Wellington - EPSG:5770

WKT

VERTCS["Wellington", 
	VDATUM["Wellington"], 
	PARAMETER["Vertical_Shift", 0.0], 
	PARAMETER["Direction", 1.0], 
	UNIT["Meter", 1.0]]

WKT2

VERTCRS["Wellington", 
	VDATUM["Wellington"], 
	CS[vertical, 1], 
	AXIS["Gravity-related height (H)", up, 
		LENGTHUNIT["Meter", 1.0]], 
	ID["EPSG", "5770"]]

Data from @esri/proj-codes

{
  "wkid": 5770,
  "latestWkid": 5770,
  "macro": "PE_VCS_WELLINGTON",
  "name": "Wellington",
  "wkt": "VERTCS[\"Wellington\",VDATUM[\"Wellington\"],PARAMETER[\"Vertical_Shift\",0.0],PARAMETER[\"Direction\",1.0],UNIT[\"Meter\",1.0]]",
  "wkt2": "VERTCRS[\"Wellington\",VDATUM[\"Wellington\"],CS[vertical,1],AXIS[\"Gravity-related height (H)\",up,LENGTHUNIT[\"Meter\",1.0]],ID[\"EPSG\",\"5770\"]]",
  "description": "Wellington",
  "authority": "EPSG",
  "version": "6.15(9.3.0.0)",
  "deprecated": "no",
  "areaname": "New Zealand - North Island - Wellington vcrs",
  "extent": {
    "slat": -41.67,
    "nlat": -40.12,
    "llon": 174.52,
    "rlon": 176.55
  }
}