrace{\frac{d \vec{V}}{dt}}_{1}^{}\, \underbrace{-\frac{\nabla P}{\rho}}_{2}^{}\, \underbrace{f \hat{k} \times \vec{V}}_{3}^{}\, \underbrace{\frac{1}{\rho}\frac{d\tau}{dz}}_{4}^{}\,$$ (1)

  where: $\vec{V}\,$ is the vector wind, t is time, $\rho$ is density, P is pressure, f is the Coriolis parameter, $\hat{k}$ is a unit vector directed upwards, and $\tau$ is the eddy stress. In this equation: term 1 is the Lagrangian acceleration (ie. the acceleration following the flow); term 2 is the pressure gradient force, term 3 is the Coriolis force, and term 4 is friction.

• assume that:
  • the channel is straight along x
  • the flow is in approximate steady-state so that term 1, the Lagrangian time derivative, can be approximated by the advection of velocity down the channel
  • the effects of rotation in the channel are negligible so that term 3 can be ignored
  • the friction term (4) can be parameterized as ${\frac{-C u^{2}}{h}}$
• Therefore equation 1 can be simplified to:
  

  $${\frac{du}{dt}} {\frac{d\; }{dx}} {\frac{u^2}{2}} \eta {\frac{Cu^{2}}{h}}$$ = (2)

  $$\eta {\frac{-1}{\rho}} {\frac{dP}{dx}}$$ =

  
  

• This balance between pressure gradient, acceleration, and friction represents an antitriptic wind
• If the friction term is neglected altogether, the result is a form of the Bernoulli equation, which can be integrated to give:
  

  $${\frac{u^{2}}{2}} {\frac{u_{0}^{2}}{2}} {\frac{\Delta P}{\rho}}$$ = (3)

  $$\sqrt{u_0^2 - 2 \frac{\Delta P}{\rho}} \sqrt{u_0^2 - 2 \frac{dP}{dx} \frac{\Delta x}{\rho}}$$ u = (4)

  
  
  where u₀ is the an upstream speed, $\Delta$P is the pressure difference between the end point and the initial point, and u is the Lagrangian velocity. For u₀ = 0 this then reduces to:

  $$\sqrt{\frac{-2\Delta P}{\rho}}$$ (5)

• This result describes the flow neglecting rotation, friction and changes in elevation and will be called the ``Bernoulli'' model.
• Incorporating friction, equation 2 can be solved analytically:

  $$\sqrt{\frac{\eta h}{C} + (u^2_0 - \frac{\eta h}{C})e^{\frac{-2Cx}{h}}}$$ (6)

  where u₀ is the upstream gap wind speed. This is called the ``Friction'' model. For u₀ = 0 this simplifies to:

  $$\sqrt{\frac{\eta h}{C}} \sqrt{1 - e^{\frac{-2Cx}{h}}}$$ (7)

  as x approaches infinity, this reduces to:

  $$\sqrt{\frac{\eta h}{C}}$$ (8)

  which represents the maximum steady-state speed. It can be shown that the above solution reduces to the Bernoulli equation in the limit as C $\rightarrow$ 0.

Depiction of Friction model.

How well do they work?

• Albert, J. 1982: Channelled winds in Juan de Fuca Strait: An Empirical Verification. Pacific Region Technical Notes, 82-024.
  
  Howe Sound and Lower Fraser Valley: Modelled (Bernoulli and friction) versus Observed.
  
  Lower Fraser Valley: Modelled (Bernoulli and friction) versus Observed.

Gap winds as hydraulic flows

• The Bernoulli and Friction models are appropriate for situations where the pressure gradient is entirely determined by synoptic scale processes
• for stratified flow situations, especially where a cold air layer is surmounted by a much less dense air layer, it is possible that pressure gradients within the flow resulting from variations in BL depth are as important as synoptically imposed gradients
• in this situation, gap flow resembles the flow of water in a channel, and thus a hydraulic analogy is appropriate.
• a hydraulic model is a superset of the Bernoulli / Friction model which allows along-channel variations in the depth of the cold lower layer and allows pressure gradients resulting from sloped channels

Variable definitions.

Hydraulic theory

The simplified momentum equation is:

$${\frac{d u}{d x}} {\frac{d h}{d x}} {\frac{d e}{d x}} {\frac{1}{\rho_1}} {\frac{d P}{d x}} {\frac{1}{\rho_1}} {\frac{d\tau}{dz}}$$ (9)

The equation of continuity in the gap wind is:

$$\overline{b}$$ (10)

Define S_P = - (g'$\rho_{1}^{}$)^-1dP/dx. Parameterize friction as:

$${\frac{1}{\rho_1}} {\frac{d\tau}{dz}} \simeq {\frac{-1}{\rho_1}} {\frac{\tau_{s}}{h}} {\frac{-C u^{2}}{h}}$$ (11)

The Energy Equation

Integrate the momentum equation (9) along x to find the Energy Equation:

$$\Delta {\frac{u^2}{2g'}} \overline{(S_{P} - \frac{Cu^2}{g'h})} \Delta$$ (12)

The Froude number, F = u/(g'D)^1/2 = ((Q²b_T)/(g'A³))^1/2, determines the flow regime:

• if the Froude number is less than 1, flow is subcritical
• if the Froude number is greater than 1, flow is supercritical
• and if it is equal to 1, flow is critical

The hydraulic jump

Energy equation breaks down in a turbulent hydraulic jump so can use conservation of momentum to find the conditions on either side of the jump:

$${\frac{Q^{2}}{g' A_{2}}} \hbar_{2}^{} {\frac{Q^{2}}{g' A_{1}}} \hbar_{1}^{}$$ (13)

Variable definitions at a hydraulic jump.

Hydraulic flow regimes.

Hydraulic model

Hydmod is subject to several assumptions:

• the flow must be steady in time, gradually varying in space, 1-dimensional, and on slopes which are not too steep;
• energy losses due to channel sinuosity are negligible;
• the gap wind should bear some resemblance to a dense fluid flowing in a channel
• no entrainment of, or interactions with air above

Hydmod requires the following data as input:

• Q (the volume flow rate)
• cross sectional area, A, by height at locations along the channel
• e, the elevation of the channel floor along the channel
• $\theta_{1}^{}$,$\theta_{2}^{}$
• dP/dx, the ``synoptic'' pressure gradient
• C, drag coefficient values
• h_f, the height of gap wind at the end of the channel

Hydraulic model.

How well does it work?

Hydraulic model: subcritical flow becoming supercritical at a contraction. (Flow is from right to left, there is a horizontal contraction in the middle of the domain.)

Hydraulic model: supercritical flow becoming subcritical upstream, and supercritical at a contraction. (Flow is from right to left, there is a horizontal contraction in the middle of the domain.)

Comparison of observations (triangle), RAMS mesoscale model output (dashed), and two hydraulic model scenarios (heavy dashed), for different scenarios.

Mesoscale models

• there has been success at simulating gap winds with mesoscale models in research settings
• the horizontal resolution of ``operational'' mesoscale models, until now, has been insufficient to adequately resolve the topographic ``gaps''
• probably (?) the greatest limitation in accuracy for operational mesoscale gap wind simulations, is accurate synoptic-scale initialization

Conclusions

• simple gap wind models (ie Bernoulli or ``friction'') based upon synoptic-scale pressure gradients have utility under some circumstances [What circumstances?]
• under some circumstances, hydraulic models may give more accurate results, but they have a disadvantage of requiring more information
• mesoscale models will eventually be able to give good gap wind forecasts, but more research and verification is required

References

There are have been several papers written on gap winds, such as:

• Overland and Walter, 1981. Mon. Wea. Rev., 2221-2233.
• Colman and Dierking, 1992. Wea. Forecasting, 49-64.
• Lackman and Overland, 1989. Mon. Wea. Rev., 1817-1833.
• Colle and Mass, 1998a,b. Mon. Wea. Rev., 28-52, 53-71.
• Jackson, 1996. Atmos. Ocean, 285-311.
• Jackson and Steyn, 1994a,b. Mon. Wea. Rev., 2645-2665, 2666-2676.
• Finnigan, Vine, Jackson, Allen, Lawrence and Steyn, 1994. Mon. Wea. Rev. 2677-2687.